CHAPTER 0 0.1 Concepts Review 1. rational numbers Preliminaries 1 ⎡ 2 1 ⎛ 1 1 ⎞⎤ 1 8. − ⎢ − ⎜ − ⎟ ⎥ = − 3 ⎣ 5 2 ⎝ 3 5 ⎠⎦ ⎡ 2 1 ⎛ 5 3 ⎞⎤ 3 ⎢ − ⎜ − ⎟⎥ ⎣ 5 2 ⎝ 15 15 ⎠ ⎦ 2. dense 1 ⎡ 2 1 ⎛ 2 ⎞⎤ 1 ⎡2 1 ⎤ = − ⎢ − ⎜ ⎟⎥ = − ⎢ − ⎥ 3 ⎣ 5 2 ⎝ 15 ⎠ ⎦ 3 ⎣ 5 15 ⎦ 1⎛ 6 1 ⎞ 1⎛ 5 ⎞ 1 =− ⎜ − ⎟=− ⎜ ⎟=− 3 ⎝ 15 15 ⎠ 3 ⎝ 15 ⎠ 9 3. If not Q then not P. 4. theorems 2 Problem Set 0.1 1. 4 − 2(8 − 11) + 6 = 4 − 2(−3) + 6 = 4 + 6 + 6 = 16 2. 3 ⎡⎣ 2 − 4 ( 7 − 12 ) ⎤⎦ = 3[ 2 − 4(−5) ] = 3[ 2 + 20] = 3(22) = 66 3. –4[5(–3 + 12 – 4) + 2(13 – 7)] = –4[5(5) + 2(6)] = –4[25 + 12] = –4(37) = –148 4. 5 [ −1(7 + 12 − 16) + 4] + 2 = 5 [ −1(3) + 4] + 2 = 5 ( −3 + 4 ) + 2 = 5 (1) + 2 = 5 + 2 = 7 5. 6. 7. 5 1 65 7 58 – = – = 7 13 91 91 91 3 3 1 3 3 1 + − = + − 4 − 7 21 6 −3 21 6 42 6 7 43 =− + − =− 42 42 42 42 1 ⎡1 ⎛ 1 1 ⎞ 1⎤ 1 ⎡1 ⎛ 3 – 4 ⎞ 1⎤ = ⎜ – ⎟+ ⎜ ⎟+ 3 ⎢⎣ 2 ⎝ 4 3 ⎠ 6 ⎥⎦ 3 ⎢⎣ 2 ⎝ 12 ⎠ 6 ⎥⎦ 1 ⎡1 ⎛ 1 ⎞ 1⎤ = ⎢ ⎜– ⎟+ ⎥ 3 ⎣ 2 ⎝ 12 ⎠ 6 ⎦ 1⎡ 1 4⎤ = ⎢– + ⎥ 3 ⎣ 24 24 ⎦ 1⎛ 3 ⎞ 1 = ⎜ ⎟= 3 ⎝ 24 ⎠ 24 Instructor’s Resource Manual 2 2 14 ⎛ 2 ⎞ 14 ⎛ 2 ⎞ 14 6 ⎜ ⎟ = ⎜ ⎟ = ⎛⎜ ⎞⎟ 9. 21 ⎜ 5 − 1 ⎟ 21 ⎜ 14 ⎟ 21 ⎝ 14 ⎠ 3⎠ ⎝ ⎝ 3 ⎠ 2 = 14 ⎛ 3 ⎞ 2⎛ 9 ⎞ 6 ⎜ ⎟ = ⎜ ⎟= 21 ⎝ 7 ⎠ 3 ⎝ 49 ⎠ 49 ⎛2 ⎞ ⎛ 2 35 ⎞ ⎛ 33 ⎞ ⎜ − 5⎟ ⎜ − ⎟ ⎜ − ⎟ 7 ⎠ = ⎝ 7 7 ⎠ = ⎝ 7 ⎠ = − 33 = − 11 10. ⎝ 6 2 ⎛ 1⎞ ⎛7 1⎞ ⎛6⎞ ⎜1 − ⎟ ⎜ − ⎟ ⎜ ⎟ ⎝ 7⎠ ⎝7 7⎠ ⎝7⎠ 7 11 – 12 11 – 4 7 7 21 7 7 = = 7 = 11. 11 + 12 11 + 4 15 15 7 21 7 7 7 1 3 7 4 6 7 5 − + − + 5 12. 2 4 8 = 8 8 8 = 8 = 1 3 7 4 6 7 3 3 + − + − 2 4 8 8 8 8 8 13. 1 – 1 1 2 3 2 1 =1– =1– = – = 1 3 3 3 3 3 1+ 2 2 14. 2 + 15. ( 3 5 1+ 2 5+ 3 3 3 = 2+ 2 5 7 − 2 2 2 6 14 6 20 = 2+ = + = 7 7 7 7 = 2+ )( ) ( 5) – ( 3) 5– 3 = 2 2 =5–3= 2 Section 0.1 1 16. ( 5− 3 ) = ( 5) 2 2 −2 ( 5 )( 3 ) + ( 3 ) 2 27. = 5 − 2 15 + 3 = 8 − 2 15 17. (3x − 4)( x + 1) = 3 x 2 + 3 x − 4 x − 4 = 3x2 − x − 4 18. (2 x − 3)2 = (2 x − 3)(2 x − 3) 12 4 2 + x + 2x x x + 2 12 4( x + 2) 2x = + + x( x + 2) x( x + 2) x( x + 2) 12 + 4 x + 8 + 2 x 6 x + 20 = = x( x + 2) x( x + 2) 2(3 x + 10) = x( x + 2) 2 + = 4 x2 − 6 x − 6 x + 9 = 4 x 2 − 12 x + 9 19. 28. (3x – 9)(2 x + 1) = 6 x 2 + 3 x –18 x – 9 2 y + 2(3 y − 1) (3 y + 1)(3 y − 1) 2(3 y + 1) 2y = + 2(3 y + 1)(3 y − 1) 2(3 y + 1)(3 y − 1) = 2 = 6 x –15 x – 9 20. (4 x − 11)(3x − 7) = 12 x 2 − 28 x − 33 x + 77 = 12 x 2 − 61x + 77 21. (3t 2 − t + 1) 2 = (3t 2 − t + 1)(3t 2 − t + 1) 4 3 2 3 2 2 = 9t − 3t + 3t − 3t + t − t + 3t − t + 1 = 9t 4 − 6t 3 + 7t 2 − 2t + 1 2 y + 6 y − 2 9 y2 −1 = 6y + 2 + 2y 8y + 2 = 2(3 y + 1)(3 y − 1) 2(3 y + 1)(3 y − 1) = 2(4 y + 1) 4y +1 = 2(3 y + 1)(3 y − 1) (3 y + 1)(3 y − 1) 0⋅0 = 0 b. 0 is undefined. 0 c. 0 =0 17 d. 3 is undefined. 0 e. 05 = 0 f. 170 = 1 29. a. 22. (2t + 3)3 = (2t + 3)(2t + 3)(2t + 3) = (4t 2 + 12t + 9)(2t + 3) = 8t 3 + 12t 2 + 24t 2 + 36t + 18t + 27 = 8t 3 + 36t 2 + 54t + 27 23. x 2 – 4 ( x – 2)( x + 2) = = x+2, x ≠ 2 x–2 x–2 24. x 2 − x − 6 ( x − 3)( x + 2) = = x+2, x ≠3 x−3 ( x − 3) 25. t 2 – 4t – 21 (t + 3)(t – 7) = = t – 7 , t ≠ −3 t +3 t +3 26. 2 2x − 2x 3 2 2 x − 2x + x = 2 x(1 − x ) 2 x( x − 2 x + 1) −2 x( x − 1) = x( x − 1)( x − 1) 2 =− x −1 Section 0.1 0 = a , then 0 = 0 ⋅ a , but this is meaningless 0 because a could be any real number. No 0 single value satisfies = a . 0 30. If 31. .083 12 1.000 96 40 36 4 Instructor’s Resource Manual 32. .285714 7 2.000000 14 60 56 40 35 50 49 10 7 30 28 2 33. .142857 21 3.000000 21 90 84 60 42 180 168 120 105 150 147 3 34. .294117... 17 5.000000... → 0.2941176470588235 34 160 153 70 68 20 17 30 17 130 119 11 Instructor’s Resource Manual 35. 3.6 3 11.0 9 20 18 2 36. .846153 13 11.000000 10 4 60 52 80 78 20 13 70 65 50 39 11 37. x = 0.123123123... 1000 x = 123.123123... x = 0.123123... 999 x = 123 123 41 x= = 999 333 38. x = 0.217171717 … 1000 x = 217.171717... 10 x = 2.171717... 990 x = 215 215 43 x= = 990 198 39. x = 2.56565656... 100 x = 256.565656... x = 2.565656... 99 x = 254 254 x= 99 40. x = 3.929292… 100 x = 392.929292... x = 3.929292... 99 x = 389 389 x= 99 Section 0.1 3 41. x = 0.199999... 100 x = 19.99999... 52. 10 x = 1.99999... 90 x = 18 18 1 x= = 90 5 54. 55. 10 x = 3.99999... 90 x = 36 36 2 x= = 90 5 56. 43. Those rational numbers that can be expressed by a terminating decimal followed by zeros. ⎛1⎞ p 1 = p ⎜ ⎟ , so we only need to look at . If q q ⎝q⎠ q = 2n ⋅ 5m , then n m 1 ⎛1⎞ ⎛1⎞ = ⎜ ⎟ ⋅ ⎜ ⎟ = (0.5)n (0.2)m . The product q ⎝ 2⎠ ⎝5⎠ of any number of terminating decimals is also a n m terminating decimal, so (0.5) and (0.2) , and hence their product, decimal. Thus 1 , is a terminating q p has a terminating decimal q expansion. 45. Answers will vary. Possible answer: 0.000001, 1 ≈ 0.0000010819... π 12 46. Smallest positive integer: 1; There is no smallest positive rational or irrational number. 47. Answers will vary. Possible answer: 3.14159101001... 48. There is no real number between 0.9999… (repeating 9's) and 1. 0.9999… and 1 represent the same real number. 49. Irrational 50. Answers will vary. Possible answers: −π and π , − 2 and 2 51. ( 3 + 1)3 ≈ 20.39230485 4 Section 0.1 2− 3 ) 4 ≈ 0.0102051443 53. 4 1.123 – 3 1.09 ≈ 0.00028307388 42. x = 0.399999… 100 x = 39.99999... 44. ( ( 3.1415 )−1/ 2 ≈ 0.5641979034 8.9π2 + 1 – 3π ≈ 0.000691744752 4 (6π 2 − 2)π ≈ 3.661591807 57. Let a and b be real numbers with a < b . Let n be a natural number that satisfies 1 / n < b − a . Let S = {k : k n > b} . Since a nonempty set of integers that is bounded below contains a least element, there is a k 0 ∈ S such that k 0 / n > b but (k 0 − 1) / n ≤ b . Then k0 − 1 k0 1 1 = − >b− > a n n n n k 0 −1 k 0 −1 Thus, a < n ≤ b . If n < b , then choose r= k 0 −1 n . Otherwise, choose r = k0 − 2 n . 1 <r. n Given a < b , choose r so that a < r1 < b . Then choose r2 , r3 so that a < r2 < r1 < r3 < b , and so on. Note that a < b − 58. Answers will vary. Possible answer: ≈ 120 in 3 ft = 21,120, 000 ft mi equator = 2π r = 2π (21,120, 000) ≈ 132, 700,874 ft 59. r = 4000 mi × 5280 60. Answers will vary. Possible answer: beats min hr day × 60 × 24 × 365 × 20 yr 70 min hr day year = 735,840, 000 beats 2 ⎛ 16 ⎞ 61. V = πr 2 h = π ⎜ ⋅12 ⎟ (270 ⋅12) ⎝ 2 ⎠ ≈ 93,807, 453.98 in.3 volume of one board foot (in inches): 1× 12 × 12 = 144 in.3 number of board feet: 93,807, 453.98 ≈ 651, 441 board ft 144 Instructor’s Resource Manual b. Every circle has area less than or equal to 9π. The original statement is true. 62. V = π (8.004) 2 (270) − π (8)2 (270) ≈ 54.3 ft.3 63. a. If I stay home from work today then it rains. If I do not stay home from work, then it does not rain. b. If the candidate will be hired then she meets all the qualifications. If the candidate will not be hired then she does not meet all the qualifications. 64. a. c. Some real number is less than or equal to its square. The negation is true. 71. a. True; If x is positive, then x 2 is positive. b. False; Take x = −2 . Then x 2 > 0 but x<0. If I pass the course, then I got an A on the final exam. If I did not pass the course, thn I did not get an A on the final exam. c. 2 2 e. 2 2 2 2 triangle, then a + b ≠ c . 72. a. c. If angle ABC is an acute angle, then its measure is 45o. If angle ABC is not an acute angle, then its measure is not 45o. 2 2 2 The statement, converse, and contrapositive are all true. True; 1/ 2n can be made arbitrarily close to 0. 73. a. If n is odd, then there is an integer k such that n = 2k + 1. Then n 2 = (2k + 1) 2 = 4k 2 + 4k + 1 = 2(2k 2 + 2k ) + 1 The statement and contrapositive are true. The converse is false. Some isosceles triangles are not equilateral. The negation is true. b. All real numbers are integers. The original statement is true. c. 70. a. True; Let x be any number. Take 1 1 y = + 1 . Then y > . x x e. b. b. The statement, converse, and contrapositive are all false. 69. a. True; x + ( − x ) < x + 1 + ( − x ) : 0 < 1 2 b. The statement, converse, and contrapositive are all true. 68. a. True; Let y be any positive number. Take y x = . Then 0 < x < y . 2 d. True; 1/ n can be made arbitrarily close to 0. b. If a < b then a < b. If a ≥ b then a ≥ b. 67. a. <x b. False; There are infinitely many prime numbers. b. If the measure of angle ABC is greater than 0o and less than 90o, it is acute. If the measure of angle ABC is less than 0o or greater than 90o, then it is not acute. 66. a. 1 4 y = x 2 + 1 . Then y > x 2 . If a triangle is a right triangle, then a + b = c . If a triangle is not a right 1 2 . Then x = 2 d. True; Let x be any number. Take b. If I take off next week, then I finished my research paper. If I do not take off next week, then I did not finish my research paper. 65. a. False; Take x = Some natural number is larger than its square. The original statement is true. Prove the contrapositive. Suppose n is even. Then there is an integer k such that n = 2k . Then n 2 = (2k )2 = 4k 2 = 2(2k 2 ) . Thus n 2 is even. Parts (a) and (b) prove that n is odd if and 74. only if n 2 is odd. 75. a. b. 243 = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 124 = 4 ⋅ 31 = 2 ⋅ 2 ⋅ 31 or 22 ⋅ 31 Some natural number is not rational. The original statement is true. Instructor’s Resource Manual Section 0.1 5 5100 = 2 ⋅ 2550 = 2 ⋅ 2 ⋅1275 c. 82. a. = 2 ⋅ 2 ⋅ 3 ⋅ 425 = 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 85 = 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 5 ⋅17 or 22 ⋅ 3 ⋅ 52 ⋅17 c. 76. For example, let A = b ⋅ c 2 ⋅ d 3 ; then A2 = b 2 ⋅ c 4 ⋅ d 6 , so the square of the number is the product of primes which occur an even number of times. 77. p p2 ;2 = ; 2q 2 = p 2 ; Since the prime 2 q q 2 factors of p must occur an even number of p times, 2q2 would not be valid and = 2 q must be irrational. 3= p p2 ; 3= ; 3q 2 = p 2 ; Since the prime q q2 factors of p 2 must occur an even number of times, 3q 2 would not be valid and e. f. 83. a. p = 3 q a b p a p aq + bp are rational. + = This q b q bq sum is the quotient of natural numbers, so it is also rational. and x = 2.4444...; 10 x = 24.4444... x = 2.4444... 9 x = 22 22 x= 9 2 3 n = 1: x = 0, n = 2: x = , n = 3: x = – , 3 2 5 n = 4: x = 4 3 The upper bound is . 2 2 Answers will vary. Possible answer: An example is S = {x : x 2 < 5, x a rational number}. Here the least upper bound is 5, which is real but irrational. must be irrational. 79. Let a, b, p, and q be natural numbers, so b. –2 d. 1 2= 78. –2 b. True 0.2 Concepts Review 1. [−1,5); (−∞, −2] 2. b > 0; b < 0 p 80. Assume a is irrational, ≠ 0 is rational, and q p r q⋅r is = is rational. Then a = q s p⋅s rational, which is a contradiction. a⋅ 81. a. – 9 = –3; rational b. 3 0.375 = ; rational 8 c. (3 2)(5 2) = 15 4 = 30; rational d. (1 + 3)2 = 1 + 2 3 + 3 = 4 + 2 3; irrational 3. (b) and (c) 4. −1 ≤ x ≤ 5 Problem Set 0.2 1. a. b. c. d. 6 Section 0.2 Instructor’s Resource Manual –3 < 1 – 6 x ≤ 4 9. –4 < –6 x ≤ 3 e. 2 1 ⎡ 1 2⎞ > x ≥ – ; ⎢– , ⎟ 3 2 ⎣ 2 3⎠ f. 2. a. c. b. (2, 7) (−∞, −2] d. [−3, 4) [−1, 3] 10. 3. x − 7 < 2 x − 5 −2 < x;( − 2, ∞) 4 < 5 − 3x < 7 −1 < −3x < 2 1 2 ⎛ 2 1⎞ > x > − ; ⎜− , ⎟ 3 3 ⎝ 3 3⎠ 4. 3x − 5 < 4 x − 6 1 < x; (1, ∞ ) 11. x2 + 2x – 12 < 0; x= 5. 7 x – 2 ≤ 9x + 3 –5 ≤ 2 x = –1 ± 13 ( 7. −4 < 3 x + 2 < 5 −6 < 3 x < 3 −2 < x < 1; (−2, −1) ) ( ) ⎡ x – –1 + 13 ⎤ ⎡ x – –1 – 13 ⎤ < 0; ⎣ ⎦⎣ ⎦ 5 ⎡ 5 ⎞ x ≥ – ; ⎢– , ∞ ⎟ 2 ⎣ 2 ⎠ 6. 5 x − 3 > 6 x − 4 1 > x; (−∞,1) –2 ± (2)2 – 4(1)(–12) –2 ± 52 = 2(1) 2 ( –1 – 13, – 1 + 13 ) 12. x 2 − 5 x − 6 > 0 ( x + 1)( x − 6) > 0; (−∞, −1) ∪ (6, ∞) 13. 2x2 + 5x – 3 > 0; (2x – 1)(x + 3) > 0; ⎛1 ⎞ (−∞, −3) ∪ ⎜ , ∞ ⎟ ⎝2 ⎠ 8. −3 < 4 x − 9 < 11 6 < 4 x < 20 3 ⎛3 ⎞ < x < 5; ⎜ ,5 ⎟ 2 ⎝2 ⎠ 14. ⎛ 3 ⎞ (4 x + 3)( x − 2) < 0; ⎜ − , 2 ⎟ ⎝ 4 ⎠ 15. Instructor’s Resource Manual 4 x2 − 5x − 6 < 0 x+4 ≤ 0; [–4, 3) x–3 Section 0.2 7 16. 3x − 2 2⎤ ⎛ ≥ 0; ⎜ −∞, ⎥ ∪ (1, ∞) x −1 3⎦ ⎝ 3 >2 x+5 20. 3 −2 > 0 x+5 17. 2 −5 < 0 x 2 − 5x < 0; x ⎛2 ⎞ (– ∞, 0) ∪ ⎜ , ∞ ⎟ ⎝5 ⎠ 18. 3 − 2( x + 5) >0 x+5 2 <5 x 7 ≤7 4x 7 −7 ≤ 0 4x 7 − 28 x ≤ 0; 4x 1 ( −∞, 0 ) ∪ ⎡⎢ , ∞ ⎞⎟ ⎣4 ⎠ −2 x − 7 7⎞ ⎛ > 0; ⎜ −5, − ⎟ 2⎠ x+5 ⎝ 21. ( x + 2)( x − 1)( x − 3) > 0;(−2,1) ∪ (3,8) 3⎞ ⎛1 ⎞ ⎛ 22. (2 x + 3)(3x − 1)( x − 2) < 0; ⎜ −∞, − ⎟ ∪ ⎜ , 2 ⎟ 2⎠ ⎝3 ⎠ ⎝ 3⎤ ⎛ 23. (2 x - 3)( x -1)2 ( x - 3) ≥ 0; ⎜ – ∞, ⎥ ∪ [3, ∞ ) 2⎦ ⎝ 24. (2 x − 3)( x − 1) 2 ( x − 3) > 0; 19. ( −∞,1) ∪ ⎛⎜1, 3⎞ ⎟ ∪ ( 3, ∞ ) ⎝ 2⎠ 1 ≤4 3x − 2 1 −4≤ 0 3x − 2 1 − 4(3 x − 2) ≤0 3x − 2 9 − 12 x 2 ⎞ ⎡3 ⎞ ⎛ ≤ 0; ⎜ −∞, ⎟ ∪ ⎢ , ∞ ⎟ 3x − 2 3 ⎠ ⎣4 ⎠ ⎝ 25. x3 – 5 x 2 – 6 x < 0 x( x 2 – 5 x – 6) < 0 x( x + 1)( x – 6) < 0; (−∞, −1) ∪ (0, 6) 26. x3 − x 2 − x + 1 > 0 ( x 2 − 1)( x − 1) > 0 ( x + 1)( x − 1) 2 > 0; (−1,1) ∪ (1, ∞) 8 Section 0.2 27. a. False. c. False. b. True. Instructor’s Resource Manual 28. a. True. c. False. 29. a. b. True. 33. a. ( x + 1)( x 2 + 2 x – 7) ≥ x 2 – 1 x3 + 3 x 2 – 5 x – 7 ≥ x 2 – 1 x3 + 2 x 2 – 5 x – 6 ≥ 0 ( x + 3)( x + 1)( x – 2) ≥ 0 [−3, −1] ∪ [2, ∞) ⇒ Let a < b , so ab < b 2 . Also, a 2 < ab . Thus, a 2 < ab < b 2 and a 2 < b 2 . ⇐ Let a 2 < b 2 , so a ≠ b Then 0 < ( a − b ) = a 2 − 2ab + b 2 2 x4 − 2 x2 ≥ 8 b. < b 2 − 2ab + b 2 = 2b ( b − a ) x4 − 2 x2 − 8 ≥ 0 Since b > 0 , we can divide by 2b to get b−a > 0. ( x 2 − 4)( x 2 + 2) ≥ 0 ( x 2 + 2)( x + 2)( x − 2) ≥ 0 b. We can divide or multiply an inequality by any positive number. a 1 1 a < b ⇔ <1⇔ < . b b a (−∞, −2] ∪ [2, ∞) c. [( x 2 + 1) − 5][( x 2 + 1) − 2] < 0 30. (b) and (c) are true. (a) is false: Take a = −1, b = 1 . (d) is false: if a ≤ b , then −a ≥ −b . 31. a. 3x + 7 > 1 and 2x + 1 < 3 3x > –6 and 2x < 2 x > –2 and x < 1; (–2, 1) ( x 2 − 4)( x 2 − 1) < 0 ( x + 2)( x + 1)( x − 1)( x − 2) < 0 (−2, −1) ∪ (1, 2) 34. a. 32. a. 3x + 7 > 1 and 2x + 1 < –4 5 x > –2 and x < – ; ∅ 2 1 ⎞ ⎛ 1 , ⎟ ⎜ 2 . 01 1 . 99 ⎠ ⎝ 2 x − 7 > 1 or 2 x + 1 < 3 b. x > 4 or x < 1 (−∞,1) ∪ (4, ∞) 2.99 < 1 < 3.01 x+2 2.99( x + 2) < 1 < 3.01( x + 2) 2.99 x + 5.98 < 1 and 1 < 3.01x + 6.02 − 4.98 and − 5.02 x< x> 2 x − 7 ≤ 1 or 2 x + 1 < 3 2 x ≤ 8 or 2 x < 2 2.99 5.02 4.98 − <x<− 3.01 2.99 ⎛ 5.02 4.98 ⎞ ,− ⎜− ⎟ ⎝ 3.01 2.99 ⎠ x ≤ 4 or x < 1 (−∞, 4] c. 1 < 2.01 x 1 1 <x< 2.01 1.99 2 x > 8 or 2 x < 2 b. 1.99 < 1.99 x < 1 < 2.01x 1.99 x < 1 and 1 < 2.01x 1 and x > 1 x< 1.99 2.01 b. 3x + 7 > 1 and 2x + 1 > –4 3x > –6 and 2x > –5 5 x > –2 and x > – ; ( −2, ∞ ) 2 c. ( x 2 + 1)2 − 7( x 2 + 1) + 10 < 0 2 x − 7 ≤ 1 or 2 x + 1 > 3 3.01 2 x ≤ 8 or 2 x > 2 x ≤ 4 or x > 1 (−∞, ∞) 35. x − 2 ≥ 5; x − 2 ≤ −5 or x − 2 ≥ 5 x ≤ −3 or x ≥ 7 (−∞, −3] ∪ [7, ∞) Instructor’s Resource Manual Section 0.2 9 36. x + 2 < 1; –1 < x + 2 < 1 43. –3 < x < –1 (–3, –1) 37. 4 x + 5 ≤ 10; −10 ≤ 4 x + 5 ≤ 10 −15 ≤ 4 x ≤ 5 − 38. 15 5 ⎡ 15 5 ⎤ ≤ x ≤ ; ⎢− , ⎥ 4 4 ⎣ 4 4⎦ 2 x – 1 > 2; 2x – 1 < –2 or 2x – 1 > 2 2x < –1 or 2x > 3; 1 3 ⎛ 1⎞ ⎛3 ⎞ x < – or x > , ⎜ – ∞, – ⎟ ∪ ⎜ , ∞ ⎟ 2 2 ⎝ 2⎠ ⎝2 ⎠ 39. 40. 2x −5 ≥ 7 7 2x 2x − 5 ≤ −7 or −5 ≥ 7 7 7 2x 2x ≤ −2 or ≥ 12 7 7 x ≤ −7 or x ≥ 42; (−∞, −7] ∪ [42, ∞) x +1 < 1 4 x −1 < + 1 < 1 4 x −2 < < 0; 4 –8 < x < 0; (–8, 0) 41. 5 x − 6 > 1; 5 x − 6 < −1 or 5 x − 6 > 1 5 x < 5 or 5 x > 7 7 ⎛7 ⎞ x < 1 or x > ;(−∞,1) ∪ ⎜ , ∞ ⎟ 5 ⎝5 ⎠ 42. 2 x – 7 > 3; 2x – 7 < –3 or 2x – 7 > 3 2x < 4 or 2x > 10 x < 2 or x > 5; (−∞, 2) ∪ (5, ∞) 44. 1 − 3 > 6; x 1 1 − 3 < −6 or − 3 > 6 x x 1 1 + 3 < 0 or − 9 > 0 x x 1 + 3x 1− 9x < 0 or > 0; x x ⎛ 1 ⎞ ⎛ 1⎞ ⎜ − , 0 ⎟ ∪ ⎜ 0, ⎟ ⎝ 3 ⎠ ⎝ 9⎠ 5 > 1; x 5 5 2 + < –1 or 2 + > 1 x x 5 5 3 + < 0 or 1 + > 0 x x 3x + 5 x+5 < 0 or > 0; x x ⎛ 5 ⎞ (– ∞, – 5) ∪ ⎜ – , 0 ⎟ ∪ (0, ∞) ⎝ 3 ⎠ 2+ 45. x 2 − 3x − 4 ≥ 0; x= 3 ± (–3)2 – 4(1)(–4) 3 ± 5 = = –1, 4 2(1) 2 ( x + 1)( x − 4) = 0;(−∞, −1] ∪ [4, ∞) 4 ± (−4)2 − 4(1)(4) =2 2(1) ( x − 2)( x − 2) ≤ 0; x = 2 46. x 2 − 4 x + 4 ≤ 0; x = 47. 3x2 + 17x – 6 > 0; x= –17 ± (17) 2 – 4(3)(–6) –17 ± 19 1 = = –6, 2(3) 6 3 ⎛1 ⎞ (3x – 1)(x + 6) > 0; (– ∞, – 6) ∪ ⎜ , ∞ ⎟ ⎝3 ⎠ 48. 14 x 2 + 11x − 15 ≤ 0; −11 ± (11) 2 − 4(14)(−15) −11 ± 31 = 2(14) 28 3 5 x=− , 2 7 3 ⎞⎛ 5⎞ ⎛ ⎡ 3 5⎤ ⎜ x + ⎟ ⎜ x − ⎟ ≤ 0; ⎢ − , ⎥ 2 ⎠⎝ 7⎠ ⎝ ⎣ 2 7⎦ x= 49. x − 3 < 0.5 ⇒ 5 x − 3 < 5(0.5) ⇒ 5 x − 15 < 2.5 50. x + 2 < 0.3 ⇒ 4 x + 2 < 4(0.3) ⇒ 4 x + 18 < 1.2 10 Section 0.2 Instructor’s Resource Manual 51. x−2 < 52. x+4 < ε 6 ε 2 ⇒ 6 x − 2 < ε ⇒ 6 x − 12 < ε 59. x –1 < 2 x – 6 ( x –1) 2 < (2 x – 6)2 ⇒ 2 x + 4 < ε ⇒ 2x + 8 < ε x 2 – 2 x + 1 < 4 x 2 – 24 x + 36 3x 2 – 22 x + 35 > 0 53. 3x − 15 < ε ⇒ 3( x − 5) < ε (3x – 7)( x – 5) > 0; ⇒ 3 x−5 < ε ⇒ x−5 < 54. ε 3 ;δ = 7⎞ ⎛ ⎜ – ∞, ⎟ ∪ (5, ∞) 3⎠ ⎝ ε 3 4 x − 8 < ε ⇒ 4( x − 2) < ε 60. 55. ε 4 ;δ = 4 x2 − 4 x + 1 ≥ x2 + 2 x + 1 4 3x2 − 6 x ≥ 0 3 x( x − 2) ≥ 0 (−∞, 0] ∪ [2, ∞) ⇒ 6 x+6 <ε ε 6 ;δ = 2 ε 6 x + 36 < ε ⇒ 6( x + 6) < ε ⇒ x+6 < 2x −1 ≥ x + 1 (2 x − 1)2 ≥ ( x + 1) ⇒ 4 x−2 <ε ⇒ x−2 < x –1 < 2 x – 3 ε 6 61. 2 2 x − 3 < x + 10 4 x − 6 < x + 10 56. 5 x + 25 < ε ⇒ 5( x + 5) < ε (4 x − 6) 2 < ( x + 10)2 ⇒ 5 x+5 <ε 16 x 2 − 48 x + 36 < x 2 + 20 x + 100 ⇒ x+5 < ε 5 ;δ = ε 15 x 2 − 68 x − 64 < 0 5 (5 x + 4)(3 x − 16) < 0; 57. C = π d C – 10 ≤ 0.02 πd – 10 ≤ 0.02 10 ⎞ ⎛ π ⎜ d – ⎟ ≤ 0.02 π⎠ ⎝ 10 0.02 d– ≤ ≈ 0.0064 π π We must measure the diameter to an accuracy of 0.0064 in. 58. C − 50 ≤ 1.5, 5 ( F − 32 ) − 50 ≤ 1.5; 9 5 ( F − 32 ) − 90 ≤ 1.5 9 F − 122 ≤ 2.7 ⎛ 4 16 ⎞ ⎜– , ⎟ ⎝ 5 3⎠ 3x − 1 < 2 x + 6 62. 3x − 1 < 2 x + 12 (3x − 1) 2 < (2 x + 12)2 9 x 2 − 6 x + 1 < 4 x 2 + 48 x + 144 5 x 2 − 54 x − 143 < 0 ( 5 x + 11)( x − 13) < 0 ⎛ 11 ⎞ ⎜ − ,13 ⎟ ⎝ 5 ⎠ We are allowed an error of 2.7 F. Instructor’s Resource Manual Section 0.2 11 63. x < y ⇒ x x ≤ x y and x y < y y 2 ⇒ x < y Order property: x < y ⇔ xz < yz when z is positive. 2 Transitivity (x ⇒ x2 < y 2 Conversely, 2 2 (x 2 x2 < y 2 ⇒ x < y 2 = x 2 2 = x2 ) ) 2 2 ⇒ x – y <0 Subtract y from each side. ⇒ ( x – y )( x + y ) < 0 Factor the difference of two squares. ⇒ x – y <0 ⇒ x < y 64. 0 < a < b ⇒ a = ( a) < ( b) 2 a < 2 This is the only factor that can be negative. Add y to each side. ( a) 2 and b = ( b) 2 , so 67. , and, by Problem 63, x +9 x–2 a + b + c = ( a + b) + c ≤ a + b + c ≤ a+b+c 66. 1 x2 + 3 − ⎛ 1 1 1 ⎞ = +⎜− ⎟ ⎜ 2 x +2 x + 2 ⎟⎠ x +3 ⎝ ≤ 1 2 x +3 +− 68. 1 x +2 1 = + 2 x +2 x +3 1 2 + x 2 + 3 ≥ 3 and x + 2 ≥ 2, so 1 2 x +3 1 2 x +3 12 ≤ 1 1 1 ≤ , thus, and x +2 2 3 + 1 1 1 ≤ + x +2 3 2 Section 0.2 x2 + 9 x 2 x +9 x + –2 2 x ≤ 2 ⇒ x2 + 2 x + 7 ≤ x2 + 2 x + 7 Thus, x2 + 2 x + 7 2 x +1 = x2 + 2 x + 7 1 2 x +1 ≤ 15 ⋅1 = 15 1 x +2 x +3 by the Triangular Inequality, and since 1 1 x 2 + 3 > 0, x + 2 > 0 ⇒ > 0, > 0. 2 x +2 x +3 ≤ x + (–2) ≤ 4 + 4 + 7 = 15 1 and x 2 + 1 ≥ 1 so ≤ 1. 2 x +1 1 = = x +9 x +2 2 ≤ + = 2 2 2 x + 9 x + 9 x + 9 x2 + 9 1 1 Since x 2 + 9 ≥ 9, ≤ 2 9 x +9 x +2 x +2 ≤ 9 x2 + 9 x +2 x–2 ≤ 2 9 x +9 b ⇒ a< b. of absolute values. c. x +9 2 a – b ≥ a – b ≥ a – b Use Property 4 b. 2 x–2 a – b = a + (–b) ≤ a + –b = a + b 65. a. x–2 69. 1 3 1 2 1 1 x + x + x+ 2 4 8 16 1 1 1 1 ≤ x 4 + x3 + x 2 + x + 2 4 8 16 1 1 1 1 ≤ 1+ + + + since x ≤ 1. 2 4 8 16 1 1 1 1 ≤ 1.9375 < 2. So x 4 + x3 + x 2 + x + 2 4 8 16 x4 + Instructor’s Resource Manual x < x2 70. a. 77. x − x2 < 0 x(1 − x) < 0 x < 0 or x > 1 1 11 ≤ R 60 2 x <x b. 2 x −x<0 x( x − 1) < 0 0 < x <1 R≥ 60 11 1 1 1 1 ≥ + + R 20 30 40 1 6+4+3 ≥ R 120 120 R≤ 13 71. a ≠ 0 ⇒ 2 1⎞ 1 ⎛ 0 ≤ ⎜ a – ⎟ = a2 – 2 + a ⎝ ⎠ a2 1 1 or a 2 + ≥2. so, 2 ≤ a 2 + 2 a a2 Thus, 72. a < b a + a < a + b and a + b < b + b 2a < a + b < 2b a+b a< <b 2 60 120 ≤R≤ 11 13 78. A = 4π r 2 ; A = 4π (10)2 = 400π 4π r 2 − 400π < 0.01 4π r 2 − 100 < 0.01 73. 0 < a < b r 2 − 100 < a 2 < ab and ab < b 2 74. 1 1 1 1 ≤ + + R 10 20 30 1 6+3+ 2 ≤ R 60 0.01 4π 0.01 2 0.01 < r − 100 < 4π 4π a 2 < ab < b 2 − a < ab < b 0.01 0.01 < r < 100 + 4π 4π δ ≈ 0.00004 in 100 − ( ) 1 1 ( a + b ) ⇔ ab ≤ a 2 + 2ab + b2 2 4 1 2 1 1 2 1 2 ⇔ 0 ≤ a − ab + b = a − 2ab + b 2 4 2 4 4 1 2 ⇔ 0 ≤ (a − b) which is always true. 4 ab ≤ ( ) 0.3 Concepts Review 1. 75. For a rectangle the area is ab, while for a 2 ⎛ a+b⎞ square the area is a 2 = ⎜ ⎟ . From ⎝ 2 ⎠ 1 ⎛ a+b⎞ (a + b) ⇔ ab ≤ ⎜ ⎟ 2 ⎝ 2 ⎠ so the square has the largest area. Problem 74, ab ≤ 76. 1 + x + x 2 + x3 + … + x99 ≤ 0; (−∞, −1] Instructor’s Resource Manual ( x + 2)2 + ( y − 3)2 2. (x + 4)2 + (y – 2)2 = 25 2 ⎛ −2 + 5 3 + 7 ⎞ 3. ⎜ , ⎟ = (1.5,5) 2 ⎠ ⎝ 2 4. d −b c−a Section 0.3 13 Problem Set 0.3 5. d1 = (5 + 2) 2 + (3 – 4)2 = 49 + 1 = 50 1. d 2 = (5 − 10)2 + (3 − 8)2 = 25 + 25 = 50 d3 = (−2 − 10)2 + (4 − 8)2 = 144 + 16 = 160 d1 = d 2 so the triangle is isosceles. 6. a = (2 − 4)2 + (−4 − 0) 2 = 4 + 16 = 20 b = (4 − 8)2 + (0 + 2)2 = 16 + 4 = 20 c = (2 − 8)2 + (−4 + 2) 2 = 36 + 4 = 40 d = (3 – 1)2 + (1 – 1)2 = 4 = 2 a 2 + b 2 = c 2 , so the triangle is a right triangle. 2. 7. (–1, –1), (–1, 3); (7, –1), (7, 3); (1, 1), (5, 1) 8. ( x − 3) 2 + (0 − 1) 2 = ( x − 6)2 + (0 − 4) 2 ; x 2 − 6 x + 10 = x 2 − 12 x + 52 6 x = 42 x = 7 ⇒ ( 7, 0 ) d = (−3 − 2)2 + (5 + 2)2 = 74 ≈ 8.60 3. ⎛ –2 + 4 –2 + 3 ⎞ ⎛ 1 ⎞ 9. ⎜ , ⎟ = ⎜1, ⎟ ; 2 ⎠ ⎝ 2⎠ ⎝ 2 2 25 ⎛1 ⎞ d = (1 + 2)2 + ⎜ – 3 ⎟ = 9 + ≈ 3.91 2 4 ⎝ ⎠ ⎛1+ 2 3 + 6 ⎞ ⎛ 3 9 ⎞ 10. midpoint of AB = ⎜ , ⎟=⎜ , ⎟ 2 ⎠ ⎝2 2⎠ ⎝ 2 ⎛ 4 + 3 7 + 4 ⎞ ⎛ 7 11 ⎞ midpoint of CD = ⎜ , ⎟=⎜ , ⎟ 2 ⎠ ⎝2 2 ⎠ ⎝ 2 2 ⎛ 3 7 ⎞ ⎛ 9 11 ⎞ d = ⎜ − ⎟ +⎜ − ⎟ ⎝2 2⎠ ⎝2 2 ⎠ 2 = 4 + 1 = 5 ≈ 2.24 d = (4 – 5)2 + (5 + 8) 2 = 170 ≈ 13.04 4. 11 (x – 1)2 + (y – 1)2 = 1 12. ( x + 2)2 + ( y − 3)2 = 42 ( x + 2)2 + ( y − 3)2 = 16 13. ( x − 2) 2 + ( y + 1) 2 = r 2 (5 − 2)2 + (3 + 1) 2 = r 2 r 2 = 9 + 16 = 25 ( x − 2) 2 + ( y + 1) 2 = 25 d = (−1 − 6)2 + (5 − 3) 2 = 49 + 4 = 53 ≈ 7.28 14 Section 0.3 Instructor’s Resource Manual 14. ( x − 4) 2 + ( y − 3) 2 = r 2 21. 4 x 2 + 16 x + 15 + 4 y 2 + 6 y = 0 (6 − 4) 2 + (2 − 3) 2 = r 2 3 9⎞ 9 ⎛ 4( x 2 + 4 x + 4) + 4 ⎜ y 2 + y + ⎟ = −15 + 16 + 2 16 ⎠ 4 ⎝ 2 r = 4 +1 = 5 2 3⎞ 13 ⎛ 4( x + 2)2 + 4 ⎜ y + ⎟ = 4⎠ 4 ⎝ ( x − 4)2 + ( y − 3)2 = 5 ⎛ 1+ 3 3 + 7 ⎞ 15. center = ⎜ , ⎟ = (2, 5) 2 ⎠ ⎝ 2 1 1 radius = (1 – 3)2 + (3 – 7)2 = 4 + 16 2 2 1 = 20 = 5 2 2 2 ( x – 2) + ( y – 5) = 5 2 3⎞ 13 ⎛ ( x + 2)2 + ⎜ y + ⎟ = 4⎠ 16 ⎝ 3⎞ ⎛ center = ⎜ −2, − ⎟ ; radius = 4⎠ ⎝ 105 + 4 y2 + 3 y = 0 16 3 9 ⎞ ⎛ 2 4( x + 4 x + 4) + 4 ⎜ y 2 + y + ⎟ 4 64 ⎠ ⎝ 105 9 =− + 16 + 16 16 22. 4 x 2 + 16 x + 16. Since the circle is tangent to the x-axis, r = 4. ( x − 3)2 + ( y − 4) 2 = 16 17. x 2 + 2 x + 10 + y 2 – 6 y –10 = 0 2 3⎞ ⎛ 4( x + 2)2 + 4 ⎜ y + ⎟ = 10 8 ⎝ ⎠ x2 + 2 x + y 2 – 6 y = 0 2 ( x 2 + 2 x + 1) + ( y 2 – 6 y + 9) = 1 + 9 3⎞ 5 ⎛ ( x + 2)2 + ⎜ y + ⎟ = 8⎠ 2 ⎝ ( x + 1) 2 + ( y – 3) 2 = 10 3⎞ 5 10 ⎛ center = ⎜ −2, − ⎟ ; radius = = 8 2 2 ⎝ ⎠ center = (–1, 3); radius = 10 x 2 + y 2 − 6 y = 16 18. x 2 + ( y 2 − 6 y + 9) = 16 + 9 23. 2 –1 =1 2 –1 24. 7−5 =2 4−3 25. –6 – 3 9 = –5 – 2 7 26. −6 + 4 =1 0−2 27. 5–0 5 =– 0–3 3 28. 6−0 =1 0+6 x 2 + ( y − 3) 2 = 25 center = (0, 3); radius = 5 19. x 2 + y 2 –12 x + 35 = 0 x 2 –12 x + y 2 = –35 ( x 2 –12 x + 36) + y 2 = –35 + 36 ( x – 6) 2 + y 2 = 1 center = (6, 0); radius = 1 x 2 + y 2 − 10 x + 10 y = 0 20. 2 29. y − 2 = −1( x − 2) y − 2 = −x + 2 x+ y−4 = 0 30. y − 4 = −1( x − 3) y − 4 = −x + 3 2 ( x − 10 x + 25) + ( y + 10 y + 25) = 25 + 25 x+ y−7 = 0 ( x − 5) 2 + ( y + 5)2 = 50 center = ( 5, −5 ) ; radius = 50 = 5 2 13 4 31. y = 2x + 3 2x – y + 3 = 0 32. Instructor’s Resource Manual y = 0x + 5 0x + y − 5 = 0 Section 0.3 15 33. m = 8–3 5 = ; 4–2 2 5 y – 3 = ( x – 2) 2 2 y – 6 = 5 x – 10 c. 3 y = –2 x + 6 y=– x − 4y + 0 = 0 2 y + 3 = – ( x – 3) 3 2 y = – x –1 3 d. 37. 6 – 2 y = 10 x – 2 –2 y = 10 x – 8 y = –5 x + 4; slope = –5; y-intercept = 4 38. 4 x + 5 y = −20 5 y = −4 x − 20 4 y = − x−4 5 4 slope = − ; y -intercept = − 4 5 39. a. b. m = 2; y + 3 = 2( x – 3) y = 2x – 9 1 m=– ; 2 1 y + 3 = – ( x – 3) 2 1 3 y=– x– 2 2 16 Section 0.3 3 m= ; 2 3 ( x – 3) 2 3 15 y= x– 2 2 y+3= 2 1 2 35. 3y = –2x + 1; y = – x + ; slope = – ; 3 3 3 1 y -intercept = 3 36. −4 y = 5 x − 6 5 3 y = − x+ 4 2 5 3 slope = − ; y -intercept = 4 2 2 x + 2; 3 2 m=– ; 3 5x – 2 y – 4 = 0 2 −1 1 34. m = = ; 8−4 4 1 y − 1 = ( x − 4) 4 4y − 4 = x − 4 2x + 3 y = 6 –1 – 2 3 =– ; 3 +1 4 3 y + 3 = – ( x – 3) 4 3 3 y=– x– 4 4 e. m= f. x=3 40. a. g. y = –3 3 x + cy = 5 3(3) + c(1) = 5 c = −4 b. c=0 c. 2 x + y = −1 y = −2 x − 1 m = −2; 3x + cy = 5 cy = −3x + 5 3 5 y = − x+ c c 3 −2 = − c 3 c= 2 d. c must be the same as the coefficient of x, so c = 3. Instructor’s Resource Manual e. y − 2 = 3( x + 3); 1 perpendicular slope = − ; 3 1 3 − =− 3 c c=9 3 ( x + 2) 2 3 y = x+2 2 y +1 = b. c. 2x + 3 y = 4 9 x – 3 y = –15 = –11 11x x = –1 –3(–1) + y = 5 3 41. m = ; 2 42. a. 45. 2 x + 3 y = 4 –3x + y = 5 m = 2; kx − 3 y = 10 −3 y = − kx + 10 10 k y = x− 3 3 k = 2; k = 6 3 1 m=− ; 2 k 1 =− 3 2 3 k =− 2 2x + 3 y = 6 3 y = −2 x + 6 2 y = − x + 2; 3 3 k 3 9 m= ; = ; k= 2 3 2 2 y=2 Point of intersection: (–1, 2) 3 y = –2 x + 4 2 4 y = – x+ 3 3 3 m= 2 3 y − 2 = ( x + 1) 2 3 7 y = x+ 2 2 46. 4 x − 5 y = 8 2 x + y = −10 4x − 5 y = 8 −4 x − 2 y = 20 − 7 y = 28 y = −4 4 x − 5(−4) = 8 4 x = −12 x = −3 Point of intersection: ( −3, −4 ) ; 4x − 5 y = 8 −5 y = −4 x + 8 y= 43. y = 3(3) – 1 = 8; (3, 9) is above the line. b−0 b =− 0−a a b bx x y y = − x + b; + y = b; + = 1 a a a b 44. (a, 0), (0, b); m = Instructor’s Resource Manual m=− 4 8 x− 5 5 5 4 5 y + 4 = − ( x + 3) 4 5 31 y = − x− 4 4 Section 0.3 17 47. 3x – 4 y = 5 2x + 3y = 9 9 x – 12 y = 15 8 x + 12 y = 36 17 x = 51 x=3 3(3) – 4 y = 5 –4 y = –4 y =1 Point of intersection: (3, 1); 3x – 4y = 5; –4 y = –3x + 5 3 5 y= x– 4 4 4 m=– 3 4 y – 1 = – ( x – 3) 3 4 y = – x+5 3 48. 5 x – 2 y = 5 2x + 3y = 6 15 x – 6 y = 15 4 x + 6 y = 12 19 x = 27 27 x= 19 ⎛ 27 ⎞ 2⎜ ⎟ + 3y = 6 ⎝ 19 ⎠ 60 3y = 19 20 y= 19 ⎛ 27 20 ⎞ Point of intersection: ⎜ , ⎟ ; ⎝ 19 19 ⎠ 5x − 2 y = 5 –2 y = –5 x + 5 5 5 y= x– 2 2 2 m=– 5 20 2⎛ 27 ⎞ y– = – ⎜x− ⎟ 19 5⎝ 19 ⎠ 2 54 20 y = – x+ + 5 95 19 2 154 y = − x+ 5 95 18 Section 0.3 ⎛ 2 + 6 –1 + 3 ⎞ , 49. center: ⎜ ⎟ = (4, 1) 2 ⎠ ⎝ 2 ⎛ 2+ 6 3+3⎞ midpoint = ⎜ , ⎟ = (4, 3) 2 ⎠ ⎝ 2 inscribed circle: radius = (4 – 4)2 + (1 – 3)2 = 4=2 2 ( x – 4) + ( y – 1)2 = 4 circumscribed circle: radius = (4 – 2)2 + (1 – 3)2 = 8 ( x – 4)2 + ( y –1)2 = 8 50. The radius of each circle is 16 = 4. The centers are (1, −2 ) and ( −9,10 ) . The length of the belt is the sum of half the circumference of the first circle, half the circumference of the second circle, and twice the distance between their centers. 1 1 L = ⋅ 2π (4) + ⋅ 2π (4) + 2 (1 + 9)2 + (−2 − 10)2 2 2 = 8π + 2 100 + 144 ≈ 56.37 51. Put the vertex of the right angle at the origin with the other vertices at (a, 0) and (0, b). The ⎛a b⎞ midpoint of the hypotenuse is ⎜ , ⎟ . The ⎝ 2 2⎠ distances from the vertices are 2 2 a⎞ ⎛ b⎞ ⎛ ⎜a – ⎟ +⎜0 – ⎟ = 2 2⎠ ⎝ ⎠ ⎝ = 2 2 a⎞ ⎛ b⎞ ⎛ ⎜0 – ⎟ + ⎜b – ⎟ = 2⎠ ⎝ 2⎠ ⎝ = 2 2 a⎞ ⎛ b⎞ ⎛ ⎜0 – ⎟ + ⎜0 – ⎟ = 2⎠ ⎝ 2⎠ ⎝ = a 2 b2 + 4 4 1 2 a + b2 , 2 a 2 b2 + 4 4 1 2 a + b 2 , and 2 a 2 b2 + 4 4 1 2 a + b2 , 2 which are all the same. 52. From Problem 51, the midpoint of the hypotenuse, ( 4,3, ) , is equidistant from the vertices. This is the center of the circle. The radius is 16 + 9 = 5. The equation of the circle is ( x − 4) 2 + ( y − 3) 2 = 25. Instructor’s Resource Manual 53. x 2 + y 2 – 4 x – 2 y – 11 = 0 ( x 2 – 4 x + 4) + ( y 2 – 2 y + 1) = 11 + 4 + 1 ( x – 2)2 + ( y – 1)2 = 16 x 2 + y 2 + 20 x – 12 y + 72 = 0 ( x 2 + 20 x + 100) + ( y 2 – 12 y + 36) = –72 + 100 + 36 2 2 ( x + 10) + ( y – 6) = 64 center of first circle: (2, 1) center of second circle: (–10, 6) d = (2 + 10)2 + (1 – 6) 2 = 144 + 25 = 169 = 13 However, the radii only sum to 4 + 8 = 12, so the circles must not intersect if the distance between their centers is 13. 54. x 2 + ax + y 2 + by + c = 0 ⎛ 2 a2 ⎞ ⎛ 2 b2 ⎞ ⎜ x + ax + ⎟ + ⎜ y + by + ⎟ ⎜ 4 ⎟⎠ ⎜⎝ 4 ⎟⎠ ⎝ = −c + a 2 b2 + 4 4 2 2 a⎞ ⎛ b⎞ a 2 + b 2 − 4c ⎛ ⎜x+ ⎟ +⎜ y+ ⎟ = 2⎠ ⎝ 2⎠ 4 ⎝ 2 2 a + b − 4c > 0 ⇒ a 2 + b 2 > 4c 4 55. Label the points C, P, Q, and R as shown in the figure below. Let d = OP , h = OR , and a = PR . Triangles ΔOPR and ΔCQR are similar because each contains a right angle and they share angle ∠QRC . For an angle of 30 , a 1 d 3 and = ⇒ h = 2a . Using a = h 2 h 2 56. The equations of the two circles are ( x − R)2 + ( y − R)2 = R 2 ( x − r )2 + ( y − r )2 = r 2 Let ( a, a ) denote the point where the two circles touch. This point must satisfy (a − R)2 + (a − R)2 = R 2 R2 2 ⎛ 2⎞ a = ⎜⎜ 1 ± ⎟R 2 ⎟⎠ ⎝ (a − R)2 = ⎛ 2⎞ Since a < R , a = ⎜⎜1 − ⎟ R. 2 ⎟⎠ ⎝ At the same time, the point where the two circles touch must satisfy (a − r )2 + (a − r )2 = r 2 ⎛ 2⎞ a = ⎜⎜ 1 ± ⎟r 2 ⎟⎠ ⎝ ⎛ 2⎞ Since a > r , a = ⎜⎜ 1 + ⎟ r. 2 ⎟⎠ ⎝ Equating the two expressions for a yields ⎛ ⎛ 2⎞ 2⎞ ⎜⎜1 − 2 ⎟⎟ R = ⎜⎜ 1 + 2 ⎟⎟ r ⎝ ⎠ ⎝ ⎠ 2 ⎛ 2⎞ 2 ⎜⎜ 1 − ⎟⎟ 1− 2 ⎝ ⎠ 2 R= r= R ⎛ ⎞⎛ 2 2 2⎞ 1+ ⎜⎜ 1 + ⎟⎜ 1 − ⎟ 2 2 ⎟⎜ 2 ⎟⎠ ⎝ ⎠⎝ 1 1− 2 + 2R r= 1 1− 2 r = (3 − 2 2) R ≈ 0.1716 R property of similar triangles, QC / RC = 3 / 2 , 2 3 4 = → a = 2+ a−2 2 3 By the Pythagorean Theorem, we have d = h 2 − a 2 = 3a = 2 3 + 4 ≈ 7.464 Instructor’s Resource Manual Section 0.3 19 57. Refer to figure 15 in the text. Given ine l1 with slope m, draw ABC with vertical and horizontal sides m, 1. Line l2 is obtained from l1 by rotating it around the point A by 90° counter-clockwise. Triangle ABC is rotated into triangle AED . We read off 1 1 slope of l2 = =− . m −m 60. See the figure below. The angle at T is a right angle, so the Pythagorean Theorem gives ( PM + r )2 = ( PT )2 + r 2 ⇔ ( PM )2 + 2rPM + r 2 = ( PT )2 + r 2 ⇔ PM ( PM + 2r ) = ( PT )2 PM + 2r = PN so this gives ( PM )( PN ) = ( PT ) 2 58. 2 ( x − 1)2 + ( y − 1)2 = ( x − 3) 2 + ( y − 4)2 4( x 2 − 2 x + 1 + y 2 − 2 y + 1) = x 2 − 6 x + 9 + y 2 − 8 y + 16 3x 2 − 2 x + 3 y 2 = 9 + 16 − 4 − 4; 2 17 x + y2 = ; 3 3 1⎞ 17 1 ⎛ 2 2 2 ⎜x − x+ ⎟+ y = + 3 9⎠ 3 9 ⎝ 3x 2 − 2 x + 3 y 2 = 17; x 2 − 2 1⎞ 52 ⎛ 2 ⎜x− ⎟ + y = 3⎠ 9 ⎝ B = (6)2 + (8)2 = 100 = 10 ⎛ 52 ⎞ ⎛1 ⎞ center: ⎜ , 0 ⎟ ; radius: ⎜⎜ ⎟⎟ ⎝3 ⎠ ⎝ 3 ⎠ 59. Let a, b, and c be the lengths of the sides of the right triangle, with c the length of the hypotenuse. Then the Pythagorean Theorem says that a 2 + b 2 = c 2 Thus, πa 2 πb 2 πc 2 + = or 8 8 8 2 61. The lengths A, B, and C are the same as the corresponding distances between the centers of the circles: A = (–2)2 + (8)2 = 68 ≈ 8.2 2 1 ⎛a⎞ 1 ⎛b⎞ 1 ⎛c⎞ π⎜ ⎟ + π⎜ ⎟ = π⎜ ⎟ 2 ⎝2⎠ 2 ⎝2⎠ 2 ⎝2⎠ C = (8)2 + (0)2 = 64 = 8 Each circle has radius 2, so the part of the belt around the wheels is 2(2π − a − π ) + 2(2π − b − π ) + 2(2π − c − π ) = 2[3π - (a + b + c)] = 2(2π ) = 4π Since a + b + c = π , the sum of the angles of a triangle. The length of the belt is ≈ 8.2 + 10 + 8 + 4π ≈ 38.8 units. 2 2 1 ⎛ x⎞ π ⎜ ⎟ is the area of a semicircle with 2 ⎝2⎠ diameter x, so the circles on the legs of the triangle have total area equal to the area of the semicircle on the hypotenuse. From a 2 + b 2 = c 2 , 3 2 3 2 3 2 a + b = c 4 4 4 3 2 x is the area of an equilateral triangle 4 with sides of length x, so the equilateral triangles on the legs of the right triangle have total area equal to the area of the equilateral triangle on the hypotenuse of the right triangle. 20 Section 0.3 62 As in Problems 50 and 61, the curved portions of the belt have total length 2π r. The lengths of the straight portions will be the same as the lengths of the sides. The belt will have length 2π r + d1 + d 2 + … + d n . Instructor’s Resource Manual 63. A = 3, B = 4, C = –6 3(–3) + 4(2) + (–6) 7 d= = 5 (3) 2 + (4)2 64. A = 2, B = −2, C = 4 d= 2(4) − 2(−1) + 4) 2 (2) + (2) 2 = 14 8 = 7 2 2 65. A = 12, B = –5, C = 1 12(–2) – 5(–1) + 1 18 d= = 13 (12) 2 + (–5) 2 66. A = 2, B = −1, C = −5 d= 2(3) − 1(−1) − 5 2 (2) + (−1) 2 = 2 5 = 2 5 5 67. 2 x + 4(0) = 5 5 x= 2 d= 2 ( 52 ) + 4(0) – 7 = (2)2 + (4) 2 2 20 = 5 5 68. 7(0) − 5 y = −1 1 y= 5 ⎛1⎞ 7(0) − 5 ⎜ ⎟ − 6 7 7 74 ⎝5⎠ d= = = 2 2 74 74 (7) + (−5) −2 − 3 5 3 = − ; m = ; passes through 1+ 2 3 5 ⎛ −2 + 1 3 − 2 ⎞ ⎛ 1 1 ⎞ , ⎜ ⎟ = ⎜− , ⎟ 2 ⎠ ⎝ 2 2⎠ ⎝ 2 1 3⎛ 1⎞ y− = ⎜x+ ⎟ 2 5⎝ 2⎠ 3 4 y = x+ 5 5 69. m = Instructor’s Resource Manual 0–4 1 = –2; m = ; passes through 2–0 2 ⎛0+2 4+0⎞ , ⎜ ⎟ = (1, 2) 2 ⎠ ⎝ 2 1 y – 2 = ( x – 1) 2 1 3 y = x+ 2 2 6–0 1 m= = 3; m = – ; passes through 4–2 3 ⎛ 2+4 0+6⎞ , ⎜ ⎟ = (3, 3) 2 ⎠ ⎝ 2 1 y – 3 = – ( x – 3) 3 1 y = – x+4 3 1 3 1 x+ = – x+4 2 2 3 5 5 x= 6 2 x=3 1 3 y = (3) + = 3 2 2 center = (3, 3) 70. m = 71. Let the origin be at the vertex as shown in the figure below. The center of the circle is then ( 4 − r , r ) , so it has equation ( x − (4 − r ))2 + ( y − r )2 = r 2 . Along the side of length 5, the y-coordinate is always 3 4 times the x-coordinate. Thus, we need to find the value of r for which there is exactly one x2 ⎛3 ⎞ solution to ( x − 4 + r ) 2 + ⎜ x − r ⎟ = r 2 . ⎝4 ⎠ Solving for x in this equation gives 16 ⎛ ⎞ x = ⎜16 − r ± 24 −r 2 + 7r − 6 ⎟ . There is 25 ⎝ ⎠ ( ) exactly one solution when −r 2 + 7 r − 6 = 0, that is, when r = 1 or r = 6 . The root r = 6 is extraneous. Thus, the largest circle that can be inscribed in this triangle has radius r = 1. Section 0.3 21 72. The line tangent to the circle at ( a, b ) will be The slope of PS is 1 [ y1 + y4 − ( y1 + y2 )] y − y 2 2 = 4 . The slope of 1 x − x 4 2 x + x − x + x ( ) [1 4 1 2] 2 1 [ y3 + y4 − ( y2 + y3 )] y − y 2 . Thus QR is 2 = 4 1 x − x [ x3 + x4 − ( x2 + x3 )] 4 2 2 PS and QR are parallel. The slopes of SR and y −y PQ are both 3 1 , so PQRS is a x3 − x1 parallelogram. perpendicular to the line through ( a, b ) and the center of the circle, which is ( 0, 0 ) . The line through ( a, b ) and ( 0, 0 ) has slope 0−b b a r2 = ; ax + by = r 2 ⇒ y = − x + 0−a a b b a so ax + by = r 2 has slope − and is b perpendicular to the line through ( a, b ) and m= ( 0, 0 ) , so it is tangent to the circle at ( a, b ) . 73. 12a + 0b = 36 a=3 32 + b 2 = 36 b = ±3 3 3x – 3 3 y = 36 x – 3 y = 12 3x + 3 3 y = 36 x + 3 y = 12 74. Use the formula given for problems 63-66, for ( x, y ) = ( 0, 0 ) . A = m, B = −1, C = B − b;(0, 0) d= m(0) − 1(0) + B − b m2 + (−1) 2 = B−b m2 + 1 75. The midpoint of the side from (0, 0) to (a, 0) is ⎛0+a 0+0⎞ ⎛ a ⎞ , ⎜ ⎟ = ⎜ , 0⎟ 2 ⎠ ⎝2 ⎠ ⎝ 2 The midpoint of the side from (0, 0) to (b, c) is ⎛0+b 0+c⎞ ⎛b c ⎞ , ⎜ ⎟=⎜ , ⎟ 2 ⎠ ⎝2 2⎠ ⎝ 2 c–0 c m1 = = b–a b–a c –0 c m2 = 2 = ; m1 = m2 b–a b–a 2 77. x 2 + ( y – 6) 2 = 25; passes through (3, 2) tangent line: 3x – 4y = 1 The dirt hits the wall at y = 8. 0.4 Concepts Review 1. y-axis 2. ( 4, −2 ) 3. 8; –2, 1, 4 4. line; parabola Problem Set 0.4 1. y = –x2 + 1; y-intercept = 1; y = (1 + x)(1 – x); x-intercepts = –1, 1 Symmetric with respect to the y-axis 2 76. See the figure below. The midpoints of the sides are ⎛ x + x y + y3 ⎞ ⎛ x + x y + y2 ⎞ P⎜ 1 2 , 1 , Q⎜ 2 3 , 2 , ⎟ 2 ⎠ 2 ⎟⎠ ⎝ 2 ⎝ 2 ⎛ x + x y + y4 ⎞ R⎜ 3 4 , 3 , and 2 ⎟⎠ ⎝ 2 ⎛ x + x y + y4 ⎞ S⎜ 1 4 , 1 . 2 ⎟⎠ ⎝ 2 22 Section 0.4 Instructor’s Resource Manual 2. x = − y 2 + 1; y -intercepts = −1,1; x-intercept = 1 . Symmetric with respect to the x-axis. 3. x = –4y2 – 1; x-intercept = –1 Symmetric with respect to the x-axis 4. y = 4 x 2 − 1; y -intercept = −1 1 1 y = (2 x + 1)(2 x − 1); x-intercepts = − , 2 2 Symmetric with respect to the y-axis. 5. x2 + y = 0; y = –x2 x-intercept = 0, y-intercept = 0 Symmetric with respect to the y-axis 6. y = x 2 − 2 x; y -intercept = 0 y = x(2 − x); x-intercepts = 0, 2 7 7. 7x2 + 3y = 0; 3y = –7x2; y = – x 2 3 x-intercept = 0, y-intercept = 0 Symmetric with respect to the y-axis 8. y = 3x 2 − 2 x + 2; y -intercept = 2 Instructor’s Resource Manual Section 0.4 23 9. x2 + y2 = 4 x-intercepts = -2, 2; y-intercepts = -2, 2 Symmetric with respect to the x-axis, y-axis, and origin 10. 3x 2 + 4 y 2 = 12; y-intercepts = − 3, 3 x-intercepts = −2, 2 Symmetric with respect to the x-axis, y-axis, and origin 11. y = –x2 – 2x + 2: y-intercept = 2 2± 4+8 2±2 3 x-intercepts = = = –1 ± 3 –2 –2 24 Section 0.4 12. 4 x 2 + 3 y 2 = 12; y -intercepts = −2, 2 x-intercepts = − 3, 3 Symmetric with respect to the x-axis, y-axis, and origin 13. x2 – y2 = 4 x-intercept = -2, 2 Symmetric with respect to the x-axis, y-axis, and origin 14. x 2 + ( y − 1)2 = 9; y -intercepts = −2, 4 x-intercepts = −2 2, 2 2 Symmetric with respect to the y-axis Instructor’s Resource Manual 15. 4(x – 1)2 + y2 = 36; y-intercepts = ± 32 = ±4 2 x-intercepts = –2, 4 Symmetric with respect to the x-axis 18. x 4 + y 4 = 1; y -intercepts = −1,1 x-intercepts = −1,1 Symmetric with respect to the x-axis, y-axis, and origin 16. x 2 − 4 x + 3 y 2 = −2 19. x4 + y4 = 16; y-intercepts = −2, 2 x-intercepts = −2, 2 Symmetric with respect to the y-axis, x-axis and origin x-intercepts = 2 ± 2 Symmetric with respect to the x-axis 17. x2 + 9(y + 2)2 = 36; y-intercepts = –4, 0 x-intercept = 0 Symmetric with respect to the y-axis Instructor’s Resource Manual 20. y = x3 – x; y-intercepts = 0; y = x(x2 – 1) = x(x + 1)(x – 1); x-intercepts = –1, 0, 1 Symmetric with respect to the origin Section 0.4 25 21. y = 1 2 ; y-intercept = 1 x +1 Symmetric with respect to the y-axis 22. y = 2 24. 4 ( x − 5 ) + 9( y + 2) 2 = 36; x-intercept = 5 25. y = (x – 1)(x – 2)(x – 3); y-intercept = –6 x-intercepts = 1, 2, 3 x ; y -intercept = 0 x +1 x-intercept = 0 Symmetric with respect to the origin 2 26. y = x2(x – 1)(x – 2); y-intercept = 0 x-intercepts = 0, 1, 2 23. 2 x 2 – 4 x + 3 y 2 + 12 y = –2 2( x 2 – 2 x + 1) + 3( y 2 + 4 y + 4) = –2 + 2 + 12 2( x – 1)2 + 3( y + 2)2 = 12 y-intercepts = –2 ± 30 3 x-intercept = 1 27. y = x 2 ( x − 1)2 ; y-intercept = 0 x-intercepts = 0, 1 26 Section 0.4 Instructor’s Resource Manual 28. y = x 4 ( x − 1)4 ( x + 1)4 ; y -intercept = 0 x-intercepts = −1, 0,1 Symmetric with respect to the y-axis Intersection points: (0, 1) and (–3, 4) 32. 2 x + 3 = −( x − 1) 2 29. x + y = 1; y-intercepts = –1, 1; x-intercepts = –1, 1 Symmetric with respect to the x-axis, y-axis and origin 2 x + 3 = − x2 + 2 x − 1 x2 + 4 = 0 No points of intersection 33. −2 x + 3 = −2( x − 4)2 30. x + y = 4; y-intercepts = –4, 4; x-intercepts = –4, 4 Symmetric with respect to the x-axis, y-axis and origin −2 x + 3 = −2 x 2 + 16 x − 32 2 x 2 − 18 x + 35 = 0 x= 18 ± 324 – 280 18 ± 2 11 9 ± 11 = = ; 4 4 2 ⎛ 9 – 11 ⎞ , – 6 + 11 ⎟⎟ , Intersection points: ⎜⎜ ⎝ 2 ⎠ ⎛ 9 + 11 ⎞ , – 6 – 11 ⎟⎟ ⎜⎜ ⎝ 2 ⎠ 31. − x + 1 = ( x + 1)2 − x + 1 = x2 + 2 x + 1 x 2 + 3x = 0 x( x + 3) = 0 x = 0, −3 Instructor’s Resource Manual Section 0.4 27 37. y = 3x + 1 34. −2 x + 3 = 3x 2 − 3 x + 12 x 2 + 2 x + (3 x + 1) 2 = 15 3x 2 − x + 9 = 0 No points of intersection x 2 + 2 x + 9 x 2 + 6 x + 1 = 15 10 x 2 + 8 x − 14 = 0 2(5 x 2 + 4 x − 7) = 0 −2 ± 39 ≈ −1.65, 0.85 5 Intersection points: ⎛ −2 − 39 −1 − 3 39 ⎞ , ⎜ ⎟ and ⎜ ⎟ 5 5 ⎝ ⎠ ⎛ −2 + 39 −1 + 3 39 ⎞ , ⎜ ⎟ ⎜ ⎟ 5 5 ⎝ ⎠ [ or roughly (–1.65, –3.95) and (0.85, 3.55) ] x= 35. x 2 + x 2 = 4 x2 = 2 x=± 2 ( )( Intersection points: – 2, – 2 , 2, 2 ) 38. x 2 + (4 x + 3) 2 = 81 x 2 + 16 x 2 + 24 x + 9 = 81 17 x 2 + 24 x − 72 = 0 −12 ± 38 ≈ −2.88, 1.47 17 Intersection points: ⎛ −12 − 38 3 − 24 38 ⎞ , ⎜ ⎟ and ⎜ ⎟ 17 17 ⎝ ⎠ ⎛ −12 + 38 3 + 24 38 ⎞ , ⎜ ⎟ ⎜ ⎟ 17 17 ⎝ ⎠ [ or roughly ( −2.88, −8.52 ) , (1.47,8.88 ) ] x= 36. 2 x 2 + 3( x − 1)2 = 12 2 x 2 + 3 x 2 − 6 x + 3 = 12 5x2 − 6 x − 9 = 0 6 ± 36 + 180 6 ± 6 6 3 ± 3 6 = = 10 10 5 Intersection points: ⎛ 3 − 3 6 −2 − 3 6 ⎞ ⎛ 3 + 3 6 −2 + 3 6 ⎞ , , ⎜⎜ ⎟⎟ , ⎜⎜ ⎟⎟ 5 5 ⎝ 5 ⎠ ⎝ 5 ⎠ x= 28 Section 0.4 Instructor’s Resource Manual 39. a. y = x 2 ; (2) ( b. ax3 + bx 2 + cx + d , with a > 0 : (1) c. ax3 + bx 2 + cx + d , with a < 0 : (3) d. y = ax3 , with a > 0 : (4) 40. x 2 + y 2 = 13;(−2, −3), (−2,3), (2, −3), (2,3) 2 2 2 2 d1 = (2 + 2) + (−3 + 3) = 4 2 2 d3 = (2 − 2) + (3 + 3) = 6 Three such distances. ( ) )( ) ( ) d1 = (–2 – 2) 2 + ⎡1 + 21 – 1 + 13 ⎤ ⎣ ⎦ ( 21 – 13 ) ) ) 2 2 ( ) d3 = (−2 + 2)2 + ⎡1 + 21 − 1 − 21 ⎤ ⎣ ⎦ ( 21 + 21 ) 2. 2 = ( 2 21 ) 2 ) 2 2 f (1) = 1 – 12 = 0 f (–2) = 1 – (–2)2 = –3 c. f (0) = 1 – 02 = 1 d. f (k ) = 1 – k 2 e. f (–5) = 1 – (–5) 2 = –24 f. 1 15 ⎛1⎞ ⎛1⎞ f ⎜ ⎟ =1– ⎜ ⎟ =1– = 16 16 ⎝4⎠ ⎝4⎠ g. f (1 + h ) = 1 − (1 + h ) = −2h − h 2 h. f (1 + h ) − f (1) = −2h − h 2 − 0 = −2h − h 2 i. f ( 2 + h ) − f ( 2) = 1 − ( 2 + h ) + 3 2 2 d 4 = (−2 − 2)2 + ⎡⎣1 − 21 − (1 + 13) ⎤⎦ ( 2 2 = −4h − h 2 = 50 + 2 273 ≈ 9.11 ( ) d5 = (−2 − 2)2 + ⎡1 − 21 − 1 − 13 ⎤ ⎣ ⎦ = 16 + ( 13 − 21 ) 2 2 2. a. b. F (1) = 13 + 3 ⋅1 = 4 F ( 2) = ( 2)3 + 3( 2) = 2 2 + 3 2 =5 2 = 50 − 2 273 ≈ 4.12 3 c. Instructor’s Resource Manual 2 b. = 2 21 ≈ 9.17 = 16 + − 21 − 13 ( 2 13 ) f (2u ) = 3(2u ) 2 = 12u 2 ; f ( x + h) = 3( x + h)2 1. a. = 50 + 2 273 ≈ 9.11 = 0+ = 0.5 Concepts Review 2 ( 21 + 13 2 Problem Set 0.5 d 2 = (–2 – 2)2 + ⎡1 + 21 – 1 – 13 ⎤ ⎣ ⎦ ( ) = 2 13 ≈ 7.21 Four such distances ( d 2 = d 4 and d1 = d5 ). 2 = 50 – 2 273 ≈ 4.12 = 16 + 13 + 13 4. even; odd; y-axis; origin 21 , 2, 1 + 13 , 2, 1 – 13 = 16 + ( 3. asymptote 41. x2 + 2x + y2 – 2y = 20; –2, 1 + 21 , )( = 0+ 2 1. domain; range d 2 = (2 + 2) + (−3 − 3) = 52 = 2 13 ( –2, 1 – ) d6 = (2 − 2)2 + ⎡1 + 13 − 1 − 13 ⎤ ⎣ ⎦ ⎛1⎞ ⎛1⎞ ⎛ 1 ⎞ 1 3 49 F ⎜ ⎟ = ⎜ ⎟ + 3⎜ ⎟ = + = ⎝4⎠ ⎝4⎠ ⎝ 4 ⎠ 64 4 64 Section 0.5 29 d. F (1 + h ) = (1 + h ) + 3 (1 + h ) 3 f. Φ ( x 2 + x) = = 1 + 3h + 3h 2 + h3 + 3 + 3h = 4 + 6h + 3h 2 + h3 e. F (1 + h ) − 1 = 3 + 6h + 3h + h f. F ( 2 + h) − F ( 2) 2 = 3 2 = 15h + 6h + h 3. a. b. G (0) = d. e. f. 4. a. b. 30 0.25 − 3 1 f ( x) = c. f (3 + 2) = 3 1 = –1 0 –1 1 f (0.25) = b. π −3 G( y ) = G (– x) = c. 1 1 =– – x –1 x +1 7. a. 2 1 x = – 1 1 – x2 1 2 ≈ 0.841 ≈ −3.293 (12.26) 2 + 9 12.26 – 3 ≈ 1.199 1 –t 1 2 c. ⎛1⎞ Φ⎜ ⎟ = ⎝2⎠ d. Φ (u + 1) = e. Φ( x2 ) = + ( 12 ) = 2 1 2 = x2 = x2 + y2 = 1 c. x = 2 y +1 x2 = 2 y + 1 ≈ 1.06 (u + 1) + (u + 1) 2 ( x2 ) + ( x2 )2 Section 0.5 –t u +1 ; undefined b. xy + y + x = 1 y(x + 1) = 1 – x 1– x 1– x y= ; f ( x) = x +1 x +1 t2 – t 3 4 1 2 3– 3 y = ± 1 – x 2 ; not a function =2 –t + (– t ) 2 ( 3)2 + 9 f ( 3) = y 2 = 1– x 2 1 x2 1 + 12 Φ (–t ) = is not y 2 –1 ⎛ 1 ⎞ G⎜ ⎟ = ⎝ x2 ⎠ Φ (1) = = 3+ 2 −3 −0.25 0.79 – 3 b. f(12.26) = 1 − 2.75 ≈ 2.658 (0.79) 2 + 9 f(0.79) = 1 G (1.01) = = 100 1.01 – 1 2 1 = 1 =2 1 G (0.999) = = –1000 0.999 –1 6. a. c. x2 + x defined = ( 2 + h ) + 3 ( 2 + h ) − ⎡ 23 − 3 ( 2 ) ⎤ ⎣ ⎦ = 8 + 12h + 6h 2 + h3 + 6 + 3h − 14 x2 + x x 4 + 2 x3 + 2 x 2 + x 3 5. a. ( x 2 + x) + ( x 2 + x) 2 = y= u 2 + 3u + 2 x2 + x4 x u +1 d. x2 – 1 x2 – 1 ; f ( x) = 2 2 y y+1 xy + x = y x = y – xy x = y(1 – x) x x ; f ( x) = y= 1– x 1– x x= Instructor’s Resource Manual 8. The graphs on the left are not graphs of functions, the graphs on the right are graphs of functions. 9. f (a + h) – f (a) [2(a + h) 2 – 1] – (2a 2 – 1) = h h 4ah + 2h 2 = = 4a + 2h h x 2 – 9 ≥ 0; x 2 ≥ 9; x ≥ 3 Domain: {x ∈ d. F (a + h) – F (a ) 4(a + h) – 4a = h h = 4a3 + 12a 2 h + 12ah 2 + 4h3 – 4a3 h = 12a 2 h + 12ah 2 + 4h3 h = x − 4 x + hx − 2h + 4 h −3h = 2 h( x − 4 x + hx − 2h + 4) 3 =– 2 x – 4 x + hx – 2h + 4 a+h a + h+ 4 : y ≤ 5} 14. a. b. f ( x) = 4 – x2 = 4 – x2 ( x – 3)( x + 2) x2 – x – 6 Domain: {x ∈ : x ≠ −2, 3} G ( y ) = ( y + 1) –1 1 ≥ 0; y > –1 y +1 3 3 g ( x + h) – g ( x) x + h –2 – x –2 = h h 3x − 6 − 3x − 3h + 6 G ( a + h) – G ( a ) = h H ( y ) = – 625 – y 4 Domain: { y ∈ Domain: { y ∈ : y > −1} c. φ (u ) = 2u + 3 (all real numbers) Domain: d. F (t ) = t 2 / 3 – 4 (all real numbers) Domain: 2 12. : x ≥ 3} 3 = 12a 2 + 12ah + 4h 2 11. ψ ( x) = x 2 – 9 625 – y 4 ≥ 0; 625 ≥ y 4 ; y ≤ 5 3 10. c. 15. f(x) = –4; f(–x) = –4; even function – a +a 4 h 2 a + 4a + ah + 4h − a 2 − ah − 4a a 2 + 8a + ah + 4h + 16 h 4h = = = 13. a. h(a 2 + 8a + ah + 4h + 16) 4 a 2 + 8a + ah + 4h + 16 F ( z) = 2 z + 3 2z + 3 ≥ 0; z ≥ – ⎧ Domain: ⎨ z ∈ ⎩ b. 16. f(x) = 3x; f(–x) = –3x; odd function g (v ) = 3 2 3⎫ :z≥− ⎬ 2⎭ 1 4v – 1 4v – 1 = 0; v = ⎧ Domain: ⎨v ∈ ⎩ 1 4 1⎫ :v≠ ⎬ 4⎭ Instructor’s Resource Manual Section 0.5 31 17. F(x) = 2x + 1; F(–x) = –2x + 1; neither 20. g (u ) = 18. F ( x) = 3x – 2; F (– x) = –3x – 2; neither 21. g ( x) = 19. g ( x) = 3x 2 + 2 x – 1; g (– x) = 3 x 2 – 2 x – 1 ; neither 32 Section 0.5 22. φ ( z ) = u3 u3 ; g (– u ) = – ; odd function 8 8 x 2 x –1 ; g (– x) = –x 2 x –1 ; odd 2z +1 –2 z + 1 ; φ (– z ) = ; neither z –1 –z –1 Instructor’s Resource Manual 23. f ( w) = w – 1; f (– w) = – w – 1; neither 2 2 24. h( x ) = x + 4; h(– x) = x + 4; even function 26. F (t ) = – t + 3 ; F (– t ) = – –t + 3 ; neither 27. g ( x) = x x ; g (− x ) = − ; neither 2 2 28. G ( x) = 2 x − 1 ; G (− x) = −2 x + 1 ; neither 25. f ( x) = 2 x ; f (– x) = –2 x = 2 x ; even function ⎧1 if t ≤ 0 ⎪ 29. g (t ) = ⎨t + 1 if 0 < t < 2 ⎪2 ⎩t – 1 if t ≥ 2 Instructor’s Resource Manual neither Section 0.5 33 ⎧⎪ – x 2 + 4 if x ≤ 1 30. h( x ) = ⎨ if x > 1 ⎪⎩3x neither 35. Let y denote the length of the other leg. Then x2 + y 2 = h2 y 2 = h2 − x 2 y = h2 − x 2 L ( x ) = h2 − x 2 36. The area is 1 1 A ( x ) = base × height = x h 2 − x 2 2 2 37. a. 31. T(x) = 5000 + 805x Domain: {x ∈ integers: 0 ≤ x ≤ 100} T ( x) 5000 u ( x) = = + 805 x x Domain: {x ∈ integers: 0 < x ≤ 100} P ( x) = 6 x – (400 + 5 x( x – 4)) 32. a. = 6 x – 400 – 5 x( x – 4) E(x) = 24 + 0.40x b. 120 = 24 + 0.40x 0.40x = 96; x = 240 mi 38. The volume of the cylinder is πr 2 h, where h is the height of the cylinder. From the figure, 2 2 2 ⎛ h⎞ 2 h 2 = 3r ; r + ⎜ ⎟ = (2r ) ; ⎝ 2⎠ 4 h = 12r 2 = 2r 3. V (r ) = πr 2 (2r 3) = 2πr 3 3 P(200) ≈ −190 ; P (1000 ) ≈ 610 b. c. ABC breaks even when P(x) = 0; 6 x – 400 – 5 x( x – 4) = 0; x ≈ 390 33. E ( x) = x – x 2 y 0.5 39. The area of the two semicircular ends is 0.5 1 x −0.5 1 exceeds its square by the maximum amount. 2 34. Each side has length p . The height of the 3 πd 2 . 4 1 – πd . 2 2 πd 2 d – πd 2 ⎛ 1 – πd ⎞ πd A(d ) = +d⎜ + ⎟= 4 4 2 ⎝ 2 ⎠ The length of each parallel side is 2d – πd 2 4 Since the track is one mile long, π d < 1, so 1 1⎫ ⎧ d < . Domain: ⎨d ∈ : 0 < d < ⎬ π π⎭ ⎩ = 3p . 6 1 ⎛ p ⎞⎛ 3p ⎞ 3 p2 A( p ) = ⎜ ⎟ ⎜⎜ ⎟⎟ = 2 ⎝ 3 ⎠⎝ 6 ⎠ 36 triangle is 34 Section 0.5 Instructor’s Resource Manual 40. a. 1 3 A(1) = 1(1) + (1)(2 − 1) = 2 2 42. a. f(x + y) = 2(x + y) = 2x + 2y = f(x) + f(y) b. f ( x + y ) = ( x + y )2 = x 2 + 2 xy + y 2 ≠ f ( x) + f ( y ) c. f(x + y) = 2(x + y) + 1 = 2x + 2y + 1 ≠ f(x) + f(y) d. f(x + y) = –3(x + y) = –3x – 3y = f(x) + f(y) b. 1 A(2) = 2(1) + (2)(3 − 1) = 4 2 c. A(0) = 0 d. 1 1 A(c) = c(1) + (c)(c + 1 − 1) = c 2 + c 2 2 e. 43. For any x, x + 0 = x, so f(x) = f(x + 0) = f(x) + f(0), hence f(0) = 0. Let m be the value of f(1). For p in N, p = p ⋅1 = 1 + 1 + ... + 1, so f(p) = f(1 + 1 + ... + 1) = f(1) + f(1) + ... + f(1) = pf(1) = pm. ⎛1⎞ 1 1 1 1 = p ⎜ ⎟ = + + ... + , so p ⎝ p⎠ p p ⎛1 1 1⎞ m = f (1) = f ⎜ + + ... + ⎟ p p p⎠ ⎝ ⎛1⎞ ⎛1⎞ ⎛1⎞ ⎛1⎞ = f ⎜ ⎟ + f ⎜ ⎟ + ... + f ⎜ ⎟ = pf ⎜ ⎟ , ⎝ p⎠ ⎝ p⎠ ⎝ p⎠ ⎝ p⎠ ⎛1⎞ m hence f ⎜ ⎟ = . Any rational number can ⎝ p⎠ p be written as f. Domain: {c ∈ Range: { y ∈ 41. a. b. : c ≥ 0} : y ≥ 0} B (0) = 0 1 1 1 ⎛1⎞ 1 B ⎜ ⎟ = B (1) = ⋅ = 2 6 12 ⎝2⎠ 2 p with p, q in N. q ⎛1⎞ 1 1 p 1 = p ⎜ ⎟ = + + ... + , q q ⎝q⎠ q q ⎛ p⎞ ⎛1 1 1⎞ so f ⎜ ⎟ = f ⎜ + + ... + ⎟ q q q q ⎝ ⎠ ⎝ ⎠ ⎛1⎞ ⎛1⎞ ⎛1⎞ = f ⎜ ⎟ + f ⎜ ⎟ + ... + f ⎜ ⎟ ⎝q⎠ ⎝q⎠ ⎝q⎠ ⎛1⎞ ⎛m⎞ ⎛ p⎞ = pf ⎜ ⎟ = p ⎜ ⎟ = m ⎜ ⎟ ⎝q⎠ ⎝q⎠ ⎝q⎠ c. Instructor’s Resource Manual Section 0.5 35 44. The player has run 10t feet after t seconds. He reaches first base when t = 9, second base when t = 18, third base when t = 27, and home plate when t = 36. The player is 10t – 90 feet from first base when 9 ≤ t ≤ 18, hence 46. a. x f(x) –4 –6.1902 –3 0.4118 –2 13.7651 –1 9.9579 0 0 1 –7.3369 2 –17.7388 if 18 < t ≤ 27 3 –0.4521 if 27 < t ≤ 36 4 4.4378 b. 902 + (10t − 90)2 feet from home plate. The player is 10t – 180 feet from second base when 18 ≤ t ≤ 27, thus he is 90 – (10t – 180) = 270 – 10t feet from third base and 902 + (270 − 10t ) 2 feet from home plate. The player is 10t – 270 feet from third base when 27 ≤ t ≤ 36, thus he is 90 – (10t – 270) = 360 – 10t feet from home plate. a. b. 45. a. b. ⎧10t ⎪ 2 2 ⎪ 90 + (10t − 90) s=⎨ ⎪ 902 + (270 − 10t ) 2 ⎪ ⎪⎩360 – 10t ⎧180 − 180 − 10t ⎪ ⎪ ⎪ s = ⎨ 902 + (10t − 90) 2 ⎪ 2 2 ⎪ 90 + (270 − 10t ) ⎪ ⎪⎩ if 0 ≤ t ≤ 9 if 9 < t ≤ 18 if 0 ≤ t ≤ 9 or 27 < t ≤ 36 47. if 9 < t ≤ 18 if 18 < t ≤ 27 f(1.38) ≈ 0.2994 f(4.12) ≈ 3.6852 x f(x) –4 –4.05 –3 –3.1538 a. –2 –2.375 b. f(x) = 0 when x ≈ –1.1, 1.7, 4.3 f(x) ≥ 0 on [–1.1, 1.7] ∪ [4.3, 5] –1 –1.8 0 –1.25 1 –0.2 2 1.125 3 2.3846 4 3.55 Section 0.5 Range: {y ∈ R: –22 ≤ y ≤ 13} 48. a. 36 f(1.38) ≈ –76.8204 f(4.12) ≈ 6.7508 f(x) = g(x) at x ≈ –0.6, 3.0, 4.6 Instructor’s Resource Manual b. f(x) ≥ g(x) on [-0.6, 3.0] ∪ [4.6, 5] c. f ( x) – g ( x) = x3 – 5 x 2 + x + 8 – 2 x 2 + 8 x + 1 = x3 – 7 x 2 + 9 x + 9 b. On ⎡⎣ −6, −3) , g increases from 13 g ( −6 ) = ≈ 4.3333 to ∞ . On ( 2, 6⎤⎦ , g 3 26 decreased from ∞ to ≈ 2.8889 . On 9 ( −3, 2 ) the maximum occurs around Largest value f (–2) – g (–2) = 45 x = 0.1451 with value 0.6748 . Thus, the range is ( −∞, 0.6748⎦⎤ ∪ ⎣⎡ 2.8889, ∞ ) . 49. c. x 2 + x – 6 = 0; (x + 3)(x – 2) = 0 Vertical asymptotes at x = –3, x = 2 d. Horizontal asymptote at y = 3 0.6 Concepts Review 1. ( x 2 + 1)3 a. x-intercept: 3x – 4 = 0; x = 4 3 3⋅ 0 – 4 2 = y-intercept: 2 0 +0–6 3 3. 2; left 4. a quotient of two polynomial functions b. c. 2. f(g(x)) x 2 + x – 6 = 0; (x + 3)(x – 2) = 0 Vertical asymptotes at x = –3, x = 2 Problem Set 0.6 1. a. ( f + g )(2) = (2 + 3) + 22 = 9 d. Horizontal asymptote at y = 0 50. a. ( f ⋅ g )(0) = (0 + 3)(02 ) = 0 c. ( g f )(3) = d. ( f g )(1) = f (12 ) = 1 + 3 = 4 e. ( g f )(1) = g (1 + 3) = 4 2 = 16 f. ( g f )(–8) = g (–8 + 3) = (–5) 2 = 25 2. a. x-intercepts: 3x 2 – 4 = 0; x = ± b. 4 2 3 =± 3 3 2 y-intercept: 3 32 9 3 = = 3+3 6 2 ( f – g )(2) = (22 + 2) – 12 + 1 c. 1 ⎡ 2 ⎤ ⎛1⎞ g 2 (3) = ⎢ ⎥ = ⎜ 3⎟ = 9 + 3 3 ⎣ ⎦ ⎝ ⎠ 2 Instructor’s Resource Manual 2 b. ( f g )(1) = 2 1+ 3 = 2 2 28 =6– = 2+3 5 5 2 4 =4 2 Section 0.6 37 2 d. (f ⎛ 2 ⎞ ⎛1⎞ 1 3 g )(1) = f ⎜ ⎟=⎜ ⎟ + = ⎝1+ 3 ⎠ ⎝ 2 ⎠ 2 4 ( g f )(1) = g (12 + 1) = 2 2 = 2+3 5 f. ⎛ 2 ⎞ ( g g )(3) = g ⎜ ⎟= ⎝ 3+3⎠ 2 2 3 = 10 = 1 +3 3 5 3 )(t ) = t 3 + 1 + (Φ 1 ⎛1⎞ ⎛1⎞ )(r ) = Φ⎜ ⎟ = ⎜ ⎟ + 1 = 3 + 1 r r r ⎝ ⎠ ⎝ ⎠ Φ )(r ) = (r + 1) = c. ( d. Φ 3 ( z ) = ( z 3 + 1) 3 e. (Φ – f. ((Φ – 1 6. g 3 (x) = (x 2 +1) 3 = (x 4 + 2x 2 + 1)(x 2 + 1) 6 ) = g[(x 2 + 1) 2 + 1] = g( x 4 + 2x 2 + 2) = ( x 4 + 2x 2 + 2) 2 + 1 = x 8 + 4x 6 + 8x 4 + 8x 2 + 5 7. g(3.141) ≈ 1.188 9. ⎡ g 2 (π ) − g (π ) ⎤ ⎣ ⎦ ≈ 4.789 1 5t ⎛1⎞ )⎜ ⎟ ⎝t⎠ 11. a. b. 12. a. b. 2 x2 – 1 x Domain: (– ∞, – 1] ∪ [1, ∞) d. 38 g ( x) = x , f ( x) = x + 7 g (x) = x15 , f (x) = x 2 + x f ( x) = ( f ⋅ g )( x) = b. 4 ⎛2⎞ f 4 ( x) + g 4 ( x) = ⎛⎜ x 2 – 1 ⎞⎟ + ⎜ ⎟ ⎝ ⎠ ⎝x⎠ 16 = ( x 2 – 1)2 + x4 Domain: (– ∞, 0 ) ∪ (0, ∞ ) 2 c. ⎛2⎞ ⎛2⎞ g )( x) = f ⎜ ⎟ = ⎜ ⎟ – 1 = ⎝ x⎠ ⎝ x⎠ Domain: [–2, 0) ∪ (0, 2] (f ( g f )( x) = g ⎛⎜ x 2 – 1 ⎞⎟ = ⎝ ⎠ 1/ 3 2 = ⎡⎢(11 − 7π ) − 11 − 7π ⎤⎥ ⎣ ⎦ 10. [ g 3 (π) – g (π)]1/ 3 = [(6π – 11)3 – (6π – 11)]1/ 3 ≈ 7.807 3 1 1 ⎛ 1⎞ = ⎜ ⎟ + 1– 1 = 3 + 1 – t ⎝ t⎠ t t 4. a. 2 1/ 3 1 5t )(t ) = (Φ – 4 = x + 3x + 3x + 1 ( g g g )( x) = ( g g )( x 2 + 1) 8. g(2.03) ≈ 0.000205 r 3 +1 )(5t) = [(5t) 3 +1] – = 125t3 + 1 – f ) ( x) = g ⎛⎜ x 2 − 4 ⎞⎟ = 1 + x 2 − 4 ⎝ ⎠ = 1 + x2 – 4 1 t (Φ + 3 2 = x2 + 2 x – 3 3 b. g ) ( x) = f ( 1 + x ) = 1 + x − 4 (f (g e. 3. a. 5. 4 f ( x) = 13. p = f 2 x 3 , g ( x) = x 2 + x + 1 1 , g (x) = x 3 + 3 x x g h if f(x) =1/ x , g ( x) = x , h( x ) = x 2 + 1 p= f g h if f ( x) = 1/ x , g(x) = x + 1, h( x) = x 2 4 –1 x2 14. p = f g h l if f ( x) = 1/ x , g ( x) = x , 2 h(x) = x + 1, l( x) = x 2 2 x –1 Domain: (– ∞ , –1) ∪ (1, ∞ ) Section 0.6 Instructor’s Resource Manual 15. Translate the graph of g ( x) = x to the right 2 units and down 3 units. 17. Translate the graph of y = x 2 to the right 2 units and down 4 units. 18. Translate the graph of y = x 3 to the left 1 unit and down 3 units. 16. Translate the graph of h( x) = x to the left 3 units and down 4 units. 19. ( f + g )( x) = x–3 + x 2 20. ( f + g )( x) = x + x Instructor’s Resource Manual Section 0.6 39 21. F (t ) = t –t 24. a. t F(x) – F(–x) is odd because F(–x) – F(x) = –[F(x) – F(–x)] b. F(x) + F(–x) is even because F(–x) + F(–(–x)) = F(–x) + F(x) = F(x) + F(–x) c. 22. G (t ) = t − t 25. Not every polynomial of even degree is an even function. For example f ( x) = x 2 + x is neither even nor odd. Not every polynomial of odd degree is an odd function. For example g ( x) = x 3 + x 2 is neither even nor odd. 26. a. 23. a. Even; (f + g)(–x) = f(–x) + g(–x) = f(x) + g(x) = (f + g)(x) if f and g are both even functions. b. Odd; (f + g)(–x) = f(–x) + g(–x) = –f(x) – g(x) = –(f + g)(x) if f and g are both odd functions. c. Even; ( f ⋅ g )(− x) = [ f (− x)][ g (− x)] = [ f ( x)][ g ( x)] = ( f ⋅ g )( x) if f and g are both even functions. d. Even; ( f ⋅ g )(− x) = [ f (− x)][ g (− x)] = [− f ( x)][− g ( x)] = [ f ( x)][ g ( x)] = ( f ⋅ g )( x) if f and g are both odd functions. e. 40 F ( x ) – F (– x) F ( x ) + F (– x) is odd and is 2 2 even. F ( x ) − F (− x) F ( x ) + F (− x) 2 F ( x) + = = F ( x) 2 2 2 Neither b. PF c. RF d. PF e. RF f. Neither 27. a. P = 29 – 3(2 + t ) + (2 + t )2 = t + t + 27 b. When t = 15, P = 15 + 15 + 27 ≈ 6.773 28. R(t) = (120 + 2t + 3t2 )(6000 + 700t ) = 2100 t3 + 19, 400t 2 + 96, 000t + 720, 000 ⎧⎪400t 29. D(t ) = ⎨ 2 2 ⎪⎩ (400t ) + [300(t − 1)] if 0 < t < 1 if t ≥ 1 if 0 < t < 1 ⎧⎪ 400t D(t ) = ⎨ 2 ⎪⎩ 250, 000t − 180, 000t + 90, 000 if t ≥ 1 30. D(2.5) ≈ 1097 mi Odd; ( f ⋅ g )(− x) = [ f (− x)][ g (− x)] = [ f ( x)][− g ( x)] = −[ f ( x)][ g ( x)] = −( f ⋅ g )( x) if f is an even function and g is an odd function. Section 0.6 Instructor’s Resource Manual 31. (axcx –+ab ) + b (axcx –+ab ) – a 36. ⎛ ax + b ⎞ a f ( f ( x)) = f ⎜ ⎟= ⎝ cx – a ⎠ c = a 2 x + ab + bcx – ab = x(a 2 + bc) =x acx + bc – acx + a 2 a 2 + bc 2 If a + bc = 0 , f(f(x)) is undefined, while if x = a , f(x) is undefined. c ⎛ x –3 – 3 ⎞ ⎛ ⎛ x – 3 ⎞⎞ x +1 ⎟ 32. f ( f ( f ( x))) = f ⎜ f ⎜ ⎟ ⎟ = f ⎜⎜ x –3 ⎟ + 1 x 1 + ⎠⎠ ⎝ ⎝ ⎝ x +1 ⎠ ⎛ x – 3 – 3x – 3 ⎞ ⎛ –2 x – 6 ⎞ ⎛ –x – 3 ⎞ = f⎜ ⎟= f ⎜ ⎟= f ⎜ ⎟ ⎝ x – 3 + x +1 ⎠ ⎝ 2x – 2 ⎠ ⎝ x –1 ⎠ – x–3 – 3 – x – 3 – 3x + 3 – 4 x = –xx––13 = = =x – x – 3+ x –1 –4 +1 x –1 If x = –1, f(x) is undefined, while if x = 1, f(f(x)) is undefined. 33. a. b. ⎛1⎞ f⎜ ⎟= ⎝ x⎠ 34. a. b. 1 x –1 = 1 1– x 1 f1 ( f 2 ( x)) = ; x f1 ( f3 ( x)) = 1 − x; 1 ; 1− x x −1 f1 ( f5 ( x)) = ; x x ; f1 ( f6 ( x)) = x −1 f1 ( f 4 ( x)) = f 2 ( f1 ( x)) = f 2 ( f 2 ( x)) = x x –1 x –1 x –1 = x; 1 ; 1− x 1 f 2 ( f 4 ( x)) = = 1 − x; f 2 ( f 6 ( x)) = x =x x – x +1 1 x −1 x 1 x x −1 = x ; x –1 = x –1 ; x f3 ( f1 ( x)) = 1 − x; ⎛ 1 ⎞ ⎛ x – 1⎞ ⎟⎟ = f ⎜ f ⎜⎜ ⎟= f ( x ) ⎝ x ⎠ ⎝ ⎠ =1–x 1/ x 1 / x −1 x –1 x x –1 –1 x = x –1 x –1– x 1 x −1 ; = x x f3 ( f3 ( x)) = 1 – (1 – x) = x; f3 ( f 2 ( x)) = 1 − 1 x = ; 1 – x x –1 x –1 1 f3 ( f5 ( x)) = 1 – = ; x x x 1 f3 ( f 6 ( x)) = 1 – = ; x –1 1– x f3 ( f 4 ( x)) = 1 – 1 = x−x f ( f ( x)) = f ( x /( x − 1)) = = 1 x f 2 ( f3 ( x)) = f 2 ( f 5 ( x )) = f (1 / x) = 1 ; x 1 1 1− x ⎛ x ⎞ f ( f ( x)) = f ⎜ ⎟= ⎝ x – 1⎠ = c. 1 x f1 ( f1 ( x)) = x; x /( x − 1) x x( x − 1) + 1 − x 35. ( f1 ( f 2 f3 ))( x) = f1 (( f 2 f3 )( x)) = f1 ( f 2 ( f3 ( x))) (( f1 f 2 ) f3 )( x) = ( f1 f 2 )( f3 ( x)) = f1 ( f 2 ( f3 ( x))) = ( f1 ( f 2 f3 ))( x) x −1 x −1 1 ; 1− x 1 x ; f 4 ( f 2 ( x)) = = 1 1− x x −1 f 4 ( f1 ( x)) = f 4 ( f3 ( x)) = f 4 ( f 4 ( x)) = f 4 ( f5 ( x)) = f 4 ( f 6 ( x)) = Instructor’s Resource Manual 1 1 = ; 1 – (1 – x) x 1 1 – 1–1x 1 1– x –1 x = 1− x x –1 = ; 1− x −1 x = x = x; x − ( x − 1) 1 x −1 = = 1 – x; x 1 – x –1 x − 1 − x Section 0.6 41 a. x −1 f5 ( f1 ( x)) = ; x 1 −1 f5 ( f 2 ( x)) = x = 1 − x; 42 b. = f 5 ( f 5 ( x)) = x –1 –1 x x –1 x x −1− x 1 = = ; x −1 1– x f 5 ( f 6 ( x)) = x –1 x –1 x x –1 = f3 f4 f5 f6 f3 ) f4 ) f5 ) f6 ) = ( f 4 f 4 ) ( f5 f 6 ) = f5 f 2 = f3 c. x − ( x − 1) 1 = ; x x If F f 6 = f 1 , then F = f 6 . d. If G f 3 G = f5. 1 ; 1– x f 6 = f 1 , then G f 4 = f 1 so If f 2 f 5 H = f 5 , then f 6 H = f 5 so H = f3. 37. f 6 ( f3 ( x)) = x –1 1– x ; = 1– x –1 x f 6 ( f 4 ( x)) = 1 1– x 1 –1 1– x = 1 1 = ; 1 − (1 − x) x f 6 ( f 5 ( x)) = x –1 x x –1 –1 x = x −1 = 1 – x; x −1− x f 6 ( f 6 ( x )) = x x –1 x –1 x –1 = x =x x − ( x − 1) 38. f1 f2 f3 f4 f5 f6 f1 f1 f2 f3 f4 f5 f6 f2 f2 f1 f4 f3 f6 f5 f3 f3 f5 f1 f6 f2 f4 f4 f4 f6 f2 f5 f1 f3 f5 f5 f3 f6 f1 f4 f2 f6 f6 f4 f5 f2 f3 f1 Section 0.6 f1 f 2 = (((( f 2 e. = f3 ) f3 ) f3 ) f3 ) = ((((( f 1 f 2 ) f 3 ) f 4 ) f 5 ) f 6 ) 1 − (1 − x) = x; 1 x ; x –1 –1 f3 = f1 f 3 = f 3 1 –1 1– x 1 1– x 1 x f3 = (( f3 f3 ) f3 ) f 5 ( f 4 ( x)) = f 6 ( f 2 ( x)) = f3 = ((( f1 f3 ) f3 ) f3 ) 1 – x –1 x f5 ( f3 ( x)) = = ; 1– x x –1 1 x f3 = (((( f 3 1 x f 6 ( f1 ( x)) = f3 39. Instructor’s Resource Manual Problem Set 0.7 40. 41. a. b. 1. a. ⎛ π ⎞ π 30 ⎜ ⎟= ⎝ 180 ⎠ 6 b. ⎛ π ⎞ π 45 ⎜ ⎟= ⎝ 180 ⎠ 4 c. π ⎛ π ⎞ –60 ⎜ ⎟=– 3 ⎝ 180 ⎠ d. ⎛ π ⎞ 4π 240 ⎜ ⎟= ⎝ 180 ⎠ 3 e. 37 π ⎛ π ⎞ –370 ⎜ ⎟=– 18 ⎝ 180 ⎠ f. ⎛ π ⎞ π 10 ⎜ ⎟= ⎝ 180 ⎠ 18 2. a. c. 3 4 c. 1 ⎛ 180 ⎞ ⎟ = –60° – π⎜ 3 ⎝ π ⎠ d. 4 ⎛ 180 ⎞ ⎟ = 240° π⎜ 3 ⎝ π ⎠ e. – f. 3 ⎛ 180 ⎞ π⎜ ⎟ = 30° 18 ⎝ π ⎠ 3. a. 4 x 4. r = (–4) + 3 = 5; cos = = – 5 r 2 Instructor’s Resource Manual ⎛ π ⎞ 33.3 ⎜ ⎟ ≈ 0.5812 ⎝ 180 ⎠ ⎛ π ⎞ 46 ⎜ ⎟ ≈ 0.8029 ⎝ 180 ⎠ c. ⎛ π ⎞ –66.6 ⎜ ⎟ ≈ –1.1624 ⎝ 180 ⎠ d. ⎛ π ⎞ 240.11⎜ ⎟ ≈ 4.1907 ⎝ 180 ⎠ e. ⎛ π ⎞ –369 ⎜ ⎟ ≈ –6.4403 ⎝ 180 ⎠ f. ⎛ π ⎞ 11⎜ ⎟ ≈ 0.1920 ⎝ 180 ⎠ 3. odd; even 2 35 ⎛ 180 ⎞ ⎟ = –350° π⎜ 18 ⎝ π ⎠ b. 1. (– ∞ , ∞ ); [–1, 1] 2. 2 π ; 2 π ; π ⎛ 180 ⎟⎞ π⎜ = 135° ⎝ π ⎠ b. 42. 0.7 Concepts Review 7 ⎛ 180 ⎞ ⎟ = 210° π⎜ 6 ⎝ π ⎠ Section 0.7 43 4. a. ⎛ 180 ⎟⎞ 3.141⎜ ≈ 180° ⎝ π ⎠ Thus b. ⎛ 180 ⎟⎞ 6. 28⎜ ≈ 359. 8° ⎝ π ⎠ c. ⎛ 180 ⎟⎞ ≈ 286.5° 5. 00⎜ ⎝ π ⎠ d. ⎛ 180 ⎟⎞ 0. 001⎜ ≈ 0 .057° ⎝ π ⎠ e. ⎛ 180 ⎟⎞ –0.1⎜ ≈ –5.73° ⎝ π ⎠ f. ⎛ 180 ⎟⎞ 36. 0⎜ ≈ 2062.6 ° ⎝ π ⎠ 5. a. 56. 4 tan34. 2° ≈ 68.37 sin 34.1° cos b. 5.34 tan 21.3° ≈ 0.8845 sin 3.1°+ cot 23.5° c. tan (0.452) ≈ 0.4855 d. sin (–0.361) ≈ –0.3532 6. a. b. 7. a. = cos π = and by the Pythagorean Identity, sin π 3 = 3 . 2 234.1sin(1.56) ≈ 248.3 cos(0.34 ) sin 2 (2.51) + cos(0.51) ≈ 1.2828 56. 3 tan34. 2° ≈ 46.097 sin 56.1° Referring to Figure 2, it is clear that sin sin 35° ⎛⎜ ⎞⎟ ≈ 0. 0789 ⎝ sin 26° + cos 26° ⎠ Identity, cos 2 π 6 Section 0.7 = 1 − sin 2 π 9. a. 2 3 ⎛1⎞ = 1− ⎜ ⎟ = . 6 4 ⎝2⎠ π π 2 =1 = 0 . The rest of the values are 2 obtained using the same kind of reasoning in the second quadrant. and cos 8. Referring to Figure 2, it is clear that sin 0 = 0 and cos 0 = 1 . If the angle is π / 6 , then the triangle in the figure below is 1 1 equilateral. Thus, PQ = OP = . This 2 2 π 1 implies that sin = . By the Pythagorean 6 2 44 6 π 3 . The results 2 = 2 were derived in the text. 4 4 2 If the angle is π / 3 then the triangle in the π 1 figure below is equilateral. Thus cos = 3 2 sin 3 b. π ⎛π ⎞ sin ⎜ ⎟ ⎛π⎞ ⎝6⎠ = 3 tan ⎜ ⎟ = 3 ⎝ 6 ⎠ cos ⎛ π ⎞ ⎜ ⎟ 6 ⎝ ⎠ 1 = –1 cos(π) b. sec(π) = c. 1 ⎛ 3π ⎞ sec ⎜ ⎟ = =– 2 4 ⎝ ⎠ cos 3π 4 d. 1 ⎛π⎞ csc ⎜ ⎟ = =1 2 ⎝ ⎠ sin π ( ) (2) Instructor’s Resource Manual e. f. 10. a. b. ( ) ( ) b. cos 3t = cos(2t + t ) = cos 2t cos t – sin 2t sin t π ⎛ π ⎞ cos 4 cot ⎜ ⎟ = =1 ⎝ 4 ⎠ sin π 4 = (2 cos 2 t – 1) cos t – 2sin 2 t cos t = 2 cos3 t – cos t – 2(1 – cos 2 t ) cos t ( ) ( ) π ⎛ π ⎞ sin – 4 tan ⎜ – ⎟ = = –1 ⎝ 4 ⎠ cos – π 4 = 2 cos3 t – cos t – 2 cos t + 2 cos3 t = 4 cos3 t – 3cos t c. ( ) ( ) π ⎛ π ⎞ sin 3 tan ⎜ ⎟ = = 3 ⎝ 3 ⎠ cos π 3 = 2(2sin x cos x)(2 cos2 x –1) = 2(4sin x cos3 x – 2sin x cos x) = 8sin x cos3 x – 4sin x cos x 1 ⎛π⎞ sec ⎜ ⎟ = =2 ⎝ 3 ⎠ cos π 3 ( ) d. ( ) ( ) c. π 3 ⎛ π ⎞ cos 3 cot ⎜ ⎟ = = 3 ⎝ 3 ⎠ sin π 3 d. 1 ⎛π⎞ csc ⎜ ⎟ = = 2 ⎝ 4 ⎠ sin π e. π 3 ⎛ π ⎞ sin – 6 tan ⎜ – ⎟ = =– π 6 3 ⎝ ⎠ cos – 6 f. ⎛ π⎞ 1 cos ⎜ – ⎟ = ⎝ 3⎠ 2 13. a. b. (4) ( ) ( ) c. d. 11. a. (1 + sin z )(1 – sin z ) = 1 – sin 2 z = cos 2 z = 1 sec z (sec t – 1)(sec t + 1) = sec 2 t – 1 = tan 2 t c. sec t – sin t tan t = d. 12. a. (1 + cos )(1 − cos ) = 1 − cos 2 14. a. = sin 2 sin u cos u + = sin 2 u + cos 2 u = 1 csc u sec u (1 − cos 2 x)(1 + cot 2 x ) = (sin 2 x)(csc2 x) 2 ⎛ 1 ⎞ = sin x ⎜ 2 ⎟ = 1 ⎝ sin x ⎠ ⎛ 1 ⎞ sin t (csc t – sin t ) = sin t ⎜ – sin t ⎟ ⎝ sin t ⎠ 2 2 = 1– sin t = cos t 1 – csc2 t csc2 t =– = – cos 2 t = – 2 b. = sin 4 x = sin[2(2 x)] = 2sin 2 x cos 2 x cot 2 t csc2 t =– cos 2 t sin 2 t 1 sin 2 t 1 sec 2 t y = sin 2x 1 sin 2 t – cos t cos t 1 – sin 2 t cos 2 t = = cos t cos t cos t sec2 t – 1 sec 2 t sin 2 v + = tan 2 t sec 2 t 1 2 sec v = sin 2 t cos 2 t 1 cos 2 t = sin 2 t = sin 2 v + cos 2 v = 1 Instructor’s Resource Manual Section 0.7 45 b. y = 2 sin t b. y = 2 cos t c. π⎞ ⎛ y = cos ⎜ x − ⎟ 4⎠ ⎝ c. y = cos 3t d. y = sec t d. ⎛ π⎞ y = cos ⎜ t + ⎟ ⎝ 3⎠ 15. a. y = csc t 46 Section 0.7 x 2 Period = 4π , amplitude = 3 16. y = 3 cos Instructor’s Resource Manual 17. y = 2 sin 2x Period = π , amplitude = 2 21. y = 21 + 7 sin( 2 x + 3) Period = π , amplitude = 7, shift: 21 units up, 3 units left 2 π⎞ ⎛ 22. y = 3cos ⎜ x – ⎟ – 1 2⎠ ⎝ 18. y = tan x Period = π Period = 2 π , amplitude = 3, shifts: π units 2 right and 1 unit down. 19. y = 2 + 1 cot(2 x) 6 Period = π 2 , shift: 2 units up π⎞ ⎛ 23. y = tan ⎜ 2 x – ⎟ ⎝ 3⎠ π π units right Period = , shift: 6 2 20. y = 3 + sec( x − π ) Period = 2π , shift: 3 units up, π units right Instructor’s Resource Manual Section 0.7 47 π⎞ ⎛ 24. a. and g.: y = sin ⎜ x + ⎟ = cos x = – cos(π – x) 2⎠ ⎝ π⎞ ⎛ b. and e.: y = cos ⎜ x + ⎟ = sin( x + π) 2⎠ ⎝ = − sin(π − x ) π⎞ ⎛ c. and f.: y = cos ⎜ x − ⎟ = sin x 2⎠ ⎝ = − sin( x + π) π⎞ ⎛ d. and h.: y = sin ⎜ x − ⎟ = cos( x + π) 2⎠ ⎝ = cos( x − π) –t sin (–t) = t sin t; even 25. a. 2 sin (– t ) = sin t ; even c. 1 csc(– t ) = = – csc t; odd sin(– t ) ( π) = e. sin(cos(–t)) = sin(cos t); even f. –x + sin(–x) = –x – sin x = –(x + sin x); odd cot(–t) + sin(–t) = –cot t – sin t = –(cot t + sin t); odd 26. a. b. sin 3 (–t ) = – sin 3 t ; odd c. sec(– t) = 1 = sec t; even cos(–t ) sin 4 (– t ) = sin 4 t ; even d. e. cos(sin(–t)) = cos(–sin t) = cos(sin t); even f. (– x )2 + sin(– x ) = x 2 – sin x; neither 27. cos 2 28. sin 2 2 2 π ⎛ π ⎞ ⎛1⎞ 1 = ⎜ cos ⎟ = ⎜ ⎟ = 3 ⎝ 3⎠ 4 ⎝2⎠ 2 2 π ⎛ π⎞ 1 ⎛1⎞ = ⎜ sin ⎟ = ⎜ ⎟ = 6 ⎝ 6⎠ 4 ⎝2⎠ 3 3 π 2 32. a. 2– 2 4 sin(x – y) = sin x cos(–y) + cos x sin(–y) = sin x cos y – cos x sin y b. cos(x – y) = cos x cos(–y) – sin x sin (–y) = cos x cos y + sin x sin y c. tan( x – y ) = = sin(−t ) = – sin t = sin t ; even d. π 1 – cos 4 1 – 2 π 1 – cos 2 8 31. sin = = = 8 2 2 2 2 2 b. (π) 1 + cos 6 1 + 2 π 1 + cos 2 12 = = = 30. cos 12 2 2 2 2+ 3 = 4 2 tan x + tan(– y ) 1 – tan x tan(– y ) tan x – tan y 1 + tan x tan y tan t + tan π tan t + 0 = 1 – tan t tan π 1 – (tan t )(0) = tan t 33. tan(t + π) = 34. cos( x − π ) = cos x cos(−π ) − sin x sin(−π ) = –cos x – 0 · sin x = –cos x 35. s = rt = (2.5 ft)( 2π rad) = 5π ft, so the tire goes 5π feet per revolution, or 1 revolutions 5π per foot. ft ⎞ ⎛ 1 rev ⎞ ⎛ mi ⎞ ⎛ 1 hr ⎞ ⎛ ⎜ ⎟ ⎜ 60 ⎟ ⎜ ⎟ ⎜ 5280 ⎟ 5 ft hr 60 min mi π ⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠ ≈ 336 rev/min 36. s = rt = (2 ft)(150 rev)( 2π rad/rev) ≈ 1885 ft 37. r1t1 = r2 t2 ; 6(2π)t1 = 8(2π)(21) t1 = 28 rev/sec 38. Δy = sin and Δx = cos Δy sin m= = = tan Δx cos 3 1 π⎞ ⎛1⎞ 3 π ⎛ 29. sin 6 = ⎜ sin 6 ⎟ = ⎜ 2 ⎟ = 8 ⎝ ⎠ ⎝ ⎠ 48 Section 0.7 Instructor’s Resource Manual 39. a. b. = 3 π = 3 tan 3x + 3 y = 6 3 y = – 3x + 6 3 3 x + 2; m = – 3 3 y=– tan 3 3 =– = 5π 6 40. m1 = tan 1 44. Divide the polygon into n isosceles triangles by drawing lines from the center of the circle to the corners of the polygon. If the base of each triangle is on the perimeter of the polygon, then 2π . the angle opposite each base has measure n Bisect this angle to divide the triangle into two right triangles (See figure). and m2 = tan 2 tan 2 + tan(− 1 ) tan = tan( 2 − 1 ) = 1 − tan 2 tan(− 1 ) = 41. a. b. tan 2 − tan 1 m − m1 = 2 1 + tan 2 tan 1 1 + m1m2 3–2 1 = 1 + 3(2 ) 7 ≈ 0.1419 tan = tan = –1 – 12 1+ ( 12 ) (–1) = –3 ≈ 1.8925 c. 2x – 6y = 12 2x + y = 0 –6y = –2x + 12y = –2x 1 y= x–2 3 1 m1 = , m2 = –2 3 –2 – 13 = –7; ≈ 1.7127 tan = 1 + 13 (–2) () 42. Recall that the area of the circle is π r 2 . The measure of the vertex angle of the circle is 2π . Observe that the ratios of the vertex angles must equal the ratios of the areas. Thus, t A = , so 2π π r 2 1 A = r 2t . 2 43. A = π b π π h = so b = 2r sin and cos = so n 2r n n r π h = r cos . n π P = nb = 2rn sin n π π ⎛1 ⎞ A = n ⎜ bh ⎟ = nr 2 cos sin n n ⎝2 ⎠ sin 45. The base of the triangle is the side opposite the t angle t. Then the base has length 2r sin 2 (similar to Problem 44). The radius of the t semicircle is r sin and the height of the 2 t triangle is r cos . 2 A= 1⎛ t ⎞⎛ t ⎞ π⎛ t⎞ ⎜ 2r sin ⎟⎜ r cos ⎟ + ⎜ r sin ⎟ 2⎝ 2 ⎠⎝ 2⎠ 2⎝ 2⎠ 2 t t πr 2 t = r 2 sin cos + sin 2 2 2 2 2 1 (2)(5) 2 = 25cm 2 2 Instructor’s Resource Manual Section 0.7 49 x x x x 46. cos cos cos cos 2 4 8 16 1⎡ 3 1 ⎤1 ⎡ 3 1 ⎤ = ⎢cos x + cos x ⎥ ⎢cos x + cos x ⎥ ⎣ ⎦ ⎣ 2 4 4 2 16 16 ⎦ 1⎡ 3 1 ⎤⎡ 3 1 ⎤ = ⎢ cos x + cos x ⎥ ⎢cos x + cos x ⎥ 4⎣ 4 4 ⎦ ⎣ 16 16 ⎦ 1⎡ 3 3 3 1 = ⎢ cos x cos x + cos x cos x 4⎣ 4 16 4 16 3 1 1 ⎤ 1 + cos x cos x + cos x cos x ⎥ 16 4 16 ⎦ 4 1 ⎡1 ⎛ 15 9 ⎞ 1⎛ 13 11 ⎞ = ⎢ ⎜ cos + cos x ⎟ + ⎜ cos x + cos x ⎟ 4 ⎣2 ⎝ 16 16 ⎠ 2 ⎝ 16 16 ⎠ 1⎛ 7 1 ⎞ 1⎛ 5 3 ⎞⎤ + ⎜ cos x + cos x ⎟ + ⎜ cos x + cos x ⎟⎥ 2⎝ 16 16 ⎠ 2 ⎝ 16 16 ⎠ ⎦ 1⎡ 15 13 11 9 = ⎢cos x + cos x + cos x + cos x 8⎣ 16 16 16 16 7 5 3 1 ⎤ + cos x + cos x + cos x + cos x ⎥ 16 16 16 16 ⎦ 47. The temperature function is ⎛ 2π ⎛ 7 ⎞ ⎞ T (t ) = 80 + 25 sin ⎜⎜ ⎜ t − ⎟ ⎟⎟ . ⎝ 12 ⎝ 2 ⎠ ⎠ The normal high temperature for November 15th is then T (10.5) = 67.5 °F. 49. As t increases, the point on the rim of the wheel will move around the circle of radius 2. a. x(2) ≈ 1.902 y (2) ≈ 0.618 x(6) ≈ −1.176 y (6) ≈ −1.618 x(10) = 0 y (10) = 2 x(0) = 0 y (0) = 2 b. ⎛π ⎞ ⎛π ⎞ x(t ) = −2 sin ⎜ t ⎟, y (t ) = 2 cos⎜ t ⎟ ⎝5 ⎠ ⎝5 ⎠ c. The point is at (2, 0) when is , when t = π 5 t= π 2 ; that 5 . 2 2π . When 10 you add functions that have the same frequency, the sum has the same frequency. 50. Both functions have frequency a. y (t ) = 3sin(π t / 5) − 5cos(π t / 5) +2sin((π t / 5) − 3) 48. The water level function is ⎛ 2π ⎞ F (t ) = 8.5 + 3.5 sin ⎜ (t − 9) ⎟ . ⎝ 12 ⎠ The water level at 5:30 P.M. is then F (17.5) ≈ 5.12 ft . b. 50 Section 0.7 y (t ) = 3cos(π t / 5 − 2) + cos(π t / 5) + cos((π t / 5) − 3) Instructor’s Resource Manual 51. a. C sin( t + φ ) = (C cos φ )sin t + (C sin φ ) cos t. Thus A = C ⋅ cos φ and B = C ⋅ sin φ . b. A2 + B 2 = (C cos φ )2 + (C sin φ ) 2 = C 2 (cos 2 φ ) + C 2 (sin 2 φ ) = C 2 Also, c. B C ⋅ sin φ = = tan φ A C ⋅ cos φ A1 sin( t + φ1 ) + A2 sin( t + φ 2 ) + A3 (sin t + φ 3 ) = A1 (sin t cos φ1 + cos t sin φ1 ) + A2 (sin t cos φ 2 + cos t sin φ 2 ) + A3 (sin t cos φ 3 + cos t sin φ 3 ) = ( A1 cos φ1 + A2 cos φ 2 + A3 cos φ 3 ) sin t + ( A1 sin φ1 + A2 sin φ 2 + A3 sin φ 3 ) cos t = C sin ( t + φ ) where C and φ can be computed from A = A1 cos φ1 + A2 cos φ2 + A3 cos φ3 B = A1 sin φ1 + A2 sin φ2 + A3 sin φ3 as in part (b). d. Written response. Answers will vary. 52. ( a.), (b.), and (c.) all look similar to this: d. e. The windows in (a)-(c) are not helpful because the function oscillates too much over the domain plotted. Plots in (d) or (e) show the behavior of the function. Instructor’s Resource Manual 53. a. b. c. The plot in (a) shows the long term behavior of the function, but not the short term behavior, whereas the plot in (c) shows the short term behavior, but not the long term behavior. The plot in (b) shows a little of each. Section 0.7 51 54. a. h( x ) = ( f g ) ( x ) 3 cos(100 x) + 2 100 = 2 ⎛ 1 ⎞ 2 ⎜ ⎟ cos (100 x) + 1 100 ⎝ ⎠ j ( x) = ( g f )( x) = 56. ⎧ 2 f ( x ) = ⎪( x − 2n ) , ⎨ ⎪0.0625, ⎩ 1 1⎤ ⎡ x ∈ ⎢ 2n − , 2n + ⎥ 4 4⎦ ⎣ otherwise where n is an integer. y 1 3x + 2 ⎞ ⎛ cos ⎜100 ⎟ 100 x2 + 1 ⎠ ⎝ 0.5 b. 0.25 −2 c. −1 1 x 2 0.8 Chapter Review Concepts Test 1. False: 2. True: ⎧ 1⎞ ⎡ ⎪ 4 x − x + 1 : x ∈ ⎢ n, n + ⎟ 4⎠ ⎪ ⎣ 55. f ( x ) = ⎨ ⎪− 4 x − x + 7 : x ∈ ⎡ n + 1 , n + 1⎞ ⎟ ⎢ ⎪ 3 3 4 ⎣ ⎠ ⎩ where n is an integer. ( ) ( p1 p2 p1q2 − p2 q1 − = ; since q1 q2 q1q2 p1 , q1 , p2 , and q2 are integers, so are p1q2 − p2 q1 and q1q2 . 3. False: If the numbers are opposites (– π and π ) then the sum is 0, which is rational. 4. True: Between any two distinct real numbers there are both a rational and an irrational number. 5. False: 0.999... is equal to 1. 6. True: ( am ) = ( an ) 7. False: (a * b) * c = abc ; a *(b * c) = ab 8. True: Since x ≤ y ≤ z and x ≥ z , x = y = z ) y p and q must be integers. 2 1 −1 1 x 9. True: 52 Section 0.8 n m = a mn c x would 2 be a positive number less than x . If x was not 0, then ε = Instructor’s Resource Manual 10. True: y − x = −( x − y ) so ( x − y )( y − x) = ( x − y )(−1)( x − y ) 20. True: since 1 + r ≥ 1 − r , 2 = (−1)( x − y ) . 11. True: 12. True: 14. True: 15. False: 16. False: 17. True: If r > 1, r = r , and 1 − r = 1 − r , so −( x − y ) 2 ≤ 0. 1 1 1 . = ≤ 1− r 1− r 1+ r a 1 1 > 1; < b b a a < b < 0; a < b; [ a, b] and [b, c ] If r < −1, r = −r and 1 − r = 1 + r , so share point b in 21. True: If (a, b) and (c, d) share a point then c < b so they share the infinitely many points between b and c. If x and y are the same sign, then x – y = x– y . x– y ≤ x+ y opposite signs then either x – y = x – (– y ) = x + y For example, if x = −3 , then − x = − ( −3) = 3 = 3 which does (x > 0, y < 0) or x – y = –x – y = x + y not equal x. (x < 0, y > 0). In either case x – y = x+ y . For example, take x = 1 and y = −2 . 4 x < y ⇔ x < y 4 If either x = 0 or y = 0, the inequality is easily seen to be true. 4 4 22. True: 4 4 x = x and y = y , so x < y x + y = −( x + y ) 4 23. True: If r = 0, then 1 1 1 = = = 1. 1+ r 1 – r 1 – r For any r, 1 + r ≥ 1 – r . Since r < 1, 1 – r > 0 so 1 1 ; ≤ 1+ r 1 – r y satisfies = y. (3 y ) 3 =y 24. True: For example x 2 ≤ 0 has solution [0]. 25. True: x 2 + ax + y 2 + y = 0 x 2 + ax + If –1 < r < 0, then r = – r and 2 a2 1 a2 1 + y2 + y + = + 4 4 4 4 2 a⎞ ⎛ 1⎞ a2 + 1 ⎛ + + + = x y ⎜ ⎟ ⎜ ⎟ 2⎠ ⎝ 2⎠ 4 ⎝ is a circle for all values of a. 1 – r = 1 + r , so 1 1 1 = ≤ . 1+ r 1 – r 1 – r 1 – r = 1 – r , so ( y) 2 For every real number y, whether it is positive, zero, or negative, the cube root x = 3 y satisfies x3 = also, –1 < r < 1. If 0 < r < 1, then r = r and If y is positive, then x = x2 = = − x + (− y ) = x + y 19. True: 1 1 1 ≤ = . 1− r 1− r 1+ r when x and y are the same sign, so x – y ≤ x + y . If x and y have x 2 = x = − x if x < 0. 4 18. True: 1 1 . ≤ 1− r 1+ r ( x − y ) 2 ≥ 0 for all x and y, so common. 13. True: If r > 1, then 1 − r < 0. Thus, 26. False: If a = b = 0 and c < 0 , the equation does not represent a circle. 1 1 1 . ≤ = 1+ r 1 – r 1 – r Instructor’s Resource Manual Section 0.8 53 27. True; 28. True: 29. True: 30. True: 31. True: 32. True: 3 ( x − a) 4 3 3a y = x − + b; 4 4 If x = a + 4: 3 3a y = (a + 4) – +b 4 4 3a 3a = +3– +b = b+3 4 4 y −b = = –( x + 3)( x + 1) If ab > 0, a and b have the same sign, so (a, b) is in either the first or third quadrant. The domain does not include π nπ + where n is an integer. 2 43. True: The domain is ( − ∞, ∞) and the range is [−6, ∞) . 44. False: The range is ( − ∞, ∞) . 45. False: The range ( − ∞, ∞) . 46. True: If f(x) and g(x) are even functions, f(x) + g(x) is even. f(–x) + g(–x) = f(x) + g(x) 47. True: If f(x) and g(x) are odd functions, f(–x) + g(–x) = –f(x) – g(x) = –[f(x) + g(x)], so f(x) + g(x) is odd 48. False: If f(x) and g(x) are odd functions, f(–x)g(–x) = –f(x)[–g(x)] = f(x)g(x), so f(x)g(x) is even. 49. True: If f(x) is even and g(x) is odd, f(–x)g(–x) = f(x)[–g(x)] = –f(x)g(x), so f(x)g(x) is odd. 50. False: If f(x) is even and g(x) is odd, f(g(–x)) = f(–g(x)) = f(g(x)); while if f(x) is odd and g(x) is even, f(g(–x)) = f(g(x)); so f(g(x)) is even. 51. False: If f(x) and g(x) are odd functions, f ( g (− x)) = f(–g(x)) = –f(g(x)), so f(g(x)) is odd. Let x = ε / 2. If ε > 0 , then x > 0 and x < ε . If ab = 0, a or b is 0, so (a, b) lies on the x-axis or the y-axis. If a = b = 0, (a, b) is the origin. −( x 2 + 4 x + 3) ≥ 0 on −3 ≤ x ≤ −1 . y1 = y2 , so ( x1 , y1 ) and ( x2 , y2 ) d = [(a + b) – (a – b)]2 + (a – a) 2 34. False: The equation of a vertical line cannot be written in point-slope form. 35. True: This is the general linear equation. 36. True: Two non-vertical lines are parallel if and only if they have the same slope. 37. False: The slopes of perpendicular lines are negative reciprocals. 38. True: If a and b are rational and ( a, 0 ) , ( 0, b ) are the intercepts, the slope is − b which is rational. a 52. True: f (– x) = 2(– x)3 + (– x) ax + y = c ⇒ y = − ax + c ax − y = c ⇒ y = ax − c (a )(− a) ≠ −1. (unless a = ±1 ) 54 f ( x) = –( x 2 + 4 x + 3) 42. False: = (2b) 2 = 2b 39. False: 41. True: The equation is (3 + 2m) x + (6m − 2) y + 4 − 2m = 0 which is the equation of a straight line unless 3 + 2m and 6m − 2 are both 0, and there is no real number m such that 3 + 2m = 0 and 6m − 2 = 0. If the points are on the same line, they have equal slope. Then the reciprocals of the slopes are also equal. are on the same horizontal line. 33. True: 40. True: Section 0.8 =− (– x )2 + 1 = –2 x3 – x x2 + 1 2 x3 + x x2 + 1 Instructor’s Resource Manual 53. True: f (–t ) = = (sin(–t )) 2 + cos(– t ) tan(– t ) csc(– t ) (− sin t )2 + cos t (sin t )2 + cos t = – tan t (– csc t ) tan t csc t 54. False: f(x) = c has domain ( − ∞, ∞) and the only value of the range is c. 55. False: f(x) = c has domain ( − ∞, ∞) , yet the range has only one value, c. g (−1.8) = 57. True: (f g )( x) = ( x 3 ) 2 = x 6 (g f )( x) = ( x 2 )3 = x 6 (f g )( x) = ( x 3 ) 2 = x 6 2 3 f ( x) ⋅ g ( x) = x x = x 59. False: 60. True: 61. True: 62. False: 63. False: 5 b. cos x sin x cos(− x) cot(− x) = sin(− x) cos x = = − cot x − sin x 2 1 ⎞ ⎛ 1⎞ 1⎞ 25 ⎛ ⎛ ⎜ n + ⎟ ; ⎜ 1 + ⎟ = 2; ⎜ 2 + ⎟ = ; n ⎠ ⎝ 1⎠ 2⎠ 4 ⎝ ⎝ 1 ⎞ ⎛ ⎜ –2 + ⎟ –2 ⎠ ⎝ –2 = 4 25 2 (n 2 – n + 1)2 ; ⎡(1)2 – (1) + 1⎤ = 1; ⎣ ⎦ 2 ⎡(2) 2 – (2) + 1⎤ = 9; ⎣ ⎦ 2 ⎡(–2)2 – (–2) + 1⎤ = 49 ⎣ ⎦ c. 43 / n ; 43 /1 = 64; 43 / 2 = 8; 4 –3 / 2 = d. n 1 1 ; 1 = 1; n 1 −2 1 = 2 −2 f The domain of excludes any g values where g = 0. f(a) = 0 Let F(x) = f(x + h), then F(a – h) = f(a – h + h) = f(a) = 0 1 n 1. a. −1.8 = −0.9 = −1 2 56. True: 58. False: Sample Test Problems 2. a. 1 1 2 = = ; 2 2 2 1 1 ⎞⎛ 1 1⎞ ⎛ ⎜1 + + ⎟⎜ 1 − + ⎟ ⎝ m n ⎠⎝ m n ⎠ −1 cot x = b. The cosine function is periodic, so cos s = cos t does not necessarily imply s = t; e.g., cos 0 = cos 2π = 1 , but 0 ≠ 2π . Instructor’s Resource Manual = c. 1 1 + m n = 1 1 1− + m n mn + n + m = mn − n + m 1+ 2 x 2 x − − 2 x + 1 x − x − 2 x + 1 ( x − 2)( x + 1) = 3 2 3 2 − − x +1 x − 2 x +1 x − 2 = The domain of the tangent function π excludes all nπ + where n is an 2 integer. 1 8 2( x − 2) − x 3 ( x − 2) − 2( x + 1) x−4 x −8 (t 3 − 1) (t − 1)(t 2 + t + 1) 2 = = t + t +1 t −1 t −1 3. Let a, b, c, and d be integers. a+ c a c ad + bc b d which is rational. = + = 2 2b 2d 2bd Section 0.8 55 4. x = 4.1282828… 1000 x = 4128.282828… 10 x = 41.282828… 13. 21t 2 – 44t + 12 ≤ –3; 21t 2 – 44t + 15 ≤ 0; t= 990 x = 4087 4087 x= 990 ⎛ 3 ⎞⎛ 5 ⎞ ⎡3 5⎤ ⎜ t – ⎟ ⎜ t – ⎟ ≤ 0; ⎢ , ⎥ ⎝ 7 ⎠⎝ 3 ⎠ ⎣7 3⎦ 5. Answers will vary. Possible answer: 13 ≈ 0.50990... 50 2x −1 1⎞ ⎛ > 0; ⎜ −∞, ⎟ ∪ ( 2, ∞ ) x−2 2⎠ ⎝ 14. 2 ⎛ 3 8.15 × 104 − 1.32 ⎞ ⎜ ⎟ ⎠ ≈ 545.39 6. ⎝ 3.24 7. (π – 2.0 ) 2.5 44 ± 442 – 4(21)(15) 44 ± 26 3 5 = = , 2(21) 42 7 3 15. ( x + 4)(2 x − 1) 2 ( x − 3) ≤ 0;[−4,3] – 3 2.0 ≈ 2.66 16. 3x − 4 < 6; −6 < 3 x − 4 < 6; −2 < 3x < 10; 8. sin 2 ( 2.45 ) + cos ( 2.40 ) − 1.00 ≈ −0.0495 2 9. 1 – 3 x > 0 3x < 1 1 x< 3 1⎞ ⎛ ⎜ – ∞, ⎟ 3⎠ ⎝ 10. 6 x + 3 > 2 x − 5 4 x > −8 x > −2; ( −2, ∞ ) 11. 3 − 2 x ≤ 4 x + 1 ≤ 2 x + 7 3 − 2 x ≤ 4 x + 1 and 4 x + 1 ≤ 2 x + 7 6 x ≥ 2 and 2 x ≥ 6 1 ⎡1 ⎤ x ≥ and x ≤ 3; ⎢ , 3⎥ 3 ⎣3 ⎦ 12. 2 x 2 + 5 x − 3 < 0;(2 x − 1)( x + 3) < 0; 1 ⎛ 1⎞ −3 < x < ; ⎜ −3, ⎟ 2 ⎝ 2⎠ − 17. 2 10 ⎛ 2 10 ⎞ < x < ;⎜ − , ⎟ 3 3 ⎝ 3 3⎠ 3 ≤2 1– x 3 –2≤0 1– x 3 – 2(1 – x) ≤0 1– x 2x +1 ≤ 0; 1– x 1⎤ ⎛ ⎜ – ∞, – ⎥ ∪ (1, ∞ ) 2⎦ ⎝ 18. 12 − 3 x ≥ x (12 − 3 x)2 ≥ x 2 144 − 72 x + 9 x 2 ≥ x 2 8 x 2 − 72 x + 144 ≥ 0 8( x − 3)( x − 6) ≥ 0 (−∞,3] ∪ [6, ∞) 19. For example, if x = –2, −(−2) = 2 ≠ −2 − x ≠ x for any x < 0 20. If − x = x, then x = x. x≥0 56 Section 0.8 Instructor’s Resource Manual 21. |t – 5| = |–(5 – t)| = |5 – t| If |5 – t| = 5 – t, then 5 − t ≥ 0. t ≤5 ⎛ 2 + 10 0 + 4 ⎞ , 27. center = ⎜ ⎟ = (6, 2) 2 ⎠ ⎝ 2 1 1 radius = (10 – 2) 2 + (4 – 0) 2 = 64 + 16 2 2 =2 5 22. t − a = −(a − t ) = a − t If a − t = a − t , then a − t ≥ 0. t≤a circle: ( x – 6)2 + ( y – 2) 2 = 20 28. x 2 + y 2 − 8 x + 6 y = 0 23. If x ≤ 2, then x 2 − 8 x + 16 + y 2 + 6 y + 9 = 16 + 9 0 ≤ 2 x 2 + 3 x + 2 ≤ 2 x 2 + 3 x + 2 ≤ 8 + 6 + 2 = 16 ( x − 4) 2 + ( y + 3) 2 = 25; 1 1 ≤ . Thus also x + 2 ≥ 2 so 2 x +2 2 2 2 x2 + 3x + 2 x2 + 2 = 2 x2 + 3x + 2 ⎛1⎞ ≤ 16 ⎜ ⎟ 2 ⎝2⎠ x +2 1 =8 24. a. The distance between x and 5 is 3. b. The distance between x and –1 is less than or equal to 2. c. The distance between x and a is greater than b. center = ( 4, −3) , radius = 5 x2 − 2 x + y 2 + 2 y = 2 29. x2 − 2 x + 1 + y2 + 2 y + 1 = 2 + 1 + 1 ( x − 1) 2 + ( y + 1) 2 = 4 center = (1, –1) x 2 + 6 x + y 2 – 4 y = –7 x 2 + 6 x + 9 + y 2 – 4 y + 4 = –7 + 9 + 4 ( x + 3)2 + ( y – 2)2 = 6 center = (–3, 2) d = (–3 – 1) 2 + (2 + 1)2 = 16 + 9 = 5 25. 30. a. d ( A, B ) = (1 + 2) 2 + (2 − 6)2 3x + 2 y = 6 2 y = −3 x + 6 3 y = − x+3 2 3 m=− 2 3 y − 2 = − ( x − 3) 2 3 13 y = − x+ 2 2 = 9 + 16 = 5 d ( B, C ) = (5 − 1)2 + (5 − 2)2 = 16 + 9 = 5 d ( A, C ) = (5 + 2)2 + (5 − 6) 2 = 49 + 1 = 50 = 5 2 2 ( AB) + ( BC )2 = ( AC ) 2 , so ΔABC is a right triangle. ⎛1+ 7 2 + 8 ⎞ , 26. midpoint: ⎜ ⎟ = ( 4,5 ) 2 ⎠ ⎝ 2 d = (4 − 3)2 + (5 + 6)2 = 1 + 121 = 122 Instructor’s Resource Manual Section 0.8 57 b. 2 m= ; 3 b. 3x – 2 y = 5 –2 y = –3 x + 5 3 5 y = x– ; 2 2 3 m= 2 3 y –1 = ( x + 2) 2 3 y = x+4 2 c. 3 x + 4y = 9 4y = –3x + 9; 4 3 9 y = – x+ ; m = 3 4 4 4 y –1 = ( x + 2) 3 4 11 y = x+ 3 3 2 ( x − 1) 3 2 5 y = x− 3 3 y +1 = c. y=9 d. x = –2 e. contains (–2, 1) and (0, 3); m = 3 –1 ; 0+2 y=x+3 3 +1 4 11 − 3 8 4 = ; m2 = = = ; 5−2 3 11 − 5 6 3 11 + 1 12 4 m3 = = = 11 − 2 9 3 m1 = m2 = m3 , so the points lie on the same line. 32. m1 = d. x = –3 33. The figure is a cubic with respect to y. The equation is (b) x = y 3 . 34. The figure is a quadratic, opening downward, with a negative y-intercept. The equation is (c) y = ax 2 + bx + c. with a < 0, b > 0, and c < 0. 31. a. 58 3 –1 2 m= = ; 7+2 9 2 y –1 = ( x + 2) 9 2 13 y = x+ 9 9 Section 0.8 35. Instructor’s Resource Manual 36. x2 − 2 x + y 2 = 3 x2 − 2 x + 1 + y 2 = 4 ( x − 1)2 + y 2 = 4 40. 4 x − y = 2 y = 4 x − 2; 1 4 contains ( a, 0 ) , ( 0, b ) ; m=− ab =8 2 ab = 16 16 a 1 b−0 b =− =− ; 0−a 4 a a = 4b b= ⎛ 16 ⎞ a = 4⎜ ⎟ ⎝a⎠ 37. a 2 = 64 a =8 b= 41. a. b. 38. c. 39. y = x2 – 2x + 4 and y – x = 4; x + 4 = x2 − 2 x + 4 x 2 − 3x = 0 x( x – 3) = 0 points of intersection: (0, 4) and (3, 7) f (1) = 1 1 1 – =– 1+1 1 2 1 1 ⎛ 1⎞ f ⎜– ⎟ = – =4 1 1 2 – + 1 – ⎝ ⎠ 2 2 f(–1) does not exist. 1 1 1 1 = – – t –1+1 t –1 t t – 1 d. f (t – 1) = e. 1 1 t ⎛1⎞ f ⎜ ⎟= – = –t 1 1 ⎝ t ⎠ t +1 t 1+ t 42. a. b. c. Instructor’s Resource Manual 16 1 = 2; y = − x + 2 8 4 g (2) = 2 +1 3 = 2 2 ⎛1⎞ g⎜ ⎟ = ⎝ 2⎠ 1 2 +1 1 2 =3 2 + h +1 – 22+1 g ( 2 + h ) – g ( 2) = 2+ h h h h 2 h + 6 – 3h – 6 – 2( h + 2) –1 2( h + 2) = = = h h 2(h + 2) 43. a. {x ∈ : x ≠ –1, 1} b. {x ∈ : x ≤ 2} Section 0.8 59 44. a. b. f (– x) = 3(– x) 2 (– x) + 1 =– 3x 2 x +1 ; odd 46. g (– x) = sin(– x) + cos(– x) = − sin x + cos x = sin x + cos x; even c. h(– x) = (– x)3 + sin(– x) = – x3 – sin x ; odd d. k (– x) = 45. a. (– x)2 + 1 – x + (– x) 4 = x2 + 1 x + x4 2 f (x) = x – 1 ; even 47. V(x) = x(32 – 2x)(24 – 2x) Domain [0, 12] 48. a. 1⎞ 13 ⎛ ( f + g )(2) = ⎜ 2 – ⎟ + (22 + 1) = 2⎠ 2 ⎝ b. 15 ⎛3⎞ ( f ⋅ g )(2) = ⎜ ⎟ (5) = 2 ⎝2⎠ c. (f g )(2) = f (5) = 5 – d. (g 13 ⎛3⎞ ⎛3⎞ f )(2) = g ⎜ ⎟ = ⎜ ⎟ + 1 = 2 2 4 ⎝ ⎠ ⎝ ⎠ e. 1⎞ ⎛ f 3 (–1) = ⎜ –1 + ⎟ = 0 1⎠ ⎝ 1 24 = 5 5 2 b. x g(x) = 2 x +1 3 2 f. 49. a. c. 60 ⎧ x2 h(x) = ⎨ ⎩6 – x Section 0.8 ⎛3⎞ f 2 (2) + g 2 (2) = ⎜ ⎟ + (5) 2 ⎝2⎠ 9 109 = + 25 = 4 4 y= 1 2 x 4 if 0 ≤ x ≤ 2 if x > 2 Instructor’s Resource Manual b. y= 1 ( x + 2)2 4 53. a. b. sin (–t) = –sin t = –0.8 sin 2 t + cos2 t = 1 cos 2 t = 1 – (0.8)2 = 0.36 cos t = –0. 6 c. 1 y = –1 + ( x + 2) 2 4 c. sin 2t = 2 sin t cos t = 2(0.8)(–0.6) = –0.96 d. tan t = e. ⎛π ⎞ cos ⎜ – t ⎟ = sin t = 0.8 ⎝2 ⎠ f. sin(π + t ) = − sin t = −0.8 sin t 0.8 4 = = – ≈ –1.333 cos t –0.6 3 54. sin 3t = sin(2t + t ) = sin 2t cos t + cos 2t sin t = 2sin t cos 2 t + (1 – 2sin 2 t ) sin t = 2sin t (1 – sin 2 t ) + sin t – 2sin 3 t = 2sin t – 2sin 3 t + sin t – 2sin 3 t = 3sin t – 4sin 3 t 55. s = rt ⎛ rev ⎞⎛ rad ⎞ ⎛ 1 min ⎞ = 9 ⎜ 20 ⎟⎜ 2π ⎟⎜ ⎟ (1 sec) = 6π ⎝ min ⎠⎝ rev ⎠ ⎝ 60 sec ⎠ ≈ 18.85 in. 50. a. b. c. (−∞,16] f Review and Preview Problems g = 16 – x 4 ; domain [–2, 2] g f = ( 16 – x ) 4 = (16 – x) 2 ; domain (−∞,16] (note: the simplification ( 16 – x ) 4 = (16 – x) 2 is only true given the restricted domain) 51. f ( x) = x , g ( x) = 1 + x, h( x) = x 2 , k(x) = sin x, F ( x) = 1 + sin 2 x = f g h k 52. a. sin(570°) = sin(210°) = – 1 2 b. ⎛ 9π ⎞ ⎛π⎞ cos ⎜ ⎟ = cos ⎜ ⎟ = 0 ⎝ 2 ⎠ ⎝2⎠ c. 3 ⎛ 13π ⎞ ⎛ π⎞ cos ⎜ – ⎟ = cos ⎜ − ⎟ = ⎝ 6 ⎠ ⎝ 6⎠ 2 Instructor’s Resource Manual 1. a) b) 2. a) b) 0 < 2 x < 4; 0 < x < 2 −6 < x < 16 13 < 2 x < 14; 6.5 < x < 7 −4 < − x / 2 < 7; − 14 < x < 8 3. x − 7 = 3 or x − 7 = −3 x = 10 or x=4 4. x + 3 = 2 or x = −1 or x + 3 = −2 x = −5 5. x − 7 = 3 or x − 7 = −3 x = 10 or x=4 6. x − 7 = d or x − 7 = −d x = 7 + d or x = 7 − d 7. a) x − 7 < 3 and x − 7 > −3 x < 10 and x>4 4 < x < 10 Review and Preview 61 b) c) d) 8. a) x − 7 ≤ 3 and x − 7 ≥ −3 x ≤ 10 and x≥4 4 ≤ x ≤ 10 x − 7 ≤ 1 and x ≤ 8 and 6≤ x≤8 x − 2 < 1 and x − 2 > −1 x < 3 and x >1 1< x < 3 x − 2 < 0.1 and x − 2 > −0.1 x < 2.1 and x > 1.9 1.9 < x < 2.1 d) x − 2 < 0.01 and x − 2 > −0.01 x < 2.01 and x > 1.99 1.99 < x < 2.01 11. a) g ( 0.999 ) = −0.000333556 g (1.001) = 0.000333111 g (1.1) = 0.03125 x − 7 < 0.1 and x − 7 > −0.1 x < 7.1 and x > 6.9 6.9 < x < 7.1 c) 10. a) g ( 0.99 ) = −0.0033557 g (1.01) = 0.00331126 x − 2 ≥ 1 or x − 2 ≤ −1 x ≥ 3 or x ≤1 b) g ( 2) = 12. a) x − 1 ≠ 0; x ≠ 1 2 x 2 − x − 1 ≠ 0; x ≠ 1, − 0.5 x≠0 b) g ( 0 ) = −1 g ( 0.9 ) = −0.0357143 x − 7 ≥ −1 x≥6 b) 9. a) b) b) 1 = −1 −1 0.1 = −1 F ( −0.1) = −0.1 0.01 F ( −0.01) = = −1 −0.01 0.001 F ( −0.001) = = −1 −0.001 0.001 F ( 0.001) = =1 0.001 0.01 F ( 0.01) = =1 0.01 0.01 F ( 0.1) = =1 0.01 1 F (1) = = 1 1 F ( −1) = G ( −1) = 0.841471 G ( −0.1) = 0.998334 G ( −0.01) = 0.999983 x≠0 0 −1 f ( 0) = =1 0 −1 0.81 − 1 f ( 0.9 ) = = 1.9 0.9 − 1 0.9801 − 1 = 1.99 f ( 0.99 ) = 0.99 − 1 0.998001 − 1 = 1.999 f ( 0.999 ) = .999 − 1 1.002001 − 1 = 2.001 f (1.001) = 1.001 − 1 1.0201 − 1 = 2.01 f (1.01) = 1.01 − 1 1.21 − 1 = 2.1 f (1.1) = 1.1 − 1 4 −1 =3 f ( 2) = 2 −1 1 5 G ( −0.001) = 0.99999983 G ( 0.001) = 0.99999983 G ( 0.01) = 0.999983 G ( 0.1) = 0.998334 G (1) = 0.841471 13. x − 5 < 0.1 and x − 5 > −0.1 x < 5.1 and x > 4.9 4.9 < x < 5.1 14. x − 5 < ε and x − 5 > −ε x < 5 + ε and x > 5−ε 5−ε < x < 5+ε 15. a. True. b. False: Choose a = 0. c. True. d. True 16. sin ( c + h ) = sin c cos h + cos c sin h 62 Review and Preview Instructor’s Resource Manual 1 CHAPTER Limits 9. 1.1 Concepts Review x3 – 4 x 2 + x + 6 x → –1 x +1 lim ( x + 1)( x 2 – 5 x + 6) x → –1 x +1 1. L; c = lim 2. 6 = lim ( x 2 – 5 x + 6) x → –1 2 3. L; right = (–1) – 5(–1) + 6 4. lim f ( x) = M = 12 x →c Problem Set 1.1 x2 = lim( x 2 + 2 x –1) = –1 x →0 1. lim( x – 5) = –2 x →0 x →3 2. lim (1 – 2t ) = 3 11. t → –1 3. 4. lim ( x 2 + 2 x − 1) = (−2) 2 + 2(−2) − 1 = −1 lim ( x 2 + 2t − 1) = (−2) 2 + 2t − 1 = 3 + 2t x →−2 ( t →−1 t →−1 ( = –t – t = –2t 12. ) ( ( −1) − 1) = 0 2 ) ( ( −1) 6. lim t 2 − x 2 = 2 ) x2 – 4 ( x – 2)( x + 2) = lim x→2 x – 2 x→2 x–2 = lim( x + 2) =3+3=6 13. x→2 lim (t + 4)(t − 2) 4 (3t − 6) 2 t →2 = lim =2+2=4 8. x2 – 9 x →3 x – 3 ( x – 3)( x + 3) = lim x →3 x–3 = lim( x + 3) lim x →3 − x2 = 1 − x2 7. lim x2 – t 2 ( x + t )( x – t ) = lim x→–t x + t x→ – t x+t = lim ( x – t ) lim x→ –t x →−2 5. lim t 2 − 1 = x 4 + 2 x3 – x 2 10. lim (t − 2) 2 t + 4 9(t − 2) 2 t →2 t 2 + 4t – 21 t → –7 t+7 (t + 7)(t – 3) = lim t → –7 t+7 = lim (t – 3) t+4 9 = lim lim t →2 = 2+4 6 = 9 9 t → –7 = –7 – 3 = –10 14. (t − 7)3 t −7 lim t →7 + = lim t →7 + = lim t →7 + (t − 7) t − 7 t −7 t −7 = 7−7 = 0 Instructor’s Resource Manual Section 1.1 63 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 15. lim x 4 –18 x 2 + 81 ( x – 3) x →3 2 = lim x →3 ( x – 3) 2 ( x + 3) 2 = lim ( x – 3) x →3 ( x 2 – 9) 2 ( x – 3) = lim( x + 3)2 = (3 + 3)2 2 lim t →0 2 (3u + 4)(2u – 2)3 (u –1) 2 u →1 = lim 1. 0.0251314 0.1 2.775 × 10−6 8(3u + 4)(u –1)3 0.01 2.77775 × 10−10 (u –1) 2 0.001 2.77778 × 10−14 –1. –0.1 0.0251314 2.775 × 10−6 –0.01 2.77775 × 10−10 –0.001 2.77778 × 10−14 u →1 = lim 8(3u + 4)(u – 1) = 8[3(1) + 4](1 – 1) = 0 u →1 17. (2 + h) 2 − 4 4 + 4h + h 2 − 4 = lim h→0 h→0 h h lim h 2 + 4h = lim(h + 4) = 4 h →0 h →0 h = lim 18. ( x + h) 2 − x 2 x 2 + 2 xh + h 2 − x 2 = lim h→0 h →0 h h lim lim sin x 2x =0 2 x (1 − cos x ) / x 0.211322 0.1 0.00249584 0.01 0.001 0.0000249996 2.5 × 10−7 –1. 0.211322 –0.1 0.00249584 0.0000249996 2.5 × 10−7 0.420735 0.1 0.499167 0.01 0.499992 –0.01 0.001 0.49999992 –0.001 –1. 0.420735 –0.1 0.499167 –0.01 0.499992 –0.001 0.49999992 lim (1 – cos x) 2 x2 x →0 2 t 23. =0 (t − 1) /(sin(t − 1)) 2. 3.56519 1.1 2.1035 1.01 2.01003 1.001 2.001 0.229849 0 1.1884 0.0249792 0.9 1.90317 0.01 0.00249998 0.99 1.99003 0.001 0.00024999998 0.999 1.999 sin x = 0.5 x →0 2 x 1− cos t 2t t 20. 1. 0.1 –1. –0.229849 –0.1 –0.0249792 –0.01 –0.00249998 –0.001 –0.00024999998 Section 1.1 2 1. 1. lim 64 x2 22. h 2 + 2 xh = lim(h + 2 x) = 2 x h →0 h →0 h x ( x – sin x) 2 x →0 = lim 19. ( x − sin x) 2 / x 2 x 21. x →3 = 36 16. lim 1 − cos t =0 2t 2 lim t −1 =2 − 1) t →1 sin(t Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. x −sin( x − 3) − 3 x −3 x 24. 4. 1. + π4 0.1 + 0.158529 3.1 3.01 3.001 2. 0.001 + lim 4 0.1 + π 0.0500 0.01 + π 0.0050 0.001 + π 0.0005 –0.001 + π –0.0005 x−π x →π =0 −0.1 + 2 0.536908 π 2 0.00226446 π 2 π 2 0.0000213564 2.12342 × 10−7 2 − 2sin u lim =0 π u→ 3u 2 29. a. –0.896664 0.01 –0.989967 0.001 –0.999 d. –1. –1.64209 e. –0.1 –1.09666 –0.01 f. –1.00997 –0.001 –1.001 lim f ( x) = 2 x → –3 b. f(–3) = 1 c. g. = –1 h. i. Instructor’s Resource Manual 0.0000210862 2.12072 × 10−7 π −0.001 + 0.1 1 t 0.00199339 −0.01 + 0.357907 t →0 0.11921 0.001 + 1. 1 – cot t = 0.25 (2 − 2sin u ) / 3u 0.01 + π2 (1 − cot t ) /(1 / t ) lim (tan x − 1)2 0.1 + π2 t 26. (x − ) 0.2505 1. + π2 –0.0500 –0.0050 0.255008 u 28. –0.4597 1 + sin ( x − 32π ) 0.300668 π 2 4 −1. + π2 –0.01 + π lim 0.674117 −0.001 + π4 x→ π 0.4597 –0.1 + π 0.2495 4 −0.01 + π4 1. + π –1. + π 0.245009 π −0.1 + π4 (1 + sin( x − 3π / 2)) /( x − π ) x 4 −1. + π4 x – sin( x – 3) – 3 =0 lim x →3 x–3 25. 0.201002 π 0.0000166666 1.66667 × 10−7 0.0000166666 1.66667 × 10−7 2.999 4 0.01 + 0.00166583 2.99 0.0320244 π 0.00166583 0.158529 2.9 ( x − π / 4) 2 /(tan x − 1) 2 x 27. f(–1) does not exist. lim f ( x) = x → –1 5 2 f(1) = 2 lim f(x) does not exist. x→1 lim f ( x) = 2 x →1– lim f ( x) = 1 x →1+ lim f ( x ) = + x →−1 5 2 Section 1.1 65 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. lim f ( x) does not exist. b. lim f ( x) does not exist. b. f(–3) = 1 c. f(1) = 2 c. f(–1) = 1 d. 30. a. d. x → –3 lim f ( x) = 2 x → –1 e. f(1) = 1 f. lim f ( x) does not exist. g. h. i. 31. a. b. c. d. e. f. 32. a. b. c. x →1 lim f ( x) = 2 x →1+ 34. x →1 lim f ( x) = 1 x →1– lim f ( x) does not exist. x →1+ lim f ( x ) = 2 x →−1+ a. f(–3) = 2 f(3) is undefined. lim f ( x) = 2 c. lim f ( x) = 4 d. x → –3+ lim f ( x) does not exist. x →1 b. g(1) does not exist. x → –3− x → –3 lim g ( x) = 0 35. lim g ( x) = 1 x→2 lim g ( x) = 1 x → 2+ f ( x) = x – ⎣⎡[ x ]⎦⎤ lim f ( x) does not exist. x →3+ lim f ( x) = −2 x → –1− lim f ( x) = −2 x → –1+ lim f ( x) = −2 x → –1 d. f (–1) = –2 e. lim f ( x) = 0 f. f (1) = 0 x →1 33. a. b. c. d. a. 66 f(0) = 0 lim f ( x) does not exist. x →0 lim f ( x) = 1 x →0 – lim f ( x) = x→ 1 2 1 2 lim f ( x) = 0 x →0 Section 1.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. f ( x) = 36. 41. lim f ( x) exists for a = –1, 0, 1. x x x→a 42. The changed values will not change lim f ( x) at x→a any a. As x approaches a, the limit is still a 2 . 43. a. x −1 lim x →1 lim x −1 x −1 − x →1 b. f (0) does not exist. a. lim f ( x) does not exist. b. lim − x →1 lim c. x →0 does not exist. x −1 x −1 x −1 = −1 and lim + x →1 x −1 =1 = −1 x2 − x − 1 − 1 x −1 x →1− x −1 = −3 lim f ( x) = –1 c. x →0 – d. ⎡ 1 1 ⎤ lim ⎢ − ⎥ does not exist. − x −1 x − 1 ⎥⎦ x →1 ⎢ ⎣ d. lim f ( x) = 1 x→ 1 2 44. a. x2 − 1 37. lim does not exist. x →1 x − 1 lim x →1− x2 − 1 x2 − 1 =2 = −2 and lim x −1 x →1+ x − 1 38. lim x →0 = lim x →0 c. ( x + 2 − 2)( x + 2 + 2) d. x( x + 2 + 2) x →0 b. lim x →0 = lim x →0 1 x− x =0 x 1 does not exist. x lim x(−1) 1/ x lim x (−1) 1 2 = = = 4 0+2 + 2 2 2 x+2+ 2 46. a) Does not exist c) lim f ( x) does not exist. =0 b) 0 −1 c) =0 1/ x x →0+ 45. a) 1 x( x + 2 + 2) 1 + x → 0+ x( x + 2 + 2) x+2−2 = lim 39. a. b. x+2− 2 x x →0 = lim lim x →1+ 1 d) −1 b) 0 d) 0.556 x →1 lim f ( x) = 0 47. lim x does not exist since x →0 for x < 0. x →0 40. x is not defined 48. lim x x = 1 x → 0+ 49. lim x →0 x =0 x 50. lim x = 1 x →0 sin 2 x 1 = x →0 4 x 2 51. lim Instructor’s Resource Manual Section 1.2 67 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 52. lim x →0 7. If x is within 0.001 of 2, then 2x is within 0.002 of 4. sin 5 x 5 = 3x 3 ⎛1⎞ 53. lim cos ⎜ ⎟ does not exist. x →0 ⎝x⎠ ⎛1⎞ 54. lim x cos ⎜ ⎟ = 0 x →0 ⎝x⎠ x3 − 1 55. lim 56. lim x →0 57. =6 2x + 2 − 2 x →1 x sin 2 x sin( x 2 ) lim x →2– 58. lim + x →1 8. If x is within 0.0005 of 2, then x2 is within 0.002 of 4. =2 x2 – x – 2 = –3 x–2 2 1/( x −1) 1+ 2 =0 59. lim x ; The computer gives a value of 0, but x →0 lim x →0− 9. If x is within 0.0019 of 2, then 0.002 of 4. 8 x is within x does not exist. 1.2 Concepts Review 1. L – ε ; L + ε 2. 0 < x – a < δ ; f ( x) – L < ε 3. ε 10. If x is within 0.001 of 2, then 8 is within 0.002 x of 4. 3 4. ma + b Problem Set 1.2 1. 0 < t – a < δ ⇒ f (t ) – M < ε 2. 0 < u – b < δ ⇒ g (u ) – L < ε 2x – 1+ 1 < ε ⇔ 2x < ε 3. 0 < z – d < δ ⇒ h( z ) – P < ε ⇔ 2 x <ε 4. 0 < y – e < δ ⇒ φ ( y ) – B < ε ⇔ x < 5. 0 < c – x < δ ⇒ f ( x) – L < ε 6. 0 < t – a < δ ⇒ g (t ) – D < ε 68 11. 0 < x – 0 < δ ⇒ (2 x – 1) – (–1) < ε Section 1.2 ε 2 ε δ = ;0 < x –0 <δ 2 (2 x – 1) – (–1) = 2 x = 2 x < 2δ = ε Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 12. 0 < x + 21 < δ ⇒ (3x – 1) – (–64) < ε 3 x – 1 + 64 < ε ⇔ 3x + 63 < ε ⇔ 3( x + 21) < ε 2 x 2 – 11x + 5 (2 x – 1)( x – 5) –9 <ε ⇔ –9 <ε x–5 x–5 ⇔ 3 x + 21 < ε ⇔ x + 21 < ε ⇔ 2x – 1 – 9 < ε 3 ⇔ 2( x – 5) < ε ε δ = ; 0 < x + 21 < δ 3 (3 x – 1) – (–64) = 3 x + 63 = 3 x + 21 < 3δ = ε x 2 – 25 13. 0 < x – 5 < δ ⇒ – 10 < ε x–5 x 2 – 25 ( x – 5)( x + 5) – 10 < ε ⇔ – 10 < ε x–5 x–5 ⇔ x + 5 – 10 < ε ⇔ x–5 < ε 2 ε δ = ;0 < x –5 <δ 2 2 2 x – 11x + 5 (2 x – 1)( x – 5) –9 = –9 x–5 x–5 = 2 x – 1 – 9 = 2( x – 5) = 2 x – 5 < 2δ = ε 16. 0 < x – 1 < δ ⇒ ⇔ x–5 <ε 2x – 2 < ε 2x – 2 < ε δ = ε; 0 < x – 5 < δ ( 2 x – 2 )( 2 x + 2 ) ⇔ 2x + 2 2 x – 25 ( x – 5)( x + 5) – 10 = – 10 = x + 5 – 10 x–5 x–5 2x – 2 ⇔ 2x + 2 = x–5 <δ =ε 2 ⇔2 2x – x 14. 0 < x – 0 < δ ⇒ − (−1) < ε x 2 x2 – x x(2 x – 1) +1 < ε ⇔ +1 < ε x x ⇔ 2x < ε ⇔ 2 x <ε ε 2 ε δ = ;0 < x –0 <δ 2 2 x2 – x x(2 x – 1) − (−1) = + 1 = 2x – 1+ 1 x x <ε x –1 2x + 2 <ε 2ε ; 0 < x –1 < δ 2 ( 2 x – 2)( 2 x + 2) 2x − 2 = 2x + 2 = 2x – 2 2x + 2 2 x –1 2 x – 1 2δ ≤ < =ε 2x + 2 2 2 17. 0 < x – 4 < δ ⇒ 2x – 1 x–3 = 2 x = 2 x < 2δ = ε ⇔ ⇔ ⇔ Instructor’s Resource Manual <ε δ= ⇔ 2x – 1 +1 < ε ⇔ x < 2 x 2 – 11x + 5 –9 <ε x–5 15. 0 < x – 5 < δ ⇒ 2x – 1 x–3 – 7 <ε ⇔ – 7 <ε 2 x – 1 – 7( x – 3) x–3 <ε ( 2 x – 1 – 7( x – 3))( 2 x – 1 + 7( x – 3)) x – 3( 2 x – 1 + 7( x – 3)) 2 x – 1 – (7 x – 21) x – 3( 2 x – 1 + 7( x – 3)) –5( x – 4) x – 3( 2 x – 1 + 7( x – 3)) <ε <ε <ε Section 1.2 69 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 5 ⇔ x−4 ⋅ x − 3( 2 x − 1 + 7( x − 3)) 5 To bound x – 3( 2 x – 1 + 7( x – 3)) 1 2 1 2 δ ≤ . If δ ≤ , then x – 3( 2 x – 1 + 7( x – 3)) hence x − 4 ⋅ 19. 0 < x – 1 < δ ⇒ , agree that 7 9 < x < , so 2 2 5 0.65 < <ε x − 3( 2 x − 1 + 7( x − 3)) <ε 2x −1 5 − 7 = x−4 ⋅ x −3 x − 3( 2 x − 1 + 7( x − 3)) < x – 4 (1.65) < 1. 65δ ≤ ε 1 1 ε since δ = only when ≤ so 1.65δ ≤ ε . 2 2 1. 65 14 x 2 – 20 x + 6 –8 < ε x –1 14 x 2 – 20 x + 6 2(7 x – 3)( x – 1) –8 <ε ⇔ –8 <ε x –1 x –1 ⇔ 2(7 x – 3) – 8 < ε ⇔ 14( x – 1) < ε ⇔ 14 x – 1 < ε ⇔ x –1 < δ= ε 14 (10 x – 6)( x – 1)2 ( x – 1)2 –4 <ε –4 <ε ⇔ 10 x – 6 – 4 < ε ⇔ 10 x – 1 < ε ⇔ x –1 < 1.65 For whatever ε is chosen, let δ be the smaller of 1 ε and . 1.65 2 ⎧1 ε ⎫ δ = min ⎨ , ⎬, 0 < x – 4 < δ ⎩ 2 1. 65 ⎭ 18. 0 < x – 1 < δ ⇒ ( x –1)2 –4 <ε ⇔ 10( x – 1) < ε 5 ε ⇔ x–4 < ( x – 1) 2 10 x3 – 26 x 2 + 22 x – 6 ⇔ < 1.65 and 10 x3 – 26 x 2 + 22 x – 6 ε 14 δ= ε 10 ε 10 ; 0 < x –1 < δ 10 x3 – 26 x 2 + 22 x – 6 ( x – 1) 2 –4 = (10 x – 6)( x – 1) 2 ( x – 1) 2 –4 = 10 x − 6 − 4 = 10( x − 1) = 10 x − 1 < 10δ = ε 20. 0 < x – 1 < δ ⇒ (2 x 2 + 1) – 3 < ε 2 x2 + 1 – 3 = 2 x2 – 2 = 2 x + 1 x – 1 To bound 2 x + 2 , agree that δ ≤ 1 . x – 1 < δ implies 2x + 2 = 2x – 2 + 4 ≤ 2x – 2 + 4 <2+4=6 ε ⎧ ε⎫ δ ≤ ; δ = min ⎨1, ⎬; 0 < x – 1 < δ 6 ⎩ 6⎭ 2 (2 x + 1) – 3 = 2 x 2 – 2 ⎛ε ⎞ = 2x + 2 x −1 < 6 ⋅ ⎜ ⎟ = ε ⎝6⎠ ; 0 < x –1 < δ 14 x 2 – 20 x + 6 2(7 x – 3)( x – 1) –8 = –8 x –1 x –1 = 2(7 x – 3) – 8 = 14( x – 1) = 14 x – 1 < 14δ = ε 70 Section 1.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 21. 0 < x + 1 < δ ⇒ ( x 2 – 2 x – 1) – 2 < ε x2 – 2 x – 1 – 2 = x2 – 2 x – 3 = x + 1 x – 3 To bound x – 3 , agree that δ ≤ 1 . x + 1 < δ implies ⎛1⎞ 25. For all x ≠ 0 , 0 ≤ sin 2 ⎜ ⎟ ≤ 1 so ⎝x⎠ 1 ⎛ ⎞ x 4 sin 2 ⎜ ⎟ ≤ x 4 for all x ≠ 0 . By Problem 18, ⎝ x⎠ 4 lim x = 0, so, by Problem 20, x→0 4 2 ⎛ 1⎞ lim x sin ⎜ ⎟ = 0. ⎝ x⎠ x→0 x – 3 = x + 1 – 4 ≤ x + 1 + –4 < 1 + 4 = 5 ε ⎧ ε⎫ δ ≤ ; δ = min ⎨1, ⎬ ; 0 < x + 1 < δ 5 ⎩ 5⎭ 26. 0 < x < δ ⇒ 2 ( x – 2 x – 1) – 2 = x 2 – 2 x – 3 = x +1 x – 3 < 5⋅ ε 5 x –0 = x = x <ε 2 For x > 0, ( x ) = x. =ε x < ε ⇔ ( x )2 = x < ε 2 δ = ε 2; 0 < x < δ ⇒ x < δ = ε 2 = ε 22. 0 < x < δ ⇒ x 4 – 0 = x 4 < ε x 4 = x x3 . To bound x3 , agree that 3 δ ≤ 1. x < δ ≤ 1 implies x3 = x ≤ 1 so δ ≤ ε. δ = min{1, ε }; 0 < x < δ ⇒ x 4 = x x3 < ε ⋅1 27. lim x : 0 < x < δ ⇒ x – 0 < ε x →0 + For x ≥ 0 , x = x . δ = ε; 0 < x < δ ⇒ x – 0 = x = x < δ = ε Thus, lim+ x = 0. x→0 lim x : 0 < 0 – x < δ ⇒ x – 0 < ε =ε 23. Choose ε > 0. Then since lim f ( x) = L, there is x →c some δ1 > 0 such that 0 < x – c < δ1 ⇒ f ( x ) – L < ε . x →0 – For x < 0, x = – x; note also that since x ≥ 0. x = x δ = ε ;0 < − x < δ ⇒ x = x = − x < δ = ε Since lim f (x) = M, there is some δ 2 > 0 such Thus, lim– x = 0, that 0 < x − c < δ 2 ⇒ f ( x) − M < ε . since lim x = lim x = 0, lim x = 0. x→c Let δ = min{δ1 , δ2 } and choose x 0 such that 0 < x0 – c < δ . Thus, f ( x0 ) – L < ε ⇒ −ε < f ( x0 ) − L < ε ⇒ − f ( x0 ) − ε < − L < − f ( x0 ) + ε ⇒ f ( x0 ) − ε < L < f ( x0 ) + ε . Similarly, f ( x0 ) − ε < M < f ( x0 ) + ε . Thus, −2ε < L − M < 2ε . As ε ⇒ 0, L − M → 0, so L = M. 24. Since lim G(x) = 0, then given any ε > 0, we x→0 x →0 + x →0 – x →0 28. Choose ε > 0. Since lim g( x) = 0 there is some x→ a δ1 > 0 such that 0 < x – a < δ1 ⇒ g(x ) − 0 < ε. B Let δ = min{1, δ1} , then f ( x) < B for x − a < δ or x − a < δ ⇒ f ( x) < B. Thus, x − a < δ ⇒ f ( x) g ( x) − 0 = f ( x) g ( x) = f ( x) g ( x) < B ⋅ ε B = ε so lim f ( x)g(x) = 0. x→ a x→c can find δ > 0 such that whenever x – c < δ , G ( x) < ε . Take any ε > 0 and the corresponding δ that works for G(x), then x – c < δ implies F ( x) – 0 = F ( x) ≤ G ( x ) < ε since lim G(x) = 0. x→c Thus, lim F( x) = 0. x→c Instructor’s Resource Manual Section 1.2 71 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 29. Choose ε > 0. Since lim f ( x) = L, there is a x→ a δ > 0 such that for 0 < x – a < δ , f ( x) – L < ε . That is, for a − δ < x < a or a < x < a + δ L − ε < f ( x) < L + ε Let f(a) = A, M = max { L − ε , L + ε , A } , c = a – δ 1.3 Concepts Review 1. 48 2. 4 3. – 8; – 4 + 5c 4. 0 d = a + δ. Then for x in (c, d), f ( x) ≤ M , since either x = a, in which case f ( x) = f (a ) = A ≤ M or 0 < x – a < δ so Problem Set 1.3 1. lim (2 x + 1) = lim 2 x + lim 1 3 = 2 lim x + lim 1 2,1 x→1 30. Suppose that L > M. Then L – M = > 0. Now and δ = min{δ1 , δ 2} where 2 0 < x – a < δ1 ⇒ f ( x) – L < ε and 2. Thus, for 0 < x – a < δ , 8 x+6 3 2 x→–1 2, 1 2 3. lim [(2 x +1)( x – 3)] x→0 = lim (2 x +1) ⋅ lim (x – 3) x→ 0 x→ 0 ⎞ ⎛ ⎞ ⎛ = ⎜ lim 2 x + lim 1⎟ ⋅ ⎜ lim x – lim 3⎟ ⎝ x→ 0 x→ 0 ⎠ ⎝ x→0 x→ 0 ⎠ ⎞ ⎛ ⎞ ⎛ = ⎜ 2 lim x + lim 1⎟ ⋅ ⎜ lim x – lim 3⎟ ⎠ ⎝ x →0 ⎝ x→ 0 x→0 x→ 0 ⎠ = [2(0) +1](0 – 3) = –3 lim [(2 x 2 + 1)(7 x 2 + 13)] x→ 2 = lim (2 x 2 + 1) ⋅ lim (7 x 2 + 13) x→ 2 + 1 has 1 , then 2.75 < x < 3 4 or 3 < x < 3.25 and by graphing If δ ≤ y = g ( x) = = 3 lim x 2 – lim 1 = 3(–1) – 1 = 2 4. x 4 – 4x 3 + x 2 + x + 6 x – 4x + x 2 + x + 6 an asymptote at x ≈ 3.49. c. 3 x→–1 ⎛ ⎞ = 3⎜ lim x ⎟ – lim 1 ⎝ x→ –1 ⎠ x →–1 32. For every ε > 0 and δ > 0 there is some x with 0 < x – c < δ such that f ( x ) – L > ε . x 3 – x 2 – 2x – 4 5 = lim 3x 2 – lim 1 x→ –1 31. (b) and (c) are equivalent to the definition of limit. 4 lim (3x 2 – 1) x→ –1 x→ –1 L – ε < f(x) < L + ε and M – ε < g(x) < M + ε. Combine the inequalities and use the fact that f ( x) ≤ g ( x) to get L – ε < f(x) ≤ g(x) < M + ε which leads to L – ε < M + ε or L – M < 2ε. However, L – M = > 2ε which is a contradiction. Thus L ≤ M . b. No, because x→1 = 2(1) + 1 = 3 0 < x – a < δ 2 ⇒ g ( x) – M < ε . g(x) = x→1 x →1 take ε < 33. a. 4 x→1 L − ε < f ( x) < L + ε and f ( x) < M . x→ 2 6 4, 5 3 2, 1 6 4, 3 ⎛ ⎞ ⎛ ⎞ = ⎜ 2 lim x 2 + lim 1⎟ ⋅ ⎜ 7 lim x 2 + lim 13 ⎟ 8,1 x→ 2 ⎠ ⎝ x→ 2 x→ 2 ⎠ ⎝ x→ 2 2 2 ⎡ ⎛ ⎤ ⎡ ⎤ ⎞ ⎛ ⎞ = ⎢2⎜ lim x ⎟ + 1⎥ ⎢7⎜ lim x ⎟ + 13⎥ 2 ⎢⎣ ⎝ x → 2 ⎠ ⎥⎦ ⎢⎣ ⎝ x → 2 ⎠ ⎥⎦ = [2( 2 ) 2 + 1][7( 2 ) 2 + 13] = 135 x3 − x 2 − 2 x − 4 x 4 − 4 x3 + x 2 + x + 6 on the interval [2.75, 3.25], we see that 0< x3 – x 2 – 2 x – 4 <3 x 4 – 4 x3 + x 2 + x + 6 so m must be at least three. 72 Section 1.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2x + 1 x→2 5 – 3x lim (2 x + 1) = x→2 lim (5 – 3 x) 9. 7 5. lim 4, 5 x→2 = 3, 1 lim 5 – lim 3 x x→2 13 4, 3 ⎡ ⎤ = ⎢2 lim t 3 + lim 15⎥ ⎣ t→ –2 t→ –2 ⎦ lim 2 x + lim 1 x→2 8 t→ –2 ⎡ ⎤ = ⎢ lim (2t3 + 15) ⎥ ⎣t→–2 ⎦ x→2 = lim (2t 3 +15)13 13 2 lim x + 1 x→2 8 3 ⎡ ⎤ = ⎢ 2 ⎛⎜ lim t ⎞⎟ + lim 15⎥ ⎣⎢ ⎝ t → –2 ⎠ t → –2 ⎦⎥ x→2 2, 1 = [2(–2) 3 + 15]13 = –1 2 5 – 3 lim x 13 x→2 2(2) + 1 = = –5 5 – 3(2 ) 10. = 3 6. 4x +1 lim 7 x → –3 7 – 2 x 2 lim (4 x + 1) lim (7 – 2 x ) lim 4 x 3 + lim 1 x → –3 lim 7 – lim 2 x 2 x → –3 = x → –3 2 x →3 13 5, 3 = 3 lim x – lim 5 2, 1 x →3 lim x → –3 = 5x2 + 2 x 9 2 lim (5 x + 2 x ) x → –3 = 5 lim x 2 + 2 lim x x → –3 x → –3 ⎛ 4 lim y 3 + 8 lim y ⎞ ⎜ y →2 y →2 ⎟ =⎜ ⎟ y + lim 4 ⎟ ⎜ ylim → y → 2 2 ⎝ ⎠ 8, 1 1/ 3 = 3(3) – 5 = 2 8. 4, 3 13 lim (3 x – 5) x →3 x →3 7 ⎡ lim (4 y 3 + 8 y ) ⎤ ⎢ y →2 ⎥ =⎢ ( y + 4) ⎥⎥ ⎢ ylim → 2 ⎣ ⎦ 9 7. lim 3 x – 5 2 9 ⎛ 4 y3 + 8 y ⎞ = ⎜ lim ⎟ ⎜ y →2 y + 4 ⎟ ⎝ ⎠ 3 = ⎛ 4 y3 + 8 y ⎞ lim ⎜ ⎟ y →2 ⎜ y + 4 ⎟ ⎝ ⎠ 1/ 3 x → –3 4⎛⎜ lim x ⎞⎟ + 1 x → –3 ⎠ = ⎝ 2 7 – 2⎛⎜ lim x ⎞⎟ ⎝ x → –3 ⎠ 4(–3)3 + 1 107 = = 11 7 – 2(–3) 2 2 1/ 3 11. 8 7 – 2 lim x 2 8 w→ –2 = –3(–2)3 + 7(–2) 2 = 2 13 3, 1 x → –3 4 lim x 3 + 1 4, 3 = –3 ⎛⎜ lim w ⎞⎟ + 7 ⎛⎜ lim w ⎞⎟ ⎝ w→ –2 ⎠ ⎝ w→ –2 ⎠ x → –3 = lim (–3w3 + 7 w2 ) 3 4, 5 2 9 w→ –2 w→ –2 x → –3 x → –3 –3w3 + 7 w2 = –3 lim w3 + 7 lim w2 3 = lim w→ –2 4, 3 3 ⎡ ⎛ ⎤ ⎞ ⎢ 4 ⎜ lim y ⎟ + 8 lim y ⎥ y →2 ⎥ ⎢ y →2 ⎠ =⎢ ⎝ ⎥ lim y + 4 ⎢ ⎥ y →2 ⎢ ⎥ ⎣ ⎦ 2 1/ 3 ⎡ 4(2)3 + 8(2) ⎤ =⎢ ⎥ 2+4 ⎣⎢ ⎦⎥ =2 8 2 = 5 ⎛⎜ lim x ⎞⎟ + 2 lim x x → –3 ⎝ x→ –3 ⎠ 2 = 5(–3)2 + 2(–3) = 39 Instructor’s Resource Manual Section 1.3 73 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 12. lim (2 w 4 – 9 w 3 +19)–1 /2 1 = lim w→ 5 7 4 2w − 9 w3 + 19 lim 1 w→5 4 = 1, 9 3 2w – 9 w + 19 lim w→ 5 x→2 19. lim lim (2w – 9 w3 + 19) w→ 5 1 lim 2 w4 − lim 9 w3 + lim 19 w→5 w→5 4 2 lim w − 9 lim w3 + 19 w→5 1 4 = 13. 3 1 144 lim x →2 = x2 − 4 2 x +4 = ( lim ( x 2 ) 4−4 = =0 + 4) 4 + 4 lim x − 4 x→2 2 x→2 x→2 ( x − 3)( x + 1) x2 − 2 x − 3 = lim x →−1 x →−1 x +1 ( x + 1) lim = lim ( x − 3) = −4 17. lim x →−1 x2 + x x2 + 1 ( lim ( x 2 = ) 0 = =0 + 1) 2 lim x + x x →−1 x →−1 2 ( x − 1)( x − 2)( x − 3) x−3 = lim x →−1 ( x − 1)( x − 2)( x + 7) x →−1 x + 7 lim = −1 − 3 2 =− −1 + 7 3 2 2 u –u– 6 u– x x+2 = lim = 5 u → –2 u – 3 u →–2 ( u + 2 )( u – x ) u→ –2 ( u + 2)(u – 3) = lim x 2 + ux – x – u ( x – 1)( x + u) = lim 2 x→1 x + 2 x – 3 x →1 ( x – 1)( x + 3) x + u 1+ u u + 1 = lim = = 4 x→1 x + 3 1+ 3 2 x2 – 6 xπ + 4 π2 2( x – π)( x – 2 π) x→ π x –π x→ π ( x – π)( x + π) 2( x – 2π) 2(π – 2 π) = lim = = –1 π+π x→ π x + π 23. lim 24. 2 lim 2 = lim (w + 2)(w 2 – w – 6) w 2 + 4w + 4 ( w + 2) 2 ( w – 3) = lim = lim ( w – 3) ( w + 2 )2 w→ –2 w→ –2 = –2 – 3 = –5 w→ –2 25. lim x→a f 2 ( x) + g 2 ( x) lim f 2 ( x) + lim g 2 ( x) x→a x→a 2 = ⎛⎜ lim f ( x) ⎞⎟ + ⎛⎜ lim g ( x) ⎞⎟ ⎝ x →a ⎠ ⎝ x→a ⎠ 2 = (3) 2 + (–1)2 = 10 [2 f ( x) – 3 g ( x)] 2 f ( x) – 3g ( x ) xlim = →a x → a f ( x) + g ( x) lim [ f ( x) + g ( x)] 26. lim x→a 2 lim f ( x) – 3 lim g ( x) x→a x→a lim f ( x) + lim g ( x) x→a Section 1.3 = lim u2 – ux + 2u – 2 x lim = 74 ( x + 2)( x − 1) ( x + 1)( x − 1) ( x + 3)( x – 17) x→ –3 x – 4 x – 21 x→ –3 ( x + 3)( x – 7) x – 17 –3 – 17 = lim = =2 –3 – 7 x→ –3 x – 7 = x →−1 x →1 22. lim 1 12 = lim ( x − 3) = −1 16. 2 2(5)4 − 9(5)3 + 19 ( x − 3)( x − 2 ) x2 − 5x + 6 14. lim = lim x→2 x→2 x−2 ( x − 2) 15. 21. 2 ⎛⎜ lim w ⎞⎟ − 9 ⎛⎜ lim w ⎞⎟ + 19 ⎝ w→ 5 ⎠ ⎝ w→5 ⎠ 1 = lim x 2 – 14 x – 51 lim 8 w→5 = = 1,3 20. w→5 1 = 2 x −1 x + 2 1+ 2 3 = lim = = x →1 x + 1 1+1 2 4,5 4 = x2 + x − 2 x →1 1 = x 2 + 7 x + 10 ( x + 2)( x + 5) = lim x→2 x → 2 x+2 x+2 = lim( x + 5) = 7 18. lim w→ 5 x→a = 2(3) – 3(–1) 9 = 3 + (–1) 2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 27. lim 3 g ( x) [ f ( x) + 3] = lim 3 g ( x) ⋅ lim [ f ( x) + 3] x→a x→a x→a = 3 lim g ( x) ⋅ ⎡ lim f ( x) + lim 3⎤ = 3 – 1 ⋅ (3 + 3) ⎢⎣ x → a x→a x→a ⎥ ⎦ = –6 28. lim [ f ( x) – 3]4 = ⎡⎢ lim ( f ( x) – 3) ⎤⎥ x→a ⎣ x→a ⎦ 4 = lim f (t ) + 3 lim g (t ) t →a ⎛ ⎞ 30. lim [ f (u) + 3g(u)] = ⎜ lim [ f (u) + 3g(u)]⎟ ⎝ ⎠ u →a u →a 3 3 3 ⎡ ⎤ = ⎢ lim f (u ) + 3 lim g(u) ⎥ = [3 + 3( –1)]3 = 0 ⎣u→ a ⎦ u →a 3x 2 – 12 3( x – 2 )(x + 2) 31. lim = lim x – 2 x –2 x→2 x→2 = 3 lim (x + 2) = 3(2 + 2) = 12 x→2 (3x 2 + 2 x + 1) – 17 3x 2 + 2 x – 16 = lim x–2 x–2 x→2 x →2 (3 x + 8)( x – 2) = lim = lim (3 x + 8) x–2 x→2 x →2 = 3 lim x + 8 = 3(2) + 8 = 14 x→2 1 2 = lim 2– x 2x = lim – 34. – 3 4 x–2 3( 4 – x 2 ) = lim 4x2 –3( x + 2 )( x – 2 ) = lim 4x2 x→2 x–2 x–2 –3 ⎛⎜ lim x + 2 ⎞⎟ –3( x + 2) ⎠ = –3(2 + 2) = lim = ⎝ x →2 2 2 x→2 4x 4(2)2 4 ⎛⎜ lim x ⎞⎟ ⎝ x→2 ⎠ 3 =– 4 x→2 x→c x→c exist δ 2 and δ 3 such that 0 < x – c < δ 2 ⇒ g ( x) – M < ε and 0 < x – c < δ 3 ⇒ L + M +1 ε L + M +1 . Let δ = min{δ1 , δ 2 , δ 3 }, then 0 < x – c < δ ⇒ f ( x) g ( x) – LM ≤ g ( x) f ( x) – L + L g ( x ) – M < ( M + 1) ε L + M +1 +L ε L + M +1 =ε Hence, lim f ( x) g ( x) = LM = ⎛⎜ lim f ( x) ⎞⎟ ⎛⎜ lim g ( x) ⎞⎟ ⎝ x →c ⎠ ⎝ x →c ⎠ x→2 36. Say lim g ( x ) = M , M ≠ 0 , and choose x →c 1 M . 2 There is some δ1 > 0 such that ε1 = 1 M or 2 1 1 M < g ( x) < M + M . 2 2 1 1 1 1 M − M ≥ M and M + M ≥ M 2 2 2 2 1 2 1 so g ( x) > M and < g ( x) M 2 M− x →2 lim 0 < x – c < δ1 , g ( x) < M + 1. Choose ε > 0. Since lim f (x) = L and lim g(x) = M, there 0 < x − c < δ1 ⇒ g ( x) − M < ε1 = x–2 2x x – 2 x→2 x – 2 x →2 x – 2 1 –1 –1 1 = lim – = = =– 2 lim x 2(2) 4 x→2 2 x x→2 3 x2 x →c x →c 32. lim – f ( x) g ( x) – LM ≤ g ( x) f ( x) – L + L g ( x ) – M as shown in the text. Choose ε 1 = 1. Since lim g ( x) = M , there is some δ1 > 0 such that if f ( x) – L < = 3 + 3 –1 = 6 33. lim x→c M – 1 ≤ M + 1 and M + 1 ≤ M + 1 so for 29. lim ⎡⎣ f (t ) + 3g (t ) ⎤⎦ = lim f (t ) + 3 lim g (t ) t →a t →a t →a 1 x x→c 0 < x – c < δ1 , g ( x) – M < ε1 = 1 or M – 1 < g(x) < M + 1 4 = ⎡⎢ lim f ( x) – lim 3⎤⎥ = (3 – 3) 4 = 0 x →a ⎦ ⎣ x→a t →a 35. Suppose lim f (x) = L and lim g(x) = M. Choose ε > 0. Since lim g(x) = M there is δ 2 > 0 such that x→c 0 < x − c < δ 2 ⇒ g ( x) − M < Let δ = min{δ1 , δ 2}, then 0< x–c <δ ⇒ = 1 2 M . 2 1 1 M – g ( x) – = g ( x) M g ( x) M 1 2 2 1 2 ⋅ M ε g ( x) − M < g ( x) − M = 2 M g ( x) M M2 2 =ε Instructor’s Resource Manual Section 1.3 75 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Thus, lim x→c 1 1 1 = = . g(x) M lim g (x) 43. x→c Using statement 6 and the above result, f ( x) 1 lim = lim f ( x) ⋅ lim x →c g ( x ) x →c x →c g ( x ) lim f ( x ) 1 . = lim f ( x) ⋅ = x →c lim g ( x ) lim g ( x) x →c x →c x→c x →3+ x→ c ⇔ lim f (x) – lim L = 0 45. x→c 2 ⎡ ⎤ 38. lim f (x) = 0 ⇔ ⎢ lim f (x) ⎥ = 0 ⎣ ⎦ x→c x→c 46. 2 ⇔ lim f ( x) = 0 lim x →1– lim x → 2+ lim f 2 ( x) = 0 47. x →c ⇔ lim x →c f 2 ( x) = 0 48. ⇔ lim f ( x) = 0 x→c 2 39. lim x = ⎛⎜ lim x ⎞⎟ = x →c ⎝ x →c ⎠ lim x x →c 2 = lim x 2 x →c 2 x +1 x–5 , g ( x) = and c = 2, then x–2 x–2 lim [ f (x) + g (x)] exists, but neither x→c lim f (x) nor lim g(x) exists. x→c 2 , g ( x) = x, and c = 0, then x lim [ f (x) ⋅ g( x)] exists, but lim f (x) does b. If f ( x) = x→c x→c not exist. 41. 42. 76 lim x → –3+ 3+ x 3–3 = =0 x –3 π3 + x3 = x x → – π+ lim Section 1.3 (3 x − 1) = 2 π3 + (– π)3 =0 –π (22 + 1) 2 (3 ⋅ 2 − 1) 2 lim ( x − x ) = lim x − lim x →3− x →3− = x = –1 x lim x 2 + 2 x = 32 + 2 ⋅ 3 = 15 x →0 – x →3+ f ( x) g ( x) = 1; g ( x) = 1 f ( x) lim g ( x) = 0 ⇔ lim 1 =0 f ( x) x →a 5⋅ 2 5 2 = 2 5 x = 3− 2 =1 lim ⇔ If f ( x) = x→c ( x 2 + 1) x x →a = ⎛⎜ lim x ⎞⎟ = c 2 = c ⎝ x →c ⎠ 40. a. 49. x2 – 9 x+3 1+ x 1+1 2 = = 4 + 4 x 4 + 4(1) 8 x →3− x→c ⇔ x2 – 9 x →3+ 32 – 9 =0 3+3 = x →c x→c ⇔ lim [ f (x) – L] = 0 ( x – 3) x 2 – 9 ( x – 3) x 2 – 9 = lim x →3+ ( x – 3)( x + 3) x →3+ 44. x→ c x2 – 9 = lim = lim x →c 37. lim f (x) = L ⇔ lim f ( x) = lim L x–3 lim 1 =0 lim f ( x) x→a No value satisfies this equation, so lim f ( x) x→ a must not exist. 1⎞ ⎛ x 50. R has the vertices ⎜ ± , ± ⎟ ⎝ 2 2⎠ Each side of Q has length x 2 + 1 so the perimeter of Q is 4 x 2 + 1. R has two sides of length 1 and two sides of length x 2 so the perimeter of R is 2 + 2 x 2 . lim x →0 + = perimeter of R 2 x2 + 2 = lim perimeter of Q x →0+ 4 x 2 + 1 2 02 + 2 2 4 0 +1 = 2 1 = 4 2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. NO = (0 – 0)2 + (1 – 0)2 = 1 51. a. 4. lim OP = ( x – 0)2 + ( y – 0) 2 = x 2 + y 2 = x2 + x NP = ( x – 0)2 + ( y – 1)2 = x 2 + y 2 – 2 y + 1 = x + x − 2 x +1 MO = (1 – 0) 2 + (0 – 0) 2 = 1 y2 + x2 – 2 x + 1 = x2 − x + 1 perimeter of ΔNOP lim x →0+ perimeter of ΔMOP = lim = 1+ 1 1+ 1 3 x tan x 3 x (sin x / cos x) 3x = lim = lim x →0 x → 0 cos x sin x sin x 0 =0 1 5. lim sin x 1 sin x 1 1 = lim = ⋅1 = 2x 2 x →0 x 2 2 sin 3 3 sin 3 3 sin 3 = lim ⋅ = lim 3 2 →0 3 →0 2 →0 2 3 3 = ⋅1 = 2 2 6. lim sin 3 sin 3 cos sin 3 = lim sin = lim → 0 tan →0 →0 sin cos 7. lim 1 + x2 + x + x2 + x – 2 x + 1 x → 0+ = x →0 2 MP = ( x – 1)2 + ( y – 0) 2 = x →0 1 + x2 + x + x2 – x + 1 =1 x 1 (1)( x) = 2 2 1 x Area of ΔMOP = (1)( y ) = 2 2 ⎡ sin 3 1 ⎤ = lim ⎢cos ⋅ 3 ⋅ ⋅ sin ⎥ →0⎢ 3 ⎥⎦ ⎣ ⎡ sin 3 1 = 3 lim ⎢cos ⋅ ⋅ sin →0 ⎢ 3 ⎣ b. Area of ΔNOP = x area of ΔNOP x = lim 2 = lim x + + + MOP area of Δ x x →0 x →0 x →0 lim 2 = lim x →0+ x =0 1.4 Concepts Review 1. 0 sin 5 tan 5 sin 5 = lim cos 5 = lim → 0 sin 2 → 0 sin 2 → 0 cos 5 sin 2 sin 5 1 2 ⎤ ⎡ 1 = lim ⎢ ⋅5⋅ ⋅ ⋅ 5 2 sin 2 ⎥⎦ →0 ⎣ cos 5 5 sin 5 2 ⎤ ⎡ 1 = lim ⎢ ⋅ ⋅ 2 →0 ⎣ cos 5 5 sin 2 ⎥⎦ 5 5 = ⋅1⋅1⋅1 = 2 2 8. lim cot π sin →0 2 sec 9. lim = lim →0 cos π sin π sin 2 cos cos π sin cos 2sin π ⎡ cos π cos sin 1 π = lim ⎢ ⋅ ⋅ ⋅ π sin π 2 →0 ⎣ 1 sin π ⎡ = ⋅ lim ⎢cos π cos ⋅ 2 π →0 ⎣ sin π 1 1 ⋅1⋅1⋅1⋅1 = = 2π 2π = lim 2. 1 →0 3. the denominator is 0 when t = 0 . 4. 1 Problem Set 1.4 cos x 1 = =1 x →0 x + 1 1 lim →π / 2 cos = π 2 ⎤ ⎥ ⎦ ⎤ ⎥ ⎦ sin 2 3t 9t sin 3t sin 3t = lim ⋅ ⋅ = 0 ⋅1 ⋅1 = 0 t →0 t → 0 2t 2 3t 3t 1. lim 2. ⎤ ⎥ = 3 ⋅1 ⋅1 ⋅1 = 3 ⎥⎦ 10. lim ⋅0 = 0 cos 2 t cos 2 0 1 = = =1 t →0 1 + sin t 1 + sin 0 1 + 0 3. lim tan 2 3t sin 2 3t = lim t →0 t →0 (2t )(cos 2 3t ) 2t 11. lim = lim t →0 3(sin 3t ) sin 3t ⋅ = 0 ⋅1 = 0 3t 2 cos 2 3t tan 2t 0 = =0 t → 0 sin 2t − 1 −1 12. lim Instructor’s Resource Manual Section 1.4 77 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. sin(3t ) + 4t 4t ⎞ ⎛ sin 3t = lim ⎜ + ⎟ t →0 ⎝ t sec t t sec t t sec t ⎠ sin 3t 4t = lim + lim t →0 t sec t t →0 t sec t sin 3t = lim 3cos t ⋅ + lim 4 cos t t →0 t →0 3t = 3 ⋅1 + 4 = 7 13. lim t →0 14. lim →0 sin 2 2 = lim →0 = lim →0 sin × lim →0 sin sin sin = 1× 1 = 1 19. lim 1 + x →0 sin x =2 x 20. The result that lim cos t = 1 was established in t →0 the proof of the theorem. Then lim cos t = lim cos(c + h) t →c h →0 = lim (cos c cos h − sin c sin h) h →0 = lim cos c lim cos h − sin c lim sin h 15. lim x sin (1/ x ) = 0 h →0 x →0 h →0 h→0 = cos c lim sin t sin t t →c sin c = = = tan c t → c cos t lim cos t cos c 21. lim tan t = lim t →c t →c lim cot t = lim t →c ( ) 16. lim x sin 1/ x 2 = 0 x →0 t →c lim cos t cos t t →c cos c = = = cot c sin t lim sin t sin c t →c 1 1 = = sec c cos t cos c 1 1 lim csc t = lim = = csc c t →c t →c sin t sin c 22. lim sec t = lim t →c t →c 23. BP = sin t , OB = cos t area( ΔOBP) ≤ area (sector OAP ) ≤ area (ΔOBP) + area( ABPQ) ( ) 17. lim 1 − cos 2 x / x = 0 x →0 1 1 1 OB ⋅ BP ≤ t (1) 2 ≤ OB ⋅ BP + (1 – OB) BP 2 2 2 1 1 1 sin t cos t ≤ t ≤ sin t cos t + (1 – cos t ) sin t 2 2 2 t ≤ 2 – cos t sin t 1 sin t 1 π π ≤ ≤ for − < t < . 2 – cos t t cos t 2 2 1 sin t 1 lim ≤ lim ≤ lim t →0 2 – cos t t →0 t t →0 cos t sin t 1 ≤ lim ≤1 t →0 t sin t = 1. Thus, lim t →0 t cos t ≤ 18. lim cos 2 x = 1 x →0 78 Section 1.4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 16. lim n →∞ n2 n2 + 1 1 = lim n →∞ 1+ = 1 n2 1 =1 1+ 0 23. lim n n2 n ∞ n →∞ = lim = = =∞ 1 n →∞ n + 1 n →∞ 1 ⎞ 1+ 0 ⎛ 1+ lim 1 + n n →∞ ⎜⎝ n ⎟⎠ 17. lim 18. lim n →∞ n n2 + 1 n →∞ 1+ lim = 1 n2 x →∞ n b0 x + b1 x x →∞ 0 =0 1+ 0 x →∞ = 20. 21. 2 1 x2 + 3 = lim 2 + 1x n +1 x →∞ x 2 +3 x 2 + 1x = lim x →∞ 1+ 2 3 x2 lim x →∞ 26. lim lim ⎛⎜ 2 x 2 + 3 – 2 x 2 – 5 ⎞⎟ ⎠ x →∞ ⎝ = lim x →∞ = lim x →∞ 2 x 2 + 3 – (2 x 2 – 5) 2 x2 + 3 + 2 x2 – 5 8 2 2 x + 3 + 2 x2 − 5 8 x 2+ 3 x2 x →∞ 8 x 2 x 2 +3 + 2 x 2 –5 x2 + 2– 22. lim ⎛⎜ x 2 + 2 x − x ⎞⎟ ⎠ x →∞ ⎝ ⎛ x 2 + 2 x – x ⎞⎛ x 2 + 2 x + x ⎞ ⎜ ⎟⎜ ⎟ ⎠⎝ ⎠ = lim ⎝ 2 x →∞ x + 2x + x = lim x →∞ = lim x →∞ 80 2 x + 2x – x 2 = lim 2x x 2 + 2 x + x x→∞ x 2 + 2 x + x 2 2 = =1 1+ 2 +1 2 + bn –1 x n –1 n →∞ + an xn + bn xn 1 1 1+ 2 n = = a0 b0 1 1+ 0 =1 t → –3+ 29. As t → 3– , t 2 → 9 while 9 – t 2 → 0+. t →3– t2 9 – t2 =∞ + 30. As x → 3 5 , x 2 → 52 / 3 while 5 – x3 → 0 – . x→3 5 5 x2 = lim + t2 – 9 (t + 3)(t – 3) = lim + + t +3 t → –3 t + 3 t → –3 = lim (t – 3) = –6 lim =0 + + bn –1 x + bn an –1 x n –1 lim lim = lim + + + an –1 x + an 27. As x → 4+ , x → 4 while x – 4 → 0 . x lim =∞ + x – 4 x →4 28. ⎛ 2 x 2 + 3 – 2 x 2 – 5 ⎞⎛ 2 x 2 + 3 + 2 x 2 – 5 ⎞ ⎜ ⎟⎜ ⎟ ⎠⎝ ⎠ = lim ⎝ 2 2 x →∞ 2x + 3 + 2x – 5 x →∞ n –1 = –∞ + + 12 2x +1 x = lim = lim =0 x+4 x →∞ 1 + 4 x →∞ 1 + 4 x x = lim y→–∞ 1 – n2 ∞ n n3/ 2 = lim = =∞ 3 n →∞ 1 2 1 n + 2n + 1 1+ 2 + 3 n n n →∞ 2 x = lim 2 =2 2 x +1 x2 b0 + b1 x 2 19. For x > 0, x = x 2 . 2x + 1 a0 + a1 x n 25. lim n →∞ lim y2 – 2 y + 2 a0 x n + a1 x n –1 + = lim 1 n = lim 24. lim y→– ∞ 1 y2 2+ 2 y y2 9y + 9 y3 + 1 x2 + 5 – x3 = –∞ 31. As x → 5– , x 2 → 25, x – 5 → 0 – , and 3 – x → –2. x2 lim =∞ x →5 – ( x – 5)(3 – x ) 32. As − → π+ , 2 → π2 while sin → 0 . 2 lim →π+ sin = −∞ x Section 1.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. − 33. As x → 3− , x3 → 27, while x − 3 → 0 . 43. 3 lim x →3− x = −∞ x−3 π+ π2 ,π → while cos → 0 – . 2 2 π = –∞ lim π + cos → 34. As → 35. 3 3 = 0, lim = 0; x +1 x→ – ∞ x + 1 Horizontal asymptote y = 0. 3 3 lim = ∞, lim = – ∞; x → –1+ x + 1 x → –1– x + 1 Vertical asymptote x = –1 lim x →∞ 2 lim x →3– x2 – x – 6 ( x + 2)( x – 3) = lim x–3 x–3 x →3– = lim ( x + 2) = 5 x →3 – 36. x2 + 2 x – 8 lim 2 x → 2+ = lim x → 2+ x –4 x+4 6 3 = lim = = x → 2+ x + 2 4 2 ( x + 4)( x – 2) ( x + 2)( x – 2) 37. For 0 ≤ x < 1 , x = 0 , so for 0 < x < 1, thus lim x →0 + x x 44. x x lim 3 x →∞ ( x + 1) 2 3 = 0, lim x → – ∞ ( x + 1) 2 = 0; Horizontal asymptote y = 0. 3 3 lim = ∞, lim = ∞; 2 2 – + x → –1 ( x + 1) x → –1 ( x + 1) Vertical asymptote x = –1 =0 =0 38. For −1 ≤ x < 0 , x = −1 , so for –1 < x < 0, x x 1 thus lim = ∞. − x x x x →0 1 (Since x < 0, – > 0. ) x =− 39. For x < 0, x = – x, thus lim x →0 – x x = lim x →0 – 45. –x = –1 x 40. For x > 0, x = x, thus lim x →0 + x x = lim x →0 + x =1 x 41. As x → 0 – , 1 + cos x → 2 while sin x → 0 –. 1 + cos x lim = –∞ – sin x x →0 lim x →∞ 2x 2 = lim = 2, x – 3 x→∞ 1 – 3 x 2x 2 lim = lim = 2, x →−∞ x – 3 x →−∞ 1 – 3 x Horizontal asymptote y = 2 2x 2x lim = ∞, lim = – ∞; x →3+ x – 3 x →3– x – 3 Vertical asymptote x = 3 42. –1 ≤ sin x ≤ 1 for all x, and 1 sin x lim = 0, so lim = 0. x →∞ x x →∞ x Instructor’s Resource Manual Section 1.5 81 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 46. lim 3 2 x →∞ 9 – 3 = 0, lim 2 49. f ( x ) = 2 x + 3 – = 0; x→ – ∞ 9 – x x Horizontal asymptote y = 0 3 3 lim = – ∞, lim = ∞, 2 2 – + x →3 9 – x x →3 9 – x 3 3 lim = ∞, lim = – ∞; 2 2 x → –3+ 9 – x x → –3– 9 – x Vertical asymptotes x = –3, x = 3 50. 14 x →∞ 2 x 2 f ( x) = 3x + 4 – 4x + 3 x2 + 1 , thus We say that lim f ( x) = – ∞ if to each x →c + negative number M there corresponds a δ > 0 such that 0 < x – c < δ ⇒ f(x) < M. 14 = 0, lim , thus ⎡ 4x + 3⎤ lim [ f ( x) – (3 x + 4)] = lim ⎢ – ⎥ x →∞ x →∞ ⎣ x 2 + 1 ⎦ ⎡ 4+ 3 ⎤ x x2 ⎥ = lim ⎢ – =0. ⎢ x →∞ 1 + 12 ⎥ ⎢⎣ ⎥ x ⎦ The oblique asymptote is y = 3x + 4. = 0; x→ – ∞ 2 x2 + 7 +7 Horizontal asymptote y = 0 2 Since 2x + 7 > 0 for all x, g(x) has no vertical asymptotes. lim x –1 1 ⎤ ⎡ lim [ f ( x) – (2 x + 3)] = lim ⎢ – ⎥=0 3 x →∞ x →∞ ⎣ x –1 ⎦ The oblique asymptote is y = 2x + 3. 51. a. 47. 1 3 b. We say that lim f ( x) = ∞ if to each x →c – positive number M there corresponds a δ > 0 such that 0 < c – x < δ ⇒ f(x) > M. We say that lim f ( x) = ∞ if to each 52. a. x →∞ positive number M there corresponds an N > 0 such that N < x ⇒ f(x) > M. b. We say that lim f ( x ) = ∞ if to each x → –∞ positive number M there corresponds an N < 0 such that x < N ⇒ f(x) > M. 53. Let ε > 0 be given. Since lim f ( x ) = A, there is x →∞ 48. lim x →∞ lim x→ – ∞ 2x 2 x +5 = lim 2x x2 + 5 x →∞ = lim 2 1+ x→ – ∞ 5 x2 = 2 – 1+ 5 x2 2 1 = a corresponding number M1 such that = 2, 2 – 1 ε x > M1 ⇒ f ( x) – A < . Similarly, there is a 2 = –2 Since x 2 + 5 > 0 for all x, g(x) has no vertical asymptotes. ε number M2 such that x > M 2 ⇒ g ( x) – B < . 2 Let M = max{M1 , M 2 } , then x > M ⇒ f ( x) + g ( x) – ( A + B) = f ( x) – A + g ( x) – B ≤ f ( x) – A + g ( x) – B ε ε =ε 2 2 Thus, lim [ f ( x) + g ( x)] = A + B < + x →∞ 54. Written response 82 Section 1.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 55. a. lim sin x does not exist as sin x oscillates x →∞ 56. between –1 and 1 as x increases. 1 , then as x → ∞, u → 0+. x 1 lim sin = lim sin u = 0 x u →0 + x →∞ b. Let u = c. 1 Let u = , then as x → ∞, u → 0+. x 1 1 sin u lim x sin = lim sin u = lim =1 + x u → 0+ u u x →∞ u →0 d. Let u = lim x 3/ 2 x →∞ e. h. 59. 60. 3/ 2 1 − v2 / c2 v →c 3x 2 + x +1 3 = 2 2 x →∞ 2x –1 2 x 2 – 3x lim = 2 5x + 1 x→ – ∞ 2 5 3 lim ⎛⎜ 2 x 2 + 3x – 2 x 2 – 5 ⎞⎟ = – ⎠ 2 2 x→ – ∞ ⎝ 2x +1 lim x →∞ 3x 2 + 1 sin u 2 = 3 10 ⎡⎛ 1 ⎞⎛ sin u ⎞⎤ ⎟⎟⎜ = lim+ ⎢⎜⎜ ⎟⎥ = ∞ u →0 ⎣ ⎢⎝ u ⎠⎝ u ⎠⎦⎥ As x → ∞, sin x oscillates between –1 and 1, 1 while x –1/ 2 = → 0. x 62. ⎛ 1⎞ lim ⎜1 + ⎟ = e ≈ 2.718 x⎠ x →∞ ⎝ –1/ 2 sin x = 0 1 , then x ⎛π 1⎞ ⎛π ⎞ lim sin ⎜ + ⎟ = lim+ sin ⎜ + u ⎟ x→∞ x 6 6 u → 0 ⎝ ⎠ ⎝ ⎠ π 1 = sin = 6 2 Let u = 1 1⎞ ⎛ → ∞, so lim sin ⎜ x + ⎟ x x⎠ x →∞ ⎝ does not exist. (See part a.) As x → ∞, x + 1⎞ 1 1 ⎛ sin ⎜ x + ⎟ = sin x cos + cos x sin x⎠ x x ⎝ ⎡ ⎛ 1⎞ ⎤ lim ⎢sin ⎜ x + ⎟ – sin x ⎥ x x →∞ ⎣ ⎝ ⎠ ⎦ ⎡ 1 ⎞ 1⎤ ⎛ = lim ⎢sin x ⎜ cos –1⎟ + cos x sin ⎥ x ⎠ x⎦ x →∞ ⎣ ⎝ 1 1 As x → ∞, cos → 1 so cos –1 → 0. x x 1 From part b., lim sin = 0. x x →∞ As x → ∞ both sin x and cos x oscillate between –1 and 1. ⎡ ⎛ 1⎞ ⎤ lim ⎢sin ⎜ x + ⎟ – sin x ⎥ = 0. x⎠ x →∞ ⎣ ⎝ ⎦ Instructor’s Resource Manual =∞ lim ⎛ 1⎞ lim ⎜1 + ⎟ x⎠ x →∞ ⎝ lim x g. 1 ⎛1⎞ sin = lim+ ⎜ ⎟ x u →0 ⎝ u ⎠ 58. v →c 61. x →∞ f. 1 , then x 57. m0 lim− m(v) = lim− =1 x ⎛ 1⎞ 63. lim ⎜1 + ⎟ x⎠ x →∞ ⎝ 64. 65. 66. 67. 68. ⎛ 1⎞ lim ⎜1 + ⎟ x⎠ x →∞ ⎝ 70. 71. =∞ sin x =1 sin x – 3 lim x →3– = –1 x–3 sin x – 3 lim x →3– tan( x – 3) = –1 lim cos( x – 3) = –∞ x–3 lim cos x = –1 x – π2 x →3– x→ π 2 69. x2 + lim (1 + x ) x →0 + 1 x = e ≈ 2.718 lim (1 + x )1/ x = ∞ x → 0+ lim (1 + x ) x = 1 x →0+ Section 1.5 83 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 13. 1.6 Concepts Review t →3+ lim f (t ) = lim (t – 3) = 0 t →3 – 1. lim f ( x) x →c t →3 – lim f (t ) = f (3); continuous t →3 2. every integer 3. lim f (t ) = lim (3 – t ) = 0 t →3+ lim f ( x) = f (a); lim f ( x) = f (b) x→a+ 14. x →b – lim f (t ) = lim (3 – t )2 = 0 t →3+ t →3+ lim f (t ) = lim (t 2 – 9) = 0 t →3– 4. a; b; f(c) = W t →3 – lim f (t ) = f (3); continuous t →3 15. lim f ( x) = −2 = f (3); continuous Problem Set 1.6 t →3 1. lim[( x – 3)( x – 4)] = 0 = f (3); continuous x →3 2. lim ( x 2 – 9) = 0 = g (3); continuous x →3 3 3. lim x →3 x – 3 and h(3) do not exist, so h(x) is not continuous at 3. 16. g is discontinuous at x = –3, 4, 6, 8; g is left continuous at x = 4, 8; g is right continuous at x = –3, 6 17. h is continuous on the intervals (−∞, −5), [ −5, 4] , (4, 6), [ 6,8] , (8, ∞) x 2 – 49 ( x – 7)( x + 7) = lim = lim ( x + 7) x–7 x →7 x – 7 x →7 x →7 = 7 + 7 = 14 Define f(7) = 14. 18. lim 4. lim t – 4 and g(3) do not exist, so g(t) is not t →3 continuous at 3. t –3 and h(3) do not exist, so h(t) is not t –3 continuous at 3. 5. lim 2 x 2 –18 2( x + 3)( x – 3) = lim 3– x x →3 3 – x x →3 = lim[–2( x + 3)] = –2(3 + 3) = –12 19. lim t →3 x →3 Define f(3) = –12. 6. h(3) does not exist, so h(t) is not continuous at 3. 7. lim t = 3 = f (3); continuous t →3 20. lim →0 t →3 21. lim t →1 t 3 – 27 (t – 3)(t 2 + 3t + 9) = lim t –3 t →3 t – 3 t →3 = lim(t 2 + 3t + 9) = (3)2 + 3(3) + 9 = 27 = r (3) 22. 12. From Problem 11, lim r (t ) = 27, so r(t) is not t →3 continuous at 3 because lim r (t ) ≠ r (3). t →3 t –1 –1)( t + 1) 1 Define H(1) = . 2 t →1 (t 11. lim continuous t –1 ( t –1)( t + 1) = lim t –1 t →1 (t –1)( t + 1) = lim 10. f(3) does not exist, so f(x) is not continuous at 3. t →3 =1 Define g(0) = 1 8. lim t – 2 = 1 = g (3); continuous 9. h(3) does not exist, so h(t) is not continuous at 3. sin( ) = lim t →1 1 t +1 = 1 2 x4 + 2 x2 – 3 ( x 2 –1)( x 2 + 3) = lim x +1 x +1 x → –1 x → –1 lim ( x + 1)( x – 1)( x 2 + 3) x +1 x → –1 = lim = lim [( x – 1)( x 2 + 3)] x → –1 = (–1 – 1)[(–1)2 + 3] = –8 Define φ(–1) = –8. 84 Section 1.6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 23. ⎛ x2 – 1 ⎞ ⎛ ( x – 1)( x + 1) ⎞ lim sin ⎜ ⎟ = lim sin ⎜ ⎟ ⎜ ⎟ x +1 x → –1 ⎠ ⎝ x + 1 ⎠ x→ –1 ⎝ = lim sin( x –1) = sin(–1 – 1) = sin(−2) = – sin 2 37. x → –1 Define F(–1) = –sin 2. 24. Discontinuous at x = π ,30 25. 33 – x 2 (π – x)( x – 3) Discontinuous at x = 3, π f ( x) = 38. 26. Continuous at all points 27. Discontinuous at all = nπ + π where n is any 2 integer. 28. Discontinuous at all u ≤ −5 39. 29. Discontinuous at u = –1 30. Continuous at all points 31. G ( x) = 1 (2 – x)(2 + x) Discontinuous on (−∞, −2] ∪ [2, ∞) 32. Continuous at all points since lim f ( x) = 0 = f (0) and lim f ( x) = 1 = f (1). x →0 40. x →1 33. lim g ( x ) = 0 = g (0) x →0 lim g ( x) = 1, lim g ( x) = –1 x →1+ x →1– lim g(x ) does not exist, so g(x) is discontinuous x→1 at x = 1. 34. Discontinuous at every integer 35. Discontinuous at t = n + 1 where n is any integer 2 Discontinuous at all points except x = 0, because lim f ( x ) ≠ f (c) for c ≠ 0 . lim f ( x ) exists only x →c x →c at c = 0 and lim f ( x) = 0 = f (0) . x →0 36. 41. Continuous. 42. Discontinuous: removable, define f (10) = 20 43. Discontinuous: removable, define f (0) = 1 44. Discontinuous: nonremovable. 45. Discontinuous, removable, redefine g (0) = 1 46. Discontinuous: removable, define F (0) = 0 Instructor’s Resource Manual Section 1.6 85 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 47. Discontinuous: nonremovable. 48. Discontinuous: removable, define f (4) = 4 49. The function is continuous on the intervals ( 0,1] , (1, 2], (2,3], … 52. Let f ( x) = x3 + 3 x − 2. f is continuous on [0, 1]. f(0) = –2 < 0 and f(1) = 2 > 0. Thus, there is at least one number c between 0 and 1 such that x 3 + 3x − 2 = 0. 53. Because the function is continuous on [ 0,2π ] and (cos 0)03 + 6sin 5 0 – 3 = –3 < 0, Cost $ 0.60 (cos 2π)(2π)3 + 6sin 5 (2π) – 3 = 8π3 – 3 > 0, there is at least one number c between 0 and 2π such 0.48 that (cos t )t 3 + 6sin 5 t – 3 = 0. 0.72 0.36 54. Let f ( x ) = x − 7 x + 14 x − 8 . f(x) is continuous at all values of x. f(0) = –8, f(5) = 12 Because 0 is between –8 and 12, there is at least one number c between 0 and 5 such that 3 0.24 0.12 1 3 5 2 4 6 Length of call in minutes 50. The function is continuous on the intervals [0, 200], (200,300], (300, 400], … 2 f ( x ) = x 3 − 7 x 2 + 14 x − 8 = 0 . This equation has three solutions (x = 1,2,4) Cost $ 80 60 40 55. Let f ( x ) = x − cos x. . f(x) is continuous at all 20 100 200 300 400 500 Miles Driven 51. The function is continuous on the intervals (0, 0.25], (0.25, 0.375], (0.375, 0.5], … values of x 0. f(0) = –1, f(π/2) = π / 2 Because 0 is between –1 and π / 2 , there is at least one number c between 0 and π/2 such that f ( x ) = x − cos x = 0. The interval [0.6,0.7] contains the solution. Cost $ 4 3 2 1 0.25 0.5 0.75 Miles Driven 1 56. Let f ( x) = x5 + 4 x3 – 7 x + 14 f(x) is continuous at all values of x. f(–2) = –36, f(0) = 14 Because 0 is between –36 and 14, there is at least one number c between –2 and 0 such that f ( x) = x5 + 4 x3 – 7 x + 14 = 0. 86 Section 1.6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 57. Suppose that f is continuous at c, so lim f ( x) = f (c). Let x = c + t, so t = x – c, then x →c as x → c , t → 0 and the statement lim f ( x) = f (c) becomes lim f (t + c ) = f (c). x →c t →0 Suppose that lim f (t + c) = f (c) and let x = t + t→ 0 c, so t = x – c. Since c is fixed, t → 0 means that x → c and the statement lim f (t + c) = f (c) t →0 becomes lim f ( x) = f (c) , so f is continuous at x →c c. 58. Since f(x) is continuous at c, lim f ( x) = f (c) > 0. Choose ε = f ( c ) , then x →c there exists a δ > 0 such that 0 < x − c < δ ⇒ f ( x) − f (c) < ε . Thus, f ( x ) − f ( c ) > −ε = − f ( c ) , or f ( x ) > 0 . Since also f ( c ) > 0 , f ( x ) > 0 for all x in (c − δ , c + δ ). 59. Let g(x) = x – f(x). Then, g(0) = 0 – f(0) = –f(0) ≤ 0 and g(1) = 1 – f(1) ≥ 0 since 0 ≤ f(x) ≤ 1 on [0, 1] . If g(0) = 0, then f(0) = 0 and c = 0 is a fixed point of f. If g(1) = 0, then f(1) = 1 and c = 1 is a fixed point of f. If neither g(0) = 0 nor g(1) = 0, then g(0) < 0 and g(1) > 0 so there is some c in [0, 1] such that g(c) = 0. If g(c) = 0 then c – f(c) = 0 or f(c) = c and c is a fixed point of f. 60. For f(x) to be continuous everywhere, f(1) = a(1) + b = 2 and f(2) = 6 = a(2) + b a+b=2 2a + b = 6 – a = –4 a = 4, b = –2 63. Let f(x) be the difference in times on the hiker’s watch where x is a point on the path, and suppose x = 0 at the bottom and x = 1 at the top of the mountain. So f(x) = (time on watch on the way up) – (time on watch on the way down). f(0) = 4 – 11 = –7, f(1) = 12 – 5 = 7. Since time is continuous, f(x) is continuous, hence there is some c between 0 and 1 where f(c) = 0. This c is the point where the hiker’s watch showed the same time on both days. ⎡ π⎤ 64. Let f be the function on ⎢0, 2 ⎥ such that f( ) is ⎣ ⎦ the length of the side of the rectangle which makes angle with the x-axis minus the length of the sides perpendicular to it. f is continuous on ⎡ π⎤ ⎢0, 2 ⎥ . If f(0) = 0 then the region is ⎣ ⎦ circumscribed by a square. If f(0) ≠ 0, then ⎛π ⎞ observe that f (0) = − f ⎜ ⎟ . Thus, by the ⎝2⎠ Intermediate Value Theorem, there is an angle π such that f ( 0 ) = 0. 2 Hence, D can be circumscribed by a square. 0 between 0 and 65. Yes, g is continuous at R . lim g ( r ) = r →R− GMm = lim g ( r ) r →R+ R2 66. No. By the Intermediate Value Theorem, if f were to change signs on [a,b], then f must be 0 at some c in [a,b]. Therefore, f cannot change sign. 67. a. f(x) = f(x + 0) = f(x) + f(0), so f(0) = 0. We want to prove that lim f (x) = f (c), or, x→c equivalently, lim [ f (x) – f (c)] = 0. But x→c 61. For x in [0, 1], let f(x) indicate where the string originally at x ends up. Thus f(0) = a, f(1) = b. f(x) is continuous since the string is unbroken. Since 0 ≤ a, b ≤ 1 , f(x) satisfies the conditions of Problem 59, so there is some c in [0, 1] with f(c) = c, i.e., the point of string originally at c ends up at c. 62. The Intermediate Value Theorem does not imply the existence of a number c between –2 and 2 such that f (c ) = 0. The reason is that the function f ( x ) is not continuous on [ −2, 2] . Instructor’s Resource Manual f(x) – f(c) = f(x – c), so lim[ f ( x) – f (c)] = lim f ( x – c). Let x →c x →c h = x – c then as x → c, h → 0 and lim f ( x – c) = lim f (h) = f (0) = 0. Hence x →c h →0 lim f (x) = f (c) and f is continuous at c. x→c Thus, f is continuous everywhere, since c was arbitrary. b. By Problem 43 of Section 0.5, f(t) = mt for all t in Q. Since g(t) = mt is a polynomial function, it is continuous for all real numbers. f(t) = g(t) for all t in Q, thus f(t) = g(t) for all t in R, i.e. f (t ) = mt. Section 1.6 87 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 68. If f(x) is continuous on an interval then lim f ( x) = f (c) for all points in the interval: x →c lim f ( x) = f (c) ⇒ lim f ( x) x →c x →c = lim x →c f 2 ( x) = ⎛⎜ lim f ( x) ⎞⎟ ⎝ x →c ⎠ ⎡ 3 3⎤ Domain: ⎢ – , ⎥ ; ⎣ 4 4⎦ 3⎫ ⎧ 3 Range: ⎨ – , 0, ⎬ 4 4⎭ ⎩ 2 = ( f (c))2 = f (c ) ⎧ 1 if x ≥ 0 69. Suppose f ( x) = ⎨ . f(x) is ⎩−1 if x < 0 discontinuous at x = 0, but g(x) = f ( x) = 1 is b. At x = 0 3 3 c. If x = 0, f ( x) = 0 , if x = – , f ( x) = – and 4 4 3 3 3 3 if x = , f ( x) = , so x = − , 0, are 4 4 4 4 fixed points of f. continuous everywhere. 70. a. 1.7 Chapter Review Concepts Test 1. False. Consider f ( x ) = x at x = 2. 2. False: c may not be in the domain of f(x), or it may be defined separately. 3. False: c may not be in the domain of f(x), or it may be defined separately. b. If r is any rational number, then any deleted interval about r contains an irrational 1 number. Thus, if f (r ) = , any deleted q interval about r contains at least one point c 1 1 such that f (r ) – f (c) = – 0 = . Hence, q q lim f (x) does not exist. 4. True. By definition, where c = 0, L = 0. 5. False: If f(c) is not defined, lim f ( x ) might exist; e.g., f ( x) = x →c x2 – 4 = −4. x →−2 x + 2 6. True: 88 Suppose the block rotates to the left. Using 3 geometry, f ( x) = – . Suppose the block 4 rotates to the right. Using geometry, 3 f ( x) = . If x = 0, the block does not rotate, 4 so f(x) = 0. Section 1.7 x 2 − 25 ( x − 5)( x + 5) = lim x−5 x →5 x − 5 x →5 = lim ( x + 5) = 5 + 5 = 10 lim x →5 7. True: 8. False: number c. 71. a. x –4 . x+2 f(–2) does not exist, but lim x→r If c is any irrational number in (0, 1), then as p p x = → c (where is the reduced form q q of the rational number) q → ∞, so f ( x) → 0 as x → c. Thus, lim f ( x) = 0 = f (c) for any irrational x→c 2 9. False: 10. True: Substitution Theorem lim x →0 sin x =1 x The tangent function is not defined for all values of c. sin x , cos x then cos x ≠ 0 , and Theorem A.7 applies.. If x is in the domain of tan x = Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 11. True: Since both sin x and cos x are continuous for all real numbers, by Theorem C we can conclude that f ( x) = 2 sin 2 x − cos x is also continuous for all real numbers. 12. True. By definition, lim f ( x ) = f ( c ) . 13. True. 2 ∈ [1,3] 14. False: lim may not exist x →0 − Consider f ( x) = sin x. 16. True. By the definition of continuity on an interval. 17. False: Since −1 ≤ sin x ≤ 1 for all x and 1 sin x lim = 0 , we get lim =0. x →∞ x x →∞ x It could be the case where lim f ( x ) = 2 x→2 26. False: The graph has many vertical asymptotes; e.g., x = ± π/2, ± 3π/2, ± 5π/2, … 20. True: x = 2 ; x = –2 21. True: As x → 1+ both the numerator and denominator are positive. Since the numerator approaches a constant and the denominator approaches zero, the limit goes to + ∞ . 22. False: lim f ( x) must equal f(c) for f to be 24. True: lim g( x ) exist; e.g., f ( x) = x→c g ( x) = 28. True: A function has only one limit at a point, so if lim f ( x ) = L and x→ a lim f ( x ) = M , L = M x→ a 29. False: That f(x) ≠ g(x) for all x does not imply that lim f ( x) ≠ lim g ( x). For x →c x →c example, if f ( x) = 2 x +x–6 and x–2 5 x, then f(x) ≠ g(x) for all x, 2 but lim f ( x ) = lim g ( x ) = 5. g ( x) = x→2 30. False: x→ 2 If f(x) < 10, lim f ( x ) could equal 10 x→2 if there is a discontinuity point (2, 10). For example, – x3 + 6 x 2 − 2 x − 12 f ( x) = < 10 for x–2 all x, but lim f ( x) = 10. lim f ( x) = f ⎛⎜ lim x ⎞⎟ = f (c), so f is ⎝ x →c ⎠ continuous at x = c. x →2.3 x+7 for c = −2 . x+2 Squeeze Theorem x →c x = 1 = f ( 2.3) 2 x–3 and x+2 27. True: x →c lim x→c x→c continuous at x = c. 23. True: That lim [ f ( x ) + g ( x )] exists does not imply that lim f ( x ) and x →−∞ 19. False: Choose ε = 0. 001 f (2) then since lim f ( x ) = f (2), there is some δ such that 0 < x − 2 < δ ⇒ f ( x ) − f (2) < 0. 001 f (2), or −0. 001 f (2 ) < f ( x ) − f (2 ) < 0.001f(2) Thus, 0.999f(2) < f(x) < 1.001f(2) and f(x) < 1.001f(2) for 0 < x − 2 < δ . Since f(2) < 1.001f(2), as f(2) > 0, f(x) < 1.001f(2) on (2 − δ , 2 + δ ). x →c 15. False: 18. False. 25. True: x →2 31. True: lim f ( x) = lim x →a x →a f 2 ( x) 2 = ⎡⎢ lim f ( x) ⎤⎥ = (b)2 = b ⎣ x→a ⎦ 32. True: Instructor’s Resource Manual If f is continuous and positive on [a, b], the reciprocal is also continuous, so it will assume all 1 1 and . values between f ( a) f (b ) Section 1.7 89 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Sample Test Problems 14. x−2 2−2 0 = = =0 x →2 x + 2 2+2 4 1. lim u 2 – 1 12 − 1 = =0 1+1 u →1 u + 1 u2 – 1 (u – 1)(u + 1) = lim = lim (u + 1) u –1 u →1 u – 1 u →1 u →1 =1+1=2 u +1 u +1 = lim u →1 (u + 1)(u – 1) –1 does not exist 1 – 2x 5. lim = lim x –2 x = lim x →2 ( x – 2)( x + 2) x→2 x 2 –4 1 1 = = 2 (2 + 2 ) 8 z2 – 4 1 u →1 u – 1 = lim x→2 1 − cos 2 x 2 1 − cos 2 x = lim 3x 2x x →0 x →0 3 2 1 − cos 2 x 2 = lim = ×0 = 0 3 x →0 2x 3 1 x ( x + 2) 1 1− x −1 x = 1+ 0 = 1 17. lim = lim 2 1+ 0 x →∞ x + 2 x →∞ 1+ x 18. Since −1 ≤ sin t ≤ 1 for all t and lim = lim get lim t →∞ 19. lim sin x tan x 1 cos x = lim = lim x → 0 sin 2 x x → 0 2 sin x cos x x → 0 2 cos 2 x 1 1 = = 2 2 cos 0 2 y →1 y 2 –1 ( y – 1)( y 2 + y + 1) y →1 ( y – 1)( y + 1) = lim t+2 ( t − 2 )2 20. 21. y 2 + y + 1 12 + 1 + 1 3 = = 1+1 2 y +1 y →1 x–4 x –2 x→4 = lim ( x – 2)( x + 2) x →4 x –2 = lim ( x + 2) = 4 + 2 = 4 x→4 12. 13. 90 x lim x x →0 – lim x →(1/ 2)+ lim t →2 – ( = lim x →0 – –x = lim (–1) = –1 x x →0 – 4x = 2 cos x = ∞ , because as x → 0+ , cos x → 1 x →0 + x while the denominator goes to 0 from the right. lim Section 1.7 − x →π / 4− tan 2 x = ∞ because as x → (π / 4 ) , − 2 x → (π / 2 ) , so tan 2 x → ∞. 22. 1 + sin x = ∞ , because as x → 0+ , + x x →0 lim 1 + sin x → 1 while the denominator goes to 0 from the right. | 2 x − 6 |< ε ⇔ 2 | x − 3 |< ε ⇔| x − 3 |< ε 2 . Choose δ = ε 2 . Let > 0. Choose δ = ε / 2. Thus, t − t ) = lim t − lim t = 1 − 2 = −1 t →2 – = ∞ because as t → 0, t + 2 → 4 23. Preliminary analysis: Let > 0. We need to find a > 0 such that 0 <| x − 3 |< δ ⇒| ( 2 x + 1) − 7 |< ε . cos x does not exist. x →0 x 10. lim 11. sin t =0. t lim = lim 9. lim = 0 , we while the denominator goes to 0 from the right. 7. lim y3 – 1 1 t →∞ t t →2 8. lim 1− x = −1 since x − 1 < 0 as x −1 16. lim z → 2 ( z + 3)( z z →2 z 2 x →1− sin 5 x 5 sin 5 x = lim x →0 3 x x →0 3 5 x 5 sin 5 x 5 5 = lim = ×1 = 3 x →0 5 x 3 3 ; ( z + 2)( z – 2) – 2) +z–6 z +2 2 + 2 4 = = = lim 5 z→ 2 z + 3 2 + 3 6. lim x −1 = lim 15. lim 3. lim u →1 u 2 x →1− x → 1− 2. lim 4. lim x −1 lim t →2 – ( 2 x + 1) − 7 = 2 x − 6 = 2 x − 3 < 2 (ε / 2 ) = ε . Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 24. a. f(1) = 0 b. c. d. 28. lim f ( x) = lim (1 – x) = 0 x →1+ x →1+ lim f ( x) = lim x = 1 x →1– x →1– lim f ( x) = –1 because x → –1 lim f ( x) = lim x3 = –1 and x → –1– x → –1– lim f ( x) = lim x = –1 x → –1+ 25. a. x → –1+ f is discontinuous at x = 1 because f(1) = 0, but lim f (x ) does not exist. f is x→1 discontinuous at x = –1 because f(–1) does not exist. b. Define f(–1) = –1 26. a. b. 27. a. 0 < u – a < δ ⇒ g (u ) – M < ε 0 < a – x < δ ⇒ f ( x) – L < ε lim[2 f ( x) – 4 g ( x)] Horizontal: lim c. d. x →3 x2 – 9 lim g ( x) = lim g ( x )( x + 3) x – 3 x →3 x →3 = lim g ( x ) ⋅ lim ( x + 3) = –2 ⋅ (3 + 3) = –12 x →3 +1 Horizontal: lim x2 x →∞ x 2 = lim x2 + 1 x →−∞ x 2 + 1 y = 1 is a horizontal asymptote. g(3) = –2 lim g ( f ( x)) = g ⎛⎜ lim f ( x) ⎞⎟ = g (3) = –2 x →3 ⎝ x →3 ⎠ lim x →3 2 f ( x) – 8 g ( x) 2 = ⎡⎢ lim f ( x) ⎤⎥ – 8 lim g ( x ) x →3 ⎣ x →3 ⎦ 33. Vertical: x = 1, x = −1 because lim x →1+ lim x →3 g ( x) – g (3) f ( x) = –2 – g (3) 3 lim x →−1− x2 x2 − 1 Horizontal: lim = 1 , so x2 x2 − 1 x2 = lim x2 − 1 x→−∞ x 2 − 1 y = 1 is a horizontal asymptote. = = 0 , so =∞ =∞ x →∞ x 2 = (3) 2 – 8(–2) = 5 f. x x →−∞ x 2 32. Vertical: None, denominator is never 0. and e. = lim +1 y = 0 is a horizontal asymptote. = 2(3) – 4(–2) = 14 x →3 x x →∞ x 2 = 2 lim f ( x) – 4 lim g ( x ) b. 30. Let f ( x) = x5 – 4 x3 – 3 x + 1 f(2) = –5, f(3) = 127 Because f(x) is continuous on [2, 3] and f(2) < 0 < f(3), there exists some number c between 2 and 3 such that f(c) = 0. 31. Vertical: None, denominator is never 0. x →3 x →3 29. a(0) + b = –1 and a(1) + b = 1 b = –1; a + b = 1 a–1=1 a=2 = 1 , so −2 − (−2) 3 =0 Instructor’s Resource Manual Section 1.7 91 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 34. Vertical: x = 2, x = −2 because x lim 3 2 x → 2+ x −4 = ∞ and x →∞ x 2 x lim 3 x →−∞ x 2 −4 asymptotes. x lim x →−2− x3 Horizontal: lim −4 2. a. 3 2 x −4 =∞ = ∞ and = −∞ , so there are no horizontal 35. Vertical: x = ±π / 4, ± 3π / 4, ± 5π / 4,… because lim tan 2 x = ∞ and similarly for other odd x →π / 4− g ( 2 ) = 1/ 2 b. g ( 2.1) = 1/ 2.1 ≈ 0.476 c. g ( 2.1) − g ( 2 ) = 0.476 − 0.5 = −0.024 d. g ( 2.1) − g ( 2 ) 2.1 − 2 −0.024 = −0.24 0.1 = e. g ( a + h ) = 1/ ( a + h ) f. g ( a + h ) − g ( a ) = 1/ ( a + h ) − 1/ a = multiples of π / 4. Horizontal: None, because lim tan 2 x and x →∞ g. 36. Vertical: x = 0, because sin x 1 sin x lim = lim =∞. 2 + + x x →0 x →0 x x h. lim tan 2 x do not exist. g (a + h) − g (a) (a + h) − a x →−∞ Horizontal: y = 0, because lim x →∞ sin x x2 = lim sin x x →−∞ x2 f ( 2.1) = 2.12 = 4.41 c. f ( 2.1) − f ( 2 ) = 4.41 − 4 = 0.41 d. e. f. g. h. f ( 2.1) − f ( 2 ) 2.1 − 2 0.41 = = 4.1 0.1 f ( a + h ) = ( a + h ) = a 2 + 2ah + h 2 2 f ( a + h ) − f ( a ) = a 2 + 2ah + h 2 − a 2 = 2ah + h 2 f (a + h) − f (a) (a + h) − a lim h→0 2ah + h 2 = = 2a + h h f (a + h) − f ( a) (a + h) − a g (a + h) − g (a) (a + h) − a h→0 h →0 c. F ( 2.1) − F ( 2 ) = 1.449 − 1.414 = 0.035 d. F ( 2.1) − F ( 2 ) 2.1 − 2 Review and Preview = 0.035 = 0.35 0.1 e. F (a + h) = a + h f. F (a + h) − F (a) = a + h − a g. h. F (a + h) − F (a) lim a+h − a h = ( a + h) − a F (a + h) − F (a) h→0 = lim ( (a + h) − a a+h − a h →0 = lim h →0 h →0 = lim h →0 92 −1 a2 F ( 2 ) = 2 ≈ 1.414 = lim = lim ( 2a + h ) = 2a = −1 a (a + h) F ( 2.1) = 2.1 ≈ 1.449 f ( 2 ) = 22 = 4 b. lim h = b. = 0. Review and Preview Problems 1. a. 3. a. = −h a (a + h) −h a (a + h) h h h ( ( ( h→0 )( a+h + a a+h + a a+h−a a+h + a h a+h + a 1 a+h + a a+h − a h = lim = ) ) ) ) 1 2 a = a 2a Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4. a. G ( 2) = ( 2) + 1 = 8 + 1 = 9 3 10. b. G ( 2.1) = ( 2.1) + 1 = 9.261 + 1 = 10.261 c. G ( 2.1) − G ( 2 ) = 10.261 − 9 = 1.261 d. e. 3 G ( 2.1) − G ( 2 ) 2.1 − 2 = 1.261 = 12.61 0.1 4 32π 3 cm3 V0 = π ( 2 ) = 3 3 4 62.5π 125π 3 = cm3 V1 = π ( 2.5 ) = 3 3 6 125π 32π ΔV = V1 − V0 = cm3 − cm3 6 3 61 = π cm3 ≈ 31.940 cm3 6 11. a. G ( a + h) = ( a + h) + 1 3 = a 3 + 3a 2 h + 3ah 2 + h3 + 1 f. G ( a + h ) − G ( a ) = ⎡( a + h ) + 1⎤ − ⎡⎣ a + 1⎤⎦ ⎣ ⎦ 3 3 ) ( ) ( b. d = 6002 + 4002 = 721 miles c. d = 6752 + 5002 = 840 miles = a3 + 3a 2 h + 3ah 2 + h3 + 1 − a 3 + 1 2 2 = 3a h + 3ah + h g. G ( a + h) − G ( a) (a + h) − a 3 = North plane has traveled 600miles. East plane has traveled 400 miles. 3a 2 h + 3ah 2 + h3 h = 3a 2 + 3ah + h 2 h. lim h→0 G ( a + h) − G ( a) (a + h) − a = lim 3a 2 + 3ah + h 2 h →0 = 3a 2 5. a. ( a + b )3 = a3 + 3a 2b + b. ( a + b ) 4 = a 4 + 4 a 3b + c. ( a + b )5 = a 5 + 5 a 4 b + 6. ( a + b )n = a n + na n −1b + 7. sin ( x + h ) = sin x cos h + cos x sin h 8. cos ( x + h ) = cos x cos h − sin x sin h 9. a. The point will be at position (10, 0 ) in all three cases ( t = 1, 2,3 ) because it will have made 4, 8, and 12 revolutions respectively. b. Since the point is rotating at a rate of 4 revolutions per second, it will complete 1 1 revolution after second. Therefore, the 4 point will first return to its starting position 1 at time t = . 4 Instructor’s Resource Manual Review and Preview 93 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2 CHAPTER 2.1 Concepts Review The Derivative 4. 1. tangent line 2. secant line 3. f (c + h ) − f ( c ) h 4. average velocity Problem Set 2.1 1. Slope = 2. Slope = 5–3 2 – 32 =4 6–4 = –2 4–6 Slope ≈ 1.5 5. 3. Slope ≈ Slope ≈ −2 5 2 6. Slope ≈ – 94 Section 2.1 3 2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 7. y = x 2 + 1 [(2.01)3 − 1.0] − 7 2.01 − 2 0.120601 = 0.01 = 12.0601 d. msec = e. mtan = lim a., b. f (2 + h) – f (2) h h →0 [(2 + h)3 – 1] – (23 − 1) h h →0 = lim 12h + 6h 2 + h3 h h →0 = lim c. m tan = 2 d. msec = e. (1.01)2 + 1.0 − 2 1.01 − 1 0.0201 = .01 = 2.01 f (1 + h) – f (1) h h →0 mtan = lim [(1 + h)2 + 1] – (12 + 1) h h →0 = lim 2 + 2h + h 2 − 2 h h →0 h(2 + h) = lim h h →0 = lim (2 + h) = 2 = lim h →0 3 8. y = x – 1 a., b. h(12 + 6h + h 2 ) h h →0 = 12 = lim 9. f (x) = x 2 – 1 f (c + h ) – f (c ) mtan = lim h h →0 [(c + h)2 – 1] – (c 2 – 1) h h →0 = lim c 2 + 2ch + h 2 – 1 – c 2 + 1 h h →0 h(2c + h) = lim = 2c h h →0 At x = –2, m tan = –4 x = –1, m tan = –2 x = 1, m tan = 2 x = 2, m tan = 4 = lim 10. f (x) = x 3 – 3x f (c + h ) – f (c ) mtan = lim h h →0 [(c + h)3 – 3(c + h)] – (c3 – 3c) h h →0 = lim c3 + 3c 2 h + 3ch 2 + h3 – 3c – 3h – c3 + 3c h h →0 = lim h(3c 2 + 3ch + h 2 − 3) = 3c 2 – 3 h h →0 At x = –2, m tan = 9 x = –1, m tan = 0 x = 0, m tan = –3 x = 1, m tan = 0 x = 2, m tan = 9 = lim c. m tan = 12 Instructor’s Resource Manual Section 2.1 95 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 11. 13. a. 16(12 ) –16(02 ) = 16 ft b. 16(22 ) – 16(12 ) = 48 ft c. Vave = d. f ( x) = mtan 1 x +1 f (1 + h) – f (1) = lim h h →0 1 e. 14. a. b. 1 4 1 1 y – = – ( x –1) 2 4 =– 1 x –1 f (0 + h) − f (0) = lim h h →0 1 +1 = lim h −1 h h →0 12. f (x) = mtan = lim 16(3.01) 2 − 16(3)2 3.01 − 3 0.9616 = 0.01 = 96.16 ft/s Vave = f (t ) = 16t 2 ; v = 32c v = 32(3) = 96 ft/s 1 − = lim 2+ h 2 h h →0 − 2(2h+ h) = lim h h →0 1 = lim − h→0 2(2 + h) h h −1 h 1 = lim h →0 h − 1 = −1 y + 1 = –1(x – 0); y = –x – 1 h →0 Vave = d. (32 + 1) – (22 + 1) = 5 m/sec 3– 2 [(2.003)2 + 1] − (22 + 1) 2.003 − 2 0.012009 = 0.003 = 4.003 m/sec Vave = Vave = c. 144 – 64 = 80 ft/sec 3–2 [(2 + h) 2 + 1] – (22 + 1) 2+h–2 4h + h 2 h = 4 +h = f (t ) = t2 + 1 f (2 + h) – f (2) v = lim h h →0 [(2 + h)2 + 1] – (22 + 1) h h →0 = lim 4h + h 2 h h →0 = lim (4 + h) = lim h →0 =4 96 Section 2.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 15. a. f ( + h) – f ( ) h v = lim h →0 2( + h) + 1 – 2 + 1 h h →0 = lim 2 + 2h + 1 – 2 + 1 h = lim h →0 = lim ( 2 + 2h + 1 – 2 + 1)( 2 + 2h + 1 + 2 + 1) h( 2 + 2h + 1 + 2 + 1) h →0 2h = lim 2 + 2h + 1 + 2 + 1) h →0 h( 2 = 2 +1 + 2 +1 1 b. 2 +1 = = 1 2 +1 ft/s 1 2 2 +1 = 2 3 2 The object reaches a velocity of 1 ft/s when t = 3 . 2 2 2 + 1= 4; = 16. f (t ) = – t2 + 4 t 18. a. [–(c + h)2 + 4(c + h)] – (– c 2 + 4c) h h →0 v = lim – c 2 – 2ch – h 2 + 4c + 4h + c 2 – 4c h h →0 h(–2c – h + 4) = lim = –2c + 4 h h →0 –2c + 4 = 0 when c = 2 The particle comes to a momentary stop at t = 2. b. 1000(2.5)2 – 1000(2)2 2250 = = 4500 2.5 – 2 0.5 c. f (t ) = 1000t 2 = lim 17. a. b. c. 1000(2 + h)2 − 1000(2) 2 h h→0 r = lim 4000 + 4000h + 1000h 2 – 4000 h h→0 h(4000 + 1000h) = lim = 4000 h h→0 = lim ⎡1 ⎤ ⎡1 2 ⎤ 2 ⎢ 2 (2.01) + 1⎥ – ⎢ 2 (2) + 1⎥ = 0.02005 g ⎣ ⎦ ⎣ ⎦ rave 0.02005 = = 2.005 g/hr 2.01 – 2 1 2 t +1 2 ⎡ 1 (2 + h)2 + 1⎤ – ⎡ 1 22 + 1⎤ 2 ⎦ ⎣2 ⎦ r = lim ⎣ h h →0 f (t ) = = lim h→0 = lim 2 + 2h + 12 h 2 + 1 − 2 − 1 ( h 2 + 12 h h h→0 At t = 2, r = 2 h ) =2 Instructor’s Resource Manual 1000(3)2 – 1000(2)2 = 5000 19. a. b. dave = 53 – 33 98 = = 49 g/cm 5–3 2 f (x) = x 3 (3 + h)3 – 33 h h →0 d = lim 27 + 27h + 9h 2 + h3 – 27 h h→0 = lim h(27 + 9h + h 2 ) = 27 g/cm h h→0 = lim Section 2.1 97 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 20. MR = lim h→0 R (c + h ) – R (c ) h [0.4(c + h) – 0.001(c + h)2 ] – (0.4c – 0.001c 2 ) h h→0 = lim 0.4c + 0.4h – 0.001c 2 – 0.002ch – 0.001h 2 – 0.4c + 0.001c 2 h h→0 h(0.4 – 0.002c – 0.001h) = lim = 0.4 – 0.002c h h→0 When n = 10, MR = 0.38; when n = 100, MR = 0.2 = lim 2(1 + h)2 – 2(1) 2 h h →0 21. a = lim 2 + 4h + 2h 2 – 2 h h→0 h(4 + 2h) = lim =4 h h→0 = lim 22. r = lim h→0 p (c + h ) – p (c ) h [120(c + h)2 – 2(c + h)3 ] – (120c 2 – 2c3 ) h h →0 = lim h(240c – 6c 2 + 120h – 6ch – 2h 2 ) h h →0 = lim = 240c – 6c 2 When t = 10, r = 240(10) – 6(10) 2 = 1800 t = 20, r = 240(20) – 6(20)2 = 2400 t = 40, r = 240(40) – 6(40)2 = 0 100 – 800 175 =– ≈ –29.167 24 – 0 6 29,167 gal/hr 700 – 400 ≈ −75 At 8 o’clock, r ≈ 6 − 10 75,000 gal/hr 23. rave = 24. a. The elevator reached the seventh floor at time t = 80 . The average velocity is v avg = (84 − 0) / 80 = 1.05 feet per second b. The slope of the line is approximately 60 − 12 = 1.2 . The velocity is 55 − 15 approximately 1.2 feet per second. 98 Section 2.1 c. The building averages 84/7=12 feet from floor to floor. Since the velocity is zero for two intervals between time 0 and time 85, the elevator stopped twice. The heights are approximately 12 and 60. Thus, the elevator stopped at floors 1 and 5. 25. a. A tangent line at t = 91 has slope approximately (63 − 48) /(91 − 61) = 0.5 . The normal high temperature increases at the rate of 0.5 degree F per day. b. A tangent line at t = 191 has approximate slope (90 − 88) / 30 ≈ 0.067 . The normal high temperature increases at the rate of 0.067 degree per day. c. There is a time in January, about January 15, when the rate of change is zero. There is also a time in July, about July 15, when the rate of change is zero. d. The greatest rate of increase occurs around day 61, that is, some time in March. The greatest rate of decrease occurs between day 301 and 331, that is, sometime in November. 26. The slope of the tangent line at t = 1930 is approximately (8 − 6) /(1945 − 1930) ≈ 0.13 . The rate of growth in 1930 is approximately 0.13 million, or 130,000, persons per year. In 1990, the tangent line has approximate slope (24 − 16) /(20000 − 1980) ≈ 0.4 . Thus, the rate of growth in 1990 is 0.4 million, or 400,000, persons per year. The approximate percentage growth in 1930 is 0.107 / 6 ≈ 0.018 and in 1990 it is approximately 0.4 / 20 ≈ 0.02 . 27. In both (a) and (b), the tangent line is always positive. In (a) the tangent line becomes steeper and steeper as t increases; thus, the velocity is increasing. In (b) the tangent line becomes flatter and flatter as t increases; thus, the velocity is decreasing. Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 28. 2.2 Concepts Review 1 f (t ) = t 3 + t 3 f ( c + h ) – f (c ) h h →0 3 1 ⎡ ( c + h ) + (c + h ) ⎤ – 1 c 3 + c 3 ⎦ 3 = lim ⎣ h h→0 current = lim ( = lim ( ) ) = c2 + 1 h c 2 + ch + 13 h 2 + 1 h h →0 When t = 3, the current =10 c 2 + 1 = 20 2 c = 19 c = 19 ≈ 4.4 A 20-amp fuse will blow at t = 4.4 s. 29. A = πr 2 , r = 2t A = 4πt2 4π(3 + h)2 – 4π(3)2 rate = lim h h →0 h(24π + 4πh) = lim = 24π km2/day h h →0 1. f (c + h) – f (c) f (t ) – f (c) ; h t –c 2. f (c ) 3. continuous; f ( x) = x 4. = dy dx Problem Set 2.2 1. f (1) = lim h →0 f (1 + h) – f (1) h (1 + h)2 – 12 2h + h 2 = lim = lim h h h →0 h →0 = lim (2 + h ) = 2 h→0 2. f (2) = lim h →0 f (2 + h) – f (2) h [2(2 + h)]2 – [2(2)]2 h h →0 4 1 30. V = π r 3 , r = t 3 4 1 V = π t3 48 rate = f '( x); = lim 16h + 4h 2 = lim (16 + 4h) = 16 h h →0 h →0 = lim 1 (3 + h)3 − 33 27 π lim = π h 48 h→0 48 9 π inch 3 / sec 16 3. m tan = 7 b. m tan = 0 c. m tan = –1 d. m tan = 17. 92 h →0 5h + h 2 = lim (5 + h) = 5 h h →0 h →0 = lim 4. f (4) = lim h →0 2 32. y = f ( x) = sin x sin 2 x a. m tan = –1.125 b. m tan ≈ –1.0315 c. m tan = 0 d. m tan ≈ 1.1891 33. s = f (t ) = t + t cos 2 t At t = 3, v ≈ 2.818 (t + 1)3 t+2 At t = 1.6, v ≈ 4.277 34. s = f (t ) = Instructor’s Resource Manual f (3 + h) – f (3) h [(3 + h)2 – (3 + h)] – (32 – 3) = lim h h →0 31. y = f ( x) = x 3 – 2 x 2 + 1 a. f (3) = lim = lim h →0 1 3+ h f (4 + h) – f (4) h 1 – 4–1 h = lim h →0 3–(3+ h ) 3(3+ h ) h –1 h →0 3(3 + h) = lim 1 =– 9 s ( x + h) – s ( x ) h h →0 [2( x + h) + 1] – (2 x + 1) = lim h h →0 2h = lim =2 h →0 h 5. s ( x) = lim Section 2.2 99 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6. f ( x + h) – f ( x ) h [ ( x + h) + ] – ( x + ) = lim h h →0 h = lim = h →0 h f ( x) = lim 12. g ( x) = lim h →0 h →0 [( x + h)4 + ( x + h) 2 ] – ( x 4 + x 2 ) h h →0 = lim 4hx3 + 6h 2 x 2 + 4h3 x + h 4 + 2hx + h 2 h h →0 = lim = lim (4 x3 + 6hx 2 + 4h 2 x + h3 + 2 x + h) r ( x + h) – r ( x ) 7. r ( x) = lim h h →0 h →0 3 = 4x + 2x [3( x + h)2 + 4] – (3 x 2 + 4) = lim h h →0 6 xh + 3h 2 = lim (6 x + 3h) = 6 x h h →0 h →0 = lim 8. f ( x) = lim h →0 f ( x + h) – f ( x ) h [( x + h)2 + ( x + h) + 1] – ( x 2 + x + 1) h h →0 = lim = lim h →0 9. 2 2 xh + h + h = lim (2 x + h + 1) = 2 x + 1 h h →0 f ( x) = lim h →0 f ( x + h) – f ( x ) h [a( x + h) 2 + b( x + h) + c] – (ax 2 + bx + c) h h →0 = lim 2axh + ah 2 + bh = lim (2ax + ah + b) h h →0 h →0 = 2ax + b h( x + h) – h( x ) h h →0 ⎡⎛ 2 2 ⎞ 1⎤ = lim ⎢⎜ – ⎟⋅ ⎥ h → 0 ⎣⎝ x + h x ⎠ h ⎦ 13. h ( x) = lim ⎡ –2h 1 ⎤ –2 2 = lim ⎢ ⋅ ⎥ = lim =– h → 0 ⎣ x ( x + h ) h ⎦ h →0 x ( x + h ) x2 S ( x + h) – S ( x ) h h →0 ⎡⎛ 1 1 ⎞ 1⎤ = lim ⎢⎜ – ⎟⋅ ⎥ h →0 ⎣⎝ x + h + 1 x + 1 ⎠ h ⎦ 14. S ( x) = lim ⎡ –h 1⎤ = lim ⎢ ⋅ ⎥ h→0 ⎣ ( x + 1)( x + h + 1) h ⎦ –1 1 = lim =− h→0 ( x + 1)( x + h + 1) ( x + 1) 2 = lim 10. f ( x) = lim h →0 f ( x + h) – f ( x ) h ( x + h) 4 – x 4 h h →0 = lim 4hx3 + 6h 2 x 2 + 4h3 x + h 4 h h →0 = lim = lim (4 x3 + 6hx 2 + 4h 2 x + h3 ) = 4 x3 h →0 11. F ( x + h) – F ( x ) h h →0 15. F ( x) = lim ⎡⎛ 6 6 ⎞ 1⎤ – = lim ⎢⎜ ⎟⋅ ⎥ h→0 ⎢⎜⎝ ( x + h) 2 + 1 x 2 + 1 ⎟⎠ h ⎥ ⎣ ⎦ 2 2 ⎡ 6( x + 1) – 6( x + 2hx + h 2 + 1) 1 ⎤ = lim ⎢ ⋅ ⎥ h ⎥⎦ h→0 ⎢⎣ ( x 2 + 1)( x 2 + 2hx + h 2 + 1) ⎡ –12hx – 6h 2 1⎤ = lim ⎢ ⋅ ⎥ h→0 ⎢ ( x 2 + 1)( x 2 + 2hx + h 2 + 1) h ⎥ ⎣ ⎦ = lim h →0 ( x 2 f ( x + h) – f ( x ) f ( x) = lim h h →0 [( x + h)3 + 2( x + h)2 + 1] – ( x3 + 2 x 2 + 1) = lim h h →0 3hx 2 + 3h 2 x + h3 + 4hx + 2h 2 h h →0 = lim 2 2 2 = lim (3x + 3hx + h + 4 x + 2h) = 3 x + 4 x h →0 g ( x + h) – g ( x ) h –12 x – 6h 2 2 + 1)( x + 2hx + h + 1) =− 12 x 2 ( x + 1)2 F ( x + h) – F ( x ) h h →0 ⎡⎛ x + h –1 x –1 ⎞ 1 ⎤ = lim ⎢⎜ – ⎟⋅ ⎥ h →0 ⎣ ⎝ x + h + 1 x + 1 ⎠ h ⎦ 16. F ( x) = lim ⎡ x 2 + hx + h –1 – ( x 2 + hx – h –1) 1 ⎤ = lim ⎢ ⋅ ⎥ h ⎦⎥ ( x + h + 1)( x + 1) h →0 ⎢ ⎣ ⎡ 2h 1⎤ 2 = lim ⎢ ⋅ ⎥= h→0 ⎣ ( x + h + 1)( x + 1) h ⎦ ( x + 1) 2 100 Section 2.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. G ( x + h) – G ( x ) h ⎡⎛ 2( x + h) –1 2 x –1 ⎞ 1 ⎤ = lim ⎢⎜ – ⎟⋅ x – 4 ⎠ h ⎥⎦ h→0 ⎣⎝ x + h – 4 17. G ( x ) = lim h →0 ⎡ 2 x 2 + 2hx − 9 x − 8h + 4 − (2 x 2 + 2hx − 9 x − h + 4) 1 ⎤ ⎡ –7h 1⎤ ⋅ ⎥ = lim ⎢ ⋅ ⎥ h ( x + h − 4)( x − 4) ⎥⎦ h→0 ⎣ ( x + h – 4)( x – 4) h ⎦ –7 7 = lim =– h→0 ( x + h – 4)( x – 4) ( x – 4)2 = lim ⎢ h →0 ⎢⎣ G ( x + h) – G ( x ) h h →0 18. G ( x ) = lim ⎡⎛ ⎡ (2 x + 2h)( x 2 – x ) – 2 x( x 2 + 2 xh + h 2 – x – h) 1 ⎤ 2( x + h) 2x ⎞ 1 ⎤ = lim ⎢⎜ ⋅ ⎥ = lim ⎢ – ⋅ ⎥ ⎟ h ⎦⎥ h→0 ⎢⎝⎜ ( x + h) 2 – ( x + h) x 2 – x ⎠⎟ h ⎥ ( x 2 + 2hx + h 2 – x – h)( x 2 – x) ⎣ ⎦ h→0 ⎣⎢ ⎡ –2h 2 x – 2hx 2 1⎤ = lim ⎢ ⋅ ⎥ 2 2 2 h→0 ⎣⎢ ( x + 2hx + h – x – h)( x – x ) h ⎦⎥ –2hx – 2 x 2 = lim h→0 ( x 2 = + 2hx + h 2 – x – h)( x 2 – x) –2 x 2 2 ( x – x) 2 19. g ( x) = lim h →0 =– 2 ( x – 1) 2 g ( x + h) – g ( x ) h 3( x + h) – 3x h h→0 = lim = lim ( 3x + 3h – 3x )( 3 x + 3h + 3 x ) h( 3 x + 3h + 3x ) h→0 3h = lim h→0 h( 3 x + 3h + 3 x ) 20. g ( x) = lim h →0 = lim h →0 3 3 x + 3h + 3 x = 3 2 3x g ( x + h) – g ( x ) h ⎡⎛ 1 1 ⎞ 1⎤ = lim ⎢⎜ – ⎟⋅ ⎥ h→0 ⎢⎜⎝ 3( x + h) 3 x ⎟⎠ h ⎦⎥ ⎣ ⎡ 3x – 3x + 3h 1 ⎤ = lim ⎢ ⋅ ⎥ h ⎦⎥ h→0 ⎣⎢ 9 x ( x + h) ⎡ ( 3 x – 3 x + 3h )( 3 x + 3 x + 3h ) 1 ⎤ = lim ⎢ ⋅ ⎥ h ⎦⎥ h→0 ⎣⎢ 9 x( x + h)( 3 x + 3 x + 3h ) = lim h→0 h –3h 9 x( x + h)( 3x + 3x + 3h ) Instructor’s Resource Manual = –3 3x ⋅ 2 3x =– 1 2 x 3x Section 2.2 101 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 21. H ( x ) = lim h →0 H ( x + h) – H ( x ) h ⎡⎛ 3 – = lim ⎢⎜ h→0 ⎣⎝ x + h – 2 ⎞ 1⎤ ⎟⋅ ⎥ x – 2 ⎠ h⎦ 3 ⎡3 x – 2 – 3 x + h – 2 1 ⎤ = lim ⎢ ⋅ ⎥ h→0 ⎣⎢ ( x + h – 2)( x – 2) h ⎦⎥ = lim h→0 3( x – 2 – x + h – 2)( x – 2 + x + h – 2) h ( x + h – 2)( x – 2)( x – 2 + x + h – 2) −3h = lim h→0 h[( x – 2) –3 = lim h→0 ( x – 2) =– x + h – 2 + ( x + h – 2) x – 2] x + h – 2 + ( x + h – 2) x – 2 3 2( x – 2) x – 2 22. H ( x) = lim h →0 =− 3 2( x − 2)3 2 H ( x + h) – H ( x ) h ( x + h) 2 + 4 – x 2 + 4 h h→0 = lim ⎛ x 2 + 2hx + h 2 + 4 – x 2 + 4 ⎞ ⎛ x 2 + 2hx + h 2 + 4 + x 2 + 4 ⎞ ⎜ ⎟⎜ ⎟ ⎠⎝ ⎠ = lim ⎝ h →0 2 2 2 ⎛ ⎞ h ⎜ x + 2hx + h + 4 + x + 4 ⎟ ⎝ ⎠ = lim h →0 = lim 2hx + h 2 h ⎛⎜ x 2 + 2hx + h 2 + 4 + x 2 + 4 ⎞⎟ ⎝ ⎠ 2x + h h →0 x 2 + 2hx + h 2 + 4 + x 2 + 4 2x x = = 2 2 2 x +4 x +4 23. f ( x) = lim t→x 2 f (t ) – f ( x) t–x 2 (t − 3t ) – ( x – 3 x) t–x t→x = lim t 2 – x 2 – (3t – 3x) t–x t→x (t – x)(t + x) – 3(t – x) = lim t–x t→x (t – x)(t + x – 3) = lim = lim (t + x – 3) t–x t→x t→x =2x–3 = lim 24. f ( x) = lim t→x 3 f (t ) – f ( x) t–x (t + 5t ) – ( x3 + 5 x) t–x t→x = lim t 3 – x3 + 5t – 5 x t–x t→x = lim (t – x)(t 2 + tx + x 2 ) + 5(t – x) t–x t→x = lim (t – x)(t 2 + tx + x 2 + 5) t–x t→x = lim = lim (t 2 + tx + x 2 + 5) = 3x 2 + 5 t→x 102 Section 2.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 25. f (t ) – f ( x) t–x t→x ⎡⎛ t x ⎞⎛ 1 ⎞ ⎤ = lim ⎢⎜ – ⎟⎜ ⎟⎥ t → x ⎣⎝ t – 5 x – 5 ⎠⎝ t – x ⎠ ⎦ f ( x) = lim 38. The slope of the tangent line is always −1 . tx – 5t – tx + 5 x t → x (t – 5)( x – 5)(t – x ) = lim = lim t → x (t =− 26. –5(t – x) –5 = lim – 5)( x – 5)(t – x) t → x (t – 5)( x – 5) 5 ( x − 5) 2 39. The derivative is positive until x = 0 , then becomes negative. f (t ) – f ( x) t–x t→x ⎡⎛ t + 3 x + 3 ⎞ ⎛ 1 ⎞ ⎤ = lim ⎢⎜ – ⎟⎜ ⎟ x ⎠ ⎝ t – x ⎠ ⎥⎦ t → x ⎣⎝ t f ( x) = lim 3x – 3t –3 3 = lim =– t → x xt (t – x ) t → x xt x2 = lim 27. f (x) = 2 x 3 at x = 5 40. The derivative is negative until x = 1 , then becomes positive. 28. f (x) = x 2 + 2 x at x = 3 29. f (x) = x 2 at x = 2 30. f (x) = x 3 + x at x = 3 31. f (x) = x 2 at x 32. f (x) = x 3 at x 33. f (t ) = 2 at t t 41. The derivative is −1 until x = 1 . To the right of x = 1 , the derivative is 1. The derivative is undefined at x = 1 . 34. f(y) = sin y at y 35. f(x) = cos x at x 36. f(t) = tan t at t 37. The slope of the tangent line is always 2. 42. The derivative is −2 to the left of x = −1 ; from −1 to 1, the derivative is 2, etc. The derivative is not defined at x = −1, 1, 3 . Instructor’s Resource Manual Section 2.2 103 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 43. The derivative is 0 on ( −3, −2 ) , 2 on ( −2, −1) , 0 on ( −1, 0 ) , −2 on ( 0,1) , 0 on (1, 2 ) , 2 on ( 2,3) 53. and 0 on ( 3, 4 ) . The derivative is undefined at x = −2, − 1, 0, 1, 2, 3 . 1 1 Δy x +Δx +1 – x +1 = Δx Δx ⎛ x + 1 – ( x + Δx + 1) ⎞ ⎛ 1 ⎞ =⎜ ⎟⎜ ⎟ ⎝ ( x + Δx + 1)( x + 1) ⎠ ⎝ Δx ⎠ = – Δx ( x + Δx + 1)( x + 1)Δx =– 1 ( x + Δx + 1)( x + 1) ⎡ ⎤ dy 1 1 = lim − =− dx Δx →0 ⎢⎣ ( x + Δx + 1)( x + 1) ⎥⎦ ( x + 1) 2 44. The derivative is 1 except at x = −2, 0, 2 where it is undefined. 1 ⎛ 1⎞ − ⎜1 + ⎟ x + Δx ⎝ x ⎠ Δx −Δx 1 1 − x ( x + Δx ) 1 = x + Δx x = =− Δx Δx x ( x + Δx ) Δy 54. = Δx 1+ dy 1 1 = lim − =− 2 dx Δx →0 x ( x + Δx ) x 55. 45. Δy = [3(1.5) + 2] – [3(1) + 2] = 1.5 46. Δy = [3(0.1) 2 + 2(0.1) + 1] – [3(0.0) 2 + 2(0.0) + 1] = 0.23 x 2 + xΔx − x + x + Δx − 1 − ⎡ x 2 + xΔx − x + x − Δx − 1⎤ 1 ⎣ ⎦× 2 Δx x + x Δx + x + x + Δ x + 1 2Δx 1 2 = 2 × = 2 Δ x x + xΔx + x + x + Δx + 1 x + x Δx + x + x + Δ x + 1 2 2 2 dy = lim = = dx Δx →0 x 2 + xΔx + x + x + Δx + 1 x 2 + 2 x + 1 ( x + 1)2 47. Δy = 1/1.2 – 1/1 = – 0.1667 = 48. Δy = 2/(0.1+1) – 2/(0+1) = – 0.1818 49. Δy = 3 3 – ≈ 0.0081 2.31 + 1 2.34 + 1 50. Δy = cos[2(0.573)] – cos[2(0.571)] ≈ –0.0036 51. 52. Δy ( x + Δx) 2 – x 2 2 xΔx + (Δx) 2 = = = 2 x + Δx Δx Δx Δx dy = lim (2 x + Δx) = 2 x dx Δx →0 56. Δy = Δx ( x + Δx ) 2 − 1 − x 2 − 1 x + Δx Δx x ( ) 3 x 2 Δx + 3x(Δx)2 – 6 xΔx – 3(Δx) 2 + Δx3 = Δx ⎡ x ( x + Δx )2 − x − ( x + Δx ) x 2 − 1 ⎤ ⎥× 1 =⎢ ⎢ ⎥ Δx x ( x + Δx ) ⎣ ⎦ 2 2 3 ⎡ x x + 2 xΔx + Δx − x − x + x 2 Δx − x − Δx =⎢ ⎢ x 2 + x Δx ⎣⎢ = 3x 2 + 3xΔx – 6 x – 3Δx + (Δx)2 = Δy [( x + Δx)3 – 3( x + Δx) 2 ] – ( x3 – 3 x 2 ) = Δx Δx dy = lim (3 x 2 + 3 xΔx – 6 x – 3Δx + (Δx)2 ) dx Δx→0 = 3x2 – 6 x 104 x + Δx − 1 x − 1 − Δy x + Δx + 1 x + 1 = Δx Δx ( x + 1)( x + Δx − 1) − ( x − 1)( x + Δx + 1) 1 = × Δx ( x + Δx + 1)( x + 1) Section 2.2 ( ( )) x 2 Δx + x ( Δx ) + Δx 2 2 x + x Δx × ( ) ⎤⎥ × 1 ⎥ Δx ⎦⎥ 1 x 2 + x Δx + 1 = 2 Δx x + x Δx dy x 2 + xΔx + 1 x 2 + 1 = lim = 2 Δ x → 0 dx x 2 + x Δx x Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 57. 1 f (0) ≈ – ; f (2) ≈ 1 2 2 f (5) ≈ ; f (7) ≈ –3 3 58. g (–1) ≈ 2; g (1) ≈ 0 g (4) ≈ –2; g (6) ≈ – 1 3 63. The derivative is 0 at approximately t = 15 and t = 201 . The greatest rate of increase occurs at about t = 61 and it is about 0.5 degree F per day. The greatest rate of decrease occurs at about t = 320 and it is about 0.5 degree F per day. The derivative is positive on (15,201) and negative on (0,15) and (201,365). 59. 64. The slope of a tangent line for the dashed function is zero when x is approximately 0.3 or 1.9. The solid function is zero at both of these points. The graph indicates that the solid function is negative when the dashed function has a tangent line with a negative slope and positive when the dashed function has a tangent line with a positive slope. Thus, the solid function is the derivative of the dashed function. 60. 61. a. 5 3 f (2) ≈ ; f (2) ≈ 2 2 f (0.5) ≈ 1.8; f (0.5) ≈ –0.6 b. 2.9 − 1.9 = 0.5 2.5 − 0.5 c. x=5 d. x = 3, 5 e. x = 1, 3, 5 f. x=0 g. x ≈ −0.7, 3 and 5 < x < 7 2 62. The derivative fails to exist at the corners of the graph; that is, at t = 10, 15, 55, 60, 80 . The derivative exists at all other points on the interval (0,85) . Instructor’s Resource Manual 65. The short-dash function has a tangent line with zero slope at about x = 2.1 , where the solid function is zero. The solid function has a tangent line with zero slope at about x = 0.4, 1.2 and 3.5. The long-dash function is zero at these points. The graph shows that the solid function is positive (negative) when the slope of the tangent line of the short-dash function is positive (negative). Also, the long-dash function is positive (negative) when the slope of the tangent line of the solid function is positive (negative). Thus, the short-dash function is f, the solid function is f ' = g , and the dash function is g ' . 66. Note that since x = 0 + x, f(x) = f(0 + x) = f(0)f(x), hence f(0) = 1. f ( a + h) – f ( a ) f (a ) = lim h h →0 f ( a ) f ( h) – f ( a ) = lim h h →0 f ( h) – 1 f (h) – f (0) = f (a ) lim = f (a) lim h h h →0 h →0 = f (a ) f (0) f ( a) exists since f (0) exists. Section 2.2 105 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 67. If f is differentiable everywhere, then it is continuous everywhere, so lim − f ( x ) = lim− ( mx + b ) = 2 m + b = f (2) = 4 x→2 b. If f is an even function, f (t ) − f ( x0 ) f (– x0 ) = lim . Let u = –t, as t + x0 t →− x0 x→ 2 and b = 4 – 2m. For f to be differentiable everywhere, f ( x) − f (2) f (2) = lim must exist. x−2 x→2 f ( x) − f (2) x2 − 4 = lim = lim ( x + 2) = 4 x−2 x → 2+ x → 2+ x − 2 x → 2+ f ( x) − f (2) mx + b − 4 lim = lim − − x x−2 − 2 x→2 x→2 f (−u ) − f ( x0 ) −u + x0 u → x0 above, then f (− x0 ) = lim f (u ) − f ( x0 ) f (u ) − f ( x0 ) = − lim u − x0 u → x0 −(u − x0 ) u → x0 = − f (x 0 ) = −m. = lim lim mx + 4 − 2m − 4 m( x − 2) = lim =m − x−2 x−2 x →2 Thus m = 4 and b = 4 – 2(4) = –4 = lim x → 2− 68. 69. f ( x + h) – f ( x ) + f ( x ) – f ( x – h ) 2h h →0 ⎡ f ( x + h ) – f ( x ) f ( x – h) – f ( x ) ⎤ = lim ⎢ + ⎥ 2h –2h h →0 ⎣ ⎦ 1 f ( x + h) – f ( x ) 1 f [ x + (– h)] – f ( x) = lim + lim h 2 h →0 2 – h →0 –h 1 1 = f ( x) + f ( x) = f ( x). 2 2 For the converse, let f (x) = x . Then h – –h h–h f s (0) = lim = lim =0 2h h→0 h →0 2h but f (0) does not exist. f s ( x) = lim 70. Say f(–x) = –f(x). Then f (– x + h) – f (– x) f (– x) = lim h h →0 – f ( x – h) + f ( x ) f ( x – h) – f ( x ) = lim = – lim h h h →0 h →0 f [ x + (– h)] − f ( x) = lim = f ( x) so f ( x ) is –h – h →0 an even function if f(x) is an odd function. Say f(–x) = f(x). Then f (– x + h) – f (– x) f (– x) = lim h h →0 f ( x – h) – f ( x ) = lim h h →0 f [ x + (– h)] – f ( x) = – lim = – f ( x) so f (x) –h – h →0 is an odd function if f(x) is an even function. 71. f (t ) − f ( x0 ) , so t − x0 0 f (t ) − f (− x0 ) f (− x0 ) = lim t − (− x0 ) t →− x 0 f ( x0 ) = lim t→x f (t ) − f (− x0 ) = lim t + x0 t →− x 0 a. If f is an odd function, f (t ) − [− f (− x0 )] f (− x0 ) = lim t + x0 t →− x0 f (t ) + f (− x0 ) . t + x0 Let u = –t. As t → − x0 , u → x 0 and so f (−u ) + f ( x0 ) f (− x0 ) = lim −u + x0 u → x0 = lim t →− x0 a. 0< x< 8 ⎛ 8⎞ ; ⎜ 0, ⎟ 3 ⎝ 3⎠ b. 0≤ x≤ 8 ⎡ 8⎤ ; 0, 3 ⎢⎣ 3 ⎥⎦ c. A function f(x) decreases as x increases when f ( x ) < 0. a. π < x < 6.8 c. A function f(x) increases as x increases when f ( x ) > 0. 72. − f (u ) + f ( x0 ) −[ f (u ) − f ( x0 )] = lim u x − ( − ) −(u − x0 ) u → x0 u → x0 0 = lim f (u ) − f ( x0 ) = f ( x0 ) = m. u − x0 u → x0 = lim 106 Section 2.2 b. π < x < 6.8 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2.3 Concepts Review 13. Dx ( x 4 + x3 + x 2 + x + 1) 1. the derivative of the second; second; f (x) g (x ) + g(x) f ( x) 2. denominator; denominator; square of the g ( x) f ( x) – f ( x) g ( x) denominator; g 2 ( x) 3. nx n– 1 h ; nx = 3Dx ( x 4 ) – 2 Dx ( x3 ) – 5Dx ( x 2 ) = 12 x3 – 6 x 2 –10 x + π 15. Dx (πx 7 – 2 x5 – 5 x –2 ) = πDx ( x7 ) – 2 Dx ( x5 ) – 5 Dx ( x –2 ) 2 1. Dx (2 x ) = 2 Dx ( x ) = 2 ⋅ 2 x = 4 x = π(7 x6 ) – 2(5 x 4 ) – 5(–2 x –3 ) = 7 πx 6 –10 x 4 + 10 x –3 2. Dx (3x3 ) = 3Dx ( x3 ) = 3 ⋅ 3x 2 = 9 x 2 16. Dx ( x12 + 5 x −2 − πx −10 ) 3. Dx (πx ) = πDx ( x) = π ⋅1 = π 3 2 4. Dx (πx ) = πDx ( x ) = π ⋅ 3 x = 3πx 5. 14. Dx (3x 4 – 2 x3 – 5 x 2 + πx + π2 ) = 3(4 x3 ) – 2(3 x 2 ) – 5(2 x) + π(1) + 0 Problem Set 2.3 3 = 4 x3 + 3 x 2 + 2 x + 1 + πDx ( x) + Dx (π2 ) n –1 4. kL(f); L(f) + L(g); Dx 2 = Dx ( x 4 ) + Dx ( x3 ) + Dx ( x 2 ) + Dx ( x) + Dx (1) 2 Dx (2 x –2 ) = 2 Dx ( x –2 ) = 2(–2 x –3 ) = –4 x –3 6. Dx (–3 x –4 ) = –3Dx ( x –4 ) = –3(–4 x –5 ) = 12 x –5 ⎛π⎞ 7. Dx ⎜ ⎟ = πDx ( x –1 ) = π(–1x –2 ) = – πx –2 ⎝x⎠ π =– 2 x ⎛ ⎞ 8. Dx ⎜ ⎟ = Dx ( x –3 ) = (–3x –4 ) = –3 x –4 ⎝ x3 ⎠ 3 =– x4 ⎛ 100 ⎞ 9. Dx ⎜ = 100 Dx ( x –5 ) = 100(–5 x –6 ) 5 ⎟ ⎝ x ⎠ 500 = –500 x –6 = – x6 ⎛ 3 ⎞ 3 3 10. Dx ⎜ Dx ( x –5 ) = = (–5 x –6 ) 5⎟ 4 ⎝ 4x ⎠ 4 15 –6 15 =– x =– 4 4 x6 11. Dx ( x 2 + 2 x) = Dx ( x 2 ) + 2 Dx ( x) = 2 x + 2 = Dx ( x12 ) + 5Dx ( x −2 ) − πDx ( x −10 ) = 12 x11 + 5(−2 x −3 ) − π(−10 x −11 ) = 12 x11 − 10 x −3 + 10πx −11 ⎛ 3 ⎞ 17. Dx ⎜ + x –4 ⎟ = 3Dx ( x –3 ) + Dx ( x –4 ) 3 ⎝x ⎠ 9 = 3(–3 x –4 ) + (–4 x –5 ) = – – 4 x –5 x4 18. Dx (2 x –6 + x –1 ) = 2 Dx ( x –6 ) + Dx ( x –1 ) = 2(–6 x –7 ) + (–1x –2 ) = –12 x –7 – x –2 ⎛2 1 ⎞ 19. Dx ⎜ – = 2 Dx ( x –1 ) – Dx ( x –2 ) 2⎟ x x ⎠ ⎝ 2 2 = 2(–1x –2 ) – (–2 x –3 ) = – + 2 x x3 ⎛ 3 1 ⎞ –3 –4 20. Dx ⎜ – ⎟ = 3 Dx ( x ) – Dx ( x ) 3 x4 ⎠ ⎝x 9 4 = 3(–3 x –4 ) – (–4 x –5 ) = – + 4 x x5 ⎛ 1 ⎞ 1 21. Dx ⎜ + 2 x ⎟ = Dx ( x –1 ) + 2 Dx ( x) ⎝ 2x ⎠ 2 1 1 = (–1x –2 ) + 2(1) = – +2 2 2 x2 12. Dx (3x 4 + x3 ) = 3Dx ( x 4 ) + Dx ( x3 ) = 3(4 x3 ) + 3x 2 = 12 x3 + 3 x 2 Instructor’s Resource Manual Section 2.3 107 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ⎛ 2 2⎞ 2 ⎛2⎞ 22. Dx ⎜ – ⎟ = Dx ( x –1 ) – Dx ⎜ ⎟ ⎝ 3x 3 ⎠ 3 ⎝3⎠ 2 2 = (–1x –2 ) – 0 = – 3 3x 2 23. Dx [ x( x 2 + 1)] = x Dx ( x 2 + 1) + ( x 2 + 1) Dx ( x) 2 2 = x(2 x) + ( x + 1)(1) = 3 x + 1 26. Dx [(–3 x + 2)2 ] = (–3 x + 2) Dx (–3 x + 2) + (–3 x + 2) Dx (–3x + 2) = (–3x + 2)(–3) + (–3x + 2)(–3) = 18x – 12 27. Dx [( x 2 + 2)( x3 + 1)] = ( x 2 + 2) Dx ( x3 + 1) + ( x3 + 1) Dx ( x 2 + 2) = ( x 2 + 2)(3x 2 ) + ( x3 + 1)(2 x) 24. Dx [3 x( x3 –1)] = 3 x Dx ( x3 –1) + ( x3 –1) Dx (3 x) = 3x(3 x 2 ) + ( x3 –1)(3) = 12 x3 – 3 2 25. Dx [(2 x + 1) ] = (2 x + 1) Dx (2 x + 1) + (2 x + 1) Dx (2 x + 1) = (2 x + 1)(2) + (2 x + 1)(2) = 8 x + 4 = 3x 4 + 6 x 2 + 2 x 4 + 2 x = 5x4 + 6 x2 + 2 x 28. Dx [( x 4 –1)( x 2 + 1)] = ( x 4 –1) Dx ( x 2 + 1) + ( x 2 + 1) Dx ( x 4 –1) = ( x 4 –1)(2 x) + ( x 2 + 1)(4 x3 ) = 2 x5 – 2 x + 4 x5 + 4 x3 = 6 x 5 + 4 x3 – 2 x 29. Dx [( x 2 + 17)( x3 – 3 x + 1)] = ( x 2 + 17) Dx ( x3 – 3 x + 1) + ( x3 – 3x + 1) Dx ( x 2 + 17) = ( x 2 + 17)(3 x 2 – 3) + ( x3 – 3x + 1)(2 x) = 3x 4 + 48 x 2 – 51 + 2 x 4 – 6 x 2 + 2 x = 5 x 4 + 42 x 2 + 2 x – 51 30. Dx [( x 4 + 2 x )( x3 + 2 x 2 + 1)] = ( x 4 + 2 x) Dx ( x3 + 2 x 2 + 1) + ( x3 + 2 x 2 + 1) Dx ( x 4 + 2 x) = ( x 4 + 2 x)(3 x 2 + 4 x) + ( x3 + 2 x 2 + 1)(4 x3 + 2) = 7 x6 + 12 x5 + 12 x3 + 12 x 2 + 2 31. Dx [(5 x 2 – 7)(3x 2 – 2 x + 1)] = (5 x 2 – 7) Dx (3x 2 – 2 x + 1) + (3x 2 – 2 x + 1) Dx (5 x 2 – 7) = (5 x 2 – 7)(6 x – 2) + (3 x 2 – 2 x + 1)(10 x) = 60 x3 – 30 x 2 – 32 x + 14 32. Dx [(3 x 2 + 2 x)( x 4 – 3 x + 1)] = (3 x 2 + 2 x ) Dx ( x 4 – 3 x + 1) + ( x 4 – 3 x + 1) Dx (3x 2 + 2 x) = (3 x 2 + 2 x)(4 x3 – 3) + ( x 4 – 3 x + 1)(6 x + 2) = 18 x5 + 10 x 4 – 27 x 2 – 6 x + 2 ⎛ 1 ⎞ (3x 2 + 1) Dx (1) – (1) Dx (3 x 2 + 1) 33. Dx ⎜ ⎟= (3 x 2 + 1)2 ⎝ 3x2 + 1 ⎠ = (3 x 2 + 1)(0) – (6 x) 2 (3 x + 1) 2 =– 6x 2 (3x + 1) 2 ⎛ 2 ⎞ (5 x 2 –1) Dx (2) – (2) Dx (5 x 2 –1) 34. Dx ⎜ ⎟= (5 x 2 – 1) 2 ⎝ 5x2 – 1 ⎠ = 108 (5 x 2 –1)(0) – 2(10 x ) 2 (5 x –1) Section 2.3 2 =– 20 x (5 x 2 –1)2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ⎛ ⎞ (4 x 2 – 3x + 9) Dx (1) – (1) Dx (4 x 2 – 3 x + 9) 1 35. Dx ⎜ ⎟= (4 x 2 – 3x + 9)2 ⎝ 4 x 2 – 3x + 9 ⎠ = = (4 x 2 – 3x + 9)(0) – (8 x – 3) 2 (4 x – 3 x + 9) −8 x + 3 =– 2 8x − 3 2 (4 x – 3x + 9)2 (4 x 2 – 3x + 9)2 ⎛ ⎞ (2 x3 – 3 x) Dx (4) – (4) Dx (2 x3 – 3 x) 4 36. Dx ⎜ ⎟ = (2 x3 – 3 x)2 ⎝ 2 x3 – 3x ⎠ = (2 x3 – 3 x)(0) – 4(6 x 2 – 3) (2 x3 – 3 x)2 = –24 x 2 + 12 (2 x3 – 3x) 2 ⎛ x –1 ⎞ ( x + 1) Dx ( x – 1) – ( x –1) Dx ( x + 1) 37. Dx ⎜ ⎟= ⎝ x +1⎠ ( x + 1)2 = ( x + 1)(1) – ( x – 1)(1) ( x + 1) 2 = 2 ( x + 1)2 ⎛ 2 x –1 ⎞ ( x –1) Dx (2 x –1) – (2 x –1) Dx ( x –1) 38. Dx ⎜ ⎟= ⎝ x –1 ⎠ ( x –1) 2 = ( x –1)(2) – (2 x –1)(1) ( x –1) 2 =– 1 ( x –1) 2 ⎛ 2 x 2 – 1 ⎞ (3 x + 5) Dx (2 x 2 –1) – (2 x 2 –1) Dx (3 x + 5) 39. Dx ⎜ ⎟ = ⎜ 3x + 5 ⎟ (3 x + 5)2 ⎝ ⎠ = = (3 x + 5)(4 x) – (2 x 2 – 1)(3) (3x + 5) 2 6 x 2 + 20 x + 3 (3x + 5)2 ⎛ 5 x – 4 ⎞ (3 x 2 + 1) Dx (5 x – 4) – (5 x – 4) Dx (3x 2 + 1) 40. Dx ⎜ ⎟= (3 x 2 + 1) 2 ⎝ 3x2 + 1 ⎠ = = (3 x 2 + 1)(5) – (5 x – 4)(6 x) (3x 2 + 1)2 −15 x 2 + 24 x + 5 (3x 2 + 1)2 ⎛ 2 x 2 – 3 x + 1 ⎞ (2 x + 1) Dx (2 x 2 – 3x + 1) – (2 x 2 – 3x + 1) Dx (2 x + 1) 41. Dx ⎜ ⎟ = ⎜ 2x +1 ⎟ (2 x + 1)2 ⎝ ⎠ = = (2 x + 1)(4 x – 3) – (2 x 2 – 3 x + 1)(2) (2 x + 1)2 4 x2 + 4 x – 5 (2 x + 1) 2 Instructor’s Resource Manual Section 2.3 109 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ⎛ 5 x 2 + 2 x – 6 ⎞ (3 x – 1) Dx (5 x 2 + 2 x – 6) – (5 x 2 + 2 x – 6) Dx (3 x – 1) 42. Dx ⎜ ⎟= ⎜ 3 x – 1 ⎟⎠ (3 x – 1)2 ⎝ = = (3 x – 1)(10 x + 2) – (5 x 2 + 2 x – 6)(3) (3x – 1)2 15 x 2 – 10 x + 16 (3x – 1)2 ⎛ x 2 – x + 1 ⎞ ( x 2 + 1) Dx ( x 2 – x + 1) – ( x 2 – x + 1) Dx ( x 2 + 1) 43. Dx ⎜ ⎟ = ⎜ x2 + 1 ⎟ ( x 2 + 1)2 ⎝ ⎠ = = ( x 2 + 1)(2 x – 1) – ( x 2 – x + 1)(2 x) ( x 2 + 1)2 x2 – 1 ( x 2 + 1)2 ⎛ x 2 – 2 x + 5 ⎞ ( x 2 + 2 x – 3) Dx ( x 2 – 2 x + 5) – ( x 2 – 2 x + 5) Dx ( x 2 + 2 x – 3) 44. Dx ⎜ ⎟= ⎜ x2 + 2 x – 3 ⎟ ( x 2 + 2 x – 3) 2 ⎝ ⎠ = = 45. a. ( x 2 + 2 x – 3)(2 x – 2) – ( x 2 – 2 x + 5)(2 x + 2) ( x 2 + 2 x – 3) 2 4 x 2 –16 x – 4 ( x 2 + 2 x – 3) 2 ( f ⋅ g ) (0) = f (0) g (0) + g (0) f (0) = 4(5) + (–3)(–1) = 23 b. ( f + g ) (0) = f (0) + g (0) = –1 + 5 = 4 c. ( f g ) (0) = = 46. a. g (0) f (0) – f (0) g (0) g 2 (0) –3(–1) – 4(5) (–3) 2 =– 17 9 ( f – g ) (3) = f (3) – g (3) = 2 – (–10) = 12 b. ( f ⋅ g ) (3) = f (3) g (3) + g (3) f (3) = 7(–10) + 6(2) = –58 c. ( g f ) (3) = f (3) g (3) – g (3) f (3) 2 f (3) = 7(–10) – 6(2) (7) 2 =– 82 49 47. Dx [ f ( x )]2 = Dx [ f ( x ) f ( x)] = f ( x) Dx [ f ( x)] + f ( x ) Dx [ f ( x)] = 2 ⋅ f ( x ) ⋅ Dx f ( x ) 48. Dx [ f ( x) g ( x)h( x)] = f ( x) Dx [ g ( x)h( x)] + g ( x)h( x) Dx f ( x) = f ( x)[ g ( x) Dx h( x) + h( x) Dx g ( x)] + g ( x)h( x) Dx f ( x) = f ( x) g ( x) Dx h( x) + f ( x)h( x) Dx g ( x) + g ( x)h( x) Dx f ( x) 110 Section 2.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 54. Proof #1: 49. Dx ( x 2 – 2 x + 2) = 2 x – 2 At x = 1: m tan = 2 (1) – 2 = 0 Tangent line: y = 1 Dx [ f ( x) − g ( x) ] = Dx [ f ( x) + (−1) g ( x) ] = Dx [ f ( x) ] + Dx [ (−1) g ( x) ] 2 2 ⎛ 1 ⎞ ( x + 4) Dx (1) – (1) Dx ( x + 4) 50. Dx ⎜ ⎟= ( x 2 + 4)2 ⎝ x2 + 4 ⎠ = ( x 2 + 4)(0) – (2 x) ( x 2 + 4)2 At x = 1: mtan = − =– 2x ( x 2 + 4) 2 2(1) 2 2 =– 2 25 (1 + 4) 1 2 Tangent line: y – = – ( x –1) 5 25 2 7 y = – x+ 25 25 51. Dx ( x3 – x 2 ) = 3 x 2 – 2 x The tangent line is horizontal when m tan = 0: mtan = 3x 2 – 2 x = 0 x(3 x − 2) = 0 2 x = 0 and x = 3 4 ⎞ ⎛2 (0, 0) and ⎜ , – ⎟ ⎝ 3 27 ⎠ ⎛1 ⎞ 52. Dx ⎜ x3 + x 2 – x ⎟ = x 2 + 2 x –1 3 ⎝ ⎠ 2 mtan = x + 2 x –1 = 1 x2 + 2 x – 2 = 0 –2 ± 4 – 4(1)(–2) –2 ± 12 x= = 2 2 = –1 – 3, –1 + 3 x = –1 ± 3 5 5 ⎛ ⎞ ⎛ ⎞ ⎜ −1 + 3, − 3 ⎟ , ⎜ −1 − 3, + 3 ⎟ 3 3 ⎝ ⎠ ⎝ ⎠ 53. y = 100 / x5 = 100 x −5 y ' = −500 x −6 Set y ' equal to −1 , the negative reciprocal of the slope of the line y = x . Solving for x gives x = ±5001/ 6 ≈ ±2.817 y = ±100(500)−5 / 6 ≈ ±0.563 = Dx f ( x) − Dx g ( x) Proof #2: Let F ( x) = f ( x) − g ( x) . Then F '( x) = lim [ f ( x + h) − g ( x + h) ] − [ f ( x ) − g ( x ) ] h →0 h ⎡ f ( x + h) − f ( x ) g ( x + h ) − g ( x ) ⎤ = lim ⎢ − ⎥ h →0 ⎣ h h ⎦ = f '( x) − g '( x) 55. a. Dt (–16t 2 + 40t + 100) = –32t + 40 v = –32(2) + 40 = –24 ft/s b. v = –32t + 40 = 0 t=5 s 4 56. Dt (4.5t 2 + 2t ) = 9t + 2 9t + 2 = 30 28 t= s 9 57. mtan = Dx (4 x – x 2 ) = 4 – 2 x The line through (2,5) and (x 0 , y0 ) has slope y0 − 5 . x0 − 2 4 – 2 x0 = 4 x0 – x0 2 – 5 x0 – 2 –2 x02 + 8 x0 – 8 = – x0 2 + 4 x0 – 5 x0 2 – 4 x0 + 3 = 0 ( x0 – 3)( x0 –1) = 0 x 0 = 1, x0 = 3 At x 0 = 1: y0 = 4(1) – (1)2 = 3 mtan = 4 – 2(1) = 2 Tangent line: y – 3 = 2(x – 1); y = 2x + 1 At x0 = 3 : y0 = 4(3) – (3)2 = 3 mtan = 4 – 2(3) = –2 Tangent line: y – 3 = –2(x – 3); y = –2x + 9 The points are (2.817,0.563) and (−2.817,−0.563) . Instructor’s Resource Manual Section 2.4 111 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 58. Dx ( x 2 ) = 2 x The line through (4, 15) and ( x0 , y0 ) has slope 61. The watermelon has volume of the rind is y0 − 15 . If (x 0 , y0 ) is on the curve y = x 2 , then x0 − 4 mtan = 2 x0 = 3 V= x02 –15 . x0 – 4 r ⎞ 4 3 4 ⎛ 271 3 πr – π ⎜ r – ⎟ = πr . 3 3 ⎝ 10 ⎠ 750 At the end of the fifth week r = 10, so 271 2 271 542π DrV = πr = π(10)2 = ≈ 340 cm3 250 250 5 per cm of radius growth. Since the radius is growing 2 cm per week, the volume of the rind is 542π (2) ≈ 681 cm3 per growing at the rate of 5 week. 2 x0 2 – 8 x0 = x02 –15 x0 2 – 8 x0 + 15 = 0 ( x0 – 3)( x0 – 5) = 0 At x0 = 3 : y0 = (3)2 = 9 She should shut off the engines at (3, 9). (At x 0 = 5 she would not go to (4, 15) since she is moving left to right.) 4 3 πr ; the volume 3 2.4 Concepts Review 2 59. Dx (7 – x ) = –2 x The line through (4, 0) and ( x0 , y0 ) has slope 1. y0 − 0 . If the fly is at ( x0 , y0 ) when the x0 − 4 spider sees it, then mtan = –2 x0 = 2 7 – x0 – 0 . x0 – 4 –2 x02 + 8 x0 = 7 – x02 x 02 – 8x 0 + 7 = 0 ( x0 – 7)( x0 –1) = 0 At x0 = 1: y0 = 6 d = (4 – 1)2 + (0 – 6) 2 = 9 + 36 = 45 = 3 5 ≈ 6. 7 They are 6.7 units apart when they see each other. 1 ⎛ 1⎞ 60. P(a, b) is ⎜ a, ⎟ . Dx y = – so the slope of a ⎝ ⎠ x2 1 the tangent line at P is – . The tangent line is a2 1 1 1 y– =– ( x – a ) or y = – ( x – 2a ) which 2 a a a2 has x-intercept (2a, 0). 1 1 d (O, P ) = a 2 + , d ( P, A) = (a – 2a )2 + 2 a a2 = a2 + 1 = d (O, P ) so AOP is an isosceles a2 triangle. The height of AOP is a while the base, 1 OA has length 2a, so the area is (2 a)(a) = a2 . 2 112 Section 2.4 sin( x + h) – sin( x) h 2. 0; 1 3. cos x; –sin x 4. cos π 1 3 1⎛ π⎞ = ;y– = ⎜x– ⎟ 3 2 2 2⎝ 3⎠ Problem Set 2.4 1. Dx(2 sin x + 3 cos x) = 2 Dx(sin x) + 3 Dx(cos x) = 2 cos x – 3 sin x 2. Dx (sin 2 x) = sin x Dx (sin x ) + sin x Dx (sin x) = sin x cos x + sin x cos x = 2 sin x cos x = sin 2x 3. Dx (sin 2 x + cos 2 x ) = Dx (1) = 0 4. Dx (1 – cos 2 x) = Dx (sin 2 x) = sin x Dx (sin x) + sin x Dx (sin x) = sin x cos x + sin x cos x = 2 sin x cos x = sin 2x ⎛ 1 ⎞ 5. Dx (sec x) = Dx ⎜ ⎟ ⎝ cos x ⎠ cos x Dx (1) – (1) Dx (cos x ) = cos 2 x sin x 1 sin x = = ⋅ = sec x tan x 2 x cos x cos cos x Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ⎛ 1 ⎞ 6. Dx (csc x) = Dx ⎜ ⎟ ⎝ sin x ⎠ sin x Dx (1) − (1) Dx (sin x) = sin 2 x – cos x –1 cos x = = ⋅ = – csc x cot x 2 sin x sin x sin x ⎛ sin x ⎞ 7. Dx (tan x) = Dx ⎜ ⎟ ⎝ cos x ⎠ cos x Dx (sin x) − sin x Dx (cos x) = cos 2 x = cos 2 x + sin 2 x cos2 x 1 = cos 2 x = sec2 x = 2 − sin x – cos x 2 = 2 2 2 sin x = cos x(cos x – sin x) – (– sin 2 x – sin x cos x) cos 2 x + sin 2 x cos2 x = cos 2 x 1 cos 2 x tan x(cos x – sin x) – sec2 x(sin x + cos x) tan 2 x ⎛ sin 2 x sin x 1 ⎞ ⎛ sin 2 x ⎞ – – = ⎜ sin x – ⎟ ⎜ ⎟ 2 cos x cos x cos x ⎠ ⎝ cos 2 x ⎠ ⎝ ⎛ sin 2 x sin x 1 ⎞⎛ cos 2 x ⎞ = ⎜ sin x − − − ⎟⎜ ⎟ 2 cos x cos x cos x ⎠⎝ sin 2 x ⎠ ⎝ cos 2 x 1 cos x − cos x − − sin x sin x sin 2 x = sin x ( − sin x ) + cos x ( cos x ) = cos 2 x − sin 2 x ⎛ sin x + cos x ⎞ 9. Dx ⎜ ⎟ cos x ⎝ ⎠ cos x Dx (sin x + cos x) − (sin x + cos x) Dx (cos x) = cos 2 x = = 11. Dx ( sin x cos x ) = sin xDx [ cos x ] + cos xDx [sin x ] –(sin x + cos x) sin x 1 =– = – csc2 x 2 sin x ⎛ sin x + cos x ⎞ Dx ⎜ ⎟ tan x ⎝ ⎠ tan x Dx (sin x + cos x) − (sin x + cos x) Dx (tan x ) = tan 2 x = ⎛ cos x ⎞ 8. Dx (cot x) = Dx ⎜ ⎟ ⎝ sin x ⎠ sin x Dx (cos x) − cos x Dx (sin x) = sin 2 x 2 10. = sec2 x 12. Dx ( sin x tan x ) = sin xDx [ tan x ] + tan xDx [sin x ] ( ) = sin x sec2 x + tan x ( cos x ) ⎛ 1 ⎞ sin x = sin x ⎜ ( cos x ) ⎟+ ⎝ cos 2 x ⎠ cos x = tan x sec x + sin x ⎛ sin x ⎞ xDx ( sin x ) − sin xDx ( x ) 13. Dx ⎜ ⎟= x2 ⎝ x ⎠ x cos x − sin x = x2 ⎛ 1 − cos x ⎞ xDx (1 − cos x ) − (1 − cos x ) Dx ( x ) 14. Dx ⎜ ⎟= x x2 ⎝ ⎠ x sin x + cos x − 1 = x2 15. Dx ( x 2 cos x) = x 2 Dx (cos x) + cos x Dx ( x 2 ) = − x 2 sin x + 2 x cos x ⎛ x cos x + sin x ⎞ 16. Dx ⎜ ⎟ x2 + 1 ⎝ ⎠ = = = ( x 2 + 1) Dx ( x cos x + sin x) − ( x cos x + sin x) Dx ( x 2 + 1) ( x 2 + 1) 2 ( x 2 + 1)(– x sin x + cos x + cos x) – 2 x( x cos x + sin x) ( x 2 + 1)2 – x3 sin x – 3 x sin x + 2 cos x ( x 2 + 1) 2 Instructor’s Resource Manual Section 2.4 113 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. Dt(20 sin t) = 20 cos t π π At t = : rate = 20 cos = 10 3 ≈ 17. 32 ft/s 6 6 y = tan 2 x = (tan x)(tan x) 17. Dx y = (tan x)(sec 2 x) + (tan x)(sec 2 x) = 2 tan x sec 2 x 3 y = sec x = (sec x)(sec x) 18. Dx y = (sec2 x) sec x tan x + (sec x) Dx (sec2 x) 25. When y = 0 , y = tan 0 = 0 and y ' = sec 2 0 = 1 . The tangent line at x = 0 is y = x . = sec3 x tan x + 2sec3 x tan x = 3sec2 x tan x 26. 19. Dx(cos x) = –sin x At x = 1: mtan = – sin1 ≈ –0.8415 y = cos 1 ≈ 0.5403 Tangent line: y – 0.5403 = –0.8415(x – 1) 20. Dx (cot x) = – csc2 x = 2 tan x sec 2 x Now, sec2 x is never 0, but tan x = 0 at x = kπ where k is an integer. = 9 ⎡sin 2 x − cos 2 x ⎤ ⎣ ⎦ = 9 [ − cos 2 x ] The tangent line is horizontal when y ' = 0 or, in this case, where cos 2 x = 0 . This occurs when 21. Dx sin 2 x = Dx (2sin x cos x) = 2 ⎣⎡sin x Dx cos x + cos x Dx sin x ⎦⎤ 2 x= 2 = −2sin x + 2 cos x 2 2 22. Dx cos 2 x = Dx (2 cos x − 1) = 2 Dx cos x − Dx 1 = −2sin x cos x 23. Dt (30sin 2t ) = 30 Dt (2sin t cos t ) ( = 30 −2sin 2 t + 2 cos 2 t ) = 60 cos 2t 30sin 2t = 15 1 sin 2t = 2 π → t= π 6 12 π ⎛ π ⎞ At t = ; 60 cos ⎜ 2 ⋅ ⎟ = 30 3 ft/sec 12 ⎝ 12 ⎠ The seat is moving to the left at the rate of 30 3 ft/s. 24. The coordinates of the seat at time t are (20 cos t, 20 sin t). a. π π⎞ ⎛ ⎜ 20 cos , 20sin ⎟ = (10 3, 10) 6 6⎠ ⎝ ≈ (17.32, 10) Section 2.4 y = 9sin x cos x y ' = 9 [sin x(− sin x) + cos x(cos x) ] π⎞ ⎛ Tangent line: y –1 = –2 ⎜ x – ⎟ 4⎠ ⎝ 2t = y = tan 2 x = (tan x)(tan x) y ' = (tan x)(sec 2 x) + (tan x)(sec2 x) 27. π At x = : mtan = –2; 4 y=1 y = tan x y ' = sec2 x = sec3 x tan x + sec x(sec x ⋅ sec x tan x + sec x ⋅ sec x tan x) 114 The fastest rate 20 cos t can obtain is 20 ft/s. c. 2 28. π 4 +k π 2 where k is an integer. f ( x) = x − sin x f '( x) = 1 − cos x f '( x) = 0 when cos x = 1 ; i.e. when x = 2kπ where k is an integer. f '( x) = 2 when x = (2k + 1)π where k is an integer. 29. The curves intersect when 2 sin x = 2 cos x, sin x = cos x at x = π for 0 < x < π . 4 2 π Dx ( 2 sin x) = 2 cos x ; 2 cos = 1 4 π Dx ( 2 cos x) = – 2 sin x ; − 2 sin = −1 4 1(–1) = –1 so the curves intersect at right angles. 30. v = Dt (3sin 2t ) = 6 cos 2t At t = 0: v = 6 cm/s π t = : v = −6 cm/s 2 t = π : v = 6 cm/s Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. sin( x + h) 2 – sin x 2 h h→0 31. Dx (sin x 2 ) = lim sin( x 2 + 2 xh + h 2 ) – sin x 2 h h→0 sin x 2 cos(2 xh + h 2 ) + cos x 2 sin(2 xh + h 2 ) – sin x 2 sin x 2 [cos(2 xh + h 2 ) – 1] + cos x 2 sin(2 xh + h 2 ) = lim = lim h →0 h h h→0 = lim 2 ⎡ cos(2 xh + h 2 ) – 1 2 sin(2 xh + h ) ⎤ cos x = lim(2 x + h) ⎢sin x 2 + ⎥ = 2 x(sin x 2 ⋅ 0 + cos x 2 ⋅1) = 2 x cos x 2 h →0 2 xh + h 2 2 xh + h 2 ⎥⎦ ⎢⎣ sin(5( x + h)) – sin 5 x h h →0 sin(5 x + 5h) – sin 5 x = lim h h →0 sin 5 x cos 5h + cos 5 x sin 5h – sin 5 x = lim h h →0 cos 5h – 1 sin 5h ⎤ ⎡ = lim ⎢sin 5 x + cos 5 x h h ⎥⎦ h →0 ⎣ cos 5h – 1 sin 5h ⎤ ⎡ = lim ⎢5sin 5 x + 5cos 5 x 5h 5h ⎥⎦ h→0 ⎣ = 0 + 5cos 5 x ⋅1 = 5cos 5 x 32. Dx (sin 5 x) = lim 33. f(x) = x sin x a. 34. f ( x ) = cos3 x − 1.25cos 2 x + 0.225 x0 ≈ 1.95 f (x 0 ) ≈ –1. 24 2.5 Concepts Review 1. Dt u; f ( g (t )) g (t ) 2. Dv w; G ( H ( s )) H ( s ) b. f(x) = 0 has 6 solutions on [π , 6π ] f (x) = 0 has 5 solutions on [π , 6π ] c. f(x) = x sin x is a counterexample. Consider the interval [ 0, π ] . f ( −π ) = f (π ) = 0 and f ( x ) = 0 has exactly two solutions in the interval (at 0 and π ). However, f ' ( x ) = 0 has two solutions in the interval, not 1 as the conjecture indicates it should have. d. The maximum value of f ( x) – f ( x) on [π , 6π ] is about 24.93. 3. ( f ( x)) 2 ;( f ( x)) 2 2 2 4. 2 x cos( x );6(2 x + 1) Problem Set 2.5 1. y = u15 and u = 1 + x Dx y = Du y ⋅ Dx u = (15u14 )(1) = 15(1 + x )14 2. y = u5 and u = 7 + x Dx y = Du y ⋅ Dx u = (5u 4 )(1) = 5(7 + x)4 3. y = u5 and u = 3 – 2x Dx y = Du y ⋅ Dx u = (5u 4 )(–2) = –10(3 – 2 x) 4 Instructor’s Resource Manual Section 2.4 115 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4. y = u7 and u = 4 + 2 x 2 Dx y = Du y ⋅ Dx u 10. y = cos u and u = 3 x 2 – 2 x Dx y = Du y ⋅ Dx u = (–sin u)(6x – 2) = (7u 6 )(4 x) = 28 x(4 + 2 x 2 )6 = –(6 x – 2) sin(3x 2 – 2 x) 5. y = u11 and u = x3 – 2 x 2 + 3 x + 1 Dx y = Du y ⋅ Dx u 11. y = u 3 and u = cos x Dx y = Du y ⋅ Dx u = (11u10 )(3x 2 – 4 x + 3) = (3u 2 )(– sin x) = 11(3 x 2 – 4 x + 3)( x3 – 2 x 2 + 3x + 1)10 = –3sin x cos 2 x 6. y = u –7 and u = x 2 – x + 1 Dx y = Du y ⋅ Dx u 12. y = u 4 , u = sin v, and v = 3 x 2 Dx y = Du y ⋅ Dv u ⋅ Dx v = (–7u –8 )(2 x – 1) = (4u 3 )(cos v )(6 x) = –7(2 x – 1)( x 2 – x + 1) –8 = 24 x sin 3 (3 x 2 ) cos(3 x 2 ) 7. y = u –5 and u = x + 3 Dx y = Du y ⋅ Dx u = (–5u –6 )(1) = –5( x + 3) –6 = – –9 x +1 x –1 Dx y = Du y ⋅ Dx u 13. y = u 3 and u = 5 ( x + 3)6 = (3u 2 ) ( x –1) 2 2 8. y = u and u = 3x + x – 3 Dx y = Du y ⋅ Dx u ⎛ x +1⎞ = 3⎜ ⎟ ⎝ x –1⎠ = (–9u –10 )(6 x + 1) = –9(6 x + 1)(3 x 2 + x – 3) –10 =– ( x –1) Dx ( x + 1) – ( x + 1) Dx ( x –1) –2 ⎜⎜ 2 ⎝ ( x – 1) 14. y = u −3 and u = 9(6 x + 1) ⎞ 6( x + 1) 2 ⎟⎟ = − ( x – 1)4 ⎠ x−2 x−π Dx y = Du y ⋅ Dx u (3 x 2 + x – 3)10 = (−3u −4 ) ⋅ 2 9. y = sin u and u = x + x Dx y = Du y ⋅ Dx u = (cos u)(2x + 1) ( x − π) Dx ( x − 2) − ( x − 2) Dx ( x − π) ⎛ x−2⎞ = −3 ⎜ ⎟ ⎝ x−π⎠ 2 = (2 x + 1) cos( x + x) 15. y = cos u and u = 2⎛ ( x − π) 2 −4 (2 − π) ( x − π) 2 = −3 ( x − π)2 ( x − 2)4 (2 − π) 3x 2 x+2 Dx y = Du y ⋅ Dx u = (– sin u ) ( x + 2) Dx (3 x 2 ) – (3x 2 ) Dx ( x + 2) ( x + 2) 2 ⎛ 3x 2 ⎞ ( x + 2)(6 x) – (3x 2 )(1) ⎛ 3x 2 ⎞ 3 x 2 + 12 x = – sin ⎜ =– sin ⎜ ⎟ ⎟ ⎜ x+2⎟ ⎜ x+2⎟ ( x + 2)2 ( x + 2)2 ⎝ ⎠ ⎝ ⎠ 16. y = u 3 , u = cos v, and v = x2 1– x Dx y = Du y ⋅ Dv u ⋅ Dx v = (3u 2 )(− sin v) (1 – x) Dx ( x 2 ) – ( x 2 ) Dx (1 − x) (1 – x) 2 ⎛ x2 ⎞ ⎛ x2 ⎞ ⎛ x 2 ⎞ ⎛ x 2 ⎞ (1 – x)(2 x) – ( x 2 )(–1) –3(2 x – x 2 ) = = –3cos 2 ⎜ cos 2 ⎜ ⎟ sin ⎜ ⎟ ⎟ sin ⎜ ⎟ ⎜1– x ⎟ ⎜1– x ⎟ ⎜1– x ⎟ ⎜1– x ⎟ (1 – x)2 (1 – x)2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 116 Section 2.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 17. Dx [(3 x – 2)2 (3 – x 2 ) 2 ] = (3 x – 2)2 Dx (3 – x 2 ) 2 + (3 – x 2 ) 2 Dx (3 x – 2) 2 = (3 x – 2)2 (2)(3 – x 2 )(–2 x) + (3 – x 2 ) 2 (2)(3 x – 2)(3) = 2(3 x − 2)(3 − x 2 )[(3 x − 2)(−2 x) + (3 − x 2 )(3)] = 2(3 x − 2)(3 − x 2 )(9 + 4 x − 9 x 2 ) 18. Dx [(2 – 3x 2 )4 ( x7 + 3)3 ] = (2 – 3 x 2 )4 Dx ( x 7 + 3)3 + ( x 7 + 3)3 Dx (2 – 3 x 2 ) 4 = (2 – 3 x 2 )4 (3)( x 7 + 3) 2 (7 x 6 ) + ( x 7 + 3)3 (4)(2 – 3 x 2 )3 (–6 x) = 3x (3 x 2 – 2)3 ( x7 + 3) 2 (29 x7 – 14 x5 + 24) ⎡ ( x + 1)2 ⎤ (3x – 4) Dx ( x + 1)2 – ( x + 1)2 Dx (3x – 4) (3 x – 4)(2)( x + 1)(1) – ( x + 1)2 (3) 3 x 2 – 8 x – 11 19. Dx ⎢ = = ⎥= (3x – 4) 2 (3x – 4) 2 (3 x – 4)2 ⎣⎢ 3x – 4 ⎦⎥ = ( x + 1)(3 x − 11) (3x − 4)2 ⎡ 2 x – 3 ⎤ ( x 2 + 4) 2 Dx (2 x – 3) – (2 x – 3) Dx ( x 2 + 4)2 20. Dx ⎢ ⎥= 2 2 ( x 2 + 4) 4 ⎣⎢ ( x + 4) ⎦⎥ = ( x 2 + 4) 2 (2) – (2 x – 3)(2)( x 2 + 4)(2 x) ( x 2 + 4) 4 ( )( ) ( = ) −6 x 2 + 12 x + 8 ( x 2 + 4)3 ( 21. y = 2 x 2 + 4 x 2 + 4 = 2 x 2 + 4 (2 x ) = 4 x x 2 + 4 ) 22. y = 2(x + sin x )(x + sin x ) = 2(x + sin x )(1 + cos x ) 3 2 ⎛ 3t – 2 ⎞ ⎛ 3t – 2 ⎞ (t + 5) Dt (3t – 2) – (3t – 2) Dt (t + 5) 23. Dt ⎜ `⎟ = 3 ⎜ ⎟ t + 5 ⎝ ⎠ ⎝ t +5 ⎠ (t + 5)2 2 51(3t – 2)2 ⎛ 3t – 2 ⎞ (t + 5)(3) – (3t – 2)(1) = = 3⎜ ⎟ ⎝ t +5 ⎠ (t + 5) 4 (t + 5)2 ⎛ s 2 – 9 ⎞ ( s + 4) Ds ( s 2 – 9) – ( s 2 – 9) Ds ( s + 4) ( s + 4)(2s ) – ( s 2 – 9)(1) s 2 + 8s + 9 24. Ds ⎜ = = ⎟= ⎜ s+4 ⎟ ( s + 4)2 ( s + 4) 2 ( s + 4)2 ⎝ ⎠ d ⎛ (3t − 2)3 ⎞ 25. ⎜ ⎟= dt ⎜⎝ t + 5 ⎟⎠ = 26. (t + 5) d d (3t − 2)3 − (3t − 2)3 (t + 5) (t + 5)(3)(3t – 2)2 (3) – (3t – 2)3 (1) dt dt = (t + 5)2 (t + 5) 2 (6t + 47)(3t – 2)2 (t + 5) 2 d (sin 3 ) = 3sin 2 cos d 3 2 2 dy d ⎛ sin x ⎞ ⎛ sin x ⎞ d sin x ⎛ sin x ⎞ = ⎜ = 3⎜ 27. ⎟ = 3⎜ ⎟ ⋅ ⎟ ⋅ dx dx ⎝ cos 2 x ⎠ ⎝ cos 2 x ⎠ dx cos 2 x ⎝ cos 2 x ⎠ (cos 2 x) d d (sin x) − (sin x) (cos 2 x) dx dx cos 2 2 x 2 2 3 ⎛ sin x ⎞ cos x cos 2 x + 2sin x sin 2 x 3sin x cos x cos 2 x + 6sin x sin 2 x = 3⎜ = ⎟ ⎝ cos 2 x ⎠ cos 2 2 x cos 4 2 x = 3(sin 2 x)(cos x cos 2 x + 2sin x sin 2 x) cos 4 2 x Instructor’s Resource Manual Section 2.5 117 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 28. dy d d d = [sin t tan(t 2 + 1)] = sin t ⋅ [tan(t 2 + 1)] + tan(t 2 + 1) ⋅ (sin t ) dt dt dt dt = (sin t )[sec 2 (t 2 + 1)](2t ) + tan(t 2 + 1) cos t = 2t sin t sec2 (t 2 + 1) + cos t tan(t 2 + 1) 2 29. 2 ⎛ x 2 + 1 ⎞ ( x + 2) Dx ( x 2 + 1) – ( x 2 + 1) Dx ( x + 2) ⎛ x 2 + 1 ⎞ 2 x 2 + 4 x – x 2 –1 3( x 2 + 1)2 ( x 2 + 4 x – 1) f ( x) = 3 ⎜ = 3⎜ = ⎟ ⎟ ⎜ x+2 ⎟ ⎜ x+2 ⎟ ( x + 2) 2 ( x + 2) 2 ( x + 2)4 ⎝ ⎠ ⎝ ⎠ f (3) = 9.6 30. G (t ) = (t 2 + 9)3 Dt (t 2 – 2)4 + (t 2 – 2) 4 Dt (t 2 + 9)3 = (t 2 + 9)3 (4)(t 2 – 2)3 (2t ) + (t 2 – 2) 4 (3)(t 2 + 9)2 (2t ) = 2t (7t 2 + 30)(t 2 + 9)2 (t 2 – 2)3 G (1) = –7400 31. F (t ) = [cos(t 2 + 3t + 1)](2t + 3) = (2t + 3) cos(t 2 + 3t + 1) ; F (1) = 5cos 5 ≈ 1.4183 32. g ( s ) = (cos πs ) Ds (sin 2 πs ) + (sin 2 πs ) Ds (cos πs ) = (cos πs )(2sin πs )(cos πs )(π) + (sin 2 πs )(– sin πs )(π) = π sin πs[2 cos 2 πs – sin 2 πs ] ⎛1⎞ g ⎜ ⎟ = –π ⎝2⎠ 33. Dx [sin 4 ( x 2 + 3x)] = 4sin 3 ( x 2 + 3x) Dx sin( x 2 + 3 x) = 4sin 3 ( x 2 + 3 x) cos( x 2 + 3 x) Dx ( x 2 + 3 x) = 4sin 3 ( x 2 + 3 x) cos( x 2 + 3x)(2 x + 3) = 4(2 x + 3) sin 3 ( x 2 + 3x) cos( x 2 + 3 x) 34. Dt [cos5 (4t – 19)] = 5cos 4 (4t – 19) Dt cos(4t – 19) = 5cos 4 (4t – 19)[– sin(4t – 19)]Dt (4t – 19) 4 = –5cos 4 (4t – 19) sin(4t – 19)(4) = –20 cos (4t – 19) sin(4t – 19) 35. Dt [sin 3 (cos t )] = 3sin 2 (cos t ) Dt sin(cos t ) = 3sin 2 (cos t ) cos(cos t ) Dt (cos t ) 2 = 3sin 2 (cos t ) cos(cos t )(– sin t ) = –3sin t sin (cos t ) cos(cos t ) ⎡ ⎛ u + 1 ⎞⎤ ⎛ u +1⎞ ⎛ u + 1 ⎞⎤ ⎛ u + 1 ⎞ 3 ⎛ u +1⎞ 3 ⎛ u +1⎞ ⎡ 36 . Du ⎢cos4 ⎜ ⎟ ⎥ = 4 cos ⎜ ⎟ Du cos ⎜ ⎟ = 4 cos ⎜ ⎟ ⎢ – sin ⎜ ⎟ ⎥ Du ⎜ ⎟ ⎝ u –1 ⎠ ⎦ ⎝ u –1 ⎠ ⎝ u –1 ⎠ ⎝ u –1 ⎠ ⎣ ⎝ u –1 ⎠ ⎦ ⎝ u –1 ⎠ ⎣ 8 ⎛ u +1⎞ ⎛ u +1⎞ ⎛ u + 1 ⎞ ⎛ u + 1 ⎞ (u –1) Du (u + 1) – (u + 1) Du (u –1) cos3 ⎜ = = –4 cos3 ⎜ ⎟ sin ⎜ ⎟ ⎟ sin ⎜ ⎟ 2 2 u –1 u –1 ⎝ u –1 ⎠ ⎝ u –1 ⎠ ⎝ ⎠ ⎝ ⎠ (u –1) (u –1) 37. D [cos 4 (sin 2 3 = –4 cos (sin )] = 4 cos3 (sin 2 ) sin(sin 2 2 )(cos ) D cos(sin 2 )D ( 2 2 ) = 4 cos3 (sin 3 ) = –8 cos (sin 2 2 )[– sin(sin ) sin(sin 2 2 )]D (sin )(cos 2 2 ) ) 38. Dx [ x sin 2 (2 x)] = x Dx sin 2 (2 x) + sin 2 (2 x) Dx x = x[2sin(2 x) Dx sin(2 x)] + sin 2 (2 x)(1) = x[2sin(2 x) cos(2 x) Dx (2 x)] + sin 2 (2 x) = x[4sin(2 x) cos(2 x)] + sin 2 (2 x) = 2 x sin(4 x) + sin 2 (2 x) 39. Dx {sin[cos(sin 2 x)]} = cos[cos(sin 2 x)]Dx cos(sin 2 x ) = cos[cos(sin 2 x)][– sin(sin 2 x)]Dx (sin 2 x) = – cos[cos(sin 2 x)]sin(sin 2 x)(cos 2 x) Dx (2 x) = –2 cos[cos(sin 2 x)]sin(sin 2 x)(cos 2 x) 40. Dt {cos 2 [cos(cos t )]} = 2 cos[cos(cos t )]Dt cos[cos(cos t )] = 2 cos[cos(cos t )]{– sin[cos(cos t )]}Dt cos(cos t ) = –2 cos[cos(cos t )]sin[cos(cos t )][– sin(cos t )]Dt (cos t ) = 2 cos[cos(cos t )]sin[cos(cos t )]sin(cos t )(– sin t ) = –2sin t cos[cos(cos t )]sin[cos(cos t )]sin(cos t ) 118 Section 2.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 41. ( f + g ) (4) = f (4) + g (4) ≈ 42. (f 53. 1 3 + ≈2 2 2 − 2g ) ( 2) = f ( 2) − ( 2g ) ( 2) 54. = f ( 2) − 2g ( 2) = 1 − 2 ( 0) = 1 43. ( fg ) (2 ) = ( fg + gf 44. ( f g ) (2) = ≈ d d F ( cos x ) = F ( cos x ) ( cos x ) dx dx = − sin xF ( cos x ) d d cos ( F ( x ) ) = − sin ( F ( x ) ) F ( x ) dx dx = − F ( x ) sin ( F ( x ) ) 55. Dx ⎣⎡ tan ( F ( 2 x ) ) ⎦⎤ = sec 2 ( F ( 2 x ) ) Dx ⎡⎣ F ( 2 x ) ⎤⎦ )(2) = 2(0) + 1(1) = 1 = sec 2 ( F ( 2 x ) ) × F ( 2 x ) × Dx [ 2 x ] = 2 F ( 2 x ) sec2 ( F ( 2 x ) ) g (2) f (2) – f (2) g (2) 2 g (2) (1)(1) – (3)(0) (1) 2 56. =1 d d ⎡⎣ g ( tan 2 x ) ⎤⎦ = g ' ( tan 2 x ) ⋅ tan 2 x dx dx ( 45. ( f g ) (6) = f ( g (6)) g (6) = f (2) g (6) ≈ (1)(−1) = –1 = 2 g ' ( tan 2 x ) sec 2 2 x 57. Dx ⎡⎣ F ( x ) sin 2 F ( x ) ⎤⎦ = F ( x ) × Dx ⎡⎣sin 2 F ( x ) ⎤⎦ + sin 2 F ( x ) × Dx F ( x ) = F ( x ) × 2sin F ( x ) × Dx ⎡⎣sin F ( x ) ⎤⎦ 46. ( g f ) (3) = g ( f (3)) f (3) 3 ⎛3⎞ = g (4) f (3) ≈ ⎜ ⎟ (1) = 2 2 ⎝ ⎠ 47. D x F (2 x ) = F (2 x )D x (2 x ) = 2 F (2 x ) 48. ( 52. ) ( = F ( x ) × 2sin F ( x ) × cos ( F ( x ) ) × Dx ⎣⎡ F ( x ) ⎦⎤ ) ( ) + F ( x ) sin 2 F ( x ) = 2 xF x 2 + 1 [ 51. ( + F ( x ) sin 2 F ( x ) Dx F x 2 +1 = F x 2 +1 Dx x 2 +1 49. Dt (F (t )) 50. ) −2 ] = −2(F (t )) −3 = 2 F ( x ) F ( x ) sin F ( x ) cos F ( x ) F (t ) + F ( x ) sin 2 F ( x ) d ⎡ 1 ⎤ −3 ⎢ ⎥ = −2(F (z )) F (z ) dz ⎣⎢ (F (z ))2 ⎦⎥ 58. Dx ⎡⎣sec3 F ( x ) ⎤⎦ = 3sec 2 ⎡⎣ F ( x ) ⎤⎦ Dx ⎡⎣sec F ( x ) ⎤⎦ = 3sec 2 ⎡⎣ F ( x ) ⎤⎦ sec F ( x ) tan F ( x ) Dx [ x ] d ⎡ d 2 1 + F ( 2 z ) ) ⎤ = 2 (1 + F ( 2 z ) ) (1 + F ( 2 z ) ) ( ⎢ ⎥ ⎦ dz ⎣ dz = 2 (1 + F ( 2 z ) ) ( 2 F ( 2 z ) ) = 4 (1 + F ( 2 z ) ) F ( 2 z ) ⎡ d ⎢ 2 1 y + ⎢ dy F y2 ⎣ ( ) ⎤ ⎥ = 2 y + d ⎡ F y2 ⎥ dy ⎢⎣ ⎦ ( ( )) ( ) dyd y = 2 y − F y2 ⎛ F y2 ⎜ = 2 y ⎜1 − ⎜ F y2 ⎝ ( ) ( ( )) 2 2 yF y 2 = 3F ( x ) sec3 F ( x ) tan F ( x ) 59. g ' ( x ) = − sin f ( x ) Dx f ( x ) = − f ⎥ ⎦ 60. G ( x ) = 2 2 = d d x − x (1 + sec F ( 2 x ) ) (1 + sec F ( 2 x ) ) dx dx (1 + sec F ( 2 x ) ) 2 (1 + sec F ( 2 x ) ) − 2 xF ( 2 x ) sec F ( 2 x ) tan F ( 2 x ) 2 (1 + sec F ( 2 x ) ) G ( 0) = = Instructor’s Resource Manual ( x ) sin f ( x ) g ( 0 ) = − f ( 0 ) sin f ( 0 ) = −2sin1 ≈ −1.683 −1 ⎤ ( ) = 2y − ( F ( y )) ⎞ ⎟ 2 ⎟ ⎟ ⎠ ) = g ' ( tan 2 x ) sec 2 2 x ⋅ 2 1 + sec F ( 0 ) − 0 (1 + sec F ( 0 ) ) 2 = 1 + sec F ( 0 ) (1 + sec F ( 0 ) ) 2 1 1 = ≈ −0.713 1 + sec F ( 0 ) 1 + sec 2 Section 2.5 119 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 61. F ( x ) = − f ( x ) g ( x ) sin g ( x ) + f F (1) = − f (1) g (1) sin g (1) + f ( x ) cos g ( x ) (1) cos g (1) c. Dt L = = −2 (1) sin 0 + −1cos 0 = −1 = 62. y = 1 + x sin 3 x; y = 3x cos 3 x + sin 3 x y (π / 3) = 3 π cos 3 π + sin 3 3 y − 1 = −π x − π / 3 π 3 = y = −π x − π / 3 + 1 The line crosses the x-axis at x = = 3−π . 3 ( ) ( ) ( ) ( ) ( x + 1)( x + 1) + 3x ( x + 1) ( x + 1) 3 2 64. y = x 2 + 1 2 x 4 + 1 x3 + 3 x 2 + 1 x x 4 + 1 = 2 x3 4 3 2 y (1) = 2 ( 2 )( 2 ) + 3 (1)( 2 ) 3 2 4 2 2 2 2 69. a. b. ( ) −3 ( 2 x ) = −4 x ( x 2 + 1) y (1) = −4 (1)(1 + 1) 1 1 1 y− = − x+ , 4 2 2 66. y = 3 ( 2 x + 1) 2 −3 70. a. −3 b. = −1/ 2 ( 2 ) = 6 ( 2 x + 1) c. 2 2 y − 1 = 6 x − 0, y = 6 x + 1 The line crosses the x-axis at x = −1/ 6 . ( ) y (1) = −4 ( 2 ) −3 −3 ( 2 x ) = −4 x ( x 2 + 1) −3 x-axis at x = 3 / 2 . 2 2 ⎛x⎞ ⎛ y⎞ ⎛ 4 cos 2t ⎞ ⎛ 7 sin 2t ⎞ ⎜ ⎟ +⎜ ⎟ = ⎜ ⎟ +⎜ ⎟ ⎝4⎠ ⎝7⎠ ⎝ 4 ⎠ ⎝ 7 ⎠ = cos 2 2t + sin 2 2t = 1 b. L = ( x – 0)2 + ( y – 0) 2 = x 2 + y 2 = (4 cos 2t )2 + (7 sin 2t )2 = 16 cos 2 2t + 49sin 2 2t 120 Section 2.5 π 33 : rate = ≈ 5.8 ft/sec. 8 16 ⋅ 12 + 49 ⋅ 12 (10 cos8π t ,10sin 8π t ) Dt (10sin 8πt ) = 10 cos(8πt ) Dt (8πt ) (cos 2t, sin 2t) (0 – cos 2t ) 2 + ( y – sin 2t )2 = 52 , so Dt ⎛⎜ sin 2t + 25 − cos 2 2t ⎞⎟ ⎝ ⎠ 1 = 2 cos 2t + ⋅ 4 cos 2t sin 2t 2 25 − cos 2 2t ⎞ ⎟ ⎟ ⎠ 71. 60 revolutions per minute is 120π radians per minute or 2π radians per second. = −1/ 2 1 1 1 1 3 = − x+ , y = − x+ 4 2 2 2 4 Set y = 0 and solve for x. The line crosses the 2 16 cos 2 2t + 49sin 2 2t ⎛ sin 2t = 2 cos 2t ⎜1 + ⎜ 25 – cos 2 2t ⎝ y− 68. a. 16 cos 2 2t + 49sin 2 2t 33sin 4t y = sin 2t + 25 – cos 2 2t 1 3 y = − x+ 2 4 y ( 0 ) = 6 (1) = 6 67. y = −2 x 2 + 1 2 16 cos 2 2t + 49sin 2 2t −16sin 4t + 49sin 4t = 80π cos(8πt ) At t = 1: rate = 80π ≈ 251 cm/s P is rising at the rate of 251 cm/s. ( 2 )2 = 32 + 48 = 80 y − 32 = 80 x − 1, y = 80 x + 31 65. y = −2 x 2 + 1 2 16 cos 2 2t + 49sin 2 2t –64 cos 2t sin 2t + 196sin 2t cos 2t At t = 63. y = sin 2 x; y = 2sin x cos x = sin 2 x = 1 x = π / 4 + kπ , k = 0, ± 1, ± 2,... Dt (16cos2 2t + 49sin 2 2t ) 32 cos 2t Dt (cos 2t ) + 98sin 2t Dt (sin 2t ) = −π + 0 = −π = 1 2 16cos2 2t + 49sin 2 2t a. (cos 2π t ,sin 2π t ) b. (0 – cos 2πt ) 2 + ( y – sin 2πt )2 = 52 , so y = sin 2πt + 25 – cos 2 2πt 2 c. Dt ⎛⎜ sin 2πt + 25 − cos 2 2πt ⎞⎟ ⎝ ⎠ = 2π cos 2πt + 1 2 25 − cos 2 2πt ⋅ 4π cos 2πt sin 2πt ⎛ sin 2πt = 2π cos 2πt ⎜1 + ⎜ 25 – cos 2 2πt ⎝ ⎞ ⎟ ⎟ ⎠ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 72. The minute hand makes 1 revolution every hour, so at t minutes after the hour, it makes an angle πt of radians with the vertical. By the Law of 30 Cosines, the length of the elastic string is πt s = 102 + 102 – 2(10)(10) cos 30 = 10 2 – 2 cos ds = 10 ⋅ dt = 74. From Problem 73, Using a computer algebra system or graphing ds ds for 0 ≤ t ≤ 60 , is largest utility to view dt dt when t ≈ 7.5. Thus, the distance between the tips of the hands is increasing most rapidly at about 12:08. πt 30 π πt sin 15 30 πt 2 2 – 2 cos 30 1 75. ⋅ π 3 Dx(sin x) = cos x, Dx(sin 2x) = 2cos 2x, so at x 0 , the tangent lines to y = sin x and y = sin 2x have 1 ⎛ 1⎞ slopes of m1 = and m2 = 2 ⎜ – ⎟ = –1, 2 ⎝ 2⎠ respectively. From Problem 40 of Section 0.7, m – m1 where is the angle between tan = 2 1 + m1m2 At 12:15, the string is stretching at the rate of π sin π2 π = ≈ 0.74 cm/min 3 2 – 2 cos π2 3 2 73. The minute hand makes 1 revolution every hour, so at t minutes after noon it makes an angle of πt radians with the vertical. Similarly, at t 30 minutes after noon the hour hand makes an angle πt of with the vertical. Thus, by the Law of 360 Cosines, the distance between the tips of the hands is ⎛ πt πt ⎞ s = 62 + 82 – 2 ⋅ 6 ⋅ 8cos ⎜ – ⎟ ⎝ 30 360 ⎠ 11πt 360 ds 1 44π 11πt = ⋅ sin 360 dt 2 100 – 96 cos 11πt 15 360 = 1 [if sin x0 ≠ 0] 2 x0 = πt 2 – 2 cos 30 = 100 – 96 cos sin x0 = sin 2 x0 sin x0 = 2sin x0 cos x0 cos x0 = πt π sin 30 3 πt 22π sin 11 ds 360 = . dt 15 100 – 96 cos 11πt 360 πt 22π sin 11 360 πt 15 100 – 96 cos 11 360 At 12:20, π 22π sin 11 ds 18 = ≈ 0.38 in./min dt 15 100 – 96 cos 11π 18 the tangent lines. tan = ( ) 1 + 12 (–1) = – 23 1 2 = –3, so ≈ –1.25. The curves intersect at an angle of 1.25 radians. 76. 1 t AB = OA sin 2 2 2 1 t t t D = OA cos ⋅ AB = OA cos sin 2 2 2 2 E = D + area (semi-circle) 2 2 t t 1 ⎛1 ⎞ = OA cos sin + π ⎜ AB ⎟ 2 2 2 ⎝2 ⎠ 2 2 t t 1 t = OA cos sin + πOA sin 2 2 2 2 2 2 t⎛ t 1 t⎞ = OA sin ⎜ cos + π sin ⎟ 2⎝ 2 2 2⎠ t cos 2 D = E cos t + 1 π sin t 2 2 2 D 1 = =1 lim + E + 1 0 t →0 lim t →π − Instructor’s Resource Manual –1 – 12 D cos(t / 2) = lim − E t →π cos(t / 2) + π sin(t / 2) 2 0 = =0 π 0+ 2 Section 2.5 121 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 81. [ f ( f ( f ( f (0))))] = f ( f ( f ( f (0)))) ⋅ f ( f ( f (0))) ⋅ f ( f (0)) ⋅ f (0) = 2 ⋅ 2 ⋅2 ⋅ 2 = 16 77. y = u and u = x 2 Dx y = Du y ⋅ Dx u = 1 2 u 2 x x2 – 1 2 78. Dx x – 1 = = x2 – 1 2 x –1 80. a. b. sin x sin x x2 – 1 (2 x) = 79. Dx sin x = = 2x ⋅ 2x = = 2 x x = x x 82. a. Dx ( x 2 –1) 2 x x 2 –1 sin x sin x b. 2 x –1 Dx (sin x) cos x = cot x sin x c. ( ) ( ) ( ) Dx L x 2 = L ' x 2 Dx x 2 = 1 x 2 ⋅ 2x = 2 x d [2] f = f '( f ( x)) ⋅ f '( x) dx d [1] = f '( f [1] ) ⋅ f ( x) dx d [3] f = f '( f ( f ( x))) ⋅ f '( f ( x)) ⋅ f '( x) dx d [1] = f '( f [2] ( x)) ⋅ f '( f [1] ( x)) ⋅ f ( x) dx d [2] = f '( f [2] ( x)) ⋅ f ( x) dx Conjecture: d [n] d [ n −1] f ( x) = f '( f [ n −1] ( x)) ⋅ f ( x) dx dx Dx L(cos 4 x) = sec4 x Dx (cos 4 x ) = sec4 x(4 cos3 x) Dx (cos x ) = 4sec4 x cos3 x(− sin x) 1 ⋅ cos3 x ⋅ ( − sin x ) cos 4 x = –4sec x sin x = −4 tan x = 4⋅ ⎛ f ( x) ⎞ ⎛ 1 ⎞ −1 −1 −1 83. Dx ⎜ ⎟ = Dx ⎜ f ( x ) ⋅ ⎟ = Dx f ( x) ⋅ ( g ( x)) = f ( x) Dx ( g ( x)) + ( g ( x)) Dx f ( x) g ( x) ⎠ ⎝ g ( x) ⎠ ⎝ ( ) ( ) = f ( x) ⋅ (−1)( g ( x)) −2 Dx g ( x) + ( g ( x)) −1 Dx f ( x) = − f ( x)( g ( x)) −2 Dx g ( x) + ( g ( x)) −1 Dx f ( x) = = D x f ( x ) − f ( x ) D x g ( x ) g ( x ) D x f ( x ) − f ( x ) D x g ( x ) g ( x ) Dx f ( x ) + ⋅ = + = g ( x) g ( x) g ( x) g 2 ( x) g 2 ( x) g ( x) g 2 ( x) g ( x) Dx f ( x) − f ( x) Dx g ( x) − f ( x) Dx g ( x) 2 + g 2 ( x) ( f ( f ( f ( x )))) f ( f ( f ( x ))) f ( f ( x )) f ( x ) g ( x ) = f ( f ( f ( f ( x )))) f ( f ( f ( x ))) f ( f ( x )) f ( x ) 84. g ( x ) = f 1 1 = f = ⎡⎣ f g ( x2 ) = f = f = ⎡⎣ f 122 1 ( f ( f ( x2 ) ) ) f ( f ( x2 ) ) f ( x1 )⎤⎦ 2 ⎡⎣ f ( x2 )⎤⎦ 1 1 ( x2 ) f ( x1 ) = f ( f ( x1 ) ) f ( x1 ) f ( x2 ) f ( x1 ) 2 ( f ( f ( f ( x )))) f ( f ( f ( x ))) f ( f ( x2 ) ) f ( x2 ) ( f ( f ( x1 ) ) ) f ( f ( x1 ) ) f ( x1 ) f ( x2 ) = f ( f ( x2 ) ) f ( x2 ) f ( x1 ) f ( x2 ) 2 ( x1 )⎤⎦ Section 2.5 2 ⎡⎣ f ( x2 )⎤⎦ 2 2 = g ( x1 ) Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2.6 Concepts Review 1. f ( x), Dx3 y, 2. ds ds d 2 s ; ; dt dt dt 2 3. f (t ) > 0 d3y dx 3 3. d2y , y ''' dx 2 d3y dx3 4. dx 2 d3y Problem Set 2.6 dx dy 1. = 3x2 + 6 x + 6 dx d y dx 2 d3y dx3 5. = 6x + 6 = 162 = –100(3 – 5 x)3 (–5) = 500(3 – 5 x)3 = 1500(3 – 5 x) 2 (–5) = –7500(3 – 5 x)2 dy = 7 cos(7 x) dx dx 2 d3y dx dy 2. = 5 x 4 + 4 x3 dx d2y = 20x 3 +12 x 2 dx 2 d3y = 60 x 2 + 24 x 3 dx 6. 3 d2y =6 = 18(3 x + 5)(3) = 162 x + 270 dy = 5(3 – 5 x )4 (–5) = –25(3 – 5 x)4 dx d2y 4. 0; < 0 2 dy = 3(3 x + 5) 2 (3) = 9(3x + 5) 2 dx 3 = –7 2 sin(7 x) = –73 cos(7 x) = –343cos(7 x) dy = 3x 2 cos( x3 ) dx d2y dx 2 d3y dx 3 = 3 x 2 [–3x 2 sin( x3 )] + 6 x cos( x3 ) = –9 x 4 sin( x3 ) + 6 x cos( x3 ) = –9 x 4 cos( x3 )(3 x 2 ) + sin( x3 )(–36 x3 ) + 6 x[– sin( x3 )(3 x 2 )] + 6 cos( x3 ) = –27 x 6 cos( x3 ) – 36 x3 sin( x3 ) –18 x3 sin( x3 ) + 6 cos( x3 ) = (6 – 27 x 6 ) cos( x3 ) – 54 x3 sin( x3 ) 7. dy ( x –1)(0) – (1)(1) 1 = =– 2 dx ( x –1) ( x –1)2 d2y dx 2 d3y dx3 =− = ( x –1)2 (0) – 2( x –1) ( x –1) 4 = 8. d2y 2 ( x –1) dy (1 – x)(3) – (3x)(–1) 3 = = 2 dx (1 – x) ( x – 1)2 3 dx 2 ( x − 1)3 (0) − 2[3( x − 1) 2 ] d3y ( x − 1)6 dx3 =− 6 ( x − 1) 4 Instructor’s Resource Manual = ( x – 1) 2 (0) – 3[2( x – 1)] ( x – 1) =− = 4 =– 6 ( x – 1)3 ( x − 1)3 (0) − 6(3)( x − 1) 2 ( x − 1)6 18 ( x − 1) 4 Section 2.6 123 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 9. 10. f ( x) = 2 x; f ( x) = 2; f (2) = 2 12. f ( x) = 15 x 2 + 4 x + 1 f (u ) = f ( x) = 30 x + 4 f (2) = 64 11. f (u ) = f (t ) = – f (t ) = = 2 t2 4 (5 – u )(4u ) – (2u 2 )(–1) (5 – u ) 2 = 20u – 2u 2 (5 – u ) 2 (5 – u )2 (20 – 4u ) – (20u – 2u 2 )2(5 – u )(–1) (5 – u )4 100 (5 – u )3 f (2) = 100 3 3 = 100 27 3 t 4 1 f (2) = = 8 2 13. f ( ) = –2(cos π) –3 (– sin π)π = 2 π(cos π) –3 (sin π) f ( ) = 2π[(cos π) –3 (π)(cos π) + (sin π)(–3)(cos π) –4 (– sin π)(π)] = 2π2 [(cos π)−2 + 3sin 2 π(cos π) −4 ] f (2) = 2π2 [1 + 3(0)(1)] = 2π2 14. ⎛ π ⎞⎛ π ⎞ ⎛π⎞ ⎛ π⎞ ⎛π⎞ ⎛π⎞ f (t ) = t cos ⎜ ⎟ ⎜ – ⎟ + sin ⎜ ⎟ = ⎜ – ⎟ cos ⎜ ⎟ + sin ⎜ ⎟ 2 ⎝ t ⎠⎝ t ⎠ ⎝t⎠ ⎝ t⎠ ⎝t⎠ ⎝t⎠ π2 ⎛ π⎞⎡ ⎛ π ⎞ ⎛ π ⎞⎤ ⎛ π ⎞ ⎛π⎞ ⎛ π ⎞ ⎛π⎞ ⎛π⎞ f (t ) = ⎜ – ⎟ ⎢ – sin ⎜ ⎟ ⎜ – ⎟ ⎥ + ⎜ ⎟ cos ⎜ ⎟ + ⎜ – ⎟ cos ⎜ ⎟ = – sin ⎜ ⎟ 2 2 2 3 ⎝ t ⎠⎣ ⎝ t ⎠ ⎝ t ⎠⎦ ⎝ t ⎠ ⎝t⎠ ⎝ t ⎠ ⎝t⎠ ⎝t⎠ t f (2) = – 15. π2 π2 ⎛π⎞ ≈ –1.23 sin ⎜ ⎟ = – 8 8 ⎝2⎠ f ( s ) = s (3)(1 – s 2 )2 (–2 s ) + (1 – s 2 )3 = –6s 2 (1 – s 2 ) 2 + (1 – s 2 )3 = –7 s 6 + 15s 4 – 9 s 2 + 1 f ( s ) = –42 s5 + 60 s3 –18s f (2) = –900 16. f ( x) = f ( x) = f (2) = ( x –1)2( x + 1) – ( x + 1)2 ( x –1)2 = x2 – 2 x – 3 ( x –1)2 ( x –1) 2 (2 x – 2) – ( x 2 – 2 x – 3)2( x –1) ( x –1) 8 13 4 ( x –1)(2 x – 2) – ( x 2 – 2 x – 3)(2) ( x –1) 3 = 8 ( x –1)3 =8 17. Dx ( x n ) = nx n –1 Dx2 ( x n ) = n(n –1) x n –2 Dx3 ( x n ) = n(n –1)(n – 2) x n –3 Dx4 ( x n ) = n(n – 1)(n – 2)(n – 3) x n –4 Dxn −1 ( x n ) = n(n –1)(n – 2)(n – 3)...(2) x Dxn ( x n ) = n(n –1)(n – 2)(n – 3)...2(1) x 0 = n! 124 = Section 2.6 18. Let k < n. Dxn ( x k ) = Dxn − k [ Dxk ( x k )] = Dx (k !) = 0 so Dxn [an x n –1 + 19. a. + a1 x + a0 ] = 0 Dx4 (3x3 + 2 x –19) = 0 b. 11 10 D12 x (100 x − 79 x ) = 0 c. 2 5 D11 x ( x – 3) = 0 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 ⎛1⎞ 20. Dx ⎜ ⎟ = – ⎝ x⎠ x2 2 ⎛1⎞ Dx2 ⎜ ⎟ = Dx (– x –2 ) = 2 x –3 = x ⎝ ⎠ x3 3(2) ⎛1⎞ Dx3 ⎜ ⎟ = Dx (2 x –3 ) = – ⎝x⎠ x4 ⎛ 1 ⎞ 4(3)(2) Dx4 ⎜ ⎟ = ⎝ x⎠ x5 n ⎛ 1 ⎞ (−1) n ! Dxn ⎜ ⎟ = ⎝x⎠ x n +1 21. f ( x) = 3 x 2 + 6 x – 45 = 3( x + 5)( x − 3) 3(x + 5)(x – 3) = 0 x = –5, x = 3 f ( x) = 6 x + 6 f (–5) = –24 f (3) = 24 22. g (t ) = 2at + b g (t ) = 2a g (1) = 2a = −4 a = −2 g (1) = 2a + b = 3 2(–2) + b = 3 b=7 g (1) = a + b + c = 5 ( −2 ) + ( 7 ) + c = 5 c=0 23. a. v(t ) = a(t ) = b. 3t 2 – 12t > 0 3t(t – 4) > 0; (−∞, 0) ∪ (4, ∞) c. 3t 2 – 12t < 0 (0, 4) d. 6t – 12 < 0 6t < 12 t < 2; (−∞, 2) e. 25. a. a(t ) = d 2s dt 2 = –4 e. a(t ) = dt 2 = 6t – 18 3t 2 –18t + 24 > 0 3(t – 2)(t – 4) > 0 (−∞, 2) ∪ (4, ∞) c. 3t 2 –18t + 24 < 0 (2, 4) d. 6t – 18 < 0 6t < 18 t < 3; (−∞,3) ds = 12 – 4t dt 12 – 4t < 0 t > 3; (3, ∞) v(t ) = d 2s e. d. a(t) = –4 < 0 for all t 24. a. ds = 3t 2 –18t + 24 dt b. 26. a. b. 12 – 4t > 0 4t < 12 t < 3; ( −∞,3) c. v(t ) = ds = 3t 2 –12t dt d 2s dt 2 v(t ) = a(t ) = ds = 6t 2 – 6 dt d 2s dt 2 = 12t b. 6t 2 – 6 > 0 6(t + 1)(t – 1) > 0 (−∞, −1) ∪ (1, ∞) c. 6t 2 – 6 < 0 (–1, 1) d. 12t < 0 t<0 The acceleration is negative for negative t. = 6t –12 Instructor’s Resource Manual Section 2.6 125 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. e. 29. v(t ) = ds = 2t 3 –15t 2 + 24t dt d 2s a(t ) = 27. a. ds 16 = 2t – dt t2 v(t ) = d 2s a(t ) = b. dt 2 = 2+ 32 t3 16 >0 t2 2t 3 – 16 > 0; (2, ∞) t2 2t – 16 < 0; (0, 2) c. 2t – d. <0 t3 2t3 + 32 < 0; The acceleration is not t3 negative for any positive t. t2 32 2+ e. v(t ) = a(t ) = 30. v(t ) = b. 1– 4 t2 t2 – 4 t2 c. d. 1– 8 3 4 t 2 dt 2 = < 0; (0, 2) < 0; The acceleration is not negative for t any positive t. e. 2 = 1 (12t 2 – 84t + 120) 10 ds1 = 4 – 6t dt ds v2 (t ) = 2 = 2t – 2 dt 31. v1 (t ) = b. 4 – 6t = 2t – 2 8t = 6 3 t = sec 4 4 – 6t = 2t – 2 ; 4 – 6t = –2t + 2 t= >0 > 0; (2, ∞) d 2s dt 1 (12t 2 – 84t + 120) = 0 10 12 (t − 2)(t − 5) = 0 10 t = 2, t = 5 v(2) = 10.4, v(5) = 5 8 t3 ds 1 = (4t 3 – 42t 2 + 120t ) dt 10 a(t ) = ds 4 =1– dt t2 d 2s = 6t 2 – 30t + 24 6t 2 – 30t + 24 = 0 6(t – 4)(t – 1) = 0 t = 4, 1 v(4) = –16, v(1) = 11 a. 28. a. dt 2 c. 1 3 sec and t = sec 2 4 4t – 3t 2 = t 2 – 2t 4t 2 – 6t = 0 2t(2t – 3) = 0 t = 0 sec and t = 3 sec 2 ds1 = 9t 2 – 24t + 18 dt ds v2 (t ) = 2 = –3t 2 + 18t –12 dt 32. v1 (t ) = 9t 2 – 24t + 18 = –3t 2 + 18t –12 12t 2 – 42t + 30 = 0 2t 2 – 7t + 5 = 0 (2t – 5)(t – 1) = 0 5 t = 1, 2 126 Section 2.6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 33. a. v(t) = –32t + 48 initial velocity = v0 = 48 ft/sec b. –32t + 48 = 0 3 t = sec 2 c. d. s = –16(1.5) 2 + 48(1.5) + 256 = 292 ft 2 –16t + 48t + 256 = 0 2 –48 ± 48 – 4(–16)(256) ≈ –2.77, 5.77 –32 The object hits the ground at t = 5.77 sec. t= e. v(5.77) ≈ –137 ft/sec; speed = −137 = 137 ft/sec. 48 – 32t = 0 t = 1.5 b. v(1) = 16 ft/sec upward 2 48t –16t = 0 –16t(–3 + t) = 0 t = 3 sec 35. v(t ) = v0 – 32t v0 – 32t = 0 t= (t – 4)(t + 2) (6t – 6) (t – 4)(t + 2) v0 32 2 ⎛v ⎞ ⎛v ⎞ v0 ⎜ 0 ⎟ –16 ⎜ 0 ⎟ = 5280 ⎝ 32 ⎠ ⎝ 32 ⎠ v02 v0 2 – = 5280 32 64 v02 = 5280 64 v0 = 337,920 ≈ 581 ft/sec 36. v(t ) = v0 + 32t v0 + 32t = 140 v0 + 32(3) = 140 v0 = 44 <0 t < –2, 1 < t < 4; (−∞, −2) ∪ (1, 4) 38. Point slowing down when d v(t ) < 0 dt v(t ) a (t ) d v(t ) = dt v(t ) v (t ) signs. s = 48(1.5) –16(1.5)2 = 36 ft c. 3t 2 – 6t – 24 d 2 3t – 6t – 24 = (6t – 6) dt 3t 2 – 6t – 24 (t – 4)(t + 2) = (6t – 6) (t – 4)(t + 2) v (t ) a(t ) 34. v(t) = 48 –32t a. 37. v(t ) = 3t 2 – 6t – 24 < 0 when a(t) and v(t) have opposite 39. Dx (uv) = uv + u v Dx2 (uv) = uv + u v + u v + u v = uv + 2u v + u v Dx3 (uv) = uv + u v + 2(u v + u v ) + u v + u v = uv + 3u v + 3u v + u v n n ⎛ ⎞ n−k k Dxn (uv) = ⎜ ⎟ Dx (u ) Dx (v) k ⎝ ⎠ k =0 ⎛ n⎞ where ⎜ ⎟ is the binomial coefficient ⎝ k⎠ n! . (n – k )!k ! ⎛ 4⎞ 40. Dx4 ( x 4 sin x ) = ⎜ ⎟ Dx4 ( x 4 ) Dx0 (sin x) ⎝0⎠ ⎛ 4⎞ ⎛ 4⎞ + ⎜ ⎟ Dx3 ( x 4 ) D1x (sin x) + ⎜ ⎟ Dx2 ( x 4 ) Dx2 (sin x) ⎝1⎠ ⎝ 2⎠ ⎛ 4⎞ ⎛ 4⎞ + ⎜ ⎟ D1x ( x 4 ) Dx3 (sin x) + ⎜ ⎟ Dx0 ( x 4 ) Dx4 (sin x) ⎝ 3⎠ ⎝ 4⎠ = 24sin x + 96 x cos x − 72 x 2 sin x −16 x3 cos x + x 4 sin x s = 44(3) + 16(3) 2 = 276 ft Instructor’s Resource Manual Section 2.6 127 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 41. a. 4. 2 x + 2 2 yDx y = 0 Dx y = – 2x 2 2 y =– x 2 y 5. x(2 y ) Dx y + y 2 = 1 Dx y = f (2.13) ≈ –1. 2826 b. 1 – y2 2 xy 6. 2 x + 2 x 2 Dx y + 4 xy + 3 x Dx y + 3 y = 0 Dx y (2 x 2 + 3 x) = –2 x – 4 xy – 3 y 42. a. Dx y = –2 x – 4 xy – 3 y 2 x2 + 3x 7. 12 x 2 + 7 x(2 y ) Dx y + 7 y 2 = 6 y 2 Dx y 12 x 2 + 7 y 2 = 6 y 2 Dx y – 14 xyDx y f (2.13) ≈ 0.0271 b. 2.7 Concepts Review 1. Dx y = x –3 3. x(2 y ) 9. dy dy dy + y2 + 3y2 – = 3x2 dx dx dx p p q –1 5 2 x ; ( x – 5 x)2 / 3 (2 x – 5) q 3 6 y 2 – 14 xy x 2 Dx y – 2 xyDx y = y 2 – 2 xy 9 dy dx 12 x 2 + 7 y 2 8. x 2 Dx y + 2 xy = y 2 + x(2 y ) Dx y 3 2. 3 y 2 4. Dx y = 1 2 5 xy y 2 – 2 xy x 2 – 2 xy ⋅ (5 x Dx y + 5 y ) + 2 Dx y = 2 y Dx y + x(3 y 2 ) Dx y + y 3 5x 2 5 xy = y3 – Dx y + 2 Dx y – 2 y Dx y – 3 xy 2 Dx y 5y 2 5 xy Problem Set 2.7 1. 2 y Dx y – 2 x = 0 Dx y = –18 x 9x =– 8y 4y 3. x Dx y + y = 0 y Dx y = – x 128 Dx y = 2x x = 2y y 2. 18 x + 8 y Dx y = 0 Dx y = y3 – Section 2.7 10. x 5x 2 5 xy 1 2 y +1 x 2 y +1 Dx y = 5y 2 5 xy + 2 – 2 y – 3 xy 2 Dx y + y + 1 = x Dx y + y Dx y – x Dx y = y – y + 1 y – y +1 x 2 y +1 –x Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 11. x Dx y + y + cos( xy )( x Dx y + y ) = 0 x Dx y + x cos( xy ) Dx y = – y – y cos( xy ) Dx y = 17. – y – y cos( xy ) y =– x + x cos( xy ) x 2 –1/ 3 2 –1/ 3 x – y y – 2y = 0 3 3 2 –1/ 3 ⎛2 ⎞ = y ⎜ y –1/ 3 + 2 ⎟ x 3 ⎝3 ⎠ y = 12. – sin( xy 2 )(2 xy Dx y + y 2 ) = 2 yDx y + 1 –2 xy sin( xy 2 ) Dx y – 2 y Dx y = 1 + y 2 sin( xy 2 ) Dx y = 3 2 2 At (1, –1), y = 1 + y sin( xy ) –2 xy sin( xy 2 ) – 2 y 2 3 y ( x3 + 3xy 2 ) = –3 x 2 y – y 3 2 –3 x y – y 3 18. x3 + 3 xy 2 36 9 =– 28 7 9 Tangent line: y – 3 = – ( x – 1) 7 y = – y cos( xy ) y cos( xy ) = x cos( xy ) – 1 1 – x cos( xy ) ⎛π ⎞ At ⎜ , 1⎟ , y = 0 ⎝2 ⎠ π⎞ ⎛ Tangent line: y – 1 = 0 ⎜ x – ⎟ 2⎠ ⎝ y=1 – y2 1 + 2 xy –1 17 2 =– 2 17 2 ( x – 4) 17 dy 1 = 5x2 / 3 + dx 2 x 20. dy 1 –2 / 3 1 = x – 7 x5 / 2 = – 7 x5 / 2 3 2 dx 3 3 x 21. 1 1 dy 1 –2 / 3 1 –4 / 3 = x = – x – 3 3 3 dx 3 3 x2 3 x4 22. dy 1 1 = (2 x + 1) –3 / 4 (2) = 4 dx 4 2 (2 x + 1)3 23. y 2 sin( xy 2 ) – 6 x 6 = –6 1 Tangent line: y – 0 = –6(x – 1) y + 2 xyy + y 2 = 0 19. y [1 – 2 xy sin( xy 2 )] = y 2 sin( xy 2 ) – 6 x 1 – 2 xy sin( xy 2 ) 1 ( x –1) 2 Tangent line: y –1 = – 16. y + [– sin( xy 2 )][2 xyy + y 2 ] + 6 x = 0 y = 2 y At (4, 1), y = – xy 2 – 2 y 15. cos( xy )( xy + y ) = y y [ x cos( xy ) – 1] = – y cos( xy ) 1 2 2 y y (2 x 2 y + 4 x – 12) = –2 xy 2 – 4 y = 2 x 2 y + 4 x – 12 x 2 y + 2 x – 6 At (2, 1), y = –2 Tangent line: y – 1 = –2( x – 2) 1 y = 14. x 2 (2 y ) y + 2 xy 2 + 4 xy + 4 y = 12 y y = = ⎛ 1 ⎞ + 2 xy ⎟ = – y 2 y⎜ ⎜2 y ⎟ ⎝ ⎠ At (1, 3), y = – –2 xy 2 – 4 y 2 3 4 3 Tangent line: y + 1 = 2 13. x y + 3 x y + y + 3xy y = 0 y = 2 x –1/ 3 3 2 y –1/ 3 + 2 3 dy 1 = (3 x 2 – 4 x) –3 / 4 (6 x – 4) dx 4 6x – 4 3x – 2 = = 2 3 4 4 4 (3 x – 4 x) 2 (3 x 2 – 4 x)3 24. dy 1 3 = ( x – 2 x) –2 / 3 (3 x 2 – 2) dx 3 25. dy d = [( x3 + 2 x)−2 / 3 ] dx dx At (1, 0), y = – 2 6 x2 + 4 = – ( x3 + 2 x) –5 / 3 (3 x 2 + 2) = − 3 3 3 ( x 3 + 2 x )5 Instructor’s Resource Manual Section 2.7 129 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 26. 5 dy = – (3 x – 9) –8 / 3 (3) = –5(3 x – 9) –8 / 3 3 dx dy 1 27. (2 x + cos x) = dx 2 x 2 + sin x 2 x + cos x = 2 x 2 + sin x 28. 31. y (x + 2)2 + y2 = 1 5 x − −5 dy d = [( x 2 sin x) –1/ 3 ] dx dx 1 = – ( x 2 sin x) –4 / 3 ( x 2 cos x + 2 x sin x) 3 x 2 cos x + 2 x sin x 2 33 ( x sin x) dy =0 dx dy 2x + 4 x+2 =− =− dx 2y y 2x + 4 + 2 y The tangent line at ( x0 , y0 ) has equation x +2 y – y0 = − 0 ( x – x0 ) which simplifies to y0 4 dy 1 = (1 + sin 5 x) –3 / 4 (cos 5 x)(5) dx 4 5cos 5 x = 4 4 (1 + sin 5 x)3 ( x0 , y0 ) is on the circle, x0 2 + y02 = –3 – 4 x0 , so the equation of the tangent line is – yy0 – 2 x0 – 2 x – xx0 = 3. dy [1 + cos( x 2 + 2 x)]–3 / 4 [– sin( x 2 + 2 x)(2 x + 2)] = dx 4 3 If (0, 0) is on the tangent line, then x0 = – . 2 Solve for y0 in the equation of the circle to get ( x + 1) sin( x 2 + 2 x) 2 4 3 2 [1 + cos( x + 2 x )] dy (tan 2 x + sin 2 x) –1/ 2 (2 tan x sec 2 x + 2 sin x cos x) = dx 2 = tan x sec 2 x + sin x cos x 33. s 2 + 2 st 2 ds – s – 3t s + 3t = =− dt 2 st 2 st dt dt s 2 + 2st + 3t 2 =0 ds ds dt 2 ( s + 3t 2 ) = –2 st ds dt 2 st =− 2 ds s + 3t 2 Section 2.7 3 . Put these values into the equation of 2 the tangent line to get that the tangent lines are 3 y + x = 0 and 3 y – x = 0. y0 = ± 36. 16( x 2 + y 2 )(2 x + 2 yy ) = 100(2 x – 2 yy ) y (4 x 2 y + 4 y 3 + 25 y ) = 25 x – 4 x3 – 4 xy 2 ds + 3t 2 = 0 dt 2 2 x0 – yy0 – 2 x – xx0 + y02 + x02 = 0. Since 32 x3 + 32 x 2 yy + 32 xy 2 + 32 y 3 y = 200 x – 200 yy tan 2 x + sin 2 x 2 130 5 2 x 2 cos x =− 32. 35. 2 x cos x – x 2 sin x =– 30. dx dx + 6x2 dy dy dx 1 = dy 2 x cos( x 2 ) + 6 x 2 dy 1 [ x 2 (– sin x) + 2 x cos x] = dx 2 x 2 cos x = 29. 34. 1 = cos( x 2 )(2 x ) 2 y = 25 x – 4 x3 – 4 xy 2 4 x 2 y + 4 y 3 + 25 y The slope of the normal line = – = 1 y 4 x 2 y + 4 y 3 + 25 y 4 x3 + 4 xy 2 – 25 x 65 13 = 45 9 13 Normal line: y – 1 = ( x – 3) 9 At (3, 1), slope = Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. xy + y + 3 y 2 y = 0 37. a. y (6 y 2 – x 2 ) = y (x + 3y2 ) = – y y =– y y = x + 3y2 ⎛ –y xy + ⎜ ⎜ x + 3y2 ⎝ b. ⎞ ⎛ –y ⎞ 2 ⎟⎟ + ⎜⎜ ⎟ + 3y y 2⎟ ⎠ ⎝ x + 3y ⎠ ⎛ –y +6 y ⎜ ⎜ x + 3 y2 ⎝ ⎞ ⎟⎟ = 0 ⎠ 2y 6 y3 xy + 3 y 2 y – + x + 3 y2 2y y (x + 3y2 ) = y (x + 3y2 ) = y = 2 x + 3y2 2 xy – ( x + 3 y 2 )2 6 y3 ( x + 3 y 2 )2 2 2 (x + 3y ) 2 xy ( x + 3 y 2 )3 2x x =– y 2y 2 + 2[ yy + ( y )2 ] = 0 2 ⎛ x⎞ 2 + 2 yy + 2 ⎜ – ⎟ = 0 ⎝ y⎠ 2 yy = −2 − 2 x2 y2 41. 3x 2 + 3 y 2 y = 3( xy + y ) y (3 y 2 – 3x) = 3 y – 3 x 2 6 x – 8( yy + ( y )2 ) = 0 ⎛ 3x2 6 x – 8 yy – 8 ⎜ ⎜ 8y ⎝ 6 x – 8 yy – 9 x4 8 y2 48 xy 2 − 9 x 4 8 y2 y = 2 ⎞ ⎟ =0 ⎟ ⎠ y – x2 y2 – x ⎛3 3⎞ At ⎜ , ⎟ , y = –1 ⎝2 2⎠ Slope of the normal line is 1. 3 3⎞ ⎛ Normal line: y – = 1⎜ x – ⎟ ; y = x 2 2⎠ ⎝ This line includes the point (0, 0). =0 = 8 yy 48 xy 2 – 9 x 4 42. xy + y = 0 64 y 3 y x 2 x − 2 yy = 0 y =– 39. 2( x 2 y + 2 xy ) – 12 y 2 y = 0 2 x 2 y – 12 y 2 y = –4 xy y = 40. 2 x + 2 yy = 0 1 x2 y 2 + x2 − =− y y3 y3 25 At (3, 4), y = − 64 3x2 8y y = (6 y 2 – x 2 )3 −120 At (2, 1), y = = −15 8 y =− 38. 3x 2 – 8 yy = 0 y = (6 y 2 – x 2 ) 2 72 y 5 − 6 x 4 y − 24 x 2 y 3 y =– =0 72 y 5 − 6 x 4 y − 24 x 2 y 3 x y The slopes of the tangents are negative reciprocals, so the hyperbolas intersect at right angles. y = 2 xy 6 y2 – x2 2( x 2 y + 2 xy + 2 xy + 2 y ) – 12[ y 2 y + 2 y ( y ) 2 ] = 0 2 x 2 y − 12 y 2 y = −8 xy − 4 y + 24 y ( y )2 y (2 x 2 – 12 y 2 ) = − y (2 x 2 – 12 y 2 ) = 16 x 2 y 6 y2 – x2 – 4y + 96 x 2 y3 (6 y 2 − x 2 )2 12 x 4 y + 48 x 2 y 3 − 144 y 5 (6 y 2 – x 2 ) 2 Instructor’s Resource Manual Section 2.7 131 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 43. Implicitly differentiate the first equation. 4 x + 2 yy = 0 45. x 2 – x(2 x) + 2(2 x) 2 = 28 7 x 2 = 28 2x y =– y Implicitly differentiate the second equation. 2 yy = 4 x2 = 4 x = –2, 2 Intersection point in first quadrant: (2, 4) y1 = 2 2 x – xy2 – y + 4 yy2 = 0 y2 (4 y – x) = y – 2 x 2 y Solve for the points of intersection. y = y2 = 2 x2 + 4 x = 6 2( x 2 + 2 x – 3) = 0 (x + 3)(x – 1) = 0 x = –3, x = 1 x = –3 is extraneous, and y = –2, 2 when x = 1. The graphs intersect at (1, –2) and (1, 2). At (1, –2): m1 = 1, m2 = –1 At (1, 2): m1 = –1, m2 = 1 44. Find the intersection points: x2 + y 2 = 1 → y 2 = 1 − x2 ( x − 1)2 + y 2 = 1 ( x − 1)2 + (1 − x 2 ) = 1 x2 − 2 x + 1 + 1 − x2 = 1 ⇒ x= 1 2 x y Implicitly differentiate the second equation. 2( x –1) + 2 yy = 0 tan = 0–2 = –2; = π + tan –1 (–2) ≈ 2.034 1 + (0)(2) 46. The equation is mv 2 – mv02 = kx02 – kx 2 . Differentiate implicitly with respect to t to get dx dv dx 2mv = –2kx . Since v = this simplifies dt dt dt dv dv to 2mv = –2kxv or m = – kx. dt dt ( )( ) = y – 2x 2y – x At (–4, 0), y = 2 At (4, 0), y = 2 Tangent lines: y = 2(x + 4) and y = 2(x – 4) 1– x y ⎛1 3⎞ 1 1 At ⎜⎜ , ⎟⎟ : m1 = – 3 , m2 = 3 2 2 ⎝ ⎠ 1 + 1 2 3 3 tan = = 3 = 3 → 2 1+ 1 − 1 3 x 2 = 16 x = –4, 4 The ellipse intersects the x-axis at (–4, 0) and (4, 0). 2 x – xy – y + 2 yy = 0 y (2 y – x) = y – 2 x y = y =– 3 At (2, 4): m1 = 2, m2 = 0 47. x 2 – xy + y 2 = 16 , when y = 0, ⎛1 3⎞ ⎛1 3⎞ Points of intersection: ⎜⎜ , ⎟⎟ and ⎜⎜ , – ⎟ 2 ⎟⎠ ⎝2 2 ⎠ ⎝2 Implicitly differentiate the first equation. 2 x + 2 yy = 0 y = y – 2x 4y – x π 3 3 ⎛1 3⎞ 1 1 , m2 = – At ⎜⎜ , – ⎟⎟ : m1 = 2 ⎠ 3 3 ⎝2 1 1 2 − − − 3 3 3 tan = = =− 3 2 1 1 1+ – 3 ( )( ) 3 = 132 3 2π 3 Section 2.7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 48. x 2 + 2 xy Problem Set 2.8 dx dx – 2 xy – y 2 =0 dy dy dx 2 xy – x 2 = dy 2 xy – y 2 2 xy – x 2 2 xy – y 2 = 0 if x(2y – x) = 0, which occurs x when x = 0 or y = . There are no points on 2 x 2 2 x y – xy = 2 where x = 0. If y = , then 2 2 x⎞ x3 x3 x3 ⎛ x⎞ = 2 = x ⎜ ⎟ – x⎜ ⎟ = – so x = 2, 2 4 4 ⎝2⎠ ⎝2⎠ 2 y = = 1. 2 The tangent line is vertical at (2, 1). 2⎛ 49. 2 x + 2 y dy dy x = 0; =– dx dx y The tangent line at ( x0 , y0 ) has slope – dx =3 dt dV dx = 3x2 dt dt dV When x = 12, = 3(12)2 (3) = 1296 in.3/s. dt 1. V = x3 ; dx (2 xy – y 2 ) = 2 xy – x 2 ; dy x0 , y0 hence the equation of the tangent line is x y – y0 = – 0 ( x – x0 ) which simplifies to y0 yy0 + xx0 – ( x02 + y02 ) = 0 or yy0 + xx0 = 1 since ( x0 , y0 ) is on x 2 + y 2 = 1 . If (1.25, 0) is on the tangent line through ( x0 , y0 ) , x0 = 0.8. Put this into x 2 + y 2 = 1 to get y0 = 0.6, since y0 > 0. The line is 6y + 8x = 10. When x = –2, 13 13 y = , so the light bulb must be units high. 3 3 4 3 dV πr ; =3 3 dt dV dr = 4πr 2 dt dt 2. V = When r = 3, 3 = 4π(3)2 dr 1 = ≈ 0.027 in./s dt 12π dx = 400 dt dy dx 2y = 2x dt dt dy x dx = mi/hr dt y dt 3. y 2 = x 2 + 12 ; When x = 5, y = 26, 1. du ;t = 2 dt 2. 400 mi/hr 3. negative 4. negative; positive Instructor’s Resource Manual dy 5 = (400) dt 26 ≈ 392 mi/h. 1 r 3 3h 4. V = πr 2 h; = ; r = 3 h 10 10 2 1 ⎛ 3h ⎞ 3πh3 dV V = π⎜ ⎟ h = ; = 3, h = 5 3 ⎝ 10 ⎠ 100 dt dV 9πh 2 dh = dt 100 dt When h = 5, 3 = 2.8 Concepts Review dr . dt 9π(5)2 dh 100 dt dh 4 = ≈ 0.42 cm/s dt 3π dx dy = 300, = 400, dt dt ds dx dy 2s = 2( x + 300) + 2 y dt dt dt ds dx dy s = ( x + 300) + y dt dt dt 5. s 2 = ( x + 300)2 + y 2 ; When x = 300, y = 400, s = 200 13 , so ds 200 13 = (300 + 300)(300) + 400(400) dt ds ≈ 471 mi/h dt Section 2.8 133 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6. y 2 = x 2 + (10)2 ; dy =2 dt dy dx = 2x dt dt When y = 25, x ≈ 22.9, so dx y dy 25 = ≈ (2) ≈ 2.18 ft/s dt x dt 22.9 2y dx =1 dt dx dy 0 = 2x + 2 y dt dt 7. 202 = x 2 + y 2 ; When x = 5, y = 375 = 5 15 , so dy x dx 5 =– =– (1) ≈ –0.258 ft/s dt y dt 5 15 The top of the ladder is moving down at 0.258 ft/s. 8. dV dh = –4 ft3/h; V = πhr 2 ; = –0.0005 ft/h dt dt V dA dV V dh A = πr 2 = = Vh –1 , so . – = h –1 h dt dt h 2 dt When h = 0.001 ft, V = π(0.001)(250) 2 = 62.5π dA = 1000(–4) –1, 000, 000(62.5π)(–0.0005) dt = –4000 + 31,250 π ≈ 94,175 ft2/h. (The height is decreasing due to the spreading of the oil rather than the bacteria.) and 1 d r 9. V = πr 2 h; h = = , r = 2h 3 4 2 1 4 dV V = π(2h) 2 h = πh3 ; = 16 3 3 dt dV dh = 4πh 2 dt dt dh When h = 4, 16 = 4π(4) 2 dt dh 1 = ≈ 0.0796 ft/s dt 4π 10. y 2 = x 2 + (90)2 ; hx 40 x (20); = , x = 8h 2 5 h dV V = 10h(8h) = 80h 2 ; = 40 dt dV dh = 160h dt dt dh When h = 3, 40 = 160(3) dt dh 1 = ft/min dt 12 11. V = 12. y = x 2 – 4; dx =5 dt dy 1 dx x dx = (2 x ) = dt 2 x 2 – 4 dt x 2 – 4 dt dy 3 15 When x = 3, = ≈ 6.7 units/s (5) = 2 dt 5 3 –4 dr = 0.02 dt dA dr = 2πr dt dt dA When r = 8.1, = 2π(0.02)(8.1) = 0.324π dt ≈ 1.018 in.2/s 13. A = πr 2 ; dx dy = 30, = 24 dt dt ds dx dy 2s = 2 x + 2( y + 48) dt dt dt ds dx dy s = x + ( y + 48) dt dt dt At 2:00 p.m., x = 3(30) = 90, y = 3(24) = 72, so s = 150. ds (150) = 90(30) + (72 + 48)(24) dt ds 5580 = = 37.2 knots/h dt 150 14. s 2 = x 2 + ( y + 48) 2 ; dx =5 dt dy dx = 2x dt dt When y = 150, x = 120, so dy x dx 120 = = (5) = 4 ft/s dt y dt 150 2y 134 Section 2.8 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 15. Let x be the distance from the beam to the point opposite the lighthouse and be the angle between the beam and the line from the lighthouse to the point opposite. x d tan = ; = 2(2π) = 4π rad/min, 1 dt d dx sec2 = dt dt 1 1 5 At x = , = tan –1 and sec2 = . 2 2 4 dx 5 = (4π) ≈ 15.71 km/min dt 4 4000 x d 4000 dx sec2 =− dt x 2 dt 1 d 1 4000 and x = = ≈ 7322. When = , 2 dt 10 tan 12 16. tan = dx ≈ sec 2 dt 1 ⎛ 1 ⎞ ⎡ (7322) 2 ⎤ ⎥ ⎜ ⎟ ⎢− 2 ⎝ 10 ⎠ ⎣⎢ 4000 ⎦⎥ ≈ –1740 ft/s or –1186 mi/h The plane’s ground speed is 1186 mi/h. 17. a. Let x be the distance along the ground from the light pole to Chris, and let s be the distance from Chris to the tip of his shadow. 6 30 x By similar triangles, = , so s = s x+s 4 ds 1 dx dx and = . = 2 ft/s, hence dt 4 dt dt ds 1 = ft/s no matter how far from the light dt 2 pole Chris is. b. Let l = x + s, then dl dx ds 1 5 = + = 2 + = ft/s. dt dt dt 2 2 c. The angular rate at which Chris must lift his head to follow his shadow is the same as the rate at which the angle that the light makes with the ground is decreasing. Let be the angle that the light makes with the ground at the tip of Chris' shadow. 6 d 6 ds tan = so sec2 =– and s dt s 2 dt 6 cos 2 d =– dt s2 When s = 6, 6 ( ) 1 2 2 2 1 d ⎛1⎞ =– ⎜ ⎟=– . 2 24 dt ⎝ ⎠ 6 Chris must lift his head at the rate of 1 rad/s. 24 18. Let be the measure of the vertex angle, a be the measure of the equal sides, and b be the measure of the base. Observe that b = 2a sin 2 and the height of the triangle is a cos . 2 1⎛ ⎞⎛ ⎞ 1 A = ⎜ 2a sin ⎟ ⎜ a cos ⎟ = a 2 sin 2⎝ 2 ⎠⎝ 2⎠ 2 d 1 1 A = (100)2 sin = 5000sin ; = dt 10 2 dA d = 5000 cos dt dt π dA π ⎞⎛ 1 ⎞ ⎛ When = , = 5000 ⎜ cos ⎟ ⎜ ⎟ = 250 3 6 dt 6 ⎠ ⎝ 10 ⎠ ⎝ ≈ 433 cm 2 min . 19. Let p be the point on the bridge directly above the railroad tracks. If a is the distance between p da and the automobile, then = 66 ft/s. If l is the dt distance between the train and the point directly dl = 88 ft/s. The distance from the below p, then dt train to p is 1002 + l 2 , while the distance from p to the automobile is a. The distance between the train and automobile is 2 D = a 2 + ⎛⎜ 1002 + l 2 ⎞⎟ = a 2 + l 2 + 1002 . ⎝ ⎠ dD 1 dl ⎞ ⎛ da = ⋅ ⎜ 2a + 2l ⎟ dt 2 a 2 + l 2 + 1002 ⎝ dt dt ⎠ = a da + l dl dt dt . After 10 seconds, a = 660 a 2 + l 2 + 1002 and l = 880, so dD 660(66) + 880(88) = ≈ 110 ft/s. dt 6602 + 8802 + 1002 ds ds 1 = ft/s . dt dt 2 π = , so 4 Instructor’s Resource Manual Section 2.8 135 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 h 20. V = πh ⋅ (a 2 + ab + b 2 ); a = 20, b = + 20, 3 4 2 ⎞ 1 ⎛ h V = πh ⎜ 400 + 5h + 400 + + 10h + 400 ⎟ ⎜ ⎟ 3 ⎝ 16 ⎠ 3 1 ⎛ h ⎞ = π ⎜1200h + 15h 2 + ⎟ ⎜ 3 ⎝ 16 ⎟⎠ dV 1 ⎛ 3h 2 ⎞ dh = π ⎜ 1200 + 30h + ⎟ dt 3 ⎜⎝ 16 ⎟⎠ dt dV = 2000, When h = 30 and dt 1 ⎛ 675 ⎞ dh 3025π dh 2000 = π ⎜1200 + 900 + = ⎟ 3 ⎝ 4 ⎠ dt 4 dt dh 320 = ≈ 0.84 cm/min. dt 121π ⎡ h ⎤ dV 21. V = πh 2 ⎢ r – ⎥ ; = –2, r = 8 ⎣ 3 ⎦ dt πh3 πh3 = 8πh 2 – 3 3 dV dh dh = 16πh – πh 2 dt dt dt dh When h = 3, –2 = [16π(3) – π(3)2 ] dt dh –2 = ≈ –0.016 ft/hr dt 39π V = πrh 2 – 22. s 2 = a 2 + b 2 − 2ab cos ; d π 11π rad/h = 2π – = a = 5, b = 4, dt 6 6 s 2 = 41 – 40 cos ds d 2s = 40sin dt dt π At 3:00, = and s = 41 , so 2 ds ⎛ π ⎞ ⎛ 11π ⎞ 220π 2 41 = 40sin ⎜ ⎟ ⎜ ⎟= dt 3 ⎝ 2 ⎠⎝ 6 ⎠ ds ≈ 18 in./hr dt 23. Let P be the point on the ground where the ball hits. Then the distance from P to the bottom of the light pole is 10 ft. Let s be the distance between P and the shadow of the ball. The height of the ball t seconds after it is dropped is 64 –16t 2 . 136 Section 2.8 By similar triangles, (for t > 1), so s = 48 64 – 16t 10t 2 – 40 1 – t2 2 = 10 + s s . ds 20t (1 – t 2 ) – (10t 2 – 40)(–2t ) 60t = =– 2 2 dt (1 – t ) (1 – t 2 )2 The ball hits the ground when t = 2, The shadow is moving ds 120 =– . dt 9 120 ≈ 13.33 ft/s. 9 h⎞ ⎛ 24. V = πh 2 ⎜ r – ⎟ ; r = 20 3⎠ ⎝ h⎞ π ⎛ V = πh 2 ⎜ 20 – ⎟ = 20πh 2 − h3 3⎠ 3 ⎝ dV dh = (40πh − πh 2 ) dt dt dh At 7:00 a.m., h = 15, ≈ −3, so dt dV = (40π(15) − π(15) 2 )(−3) ≈ −1125π ≈ −3534. dt Webster City residents used water at the rate of 2400 + 3534 = 5934 ft3/h. 25. Assuming that the tank is now in the shape of an upper hemisphere with radius r, we again let t be the number of hours past midnight and h be the height of the water at time t. The volume, V, of water in the tank at that time is given by 2 π V = π r 3 − ( r − h) 2 ( 2r + h ) 3 3 16000 π and so V = π − (20 − h)2 ( 40 + h ) 3 3 from which dV π dh 2π dh = − (20 − h)2 + (20 − h) ( 40 + h ) dt 3 dt 3 dt dV At t = 7 , ≈ −525π ≈ −1649 dt Thus Webster City residents were using water at the rate of 2400 + 1649 = 4049 cubic feet per hour at 7:00 A.M. 26. The amount of water used by Webster City can be found by: usage = beginning amount + added amount − remaining amount Thus the usage is ≈ π (20)2 (9) + 2400(12) − π (20)2 (10.5) ≈ 26,915 ft 3 over the 12 hour period. Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 27. a. dx = 2 ft/s. Let y dt y 18 216 = be the height of the opposite end of the ladder. By similar triangles, , so y = . 2 12 144 + x 144 + x 2 dy 216 dx 216 x dx =– 2x =– 2 3/ 2 2 3 / 2 dt dt dt 2(144 + x ) (144 + x ) Let x be the distance from the bottom of the wall to the end of the ladder on the ground, so When the ladder makes an angle of 60° with the ground, x = 4 3 and b. d2y dt 2 Since d2y dt 2 = = d ⎛ 216 x dx ⎞ d ⎛ 216 x ⎜⎜ – ⎟ = ⎜– dt ⎝ (144 + x 2 )3 / 2 dt ⎟⎠ dt ⎜⎝ (144 + x 2 )3 / 2 dy 216(4 3) =– ⋅ 2 = –1.125 ft/s. dt (144 + 48)3 / 2 ⎞ dx 216 x d2x ⋅ ⎟⎟ – 2 3/ 2 dt 2 ⎠ dt (144 + x ) dx d2x = 2, = 0, thus dt dt 2 ( ) ⎡ –216(144 + x 2 )3 / 2 dx + 216 x 3 2 dt =⎢ 2 3 ⎢ (144 + x ) ⎢⎣ –216(144 + x 2 ) + 648 x 2 ⎛ dx ⎞ ⎜ ⎟ ⎝ dt ⎠ (144 + x 2 )5 / 2 2 = ⎤ 144 + x 2 (2 x) dx dt ⎥ dx ⎥ dt ⎥⎦ 432 x 2 – 31,104 ⎛ dx ⎞ ⎜ ⎟ (144 + x 2 )5 / 2 ⎝ dt ⎠ 2 When the ladder makes an angle of 60° with the ground, d 2 y 432 ⋅ 48 – 31,104 2 = (2) ≈ –0.08 ft/s2 dt 2 (144 + 48)5 / 2 28. a. If the ball has radius 6 in., the volume of the water in the tank is V = 8πh 2 – πh3 4 ⎛ 1 ⎞ – π⎜ ⎟ 3 3 ⎝2⎠ 3 πh3 π – 3 6 dV dh dh = 16πh – πh 2 dt dt dt V= 4 3 πr 3 dV dr = 4πr 2 dt dt k (4πr 2 ) = 4πr 2 dh is dt again –0.016 ft/hr. b. If the ball has radius 2 ft, and the height of the water in the tank is h feet with 2 ≤ h ≤ 3 , the part of the ball in the water has volume 4 4 – h ⎤ (6 – h)h 2 π ⎡ π(2)3 – π(4 – h) 2 ⎢ 2 – = . 3 3 ⎥⎦ 3 ⎣ The volume of water in the tank is πh3 (6 – h)h 2 π V = 8πh 2 – – = 6h 2 π 3 3 dV dh = 12hπ dt dt dh 1 dV = dt 12hπ dt dh 1 When h = 3, = (–2) ≈ –0.018 ft/hr. dt 36π Instructor's Resource Manual dV = k (4πr 2 ) dt a. = 8πh 2 – This is the same as in Problem 21, so 29. dr dt dr =k dt b. If the original volume was V0 , the volume after 1 hour is was r0 = 3 8 V0 . The original radius 27 3 V0 while the radius after 1 4π 8 3 2 dr is V0 ⋅ = r0 . Since dt 27 4π 3 dr 1 constant, = – r0 unit/hr. The snowball dt 3 will take 3 hours to melt completely. hour is r1 = 3 Section 2.8 137 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 30. PV = k dV dP P +V =0 dt dt dP ≈ –30, V = 300 dt dV V dP 300 =– =– (–30) ≈ 134 in.3/min dt P dt 67 At t = 6.5, P ≈ 67, 31. Let l be the distance along the ground from the brother to the tip of the shadow. The shadow is 3 5 or controlled by both siblings when = l l+4 l = 6. Again using similar triangles, this occurs y 6 when = , so y = 40. Thus, the girl controls 20 3 the tip of the shadow when y ≥ 40 and the boy controls it when y < 40. Let x be the distance along the ground from the dx = –4 light pole to the girl. dt 4 20 5 When y ≥ 40, = or y = x. 3 y y–x 20 20 3 ( x + 4). = or y = 17 y y – ( x + 4) x = 30 when y = 40. Thus, ⎧ 4 if x ≥ 30 ⎪⎪ 3 x y=⎨ ⎪ 20 ( x + 4) if x < 30 ⎪⎩ 17 and ⎧ 4 dx if x ≥ 30 dy ⎪⎪ 3 dt =⎨ dt ⎪ 20 dx if x < 30 ⎪⎩ 17 dt Hence, the tip of the shadow is moving at the rate 4 16 ft/s when the girl is at least 30 feet of (4) = 3 3 from the light pole, and it is moving 20 80 ft/s when the girl is less than 30 ft (4) = 17 17 from the light pole. When y < 40, Problem Set 2.9 1. dy = (2x + 1)dx 2. dy = (21x 2 + 6 x)dx 3. dy = –4(2 x + 3) –5 (2)dx = –8(2 x + 3) –5 dx 4. dy = –2(3 x 2 + x + 1) –3 (6 x + 1)dx = –2(6 x + 1)(3x 2 + x + 1) –3 dx 5. dy = 3(sin x + cos x)2 (cos x – sin x)dx 6. dy = 3(tan x + 1) 2 (sec2 x)dx = 3sec2 x(tan x + 1)2 dx 3 7. dy = – (7 x 2 + 3x –1) –5 / 2 (14 x + 3)dx 2 3 = − (14 x + 3)(7 x 2 + 3 x − 1) −5 2 dx 2 8. dy = 2( x10 + sin 2 x )[10 x9 + 1 2 sin 2 x ⋅ (cos 2 x)(2)]dx ⎛ cos 2 x ⎞ 10 = 2 ⎜10 x9 + ⎟ ( x + sin 2 x )dx sin 2 x ⎠ ⎝ 9. ds = = 10. a. b. 3 2 (t – cot t + 2)1/ 2 (2t + csc 2 t )dt 2 3 (2t + csc2 t ) t 2 – cot t + 2dt 2 dy = 3 x 2 dx = 3(0.5)2 (1) = 0.75 dy = 3x 2 dx = 3(–1)2 (0.75) = 2.25 11. 2.9 Concepts Review 1. f ( x)dx 2. Δy; dy 12. a. dy = – b. dy = – 3. Δx is small. 4. larger ; smaller 138 Section 2.9 dx x 2 dx x 2 =– =– 0.5 (1)2 = –0.5 0.75 (–2)2 = –0.1875 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 13. 20. y = 3 x ; dy = 1 –2 / 3 1 x dx = dx; 3 2 3 3 x x = 27, dx = –0.09 1 dy = (–0.09) ≈ –0.0033 33 (27)2 3 26.91 ≈ 3 27 + dy = 3 – 0.0033 = 2.9967 21. V = 4 3 πr ; r = 5, dr = 0.125 3 dV = 4πr 2 dr = 4π(5)2 (0.125) ≈ 39.27 cm3 Δy = (1.5)3 – (0.5)3 = 3.25 14. a. Δy = (–0.25)3 – (–1)3 = 0.984375 b. Δy = 15. a. b. V≈ 2 Δy = [(2.88) – 3] – [(3) – 3] = –0.7056 dy = 2xdx = 2(3)(–0.12) = –0.72 Δy = [(3) 4 + 2(3)] – [(2)4 + 2(2)] = 67 17. a. dy = (4 x3 + 2)dx = [4(2)3 + 2](1) = 34 b. Δy = [(2.005)4 + 2(2.005)] – [(2)4 + 2(2)] ≈ 0.1706 dy = (4 x3 + 2)dx = [4(2)3 + 2](0.005) = 0.17 18. y = x ; dy = dy = 1 2 400 1 2 x dx; x = 400, dx = 2 (2) = 0.05 402 ≈ 400 + dy = 20 + 0.05 = 20.05 19. y = x ; dy = dy = 1 2 36 1 2 x dx; x = 36, dx = –0.1 (–0.1) ≈ –0.0083 35.9 ≈ 36 + dy = 6 – 0.0083 = 5.9917 Instructor’s Resource Manual 4 3 πr ; r = 6 ft = 72in., dr = –0.3 3 dV = 4πr 2 dr = 4π(72)2 (–0.3) ≈ –19,543 Δy = [(2.5) 2 – 3] – [(2) 2 – 3] = 2.25 dy = 2xdx = 2(2)(0.5) = 2 2 dV = 3 x 2 dx = 3( 3 40)2 (0.5) ≈ 17.54 in.3 23. V = 1 1 Δy = + = –0.3 –1.25 2 b. 16. a. 1 1 1 – =– 1.5 1 3 22. V = x3 ; x = 3 40, dx = 0.5 4 π(72)3 –19,543 3 ≈ 1,543,915 in 3 ≈ 893 ft 3 24. V = πr 2 h; r = 6 ft = 72in., dr = −0.05, h = 8ft = 96in. dV = 2πrhdr = 2π(72)(96)(−0.05) ≈ −2171in.3 About 9.4 gal of paint are needed. 25. C = 2π r ; r = 4000 mi = 21,120,000 ft, dr = 2 dC = 2π dr = 2π (2) = 4π ≈ 12.6 ft L ; L = 4, dL = –0.03 32 π 2π 1 dT = ⋅ ⋅ dL = dL 32 L 2 L 32 26. T = 2π 32 dT = π (–0.03) ≈ –0.0083 32(4) The time change in 24 hours is (0.0083)(60)(60)(24) ≈ 717 sec 27. V = 4 3 4 πr = π(10)3 ≈ 4189 3 3 dV = 4πr 2 dr = 4π(10) 2 (0.05) ≈ 62.8 The volume is 4189 ± 62.8 cm3. The absolute error is ≈ 62.8 while the relative error is 62.8 / 4189 ≈ 0.015 or 1.5% . Section 2.9 139 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 28. V = πr 2 h = π(3) 2 (12) ≈ 339 dV = 24πrdr = 24π(3)(0.0025) ≈ 0.565 The volume is 339 ± 0.565 in.3 The absolute error is ≈ 0.565 while the relative error is 0.565 / 339 ≈ 0.0017 or 0.17% . 29. s = a 2 + b 2 – 2ab cos = 1512 + 1512 – 2(151)(151) cos 0.53 ≈ 79.097 s = 45, 602 – 45, 602 cos ds = 1 2 45, 602 – 45, 602 cos 22,801sin = 45, 602 – 45, 602 cos d 22,801sin 0.53 = ⋅ 45, 602sin d 45, 602 – 45, 602 cos 0.53 (0.005) ≈ 0.729 s ≈ 79.097 ± 0.729 cm The absolute error is ≈ 0.729 while the relative error is 0.729 / 79.097 ≈ 0.0092 or 0.92% . 1 1 ab sin = (151)(151) sin 0.53 ≈ 5763.33 2 2 22,801 A= sin ; = 0.53, d = 0.005 2 22,801 dA = (cos )d 2 22,801 = (cos 0.53)(0.005) ≈ 49.18 2 A ≈ 5763.33 ± 49.18 cm2 The absolute error is ≈ 49.18 while the relative error is 49.18 / 5763.33 ≈ 0.0085 or 0.85% . 30. A = 31. y = 3 x 2 – 2 x + 11; x = 2, dx = 0.001 dy = (6x – 2)dx = [6(2) – 2](0.001) = 0.01 d2y = 6, so with Δx = 0.001, dx 2 1 Δy – dy ≤ (6)(0.001) 2 = 0.000003 2 32. Using the approximation f ( x + Δx) ≈ f ( x) + f '( x)Δx we let x = 1.02 and Δx = −0.02 . We can rewrite the above form as f ( x) ≈ f ( x + Δx) − f '( x)Δx which gives f (1.02) ≈ f (1) − f '(1.02)( −0.02) = 10 + 12(0.02) = 10.24 140 Section 2.9 33. Using the approximation f ( x + Δx) ≈ f ( x) + f '( x)Δx we let x = 3.05 and Δx = −0.05 . We can rewrite the above form as f ( x) ≈ f ( x + Δx) − f '( x)Δx which gives f (3.05) ≈ f (3) − f '(3.05)(−0.05) 1 = 8 + (0.05) = 8.0125 4 34. From similar triangles, the radius at height h is 2 1 4 h. Thus, V = πr 2 h = πh3 , so 5 3 75 4 dV = πh 2 dh. h = 10, dh = –1: 25 4 dV = π(100)(−1) ≈ −50 cm3 25 The ice cube has volume 33 = 27 cm3 , so there is room for the ice cube without the cup overflowing. 4 35. V = πr 2 h + πr 3 3 4 V = 100πr 2 + πr 3 ; r = 10, dr = 0.1 3 dV = (200πr + 4πr 2 )dr = (2000π + 400π)(0.1) = 240π ≈ 754 cm3 36. The percent increase in mass is m ⎛ v2 ⎞ dm = – 0 ⎜ 1 – ⎟ 2 ⎜⎝ c 2 ⎟⎠ m v ⎛ v2 = 0 ⎜1 – c 2 ⎜⎝ c 2 ⎞ ⎟ ⎟ ⎠ –3 / 2 dm . m ⎛ 2v ⎞ ⎜ – 2 ⎟ dv ⎝ c ⎠ –3 / 2 dv –1 dm v ⎛ v 2 ⎞ v ⎛ c2 ⎞ = 1 – ⎟ dv = ⎜ ⎜ ⎟ dv m c 2 ⎜⎝ c 2 ⎟⎠ c 2 ⎜⎝ c 2 − v 2 ⎟⎠ v = dv 2 c − v2 v = 0.9c, dv = 0.02c 0.9c 0.018 dm (0.02c) = = ≈ 0.095 0.19 m c 2 − 0.81c 2 The percent increase in mass is about 9.5. Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 37. f ( x) = x 2 ; f '( x) = 2 x; a = 2 The linear approximation is then L( x) = f (2) + f '(2)( x − 2) 39. h( x) = sin x; h '( x) = cos x; a = 0 The linear approximation is then L( x) = 0 + 1( x − 0) = x = 4 + 4( x − 2) = 4 x − 4 2 2 38. g ( x) = x cos x; g '( x) = − x sin x + 2 x cos x a =π /2 The linear approximation is then 40. F ( x ) = 3x + 4; F '( x) = 3; a = 3 The linear approximation is then L( x) = 13 + 3( x − 3) = 13 + 3 x − 9 = 3x + 4 2 π⎞ ⎛π ⎞ ⎛ L( x) = 0 + − ⎜ ⎟ ⎜ x − ⎟ 2⎠ ⎝2⎠ ⎝ =− π2 4 L( x) = 0 + − =− π2 4 x+ π3 8 2 π ⎛ π⎞ ⎜x− ⎟ 4 ⎝ 2⎠ x+ π3 8 Instructor’s Resource Manual Section 2.9 141 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 41. 45. f ( x ) = 1 − x2 ; ( ) −1/ 2 1 1 − x2 (−2 x) 2 −x = , a=0 1 − x2 The linear approximation is then L ( x ) = 1 + 0 ( x − 0) = 1 f ( x) = f (x ) = mx + b; f (x ) = m The linear approximation is then L(x ) = ma + b + m(x − a ) = am + b + mx − ma f ( x ) = L(x ) = mx + b 46. L ( x ) − f ( x ) = a + = x 2 a ( = ) 2 a 2 a ( x − a) − x a x−2 a x +a = 2 2 a − x+ x− a 1 2 ≥0 47. The linear approximation to f ( x ) at a is L( x) = f (a) + f '(a)( x − a) 42. g ( x ) = x 1 − x2 = a 2 + 2a ( x − a ) ; (1 − x ) − x ( −2 x ) = 1 + x g '( x) = (1 − x ) (1 − x ) 2 2 2 2 2 2 ,a = = 2ax − a 2 Thus, 1 2 ( f ( x) − L( x) = x 2 − 2ax − a 2 ) = x 2 − 2ax + a 2 The linear approximation is then 2 20 ⎛ 1 ⎞ 20 4 L(x ) = + x− ⎜x− ⎟ = 3 9 ⎝ 2⎠ 9 9 = ( x − a)2 ≥0 48. f (x ) = (1 + x ) , f (x ) = (1 + x ) −1 , a = 0 The linear approximation is then L(x ) = 1 + (x ) = x + 1 y 5 43. h(x ) = x sec x; h (x ) = sec x + x sec x tan x, a = 0 The linear approximation is then L(x ) = 0 + 1(x − 0) = x −5 5 x −5 = −2 y 44. G (x ) = x + sin 2 x; G (x ) = 1 + 2 cos 2 x , a = π / 2 The linear approximation is then π π⎞ ⎛ L(x ) = + (− 1)⎜ x − ⎟ = − x + π 2 2⎠ ⎝ 5 −5 5 −5 142 Section 2.9 x = −1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. y 49. a. lim ε ( h ) = lim ( f ( x + h ) − f ( x ) − f h→0 = f ( x) − f ( x) − f 5 −5 5 x −5 ( x) h) ( x) 0 = 0 ⎡ f ( x + h) − f ( x) = lim ⎢ −f h→0 h h ⎣ = f ( x) − f ( x) = 0 b. lim ε (h) ⎤ ( x )⎥ ⎦ = −0.5 y 2.10 Chapter Review Concepts Test 5 −5 5 1. False: If f ( x) = x3 , f '( x) = 3 x 2 and the tangent line y = 0 at x = 0 crosses the curve at the point of tangency. 2. False: The tangent line can touch the curve at infinitely many points. 3. True: mtan = 4 x3 , which is unique for each value of x. 4. False: mtan = – sin x, which is periodic. 5. True: If the velocity is negative and increasing, the speed is decreasing. 6. True: If the velocity is negative and decreasing, the speed is increasing. 7. True: If the tangent line is horizontal, the slope must be 0. x −5 =0 y 5 −5 5 x −5 = 0.5 y 8. False: 5 f ( x) = ax 2 + b, g ( x) = ax 2 + c, b ≠ c . Then f ( x) = 2ax = g ( x), but f(x) ≠ g(x). −5 5 9. True: x −5 =1 5 5 −5 Instructor’s Resource Manual Dx f ( g ( x)) = f ( g ( x)) g ( x); since g(x) = x, g ( x) = 1, so Dx f ( g ( x)) = f ( g ( x)). 10. False: Dx y = 0 because π is a constant, not a variable. 11. True: Theorem 3.2.A 12. True: The derivative does not exist when the tangent line is vertical. 13. False: ( f ⋅ g ) ( x) = f ( x) g ( x) + g ( x) f ( x) 14. True: Negative acceleration indicates decreasing velocity. y −5 h →0 x =2 Section 2.10 143 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 15. True: 29. True: If f ( x) = x3 g ( x), then Dx2 (sin x ) = – sin x; Dx f ( x) = x3 g ( x) + 3x 2 g ( x) Dx3 (sin x) = – cos x; = x 2 [ xg ( x) + 3 g ( x)]. 16. False: Dx4 (sin x) = sin x; Dx y = 3 x 2 ; At (1, 1): Dx5 (sin x) = cos x 2 mtan = 3(1) = 3 Tangent line: y – 1 = 3(x – 1) 17. False: 30. False: Dx3 (cos x) = sin x; Dx4 (cos x ) = Dx [ Dx3 (cos x)] = Dx (sin x) + g ( x) f ( x) + f ( x) g ( x) = f ( x) g ( x) + 2 f ( x) g ( x) + f ( x) g ( x) 3 Dxn +3 (cos x ) = Dxn (sin x). The degree of y = ( x + x) is 24, so 31. True: 19. True: f ( x) = ax n ; f ( x) = anx n –1 20. True: Dx 21. True: h ( x) = f ( x) g ( x) + g ( x) f ( x) h ( c ) = f (c ) g (c ) + g (c ) f (c ) = f(c)(0) + g(c)(0) = 0 22. True: Since D1x+3 (cos x) = D1x (sin x), 8 Dx25 y = 0. f ( x) g ( x) f ( x) – f ( x) g ( x) = g ( x) g 2 ( x) sin x – sin ⎛π⎞ f ⎜ ⎟ = lim x – π2 ⎝ 2 ⎠ x→ π 32. True: 33. True: x→ π 2 23. True: tan x 1 sin x = lim 3 x →0 x cos x x →0 3 x 1 1 = ⋅1 = 3 3 lim ds = 15t 2 + 6 which is greater dt than 0 for all t. v= V= 4 3 πr 3 dV dr = 4πr 2 dt dt dV dr 3 = = 3, then so If dt 4πr 2 dt dr > 0. dt ( π2 ) 2 = lim Dx (cos x ) = – sin x; Dx2 (cos x) = – cos x; Dx y = f ( x) g ( x ) + g ( x ) f ( x) Dx2 y = f ( x) g ( x) + g ( x) f ( x) 18. True: Dx (sin x ) = cos x; sin x –1 x – π2 d 2r D 2 (kf ) = kD 2 f and dt 2 D2 ( f + g ) = D2 f + D2 g =– d 2r dr so <0 dt 2 2πr 3 dt 3 24. True: h ( x) = f ( g ( x)) ⋅ g ( x) h (c) = f ( g (c)) ⋅ g (c) = 0 34. True: When h > r, then 25. True: ( f g ) (2) = f ( g (2)) ⋅ g (2) = f (2) ⋅ g (2) = 2 ⋅ 2 = 4 35. True: V= 26. False: 27. False: 28. True: 144 Consider f ( x) = x . The curve always lies below the tangent. The rate of volume change depends on the radius of the sphere. dr =4 dt dc dr = 2π = 2π(4) = 8π dt dt c = 2π r ; Section 2.10 d 2h dt 2 >0 4 3 πr , S = 4πr 2 3 dV = 4πr 2 dr = S ⋅ dr If Δr = dr, then dV = S ⋅ Δr 36. False: dy = 5 x 4 dx, so dy > 0 when dx > 0, but dy < 0 when dx < 0. 37. False: The slope of the linear approximation is equal to f '(a ) = f '(0) = − sin(0) = 0 . Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Sample Test Problems 3( x + h)3 – 3x3 9 x 2 h + 9 xh 2 + 3h3 = lim = lim (9 x 2 + 9 xh + 3h 2 ) = 9 x 2 h h h →0 h →0 h →0 1. a. f ( x) = lim b. f ( x) = lim [2( x + h)5 + 3( x + h)] – (2 x5 + 3 x) 10 x 4 h + 20 x3 h 2 + 20 x 2 h3 + 10 xh 4 + 2h5 + 3h = lim h h h →0 h →0 = lim (10 x 4 + 20 x3 h + 20 x 2 h 2 + 10 xh3 + 2h 4 + 3) = 10 x 4 + 3 h →0 1 3( x + h ) – 31x ⎛ ⎞ ⎡ ⎤1 1 1 h = lim ⎢ – ⎥ = lim – ⎜ 3x( x + h) ⎟ = – 2 h →0 ⎝ h →0 ⎣ 3( x + h) x ⎦ h 3x ⎠ c. f ( x) = lim d. ⎡⎛ ⎡ 3x 2 + 2 – 3( x + h) 2 – 2 1 ⎤ 1 1 ⎞ 1⎤ f ( x) = lim ⎢⎜ = lim ⎢ ⋅ ⎥ – ⎥ ⎟ 2 2 h →0 ⎢⎜⎝ 3( x + h) 2 + 2 3 x 2 + 2 ⎟⎠ h ⎥ ⎣ ⎦ h→0 ⎣⎢ (3( x + h) + 2)(3x + 2) h ⎦⎥ h h →0 ⎡ –6 xh – 3h 2 1⎤ –6 x – 3h 6x =– = lim ⎢ ⋅ ⎥ = lim 2 2 2 2 2 h→0 ⎢ (3( x + h) + 2)(3 x + 2) h ⎥ 0 h → (3( x + h) + 2)(3x + 2) (3 x + 2)2 ⎣ ⎦ e. f ( x) = lim h →0 3h = lim h →0 h( f. g. h →0 3 3x + 3h + 3x = 3 2 3x sin[3( x + h)] – sin 3x sin(3x + 3h) – sin 3 x = lim h h h →0 sin 3 x cos 3h + sin 3h cos 3x – sin 3x sin 3 x(cos 3h –1) sin 3h cos 3 x = lim + lim = lim h h h h →0 h →0 h →0 cos 3h –1 sin 3h sin 3h = 3sin 3 x lim + cos 3 x lim = (3sin 3 x)(0) + (cos 3 x)3 lim = (cos 3x)(3)(1) = 3cos 3 x 3h h →0 h →0 h h →0 3h h →0 ⎛ ( x + h) 2 + 5 – x 2 + 5 ⎞ ⎛ ( x + h) 2 + 5 + x 2 + 5 ⎞ ⎜ ⎟⎜ ⎟ ( x + h) 2 + 5 – x 2 + 5 ⎠⎝ ⎠ f ( x) = lim = lim ⎝ h h →0 h →0 h ⎛⎜ ( x + h)2 + 5 + x 2 + 5 ⎞⎟ ⎝ ⎠ h →0 2. a. 3 x + 3h + 3 x ) = lim f ( x) = lim = lim h. 3( x + h) – 3 x ( 3x + 3h – 3x )( 3x + 3h + 3 x ) = lim h h →0 h( 3 x + 3h + 3x ) 2 xh + h 2 h ⎛⎜ ( x + h) 2 + 5 + x 2 + 5 ⎞⎟ ⎝ ⎠ = lim h →0 2x + h 2 2 ( x + h) + 5 + x + 5 = 2x 2 2 x +5 = x 2 x +5 cos[π( x + h)] – cos πx cos(πx + πh) – cos πx cos πx cos πh – sin πx sin πh – cos πx = lim = lim h h h h →0 h →0 1 – cos πh ⎞ sin πh ⎞ ⎛ ⎛ = lim ⎜ – π cos πx ⎟ − lim ⎜ π sin πx ⎟ = (–π cos πx)(0) – (π sin πx) = – π sin πx πh πh ⎠ h →0 ⎝ ⎠ h→0 ⎝ f ( x) = lim h →0 2t 2 – 2 x 2 2(t – x)(t + x) = lim t–x t–x t→x t→x = 2 lim (t + x) = 2(2 x) = 4 x g ( x) = lim t→x Instructor’s Resource Manual Section 2.10 145 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. (t 3 + t ) – ( x3 + x) t–x t→x g ( x) = lim = lim t→x 2 g. 2 (t – x)(t + tx + x ) + (t – x) t–x t→x c. g ( x) = lim t→x t–x = lim x–t t → x tx(t – x) t→x = lim –1 1 =– t → x tx x2 = lim d. h. + 1)( x 2 + 1)(t – x) –( x + t )(t – x) = lim t → x (t 2 + 1)( x 2 + 1)(t – x) –( x + t ) 2x = lim =– 2 t → x (t 2 + 1)( x 2 + 1) ( x + 1)2 e. t– x t→x t – x g ( x ) = lim = lim ( t – x )( t + x ) t→x = lim t → x (t = f. (t – x)( t + x ) t–x – x)( t + x ) = lim t→x 1 3. a. t+ x 1 sin πt – sin πx t–x t→x Let v = t – x, then t = v + x and as t → x, v → 0. sin πt – sin πx sin π(v + x) – sin πx lim = lim t–x v t→x v →0 sin πv cos πx + sin πx cos πv – sin πx = lim v v →0 sin πv cos πv –1 ⎤ ⎡ = lim ⎢ π cos πx + π sin πx πv πv ⎥⎦ v →0 ⎣ = π cos πx ⋅1 + π sin πx ⋅ 0 = π cos πx Other method: Use the subtraction formula π(t + x) π(t − x) sin πt – sin πx = 2 cos sin 2 2 Section 2.10 t 3 + C + x3 + C = 3x 2 2 x3 + C cos 2t – cos 2 x t–x Let v = t – x, then t = v + x and as t → x, v → 0. cos 2t – cos 2 x cos 2(v + x) – cos 2 x lim = lim t–x v t→x v →0 cos 2v cos 2 x – sin 2v sin 2 x – cos 2 x = lim v v →0 cos 2v –1 sin 2v ⎤ ⎡ – 2sin 2 x = lim ⎢ 2 cos 2 x 2v 2v ⎥⎦ v →0 ⎣ = 2 cos 2 x ⋅ 0 – 2sin 2 x ⋅1 = –2sin 2 x Other method: Use the subtraction formula cos 2t − cos 2 x = −2sin(t + x) sin(t − x). g ( x) = lim t→x f(x) = 3x at x = 1 f ( x) = 4 x3 at x = 2 c. f ( x) = x3 at x = 1 d. f(x) = sin x at x = π e. f ( x) = 4 at x x f. f(x) = –sin 3x at x g. f(x) = tan x at x = h. 4. a. b. 146 t 2 + tx + x 2 b. 2 x g ( x) = lim (t – x) ⎛⎜ t 3 + C + x3 + C ⎞⎟ ⎝ ⎠ t→x x2 – t 2 t → x (t 2 t 3 – x3 = lim ⎡⎛ 1 1 ⎞ ⎛ 1 ⎞⎤ g ( x) = lim ⎢⎜ – ⎟⎜ ⎟⎥ t → x ⎣⎝ t 2 + 1 x 2 + 1 ⎠ ⎝ t – x ⎠ ⎦ = lim t→x t 3 + C – x3 + C t–x ⎛ t 3 + C – x3 + C ⎞ ⎛ t 3 + C + x3 + C ⎞ ⎜ ⎟⎜ ⎟ ⎠⎝ ⎠ = lim ⎝ t→x 3 3 ⎛ ⎞ (t – x) ⎜ t + C + x + C ⎟ ⎝ ⎠ = lim (t 2 + tx + x 2 + 1) = 3x 2 + 1 1– 1 t x g ( x) = lim f ( x) = 1 x f (2) ≈ – f (6) ≈ π 4 at x = 5 3 4 3 2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. c. Vavg = 6 – 32 7–3 = 9 8 11. d d. f (t 2 ) = f (t 2 )(2t ) dt ⎛2⎞ 8 At t = 2, 4 f (4) ≈ 4 ⎜ ⎟ = ⎝3⎠ 3 e. f. d 2 [ f (t )] = 2 f (t ) f (t ) dt At t = 2, ⎛ 3⎞ 2 f (2) f (2) ≈ 2(2) ⎜ – ⎟ = –3 ⎝ 4⎠ = = 13. ⎛ 3 ⎞⎛ 3 ⎞ 9 ≈ ⎜ – ⎟⎜ – ⎟ = ⎝ 4 ⎠ ⎝ 4 ⎠ 16 5. Dx (3x ) = 15 x 14. 4 6. Dx ( x3 – 3 x 2 + x –2 ) = 3 x 2 – 6 x + (–2) x –3 2 (2) + 2t + 6 + 2t + 6 ⎞ d 2 d ⎛ 1 ⎜ ⎟ = ( x + 4) –1/ 2 ⎜ 2 dx ⎝ x + 4 ⎟⎠ dx 1 = – ( x 2 + 4) –3 / 2 (2 x) 2 x =– 2 ( x + 4)3 d dx x2 – 1 3 x –x = d 1 d −1 2 1 = =− x dx x dx 2 x3 2 = – sin – 3[sin (2)(cos )(– sin ) + cos3 ] = – sin + 6sin 2 cos – 3cos3 16. −3 x 2 + 10 x + 3 ( x 2 + 1) 2 d [sin(t 2 ) – sin 2 (t )] = cos(t 2 )(2t ) – (2sin t )(cos t ) dt = 2t cos(t 2 ) – sin(2t ) ⎛ 4t − 5 ⎞ (6t 2 + 2t )(4) – (4t – 5)(12t + 2) 9. Dt ⎜ ⎟= (6t 2 + 2t )2 ⎝ 6t 2 + 2t ⎠ −24t 2 + 60t + 10 (6t 2 + 2t ) 2 10. Dx (3x + 2) 2 / 3 = 2t + 6 2 2t + 6 D 2 (sin + cos3 ) 2 ⎛ 3 x – 5 ⎞ ( x 2 + 1)(3) – (3 x – 5)(2 x) 8. Dx ⎜ ⎟= ( x 2 + 1)2 ⎝ x2 + 1 ⎠ = t 1 = cos – 3sin cos 2 7. Dz ( z + 4 z + 2 z ) = 3z + 8 z + 2 = ( x3 + x ) 2 15. D (sin + cos3 ) = cos + 3cos 2 (– sin ) = 3x 2 – 6 x – 2 x –3 3 −4 x 4 + 10 x 2 + 2 12. Dt (t 2t + 6) = t d ( f ( f (t ))) = f ( f (t )) f (t ) dt At t = 2, f ( f (2)) f (2) = f (2) f (2) 5 d ⎛ 4 x 2 – 2 ⎞ ( x3 + x)(8 x) – (4 x 2 – 2)(3 x 2 + 1) ⎜ ⎟= dx ⎜⎝ x3 + x ⎟⎠ ( x3 + x ) 2 17. D [sin( 18. 2 )] = cos( 2 )(2 ) = 2 cos( 2 ) d (cos3 5 x) = (3cos2 5 x )(– sin 5 x )(5) dx = –15cos 2 5 x sin 5 x 2 (3 x + 2) –1/ 3 (3) 3 = 2(3 x + 2) –1/ 3 2 Dx2 (3x + 2) 2 / 3 = – (3x + 2) –4 / 3 (3) 3 = –2(3x + 2) –4 / 3 Instructor’s Resource Manual Section 2.10 147 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 19. d [sin 2 (sin(π ))] = 2sin(sin(π )) cos(sin(π ))(cos(π ))(π) = 2π sin(sin(π )) cos(sin(π )) cos(π ) d 20. d [sin 2 (cos 4t )] = 2sin(cos 4t ) ( cos(cos 4t ) ) (– sin 4t )(4) = –8sin(cos 4t ) cos(cos 4t ) sin 4t dt 21. D tan 3 = (sec2 3 )(3) = 3sec2 3 22. 23. d ⎛ sin 3 x ⎞ (cos 5 x 2 )(cos 3x)(3) – (sin 3x)(– sin 5 x 2 )(10 x ) 3cos 5 x 2 cos 3 x + 10 x sin 3x sin 5 x 2 = ⎜ ⎟= dx ⎝ cos 5 x 2 ⎠ cos 2 5 x 2 cos 2 5 x 2 f ( x) = ( x 2 –1)2 (9 x 2 – 4) + (3 x3 – 4 x)(2)( x 2 –1)(2 x) = ( x 2 –1)2 (9 x 2 – 4) + 4 x( x 2 –1)(3 x3 – 4 x) f (2) = 672 24. g ( x) = 3cos 3 x + 2(sin 3 x)(cos 3 x)(3) = 3cos 3x + 3sin 6 x g ( x) = –9sin 3 x + 18cos 6 x g (0) = 18 25. d ⎛ cot x ⎞ (sec x 2 )(– csc 2 x) – (cot x)(sec x 2 )(tan x 2 )(2 x) – csc2 x – 2 x cot x tan x 2 = ⎜ ⎟= dx ⎝ sec x 2 ⎠ sec x 2 sec 2 x 2 ⎛ 4t sin t ⎞ (cos t – sin t )(4t cos t + 4sin t ) – (4t sin t )(– sin t – cos t ) 26. Dt ⎜ ⎟= ⎝ cos t – sin t ⎠ (cos t – sin t )2 = 27. 4t cos 2 t + 2sin 2t – 4sin 2 t + 4t sin 2 t (cos t – sin t ) 2 = 4t + 2sin 2t – 4sin 2 t (cos t – sin t )2 f ( x) = ( x – 1)3 2(sin πx – x)(π cos πx – 1) + (sin πx – x) 2 3( x – 1)2 = 2( x – 1)3 (sin πx – x )(π cos πx – 1) + 3(sin πx – x)2 ( x – 1)2 f (2) = 16 − 4π ≈ 3.43 28. h (t ) = 5(sin(2t ) + cos(3t )) 4 (2 cos(2t ) – 3sin(3t )) h (t ) = 5(sin(2t ) + cos(3t )) 4 (−4sin(2t ) – 9 cos(3t )) + 20(sin(2t ) + cos(3t ))3 (2 cos(2t ) – 3sin(3t ))2 h (0) = 5 ⋅14 ⋅ (−9) + 20 ⋅13 ⋅ 22 = 35 29. g (r ) = 3(cos 2 5r )(– sin 5r )(5) = –15cos 2 5r sin 5r g (r ) = –15[(cos 2 5r )(cos 5r )(5) + (sin 5r )2(cos 5r )(– sin 5r )(5)] = –15[5cos3 5r – 10(sin 2 5r )(cos 5r )] g (r ) = –15[5(3)(cos 2 5r )(– sin 5r )(5) − (10sin 2 5r )(− sin 5r )(5) − (cos 5r )(20sin 5r )(cos 5r )(5)] = –15[−175(cos 2 5r )(sin 5r ) + 50sin 3 5r ] g (1) ≈ 458.8 30. f (t ) = h ( g (t )) g (t ) + 2 g (t ) g (t ) 31. G ( x ) = F (r ( x) + s ( x))(r ( x) + s ( x)) + s ( x) G ( x) = F (r ( x) + s ( x))(r ( x) + s ( x)) + (r ( x ) + s ( x)) F (r ( x) + s ( x))(r ( x) + s ( x)) + s ( x) = F (r ( x) + s ( x))(r ( x ) + s ( x)) + (r ( x) + s ( x))2 F (r ( x) + s ( x)) + s ( x) 148 Section 2.10 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 32. F ( x) = Q ( R ( x)) R ( x) = 3[ R( x)]2 (– sin x) b. = –3cos 2 x sin x 33. F ( z ) = r ( s ( z )) s ( z ) = [3cos(3s ( z ))](9 z 2 ) = 27 z 2 cos(9 z 3 ) 34. dy = 2( x – 2) dx 2x – y + 2 = 0; y = 2x + 2; m = 2 1 2( x – 2) = – 2 7 x= 4 39. s = t 3 – 6t 2 + 9t ds v(t ) = = 3t 2 – 12t + 9 dt a(t ) = 35. V = dV = 4π(5) 2 = 100π ≈ 314 m3 per dr meter of increase in the radius. When r = 5, 4 dV 36. V = πr 3 ; = 10 3 dt dV dr = 4πr 2 dt dt dr When r = 5, 10 = 4π(5) dt dr 1 = ≈ 0.0318 m/h dt 10π 2 1 6 b 3h bh(12); = ; b = 2 4 h 2 dV ⎛ 3h ⎞ V = 6 ⎜ ⎟ h = 9h 2 ; =9 2 dt ⎝ ⎠ dV dh = 18h dt dt dh When h = 3, 9 = 18(3) dt dh 1 = ≈ 0.167 ft/min dt 6 37. V = 38. a. v = 128 – 32t v = 0, when t = 4s s = 128(4) – 16(4) 2 = 256 ft Instructor’s Resource Manual dt 2 = 6t –12 3t 2 – 12t + 9 < 0 3(t – 3)(t – 1) < 0 1 < t < 3; (1,3) b. 3t 2 – 12t + 9 = 0 3(t – 3)(t – 1) = 0 t = 1, 3 a(1) = –6, a(3) = 6 c. 6t – 12 > 0 t > 2; (2, ∞) 4 3 πr 3 dV = 4πr 2 dr d 2s a. 2 1 ⎛7 1 ⎞ ⎛7 ⎞ y = ⎜ – 2⎟ = ; ⎜ , ⎟ 16 ⎝ 4 16 ⎠ ⎝4 ⎠ 128t – 16t 2 = 0 –16t(t – 8) = 0 The object hits the ground when t = 8s v = 128 – 32(8) = –128 ft/s 40. a. Dx20 ( x19 + x12 + x5 + 100) = 0 b. Dx20 ( x 20 + x19 + x18 ) = 20! c. Dx20 (7 x 21 + 3 x 20 ) = (7 ⋅ 21!) x + (3 ⋅ 20!) d. Dx20 (sin x + cos x) = Dx4 (sin x + cos x) = sin x + cos x e. Dx20 (sin 2 x) = 220 sin 2 x = 1,048,576 sin 2x f. 41. a. b. 20 ⎛ 1 ⎞ (–1) (20!) 20! Dx20 ⎜ ⎟ = = ⎝ x⎠ x 21 x 21 dy =0 dx dy –( x – 1) 1 – x = = dx y y 2( x –1) + 2 y x(2 y ) dy dy + y 2 + y (2 x) + x 2 =0 dx dx dy (2 xy + x 2 ) = –( y 2 + 2 xy ) dx dy y 2 + 2 xy =− dx x 2 + 2 xy Section 2.10 149 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. c. 3x 2 + 3 y 2 dy dy = x3 (3 y 2 ) + 3 x 2 y 3 dx dx a. dy = – b. dy = – dy (3 y 2 – 3 x3 y 2 ) = 3x 2 y3 – 3x 2 dx dy 3x 2 y 3 – 3x 2 x 2 y 3 – x 2 = = dx 3 y 2 – 3 x3 y 2 y 2 – x3 y 2 d. e. ⎡ dy ⎤ x cos( xy ) ⎢ x + y ⎥ + sin( xy ) = 2 x ⎣ dx ⎦ dy x 2 cos( xy ) = 2 x – sin( xy ) – xy cos( xy ) dx dy 2 x – sin( xy ) – xy cos( xy ) = dx x 2 cos( xy ) ⎛ dy ⎞ x sec 2 ( xy ) ⎜ x + y ⎟ + tan( xy ) = 0 dx ⎝ ⎠ dy x 2 sec2 ( xy ) = –[tan( xy ) + xy sec2 ( xy )] dx 2 dy tan( xy ) + xy sec ( xy ) =– dx x 2 sec2 ( xy ) 45. a. 43. dy = [π cos(π x) + 2 x]dx ; x = 2, dx = 0.01 dy = [π cos(2π ) + 2(2)](0.01) = (4 + π )(0.01) ≈ 0.0714 dy dy + y 2 + 2 y[2( x + 2)] + ( x + 2)2 (2) =0 dx dx dy [2 xy + 2( x + 2) 2 ] = –[ y 2 + 2 y (2 x + 4)] dx dy –( y 2 + 4 xy + 8 y ) = dx 2 xy + 2( x + 2) 2 = 2(3)(4) + 3(2) 2 (5) = 84 b. c. Section 2.10 d [ f ( x) g ( x)] = f ( x) g ( x) + g ( x) f ( x) dx f (2) g (2) + g (2) f (2) = (3)(5) + (2)(4) = 23 d [ f ( g ( x))] = f ( g ( x )) g ( x) dx f ( g (2)) g (2) = f (2) g (2) = (4)(5) = 20 Dx [ f 2 ( x)] = 2 f ( x) f ( x) dx =2 dt dx dy 0 = 2x + 2 y dt dt dy x dx =– dt y dt When y= 5, x = 12, so 12 24 dy = – (2) = – = –4.8 ft/s dt 5 5 dx y dx , = 400 x dt y = x sin15° 47. sin15° = dy dx = sin15° dt dt dy = 400sin15° ≈ 104 mi/hr dt 48. a. b. 150 d 2 [ f ( x) + g 3 ( x)] dx 46. (13) 2 = x 2 + y 2 ; 1 3 Since ( y1 )( y2 ) = –1 at (1, 2), the tangents are perpendicular. 2 xy + 2( x + 2)2 When x = –2, y = ±1 (–0.01) = 2(3)(–1) + 2(4)2 = 26 At (1, 2): y2 = – dy = – 2(–2)(–1) + 2(–2 + 2) 2 = 0.0025 = 2 f (2) f (2) + 2[ f (2)]2 2x 3y y 2 + 4 xy + 8 y (–1)2 + 4(–2)(–1) + 8(–1) Dx2 [ f 2 ( x)] = 2[ f ( x) f ( x) + f ( x) f ( x)] At (1, 2): y1 = 3 4 x + 6 yy2 = 0 44. x(2 y ) (–0.01) 2 f (2) f (2) + 3 g 2 (2) g (2) 6x 2 y1 = y y2 = – 2(–2)(1) + 2(–2 + 2)2 = –0.0025 = 2 f ( x) f ( x) + 3g 2 ( x) g ( x) d. 42. 2 yy1 = 12 x 2 (1)2 + 4(–2)(1) + 8(1) 2 Dx ( x ) = 2 x ⋅ x x 2 = 2( x ) x = 2x2 = 2x x x x ⎛⎜ x ⎞⎟ − x ⎛ x ⎞ x−x = =0 Dx2 x = Dx ⎜ ⎟ = ⎝ ⎠ 2 x x2 ⎝ x⎠ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. c. Dx3 x = Dx ( Dx2 x ) = Dx (0) = 0 d. Dx2 ( 49. a. x ( x − 1)( x − 2 ) = 0 2 x = 0, x = 1 or x = 2 The split points are 0, 1, and 2. The expression on the left can only change signs at the split points. Check a point in the intervals ( −∞, 0 ) , x ) = Dx (2 x) = 2 = D sin b. 3. x ( x − 1)( x − 2 ) ≤ 0 = D cos sin sin cos cos cos = cot sin (− sin ) = − tan ( 0,1) , (1, 2 ) , and ( 2, ∞ ) . The solution set is { x | x ≤ 0 or 1 ≤ x ≤ 2} , or ( −∞, 0] ∪ [1, 2] . cos 1 ( x + 1)−1/ 2 ; a = 3 2 L( x) = f (3) + f '(3)( x − 3) 4. ( and ( 3, ∞ ) . The solution set is { x | 2 < x < 3} or ( 2,3) . − x = 0, x = −1, x = −2 The split points are 0, −1 , and −2 . The expression on the left can only change signs at the split points. Check a point in the intervals ( −∞, −2 ) , ( −2, −1) , ( −1, 0 ) , and ( 0, ∞ ) . The solution set is { x | −2 ≤ x ≤ −1 or x ≥ 0} , or [ −2, −1] ∪ [0, ∞ ) . − − − x ( x − 2) 5. x2 − 4 x ( x − 2) ( x − 2 )( x + 2 ) − ≥0 ≥0 The expression on the left is equal to 0 or undefined at x = 0 , x = 2 , and x = −2 . These are the split points. The expression on the left can only change signs at the split points. Check a point in the intervals: ( −∞, −2 ) , ( −2, 0 ) , ( 0, 2 ) , and ( 2, ∞ ) . The solution set is − x2 − x − 6 > 0 2. ) x ( x + 1)( x + 2 ) = 0 − x = 2 or x = 3 The split points are 2 and 3. The expression on the left can only change signs at the split points. Check a point in the intervals ( −∞, 2 ) , ( 2,3) , − x ( x + 1)( x + 2 ) ≥ 0 f ( x) = x cos x; f '( x) = − x sin x + cos x; a = 1 L( x) = f (1) + f '(1)( x − 1) = cos1 + (− sin1 + cos1)( x − 1) = cos1 − (sin1) x + sin1 + (cos1) x − cos1 = (cos1 − sin1) x + sin1 ≈ −0.3012 x + 0.8415 ( x − 2 )( x − 3) < 0 ( x − 2 )( x − 3) = 0 − x3 + 3x 2 + 2 x ≥ 0 Review and Preview Problems 1. − x x 2 + 3x + 2 ≥ 0 1 = 4 + − (4) −1/ 2 ( x − 3) 2 1 3 1 11 = 2− x+ = − x+ 4 4 4 4 b. − − f ( x) = x + 1; f '( x) = − 50. a. ( x − 3)( x + 2 ) > 0 ( x − 3)( x + 2 ) = 0 { x | x < −2 or 0 ≤ x < 2 or x > 2} , or ( −∞, −2 ) ∪ [0, 2 ) ∪ ( 2, ∞ ) . − − − − − x = 3 or x = −2 The split points are 3 and −2 . The expression on the left can only change signs at the split points. Check a point in the intervals ( −∞, −2 ) , ( −2,3) , and ( 3, ∞ ) . The solution set is { x | x < −2 or x > 3} , or ( −∞, −2 ) ∪ ( 3, ∞ ) . − − − − − Instructor’s Resource Manual Review and Preview 151 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6. x2 − 9 x2 + 2 ( x − 3)( x + 3) >0 >0 x2 + 2 The expression on the left is equal to 0 at x = 3 , and x = −3 . These are the split points. The expression on the left can only change signs at the split points. Check a point in the intervals: ( −∞, −3) , ( −3, 3) , and ( 3, ∞ ) . The solution set is { x | x < −3 or x > 3} , or ( −∞, −3) ∪ ( 3, ∞ ) . − − − − − 7. f ' ( x ) = 4 ( 2 x + 1) ( 2 ) = 8 ( 2 x + 1) 8. f ' ( x ) = cos (π x ) ⋅ π = π cos (π x ) 9. f ' ( x ) = x 2 − 1 ⋅ − sin ( 2 x ) ⋅ 2 + cos ( 2 x ) ⋅ ( 2 x ) 3 ( ( ) 3 ) = −2 x 2 − 1 sin ( 2 x ) + 2 x cos ( 2 x ) 10. f '( x) = = 11. x ⋅ sec x tan x − sec x ⋅1 x2 sec x ( x tan x − 1) f '( x) = ) = 13. ( 1 1 + sin 2 x 2 sin x cos x ) −1/ 2 17. The line y = 2 + x has slope 1, so any line parallel to this line will also have a slope of 1. For the tangent line to y = x + sin x to be parallel to the given line, we need its derivative to equal 1. y ' = 1 + cos x = 1 cos x = 0 The tangent line will be parallel to y = 2 + x ( x ) ⋅ 12 x . 1 −1/ 2 = x cos x 2 x 1 cos 2 x −1/ 2 f ' ( x ) = ( sin 2 x ) ⋅ cos 2 x ⋅ 2 = 2 sin 2 x 4− x His distance swimming will be 12 + x 2 = x 2 + 1 kilometers. His distance running will be 4 − x kilometers. Using the distance traveled formula, d = r ⋅ t , we d solve for t to get t = . Andy can swim at 4 r kilometers per hour and run 10 kilometers per hour. Therefore, the time to get from A to D will be 152 2 19. Consider the diagram: 1 + sin 2 x f ' ( x ) = cos π = x ( 9 − 2 x )( 24 − 2 x ) ( 2sin x )( cos x ) (note: you cannot cancel the x here because it is not a factor of both the numerator and denominator. It is the argument for the cosine in the numerator.) 14. an integer. 18. Length: 24 − 2x Width: 9 − 2x Height: x Volume: l ⋅ w ⋅ h = ( 24 − 2 x )( 9 − 2 x ) x f ' ( x ) = 2 ( tan 3 x ) ⋅ sec 2 3 x ⋅ 3 = 6 sec 2 3 x ( tan 3 x ) 12. 16. The tangent line is horizontal when the derivative is 0. y ' = 1 + cos x The tangent line is horizontal whenever cos x = −1 . That is, for x = ( 2k + 1) π where k is whenever x = ( 2k + 1) x2 ( 15. The tangent line is horizontal when the derivative is 0. y ' = 2 tan x ⋅ sec 2 x 2 tan x sec x = 0 2sin x =0 cos 2 x The tangent line is horizontal whenever sin x = 0 . That is, for x = kπ where k is an integer. Review and Preview x2 + 1 4 − x + hours. 4 10 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 20. a. f ( 0 ) = 0 − cos ( 0 ) = 0 − 1 = −1 f (π ) = π − cos (π ) = π − ( −1) = π + 1 Since x − cos x is continuous, f ( 0 ) < 0 , and f (π ) > 0 , there is at least one point c .in the interval ( 0, π ) where f ( c ) = 0 . (Intermediate Value Theorem) b. ⎛π ⎞ π ⎛π ⎞ π f ⎜ ⎟ = − cos ⎜ ⎟ = ⎝2⎠ 2 ⎝2⎠ 2 f ' ( x ) = 1 + sin x ⎛π ⎞ ⎛π ⎞ f ' ⎜ ⎟ = 1 + sin ⎜ ⎟ = 1 + 1 = 2 ⎝2⎠ ⎝2⎠ The slope of the tangent line is m = 2 at the ⎛π π ⎞ point ⎜ , ⎟ . Therefore, ⎝2 2⎠ y− c. π⎞ π ⎛ = 2 ⎜ x − ⎟ or y = 2 x − . 2 2⎠ 2 ⎝ π 2x − 2x = x= π 2 = 0. π 2 π 4 The tangent line will intersect the x-axis at x= 21. a. π 4 . The derivative of x 2 is 2x and the derivative of a constant is 0. Therefore, one possible function is f ( x ) = x 2 + 3 . b. The derivative of − cos x is sin x and the derivative of a constant is 0. Therefore, one possible function is f ( x ) = − ( cos x ) + 8 . c. The derivative of x3 is 3x 2 , so the 1 derivative of x3 is x 2 . The derivative of 3 1 x 2 is 2x , so the derivative of x 2 is x . 2 The derivative of x is 1, and the derivative of a constant is 0. Therefore, one possible 1 1 function is x3 + x 2 + x + 2 . 3 2 22. Yes. Adding 1 only changes the constant term in the function and the derivative of a constant is 0. Therefore, we would get the same derivative regardless of the value of the constant. Instructor’s Resource Manual Review and Preview 153 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. CHAPTER Applications of the Derivative 3 7. 3.1 Concepts Review 1. continuous; closed and bounded 2. extreme 3. endpoints; stationary points; singular points 4. 3 ( x) = 2 x + 3; 2x + 3 = 0 when x = – . 2 3 Critical points: –2, – , 1 2 9 ⎛ 3⎞ (–2) = –2, ⎜ – ⎟ = – , (1) = 4 2 4 ⎝ ⎠ Maximum value = 4, minimum value = – f (c) = 0; f (c) does not exist 9 4 1 6 8. G ( x) = (6 x 2 + 6 x –12) = ( x 2 + x – 2); 5 5 Problem Set 3.1 x 2 + x – 2 = 0 when x = –2, 1 Critical points: –3, –2, 1, 3 9 7 G (–3) = , G (–2) = 4, G (1) = – , G (3) = 9 5 5 Maximum value = 9, 7 minimum value = – 5 1. Endpoints: −2 , 4 Singular points: none Stationary points: 0, 2 Critical points: −2, 0, 2, 4 2. Endpoints: −2 , 4 Singular points: 2 Stationary points: 0 Critical points: −2, 0, 2, 4 9. 3. Endpoints: −2 , 4 Singular points: none Stationary points: −1, 0,1, 2,3 Critical points: −2, −1, 0,1, 2,3, 4 f ( x) = 3 x 2 – 3; 3x 2 – 3 = 0 when x = –1, 1. Critical points: –1, 1 f(–1) = 3, f(1) = –1 No maximum value, minimum value = –1 (See graph.) 4. Endpoints: −2 , 4 Singular points: none Stationary points: none Critical points: −2, 4 5. f ( x) = 2 x + 4; 2 x + 4 = 0 when x = –2. Critical points: –4, –2, 0 f(–4) = 4, f(–2) = 0, f(0) = 4 Maximum value = 4, minimum value = 0 10. 1 6. h ( x) = 2 x + 1; 2 x + 1 = 0 when x = – . 2 1 Critical points: –2, – , 2 2 1 ⎛ 1⎞ h(–2) = 2, h ⎜ – ⎟ = – , h(2) = 6 2 4 ⎝ ⎠ Maximum value = 6, minimum value = – 154 Section 3.1 f ( x) = 3 x 2 – 3; 3x 2 – 3 = 0 when x = –1, 1. 3 Critical points: – , –1, 1, 3 2 ⎛ 3 ⎞ 17 f ⎜ – ⎟ = , f (–1) = 3, f (1) = –1, f (3) = 19 ⎝ 2⎠ 8 Maximum value = 19, minimum value = –1 1 4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 11. h (r ) = − 1 2 ; h (r ) is never 0; h (r ) is not defined r when r = 0, but r = 0 is not in the domain on [–1, 3] since h(0) is not defined. Critical points: –1, 3 Note that lim h(r ) = −∞ and lim h( x) = ∞. x → 0− 15. g ( x) = − ; − 2 2 2x (1 + x ) (1 + x 2 ) 2 Critical points: –3, 0, 1 1 1 g(–3) = , g(0) = 1, g(1) = 10 2 f '( x) = 4x − 4x 1 10 16. 3 ( ) = 4 x x2 − 1 = 4 x ( x − 1)( x + 1) 4 x ( x − 1)( x + 1) = 0 when x = 0,1, −1 . Critical points: −2, −1, 0,1, 2 f ( −2 ) = 10 ; f ( −1) = 1 ; f ( 0 ) = 2 ; f (1) = 1 ; f ( 2 ) = 10 Maximum value: 10 Minimum value: 1 14. f ' ( x ) = 5 x 4 − 25 x 2 + 20 ( ) = 5 ( x 2 − 4 )( x 2 − 1) 4 2 = 5 x − 5x + 4 = 5 ( x − 2 )( x + 2 )( x − 1)( x + 1) 5 ( x − 2 )( x + 2 )( x − 1)( x + 1) = 0 when x = −2, −1,1, 2 Critical points: −3, −2, −1,1, 2 19 41 f ( −3) = −79 ; f ( −2 ) = − ; f ( −1) = − ; 3 3 35 13 f (1) = ; f ( 2) = 3 3 35 Maximum value: 3 Minimum value: −79 Instructor’s Resource Manual 2x (1 + x 2 ) 2 = 0 when x = 0. = 0 when x = 0 Maximum value = 1, minimum value = 13. ; − As x → ∞, g ( x) → 0+ ; as x → −∞, g ( x) → 0+. Maximum value = 1, no minimum value (See graph.) x → 0+ 2x 2 2 (1 + x ) Critical point: 0 g(0) = 1 No maximum value, no minimum value. 12. g ( x) = − 2x f ( x) = 1 − x2 (1 + x 2 )2 ; 1 − x2 = 0 when x = –1, 1 (1 + x 2 )2 Critical points: –1, 1, 4 1 1 4 f (−1) = − , f (1) = , f (4) = 2 2 17 1 Maximum value = , 2 1 minimum value = – 2 17. r ( ) = cos ; cos = 0 when = π + kπ 2 π π Critical points: – , 4 6 1 ⎛ π⎞ ⎛ π⎞ 1 r⎜− ⎟ = − , r⎜ ⎟ = 2 ⎝ 4⎠ ⎝6⎠ 2 1 1 Maximum value = , minimum value = – 2 2 18. s (t ) = cos t + sin t ; cos t + sin t = 0 when π + k π. 4 3π Critical points: 0, ,π 4 ⎛ 3π ⎞ s(0) = –1, s ⎜ ⎟ = 2, s (π ) = 1 . ⎝ 4 ⎠ Maximum value = 2, minimum value = –1 tan t = –1 or t = – Section 3.1 155 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. x –1 ; a ( x) does not exist when x = 1. x –1 19. a ( x) = Critical points: 0, 1, 3 a(0) = 1, a(1) = 0, a(3) = 2 Maximum value = 2, minimum value = 0 20. f ( s) = 25. g ' ( sec y 3(3s – 2) ; f ( s) does not exist when s = 2 . 3 3s – 2 1 ( ) 21. g ( x) = 2 22. s (t ) = − π4 ; s (t ) does not exist when t = 0. 5t Critical points: –1, 0, 32 s(–1) = 1, s(0) = 0, s(32) = 4 Maximum value = 4, minimum value = 0 Critical points: − − sin t = 0 when t = 0, π , 2π ,3π , 4π ,5π , 6π , 7π ,8π Critical points: 0, π , 2π ,3π , 4π ,5π , 6π , 7π ,8π H ( 0 ) = 1 ; H (π ) = −1 ; H ( 2π ) = 1 ; 24. g ' ( x ) = 1 − 2 cos x 3 π π 5π , 1 when 2 3 3 3 π π π 5π , , , 2π 3 3 3 ⎛ 5π ⎞ −5π g ( −2π ) = −2π ; g ⎜ − − 3; ⎟= 3 ⎝ 3 ⎠ π ⎛ π⎞ ⎛π ⎞ π g⎜− ⎟ = − + 3 ; g⎜ ⎟ = − 3 ; 3 ⎝ 3⎠ ⎝3⎠ 3 5 π 5 π ⎛ ⎞ g⎜ + 3 ; g ( 2π ) = 2π ⎟= ⎝ 3 ⎠ 3 5π + 3 Maximum value: 3 5π − 3 Minimum value: − 3 Section 3.1 4 26. h ' ( t ) = π2 2 16 ; Minimum value: 0 5 ( 2 + t ) ⎛⎜ t 2 / 3 ⎞⎟ − t 5/ 3 (1) ⎝3 ⎠ ( 2 + t )2 ⎛5 ⎞ ⎛ 10 2 ⎞ t2/3 ⎜ (2 + t ) − t ⎟ t2/3 ⎜ + t ⎟ ⎝3 ⎠= ⎝ 3 3 ⎠ = ( 2 + t )2 ( 2 + t )2 = 2t 2 / 3 ( t + 5 ) 3( 2 + t ) 2 when t = 0 or t = −5 . Since −5 is not in the interval of interest, it is not a critical point. Critical points: −1, 0,8 h ( −1) = −1 ; h ( 0 ) = 0 ; h ( 8 ) = 16 5 Maximum value: 16 ; Minimum value: −1 5 , Critical points: −2π , − π h ' ( t ) is undefined when t = −2 and h ' ( t ) = 0 Maximum value: 1 Minimum value: −1 ,− , 0, 2 Maximum value: H ( 6π ) = 1 ; H ( 7π ) = −1 ; H ( 8π ) = 1 π 4 ⎛ π⎞ π 2 ⎛π ⎞ π 2 g⎜− ⎟ = ; g ( 0) = 0 ; g ⎜ ⎟ = 4 16 16 ⎝ ⎠ ⎝4⎠ H ( 3π ) = −1 ; H ( 4π ) = 1 ; H ( 5π ) = −1 ; 1 − 2 cos x = 0 → cos x = π 2 3/ 5 23. H ' ( t ) = − sin t x π 4 −1 1 ; f ( x) does not exist when x = 0. 3x2 / 3 Critical points: –1, 0, 27 g(–1) = –1, g(0) = 0, g(27) = 3 Maximum value = 3, minimum value = –1 156 ( sec tan ) + 2 sec = sec ( tan + 2 ) ( tan + 2 ) = 0 when = 0 . 2 Consider the graph: Critical points: −1, 2 , 4 3 2 = 0, f(4) = 10 f(–1) = 5, f 3 Maximum value = 10, minimum value = 0 x=− )= 3 ,− 27. a. f ( x) = 3 x 2 –12 x + 1;3x 2 –12 x + 1 = 0 when x = 2 – 33 33 and x = 2 + . 3 3 Critical points: –1, 2 – 33 33 ,2+ ,5 3 3 ⎛ 33 ⎞ f(–1) = –6, f ⎜⎜ 2 – ⎟ ≈ 2.04, 3 ⎟⎠ ⎝ ⎛ 33 ⎞ f ⎜⎜ 2 + ⎟ ≈ –26.04, f(5) = –18 3 ⎟⎠ ⎝ Maximum value ≈ 2.04; minimum value ≈ −26.04 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. g ( x) = ( x3 – 6 x 2 + x + 2)(3x 2 –12 x + 1) 3 2 x – 6x + x + 2 ; 29. Answers will vary. One possibility: y 5 33 g '( x) = 0 when x = 2 – and 3 33 . g ( x) does not exist when 3 f(x) = 0; on [–1, 5], f(x) = 0 when x ≈ –0.4836 and x ≈ 0.7172 33 , Critical points: –1, –0.4836, 2 – 3 x = 2+ 33 , 5 3 g(–1) = 6, g(–0.4836) = 0, ⎛ 33 ⎞ g ⎜⎜ 2 – ⎟ ≈ 2.04, g(0.7172) = 0, 3 ⎟⎠ ⎝ ⎛ 33 ⎞ g ⎜⎜ 2 + ⎟ ≈ 26.04, g(5) = 18 3 ⎟⎠ ⎝ Maximum value ≈ 26.04, minimum value = 0 0.7172, 2 + 28. a. f ( x) = x cos x; on [–1, 5], x cos x = 0 when π 3π x = 0, x = , x = 2 2 π 3π Critical points: –1, 0, , , 5 2 2 ⎛π⎞ f(–1) ≈3.38, f(0) = 3, f ⎜ ⎟ ≈ 3.57, ⎝2⎠ ⎛ 3π ⎞ f ⎜ ⎟ ≈ –2.71, f(5) ≈ −2.51 ⎝ 2 ⎠ Maximum value ≈ 3.57, minimum value ≈–2.71 b. g ( x) = (cos x + x sin x + 2)( x cos x) ; cos x + x sin x + 2 π 3π , x= 2 2 g ( x) does not exist when f(x) = 0; on [–1, 5], f(x) = 0 when x ≈ 3.45 π 3π Critical points: –1, 0, , 3.45, , 5 2 2 ⎛π⎞ g(–1) ≈ 3.38, g(0) = 3, g ⎜ ⎟ ≈ 3.57, ⎝2⎠ ⎛ 3π ⎞ g(3.45) = 0, g ⎜ ⎟ ≈ 2.71, g(5) ≈ 2.51 ⎝ 2 ⎠ Maximum value ≈ 3.57; minimum value = 0 g ( x ) = 0 when x = 0, x = Instructor’s Resource Manual −5 5 x −5 30. Answers will vary. One possibility: y 5 5 x −5 31. Answers will vary. One possibility: y 5 5 x −5 32. Answers will vary. One possibility: y 5 5 x −5 Section 3.1 157 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 33. Answers will vary. One possibility: y 3.2 Concepts Review 1. Increasing; concave up 5 2. f ( x) > 0; f ( x) < 0 3. An inflection point 5 x −5 4. f (c) = 0; f (c) does not exist. Problem Set 3.2 34. Answers will vary. One possibility: y 5 5 x −5 35. Answers will vary. One possibility: y 1. 1 2. g ( x) = 2 x – 1; 2x – 1 > 0 when x > . g(x) is 2 ⎡1 ⎞ increasing on ⎢ , ∞ ⎟ and decreasing on ⎣2 ⎠ 1⎤ ⎛ ⎜ – ∞, ⎥ . 2⎦ ⎝ 3. h (t ) = 2t + 2; 2t + 2 > 0 when t > –1. h(t) is increasing on [–1, ∞ ) and decreasing on ( −∞ , –1]. 4. 5 5 x −5 36. Answers will vary. One possibility: y 5 −5 x f ( x) = 3x 2 ; 3 x 2 > 0 for x ≠ 0 . f(x) is increasing for all x. 5. G ( x ) = 6 x 2 – 18 x + 12 = 6( x – 2)( x – 1) Split the x-axis into the intervals (– ∞ , 1), (1, 2), (2, ∞ ). 3 3 ⎛3⎞ Test points: x = 0, , 3; G (0) = 12, G ⎜ ⎟ = – , 2 2 ⎝2⎠ G (3) = 12 G(x) is increasing on (– ∞ , 1] ∪ [2, ∞ ) and decreasing on [1, 2]. 6. 5 f ( x) = 3; 3 > 0 for all x. f(x) is increasing for all x. f (t ) = 3t 2 + 6t = 3t (t + 2) Split the x-axis into the intervals (– ∞ , –2), (–2, 0), (0, ∞ ). Test points: t = –3, –1, 1; f (–3) = 9, f (–1) = –3, f (1) = 9 f(t) is increasing on (– ∞ , –2] ∪ [0, ∞ ) and decreasing on [–2, 0]. 7. h ( z ) = z 3 – 2 z 2 = z 2 ( z – 2) Split the x-axis into the intervals (– ∞ , 0), (0, 2), (2, ∞ ). Test points: z = –1, 1, 3; h (–1) = –3, h (1) = –1, h (3) = 9 h(z) is increasing on [2, ∞ ) and decreasing on (– ∞ , 2]. 158 Section 3.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 8. f ( x) = 2– x 16. 3 x Split the x-axis into the intervals (– ∞ , 0), (0, 2), (2, ∞ ). Test points: –1, 1, 3; f (–1) = –3, f (1) = 1, 1 27 f(x) is increasing on (0, 2] and decreasing on (– ∞ , 0) ∪ [2, ∞ ). f (3) = – 9. H (t ) = cos t ; H (t ) > 0 when 0 ≤ t < 17. F ( x) = 2sin 2 x – 2 cos 2 x + 4 = 6 – 4 cos 2 x; 6 – 4 cos 2 x > 0 for all x since 0 ≤ cos 2 x ≤ 1. F(x) is concave up for all x; no inflection points. π and 2 18. G ( x) = 48 + 24 cos 2 x – 24sin 2 x = 24 + 48cos 2 x; 24 + 48cos 2 x > 0 for all x. G(x) is concave up for all x; no inflection points. 3π < t ≤ 2π. 2 ⎡ π ⎤ ⎡ 3π ⎤ H(t) is increasing on ⎢ 0, ⎥ ∪ ⎢ , 2π ⎥ and ⎣ 2⎦ ⎣ 2 ⎦ ⎡ π 3π ⎤ decreasing on ⎢ , ⎥ . ⎣2 2 ⎦ 10. R ( ) = –2 cos sin ; R ( ) > 0 when and 3π < 2 π < 2 f ( x) = 12 x 2 + 48 x = 12 x( x + 4); f ( x) > 0 when x < –4 and x > 0. f(x) is concave up on (– ∞ , –4) ∪ (0, ∞ ) and concave down on (–4, 0); inflection points are (–4, –258) and (0, –2). 19. <π f ( x) = 3 x 2 – 12; 3 x 2 – 12 > 0 when x < –2 or x > 2. f(x) is increasing on (– ∞ , –2] ∪ [2, ∞ ) and decreasing on [–2, 2]. f ( x) = 6 x; 6x > 0 when x > 0. f(x) is concave up on (0, ∞ ) and concave down on (– ∞ , 0). < 2π. ⎡ π ⎤ ⎡ 3π ⎤ R( ) is increasing on ⎢ , π ⎥ ∪ ⎢ , 2π ⎥ and ⎣2 ⎦ ⎣ 2 ⎦ ⎡ π ⎤ ⎡ 3π ⎤ decreasing on ⎢ 0, ⎥ ∪ ⎢ π, ⎥ . ⎣ 2⎦ ⎣ 2 ⎦ 11. f ( x) = 2; 2 > 0 for all x. f(x) is concave up for all x; no inflection points. 12. G ( w) = 2; 2 > 0 for all w. G(w) is concave up for all w; no inflection points. 13. T (t ) = 18t ; 18t > 0 when t > 0. T(t) is concave up on (0, ∞ ) and concave down on (– ∞ , 0); (0, 0) is the only inflection point. 14. f ( z) = 2 – 6 z 4 = 2 z4 ( z 4 – 3); z 4 – 3 > 0 for z < – 4 3 and z > 4 3. f(z) is concave up on (– ∞, – 4 3) ∪ ( 4 3, ∞) and concave down on (– 4 3, 0) ∪ (0, 4 3); inflection 20. g ( x) = 12 x 2 – 6 x – 6 = 6(2 x + 1)( x – 1); g ( x) > 0 1 or x > 1. g(x) is increasing on 2 1⎤ ⎛ ⎡ 1 ⎤ ⎜ – ∞, – ⎥ ∪ [1, ∞) and decreasing on ⎢ – , 1⎥ . 2⎦ ⎣ 2 ⎦ ⎝ g ( x) = 24 x – 6 = 6(4 x – 1); g ( x) > 0 when when x < – 1 x> . 4 ⎛1 ⎞ g(x) is concave up on ⎜ , ∞ ⎟ and concave down ⎝4 ⎠ 1⎞ ⎛ on ⎜ – ∞, ⎟ . 4⎠ ⎝ 1 ⎞ 1 ⎞ ⎛ ⎛4 points are ⎜ – 4 3, 3 – ⎟ and ⎜ 3, 3 – ⎟. 3⎠ 3⎠ ⎝ ⎝ 15. q ( x ) = 12 x 2 – 36 x – 48; q ( x) > 0 when x < –1 and x > 4. q(x) is concave up on (– ∞ , –1) ∪ (4, ∞ ) and concave down on (–1, 4); inflection points are (–1, –19) and (4, –499). Instructor’s Resource Manual Section 3.2 159 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 21. g ( x) = 12 x3 – 12 x 2 = 12 x 2 ( x – 1); g ( x) > 0 when x > 1. g(x) is increasing on [1, ∞ ) and decreasing on (−∞,1]. 23. G ( x ) = 15 x 4 – 15 x 2 = 15 x 2 ( x 2 – 1); G ( x) > 0 when x < –1 or x > 1. G(x) is increasing on (– ∞ , –1] ∪ [1, ∞ ) and decreasing on [–1, 1]. g ( x) = 36 x 2 – 24 x = 12 x(3 x – 2); g ( x) > 0 G ( x) = 60 x3 – 30 x = 30 x(2 x 2 – 1); 2 when x < 0 or x > . g(x) is concave up on 3 ⎛2 ⎞ ⎛ 2⎞ (– ∞, 0) ∪ ⎜ , ∞ ⎟ and concave down on ⎜ 0, ⎟ . 3 ⎝ ⎠ ⎝ 3⎠ 1 ⎞ ⎛ Split the x-axis into the intervals ⎜ −∞, − ⎟, 2⎠ ⎝ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ , 0 ⎟ , ⎜ 0, , ∞ ⎟. ⎜− ⎟, ⎜ 2 ⎠ ⎝ 2⎠ ⎝ 2 ⎝ ⎠ 1 1 Test points: x = –1, – , , 1; G (–1) = –30, 2 2 15 ⎛ 1 ⎞ 15 ⎛1⎞ G ⎜ – ⎟ = , G ⎜ ⎟ = – , G (1) = 30. 2 ⎝ 2⎠ 2 ⎝2⎠ ⎛ 1 ⎞ ⎛ , 0⎟ ∪ ⎜ G(x) is concave up on ⎜ – 2 ⎠ ⎝ ⎝ 1 ⎛ ⎞ ⎛ concave down on ⎜ – ∞, – ⎟ ∪ ⎜ 0, 2⎠ ⎝ ⎝ ⎞ , ∞ ⎟ and 2 ⎠ 1 ⎞ ⎟. 2⎠ 1 22. F ( x) = 6 x5 – 12 x3 = 6 x3 ( x 2 – 2) Split the x-axis into the intervals (– ∞ , − 2) , (− 2, 0), (0, 2), ( 2, ∞) . Test points: x = –2, –1, 1, 2; F (–2) = –96, F (–1) = 6, F (1) = –6, F (2) = 96 F(x) is increasing on [– 2, 0] ∪ [ 2, ∞) and decreasing on (– ∞, – 2] ∪ [0, 2] 4 2 2 2 2 F ( x) = 30 x – 36 x = 6 x (5 x – 6); 5 x – 6 > 0 6 6 . or x > 5 5 ⎛ 6⎞ ⎛ 6 ⎞ , ∞ ⎟ and F(x) is concave up on ⎜⎜ – ∞, – ⎟∪⎜ 5 ⎟⎠ ⎜⎝ 5 ⎟⎠ ⎝ ⎛ 6 6⎞ concave down on ⎜⎜ – , ⎟⎟ . ⎝ 5 5⎠ when x < – 160 Section 3.2 24. H ( x ) = 2x ; H ( x) > 0 when x > 0. ( x + 1)2 H(x) is increasing on [0, ∞ ) and decreasing on (– ∞ , 0]. 2(1 – 3 x 2 ) H ( x) = ; H ( x) > 0 when ( x 2 + 1)3 – 1 3 2 <x< 1 3 . ⎛ H(x) is concave up on ⎜ – ⎝ 1 ⎞ ⎛ ⎛ down on ⎜ – ∞, – ⎟∪⎜ 3⎠ ⎝ ⎝ 1 ⎞ ⎟ and concave 3⎠ 1 ⎞ , ∞ ⎟. 3 ⎠ 1 3 , Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 25. f ( x) = cos x 2 sin x ; f ( x) > 0 when 0 < x < π . f(x) 2 ⎡ π⎤ is increasing on ⎢ 0, ⎥ and decreasing on ⎣ 2⎦ ⎡π ⎤ ⎢ 2 , π⎥ . ⎣ ⎦ f ( x) = – cos 2 x – 2sin 2 x ; f ( x) < 0 for all x in 4sin 3 / 2 x (0, ∞ ). f(x) is concave down on (0, π ). 4 ; 3x – 4 > 0 when x > . 3 2 x–2 g(x) is increasing on [2, ∞ ). 3x – 8 8 g ( x) = ; 3x – 8 > 0 when x > . 3/ 2 3 4( x – 2) 26. g ( x) = f ( x) = –2(5 x + 1) 9 x4 / 3 ; –2(5x + 1) > 0 when 1 x < – , f ( x) does not exist at x = 0. 5 1 8 Test points: –1, – , 1; f (–1) = , 10 9 104 / 3 4 ⎛ 1⎞ f ⎜– ⎟ = – , f (1) = – . 10 9 3 ⎝ ⎠ 1⎞ ⎛ f(x) is concave up on ⎜ – ∞, – ⎟ and concave 5⎠ ⎝ ⎛ 1 ⎞ down on ⎜ – , 0 ⎟ ∪ (0, ∞). ⎝ 5 ⎠ 3x – 4 ⎛8 ⎞ g(x) is concave up on ⎜ , ∞ ⎟ and concave down ⎝3 ⎠ ⎛ 8⎞ on ⎜ 2, ⎟ . ⎝ 3⎠ 28. g ( x) = 4( x + 2) ; x + 2 > 0 when x > –2, g ( x) 3x 2 / 3 does not exist at x = 0. Split the x-axis into the intervals ( −∞, −2 ) , (–2, 0), (0, ∞ ). Test points: –3, –1, 1; g (–3) = – 27. 4 5/3 3 , 2 ; 2 – 5x > 0 when x < , f ( x ) 5 3x does not exist at x = 0. Split the x-axis into the intervals ( − ∞, 0), 4 g (–1) = , g (1) = 4. 3 g(x) is increasing on [–2, ∞ ) and decreasing on (– ∞ , –2]. 4( x – 4) g ( x) = ; x – 4 > 0 when x > 4, g ( x) 9 x5 / 3 does not exist at x = 0. 20 Test points: –1, 1, 5; g (–1) = , 9 4 4 g (1) = – , g (5) = . 3 9(5)5 / 3 ⎛ 2⎞ ⎛2 ⎞ ⎜ 0, ⎟ , ⎜ , ∞ ⎟ . ⎝ 5⎠ ⎝5 ⎠ g(x) is concave up on (– ∞ , 0) ∪ (4, ∞ ) and concave down on (0, 4). f ( x) = 2 – 5x 1/ 3 1 7 Test points: –1, , 1; f (−1) = – , 5 3 ⎛1⎞ 35 f ⎜ ⎟= , f (1) = –1. ⎝5⎠ 3 ⎡ 2⎤ f(x) is increasing on ⎢ 0, ⎥ and decreasing on ⎣ 5⎦ ⎡2 ⎞ (– ∞, 0] ∪ ⎢ , ∞ ⎟ . ⎣5 ⎠ Instructor’s Resource Manual Section 3.2 161 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 29. 35. f ( x) = ax 2 + bx + c; f ( x) = 2ax + b; f ( x ) = 2a An inflection point would occur where f ( x) = 0 , or 2a = 0. This would only occur when a = 0, but if a = 0, the equation is not quadratic. Thus, quadratic functions have no points of inflection. 36. f ( x) = ax3 + bx 2 + cx + d ; f ( x) = 3ax 2 + 2bx + c; f ( x) = 6ax + 2b 30. An inflection point occurs where f ( x) = 0 , or 6ax + 2b = 0. The function will have an inflection point at b x = – , a ≠ 0. 3a 31. 37. Suppose that there are points x1 and x2 in I where f ( x1 ) > 0 and f ( x2 ) < 0. Since f is continuous on I, the Intermediate Value Theorem says that there is some number c between x1 and x2 such that f (c) = 0, which is a contradiction. Thus, either f ( x) > 0 for all x in I and f is increasing throughout I or f ( x) < 0 for all x in I and f is decreasing throughout I. 32. 38. Since x 2 + 1 = 0 has no real solutions, f ( x ) exists and is continuous everywhere. x 2 – x + 1 = 0 has no real solutions. x 2 – x + 1 > 0 and x 2 + 1 > 0 for all x, so f ( x) > 0 for all x. Thus f is increasing everywhere. 39. a. Let f ( x) = x 2 and let I = [ 0, a ] , a > y . f ( x) = 2 x > 0 on I. Therefore, f(x) is increasing on I, so f(x) < f(y) for x < y. 33. b. Let f ( x) = x and let I = [ 0, a ] , a > y . 1 > 0 on I. Therefore, f(x) is 2 x increasing on I, so f(x) < f(y) for x < y. f ( x) = c. 34. 40. 1 and let I = [0, a], a > y. x 1 f ( x) = − < 0 on I. Therefore f(x) is x2 decreasing on I, so f(x) > f(y) for x < y. Let f ( x) = f ( x) = 3ax 2 + 2bx + c In order for f(x) to always be increasing, a, b, and c must meet the condition 3ax 2 + 2bx + c > 0 for all x. More specifically, a > 0 and b 2 − 3ac < 0. 162 Section 3.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 41. f ( x) = 3b – ax 4x 5/ 2 . If (4, 13) is an inflection point 45. a. 3b – 4a b = 0. Solving these and 4 ⋅ 32 2 39 13 equations simultaneously, a = and b = . 2 8 then 13 = 2a + 42. f ( x) = a ( x − r1 )( x − r2 )( x − r3 ) f ( x) = a[( x − r1 )(2 x − r2 − r3 ) + ( x − r2 )( x − r3 )] b. f ( x) < 0 : (1.3, 5.0) c. f ( x) < 0 : (−0.25, 3.1) ∪ (6.5, 7] d. 1 x f ( x) = cos x – sin 2 2 e. 1 x f ( x) = − sin x − cos 4 2 2 f ( x) = a[3 x − 2 x(r1 + r2 + r3 ) + r1r2 + r2 r3 + r1r3 ] f ( x) = a[6 x − 2(r1 + r2 + r3 )] a[6 x − 2(r1 + r2 + r3 )] = 0 r +r +r 6 x = 2(r1 + r2 + r3 ); x = 1 2 3 3 43. a. b. [ f ( x ) + g ( x)] = f ( x) + g ( x). Since f ( x) > 0 and g ( x) > 0 for all x, f ( x) + g ( x ) > 0 for all x. No additional conditions are needed. [ f ( x) ⋅ g ( x)] = f ( x) g ( x) + f ( x) g ( x). f ( x) g ( x) + f ( x) g ( x) > 0 if f ( x) > − c. 44. a. b. f ( x) g ( x) for all x. g ( x) [ f ( g ( x))] = f ( g ( x)) g ( x). Since f ( x) > 0 and g ( x) > 0 for all x, f ( g ( x)) g ( x) > 0 for all x. No additional conditions are needed. [ f ( x) + g ( x)] = f ( x) + g ( x). Since f ( x) > 0 and g > 0 for all x, f ( x) + g ( x) > 0 for all x. No additional conditions are needed. b. f ( x) < 0 : (2.0, 4.7) ∪ (9.9, 10] [ f ( x ) ⋅ g ( x)] = [ f ( x) g ( x) + f ( x) g ( x)] c. f ( x) < 0 : [0, 3.4) ∪ (7.6, 10] = f ( x) g ( x) + f ( x) g ( x) + 2 f ( x) g ( x). The additional condition is that f ( x) g ( x) + f ( x) g ( x) + 2 f ( x) g ( x) > 0 for all x is needed. c. 46. a. [ f ( g ( x))] = [ f ( g ( x)) g ( x)] ⎡ 2 ⎛ x ⎞ ⎛ x ⎞⎤ ⎛ x⎞ d. f ( x) = x ⎢ − cos ⎜ ⎟ sin ⎜ ⎟ ⎥ + cos2 ⎜ ⎟ ⎝ 3 ⎠ ⎝ 3 ⎠⎦ ⎝3⎠ ⎣ 3 ⎛x⎞ x ⎛ 2x ⎞ = cos 2 ⎜ ⎟ − sin ⎜ ⎟ 3 3 ⎝ ⎠ ⎝ 3 ⎠ = f ( g ( x)) g ( x) + f ( g ( x))[ g ( x)]2 . The additional condition is that f ( g ( x))[ g ( x)]2 for all x. f ( g ( x)) > − g ( x) Instructor’s Resource Manual Section 3.2 163 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. e. f ( x) = − 2x ⎛ 2x ⎞ 2 ⎛ 2x ⎞ cos ⎜ ⎟ − sin ⎜ ⎟ 9 ⎝ 3 ⎠ 3 ⎝ 3 ⎠ c. d 3s dt 3 < 0, d 2s dt 2 >0 s 47. f ( x) > 0 on (–0.598, 0.680) f is increasing on [–0.598, 0.680]. 48. f ( x) < 0 when x > 1.63 in [–2, 3] f is concave down on (1.63, 3). 49. Let s be the distance traveled. Then speed of the car. a. t Concave up. ds is the dt d. d 2s dt 2 = 10 mph/min s ds = ks, k a constant dt s t Concave up. Concave up. b. d 2s dt 2 e. t ds d 2s are approaching zero. and dt dt 2 s >0 s Concave down. Concave up. 164 Section 3.2 t t Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. f. ds is constant. dt c. s Neither concave up nor down. 50. a. dV dh d 2h = k, > 0, <0 dt dt dt 2 Concave down. h ( t) t dV = k <0, V is the volume of water in the dt tank, k is a constant. Neither concave up nor down. t d. dI d 2 I , > 0 in the future dt dt 2 where I is inflation. I ( t) = k now, but I ( t) v(t) t t b. e. dV 1 1 = 3 – = 2 gal/min dt 2 2 Neither concave up nor down. P ( t) v(t) t Instructor’s Resource Manual dp d2 p dp < 0, but > 0 and at t = 2: >0. 2 dt dt dt where p is the price of oil. Concave up. 2 t Section 3.2 165 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. f. dT d 2T > 0, < 0 , where T is David’s dt dt 2 temperature. Concave down. c. T ( t) dP d 2P > 0, < 0 , where P is world dt dt 2 population. Concave down. P ( t) t 51. a. dC d 2C > 0, > 0 , where C is the car’s cost. dt dt 2 Concave up. t d. C ( t) d d2 > 0, > 0 , where is the angle that dt dt 2 the tower makes with the vertical. Concave up. ( t) t b. f(t) is oil consumption at time t. df d2 f < 0, >0 dt dt 2 Concave up. f( t) t e. P = f(t) is profit at time t. dP d 2P > 0, <0 dt dt 2 Concave down. P ( t) t t 166 Section 3.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. f. R is revenue at time t. dP >0 P < 0, dt Could be either concave up or down. P t 54. The height is always increasing so h '(t ) > 0 . The rate of change of the height decreases for the first 50 minutes and then increases over the next 50 minutes. Thus h ''(t ) < 0 for 0 ≤ t ≤ 50 and h ''(t ) > 0 for 50 < t ≤ 100 . P t 52. a. R(t) ≈ 0.28, t < 1981 b. On [1981, 1983], R(1983) ≈ 0.36 53. dR d 2R > 0, >0, dt dt 2 dV = 2 in 3 / sec dt The cup is a portion of a cone with the bottom cut off. If we let x represent the height of the missing cone, we can use similar triangles to show that x x+5 = 3 3.5 3.5 x = 3x + 15 0.5 x = 15 x = 30 Similar triangles can be used again to show that, at any given time, the radius of the cone at water level is h + 30 r= 20 Therefore, the volume of water can be expressed as 55. V = 3t , 0 ≤ t ≤ 8 . The height is always increasing, so h '(t ) > 0. The rate of change of the height decreases from time t = 0 until time t1 when the water reaches the middle of the rounded bottom part. The rate of change then increases until time t2 when the water reaches the middle of the neck. Then the rate of change decreases until t = 8 and the vase is full. Thus, h ''(t ) > 0 for t1 < t < t2 and h ''(t ) < 0 for t2 < t < 8 . h ( t) 24 t1 t2 8 t π (h + 30) 3 45π − . 1200 2 We also know that V = 2t from above. Setting the two volume equations equal to each other and 2400 t + 27000 − 30 . solving for h gives h = 3 V= π Instructor’s Resource Manual Section 3.2 167 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 56. V = 20 − .1t , 0 ≤ t ≤ 200 . The height of the water is always decreasing so h '(t ) < 0 . The rate of change in the height increases (the rate is negative, and its absolute value decreases) for the first 100 days and then decreases for the remaining time. Therefore we have h ''(t ) > 0 for 0 < t < 100 , and h ''(t ) < 0 for 100 < t < 200 . 3.3 Concepts Review 1. maximum 2. maximum; minimum 3. maximum 4. local maximum, local minimum, 0 Problem Set 3.3 1. f ( x) = 3 x 2 –12 x = 3 x( x – 4) Critical points: 0, 4 f ( x) > 0 on (– ∞ , 0), f ( x) < 0 on (0, 4), f ( x) > 0 on (4, ∞ ) f ( x) = 6 x –12; f (0) = –12, f (4) = 12. Local minimum at x = 4; local maximum at x = 0 2. f ( x) = 3 x 2 –12 = 3( x 2 – 4) Critical points: –2, 2 f ( x) > 0 on (– ∞ , –2), f ( x) < 0 on (–2, 2), f ( x) > 0 on (2, ∞ ) f ( x) = 6 x; f (–2) = –12, f (2) = 12 Local minimum at x = 2; local maximum at x = –2 57. a. The cross-sectional area of the vase is approximately equal to ΔV and the corresponding radius is r = ΔV / π . The table below gives the approximate values for r. The vase becomes slightly narrower as you move above the base, and then gets wider as you near the top. Depth V A ≈ ΔV r = ΔV / π 1 4 4 1.13 2 8 4 1.13 3 11 3 0.98 4 14 3 0.98 5 20 6 1.38 6 28 8 1.60 b. Near the base, this vase is like the one in part (a), but just above the base it becomes larger. Near the middle of the vase it becomes very narrow. The top of the vase is similar to the one in part (a). 168 Depth V A ≈ ΔV r = ΔV / π 1 4 4 1.13 2 9 5 1.26 3 12 3 0.98 4 14 2 0.80 5 20 6 1.38 6 28 8 1.60 Section 3.3 3. 4. ⎛ π⎞ f ( ) = 2 cos 2 ; 2 cos 2 ≠ 0 on ⎜ 0, ⎟ ⎝ 4⎠ No critical points; no local maxima or minima on ⎛ π⎞ ⎜ 0, ⎟ . ⎝ 4⎠ 1 1 1 + cos x; + cos x = 0 when cos x = – . 2 2 2 2π 4π , Critical points: 3 3 ⎛ 2π ⎞ ⎛ 2π 4π ⎞ f ( x) > 0 on ⎜ 0, ⎟ , f ( x) < 0 on ⎜ , ⎟, 3 ⎝ ⎠ ⎝ 3 3 ⎠ ⎛ 4π ⎞ f ( x) > 0 on ⎜ , 2π ⎟ ⎝ 3 ⎠ f ( x) = 3 3 ⎛ 2π ⎞ ⎛ 4π ⎞ f ( x) = – sin x; f ⎜ ⎟ = – , f ⎜ ⎟= 3 2 3 2 ⎝ ⎠ ⎝ ⎠ 4π ; local maximum at Local minimum at x = 3 2π x= . 3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ( ) = 2sin cos 5. − π < < 9. h ' ( y ) = 2 y + π 2 2 Critical point: 0 ⎛ π ⎞ ( ) < 0 on ⎜ − , 0 ⎟ , ⎝ 2 ⎠ ( ) = 2 cos 2 – 2sin 2 ; Local minimum at x = 0 ⎛ π⎞ ( ) > 0 on ⎜ 0, ⎟ , ⎝ 2⎠ (0) = 2 ( x 2 + 4 ) ⋅1 − x ( 2 x ) = 4 − x 2 2 2 ( x2 + 4) ( x2 + 4) Critical points: −2, 2 f ' ( x ) < 0 on ( −∞, −2 ) and ( 2, ∞ ) ; f ' ( x ) > 0 on ( −2, 2 ) f '' ( x ) = ( 2 x x 2 − 12 ( x2 + 4) ) 1+ z2 ) ( 2z ) − z2 ( 2z ) ( 2z g '( z ) = = 2 2 2 (1 + z ) (1 + z 2 ) Critical point: z = 0 g ' ( z ) < 0 on ( −∞, 0 ) g ' ( z ) > 0 on ( 0, ∞ ) g '' ( z ) = ( ) −2 3 z 2 − 1 ( 2 ) z +1 3 g '' ( 0 ) = 2 Local minima at z = 0 . Instructor’s Resource Manual 4 2 ⎛ 34⎞ 2 h⎜ − ⎟ = 2− 3 ⎝ 2 ⎠ − 24 ( ) Local minima at − 10. f '( x) = (x 2 ) 3 16 =6 4 = 2+ 3 4 2 + 1 ( 3) − ( 3 x + 1)( 2 x ) (x 2 ) +1 2 = 3 − 2 x − 3x 2 (x 2 ) +1 2 The only critical points are stationary points. Find these by setting the numerator equal to 0 and solving. 3 − 2 x − 3x2 = 0 a = −3, b = −2, c = 3 3 1 1 f '' ( −2 ) = ; f '' ( 2 ) = − 16 16 Local minima at x = −2 ; Local maxima at x = 2 8. 3 3 ⎛ 4⎞ h ' ( y ) < 0 on ⎜ −∞, − ⎟ 2 ⎠ ⎝ ⎛ 34 ⎞ , 0 ⎟ and ( 0, ∞ ) h ' ( y ) > 0 on ⎜ − ⎝ 2 ⎠ 2 h '' ( y ) = 2 − 3 y r ( x ) = 12 x 2 ; r (0) = 0; the Second Derivative Test fails. Local minimum at z = 0; no local maxima f '( x) = y2 Critical point: − 6. r ( z ) = 4 z 3 Critical point: 0 r ( z ) < 0 on (−∞, 0); r ( z ) > 0 on (0, ∞) 7. 1 x= 2± ( −2 )2 − 4 ( −3)( 3) 2 ± 40 = 2 ( −3) −6 = −1 ± 10 3 −1 − 10 −1 + 10 and 3 3 ⎛ −1 − 10 ⎞ f ' ( x ) < 0 on ⎜⎜ −∞, ⎟⎟ and 3 ⎝ ⎠ ⎛ −1 + 10 ⎞ , ∞ ⎟⎟ . ⎜⎜ 3 ⎝ ⎠ ⎛ −1 − 10 −1 + 10 ⎞ f ' ( 0 ) > 0 on ⎜⎜ , ⎟⎟ 3 3 ⎝ ⎠ Critical points: f '' ( x ) = ( ) 2 3x3 + 3x 2 − 9 x − 1 (x 2 ) +1 3 ⎛ −1 − 10 ⎞ f '' ⎜⎜ ⎟⎟ ≈ 0.739 3 ⎝ ⎠ ⎛ −1 + 10 ⎞ f '' ⎜⎜ ⎟⎟ ≈ −2.739 3 ⎝ ⎠ Local minima at x = −1 − 10 ; 3 Local maxima at x = −1 + 10 3 Section 3.3 169 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 11. f ( x) = 3 x 2 – 3 = 3( x 2 – 1) Critical points: –1, 1 f ( x) = 6 x; f (–1) = –6, f (1) = 6 Local minimum value f(1) = –2; local maximum value f(–1) = 2 8/3 ⎛ ⎛ 2 ⎞5 / 3 ⎞ 6 ⎛ 15 ⎞ ;r ⎜–⎜ ⎟ ⎟ = – ⎜ ⎟ ⎜ ⎝ 15 ⎠ ⎟ 25 ⎝ 2 ⎠ 25s8 / 5 ⎝ ⎠ ⎛ ⎛ 2 ⎞5 3 ⎞ r ( s ) < 0 on ⎜ − ⎜ ⎟ , 0 ⎟ , r ( s ) > 0 on (0, ∞) ⎜ ⎝ 15 ⎠ ⎟ ⎝ ⎠ 12. g ( x) = 4 x3 + 2 x = 2 x(2 x 2 + 1) Critical point: 0 Local minimum value r(0) = 0; local maximum value 5/3 2/3 2/3 ⎛ ⎛ 2 ⎞5 / 3 ⎞ 3⎛ 2 ⎞ ⎛ 2⎞ ⎛ 2⎞ r ⎜ – ⎜ ⎟ ⎟ = –3 ⎜ ⎟ + ⎜ ⎟ = ⎜ ⎟ ⎜ ⎝ 15 ⎠ ⎟ 5 ⎝ 15 ⎠ ⎝ 15 ⎠ ⎝ 15 ⎠ ⎝ ⎠ g ( x) = 12 x 2 + 2; g (0) = 2 Local minimum value g(0) = 3; no local maximum values 13. H ( x ) = 4 x3 – 6 x 2 = 2 x 2 (2 x – 3) 17. 3 Critical points: 0, 2 H ( x) = 12 x 2 – 12 x = 12 x( x – 1); H (0) = 0, ⎛3⎞ H ⎜ ⎟=9 ⎝2⎠ 18. ⎛ 3⎞ H ( x) < 0 on (−∞, 0), H ( x) < 0 on ⎜ 0, ⎟ ⎝ 2⎠ 27 ⎛3⎞ Local minimum value H ⎜ ⎟ = – ; no local 2 16 ⎝ ⎠ maximum values (x = 0 is neither a local minimum nor maximum) 14. 19. f ( x) > 0 on (−∞, 2), f ( x) > 0 on (2, ∞) No local minimum or maximum values 20. 2 ; g (t ) does not exist at t = 2. 3(t – 2)1/ 3 Critical point: 2 2 2 g (1) = , g (3) = – 3 3 No local minimum values; local maximum value g(2) = π . ⎛ 2⎞ s = –⎜ ⎟ ⎝ 15 ⎠ 2 5s 3 / 5 = 15s3 / 5 + 2 5s 3 / 5 ; r ( s ) = 0 when 5/3 , r ( s ) does not exist at s = 0. ⎛ 2⎞ Critical points: – ⎜ ⎟ ⎝ 15 ⎠ 170 Section 3.3 1 t2 No critical points No local minimum or maximum values f ( x) = x( x 2 + 8) ( x 2 + 4)3 / 2 Critical point: 0 f ( x) < 0 on (−∞, 0), f ( x) > 0 on (0, ∞) Local minimum value f(0) = 0, no local maximum values ( )=– 1 ; 1 + sin ( ) does not exist at = f ( x) = 20( x – 2)3 ; f (2) = 0 16. r ( s ) = 3 + f (t ) = 1 + 3π , but ( ) does not exist at that point 2 either. No critical points No local minimum or maximum values f ( x) = 5( x – 2) 4 Critical point: 2 15. g (t ) = – 6 r (s) = – 5/3 ,0 g( )= sin cos ; g ( ) = 0 when sin = π 3π ; , 2 2 g ( ) does not exist at x = π . ⎛ π⎞ Split the x -axis into the intervals ⎜ 0, ⎟ , ⎝ 2⎠ ⎛ π ⎞ ⎛ 3π ⎞ ⎛ 3π ⎞ ⎜ , π ⎟ , ⎜ π, ⎟ , ⎜ , 2π ⎟ . 2 2 2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ π 3π 5π 7 π ⎛π⎞ 1 Test points: , , , ; g ⎜ ⎟ = , 4 4 4 4 2 ⎝4⎠ 1 1 ⎛ 3π ⎞ ⎛ 5π ⎞ 1 ⎛ 7π ⎞ g ⎜ ⎟=– ,g ⎜ ⎟= ,g ⎜ ⎟=– 2 2 2 ⎝ 4 ⎠ ⎝ 4 ⎠ ⎝ 4 ⎠ Local minimum value g( π ) = 0; local maximum ⎛π⎞ ⎛ 3π ⎞ values g ⎜ ⎟ = 1 and g ⎜ ⎟ = 1 ⎝2⎠ ⎝ 2 ⎠ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 21. f ' ( x ) = 4 ( sin 2 x )( cos 2 x ) 4 ( sin 2 x )( cos 2 x ) = 0 when x = ( 2k − 1) π kπ where k is an integer. x= 2 Critical points: 0, π4 , π2 , 2 4 ⎛π ⎞ Minimum value: f ( 0 ) = f ⎜ ⎟ = 0 ⎝2⎠ π ⎛ ⎞ Maximum value: f ⎜ ⎟ = 1 ⎝4⎠ 22. f '( x) = ( (x 2 +4 ) f ' ( x ) = 0 when x = 2 or x = −2 . (there are no 23. g ' ( x ) = ( − x x3 − 64 ( x3 + 32 ) 1 2 ) 2 g ' ( x ) = 0 when x = 0 or x = 4 . Critical points: 0, 4 1 g ( 0) = 0 ; g ( 4) = 6 As x approaches ∞ , the value of g approaches 0 but never actually gets there. 1 Maximum value: g ( 4 ) = 6 Minimum value: g ( 0 ) = 0 24. h ' ( x ) = – 4 = 0 when x = 9 16 9 . 16 ⎛ 9⎞ ⎛9 ⎞ F ( x) > 0 on ⎜ 0, ⎟ , F ( x) < 0 on ⎜ , ∞ ⎟ ⎝ 16 ⎠ ⎝ 16 ⎠ 9 ⎛ ⎞ F decreases without bound on ⎜ , ∞ ⎟ . No ⎝ 16 ⎠ ⎛9⎞ 9 minimum values; maximum value F ⎜ ⎟ = ⎝ 16 ⎠ 4 ) Maximum value: f ( 2 ) = 3 26. From Problem 25, the critical points are 0 and 2 singular points) Critical points: 0, 2 (note: −2 is not in the given domain) 1 f ( 0 ) = 0 ; f ( 2 ) = ; f ( x ) → 0 as x → ∞ . 2 Minimum value: f ( 0 ) = 0 x – 4; x 9 Critical points: 0, , 4 16 ⎛9⎞ 9 F(0) = 0, F ⎜ ⎟ = , F(4) = –4 ⎝ 16 ⎠ 4 Minimum value F(4) = –4; maximum value ⎛9⎞ 9 F⎜ ⎟= ⎝ 16 ⎠ 4 or ⎛π ⎞ ⎛π ⎞ f (0) = 0 ; f ⎜ ⎟ = 1 ; f ⎜ ⎟ = 0 ; ⎝4⎠ ⎝2⎠ f ( 2 ) ≈ 0.5728 −2 x 2 − 4 3 25. F ( x) = 27. f ( x) = 64(−1)(sin x)−2 cos x +27(−1)(cos x)−2 (− sin x) =− = 64 cos x 2 sin x + 27 sin x cos 2 x 2 2 (3sin x − 4 cos x)(9 sin x + 12 cos x sin x + 16 cos x) 2 2 sin x cos x ⎛ π⎞ On ⎜ 0, ⎟ , f ( x) = 0 only where 3sin x = 4cos x; ⎝ 2⎠ 4 tan x = ; 3 4 x = tan −1 ≈ 0.9273 3 Critical point: 0.9273 For 0 < x < 0.9273, f ( x) < 0, while for 0.9273 < x < π 2 , f '( x) > 0 4 ⎞ 64 27 ⎛ Minimum value f ⎜ tan −1 ⎟ = + = 125; 3 3⎠ 4 ⎝ 5 5 no maximum value −2 x ( x2 + 4) 2 h ' ( x ) = 0 when x = 0 . (there are no singular points) Critical points: 0 Since h ' ( x ) < 0 for x > 0 , the function is always decreasing. Thus, there is no minimum value. 1 Maximum value: h ( 0 ) = 4 Instructor’s Resource Manual Section 3.3 171 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 28. g ( x) = 2 x + = 2x + (8 − x)2 (32 x) − (16 x 2 )2(8 − x)(−1) (8 − x) 256 x = (8 − x )3 32. (1, 2) ∪ (3, 4). Thus, the function has a local minimum at x = 1,3 and a local maximum at x = 2, 4 . 2 x[(8 − x)3 + 128] (8 − x)3 For x > 8, g ( x) = 0 when (8 − x)3 + 128 = 0; (8 − x)3 = −128; 8 − x = − 3 128 ; 33. (−∞,1) ∪ (1, 2) ∪ (2,3) ∪ (4, ∞) Thus, the function has a local minimum at x = 4 and a local maximum at x = 3 . g ( x) < 0 on (8, 8 + 4 3 2), 29. H ' ( x ) = ( ) f '( x) = 0 at x = 1, 2,3, 4 ; f '( x) is negative on (3, 4) and positive on x = 8 + 4 3 2 ≈ 13.04 g ( x) > 0 on (8 + 4 3 2, ∞) g(13.04) ≈ 277 is the minimum value f '( x) = 0 at x = 1, 2,3, 4 ; f '( x) is negative on (−∞,1) ∪ (2,3) ∪ (4, ∞) and positive on 4 34. Since f ' ( x ) ≥ 0 for all x, the function is always 2 x x2 − 1 increasing. Therefore, there are no local extrema. x2 − 1 35. Since f ' ( x ) ≥ 0 for all x, the function is always H ' ( x ) = 0 when x = 0 . increasing. Therefore, there are no local extrema. H ' ( x ) is undefined when x = −1 or x = 1 Critical points: −2 , −1 , 0, 1, 2 H ( −2 ) = 3 ; H ( −1) = 0 ; H ( 0 ) = 1 ; H (1) = 0 ; H ( 2) = 3 Minimum value: H ( −1) = H (1) = 0 Maximum value: H ( −2 ) = H ( 2 ) = 3 36. f ' ( x ) = 0 at x = 0, A, and B . f ' ( x ) is negative on ( −∞, 0 ) and ( A, B ) f ' ( x ) is positive on ( 0, A ) and ( B, ∞ ) Therefore, the function has a local minimum at x = 0 and x = B , and a local maximum at x = A . 37. Answers will vary. One possibility: y 30. h ' ( t ) = 2t cos t 2 h ' ( t ) = 0 when t = 0 , t = t= 10π 2 3π 5π (Consider t = , t = , and t 2 = ) 2 2 2 2π 6π 10π Critical points: 0, , , ,π 2 2 2 ⎛ 2π ⎞ ⎛ 6π ⎞ h ( 0) = 0 ; h ⎜ ⎟ = 1; h⎜ ⎟ = −1 ; ⎝ 2 ⎠ ⎝ 2 ⎠ 2 π 2π 6π , t= , and 2 2 3 6 x 2 ⎛ 10π ⎞ h⎜ ⎟ = 1 ; h (π ) ≈ −0.4303 ⎝ 2 ⎠ ⎛ 6π ⎞ Minimum value: h ⎜ ⎟ = −1 ⎝ 2 ⎠ ⎛ 2π ⎞ ⎛ 10π ⎞ Maximum value: h ⎜ ⎟ = h⎜ ⎟ =1 ⎝ 2 ⎠ ⎝ 2 ⎠ 31. 5 −5 38. Answers will vary. One possibility: y 5 3 6 x −5 f '( x) = 0 when x = 0 and x = 1 . On the interval (−∞, 0) we get f '( x) < 0 . On (0, ∞) , we get f '( x) > 0 . Thus there is a local min at x = 0 but no local max. 172 Section 3.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 39. Answers will vary. One possibility: y 5 3 6 x 43. The graph of f is a parabola which opens up. B f ' ( x ) = 2 Ax + B = 0 → x = − 2A f '' ( x ) = 2 A Since A > 0 , the graph of f is always concave up. There is exactly one critical point which yields the minimum of the graph. 2 ⎛ B ⎞ ⎛ B ⎞ ⎛ B ⎞ f ⎜− ⎟ = A⎜ − ⎟ + B⎜− ⎟+C ⎝ 2A ⎠ ⎝ 2A ⎠ ⎝ 2A ⎠ B2 B2 − +C 4A 2A B 2 − 2 B 2 + 4 AC = 4A B 2 − 4 AC 4 AC − B 2 = =− 4A 4A −5 = 40. Answers will vary. One possibility: y 5 ( ) If f ( x ) ≥ 0 with A > 0 , then − B 2 − 4 AC ≥ 0 , 3 6 x −5 41. Answers will vary. One possibility: y or B 2 − 4 AC ≤ 0 . ⎛ B ⎞ If B 2 − 4 AC ≤ 0 , then we get f ⎜ − ⎟≥0 ⎝ 2A ⎠ ⎛ B ⎞ Since 0 ≤ f ⎜ − ⎟ ≤ f ( x ) for all x, we get ⎝ 2A ⎠ f ( x ) ≥ 0 for all x. 44. A third degree polynomial will have at most two extrema. f ' ( x ) = 3 Ax 2 + 2 Bx + C 5 f '' ( x ) = 6 Ax + 2 B 3 6 x Critical points are obtained by solving f ' ( x ) = 0 . 3 Ax 2 + 2 Bx + C = 0 −5 x= 42. Answers will vary. One possibility: y = 5 −2 B ± 4 B 2 − 12 AC 6A −2 B ± 2 B 2 − 3 AC 6A − B ± B 2 − 3 AC 3A To have a relative maximum and a relative minimum, we must have two solutions to the above quadratic equation. That is, we must have B 2 − 3 AC > 0 . = 3 −5 6 x The two solutions would be − B − B 2 − 3 AC 3A − B + B 2 − 3 AC . Evaluating the second 3A derivative at each of these values gives: and Instructor’s Resource Manual Section 3.3 173 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ⎛ − B − B 2 − 3 AC f '' ⎜ ⎜ 3A ⎝ Problem Set 3.4 ⎞ ⎟ ⎟ ⎠ ⎛ − B − B 2 − 3 AC = 6A⎜ ⎜ 3A ⎝ ⎞ ⎟ + 2B ⎟ ⎠ = −2 B − 2 B 2 − 3 AC + 2 B = −2 B 2 − 3 AC and ⎛ − B + B 2 − 3 AC f '' ⎜ ⎜ 3A ⎝ ⎞ ⎟ ⎟ ⎠ ⎛ − B + B 2 − 3 AC = 6A⎜ ⎜ 3A ⎝ ⎞ ⎟ + 2B ⎟ ⎠ = −2 B + 2 B 2 − 3 AC + 2 B = 2 B 2 − 3 AC If B 2 − 3 AC > 0 , then −2 B 2 − 3 AC exists and is negative, and 2 B 2 − 3 AC exists and is positive. Thus, from the Second Derivative Test, − B − B 2 − 3 AC would yield a local maximum 3A − B + B 2 − 3 AC would yield a local 3A minimum. and 45. f (c) > 0 implies that f is increasing at c, so f is concave up to the right of c (since f ( x) > 0 to the right of c) and concave down to the left of c (since f ( x) < 0 to the left of c). Therefore f has a point of inflection at c. 3.4 Concepts Review 1. 0 < x < ∞ 200 2. 2x + x 3. S = n i =1 ( yi − bxi )2 4. marginal revenue; marginal cost 174 Section 3.4 1. Let x be one number, y be the other, and Q be the sum of the squares. xy = –16 16 y=– x The possible values for x are in (– ∞ , 0) or (0, ∞) . 256 Q = x2 + y 2 = x2 + x2 dQ 512 = 2x – dx x3 512 =0 2x – x3 x 4 = 256 x = ±4 The critical points are –4, 4. dQ dQ < 0 on (– ∞ , –4) and (0, 4). > 0 on dx dx (–4, 0) and (4, ∞ ). When x = –4, y = 4 and when x = 4, y = –4. The two numbers are –4 and 4. 2. Let x be the number. Q = x – 8x x will be in the interval (0, ∞ ). dQ 1 –1/ 2 = x –8 dx 2 1 –1/ 2 x –8 = 0 2 x –1/ 2 = 16 1 x= 256 dQ > 0 on dx 1 ⎞ dQ ⎛ ⎛ 1 ⎞ < 0 on ⎜ , ∞ ⎟. ⎜ 0, ⎟ and dx ⎝ 256 ⎠ ⎝ 256 ⎠ 1 . Q attains its maximum value at x = 256 3. Let x be the number. Q = 4 x – 2x x will be in the interval (0, ∞ ). dQ 1 –3 / 4 = x –2 dx 4 1 –3 / 4 x –2=0 4 x –3 / 4 = 8 1 x= 16 dQ dQ ⎛ 1⎞ ⎛1 ⎞ > 0 on ⎜ 0, ⎟ and < 0 on ⎜ , ∞ ⎟ 16 16 dx dx ⎝ ⎠ ⎝ ⎠ 1 Q attains its maximum value at x = . 16 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4. Let x be one number, y be the other, and Q be the sum of the squares. xy = –12 12 y=– x The possible values for x are in (– ∞ , 0) or (0, ∞) . Q = x2 + y2 = x2 + 144 x2 dQ 288 = 2x – dx x3 288 2x – =0 x3 x 4 = 144 x = ±2 3 The critical points are –2 3, 2 3 dQ < 0 on (– ∞, – 2 3) and (0, 2 3). dx dQ > 0 on (–2 3, 0) and (2 3, ∞). dx When x = –2 3, y = 2 3 and when x = 2 3, y = –2 3. The two numbers are –2 3 and 2 3. 5. Let Q be the square of the distance between (x, y) and (0, 5). Q = ( x – 0)2 + ( y – 5)2 = x 2 + ( x 2 – 5)2 = x 4 – 9 x 2 + 25 dQ = 4 x3 – 18 x dx 4 x3 – 18 x = 0 2 x(2 x 2 – 9) = 0 x = 0, ± 3 2 3 ⎞ dQ ⎛ < 0 on ⎜ – ∞, – ⎟ and dx 2⎠ ⎝ dQ ⎛ 3 ⎞ ⎛ > 0 on ⎜ – , 0 ⎟ and ⎜ dx 2 ⎝ ⎠ ⎝ 3 ⎞ ⎛ ⎜ 0, ⎟. 2⎠ ⎝ 3 ⎞ , ∞ ⎟. 2 ⎠ 3 9 3 When x = – , y = and when x = , 2 2 2 9 y= . 2 ⎛ 3 9⎞ ⎛ 3 9⎞ The points are ⎜ – , ⎟ and ⎜ , ⎟. 2 2⎠ ⎝ ⎝ 2 2⎠ Instructor’s Resource Manual 6. Let Q be the square of the distance between (x, y) and (10, 0). Q = ( x – 10)2 + ( y – 0) 2 = (2 y 2 – 10) 2 + y 2 = 4 y 4 – 39 y 2 + 100 dQ = 16 y 3 – 78 y dy 16 y 3 – 78 y = 0 2 y (8 y 2 – 39) = 0 y = 0, ± dQ dy dQ dy 39 2 2 ⎛ ⎛ 39 ⎞ 39 ⎞ < 0 on ⎜⎜ – ∞, – ⎟⎟ and ⎜⎜ 0, ⎟⎟ . 2 2⎠ ⎝ ⎝ 2 2⎠ ⎛ ⎛ 39 ⎞ 39 ⎞ , 0 ⎟⎟ and ⎜⎜ , ∞ ⎟⎟ . > 0 on ⎜⎜ – ⎝ 2 2 ⎠ ⎝2 2 ⎠ When y = – y= 39 2 2 39 2 2 ,x= ,x= 39 and when 4 39 . 4 ⎛ 39 ⎛ 39 39 ⎞ 39 ⎞ The points are ⎜⎜ , – ⎟⎟ and ⎜⎜ , ⎟⎟ . 2 2⎠ ⎝ 4 ⎝ 4 2 2⎠ 7. x ≥ x 2 if 0 ≤ x ≤ 1 f ( x) = x − x 2 ; f ( x) = 1 − 2 x; f ( x) = 0 when x = 1 2 1 Critical points: 0, , 1 2 1 ⎛1⎞ 1 f(0) = 0, f(1) = 0, f ⎜ ⎟ = ; therefore, 2 4 2 ⎝ ⎠ exceeds its square by the maximum amount. 8. For a rectangle with perimeter K and width x, the K − x . Then the area is length is 2 ⎛K ⎞ Kx A = x⎜ − x⎟ = − x2 . ⎝2 ⎠ 2 dA K dA K = − 2 x; = 0 when x = dx 2 dx 4 K K Critical points: 0, , 4 2 K2 K K , A = 0; at x = , A = . At x = 0 or 16 2 4 The area is maximized when the width is one fourth of the perimeter, so the rectangle is a square. Section 3.4 175 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 9. Let x be the width of the square to be cut out and V the volume of the resulting open box. V = x (24 − 2 x)2 = 4 x3 − 96 x 2 + 576 x dV = 12 x 2 − 192 x + 576 = 12( x − 12)( x − 4); dx 12(x – 12)(x – 4) = 0; x = 12 or x = 4. Critical points: 0, 4, 12 At x = 0 or 12, V = 0; at x = 4, V = 1024. 3 The volume of the largest box is 1024 in. 10. Let A be the area of the pen. dA A = x(80 − 2 x) = 80 x − 2 x 2 ; = 80 − 4 x; dx 80 − 4 x = 0; x = 20 Critical points: 0, 20, 40. At x = 0 or 40, A = 0; at x = 20, A = 800. The dimensions are 20 ft by 80 – 2(20) = 40 ft, with the length along the barn being 40 ft. 11. Let x be the width of each pen, then the length along the barn is 80 – 4x. dA A = x(80 − 4 x) = 80 x − 4 x 2 ; = 80 − 8 x; dx dA = 0 when x = 10. dx Critical points: 0, 10, 20 At x = 0 or 20, A = 0; at x = 10, A = 400. The area is largest with width 10 ft and length 40 ft. 12. Let A be the area of the pen. The perimeter is 100 + 180 = 280 ft. y + y – 100 + 2x = 180; y = 140 – x dA A = x(140 − x ) = 140 x − x 2 ; = 140 − 2 x; dx 140 − 2 x = 0; x = 70 Since 0 ≤ x ≤ 40 , the critical points are 0 and 40. When x = 0, A = 0. When x = 40, A = 4000. The dimensions are 40 ft by 100 ft. 900 x The possible values for x are in (0, ∞ ). 2700 ⎛ 900 ⎞ Q = 4x + 3 y = 4x + 3⎜ ⎟ = 4x + x ⎝ x ⎠ dQ 2700 = 4− dx x2 2700 4– =0 x2 13. xy = 900; y = x 2 = 675 x = ±15 3 x = 15 3 is the only critical point in (0, ∞ ). 176 Section 3.4 dQ < 0 on (0, 15 3) and dx dQ > 0 on (15 3, ∞). dx 900 When x = 15 3, y = = 20 3. 15 3 Q has a minimum when x = 15 3 ≈ 25.98 ft and y = 20 3 ≈ 34.64 ft. 300 x The possible values for x are in (0, ∞ ). 1200 Q = 6x + 4 y = 6x + x dQ 1200 =6– dx x2 1200 =0 6– x2 14. xy = 300; y = x 2 = 200 x = ±10 2 x = 10 2 is the only critical point in (0, ∞ ). dQ dQ < 0 on (0, 10 2) and > 0 on (10 2, ∞) dx dx 300 When x = 10 2, y = = 15 2. 10 2 Q has a minimum when x = 10 2 ≈ 14.14 ft and y = 15 2 ≈ 21.21 ft. 300 x The possible values for x are in (0, ). 15. xy = 300; y = Q = 3(6x + 2y) + 2(2y) = 18x + 10y = 18x + 3000 x dQ 3000 = 18 – dx x2 3000 =0 18 – x2 500 x2 = 3 x=± x= 10 5 3 10 5 3 is the only critical point in (0, ). ⎛ 10 5 ⎞ ⎜⎜ 0, ⎟ and 3 ⎟⎠ ⎝ ⎛ 10 5 ⎞ dQ , ∞ ⎟⎟ . > 0 on ⎜⎜ dx ⎝ 3 ⎠ dQ < 0 on dx Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. When x = 10 5 3 ,y= 300 10 5 3 = 6 15 Q has a minimum when x = 10 5 3 ≈ 12.91 ft and x = y y = 6 15 ≈ 23.24 ft. 900 16. xy = 900; y = x The possible values for x are in (0, ∞ ). 3600 Q = 6x + 4 y = 6x + x dQ 3600 =6– dx x2 3600 6– =0 x2 x 2 = 600 x = ±10 6 x = 10 6 is the only critical point in (0, ∞ ). dQ dQ < 0 on (0, 10 6) and > 0 on (10 6, ∞). dx dx 900 When x = 10 6, y = = 15 6 10 6 Q has a minimum when x = 10 6 ≈ 24.49 ft and y = 15 6 ≈ 36.74. x 2 = . y 3 Suppose that each pen has area A. A xy = A; y = x The possible values for x are in (0, ∞ ). 4A Q = 6x + 4 y = 6x + x dQ 4A =6– dx x2 4A 6– =0 x2 2A x2 = 3 It appears that x=± x= dQ dx dQ dx 2A ,y= 3 When x = 2A 3 3A 2 = A 2A 3 = 3A 2 2 3 17. Let D be the square of the distance. ⎛ x2 ⎞ D = ( x − 0) + ( y − 4) = x + ⎜ − 4⎟ ⎜ 4 ⎟ ⎝ ⎠ 2 = 2 2 2 x4 − x 2 + 16 16 dD x3 x3 = − 2 x; − 2 x = 0; x( x 2 − 8) = 0 dx 4 4 x = 0, x = ± 2 2 Critical points: 0, 2 2, 2 3 Since D is continuous and we are considering a closed interval for x, there is a maximum and minimum value of D on the interval. These extrema must occur at one of the critical points. At x = 0, y = 0, and D = 16. At x = 2 2, y = 2, and D = 12. At x = 2 3 , y = 3, and D = 13. Therefore, the point on y = ( ) x2 closest to ( 0, 4 ) is 4 P 2 2, 2 and the point farthest from ( 0, 4 ) is Q ( 0, 0 ) . 18. Let r1 and h1 be the radius and altitude of the outer cone; r2 and h2 the radius and altitude of the inner cone. 3V1 1 V1 = πr12 h1 is fixed. r1 = πh1 3 By similar triangles h1 – h2 r2 = (see figure). h1 r1 2A 3 2A is the only critical point on (0, ∞ ). 3 ⎛ 2A ⎞ < 0 on ⎜⎜ 0, ⎟ and 3 ⎟⎠ ⎝ ⎛ 2A ⎞ , ∞ ⎟⎟ . > 0 on ⎜⎜ ⎝ 3 ⎠ Instructor’s Resource Manual ⎛ h r2 = r1 ⎜ 1 – 2 h1 ⎝ ⎞ ⎟= ⎠ 3V1 πh1 ⎛ h2 ⎞ ⎜1 – ⎟ h1 ⎠ ⎝ Section 3.4 177 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2 1 1 ⎡ 3V1 ⎛ h2 ⎞ ⎤ V2 = πr22 h2 = π ⎢ ⎜ 1 – ⎟ ⎥ h2 3 3 ⎢⎣ πh1 ⎝ h1 ⎠ ⎦⎥ 2 2 h2 ⎛ h2 ⎞ π 3V1h2 ⎛ h2 ⎞ ⋅ ⎜ 1 – ⎟ = V1 ⎜ 1 – ⎟ 3 πh1 ⎝ h1 ⎠ h1 ⎝ h1 ⎠ h Let k = 2 , the ratio of the altitudes of the cones, h1 = then V2 = V1k (1 – k ) 2 . dV2 = V1 (1 – k ) 2 – 2kV1 (1 – k ) = V1 (1 – k )(1 – 3k ) dk dV2 1 0 < k < 1 so = 0 when k = . 3 dk d 2V2 d 2V2 1 = V1 (6k − 4); < 0 when k = 2 2 3 dk dk 1 The altitude of the inner cone must be the 3 altitude of the outer cone. 19. Let x be the distance from P to where the woman lands the boat. She must row a distance of x 2 + 4 miles and walk 10 – x miles. This will x 2 + 4 10 – x + hours; 3 4 1 x 0 ≤ x ≤ 10. T ( x) = – ; T ( x) = 0 3 x2 + 4 4 take her T ( x) = when x = T (0) = 6 7 . 19 hr = 3 hr 10 min ≈ 3.17 hr , 6 ⎛ 6 ⎞ 15 + 7 T⎜ ≈ 2.94 hr, ⎟= 6 ⎝ 7⎠ shore from P. 6 x= 2491 13 ⎛ 6 ⎞ T (0) = ≈ 0.867 hr; T ⎜ ⎟ ≈ 0.865 hr; 15 ⎝ 2491 ⎠ T (10) ≈ 3.399 hr ≈ 0.12 mi down x 2 + 4 10 – x + , 0 ≤ x ≤ 10. 20 4 x 1 T ( x) = – ; T ( x) = 0 has no solution. 2 20 x + 4 4 21. T ( x) = T (0) = 2 10 13 + = hr = 2 hr, 36 min 20 4 5 104 ≈ 0.5 hr 20 She should take the boat all the way to town. T (10) = 22. Let x be the length of cable on land, 0 ≤ x ≤ L. Let C be the cost. C = a ( L − x) 2 + w2 + bx dC a( L − x) =− +b dx ( L − x ) 2 + w2 − a( L − x) ( L − x ) 2 + w2 + b = 0 when (a 2 − b 2 )( L − x) 2 = b 2 w2 6 7 ≈ 2.27 mi down the x = L− aw d 2C = bw 2 a – b2 ft on land; ft under water aw2 dx 2 [( L − x)2 + w2 ]3 2 minimizes the cost. Section 3.4 2491 the shore from P. a 2 – b2 178 6 She should land the boat b 2 [( L − x) 2 + w2 ] = a 2 ( L − x )2 104 T (10) = ≈ 3.40 hr 3 She should land the boat x 2 + 4 10 – x + , 0 ≤ x ≤ 10. 3 50 x 1 T ( x) = – ; T ( x) = 0 when 3 x 2 – 4 50 20. T ( x) = > 0 for all x, so this Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 23. Let the coordinates of the first ship at 7:00 a.m. be (0, 0). Thus, the coordinates of the second ship at 7:00 a.m. are (–60, 0). Let t be the time in hours since 7:00 a.m. The coordinates of the first and second ships at t are (–20t, 0) and ( −60 + 15 = x= ) + ( 0 + 15 2t ) = (1300 + 600 2 ) t 2 − ( 2400 + 1800 2 ) t + 3600 dD = 2 (1300 + 600 2 ) t − ( 2400 + 1800 2 ) dt 2 (1300 + 600 2 ) t − ( 2400 + 1800 2 ) = 0 when t= 12 + 9 2 13 + 6 2 2 D is the minimum at t = 12 + 9 2 13 + 6 2 since d D dt 2 = 2 a a −x ab 2 bx 2 2 a a −x 2 + >0 ⎛ a ⎞ b⎜ ⎟ ⎝ 2⎠ dA < 0 on dx 2 =− b a ⎛ a ⎞ ⎜ 0, ⎟ and 2⎠ ⎝ 2 2 h ⎛h⎞ V = πx 2 h; r 2 = x 2 + ⎜ ⎟ ; x 2 = r 2 − 4 ⎝2⎠ 2 3 ⎛ h ⎞ πh V = π ⎜ r 2 − ⎟ h = πhr 2 − ⎜ ⎟ 4 4 ⎝ ⎠ dV 3πh 2 2 3r = πr 2 − ; V = 0 when h = ± dh 4 3 Since d 2V dh 2 when h = a2 − x2 Find the x-intercept, x0 , of the tangent line through the point (x, y): y–0 bx =– x – x0 a a2 – x2 =− 3πh , the volume is maximized 2 2 3r . 3 ⎛2 3 ⎞ 2 π V = π ⎜⎜ r ⎟⎟ r − ⎝ 3 ⎠ = ay a 2 – x 2 a2 – x2 a2 +x= +x= bx x x Compute the Area A of the resulting triangle and maximize: −1 1 a 3b a 3b ⎛ 2 2⎞ = A = x0 y0 = ⎜x a −x ⎟ ⎠ 2 2 ⎝ 2 x a2 − x2 x0 = Instructor’s Resource Manual b 2 ⎛ a ⎞ b a −⎜ ⎟ = a 2 ⎝ 2⎠ 25. Let x be the radius of the base of the cylinder and h the height. b 2 a − x2 a −2 ⎛ dA a 3b ⎛ x2 2 2⎞ 2 2 =− ⎜ x a − x ⎟ ⎜⎜ a − x − ⎠ ⎝ dx 2 ⎝ a2 − x2 2 ;y= dA ⎛ a ⎞ , a ⎟ , so A is a minimum at > 0 on ⎜ dx ⎝ 2 ⎠ a x= . Then the equation of the tangent line is 2 b⎛ a ⎞ b y = − ⎜x− or bx + ay − ab 2 = 0 . ⎟+ a⎝ 2⎠ 2 b 2 a − x 2 (positive a square root since the point is in the first quadrant). Compute the slope of the tangent line: bx y =− . a a2 − x2 Find the y-intercept, y0 , of the tangent line through the point (x, y): y0 − y bx =− 0− x a a2 − x2 +y= (2 x 2 − a 2 ) = 0 when 2 a Note that 24. Write y in terms of x: y = bx 2 2 3/ 2 (2 x 2 – a 2 ) ⎛ a ⎞ a a2 − ⎜ ⎟ ⎝ 2⎠ for all t. The ships are closest at 8:09 A.M. y0 = 2 y =− ≈ 1.15 hrs or 1 hr, 9 min 2 2 3/ 2 2 x (a − x ) square of the distances at t. 2 2 2 x (a − x ) 2 ) ( 2 a 3b 2t , − 15 2t respectively. Let D be the D = −20t + 60 − 15 2t a 3b ( r) 2 3 3 3 4 2 π 3 3 2 π 3 3 4π 3 3 r − r = r 3 9 9 26. Let r be the radius of the circle, x the length of the rectangle, and y the width of the rectangle. 2 2 x2 y2 ⎛x⎞ ⎛ y⎞ ; P = 2x + 2y; r 2 = ⎜ ⎟ + ⎜ ⎟ ; r 2 = + 4 4 ⎝2⎠ ⎝ 2⎠ ⎞ ⎟ ⎟ ⎠ y = 4r 2 − x 2 ; P = 2 x + 2 4r 2 − x 2 dP 2x = 2− ; dx 4r 2 − x 2 Section 3.4 179 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2x 2− 2 4r − x = 0; 2 4r 2 − x 2 = 2 x; 2 2 16r − 4 x 2 = 4 x 2 ; x = ± 2r d 2P dx 2 8r 2 =− (4r 2 − x 2 )3 2 2 300 3 400 − ≈ 10.874 . 11 11 Critical points: x = 0, 10.874, 25 At x = 0, A ≈ 481; at x = 10.874, A ≈ 272; at x = 25, A = 625. A ' ( x ) = 0 when x = < 0 when x = 2r ; a. 2 y = 4 r − 2r = 2 r The rectangle with maximum perimeter is a square with side length 2r . 27. Let x be the radius of the cylinder, r the radius of the sphere, and h the height of the cylinder. A = 2π xh ; r 2 = x 2 + A = 2π r 2 − ( h2 h2 ; x = r2 − 4 4 h2 h4 h = 2π h 2 r 2 − 4 4 2 3 ) so A is a maximum when h = 2r. r 2 . 28. Let x be the distance from I1. kI kI 2 Q= 1+ 2 x ( s − x)2 2kI 2 dQ −2kI1 = + 3 dx ( s − x )3 x 2kI1 x x= 3 + 2kI 2 ( s − x) 3 = 0; x3 I = 1; I2 ( s − x) 3 s 3 I1 3 d 2Q 6kI1 + ⎛V ⎞ 4kV = 2.7 kx 2 + 4kx ⎜ ⎟ = 2.7 kx 2 + 2 x ⎝x ⎠ dC 4kV dC = 5.4kx − ; = 0 when x ≈ 0.9053 V dx x 2 dx V y≈ ≈ 1.22 3 V (0.9053 V )2 31. Let r be the radius of the cylinder and h the height of the cylinder. V − 23 πr 3 V 2 2 V = πr 2 h + πr 3 ; h = = − r 2 2 3 3 πr πr Let k be the cost per square foot of the cylindrical wall. The cost is C = k (2πrh) + 2k (2πr 2 ) ⎛ 2V 8πr 2 ⎛ ⎞ 2 ⎞ ⎛ V = k ⎜ 2πr ⎜ − r ⎟ + 4πr 2 ⎟ = k ⎜ + ⎜ 3 ⎝ πr 2 3 ⎠ ⎝ ⎠ ⎝ r ⎞ ⎟ ⎟ ⎠ dC ⎛ 2V 16πr ⎞ ⎛ 2V 16πr ⎞ = k ⎜− + + ⎟; k ⎜ − ⎟=0 dr 3 ⎠ ⎝ r2 3 ⎠ ⎝ r2 I1 + 3 I 2 = 30. Let x be the length of the sides of the base, y be the height of the box, and k be the cost per square inch of the material in the sides of the box. The cost is C = 1.2kx 2 + 1.5kx 2 + 4kxy dA dA > 0 on (0, 2r ) and < 0 on ( 2r , 2r ), dh dh − b. For maximum area, the wire should not be cut; it should be bent to form a square. V = x 2 y; dA π 2r h − h ; A = 0 when h = 0, ± 2r = dh 2 2 h4 h r − 4 The dimensions are h = 2r , x = For minimum area, the cut should be approximately 4(10.874) ≈ 43.50 cm from one end and the shorter length should be bent to form the square. 6kI 2 dx 2 x 4 ( s − x) 4 minimizes the sum. > 0, so this point 29. Let x be the length of a side of the square, so 100 − 4 x is the side of the triangle, 0 ≤ x ≤ 25 3 1 ⎛ 100 − 4 x ⎞ 3 ⎛ 100 − 4 x ⎞ A = x2 + ⎜ ⎟ ⎜ ⎟ 2⎝ 3 3 ⎠ 2 ⎝ ⎠ 2⎞ ⎛ 3 10, 000 − 800 x + 16 x = x2 + ⎜ ⎟ ⎟ 4 ⎜⎝ 9 ⎠ when r 3 = h= 1/ 3 3V 1 ⎛ 3V ⎞ ,r = ⎜ ⎟ 8π 2⎝ π ⎠ 1/ 3 1 ⎛ 3V ⎞ − ⎜ ⎟ 2 / 3 3⎝ π ⎠ π 3πV 4V ( ) 13 ⎛ 3V ⎞ =⎜ ⎟ ⎝ π ⎠ For a given volume V, the height of the cylinder is 1/ 3 ⎛ 3V ⎞ ⎜ ⎟ ⎝ π ⎠ 1/ 3 and the radius is 1 ⎛ 3V ⎞ ⎜ ⎟ 2⎝ π ⎠ . dA 200 3 8 3 = 2x − + x dx 9 9 180 Section 3.4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 32. dx = 2 cos 2t − 2 3 sin 2t ; dt dx 1 = 0 when tan2t= ; dt 3 π 2t = + πn for any integer n 6 π π t= + n 12 2 π π When t = + n , 12 2 tan = cos = r ⎛ x = h⎜ ⎜ ⎝ dr r3 > 0, so this minimizes the perimeter. 34. The distance from the fence to the base of the h . ladder is tan The length of the ladder is x. h +w h cos = tan ; x cos = + w; x tan h w + x= sin cos dx h cos w sin w sin 3 − h cos3 =− + ; d sin 2 cos 2 sin 2 cos2 h when tan 3 = w = tan −1 3 3 h + w2 / 3 , w 2/3 + w2 / 3 ⎛ h 2 / 3 + w2 / 3 h 2 / 3 + w2 / 3 ⎞⎟ + w⎜ 3 3 ⎟ ⎜ h w ⎠ ⎝ ⎞ ⎟ ⎟ ⎠ A = 2 x(12 − x 2 ) = 24 x − 2 x3 ; dA = 24 − 6 x 2 ; dx 24 − 6 x 2 = 0; x = −2, 2 Critical points: 0, 2, 12. When x = 0 or 12, A = 0. When x = 2, y = 12 − (2)2 = 8. The dimensions are 2x = 2(2) = 4 by 8. 36. Let the x-axis lie on the diameter of the semicircle and the y-axis pass through the middle. A = 2 xy = 2 x r 2 − x 2 2A = 2r + Q = 2r + r = 2r + 2 r r dQ 2A = 2− ; Q = 0 when r = A dr r2 2A = =2 ( A )2 4A h 2/3 Then the equation y = r 2 − x 2 describes the semicircle. Let (x, y) be the upper-right corner of the rectangle. x is limited by 0 ≤ x ≤ r . 2 = w 35. x is limited by 0 ≤ x ≤ 12 . 2 Ar 2 3 3 ; sin = = ( h 2 / 3 + w 2 / 3 )3 / 2 2A ; = 2 r2 The perimeter is d 2Q h h ⎛π ⎞ ⎛π ⎞ x = sin ⎜ + πn ⎟ + 3 cos ⎜ + πn ⎟ ⎝6 ⎠ ⎝6 ⎠ π π = sin cos πn + cos sin πn 6 6 π π + 3(cos cos πn − sin sin πn) 6 6 1 3 = (−1)n + (−1)n = 2. 2 2 The farthest the weight gets from the origin is 2 units. 33. A = 3 dA 2x2 2 = 2 r 2 − x2 − = (r 2 − 2 x 2 ) 2 2 2 2 dx r −x r −x 2 r (r 2 − 2 x 2 ) = 0; x = 2 2 2 r −x r Critical points: 0, ,r 2 r When x = 0 or r, A = 0. When x = , A = r2. 2 2 r ⎛ r ⎞ y = r2 − ⎜ ⎟ = 2 2 ⎝ ⎠ The dimensions are r 2 by 2r 2 . =0 h w Instructor’s Resource Manual Section 3.4 181 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 37. If the end of the cylinder has radius r and h is the height of the cylinder, the surface area is A A = 2πr 2 + 2πrh so h = – r. 2πr The volume is ⎛ A ⎞ Ar –r⎟ = – πr 3 . V = πr 2 h = πr 2 ⎜ 2 2 π r ⎝ ⎠ A A V (r ) = – 3πr 2 ; V (r ) = 0 when r = , 6π 2 V (r ) = −6πr , so the volume is maximum when r= h= A . 6π sin t = h , so h = r cos t, r 1 2 r – h 2 , and r r 2 – h 2 = r sin t Area of submerged region = tr 2 – h r 2 – h 2 = tr 2 – (r cos t )(r sin t ) = r 2 (t – cos t sin t ) A = area of exposed wetted region = r 2 (π – π cos 2 t – t + cos t sin t ) dA = r 2 (2π cos t sin t –1 + cos 2 t – sin 2 t ) dt 38. The ellipse has equation y = ± b2 – b2 x2 a2 =± = r 2 (2π cos t sin t – 2sin 2 t ) b 2 a – x2 a = 2r 2 sin t (π cos t – sin t ) ⎛ b 2 ⎞ Let ( x, y ) = ⎜ x, a – x 2 ⎟ be the upper right⎝ a ⎠ hand corner of the rectangle (use a and b positive). Then the dimensions of the rectangle are 2x by 2b 2 a – x 2 and the area is a 4bx 2 A( x) = a – x2 . a 4b 2 4bx 2 4b(a 2 – 2 x 2 ) = A ( x) = a – x2 – ; a a a2 – x2 a a2 – x2 a A ( x) = 0 when x = , so the corner is at 2 ⎛ a b ⎞ , ⎜ ⎟ . The corners of the rectangle are at ⎝ 2 2⎠ b ⎞ ⎛ a b ⎞ ⎛ a b ⎞ ⎛ a , , ,– ⎜ ⎟, ⎜ – ⎟, ⎜ – ⎟, 2 2⎠ ⎝ 2 2⎠ ⎝ 2 2⎠ ⎝ b ⎞ ⎛ a ,− ⎜ ⎟. 2⎠ ⎝ 2 diagonal is d = l 2 + w2 , so l = d 2 – w2 . The area is A = lw = w d 2 – w2 . w2 d 2 – w2 d A ( w) = 0 when w = and so 2 = d 2 – 2 w2 d 2 – w2 ; d2 d ⎛ d ⎞ = . A ( w) > 0 on ⎜ 0, ⎟ and 2 2 2⎠ ⎝ Section 3.4 this is r 2 – h2 1 r h r = π or h = r 1 + π2 . 41. The carrying capacity of the gutter is maximized when the area of the vertical end of the gutter is maximized. The height of the gutter is 3sin . The area is ⎛1⎞ A = 3(3sin ) + 2 ⎜ ⎟ (3cos )(3sin ) ⎝2⎠ = 9sin + 9 cos sin . dA = 9 cos + 9(− sin ) sin + 9 cos cos d = 9(2 cos 2 + cos − 1) 39. If the rectangle has length l and width w, the A ( w) = d 2 – w2 – dA = 0 only when dt π cos t = sin t or tan t = π . In terms of r and h, Since 0 < t < π , = 9(cos − sin 2 + cos 2 ) The dimensions are a 2 and b 2 . 182 40. Note that cos t = = πr 2 – πh 2 – r 2 (t – cos t sin t ) A A –r =2 = 2r 2πr 6π l = d2 – ⎛ d ⎞ A ( w) < 0 on ⎜ , d ⎟ . Maximum area is for a ⎝ 2 ⎠ square. 1 π 2 cos 2 + cos − 1 = 0; cos = −1, ; = π, 2 3 π Since 0 ≤ ≤ , the critical points are 2 π π 0, , and . 3 2 When = 0 , A = 0. When When π 27 3 ,A= ≈ 11.7. 3 4 π = , A = 9. 2 = The carrying capacity is maximized when = π . 3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 42. The circumference of the top of the tank is the circumference of the circular sheet minus the arc length of the sector, 20π − 10 meters. The radius of the top of the 20π − 10 5 = (2π − ) meters. The tank is r = 2π π slant height of the tank is 10 meters, so the height of the tank is 2 5 ⎞ 5 ⎛ h = 102 − ⎜10 − ⎟ = 4π − π ⎠ π ⎝ 2 1 1 ⎡5 ⎤ V = πr 2 h = π ⎢ (2π − ) ⎥ 3 3 ⎣π ⎦ 125 (2π − )2 4π − 2 = 3π2 dV 125 ⎛ = ⎜ 2(2π − )(−1) 4π d 3π2 ⎝ + = (2π − )2 125(2π − ) 3π 2 4π − 125(2π − ) 3π 2 4π − 2 2 (3 (3 2π − = 0 or 3 2 2 2 2 meters. ⎡5 ⎢ π 4π − ⎣ − ( 12 ) (4π − 2 4π − 2 2⎤ ⎥ ⎦ 2 )⎞ ⎟ ⎟ ⎠ − 12π + 4π2 ) ; − 12π + 4π2 ) = 0 − 12π + 4π2 = 0 2 6 2 6 π, = 2π + π 3 3 < 2π, the only critical point is = 2π, = 2π − Since 0 < 2 6 π . A graph shows that this maximizes 3 the volume. 2π − 44. Let x be the length of the edges of the cube. The 1 surface area of the cube is 6x 2 so 0 ≤ x ≤ . 6 The surface area of the sphere is 4πr 2 , so 1 – 6x2 4π 4 1 V = x3 + πr 3 = x3 + (1 – 6 x 2 )3 / 2 3 6 π ⎛ dV 3 1 – 6x2 = 3x2 – x 1 – 6 x 2 = 3x ⎜ x – ⎜ dx π π ⎝ dV 1 = 0 when x = 0, dx 6+π 1 V (0) = ≈ 0.094 m3 . 6 π 6 x 2 + 4πr 2 = 1, r = ⎞ ⎟ ⎟ ⎠ 3/ 2 1 ⎛ 6 ⎞ ⎛ 1 ⎞ –3 / 2 V⎜ + ⎟ = (6 + π) ⎜1 – ⎟ 6 + π⎠ 6 π⎝ ⎝ 6+π ⎠ 1 ⎛ π⎞ = ⎜ 1 + ⎟ (6 + π) –3 / 2 = ≈ 0.055 m3 6 6+π ⎝ 6⎠ For maximum volume: no cube, a sphere of radius 1 ≈ 0.282 meters. 2 π For minimum volume: cube with sides of length 1 ≈ 0.331 meters, 6+π 1 sphere of radius ≈ 0.165 meters 2 6+π 45. Consider the figure below. 43. Let V be the volume. y = 4 – x and z = 5 – 2x. x is limited by 0 ≤ x ≤ 2.5 . V = x(4 − x)(5 − 2 x) = 20 x − 13 x 2 + 2 x3 dV = 20 − 26 x + 6 x 2 ; 2(3x 2 − 13 x + 10) = 0; dx 2(3 x − 10)( x − 1) = 0; 10 x = 1, 3 Critical points: 0, 1, 2.5 At x = 0 or 2.5, V = 0. At x = 1, V = 9. Maximum volume when x = 1, y = 4 – 1 = 3, and z = 5 – 2(1) = 3. a. y = x 2 − (a − x )2 = 2ax − a 2 Area of A = A = = 1 (a − x) 2ax − a 2 2 ( ) 1 (a − x) 1 (2a ) dA 1 2 2 =− 2ax − a + 2 dx 2 2ax − a 2 = Instructor’s Resource Manual 1 (a − x) y 2 a 2 − 32 ax 2ax − a 2 Section 3.4 183 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. a 2 − 32 ax = 0 when x = 2a . 3 2ax − a 2 dA dA ⎛ a 2a ⎞ ⎛ 2a ⎞ > 0 on ⎜ , ⎟ and < 0 on ⎜ , a ⎟ , dx dx ⎝2 3 ⎠ ⎝ 3 ⎠ 2a so x = maximizes the area of triangle A. 3 c. b. Triangle A is similar to triangle C, so ax ax w= = y 2ax − a 2 Area of B = B = 1 ax 2 xw = 2 2 2ax − a 2 ⎛ 2 x 2ax − a 2 − x 2 dB a ⎜ = ⎜ dx 2 ⎜ 2ax − a 2 ⎜ ⎝ a 2 ax − a 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ z = x 2 + w2 = x 2 + = 2ax – a 2 2ax3 2ax – a 2 dz 1 2ax − a 2 = dx 2 2ax3 = a2 x2 ⎛ 6ax 2 (2ax − a 2 ) − 2ax3 (2a) ⎞ ⎜ ⎟ ⎜ ⎟ (2ax − a 2 ) 2 ⎝ ⎠ 4a 2 x3 − 3a3 x 2 2ax3 (2ax − a 2 )3 dz 3a 3a = 0 when x = 0, → x= dx 4 4 dz dz ⎛ a 3a ⎞ ⎛ 3a ⎞ < 0 on ⎜ , ⎟ and > 0 on ⎜ , a ⎟ , dx 2 4 dx ⎝ ⎠ ⎝ 4 ⎠ 3a so x = minimizes length z. 4 a ⎛ 2 x(2ax − a 2 ) − ax 2 ⎞ a ⎛ 3ax 2 − 2 xa 2 ⎞ ⎜ ⎟= ⎜ ⎟ 2 ⎜⎝ (2ax − a 2 )3 / 2 ⎟⎠ 2 ⎜⎝ (2ax − a 2 )3 / 2 ⎟⎠ a 2 ⎛ 3x 2 − 2 xa ⎞ 2a ⎜ ⎟ = 0 when x = 0, 2 3 / 2 ⎜ ⎟ 2 ⎝ (2ax − a ) 3 ⎠ 2a . Since x = 0 is not possible, x = 3 dB dB ⎛ a 2a ⎞ ⎛ 2a ⎞ < 0 on ⎜ , ⎟ and > 0 on ⎜ , a ⎟ , dx dx ⎝2 3 ⎠ ⎝ 3 ⎠ 2a minimizes the area of triangle B. so x = 3 = 46. Let 2x be the length of a bar and 2y be the width of a bar. 1 ⎛ 1 ⎞ a ⎛ ⎛π ⎞ ⎞ x = a cos ⎜ − ⎟ = a ⎜ cos + sin ⎟ = ⎜ cos + sin ⎟ 2 2⎠ 2 2⎠ 2 2⎝ ⎝4 2⎠ ⎝ 2 1 ⎛ 1 ⎞ a ⎛ ⎛π ⎞ ⎞ y = a sin ⎜ − ⎟ = a ⎜ cos − sin ⎟ = ⎜ cos − sin ⎟ 4 2 2 2 2 2 2 2⎝ ⎝ ⎠ ⎠ ⎝ 2 ⎠ Compute the area A of the cross and maximize. ⎡ a ⎛ ⎡ a ⎛ ⎞⎤ ⎡ a ⎛ ⎞⎤ ⎞⎤ A = 2(2 x)(2 y ) − (2 y ) 2 = 8 ⎢ ⎜ cos + sin ⎟ ⎥ ⎢ ⎜ cos − sin ⎟ ⎥ − 4 ⎢ ⎜ cos − sin ⎟ ⎥ 2 2 ⎠⎦ ⎣ 2 ⎝ 2 2 ⎠⎦ 2 2 ⎠⎦ ⎣ 2⎝ ⎣ 2⎝ ⎛ ⎞ ⎛ ⎞ = 4a 2 ⎜ cos 2 − sin 2 ⎟ − 2a 2 ⎜1 − 2 cos sin ⎟ = 4a 2 cos − 2a 2 (1 − sin ) 2 2⎠ 2 2⎠ ⎝ ⎝ dA 1 = −4a 2 sin + 2a 2 cos ; −4a 2 sin + 2a 2 cos = 0 when tan = ; d 2 1 2 sin = , cos = 5 5 d2A d 2 < 0 when tan = 2 1 , so this maximizes the area. 2 ⎛ 2 ⎞ 1 ⎞ 10a 2 2⎛ A = 4a 2 ⎜ – 2 a 1 – – 2a 2 = 2a 2 ( 5 – 1) ⎟ ⎜ ⎟= 5⎠ 5 ⎝ 5⎠ ⎝ 184 Section 3.4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 47. a. L ( ) = 15(9 + 25 − 30 cos )−1/ 2 sin = 15(34 − 30 cos )−1/ 2 sin L ( )=− 15 (34 − 30 cos )−3 / 2 (30sin ) sin + 15(34 − 30 cos )−1/ 2 cos 2 = −225(34 − 30 cos ) −3 / 2 sin 2 + 15(34 − 30 cos ) −1/ 2 cos = 15(34 − 30 cos )−3 / 2 [−15sin 2 + (34 − 30 cos ) cos ] = 15(34 − 30 cos ) −3 / 2 [−15sin 2 + 34 cos − 30 cos 2 ] = 15(34 − 30 cos ) −3 / 2 [−15 + 34 cos − 15cos 2 ] = −15(34 − 30 cos )−3 / 2 [15cos 2 − 34 cos + 15] L = 0 when cos = 34 ± (34)2 − 4(15)(15) 5 3 = , 2(15) 3 5 ⎛3⎞ = cos −1 ⎜ ⎟ ⎝5⎠ ⎛ ⎛ ⎛ 3 ⎞⎞ ⎛ 3 ⎞⎞ L ⎜ cos −1 ⎜ ⎟ ⎟ = 15 ⎜ 9 + 25 − 30 ⎜ ⎟ ⎟ 5 ⎝ ⎠⎠ ⎝ 5 ⎠⎠ ⎝ ⎝ −1/ 2 ⎛4⎞ ⎜ ⎟=3 ⎝5⎠ 1/ 2 ⎛ ⎛ 3 ⎞⎞ ⎛ ⎛ 3 ⎞⎞ L ⎜ cos−1 ⎜ ⎟ ⎟ = ⎜ 9 + 25 − 30 ⎜ ⎟ ⎟ =4 ⎝ 5 ⎠⎠ ⎝ ⎝ 5 ⎠⎠ ⎝ φ = 90° since the resulting triangle is a 3-4-5 right triangle. b. L ( ) = 65(25 + 169 − 130 cos )−1/ 2 sin = 65(194 − 130 cos )−1/ 2 sin L ( )=− 65 (194 − 130 cos )−3 / 2 (130sin ) sin + 65(194 − 130 cos ) −1/ 2 cos 2 = −4225(194 − 130 cos )−3 / 2 sin 2 + 65(194 − 130 cos ) −1/ 2 cos = 65(194 − 130 cos )−3 / 2 [−65sin 2 + (194 − 130 cos ) cos ] = 65(194 − 130 cos )−3 / 2 [−65sin 2 + 194 cos − 130 cos 2 ] = 65(194 − 130 cos )−3 / 2 [−65cos 2 + 194 cos − 65] = −65(194 − 130 cos )−3 / 2 [65cos 2 − 194 cos + 65] L = 0 when cos = 194 ± (194)2 − 4(65)(65) 13 5 = , 2(65) 5 13 ⎛5⎞ = cos −1 ⎜ ⎟ ⎝ 13 ⎠ 1/ 2 ⎛ ⎛ ⎛ 5 ⎞⎞ ⎛ 5 ⎞⎞ L ⎜ cos −1 ⎜ ⎟ ⎟ = 65 ⎜ 25 + 169 − 130 ⎜ ⎟ ⎟ 13 ⎝ ⎠ ⎝ 13 ⎠ ⎠ ⎝ ⎠ ⎝ ⎛ 12 ⎞ ⎜ ⎟=5 ⎝ 13 ⎠ 1/ 2 ⎛ ⎛ 5 ⎞⎞ ⎛ ⎛ 5 ⎞⎞ L ⎜ cos−1 ⎜ ⎟ ⎟ = ⎜ 25 + 169 − 130 ⎜ ⎟ ⎟ = 12 ⎝ 13 ⎠ ⎠ ⎝ ⎝ 13 ⎠ ⎠ ⎝ φ = 90° since the resulting triangle is a 5-12-13 right triangle. c. When the tips are separating most rapidly, φ = 90°, L = m 2 − h 2 , L = h Instructor’s Resource Manual Section 3.4 185 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. d. L ( ) = hm(h 2 + m2 − 2hm cos )−1/ 2 sin L ( ) = −h 2 m2 (h 2 + m 2 − 2hm cos ) −3 / 2 sin 2 + hm(h 2 + m 2 − 2hm cos )−1/ 2 cos = hm(h 2 + m 2 − 2hm cos )−3 / 2 [− hm sin 2 + (h 2 + m 2 ) cos − 2hm cos 2 ] = hm(h 2 + m 2 − 2hm cos ) −3 / 2 [− hm cos 2 + (h 2 + m 2 ) cos − hm] = −hm(h 2 + m2 − 2hm cos )−3 / 2 [hm cos 2 − (h 2 + m2 ) cos + hm] L = 0 when hm cos 2 − (h 2 + m2 ) cos + hm = 0 (h cos − m)(m cos − h) = 0 cos = m h , h m Since h < m, cos = h so m ⎛h⎞ = cos −1 ⎜ ⎟ . ⎝m⎠ ⎛ ⎛ ⎛ h ⎞⎞ ⎛ h ⎞⎞ L ⎜ cos −1 ⎜ ⎟ ⎟ = hm ⎜ h 2 + m 2 − 2hm ⎜ ⎟ ⎟ ⎝ m ⎠⎠ ⎝ m ⎠⎠ ⎝ ⎝ 1/ 2 ⎛ ⎛ h ⎞⎞ ⎛ ⎛ h ⎞⎞ L ⎜ cos −1 ⎜ ⎟ ⎟ = ⎜ h 2 + m2 − 2hm ⎜ ⎟ ⎟ m ⎝ ⎠⎠ ⎝ ⎝ m ⎠⎠ ⎝ −1/ 2 m2 − h2 = hm(m2 − h 2 ) −1/ 2 m m2 − h2 =h m = m2 − h2 Since h 2 + L2 = m 2 , φ = 90°. 48. We are interested in finding the global extrema for the distance of the object from the observer. We will obtain this result by considering the squared distance instead. The squared distance can be expressed as 2 1 ⎛ ⎞ D( x) = ( x − 2)2 + ⎜ 100 + x − x 2 ⎟ 10 ⎝ ⎠ The first and second derivatives are given by 1 3 3 2 D '( x) = x − x − 36 x + 196 and 25 5 3 2 D ''( x) = x − 10 x − 300 25 Using a computer package, we can solve the equation D '( x) = 0 to find the critical points. The critical points are x ≈ 5.1538,36.148 . Using the second derivative we see that D ''(5.1538) ≈ −38.9972 (max) and ( ) D ''(36.148) ≈ 77.4237 (min) Therefore, the position of the object closest to the observer is ≈ ( 36.148,5.48 ) while the position of the object farthest from the person is ≈ (5.1538,102.5) . (Remember to go back to the original equation for the path of the object once you find the critical points.) 186 Section 3.4 49. Here we are interested in minimizing the distance between the earth and the asteroid. Using the coordinates P and Q for the two bodies, we can use the distance formula to obtain a suitable equation. However, for simplicity, we will minimize the squared distance to find the critical points. The squared distance between the objects is given by D(t ) = (93cos(2π t ) − 60 cos[2π (1.51t − 1)]) 2 + (93sin(2π t ) − 120sin[2π (1.51t − 1)])2 The first derivative is D '(t ) ≈ −34359 [ cos(2π t )][ sin(9.48761t )] + [ cos(9.48761t ) ][(204932sin(9.48761t ) −141643sin(2π t ))] Plotting the function and its derivative reveal a periodic relationship due to the orbiting of the objects. Careful examination of the graphs reveals that there is indeed a minimum squared distance (and hence a minimum distance) that occurs only once. The critical value for this occurrence is t ≈ 13.82790355 . This value gives a squared distance between the objects of ≈ 0.0022743 million miles. The actual distance is ≈ 0.047851 million miles ≈ 47,851 miles. Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 50. Let x be the width and y the height of the flyer. 51. Consider the following sketch. 1 inch 1 inch 2 inches 2 inches We wish to minimize the area of the flyer, A = xy . As it stands, A is expressed in terms of two variables so we need to write one in terms of the other. The printed area of the flyer has an area of 50 square inches. The equation for this area is ( x − 2 )( y − 4 ) = 50 We can solve this equation for y to obtain 50 y= +4 x−2 Substituting this expression for y in our equation for A, we get A in terms of a single variable, x. A = xy ⎛ 50 ⎞ 50 x = x⎜ + 4⎟ = + 4x ⎝ x−2 ⎠ x−2 The allowable values for x are 2 < x < ∞ ; we want to minimize A on the open interval ( 2, ∞ ) . dA ( x − 2 ) 50 − 50 x −100 = +4= +4 2 dx ( x − 2 )2 ( x − 2) = 4 x 2 − 16 x − 84 ( x − 2) 2 = 4 ( x − 7 )( x + 3) ( x − 2 )2 The only critical points are obtained by solving dA = 0 ; this yields x = 7 and x = −3 . We reject dx x = −3 because it is not in the feasible domain dA dA ( 2, ∞ ) . Since < 0 for x in ( 2, 7 ) and > 0 dx dx for x in ( 7, ∞ ) , we conclude that A attains its minimum value at x = 7 . This value of x makes y = 14 . So, the dimensions for the flyer that will use the least amount of paper are 7 inches by 14 inches. Instructor’s Resource Manual By similar triangles, 27t x= t + 64 2 t + 64 1728 (t + 64)3 / 2 dt 2 27 − t + 64 27t 2 t 2 + 64 27 t 2 + 64 − 2 d2x 2 = −1 = −5184t ( 27 4 5 (4 5) t 2 t + 64 . 1728 2 (t + 64)3 / 2 −1 − 1 = 0 when t = 4 5 ; d2x (t 2 + 64)5 / 2 dt 2 t = 4 Therefore x= = −t 2 dx = dt x 2 ) <0 5 − 4 5 = 5 5 ≈ 11.18 ft is the + 64 maximum horizontal overhang. 52. a. b. There are only a few data points, but they do seem fairly linear. c. The data values can be entered into most scientific calculators to utilize the Least Squares Regression feature. Alternately one could use the formulas for the slope and intercept provided in the text. The resulting line should be y = 0.56852 + 2.6074 x Section 3.4 187 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. d. Using the result from c., the predicted number of surface imperfections on a sheet with area 2.0 square feet is y = 0.56852 + 2.6074(2.0) = 5.7833 ≈ 6 since we can't have partial imperfections 53. a. dS d n = [ yi − (5 + bxi )]2 db db i =1 54. C(x) = 7000 + 100x 55. n = 100 + 10 R (n) = np (n) = 300n – d = [ yi − (5 + bxi )]2 db i =1 = = 300n – ⎡n = 2⎢ − xi yi + 5 xi + bxi 2 ⎢⎣ i =1 ( n = −2 i =1 Setting 0 = −2 n i =1 n b i =1 b= i =1 xi yi + 10 xi yi + 5 xi 2 = n n i =1 xi + 2b n i =1 n i =1 i =1 i =1 n i =1 xi + 2b xi + b xi yi − 5 xi yi − 5 n n xi 2 n i =1 n i =1 xi xi 2 n i =1 xi xi 2 b. Using the formula from a., we get that (2037) − 5(52) b= ≈ 3.0119 590 188 The Least Squares Regression line is y = 5 + 3.0119 x Using this line, the predicted total number of labor hours to produce a lot of 15 brass bookcases is y = 5 + 3.0119(15) ≈ 50.179 hours Section 3.4 58. xi You should check that this is indeed the value of b that minimizes the sum. Taking the second derivative yields n d 2S = 2 xi 2 db 2 i =1 which is always positive (unless all the x values are zero). Therefore, the value for b above does minimize the sum as required. c. Estimate n ≈ 200 P (n) = 200 – n; 200 – n = 0 when n = 200. P (n) = –1, so profit is maximum at n = 200. 2 n i =1 n2 2 57. ⎤ )⎥⎥⎦ dS = 0 gives db i =1 n 0=− xi yi + 10 n2 – (7000 + 100n) 2 = −7000 + 200n – 2( yi − 5 − bxi )(− xi ) i =1 n2 2 56. P (n) = R (n) – C (n) n n 250 – p (n) n so p(n) = 300 – 5 2 59. C ( x) 100 = + 3.002 – 0.0001x x x C ( x) = 2.9045 or $2.90 per unit. When x = 1600, x dC = 3.002 − 0.0002 x dx C (1600) = 2.682 or $2.68 C (n) 1000 n = + n n 1200 C ( n) ≈ 1.9167 or $1.92 per unit. When n = 800, n dC n = dn 600 C (800) ≈ 1.333 or $1.33 60. a. dC = 33 − 18 x + 3 x 2 dx d 2C d 2C = 0 when x = 3 dx 2 d 2C < 0 on (0, 3), > 0 on (3, ∞) dx 2 dx 2 Thus, the marginal cost is a minimum when x = 3 or 300 units. dx 2 d 2C b. = −18 + 6 x; 33 − 18(3) + 3(3) 2 = 6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 61. a. R ( x) = xp( x) = 20 x + 4 x 2 − 65. The revenue function would be R ( x ) = x ⋅ p ( x ) = 200 x − 0.15 x 2 . This, together x3 3 with the cost function yields the following profit function: ⎧⎪−5000 + 194 x − 0.148 x 2 if 0 ≤ x ≤ 500 P ( x) = ⎨ 2 ⎪⎩−9000 + 194 x − 0.148 x if 500 < x ≤ 750 dR = 20 + 8 x − x 2 = (10 − x )( x + 2 ) dx b. Increasing when dR >0 dx 20 + 8 x − x 2 > 0 on [0, 10) Total revenue is increasing if 0 ≤ x ≤ 10. c. d 2R dx 2 d 3R dx 3 = 8 – 2 x; = −2; d 2R dx 2 a. = 0 when x = 4 dR is maximum at x = 4. dx 1/ 2 x ⎞ ⎛ 62. R ( x) = x ⎜182 − ⎟ 36 ⎠ ⎝ 1⎛ dR x ⎞ = x ⎜182 − ⎟ 2⎝ 36 ⎠ dx −1/ 2 1/ 2 x ⎞ ⎛ 1 ⎞ ⎛ ⎜ − ⎟ + ⎜182 − ⎟ 36 ⎠ ⎝ 36 ⎠ ⎝ −1 2 x ⎞ x ⎞ ⎛ ⎛ = ⎜182 − ⎟ ⎜ 182 − ⎟ 36 ⎠ 24 ⎠ ⎝ ⎝ dR = 0 when x = 4368 dx x1 = 4368; R(4368) ≈ 34, 021.83 At x1 , At x1 , P ( 655 ) = 54,574.30 ; P ( 750 ) = 53, 250 The profit is maximized if the company produces 500 chairs. The current machine can handle this work, so they should not buy the new machine. dR =0. dx 800 x − 3x 63. R( x) = x+3 dR ( x + 3)(800) − 800 x 2400 = −3 = − 3; 2 dx ( x + 3) ( x + 3)2 dR = 0 when x = 20 2 − 3 ≈ 25 dx x1 = 25; R (25) ≈ 639.29 dR =0. dx 64. p( x) = 12 − (0.20) ( x − 400) = 20 − 0.02 x 10 R ( x) = 20 x − 0.02 x 2 dR dR = 20 − 0.04 x; = 0 when x = 500 dx dx Total revenue is maximized at x1 = 500 . Instructor’s Resource Manual The only difference in the two pieces of the profit function is the constant. Since the derivative of a constant is 0, we can say that on the interval 0 < x < 750 , dP = 194 − 0.296 x dx There are no singular points in the given interval. To find stationary points, we solve dP =0 dx 194 − 0.296 x = 0 −0.296 x = −194 x ≈ 655 Thus, the critical points are 0, 500, 655, and 750. P ( 0 ) = −5000 ; P ( 500 ) = 55, 000 ; b. Without the new machine, a production level of 500 chairs would yield a maximum profit of $55,000. 66. The revenue function would be R ( x ) = x ⋅ p ( x ) = 200 x − 0.15 x 2 . This, together with the cost function yields the following profit function: ⎧⎪−5000 + 194 x − 0.148 x 2 if 0 ≤ x ≤ 500 P ( x) = ⎨ 2 ⎪⎩−8000 + 194 x − 0.148 x if 500 < x ≤ 750 a. The only difference in the two pieces of the profit function is the constant. Since the derivative of a constant is 0, we can say that on the interval 0 < x < 750 , dP = 194 − 0.296 x dx There are no singular points in the given interval. To find stationary points, we solve dP =0 dx 194 − 0.296 x = 0 −0.296 x = −194 x ≈ 655 Section 3.4 189 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Thus, the critical points are 0, 500, 655, and 750. P ( 0 ) = −5000 ; P ( 500 ) = 55, 000 ; b. F (b) = a 2 + 2ab + b 2 4b P ( 655 ) = 55,574.30 ; P ( 750 ) = 54, 250 As b → 0+ , a 2 + 2ab + b 2 → a 2 while The profit is maximized if the company produces 655 chairs. The current machine cannot handle this work, so they should buy the new machine. 4b → 0+ , thus lim F (b) = ∞ which is not b → 0+ close to a. 2 a + 2a + b a 2 + 2ab + b 2 b lim = =∞ , 4b 4 b →∞ b →∞ so when b is very large, F(b) is not close to a. 2(a + b)(4b) – 4(a + b)2 F (b) = 16b 2 4b 2 – 4a 2 b 2 – a 2 = = ; 16b 2 4b 2 b. With the new machine, a production level of 655 chairs would yield a maximum profit of $55,574.30. lim 67. R ( x) = 10 x − 0.001x 2 ; 0 ≤ x ≤ 300 P ( x) = (10 x – 0.001x 2 ) – (200 + 4 x – 0.01x 2 ) = –200 + 6 x + 0.009 x 2 dP dP = 6 + 0.018 x; = 0 when x ≈ −333 dx dx Critical numbers: x = 0, 300; P(0) = –200; P(300) = 2410; Maximum profit is $2410 at x = 300. F (b) = 0 when b 2 = a 2 or b = a since a and b are both positive. ( a + a ) 2 4a 2 F (a) = = =a 4a 4a ⎧⎪200 + 4 x − 0.01x 2 if 0 ≤ x ≤ 300 68. C ( x) = ⎨ 2 if 300 < x ≤ 450 ⎪⎩800 + 3 x − 0.01x ⎧⎪−200 + 6 x + 0.009 x 2 if 0 ≤ x ≤ 300 P( x) = ⎨ 2 if 300 < x ≤ 450 ⎪⎩−800 + 7 x + 0.009 x There are no stationary points on the interval [0, 300]. On [300, 450]: dP dP = 7 + 0.018 x; = 0 when x ≈ −389 dx dx The critical numbers are 0, 300, 450. P(0) = –200, P(300) = 2410, P(450) = 4172.5 Monthly profit is maximized at x = 450, P(450) = 4172.50 Thus a ≤ ab ≤ ( a + b) 2 for all b > 0 or 4b ( a + b) 2 which leads to 4 ab ≤ 3 c. 1⎛ a+b+c⎞ ( a + b + c )3 Let F (b) = ⎜ ⎟ = b⎝ 3 27b ⎠ F (b) = = = 3(a + b + c)2 (27b) – 27(a + b + c)3 27 2 b 2 (a + b + c) 2 [3b – (a + b + c)] 27b 2 (a + b + c)2 (2b – a – c) 27b 2 F (b) = 0 when b = ; a+c . 2 2 ⎛a+c a+c⎞ ⎛a+c⎞ F⎜ ⋅⎜ + ⎟= ⎟ 6 ⎠ ⎝ 2 ⎠ a+c ⎝ 3 3 = Section 3.4 3 2 3 2 a2 1 b2 ⎛ a b ⎞ ⎛a–b⎞ – ab + =⎜ – ⎟ =⎜ ⎟ 4 2 4 ⎝ 2 2⎠ ⎝ 2 ⎠ Since a square can never be negative, this is always true. 190 2 ⎛a+c⎞ From (b), ac ≤ ⎜ ⎟ , thus ⎝ 2 ⎠ a2 1 b2 = + ab + 4 2 4 This is true if 2 3 2 ⎛ 3(a + c) ⎞ 2 ⎛a+c⎞ ⎛a+c⎞ ⎜ ⎟ = ⎜ ⎟ =⎜ ⎟ a+c⎝ 6 ⎠ a+c⎝ 2 ⎠ ⎝ 2 ⎠ 2 a 2 + 2ab + b 2 ⎛ a+b⎞ ab ≤ ⎜ ⎟ = 4 ⎝ 2 ⎠ 0≤ 3 1⎛ a+b+c⎞ ⎛a+c⎞ Thus ⎜ ⎟ ≤ ⎜ ⎟ for all b > 0. b⎝ 3 ⎝ 2 ⎠ ⎠ 2 69. a. a+b . 2 1⎛ a+b+c⎞ ⎛ a+b+c⎞ ac ≤ ⎜ ⎟ or abc ≤ ⎜ ⎟ b⎝ 3 3 ⎠ ⎝ ⎠ which gives the desired result a+b+c (abc)1/ 3 ≤ . 3 3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 70. Let a = lw, b = lh, and c = hw, then S = 2(a + b + c) while V 2 = abc. By problem a+b+c 69(c), (abc)1/ 3 ≤ so 3 2(a + b + c) S (V 2 )1/ 3 ≤ = . 2⋅3 6 In problem 1c, the minimum occurs, hence a+c equality holds, when b = . In the result used 2 from Problem 69(b), equality holds when c = a, a+a = a, so a = b = c. For the boxes, thus b = 2 this means l = w = h, so the box is a cube. 3.5 Concepts Review 1. f(x); –f(x) Critical points: – 1 2 , f ( x) > 0 when x < – 1 2 1 2 or x > 1 2 1 ⎤ ⎡ 1 ⎛ ⎞ , ∞ ⎟ and f(x) is increasing on ⎜ – ∞, – ⎥∪⎢ 2⎦ ⎣ 2 ⎝ ⎠ ⎡ 1 1 ⎤ decreasing on ⎢ – , ⎥. ⎣ 2 2⎦ ⎛ 1 ⎞ Local minimum f ⎜ ⎟ = – 2 –10 ≈ –11.4 ⎝ 2⎠ ⎛ 1 ⎞ Local maximum f ⎜ – ⎟ = 2 –10 ≈ –8.6 2⎠ ⎝ f ( x) = 12 x; f ( x) > 0 when x > 0. f(x) is concave up on (0, ∞ ) and concave down on (– ∞ , 0); inflection point (0, –10). 2. decreasing; concave up 3. x = –1, x = 2, x = 3; y = 1 4. polynomial; rational. Problem Set 3.5 1. Domain: (−∞, ∞) ; range: (−∞, ∞) Neither an even nor an odd function. y-intercept: 5; x-intercept: ≈ –2.3 f ( x) = 3x 2 – 3; 3 x 2 – 3 = 0 when x = –1, 1 Critical points: –1, 1 f ( x) > 0 when x < –1 or x > 1 f(x) is increasing on (– ∞ , –1] ∪ [1, ∞ ) and decreasing on [–1, 1]. Local minimum f(1) = 3; local maximum f(–1) = 7 f ( x) = 6 x; f ( x) > 0 when x > 0. f(x) is concave up on (0, ∞ ) and concave down on (– ∞ , 0); inflection point (0, 5). 3. Domain: (– ∞ , ∞ ); range: (– ∞ , ∞ ) Neither an even nor an odd function. y-intercept: 3; x-intercepts: ≈ –2.0, 0.2, 3.2 f ( x) = 6 x 2 – 6 x –12 = 6( x – 2)( x + 1); f ( x) = 0 when x = –1, 2 Critical points: –1, 2 f ( x) > 0 when x < –1 or x > 2 f(x) is increasing on (– ∞ , –1] ∪ [2, ∞ ) and decreasing on [–1, 2]. Local minimum f(2) = –17; local maximum f(–1) = 10 1 f ( x) = 12 x − 6 = 6(2 x − 1); f ( x) > 0 when x > . 2 ⎛1 ⎞ f(x) is concave up on ⎜ , ∞ ⎟ and concave down ⎝2 ⎠ 1 ⎛ ⎞ ⎛1 7⎞ on ⎜ – ∞, ⎟ ; inflection point: ⎜ , – ⎟ 2⎠ ⎝ ⎝2 2⎠ 2. Domain: (– ∞ , ∞ ); range: (– ∞ , ∞ ) Neither an even nor an odd function. y-intercept: –10; x-intercept: 2 f ( x) = 6 x 2 – 3 = 3(2 x 2 –1); 2 x 2 –1 = 0 when x=– 1 2 , 1 2 Instructor’s Resource Manual Section 3.5 191 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4. Domain: (– ∞ , ∞ ); range: (– ∞ , ∞ ) Neither an even nor an odd function y-intercept: –1; x-intercept: 1 H (t ) > 0 for – 1 2 < t < 0 or 1 2 < t. ⎡ 1 ⎤ ⎡ 1 ⎞ H(t) is increasing on ⎢ – , 0⎥ ∪ ⎢ , ∞ ⎟ and 2 2 ⎣ ⎦ ⎣ ⎠ 1 ⎤ ⎡ 1 ⎤ ⎛ decreasing on ⎜ – ∞, – ⎥ ∪ ⎢0, ⎥ 2⎦ ⎣ 2⎦ ⎝ f ( x) = 3( x –1)2 ; f ( x) = 0 when x = 1 Critical point: 1 f ( x) > 0 for all x ≠ 1 f(x) is increasing on (– ∞ , ∞ ) No local minima or maxima f ( x) = 6( x –1); f ( x) > 0 when x > 1. f(x) is concave up on (1, ∞ ) and concave down on (– ∞ , 1); inflection point (1, 0) 1 ⎛ 1 ⎞ 1 ⎛ 1 ⎞ Global minima f ⎜ – ⎟=– , f ⎜ ⎟=– ; 4 ⎝ 2⎠ 4 2⎠ ⎝ Local maximum f(0) = 0 H (t ) = 12t 2 – 2 = 2(6t 2 –1); H > 0 when t<– 1 6 or t > 1 6 ⎛ H(t) is concave up on ⎜ – ∞, – ⎝ ⎛ 1 and concave down on ⎜ – , 6 ⎝ ⎞ 1 ⎞ ⎛ 1 , ∞⎟ ⎟∪⎜ 6⎠ ⎝ 6 ⎠ 1 ⎞ ⎟ ; inflection 6⎠ ⎛ 1 5 ⎞ ⎛ 1 5 ⎞ points H ⎜ – , − ⎟ and H ⎜ , ⎟ 36 6 ⎝ ⎠ ⎝ 6 36 ⎠ 5. Domain: (– ∞ , ∞ ); range: [0, ∞ ) Neither an even nor an odd function. y-intercept: 1; x-intercept: 1 G ( x ) = 4( x – 1)3 ; G ( x) = 0 when x = 1 Critical point: 1 G ( x) > 0 for x > 1 G(x) is increasing on [1, ∞ ) and decreasing on (– ∞ , 1]. Global minimum f(1) = 0; no local maxima G ( x) = 12( x –1) 2 ; G ( x ) > 0 for all x ≠ 1 G(x) is concave up on (– ∞ , 1) ∪ (1, ∞ ); no inflection points ⎡ 1 ⎞ 6. Domain: (– ∞ , ∞ ); range: ⎢ – , ∞ ⎟ ⎣ 4 ⎠ 7. Domain: (– ∞ , ∞ ); range: (– ∞ , ∞ ) Neither an even nor an odd function. y-intercept: 10; x-intercept: 1 –111/ 3 ≈ –1.2 f ( x) = 3 x 2 – 6 x + 3 = 3( x –1) 2 ; f ( x) = 0 when x = 1. Critical point: 1 f ( x) > 0 for all x ≠ 1. f(x) is increasing on (– ∞ , ∞ ) and decreasing nowhere. No local maxima or minima f ( x) = 6 x – 6 = 6( x –1); f ( x) > 0 when x > 1. f(x) is concave up on (1, ∞ ) and concave down on (– ∞ , 1); inflection point (1, 11) H (–t ) = (– t )2 [(– t )2 – 1] = t 2 (t 2 – 1) = H (t ); even function; symmetric with respect to the y-axis. y-intercept: 0; t-intercepts: –1, 0, 1 H (t ) = 4t 3 – 2t = 2t (2t 2 – 1); H (t ) = 0 when t=– 1 2 , 0, 1 2 Critical points: – 192 Section 3.5 1 2 , 0, 1 2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 8. ⎡ 16 ⎞ Domain: (– ∞ , ∞ ); range: ⎢ – , ∞ ⎟ ⎣ 3 ⎠ 4(– s )4 – 8(– s ) 2 –12 4s 4 – 8s 2 –12 = 3 3 = F ( s ); even function; symmetric with respect to the y-axis y-intercept: –4; s-intercepts: – 3, 3 16 16 16 F ( s ) = s3 – s = s ( s 2 –1); F ( s ) = 0 3 3 3 when s = –1, 0, 1. Critical points: –1, 0, 1 F ( s ) > 0 when –1 < x < 0 or x > 1. F(s) is increasing on [–1, 0] ∪ [1, ∞ ) and decreasing on (– ∞ , –1] ∪ [0, 1] 16 16 Global minima F (−1) = − , F (1) = − ; local 3 3 maximum F(0) = –4 16 1⎞ ⎛ F ( s ) = 16 s 2 − = 16 ⎜ s 2 − ⎟ ; F ( s ) > 0 3 3⎠ ⎝ 1 1 when s < – or s > 3 3 1 ⎞ ⎛ 1 ⎛ ⎞ F(s) is concave up on ⎜ −∞, − , ∞⎟ ⎟∪⎜ 3⎠ ⎝ 3 ⎠ ⎝ ⎛ 1 1 ⎞ , and concave down on ⎜ – ⎟; 3 3⎠ ⎝ inflection points 128 ⎞ ⎛ 1 128 ⎞ ⎛ 1 F⎜– ,− ,− ⎟, F ⎜ ⎟ 27 27 ⎠ 3 3 ⎝ ⎠ ⎝ F (– s ) = down on (–1, ∞ ); no inflection points (–1 is not in the domain of g). x 1 lim = lim = 1; x →∞ x + 1 x →∞ 1 + 1 x x 1 = lim = 1; x→ – ∞ x + 1 x→ – ∞ 1 + 1 lim x horizontal asymptote: y = 1 As x → –1– , x + 1 → 0 – so as x → –1+ , x + 1 → 0+ so lim x = ∞; x +1 lim x = – ∞; x +1 x → –1– x → –1+ vertical asymptote: x = –1 10. Domain: (– ∞ , 0) ∪ (0, ∞ ); range: (– ∞ , –4 π ] ∪ [0, ∞ ) Neither an even nor an odd function No y-intercept; s-intercept: π g (s) = s 2 – π2 s2 ; g ( s ) = 0 when s = – π , π Critical points: −π , π g ( s ) > 0 when s < – π or s > π g(s) is increasing on (−∞, −π ] ∪ [π , ∞) and decreasing on [– π , 0) ∪ (0, π ]. Local minimum g( π ) = 0; local maximum g(– π ) = –4 π 9. Domain: (– ∞ , –1) ∪ (–1, ∞ ); range: (– ∞ , 1) ∪ (1, ∞ ) Neither an even nor an odd function y-intercept: 0; x-intercept: 0 1 g ( x) = ; g ( x) is never 0. ( x + 1) 2 No critical points g ( x) > 0 for all x ≠ –1. g(x) is increasing on (– ∞ , –1) ∪ (–1, ∞ ). No local minima or maxima 2 g ( x) = – ; g ( x) > 0 when x < –1. ( x + 1)3 g(x) is concave up on (– ∞ , –1) and concave Instructor’s Resource Manual g ( s) = 2π2 s3 ; g ( s ) > 0 when s > 0 g(s) is concave up on (0, ∞ ) and concave down on (– ∞ , 0); no inflection points (0 is not in the domain of g(s)). g ( s ) = s – 2π + π2 ; y = s – 2π is an oblique s asymptote. As s → 0 – , ( s – π) 2 → π2 , so lim g ( s ) = – ∞; s →0 – Section 3.5 193 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. as s → 0+ , ( s – π)2 → π2 , so lim g ( s ) = ∞; No vertical asymptotes s →0 + s = 0 is a vertical asymptote. ⎡ 1 1⎤ 11. Domain: (– ∞ , ∞ ); range: ⎢ – , ⎥ ⎣ 4 4⎦ –x x f (– x) = =– = – f ( x); odd x2 + 4 (– x) 2 + 4 function; symmetric with respect to the origin. y-intercept: 0; x-intercept: 0 4 – x2 f ( x) = ; f ( x) = 0 when x = –2, 2 ( x 2 + 4)2 Critical points: –2, 2 f ( x) > 0 for –2 < x < 2 f(x) is increasing on [–2, 2] and decreasing on (– ∞ , –2] ∪ [2, ∞ ). 1 Global minimum f (–2) = – ; global maximum 4 1 f (2) = 4 2 x( x 2 – 12) f ( x) = ; f ( x) > 0 when ( x 2 + 4)3 –2 3 < x < 0 or x > 2 3 f(x) is concave up on (–2 3, 0) ∪ (2 3, ∞) and concave down on (– ∞, – 2 3) ∪ (0, 2 3); ⎛ 3⎞ inflection points ⎜⎜ −2 3, − ⎟ , (0, 0) , 8 ⎟⎠ ⎝ ⎛ 3⎞ ⎜⎜ 2 3, ⎟ 8 ⎟⎠ ⎝ lim x →∞ x 2 lim x +4 x→ – ∞ x 2 = lim x +4 1 x x →∞ 1 + 4 x2 1 x = lim = 0; x→ – ∞ 1 + 4 x2 12. Domain: (– ∞ , ∞ ); range: [0, 1) 2 (– ) 2 = = ( ); even (– ) = 2 +1 (– )2 + 1 function; symmetric with respect to the y-axis. y-intercept: 0; -intercept: 0 2 ( )= ; ( ) = 0 when = 0 2 ( + 1)2 Critical point: 0 ( ) > 0 when > 0 ( ) is increasing on [0, ∞ ) and decreasing on (– ∞ , 0]. Global minimum (0) = 0; no local maxima 2(1 – 3 2 ) ( )= ; ( ) > 0 when ( 2 + 1)3 1 1 < < – 3 3 ⎛ 1 1 ⎞ ( ) is concave up on ⎜ – , ⎟ and 3 3⎠ ⎝ 1 ⎞ ⎛ 1 ⎛ ⎞ concave down on ⎜ – ∞, – , ∞ ⎟; ⎟∪⎜ 3⎠ ⎝ 3 ⎠ ⎝ ⎛ 1 1⎞ ⎛ 1 1⎞ inflection points ⎜ – , ⎟, ⎜ , ⎟ 3 4⎠ ⎝ 3 4⎠ ⎝ 2 lim →∞ 2 +1 = lim 2 lim →–∞ 2 +1 1 →∞ 1 + 1 2 = 1; 1 = 1; →–∞ 1 + 1 2 = lim y = 1 is a horizontal asymptote. No vertical asymptotes = 0; y = 0 is a horizontal asymptote. 194 Section 3.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 13. Domain: (– ∞ , 1) ∪ (1, ∞ ); range (– ∞ , 1) ∪ (1, ∞ ) Neither an even nor an odd function y-intercept: 0; x-intercept: 0 1 ; h ( x) is never 0. h( x ) = − ( x − 1)2 No critical points h ( x) < 0 for all x ≠ 1. h(x) is increasing nowhere and decreasing on (– ∞ , 1) ∪ (1, ∞ ). No local maxima or minima 2 ; h ( x ) > 0 when x > 1 h ( x) = ( x – 1)3 (−∞, 0] and decreasing on [0, ∞) . Global maximum P (0) = 1 ; no local minima. P ''( x) = 6x2 − 2 ( x 2 + 1)3 P ''( x) > 0 on (−∞, −1/ 3) ∪ (1/ 3, ∞) (concave up) and P ''( x) < 0 on (−1/ 3,1/ 3) (concave down). ⎛ 1 3⎞ , ⎟ Inflection points: ⎜ ± 3 4⎠ ⎝ No vertical asymptotes. lim P ( x) = 0; lim P ( x) = 0 x →∞ x →−∞ y = 0 is a horizontal asymptote. h( x) is concave up on (1, ∞ ) and concave down on (– ∞ , 1); no inflection points (1 is not in the domain of h( x) ) lim x →∞ x 1 = lim = 1; x – 1 x→∞ 1 – 1 x x 1 lim = lim = 1; x →−∞ x − 1 x →−∞ 1 − 1 x y = 1 is a horizontal asymptote. As x → 1– , x – 1 → 0 – so lim x = – ∞; x –1 as x → 1+ , x – 1 → 0+ so lim x = ∞; x –1 x →1– x →1+ x = 1 is a vertical asymptote. 14. Domain: ( −∞, ∞ ) Range: ( 0,1] Even function since 1 1 P(− x) = = = P( x) 2 2 (− x) + 1 x + 1 so the function is symmetric with respect to the y-axis. y-intercept: y = 1 x-intercept: none −2 x P '( x) = ; P '( x) is 0 when x = 0 . 2 ( x + 1) 2 critical point: x = 0 P '( x) > 0 when x < 0 so P ( x) is increasing on Instructor’s Resource Manual 15. Domain: (– ∞ , –1) ∪ (–1, 2) ∪ (2, ∞ ); range: (– ∞ , ∞ ) Neither an even nor an odd function 3 y-intercept: – ; x-intercepts: 1, 3 2 2 3x –10 x + 11 ; f ( x) is never 0. f ( x) = ( x + 1) 2 ( x – 2) 2 No critical points f ( x) > 0 for all x ≠ –1, 2 f(x) is increasing on (– ∞ , –1) ∪ (–1, 2) ∪ ( 2, ∞ ) . No local minima or maxima –6 x3 + 30 x 2 – 66 x + 42 ; f ( x) > 0 when f ( x) = ( x + 1)3 ( x – 2)3 x < –1 or 1 < x < 2 f(x) is concave up on (– ∞ , –1) ∪ (1, 2) and concave down on (–1, 1) ∪ (2, ∞ ); inflection point f(1) = 0 ( x – 1)( x – 3) x2 – 4 x + 3 lim = lim x →∞ ( x + 1)( x – 2) x →∞ x 2 – x – 2 = lim 1 – 4x + x →∞ 1 – 1 x – 3 x2 2 x2 = 1; 1 – 4x + ( x – 1)( x – 3) lim = lim x → – ∞ ( x + 1)( x – 2) x → – ∞ 1 – 1 – x 3 x2 2 x2 = 1; y =1 is a horizontal asymptote. As x → –1– , x – 1 → –2, x – 3 → –4, x – 2 → –3, and x + 1 → 0 – so lim f ( x) = ∞; x → –1– Section 3.5 195 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. as x → –1+ , x – 1 → –2, x – 3 → –4, x – 2 → –3, and x + 1 → 0+ , so lim f ( x) = – ∞ x → –1+ As x → 2 – , x – 1 → 1, x – 3 → –1, x + 1 → 3, and x – 2 → 0 – , so lim f ( x) = ∞; as x→2– x → 2+ , x – 1 → 1, x – 3 → –1, x + 1 → 3, and x – 2 → 0+ , so lim f ( x) = – ∞ x → 2+ x = –1 and x = 2 are vertical asymptotes. 17. Domain: ( −∞,1) ∪ (1, ∞ ) Range: ( −∞, ∞ ) Neither even nor odd function. y-intercept: y = 6 ; x-intercept: x = −3, 2 g '( x) = x2 − 2 x + 5 ; g '( x) is never zero. No ( x − 1) 2 critical points. g '( x) > 0 over the entire domain so the function is always increasing. No local extrema. −8 f ''( x) = ; f ''( x) > 0 when ( x − 1)3 x < 1 (concave up) and f ''( x) < 0 when x > 1 (concave down); no inflection points. No horizontal asymptote; x = 1 is a vertical asymptote; the line y = x + 2 is an oblique (or slant) asymptote. 16. Domain: ( −∞, 0 ) ∪ ( 0, ∞ ) Range: (−∞, −2] ∪ [2, ∞) Odd function since (− z )2 + 1 z2 +1 w(− z ) = =− = − w( z ) ; symmetric −z z with respect to the origin. y-intercept: none x-intercept: none 1 w '( z ) = 1 − ; w '( z ) = 0 when z = ±1 . z2 critical points: z = ±1 . w '( z ) > 0 on (−∞, −1) ∪ (1, ∞) so the function is increasing on (−∞, −1] ∪ [1, ∞) . The function is decreasing on [−1, 0) ∪ (0,1) . local minimum w(1) = 2 and local maximum w(−1) = −2 . No global extrema. w ''( z ) = 2 > 0 when z > 0 . Concave up on z3 (0, ∞) and concave down on ( −∞, 0 ) . 18. Domain: (– ∞ , ∞ ); range [0, ∞ ) 3 3 f (– x) = – x = x = f ( x); even function; symmetric with respect to the y-axis. y-intercept: 0; x-intercept: 0 2⎛ x ⎞ f ( x) = 3 x ⎜⎜ ⎟⎟ = 3x x ; f ( x) = 0 when x = 0 ⎝ x⎠ Critical point: 0 f ( x) > 0 when x > 0 f(x) is increasing on [0, ∞ ) and decreasing on (– ∞ , 0]. Global minimum f(0) = 0; no local maxima 3x 2 2 f ( x) = 3 x + = 6 x as x 2 = x ; x f ( x) > 0 when x ≠ 0 f(x) is concave up on (– ∞ , 0) ∪ (0, ∞ ); no inflection points No horizontal asymptote; x = 0 is a vertical asymptote; the line y = z is an oblique (or slant) asymptote. No inflection points. 196 Section 3.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 19. Domain: (– ∞ , ∞ ); range: (– ∞ , ∞ ) R (– z ) = – z – z = – z z = – R( z ); odd function; symmetric with respect to the origin. y-intercept: 0; z-intercept: 0 z2 2 R ( z) = z + = 2 z since z 2 = z for all z; z R ( z ) = 0 when z = 0 Critical point: 0 R ( z ) > 0 when z ≠ 0 R(z) is increasing on (– ∞ , ∞ ) and decreasing nowhere. No local minima or maxima 2z R ( z ) = ; R ( z ) > 0 when z > 0. z R(z) is concave up on (0, ∞ ) and concave down on (– ∞ , 0); inflection point (0, 0). 21. Domain: (– ∞ , ∞ ); range: [0, ∞ ) Neither an even nor an odd function. Note that for x ≤ 0, x = – x so x + x = 0, while for x > 0, x = x so x +x 2 if x ≤ 0 = x. ⎧⎪0 g ( x) = ⎨ 2 ⎪⎩3x + 2 x if x > 0 y-intercept: 0; x-intercepts: ( − ∞, 0] if x ≤ 0 ⎧0 g ( x) = ⎨ x 6 2 if x > 0 + ⎩ No critical points for x > 0. g(x) is increasing on [0, ∞ ) and decreasing nowhere. ⎧0 if x ≤ 0 g ( x) = ⎨ ⎩6 if x > 0 g(x) is concave up on (0, ∞ ); no inflection points 20. Domain: (– ∞ , ∞ ); range: [0, ∞ ) H (– q) = (– q )2 – q = q 2 q = H (q); even function; symmetric with respect to the y-axis. y-intercept: 0; q-intercept: 0 q3 3q3 2 H ( q ) = 2q q + = = 3q q as q = q 2 q q for all q; H (q ) = 0 when q = 0 Critical point: 0 H (q ) > 0 when q > 0 H(q) is increasing on [0, ∞ ) and decreasing on (– ∞ , 0]. Global minimum H(0) = 0; no local maxima 3q 2 H (q) = 3 q + = 6 q ; H (q ) > 0 when q q ≠ 0. H(q) is concave up on (– ∞ , 0) ∪ (0, ∞ ); no inflection points. Instructor’s Resource Manual 22. Domain: (– ∞ , ∞ ); range: [0, ∞ ) Neither an even nor an odd function. Note that x –x = – x , while for for x < 0, x = – x so 2 x −x x ≥ 0, x = x so = 0. 2 ⎧⎪− x3 + x 2 − 6 x if x < 0 h( x ) = ⎨ if x ≥ 0 ⎪⎩0 y-intercept: 0; x-intercepts: [0, ∞ ) ⎧⎪−3 x 2 + 2 x − 6 if x < 0 h ( x) = ⎨ if x ≥ 0 ⎪⎩0 No critical points for x < 0 h(x) is increasing nowhere and decreasing on (– ∞ , 0]. ⎧−6 x + 2 if x < 0 h ( x) = ⎨ if x ≥ 0 ⎩0 h(x) is concave up on (– ∞ , 0); no inflection Section 3.5 197 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. points 23. Domain: (– ∞ , ∞ ); range: [0, 1] f (− x) = sin(− x) = − sin x = sin x = f ( x); even function; symmetric with respect to the y-axis. y-intercept: 0; x-intercepts: k π where k is any integer. sin x π f ( x) = cos x; f ( x) = 0 when x = + k π sin x 2 and f ( x) does not exist when x = k π , where k is any integer. π kπ and kπ + , where k is any Critical points: 2 2 integer; f ( x) > 0 when sin x and cos x are either both positive or both negative. π⎤ ⎡ f(x) is increasing on ⎢ k π, k π + ⎥ and decreasing 2⎦ ⎣ π ⎡ ⎤ on ⎢ k π + , (k + 1)π ⎥ where k is any integer. 2 ⎣ ⎦ Global minima f(k π ) = 0; global maxima π⎞ ⎛ f ⎜ k π + ⎟ = 1, where k is any integer. 2⎠ ⎝ f ( x) = 24. Domain: [2k π , (2k + 1) π ] where k is any integer; range: [0, 1] Neither an even nor an odd function y-intercept: 0; x-intercepts: k π , where k is any integer. π cos x f ( x) = ; f ( x) = 0 when x = 2k π + 2 2 sin x while f ( x) does not exist when x = k π , k any integer. π Critical points: k π, 2k π + where k is any 2 integer π f ( x) > 0 when 2k π < x < 2k π + 2 π⎤ ⎡ f(x) is increasing on ⎢ 2k π, 2k π + ⎥ and 2⎦ ⎣ π ⎡ ⎤ decreasing on ⎢ 2k π + , (2k + 1)π ⎥ , k any 2 ⎣ ⎦ integer. Global minima f(k π ) = 0; global maxima π⎞ ⎛ f ⎜ 2k π + ⎟ = 1, k any integer 2⎠ ⎝ f ( x) = – cos 2 x – 2sin 2 x = –1 – sin 2 x 4sin 3 / 2 x 4sin 3 / 2 x 1 + sin 2 x =– ; 4sin 3 / 2 x f ( x) < 0 for all x. f(x) is concave down on (2k π , (2k + 1) π ); no inflection points cos 2 x sin 2 x − sin x sin x ⎛ 1 ⎞⎟ ⎛ sin x ⎞ + sin x cos x ⎜ − ⎜ ⎟ (cos x) ⎜ sin x 2 ⎟ ⎜⎝ sin x ⎟⎠ ⎝ ⎠ = cos 2 x sin 2 x cos 2 x sin 2 x − − =− = − sin x sin x sin x sin x sin x f ( x) < 0 when x ≠ k π , k any integer f(x) is never concave up and concave down on (k π , (k + 1) π ) where k is any integer. No inflection points 25. Domain: (−∞, ∞) Range: [0,1] Even function since h(−t ) = cos 2 (−t ) = cos 2 t = h(t ) so the function is symmetric with respect to the y-axis. y-intercept: y = 1 ; t-intercepts: x = π 2 + kπ where k is any integer. h '(t ) = −2 cos t sin t ; h '(t ) = 0 at t = Critical points: t = 198 Section 3.5 kπ 2 kπ . 2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. h '(t ) > 0 when kπ + π < t < (k + 1)π . The 2 function is increasing on the intervals [ kπ + (π / 2), (k + 1)π ] and decreasing on the g '(t ) = 2 =2 intervals [ kπ , kπ + (π / 2) ] . Global maxima h ( kπ ) = 1 =2 ⎛π ⎞ Global minima h ⎜ + kπ ⎟ = 0 ⎝2 ⎠ h ''(t ) = 2sin 2 t − 2 cos 2 t = −2(cos 2t ) π π⎞ ⎛ h ''(t ) < 0 on ⎜ kπ − , kπ + ⎟ so h is concave 4 4⎠ ⎝ π 3π ⎞ ⎛ down, and h ''(t ) > 0 on ⎜ kπ + , kπ + ⎟ so h 4 4 ⎠ ⎝ is concave up. ⎛ kπ π 1 ⎞ + , ⎟ Inflection points: ⎜ ⎝ 2 4 2⎠ No vertical asymptotes; no horizontal asymptotes. cos4 t + sin t (3) cos 2 t sin t cos6 t cos 2 t + 3sin 2 t cos 4 t 1 + 2sin 2 t >0 cos 4 t over the entire domain. Thus the function is π π⎞ ⎛ concave up on ⎜ kπ − , kπ + ⎟ ; no inflection 2 2⎠ ⎝ points. No horizontal asymptotes; t = π 2 + kπ are vertical asymptotes. 27. Domain: ≈ (– ∞ , 0.44) ∪ (0.44, ∞ ); range: (– ∞ , ∞ ) Neither an even nor an odd function 26. Domain: all reals except t = π 2 + kπ Range: [0, ∞) y-intercepts: y = 0 ; t-intercepts: t = kπ where k is any integer. Even function since g (−t ) = tan 2 (−t ) = (− tan t )2 = tan 2 t so the function is symmetric with respect to the y-axis. 2sin t g '(t ) = 2sec 2 t tan t = ; g '(t ) = 0 when cos3 t t = kπ . Critical points: kπ π⎞ ⎡ g ( t ) is increasing on ⎢ kπ , kπ + ⎟ and 2⎠ ⎣ π ⎛ ⎤ decreasing on ⎜ kπ − , kπ ⎥ . 2 ⎝ ⎦ Global minima g (kπ ) = 0 ; no local maxima y-intercept: 0; x-intercepts: 0, ≈ 0.24 f ( x) = 74.6092 x3 – 58.2013 x 2 + 7.82109 x (7.126 x – 3.141) 2 ; f ( x) = 0 when x = 0, ≈ 0.17, ≈ 0.61 Critical points: 0, ≈ 0.17, ≈ 0.61 f ( x) > 0 when 0 < x < 0.17 or 0.61 < x f(x) is increasing on ≈ [0, 0.17] ∪ [0.61, ∞ ) and decreasing on (– ∞ , 0] ∪ [0.17, 0.44) ∪ (0.44, 0.61] Local minima f(0) = 0, f(0.61) ≈ 0.60; local maximum f(0.17) ≈ 0.01 f ( x) = 531.665 x3 – 703.043 x 2 + 309.887 x – 24.566 (7.126 x – 3.141) 3 ; f ( x) > 0 when x < 0.10 or x > 0.44 Instructor’s Resource Manual Section 3.5 199 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. f(x) is concave up on (– ∞ , 0.10) ∪ (0.44, ∞ ) 30. and concave down on (0.10, 0.44); inflection point ≈ (0.10, 0.003) 5.235 x3 − 1.245 x 2 5.235 x 2 − 1.245 x = lim =∞ x →∞ 7.126 x − 3.141 x →∞ 7.126 − 3.141 lim x so f(x) does not have a horizontal asymptote. As x → 0.44 – , 5.235 x3 – 1.245 x 2 → 0.20 while 7.126 x – 3.141 → 0 – , so lim x →0.44 – 31. f ( x) = – ∞; as x → 0.44+ , 5.235 x3 – 1.245 x 2 → 0.20 while 7.126 x – 3.141 → 0+ , so lim x →0.44+ f ( x) = ∞; 32. x ≈ 0.44 is a vertical asymptote of f(x). 33. 28. 34. 29. 200 Section 3.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 35. 40. Let f ( x) = ax 2 + bx + c, then f ( x) = 2ax + b and f ( x) = 2a. An inflection point occurs where f ( x) changes from positive to negative, but 2a is either always positive or always negative, so f(x) does not have any inflection points. ( f ( x) = 0 only when a = 0, but then f(x) is not a quadratic curve.) 36. y = 5( x – 1) 4 ; y = 20( x – 1)3 ; y ( x) > 0 when x > 1; inflection point (1, 3) At x = 1, y = 0, so the linear approximation is a horizontal line. 41. Let f ( x ) = ax3 + bx 2 + cx + d , then f ( x) = 3ax 2 + 2bx + c and f ( x) = 6ax + 2b. As long as a ≠ 0 , f ( x) will be positive on one side of x = x= 3 b and negative on the other side. 3a b is the only inflection point. 3a 42. Let f ( x) = ax 4 + bx3 + cx 2 + dx + c, then f ( x) = 4ax3 + 3bx 2 + 2cx + d and f ( x) = 12ax 2 + 6bx + 2c = 2(6ax 2 + 3bx + c) 37. Inflection points can only occur when f ( x) changes sign from positive to negative and f ( x) = 0. f ( x) has at most 2 zeros, thus f(x) has at most 2 inflection points. 43. Since the c term is squared, the only difference occurs when c = 0. When c = 0, y = x2 x2 = x 38. 3 which has domain (– ∞ , ∞ ) and range [0, ∞ ). When c ≠ 0, y = x 2 x 2 – c 2 has domain (– ∞ , –|c|] ∪ [|c|, ∞ ) and range [0, ∞ ). 39. Instructor’s Resource Manual Section 3.5 201 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. The only extremum points are ± c . For c = 0 , there is one minimum, for c ≠ 0 there are two. No maxima, independent of c. No inflection points, independent of c. 44. f ( x) = cx 4 + (cx) f ( x) = 2 = cx ; f ( x ) = 0 when x = ± (4 + c 2 x 2 )2 unless c = 0, in which case f(x) = 0 and f ( x) = 0. 2 c ⎡ 2 2⎤ If c > 0, f(x) is increasing on ⎢ – , ⎥ and ⎣ c c⎦ 2⎤ ⎡2 ⎞ ⎛ decreasing on ⎜ – ∞, – ⎥ ∪ ⎢ , ∞ ⎟ , thus, f(x) has c⎦ ⎣c ⎠ ⎝ 1 ⎛ 2⎞ a global minimum at f ⎜ – ⎟ = – and a global 4 ⎝ c⎠ 2 1 ⎛ ⎞ maximum of f ⎜ ⎟ = . ⎝c⎠ 4 2⎤ ⎡ 2 ⎞ ⎛ If c < 0, f(x) is increasing on ⎜ – ∞, ⎥ ∪ ⎢ – , ∞ ⎟ c⎦ ⎣ c ⎠ ⎝ ⎡2 2⎤ and decreasing on ⎢ , – ⎥ . Thus, f(x) has a ⎣c c⎦ 1 ⎛ 2⎞ global minimum at f ⎜ – ⎟ = – and a global 4 ⎝ c⎠ 2 1 ⎛ ⎞ maximum at f ⎜ ⎟ = . ⎝c⎠ 4 f ( x) = 2c3 x(c 2 x 2 – 12) (4 + c 2 x 2 )3 points at x = 0, ± 202 Section 3.5 f ( x) = f ( x) = 1 2 (cx – 4)2 + cx 2 , then 2cx(7 − 2cx 2 ) [(cx 2 – 4) 2 + cx 2 ]2 ; If c > 0, f ( x) = 0 when x = 0, ± 4 + c2 x2 c(4 – c 2 x 2 ) 45. 7 . 2c If c < 0, f ( x) = 0 when x = 0. Note that f(x) = 1 (a horizontal line) if c = 0. 16 If c > 0, f ( x) > 0 when x < − 7 and 2c 7 , so f(x) is increasing on 2c ⎛ 7 ⎤ ⎡ 7 ⎤ ⎜⎜ −∞, − ⎥ ∪ ⎢ 0, ⎥ and decreasing on 2c ⎦ ⎣ 2c ⎦ ⎝ ⎡ ⎞ 7 ⎤ ⎡ 7 , 0⎥ ∪ ⎢ , ∞ ⎟⎟ . Thus, f(x) has local ⎢− ⎣ 2c ⎦ ⎣ 2 c ⎠ ⎛ ⎛ 7 ⎞ 4 7 ⎞ 4 maxima f ⎜⎜ − and ⎟⎟ = , f ⎜⎜ ⎟⎟ = ⎝ 2c ⎠ 15 ⎝ 2c ⎠ 15 1 . If c < 0, f ( x) > 0 local minimum f (0) = 16 when x < 0, so f(x) is increasing on (– ∞ , 0] and decreasing on [0, ∞ ). Thus, f(x) has a local 1 . Note that f(x) > 0 and has maximum f (0) = 16 horizontal asymptote y = 0. 0< x< , so f(x) has inflection 2 3 , c ≠ 0 c Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 46. f ( x) = 1 2 x + 4x + c . By the quadratic formula, x 2 + 4 x + c = 0 when x = –2 ± 4 – c . Thus f(x) has vertical asymptote(s) at x = –2 ± 4 – c –2 x – 4 ; f ( x) = 0 when c ≤ 4. f ( x) = 2 ( x + 4 x + c) 2 when x = –2, unless c = 4 since then x = –2 is a vertical asymptote. For c ≠ 4, f ( x) > 0 when x < –2, so f(x) is increasing on (– ∞ , –2] and decreasing on [–2, ∞ ) (with the asymptotes excluded). Thus 1 f(x) has a local maximum at f (–2) = . For c–4 2 c = 4, f ( x) = – so f(x) is increasing on ( x + 2)3 (– ∞ , –2) and decreasing on (–2, ∞ ). If c < 0 : ⎡ (4k + 1)π ( 4k − 1) π ⎤ f ( x ) is decreasing on ⎢ , ⎥ 2c ⎣ 2c ⎦ ⎡ ( 4k − 1) π ( 4k − 3) π ⎤ f ( x ) is increasing on ⎢ , ⎥ 2c 2c ⎣ ⎦ f ( x ) has local minima at x = maxima at x = ( 4k − 3 ) π 2c ( 4k − 1) 2c π and local where k is an integer. If c = 0 , f ( x ) = 0 and there are no extrema. If c > 0 : ⎡ ( 4k − 3) π ( 4k − 1) π ⎤ f ( x ) is decreasing on ⎢ , ⎥ 2c 2c ⎣ ⎦ ⎡ ( 4k − 1) π (4k + 1)π ⎤ f ( x ) is increasing on ⎢ , 2c 2c ⎥⎦ ⎣ ( 4k − 1) f ( x ) has local minima at x = π and 2c ( 4k − 3 ) π where k is an local maxima at x = 2c integer. y − π − π π −π π π x c = −2 c y 47. f ( x ) = c + sin cx . Since c is constant for all x and sin cx is continuous everywhere, the function f ( x ) is continuous everywhere. f ' ( x ) = c ⋅ cos cx − π − π ( f ' ( x ) = 0 when cx = k + 12 where k is an integer. f '' ( x ) = −c 2 ⋅ sin cx (( ) ) )π ( ( ( ) or x = k + 12 πc π −π c π π x c = −1 ) ) k f '' k + 12 πc = −c 2 ⋅ sin c ⋅ k + 12 πc = −c 2 ⋅ ( −1) In general, the graph of f will resemble the graph of y = sin x . The period will decrease as c increases and the graph will shift up or down depending on whether c is positive or negative. If c = 0 , then f ( x ) = 0 . Instructor’s Resource Manual Section 3.5 203 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. y Justification: f (1) = g (1) = 1 c=0 − π − π π −π π π x f (− x) = g ((− x)4 ) = g ( x 4 ) = f ( x) f is an even function; symmetric with respect to the y-axis. f f f f y c −π '( x) = 0 for x = −1, 0,1 since f ' is continuous. f ''( x) < 0 for x on ( x0 ,1) π π x y − π − π '( x) < 0 for x on (−∞, −1) ∪ (−1, 0) f ''( x) = 0 for x = −1, 0,1 f ''( x) > 0 for x on (0, x0 ) ∪ (1, ∞) c=1 π −π '( x) > 0 for x on (0,1) ∪ (1, ∞) f ''( x) = g ''( x 4 )16 x6 + g '( x)12 x 2 c − π − π '( x) = g '( x 4 )4 x3 c=2 π π π x 48. Since we have f ''' ( c ) > 0 , we know that f ' ( x ) is concave up in a neighborhood around x = c . Since f ' ( c ) = 0 , we then know that the graph of Where x0 is a root of f ''( x) = 0 (assume that there is only one root on (0, 1)). 50. Suppose H (1) < 0, then H ( x) is decreasing in a neighborhood around x = 1. Thus, H ( x) > 0 to the left of 1 and H ( x) < 0 to the right of 1, so H(x) is concave up to the left of 1 and concave down to the right of 1. Suppose H (1) > 0, then H ( x) is increasing in a neighborhood around x = 1. Thus, H ( x) < 0 to the left of 1 and H ( x) > 0 to the right of 1, so H(x) is concave up to the right of 1 and concave down to the left of 1. In either case, H(x) has a point of inflection at x = 1 and not a local max or min. 51. a. f ' ( x ) must be positive in that neighborhood. This means that the graph of f must be increasing to the left of c and increasing to the right of c. Therefore, there is a point of inflection at c. Not possible; F ( x) > 0 means that F(x) is increasing. F ( x) > 0 means that the rate at which F(x) is increasing never slows down. Thus the values of F must eventually become positive. b. Not possible; If F(x) is concave down for all x, then F(x) cannot always be positive. 49. c. 204 Section 3.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 52. a. 53. a. No global extrema; inflection point at (0, 0) f ( x) = 2 cos x – 2 cos x sin x = 2 cos x(1 – sin x); b. π π f ( x) = 0 when x = – , 2 2 f ( x) = –2sin x – 2 cos 2 x + 2sin 2 x = 4sin 2 x – 2sin x – 2; f ( x) = 0 when 1 or sin x = 1 which occur when 2 π 5π π x= – ,– , 6 6 2 ⎛ π⎞ Global minimum f ⎜ – ⎟ = –2; global ⎝ 2⎠ ⎛π⎞ maximum f ⎜ ⎟ = 2; inflection points ⎝2⎠ π 1 1 ⎛ ⎞ ⎛ 5π ⎞ f ⎜– ⎟ = – , f ⎜– ⎟ = – 4 ⎝ 6 ⎠ 4 ⎝ 6⎠ sin x = – No global maximum; global minimum at (0, 0); no inflection points c. Global minimum f(– π ) = –2 π + sin (– π ) = –2 π ≈ –6.3; b. global maximum f (π ) = 2π + sin π = 2π ≈ 6.3 ; inflection point at (0, 0) f ( x) = 2 cos x + 2sin x cos x = 2 cos x(1 + sin x); f ( x) = 0 when d. π π x=– , 2 2 f ( x) = –2sin x + 2 cos 2 x – 2sin 2 x Global minimum sin(– π) = – π ≈ 3.1; global f (– π) = – π – 2 sin π = π ≈ 3.1; maximum f (π) = π + 2 inflection point at (0, 0). Instructor’s Resource Manual = –4sin 2 x – 2sin x + 2; f ( x) = 0 when sin x = –1 or sin x = 1 which occur when 2 π π 5π x=– , , 2 6 6 ⎛ π⎞ Global minimum f ⎜ – ⎟ = –1; global ⎝ 2⎠ ⎛π⎞ maximum f ⎜ ⎟ = 3; inflection points ⎝2⎠ π 5 ⎛ ⎞ ⎛ 5π ⎞ 5 f ⎜ ⎟= , f ⎜ ⎟= . ⎝6⎠ 4 ⎝ 6 ⎠ 4 Section 3.5 205 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. f ( x) = 2 cos 2 x + 3sin 3 x Using the graphs, f(x) has a global minimum at f(2.17) ≈ –1.9 and a global maximum at f(0.97) ≈ 1.9 f ( x) = –4sin 2 x + 9 cos 3 x; f ( x) = 0 when c. π π x = – , and when 2 2 x ≈ –2.469, –0.673, 0.413, 2.729. ⎛ π ⎞ ⎛π ⎞ Inflection points: ⎜ − , 0 ⎟ , ⎜ , 0 ⎟ , ⎝ 2 ⎠ ⎝2 ⎠ ≈ ( −2.469, 0.542 ) , ( −0.673, −0.542 ) , f ( x) = –2sin 2 x + 2sin x = –4sin x cos x + 2sin x = 2sin x(1 – 2 cos x); π π , 0, , π 3 3 f ( x) = –4 cos 2 x + 2 cos x; f ( x) = 0 when x ≈ –2.206, –0.568, 0.568, 2.206 ⎛ π⎞ ⎛π⎞ Global minimum f ⎜ – ⎟ = f ⎜ ⎟ = –1.5; 3 ⎝ ⎠ ⎝3⎠ Global maximum f(– π ) = f( π ) = 3; Inflection points: ≈ ( −2.206, 0.890 ) , f ( x) = 0 when x = – π, – ( 0.413, 0.408 ) , ( 2.729, −0.408) 54. ( −0.568, −1.265 ) , ( 0.568, −1.265) , ( 2.206, 0.890 ) d. y 55. 5 f ( x) = 3cos 3 x – cos x; f ( x) = 0 when 3cos 3x = cos x which occurs when π π x = – , and when 2 2 x ≈ –2.7, –0.4, 0.4, 2.7 f ( x) = –9sin 3 x + sin x which occurs when x = – π , 0, π and when x ≈ –2.126, –1.016, 1.016, 2.126 ⎛π⎞ Global minimum f ⎜ ⎟ = –2; ⎝2⎠ ⎛ π⎞ global maximum f ⎜ – ⎟ = 2; ⎝ 2⎠ Inflection points: ≈ ( −2.126, 0.755 ) , ( −1.016, 0.755 ) , ( 0, 0 ) , (1.016, −0.755 ) , ( 2.126, −0.755) −5 Section 3.5 x −5 a. f is increasing on the intervals ( −∞, −3] and [ −1, 0] . f is decreasing on the intervals [ −3, −1] and [ 0, ∞ ) . b. f is concave down on the intervals ( −∞, −2 ) and ( 2, ∞ ) . f is concave up on the intervals ( −2, 0 ) and ( 0, 2 ) . e. 206 5 c. f attains a local maximum at x = −3 and x = 0. f attains a local minimum at x = −1 . d. f has a point of inflection at x = −2 and x = 2. Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. y 56. 59. a. 5 −5 5 x f ( x) = ; f ( x) is never 0, x 2 – 6 x + 40 and always positive, so f(x) is increasing for all x. Thus, on [–1, 7], the global minimum is f(–1) ≈ –6.9 and the global maximum if f(7) ≈ 48.0. 2 x3 − 18 x 2 + 147 x – 240 f ( x) = ; f ( x) = 0 ( x 2 – 6 x + 40)3 / 2 when x ≈ 2.02; inflection point f(2.02) ≈ 11.4 −5 a. f is increasing on the interval [ −1, ∞ ) . f is decreasing on the interval ( −∞, −1] b. f is concave up on the intervals ( −2, 0 ) and ( 2, ∞ ) . f is concave down on the interval ( 0, 2 ) . c. f does not have any local maxima. f attains a local minimum at x = −1 . d. f has inflection points at x = 0 and x = 2. 2 x 2 – 9 x + 40 b. 57. Global minimum f(0) = 0; global maximum f(7) ≈ 124.4; inflection point at x ≈ 2.34, f(2.34) ≈ 48.09 c. 58. No global minimum or maximum; no inflection points d. Global minimum f(3) ≈ –0.9; global maximum f(–1) ≈ 1.0 or f(7) ≈ 1.0; Inflection points at x ≈ 0.05 and x ≈ 5.9, f(0.05) ≈ 0.3, f(5.9) ≈ 0.3. Instructor’s Resource Manual Section 3.5 207 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 3.6 Concepts Review 60. a. 1. continuous; (a, b); f (b) – f (a) = f (c)(b – a) 2. f ( x) = 3 x 2 –16 x + 5; f ( x) = 0 when 1 x = , 5. 3 Global minimum f(5) = –46; ⎛1⎞ global maximum f ⎜ ⎟ ≈ 4.8 ⎝3⎠ 8 f ( x) = 6 x –16; f ( x) = 0 when x = ; 3 inflection point: ( 8 , −20.6 3 f (0) does not exist. 3. F(x) = G(x) + C 4. x 4 + C Problem Set 3.6 1. f ( x) = x x f (2) – f (1) 2 –1 = =1 2 –1 1 c = 1 for all c > 0, hence for all c in (1, 2) c ) b. Global minimum when x ≈ –0.5 and x ≈ 1.2, f(–0.5) ≈ 0, f(1.2) ≈ 0; global maximum f(5) = 46 Inflection point: ( −0.5, 0 ) , (1.2, 0 ) , ( 83 , 20.6) 2. The Mean Value Theorem does not apply because g (0) does not exist. c. No global minimum or maximum; inflection point at x ≈ −0.26, f (−0.26) ≈ −1.7 3. d. f ( x) = 2 x + 1 f (2) – f (–2) 6 – 2 = =1 2 – (–2) 4 2c + 1 = 1 when c = 0 No global minimum, global maximum when x ≈ 0.26, f(0.26) ≈ 4.7 Inflection points when x ≈ 0.75 and x ≈ 3.15, f(0.75) ≈ 2.6, f(3.15) ≈ –0.88 208 Section 3.6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4. g ( x) = 3( x + 1)2 7. g (1) – g (–1) 8 – 0 = =4 1 – (–1) 2 3(c + 1)2 = 4 when c = –1 + 2 3 ≈ 0.15 1 1 f ( z ) = (3 z 2 + 1) = z 2 + 3 3 f (2) – f (–1) 2 – (–2) 4 = = 2 – (–1) 3 3 1 4 = when c = –1, 1, but −1 is not in 3 3 (−1, 2) so c = 1 is the only solution. c2 + 5. H ( s ) = 2s + 3 H (1) – H (–3) 3 – (–1) = =1 1 – (–3) 1 – (–3) 2c + 3 = 1 when c = –1 8. The Mean Value Theorem does not apply because F(t) is not continuous at t = 1. 9. h ( x) = – 6. F ( x) = x 2 ( ) 8 8 F (2) – F (–2) 3 – – 3 4 = = 2 – (–2) 4 3 c2 = 4 2 ≈ ±1.15 when c = ± 3 3 Instructor’s Resource Manual 3 ( x – 3) 2 h(2) – h(0) –2 – 0 = = –1 2–0 2 3 – = –1 when c = 3 ± 3, (c – 3) 2 c = 3 – 3 ≈ 1.27 (3 + 3 is not in (0, 2).) Section 3.6 209 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 10. The Mean Value Theorem does not apply because f(x) is not continuous at x = 3. 5 2/3 x 3 g (1) – g (–1) 1 – (–1) = =1 1 – (–1) 2 14. g ( x) = 5 2/3 ⎛3⎞ c = 1 when c = ± ⎜ ⎟ 3 ⎝5⎠ 11. h (t ) = 3/ 2 ≈ ±0.46 2 1/ 3 3t h(2) – h(0) 22 / 3 − 0 = = 2 –1/ 3 2–0 2 16 2 = 2 –1/ 3 when c = ≈ 0.59 1/ 3 27 3c 15. S ( ) = cos S (π) – S (– π) 0 – 0 = =0 π – (– π) 2π π cos c = 0 when c = ± . 2 12. The Mean Value Theorem does not apply because h (0) does not exist. 16. The Mean Value Theorem does not apply because C ( ) is not continuous at = −π , 0, π . 5 2/3 x 3 g (1) – g (0) 1 – 0 = =1 1– 0 1 13. g ( x) = 5 2/3 ⎛3⎞ c = 1 when c = ± ⎜ ⎟ 3 ⎝5⎠ ⎛3⎞ c=⎜ ⎟ ⎝5⎠ 210 3/ 2 3/ 2 , ⎛ ⎛ 3 ⎞3 / 2 ⎞ is not in (0, 1). ⎟ ≈ 0.46, ⎜ – ⎜ ⎟ ⎜ ⎝5⎠ ⎟ ⎝ ⎠ Section 3.6 17. The Mean Value Theorem does not apply π because T ( ) is not continuous at = . 2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 18. The Mean Value Theorem does not apply because f(x) is not continuous at x = 0. 22. By the Mean Value Theorem f (b) − f (a) = f (c) for some c in (a, b). b−a 0 Since f(b) = f(a), = f (c); f (c) = 0 . b−a 23. 19. f ( x) = 1 – 1 x2 5 f (2) – f (1) 2 – 2 1 = = 2 –1 1 2 1 1 1– = when c = ± 2, c = 2 ≈ 1.41 c2 2 (c = – 2 is not in (1, 2).) 24. f (8) − f (0) 1 =− 8−0 4 There are three values for c such that 1 f (c ) = − . 4 They are approximately 1.5, 3.75, and 7. f ( x) = 2 x + f (b) – f (a ) 1 [ (b 2 – a 2 ) + (b – a )] = b–a b–a = ( a + b) + 2 c+ = ( a + b) + when c = a+b which is 2 the midpoint of [a, b]. 20. The Mean Value Theorem does not apply because f(x) is not continuous at x = 2. 25. By the Monotonicity Theorem, f is increasing on the intervals (a, x0 ) and ( x0 , b) . To show that f ( x0 ) > f ( x) for x in (a, x0 ) , consider f on the interval (a, x0 ] . f satisfies the conditions of the Mean Value Theorem on the interval [ x, x0 ] for x in (a, x0 ) . So for some c in ( x, x0 ), f ( x0 ) − f ( x) = f (c)( x0 − x) . Because f (c) > 0 and x0 − x > 0, f ( x0 ) − f ( x) > 0, so f ( x0 ) > f ( x ) . Similar reasoning shows that f ( x) > f ( x0 ) for x in ( x0 , b) . Therefore, f is increasing on (a, b). 26. a. 21. The Mean Value Theorem does not apply because f is not differentiable at x = 0 . f ( x) = 3 x 2 > 0 except at x = 0 in (– ∞ , ∞ ). f ( x) = x3 is increasing on (– ∞ , ∞ ) by Problem 25. b. f ( x) = 5 x 4 > 0 except at x = 0 in (– ∞ , ∞ ). f ( x) = x5 is increasing on (– ∞ , ∞ ) by Problem 25. c. ⎪⎧3 x 2 x ≤ 0 > 0 except at x = 0 in f ( x) = ⎨ x>0 ⎪⎩1 (– ∞ , ∞ ). ⎧⎪ x3 x ≤ 0 is increasing on f ( x) = ⎨ x>0 ⎪⎩ x (– ∞ , ∞ ) by Problem 25. Instructor’s Resource Manual Section 3.6 211 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 27. s(t) is defined in any interval not containing t = 0. 1 s (c ) = – < 0 for all c ≠ 0. For any a, b with c2 a < b and both either positive or negative, the Mean Value Theorem says s (b) – s (a) = s (c)(b – a ) for some c in (a, b). Since a < b, b – a > 0 while s (c) < 0, hence s(b) – s(a) < 0, or s(b) < s(a). Thus, s(t) is decreasing on any interval not containing t = 0. 28. s (c) = – 2 < 0 for all c > 0. If 0 < a < b, the c3 Mean Value Theorem says s (b) – s (a) = s (c)(b – a ) for some c in (a, b). Since a < b, b – a > 0 while s (c) < 0, hence s(b) – s(a) < 0, or s(b) < s(a). Thus, s(t) is decreasing on any interval to the right of the origin. 29. F ( x) = 0 and G ( x ) = 0; G ( x) = 0 . By Theorem B, F(x) = G(x) + C, so F(x) = 0 + C = C. 30. F ( x ) = cos 2 x + sin 2 x; F (0) = 12 + 02 = 1 F ( x) = 2 cos x(− sin x) + 2sin x(cos x) = 0 By Problem 29, F(x) = C for all x. Since F(0) = 1, C = 1, so sin 2 x + cos 2 x = 1 for all x. 31. Let G ( x) = Dx; F ( x) = D and G ( x) = D . By Theorem B, F(x) = G(x) + C; F(x) = Dx + C. 32. F ( x) = 5; F (0) = 4 F(x) = 5x + C by Problem 31. F(0) = 4 so C = 4. F(x) = 5x + 4 33. Since f(a) and f(b) have opposite signs, 0 is between f(a) and f(b). f(x) is continuous on [a, b], since it has a derivative. Thus, by the Intermediate Value Theorem, there is at least one point c, a < c < b with f(c) = 0. Suppose there are two points, c and c , c < c in (a, b) with f (c) = f (c ) = 0. Then by Rolle’s Theorem, there is at least one number d in (c, c ) with f (d ) = 0. This contradicts the given information that f ( x) ≠ 0 for all x in [a, b], thus there cannot be more than one x in [a, b] where f(x) = 0. 212 Section 3.6 34. f ( x) = 6 x 2 – 18 x = 6 x( x – 3); f ( x) = 0 when x = 0 or x = 3. f(–1) = –10, f(0) = 1 so, by Problem 33, f(x) = 0 has exactly one solution on (–1, 0). f(0) = 1, f(1) = –6 so, by Problem 33, f(x) = 0 has exactly one solution on (0, 1). f(4) = –15, f(5) = 26 so, by Problem 33, f(x) = 0 has exactly one solution on (4, 5). 35. Suppose there is more than one zero between successive distinct zeros of f . That is, there are a and b such that f(a) = f(b) = 0 with a and b between successive distinct zeros of f . Then by Rolle’s Theorem, there is a c between a and b such that f (c) = 0 . This contradicts the supposition that a and b lie between successive distinct zeros. 36. Let x1 , x2 , and x3 be the three values such that g ( x1 ) = g ( x2 ) = g ( x3 ) = 0 and a ≤ x1 < x2 < x3 ≤ b . By applying Rolle’s Theorem (see Problem 22) there is at least one number x4 in ( x1 , x2 ) and one number x5 in ( x2 , x3 ) such that g ( x4 ) = g ( x5 ) = 0 . Then by applying Rolle’s Theorem to g ( x) , there is at least one number x6 in ( x4 , x5 ) such that g ( x6 ) = 0 . 37. f(x) is a polynomial function so it is continuous on [0, 4] and f ( x) exists for all x on (0, 4). f(1) = f(2) = f(3) = 0, so by Problem 36, there are at least two values of x in [0, 4] where f ( x) = 0 and at least one value of x in [0, 4] where f ( x) = 0. 38. By applying the Mean Value Theorem and taking the absolute value of both sides, f ( x2 ) − f ( x1 ) = f (c) , for some c in ( x1 , x2 ) . x2 − x1 Since f ( x) ≤ M for all x in (a, b), f ( x2 ) − f ( x1 ) ≤ M ; f ( x2 ) − f ( x1 ) ≤ M x2 − x1 . x2 − x1 39. f ( x) = 2 cos 2 x; f ( x) ≤ 2 f ( x2 ) − f ( x1 ) x2 − x1 = f ( x) ; f ( x2 ) − f ( x1 ) x2 − x1 ≤2 f ( x2 ) − f ( x1 ) ≤ 2 x2 − x1 ; sin 2 x2 − sin 2 x1 ≤ 2 x2 − x1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 40. a. b. 44. Let f ( x) = x so f ( x) = 1 . Apply the Mean 2 x Value Theorem to f on the interval [x, x + 2] for x > 0. 1 1 for some c in Thus x + 2 − x = (2) = 2 c c 1 1 1 (x, x + 2). Observe < < . x+2 c x 1 Thus as x → ∞, → 0. c 1 Therefore lim x + 2 − x = lim =0. x →∞ x →∞ c ( ) 45. Let f(x) = sin x. f ( x) = cos x, so f ( x) = cos x ≤ 1 for all x. 41. Suppose f ( x) ≥ 0 . Let a and b lie in the interior of I such that b > a. By the Mean Value Theorem, there is a point c between a and b such that f (b) − f (a ) f (b) − f (a) f (c ) = ; ≥0. b−a b−a Since a < b, f(b) ≥ f(a), so f is nondecreasing. Suppose f ( x) ≤ 0. Let a and b lie in the interior of I such that b > a. By the Mean Value Theorem, there is a point c between a and b such that f (b) − f (a ) f (b) − f (a) f (c ) = ; ≤ 0 . Since b−a b−a a < b, f(a) ≥ f(b), so f is nonincreasing. 42. [ f 2 ( x)] = 2 f ( x) f ( x) Because f(x) ≥ 0 and f ( x) ≥ 0 on I , [ f 2 ( x)] ≥ 0 on I. As a consequence of the Mean Value Theorem, f 2 ( x2 ) − f 2 ( x1 ) ≥ 0 for all x2 > x1 on I. Therefore f 2 is nondecreasing. 43. Let f(x) = h(x) – g(x). f ( x) = h ( x) − g ( x); f ( x) ≥ 0 for all x in (a, b) since g ( x) ≤ h ( x ) for all x in (a, b), so f is nondecreasing on (a, b) by Problem 41. Thus x1 < x2 ⇒ f ( x1 ) ≤ f ( x2 ); h( x1 ) − g ( x1 ) ≤ h( x2 ) − g ( x2 ); g ( x2 ) − g ( x1 ) ≤ h( x2 ) − h( x1 ) for all x1 and x2 in (a, b). Instructor’s Resource Manual By the Mean Value Theorem, f ( x) − f ( y ) = f (c) for some c in (x, y). x− y Thus, f ( x) − f ( y ) x− y = f (c) ≤ 1; sin x − sin y ≤ x − y . 46. Let d be the difference in distance between horse A and horse B as a function of time t. Then d is the difference in speeds. Let t0 and t1 and be the start and finish times of the race. d (t0 ) = d (t1 ) = 0 By the Mean Value Theorem, d (t1 ) − d (t0 ) = d (c) for some c in (t0 , t1 ) . t1 − t0 Therefore d (c) = 0 for some c in (t0 , t1 ) . 47. Let s be the difference in speeds between horse A and horse B as function of time t. Then s is the difference in accelerations. Let t2 be the time in Problem 46 at which the horses had the same speeds and let t1 be the finish time of the race. s (t2 ) = s (t1 ) = 0 By the Mean Value Theorem, s (t1 ) − s (t2 ) = s (c) for some c in (t2 , t1 ) . t1 − t2 Therefore s (c) = 0 for some c in (t2 , t1 ) . Section 3.6 213 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 48. Suppose x > c. Then by the Mean Value Theorem, f ( x) − f (c ) = f (a )( x − c) for some a in (c, x) . Since f is concave up, f > 0 and by the Monotonicity Theorem f is increasing. Therefore f (a ) > f (c) and f ( x) − f (c ) = f (a )( x − c) > f (c)( x − c) f ( x) > f (c ) + f (c )( x − c ), x > c Suppose x < c. Then by the Mean Value Theorem, f (c) − f ( x) = f (a )(c − x) for some a in ( x, c) . Since f is concave up, f > 0 , and by the Monotonicity Theorem f is increasing. Therefore, f (c) > f (a) and f (c) − f ( x ) = f (a )(c − x) < f (c)(c − x) . − f ( x) < − f (c ) + f (c)(c − x) f ( x) > f (c) − f (c)(c − x ) f ( x) > f (c ) + f (c )( x − c ), x < c Therefore f ( x) > f (c ) + f (c )( x − c ), x ≠ c . 49. Fix an arbitrary x. f ( y) − f ( x) f ' ( x ) = lim = 0 , since y→x y−x f ( y) − f ( x) y−x So, f ' 50. ≤M y−x . 0 → f = constant . 1/ 3 f ( x) = x on [0, a] or [–a, 0] where a is any positive number. f (0) does not exist, but f(x) has a vertical tangent line at x = 0. 51. Let f(t) be the distance traveled at time t. f (2) − f (0) 112 − 0 = = 56 2−0 2 By the Mean Value Theorem, there is a time c such that f (c) = 56. At some time during the trip, Johnny must have gone 56 miles per hour. 214 Section 3.6 52. s is differentiable with s (0) = 0 and s (18) = 20 so we can apply the Mean Value Theorem. There exists a c in the interval ( 0,18 ) such that v(c) = s '(c) = (20 − 0) ≈ 1.11 miles per minute (18 − 0 ) ≈ 66.67 miles per hour 53. Since the car is stationary at t = 0 , and since v is 1 continuous, there exists a δ such that v(t ) < 2 for all t in the interval [0, δ ] . v(t ) is therefore 1 1 δ and s (δ ) < δ ⋅ = . By the Mean 2 2 2 Value Theorem, there exists a c in the interval (δ , 20) such that less than δ⎞ ⎛ ⎜ 20 − ⎟ 2⎠ ⎝ v(c) = s '(c) = (20 − δ ) 20 − δ > 20 − δ = 1 mile per minute = 60 miles per hour Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 54. Given the position function s ( t ) = at 2 + bt + c , the car’s instantaneous velocity is given by the function s ' ( t ) = 2at + b . A+ B . 2 Thus, the car’s instantaneous velocity at the midpoint of the interval is given by ⎛ A+ B ⎞ ⎛ A+ B ⎞ s '⎜ ⎟ = 2a ⎜ ⎟+b ⎝ 2 ⎠ ⎝ 2 ⎠ = a ( A + B) + b The midpoint of the interval [ A, B ] is The car’s average velocity will be its change in position divided by the length of the interval. That is, 2 2 s ( B ) − s ( A ) ( aB + bB + c ) − ( aA + bA + c ) = B− A B− A 2 2 aB − aA + bB − bA = B− A = = ( ) a B 2 − A2 + b ( B − A ) B− A a ( B − A )( B + A) + b ( B − A ) B− A = a ( B + A) + b = a ( A + B) + b This is the same result as the instantaneous velocity at the midpoint. 3.7 Concepts Review 1. slowness of convergence 2. root; Intermediate Value 3. algorithms 4. fixed point Problem Set 3.7 1. Let f ( x) = x3 + 2 x – 6. f(1) = –3, f(2) = 6 n hn mn f (mn ) 1 0.5 1.5 0.375 2 0.25 1.25 –1.546875 3 0.125 1.375 –0.650391 4 0.0625 1.4375 –0.154541 5 0.03125 1.46875 0.105927 6 0.015625 1.45312 –0.0253716 7 0.0078125 1.46094 0.04001 8 0.00390625 1.45703 0.00725670 9 0.00195312 r ≈ 1.46 1.45508 –0.00907617 2. Let f ( x) = x 4 + 5 x3 + 1. f(–1) = –3, f(0) = 1 n hn mn f (mn ) 1 0.5 –0.5 0.4375 2 0.25 –0.75 –0.792969 3 0.125 –0.625 –0.0681152 4 0.0625 –0.5625 0.21022 5 0.03125 –0.59375 0.0776834 6 0.015625 –0.609375 0.00647169 7 0.0078125 –0.617187 –0.0303962 8 0.00390625 –0.613281 –0.011854 9 0.00195312 r ≈ –0.61 –0.611328 –0.00266589 3. Let f ( x ) = 2 cos x − sin x . f (1) ≈ 0.23913 ; f ( 2 ) ≈ −1.74159 n hn mn f ( mn ) 1 0.5 1.5 −0.856021 2 0.25 1.25 −0.318340 3 0.125 1.125 −0.039915 4 0.0625 1.0625 0.998044 5 0.03125 1.09375 0.029960 6 0.01563 1.109375 −0.004978 r ≈ 1.11 Instructor’s Resource Manual Section 3.7 215 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4. Let f ( x ) = x − 2 + 2 cos x f (1) = 1 − 2 + 2 cos (1) ≈ 0.080605 f ( 2 ) = 2 − 2 + 2 cos ( 2 ) ≈ −0.832294 n xn 1 1 2 0.8636364 n hn mn f ( mn ) 3 0.8412670 1 0.5 1.5 −0.358526 4 0.8406998 5 0.8406994 2 0.25 1.25 −0.119355 3 0.125 1.125 −0.012647 4 0.0625 1.0625 0.035879 5 0.03125 1.09375 0.012065 6 0.01563 1.109375 −0.000183 r ≈ 1.11 6 r ≈ 0.84070 0.8406994 7. Let f ( x ) = x − 2 + 2 cos x . y 5 5. Let f ( x) = x3 + 6 x 2 + 9 x + 1 = 0 . −5 5 x −5 f ' ( x ) = 1 − 2sin x f ( x) = 3x 2 + 12 x + 9 n xn 1 4 2 3.724415 n xn 1 0 4 3.698154 2 –0.1111111 3 –0.1205484 5 3.698154 r ≈ 3.69815 4 –0.1206148 5 r ≈ –0.12061 –0.1206148 3 3.698429 8. Let f ( x ) = 2 cos x − sin x . y 5 6. Let f ( x) = 7 x3 + x – 5 −5 5 x −5 f ' ( x ) = −2sin x − cos x f ( x) = 21x 2 + 1 n xn 1 0.5 2 1.1946833 3 1.1069244 4 1.1071487 5 1.1071487 r ≈ 1.10715 216 Section 3.7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 9. Let f(x) = cos x – 2x. f ( x) = 4 x3 – 24 x 2 + 44 x – 24 Note that f(2) = 0. f ( x) = – sin x – 2 n xn 1 0.5 2 0.4506267 3 0.4501836 4 r ≈ 0.45018 0.4501836 n xn 1 0.5 2 0.575 3 0.585586 4 0.585786 n xn 1 3.5 2 3.425 3 3.414414 4 3.414214 5 3.414214 r = 2, r ≈ 0.58579, r ≈ 3.41421 12. Let f ( x) = x 4 + 6 x3 + 2 x 2 + 24 x – 8. 10. Let f ( x ) = 2 x − sin x − 1 . y 5 −5 5 x −5 f ' ( x ) = 2 − cos x f ( x) = 4 x3 + 18 x 2 + 4 x + 24 n xn n xn 1 –6.5 1 1 2 –6.3299632 3 –6.3167022 4 –6.3166248 5 –6.3166248 n xn 1 0.5 2 0.3286290 3 0.3166694 4 0.3166248 2 0.891396 3 0.887866 4 0.887862 5 0.887862 r ≈ 0.88786 11. Let f ( x) = x 4 – 8 x3 + 22 x 2 – 24 x + 8. 5 0.3166248 r ≈ –6.31662, r ≈ 0.31662 Instructor’s Resource Manual Section 3.7 217 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 13. Let f ( x ) = 2 x 2 − sin x . 16. Let f ( x) = x 4 – 47 . y f ( x ) = 4 x3 2 −2 x 2 −2 4 f ' ( x ) = 4 x − cos x n 1 xn 0.5 17. n xn 1 2.5 2 2.627 3 2.618373 4 2.618330 5 2.618330 47 ≈ 2.61833 f ( x ) = x 4 + x3 + x 2 + x is continuous on the given interval. 2 0.481670 3 0.480947 4 0.480946 r ≈ 0.48095 From the graph of f, we see that the maximum value of the function on the interval occurs at the right endpoint. The minimum occurs at a stationary point within the interval. To find where the minimum occurs, we solve f ' ( x ) = 0 14. Let f ( x ) = 2 cot x − x . y 2 on the interval [ −1,1] . −2 2 x f ' ( x ) = −2 csc 2 x − 1 xn 1 1 Using Newton’s Method to solve g ( x ) = 0 , we get: −2 n f ' ( x ) = 4 x3 + 3x 2 + 2 x + 1 = g ( x ) n xn 1 0 2 −0.5 3 −0.625 4 −0.60638 2 1.074305 5 −0.60583 3 1.076871 6 −0.60583 4 1.076874 r ≈ 1.07687 Minimum: f ( −0.60583) ≈ −0.32645 Maximum: f (1) = 4 15. Let f ( x) = x3 – 6. f ( x) = 3 x 2 3 218 n xn 1 1.5 2 1.888889 3 1.819813 4 1.817125 5 1.817121 6 1.817121 6 ≈ 1.81712 Section 3.7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 18. f ( x) = x3 + 1 x4 + 1 n is continuous on the given interval. xn n xn 1 4.712389 1 7.853982 2 4.479179 2 7.722391 3 4.793365 3 7.725251 4 4.493409 4 7.725252 5 4.493409 5 7.725252 Minimum: f ( 4.493409 ) ≈ −0.21723 From the graph of f, we see that the maximum and minimum will both occur at stationary points within the interval. The minimum appears to occur at about x = −1.5 while the maximum appears to occur at about x = 0.8 . To find the stationary points, we solve f ' ( x ) = 0 . f '( x) = ( (x − x2 x4 + 4 x − 3 4 ) +1 2 20. f ( x ) = x 2 sin x is continuous on the given 2 interval. ) = g ( x) Using Newton’s method to solve g ( x ) = 0 on the interval, we use the starting values of −1.5 and 0.8 . n xn n xn 1 −1.5 1 0.8 From the graph of f, we see that the minimum value and maximum value on the interval will occur at stationary points within the interval. To find these points, we need to solve f ' ( x ) = 0 on the interval. 2 −1.680734 2 0.694908 3 −1.766642 3 0.692512 4 −1.783766 4 0.692505 x x x 2 cos + 4 x sin 2 2 = g ( x) f '( x) = 2 Using Newton’s method to solve g ( x ) = 0 on 5 −1.784357 5 0.692505 the interval, we use the starting values of 6 −1.784358 13π . 4 7 −1.784358 Maximum: f ( 0.692505 ) ≈ 1.08302 n Minimum: f ( −1.78436 ) ≈ −0.42032 19. Maximum: f ( 7.725252 ) ≈ 0.128375 f ( x) = sin x is continuous on the given interval. x xn n 3π and 2 xn 1 4.712389 1 10.210176 2 4.583037 2 10.174197 3 4.577868 3 10.173970 4 4.577859 4 10.173970 5 4.577859 Minimum: f (10.173970 ) ≈ −96.331841 From the graph of f, we see that the minimum value and maximum value on the interval will occur at stationary points within the interval. To find these points, we need to solve f ' ( x ) = 0 on Maximum: f ( 4.577859 ) ≈ 15.78121 the interval. x cos x − sin x = g ( x) f '( x) = x2 Using Newton’s method to solve g ( x ) = 0 on the interval, we use the starting values of 5π . 2 Instructor’s Resource Manual 3π and 2 Section 3.7 219 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 21. Graph y = x and y = 0.8 + 0.2 sin x. b. Let f (i ) = 20i (1 + i ) 24 − (1 + i )24 + 1 = (1 + i )24 (20i − 1) + 1 . Then f (i ) = 20(1 + i ) 24 + 480i (1 + i )23 − 24(1 + i )23 = (1 + i )23 (500i − 4), so in +1 = in − xn +1 = 0.8 + 0.2sin xn Let x1 = 1. n ⎡ 20i 2 + 19in − 1 + (1 + in ) −23 ⎤ = in − ⎢ n ⎥. 500in − 4 ⎣⎢ ⎦⎥ xn 1 2 3 4 5 6 7 x ≈ 0.9643 1 0.96829 0.96478 0.96439 0.96434 0.96433 0.96433 f (in ) (1 + in ) 24 (20in − 1) + 1 = in − f (in ) (1 + in )23 (500in − 4) c. 22. n in 1 0.012 2 0.0165297 3 0.0152651 4 0.0151323 5 0.0151308 6 0.0151308 i = 0.0151308 r = 18.157% 24. From Newton’s algorithm, xn +1 – xn = − f ( xn ) . f ( xn ) lim ( xn +1 – xn ) = lim xn +1 – lim xn xn → x xn +1 = xn – f ( xn ) x 1/ 3 = xn – n 1 x –2 / 3 f ( xn ) 3 n = xn – 3 xn = –2 xn Thus, every iteration of Newton’s Method gets further from zero. Note that xn +1 = (–2) n +1 x0 . Newton’s Method is based on approximating f by its tangent line near the root. This function has a vertical tangent at the root. 23. a. For Tom’s car, P = 2000, R = 100, and k = 24, thus 100 ⎡ 1 ⎤ 2000 = ⎢1 − ⎥ or i ⎣⎢ (1 + i ) 24 ⎦⎥ 20i = 1 − 1 (1 + i )24 , which is equivalent to xn → x xn → x =x–x =0 f ( xn ) exists if f and f are continuous at lim xn → x f ( xn ) x and f ( x ) ≠ 0. f ( xn ) f (x ) = = 0, so f ( x ) = 0. f (x ) xn → x f ( xn ) x is a solution of f(x) = 0. Thus, lim 25. xn +1 = n 1 xn + 1.5cos xn 2 xn n 1 xn 5 0.914864 2 0.905227 6 0.914856 3 0.915744 4 0.914773 x ≈ 0.91486 7 0.914857 20i (1 + i ) 24 − (1 + i )24 + 1 = 0 . 220 Section 3.7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 26. xn +1 = 2 − sin x 29. a. n xn n 1 2 xn n xn 5 1.10746 9 1.10603 2 1.09070 6 1.10543 10 1.10607 3 1.11305 4 1.10295 x ≈ 1.10606 7 1.10634 8 1.10612 11 1.10606 12 1.10606 27. xn +1 = 2.7 + xn x ≈ 0.5 n xn 1 1 2 1.923538 3 2.150241 4 2.202326 5 2.214120 6 2.216781 7 2.217382 8 2.217517 9 2.217548 10 2.217554 11 2.217556 12 2.217556 b. c. xn +1 = 2( xn – xn2 ) n xn 1 0.7 2 0.42 3 0.4872 4 0.4996723 5 0.4999998 6 0.5 7 0.5 x = 2( x – x 2 ) 2 x2 – x = 0 x(2x – 1) = 0 1 x = 0, x = 2 x ≈ 2.21756 30. a. 28. xn +1 = 3.2 + xn n xn 1 47 2 7.085196 3 3.207054 4 2.531216 5 2.393996 6 2.365163 7 2.359060 8 2.357766 9 2.357491 10 2.357433 11 2.357421 12 2.357418 13 2.357418 x ≈ 2.35742 Instructor’s Resource Manual x ≈ 0.8 b. xn +1 = 5( xn – xn2 ) n xn 1 0.7 2 1.05 3 –0.2625 4 –1.657031 5 –22.01392 6 –2533.133 Section 3.7 221 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. c. 33. a. x = 5( x – x 2 ) 1 x2 = 1 + = 2 1 1 3 x3 = 1 + = = 1.5 1 + 11 2 5x2 – 4 x = 0 x(5x – 4) = 0 4 x = 0, x = 5 31. a. x1 = 1 x1 = 0 x4 = 1 + x2 = 1 = 1 x3 = 1 + 1 = 2 ≈ 1.4142136 x5 = 1 + x4 = 1 + 1 + 1 ≈ 1.553774 1 1 1+ 1 1+ 1 x = 1+ x x2 = 1 + x 1+ 1 ± 1 + 4 ⋅1⋅1 1 ± 5 = 2 2 Taking the minus sign gives a negative solution for x, violating the requirement that 1+ 5 x ≥ 0 . Hence, x = ≈ 1.618034 . 2 . Then x satisfies c. (1 + 5 ) / 2 ≈ 1.618034 . x1 = 0 x2 = 5 ≈ 2.236068 1 x x4 = 5 + 5 + 5 ≈ 2.7730839 x5 = 5 + 5 + 5 + 5 ≈ 2.7880251 x = 5 + x , and x must satisfy x ≥ 0 x2 = 5 + x 1 ± 1 + 4 ⋅1⋅1 1 ± 5 = 2 2 Taking the minus sign gives a negative solution for x, violating the requirement that 1+ 5 ≈ 1.618034 . x ≥ 0 . Hence, x = 2 Let 1 x = 1+ . 1 1+ 1+ 1 Then x satisfies the equation x = 1 + . x From part (b) we know that x must equal (1 + 5 ) / 2 ≈ 1.618034 . x3 = 5 + 5 ≈ 2.689994 b. 1 1 + 11 x= the equation x = 1 + x . From part (b) we know that x must equal 32. a. 34. a. Suppose r is a root. Then r = r – Suppose f(r) = 0. Then r – 1 ± 1 + 4 ⋅1 ⋅ 5 1 ± 21 x= = 2 2 Taking the minus sign gives a negative solution for x, violating the requirement that x ≥ 0 . Hence, f (r ) . f (r ) f (r ) = 0, so f(r) = 0. f (r ) x2 − x − 5 = 0 ( 8 = 1.6 5 x2 − x − 1 = 0 x= Let x = 1 + 1 + 1 + x = 1+ = x2 = x + 1 x2 − x − 1 = 0 c. 1 1+ b. 5 ≈ 1.6666667 3 1 x5 = 1 + 1 + 1 + 1 ≈ 1.5980532 b. = so r is a root of x = x – f (r ) = r – 0 = r, f (r ) f ( x) . f ( x) ) x = 1 + 21 / 2 ≈ 2.7912878 c. Let x = 5 + 5 + 5 + . Then x satisfies the equation x = 5 + x . From part (b) we know that x must equal (1 + 222 ) 21 / 2 ≈ 2.7912878 Section 3.7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. If we want to solve f(x) = 0 and f ( x) ≠ 0 in f ( x) = 0 or f ( x) [a, b], then x= x– f ( x) = g ( x) . f ( x) = f ( x) f ( x) [ f ( x)]2 f (r ) f (r ) and g '(r ) = 35. a. starting with x = f ( x) f ( x) + f ( x) f ( x) [ f ( x)]2 g ( x) = 1 – [ f (r )]2 = 0. The algorithm computes the root of 1 1 – a = 0 for x1 close to . a x b. Let f ( x) = ⎛ π⎞ On the interval ⎜ 0, ⎟ , there is only one ⎝ 2⎠ stationary point (check graphically). We will use Newton’s Method to find the stationary point, π 4 ≈ 0.785398 . n xn π 1 4 ≈ 0.785398 2 3 0.862443 0.860335 4 0.860334 5 0.860334 x ≈ 0.860334 will maximize the area of the rectangle in quadrant I, and subsequently the larger rectangle as well. y = cos x = cos ( 0.860334 ) ≈ 0.652184 The maximum area of the larger rectangle is AL = ( 2 x ) y ≈ 2 ( 0.860334 )( 0.652184 ) 1 – a. x ≈ 1.122192 square units x2 f ( x) = – x + ax 2 f ( x) The recursion formula is f ( xn ) xn +1 = xn – = 2 xn – axn 2 . f ( xn ) 37. The rod that barely fits around the corner will touch the outside walls as well as the inside corner. F E 36. We can start by drawing a diagram: C y b 2 D (x, y) − π2 0 x cos x − x sin x = 0 Instructor’s Resource Manual B a π 2 x From symmetry, maximizing the area of the entire rectangle is equivalent to maximizing the area of the rectangle in quadrant I. The area of the rectangle in quadrant I is given by A = xy = x cos x To find the maximum area, we first need the ⎛ π⎞ stationary points on the interval ⎜ 0, ⎟ . ⎝ 2⎠ A ' ( x ) = cos x − x sin x Therefore, we need to solve A'( x) = 0 6.2 feet f ( x) = – 1 A 8.6 feet As suggested in the diagram, let a and b represent the lengths of the segments AB and BC, and let denote the angles ∠DBA and ∠FCB . Consider the two similar triangles ΔADB and ΔBFC ; these have hypotenuses a and b respectively. A little trigonometry applied to these angles gives 8.6 6.2 = 8.6sec and b = = 6.2 csc a= cos sin Note that the angle determines the position of the rod. The total length of the rod is then L = a + b = 8.6sec + 6.2 csc ⎛ π⎞ The domain for is the open interval ⎜ 0, ⎟ . ⎝ 2⎠ Section 3.7 223 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. The derivative of L is 8.6sin 3 − 6.2 cos3 L '( ) = sin 2 ⋅ cos 2 Thus, L ' ( ) = 0 provided 8.6sin 3 − 6.2 cos3 8.6sin =0 3 = 6.2 cos3 sin 3 6.2 8.6 6.2 = 8.6 = cos3 tan 3 tan = 3 Note that the angle determines the position of the rod. The total length of the rod is then L = a + b = 8csc + 8csc ( 75 − ) 6.2 8.6 ⎛ π⎞ On the interval ⎜ 0, ⎟ , there will only be one ⎝ 2⎠ solution to this equation. We will use Newton’s 6.2 = 0 starting with method to solve tan − 3 8.6 1 = π 4 As suggested in the diagram, let a and b represent the lengths of the segments AB and BC, and let denote the angle ∠ABD . Consider the two right triangles ΔADB and ΔCEB ; these have hypotenuses a and b respectively. A little trigonometry applied to these angles gives 8 = 8csc and a= sin 8 b= = 8csc ( 75 − ) sin ( 75 − ) The domain for is the open interval ( 0, 75 ) . A graph of of L indicates there is only one extremum (a minimum) on the interval. . The derivative of L is n n 1 π4 ≈ 0.78540 2 3 0.73373 0.73098 4 0.73097 L '( )= ( 8 sin 2 ⋅ cos ( − 75 ) − cos ⋅ sin 2 ( − 75 ) ) sin ⋅ sin ( − 75 ) We will use Newton’s method to solve L ' ( ) = 0 starting with 1 = 40 . 5 0.73097 Note that ≈ 0.73097 minimizes the length of the rod that does not fit around the corner, which in turn maximizes the length of the rod that will fit around the corner (verify by using the Second Derivative Test). L ( 0.73097 ) = 8.6sec ( 0.73097 ) + 6.2 csc ( 0.73097 ) ≈ 20.84 Thus, the length of the longest rod that will fit around the corner is about 20.84 feet. 38. The rod that barely fits around the corner will touch the outside walls as well as the inside corner. C 8 feet 2 n n 1 40 2 2 37.54338 3 37.50000 4 37.5 Note that = 37.5° minimizes the length of the rod that does not fit around the corner, which in turn maximizes the length of the rod that will fit around the corner (verify by using the Second Derivative Test). L ( 37.5 ) = 8csc ( 37.5 ) + 8csc ( 75 − 37.5 ) = 16 csc ( 37.5 ) ≈ 26.28 Thus, the length of the longest rod that will fit around the corner is about 26.28 feet. b E 75 − B 105 o a A 224 8 feet Section 3.7 D Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2x2 + x + 42 = 0 to 25 find the value for x when the object hits the ground. We want the value to be positive, so we use the quadratic formula, keeping only the positive solution. 39. We can solve the equation − x= −1 − 12 − 4 ( −0.08 )( 42 ) 2 ( −0.08 ) = 30 3.8 Concepts Review 1. rx r −1 ; r −1 2. r [ f ( x) ] f ( x); [ f ( x) ] f ( x) r 3. u = x 4 + 3 x 2 + 1, du = (4 x3 + 6 x)dx ( x 4 + 3x 2 + 1)8 (4 x3 + 6 x )dx = u8 du We are interested in the global extrema for the distance of the object from the observer. We obtain the same extrema by considering the squared distance D( x) = ( x − 3)2 + (42 + x − .08 x 2 ) 2 A graph of D will help us identify a starting point for our numeric approach. x r +1 + C , r ≠ −1 r +1 = u9 ( x 4 + 3 x 2 + 1)9 +C = +C 9 9 4. c1 f ( x)dx + c2 g ( x)dx Problem Set 3.8 From the graph, it appears that D (and thus the distance from the observer) is maximized at about x = 7 feet and minimized just before the object hits the ground at about x = 28 feet. The first derivative is given by 16 3 12 2 236 D '( x) = x − x − x + 78 . 625 25 25 a. We will use Newton’s method to find the stationary point that yields the minimum distance, starting with x1 = 28 . n xn 1 28 2 28.0280 3 28.0279 4 28.0279 x ≈ 28.0279; y ≈ 7.1828 The object is closest to the observer when it is at the point ( 28.0279, 7.1828 ) . b. We will use Newton’s method to find the stationary point that yields the maximum distance, starting with x1 = 7 . n xn 1 7 2 6.7726 3 6.7728 4 6.7728 x ≈ 6.7728; y ≈ 45.1031 The object is closest to the observer when it is at the point ( 6.7728, 45.1031) . Instructor’s Resource Manual 1. 5dx = 5 x + C 2. ( x − 4)dx = xdx − 4 1dx = x2 − 4x + C 2 3. ( x 2 + π)dx = x 2 dx + π 1dx = 4. ( 3x2 + 3 ) dx = 3 =3 5. 6. 7. x3 + πx + C 3 x 2 dx + 3 1dx x3 + 3 x + C = x3 + 3 x + C 3 x5 / 4 dx = x9 / 4 9 4 +C = 4 9/4 x +C 9 ⎛ x5 / 3 ⎞ 3x 2 / 3 dx = 3 x 2 / 3 dx = 3 ⎜ + C1 ⎟ ⎜ 5 ⎟ ⎝ 3 ⎠ 9 = x5 / 3 + C 5 1 3 2 dx = x −2 / 3 dx = 3 x1/ 3 + C = 33 x + C x 8. 7 x −3 / 4 dx = 7 x −3 / 4 dx = 7(4 x1/ 4 + C1 ) = 28 x1/ 4 + C 9. ( x 2 − x)dx = x 2 dx − x dx = x3 x 2 − +C 3 2 Section 3.8 225 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. (3 x 2 − πx)dx = 3 x 2 dx − π x dx 10. 4 x6 + 3x 4 17. x dx = (4 x3 + 3 x) dx ⎛ x3 ⎞ ⎛ x2 ⎞ = 3 ⎜ + C1 ⎟ − π ⎜ + C2 ⎟ ⎜ 3 ⎟ ⎜ 2 ⎟ ⎝ ⎠ ⎝ ⎠ = 4 x3 dx + 3 x dx πx 2 =x − +C 2 = x4 + 3 (4 x5 − x3 )dx = 4 x5 dx − x3 dx 11. = = x 2 x6 x 4 − +C 3 4 = x101 x100 + +C 101 100 = 6 4 20. ( x3 + x ) dx = = ( ) = ( x5 + 5 x 4 − 3 x3 + ) 3 x 2 dx =3 x dx − 2 x −1 ( 16. = 2 x1/ 2 1 2 = 2 2x − + dx = ( 4x 4 +C Section 3.8 2 2 ( ) 2 1 + 2 ) z3 ( 2 +C z dz = 2 dz = ⎡ 1 + 2 z ⎤ dz ⎣ ⎦ 3 dz = z4 + 2z2 + 1 z dz 2 9/ 2 4 5/ 2 z + z + 2 z1/ 2 + C 9 5 s( s + 1)2 24. 3x −4 +C −4 3 z ) = z 7 / 2 dz + 2 z 3 / 2 + z −1/ 2 dz −2 ⎛ 2x 3 ⎞ + ⎟⎟ dx = ⎜⎜ x5 ⎠ ⎝ x ) ( z 2 + 1)2 23. 3x 2x − +C −1 −2 3 1 =− + +C x x2 = 2z = 1+ 2 ⎛ 3 2 ⎞ −2 −3 ⎜ 2 − 3 ⎟ dx = (3 x − 2 x ) dx x ⎠ ⎝x −3 x 4 x3 / 2 x 4 2 x3 + +C = + +C 3 4 4 3 (z + 22. x6 3x4 3 x3 = + x5 − + +C 6 4 3 −2 x3 dx + x1/ 2 dx 2 = x5 dx + 5 x 4 dx −3 x3 dx + 3 x 2 dx 15. x3 x 2 + +C 3 2 21. Let u = x + 1; then du = dx. u3 ( x + 1)3 ( x + 1)2 dx = u 2 du = +C = +C 3 3 ⎡ x 2 x3 + 5 x 2 − 3x + 3 ⎤ dx ⎥⎦ ⎣⎢ 14. x4 1 + +C 4 x 2 27 x x 45 x 2x + − + +C 8 2 4 2 x 4 x −1 − +C 4 −1 ( x 2 + x) dx = x 2 dx + x dx = = 27 x7 dx + 3 x5 dx − 45 x3 dx + 2 x dx 8 dx = ( x3 − x −2 ) dx 19. (27 x7 + 3x5 − 45 x3 + 2 x)dx 13. 3 = x3 dx − x −2 dx = ( x100 + x99 )dx = x100 dx + x99 dx 12. 3x 2 +C 2 x6 − x 18. ⎛ x6 ⎞ ⎛ x4 ⎞ = 4 ⎜ + C1 ⎟ − ⎜ + C2 ⎟ ⎜ 6 ⎟ ⎜ 4 ⎟ ⎝ ⎠ ⎝ ⎠ 226 3 s ds = s3 + 2 s 2 + s s ds = s5 / 2 ds + 2 s3 / 2 ds + s1/ 2 ds ) 2 x −1/ 2 + 3 x −5 dx = 25. 2s 7 / 2 4s5 / 2 2s3 / 2 + + +C 7 5 3 (sin − cos )d = sin d − cos d = − cos − sin + C Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. (t 2 − 2 cos t )dt = t 2 dt − 2 cos t dt 26. = 3y t3 − 2sin t + C 3 2 2y + 5y 27. Let g ( x) = 2 x + 1 ; then g ( x) = 2 . ( ) 2 x +1 3 [ g ( x)]3 g ( x)dx 2 dx = [ g ( x)]4 + C = ( = ) 2 x +1 4 4 4 +C 28. Let g ( x) = πx3 + 1 ; then g ( x) = 3πx 2 . (πx3 + 1)4 3πx 2 dx = = [ g ( x)]4 g ( x) dx [ g ( x)]5 + C = (πx3 + 1)5 + C 5 5 29. Let u = 5 x3 + 3x − 8 ; then du = (15 x 2 + 3) dx . (5 x 2 + 1)(5 x3 + 3x − 8)6 dx 1 (15 x 2 + 3)(5 x3 + 3 x − 8)6 dx 3 ⎞ 1 6 1 ⎛ u7 u du = ⎜ = + C1 ⎟ ⎟ 3 3 ⎜⎝ 7 ⎠ = = 32. Let u = 2 y 2 + 5; then du = 4 y dy (5 x3 + 3x − 8)7 +C 21 30. Let u = 5 x3 + 3x − 2; then du = (15 x 2 + 3)dx . (5 x 2 + 1) 5 x3 + 3 x − 2 dx 1 (15 x 2 + 3) 5 x3 + 3 x − 2 dx = 3 1 1/ 2 1⎛ 2 ⎞ u du = ⎜ u 3 / 2 + C1 ⎟ = 3 3⎝ 3 ⎠ 2 = (5 x3 + 3x − 2)3 / 2 + C 9 2 (5 x3 + 3x − 2)3 + C = 9 31. Let u = 2t 2 − 11; then du = 4t dt . 3 (4t )(2t 2 − 11)1/ 3 dt 4 3 1/ 3 3⎛3 ⎞ u du = ⎜ u 4 / 3 + C1 ⎟ = 4 4⎝4 ⎠ 9 = (2t 2 − 11) 4 / 3 + C 16 9 = 3 (2t 2 − 11) 4 + C 16 3 (4 y )(2 y 2 + 5) −1/ 2 dy 4 3 −1/ 2 3 u du = (2u1/ 2 + C1 ) 4 4 3 2 2y + 5 + C = 2 = 33. Let u = x3 + 4 ; then du = 3 x 2 dx . 1 2 3 3x x + 4 dx x 2 x3 + 4 dx = 3 1 1 = u du = u1/ 2 du 3 3 1⎛ 2 ⎞ = ⎜ u 3/ 2 + C1 ⎟ 3⎝ 3 ⎠ 3/ 2 2 = x3 + 4 +C 9 ( ) 34. Let u = x 4 + 2 x 2 ; then ( ) ( ) du = 4 x3 + 4 x dx = 4 x 3 + x dx . ( x3 + x ) x4 + 2 x2 dx 1 = ⋅ 4 ( x3 + x ) x 4 + 2 x 2 dx 4 1 1 u du = u1/ 2 du 4 4 1⎛2 ⎞ = ⎜ u 3/ 2 + C1 ⎟ 4⎝3 ⎠ 3/ 2 1 4 = x + 2 x2 +C 6 = ( ) 35. Let u = 1 + cos x ; then du = − sin x dx . sin x (1 + cos x ) dx = − − sin x (1 + cos x ) dx 4 4 ⎛1 ⎞ = − u 4 du = − ⎜ u 5 + C1 ⎟ ⎝5 ⎠ 1 5 = − (1 + cos x ) + C 5 36. Let u = 1 + sin 2 x ; then du = 2sin x cos x dx . sin x cos x 1 + sin 2 x dx 3 3t 2t 2 − 11 dt = Instructor’s Resource Manual dy = 1 ⋅ 2sin x cos x 1 + sin 2 x dx 2 1 1 = u du = u1/ 2 du 2 2 1⎛2 ⎞ = ⎜ u 3/ 2 + C1 ⎟ 2⎝3 ⎠ 3/ 2 1 = 1 + sin 2 x +C 3 = ( ) Section 3.8 227 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 37. 38. 3 2 x + x + C1 2 ⎛3 ⎞ f ( x) = ⎜ x 2 + x + C1 ⎟ dx 2 ⎝ ⎠ 1 3 1 2 = x + x + C1 x + C2 2 2 f ( x) = (3 x + 1)dx = f ( x) = (−2 x + 3) dx = − x 2 + 3x + C1 44. The Quotient Rule for derivatives says ⎤ g ( x) f ( x) − f ( x) g ( x) d ⎡ f ( x) + C⎥ = . ⎢ dx ⎣ g ( x) g 2 ( x) ⎦ Thus, 40. 41. = 228 =− ( x) g ( x) ] dx = f ( x) g ( x) + C 1 (2 x + 5)3 / 2 ⎡ − x3 3x2 ⎤ + ⎢ ⎥ dx 3/ 2 2 x + 5 ⎦⎥ ⎣⎢ (2 x + 5) = [ f ( x) g ( x) + g ( x) f ( x) ] dx = f ( x) g ( x) + C = x3 (2 x + 5)−1/ 2 + C f ( x) = x + x −3 Section 3.8 [ f ( x) g ( x) + f f ( x) = 3 x 2 , g ( x) = −(2 x + 5)−3 / 2 3 7/3 x + C1 7 9 10 / 3 ⎛3 ⎞ f ( x) = ⎜ x7 / 3 + C1 ⎟ dx = x + C1 x + C2 70 ⎝7 ⎠ 43. The Product Rule for derivatives says d [ f ( x) g ( x) + C ] = f ( x) g ( x) + f ( x) g ( x) . dx Thus, [ f ( x) g ( x) + f ( x) g ( x)]dx = f ( x) g ( x) + C . 1 46. Let f ( x) = x3 , g ( x) = (2 x + 5)−1/ 2 . f ( x) = x 4 / 3 dx = 3 f ( x) = 2 ( x + 1)1/ 3 dx = ( x + 1)4 / 3 + C1 2 ⎡3 ⎤ f ( x) = ⎢ ( x + 1)4 / 3 + C1 ⎥ dx ⎣2 ⎦ 9 = ( x + 1)7 / 3 + C1 x + C2 14 f ( x) +C . g ( x) = x2 x − 1 + C f ( x) = ( x + x −3 )dx = 42. dx = 2 x −1 ⎡ x2 ⎤ + 2 x x − 1 ⎥ dx ⎢ ⎢⎣ 2 x − 1 ⎦⎥ f ( x) = x1/ 2 dx = x 2 x −2 − + C1 2 2 1 ⎛1 ⎞ f ( x) = ⎜ x 2 − x −2 + C1 ⎟ dx 2 ⎝2 ⎠ 1 3 1 −1 = x + x + C1 x + C2 6 2 1 3 1 = x + + C1 x + C2 6 2x g ( x) f ( x ) = 2 x, g ( x ) = 1 3 = − x3 + x 2 + C1 x + C2 3 2 2 3/ 2 x + C1 3 ⎛2 ⎞ f ( x) = ⎜ x3 / 2 + C1 ⎟ dx 3 ⎝ ⎠ 4 5/ 2 = x + C1 x + C2 15 2 45. Let f ( x) = x 2 , g ( x) = x − 1 . f ( x) = (− x 2 + 3 x + C1 )dx 39. g ( x) f ( x) − f ( x) g ( x) = x3 2x + 5 +C f ( x)dx = 47. d f ( x)dx = f ( x) + C dx f ( x ) = x3 + 1 + f ( x)dx = 48. 3 x3 2 x3 + 1 5 x3 + 2 2 x3 + 1 = 5 x3 + 2 2 x3 + 1 so +C . ⎞ d ⎛ f ( x) +C⎟ ⎜⎜ ⎟ dx ⎝ g ( x) ⎠ = = g ( x) f ( x) − f ( x) 12 [ g ( x)]−1/ 2 g ( x) g ( x) 2 g ( x) f ( x) − f ( x) g ( x) 2[ g ( x)]3 / 2 Thus, 2 g ( x) f '( x) − f ( x) g '( x) 2 [ g ( x)] 3/ 2 = f ( x) g ( x) +C Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 49. The Product Rule for derivatives says that d m [ f ( x) g n ( x) + C ] dx 54. a. F1 ( x) = ( x sin x)dx = sin x − x cos x + C1 F2 ( x) = (sin x − x cos x + C1 )dx = f m ( x )[ g n ( x)] + [ f m ( x)] g n ( x) = −2 cos x − x sin x + C1 x + C2 = f m ( x)[ng n −1 ( x) g ( x)] + [mf m −1 ( x) f ( x)]g n ( x) F3 ( x) = (−2 cos x − x sin x + C1 x + C2 )dx = f m −1 ( x) g n −1 ( x)[nf ( x) g ( x) + mg ( x) f ( x)] . Thus, 1 = x cos x − 3sin x + C1 x 2 + C2 x + C3 2 1 F4 ( x) = ( x cos x − 3sin x + C1 x 2 + C2 x + C3 )dx 2 1 1 = x sin x + 4 cos x + C1 x3 + C2 x 2 + C3 x + C4 6 2 f m −1 ( x) g n −1 ( x)[nf ( x) g ( x) + mg ( x) f ( x)]dx = f m ( x) g n ( x) + C . 50. Let u = sin[( x 2 + 1)4 ]; b. then du = cos ⎡( x 2 + 1)4 ⎤ 4( x 2 + 1)3 (2 x)dx . ⎣ ⎦ F16 ( x) = x sin x + 16 cos x + Cn x16− n n =1 (16 − n)! 16 du = 8 x cos ⎡ ( x 2 + 1)4 ⎤ ( x 2 + 1)3 dx ⎣ ⎦ sin 3 ⎡( x 2 + 1)4 ⎤ cos ⎡( x 2 + 1) 4 ⎤ ( x 2 + 1)3 x dx ⎣ ⎦ ⎣ ⎦ 4 ⎛ ⎞ 1 1 3 1 u = u 3 ⋅ du = u du = ⎜ + C1 ⎟ ⎟ 8 8 8 ⎜⎝ 4 ⎠ If x < 0, then x = − x and 1. differential equation 2. function 3. separate variables sin 4 ⎡ ( x 2 + 1)4 ⎤ ⎣ ⎦ +C = 32 51. If x ≥ 0, then x = x and 3.9 Concepts Review 4. −32t + v0 ; − 16t 2 + v0t + s0 1 2 x +C . 2 1 x dx = − x 2 + C . 2 x dx = Problem Set 3.9 1. ⎧1 2 if x ≥ 0 ⎪⎪ 2 x + C x dx = ⎨ ⎪− 1 x 2 + C if x < 0 ⎪⎩ 2 u 1 − cos u , = 2 2 1 − cos 2 x 1 1 sin 2 x dx = dx = x − sin 2 x + C . 2 2 4 2. 52. Using sin 2 53. Different software may produce different, but equivalent answers. These answers were produced by Mathematica. a. 6sin ( 3( x − 2) ) dx = −2 cos ( 3( x − 2) ) + C b. 1 ⎛ x⎞ ⎛x⎞ 9 ⎛ x⎞ sin 3 ⎜ ⎟ dx = cos ⎜ ⎟ − cos ⎜ ⎟ + C 6 2 2 2 ⎝ ⎠ ⎝ ⎠ ⎝6⎠ c. ( x 2 cos 2 x + x sin 2 x)dx = Instructor’s Resource Manual 3. −2 x −x dy = = dx 2 1 − x 2 1 − x2 dy x −x x + = + =0 dx y 1 − x2 1 − x2 dy =C dx dy − x + y = −Cx + Cx = 0 dx dy = C1 cos x − C2 sin x; dx d2y dx 2 d2y = −C1 sin x − C2 cos x +y dx 2 = (−C1 sin x − C2 cos x) + (C1 sin x + C2 cos x) = 0 x 2 sin 2 x +C 2 Section 3.9 229 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4. For y = sin(x + C), dy = cos( x + C ) dx 7. 2 y dy = x dx ⎛ dy ⎞ 2 2 2 ⎜ ⎟ + y = cos ( x + C ) + sin ( x + C ) = 1 dx ⎝ ⎠ dy = 0. For y = ±1, dx y2 x2 + C1 = + C2 2 2 y 2 = x2 + C 2 ⎛ dy ⎞ 2 2 2 ⎜ ⎟ + y = 0 + (±1) = 1 dx ⎝ ⎠ 5. dy = x2 + 1 dx y = ± x2 + C At x = 1, y = 1: 1 = ± 1 + C ; C = 0 and the square root is positive. dy = ( x 2 + 1) dx y = x 2 or y = x dy = ( x 2 + 1) dx x3 y + C1 = + x + C2 3 8. y= y=− 1 2 x −2 + 2 x + C2 2 + 2x + C 2x At x = 1, y = 3: 1 3 3 = − + 2 + C; C = 2 2 1 3 y=− + 2x + 2 2 x2 230 y = ( x3 / 2 + C ) 2 / 3 At x = 1, y = 4: 4 = (1 + C )2 / 3 ; C = 7 y = ( x3 / 2 + 7)2 / 3 dy = x −3 + 2 dx y + C1 = − Section 3.9 x dx y 3 / 2 = x3 / 2 + C x 1 +x− 3 3 dy = ( x −3 + 2) dx x y 2 3/ 2 2 y + C1 = x3 / 2 + C2 3 3 3 dy = ( x −3 + 2) dx dy = dx y dy = x3 y= + x+C 3 At x = 1, y = 1: 1 1 1 = + 1 + C; C = − 3 3 6. dy x = dx y 9. dz 2 2 =t z dt z −2 dz = t 2 dt − z −1 + C1 = t3 + C2 3 1 t3 C − t3 = − + C3 = z 3 3 3 z= C − t3 1 At t = 1, z = : 3 1 3 = ; C − 1 = 9; C = 10 3 C −1 3 z= 10 − t 3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 10. dy = y4 dt 13. y −4 dy = dt − 1 3 y3 y=− 1 (2 x + 1)4 2 dx 2 1 (2 x + 1)5 (2 x + 1)5 = +C = +C 2 5 10 At x = 0, y = 6: 1 59 6 = + C; C = 10 10 y = (2 x + 1)4 dx = + C1 = t + C2 1 3 3t + C At t = 0, y = 1: C = –1 1 y=− 3 3t − 1 y= 14. ds 11. = 16t 2 + 4t − 1 dt ds = (16t 2 + 4t − 1) dt s + C1 = 16 3 t + 2t 2 − t + C2 3 16 3 t + 2t 2 − t + C 3 At t = 0, s = 100:C = 100 16 s = t 3 + 2t 2 − t + 100 3 u 2u 2 −2 + C1 = t4 t2 − + C2 4 2 15. t4 = t − +C 2 2 −1/ 2 ⎛ ⎞ t4 u = ⎜t2 − + C ⎟ ⎜ ⎟ 2 ⎝ ⎠ At t = 0, u = 4: 1 4 = C −1/ 2 ; C = 16 ⎛ t4 1 ⎞ u = ⎜t2 − + ⎟ ⎜ 2 16 ⎟⎠ ⎝ −1/ 2 Instructor’s Resource Manual 10 ( x + 2)5 + C At x = 0, y = 1: 10 1= ; C = 10 − 32 = −22 32 + C 10 y= 2 ( x + 2)5 − 22 u −3 du = (t 3 − t ) dt 1 dy = − y 2 x( x 2 + 2)4 dx 1 − y −2 dy = 2 x( x 2 + 2)4 dx 2 1 1 ( x 2 + 2)5 + C1 = + C2 y 2 5 y= du = u 3 (t 3 − t ) dt − (2 x + 1)5 59 (2 x + 1)5 + 59 + = 10 10 10 1 ( x 2 + 2)5 + C = y 10 s= 12. dy = (2 x + 1) 4 dx 2 dy = 3x dx y = 3 x dx = 3 2 x +C 2 At (1, 2): 3 2 = +C 2 1 C= 2 3 2 1 3x 2 + 1 y= x + = 2 2 2 Section 3.9 231 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 16. dy = 3y2 dx 19. v = (2t + 1)1/ 3 dt = y −2 dy = 3 dx − 3 = (2t + 1)4 / 3 + C1 8 3 3 v0 = 0 : 0 = + C1 ; C1 = − 8 8 3 3 v = (2t + 1) 4 / 3 − 8 8 3 3 (2t + 1) 4 / 3 dt − 1dt s= 8 8 3 3 = (2t + 1)4 3 2dt − 1dt 16 8 9 3 (2t + 1)7 3 − t + C2 = 112 8 9 1111 s0 = 10 :10 = + C2 ; C2 = 112 112 9 3 1111 (2t + 1)7 3 − t + s= 112 8 112 3 3 At t = 2: v = (5) 4 3 − ≈ 2.83 8 8 9 6 1111 s= (5)7 3 − + ≈ 12.6 112 8 112 1 + C1 = 3 x + C2 y 1 = −3x + C y 1 C − 3x At (1, 2): 1 2= C −3 7 C= 2 1 2 = y= 7 − 3x 7 − 6x 2 y= 17. v = t dt = t2 + v0 2 t2 +3 2 ⎛ t2 ⎞ t3 s = ⎜ + 3 ⎟ dt = + 3t + s0 ⎜2 ⎟ 6 ⎝ ⎠ v= 20. v = (3t + 1) −3 dt = t3 t3 + 3t + 0 = + 3t 6 6 At t = 2: v = 5 cm/s 22 s= cm 3 s= 18. v = (1 + t )−4 dt = − v0 = 0 : 0 = − v=− 3(1 + t )3 1 3(1 + 0) 1 3(1 + t ) 1 3 + 3 +C + C; C = 1 3 1 3 ⎛ 1 1⎞ 1 1 s = ⎜− + ⎟ dt = + t +C 2 3 ⎜ 3(1 + t )3 3 ⎟ 6(1 + t ) ⎝ ⎠ 1 1 59 s0 = 10 :10 = + (0) + C ; C = 2 3 6 6(1 + 0) 1 59 + t+ 3 6 6(1 + t ) At t = 2: 1 1 26 cm/s v=− + = 81 3 81 1 2 59 284 cm s= + + = 54 3 6 27 s= 232 1 2 Section 3.9 1 (2t + 1)1/ 3 2dt 2 1 (3t + 1) −3 3dt 3 1 = − (3t + 1)−2 + C1 6 1 25 v0 = 4 : 4 = − + C1; C1 = 6 6 1 25 v = − (3t + 1)−2 + 6 6 1 25 s = − (3t + 1)−2 dt + dt 6 6 1 25 (3t + 1) −2 3dt + =− dt 18 6 1 25 = (3t + 1) −1 + t + C2 18 6 1 1 s0 = 0 : 0 = + C2 ; C2 = − 18 18 1 25 1 s = (3t + 1)−1 + t − 18 6 18 1 −2 25 ≈ 4.16 At t = 2: v = − (7) + 6 6 1 25 1 s = (7) −1 + − ≈ 8.29 18 3 18 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 21. v = –32t + 96, 2 27. vesc = 2 gR 2 s = −16t + 96t + s0 = −16t + 96t v = 0 at t = 3 At t = 3, s = −16(32 ) + 96(3) = 144 ft 22. a = dv =k dt v = k dt = kt + v0 = ds ; dt k k s = (kt + v0 )dt = t 2 + v0t + s0 = t 2 + v0t 2 2 v0 v = 0 when t = − . Then k s= 23. 2 v2 k ⎛ v0 ⎞ ⎛ v02 ⎞ − ⎟ +⎜− ⎟ = − 0 . ⎜ 2 ⎝ k ⎠ ⎜⎝ k ⎟⎠ 2k dv = −5.28 dt 28. v0 = 60 mi/h = 88 ft/s v = 0 = –11t + 88; t = 8 sec 11 s ( t ) = − t 2 + 88t 2 11 2 s ( 8 ) = − ( 8 ) + 88 ( 8 ) = 352 feet 2 The shortest distance in which the car can be braked to a halt is 352 feet. 29. a = dv = − 5.28dt v= For the Moon, vesc ≈ 2(0.165)(32)(1080 ⋅ 5280) ≈ 7760 ft/s ≈ 1.470 mi/s. For Venus, vesc ≈ 2(0.85)(32)(3800 ⋅ 5280) ≈ 33,038 ft/s ≈ 6.257 mi/s. For Jupiter, vesc ≈ 194,369 ft/s ≈ 36.812 mi/s. For the Sun, vesc ≈ 2,021,752 ft/s ≈ 382.908 mi/s. ds = −5.28t + v0 = –5.28t + 56 dt 30. 75 = ds = (−5.28t + 56)dt s = −2.64t 2 + 56t + s0 = −2.64t 2 + 56t + 1000 When t = 4.5, v = 32.24 ft/s and s = 1198.54 ft 24. v = 0 when t = −56 ≈ 10.6061 . Then −5.28 s ≈ −2.64(10.6061) 2 + 56(10.6061) + 1000 ≈ 1296.97 ft 25. 4 3 πr and S = 4πr 2 , 3 dr dr 4πr 2 = −k 4πr 2 so = −k . dt dt dr = − k dt Since V = r = –kt + C 2 = –k(0) + C and 0.5 = –k(10) + C, so 3 3 C = 2 and k = . Then, r = − t + 2 . 20 20 8 (3.75) 2 + v0 (3.75) + 0; v0 = 5 ft/s 2 31. For the first 10 s, a = dv = 6t , v = 3t 2 , and dt s = t 3 . So v(10) = 300 and s(10) = 1000. After dv 10 s, a = = −10 , v = –10(t – 10) + 300, and dt s = −5(t − 10)2 + 300(t − 10) + 1000. v = 0 at t = 40, at which time s = 5500 m. 32. a. dV = −kS dt 26. Solving v = –136 = –32t yields t = dv Δv 60 − 45 = = = 1.5 mi/h/s = 2.2 ft/s2 dt Δt 10 After accelerating for 8 seconds, the velocity is 8 · 3 = 24 m/s. b. Since acceleration and deceleration are constant, the average velocity during those times is 24 = 12 m/s . Solve 0 = –4t + 24 to get the 2 24 time spent decelerating. t = = 6 s; 4 d = (12)(8) + (24)(100) + (12)(6) = 2568 m. 17 . 4 2 ⎛ 17 ⎞ ⎛ 17 ⎞ Then s = 0 = −16 ⎜ ⎟ + (0) ⎜ ⎟ + s0 , so ⎝ 4⎠ ⎝ 4⎠ s0 = 289 ft. Instructor’s Resource Manual Section 3.9 233 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 3.10 Chapter Review 15. True: Concepts Test 1. True: Max-Min Existence Theorem 2. True: Since c is an interior point and f is differentiable ( f (c) exists), by the Critical Point Theorem, c is a stationary point ( f (c) = 0). 3. True: 4. False: increasing for all x, but f ' ( x ) does f ( x) = 18 x5 + 16 x3 + 4 x; 4 2 6. False: When f ( x) > 0, f ( x) is increasing. 8. False: If f (c) = 0 , c is a candidate, but not necessarily an inflection point. For = 16. True: 17. True: 18. False: 10. True: f ( x) = ax 2 + bx + c; f ( x) = 2ax + b; f ( x) = 2a 12. True: 23. False: x →−∞ 3x 2 + 2 x + sin x sin x – (3 x + 2) = ; x x sin x sin x lim = 0 and lim = 0. x →∞ x x→ – ∞ x The function is differentiable on (0, 2). f ( x) = lim (2 x3 + x + tan x) = ∞ while + Instructor's Resource Manual 3 3 , . 3 3 The rectangle will have minimum perimeter if it is a square. K A = xy = K; y = x 2 K dP 2K d 2 P 4K = 2− = ; ; x dx x 2 dx 2 x3 dP d 2P = 0 and >0 dx dx 2 when x = K , y = K . − At x = 3 there is a removable discontinuity. x so f (0) does not exist. x dy d2y = cos x; = − sin x; –sin x = 0 dx dx 2 has infinitely many solutions. P = 2x + lim (2 x3 + x + tan x) = −∞. 14. False: 1 = –1. –1 For example if f ( x) = x 4 , f (0) = f (0) = 0 but f has a minimum at x = 0. x →∞ x →− π 2 x –1 21. False: lim (2 x3 + x) = ∞ while x→ π 2 x→ – ∞ 1 x2 Let g(x) = D where D is any number. Then g ( x) = 0 and so, by Theorem B of Section 3.6, f(x) = g(x) + C = D + C, which is a constant, for all x in (a, b). lim (2 x3 + x) = −∞ 13. True: = lim 1 + 12 20. True: If f(x) is increasing for all x in [a, b], the maximum occurs at b. tan 2 x has a minimum value of 0. This occurs whenever x = kπ where k is an integer. –1 There are two points: x = − 22. True: 11. False: x →∞ 1 x2 19. False: example, if f ( x) = x 4 , P (0) = 0 but x = 0 is not an inflection point. 9. True: x2 + 1 For example, f ( x) = x is increasing on [–1, 1] but f (0) = 0. 7. True: = lim 1 x2 1 = –1 and –1 lim f ( x) = 90 x + 48 x + 4 , which is greater than zero for all x. 3 1+ x→ – ∞ 1 – x 2 not exist at x = 0. 5. True: x →∞ 1 – x 2 = For example, let f(x) = sin x. f ( x) = x1/ 3 is continuous and lim x2 + 1 24. True: By the Mean Value Theorem, the derivative must be zero between each pair of distinct x-intercepts. Section 3.10 235 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 25. True: 26. False: If f ( x1 ) < f ( x2 ) and g ( x1 ) < g ( x2 ) for x1 < x2 , f ( x1 ) + g ( x1 ) < f ( x2 ) + g ( x2 ), so f + g is increasing. 28. False: 29. True: 30. True: 31. False: b 2 – 3ac < 0 there are no critical points.) On an open interval, no local maxima can come from endpoints, so there can be at most one local maximum in an open interval. Let f(x) = g(x) = 2x, f ( x) > 0 and g ( x) > 0 for all x, but f ( x) g ( x) = 4 x 2 is decreasing on (– ∞ , 0). 27. True: is an inflection point while if Since f ( x) > 0, f ( x) is increasing for x ≥ 0. Therefore, f ( x) > 0 for x in [0, ∞ ), so f(x) is increasing. If f(3) = 4, the Mean Value Theorem requires that at some point c in [0, 3], f (3) – f (0) 4 –1 = = 1 which f (c ) = 3–0 3–0 does not contradict that f ( x) ≤ 2 for all x in [0, 3]. If the function is nondecreasing, f ( x) must be greater than or equal to zero, and if f ( x) ≥ 0, f is nondecreasing. This can be seen using the Mean Value Theorem. 34. True: f ( x) = a ≠ 0 so f(x) has no local minima or maxima. On an open interval, no local minima or maxima can come from endpoints, so f(x) has no local minima. 35. True: Intermediate Value Theorem 36. False: The Bisection Method can be very slow to converge. 37. False: xn +1 = xn – 38. False: Newton’s method can fail to exist for several reasons (e.g. if f’(x) is 0 at or near r). It may be possible to achieve convergence by selecting a different starting value. 39. True: From the Fixed-point Theorem, if g is continuous on [ a, b ] and However, if the constant is 0, the functions are the same. a ≤ g ( x ) ≤ b whenever a ≤ x ≤ b , then there is at least one fixed point on [ a, b ] . The given conditions satisfy x For example, let f ( x) = e . these criteria. lim e x = 0, so y = 0 is a horizontal x →−∞ 32. True: 33. True: f ( xn ) = –2 xn . f ( xn ) asymptote. 40. True: If f(c) is a global maximum then f(c) is the maximum value of f on S where (a, b) is any interval (a, b) containing c and S is the domain of f. Hence, f(c) is a local maximum value. The Bisection Method always converges as long as the function is continuous and the values of the function at the endpoints are of opposite sign. 41. True: Theorem 3.8.C 42. True: Obtained by integrating both sides of the Product Rule 43. True: (− sin x) 2 = sin 2 x = 1 − cos 2 x 44. True: If F ( x) = f ( x) = 3ax 2 + 2bx + c; f ( x) = 0 –b ± b 2 – 3ac by the 3a Quadratic Formula. f ( x) = 6ax + 2b so ⎛ – b ± b 2 – 3ac ⎞ ⎟ = ±2 b 2 – 3ac . f ⎜ ⎜ ⎟ 3a ⎝ ⎠ when x = Thus, if b 2 – 3ac > 0, one critical point is a local maximum and the other is a local minimum. f ( x) dx, f ( x) is a derivative of F(x). 45. False: f ( x) = x 2 + 2 x + 1 and g ( x) = x 2 + 7 x − 5 are a counterexample. (If b2 – 3ac = 0 the only critical point 236 Section 3.10 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 46. False: The two sides will in general differ by a constant term. 47. True: At any given height, speed on the downward trip is the negative of speed on the upward. 5. 2. 3. f ( x) = 2 x – 2; 2x – 2 = 0 when x = 1. Critical points: 0, 1, 4 f(0) = 0, f(1) = –1, f(4) = 8 Global minimum f(1) = –1; global maximum f(4) = 8 f (t ) = – 1 f ( z) = – ;– 2 z 3 ;– 6. 2 z3 7. 8. 2 ;– 1 Global minimum f (–2) = ; no global 4 maximum. Instructor's Resource Manual f (u ) = u (7u – 12) 2/3 ; f (u ) = 0 when u = 0, 3(u – 2) f (2) does not exist. 12 7 12 , 2, 3 7 f (–1) = 3 –3 ≈ –1.44, f (0) = 0, ⎛ 12 ⎞ 144 3 2 f ⎜ ⎟= – ≈ –1.94, f(2) = 0, f(3) = 9 7 ⎝ 7 ⎠ 49 ⎛ 12 ⎞ Global minimum f ⎜ ⎟ ≈ –1.94; ⎝7⎠ global maximum f(3) = 9 2 is never 0. x x3 Critical point: –2 1 f (–2) = 4 f ( x) > 0 for x < 0, so f is increasing. 3 f ( x) = 12 x3 – 12 x 2 = 12 x 2 ( x – 1); f ( x) = 0 when x = 0, 1 Critical points: –2, 0, 1, 3 f(–2) = 80, f(0) = 0, f(1) = –1, f(3) = 135 Global minimum f(1) = –1; global maximum f(3) = 135 Critical points: –1, 0, 1 Global minimum f (–2) = ; 4 ⎛ 1⎞ global maximum f ⎜ – ⎟ = 4. ⎝ 2⎠ f ( x) = – s ; f ( s ) does not exist when s = 0. s f ( s ) = 1 – 1 = 0. Critical points: 1 and all s in [–1, 0] f(1) = 2, f(s) = 0 for s in [–1, 0] Global minimum f(s) = 0, –1 ≤ s ≤ 0; global maximum f(1) = 2. is never 0. 1 Critical points: –2, – 2 1 ⎛ 1⎞ f (–2) = , f ⎜ – ⎟ = 4 4 ⎝ 2⎠ 4. f ( s) = 1 + For s < 0, s = – s so f(s) = s – s = 0 and 1 is never 0. t t2 Critical points: 1, 4 1 f(1) = 1, f (4) = 4 1 Global minimum f (4) = ; 4 global maximum f(1) = 1. 2 x ; f ( x) does not exist at x = 0. x 1 Critical points: – , 0, 1 2 ⎛ 1⎞ 1 f ⎜ – ⎟ = , f (0) = 0, f (1) = 1 ⎝ 2⎠ 2 Global minimum f(0) = 0; global maximum f(1) = 1 Sample Test Problems 1. f ( x) = 9. f ( x) = 10 x 4 – 20 x3 = 10 x3 ( x – 2); f ( x) = 0 when x = 0, 2 Critical points: –1, 0, 2, 3 f(–1) = 0, f(0) = 7, f(2) = –9, f(3) = 88 Global minimum f(2) = –9; global maximum f(3) = 88 Section 3.10 237 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 10. f ( x) = 3( x – 1)2 ( x + 2)2 + 2( x – 1)3 ( x + 2) 14. = ( x – 1)2 ( x + 2)(5 x + 4); f ( x) = 0 when f ( x) = 72 x 7 ; f ( x) < 0 when x < 0. f(x) is increasing on (– ∞ , ∞ ) and concave down on (– ∞ , 0). 4 x = –2, – , 1 5 4 Critical points: –2, – , 1 , 2 5 26, 244 ⎛ 4⎞ ≈ –8.40, f(–2) = 0, f ⎜ – ⎟ = – 3125 ⎝ 5⎠ f(1) = 0, f(2) = 16 ⎛ 4⎞ Global minimum f ⎜ – ⎟ ≈ –8.40; ⎝ 5⎠ global maximum f(2) = 16 11. f ( ) = cos ; f ( ) = 0 when 15. 16. 1 x>− . 2 f(x) is increasing on [–2, 1] and concave down on ⎛ 1 ⎞ ⎜− , ∞⎟ . ⎝ 2 ⎠ Critical points: 17. π π 5π in [0, π ] , , 6 2 6 π π 5π Critical points: 0, , , , π 6 2 6 1 ⎛π⎞ ⎛π⎞ f(0) = 0, f ⎜ ⎟ = – , f ⎜ ⎟ = 0, 4 ⎝2⎠ ⎝6⎠ 1 ⎛ 5π ⎞ f ⎜ ⎟ = – , f( π ) = 0 6 4 ⎝ ⎠ 1 1 ⎛π⎞ ⎛ 5π ⎞ Global minimum f ⎜ ⎟ = – or f ⎜ ⎟ = – ; 4 4 ⎝6⎠ ⎝ 6 ⎠ ⎛π⎞ global maximum f(0) = 0, f ⎜ ⎟ = 0, or ⎝2⎠ f( π ) = 0 13. 1 . 5 f ( x) = 12 x 2 – 80 x3 = 4 x 2 (3 – 20 x); f ( x) < 0 when x > 3 . 20 f(x) is increasing on ⎡ 0, 1 ⎤ and concave down on ⎣ 5⎦ ⎛ 3 ⎞ ⎜ , ∞ ⎟. 20 ⎝ ⎠ f ( ) = 2sin cos – cos = cos (2sin – 1); = f ( x) = 4 x3 – 20 x 4 = 4 x3 (1 – 5 x); f ( x) > 0 when 0 < x < 3 ⎛ 4π ⎞ f ⎜ ⎟=– ≈ –0.87 2 ⎝ 3 ⎠ ⎛ 4π ⎞ Global minimum f ⎜ ⎟ ≈ –0.87; ⎝ 3 ⎠ ⎛π⎞ global maximum f ⎜ ⎟ = 1 ⎝2⎠ f ( ) = 0 when f ( x) = −6 x 2 − 6 x + 12 = –6(x + 2)(x – 1); f ( x) = −12 x − 6 = –6(2x + 1); f ( x) < 0 when ⎡ π 4π ⎤ ⎢4 , 3 ⎥ ⎣ ⎦ 12. f ( x) = 3 x 2 – 3 = 3( x 2 – 1); f ( x) > 0 when x < –1 or x > 1. f ( x) = 6 x; f ( x) < 0 when x < 0. f(x) is increasing on (– ∞ , –1] ∪ [1, ∞ ) and concave down on (– ∞ , 0). f ( x) > 0 when –2 < x < 1. π = in 2 π π 4π , , 4 2 3 ⎛π⎞ 1 ⎛π⎞ f ⎜ ⎟= ≈ 0.71, f ⎜ ⎟ = 1, 4 2 ⎝ ⎠ ⎝2⎠ f ( x) = 9 x8 ; f ( x) > 0 for all x ≠ 0. 18. f ( x) = 3 x 2 – 6 x 4 = 3 x 2 (1 – 2 x 2 ); f ( x) > 0 when – 1 2 < x < 0 and 0 < x < 1 2 . f ( x) = 6 x – 24 x3 = 6 x(1 – 4 x 2 ); f ( x) < 0 when 1 1 < x < 0 or x > . 2 2 ⎡ 1 1 ⎤ f(x) is increasing on ⎢ – , ⎥ and concave 2 2⎦ ⎣ ⎛ 1 ⎞ ⎛1 ⎞ down on ⎜ – , 0 ⎟ ∪ ⎜ , ∞ ⎟ . ⎝ 2 ⎠ ⎝2 ⎠ – 3 f ( x) = 3 – 2 x; f ( x) > 0 when x < . 2 f ( x) = –2; f ( x) is always negative. 3⎤ ⎛ f(x) is increasing on ⎜ – ∞, ⎥ and concave down 2⎦ ⎝ on (– ∞ , ∞ ). 238 Section 3.10 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 19. f ( x) = 3 x 2 – 4 x3 = x 2 (3 – 4 x); f ( x) > 0 when 4 f ( x) = 6 x – 8; f ( x) > 0 when x > . 3 ⎛4 ⎞ f(x) is concave up on ⎜ , ∞ ⎟ and concave down ⎝3 ⎠ 4⎞ ⎛ 4 128 ⎞ ⎛ on ⎜ – ∞, ⎟ ; inflection point ⎜ , − ⎟ 27 ⎠ 3⎠ ⎝3 ⎝ 3 x< . 4 f ( x) = 6 x – 12 x 2 = 6 x(1 – 2 x); f ( x) < 0 when x < 0 or x > 1 . 2 3⎤ ⎛ f(x) is increasing on ⎜ – ∞, ⎥ and concave down 4⎦ ⎝ ⎛1 ⎞ on (– ∞, 0) ∪ ⎜ , ∞ ⎟ . ⎝2 ⎠ 20. g (t ) = 3t 2 – 1 t 2 ; g (t ) > 0 when 3t 2 > 1 t2 or 1 1 1 t 4 > , so t < – or t > . 1/ 4 1/ 4 3 3 3 1 ⎤ ⎡ 1 ⎛ ⎞ g (t ) is increasing on ⎜ – ∞, – ∪ , ∞⎟ 1/ 4 ⎥ ⎢ 1/ 4 3 ⎦ ⎣3 ⎝ ⎠ ⎡ 1 ⎞ ⎛ 1 ⎤ , 0 ⎟ ∪ ⎜ 0, and decreasing on ⎢ – . 1/ 4 1/ 4 ⎥ ⎣ 3 ⎠ ⎝ 3 ⎦ ⎛ 1 ⎞ 1 = + 31/ 4 ≈ 1.75; Local minimum g ⎜ 1/ 4 ⎟ 3/ 4 ⎝3 ⎠ 3 local maximum ⎛ 1 ⎞ 1 =– g⎜– – 31/ 4 ≈ –1.75 1/ 4 ⎟ 3/ 4 3 ⎝ 3 ⎠ 2 g (t ) = 6t + ; g (t ) > 0 when t > 0. g(t) has no t3 inflection point since g(0) does not exist. 21. 22. f ( x) = – f ( x) = 8x 2 ( x + 1)2 ; f ( x) = 0 when x = 0. 8(3 x 2 – 1) ; f (0) = –8, so f(0) = 6 is a ( x 2 + 1)3 local maximum. f ( x) > 0 for x < 0 and f ( x) < 0 for x > 0 so f(0) = 6 is a global maximum value. f(x) has no minimum value. 23. f ( x) = 4 x3 – 2; f ( x) = 0 when x = 1 3 2 . f ( x) = 12 x 2 ; f ( x) = 0 when x = 0. ⎛ 1 ⎞ 12 f ⎜ = > 0, so 3 ⎟ 2/3 ⎝ 2⎠ 2 ⎛ 1 ⎞ 1 2 3 – f⎜ is a global = =– 3 ⎟ 4/3 1/ 3 4/3 2 2 ⎝ 2⎠ 2 minimum. f ( x) > 0 for all x ≠ 0; no inflection points No horizontal or vertical asymptotes f ( x) = 2 x ( x – 4) + x 2 = 3 x 2 – 8 x = x(3 x – 8); 8 3 ⎡8 ⎞ f(x) is increasing on (– ∞, 0] ∪ ⎢ , ∞ ⎟ and ⎣3 ⎠ ⎡ 8⎤ decreasing on ⎢ 0, ⎥ ⎣ 3⎦ 256 ⎛8⎞ Local minimum f ⎜ ⎟ = – ≈ –9.48; 27 ⎝3⎠ local maximum f(0) = 0 f ( x) > 0 when x < 0 or x > Instructor's Resource Manual Section 3.10 239 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 24. Vertical asymptote x = 3 f ( x) = 2( x 2 – 1)(2 x) = 4 x( x 2 – 1) = 4 x3 – 4 x; f ( x) = 0 when x = –1, 0, 1. f ( x) = 12 x 2 – 4 = 4(3 x 2 – 1); f ( x) = 0 when x=± 1 . 3 f (–1) = 8, f (0) = –4, f (1) = 8 Global minima f(–1) = 0, f(1) = 0; local maximum f(0) = 1 ⎛ 1 4⎞ Inflection points ⎜ ± , ⎟ 3 9⎠ ⎝ No horizontal or vertical asymptotes 27. f ( x) = 12 x3 – 12 x 2 = 12 x 2 ( x – 1); f ( x) = 0 when x = 0, 1. f ( x) = 36 x 2 – 24 x = 12 x(3x – 2); f ( x) = 0 2 . 3 f (1) = 12, so f(1) = –1 is a minimum. Global minimum f(1) = –1; no local maxima ⎛ 2 16 ⎞ Inflection points (0, 0), ⎜ , − ⎟ ⎝ 3 27 ⎠ No horizontal or vertical asymptotes. when x = 0, 25. 26. f ( x) = 3x – 6 ; f ( x) = 0 when x = 2, but x = 2 2 x–3 is not in the domain of f(x). f ( x ) does not exist when x = 3. 3( x – 4) f ( x) = ; f ( x) = 0 when x = 4. 4( x – 3)3 / 2 Global minimum f(3) = 0; no local maxima Inflection point (4, 4) No horizontal or vertical asymptotes. f ( x) = – f ( x) = 1 ( x – 3) 2 ; f ( x) < 0 for all x ≠ 3. 28. 1 ; f ( x) > 0 for all x ≠ 0. x2 2 f ( x) = – ; f ( x) > 0 when x < 0 and x3 f ( x) < 0 when x > 0. No local minima or maxima No inflection points 1 f ( x) = x – , so x ⎛ 1⎞ lim [ f ( x) − x] = lim ⎜ − ⎟ = 0 and y = x is an x →∞ x →∞ ⎝ x ⎠ oblique asymptote. Vertical asymptote x = 0 f ( x) = 1 + 2 ; f ( x) > 0 when x > 3. ( x – 3)3 No local minima or maxima No inflection points 1 – 2x x–2 lim = lim =1 x →∞ x – 3 x →∞ 1 – 3 x Horizontal asymptote y = 1 240 Section 3.10 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 29. f ( x) = 3 + f ( x) = – 1 x 2 2 ; f ( x ) > 0 for all x ≠ 0. 31. π 3π x=– , . 4 4 f ( x) = – cos x + sin x; f ( x) = 0 when ; f ( x) > 0 when x < 0 and x3 f ( x) < 0 when x > 0 No local minima or maxima No inflection points 1 f ( x) = 3x – , so x ⎛ 1⎞ lim [ f ( x) − 3 x] = lim ⎜ − ⎟ = 0 and y = 3x is an x →∞ x →∞ ⎝ x ⎠ oblique asymptote. Vertical asymptote x = 0 30. f ( x) = – 12 ; f ( x) > 0 for all x ≠ −1 . ( x + 1)4 No local minima or maxima No inflection points lim f ( x) = 0, lim f ( x) = 0, so y = 0 is a x →∞ 3π π , . 4 4 ⎛ π⎞ ⎛ 3π ⎞ f ⎜ – ⎟ = – 2, f ⎜ ⎟ = 2 ⎝ 4⎠ ⎝ 4 ⎠ ⎛ 3π ⎞ Global minimum f ⎜ ⎟ = – 2; ⎝ 4 ⎠ ⎛ π⎞ global maximum f ⎜ – ⎟ = 2 ⎝ 4⎠ ⎛ 3π ⎞ ⎛ π ⎞ Inflection points ⎜ − , 0 ⎟ , ⎜ , 0 ⎟ ⎝ 4 ⎠ ⎝4 ⎠ x=– 4 ; f ( x ) > 0 when x < −1 and ( x + 1)3 f ( x) < 0 when x > –1. f ( x) = f ( x) = – sin x – cos x; f ( x) = 0 when x→ – ∞ horizontal asymptote. Vertical asymptote x = –1 Instructor's Resource Manual 32. f ( x) = cos x – sec 2 x; f ( x ) = 0 when x = 0 f ( x) = – sin x – 2sec2 x tan x = – sin x(1 + 2sec3 x ) f ( x) = 0 when x = 0 No local minima or maxima Inflection point f(0) = 0 π π Vertical asymptotes x = – , 2 2 Section 3.10 241 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 33. f ( x) = x sec 2 x + tan x; f ( x ) = 0 when x = 0 f ( x) = 2sec 2 x(1 + x tan x); f ( x) is never 0 on ⎛ π π⎞ ⎜– , ⎟. ⎝ 2 2⎠ f (0) = 2 Global minimum f(0) = 0 36. f ( x) = –2sin x – 2 cos x; f ( x) = 0 when π 3π x=– , . 4 4 f ( x) = –2 cos x + 2sin x; f ( x) = 0 when 3π π , . 4 4 ⎛ π⎞ ⎛ 3π ⎞ f ⎜ – ⎟ = –2 2, f ⎜ ⎟ = 2 2 ⎝ 4⎠ ⎝ 4 ⎠ ⎛ 3π ⎞ Global minimum f ⎜ ⎟ = –2 2; ⎝ 4 ⎠ ⎛ π⎞ global maximum f ⎜ – ⎟ = 2 2 ⎝ 4⎠ ⎛ 3π ⎞ ⎛ π ⎞ Inflection points ⎜ − , 0 ⎟ , ⎜ , 0 ⎟ ⎝ 4 ⎠ ⎝4 ⎠ x=– 34. f ( x) = 2 + csc2 x; f ( x ) > 0 on (0, π ) f ( x) = –2 cot x csc2 x; f ( x) = 0 when π ⎛π ⎞ ; f ( x) > 0 on ⎜ , π ⎟ 2 ⎝2 ⎠ ⎛π ⎞ Inflection point ⎜ , π ⎟ ⎝2 ⎠ x= 35. f ( x) = cos x – 2 cos x sin x = cos x(1 – 2sin x); π π π 5π f ( x) = 0 when x = – , , , 2 6 2 6 f ( x) = – sin x + 2sin 2 x – 2 cos 2 x; f ( x ) = 0 when x ≈ –2.51, –0.63, 1.00, 2.14 3 ⎛ π⎞ ⎛π⎞ ⎛π⎞ f ⎜ – ⎟ = 3, f ⎜ ⎟ = – , f ⎜ ⎟ = 1, 2 6 2 ⎝ ⎠ ⎝ ⎠ ⎝2⎠ 3 ⎛ 5π ⎞ f ⎜ ⎟=– 6 2 ⎝ ⎠ ⎛ π⎞ Global minimum f ⎜ – ⎟ = –2, ⎝ 2⎠ π ⎛ ⎞ local minimum f ⎜ ⎟ = 0; ⎝2⎠ ⎛ π ⎞ 1 ⎛ 5π ⎞ 1 global maxima f ⎜ ⎟ = , f ⎜ ⎟ = ⎝6⎠ 4 ⎝ 6 ⎠ 4 Inflection points (–2.51, –0.94), (–0.63, –0.94), (1.00, 0.13), (2.14, 0.13) 242 Section 3.10 37. Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 38. = = ⎛ ⎛ 1⎞ ⎞ –( x 2 + 64) + ⎜1 + ⎟ x3 ⎟ ⎜ ⎝ x⎠ ⎠ x 2 + 64 ⎝ 1 x2 x3 − 64 x 2 x 2 + 64 x3 − 64 = 0; x = 4 x 2 x 2 + 64 dp dp < 0 if x < 4, > 0 if x > 4 dx dx ⎛ 1⎞ When x = 4, p = ⎜ 1 + ⎟ 16 + 64 ≈ 11.18 ft. ⎝ 4⎠ 39. 40. Let x be the length of a turned up side and let l be the (fixed) length of the sheet of metal. V = x (16 − 2 x )l = 16 xl − 2 x 2 l 42. Let x be the width and y the height of a page. A = xy. Because of the margins, 27 (y – 4)(x – 3) = 27 or y = +4 x−3 27 x A= + 4 x; x −3 dA ( x − 3)(27) − 27 x 81 = +4= − +4 2 dx ( x − 3) ( x − 3) 2 dA 3 15 = 0 when x = − , dx 2 2 dV = 16l − 4 xl ; V = 0 when x = 4 dx d 2V d2A = −4l ; 4 inches should be turned up for dx dx 2 each side. 41. Let p be the length of the plank and let x be the distance from the fence to where the plank touches the ground. See the figure below. 2 x= 43. p x 2 + 64 = x +1 x ⎛ 1⎞ p = ⎜ 1 + ⎟ x 2 + 64 ⎝ x⎠ Minimize p: dp 1 x ⎛ 1⎞ =− x 2 + 64 + ⎜ 1 + ⎟ 2 2 dx ⎝ x ⎠ x + 64 x Instructor's Resource Manual 162 ( x − 3) 3 ; d2A dx 2 > 0 when x = 15 2 15 ; y = 10 2 1 2 πr h = 128π 2 256 h= r2 Let S be the surface area of the trough. 256π S = πr 2 + πrh = πr 2 + r dS 256π = 2πr − dr r2 256π 2πr − = 0; r 3 = 128, r = 4 3 2 2 r Since By properties of similar triangles, = d 2S 2 > 0 when r = 4 3 2 , r = 4 3 2 dr minimizes S. 256 h= = 83 2 2 43 2 ( ) Section 3.10 243 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ⎧x 3 if − 2 < x < 0 ⎪⎪ 2 + 2 44. f ( x) = ⎨ ⎪− x + 2 if 0 < x < 2 ⎪⎩ 3 x 3 + = 0; x = −3 , which is not in the domain. 2 2 x+2 − = 0; x = −2, which is not in the domain. 3 g ( x) = c. ( x − 1) − ( x + 1) 2 = −2 ( x − 1) ( x − 1) 2 g (3) − g (2) 2 − 3 = = −1 3− 2 1 −2 = –1; c = 1 ± 2 (c − 1)2 Only c = 1 + 2 is in the interval (2, 3). Critical points: x = –2, 0, 2 f(–2) = 0, f(0) = 2, f(2) = 0 Minima f(–2) = 0, f(2) = 0, maximum f(0) = 2. ⎧1 if − 2 < x < 0 ⎪⎪ 2 f ( x) = ⎨ ⎪− 1 if 0 < x < 2 ⎪⎩ 3 Concave up on (–2, 0), concave down on (0, 2) 46. dy = 4 x3 − 18 x 2 + 24 x − 3 dx d2y 2 45. a. = 12 x 2 − 36 x + 24; 12( x 2 − 3x + 2) = 0 when dx x = 1, 2 Inflection points: x = 1, y = 5 and x = 2, y = 11 dy =7 Slope at x = 1: dx x =1 Tangent line: y – 5 = 7(x – 1); y = 7x – 2 dy =5 Slope at x = 2: dx x = 2 Tangent line: y – 11 = 5(x – 2); y = 5x + 1 f ( x) = x 2 f (3) − f (−3) 9 + 9 = =3 3 − (−3) 6 c 2 = 3; c = − 3, 3 47. b. The Mean Value Theorem does not apply because F (0) does not exist. 244 Section 3.10 48. Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 49. Let f ( x ) = 3 x − cos 2 x ; a1 = 0 , b1 = 1 . f ( 0 ) = −1 ; f (1) ≈ 3.4161468 n hn 1 0.5 2 0.25 3 0.125 4 0.0625 5 0.03125 6 0.015625 7 0.0078125 8 0.0039063 9 0.0019532 10 0.0009766 11 0.0004883 12 0.0002442 13 0.0001221 14 0.0000611 15 0.0000306 16 0.0000153 17 0.0000077 18 0.0000039 19 0.0000020 20 0.0000010 21 0.0000005 22 0.0000003 23 0.0000002 x ≈ 0.281785 mn 0.5 0.25 0.375 0.3125 0.28125 0.296875 0.2890625 0.2851563 0.2832031 0.2822266 0.2817383 0.2819824 0.2818604 0.2817994 0.2817689 0.2817842 0.2817918 0.2817880 0.2817861 0.2817852 0.2817847 0.2817845 0.2817846 f ( mn ) 0.9596977 −0.1275826 0.3933111 0.1265369 −0.0021745 0.0617765 0.0296988 0.0137364 0.0057745 0.0017984 −0.0001884 0.0008049 0.0003082 0.0000600 −0.0000641 −0.0000018 0.0000293 0.0000138 0.0000061 0.0000022 0.0000004 −0.0000006 −0.0000000 50. f(x) = 3x – cos 2x, f ( x) = 3 + 2sin 2 x Let x1 = 0.5 . n xn 1 2 3 4 5 0.5 0.2950652 0.2818563 0.2817846 0.2817846 x ≈ 0.281785 51. xn +1 = cos 2 xn 3 n xn 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.5 0.18010 0.311942 0.270539 0.285718 0.280375 0.282285 0.281606 0.281848 0.281762 0.281793 0.281782 0.281786 0.281784 0.281785 0.281785 x ≈ 0.2818 52. y = x and y = tan x Let x1 = 11π . 8 f(x) = x – tan x, f ( x) = 1 – sec 2 x . n 1 2 3 4 5 6 7 8 xn 11π 8 4.64661795 4.60091050 4.54662258 4.50658016 4.49422443 4.49341259 4.49340946 x ≈ 4.4934 Instructor's Resource Manual Section 3.10 245 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ( x − 3x + 3 x ) dx ( x − 3x + 3x ) dx 3 53. = 58. Let u = cos x ; then du = − sin x dx or −du = sin x dx . 2 3 2 1/ 2 1 4 2 x − x3 + 3 ⋅ x3/ 2 + C 4 3 1 = x 4 − x3 + 2 x3/ 2 + C 4 = 2 x 4 − 3x 2 + 1 54. x2 = = − u 4 du 1 = − u5 + C 5 1 = − cos5 x + C 5 dx ( 2 x2 − 3 + x−2 ) dx 2 3 x − 3x − x −1 + C 3 2 x3 1 = − 3x − + C or x 3 59. u = tan(3 x 2 + 6 x), du = (6 x + 6) sec 2 (3x 2 + 6 x) 2 x4 − 9 x2 − 3 +C 3x y 3 − 9 y sin y + 26 y −1 dy y = ( y 2 − 9sin y + 26 ) dy = 1 3 y + 9 cos y + 26 y + C 3 ( x + 1) tan 2 ( 3x 2 + 6 x ) sec2 ( 3x 2 + 6 x ) dx 1 2 1 u du = u 3 + C 6 18 1 = tan 3 3 x 2 + 6 x + C 18 = ( 56. Let u = y 2 − 4 ; then du = 2 ydy or 1 du = ydy . 2 ) 60. u = t 4 + 9, du = 4t 3 dt 1 du t3 dt = 4 u t4 + 9 1 u −1/ 2 du 4 1 = ⋅ 2u1/ 2 + C 4 1 4 t +9 +C = 2 = 1 u ⋅ du 2 y y 2 − 4 dy = 1 u1/ 2 du 2 1 2 3/ 2 = ⋅ u +C 2 3 3/ 2 1 2 = y −4 +C 3 = ( ( ) 1/ 3 dz = 61. Let u = t 5 + 5 ; then du = 5t 4 dt or ) 57. Let u = 2 z 2 − 3 ; then du = 4 zdz or z 2z2 − 3 1 du = zdz . 4 1 u1/ 3 du 4 1 3 = ⋅ u4/3 + C 4 4 4/3 3 = +C 2z2 − 3 16 ( Section 3.10 ( t 4 t5 + 5 1 u1/ 3 ⋅ du 4 = 246 u 4 ⋅ −du = = 55. ( cos x )4 sin x dx cos 4 x sin x dx = ) 2/3 dt = 1 du = t 4 dt . 5 1 2/3 u du 5 1 u 2 / 3 du 5 1 3 5/3 = ⋅ u +C 5 5 5/ 3 3 5 = +C t +5 25 = ( ) ) Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 62. Let u = x 2 + 4 ; then du = 2 x dx or x 1 dx = 2 2 x +4 1 du = xdx . 2 66. Let u = y 3 − 3 y ; then ( ) ( y2 −1 u (y 1 u −1/ 2 du 2 1 = ⋅ 2u1/ 2 + C 2 = 3 − 3y ) 2 dy = x2 1 dx = 3 3 x +9 1 du = x 2 dx . 3 =− u 1 u −1/ 2 du 3 1 = ⋅ 2u1/ 2 + C 3 2 3 = x +9 +C 3 ( y + 1) dy = = 1 u2 du ( 2 y − 1) 3 dy = = du u3 1 dy = 69. x +1 dx y = 2 x +1 + C y = 2 x + 1 + 14 sin y dy = dx 70. − cos y = x + C x = –1 – cos y dy = 71. 2t − 1 dt 1 y = (2t − 1)3 2 + C 3 1 y = (2t − 1)3 2 − 1 3 y −4 dy = t 2 dt 72. − − 1 3 y3 1 3 y3 y=3 Instructor's Resource Manual +C y = − cos x + C y = –cos x + 3 u −3 du 1 = − u −2 + C 2 1 =− +C 2 2 ( 2 y − 1) 3y − 9 y dy = sin x dx 68. 65. Let u = 2 y − 1 ; then du = 2dy . 2 1 3 1 5 u −1/ 5 du = (2 y 3 + 3 y 2 + 6 y )4 / 5 + C 6 24 u −2 du = −u −1 + C 1 =− +C y +1 u2 67. u = 2 y 3 + 3 y 2 + 6 y , du = (6 y 2 + 6 y + 6) dy 64. Let u = y + 1 ; then du = dy . 2 du 1 u −2 du 3 1 = ⋅ −u −1 + C 3 1 1 =− ⋅ 3 +C 3 y − 3y du = 1 1 3 = = x2 + 4 + C 63. Let u = x3 + 9 ; then du = 3 x 2 dx or ) du = 3 y 2 − 3 dy = 3 y 2 − 1 dy . du = t3 +C 3 = t3 2 − 3 3 1 2 − t3 Section 3.10 247 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 7. Aregion = 0.5 (1 + 1.5 + 2 + 2.5) = 3.5 2 y dy = (6 x − x3 )dx 73. 1 4 x +C 4 1 y 2 = 3x 2 − x 4 + 9 4 1 y = 3x 2 − x 4 + 9 4 y 2 = 3x 2 − 8. Aregion = 0.5 (1.5 + 2 + 2.5 + 3) = 4.5 9. Aregion = Arect + Atri = 1x + 1 1 1 10. Aregion = bh = x ⋅ xt = x 2t 2 2 2 cos y dy = x dx 74. 1 1 x ⋅ x = x2 + x 2 2 11. y = 5 − x; Aregion = Arect + Atri x2 +C 2 ⎛ x2 ⎞ y = sin −1 ⎜ ⎟ ⎜ 2 ⎟ ⎝ ⎠ sin y = = 2 ( 2) + 1 ( 2 )( 2 ) = 6 2 12. Aregion = Arect + Atri 75. s ( t ) = −16t 2 + 48t + 448; s = 0 at t = 7; v ( t ) = s ' ( t ) = −32t + 48 = 1(1) + 1 (1)( 7 ) = 4.5 2 when t = 7, v = –32(7) + 48 = –176 ft/s Review and Preview Problems 1 1 3 2 1. Aregion = bh = aa sin 60o = a 2 2 4 ⎛1 ⎞ ⎛ 1 ⎞⎛ 3 ⎞ 2. Aregion = 6 ⎜ base × height ⎟ = 6 ⎜ a ⎟ ⎜⎜ a⎟ ⎝2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎟⎠ = 3 3 2 a 2 2 a ⎛1 ⎞ 3. Aregion = 10 ⎜ base × height ⎟ = 5 cot 36 2 4 ⎝ ⎠ 5 = a 2 cot 36 4 1 ⎛ 8.5 ⎞ 4. Aregion = Arect + Atri = 17 ( 8.5 ) + 17 ⎜ 2 ⎝ tan 45o ⎟⎠ = 216.75 1 2 5. Aregion = Arect + Asemic. = 3.6 ⋅ 5.8 + π (1.8 ) 2 ≈ 25.97 ⎛1 ⎞ 6. Aregion = A#5 + 2 Atri = 25.97 + 2 ⎜ ⋅1.2 ⎟ 5.8 2 ⎝ ⎠ = 32.93 248 Review and Preview Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 33. a. b. 10V −1/ 2 dV = C1dt ; 20 V = C1t + C2 ; V(0) = 1600: C2 = 20 ⋅ 40 = 800; V(40) = 0: C1 = − V (t ) = b. Since the trip that involves 1 min more travel time at speed vm is 0.6 mi longer, vm = 0.6 mi/min = 36 mi/h. c. From part b, vm = 0.6 mi/min. Note that the average speed during acceleration and v deceleration is m = 0.3 mi/min. Let t be the 2 time spent between stop C and stop D at the constant speed vm , so 0.6t + 0.3(4 – t)= 2 miles. Therefore, 2 t = 2 min and the time spent accelerating 3 is 4 − 2 23 a= = 2 0.6 − 0 2 3 2 min. 3 c. 36. a. b. 1 2 (−20t + 800)2 = ( 40 − t ) 400 2 V (10) = ( 40 − 10 ) = 900 cm3 dP = C1 3 P , P(0) = 1000, P(10) = 1700 dt where t is the number of years since 1980. 3 P −1/ 3 dP = C1dt ; P 2 / 3 = C1t + C2 2 3 P(0) = 1000: C2 = ⋅10002 / 3 = 150 2 P(10) = 1700: C1 = 3 ⋅17002 / 3 2 − 150 10 ≈ 6.3660 P = (4.2440t + 100)3 / 2 c. 4000 = (4.2440t + 100)3 / 2 40002 / 3 − 100 ≈ 35.812 4.2440 t ≈ 36 years, so the population will reach 4000 by 2016. = 0.9 mi/min 2 . t= dh = 4 , so h(t ) = 4t + C1 . Set dt t = 0 at the time when Victoria threw the ball, and height 0 at the ground, then h(t) = 4t + 64. The 34. For the balloon, height of the ball is given by s (t ) = −16t 2 + v0t , since s0 = 0 . The maximum height of the ball is v when t = 0 , since then s (t ) = 0 . At this time 32 2 ⎛v ⎞ ⎛v ⎞ ⎛v ⎞ h(t) = s(t) or 4 ⎜ 0 ⎟ + 64 = −16 ⎜ 0 ⎟ + v0 ⎜ 0 ⎟ . 32 32 ⎝ ⎠ ⎝ ⎠ ⎝ 32 ⎠ Solve this for v0 to get v0 ≈ 68.125 feet per second. 35. a. 800 = −20 40 dV = C1 h where h is the depth of the dt V . water. Here, V = πr 2 h = 100h , so h = 100 37. Initially, v = –32t and s = −16t 2 + 16 . s = 0 when t = 1. Later, the ball falls 9 ft in a time given by 3 0 = −16t 2 + 9 , or s, and on impact has a 4 ⎛3⎞ velocity of −32 ⎜ ⎟ = −24 ft/s. By symmetry, ⎝4⎠ 24 ft/s must be the velocity right after the first bounce. So a. b. for 0 ≤ t < 1 ⎧−32t v(t ) = ⎨ 32( 1) 24 for 1 < t ≤ 2.5 − t − + ⎩ 9 = −16t 2 + 16 ⇒ t ≈ 0.66 sec; s also equals 9 at the apex of the first rebound at t = 1.75 sec. dV V = C1 , V(0) = 1600, dt 10 V(40) = 0. Hence 234 Section 3.9 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4 CHAPTER 4.1 Concepts Review 1. 2 ⋅ The Definite Integral 8 m =1 5(6) = 30; 2(5) = 10 2 = (−1)1 2−1 + (−1) 2 20 + (−1)3 21 +(−1) 4 22 + (−1)5 23 + (−1)6 24 2. 3(9) – 2(7) = 13; 9 + 4(10) = 49 +(−1)7 25 + (−1)8 26 3. inscribed; circumscribed 1 = − + 1 − 2 + 4 − 8 + 16 − 32 + 64 2 85 = 2 4. 0 + 1 + 2 + 3 = 6 Problem Set 4.1 6 (k − 1) = 1. k =1 6 k =1 k− 6 1 k =1 i =1 i2 = (−1)5 25 (−1)6 26 (−1)7 27 + + 6 7 8 1154 =− 105 + 6(7)(13) = 91 6 6 7 1 1 1 1 3. = + + k =1 k + 1 1 + 1 2 + 1 3 + 1 1 1 1 1 + + + 4 +1 5 +1 6 +1 7 +1 1 1 1 1 1 1 1 = + + + + + + 2 3 4 5 6 7 8 1443 = 840 481 = 280 7. 8 (l + 1)2 = 42 + 52 + 62 + 7 2 + 82 + 92 = 271 6 n cos(nπ) = n =1 n =1 ( −1)n ⋅ n = –1 + 2 – 3 + 4 – 5 + 6 =3 + 4. (−1)3 23 (−1) 4 24 + 4 5 = 6(7) − 6(1) 2 = 15 6 (−1)k 2k k =3 ( k + 1) 7 6. = 2. (−1)m 2m − 2 5. 6 8. ⎛ kπ ⎞ k sin ⎜ ⎟ ⎝ 2 ⎠ k =−1 ⎛ π⎞ ⎛π⎞ = − sin ⎜ − ⎟ + sin ⎜ ⎟ + 2sin(π) ⎝ 2⎠ ⎝2⎠ π 3 ⎛ ⎞ ⎛ 5π ⎞ +3sin ⎜ ⎟ + 4sin(2π) +5sin ⎜ ⎟ + 6sin(3π) ⎝ 2 ⎠ ⎝ 2 ⎠ =1+1+0–3+0+5+0 =4 l =3 9. 1 + 2 + 3 + + 41 = 41 i i =1 10. 2 + 4 + 6 + 8 + + 50 = 25 2i i =1 11. 1 + 1 1 + + 2 3 + 1 100 1 = 100 i =1 i Instructor’s Resource Manual Section 4.1 249 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 1 1 12. 1 − + − + 2 3 4 1 100 (−1)i +1 = 100 i =1 i − 13. a1 + a3 + a5 + a7 + 14. + a99 = f ( w1 )Δx + f ( w2 )Δx + n = 10 20. = 50 i =1 a2i −1 =4 + f ( wn )Δx i =1 10 = i =1 i =1 bi 10 10 k =1 n =1 an + 2 17. p =0 p =1 ap − 10 n =1 bn n 10 q =1 bp = q =1 aq − q =1 = 40 − 50 − bq − 19. 25. (3i − 2) i =1 100 =3 i =1 i− 100 2 i =1 = 3(5050) − 2(100) = 14,950 i+ i =1 4n3 − 3n 2 − n 6 (2i − 3) 2 = n i 2 − 12 n n 1 i =1 = i =1 10(11) 2 n 2n3 + 3n 2 + n 3n 2 + 3n − +n 3 2 10 q i2 − 3 = =4 q =1 n i =1 i =1 = −65 100 k =1 2n(n + 1)(2n + 1) 3n(n + 1) − +n 6 2 n 10 k2 = 24. (aq − bq − q) 10 k 3 + 20 10 i =1 = 40 − 50 = –10 18. (5k 3 + 20k 2 ) (2i 2 − 3i + 1) = 2 23. 10 p =1 k =1 k =1 10 k =1 (a p +1 − b p +1 ) 10 = k2 = 5(3025) + 20(385) = 22,825 = 3(40) + 2(50) = 220 9 10 k3 − 10 5k 2 (k + 4) = 22. (3an + 2bn ) =3 10 k =1 =5 n =1 3 i =1 = 3025 − 385 = 2640 = 90 16. i =1 k =1 = 40 + 50 10 10 i− (k 3 − k 2 ) = 21. 10 ai + 10 = 4(385) – 55 – 3(10) = 1455 (ai + bi ) 15. i2 − i =1 10 10 (4i 2 − i − 3) i =1 10 f ( wi )Δx i =1 [(i − 1)(4i + 3)] i =1 10 (4i 2 − 12i + 9) i =1 n i+ i =1 n 9 i =1 = 4n(n + 1)(2n + 1) 12n(n + 1) − + 9n 6 2 = 4n3 − 12n 2 + 11n 3 S = 1 + 2 + 3 + + (n − 2) + (n − 1) + n + S = n + (n − 1) + (n − 2) + + 3 + 2 + 1 2S = (n + 1) + (n + 1) + (n + 1) + 2S = n(n + 1) n(n + 1) S= 2 + (n + 1) + (n + 1) + (n + 1) 250 Section 4.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 26. S − rS = a + ar + ar 2 + + ar n 27. a. + ar n + ar n +1 ) − (ar + ar 2 + = a − ar n +1 (1) 1− 2 ⎛1⎞ ⎜ ⎟ = 1 k =0 ⎝ 2 ⎠ 2 10 k 10 k 11 10 ⎛1⎞ ⎛1⎞ ⎜ ⎟ = 1− ⎜ ⎟ 2 ⎝ ⎠ ⎝2⎠ k =1 n +1 a − ar = S (1 − r ); S = 1− r 10 2k = b. k =0 10 10 ⎛1⎞ = 2 − ⎜ ⎟ , so ⎝2⎠ = 1023 . 1024 1 − 211 = 211 − 1, so −1 2k = 211 − 2 = 2046 . k =1 28. + [ a + (n − 2)d ] + [ a + (n − 1)d ] + (a + nd ) S = a + ( a + d ) + ( a + 2d ) + + S = (a + nd ) + [ a + (n − 1)d ] + [ a + (n − 2)d ] + 2S = (2a + nd ) + (2a + nd ) + (2a + nd ) + 2S = (n + 1)(2a + nd) (n + 1)(2a + nd ) S= 2 + ( a + 2d ) + ( a + d ) + a + (2a + nd ) + (2a + nd ) + (2a + nd ) ( i + 1)3 − i3 = 3i 2 + 3i + 1 29. n ⎡( i + 1)3 − i 3 ⎤ = ⎣ ⎦ i =1 n i =1 ( 3i n ( n + 1)3 − 13 = 3 2 i2 + 3 i =1 n n3 + 3n 2 + 3n = 3 i2 + 3 n n i+ i =1 i =1 2n3 + 6n 2 + 6n = 6 ) + 3i + 1 n 1 i =1 n ( n + 1) 2 +n i 2 + 3n 2 + 3n + 2n i =1 3 2 2n + 3n + n = 6 n ( n + 1)( 2n + 1) 6 = n i2 i =1 n i2 i =1 Instructor’s Resource Manual Section 4.1 251 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ( i + 1)4 − i 4 = 4i3 + 6i 2 + 4i + 1 30. n i =1 ⎡(i + 1)4 − i 4 ⎤ = ⎣ ⎦ n 4i3 + 6i 2 + 4i + 1) ( i =1 n (n + 1)4 − 14 = 4 n 4 + 4 n3 + 6 n 2 + 4 n = 4 i =1 n i3 + 6 i3 + 6 Solving for n i2 + 4 i =1 i =1 n n n i+ i =1 1 i =1 n(n + 1)(2n + 1) n(n + 1) +4 +n 6 2 i3 gives i =1 n 4 i =1 n 4 ( ) ( ) i3 = n 4 + 4n3 + 6n 2 + 4n − 2n3 + 3n 2 + n − 2n 2 + 2n − n i 3 = n 4 + 2 n3 + n 2 i =1 n 4 + 2n3 + n 2 ⎡ n ( n + 1) ⎤ i = =⎢ ⎥ 4 ⎣ 2 ⎦ i =1 n 2 3 ( i + 1)5 − i5 = 5i 4 + 10i3 + 10i 2 + 5i + 1 31. n i =1 ⎡( i + 1)5 − i5 ⎤ =5 ⎣⎢ ⎦⎥ n i 4 + 10 i =1 ( n + 1)5 − 15 = 5 n5 + 5n 4 + 10n3 + 10n 2 + 5n = 5 n i =1 n i =1 n Solving for n i =1 i 4 + 10 n i3 + 10 i =1 2 n 2 ( n + 1) 4 i 4 + 52 n 2 ( n + 1) 2 n i2 + 5 i+ i =1 + 10 n 1 i =1 n(n + 1)(2n + 1) n(n + 1) +5 +n 6 2 5 + 10 n ( n + 1) (2n + 1) + n(n + 1) + n 6 2 i 4 yields i =1 n i =1 n(n + 1)(2n + 1)(3n 2 + 3n − 1) i 4 = 15 ⎡ n5 + 52 n 4 + 53 n3 − 16 n ⎤ = ⎣ ⎦ 30 32. Suppose we have a ( n + 1) × n grid. Shade in n + 1 – k boxes in the kth column. There are n columns, and the shaded area is 1 + 2 + also half the area of the grid or n(n + 1) . Thus, 1 + 2 + 2 33. x = 2 ⎡ n(n + 1) ⎤ 3 3 + n3 or ⎢ ⎥ . Thus, 1 + 2 + ⎣ 2 ⎦ n(n + 1) . 2 n(n + 1) +n= . From the diagram the area is 2 +n = Suppose we have a square grid with sides of length 1 + 2 + 13 + 23 + + n . The shaded area is 2 ⎡ n(n + 1) ⎤ + n3 = ⎢ ⎥ . ⎣ 2 ⎦ 1 55 (2 + 5 + 7 + 8 + 9 + 10 + 14) = ≈ 7.86 7 7 2 1 ⎡⎛ 55 ⎞ ⎛ 55 ⎞ ⎛ 55 ⎞ ⎛ 55 ⎞ 55 ⎞ ⎛ 55 ⎞ ⎛ 55 ⎞ ⎛ s = ⎢⎜ 2 − ⎟ + ⎜ 5 − ⎟ + ⎜ 7 − ⎟ + ⎜ 8 − ⎟ + ⎜ 9 − ⎟ + ⎜ 10 − ⎟ + ⎜ 14 − ⎟ 7 ⎢⎝ 7 ⎠ ⎝ 7 ⎠ ⎝ 7 ⎠ ⎝ 7 ⎠ 7 7 7 ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎣ 2 2 2 2 2 2 2⎤ 608 ⎥ = ≈ 12.4 49 ⎥⎦ 252 Section 4.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 34. a. x = 1, s 2 = 0 b. x = 1001, s 2 = 0 c. x=2 2 s 2 = 13 ⎡(1 − 2)2 + (2 − 2) 2 + (3 − 2)2 ⎤ = 13 ⎡⎢( −1) + 02 + 12 ⎤⎥ = 13 ( 2 ) = 32 ⎣ ⎦ ⎣ ⎦ d. x = 1, 000, 002 s 2 = 13 ⎡(−1)2 + 02 + 12 ⎤ = 32 ⎣ ⎦ n 35. a. i =1 b. ( xi − x ) = n i =1 xi − n x = nx − nx = 0 i =1 1 n 1 n 2 ( xi − x )2 = ( xi − 2 x xi + x 2 ) n i =1 n i =1 1 n 2 2x n 1 n 2 = xi − xi + x n i =1 n i =1 n i =1 s2 = = 1 n 2 2x 1 xi − (nx ) + (nx 2 ) n i =1 n n ⎛1 n 2⎞ ⎛1 n 2⎞ xi ⎟ − 2 x 2 + x 2 = ⎜ xi ⎟ − x 2 =⎜ ⎜n ⎟ ⎜n ⎟ ⎝ i =1 ⎠ ⎝ i =1 ⎠ 36. The variance of n identical numbers is 0. Let c be the constant. Then 2 2 s 2 = 1n ⎡⎢( c − c ) + ( c − c ) + + (c − c)2 ⎤⎥ = 0 ⎣ ⎦ 37. Let S (c) = n i =1 S '(c) = = = ( xi − c)2 . Then d n ( xi − c )2 dc i =1 n d ( xi − c )2 dc i =1 n i =1 = −2 2( xi − c)(−1) n i =1 xi + 2nc S ''(c) = 2n Set S '(c) = 0 and solve for c : −2 c n xi + 2nc i =1 n = 1n xi = x i =1 =0 Since S ''( x) = 2n > 0 we know that x minimizes S (c) . Instructor’s Resource Manual Section 4.1 253 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. i The number of gifts given on the nth day is 38. a. m= m =1 i (i + 1) . 2 12 The total number of gifts is i (i + 1) = 364 . i =1 2 n b. For n days, the total number of gifts is i (i + 1) . i =1 2 i (i + 1) n i 2 n i 1 n 2 1 n 1 ⎡ n(n + 1)(2n + 1) ⎤ 1 ⎡ n(n + 1) ⎤ = + = i + i= ⎢ ⎥+ 2⎢ 2 ⎥ 2 2 2 2 2 2⎣ 6 ⎦ ⎣ ⎦ i =1 i =1 i =1 i =1 i =1 1 1 ⎛ 2n + 1 ⎞ 1 = n(n + 1) ⎜ + 1⎟ = n(n + 1)(2n + 4) = n(n + 1)(n + 2) 4 6 ⎝ 3 ⎠ 12 n 39. The bottom layer contains 10 · 16 = 160 oranges, the next layer contains 9 · 15 = 135 oranges, the third layer contains 8 · 14 = 112 oranges, and so on, up to the top layer, which contains 1 · 7 = 7 oranges. The stack contains 1 · 7 + 2 · 8+ ... + 9· 15 + 10 · 16 10 i (6 + i ) = 715 oranges. = i =1 50 40. If the bottom layer is 50 oranges by 60 oranges, the stack contains i (10 + i ) = 55, 675. i =1 41. For a general stack whose base is m rows of n oranges with m ≤ n, the stack contains m i ( n − m + i ) = ( n − m) i =1 m i =1 i+ m i2 i =1 m(m + 1) m(m + 1)(2m + 1) = ( n − m) + 2 6 m(m + 1)(3n − m + 1) = 6 42. 1 1 1 + + + 1⋅ 2 2 ⋅ 3 3 ⋅ 4 + 1 n(n + 1) ⎛ 1⎞ ⎛ 1 1⎞ ⎛1 1⎞ = ⎜1 − ⎟ + ⎜ − ⎟ + ⎜ − ⎟ + ⎝ 2⎠ ⎝ 2 3⎠ ⎝3 4⎠ 1 = 1− n +1 1 ⎞ ⎛1 +⎜ − ⎟ n n +1⎠ ⎝ 43. A = 1⎡ 3 5⎤ 7 1+ + 2 + ⎥ = 2 ⎢⎣ 2 2⎦ 2 44. A = 1⎡ 5 3 7 9 5 11 ⎤ 15 1+ + + + 2 + + + ⎥ = ⎢ 4⎣ 4 2 4 4 2 4⎦ 4 45. A = 1 ⎡3 5 ⎤ 9 + 2 + + 3⎥ = ⎢ 2 ⎣2 2 ⎦ 2 46. A = 1 ⎡5 3 7 9 5 11 ⎤ 17 + + + 2 + + + + 3⎥ = 4 ⎢⎣ 4 2 4 4 2 4 ⎦ 4 254 Section 4.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 47. A = 2 2 ⎞⎤ ⎞ ⎛1 1 ⎡⎛ 1 2 ⎞ ⎛ 1 ⎛ 1 ⎞ 1 ⎛ 9 3 17 ⎞ 23 ⎞ ⎛1 ⎛3⎞ 2 ⎜ ⎜ ⎟ + ⋅ + 1⎟ ⎥ = ⎜ 1 + + + ⎟ = ⋅ + + ⋅ + + ⋅ + 0 1 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎢ ⎜ ⎟ ⎟ 2⎝ 8 2 8 ⎠ 8 2 2 2 ⎣⎝ 2 2 2 2 ⎠ ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠ ⎦⎥ ⎠ ⎝ 48. A = 2 ⎤ ⎞ 1 ⎛1 3 2 ⎞ 1 1 ⎡⎛ 1 ⎛ 1 ⎞ 1 9 3 17 31 ⎢⎜ ⋅ ⎜ ⎟ + 1⎟ + ⎛⎜ ⋅12 + 1⎞⎟ + ⎜ ⋅ ⎜⎛ ⎟⎞ + 1⎟ + ⎜⎛ ⋅ 22 + 1⎟⎞ ⎥ = ⎛⎜ + + + 3 ⎞⎟ = ⎜ ⎟ ⎜ ⎟ 2⎢ 2 ⎝2⎠ 2 2 2⎝8 2 8 ⎠ ⎥⎦ ⎠ ⎝2 ⎝2⎠ ⎠ 8 ⎠ ⎝ ⎠ ⎝ ⎣⎝ 49. A = 1(1 + 2 + 3) = 6 50. A= 1 ⎡⎛ 3 ⎞ ⎛ 5 ⎞ ⎜ 3 ⋅ − 1⎟ + ( 3 ⋅ 2 − 1) + ⎜ 3 ⋅ − 1⎟ + (3 · 3 – 1)] = 2 ⎢⎣⎝ 2 ⎠ ⎝ 2 ⎠ 1⎛7 13 ⎞ 23 ⎜ + 5 + + 8⎟ = 2⎝2 2 ⎠ 2 51. 2 ⎤ ⎞ ⎛ 7 2 ⎞ ⎛ 5 2 ⎞ ⎛ 8 2 ⎞ ⎛ 17 2 ⎞ 1 ⎡⎛ ⎛ 13 ⎞ ⎢⎜ ⎜ ⎟ − 1⎟ + ⎜ ⎛⎜ ⎞⎟ − 1⎟ + ⎜ ⎛⎜ ⎞⎟ − 1⎟ + ⎜ ⎛⎜ ⎞⎟ − 1⎟ + ⎜ ⎛⎜ ⎞⎟ − 1⎟ + (32 − 1) ⎥ ⎟ ⎜⎝ 3 ⎠ ⎟ ⎜⎝ 2 ⎠ ⎟ ⎜⎝ 3⎠ ⎟ ⎜⎝ 6 ⎠ ⎟ 6 ⎢⎜ ⎝ 6 ⎠ ⎥⎦ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣⎝ 1 ⎛ 133 40 21 55 253 ⎞ 1243 = ⎜ + + + + + 8⎟ = 6 ⎝ 36 9 4 9 36 ⎠ 216 A= Instructor’s Resource Manual Section 4.1 255 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 52. 2 2 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1⎡ 4 4 3 3 2 2 ⎢(3(−1)2 + (−1) + 1) + ⎜ 3 ⎛⎜ − ⎞⎟ + ⎛⎜ − ⎞⎟ + 1⎟ + ⎜ 3 ⎛⎜ − ⎞⎟ + ⎛⎜ − ⎞⎟ + 1⎟ + ⎜ 3 ⎛⎜ − ⎞⎟ + ⎛⎜ − ⎞⎟ + 1⎟ + (3(0)2 + 0 + 1) ⎜ ⎝ 5⎠ ⎝ 5⎠ ⎟ ⎜ ⎝ 5⎠ ⎝ 5⎠ ⎟ ⎜ ⎝ 5⎠ ⎝ 5⎠ ⎟ 5⎢ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ 2 2 2 2 ⎤ ⎛ ⎛1⎞ 1 ⎞ ⎛ ⎛2⎞ 2 ⎞ ⎛ ⎛3⎞ 3 ⎞⎛ ⎛ 4 ⎞ 4 ⎞ + ⎜ 3 ⎜ ⎟ + + 1⎟ + ⎜ 3 ⎜ ⎟ + + 1⎟ + ⎜ 3 ⎜ ⎟ + + 1⎟ ⎜ 3 ⎜ ⎟ + + 1⎟ + (3(1) 2 + 1 + 1) ⎥ ⎜ ⎝5⎠ 5 ⎟ ⎜ ⎝ 5⎠ 5 ⎟ ⎜ ⎝5⎠ 5 ⎟⎜ ⎝ 5 ⎠ 5 ⎟ ⎥⎦ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ 1 = [3 + 2.12 + 1.48 + 1.08 + 1 + 1.32 + 1.88 + 2.68 + 3.72 + 5] = 4.656 5 A= 1 i , xi = n n i ⎛ ⎞⎛ 1 ⎞ i 2 f ( xi )Δx = ⎜ + 2 ⎟ ⎜ ⎟ = + ⎝n ⎠ ⎝ n ⎠ n2 n ⎡⎛ 1 2 ⎞ ⎛ 2 2 ⎞ + + + + A( Sn ) = ⎢⎜ 2 n⎟ ⎜ 2 n⎟ ⎠ ⎝n ⎠ ⎣⎝ n 53. Δx = ⎛ n 2 ⎞⎤ 1 +⎜ + = (1 + 2 + 3 + 2 n ⎟⎥ ⎝n ⎠ ⎦ n2 + n) + 2 = n(n + 1) 2n 2 +2 = 1 5 + 2n 2 ⎛ 1 5⎞ 5 lim A( Sn ) = lim ⎜ + ⎟ = n →∞ n→∞ ⎝ 2n 2 ⎠ 2 1 i , xi = n n ⎡ 1 ⎛ i ⎞2 ⎤ ⎛ 1 ⎞ i 2 1 f ( xi )Δx = ⎢ ⋅ ⎜ ⎟ + 1⎥ ⎜ ⎟ = + 3 2 n n n ⎢⎣ ⎝ ⎠ ⎥⎦ ⎝ ⎠ 2n ⎡⎛ 12 ⎛ n2 1 ⎞⎤ 1 ⎞ ⎛ 22 1 ⎞ 1 2 + ⎟+⎜ + ⎟+ +⎜ + ⎟⎥ = A( Sn ) = ⎢⎜ (1 + 22 + 32 + + n 2 ) + 1 3 3 n⎟ ⎜ 3 n⎟ 3 n⎟ ⎜ ⎜ ⎢⎣⎝ 2n ⎠ ⎝ 2n ⎠ ⎝ 2n ⎠ ⎥⎦ 2n 1 ⎡ n(n + 1)(2n + 1) ⎤ 1 ⎡ 2n3 + 3n 2 + n ⎤ 1 ⎡ 3 1 ⎤ = = + 1 ⎢ ⎥ + 1 = ⎢2 + + 2 ⎥ + 1 ⎥ 3 ⎢⎣ 3 6 12 ⎢⎣ n n ⎦ 12 ⎣ ⎦ n 2n ⎥⎦ ⎡1⎛ 3 1 ⎞ ⎤ 7 lim A( Sn ) = lim ⎢ ⎜ 2 + + ⎟ + 1⎥ = n n2 ⎠ ⎦ 6 n →∞ n →∞ ⎣12 ⎝ 54. Δx = 256 Section 4.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2 2i , xi = −1 + n n ⎡ ⎛ 2i ⎞ ⎤ ⎛ 2 ⎞ 8i f ( xi )Δx = ⎢ 2 ⎜ −1 + ⎟ + 2 ⎥ ⎜ ⎟ = n ⎠ ⎦ ⎝ n ⎠ n2 ⎣ ⎝ 1 i , xi = n n ⎡⎛ i ⎞3 i ⎤ ⎛ 1 ⎞ i3 i f ( xi )Δx = ⎢⎜ ⎟ + ⎥ ⎜ ⎟ = + 4 n n n ⎝ ⎠ ⎝ ⎠ n n2 ⎣⎢ ⎦⎥ 55. Δx = 58. Δx = ⎡⎛ 8 ⎞ ⎛ 16 ⎞ ⎛ 8n ⎞ ⎤ A( Sn ) = ⎢⎜ ⎟ + ⎜ ⎟ + + ⎜ ⎟ ⎥ 2 2 ⎝ n2 ⎠⎦ ⎣⎝ n ⎠ ⎝ n ⎠ 8 8 ⎡ n(n + 1) ⎤ = (1 + 2 + 3 + + n) = ⎢ ⎥ 2 n n2 ⎣ 2 ⎦ ⎡ n2 + n ⎤ 4 = 4⎢ ⎥ = 4+ 2 n ⎣⎢ n ⎦⎥ A( Sn ) = 59. ⎛8 4 4 ⎞ 8 lim A( Sn ) = lim ⎜ + + ⎟= . n →∞ n→∞ ⎝ 3 n 3n 2 ⎠ 3 ⎛ 8 ⎞ 16 By symmetry, A = 2 ⎜ ⎟ = . ⎝3⎠ 3 60. 1 i , xi = n n 3 = n 4 (13 + 23 + + n3 ) = ⎤ ( n3 ) ⎥ n ⎦ 1 4 1 ⎡ n(n + 1) ⎤ ⎢ ⎥ n4 ⎣ 2 ⎦ (1 + 2 + + n) 2 i 2 ⎡i ⎤1 + f (ti )Δt = ⎢ + 2 ⎥ = ⎣n ⎦ n n2 n n ⎛1 1 ⎞ =⎜ + ⎟+2 ⎝ 2 2n ⎠ 1 5 lim A( Sn ) = + 2 = 2 2 n →∞ 1 The object traveled 2 ft. 2 4 ⎡ 2n3 + 3n 2 + n ⎤ 8 4 4 = ⎢ ⎥= + + 2 3 n 3 ⎣⎢ 3 n 3n ⎦⎥ 1 n2 n 2 ⎛ i 2⎞ 1 n ⎜ 2 + ⎟= 2 i+ n ⎠ n i =1 i =1 n i =1 ⎝ n 1 ⎡ n(n + 1) ⎤ = ⎢ ⎥+2 n2 ⎣ 2 ⎦ ⎡ n2 + n ⎤ =⎢ ⎥+2 2 ⎣⎢ 2n ⎦⎥ ⎛ 2i ⎞ ⎛ 2 ⎞ 8i f ( xi )Δx = ⎜ ⎟ ⎜ ⎟ = ⎝ n ⎠ ⎝ n ⎠ n3 ⎡⎛ 8 ⎞ ⎛ 8(22 ) ⎞ ⎛ 8n 2 ⎞ ⎤ A( Sn ) = ⎢⎜ ⎟ + ⎜ + ⋅⋅⋅+ ⎜ ⎟ ⎟⎥ 3 ⎜ 3 ⎟ ⎜ n3 ⎟ ⎥ ⎝ ⎠⎦ ⎣⎢⎝ n ⎠ ⎝ n ⎠ 8 2 8 ⎡ n(n + 1)(2n + 1) ⎤ = (1 + 22 + ⋅⋅⋅ + n 2 ) = ⎢ ⎥ 3 6 ⎦ n n3 ⎣ = 1 1 ⎡ n(n + 1) ⎤ 1 ⎡ n(n + 1) ⎤ ⎥ + 2⎢ 2 ⎥ 4 ⎢⎣ 2 ⎦ ⎦ n n ⎣ A( Sn ) = 2 + + n3 ) + n 2 + 2n + 1 n 2 + n 3n 2 + 4n + 1 3 1 1 + = = + + 2 2 2 4 n 4n 2n 4n 4n 2 3 lim A( Sn ) = 4 n →∞ 56. First, consider a = 0 and b = 2. 2 2i Δx = , xi = n n 3 ⎛ i ⎞ ⎛1⎞ i f ( xi )Δx = ⎜ ⎟ ⎜ ⎟ = ⎝ n ⎠ ⎝ n ⎠ n4 ⎡1 1 3 A( Sn ) = ⎢ (13 ) + (2 ) + 4 n4 ⎣n (13 + 23 + = 4⎞ ⎛ lim A( Sn ) = lim ⎜ 4 + ⎟ = 4 n⎠ n→∞ ⎝ 57. Δx = n 4 2 = n →∞ 2 1 ⎡ 1 ⎛ i ⎞2 ⎤ 1 i2 1 + f (ti )Δt = ⎢ ⎜ ⎟ + 1⎥ = 3 n ⎢⎣ 2 ⎝ n ⎠ ⎥⎦ n 2n n ⎛ 2 1i 1⎞ 1 n 2 n 1 A( Sn ) = ⎜ + ⎟= i + ⎜ 3 n ⎟ 2 n3 i =1 ⎝ 2n i =1 i =1 n ⎠ 1 ⎡ n(n + 1)(2n + 1) ⎤ 1 ⎡ 3 1 ⎤ = ⎥ + 1 = 12 ⎢ 2 + n + 2 ⎥ + 1 3 ⎢⎣ 6 ⎦ 2n n ⎦ ⎣ 1 7 (2) + 1 = ≈ 1.17 12 6 n →∞ The object traveled about 1.17 feet. lim A( Sn ) = 1⎡ 2 1 ⎤ 1 ⎡ n 4 + 2n3 + n 2 ⎤ ⎢ ⎥ = ⎢1 + + 2 ⎥ 4 4 4⎣ n n ⎦ n ⎣⎢ ⎦⎥ 1⎡ 2 1 ⎤ 1 ⎢1 + n + 2 ⎥ = 4 n →∞ 4 ⎣ n ⎦ lim A( Sn ) = lim n →∞ Instructor’s Resource Manual Section 4.1 257 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 65. a. A02 ( x3 ) = 23+1 =4 3 +1 b. A12 ( x3 ) = 23+1 13+1 1 15 − = 4− = 3 +1 3 +1 4 4 c. A12 ( x5 ) = 25+1 15+1 32 1 63 − = − = 5 +1 5 +1 3 6 6 2 61. a. 3 2 ⎛ ib ⎞ ⎛ b ⎞ b i f ( xi )Δx = ⎜ ⎟ ⎜ ⎟ = ⎝ n ⎠ ⎝ n ⎠ n3 A0b = b3 n i2 = b3 ⎡ n(n + 1)(2n + 1) ⎤ ⎢ ⎥ 6 ⎦ n3 ⎣ n3 i =1 b3 ⎡ 3 1 ⎤ = 2+ + ⎥ ⎢ 6 ⎣ n n2 ⎦ lim A0b = n →∞ 2b3 b3 = 6 3 = 21 = 10.5 2 b. Since a ≥ 0, A0b = A0a + Aab , or Aab = A0b − A0a b3 a 3 = − . 3 3 d. A05 = 53 125 = 3 3 b. A14 = 43 13 63 − = = 21 3 3 3 c. A25 = 53 23 117 − = = 39 3 3 3 64. a. Δx = b bi , xi = n n m m +1 m ⎛ bi ⎞ ⎛ b ⎞ b i f ( xi )Δx = ⎜ ⎟ ⎜ ⎟ = ⎝ n ⎠ ⎝n⎠ n m +1 b m +1 n m A( Sn ) = i n m +1 i =1 = ⎤ b m +1 ⎡ n m +1 + Cn ⎥ ⎢ m +1 m + 1 n ⎢⎣ ⎥⎦ b m +1 b m +1Cn = + m +1 n m +1 A0b ( x m ) = lim A( Sn ) = n →∞ lim Cn n →∞ n m +1 29+1 1024 = = 102.4 9 +1 10 66. Inscribed: Consider an isosceles triangle formed by one side of the polygon and the center of the circle. The 2π . The length of the base angle at the center is n π π is 2r sin . The height is r cos . Thus the area n n π π 1 2 2π 2 of the triangle is r sin cos = r sin . n n 2 n 2π ⎞ 1 2π ⎛1 An = n ⎜ r 2 sin ⎟ = nr 2 sin n ⎠ 2 n ⎝2 53 33 98 62. = − = ≈ 32.7 3 3 3 The object traveled about 32.7 m. A35 63. a. A02 ( x9 ) = Circumscribed: Consider an isosceles triangle formed by one side of the polygon and the center of the circle. The 2π . The length of the base angle at the center is n π is 2r tan . The height is r. Thus the area of the n π triangle is r 2 tan . n π π ⎛ ⎞ Bn = n ⎜ r 2 tan ⎟ = nr 2 tan n⎠ n ⎝ ⎛ sin 2π ⎞ 1 2 2π n ⎟ nr sin = lim πr 2 ⎜ ⎜ 2π ⎟ n n→∞ n→∞ 2 ⎝ n ⎠ lim An = lim n →∞ b m +1 m +1 = πr 2 = 0 since Cn is a polynomial in n of degree m. lim Bn = lim nr 2 tan n →∞ n→∞ π π πr 2 ⎛ sin n ⎞ ⎜ ⎟ = lim n n→∞ cos π ⎜ π ⎟ n⎝ n ⎠ = πr 2 b. Notice that Aab ( x m ) = A0b ( x m ) − A0a ( x m ) . Thus, using part a, Aab ( x m ) = b m +1 a m +1 − . m +1 m +1 258 Section 4.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4.2 Concepts Review 1. Riemann sum 2. definite integral; b a f ( x )dx 3. Aup − Adown 4. 8 − 1 15 = 2 2 Problem Set 4.2 1. RP = f (2)(2.5 − 1) + f (3)(3.5 − 2.5) + f (4.5)(5 − 3.5) = 4(1.5) + 3(1) + (–2.25)(1.5) = 5.625 2. RP = f(0.5)(0.7 – 0) + f(1.5)(1.7 – 0.7) + f(2)(2.7 – 1.7) + f(3.5)(4 – 2.7) = 1.25(0.7) + (–0.75)(1) + (–1)(1) + 1.25(1.3) = 0.75 3. RP = 5 i =1 f ( xi )Δxi = f (3)(3.75 − 3) + f (4)(4.25 − 3.75) + f (4.75)(5.5 − 4.25) + f (6)(6 − 5.5) + f (6.5)(7 − 6) = 2(0.75) + 3(0.5) + 3.75(1.25) + 5(0.5) + 5.5(1) = 15.6875 4. RP = 4 i =1 f ( xi )Δxi = f (−2)(−1.3 + 3) + f (−0.5)(0 + 1.3) + f (0)(0.9 − 0) + f (2)(2 − 0.9) = 4(1.7) + 3.25(1.3) + 3(0.9) + 2(1.1) = 15.925 5. RP = 8 i =1 f ( xi )Δxi = [ f (−1.75) + f (−1.25) + f (−0.75) + f (−0.25) + f (0.25) + f (0.75) + f (1.25) + f (1.75) ] (0.5) = [–0.21875 – 0.46875 – 0.46875 – 0.21875 + 0.28125 + 1.03125 + 2.03125 + 3.28125](0.5) = 2.625 6. RP = 6 i =1 f ( xi )Δxi = [ f (0.5) + f (1) + f (1.5) + f (2) + f (2.5) + f (3) ] (0.5) = [1.5 + 5 + 14.5 + 33 + 63.5 + 109](0.5) = 113.25 7. 8. 9. 10. 3 1 2 0 1 x3 dx ( x + 1)3 dx x π dx i =1 = (sin x)2 dx 2 2i , xi = n n f ( xi ) = xi + 1 = n 2 −1 1 + x 0 11. Δx = f ( xi )Δx = 2i +1 n n ⎡ ⎛ 2 ⎞⎤ 2 ⎢1 + i ⎜ n ⎟ ⎥ n ⎝ ⎠⎦ i =1 ⎣ 2 n 4 n 2 4 ⎡ n(n + 1) ⎤ i = (n) + 1+ ⎢ ⎥ n i =1 n 2 i =1 n n2 ⎣ 2 ⎦ ⎛ 1⎞ = 2 + 2 ⎜1 + ⎟ ⎝ n⎠ 2 0 ⎡ ⎛ 1 ⎞⎤ ( x + 1)dx = lim ⎢ 2 + 2 ⎜1 + ⎟ ⎥ = 4 n →∞ ⎣ ⎝ n ⎠⎦ Instructor’s Resource Manual Section 4.2 259 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 12. Δx = 2 2i , xi = n n 14. Δx = 2 2 4i ⎛ 2i ⎞ f ( xi ) = ⎜ ⎟ + 1 = +1 ⎝n⎠ n2 n n ⎡ ⎛ 4 ⎞⎤ 2 f ( xi )Δx = ⎢1 + i 2 ⎜ ⎟ ⎥ ⎝ n2 ⎠⎦ n i =1 i =1 ⎣ = 2 n 8 n 2 2 8 i = ( n) + 1+ 3 n i =1 n i =1 n n3 ⎡ n(n + 1)(2n + 1) ⎤ ⎢ ⎥ 6 ⎣ ⎦ 4⎛ 3 1 ⎞ = 2+ ⎜2+ + ⎟ 3⎝ n n2 ⎠ 2 0 3 3i , xi = −2 + n n 3i ⎞ 6i ⎛ f ( xi ) = 2 ⎜ −2 + ⎟ + π = π − 4 + n⎠ n ⎝ 13. Δx = i =1 = f ( xi )Δx = 2 3i ⎞ 36i 27i 2 ⎛ f ( xi ) = 3 ⎜ −2 + ⎟ + 2 = 14 − + n⎠ n ⎝ n2 n n ⎡ ⎛ 36 ⎞ ⎛ 27 ⎞ ⎤ 3 f ( xi )Δx = ⎢14 − ⎜ ⎟ i + ⎜ ⎟ i 2 ⎥ ⎝ n ⎠ ⎝ n2 ⎠ ⎦ n i =1 i =1 ⎣ = 3 n 108 n 81 n 2 i+ i 14 − n i =1 n 2 i =1 n3 i =1 = 42 − ⎡ 4⎛ 3 1 ⎞ ⎤ 14 ( x 2 + 1) dx = lim ⎢ 2 + ⎜ 2 + + ⎟⎥ = n n2 ⎠ ⎦ 3 3⎝ n →∞ ⎣ n n 6i ⎤ 3 ⎡ ⎢π − 4 + n ⎥ n ⎦ i =1 ⎣ 3 n 18 n 18 ⎡ n(n + 1) ⎤ i = 3(π − 4) + (π − 4) + ⎢ ⎥ 2 n i =1 n i =1 n2 ⎣ 2 ⎦ ⎛ 1⎞ = 3π − 12 + 9 ⎜ 1 + ⎟ ⎝ n⎠ ⎡ ⎛ 1 ⎞⎤ (2 x + π) dx = lim ⎢3π − 12 + 9 ⎜ 1 + ⎟ ⎥ −2 n →∞ ⎣ ⎝ n ⎠⎦ = 3π − 3 108 ⎡ n(n + 1) ⎤ 81 ⎡ n(n + 1)(2n + 1) ⎤ ⎢ ⎥+ ⎢ ⎥ 6 ⎦ n 2 ⎣ 2 ⎦ n3 ⎣ 3 1 ⎞ ⎛ 1 ⎞ 27 ⎛ = 42 − 54 ⎜1 + ⎟ + ⎜ 2 + + ⎟ n n2 ⎠ ⎝ n⎠ 2 ⎝ 1 −2 (3 x 2 + 2) dx ⎡ 3 1 ⎞⎤ ⎛ 1 ⎞ 27 ⎛ = lim ⎢ 42 − 54 ⎜1 + ⎟ + ⎜ 2 + + ⎟ ⎥ = 15 n n2 ⎠ ⎦ n→∞ ⎣ ⎝ n⎠ 2 ⎝ 5 5i , xi = n n 5i f ( xi ) = 1 + n 15. Δx = n i =1 1 = f ( xi )Δx = n ⎡ ⎛ 5 ⎞⎤ 5 ⎢1 + i ⎜ n ⎟ ⎥ n ⎝ ⎠⎦ i =1 ⎣ 5 n 25 n 25 ⎡ n(n + 1) ⎤ i = 5+ 1+ ⎢ ⎥ n i =1 n 2 i =1 n2 ⎣ 2 ⎦ = 5+ 5 0 16. Δx = 3 3i , xi = −2 + n n 25 ⎛ 1 ⎞ ⎜1 + ⎟ 2 ⎝ n⎠ ⎡ 25 ⎛ 1 ⎞ ⎤ 35 ( x + 1) dx = lim ⎢5 + ⎜1 + ⎟ ⎥ = 2 ⎝ n ⎠⎦ 2 n →∞ ⎣ 20 20i , xi = −10 + n n 2 20i ⎞ ⎛ 20i ⎞ 380i 400i 2 ⎛ f ( xi ) = ⎜ −10 + + ⎟ + ⎜ −10 + ⎟ = 90 − n ⎠ ⎝ n ⎠ n ⎝ n2 n n ⎡ 20 n 7600 n 8000 n 2 ⎛ 380 ⎞ 2 ⎛ 400 ⎞ ⎤ 20 f ( xi )Δx = ⎢90 − i ⎜ i = − i + i 90 + ⎜ 2 ⎟⎥ ⎟ n i =1 ⎝ n ⎠ n 2 i =1 n3 i =1 ⎝ n ⎠⎦ n i =1 i =1 ⎣ = 1800 − 7600 ⎡ n(n + 1) ⎤ 8000 ⎡ n(n + 1)(2n + 1) ⎤ 3 1 ⎞ ⎛ 1 ⎞ 4000 ⎛ ⎥+ 3 ⎢ ⎥ = 1800 − 3800 ⎜ 1 + n ⎟ + 3 ⎜ 2 + n + 2 ⎟ 2 ⎢⎣ 2 6 ⎝ ⎠ ⎦ ⎣ ⎦ n n n ⎠ ⎝ ⎡ 3 1 ⎞ ⎤ 2000 ⎛ 1 ⎞ 4000 ⎛ 2+ + ( x 2 + x) dx = lim ⎢1800 − 3800 ⎜1 + ⎟ + ⎜ ⎟⎥ = −10 n n2 ⎠ ⎦ 3 ⎝ 3 n→∞ ⎣ ⎝ n⎠ 10 260 Section 4.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 21. The area under the curve is equal to the area of a 17. semi-circle: 5 0 1 1 27 (1)(2) + 1(2) + 3(2) + (3)(3) = 2 2 2 = 18. f ( x) dx A −A A2 − x 2 dx = 12 π A2 . 22. The area under the curve is equal to the area of a triangle: y 4 2 4 −4 2 2 0 f ( x) dx = 4 x 1 1 9 (1)( 3) + (1)( 2 ) + (1)( 2 ) = 2 2 2 ⎛1⎞ f ( x ) dx = 2 ⎜ ⎟ 4 ⋅ 4 = 16 ⎝2⎠ 23. s ( 4 ) = 24. s ( 4 ) = 19. 25. s ( 4 ) = 26. s ( 4 ) = 27. 2 0 f ( x) dx = 1 1 1 π (π ⋅12 ) + (1)(1) = + 4 2 2 4 20. 4 0 4 0 4 0 4 0 v ( t ) dt = 1 ⎛ 4 ⎞ 2 4⎜ ⎟ = 2 ⎝ 60 ⎠ 15 1 v ( t ) dt = 4 + 4 ( 9 − 1) = 20 2 v ( t ) dt = 1 2 (1) + 2 (1) = 3 2 1 2 v ( t ) dt = π ( 2 ) + 0 = π 4 t s(t) 20 40 40 80 60 120 80 160 100 200 120 240 1 1 f ( x) dx = − (π ⋅ 22 ) − (2)(2) − (2)(4) 4 2 = −π − 8 2 −2 Instructor’s Resource Manual Section 4.2 261 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. t 28. s(t) 20 10 40 40 60 90 80 160 100 250 120 360 3 e. 3 f. g. 2 30. 31. a. b. t s(t) 20 20 40 80 60 160 80 240 100 320 120 400 s(t) 20 20 −1 2 d. 0 120 -100 0 x dx + 2 1 x dx x 2 x dx = − 0 −1 x 2 dx + 0 1 2 0 x dx 2 2 1 x dx 13 ⎛ 23 13 ⎞ +⎜ − ⎟ = 2 3 ⎜⎝ 3 3 ⎟⎠ f ( x) dx = 0 because this is an odd −1 1 −1 1 −1 g ( x ) dx = 3 + 3 = 6 f ( x) dx = 3 + 3 = 6 [ − g ( x)] dx = −3 + (−3) = −6 xg ( x) dx = 0 because xg(x) is an odd function. 1 f. −1 f 3 ( x) g ( x) dx = 0 because f 3 ( x) g ( x) is an odd function. 33. RP = x dx = (−3 − 2 − 1 + 0 + 1 + 2)(1) = −3 2 (x − (x − −3 3 1 e. 100 3 1 −1 d. 60 −3 −1 c. ⎡1 ⎤ x ) dx = 6 ⎢ (1)(1) ⎥ = 3 ⎣2 ⎦ x ) 2 1 dx = 6 x 2 dx = 6 ⋅ 0 = 1 n ( xi + xi −1 )( xi − xi −1 ) 2 i =1 1 n 2 xi − xi2−1 2 i =1 ( ) 1⎡ 2 ( x1 − x02 ) + ( x22 − x12 ) + ( x32 − x22 ) ⎣ 2 + + ( xn2 − xn2−1 ) ⎤ ⎦ 1 2 = ( xn − x02 ) 2 1 2 = (b − a 2 ) 2 1 1 lim (b 2 − a 2 ) = (b 2 − a 2 ) 2 n →∞ 2 = dx = [(−3)2 + (−2) 2 +(−1) 2 + 0 + 1 + 4](1) = 19 c. 1 b. 80 −3 1 function. 80 x −1 32. a. 60 3 −1 x dx + 0 + 60 −3 0 1 1 = − (1)(1) + 1(1) + (1)(1) = 1 2 2 40 3 (−3)3 (3)3 + =0 3 3 x x dx = − =− t 1 1 (3)(3) + (3)(3) = 9 2 2 x x dx = −3 h. 29. x dx = −3 13 =2 3 262 Section 4.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ( ) 12 ⎡1 ⎤ 34. Note that xi = ⎢ xi2−1 + xi −1 xi + xi2 ⎥ ⎣3 ⎦ 1/ 2 ⎡1 ⎤ ≥ ⎢ ( xi2−1 + xi2−1 + xi2−1 ⎥ ⎣3 ⎦ = xi −1 and 1/ 2 ⎡1 ⎤ xi = ⎢ ( xi2−1 + xi −1 xi + xi2 ⎥ ⎣3 ⎦ 1/ 2 ⎡1 ⎤ ≤ ⎢ ( xi2 + xi2 + xi2 ) ⎥ ⎣3 ⎦ Rp = n = xi . xi2 Δxi i =1 n 1 2 = ( xi + xi −1 xi + xi2−1 )( xi − xi −1 ) 3 i =1 = n 1 ( xi3 − xi3−1 ) 3 i =1 1 = ⎡ ( x13 − x03 ) + ( x23 − x13 ) + ( x33 − x23 ) 3⎣ + + ( xn3 − xn3−1 ) ⎤ ⎦ 1 3 1 = ( xn − x03 ) = (b3 − a3 ) 3 3 35. Left: 2 0 ( x3 + 1) dx = 5.24 2 Right: 0 Midpoint: 36. Left: Right: 1 0 ( x3 + 1) dx = 6.84 2 0 1 37. Left: 0 cos x dx ≈ 0.8638 1 Right: cos x dx ≈ 0.8178 0 Midpoint: 1 0 cos x dx ≈ 0.8418 ⎛1⎞ ⎜ ⎟ dx ≈ 1.1682 ⎝x⎠ 3 ⎛1⎞ Right: dx ≈ 1.0349 1 ⎜⎝ x ⎟⎠ 3 ⎛1⎞ Midpoint: dx ≈ 1.0971 1 ⎜⎝ x ⎟⎠ 38. Left: 3 1 39. Partition [0, 1] into n regular intervals, so 1 P = . n i 1 If xi = + , f ( xi ) = 1 . n 2n n f ( xi )Δxi = lim If xi = i 1 + , f ( xi ) = 0 . n πn P →0 i =1 n lim P →0 i =1 n 1 =1 n →∞ i =1 n lim f ( xi )Δxi = lim n n →∞ i =1 0=0 Thus f is not integrable on [0, 1]. ( x3 + 1) dx = 5.98 tan x dx ≈ 0.5398 1 0 Midpoint: tan x dx ≈ 0.6955 1 0 tan x dx ≈ 0.6146 Instructor’s Resource Manual Section 4.2 263 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4.3 Concepts Review 5. A( x ) = 1 ax 2 x ( ax ) = 2 2 6. A( x ) = 1 1 2 ( x − 2)(−1 + x / 2) = ( x − 2 ) , x ≥ 2 2 4 1. 4(4 – 2) = 8; 16(4 – 2) = 32 2. sin 3 x 3. 4 1 f ( x) dx ; 5 2 x dx 4. 5 Problem Set 4.3 1. A( x) = 2 x 7. 2. A( x) = ax 3. A( x) = 1 2 ( x − 1)2 , ⎧2 x ⎪ ⎪⎪2 + ( x − 1) A( x ) = ⎨3 + 2( x − 2) ⎪5 + ( x − 3) ⎪ ⎪⎩etc. 0 ≤ x ≤1 1< x ≤ 2 2< x≤3 3< x≤ 4 x ≥1 8. 4. If 1 ≤ x ≤ 2 , then A( x) = If 2 ≤ x , then A( x) = x − 1 2 3 2 ( x − 1)2 . ⎧ 1 x2 ⎪2 ⎪ 1 + 1 (3 − x)( x − 1) ⎪2 2 ⎪1 + 1 ( x − 2) 2 ⎪ A( x ) = ⎨ 2 ⎪ 3 + 1 (5 − x)( x − 3) ⎪2 2 ⎪2+ 1 ( x − 4)2 ⎪ 2 ⎪⎩etc. 0 ≤ x ≤1 1< x ≤ 2 2< x≤3 3< x≤ 4 4< x≤5 264 Section 4.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2 9. 10. 2 f ( x) dx = 2 1 2 0 2 f ( x) dx = 2 = 2 ⎡⎢ ⎣ 11. 2 0 1 0 f ( x) dx + 2 f ( x ) dx = 2(3) = 6 1 2 0 f ( x) dx 2 1 f ( x)dx ⎤⎥ = 2(2 + 3) = 10 ⎦ 2 [ 2 f ( x) + g ( x)] dx = 2 f ( x) dx + 0 1 2 = 2 ⎡⎢ f ( x) dx + f ( x) dx ⎤⎥ + 0 1 ⎣ ⎦ = 2(2 + 3) + 4 = 14 12. 1 1 0 0 [2 f ( s ) + g ( s )] ds = 2 2 0 f ( s ) ds + 2 0 g ( x) dx g ( x) dx 1 0 1 2 [2 f ( s ) + 5 g ( s )] ds = −2 2 = −2(3) − 5 ⎡⎢ g ( s ) ds − ⎣ 0 = –6 – 5[4 + 1] = –31 14. 15. 1 1 2 0 2 1 1 0 x x 22. G ( x) = Dx ⎡⎢ xt dt ⎤⎥ = Dx ⎡⎢ x t dt ⎤⎥ ⎣1 ⎦ ⎣ 1 ⎦ x ⎡ ⎡ 2⎤ ⎤ ⎡ ⎛ x2 − 1 ⎞⎤ t = Dx ⎢ x ⎢ ⎥ ⎥ = Dx ⎢ x ⎜ ⎟⎥ ⎜ 2 ⎟⎥ ⎢ ⎢2⎥ ⎥ ⎢ ⎣ ⎦ ⎝ ⎠ ⎣ ⎦ 1 ⎣⎢ ⎦⎥ ⎛ x3 x ⎞ 3 1 = Dx ⎜ − ⎟ = x 2 − ⎜ 2 2⎟ 2 2 ⎝ ⎠ g ( s ) ds = 2(2) + (–1) = 3 13. π/4 21. G ( x) = Dx ⎡⎢ ( s − 2) cot(2 s)ds ⎤⎥ x ⎣ ⎦ x ⎡ ⎤ = Dx ⎢ − ( s − 2) cot(2 s )ds ⎥ ⎣ π/4 ⎦ = −( x − 2) cot(2 x) f ( s ) ds − 5 2 1 g ( s ) ds ⎡ 23. G ( x ) = Dx ⎢ ⎢⎣ ⎡ 24. G ( x ) = Dx ⎢ ⎣⎢ g ( s ) ds ⎤⎥ ⎦ 2 0 [3 f ( x) + 2 g ( x)] dx = 0 25. = dt x 1 = 2t dt ⎤⎥ = 2 x ⎦ 1 18. G ( x) = Dx ⎡⎢ 2t dt ⎤⎥ = Dx ⎡⎢ − x ⎣ ⎦ ⎣ x 1 2t dt ⎤⎥ = −2 x ⎦ ( 2t 2 + t ) dt ⎤⎥⎦ = 2x2 + 19. G ( x) = Dx ⎡⎢ ⎣ 0 20. G ( x) = Dx ⎡⎢ ⎣ cos3 (2t ) tan(t ) dt ⎤⎥ 1 ⎦ x x = cos3 (2 x) tan( x) t2 1+ t2 t2 0 − x2 1 + t 2 dt dt + t2 − x2 1+ t2 0 x t2 0 1+ t2 dt + dt x t2 0 1+ t2 dt 2 = 3 (2 + 3) + 2(4) + 2π = 5 3 + 4 2 + 2π 17. G ( x) = Dx ⎡⎢ ⎣ 1 − x2 ) ( x2 G '( x) = − −2 x ) + ( 2 1 + x2 1 + ( − x2 ) 1 2 2 = 3 ⎡⎢ f (t ) dt + f (t ) dt ⎤⎥ + 2 g (t ) dt 0 1 0 ⎣ ⎦ 2 − x2 =− ⎡ 3 f (t ) + 2 g (t ) + π ⎤ dt ⎣ ⎦ 0 ⎤ 2 z + sin z dz ⎥ ⎦⎥ x2 + x x G ( x) = [3 f (t ) + 2 g (t )] dt +π 1 = (2 x + 1) 2( x 2 + x) + sin( x 2 + x) 1 2 2 = 3 ⎡⎢ f (t ) dt + f (t ) dt ⎤⎥ + 2 g (t ) dt 1 0 ⎣ 0 ⎦ = 3(2 + 3) + 2(4) = 23 16. ⎤ sin t dt ⎥ = 2 x sin( x 2 ) ⎥⎦ x2 x 2 x5 1 + x4 + x2 1 + x2 sin x 5 ⎤ 26. G ( x) = Dx ⎡⎢ t dt ⎥ cos x ⎣ ⎦ sin x 5 0 ⎡ = Dx ⎢ t dt + t 5 dt ⎤⎥ 0 cos x ⎣ ⎦ sin x 5 cos x 5 ⎤ ⎡ = Dx ⎢ t dt − t dt ⎥ 0 ⎣ 0 ⎦ = sin 5 x cos x + cos5 x sin x 27. f ( x) = x 1 + x2 ; f ( x) = 1 ( x + 1) 2 3/ 2 So, f(x) is increasing on [0, ∞) and concave up on (0, ∞ ). Instructor’s Resource Manual Section 4.3 265 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 28. 35. 1+ x f ( x) = f (1 + x ) − (1 + x ) 2 x = − x + 2 x − 1 ( x) = ( x + 1) ( x + 1) 1+ x 2 2 2 2 2 2 2 So, f(x) is increasing on [0, ∞ ) and concave up on ( 0, −1 + 2 ) . 29. ( x ) = cos x; f f ( x ) = − sin x 4 ⎡ π ⎤ ⎡ 3π 5π ⎤ So, f(x) is increasing on ⎢0, ⎥ , ⎢ , ⎥ ,... and ⎣ 2⎦ ⎣ 2 2 ⎦ concave up on (π , 2π ) , ( 3π , 4π ) ,... . 30. ( x ) = x + sin x; f f f ( x) dx = 0 2 0 (2 − x)dx + 4 2 ( x − 2) dx = 2+2 = 4 36. ( x ) = 1 + cos x So, f(x) is increasing on ( 0, ∞ ) and concave up on ( 0, ∞ ) . 31. 1 1 ; f ( x) = − 2 x x So, f(x) is increasing on (0, ∞) and never concave up. ( x) = f 32. f(x) is increasing on x ≥ 0 and concave up on ( 0,1) , ( 2,3) ,... 4 0 = = 37. a. 33. ( 3 + x − 3 ) dx 3 4 ( 3 + x − 3 ) dx + 3 ( 3 + x − 3 ) dx 0 3 0 ( 6 − x ) dx + 4 3 x dx = 27 7 + = 17 2 2 Local minima at 0, ≈ 3.8, ≈ 5.8, ≈ 7.9, ≈ 9.9; local maxima at ≈ 3.1, ≈ 5, ≈ 7.1, ≈ 9, 10 b. Absolute minimum at 0, absolute maximum at ≈ 9 c. ≈ (0.7, 1.5), (2.5, 3.5), (4.5, 5.5), (6.5, 7.5), (8.5, 9.5) d. 4 0 f ( x) dx = 2 0 4 2 dx + 2 x dx = 4 + 6 = 10 34. 38. a. Local minima at 0, ≈ 1.8, ≈ 3.8, ≈ 5.8; local maxima at ≈ 1, ≈ 2.9, ≈ 5.2, ≈ 10 b. Absolute minimum at 0, absolute maximum at 10 c. 4 0 f ( x) dx = 1 0 dx + 2 1 x dx + 4 2 (4 − x) dx (0.5, 1.5), (2.2, 3.2), (4.2,5.2), (6.2,7.2), (8.2, 9.2) = 1 + 1.5 + 2.0 = 4.5 266 Section 4.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. d. 39. a. b. f. 0 F (0) = 0 ( t 4 + 1) dt = 0 41. 1 ≤ 1 + x4 ≤ 1 + x4 . y = F ( x) 1 dy = F '( x) = x 4 + 1 dx 0 ( 1 dx ≤ dy = x + 1 dx y = 15 x5 + x + C c. 42. 0 0 = 15 05 + 0 + C 1 40. a. G ( x) = G (0) = 0 0 0 G (2π ) = b. 1 0 ( x4 + 1) dx = F (1) = 15 15 + 1 = 65 . x 1 0 1 0 4 + x 2 dx ≤ 1 sin t dt 43. sin t dt = 0 0 1 + x 4 dx ≤ 6 5 1 0 ( 4 + x2 ) dx 4 + x 2 dx ≤ ( 4 + x4 ) dx = 01( 3 + 1 + x4 ) dx 1 1 = 3 dx + (1 + x 4 ) dx 0 0 = 3 + 65 = 2π (1 + x 4 ) dx 21 0 5 Here, we have used the result from problem 39: Thus y = F ( x) = 15 x5 + x 0 2 dx ≤ 2≤ C=0 d. 0 On the interval [0,1], 2 ≤ 4 + x 4 ≤ 4 + x 4 . Thus 1 Now apply the initial condition y (0) = 0 : 1 1 + x 4 dx ≤ 0 By problem 39d, 1 ≤ ) 4 t ≤ t . Since 1 + x 4 ≥ 1 for all x, For t ≥ 1 , 5 ≤ f ( x) ≤ 69 so 4⋅5 ≤ sin t dt = 0 21 5 20 ≤ ( 5 + x3 ) dx ≤ 4 ⋅ 69 5 + x3 ) dx ≤ 276 0( 4 0 4 Let y = G ( x) . Then dy = G '( x) = sin x . dx dy = sin x dx y = − cos x + C c. d. Apply the initial condition 0 = y (0) = − cos 0 + C . Thus, C = 1 , and hence y = G ( x) = 1 − cos x . π 0 e. sin x dx = G (π ) = 1 − cos π = 2 44. 5 On [2,4], 85 ≤ ( x + 6 ) ≤ 105 . Thus, 2 ⋅ 85 ≤ 4 2 65,536 ≤ ( x + 6 )5 dx ≤ 2 ⋅105 4 2 ( x + 6 )5 dx ≤ 200, 000 G attains the maximum of 2 when x = π ,3π . G attains the minimum of 0 when x = 0, 2π , 4π Inflection points of G occur at π 3π 5π 7π x= , , , 2 2 2 2 Instructor’s Resource Manual Section 4.3 267 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 45. On [1,5], 2 2 2 3+ ≤ 3+ ≤ 3+ 5 1 x 5⎛ 2⎞ ⎛ 17 ⎞ 4 ⎜ ⎟ ≤ ⎜ 3 + ⎟ dx ≤ 4 ⋅ 5 1 5 x⎠ ⎝ ⎠ ⎝ 5⎛ 68 2⎞ ≤ ⎜ 3 + ⎟ dx ≤ 20 1 5 x⎠ ⎝ 48. On [0.2,0.4], 0.002 + 0.0001cos 2 0.4 ≤ 0.002 + 0.0001cos 2 x ≤ 0.002 + 0.0001cos 2 0.2 ( 0.2 0.002 + 0.0001cos 2 0.4 ) ( 0.002 + 0.0001cos2 x ) dx ≤ 0.2 ( 0.002 + 0.0001cos 2 0.2 ) ≤ 0.4 0.2 Thus, 0.000417 ≤ 0.4 0.2 ( 0.002 + 0.0001cos2 x ) dx ≤ 0.000419 46. On [10, 20], 5 5 1 ⎞ ⎛ 1⎞ ⎛ 1⎞ ⎛ ⎜1 + ⎟ ≤ ⎜ 1 + ⎟ ≤ ⎜1 + ⎟ 20 x 10 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 5 5 5 20 ⎛ 1⎞ ⎛ 21 ⎞ ⎛ 11 ⎞ 10 ⎜ ⎟ ≤ ⎜ 1 + ⎟ dx ≤ 10 ⎜ ⎟ 10 ⎝ 20 ⎠ ⎝ x⎠ ⎝ 10 ⎠ 5 5 20 ⎛ 4, 084,101 1⎞ 161, 051 ≤ 1 + ⎜ ⎟ dx ≤ 10 320, 000 10, 000 ⎝ x⎠ 12.7628 ≤ 20 ⎛ 49. lim On [ 4π ,8π ] 1 sin 2 x ≤ 5 + 1 5 ≤ 5 + 20 20 ( 4π ) (5) ≤ 20π ≤ (5 + 201 sin 2 x ) dx ≤ ( 4π ) (5 + 201 ) 8π 1 sin 2 x dx ≤ 101 π 5 + 20 ) 5 4π ( 8π 4π 51. lim x 1+ t 0 x 1+ t 1 x →0 x 5 1⎞ 1 + ⎟ dx ≤ 16.1051 ⎜ 10 ⎝ x⎠ 50. 47. Let F ( x) = 0 1 2+t 2+t dt . Then F ( x) − F (0) x−0 x →0 1+ 0 1 = F '(0) = = 2+0 2 dt = lim x 1+ t dt 2+t 1 ⎡ x 1+ t = lim dt − ⎢ x →1 x − 1 ⎣ 0 2 + t F ( x) − F (1) = lim x −1 1 x→ 1+1 2 = F '(1) = = 2 +1 3 x →1 x − 1 1 x 1 1 1+ t 0 2+t ⎤ dt ⎥ ⎦ f (t ) dt = 2 x − 2 Differentiate both sides with respect to x: d x d f (t ) dt = ( 2 x − 2 ) dx 1 dx f ( x) = 2 If such a function exists, it must satisfy f ( x) = 2 , but both sides of the first equality may differ by a constant yet still have equal derivatives. When x = 1 the left side is 1 1 f (t ) dt = 0 and the right side is 2 ⋅1 − 2 = 0 . Thus the function f ( x) = 2 satisfies x 1 f (t ) dt = 2 x − 2 . 268 Section 4.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 52. x 0 59. f (t ) dt = x 2 = Differentiate both sides with respect to x: d x d 2 f (t ) dt = x 0 dx dx f ( x) = 2 x 53. x2 0 f (t ) dt = 60. 61. 1 x3 3 Differentiate both sides with respect to x: d x2 d 1 3 f (t ) dt = x 0 dx dx 3 ( ) 2 f ( x) = No such function exists. When x = 0 the left side is 0, whereas the right side is 1 55. True; by Theorem B (Comparison Property) 56. False. a = –1, b = 2, f(x) = x is a counterexample. 57. False. a = –1, b = 1, f(x) = x is a counterexample. 62. not identically zero. ⎧ t ⎪ 5 du, ⎪ 0 ⎪⎪ 100 5 du + a. s ( t ) = ⎨ 0 ⎪ ⎪ 100 5 du + ⎪ ⎩⎪ 0 2 0 f ( x)dx − b a g ( x ) dx ( x) − g ( x )]dx ⎧⎪2 + ( t − 2 ) , t ≤ 2 v (t ) = ⎨ ⎪⎩ 2 − ( t − 2 ) , t > 2 t≤2 ⎧ t, =⎨ − >2 4 , t t ⎩ t 0 v ( u ) du ⎧ t 0≤t ≤2 ⎪ 0 u du , =⎨ t 2 ⎪ u du + ( 4 − u ) du, t > 2 2 ⎩ 0 ⎧t2 0≤t ≤2 ⎪ , ⎪2 =⎨ 2 ⎪2 + ⎡⎢ 4t − t ⎤⎥ , t > 2 ⎪ ⎢ 2 ⎥⎦ ⎩ ⎣ ⎧ t2 , ⎪ ⎪ 2 =⎨ t2 ⎪ ⎪⎩−4 + 4t − 2 0≤t ≤2 t>2 t2 − 4t + 4 = 0; t = 4 + 2 2 ≈ 6.83 2 False; A counterexample is f ( x ) = 0 for all x, except f (1) = 1 . Thus, b a False. a = 0, b = 1, f(x) = 0, g(x) = –1 is a counterexample. x 2 54. 58. b a[ f s (t ) = ( ) ( 2x) = x x f ( x2 ) = 2 f x 2 True. f ( x ) dx = 0 , but f is 0 ≤ t ≤ 100 u ⎞ ⎛ 6− du ⎜ 100 ⎝ 100 ⎟⎠ 700 ⎛ u ⎞ 6− ⎟ du + 100 ⎜ ⎝ 100 ⎠ t 100 < t ≤ 700 t 700 ( −1) du, t > 700 ⎧ ⎪ ⎪5t , 0 ≤ t ≤ 100 ⎪ t ⎪ ⎡ u2 ⎤ ⎪ = ⎨500 + ⎢6u − 100 < t ≤ 700 ⎥ 200 ⎦⎥ ⎪ ⎣⎢ 100 ⎪ 700 2 ⎪ ⎡ u ⎤ ⎪500 + ⎢6u − ⎥ − ( t − 700 ) t > 700 200 ⎦⎥ ⎪⎩ ⎣⎢ 100 0 ≤ t ≤ 100 ⎧5t , ⎪ 2 t ⎪ = ⎨−50 + 6t − , 100 < t ≤ 700 200 ⎪ ⎪2400 − t , t > 700 ⎩ Instructor’s Resource Manual Section 4.3 269 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. v ( t ) > 0 for 0 ≤ t < 600 and v ( t ) < 0 for t > 600 . So, t = 600 is the point at which the object is farthest to the right of the origin. At t = 600 , s ( t ) = 1750 . 3. c. s ( t ) = 0 = 2400 − t ; t = 2400 − f ( x ) ≤ f ( x) ≤ f ( x) , so 63. b a b − f ( x) dx ≤ b a a f ( x ) dx ≥ − b a 4. f ( x) dx ⇒ f ( x) dx and combining this with b a f ( x ) dx ≥ b a b f ( x) dx ≤ If x > a , 64. x a b x (3 x 2 − 2 x + 3) dx = ⎡ x3 − x 2 + 3 x ⎤ ⎣ ⎦ −1 −1 = (8 – 4 + 6) – (–1 –1 – 3) = 15 2 2 (4 x3 + 7) dx = ⎡ x 4 + 7 x ⎤ ⎣ ⎦1 1 = (16 + 14) – (1 + 7) = 22 6. ⎡ 1⎤ 8 ⎛ 1⎞ dt = ⎢ − ⎥ = ⎜ − ⎟ − (−1) = 2 1 t3 9 ⎝ 9⎠ ⎣ t ⎦1 1 4 f ( x ) dx ≤ M ( x − a) by the 7. x a f ( x) dx ≥ − M ( x − a ) by 8. x f ( x) dx ≤ M x − a . From Problem 63, x a x a f ( x) dx ≥ 9. x a f ( x) dx . f ( x ) dx = f ( x) − f (a) ≥ f ( x) − f (a) Therefore, f ( x) − f (a) ≤ M x − a or f ( x) ≤ f (a ) + M x − a . 10. 3 2 3 4 16 ⎡2 ⎤ ⎛2 ⎞ t dt = ⎢ t 3 / 2 ⎥ = ⎜ ⋅ 8 ⎟ − 0 = 3 ⎣3 ⎦0 ⎝ 3 ⎠ 4 0 8 3 1 the Boundedness Property. Thus a 4 3 ⎡ 1⎤ ⎛ 1⎞ dw = ⎢ − ⎥ = ⎜ − ⎟ − (−1) = 1 w2 4 ⎝ 4⎠ ⎣ w ⎦1 f ( x) dx a f ( x) dx = − 2 2 5. Boundedness Property. If x < a , a 2 ⎡ x5 ⎤ 32 1 33 + = x 4 dx = ⎢ ⎥ = −1 5 ⎣⎢ ⎦⎥ −1 5 5 5 2 f ( x) dx, we can conclude that a 2. 8 ⎡3 ⎤ ⎛3 ⎞ ⎛ 3 ⎞ 45 w dw = ⎢ w4 / 3 ⎥ = ⎜ ⋅16 ⎟ − ⎜ ⋅1⎟ = ⎝4 ⎠ ⎝4 ⎠ 4 ⎣4 ⎦1 −2 ⎡ y3 ⎛ 2 1 ⎞ 1 ⎤ ⎜⎜ y + 3 ⎟⎟ dy = ⎢ − 2 ⎥ −4 y ⎠ ⎝ ⎣⎢ 3 2 y ⎦⎥ −4 ⎛ 8 1 ⎞ ⎛ 64 1 ⎞ 1783 = ⎜− − ⎟−⎜− − ⎟ = 96 ⎝ 3 8 ⎠ ⎝ 3 32 ⎠ −2 4 s4 − 8 1 s2 ds = 4 1 4 2 ( s − 8s −2 ⎡ s3 8 ⎤ ) ds = ⎢ + ⎥ ⎢⎣ 3 s ⎥⎦1 ⎛ 64 ⎞ ⎛1 ⎞ = ⎜ + 2 ⎟ − ⎜ + 8 ⎟ = 15 ⎝ 3 ⎠ ⎝3 ⎠ 4.4 Concepts Review 11. 1. antiderivative; F(b) – F(a) 12. 2. F(b) – F(a) 3. F (d ) − F (c ) 4. 2 1 13. 1 4 u du 3 Problem Set 4.4 1. 14. 2 ⎡ x4 ⎤ x dx = ⎢ ⎥ = 4 − 0 = 4 0 ⎣⎢ 4 ⎦⎥ 0 2 3 π/2 0 π/2 π/6 π/2 cos x dx = [sin x ]0 =1–0=1 π/2 2sin t dt = [ −2 cos t ]π / 6 = 0 + 3 = 3 1 ⎡2 ⎤ (2 x 4 − 3x 2 + 5) dx = ⎢ x5 − x3 + 5 x ⎥ 0 ⎣5 ⎦0 22 ⎛2 ⎞ = ⎜ −1+ 5⎟ − 0 = 5 5 ⎝ ⎠ 1 1 3 ⎡3 ⎤ ( x 4 / 3 − 2 x1/ 3 ) dx = ⎢ x 7 / 3 − x 4 / 3 ⎥ 0 2 ⎣7 ⎦0 15 ⎛3 3⎞ = ⎜ − ⎟−0 = − 7 2 14 ⎝ ⎠ 1 270 Section 4.4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 15. u = 3x + 2, du = 3 dx 1 2 2 u ⋅ du = u 3 / 2 + C = (3x + 2)3 / 2 + C 3 9 9 16. u = 2x – 4, du = 2 dx 1 3 3 u1/ 3 ⋅ du = u 4 / 3 + C = (2 x − 4)4 / 3 + C 2 8 8 25. u = x 2 + 4, du = 2 x dx 1 1 sin(u ) ⋅ du = − cos u + C 2 2 1 = − cos( x 2 + 4) + C 2 26. u = x3 + 5, du = 3 x 2 dx 17. u = 3x + 2, du = 3 dx 1 1 1 cos(u ) ⋅ du = sin u + C = sin(3 x + 2) + C 3 3 3 18. u = 2x – 4, du = 2 dx 1 1 sin u ⋅ du = − cos u + C 2 2 1 = − cos(2 x − 4) + C 2 19. u = 6x – 7, du = 6dx 1 1 sin u ⋅ du = − cos u + C 6 6 1 = − cos(6 x − 7) + C 6 20. u = πv − 7, du = π dv 1 1 1 cos u ⋅ du = sin u + C = sin(πv − 7) + C π π π 1 1 1 cos u ⋅ du = sin u + C = sin( x3 + 5) + C 3 3 3 27. u = x 2 + 4, du = 1 1 1 u ⋅ du = u 3 / 2 + C = ( x 2 + 4)3 / 2 + C 2 3 3 22. u = x3 + 5, du = 3 x 2 dx 1 1 1 u ⋅ du = u10 + C = ( x3 + 5)10 + C 3 30 30 9 23. u = x + 3, du = 2 x dx 1 7 u −12 / 7 ⋅ du = − u −5 / 7 + C 2 10 7 2 = − ( x + 3)−5 / 7 + C 10 2 24. u = 3 v + π, du = 2 3v dv u7 / 8 ⋅ = 4 15 2 3 ( 3 du = 4 15 3 3 v2 + π ) dx 3 28. u = z 2 + 3, du = 2z dz 2 3 2 ⎛ ⎞ 3⎜ z + 3 ⎟ ⎝ ⎠ 3 3 3 3 cos u ⋅ du = sin u + C = sin z 2 + 3 + C 2 2 2 29. u = ( x3 + 5)9 , du = 9( x3 + 5)8 (3x 2 )dx = 27 x 2 ( x3 + 5)8 dx cos u ⋅ 1 1 du = sin u + C 27 27 1 sin ⎡( x3 + 5)9 ⎤ + C ⎣ ⎦ 27 30. u = (7 x 7 + π)9 , du = 441x 6 (7 x 7 + π)8 dx sin u ⋅ =− 1 1 du = − cos u + C 441 441 1 cos(7 x 7 + π)9 + C 441 31. u = sin( x 2 + 4), du = 2 x cos( x 2 + 4) dx 2 1 x +4 sin u du = − cos u + C = − cos x 2 + 4 + C = 21. u = x 2 + 4, du = 2 x dx x 2 15 / 8 u15 / 8 + C +C 1 1 u ⋅ du = u 3 / 2 + C 2 3 3/ 2 1⎡ = sin( x 2 + 4) ⎤ +C ⎣ ⎦ 3 32. u = cos(3 x7 + 9) du = −21x 6 sin(3 x7 + 9) dx 1 ⎛ 1 ⎞ u ⋅ ⎜ − ⎟ du = − u 4 / 3 + C 21 28 ⎝ ⎠ 4/3 1 ⎡ =− +C cos(3x 7 + 9) ⎤ ⎦ 28 ⎣ 3 Instructor’s Resource Manual Section 4.4 271 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 33. u = cos( x3 + 5), du = −3 x 2 sin( x3 + 5) dx 41. 1 ⎛ 1⎞ u 9 ⋅ ⎜ − ⎟ du = − u10 + C 3 30 ⎝ ⎠ 1 = − cos10 ( x3 + 5) + C 30 34. 3 42. u = x 2 + 1, du = 2 x dx 0 ( x 2 + 1)10 (2 x)dx = 2 10 u du 1 25 2 ⎡ u11 ⎤ =⎢ ⎥ ⎣⎢ 11 ⎦⎥1 3 x2 + 1 1 x3 + 3 x 1 3 16 −1/ 2 u du 4 43. −1 0 1 0 1 u = t + 2, du = dt 3 −1 1 2 5 −2 u du 1 dt = (t + 2) 4 ⎡ 1⎤ = ⎢ − ⎥ − [ −1] = 5 ⎣ 5⎦ 38. 44. 5 ⎡ 1⎤ = ⎢− ⎥ ⎣ u ⎦1 39. 9 π/2 = dx 36 cos 2 x sin x dx = − π /2 0 cos 2 x ( − sin x ) dx 0 sin 2 3 x cos 3x dx 1 π /2 2 1 sin 3 x ( 3cos 3 x ) dx = 3 0 3 −1 2 u du 0 −1 ⎡ u3 ⎤ 1 ⎛ 1⎞ = ⎢ ⎥ = ⎜− ⎟−0 = − 9 ⎝ 9⎠ ⎢⎣ 9 ⎥⎦ 0 9 u = 3x + 1, du = 3 dx 8 1 8 1 3x + 1 dx = 3 x + 1 ⋅ 3dx = 5 5 3 3 45. u = x 2 + 2 x, du = (2 x + 2) dx = 2( x + 1) dx 1 ( x + 1)( x 2 + 2 x)2 dx 0 11 = 25 16 u du = 25 ⎡2 ⎤ ⎡2 ⎤ ⎡2 ⎤ 122 = ⎢ u 3 / 2 ⎥ = ⎢ (125) ⎥ − ⎢ (64) ⎥ = 9 9 9 9 ⎣ ⎦16 ⎣ ⎦ ⎣ ⎦ 40. x3 + 3 x u = sin 3 x, du = 3cos 3 x dx 0 ⎡2 ⎤ udu = ⎢ u 3 / 2 ⎥ 2 1 3 ⎣ ⎦1 2 2 52 ⎡ ⎤ ⎡ ⎤ = ⎢ (27) ⎥ − ⎢ (1) ⎥ = ⎣3 ⎦ ⎣3 ⎦ 3 y − 1 dy = 1 ⎡2 ⎤ = ⎢ u1/ 2 ⎥ ⎣3 ⎦4 0 2 u du 1 u = y – 1, du = dy 10 3x2 + 3 3 ⎡ u3 ⎤ =− = ⎢− ⎥ ⎢⎣ 3 ⎥⎦1 ⎛ 1⎞ 1 = 0−⎜− ⎟ = ⎝ 3⎠ 3 ⎡2 ⎤ udu = ⎢ u 3 / 2 ⎥ 3 ⎣ ⎦0 ⎛2 ⎞ ⎛2 ⎞ 2 = ⎜ ⋅13 / 2 ⎟ − ⎜ ⋅ 0 ⎟ = ⎝3 ⎠ ⎝3 ⎠ 3 37. 1 3 u = cos x, du = − sin x dx π/2 x3 + 1 (3x 2 ) dx = dx = ⎛2 ⎞ ⎛2 ⎞ 8 = ⎜ ⋅6⎟ − ⎜ ⋅ 2⎟ = ⎝3 ⎠ ⎝3 ⎠ 3 u = x3 + 1, du = 3 x 2 dx 0 ⎡4 ⎤ u du = ⎢ u 3 / 2 ⎥ 3 ⎣ ⎦ 25 u = x3 + 3x, du = (3 x 2 + 3) dx = 2047 ⎡1 ⎤ ⎡1 ⎤ = ⎢ (2)11 ⎥ − ⎢ (1)11 ⎥ = 11 11 11 ⎣ ⎦ ⎣ ⎦ 36. 7 + 2t 2 ⋅ ( 4t ) dt −3 25 ⎡4 ⎤ ⎡4 ⎤ = ⎢ (125) ⎥ − ⎢ (125) ⎥ = 0 ⎣3 ⎦ ⎣3 ⎦ u = tan( x −3 + 1) , du = −3 x −4 sec2 ( x −3 + 1) dx 1 25 =2 5 ⎛ 1⎞ u ⋅ ⎜ − ⎟ du = − u 6 / 5 + C 3 18 ⎝ ⎠ 6/5 5 ⎡ =− tan( x −3 + 1) ⎤ +C ⎣ ⎦ 18 3 7 + 2t 2 (8t ) dt = 2 −3 5 35. u = 7 + 2t 2 , du = 4t dt u = 2x + 2, du = 2 dx 7 1 1 7 2 dx = dx 1 2x + 2 2 1 2x + 2 16 1 16 −1/ 2 u du = ⎡⎣ u ⎤⎦ = 4 − 2 = 2 = 4 2 4 46. 02 1 2 ( x 2 + 2 x)2 2( x + 1) dx 3 2 u 0 3 ⎡ u3 ⎤ 9 du = ⎢ ⎥ = 6 ⎣⎢ ⎦⎥ 0 2 u = x − 1, du = 4 1 =2 ( x − 1)3 x 1 3 u du 0 1 2 x dx = 2 4 1 dx ( x − 1)3 2 x dx 1 ⎡u4 ⎤ 1 = 2⎢ ⎥ = ⎢⎣ 4 ⎥⎦ 0 2 272 Section 4.4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 47. u = sin , du = cos d 1/ 2 3 u du 0 48. 50. 51. ⎡u =⎢ ⎥ ⎣⎢ 4 ⎦⎥ 0 = 56. 3/2 1 1 π 1 π cos u du = [ sin u ]−π = 0 π −π π 57. u = 3 x − 3, du = 3dx 1 0 1 1 0 cos u du = [sin u ]−3 = (0 − sin(−3)) 3 −3 3 3 sin 3 = 3 u = 2πx, du = 2πdx 1 π 1 1 π sin u du = − [ cos u ]0 = − (−1 − 1) 2π 0 2π 2π 1 = π = 58. cos1 3 u du 1 1 − cos 1 8 3 2sin 3 ⎛⎜ π8 ⎞⎟ ⎝ ⎠ = 9 59. a. Between 0 and 3, f ( x) > 0 . Thus, 3 3 0 c. 3 0 − 1 f ''( x) dx = f '(3) − f '(0) = −1 − 0 = −1 < 0 d. Since f is concave down at 0, f ''(0) < 0 . 3 0 f '''( x) dx = f ''(3) − f ''(0) = 0 − (negative number) > 0 60. a. On [ 0, 4] , f ( x) > 0 . Thus, 4 0 f ( x) dx > 0 . b. Since f is an antiderivative of f ' , 4 u = 3 x, du = 3dx; v = 5 x, dv = 5dx 0 f '( x) dx = f (3) − f (0) = 0 − 2 = −2 < 0 2 π5 u = cos x, du = − sin x dx f ( x) dx > 0 . b. Since f is an antiderivative of f ' , 0 55. 1 cos 4 1 1 + 8 8 3 u = 2 x5 , du = 10 x 4 dx 1 3π / 2 1 5π / 2 cos u du + sin v dv 3 / 2 − π 3 5 −5π / 2 1 1 3π / 2 5π / 2 = [sin u ]−3π / 2 − [ cos v ]−5π / 2 3 5 1 1 2 = [(−1) − 1] − [0 − 0] = − 3 5 3 =− 1 sin( π3 / 8) 2 1 ⎡ 3 ⎤ sin( π / 8) 3 / 8) u du = ⎣u ⎦ sin( − π − sin( π3 / 8) 3 9 1 2 π5 cos u du = [sin u ]0 0 10 1 1 = (sin(2π5 ) − 0) = sin(2π5 ) 10 10 54. cos1 4 0 u = 2 x, du = 2dx 1 π/ 2 1 π/2 cos u du + sin u du 2 0 2 0 1 π/2 1 π/2 = [sin u ]0 − [ cos u ]0 2 2 1 1 = (1 − 0) − (0 − 1) = 1 2 2 1 ⎡u4 ⎤ =− ⎢ ⎥ 2 ⎣⎢ 4 ⎦⎥ u = sin( x3 ), du = 3x 2 cos( x3 )dx u = πx 2 , du = 2πx dx 1 10 53. u = cos( x 2 ), du = −2 x sin( x 2 )dx 1 − 2 3/2 1 1⎛4 ⎞ 1 u −3 du = ⎡u −2 ⎤ = ⎜ − 1⎟ = ⎣ ⎦ 1 2 2⎝3 ⎠ 6 1 π 1 1 π sin u du = − [ cos u ]0 = − (−1 − 1) 2π 0 2π 2π 1 = π 52. u = π sin , du = π cos d 1 1 −0 = 64 64 u = cos , du = − sin d − 49. 4 ⎤1/ 2 f '( x) dx = f (4) − f (0) = 1 − 2 = −1 < 0 c. 4 0 f ''( x) dx = f '(4) − f '(0) = d. 4 0 1 9 − (−2) = > 0 4 4 f '''( x) dx = f ''(4) − f ''(0) = ( negative ) − ( positive ) < 0 sin u du = [ cos u ] = 1 − cos1 0 1 Instructor’s Resource Manual Section 4.4 273 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 61. 1 2 t +C 2 V ( 0 ) = C = 0 since no water has leaked out at V (t ) = V (t ) = ( 20 − t ) dt = 20t − 67. a. b n bn x dx + n n a a b. ⎡ t2 ⎤ 219 V ( t ) dt = ⎢t − ⎥ = 0 220 ⎣⎢ ⎦⎥ 0 220 V (10 ) − V ( 9 ) = t ⎞ T ⎜1 − 110 ⎟ dt = T − 220 ⎝ ⎠ = T ≈ 110 hrs 63. Use a midpoint Riemann sum with n = 12 partitions. V= 12 = ≈ 1(5.4 + 6.3 + 6.4 + 6.5 + 6.9 + 7.5 + 8.4 + 8.4 + 8.0 + 7.5 + 7.0 + 6.5) = 84.8 64. 10 i =1 65. 68. 12 i =0 P ( ti ) Δti mass = 2 0 ( bn ⎡ n ( n +1) / n ⎤ y dy = ⎢ y ⎥ n ⎣ n +1 ⎦a ) x Let y = G ( x) = a f (t ) dt . Then G ( x ) = F ( x) − F (a) . Now choose x = b to obtain b a δ ( x) = m ( x) = 1+ bn n an G ( a ) = 0 . Thus, C = − F (a) and ≈ 2(3.0 + 3.0 + 3.8 + 5.8 + 7.8 + 6.9 + 6.5 + 6.3 + 7.2 + 8.2 + 8.7 + 5.4) = 145.2 66. 1 ⎡ n +1 ⎤ b x ⎦a n +1 ⎣ dy = G '( x) = f ( x ) dx dy = f ( x) dx Let F be any antiderivative of f . Then G ( x ) = F ( x ) + C . When x = a , we must have Use a midpoint Riemann sum with n = 12 partitions. E= x n dx = n b n +1 − a n +1 n +1 n (b n +1 − a n +1 ) = An nBn = n +1 = f ( xi ) Δxi ⎛ 6200 + 6300 + 6500 + 6500 + 6600 ⎞ ≈ 1⎜ ⎟ ⎝ + 6700 + 6800 + 7000 + 7200 + 7200 ⎠ = 67, 000 b a 1 (b n +1 − a n +1 ) n +1 An = Use a midpoint Riemann sum with n = 10 partitions. V= n (n + 1)b n +1 − (n + 1)a n +1 = b n +1 − a n +1 n +1 c. Bn = f ( xi ) Δxi i =1 y dy ⎛ b n +1 a n +1 ⎞ ⎛ n n +1 n n +1 ⎞ b − a ⎟ =⎜ − ⎟+ ⎜ n + 1 n + 1 ⎟ ⎜⎝ n + 1 n +1 ⎠ ⎝ ⎠ 2 T⎛ 0 a bn n an x n dx + b t ⎞ 201 ⎜1 − 110 ⎟ dt = 220 ⎝ ⎠ 55 = V (T ) − V ( 0 ) = b y dy = b n +1 − a n +1 b ⎡ x n +1 ⎤ ⎡ n ( n +1) / n ⎤ y =⎢ ⎥ +⎢ ⎥ n ⎦a ⎢⎣ n + 1 ⎥⎦ a ⎣ n + 1 10 ⎛ 9 y dy = An Bn + An = b n +1 − a n +1 . Thus 1 1 bn n an (a )(a n ) + An + Bn = (b)(b n ) or 1 Time to drain: 20t − t 2 = 200; t = 20 hours. 2 V (1) − V ( 0 ) = a x n dx = Bn ; Using Figure 3 of the text, 1 time t = 0 . Thus, V ( t ) = 20t − t 2 , so 2 V ( 20 ) − V (10 ) = 200 − 150 = 50 gallons. 62. b x 4 δ ( x ) dx = m ( 2 ) = 5 2 f (t ) dt = G ( b ) = F (b) − F (a) 3 69. 3 2 x dx 0 ⎡ x3 ⎤ = ⎢ ⎥ = 9−0 = 9 ⎣⎢ 3 ⎦⎥ 0 70. 2 3 x dx 0 ⎡ x4 ⎤ = ⎢ ⎥ = 4−0 = 4 ⎢⎣ 4 ⎥⎦ 0 2 274 Section 4.4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 71. 72. 73. π 0 77. π sin x dx = [ − cos x ]0 = 1 + 1 = 2 First Fundamental Theorem of Calculus. Since G is differentiable at c , G is continuous there. Now suppose c = a . 2 1 1 ⎤ ⎡ (1 + x + x 2 ) dx = ⎢ x + x 2 + x3 ⎥ 0 2 3 ⎦0 ⎣ 8⎞ 20 ⎛ = ⎜2+ 2+ ⎟−0 = 3⎠ 3 ⎝ 2 Then lim G ( x) = lim 2 74 4 (2 −2 1 1 ⎡1 ⎤ = ⎢ x3 ⎥ = = 0.333 ⎣ 3 ⎦0 3 x − 3 x ) dx = 2 4 −2 x dx − 3 = 2 [ (−2 − 1 + 0 + 1 + 2 + 3)(1) ] −2 x dx b 1 ⎡1 ⎤ x dx = ⎢ x x ⎥ = ( b b − a a ) a 2 ⎣ ⎦a 2 76. For b > 0, if b is an integer, b = i =1 i= (b − 1)b . 2 If b is not an integer, let n = b . Then 0 x dx = 0 + 1 + 2 + ⋅⋅⋅ + (n − 1) + n(b − n) (n − 1)n + n(b − n) 2 ( b − 1) b = + b (b − b ) . 2 = x a f (m) dt ≤ x a f (t ) dt ≤ x a f ( M ) dt ( x − a ) f ( m) ≤ G ( x ) ≤ ( x − a ) f ( M ) By the Squeeze Theorem lim ( x − a ) f (m) ≤ lim G ( x ) x →a + x →a + Thus, a lim G ( x) = 0 = x→a+ a f (t ) dt = G (a) Therefore G is right-continuous at x = a . Now, suppose c = b . Then b lim G ( x) = lim x →b − x x →b − f (t ) dt As before, (b − x) f (m) ≤ G ( x) ≤ (b − x) f ( M ) so we can apply the Squeeze Theorem again to obtain lim (b − x) f (m) ≤ lim G ( x) x →b − x →b − ≤ lim (b − x) f ( M ) x dx = 0 + 1 + 2 + ⋅⋅⋅ + (b − 1) 0 b −1 b Then ≤ lim ( x − a ) f ( M ) 4 d ⎛1 ⎞ 1 ⎛ x⎞ x ⎜ x x ⎟ = x⎜ ⎟ + = x dx ⎝ 2 ⎠ 2 ⎝ x⎠ 2 b Min-Max Existence Theorem) m and M such that f (m) ≤ f ( x) ≤ f ( M ) for all x in [ a, b ] . x→a+ 1 ⎡1 ⎤ −3 ⎢ (2)(2) + (4)(4) ⎥ 2 ⎣2 ⎦ = –24 75. f (t ) dt . Since f is continuous on [ a, b ] , there exist (by the n ⎛ 1− 0 ⎞ ⎛ 1 ⎞ 1 0 i i 2 , which for + = ⎜ ⎟ ⎜ ⎟ 3 n n ⎝ ⎠ ⎝ ⎠ n i =1 i =1 77 = 0.385 . n = 10 equals 200 1 2 x dx 0 x x→a a x →c The right-endpoint Riemann sum is n a. Let c be in ( a, b ) . Then G '(c) = f (c) by the x →b − Thus lim G ( x) = 0 = x →b − b b f (t ) dt = G (b) Therefore, G is left-continuous at x = b . b. Let F be any antiderivative of f. Note that G is also an antiderivative of f. Thus, F ( x) = G ( x) + C . We know from part (a) that G ( x) is continuous on [ a, b ] . Thus F ( x ) , being equal to G ( x) plus a constant, is also continuous on [ a, b ] . Instructor’s Resource Manual Section 4.4 275 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 78. x ⎧1, x > 0 Let f ( x) = ⎨ and F ( x) = f (t ) dt . −1 ⎩0, x ≤ 0 If x < 0 , then F ( x) = 0 . If x ≥ 0 , then x F ( x) = f (t ) dt −1 0 = −1 6. x 0 dt + 0 1 dt = 0+ x = x Thus, ⎧ x, x ≥ 0 F ( x) = ⎨ ⎩0, x < 0 which is continuous everywhere even though f ( x) is not continuous everywhere. 7. 1 b−a b a 8. 0; 2 π 0 f ( x ) dx f ( x ) dx 10. 2. 1 2−0 1 = 3 5. 4 x3 dx ( 3 0 11. = 40 0 3 x + 16 2 24 − 4 = 3 ) ( 1 [sin π − sin 0] = 0 π x cos x 2 dx = 12. ( 1 ⎡2 3 ⎤ x + 16 ⎥ ⎢ 2 ⎣3 ⎦0 6 −2 π/2 0 sin 2 x cos x dx 2 ⎡1 3 ⎤ sin x ⎥ π ⎢⎣ 3 ⎦0 ( 2 y 1 + y2 1 2 3π = ) 3 ( ⎡1 dy = ⎢ 1 + y 2 ⎣8 ) ⎥⎦1 625 609 −2 = = 76.125 8 8 2 π −4 π /4 0 tan x sec2 x = (1 − 0 ) = π /4 1 ⎡1 ⎤ tan 2 x ⎥ π / 4 − 1 ⎢⎣ 2 ⎦0 2 π −4 1 π / 2 sin z 4 dz = ⎡ −2 cos z ⎤ ⎦ π / 4 π /4 π⎣ z π /4 8 cos π / 4 − cos π / 2 ≈ 0.815 = π ) 4 ⎤2 π /2 13. ) 1 17 −2(−2) + 12 (−2)2 + 2 + 12 = 3 6 π (0 − 0) = 0 1 π / 4 −1 = 1 ⎛1 2⎞ ⎜ sin x ⎟ π ⎝2 ⎠0 2 dx = 1 1 ( 2 + x ) dx 1 + 2 −2 1 1 0 = ⎡⎢ ( 2 − x ) dx + ( 2 + x ) dx ⎤⎥ − 2 0 3⎣ ⎦ 0 1⎫ 1 ⎧⎡ = ⎨ 2 x − 12 x 2 ⎤ + ⎡ 2 x + 12 x 2 ⎤ ⎬ ⎣ ⎦ ⎣ ⎦ −2 0⎭ 3⎩ = π 1 2 −1 = 3 1 1 dx = ⎡ x 2 + 16 ⎤ = ⎢ ⎥0 3 2 ⎣ ⎦ 3 x + 16 x2 [sin x ]π0 π π/ 2 x 2 1 0 1 π /2−0 = 4 1 4 2 1 ⎡5 ⎤ 5 x dx = ⎢ x3 ⎥ = 35 4 −1 1 3 ⎣ 3 ⎦1 1 3. 3−0 4. 3 1 3 1 = ⎡ x4 ⎤ 2 ⎣ ⎦1 π π −0 Problem Set 4.5 1 3 −1 1 2 x dx ⎞⎟ ⎠ π 1 9. 4. f ( x + p ) = f ( x ) ; period 1. cos x dx = 2 0 1 π 1 π sin x dx = ( − cos x )0 π −0 0 π 1 2 = − ( −1 − 1) = = 2 0 π 2. f ( c ) 3. 1 π = 4.5 Concepts Review 1. 1 2 ( x + x ) dx 2 + 3 −3 1 0 = ⎛⎜ ( − x + x ) dx + 5 ⎝ −3 2 1 4 = ⎡ x2 ⎤ = ⎣ ⎦ 0 5 5 14. ( ) 1 π / 2 sin v cos v dv π /2 0 1 + cos 2 v 2⎡ − 1 + cos 2 v π ⎢⎣ 2 = −1 + 2 = π ( π /2 ⎤ ⎥⎦ 0 ) 276 Section 4.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 15. 3 x + 1 dx = c + 1 ( 3 − 0 ) 0 24. 3 ⎡2 3/ 2 ⎤ ⎢ 3 ( x + 1) ⎥ = 3 c + 1 ⎣ ⎦0 115 ≈ 1.42 14 / 3 = 3 c + 1; c = 81 2 3 x dx 0 2 ⎡1 ⎤ = c3 ( 2 − 0 ) ; ⎢ x 4 ⎥ = 2c3 ⎣ 4 ⎦0 c = 3 2 ≈ 1.26 25. 4 1 ( ax + b ) dx = ( ac + b )( 4 − 1) 4 16. 1 −1 5 ⎡a 2 ⎤ ⎢ 2 x + bx ⎥ = 3ac + 3b; c = 2 ⎣ ⎦1 x 2 dx = c 2 (1 − ( −1) ) 1 3 ⎡1 3 ⎤ 2 ⎢ 3 x ⎥ = 2c ; c = ± 3 ≈ ±0.58 ⎣ ⎦ −1 17. 3 (1 − x ) dx = (1 − c ) (3 + 4) 2 −4 26. 2 b 2 y dy 0 c= b 3 3 ⎡ 1 3⎤ 2 ⎢ x − 3 x ⎥ = 7 − 7c ⎣ ⎦ −4 B 39 c=± ≈ ±2.08 3 18. 1 0 A 27. B 1 20. 21. 3± 3 ≈ 0.21 or 0.79 6 2 ⎡x x ⎤ x dx = c ( 2 − 0 ) ; ⎢ ⎥ = 2 c ; c =1 ⎣ 2 ⎦0 2 0 −2 π −π ⎡x x ⎤ x dx = c ( 2 + 2 ) ; ⎢ ⎥ = 4 c ; c = −1,1 ⎣ 2 ⎦ −2 sin z dz = sin c (π + π ) [ − cos z ]π−π 22. π 0 28. 2 2 = f (c) ⎡a 2 ⎤ ⎢ 2 x + bx ⎥ ⎣ ⎦A = ac + b B−A a ( B − A)( B + A) + b ( B − A) 2 = ac + b B− A a a B + A + b = ac + b; 2 2 1 1 c = B + A = ( A + B) / 2 2 2 ⎡ − x 2 ( 2 x − 3) ⎤ ⎢ ⎥ = c − c2 6 ⎢⎣ ⎥⎦ 0 19. ( ax + b ) dx B−A x (1 − x ) dx = c (1 − c )(1 − 0 ) c= b ⎡1 ⎤ = c 2 ( b − 0 ) ; ⎢ y 3 ⎥ = bc 2 ⎣ 3 ⎦0 b ⎡1 ⎤ ay 2 dy = ac 2 ( b − 0 ) ; ⎢ ay 3 ⎥ = abc 2 0 ⎣3 ⎦0 b c= b 3 3 29.. Using c = π yields 2π (5) 4 = 1250π ≈ 3927 = 2π sin c; c = 0 cos 2 y dy = ( cos 2c )(π − 0 ) π π 3π ⎡ sin 2 y ⎤ ⎢ 2 ⎥ = π cos 2c; c = 4 , 4 ⎣ ⎦0 23. 2 0 (v 2 ) ( ) − v dv = c − c ( 2 − 0 ) 2 30. ( ) Using c = 0.8 yields 2 3 + sin 0.82 ≈ 7.19 2 ⎡1 3 1 2 ⎤ 2 ⎢ 3 v − 2 v ⎥ = 2c − 2c ⎣ ⎦0 c= 21 + 3 ≈ 1.26 6 Instructor’s Resource Manual Section 4.5 277 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 31. Using c = 0.5 yields 2 2 1 + 0.5 2 = 3.2 38. 3π π ( π = 32. −π π = 5 ) (sin 2 x + 2sin x cos x + cos 2 x) dx (1 + 2sin x cos x) dx = −π π =2 40. x cos( x3 ) dx (sin x + cos x)2 dx −π ⎛ 16 ⎞ Using c = 15 yields ⎜ ⎟ (20 − 10) ≈ 13.8 . ⎝ 15 ⎠ 3π 2 0 3π 2⎡ 2 = sin 3 3 π3 sin( x3 ) ⎤ ⎣ ⎦ 0 3 3 = 39. x 2 cos( x3 ) dx = 2 − 3π 0 π/2 π −π dx + π −π sin 2 x dx dx + 0 = 2[ x]0π = 2π z sin 2 ( z 3 ) cos( z 3 )dz = 0 , since −π / 2 (− z ) sin 2 [(− z )3 ]cos[(− z )3 ] = − z sin 2 (− z 3 ) cos(− z 3 ) = − z[− sin( z 3 )]2 cos( z 3 ) 33. 34. A rectangle with height 25 and width 7 has approximately the same area as that under the curve. Thus 1 7 H (t ) dt ≈ 25 7 0 a. A rectangle with height 28 and width 24 has approximately the same area as that under the curve. Thus, 24 1 T (t ) dt ≈ 28 24 − 0 0 b. Yes. The Mean Value Theorem for Integrals guarantees the existence of a c such that 24 1 T (t ) dt = T (c) 0 24 − 0 The figure indicates that there are actually two such values of c, roughly, c = 11 and c = 16 . 35. π −π (sin x + cos x) dx = π −π sin x dx + 2 π 0 = − z sin 2 ( z 3 ) cos( z 3 ) 41. 36. 37. −1 x 3 2 4 (1 + x ) odd. π/2 −π / 2 1 dx + −1 1 x dx + −1 x 2 dx + 1 −1 x3 dx 1 = 2 [ x] 42. 100 ⎡ x3 ⎤ 8 +0+ 2⎢ ⎥ +0 = 3 3 ⎣⎢ ⎦⎥ 0 (v + sin v + v cos v + sin 3 v)5 dv = 0 −100 since (−v + sin(−v) − v cos(−v) + sin 3 (−v))5 = (−v − sin v − v cos v − sin 3 v)5 = −(v + sin v + v cos v + sin 3 v)5 43. (x 1 −1 3 ) + x3 dx = 2 1 0 x3 dx + 1 −1 x3 dx 1 ⎡ x4 ⎤ 1 = 2⎢ ⎥ +0 = 4 2 ⎣⎢ ⎦⎥ 0 44. ( x sin π/4 −π / 4 5 ) 2 x + x tan x dx = 0 2 dx = 0 , since the integrand is sin x dx = 0 , since the integrand is odd. 1 + cos x 1 −1 1 0 π 1 (1 + x + x 2 + x3 ) dx = cos x dx = 0 + 2 [sin x ]0 = 0 1 −1 since − x sin 5 (− x) + − x tan(− x) 2 = − x sin 5 x − x tan x 45. −a f ( x) dx = −b −a −b b a f ( x) dx when f is even. f ( x) dx = − b a f ( x) dx when f is odd. 278 Section 4.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 46. u = − x, du = − dx b a = f ( − x ) dx = − −a −b b. ku = −b −a −a f (u ) du = −b f (u ) du c. Note that 1 b 1 a u= u ( x ) dx = u ( x ) dx , so b−a a a−b b we can assume a < b . 1 b 1 b u= u dx ≤ v dx = v b−a a b−a a f ( x) dx since the variable used in the integration is not important. 47. 4π π/2 cos x dx = 8 0 0 cos x dx π/ 2 = 8 [sin x ]0 =8 48. Since sin x is periodic with period 2π , sin 2x is periodic with period π . 4π 0 sin 2 x dx = 8 0 π2 ⎡ 1 ⎤ = 8 ⎢ − cos 2 x ⎥ ⎣ 2 ⎦0 49. 1+π 1 π2 sin x dx = π 0 54. a. V = 0 by periodicity. b. V = 0 by periodicity. sin 2 x dx 2 c. Vrms = = –4(–1 – 1) = 8 sin x dx = k b 1 b u dx = ku dx = ku a b−a b−a a π 0 = φ +1 Vˆ 2 sin 2 (120π t + φ ) dt φ 1 0 Vˆ 2 sin 2 (120π t ) dt by periodicity. u = 120π t , du = 120π dt sin x dx 1 120π ˆ 2 2 V sin u du 120π 0 120π 1 ⎤ Vˆ 2 ⎡ 1 cos sin = − u u + u 120π ⎢⎣ 2 2 ⎥⎦ 0 1 = Vˆ 2 2 2 Vrms = π = [ − cos x ]0 = 2 Vˆ 2 2 ˆ V = 120 2 ≈ 169.71 Volts d. 120 = 50. 2 +π / 2 2 = 51. 1+π 1 sin 2 x dx = cos x dx = = 2 [sin x ]0 53. 55. sin 2 x dx π 0 cos x dx = 2 Since f is continuous on a closed interval [ a, b ] there exist (by the Min-Max Existence Theorem) an m and M in [ a, b ] such that 1 [ − cos 2 x ]0π / 2 = 1 2 π /2 52. π/2 0 π /2 0 f (m) ≤ f ( x) ≤ f ( M ) for all x in [ a, b ] . Thus b cos x dx = 2 (1 − 0 ) = 2 The statement is true. Recall that 1 b f = f ( x) dx . b−a a b b b 1 b fdx = f dx = f ( x)dx ⋅ dx a a a a b−a b 1 b = f ( x)dx ⋅ (b − a ) = f ( x) dx a a b−a a (b − a) f (m) ≤ b a b a f ( x) dx ≤ b a f ( M ) dx f ( x) dx ≤ (b − a ) f ( M ) 1 b f ( x) dx ≤ f ( M ) b−a a Since f is continuous, we can apply the Intermediate Value Theorem and say that f takes on every value between f (m) and f ( M ) . Since f ( m) ≤ 1 b f ( x) dx is between f (m) and f ( M ) , b−a a there exists a c in [ a, b ] such that All the statements are true. 1 b 1 b a. u + v = u dx + v dx b−a a b−a a 1 b (u + v) dx = u + v = b−a a f (m) dx ≤ f (c ) = 56. a. 2π 0 1 b f ( x) dx . b−a a (sin 2 x + cos 2 x) dx = 2π 0 2π dx = [ x ]0 = 2π Instructor’s Resource Manual Section 4.5 279 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 58. b. a. Odd b. 2π c. This function cannot be integrated in closed form. We can only simplify the integrals using symmetry and periodicity, and approximate them numerically. a Note that π +a f ( x ) dx = 0 since f is odd, and −a f ( x ) dx = 0 since π −a f (π + x ) = − f (π − x ) . π /2 0 f ( x ) dx = π J1 (1) ≈ 0.69 (Bessel 2 function) π /2 f ( x ) dx = 0 −π / 2 3π / 2 c. 2π = =2 = 57. 2π 0 2π 0 2π 0 2π 2 cos x dx + 0 cos 2 x dx, thus sin x dx 0 3π / 2 2π cos 2 x dx −3π / 2 13π / 6 0 π /6 4π / 3 2 sin x dx = π π /6 10π / 3 a. Even c. On [ 0, π ] , sin x = sin x . 59. u = cos x , du = − sin x dx f ( x ) dx = =− sin u du = cos u + C f ( x ) dx = − cos ( cos x ) + C −π / 2 π /2 f ( x ) dx = 2 3π / 2 0 0 f ( x ) dx π 0 f ( x ) dx + 3π / 2 π −3π / 2 0 f ( x ) dx = 2 3π / 2 0 f ( x ) dx f ( x ) dx = 2 ( cos1 − 1) ≈ −0.92 4π / 3 10π / 3 13π / 6 f ( x ) dx = 4π / 3 π /6 a a g ( x) dx = 0 a ⎛c ⎞ f ⎜ x ⎟ dx c ⎝a ⎠ a2 c f ( x ) dx ≈ −0.44 0 f ( x) dx ⎛c ⎞ f ⎜ x ⎟ dx ⎝b ⎠ a 0 a2 c c2 0 g ( x ) dx + f ( x) dx + a 2 + b2 c c 2 0 b h( x) dx 0 2 c b c2 0 f ( x) dx = f ( x) dx c 0 f ( x) dx since a 2 + b 2 = c 2 from the triangle. 60. If f is odd, then f (− x ) = − f ( x) and we can write −a ⎛ 3⎞ ⎛1⎞ f ( x ) dx = 2 cos1 − cos ⎜⎜ ⎟⎟ + cos ⎜ ⎟ ⎝2⎠ ⎝ 2 ⎠ ≈ −0.44 f ( x ) dx ≈ 1.055 b b b2 c f ( x) dx = f ( x) dx 0 c c c2 0 = 0 f ( x ) dx = 0 π /6 π /6 c = = 2 (1 − cos1) ≈ 0.92 f ( x ) dx = 4π / 3 f ( x ) dx = 0 Thus, = cos1 − 1 ≈ −0.46 3π / 2 2π f ( x ) dx ≈ 1.055 (numeric integration) a a f ( x) dx = 0 c c = f ( x ) dx = 1 − cos1 ≈ 0.46 π /2 f ( x ) dx = 0 f ( x ) dx = 0 0 c2 b b b B= h( x) dx = 0 0 c Likewise, on [π , 2π ] , π /2 2π f ( x ) dx = c = = cos ( cos x ) + C 0 2π 0 a. Written response. b. A = sin x ⋅ sin ( cos x ) dx f ( x ) dx ≈ 0.69 0 f ( x ) dx = 0 ; 13π / 6 b. 2π π /2 f ( x ) dx = 2 =− f ( x) dx = a 0 0 −a [ − f (− x)] dx = f (u ) du = − a 0 0 a f (u ) du f ( x) dx On the second line, we have made the substitution u = − x . 280 Section 4.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4.6 Concepts Review 1. 1, 2, 2, 2, …, 2, 1 2. 1, 4, 2, 4, 2, …, 4, 1 3. n 4 4. large Problem Set 4.6 1. f ( x) = 1 2 x x0 = 1.00 ;h = 3 –1 = 0.25 8 f ( x0 ) = 1 x5 = 2.25 f ( x5 ) ≈ 0.1975 x1 = 1.25 f ( x1 ) = 0.64 x6 = 2.50 f ( x6 ) = 0.16 x2 = 1.50 f ( x2 ) ≈ 0.4444 x7 = 2.75 f ( x7 ) ≈ 0.1322 x3 = 1.75 f ( x3 ) ≈ 0.3265 x8 = 3.00 f ( x8 ) ≈ 0.1111 x4 = 2.00 f ( x4 ) = 0.25 Left Riemann Sum: 3 1 1 x2 Right Riemann Sum: dx ≈ 0.25[ f ( x0 ) + f ( x1 ) + 3 1 1 x2 3 1 2 dx ≈ 0.25[ f ( x1 ) + f ( x2 ) + ... + f ( x8 )] ≈ 0.5655 0.25 [ f ( x0 ) + 2 f ( x1 ) + + 2 f ( x7 ) + f ( x8 )] ≈ 0.6766 2 x 3 1 0.25 Parabolic Rule: dx ≈ [ f ( x0 ) + 4 f ( x1 ) + 2 f ( x2 ) + + 4 f ( x7 ) + f ( x8 )] ≈ 0.6671 1 x2 3 Trapezoidal Rule: 1 + f ( x7 )] ≈ 0.7877 dx ≈ Fundamental Theorem of Calculus: 2. f ( x) = 1 3 x x0 = 1.00 ;h = 3 1 2 ⎡ 1⎤ dx = ⎢ – ⎥ = – + 1 = ≈ 0.6667 1 x2 3 3 ⎣ x ⎦1 3 1 3 –1 = 0.25 8 f ( x0 ) = 1 x5 = 2.25 f ( x5 ) ≈ 0.0878 x1 = 1.25 f ( x1 ) = 0.5120 x6 = 2.50 f ( x6 ) = 0.0640 x2 = 1.50 f ( x2 ) ≈ 0.2963 x7 = 2.75 f ( x7 ) ≈ 0.0481 x3 = 1.75 f ( x3 ) ≈ 0.1866 x8 = 3.00 f ( x8 ) ≈ 0.0370 x4 = 2.00 f ( x4 ) = 0.1250 Left Riemann Sum: 3 1 1 x3 Right Riemann Sum: Trapezoidal Rule: Parabolic Rule: 3 1 1 x3 3 1 1 3 3 1 1 3 x x dx ≈ 0.25[ f ( x0 ) + f ( x1 ) + dx ≈ 0.25[ f ( x1 ) + f ( x2 ) + ... + f ( x8 )] ≈ 0.3392 dx ≈ dx ≈ + f ( x7 )] ≈ 0.5799 0.25 [ f ( x0 ) + 2 f ( x1 ) + 2 + 2 f ( x7 ) + f ( x8 )] ≈ 0.4596 0.25 [ f ( x0 ) + 4 f ( x1 ) + 2 f ( x2 ) + 3 Fundamental Theorem of Calculus: 3 1 + 4 f ( x7 ) + f ( x8 )] ≈ 0.4455 3 4 ⎡ 1 ⎤ dx = ⎢ − = ≈ 0.4444 3 2⎥ 9 x ⎣ 2 x ⎦1 1 Instructor's Resource Manual Section 4.6 281 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. f ( x) = x ; h = 3. x0 = 0.00 x1 = 0.25 x2 = 0.50 x3 = 0.75 x4 = 1.00 2–0 = 0.25 8 f ( x0 ) = 0 f ( x1 ) = 0.5 f ( x2 ) ≈ 0.7071 f ( x3 ) ≈ 0.8660 f ( x4 ) = 1 2 Left Riemann Sum: Trapezoidal Rule: Parabolic Rule: x dx ≈ 0.25[ f ( x0 ) + f ( x1 ) + 0 2 Right Riemann Sum: 0 f ( x5 ) ≈ 1.1180 f ( x6 ) ≈ 1.2247 f ( x7 ) ≈ 1.3229 f ( x8 ) ≈ 1.4142 + f ( x7 )] ≈ 1.6847 0.25 [ f ( x0 ) + 2 f ( x1 ) + + 2 f ( x7 ) + f ( x8 )] ≈ 1.8615 2 2 0.25 xdx ≈ [ f ( x1 ) + 4 f ( x2 ) + 2 f ( x3 ) + + 4 f ( x7 ) + f ( x8 )] ≈ 1.8755 0 3 2 xdx ≈ 0 2 0 2 4 2 ⎡2 ⎤ xdx = ⎢ x3 / 2 ⎥ = ≈ 1.8856 3 3 ⎣ ⎦0 x0 = 1.00 3 –1 = 0.25 8 f ( x0 ) ≈ 1.4142 x5 = 2.25 f ( x5 ) ≈ 5.5400 x1 = 1.25 f ( x1 ) ≈ 2.0010 x6 = 2.50 f ( x6 ) ≈ 6.7315 x2 = 1.50 f ( x2 ) ≈ 2.7042 x3 = 1.75 f ( x3 ) ≈ 3.5272 x7 = 2.75 x8 = 3.00 f ( x7 ) ≈ 8.0470 f ( x8 ) ≈ 9.4868 x4 = 2.00 f ( x4 ) ≈ 4.4721 f ( x) = x x 2 + 1; h = 3 Left Riemann Sum: 1 Trapezoidal Rule: Parabolic Rule: x x 2 + 1 dx ≈ 0.25[ f ( x0 ) + f ( x1 ) + 3 Right Riemann Sum: 1 ( ) 5 0.25 [ f ( x0 ) + 2 f ( x1 ) + + 2 f ( x7 ) + f ( x8 )] ≈ 9.6184 2 3 0.25 x x 2 + 1dx ≈ [ f ( x0 ) + 4 f ( x1 ) + 2 f ( x2 ) + + 4 f ( x7 ) + f ( x8 )] ≈ 9.5981 1 3 3 1 f ( x) = x x 2 + 1 ; h = x0 = 0.00 x1 = 0.125 x2 = 0.250 x3 = 0.375 x4 = 0.500 + f ( x7 )] ≈ 8.6093 x x 2 + 1 dx ≈ 0.25[ f ( x1 ) + f ( x2 ) + ... + f ( x8 )] ≈ 10.6274 x x 2 + 1 dx ≈ Fundamental Theorem of Calculus: 5. = 1.25 = 1.50 = 1.75 = 2.00 x dx ≈ 0.25[ f ( x1 ) + f ( x2 ) + ... + f ( x8 )] ≈ 2.0383 Fundamental Theorem of Calculus: 4. x5 x6 x7 x8 3 ( ) 1 ⎡1 ⎤ x x 2 + 1dx = ⎢ ( x 2 + 1)3 / 2 ⎥ = 10 10 – 2 2 ≈ 9.5981 1 ⎣3 ⎦1 3 3 1− 0 = 0.125 8 f ( x0 ) = 0 f ( x1 ) ≈ 0.1351 f ( x2 ) ≈ 0.3385 f ( x3 ) ≈ 0.7240 f ( x4 ) ≈ 1.5259 x5 = 0.625 f ( x5 ) ≈ 3.2504 x6 = 0.750 f ( x6 ) ≈ 6.9849 x7 = 0.875 x8 = 1.000 f ( x7 ) ≈ 15.0414 f ( x8 ) = 32 282 Section 4.6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ) 5 x x 2 + 1 dx ≈ 0.125[ f ( x0 ) + f ( x1 ) + 0 1 Right Riemann Sum: Trapezoidal Rule: ( 1 Left Riemann Sum: 0 ( ) 5 x x 2 + 1 dx ≈ 0.125[ f ( x1 ) + f ( x2 ) + ... + f ( x8 )] ≈ 7.4966 ( ) 5 0.125 [ f ( x0 ) + 2 f ( x1 ) + + 2 f ( x7 ) + f ( x8 )] ≈ 5.4966 2 5 1 0.125 x x 2 + 1 dx ≈ [ f ( x0 ) + 4 f ( x1 ) + 2 f ( x2 ) + + 4 f ( x7 ) + f ( x8 )] ≈ 5.2580 0 3 1 0 x x 2 + 1 dx ≈ ( Parabolic Rule: ) Fundamental Theorem of Calculus: 6. + f ( x7 )] ≈ 3.4966 ( ) ( ) 1 5 6⎤ ⎡1 2 x x 2 + 1 dx = ⎢ x + 1 ⎥ = 5.25 0 ⎣12 ⎦0 1 x0 = 1.000 4 −1 = 0.375 8 f ( x0 ) ≈ 2.8284 x5 = 2.875 f ( x5 ) ≈ 7.6279 x1 = 1.375 f ( x1 ) ≈ 3.6601 x6 = 3.250 f ( x6 ) ≈ 8.7616 x2 = 1.750 f ( x2 ) ≈ 4.5604 x3 = 2.125 f ( x3 ) ≈ 5.5243 x7 = 3.625 x8 = 4.000 f ( x7 ) ≈ 9.9464 f ( x8 ) ≈ 11.1803 x4 = 2.500 f ( x4 ) ≈ 6.5479 3/ 2 f ( x) = ( x + 1) ;h = 4 Left Riemann Sum: 1 4 Right Riemann Sum: Trapezoidal Rule: Parabolic Rule: ( x + 1)3 / 2 dx ≈ 0.375[ f ( x0 ) + f ( x1 ) + 1 + f ( x7 )] ≈ 18.5464 ( x + 1)3 / 2 dx ≈ 0.375[ f ( x1 ) + f ( x2 ) + ... + f ( x8 )] ≈ 21.6784 0.375 [ f ( x0 ) + 2 f ( x1 ) + + 2 f ( x7 ) + f ( x8 )] ≈ 20.1124 2 4 0.375 [ f ( x0 ) + 4 f ( x1 ) + 2 f ( x2 ) + + 4 f ( x7 ) + f ( x8 )] ≈ 20.0979 ( x + 1)3 / 2 dx ≈ 1 3 4 1 ( x + 1)3 / 2 dx ≈ Fundamental Theorem of Calculus: 4 1 3/ 2 ( x + 1) 4 ⎡2 5/ 2 ⎤ dx = ⎢ ( x + 1) ⎥ ≈ 20.0979 5 ⎣ ⎦1 7. LRS RRS MRS Trap Parabolic n=4 0.5728 0.3728 0.4590 0.4728 0.4637 n=8 0.5159 0.4159 0.4625 0.4659 0.4636 n = 16 0.4892 0.4392 0.4634 0.4642 0.4636 LRS RRS MRS Trap Parabolic n=4 1.2833 0.9500 1.0898 1.1167 1.1000 n=8 1.1865 1.0199 1.0963 1.1032 1.0987 n = 16 1.1414 1.0581 1.0980 1.0998 1.0986 8. Instructor’s Resource Manual Section 4.6 283 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 9. LRS RRS MRS Trap Parabolic n=4 2.6675 3.2855 2.9486 2.9765 2.9580 n=8 2.8080 3.1171 2.9556 2.9625 2.9579 n = 16 2.8818 3.0363 2.9573 2.9591 2.9579 10. 11. f LRS RRS MRS Trap Parabolic n=4 10.3726 17.6027 13.6601 13.9876 13.7687 n=8 12.0163 15.6314 13.7421 13.8239 13.7693 n = 16 12.8792 14.6867 13.7625 13.7830 13.7693 ( x) = − 1 x 2 ( x) = ; f The largest that f (c) 2 x3 can be on [1,3] occurs when c = 1 , and f (1) = 2 ( 3 − 1)3 400 Round up: n = 12 ( 2 ) ≤ 0.01; n ≥ 3 12n 2 31 0.167 dx ≈ [ f ( x0 ) + 2 f ( x1 ) + + 2 f ( x11 ) + f ( x12 )] 1 x 2 ≈ 1.1007 12. f ( x) = − 1 (1 + x ) 2 The largest that f ; f (c) ( x) = 2 (1 + x )3 can be on [1,3] occurs when c = 1 , and f (1) = 1 . 4 ( 3 − 1)3 ⎛ 1 ⎞ 100 ⎜ 4 ⎟ ≤ 0.01; n ≥ 6 Round up: n = 5 12n ⎝ ⎠ 3 1 0.4 dx ≈ [ f ( x0 ) + 2 f ( x1 ) + + 2 f ( x4 ) + f ( x5 )] 1 1+ x 2 ≈ 0.6956 2 13. f ( x) = 1 2 x ; f The largest that f ( x) = − (c) 1 4 x3 / 2 can be on [1, 4] occurs when c = 1 , and f (1) = 1 . 4 ( 4 − 1)3 ⎛ 1 ⎞ 900 ⎜ ⎟ ≤ 0.01; n ≥ 16 Round up: n = 8 12n 2 ⎝ 4 ⎠ 4 0.375 [ f ( x0 ) + 2 f ( x1 ) + + 2 f ( x7 ) + f ( x8 )] x dx ≈ 1 2 ≈ 4.6637 284 Section 4.6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 14. f 1 ( x) = 2 x +1 ; f The largest that f ( 3 − 1)3 ⎛ 12n 15. f 2 4 f ( ) ( x) = 1 2 x 24 (c) 1 4 ( x + 1) 3/ 2 can be on [1,3] occurs when c = 1 , and f (1) = 1 4 × 23 / 2 . 100 ⎞ Round up: n = 3 ⎟ ≤ 0.01; n ≥ 12 2 ⎠ 3 0.667 [ f ( x0 ) + 2 f ( x1 ) + 2 f ( x2 ) + f ( x3 )] x + 1 dx ≈ 1 2 ≈ 3.4439 1 ⎜ ⎝ 4 × 23 / 2 ( x) = − ( x) = − ( x) = ; f 2 x 3 ; f 6 ( x) = − x4 ; x5 4 4 The largest that f ( ) ( c ) can be on [1,3] occurs when c = 1 , and f ( ) (1) = 24 . ( 4 − 1)5 ( 24 ) ≤ 0.01; n ≈ 4.545 Round up to even: n = 6 180n 4 31 0.333 dx ≈ [ f ( x0 ) + 4 f ( x1 ) + ... + 4 f ( x5 ) + f ( x6 )] 1 x 3 ≈ 1.0989 16. f ( x) = f ( x) = 1 2 x +1 ; f 3 8 ( x + 1) 5/ 2 ( x) = − 1 4 ( x + 1) 4 ; f ( ) ( x) = − 3/ 2 ; 15 16 ( x + 1) 7/2 4 4 The largest that f ( ) ( c ) can be on [ 4,8] occurs when c = 4 , and f ( ) ( 4 ) = 3 400 5 . (8 − 4 )5 ⎛ 3 ⎞ ⎜ ⎟ ≤ 0.01; n ≈ 1.1753 Round up to even: n = 2 180n ⎝ 400 5 ⎠ 8 2 x + 1 dx ≈ ⎡⎣ f ( x0 ) + 4 f ( x1 ) + f ( x2 ) ⎤⎦ ≈ 10.5464 4 3 4 Instructor’s Resource Manual Section 4.6 285 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 17. m+ h b ⎡a ⎤ (ax 2 + bx + c)dx = ⎢ x3 + x 2 + cx ⎥ m–h 2 ⎣3 ⎦m–h a b a b = ( m + h )3 + ( m + h ) 2 + c ( m + h ) – ( m – h ) 3 – ( m – h ) 2 – c ( m – h ) 3 2 3 2 a b h = (6m 2 h + 2h3 ) + (4mh) + c(2h) = [a(6m2 + 2h 2 ) + b(6m) + 6c] 3 2 3 m+ h h [ f (m − h) + 4 f (m) + f (m + h)] 3 h = [a(m – h)2 + b(m – h) + c + 4am 2 + 4bm + 4c + a (m + h)2 + b(m + h) + c] 3 h = [a(6m2 + 2h 2 ) + b(6m) + 6c] 3 18. a. To show that the Parabolic Rule is exact, examine it on the interval [m – h, m + h]. Let f ( x) = ax3 + bx 2 + cx + d , then m+h m−h f ( x) dx a⎡ b c (m + h)4 − (m − h) 4 ⎤ + ⎡(m + h)3 − (m − h)3 ⎤ + ⎡ (m + h) 2 − (m − h) 2 ⎤ + d [(m + h) − (m − h)] ⎣ ⎦ ⎣ ⎦ ⎦ 4 3 2⎣ a b c = (8m3 h + 8h3 m) + (6m2 h + 2h3 ) + (4mh) + d (2h). 4 3 2 The Parabolic Rule with n = 2 gives m+ h h 2 f ( x) dx = [ f (m − h) + 4 f (m) + f (m + h)] = 2am3 h + 2amh3 + 2bm2 h + bh3 + 2chm + 2dh m−h 3 3 a b c = (8m3 h + 8mh3 ) + (6m 2 h + 2h3 ) + (4mh) + d (2h) 4 3 2 which agrees with the direct computation. Thus, the Parabolic Rule is exact for any cubic polynomial. = b. The error in using the Parabolic Rule is given by En = − 2 However, f ( x) = 3ax + 2bx + c, f ( x) = 6ax + 2b, f 19. The left Riemann sum will be smaller than b a (3) (l − k )5 180n 4 f (4) (m) for some m between l and k. ( x) = 6a, and f (4) ( x) = 0 , so En = 0. f ( x ) dx . If the function is increasing, then f ( xi ) < f ( xi +1 ) on the interval [ xi , xi +1 ] . Therefore, the left Riemann sum will underestimate the value of the definite integral. The following example illustrates this behavior: If f is increasing, then f ( c ) > 0 for any c ∈ ( a, b ) . Thus, the error En = ( b − a )2 2n f ( c ) > 0 . Since the error is positive, then the Riemann sum must be less than the integral. 286 Section 4.6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 20. The right Riemann sum will be larger than b a f ( x ) dx . If the function is increasing, then f ( xi ) < f ( xi +1 ) on the interval [ xi , xi +1 ] . Therefore, the right Riemann sum will overestimate the value of the definite integral. The following example illustrates this behavior: If f is increasing, then f ( c ) > 0 for any c ∈ ( a, b ) . Thus, the error En = − ( b − a )2 2n f ( c ) < 0 . Since the error is negative, then the Riemann sum must be greater than the integral. 21. The midpoint Riemann sum will be larger than If f is concave down, then f b a f ( x ) dx . ( c ) < 0 for any c ∈ ( a, b ) . Thus, the error En = ( b − a )3 24n 2 (c) < 0 . f Since the error is negative, then the Riemann sum must be greater than the integral. 22. The Trapezoidal Rule approximation will be smaller than If f is concave down, then f (c) < 0 b a f ( x ) dx . for any c ∈ ( a, b ) . Thus, the error En = − ( b − a )3 12n 2 error is positive, then the Trapezoidal Rule approximation must be less than the integral. f (c) > 0 . Since the 23. Let n = 2. f ( x) = x k ; h = a x0 = – a f ( x0 ) = – a k x1 = 0 f ( x1 ) = 0 x2 = a f ( x2 ) = a k a –a a x k dx ≈ [– a k + 2 ⋅ 0 + a k ] = 0 2 a 1 1 ⎡ 1 k +1 ⎤ [a k +1 – a k +1 ] = 0 x k dx = ⎢ x ⎥ = [a k +1 – (– a) k +1 ] = –a 1 k + k k 1 1 + + ⎣ ⎦ –a A corresponding argument works for all n. a 24. a. T ≈ 48.9414; f ( x) = 4 x3 [4(3)3 – 4(1)3 ](0.25)2 ≈ 48.9414 – 0.5417 = 48.3997 12 The correct value is 48.4 . T– Instructor’s Resource Manual Section 4.6 287 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. T ≈ 1.9886; f ( x) = cos x T– ( ) π [cos π – cos 0] 12 2 ≈ 1.999987 12 The correct value is 2. 25. The integrand is increasing and concave down. By problems 19-22, LRS < TRAP < MRS < RRS. 26. The integrand is increasing and concave up. By problems 19-22, LRS < MRS < TRAP < RRS 27. A ≈ 10 [75 + 2 ⋅ 71 + 2 ⋅ 60 + 2 ⋅ 45 + 2 ⋅ 45 + 2 ⋅ 52 + 2 ⋅ 57 + 2 ⋅ 60 + 59] = 4570 ft2 2 3 28. A ≈ [23 + 4 ⋅ 24 + 2 ⋅ 23 + 4 ⋅ 21 + 2 ⋅ 18 + 4 ⋅15 + 2 ⋅ 12 + 4 ⋅11 + 2 ⋅ 10 + 4 ⋅ 8 + 0] = 465 ft2 3 V = A ⋅ 6 ≈ 2790 ft3 20 [0 + 4 ⋅ 7 + 2 ⋅12 + 4 ⋅18 + 2 ⋅ 20 + 4 ⋅ 20 + 2 ⋅17 + 4 ⋅10 + 0] = 2120 ft2 3 4 mi/h = 21,120 ft/h (2120)(21,120)(24) = 1,074,585,600 ft3 29. A ≈ 30. Using a right-Riemann sum, 24 Distance = 0 v(t ) dt ≈ 8 i =1 4. False: = g ( x) = x 2 + 7 x − 5 are a counterexample. v(ti ) Δt = ( 31 + 54 + 53 + 52 + 35 + 31 + 28 ) 3 60 852 = 14.2 miles 60 5. False: The two sides will in general differ by a constant term. 6. True: At any given height, speed on the downward trip is the negative of speed on the upward. 7. True: a1 + a0 + a2 + a1 + a3 + a2 31. Using a right-Riemann sum, Water Usage = ≈ 10 i =1 120 0 F (t ) dt F (ti ) Δt = 12(71 + 68 + f ( x) = x 2 + 2 x + 1 and + + an −1 + an − 2 + an + an −1 = a0 + 2a1 + 2a2 + + 2an −1 + an + 148) = 13, 740 gallons 100 8. True: 100 (2i − 1) = 2 i =1 4.7 Chapter Review = Concepts Test Theorem 4.3.D 9. True: i =1 2. True: Obtained by integrating both sides of the Product Rule 3. True: If F ( x) = 288 Section 4.7 1 i =1 (ai + 1) 2 = 10 i =1 ai2 + 2 10 i =1 ai + 100 1 i =1 = 100 + 2(20) + 10 = 150 10. False: f must also be continuous except at a finite number of points on [a, b]. 11. True: The area of a vertical line segment is 0. f ( x) dx, f ( x) is a derivative of F(x). 100 2(100)(100 + 1) − 100 = 10, 000 2 10 1. True: i =1 i− Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 12. False: 13. False: 1 −1 x dx is a counterexample. A counterexample is ⎧0, x ≠ 0 f ( x) = ⎨ ⎩1, x = 0 with 1 28. False: f ( x) = x3 is a counterexample. 29. False: f ( x) = x is a counterexample. 30. True: All rectangles have height 4, regardless of xi . 31. True: F (b) − F (a ) = 2 ⎡ f ( x ) ⎤⎦ dx = 0 . −1 ⎣ If f ( x ) is continuous, then 2 2 [ f ( x)] ≥ 0 , and if [ f ( x)] is greater than 0 on [a, b], the integral will be also. 14. False: Dx ⎡⎢ f ( z ) dz ⎤⎥ = f ( x) ⎣ a ⎦ 15. True: sin x + cos x has period 2π , so x x + 2π x 32. False: 33. False: 34. False: lim kf ( x) = k lim f ( x) and x →a x→a x →a sin13 x is an odd function. 18. True: Theorem 4.2.B 19. False: The statement is not true if c > d. 20. False: ⎡ x2 1 ⎤ 2x Dx ⎢ dt ⎥ = 0 1+ t2 ⎥⎦ 1 + x 2 ⎣⎢ f ( x) dx = 2 a 0 f ( x) dx because f z (t ) = t 2 is a counterexample. b 0 f ( x ) dx = F (b) − F (0) 36. False: a = 0, b = 1, f(x) = –1, g(x) = 0 is a counterexample. 37. False: a = 0, b = 1, f(x) = –1, g(x) = 0 is a counterexample. limits exist. 17. True: a −a G ( x ) dx = G (b) − G (a ) Odd-exponent terms cancel themselves out over the interval, since they are odd. lim [ f ( x) + g ( x) ] x→a a 35. True: x→a = lim f ( x) + lim g ( x ) when all the b F ( x ) dx is even. (sin x + cos x) dx is independent of x. 16. True: = b a 38. True: a1 + a2 + a3 + + an ≤ a1 + a2 + a3 + + an because any negative values of ai make the left side smaller than the right side. 39. True: Note that − f ( x ) ≤ f ( x ) ≤ f ( x ) 21. True: Both sides equal 4. 22. True: Both sides equal 4. 40. True: Definition of Definite Integral 23. True: If f is odd, then the accumulation 41. True: Definition of Definite Integral 42. False: Consider 43. True. Right Riemann sum always bigger. 44. True. Midpoint of x coordinate is midpoint of y coordinate. 45. False. Trapeziod rule overestimates integral. 46. True. Parabolic Rule gives exact value for quadratic and cubic functions. function F ( x ) = x 0 and use Theorem 4.3.B. f ( t ) dt is even, and so is F ( x ) + C for any C. 24. False: f ( x) = x 2 is a counterexample. 25. False: f ( x) = x 2 is a counterexample. 26. False: f ( x) = x 2 is a counterexample. 27. False: f ( x) = x 2 , v(x) = 2x + 1 is a counterexample. Instructor’s Resource Manual ( ) cos x 2 dx Section 4.7 289 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Sample Test Problems 12. u = 2 y 3 + 3 y 2 + 6 y, du = (6 y 2 + 6 y + 6) dy 1 ( y2 + y +1) 5 5 ⎡1 ⎤ 1. ⎢ x 4 − x3 + 2 x3 / 2 ⎥ = ⎣4 ⎦0 4 dy = 1 5 3 2 y +3 y 2 + 6 y 355 = 2 1⎤ 13 ⎡2 2. ⎢ x3 − 3 x − ⎥ = 3 x ⎣ ⎦1 6 1 355 −1/ 5 u du 6 11 ( 1 ⎡5 4/5 ⎤ 5 3554 / 5 − 114 / 5 u ⎥ = ⎢ 6 ⎣4 24 ⎦11 13. ⎡⎛ i ⎞ 2 ⎤ ⎛ 1 ⎞ 7 ⎢⎜ ⎟ − 1⎥ ⎜ ⎟ = ⎥⎦ ⎝ 2 ⎠ 4 i =1 ⎢⎣⎝ 2 ⎠ 14. f ( x) = ) 4 π 3 ⎡1 26 ⎤ 50 26 π 3. ⎢ y 3 + 9 cos y − ⎥ = − + − 9 cos1 3 y 3 π 3 ⎣ ⎦1 9 77 77 ⎡1 ⎤ 4. ⎢ ( y 2 − 4)3 / 2 ⎥ = −8 3 + 3 3 ⎣ ⎦4 ( 8 −15 −125 + 3 5 ⎡3 ⎤ 5. ⎢ (2 z 2 − 3)4 / 3 ⎥ = 16 ⎣16 ⎦2 π/2 ⎡ 1 ⎤ 6. ⎢ − cos5 x ⎥ ⎣ 5 ⎦0 = ) 1 5 7. u = tan(3 x 2 + 6 x ), du = (6 x + 6) sec 2 (3 x 2 + 6 x) 1 2 1 u du = u 3 + C 6 18 π 1 ⎡ 3 2 1 tan (3 x + 6 x) ⎤ = tan 3 (3π2 + 6π ) ⎦ 0 18 18 ⎣ 8. u = t 4 + 9, du = 4t 3 dt 25 1 25 −1/ 2 1 u du = ⎡u1/ 2 ⎤ = 1 ⎦9 4 9 2⎣ 15. 11. 16. ) ( ( 290 Section 4.7 1 ⎞ 5 − ⎟ dx = 2 ⎜⎝ x2 ⎠ 4 1⎤ 39 ⎡ ⎢5 x + x ⎥ = 4 ⎣ ⎦2 (3i − 3i −1 ) i =1 = (3 − 1) + (32 − 3) + (33 − 32 ) + ) ) ) x + 1 dx 4⎛ 18. ( x + 1) sin x 2 + 2 x + 3 dx 1 = sin x 2 + 2 x + 3 ( 2 x + 2 ) dx 2 1 = sin u du 2 1 = − cos x 2 + 2 x + 3 + C 2 (x+5−4 5 1 5 2 3 1 ⎡2 ⎤ 3x x − 4 dx = ⎢ ( x3 − 4)3 / 2 ⎥ 5−2 2 3 ⎣3 ⎦2 = 294 n ( 0 3 17. 3 3 8 5 ⎡1 ⎤ = ⎢ x 2 + 5 x − ( x + 1)3 / 2 ⎥ = 2 3 ⎣ ⎦0 6 1 ⎡3 5 3 ⎡ 5/3 5/3 ⎤ ⎤ (t + 5)5 / 3 ⎥ = 37 −6 ≈ 46.9 ⎢ ⎦ 5 ⎣5 ⎦1 25 ⎣ ⎡ ⎤ 1 4 10. ⎢ ⎥ = 3 ⎢⎣ 9 y − 3 y ⎥⎦ 2 27 (2 − x + 1) 2 dx = 2 9. 3 0 1 1 , f (7) = x+3 10 + (3n − 3n−1 ) = 3n − 1 10 19. i =1 (6i 2 − 8i ) = 6 10 i =1 i2 − 8 10 i i =1 ⎡10(11)(21) ⎤ ⎡10(11) ⎤ = 6⎢ ⎥ − 8 ⎢ 2 ⎥ = 1870 6 ⎣ ⎦ ⎣ ⎦ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 20. a. 1 1 1 13 + + = 2 3 4 12 e. b. 1 + 0 + (–1) + (–2) + (–3) + (–4) = –9 c. −2 0 2 f (− x) dx = − f ( x) dx = −2 0 24. a. 2 2 +0− −1 = 0 2 2 1+ 78 21. a. 1 n=2 n 50 nx 2n b. 4 n =1 x − 1 dx = 0 22. 1 1 (1)(1) + (3)(3) = 5 2 2 b. 2 ⎡ ⎛ 3i ⎞ ⎤ ⎛ 3 ⎞ ⎢16 − ⎜ ⎟ ⎥ ⎜ ⎟ n →∞ i =1 ⎢ ⎝ n ⎠ ⎥⎦ ⎝ n ⎠ ⎣ ⎧⎪ n ⎡ 48 27 ⎤ ⎫⎪ = lim ⎨ ⎢ − i 2 ⎥ ⎬ 3 n→∞ ⎩ ⎦ ⎭⎪ ⎪ i =1 ⎣ n n n A = lim 4 c. ⎧ 9⎡ 3 1 ⎤⎫ = lim ⎨48 − ⎢ 2 + + ⎥ ⎬ 2⎣ n n2 ⎦ ⎭ n→∞ ⎩ = 48 – 9 = 39 f ( x) dx = 1 0 f ( x) dx + 1 b. c. d. f ( x) dx = − 1 2 2 0 0 3 f (u ) du = 3 0 2 2 0 0 2 0 g ( x) − 3 2 0 f ( x) dx = 2 f ( x ) dx = − x dx − 2 4 0 x dx f ( x) dx = 2(−4) = −8 0 f (u ) du = 3(2) = 6 2 −2 2 = −2 f ( x) dx d. 2 2 −2 −2 f ( x) dx = −4 [ 2 g ( x) − 3 f ( x)] dx =2 25. a. c. 1 0 b. Since f ( x ) ≤ 0 , f ( x ) = − f ( x ) and = –4 + 2 = –2 0 4 ( x − x ) dx = 4 ⎧ 27 ⎡ n(n + 1)(2n + 1) ⎤ ⎫ = lim ⎨48 − ⎢ ⎥⎬ 6 n→∞ ⎩ ⎦⎭ n3 ⎣ 2 4 0 ⎡1 2 ⎤ ⎢2 x ⎥ − 6 = 8 – 6 = 2 ⎣ ⎦0 ⎧⎪ 48 n 27 n 2 ⎫⎪ 1− i ⎬ = lim ⎨ 3 n→∞ ⎩ ⎪ n i =1 n i =1 ⎭⎪ 23. a. x dx = 1 + 2 + 3 = 6 0 2 −2 2 −2 =2 0 f ( x ) dx f ( x ) dx = 8 g ( x) dx = 0 [ f ( x) + f (− x)] dx 2 0 f ( x) dx + 2 2 0 f ( x) dx = 4(–4) = –16 f ( x) dx = 2(–3) – 3(2) = –12 Instructor’s Resource Manual Section 4.7 291 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2 e. 0 [ 2 g ( x) + 3 f ( x)] dx =2 2 0 2 g ( x) dx + 3 f ( x) dx 0 = 2(5) + 3(–4) = –2 0 f. 26. 27. −2 100 −100 −1 −4 g ( x) dx = − 2 g ( x) dx = −5 0 c. G ( x) = − d. G ( x) = e. G ( x) = G ( x) = f. =− −1 ⎡ x3 ⎤ = 9c 2 ⎣ ⎦ −4 G ( x) = G ( x) = c. g ( x ) dg (u ) 0 du = [ g (u )]0g ( x ) x 0 −x 0 f (−t ) dt = x 0 f (u )(− du ) f (u ) du 4 0 30. a. x dx = 1 2 x +1 3 2 1 x dx b. 2 ⎡ 3 / 2 ⎤ 4 16 = x ⎦0 3 3⎣ 3 1 26 = ⎡ x3 ⎤ = ⎣ ⎦ 1 3 3 2x 4 x +1 3x2 6 x +1 31. − 2x 1 dt − dt 1 t t 1 1 ⋅5 − ⋅2 = 0 f (x) = 5x 2x f ( x) = 1 2 x +1 5x 1 2x t dt = 5x 1 1 G ( x ) = f ( x + 1) − f ( x ) b. 32. Left Riemann Sum: 2 1 1 1 + x4 Right Riemann Sum: 2 1 1+ x 4 f ''(c) = En = − dx ≈ dx ≈ 0.125[ f ( x0 ) + f ( x1 ) + 2 1 1 1 + x4 Midpoint Riemann Sum: 1 f (t ) dt G ( x ) = sin 2 x 29. a. 33. 1 f ( x) x G ( x) = − f ( x) c2 = 7 c = − 7 ≈ −2.65 b. x 0 f ( z ) dz + G ( x) = g ( g ( x)) g ( x) 3 x 2 dx = 3c 2 (−1 + 4) G ( x) = x x2 0 du = g ( g ( x)) − g (0) ( x3 + sin 5 x) dx = 0 28. a. 1 dx ≈ 0.125[ f ( x1 ) + f ( x2 ) + 2 1 1 1 + x4 (1 + c 4 )3 (2 − 1)3 (12)82 ≤ f ''(c) = + f ( x8 )] ≈ 0.1767 dx ≈ 0.125[ f ( x0.5 ) + f ( x1.5 ) + 0.125 [ f ( x0 ) + 2 f ( x1 ) + 2 4c 2 (5c 4 − 3) + f ( x7 )] ≈ 0.2319 + 2 f ( x7 ) + f ( x8 )] ≈ 0.2043 ( (4)(22 ) (5)(24 ) − 3 (1 + 1 ) 4 + f ( x7.5 )] ≈ 0.2026 3 ) = 154 1 154 f ''(c) ≤ ≈ 0.2005 (12)(64) 768 Remark: A plot of f '' shows that in fact f '' ( c ) < 1.5 , so En < 0.002 . 292 Section 4.7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 34. 4 0 1 0.5 dx ≈ [ f ( x0 ) + 4 f ( x1 ) + 2 f ( x2 ) + 1 + 2x 3 4 f ( ) (c) = En = − 35. n2 > (1 + 2c )5 ≤ 384 ( 4 − 0 )5 5 ( 4 ) c ≤ 4 ⋅ 384 = 8 f ⋅ ( ) 180 ⋅ 84 180 ⋅ 84 15 4c 2 (5c 4 − 3) f ''(c) = En = − 384 + 4 f ( x7 ) + f ( x8 )] ≈ 1.1050 4 3 (1 + c ) (2 − 1)3 12n 2 ( (4)(22 ) (5)(24 ) + 3 ≤ f ''(c) = (1 + 1 ) 4 1 12n 2 3 f ''(c) ≤ 166 12n 2 ) = 166 < 0.0001 166 ≈ 138,333 so n > 138,333 ≈ 371.9 Round up to n = 372 . (12)(0.0001) Remark: A plot of f '' shows that in fact f '' ( c ) < 1.5 which leads to n = 36 . 36. 4 f ( ) (c) = En = − n4 > 384 (1 + 2c )5 ≤ 384 ( 4 − 0 )5 5 ( 4 ) c ≤ 4 ⋅ 384 < 0.0001 f ⋅ ( ) 180 ⋅ n 4 180 ⋅ n 4 45 ⋅ 384 ≈ 21,845,333 , so n ≈ 68.4 . Round up to n = 69 . 180 ( 0.0001) 37. The integrand is decreasing and concave up. Therefore, we get: Midpoint Rule, Trapezoidal rule, Left Riemann Sum Review and Preview Problems 2 1. 1 ⎛1⎞ 1 1 1 −⎜ ⎟ = − = 2 ⎝2⎠ 2 4 4 2. x − x 2 6. ( ( x + h − x )2 + ( x + h )2 − x 2 ( = h 2 + 2 xh + h 2 ) ) 2 2 7. V = (π ⋅ 22 )0.4 = 1.6π 3. the distance between (1, 4 ) and ( 3 4, 4) is 3 4 -1 4. the distance between ⎛y ⎞ 3 y , y is 3 y − y ⎜ , y ⎟ and 4 ⎝4 ⎠ ( ) 8. V = [π (42 − 12 )]1 = 15π 9. V = [π (r22 − r12 )]Δx 10. V = [π (52 − 4.52 )]6 = 28.5π 5. the distance between (2,4) and (1,1) is (2 − 1)2 + (4 − 1) 2 = 10 Instructor’s Resource Manual Review and Preview 293 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 11. = 12. 13. 2 ⎡ x5 x 4 ⎤ x − 2 x + 2 dx = ⎢ − + 2x⎥ −1 2 ⎢⎣ 5 ⎥⎦ −1 2 ( 4 ) 3 12 ⎛ 27 ⎞ 51 −⎜− ⎟ = 5 ⎝ 10 ⎠ 10 3 23 y dy 0 3 3 3 = ⋅ y 5 3 = ⋅ 35 3 ≈ 3.74 0 5 5 2 2⎛ ⎡ x2 x4 ⎞ x3 x5 ⎤ 16 + ⎜1 − ⎟ dx = ⎢ x − + ⎥ = ⎟ 0⎜ 2 16 6 80 ⎝ ⎠ ⎣⎢ ⎦⎥ 0 15 14. Let u = 1 + 94 x; then du = 94 dx and 1+ 9 4 x dx = 4 9 8 ⎛ 9 ⎞ = ⎜1 + x ⎟ 27 ⎝ 4 ⎠ 3 2 u du = 4 2 32 u +C 93 +C 4 Thus, = 294 4 1 3 ⎤ ⎡ 9 8 ⎛ 9 ⎞ 2⎥ ⎢ 1 + x dx = 1+ x ⎢ 27 ⎜⎝ 4 ⎟⎠ ⎥ 4 ⎣ ⎦1 8 ⎛ 3 2 133 2 ⎞ ⎜ 10 − ⎟ ≈ 7.63 27 ⎜⎝ 8 ⎟⎠ Review and Preview Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 5 CHAPTER Applications of the Integral 6. To find the intersection points, solve 5.1 Concepts Review 1. b f ( x)dx; − a b a x + 4 = x2 − 2 . f ( x)dx x2 − x − 6 = 0 (x + 2)(x – 3) = 0 x = –2, 3 Slice vertically. ΔA ≈ ⎡ ( x + 4) − ( x 2 − 2) ⎤ Δx = (− x 2 + x + 6)Δx ⎣ ⎦ 2. slice, approximate, integrate 3. g ( x) − f ( x); f ( x) = g ( x) 4. d c [ q( y) − p( y )] dy A= Problem Set 5.1 1. Slice vertically. 9 ⎛ ⎞ ⎛8 ⎞ 125 ⎜ −9 + + 18 ⎟ − ⎜ + 2 − 12 ⎟ = 2 6 ⎝ ⎠ ⎝3 ⎠ ΔA ≈ ( x 2 + 1)Δx A= 2 ⎡1 ⎤ ( x 2 + 1)dx = ⎢ x3 + x ⎥ = 6 –1 3 ⎣ ⎦ −1 2 7. Solve x3 − x 2 − 6 x = 0 . 2. Slice vertically. ΔA ≈ ( x3 − x + 2)Δx A= 2 33 1 ⎡1 ⎤ ( x3 − x + 2)dx = ⎢ x 4 − x 2 + 2 x ⎥ = −1 2 ⎣4 ⎦ −1 4 2 A= −2 = 2 ΔA ≈ −( x 2 + 2 x − 3)Δx = (− x 2 − 2 x + 3)Δx 1 32 ⎡ 1 ⎤ (− x 2 − 2 x + 3)dx = ⎢ − x3 − x 2 + 3 x ⎥ = −3 ⎣ 3 ⎦ −3 3 5. To find the intersection points, solve 2 – x 2 = x . 2 1 1 ⎡ 1 ⎤ (– x 2 – x + 2)dx = ⎢ – x3 – x 2 + 2 x ⎥ –2 3 2 ⎣ ⎦ −2 1 1 8 9 ⎛ ⎞ ⎛ ⎞ = ⎜ − − + 2⎟ − ⎜ − 2 − 4⎟ = ⎝ 3 2 ⎠ ⎝3 ⎠ 2 1 Instructor’s Resource Manual 3 0 (− x3 + x 2 + 6 x)dx 3 ⎡ ⎛ 8 ⎞ ⎤ ⎡ 81 ⎤ = ⎢ 0 − ⎜ 4 + − 12 ⎟ ⎥ + ⎢ − + 9 + 27 − 0 ⎥ 3 ⎠⎦ ⎣ 4 ⎦ ⎣ ⎝ 16 63 253 = + = 3 4 12 8. To find the intersection points, solve − x + 2 = x2 . x2 + x − 2 = 0 (x + 2)(x – 1) = 0 x = –2, 1 Slice vertically. ΔA ≈ ⎡ (− x + 2) − x 2 ⎤ Δx = (− x 2 − x + 2)Δx ⎣ ⎦ x + x−2 = 0 (x + 2)(x – 1) = 0 x = –2, 1 Slice vertically. ΔA ≈ ⎡(2 − x 2 ) − x ⎤ Δx = (− x 2 − x − 2)Δx ⎣ ⎦ A= ( x3 − x 2 − 6 x)dx + 0 4. Slice vertically. 1 0 −2 1 1 ⎡ 1 ⎤ ⎡1 ⎤ + ⎢ − x 4 + x3 + 3 x 2 ⎥ = ⎢ x 4 − x3 − 3 x 2 ⎥ 4 3 4 3 ⎣ ⎦ −2 ⎣ ⎦0 ⎛8 ⎞ ⎛ 8 ⎞ 40 = ⎜ + 2 + 4⎟ − ⎜ − + 2 − 4⎟ = 3 3 ⎝ ⎠ ⎝ ⎠ 3 A= ΔA1 ≈ ( x3 − x 2 − 6 x)Δx A = A1 + A2 1 ⎡1 ⎤ ( x + x + 2)dx = ⎢ x3 + x 2 + 2 x ⎥ 2 ⎣3 ⎦ −2 2 x( x 2 − x − 6) = 0 x(x + 2)(x – 3) = 0 x = –2, 0, 3 Slice vertically. ΔA2 ≈ −( x3 − x 2 − 6 x)Δx = (− x3 + x 2 + 6 x)Δx 3. Slice vertically. ΔA ≈ ⎡ ( x 2 + 2) − (− x) ⎤ Δx = ( x 2 + x + 2)Δx ⎣ ⎦ 2 3 1 ⎡ 1 ⎤ (− x 2 + x + 6) dx = ⎢ − x3 + x 2 + 6 x ⎥ −2 2 ⎣ 3 ⎦ −2 3 1 1 ⎡ 1 ⎤ (− x 2 − x + 2)dx = ⎢ − x3 − x 2 + 2 x ⎥ −2 3 2 ⎣ ⎦ −2 1 1 8 9 ⎛ ⎞ ⎛ ⎞ = ⎜ − − + 2⎟ − ⎜ − 2 − 4⎟ = ⎝ 3 2 ⎠ ⎝3 ⎠ 2 A= 1 Section 5.1 295 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 9. To find the intersection points, solve 13. 2 y +1 = 3 – y . y2 + y − 2 = 0 (y + 2)(y – 1) = 0 y = –2, 1 Slice horizontally. ΔA ≈ ⎡ (3 − y 2 ) − ( y + 1) ⎤ Δy = (− y 2 − y + 2)Δy ⎣ ⎦ 1 1 1 (– y 2 – y + 2)dy = ⎡⎢ – y 3 – y 2 + 2 y ⎤⎥ –2 2 ⎣ 3 ⎦ −2 ⎛ 1 1 ⎞ ⎛8 ⎞ 9 = ⎜ − − + 2⎟ − ⎜ − 2 − 4⎟ = 3 2 3 ⎝ ⎠ ⎝ ⎠ 2 A= 1 ΔA ≈ −( x − 4)( x + 2)Δx = (− x 2 + 2 x + 8)Δx 10. To find the intersection point, solve y 2 = 6 − y . y2 + y − 6 = 0 (y + 3)(y – 2) = 0 y = –3, 2 Slice horizontally. ΔA ≈ ⎡ (6 − y ) − y 2 ⎤ Δy = (− y 2 − y + 6)Δy ⎣ ⎦ A= 2 0 3 ⎡ 1 ⎤ (− x 2 + 2 x + 8)dx = ⎢ − x3 + x 2 + 8 x ⎥ 0 ⎣ 3 ⎦0 = –9 + 9 + 24 = 24 Estimate the area to be (3)(8) = 24. A= 3 14. 2 22 1 ⎡ 1 ⎤ ( − y 2 − y + 6)dy = ⎢ − y 3 − y 2 + 6 y ⎥ = 3 2 ⎣ 3 ⎦0 ΔA ≈ −( x 2 − 4 x − 5)Δx = (− x 2 + 4 x + 5)Δx 11. 4 ⎡ 1 ⎤ (− x 2 + 4 x + 5)dx = ⎢ − x3 + 2 x 2 + 5 x ⎥ −1 ⎣ 3 ⎦ −1 ⎛ 64 ⎞ ⎛1 ⎞ 100 = ⎜ − + 32 + 20 ⎟ − ⎜ + 2 − 5 ⎟ = ≈ 33.33 3 3 3 ⎝ ⎠ ⎝ ⎠ 1 ⎛ 1⎞ Estimate the area to be (5) ⎜ 6 ⎟ = 32 . 2 ⎝ 2⎠ A= 15. 1 ⎞ ⎛ ΔA ≈ ⎜ 3 − x 2 ⎟ Δx 3 ⎠ ⎝ A= 4 3 1 ⎤ ⎛ 1 2⎞ ⎡ 3 − x ⎟ dx = ⎢3 x − x3 ⎥ = 9 − 3 = 6 0 ⎜⎝ 3 ⎠ 9 ⎦0 ⎣ 3 Estimate the area to be (3)(2) = 6. 12. 1 ΔA ≈ − ( x 2 − 7)Δx 4 A= ΔA ≈ (5 x − x 2 )Δx 3 1 ⎤ ⎡5 (5 x − x 2 )dx = ⎢ x 2 − x3 ⎥ ≈ 11.33 1 3 ⎦1 ⎣2 ⎛ 1⎞ Estimate the area to be (2) ⎜ 5 ⎟ = 11 . ⎝ 2⎠ A= 296 3 Section 5.1 2 1 1 ⎡1 ⎤ − ( x 2 − 7)dx = − ⎢ x3 − 7 x ⎥ 0 4 4 ⎣3 ⎦0 2 1⎛8 ⎞ 17 = − ⎜ − 14 ⎟ = ≈ 2.83 4⎝3 ⎠ 6 ⎛ 1⎞ Estimate the area to be (2) ⎜ 1 ⎟ = 3 . ⎝ 2⎠ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 16. 19. ΔA ≈ [ x − ( x − 3)( x − 1) ] Δx ΔA1 ≈ − x3 Δx = ⎡ x − ( x 2 − 4 x + 3) ⎤ Δx = (− x 2 + 5 x − 3)Δx ⎣ ⎦ To find the intersection points, solve x = (x – 3)(x – 1). 3 ΔA2 ≈ x Δx A = A1 + A2 = 0 −3 0 − x3 dx + 3 0 x3 dx x2 − 5x + 3 = 0 3 ⎡ 1 ⎤ ⎡1 ⎤ ⎛ 81 ⎞ ⎛ 81 ⎞ 81 = ⎢− x4 ⎥ + ⎢ x4 ⎥ = ⎜ ⎟ + ⎜ ⎟ = ⎣ 4 ⎦ −3 ⎣ 4 ⎦ 0 ⎝ 4 ⎠ ⎝ 4 ⎠ 2 = 40.5 Estimate the area to be (3)(7) + (3)(7) = 42. 5 ± 25 − 12 2 5 ± 13 x= 2 x= 17. A= 5+ 13 2 5− 13 2 (− x 2 + 5 x − 3)dx 5+ 13 5 13 13 ⎡ 1 ⎤ 2 = ⎢ − x3 + x 2 − 3 x ⎥ = ≈ 7.81 − 5 13 2 6 ⎣ 3 ⎦ 2 3 ΔA1 ≈ − x Δx Estimate the area to be ΔA2 ≈ 3 x Δx A = A1 + A2 = 0 −2 − 3 x dx + 0 2 3 0 x dx 1 (4)(4) = 8 . 2 20. 2 ⎛ 3 3 2 ⎞ ⎛ 33 2 ⎞ ⎡ 3 ⎤ ⎡3 ⎤ = ⎢− x4 / 3 ⎥ + ⎢ x4 / 3 ⎥ = ⎜ ⎟+⎜ ⎟ ⎣ 4 ⎦ −2 ⎣ 4 ⎦ 0 ⎜⎝ 2 ⎟⎠ ⎜⎝ 2 ⎟⎠ = 33 2 ≈ 3.78 Estimate the area to be (2)(1) + (2)(1) = 4. 18. ΔA ≈ ⎣⎡ x − ( x − 4) ⎦⎤ Δx = ( ) x − x + 4 Δx To find the intersection point, solve x = ( x − 4) . x = ( x − 4)2 x 2 − 9 x + 16 = 0 ΔA ≈ −( x − 10)Δx = (10 − x )Δx 9 2 ⎡ ⎤ (10 − x ) dx = ⎢10 x − x3 2 ⎥ 0 3 ⎣ ⎦0 = 90 – 18 = 72 Estimate the area to be 9 · 8 =72. A= 9 Instructor’s Resource Manual 9 ± 81 − 64 2 9 ± 17 x= 2 ⎛ 9 − 17 9 + 17 ⎞ is extraneous so x = . ⎟⎟ ⎜⎜ x = 2 2 ⎝ ⎠ x= Section 5.1 297 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. A= 9+ 17 2 0 ( x2 + 5x + 6 = 0 (x + 3)(x + 2) = 0 x = –3, –2 ) x − x + 4 dx 9+ 17 1 ⎡2 ⎤ 2 = ⎢ x3 / 2 − x 2 + 4 x ⎥ 2 ⎣3 ⎦0 2 ⎛ 9 + 17 ⎞ = ⎜⎜ ⎟ 3 ⎝ 2 ⎟⎠ 3/ 2 + (− x 2 − 5 x − 6)dx −2 23 17 − ≈ 15.92 4 4 Estimate the area to be −2 −3 5 ⎡ 1 ⎤ = ⎢ − x3 − x 2 − 6 x ⎥ 2 ⎣ 3 ⎦ −3 45 ⎛8 ⎞ ⎛ ⎞ 1 = ⎜ − 10 + 12 ⎟ − ⎜ 9 − + 18 ⎟ = ≈ 0.17 3 2 ⎝ ⎠ ⎝ ⎠ 6 1 ⎛ 2⎞ 1 Estimate the area to be (1) ⎜ 5 − 4 ⎟ = . 2 ⎝ 3⎠ 6 2 ⎛ 9 + 17 ⎞ 1 ⎛ 9 + 17 ⎞ − ⎜⎜ ⎟⎟ + 4 ⎜⎜ ⎟⎟ 2⎝ 2 ⎠ ⎝ 2 ⎠ 3/ 2 2 ⎛ 9 + 17 ⎞ = ⎜⎜ ⎟ 3 ⎝ 2 ⎟⎠ A= 1 ⎛ 1 ⎞⎛ 1 ⎞ 1 ⎜ 5 ⎟ ⎜ 5 ⎟ = 15 . 2 ⎝ 2 ⎠⎝ 2 ⎠ 8 23. 21. ΔA ≈ (8 y − y 2 )Δy To find the intersection points, solve ΔA ≈ ⎡ − x 2 − ( x 2 − 2 x) ⎤ Δx = (−2 x 2 + 2 x)Δx ⎣ ⎦ To find the intersection points, solve 8 y − y2 = 0 . y(8 – y) = 0 y = 0, 8 − x2 = x2 − 2 x . 2 x2 − 2 x = 0 2x(x – 1) = 0 x = 0, x = 1 1 ⎡ 2 ⎤ (−2 x 2 + 2 x )dx = ⎢ − x3 + x 2 ⎥ 0 ⎣ 3 ⎦0 2 1 = − + 1 = ≈ 0.33 3 3 ⎛ 1 ⎞⎛ 1 ⎞ 1 Estimate the area to be ⎜ ⎟⎜ ⎟ = . ⎝ 2 ⎠⎝ 2 ⎠ 4 A= 1 8 1 ⎤ ⎡ (8 y − y 2 ) dy = ⎢ 4 y 2 − y 3 ⎥ 0 3 ⎦0 ⎣ 512 256 = 256 − = ≈ 85.33 3 3 Estimate the area to be (16)(5) = 80. A= 8 24. 22. ΔA ≈ (3 − y )( y + 1)Δy = (− y 2 + 2 y + 3)Δy ΔA ≈ ⎡( x 2 − 9) − (2 x − 1)( x + 3) ⎤ Δx ⎣ ⎦ = ⎡( x 2 − 9) − (2 x 2 + 5 x − 3) ⎤ Δx ⎣ ⎦ = (− x 2 − 5 x − 6)Δx To find the intersection points, solve 3 ⎡ 1 ⎤ (− y 2 + 2 y + 3)dy = ⎢ − y 3 + y 2 + 3 y ⎥ −1 ⎣ 3 ⎦ −1 ⎛1 ⎞ 32 = (−9 + 9 + 9) − ⎜ + 1 − 3 ⎟ = ≈ 10.67 ⎝3 ⎠ 3 ⎛ 1⎞ Estimate the area to be (4) ⎜ 2 ⎟ = 10 . ⎝ 2⎠ A= 3 (2 x − 1)( x + 3) = x 2 − 9 . 298 Section 5.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 25. 27. ΔA ≈ ⎡(3 − y 2 ) − 2 y 2 ⎤ Δy = (−3 y 2 + 3)Δy ⎣ ⎦ To find the intersection points, solve ΔA ≈ ⎡(−6 y 2 + 4 y ) − (2 − 3 y ) ⎤ Δy ⎣ ⎦ 2 y2 = 3 − y2 . = (−6 y 2 + 7 y − 2)Δy To find the intersection points, solve 3y2 − 3 = 0 3(y + 1)(y – 1) = 0 y = –1, 1 −6 y 2 + 4 y = 2 − 3 y. 6 y2 − 7 y + 2 = 0 (2 y − 1)(3 y − 2) = 0 1 2 y= , 2 3 2/3 7 ⎡ ⎤ (−6 y 2 + 7 y − 2)dy = ⎢ −2 y 3 + y 2 − 2 y ⎥ 1/ 2 2 ⎣ ⎦1/ 2 1 16 14 4 1 7 ⎛ ⎞ ⎛ ⎞ ≈ 0.0046 = ⎜ − + − ⎟ − ⎜ − + − 1⎟ = 216 ⎝ 27 9 3 ⎠ ⎝ 4 8 ⎠ Estimate the area to be 1 ⎛ 1 ⎞⎛ 1 ⎞ 1 ⎛ 1 ⎞⎛ 1 ⎞ 1 . ⎜ ⎟⎜ ⎟ − ⎜ ⎟⎜ ⎟ = 2 ⎝ 2 ⎠⎝ 5 ⎠ 2 ⎝ 2 ⎠⎝ 6 ⎠ 120 A= 1 1 (−3 y 2 + 3)dy = ⎡ − y 3 + 3 y ⎤ ⎣ ⎦ −1 = (–1 + 3) – (1 – 3) = 4 Estimate the value to be (2)(2) = 4. A= 2/3 26. −1 28. ΔA ≈ ⎡(8 − 4 y 4 ) − (4 y 4 ) ⎤ Δy = (8 − 8 y 4 )Δy ⎣ ⎦ To find the intersection points, solve 4 y4 = 8 − 4 y4 . 8 y4 = 8 y4 = 1 y = ±1 1 8 ⎤ ⎡ (8 − 8 y 4 )dy = ⎢8 y − y 5 ⎥ −1 5 ⎦ −1 ⎣ 8 ⎞ 64 ⎛ 8⎞ ⎛ = ⎜ 8 − ⎟ − ⎜ −8 + ⎟ = = 12.8 5⎠ ⎝ 5⎠ 5 ⎝ ⎛ 1⎞ Estimate the area to be (8) ⎜ 1 ⎟ = 12 . ⎝ 2⎠ ΔA ≈ ⎡( y + 4) − ( y 2 − 2 y ) ⎤ Δy = (− y 2 + 3 y + 4)Δy ⎣ ⎦ To find the intersection points, solve A= y2 − 2 y = y + 4 . y2 − 3y − 4 = 0 (y + 1)(y – 4) = 0 y = –1, 4 1 4 3 ⎡ 1 ⎤ (− y 2 + 3 y + 4)dy = ⎢ − y 3 + y 2 + 4 y ⎥ −1 2 ⎣ 3 ⎦ −1 ⎛ 64 ⎞ ⎛1 3 ⎞ 125 = ⎜ − + 24 + 16 ⎟ − ⎜ + − 4 ⎟ = ≈ 20.83 3 3 2 6 ⎝ ⎠ ⎝ ⎠ Estimate the area to be (7)(3) = 21. A= 4 Instructor’s Resource Manual Section 5.1 299 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 29. 30. y = x3 y = x+6 2y + x = 0 An equation of the line through (–1, 4) and (5, 1) 1 7 is y = − x + . An equation of the line through 2 2 (–1, 4) and (2, –2) is y = –2x + 2. An equation of the line through (2, –2) and (5, 1) is y = x – 4. Two integrals must be used. The left-hand part of the triangle has area 2 ⎡ 1 2 ⎛3 7 3⎞ ⎤ − x + − (−2 x + 2) ⎥ dx = ⎜ x + ⎟ dx . −1 ⎢⎣ 2 − 1 2 2⎠ ⎝2 ⎦ The right-hand part of the triangle has area 5⎡ 1 5⎛ 3 7 15 ⎞ ⎤ − x + − ( x − 4) ⎥ dx = ⎜ − x + ⎟ dx . ⎢ 2⎣ 2 2 ⎝ 2 2 2⎠ ⎦ The triangle has area 2 ⎛3 5⎛ 3 3⎞ 15 ⎞ x + ⎟ dx + ⎜ − x + ⎟ dx ⎜ −1 ⎝ 2 2⎝ 2 2⎠ 2⎠ Let R1 be the region bounded by 2y + x = 0, y = x + 6, and x = 0. 0 ⎡ ⎛ 1 ⎞⎤ A( R1 ) = ( x + 6) − ⎜ − x ⎟ ⎥ dx −4 ⎢⎣ ⎝ 2 ⎠⎦ ⎛3 ⎞ ⎜ x + 6 ⎟ dx ⎝2 ⎠ Let R2 be the region bounded by y = x + 6, 0 = −4 y = x3 , and x = 0. 2 2 0 0 ⎡( x + 6) − x3 ⎤ dx = ⎣ ⎦ A( R ) = A( R1 ) + A( R2 ) A( R2 ) = 0 = −4 ⎛3 ⎞ ⎜ x + 6 ⎟ dx + ⎝2 ⎠ 2 0 (− x3 + x + 6)dx (− x3 + x + 6)dx 0 2 31. 5 3 ⎤ 15 ⎤ ⎡3 ⎡ 3 = ⎢ x2 + x ⎥ + ⎢− x2 + x ⎥ 2 ⎦ −1 ⎣ 4 2 ⎦2 ⎣4 27 27 27 = + = = 13.5 4 4 2 2 1 ⎡3 ⎤ ⎡ 1 ⎤ = ⎢ x2 + 6 x ⎥ + ⎢− x4 + x2 + 6 x ⎥ 2 ⎣4 ⎦ −4 ⎣ 4 ⎦0 = 12 + 10 = 22 9 9 (3t 2 − 24t + 36)dt = ⎡t 3 − 12t 2 + 36t ⎤ = (729 – 972 + 324) – (–1 – 12 – 36) = 130 ⎣ ⎦ −1 −1 The displacement is 130 ft. Solve 3t 2 − 24t + 36 = 0 . 3(t – 2)(t – 6) = 0 t = 2, 6 ⎧⎪3t 2 − 24t + 36 t ≤ 2, t ≥ 6 V (t ) = ⎨ 2 ⎪⎩−3t + 24t − 36 2 < t < 6 9 −1 3t 2 − 24t + 36 dt = 2 2 −1 (3t 2 − 24t + 36) dt + 6 6 2 (−3t 2 + 24t − 36) dt + 9 6 (3t 2 − 24t + 36) dt 9 = ⎡t 3 − 12t 2 + 36t ⎤ + ⎡ −t 3 + 12t 2 − 36t ⎤ + ⎡t 3 − 12t 2 + 36t ⎤ = 81 + 32 + 81 = 194 ⎣ ⎦ −1 ⎣ ⎦2 ⎣ ⎦6 The total distance traveled is 194 feet. 32. 300 3π / 2 1 ⎞ 3π ⎛1 ⎞ ⎡1 1 ⎤ ⎛ 3π 1 ⎞ ⎛ + sin 2t ⎟ dt = ⎢ t − cos 2t ⎥ = ⎜ + ⎟ −⎜0 − ⎟ = +1 ⎜ 0 2⎠ 4 ⎝2 ⎠ ⎣2 2 ⎦0 ⎝ 4 2⎠ ⎝ 3π 1 3π + 1 ≈ 3.36 feet . Solve + sin 2t = 0 for 0 ≤ t ≤ . The displacement is 4 2 2 1 7π 11π 7π 11π , sin 2t = − ⇒ 2t = , ⇒t= 6 6 2 12 12 3π / 2 Section 5.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 7 π 11π 3π ⎧1 0≤t≤ , ≤t≤ ⎪⎪ 2 + sin 2t 1 12 12 2 + sin 2t = ⎨ 1 7 11 π π 2 ⎪− − sin 2t <t < ⎪⎩ 2 12 12 3π / 2 1 7 π /12 ⎛ 1 11π /12 ⎛ 1 ⎞ ⎞ + sin 2t dt = ⎜ + sin 2t ⎟ dt + 7 π /12 ⎜ − − sin 2t ⎟ dt + 0 0 2 ⎝2 ⎠ ⎝ 2 ⎠ 7 π /12 ⎡1 1 ⎤ = ⎢ t − cos 2t ⎥ ⎣2 2 ⎦0 11π /12 3π / 2 11π /12 ⎛1 ⎞ ⎜ + sin 2t ⎟ dt ⎝2 ⎠ 3π / 2 ⎡ 1 1 ⎤ ⎡1 1 ⎤ + ⎢ − t + cos 2t ⎥ + ⎢ t − cos 2t ⎥ ⎣ 2 2 ⎦ 7 π /12 ⎣ 2 2 ⎦11π /12 ⎛ 7π 3 1⎞ ⎛ π 3 ⎞ ⎛ 7π 3 1 ⎞ 5π = ⎜⎜ + + ⎟⎟ + ⎜⎜ − + + + ⎟⎟ = + 3 +1 ⎟⎟ + ⎜⎜ ⎝ 24 4 2 ⎠ ⎝ 6 2 ⎠ ⎝ 24 4 2 ⎠ 12 5π The total distance traveled is + 3 + 1 ≈ 4.04 feet. 12 33. s (t ) = v(t )dt = (2t − 4)dt = t 2 − 4t + C c. Since s(0) = 0, C = 0 and s (t ) = t 2 − 4t. s = 12 when t = 6, so it takes the object 6 seconds to get s = 12. ⎧4 − 2t 0 ≤ t < 2 2t − 4 = ⎨ ⎩2t − 4 2 ≤ t line that bisects the area is between y = 2t − 4 dt = ⎡ −t 2 + 4t ⎤ = 4, so the object ⎣ ⎦0 travels a distance of 4 cm in the first two seconds. 0 x 2 = 2−2 d ; 2−2 d = x 2t − 4 dt = ⎡t 2 − 4t ⎤ = x 2 − 4 x + 4 ⎣ ⎦2 361 ≈ 0.627 . 576 The line y = 0.627 approximately bisects the area. takes 2 + 2 2 ≈ 4.83 seconds to travel a total distance of 12 centimeters. A= 6 −2 x dx 1 b. Find c so that 6 1 5 ⎡ 1⎤ = ⎢− ⎥ = − + 1 = 6 6 ⎣ x ⎦1 c −2 1 x dx = 5 . 12 1 ⎡ 1⎤ x dx = ⎢ − ⎥ = 1 − c ⎣ x ⎦1 1 5 12 1− = , c = c 12 7 12 The line x = bisects the area. 7 c −2 35. Equation of line through (–2, 4) and (3, 9): y=x+6 Equation of line through (2, 4) and (–3, 9): y = –x + 6 A( A) = c 1 5 ; 12 d= x 2 − 4 x + 4 = 8 when x = 2 + 2 2, so the object 34. a. 1 36 and y = 1, so we find d such that 1 1 1 5 1 1 dy = ; dy = ⎡⎣ 2 y ⎤⎦ d y d 12 d y 2 2 Slicing the region horizontally, the area is 1 1 5 5 ⎛ 1 ⎞ < dy + ⎜ ⎟ (5) . Since the 1/ 36 y 36 12 ⎝ 36 ⎠ = 0 –3 0 –3 [9 – (– x + 6)]dx + (3 + x)dx + 3 0 3 0 [9 – ( x + 6)]dx (3 – x)dx 0 3 1 ⎤ 1 ⎤ 9 9 ⎡ ⎡ = ⎢3 x + x 2 ⎥ + ⎢3 x – x 2 ⎥ = + = 9 2 2 ⎣ ⎦ −3 ⎣ ⎦0 2 2 A( B ) = –2 –3 [(– x + 6) – x 2 ]dx + = –2 –3 0 –2 [(– x + 6) – ( x + 6)]dx (– x 2 – x + 6)dx + 0 –2 (–2 x)dx −2 0 1 37 ⎡ 1 ⎤ = ⎢ – x3 – x 2 + 6 x ⎥ + ⎡ – x 2 ⎤ = ⎣ ⎦ −2 2 6 ⎣ 3 ⎦ −3 Instructor’s Resource Manual Section 5.1 301 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. A(C ) = A( B) = A( D) = 0 –2 π/6 37 (by symmetry) 6 [( x + 6) – x 2 ]dx + 2 0 [(– x + 6) – x 2 ]dx 0 2 1 1 ⎡ 1 ⎤ ⎡ 1 ⎤ = ⎢ – x3 + x 2 + 6 x ⎥ + ⎢ – x3 – x 2 + 6 x ⎥ 2 2 ⎣ 3 ⎦ −2 ⎣ 3 ⎦0 44 = 3 A(A) + A(B) + A(C) + A(D) = 36 A( A + B + C + D) = 3 ⎡1 ⎤ = ⎢ x + cos x ⎥ ⎣2 ⎦0 5π / 6 1 ⎤ ⎡ + ⎢ − cos x − x ⎥ 2 ⎦π / 6 ⎣ 13π / 6 17 π / 6 1 ⎤ ⎡1 ⎤ ⎡ + ⎢ x + cos x ⎥ + ⎢ − cos x − x ⎥ 2 ⎦13π / 6 ⎣2 ⎦ 5π / 6 ⎣ ⎛π 3 ⎞ ⎛ π⎞ ⎛ 2π ⎞ = ⎜⎜ + − 1⎟⎟ + ⎜ 3 − ⎟ + ⎜ 3 + ⎟ 12 2 3 3 ⎠ ⎝ ⎠ ⎝ ⎝ ⎠ π⎞ π 7 3 ⎛ +⎜ 3 − ⎟ = + − 1 ≈ 5.32 3 ⎠ 12 2 ⎝ 1 ⎤ ⎡ (9 – x 2 )dx = ⎢9 x – x3 ⎥ –3 3 ⎦ −3 ⎣ 3 = 36 5.2 Concepts Review 36. Let f(x) be the width of region 1 at every x. ΔA1 ≈ f ( x)Δx, so A1 = b a f ( x)dx . Let g(x) be the width of region 2 at every x. ΔA2 ≈ g ( x)Δx, so A2 = b a g ( x)dx . Since f(x) = g(x) at every x in [a, b], A1 = b a f ( x)dx = b a g ( x)dx = A2 . 37. The height of the triangular region is given by for 0 ≤ x ≤ 1 . We need only show that the height of the second region is the same in order to apply Cavalieri'’s Principle. The height of the second region is h2 = ( x 2 − 2 x + 1) − ( x 2 − 3 x + 1) = x2 − 2 x + 1 − x2 + 3x − 1 = x for 0 ≤ x ≤ 1. Since h1 = h2 over the same closed interval, we can conclude that their areas are equal. 38. Sketch the graph. 1. πr 2 h 2. π( R 2 − r 2 )h 3. πx 4 Δx 4. π[( x 2 + 2)2 − 4]Δx Problem Set 5.2 1. Slice vertically. ΔV ≈ π( x 2 + 1) 2 Δx = π( x 4 + 2 x 2 + 1)Δx V =π 2 0 ( x 4 + 2 x 2 + 1)dx 2 2 ⎡1 ⎤ ⎛ 32 16 ⎞ 206π = π ⎢ x5 + x3 + x ⎥ = π ⎜ + + 2 ⎟ = 3 15 ⎣5 ⎦0 ⎝ 5 3 ⎠ ≈ 43.14 2. Slice vertically. ΔV ≈ π(− x 2 + 4 x)2 Δx = π( x 4 − 8 x3 + 16 x 2 )Δx V =π 3 0 ( x 4 − 8 x3 + 16 x 2 )dx 3 16 ⎤ ⎡1 = π ⎢ x5 − 2 x 4 + x3 ⎥ 3 ⎦0 ⎣5 1 17π . for 0 ≤ x ≤ 2 6 π 5π 13π 17π x= , , , 6 6 6 6 The area of the trapped region is π/6 ⎛ 1 5π / 6 ⎛ 1⎞ ⎞ ⎜ − sin x ⎟ dx + π / 6 ⎜ sin x − ⎟ dx 0 2 2⎠ ⎝ ⎠ ⎝ 13π / 6 ⎛ 1 17 π / 6 ⎛ 1⎞ ⎞ + − sin x ⎟ dx + ⎜ sin x − ⎟ dx π 5π / 6 ⎜⎝ 2 13 / 6 2⎠ ⎠ ⎝ Solve sin x = 302 Section 5.2 ⎛ 243 ⎞ = π⎜ − 162 + 144 ⎟ 5 ⎝ ⎠ 153π = ≈ 96.13 5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 3. a. Slice vertically. 6. 2 2 2 4 ΔV ≈ π(4 − x ) Δx = π(16 − 8 x + x )Δx V =π 2 0 (16 – 8 x 2 + x 4 )dx 256π ≈ 53.62 15 = b. Slice horizontally. x = 4– y Note that when x = 0, y = 4. ΔV ≈ π ( 4− y ) 2 ΔV ≈ π( x3 )2 Δx = πx6 Δx Δy = π(4 − y )Δy V =π 4 1 ⎤ ⎡ (4 – y )dy = π ⎢ 4 y – y 2 ⎥ 0 2 ⎦0 ⎣ = π(16 − 8) = 8π ≈ 25.13 4 V =π 4. a. 3 0 3 2187π ⎡1 ⎤ x 6 dx = π ⎢ x7 ⎥ = ≈ 981.52 7 7 ⎣ ⎦0 7. Slice vertically. ΔV ≈ π(4 − 2 x)2 Δx 0≤ x≤2 V =π = 2 ⎡ 1 ⎤ (4 − 2 x)2 dx = π ⎢ − (4 − 2 x)3 ⎥ 0 ⎣ 6 ⎦0 2 32π ≈ 33.51 3 2 ⎛ 1 ⎞ ⎛1⎞ ΔV ≈ π ⎜ ⎟ Δx = π ⎜ ⎟ Δx ⎝x⎠ ⎝ x2 ⎠ b. Slice vertically. y x = 2− 2 V =π 2 4 2 ≈ 0.79 y⎞ ⎛ ΔV ≈ π ⎜ 2 − ⎟ Δy 2 ⎝ ⎠ 0 ≤ y ≤ 4 4 ⎡ 1⎤ ⎛ 1 1⎞ π dx = π ⎢ − ⎥ = π ⎜ − + ⎟ = 2 ⎣ x ⎦2 ⎝ 4 2⎠ 4 x 1 8. 4 2 3 ⎡ 2⎛ y⎞ y⎞ ⎤ V = π ⎜ 2 − ⎟ dy = π ⎢− ⎜ 2 − ⎟ ⎥ 0⎝ 2⎠ 3 2⎠ ⎥ ⎣⎢ ⎝ ⎦0 16π = ≈ 16.76 3 4⎛ 5. ΔV ≈ π( x3 / 2 ) 2 Δx = πx3Δx V =π = ⎛ x2 ΔV ≈ π ⎜ ⎜ π ⎝ V= 4 0 3 2 3 ⎡1 ⎤ ⎛ 81 16 ⎞ x3 dx = π ⎢ x 4 ⎥ = π ⎜ − ⎟ ⎣ 4 ⎦2 ⎝ 4 4⎠ 65π ≈ 51.05 4 2 ⎞ x4 Δx ⎟ Δx = ⎟ π ⎠ 4 x4 1 ⎡1 ⎤ 1024 dx = ⎢ x5 ⎥ = ≈ 65.19 π π ⎣ 5 ⎦0 5π Instructor’s Resource Manual Section 5.2 303 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 12. 9. 2 2 ⎛ 1 ⎞ ⎛2⎞ ΔV ≈ π ⎜ ⎟ Δy = 4π ⎜ Δy ⎜ y 2 ⎟⎟ ⎝ y⎠ ⎝ ⎠ ΔV ≈ π ⎛⎜ 9 − x 2 ⎞⎟ Δx = π(9 − x 2 )Δx ⎝ ⎠ V =π 3 −2 3 1 ⎤ ⎡ (9 − x 2 )dx = π ⎢9 x − x3 ⎥ 3 ⎦ −2 ⎣ V = 4π ⎡ 8 ⎞ ⎤ 100π ⎛ = π ⎢(27 − 9) − ⎜ −18 + ⎟ ⎥ = ≈ 104.72 3 ⎠⎦ 3 ⎝ ⎣ 10. = 6 2 6 ⎡ 1⎤ ⎛ 1 1⎞ dy = 4π ⎢ − ⎥ = 4π ⎜ − + ⎟ 2 ⎝ 6 2⎠ y ⎣ y ⎦2 1 4π ≈ 4.19 3 13. ΔV ≈ π( x 2 / 3 ) 2 Δx = πx 4 / 3Δx V =π = 27 1 x 4/3 ( ΔV ≈ π 2 y 27 ⎡3 ⎤ ⎛ 6561 3 ⎞ dx = π ⎢ x 7 / 3 ⎥ = π ⎜ − ⎟ 7 7⎠ ⎣ ⎦1 ⎝ 7 6558π ≈ 2943.22 7 V = 4π 4 0 ) 2 Δy = 4πy Δy 4 ⎡1 ⎤ y dy = 4π ⎢ y 2 ⎥ = 32π ≈ 100.53 ⎣ 2 ⎦0 14. 11. ΔV ≈ π( y 2 ) 2 Δy = πy 4 Δy V =π 304 3 4 y dy 0 3 243π ⎡1 ⎤ = π ⎢ y5 ⎥ = ≈ 152.68 5 ⎣ 5 ⎦0 Section 5.2 ΔV ≈ π( y 2 / 3 )2 Δy = πy 4 / 3 Δy 27 6561π ⎡3 ⎤ y 4 / 3 dy = π ⎢ y 7 / 3 ⎥ = 0 7 ⎣7 ⎦0 ≈ 2944.57 V =π 27 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 18. Sketch the region. 15. y y = 6 x2 8 y = 6x −1 1 ΔV ≈ π( y 3 / 2 )2 Δy = πy 3Δy V =π 9 0 9 6561π ⎡1 ⎤ y dy = π ⎢ y 4 ⎥ = ≈ 5153.00 4 4 ⎣ ⎦0 3 16. x −4 To find the intersection points, solve 6 x = 6 x 2 . 6( x 2 − x) = 0 6x(x – 1) = 0 x = 0, 1 ΔV ≈ π ⎡ (6 x) 2 − (6 x 2 ) 2 ⎤ Δx = 36π( x 2 − x 4 )Δx ⎣ ⎦ 1 1 ⎤ ⎡1 ( x 2 − x 4 )dx = 36π ⎢ x3 − x5 ⎥ 0 3 5 ⎦0 ⎣ ⎛ 1 1 ⎞ 24π = 36π ⎜ − ⎟ = ≈ 15.08 5 ⎝3 5⎠ 1 V = 36π 19. Sketch the region. 2 ΔV ≈ π ⎛⎜ 4 − y 2 ⎞⎟ Δy = π(4 − y 2 )Δy ⎝ ⎠ V =π 2 1 ⎤ ⎡ (4 − y 2 )dy = π ⎢ 4 y − y 3 ⎥ −2 3 ⎦ −2 ⎣ 2 8 ⎞ ⎤ 32π ⎡⎛ 8 ⎞ ⎛ = π ⎢⎜ 8 − ⎟ − ⎜ −8 + ⎟ ⎥ = ≈ 33.51 3 3 ⎠⎦ 3 ⎠ ⎝ ⎣⎝ 17. The equation of the upper half of the ellipse is y = b 1− V =π a x2 a2 b2 −a a2 or y = b 2 a − x2 . a (a 2 − x 2 )dx 3 ⎤a b2 π ⎡ 2 x ⎢a x − ⎥ 2 3 ⎦⎥ a ⎣⎢ −a 2 ⎡⎛ 3⎞ ⎛ b π a a3 ⎞ ⎤ 4 a3 − ⎟ − ⎜ − a3 + ⎟ ⎥ = ab 2 π = ⎢ ⎜ 3 ⎟⎠ ⎜⎝ 3 ⎟⎠ ⎥⎦ 3 a 2 ⎢⎣⎜⎝ = Instructor’s Resource Manual To find the intersection points, solve x =2 x. 2 x2 = 4x 4 x 2 − 16 x = 0 x(x – 16) = 0 x = 0, 16 ⎡ ΔV ≈ π ⎢ 2 x ⎢⎣ ( ) 2 ⎛ x⎞ −⎜ ⎟ ⎝2⎠ 2⎤ ⎛ x2 ⎞ ⎥ Δx = π ⎜ 4 x − ⎟ Δx ⎜ 4 ⎟⎠ ⎥⎦ ⎝ 16 ⎛ ⎡ x2 ⎞ x3 ⎤ V =π ⎜ 4 x − ⎟ dx = π ⎢ 2 x 2 − ⎥ 0 ⎜ 4 ⎟⎠ 12 ⎦⎥ ⎝ ⎣⎢ 0 1024 ⎞ 512π ⎛ = π ⎜ 512 − ≈ 536.17 ⎟= 3 ⎠ 3 ⎝ 16 Section 5.2 305 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 20. Sketch the region. V =π 3 2 1 x + 3, y = x 2 + 5 16 16 Sketch the region. 22. y = r 1 ⎤ ⎡ (r 2 − x 2 )dx = π ⎢ r 2 x − x3 ⎥ r −h 3 ⎦ r −h ⎣ r To find the intersection point, solve 3 2 1 x + 3 = x2 + 5 . 16 16 1 2 x −2 = 0 8 1 = πh 2 (3r − h) 3 21. Sketch the region. y x 2 − 16 = 0 (x + 4)(x – 4) = 0 x = –4, 4 2 2 4 ⎡⎛ 1 ⎞ ⎛ 3 ⎞ ⎤ V = π ⎢⎜ x 2 + 5 − 2 ⎟ − ⎜ x 2 + 3 − 2 ⎟ ⎥ dx 0 ⎢⎝ 16 ⎠ ⎝ 16 ⎠ ⎦⎥ ⎣ 4 2 =π −1 1 2 x V =π = 4⎛ ⎡⎛ 1 4 3 2 ⎞ ⎢⎜ 256 x − 8 x + 9 ⎟ ⎠ ⎣⎝ ⎛ 9 4 3 2 ⎞⎤ x − x + 1⎟⎥ dx −⎜ 8 ⎝ 256 ⎠⎦ y y . To find the intersection points, solve = 4 2 y2 y = 16 4 y2 − 4 y = 0 y(y – 4) = 0 y = 0, 4 2 ⎡⎛ ⎤ ⎛ y y2 ⎞ y ⎞ ⎛ y ⎞2 ⎥ ⎢ ΔV ≈ π ⎜ − ⎜ ⎟ Δy = π ⎜ − ⎟ Δy ⎟ ⎜ 4 16 ⎟ ⎢⎜ 2 ⎟ ⎝ 4 ⎠ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ 4 0 4 1 5⎤ 1 ⎡ ⎞ x 8 − x 4 ⎟ dx = π ⎢8 x − 0 ⎜⎝ 32 ⎠ 160 ⎥⎦ 0 ⎣ 32 ⎞ 128π ⎛ = π ⎜ 32 − ⎟ = ≈ 80.42 5 ⎠ 5 ⎝ =π 4⎛ 23. 4 ⎡ y 2 y3 ⎤ y y2 ⎞ ⎜ − ⎟ dy = π ⎢ − ⎥ 0 ⎜ 4 16 ⎟ ⎝ ⎠ ⎣⎢ 8 48 ⎥⎦ 0 2π ≈ 2.0944 3 The square at x has sides of length 2 4 − x 2 , as shown. V= 2 ⎛ 2 4 − x 2 ⎞ dx = 2 4(4 − x 2 )dx ⎜ ⎟ −2 ⎝ −2 ⎠ 2 2 ⎡ x3 ⎤ ⎡⎛ 8 ⎞ ⎛ 8 ⎞ ⎤ 128 = 4 ⎢ 4 x − ⎥ = 4 ⎢ ⎜ 8 − ⎟ − ⎜ −8 + ⎟ ⎥ = 3 3 3 ⎠⎦ 3 ⎠ ⎝ ⎣⎝ ⎣⎢ ⎦⎥ −2 ≈ 42.67 306 Section 5.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 24. The area of each cross section perpendicular to 1 the x-axis is (4) ⎛⎜ 2 4 − x 2 ⎞⎟ = 4 4 − x 2 . ⎠ 2 ⎝ The area of a semicircle with radius 2 is 2 −2 V= 2 −2 4 4 − x 2 dx = 4(2π) = 8π ≈ 25.13 π/2 −π / 2 cos x . cos xdx = [sin x ]π−π/ 2/ 2 = 2 26. The area of each cross section perpendicular to the x-axis is [(1 − x 2 ) − (1 − x 4 )]2 = x8 − 2 x 6 + x 4 . V= 1 −1 ( x8 − 2 x6 + x 4 )dx 1 16 2 1 ⎤ ⎡1 = ⎢ x9 − x 7 + x5 ⎥ = ≈ 0.051 7 5 ⎦ −1 315 ⎣9 27. The square at x has sides of length 1 − x 2 . V= ⎛2 ⎞ = 2(π r 2 L) − 8 ⎜ r 3 ⎟ ⎝3 ⎠ 16 = 2π r 2 L − r 3 3 4 − x 2 dx = 2π . 25. The square at x has sides of length V= Thus, the volume inside the “+” for two cylinders of radius r and length L is V = vol. of cylinders - vol. of common region 30. From Problem 28, the volume of one octant of 2 the common region is r 3 . We can find the 3 volume of the “T” similarly. Since the “T” has one-half the common region of the “+” in Problem 28, the volume of the “T” is given by V = vol. of cylinders - vol. of common region ⎛2 ⎞ =(π r 2 )( L1 + L2 ) − 4 ⎜ r 3 ⎟ ⎝3 ⎠ With r = 2, L1 = 12, and L2 = 8 (inches), the volume of the “T” is V = vol. of cylinders - vol. of common region ⎛2 ⎞ =(π r 2 )( L1 + L2 ) − 4 ⎜ r 3 ⎟ ⎝3 ⎠ ⎛2 ⎞ = (π 22 )(12 + 8) − 4 ⎜ 23 ⎟ ⎝3 ⎠ 64 = 80π − in 3 3 3 ⎤1 ⎡ x 2 (1 − x 2 )dx = ⎢ x − ⎥ = ≈ 0.67 0 3 ⎦⎥ 3 ⎣⎢ 0 1 28. From Problem 27 we see that horizontal cross sections of one octant of the common region are squares. The length of a side at height y is 2 ≈ 229.99 in 3 2 r − y where r is the common radius of the cylinders. The volume of the “+” can be found by adding the volumes of each cylinder and subtracting off the volume of the common region (which is counted twice). The volume of one octant of the common region is r 2 1 (r − y 2 )dy = r 2 y − y 2 |0r 0 3 2 3 1 3 = r − r = r3 3 3 Thus, the volume of the “+” is V = vol. of cylinders - vol. of common region ⎛2 ⎞ =2(π r 2 l ) − 8 ⎜ r 3 ⎟ ⎝3 ⎠ 128 ⎛2 ⎞ = 2π (22 )(12) − 8 ⎜ (2)3 ⎟ = 96π − 3 3 ⎝ ⎠ ≈ 258.93 in 2 29. Using the result from Problem 28, the volume of one octant of the common region in the “+” is r 2 1 (r − y 2 )dy = r 2 y − y 2 |0r 0 3 1 2 = r3 − r3 = r3 3 3 Instructor’s Resource Manual 31. From Problem 30, the general form for the volume of a “T” formed by two cylinders with the same radius is V = vol. of cylinders - vol. of common region ⎛2 ⎞ =(π r 2 )( L1 + L2 ) − 4 ⎜ r 3 ⎟ ⎝3 ⎠ 8 = π r 2 ( L1 + L2 ) − r 3 3 32. The area of each cross section perpendicular to the x-axis is 1 ⎡1 π 2 ⎢⎣ 2 ( ) ⎤ x − x2 ⎥ ⎦ 2 π 4 ( x − 2 x5 / 2 + x). 8 π 1 4 V= ( x − 2 x5 / 2 + x)dx 8 0 = 1 = π ⎡1 5 4 7 / 2 1 2 ⎤ 9π x − x + x ⎥ = ≈ 0.050 ⎢ 8 ⎣5 7 2 ⎦ 0 560 Section 5.2 307 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 33. Sketch the region. 8 3 ⎡ 24 ⎤ = π ⎢ y5 / 3 − y 7 / 3 ⎥ 7 ⎣5 ⎦0 ⎛ 768 384 ⎞ 3456π = π⎜ − ≈ 310.21 ⎟= 7 ⎠ 35 ⎝ 5 b. Revolving about the line y = 8, the radius of the disk at x is 8 − x3 = 8 − x3 / 2 . V =π a. Revolving about the line x = 4, the radius of =π the disk at y is 4 − 3 y 2 = 4 − y 2 / 3 . V =π =π 8 0 8 0 8 b. Revolving about the line y = 8, the inner radius of the disk at x is 8 − x3 = 8 − x3 / 2 . =π 4 0 0 (64 − 16 x3 / 2 + x3 )dx 4 (16 − 8 y 2 / 3 + y 4 / 3 )dy 4 (8 − x3 / 2 )2 dx 32 1 ⎤ ⎡ = π ⎢64 x − x5 / 2 + x 4 ⎥ 5 4 ⎦0 ⎣ 1024 ⎡ ⎤ 576π = π ⎢ 256 − + 64 ⎥ = ≈ 361.91 5 5 ⎣ ⎦ (4 − y 2 / 3 ) 2 dy 24 5 / 3 3 7 / 3 ⎤ ⎡ = π ⎢16 y − y + y ⎥ 5 7 ⎣ ⎦0 768 384 ⎞ ⎛ = π ⎜ 128 − + ⎟ 5 7 ⎠ ⎝ 1024π = ≈ 91.91 35 V =π 4 0 4 0 ⎡82 − (8 − x3 / 2 )2 ⎤ dx ⎣ ⎦ (16 x3 / 2 − x3 )dx 4 1 ⎤ ⎡ 32 ⎛ 1024 ⎞ = π ⎢ x5 / 2 − x 4 ⎥ = π ⎜ − 64 ⎟ 5 4 ⎦0 ⎣5 ⎝ ⎠ 704π = ≈ 442.34 5 34. Sketch the region. 35. The area of a quarter circle with radius 2 is 2 0 2 0 4 − y 2 dy = π . ⎡ 2 4 − y 2 + 4 − y 2 ⎤ dy ⎢⎣ ⎥⎦ =2 2 0 4 − y 2 dy + 2 0 (4 − y 2 )dy 2 1 ⎤ ⎡ ⎛ 8⎞ = 2 π + ⎢ 4 y − y 3 ⎥ = 2π + ⎜ 8 − ⎟ 3 ⎦0 3⎠ ⎣ ⎝ 16 = 2π + ≈ 11.62 3 36. Let the x-axis lie along the diameter at the base perpendicular to the water level and slice perpendicular to the x-axis. Let x = 0 be at the center. The slice has base length 2 r 2 − x 2 and hx height . r 2h r V= x r 2 − x 2 dx r 0 r 2h ⎡ 1 2 2h ⎛ 1 3 ⎞ 2 2 2 3/ 2 ⎤ = − r −x ⎥ = r ⎜3r ⎟ = 3r h r ⎢⎣ 3 ⎝ ⎠ ⎦0 ( ) 37. Let the x-axis lie on the base perpendicular to the diameter through the center of the base. The slice a. Revolving about the line x = 4, the inner radius of the disk at y is 4 − 3 y 2 = 4 − y 2 / 3 . V =π =π 308 8 0 8⎡ 2 4 0 ⎢⎣ ( − 4− y ) ⎥⎦ dy 2/3 2 ⎤ (8 y 2 / 3 − y 4 / 3 )dy Section 5.2 at x is a rectangle with base of length 2 r 2 − x 2 and height x tan . V= r 0 2 x tan r 2 − x 2 dx r ⎡ 2 ⎤ = ⎢ − tan (r 2 − x 2 )3 / 2 ⎥ ⎣ 3 ⎦0 2 = r 3 tan 3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 38. a. y k Slice horizontally. 1 3 3 2 r⋅ r= r . 2 2 4 The center of an equilateral triangle is is A = x=4 2 2 3 1 ⋅ r= r from a vertex. Then the 3 2 3 height of a regular tetrahedron is ⎛ y⎞ ⎛ y⎞ ΔV ≈ π ⎜⎜ 4 ⎟⎟ Δy = π ⎜⎜ ⎟⎟ Δy ⎝ k⎠ ⎝ k⎠ If the depth of the tank is h, then V =π = 2π h 0 2 ⎛ 1 ⎞ h = r2 − ⎜ r⎟ = ⎝ 3 ⎠ h y π ⎡ 2 3/ 2 ⎤ dy = y ⎥ k k ⎢⎣ 3 ⎦0 h3 / 2 . 3 k The volume as a function of the depth of the 2π 3 / 2 y tank is V ( y ) = 3 k dy −m k dy = = − m y and dt π dt k which is constant. π y 39. Let A lie on the xy-plane. Suppose ΔA = f ( x)Δx where f(x) is the length at x, so A = f ( x)dx . Slice the general cone at height z parallel to A. The slice of the resulting region is Az and ΔAz is a region related to f(x) and Δ x by similar triangles: ⎛ z⎞ ⎛ z⎞ ΔAz = ⎜1 − ⎟ f ( x) ⋅ ⎜1 − ⎟ Δx ⎝ h⎠ ⎝ h⎠ 2 ⎛ z⎞ = ⎜ 1 − ⎟ f ( x ) Δx ⎝ h⎠ ⎛ z⎞ Therefore, Az = ⎜ 1 − ⎟ ⎝ h⎠ 2 2 ⎛ z⎞ f ( x)dx = ⎜ 1 − ⎟ A. ⎝ h⎠ 2 ⎛ z⎞ ΔV ≈ Az Δz = A ⎜1 − ⎟ Δz V = A ⎝ h⎠ 2 3 r. 1 2 3 Ah = r 3 12 40. If two solids have the same cross sectional area at every x in [a, b], then they have the same volume. 41. First we examine the cross-sectional areas of each shape. Hemisphere: cross-sectional shape is a circle. dV = −m y . b. It is given that dt dV π 1/ 2 dy From part a, y . = dt dt k Thus, V= 2 2 r = 3 h⎛ 2 z⎞ 1 − ⎟ dz ⎜ 0⎝ h⎠ The radius of the circle at height y is r 2 − y 2 . Therefore, the cross-sectional area for the hemisphere is Ah = π ( r 2 − y 2 )2 = π (r 2 − y 2 ) Cylinder w/o cone: cross-sectional shape is a washer. The outer radius is a constant , r. The inner radius at height y is equal to y. Therefore, the cross-sectional area is A2 = π r 2 − π y 2 = π (r 2 − y 2 ) . Since both cross-sectional areas are the same, we can apply Cavaleri’s Principle. The volume of the hemisphere of radius r is V = vol. of cylinder - vol. of cone 1 = π r 2h − π r 2h 3 2 2 = πr h 3 With the height of the cylinder and cone equal to r, the volume of the hemisphere is 2 2 V = π r 2 (r ) = π r 3 3 3 h ⎡ h ⎛ z ⎞3 ⎤ 1 = A ⎢− ⎜ 1 − ⎟ ⎥ = Ah. 3 ⎢⎣ 3 ⎝ h ⎠ ⎥⎦ 0 a. A = πr 2 1 1 V = Ah = πr 2 h 3 3 b. A face of a regular tetrahedron is an equilateral triangle. If the side of an equilateral triangle has length r, then the area Instructor’s Resource Manual Section 5.2 309 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 5.3 Concepts Review 3. a, b. 1. 2πx f ( x)Δx 2. 2π 3. 2π 4. 2π 2 0 2 0 2 0 x 2 dx; π 2 0 (4 − y 2 )dy (1 + x) x dx (1 + y )(2 − y ) dy c. ΔV ≈ 2πx x Δx = 2πx3 / 2 Δx d, e. V = 2π Problem Set 5.3 = 1. a, b. 3 3/ 2 x dx 0 3 ⎡2 ⎤ = 2π ⎢ x 5 / 2 ⎥ ⎣5 ⎦0 36 3 π ≈ 39.18 5 4. a,b. ⎛1⎞ c. ΔV ≈ 2πx ⎜ ⎟ Δx = 2πΔx ⎝x⎠ d,e. V = 2π 4 1 c. ΔV ≈ 2πx(9 − x 2 )Δx = 2π(9 x − x3 )Δx dx = 2π [ x ]1 = 6π ≈ 18.85 4 3 1 ⎤ ⎡9 (9 x − x3 )dx = 2π ⎢ x 2 − x 4 ⎥ 0 4 ⎦0 ⎣2 81 81 81 π ⎛ ⎞ = 2π ⎜ − ⎟ = ≈ 127.23 2 ⎝ 2 4⎠ d, e. V = 2π 2. a, b. 3 5. a, b. c. ΔV ≈ 2πx ( x 2 )Δx = 2πx3Δx d, e. V = 2π 1 3 x dx 0 1 π ⎡1 ⎤ = 2π ⎢ x 4 ⎥ = ≈ 1.57 ⎣ 4 ⎦0 2 c. ΔV ≈ 2π(5 − x) x Δx = 2π(5 x1/ 2 − x3 / 2 )Δx d, e. V = 2π 5 0 (5 x1/ 2 − x3 / 2 )dx 5 2 ⎡10 ⎤ = 2π ⎢ x3 / 2 − x5 / 2 ⎥ 5 ⎣3 ⎦0 ⎛ 50 5 ⎞ 40 5 = 2π ⎜⎜ − 10 5 ⎟⎟ = π ≈ 93.66 3 ⎝ 3 ⎠ 310 Section 5.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6. a, b. c. ΔV ≈ 2πx (3 x − x 2 )Δx = 2π(3 x 2 − x3 )Δx 3 1 ⎤ ⎡ (3 x 2 − x3 )dx = 2π ⎢ x3 − x 4 ⎥ 0 4 ⎦0 ⎣ 81 ⎞ 27π ⎛ = 2π ⎜ 27 − ⎟ = ≈ 42.41 4⎠ 2 ⎝ d, e. V = 2π 3 9. a, b. c. ΔV ≈ 2π(3 − x)(9 − x 2 )Δx = 2π(27 − 9 x − 3 x 2 + x3 )Δx d, e. V = 2π 3 0 (27 − 9 x − 3 x 2 + x3 )dx 3 9 1 ⎤ ⎡ = 2π ⎢ 27 x − x 2 − x3 + x 4 ⎥ 2 4 ⎦0 ⎣ 81 81 ⎞ 135π ⎛ = 2π ⎜ 81 − − 27 + ⎟ = ≈ 212.06 2 4⎠ 2 ⎝ c. ΔV ≈ 2πy ( y 2 )Δy = 2πy 3Δy 7. a, b. d, e. V = 2π 1 0 1 π ⎡1 ⎤ y 3 dy = 2π ⎢ y 4 ⎥ = ≈ 1.57 ⎣ 4 ⎦0 2 10. a, b. ⎡⎛ 1 ⎤ ⎞ c. ΔV ≈ 2πx ⎢⎜ x3 + 1⎟ − (1 − x) ⎥ Δx 4 ⎠ ⎣⎝ ⎦ 1 ⎛ ⎞ = 2 π ⎜ x 4 + x 2 ⎟ Δx ⎝4 ⎠ d, e. V = 2π 1⎛1 4 x 0 ⎜⎝ 4 c. ΔV ≈ 2πy 2⎞ + x ⎟ dx ⎠ 1 1 ⎤ ⎡1 ⎛ 1 1⎞ = 2π ⎢ x 5 + x 3 ⎥ = 2π ⎜ + ⎟ 3 ⎦0 ⎣ 20 ⎝ 20 3 ⎠ 23π = ≈ 2.41 30 d, e. V = 2π 4 0 ( ) y + 1 Δ y = 2 π( y 3 / 2 + y ) Δ y ( y 3 / 2 + y ) dy 4 1 ⎤ ⎡2 ⎛ 64 ⎞ = 2π ⎢ y 5 / 2 + y 2 ⎥ = 2π ⎜ + 8 ⎟ 2 ⎦0 ⎣5 ⎝ 5 ⎠ 208π = ≈ 130.69 5 8. a, b. 11. a, b. Instructor’s Resource Manual Section 5.3 311 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 15. c. ΔV ≈ 2π(2 − y ) y 2 Δy = 2π(2 y 2 − y 3 )Δy 2 1 ⎤ ⎡2 (2 y 2 − y 3 )dy = 2π ⎢ y 3 − y 4 ⎥ 0 4 ⎦0 ⎣3 ⎛ 16 ⎞ 8π = 2π ⎜ − 4 ⎟ = ≈ 8.38 3 ⎝ ⎠ 3 d, e. V = 2π 2 12. a, b. c. ΔV ≈ 2π(3 − y ) ( ( ) 2 y + 1 Δy V = 2π 2 0 (3 + 3 V = 2π c. V =π d. V = 2π a. A= b. V = 2π c. V =π ) 2 y1/ 2 − y − 2 y 3 / 2 dy 2 16 ⎞ 88π ⎛ = 2π ⎜ 6 + 8 − 2 − ⎟ = ≈ 55.29 5⎠ 5 ⎝ 13. a. π b 2π c. 2π d. 2π 14. a. b. c. π b a b a b a d 2π 3 1 1 x2 dx 2 ⎡⎛ 1 ⎤ ⎞ ⎢⎜ + 1⎟ − (−1)2 ⎥ dx 1 ⎢⎝ x 3 ⎥⎦ ⎠ ⎣ 3⎛ 1 2 ⎞ =π ⎜ + ⎟ dx 1 ⎝ x6 x3 ⎠ 3 ⎛ 1 ⎞ (4 − x) ⎜ ⎟ dx ⎝ x3 ⎠ 3⎛ 4 1 ⎞ = 2π ⎜ − ⎟ dx 1 ⎝ x3 x 2 ⎠ 3 1 16. x [ f ( x) − g ( x) ] dx ( x − a ) [ f ( x) − g ( x) ] dx (b − x) [ f ( x) − g ( x) ] dx d c d c y [ f ( y ) − g ( y ) ] dy =π (3 − y ) [ f ( y ) − g ( y ) ] dy d. 2 Section 5.3 ( x3 + 1) dx 0 2 0 2 0 2 0 x( x3 + 1)dx = 2π 2 0 ( x 4 + x)dx ⎡ ( x3 + 2)2 − (−1)2 ⎤ ⎣ ⎦ ( x 6 + 4 x3 + 3)dx V = 2π = 2π 312 ⎛ 1 ⎞ x ⎜ ⎟ dx = 2π ⎝ x3 ⎠ 2⎤ ⎡ f ( y ) 2 − g ( y ) 2 ⎤ dy ⎣ ⎦ c 2π 3 1 dx ⎡ f ( x ) − g ( x) dx ⎣ ⎦ 2 a b. x3 b. ) ⎡ 1 2 2 5/ 2 ⎤ = 2π ⎢3 y + 2 2 y 3 / 2 − y 2 − y ⎥ 2 5 ⎣ ⎦0 1 A= = 2π 3 + 3 2 y1/ 2 − y − 2 y 3 / 2 Δy d, e. 3 1 a. 2 0 2 0 (4 − x)( x3 + 1)dx (− x 4 + 4 x3 − x + 4)dx Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 17. To find the intersection point, solve y= y= y3 . 32 π 4 π x= 2 x2 = 6 y − 1024 y = 0 y ( y 5 − 1024) = 0 y = 0, 4 4 ⎛ y3 ⎞ V = 2π y ⎜ y − ⎟ dy 0 ⎜ 32 ⎟⎠ ⎝ 4 0 sin( x 2 ) = cos( x 2 ) . tan( x 2 ) = 1 y6 1024 = 2π 21. To find the intersection point, solve = 2π ⎛ 3/ 2 y ⎞ − ⎜y ⎟ dy ⎜ 32 ⎟⎠ ⎝ 4 4 = 2π = 2π 23. a. 20. y = ± a 2 − x 2 , − a ≤ x ≤ a −a a 2 − x 2 dx − 4π ⎞ dx ⎟ ⎠ a −a x sin x dx 2π The curves intersect when x = 0 and x = 1. 1 0 [ x 2 − ( x 2 )2 ] dx = π 1 0 ( x 2 − x 4 )dx 1 b. V = 2π 1 0 x( x − x 2 )dx = 2π 1 0 ( x 2 − x3 ) dx 1 1 ⎤ ⎡1 ⎛1 1⎞ π = 2π ⎢ x3 − x 4 ⎥ = 2π ⎜ − ⎟ = 4 ⎦0 ⎣3 ⎝3 4⎠ 6 ≈ 0.52 c. Slice perpendicular to the line y = x. At (a, a), the perpendicular line has equation y = −( x − a ) + a = − x + 2a . Substitute y = –x + 2a into y = x 2 and solve for x ≥ 0 . x a 2 − x 2 dx a ⎛1 ⎞ ⎡ 1 ⎤ = 4πb ⎜ πa 2 ⎟ − 4π ⎢ − (a 2 − x 2 )3 2 ⎥ = 2π2 a 2b 2 3 ⎝ ⎠ ⎣ ⎦ −a (Note that the area of a semicircle with radius a is a 1 a 2 − x 2 dx = πa 2 .) −a 2 Instructor’s Resource Manual 2π 0 1 ⎤ ⎡1 ⎛ 1 1 ⎞ 2π = π ⎢ x3 − x5 ⎥ = π ⎜ − ⎟ = ≈ 0.42 5 ⎦0 ⎣3 ⎝ 3 5 ⎠ 15 b b a 2 x dx + 2π V =π 2 ⎡ 1 ⎤ x b 2 − x 2 dx = 4π ⎢ − (b 2 − x 2 )3 / 2 ⎥ a 3 ⎣ ⎦a 4 π ⎡1 2 ⎤ = 4π ⎢ (b − a 2 )3 / 2 ⎥ = (b 2 − a 2 )3 / 2 ⎣3 ⎦ 3 = 4πb 0 (2 x + x sin x)dx = 2π(4π2 ) + 2π(−2π) = 4π 2 (2π − 1) ≈ 208.57 y = − b 2 − x 2 , and x = a. When R is revolved about the y-axis, it produces the desired solid. b V = 2π x ⎛⎜ b 2 − x 2 + b 2 − x 2 ⎞⎟ dx a ⎝ ⎠ (b − x) ⎛⎜ 2 a 2 − x 2 ⎝ 2π ) 2 − 1 ≈ 1.30 x(2 + sin x)dx 0 0 2π ( = 2π ⎡ x 2 ⎤ + 2π [sin x − x cos x ]0 ⎣ ⎦0 19. Let R be the region bounded by y = b − x , a 2π π/2 2π 4 ⎡8 2 y 4 y5 ⎤ = 2π ⎢ y 3 / 2 − y 5 / 2 − + ⎥ 5 32 160 ⎥⎦ ⎣⎢ 3 0 32 ⎞ 208π ⎛ 64 64 = 2π ⎜ − − 8 + ⎟ = ≈ 43.56 5 5 ⎠ 15 ⎝ 3 2 ⎡ x cos( x 2 ) − x sin( x 2 ) ⎤ dx ⎣ ⎦ 0 22. V = 2π ⎛ y3 ⎞ 18. V = 2π (4 − y ) ⎜ y − ⎟ dy ⎜ 0 32 ⎟⎠ ⎝ 4⎛ y3 y 4 ⎞ = 2π ⎜ 4 y1/ 2 − y 3 / 2 − + ⎟ dy 0 ⎜ 8 32 ⎟⎠ ⎝ −a π /2 x ⎡ cos( x 2 ) − sin( x 2 ) ⎤ dx ⎣ ⎦ ⎡⎛ 1 1 ⎞ 1⎤ = 2π ⎢⎜ + ⎟− ⎥ = π ⎣⎝ 2 2 2 2 ⎠ 2 ⎦ 4 V = 2π 0 1 ⎡1 ⎤ = 2π ⎢ sin( x 2 ) + cos( x 2 ) ⎥ 2 2 ⎣ ⎦0 ⎡2 y5 ⎤ ⎛ 64 32 ⎞ 64π = 2π ⎢ y 5 / 2 − ⎥ = 2π ⎜ − ⎟ = 5 160 5 ⎠ 5 ⎝ 5 ⎣⎢ ⎦⎥ 0 ≈ 40.21 = 4π π/2 V = 2π x 2 + x − 2a = 0 −1 ± 1 + 8a 2 −1 + 1 + 8a x= 2 x= Section 5.3 313 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Substitute into y = –x + 2a, so 1 + 4a − 1 + 8a . Find an expression for y= 2 r 2 , the square of the distance from (a, a) to ⎛ −1 + 1 + 8a 1 + 4a − 1 + 8a ⎞ , ⎜⎜ ⎟⎟ . 2 2 ⎝ ⎠ ⎡ −1 + 1 + 8a ⎤ r = ⎢a − ⎥ 2 ⎣ ⎦ 24. ΔV ≈ 4πx 2 Δx V = 4π 2 V= 2 ⎡ 1 + 4a − 1 + 8a ⎤ + ⎢a − ⎥ 2 ⎣ ⎦ ⎡ 2a + 1 − 1 + 8a ⎤ =⎢ ⎥ 2 ⎣ ⎦ r2 S Δx S r r2 0 x 2 dx = r S ⎡1 3 ⎤ 1 x ⎥ = rS 2 ⎢⎣ 3 ⎦0 3 r 2 5.4 Concepts Review 2 1. Circle ⎡ 2a + 1 − 1 + 8a ⎤ + ⎢− ⎥ 2 ⎣ ⎦ ⎡ 2 a + 1 − 1 + 8a ⎤ = 2⎢ ⎥ 2 ⎣ ⎦ x2 25. ΔV ≈ r 4 ⎡1 ⎤ dx = 4π ⎢ x3 ⎥ = πr 3 ⎣ 3 ⎦0 3 r 2 x 0 x 2 + y 2 = 16 cos 2 t + 16sin 2 t = 16 2 2. x ; x 2 + 1 2 3. = 2a 2 + 6a + 1 − 2a 1 + 8a − 1 + 8a b [f a (t )] + [ g (t ) ] dt 2 2 4. Mean Value Theorem (for derivatives) 2 ΔV ≈ πr Δa V =π 1 (2a 2 + 6a + 1 0 Problem Set 5.4 − 2a 1 + 8a − 1 + 8a ) da 1 1 ⎡2 ⎤ = π ⎢ a3 + 3a 2 + a − (1 + 8a )3 / 2 ⎥ 12 ⎣3 ⎦0 −π 1 0 1. L= 2a 1 + 8a da = 1 0 5π −π 2 0 To integrate 2a 1 + 8a da 1 0 2a 1 + 8a da , use the substitution u = 1 + 8a. 1 9 1 1 2a 1 + 8a da = (u − 1) u du 0 1 4 8 1 9 3 / 2 1/ 2 = (u − u )du 32 1 5 1 + (6 x1/ 2 ) 2 dx = 1/ 3 1 + 36 x dx 5 ( 2a 1 + 8a da 1 5 1/ 3 ⎡1 2 ⎤ = ⎢ ⋅ (1 + 36 x)3 / 2 ⎥ ⎣ 36 3 ⎦1/ 3 1 = 181 181 − 13 13 ≈ 44.23 54 ⎡⎛ 2 9 ⎞ ⎛ 1 ⎞⎤ = π ⎢⎜ + 3 + 1 − ⎟ − ⎜ − ⎟ ⎥ 4 ⎠ ⎝ 12 ⎠ ⎦ ⎣⎝ 3 −π f ( x) = 4 x3 / 2 , f ( x ) = 6 x1/ 2 2. f ( x) = L= = 2 1 2 1 ) 2 2 ( x + 1)3 / 2 , f ( x) = 2 x( x 2 + 1)1/ 2 3 2 1 + ⎡ 2 x ( x 2 + 1)1/ 2 ⎤ dx ⎣ ⎦ 4 x 4 + 4 x 2 + 1 dx = 2 1 (2 x 2 + 1)dx 2 ⎡2 ⎤ ⎛ 16 ⎞ ⎛ 2 ⎞ 17 = ⎢ x3 + x ⎥ = ⎜ + 2 ⎟ − ⎜ + 1⎟ = ≈ 5.67 ⎣3 ⎦1 ⎝ 3 ⎠ ⎝3 ⎠ 3 9 = 1 ⎡ 2 5/ 2 2 3/ 2 ⎤ u − u ⎥ 32 ⎢⎣ 5 3 ⎦1 = 1 ⎡⎛ 486 ⎞ ⎛ 2 2 ⎞ ⎤ 149 − 18 ⎟ − ⎜ − ⎟ ⎥ = ⎜ ⎢ 32 ⎣⎝ 5 ⎠ ⎝ 5 3 ⎠ ⎦ 60 V= 314 5π 149π π − = ≈ 0.052 2 60 60 Section 5.4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 3. f ( x) = (4 − x 2 / 3 )3 / 2 , f ( x) = 3 ⎛ 2 ⎞ (4 − x 2 / 3 )1/ 2 ⎜ − x −1/ 3 ⎟ 2 ⎝ 3 ⎠ g ( y) = = − x −1/ 3 (4 − x 2 / 3 )1/ 2 L= = 2 8 1 + ⎡ − x −1/ 3 (4 − x 2 / 3 )1/ 2 ⎤ dx ⎣ ⎦ 1 8 4x 1 −2 / 3 dx = 8 1 2x −1/ 3 L= ⎡3 ⎤ = 2 ⎢ x 2 / 3 ⎥ = 3(4 − 1) = 9 2 ⎣ ⎦1 = L= = 1 2 1 3 1 ⎛ x2 1 + ⎜ ⎜ 2 2 x2 ⎝ 1 2 ⎛ y4 3 ⎞ + ⎜ ⎟ dy ⎜ 6 2 y4 ⎟ ⎝ ⎠ 3 1 3 3 2 dx 2 dy =t , =t dt dt L= y4 1 y3 1 5. g ( y ) = + , g ( y) = − 16 2 y 2 4 y3 = y8 1 9 + + dy = 36 2 4 y8 3 3 3 L= 2 ⎞ ⎟ dy ⎟ ⎠ ⎞ ⎟ dx ⎟ ⎠ ⎛ x2 ⎡ x3 1 ⎤ 1 ⎞ dx = ⎢ − ⎥ + ⎜ ⎟ 1 ⎜ 2 2 x 2 ⎟⎠ ⎝ ⎣⎢ 6 2 x ⎦⎥1 ⎛ 9 1 ⎞ ⎛ 1 1 ⎞ 14 = ⎜ − ⎟−⎜ − ⎟ = ≈ 4.67 ⎝2 6⎠ ⎝6 2⎠ 3 = ⎛ y4 3 1+ ⎜ − ⎜ 6 2 y4 ⎝ 7. ⎞ ⎟ dx ⎟ ⎠ x4 1 1 + + dx = 4 2 4 x4 3 = x2 1 − 2 2 x2 ⎛ x2 1 1+ ⎜ − ⎜ 2 2 x2 ⎝ 3 1 y5 1 y4 3 + , g ( y) = − 3 30 2 y 6 2 y4 ⎛ y4 ⎡ y5 3 ⎞ 1 ⎤ + dy = ⎢ − ⎜ ⎟ ⎥ 4 3 1 ⎜ 6 2 y ⎟⎠ ⎝ ⎣⎢ 30 2 y ⎦⎥1 ⎛ 81 1 ⎞ ⎛ 1 1 ⎞ 1154 = ⎜ − ⎟−⎜ − ⎟ = ≈ 8.55 ⎝ 10 54 ⎠ ⎝ 30 2 ⎠ 135 x 4 + 3 x3 1 f ( x) = = + 6x 6 2x f ( x) = 3 dx 8 4. y5 1 + 30 2 y 3 6. x = −2 −3 −2 −3 2 −2 −3 (t 2 )2 + (t )2 dt = 1 0 t 4 + t 2 dt 1 ( ) 1 ⎡1 ⎤ t t 2 + 1 dt = ⎢ (t 2 + 1)3 / 2 ⎥ = 2 2 − 1 0 ⎣3 ⎦0 3 ≈ 0.61 = ⎛ y3 1 ⎞ 1+ ⎜ − ⎟ dy ⎜ 4 y3 ⎟ ⎝ ⎠ y6 1 1 + + dy = 16 2 y 6 1 0 2 ⎛ y3 1 ⎞ + ⎜ ⎟ dy ⎜ 4 y3 ⎟ ⎝ ⎠ 1 8. −2 ⎛ y3 1 ⎞ ⎡ y4 1 ⎤ dy = − ⎢ − = −⎜ + ⎟ ⎥ 3⎟ 2 −3 ⎜ 4 y ⎠ ⎝ ⎣⎢ 16 2 y ⎦⎥ −3 −2 ⎡⎛ 1 ⎞ ⎛ 81 1 ⎞ ⎤ 595 = − ⎢⎜ 1 − ⎟ − ⎜ − ⎟ ⎥ = ≈ 4.13 ⎣⎝ 8 ⎠ ⎝ 16 18 ⎠ ⎦ 144 dx dy = 6t , = 6t 2 dt dt L= = 4 1 ( 4 1 (6t ) 2 + (6t 2 ) 2 dt = 4 1 36t 2 + 36t 4 dt 4 6t 1 + t 2 dt = ⎡ 2(1 + t 2 )3 / 2 ⎤ ⎣ ⎦1 ) = 2 17 17 − 2 2 ≈ 134.53 Instructor’s Resource Manual Section 5.4 315 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 9. 12. x = y + 3 2 3 g ( y) = y + , g ( y) = 1 2 L= 3 1 1 + (1)2 = 2 3 1 dy = 2 2 3 5 = . 2 2 3 9 At y = 3, x = 3 + = . 2 2 At y = 1, x = 1 + dx dy = 4 cos t , = −4sin t dt dt L= = π 0 π 16 cos 2 t + 16sin 2 t dt = 0 2 ⎛9 5⎞ d = ⎜ − ⎟ + (3 − 1) 2 = 8 = 2 2 ⎝2 2⎠ (4 cos t ) 2 + (−4sin t )2 dt π 0 4dt 13. = 4π ≈ 12.57 dx dy = 1, = 2t dt dt 10. L= − −2 2−0 ⎡ ⎛1⎞ ⎛1⎞ ⎛3⎞ f (0) + 4 f ⎜ ⎟ + 2 f ⎜ ⎟ + 4 f ⎜ ⎟ 3 × 8 ⎢⎣ ⎝4⎠ ⎝2⎠ ⎝4⎠ ⎛5⎞ ⎛3⎞ ⎛7⎞ +2 f (1) + 4 f ⎜ ⎟ + 2 f ⎜ ⎟ + 4 f ⎜ ⎟ + f (2)] ⎝4⎠ ⎝2⎠ ⎝4⎠ 1 ≈ [1 + 4 × 1.118 + 2 × 1.4142 12 + 4 × 1.8028 + 2 × 2.2361 L≈ dx dy = 2 5 cos 2t , = −2 5 sin 2t dt dt = 0 (2 5 cos 2t ) + ( −2 2 5 sin 2t 20 cos 2 2t + 20sin 2 2t dt = 5π = ≈ 3.51 2 11. 3 1 1 + (2)2 dx = 5 3 1 dx = 2 5 At x = 1, y = 2(1) + 3 = 5. At x = 3, y = 2(3) + 3 = 9. 2 ) 2 π/ 4 0 dt + 4 × 2.6926 + 2 × 3.1623 2 5 dt + 4 × 3.6401 + 4.1231] ≈ 4.6468 14. f ( x) = 2 x + 3, f ( x) = 2 L= 1 + 4t 2 dt with n = 8, − π/4 0 Let f (t ) = 1 + 4t 2 . Using the Parabolic Rule −2 0 π/ 4 2 12 + (2t )2 dt = −1 −1 L= 2 0 2 d = (3 − 1) + (9 − 5) = 20 = 2 5 dx dy 1 = 2t , = dt dt 2 t L≈ 4 1 2 ⎛ 1 ⎞ (2t ) 2 + ⎜ ⎟ dt = ⎝2 t ⎠ Let f (t ) = 4t 2 + 4 1 4t 2 + 1 dt 4t 1 . Using the Parabolic Rule 4t with n = 8, 4 −1 ⎡ ⎛ 11 ⎞ ⎛ 14 ⎞ f (1) + 4 f ⎜ ⎟ + 2 f ⎜ ⎟ ⎢ 3× 8 ⎣ ⎝8⎠ ⎝8⎠ ⎛ 17 ⎞ ⎛ 20 ⎞ ⎛ 23 ⎞ ⎛ 26 ⎞ +4 f ⎜ ⎟ + 2 f ⎜ ⎟ + 4 f ⎜ ⎟ + 2 f ⎜ ⎟ 8 8 8 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ 8 ⎠ ⎤ 1 ⎛ 29 ⎞ +4 f ⎜ ⎟ + f (4) ⎥ ≈ ( 2.0616 + 4 × 2.8118 ⎝ 8 ⎠ ⎦ 8 +2 × 3.562 + 4 × 4.312 + 2 × 5.0621 + 4 × 5.8122 2 × 6.5622 + 4 × 7.3122 + 8.0623 ) ≈ 15.0467 L≈ 316 Section 5.4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 15. dx dy = cos t , = −2sin 2t dt dt L= = π 2 0 π 2 0 = = ( cos t )2 + ( −2sin 2t )2 dt cos 2 t + 4sin 2 2t dt ⎛π ⎞ ⎛ 2π ⎞ f (0) + 4 f ⎜ ⎟ + 2 f ⎜ ⎟ 3 × 8 ⎢⎣ 16 ⎝ ⎠ ⎝ 16 ⎠ ⎛ 3π ⎞ ⎛ 4π ⎞ ⎛ 5π ⎞ ⎛ 6π ⎞ +4 f ⎜ ⎟ + 2 f ⎜ ⎟+4f ⎜ ⎟+2f ⎜ ⎟ ⎝ 16 ⎠ ⎝ 16 ⎠ ⎝ 16 ⎠ ⎝ 16 ⎠ ⎛ 7π ⎞ ⎛ π ⎞⎤ π +4 f ⎜ ⎟ + f ⎜ ⎟⎥ ≈ [1 + 4 × 1.2441 16 ⎝ ⎠ ⎝ 2 ⎠ ⎦ 48 + 2 × 1.6892 + 4 × 2.0262 + 2 × 2.1213 + 4 × 1.9295 +2 × 1.4651 + 4 × 0.7898 + 0) ≈ 2.3241 L≈ 16. π 2−0 ⎡ dx dy = 1, = sec 2 t dt dt L= π 4 0 π 4 0 π 4−0 ⎡ ⎛π ⎞ f (0) + 4 f ⎜ ⎟ ⎢ 3× 8 ⎣ ⎝ 32 ⎠ ⎛ 2π ⎞ ⎛ 3π ⎞ ⎛ 4π ⎞ ⎛ 5π ⎞ +2 f ⎜ ⎟+4f ⎜ ⎟+2f ⎜ ⎟+4f ⎜ ⎟ ⎝ 32 ⎠ ⎝ 32 ⎠ ⎝ 32 ⎠ ⎝ 32 ⎠ ⎛ 6π ⎞ ⎛ 7π ⎞ ⎛ π ⎞⎤ +2 f ⎜ ⎟+4f ⎜ ⎟ + f ⎜ ⎟⎥ ⎝ 32 ⎠ ⎝ 32 ⎠ ⎝ 4 ⎠⎦ π [1.4142 + 4 × 1.4211 + 2 × 1.4425 + 4 × 1.4807 96 +2 × 1.5403 + 4 × 1.6288 + 2 × 1.7585 +4 × 1.9495 + 2.2361] ≈ 1.278 π/ 2 = 3a 2 π/ 2 ⎡1 ⎤ 3a ⎢ sin 2 t ⎥ with the same result.) ⎣2 ⎦0 The total length is 6a. 18. a. ( ) OT = length PT = a b. From Figure 18 of the text, PQ PQ QC QC sin = = and cos = = . a a PC PC Therefore PQ = a sin and QC = a cos . x = OT − PQ = a − a sin = a ( − sin ) y = CT − CQ = a − a cos = a (1 − cos ) 19. From Problem 18, x = a( − sin ), y = a (1 − cos ) dx dy = a(1 − cos ), = a sin d d 2 so 2 ⎛ dx ⎞ ⎛ dy ⎞ 2 ⎜ ⎟ +⎜ ⎟ = [ a (1 − cos ) ] + [ a sin ⎝d ⎠ ⎝d ⎠ ]2 = a 2 − 2a 2 cos + a 2 cos 2 + a 2 sin 2 = 2a 2 − 2a 2 cos = 2a 2 (1 − cos ) 1 − cos ⎛ ⎞ = 4a 2 sin 2 ⎜ ⎟ . 2 ⎝2⎠ The length of one arch of the cycloid is = 4a 2 2π 0 17. 9a 2 cos 2 t sin 2 t (sin 2 t + cos 2 t ) dt π/ 2 1 + sec 4 t dt Let f (t ) = 1 + sec4 t . Using the Parabolic ≈ 0 c. 12 + (sec 2 t )2 dt = Rule with n = 8, L ≈ π/ 2 9a 2 cos 2 t sin 4 t + 9a 2 sin 2 t cos 4 t dt ⎡ 1 ⎤ 3a cos t sin tdt = 3a ⎢ − cos 2 t ⎥ 0 2 ⎣ ⎦0 (The integral can also be evaluated as = Let f (t ) = cos 2 t + 4sin 2 2t . Using the Parabolic Rule with n = 8, π/ 2 0 ⎛ ⎞ 4a 2 sin 2 ⎜ ⎟ d = ⎝2⎠ 2π 0 ⎛ ⎞ 2a sin ⎜ ⎟ d ⎝2⎠ 2π ⎡ ⎤ = 2a ⎢ −2 cos ⎥ = 2a(2 + 2) = 8a 2 ⎦0 ⎣ dx dy = 3a cos t sin 2 t , = −3a sin t cos 2 t dt dt The first quadrant length is L = π/ 2 0 (3a cos t sin 2 t ) 2 + (−3a sin t cos 2 t )2 dt Instructor’s Resource Manual Section 5.4 317 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 20. a. = t , the point P is at x = a t − a sin( t ), y = a − a cos( t ) at time t. Using dx = a − a cos( t ) = a (1 − cos( t )) dt dy = a sin( t ) dt 2 ds ⎡ dy ⎤ ⎡ dx ⎤ = ⎢ ⎥ +⎢ ⎥ dt ⎣ dt ⎦ ⎣ dt ⎦ 2 = a2 2 sin 2 ( t ) + a 2 2 − 2a 2 = 2a 1 (1 − cos( t )) = 2a 2 2 cos( t ) + a 2 sin 2 t = 2a 2 2 sin cos 2 ( t ) = 2a 2 2 − 2a 2 2 cos( t ) t 2 π t = 1, which occurs when t = (2k + 1). The speed is a minimum when 2 2k π t . sin = 0, which occurs when t = 2 b. The speed is a maximum when sin c. 21. a. From Problem 18a, the distance traveled by the wheel is a , so at time t, the wheel has gone a = a t miles. Since the car is going 60 miles per hour, the wheel has gone 60t miles at time t. Thus, a = 60 and the maximum speed of the bug on the wheel is 2a = 2(60) = 120 miles per hour. dy = x3 − 1 dx L= 2 1 + x3 − 1 dx = 1 2 2 3/ 2 x dx 1 ( ) 2 ⎡2 ⎤ = ⎢ x5 / 2 ⎥ = 4 2 − 1 ≈ 1.86 ⎣5 ⎦1 5 b. L= 0 2 − 2 cos t dt = 4π 0 L= π/6 2 23. 318 Section 5.4 0 at dt − 0 −1 at dt 0 f ( x ) = 6 x, f ( x ) = 6 1 1 0 0 A = 2π 6 x 1 + 36 dx = 12 37 π x dx 1 24. π/3 x f ( x) = 25 − x 2 , f ( x) = − A = 2π 4 1 + 64sin cos x − 1 dx π/3 1 ⎡1 ⎤ = 12 37 π ⎢ x 2 ⎥ = 6 37π ≈ 114.66 ⎣ 2 ⎦0 ⎡ 8 ⎤ 8sin x cos 2 xdx = ⎢ − cos3 x ⎥ π/6 ⎣ 3 ⎦π / 6 1 = − + 3 ≈ 1.40 3 = at dt = 1 2π dy = 64sin 2 x cos 4 x − 1 dx π/3 −1 a 2 t 2 cos 2 t + a 2t 2 sin 2 tdt a a ⎡a ⎤ ⎡a ⎤ = ⎢ t2 ⎥ − ⎢ t2 ⎥ = + = a ⎣ 2 ⎦ 0 ⎣ 2 ⎦ −1 2 2 ⎛t⎞ 2 sin ⎜ ⎟ dt ⎝2⎠ t⎤ ⎛t⎞ ⎡ L=4 sin ⎜ ⎟ dt = ⎢ −8cos ⎥ 0 2 ⎦0 ⎝2⎠ ⎣ = 8 + 8 = 16 22. a. −1 1 = ⎛t⎞ sin ⎜ ⎟ is positive for 0 < t < 2 π , and ⎝2⎠ by symmetry, we can double the integral from 0 to 2 π . 2π 1 L= f (t ) = 1 − cos t , g (t ) = sin t 4π dx = − a sin t + a sin t + at cos t = at cos t dt dy = a cos t − a cos t + at sin t = at sin t dt b. = 2π = 2π 3 −2 3 −2 3 −2 25 − x 2 1 + 25 − x 2 x2 25 − x 2 dx 25 − x 2 + x 2 dx 5dx = 10π[ x]3−2 = 50π ≈ 157.08 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 25. f ( x) = A = 2π x3 , f ( x) = x 2 3 7 1 3 x 3 26. f '( x ) = − x(r 2 − x 2 )−1/ 2 1 + x 4 dx ⎡1 ⎤ = 2π ⎢ (1 + x 4 )3 / 2 ⎥ 18 ⎣ ⎦1 = 29. y = f ( x) = r 2 − x 2 7 π = 250 2 − 2 2 9 ( ) f ( x) = A = 2π x6 + 2 8x 3 1 2 = x4 1 x3 1 + , f ( x) = − 8 4 x2 2 2 x3 ⎛ x4 ⎛ x3 1 ⎞ 1 ⎞ + 1+ ⎜ − ⎜ ⎟ ⎟ ⎜ 8 4x2 ⎟ ⎜ 2 2 x3 ⎟ ⎝ ⎠ ⎝ ⎠ 2 3 r −r r = 2π −r r 2 − x 2 1 + ⎡ − x(r 2 − x 2 )−1/ 2 ⎤ dx ⎣ ⎦ r 2 − x 2 1 + x 2 (r 2 − x 2 )−1 dx ( r 2 − x2 )(1 + x2 (r 2 − x2 )−1 )dx r 2 − x 2 + x 2 dx r 2 dx = 2π r −r rdx = 2π rx |−r r = 4π r 2 30. x = f (t ) = r cos t y = g (t ) = r sin t f '(t ) = −r sin t g '(t ) = r cos t π A = 2π = 2π = 2π = 2π 3 π 0 π 0 0 r sin t (−r sin t ) 2 + (r cos t )2 dt r sin t r 2 sin 2 t + r 2 cos 2 tdt r sin t r 2 dt π 2 0 2 r sin tdt = −2π r 2 cos t |π0 = −2r (−1 − 1) = 4π r 2 ⎡⎛ 6561 27 1 ⎞ ⎛ 1 3 1 ⎞⎤ = 2π ⎢⎜ + − + − ⎟⎥ ⎟−⎜ ⎣⎝ 128 32 2592 ⎠ ⎝ 128 32 32 ⎠ ⎦ 8429π = ≈ 326.92 81 31. a. The base circumference is equal to the arc length of the sector, so 2πr = l. Therefore, 2πr = . l b. The area of the sector is equal to the lateral surface area. Therefore, the lateral surface 1 1 ⎛ 2πr ⎞ area is l 2 = l 2 ⎜ ⎟ = πrl . 2 2 ⎝ l ⎠ dx dy = 1, = 3t 2 dt dt 1 A = 2π t 3 1 + 9t 4 dt 0 1 π ⎡1 ⎤ = 2π ⎢ (1 + 9t 4 )3 / 2 ⎥ = 10 10 − 1 ⎣ 54 ⎦ 0 27 ≈ 3.56 28. −r = 2π ⎡ x8 3 x 2 1 ⎤ = 2π ⎢ + − ⎥ 4 ⎣⎢128 32 32 x ⎦⎥1 27. r −r r = 2π ⎛ x4 1 ⎞ x6 1 1 + + + dx ⎜ ⎟ 2⎟ 4 1 ⎜ 8 2 4x ⎠ 4 x6 ⎝ 3 ⎛ x4 1 ⎞ ⎛ x3 1 ⎞ = 2π ⎜ + ⎟ ⎜ + 3 ⎟ dx 2 1 ⎜ 8 4 x ⎟⎠ ⎜⎝ 2 2 x ⎟⎠ ⎝ 3 ⎛ x7 3x 1 ⎞ = 2π ⎜ + + ⎟ dx 1 ⎜ 16 16 8 x5 ⎟ ⎝ ⎠ = 2π −r = 2π 248π 2 ≈ 122.43 9 2 r A = 2π ( ) dx dy = −2t , =2 dt dt c. Assume r2 > r1 . Let l1 and l2 be the slant heights for r1 and r2 , respectively. Then A = πr2l2 − πr1l1 = πr2 (l1 + l ) − πr1l1 . From part a, 1 1 0 0 A = 2π 2t 4t 2 + 4 dt = 8π t t 2 + 1 dt 1 8π ⎡1 ⎤ = 8π ⎢ (t 2 + 1)3 2 ⎥ = (2 2 − 1) ≈ 15.32 ⎣3 ⎦0 3 = 2πr2 2πr2 2πr1 = = . l2 l1 + l l1 Solve for l1 : l1r2 = l1r1 + lr1 l1 (r2 − r1 ) = lr1 l1 = lr1 r2 − r1 ⎛ lr ⎞ ⎛ lr ⎞ A = πr2 ⎜ 1 + l ⎟ − πr1 ⎜ 1 ⎟ r − r ⎝ 2 1 ⎠ ⎝ r2 − r1 ⎠ ⎡r + r ⎤ = π(lr1 + lr2 ) = 2π ⎢ 1 2 ⎥ l ⎣ 2 ⎦ Instructor’s Resource Manual Section 5.4 319 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 32. Put the center of a circle of radius a at (a, 0). Revolving the portion of the circle from x = b to x = b + h about the x-axis results in the surface in question. (See figure.) 34. dx dy = −a sin t , = a cos t dt dt Since the circle is being revolved about the line x = b, the surface area is A = 2π = 2πa 2π 0 2π 0 (b − a cos t ) a 2 sin 2 t + a 2 cos 2 tdt (b − a cos t )dt = 2πa[bt − a sin t ]02π = 4π2 ab 35. a. The equation of the top half of the circle is y = a 2 − ( x − a)2 . −( x − a ) dy = dx a − ( x − a)2 A = 2π = 2π = 2π b. 2 b+h a 2 − ( x − a)2 1 + b b+h b b+ h ( x − a)2 a 2 − ( x − a)2 dx a 2 − ( x − a )2 + ( x − a) 2 dx a dx = 2πa[ x ]bb + h = 2 π ah b c. A right circular cylinder of radius a and height h has surface area 2 π ah. 33. a. dx dy = a (1 − cos t ), = a sin t dt dt A = 2π 2π 0 d. a(1 − cos t ) ⋅ a 2 (1 − cos t )2 + a 2 sin 2 t dt = 2πa 2π 0 (1 − cos t ) 2a 2 − 2a 2 cos t dt = 2 2πa 2 b. 2π 0 (1 − cos t )3 / 2 dt e. ⎛t⎞ 1 − cos t = 2sin 2 ⎜ ⎟ , so ⎝2⎠ ⎛t⎞ sin 3 ⎜ ⎟ dt ⎝2⎠ 2π t t ⎛ ⎞ ⎛ ⎞ sin ⎜ ⎟ sin 2 ⎜ ⎟ dt 0 ⎝2⎠ ⎝2⎠ 2π ⎛ t ⎞⎡ ⎛ t ⎞⎤ sin ⎜ ⎟ ⎢1 − cos 2 ⎜ ⎟ ⎥ dt 0 ⎝ 2⎠⎣ ⎝ 2 ⎠⎦ A = 2 2πa 2 = 8πa 2 = 8πa 2 2π 3 / 2 0 2 f. 2π ⎡ ⎛t⎞ 2 ⎛ t ⎞⎤ = 8πa 2 ⎢ −2 cos ⎜ ⎟ + cos3 ⎜ ⎟ ⎥ ⎝2⎠ 3 ⎝ 2 ⎠⎦0 ⎣ ⎡⎛ 2⎞ ⎛ 2 ⎞⎤ 64 2 = 8πa 2 ⎢⎜ 2 − ⎟ − ⎜ −2 + ⎟ ⎥ = πa 3 3⎠ ⎝ 3 ⎠⎦ ⎣⎝ 320 Section 5.4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. f (t ) = −3sin t , g (t ) = 3cos t 36. a. L= 2π 0 2π = 0 9sin 2 t + 9 cos 2 tdt 3dt = 3[t ]02 π = 6π ≈ 18.850 f (t ) = −3sin t , g (t ) = cos t b. L= 2π 0 9sin 2 t + cos 2 tdt ≈ 13.365 f (t ) = cos t − t sin t , g (t ) = t cos t + sin t c. L= 6π 0 6π = 0 (cos t − t sin t )2 + (t cos t + sin t )2 dt 1 + t 2 dt ≈ 179.718 f (t ) = − sin t , g (t ) = 2 cos 2t d. L= 2π 0 sin 2 t + 4 cos 2 2t dt ≈ 9.429 f (t ) = −3sin 3t , g (t ) = 2 cos 2t e. L= 2π 0 9sin 2 3t + 4 cos 2 2t dt ≈ 15.289 f (t ) = − sin t , g (t ) = π cos πt f. L= 40 0 2 2 5.5 Concepts Review 1. F ⋅ (b − a); b a F ( x) dx 2. 30 · 10 = 300 3. the depth of that part of the surface 4. δ hA Problem Set 5.5 1 ⎛1⎞ 1. F ⎜ ⎟ = 6; k ⋅ = 6, k = 12 2 2 ⎝ ⎠ F(x) = 12x W= 1/ 2 0 1/ 2 12 x dx = ⎡ 6 x 2 ⎤ ⎣ ⎦0 = 3 = 1.5 ft-lb 2 2. From Problem 1, F(x) = 12x. W= 2 0 2 12 x dx = ⎡6 x 2 ⎤ = 24 ft-lb ⎣ ⎦0 3. F(0.01) = 0.6; k = 60 F(x) = 60x W= 2 sin t + π cos πt dt ≈ 86.58 0.02 0 0.02 60 x dx = ⎡30 x 2 ⎤ ⎣ ⎦0 = 0.012 Joules 4. F(x) = kx and let l be the natural length of the spring. 37. W= 9 −l 8− l 9 −l ⎡1 ⎤ kx dx = ⎢ kx 2 ⎥ 2 ⎣ ⎦ 8−l 1 ⎡ k (81 − 18l + l 2 ) − (64 − 16l + l 2 ) ⎤ ⎦ 2 ⎣ 1 = k (17 − 2l ) = 0.05 2 0.1 . Thus, k = 17 − 2l = y = x, y = 1 , L= 1 0 1 2dx = ⎡⎣ 2 x ⎤⎦ = 2 ≈ 1.41421 0 y = x2 , y = 2 x , L = y = x 4 , y = 4 x3 , L = 10 1 1 + 4 x 2 dx ≈ 1.47894 0 1 0 1 + 16 x6 dx ≈ 1.60023 9 y = x , y = 10 x , L= 1 y=x L= 1 + 10018 dx ≈ 1.75441 0 100 1 0 , y = 100 x99 , 1 + 10, 000 x198 dx ≈ 1.95167 When n = 10,000 the length will be close to 2. Instructor’s Resource Manual W= 10 −l 9 −l 10−l ⎡1 ⎤ kx dx = ⎢ kx 2 ⎥ 2 ⎣ ⎦ 9 −l 1 ⎡ k (100 − 20l + l 2 ) − (81 − 18l + l 2 ) ⎤ ⎦ 2 ⎣ 1 = k (19 − 2l ) = 0.1 2 0.2 . Thus, k = 19 − 2l 0.1 0.2 15 Solving = ,l = . 17 − 2l 19 − 2l 2 Thus k = 0.05, and the natural length is 7.5 cm. = Section 5.5 321 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 11. A slab of thickness Δy at height y has width d ⎡1 ⎤ kxdx = ⎢ kx 2 ⎥ 0 2 ⎣ ⎦0 1 1 = k (d 2 − 0) = kd 2 2 2 5. W = d 6. F (8) = 2; k16 = 2, k = 1 8 27 1 4/3 1 ⎡3 6561 ⎤ s ds = ⎢ s 7 / 3 ⎥ = 0 8 8 ⎣7 56 ⎦0 ≈ 117.16 inch-pounds W= 7. W = 27 2 ⎡1 ⎤ 9 s ds = 9 ⎢ s 2 ⎥ = 18 ft-lb 0 ⎣ 2 ⎦0 2 3 1 3 1 6 s ds = ⎡3s 2 ⎤ + ⎡3s 2 ⎤ ⎣ ⎦2 ⎣ ⎦2 2 2 = 3(9 – 4) + 3(1 – 4) = 6 ft-lb 6 s ds + 9. A slab of thickness Δy at height y has width 4 4 − y and length 10. The slab will be lifted a 5 distance 10 – y. 4 ⎞ ⎛ ΔW ≈ δ ⋅10 ⋅ ⎜ 4 − y ⎟ Δy (10 − y ) 5 ⎠ ⎝ 5 0 15 5 1 ⎤ ⎡ (62.4) ⎢36 y + y 2 − y 3 ⎥ 2 2 3 ⎦0 ⎣ 15 64 ⎞ ⎛ (62.4) ⎜144 + 40 − ⎟ 2 3 ⎠ ⎝ = 76,128 ft-lb 12. A slab of thickness Δy at height y has width 2 6 y − y 2 and length 10. The slab will be lifted a distance 8 – y. ΔW ≈ δ ⋅10 ⋅ 2 6 y − y 2 Δy (8 − y ) = 20δ 6 y − y 2 (8 − y )Δy W= = 20δ 3 0 20δ 6 y − y 2 (8 − y )dy 3 0 6 y − y 2 (3 − y ) dy +20δ 3 0 6 y − y 2 (5)dy 3 ⎡1 ⎤ = 20δ ⎢ (6 y − y 2 )3 / 2 ⎥ +100δ ⎣3 ⎦0 = 8δ ( y 2 − 15 y + 50)Δy W= 4 = = 8. One spring will move from 2 feet beyond its natural length to 3 feet beyond its natural length. The other will move from 2 feet beyond its natural length to 1 foot beyond its natural length. W= 3 y + 3 and length 10. The slab will be lifted a 4 ⎛3 ⎞ distance 9 – y. ΔW ≈ δ ⋅10 ⋅ ⎜ y + 3 ⎟ Δy (9 − y ) ⎝4 ⎠ 15 = δ (36 + 5 y − y 2 )Δy 2 4 15 δ (36 + 5 y − y 2 )dy W= 0 2 8δ ( y 2 − 15 y + 50) dy Notice that 5 15 ⎡1 ⎤ = 8(62.4) ⎢ y 3 − y 2 + 50 y ⎥ 3 2 ⎣ ⎦0 125 375 ⎛ ⎞ = 8(62.4) ⎜ − + 250 ⎟ = 52,000 ft-lb 2 ⎝ 3 ⎠ 10. A slab of thickness Δy at height y has width 4 y and length 10. The slab will be lifted a 3 distance 8 – y. 4 ⎞ ⎛ ΔW ≈ δ ⋅10 ⋅ ⎜ 4 − y ⎟ Δy (8 − y ) 3 ⎠ ⎝ 40 = δ (24 − 11y + y 2 )Δy 3 3 40 W= δ (24 − 11 y + y 2 )dy 0 3 4− 3 0 3 0 6 y − y 2 dy 6 y − y 2 dy is the area of a quarter of a circle with radius 3. ⎛1 ⎞ W = 20δ (9) + 100δ ⎜ π9 ⎟ ⎝4 ⎠ = (62.4)(180 + 225 π ) ≈ 55,340 ft-lb 13. The volume of a disk with thickness Δy is 16πΔy . If it is at height y, it will be lifted a distance 10 – y. ΔW ≈ δ 16πΔy (10 − y ) = 16πδ (10 − y )Δy 10 1 ⎤ ⎡ 16πδ (10 − y )dy = 16π(50) ⎢10 y − y 2 ⎥ 0 2 ⎦0 ⎣ = 16 π (50)(100 – 50) ≈ 125,664 ft-lb W= 10 3 322 = 40 11 1 ⎤ ⎡ (62.4) ⎢ 24 y − y 2 + y 3 ⎥ 3 2 3 ⎦0 ⎣ = 40 99 ⎛ ⎞ (62.4) ⎜ 72 − + 9 ⎟ = 26,208 ft-lb 3 2 ⎝ ⎠ Section 5.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 14. The volume of a disk with thickness Δx at height 2 x is π(4 + x) Δx . It will be lifted a distance of 10 – x. ΔW ≈ δπ(4 + x)2 Δx (10 − x ) = πδ (160 + 64 x + 2 x 2 − x3 )Δx W= 10 19. The total work is equal to the work W1 to haul the load by itself and the work W2 to haul the rope by itself. W1 = 200 ⋅ 500 = 100, 000 ft-lb Let y = 0 be the bottom of the shaft. When the rope is at y, ΔW2 ≈ 2Δy (500 − y ) . πδ (160 + 64 x + 2 x 2 − x3 )dx 0 10 2 1 ⎤ ⎡ = π(50) ⎢160 x + 32 x 2 + x3 − x 4 ⎥ 3 4 ⎦0 ⎣ 2000 ⎛ ⎞ = π(50) ⎜ 1600 + 3200 + − 2500 ⎟ 3 ⎝ ⎠ ≈ 466,003 ft-lb 15. The total force on the face of the piston is A · f(x) if the piston is x inches from the cylinder head. The work done by moving the piston from x1 to x2 is W = x2 x1 x A ⋅ f ( x)dx = A 2 f ( x)dx . x1 This is the work done by the gas in moving the piston. The work done by the piston to compress the gas is the opposite of this or A x1 x2 20. The total work is equal to the work W1 to lift the monkey plus the work W2 to lift the chain. W1 = 10 ⋅ 20 = 200 ft-lb Let y = 20 represent the top. As the monkey climbs the chain, the piece of chain at height y (0 ≤ y ≤ 10) will be lifted 20 – 2y ft. 1 ΔW2 ≈ Δy (20 − 2 y ) = (10 − y )Δy 2 A = 1; p(v) = cv −1.4 f ( x) = cx −1.4 W= 21. 16 2 = 16, x2 = = 2 1 1 16 cx −1.4 dx 2 1.4 = 40(16) (−2.5)(16 ≈ 2075.83 in.-lb 40002 −2 W= ) f ( x) = c(2 x) −1.4 1 c(2 x) 1.4 dx 8 = 2c ⎡ −1.25(2 x)−0.4 ⎤ ⎣ ⎦1 −0.4 −0.4 = 80(16) (−1.25)(16 ≈ 2075.83 in.-lb −2 ; f (4000) = 5000 = 5000 , k = 80,000,000,000 4200 80, 000, 000, 000 4000 22. F ( x) = 16 2 x1 = = 8, x2 = = 1 2 2 W =2 x2 x2 dx 4200 A = 2; p (v) = cv −1.4 −1.4 k ⎡ 1⎤ = 80, 000, 000, 000 ⎢ − ⎥ ⎣ x ⎦ 4000 20, 000, 000 = ≈ 952,381 mi-lb 21 17. c = 40(16)1.4 8 f ( x) = k 16 = c ⎡ −2.5 x −0.4 ⎤ ⎣ ⎦2 −0.4 −0.4 10 1 ⎤ ⎡ (10 − y )dy = ⎢10 y − y 2 ⎥ 0 2 ⎦0 ⎣ = 100 – 50 = 50 ft-lb W = W1 + W2 = 250 ft-lb 10 W2 = f ( x)dx . 16. c = 40(16)1.4 x1 = 500 1 ⎤ ⎡ 2(500 − y )dy = 2 ⎢500 y − y 2 ⎥ 0 2 ⎦0 ⎣ = 2(250,000 – 125,000) = 250,000 ft-lb W = W1 + W2 = 100, 000 + 250, 000 = 350,000 ft-lb 500 W2 = ) k x2 where x is the distance between the charges. F (2) = 10; W= k = 10, k = 40 4 5 ⎡ 40 ⎤ dx = ⎢ − ⎥ = 32 ergs 1 x2 ⎣ x ⎦1 5 40 18. 80 lb/in.2 = 11,520 lb/ft2 c =11,520(1)1.4 = 11,520 ΔW ≈ p (v)Δv = 11,520v −1.4 Δv W= 4 4 11,520v −1.4 dv = ⎡ −28,800v −0.4 ⎤ ⎣ ⎦1 1 = −28,800(4−0.4 − 1−0.4 ) ≈ 12,259 ft-lb Instructor’s Resource Manual Section 5.5 323 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 23. The relationship between the height of the bucket 1 and time is y = 2t, so t = y . When the bucket is 2 a height y, the sand has been leaking out of the 1 bucket for y seconds. The weight of the bucket 2 3 ⎛1 ⎞ and sand is 100 + 500 − 3 ⎜ y ⎟ = 600 − y. 2 ⎝2 ⎠ 3 ⎞ ⎛ ΔW ≈ ⎜ 600 − y ⎟ Δy 2 ⎠ ⎝ 80 3 ⎞ 3 ⎤ ⎡ 600 − y ⎟ dy = ⎢600 y − y 2 ⎥ 0 ⎜⎝ 2 ⎠ 4 ⎦0 ⎣ = 48,000 – 4800 = 43,200 ft-lb W= 80 ⎛ 24. The total work is equal to the work W1 needed to fill the pipe plus the work W2 needed to fill the tank. 26. Let y measure the height of a narrow rectangle with 0 ≤ y ≤ 3. The force against this rectangle at depth 5 – y is ΔF ≈ δ (5 − y )(6)Δy . Thus, F= 3 0 ⎣⎢ = 6 ⋅ 62.4 ⋅10.5 = 3931.2 pounds W1 = 30 δπy 0 4 dy = ( 62.4 ) π ⎡ 1 4 ( 0 −3 ⎞⎞ + 3 ⎟ ⎟ dy ⎠⎠ ⎝ ⎝ 3 δ (− y ) ⎜ 2 ⎜ ⎛ y2 ⎞ + 3 y ⎟ dy ⎜ ⎜ ⎟ 3 ⎝ 3 ⎠ 0 r = 102 − (40 − y )2 = − y 2 + 80 y − 1500 . = 1684.8 pounds 30 3 2 δπ(− y + 80 y − 1500 y )dy 50 80 3 ⎡ 1 ⎤ = (62.4)π ⎢ − y 4 + y − 750 y 2 ⎥ 3 ⎣ 4 ⎦ 30 10, 000, 000 ⎡⎛ ⎞ = (62.4)π ⎢⎜ −1,562,500 + − 1,875, 000 ⎟ 3 ⎠ ⎣⎝ − ( −202,500 + 720, 000 − 675, 000 ) ⎤⎦ ≈ 10,455,220 ft-lb W = W1 + W2 ≈ 10, 477, 274 ft-lb with 0 ≤ y ≤ 3. The force against this rectangle at depth 3 – y is ΔF ≈ δ (3 − y )(6)Δy . Thus, 3 0 ⎡ δ (3 − y )(6) dy = 6δ ⎢3 y − ⎢⎣ = 6 ⋅ 62.4 ( 4.5 ) = 1684.8 pounds Section 5.5 = −2 ⋅ 62.4(0 − 13.5) 3 28. Place the right triangle in the coordinate system such that the vertices are (0,0), (3,0) and (0,-4). The equation of the line in Quadrant IV is 4 3 y = x − 4 or x = y + 3. 3 4 ⎛3 ⎞ ΔF ≈ δ (3 − y ) ⎜ y + 3 ⎟ Δy and ⎝4 ⎠ 0 ⎛ 3 3 ⎞ F= δ 9 − y − y 2 ⎟ dy −4 ⎜⎝ 4 4 ⎠ 0 25. Let y measure the height of a narrow rectangle F= + 3. 3 ⎛ ⎛ y 0 −3 3 (30 ≤ y ≤ 50) is π r 2 where 50 y ⎛ ⎛ y ⎞⎞ ΔF ≈ δ (− y ) ⎜ 2 ⎜ + 3 ⎟ ⎟ Δy and ⎠⎠ ⎝ ⎝ 3 ⎡ y3 3 y 2 ⎤ = −2δ ⎢ + ⎥ 2 ⎥⎦ ⎢⎣ 3 3 −3 W2 = 324 ) y= 3 ⋅ x − 3 3 or x = = −2δ ΔW2 = δπr 2 Δy y = δπ(− y3 + 80 y 2 − 1500 y )Δy 0 The equation of the line in Quadrant I is 30 ≈ 22, 054 ft-lb The cross sectional area at height y feet 3 (−3, 0), (3, 0) and 0, −3 3 . F= 2⎤ ⎢2 y ⎥ ⎣ ⎦0 y2 ⎤ ⎥ 2 ⎦⎥ 27. Place the equilateral triangle in the coordinate system such that the vertices are 2 δπy ⎛1⎞ ΔW1 = δπ ⎜ ⎟ Δy ( y ) = Δy 2 4 ⎝ ⎠ ⎡ δ (5 − y )(6) dy = 6δ ⎢5 y − 2 ⎤3 y ⎥ 2 ⎥⎦ 0 ⎡ 3 y 2 y3 ⎤ = δ ⎢9 y − − ⎥ = 62.4 ⋅ 26 8 4 ⎥⎦ ⎣⎢ −4 = 1622.4 pounds 29. ΔF ≈ δ (1 − y ) =δ 1 0 ( y ) Δy; F = ( y1 2 − y3 2 ) dy 1 0 δ (1 − y ) ( y ) dy 1 2 ⎡2 ⎤ ⎛ 4⎞ = δ ⎢ y3 2 − y5 2 ⎥ = 62.4 ⎜ ⎟ 5 ⎣3 ⎦0 ⎝ 15 ⎠ = 16.64 pounds Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 30. Place the circle in the coordinate system so that the center is (0.0). The equation of the circle is 33. We can position the x-axis along the bottom of the pool as shown: 20 x 2 + y 2 = 16 and in Quadrants I and IV, x = 16 − y 2 . ΔF ≈ δ (6 − y ) ⎛⎜ 2 16 − y 2 ⎞⎟ Δy ⎝ ⎠ 4 2⎞ ⎛ F= δ (6 − y ) ⎜ 2 16 − y ⎟ dy −4 ⎝ ⎠ Using a CAS, F ≈ 18,819 pounds. ⎛a ⎞ ΔF ≈ δ (b − y ) ⎜ y ⎟ Δy and ⎝b ⎠ b ⎛a ⎞ F = δ (b − y ) ⎜ y ⎟ dy 0 ⎝b ⎠ a ⎞ = δ ⎜ y − y 2 ⎟ dy = δ 0⎝ b ⎠ ⎛ 0 ⎝ b δ ⎜ ab − 2ay + x From the diagram, we let h = the depth of an arbitrary slice along the width of the bottom of the pool. 20 Using the Pythagorean Theorem, we can find that the length of the bottom of the pool is b 202 + 42 = 416 = 4 26 Next, we need to get h in terms of x. This can be done by using similar triangles to set up a proportion. ⎡ ay 2 ay 3 ⎤ − ⎢ ⎥ 3b ⎥⎦ ⎢⎣ 2 0 h−4 b 4 x 4 26 a 2⎞ y ⎟ dy b ⎠ 2 ⎡ ⎛ 2 ay 3 ⎤ 2 ab = δ ⎢ aby − ay 2 + ⎥ = δ ⎜⎜ ab − ab + 3b ⎦⎥ 3 ⎣⎢ ⎝ 0 ⎞ ⎟ ⎟ ⎠ 2 ab 3 The total force on one half of the dam is twice the =δ 10 4 ⎛ ab 2 ab 2 ⎞ ab 2 =δ ⎜ − ⎟ =δ ⎜ 2 3 ⎟⎠ 6 ⎝ For the lower right triangle II, a ⎞ ⎛ ΔF ≈ δ (b − y ) ⎜ a − y ⎟ dy and b ⎠ ⎝ b a ⎞ ⎛ F = δ (b − y ) ⎜ a − y ⎟ dy 0 b ⎠ ⎝ = h Δx 31. Place a rectangle in the coordinate system such that the vertices are (0,0), (0,b), (a,0) and (a,b). The equation of the diagonal from (0,0) to (a,b) b a is y = x or x = y. For the upper left triangle I, a b b⎛ 8 4 ab 2 δ 3 = 2. total force on the other half since ab 2 δ 6 h−4 x = 4 4 26 → h = 4+ x 26 ΔF = δ ⋅ h ⋅ ΔA F= 4 26 ⎛ δ ⎜4+ ⎝ 0 x ⎞ ⎟ (10 ) dx 26 ⎠ x ⎞ ⎛ 62.4 ⎜ 4 + ⎟ (10 ) dx 0 26 ⎠ ⎝ 4 26 ⎛ x ⎞ = 624 ⎜4+ ⎟ dx 0 26 ⎠ ⎝ = 4 26 4 26 32. Consider one side of the cube and place the vertices of this square on (0,0), (0,2), (2,0) and (2,2). ΔF ≈ δ (102 − y )(2)Δy; F = 2 ⎤2 2 0 2δ (102 − y ) dy ⎡ x2 ⎤ = 624 ⎢ 4 x + ⎥ 2 26 ⎦ 0 ⎣ ( ) ( = 624 16 26 + 8 26 = 624 24 26 = 14,976 26 lb ) ( ≈ 76,362.92 lb ) ⎡ y = 2δ ⎢102 y − ⎥ = 2 ⋅ 62.4 ⋅ 202 = 25, 209.6 2 ⎦⎥ ⎣⎢ 0 The force on all six sides would be 6(25,209.6) = 151,257.6 pounds. Instructor’s Resource Manual Section 5.5 325 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 34. If we imagine unrolling the cylinder so we have a flat sheet, then we need to find the total force against one side of a rectangular plate as if it had been submerged in the oil. The rectangle would be 2π ( 5 ) = 10π feet wide and 6 feet high. Thus, the total lateral force is given by F= 6 0 6 0 y dy = ⎡⎣ 250π y 2 ⎤⎦ 6 0 = 250π ( 36 ) = 9000π lbs ( ≈ 28, 274.33 lb) 35. Let W1 be the work to lift V to the surface and W2 be the work to lift V from the surface to 15 feet above the surface. The volume displaced by the buoy y feet above its original position is 1 ⎛ a π⎜ a − 3 ⎝ h 2 3 1 y⎞ ⎞ ⎛ y ⎟ (h − y ) = πa 2 h ⎜ 1 − ⎟ . 3 ⎠ ⎝ h⎠ δ The weight displaced is 3 a2 h = 3 y⎞ ⎛ πa 2 h ⎜ 1 − ⎟ . ⎝ h⎠ Note by Archimede’s Principle m = δ 3 πa 2 h or 3m , so the displaced weight is δπ 3 y⎞ ⎛ m ⎜1 − ⎟ . ⎝ h⎠ 3 3 ⎛ ⎛ ⎛ y⎞ ⎞ y⎞ ⎞ ⎛ ΔW1 ≈ ⎜ m − m ⎜1 − ⎟ ⎟ Δy = m ⎜ 1 − ⎜ 1 − ⎟ ⎟ Δy ⎜ ⎜ ⎝ h⎠ ⎟ ⎝ h ⎠ ⎟⎠ ⎝ ⎝ ⎠ 3⎞ ⎛ h y⎞ ⎛ W1 = m ⎜1 − ⎜1 − ⎟ ⎟ dy 0⎜ h⎠ ⎟ ⎝ ⎝ ⎠ h 4 ⎡ h⎛ y⎞ ⎤ 3mh = m ⎢ y + ⎜1 − ⎟ ⎥ = 4⎝ h⎠ ⎥ 4 ⎢⎣ ⎦0 W2 = m ⋅15 = 15m W = W1 + W2 = 326 Section 5.5 3mh + 15m 4 W1 = 4 0 δ 40(10 − y )dy 4 50 ⋅ y ⋅10π dy = 500π 36. First calculate the work W1 needed to lift the contents of the bottom tank to 10 feet. ΔW1 ≈ δ 40Δy (10 − y ) ⎡ 1 ⎤ = (62.4)(40) ⎢ − (10 − y )2 ⎥ ⎣ 2 ⎦0 = (62.4)(40)(–18 + 50) = 79,872 ft-lb Next calculate the work W2 needed to fill the top tank. Let y be the distance from the bottom of the top tank. ΔW2 ≈ δ (36π)Δy y Solve for the height of the top tank: 160 40 36πh = 160; h = = 36π 9π W2 = 40 / 9 π 0 δ 36πy dy 40 / 9 π ⎡1 ⎤ = (62.4)(36π) ⎢ y 2 ⎥ ⎣ 2 ⎦0 ⎛ 800 ⎞ = (62.4)(36π) ⎜ ⎟ ≈ 7062 ft-lbs ⎝ 81π2 ⎠ W = W1 + W2 ≈ 86,934 ft-lbs 225 ⎛1 ⎞ 37. Since δ ⎜ πa 2 ⎟ (8) = 300, a = . 2πδ ⎝3 ⎠ When the buoy is at z feet (0 ≤ z ≤ 2) below floating position, the radius r at the water level is 225 ⎛ 8 + z ⎞ ⎛8+ z ⎞ r =⎜ ⎟a = ⎜ ⎟. 2πδ ⎝ 8 ⎠ ⎝ 8 ⎠ ⎛1 ⎞ F = δ ⎜ πr 2 ⎟ (8 + z ) − 300 3 ⎝ ⎠ 75 3 = (8 + z ) − 300 128 2 ⎡ 75 ⎤ W= ⎢ (8 + z ) − 300 ⎥ dz 0 ⎣128 ⎦ 2 ⎡ 75 ⎤ =⎢ (8 + z ) 4 − 300 z ⎥ ⎣ 512 ⎦0 ⎛ 46,875 ⎞ =⎜ − 600 ⎟ − (600 − 0) ⎝ 32 ⎠ 8475 = ≈ 264.84 ft-lb 32 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6. M y = (−3) ⋅ 5 + (−2) ⋅ 6 + 3 ⋅ 2 + 4 ⋅ 7 + 7 ⋅1 = 14 5.6 Concepts Review M x = 2 ⋅ 5 + (−2) ⋅ 6 + 5 ⋅ 2 + 3 ⋅ 7 + (−1) ⋅1 = 28 m = 5 + 6 + 2 + 7 + 1 = 21 My 2 M 4 = ,y= x = x= 3 3 m m 4 ⋅1 + 6 ⋅ 3 = 2.2 4+6 1. right; 2. 2.5; right; x(1+x); 1 + x 3. 1; 3 4. 24 40 ; 16 16 The second lamina balances at x = 3, y = 1 . The first lamina has area 12 and the second lamina has area 4. 12 ⋅1 + 4 ⋅ 3 24 12 ⋅ 3 + 4 ⋅1 40 x= = ,y= = 12 + 4 16 12 + 4 16 Problem Set 5.6 1. x = 7. Consider two regions R1 and R2 such that R1 is bounded by f(x) and the x-axis, and R2 is bounded by g(x) and the x-axis. Let R3 be the region formed by R1 − R2 . Make a regular partition of the homogeneous region R3 such that each sub-region is of width , Δx and let x be the distance from the y-axis to the center of mass of a sub-region. The heights of R1 and R2 at x are approximately f(x) and g(x) respectively. The mass of R3 is approximately Δm = Δm1 − Δm2 ≈ δ f ( x ) Δx − δ g ( x ) Δ x 2 ⋅ 5 + (−2) ⋅ 7 + 1 ⋅ 9 5 = 5+7+9 21 = δ [ f ( x) − g ( x)]Δx where δ is the density. The moments for R3 are approximately M x = M x ( R1 ) − M x ( R2 ) 2. Let x measure the distance from the end where John sits. 180 ⋅ 0 + 80 ⋅ x + 110 ⋅12 =6 180 + 80 + 110 80x + 1320 = 6 · 370 80x = 900 x = 11.25 Tom should be 11.25 feet from John, or, equivalently, 0.75 feet from Mary. 7 3. x = 0 7 0 7 4. x = 0 7 0 ( = 49 2 x x dx = x dx 7 ⎡ 2 x3 / 2 ⎤ ⎣3 ⎦0 x(1 + x3 )dx (1 + x3 )dx + 16,807 5 (7 + 7 ⎡ 2 x5 / 2 ⎤ ⎣5 ⎦0 2401 4 ) )= = = δ [ f ( x)]2 Δx − [ g ( x)]2 Δx 2 2 δ⎡ ( f ( x)) 2 − ( g ( x))2 ⎤ Δx ≈ xδ f ( x)Δx − xδ g ( x)Δx = xδ [ f ( x) − g ( x)]Δx Taking the limit of the regular partition as Δx → 0 yields the resulting integrals in Figure 10. ( 49 7 ) = 21 (7 7 ) 5 ⎡ 1 x 2 + 1 x5 ⎤ 5 ⎣2 ⎦0 δ ⎦ 2⎣ M y = M y ( R1 ) − M y ( R2 ) 2 5 2 3 7 = ≈ 8. 7 ⎡ x + 1 x4 ⎤ 4 ⎣ ⎦0 33,859 10 2429 4 = 9674 ≈ 5.58 1735 5. M y = 1 ⋅ 2 + 7 ⋅ 3 + (−2) ⋅ 4 + (−1) ⋅ 6 + 4 ⋅ 2 = 17 M x = 1 ⋅ 2 + 1 ⋅ 3 + (−5) ⋅ 4 + 0 ⋅ 6 + 6 ⋅ 2 = −3 m = 2 + 3 + 4 + 6 + 2 = 17 My M 3 = 1, y = x = − x= 17 m m f ( x) = 2 − x; g ( x) = 0 2 x= x[(2 − x) − 0]dx 0 2 0 2 = 0 [(2 − x) − 0]dx [2 x − x 2 ]dx 2 0 Instructor’s Resource Manual [2 − x]dx Section 5.6 327 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 10. 2 ⎛ 2 1 3⎞ ⎜x − x ⎟ 3 ⎠0 ⎝ 8 3 = = 2 4 − 2 1 2⎞ ⎛ ⎜ 2x − x ⎟ 2 ⎠0 ⎝ 2 = 3 1 y= 2 2 2 = 0 [(2 − x )2 − 02 ]dx 0 2 0 4− [(2 − x ) − 0]dx 4 x= [4 − 4 x + x 2 ]dx 4 2 ⎛ 2 1 3⎞ 8 ⎜ 4x − 2x + x ⎟ 3 ⎠0 8 − 8 + 3 ⎝ = = 4 4 2 = 3 = x ( 13 x2 ) dx = 13 04 x3dx 4 1 2 x dx 0 3 4 1 ⎡ 1 x4 ⎤ 64 3 ⎣4 ⎦0 = 3 4 64 1 ⎡ 1 x3 ⎤ 9 3 ⎣3 ⎦0 y= 9. = 0 ( ) 1 4 3 0 =3 2 1 4 1 x 2 dx 2 0 3 4 1 2 x dx 0 3 512 45 64 9 = x 2 dx = 1 4 x 4 dx 18 0 64 9 = 4 1 ⎡ 1 x5 ⎤ 18 ⎣ 5 ⎦0 64 9 8 5 11. x = 0 (by symmetry) y= 2 1 2 − 2 2 − 2 2 = 1 2 − 2 (2 − x 2 )2 dx (2 − x 2 )dx 1 (4 − 4 x 2 + x 4 )dx 1 3⎤ ⎡ ⎢⎣ 2 x − 3 x ⎥⎦ − = 328 1 ⎡ 4 x − 4 x3 2⎣ 3 Section 5.6 0 1 3 x dx 0 2 1 y= 2 2 + 15 x5 ⎤ ⎦− 8 2 3 x= 2 2 = 32 2 15 8 2 3 4 = 5 = x( x3 )dx 1 14 1 4 = 1 3 2 ( x ) dx 0 1 3 x dx 0 = 1 4 x 0 dx 1 ⎡ 1 x4 ⎤ ⎣ 4 ⎦0 1 = 2 1 = ⎡ 1 x5 ⎤ ⎣ 5 ⎦0 1 4 x dx 1 4 1 5 1 4 = 4 5 1 1 6 0 = = ⎡ 1 x7 ⎤ ⎣ 14 ⎦ 0 1 4 2 7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 12. y= ( ⎡ ⎢⎣ 2 x 1 4 2 1 4 = 4 1 ⎤ − (2 x − 4)2 ⎥ dx ⎦ 2 ⎡ 2 x − ( 2 x − 4 ) ⎤ dx ⎣ ⎦ 1 2 ) (− x 2 + 5 x − 4)dx 19 3 4 = x = 0 (by symmetry) 2 1 2 ⎡ − 1 ( x 2 − 10) ⎤ dx ⎥ 2 −2 ⎢ 2 ⎣ ⎦ y= 2 ⎡ − 1 ( x 2 − 10) ⎤ dx ⎦ −2 ⎣ 2 ( = 2 − 18 −2 ) 19 3 = 9 19 3 = 27 19 14. ( x 4 − 20 x 2 + 100)dx − 12 2 −2 ( x 2 − 10)dx 2 = 2 ⎡ − 13 x3 + 52 x 2 − 4 x ⎤ ⎣ ⎦1 x3 + 100 x ⎤ − 18 ⎡ 15 x5 − 20 3 ⎣ ⎦ −2 2 − 12 ⎡ 13 x3 − 10 x ⎤ ⎣ ⎦ −2 = − 574 15 52 3 =− 287 130 To find the intersection points, x 2 = x + 3 . 13. x2 − x − 3 = 0 1 ± 13 2 x= (1+ 13 ) x( x + 3 − x 2 )dx 2 (1− 13 ) x= 2 (1+ 13 ) 2 (1− 13 ) To find the intersection point, solve 2x − 4 = 2 x . x−2 = x x2 − 4 x + 4 = x 4 x= = = 1 4 2 (1+ 13 ) ( x 2 + 3x − x3 )dx 2 (1− 13 ) 2 = x2 − 5x + 4 = 0 (x – 4)(x – 1) = 0 x = 4 (x = 1 is extraneous.) ( x + 3 − x 2 )dx (1+ 13 ) ⎡ 1 x 2 + 3 x − 1 x3 ⎤ 2 3 ⎣2 ⎦ (1− 13 ) 2 x ⎡⎣ 2 x − (2 x − 4) ⎤⎦ dx (1+ 13 ) ⎡ 1 x3 + 3 x 2 − 1 x 4 ⎤ 2 2 4 ⎣3 ⎦ (1− 13 ) ⎡ 2 x − (2 x − 4) ⎤ dx ⎦ 1 ⎣ 4 3/ 2 2 2 (x − x + 2 x)dx 1 4 1/ 2 2 ( x − x + 2)dx 1 4 2 ⎡ 52 x5 / 2 − 13 x3 + x 2 ⎤ ⎣ ⎦1 4 2 ⎡ 23 x3 / 2 − 12 x 2 + 2 x ⎤ ⎣ ⎦1 2 = 13 13 6 = (1+ 13 ) = 64 5 19 3 = 192 95 13 3 12 13 13 6 = 1 2 1 2 ⎡ ( x + 3)2 − ( x 2 ) 2 ⎤ dx ⎦ 2 (1− 13 ) ⎣ y= 2 1+ 13 ( ) 2 1− 13 ) ( ( x + 3 − x 2 )dx 2 Instructor’s Resource Manual Section 5.6 329 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. (1+ 13 ) 1 2 2 1− 13 ( ) y = 1± 5 ( x2 + 6 x + 9 − x4 ) x= 2 = 13 13 6 1 ⎡ 1 x3 2 ⎣3 + 3x 2 ( ) = ) 1 1+ 5 (− y 4 2 1− 5 1+ 5 1− 5 13 13 6 143 13 30 13 13 6 = 11 = 5 1 ⎡ − 1 y5 2⎣ 5 1− 5 1+ 5 1− 5 1+ 5 = = To find the intersection points, solve y 2 = 2 . y=± 2 2 1 ⎡ 22 2 − 2 ⎣ 2 − 2 − ( y 2 )2 ⎤ dy ⎦ (2 − y 2 )dy 2 1 ⎡ 4 y − 1 y5 ⎤ 2⎣ 5 ⎦− 2 8 2 3 = 16 2 5 8 2 3 2 = 1 2 − 2 = 6 5 16. To find the intersection points, solve y2 − 3y − 4 = − y . y2 − 2 y − 4 = 0 2 ± 20 2 Section 5.6 (4 − y 4 )dy ⎡2 y − 1 y3 ⎤ 3 ⎣ ⎦− y = 0 (by symmetry) 330 1+ 5 + 23 y 4 − 12 y 2 − 16 y ⎤ ⎦1− 5 = −20 5 5 20 5 3 = −3 y= y= (− y 2 + 2 y + 4)dy 1+ 5 1+ 5 = + 6 y 3 − 24 y − 16)dy ⎡ − 1 y3 + y 2 + 4 y ⎤ ⎣ 3 ⎦1− 15. x= − ( y 2 − 3 y − 4) 2 ⎤ dy ⎦ ⎡ (− y ) − ( y 2 − 3 y − 4) ⎤ dy ⎣ ⎦ 1− 5 2 = = ( 1+ 13 5⎤ 1 2 + 9x − 5 x ⎦ 1− 13 1 1+ 5 ⎡ ( − y ) 2 2 1− 5 ⎣ 1+ 5 2 2 1− 5 y ⎡ (− y ) − ( y 2 − 3 y − 4) ⎤ dy ⎣ ⎦ ⎡ (− y ) − ( y 2 − 3 y − 4) ⎤ dy ⎣ ⎦ (− y 3 + 2 y 2 + 4 y )dy 20 5 3 1+ 5 ⎡ − 1 y 4 + 2 y3 + 2 y 2 ⎤ 3 ⎣ 4 ⎦1− 5 20 5 3 = 20 5 3 20 5 3 =1 17. We let δ be the density of the regions and Ai be the area of region i. Region R1 : 1 m( R1 ) = δ A1 = δ (1/ 2)(1)(1) = δ 2 1 1 3 1 1 x( x)dx 3 x | 2 0 = = 3= x1 = 0 1 1 1 3 1 2 xdx x | 0 2 2 0 Since R1 is symmetric about the line y = 1 − x , the centroid must lie on this line. Therefore, 2 1 y1 = 1 − x1 = 1 − = ; and we have 3 3 1 M y ( R1 ) = x2 ⋅ m( R1 ) = δ 3 1 M x ( R1 ) = y2 ⋅ m( R1 ) = δ 6 Region R2 : m( R2 ) = δ A2 = δ (2)(1) = 2δ By symmetry we get 1 x 2 = 2 and y2 = . 2 Thus, M y ( R2 ) = x2 ⋅ m( R2 ) = 4δ M x ( R2 ) = y2 ⋅ m( R2 ) = δ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 18. We can obtain the mass and moments for the whole region by adding the individual regions. Using the results from Problem 17 we get that 1 5 m = m( R1 ) + m( R2 ) = δ + 2δ = δ 2 2 1 13 M y = M y ( R1 ) + M y ( R2 ) = δ + 4δ = δ 3 3 1 7 M x = M x ( R1 ) + M x ( R2 ) = δ + δ = δ 6 6 Therefore, the centroid is given by 13 δ My 26 = 3 = x= 5 15 m δ 2 7 δ Mx 6 7 = = y= 5 15 m δ 2 b 19. m( R1 ) = δ a c m( R2 ) = δ M x ( R1 ) = b 2 a c 2 b b M y ( R2 ) = δ =δ c a b a M x ( R1 ) = a c 2 2 2 2 = δ δ 2 a δ c M y ( R3 ) = δ =δ +δ c a b a c b x( g ( x) − f ( x))dx x( g ( x) − f ( x))dx x( g ( x) − f ( x))dx = M y ( R1 ) + M y ( R2 ) b m( R3 ) = δ =δ =δ b a b a b a a b (( g ( x))2 − ( f ( x)) 2 )dx x(h( x) − g ( x))dx x( g ( x) − f ( x))dx a (h( x) − f ( x))dx (h( x) − g ( x) + g ( x) − f ( x))dx (h( x) − g ( x))dx +δ = = δ b 2 a δ b 2 a δ b 2 a δ =δ c b =δ ( g ( x) − f ( x))dx (( g ( x)) 2 − ( f ( x))2 )dx b 2 M y ( R3 ) = δ (( g ( x))2 − ( f ( x)) 2 )dx 2 = M x ( R1 ) + M x ( R2 ) + b a b a ( g ( x) − f ( x))dx ((h( x))2 − ( f ( x))2 )dx ((h( x))2 − ( g ( x)) 2 + ( g ( x))2 − ( f ( x)) 2 )dx ((h( x))2 − ( g ( x)) 2 )dx b (( g ( x))2 − ( f ( x )) 2 )dx 2 = M x ( R1 ) + M x ( R2 ) (( g ( x)) 2 − ( f ( x))2 )dx b δ ((h( x))2 − ( g ( x)) 2 )dx Now, ( g ( x) − f ( x))dx c a + x( g ( x) − f ( x))dx 2 a 2 M y ( R2 ) = δ a b a b a b a x(h( x) − f ( x))dx x(h( x) − g ( x) + g ( x) − f ( x))dx x(h( x) − g ( x))dx +δ b a x( g ( x) − f ( x))dx = M y ( R1 ) + M y ( R2 ) = m( R1 ) + m( R2 ) M x ( R3 ) = b M y ( R1 ) = δ (( g ( x)) − ( f ( x)) )dx ( g ( x) − f ( x))dx + δ ( g ( x) − f ( x))dx δ M x ( R2 ) = x( g ( x) − f ( x))dx b b M x ( R3 ) = Now, m( R3 ) = δ a (( g ( x)) − ( f ( x)) )dx δ M y ( R1 ) = δ m( R2 ) = δ (h( x) − g ( x))dx = m( R1 ) + m( R2 ) ( g ( x) − f ( x))dx b M x ( R2 ) = a ( g ( x) − f ( x))dx δ b 20. m( R1 ) = δ 21. Let region 1 be the region bounded by x = –2, x = 2, y = 0, and y = 1, so m1 = 4 ⋅1 = 4 . 1 . Therefore 2 = y1m1 = 2 . By symmetry, x1 = 0 and y1 = M1 y = x1m1 = 0 and M1x Let region 2 be the region bounded by x = –2, x = 1, y = –1, and y = 0, so m2 = 3 ⋅1 = 3 . 1 1 and y2 = − . Therefore 2 2 3 3 M 2 y = x2 m2 = − and M 2 x = y2 m2 = − . 2 2 M1 y + M 2 y − 32 3 x= = =− m1 + m2 7 14 By symmetry, x2 = − 1 y= Instructor’s Resource Manual M 1x + M 2 x 2 1 = = m1 + m2 7 14 Section 5.6 331 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 22. Let region 1 be the region bounded by x = –3, x = 1, y = –1, and y = 4, so m1 = 20 . By 3 . Therefore, 2 M1 y = x1 m1 = −20 and M1x = y1 m1 = 30 . Let region 2 be the region bounded by x = –3, x = –2, y = –3, and y = –1, so m2 = 2 . By symmetry, symmetry, x = −1 and y1 = 5 and y2 = −2 . Therefore, 2 M 2 y = x2 m2 = −5 and M 2 x = y2 m2 = −4 . Let region 3 be the region bounded by x = 0, x = 1, y = –2, and y = –1, so m3 = 1 . By symmetry, x2 = − 1 3 and y3 = − . Therefore, 2 2 1 3 M 3 y = x3 m3 = and M 3 x = y3 m3 = − . 2 2 49 M1 y + M 2 y + M 3 y − 2 49 x= = =− 23 46 m1 + m2 + m3 x3 = 49 M + M 2 x + M 3x 49 y = 1x = 2 = 23 46 m1 + m2 + m3 23. Let region 1 be the region bounded by x = –2, x = 2, y = 2, and y = 4, so m1 = 4 ⋅ 2 = 8 . By symmetry, x1 = 0 and y1 = 3 . Therefore, M1 y = x1m1 = 0 and M1x = y1m1 = 24 . Let region 2 be the region bounded by x = –1, x = 2, y = 0, and y = 2, so m2 = 3 ⋅ 2 = 6 . By 1 symmetry, x2 = and y2 = 1 . Therefore, 2 M 2 y = x2 m2 = 3 and M 2 x = y2 m2 = 6 . Let region 3 be the region bounded by x = 2, x = 4, y = 0, and y = 1, so m3 = 2 ⋅1 = 2 . By symmetry, 1 . Therefore, M 3 y = x3 m3 = 6 2 and M 3 x = y3 m3 = 1 . M1 y + M 2 y + M 3 y 9 = x= m1 + m2 + m3 16 x3 = 3 and y2 = y= M1x + M 2 x + M 3 x 31 = m1 + m2 + m3 16 24. Let region 1 be the region bounded by x = –3, x = –1, y = –2, and y = 1, so m1 = 6 . By 1 . Therefore, 2 M1 y = x1 m1 = −12 and M1x = y1 m1 = −3 . Let region 2 be the region bounded by x = –1, x = 0, y = –2, and y = 0, so m2 = 2 . By symmetry, symmetry, x1 = −2 and y1 = − x2 = − 332 1 and y2 = −1 . Therefore, 2 Section 5.6 M 2 y = x2 m2 = −1 and M 2 x = y2 m2 = −2 . Let region 3 be the remaining region, so m3 = 22 . 1 . Therefore, 2 M 3 y = x3 m3 = 44 and M 3 x = y3 m3 = −11 . M1 y + M 2 y + M 3 y 31 = x= m1 + m2 + m3 30 By symmetry, x3 = 2 and y3 = − y= M 1x + M 2 x + M 3 x 16 8 =− =− m1 + m2 + m3 30 15 1 1 ⎡1 ⎤ = ⎢ x4 ⎥ = 4 ⎣ ⎦0 4 4 From Problem 11, x = . 5 1⎛ 4 ⎞ 2π V = A(2πx ) = ⎜ 2π ⋅ ⎟ = 4⎝ 5⎠ 5 Using cylindrical shells: 25. A = 1 3 x dx 0 V = 2π 1 0 x ⋅ x3 dx = 2π 1 4 x dx 0 1 2π ⎡1 ⎤ = 2π ⎢ x 5 ⎥ = 5 ⎣ 5 ⎦0 26. The area of the region is πa 2 . The centroid is the center (0, 0) of the circle. It travels a distance of 2 π (2a) = 4 π a. V = 4π2 a3 27. The volume of a sphere of radius a is 4 3 πa . If 3 the semicircle y = a 2 − x 2 is revolved about the x-axis the result is a sphere of radius a. The centroid of the region travels a distance of 2πy . The area of the region is 1 2 πa . Pappus's 2 Theorem says that 4 ⎛1 ⎞ (2πy ) ⎜ πa 2 ⎟ = π2 a 2 y = πa3 . 2 3 ⎝ ⎠ 4a y= , x = 0 (by symmetry) 3π 28. Consider a slice at x rotated about the y-axis. ΔV = 2π xh( x)Δx , so V = 2π Δm ≈ h( x)Δx , so m = b a b a xh( x)dx . h( x)dx = A . b ΔM y ≈ xh( x)Δx so M y = xh( x)dx . a , x= My b = a xh( x)dx m A The distance traveled by the centroid is 2πx . (2πx ) A = 2π b a xh( x)dx Therefore, V = 2πxA . Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 29. a. ΔV ≈ 2π( K − y ) w( y )Δy V = 2π b. d c π , so the centroid travels a distance 2n π of 2πr cos . 2n Thus, by Pappus's Theorem, the volume of the resulting solid is π ⎞⎛ 2 π π ⎞ ⎛ ⎜ 2πr cos ⎟ ⎜ 2r n sin cos ⎟ 2 n 2 n 2 n⎠ ⎝ ⎠⎝ π π = 4πr 3 n sin cos 2 . 2n 2n r cos ( K − y ) w( y )dy Δm ≈ w( y )Δy , so m = d c w( y )dy = A . d ΔM x ≈ yw( y )Δy , so M x = yw( y )dy . c d y= c yw( y )dy A The distance traveled by the centroid is 2π( K − y ) . 2π( K − y ) A = 2π( KA − M x ) b. d d = 2π ⎛⎜ Kw( y )dy − yw( y )dy ⎞⎟ c c ⎝ ⎠ = 2π d c 32. a. π since 2 π f (sin x) = f ( sin(π − x ) ) . Thus x = . 2 π x= 0 x f (sin x) dx π 0 = f (sin x)dx π 2 Therefore π A= Instructor’s Resource Manual The graph of f (sin x) on [0, π ] is symmetric about the line x = h The area of a regular polygon P of 2n sides π π is 2r 2 n sin cos . (To find this consider 2n 2n the isosceles triangles with one vertex at the center of the polygon and the other vertices on adjacent corners of the polygon. Each π such triangle has base of length 2r sin 2n π ⎞ and height r cos . ⎟ Since P is a regular 2n ⎠ polygon the centroid is at its center. The distance from the centroid to any side is π = 2π2 r 3 2n solid is (πr 2 )(2πr ) = 2π2 r 3 which agrees with the results from the polygon. b 3⎤ 1 ⎡1 = ⎢ by 2 − y ⎥ = bh 2 3h ⎦ 0 6 ⎣2 M h y= x = m 3 31. a. 2π2 r 3 cos 2 circle of area πr 2 whose centroid (= center) travels a distance of 2πr, the volume of the 1 m = bh 2 1 bh ; the distance traveled by the 2 h⎞ ⎛ centroid is 2π ⎜ k − ⎟ . 3⎠ ⎝ h ⎞ ⎛ 1 ⎞ πbh ⎛ V = 2π ⎜ k − ⎟ ⎜ bh ⎟ = ( 3k − h ) 3 ⎠⎝ 2 ⎠ 3 ⎝ π 2n π π cos 2 2n 2n As n → ∞ , the regular polygon approaches a circle. Using Pappus's Theorem on the The length of a segment at y is b − b. sin 2πn n →∞ ( K − y ) w( y )dy b y. h b ⎞ b ⎞ ⎛ ⎛ ΔM x ≈ y ⎜ b − y ⎟ Δy = ⎜ by − y 2 ⎟ Δy h ⎠ h ⎠ ⎝ ⎝ h⎛ b 2⎞ M x = ⎜ by − y ⎟ dy 0⎝ h ⎠ n →∞ lim Therefore, V = 2π( K − y ) A . 30. a. lim 4πr 3 n sin 0 b. x f (sin x)dx = π π f (sin x)dx 2 0 sin x cos 4 x = sin x(1 − sin 2 x) 2 , so f ( x) = x(1 − x 2 )2 . π 0 x sin x cos 4 x dx = π = π π sin x cos 4 x dx 2 0 π⎡ 1 π ⎤ − cos5 x ⎥ = 2 ⎢⎣ 5 ⎦0 5 Section 5.6 333 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 33. Consider the region S – R. 1 1 ⎡ g 2 ( x) − 2 0⎣ f ( x) ⎤ dx ⎦ ≥y R S−R yS − R = 1 1 2 0 = 2 f 2 ( x)dx 2 g ( x) dx S yS ≥ y R x≈ ≥ 1 1 2 0 x≈ = i =1 = 8 i =1 = 6.5 + 8 + + (35)(10) + (40)(8) + 10 + 8 hi = ( −25)(6.5) + ( −15)(8) + 6.5 + 8 + + (5)(10) + (10)(8) + 10 + 8 y≈ 1 8 ((hi − 4) 2 − (−4) 2 ) 2 i =1 8 i =1 2 hi (1/ 2)((2.5 − (−4) 2 ) + + (42 − (−4) 2 )) 6.5 + 8 + + 10 + 8 45.875 = ≈ 0.633 72.5 A quick computation will show that these values agree with those in Problem 34 (using a different reference point). Now consider the whole lamina as R3 , the circular hole as R2 , and the remaining lamina as R1 . We can find the centroid of R1 by noting that M x ( R1 ) = M x ( R3 ) − M x ( R2 ) and similarly for M y ( R1 ) . From symmetry, we know that the centroid of a circle is at the center. Therefore, both M x ( R2 ) and M y ( R2 ) must be zero in our case. 1695 ≈ 23.38 72.5 y≈ hi −480 = ≈ −6.62 72.5 = (5)(6.5) + (10)(8) + 1 8 (hi ) 2 2 i =1 = R hi xi hi i =1 f ( x)dx 8 xi hi i =1 8 2 34. To approximate the centroid, we can lay the figure on the x-axis (flat side down) and put the shortest side against the y-axis. Next we can use the eight regions between measurements to approximate the centroid. We will let hi ,the height of the ith region, be approximated by the height at the right end of the interval. Each interval is of width Δx = 5 cm. The centroid can be approximated as i =1 8 8 R 1 1 1⎡ 2 1 R g ( x) − f 2 ( x) ⎤ dx ≥ ( S − R) f 2 ( x)dx ⎣ ⎦ 0 0 2 2 1 1 1⎡ 2 1 2 R g ( x) − f ( x) ⎤ dx + R f 2 ( x)dx ⎣ ⎦ 0 0 2 2 1 2 1 1 1 2 ≥ ( S − R ) f ( x)dx + R f ( x)dx 0 2 2 0 1 1 1 1 R g 2 ( x)dx ≥ S f 2 ( x)dx 2 0 2 0 1 1 2 0 35. First we place the lamina so that the origin is centered inside the hole. We then recompute the centroid of Problem 34 (in this position) as (1/ 2)(6.52 + 82 + + 102 + 82 ) (6.5 + 8 + + 10 + 8) 335.875 ≈ 4.63 72.5 This leads to the following equations M y ( R3 ) − M y ( R2 ) x= m( R3 ) − m( R2 ) = = δΔx(−480) δΔx(72.5) − δπ (2.5)2 −2400 ≈ −7 342.87 y= = M x ( R3 ) − M x ( R2 ) m( R3 ) − m( R2 ) δΔx(45.875) δΔx(72.5) − δπ (2.5) 2 229.375 ≈ 0.669 342.87 Thus, the centroid is 7 cm above the center of the hole and 0.669 cm to the right of the center of the hole. = 334 Section 5.6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 36. This problem is much like Problem 34 except we don’t have one side that is completely flat. In this problem, it will be necessary, in some regions, to find the value of g(x) instead of just f(x) – g(x). We will use the 19 regions in the figure to approximate the centroid. Again we choose the height of a region to be approximately the value at the right end of that region. Each region has a width of 20 miles. We will place the north-east corner of the state at the origin. The centroid is approximately 19 x≈ xi ( f ( xi ) − g ( xi )) i =1 19 i =1 a. P ( X ≥ 2) = P (2) + P(3) + P(4) = 0.05 + 0.05 + 0.05 = 0.15 b. E( X ) = (20)(145 − 13) + (40)(149 − 10) + (380)(85 − 85) (145 − 13) + (149 − 19) + (85 − 85) 482,860 = ≈ 173.69 2780 i =1 ( f ( xi ) − g ( xi )) 1⎡ (1452 − 132 ) + (1492 − 102 ) + + (852 − 852 ) ⎤ ⎣ ⎦ 2 = (145 − 13) + (149 − 19) + + (85 − 85) 230,805 = ≈ 83.02 2780 This would put the geographic center of Illinois just south-east of Lincoln, IL. + 3(0.05) + 4(0.5) = 0.6 3. a. P ( X ≥ 2) = P (2) = 0.2 b. E ( X ) = −2(0.2) + (−1)(0.2) + 0(0.2) + 1(0.2) + 2(0.2) =0 4. a. P ( X ≥ 2) = P (2) = 0.1 b. E ( X ) = −2(0.1) + (−1)(0.2) + 0(0.4) +1(0.2) + 2(0.1) =0 5. a. b. 2. sum, integral 3. 4. 5 0 E ( X ) = 1(0.4) + 2(0.2) + 3(0.2) + 4(0.2) = 2.2 6. a. P ( X ≥ 2) = P (100) + P(1000) = 0.018 + 0.002 = 0.02 5.7 Concepts Review discrete, continuous P ( X ≥ 2) = P (2) + P(3) + P(4) = 0.2 + 0.2 + 0.2 = 0.6 b. 1. E ( X ) = −0.1(0.98) + 100(0.018) + 1000(0.002) = 3.702 7. a. P ( X ≥ 2) = P (2) + P(3) + P(4) = f ( x) dx xi pi = 0(0.7) + 1(0.15) + 2(0.05) 1 19 [( f ( xi )) 2 − ( g ( xi )) 2 ] 2 i =1 19 5 i =1 ( f ( xi ) − g ( xi )) = y≈ 2. b. 3 2 1 6 + + = = 0.6 10 10 10 10 E ( X ) = 1(0.4) + 2(0.3) + 3(0.2) + 4(0.1) = 2 cumulative distribution function Problem Set 5.7 1. a. P ( X ≥ 2) = P (2) + P(3) = 0.05 + 0.05 = 0.1 b. E( X ) = 4 i =1 xi pi = 0(0.8) + 1(0.1) + 2(0.05) + 3(0.05) = 0.35 Instructor’s Resource Manual Section 5.7 335 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 8. a. P ( X ≥ 2) = P (2) + P(3) + P(4) = 2 2 b. 2 0 (−1) (−2) 5 + + = = 0.5 10 10 10 10 b. E ( X ) = 0(0.4) + 1(0.1) + 2(0) + 3(0.1) + 4(0.4) =2 9. 20 P( X ≥ 2) = b. ⎡ x2 ⎤ 1 E( X ) = x ⋅ dx = ⎢ ⎥ 0 20 ⎢⎣ 40 ⎥⎦ c. 2 20 20 0 x⋅ 3 x (20 − x) dx 4000 ( 20x2 − x3 ) dx = 3 4000 = 3 ⎡ 20 x3 x 4 ⎤ − ⎥ = 10 ⎢ 4000 ⎣⎢ 3 4 ⎥⎦ 0 20 0 20 1 1 dx = ⋅18 = 0.9 20 20 a. E( X ) = c. 20 = 10 For 0 ≤ x ≤ 20 x 3 F ( x) = t (20 − t ) dt 0 4000 x 3 ⎡ 2 t3 ⎤ 3 2 1 3 = x − x ⎢10t − ⎥ = 4000 ⎢⎣ 3 ⎥⎦ 400 4000 0 0 For x between 0 and 20, x 1 1 x F ( x) = dt = ⋅x = 0 20 20 20 13. a. 4 P ( X ≥ 2) = 2 3 2 x (4 − x) dx 64 4 10. a. 1 1 P ( X ≥ 2) = dx = ⋅18 = 0.45 2 40 40 b. E( X ) = c. 11. a. 3 ⎡ 4 x3 x 4 ⎤ = − ⎥ = 0.6875 ⎢ 64 ⎢⎣ 3 4 ⎥⎦ 2 20 20 −20 x⋅ ⎡ x2 ⎤ 1 dx = ⎢ ⎥ 40 ⎣⎢ 80 ⎦⎥ 20 b. = 5−5 = 0 −20 3 = 64 For −20 ≤ x ≤ 20 , x 1 1 1 1 ( x + 20) = F ( x) = dt = x+ −20 40 40 40 2 P ( X ≥ 2) = 8 2 c. b. 8 0 x⋅ 3 x(8 − x) dx 256 14. a. 3 256 = 3 ⎡ 8 x3 x 4 ⎤ − ⎥ =4 ⎢ 256 ⎣⎢ 3 4 ⎥⎦ 0 b. For 0 ≤ x ≤ 8 12. a. = 336 20 2 3 x(20 − x) dx 4000 3 ⎤ 20 3 ⎡ 2 x ⎢10 x − ⎥ = 0.972 4000 ⎢⎣ 3 ⎥⎦ 2 Section 5.7 ) 4 3 ⎡ 4 x5 ⎤ dx = ⎢ x − ⎥ = 2.4 64 ⎣⎢ 5 ⎦⎥ 0 For 0 ≤ x ≤ 4 x x P ( X ≥ 2) = 8 2 1 (8 − x) dx 32 E( X ) = 8 0 x⋅ 1 (8 − x) dx 32 8 x P ( X ≥ 2) = −x 4 1 ⎡ 2 x3 ⎤ 8 = ⎢4 x − ⎥ = 32 ⎢⎣ 3 ⎥⎦ 3 3 ⎤x 3 3 ⎡ 2 t t (8 − t ) dt = ⎢ 4t − ⎥ 0 256 256 ⎢⎣ 3 ⎦⎥ 0 3 2 1 3 = x − x 64 256 F ( x) = (4x 3 3 2 x (4 − x) dx 64 1 ⎡ x2 ⎤ 9 = ⎢8 x − ⎥ = 32 ⎢⎣ 2 ⎥⎦ 16 2 8 8 c. x⋅ 8 8 x 2 − x3 ) dx 0( = 0 F ( x) = 3 ⎡ 2 x3 ⎤ 3 27 ⋅ 72 = ⎢4 x − ⎥ = 256 ⎣⎢ 3 ⎦⎥ 256 32 2 E( X ) = 4 4 0 3 2 3 ⎡ 4t 3 t 4 ⎤ t (4 − t ) dt = − ⎥ ⎢ 0 64 64 ⎢⎣ 3 4 ⎥⎦ 0 1 3 3 4 = x − x 16 256 3 x(8 − x) dx 256 8 = E( X ) = 0 c. For 0 ≤ x ≤ 8 F ( x) = = x 0 x 1 1 ⎡ t2 ⎤ (8 − t ) dt = ⎢8t − ⎥ 32 32 ⎣⎢ 2 ⎦⎥ 0 1 1 x − x2 4 64 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 15. a. 4π P ( X ≥ 2) = ⎛πx ⎞ sin ⎜ ⎟ dx 8 ⎝ 4 ⎠ 2 9 b. E( X ) = c. For 1 ≤ x ≤ 9 1 4 = π⎡ 4 π x⎤ 1 1 − cos ⎥ = − (−1 − 0) = ⎢ 8⎣ π 4 ⎦2 2 2 c. ⎛πx ⎞ sin ⎜ ⎟ dx 8 ⎝ 4 ⎠ Using integration by parts or a CAS, E(X ) = 2 . E( X ) = 4 0 π x⋅ =− 19. For 0 ≤ x ≤ 4 x 16. a. xπ F ( B) = 4 c. ⎛π x ⎞ cos ⎜ ⎟ dx 8 ⎝ 8 ⎠ Using a CAS, E ( X ) ≈ 1.4535 E( X ) = 0 20. a. a. ⎡ ⎛ π t ⎞⎤ ⎛ πt ⎞ cos ⎜ ⎟ dt = ⎢sin ⎜ ⎟ ⎥ 8 ⎝ 8 ⎠ ⎣ ⎝ 8 ⎠⎦0 b. E( X ) = = c. 4 P ( X ≥ 2) = 4 1 2 4 a. x c. 1 P ( X ≥ 2) = = 4 2 9 2 9 ⎡ 81 ⎤ dx = ⎢ − ⎥ 3 40 x ⎣ 80 x 2 ⎦ 2 81 77 ≈ 0.24 320 Instructor’s Resource Manual E( X ) = = 4 ln 4 ≈ 1.85 3 1 f ( x) dx = F (b) − F (a) due to a+b . 2 a +b 2 a 4 −4 4 ⎡ 4⎤ + dt = ⎢ − ⎥ = t 3 ⎣ ⎦1 3x 3 3t 4x − 4 = 3x 18. b. ⎡4 ⎤ x⋅ dx = ⎢ ln x ⎥ 2 ⎣3 ⎦1 3x 4 b a 1 1 ⎛ a+b ⎞ dx = − a⎟ ⎜ b−a b−a⎝ 2 ⎠ 1 b−a 1 = ⋅ = b−a 2 2 For 1 ≤ x ≤ 4 F ( x) = f ( x) dx = 1 A The midpoint of the interval [a,b] is = 1 ⎡ 4⎤ dx = ⎢ − ⎥ = 2 ⎣ 3x ⎦ 2 3 3x 4 f ( x) dx = 0; a+b⎞ a+b⎞ ⎛ ⎛ P⎜ X < ⎟ = P⎜ X ≤ ⎟ 2 ⎠ 2 ⎠ ⎝ ⎝ x ⎛π x ⎞ = sin ⎜ ⎟ ⎝ 8 ⎠ 17. f (t ) dt. By the First the Second Fundamental Theorem of Calculus. xπ 0 A A B P ( a ≤ X ≤ b) = For 0 ≤ x ≤ 4 F ( x) = x A Proof of P (a ≤ X ≤ b) = F (b) − F (a ) : π x⋅ 81 81x 2 − 81 = 80 80 x 2 Proof of F ( x) = f ( x) : F ( A) = ⎛πx⎞ P ( X ≥ 2) = cos ⎜ ⎟ dx 2 8 ⎝ 8 ⎠ ⎡ ⎛ π x ⎞⎤ π π 1 = ⎢sin ⎜ ⎟ ⎥ = sin − sin = 1 − 2 4 2 ⎣ ⎝ 8 ⎠⎦ 2 b. 80 x 2 + Fundamental Theorem of Calculus, F ( x) = f ( x). Proof of F ( A) = 0 and F ( B) = 1: 4π 4 81 By definition, F ( x ) = π ⎡ −4 πt ⎤ ⎛πt ⎞ sin ⎜ ⎟ dt = ⎢ cos ⎥ 0 8 π 4 8 4 ⎦0 ⎝ ⎠ ⎣ 1⎛ πx ⎞ 1 πx 1 = − ⎜ cos − 1⎟ = − cos + 2⎝ 4 2 4 2 ⎠ F ( x) = x ⎡ 81 ⎤ F ( x) = dt = ⎢ − ⎥ 1 40t 3 ⎣ 80t 2 ⎦1 81 x b. 9 ⎡ 81 ⎤ dx = ⎢ − ⎥ = 1.8 3 ⎣ 40 x ⎦1 40 x 81 x⋅ 21. F ( x) = b a 2 x⋅ 1 1 ⎡ x2 ⎤ dx = ⎢ ⎥ b−a b − a ⎣⎢ 2 ⎦⎥ b a 2 b −a a+b = 2(b − a ) 2 x a 1 1 x−a dt = ( x − a) = b−a b−a b−a The median will be the solution to the x 1 dx = 0.5 . equation 0 a b−a 1 ( x0 − a ) = 0.5 b−a b−a x0 − a = 2 a+b x0 = 2 Section 5.7 337 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 22. 23. 15 2 x (4 − x) 2 is 512 symmetric about the line x = 2. Consequently, P ( X ≤ 2) = 0.5 and 2 must be the median of X. The graph of f ( x) = c. 5 = 5 ⎡ 5kx 2 kx3 ⎤ − ⎢ ⎥ =1 3 ⎥⎦ ⎢⎣ 2 0 125k 125k − =1 2 3 375k − 250k = 6 24. Solve k 5 0 5 0 d. 6 125 4 e. k ( 2 − x − 2 ) dx = 1 Due to the symmetry about the line x = 2, the solution can be found by solving 2 2 0 kx dx = 1 k ⋅ x2 2 0 =1 3 4 338 41 3 4 ( 2 − x − 2 ) dx (2 − ( x − 2)) dx = 4 = 1⎡ 1 x2 ⎤ ⎢4 x − ⎥ = 4 ⎣⎢ 2 ⎦⎥ 8 3 Section 5.7 x dx + x1 2 4 (4 − t ) dt x t2 ⎤ x2 3 ⎞ 1⎡ 1 ⎛ + ⎢ 4t − ⎥ = + ⎜ x − − ⎟ 4 ⎣⎢ 2 ⎥⎦ 2 ⎝⎜ 8 2 ⎠⎟ 2 if x < 0 if 0 ≤ x ≤ 2 if 2 < x ≤ 4 if x > 4 Using a similar procedure as shown in part (a), the PDF for Y is 1 f ( y) = (120 − y − 120 ) 14, 400 y 1 t dt If 0 ≤ y < 120, F ( y ) = 0 14, 400 If 120 < y ≤ 240, y 1 1 F ( y) = + (240 − t ) dt 2 120 14, 400 P (3 ≤ X ≤ 4) = 41 0 2 4 y 1 k= 4 = 21 0 ⎡ t2 ⎤ y2 =⎢ ⎥ = ⎣⎢ 28,800 ⎦⎥ 0 28,800 4k = 1 b. 0 x + x −1 8 0 ⎧ ⎪ x2 ⎪ ⎪⎪ 8 F ( x) = ⎨ 2 ⎪ x ⎪− 8 + x − 1 ⎪ ⎩⎪1 =− ( 25x2 − 10x3 + x4 ) dx = 1 0 2 x ⎡t2 ⎤ x2 t dt = ⎢ ⎥ = 4 8 ⎣⎢ 8 ⎦⎥ 0 x1 If 0 ≤ x ≤ 2, F ( x) = x2 = 8 kx 2 (5 − x) 2 dx = 1 Solve 1 3 2 1 ⎡ 2 x3 ⎤ 2 4 x + ⎢2 x − ⎥ = + = 2 12 0 4 ⎣⎢ 3 ⎦⎥ 3 3 If 2 < x ≤ 4, F ( x) = 5 a. 2 0 2 ⎡ 25 x3 5 x 4 x5 ⎤ k⎢ − + ⎥ =1 2 5 ⎥⎦ ⎢⎣ 3 0 625 k =1 6 6 k= 625 25. 1 x ⋅ (2 − x − 2 ) dx 4 4 kx(5 − x) dx = 1. k= 4 0 4 1 1 x ⋅ (2 + ( x − 2)) dx + x ⋅ (2 − ( x − 2)) dx 2 4 4 1 2 2 1 4 (4 x − x 2 ) dx = x dx + 4 0 4 2 = Since the PDF must integrate to one, solve 0 E( X ) = 1 4 4 3 1 1 ⎡ t2 ⎤ = + ⎢ 240t − ⎥ 2 14, 400 ⎣⎢ 2 ⎦⎥ y 120 (4 − x) dx = 2 1 y y 3 y2 y + − − =− + −1 2 60 28,800 2 28,800 60 0 ⎧ if y < 0 ⎪ y2 ⎪ if 0 ≤ y ≤ 120 ⎪⎪ 28,800 F ( x) = ⎨ y2 y ⎪ − ⎪ 28,800 + 60 − 1 if 120 < y ≤ 240 ⎪ if y > 240 ⎪⎩1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 26. a. 180 Solve 0 Alternatively, we can proceed as follows: kx 2 (180 − x) dx = 1. Solve 180 b. ⎡ x4 ⎤ =1 k ⎢60 x3 − ⎥ 4 ⎥⎦ ⎢⎣ 0 1 k= 87, 480, 000 P (100 ≤ X ≤ 150) = 150 100 k ≈ 1.132096857 × 1029 FY ( y ) ≈ (7.54731× 1027 ) y 7 ( y8 − 0.202475 y 7 150 180 0 x⋅ 8 ⎛ 3 ⎞ − t ⎟ dt k ⋅ t6 ⎜ 0 ⎝ 127 ⎠ Using a CAS, y FY ( y ) = 1 x 2 (180 − x) dx 87, 480, 000 E( X ) = 0 8 ⎛ 3 ⎞ k ⋅ y6 ⎜ − y ⎟ dy = 1 using a 127 ⎝ ⎠ CAS. +0.01802 y 6 − 0.000923 y 5 4⎤ ⎡ 1 3 x = ≈ 0.468 ⎢ 60 x − ⎥ 87, 480, 000 ⎢⎣ 4 ⎥⎦ 100 c. 3 127 +0.00003 y 4 − (6.17827 × 10−7 ) y 3 +(8.108 × 10−9 ) y 2 − (6.156 × 10−11 ) y 1 x 2 (180 − x) dx 87, 480, 000 +2.07746 × 10−13 ) 180 = 27. a. 5⎤ ⎡ 1 4 x ⎢ 45 x − ⎥ 87, 480, 000 ⎣⎢ 5 ⎦⎥ Solve k 0.6 6 = 108 0 28. a. 0.6 6 x (0.6 − x)8 dx 0 =1 = 1− k P ( X ≥ 100) = k c. x (0.6 − x)8 dx d. 0.6 E( X ) = =k 0 x ⋅ kx 6 (0.6 − x)8 dx F ( x) = 0 x F ( x) = x ⋅ x 2 (200 − x)8 dx 0 (2.417 × 10−23 )t 2 (200 − t )8 dx +(1.76 × 1011 ) x 4 − (3.2853 × 1013 ) x3 +(3.942 × 1015 ) x 2 − (2.816 × 1017 ) x 95,802, 719t 6 (0.6 − t )8 dt +9.39 × 1018 ) Using a CAS, F ( x ) ≈ 6,386,850 x 7 ( x8 − 5.14286 x 7 e. Solve 100 0 kx 2 (100 − x)8 dx. Using a CAS, + 11.6308 x 6 − 15.12 x5 + 12.3709 x 4 k = 4.95 × 10−20 − 6.53184 x3 + 2.17728 x 2 F ( x) = − 0.419904 x + 0.36) e. 0 ( x8 − 1760 x 7 + 136889 x 6 − (6.16 × 108 ) x5 ≈ 0.2625 d. 200 E( X ) = k Using a CAS, F(x) ≈ (2.19727 ×10−24 ) x3 ⋅ 0.6 7 x (0.6 − x)8 dx 0 x x (200 − x)8 dx = 50 using a CAS ≈ 0.884 using a CAS c. 200 2 100 ≈ 0.0327 using a CAS. 0.45 6 0.35 kx 2 (200 − x)8 dx = 1. b. The probability that a batch is not accepted is Using a CAS, k ≈ 95,802,719 b. The probability that a unit is scrapped is 1 − P (0.35 ≤ X ≤ 0.45) 200 0 Using a CAS, k ≈ 2.417 × 10−23 8 kx (0.6 − x) dx = 1. 0 Solve If X = measurement in mm, and Y = measurement in inches, then Y = X / 25.4 . Thus, FY ( y ) = P (Y ≤ y ) = P ( X / 25.4 ≤ y ) = P ( X ≤ 25.4 y ) = F ( 25.4 y ) where F ( x ) is given in part (d). Instructor’s Resource Manual y 0 ( 4.95 ×10−20 ) t 2 (100 − t )8 dt Using a CAS, F ( x ) ≈ (4.5 × 10−21 ) x3 ( x8 − 880 x 7 + 342, 222 x 6 − (7.7 × 107 ) x5 + (1.1× 1010 ) x 4 − (1.027 × 1012 ) x3 + (6.16 × 1013 ) x 2 − (2.2 × 1015 ) x + 3.667 × 1016 ) Section 5.7 339 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 29. The PDF for the random variable X is ⎧1 if 0 ≤ x ≤ 1 f ( x) = ⎨ ⎩0 otherwise From Problem 20, the CDF for X is F ( x ) = x 33. Y is the distance from (1, X ) to the origin, so Y= (1 − 0 )2 + ( X − 0 )2 = 1 + X 2 if x < 0 ⎧ 0 ⎪ 0.8 if 0 ≤ x < 1 ⎪⎪ F ( x) = ⎨ 0.9 if 1 ≤ x < 2 ⎪0.95 if 2 ≤ x < 3 ⎪ if x > 3 ⎪⎩ 1 F(x) 1.0 Here we have a one-to-one transformation from { 0.8 } the set { x : 0 ≤ x ≤ 1} to y :1 ≤ y ≤ 2 . For 0.6 every 1 < a < b < 2 , the event a < Y < b will occur when, and only when, 0.4 0.2 a2 − 1 < X < b2 − 1 . If we let a = 1 and b = y , we can obtain the CDF for Y. ( 1 − 1 ≤ X ≤ y −1) = P (0 ≤ X ≤ y − 1) = F ( y − 1) = y − 1 P (1 ≤ Y ≤ y ) = P 2 0 2 2 2 34. 2 To find the PDF, we differentiate the CDF with respect to y. d 1 1 y PDF = y2 −1 = ⋅ ⋅2y = dy 2 y2 −1 y2 −1 x P ( X = x) = x expressions, P( a < X < b), P( a ≤ X ≤ b), P (a < X ≤ b) and P (a ≤ X < b), are if 1 ≤ x < 2 if 2 ≤ x < 3 if 3 ≤ x < 4 if x ≥ 4 0.8 0.6 0.4 0.2 0 35. 1 b. P (0.5 < Y < 0.6) = F (0.6) − F (0.5) c. f ( y) = F ( y) = d. E (Y ) = Since P( S ) = 1, P( A ∪ A ) = 1. c P ( A) + P( A ) = 1 and P( A ) = 1 − P ( A). 32. 5 x 1.2 1 1 − = 1.6 1.5 12 c c 4 P (Y < 2) = P(Y ≤ 2) = F (2) = 1 By the defintion of a complement of a set, Since P( A ∪ Ac ) = P( A) + P( Ac ), 3 2 a. = A ∪ Ac = S , where S denotes the sample space. if 0 ≤ x < 1 1.0 equivalent. 31. if x < 0 ⎧ 0 ⎪ 0.7 ⎪ ⎪⎪0.85 F ( x) = ⎨ ⎪ 0.9 ⎪0.95 ⎪ ⎪⎩1 f (t ) dt = 0. Consequently, P ( X < c) = P ( X ≤ c). As a result, all four x 3 2 F (x ) Therefore, for 0 ≤ y ≤ 2 the PDF and CDF are respectively y g ( y) = and G ( y ) = y 2 − 1 . 2 y −1 30. 1 1 0 y⋅ 2 ( y + 1)2 2 ( y + 1) 2 , 0 ≤ y ≤1 dy ≈ 0.38629 P ( X ≥ 1) = 1 − P ( X < 1) For Problem 1, 1 − P( X < 1) = 1 − P( X = 0) = 1 − 0.8 = 0.2 For Problem 2, 1 − P( X < 1) = 1 − P( X = 0) = 1 − 0.7 = 0.3 For Problem 5, 1 − P ( X < 1) = 1 − 0 = 1 340 Section 5.7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 36. a. P ( Z > 1) = 1 − P ( Z ≤ 1) = 1 − F (1) = 1− b. 38. Concepts Test P (1 < Z < 2) = P (1 ≤ Z ≤ 2) = F (2) − F (1) = 37. 1 8 = 9 9 4 1 1 − = 9 9 3 f ( z) = F ( z) = d. E (Z ) = 0 2z ,0≤ z≤3 9 3 z⋅ ⎡ 2 z3 ⎤ 2z dz = ⎢ ⎥ =2 9 ⎣⎢ 27 ⎦⎥ 0 15 2 x (4 − x)2 dx = 2 512 4 15 2 x (4 − x)2 dx and E(X 2 ) = x 2 ⋅ 0 512 32 ≈ 4.57 using a CAS = 7 E( X ) = E( X 2 ) = 3 E( X ) = 4 0 x⋅ 8 2 3 x ⋅ 0 256 x(8 − x) dx = 19.2 and 8 3 3 x ⋅ 0 256 x (8 − x) dx = 102.4 using a CAS 39. 40. 41. V ( X ) = E ⎡( X − ) 2 ⎤ , where = E ( X ) = 2 ⎣ ⎦ 4 15 2 4 V ( X ) = ( x − 2) 2 ⋅ x (4 − x) 2 dx = 0 512 7 3 x ( 8 − x ) dx = 4 256 8 3 16 V ( X ) = ( x − 4) 2 ⋅ x(8 − x) dx = 0 256 5 8 = E(X ) = E ⎡⎢( X − ⎣ 0 x⋅ )2 ⎤⎥⎦ = E ( X 2 − 2 X = E( X 2 ) − E ( 2 X ) + E( = E( X 2 ) − 2 ⋅ E( X ) + 2 2 = E( X ) − 2 = E( X 2 ) − 1. False: π 0 cos x dx = 0 because half of the area lies above the x-axis and half below the xaxis. c. 3 5.8 Chapter Review + 2 2 + 2) ) 2. True: The integral represents the area of the region in the first quadrant if the center of the circle is at the origin. 3. False: The statement would be true if either f(x) ≥ g(x) or g(x) ≥ f(x) for a ≤ x ≤ b. Consider Problem 1 with f(x) = cos x and g(x) = 0. 4. True: The area of a cross section of a cylinder will be the same in any plane parallel to the base. 5. True: Since the cross sections in all planes parallel to the bases have the same area, the integrals used to compute the volumes will be equal. 6. False: The volume of a right circular cone of 1 radius r and height h is πr 2 h . If the 3 radius is doubled and the height halved 2 the volume is πr 2 h. 3 7. False: Using the method of shells, V = 2π 1 0 x(− x 2 + x)dx . To use the method of washers we need to solve y = − x 2 + x for x in terms of y. 8. True: The bounded region is symmetric about 1 the line x = . Thus the solids obtained 2 by revolving about the lines x = 0 and x = 1 have the same volume. 2 since E ( X ) = 2 For Problem 37, V ( X ) = E ( X 2 ) − 2 and 32 2 4 using previous results, V ( X ) = −2 = 7 7 9. False: Consider the curve given by x = y= cos t , t sin t ,2≤t <∞. t 10. False: The work required to stretch a spring 2 inches beyond its natural length is 2 0 kx dx = 2k , while the work required to stretch it 1 inch beyond its natural length 1 1 is kx dx = k . 0 2 Instructor’s Resource Manual Section 5.8 341 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 11. False: If the cone-shaped tank is placed with the point downward, then the amount of water that needs to be pumped from near the bottom of the tank is much less than the amount that needs to be pumped from near the bottom of the cylindrical tank. 12. False: The force depends on the depth, but the force is the same at all points on a surface as long as they are at the same depth. 13. True: This is the definition of the center of mass. 14. True: The region is symmetric about the point ( π , 0). 15. True: By symmetry, the centroid is on the line π x = , so the centroid travels a distance 2 ⎛π⎞ of 2π ⎜ ⎟ = π2 . ⎝2⎠ 1 2. V = π ( x − x 2 )2 dx 0 1 = π ( x 2 − 2 x3 + x 4 )dx 0 1 1 1 ⎤ π ⎡1 = π ⎢ x3 − x 4 + x 5 ⎥ = 2 5 ⎦ 0 30 ⎣3 20. True: The computation of E(X) would be the same as the computation for the center of mass of the wire. x A 1 1 4. V = π ⎡( x − x 2 + 2)2 − (2)2 ⎤ dx ⎦ 0⎣ =π f (t ) dt , then F ( x) = f ( x) P ( X = 1) = P (1 ≤ X ≤ 1) = 1 1 f ( x) dx = 0 Sample Test Problems 1. A = 342 1 1 ⎤ 1 ⎡1 ( x − x 2 )dx = ⎢ x 2 − x3 ⎥ = 0 3 ⎦0 6 ⎣2 1 Section 5.8 1 0 ( x 4 − 2 x3 − 3x 2 + 4 x) dx 1 1 7π ⎡1 ⎤ = π ⎢ x5 − x 4 − x3 + 2 x 2 ⎥ = 2 ⎣5 ⎦ 0 10 1 = 2π ( x3 − 4 x 2 + 3 x) dx 0 1 4 3 ⎤ 5π ⎡1 = 2π ⎢ x 4 − x3 + x 2 ⎥ = 4 3 2 ⎣ ⎦0 6 1 6. x = x( x − x 2 )dx 0 1 0 y= ( x − x )dx = 1 1 ( x − x 2 ) 2 dx 2 0 1 2 0 = 2 1 ( x − x )dx ⎡ 1 x3 − 1 x 4 ⎤ 4 ⎣3 ⎦0 1 ⎡ 1 x 2 − 1 x3 ⎤ 3 ⎣2 ⎦0 = 1 ⎡ 1 x3 2 ⎣3 = 1 2 1 − 12 x 4 + 15 x5 ⎤ ⎦0 1 ⎡ 1 x 2 − 1 x3 ⎤ 3 ⎣2 ⎦0 1 10 1 . 6 1 1 From Problem 6, x = and y = . 2 10 1 1 π ⎛ ⎞⎛ ⎞ V ( S1 ) = 2π ⎜ ⎟ ⎜ ⎟ = ⎝ 10 ⎠ ⎝ 6 ⎠ 30 ⎛ 1 ⎞⎛ 1 ⎞ π V ( S2 ) = 2π ⎜ ⎟⎜ ⎟ = ⎝ 2 ⎠⎝ 6 ⎠ 6 ⎛1 ⎞ ⎛ 1 ⎞ 7π V ( S3 ) = 2 π ⎜ + 2 ⎟ ⎜ ⎟ = 10 ⎝ ⎠ ⎝ 6 ⎠ 10 1 ⎞⎛ 1 ⎞ 5π ⎛ V ( S4 ) = 2π ⎜ 3 − ⎟⎜ ⎟ = 2 ⎠⎝ 6 ⎠ 6 ⎝ 7. From Problem 1, A = by the First Fundamental Theorem of Calculus. 23. True: ( x 2 − x3 )dx 1 19. True: A discrete random variable takes on a finite number of possible values, or an infinite set of possible outcomes provided that these outcomes can be put in a list such as {x1, x2, …}. If F ( x) = 1 0 1 ⎤ π ⎡1 = 2π ⎢ x3 − x 4 ⎥ = 4 ⎦0 6 ⎣3 18. True: See Problem 30 in Section 5.6. 22. True: x( x − x 2 )dx = 2π 0 17. True: Since the density is proportional to the square of the distance from the midpoint, equal masses are on either side of the midpoint. E ( X ) = 5 ⋅1 = 5 0 5. V = 2π (3 − x)( x − x 2 )dx 16. True: At slice y, ΔA ≈ (9 − y 2 )Δy . 21. True: 1 3. V = 2π Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 8. 8 = F(8) = 8k, k = 1 a. 4 12. x = 8 1 ⎡1 ⎤ x dx = ⎢ x 2 ⎥ = (64 − 4) 2 2 ⎣ ⎦2 2 = 30 in.-lb 8 W= 6 9. W = 0 4 = 10. The total work is equal to the work W1 to pull up the object to the top without the cable and the work W2 to pull up the cable. W1 = 200 ⋅100 = 20,000 ft-lb 120 6 = lb/ft. The cable weighs 100 5 6 6 ΔW2 = Δy ⋅ y = yΔy 5 5 11. a. 4x = x 2 . x2 − 4 x = 0 x(x – 4) = 0 x = 0, 4 0 b. To find the intersection points, solve y = y. 4 y2 =y 16 16 y⎞ 1 ⎤ ⎛ ⎡2 y − ⎟ dy = ⎢ y 3 / 2 − y 2 ⎥ 0 ⎜⎝ 4⎠ 8 ⎦0 ⎣3 ⎛ 128 ⎞ 32 =⎜ − 32 ⎟ = ⎝ 3 ⎠ 3 Instructor’s Resource Manual =2 − ( x 2 )2 ⎤ dx ⎦ (4 x − x 2 )dx (16 x 2 − x 4 ) dx 1 ⎡ 16 x3 2⎣3 4 − 15 x5 ⎤ ⎦0 32 3 4 0 4 = 1024 15 32 3 = 32 5 ⎡ (4 x) 2 − ( x 2 ) 2 ⎤ dx ⎣ ⎦ (16 x 2 − x 4 ) dx 0 (See example 4, section 5.5). Think of cutting the barrel vertically and opening the lateral surface into a rectangle as shown in the sketch below. At depth 3 – y, a narrow rectangle has width 16π , so the total force on the lateral surface is (δ = density of water = 62.4 lbs 3 ) ft y 2 − 16 y = 0 y(y – 16) = 0 y = 0, 16 16 64 3 32 3 32 3 32 3 13. V = π 14. a. 4 1 ⎤ ⎡ (4 x − x 2 )dx = ⎢ 2 x 2 − x3 ⎥ 3 ⎦0 ⎣ 64 ⎞ 32 ⎛ = ⎜ 32 − ⎟ = 3 ⎠ 3 ⎝ A= = (4 x 2 − x3 ) dx 4 100 4 = = 4 0 1 ⎤ 2048π ⎡16 = π ⎢ x3 − x5 ⎥ = 5 ⎦0 15 ⎣3 Using Pappus’s Theorem: 32 . From Problem 11, A = 3 32 From Problem 12, y = . 5 ⎛ 32 ⎞ ⎛ 32 ⎞ 2048π V = 2πy ⋅ A = 2π ⎜ ⎟ ⎜ ⎟ = 15 ⎝ 5 ⎠⎝ 3 ⎠ To find the intersection points, solve A= = 1 4 2 0 =π 6 6 ⎡1 ⎤ y dy = ⎢ y 2 ⎥ 0 5 5 ⎣ 2 ⎦0 = 6000 ft-lb W = W1 + W2 = 26,000 ft-lb 100 32 3 4 1 ⎡ (4 x) 2 2 0 ⎣ 4 0 6 1 ⎤ ⎡ (10 − y )dy = 1560π ⎢10 y − y 2 ⎥ 0 2 ⎦0 ⎣ = 65,520π ≈ 205,837 ft-lb W2 = ⎡ 4 x3 − 1 x 4 ⎤ 4 ⎣3 ⎦0 y= 6 (4 x − x 2 ) dx 4 (62.4)(52 )π(10 − y )dy = 1560π x(4 x − x 2 )dx 0 ⎡1 ⎤ x dx = ⎢ x 2 ⎥ = 8 in.-lb 0 ⎣ 2 ⎦0 −4 W= b. 0 4 3 0 δ (3 − y )(16π ) dy =16πδ 3 0 (3 − y ) dy 3 ⎡ y2 ⎤ = 16πδ ⎢3 y − ⎥ = 16πδ (4.5) ≈ 14,114.55 lbs. 2 ⎦⎥ ⎣⎢ 0 Section 5.8 343 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. All points on the bottom of the barrel are at the same depth; thus the total force on the bottom is simply the weight of the column of water in the barrel, namely F = π (82 )(3)δ ≈ 37, 638.8 lbs. = 2 1 1 1 x + + dx = 2 16 x 4 3 3 4 1 L2 = 1 ⎛ 2 1 ⎜x + 2 4x ⎝ ⎞ ⎟ dx ⎠ 3 3 0 b. = 3 17. V = 2 ⎛ 9 − x 2 ⎞ dx = 3 (9 − x 2 ) dx ⎜ ⎟ −3 ⎝ −3 ⎠ 3 18. A = b a 19. V = π ⎡ f 2 ( x) − g 2 ( x) ⎤ dx ⎦ a⎣ 20. V = 2π b a ( x − a ) [ f ( x) − g ( x) ] dx 2 1 d. ( ) 17 ≈ 0.354 48 0.5 0 b. ( ) 85 ≈ 0.332 256 2 0 x⋅ 21. M y = δ Mx = 344 a x [ f ( x) − g ( x) ] dx δ b ⎡ f 2 ( x) − g 2 ( x ) ⎤ dx ⎦ 2 a⎣ Section 5.8 2 1 1 ⎡ x5 ⎤ 8 − x3 dx = ⎢ 4 x 2 − ⎥ 12 12 ⎣⎢ 5 ⎦⎥ 0 ( ) x 1 1 ⎡ t4 ⎤ 8 − t 3 dt = ⎢8t − ⎥ 0 12 12 ⎢⎣ 4 ⎦⎥ 0 x ( ) 1⎛ x4 ⎞ 2 x4 ⎜ 8x − ⎟ = x − 12 ⎜⎝ 4 ⎟⎠ 3 48 P ( X ≤ 3) = F (3) = 1 − f ( x) = F ( x) = 2 ⋅ 0≤ x≤6 b 2 1 1 ⎡ x4 ⎤ 8 − x3 dx = ⎢8 x − ⎥ 12 12 ⎢⎣ 4 ⎥⎦ 1 F ( x) = = 25. a. 2 1 1 ⎡ x4 ⎤ 8 − x3 dx = ⎢8 x − ⎥ 12 12 ⎢⎣ 4 ⎥⎦ 1 = 0.8 [ f ( x) − g ( x)] dx b 2 c. E ( X ) = 3 1 ⎤ ⎡ = ⎢9 x − x3 ⎥ = (27 − 9) − (−27 + 9) = 36 3 ⎦ −3 ⎣ g ( x) 1 + [ g ( x) ] dx P (0 ≤ X < 0.5) = P(0 ≤ X ≤ 0.5) = (t 2 + 1)dt ⎡1 ⎤ = 2 ⎢ t3 + t ⎥ = 4 3 ⎣3 ⎦0 2 P ( X ≥ 1) = P(1 ≤ X ≤ 2) = The loop is − 3 ≤ t ≤ 3 . By symmetry, we can double the length of the loop from t = 0 to dx dy 2 = 2t ; = t −1 t = 3, dt dt a f ( x) 1 + [ f ( x) ] dx A4 = π ⎡ f 2 (b) − g 2 (b) ⎤ ⎣ ⎦ Total surface area = A1 + A2 + A3 + A4 . = t 4 + 2t 2 + 1 dt = 2 b a b A3 = π ⎡ f 2 (a) − g 2 (a) ⎤ ⎣ ⎦ 24. a. 3 2 A2 = 2π 16. 0 1 + [ g ( x) ] dx a 23. A1 = 2π 1⎤ 1 ⎞ ⎛ 1 1 ⎞ 53 ⎡1 ⎛ = ⎢ x3 − ⎥ = ⎜ 9 − ⎟ − ⎜ − ⎟ = 3 4 x 12 ⎣ ⎦1 ⎝ ⎠ ⎝3 4⎠ 6 L=2 2 L4 = f (b) − g (b) Total length = L1 + L2 + L3 + L4 ⎛ 1 ⎞ 1 + ⎜ x2 − ⎟ dx 4 x2 ⎠ ⎝ 3 1 + [ f ( x) ] dx a b L3 = f (a) − g (a) dy 1 15. = x2 − dx 4 x2 L= b 22. L1 = c. E ( X ) = 6 0 (6 − 3) 2 3 = 36 4 1 6− x , (6 − x) = 36 18 ⎛6− x⎞ x ⋅⎜ ⎟ dx ⎝ 18 ⎠ 6 1 ⎡ x3 ⎤ = ⎢3 x 2 − ⎥ = 2 18 ⎢⎣ 3 ⎥⎦ 0 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Review and Preview Problems 1. By the Power Rule 1 x −2 +1 x −1 1 −2 = = = = − +C dx x dx 2 −2 + 1 −1 x x 2. By the Power Rule 1 2 x −1.5+1 x −0.5 −1.5 = = = =− +C dx x dx 1.5 −1.5 + 1 −0.5 x x 3. By the Power Rule 1 x −1.01+1 x −0.01 100 −1.01 = = = = − 0.01 + C dx x dx 1.01 −1.01 + 1 −0.01 x x 4. By the Power Rule 1 x −0.99 +1 x 0.01 −0.99 = = = = 100 x 0.01 + C dx x dx −0.99 + 1 0.01 x 0.99 1 10. a. (1 + )1 = 21 = 2 1 10 b. (1 + 1 10 ⎛ 11 ⎞ ) = ⎜ ⎟ ≈ 2.593742 10 ⎝ 10 ⎠ c. (1 + 1 100 ⎛ 101 ⎞ ) =⎜ ⎟ 100 ⎝ 100 ⎠ d. (1 + 1 1000 ⎛ 1001 ⎞ ) =⎜ ⎟ 1000 ⎝ 1000 ⎠ 100 1000 5. F (1) = 1 t 2 b. (1 + dt = 0 6. By the First Fundamental Theorem of Calculus d x1 1 F ( x) = dt = 1 dx t x 7. Let g ( x ) = x 2 ; then by the Chain Rule and problem 6, Dx F ( x 2 ) = Dx F ( g ( x )) = F ( g ( x )) g ( x ) ⎛ 1 =⎜ 2 ⎝x 1 1 1 1 5 1 1 c. (1 + ) 10 5 ⎛6⎞ = ⎜ ⎟ = 2.48832 ⎝5⎠ 1 10 1 1 d. (1 + ) 50 1 50 10 ⎛ 11 ⎞ = ⎜ ⎟ ≈ 2.593742 ⎝ 10 ⎠ ⎛ 51 ⎞ =⎜ ⎟ ⎝ 50 ⎠ 50 1 1 e. (1 + ) 100 1 100 2 1 10 2 2 ) 10 1 10 1 ⎞ ⎛ = ⎜1 + ⎟ ⎝ 20 ⎠ d. (1 + 2 1 1 50 ) 50 e. (1 + 2 1 1 100 ) 100 2 ≈ 2.70481 2 ≈ 2.71152 10 1⎞ ⎛ ⎛ 11 ⎞ = ⎜ 1 + ⎟ = ⎜ ⎟ ≈ 2.593742 10 ⎝ ⎠ ⎝ 10 ⎠ 20 ⎛ 21 ⎞ =⎜ ⎟ ⎝ 20 ⎠ 100 1 ⎞ ⎛ = ⎜1 + ⎟ ⎝ 100 ⎠ 1 ⎞ ⎛ = ⎜1 + ⎟ ⎝ 200 ⎠ 20 ≈ 2.6533 100 ⎛ 101 ⎞ =⎜ ⎟ ⎝ 100 ⎠ 200 ⎛ 201 ⎞ =⎜ ⎟ ⎝ 200 ⎠ 200 b. (1 + 2 10 2 5 ) = (1.2 ) ≈ 2.48832 10 c. (1 + 2 100 2 50 ) = (1.02 ) ≈ 2.691588 100 d. (1 + 2 1000 2 500 ) = (1.002 ) ≈ 2.715569 1000 13. We know from trigonometry that, for any x and any integer k , sin( x + 2kπ ) = sin( x) . Since = 21 = 2 1 b. (1 + ) 5 2 1 1 5) 5 2 1 12. a. (1 + ) 2 = 3 ≈ 1.732051 1 2 ⎞ ⎟ (2 x) = x ⎠ 8. Let h( x ) = x3 ; then by the Chain Rule and problem 6, x3 1 Dx 1 dt = Dx F ( h( x )) = F ( h( x ))h ( x ) t 3 ⎛ 1 ⎞ = ⎜ 3 ⎟ (3 x 2 ) = x ⎝x ⎠ 9. a. (1 + 1) ≈ 2.7169239 1 2 ⎛3⎞ 11. a. (1 + ) 1 = ⎜ ⎟ = 2.25 2 ⎝2⎠ c. (1 + 11 ≈ 2.704814 ≈ 2.691588 100 ⎛ 101 ⎞ =⎜ ⎟ ⎝ 100 ⎠ Instructor’s Resource Manual ⎛π ⎞ 1 ⎛ 5π ⎞ 1 and sin ⎜ ⎟ = , sin ⎜ ⎟ = ⎝6⎠ 2 ⎝ 6 ⎠ 2 1 π 12k + 1 π sin( x) = if x = + 2kπ = 2 6 6 5π 12k + 5 or x = π + 2kπ = 6 6 where k is any integer. ≈ 2.704814 Review and Preview 345 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 14. We know from trigonometry that, for any x and any integer k , cos( x + 2kπ ) = cos( x) . Since cos(π ) = -1, cos( x) = −1 if x = π + 2kπ = (2k + 1)π where k is any integer. 15. We know from trigonometry that, for any x and any integer k , tan( x + kπ ) = tan( x) . Since π 4k + 1 ⎛π ⎞ π tan ⎜ ⎟ = 1, tan( x ) = 1 if x = + kπ = 4 4 4 ⎝ ⎠ where k is any integer. 16. Since sec( x) = 1 , sec( x) is never 0 . cos( x) 17. In the triangle, relative to , 1 x2 − 1 sin = cos = tan = x 2 − 1 x x x 1 cot = sec = x csc = x2 − 1 x2 − 1 , opp = x , adj = 1 − x 2 , hypot = 1 so that x sin = x cos = 1 − x 2 tan = 1 − x2 cot = 1 − x2 x 1 sec = 1− x 19. In the triangle, relative to 1 y2 dy = xdx dy y2 2 csc = = xdx 1 1 2 = x +C y 2 When x = 0 and y = 1 we get C = −1 . Thus, − − x2 − 2 1 1 2 = x −1 = y 2 2 2 y=− 2 x −2 22. y ' = opp = x 2 − 1 , adj = 1 , hypot = x so that 18. In the triangle, relative to y ' = xy 2 → dy = xy 2 dx 21. cos x y → dy = cos x dx y y dy = cos x dx ydy = cos x dx 1 2 y = sin x + C 2 When x = 0 and y = 4 we get C = 8 . Thus, 1 2 y = sin x + 8 2 y 2 = 2sin x + 16 1 x , opp = 1 , adj = x , hypot = 1 + x 2 so that sin = 1 1+ x 2 x cos = cot = x sec = 1+ x 1 + x2 x 20. In the triangle, relative to 2 tan = 1 x csc = 1 + x 2 , opp = 1 − x 2 , adj = x , hypot = 1 so that sin = 1 − x 2 cot = 346 x 1 − x2 cos = x tan = sec = 1 − x2 x 1 1 csc = x 1 − x2 Review and Preview Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6 CHAPTER Transcendental Functions 6.1 Concepts Review 1. 2. 3. x1 1 t 1 ln(3x – 2) 2 1 1 3 = ⋅ Dx (3 x – 2) = 2 3x – 2 2(3 x – 2) 6. Dx ln 3 x – 2 = Dx dt ; (0, ∞); (– ∞, ∞) 1 x 1 ; ln x + C x 7. dy 1 3 = 3⋅ = dx x x 8. dy 1 = x 2 ⋅ + 2 x ⋅ ln x = x(1 + 2 ln x) dx x 4. ln x + ln y; ln x – ln y; r ln x 9. z = x 2 ln x 2 + (ln x)3 = x 2 ⋅ 2 ln x + (ln x)3 Problem Set 6.1 1. a. b. ⎛3⎞ ln1.5 = ln ⎜ ⎟ = ln 3 − ln 2 = 0.406 ⎝2⎠ 4 c. ln 81 = ln 3 = 4 ln 3 = 4(1.099) = 4.396 d. ln 2 = ln 21/ 2 = e. f. dz 2 1 = x 2 ⋅ + 2 x ⋅ 2 ln x + 3(ln x) 2 ⋅ dx x x 3 = 2 x + 4 x ln x + (ln x)2 x ln 6 = ln (2 · 3) = ln 2 + ln 3 = 0.693 + 1.099 = 1.792 10. r = 1 –2 x – (ln x)3 2 dr –2 –3 1 1 3(ln x) 2 = x – 3(ln x)2 ⋅ = − − x dx 2 x x3 1 1 ln 2 = (0.693) = 0.3465 2 2 ⎛ 1 ⎞ ln ⎜ ⎟ = – ln 36 = – ln(22 ⋅ 32 ) ⎝ 36 ⎠ = −2 ln 2 − 2 ln 3 = −3.584 = 11. g ( x) = ln 48 = ln(24 ⋅ 3) = 4 ln 2 + ln 3 = 3.871 2. a. 1.792 b. 0.405 c. 4.394 d. 0.3466 e. –3.584 f. 3.871 = 1 2 x + 3x + π ⋅ Dx ( x 2 + 3 x + π ) = = 2x + 3 2 x + 3x + π 13. 4. Dx ln(3x3 + 2 x) = = 1 3 3x + 2 x Dx (3x3 + 2 x) 9 x2 + 2 3 x3 + 2 x 5. Dx ln( x – 4)3 = Dx 3ln( x – 4) 1 3 = 3⋅ Dx ( x – 4) = x–4 x–4 Instructor’s Resource Manual 14. ⎡ 1 2 ⎤ 1 + ( x + 1) –1/ 2 ⋅ 2 x ⎥ ⎢ ⎦ x + x2 + 1 ⎣ 2 1 1 x2 + 1 12. h ( x) = 3. Dx ln( x 2 + 3 x + π) = 3 ln x ⎛ 1⎞ + ln = + (– ln x)3 2 2 ⎜⎝ x ⎟⎠ 2 x ln x x ⋅ 2 ln x ln x ⎡ 1 2 ⎤ –1/ 2 ⋅ 2x⎥ ⎢1 + 2 ( x – 1) ⎣ ⎦ x + x –1 1 2 1 2 x −1 1 f ( x) = ln 3 x = ln x 3 1 1 1 f ( x) = ⋅ = 3 x 3x 1 1 = f (81) = 3 ⋅ 81 243 1 (– sin x ) = – tan x cos x ⎛π⎞ ⎛π⎞ f ⎜ ⎟ = − tan ⎜ ⎟ = −1 . ⎝4⎠ ⎝4⎠ f ( x) = Section 6.1 347 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 15. Let u = 2x + 1 so du = 2 dx. 1 1 1 dx = du 2x +1 2 u 1 1 = ln u + C = ln 2 x + 1 + C 2 2 22. Let u = 2t 2 + 4t + 3 so du = (4t + 4)dt . t +1 17. Let u = 3v 2 + 9v so du = 6v + 9. 6v + 9 1 dv = du = ln u + C 2 u 3v + 9v 18. Let u = 2 z 2 + 8 so du = 4z dz. 1 1 z dz = du 2 4 u 2z + 8 1 1 = ln u + C = ln 2 z 2 + 8 + C 4 4 23. By long division, so 2 1 dx x 1 dx . x dx = – u –2 du 24. By long division, 1 1 dx = du 5 10 u 2x + π 1 1 = ln u + C = ln 2 x5 + π + C 10 10 3 ⎡1 ⎤ dx = ⎢ ln 2 x5 + π ⎥ 5 10 ⎣ ⎦0 2x + π 1 486 + π = [ln(486 + π) – ln π] = ln 10 ≈ 0.5048 10 π 348 Section 6.1 so 3 1 x2 3 dx + x+ 4 4 4 2x −1 Let u = 2 x − 1 ; then du = 2dx . Hence 1 1 1 1 dx = du = ln u + C 2x −1 2 u 2 1 = ln 2 x − 1 + C 2 2 x +x x2 3 3 + x + ln 2 x − 1 + C and dx = 2x −1 4 4 8 so x4 dx = x+4 x3 dx − 4 x 2 dx + 16 xdx − 64dx + 256 x4 0 x2 + x x 3 3 = + + 2 x − 1 2 4 4(2 x − 1) 3 3 x2 + x x dx = dx + dx + dx 2x −1 2 4 4(2 x − 1) 21. Let u = 2 x5 + π so du = 10 x 4 dx . x4 x2 1 dx = x dx + 1 dx + dx x −1 x −1 x2 = + x + ln x − 1 + C 2 25. By long division, x4 256 = x3 − 4 x 2 + 16 x − 64 + x+4 x+4 x(ln x) 1 1 = +C = +C u ln x 3 x2 1 = x +1+ x −1 x −1 = = u 2 + C = (ln x) 2 + C –1 1 1 9 1 ln 9 – ln 3 = ln 4 = ln 4 3 = ln 3 4 4 3 4 = ) 2 ln x dx = 2 udu x 20. Let u = ln x, so du = 1 t +1 ⎡1 ⎤ dt = ⎢ ln 2t 2 + 4t + 3 ⎥ 0 2t 2 + 4t + 3 4 ⎣ ⎦0 1 = ln 3v 2 + 9v + C 19. Let u = ln x so du = 1 1 du 4 u 2t + 4t + 3 1 1 = ln u + C = ln 2t 2 + 4t + 3 + C 4 4 16. Let u = 1 – 2x so du = –2dx. 1 1 1 dx = – du 1 – 2x 2 u 1 1 = – ln u + C = – ln 1 – 2 x + C 2 2 ( dt = 2 = 1 dx x+4 x 4 4 x3 − + 8 x 2 − 64 x + 256 ln x + 4 + C 4 3 26. By long division, x3 + x 2 4 = x2 − x + 2 − so x+2 x+2 x3 + x 2 1 dx = x 2 dx − xdx + 2dx − 4 dx x+2 x+2 x3 x 2 = − + 2 x − 4 ln x + 2 + C 3 2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 27. 2 ln( x + 1) – ln x = ln( x + 1) 2 – ln x = ln 28. ( x + 1)2 x 1 1 ln( x – 9) + ln x = ln x – 9 – ln x 2 2 x–9 x–9 = ln = ln x x 1 31. ln y = ln( x + 11) – ln( x3 – 4) 2 1 dy 1 1 1 = ⋅1 – ⋅ ⋅ 3x2 3 y dx x + 11 2 x –4 = ⎡ 1 3x2 ⎤ dy = y⋅⎢ – ⎥ 3 dx ⎣⎢ x + 11 2( x – 4) ⎦⎥ 29. ln(x – 2) – ln(x + 2) + 2 ln x = ln( x – 2) – ln( x + 2) + ln x 2 = ln 1 3x2 – x + 11 2( x3 – 4) x 2 ( x – 2) x+2 = 30. ln( x 2 – 9) – 2 ln( x – 3) – ln( x + 3) = ln( x 2 − 9) − ln( x − 3)2 − ln( x + 3) =– x2 – 9 1 = ln = ln 2 x–3 ( x – 3) ( x + 3) x + 11 ⎡ 1 3x2 ⎤ – ⎢ ⎥ 3 x3 – 4 ⎣⎢ x + 11 2( x – 4) ⎦⎥ x3 + 33x 2 + 8 2( x3 – 4)3 / 2 32. ln y = ln( x 2 + 3x ) + ln( x – 2) + ln( x 2 + 1) 1 dy 2x + 3 1 2x = + + y dx x 2 + 3 x x – 2 x 2 + 1 dy ⎛ 2x + 3 1 2x ⎞ 4 3 2 = ( x 2 + 3 x)( x – 2)( x 2 + 1) ⎜ + + ⎟ = 5 x + 4 x –15 x + 2 x – 6 2 2 dx x – 2 + + x 3 x x 1 ⎝ ⎠ 1 1 ln( x + 13) – ln( x – 4) – ln(2 x + 1) 2 3 1 dy 1 1 2 = – – y dx 2( x + 13) x – 4 3(2 x + 1) 33. ln y = ⎡ ⎤ dy x + 13 1 1 2 10 x 2 + 219 x – 118 = – – = – dx ( x – 4) 3 2 x + 1 ⎢⎣ 2( x + 13) x – 4 3(2 x + 1) ⎥⎦ 6( x – 4)2 ( x + 13)1/ 2 (2 x + 1)4 / 3 2 1 ln( x 2 + 3) + 2 ln(3 x + 2) – ln( x + 1) 3 2 1 dy 2 2 x 2⋅3 1 = ⋅ + – y dx 3 x 2 + 3 3x + 2 2( x + 1) 34. ln y = dy ( x 2 + 3)2 / 3 (3 x + 2)2 = dx x +1 ⎡ 4x 6 1 ⎤ (3 x + 2)(51x3 + 70 x 2 + 97 x + 90) – + ⎢ 2 ⎥= 6( x 2 + 3)1/ 3 ( x + 1)3 / 2 ⎢⎣ 3( x + 3) 3 x + 2 2( x + 1) ⎥⎦ 35. 36. y = ln x is reflected across the y-axis. The y-values of y = ln x are multiplied by since ln x = Instructor’s Resource Manual 1 , 2 1 ln x. 2 Section 6.1 349 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 37. y = ln x is reflected across the x-axis since ⎛1⎞ ln ⎜ ⎟ = – ln x. ⎝x⎠ 38. 42. Let r(x) = rate of transmission 1 = kx 2 ln = −kx 2 ln x. x ⎛1⎞ r ( x) = −2kx ln x − kx 2 ⎜ ⎟ = −kx(2 ln x + 1) ⎝x⎠ 1 1 r ( x) = 0 if ln x = − , or − ln x = , so 2 2 1 1 ln = . x 2 1 1 ln1.65 ≈ , so x ≈ ≈ 0.606. 1.65 2 ⎛ 1⎞ r ( x) = −k (2 ln x + 1) − kx ⎜ 2 ⋅ ⎟ = –k(2 ln x + 3) ⎝ x⎠ r (0.606) ≈ −2k < 0 since k > 0, so x ≈ 0.606 gives the maximum rate of transmission. 43. ln 4 > 1 so ln 4m = m ln 4 > m ⋅1 = m Thus x > 4m ⇒ ln x > m so lim ln x = ∞ x →∞ y = ln x is shifted two units to the right. 1 so z → ∞ as x → 0+ x ⎛1⎞ Then lim ln x = lim ln ⎜ ⎟ = lim (– ln z ) + z →∞ ⎝ z ⎠ z →∞ x →0 = – lim ln z = – ∞ 44. Let z = 39. z →∞ 45. y = ln cos x + ln sec x = ln cos x + ln 1 cos x ⎛ π π⎞ = ln cos x − ln cos x = 0 on ⎜ − , ⎟ ⎝ 2 2⎠ 40. Since ln is continuous, sin x sin x lim ln = ln lim = ln1 = 0 x x →0 x →0 x 41. The domain is ( 0, ∞ ) . ⎛1⎞ f ( x) = 4 x ln x + 2 x 2 ⎜ ⎟ − 2 x = 4 x ln x ⎝x⎠ f ' ( x ) = 0 if ln x = 0 , or x = 1 . f ' ( x ) < 0 for x < 1 and f ' ( x ) > 0 for x > 1 so f(1) = –1 is a minimum. 350 Section 6.1 x 1 1/ 3 t 1 1 dt = 2 t dt x1 dt = 2 t 1 1 x1 dt = dt 1/ 3 t 1 t 1/ 3 1 x1 – dt = dt 1 t 1 t 1 – ln = ln x 3 ln 3 = ln x x=3 1/ 3 t 46. a. dt + x1 1 1 x1 1 t 1 1 < for t > 1, t t x1 so ln x = dt < 1 t 1 x dt 1 t dt = x –1/ 2 1 t dt x = ⎡⎣ 2 t ⎤⎦ = 2( x –1) 1 so ln x < 2( x – 1) Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. If x > 1, 0 < ln x < 2( x – 1) , x →∞ and lim x →∞ π ln x 2( x + 1) ≤ lim =0 x x x →∞ 1, 000, 000 ≈ 72,382 ln1, 000, 000 c ⎛ ax – b ⎞ ⎛ ax − b ⎞ f ( x) = ln ⎜ ⎟ = c ln ⎜ ⎟ + ax b ⎝ ⎠ ⎝ ax + b ⎠ = a 2 – b2 [ln(ax – b) – ln(ax + b)] 2ab f ( x) = = f (1) = b. a 2 – b2 ⎡ a a ⎤ – ⎢ 2ab ⎣ ax – b ax + b ⎦⎥ a 2 − b2 2ab ⎡ ⎤ 2ab a 2 – b2 ⎢ (ax − b)(ax + b) ⎥ = 2 2 ⎣ ⎦ a x – b2 a 2 − b2 a 2 − b2 f ( x) = cos 2 u ⋅ 3 4 π = ⎡⎣ln tan x ⎤⎦π 3 = ln tan π 3 − ln tan π 4 4 cos x 1 dx = du = ln u + C 1 + sin x u = ln 1 + sin x + C = ln(1 + sin x) + C (since 1 + sin x ≥ 0 for all x ). 53. V = 2π 4 2πx 1 2 x + x –1 2 ⋅1 + 1 f (1) = cos [ln(1 + 1 –1)] ⋅ 2 1 + 1 –1 = 3cos 2 (0) = 3 2πx x2 + 4 dx x2 x2 1 – ln x = – ln x 4 4 2 dy 2 x 1 1 x 1 = – ⋅ = – dx 4 2 x 2 2x 54. y = = 2 4 1 4 dx = ⎡⎢ π ln x 2 + 4 ⎤⎥ ⎣ ⎦ 1 x +4 = π ln 20 − π ln 5 = π ln 4 ≈ 4.355 du dx 2x + 1 xf ( x)dx = = π ln x 2 + 4 + C L= 2 4 1 Let u = x 2 + 4 so du = 2x dx. 2πx 1 dx = π du = π ln u + C 2 u x +4 =1 = cos 2 [ln( x 2 + x –1)] ⋅ 2 4 = ⎡− ⎣ ln cos x + ln sin x ⎤⎦π 52. Let u = 1 + sin x ; then du = cos x dx so that ⎡ 1 1 1 ⎤ 1 ⎥⋅ = lim ⎢ + + ⋅⋅⋅ + n⎥ n n→∞ ⎢1 + 1 1 + 2 + 1 n n⎦ ⎣ n n ⎛ ⎞ 21 1 1 ⎜ ⎟⋅ = = lim dx = ln 2 ≈ 0.693 i 1 x n→∞ i =1 ⎜ 1 + ⎟ n n⎠ ⎝ 49. a. 3 sec x csc x dx = ln( 3) − ln1 = 0.5493 − 0 = 0.5493 ln x = 0. x 1 1⎤ ⎡ 1 + + ⋅⋅⋅ + ⎥ 47. lim ⎢ 2n ⎦ n →∞ ⎣ n + 1 n + 2 48. π π ln x 2( x – 1) so 0 < < . x x Hence 0 ≤ lim 51. From Ex 10, 2 2 2 ⎛ dy ⎞ ⎛x 1 ⎞ 1 + ⎜ ⎟ dx = 1 + ⎜ – ⎟ dx 1 1 ⎝ dx ⎠ ⎝ 2 2x ⎠ 2 2 1 2 2⎛ x 1 ⎞ ⎛x 1 ⎞ ⎜ + ⎟ dx = 1 ⎜ + ⎟ dx ⎝ 2 2x ⎠ ⎝ 2 2x ⎠ 2 ⎤ 1 ⎡ x2 ⎡ ⎛1 ⎞⎤ = ⎢ + ln x ⎥ = ⎢ 2 + ln 2 − ⎜ + ln1⎟ ⎥ 2 ⎢⎣ 2 ⎝2 ⎠⎦ ⎥⎦1 2 ⎣ 3 1 = + ln 2 ≈ 1.097 4 2 50. From Ex 9, π 0 3 tan x dx π 3 = ⎡− ⎣ ln cos x ⎤⎦ 0 = ln cos 0 − ln cos π 3 ⎛ 1 ⎞ = ln(1) − ln(0.5) = ln ⎜ ⎟ ⎝ 0.5 ⎠ = ln 2 ≈ 0.69315 Instructor’s Resource Manual Section 6.1 351 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 55. b. c. 1 1 1 + + ⋅⋅⋅ + = the lower approximate area 2 3 n 1 1 1 + + ⋅⋅⋅ + = the upper approximate area 2 n –1 ln n = the exact area under the curve 58. a. Thus, 1 1 1 1 1 1 + + ⋅⋅⋅ + < ln n < 1 + + + ⋅⋅⋅ + . 2 3 n 2 3 n −1 y1 56. ln y – ln x = y–x 1 t dt – x1 1 y–x = the average value of t dt dt t y–x x 1 on [x, y]. t 1 Since is decreasing on the interval [x, y], the t average value is between the minimum value of 1 1 and the maximum value of . y x 57. a. 1 + 1.5sin x (1.5 + sin x)2 On [0,3π ], f ( x) = 0 when x ≈ 3.871, 5.553. Inflection points are (3.871, –0.182), (5.553, –0.182). 3π 0 ln(1.5 + sin x)dx ≈ 4.042 sin(ln x) x On [0.1, 20], f ( x) = 0 when x = 1. Critical points: 0.1, 1, 20 f(0.1) ≈ –0.668, f(1) = 1, f(20) ≈ –0.989 On [0.1, 20], the maximum value point is (1, 1) and minimum value point is (20, –0.989). f ( x) = − b. On [0.01, 0.1], f ( x) = 0 when x ≈ 0.043. f(0.01) ≈ –0.107, f(0.043) ≈ –1 On [0.01, 20], the maximum value point is (1, 1) and the minimum value point is (0.043, –1). y1 = f ( x) = − c. 20 0.1 cos(ln x)dx ≈ −8.37 59. 1 cos x ⋅ cos x = 1.5 + sin x 1.5 + sin x f ( x) = 0 when cos x = 0. f ( x) = π 3π 5π Critical points: 0, , , , 3π 2 2 2 f(0) ≈ 0.405, ⎛π⎞ ⎛ 3π ⎞ f ⎜ ⎟ ≈ 0.916, f ⎜ ⎟ ≈ −0.693, ⎝2⎠ ⎝ 2 ⎠ ⎛ 5π ⎞ f ⎜ ⎟ ≈ 0.916, f (3π) ≈ 0.405. ⎝ 2 ⎠ On [0,3π ], the maximum value points are ⎛π ⎞ ⎛ 5π ⎞ ⎜ , 0.916 ⎟ , ⎜ , 0.916 ⎟ and the minimum 2 2 ⎝ ⎠ ⎝ ⎠ ⎛ 3π ⎞ value point is ⎜ , −0.693 ⎟ . 2 ⎝ ⎠ a. 1⎡ 0 ⎢⎣ 5 ⎛1⎞ ⎛ 1 ⎞⎤ x ln ⎜ ⎟ − x 2 ln ⎜ ⎟ ⎥ dx = ≈ 0.139 x x 36 ⎝ ⎠ ⎝ ⎠⎦ b. Maximum of ≈ 0.260 at x ≈ 0.236 60. a. 1 0 [ x ln x − x ln x]dx = 7 ≈ 0.194 36 b. Maximum of ≈ 0.521 at x ≈ 0.0555 352 Section 6.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6.2 Concepts Review 1. 11. f ( z ) = 2( z – 1) > 0 for z > 1 f(z) is increasing at z = 1 because f(1) = 0 and f(z) > 0 for z > 1. Therefore, f(z) is strictly increasing on z ≥ 1 and so it has an inverse. 12. f ( x) = 2 x + 1 > 0 for x ≥ 2 . f(x) is strictly increasing on x ≥ 2 and so it has an inverse. 13. f ( x) = x 4 + x 2 + 10 > 0 for all real x. f(x) is strictly increasing and so it has an inverse. f ( x1 ) ≠ f ( x2 ) 2. x; f –1 ( y ) 3. monotonic; strictly increasing; strictly decreasing 4. ( f –1 ) ( y ) = 1 f ( x) Problem Set 6.2 1. f(x) is one-to-one, so it has an inverse. Since f (4) = 2, f −1 (2) = 4 . 2. f(x) is one-to-one, so it has an inverse. Since f(1) = 2, f −1 (2) = 1 . 3. f(x) is not one-to-one, so it does not have an inverse. 4. f(x) is not one-to-one, so it does not have an inverse. 5. f(x) is one-to-one, so it has an inverse. Since f(–1.3) ≈ 2, f −1 (2) ≈ −1.3 . 6. f(x) is one-to-one, so it has an inverse. Since 1 ⎛1⎞ f ⎜ ⎟ = 2, f −1 (2) = . 2 2 ⎝ ⎠ 7. 8. 9. 10. f ( x) = –5 x 4 – 3x 2 = –(5 x 4 + 3 x 2 ) < 0 for all x ≠ 0. f(x) is strictly decreasing at x = 0 because f(x) > 0 for x < 0 and f(x) < 0 for x > 0. Therefore f(x) is strictly decreasing for x and so it has an inverse. f ( x) = 7 x 6 + 5 x 4 > 0 for all x ≠ 0. f(x) is strictly increasing at x = 0 because f(x) > 0 for x > 0 and f(x) < 0 for x < 0. Therefore f(x) is strictly increasing for all x and so it has an inverse. <π f ( ) is decreasing at = 0 because f(0) = 1 and f( ) < 1 for 0 < < π . f( ) is decreasing at = π because f( π ) = –1 and f( ) > –1 for 0 < < π . Therefore f( ) is strictly decreasing on 0 ≤ ≤ π and so it has an inverse. f ( ) = – sin < 0 for 0 < f ( x) = – csc 2 x < 0 for 0 < x < f(x) is decreasing on 0 < x < 14. f (r ) = 1 r cos 4 tdt = – r 1 cos 4 tdt π f (r ) = – cos 4 r < 0 for all r ≠ k π + , k any 2 integer. π f(r) is decreasing at r = k π + since f (r ) < 0 2 on the deleted neighborhood π π ⎛ ⎞ ⎜ k π + − ε , k π + + ε ⎟ . Therefore, f(r) is 2 2 ⎝ ⎠ strictly decreasing for all r and so it has an inverse. 15. Step 1: y=x+1 x=y–1 Step 2: f –1 ( y ) = y – 1 Step 3: f –1 ( x) = x – 1 Check: f –1 ( f ( x)) = ( x + 1) – 1 = x f ( f –1 ( x)) = ( x – 1) + 1 = x 16. Step 1: x y = – +1 3 x – = y –1 3 x = –3(y – 1) = 3 – 3y Step 2: f –1 ( y ) = 3 – 3 y Step 3: f –1 ( x) = 3 – 3 x Check: ⎛ x ⎞ f –1 ( f ( x)) = 3 – 3 ⎜ – + 1⎟ = 3 + ( x – 3) = x ⎝ 3 ⎠ –(3 – 3 x) + 1 = (–1 + x) + 1 = x f ( f –1 ( x)) = 3 π 2 π and so it has an 2 inverse. Instructor’s Resource Manual Section 6.2 353 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 17. Step 1: y = x + 1 (note that y ≥ 0 ) x + 1 = y2 x = 2+ x = y 2 – 1, y ≥ 0 Step 2: f –1 ( y ) = y 2 – 1, y ≥ 0 Step 3: f Check: f –1 –1 x–2= 1 y2 1 y2 ,y>0 Step 2: f –1 ( y ) = 2 + 2 ( x) = x – 1, x ≥ 0 Step 3: f –1 ( x) = 2 + 2 ( f ( x)) = ( x + 1) – 1 = ( x + 1) – 1 = x f ( f –1 ( x)) = ( x 2 – 1) + 1 = x 2 = x = x 18. Step 1: y = – 1 – x (note that y ≤ 0 ) 1– x = – y f –1 ( f ( x)) = 2 + Step 2: f ( y) = 1 – y , y ≤ 0 Step 3: f Check: –1 ( x) = 1 – x 2 , x ≤ 0 f ( f –1 ( x)) = – 1 – (1 – x 2 ) = – x 2 = – x = –(–x) = x 1 x–3 1 x–3= – y 1 y ) 1 ⎛2+ ⎜ ⎝ 1 ( x 1–2 ) 1 ⎞–2 ⎟ x2 ⎠ = 1 = x2 ⎛ 1 ⎞ ⎜ 2⎟ ⎝x ⎠ 21. Step 1: y = 4 x 2 , x ≤ 0 (note that y ≥ 0 ) x2 = y 4 x=– y y =− , negative since x ≤ 0 4 2 f –1 ( f ( x)) = – f(f 1 x ( x)) = – 1 – x1–3 1 (3 – ) 1 x ⎛ x⎞ x ( x)) = 4 ⎜⎜ – ⎟⎟ = 4 ⋅ = x 4 ⎝ 2 ⎠ y = ( x – 3)2 , x ≥ 3 (note that y ≥ 0 ) = 3 + ( x – 3) = x x–3= y x = 3+ y 1 =– =x 1 – –3 x 20. Step 1: 1 (note that y > 0) y= x–2 1 y2 = x–2 Section 6.2 –1 22. Step 1: Check: f –1 ( f ( x)) = 3 – 4 x2 = – x 2 = – x = –(– x) = x 2 2 1 ( y) = 3 – y Step 3: f –1 ( x) = 3 – 354 1 x –2 = 2+ Step 2: f –1 ( y ) = − y=– f(f ( 2 y 2 x Step 3: f –1 ( x) = − 2 Check: 19. Step 1: –1 1 2 f –1 ( f ( x)) = 1 – (– 1 – x ) 2 = 1 – (1 – x) = x Step 2: f ,x>0 x2 = x =x –1 –1 1 = 2 + (x – 2) = x 1 – x = (– y ) 2 = y 2 x = 3– ,y>0 y2 Check: f ( f –1 ( x)) = x = 1 – y2 , y ≤ 0 1 Step 2: f –1 ( y ) = 3 + y Step 3: f –1 ( x) = 3 + x Check: f –1 ( f ( x)) = 3 + ( x – 3)2 = 3 + x – 3 = 3 + ( x – 3) = x f ( f –1 ( x)) = [(3 + x ) – 3]2 = ( x )2 = x Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 23. Step 1: y = ( x –1) x –1 = 3 y Step 3: f –1 ( x) = x = 1+ 3 y Step 3: f –1 ( x) = 1 + 3 x f(f –1 –1 3 ( f ( x)) = 1 + 3 ( x –1) = 1 + ( x –1) = x 3 3 3 3 ( x)) = [(1 + x ) –1] = ( x ) = x 24. Step 1: y = x5 / 2 , x ≥ 0 x= y Step 3: f –1 ( x) = x 2 / 5 Check: f ( f –1 ( x)) = ( x 2 / 5 )5 / 2 = x 25. Step 1: x –1 y= x +1 xy + y = x –1 x – xy = 1 + y 1+ y x= 1– y = 1+ 1– x –1 x +1 x –1 x +1 ⎛ 2 x1/ 3 ⎞ =⎜ ⎟ = ( x1/ 3 )3 = x ⎜ 2 ⎟ ⎝ ⎠ 27. Step 1: x3 + 2 y= x3 + 1 1+ 1– 1+ x 1– x 1+ x 1– x 2– y y –1 1/ 3 ⎛2– y⎞ x=⎜ ⎟ ⎝ y –1 ⎠ 1/ 3 1+ x 1– x ⎛2– y⎞ Step 2: f –1 ( y ) = ⎜ ⎟ ⎝ y –1 ⎠ x –1 x +1 x –1 x +1 ⎛2– x⎞ Step 3: f –1 ( x) = ⎜ ⎟ ⎝ x –1 ⎠ Check: 1/ 3 = x + 1 + x –1 2 x = =x x +1 – x +1 2 –1 1 + x –1 + x 2 x = = =x +1 1+ x +1 – x 2 26. Step 1: 3 ⎛ x –1 ⎞ y=⎜ ⎟ ⎝ x +1⎠ x –1 y1/ 3 = x +1 xy1/ 3 + y1/ 3 = x –1 x – xy1/ 3 = 1 + y1/ 3 x= ( ) 1/ 3 ⎥⎦ 3⎤ ⎡ 1 – ⎢ xx +–11 ⎥ ⎣ ⎦ x +1 + x – 1 2x = = =x x +1 – x +1 2 x3 = 1+ y ( y) = 1− y ( x)) = 3 ⎤1/ 3 x3 y – x3 = 2 – y Check: f(f ( xx+–11 ) x3 y + y = x3 + 2 Step 3: f –1 ( x) = –1 ⎡ 1+ ⎢ ⎣ f –1 ( f ( x)) = 3 f –1 ( f ( x)) = ( x5 / 2 ) 2 / 5 = x f –1 ( f ( x)) = 1 – x1/ 3 3 Step 2: f –1 ( y ) = y 2 / 5 Step 2: f 1 + x1/ 3 ⎛ 1+ x1/ 3 – 1 ⎞ 3 ⎛ 1 + x1/ 3 – 1 + x1/ 3 ⎞ ⎜ 1– x1/ 3 ⎟ –1 f ( f ( x)) = ⎜ ⎟ ⎟ = ⎜⎜ 1/ 3 1 + x1/ 3 + 1 – x1/ 3 ⎟⎠ ⎜⎜ 1+ x + 1 ⎟⎟ ⎝ ⎝ 1– x1/ 3 ⎠ 2/5 –1 1 – y1/ 3 Check: Step 2: f –1 ( y ) = 1 + 3 y Check: f 1 + y1/ 3 Step 2: f –1 ( y ) = 3 1 + y1/ 3 1 – y1/ 3 Instructor’s Resource Manual 1/ 3 ⎛ 2 – x3 + 2 ⎞ ⎜ x3 +1 ⎟ f –1 ( f ( x)) = ⎜ ⎟ 3 ⎜⎜ x + 2 –1 ⎟⎟ ⎝ x3 +1 ⎠ 1/ 3 ⎛ 2 x3 + 2 – x3 – 2 ⎞ =⎜ ⎟ ⎜ x3 + 2 – x3 –1 ⎟ ⎝ ⎠ 1/ 3 ⎛ x3 ⎞ =⎜ ⎟ ⎜ 1 ⎟ ⎝ ⎠ =x ( ) 3 ⎡ 2– x 1/ 3 ⎤ x +2 ⎢⎣ x –1 ⎥⎦ + 2 2– –1 = x –1 f ( f ( x)) = 3 2– x + 1 ⎡ 2– x 1/ 3 ⎤ x –1 1 + ⎢⎣ x –1 ⎥⎦ 2 – x + 2x – 2 x = = =x 2 – x + x –1 1 ( ) Section 6.2 355 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 28. Step 1: ⎛ x3 + 2 ⎞ y=⎜ ⎟ ⎜ x3 + 1 ⎟ ⎝ ⎠ 1/ 5 y 29. By similar triangles, 5 This gives x3 + 2 = V= 3 V 27 h3 = 4π x3 + 1 x3 y1/ 5 + y1/ 5 = x3 + 2 x3 y1/ 5 – x3 = 2 – y1/ 5 x3 = h = 33 2 – y1/ 5 y1/ 5 – 1 ⎛2– y Step 2: f –1 ( y ) = ⎜ ⎟ ⎜ y1/ 5 – 1 ⎟ ⎝ ⎠ 1/ 3 3 = 4π h3 27 V 4π v v2 v2 H = s (v0 / 32) = v0 0 − 16 0 = 0 32 322 64 Check: v02 = 64 H 1/ 3 5 ⎤1/ 5 ⎫ ⎧ ⎡ ⎪ 2 – ⎢⎛ x3 + 2 ⎞ ⎥ ⎪ ⎜ 3 ⎟ ⎪⎪ ⎢⎣⎝ x +1 ⎠ ⎥⎦ ⎪⎪ –1 f ( f ( x)) = ⎨ ⎬ 1/ 5 ⎪ ⎡ ⎛ 3 ⎞5 ⎤ ⎪ ⎪ ⎢ ⎜ x 3+ 2 ⎟ ⎥ – 1 ⎪ ⎪⎩ ⎢⎣⎝ x +1 ⎠ ⎥⎦ ⎪⎭ v0 = 8 H 31. f ( x) = 4 x + 1; f ( x) > 0 when x > − 1 and 4 1 f ( x) < 0 when x < − . 4 1/ 3 1/ 3 ⎛ 2 x3 + 2 – x3 – 2 ⎞ =⎜ ⎟ 3 ⎜ 3 ⎟ ⎝ x + 2 – x –1 ⎠ 1/ 3 ⎛ x3 ⎞ =⎜ ⎟ ⎜ 1 ⎟ ⎝ ⎠ ) s (t ) = v0t − 16t 2 . The ball then reaches a height of ⎛ 2 – x1/ 5 ⎞ ( x) = ⎜ ⎟ ⎜ x1/ 5 – 1 ⎟ ⎝ ⎠ ⎛ 2 – x3 + 2 ⎞ ⎜ x3 +1 ⎟ =⎜ ⎟ 3 ⎜⎜ x + 2 – 1 ⎟⎟ ⎝ x3 +1 ⎠ ( v t = 0 . The position function is 32 1/ 5 ⎞1/ 3 Step 3: f = π 4h 2 / 9 h 30. v = v0 − 32t v = 0 when v0 = 32t , that is, when 1/ 3 ⎛ 2 – y1/ 5 ⎞ x=⎜ ⎟ ⎜ y1/ 5 – 1 ⎟ ⎝ ⎠ –1 π r 2h r 4 2h = . Thus, r = h 6 3 =x 5 ⎧⎡ ⎫ 1/ 3 ⎤ 3 ⎪ ⎢⎛ 2– x1/ 5 ⎞ ⎥ + 2 ⎪ ⎜ ⎟ ⎪⎪ ⎢⎝ x1/ 5 –1 ⎠ ⎥ ⎪⎪ ⎦ f ( f –1 ( x)) = ⎨ ⎣ ⎬ 3 ⎪ ⎡⎛ 1/ 5 ⎞1/ 3 ⎤ ⎪ 2– x ⎪ ⎢⎜ 1/ 5 ⎟ ⎥ + 1 ⎪ ⎩⎪ ⎣⎢⎝ x –1 ⎠ ⎦⎥ ⎭⎪ 1⎤ ⎛ The function is decreasing on ⎜ −∞, − ⎥ and 4⎦ ⎝ ⎡ 1 ⎞ increasing on ⎢ − , ∞ ⎟ . Restrict the domain to ⎣ 4 ⎠ 1⎤ ⎛ ⎡ 1 ⎞ ⎜ −∞, − ⎥ or restrict it to ⎢ − , ∞ ⎟ . 4⎦ ⎣ 4 ⎠ ⎝ Then f −1 ( x) = f −1 ( x) = 1 (−1 − 8 x + 33) or 4 1 (−1 + 8 x + 33). 4 5 ⎛ 2– x1/ 5 ⎞ 5 ⎛ 2 – x1/ 5 + 2 x1/ 5 – 2 ⎞ ⎜ x1/ 5 –1 + 2 ⎟ =⎜ = ⎜ ⎟ ⎟ 1/ 5 ⎜ 2 – x1/ 5 + x1/ 5 – 1 ⎟ ⎜⎜ 2– x + 1 ⎟⎟ ⎝ ⎠ 1/ 5 ⎝ x –1 ⎠ 5 ⎛ x1/ 5 ⎞ =⎜ ⎟ =x ⎜ 1 ⎟ ⎝ ⎠ 356 Section 6.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 32. f ( x) = 2 x − 3; f ( x) > 0 when x > 3 2 36. ( f −1 ) (3) ≈ 1 2 3 and f ( x) < 0 when x < . 2 ⎛ 3⎤ The function is decreasing on ⎜ −∞, ⎥ and 2⎦ ⎝ ⎡3 ⎞ increasing on ⎢ , ∞ ⎟ . Restrict the domain to ⎣2 ⎠ ⎛ 3⎤ ⎡3 ⎞ ⎜ −∞, ⎥ or restrict it to ⎢ , ∞ ⎟ . Then 2⎦ ⎣2 ⎠ ⎝ 1 (3 − 4 x + 5) or 2 1 f −1 ( x) = (3 + 4 x + 5). 2 f −1 ( x) = 33. 37. f ( x) = 15 x 4 + 1 and y = 2 corresponds to x = 1, so ( f −1 ) (2) = 38. f ( x) = 5 x 4 + 5 and y = 2 corresponds to x = 1, so ( f −1 ) (2) = 39. 1 1 1 = = . f (1) 15 + 1 16 1 1 1 = = f (1) 5 + 5 10 π , 4 1 1 1 ⎛π⎞ so ( f −1 ) (2) = = = cos 2 ⎜ ⎟ 2 π π 2 ⎝4⎠ f 4 2sec 4 f ( x) = 2sec2 x and y = 2 corresponds to x = ( ) (f 34. −1 = 1 ) (3) ≈ 3 ( f −1 ) (3) ≈ − 1 2 40. ( ) 1 . 4 f ( x) = 1 2 x +1 so ( f −1 ) (2) = and y = 2 corresponds to x = 3, 1 = 2 3 +1 = 4 . f (3) 41. ( g –1 f –1 )(h( x)) = ( g –1 f –1 )( f ( g ( x))) = g –1 [ f –1 ( f ( g ( x)))] = g –1 [ g ( x)] = x Similarly, h(( g –1 f –1 )( x)) = f ( g (( g –1 f –1 )( x))) = f ( g ( g –1 ( f –1 ( x )))) = f ( f –1 ( x)) = x Thus h –1 = g –1 f –1 35. ( f −1 ) (3) ≈ − 1 3 Instructor’s Resource Manual Section 6.2 357 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 42. Find f −1 ( x) : y= ax + b cx + d cxy + dy = ax + b (cy – a)x = b – dy b − dy dy − b x= =− cy − a cy − a y= 44. a. 1 1 , x= x y f −1 ( y ) = 1 y f −1 ( x ) = 1 x Find g −1 ( x) : y = 3x + 2 y−2 x= 3 y−2 g −1 ( y ) = 3 x − 2 g −1 ( x) = 3 f −1 ( x) = − dx − b cx − a If f = f −1 , then for all x in the domain we have: ax + b dx − b + =0 cx + d cx − a (ax + b)(cx – a) + (dx – b)(cx + d) = 0 c. 1 3x + 2 (1) −2 ⎛1⎞ h −1 ( x) = g −1 ( f −1 ( x)) = g −1 ⎜ ⎟ = x 3 ⎝x⎠ ⎛ 1 ⎞ (3x + 2) − 2 3x h −1 (h( x)) = h −1 ⎜ = =x ⎟= 3 3 ⎝ 3x + 2 ⎠ ( ) acx 2 + (bc − a 2 ) x − ab + dcx 2 +(d 2 − bc) x − bd = 0 (ac + dc) x 2 + (d 2 − a 2 ) x + (− ab − bd ) = 0 Setting the coefficients equal to 0 gives three requirements: (1) a = –d or c = 0 (2) a = ±d (3) a = –d or b = 0 ⎛ 1 −2⎞ 1 1 ⎟= h(h −1 ( x)) = h ⎜ x = =x 1 ⎜ 3 ⎟ ⎡ 1 − 2⎤ + 2 x ⎝ ⎠ ⎣ x ⎦ ( ) ( ) 43. f has an inverse because it is monotonic (increasing): If a = d, then f = f −1 requires b = 0 and ax = x . If a = –d, there are d no requirements on b and c (other than c = 0, so f ( x) = f ( x) = 1 + cos 2 x > 0 a. ( f −1 ) ( A) = b. ( f −1 ) ( B ) = c. dy − b cy − a b. If bc – ad = 0, then f(x) is either a constant function or undefined. h( x ) = f ( g ( x)) = f (3x + 2) = = f −1 ( y ) = − 1 f ( ) π 2 1 f = ( 56π ) = 1 ( ) 1 + cos 2 π2 1 ( ) 1 + cos 2 56π bc − ad ≠ 0 ). Therefore, f = f −1 if a = –d or if f is the identity function. =1 = 1 45. 7 4 2 7 ( f −1 ) (0) = 1 1 1 = = 2 f (0) 2 1 + cos (0) 1 −1 0 f ( y ) dy = (Area of region B) = 1 – (Area of region A) 1 2 3 = 1 − f ( x) dx = 1 − = 0 5 5 358 Section 6.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 46. a 0 f ( x)dx = the area bounded by y = f(x), y = 0, 47. Given p > 1, q > 1, and x = a [the area under the curve]. f –1 ( y )dy = the area bounded by x = f –1 ( y ) solving x = 0, and y = b. ab = the area of the rectangle bounded by x = 0, x = a, y = 0, and y = b. Case 1: b > f(a) 1 = p –1 b 0 1 1 + = 1, and f ( x) = x p –1 , p q 1 1 q + = 1 for p gives p = , so p q q –1 1 q –1 q –1 = 1 = ⎡ q –( q –1) ⎤ ⎣⎢ q –1 ⎦⎥ q –1 = q – 1. 1 1 Thus, if y = x p –1 then x = y p –1 = y q –1 , so f –1 ( y ) = y q –1. By Problem 44, since f ( x) = x p −1 is strictly increasing for p > 1, ab ≤ a The area above the curve is greater than the area of the part of the rectangle above the curve, so the total area represented by the sum of the two integrals is greater than the area ab of the rectangle. Case 2: b = f(a) a p –1 b q –1 0 0 x dx + y dy b ⎡xp ⎤ ⎡ yq ⎤ ab ≤ ⎢ ⎥ + ⎢ ⎥ ⎣⎢ p ⎦⎥ 0 ⎣⎢ q ⎥⎦ 0 a p bq ab ≤ + p q 6.3 Concepts Review 1. increasing; exp 2. ln e = 1; 2.72 3. x; x The area represented by the sum of the two integrals = the area ab of the rectangle. Case 3: b < f(a) 4. e x ; e x + C Problem Set 6.3 1. a. 20.086 b. 8.1662 c. e 2 ≈ e1.41 ≈ 4.1 d. ecos(ln 4) ≈ e0.18 ≈ 1.20 2. a. The area below the curve is greater than the area of the part of the rectangle which is below the curve, so the total area represented by the sum of the two integrals is greater than the area ab of the rectangle. ab ≤ a 0 f ( x) dx + b 0 f −1 ( y ) dy with equality holding when b = f(a). Instructor’s Resource Manual b. 3 e3ln 2 = eln(2 ) = eln 8 = 8 e ln 64 2 1/ 2 ) = eln(64 = eln 8 = 8 3 3. e3ln x = eln x = x3 4. e –2 ln x = eln x −2 = x −2 = 1 x2 Section 6.3 359 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 5. ln ecos x = cos x 19. 6. ln e –2 x –3 = –2 x – 3 ex ex x = eln x 2 9. eln 3+ 2 ln x = eln 3 ⋅ e2 ln x = 3 ⋅ eln x = 3 x 2 10. eln x 2 – y ln x = eln x e 2 x2 = y ln x e ln x y x2 = x y = x 2– y 11. Dx e x + 2 = e x + 2 Dx ( x + 2) = e x + 2 12. Dx e =e 2 x2 – x 2 x2 – x 14. Dx – 1 2 e x 2 x2 – x =e x+ 2 Dx (2 x – x) Dx x + 2 = e x+2 2 x+2 ⋅ 2x ln x 2 x2 –3 = x2 x2 xe x + −2 2 1 ⎤ ⎥ = Dx e x + Dx e − x 2 e x ⎥⎦ −2 Dx x −2 + e− x Dx [− x 2 ] −2 ⋅ (−2 x −3 ) + e− x ⋅ (−2 x) 2 2 x2 2e1 x 3 − 2x ex 2 21. Dx [e xy + xy ] = Dx [2] e xy ( xDx y + y ) + ( xDx y + y ) = 0 xe xy Dx y + ye xy + xDx y + y = 0 xe xy Dx y + xDx y = – ye xy – y – 1 2 2e x x3 = Dx x 2 = 2 x 1 x x x (ln x ) ⋅1 – x ⋅ x x = e ln x ⋅ 16. Dx e ln x = e ln x Dx 2 ln x (ln x ) = ⎡ 20. Dx ⎢e1 ⎢⎣ =− ⎞ ⎟ ⎝ x ⎠ 15. Dx e2 ln x = Dx e 2 = x ex + = ex 2 – 1 ⎛ 1 2 = e x Dx ⎜ – 2 – 1 2 =e x 2 ] = Dx (e x )1 2 + Dx e 2 2 1 x2 −1 2 Dx e x + e x Dx x 2 (e ) 2 2 2 1 2 x = (e x )−1 2 e x Dx x 2 + e x ⋅ 2 x2 2 x 1 2 = (e x )1 2 2 x + e x ⋅ x 2 = ex (4 x –1) x+2 13. Dx e =e x2 = 7. ln( x3e –3 x ) = ln x3 + ln e –3 x = 3ln x – 3 x 8. e x –ln x = 2 Dx [ e x + e x e ln x (ln x –1) Dx y = − ye xy – y xe xy +x =– y (e xy + 1) x (e xy + 1) =– y x 22. Dx [e x + y ] = Dx [4 + x + y ] e x + y (1 + Dx y ) = 1 + Dx y e x + y + e x + y Dx y = 1 + Dx y e x + y Dx y – Dx y = 1 – e x + y Dx y = 1 – e x+ y e x + y –1 = –1 23. a. (ln x) 2 17. Dx ( x3e x ) = x3 Dx e x + e x Dx ( x3 ) = x3e x + e x ⋅ 3x 2 = x 2 e x ( x + 3) 18. Dx e x = ex = ex 3 ln x 3 ln x Dx ( x3 ln x) 3 ln x ⎛ 3 1 2⎞ ⎜ x ⋅ + ln x ⋅ 3x ⎟ x ⎝ ⎠ 3 ln x = x2e x 360 = ex ( x 2 + 3x 2 ln x) 3 ln x The graph of y = e x is reflected across the x-axis. (1 + 3ln x) Section 6.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. 26. f ( x) = e −x 2 Domain = (−∞, ∞) 1 1 −x f ( x) = − e 2 , f ( x) = e 2 2 4 Since f ( x) < 0 for all x, f is decreasing on (−∞, ∞) . Since f ( x) > 0 for all x, f is concave upward on (−∞, ∞) . Since f and f are both monotonic, there are no extreme values or points of inflection. −x The graph of y = e x is reflected across the y-axis. y 24. a < b ⇒ – a > – b ⇒ e increasing function. 25. f ( x) = e 2x –a >e –b x , since e is an 8 Domain = (−∞, ∞) 2x f ( x) = 2e , f ( x) = 4e2 x Since f ( x) > 0 for all x, f is increasing on (−∞, ∞) . Since f ( x) > 0 for all x, f is concave upward on (−∞, ∞) . Since f and f are both monotonic, there are no extreme values or points of inflection. 4 −5 27. f ( x) = xe − x Domain = (−∞, ∞) f ( x) = (1 − x)e− x , y (−∞,1) + − x f f 8 x 5 f ( x) = ( x − 2)e − x 1 0 − (1, 2) − − (2, ∞) − + 2 − 0 f is increasing on (−∞,1] and decreasing on [1, ∞) . f has a maximum at (1, 1 ) e f is concave up on (2, ∞) and concave down on 4 −2 2 x (−∞, 2) . f has a point of inflection at (2, 2 y e2 ) 5 −3 8 x −5 Instructor’s Resource Manual Section 6.3 361 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 28. f ( x) = e x + x Domain = (−∞, ∞) f ( x) = e x + 1 , 30. f ( x) = e x f ( x) = ln(2 x − 1) . Since 2 x − 1 > 0 if and only if x> Since f ( x) > 0 for all x, f is increasing on (−∞, ∞) . Since f ( x) > 0 for all x, f is concave upward on (−∞, ∞) . Since f and f are both monotonic, there are no extreme values or points of inflection. 1 2 1 2 , domain = ( , ∞) f ( x) = 2 , 2x −1 f ( x) = −4 (2 x − 1) 2 Since f ( x) > 0 for all domain values, f is 1 2 increasing on ( , ∞) . Since f ( x) < 0 for all domain values, f is 1 2 concave downward on ( , ∞) . y Since f and f are both monotonic, there are no extreme values or points of inflection. 5 y 5 −5 5 x −5 x 8 29. f ( x) = ln( x 2 + 1) Since x 2 + 1 > 0 for all x, domain = (−∞, ∞) f ( x) = x f f 2x , x +1 2 f ( x) = −5 2 −2( x − 1) ( x 2 + 1) 2 ( −∞ , −1) −1 ( −1,0) 0 (0,1) 1 (1,∞ ) 0 − − − + + + 0 0 − + + + − f is increasing on (0, ∞) and decreasing on (−∞, 0) . f has a minimum at (0, 0) f is concave up on (−1,1) and concave down on (−∞, −1) ∪ (1, ∞) . f has points of inflection at (−1, ln 2) and (1, ln 2) y 31. f ( x) = ln(1 + e x ) Since 1 + e x > 0 for all x, domain = (−∞, ∞) f ( x) = ex , f ( x) = ex 1 + ex (1 + e x ) 2 Since f ( x) > 0 for all x, f is increasing on (−∞, ∞) . Since f ( x) > 0 for all x, f is concave upward on (−∞, ∞) . Since f and f are both monotonic, there are no extreme values or points of inflection. y 5 5 5 −5 x 5 −5 x −5 −5 362 Section 6.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 32. f ( x) = e1− x 2 f ( x) = −2 xe1− x , f ( x) = (4 x 2 − 2)e1− x 2 x ( −∞ , − f + y Domain = (−∞, ∞) 2 2 ) − 2 2 (− + 2 2 ,0) 0 (0, ) 2 2 2 2 + − − 0 3 2 ( 2 ,∞ ) 2 − −1 f + − 0 − − + 0 f is increasing on (−∞, 0] and decreasing on [0, ∞) . f has a maximum at (0, e) f is concave up on (−∞, − concave down on (− inflection at (− 2 2 2 2 , 2 2 ) ∪ ( , ∞ ) and 2 2 2 ). 2 , e ) and ( f has points of 2 2 x 4 2 , e) y −3 34. f ( x) = e x − e− x x f ( x) = e + e x f f −x Domain = (−∞, ∞) , f ( x) = e x − e− x (−∞, 0) 0 (0, ∞) + + + − + 0 f is increasing on (−∞, ∞) and so has no extreme values. f is concave up on (0, ∞) and concave down on (−∞, 0) . f has a point of inflection at (0, 0) 3 −3 3 y x 3 −3 33. f ( x) = e −( x − 2) 2 Domain = (−∞, ∞) 3 −3 x f ( x) = (4 − 2 x)e −( x − 2) , 2 f ( x) = (4 x 2 − 16 x + 14)e − ( x − 2) 2 −3 Note that 4 x 2 − 16 x + 14 = 0 when x= x f f 4± 2 ≈ 2 ± 0.707 2 ( −∞ ,1.293) ≈1.293 (1.293,2) 2 (2,2.707) ≈ 2.707 (2.707,∞ ) + + + 0 − − − + − − − + 0 0 f is increasing on (−∞, 2] and decreasing on [2, ∞) . f has a maximum at (2,1) f is concave up on (−∞, 4−2 2 ) ∪ ( 4+2 2 , ∞) and concave down on ( 4− 2 4+ 2 , ) . f has points 2 2 4− 2 of inflection at ( 2 , 1 ) and e Instructor’s Resource Manual ( 4+ 2 2 , 1 ). e Section 6.3 363 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 35. f ( x) = x − t2 0 e f ( x) = e− x , 2 37. Let u = 3x + 1, so du = 3dx. 1 3 x +1 1 u 1 3dx = e3 x +1dx = e e du = eu + C 3 3 3 1 3 x +1 = e +C 3 Domain = (−∞, ∞) dt f ( x) = −2 xe− x 2 (−∞, 0) 0 (0, ∞) + + + + 0 − x f f f is increasing on (−∞, ∞) and so has no extreme values. f is concave up on (−∞, 0) and concave down on (0, ∞) . f has a point of inflection at (0, 0) y 39. Let u = x 2 + 6 x , so du = (2x + 6)dx. 2 1 u 1 e du = eu + C ( x + 3)e x + 6 x dx = 2 2 1 x2 +6 x = e +C 2 3 3 −3 38. Let u = x 2 − 3, so du = 2x dx. 2 1 x 2 −3 1 u xe x −3 dx = e e du 2 x dx = 2 2 1 1 2 = eu + C = e x −3 + C 2 2 x 40. Let u = e x − 1, so du = e x dx . ex 1 dx = du = ln u + C = ln e x − 1 + C u e −1 −3 36. f ( x) = x 0 te− t dt f ( x) = xe −x , ( −∞ ,0) − + x f f x Domain = (−∞, ∞) f ( x) = (1 − x)e 0 0 + (0,1) + + 1 1 41. Let u = − , so du = dx . x x2 e−1/ x dx = eu du = eu + C = e−1/ x + C x2 −x 1 + 0 (1,∞ ) + − f is increasing on [0, ∞) and decreasing on (−∞, 0] . f has a minimum at (0, 0) f is concave up on (−∞,1) and concave down on (1, ∞) . f has a point of inflection at (1, 1 0 x 43. Let u = 2x + 3, so du = 2dx 1 u 1 1 e2 x +3 dx = e du = eu + C = e 2 x +3 + C 2 2 2 = −2 x 1 1 1 ⎡1 ⎤ = ⎢ e2 x +3 ⎥ = e5 – e3 2 ⎣2 ⎦0 2 1 3 2 e (e − 1) ≈ 64.2 2 44. Let u = 9 x e x ⋅ ee dx = eu du = eu + C = ee + C 1 2 x +3 e dx 0 4 −3 x Let u = e x , so du = e x dx. te−t dt ) . Note: It can be shown with techniques in 2 1 Chapter 7 that 0 te− t dt = 1 − ≈ 0.264 e y x e x + e dx = e x ⋅ ee dx 42. e3 / x 2 3 3 , so du = − dx. x x2 dx = – 1 u 1 e du = – eu + C 3 3 x 1 = – e3 / x + C 3 2 e3 / x 2 1 1 ⎡ 1 ⎤ dx = ⎢ – e3 / x ⎥ = – e3 / 2 + e3 ≈ 5.2 1 x2 3 3 3 ⎣ ⎦1 364 Section 6.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ln 3 45. V = π 0 (e x )2 dx = π ln 3 ⎡1 ⎤ = π ⎢ e2 x ⎥ ⎣2 ⎦0 46. V = ln 3 2 x e dx 0 1 ⎞ ⎛1 = π ⎜ e2 ln 3 − e0 ⎟ = 4π ≈ 12.57 2 ⎠ ⎝2 2 1 2πxe− x dx . 0 y = et cos t , so dy = (et cos t − et sin t )dt 2 ds = dx 2 + dy 2 2 2πxe− x dx = −π e− x (−2 x)dx = −π eu du = et (sin t + cos t )2 + (cos t − sin t )2 dt 2 = −πeu + C = −πe− x + C 0 2πxe − x2 = et 2sin 2 t + 2 cos 2 tdt = 2et dt The length of the curve is 1 ⎡ 2⎤ dx = −π ⎢ e− x ⎥ = – π(e−1 − e0 ) ⎣ ⎦0 π −1 = π(1 − e ) ≈ 1.99 0 1 1− e 1− e ( x − 0); −1 = ⇒ y −1 = 1− 0 e e e 1− e y= x +1 e = 53. a. 1 1 ⎡⎛ 1 − e 48. f ( x) = = = (e x –1)2 e x − 1 − xe x (e x − 1) 2 e x − 1 − xe x (e x − 1) 2 =− − − 1 1 − e− x 1 ex −1 – 1 1 – e– x (– e = lim x →0+ b. e x − 1 − xe x − (e x − 1) (e x − 1) 2 Dx [1 + (ln x) 2 ] ln x f ( x) = = = lim x →0 + ∞ . ∞ 1 x 2 ln x ⋅ 1x 1 =0 2 ln x x →∞ 1 + (ln x) 2 )(–1) ⎛ 1 ⎞ ⎜ x⎟ ⎝e ⎠ = Dx ln x lim –x is of the form 2 x →0+ 1 + (ln x) x →0+ e (e x –1)(1) – x(e x ) ln x lim = lim ⎤ ⎞ ⎡1 − e 2 ⎤ x + 1⎟ − e− x ⎥ dx = ⎢ x + x + e− x ⎥ 2 e ⎠ ⎣ ⎦0 ⎦ 1− e 1 3−e = +1+ −1 = ≈ 0.052 2e e 2e 0 ⎢⎣⎜⎝ π 2et dt = 2 ⎡ et ⎤ = 2(eπ − 1) ≈ 31.312 ⎣ ⎦0 52. Use x = 30, n = 8, and k = 0.25. (kx) n e− kx (0.25 ⋅ 30)8 e−0.25⋅30 ≈ 0.14 Pn ( x) = = n! 8! ⎛ 1⎞ 47. The line through (0, 1) and ⎜1, ⎟ has slope ⎝ e⎠ 1 −1 e e0.3 ≈ 1.3498588 by direct calculation 51. x = et sin t , so dx = (et sin t + et cos t )dt Let u = − x 2 , so du = –2x dx. 1 ⎧ ⎡⎛ 0.3 ⎞ 0.3 ⎤ 0.3 ⎫ 50. e0.3 ≈ ⎨ ⎢⎜ + 1⎟ + 1⎥ + 1⎬ ( 0.3) + 1 ⎠ 3 ⎦ 2 ⎩ ⎣⎝ 4 ⎭ = 1.3498375 1 = lim =0 x →∞ 2 ln x [1 + (ln x) 2 ] ⋅ 1x – ln x ⋅ 2 ln x ⋅ 1x [1 + (ln x) 2 ]2 1 – (ln x)2 x[1 + (ln x) 2 ]2 f ( x) = 0 when ln x = ±1 so x = e1 = e xe x (e x − 1) 2 When x > 0, f ( x) < 0, so f(x) is decreasing for x > 0. 49. a. Exact: 10! = 10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 = 3, 628,800 Approximate: 10 ⎛ 10 ⎞ 10! ≈ 20π ⎜ ⎟ ⎝ e ⎠ ⎛ 60 ⎞ b. 60! ≈ 120π ⎜ ⎟ ⎝ e ⎠ ≈ 3,598, 696 1 e ln e or x = e –1 = f (e) = 1 + (ln e) = 1 2 1+1 = 1 2 ln 1e –1 1 ⎛1⎞ f ⎜ ⎟= = =– 2 2 2 ⎝ e ⎠ 1 + ln 1 1 + (–1) e ( ) Maximum value of value of − 60 2 1 at x = e; minimum 2 1 at x = e−1. 2 ≈ 8.31× 1081 Instructor’s Resource Manual Section 6.3 365 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. x2 F ( x) = c. ln t 1 + (ln t )2 1 ln x 2 F ( x) = 1 + (ln x 2 )2 F ( e) = ln( e ) y-axis so the area is ⎧⎪ 3 2 2 2 ⎨ 2 [e− x − 2e x (2 x 2 − 1)] dx 0 ⎪⎩ dt ⋅ 2x 2 2 2 1 + [ln( e ) ] ⋅2 e = + 1 1 + 12 ⋅2 e = e ≈ 1.65 e x0 – 0 = f ( x0 ) = e x0 ⇒ e x0 = x0 e x0 ⇒ x0 = 1 x0 – 0 a. b. 1 2 ⎫⎪ (2 x 2 − 1) − e− x ] dx ⎬ ⎪⎭ x →∞ f ( x) = x p e – x (–1) + e – x ⋅ px p –1 b. = x p –1e – x ( p – x) f ( x) = 0 when x = p so the line is y = e x0 x or y = ex. 59. ⎡ ex 2 ⎤ A = (e – ex)dx = ⎢ e x – ⎥ 0 2 ⎥⎦ ⎣⎢ 0 e e = e − − (e0 − 0) = – 1 ≈ 0.36 2 2 − x2 lim x p e – x = 0 58. a. x 3 [2e 2 ≈ 4.2614 54. Let ( x0 , e x0 ) be the point of tangency. Then 1 3 lim ln( x 2 + e – x ) = ∞ (behaves like − x ) x→ – ∞ lim ln( x 2 + e – x ) = ∞ (behaves like 2 ln x ) x →∞ 60. 1 V = π [(e x )2 – (ex) 2 ]dx 0 2 3 ⎤1 ⎡1 1 e x = π (e2 x – e2 x 2 )dx = π ⎢ e2 x – ⎥ 0 3 ⎥⎦ ⎢⎣ 2 0 –1 –1 f ( x) = –(1 + e x ) –2 ⋅ e x (– x –2 ) = e1/ x x 2 (1 + e1/ x )2 ⎡1 π e2 ⎛ 1 ⎞ ⎤ = π ⎢ e2 − − ⎜ e 0 ⎟ ⎥ = (e2 – 3) ≈ 2.30 6 3 ⎝ 2 ⎠ ⎥⎦ ⎢⎣ 2 3 55. a. −3 3 ⎞ ⎛ 1 ⎞ ⎟ dx = 2 0 exp ⎜ − 2 ⎟ dx ≈ 3.11 ⎠ ⎝ x ⎠ 8π −0.1x e sin x dx 0 b. ≈ 0.910 lim (1 + x)1 x = e ≈ 2.72 56. a. x →0 lim (1 + x)−1 x = b. 57. ⎛ 1 exp ⎜ − ⎝ x2 x →0 f ( x) = e 1 ≈ 0.368 e a. − x2 f ( x) = −2 xe− x b. 2 2 2 2 366 c. e− x = 2e− x (2 x 2 − 1); 1 = 4 x 2 − 2; d. 3 2 Both graphs are symmetric with respect to the e. 4 x 2 − 3 = 0, x = ± Section 6.3 lim f ( x) = 1 x →0 – 2 f ( x) = −2e− x + 4 x 2 e− x = 2e− x (2 x 2 − 1) y = f(x) and y = f ( x) intersect when 2 lim f ( x) = 0 x →0 + lim f ( x) = x →±∞ 1 2 lim f ( x) = 0 x →0 f has no minimum or maximum values. Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. x + 3 = 5x 3 x= 4 6.4 Concepts Review 3 ln π 1. e ; e x ln a 2. e 3. 9. log5 12 = ln x ln a ln12 ≈ 1.544 ln 5 10. log 7 0.11 = 4. ax a −1 ; a x ln a Problem Set 6.4 ln 0.11 ≈ –1.1343 ln 7 11. log11 (8.12)1/ 5 = 1 ln 8.12 ≈ 0.1747 5 ln11 12. log10 (8.57)7 = 7 ln 8.57 ≈ 6.5309 ln10 1. 2 x = 8 = 23 ; x = 3 2. x = 52 = 25 3. x = 43 / 2 = 8 4 4. x = 64 x = 4 64 = 2 2 ⎛ x⎞ 1 5. log9 ⎜ ⎟ = ⎝3⎠ 2 x = 91/ 2 = 3 3 x= 1 2x 1 3 2⋅4 = 1 128 7. log 2 ( x + 3) – log 2 x = 2 log 2 x+3 =2 x x+3 = 22 = 4 x x + 3 = 4x x=1 8. log5 ( x + 3) – log5 x = 1 log5 14. x ln 5 = ln 13 ln13 x= ≈ 1.5937 ln 5 15. (2s – 3) ln 5 = ln 4 ln 4 2s – 3 = ln 5 1⎛ ln 4 ⎞ s = ⎜3 + ⎟ ≈ 1.9307 2⎝ ln 5 ⎠ 16. x=9 6. 43 = 13. x ln 2 = ln 17 ln17 x= ≈ 4.08746 ln 2 x+3 =1 x x+3 1 =5 =5 x Instructor’s Resource Manual 1 ln12 = ln 4 –1 ln12 = –1 ln 4 ln12 = 1+ ≈ 2.7925 ln 4 17. Dx (62 x ) = 62 x ln 6 ⋅ Dx (2 x) = 2 ⋅ 62 x ln 6 18. Dx (32 x 2 –3 x ) = 32 x = (4 x – 3) ⋅ 32 x 19. Dx log3 e x = 2 –3 x 2 –3 x ln 3 ⋅ Dx (2 x 2 – 3 x) ln 3 1 ⋅ Dx e x e ln 3 ex 1 = = ≈ 0.9102 x e ln 3 ln 3 Alternate method: x Dx log3 e x = Dx ( x log3 e) = log3 e = ln e 1 = ≈ 0.9102 ln 3 ln 3 Section 6.4 367 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 20. Dx log10 ( x3 + 9) = 3x = 21. 3 ( x + 9) ln10 ⋅ Dx ( x3 + 9) 26. 1 (1) + ln( z + 5) ⋅ 3z ln 3 z +5 ⎡ 1 ⎤ = 3z ⎢ + ln( z + 5) ln 3⎥ ⎣z +5 ⎦ ( =D = = 2 )=D 2 2 –1 ln 3 ln10 2 ( 2 – =– 2 2 – – ) –1/ 2 (2 –1) 27. 1 u 1 2u 2 du = ⋅ +C 2 2 ln 2 2 = 5 x x = 1 2 x 28. dx. 5u +C ln 5 2⋅5 x +C ln 5 4 ⎡5 x ⎤ 25 5 ⎞ ⎥ = 2 ⎛⎜ dx = 2 ⎢ − ⎟ 1 ln 5 ln 5 ln 5⎠ ⎢ ⎥ x ⎝ ⎣ ⎦1 40 = ≈ 24.85 ln 5 368 Section 6.4 2 2 d ( x2 ) d 10 = 10( x ) ln10 x 2 = 10( x ) 2 x ln10 dx dx d 2 10 d 20 (x ) = x = 20 x19 dx dx 2 dy d = [10( x ) + ( x 2 )10 ] dx dx d d sin 2 x = 2sin x sin x = 2sin x cos x dx dx d sin x d 2 = 2sin x ln 2 sin x = 2sin x ln 2 cos x dx dx dy d = (sin 2 x + 2sin x ) dx dx = 2sin x cos x + 2sin x cos x ln 2 dx = 2 5u du = 2 ⋅ 45 x –3 x 2 105 x –1 +C 5ln10 25. Let u = x , so du = 3x = 10( x ) 2 x ln10 + 20 x19 2 2x 2 x –1 = +C = +C 2 ln 2 ln 2 24. Let u = 5x – 1, so du = 5 dx. 1 1 10u 105 x –1 dx = 10u du = ⋅ +C 5 5 ln10 1 ⎡103 x –10 –3 x ⎤ Thus, (10 + 10 )dx = ⎢ ⎥ 0 ⎢⎣ 3ln10 ⎥⎦ 0 1 ⎛ 1 ⎞ 999,999 = ⎜1000 – ⎟= 3ln10 ⎝ 1000 ⎠ 3000 ln10 ≈ 144.76 23. Let u = x 2 so du = 2xdx. x ⋅ 2 x dx = 10 –3 x +C 3ln10 1 – ) log10 3 – ) ln 3 ln 3 = ⋅D ln10 ln10 ln 3 1 ⋅ ( ln10 2 2 1 0 103 x dx + 10 –3 x dx = = 3z ⋅ 2– 1 0 103 x +C 3ln10 Now let u = –3x, so du = –3dx. 1 1 10u 10 –3 x dx = – 10u du = – ⋅ +C 3 3 ln10 Dz [3z ln( z + 5)] log10 (3 (103 x + 10 –3 x )dx = Let u = 3x, so du = 3dx. 1 1 10u 103 x dx = 10u du = ⋅ +C 3 3 ln10 2 ( x3 + 9) ln10 22. D 1 0 29. d π+1 x = (π + 1) x π dx d (π + 1) x = (π + 1) x ln(π + 1) dx dy d π+1 = [x + (π + 1) x ] dx dx = (π + 1) x π + (π + 1) x ln(π + 1) Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 30. x x d (e x ) d 2 = 2(e ) ln 2 e x = 2(e ) e x ln 2 dx dx d e x (2 ) = (2e ) x ln 2e = (2e ) x e ln 2 dx dy d (e x ) = [2 + (2e ) x ] dx dx 35. f ( x) = (− ln 2)2 x 2 +1) (2 x +3) ln(ln x 2 ) f ( x) = xsin x = esin x ln x d (sin x ln x ) dx ⎡ ⎤ ⎛1⎞ = esin x ln x ⎢ (sin x) ⎜ ⎟ + (cos x)(ln x ) ⎥ ⎝x⎠ ⎣ ⎦ f ( x) = esin x ln x ⎛ sin x ⎞ = xsin x ⎜ + cos x ln x ⎟ x ⎝ ⎠ sin1 ⎛ ⎞ + cos1ln1⎟ = sin1 ≈ 0.8415 f (1) = 1sin1 ⎜ ⎝ 1 ⎠ 34. f (e) = πe ≈ 22.46 g (e) = e π ≈ 23.14 g(e) is larger than f(e). d f ( x) = π x = π x ln π dx 9 −3 2) d [(2 x + 3) ln(ln x 2 )] dx 2 ⎡ 1 1 ⎤ = e(2 x +3) ln(ln x ) ⎢ 2 ln(ln x 2 ) + (2 x + 3) (2 x) ⎥ 2 2 ln x x ⎣ ⎦ ⎡ ⎤ 2 x +3 ⎢ 2x + 3 ⎥ 2 ln (2 ln x ) = (2 ln x) + ⎢ x ln x ⎥ ⎢ ⎥⎦ 2 2 ln x ln x ⎣ 33. f ( x) = (ln 2) 2 2− x 4 ⎛ ln( x 2 + 1) 2 x ln x ⎞ = ( x 2 + 1)ln x ⎜ + ⎟ ⎜ x x 2 + 1 ⎟⎠ ⎝ dy =e dx , Domain = (−∞, ∞) y 2 dy d = e(ln x ) ln( x +1) [(ln x) ln( x 2 + 1)] dx dx 2 1 2x ⎤ ⎡ = e(ln x ) ln( x +1) ⎢ ln( x 2 + 1) + ln x ⎥ 2 x + 1⎦ ⎣x 32. y = (ln x 2 ) 2 x +3 = e(2 x +3) ln(ln x −x Since f ( x) < 0 for all x, f is decreasing on (−∞, ∞) . Since f ( x) > 0 for all x, f is concave upward on (−∞, ∞) . Since f and f are both monotonic, there are no extreme values or points of inflection. = 2(e ) e x ln 2 + (2e ) x e ln 2 31. y = ( x 2 + 1)ln x = e(ln x ) ln( x f ( x) = 2− x = e(ln 2)( − x ) x −2 36. f ( x) = x 2− x Domain = (−∞, ∞) f ( x) = [1 − (ln 2) x]2− x , f ( x) = (ln 2)[(ln 2) x − 2]2− x 1 ) ln 2 x ( −∞ , f + f − 1 ln 2 1 2 , ) ln 2 ln 2 2 ln 2 0 − − − − − 0 + ( ( 2 ,∞ ) ln 2 ⎛ 1 ⎤ f is increasing on ⎜ −∞, ⎥ and decreasing on ln 2⎦ ⎝ ⎡ 1 ⎞ 1 1 , ∞ ⎟ . f has a maximum at ( , ) ⎢ ln 2 ln 2 (e ln 2) ⎣ ⎠ f is concave up on ( (−∞, ( 2 , ∞) and concave down on ln 2 2 ) . f has a point of inflection at ln 2 2 2 , ) (e 2 ln 2) ln 2 y 3 f (e) = πe ln π ≈ 25.71 g ( x) = d π x = πx π−1 dx 8 −2 x g (e) = πe π−1 ≈ 26.74 g (e) is larger than f (e) . −3 Instructor’s Resource Manual Section 6.4 369 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 37. y ln( x 2 + 1) . Since f ( x) = log 2 ( x + 1) = ln 2 2 5 x 2 + 1 > 0 for all x, domain = (−∞, ∞) ⎛ 2 ⎞⎛ x ⎞ ⎛ 2 ⎞ ⎛ 1 − x2 f ( x) = ⎜ ⎟⎜ 2 ⎟ ⎜⎜ 2 ⎟ , f ( x) = ⎜ 2 ⎝ ln 2 ⎠ ⎝ x + 1 ⎠ ⎝ ln 2 ⎠ ⎝ ( x + 1) x ⎞ ⎟⎟ ⎠ −5 f − − − 0 + + + f − 0 + + + 0 − y −5 39. f ( x) = 2 dt f ( x) = 2− x , Domain = (−∞, ∞) f ( x) = −2(ln 2) x 2− x 2 2 (−∞, 0) 0 (0, ∞) x f + + + f + 0 − (0, −5 5 x 0 − t2 1 2 dt ) ≈ (0, −0.81) y 5 −5 f ( x) = x log3 ( x 2 + 1) = x ln( x 2 + 1) . Since ln 3 x 2 + 1 > 0 for all x, domain = (−∞, ∞) f ( x) = x − t2 1 f is increasing on (−∞, ∞) and so has no extreme values. f is concave up on (−∞, 0) and concave down on (0, ∞) . f has a point of inflection at 5 x x (−∞, −1) −1 (−1, 0) 0 (0,1) 1 (1, ∞) f is increasing on [0, ∞) and decreasing on (−∞, 0] . f has a minimum at (0, 0) f is concave up on (−1,1) and concave down on (−∞, −1) ∪ (1, ∞) . f has points of inflection at (−1,1) and (1,1) 38. 5 ⎤ 1 ⎡ 2 x2 2 ⎡ x3 + 3 x ⎤ ⎢ ⎢ ⎥ + ln( x 2 +1) ⎥ , f ( x ) = ln 3 ⎢ x 2 +1 ln 3 ⎢ x 2 +1 ⎥ ⎥⎦ ⎣ ⎣ ⎦ −5 5 x −5 (−∞, 0) 0 (0, ∞) f + 0 + f − 0 + f is increasing on (−∞, ∞) and so has no extreme values. f is concave up on (0, ∞) and concave down on (−∞, 0) . f has a point of inflection at (0, 0) 370 Section 6.4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 40. f ( x) = x 0 log10 (t 2 + 1)dt . Since log10 (t 2 + 1) has domain = (−∞, ∞) , f also has domain = (−∞, ∞) ln( x 2 + 1) f ( x) = log10 ( x + 1) = , ln10 2 ⎛ 1 ⎞ ⎛ 2x ⎞ f ( x) = ⎜ ⎟⎜ 2 ⎟ ⎝ ln10 ⎠ ⎝ x + 1 ⎠ x 44. 115 = 20 log10 (121.3P ) log10 (121.3P ) = 5.75 P= 105.75 ≈ 4636 lb/in.2 121.3 45. If r is the ratio between the frequencies of successive notes, then the frequency of C = r12 (the frequency of C). Since C has twice the (−∞, 0) 0 (0, ∞) f + 0 + frequency of C, r = 21/12 ≈ 1.0595 f − 0 + Frequency of C = 440(21/12 )3 = 440 4 2 ≈ 523.25 f is increasing on (−∞, ∞) and so has no extreme values. f is concave up on (0, ∞) and concave down on (−∞, 0) . f has a point of inflection at (0, 0) 2 (p times) and has only powers of 2 as factors and 3q = 3 ⋅ 3 3 (q times) and has only powers of 3 as factors. 5 2 p = 3q only for p = q = 0 which contradicts our assumption, so log 2 3 cannot be rational. 5 x 47. If y = A ⋅ b x , then ln y = ln A + x ln b, so the ln y vs. x plot will be linear. If y = C ⋅ x d , then ln y = ln C + d ln x, so the ln y vs. ln x plot will be linear. −5 41. log1/ 2 x = p where p and q are integers, q q ≠ 0 . Then 2 p q = 3 or 2 p = 3q. But 2p = 2⋅2 y −5 46. Assume log 2 3 = ln x ln x = = − log 2 x 1 ln 2 − ln 2 42. 48. WRONG 1: y = f ( x) g ( x ) y = g ( x) f ( x) g ( x ) −1 f ( x) WRONG 2: y = f ( x) g ( x ) y = f ( x) g ( x ) (ln f ( x)) ⋅ g ( x) = f ( x ) g ( x ) g ( x) ln f ( x) RIGHT: y = f ( x) g ( x ) = e g ( x ) ln f ( x ) 43. M = 0.67 log10 (0.37 E ) + 1.46 log10 (0.37 E ) = E= 10 M − 1.46 0.67 M −1.46 0.67 0.37 Evaluating this expression for M = 7 and M = 8 y = e g ( x ) ln f ( x ) d [ g ( x) ln f ( x)] dx ⎡ ⎤ 1 = f ( x) g ( x ) ⎢ g ( x) ln f ( x) + g ( x) f ( x) ⎥ f ( x) ⎣ ⎦ = f ( x) g ( x ) g ( x) ln f ( x) + f ( x) g ( x ) −1 g ( x) f ( x) Note that RIGHT = WRONG 2 + WRONG 1. gives E ≈ 5.017 × 108 kW-h and E ≈ 1.560 × 1010 kW-h, respectively. Instructor’s Resource Manual Section 6.4 371 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 49. f ( x) = ( x x ) x = x ( x f ( x) = x( x 2) 2) ≠ x( x x) = g ( x) 51. a. 2 ln x = ex d 2 ( x ln x) dx 2 1⎞ ⎛ = e x ln x ⎜ 2 x ln x + x 2 ⋅ ⎟ x⎠ ⎝ f ( x) = e x 2 ln x 2 = x ( x ) (2 x ln x + x) x x g ( x) = x ( x ) = e x ln x Using the result from Example 5 ⎛d x ⎞ x ⎜ x = x (1 + ln x) ⎟ : dx ⎝ ⎠ x x ln x d x g ( x) = e ( x ln x) dx x 1⎤ ⎡ = e x ln x ⎢ x x (1 + ln x) ln x + x x ⋅ ⎥ x⎦ ⎣ x 1 ⎡ ⎤ = x ( x ) x x ⎢(1 + ln x ) ln x + ⎥ x⎦ ⎣ x 1 ⎡ ⎤ = x x + x ⎢ ln x + (ln x)2 + ⎥ x⎦ ⎣ 50. f ( x) = f ( x) = lim f ( x) = lim e x →∞ c. ax +1 (a x + 1) 2 = 2a x ln a (a x + 1)2 Since a is positive, a x is always positive. (a x + 1) 2 is also always positive, thus f ( x) > 0 if ln a > 0 and f ( x) < 0 if ln a < 0. f(x) is either always increasing or always decreasing, depending on a, so f(x) has an inverse. ax −1 y= ax +1 y (a x + 1) = a x − 1 a x ( y − 1) = −1 − y ax = 1+ y 1− y x ln a = ln 1+ y 1− y 1+ y x= 372 ln 1− y ln a = log a 1+ y 1− y f −1 ( y ) = log a 1+ y 1− y f −1 ( x) = log a 1+ x 1− x Section 6.4 x →∞ g ( x) x →∞ = 0. b. Again let g(x) = ln f(x) = a ln x – x ln a. Since y = ln x is an increasing function, f(x) is maximized when g(x) is maximized. a ⎞ ⎛a⎞ ⎛ g ( x) = ⎜ ⎟ − ln a, so g ( x) > 0 on ⎜ 0, ⎟ ⎝ x⎠ ⎝ ln a ⎠ ⎛ a ⎞ and g ( x) < 0 on ⎜ , ∞ ⎟. ⎝ ln a ⎠ Therefore, g(x) (and hence f(x)) is a . maximized at x0 = ln a a x −1 (a x + 1)a x ln a − (a x − 1)a x ln a ⎛ xa ⎞ Let g(x) = ln f(x) = ln ⎜ ⎟ = a ln x − x ln a . ⎜ ax ⎟ ⎝ ⎠ a ⎛ ⎞ g ( x) = ⎜ ⎟ − ln a ⎝ x⎠ a g ( x) < 0 when x > , so as x → ∞ g(x) ln a a is decreasing. g ( x) = − , so g(x) is x2 concave down. Thus, lim g ( x) = −∞, so Note that x a = a x is equivalent to g(x) = 0. a By part b., g(x) is maximized at x0 = . ln a If a = e, then ⎛ e ⎞ g ( x0 ) = g ⎜ ⎟ = g (e) = e ln e − e ln e = 0. ⎝ ln e ⎠ Since g ( x) < g ( x0 ) = 0 for all x ≠ x0 , the equation g(x) = 0 (and hence x a = a x ) has just one positive solution. If a ≠ e , then ⎛ a ⎞ ⎛ a ⎞ a g ( x0 ) = g ⎜ (ln a ) ⎟ = a ln ⎜ ⎟− ⎝ ln a ⎠ ⎝ ln a ⎠ ln a ⎡ ⎛ a ⎞ ⎤ = a ⎢ ln ⎜ ⎟ − 1⎥ . ⎣ ⎝ ln a ⎠ ⎦ a Now > e (justified below), so ln a a ⎡ ⎤ g ( x0 ) = a ⎢ln − 1⎥ > a (ln e − 1) = 0. Since ⎣ ln a ⎦ g ( x) > 0 on (0, x0 ), g ( x0 ) > 0, and lim g ( x) = −∞, g(x) = 0 has exactly one x →0 solution on (0, x0 ). Since g ( x) < 0 on ( x0 , ∞) , g ( x0 ) > 0, and lim g ( x) = −∞, g(x) = 0 has x →∞ exactly one solution on ( x0 , ∞). Therefore, Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. the equation g(x) = 0 (and hence x a = a x ) has exactly two positive solutions. a To show that > e when a ≠ e : ln a x Consider the function h( x ) = , for x > 1. ln x h ( x) = ln( x)(1) − x ( ) = ln x − 1 u →∞ u + 1 u 53. 1 x 1 + − x →0 x2 = lim (− x) = 0 x →0 + x →0 + g ( x) = 1 + ln x Since g ( x) < 0 on ( 0,1/ e ) and g ( x) > 0 on π implies x < e . In particular, π < e . (1/ e, ∞ ) , g(x) has its minimum at u −x fu ( x ) = x e Since fu ( x ) > 0 on (0, u) and fu ( x ) < 0 on (u, ∞ ), fu ( x) attains its maximum at x0 = u. fu (u ) > fu (u + 1) means x = 1e . Therefore, f(x) has its minimum at (e −1 , e −1/ e ) . Note: this point could also be written as 1⎞ ⎛1 e ⎜ , 1e ⎟ . ⎜e ⎟ ⎝ ⎠ fu ( x ) = uxu −1e − x − xu e− x = (u − x) xu −1e− x b. 1 x Therefore, lim x x = e0 = 1 . Therefore, when x ≠ e , ln x < ln e , which 52. a. x →0 + = lim x e f ( x) = x x = e x ln x x →0 + d. For the case a = e, part c. shows that g ( x) = e ln x − x ln e < 0 for x ≠ e . x u Let g ( x) = x ln x. Using L’Hôpital’s Rule, ln x lim g ( x) = lim (ln x) 2 (ln x)2 Note that h ( x) < 0 on (1, e) and h ( x) > 0 on (e, ∞ ), so h(x) has its minimum at (e, e). x > e for all x ≠ e , x > 1. Therefore ln x e e = e , this implies that ⎛ u +1⎞ ⎛ 1⎞ lim ⎜ ⎟ = e, i.e., lim ⎜ 1 + ⎟ = e . u⎠ u →∞ ⎝ u ⎠ u →∞ ⎝ 1 x e u Since lim () 54. u u e −u > (u + 1)u e −(u +1) . eu +1 u ⎛ u +1⎞ gives e > ⎜ ⎟ . u ⎝ u ⎠ u fu +1 (u + 1) > fu +1 (u ) means Multiplying by (u + 1)u +1 e −(u +1) > u u +1e−u . eu +1 ⎛ u +1⎞ gives ⎜ Multiplying by ⎟ u +1 ⎝ u ⎠ u Combining the two inequalities, u ⎛ u +1⎞ ⎛ u +1⎞ ⎜ ⎟ <e<⎜ ⎟ ⎝ u ⎠ ⎝ u ⎠ (2.4781, 15.2171), (3, 27) u +1 u +1 . > e. 55. 4 π sin x x dx 0 ≈ 20.2259 56. u +1 c. ⎛ u +1⎞ From part b., e < ⎜ ⎟ . ⎝ u ⎠ u Multiplying by gives u +1 u u ⎛ u +1⎞ e<⎜ ⎟ . u +1 ⎝ u ⎠ u ⎛ u +1⎞ We showed ⎜ ⎟ < e in part b., so ⎝ u ⎠ u u ⎛ u +1⎞ e<⎜ ⎟ < e. u +1 ⎝ u ⎠ Instructor’s Resource Manual Section 6.4 373 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 57. a. In order of increasing slope, the graphs represent the curves y = 2 x , y = 3x , and y = 4 x. b. ln y is linear with respect to x, and at x = 0, y = 1 since C = 1. c. 6.5 Concepts Review 1. ky; ky ( L − y ) 2. 23 = 8 3. half-life 4. (1 + h )1/ h The graph passes through the points (0.2, 4) and (0.6, 8). Thus, 4 = Cb0.2 and 8 = Cb0.6 . Dividing the second equation by the first, gets 2 = b0.4 so b = 25 2. Therefore C = 23 2. Problem Set 6.5 1. k = −6 , y0 = 4, so y = 4e −6t 2. k = 6, y0 = 1, so y = e6t 58. The graph of the equation whose log-log plot has negative slope contains the points (2, 7) and (7, 2). r 7 ⎛2⎞ Thus, 7 = C 2 and 2 = C 7 , so = ⎜ ⎟ . 2 ⎝7⎠ 7 2 ln 7 − ln 2 ln = r ln ⇒ r = = −1 and C = 14. 2 7 ln 2 − ln 7 r r Hence, one equation is y = 14 x −1. The graph of one equation contains the points r (7, 30) and (10, 70). Thus, 30 = C 7 and r 3 ⎛7⎞ 70 = C10r , so = ⎜ ⎟ 7 ⎝ 10 ⎠ 3 7 ln 3 − ln 7 ln = r ln ⇒ r = ≈ 2.38 and 7 10 ln 7 − ln10 C ≈ 30 ⋅ 7 −2.38 ≈ 0.29 . Hence, another equation is y = 0.29 x 2.38 . The graph of another equation contains the points r r (1, 2) and (7, 5). Thus, 2 = C1 and 5 = C 7 , so C = 2 and ln 5 − ln 2 ln 5 − ln 2 = r ln 7 ⇒ r = ≈ 0.47. ln 7 0.47 . Hence, the last equation is y = 2 x The given answers are only approximate. Student answers may also vary. 3. k = 0.005, so y = y0 e0.005t y (10) = y0 e0.005(10) = y0 e0.05 2 y (10) = 2 ⇒ y0 = 0.05 e 2 0.005t y= e = 2e0.005t −0.05 = 2e0.005(t −10) e0.05 4. k = –0.003, so y = y0 e –0.003t y (–2) = y0 e(–0.003)(–2) = y0 e0.006 3 y (−2) = 3 ⇒ y0 = 0.006 e 3 y= e –0.003t = 3e –0.003t –0.006 = 3e –0.003(t + 2) e0.006 5. y0 = 10, 000, y(10) = 20,000 20, 000 = 10, 000e k (10) 2 = e10k ln 2 = 10k; k= ln 2 10 y = 10, 000e((ln 2) /10)t = 10, 000 ⋅ 2t /10 After 25 days, y = 10, 000 ⋅ 22.5 ≈ 56,568. 6. Since the growth is exponential and it doubles in 10 days (from t = 0 to t = 10), it will always double in 10 days. 7. 3 y0 = y0 e((ln 2) /10)t 3 = e((ln 2) /10)t ln 2 ln 3 = t 10 10 ln 3 t= ≈ 15.8 days ln 2 374 Section 6.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 8. Let P(t) = population (in millions) in year 1790 + t. In 1960, t = 170. P (t ) = P0 ekt 178 = 3.9e170k 45.64 = e170k ln 45.64 k= ≈ 0.02248 170 In 2000, t = 210 P (210) ≈ 3.9e0.02248⋅210 ≈ 438 The model predicts that the population will be about 438 million. The actual number, 275 million, is quite a bit smaller because the rate of growth has declined in recent decades. 9. 1 year: (4.5 million) (1.032) ≈ 4.64 million 2 years: (4.5 million) (1.032) 2 ≈ 4.79 million 10 years: (4.5 million) (1.032)10 ≈ 6.17 million 100 years: (4.5 million) (1.032)100 ≈ 105 million 10. y = y0 ekt 1.032 A = Aek (1) k = ln1.032 ≈ 0.03150 At t = 100, y = 4.5e(0.03150)(100) ≈ 105 . After 100 years, the population will be about 105 million. 11. The formula to use is y = y0 e kt , where y = population after t years, y0 =population at time t = 0, and k is the rate of growth. We are given 235, 000 = y0 e k (12) and 164, 000 = y0 e k (5) Dividing one equation by the other yields 1.43293 = e12 k −5k = e7 k or ln(1.43293) k= ≈ 0.0513888 7 235, 000 Thus y0 = 12(0.0513888) = 126,839. e 12. The formula to use is y = y0 e kt , where y = mass t months after initial measurement, y0 = mass at time of initial measurement, and k is the rate of growth. We are given 6.76 = 4e k (4) so that 1 ⎛ 6.76 ⎞ 0.5247 k = ln ⎜ ≈ 0.1312 ⎟= 4 ⎝ 4 ⎠ 4 13. 1 = e k (700) and y0 = 10 2 –ln 2 = 700k ln 2 k =− ≈ −0.00099 700 y = 10e−0.00099t At t = 300, y = 10e−0.00099⋅300 ≈ 7.43. After 300 years there will be about 7.43 g. 14. 0.85 = e k (2) ln 0.85 = 2k ln 0.85 k= ≈ −0.0813 2 1 = e −0.0813t 2 – ln 2 = −0.0813t ln 2 t= ≈ 8.53 0.0813 The half-life is about 8.53 days. 15. The basic formula is y = y0 e kt . If t* denotes the half-life of the material, then (see Example 3) 1 ln(0.5) = e kt* or k = . Thus 2 t* −0.693 −0.693 = −0.0229 and k S = = −0.0241 30.22 28.8 To find when 1% of each material will remain, we ln(0.01) . Thus use 0.01y0 = y0 ekt or t = k −4.6052 tC = ≈ 201 years (2187) and −0.0229 −4.6052 tS = ≈ 191 years (2177) −0.0241 kC = 16. The basic formula is y = y0 e kt . We are given 15.231 = y0 ek (2) and 9.086 = y0 ek (8) Dividing one equation by the other gives 15.231 k (2) − k (8) =e = ek ( −6) so k = −0.0861 9.086 15.231 Thus y0 = ( −.0861)(2) ≈ 18.093 grams. e To find the half-life: t* = ln(0.5) −0.693 = ≈ 8 days −0.0861 k Thus, 6 months before the initial measurement, the mass was y = 4e(0.1312)( −6) ≈ 1.82 grams. The tumor would have been detectable at that time. Instructor’s Resource Manual Section 6.5 375 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 17. 22. From Example 4, T (t ) = T1 + (T0 − T1 )e kt . In this 1 = e5730 k 2 ln 12 k= ≈ −1.210 × 10−4 5730 problem, 250 = T (15) = 40 + (350 − 40)e k (15) so ( ) 0.7 y0 = y0 e( −1.210×10 t= ⎛ 210 ⎞ ln ⎜ ⎟ 310 ⎠ k= ⎝ = −0.026 ; the brownies will be 15 −4 )t 110 F when 110 = 40 + (310)e−0.026 t or ln 0.7 ≈ 2950 −1.210 × 10−4 The fort burned down about 2950 years ago. 18. ⎛ 70 ⎞ ln ⎜ ⎟ 310 ⎠ t= ⎝ = 57.2 min. −0.026 1 = e5730 k 2 ln 12 ≈ −1.210 × 10−4 k= 5730 23. From Example 4, T (t ) = T1 + (T0 − T1 )e kt . Let w = the time of death; then 82 = T (10 − w) = 70 + (98.6 − 70)ek (10 − w) ( ) 0.51 y0 = y0 e( −1.210×10 t= 76 = T (11 − w) = 70 + (98.6 − 70)ek (11− w) −4 )t ≈ 5565 −1.210 × 10−4 The body was buried about 5565 years ago. 6 = 28.6ek (11− w) Dividing: 2 = e k ( −1) or k = ln (0.5) = −0.693 19. From Example 4, T (t ) = T1 + (T0 − T1 )e kt . In this problem, 200 = T (0.5) = 75 + (300 − 75)e k (0.5) 12 = 28.6ek (10 − w) or ln 0.51 so ⎛ 125 ⎞ ln ⎜ ⎟ 225 ⎠ k= ⎝ = −1.1756 and 0.5 T (3) = 75 + 225e( −1.1756)(3) = 81.6 F 20. From Example 4, T (t ) = T1 + (T0 − T1 )e kt . In this problem, 0 = T (5) = 24 + (−20 − 24)ek (5) so ⎛ −24 ⎞ ln ⎜ ⎟ −44 ⎠ k= ⎝ = −0.1212 ; the thermometer will 5 register 20 C when 20 = 24 + (−44)e −0.1212 t or ⎛ −4 ⎞ ln ⎜ ⎟ −44 ⎠ t= ⎝ = 19.78 min. −0.1212 21. From Example 4, T (t ) = T1 + (T0 − T1 )e kt . In this problem, 70 = T (5) = 90 + (26 − 90)e k (5) so ⎛ −20 ⎞ ln ⎜ ⎟ −64 ⎠ k= ⎝ = −0.2326 and 5 T (10) = 90 − 64e( −0.2326)(10) = 90 − 64(0.0977) = 83.7 C To find w : ⎛ 12 ⎞ ln ⎜ ⎟ 28.6 ⎠ = 1.25 12 = 28.6e−0.693(10 − w) so 10 − w = ⎝ −0.693 Therefore w = 10 − 1.25 = 8.75 = 8 : 45 pm . 24. a. From example 4 of this section, dT = k (T − T1 ) or dt dT = k dt or ln T(t)-T1 = kt + C T − T1 This gives T (t ) − T1 = e kt eC . Now, if T0 is the temperature at t = 0, T0 − T1 = eC and the Law of Cooling becomes T (t ) − T1 = T0 − T1 e kt . Note that T (t ) is always between T0 and T1 so that T (t ) − T1 and T0 − T1 always have the same sign; this simplifies the Law of Cooling to T (t ) − T1 = (T0 − T1 )e kt or T (t ) = T1 + (T0 − T1 )e kt b. Since T (t ) is always between T0 and T1 , it follows that e kt = T (t ) − T1 < 1 so that k < 0 . T0 − T1 Hence lim T (t ) = T1 + (T0 − T1 ) lim e kt = T1 + 0 = T1 t →∞ 376 Section 6.5 t →∞ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 25. a. ($375)(1.035) 2 ≈ $401.71 ⎛ 0.035 ⎞ ($375) ⎜ 1 + ⎟ 12 ⎠ ⎝ 24 b. ⎛ 0.035 ⎞ ($375) ⎜ 1 + ⎟ 365 ⎠ ⎝ 730 c. d. ($375)e0.035⋅2 ≈ $402.19 26. a. ≈ $402.15 ≈ $402.19 ($375)(1.046)2 = $410.29 ⎛ 0.046 ⎞ ($375) ⎜ 1 + ⎟ 12 ⎠ ⎝ 24 b. ⎛ 0.046 ⎞ ($375) ⎜ 1 + ⎟ 365 ⎠ ⎝ 730 c. d. ($375)e0.046⋅2 ≈ $411.14 ≈ $411.06 ≈ $411.13 12t 27. a. ⎛ 0.06 ⎞ ⎜1 + ⎟ 12 ⎠ ⎝ =2 12t =2 ln 2 ln 2 12t = ≈ 11.58 so t = ln1.005 12 ln1.005 It will take about 11.58 years or 11 years, 6 months, 29 days. 1.005 ln 2 b. e ≈ 11.55 =2 ⇒ t= 0.06 It will take about 11.55 years or 11 years, 6 months, and 18 days. 0.06t 28. $20, 000(1.025)5 ≈ $22, 628.16 29. 1626 to 2000 is 374 years. y = 24e 0.06 ⋅ 374 34. dy = ky ( L – y ) dt 1 dy = kdt y( L – y) ⎡1 ⎤ 1 ⎢ Ly + L( L – y ) ⎥ dy = kdt ⎣ ⎦ 1 ⎛1 1 ⎞ ⎜ + ⎟ dy = kdt L ⎝ y L– y⎠ 1 [ln y – ln L − y ] = kt + C1 L y ln = Lkt + LC1 L– y y y = e Lkt + LC1 = e LC1 ⋅ e Lkt , so = Ce Lkt L– y L− y ⎛ Note that: C = Ce0 = Ce Lk ⋅0 ⎞ ⎜ ⎟ y0 ⎟ y (0) ⎜ = = . ⎜ L – y (0) L – y0 ⎟⎠ ⎝ y = LCe Lkt – yCe Lkt y + yCe Lkt = LCe Lkt y= LCe Lkt 1 + Ce Lkt = LC LC = + C C + e – Lkt 1 e Lkt y = L ⋅ L –0y y0 L – y0 35. y = = ≈ $133.6 billion +e 0 – Lkt = Ly0 y0 + ( L – y0 )e – Lkt 16 ( 6.4 ) 6.4 + (16 − 6.4)e−16(0.00186)t 102.4 6.4 + 9.6e −0.02976t y 30. $100(1.04)969 ≈ $3.201× 1018 20 31. 1000e(0.05)(1) = $1051.27 10 32. A0 e(0.05)(1) = 1000 A0 = 1000e −0.05 ≈ $951.23 33. If t is the doubling time, then −50 150 t t p ⎞ ⎛ ⎜1 + ⎟ =2 ⎝ 100 ⎠ p ⎞ ⎛ t ln ⎜ 1 + ⎟ = ln 2 100 ⎝ ⎠ ln 2 ln 2 100 ln 2 70 t= ≈ = ≈ p p p p ln 1 + 100 100 ( ) Instructor’s Resource Manual Section 6.5 377 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 39. Let y = population in millions, t = 0 in 1985, a = 0.012, b = 0.06 , y0 = 10 lim (1 + x)1000 = 11000 = 1 36. a. x →0 dy = 0.012 y + 0.06 dt 0.06 ⎞ 0.012t 0.06 ⎛ y = ⎜ 10 + = 15e0.012t – 5 – ⎟e 0.012 0.012 ⎝ ⎠ From 1985 to 2010 is 25 years. At t = 25, lim 11/ x = lim 1 = 1 b. x →0 x →0 lim (1 + ε )1/ x = lim (1 + ε )n = ∞ c. x →0 + n →∞ 1 lim (1 + ε )1/ x = lim d. y = 15e0.012⋅25 − 5 ≈ 15.25. The population in 2010 will be about 15.25 million. =0 n→∞ (1 + ε ) n x →0 − lim (1 + x)1/ x = e e. x →0 1 lim (1 − x)1/ x = lim 37. a. 1 (− x) x →0 [1 + ( − x )] x →0 3 lim (1 + 3 x) x →0 n ⎛n+2⎞ ⎛ 2⎞ lim ⎜ ⎟ = lim ⎜ 1 + ⎟ n⎠ n →∞ ⎝ n ⎠ n→∞ ⎝ c. 1 e 1 ⎤ ⎡ = lim ⎢(1 + 3x) 3 x ⎥ = e3 x →0 ⎣⎢ ⎦⎥ 1/ x b. = 40. Let N(t) be the number of people who have heard dN = k (L − N ) . the news after t days. Then dt 1 dN = k dt L−N –ln(L – N) = kt + C L − N = e− kt −C N = L − Ae− kt N(0) = 0, ⇒ A = L n N (t ) = L(1 − e− kt ) . 1/ x = lim (1 + 2 x) x →0 + N (5) = 2 1 ⎤ ⎡ = lim ⎢(1 + 2 x) 2 x ⎥ = e 2 x →0+ ⎣⎢ ⎦⎥ ⎛ n −1 ⎞ lim ⎜ ⎟ n →∞ ⎝ n ⎠ d. 2n ⎛ 1⎞ = lim ⎜ 1 − ⎟ n⎠ n →∞ ⎝ 1 = e −5k 2 ln 1 k = 2 ≈ 0.1386 −5 2n N (t ) = L(1 − e−0.1386t ) = lim (1 − x)2 / x 0.99 L = L(1 − e−0.1386t ) x →0+ 1 ⎤ ⎡ = lim ⎢ (1 − x) − x ⎥ x →0+ ⎣⎢ ⎦⎥ 38. −2 = dy = ay + b dt dy = a dt y + ba ln y + b = at + C a b b = e at +C ; y + = Aeat a a b y = Aeat − a b b y0 = A − ⇒ A = y0 + a a b ⎞ at b ⎛ y = ⎜ y0 + ⎟ e − a⎠ a ⎝ y+ 378 Section 6.5 L L ⇒ = L(1 − e −5k ) 2 2 0.01 = e −0.1386t ln 0.01 ≈ 33 t= −0.1386 99% of the people will have heard about the scandal after 33 days. 1 e2 41. If f(t) = e kt , then 42. f (t ) kekt = =k. f (t ) e kt f ( x) = an x n + an –1 x n –1 + ⋅⋅⋅ + a1 x + a0 lim x →∞ f ( x) f ( x) = lim nan x n –1 + (n – 1)an –1 x n –2 + ⋅⋅⋅ + a1 an x n + an –1 x + ⋅⋅⋅ + a1 x + a0 x →∞ = lim nan x x →∞ a n + + ( n –1) an –1 x2 an –1 x + ⋅⋅⋅ + + ⋅⋅⋅ + a1 n x –1 + a1 xn a0 xn =0 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 43. e. f ( x) = k > 0 can be written as f ( x) 1 dy = k where y = f(x). y dx t = 66 , which is year 2070. The population will equal the 2004 value of 6.4 billion when 0.0132t − 0.0001t 2 = 0 f ( x) 1 dy = k < 0 can be written as = k where f ( x) y dx dy y = f(x). = k dx has the solution y = Cekx . y Thus, f ( x) = Cekx which represents exponential decay since k < 0. 45. Maximum population: 640 acres 1 person ⋅ 13,500, 000 mi 2 ⋅ 1 acre 1 mi 2 2 t = 0 or t = 132 . The model predicts that the population will return to the 2004 level in year 2136. 47. a. b. c. = 1.728 ×1010 people Let t = 0 be in 2004. y ' = ( 0.0132 − 0.0001t ) y dy = ( 0.0132 − 0.0001t ) y dt dy = ( 0.0132 − 0.0001t ) dt y ⎛ 1.728 ⋅10 ⎞ ln ⎜ ⎜ 6.4 ⋅109 ⎟⎟ ⎠ ≈ 75.2 years from 2004, or t= ⎝ 0.0132 sometime in the year 2079. y = C1e0.0132t −0.00005t 10 c. k = 0.0132 − 0.0001t ln y = 0.0132t − 0.00005t 2 + C0 (6.4 × 109 )e0.0132t = 1.728 × 1010 b. ) t = 0.0132 / 0.0002 = 66 Thus, the equation f ( x) = Ce kx represents exponential growth since k > 0. 46. a. ( 0.0132 = 0.0002t dy = k dx has the solution y = Cekx . y 44. The maximum population will occur when d 0.0132t − 0.0001t 2 = 0 dt 2 The initial condition y (0) = 6.4 implies that C1 = 6.4 . Thus y = 6.4e0.0132t −0.00005t 2 y d. k = 0.0132 − 0.0002t 20 y ' = ( 0.0132 − 0.0002t ) y dy = ( 0.0132 − 0.0002t ) y dt dy = ( 0.0132 − 0.0002t ) dt y 10 100 ln y = 0.0132t − 0.0001t 2 + C0 y = C1e0.0132t −0.0001t e. 2 The initial condition y (0) = 6.4 implies that C1 = 6.4 . Thus y = 6.4e0.0132t −0.0001t d. y 10 2 200 t 300 The maximum population will occur when d 0.0132t − 0.00005t 2 = 0 dt 0.0132 = 0.0001t ( ) t = 0.0132 / 0.0001 = 132 t = 132 , which is year 2136 The population will equal the 2004 value of 6.4 billion when 0.0132t − 0.00005t 2 = 0 t = 0 or t = 264 . The model predicts that the population will return to the 2004 level in year 2268. 5 50 100 150 Instructor’s Resource Manual t Section 6.5 379 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6.6 Concepts Review E ( x + h) – E ( x ) h h →0 E ( x ) E ( h) – E ( x ) = lim h h →0 E ( h) – 1 E ( h) – 1 = lim E ( x) ⋅ = E ( x) lim h h h →0 h →0 E ( x) = E ( x + 0) = E ( x ) ⋅ E (0) 48. E ( x) = lim 1. exp 2. y exp 3. so E (0) = 1. E (h) – E (0) h E (0 + h) – E (0) = E ( x) lim = E ( x) ⋅ E (0) h h →0 = kE(x) where k = E (0) . Thus, E ( x) = E ( x ) lim Hence, E ( x) = E 0 e Check: E (u + v) = e = E (0)e k (u + v ) = e ku ⋅ ekv = E (u ) ⋅ E (v) 49. =e kx = 1⋅ e ku + kv kx P ( x)dx ( ) P( x)dx 1 d ⎛ ; ⎜ x dx ⎝ ) y⎞ 2 ⎟ = 1; x + Cx x⎠ 4. particular h →0 kx ( Problem Set 6.6 kx =e . 1. Integrating factor is e x . D ( ye x ) = 1 y = e– x ( x + C) 2. The left-hand side is already an exact derivative. D[ y ( x + 1)] = x 2 – 1 y= 3. y + Exponential growth: In 2010 (t = 6): 6.93 billion In 2040 (t = 36): 10.29 billion In 2090 (t = 86): 19.92 billion Logistic growth: In 2010 (t = 6): 7.13 billion In 2040 (t = 36): 10.90 billion In 2090 (t = 86): 15.15 billion 50. a. b. lim (1 + x)1/ x = e x →0 lim (1 – x)1/ x = x →0 x3 – 3 x + C 3( x + 1) x 2 y= ax 1– x 1 – x2 Integrating factor: x exp dx = exp ⎡ ln(1 – x 2 ) –1/ 2 ⎤ ⎣ ⎦ 1 – x2 = (1 – x 2 ) –1/ 2 D[ y (1 – x 2 ) –1/ 2 ] = ax(1 – x 2 ) –3 / 2 Then y (1 – x 2 ) –1/ 2 = a (1 – x 2 ) –1/ 2 + C , so y = a + C (1 – x 2 )1/ 2 . 4. Integrating factor is sec x. 1 e D[ y sec x] = sec 2 x y = sin x + C cos x 5. Integrating factor is 1 . x ⎡ y⎤ D ⎢ ⎥ = ex ⎣x⎦ y = xe x + Cx 380 Section 6.6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6. y – ay = f ( x) 14. Integrating factor is sin 2 x. Integrating factor: e – adx D[ ye – ax ] = e – ax f ( x) Then ye – ax = e – ax f ( x)dx, so y = e ax e – ax f ( x)dx . 7. Integrating factor is x. D[yx] = 1; y = 1 + Cx –1 8. Integrating factor is ( x + 1) 2 . D[ y ( x + 1) 2 ] = ( x + 1)5 ⎛1⎞ y = ⎜ ⎟ ( x + 1)4 + C ( x + 1) –2 ⎝6⎠ 9. y + f ( x) y = f ( x) f ( x ) dx Integrating factor: e f ( x ) dx ⎤ D ⎡ ye = f ( x )e ⎣⎢ ⎦⎥ Then ye f ( x ) dx y = 1 + Ce =e – f ( x ) dx D[ y sin 2 x] = 2sin 2 x cos x = e – ax f ( x ) dx f ( x ) dx + C , so 2 y sin 2 x = sin 3 x + C 3 2 C y = sin x + 3 sin 2 x 2 5 y = sin x + csc2 x 3 12 ⎛π ⎞ goes through ⎜ , 2 ⎟ . ⎝6 ⎠ 15. Let y denote the number of pounds of chemical A after t minutes. dy ⎛ lbs ⎞ ⎛ gal ⎞ ⎛ y lbs ⎞ ⎛ 3 gal ⎞ = ⎜2 ⎟⎜3 ⎟⎜ ⎟–⎜ ⎟ dt ⎝ gal ⎠ ⎝ min ⎠ ⎝ 20 gal ⎠ ⎝ min ⎠ 3y =6– lb/min 20 3 y + y=6 20 ( 3 / 20 ) dt = e3t / 20 Integrating factor: e D[ ye3t / 20 ] = 6e3t / 20 . Then ye3t / 20 = 40e3t / 20 + C. t = 0, y = 10 ⇒ C = –30. 2x 10. Integrating factor is e . D[ ye 2 x ] = xe2 x Therefore, y (t ) = 40 – 30e –3t / 20 , so ⎛1⎞ ⎛1⎞ y = ⎜ ⎟ x – ⎜ ⎟ + Ce –2 x ⎝2⎠ ⎝4⎠ y (20) = 40 – 30e –3 ≈ 38.506 lb. 1 ⎡ y⎤ 11. Integrating factor is . D ⎢ ⎥ = 3 x 2 ; y = x 4 + Cx x ⎣x⎦ 16. Integrating factor is et / 50 . y = x 4 + 2 x goes through (1, 3). D[ yet / 50 ] = 8et / 50 y (t ) = 400 + Ce – t / 50 12. y + 3 y = e2 x Integrating factor: e 3dx y (t ) = 400 – 350e – t / 50 goes through (0, 50). = e3 x y (40) = 400 – 350e –0.8 ≈ 242.735 lb of salt D[ ye3 x ] = e5 x Then ye3 x = ye3 x = dy y ⎛ y ⎞ = (2)(4) – ⎜ =8 ⎟ (4) or y + dt 50 ⎝ 200 ⎠ e5 x 4 + C. x = 0, y = 1 ⇒ C = , so 5 5 e5 x 4 + . 5 5 ⎡ ⎤ ⎡ 3 ⎤ dy y = 4–⎢ ⎥ (6) or y + ⎢ ⎥y=4 dt ⎣ (120 – 2t ) ⎦ ⎣ (60 – t ) ⎦ Integrating factor is (60 – t ) –3 . e 2 x + 4e –3 x is the particular 5 solution through (0, 1). Therefore, y = 13. Integrating factor: xe 17. x d [ yxe x ] = 1 ; y = e – x (1 + Cx –1 ); y = e – x (1 – x –1 ) goes through (1, 0). Instructor’s Resource Manual D[ y (60 – t ) –3 ] = 4(60 – t ) –3 y (t ) = 2(60 – t ) + C (60 – t )3 ⎛ 1 ⎞ 3 y (t ) = 2(60 – t ) – ⎜ ⎟ (60 − t ) goes through ⎝ 1800 ⎠ (0, 0). Section 6.6 381 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 18. dy –2 y 2 = y = 0. or y + dt 50 + t 50 + t Integrating factor: 2 ⎛ ⎞ exp ⎜ dt ⎟ = e2 ln(50+t ) = (50 + t )2 ⎝ 50 + t ⎠ 23. Let y be the number of gallons of pure alcohol in the tank at time t. a. Integrating factor is e0.05t . D[ y (50 + t )2 ] = 0 y (t ) = 25 + Ce –0.05t ; y = 100, t = 0, C = 75 Then y (50 + t )2 = C. t = 0, y = 30 ⇒ C = 75000 y (t ) = 25 + 75e –0.05t ; y = 50, t = T, T = 20(ln 3) ≈ 21.97 min Thus, y (50 + t )2 = 75, 000. If y = 25, 25(50 + t ) 2 = 75, 000, so t = 3000 – 50 ≈ 4.772 min. 19. I + 106 I = 1 Integrating factor = exp(106 t ) b. Let A be the number of gallons of pure alcohol drained away. 200 (100 – A) + 0.25A = 50 ⇒ A = 3 It took I (t ) = 10 –6 + C exp(–106 t ) ( ) I (t ) = 10 –6 [1 – exp(–106 t )] goes through (0, 0). c. ⎛ 240 ⎞ I =⎜ ⎟ sin 377t ⎝ 7 ⎠ ⎛ 240 ⎞ I = ⎜– ⎟ cos 377t + C ⎝ 2639 ⎠ ⎛ 240 ⎞ I (t ) = ⎜ ⎟ (1 – cos 377t ) through (0, 0). ⎝ 2639 ⎠ 21. 1000 I = 120 sin 377t I(t) = 0.12 sin 377t D[ xe t / 50 ]=0 x = Ce – t / 50 x(t ) = 50e – t / 50 satisfies t = 0, x = 50. ⎛ 50e – t / 50 ⎞ dy ⎛ y ⎞ = 2⎜ ⎟ – 2⎜ ⎟ ⎜ ⎟ dt ⎝ 200 ⎠ ⎝ 100 ⎠ ⎛ 1 ⎞ – t / 50 y +⎜ ⎟y=e ⎝ 100 ⎠ c would need to satisfy 200 3 5 c> d. + 200 3 c < 20(ln 3). 10 ≈ 7.7170 (3ln 3 – 2) y = 4(0.25) – 0.05 y = 1 – 0.05 y Solving for y, as in part a, yields y = 20 + 80e –0.05t . The drain is closed when t = 0.8T . We require that dx 2x =– 22. dt 100 ⎛ 1 ⎞ x +⎜ ⎟x = 0 ⎝ 50 ⎠ Integrating factor is et / 50 . 200 3 minutes for the draining and the 5 same amount of time to refill, so 2 200 80 3 T= = ≈ 26.67 min. 5 3 D[ I exp(106 t )] = exp(106 t ) 20. 3.5I = 120sin 377t dy ⎛ 5 ⎞ = 5(0.25) – ⎜ ⎟ y = 1.25 – 0.05 y dt ⎝ 100 ⎠ y = (20 + 80e −0.05⋅0.8T ) + 4 ⋅ 0.25 ⋅ 0.2T = 50, or 400e –0.04T + T = 150. 24. a. v + av = – g Integrating factor: eat d (veat ) = – geat dt – g at –g veat = – geat dt = e + C; v = + Ce – at a a v = v0 , t = 0 e at (v + av ) = – ge at ; D[ yet /100 ] = e – t /100 –g g + C ⇒ C = v0 + a a –g ⎛ g⎞ Therefore, v = + ⎜ v0 + ⎟ e – at , so a ⎝ a⎠ y (t ) = e – t /100 (C – 100e – t /100 ) v(t ) = v∞ + (v0 – v∞ )e – at . Integrating factor is et /100 . v0 = y (t ) = e – t /100 (250 – 100e – t /100 ) satisfies t = 0, y = 150. 382 Section 6.6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. dy = v∞ + (v0 – v∞ )e – at , so dt (v0 − v∞ )e− at + C. a –(v0 – v∞ ) +C y = y0 , t = 0 ⇒ y0 = a v –v ⇒ C = y0 + 0 ∞ a y = v∞ ⋅ t − (v0 – v∞ )e – at ⎛ v –v ⎞ + ⎜ y0 + 0 ∞ ⎟ a a ⎠ ⎝ v –v = y0 + v∞ t + 0 ∞ (1 – e – at ) a y = v∞ t – 25. a. 32 v∞ = – = –640 0.05 26. For t in [0, 15], –32 v∞ = = –320. 0.10 v(t ) = (0 + 320)e –0.1t – 320 = 320(e –0.1t –1); v(15) = 320(e –1.5 –1) ≈ –248.6 y (t ) = 8000 – 320t + 10(320)(1 – e –0.1t ); y (15) = 3200(2 – e –1.5 ) ≈ 5686 Let t be the number of seconds after the parachute opens that it takes Megan to reach the ground. 32 = –20. For t in [15, 15+T], v∞ = – 1.6 0 = y (T + 15) = [3200(2 – e –1.5 )] –20T + (0.625)[320(e –1.5 – 1) + 20](1 – e –1.6T ) v(t ) = [120 − (−640)]e−0.05t + (−640) = 0 if ≈ 5543 – 20T –142.9e−1.6T ≈ 5543 – 20T [since ⎛ 19 ⎞ t = 20 ln ⎜ ⎟ . ⎝ 16 ⎠ y (t ) = 0 + (–640)t T > 50, so e –1.6T < 10 –35 (very small)] Therefore, T ≈ 277, so it takes Megan about 292 s (4 min, 52 s) to reach the ground. ⎛ 1 ⎞ –0.05t +⎜ ) ⎟ [120 – (–640)](1 – e ⎝ 0.05 ⎠ 27. a. = –640t + 15, 200(1 – e –0.05t ) Therefore, the maximum altitude is ⎛ ⎛ 19 ⎞ ⎞ ⎛ 19 ⎞ 45, 600 y ⎜ 20 ln ⎜ ⎟ ⎟ = −12,800 ln ⎜ ⎟ + 19 ⎝ 16 ⎠ ⎠ ⎝ 16 ⎠ ⎝ ≈ 200.32 ft b. –640T + 15, 200(1 – e –0.05T ) = 0; 95 – 4T – 95e –0.05T = 0 b. ⎛ dy y ⎞ e − ln x +C ⎜ − ⎟ = x 2 e − ln x +C ⎝ dx x ⎠ ⎛ dy y ⎞ e − ln x eC ⎜ − ⎟ = x 2 eC e− ln x ⎝ dx x ⎠ 1 C dy 1 1 − yeC = x 2 eC e 2 x dx x x d ⎛ C1 ⎞ y ⎟ = xeC ⎜e dx ⎝ x ⎠ y eC = eC x dx x y x2 = + C1 x 2 y= 28. e x3 + C1 x 2 P ( x ) dx +C P ( x ) dx +C dy + P ( x )e y dx P ( x ) dx +C = Q ( x )e P ( x ) dx +C d ⎛ P ( x ) dx +C ⎞ y ⎟ = Q ( x )e ⎜e ⎟ dx ⎜⎝ ⎠ ye P ( x ) dx +C y=e − = P ( x ) dx + C2 e Instructor’s Resource Manual P ( x ) dx C e dx + C1 Q ( x )e Q ( x )e − P ( x ) dx dx P ( x ) dx Section 6.6 383 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6.7 Concepts Review 5. 1. slope field 2. tangent line 3. yn −1 + hf ( xn −1 , yn −1 ) The oblique asymptote is y = x . 4. underestimate 6. Problem Set 6.7 1. The oblique asymptote is y = 3 + x / 2 . lim y ( x ) = 12 and y (2) ≈ 10.5 7. x →∞ 2. lim y ( x) = ∞ and y (2) ≈ 16 x →∞ dy 1 1 = y; y (0) = dx 2 2 dy 1 = dx y 2 ln y = 3. x +C 2 y = C1e x / 2 To find C1 , apply the initial condition: lim y ( x) = 0 and y (2) ≈ 6 x →∞ 1 = y (0) = C1e0 = C1 2 1 y = ex / 2 2 4. lim y ( x) = ∞ and y (2) ≈ 13 x →∞ 384 Section 6.7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 3 2 3⎞ ⎛ e x y '+ ye x = ⎜ 2 x + ⎟ e x 2⎠ ⎝ d x 3⎞ ⎛ e y = ⎜ 2x + ⎟ ex dx 2⎠ ⎝ 3⎞ ⎛ e x y = ⎜ 2 x + ⎟ e x dx 2⎠ ⎝ y '+ y = 2 x + 8. ( ) dy = − y; dx dy = −dx y y (0) = 4 3 Integrate by parts: let u = 2 x + , 2 dv = e x dx . Then du = 2dx and v = e x . Thus, 3⎞ ⎛ e x y = ⎜ 2 x + ⎟ e x − 2e x dx 2⎠ ⎝ 3⎞ ⎛ e x y = ⎜ 2 x + ⎟ e x − 2e x + C 2⎠ ⎝ 1 y = 2 x − + Ce− x 2 To find C, apply the initial condition: 1 1 3 = y (0) = 0 − + Ce −0 = C − 2 2 7 Thus C = , so the solution is 2 1 7 −x y = 2x − + e 2 2 ln y = − x + C y = C1e− x To find C1 , apply the initial condition: 4 = y (0) = C1e −0 = C1 y = 4e − x 9. y '+ y = x + 2 The integrating factor is e 1dx = ex . e x y '+ ye x = e x ( x + 2) ( ) d x e y = ( x + 2) e x dx x Note: Solutions to Problems 22-28 are given along with the corresponding solutions to 11-16. 11., 22. e y = ( x + 2) e dx 0.0 Integrate by parts: let u = x + 2, dv = e x dx . Then du = dx and v = e x . Thus e x y = ( x + 2)e x − e x dx x xn x x x e y = ( x + 2)e − e + C −x y = x + 2 − 1 + Ce To find C , apply the initial condition: 4 = y (0) = 0 + 1 + Ce−0 = 1 + C → C = 3 Thus, y = x + 1 + 3e −x . 10. Instructor’s Resource Manual 12., 23. Euler's Method yn 3.0 Improved Euler Method yn 3.0 0.2 4.2 4.44 0.4 5.88 6.5712 0.6 8.232 9.72538 0.8 11.5248 14.39356 1.0 16.1347 21.30246 xn Euler's Method yn 2.0 Improved Euler Method yn 2.0 0.0 0.2 1.6 1.64 0.4 1.28 1.3448 0.6 1.024 1.10274 0.8 0.8195 0.90424 1.0 0.65536 0.74148 Section 6.7 385 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 13., 24. xn 0.0 Euler's Method yn 0.0 Improved Euler Method yn 0.0 0.2 0.0 0.02 0.4 0.04 0.08 0.6 0.12 0.18 0.8 0.24 0.32 1.0 0.40 0.50 17. a. y0 = 1 y1 = y0 + hf ( x0 , y0 ) = y0 + hy0 = (1 + h) y0 y2 = y1 + hf ( x1 , y1 ) = y1 + hy1 = (1 + h) y1 = (1 + h)2 y0 y3 = y2 + hf ( x2 , y2 ) = y2 + hy2 = (1 + h) y2 = (1 + h)3 y0 yn = yn −1 + hf ( xn −1 , yn −1 ) = yn −1 + hyn −1 14., 25. xn = (1 + h) yn −1 = (1 + h) n y0 = (1 + h ) n 0.2 0.0 0.004 0.4 0.008 0.024 0.6 0.040 0.076 (1 + 1/ N ) N 0.8 0.112 0.176 we know that lim (1 + 1/ N ) 1.0 xn 1.0 16., 27. Improved Euler Method yn 0.0 Let N = 1/ h . Then y N is an approximation to the solution at x = Nh = (1/ h)h = 1 . The exact solution is y (1) = e . Thus, 0.0 15., 26. Euler's Method yn 0.0 0.240 Euler's Method yn 1.0 Improved Euler Method yn 1.0 1.2 1.244 1.4 1.488 1.60924 1.6 1.90464 2.16410 1.8 2.51412 3.02455 2.0 3.41921 4.391765 1.0 Euler's Method yn 2.0 Improved Euler Method yn 2.0 ≈ e for large N. From Chapter 7, N N →∞ 0.340 1.2 xn b. =e. 18. y0 = y ( x0 ) = 0 y1 = y0 + hf ( x0 ) = 0 + hf ( x0 ) = hf ( x0 ) y2 = y1 + hf ( x1 ) = hf ( x0 ) + hf ( x1 ) = h ( f ( x0 ) + f ( x1 ) ) y3 = y2 + hf ( x2 ) = h [ f ( x0 ) + f ( x1 ) ] + hf ( x2 ) = h [ f ( x0 ) + f ( x1 ) + f ( x2 ) ] = h 3−1 i =0 f ( xi ) At the nth step of Euler's method, yn = yn −1 + hf ( xn −1 ) = h 19. a. x1 x0 y '( x)dx = n −1 i =0 f ( xi ) x1 sin x 2 dx x0 1.2 1.2 1.312 y ( x1 ) − y ( x0 ) ≈ ( x1 − x0 ) sin x02 1.4 0.624 0.80609 y ( x1 ) − y (0) = h sin x02 1.6 0.27456 0.46689 1.8 0.09884 0.25698 y ( x1 ) − 0 ≈ 0.1sin 02 y ( x1 ) ≈ 0 2.0 0.02768 0.13568 b. x2 x0 y '( x)dx = x2 sin x 2 dx x0 y ( x2 ) − y ( x0 ) ≈ ( x1 − x0 ) sin x02 + ( x2 − x1 ) sin x12 y ( x2 ) − y (0) = h sin x02 + h sin x12 y ( x2 ) − 0 ≈ 0.1sin 02 + 0.1sin 0.12 y ( x2 ) ≈ 0.00099998 386 Section 6.7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. c. x3 x0 x3 sin x 2 dx x0 y '( x)dx = x3 x0 c. + ( x2 − x1 ) sin x12 + ( x3 − x2 ) sin x12 y ( x3 ) − y (0) = h sin x02 + h sin x12 + h sin x22 y ( x3 ) − y (0) = 0.1 0 + 1 + 0.1 0.1 + 1 y ( x3 ) − 0 ≈ 0.1sin 02 + 0.1sin 0.12 + 0.1 0.2 + 1 y ( x3 ) ≈ 0.314425 Continuing in this fashion, we have y ( x3 ) ≈ 0.004999 Continuing in this fashion, we have xn x0 y '( x)dx = y ( xn ) ≈ h xn x0 xn sin x 2 dx x0 n −1 y ( xn ) − y ( x0 ) ≈ i =0 ( xi +1 − xi ) sin xi2 y ( xn ) ≈ h n −1 i =0 f ( xi −1 ) The result y ( xn ) ≈ h b. that given in Problem 18. Thus, when f ( x, y ) depends only on x , then the two methods (1) Euler's method for approximating the solution to y ' = f ( x) at xn , and (2) the left-endpoint xn 0 f ( x) dx , c. are equivalent. x1 x0 y '( x)dx = x1 x0 y ( x1 ) − 0 ≈ 0.1 0 + 1 y ( x1 ) ≈ 0.1 x2 x0 x + 1 dx y ( x2 ) − y ( x0 ) ≈ ( x1 − x0 ) x0 + 1 + ( x2 − x1 ) x1 + 1 y ( x2 ) − y (0) = h x0 + 1 + h x1 + 1 y ( x2 ) − 0 ≈ 0.1 0 + 1 + 0.1 0.1 + 1 y ( x2 ) ≈ 0.204881 xi −1 + 1 2. yn −1 + hf ( xn −1 , yn −1 ) x + 1 dx y ( x1 ) − y (0) = h x0 + 1 y '( x)dx = i =0 ( xi +1 − xi ) xi −1 + 1 h y1 − y0 = [ f ( x0 , y0 ) + f ( x1 + yˆ1 )] ⇒ 2 h y1 = y0 + [ f ( x0 , y0 ) + f ( x1 + yˆ1 )] 2 1. xn −1 + h h 3. yn −1 + [ f ( xn −1 , yn −1 ) + f ( xn , yˆ n )] 2 y ( x1 ) − y ( x0 ) ≈ ( x1 − x0 ) x0 + 1 x2 x0 n −1 i =0 Δy 1 = [ f ( x0 , y0 ) + f ( x1 + yˆ1 )] Δx 2 y1 − y0 Δy 1 = = [ f ( x0 , y0 ) + f ( x1 + yˆ1 )] ⇒ h Δx 2 2( y1 − y0 ) = h[ f ( x0 , y0 ) + f ( x1 + yˆ1 )] ⇒ 21. a. f ( xi −1 ) is the same as Riemann sum for approximating xn x + 1 dx x0 n −1 When n = 10 , this becomes y ( x10 ) = y (1) ≈ 1.198119 n −1 i =0 y '( x)dx = y ( xn ) − y ( x0 ) ≈ When n = 10 , this becomes y ( x10 ) = y (1) ≈ 0.269097 b. x + 1 dx + ( x2 − x1 ) x1 + 1 + ( x3 − x2 ) x2 + 1 + 0.1sin 0.22 20. a. x3 x0 y ( x3 ) − y ( x0 ) ≈ ( x1 − x0 ) x0 + 1 y ( x3 ) − y ( x0 ) ≈ ( x1 − x0 ) sin x02 d. y '( x)dx = 22-27. See problems 11-16 h 0.2 Error from Euler's Method 0.229962 Error from Improved Euler Method 0.015574 0.1 0.124539 0.004201 0.05 0.064984 0.001091 0.01 0.013468 0.000045 0.005 0.006765 0.000011 28. For Euler's method, the error is halved as the step size h is halved. Thus, the error is proportional to h. For the improved Euler method, when h is halved, the error decreases to approximately one-fourth of what is was. Hence, for the improved Euler method, the error is proportional to h 2 Instructor’s Resource Manual Section 6.7 387 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6.8 Concepts Review ⎡ π π⎤ 1. ⎢ – , ⎥ ; arcsin ⎣ 2 2⎦ 1 ⎞ ⎛ 13. cos(arccot 3.212) = cos ⎜ arctan ⎟ 3.212 ⎠ ⎝ ≈ cos 0.3018 ≈ 0.9548 14. sec(arccos 0.5111) = ⎛ π π⎞ 2. ⎜ – , ⎟ ; arctan ⎝ 2 2⎠ 3. 1 1 ≈ 1.957 0.5111 ⎛ 1 ⎞ 15. sec –1 (–2.222) = cos –1 ⎜ ⎟ ≈ 2.038 ⎝ –2.222 ⎠ 4. π Problem Set 6.8 16. tan −1 (−60.11) ≈ –1.554 ⎛ 2⎞ π π 2 1. arccos ⎜⎜ ⎟⎟ = since cos = 4 2 ⎝ 2 ⎠ 4 ⎛ 3⎞ π 3 ⎛ π⎞ 2. arcsin ⎜⎜ – ⎟⎟ = – since sin ⎜ – ⎟ = – 3 2 ⎝ 3⎠ ⎝ 2 ⎠ ⎛ 3⎞ π 3 ⎛ π⎞ 3. sin –1 ⎜⎜ – ⎟⎟ = – since sin ⎜ – ⎟ = – 2 3 3 2 ⎝ ⎠ ⎝ ⎠ ⎛ 2⎞ π 2 ⎛ π⎞ 4. sin –1 ⎜⎜ – ⎟⎟ = – 4 since sin ⎜⎝ – 4 ⎟⎠ = – 2 2 ⎝ ⎠ 5. arctan( 3) = π ⎛π⎞ since tan ⎜ ⎟ = 3 3 ⎝3⎠ ⎛1⎞ π ⎛π⎞ 1 6. arcsec(2) = arccos ⎜ ⎟ = since cos ⎜ ⎟ = , so ⎝2⎠ 3 ⎝3⎠ 2 ⎛π⎞ sec ⎜ ⎟ = 2 ⎝3⎠ π 1 ⎛ 1⎞ ⎛ π⎞ 7. arcsin ⎜ – ⎟ = – since sin ⎜ – ⎟ = – 6 2 ⎝ 2⎠ ⎝ 6⎠ ⎛ 3⎞ π 3 ⎛ π⎞ 8. tan –1 ⎜⎜ – ⎟⎟ = – since tan ⎜ – ⎟ = – 3 6 6 3 ⎝ ⎠ ⎝ ⎠ 9. sin(sin –1 0.4567) = 0.4567 by definition 10. cos(sin –1 0.56) = 1 − sin 2 (sin −1 0.56) = 1 – (0.56) 2 ≈ 0.828 11. sin −1 (0.1113) ≈ 0.1115 12. arccos(0.6341) ≈ 0.8840 388 = 1 cos(arccos 0.5111) Section 6.8 17. cos(sin(tan −1 2.001)) ≈ 0.6259 18. sin 2 (ln(cos 0.5555)) ≈ 0.02632 19. = sin −1 x 8 20. = tan −1 x 6 21. = sin −1 5 x 22. = cos −1 9 or x = sec−1 x 9 23. Let 1 be the angle opposite the side of length 3, and 2 = and tan 24. Let and 1 2 1 = . x 1 = tan –1 – 2. Then tan 1 = 3 x 3 1 – tan –1 . x x be the angle opposite the side of length 5, 2 = 1 − , and y the length of the unlabeled 1 = side. Then tan = – , so 1 = 5 = y = tan −1 1− 2 5 2 and y = x 2 − 25. , tan 2 = 2 = y 2 2 x − 25 x − 25 5 2 − tan −1 2 2 x − 25 x − 25 , ⎡ ⎡ ⎛ 2 ⎞⎤ ⎛ 2 ⎞⎤ 25. cos ⎢ 2sin –1 ⎜ – ⎟ ⎥ = 1 – 2sin 2 ⎢sin –1 ⎜ – ⎟ ⎥ ⎝ 3 ⎠⎦ ⎝ 3 ⎠⎦ ⎣ ⎣ 2 1 ⎛ 2⎞ = 1– 2⎜ – ⎟ = 9 ⎝ 3⎠ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. () () 2 tan ⎡ tan –1 13 ⎤ ⎡ ⎣ ⎦ –1 ⎛ 1 ⎞ ⎤ 26. tan ⎢ 2 tan ⎜ ⎟ ⎥ = ⎝ 3 ⎠ ⎦ 1 – tan 2 ⎡ tan –1 1 ⎤ ⎣ 3 ⎦ ⎣ = 2 ⋅ 13 () 1 – 13 2 = 3 4 ⎡ ⎡ ⎡ ⎡ ⎛3⎞ ⎛ 5 ⎞⎤ ⎛ 3 ⎞⎤ ⎛ 5 ⎞⎤ ⎛ 3 ⎞⎤ ⎡ ⎛ 5 ⎞⎤ 27. sin ⎢cos –1 ⎜ ⎟ + cos –1 ⎜ ⎟ ⎥ = sin ⎢cos –1 ⎜ ⎟ ⎥ cos ⎢cos –1 ⎜ ⎟ ⎥ + cos ⎢cos –1 ⎜ ⎟ ⎥ sin ⎢ cos –1 ⎜ ⎟ ⎥ 5 13 5 13 5 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ 13 ⎠ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ 2 2 56 ⎛3⎞ 5 3 ⎛5⎞ = 1– ⎜ ⎟ ⋅ + 1– ⎜ ⎟ = 65 ⎝ 5 ⎠ 13 5 ⎝ 13 ⎠ ⎡ ⎡ ⎡ ⎡ ⎛4⎞ ⎛ 12 ⎞ ⎤ ⎛ 4 ⎞⎤ ⎛ 12 ⎞ ⎤ ⎛ 4 ⎞⎤ ⎡ ⎛ 12 ⎞ ⎤ 28. cos ⎢cos –1 ⎜ ⎟ + sin –1 ⎜ ⎟ ⎥ = cos ⎢cos –1 ⎜ ⎟ ⎥ cos ⎢sin –1 ⎜ ⎟ ⎥ – sin ⎢ cos –1 ⎜ ⎟ ⎥ sin ⎢sin –1 ⎜ ⎟ ⎥ 5 13 5 13 5 ⎝ ⎠ ⎝ ⎠⎦ ⎝ ⎠⎦ ⎝ ⎠⎦ ⎝ ⎠⎦ ⎣ ⎝ 13 ⎠ ⎦ ⎣ ⎣ ⎣ ⎣ 2 = 29. tan(sin –1 x) = 30. sin(tan –1 x) = = 2 4 16 ⎛ 12 ⎞ ⎛ 4 ⎞ 12 ⋅ 1– ⎜ ⎟ – 1– ⎜ ⎟ ⋅ = – 5 13 5 13 65 ⎝ ⎠ ⎝ ⎠ sin(sin –1 x) cos(sin –1 x) 1 csc(tan 1 1+ 1 tan 2 (tan –1 x ) = –1 x) 1 x2 34. a. 1 – x2 1 = 1 1+ x = 1 + cot 2 (tan –1 x) = b. x x2 + 1 31. cos(2sin –1 x) = 1 – 2sin 2 (sin –1 x) = 1 – 2 x 2 32. tan(2 tan 33. a. b. –1 x) = 2 tan(tan –1 x) 1 – tan 2 (tan –1 x ) lim tan –1 x = x →∞ π since 2 lim →π / 2− π lim tan x = – since 2 x→ – ∞ lim tan = −∞ –1 →−π / 2+ = 35. a. ⎛1⎞ lim sec –1 x = lim cos –1 ⎜ ⎟ x →∞ x →∞ ⎝x⎠ π = lim cos –1 z = 2 z → 0+ ⎛1⎞ lim sec –1 x = lim cos –1 ⎜ ⎟ x→ – ∞ x→ – ∞ ⎝x⎠ π = lim cos –1 z = – 2 z →0 Let L = lim sin −1 x . Since 2x x →1− 1 – x2 sin(sin −1 x) = x, tan = ∞ Thus, since sin is continuous, the Composite lim sin(sin −1 x) = lim x = 1 . x →1− x →1− Limit Theorem gives us lim sin(sin −1 x) = lim sin( L) ; hence x →1− x →1− sin L = 1 and since the range of sin −1 is π ⎡ π π⎤ ⎢− 2 , 2 ⎥ , L = 2 . ⎣ ⎦ Instructor’s Resource Manual Section 6.8 389 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. Let L = lim sin −1 x . Since sin(sin x →−1+ −1 39. y = ln(2 + sin x) . Let u = 2 + sin x ; then y = ln u so by the Chain Rule x ) = x, lim sin(sin −1 x →−1+ dy dy du ⎛ 1 ⎞ du ⎛ 1 ⎞ = =⎜ ⎟ =⎜ ⎟ ⋅ cos x dx du dx ⎝ u ⎠ dx ⎝ 2 + sin x ⎠ cos x = 2 + sin x x) = lim x = −1 . x →−1+ Thus, since sin is continuous, the Composite Limit Theorem gives us lim sin(sin −1 x) = lim sin( L) ; x →−1+ x →−1+ 40. hence sin L = −1 and since the range of sin −1 is π ⎡ π π⎤ ⎢− 2 , 2 ⎥ , L = − 2 . ⎣ ⎦ 41. 36. No. Since sin −1 x is not defined on (1, ∞) , 42. lim sin −1 x does not exist so neither can the x →1+ two-sided limit lim sin −1 x . x →1 f (c ) = 1 1 − c2 d – csc x cot x – csc2 x [– ln(csc x + cot x)] = – dx csc x + cot x csc x(cot x + csc x) = = csc x cot x + csc x d 1 4x sin –1 (2 x 2 ) = ⋅ 4x = dx 1 – (2 x 2 ) 2 1 – 4 x4 44. d 1 ex arccos(e x ) = – ⋅ ex = – dx 1 – (e x ) 2 1 – e2 x 45. d 3 ex [ x tan –1 (e x )] = x3 ⋅ + 3x 2 tan –1 (e x ) dx 1 + (e x ) 2 . Hence, lim f (c) = ∞ so c →1− d sec x tan x + sec 2 x ln(sec x + tan x) = dx sec x + tan x (sec x)(tan x + sec x) = = sec x sec x + tan x 43. 37. Let f ( x) = y = sin −1 x ; then the slope of the tangent line to the graph of y at c is d tan x d e = e tan x tan x = e tan x sec 2 x dx dx that the tangent lines approach the vertical. ⎡ xe x ⎤ = x2 ⎢ + 3 tan –1 (e x ) ⎥ 2x ⎣⎢1 + e ⎦⎥ 38. 46. d x 2x (e arcsin x 2 ) = e x ⋅ + e x arcsin x 2 2 2 dx 1 – (x ) ⎛ 2x ⎞ = ex ⎜ + arcsin x 2 ⎟ ⎜ ⎟ 4 ⎝ 1– x ⎠ 47. 3(tan –1 x) 2 d 1 = (tan –1 x)3 = 3(tan –1 x)2 ⋅ dx 1 + x2 1 + x2 48. d d sin(cos −1 x) d 1 – x 2 tan(cos –1 x) = = dx dx cos(cos −1 x) dx x = x ⋅ 12 ⋅ 1 1– x 2 (–2 x) – 1 – x 2 ⋅1 x2 – x – (1 – x ) 1 = =– 2 2 2 x 1– x x 1 – x2 2 390 Section 6.8 2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 49. 50. d sec –1 ( x3 ) = dx x3 ( x3 ) 2 – 1 3 ⋅ 3x2 = d (sec –1 x)3 = 3(sec –1 x) 2 ⋅ dx x = x 54. y = x arc sec( x 2 + 1) x6 – 1 dy ⎡d ⎤ ⎛d ⎞ = x ⎢ arcsec(x 2 + 1) ⎥ + ⎜ x ⎟ ⋅ arcsec(x 2 + 1) dx ⎣ dx ⎦ ⎝ dx ⎠ ⎡ ⎢ = x⎢ 2 ⎢⎣ x + 1 1 2 x –1 ( 3(sec –1 x)2 x 51. 1 ⎡ 2x2 ⎢ =⎢ 2 4 2 ⎢⎣ x + 1 x + 2 x x 2 –1 ( d 1 (1 + sin –1 x)3 = 3(1 + sin –1 x)2 ⋅ dx 1 – x2 = 1 2 3(1 + sin –1 x)2 ( 1 – x2 x +4 Chain Rule: dy dy du 1 du = = ⋅ = 2 dx dx du dx 1− u ⎛ −2 x ⋅⎜ 2 2 ⎜ 2 ⎛ 1 ⎞ ⎝ ( x + 4) 1− ⎜ 2 ⎟ ⎝ x +4⎠ 1 ⎞ ⎟⎟ = ⎠ ⎛ ⎞ ⎛ −2 x ⎞ ( x 2 + 4) ⎜ ⎟⋅⎜ ⎟= 2⎟ ⎜ 4 ⎟ ⎜ 2 2 ⎝ x + 8 x + 15 ⎠ ⎝ ( x + 4) ⎠ −2 x ( x 2 + 4) x 4 + 8 x 2 + 15 ( 53. y = tan −1 ln x 2 ) Let u = x , v = ln u ; then y = tan −1 ( v(u ( x)) ) so 2 by the Chain Rule: dy dy dv du 1 1 = = ⋅ ⋅ 2x = dx dv du dx 1 + v 2 u 1 1 ⋅ ⋅ 2x = 2 2 1 + (ln x ) x 2 2 x[1 + (ln x 2 )2 ] ) ⎡ ⎤ 2x ⎢ ⎥ 2 =⎢ ⎥ + arcsec(x + 1) 2 2 ⎢⎣ x + 1 x + 2 ⎥⎦ ( ; then y = sin −1 ( u ( x) ) so by the ) ⎤ ⎥ 2 ⎥ + arcsec(x + 1) ⎥⎦ ⎡ ⎤ 2 x2 ⎢ ⎥ 2 =⎢ ⎥ + arcsec(x + 1) 2 2 ⎢⎣ x + 1 ⋅ x x + 2 ⎥⎦ ⎛ 1 ⎞ 52. y = sin −1 ⎜ ⎟ ⎝ x2 + 4 ⎠ Let u = ) ⎤ ⎥ 2 ⎥ + 1 ⋅ arcsec(x + 1) 2 2 ( x + 1) − 1 ⎥ ⎦ 2x 55. ) cos 3x dx Let u = 3 x, du = 3dx ; then 1 cos 3 x (3dx) = 3 1 1 1 cos u du = sin u + C = sin 3 x + C 3 3 3 cos 3 x dx = 56. Let u = x 2 , so du = 2 x dx . 1 sin( x 2 ) ⋅ 2 x dx 2 1 1 = sin u du = − cos u + C 2 2 1 2 = − cos( x ) + C 2 x sin( x 2 )dx = 57. Let u = sin 2x, so du = 2 cos 2x dx. 1 sin 2 x cos 2 x dx = sin 2 x(2 cos 2 x)dx 2 1 = u du 2 u2 1 = + C = sin 2 2 x + C 4 4 58. Let u = cos x, so du = − sin x dx . tan x dx = sin x 1 (− sin x)dx dx = − cos x cos x 1 du = − ln u + C = − ln cos x + C u = ln sec x + C =− Instructor’s Resource Manual Section 6.8 391 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 65. Let u = 2x, so du = 2 dx. 1 1 1 dx = 2dx 2 2 1 + (2 x)2 1+ 4x 59. Let u = e2 x , so du = 2e2 x dx . 1 cos(e2 x )(2e2 x )dx e2 x cos(e2 x )dx = 2 1 = cos u du 2 1 1 = sin u + C = sin(e2 x ) + C 2 2 1 1 1 du = arctan u + C 2 2 1+ u 2 1 = arctan 2 x + C 2 = 1 ⎡1 ⎤ cos(e2 x ) dx = ⎢ sin(e2 x ) ⎥ ⎣2 ⎦0 1 ⎡1 ⎤ = ⎢ sin(e2 ) − sin(e0 ) ⎥ 2 ⎣2 ⎦ 1 2x e 0 66. Let u = e x , so du = e x dx . ex 2x 60. Let u = sin x, so du = cos x dx. u3 sin 3 x sin 2 x cos x dx = u 2 du = +C = +C 3 3 π/2 0 61. π/2 ⎡ sin 3 x ⎤ sin 2 x cos x dx = ⎢ ⎥ ⎢⎣ 3 ⎥⎦ 0 1 2/2 0 1– x dx 2 2 x x2 − 1 = sec −1 = cos 63. 2 − sec 1 1 −0 = 3 3 Let u = 1 = dx 2 2 −1 x2 − 1 x = ⎡sec−1 x ⎤ ⎣ ⎦ 2⎞ π π π ⎜⎜ ⎟⎟ = − = ⎝ 2 ⎠ 3 4 12 1 −1 d =− dx ⎛ 3 ⎞ 12⎜ 1− x 2 ⎟ ⎝ 4 ⎠ 1 1 2 3 ⎛ 3 ⎞ x ⎟⎟ 1−⎜⎜ ⎝ 2 ⎠ 2 dx 3 3 x, du = dx ; then 2 2 1 1 ⎛ 2 ⎞ dx = ⎜ ⎟ 2 2 3⎝ 3⎠ ⎛ 3 ⎞ x 12 − 9 x 1 1− u2 du 2 dx . Let u = 12 − 9 x 2 , du = −18 x dx; dx = − 1 18 1 (−18 dx) 12 − 9 x 2 12 − 9 x 2 1 1 ⎛ 1⎞ =− du = ⎜ − ⎟ (2 u ) + C 18 u ⎝ 18 ⎠ =− 1 + cos 1 + cos 2 1 du = − tan −1 u + C =− 2 1+ u 1 dx = 1−⎜⎜ x ⎟⎟ ⎝ 2 ⎠ x x ⎤ = tan 1 − tan (−1) ⎦ −1 1 2 du then −1 64. Let u = cos , so du = − sin d . 2 2 −1 ⎛ dx = ⎡ tan ⎣ sin 2 2 −1 1 dx = ⎛ 3 ⎞ 1 1 x ⎟⎟ + C = sin −1 u + C = sin −1 ⎜⎜ 3 3 ⎝ 2 ⎠ 68. π ⎛ π⎞ π −⎜− ⎟ = 4 ⎝ 4⎠ 2 = 12 − 9 x 2 3 ⎜ ⎟ − cos ⎝2⎠ −1 1 + x 2 1 = dx = [arcsin x]0 2 / 2 −1 ⎛ 1 ⎞ 1 1 = 67. π 2 – arcsin 0 = 2 4 = arcsin 62. 2 x 2 1+ e 1 + (e ) 1 + u2 = arctan u + C = arctan ex + C sin e2 − sin1 ≈ 0.0262 2 = ex dx = 12 − 9 x 2 +C 9 (− sin )d = − tan −1 (cos ) + C π/2 0 sin 1 + cos 2 = − tan −1 0 + tan −1 1 = −0 + 392 π/2 d = ⎡ − tan −1 (cos ) ⎤ ⎣ ⎦0 Section 6.8 π π = 4 4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 69. 2 = x − 6 x + 13 1 1 dx = 2 ( x − 6 x + 9) + 4 73. The top of the picture is 7.6 ft above eye level, and the bottom of the picture is 2.6 ft above eye level. Let 1 be the angle between the viewer’s line of sight to the top of the picture and the horizontal. Then call 2 = 1 − , so = 1 − 2 . dx dx ( x − 3)2 + 4 Let u = x − 3, du = dx, a = 2; then 1 ⎛u⎞ dx = du = tan −1 ⎜ ⎟ + C 2 2 2 a ⎝a⎠ ( x − 3) + 4 u +a 1 = 1 1 ⎛ x−3⎞ tan −1 ⎜ ⎟+C 2 ⎝ 2 ⎠ 1 70. 2 2 x + 8 x + 25 1 2 dx = 1 ⎛ 17 ⎞ ⎟ ⎝ 2 ⎠ ( x + 2) 2 + ⎜ 2 1 2 2( x + 4 x + 4 + 17 ) 2 dx = dx 17 ; then 2 1 1 dx = du = 2 u2 + a2 Let u = x + 2, du = dx, a = 1 1 2 ( x + 2) 2 + 17 2 ⎛ x+2⎞ 1 1 1 2 ⎛u⎞ ⋅ tan −1 ⎜ ⎟ + C = ⋅ tan −1 ⎜ ⎟+C ⎜ 17 ⎟ 2 a 2 17 ⎝a⎠ 2 ⎠ ⎝ = 71. ⎡ 34 ⋅ ( x + 2) ⎤ 34 tan −1 ⎢ ⎥+C 34 17 ⎣ ⎦ 1 x 4 x2 − 9 then dx . Let u = 2 x, du = 2 dx, a = 3 ; 1 x 4x − 9 1 du = u u 2 − a2 1 −1 ⎛ 2 x sec ⎜ 3 ⎝ 3 72. x +1 dx = 1 dx = 2 2 x 4 x2 − 9 (2 dx) = 1 −1 ⎛ u ⎞ sec ⎜ ⎟ + C = a ⎝a⎠ ⎞ ⎟+C ⎠ x 4 − 9x 2 dx = − ⎡ π⎤ 74. a. Restrict 2x to [0, π ] , i.e., restrict x to ⎢0, ⎥ . ⎣ 2⎦ Then y = 3 cos 2x y = cos 2 x 3 y 2 x = arccos 3 1 y x = f –1 ( y ) = arccos 2 3 1 x f –1 ( x) = arccos 2 3 π π⎤ ⎡ b. Restrict 3x to ⎢ – , ⎥ , i.e., restrict x to ⎣ 2 2⎦ ⎡ π π⎤ ⎢– 6 , 6 ⎥ ⎣ ⎦ Then y = 2 sin 3x y = sin 3 x 2 y 3x = arcsin 2 1 y x = f –1 ( y ) = arcsin 3 2 1 x f –1 ( x) = arcsin 3 2 c. dx + 1 dx 4 − 9 x2 4 − 9 x2 4 − 9 x2 These integrals are evaluated the same as those in problems 67 and 68 (with a constant of 4 rather than 12). Thus x +1 7.6 2.6 ; tan 2 = ; b b 7.6 2.6 = tan −1 − tan −1 b b If b = 12.9, ≈ 0.3335 or 19.1° . tan 1 = ⎛ π π⎞ Restrict x to ⎜ – , ⎟ ⎝ 2 2⎠ 1 y = tan x 2 2y = tan x x = f –1 ( y ) = arctan 2 y f –1 ( x) = arctan 2 x 1 1 ⎛ 3x ⎞ 4 − 9 x 2 + sin −1 ⎜ ⎟ + C 9 3 ⎝ 2 ⎠ Instructor’s Resource Manual Section 6.8 393 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 2⎞ ⎛2 ⎞ ⎛ d. Restrict x to ⎜ – ∞, – ⎟ ∪ ⎜ , ∞ ⎟ so is x π⎠ ⎝π ⎠ ⎝ ⎛ π ⎞ ⎛ π⎞ restricted to ⎜ – , 0 ⎟ ∪ ⎜ 0, ⎟ ⎝ 2 ⎠ ⎝ 2⎠ 1 then y = sin x 1 = arcsin y x 1 x = f −1 ( y ) = arcsin y f −1 ( x) = 2 ⋅ 14 1– ( ) 1 2 4 = 47 = 52 ( )⎤⎦ + tan ⎡⎣ tan –1 ( 995 )⎤⎦ ( 14 )⎤⎦ tan ⎡⎣ tan –1 ( 995 )⎤⎦ tan ⎡3 tan –1 14 ⎣ = 1 – tan ⎡3 tan –1 ⎣ 5 + 99 4913 π = = = 1 = tan 47 5 4 1 – 52 ⋅ 99 4913 π ⎛1⎞ ⎛ 5 ⎞ Thus, 3 tan –1 ⎜ ⎟ + tan –1 ⎜ ⎟ = tan –1 (1) = . 4 ⎝4⎠ ⎝ 99 ⎠ ⎡ ⎛ 1 ⎞⎤ 2 tan ⎢ tan –1 ⎜ ⎟ ⎥ ⎡ ⎛ 1 ⎞⎤ ⎝ 5 ⎠⎦ ⎣ 76. tan ⎢ 2 tan –1 ⎜ ⎟ ⎥ = 5 ⎡ ⎛ 1 ⎞⎤ ⎝ ⎠⎦ ⎣ 1 – tan 2 ⎢ tan –1 ⎜ ⎟ ⎥ ⎝ 5 ⎠⎦ ⎣ 2 ⋅ 15 () 1 – 15 2 = 5 12 ⎡ ⎡ ⎛ 1 ⎞⎤ ⎛ 1 ⎞⎤ tan ⎢ 4 tan –1 ⎜ ⎟ ⎥ = tan ⎢ 2 ⋅ 2 tan –1 ⎜ ⎟ ⎥ 5 ⎝ ⎠ ⎝ 5 ⎠⎦ ⎣ ⎦ ⎣ 394 1 – 28,561 π = 119 239 = = 1 = tan 120 1 4 1 + 119 ⋅ 239 28,561 π ⎛1⎞ ⎛ 1 ⎞ −1 Thus, 4 tan –1 ⎜ ⎟ – tan –1 ⎜ ⎟ = tan (1) = 4 ⎝5⎠ ⎝ 239 ⎠ 77. ( ) ( ) ⎡ ⎛1⎞ ⎛ 5 ⎞⎤ tan ⎢3 tan –1 ⎜ ⎟ + tan –1 ⎜ ⎟ ⎥ 4 ⎝ ⎠ ⎝ 99 ⎠ ⎦ ⎣ = ⎡ ⎡ ⎛ 1 ⎞⎤ ⎛ 1 ⎞⎤ tan ⎢ 4 tan –1 ⎜ ⎟ ⎥ – tan ⎢ tan –1 ⎜ ⎟⎥ ⎝ 5 ⎠⎦ ⎝ 239 ⎠ ⎦ ⎣ ⎣ = ⎡ ⎡ ⎛ 1 ⎞⎤ ⎛ 1 ⎞⎤ 1 + tan ⎢ 4 tan –1 ⎜ ⎟ ⎥ tan ⎢ tan –1 ⎜ ⎟⎥ 5 ⎝ ⎠ ⎝ 239 ⎠ ⎦ ⎣ ⎦ ⎣ 8 15 ( ) ( ) 47 52 ⎡ ⎛1⎞ ⎛ 1 ⎞⎤ tan ⎢ 4 tan –1 ⎜ ⎟ – tan –1 ⎜ ⎟⎥ 5 ⎝ ⎠ ⎝ 239 ⎠ ⎦ ⎣ 120 ⎡ ⎡ ⎛ 1 ⎞⎤ ⎛1⎞ ⎛ 1 ⎞⎤ tan ⎢3 tan –1 ⎜ ⎟ ⎥ = tan ⎢ 2 tan –1 ⎜ ⎟ + tan –1 ⎜ ⎟ ⎥ 4 4 ⎝ ⎠⎦ ⎝ ⎠ ⎝ 4 ⎠⎦ ⎣ ⎣ tan ⎡ 2 tan –1 14 ⎤ + tan ⎡ tan –1 14 ⎤ ⎣ ⎦ ⎣ ⎦ = − 1 –1 1 1 ⎤ tan ⎡ tan ⎤ 1 – tan ⎡ 2 tan 4 ⎦ 4 ⎦ ⎣ ⎣ 8 +1 = 15 4 8 ⋅1 1 – 15 4 ( ) 1 arcsin x ⎡ ⎛ 1 ⎞⎤ 2 tan ⎢ tan –1 ⎜ ⎟ ⎥ ⎡ ⎛ 1 ⎞⎤ ⎝ 4 ⎠⎦ ⎣ 75. tan ⎢ 2 tan –1 ⎜ ⎟ ⎥ = 4 ⎡ ⎛ 1 ⎞⎤ ⎝ ⎠⎦ ⎣ 1 – tan 2 ⎢ tan –1 ⎜ ⎟ ⎥ ⎝ 4 ⎠⎦ ⎣ = ⎡ ⎛ 1 ⎞⎤ 2 tan ⎢ 2 tan –1 ⎜ ⎟ ⎥ 5 2 ⋅ 12 120 ⎝ 5 ⎠⎦ ⎣ = = = 2 119 ⎡ ⎛ 1 ⎞⎤ 5 1 – tan 2 ⎢ 2 tan –1 ⎜ ⎟ ⎥ 1 – 12 5 ⎝ ⎠⎦ ⎣ Section 6.8 represent ∠DAB, then ∠CAB is Let 2 . Since b ΔABC is isosceles, AE = b b , cos = 2 = and 2 2 a 2a b . Thus sector ADB has area 2a 1⎛ –1 b ⎞ 2 2 –1 b . Let φ represent ⎜ 2 cos ⎟ b = b cos 2⎝ 2a ⎠ 2a = 2 cos –1 ∠DCB, then ∠ACB is φ φ 2 and ∠ECA is φ 4 , so b b b = 2 = and φ = 4sin –1 . Thus sector 4 a 2a 2a 1⎛ b b ⎞ DCB has area ⎜ 4sin –1 ⎟ a 2 = 2a 2 sin –1 . 2⎝ 2a ⎠ 2a These sectors overlap on the triangles ΔDAC and ΔCAB, each of which has area sin 2 1 1 1 4a 2 – b 2 ⎛b⎞ AB h = b a 2 – ⎜ ⎟ = b . 2 2 2 2 ⎝2⎠ The large circle has area πb 2 , hence the shaded region has area b b 1 – 2a 2 sin –1 πb 2 – b 2 cos –1 + b 4a 2 – b 2 2a 2a 2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. y = sin(arcsin x) is the line y = x, but only defined for −1 ≤ x ≤ 1 . 78. y = arcsin(sin x) is defined for all x, but only the π π portion for – ≤ x ≤ is the line y = x. 2 2 They have the same graph. Conjecture: arcsin x = arctan x 1 – x2 –1 < x < 1 Proof: Let = arcsin x, so x = sin x sin sin Then = = cos 1 – x2 1 – sin 2 x so = arctan . 1 – x2 for dx 81. a2 – x2 = = tan dx = ( ) 2⎤ ⎡ a 2 ⎢1 – ax ⎥ ⎣ ⎦ 1 dx 1 dx ⋅ = ⋅ 2 a a 1 – ax 1 – ax ( ) ( ) 2 since a > 0 x 1 , so du = dx. a a 1 dx du ⋅ = = sin −1 u + C 2 2 a 1− u 1 – ax Let u = ( ) 79. = sin −1 x +C a 82. Dx sin −1 It is the same graph as y = arccos x. π Conjecture: – arcsin x = arccos x 2 π Proof: Let = – arcsin x 2 ⎛π ⎞ Then x = sin ⎜ − ⎟ = cos 2 ⎝ ⎠ so = arccos x. = = = 80. x = a 1 a2 – x2 a2 ⋅ 1 1– 1 = a ( ) x 2 a ⋅ 1 a a a2 – x2 ⋅ 1 a 1 ⋅ , since a > 0 a a –x 1 a 2 2 a2 – x2 x 1 , so du = dx a a dx 1 1 1 = dx 2 2 2 a 1+ x a a +x 83. Let u = (a) 1 1 1 du = tan −1 u + C a 1+ u2 a 1 ⎛ x⎞ = tan −1 ⎜ ⎟ + C a ⎝a⎠ = Instructor’s Resource Manual Section 6.8 395 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. x , so du = (1/ a ) dx. Since a > 0, a dx 1 1 1 = dx 2 2 2 a a x x x x −a −1 a a 84. Let u = 86. 1 a − x2 πa 2 2 This result is expected because the integral should be half the area of a circle with radius a. = a 2 sin −1 (1) = (See 87. Let be the angle subtended by viewer’s eye. ⎛ 12 ⎞ ⎛2⎞ = tan −1 ⎜ ⎟ − tan −1 ⎜ ⎟ b ⎝ ⎠ ⎝b⎠ ⎛ 12 ⎞ d 1 1 ⎛ 2 ⎞ = ⎜− ⎟− ⎜− ⎟ db 1 + 12 2 ⎜⎝ b 2 ⎟⎠ 1 + 2 2 ⎝ b 2 ⎠ b b Problem 67). ⎤ d ⎡x 2 a2 x a − x2 + sin −1 + C ⎥ ⎢ dx ⎢⎣ 2 2 a ⎥⎦ 1 2 x 1 = a − x2 + (−2 x) 2 2 2 a2 − x2 a2 + 2 = 1 a2 − x2 a 2 − x 2 dx ⎡a ⎤ a2 0 2 a2 sin −1 (1) − a − sin −1 (0) ⎥ = 2 ⎢ (0) + 2 2 2 ⎣⎢ 2 ⎦⎥ 1 1 du a u u2 −1 x 1 1 = sec −1 u + C = sec −1 + C a a a 2 a 0 a = d ⎛ x⎞ sin −1 ⎜ ⎟ = dx ⎝a⎠ a 2 − x 2 dx = 2 ⎡x 2 a2 x⎤ a − x2 + sin −1 ⎥ = 2⎢ a ⎦⎥ 2 ⎣⎢ 2 0 ( ) ( ) 85. Note that a −a ( ) = +0 2 − ( ) 12 = 10(24 − b 2 ) b 2 + 4 b 2 + 144 (b 2 + 4)(b 2 + 144) d Since > 0 for b in ⎡⎣0, 2 6 db d and < 0 for b in 2 6, ∞ , the angle is db ) 1 2 1 − x2 + a2 a − x2 + = a2 − x2 2 2 a2 − x2 ( ) maximized for b = 2 6 ≈ 4.899 . The ideal distance is about 4.9 ft from the wall. 88. a. ⎛x⎞ ⎛ x⎞ = cos −1 ⎜ ⎟ − cos −1 ⎜ ⎟ ⎝b⎠ ⎝a⎠ ⎛ ⎞ ⎛ ⎜ ⎟ ⎛ 1 ⎞⎛ dx ⎞ ⎜ −1 −1 d =⎜ ⎟ ⎜ ⎟⎜ ⎟ − ⎜ 2 ⎝ b ⎠⎝ dt ⎠ ⎜ dt ⎜ ⎜ 1 − bx ⎟⎟ ⎜ 1 − ax ⎝ ⎠ ⎝ ( ) b. ( ) ⎛ a+x = tan −1 ⎜ ⎜ 2 2 ⎝ b −x ⎛ ⎜ ⎜ d 1 =⎜ dt ⎜ ⎛ a+ x ⎜⎜ 1 + ⎜ 2 2 ⎝ ⎝ b −x ⎞ ⎟ ⎠ 2 ⎞ ⎟ ⎛ b2 − x2 + (a + x) x ⎟⎜ b2 − x2 ⎟⎜ 2 2 b −x ⎟⎜ ⎟⎟ ⎜⎝ ⎠ ⎞ ⎛ ⎞ ⎟ ⎛ dx ⎞ ⎜ ⎟ ⎛ 1 ⎞ ⎛ dx ⎞ 1 ⎟⎜ ⎟− ⎜ ⎟⎜ ⎟⎜ ⎟ ⎟ ⎝ dt ⎠ ⎜ x 2 ⎟⎟ ⎝ b ⎠ ⎝ dt ⎠ ⎜ − 1 ⎟ b ⎝ ⎠ ⎠ ⎞⎛ b 2 + ax ⎞ 1 ⎟⎜ 2 ⎟− ⎟⎜ (b − x 2 )3 / 2 ⎟ 2 b − x2 ⎠⎝ ⎠ ⎡ b 2 + ax 1 =⎢ − ⎢ (b 2 + a 2 + 2ax) b 2 − x 2 b2 − x2 ⎣ Section 6.8 ⎞ dx ⎟ ⎟ dt ⎠ ⎞ ⎛ x⎞ ⎟ − sin −1 ⎜ ⎟ ⎟ ⎝b⎠ ⎠ ⎡⎛ b2 − x2 = ⎢⎜ 2 2 ⎜ 2 ⎣⎢⎝ b − x + (a + x) 396 2 ⎞ ⎟ ⎛ 1 ⎞ ⎛ dx ⎞ ⎛ 1 1 − ⎟⎜ ⎟⎜ ⎟ = ⎜ ⎟⎟ ⎝ a ⎠ ⎝ dt ⎠ ⎜⎝ a 2 − x 2 b2 − x2 ⎠ ( ) ⎤ dx ⎥ ⎦⎥ dt ⎤ dx ⎡ a 2 + ax ⎥ = ⎢− ⎥ dt ⎢ (b 2 + a 2 + 2ax) b 2 − x 2 ⎦ ⎣ ⎤ dx ⎥ ⎥ dt ⎦ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 89. Let h(t) represent the height of the elevator (the number of feet above the spectator’s line of sight) t seconds after the line of sight passes horizontal, and let (t ) denote the angle of elevation. ⎛ 15t ⎞ −1 ⎛ t ⎞ Then h(t) = 15t, so (t ) = tan −1 ⎜ ⎟ = tan ⎜ ⎟ . ⎝ 60 ⎠ ⎝4⎠ d 1 ⎛1⎞ 4 = ⎜ ⎟= dt 1 + t 2 ⎝ 4 ⎠ 16 + t 2 (4) d 4 1 = = radians per second or dt 16 + 62 13 about 4.41° per second. At t = 6, 90. Let x(t) be the horizontal distance from the observer to the plane, in miles, at time t., in minutes. Let t = 0 when the distance to the plane is 3 miles. Then x(0) = 32 − 22 = 5 . The speed of the plane is 10 miles per minute, so x(t ) = 5 − 10t. The angle of ⎛ 2 ⎞ −1 ⎛ elevation is (t ) = tan −1 ⎜ ⎟ = tan ⎜ x ( t ) ⎝ ⎝ ⎠ ⎛ −2 1 d so = ⎜⎜ 2 2 dt ⎝ ( 5 − 10t ) 1 + 2 / 5 − 10t ( ( = 20 ( 5 − 10t ) 2 + 4 When t = 0, 91. )) ⎞ ⎟, 5 − 10t ⎠ ⎞ ⎟⎟ (−10) ⎠ 2 . d 20 = ≈ 2.22 radians per minute. dt 9 Let x represent the position on the shoreline and let represent the angle of the beam (x = 0 and = 0 when the light is pointed at P). Then d 1 1 dx 2 dx ⎛x⎞ = tan −1 ⎜ ⎟ , so = = 2 dt 1 + x 2 dt 4 + x 2 dt ⎝2⎠ 92 Let x represent the length of the rope and let represent the angle of depression of the rope. ⎛8⎞ Then = sin −1 ⎜ ⎟ , so ⎝x⎠ d 1 8 dx 8 dx = − =− . 2 dt 2 2 dt dt x 8 − x x 64 1− x ( ) dx = −5 , we obtain dt d 8 8 =− (−5) = . 2 dt 51 17 17 − 64 The angle of depression is increasing at a rate of 8 / 51 ≈ 0.16 radians per second. When x = 17 and 93. Let x represent the distance to the center of the earth and let represent the angle subtended by the ⎛ 6376 ⎞ earth. Then = 2sin −1 ⎜ ⎟ , so ⎝ x ⎠ d 1 ⎛ 6376 ⎞ dx =2 ⎜− 2 ⎟ 2 dt ⎝ x ⎠ dt 1 − 6376 x ( ) dx x x 2 − 63762 dt When she is 3000 km from the surface dx x = 3000 + 6376 = 9376 and = −2 . Substituting dt d ≈ 3.96 × 10−4 radians these values, we obtain dt per second. =− 12, 752 (2) When x = 1, dx d 2 = 5π, so = (5π) = 2π The beacon dt dt 4 + 12 revolves at a rate of 2π radians per minute or 1 revolution per minute. Instructor’s Resource Manual Section 6.8 397 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6.9 Concepts Review 1. e x – e– x e x + e– x ; 2 2 2. cosh 2 x − sinh 2 x = 1 3. the graph of x 2 − y 2 = 1 , a hyperbola 4. catenary; a hanging cable or chain Problem Set 6.9 1. cosh x + sinh x = = e x + e– x e x – e– x + 2 2 2e x = ex 2 2. cosh 2 x + sinh 2 x = = 2e 2 x = e2 x 2 3. cosh x – sinh x = = e 2 x + e –2 x e 2 x – e –2 x + 2 2 e x + e– x e x – e– x – 2 2 2e – x = e– x 2 4. cosh 2 x – sinh 2 x = e 2 x + e –2 x e 2 x – e –2 x 2e –2 x – = = e –2 x 2 2 2 ex – e– x e y + e– y ex + e– x e y – e– y ⋅ + ⋅ 2 2 2 2 x+ y x– y – x+ y – x– y x+ y x– y – x+ y – x– y e +e –e –e e –e +e –e = + 4 4 2e x + y – 2e –( x + y ) e x + y – e –( x + y ) = = = sinh( x + y ) 4 2 5. sinh x cosh y + cosh x sinh y = ex – e– x e y + e– y ex + e– x e y – e– y ⋅ – ⋅ 2 2 2 2 e x+ y + e x – y – e – x+ y – e – x – y e x+ y – e x – y + e – x+ y – e – x – y = – 4 4 x– y – x+ y x– y –( x – y ) 2e – 2e e –e = = = sinh( x – y ) 4 2 6. sinh x cosh y – cosh x sinh y = 398 Section 6.9 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ex + e– x e y + e– y ex – e– x e y – e– y ⋅ + ⋅ 2 2 2 2 e x+ y + e x – y + e – x+ y + e– x – y e x+ y – e x – y – e – x+ y + e – x – y = + 4 4 x+ y – x– y x+ y –( x + y ) 2e + 2e e +e = = = cosh( x + y ) 4 2 7. cosh x cosh y + sinh x sinh y = ex + e– x e y + e– y ex – e– x e y – e– y ⋅ – ⋅ 2 2 2 2 x+ y x– y – x+ y – x– y x+ y x– y – x+ y – x– y e +e +e +e e –e –e +e = – 4 4 2e x – y + 2e – x + y e x – y + e –( x – y ) = = = cosh( x – y ) 4 2 8. cosh x cosh y – sinh x sinh y = sinh y 9. sinh x + tanh x + tanh y cosh x cosh y = 1 + tanh x tanh y 1 + sinh x ⋅ sinh y cosh x cosh y sinh x cosh y + cosh x sinh y sinh( x + y ) = cosh x cosh y + sinh x sinh y cosh( x + y ) = tanh (x + y) = sinh x 16. Dx cosh 3 x = 3cosh 2 x sinh x 17. Dx cosh(3x + 1) = sinh(3 x + 1) ⋅ 3 = 3sinh(3 x + 1) 18. Dx sinh( x 2 + x) = cosh( x 2 + x) ⋅ (2 x + 1) = (2 x + 1) cosh( x 2 + x) sinh y – tanh x – tanh y cosh x cosh y 10. = 1 – tanh x tanh y 1 – sinh x ⋅ sinh y cosh x cosh y sinh x cosh y – cosh x sinh y sinh( x – y ) = = cosh x cosh y – sinh x sinh y cosh( x – y ) = tanh(x – y) 11. 2 sinh x cosh x = sinh x cosh x + cosh x sinh x = sinh (x + x) = sinh 2x 12. cosh 2 x + sinh 2 x = cosh x cosh x + sinh x sinh x = cosh( x + x) = cosh 2 x 19. Dx ln(sinh x) = 1 cosh x ⋅ cosh x = sinh x sinh x = coth x 1 (–csch 2 x) coth x sinh x 1 1 =– ⋅ =– 2 cosh x sinh x sinh x cosh x = − csch x sech x 20. Dx ln(coth x) = 21. Dx ( x 2 cosh x) = x 2 ⋅ sinh x + cosh x ⋅ 2 x = x 2 sinh x + 2 x cosh x 2 13. Dx sinh x = 2sinh x cosh x = sinh 2 x 14. Dx cosh 2 x = 2 cosh x sinh x = sinh 2 x 22. Dx ( x –2 sinh x) = x –2 ⋅ cosh x + sinh x ⋅ (–2 x –3 ) = x −2 cosh x − 2 x −3 sinh x 15. Dx (5sinh 2 x) = 10sinh x ⋅ cosh x = 5sinh 2 x 23. Dx (cosh 3x sinh x) = cosh 3x ⋅ cosh x + sinh x ⋅ sinh 3x ⋅ 3 = cosh 3x cosh x + 3sinh 3x sinh x 24. Dx (sinh x cosh 4 x) = sinh x ⋅ sinh 4 x ⋅ 4 + cosh 4 x ⋅ cosh x = 4 sinh x sinh 4x + cosh x cosh 4x 25. Dx (tanh x sinh 2 x) = tanh x ⋅ cosh 2 x ⋅ 2 + sinh 2 x ⋅ sech 2 x = 2 tanh x cosh 2 x + sinh 2 x sech 2 x 26. Dx (coth 4 x sinh x) = coth 4 x ⋅ cosh x + sinh x(–csch 2 4 x) ⋅ 4 = cosh x coth 4 x – 4sinh x csch 2 4 x Instructor’s Resource Manual Section 6.9 399 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 27. Dx sinh –1 ( x 2 ) = 2 2 (x ) +1 1 28. Dx cosh –1 ( x3 ) = ( x3 ) 2 –1 29. Dx tanh –1 (2 x – 3) = 2x ⋅ 2x = x4 + 1 1 1 – (2 x – 3) 2 1 = 1 – (4 x –12 x + 9) ⎛ 5 ⋅ − 2 ⎜ ⎝ x6 1 – ⎛⎜ 15 ⎞⎟ ⎝x ⎠ 1 (3 x) –1 1 5 2 (x ) +1 1 cosh 2 2 –1 x ⋅ = 2 2 –4 x + 12 x – 8 =– 1 2 2( x – 3 x + 2) 5x4 x10 ⎛ 5 ⎞ ⎞ ⋅⎜ − ⎟ = – ⎟ = 10 x10 –1 ⎠ x – 1 ⎝ x6 ⎠ ⋅ 3 + cosh –1 (3 x) ⋅1 = 2 32. Dx ( x 2 sinh –1 x5 ) = x 2 ⋅ 33. Dx ln(cosh –1 x) = x6 –1 ⋅2 = ⎛ 1 ⎞ 30. Dx coth –1 ( x5 ) = Dx tanh −1 ⎜ ⎟ = ⎝ x5 ⎠ 31. Dx [ x cosh –1 (3 x)] = x ⋅ 3x2 ⋅ 3x2 = 3x + cosh –1 3 x 2 9x – 1 ⋅ 5 x 4 + sinh –1 x5 ⋅ 2 x = 5 x6 10 x +1 + 2 x sinh –1 x5 38. Let u = 3x + 2, so du = 3 dx. 1 1 sinh(3 x + 2)dx = sinh u du = cosh u + C 3 3 1 = cosh(3x + 2) + C 3 1 x2 – 1 1 x 2 – 1 cosh –1 x 34. cosh –1 (cos x) does not have a derivative, since Du cosh −1 u is only defined for u > 1 while cos x ≤ 1 for all x. 35. Dx tanh(cot x ) = sech 2 (cot x ) ⋅ (– csc2 x) 39. Let u = πx 2 + 5, so du = 2πxdx . x cosh(πx 2 + 5)dx = = 1 1 sinh u + C = sinh(πx 2 + 5) + C 2π 2π = – csc2 x sech 2 (cot x) ⎛ 1 ⎞ 36. Dx coth –1 (tanh x) = Dx tanh –1 ⎜ ⎟ ⎝ tanh x ⎠ –1 = 1 1 – (coth x)2 (–csch 2 x) = –csch 2 x –csch 2 x =1 ⎡1 ⎤ ln 3 cosh 2 xdx = ⎢ sinh 2 x ⎥ 0 ⎣2 ⎦ 0 2 ln 3 –2 ln 3 0 –0 1⎛e –e e –e ⎞ – = ⎜ ⎟ ⎟ 2 ⎜⎝ 2 2 ⎠ 37. Area = = 400 40. Let u = z , so du = cosh z z = Dx tanh (coth x) ln 3 1 cosh u du 2π 1 2 z dz . dz = 2 cosh u du = 2sinh u + C = 2sinh z + C 41. Let u = 2 z1/ 4 , so du = sinh(2 z1/ 4 ) 4 3 1 1 ⋅ 2 z –3 / 4 dz = dz. 4 4 2 z3 dz = 2 sinh u du = 2 cosh u + C z = 2 cosh(2 z1/ 4 ) + C 1 ln 9 ln 19 1⎛ 1 ⎞ 20 (e − e ) = ⎜ 9 − ⎟ = 4 4⎝ 9⎠ 9 Section 6.9 Instructor's Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 48. tanh x = 0 when sinh x = 0, which is when x = 0. 42. Let u = e x , so du = e x dx . e x sinh e x dx = sinh u du = cosh u + C = cosh e x + C = cosh(sin x) + C 44. Let u = ln(cosh x), so 1 ⋅ sinh x = tanh x dx . cosh x tanh x ln(cosh x )dx = u du = u2 +C 2 sinh x 2 1 1 u2 u du = ⋅ +C 2 2 2 1 = [ln(sinh x 2 )]2 + C 4 46. Area = ln 5 – ln 5 cosh 2 x dx = 2 0 47. Note that the graphs of y = sinh x and y = 0 intersect at the origin. = ln 2 tanh x dx 8 tanh x dx = 2 1 0 0 π cosh 2 x dx = π 1 (1 + cosh 2 x)dx 2 0 π ⎛ sinh 2 ⎞ − 0⎟ ⎜1 + 2⎝ 2 ⎠ π π sinh 2 = + ≈ 4.42 2 4 50. Volume = cosh 2 x dx ⎡1 ⎤ = 2 ⎢ sinh 2 x ⎥ 2 ⎣ ⎦0 1 = sinh(2 ln 5) = (e2 ln 5 − e−2 ln 5 ) 2 1 ln 1 1⎛ 1 ⎞ = (eln 25 − e 25 ) = ⎜ 25 − ⎟ 2 2⎝ 25 ⎠ 312 = 12.48 = 25 0 0 sinh 2 x ⎤ π⎡ x+ ⎢ 2⎣ 2 ⎥⎦ 0 =π ln 5 ln 5 Area = 8 = ⋅ cosh x 2 ⋅ 2 xdx = 2 x coth x 2 dx . x coth x 2 ln(sinh x 2 )dx = 8 0 49. Volume = = 45. Let u = ln(sinh x 2 ) , so 1 −8 (− tanh x) dx + 1 1 = [ln(cosh x)]2 + C 2 du = 0 sinh x dx cosh x Let u = cosh x, so du = sinh xdx. sinh x 1 2 dx = 2 du = 2 ln u + C cosh x u 8 sinh x 8 2 dx = ⎡⎣ 2 ln cosh x ⎤⎦ 0 0 cosh x = 2(ln cosh 8 − ln1) = 2 ln(cosh 8) ≈ 14.61 =2 43. Let u = sin x, so du = cos x dx cos x sinh(sin x)dx = sinh u du = cosh u + C du = Area = 2 sinh x dx = [cosh x]ln 0 eln 2 + e− ln 2 e0 + e0 1 ⎛ 1⎞ 1 − = ⎜ 2 + ⎟ −1 = 2⎝ 2⎠ 4 2 2 =π ln10 0 ln10 ⎛ e x 0 ⎜ ⎜ ⎝ ln10 e 2 x 0 π sinh 2 xdx − e− x 2 2 ⎞ ⎟ dx ⎟ ⎠ – 2 + e –2 x π dx = 4 4 ln10 0 (e 2 x – 2 + e –2 x )dx ln10 = 1 π ⎡ 1 2x ⎤ e – 2 x – e –2 x ⎥ ⎢ 4 ⎣2 2 ⎦0 π = [e2 x – 4 x – e –2 x ]ln10 0 8 π⎛ 1 ⎞ = ⎜ 100 – 4 ln10 – ⎟ ≈ 35.65 8⎝ 100 ⎠ 51. Note that 1 + sinh 2 x = cosh 2 x and 1 + cosh 2 x cosh 2 x = 2 Surface area = = = = 1 0 1 0 1 0 1 0 ( ) dx dy 2 2πy 1 + dx 2π cosh x 1 + sinh 2 x dx 2π cosh x cosh x dx π(1 + cosh 2 x)dx 1 π π ⎡ ⎤ = ⎢ πx + sinh 2 x ⎥ = π + sinh 2 ≈ 8.84 2 2 ⎣ ⎦0 Instructor’s Resource Manual Section 6.9 401 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2 1 ⎛ dy ⎞ 2πy 1 + ⎜ ⎟ dx = 2π sinh x 1 + cosh 2 xdx 0 0 ⎝ dx ⎠ Let u = cosh x, so du = sinh x dx 1 ⎡u ⎤ 2π sinh x 1 + cosh 2 xdx = 2π 1 + u 2 du = 2π ⎢ 1 + u 2 + ln u + 1 + u 2 + C ⎥ 2 ⎣2 ⎦ 52. Surface area = 1 = π cosh x 1 + cosh 2 x + π ln cosh x + 1 + cosh 2 x + C (The integration of 1 + u 2 du is shown in Formula 44 of the Tables in the back of the text, which is covered in Chapter 8.) 1 ⎡ ⎤ 2π sinh x 1 + cosh 2 xdx = π ⎢cosh x 1 + cosh 2 x + ln cosh x + 1 + cosh 2 x ⎥ 0 ⎣ ⎦0 ⎡ ⎤ = π ⎢cosh1 1 + cosh 2 1 + ln cosh1 + 1 + cos 2 1 − 2 + ln 1 + 2 ⎥ 5.53 ⎣ ⎦ 1 ( ) ⎛x⎞ 53. y = a cosh ⎜ ⎟ + C ⎝a⎠ dy ⎛x⎞ = sinh ⎜ ⎟ dx ⎝a⎠ d2y dx 2 = 1 ⎛x⎞ cosh ⎜ ⎟ a ⎝a⎠ d2y 2 1 ⎛ dy ⎞ 1+ ⎜ ⎟ . 2 a ⎝ dx ⎠ dx ⎛x⎞ ⎛ x⎞ ⎛ x⎞ Note that 1 + sinh 2 ⎜ ⎟ = cosh 2 ⎜ ⎟ and cosh ⎜ ⎟ > 0. Therefore, ⎝a⎠ ⎝a⎠ ⎝a⎠ We need to show that = 2 2 1 1 ⎛ dy ⎞ ⎛ x⎞ 1 ⎛ x⎞ 1 ⎛x⎞ d y 1+ ⎜ ⎟ = 1 + sinh 2 ⎜ ⎟ = cosh 2 ⎜ ⎟ = cosh ⎜ ⎟ = a a ⎝ dx ⎠ ⎝a⎠ a ⎝a⎠ a ⎝ a ⎠ dx 2 54. a. ⎛ x⎞ The graph of y = b − a cosh ⎜ ⎟ is symmetric about the y-axis, so if its width along the ⎝a⎠ ⎛a⎞ x-axis is 2a, its x-intercepts are (±a, 0). Therefore, y (a ) = b − a cosh ⎜ ⎟ = 0, so b = a cosh1 ≈ 1.54308a. ⎝a⎠ b. The height is y (0) ≈ 1.54308a − a cosh 0 = 0.54308a . c. If 2a = 48, the height is about 0.54308a = (0.54308)(24) ≈ 13 . 55. a. b. Area under the curve is 24 −24 24 ⎡ ⎡ ⎛ x ⎞⎤ ⎛ x ⎞⎤ ≈ 422 ⎢37 − 24 cosh ⎜ 24 ⎟ ⎥dx = ⎢37 x − 576sinh ⎜ 24 ⎟ ⎥ ⎝ ⎠⎦ ⎝ ⎠ ⎦ −24 ⎣ ⎣ Volume is about (422)(100) = 42,200 ft3. 402 Section 6.9 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. c. Length of the curve is 24 −24 2 ⎛ dy ⎞ 1 + ⎜ ⎟ dx = ⎝ dx ⎠ 24 −24 2⎛ x ⎞ 1 + sinh ⎜ ⎟dx = ⎝ 24 ⎠ 24 ⎡ ⎛ x ⎞ ⎛ x ⎞⎤ cosh ⎜ ⎟ dx = ⎢ 24sinh ⎜ ⎟ ⎥ = 48sinh1 ≈ 56.4 −24 ⎝ 24 ⎠ ⎦ −24 ⎝ 24 ⎠ ⎣ 24 Surface area ≈ (56.4)(100) = 5640 ft 2 56. Area = 1 cosh t sinh t − 2 cosh t 1 x 2 − 1 dx = cosh t 1 1 ⎡1 ⎤ cosh t sinh t − ⎢ x x 2 − 1 − ln x + x 2 − 1 ⎥ 2 2 ⎣2 ⎦1 1 1 ⎡1 ⎤ cosh t sinh t − ⎢ cosh t cosh 2 t − 1 − ln cosh t + cosh 2 t − 1 − 0 ⎥ 2 2 ⎣2 ⎦ 1 1 1 1 t = cosh t sinh t − cosh t sinh t + ln cosh t + sinh t = ln et = 2 2 2 2 2 = 57. a. ⎛ ex – e– x ex + e– x (sinh x + cosh x) = ⎜ + ⎜ 2 2 ⎝ r sinh rx + cosh rx = b. c. ( cos x + i sin x ) cos rx + i sin rx = d. ( cos x − i sin x ) r r ⎞ ⎟ = e – rx ⎟ ⎠ erx + e – rx e rx – e – rx 2e – rx – = = e – rx 2 2 2 r ⎞ ⎛ 2eix ⎟ =⎜ ⎜ 2 ⎟ ⎝ ⎠ r ⎞ ⎟ = eirx ⎟ ⎠ eirx + e−irx eirx − e −irx 2eirx +i = = eirx 2 2i 2 ⎛ eix + e −ix eix − e −ix =⎜ −i ⎜ 2 2i ⎝ cos rx − i sin rx = 58. a. r ⎞ ⎛ 2e – x ⎟ =⎜ ⎜ 2 ⎟ ⎝ ⎠ ⎛ eix + e−ix eix − e −ix =⎜ +i ⎜ 2 2i ⎝ r r erx – e – rx e rx + e – rx 2e rx + = = e rx 2 2 2 ⎛ e x + e– x e x – e– x (cosh x – sinh x)r = ⎜ – ⎜ 2 2 ⎝ cosh rx – sinh rx = r ⎞ ⎛ 2e x ⎞ ⎟ =⎜ ⎟ = erx ⎟ ⎜ 2 ⎟ ⎠ ⎝ ⎠ r ⎞ ⎛ 2e −ix ⎟ =⎜ ⎜ 2 ⎟ ⎝ ⎠ r ⎞ ⎟ = e−irx ⎟ ⎠ eirx + e −irx eirx − e−irx 2e−irx = = e −irx −i 2 2i 2 gd (– t ) = tan –1[sinh(–t )] = tan –1 (– sinh t ) = – tan –1 (sinh t ) = − gd (t ) so gd is odd. 1 cosh t Dt [ gd (t )] = ⋅ cosh t = 2 1 + sinh t cosh 2 t = sech t > 0 for all t, so gd is increasing. Dt2 [ gd (t )] = Dt (sech t ) = −sech t tanh t Dt2 [ gd (t )] = 0 when tanh t = 0, since sech t > 0 for all t. tanh t = 0 at t = 0 and tanh t < 0 for t < 0, thus Dt2 [ gd (t )] > 0 for t < 0 and Dt2 [ gd (t )] < 0 for t > 0. Hence gd(t) has an inflection point at b. If y = tan –1 (sinh t ) then tan y = sinh t so tan y sin y = 2 tan y + 1 = sinh t sinh 2 t + 1 sinh t = tanh t so y = sin –1 (tanh t ) cosh t 1 Also, Dt y = ⋅ cosh t 1 + sinh 2 t cosh t 1 = = = sech t , 2 cosh t cosh t = so y = t 0 sech u du by the Fundamental Theorem of Calculus. (0, gd(0)) = (0, tan −1 0) = (0, 0). Instructor’s Resource Manual Section 6.9 403 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 59. Area = x 0 cosh t dt = [sinh t ]0x = sinh x 61. Arc length = x 1 + [ Dt cosh t ]2 dt = 0 = x 0 x 0 1 + sinh 2 tdt cosh t dt = [ sinh t ]0 = sinh x x 60. From Problem 54, the equation of an inverted x catenary is y = b − a cosh . Given the a information about the Gateway Arch, the curve passes through the points (±315, 0) and (0, 630). 315 Thus, b = a cosh and 630 = b – a, so a b = a + 630. 315 a + 630 = a cosh ⇒ a ≈ 128, so b ≈ 758 . a x The equation is y = 758 − 128cosh . 128 6.10 Chapter Review The functions y = sinh x and y = ln( x + x 2 + 1) are inverse functions. 62. y = gd ( x) = tan –1 (sinh x) tan y = sinh x x = gd –1 ( y ) = sinh –1 (tan y ) Thus, y = gd –1 ( x) = sinh –1 (tan x) 9. True: = 4 + ( x − 4) = x and Concepts Test 1. False: 2. True: 3. True: ln 0 is undefined. d2y dx 2 e3 1 =− 1 x2 g ( f ( x)) = ln(4 + e x − 4) = ln e x = x < 0 for all x > 0. e3 1 dt = ⎡⎣ ln t ⎤⎦ = ln e3 − ln1 = 3 t 1 4. False: The graph is intersected at most once by every horizontal line. 5. True: The range of y = ln x is the set of all real numbers. 6. False: ⎛x⎞ ln x − ln y = ln ⎜ ⎟ ⎝ y⎠ 7. False: 4 ln x = ln( x 4 ) 8. True: x +1 ln(2e Section 6.10 x ) – ln(2e ) = ln = ln e = 1 404 f ( g ( x)) = 4 + eln( x − 4) 10. False: exp( x + y ) = exp x exp y 11. True: ln x is an increasing function. 12. False: Only true for x > 1, or ln x > 0. 13. True: e z > 0 for all z. 14. True: e x is an increasing function. 15. True: lim (ln sin x − ln x) x →0 + ⎛ sin x ⎞ = lim ln ⎜ ⎟ = ln1 = 0 + ⎝ x ⎠ x →0 2e x +1 16. True: π 2 = e 2 ln π 17. False: ln π is a constant so 2e x 18. True: d ln π = 0. dx d (ln 3 x + C ) dx 1 d = (ln x + ln 3 + C ) = dx x Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 19. True: e is a number. e x > 1 and e− x < 1 < e x , thus 20. True: exp[ g ( x)] ≠ 0 because 0 is not in the e x − e− x = e x − e− x < e x = e . x range of the function y = e x . If x < 0, e − x > 1 and e x < 1 < e − x , thus 21. False: Dx ( x x ) = x x (1 + ln x) 22. True: 2 ( tan x + sec x ) '− ( tan x + sec x ) ( = 2 sec2 x + sec x tan x e x − e − x = −(e x − e − x ) 2 ) 2 2 − tan x − 2 tan x sec x − sec x 23. True: x 32. False: = sec 2 x − tan 2 x = 1 The integrating factor is e 24. True: = e− x − e x < e− x = e . ( ) 4 / x dx = e4 ln x = eln x but 4 = x4 The solution is y ( x ) = e−4 ⋅ e2x . Thus, The solution is y ( x ) = e2x , so 26. False: sin ( arcsin(2) ) is undefined 27. False: arcsin(sin 2π) = arcsin 0 = 0 28. True: sinh x is increasing. 29. False: cosh x is not increasing. 30. True: cosh(0) = 1 = e0 If x > 0, e x > 1 while e− x < 1 < e x so 1 1 cosh x = (e x + e− x ) < (2e x ) 2 2 ⎛ sin x ⎞ lim ln ⎜ ⎟ = ln1 = 0 ⎝ x ⎠ x →0 35. True: 36. False: 37. True: sinh x ≤ 1 x e is equivalent to 2 e x − e− x ≤ e . When x = 0, x sinh x = 0 < 1 0 1 e = . If x > 0, 2 2 Instructor’s Resource Manual + cosh x > 1 for x ≠ 0 , while sin −1 u is only defined for −1 ≤ u ≤ 1. sinh x ; sinh x is an odd cosh x function and cosh x is an even function. tanh x = 38. False: Both functions satisfy y − y = 0 . 39. True: ln 3100 = 100 ln 3 > 100 ⋅1 since ln 3 > 1. 40. False: ln(x – 3) is not defined for x < 3. 41. True: y triples every time t increases by t1. 42. False: x(0) = C; x 31. True: π , since 2 lim tan x = −∞ . lim tan −1 x = − x →−∞ x →− π 2 x = e− x = e . eln 3 + e − ln 3 2 1⎛ 1⎞ 5 = ⎜3+ ⎟ = 2⎝ 3⎠ 3 34. False: = e x = e . If x < 0, –x > 0 and e− x > 1 while e x < 1 < e− x so 1 1 cosh x = (e x + e− x ) < (2e− x ) 2 2 cos −1 12 cosh(ln 3) = y ' ( x ) = 2e2 x . In general, Euler’s method will underestimate the solution if the slope of the solution is increasing as it is in this case. ( )=1 ( ) 2 sin −1 12 33. False: slope = 2e −4 ⋅ e 2 x and at x = 2 the slope is 2. 25. False: ⎛1⎞ tan −1 ⎜ ⎟ ≈ 0.4636 ⎝ 2⎠ 1 C = Ce− kt when 2 1 1 = e− kt , so ln = −kt or 2 2 1 ln − ln 2 ln 2 t= 2 = = −k −k k Section 6.10 405 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 43. True: ( y (t ) + z (t )) = y (t ) + z (t ) = ky (t ) + kz (t ) = k ( y (t ) + z (t )) 44. False: Only true if C = 0; ( y1 (t ) + y2 (t )) = y1 (t ) + y2 (t ) = ky1 (t ) + C + ky2 (t ) + C = k ( y1 (t ) + y2 (t )) + 2C . 45. False: 8. 9. Use the substitution u = –h. 10. 12 ⎛ 0.06 ⎞ e0.05 ≈ 1.051 < ⎜1 + ⎟ 12 ⎠ ⎝ Sample Test Problems x4 1. ln = 4 ln x − ln 2 2 d x4 d 4 ln = (4 ln x − ln 2) = dx 2 dx x 11. 4. x2 −4 x d d 5 1 log10 ( x5 − 1) = ( x − 1) 5 dx ( x − 1) ln10 dx = 12. 6. d ln cot x d e = cot x = − csc2 x dx dx 7. 406 d d sech 2 x 2 tanh x = 2sech 2 x x= dx dx x Section 6.10 2 1− 3 1 − 3x 2 3x = ex e x e2 x − 1 = ( 3x 1 ( ) sin 2 2x 1 ( ) sin 2 2x ) 2 d 3x dx 3 3x − 9 x 2 1 d x e (e x ) 2 − 1 dx 1 e2 x − 1 ( ) = x 2 d ⎛x⎞ sin 2 ⎜ ⎟ dx ⎝2⎠ ⎛x⎞ d ⎛ x⎞ 2sin ⎜ ⎟ sin ⎜ ⎟ ⎝ 2 ⎠ dx ⎝2⎠ ⎡ ⎛ x ⎞⎤ 1 ⎛ x⎞ ⎛ x⎞ ⎢ 2sin ⎜ 2 ⎟ ⎥ 2 cos ⎜ 2 ⎟ = cot ⎜ 2 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ 13. d 3 15e5 x 3ln(e5 x + 1) = (5e5 x ) = dx e5 x + 1 e5 x + 1 14. d ln(2 x3 − 4 x + 5) dx = ( x5 − 1) ln10 d d tan(ln e x ) = tan x = sec2 x dx dx 2 = sec x sec 2 x d 1 ⎛x⎞ ln sin 2 ⎜ ⎟ = 2 2 dx ⎝ ⎠ sin = 5x4 5. sec 2 x = d sec−1 e x = dx ex = 2 d x2 −4 x d 2 e = e x −4 x ( x − 4 x) dx dx = (2 x − 4)e d tan x dx tan x + 1 2 d 2sin −1 3 x = dx = d d 2. sin 2 ( x3 ) = 2sin( x3 ) sin( x3 ) dx dx d = 2sin( x3 ) cos( x3 ) x3 = 6 x 2 sin( x3 ) cos( x3 ) dx 3. tan 2 x + 1 1 ≈ 1.062 If Dx (a x ) = a x ln a = a x , then ln a = 1, so a = e. 47. True: sec2 x u →0 by Theorem 6.5.A. 46. False: d sinh −1 (tan x) = dx = lim (1 – h) –1/ h = lim (1 + u )1 u = e h →0 d d 1 cos x tanh −1 (sin x) = sin x = 2 dx dx 1 − sin x 1 − sin 2 x cos x = = sec x cos 2 x 15. d 6 x2 − 4 (2 x3 − 4 x + 5) = 2 x3 − 4 x + 5 dx 2 x3 − 4 x + 5 1 d d cos e x = − sin e x e x dx dx d = (− sin e x )e x x dx =− e x sin e x 2 x Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 16. 17. d 1 d ln(tanh x) = tanh x dx tanh x dx 1 sech 2 x = csch x sech x = tanh x 23. d −2 d x 2 cos −1 x = dx 1 − ( x ) 2 dx = −2 1 1− x 2 x 1⎞ ⎛ = x1+ x ⎜ ln x + 1 + ⎟ x⎠ ⎝ 1 =− x − x2 24. 18. d ⎡ 3x d 4 + (3 x)4 ⎤ = (64 x + 81x 4 ) ⎣ ⎦ dx dx 20. 21. 25. Let u = 3x – 1, so du = 3 dx. 1 3 x −1 1 u 3dx = e3 x −1dx = e e du 3 3 1 1 = eu + C = e3 x −1 + C 3 3 Check: d ⎛ 1 3 x −1 d ⎞ 1 + C ⎟ = e3 x −1 (3 x − 1) = e3 x −1 ⎜ e dx ⎝ 3 dx ⎠ 3 d d 2 csc eln x = 2 csc x dx dx d = −2 csc x cot x x dx =− csc x cot x x d (log10 2 x)2 / 3 dx 2 d = (log10 2 x) −1/ 3 (log10 2 + log10 x) dx 3 2 1 = (log10 2 x) −1/ 3 3 x ln10 2 = 3 3x ln10 log10 2 x 26. Let u = sin 3x, so du = 3 cos 3x dx. 1 1 6 cot 3x dx = 2 3cos 3x dx = 2 du sin 3 x u = 2 ln u + C = 2 ln sin 3 x + C d 4 tan 5 x sec5 x dx 27. Let u = e x , so du = e x dx . = 20sec 2 5 x sec 5 x + 20 tan 5 x sec 5 x tan 5 x = 20sec 5 x(sec2 5 x + tan 2 5 x) = 20sec 5 x(2sec 2 5 x − 1) 22 d d (1 + x 2 )e = e(1 + x 2 )e −1 (1 + x 2 ) dx dx = 2 xe(1 + x 2 )e −1 = 64 x ln 64 + 324 x3 19. d 1+ x d (1+ x ) ln x x = e dx dx d = e(1+ x ) ln x [(1 + x) ln x] dx ⎡ ⎛ 1 ⎞⎤ = x1+ x ⎢(1)(ln x) + (1 + x) ⎜ ⎟ ⎥ ⎝ x ⎠⎦ ⎣ ⎛ x2 d tan −1 ⎜ ⎜ 2 dx ⎝ ⎞ 1 d ⎛ x2 ⎞ ⎟= ⎜ ⎟ 2 ⎟ ⎜ ⎟ ⎠ ⎛ x 2 ⎞ + 1 dx ⎝ 2 ⎠ ⎜ 2 ⎟ ⎝ ⎠ 4x x = = ⎛ x4 ⎞ + 1 x4 + 4 ⎜ 4 ⎟ ⎝ ⎠ ⎛x = tan −1 ⎜ ⎜ 2 ⎝ Instructor’s Resource Manual e x sin e x dx = sin u du = − cos u + C = − cos e x + C Check: d d (− cos e x + C ) = (sin e x ) e x = e x sin e x dx dx 28. Let u = x 2 + x − 5, so du = (2 x + 1)dx . 6x + 3 ⎞ 4x ⎟+ 4 ⎟ x +4 ⎠ 2 ( dx = 3 1 (2 x + 1)dx x + x −5 x + x −5 1 = 3 du = 3ln u + C = 3ln x 2 + x − 5 + C u Check: 3 d d 2 3ln x 2 + x − 5 + C = ( x + x − 5) 2 dx x + x − 5 dx 6x + 3 = 2 x + x −5 2 ⎛ x2 ⎞⎤ ⎛ x2 ⎞ ⎛ 4x ⎞ d ⎡ ⎢ x tan −1 ⎜ ⎟ ⎥ = (1) tan −1 ⎜ ⎟ + ( x) ⎜ 4 ⎟ ⎜ ⎟ ⎜ ⎟ dx ⎢⎣ ⎝ x +4⎠ ⎝ 2 ⎠ ⎥⎦ ⎝ 2 ⎠ 2 Check: d 2 d (2 ln sin 3x + C ) = sin 3 x sin 3 x dx dx 2(3cos 3x) = = 6 cot 3 x sin 3 x 2 ) Section 6.10 407 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 29. Let u = e x +3 + 1, so du = e x +3 dx . e x+2 33. Let u = ln x, so du = 1 1 1 1 e x +3 dx = du 3 x + e e e u +1 +1 e 1 ln(e x +3 + 1) = ln u + C = +C e e Check: ⎞ 1 d ⎛ ln(e x +3 + 1) 1 d x +3 +C⎟ = + 1) (e ⎜ ⎟ e e x +3 + 1 dx dx ⎜⎝ e ⎠ dx = x +3 = e x +3e−1 e x +3 + 1 = −1 2 = e x +3 + 1 2 = tanh( x − 3) + C Check: d d [tanh( x − 3)] = sech 2 ( x − 3) ( x − 3) dx dx = sech 2 ( x − 3) 31. Let u = 2x, so du = 2 dx. 4 1 dx = 2 2dx 1 − 4 x2 1 − (2 x) 2 1− u2 35. π 4 f ( x) > 0 when cos x > sin x which occurs when ⎞d ⎟ 2x ⎟ dx ⎠ = tan −1 (sin x) + C Check: d ⎡ −1 1 d tan (sin x ) + C ⎤ = sin x 2 ⎣ ⎦ dx 1 + sin x dx cos x = 1 + sin 2 x Section 6.10 π π ≤x< . 2 4 f ( x) = – sin x – cos x; f ( x) = 0 when – 32. Let u = sin x, so du = cos x dx. cos x 1 dx = du = tan −1 u + C 2 1 + sin x 1+ u2 408 f ( x) = cos x – sin x; f ( x) = 0 when tan x = 1, x= du = 2sin −1 u + C = 2sin −1 2 x + C Check: ⎛ d 1 (2sin −1 2 x + C ) = 2 ⎜ ⎜ 1 − (2 x) 2 dx ⎝ 4 = 1 − 4x 2 x + x(ln x) 2 sech 2 ( x − 3)dx = sech 2 u du = tanh u + C = 2sin u + C = 2sin x 2 + C Check: d d 2 (2sin x 2 + C ) = 2 cos x 2 x = 4 x cos x 2 dx dx =2 −1 34. Let u = x – 3, so du = dx. 2 4 x cos x dx = 2 (cos x )2 x dx = 2 cos u du 1 1 ⋅ dx 1 + (ln x) x 2 x + x(ln x) 1 =− du = − tan −1 u + C = − tan −1 (ln x ) + C 1+ u2 Check: d 1 d [− tan −1 (ln x) + C ] = − ln x 2 dx 1 + (ln x) dx e x+ 2 30. Let u = x 2 , so du = 2x dx. dx = − 1 dx . x 1 π 4 f ( x) > 0 when cos x < –sin x which occurs tan x = –1, x = – π π ≤x<– . 2 4 ⎡ π π⎤ Increasing on ⎢ – , ⎥ ⎣ 2 4⎦ ⎡π π⎤ Decreasing on ⎢ , ⎥ ⎣4 2⎦ ⎛ π π⎞ Concave up on ⎜ – , – ⎟ ⎝ 2 4⎠ ⎛ π π⎞ Concave down on ⎜ – , ⎟ ⎝ 4 2⎠ ⎛ π ⎞ Inflection point at ⎜ – , 0 ⎟ ⎝ 4 ⎠ ⎛π ⎞ Global maximum at ⎜ , 2 ⎟ ⎝4 ⎠ when – Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ⎛ π ⎞ Global minimum at ⎜ – , –1⎟ ⎝ 2 ⎠ 37. a. f ( x) = 5 x 4 + 6 x 2 + 4 ≥ 4 > 0 for all x, so f(x) is increasing. b. f(1) = 7, so g(7) = f −1 (7) = 1. c. 38. 36. f ( x) = f ( x) = 1 1 = f (1) 15 1 = e10 k 2 ln 12 k= ≈ −0.06931 10 ( ) x2 ex e x (2 x) − x 2 (e x ) x 2 = 2x − x y = 100e−0.06931t 2 1 = 100e−0.06931t x (e ) e f is increasing on [0, 2] because f ( x) > 0 on (0, 2). f is decreasing on (−∞, 0] ∪ [2, ∞) because f ( x) < 0 on (−∞, 0) ∪ (2, ∞). f ( x) = g (7) = e x (2 − 2 x) − (2 x − x 2 )e x x2 − 4x + 2 = (e x ) 2 ex Inflection points are at 4 ± 16 − 4 ⋅ 2 x= = 2± 2 . 2 The graph of f is concave up on (−∞, 2 − 2) ∪ (2 + 2, ∞) because f ( x) > 0 on these intervals. The graph of f is concave down on (2 − 2, 2 + 2) because f ( x) < 0 on this interval. The absolute minimum value is f(0) = 0. 4 The relative maximum value is f (2) = . e2 The inflection points are ⎛ ⎛ 6−4 2 ⎞ 6+4 2 ⎞ ⎜⎜ 2 − 2, ⎟⎟ and ⎜⎜ 2 + 2, ⎟⎟ . e 2− 2 ⎠ e 2+ 2 ⎠ ⎝ ⎝ t= ln ( 1001 ) ≈ 66.44 −0.06931 It will take about 66.44 years. 39. xn yn 1.0 2.0 1.2 2.4 1.4 2.976 1.6 3.80928 1.8 5.02825 2.0 6.83842 40. Let x be the horizontal distance from the airplane dx = 300. to the searchlight, dt 500 500 tan = , so = tan −1 . x x d 1 ⎛ 500 ⎞ dx = ⎜− ⎟ 2 dt 1 + 500 ⎝ x 2 ⎠ dt (x) =− 500 dx x + 250, 000 dt 2 When = 30°, x = 500 = 500 3 and tan 30° d 500 (300) =− dt (500 3)2 + (500)2 300 3 =− = − . The angle is decreasing at the 2000 20 rate of 0.15 rad/s ≈ 8.59°/s. Instructor’s Resource Manual Section 6.10 409 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 45. (Linear first-order) y + 2 xy = 2 x 41. y = (cos x)sin x = esin x ln(cos x ) dy d = esin x ln(cos x ) [sin x ln(cos x)] dx dx ⎡ ⎤ ⎛ 1 ⎞ = esin x ln(cos x ) ⎢cos x ln(cos x) + (sin x ) ⎜ ⎟ ( − sin x) ⎥ ⎝ cos x ⎠ ⎣ ⎦ 2 ⎡ sin x ⎤ = (cos x)sin x ⎢cos x ln(cos x) − ⎥ cos x ⎥⎦ ⎢⎣ dy 0 At x = 0, = 1 (1ln1 − 0) = 0 . dx The tangent line has slope 0, so it is horizontal: y = 1. 42. Let t represent the number of years since 1990. Integrating factor: e 2 2 2 Therefore, y = 1 + 2e – x . 46. Integrating factor is e – ax . D[ ye – ax ] = 1; y = e ax ( x + C ) 47. Integrating factor is e –2 x . 48. a. Q (t ) = 3 – 0.02Q b. Q (t ) + 0.02Q = 3 y (20) = 10, 000e ≈ 19, 601 The population will be about 19,600. Integrating factor is e0.02t D[Qe0.02t ] = 3e0.02t −1 Q(t ) = 150 + Ce –0.02t Q(t ) = 150 – 30e –0.02t goes through (0, 120). 44. Integrating factor is x 2 . ⎛1⎞ D[ yx 2 ] = x3 ; y = ⎜ ⎟ x 2 + Cx –2 ⎝4⎠ Review and Preview Problems u = 2x du = 2 dx 3. e3t dt = u = 3t du = 3 dt sin t cos t 5. 1 1 sin u du = − cos u + C = 2 2 ln 1 u 1 1 e du = eu + C = e3t + C 3 3 3 1 1 x sin x dx = sin u du = − cos u + C = 2 2 2 u=x 2 2 u = 3 x2 du = 6 x dx dx = sin 2 x cos x dx = u 2 du = 7. x x 2 + 2 dx = u = sin x du = cos x dx u = x2 + 2 du = 2 x dx = 1 u 1 1 2 e du = eu + C = e3 x + C 6 6 6 1 1 + C = ln + C = ln sec t + C u cos t 6. 1 − cos x 2 + C 2 xe3 x 1 du = − ln u + C = u dt = − u = cos t du = −sin t dt du = 2 x dx 4. Q → 150 g, as t → ∞ . c. 1 − cos 2 x + C 2 2. 2 y = 1 + Ce – x If x = 0, y = 3, then 3 = 1 + C, so C = 2. (0.03365)(20) sin 2 x dx = 2 2 D[ ye x ] = 2 xe x ; ye x = e x + C ; y = 10, 000e0.03365t 1. 2 = ex D[ ye –2 x ] = e – x ; y = – e x + Ce2 x 14, 000 = 10, 000e10 k ln(1.4) k= ≈ 0.03365 10 43. Integrating factor is x . D [ yx ] = 0; y = Cx 2 xdx 8. ( 1 2 x +2 3 x 2 x +1 u = x 2 +1 du = 2 x dx 1 2 u3 sin 3 x +C = +C 3 3 1 3 u du = u 2 + C 3 3 ) 2 +C dx = 1 1 1 du = ln u + C 2 u 2 = ln u + C = ln x 2 + 1 + C 410 Review and Preview Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ⎡ ⎛1⎞ ⎤ f ( x) = ⎢ x ⎜ ⎟ + (ln x)(1) ⎥ − 1 = ln x x ⎣ ⎝ ⎠ ⎦ 21. ⎡ x ⎤ −2 x + (1) arcsin x ⎥ + 10. f '( x) = ⎢ 2 ⎢⎣ 1 − x ⎥⎦ 2 1 − x 2 = arcsin x 22. 9. 11. a f ( x) = ⎡(−2 x)(cos x) + (− x 2 )(− sin x) ⎤ + ⎣ ⎦ [(2)(sin x) + (2 x)(cos x)] + [ 2(− sin x)] f ( x) = e x ( cos x + sin x ) + e x (sin x − cos x) x = 2e sin x 13. cos 2 x = 1 − 2sin 2 x ; thus sin 2 x = ⎛ 1 + cos 2 x ⎞ 15. cos 4 x = (cos 2 x)2 = ⎜ ⎟ 2 ⎝ ⎠ 24. 1 − cos 2 x 2 14. cos 2 x = 2 cos 2 x − 1 ; thus cos 2 x = 1 + cos 2 x 2 2 a 1 1 ⇒ ⎡ −e − x ⎤ = ⇒ ⎣ ⎦0 2 2 ⎡ −e − a + 1⎤ = 1 ⇒ 1 = 1 ⇒ ⎣ ⎦ 2 ea 2 a −x e 0 dx = 1 1 x − (1 − x) 2x −1 − = = 1− x x (1 − x) x x(1 − x) 7 8 7( x − 3) + 8( x + 2) + = = 5( x + 2) 5( x − 3) 5( x + 2)( x − 3) 15 x − 5 5(3 x − 1) = = 5( x + 2)( x − 3) 5( x + 2)( x − 3) (3 x − 1) ( x + 2)( x − 3) 1 1 3 25. − − + x 2( x + 1) 2( x − 3) = sin(u + v) + sin(u − v) ⇒ 2 sin(7 x) + sin(− x) sin 7 x − sin x = sin 3 x cos 4 x = 2 2 −2( x + 1)( x − 3) − x( x − 3) + 3 x( x + 1) 2 x( x + 1)( x − 3) −2( x 2 − 2 x − 3) − ( x 2 − 3 x) + (3 x 2 + 3 x) 2 x( x + 1)( x − 3) 10 x + 6 2(5 x + 3) = = = 2 x( x + 1)( x − 3) 2 x( x + 1)( x − 3) (5 x + 3) x( x + 1)( x − 3) 16. sin u cos v = cos(u + v) + cos(u − v) ⇒ 2 cos(8 x) + cos(−2 x) cos 3x cos 5 x = 2 cos8 x + cos 2 x = 2 tan 2 t = a ⋅ tan t e a = 2 ⇒ a = ln 2 23. = x 2 sin x 12. (a sec t )2 − a 2 − = a 2 (sec2 t − 1) = = 17. cos u cos v = 26. 1 1 (2000 − y ) + y 2000 + = = y 2000 − y y (2000 − y ) y (2000 − y ) cos(u − v) − cos(u + v) ⇒ 2 cos(− x) − cos(5 x) sin 2 x sin 3x = 2 cos x − cos 5 x = 2 18. sin u sin v = 19. a 2 − (a sin t )2 = a 2 (1 − sin 2 t ) = a cos 2 t = a cos t 20. a 2 + (a tan t )2 = a 2 (1 + tan 2 t ) = a sec 2 t = a sec t Instructor’s Resource Manual Review and Preview 411 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 7 CHAPTER Techniques of Integration 7.1 Concepts Review 5. 1. elementary function x +4 ex x 1 ⎛x⎞ tan –1 ⎜ ⎟ + C 2 ⎝2⎠ dx = 2+e = ln u + C 3. e x 4. = 6. u = 2 + e x , du = e x dx u 5 du 2. dx 2 2 3 u du 1 du u = ln 2 + e x + C = ln(2 + e x ) + C Problem Set 7.1 1 ( x – 2)5 dx = ( x – 2)6 + C 6 1. 3 x dx = 2. 1 3 3x ⋅ 3dx = 2 (3x)3 / 2 + C 9 3. u = x 2 + 1, du = 2 x dx When x = 0, u = 1 and when x = 2, u = 5 . 2 0 x( x 2 + 1)5 dx = 1 2 2 ( x + 1)5 (2 x dx) 2 0 1 5 5 = u du 2 1 15624 = 1302 12 4. u = 1 – x 2 , du = –2 x dx When x = 0, u = 1 and when x = 1, u = 0 . 1 0 x 1 – x 2 dx = – 1 0 1/ 2 =− u du 2 1 1 1 1/ 2 = u du 2 0 8. 2t 2 2t 2 + 1 = dt – 5 ⎡ u6 ⎤ 56 − 16 =⎢ ⎥ = 12 ⎣⎢ 12 ⎦⎥1 = 7. u = x 2 + 4, du = 2x dx x 1 du dx = 2 2 u x +4 1 = ln u + C 2 1 = ln x 2 + 4 + C 2 1 = ln( x 2 + 4) + C 2 1 2 1 0 1 − x 2 (−2 x dx) dt = 2t 2 + 1 − 1 2t 2 + 1 dt 1 dt 2t + 1 u = 2t , du = 2dt 1 1 du t– dt = t – 2 2 1 + u2 2t + 1 1 =t– tan –1 ( 2t ) + C 2 2 9. u = 4 + z 2 , du = 2z dz 6 z 4 + z 2 dz = 3 u du = 2u 3 / 2 + C = 2(4 + z 2 )3 / 2 + C 1 1 ⎡1 ⎤ = ⎢ u3 / 2 ⎥ = ⎣3 ⎦0 3 412 Section 7.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 10. u = 2t + 1, du = 2dt 5 5 du dt = 2 2t + 1 u 16. u = 1 – x , du = – 3 / 4 sin = 5 u +C = 5 2t + 1 + C tan z 11. 2 cos z 0 tan z sec2 z dz = u du 17. 12. u = cos z, du = –sin z dz 18. = − eu du = −eu + C = – ecos z + C 13. u = t , du = 1 2 t dt sin t dt = 2 sin u du t = –2 cos u + C = –2 cos t + C 14. u = x 2 , du = 2x dx 2 x dx du = 4 1– x 1 – u2 = sin –1 u + C –1 2 = sin ( x ) + C 15. u = sin x, du = cos x dx π / 4 cos x 2 / 2 du dx = 0 1 + sin 2 x 0 1 + u2 = [tan −1 u ]0 2 / 2 2 2 ≈ 0.6155 = tan −1 1/ 2 1 sin u du sin u du 1⎞ ⎛ = −2 ⎜ cos1 − cos ⎟ 2⎠ ⎝ ≈ 0.6746 u = tan z, du = sec2 z dz ecos z sin z dz = – ecos z (– sin z dz ) 1 1/ 2 dx = [−2 cos u ]11/ 2 dz = tan z sec2 z dz 1 2 u +C 2 1 = tan 2 z + C 2 dx = –2 1– x =2 = 1– x 1 2 1– x 19. 3x2 + 2 x 1 dx = (3x – 1)dx + dx x +1 x +1 3 = x 2 – x + ln x + 1 + C 2 1 x3 + 7 x dx = ( x 2 + x + 8)dx + 8 dx x –1 x –1 1 1 = x3 + x 2 + 8 x + 8ln x –1 + C 3 2 u = ln 4 x 2 , du = 2 dx x sin(ln 4 x 2 ) 1 dx = sin u du x 2 1 = – cos u + C 2 1 = – cos(ln 4 x 2 ) + C 2 20. u = ln x, du = 1 dx x sec 2 (ln x ) 1 dx = sec2 u du 2x 2 1 = tan u + C 2 1 = tan(ln x) + C 2 21. u = e x , du = e x dx 6e x 1 − e2 x dx = 6 du 1− u2 du = 6sin −1 u + C = 6sin −1 (e x ) + C Instructor’s Resource Manual Section 7.1 413 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 22. u = x 2 , du = 2x dx x 1 du dx = 4 2 4 + u2 x +4 1 u = tan −1 + C 4 2 ⎛ 1 x2 ⎞ = tan –1 ⎜ ⎟ + C ⎜ 2 ⎟ 4 ⎝ ⎠ 27. = x − ln sin x + C 23. u = 1 – e2 x , du = –2e2 x dx 3e2 x 1– e 2x dx = – 3 du 2 u = –3 u + C = –3 1 – e2 x + C 24. 25. x3 dx = 1 4 x3 dx 4 x4 + 4 x4 + 4 1 = ln x 4 + 4 + C 4 1 = ln( x 4 + 4) + C 4 1 0 2 t 3t dt = 26. 29. u = e x , du = e x dx e x sec e x dx = sec u du = ln sec e x + tan e x + C 1 1 t2 2t 3 dt 2 0 30. u = e x , du = e x dx e x sec 2 (e x )dx = sec 2 u du = tan u + C 1 π / 6 cos x 2 sin x dx 0 cos x π / 6 ⎡ 2 = ⎢– ⎣⎢ ln 2 =– 28. u = cos(4t – 1), du = –4 sin(4t – 1)dt sin(4t − 1) sin(4t − 1) dt = dt 2 1 − sin (4t − 1) cos 2 (4t − 1) 1 1 =− du 4 u2 1 1 = u −1 + C = sec(4t − 1) + C 4 4 = ln sec u + tan u + C ⎡ 3t ⎤ ⎥ = 3 – 1 =⎢ ⎢ 2 ln 3 ⎥ 2 ln 3 2 ln 3 ⎣ ⎦0 1 = ≈ 0.9102 ln 3 2 =– ⎤ ⎥ ⎦⎥ 0 1 (2 3 / 2 – 2) ln 2 2−2 3/2 ln 2 ≈ 0.2559 = sin x − cos x ⎛ cos x ⎞ dx = ⎜1 − ⎟ dx sin x ⎝ sin x ⎠ u = sin x, du = cos x dx sin x − cos x du dx = x − u sin x = x − ln u + C π / 6 cos x 2 (– sin x dx) 0 = tan(e x ) + C 31. sec3 x + esin x dx = (sec2 x + esin x cos x) dx sec x = tan x + esin x cos x dx u = sin x, du = cos x dx tan x + esin x cos x dx = tan x + eu du = tan x + eu + C = tan x + esin x + C 32. u = 3t 2 − t − 1 , 1 du = (3t 2 − t − 1)−1/ 2 (6t − 1) dt 2 (6t − 1) sin 3t 2 − t − 1 3t 2 − t − 1 = –2 cos u + C dt = 2 sin u du = −2 cos 3t 2 − t − 1 + C 414 Section 7.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 33. u = t 3 − 2 , du = 3t 2 dt 39. u = 3 y 2 , du = 6 y dy t 2 cos(t 3 − 2) y 1 cos u du 3 sin 2 u sin (t − 2) v = sin u, dv = cos u du 1 cos u 1 1 du = v −2 dv = − v −1 + C 3 sin 2 u 3 3 1 =− +C 3sin u 1 =− +C . 3sin(t 3 − 2) 2 3 34. 1 + cos 2 x 2 sin 2 x dt = 1 dx = 2 sin 2 x dx + cos 2 x sin 2 2 x = csc2 2 x dx + cot 2 x csc 2 x dx 1 1 = − cot 2 x − csc 2 x + C 2 2 35. u = t 3 − 2 , du = 3t 2 dt t 2 cos 2 (t 3 − 2) dt = 1 cos 2 u du 3 sin 2 u sin 2 (t 3 − 2) 1 1 = cot 2 u du = (csc2 u –1)du 3 3 1 = [− cot u − u ] + C1 3 1 = [− cot(t 3 − 2) − (t 3 − 2)] + C1 3 1 = − [cot(t 3 − 2) + t 3 ] + C 3 36. u = 1 + cot 2t, du = −2 csc2 2t csc2 2t 1 + cot 2t dt = − 1 2 1 u du e tan −1 ⎛ 3y2 1 = sin −1 ⎜ ⎜ 4 6 ⎝ 1 + 4t 2 dt 2t 1 u dt = e du 2 2 1 + 4t −1 1 1 = eu + C = e tan 2t + C 2 2 38. u = −t 2 − 2t − 5 , du = (–2t – 2)dt = –2(t + 1)dt 2 1 (t + 1)e−t − 2t −5 = − eu du 2 1 u 1 − t 2 − 2t − 5 = − e +C = − e +C 2 2 Instructor’s Resource Manual 1 42 − u 2 du ⎞ ⎟+C ⎟ ⎠ 40. u = 3x, du = 3 dx cosh 3 x dx dx 1 1 (cosh u )du = sinh u + C 3 3 1 = sinh 3 x + C 3 = 41. u = x3 , du = 3 x 2 dx 1 sinh u du x 2 sinh x3 dx = 3 1 = cosh u + C 3 1 = cosh x3 + C 3 42. u = 2x, du = 2 dx 5 5 dx = 2 9 − 4 x2 1 32 − u 2 du 5 ⎛u⎞ = sin −1 ⎜ ⎟ + C 2 ⎝3⎠ 5 −1 ⎛ 2 x ⎞ = sin ⎜ ⎟ + C 2 ⎝ 3 ⎠ 43. u = e3t , du = 3e3t dt 4−e 2 1 6 1 ⎛u⎞ = sin −1 ⎜ ⎟ + C 6 ⎝4⎠ e3t =− u +C = − 1 + cot 2t + C 37. u = tan −1 2t , du = 16 − 9 y 4 dy = 6t dt = 1 3 1 2 2 − u2 du 1 ⎛u⎞ = sin −1 ⎜ ⎟ + C 3 ⎝2⎠ ⎛ e3t 1 = sin −1 ⎜ ⎜ 2 3 ⎝ ⎞ ⎟+C ⎟ ⎠ 44. u = 2t, du = 2dt dt 1 1 = du 2 2t 4t − 1 2 u u 2 − 1 1 = ⎡sec −1 u ⎤ + C ⎦ 2⎣ 1 = sec −1 2t + C 2 Section 7.1 415 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 45. u = cos x, du = –sin x dx π/2 0 sin x 1 dx = − du 2 0 16 + cos x 1 16 + u 2 1 1 = du 0 16 + u 2 51. = 2(e2 x − e −2 x )dx 1 e4 + 1 1 = ln − ln 2 2 2 e2 1 2 = = x + 2x + 5 1 ( x + 1) 2 + 22 2 = = x – 4x + 9 1 ( x – 2)2 + ( 5)2 1 x – 4x + 4 + 5 = Section 7.1 dx = 1 18 x + 18 dx 2 18 9 x + 18 x + 10 ) 3– x 52. 16 + 6 x – x 2 dx = 1 2 6 – 2x 16 + 6 x – x 2 dx = 16 + 6 x – x 2 + C tan x 2 dx = sec x – 4 sin x cos x tan x dx cos x sec2 x – 4 dx 1 – 4 cos 2 x u = 2 cos x, du = –2 sin x dx sin x 1 1 dx = − du 2 2 1 − 4 cos x 1− u2 1 1 = − sin −1 u + C = – sin –1 (2 cos x) + C 2 2 55. The length is given by L= dx 2 9 x + 18 x + 9 + 1 (3x + 3)2 + 12 u = 3x + 3, du = 3 dx dx 1 du = 2 2 2 3 u + 12 (3 x + 3) + 1 416 dx d ( x – 2) 1 = tan –1 (3x + 3) + C 3 52 – ( x – 3)2 9 x + 18 x + 10 1 = ln 9 x 2 + 18 x + 10 + C 18 1 = ln 9 x 2 + 18 x + 10 + C 18 54. ⎛ x–2⎞ tan –1 ⎜ ⎟+C 5 ⎝ 5 ⎠ 9 x + 18 x + 10 dx x +1 = 2 dx = –( x – 3)2 + 52 dx 1 2 = x + 2x + 1+ 4 dx 2 d ( x + 1) dx = dx 49. 1 2 –( x – 6 x + 9 – 25) ⎛ 2t ⎞ 1 ⎟+C = sec –1 ⎜ ⎜ 3 ⎟ 3 ⎝ ⎠ 1 ⎛ x +1⎞ tan –1 ⎜ ⎟+C 2 ⎝ 2 ⎠ 1 48. dx = 2 53. u = 2t , du = 2dt dt du = 2 t 2t – 9 u u 2 – 32 1 1 1 = ln(e4 + 1) − ln(e 2 ) − ln 2 2 2 2 4 ⎛ ⎞ ⎛ ⎞ e +1 1 = ⎜ ln ⎜ ⎟ − 2 ⎟ ≈ 0.6625 2 ⎜⎝ ⎜⎝ 2 ⎟⎠ ⎟⎠ 47. dx = ( − e−2 x 1 e2 + e−2 1 dx = du 0 e 2 x + e −2 x 2 2 u 1 1 1 e2 + e −2 = ⎡⎣ ln u ⎤⎦ = ln e2 + e −2 − ln 2 2 2 2 2 2 ⎛ x –3⎞ = sin –1 ⎜ ⎟+C ⎝ 5 ⎠ 46. u = e2 x + e−2 x , du = (2e2 x − 2e−2 x )dx 1 e2 x 16 + 6 x – x = 1 ⎡1 ⎡1 ⎤ ⎛ u ⎞⎤ ⎛1⎞ 1 = ⎢ tan −1 ⎜ ⎟ ⎥ = ⎢ tan −1 ⎜ ⎟ − tan −1 0 ⎥ ⎝ 4 ⎠⎦0 ⎣ 4 ⎝4⎠ 4 ⎣4 ⎦ 1 ⎛1⎞ = tan −1 ⎜ ⎟ ≈ 0.0612 4 ⎝4⎠ dx 50. = = = 2 ⎛ dy ⎞ 1 + ⎜ ⎟ dx a ⎝ dx ⎠ b π/ 4 0 π/ 4 0 π/ 4 0 = ln 2 ⎡ 1 ⎤ 1+ ⎢ (− sin x) ⎥ dx ⎣ cos x ⎦ 1 + tan 2 x dx = π/ 4 0 sec2 x dx sec x dx = ⎡⎣ln sec x + tan x ⎤⎦ 0π / 4 2 + 1 − ln 1 = ln 2 + 1 ≈ 0.881 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 56. sec x = = 57. u = x – π , du = dx 2 π x sin x π (u + π) sin(u + π) dx = du 0 1 + cos 2 x – π 1 + cos 2 (u + π) π (u + π) sin u = du – π 1 + cos 2 u π u sin u π π sin u = du + du 2 – π 1 + cos u – π 1 + cos 2 u π u sin u du = 0 by symmetry. – π 1 + cos 2 u π π sin u π π sin u du = 2 du 2 – π 1 + cos u 0 1 + cos 2 u v = cos u, dv = –sin u du –1 π 1 1 −2 dv = 2π dv 2 1 1+ v –1 1 + v 2 ⎡ π ⎛ π ⎞⎤ = 2π[tan −1 v]1−1 = 2π ⎢ − ⎜ − ⎟ ⎥ ⎣ 4 ⎝ 4 ⎠⎦ 1 1 + sin x = cos x cos x(1 + sin x) sin x + sin 2 x + cos 2 x sin x(1 + sin x) + cos 2 x = cos x(1 + sin x) cos x(1 + sin x) sin x cos x + cos x 1 + sin x cos x ⎞ ⎛ sin x sec x = ⎜ + ⎟dx ⎝ cos x 1 + sin x ⎠ sin x cos x dx + dx = cos x 1 + sin x For the first integral use u = cos x, du = –sin x dx, and for the second integral use v = 1 + sin x, dv = cos x dx. sin x cos x du dv dx + dx = – + cos x 1 + sin x u v = – ln u + ln v + C = = – ln cos x + ln 1 + sin x + C = ln ⎛π⎞ = 2π ⎜ ⎟ = π2 ⎝2⎠ 1 + sin x +C cos x = ln sec x + tan x + C 58. V = 2π 3π 4 ⎛x+ – π ⎜⎝ 4 π⎞ ⎟ sin x – cos x dx 4⎠ π , du = dx 4 π π⎞ π⎞ π⎞ ⎛ ⎛ ⎛ V = 2π 2π ⎜ u + ⎟ sin ⎜ u + ⎟ – cos ⎜ u + ⎟ du – ⎝ 2⎠ 4⎠ 4⎠ ⎝ ⎝ 2 u= x– = 2π π 2 ⎛u + – π ⎜⎝ 2 π⎞ 2 2 2 2 sin u + cos u – cos u + sin u du ⎟ 2⎠ 2 2 2 2 π π 2 ⎛ u + π ⎞ 2 sin u du = 2 2π 2 u sin u du + ⎜ ⎟ −π ⎝ −π 2⎠ 2 2 π 2π 2π u sin u du = 0 by symmetry. Therefore, − 2 π π = 2π2 2 2 sin u du = 2 2π2 [− cos u ] 2 = 2 2π2 = 2π 2 V 0 Instructor’s Resource Manual 2π 2 π 2 −π 2 sin u du 0 Section 7.1 417 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6. u = x dv = sin 2x dx 1 du = dx v = – cos 2 x 2 1 1 x sin 2 x dx = – x cos 2 x – – cos 2 x dx 2 2 1 1 = – x cos 2 x + sin 2 x + C 2 4 7.2 Concepts Review 1. uv – v du 2. x; sin x dx 3. 1 4. reduction 7. u = t – 3 dv = cos (t – 3)dt du = dt v = sin (t – 3) (t – 3) cos(t – 3)dt = (t – 3) sin(t – 3) – sin(t – 3)dt Problem Set 7.2 1. u = x v = ex du = dx x = (t – 3) sin (t – 3) + cos (t – 3) + C dv = e x dx x x x x xe dx = xe − e dx = xe − e + C dv = e3 x dx 1 du = dx v = e3 x 3 1 1 xe3 x dx = xe3 x − e3 x dx 3 3 1 3x 1 3x = xe − e + C 3 9 2. u = x dv = e5t +π dt 1 du = dt v = e5t +π 5 1 1 5t +π te5t +π dt = te5t +π – e dt 5 5 1 1 = te5t +π – e5t +π + C 5 25 3. u = t 4. u = t + 7 dv = e2t +3 dt 1 du = dt v = e 2t + 3 2 1 1 2t + 3 (t + 7)e2t +3 dt = (t + 7)e2t +3 – e dt 2 2 1 1 = (t + 7)e2t +3 – e2t +3 + C 2 4 t 13 = e 2 t + 3 + e 2t + 3 + C 2 4 5. u = x dv = cos x dx du = dx v = sin x x cos x dx = x sin x – sin x dx = x sin x + cos x + C 418 Section 7.2 8. u = x – π dv = sin(x)dx du = dx v = –cos x ( x – π) sin( x)dx = –( x – π) cos x + cos x dx = ( π – x) cos x + sin x + C dv = t + 1 dt 9. u = t du = dt v= 2 (t + 1)3 / 2 3 2 2 t t + 1 dt = t (t + 1)3 / 2 – (t + 1)3 / 2 dt 3 3 2 4 = t (t + 1)3 / 2 – (t + 1)5 / 2 + C 3 15 dv = 3 2t + 7dt 3 v = (2t + 7)4 / 3 du = dt 8 3 3 t 3 2t + 7dt = t (2t + 7) 4 / 3 – (2t + 7)4 / 3 dt 8 8 3 9 (2t + 7)7 / 3 + C = t (2t + 7)4 / 3 – 8 112 10. u = t 11. u = ln 3x 1 du = dx x dv = dx v=x 1 ln 3x dx = x ln 3 x − x dx = x ln 3 x − x + C x 12. u = ln(7 x5 ) du = 5 dx x dv = dx v=x 5 ln(7 x5 )dx = x ln(7 x5 ) – x dx x = x ln(7 x5 ) – 5 x + C Instructor Solutions Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. 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No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 13. u = arctan x 1 du = dx 1 + x2 dv = dx 17. u = ln t 1 du = dt t v=x arctan x = x arctan x − x 1+ x 1 2x = x arctan x − dx 2 1 + x2 1 = x arctan x − ln(1 + x 2 ) + C 2 14. u = arctan 5x 5 du = dx 1 + 25 x 2 2 dx 1 e v=x 5x 1 + 25 x 2 dx 1 50 x dx = x arctan 5 x – 10 1 + 25 x 2 1 = x arctan 5 x – ln(1 + 25 x 2 ) + C 10 dv = 15. u = ln x du = 2 dv = x 2 2 3/ 2 2 ⎡4 ⎤ e ln e – ⋅1ln1 − ⎢ t 3 / 2 ⎥ 3 3 9 ⎣ ⎦1 = 2 3/ 2 4 4 2 4 e − 0 − e3 / 2 + = e3 / 2 + ≈ 1.4404 3 9 9 9 9 1 3 5 ⎡1 ⎤ 2 x ln x3 dx = ⎢ (2 x)3 2 ln x3 ⎥ − ⎣3 ⎦1 5 32 1 2 x dx 1 25 2 3 / 2 ⎛ 1 3 2 3 25 2 ⎞ = (10)3 2 ln 53 − 5 − ⎜ (2) ln1 − ⎟ ⎜3 3 3 3 ⎟⎠ ⎝ =− 4 2 32 4 2 5 + + 103 2 ln 5 ≈ 31.699 3 3 dv = z 3dz 1 v = z4 4 1 1 4 1 z 3 ln z dz = z 4 ln z − z ⋅ dz 4 4 z 1 4 1 3 = z ln z − z dz 4 4 1 1 = z 4 ln z − z 4 + C 4 16 19. u = ln z 1 du = dz z 3 1 2 x2 dx 3 5⎤ ⎡ 1 = ⎢ − ln 2 x5 − ⎥ x ⎦2 ⎣ x 5⎞ ⎛ 1 5⎞ ⎛ 1 = ⎜ − ln(2 ⋅ 35 ) − ⎟ − ⎜ − ln(2 ⋅ 25 ) − ⎟ 3⎠ ⎝ 2 2⎠ ⎝ 3 1 5 5 5 = − ln 2 − ln 3 − + 3ln 2 + 3 3 3 2 8 5 5 = ln 2 − ln 3 + ≈ 0.8507 3 3 6 Instructor's Resource Manual 5 dv = 2 xdx 1 v = (2 x)3 / 2 3 5 dx ⎡ 1 ⎤ dx = ⎢ − ln 2 x5 ⎥ + 5 ⎣ x ⎦2 18. u = ln x3 3 du = dx x 1 25 / 2 3 / 2 ⎤ = ⎡ (2 x)3 / 2 ln x3 − x 3 ⎢⎣ 3 ⎥⎦1 dx x2 1 v=− x 5 dx x 3 ln 2 x5 = 1 x2 1 1 du = dx v=– x x ln x ln x 1⎛1⎞ dx = – – – ⎜ ⎟ dx x x⎝ x⎠ x2 ln x 1 =– – +C x x 16. u = ln 2 x5 2 v = t3 / 2 3 e e 2 1/ 2 ⎡2 ⎤ t ln t dt = ⎢ t 3 / 2 ln t ⎥ – t dt ⎣3 ⎦1 1 3 e dv = dx arctan 5 x dx = x arctan 5 x – dv = t dt 20. u = arctan t 1 du = dt 1+ t2 dv = t dt 1 v = t2 2 1 1 t2 t arctan t dt = t 2 arctan t – dt 2 2 1+ t2 1 1 1+ t2 −1 = t 2 arctan t − dt 2 2 1+ t2 1 1 1 1 = t 2 arctan t − dt + dt 2 2 2 1+ t2 1 1 1 = t 2 arctan t − t + arctan t + C 2 2 2 Section 7.2 419 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ⎛1⎞ 21. u = arctan ⎜ ⎟ dv = dt ⎝t ⎠ 1 v=t du = – dt 1+ t2 t ⎛1⎞ ⎛1⎞ dt arctan ⎜ ⎟ dt = t arctan ⎜ ⎟ + ⎝t ⎠ ⎝ t ⎠ 1+ t2 ⎛1⎞ 1 = t arctan ⎜ ⎟ + ln(1 + t 2 ) + C ⎝t ⎠ 2 π/2 dv = t 5 dt 7 dt t 1 v = t6 6 du = 1 7 5 t 5 ln(t 7 )dt = t 6 ln(t 7 ) – t dt 6 6 1 7 = t 6 ln(t 7 ) – t 6 + C 6 36 dv = csc2 x dx v = − cot x 23. u = x du = dx π/6 22. u = ln(t 7 ) π/ 2 x csc2 x dx = [ − x cot x ]π / 6 + π/ 2 π/6 cot x dx = ⎣⎡− x cot x + ln sin x ⎤⎦ π2 π6 π π 1 π = − ⋅ 0 + ln1 + 3 − ln = + ln 2 ≈ 1.60 2 6 2 2 3 dv = sec2 x dx v = tan x 24. u = x du = dx π4 π6 = π4 π4 x sec2 x dx = [ x tan x ]π 6 − π6 π 2 ⎛ π 3⎞ π4 = + ln − ⎜⎜ + ln tan x dx = ⎡⎣ x tan x + ln cos x ⎤⎦ ⎟ π6 4 2 ⎝6 3 2 ⎟⎠ π π 1 2 − + ln ≈ 0.28 4 6 3 2 3 25. u = x3 dv = x 2 x3 + 4dx 2 v = ( x3 + 4)3 / 2 du = 3 x 2 dx 9 2 2 2 3 2 4 x5 x3 + 4dx = x3 ( x3 + 4)3 / 2 – x ( x + 4)3 / 2 dx = x3 ( x3 + 4)3 / 2 – ( x3 + 4)5 / 2 + C 9 3 9 45 26. u = x7 dv = x 6 x 7 + 1 dx du = 7 x 6 dx v= x13 x7 + 1dx = 2 7 7 2 6 7 2 4 7 ( x + 1)5 / 2 + C x ( x + 1)3 / 2 – x ( x + 1)3 / 2 dx = x 7 ( x 7 + 1)3 / 2 – 21 3 21 105 4 27. u = t dv = du = 4t 3 dt t7 (7 – 3t 4 )3 / 2 420 2 7 ( x + 1)3 / 2 21 v= dt = Section 7.2 t3 (7 – 3t 4 )3 / 2 1 dt 6(7 – 3t 4 )1/ 2 t4 6(7 – 3t 4 )1/ 2 – t3 t4 2 1 dt = + (7 – 3t 4 )1/ 2 + C 4 1/ 2 9 3 (7 – 3t 4 )1/ 2 6(7 – 3t ) Instructor Solutions Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 28. u = x 2 dv = x 4 – x 2 dx 1 du = 2x dx v = – (4 – x 2 )3 / 2 3 1 2 1 2 x3 4 – x 2 dx = – x 2 (4 – x 2 )3 / 2 + x(4 – x 2 )3 / 2 dx = – x 2 (4 – x 2 )3 / 2 – (4 – x 2 )5 / 2 + C 3 3 3 15 29. u = z 4 dv = du = 4 z 3 dz z7 4 2 (4 – z ) v= dz = z3 (4 – z 4 ) 2 dz 1 4(4 – z 4 ) z4 4 4(4 – z ) − z3 4– z 4 dz = z4 1 + ln 4 – z 4 + C 4(4 – z ) 4 4 30. u = x dv = cosh x dx du = dx v = sinh x x cosh x dx = x sinh x – sinh x dx = x sinh x – cosh x + C 31. u = x dv = sinh x dx du = dx v = cosh x x sinh x dx = x cosh x – cosh x dx = x cosh x – sinh x + C 32. u = ln x dv = x –1/ 2 dx 1 du = dx v = 2 x1/ 2 x ln x 1 dx = 2 x ln x – 2 dx = 2 x ln x – 4 x + C 1/ x x 2 33. u = x dv = (3 x + 10)49 dx 1 (3x + 10)50 150 x 1 x 1 (3 x + 10)50 – (3 x + 10)50 dx = x(3x + 10) 49 dx = (3x + 10)50 – (3 x + 10)51 + C 150 150 150 22,950 du = dx 34. u = t du = dt v= dv = ( t − 1) dt 12 v= 1 ( t − 1)13 13 1 1 1 ⎡t 13 ⎤ t (t − 1)12 dt = ⎢ ( t − 1) ⎥ − ( t − 1)13 dt 0 ⎣13 ⎦ 0 13 0 1 1 1 1 ⎡t 13 = ⎢ ( t − 1) − ( t − 1)14 ⎤⎥ = 13 182 182 ⎣ ⎦0 x dv = 2 dx 1 x du = dx v= 2 ln 2 x x 1 x 2 x dx = 2 – 2 x dx ln 2 ln 2 x x 1 = 2 – 2x + C 2 ln 2 (ln 2) 35. u = x Instructor's Resource Manual dv = a z dz 1 z du = dz v= a ln a z z 1 za z dz = a – a z dz ln a ln a z z 1 a – az + C = 2 ln a (ln a) 36. u = z 37. u = x 2 du = 2x dx dv = e x dx v = ex x 2 e x dx = x 2 e x − 2 xe x dx u=x dv = e x dx du = dx v = ex ( x 2 e x dx = x 2 e x − 2 xe x − e x dx ) = x 2 e x − 2 xe x + 2e x + C Section 7.2 421 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 41. u = et 2 38. u = x 4 dv = xe x dx 1 2 v = ex 2 du = 4 x3 dx du = et dt e cos t dt = e sin t − et sin t dt 2 2 1 x e dx = x 4 e x – 2 x3e x dx 2 du = e dt v = –cos t e cos t dt = e sin t − ⎡ −et cos t + et cos t dt ⎤ ⎣ ⎦ t 2 v = ex 2 2 ⎛ 2 2 1 ⎞ x5 e x dx = x 4 e x – ⎜ x 2 e x − 2 xe x dx ⎟ 2 ⎝ ⎠ 2 2 1 4 x2 = x e – x2e x + e x + C 2 2 et cos t dt = et sin t + et cos t + C et cos t dt = v=z du = ae dt at du = ae dt at v = sin t ( e sin t dt = – e cos t + a eat sin t – a e at sin t dt ) at ) eat sin t dt = – e at cos t + aeat sin t – a 2 e at sin t dt = z ln 2 z – 2 z ln z + 2 z + C (1 + a 2 ) e at sin t dt = – e at cos t + ae at sin t + C dv = dx eat sin t dt = v=x ln 2 x 20 dx = x ln 2 x 20 – 40 ln x 20 dx u = ln x 20 20 du = dx x dv = cos t dt at ( 40 ln x 20 dx x at u = e at ln z dz = z ln z – 2 z ln z – dz du = v = –cos t e sin t dt = – e cos t + a e at cos t dt v=z 40. u = ln 2 x 20 dv = sin t dt at dv = dz 2 1 t e (sin t + cos t ) + C 2 42. u = e at ln 2 z dz = z ln 2 z – 2 ln z dz 2 t et cos t dt = et sin t + et cos t − et cos t dt dv = dz u = ln z 1 du = dz z dv = sin t dt t 2 39. u = ln 2 z 2 ln z du = dz z t u = et dv = 2 xe x dx du = 2x dx v = sin t t 5 x2 u = x2 dv = cos t dt 43. u = x 2 du = 2 x dx dv = dx v=x – e at cos t a2 + 1 + aeat sin t a2 + 1 +C dv = cos x dx v = sin x 2 ( ln 2 x 20 dx = x ln 2 x 20 – 40 x ln x 20 – 20 dx ) = x ln 2 x 20 – 40 x ln x 20 + 800 x + C x cos x dx = x 2 sin x − 2 x sin x dx u = 2x du = 2dx dv = sin x dx v = − cos x ( x 2 cos x dx = x 2 sin x − −2 x cos x + 2 cos x dx ) 2 = x sin x + 2 x cos x − 2sin x + C 44. u = r 2 du = 2r dr dv = sin r dr v = –cos r r 2 sin r dr = – r 2 cos r + 2 r cos r dr u=r du = dr dv = cos r dr v = sin r ( ) r 2 sin r dr = – r 2 cos r + 2 r sin r – sin r dr = – r 2 cos r + 2r sin r + 2 cos r + C 422 Section 7.2 Instructor Solutions Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 45. u = sin(ln x) dv = dx 1 du = cos(ln x) ⋅ dx x v=x sin(ln x)dx = x sin(ln x) − cos(ln x) dx u = cos (ln x) dv = dx 1 du = − sin(ln x) ⋅ dx x v=x sin(ln x)dx = x sin(ln x) − ⎡⎣ x cos(ln x) − − sin(ln x )dx ⎤⎦ sin(ln x)dx = x sin(ln x) − x cos(ln x) − sin(ln x) dx 2 sin(ln x)dx = x sin(ln x) − x cos(ln x) + C x sin(ln x)dx = [sin(ln x) − cos(ln x)] + C 2 46. u = cos(ln x) dv = dx 1 du = – sin(ln x) dx x v=x cos(ln x )dx = x cos(ln x) + sin(ln x)dx u = sin(ln x) dv = dx 1 du = cos(ln x) dx x v=x cos(ln x)dx = x cos(ln x) + ⎡⎣ x sin(ln x) – cos(ln x) dx ⎤⎦ 2 cos(ln x)dx = x[cos(ln x ) + sin(ln x )] + C x cos(ln x)dx = [cos(ln x ) + sin(ln x)] + C 2 47. u = (ln x)3 du = 3ln 2 x dx x dv = dx v=x (ln x)3 dx = x(ln x)3 – 3 ln 2 x dx = x ln 3 x – 3( x ln 2 x – 2 x ln x + 2 x + C ) = x ln 3 x – 3x ln 2 x + 6 x ln x − 6 x + C 48. u = (ln x)4 du = 4 ln 3 x dx x dv = dx v=x (ln x) 4 dx = x(ln x )4 – 4 ln 3 x dx = x ln 4 x – 4( x ln 3 x – 3 x ln 2 x + 6 x ln x − 6 x + C ) = x ln 4 x – 4 x ln 3 x + 12 x ln 2 x − 24 x ln x + 24 x + C 49. u = sin x dv = sin(3x)dx 1 du = cos x dx v = – cos(3x) 3 1 1 sin x sin(3 x) dx = – sin x cos(3 x) + cos x cos(3 x) dx 3 3 u = cos x dv = cos(3x)dx 1 v = sin(3 x) du = –sin x dx 3 Instructor's Resource Manual Section 7.2 423 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 1 ⎡1 1 ⎤ sin x sin(3 x) dx = – sin x cos(3 x) + ⎢ cos x sin(3 x) + sin x sin(3 x) dx ⎥ 3 3 ⎣3 3 ⎦ 1 1 1 = – sin x cos(3 x) + cos x sin(3 x) + sin x sin(3 x)dx 3 9 9 8 1 1 sin x sin(3 x) dx = – sin x cos(3 x) + cos x sin(3 x) + C 9 3 9 3 1 sin x sin(3 x) dx = – sin x cos(3x) + cos x sin(3x) + C 8 8 50. u = cos (5x) dv = sin(7x)dx 1 du = –5 sin(5x)dx v = – cos(7 x) 7 1 5 cos(5 x) sin(7 x)dx = – cos(5 x ) cos(7 x) – sin(5 x ) cos(7 x )dx 7 7 u = sin(5x) dv = cos(7x)dx 1 v = sin(7 x) du = 5 cos(5x)dx 7 1 5 ⎡1 5 ⎤ cos(5 x) sin(7 x)dx = – cos(5 x) cos(7 x) – ⎢ sin(5 x ) sin(7 x ) – cos(5 x ) sin(7 x )dx ⎥ 7 7 ⎣7 7 ⎦ 1 5 25 = – cos(5 x) cos(7 x) – sin(5 x) sin(7 x) + cos(5 x) sin(7 x)dx 7 49 49 24 1 5 cos(5 x ) sin(7 x )dx = – cos(5 x ) cos(7 x ) – sin(5 x ) sin(7 x ) + C 49 7 49 7 5 cos(5 x) sin(7 x)dx = – cos(5 x) cos(7 x) – sin(5 x) sin(7 x) + C 24 24 51. u = e z du = e z dz dv = sin z dz 1 v = – cos z 1 e z sin z dz = – e z cos z + u=e z du = e z dz dv = cos z dz 1 v = sin z 1 e z sin z dz = − e z cos z + =– 2 1 e z cos z + + 2 2 2 e z sin z – ⎡1 z ⎢ e sin z − ⎣ 2 2 Section 7.2 – 2 + 2 e z cos z + ⎤ e z sin z dz ⎥ ⎦ e z sin z dz 1 e z sin z dz = – e z cos z + e z sin z dz = 424 e z cos z dz 2 e z sin z + C e z ( sin z – cos z ) e z sin z + C = +C 2 2 + 2 + 2 Instructor Solutions Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. z 52. u = e dv = cos z dz du = e z dz 1 v= sin z 1 e z cos z dz = e z sin z – u=e z dv = sin z dz 1 v = – cos z du = e z dz z e = 1 2 cos z dz = e z sin z + + 2 2 e z sin z dz 1 e 2 z sin z − ⎡ 1 ⎢− e ⎣ 2 e z cos z – e z cos z dz = 2 2 z cos z + e z ⎤ cos z dz ⎥ ⎦ e z cos z dz 1 e z cos z + e z sin z + C e z ( cos z + sin z ) e z cos z dz = +C 2 + 2 dv = x dx 53. u = ln x du = 1 dx x x ln x dx = v= ≠ –1 dv = x dx 2 ln x dx x x (ln x)2 dx = = ≠ –1 x +1 x +1 x +1 1 ln x – x dx = ln x – + C, +1 +1 +1 ( + 1)2 54. u = (ln x)2 du = x +1 , +1 v= x +1 , +1 ≠ –1 x +1 2 x +1 x +1 ⎤ 2 ⎡ x +1 (ln x)2 – x ln x dx = (ln x)2 − ln x − ⎢ ⎥+C +1 +1 +1 + 1 ⎣⎢ + 1 ( + 1) 2 ⎦⎥ x +1 x +1 x +1 (ln x) 2 – 2 ln x + 2 + C, +1 ( + 1)2 ( + 1)3 Problem 53 was used for 55. u = x du = x –1dx x ln x dx. dv = e x dx v= 1 x e x x e x dx = – 56. u = x du = x –1dx x sin x dx = – ≠ –1 e x x –1e x dx dv = sin x dx 1 v = – cos x x cos x Instructor's Resource Manual + x –1 cos x dx Section 7.2 425 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 57. u = x dv = cos x dx 1 v = sin x du = x –1dx x sin x x cos x dx = x –1 sin x dx – 58. u = (ln x) du = dv = dx –1 (ln x) x v=x dx (ln x) –1 dx (ln x) dx = x(ln x) – 59. u = (a 2 – x 2 ) dv = dx 2 2 du = –2 x(a – x ) –1 v=x dx (a 2 – x 2 ) dx = x (a 2 – x 2 ) + 2 60. u = cos –1 x 2 ( a 2 – x 2 ) –1 dx dv = cos x dx x du = –( – 1) cos –2 v = sin x x sin x dx cos x dx = cos –1 x sin x + ( – 1) cos –2 x sin 2 x dx = cos −1 x sin x + ( − 1) cos − 2 x(1 − cos 2 x) dx = cos –1 x sin x + ( – 1) cos –2 x dx – ( – 1) cos x dx −1 cos x dx = cos cos cos x dx = 61. u = cos –1 –1 x sin x = = cos –1 x sin x x= + ( – 1) cos –2 1 sin x x sin 2 + ( − 1) cos − 2 x(1 − cos 2 + ( – 1) cos –2 x dx – ( – 1) cos cos −1 x sin x x dx = v= + ( − 1) cos − 2 cos –1 x sin x + –1 cos –2 x dx x) dx x dx x dx x dx 1 4 3x 4 3 3x 1 4 ⎡1 ⎤ x e – x e dx = x 4 e3 x – ⎢ x3e3 x – x 2 e3 x dx ⎥ 3 3 3 3 ⎣3 ⎦ 4 4 ⎡1 2 1 4 4 8 ⎡1 1 3x ⎤ ⎤ – x 3 e3 x + ⎢ x 2 e3 x – xe3 x dx ⎥ = x 4 e3 x – x3e3 x + x 2 e3 x – ⎢ xe3 x – e dx ⎥ 9 3 ⎣3 3 3 9 9 9 3 3 ⎣ ⎦ ⎦ 4 3 3x 4 2 3x 8 3x 8 3x – x e + x e – xe + e + C 9 9 27 81 x 4 e3 x dx = 62. 1 4 3x x e 3 1 = x 4 e3 x 3 = 426 x dx cos –2 x dx x sin x dx cos –1 x sin x cos −1 x sin x cos –1 −2 dv = cos x dx x dx = cos + x du = – ( – 1) cos –2 cos x sin x + ( − 1) cos Section 7.2 Instructor Solutions Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 63. 64. 1 4 4 3 1 4⎡ 1 ⎤ x sin 3 x – x sin 3 x dx = x 4 sin 3 x – ⎢ – x3 cos 3x + x 2 cos 3 x dx ⎥ 3 3 3 3⎣ 3 ⎦ 1 4 4 ⎡1 2 ⎤ = x 4 sin 3 x + x3 cos 3 x – ⎢ x 2 sin 3x − x sin 3 x dx ⎥ 3 9 3 ⎣3 3 ⎦ 1 4 4 3 4 2 8⎡ 1 1 ⎤ = x sin 3 x + x cos 3 x – x sin 3 x + ⎢ – x cos 3 x + cos 3 x dx ⎥ 3 9 9 9⎣ 3 3 ⎦ 1 4 4 3 4 2 8 8 = x sin 3 x + x cos 3 x – x sin 3 x – x cos 3 x + sin 3 x + C 3 9 9 27 81 x 4 cos 3 x dx = 1 5 1 5⎡ 1 3 ⎤ cos5 3 x sin 3 x + cos 4 3x dx = cos5 3x sin 3x + ⎢ cos3 3x sin 3x + cos 2 3 x dx ⎥ 18 6 ⎣12 4 18 6 ⎦ 1 5 5 1 1 ⎡ ⎤ dx ⎥ = cos5 3 x sin 3 x + cos3 3 x sin 3 x + ⎢ cos 3 x sin 3 x + 18 72 8 ⎣6 2 ⎦ x 1 5 5 5 = cos5 3x sin 3x + cos3 3x sin 3x + cos 3x sin 3x + + C 18 72 48 16 cos6 3 x dx = 65. First make a sketch. From the sketch, the area is given by e 1 ln x dx u = ln x 1 du = dx x e 1 dv = dx v=x ln x dx = [ x ln x ]1 − e 66. V = e 1 e 1 dx = [ x ln x − x]1e = (e – e) – (1 · 0 – 1) = 1 π(ln x) 2 dx u = (ln x)2 du = 2 ln x dx x dv = dx v=x e e e ⎛ ⎞ (ln x) 2 dx = π ⎜ ⎡ x(ln x) 2 ⎤ − 2 ln x dx ⎟ = π ⎡ x(ln x)2 − 2( x ln x − x) ⎤ = π[ x (ln x) 2 − 2 x ln x + 2 x]1e ⎦1 1 1 ⎣ ⎦1 ⎝⎣ ⎠ = π[(e − 2e + 2e) − (0 − 0 + 2)] = π(e − 2) ≈ 2.26 π e Instructor's Resource Manual Section 7.2 427 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 67. 9 0 9 –x /3 ⎛ 3e – x / 3 dx = –9 68. V = 9 0 0 e 1 ⎞ 9 –x /3 9 ]0 = – + 9 ≈ 8.55 ⎜ – dx ⎟ = –9[e 3 ⎝ ⎠ e3 π(3e – x / 3 ) 2 dx = 9π 9 –2 x / 3 e dx 0 27π –2 x / 3 9 27π 27 π ⎛ 3⎞ 9 ⎛ 2 ⎞ = 9π ⎜ – ⎟ e –2 x / 3 ⎜ – dx ⎟ = – [e ]0 = – + ≈ 42.31 0 2 3 2 2 ⎝ ⎠ ⎝ ⎠ 2e 6 69. π/4 0 ( x cos x – x sin x)dx = π/ 4 0 x cos x dx – π/4 0 x sin x dx π4 π4 π4 ⎛ ⎞ π4 cos x dx ⎞⎟ sin x dx ⎟ − ⎛⎜ [ − x cos x ]0 + = ⎜ ⎣⎡ x sin x ⎦⎤ 0 − 0 0 ⎠ ⎝ ⎠ ⎝ 2π –1 ≈ 0.11 4 x sin x dx and x cos x dx. = [ x sin x + cos x + x cos x – sin x]0π / 4 = Use Problems 60 and 61 for ⎛ x⎞ x sin ⎜ ⎟ dx ⎝2⎠ x u=x dv = sin dx 2 x du = dx v = –2 cos 2 2π 2π ⎛⎡ ⎛ 2π x⎤ x ⎞ x⎤ ⎞ ⎡ V = 2π ⎜ ⎢ –2 x cos ⎥ + 2 cos dx ⎟ = 2π ⎜ 4π + ⎢ 4sin ⎥ ⎟ = 8π2 0 ⎜⎣ ⎜ 2 ⎦0 2 ⎟ 2 ⎦0 ⎟ ⎣ ⎝ ⎠ ⎝ ⎠ 70. V = 2π 71. e 1 2π 0 ln x 2 dx = 2 u = ln x 1 du = dx x e 2 1 e 1 e 1 ln x dx dv = dx v=x ⎛ ⎝ ln x dx = 2 ⎜ [ x ln x]1e − x ln x 2 dx = 2 e 1 e 1 ⎞ ⎠ ( ) dx ⎟ = 2 e − [ x]1e = 2 x ln x dx u = ln x 1 du = dx x dv = x dx 1 v = x2 2 e ⎛ e ⎡1 ⎤ 2 x ln x dx = 2 ⎜ ⎢ x 2 ln x ⎥ – 1 ⎜ ⎣2 ⎦1 ⎝ 428 Section 7.2 e ⎞ ⎛1 ⎡1 ⎤ ⎞ 1 x dx ⎟ = 2 ⎜ e2 – ⎢ x 2 ⎥ ⎟ = (e2 + 1) 1 2 ⎟ ⎜2 ⎣ 4 ⎦1 ⎟⎠ 2 ⎠ ⎝ e1 Instructor Solutions Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 e (ln x)2 dx 2 1 u = (ln x)2 du = 1 2 e 1 x= y= 72. a. dv = dx 2 ln x dx x v=x 1 (ln x)2 dx = ⎛⎜ [ x(ln x) 2 ]1e – 2 2⎝ 1 (e 2 2 + 1) = 2 1 (e – 2) 2 2 = e 1 1 ln x dx ⎞⎟ = (e – 2) ⎠ 2 e2 + 1 4 e–2 4 u = cot x dv = csc2 x dx du = – csc2 x dx v = –cot x cot x csc2 x dx = − cot 2 x − cot x csc2 x dx 2 cot x csc2 x dx = − cot 2 x + C 1 cot x csc2 x dx = − cot 2 x + C 2 b. u = csc x du = –cot x csc x dx dv = cot x csc x dx v = –csc x cot x csc2 x dx = − csc2 x − cot x csc2 x dx 2 cot x csc2 x dx = − csc2 x + C 1 cot x csc2 x dx = − csc 2 x + C 2 c. 73. a. 1 1 1 1 – cot 2 x = – (csc2 x –1) = – csc2 x + 2 2 2 2 p ( x ) = x3 − 2 x g ( x) = e x All antiderivatives of g ( x) = e x ( x3 − 2 x)e x dx = ( x3 − 2 x)e x − (3x 2 − 2)e x + 6 xe x − 6e x + C b. p( x) = x 2 − 3x + 1 g(x) = sin x G1 ( x) = − cos x G2 ( x) = − sin x G3 ( x) = cos x ( x 2 − 3 x + 1) sin x dx = ( x 2 − 3 x + 1)(− cos x) − (2 x − 3)(− sin x) + 2 cos x + C Instructor's Resource Manual Section 7.2 429 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 74. a. We note that the nth arch extends from x = 2π (n − 1) to x = π (2n − 1) , so the area of the nth arch is A(n) = π (2 n −1) 2π ( n −1) u=x x sin x dx . Using integration by parts: dv = sin x dx du = dx v = − cos x π (2 n −1) π (2 n −1) π (2 n −1) 1) 1) A(n) = 2ππ(2( nn−−1) x sin x dx = − x cos x 2π ( n −1) − 2ππ(2( nn−−1) − cos x dx = − x cos x 2π ( n −1) + sin x 2π ( n −1) [ ] [ ] [ ] = [ −π (2n − 1) cos(π (2n − 1)) + 2π (n − 1) cos(2π (n − 1)) ] + [sin(π (2n − 1)) − sin(2π ( n − 1)) ] = −π (2n − 1)(−1) + 2π (n − 1)(1) + 0 − 0 = π [ (2n − 1) + (2n − 2) ] . b. V = 2π So A(n) = (4n − 3)π 3π 2 2π x sin x dx u = x2 dv = sin x dx du = 2x dx v = –cos x 3π 3π 3π ⎛ ⎞ ⎛ ⎞ V = 2π ⎜ ⎡ – x 2 cos x ⎤ + 2 x cos x dx ⎟ = 2π ⎜ 9π2 + 4π2 + 2 x cos x dx ⎟ ⎣ ⎦ π 2 π 2 2π ⎝ ⎠ ⎝ ⎠ u = 2x dv = cos x dx du = 2 dx v = sin x 3π ⎛ ⎞ V = 2π ⎜13π2 + [2 x sin x]32ππ – 2sin x ⎟ 2π ⎝ ⎠ ( ) = 2π 13π2 + [2 cos x]32ππ = 2π(13π2 – 4) ≈ 781 dv = sin nx dx 75. u = f(x) 1 v = − cos nx n du = f ( x)dx an = π ⎤ 1⎡ ⎡ 1 1 π ⎤ cos( ) ( ) cos(nx) f ( x)dx ⎥ nx f x − + ⎢ ⎢ ⎥ π⎣ ⎣ n ⎦ −π n −π ⎦ Term 2 Term 1 Term 1 = 1 1 cos(nπ)( f (−π) − f (π)) = ± ( f (−π) − f (π)) n n Since f ( x) is continuous on [– ∞ , ∞ ], it is bounded. Thus, π –π cos(nx) f ( x)dx is bounded so π 1 ⎡ cos(nx) f ( x) dx ⎥⎤ = 0. ± ( f (−π) − f (π)) + ⎢ −π n→∞ πn ⎣ ⎦ lim an = lim n →∞ Gn [(n + 1)(n + 2) ⋅⋅⋅ (n + n)]1 n ⎡⎛ 1 ⎞⎛ 2 ⎞ = 76. = ⎢⎜1 + ⎟⎜ 1 + ⎟ n [n n ]1 n ⎣⎝ n ⎠⎝ n ⎠ ⎛ G ⎞ 1 ⎡⎛ 1 ⎞⎛ 2 ⎞ ln ⎜ n ⎟ = ln ⎢⎜1 + ⎟⎜ 1 + ⎟ ⎝ n ⎠ n ⎣⎝ n ⎠⎝ n ⎠ = ⎛ n ⎞⎤ ⎜1 + ⎟ ⎥ ⎝ n ⎠⎦ 1⎡ ⎛ 1⎞ ⎛ 2⎞ ⎛ n ⎞⎤ ln ⎜1 + ⎟ + ln ⎜ 1 + ⎟ + ⋅⋅⋅ + ln ⎜1 + ⎟ ⎥ n ⎢⎣ ⎝ n ⎠ n ⎝ ⎠ ⎝ n ⎠⎦ ⎛G lim ln ⎜ n n →∞ ⎝ n ⎛G lim ⎜ n n →∞ ⎝ n 430 1/ n ⎛ n ⎞⎤ ⎜1 + ⎟ ⎥ ⎝ n ⎠⎦ 2 ⎞ ⎟ = 1 ln x dx = 2 ln 2 – 1 ⎠ 4 ⎞ 2 ln 2–1 = 4e –1 = ⎟=e e ⎠ Section 7.2 Instructor Solutions Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 77. The proof fails to consider the constants when integrating 1t . The symbol (1 t ) dt is a family of functions, all of who whom have derivative 1t . We know that any two of these (1 t ) dt = (1 t ) dt + 1 functions will differ by a constant, so it is perfectly correct (notationally) to write 78. d 5x [e (C1 cos 7 x + C2 sin 7 x) + C3 ] = 5e5 x (C1 cos 7 x + C2 sin 7 x) + e5 x (–7C1 sin 7 x + 7C2 cos 7 x) dx = e5 x [(5C1 + 7C2 ) cos 7 x + (5C2 – 7C1 ) sin 7 x] Thus, 5C1 + 7C2 = 4 and 5C2 – 7C1 = 6. Solving, C1 = – 11 29 ; C2 = 37 37 79. u = f(x) du = f ( x)dx b a dv = dx v=x f ( x)dx = [ xf ( x)]a – b b a xf ( x)dx Starting with the same integral, u = f(x) dv = dx du = f ( x)dx v=x–a b a b f ( x) dx = [ ( x – a) f ( x) ]a – b 80. u = f ( x) du = f ( x)dx f (b) – f (a ) = a ( x – a) f ( x)dx dv = dx v=x–a b a f ( x)dx = [ ( x – a) f ( x) ]a – b b a ( x – a ) f ( x)dx = f (b)(b – a ) – b a ( x – a ) f ( x)dx Starting with the same integral, u = f ( x) dv = dx du = f ( x)dx v=x–b f (b) − f (a) = b a f ( x)dx = [ ( x – b) f ( x) ]a – b b a ( x – b) f ( x)dx = f (a)(b − a) – b a ( x – b) f ( x)dx 81. Use proof by induction. n = 1: f (a) + f (a )(t – a) + t a (t – x) f ( x )dx = f (a ) + f (a )(t – a ) + [ f ( x)(t – x)]ta + t a f ( x)dx = f (a) + f (a)(t – a) – f (a)(t – a) + [ f ( x)]ta = f (t ) Thus, the statement is true for n = 1. Note that integration by parts was used with u = (t – x), dv = f ( x)dx. Suppose the statement is true for n. n t (t – x ) n ( n +1) f (i ) ( a ) (t – a)i + ( x)dx f (t ) = f (a ) + f a i! n! i =1 Integrate (t – x)n ( n +1) ( x)dx by parts. f a n! t u = f ( n +1) ( x) dv = (t – x)n dx n! du = f ( n + 2) ( x ) v=– (t – x)n +1 (n + 1)! t ⎡ (t – x)n +1 ( n +1) ⎤ t (t – x) n +1 ( n + 2) (t – x) n ( n +1) ( x)dx = ⎢ – ( x) ⎥ + ( x)dx f f f a a ( n + 1)! n! ⎣⎢ (n + 1)! ⎦⎥ t a Instructor's Resource Manual Section 7.2 431 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. = (t – a )n +1 ( n +1) (a) + f (n + 1)! (t – x)n +1 ( n + 2) ( x)dx f a (n + 1)! t f (i ) ( a ) (t – a) n +1 ( n +1) (t – a)i + (a) + f (n + 1)! i! i =1 n +1 (i ) t (t – x ) n +1 ( n + 2) f (a) (t – a)i + ( x)dx = f (a) + f a ( n + 1)! i! i =1 Thus f (t ) = f (a ) + n (t – x)n +1 ( n + 2) ( x)dx f a ( n + 1)! t Thus, the statement is true for n + 1. 82. a. B( , ) = 1 0 x −1 (1 − x) −1 ≥ 1, dx where ≥1 x = 1 – u, dx = –du 1 0 x −1 (1 − x) −1 dx = 0 1 1 (1 − u ) −1 (u ) −1 (−du ) = (1 − u ) −1 u −1 du = B ( , ) 0 Thus, B( , ) = B( , ). b. B( , ) = u=x 1 0 x −1 (1 − x) −1 dx −1 du = ( − 1) x −2 v=− dx 1 B( , −1 dv = (1 − x) ⎡ 1 ⎤ ) = ⎢ − x −1 (1 − x) ⎥ + ⎣ ⎦0 −1 = B ( − 1, + 1) 1 dx (1 − x) −1 1 0 x −2 −1 (1 − x) dx = 1 0 x −2 (1 − x) dx (*) Similarly, B( , )= u = (1 − x) 1 x −1 0 −1 (1 − x) c. dx −2 ⎡1 ) = ⎢ x (1 − x) ⎣ v= dx 1 −1 ⎤ ⎥ + ⎦0 1 −1 dx x 1 0 x (1 − x) −2 −1 dx = 1 0 x (1 − x) −2 dx = −1 B ( + 1, − 1) Assume that n ≤ m. Using part (b) n times, ( n − 1) (n − 2) n −1 B (n, m) = B (n − 1, m + 1) = B (n − 2, m + 2) m m(m + 1) = = ( n − 1) (n − 2) ( n − 3) m(m + 1) ( m + 2 ) B(1, m + n − 1) = Thus, B (n, m) = 1 0 Section 7.2 ⋅ 2 ⋅1 (m + n − 2) (1 − x)m+ n − 2 dx = − B(1, m + n − 1). 1 1 [(1 − x)m+ n −1 ]10 = m + n −1 m + n −1 ( n − 1) (n − 2) ( n − 3) m(m + 1) ( m + 2 ) If n > m, then B (n, m) = B (m, n) = 432 −1 dv = x du = − ( − 1) (1 − x) B( , −1 ⋅ 2 ⋅1 (m + n − 2) ( m + n − 1) ( n − 1)!( m − 1)! (n + m − 1)! = ( n − 1)!( m − 1)! ( n − 1)!( m − 1)! (m + n − 1)! = (n + m − 1)! by the above reasoning. Instructor Solutions Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. dv = f (t )dt 83. u = f(t) du = f (t )dt b a v = f (t ) b = f (b) f (b) − f (a) f (a) − b [ f (t )]2 ≥ 0, so − 84. b f (t ) f (t )dt = [ f (t ) f (t ) ]a – a b a a [ f (t )]2 dt [ f (t )]2 dt = − b a [ f (t )]2 dt [ f (t )]2 ≤ 0 . x⎛ t 0 ⎞ ⎜ 0 f ( z )dz ⎟ dt ⎝ ⎠ u= t 0 f ( z )dz dv = dt du = f(t)dt v=t x x⎛ t x x x ⎞ ⎡ t ⎤ ⎜ f ( z )dz ⎟ dt = ⎢t 0 f ( z )dz ⎥ – 0 t f (t )dt = 0 x f ( z )dz – 0 t f (t )dt 0⎝ 0 ⎠ ⎣ ⎦0 x By letting z = t, 0 x x f ( z )dz = 0 x f (t )dt , so x⎛ t 0 x x x ⎞ ⎜ 0 f ( z )dz ⎟ dt = 0 x f (t )dt – 0 t f (t )dt = 0 ( x – t ) f (t )dt ⎝ ⎠ 85. Let I = x t1 t ⋅⋅⋅ n −1 0 0 0 f (tn ) dtn ...dt2 dt1 be the iterated integral. Note that for i ≥ 2, the limits of integration of the integral with respect to ti are 0 to ti −1 so that any change of variables in an outer integral affects the limits, and hence the variables in all interior integrals. We use induction on n, noting that the case n = 2 is solved in the previous problem. Assume we know the formula for n − 1 , and we want to show it for n. I= x t1 t2 t ⋅⋅⋅ n −1 0 0 0 0 where F ( tn −1 ) = f (tn ) dtn ...dt3 dt2 dt1 = tn −1 0 t1 t2 t ⋅⋅⋅ n − 2 0 0 0 F (tn −1 ) dtn −1...dt3 dt2 dt1 f ( tn ) dn . By induction, x 1 n−2 I= F ( t1 )( x − t1 ) dt1 0 − n 2 ! ( ) u = F ( t1 ) = t1 0 f ( tn ) dtn , du = f ( t1 ) dt1 , I= dv = ( x − t1 ) v=− 1 ⎧⎪ ⎡ 1 ( x − t1 )n −1 ⎨⎢− ( n − 2 )! ⎩⎪ ⎣ n − 1 t1 0 n−2 1 ( x − t1 )n −1 n −1 t =x 1 ⎤1 + f ( tn ) dtn ⎥ ⎦ t1 =0 n − 1 x 0 f ( t1 )( x − t1 ) n −1 ⎪⎫ dt1 ⎬ . ⎭⎪ x 1 f (t1 )( x − t1 ) n −1 dt1 0 (n − 1)! (note: that the quantity in square brackets equals 0 when evaluated at the given limits) = Instructor’s Resource Manual Section 7.3 433 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 86. Proof by induction. n = 1: dv = e x dx u = P1 ( x) du = dP1 ( x) dx dx v = ex e x P1 ( x)dx = e x P1 ( x) – e x Note that dP1 ( x) dP ( x ) x dP ( x) dx = e x P1 ( x) – 1 e dx = e x P1 ( x) – e x 1 dx dx dx dP1 ( x) is a constant. dx Suppose the formula is true for n. By using integration by parts with u = Pn +1 ( x) and dv = e x dx, dPn +1 ( x) dx dx e x Pn +1 ( x)dx = e x Pn +1 ( x) – e x Note that dPn +1 ( x) is a polynomial of degree n, so dx ⎡ e x Pn +1 ( x)dx = e x Pn +1 ( x) − ⎢ e x ⎢⎣ = e x Pn +1 ( x) + e x n +1 (−1) j (−1) j j =0 d j Pn +1 ( x ) j =1 (3 x 4 + 2 x 2 )e x dx = e x 87. n 4 dx j (–1) j = ex d j ⎛ dPn +1 ( x) ⎞ ⎤ x x ⎟ ⎥ = e Pn +1 ( x ) − e j ⎜ dx ⎥ ⎠⎦ dx ⎝ n +1 (−1) j j =0 n ( −1) j j =0 d j +1Pn +1 ( x ) dx j +1 d j Pn +1 ( x) dx j d j (3 x 4 + 2 x 2 ) j =0 dx j = e x [3x 4 + 2 x 2 – 12 x3 – 4 x + 36 x 2 + 4 – 72 x + 72] = e x (3x 4 – 12 x3 + 38 x 2 – 76 x + 76) 7.3 Concepts Review 1 + cos 2 x dx 2 1. 2 2. (1 – sin 2 x) cos x dx 3. sin 2 x(1 – sin 2 x) cos x dx 4. cos mx cos nx = 1 [cos(m + n) x + cos(m − n) x ] 2 Problem Set 7.3 sin 2 x dx = 1. 2. u = 6x, du = 6 dx 1 sin 4 6 x dx = sin 4 u du 6 1 – cos 2 x dx 2 1 1 dx – cos 2 x dx 2 2 1 1 = x – sin 2 x + C 2 4 1 ⎛ 1 – cos 2u ⎞ ⎜ ⎟ du 6 ⎝ 2 ⎠ 1 = (1 – 2 cos 2u + cos 2 2u )du 24 1 1 1 = du – 2 cos 2u du + (1 + cos 4u )du 24 24 48 3 1 1 = du – 2 cos 2u du + 4 cos 4u du 48 24 192 3 1 1 sin 24 x + C = (6 x) – sin12 x + 48 24 192 3 1 1 = x – sin12 x + sin 24 x + C 8 24 192 = = 434 Section 7.2 Instructor Solutions Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. sin 3 x dx = sin x(1 − cos 2 x)dx 3. 5. = sin x dx − sin x cos 2 x dx = 1 = − cos x + cos3 x + C 3 cos5 d = π/ 2 0 π/2 0 (1 – sin 2 )2 cos d (1 – 2sin 2 + sin 4 ) cos d π/2 2 1 ⎡ ⎤ = ⎢sin – sin 3 + sin 5 ⎥ 3 5 ⎣ ⎦0 2 1 8 ⎛ ⎞ = ⎜1 – + ⎟ – 0 = 15 ⎝ 3 5⎠ cos3 x dx = 4. π/2 0 = cos x (1 − sin 2 x)dx = cos x dx − cos x sin 2 x dx 1 = sin x − sin 3 x + C 3 6. π/2 0 sin 6 d = π / 2 ⎛ 1 – cos 2 0 ⎜ ⎝ 2 3 ⎞ ⎟ d ⎠ 1 π/ 2 (1 – 3cos 2 + 3cos 2 2 – cos3 2 )d 0 8 1 π/ 2 3 π/2 3 π/ 2 2 1 π/2 3 = d – 2 cos 2 d + cos 2 – cos 2 d 0 0 0 8 16 8 8 0 1 3 3 π / 2 ⎛ 1 + cos 4 ⎞ 1 π/2 2 = [ ]0π / 2 – [sin 2 ]0π / 2 + ⎜ ⎟ d – 0 (1 – sin 2 ) cos 2 d 0 8 16 8 2 8 ⎝ ⎠ 1 π 3 π/ 2 3 π/2 1 π/2 1 π/2 2 d + 4 cos 4 d – 2 cos 2 d + sin 2 ⋅ 2 cos 2 d = ⋅ + 0 0 0 8 2 16 64 16 16 0 π 3π 3 1 1 5π = + + [sin 4 ]0π / 2 – [sin 2 ]0π / 2 + [sin 3 2 ]0π / 2 = 16 32 64 16 48 32 = sin 5 4 x cos 2 4 x dx = (1 – cos 2 4 x) 2 cos 2 4 x sin 4 x dx = (1 – 2 cos 2 4 x + cos 4 4 x) cos 2 4 x sin 4 x dx 7. =– 1 1 1 1 (cos 2 4 x – 2 cos 4 4 x + cos6 4 x)(–4sin 4 x)dx = – cos3 4 x + cos5 4 x – cos7 4 x + C 12 10 28 4 (sin 3 2t ) cos 2tdt = (1 – cos 2 2t )(cos 2t )1/ 2 sin 2t dt = – 8. 1 [(cos 2t )1/ 2 – (cos 2t )5 / 2 ](–2sin 2t )dt 2 1 1 = – (cos 2t )3 / 2 + (cos 2t )7 / 2 + C 3 7 cos3 3 sin –2 3 d = (1 – sin 2 3 ) sin –2 3 cos 3 d 9. = 1 (sin −2 3 − 1)3cos 3 d 3 1 1 = – csc 3 – sin 3 + C 3 3 sin1/ 2 2 z cos3 2 z dz = (1 – sin 2 2 z ) sin1/ 2 2 z cos 2 z dz 10. = 1 1 1 (sin1/ 2 2 z – sin 5 / 2 2 z )2 cos 2 z dz = sin 3 / 2 2 z – sin 7 / 2 2 z + C 3 7 2 Instructor’s Resource Manual Section 7.3 435 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2 11. 2 1 ⎛ 1 – cos 6t ⎞ ⎛ 1 + cos 6t ⎞ (1 – 2 cos 2 6t + cos 4 6t )dt sin 4 3t cos 4 3t dt = ⎜ ⎟ ⎜ ⎟ dt = 2 2 16 ⎝ ⎠ ⎝ ⎠ 1 ⎡ 1 1 1 ⎤ = cos12t dt + (1 + 2 cos12t + cos 2 12t )dt 1 – (1 + cos12t ) + (1 + cos12t )2 ⎥ dt = – 16 ⎢⎣ 4 16 64 ⎦ 1 1 1 1 =– 12 cos12t dt + dt + 12 cos12t dt + (1 + cos 24t )dt 192 64 384 128 1 1 1 1 1 3 1 1 sin12t + t + sin12t + sin 24t + C = sin12t + sin 24t + C =– t+ t– 192 64 384 128 3072 128 384 3072 3 12. 1 ⎛ 1 + cos 2 ⎞ ⎛ 1 – cos 2 ⎞ cos6 sin 2 d = ⎜ (1 + 2 cos 2 – 2 cos3 2 – cos 4 2 )d ⎟ ⎜ ⎟d = 2 2 16 ⎝ ⎠ ⎝ ⎠ 1 1 1 1 = d + 2 cos 2 d – (1 – sin 2 2 ) cos 2 d – (1 + cos 4 )2 d 16 16 8 64 1 1 1 1 1 = d + 2 cos 2 d – 2 cos 2 d + 2sin 2 2 cos 2 d – (1 + 2 cos 4 + cos 2 4 )d 16 16 16 16 64 1 1 1 1 1 sin 2 2 ⋅ 2 cos 2 d – 4 cos 4 d – (1 + cos8 )d = d + d – 16 16 64 128 128 1 1 1 1 1 1 + sin 3 2 − − − sin 4 − sin 8 + C = 16 48 64 128 128 1024 5 1 1 1 = + sin 3 2 – sin 4 – sin 8 + C 128 48 128 1024 sin 4 y cos 5 y dy = 13. = 14. 1 2 [sin 9 y + sin(− y )] dy = 1 (sin 9 y − sin y ) dy 2 1⎛ 1 1 1 ⎞ ⎜ − cos 9 y + cos y ⎟ + C = cos y − cos 9 y + C 2⎝ 9 2 18 ⎠ cos y cos 4 y dy = 1 1 1 1 1 [cos 5 y + cos(−3 y )]dy = sin 5 y − sin(−3 y) + C = sin 5 y + sin 3 y + C 10 6 10 6 2 2 15. 16. 1 ⎛ w⎞ ⎛ w⎞ ⎛ 1 – cos w ⎞ ⎛ 1 + cos w ⎞ (1 – cos w – cos 2 w + cos3 w)dw sin 4 ⎜ ⎟ cos 2 ⎜ ⎟ dw = ⎜ ⎟ ⎜ ⎟ dw = 8 2 2 ⎝2⎠ ⎝2⎠ ⎝ ⎠ ⎝ ⎠ 1 ⎡ 1 1 ⎡1 1 ⎤ ⎤ 2 = 1 – cos w – (1 + cos 2w) + (1 – sin 2 w) cos w⎥ dw = ⎢ 2 – 2 cos 2 w – sin w cos w⎥dw 8 ⎢⎣ 2 8 ⎦ ⎣ ⎦ 1 1 1 = w – sin 2 w – sin 3 w + C 16 32 24 sin 3t sin t dt = − ( 1 [cos 4t − cos 2t ] dt 2 ) 1 cos 4tdt − cos 2tdt 2 1⎛1 1 ⎞ = − ⎜ sin 4t − sin 2t ⎟ + C 2⎝4 2 ⎠ 1 1 = − sin 4t + sin 2t + C 8 4 =− 436 Section 7.3 Instructor's Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 17. x cos 2 x sin x dx u=x tan 4 x dx = 19. du = 1 dx = 2 dv = cos x sin x dx = 1 v = − (cos x) 2 (− sin x) dx = − cos3 x 3 t =cos x Thus = tan 2 x sec2 x dx − (sec2 x − 1)dx 1 = tan 3 x − tan x + x + C 3 x cos 2 x sin x dx = 1 1 x(− cos3 x) − (1)(− cos3 x) dx = 3 3 1⎡ 3 3 − x cos x + cos x dx ⎤ = ⎦ 3⎣ 1⎡ − x cos3 x + cos x (1 − sin 2 x) dx ⎤ = ⎦ 3⎣ ⎤ 1⎡ 3 2 ⎢ − x cos x + (cos x − cos x sin x) dx ⎥ = 3⎣ t =sin x ⎦ 1⎡ 1 ⎤ − x cos3 x + sin x − sin 3 x ⎥ + C 3 ⎢⎣ 3 ⎦ 18. 3 cot 4 x dx = 20. = = = cot 2 x csc2 x dx − (csc2 x − 1)dx 1 = − cot 3 x + cot x + x + C 3 21. tan 3 x = du = 1 dx dv = sin 3 x cos x dx v = (sin x)3 (cos x) dx = t =sin x 1 4 sin x 4 Thus x sin 3 x cos x dx = 1 1 x( sin 4 x ) − (1)( sin 4 x) dx = 4 4 1⎡ 4 2 2 x sin x − (sin x) dx ⎤ = ⎦ 4⎣ 1⎡ 1 ⎤ (1 − cos 2 x) 2 dx ⎥ = x sin 4 x − 4 ⎢⎣ 4 ⎦ 1⎡ 1 ⎤ x sin 4 x − (1 − 2 cos 2 x + cos 2 2 x) dx ⎥ = 4 ⎢⎣ 4 ⎦ ( cot 2 x )( cot 2 x ) dx ( cot 2 x ) (csc2 x − 1) dx ( cot 2 x csc2 x − cot 2 x ) dx x sin x cos x dx u=x ( tan 2 x )( tan 2 x ) dx ( tan 2 x ) (sec2 x − 1) dx ( tan 2 x sec2 x − tan 2 x ) dx ( tan x ) ( tan 2 x ) dx = ( tan x ) ( sec2 x − 1) dx = 1 tan 2 x + ln cos x + C 2 cot 3 2t dt = 22. = ( cot 2t ) ( cot 2 2t ) dt ( cot 2t ) ( csc2 2t − 1)dt = cot 2t csc2 2t dt − cot 2t dt 1 1 = − cot 2 2t − ln sin 2t + C 4 2 1⎡ 1 1 1 ⎤ x sin 4 x − x + sin 2 x − (1 + cos 4 x) dx ⎥ = 4 ⎢⎣ 4 4 8 ⎦ 1⎡ 3 1 1 ⎤ x sin 4 x − x + sin 2 x − sin 4 x ⎥ + C ⎢ 4⎣ 8 4 32 ⎦ Instructor’s Resource Manual Section 7.3 437 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 23. ⎛ ⎞ tan 5 ⎜ ⎟ d ⎝2⎠ d ⎛ ⎞ u = ⎜ ⎟ ; du = 2 2 ⎝ ⎠ ⎛ ⎞ tan 5 ⎜ ⎟ d = 2 tan 5 u du ⎝2⎠ ( )( ) = 2 tan 3 u sec2 u − 1 du = 2 tan 3 u sec2 u du − 2 tan 3 u du ( ) = 2 tan 3 u sec2 u du − 2 tan u sec2 u − 1 du = 2 tan 3 u sec2 u du − 2 tan u sec2 u du + 2 tan u du = 24. 1 ⎛ ⎞ ⎛ ⎞ tan 4 ⎜ ⎟ − tan 2 ⎜ ⎟ − 2 ln cos + C 2 2 ⎝2⎠ ⎝2⎠ cot 5 2t dt u = 2t ; du = 2dt 1 cot 5 2t dt = cot 5 u du 2 1 1 = cot 3 u cot 2 u du = cot 3 u csc2 − 1 du 2 2 1 1 = cot 3 u csc 2 u du − cot 3 u du 2 2 1 1 = cot 3 u csc2 u du − ( cot u ) csc2 u − 1 du 2 2 1 1 1 3 2 cot u csc u du − = ( cot u ) csc2 u du + cot u 2 2 2 1 4 1 2 1 = − cot u + cot u + ln sin u + C 8 4 2 1 4 1 2 1 = − cot 2t + cot 2t + ln sin 2t + C 8 4 2 ( ( ( ( 25. )( )( )( )( tan −3 x sec4 xdx = = ) ) ) ) ( )( ) ( ( ) ) ( tan −3 x )(sec2 x )(sec2 x ) dx ( tan −3 x )(1 + tan 2 x )(sec2 x ) dx = tan −3 x sec2 x dx + ( tan x )−1 sec2 x dx 1 = − tan −2 x + ln tan x + C 2 26. tan −3 / 2 x sec4 x dx = = ( tan −3 / 2 x )(sec2 x )(sec2 x ) ( tan −3 / 2 x )(1 + tan 2 x )(sec2 x ) dx = tan −3 / 2 x sec2 x dx + tan1/ 2 x sec2 x dx 2 = −2 tan −1/ 2 x + tan 3 / 2 x + C 3 438 Section 7.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. tan 3 x sec2 x dx 27. Let u = tan x . Then du = sec 2 x dx . tan 3 x sec2 x dx = u 3 du = 1 4 1 u + C = tan 4 x + C 4 4 tan 3 x sec−1/ 2 x dx = tan 2 x sec−3 / 2 x(sec x tan x)dx 28. (sec2 x − 1)(sec−3 / 2 x ) (sec x tan x ) dx = = sec1/ 2 x ( sec x tan x ) dx − sec−3 / 2 x ( sec x tan x ) dx = 29. 30. 31. 2 3/ 2 sec x + 2sec−1/ 2 x + C 3 π 1 π 1⎡ 1 1 ⎤ (cos[(m + n) x] + cos[(m − n) x])dx = ⎢ sin[(m + n) x] + sin[(m − n) x]⎥ –π – π 2 2 ⎣m + n m−n ⎦ −π = 0 for m ≠ n, since sin k π = 0 for all integers k. π cos mx cos nx dx = If we let u = πx then du = π dx . Making the substitution and changing the limits as necessary, we get L L L mπ x nπ x L π dx = cos cos cos mu cos nu du = 0 (See Problem 29 −L L L π −π π 0 π( x + sin x)2 dx = π π 0 ( x 2 + 2 x sin x + sin 2 x) dx = π π π 2 x 0 dx + 2π π 0 x sin x dx + π 2 π 0 (1 − cos 2 x)dx π 1 1 π 1 5 ⎡1 ⎤ ⎤ π π⎡ = π ⎢ x3 ⎥ + 2π [sin x − x cos x ]0 + ⎢ x − sin 2 x ⎥ = π4 + 2π(0 + π − 0) + (π − 0 − 0) = π4 + π2 ≈ 57.1437 2⎣ 2 2 3 2 ⎣ 3 ⎦0 ⎦0 3 Use Formula 40 with u = x for π/2 32. V = 2π 0 x sin x dx x sin 2 ( x 2 )dx u = x 2 , du = 2x dx π/2 V =π 33. a. 0 1 π π −π sin 2 u du = π π / 2 1 – cos 2u 0 f ( x) sin(mx)dx = 2 1 π π/ 2 1 ⎡1 ⎤ du = π ⎢ u – sin 2u ⎥ 4 ⎣2 ⎦0 = π2 ≈ 2.4674 4 ⎛ N ⎞ 1 N an ⎜⎜ an sin(nx) ⎟⎟ sin(mx)dx = −π π n =1 ⎝ n =1 ⎠ π π −π sin(nx) sin(mx) dx From Example 6, ⎧0 if n ≠ m so every term in the sum is 0 except for when n = m. sin(nx) sin(mx)dx = ⎨ ⎩π if n = m If m > N, there is no term where n = m, while if m ≤ N, then n = m occurs. When n = m π −π an 1 π π −π π −π sin(nx) sin(mx) dx = am π so when m ≤ N, f ( x) sin(mx) dx = Instructor’s Resource Manual 1 ⋅ am ⋅ π = am . π Section 7.3 439 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. N ⎛ N ⎞⎛ N ⎞ π 1 N sin(nx) sin(mx) dx am sin(mx) ⎟ dx = an am ⎜⎜ an sin(nx) ⎟⎟ ⎜⎜ ⎟ −π −π −π π n =1 m =1 ⎝ n =1 ⎠ ⎝ m =1 ⎠ From Example 6, the integral is 0 except when m = n. When m = n, we obtain N 1 N an (an π) = an2 . π n =1 n =1 1 π b. π f 2 ( x)dx = 1 π π Proof by induction x x n = 1: cos = cos 2 2 Assume true for k ≤ n. 34. a. x x cos cos 2 4 cos x 2n ⋅ cos x 2n +1 ⎡ 1 3 = ⎢ cos x + cos x+ n 2 2n ⎢⎣ + cos 2n –1 ⎤ 1 x x⎥ cos n n –1 n +1 2 2 ⎥⎦ 2 Note that k ⎞⎛ 1 ⎞ 1⎡ 2k + 1 2k –1 ⎤ ⎛ ⎜ cos n x ⎟⎜ cos n +1 x ⎟ = ⎢cos n +1 x + cos n +1 x ⎥ , so 2 ⎠⎝ 2 2 2 ⎝ ⎠ 2⎣ ⎦ n ⎡ ⎡ 1 3 2 –1 ⎤ ⎛ 1 ⎞ 1 1 3 ⎢cos n x + cos n x + + cos n x ⎥ ⎜ cos n +1 x ⎟ n –1 = ⎢cos n +1 x + cos n +1 x + ⎠2 2 2 2 2 2 2 ⎣⎢ ⎦⎥ ⎝ ⎣⎢ ⎡ 1 3 lim ⎢ cos x + cos x+ n n →∞ ⎢ 2 2n ⎣ b. = 1 x c. 1 x x 0 x 0 + cos ⎡ 2n –1 ⎤ 1 1 1 3 x⎥ x + cos x+ = lim ⎢cos n n –1 n x n →∞ 2 2 2n ⎦⎥ 2 ⎣⎢ + cos + cos 2n +1 –1 ⎤ 1 x⎥ 2n +1 ⎦⎥ 2n 2n –1 ⎤ x x⎥ 2n ⎦⎥ 2n –1 cos t dt 1 sin x cos t dt = [sin t ]0x = x x 35. Using the half-angle identity cos x 1 + cos x = , we see that since 2 2 π cos π 2 = cos 2 = 4 2 2 cos π 1 + 22 π = cos 2 = = 8 4 2 cos π 1 + 2+2 2 π = cos 2 = = 16 8 2 Thus, 2+ 2+ 2 , etc. 2 2 2+ 2 2+ 2+ 2 ⋅ ⋅ 2 2 2 ⎛π⎞ ⎛π⎞ = lim cos ⎜ 2 ⎟ cos ⎜ 2 ⎟ ⎜2⎟ ⎜4⎟ n→∞ ⎝ ⎠ ⎝ ⎠ 440 2+ 2 , 2 Section 7.3 ⎛ π cos ⎜ 2 ⎜ 2n ⎝ ⎛π⎞ ⎛π⎞ ⎛π⎞ = cos ⎜ 2 ⎟ cos ⎜ 2 ⎟ cos ⎜ 2 ⎟ ⎜2⎟ ⎜4⎟ ⎜8⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ( ) ⎞ sin π2 2 ⎟= = π ⎟ π 2 ⎠ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 36. Since (k − sin x)2 = (sin x − k ) 2 , the volume of S is = πk 2 π 0 dx − 2k π π 0 sin x dx + π 0 π(k − sin x)2 = π π 0 (k 2 − 2k sin x + sin 2 x) dx π π π 1 ⎤ π π π⎡ (1 − cos 2 x) dx = πk 2 [ x ]0 + 2k π [ cos x ]0 + ⎢ x − sin 2 x ⎥ 2⎣ 2 2 0 ⎦0 π π2 (π − 0) = π2 k 2 − 4k π + 2 2 2 2 π , then f (k ) = 2π2 k − 4π and f (k ) = 0 when k = . Let f (k ) = π2 k 2 − 4k π + π 2 2 The critical points of f(k) on 0 ≤ k ≤ 1 are 0, , 1. π = π2 k 2 + 2k π(−1 − 1) + f (0) = a. π2 ≈ 4.93, 2 2 2 π π ⎛2⎞ f ⎜ ⎟ = 4−8+ ≈ 0.93, f (1) = π2 − 4π + ≈ 2.24 2 2 ⎝π⎠ S has minimum volume when k = 2 . π b. S has maximum volume when k = 0. 7.4 Concepts Review 1. 4. u = x + 4, u 2 = x + 4, 2u du = dx x 2 + 3x (u 2 – 4)2 + 3(u 2 – 4) 2u du u x+4 2 10 = 2 (u 4 – 5u 2 + 4)du = u 5 – u 3 + 8u + C 5 3 2 10 = ( x + 4)5 / 2 – ( x + 4)3 / 2 + 8( x + 4)1/ 2 + C 5 3 x–3 2. 2 sin t 3. 2 tan t 4. 2 sec t Problem Set 7.4 1. u = x + 1, u 2 = x + 1, 2u du = dx x x + 1dx = (u 2 – 1)u (2u du ) 2 2 = (2u 4 – 2u 2 )du = u 5 – u 3 + C 5 3 2 5/ 2 2 3/ 2 = ( x + 1) +C – ( x + 1) 5 3 5. u = t , u 2 = t , 2u du = dt 2 dt 2 2u du = =2 1 t +e 1 u+e 1 2 2 e =2 du – 2 du 1 1 u+e = 3 3π ( x + π)7 / 3 – ( x + π)4 / 3 + C 7 4 3. u = 3t + 4, u 2 = 3t + 4, 2u du = 3 dt 1 (u 2 3 − 4) 23 u du t dt 2 = = (u 2 – 4)du u 9 3t + 4 2 3 8 = u – u+C 27 9 2 8 = (3t + 4)3 / 2 – (3t + 4)1/ 2 + C 27 9 Instructor’s Resource Manual u+e−e du u+e 1 = 2( 2 – 1) – 2e[ln( 2 + e) – ln(1 + e)] ⎛ 2 +e⎞ = 2 2 – 2 – 2e ln ⎜⎜ ⎟⎟ ⎝ 1+ e ⎠ x 3 x + πdx = (u 3 – π)u (3u 2 du ) 3 7 3π 4 u – u +C 7 4 2 2 = 2[u ]1 2 – 2e ⎡⎣ln u + e ⎤⎦ 2. u = 3 x + π , u 3 = x + π, 3u 2 du = dx = (3u 6 – 3πu 3 )du = dx = u = t , u 2 = t , 2u du = dt 6. 1 t 0 t +1 =2 1 1 dt = u2 0 u2 +1 u 0 u2 +1 du = 2 1 1 1 0 0 u2 = 2 du – 2 (2u du ) 1 u2 +1−1 0 2 u +1 du du = 2[u ]10 – 2[tan –1 u ]10 +1 π = 2 – 2 tan 1 = 2 – ≈ 0.4292 2 –1 Section 7.4 441 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 7. u = (3t + 2)1/ 2 , u 2 = 3t + 2, 2u du = 3dt 1 2 ⎛2 ⎞ (u – 2)u 3 ⎜ u du ⎟ 3 ⎝3 ⎠ 2 2 4 = (u 6 – 2u 4 )du = u 7 − u 5 + C 9 63 45 2 4 = (3t + 2)7 / 2 – (3t + 2)5 / 2 + C 63 45 t (3t + 2)3 / 2 dt = 8. u = (1 – x)1/ 3 , u 3 = 1 – x, 3u 2 du = – dx x(1 – x) 2/3 3 2 2 dx = (1 – u )u (–3u )du 3 3 = 3 (u 7 – u 4 )du = u8 − u 5 + C 8 5 3 3 = (1 – x)8 / 3 – (1 – x)5 / 3 + C 8 5 9. x = 2 sin t, dx = 2 cos t dt 12. t = sec x, dt = sec x tan x dx π Note that 0 ≤ x < . 2 t 2 – 1 = tan x = tan x 2 2 = = 2 ln 2 − 4 – x2 + 4 – x2 + C x 10. x = 4sin t , dx = 4 cos t dt x 2 dx 16 – x 2 = 16 sin 2 t cos t dt cos t = 16 sin 2 t dt = 8 (1 – cos 2t )dt = 8t – 4sin 2t + C = 8t − 8sin t cos t + C 2 ⎛ x ⎞ x 16 – x = 8sin –1 ⎜ ⎟ – +C 2 ⎝4⎠ 2 11. x = 2 tan t , dx = 2sec t dt dx 2sec 2 t dt 1 = = cos t dt 2 3/ 2 2 3/ 2 4 ( x + 4) (4sec t ) = 442 1 x sin t + C = +C 4 4 x2 + 4 Section 7.4 = 2 t t –1 sec –1 (3) π/3 sec –1 (3) π/3 sec x tan x sec 2 x tan x dx cos x dx –1 = [sin x]sec π/3 (3) = sin[sec−1 (3)] − sin π 3 ⎡ 3 2 2 3 ⎛ 1 ⎞⎤ = sin ⎢cos −1 ⎜ ⎟ ⎥ − = – ≈ 0.0768 3 2 ⎝ 3 ⎠⎦ 2 ⎣ 13. t = sec x, dt = sec x tan x dx π Note that < x ≤ π. 2 t 2 – 1 = tan x = – tan x 4 – x2 2 cos t dx = (2 cos t dt ) x 2sin t 1 – sin 2 t dt = 2 csc t dt – 2 sin t dt =2 sin t = 2 ln csc t − cot t + 2 cos t + C dt 3 –3 t2 –1 –2 3 = t sec –1 (–3) 2π / 3 dt = sec –1 (–3) – tan x 2π / 3 sec3 x – sin 2 x dx = sec x tan x dx sec –1 (–3) ⎛ 1 2π / 3 1⎞ ⎜ cos 2 x – ⎟ dx 2⎠ ⎝2 sec –1 (–3) 1 ⎤ ⎡1 = ⎢ sin 2 x – x ⎥ 2 ⎦ 2π / 3 ⎣4 sec –1 (–3) 1 ⎤ ⎡1 = ⎢ sin x cos x – x ⎥ 2 ⎦ 2π / 3 ⎣2 =– 2 1 3 π – sec –1 (–3) + + ≈ 0.151252 9 2 8 3 14. t = sin x, dt = cos x dx t dt = sin x dx = –cos x + C 1– t2 = – 1– t2 + C 15. z = sin t, dz = cos t dt 2z – 3 dz = (2sin t – 3)dt 1 – z2 = –2 cos t – 3t + C = –2 1 – z 2 – 3sin –1 z + C Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 16. x = π tan t, dx = π sec 2 t dt πx – 1 2 x +π =π 2 2 dx = (π2 tan t – 1) sec t dt tan t sec t dt – sec t dt = π2 sec t – ln sec t + tan t + C = π x 2 + π2 − ln π πx − 1 0 x 2 + π2 1 2 x x + π2 + + C π π dx ⎡ = ⎢ π x 2 + π2 – ln ⎢ ⎣ = 3 x 2 + 2 x + 5 – 3ln ⎤ x 2 + π2 x ⎥ + π π⎥ ⎦0 = ( 2 – 1)π2 – ln( 2 + 1) ≈ 3.207 17. x 2 + 2 x + 5 = x 2 + 2 x + 1 + 4 = ( x + 1)2 + 4 u = x + 1, du = dx dx du = x2 + 2 x + 5 u2 + 4 u = 2 tan t, du = 2sec 2 t dt u2 + 4 = sec t dt = ln sec t + tan t + C1 = ln u2 + 4 u + + C1 2 2 = ln x2 + 2 x + 5 + x + 1 + C1 2 = ln x2 + 2 x + 5 + x + 1 + C 18. x 2 + 4 x + 5 = x 2 + 4 x + 4 + 1 = ( x + 2)2 + 1 u = x + 2, du = dx dx du = 2 x + 4x + 5 u2 + 1 u = tan t , du = sec2 t dt du u2 +1 dx 2 = sec t dt = ln sec t + tan t + C x + 4x + 5 = ln = 3 u 2 + 4 – 3ln u 2 + 4 + u + C = ln u 2 + 1 + u + C x2 + 4 x + 5 + x + 2 + C x2 + 2 x + 5 + x + 1 + C π = [ 2π2 – ln( 2 + 1)] – [π2 − ln1] du 19. x 2 + 2 x + 5 = x 2 + 2 x + 1 + 4 = ( x + 1)2 + 4 u = x + 1, du = dx 3x 3u – 3 dx = du 2 x + 2x + 5 u2 + 4 u du du – 3 =3 2 u +4 u2 + 4 (Use the result of Problem 17.) 20. x 2 + 4 x + 5 = x 2 + 4 x + 4 + 1 = ( x + 2)2 + 1 u = x + 2, du = dx 2x – 1 2u − 5 dx = du x2 + 4 x + 5 u2 + 1 2u du du = –5 u2 +1 u2 +1 (Use the result of Problem 18.) = 2 u 2 + 1 – 5ln u 2 + 1 + u + C = 2 x 2 + 4 x + 5 – 5ln x2 + 4 x + 5 + x + 2 + C 21. 5 − 4 x − x 2 = 9 − (4 + 4 x + x 2 ) = 9 − ( x + 2)2 u = x + 2, du = dx 5 – 4 x – x 2 dx = 9 – u 2 du u = 3 sin t, du = 3 cos t dt 9 − u 2 du =9 cos 2 t dt = 9 (1 + cos 2t )dt 2 9⎛ 1 9 ⎞ ⎜ t + sin 2t ⎟ + C = (t + sin t cos t ) + C 2⎝ 2 2 ⎠ 9 ⎛u⎞ 1 = sin –1 ⎜ ⎟ + u 9 – u 2 + C 2 ⎝3⎠ 2 9 ⎛ x+2⎞ x+2 = sin –1 ⎜ 5 – 4 x – x2 + C ⎟+ 2 2 ⎝ 3 ⎠ = 22. 16 + 6 x – x 2 = 25 − (9 − 6 x + x 2 ) = 25 – ( x – 3) 2 u = x – 3, du = dx dx du = 16 + 6 x – x 2 25 – u 2 u = 5 sin t, du = 5 cos t du ⎛u⎞ = dt = t + C = sin –1 ⎜ ⎟ + C 2 ⎝5⎠ 25 − u ⎛ x –3⎞ = sin –1 ⎜ ⎟+C ⎝ 5 ⎠ Instructor’s Resource Manual Section 7.4 443 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 23. 4 x – x 2 = 4 − (4 − 4 x + x 2 ) = 4 – ( x – 2)2 u = x – 2, du = dx dx du = 2 4x – x 4 – u2 u = 2 sin t, du = 2 cos t dt du ⎛u⎞ = dt = t + C = sin –1 ⎜ ⎟ + C 2 ⎝2⎠ 4−u ⎛ x–2⎞ = sin –1 ⎜ ⎟+C ⎝ 2 ⎠ 25. x 2 + 2 x + 2 = x 2 + 2 x + 1 + 1 = ( x + 1) 2 + 1 u = x + 1, du = dx 2x + 1 2u – 1 dx = du 2 x + 2x + 2 u2 + 1 2u du du – = 2 2 u +1 u +1 = ln u 2 + 1 – tan –1 u + C ) = ln x 2 + 2 x + 2 − tan −1 ( x + 1) + C π 8 π = 8 = 26. x 2 – 6 x + 18 = x 2 – 6 x + 9 + 9 = ( x – 3)2 + 9 u = x – 3, du = dx 2x – 1 2u + 5 dx = du 2 x – 6 x + 18 u2 + 9 2u du du = +5 2 2 u +9 u +9 5 ⎛u⎞ = ln u 2 + 9 + tan −1 ⎜ ⎟ + C 3 ⎝3⎠ 5 ⎛ x−3⎞ = ln x 2 − 6 x + 18 + tan −1 ⎜ ⎟+C 3 ⎝ 3 ⎠ ) ) 2 1⎛ 1 ⎞ 27. V = π ⎜ dx 0 ⎝ x 2 + 2 x + 5 ⎟⎠ 1⎡ 2 ⎤ =π ⎢ ⎥ dx 2 0 ( x + 1) + 4 ⎣⎢ ⎦⎥ 444 1 Section 7.4 –1 = π ⎡1 1 ⎤ t + sin 2t ⎥ 8 ⎢⎣ 2 4 ⎦ tan –1 (1/ 2) = π ⎡1 1 ⎤ t + sin t cos t ⎥ 8 ⎢⎣ 2 2 ⎦ tan –1 (1/ 2) = π ⎡⎛ π 1 ⎞ ⎛ 1 –1 1 1 ⎞ ⎤ + ⎟ ⎜ + ⎟ – ⎜ tan 8 ⎢⎣⎝ 8 4 ⎠ ⎝ 2 2 5 ⎠ ⎥⎦ = π⎛1 π –1 1 ⎞ ⎜ + – tan ⎟ ≈ 0.082811 16 ⎝ 10 4 2⎠ π/4 28. V = 2π 1 1 0 x2 + 2x + 5 1 x = 2π dx 0 ( x + 1) 2 + 4 = 2π x +1 1 0 ( x + 1) 2 +4 x dx dx – 2π 1 1 0 ( x + 1) 2 +4 dx 1 1 ⎡1 ⎡1 ⎤ ⎛ x + 1 ⎞⎤ = 2π ⎢ ln[( x + 1) 2 + 4]⎥ – 2π ⎢ tan –1 ⎜ ⎟⎥ 2 2 ⎣ ⎦0 ⎝ 2 ⎠⎦0 ⎣ 1⎤ ⎡ = π[ln 8 – ln 5] – π ⎢ tan –1 1 – tan –1 ⎥ 2⎦ ⎣ 1⎞ ⎛ 8 π = π ⎜ ln – + tan –1 ⎟ ≈ 0.465751 5 4 2⎠ ⎝ 29. a. ( ( 2 ⎛ 1 ⎞ 2sec2 t dt tan (1/ 2) ⎜⎝ 4sec 2 t ⎟⎠ π/4 1 π π/ 4 cos 2 t dt dt = –1 –1 2 tan (1/ 2) sec t 8 tan (1/ 2) π/ 4 ⎛1 1 ⎞ + cos 2t ⎟ dt tan –1 (1/ 2) ⎜⎝ 2 2 ⎠ π/4 V =π π/ 4 24. 4 x – x 2 = 4 − (4 − 4 x + x 2 ) = 4 – ( x – 2)2 u = x – 2, du = dx x u+2 dx = du 2 4x – x 4 – u2 – u du du =– +2 2 4–u 4 – u2 (Use the result of Problem 23.) ⎛u⎞ = – 4 – u 2 + 2sin –1 ⎜ ⎟ + C ⎝2⎠ ⎛ x–2⎞ = – 4 x – x 2 + 2sin –1 ⎜ ⎟+C ⎝ 2 ⎠ ( x + 1 = 2 tan t, dx = 2sec2 t dt u = x 2 + 9, du = 2 x dx x dx 1 du 1 = ln u + C 2 x +9 2 u 1 1 = ln x 2 + 9 + C = ln x 2 + 9 + C 2 2 2 = ( ) b. x = 3 tan t, dx = 3sec 2 t dt x dx x2 + 9 = tan t dt = – ln cos t + C ⎛ ⎞ 3 ⎟+C + C1 = − ln ⎜ ⎜ 2 ⎟ 1 x2 + 9 ⎝ x +9 ⎠ = ln ⎛⎜ x 2 + 9 ⎞⎟ − ln 3 + C1 ⎝ ⎠ 1 = ln ( x 2 + 9)1/ 2 + C = ln x 2 + 9 + C 2 = − ln ( 3 ) ( ) Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 32. The equation of the circle with center (–a, 0) is 30. u = 9 + x 2 , u 2 = 9 + x 2 , 2u du = 2x dx 3 x3 dx 0 9 + x2 = x2 3 0 x dx = 9 + x2 3 2 3 3 2 ⎡ u3 ⎤ (u 2 − 9) du = ⎢ – 9u ⎥ 3 ⎣⎢ 3 ⎦⎥ 3 ≈ 5.272 = 31. a. u2 − 9 udu u 3 2 = 18 – 9 2 4 − x2 x2 = u 2 du 4 – u2 1 du = −4 du + du 4 − u2 4 − u2 1 u+2 = −4 ⋅ ln +u+C 4 u−2 = − ln 4 − x2 + 2 2 + 4− x +C 4 − x2 − 2 b. x = 2 sin t, dx = 2 cos t dt 4 – x2 cos 2 t dx = 2 dt sin t x =2 b–a b 2 – ( x + a) 2 dx 0 x + a = b sin t, dx = b cos t dt x dx = – −4 + 4 − u 2 and y = b 2 – ( x + a )2 dx . A=4 u = 4 – x 2 , u 2 = 4 – x 2 , 2u du = –2x dx 4 – x2 dx = x ( x + a )2 + y 2 = b 2 , so y = ± b 2 – ( x + a )2 . By symmetry, the area of the overlap is four times the area of the region bounded by x = 0, y = 0, sin –1 ( a / b ) b 2 cos 2 t dt π/2 = 2b 2 sin –1 ( a / b ) (1 + cos 2t )dt π/2 ⎡ 1 ⎤ = 2b 2 ⎢t + sin 2t ⎥ ⎣ 2 ⎦ sin –1 ( a / b ) = 2b 2 [t + sin t cos t ]π / 2–1 sin (a / b) ⎡ 2 2 ⎞⎤ π ⎛ ⎛ a ⎞ a b – a ⎟⎥ = 2b 2 ⎢ – ⎜ sin –1 ⎜ ⎟ + ⎢2 ⎜ ⎟⎥ b ⎝b⎠ b ⎝ ⎠⎦ ⎣ a ⎛ ⎞ = πb 2 – 2b 2 sin –1 ⎜ ⎟ – 2a b 2 – a 2 ⎝b⎠ 33. a. (1 – sin 2 t ) dt sin t π/ 2 A=4 The coordinate of C is (0, –a). The lower arc of the lune lies on the circle given by the equation x 2 + ( y + a)2 = 2a 2 or = 2 csc t dt – 2 sin t dt y = ± 2a 2 – x 2 – a. The upper arc of the lune lies on the circle given by the equation = 2 ln csc t − cot t + 2 cos t + C x 2 + y 2 = a 2 or y = ± a 2 – x 2 . = 2 ln = 2 ln 2 4 − x2 − + 4 − x2 + C x x 2− 4− x x + 4 − x2 + C − ln = ln = ln 4− x +2 4 − x2 − 2 = ln 2 4− x −2 4 − x2 + 2 (2 − 4 − x ) 2 2 4− x −4 ⎛ 2 − 4 − x2 = ln ⎜ ⎜ x ⎝ = ln a For ⎛ 2a 2 – x 2 – a ⎞ dx ⎜ ⎟ ⎠ a 2a 2 – x 2 dx + 2a 2 –a a 2 – x 2 dx is the area of a –a 2a 2 – x 2 dx, let x = 2a sin t , dx = 2a cos t dt ( 4 − x 2 + 2)( 4 − x 2 − 2) 2 2 a –a a –a ⎝ semicircle with radius a, so a πa 2 a 2 – x 2 dx = . –a 2 2 ( 4 − x − 2) a 2 – x 2 dx – –a Note that To reconcile the answers, note that 2 a 2 – x 2 dx – –a a = 2 a A= a 2a 2 – x 2 dx = (2 − 4 − x ) –a − x2 = a2 2 2 2 2 ⎞ ⎟ = 2 ln 2 − 4 − x ⎟ x ⎠ = π/4 –π / 4 2a 2 cos 2 t dt π/4 ⎡ 1 ⎤ (1 + cos 2t )dt = a 2 ⎢t + sin 2t ⎥ –π / 4 ⎣ 2 ⎦–π / 4 π/4 πa 2 + a2 2 A= ⎞ πa 2 ⎛ πa 2 –⎜ + a 2 ⎟ + 2a 2 = a 2 ⎟ 2 ⎜⎝ 2 ⎠ Thus, the area of the lune is equal to the area of the square. Instructor’s Resource Manual Section 7.4 445 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. Without using calculus, consider the following labels on the figure. 7.5 Concepts Review 1. proper 2. x – 1 + 5 x +1 3. a = 2; b = 3; c = –1 4. Area of the lune = Area of the semicircle of radius a at O + Area (ΔABC) – Area of the sector ABC. 1 1⎛π⎞ A = πa 2 + a 2 – ⎜ ⎟ ( 2a )2 2 2⎝ 2⎠ 1 2 1 = πa + a 2 – πa 2 = a 2 2 2 Note that since BC has length 2a, the π measure of angle OCB is , so the measure 4 π of angle ACB is . 2 Problem Set 7.5 1. dy a2 – x2 a2 – x2 ; y= – =– dx dx x x x = a sin t, dx = a cos t dt a cos t cos 2 t y= – a cos t dt = – a dt a sin t sin t = –a 1 – sin 2 t dt = a (sin t – csc t )dt sin t = a ( – cos t − ln csc t − cot t ) + C a2 – x2 x ⎛ ⎞ a2 – x2 a a2 − x2 ⎟ y = a⎜ – − ln − +C ⎜ ⎟ a x x ⎝ ⎠ cos t = a2 – x2 a , csc t = , cot t = a x = − a 2 − x 2 − a ln 1 A B = + ( 1) x x+ x x +1 1 = A(x + 1) + Bx A = 1, B = –1 1 1 1 dx = dx – dx x( x + 1) x x +1 = ln x – ln x + 1 + C 2. 34. Using reasoning similar to Problem 33 b, the area is 1 2 1 1⎛ a⎞ πa + (2a ) b 2 – a 2 – ⎜ 2sin –1 ⎟ b 2 2 2 2⎝ b⎠ 1 2 a = πa + a b 2 – a 2 – b 2 sin –1 . 2 b 35. A B Cx + D + + x –1 ( x –1)2 x 2 + 1 3. 2 2 A B = + x + 3 x x ( x + 3) x x + 3 2 = A(x + 3) + Bx 2 2 A= ,B= – 3 3 2 2 1 2 B dx = dx – dx 2 3 x 3 x+3 x + 3x 2 2 = ln x – ln x + 3 + C 3 3 = 2 3 3 A B = + ( x + 1)( x –1) x + 1 x –1 x –1 3 = A(x – 1) + B(x + 1) 3 3 A= – ,B= 2 2 3 3 1 3 1 dx = – dx + dx 2 2 x + 1 2 x –1 x –1 3 3 = – ln x + 1 + ln x – 1 + C 2 2 2 = a − a2 − x2 +C x Since y = 0 when x = a, 0 = 0 – a ln 1 + C, so C = 0. y = – a 2 − x 2 − a ln 446 Section 7.5 a − a2 – x2 x Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4. 5. 5x 3 2 = 5x 2 = 5 2 x( x + 3) 2x + 6x 2 x ( x + 3) A B = + x x+3 5 = A( x + 3) + Bx 2 5 5 A= ,B=– 6 6 5x 5 1 5 1 = dx – dx 3 2 x 6 6 x+3 2x + 6x 5 5 = ln x – ln x + 3 + C 6 6 x − 11 x − 11 A B = + x + 3 x − 4 ( x + 4)( x − 1) x + 4 x − 1 x – 11 = A(x – 1) + B(x + 4) A =3, B = –2 x − 11 1 1 dx = 3 dx − 2 dx 2 + −1 x x 4 x + 3x − 4 2 7. x–7 8. = x–7 A B = + x – x – 12 ( x – 4)( x + 3) x – 4 x + 3 x – 7 = A(x + 3) + B(x – 4) 3 10 A= – ,B= 7 7 x–7 3 1 10 1 dx = – dx + dx 2 7 x–4 7 x+3 x – x – 12 3 10 = – ln x – 4 + ln x + 3 + C 7 7 2 = 3x − 13 A B = + ( x + 5)( x − 2) x + 5 x − 2 x + 3 x − 10 3x − 13 = A( x − 2) + B ( x + 5) A = 4, B = –1 1 1 3 x − 13 dx − dx dx = 4 2 x+5 x−2 x + 3x − 10 = 4 ln x + 5 − ln x − 2 + C x+π 2 2 = A B x+π = + ( x – 2π)( x – π) x – 2π x – π x – 3πx + 2π x + π = A( x − π ) + B ( x − 2π ) A = 3, B = –2 3 2 x+π dx = dx – dx 2 2 π π – 2 – x x x – 3πx + 2π = 3ln x – 2π – 2 ln x – π + C 9. = 3ln x + 4 − 2 ln x − 1 + C 6. 3x − 13 2 = 2 x + 21 A B 2 x + 21 = + 2 x + 9 x – 5 (2 x – 1)( x + 5) 2 x – 1 x + 5 2x + 21 = A(x + 5) + B(2x – 1) A = 4, B = –1 2 x + 21 4 1 dx = dx – dx 2 2 x – 1 x +5 2x + 9x – 5 2 = = 2 ln 2 x – 1 – ln x + 5 + C 10. 2 x 2 − x − 20 = 2( x 2 + x − 6) − 3x − 8 x2 + x − 6 x2 + x − 6 3x + 8 = 2− 2 x + x−6 A B 3x + 8 3x + 8 = + = 2 x + x − 6 ( x + 3)( x − 2) x + 3 x − 2 3x + 8 = A(x – 2) + B(x + 3) 1 14 A= ,B= 5 5 2 x 2 − x − 20 dx x2 + x − 6 1 1 14 1 = 2 dx − dx − dx 5 x+3 5 x−2 1 14 = 2 x − ln x + 3 − ln x − 2 + C 5 5 11. A B 17 x – 3 = + 3 x – 2 x +1 (3 x – 2)( x + 1) 3x + x – 2 17x – 3 = A(x + 1) + B(3x – 2) A = 5, B = 4 5 17 x – 3 5 4 dx = dx + dx = ln 3 x – 2 + 4 ln x + 1 + C 2 3 x – 2 x + 1 3 3x + x – 2 17 x – 3 2 = Instructor’s Resource Manual Section 7.5 447 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 12. 13. 5– x = 2 5– x A B = + ( x – π)( x – 4) x – π x – 4 x – x(π + 4) + 4π 5 – x = A(x – 4) + B(x – π ) 5–π 1 A= ,B= π–4 4–π 5– x 5–π 1 1 1 5–π 1 dx = dx + dx = ln x – π + ln x – 4 + C 2 π – 4 x – π 4 – π x – 4 π – 4 4 – π x − x(π + 4) + 4π 2 x2 + x − 4 x3 − x 2 − 2 x = A B C 2 x2 + x − 4 = + + x( x + 1)( x − 2) x x + 1 x − 2 2 x 2 + x − 4 = A( x + 1)( x − 2) + Bx( x − 2) + Cx( x + 1) A = 2, B = –1, C = 1 2 x2 + x − 4 2 1 1 dx = dx − dx + dx = 2 ln x − ln x + 1 + ln x − 2 + C 3 2 x x +1 x−2 x − x − 2x 14. 7 x2 + 2 x – 3 A B C = + + (2 x – 1)(3x + 2)( x – 3) 2 x – 1 3 x + 2 x – 3 7 x 2 + 2 x – 3 = A(3 x + 2)( x – 3) + B(2 x – 1)( x – 3) + C (2 x – 1)(3x + 2) A= 1 1 6 , B = – ,C = 35 7 5 7 x2 + 2 x – 3 1 1 1 1 6 1 dx = dx – dx + dx (2 x – 1)(3 x + 2)( x – 3) 35 2 x – 1 7 3x + 2 5 x–3 = 15. 1 1 6 ln 2 x –1 – ln 3 x + 2 + ln x – 3 + C 70 21 5 6 x 2 + 22 x − 23 (2 x − 1)( x 2 + x − 6) = 6 x 2 + 22 x − 23 A B C = + + (2 x − 1)( x + 3)( x − 2) 2 x − 1 x + 3 x − 2 6 x 2 + 22 x − 23 = A( x + 3)( x − 2) + B (2 x − 1)( x − 2) + C (2 x − 1)( x + 3) A = 2, B = –1, C = 3 6 x 2 + 22 x − 23 2 1 3 dx = dx − dx + dx = ln 2 x − 1 − ln x + 3 + 3ln x − 2 + C 2 2x −1 x+3 x−2 (2 x − 1)( x + x − 6) 16. ⎞ 1 ⎛ x3 − 6 x 2 + 11x − 6 ⎞ 1 ⎛ x 2 − 3x + 2 = ⎜ = ⎜1 + ⎟ ⎟ 3 2 3 2 3 2 ⎜ ⎟ ⎜ 4 x − 28 x + 56 x − 32 4 ⎝ x − 7 x + 14 x − 8 ⎠ 4 ⎝ x − 7 x + 14 x − 8 ⎠⎟ 1⎛ ( x − 1)( x − 2) ⎞ 1 ⎛ 1 ⎞ = ⎜1 + ⎟ = ⎜1 + ⎟ 4 ⎝ ( x − 1)( x − 2)( x − 4) ⎠ 4 ⎝ x − 4 ⎠ x3 − 6 x 2 + 11x − 6 x3 – 6 x 2 + 11x – 6 3 2 4 x – 28 x + 56 x – 32 448 Section 7.5 dx = 1 1 1 1 1 dx + dx = x + ln x – 4 + C 4 4 x–4 4 4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 17. x3 2 x +x–2 3x – 2 2 x +x–2 19. 2 x +x–2 3x – 2 A B = = + 2 ( 2)( – 1) 2 –1 x + x x + x x +x–2 3x – 2 = A(x – 1) + B(x + 2) 8 1 A= ,B= 3 3 x3 18. 3x – 2 = x –1 + x3 + x 2 2 dx = ( x − 1) dx + = x – 4+ 8 1 1 1 1 8 1 dx + dx = x 2 – x + ln x + 2 + ln x – 1 + C 3 x+2 3 x −1 2 3 3 14 x + 24 ( x + 3)( x + 2) x + 5x + 6 14 x + 24 A B = + ( x + 3)( x + 2) x + 3 x + 2 14x + 24 = A(x + 2) + B(x + 3) A = 18, B = –4 18 4 1 x3 + x 2 dx – dx = x 2 − 4 x + 18ln x + 3 – 4 ln x + 2 + C dx = ( x − 4) dx + 2 x+3 x+2 2 x + 5x + 6 x4 + 8x2 + 8 x3 − 4 x = x+ 12 x 2 + 8 x( x + 2)( x – 2) 12 x 2 + 8 A B C = + + x( x + 2)( x – 2) x x + 2 x – 2 12 x 2 + 8 = A( x + 2)( x – 2) + Bx( x – 2) + Cx( x + 2) A = –2, B = 7, C = 7 1 1 1 1 x4 + 8x2 + 8 dx = x dx – 2 dx + 7 dx + 7 dx = x 2 – 2 ln x + 7 ln x + 2 + 7 ln x – 2 + C 3 2 – 2 x x x + 2 x – 4x 20. x 6 + 4 x3 + 4 x3 – 4 x 2 272 x 2 + 4 2 x ( x – 4) = 272 x 2 + 4 = x3 + 4 x 2 + 16 x + 68 + x3 – 4 x 2 A B C + + x x2 x – 4 272 x 2 + 4 = Ax( x – 4) + B( x – 4) + Cx 2 1 1089 A = – , B = –1, C = 4 4 x 6 + 4 x3 + 4 1 1 1 1089 1 dx – dx + dx dx = ( x3 + 4 x 2 + 16 x + 68) dx – 2 3 2 x−4 4 x 4 x x – 4x 1 4 1 1 1089 = x 4 + x3 + 8 x 2 + 68 x – ln x + + ln x – 4 + C x 4 3 4 4 21. x +1 2 = A B + x − 3 ( x − 3)2 ( x − 3) x + 1 = A(x – 3) + B A = 1, B = 4 x +1 1 4 4 +C dx = dx + dx = ln x − 3 − 2 2 x−3 x−3 ( x − 3) ( x − 3) Instructor’s Resource Manual Section 7.5 449 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. 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No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 22. 23. 5x + 7 5x + 7 = 2 2 = A B + x + 2 ( x + 2) 2 x + 4 x + 4 ( x + 2) 5x + 7 = A(x + 2) + B A = 5, B = –3 5x + 7 5 3 3 dx = dx − dx = 5ln x + 2 + +C 2 2 + x x + 2 2 x + 4x + 4 ( x + 2) 3x + 2 3 2 x + 3x + 3x + 1 3x + 2 = ( x + 1) 3 = A B C + + x + 1 ( x + 1)2 ( x + 1)3 2 3x + 2 = A( x + 1) + B ( x + 1) + C A = 0, B = 3, C = –1 3x + 2 3 1 3 1 dx = dx − dx = − + +C 3 2 2 3 x + 1 2( x + 1) 2 x + 3x + 3x + 1 ( x + 1) ( x + 1) 24. x6 A B C D E F G + + + + + + 2 2 3 4 x – 2 ( x – 2) 1 – x (1 – x) ( x – 2) (1 – x) (1 – x) (1 – x) (1 – x)5 A = 128, B = –64, C = 129, D = –72, E = 30, F = –8, G = 1 ⎡ 128 x6 64 129 72 30 8 1 ⎤ dx = ⎢ – + – + − + ⎥ dx 2 5 2 2 3 4 1 – x (1 – x) ( x – 2) (1 – x) (1 – x) (1 – x) (1 – x)5 ⎦⎥ ⎣⎢ x – 2 ( x – 2) 2 = 5 = 128ln x – 2 + 25. 3 x 2 − 21x + 32 3 2 x − 8 x + 16 x 64 72 15 8 1 –129 ln 1 – x + − + − +C 2 3 1 – x (1 – x) x–2 3(1 – x) 4(1 – x) 4 = 3 x 2 − 21x + 32 x( x − 4) 2 = A B C + + x x − 4 ( x − 4)2 2 3x − 21x + 32 = A( x − 4) 2 + Bx( x − 4) + Cx A = 2, B = 1, C = –1 3x 2 − 21x + 32 2 1 1 1 +C dx = dx + dx − dx = 2 ln x + ln x − 4 + 3 2 2 x − x x − 4 4 x − 8 x + 16 ( x − 4) 26. 27. 450 x 2 + 19 x + 10 4 3 = x 2 + 19 x + 10 3 = A B C D + + + x x 2 x3 2 x + 5 2 x + 5x x (2 x + 5) A = –1, B = 3, C = 2, D = 2 x 2 + 19 x + 10 ⎛ 1 3 2 2 ⎞ 3 1 + ln 2 x + 5 + C dx = ⎜ – + + + dx = – ln x – – 4 3 2 3 2x + 5 ⎟ x x2 x 2 x + 5x x x ⎝ ⎠ 2 x2 + x – 8 3 = 2x2 + x – 8 2 = A Bx + C + x x2 + 4 x + 4x x( x + 4) A = –2, B = 4, C = 1 2 x2 + x – 8 1 4x +1 1 2x 1 dx = –2 dx + dx = −2 dx + 2 dx + dx 2 2 3 2 x x x + 4x x +4 x +4 x +4 1 ⎛x⎞ = –2 ln x + 2 ln x 2 + 4 + tan –1 ⎜ ⎟ + C 2 ⎝2⎠ Section 7.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. 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No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 28. 3x + 2 3x + 2 = 2 2 x( x + 2) + 16 x x( x + 4 x + 20) 1 1 13 A = , B = – ,C = 10 10 5 = A Bx + C + 2 x x + 4 x + 20 1 x + 13 – 10 1 1 1 1 14 1 1 2x + 4 5 dx + dx − dx + dx = dx 2 2 2 2 10 x 20 10 x 5 x + 4 x + 20 x + 4 x + 20 ( x + 2) + 16 x( x + 2) + 16 x 1 7 ⎛ x+2⎞ 1 2 = ln x + tan –1 ⎜ ⎟ – ln x + 4 x + 20 + C 10 10 ⎝ 4 ⎠ 20 3x + 2 29. 30. 31. dx = 2 x 2 – 3x – 36 A Bx + C + 2 x – 1 (2 x − 1)( x + 9) x2 + 9 A = –4, B = 3, C = 0 2 x 2 – 3 x – 36 1 3x 3 dx = –4 dx + dx = –2 ln 2 x – 1 + ln x 2 + 9 + C 2 2 2 x – 1 2 (2 x – 1)( x + 9) x +9 2 1 = 1 = 4 x –16 ( x − 2)( x + 2)( x 2 + 4) A B Cx + D = + + x – 2 x + 2 x2 + 4 1 1 1 A = , B = – , C = 0, D = – 32 32 8 1 1 1 1 1 1 1 1 1 1 ⎛x⎞ dx = dx – dx − dx = ln x – 2 – ln x + 2 – tan –1 ⎜ ⎟ + C 4 2 32 x – 2 32 x + 2 8 x +4 32 32 16 ⎝2⎠ x – 16 1 2 ( x – 1) ( x + 4) 2 = A B C D + + + x – 1 ( x – 1) 2 x + 4 ( x + 4)2 2 1 2 1 , B = ,C = ,D= 125 25 125 25 1 2 1 1 1 2 1 1 1 dx = – dx + dx + dx + dx 2 2 2 125 x –1 25 125 x + 4 25 ( x –1) ( x + 4) ( x –1) ( x + 4)2 2 1 2 1 =– + +C ln x – 1 – ln x + 4 – 125 25( x – 1) 125 25( x + 4) A= – 32. x3 – 8 x 2 – 1 ( x + 3)( x 2 – 4 x + 5) –7 x 2 + 7 x – 16 2 ( x + 3)( x – 4 x + 5) A= – = 1+ = –7 x 2 + 7 x – 16 ( x + 3)( x 2 − 4 x + 5) A Bx + C + 2 x + 3 x – 4x + 5 50 41 14 , B = – ,C = 13 13 13 ⎡ 50 ⎛ 1 ⎞ – 41 x + 14 ⎤ 13 13 ⎥ dx = ⎢1 – ⎜ dx ⎟+ ⎢⎣ 13 ⎝ x + 3 ⎠ x 2 – 4 x + 5 ⎥⎦ ( x + 3)( x 2 – 4 x + 5) 50 1 68 1 41 2x − 4 = dx − dx − dx − dx 13 x + 3 13 ( x − 2) 2 + 1 26 x 2 − 4 x + 5 x3 – 8 x 2 – 1 = x– 50 68 41 ln x + 3 – tan –1 ( x – 2) – ln x 2 – 4 x + 5 + C 13 13 26 Instructor’s Resource Manual Section 7.5 451 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 33. x = sin t, dx = cos t dt (sin 3 t − 8sin 2 t − 1) cos t (sin t + 3)(sin 2 t − 4sin t + 5) dt = x3 − 8 x 2 − 1 ( x + 3)( x 2 − 4 x + 5) dx 50 68 41 ln x + 3 − tan −1 ( x − 2) − ln x 2 − 4 x + 5 + C 13 13 26 which is the result of Problem 32. (sin 3 t – 8sin 2 t – 1) cos t 50 68 41 dt = sin t – ln sin t + 3 – tan –1 (sin t – 2) – ln sin 2 t – 4sin t + 5 + C 2 13 13 26 (sin t + 3)(sin t – 4sin t + 5) = x− 34. x = sin t, dx = cos t dt cos t 1 1 1 1 ⎛x⎞ dt = dx = ln x − 2 − ln x + 2 − tan −1 ⎜ ⎟ + C 4 4 32 32 16 ⎝2⎠ sin t − 16 x − 16 which is the result of Problem 30. cos t 1 1 1 ⎛ sin t ⎞ dt = ln sin t – 2 – ln sin t + 2 – tan –1 ⎜ ⎟+C 4 32 32 16 ⎝ 2 ⎠ sin t – 16 35. x3 – 4 x 2 = 2 Ax + B + Cx + D ( x + 1) x + 1 ( x 2 + 1) 2 A = 1, B = 0, C = –5, D = 0 x3 – 4 x 2 ( x + 1) 2 2 x dx = 2 x +1 dx − 5 36. x = cos t, dx = –sin t dt (sin t )(4 cos 2 t –1) (cos t )(1 + 2 cos 2 t + cos 4 t ) 4 x2 − 1 x 2 ( x + 1) dt = – 4 x2 − 1 2 dx = 1 5 ln x 2 + 1 + +C 2 2 2( x + 1) 4 x 2 –1 x(1 + 2 x 2 + x 4 ) dx A Bx + C Dx + E + + x x 2 + 1 ( x 2 + 1) 2 x(1 + 2 x + x ) x( x + 1) A = –1, B = 1, C = 0, D = 5, E = 0 ⎡ 1 x 5x ⎤ 1 5 1 5 2 − ⎢− + + + C = ln cos t − ln cos 2 t + 1 + +C ⎥ dx = ln x − ln x + 1 + 2 2 2 2 x 2 2 x + 1 ( x + 1) ⎦⎥ 2( x + 1) 2(cos 2 t + 1) ⎣⎢ 2 37. 4 2 x3 + 5 x 2 + 16 x 5 3 = 2 = = 2 x(2 x 2 + 5 x + 16) 2 2 dx = 2 2 x 2 + 5 x + 16 16sec2 4 d 2 2 = Ax + B 2 + Cx + D x + 8 x + 16 x x( x + 8 x + 16) ( x + 4) x + 4 ( x 2 + 4)2 A = 0, B = 2, C = 5, D = 8 2 x3 + 5 x 2 + 16 x 2 5x + 8 2 5x 8 dx = dx + dx = dx + dx + dx 5 3 2 2 2 2 2 2 2 x + 8 x + 16 x x +4 ( x + 4) x +4 ( x + 4) ( x + 4)2 8 dx, let x = 2 tan , dx = 2sec2 d . To integrate 2 2 ( x + 4) 8 4 = ⎛1 1 ⎞ = cos 2 d = ⎜ + cos 2 ⎟ d 2 2 ⎝ ⎠ ( x + 4) 16sec x x 1 1 1 1 1 +C = + sin 2 + C = + sin cos + C = tan –1 + 2 2 x2 + 4 2 4 2 2 2 x3 + 5 x 2 + 16 x x 5 1 x x 3 x 2x – 5 dx = tan –1 – + tan –1 + + C = tan –1 + +C 5 3 2 2 2 2( x + 4) 2 2 x +4 2 2 2( x 2 + 4) x + 8 x + 16 x 452 Section 7.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 38. x –17 x –17 A B = + ( x + 4)( x – 3) x + 4 x – 3 = 2 x + x –12 A = 3, B = –2 6 x –17 3 2 ⎞ 6 – ⎜ ⎟ dx = ⎡⎣3ln x + 4 – 2 ln x – 3 ⎤⎦ 4 = (3ln10 – 2 ln 3) – (3ln 8 – 2 ln1) ⎝ x +4 x –3⎠ x + x –12 = 3ln10 – 2 ln 3 – 3ln 8 ≈ –1.53 39. u = sin 4 du = cos cos π/4 0 6⎛ dx = 2 4 (1 – sin 2 )(sin 1 2 2 d 2 = 2 + 1) 2 d = 1 1/ 2 0 2 2 (1 – u )(u + 1) 2 du = 1/ 2 0 1 1 du + 1+ u 4 1/ 2 ⎡ 1 1 1 1⎛ u ⎞⎤ = ⎢ – ln 1 – u + ln 1 + u + tan –1 u + ⎜ tan –1 u + ⎟⎥ 2 8 4 4⎝ u + 1 ⎠⎦0 ⎣ 8 41. 1 1/ 2 0 2 u +1 du + 1 2 1 1/ 2 0 2 (u + 1) 2 du 1/ 2 ⎡1 1+ u 1 ⎤ u = ⎢ ln + tan –1 u + ⎥ 2 4(u + 1) ⎥⎦ 0 ⎢⎣ 8 1 − u 2 1 1 1 + tan –1 + ≈ 0.65 2 −1 2 2 6 2 3x + 13 2 1 2 (u + 1) 2 = x + 4x + 3 A = –2, B = 5 5 3 x + 13 1 du 2 +1 (To integrate 40. (1 – u )(1 + u )(u 2 + 1)2 A B Cu + D Eu + F + + + 1 – u 1 + u u 2 + 1 (u 2 + 1)2 (1 – u )(u + 1) 1 1 1 1 A = , B = , C = 0, D = , E = 0, F = 8 8 4 2 1/ 2 1 1 1/ 2 1 1 du = du + 2 2 2 0 8 0 1− u 8 (1 − u )(u + 1) 1 = ln 8 1 1/ 2 0 2 x + 4x + 3 du, let u = tan t.) A B 3 x + 13 = + ( x + 3)( x + 1) x + 3 x + 1 5 1 dx = ⎡⎣ –2 ln x + 3 + 5ln x + 1 ⎤⎦ = –2 ln 8 + 5 ln 6 + 2 ln 4 – 5 ln 2 = 5ln 3 − 2 ln 2 ≈ 4.11 dy = y (1 − y ) so that dt 1 dy = 1 dt = t + C1 y (1 − y ) a. Using partial fractions: 1 A B A(1 − y ) + By = + = ⇒ y (1 − y ) y 1 − y y (1 − y ) A + ( B − A) y = 1 + 0 y ⇒ A = 1, B − A = 0 ⇒ A = 1, B = 1 ⇒ 1 1 1 = + y (1 − y ) y 1 − y ⎛1 ⎛ y ⎞ 1 ⎞ Thus: t + C1 = ⎜ + ⎟ dy = ln y − ln(1 − y ) = ln ⎜ ⎟ so that ⎝ y 1− y ⎠ ⎝1− y ⎠ y et = et +C1 = Cet or y (t ) = 1 1− y +et (C =eC1 ) C Since y (0) = 0.5, 0.5 = b. y (3) = e3 1 + e3 1 1 +1 C or C = 1 ; thus y (t ) = et 1+et ≈ 0.953 Instructor’s Resource Manual Section 7.5 453 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 42. so that y = e 2.4t +8000C1 = Ce 2.4t or 8000 − y (C =e8000C1 ) dy 1 y (12 − y ) so that = dt 10 1 1 1 dy = dt = t + C1 y (12 − y ) 10 10 y (t ) = a. Using partial fractions: 1 A B A(12 − y ) + By = + = ⇒ y (12 − y ) y 12 − y y (12 − y ) thus y (t ) = 1 1 ⇒ A= , B= 12 12 1 1 1 ⇒ = + y (12 − y ) 12 y 12(12 − y ) b. y (3) = Thus: y = e1.2t +12C1 = Ce1.2t or 12 − y (C =e12C1 ) y (t ) = 44. a. 12e 1 1.2t +e C 12 1 +1 C or C = 0.2 ; 43. 12e3.6 5 + e3.6 8000e7.2 7 + e7.2 ≈ 7958.4 Using partial fractions: 1 A B = + y (4000 − y ) y 4000 − y A(4000 − y ) + By = y (4000 − y ) 1 1 , B= 4000 4000 ⎤ 1 1 ⎡1 1 ⇒ = + y (4000 − y ) 4000 ⎢⎣ y (4000 − y ) ⎥⎦ Thus: dy = 0.0003 y (8000 − y ) so that dt 1 dy = 0.0003 dt = 0.0003t + C1 y (8000 − y ) Section 7.5 8000e2.4t 7+e2.4t ⇒ A= ⎛1 ⎞ 1 1 ⎜ + ⎟ dy = 4000 ⎝ y (4000 − y ) ⎠ ⎛ ⎞ 1 1 y ln ⎜ [ln y − ln(4000 − y)] = ⎟ 4000 4000 ⎝ 4000 − y ⎠ so that y = e4t + 4000C1 = Ce4t or 4000 − y (C =e4000C1 ) 0.001t + C1 = a. Using partial fractions: 1 A B A(8000 − y ) + By = + = y (8000 − y ) y 8000 − y y (8000 − y ) ⇒ 8000 A + ( B − A) y = 1 + 0 y ⇒ 8000 A = 1, B − A = 0 1 1 ⇒ A= , B= 8000 8000 ⎤ 1 1 ⎡1 1 ⇒ = + y (8000 − y ) 8000 ⎢⎣ y (8000 − y ) ⎥⎦ Thus: ⎛1 ⎞ 1 1 0.0003t + C1 = ⎜ + ⎟ dy = 8000 ⎝ y (8000 − y ) ⎠ 454 1 ; 7 ⇒ 4000 A = 1, B − A = 0 ≈ 10.56 ⎛ ⎞ 1 1 y ln ⎜ [ln y − ln(8000 − y)] = ⎟ 8000 8000 ⎝ 8000 − y ⎠ or C = 1 +1 C ⇒ 4000 A + ( B − A) y = 1 + 0 y 12e1.2t thus y (t ) = 5+e1.2t b. y (3) = 8000 dy = 0.001y (4000 − y ) so that dt 1 dy = 0.001 dt = 0.001t + C1 y (4000 − y ) 1.2t Since y (0) = 2.0, 2.0 = 1 +e2.4t C Since y (0) = 1000, 1000 = 12 A + ( B − A) y = 1 + 0 y ⇒ 12 A = 1, B − A = 0 ⎛ 1 ⎞ 1 1 t + C1 = ⎜ + ⎟ dy = 10 ⎝ 12 y 12(12 − y ) ⎠ 1 1 ⎛ y ⎞ [ln y − ln(12 − y)] = ln ⎜ ⎟ so that 12 12 ⎝ 12 − y ⎠ 8000e 2.4t y (t ) = 4000e 4t 1 + e 4t C Since y (0) = 100, 100 = thus y (t ) = b. y (3) = 4000 1 +1 C or C = 1 ; 39 4000e 4t 39+e4t 4000e12 39 + e12 ≈ 3999.04 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 45. dy = ky ( L − y ) so that dt 1 dy = k dt = kt + C1 y( L − y) Using partial fractions: 1 A B A( L − y ) + By = + = ⇒ y( L − y) y L − y y( L − y) Thus if y0 < L , then y (t ) < L for all positive t (see note at the end of problem 45 solution) and so the graph will be concave up as long as L − 2 y > 0 ; that is, as long as the population is less than half the capacity. 49. a. LA + ( B − A) y = 1 + 0 y ⇒ LA = 1, B − A = 0 ⇒ A= 1 1 1 1 ⎡1 1 ⎤ = ⎢ + , B= ⇒ L L y ( L − y ) L ⎣ y L − y ⎥⎦ dy = kdt y (16 – y ) 1 ⎛1 1 ⎞ ⎜ + ⎟ dy = L ⎝ y L− y⎠ 1 1 ⎛ y ⎞ [ln y − ln( L − y )] = ln ⎜ ⎟ so that L L ⎝ L− y⎠ Thus: kt + C1 = 1 ⎛1 1 ⎞ ⎜ + ⎟ dy = kt + C 16 ⎝ y 16 – y ⎠ 1 ( ln y – ln 16 – y ) = kt + C 16 y ln = 16kt + C 16 – y y Le kLt = e kLt + LC1 = Ce kLt or y (t ) = 1 L− y +ekLt (C =e LC1 ) C If y0 = y (0) = L 1 +1 C then final formula is y (t ) = 1 L − y0 = ; so our C y0 Le kLt ⎛ L − y0 ⎞ kLt ⎜ ⎟+e ⎝ y0 ⎠ (Note: if y0 < L , then u = ekLt u + ekLt y = Ce16kt 16 – y . y(50) = 4: ( 1 ln 7 )t ( 1 ln 7 )t ( 1 ln 7 )t 16e 50 3 16 y= = 1 ln 7 t – ( ) 1 + 7e ( 501 ln 73 )t 7 + e 50 3 7 y = 16e 50 3 – ye 50 3 b. y (90) = c. 9= L − y0 47. If y0 < L , then y (0) = ky0 ( L − y0 ) > 0 and the population is increasing initially. 48. The graph will be concave up for values of t that make y (t ) > 0 . Now dy d y (t ) = = [ ky ( L − y ) ] = dt dt k [ − yy + ( L − y ) y ] = k [ ky ( L − y ) ][ L − 2 y ] Instructor’s Resource Manual ) ( 46. Since y (0) = ky0 ( L − y0 ) is negative if y0 > L , the population would be decreasing at time t = 0. Further, since L L lim y (t ) = lim = =L ⎛ ⎞ 0 +1 t →∞ t →∞ ⎜ L − y0 ⎟ +1 is monotonic as t → ∞ ,we conclude y0 ekLt that the population would decrease toward a limiting value of L. 1 1 800k 1 7 , so k = ln = e 3 7 800 3 7 y 1 1 ln t = e 50 3 16 – y 7 < 1 ; thus y (t ) < L for all t) (no matter how y0 and L compare), and since 1 y 1 = C; = e16 kt 7 16 – y 7 y(0) = 2: L − y0 > 0 and y0 ⎜ y ekLt ⎟ ⎝ 0 ⎠ dy = ky (16 – y ) dt dy = kdt y (16 – y ) 16 ( ) – 1 ln 7 90 1 + 7e 50 3 ≈ 6.34 billion 16 ( ) – ( 1 ln 7 )t 16 7e 50 3 = –1 1 + 7e ( ) – 1 ln 7 t 50 3 – 1 ln 7 t e 50 3 – 9 = 1 9 ( 501 ln 73 ) t = ln 91 ⎛ ln 1 ⎞ t = –50 ⎜ 9 ⎟ ≈ 129.66 ⎜ ln 7 ⎟ ⎝ 3⎠ The population will be 9 billion in 2055. Section 7.5 455 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 50. a. 51. a. Separating variables, we obtain dx = k dt (a − x)(b − x) dy = ky (10 – y ) dt dy = kdt y (10 – y ) 1 A B = + (a − x )(b − x) a − x b − x 1 ⎛1 1 ⎞ ⎜ + ⎟ dy = kdt 10 ⎝ y 10 – y ⎠ y ln = 10kt + C 10 – y 1 1 ,B= a −b a −b dx (a − x)(b − x) A=− y = Ce10kt 10 – y 1 1 ⎞ ⎛ + ⎜− ⎟ dx = k dt ⎝ a−x b−x⎠ − ln b − x = kt + C a−b 1 a−x = kt + C ln a−b b− x a−x = Ce( a −b) kt b−x a Since x = 0 when t = 0, C = , so b a ( a −b) kt a − x = (b − x) e b ⎛ a ⎞ a 1 − e( a −b ) kt = x ⎜1 − e( a −b) kt ⎟ ⎝ b ⎠ 1 a −b ln a − x = 1 y 1 = C; = e10 kt 4 10 – y 4 y(0) = 2: 2 1 500k 1 8 ,k = ln = e 3 4 500 3 y(50) = 4: ( 501 ln 83 )t y 1 = e 10 – y 4 ( 501 ln 83 )t – ye( 501 ln 83 )t ( 1 ln 8 )t 10e 50 3 10 y= = 1 ln 8 t – ( ) 1 + 4e ( 501 ln 83 )t 4 + e 50 3 4 y = 10e b. c. y (90) = 9= 10 ( ) – 1 ln 8 90 1 + 4e 50 3 10 ( ≈ 5.94 billion ) – 1 ln 8 t 1 + 4e 50 3 ) ( – 1 ln 8 t 10 4e 50 3 = – 1 9 ( 8 ) – 1 ln t 1 e 50 3 = 36 1 ⎛ 1 8⎞ – ⎜ ln ⎟ t = ln 36 ⎝ 50 3 ⎠ ⎞ ⎟ ≈ 182.68 ⎟ ⎠ The population will be 9 billion in 2108. t= ⎛ ln 1 –50 ⎜ 36 ⎜ ln 8 ⎝ 3 ( x(t ) = ) a(1 − e( a −b) kt ) 1 − ba e( a −b) kt = ab(1 − e( a −b) kt ) b − ae( a −b) kt b. Since b > a and k > 0, e( a −b ) kt → 0 as t → ∞ . Thus, ab(1) x→ =a. b−0 c. x(t ) = 8(1 − e−2kt ) 4 − 2e−2kt x(20) = 1, so 4 − 2e−40k = 8 − 8e−40k 6e−40k = 4 1 2 k = − ln 40 3 t / 20 e−2kt = et / 20 ln 2 / 3 = eln(2 / 3) ⎛2⎞ =⎜ ⎟ ⎝3⎠ t / 20 ( 23 ) ⎞⎟⎠ t / 20 ( 23 ) 3⎞ ⎛ 4 ⎜1 − ( 23 ) ⎟ ⎠ = 38 ≈ 1.65 grams x(60) = ⎝ 3 23 2 − ( 23 ) ⎛ 4 ⎜1 − x(t ) = ⎝ 2− 456 Section 7.5 t / 20 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. d. If a = b, the differential equation is, after separating variables dx = k dt (a − x) 2 dx = k dt (a − x) 2 1 = kt + C a−x 1 =a−x kt + C 1 x(t ) = a − kt + C 1 Since x = 0 when t = 0, C = , so a a 1 x(t ) = a − = a− akt + 1 kt + 1 a 1 ⎞ ⎛ ⎛ akt ⎞ = a ⎜1 − ⎟ = a⎜ ⎟. ⎝ akt + 1 ⎠ ⎝ akt + 1 ⎠ 52. Separating variables, we obtain dy = k dt . ( y − m)( M − y ) 1 A B = + ( y − m)( M − y ) y − m M − y 1 1 ,B= M −m M −m ⎛ 1 1 1 ⎞ dy = + ⎜ ⎟ dy ( y − m)( M − y ) M − m ⎝ y − m M − y ⎠ = k dt A= ln y − m − ln M − y = kt + C M −m 1 y−m = kt + C ln M −m M − y y−m = Ce( M − m) kt M−y y − m = ( M − y )Ce m + MCe( M − m) kt 1 + Ce( M − m) kt 1 C D = + ( A − y )( B + y ) A − y B + y 1 1 ,D= A+ B A+ B ⎛ 1 dy 1 1 ⎞ = + ⎜ ⎟ dy ( A − y )( B + y ) A + B ⎝ A − y B + y ⎠ = k dt C= − ln( A − y ) + ln( B + y ) = kt + C A+ B 1 B+ y ln = kt + C A+ B A− y B+ y = Ce( A+ B ) kt A− y B + y = ( A − y )Ce( A+ B ) kt y (1 + Ce( A+ B ) kt ) = ACe( A+ B ) kt − B y (t ) = ACe( A+ B ) kt − B 1 + Ce( A+ B ) kt 54. u = sin x, du = cos x dx π/2 1 cos x 1 dx = 1 du 2 2 π / 6 sin x(sin 2 x + 1) 2 2 u (u + 1) 1 A Bu + C Du + E = + + 2 2 u u 2 + 1 (u 2 + 1)2 u (u + 1) A = 1, B = –1, C = 0, D = –1, E = 0 1 1 du 1 2 2 2 u (u + 1) 11 1 u 1 u = 1 du − 1 du − 1 du 2 2 2 2u 2 u +1 2 (u + 1) 1 ⎡ ⎤ 1 1 = ⎢ln u − ln(u 2 + 1) + ⎥ 2 2(u 2 + 1) ⎦⎥ 1 ⎣⎢ 2 ( M − m) kt y (1 + Ce( M − m) kt = m + MCe( M − m) kt y= 53. Separating variables, we obtain dy = k dt ( A − y )( B + y ) = 1 1 ⎛ 1 1 5 2⎞ = 0 − ln 2 + − ⎜ ln − ln + ⎟ ≈ 0.308 2 4 ⎝ 2 2 4 5⎠ me−( M − m) kt + MC e−( M − m) kt + C − ( M − m ) kt As t → ∞, e → 0 since M > m. MC = M as t → ∞ . Thus y → C Instructor’s Resource Manual Section 7.5 457 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 7.6 Concepts Review 5. Trig identity cos 2 u = 1. substitution substitution. 2 ⎛ 1 + cos 4 x ⎞ cos 4 2 x dx = ⎜ ⎟ dx = 2 ⎝ ⎠ 2. 53 3. approximation ⎡ ⎤ 1 ⎢ ⎥ 2 1 + 2 cos 4 x + cos 4 x ⎥ dx = ⎢ 4 u = 4x ⎢⎣ du = 4 dx ⎥⎦ 4. 0 Problem Set 7.6 Note: Throughout this section, the notation Fxxx refers to integration formula number xxx in the back of the book. 1. Integration by parts. dv = e−5 x u=x 1 v = − e−5 x 5 1 1 xe −5 x dx = − xe−5 x − − e−5 x dx 5 5 1 −5 x 1 −5 x = − xe − e +C 5 25 1 1⎞ ⎛ = − e −5 x ⎜ x + ⎟ + C 5 5⎠ ⎝ du = 1 dx sin 3 x cos x dx = u 3 du = u = sin x du = cos x dx ( ) u = x 2 +9 du = 2 x dx 3. Substitution 2 ln x dx = x ln 2 u = ln x du = 1 dx 0 ln 2 ⎡ u2 ⎤ u du = ⎢ ⎥ ⎢⎣ 2 ⎥⎦ 0 = ( ln 2 )2 2 ≈ 0.2402 x 4. Partial fractions x x dx = dx 2 ( x − 3)( x − 2) x − 5x + 6 x A B = + = ( x − 3)( x − 2) ( x − 3) ( x − 2) A( x − 2) + B ( x − 3) ( A + B) x + (−2 A − 3B) = ⇒ ( x − 3)( x − 2) ( x − 3)( x − 2) A + B = 1, − 2 A − 3B = 0 ⇒ A = 3, B = −2 x 2 x − 5x + 6 dx = Section 7.6 u4 sin 4 x +C = +C 4 4 7. Partial fractions 1 1 dx = dx 2 ( x + 4)( x + 2) x + 6x + 8 1 A B = + = ( x + 4)( x + 2) ( x + 4) ( x + 2) A( x + 2) + B ( x + 4) ( A + B ) x + (2 A + 4 B) = ⇒ ( x + 4)( x + 2) ( x + 4)( x + 2) 1 1 A + B = 0, 2 A + 4 B = 1 ⇒ A = − , B = 2 2 2 1 1 2⎛ 1 1 ⎞ = − dx 1 x 2 + 6 x + 8 2 1 ⎜⎝ x + 2 x + 4 ⎟⎠ = 1 1 ⎡ ( x + 2) 2 ⎡ ln x + 2 − ln x + 4 ⎤⎦ = ⎢ ln 1 2⎣ 2 ⎣ ( x + 4) 2 ⎤ ⎥ ⎦1 1⎛ 4 3 ⎞ 1 10 = ⎜ ln − ln ⎟ = ln ≈ 0.0527 2⎝ 6 5⎠ 2 9 3 2 − dx = ( x − 3) ( x − 2) 3ln x − 3 − 2 ln x − 2 = ln 458 ⎡ ⎤ ⎢ ⎥ 1⎢ 1 ⎛ 1 + cos8 x ⎞ ⎥ x + sin 4 x + ⎜ ⎟ dx ⎥ = 4⎢ 2 2 ⎝ ⎠ ⎥ ⎢ v = 8x ⎢⎣ ⎥⎦ dv = 8dx 1⎡ 1 1 1 ⎤ x + sin 4 x + x + sin 8 x ⎥ + C = ⎢ 4⎣ 2 2 16 ⎦ 1 [ 24 x + 8sin 4 x + sin 8 x ] + C 64 6. Substitution 2. Substitution x 1 1 dx = du = ln u + C = ln x 2 + 9 + C 2 2 u x +9 1 1 + cos 2u and 2 ( x − 3)3 ( x − 2) 2 +C Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 8. Partial fractions 1 1 dt = dt 2 (1 − t )(1 + t) 1− t 1 A B = + = (1 − t )(1 + t ) (1 − t ) (1 + t ) A(1 + t ) + B (1 − t ) ( A − B)t + ( A + B) = ⇒ (1 − t )(1 + t ) (1 − t )(1 + t ) 1 1 A + B = 1, A − B = 0 ⇒ A = , B = 2 2 1 1 2 1 dt = 1 2 ⎛ 1 + 1 ⎞ dx ⎜ ⎟ 0 1− t2 2 0 ⎝ 1− t 1+ t ⎠ 1 1 ⎡− ln 1 − t + ln 1 + t ⎤⎦ 2 = ⎣ 0 2 13. a. Formula 96 x 3x + 1 dx = 2 135 F 96 a =3, b =1 ( 9 x − 2 )( 3x + 1) 9. Substitution 5 0 7 x x + 2 dx = 2 u = x+2 u2 = x+2 2u du = dx (u 2 − 2)(u )2u du = 2 ⎡ 5 3u − 10u 3 ⎤ ⎣ ⎦ 15 = 2 10. Substitution 4 1 dt = 3 t − 2t e x 3e x + 1 dx = u 3u + 1 du u = e x , du = e x dx ( )( 2 ⎣⎡ln u − 2 ⎤⎦ 8 6 = 2 3 ) 2 +C 2 9e x − 2 3e x + 1 135 14. a. Formula 96 2t (3 − 4t ) dt = 2 t (3 − 4t ) dt = F 96 a =−4, b =3 2 −π 6 u2 2 = 2 ln 1 du = 2 du = 6 u−2 8 −u 4 π 0 π 0 4 sin 2 x dx = = 4 [ − cos u ] 0 2 ex =4 dx = 2x u = 4e x , du = 4e x dx 1 du 4 9 − u2 = part a. 1 4e x + 3 +C ln 24 4e x − 3 16. a. Substitution, Formula 18 dx dx 5 du =− =− 2 2 5 5 x − 11 11 − 5 x 11 − u 2 u = 5x, du = 5 dx − 5 11 ln 5 22 = Instructor’s Resource Manual a =3 1 ⎡1 u + 3 ⎤ 1 4x + 3 +C = +C ln ln ⎢ ⎥ 4 ⎣6 u − 3 ⎦ 24 4 x − 3 u =2 x du = 2 dx π 2 sin u du 15. a. Substitution, Formula 18 dx 1 du = = 2 4 9 − u 2 F18 9 − 16 x 9 − 16e cos 2 x sin x dx = 0 sin 2 x dx = 8 1 (2 cos t + 1)(3 − 4 cos t ) 2 + C 20 ≈ 1.223 6 −2 12. Use of symmetry; substitution 0 part a. 3 b. Substitution, Formula 18 8−2 2 2π 3 ⎤ ⎡ 2 2⎢ (−12t − 6)(3 − 4t ) 2 ⎥ + C = ⎣ 240 ⎦ 3 1 − (2t + 1)(3 − 4t ) 2 + C 10 u = 4 x , du = 4 dx u 11. Use of symmetry; this is an odd function, so π = F 96 a =3, b =1 u = cos t , du = −sin t dt 7 2 ⎡ 77 7 + 8 2 ⎤⎦ ≈ 28.67 15 ⎣ 8 u = 2t , u 2 = 2t u du = dt +C b. Substitution; Formula 96 cos t 3 − 4 cos t sin t dt = − u 3 − 4u du = ⎡ u 5 2u 3 ⎤ 7 2u 4 − 4u 2 du =2 ⎢ − ⎥ 2 3 ⎥⎦ ⎢⎣ 5 7 2 b. Substitution; Formula 96 1 1 ⎡ (1 + t ) ⎤ 2 ⎢ln ⎥ ≈ 0.5493 2 ⎣ (1 − t ) ⎦ 0 3 55 ln 110 5 x + 11 5 x − 11 5 x − 11 5 x + 11 = F 18 a = 11 +C +C Section 7.6 459 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. Substitution, Formula 18 x dx 5 du =− 4 10 11 − u 2 5 x − 11 u = 5x 2 , du = 2 5 x dx = F 18 a = 11 5 x 2 − 11 55 ln 220 19. a. Substitution, Formula 45 dx 3 du = 3 5 + 3x 2 5 + u2 2 5 x + 11 u = 3x du = 3 dx a =3 du = 2 dx ⎛ 2x ⎞ 1⎛ 2 2 ⎞ 81 2 sin −1 ⎜⎜ ⎟⎟ + C ⎜ x(4 x − 9) 9 − 2 x ⎟ + ⎠ 16 ⎝ 32 ⎝ 3 ⎠ b. Substitution, Formula 57 2 2 u 9 − u 2 du sin 2 x cos x 9 − 2sin 2 x dx = 4 u = 2 sin x du = 2 cos x dx a =3 81 2 −1 ⎛ 2 sin x ⎞ sin ⎜⎜ ⎟⎟ + C 32 3 ⎝ ⎠ 2 16 − u du = u F 55 a=4 u = 3t du = 3 dt 16 − 3t 2 − 4ln 4 + 16 − 3t 2 +C 3t 16 − 3t 6 t 2 16 − 3t 6 dt = dt = t t3 u = 3t 3 du = 3 3 t 2 dt 16 − u 2 du = u F 55 a=4 ⎧ 1⎪ 4 + 16 − 3t 6 6 ⎨ 16 − 3t − 4 ln 3⎪ 3 t3 ⎩ 460 Section 7.6 a= 5 3 ln 3 x 2 + 5 + 3 x 4 + C 6 20. a. Substitution; Formula 48 5 2 t 2 3 + 5t 2 dt = u 3 + u 2 du 25 u= 5t = F 48 a= 3 ⎧⎛ 5 ⎞ ⎫ t ⎟⎟ 10t 2 + 3 ⎛⎜ 3 + 5t 2 ⎞⎟ − ⎪ ⎪⎜⎜ 5 ⎪⎝ 8 ⎠ ⎝ ⎠ ⎪ ⎨ ⎬+C = 25 ⎪ 9 ⎪ 2 ⎪ 8 ln 5 t + 3 + 5t ⎪ ⎩ ⎭ 1 5t (10t 2 + 3) 3 + 5t 2 − 9 5 ln 5t + 3 + 5t 2 + C 200 ( ) } b. Substitution; Formula 48 t 8 3 + 5t 6 dt = t 6 3 + 5t 6 t 2 dt = u = 5 t3 du = 3 5 t 2 dt 5 2 u 3 + u 2 du 75 b. Substitution, Formula 55 1 3 u = 3x 2 du = 2 3 x dx = F 45 { 18. a. Substitution, Formula 55 16 − 3t dt = t b. Substitution, Formula 45 x 3 du dx = 4 6 5 + 3x 5 + u2 du = 5 dt 1⎛ 2 2 ⎞ = ⎜ sin x(4sin x − 9) 9 − 2sin x ⎟ ⎠ F 57 16 ⎝ 2 a= 5 3 ln 3 x + 5 + 3 x 2 + C 3 +C 17. a. Substitution, Formula 57 2 2 x 2 9 − 2 x 2 dx = u 9 − u 2 du = 4 F 57 u= 2 x + = F 45 ⎫ ⎪ ⎬+C ⎪⎭ = F 48 a= 3 ⎧⎛ 5 3 ⎞ ⎫ t ⎟⎟ 10t 6 + 3 ⎛⎜ 3 + 5t 6 ⎞⎟ − ⎪ ⎪⎜⎜ 5 ⎪⎝ 8 ⎠ ⎝ ⎠ ⎪ ⎨ ⎬+C = 75 ⎪ 9 ⎪ 3 6 ⎪ 8 ln 5 t + 3 + 5t ⎪ ⎩ ⎭ ⎧5t 3 (10t 6 + 3) 3 + 5t 6 − ⎫ 1 ⎪ ⎪ ⎨ ⎬+C 600 ⎪9 5 ln 5t 3 + 3 + 5t 6 ⎪ ⎩ ⎭ ( ) Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 21. a. Complete the square; substitution; Formula 45. dt dt du = = = 2 2 t + 2t − 3 u 2 − 4 aF=452 (t + 1) − 4 b. Substitution, Formula 98 sin t cos t u = du 3sin t + 5 3u + 5 u = sin t du = cos t dt 2 (3sin t − 10) 3sin t + 5 + C 27 u = t +1 du = dt ln (t + 1) + t 2 + 2t − 3 + C 24. a. Formula 100a dz 5 = ln z 5 − 4 z F100 a 5 b. Complete the square; substitution; Formula 45. dt dt = = 3 2 29 2 t + 3t − 5 (t + ) − 2 u = t+ 32 du = dt du u2 − 29 4 = F 45 a = 29 2 ln (t + u = cos x du = −sin x dx 2 t + 3t − 5 + C 22. a. Complete the square; substitution; Formula 47. ( x + 1) − 4 dx = x +1 u = x +1 du = dx ⎛ x +1⎞ x 2 + 2 x − 3 − 2sec−1 ⎜ ⎟+C ⎝ 2 ⎠ b. Complete the square; substitution; Formula 47. ( x − 2) 2 − 4 dx = x−2 u = x−2 du = dx u2 − 4 du = F 47 u 23. a. Formula 98 y dy 3y + 5 25. Substitution; Formula 84 1 sinh 2 3t dt = sinh 2 u du = F 84 3 u = 3t 1⎛ 1 3 ⎞ 1 ⎜ sinh 6t − t ⎟ + C = ( sinh 6t − 6t ) + C 3⎝ 4 2 ⎠ 12 26. Substitution; Formula 82 sech x dx = 2 sech u du = F 82 x u= x du = 1 dx 2 x 2 tan −1 sinh x−2⎞ ⎜ ⎟+C ⎝ 2 ⎠ x +C 2 F 98 27 a =3, b =5 (3 y − 10) 3 y + 5 + C a=2 b =1 u = cos t du = −sin t dt −1 ⎛ = +C 27. Substitution; Formula 98 cos t sin t u dt = − du = 2 cos t + 1 2u + 1 F 98 a=2 x − 4 x − 2sec 5 − 4 cos x − 5 +C = du = 3 dt a=2 2 5 − 4 cos x + 5 5 − 4 cos x + 5 = F 100 a a = −4 b=5 5 − 4 cos x − 5 5 ln 5 5 ln 5 2 u2 − 4 du = F 47 u x2 − 4 x dx = x−2 +C 5 − 4z + 5 b. Substitution, Formula 100a sin x du dx = − cos x 5 − 4 cos x u 5 − 4u − x2 + 2 x − 3 dx = x +1 5 − 4z − 5 a = −4 b =5 4 3 )+ 2 = F 98 a =3, b =5 1 − (2 cos t − 2) 2 cos t + 1 + C = 6 1 (1 − cos t ) 2 cos t + 1 + C 3 28. Substitution; Formula 96 cos t sin t 4 cos t − 1 dt = − u 4u − 1 du = F 96 a=4 b = −1 u = cos t du = −sin t dt − Instructor’s Resource Manual 3 1 (6 cos t + 1)(4 cos t − 1) 2 + C 60 Section 7.6 461 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 29. Substitution; Formula 99, Formula 98 cos 2 t sin t u2 dt = − du = cos t + 1 u + 1 F 99 n=2 a =1 b =1 u = cos t du = −sin t dt 2⎡ − ⎢u 2 u + 1 − 2 5⎣ ⎤ du ⎥ = u + 1 ⎦ F 98 u 2⎡ ⎛2 ⎞⎤ − ⎢u 2 u + 1 − 2 ⎜ (u − 2) u + 1 ⎟ ⎥ + C = 5⎣ 3 ⎝ ⎠⎦ 2 4 ⎡ ⎤ cos t + 1 ⎢cos 2 t − (cos t − 2) ⎥ + C − 5 3 ⎣ ⎦ n =3 a =3 n=2 a =3 ⎡1⎛ 1 ⎡ x x dx ⎞ ⎤ ⎤ ⎢ + 3⎢ ⎜ + ⎟⎥ ⎥ 2 2 2 36 ⎢ (9 + x ) 9 + x 2 ⎟⎠ ⎦⎥ ⎥⎦ ⎢⎣18 ⎜⎝ (9 + x ) ⎣ 1 ⎧⎪ x x ⎛ x ⎞ ⎫⎪ + + tan −1 ⎜ ⎟ ⎬ + C ⎨ F 17 36 ⎪ (9 + x 2 ) 2 ⎝ 3 ⎠ ⎭⎪ 6 ⋅ (9 + x 2 ) ⎩ = a =3 31. Using a CAS, we obtain: 2 π cos x dx = π − 2 ≈ 1.14159 0 1 + sin x 32. Using a CAS, we obtain: 0 3 x dx ≈ 0.76803 33. Using a CAS, we obtain: π /2 231π sin12 x dx = ≈ 0.35435 0 2048 34. Using a CAS, we obtain: π x 3π cos 4 dx = ≈ 1.17810 0 2 8 35. Using a CAS, we obtain: 4 t dt ≈ 0.11083 1 1 + t8 36. Using a CAS, we obtain: 3 4 −x / 2 0 x e 39. Using a CAS, we obtain: 2 3 x + 2x −1 dx = 4 ln ( 2 ) + 2 ≈ 4.77259 2 x2 − 2 x + 1 40. Using a CAS, we obtain: 3 π du = 2 tan −1 5 − ≈ 0.72973 1 u 2u − 1 2 41. 1 ⎡ x dx ⎤ +3 ⎢ ⎥ = 2 2 36 ⎢⎣ (9 + x ) (9 + x 2 )2 ⎥⎦ F 95 sech 38. Using a CAS, we obtain: π /4 x3 dx ≈ −0.00921 −π / 4 4 + tan x ( ) 30. Formula 95, Formula 17 1 dx = (9 + x 2 )3 F 95 1 37. Using a CAS, we obtain: π /2 1 dx ≈ 1.10577 0 1 + 2 cos5 x dx = 768 − 3378e −3/ 2 ≈ 14.26632 1 c dx = ⎡ln x + 1 ⎤⎦ = ln(c + 1) 0 x +1 F3 ⎣ ln(c + 1) = 1 ⇒ c + 1 = e ⇒ c 0 c = e − 1 ≈ 1.71828 42. Formula 17 c c 2 dx = ⎡ 2 tan −1 x ⎤ = 2 tan −1 c ⎦0 0 x2 + 1 F 17 ⎣ 1 2 tan −1 c = 1 ⇒ tan −1 c = ⇒ 2 1 c = tan ≈ 0.5463 2 43. Substitution; Formula 65 ln( x + 1) dx = ln u du = F 65 u = x +1 du = dx ( x + 1) [ ln( x + 1) − 1] . Thus c 0 ln( x + 1) dx =( x + 1) [ ln( x + 1) − 1]0 = c (c + 1) ln(c + 1) − c and (c + 1) ln(c + 1) − c = 1 ⇒ ln(c + 1) = 1 ⇒ c + 1 = e ⇒ c = e − 1 ≈ 1.71828 44. Substitution ; Formula 3 c x 1 c 2 +1 1 dx = du = 0 x2 + 1 2 1 u u = x 2 +1 du = 2 x dx 2 1 1 [ln u ]1c +1 = ln(c 2 + 1) 2 2 1 2 ln(c + 1) = 1 ⇒ c 2 + 1 = e2 ⇒ 2 c = e2 − 1 ≈ 2.528 462 Section 7.6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 45. There is no antiderivative that can be expressed in terms of elementary functions; an approximation for the integral, as well as a process such as Newton’s Method, must be used. Several approaches are possible. c ≈ 0.59601 46. Integration by parts; partial fractions; Formula 17 a. ln( x3 + 1) dx = x ln( x3 + 1) − 3 u = ln( x3 +1) du = 3x x3 x3 + 1 2 ln( x3 + 1) dx = ⎡ x ln( x3 + 1) − 3x + ln( x + 1) − ⎢ ⎢ 1 ln( x 2 − x + 1) + 3 tan −1 2 ( x − 1 ) ⎢⎣ 2 2 3 ( ⎛ ⎞ 1 x ln( x3 + 1) − 3x + 3 ⎜ dx ⎜ ( x + 1)( x 2 − x + 1) ⎟⎟ ⎝ ⎠ 1 2 ( x + 1)( x − x + 1) = ( x + 1)( x 2 − x + 1) A + C = 1 B + C = A A = −B ⇒ 1 1 2 A= B=− C= . 3 3 3 Therefore 1 3 dx = ( x + 1)( x 2 − x + 1) 1 x−2 dx − dx = 2 x +1 x − x +1 x−2 ln x + 1 − dx = 1 3 ( x − )2 + 2 1 2 du = dx and G (c) = ln(c3 + 1) we get n an ⇒ 3 2 2 3 u + 4 c 0 3 4 5 2.0000 1.6976 1.6621 1.6615 1.6615 ln( x3 + 1) dx = 1 ⇒ c ≈ 1.6615 48. There is no antiderivative that can be expressed in terms of elementary functions; an approximation for the integral, as well as a process such as Newton’s Method, must be used. Several approaches are possible. c ≈ 0.2509 du = F 17 ( 2 47. There is no antiderivative that can be expressed in terms of elementary functions; an approximation for the integral, as well as a process such as Newton’s Method, must be used. Several approaches are possible. c ≈ 0.16668 4 1 ln x + 1 − ln x 2 − x + 1 + 3 tan −1 2 1 Therefore u = x− u− ) ) ( A Bx + C + = x + 1 x2 − x + 1 ( A + B) x 2 + ( B + C − A) x + ( A + C ) c ⎤ ⎥ = ⎥ ⎥⎦ 0 ) ( 1 ⎞ ⎛ x ln( x3 + 1) − 3 ⎜1 − ⎟ dx = 3 ⎝ x +1⎠ ln x + 1 − c 0 ⎧ ⎛ c +1 ⎞ ⎫ ⎪c(ln(c3 + 1) − 3) + ln ⎜ ⎟ +⎪ ⎜ 2 ⎟ ⎪ ⎪ c c 1 − + ⎝ ⎠ ⎬ ⎨ ⎪ ⎪ 3π −1 2 (c − 1 ) + ⎪ 3 tan ⎪ 2 3 6 ⎩ ⎭ Using Newton’s Method , with ⎧ ⎛ c +1 ⎞ ⎫ ⎪c(ln(c3 + 1) − 3) + ln ⎜ ⎟ +⎪ ⎜ 2 ⎟ ⎪ ⎪ − + c c 1 ⎝ ⎠ ⎬ G (c ) = ⎨ ⎪ ⎪ 3π −1 2 −1 (c − 1 ) + ⎪ 3 tan ⎪ 2 3 6 ⎩ ⎭ dx = x3 +1 dv = dx, v = x b. c. Summarizing 2 ( x− 1 ) 2 3 ) 49. There is no antiderivative that can be expressed in terms of elementary functions; an approximation for the integral, as well as a process such as Newton’s Method, must be used. Several approaches are possible. c ≈ 9.2365 50. There is no antiderivative that can be expressed in terms of elementary functions; an approximation for the integral, as well as a process such as Newton’s Method, must be used. Several approaches are possible. c ≈ 1.96 Instructor’s Resource Manual Section 7.6 463 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 51. f ( x) = 8 − x g ( x) = cx a = 0 b = a. b a x( f ( x) − g ( x)) dx = 8 0 8 c +1 53. c +1 8 x − (c + 1) x 2 dx = f ( x ) = 6e a. 8 b a − x3 0 ( f ( x) − g ( x)) dx = 8 0 v = −3e c +1 8 − (c + 1) x −x −c ⎡ 3 ( x + 3) ⎤ = −18e 3 (c + 3) + 54 ⎢ −18e ⎥ ⎣ ⎦0 b. c 0 c. ( f ( x) − g ( x)) dx = cx − x 2 dx = ⎛ c3 ⎞ ⎛ 2 ⎞ c x = ⎜ ⎟⎜ ⎟ = ⎜ 6 ⎟ ⎝ c2 ⎠ 3 ⎝ ⎠ x =2⇒c =6 0 6e −x 3 dx = For notational convenience, let u 1 c= e c 0 −c −1 ⇒ 3 1 =e c +1 −c 3 Let (c − x) dx = h (c ) = c ⎡ x2 ⎤ c2 ⎢cx − ⎥ = 2 ⎥⎦ 2 ⎣⎢ 0 c. c −c c a a ( f ( x) − g ( x)) dx = u = −18e 3 ; then u (c + 3) + 54 cu 3(u + 18) = + = x= u + 18 u + 18 u + 18 cu +3 u + 18 cu c = −1 ⇒ = −1 ⇒ x =2⇒ 18 u + 18 1+ ⎡ cx 2 x3 ⎤ c3 − ⎥ = ⎢ 3 ⎥⎦ 6 ⎢⎣ 2 0 b. b ⎛ −c ⎞ −18 ⎜ e 3 − 1⎟ ⎝ ⎠ 52. f ( x) = c g ( x) = x a = 0 b = c b 3 c dx = ⎛ 256 ⎞ ⎛ c + 1 ⎞ 8 c. x = ⎜ = ⎜ 3(c + 1)2 ⎟⎟ ⎝⎜ 32 ⎠⎟ 3(c + 1) ⎝ ⎠ 8 1 x =2⇒ =2⇒c= 3(c + 1) 3 x( f ( x) − g ( x)) dx = −x c 32 (c + 1) b dx = c −x −x ⎤ ⎡ 6 ⎢ −3 xe 3 ⎥ + 18 e 3 dx = 0 ⎣ ⎦0 8 a − x3 −x ⎡ 64 32 ⎛ c + 1 ⎞ 2 ⎤ c +1 = − = ⎢8 x − ⎜ 2 ⎟ x ⎥ (c + 1) (c + 1) ⎝ ⎠ ⎦0 ⎣ a. xe 0 u=x dv = e 3 du = dx 3(c + 1)2 c +1 c x( f ( x) − g ( x)) dx = 6 ⎡ 2 ⎛ c + 1 ⎞ 3 ⎤ c +1 256 512 = − = ⎢4x − ⎜ 3 ⎟ x ⎥ 2 ⎝ ⎠ ⎦0 (c + 1) 3(c + 1)2 ⎣ 256 b. g ( x) = 0 a = 0 b = c −c 1 −e 3 , c +1 1 −c 1 h (c ) = e 3 − 3 (c + 1)2 and apply Newton’s Method n an 1 2 3 4 5 6 2.0000 5.0000 5.6313 5.7103 5.7114 5.7114 c ≈ 5.7114 54. ⎛πx ⎞ f ( x) = c sin ⎜ ⎟ g ( x) = x a = 0 b = c ⎝ 2c ⎠ (Note: the value for b is obtained by setting x ⎛πx ⎞ c sin ⎜ ⎟ = x This requires that be a zero for c ⎝ 2c ⎠ ⎛π ⎞ the function h(u ) = u − sin ⎜ u ⎟ . Applying ⎝2 ⎠ Newton’s Method to h we discover that the zeros of h are -1, 0, and 1. Since we are dealing with x positive values, we conclude that =1 or x = c.) c 464 Section 7.6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b a. a x( f ( x) − g ( x )) dx = ⎛πx ⎞ 2⎤ ⎢ cx sin ⎜ 2c ⎟ − x ⎥ dx ⎝ ⎠ ⎣ ⎦ c⎡ 0 ⎛ πt2 ⎞ cos ⎜ ⎟ dt ⎜ 2 ⎟ 0 ⎝ ⎠ ⎛ π x2 ⎞ d C ( x) = cos ⎜ ⎟ ⎜ 2 ⎟ dx ⎝ ⎠ b. C ( x) = c ⎡ x3 ⎤ ⎛πx ⎞ = − cx sin ⎜ dx ⎢ ⎥ ⎟ 0 ⎝ 2c ⎠ ⎢⎣ 3 ⎦⎥ 0 π π c u= = = π 0 F 40 2c x , du = 2c dx 2 c ⎛ 2c u ⎞ sin u ⎛ 2c ⎞ du − ⎜ ⎝π ⎟ ⎠ ⎜ ⎟ ⎝π ⎠ 4c3 π π 3⎤ ⎡c ⎢ ⎥ ⎢⎣ 3 ⎥⎦ ⎡ c 3 ⎤ 4c 3 c 3 ⎥= 2 − 3 ⎢⎣ 3 ⎥⎦ π b. b a ⎛πx ⎞ ⎤ ⎢ c sin ⎜ 2c ⎟ − x ⎥ dx = ⎝ ⎠ ⎦ ⎣ c⎡ 0 2 c ⎡ 2c 2 2c 2 c 2 ⎛πx ⎞ x ⎤ − − − = cos ⎢ ⎥ = ⎜ ⎟ π 2 ⎝ 2c ⎠ 2 ⎦⎥ 0 ⎣⎢ π ⎛ 2 1⎞ c2 ⎜ − ⎟ ⎝π 2⎠ ⎛ 12 − π 2 ⎞ c3 ⎜ ⎜ 3π 2 ⎟⎟ 2 ⎝ ⎠ = c ⎡ 2(12 − π ) ⎤ c. x = ⎢ ⎥ ⎛ 4 −π ⎞ ⎢⎣ 3π (4 − π ) ⎥⎦ c2 ⎜ ⎟ ⎝ 2π ⎠ 3π (4 − π ) ≈ 3.798 x =2⇒c = 12 − π 2 2 55. a. erf ( x) = x −t 2 e 0 π dt d 2 − x2 erf ( x) = e dx π x sin t 0 t dt sin x d Si ( x) = dx x ⎛ π t2 ⎞ sin ⎜ ⎟ dt ⎜ 2 ⎟ 0 ⎝ ⎠ ⎛ π x2 ⎞ d S ( x) = sin ⎜ ⎟ ⎜ 2 ⎟ dx ⎝ ⎠ 56. a. S ( x) = b. erf ( x) = x Instructor’s Resource Manual −4 x π 2 e− x which is negative on (0, ∞) , so erf ( x) is not concave up anywhere on the interval. 58. a. ( f ( x) − g ( x)) dx = b. Si ( x) = 57. a. (See problem 55 a.) . Since erf ( x) > 0 for all x , erf ( x) is increasing on (0, ∞) . [sin u − u cos u ]0 2 − ⎢ 2 ⎛ 4 1⎞ = c3 ⎜ − ⎟ ⎝π2 3⎠ x (See problem 56 a.) Since ⎛π ⎞ S ( x) = sin ⎜ x 2 ⎟ , S ( x) > 0 when 2 ⎝ ⎠ 0< π 2 x 2 < π or 0 < x 2 < 2; thus S ( x) is increasing on ( 0, 2 ) . ⎛π ⎞ b. Since S ( x) = π x cos ⎜ x 2 ⎟ , S ( x) > 0 ⎝2 ⎠ when π π 3π π 2 < x < 2π , 0 < x2 < and 2 2 2 2 or 0 < x 2 < 1 and 3 < x 2 < 4. Thus S ( x) is concave up on (0,1) ∪ ( 3, 2) . 59. a. (See problem 56 b.) Since ⎛π ⎞ C ( x) = cos ⎜ x 2 ⎟ , C ( x) > 0 when ⎝2 ⎠ π π 3π π 2 0 < x2 < or < x < 2π ; thus 2 2 2 2 C ( x) is increasing on (0,1) ∪ ( 3, 2) . ⎛π ⎞ b. Since C ( x) = −π x sin ⎜ x 2 ⎟ , C ( x) > 0 ⎝2 ⎠ π x 2 < 2π . Thus C ( x) is concave 2 up on ( 2, 2) . when π < 60. From problem 58 we know that S ( x) is concave up on (0,1) and concave down on (1, 3) so the first point of inflection occurs at x = 1 . Now 1 ⎛π ⎞ S (1) = sin ⎜ t 2 ⎟ dt . Since the integral cannot 0 ⎝2 ⎠ be integrated directly, we must use some approximation method. Methods may vary but the result will be S (1) ≈ 0.43826 . Thus the first point of inflection is (1, 0.43826) Section 7.6 465 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 17. False: 7.7 Chapter Review x2 2 x –1 Concepts Test 1. True: The resulting integrand will be of the form sin u. 2. True: The resulting integrand will be of the 1 form . 2 a + u2 3. False: 18. True: 19. True: 20. False: Try the substitution u = x 4 , du = 4 x3 dx 4. False: Use the substitution u = x 2 – 3 x + 5, du = (2x – 3)dx. = 1+ 1 1 − 2( x − 1) 2( x + 1) x2 + 2 2 3 3 =− + + x 2( x + 1) 2( x − 1) x( x − 1) 2 x2 + 2 2 x( x + 1) = 2 –x + x x2 + 1 x+2 2 x ( x 2 − 1) 1 2 3 1 =− − + − x x 2 2( x − 1) 2( x + 1) b2 . 4a 21. False: To complete the square, add 22. False: The resulting integrand will be of the 1 form . 2 a − x2 Polynomials can be factored into products of linear and quadratic polynomials with real coefficients. 23. True: Polynomials with the same values for all x will have identical coefficients for like degree terms. 7. True: This integral is most easily solved with a partial fraction decomposition. 24. True: 8. False: This improper fraction should be reduced first, then a partial fraction decomposition can be used. Let u = 2 x ; then du = 2dx and 1 x 2 25 − 4 x 2 dx = u 2 25 − u 2 du 8 which can be evaluated using Formula 57. 9. True: Because both exponents are even positive integers, half-angle formulas are used. 25. False: It can, however, be solved by the 5. True: 6. True: 10. False: The resulting integrand will be of the 1 form . 2 a + u2 substitution u = 25 − 4 x 2 ; then du = −8 x dx and Use the substitution x 25 − 4 x 2 dx = − u = 1 + e x , du = e x dx 11. False: − Use the substitution u = – x 2 – 4 x, du = (−2 x − 4)dx 26. True: 12. True: 13. True: 14. True: This substitution eliminates the radical. Then expand and use the substitution u = sin x, du = cos x dx The trigonometric substitution x = 3sin t will eliminate the radical. dv = x dx 1 v = x3 3 15. True: Let u = ln x 1 du = dx x 16. False: Use a product identity. 466 Section 7.7 2 1 8 u du = 3 1 (25 − 4 x 2 ) 2 + C 12 Since (see Section 7.6, prob 55 a.) 2 − x2 erf ( x) = e > 0 for all x , π erf ( x) is an increasing function. 27. True: by the First Fundamental Theorem of Calculus. 28. False: Since (see Section 7.6, prob 55 b.) sin x Si ( x) = , which is negative on, x say, (π , 2π ) , Si ( x) will be decreasing on that same interval. Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Sample Test Problems 1. 4 t dt = ⎡ 9 + t 2 ⎤ = 5 − 3 = 2 ⎢⎣ ⎥⎦ 0 2 9+t 4 0 cot 2 (2 )d = 2. 10. cos 2 2 sin 2 2 1 − sin 2 2 = = (cos x + csc x − sin x)dx d = sin x + ln csc x − cot x + cos x + C 2 d = (csc 2 − 1)d sin 2 2 1 = − cot 2 − + C 2 3. 4. π / 2 cos x e sin x dx 0 = ⎛ cos 2 x ⎞ ⎜ cos x + ⎟ dx ⎜ sin x ⎟⎠ ⎝ ⎛ 1 − sin 2 x ⎞ = ⎜ cos x + ⎟ dx ⎜ sin x ⎟⎠ ⎝ sin x + cos x dx = tan x π/2 ⎡ −ecos x ⎤ ⎣ ⎦0 (Use Formula 15 for csc x dx .) 11. = e − 1 ≈ 1.718 ⎡ sin 2 x x ⎤ x sin 2 x dx = ⎢ − cos 2 x ⎥ 0 2 ⎣ 4 ⎦0 (Use integration by parts with u = x, dv = sin 2 x dx .) 2 = 1 4 x 2 e x dx = e x (2 − 2 x + x 2 ) + C Use integration by parts twice. 2 tan t , dy = 3 13. y = dy 3 ⎛ y +y 2 ⎞ 5. dy = ⎜ y 2 − y + 2 − ⎟ dy y +1 1+ y ⎠ ⎝ 1 1 = y 3 − y 2 + 2 y − 2 ln 1 + y + C 3 2 6. 2 + 3y2 = sin 3 (2t ) dt = [1 – cos 2 (2t )]sin(2t )dt = 1 1 = – cos(2t ) + cos3 (2t ) + C 2 6 = y–2 1 2y – 4 dy = dy 7. 2 2 2 y – 4y + 2 y – 4y + 2 1 = ln y 2 – 4 y + 2 + C 2 8. 9. dy 3/ 2 2 y +1 0 e 2t t e −2 = ⎡⎣ 2 y + 1 ⎤⎦ 3/ 2 0 = 14. (Use the substitution u = et − 2 , du = et dt which gives the integral Instructor’s Resource Manual u+2 ⎞ du. ⎟ u ⎠ 1 1 3 1 3 1 3 ln ln ln 15. sec2 t 2 sec t sec t dt = 3 2 sec2 t dt 3 2 3 = y 2 + 23 2 3 1 3 + 2 3 y2 + dt ln sec t + tan t + C1 y 2 + 23 + y y 2 3 + C1 + C1 2 + y +C 3 Note that tan t = = 2 −1 = 1 dt = et + 2 ln et − 2 + C ⎛ x −1 ⎞ sin –1 ⎜ ⎟+C 2 ⎝ 3 ⎠ 1 = 16 + 4 x – 2 x (Complete the square.) 12. π/4 π/4 dx y 2 3 , so sec t = y 2 + 23 2 3 . w3 1 1 dw = – w2 – ln 1 – w2 + C 2 2 1– w Divide the numerator by the denominator. 2 tan x dx = – ln ln cos x + C ln cos x Use the substitution u = ln cos x . Section 7.7 467 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 t+2 dt − dt 2 t −1 –1 t + t +1 1 1 2t + 4 dt − dt 2 t2 + t +1 t −1 1 1 2t + 1 + 3 dt − dt 2 t2 + t +1 t −1 1 1 2t + 1 3 1 dt − dt − dt t −1 2 t2 + t +1 2 t+1 2+3 2 4 3dt 16. t = = = 3 = ( 22. u = ln( y 2 + 9) du = 18. u = ln y, du = 1 dy y 5 23. 24. (ln y ) 1 dy = u 5 du = (ln y )6 + C y 6 19. u = x du = dx dv = cot 2 x dx v = –cot x – x x cot 2 x dx = – x cot x – x 2 – (– cot x – x)dx 1 = – x cot x – x 2 + ln sin x + C 2 25. Use cot 2 x = csc2 x − 1 for cot 2 x dx. 20. u = x , du = sin x x 1 −1/ 2 x dx 2 = −2 cos x + C 2 dt t ln t 2 [ln(t 2 )]2 dt = +C t 4 v=y 2 y2 y2 + 9 dy −3et / 3 (9 cos 3t − sin 3t ) +C 82 Use integration by parts twice. et / 3 sin 3t dt = t +9 1 –t + 1 dt + dt t t + 9t t2 + 9 1 t 1 dt + dt = dt − 2 2 t t +9 t +9 1 1 ⎛t⎞ = ln t – ln t 2 + 9 + tan –1 ⎜ ⎟ + C 2 3 ⎝3⎠ 3 dt = 3x x cos x cos 2 x − +C cos dx = − 2 2 2 4 Use a product identity. sin 2 26. dx = 2 sin u du 21. u = ln t 2 , du = dy ⎛ 18 ⎞ = y ln( y 2 + 9) − ⎜ 2 − ⎟ dy 2 ⎜ y + 9 ⎟⎠ ⎝ ⎛ y⎞ = y ln( y 2 + 9) – 2 y + 6 tan –1 ⎜ ⎟ + C ⎝3⎠ ) sinh x dx = cosh x + C y +9 ln( y 2 + 9)dy = y ln( y 2 + 9) – 1 ⎛ 2t + 1 ⎞ = ln t − 1 − ln t 2 + t + 1 − 3 tan −1 ⎜ ⎟+C 2 ⎝ 3 ⎠ 17. 2y 2 dv = dy 27. ⎛x⎞ ⎛ 1 + cos x ⎞ cos 4 ⎜ ⎟ dx = ⎜ ⎟ dx 2 ⎝2⎠ ⎝ ⎠ 1 1 1 = dx + 2 cos x dx + cos 2 x dx 4 4 4 1 1 1 = dx + cos x dx + (1 + cos 2 x)dx 4 2 8 3 1 1 = x + sin x + sin 2 x + C 8 2 16 tan 3 2 x sec 2 x dx = 1 (sec2 2 x – 1)d (sec 2 x) 2 1 1 = sec3 (2 x) – sec(2 x) + C 6 2 28. u = x , du = 1 2 x dx 2x ⎛ 1 ⎜ 1+ x 1+ x ⎝ 2 x (u + 1)(u − 1) + 1 =2 du = 2 u +1 x dx = u2 ⎞ du dx ⎟ = 2 1+ u ⎠ 1 ⎞ ⎛ ⎜ u −1+ ⎟ du u +1⎠ ⎝ ⎛ u2 ⎞ = 2⎜ − u + ln u + 1 ⎟ + C ⎜ 2 ⎟ ⎝ ⎠ ( ) = x − 2 x + 2 ln 1 + x + C 468 Section 7.7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. tan 3 / 2 x sec4 x dx = tan 3 / 2 x(1 + tan 2 x) sec2 x dx = tan 3 / 2 x sec2 x dx + tan 7 / 2 x sec2 x dx 29. = 2 5/ 2 2 tan x + tan 9 / 2 x + C 5 9 30. u = t1/ 6 + 1, (u – 1)6 = t , 6(u − 1)5 du = dt dt 1/ 6 t (t + 1) 6(u − 1)5 du = 6 (u − 1) u = 6du 1 1 = –6 du + 6 du = –6 ln t1/ 6 + 1 + 6 ln t1/ 6 + C u u −1 u (u – 1) 31. u = 9 − e2 y , du = −2e 2 y dy e2 y 9−e 32. 2y dy = − 1 −1/ 2 u du = − u + C = − 9 − e2 y + C 2 cos5 x sin xdx = (1 – sin 2 x) 2 (sin1/ 2 x) cos x dx = sin1/ 2 x cos x dx – 2 sin 5 / 2 x cos x dx + sin 9 / 2 x cos x dx 2 4 2 = sin 3 / 2 x – sin 7 / 2 x + sin11/ 2 x + C 3 7 11 33. eln(3cos x ) dx = 3cos x dx = 3sin x + C x2 + a2 34. y = 3 sin t, dy = 3 cos t dt 9 − y2 3cos t dy = ⋅ 3cos t dt y 3sin t 1 − sin 2 t = 3 (csc t − sin t ) dt sin t = 3 ⎡⎣ln csc t − cot t + cos t ⎤⎦ + C =3 9− y 3 = 3ln − y y Note that sin t = cot t = 2 2 + 9− y +C y 3 , so csc t = and 3 y 2 9– y . y 35. u = e4x , du = 4e4 x dx e4 x 8x 1+ e dx = 36. x = a tan t, dx = a sec 2 t dt 1 du 1 = tan −1 (e 4 x ) + C 4 1+ u2 4 a sec t a sec2 t dt a tan 4 t 1 sec3 t 1 cos t = dt = dt 2 4 a tan t a 2 sin 4 t 1 ⎛ 1 1 ⎞ 1 = +C = – – csc3 t + C 2⎜ 3 3 ⎟ 2 3a a ⎝ sin t ⎠ x =– 4 dx = 4 1 ( x 2 + a 2 )3 / 2 3a 2 x3 Note that tan t = +C x , so csc t = a x2 + a2 . x 37. u = w + 5, u 2 = w + 5, 2u du = dw w 2 dw = 2 (u 2 – 5)du = u 3 – 10u + C 3 w+5 2 = ( w + 5)3 / 2 –10( w + 5)1/ 2 + C 3 38. u = 1 + cos t, du = –sin t dt sin t dt du =– = –2 1 + cos t + C 1 + cos t u 39. u = cos 2 y, du = –2 cos y sin y dy sin y cos y 4 9 + cos y dy = – 1 du 2 9 + u2 ⎛ cos 2 y ⎞ 1 = – tan –1 ⎜ ⎟+C ⎜ 3 ⎟ 6 ⎝ ⎠ Instructor’s Resource Manual Section 7.7 469 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. dx 40. 1 – 6x – x dx = 2 10 – ( x + 3)2 ⎛ x+3⎞ = sin –1 ⎜ ⎟+C ⎝ 10 ⎠ 41. 4 x2 + 3x + 6 A B Cx + D + + x x2 x2 + 3 x ( x + 3) A = 1, B = 2, C = –1, D = 2 4 x2 + 3x + 6 1 1 –x + 2 dx = dx + 2 dx + dx 2 2 2 x x ( x + 3) x x2 + 3 2 2 = 1 1 1 2x 1 dx + 2 dx − dx + 2 dx 2 2 2 2 x +3 x x x +3 2 1 2 ⎛ x ⎞ tan –1 ⎜ = ln x – – ln x 2 + 3 + ⎟+C x 2 3 ⎝ 3⎠ = 42. x = 4 tan t, dx = 4sec2 t dt = ⎞ 1 1 1⎛ x x ⎟+C = cos t dt = sin t + C = ⎜ +C 16 16 16 ⎜⎝ x 2 + 16 ⎟⎠ 16 x 2 + 16 = A B C + + 2 x + 1 (2 x + 1) 2 (2 x + 1)3 dx (16 + x 2 )3 / 2 43. a. b. c. d. e. f. 44. a. 470 3 – 4 x2 (2 x + 1) 3 7 x – 41 2 ( x – 1) (2 – x) 3x + 1 2 ( x + x + 10) = 3 = 2 A B C D E + + + + 2 2 2 – x (2 – x) x – 1 ( x – 1) (2 – x)3 Ax + B 2 x + x + 10 ( x + 1) 2 2 2 2 2 ( x – x + 10) (1 – x ) x5 4 2 ( x + 3) ( x + 2 x + 10) (3 x 2 + 2 x –1)2 2 (2 x + x + 10) 2⎡ 3 = 2 + ( x + x + 10)2 = A B C D Ex + F Gx + H + + + + + 2 2 2 2 1 – x (1 – x) 1 + x (1 + x) x – x + 10 ( x – x + 10)2 = A B C D Ex + F Gx + H + + + + + x + 3 ( x + 3)2 ( x + 3)3 ( x + 3) 4 x 2 + 2 x + 10 ( x 2 + 2 x + 10) 2 Ax + B 2 Cx + D 2 2 x + x + 10 + Cx + D 2 (2 x + x + 10) 2 + Ex + F 2 (2 x + x + 10)3 2 ⎤ 2 1 V =π ⎢ dx ⎥ dx = π 1 ⎢ 1 3x – x2 2⎥ ⎣ 3x – x ⎦ 1 A B = + 2 x 3 –x 3x – x 1 1 A= ,B= 3 3 21⎛1 π 2π π 1 ⎞ 2 ln 2 ≈ 1.4517 V =π dx = ⎡⎣ln x – ln 3 – x ⎤⎦ = (ln 2 + ln 2) = + 1 1 3 ⎜⎝ x 3 – x ⎟⎠ 3 3 3 Section 7.7 1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. V = 2π x 2 1 3x – x dx = −π 2 2 = −π ⎡ 2 3 x − x 2 ⎤ + 3π ⎢⎣ ⎥⎦1 2 −2 x + 3 − 3 1 3x − x 2 1 9 4 2 1 3 − 2x 3x − x 2 dx + 3π 2 1 1 3x − x2 dx 2 1 2 dx = – π ( − x − 23 ) 2 ⎡ ⎛ 2 x − 3 ⎞⎤ dx = ⎢ −2π 3x − x 2 + 3π sin −1 ⎜ ⎟⎥ ⎝ 3 ⎠ ⎦1 ⎣ 1 1 ⎛ 1⎞ = −2π 2 + 3π sin −1 + 2π 2 − 3π sin −1 ⎜ − ⎟ = 6π sin −1 ≈ 6.4058 3 3 ⎝ 3⎠ 45. y = L= x2 x ,y = 16 8 2 ⎛x⎞ 1 + ⎜ ⎟ dx = ⎝8⎠ 4 0 4 0 1+ x2 dx 64 x = 8 tan t, dx = 8sec2 t L= tan –1 1 2 sec t ⋅ 8sec 2 t dt 0 =8 tan –1 1 2 sec3 t dt 0 = 4 ⎡⎣sec t tan t + ln sec t + tan t ⎤⎦ ⎡⎛ 5 ⎞ ⎛ 1 ⎞ ⎛ 1+ 5 ⎞ 1 5⎤ = 4 ⎢⎜⎜ ⎥ = 5 + 4 ln ⎜⎜ ⎟⎟ ⎜ ⎟ + ln + ⎟⎟ ≈ 4.1609 2 2 ⎥⎦ ⎝ 2 ⎠ ⎣⎢⎝ 2 ⎠ ⎝ 2 ⎠ 46. V = π 1 3 dx = π tan –1 1 2 0 Note: Use Formula 28 for sec3 t dt. 1 3 dx ( x + 5 x + 6) ( x + 3) ( x + 2) 2 1 A B C D = + + + 2 2 2 3 2 x x + + ( x + 3) ( x + 2) ( x + 3) ( x + 2)2 2 0 2 2 0 A = 2, B = 1, C = –2, D = 1 3 3⎡ 2 1 2 1 ⎤ 1 1 ⎤ ⎡ – – 2 ln x + 2 – V =π ⎢ dx = π ⎢ 2 ln x + 3 – + + ⎥ 0 x + 3 ( x + 3) 2 x + 2 ( x + 2)2 ⎥⎦ x+3 x + 2 ⎥⎦ 0 ⎣ ⎢⎣ ⎡⎛ 1 1⎞ ⎛ 1 1 ⎞⎤ 4⎞ ⎛7 = π ⎢⎜ 2 ln 6 – – 2 ln 5 – ⎟ – ⎜ 2 ln 3 – – 2 ln 2 – ⎟ ⎥ = π ⎜ + 2 ln ⎟ ≈ 0.06402 5⎠ 6 5⎠ ⎝ 3 2 ⎠⎦ ⎝ 15 ⎣⎝ 47. V = 2π x 3 dx x + 5x + 6 x A B = + 2 2 x x + +3 x + 5x + 6 A = –2, B = 3 3⎡ 2 3 ⎤ 3 V = 2π ⎢ – dx = 2π [ –2 ln( x + 2) + 3ln( x + 3) ]0 + 0 ⎣ x + 2 x + 3 ⎥⎦ 2⎞ 32 ⎛ = 2π[(–2 ln 5 + 3ln 6) – (–2 ln 2 + 3ln 3)] = 2π ⎜ 3ln 2 + 2 ln ⎟ = 2π ln ≈ 1.5511 5⎠ 25 ⎝ 48. V = 2π 0 2 0 u=2–x x=2–u V = 2π 0 2 2 4 x 2 2 – xdx du = –dx dx = –du 4(2 – u )2 u (– du ) = 8π 2 8 2 ⎡8 ⎤ (4u1/ 2 – 4u 3 / 2 + u 5 / 2 )du = 8π ⎢ u 3 / 2 – u 5 / 2 + u 7 / 2 ⎥ 0 5 7 ⎣3 ⎦0 2 ⎛ 16 2 32 2 16 2 ⎞ ⎛ 128 2 ⎞ 1024 2π = 8π ⎜⎜ – + ≈ 43.3287 ⎟⎟ = 8π ⎜⎜ ⎟⎟ = 5 7 ⎠ 105 ⎝ 3 ⎝ 105 ⎠ Instructor’s Resource Manual Section 7.7 471 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ln 3 49. V = 2π 0 2(e x –1)(ln 3 – x)dx = 4π ln 3 0 x [(ln 3)e x – xe x – ln 3 + x]dx xe x dx = xe x – e x dx = xe – e x + C by using integration by parts. Note that ln 3 1 ⎤ ⎡ V = 4π ⎢(ln 3)e x – xe x + e x – (ln 3) x + x 2 ⎥ 2 ⎦0 ⎣ 1 ⎡ ⎤ = 4π ⎢ 2 – ln 3 – (ln 3) 2 ⎥ ≈ 3.7437 2 ⎣ ⎦ 50. A = 18 3 3 3 x 2 x2 + 9 ⎡⎛ 1 ⎤ ⎞ = 4π ⎢⎜ 3ln 3 – 3ln 3 + 3 – (ln 3) 2 + (ln 3) 2 ⎟ – (ln 3 + 1) ⎥ 2 ⎠ ⎣⎝ ⎦ dx x = 3 tan t, dx = 3sec2 t dt A= π/3 18 π/6 27 tan 2 t sec t 51. A = – 0 3sec2 t dt = 2 π/3 1 ⎞ ⎛ 2 ⎞ ⎛ ⎡ 1 ⎤ = 2⎜ – + 2 ⎟ = 4 ⎜1 – dt = 2 ⎢ – ⎟ ≈ 1.6906 ⎥ π / 6 sin 2 t 3 3⎠ ⎣ sin t ⎦ π / 6 ⎝ ⎠ ⎝ π/3 cos t t dt –1)2 t A B = + 2 (t –1) (t –1)2 (t –1) A = 1, B = 1 0 0 ⎡ 1 1 ⎤ 1 ⎞⎤ 6 ⎡ 1 ⎤ ⎛ ⎡ A= – ⎢ + dt = – ⎢ln t –1 – ⎥ ⎥ = – ⎢(0 + 1) – ⎜ ln 7 + 7 ⎟ ⎥ = ln 7 – 7 ≈ 1.0888 –6 t –1 (t –1) 2 t –1 ⎣ ⎦ ⎝ ⎠ ⎣ ⎦ –6 ⎣⎢ ⎦⎥ –6 (t 52. –1 ⎛ 2 –1 ⎞ 36 dx = π dx –3 ⎜⎝ x x + 4 ⎟⎠ –3 x 2 ( x + 4) 36 A B C = + + 2 2 x+4 x ( x + 4) x x V =π 6 9 9 A = – , B = 9, C = 4 4 –1 9 9 9 ⎤ 9π –1 ⎛ 1 4 1 ⎞ 9π ⎡ 4 ⎤ – dx = – + + + + ⎟ dx = ⎢ – ln x – x + ln x + 4 ⎥ –3 ⎢ 4 x x 2 4( x + 4) ⎥ –3 ⎜⎝ x x 2 4 x 4 4 + ⎣ ⎦ –3 ⎠ ⎣ ⎦ 9π ⎡ 4 ⎤ 9 π 8 3 π ⎛ ⎞ ⎛ ⎞ = (4 + ln 3) – ⎜ – ln 3 + ⎟ ⎥ = (4 + 3ln 3) ≈ 34.3808 ⎜ + 2 ln 3 ⎟ = 4 ⎝3 4 ⎢⎣ 3 ⎠⎦ ⎝ ⎠ 2 V =π –1 ⎡ 53. The length is given by π/3 π/6 1 + [ f ( x)]2 dx = = ⎡⎣ln csc x − cot x ⎤⎦ 472 Section 7.7 π/3 π/6 π/3 π/6 = ln 1+ 2 3 − cos 2 x 2 sin x dx = π/3 π/6 sin 2 x + cos 2 x 2 sin x dx = π/3 π/6 1 dx = sin x π/3 π/6 csc x dx ⎛ 2 3 +3⎞ ⎛ 1 ⎞ − ln 2 − 3 = ln ⎜ ⎟⎟ ≈ 0.768 ⎟ − ln(2 − 3) = ln ⎜⎜ 3 ⎝ 3⎠ ⎝ 3 ⎠ 1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 81 − 4 x 2 dx = x 54. a. First substitute u = 2 x, du = 2 dx to obtain 81 − u 2 du , then use Formula 55: u 81 − 4 x 2 9 + 81 − 4 x 2 dx = 81 − 4 x 2 − 9 ln +C x 2x ( b. First substitute u = e x , du = e x dx to obtain e x 9 − e2 x ( e x 9 − e2 x x 3 ) 2 dx = e8 ( 45 − 2e2 x ) 9 − e2 x + 3 3 ) 2 dx = (9 − u 2 ) 2 du , then use Formula 62: 243 −1 ⎛ e x sin ⎜ ⎜ 3 8 ⎝ ⎞ ⎟+C ⎟ ⎠ 55. a. First substitute u = sin x, du = cos x dx to obtain cos x sin 2 x + 4 dx = cos x sin 2 x + 4 dx = sin x sin 2 x + 4 + 2 ln sin x + sin 2 x + 4 + C 2 1 b. First substitute u = 2 x , du = 2dx to obtain Then use Formula 18: u 2 + 4 du , then use Formula 44: 2 dx = 1 − 4x 1 1 2x +1 dx = ln +C . 4 2x −1 1 − 4 x2 56. By the First Fundamental Theorem of Calculus, ⎧ x cos x − sin x ⎧ sin x x≠0 ⎪ ⎪ Si ( x) = ⎨ x Si ( x) = ⎨ x2 ⎪⎩ 1 ⎪ x=0 0 ⎩ 1 2 du 1− u2 . for x ≠ 0 for x = 0 57. Using partial fractions (see Section 7.6, prob 46 b.): 1 1 A Bx + C ( A + B) x 2 + ( B + C − A) x + ( A + C ) = = + = ⇒ 1 + x3 ( x + 1)( x 2 − x + 1) x + 1 x 2 − x + 1 ( x + 1)( x 2 − x + 1) A + C = 1 B + C = A A = −B ⇒ A = 1 1 2 B=− C= . 3 3 3 Therefore: ⎡ ⎤ ⎢ ⎥ ⎥ 1 1⎡ 1 x−2 x−2 ⎤ 1⎢ ⎥ dx = ⎢ dx − dx ⎥ = ⎢ln x + 1 − dx 1 3 3 ⎣ x +1 ⎥ 1 + x3 x2 − x + 1 ⎦ 3 ⎢ ( x − )2 + 2 4 ⎢ ⎥ 1 u = x − , du = dx ⎢⎣ ⎥⎦ 2 3 ⎡ ⎤ ⎡ ⎤ u− x +1 1 −1 2 2 ⎢ ⎥ = 1 ⎢ln ⎥ = ln x + 1 − du + 3 tan ( x − ) 3 2 ⎥ 3 ⎢ ⎥ F17 3 ⎢ 2 u2 + x x 1 − + ⎣ ⎦ 4 ⎣ ⎦ ) ( so 1⎡ dx = ⎢ln 0 1 + x3 3⎢ ⎣ c 1 1⎡ Letting G (c) = ⎢ln 3⎢ ⎣ c +1 ⎡ + 3 ⎢ tan −1 2 ⎣ c − c +1 c +1 ⎡ + 3 ⎢ tan −1 2 ⎣ c − c +1 Method to find the value of c such that c ( ( 2 (c − 1 ) 2 3 2 (c − 1 ) 2 3 1 0 1 + x3 ⎤ ) + π6 ⎤⎥⎦ ⎥⎥⎦ . ⎤ ) + π6 ⎤⎥⎦ ⎥⎥⎦ − 0.5 and G (c) = 1 +1c 3 we apply Newton’s dx = 0.5 : n 1 2 3 4 5 6 an 1.0000 0.3287 0.5090 0.5165 0.5165 0.5165 Thus c ≈ 0.5165 . Instructor’s Resource Manual Section 7.7 473 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Review and Preview Problems 1. lim x2 + 1 x →2 x 2 −1 = 22 + 1 = 2 2 −1 5 3 2 x + 1 2(3) + 1 7 = = 3+5 8 x →3 x + 5 2. lim 14. Note that, if = sec−1 x, then 1 1 se c = x ⇒ cos = ⇒ = cos −1 . Hence x x 1 lim sec−1 x = lim cos −1 = 1 x x →∞ x →∞ 15. f ( x ) = xe− x y x2 − 9 ( x + 3)( x − 3) = lim = x − 3 x−3 x →3 x →3 lim ( x + 3) = 3 + 3 = 6 3. lim 2 x →3 x2 − 5x + 6 ( x − 2)( x − 3) = lim = x−2 x−2 x→2 x →2 lim ( x − 3) = 2 − 3 = −1 4. lim 5 x→2 10 x −2 sin 2 x 2sin x cos x 5. lim = lim = x x →0 x x →0 ⎛ sin x ⎞ lim 2 ⎜ ⎟ cos x = 2(1)(1) = 2 x →0 ⎝ x ⎠ We would conjecture lim xe− x = 0 . x →∞ 16. f ( x ) = x 2 e− x y tan 3x ⎛ sin 3 x ⎞ ⎛ 3 ⎞ = lim ⎜ ⎟⎜ ⎟ = x →0 x x →0 ⎝ cos 3 x ⎠ ⎝ 3 x ⎠ 6. lim ⎛ sin 3x ⎞ ⎛ 1 ⎞ lim 3 ⎜ ⎟⎜ ⎟ = 3(1)(1) = 3 x →0 ⎝ 3 x ⎠ ⎝ cos 3 x ⎠ 1+ 2 x +1 2 1 5 x 2 = 1 + 0 = 1 or: 7. lim = lim 1 x →∞ x 2 − 1 x →∞ 1− 2 1− 0 x lim x2 + 1 x →∞ x 2 −1 = lim 1 + x →∞ 2 2 x −1 1 2x + 1 x = 2+0 = 2 8. lim = lim 5 1+ 0 x →∞ x + 5 x →∞ 1+ x 9. 10. 11. 12. lim e− x = lim lim e − x2 2x lim e x →∞ 17. f ( x ) = x3e− x y 5 5 10 x =0 e x →∞ x →− ∞ 1 = lim We would conjecture lim x 2 e− x = 0 . =0 x →∞ x 2 x →∞ lim e 1 x →∞ e x x →∞ x −2 = 1+ 0 = 1 2+ 10 −5 = ∞ (has no finite value) We would conjecture lim x3e− x = 0 . x →∞ − 2x = lim e 2u (u = − x ) u → ∞ =∞ (has no finite value) 13. 474 lim tan −1 x = x →∞ π 2 Review and Preview Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 18. f ( x ) = x 4 e− x 23. y x a 0 1+ x dx = 2 u = x2 du = 2 x dx 2 a a 1⎡ ln(1 + x 2 ) ⎤ = ln ⎛⎜ 1 + a 2 ⎞⎟ ⎢ ⎥ ⎦0 ⎝ ⎠ 2⎣ 1 2 4 8 16 ⎛ ⎞ ln ⎜ 1+ a 2 ⎟ 0.3466 0.8047 1.4166 2.0872 2.7745 ⎝ ⎠ 5 10 x 24. −2 a 0 1 a dx = [ ln(1 + x) ]0 = ln (1 + a ) 1+ x a 1 2 4 8 16 ln (1+ a ) 0.6931 1.0986 1.6094 2.1972 2.8332 We would conjecture lim x10 e − x = 0 . x →∞ 25. 10 − x 19. y = x e y a 1 ⎡ 1⎤ dx = ⎢ − ⎥ = 1 − 1 x2 a ⎣ x ⎦1 a 1 a 1 1− a 480,000 26. 240,000 a 1 2 4 8 16 0.5 0.75 0.875 0.9375 a ⎡ 1 ⎤ 1⎡ 1 ⎤ dx = ⎢ − = ⎢1 − ⎥ 3 2⎥ x ⎣ 2 x ⎦1 2 ⎣ a 2 ⎦ 1 a 2 4 8 16 1⎡ 1 ⎤ ⎢1− ⎥ 0.375 0.46875 0.4921875 0.498046875 2 ⎣ a2 ⎦ 10 20 x 2 −x =0. We would conjecture lim x e x →∞ 27. 4 1 a 20. Based on the results from problems 15-19, we would conjecture lim x n e− x = 0 4 dx = ⎡⎣ 2 x ⎤⎦ = 4 − 2 a a x a 4− 2 a 1 1 2 2 2.58579 1 1 4 8 3 3.29289 1 16 3.5 x →∞ 21. a −x e 0 a 1 1−e− a 22. a 0 xe 28. a dx = ⎡ −e− x ⎤ = 1 − e− a ⎣ ⎦0 − x2 2 4 = u =− x 2 du =−2 x dx a 16 1 2 4 8 16 1⎡ 2⎤ e− a − ⎢e− x ⎥ = 1 − 2⎣ 2 ⎦ 1− dx = [ ln x ]a = ln 4 x a 8 0.632 0.865 0.982 0.9997 0.9999+ dx 41 a ln 1 1 2 4 a 1 4 1 8 1 16 4 1.38629 2.07944 2.77259 3.46574 4.15888 a 2 1 2 2 ea 0.81606028 0.93233236 0.999999944 1− (8.02×10−29 ) Instructor’s Resource Manual 1 Review and Preview 475 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 8 CHAPTER Indeterminate Forms and Improper Integrals 8.1 Concepts Review 7. The limit is not of the form 1. lim f ( x); lim g ( x) x→a 2. As x → 1– , x 2 – 2 x + 2 → 1, and x 2 – 1 → 0 – so x →a f ( x) g ( x) 9. The limit is of the form 0 1. The limit is of the form . 0 2 x – sin x 2 – cos x = lim =1 lim x 1 x →0 x →0 1 0 . 0 cos x – sin x lim = lim =1 x →π / 2 π / 2 – x x →π / 2 –1 0 . 0 1 – 2 cos 2 x 10. The limit is of the form sin –1 x = = 1– 2 = –1 1 x2 + 6 x + 8 x → –2 x 2 – 3x –10 2 2 = =– –7 7 = 3 =3 1 x →0 476 x3 – 3 x 2 + x x3 – 2 x Section 8.1 11. The limit is of the form lim x →0 2x + 6 x → –2 2 x – 3 = = lim + 7 x 2 x –1 –1 x →0 3x2 + 6 x + 1 3x2 – 2 0 . 0 7 x ln 7 2 x = lim x →0+ 2 x ln 2 = lim x →0 2 x + 7 x ln 7 2 x ln 2 ln 7 ≈ 2.81 ln 2 13. The limit is of the form = lim 0 . 0 1 – 2t –3 t – t2 3 2 t = lim = 2 =– lim 1 1 2 t →1 ln t t →1 12. The limit is of the form 0 . (Apply l’Hôpital’s 0 Rule twice.) 0 6. The limit is of the form . 0 lim 0 . 0 t 0 5. The limit is of the form . 0 lim –1 ex – e– x ex + e– x 2 = lim = =1 2 x →0 2sin x x →0 2 cos x 0 . 0 3 1+ 9 x 2 lim x →0 1 1– x 2 3sin 2 x cos x lim 3. The limit is of the form sec2 x 0 . 0 3 ln(sin x)3 = lim sin x lim x →π / 2 π / 2 – x x →π / 2 0 = =0 –1 2. The limit is of the form tan –1 3 x ln x 2 lim Problem Set 8.1 4. The limit is of the form 0 . 0 1 2x 2 1 = lim x = lim =1 x →1 x 2 – 1 x →1 2 x x →1 x 2 4. Cauchy’s Mean Value x – sin 2 x = lim x →0 tan x x →0 = –∞ 8. The limit is of the form x →0 x →0 x2 + 1 x →1– 3. sec2 x; 1; lim cos x ≠ 0 lim x2 – 2 x + 2 lim lim 0 . 0 lim = 1 1 =– –2 2 ln cos 2 x x →0 = lim 7x 2 = lim –2sin 2 x cos 2 x x →0 14 x –4 cos 2 x x →0 14 cos 2 x – 28 x sin 2 x = lim –2sin 2 x x →0 14 x cos 2 x = –4 2 =– 14 – 0 7 Instructor's Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 0 . 0 3sin x 3cos x lim = lim 1 –x x →0 – x →0 – – 14. The limit is of the form 19. The limit is of the form Rule twice.) 2 –x lim x →0 – 0 . (Apply l’Hôpital’s 0 Rule three times.) tan x – x sec2 x – 1 lim = lim x →0 sin 2 x – 2 x x →0 2 cos 2 x – 2 2sec 2 x tan x 2sec 4 x + 4sec2 x tan 2 x = lim –8cos 2 x x →0 –4sin 2 x x →0 2+0 1 = =– –8 4 = lim 0 16. The limit is of the form . (Apply l’Hôpital’s 0 Rule three times.) sin x – tan x cos x – sec2 x lim = lim x →0 x 2 sin x x →0 2 x sin x + x 2 cos x – sin x – 2sec2 x tan x = lim x →0 2sin x + 4 x cos x – x 2 sin x – cos x – 2sec4 x – 4sec2 x tan 2 x = lim x →0 6 cos x – x 2 cos x – 6 x sin x –1 – 2 – 0 1 = =– 6–0–0 2 17. The limit is of the form 0 . (Apply l’Hôpital’s 0 Rule twice.) x2 2x 2 lim = lim = lim + sin x – x + cos x – 1 + − sin x x →0 x →0 x →0 0 This limit is not of the form . As 0 x → 0+ , 2 → 2, and − sin x → 0− , so 2 lim = −∞. + sin x x →0 18. The limit is of the form 0 . (Apply l’Hôpital’s 0 e – ln(1 + x) –1 x x e + = lim x →0 2 1 (1+ x )2 2 = lim x →0 = e x – 1+1x = lim x →0 20. The limit is of the form –1 = lim –2 x (1+ x 2 ) 2 x →0 48 x 0 . (Apply l’Hôpital’s 0 Rule twice.) cosh x –1 sinh x cosh x 1 = lim = lim = lim 2` 2 x 2 2 0 0 x →0 x → x → x 21. The limit is of the form 0 . (Apply l’Hôpital’s 0 Rule twice.) 1 − cos x − x sin x lim 2 + x →0 2 − 2 cos x − sin x − x cos x = lim x →0+ 2sin x − 2 cos x sin s x sin x – cos x = lim 2 2 + x →0 2 cos x – 2 cos x + 2sin x 0 This limit is not of the form . 0 As x → 0+ , x sin x – cos x → −1 and 2 cos x – 2 cos 2 x + 2sin 2 x → 0+ , so x sin x – cos x lim = –∞ + 2 cos x – 2 cos 2 x + 2sin 2 x x →0 22. The limit is of the form lim sin x + tan x ex + e– x – 2 0 . 0 cos x + sec2 x = lim ex – e– x 0 This limit is not of the form . 0 x →0 – x →0 – As x → 0 – , cos x + sec 2 x → 2, and e x – e – x → 0 – , so lim cos x + sec2 x x →0 – 23. The limit is of the form x x x →0 8 x3 1 1+ x 2 24 x 2 1 1 = lim – =– 24 x →0 24(1 + x 2 ) 2 Rule twice.) lim tan –1 x – x x →0 = lim – 6 – x cos x = 0 15. The limit is of the form 0 . (Apply l’Hôpital’s 0 lim x →0 0 1 + sin t dt x ex – e– x = – ∞. 0 . 0 = lim 1 + sin x = 1 x →0 2x 1+1 =1 2 Instructor’s Resource Manual Section 8.1 477 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 24. The limit is of the form x 0 lim x →0 t cos t dt 2 + = lim x →0 x cos x + 2 x 26. Note that sin (1 0 ) is undefined (not zero), so 0 . 0 = lim x →0 + l'Hôpital's Rule cannot be used. 1 ⎛1⎞ As x → 0, → ∞ and sin ⎜ ⎟ oscillates rapidly x ⎝ x⎠ between –1 and 1, so x cos x 2x = lim x →0 25. It would not have helped us because we proved sin x lim = 1 in order to find the derivative of x →0 x sin x. ( ) ≤ lim x 2 sin 1x tan x x2 . x →0 tan x x2 x 2 cos x = tan x sin x x 2 cos x ⎡⎛ x ⎞ ⎤ = lim ⎢⎜ ⎟ x cos x ⎥ = 0 . x →0 sin x x →0 ⎣⎝ sin x ⎠ ⎦ lim Thus, lim x 2 sin ( 1x ) = 0 . x →0 tan x A table of values or graphing utility confirms this. 27. a. OB = cos t , BC = sin t and AB = 1 – cos t , so the area of triangle ABC is The area of the sector COA is region ABC is 1 sin t (1 – cos t ). 2 1 1 t while the area of triangle COB is cos t sin t , thus the area of the curved 2 2 1 (t – cos t sin t ). 2 1 sin t (1 – cos t ) area of triangle ABC = lim 2 1 t →0+ area of curved region ABC t →0+ 2 (t – cos t sin t ) lim sin t (1 – cos t ) cos t – cos 2 t + sin 2 t 4sin t cos t – sin t 4 cos t – 1 3 = lim = lim = lim = + t – cos t sin t + 1 – cos 2 t + sin 2 t + + 4 cos t sin t 4 cos t 4 t →0 t →0 t →0 t →0 (L’Hôpital’s Rule was applied twice.) = lim 1 1 1 t cos 2 t , so the area of the curved region BCD is cos t sin t – t cos 2 t. 2 2 2 1 cos t (sin t – t cos t ) area of curved region BCD = lim 2 lim 1 (t – cos t sin t ) + area of curved region ABC t →0 t →0+ 2 b. The area of the sector BOD is cos t (sin t – t cos t ) sin t (2t cos t – sin t ) 2t (cos 2 t – sin 2 t ) t (cos 2 t – sin 2 t ) = lim = lim = lim 2 2 t – sin t cos t 4 cos t sin t 2 cos t sin t t →0+ t →0+ 1 – cos t + sin t t →0+ t →0 + = lim cos 2 t – 4t cos t sin t – sin 2 t 1– 0 – 0 1 = 2–0 2 2 cos t – 2sin t t →0 (L’Hôpital’s Rule was applied three times.) = lim + 478 Section 8.1 2 2 = Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 28. a. Note that ∠DOE has measure t radians. Thus the coordinates of E are (cost, sint). Also, slope BC = slope CE . Thus, 0− y sin t − 0 = (1 − t ) − 0 cos t − (1 − t ) (1 − t ) sin t cos t + t − 1 (t − 1) sin t y= cos t + t –1 (t – 1) sin t lim y = lim + + cos t + t – 1 t →0 t →0 0 This limit is of the form . 0 (t – 1) sin t sin t + (t – 1) cos t 0 + (–1)(1) = = –1 lim = lim – sin t + 1 –0 + 1 t →0+ cos t + t – 1 t →0+ −y = b. Slope AF = slope EF . Thus, t t − sin t = 1 − x 1 − cos t t (1 − cos t ) = 1− x t − sin t t (1 + cos t ) x = 1− t − sin t t cos t – sin t x= t – sin t t cos t – sin t lim x = lim + + t – sin t t →0 t →0 0 The limit is of the form . (Apply l’Hôpital’s Rule three times.) 0 t cos t – sin t –t sin t = lim lim + + t – sin t t →0 t →0 1 – cos t – sin t – t cos t t sin t – 2 cos t 0 – 2 = lim = lim = = –2 + + sin t cos t 1 t →0 t →0 ex −1 ex ⎛0⎞ 29. By l’H pital’s Rule ⎜ ⎟ , we have lim f ( x) = lim = lim = 1 and x ⎝0⎠ x →0 + x →0 + x →0+ 1 ex −1 ex = lim = 1 so we define f (0) = 1 . x →0− x x →0− 1 lim f ( x) = lim x →0 − 1 ln x ⎛0⎞ 30. By l’H pital’s Rule ⎜ ⎟ , we have lim f ( x) = lim = lim x = 1 and ⎝0⎠ x →1+ x →1+ x − 1 x →1+ 1 1 ln x = lim x = 1 so we define f (1) = 1 . lim f ( x) = lim x →1− x →1− x − 1 x →1− 1 Instructor’s Resource Manual Section 8.1 479 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 31. A should approach 4πb 2 , the surface area of a sphere of radius b. 2 2 ⎡ 2πa 2 b arcsin a a– b ⎢ 2 lim ⎢ 2πb + a →b + ⎢ a 2 – b2 ⎣ Focusing on the limit, we have lim a →b a 2 – b2 a 2 a 2 arcsin + 2 2 2 ⎤ a 2 arcsin a a– b ⎥ 2 ⎥ = 2πb + 2πb lim+ a →b a 2 – b2 ⎥ ⎦ 2a arcsin = lim a →b a –b a 2 – b2 a ⎛ + a2 ⎜ ⎝a a + a 2 –b2 b 2 a –b 2 ⎞ ⎟ 2 2 ⎛ ⎞ ⎠ = lim ⎜ 2 a 2 – b 2 arcsin a – b + b ⎟ = b. ⎟ a a →b + ⎜ ⎝ ⎠ Thus, lim A = 2πb 2 + 2πb(b) = 4πb 2 . a →b + 32. In order for l’Hôpital’s Rule to be of any use, a(1)4 + b(1)3 + 1 = 0, so b = –1 – a. Using l’Hôpital’s Rule, ax 4 + bx3 + 1 4ax3 + 3bx 2 lim = lim x →1 ( x – 1) sin πx x →1 sin πx + π( x – 1) cos πx To use l’Hôpital’s Rule here, 4a(1)3 + 3b(1)2 = 0, so 4a + 3b = 0, hence a = 3, b = –4. 36 x 2 – 24 x 12 6 3 x 4 – 4 x3 + 1 12 x3 – 12 x 2 = lim = =– = lim 2 –2π π x →1 2π cos πx – π ( x – 1) sin πx x →1 ( x – 1) sin πx x →1 sin πx + π( x – 1) cos πx lim a = 3, b = –4, c = – 6 π 33. If f (a ) and g (a ) both exist, then f and g are both continuous at a. Thus, lim f ( x) = 0 = f (a ) 38. x →a and lim g ( x ) = 0 = g (a ). x →a lim x→a f ( x) f ( x) – f (a ) = lim g ( x) x→a g ( x) – g (a ) f ( x )– f ( a ) x–a lim x → a g ( x )– g ( a ) x–a = cos x – 1 + x2 2 34. lim x →0 35. lim x 36. lim x →0 4 ex – 1 – x – x →0 f ( x )– f ( a ) x–a x →a g ( x )– g ( a ) lim x–a x→a lim x x2 2 = – 4 1 – cos( x 2 ) 3 x sin x = = f (a) g (a) 1 24 x3 6 = 1 24 1 2 tan x − x sec2 x − 1 = lim 1 =2 x → 0 arcsin x − x x →0 −1 2 37. lim 1− x 480 Section 8.1 The slopes are approximately 0.02 / 0.01 = 2 and 0.01/ 0.01 = 1 . The ratio of the slopes is therefore 2 /1 = 2 , indicating that the limit of the ratio should be about 2. An application of l'Hopital's Rule confirms this. Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 41. 39. The slopes are approximately 0.005 / 0.01 = 1/ 2 and 0.01/ 0.01 = 1 . The ratio of the slopes is therefore 1/ 2 , indicating that the limit of the ratio should be about 1/ 2 . An application of l'Hopital's Rule confirms this. 40. The slopes are approximately 0.01/ 0.01 = 1 and −0.01/ 0.01 = 1 . The ratio of the slopes is therefore −1/1 = −1 , indicating that the limit of the ratio should be about −1 . An application of l'Hopital's Rule confirms this. 42. If f and g are locally linear at zero, then, since lim f ( x ) = lim g ( x ) = 0 , f ( x) ≈ px and x →0 x →0 g ( x) ≈ qx , where p = f '(0) and q = g '(0) . Then f ( x) / g ( x) ≈ px / px = p / q when x is near 0. The slopes are approximately 0.01/ 0.01 = 1 and 0.02 / 0.01 = 2 . The ratio of the slopes is therefore 1/ 2 , indicating that the limit of the ratio should be about 1/ 2 . An application of l'Hopital's Rule confirms this. Instructor’s Resource Manual Section 8.1 481 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 8.2 Concepts Review 1. f ( x) g ( x) 2. lim x →a ∞ . ∞ 3sec x tan x 5. The limit is of the form lim x→ π 2 f ( x) g ( x) or lim 1 x →a 1 g ( x) f ( x) 3sec x + 5 = lim tan x x→ π = lim x→ π 2 3. ∞ – ∞, 0°, ∞°, 1∞ 3 tan x = lim 3sin x = 3 sec x x→ π 2 ln sin 2 x = lim x →0+ 3ln tan x x →0+ lim x →0+ 1. The limit is of the form ∞ . ∞ 7. The limit is of the form 1 1000 x999 1000 ln x1000 = lim x lim x 1 x →∞ x →∞ 1000 = lim =0 x →∞ x 2. The limit is of the form ∞ . (Apply l’Hôpital’s ∞ Rule twice.) 2x x →∞ x →∞ = lim x →∞ x x ⋅ 2 ln 2(1 + x ln 2) x x →∞ 2 x →∞ 2 x e x ( 1x ) ln 2(1 + x ln 2) =0 ∞ 4. The limit is of the form . (Apply l’Hôpital’s ∞ Rule three times.) 3x 3 = lim lim = 1 x →∞ ln(100 x + e x ) x →∞ (100 + e x ) x 100 x + e = lim x →∞ = lim x →∞ 100 + e x 3e x ex =3 = lim x →∞ ) =0 8. The limit is of the form –∞ . (Apply l’Hôpital’s ∞ ln(4 – 8 x) 2 lim = lim – tan πx x→ 1 x→ 1 ( 2) 1 (4–8 x )2 ( 2) = lim (2) – x→ 1 – 2(4 – 8 x)(–8) π sec2 πx –16 cos 2 πx 32π cos πx sin πx = lim – π(4 – 8 x) –8π x→ 1 (2) 300 + 3e x ex (2) – x→ 1 ∞ . ∞ cot x – csc2 x = lim 1 – ln x x →0+ – 9. The limit is of the form lim x →0 + 2 x – ln x = lim 2 x – ln x sin 2 x ⎡ 2x ⎤ = lim ⎢ csc x – ln x ⎥ = ∞ + ⎣ sin x ⎦ x →0 x since lim = 1 while lim csc x = ∞ and + sin x x →0 + x →0 x →0 lim Section 8.2 x ln x1000 x →∞ x →0 + 482 1000 = lim ( 1 1 1000 x999 ln x1000 x1000 1 x = lim – 4 cos πx sin πx = 0 = 0 (See Example 2). 300 x + 3e x ln(ln x1000 ) = lim lim ln x x →∞ x →∞ ∞ . ∞ Rule twice.) 2 x ln 2 = lim x ⋅ 2 x ln 2 2 10000 lim 2(ln x) 1x x →∞ 2 ln x = lim 3. = lim 1 2sin x cos x sin 2 x 3 sec 2 x tan x 2 cos 2 x 2 = 3 3 = lim Problem Set 8.2 (ln x)2 –∞ . –∞ 6. The limit is of the form 4. ln x lim sec 2 x 2 + – ln x = ∞. Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 10. The limit is of the form ∞ , but the fraction can ∞ 15. The limit is of the form 00. 2 Let y = (3 x) x , then ln y = x 2 ln 3x be simplified. 2 csc2 x 2 2 = lim = =2 lim x →0 cot 2 x x →0 cos 2 x 12 1000 11. lim ( x ln x x →0 The limit is of the form lim x →0 x = lim 1 x x →0 + – x →0 x →0 1 x2 2 13. lim (csc 2 x – cot 2 x) = lim x →0 14. x →0 The limit is of the form ln(cos x) sin x 0 . 0 1 (– sin x ) ln(cos x) = lim cos x cos x x →0 sin x x →0 sin x 0 = lim – =– =0 1 x →0 cos 2 x lim sin 2 x lim (cos x)csc x = lim eln y = 1 x →0 lim (tan x – sec x) = lim x→ π 2 x→ π 2 sin x – 1 cos x 2 x →0 17. The limit is of the form 0∞ , which is not an indeterminate form. 0 The limit is of the form . 0 sin x – 1 cos x 0 = lim = =0 lim –1 x → π cos x x → π – sin x lim (5cos x) tan x = 0 – x →(π / 2 ) 2 2 2 ⎛ x 2 – sin 2 x ⎞ ⎛ 1 ⎞ ⎛ 1 1 ⎞ 18. lim ⎜ csc2 x – = lim – = lim ⎜ ⎟ ⎟ ⎜ ⎟ x →0 ⎜ x 2 sin 2 x ⎟ x →0 ⎝ x →0 ⎝ sin 2 x x 2 ⎠ x2 ⎠ ⎝ ⎠ Consider lim x 2 – sin 2 x 2 x →0 2 lim x →0 + lim csc x(ln(cos x)) = lim =1 x + x →0 1 – cos 2 x x →0 x →0 sin 2 x →0 + x2 =0 2 Let y = (cos x)csc x , then ln y = csc x(ln(cos x)) ⎛ x ⎞ 12. lim 3 x 2 csc 2 x = lim 3 ⎜ ⎟ = 3 since x →0 x →0 ⎝ sin x ⎠ x lim =1 x →0 sin x sin x = lim – 16. The limit is of the form 1∞. x →0 = lim 1 x2 1 ⋅3 3x 2 x →0 + – 3 x = lim ∞ . ∞ lim (3x) x = lim eln y = 1 = lim – 1000 x = 0 2 1 x2 + 2 1000 x999 1 x →0 ln 3 x lim ∞ . ∞ 1000 + The limit is of the form 1 x x →0 ln x1000 x →0 ln x1000 ) = lim ln 3 x lim x 2 ln 3 x = lim x sin x 2 x – sin x x →0 = lim 2 x sin x x →0 sin 2 = lim 2 . The limit is of the form 2 2 x – 2sin x cos x = lim x →0 2 x sin 2 2 x + 2 x sin x cos x 2 2 2 2 x + 4 x sin x cos x + x cos x – x sin x x →0 12 cos 0 . (Apply l’Hôpital’s Rule four times.) 0 = lim 1 – cos 2 x + sin 2 x 2 x →0 x – sin x cos x 2 x sin x + x 2 sin x cos x 4sin x cos x = lim x →0 6 x cos x 2 + 6 cos x sin x − 4 x 2 cos x sin x − 6 x sin 2 x 4 cos 2 x – 4sin 2 x 2 2 2 2 2 2 x – 4 x cos x – 32 x cos x sin x – 12sin x + 4 x sin x = 4 1 = 12 3 2 2 ⎛ x 2 – sin 2 x ⎞ 1 ⎛1⎞ =⎜ ⎟ = Thus, lim ⎜ ⎟ 2 2 ⎜ ⎟ 9 x →0 x sin x ⎝ 3⎠ ⎝ ⎠ Instructor's Resource Manual Section 8.2 483 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 19. The limit is of the form 1∞. 24. The limit is of the form 1∞. Let y = ( x + e x / 3 )3 / x , then ln y = 3 ln( x + e x / 3 ). x 3 3ln( x + e x / 3 ) ln( x + e x / 3 ) = lim x x →0 x x →0 0 The limit is of the form . 0 ( 3 + ex / 3 = lim x →0 x+e x/3 = x →0 x 2 ) x →0 20. The limit is of the form (–1)0 . The limit does not exist. 21. The limit is of the form 10 , which is not an indeterminate form. lim (sin x)cos x = 1 x2 x →0 0 . 0 (Apply l’Hôpital’s rule twice.) 1 (– sin x ) ln(cos x) − tan x lim = lim cos x = lim 2 2x x →0 x →0 x →0 2 x x − sec 2 x −1 1 = =− 2 2 2 x →0 2 lim (cos x)1/ x = lim eln y = e−1/ 2 = x →0 2 ∞ 22. The limit is of the form ∞ , which is not an indeterminate form. lim x x = ∞ 1 e 25. The limit is of the form 0∞ , which is not an indeterminate form. lim (tan x) 2 / x = 0 x →0 + 26. The limit is of the form indeterminate form. x→ π + , which is not an lim (e – x – x) = lim (e x + x) = ∞ x→ – ∞ x →∞ 27. The limit is of the form 00. Let y = (sin x) x , then ln y = x ln(sin x). x →∞ 23. The limit is of the form ∞ 0 . Let 1 y = x1/ x , then ln y = ln x. x 1 ln x lim ln x = lim x →∞ x x →∞ x –∞ . The limit is of the form ∞ 1 ln x 1 = lim x = lim = 0 lim x →∞ x x →∞ 1 x →∞ x lim x ln(cos x) . ln(cos x) ln(cos x ) = lim x →0 x →∞ x2 = lim lim ( x + e x / 3 )3 / x = lim eln y = e 4 1/ x 1 The limit is of the form 4 =4 1 x →0 1 lim lim 3 1 + 13 e x / 3 3ln( x + e x / 3 ) x +e x / 3 lim = lim x 1 x →0 x →0 2 Let y = (cos x)1/ x , then ln y = = lim e x →∞ ln y =1 ln(sin x) lim x ln(sin x) = lim x →0 + x →0 1 x + –∞ . ∞ 1 cos x sin x The limit is of the form lim x →0 ln(sin x) + 1 x = lim x →0 + – 1 x2 ⎡ x ⎤ = lim ⎢ (– x cos x) ⎥ = 1 ⋅ 0 = 0 + ⎣ sin x ⎦ x →0 lim (sin x ) x = lim eln y = 1 x →0 + x →0+ 28. The limit is of the form 1∞. Let 1 ln(cos x – sin x). x 1 ln(cos x − sin x) lim ln(cos x − sin x ) = lim x x →0 x x →0 y = (cos x – sin x)1/ x , then ln y = = lim 1 (− sin x − cos x) cos x −sin x 1 − sin x − cos x = lim = −1 x →0 cos x − sin x x →0 lim (cos x − sin x )1/ x = lim eln y = e−1 x →0 484 Section 8.2 x →0 Instructor's Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 29. The limit is of the form – . 1⎞ 1⎞ x – sin x ⎛ ⎛ 1 lim ⎜ csc x – ⎟ = lim ⎜ – ⎟ = lim x ⎠ x →0 ⎝ sin x x ⎠ x →0 x sin x x →0 ⎝ 0 The limit is of the form . (Apply l’Hôpital’s 0 Rule twice.) x – sin x 1 – cos x = lim lim x →0 x sin x x →0 sin x + x cos x sin x 0 = lim = =0 2 x →0 2 cos x – x sin x x ⎛ 1⎞ ⎛ 1⎞ Let y = ⎜1 + ⎟ , then ln y = x ln ⎜ 1 + ⎟ . x ⎝ ⎠ ⎝ x⎠ ( ln 1 + 1x ⎛ 1⎞ lim x ln ⎜ 1 + ⎟ = lim 1 x →∞ ⎝ x ⎠ x →∞ x lim ( ln 1 + 1x x →∞ 1 x ) = lim x →∞ ) 0 . 0 1 1+ 1 x (– ) – 1 x2 1 x2 1 =1 x →∞ 1 + 1 x = lim x 31. The limit is of the form 3∞ , which is not an indeterminate form. lim (1 + 2e ) x →0 + =∞ 32. The limit is of the form – . x ⎞ ln x – x 2 + x ⎛ 1 lim ⎜ – ⎟ = lim x →1 ⎝ x – 1 ln x ⎠ x →1 ( x – 1) ln x 0 . 0 Apply l’Hôpital’s Rule twice. 1 − 2x +1 ln x − x 2 + x lim = lim x x →1 ( x − 1) ln x x →1 ln x + x −1 x The limit is of the form 2 1− 2x + x −4 x + 1 −3 3 = lim = =− 2 2 x →1 x ln x + x − 1 x →1 ln x + 2 = lim 1 ln(cos x). x 1 ln(cos x) ln(cos x) = lim x x x →0 0 The limit is of the form . 0 lim x →0 1 (– sin x) ln(cos x) sin x = lim cos x = lim – =0 1 x x →0 x →0 x →0 cos x lim lim (cos x)1/ x = lim eln y = 1 x →0 34. The limit is of the form 0 ⋅ – ∞. ln x lim ( x1/ 2 ln x) = lim x →0 + The limit is of the form lim x →0 + 1 x x →0+ ln x 1 x = lim x →0 + – –∞ . ∞ 1 x 1 2 x3/ 2 = lim – 2 x = 0 x →0+ 35. Since cos x oscillates between –1 and 1 as x → ∞, this limit is not of an indeterminate form previously seen. Let y = ecos x , then ln y = (cos x)ln e = cos x ⎛ 1⎞ lim ⎜1 + ⎟ = lim eln y = e1 = e x⎠ x →∞ ⎝ x →∞ x 1/ x Let y = (cos x)1/ x , then ln y = x →0 30. The limit is of the form 1∞. The limit is of the form 33. The limit is of the form 1∞. Instructor's Resource Manual lim cos x does not exist, so lim ecos x does not x →∞ x →∞ exist. 36. The limit is of the form – . lim [ln( x + 1) – ln( x – 1)] = lim ln x →∞ x →∞ x +1 x –1 1 + 1x x +1 x +1 = lim = 1, so lim ln =0 1 x –1 x →∞ x – 1 x →∞ 1 – x →∞ lim x 37. The limit is of the form 0 , which is not an –∞ indeterminate form. x lim =0 x →0+ ln x 38. The limit is of the form – ∞ ⋅ ∞, which is not an indeterminate form. lim (ln x cot x) = – ∞ x →0 + Section 8.2 485 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 + e−t > 1 for all t, so 39. x x 1 + e−t dt > 1 1 d. dt = x − 1 . ∞ . The limit is of the form ∞ x 1 + e−t dt 1 lim x x →∞ x →∞ n →∞ −x n →∞ =1 n lim lim 1 + x →1 41. a. sin t dt x −1 = lim + x →1 = lim sin x = sin(1) 1 lim a = lim e n →∞ n →∞ 42. a. c. x →0 + n →∞ since lim n →∞ lim n n →∞ = lim n →∞ 486 a −1 1 n n n→∞ a ln a = ln a Section 8.2 x →0 ln x + 1 n2 ln x 1 x 1 x 1 x 1 x →0 + – 2 x ln y = lim lim x x = lim e x →0 + –∞ . ∞ = lim – x = 0 x →0 + =1 b. The limit is of the form 10 , since lim x x = 1 by part a. x →0 + Let y = ( x x ) x , then ln y = x ln( x x ). lim ( x x ) x = lim eln y = 1 x →0 + − − x →0+ n 1 n 1 n n2 − a ln a 1 n2 x →0 + Note that 10 is not an indeterminate form. a −1 a = 1 by part a. = lim 1 n2 lim x ln( x x ) = 0 0 This limit is of the form , 0 n ( ) (1 − ln n) n (ln n − 1) = ∞ x →0 + n = lim eln y = 1 ( n a − 1) = nlim →∞ 0 , 0 n →∞ + lim n →∞ lim n n The limit is of the form 1 n →∞ 1 n The limit is of the form 00. x →0 ln n = lim n = 0 n →∞ n n→∞ 1 n = lim lim x ln x = lim lim lim n −1 Let y = x x , then ln y = x ln x. =1 b. The limit is of the form ∞ 0 . 1 Let y = n n , then ln y = ln n . n 1 ln n lim ln n = lim n →∞ n n →∞ n ∞ . This limit is of the form ∞ n n→∞ 1 Let y = a , then ln y = ln a. n 1 lim ln a = 0 n →∞ n ln y n n = 1 by part b. n n −1 1 n n →∞ n n n since lim 0 40. This limit is of the form . 0 x ( n n − 1) = nlim →∞ This limit is of the form 1+ e 1 = lim lim n c. The limit is of the form 01 , since lim x x = 1 by part a. x →0 + x Let y = x( x ) , then ln y = x x ln x lim x x ln x = – ∞ x →0+ lim x( x x →0 + x ) = lim eln y = 0 x →0 + Note that 01 is not an indeterminate form. Instructor's Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. d. The limit is of the form 10 , since 1 ln x = lim x = 0, so lim x1/ x = lim eln y = 1 x →∞ x x →∞ 1 x →∞ x →∞ lim lim ( x x ) x = 1 by part b. x →0 + x x x x x Let y = (( x ) ) , then ln y = x ln(( x ) ). lim x ln(( x x ) x ) = 0 x →0 + lim (( x x ) x ) x = lim eln y = 1 x →0 + x →0 + Note that 10 is not an indeterminate form. e. The limit is of the form 00 , since lim ( x (xx ) x →0 + Let y = x( x ) = 0 by part c. ( xx ) ) x →0 x →0 x →0 + ln x 1 x x( x ) = lim + x x( x ) x 2 x ) Note: lim x(ln x )2 = lim x →0 + lim x( x x →0 + ( xx ) x (ln x)2 x →0 + ) x →0 + c. x ( x( x ) )2 2 ln x x 1 x →0+ – 2 x lim (1x + 2 x )1/ x = ∞ lim (1x + 2 x )1/ x = 0 1 x – x( x The limit is of the form (1 + 1)∞ = 2∞ , which is not an indeterminate form. x →0 – x x →0+ – x ( x x ) ⎡ x x (ln x +1) ln x + x ⎤ ⎢ x ⎥⎦ ⎣ = lim 44. a. –∞ . ∞ x x(ln x) + x x ln x + x 0 = =0 1⋅ 0 + 1⋅ 0 + 1 x →0 y < 0 on (e, ). When x = e, y = e1/ e . b. The limit is of the form (1 + 1) – ∞ = 2 – ∞ , which is not an indeterminate form. 1 + = lim x y is maximum at x = e since y > 0 on (0, e) and , then ln y = x( x ) ln x. The limit is of the form lim ⎛ 1 ln x ⎞ 1x ln x y =⎜ − ⎟e ⎝ x2 x2 ⎠ y = 0 when x = e. x x ln x lim x( x ) ln x = lim + 1 ln x y = x1/ x = e x 1 x = lim – 2 x ln x = 0 x →0 + = lim eln y = 1 x →0+ The limit is of the form ∞0 . Let y = (1x + 2 x )1/ x , then ln y = 1 ln(1x + 2 x ) x 1 ln(1x + 2 x ) ln(1x + 2 x ) = lim x x →∞ x x →∞ ∞ The limit is of the form . (Apply ∞ l’Hôpital’s Rule twice.) 1 (1x ln1 + 2 x ln 2) ln(1x + 2 x ) 1x + 2 x lim = lim 1 x x →∞ x →∞ lim = lim 2 x ln 2 x →∞ 1x + 2x 2 x (ln 2)2 = lim x →∞ 1x ln1 + 2 x ln 2 = ln 2 lim (1x + 2 x )1/ x = lim eln y = eln 2 = 2 x →∞ x →∞ d. The limit is of the form 10 , since 1x = 1 for all x. This is not an indeterminate form. 43. lim (1x + 2 x )1/ x = 1 x →−∞ ln x x ln x = −∞, so lim x1/ x = lim eln y = 0 lim + x x →0 x →0+ x →0+ ln y = Instructor's Resource Manual Section 8.2 487 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 45. 1k + 2k + lim 1/ t + nk c. n k +1 n →∞ 1 ⎡⎛ 1 ⎞ ⎛ 2 ⎞ ⎢⎜ ⎟ + ⎜ ⎟ + n →∞ n ⎢⎝ n ⎠ ⎝n⎠ ⎣ k k ⎛n⎞ ⎜ ⎟ ⎝n⎠ = lim k ⎤ ⎥ ⎥⎦ 9 ⎞ ⎛1 lim ⎜ 2t + 5t ⎟ + ⎝ 10 10 ⎠ t →0 = 10 2 ⋅ 10 9 5 ≈ 4.562 48. a. k n 1 ⎛i⎞ ⋅⎜ ⎟ n →∞ i =1 n ⎝ n ⎠ The summation has the form of a Reimann sum for f ( x ) = x k on the interval [ 0,1] using a = lim regular partition and evaluating the function at 1 i each right endpoint. Thus, Δxi = , xi = , and n n b. k ⎛i⎞ f ( xi ) = ⎜ ⎟ . Therefore, ⎝n⎠ 1k + 2k + lim + nk 2 n 1 ⎛i⎞ = lim ⋅⎜ ⎟ n →∞ i =1 n ⎝ n ⎠ n k +1 n →∞ lim k 1/ t ⎛ n ⎞ 46. Let y = ⎜ ci xit ⎟ ⎜ ⎟ ⎝ i =1 ⎠ 0 ⎞ 1 ⎛ n , then ln y = ln ⎜ ci xit ⎟ . ⎜ ⎟ t ⎝ i =1 ⎠ ⎛ n ⎞ ln ⎜ ci xit ⎟ ⎜ ⎟ ⎞ 1 ⎛ n ⎠ lim ln ⎜ ci xit ⎟ = lim ⎝ i =1 ⎜ ⎟ + t + t t →0 ⎝ i =1 ⎠ t →0 The limit is of the form 0 , since 0 ⎛ ⎞ ln ⎜ ci xit ⎟ ⎜ ⎟ ⎠ = lim lim ⎝ i =1 t t →0 + t →0+ n i =1 ci = 1. = i =1 ci ln xi = n = ei =1 ln xi ci ci xit i =1 ci xit ln xi ln xi ci = lim eln y t →0 + = x1c1 x2c2 xncn = n i =1 xi ci 1/ t 47. a. 1 ⎞ ⎛1 lim ⎜ 2t + 5t ⎟ +⎝2 2 ⎠ t →0 b. 4 ⎞ ⎛1 lim ⎜ 2t + 5t ⎟ +⎝5 5 ⎠ t →0 1/ t 488 Section 8.2 , so the limit is of the form = lim n →∞ lim n →∞ c. 2nx xe nx ∞ . ∞ 2nx xenx = lim 2x n →∞ x 2 enx ∞ . ∞ =0 1 2 xe− x dx = ⎡ − xe− x − e− x ⎤ = 1 − ⎣ ⎦0 0 e 1 1 3 4 xe−2 x dx = ⎡ −2 xe−2 x − e−2 x ⎤ = 1 − ⎣ ⎦0 0 e2 1 1 0 1 4 9 xe−3 x dx = ⎡ −3xe−3 x − e−3 x ⎤ = 1 − ⎣ ⎦0 e3 1 5 16 xe−4 x dx = ⎡ −4 xe−4 x − e−4 x ⎤ = 1 − ⎣ ⎦0 e4 1 0 1 6 25 xe−5 x = ⎡ −5 xe−5 x − e−5 x ⎤ = 1 − ⎣ ⎦0 0 e5 1 1 7 36e−6 x dx = ⎡ −6 xe−6 x − e−6 x ⎤ = 1 − ⎣ ⎦0 0 e6 d. Guess: lim 1 2 n →∞ 0 n xe− nx dx = 1 1 1 2 − nx n xe dx 0 i =1 1/ t ⎛ n ⎞ lim ⎜ ci xi t ⎟ ⎜ ⎟ t →0+ ⎝ i =1 ⎠ n n 1 i =1 n e nx 1 n n n2 x This limit is of the form 1 1 k n x n →∞ e nx ⎡ 1 k +1 ⎤ x dx = ⎢ x ⎥ ⎣ k +1 ⎦0 1 = k +1 = n 2 xe− nx = = 2 5 ≈ 3.162 5 = 5 2 ⋅ 54 ≈ 4.163 = ⎡ − nxe− nx − e− nx ⎤ ⎣ ⎦0 n +1 = −(n + 1)e− n + 1 = 1 − en 1 ⎛ n +1⎞ lim n 2 xe − nx dx = lim ⎜1 − ⎟ 0 n →∞ n →∞ ⎝ en ⎠ n +1 = 1 − lim if this last limit exists. The n →∞ e n ∞ . limit is of the form ∞ n +1 1 lim = lim = 0, so n →∞ e n n →∞ en 1 2 − nx n xe dx n →∞ 0 lim =1. Instructor's Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 49. Note f(x) > 0 on [0, ). ⎛ x 25 x3 ⎛ 2 ⎞ x ⎞ lim f ( x) = lim ⎜ + +⎜ ⎟ ⎟ = 0 x →∞ x →∞ ⎜ e x e x ⎝ e ⎠ ⎟⎠ ⎝ Therefore there is no absolute minimum. f ( x) = (25 x 24 + 3 x 2 + 2 x ln 2)e− x − ( x 25 + x3 + 2 x )e− x 8. = (− x 25 + 25 x 24 − x3 + 3 x 2 − 2 x + 2 x ln 2)e− x Solve for x when f ( x) = 0 . Using a numerical method, x 25. A graph using a computer algebra system verifies that an absolute maximum occurs at about x = 25. 8.3 Concepts Review 1. converge 2. 3. b lim b →∞ 0 0 –∞ 7. ∞ f ( x)dx 11. Problem Set 8.3 In this section and the chapter review, it is understood ∞ a means lim [ g ( x)] b →∞ b a 2. 3. 4. 12. and likewise for = ∞ – e100 = ∞ e x dx = ⎡ e x ⎤ ⎣ ⎦100 100 The integral diverges. 5 dx –∞ x4 ∞ 1 ⎡ 1 ⎤ 1 1 = ⎢– =– –0= 3⎥ 3(–125) 375 ⎣ 3x ⎦ – ∞ 1 x2 dx b 14. ∞ 1 xe – x dx u = x, du = dx 1 dv = e – x dx, v = – e – x ∞ 1 ∞ x dx b 2 ln 2 + 1 ⎡ ln x 1 ⎤ = lim ⎢ − − ⎥ = b →∞ ⎣ x x ⎦2 2 ∞ 2 2 1 2 xe – x dx = ⎡⎢ – e – x ⎤⎥ = 0 – (– e –1 ) = e ⎣ ⎦1 ∞ ⎡ x⎤ 2 = ⎢2 =∞ ⎥ =∞– 1 πx ⎣ π ⎦1 π The integral diverges. dx ∞ 1 ⎡1 ⎤ dx = ⎢ (ln x)2 ⎥ = ∞ – = ∞ e x 2 2 ⎣ ⎦e The integral diverges. ∞ ln x b 1 1 ⎡1 ⎤ e4 x dx = ⎢ e4 x ⎥ = e4 – 0 = e4 –∞ 4 ⎣4 ⎦ –∞ 4 ∞ 1 dx = [ln(ln x)]e∞ = ∞ – 0 = ∞ x ln x The integral diverges. ∞ e ⎡ ln x ⎤ = lim ⎢ − + lim b →∞ ⎣ x ⎥⎦ 2 b →∞ = ⎡ 1 + x 2 ⎤ = ∞ – 82 = ∞ 5. ⎢⎣ ⎥⎦ 9 9 2 1+ x The integral diverges. 6. ∞ ⎡ ⎤ 1 dx = ⎢ – ⎥ 2 1 (1 + x 2 ) 2 ⎣⎢ 2(1 + x ) ⎦⎥1 x ∞ –5 1 ∞ ∞ dx 1 1 1 dx, dv = dx, v = − . 2 x x x ∞ ln x b ln x dx = lim dx 2 x2 b →∞ 2 x 2 ∞ ∞ dx = 13. Let u = ln x, du = similar expressions. 1. x ⎛ 1⎞ 1 = 0–⎜– ⎟ = ⎝ 4⎠ 4 4. p > 1 that [ g ( x)] ∞ 1⎡ ln(1 + x 2 ) ⎤ ⎣ ⎦ 10 2 1 = ∞ – ln 101 = ∞ 2 The integral diverges. ∞ ⎡ x 0.00001 ⎤ 9. =⎢ ⎥ = ∞ – 100, 000 = ∞ 1 x 0.99999 ⎣⎢ 0.00001 ⎦⎥1 The integral diverges. 10. 0 dx 10 1 + x 2 ∞ cos x dx f ( x)dx; ∞ ⎡ 1 ⎤ = – 1 x1.00001 ⎢⎣ 0.00001x 0.00001 ⎥⎦ 1 1 1 ⎛ ⎞ = 0–⎜– = 100, 000 ⎟= ⎝ 0.00001 ⎠ 0.00001 ∞ Instructor’s Resource Manual ∞ xe – x d = ⎡ – xe – x ⎤ + ⎣ ⎦1 ∞ –x e dx 1 ∞ 2 = ⎡ – xe – x – e – x ⎤ = 0 – 0 – (– e –1 – e –1 ) = ⎣ ⎦1 e 15. 1 ⎡ ⎤ 1 = ⎢– ⎥ 3 2 – ∞ (2 x – 3) ⎣⎢ 4(2 x – 3) ⎦⎥ – ∞ 1 dx =– 1 1 – (–0) = − 4 4 Section 8.3 489 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 16. 4 1/ 3 ⎤ ∞ dx ∞ (π − x ) = ⎡ −3 (π − x ) ⎣ 2/3 = ∞ + 33 π − 4 = ∞ ⎦4 The integral diverges. 17. x ∞ –∞ x2 + 9 dx = x 0 –∞ x2 + 9 The integral diverges since both 18. dx ∞ –∞ ( x2 dx 0 = –∞ ( x2 2 0 dx = ⎡ x 2 + 9 ⎤ + ⎡ x 2 + 9 ⎤ = (3 – ∞) + (∞ – 3) 2 ⎣⎢ ⎦⎥ – ∞ ⎢⎣ ⎦⎥ 0 x +9 0 x –∞ 2 x +9 dx and ∞ x 0 2 x +9 dx diverge. dx ∞ + ∞ 0 x + 16) ( x + 16) 2 dx 1 x x by using the substitution x = 4 tan . = tan –1 + 2 2 2 128 4 32( x + 16) ( x + 16) + 16) 2 ∞ dx + 2 0 0 ⎡ 1 ⎤ x x ⎡ 1 ⎛ π⎞ ⎤ π =0–⎢ =⎢ tan –1 + ⎥ ⎜ – ⎟ + 0⎥ = 2 – ∞ ( x 2 + 16) 2 4 32( x + 16) ⎥⎦ ⎣128 ⎝ 2 ⎠ ⎦ 256 ⎢⎣128 –∞ dx 0 ∞ ⎡ 1 ⎤ x x 1 ⎛ π⎞ π =⎢ tan –1 + ⎥ = ⎜ ⎟ + 0 – (0) = 2 0 ( x 2 + 16) 2 128 ⎝ 2 ⎠ 256 4 32( x + 16) ⎥⎦ ⎢⎣128 0 ∞ dx π π π = + = – ∞ ( x 2 + 16) 2 256 256 128 dx ∞ 19. 1 ∞ dx = –∞ x2 1 ∞ dx = 0 1 dx + 2 0 x + 1⎤ 1 ⎡1 –1 1 1 ⎛ π ⎞ dx = ⎢ tan –1 ⎥ = 3 tan 3 – 3 ⎜ – 2 ⎟ = – ∞ ( x + 1) 2 + 9 3 3 ⎣ ⎦ –∞ ⎝ ⎠ 1 0 ∞ 0 1⎛ –1 1 ⎞ ⎜ π + 2 tan ⎟ 6⎝ 3⎠ ∞ 1⎛ 1⎞ 1⎛ 1⎞ π dx = ⎜ π + 2 tan –1 ⎟ + ⎜ π – 2 tan –1 ⎟ = 6⎝ 3⎠ 6 ⎝ 3⎠ 3 + 2 x + 10 1 ∞ x ∞ –∞ 2 x dx = e For dx x + 1⎤ 1⎛ π⎞ 1 1 1⎛ 1⎞ ⎡1 = ⎜ ⎟ – tan –1 = ⎜ π – 2 tan –1 ⎟ dx = ⎢ tan –1 ⎥ 2 3 ⎦0 3 ⎝ 2 ⎠ 3 3 6⎝ 3⎠ ⎣3 ( x + 1) + 9 1 –∞ x2 20. 1 ∞ – ∞ ( x + 1) + 9 – ∞ ( x + 1) + 9 0 ( x + 1) 2 + 9 + 2 x + 10 1 1 x +1 dx = tan –1 by using the substitution x + 1 = 3 tan 2 3 3 ( x + 1) + 9 2 0 x – ∞ e –2 x 0 x – ∞ e –2 x dx = 0 –∞ dx + ∞ x 0 2x e dx xe2 x dx, use u = x, du = dx, dv = e2 x dx, v = 0 1 2x e . 2 0 1 0 2x 1 1 1 ⎡1 ⎤ ⎡1 ⎤ xe2 x dx = ⎢ xe2 x ⎥ – e dx = ⎢ xe2 x – e2 x ⎥ = 0 – – (0) = – –∞ 4 4 4 ⎣2 ⎦ –∞ 2 –∞ ⎣2 ⎦ –∞ ∞ x ∞ –2 x 1 For dx = xe dx, use u = x, du = dx, dv = e –2 x dx, v = – e –2 x . 0 e2 x 0 2 0 ∞ 1 ⎡ 1 ⎤ xe –2 x dx = ⎢ – xe –2 x ⎥ + 0 ⎣ 2 ⎦0 2 ∞ x 1 1 dx = – + = 0 –∞ 2 x 4 4 e ∞ 490 Section 8.3 ∞ –2 x e dx 0 ∞ 1 1⎞ 1 ⎡ 1 ⎤ ⎛ = ⎢ – xe –2 x – e –2 x ⎥ = 0 – ⎜ 0 – ⎟ = 4 4⎠ 4 ⎣ 2 ⎦0 ⎝ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 21. ∞ –∞ 0 sech x dx = –∞ –1 0 = [tan (sinh x)]– ∞ ∞ sech x dx = 0 25. The area is given by ∞ ∞⎛ 1 2 1 ⎞ dx = ⎜ – dx 2 1 4x −1 1 ⎝ 2 x –1 2 x + 1 ⎟⎠ sech x dx + [tan –1 (sinh x)]∞ 0 ∞ ⎡ ⎛ π ⎞⎤ ⎡ π ⎤ = ⎢0 – ⎜ – ⎟ ⎥ + ⎢ – 0 ⎥ = π 2 2 ⎝ ⎠ ⎣ ⎦ ⎣ ⎦ 22. ∞ 1 = csch x dx = 2e x ∞ e2 x –1 1 ∞ 1 1 dx = sinh x = 2 ∞ ex – e– x 1 1⎛ ⎛ 1 ⎞⎞ 1 = ⎜ 0 − ln ⎜ ⎟ ⎟ = ln 3 2⎝ ⎝ 3 ⎠⎠ 2 2x −1 Note:. lim ln = = 0 since 2x + 1 x →∞ ⎛ 2x −1 ⎞ . lim ⎜ ⎟ =1 x →∞ ⎝ 2 x + 1 ⎠ dx dx Let u = e x , du = e x dx . ∞ 2e x 1 2x e –1 dx = 2 ∞ e 2 u –1 = [ln(u –1) – ln(u + 1)]∞ e du = ∞⎛ e 1 1 ⎞ – ⎜ ⎟ du ⎝ u –1 u + 1 ⎠ 26. The area is ∞ 1 dx = 1 x2 + x ∞ ⎡ u –1 ⎤ = ⎢ln ⎥ ⎣ u + 1⎦ e 24. 27. The integral would take the form ∞ 1 ∞ k dx = [ k ln x ]3960 = ∞ 3960 x which would make it impossible to send anything out of the earth's gravitational field. ∞ 28. At x = 1080 mi, F = 165, so k = 165(1080) 2 ≈ 1.925 × 108 . So the work done in mi-lb is ∞ ∞ 1 1.925 × 108 dx = 1.925 × 108 ⎡ − x −1 ⎤ ⎣ ⎦1080 1080 x 2 8 1.925 × 10 = ≈ 1.782 × 105 mi-lb. 1080 ∞ ⎡ 1 ⎤ = ⎢− (cos x + sin x) ⎥ x ⎣ 2e ⎦0 1 1 = 0 + (1 + 0) = 2 2 (Use Formula 67 with a = –1 and b = 1.) ∞ −x e sin x dx 0 1 ⎞ ⎜ – ⎟ dx ⎝ x x +1⎠ ∞ ⎡ 1 ⎤ cos x dx = ⎢ (sin x − cos x) ⎥ x ⎣ 2e ⎦0 1 1 = 0 − (0 − 1) = 2 2 (Use Formula 68 with a = –1 and b = 1.) ∞ −x e 0 ∞⎛ 1 1 x ⎤ 1 ∞ ⎡ = ⎡⎣ln x − ln x + 1 ⎤⎦ = ⎢ln = 0 − ln = ln 2 ⎥ 1 2 ⎣ x + 1 ⎦1 . e –1 = 0 – ln ≈ 0.7719 e +1 b –1 b –1 ⎞ ⎛ = 0 since lim = 1⎟ ⎜ lim ln b b +1 ⎠ + 1 b →∞ b →∞ ⎝ 23. 1 1 ⎡ 2x −1 ⎤ ∞ ⎡ ln 2 x − 1 − ln 2 x + 1 ⎤⎦ = ⎢ ln 1 2⎣ 2 ⎣ 2 x + 1 ⎥⎦1 29. FP = ∞ − rt 0 e f (t ) dt = ∞ 0 100, 000e−0.08t ∞ ⎡ 1 ⎤ = ⎢− 100, 000e−0.08t ⎥ = 1,250,000 ⎣ 0.08 ⎦0 The present value is $1,250,000. 30. FP = ∞ −0.08t 0 e (100, 000 + 1000t )dt ∞ = ⎡ −1, 250, 000e−0.08t − 12,500te−0.08t − 156, 250e−0.08t ⎤ = 1,406,250 ⎣ ⎦0 The present value is $1,406,250. 31. a. ∞ −∞ f ( x) dx = = 0+ a −∞ 0 dx + b a 1 dx + b−a ∞ b 0 dx 1 1 (b − a ) [ x ]b + 0 = b−a a b−a Instructor’s Resource Manual Section 8.3 491 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. ∞ = −∞ a = −∞ x f ( x) dx b x ⋅ 0 dx + a x 1 dx + b−a ∞ b x ⋅ 0 dx b 1 ⎡ x2 ⎤ = 0+ ⎢ ⎥ +0 b − a ⎢⎣ 2 ⎥⎦ a 2 2 = b −a 2(b − a) = (b + a)(b − a ) 2(b − a) = a+b 2 2 ∞ = −∞ a = −∞ ( x − ) 2 dx 1 ⎡( x − ⎢ = 0+ b−a ⎢ 3 ⎣ = 2 = c. a. ( x − )2 ⋅ 0 dx ⎥ ⎦a +0 2 + 3b 2 1 ( b − a )3 12 ( b − a ) ( b − a )2 12 0 = 32. b 1 ⎡ 1 b3 − 3 b 2 a + 3 ba 2 − 1 a3 ⎤ 4 4 4 ⎦ 3(b − a ) ⎣ 4 = P ( X < 2) = = ∞ b − a3 + 3a 2 − 3a 3 = (a + b) / 2 to obtain 1 b − 3b b−a Next, substitute = 1 dx + b−a 3 3 = )3 ⎤⎥ a ( x − )2 )3 − ( a − )3 1 (b − b−a 2 b ( x − )2 ⋅ 0 dx + −∞ 0 dx + 2 −∞ 2 0 f ( x) dx 1 dx 10 − 0 2 1 = 10 5 ∞ −∞ f ( x) dx = 0 −∞ 0 dx + ∞ 0 (x) −1 −( x / ) e dx In the second integral, let u = ( x / ) . Then, du = ( / )(t / ) −1 dt . When x = 0, u = 0 and when x → ∞, u → ∞ . Thus, ∞ −∞ = 492 f ( x) dx = ∞ −u 0 e ∞ 0 (x) −1 − ( x / ) e dx ∞ du = ⎡ −e−u ⎤ = −0 + e0 = 1 ⎣ ⎦0 Section 8.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. b. ∞ = −∞ 2 3 = 2 0 xf ( x) dx = −∞ ∞ 2 − ( x / 3) 2 x e 0 ∞ = −∞ −1 ⎛ x⎞ x⎜ ⎟ ⎝ ⎠ ∞ x ⋅ 0 dx + 0 e−( x / ) dx 3 π 2 dx = ( x − )2 f ( x) dx = 0 −∞ ( x − )2 ⋅ 0 dx + 2 9 ∞ 0 ( x − )2 xe−( x 2 / 9) dx 3 3 3 π− = π − π =0 2 2 2 The probability of being less than 2 is = c. 2 −∞ f ( x ) dx = 0 −∞ 0 dx + 2 0 (x) −1 − ( x / ) e dx = 0 + ⎡⎢ −e −( x / ⎣ ) 2 ⎤ ⎥⎦ 0 2 = 1 − e−(2 / ) = 1 − e−(2 / 3) ≈ 0.359 33. f ( x) = – x– 3 2π 1 f ( x) = – 3 2π 2 e –( x – ) / 2 2 2 e –( x – ) / 2 2 + ( x – )2 –( x – )2 / 2 e 5 2π 2 ⎛ ( x – )2 1 ⎞ –( x – )2 / 2 2 – =⎜ = ⎟e 3 ⎜ 5 2π 2π ⎟⎠ ⎝ 2 2 1 [( x − )2 − 2 ]e –( x – ) / 2 5 2π f ( x) = 0 when ( x – )2 = 2 so x = 34. a. b. ∞ –∞ = f ( x)dx = ∞ –∞ ± CM k and the distance from ⎡ 1 dx = CM k ⎢ – M x k +1 ⎣ kx k ∞ xf ( x)dx = ∞ M x kM k k +1 dx = kM k to each inflection point is ∞ 1 ⎞ C C ⎤ k⎛ = . Thus, = 1 when C = k. ⎥ = CM ⎜ 0 + k ⎟ k kM ⎠ k ⎦M ⎝ ∞ M b 1 ⎛ ⎞ dx = kM k ⎜ lim dx ⎟ k M x x ⎝ b→∞ ⎠ 1 k x This integral converges when k > 1. b ⎛ ⎡ ⎤ 1 k⎜ When k > 1, = kM lim ⎢ – ⎥ ⎜⎜ b→∞ ⎢ (k –1) x k –1 ⎥ ⎣ ⎦M ⎝ ⎞ ⎛ ⎞ kM 1 ⎟ = kM k ⎜ –0 + ⎟= k –1 ⎟ k –1 ⎜ ⎟⎟ (k –1) M ⎝ ⎠ ⎠ The mean is finite only when k > 1. Instructor’s Resource Manual Section 8.3 493 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. c. Since the mean is finite only when k > 1, the variance is only defined when k > 1. 2 ⎛ 2 2kM ∞ ∞⎛ kM ⎞ kM k k 2M 2 ⎞ 1 k ∞ 2 = = + kM x – x dx ( x – ) 2 f ( x)dx = x – dx ⎜ ⎟ ⎟ M⎜ –∞ M ⎜⎝ k –1 k –1 ⎠ x k +1 (k –1)2 ⎟⎠ x k +1 ⎝ 2k 2 M k +1 ∞ 1 k 3M k +2 ∞ 1 dx + dx M x k –1 M xk k –1 (k –1) 2 M x k +1 The first integral converges only when k – 1 > 1 or k > 2. The second integral converges only when k > 1, which is taken care of by requiring k > 2. = kM k 1 ∞ dx – ∞ 2 ∞ ⎡ ⎤ ⎤ 1 2k 2 M k +1 ⎡ 1 k 3M k +2 = kM k ⎢ – + – – ⎥ ⎢ ⎥ k –2 k –1 ⎣⎢ (k –1) x k –1 ⎦⎥ (k –1)2 ⎣⎢ (k – 2) x ⎦⎥ M M ∞ ⎡ 1 ⎤ ⎢– k ⎥ ⎣ kx ⎦ M ⎛ ⎞ 2k 2 M k +1 ⎛ ⎞ k 3M k +2 ⎛ 1 1 1 ⎞ = kM k ⎜ –0 + – –0 + –0 + ⎟ ⎜ ⎟+ ⎟ k –2 ⎟ k –1 ⎟ 2 ⎜ ⎜ ⎜ k –1 (k – 2) M (k –1) M kM k ⎠ ⎝ ⎠ ⎝ ⎠ (k –1) ⎝ = kM 2 2k 2 M 2 k 2 M 2 + – k – 2 (k –1) 2 (k –1) 2 ⎛ k 2 – 2k + 1 – k 2 + 2k ⎞ ⎛ 1 kM 2 k ⎞ = kM 2 ⎜ = kM 2 ⎜ = – ⎟ ⎟ ⎜ k – 2 (k –1)2 ⎟ ⎜ (k – 2)(k –1) 2 ⎟ (k – 2)(k –1)2 ⎝ ⎠ ⎝ ⎠ 35. We use the results from problem 34: a. To have a probability density function (34 a.) we need C = k ; so C = 3. Also, kM = (34 b.) and since, in our problem, k −1 = 20, 000 and k =3, we have 20000 = 3 4 × 104 M or M = . 2 3 2 b. By 34 c., 2 c. ∞ 105 = kM 2 (k − 2)(k − 1) 4 ⎞2 2 so that 36. u = Ar 37. a. 25 $100,000. 494 of one percent earn over Section 8.3 −∞ sin x dx = 0 sin x dx + −∞ a + lim 0 a →−∞ ∞ 0 sin x dx [ − cos x ]0a Both do not converge since –cos x is oscillating between –1 and 1, so the integral diverges. b. lim a a →∞ − a sin x dx = lim [− cos x]−a a a →∞ = lim [− cos a + cos(−a)] ⎛ 4 × 10 ⎡1⎤ −⎜ ⎟ lim ⎜ 3 ⎟ t →∞ ⎢⎣ x3 ⎥⎦ 5 10 ⎝ ⎠ Thus 6 ∞ a →∞ t ⎛ 4 × 10 1⎤ 64 ⎡ 1 =⎜ − = ⎟ lim ⎜ 3 ⎟ t →∞ ⎣⎢1015 t 3 ⎦⎥ 27 × 103 ⎝ ⎠ ≈ 0.0024 ∞ = lim [ − cos x ] 3 4 ⎞3 ( r + x 2 )3 / 2 ⎤ ⎞ A⎛ a ⎟ ⎥ = ⎜1 − ⎟ r ⎝⎜ ⎥⎦ a r 2 + a2 ⎠ dx x = Note that by using 2 2 3/ 2 (r + x ) r 2 r 2 + x2 the substitution x = r tan . 3 ⎛ 4 × 10 4 × 10 = ⎜ ⎟ = ⎜ ⎟ 4⎝ 3 ⎠ 3 4 ⎞3 2 A⎡ x = ⎢ r ⎣⎢ r 2 + x 2 8 ⎛ 4 × 104 ⎞ t 3 f ( x) dx = ⎜ dx = ⎟ lim ⎜ 3 ⎟ t →∞ 105 x 4 ⎝ ⎠ dx ∞ a a →∞ = lim [− cos a + cos a] = lim 0 = 0 a →∞ 38. a. b. a→∞ The total mass of the wire is ∞ 1 π dx = from Example 4. 0 1 + x2 2 ∞ ⎡1 ⎤ dx = ⎢ ln 1 + x 2 ⎥ which 0 1 + x2 ⎣2 ⎦0 diverges. Thus, the wire does not have a center of mass. ∞ x Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 39. For example, the region under the curve y = ⎡ ⎤ 1 ⎤ 1 ⎡ , n + 1⎥ ⎢ n, n + 2 ⎥ and ⎢ n + 1 – 2 2n ⎦ 2(n + 1) ⎣ ⎢⎣ ⎥⎦ 1 1 will never overlap since ≤ and 2 2 2n 1 1 ≤ . 2 8 2(n + 1) 1 x to the right of x = 1. Rotated about the x-axis the volume is ∞ 1 π dx = π . Rotated about the y-axis, the 1 x2 ∞ 1 volume is 2π x ⋅ dx which diverges. 1 x 40. a. The graph of f consists of a series of isosceles triangles, each of height 1, vertices at 1 1 ⎛ ⎞ ⎛ ⎞ ⎜ n – 2 , 0 ⎟ , (n, 1), and ⎜ n + 2 , 0 ⎟ , 2n 2n ⎝ ⎠ ⎝ ⎠ based on the x-axis, and centered over each integer n. lim f ( x) does not exist, since f(x) will be 1 Suppose lim f ( x) = M ≠ 0, so the limit x →∞ exists but is non-zero. Since lim f ( x) = M , x →∞ there is some N > 0 such that when x M f ( x) – M ≤ , or 2 M M M– ≤ f ( x) ≤ M + 2 2 Since f(x) is nonnegative, M > 0, thus M > 0 and 2 ∞ 0 N f ( x )dx = 0 f ( x)dx + ∞ N N, x →∞ at each integer, but 0 between the triangles. Each triangle has area 1 1⎡ 1 1 ⎞⎤ ⎛ bh = ⎢ n + –⎜n – ⎟ ⎥ (1) 2 2 2⎣ 2n 2n 2 ⎠ ⎦ ⎝ = f ( x)dx ∞ M ⎡ Mx ⎤ dx = f ( x)dx + ⎢ ⎥ =∞ N 2 0 0 ⎣ 2 ⎦N so the integral diverges. Thus, if the limit exists, it must be 0. ≥ b. N f ( x)dx + ∞ N For example, let f(x) be given by 1 ⎧ 2 3 ⎪2n x – 2n + 1 if n – 2 ≤ x ≤ n 2n ⎪ 1 ⎪ f ( x) = ⎨ –2n 2 x + 2n3 + 1 if n < x ≤ n + 2n 2 ⎪ ⎪0 otherwise ⎪ ⎩ for every positive integer n. ⎛ ⎛ 1 ⎞ 1 ⎞ 3 f ⎜n – = 2n 2 ⎜ n – ⎟ – 2n + 1 2⎟ 2n ⎠ 2n 2 ⎠ ⎝ ⎝ 3 3 = 2n – 1 – 2n + 1 = 0 f ( n ) = 2 n 2 ( n ) – 2 n3 + 1 = 1 1⎛ 1 ⎞ 1 ⎜ ⎟= 2 ⎝ n 2 ⎠ 2n 2 ∞ 0 f ( x)dx is the area in all of the triangles, thus ∞ 0 f ( x)dx = ∞ 1 n =1 2n 2 = 1 ∞ 1 2 n =1 n 2 = 1 1 ∞ 1 1 1 ∞ 1 + ≤ + dx 2 2 n=2 n2 2 2 1 x2 = 1 1 ⎡ 1⎤ 1 1 + – = + (–0 + 1) = 1 2 2 ⎢⎣ x ⎥⎦1 2 2 ∞ ∞ (By viewing for 1 x2 Thus, 1 n=2 n 2 as a lower Riemann sum ) ∞ 0 f ( x )dx converges, although lim f ( x) does not exist. x →∞ lim f (n) = lim (–2n 2 x + 2n3 + 1) = 1 = f (n) x→n+ x →n+ ⎛ ⎛ 1 ⎞ 1 ⎞ 3 f ⎜n+ = –2n 2 ⎜ n + ⎟ + 2n + 1 2⎟ 2n ⎠ 2n 2 ⎠ ⎝ ⎝ = –2n3 –1 + 2n3 + 1 = 0 Thus, f is continuous at 1 1 n– , n, and n + . 2 2n 2n 2 Note that the intervals Instructor’s Resource Manual Section 8.3 495 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 41. 1 100 1 1.1 x x 100 1 x 1 ⎡ ⎤ dx = ⎢ – 0.01 ⎥ ⎣ 0.01x ⎦1 100 ⎡ x 0.01 ⎤ dx = ⎢ ⎥ x 0.99 ⎣⎢ 0.01 ⎦⎥1 1 1 10 2 0 dx = ≈ 4.71 1 ⎡ −1 ⎤10 tan x ⎦0 π⎣ π(1 + x ) 1.4711 ≈ ≈ 0.468 π 50 50 1 1 dx = ⎡ tan −1 x ⎤ ⎦0 0 π(1 + x 2 ) π⎣ 1.5508 ≈ ≈ 0.494 π 100 100 1 1 dx = ⎡ tan −1 x ⎤ 2 ⎦0 0 π(1 + x ) π⎣ 1.5608 ≈ ≈ 0.497 π 43. 1 1 0 2π 1 2 2π 3 1 0 2π 4 1 0 2π 0 3. dx 10 = lim ⎡ 2 x – 3 ⎤⎦ b x – 3 b→3+ ⎣ b →3+ 4. 9 0 dx b = lim ⎡ −2 9 – x ⎤⎦ 0 9 – x b→9 – ⎣ = lim − 2 9 – b + 2 9 = 6 b →9 – 5. 6. 1 b dx = lim ⎡sin –1 x ⎤ ⎣ ⎦0 0 2 b →1– 1– x π π = lim sin –1 b – sin –1 0 = – 0 = – 2 2 b →1 b x ∞ dx = lim ⎡ 1 + x 2 ⎤ ⎥⎦100 100 2 b →∞ ⎢⎣ 1+ x exp(–0.5 x 2 )dx ≈ 0.5000 b →∞ b 1 4– x 4. p < 1 496 10 3 The integral diverges. 2. 2 b→4 3 = lim 1 + b 2 + 10, 001 = ∞ 1. unbounded – 0 dx exp(–0.5 x 2 )dx ≈ 0.4987 8.4 Concepts Review lim 3 = 2 7 – lim 2 b – 3 = 2 7 7. 3. 33 2 3(b – 1) 2 / 3 3 3 2 – lim = –0= 3 3 2 2 2 2 b→1+ ⎡ ⎤ 3 = lim ⎢ – 2. ⎥ 1/ 3 1 ( x – 1) 4 / 3 b →1+ ⎢⎣ ( x – 1) ⎥⎦ b 3 3 3 =– + lim =– +∞ 3 1/ 3 3 + 2 b→1 ( x –1) 2 The integral diverges. exp(–0.5 x 2 )dx ≈ 0.3413 exp(–0.5 x 2 )dx ≈ 0.4772 3 ⎡ 3( x – 1) 2 / 3 ⎤ = lim ⎢ ⎥ 1 ( x – 1)1/ 3 b →1+ 2 ⎣⎢ ⎦⎥ b dx 3 = ≈ 4.50 dx = [ln x]100 1 = ln100 ≈ 4.61 1 100 ≈ 3.69 1. 100 1 1 42. 1 ⎤ ⎡ dx = ⎢ – ⎥ ⎣ 0.1x 0.1 ⎦1 1.01 1 = 0.99 100 1 100 Problem Set 8.4 100 ⎡ 1⎤ dx = ⎢ – ⎥ 2 ⎣ x ⎦1 x 1 100 Section 8.4 dx 3 1 –1 x3 b dx = lim b →0 1 – –1 x3 b 3 1 + b x3 dx + lim b→0 dx 3 ⎡ 1 ⎤ ⎡ 1 ⎤ = lim ⎢ – + lim ⎢ – 2⎥ 2⎥ – + b →0 ⎣ 2 x ⎦ –1 b →0 ⎣ 2 x ⎦ b ⎛ 1 1⎞ ⎛ 1 1 ⎞ = ⎜ lim – + + – + lim 2 2 ⎟ ⎜ 18 2⎟ – + b →0 2b ⎠ ⎝ b→0 2b ⎠ ⎝ 1⎞ ⎛ 1 ⎛ ⎞ = ⎜ −∞ + ⎟ + ⎜ – + ∞ ⎟ 2⎠ ⎝ 8 ⎝ ⎠ The integral diverges. Instructor's Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 8. –5 1 5 2/3 x b →0 + 5 1/ 3 b = lim ⎡3x ⎣ b →0+ 1 b dx = lim x 2/3 b→0– b 1/ 3 –5 ⎤ + lim ⎡3 x ⎦ 5 b→0 – ⎣ 1 –5 dx + lim x 2/3 –1 x = lim ⎤ ⎦b dx b b →0 – –1 128 –5 / 7 x –5 / 7 dx + lim b →0 + b x b dx 128 ⎡7 ⎤ ⎡7 ⎤ = lim ⎢ x 2 / 7 ⎥ + lim ⎢ x 2 / 7 ⎥ – ⎣2 + ⎦ –1 b→0 ⎣ 2 ⎦b b →0 7 7 7 7 = lim b 2 / 7 – (–1)2 / 7 + (128) 2 / 7 – lim b 2 / 7 – 2 + 2 2 2 b →0 b →0 7 7 21 = 0 – + (4) – 0 = 2 2 2 = lim 3b1/ 3 – 33 5 + 33 –5 – lim 3b1/ 3 b →0+ 3 128 –5 / 7 9. dx b →0 – 3 = 0 – 3 5 + 33 5 – 0 = 33 −5 − 3 5 = −6 3 5 1 10. 03 x 1 – x2 dx = lim x b b →1– 0 3 1 – x2 dx b ⎡ 3 ⎤ = lim ⎢ – (1 – x 2 )2 / 3 ⎥ –⎣ 4 ⎦0 b →1 3 3 3 3 = lim − (1 – b 2 ) 2 / 3 + = –0 + = – 4 4 4 4 b →1 11. 4 dx 0 (2 – 3x)1/ 3 b→ 2 dx b = lim – 0 3 (2 – 3 x)1/ 3 + lim b→ 2 + b 3 b dx 4 (2 – 3 x)1/ 3 4 ⎡ 1 ⎤ ⎡ 1 ⎤ = lim ⎢ – (2 – 3 x)2 / 3 ⎥ + lim ⎢ – (2 – 3x ) 2 / 3 ⎥ – + ⎦ 0 b→ 2 ⎣ 2 ⎦b b→ 2 ⎣ 2 3 3 1 1 1 1 = lim − (2 – 3b) 2 / 3 + (2)2 / 3 – (–10) 2 / 3 + lim (2 – 3b)2 / 3 – + 2 2 2 b→ 2 b→ 2 2 3 3 1 1 1 = 0 + 22 / 3 − 102 / 3 + 0 = (22 / 3 − 102 / 3 ) 2 2 2 13. x 8 12. 5 2 2/3 (16 − 2 x ) x –4 0 16 – 2 x 2 dx = ⎡ 3 ⎤ dx = lim ⎢ − (16 − 2 x 2 )1/ 3 ⎥ −⎣ 4 ⎦ b→ 8 lim b→ – 8 x b + 0 16 – 2 x 2 dx + x –4 – b 16 – 2 x 2 dx ⎡ 1 ⎤ ⎡ 1 ⎤ – ln 16 – 2 x 2 ⎥ + lim ⎢ – ln 16 – 2 x 2 ⎥ +⎢ – 4 4 ⎣ ⎦ ⎣ ⎦b 0 b→ – 8 8 lim b→ – = b→ – 8 5 3 3 3 = lim − (16 − 2b 2 )1/ 3 + 3 6 = 3 6 − 4 4 4 b→ 8 –4 b = lim b 1 1 1 1 lim − ln 16 – 2b 2 + ln16 – ln16 + lim ln 16 – 2b 2 + 4 – 4 4 4 b→ – 8 b→ – 8 1 ⎡ ⎤ ⎡ 1 ⎤ = ⎢ –(– ∞) + ln16 ⎥ + ⎢ – ln16 + (– ∞) ⎥ 4 ⎣ ⎦ ⎣ 4 ⎦ The integral diverges. 14. 15. 3 0 b x dx = lim ⎡ – 9 – x 2 ⎤ = lim − 9 – b 2 + 9 = 3 ⎢ ⎥⎦ 0 b→3– 2 b →3 – ⎣ 9– x –1 dx –2 ( x + 1)4 / 3 b ⎡ ⎤ 3 3 3 = lim – + = –(– ∞) – 3 = lim ⎢ – ⎥ 1/ 3 1/ 3 – (–1)1/ 3 b → –1– ⎣⎢ ( x + 1) ⎦⎥ –2 b→ –1 (b + 1) The integral diverges. Instructor’s Resource Manual Section 8.4 497 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ⎡ 1 dx 1 ⎤ = ⎢ − ⎥ dx by using a partial fraction decomposition. ( x − 1)( x + 2) 3( x − 1) 3( x + 2) ⎦ x + x−2 ⎣ 3 b 3 dx dx dx = lim + lim 0 x 2 + x – 2 b →1– 0 x 2 + x – 2 b→1+ b x 2 + x – 2 dx 16. Note that = 2 3 b 1 1 ⎡1 ⎤ ⎡1 ⎤ = lim ⎢ ln x – 1 – ln x + 2 ⎥ + lim ⎢ ln x – 1 – ln x + 2 ⎥ – ⎣3 + 3 3 ⎦ 0 b→1 ⎣ 3 ⎦b b →1 3 b ⎡1 x –1 ⎤ ⎡1 x –1 ⎤ 1 b –1 1 1 1 2 1 b –1 = lim ⎢ ln ⎥ + lim+ ⎢ 3 ln x + 2 ⎥ = lim– 3 ln b + 2 – 3 ln 2 + 3 ln 5 – lim+ 3 ln b + 2 – 3 x + 2 b →1 ⎣ b →1 ⎦ 0 b→1 ⎣ ⎦ b b→1 1 1⎞ ⎛1 2 ⎛ ⎞ = ⎜ – ∞ – ln ⎟ + ⎜ ln + ∞ ⎟ 3 2⎠ ⎝3 5 ⎝ ⎠ The integral diverges. 1 17. Note that 3 2 x − x − x +1 3 dx = lim 0 x3 – x 2 – x + 1 b →1– 1 = 2 2( x − 1) b dx 0 3 − 1 1 + 4( x − 1) 4( x + 1) 2 x – x – x +1 + lim dx 3 + b b→1 3 x – x2 – x + 1 3 b ⎡ ⎤ ⎡ ⎤ 1 1 1 1 1 1 = lim ⎢ – – ln x − 1 + ln x + 1 ⎥ + lim ⎢ – – ln x − 1 + ln x + 1 ⎥ – + 4 4 b →1 ⎣ 2( x –1) 4 ⎦ 0 b→1 ⎣ 2( x –1) 4 ⎦b ⎡⎛ ⎡ 1 1 ⎛ 1 1 b +1 ⎞ ⎛ 1 1 1 b + 1 ⎞⎤ ⎞⎤ lim ⎢⎜ – + ln + ⎜ – + 0 ⎟ ⎥ + lim ⎢ – + ln 2 – ⎜ – + ln ⎟ ⎟⎥ b −1 ⎠ ⎝ 2 ⎠ ⎦ b→1+ ⎣ 4 4 b →1– ⎣⎝ 2(b –1) 4 ⎝ 2(b –1) 4 b − 1 ⎠ ⎦ 1⎞ ⎛ 1 1 ⎛ ⎞ = ⎜ ∞ + ∞ – ⎟ + ⎜ – + ln 2 + ∞ – ∞ ⎟ 2⎠ ⎝ 4 4 ⎝ ⎠ The integral diverges. 18. Note that x1/ 3 x 2/3 −9 = 1 1/ 3 x + 9 1/ 3 x ( x 2 / 3 − 9) . b x1/ 3 27 27 27 ⎡3 ⎤ ⎛3 ⎞ ⎛ ⎞ dx = lim ⎢ x 2 / 3 + ln x 2 / 3 – 9 ⎥ = lim ⎜ b 2 / 3 + ln b 2 / 3 – 9 ⎟ – ⎜ 0 + ln 9 ⎟ 0 x2 / 3 – 9 – ⎣2 – 2 2 2 2 ⎦ 0 b→27 ⎝ ⎠ ⎝ ⎠ b →27 27 27 = – ∞ – ln 9 2 2 The integral diverges. 27 19. π/4 0 b ⎡ 1 ⎤ tan 2 xdx = lim ⎢ – ln cos 2 x ⎥ – 2 ⎦0 b→ π ⎣ 4 1 1 = lim − ln cos 2b + ln1 = –(– ) + 0 – 2 2 b→ π 4 The integral diverges. 20. π/2 0 π/2 csc xdx = lim ⎡⎣ln csc x – cot x ⎤⎦ b + b →0 = ln 1 – 0 – lim ln csc b – cot b b →0 + = 0 – lim ln b →0 + 1 – cos b sin b 1 – cos b 0 is of the form . 0 b →0+ sin b 1 – cos b sin b 0 = lim = =0 lim + sin b + cos b 1 b →0 b →0 1 – cos b Thus, lim ln = – ∞ and the integral + sin b b →0 diverges. lim 498 Section 8.4 Instructor's Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 21. π/2 0 1 − cos x x = sin 2 , 2 2 1 1 x = − csc 2 . cos x − 1 2 2 sin x π/ 2 dx = lim ⎡⎣ ln 1 – cos x ⎤⎦ b + 1 – cos x b →0 25. Since = ln1 – lim ln 1 – cos b = 0 – (– ∞) b →0+ The integral diverges. 22. 23. π/2 ⎡3 ⎤ dx = lim ⎢ sin 2 / 3 x ⎥ 3 0 + 2 ⎦b sin x b →0 ⎣ 3 2/3 3 2/3 3 = (1) – (0) = 2 2 2 π/2 π/2 0 cos x π b – lim cot = 0 – ∞ 2 b →0 + 2 The integral diverges. = cot b 26. ⎡1 ⎤ tan x sec x dx = lim ⎢ tan 3 x ⎥ – 3 ⎦0 b→ π ⎣ 2 π dx x⎤ ⎡ = lim cot ⎥ 0 cos x – 1 b →0+ ⎢⎣ 2 ⎦b π 2 –1 –3 dx b = lim ⎡ 2 ln(– x) ⎤⎦ –3 x ln(– x) b→ –1– ⎣ = lim 2 ln(–b) – 2 ln 3 = 0 – 2 ln 3 2 b →−1– 1 1 = lim tan 3 b – (0)3 = ∞ – 3 3 b→ π = –2 ln 3 2 The integral diverges. 24. π/4 0 27. b sec2 x 1 ⎤ ⎡ dx = lim ⎢ – 2 – tan x – 1 ⎥⎦ 0 (tan x – 1) b→ π ⎣ ln 3 0 – b→ π 4 ln 3 = lim ⎡ 2 e x –1 ⎤ ⎥⎦ b +⎢ x e –1 b→0 ⎣ = 2 3 – 1 – lim 2 eb – 1 = 2 2 – 0 = 2 2 b →0+ 4 = lim − e x dx 1 1 + = –(– ∞) – 1 tan b – 1 0 – 1 The integral diverges. 28. Note that 4 2 29. 4 x − x 2 = 4 − ( x 2 − 4 x + 4) = 22 − ( x − 2)2 . (by completing the square) dx 4 x – x2 = lim dx b b→4 – 2 4 x – x2 b x – 2⎤ π π ⎡ –1 b – 2 – sin –1 0 = – 0 = = lim ⎢sin –1 ⎥ = lim– sin –⎣ 2 2 2 2 ⎦ 2 b →4 b→4 dx = lim [ln(ln x)]be = ln(ln e) – lim ln(ln b) = ln 1 – ln 0 = 0 + x ln x b→1+ b →1+ The integral diverges. e 1 10 ⎡ 1 ⎤ 1 1 1 = lim – 30. =– + lim =– +∞ 99 99 1 x ln100 x b →1+ ⎢⎣ 99 ln 99 x ⎥⎦ + 99 ln 10 b→1 99 ln b 99 ln 99 10 b The integral diverges. 10 31. dx dx 4c ⎡ ⎤ = lim ⎢ln x + x 2 − 4c 2 ⎥ = ln ⎡⎣ (4 + 2 3)c ⎤⎦ − lim ln b + b 2 − 4c 2 2c +⎣ 2 2 ⎦b b → 2c + b → 2c x − 4c = ln ⎡⎣ (4 + 2 3)c ⎤⎦ − ln 2c = ln(2 + 3) 4c Instructor’s Resource Manual Section 8.4 499 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 32. x dx 2c c 2 x + xc – 2c 2 x dx 2c = (x + ) c c 2 2 = – 94 c 2 2c c ( x + 2c ) dx − c 2 ( x + 2c ) − 94 c2 2 ⎡ c c = lim ⎢ x 2 + xc – 2c 2 − ln x + + x 2 + xc – 2c 2 + 2 2 b →c ⎣ dx 2c 0 ( x + 2c ) 2 − 94 c 2 2c ⎤ ⎥ ⎦b c 5c ⎡ c c ⎤ = 4c 2 − ln + 4c 2 – lim ⎢ b 2 + bc – 2c 2 − ln b + + b 2 + bc – 2c 2 ⎥ + 2 2 2 2 b →c ⎣ ⎦ c 9c ⎛ c 3c ⎞ c 9c c 3c c = 2c − ln – ⎜ 0 − ln + 0 ⎟ = 2c − ln + ln = 2c − ln 3 2 2 2 2 2 2 2 ⎝ 2 2 ⎠ 1 33. For 0 < c < 1, 1 dv = x is continuous. Let u = x (1 + x) 1 1 , du = – dx . 1+ x (1 + x) 2 dx, v = 2 x . 1 ⎡2 x ⎤ 1 1 1 2 2 c xdx 2 c xdx xdx dx = ⎢ = – +2 =1– +2 ⎥ +2 c 2 2 c x (1 + x) c c 2 1+ c 1+ c (1 + x) (1 + x)2 (1 + x) ⎣1 + x ⎦c 1 1 ⎡ 2 c 1 1 xdx xdx ⎤ +2 dx = lim ⎢1 – ⎥ =1– 0 + 2 0 2 c 1+ c c →0 ⎢⎣ x (1 + x) (1 + x) 2 (1 + x) ⎥⎦ This last integral is a proper integral. 1 1 Thus, lim c →0 c 1 34. Let u = 1+ x 1 dv = x , du = – 1 2(1 + x)3 / 2 dx dx, v = 2 x . 1 ⎡ 2 x ⎤ 1 1 2 1 2 c x x =⎢ dx = – + dx For 0 < c < 1, ⎥ + c 3 / 2 c x(1 + x) c 2 1+ c (1 + x)3 / 2 (1 + x) ⎣ 1 + x ⎦c dx 1 dx 1 Thus, 0 x(1 + x) = lim ⎡ ⎤ 1 1 x 2 c x = lim ⎢ 2 – + dx ⎥ = 2 – 0 + dx 3 / 2 c 0 1+ c x(1 + x) c→0 ⎢⎣ (1 + x) (1 + x)3 / 2 ⎥⎦ dx 1 c →0 c This is a proper integral. 35. 3 x –3 9 – x2 dx = = – 9 + lim + b → –3 36. 500 0 x –3 9 – x2 dx + 9 – b 2 – lim b →3 0 x b dx = lim ⎡ – 9 – x 2 ⎤ + lim ⎡ – 9 – x 2 ⎤ ⎢ ⎥⎦ b b→3– ⎢⎣ ⎥⎦ 0 0 2 b→ –3+ ⎣ 9– x 3 9 – b 2 + 9 = –3 + 0 – 0 + 3 = 0 – 0 b ⎡ 1 ⎤ ⎡ 1 ⎤ dx = lim ⎢ − ln 9 − x 2 ⎥ + lim ⎢ − ln 9 − x 2 ⎥ −3 9 − x 2 −3 9 − x 2 0 9 − x2 +⎣ 2 − ⎦b b→3 ⎣ 2 ⎦0 b →3 1 1 ln 9 − b 2 − lim ln 9 − b 2 + ln 3 = (− ln 3 − ∞) + (∞ + ln 3) = − ln 3 + lim + 2 − 2 b →−3 b →3 The integral diverges. 3 x dx = Section 8.4 0 x dx + 3 x Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 37. 1 4 –4 16 – x 2 1 0 dx = –4 16 – x 2 0 b ⎡1 x + 4 ⎤ ⎡1 x + 4 ⎤ dx = lim ⎢ ln ⎥ + lim– ⎢ 8 ln x – 4 ⎥ 0 16 – x 2 + 8 x – 4 b → –4 ⎣ ⎦ b b →4 ⎣ ⎦0 1 4 dx + 1 1 b+4 1 b+4 1 = ln1 – lim ln + lim ln – ln1 = (0 + ) + ( – 0) +8 –8 8 b – 4 b–4 8 b→ –4 b→4 The integral diverges. 38. 1 1 −1 x − ln x 1 −1 2 dx = −1 x − ln x dx + 1 0 −1 2 x − ln x −1 2 dx + 1 12 0 x − ln x b = lim ⎡ −2 − ln x ⎤ ⎣ ⎦b b →−1+ dx + 1 1 12 x − ln x dx 12 b + lim ⎡ −2 − ln x ⎤ + lim ⎡ −2 − ln x ⎤ + lim ⎡ −2 − ln x ⎤ ⎣ ⎦ −1 2 b→0+ ⎣ ⎦b ⎣ ⎦1 2 b →0 − b →1− = (−2 ln 2 + 0) + (−∞ + 2 ln 2) + (−2 ln 2 + ∞) + (0 + 2 ln 2) The integral diverges. 39. ∞ 1 0 xp dx = 1 1 0 xp 1 ∞ dx + 1 xp 1 = −3π + 3π lim b −1/ 3 x →0 + 0 40. 0 0 1 0 1 dx and x ∞1 1 x dx diverge. 1 ln x dx + c →0 − c 1 lim x ln x − x c + + c →0 [ b 1 ln x dx ] [ x ln x − x ]1b = −1 − lim (c ln c − c) + b ln b − b + 1 = b ln b − b Thus, b ln b – b = 0 when b = e. f ( x)dx b − 0 b →1 f ( x)dx + lim c + b where 1 < c < ∞. b→1 f ( x)dx + lim b 8 b b →∞ c ( x − 8) −2 / 3 dx = lim ⎡3( x − 8)1/ 3 ⎤ ⎣ ⎦0 0 b →8− = 3(0) – 3(–2)= 6 42. = ln x dx = lim c →0+ = lim 41. b dx = ∞ . If p = 1, both ∞ The limit tends to infinity as b → 0, so the volume is infinite. 44. Since ln x < 0 for 0 < x < 1, b > 1 x →∞ If p = 0, b →0 ∞ ⎡ 1 ⎤ 0, dx = ⎢ x − p +1 ⎥ 1 xp ⎣ − p +1 ⎦1 1 ∞ diverges since lim x − p +1 = ∞ . ∞ 1 1 V = π x −4 / 3 dx = lim π ⎡ −3x −1/ 3 ⎤ ⎣ ⎦b 0 b →0 + b. since lim x − p +1 = ∞ . If p < 1 and p 1 = lim ⎡3x1/ 3 ⎤ = 3 ⎣ ⎦b b →0 + 1 ⎡ 1 ⎤ dx = ⎢ x − p +1 ⎥ diverges 0 xp ⎣ − p +1 ⎦0 1 If p > 1, 1 −2 / 3 x dx 0 43. a. dx f ( x)dx 45. 1 sin x dx is not an improper integral since x sin x is bounded in the interval 0 x 1. x 0 46. For x 1⎛ 1 1 ⎞ dx − ⎜ 3 0 ⎝ x x + x ⎟⎠ 1 x 1 Instructor’s Resource Manual 1 1+ x 4 < 1 so 1 4 4 x (1 + x ) < 1 x4 . b 1 1 ⎡ 1 ⎤ dx = lim ⎢ – + ⎥ = – blim 1 x4 →∞ 3b3 3 b →∞ ⎣ 3 x3 ⎦1 1 1 = –0 + = 3 3 ∞ 1 Thus, by the Comparison Test dx 1 x 4 (1 + x 4 ) converges. ∞ ⎡1 ⎤ dx = lim ⎢ ln x 2 + 1 ⎥ − b 2 − ⎣2 ⎦b x +1 b →0 b →0 1 1 1 = ln 2 − lim ln b 2 + 1 = ln 2 2 2 b →0− 2 = lim 1, 1 Section 8.4 501 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1, x 2 ≥ x so – x 2 ≤ – x, thus 47. For x From Example 2 of Section 8.2, lim 51. a. 2 = –0 + = lim [– e – x ]1b = – lim 1 b →∞ eb b→∞ for a any positive real number. x n +1 = 0 for any positive real Thus lim x →∞ e x number n, hence there is a number M such x n +1 that 0 < ≤ 1 for x M. Divide the ex x n –1 1 ≤ inequality by x 2 to get that 0 < x e x2 for x M. + e –1 1 1 = e e ∞ – x2 e dx 1 Thus, by the Comparison Test, converges. 48. Since x + 2 − 1 ≤ x + 2 we know that ∞ 1 1 1 . Consider ≥ dx 0 x + 2 −1 x+2 x+2 ∞ b 1 1 dx = lim dx 2 2 b →∞ x+2 x+2 ∞ = lim ⎡⎣ 2 x + 2 ⎤⎦ = lim 2 2 b →∞ b →∞ ( ) ∞ n –1 – x b+2 −2 = ∞ 1 ≤ 1 2 . Since ∞ 1 of a b a x →b g ( x)dx implies the convergence of implies the divergence of b a b a g ( x)dx. M ∞ 1 1 x2 ∞ n –1 – x M x e dx dx x n –1e – x dx 1 ∞ n –1 – x 1 x e dx 1 1 = ⎡ – e – x ⎤ = – e –1 + 1 = 1 – , so the ⎣ ⎦0 e integral converges when n = 1. For 0 x 1, 1 –x e dx 0 0 ≤ x n –1 ≤ 1 for n > 1. Thus, e x→a f ( x)dx and the divergence of 52. x n –1 lim f ( x) = lim g ( x) = ∞, then the convergence x →b b x n –1e – x dx + x n –1e – x dx + converges. 2 50. If 0 f(x) g(x) on [a, b] and either lim f ( x) = lim g ( x) = ∞ or x→a M 1 1 integral is finite, so ∞ 1 e dx = by part a and Problem 46. The remaining ⎡ 1⎤ dx = ⎢ − ⎥ = 1 ⎣ x ⎦1 x x ln ( x + 1) x we can apply the Comparison Test of Problem 46 ∞ 1 to conclude that dx converges. 1 x 2 ln x + 1 ( ) ≤ M x = 1+ 49. Since x 2 ln ( x + 1) ≥ x 2 , we know that 1 b 1 1 ⎡ 1⎤ dx = lim ⎢ – ⎥ = – lim + 2 1 x b →∞ ⎣ x ⎦1 b →∞ b 1 = –0 + 1 = 1 1 ∞ b. Thus, by the Comparison Test of Problem 46, we ∞ 1 conclude that dx diverges. 0 x+2 2 =0 x →∞ e x e– x ≤ e– x . ∞ –x e dx 1 xa x = x n –1e – x ≤ e – x . By the comparison test from Problem 50, 53. a. (1) = f ( x)dx b. 1 n −1 – x 0 x ∞ 0 −x x e dx 0 e dx converges. ∞ = ⎡ −e − x ⎤ = 1 ⎣ ⎦0 ∞ n −x x e dx 0 n −x (n + 1) = Let u = x , dv = e dx, du = nx n −1dx, v = −e− x . (n + 1) = [− x n e− x ]∞ 0 + = 0+n c. 502 Section 8.4 ∞ 0 nx n −1e− x dx ∞ n −1 − x 0 x e dx = n (n) From parts a and b, (1) = 1, (2) = 1 ⋅ (1) = 1, (3) = 2 ⋅ (2) = 2 ⋅1 = 2! . Suppose (n) = (n − 1)!, then by part b, (n + 1) = n (n) = n[(n − 1)!] = n ! . Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ∞ –x 54. n = 1, 0 ∞ xe – x dx = 1 = 1! = (2 –1)! 0 ∞ 2 –x x e dx = 2 = 2! = (3 –1)! 0 ∞ 3 –x n = 2, n = 3, x e dx = 6 = 3! = (4 –1)! n = 4, 0 ∞ 4 –x x e dx 0 n = 5, 55. a. e dx = 1 = 0! = (1 –1)! ∞ –∞ = 24 = 4! = (5 –1)! ∞ f ( x)dx = 0 Cx –1e – y Let y = x, so x = ∞ 0 –1 – x Cx e C – –∞ ∞ dx = 0 ∞ xf ( x)dx = Let y = x, so x = dx 1 and dx = 0 y –1 ⎛ y⎞ C⎜ ⎟ ⎝ ⎠ x dy . – ( ) 1 e– y 1 ( ) = 1 when C = ∞ = b. x ( ) = ( ) x –1e – 1 and dx = x C dy = ∞ y –1e – y dy = C – 0 . ∞ dx = x e– 0 ( ) ∞ y ⎞ –y 1 1 dy = y e – y dy = ⎜ ⎟ e 0 ( ) ⎝ ⎠ ( ) (Recall that ( + 1) = ( ) for > 0.) 2 c. = –∞ ( x – ) f ( x)dx = ∞ = ∞⎛ 2 ( ) 0 x +1e – x dx – 0 ⎞ ⎟ ⎠ ⎜x– ⎝ –1 ∞ 2 0 ( ) = = = ∞⎛ y⎞ ⎜ ⎟ ( ) 0 ⎝ ⎠ = 1 2 ∞ ( ) 1 2 2 ( ) + 2 – 0 y +1 +1 – y e ( + 2) – 2 2 2 + 2 2 Instructor’s Resource Manual e– y 1 dy – 2 2 2 = ( ) ( ) y –1 – x dx + e dx = 2 –2 ( ) and dx = 1 ( ) ∞ ∞⎛ 2 ⎜x 0 ⎜ x –1e – 0 ⎝ x – 2 ( )= x+ 2 2 ⎞ –1 – ⎟x e ⎟ ⎠ x dx dx dy . –1 2 –2 ∞ ⎛ y⎞ y ⎞ –y 1 1 e dy + e – y dy 0 ⎜ ⎟ 0 ⎜ ⎟ ( ) ⎝ ⎠ ⎝ ⎠ ( ) ∞ ( ) x x 1 ( ) –1 ∞ ⎛ 2 2 dy – 2 x e– In all three integrals, let y = x, so x = 2 dx 1 ( + 1) = ( ) 0 ∞ x dy. ∞⎛ = ( ) 0 y e – y dy + ( + 1) + 2 2 ( ) 2 2 ∞ ( ) 0 ( )= y –1e – y dy 1 2 ( ) ( + 1) ( )– 2 2 ( ) ( )+ 2 2 2 Section 8.4 503 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 56. a. ∞ L{t }( s ) = 0 t e – st dt x 1 , so dt = dx, then s s Let t = ∞ 0 ∞ 1 x ⎞ –x 1 ( + 1) e dx = x e – x dx = . ⎜ ⎟ 0 ⎝s⎠ 0 s +1 s s +1 ∞⎛ t e – st dt = If s ≤ 0 , t e− st → ∞ as t → ∞, so the integral does not converge. Thus, the transform is defined only when s > 0. b. L{e t }( s ) = ∞ 0 e t e – st dt = ∞ ( – s )t e dt 0 ⎧∞ if > s lim e( – s )b = ⎨ b →∞ ⎩0 if s > −1 1 = Thus, L{e t }( s ) = –s s− c. L{sin( t )}( s ) = Let I = ∞ 0 ∞ 0 ∞ ⎡ 1 ( – s )t ⎤ =⎢ e ⎥ = ⎣ –s ⎦0 when s > . (When s ≤ 1 ⎡ lim e( – s )b – 1⎤⎥ – s ⎢⎣b→∞ ⎦ , the integral does not converge.) sin( t )e – st dt sin( t )e – st dt and use integration by parts with u = sin( t), du = cos( t)dt, 1 dv = e – st dt , and v = – e – st . s ∞ ∞ ⎡ 1 ⎤ cos( t )e – st dt Then I = ⎢ – sin( t )e – st ⎥ + 0 s s ⎣ ⎦0 Use integration by parts on this integral with 1 u = cos( t), du = – sin( t)dt, dv = e – st dt , and v = – e – st . s ∞ ∞ ⎛ ⎞ ∞ ⎡ 1 ⎤ ⎡ 1 ⎤ I = ⎢ – sin( t )e – st ⎥ + ⎜ ⎢ – cos( t )e – st ⎥ – sin( t )e – st dt ⎟ ⎟ ⎣ s ⎦ 0 s ⎜⎝ ⎣ s ⎦0 s 0 ⎠ ∞ 2 1⎡ ⎛ ⎞⎤ I = – ⎢e – st ⎜ sin( t ) + cos( t ) ⎟ ⎥ – s⎣ s ⎝ ⎠⎦0 s 2 Thus, ∞ 2⎞ ⎛ 1 ⎡ – st ⎛ ⎞⎤ I ⎜1 + ⎟ = – ⎢e ⎜ sin( t ) + cos( t ) ⎟ ⎥ ⎜ s⎣ s ⎝ ⎠⎦0 s 2 ⎟⎠ ⎝ I=– ( ∞ 1 s 1+ 2 s2 ) s ⎡ – st ⎛ ⎞⎤ ⎢ e ⎜ sin( t ) + s cos( t ) ⎟ ⎥ = – 2 ⎝ ⎠⎦0 s + ⎣ ⎡ ⎤ ⎛ ⎞ lim e – sb ⎜ sin( b) + cos( b) ⎟ – ⎥ s s ⎝ ⎠ ⎦ 2 ⎢b →∞ ⎣ ⎛ ⎞ ⎧0 if s > 0 lim e – sb ⎜ sin( b) + cos( b) ⎟ = ⎨ s b →∞ ⎝ ⎠ ⎩∞ if s ≤ 0 Thus, I = 504 s2 + Section 8.4 2 when s > 0. Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 57. a. The integral is the area between the curve 1– x y2 = and the x-axis from x = 0 to x =1. x 1– x y2 = ; xy 2 = 1 – x; x( y 2 + 1) = 1 x 1 x= 2 y +1 1– x → ∞, while x As x → 0, y = when x = 1, y = ∞ 0 1 2 y +1 1–1 = 0, thus the area is 1 dy = lim [tan –1 y ]b0 b →∞ = lim tan –1 b – tan –1 0 = b →∞ π 2 b. The integral is the area between the curve 1+ x y2 = and the x-axis from x = –1 to 1– x x = 1. 1+ x 2 y2 = ; y – xy 2 = 1 + x; y 2 –1 = x( y 2 + 1); 1– x x= 2 58. For 0 < x < 1, x p > x q so 2 x p > x p + x q and 1 1 . For 1 < x, x q > x p so > p q p 2x x +x 1 1 q p . > 2 x > x + x q and p q 2 xq x +x ∞ 1 ∞ 1 1 1 dx = dx + dx 0 x p + xq 0 x p + xq 1 x p + xq Both of these integrals must converge. 1 1 1 1 1 1 1 dx > dx = dx which 0 x p + xq 0 2x p 2 0 xp converges if and only if p < 1. ∞ ∞ 1 1 1 ∞ 1 dx > dx = dx which 1 x p + xq 1 2xq 2 1 xq converges if and only if q > 1. Thus, 0 < p < 1 and 1 < q. 8.5 Chapter Review Concepts Test 1. True: See Example 2 of Section 8.2. 2. True: Use l'Hôpital's Rule. y –1 y2 + 1 3. False: 1 + (–1) = 1 – (–1) When x = –1, y = 0 = 0, while 2 1+ x → ∞. 1– x The area in question is the area to the right of 1+ x and to the left of the the curve y = 1– x line x = 1. Thus, the area is ∞⎛ ∞ 2 y2 – 1 ⎞ dy ⎜1 – 2 ⎟ dy = ⎟ 0 ⎜ 0 y +1⎠ y2 + 1 ⎝ 4. False: as x → 1, y = b = lim ⎡ 2 tan –1 y ⎤ ⎦0 b →∞ ⎣ ⎛ π⎞ lim 2 tan –1 b – 2 tan –1 0 = 2 ⎜ ⎟ = π b →∞ ⎝2⎠ lim x →∞ 1000 x 4 + 1000 4 0.001x + 1 = 1000 = 106 0.001 lim xe −1/ x = ∞ since e−1/ x → 1 and x →∞ x → ∞ as x → ∞ . 5. False: For example, if f(x) = x and g ( x) = e x , lim x x →∞ e x = 0. 6. False: See Example 7 of Section 8.2. 7. True: Take the inner limit first. 8. True: Raising a small number to a large exponent results in an even smaller number. 9. True: Since lim f ( x) = –1 ≠ 0, it serves x→a only to affect the sign of the limit of the product. Instructor’s Resource Manual Section 8.5 505 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 10. False: Consider f ( x) = ( x – a )2 and g ( x) = 1 2 17. False: g ( x) = 4 x3 + 2 x + 3; f ( x) = 6 x + 1 , then lim f ( x ) = 0 x →a ( x – a) and lim g ( x) = ∞, while g ( x) = 12 x 2 + 2, and so x→a lim [ f ( x) g ( x)] = 1. Consider f ( x) = 3x 2 and x →∞ f ( x) 6x + 1 1 = lim = while 2 g ( x) x →0 12 x + 2 2 lim f ( x) 3x 2 + x + 1 1 = lim = g ( x ) x →0 4 x 3 + 2 x + 3 3 x →0 g ( x) = x 2 + 1, then lim lim x →0 x→a 11. False: Consider f ( x) = 3x 2 + x + 1 and 18. False: f ( x) 3x2 = lim g ( x) x→∞ x 2 + 1 19. True: 3 = lim = 3, but x →∞ 1 + 1 2 p > 1. See Example 4 of Section 8.4. 1 x 1 ∞ 0 x 1 xp ∞ 1 0 lim [ f ( x) – 3 g ( x)] x →∞ = lim [3x 2 – 3( x 2 + 1)] 12. True: 20. False: 13. True: 14. True: 21. True: See Example 7 of Section 8.2. 1/ f ( x ) Let y = [1 + f ( x)] ln[1 + f ( x)] = lim f ( x) x→a x →a lim ∞ –∞ 0 . 0 22. False: f ( x) f ( x) lim [1 + f ( x)]1/ f ( x ) = lim eln y = e1 = e 1 and 1. 1 dx . x +1 0 −∞ f ( x)dx + ∞ f ( x )dx = 0 ∞ 0 f ( x)dx f ( x)dx. x →a f ( x )dx converge so their sum See Problem 37 of Section 8.3. ∞ 0 f ( x)dx = lim b b →∞ 0 = lim [ f ( x)]b0 = lim b →∞ b →∞ 24. True: ∞ 0 f ( x )dx ≤ f ( x )dx f (b) – f (0) ∞ –x 0 Use repeated applications of l'Hôpital's Rule. = lim – e e0 = 1 and p(0) is the constant term. must converge. b →∞ 25. False: Section 8.5 dx; = 0 – f(0) = –f(0). f(0) must exist and be finite since f ( x) is continuous on [0, ). 1 =1 x →a 1 + f ( x) 506 xp 1 converges. 23. True: 1 1+ f ( x ) x 1 ∞ dx + Thus, both integrals making up = lim x→a –∞ –∞ 1 ln[1 + f ( x)] lim ln[1 + f ( x)] = lim f ( x) x→a f ( x) x →a 16. True: 0 f ( x)dx = 0 , then This limit is of the form ∞ p If f is an even function, then f(–x) = f(x) so 1 ln y = ln[1 + f ( x)]. f ( x) 15. True: ∞ Consider As x → a, f ( x) → 2 while 1 → ∞. g ( x) 1 dx diverges for p = lim [–3] = –3 x →∞ 1 0 dx diverges for p xp 1 x →∞ dx = p –b e dx = lim [– e – x ]b0 + 1 = 1, so b →∞ ∞ 0 f ( x)dx The integrand is bounded on the ⎡ π⎤ interval ⎢0, ⎥ . ⎣ 4⎦ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Sample Test Problems 1. The limit is of the form 0 . 0 2 x3 6 x2 = lim = lim 6 x3 = ∞ x →∞ ln x x →∞ 1 x →∞ lim x 4x 4 = lim =4 x →0 tan x x →0 sec 2 x lim 2. The limit is of the form 1 → ∞ . A number x less than 1, raised to a large power, is a very ⎛ ⎛ 1 ⎞32 ⎞ small number ⎜ ⎜ ⎟ = 2.328 × 10−10 ⎟ so ⎜⎝ 2 ⎠ ⎟ ⎝ ⎠ 9. As x → 0, sin x → 0 , and 0 . 0 tan 2 x 2sec2 2 x 2 = lim = 3 x →0 sin 3 x x →0 3cos 3 x lim 0 3. The limit is of the form . (Apply l’Hôpital’s 0 Rule twice.) sin x − tan x cos x − sec2 x lim = lim 2x 1 x2 x →0 x →0 3 3 = lim − sin x − 2sec x(sec x tan x) 2 3 x →0 x →0 + 10. lim x ln x = lim x →0 + lim x →0 + = ∞ (L’Hôpital’s Rule does not apply x2 since cos(0) = 1.) 2 x cos x x →0 x →0 sin x 0 The limit is of the form . 0 2 x cos x 2 cos x – 2 x sin x lim = lim cos x x →0 sin x x →0 2–0 = =2 1 5. lim 2 x cot x = lim 6. The limit is of the form ∞ . ∞ − 1−1x ln(1 − x) = lim 2 x →1 cot πx x →1− −π csc πx sin 2 πx = lim x →1− π(1 − x ) lim − The limit is of the form 0 . 0 sin 2 πx 2π sin πx cos πx lim = lim =0 − π(1 – x ) − −π x →1 x →1 7. The limit is of the form ln t t →∞ t 2 = lim 1 t t →∞ 2t ∞ . ∞ = lim 1 t →∞ 2t 2 Instructor’s Resource Manual x →0+ ln x 1 x ∞ . ∞ The limit is of the form =0 x →0 lim lim (sin x)1/ x = 0 . cos x 4. lim ∞ . ∞ 8. The limit is of the form ln x 1 x 1 x 1 x →0 + – 2 x = lim = lim – x = 0 x →0 + 11. The limit is of the form 00. Let y = x x , then ln y = x ln x. lim x ln x = lim x →0 + x →0+ ln x 1 x ∞ . ∞ The limit is of the form lim x →0 + ln x 1 x 1 x 1 x →0 + – 2 x ln y = lim lim x x = lim e x →0 + x →0 + = lim – x = 0 x →0 + =1 12. The limit is of the form 1∞. 2 ln(1 + sin x). x 2 2 ln(1 + sin x ) lim ln(1 + sin x) = lim x x →0 x x →0 0 The limit is of the form . 0 Let y = (1 + sin x)2 / x , then ln y = 2 cos x 2 ln(1 + sin x) = lim 1+sin x 1 x x →0 x →0 2 cos x 2 = lim = =2 1 x →0 1 + sin x lim lim (1 + sin x)2 / x = lim eln y = e2 x →0 x →0 =0 Section 8.5 507 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 13. x ln x = lim lim x →0 + 1 x x →0+ The limit is of the form lim ln x 1 x x →0 + = lim x →0 + – 17. The limit is of the form 1∞. ln x Let y = (sin x) tan x , then ln y = tan x ln(sin x). ∞ . ∞ 1 x 1 2 x3/ 2 lim tan x ln(sin x ) = lim x→ π 2 = lim – 2 x = 0 x →0+ The limit is of the form sin x = lim x→ π 2 lim (sin x) tan x = lim eln y = 1 18. lim t1/ t = lim eln y = 1 t →∞ ∞ 16. The limit is of the form . (Apply l’Hôpital’s ∞ Rule three times.) tan 3x 3sec2 3 x lim = lim 2 x → π tan x x → π sec x x→ π 2 = lim x→ π 2 508 cos 2 3 x x sin x – π2 cos x x→ π 2 19. 20. 21. 22. = lim x→ π 2 0 . 0 sin x + x cos x 1 = =1 sin x 1 ∞ ⎡ 1 ⎤ = ⎢– ⎥ = 0 +1 = 1 0 ( x + 1) 2 ⎣ x + 1⎦0 ∞ ∞ 0 dx ∞ π π = ⎡ tan –1 x ⎤ = – 0 = ⎣ ⎦ 0 2 2 1+ x dx 2 1 1 1 ⎡1 ⎤ e2 x dx = ⎢ e2 x ⎥ = e2 – 0 = e2 –∞ 2 ⎣2 ⎦ –∞ 2 1 dx = lim[– ln(1 – x)]b–1 x b→1 = – lim ln(1 – b) + ln 2 = ∞ 1 –1 1 – b→1 cos x sin x π cos 3 x sin 3 x x→ = lim The integral diverges. 2 2 2 2 2 cos x − sin x 3(cos 3 x − sin 3 x) Section 8.5 x sin x – π2 π ⎛ ⎞ lim ⎜ x tan x – sec x ⎟ = lim 2 ⎠ x → π2 cos x x→ π ⎝ 2 lim 2 3cos x 2 The limit is of the form 1⎞ x – sin x ⎛ 1 lim ⎜ – ⎟ = lim + ⎝ sin x + x ⎠ x →0 x sin x x →0 0 The limit is of the form . (Apply l’Hôpital’s 0 Rule twice.) x – sin x 1 – cos x = lim lim + x sin x + sin x + x cos x x →0 x →0 sin x 0 = lim = =0 + 2 cos x – x sin x 2 x →0 = lim x→ π 2 1 2 cos x(1 + ln(sin x)) 0 = =0 sin x 1 x→ π ln t 1 = lim t = lim = 0 t →∞ t t →∞ 1 t →∞ t 2 0 . 0 lim lim 15. sin x ln(sin x) cos x cos x ln(sin x) + sin x cos x sin x ln(sin x) = lim cos x sin x x→ π x→ π 2 2 14. The limit is of the form ∞0 . 1 Let y = t1/ t , then ln y = ln t. t 1 ln t lim ln t = lim t →∞ t t →∞ t ∞ The limit is of the form . ∞ t →∞ x→ π 2 =− 1 1 = 3(0 − 1) 3 23. dx = [ln( x + 1)]0∞ = ∞ – 0 = ∞ x +1 The integral diverges. ∞ 0 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 24. 2 dx 1 2 x(ln x)1/ 5 = lim b dx 1 x(ln x)1/ 5 b →1– 2 b 2 4/5 ⎞ ⎛5 5⎛ 1⎞ 5 ⎞ 5 5⎛ 1⎞ ⎛5 = ⎜ (0) – ⎜ ln ⎟ ⎟ + ⎜ (ln 2)4 / 5 – (0) ⎟ = (ln 2) 4 / 5 – ⎜ ln ⎟ ⎜4 4⎝ 2⎠ 4⎝ 2⎠ ⎟ ⎝ 4 4 ⎠ 4 ⎝ ⎠ 5 = [(ln 2) 4 / 5 – (ln 2) 4 / 5 ] = 0 4 25. ∞ dx x2 + x4 1 = ∞⎛ 1 ⎜ 2 ⎝x 1 − 2 ⎡5 ⎤ ⎡5 ⎤ = lim ⎢ (ln x)4 / 5 ⎥ + lim ⎢ (ln x) 4 / 5 ⎥ 1/ 5 1 + b – + ⎦ ⎦b x(ln x) b →1 b →1 ⎣ 4 b →1 ⎣ 4 dx 2 + lim 4/5 5 = [(ln 2)4 / 5 – (– ln 2)4 / 5 ] 4 ∞ π π π π ⎞ ⎡ 1 ⎤ dx = ⎢ – – tan –1 x ⎥ = 0 – + 1 + tan –1 1 = 1 + – = 1 – 2⎟ x 2 4 2 4 ⎣ ⎦ 1+ x ⎠ 1 1 1 26. 1 ⎡ 1 ⎤ =⎢ = −0 =1 ⎥ – ∞ (2 – x ) 2 ⎣ 2 – x ⎦ –∞ 1 27. b 0 b 0 dx dx dx ⎡1 ⎤ ⎡1 ⎤ = lim + lim = lim ⎢ ln 2 x + 3 ⎥ + lim ⎢ ln 2 x + 3 ⎥ –2 2 x + 3 – –2 2 x + 3 + b 2x + 3 – 2 ⎦ –2 b→ – 3 + ⎣ 2 ⎦b b→ – 3 b→ – 3 b→ – 3 ⎣ 2 2 2 2 dx 1 0 ⎛ ⎞ ⎛ ⎞ 1 1 1 1 ⎛1 ⎞ ln 2b + 3 – (0) ⎟ + ⎜ ln 3 – lim ln 2b + 3 ⎟ = (– ∞) + ⎜ ln 3 + ∞ ⎟ = ⎜ lim ⎜⎜ ⎟ ⎜ ⎟ – + 2 ⎟ ⎜2 2 3 2 ⎝2 ⎠ ⎟ b→ – 3 2 ⎝ b→ – 2 ⎠ ⎝ ⎠ The integral diverges. 28. 29. 30. 31. 4 dx 1 x –1 ∞ 2 b →1+ b →1+ ∞ 1 1 ⎡ 1 ⎤ = ⎢– = –0 + = ⎥ 2 ln 2 ln 2 ⎣ ln x ⎦ 2 x(ln x) dx ∞ ∞ dx 0 ex / 2 ⎡ 2 ⎤ 2 = ⎢– = –0 + = 2 x/2 ⎥ 1 ⎣ e ⎦0 dx 5 3 = lim [2 x – 1]b4 = 2 3 – lim 2 x – 1 = 2 3 – 0 = 2 3 (4 – x) 2/3 dx b = lim b→4– 3 (4 – x) 2/3 b → 4+ b b dx 5 + lim (4 – x) 2/3 5 = lim ⎡ –3(4 – x)1/ 3 ⎤ + lim ⎡ –3(4 – x)1/ 3 ⎤ ⎣ ⎦ 3 b → 4+ ⎣ ⎦b b→4– = lim − 3(4 – b)1/ 3 + 3(1)1/ 3 – 3(–1)1/ 3 + lim 3(4 – b)1/ 3 = 0 + 3 + 3 + 0 = 6 b→4+ b→4– 32. 33. ∞ 2 ∞ ∞ 2 2⎤ 1 1 ⎡ 1 xe – x dx = ⎢ – e – x ⎥ = 0 + e –4 = e –4 2 2 2 ⎣ ⎦2 x –∞ x2 +1 dx = 0 x – ∞ x2 +1 dx + ∞ 0 x 2 x +1 0 ∞ 1⎡ 1 ln( x 2 + 1) ⎤ + ⎡ ln( x 2 + 1) ⎤ = ⎣ ⎦ ⎣ ⎦ –∞ 2 0 2 (0 + ∞) + (∞ – 0) = 34. dx The integral diverges. 0 ∞ ⎡1 ⎤ ⎡1 ⎤ dx = ⎢ tan –1 x 2 ⎥ + ⎢ tan –1 x 2 ⎥ –∞ 1 + x4 –∞ 1 + x4 0 1 + x4 2 2 ⎣ ⎦ –∞ ⎣ ⎦0 π π 1 1 1 1 π π ⎛ ⎞ ⎛ ⎞ = tan –1 0 – ⎜ ⎟ + ⎜ ⎟ – tan –1 0 = 0 – + – 0 = 0 2 2⎝ 2⎠ 2⎝ 2⎠ 2 4 4 ∞ x dx = 0 x Instructor’s Resource Manual dx + ∞ x Section 8.5 509 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 35. ex = e2 x + 1 ex (e x ) 2 + 1 Let u = e x , du = e x dx ex ∞ 0 e 2x ∞ dx = +1 ∞ π π π π du = ⎡ tan –1 u ⎤ = – tan –1 1 = – = ⎣ ⎦ 1 2 2 4 4 u +1 1 2 1 36. Let u = x3 , du = 3 x 2 dx ∞ 2 – x3 ∞ 1 –u 1 x e dx = e du = –∞ –∞ 3 3 The integral diverges. 37. 0 –∞ e – u du + 0 ∞ 1 1 1 1 e du = ⎡ – e – u ⎤ + ⎡ – e – u ⎤ = (–1 + ∞) + (–0 + 1) ⎣ ⎦ ⎣ ⎦ –∞ 3 0 3 3 3 ∞ –u 0 x 3 dx = 0 9 − x2 See Problem 35 in Section 8.4. −3 38. let u = ln(cos x), then du = π 2 π 3 tan x (ln cos x) 2 dx = 39. For p ≠ 1, p ≠ 0, 1 lim b →∞ b p –1 1 ∞ xp b →0 b 1 p –1 ∞1 1 xp 1 ∞ ⎡ ⎤ 1 1 1 dx = ⎢ – + ⎥ = lim p –1 p p –1 1 x p –1 b →∞ (1 – p )b ⎣⎢ ( p – 1) x ⎦⎥1 ∞ 1 1 = ∞ when p < 1, p ≠ 0. dx = [ln x]1∞ = ∞ – 0 . The integral diverges. x ∞ 1dx = [ x]1∞ = ∞ – 1 . The integral diverges. 1 1 ⎡ ⎤ 1 1 1 dx = ⎢ – + lim ⎥ = p –1 0 xp 1 – p b → 0 ( p – 1)b p –1 ⎣⎢ ( p – 1) x ⎦⎥ 0 1 1 converges when p – 1 < 0 or p < 1. When p = 0, 1 ln 1 1 ⎡ 1⎤ 2 du = ⎢ – ⎥ =– +0 = 2 1 u ln 2 ln 2 ⎣ ⎦ –∞ u 1 dx converges when p > 1 and diverges when p ≤ 1. When p = 1, 0 du = 1 2 –∞ ln 1 40. For p ≠ 1, p ≠ 0, lim –∞ 1 1– 2 ln 2 u b →∞ b p −1 When p = 0, 1 1 ⋅ – sin x dx = – tan x dx cos x = 0 when p – 1 > 0 or p > 1, and lim When p = 1, 11 0 x dx = [ln x]10 = 0 – lim ln b = ∞ . The integral diverges. b →0+ 1 0 1dx = [ x]10 = 1 – 0 = 1 dx converges when p < 1 and diverges when 1 ≤ p. 41. For x ≥ 1, x 6 + x > x 6 , so Section 8.5 1 x 6 + x > x 6 = x3 and converges since 3 > 1 (see Problem 39). Thus 510 1 3 ∞ 1 x6 + x 1 x6 + x < 1 x 3 . Hence, ∞ 1 1 x6 + x dx < ∞ 1 1 x3 dx which dx converges. Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 42. For x > 1, ln x < e x , so ln x ln x ln x e x < 1 and 1 = < . e 2 x (e x ) 2 e x Hence, ∞ ln x ∞ 1 dx < e – x dx = [– e – x ]1∞ = –0 + e –1 = . x 2 1 e 1 e ∞ ln x Thus, dx converges. 1 e2 x 43. For x > 3, ln x > 1, so ∞ ln x 3 x dx > ∞1 3 x ln x 1 > . Hence, x x dx = [ln x]3∞ = ∞ – ln 3. The integral diverges, thus ∞ ln x 3 x dx also diverges. 44. For x ≥ 1, ln x < x, so ln x ln x 1 . < 1 and < x x3 x2 4. Original: f continuous at c ⇒ f differentiable at c Converse: f differentiable at c ⇒ f continuous at c (AT) Contrapositive: f non-differentiable at c ⇒ f discontinuous at c 5. Original: f right continuous at c ⇒ f continuous at c Converse: f continuous at c ⇒ f right continuous at c (AT) Contrapositive: f discontinuous at c ⇒ f not right continuous at c 6. Original: f ( x) 0 ⇒ f ( x) = c (AT) Converse: f ( x) = c ⇒ f ( x) 0 (AT) Contrapositive: f ( x) ≠ c ⇒ f ( x) 0 (AT) 7. Original: f ( x) = x 2 ⇒ f ( x) = 2 x (AT) Converse: f ( x) = 2 x ⇒ f ( x) = x 2 Hence, ∞ 1 (Could have f ( x) = x 2 + 3 ) ∞ ⎡ 1⎤ dx < dx = ⎢ – ⎥ = –0 + 1 = 1. 1 x3 1 x2 ⎣ x ⎦1 ∞ ln x dx converges. Thus, 1 x3 ∞ ln x Contrapositive: f ( x ) ≠ 2 x ⇒ f ( x) ≠ x 2 (AT) 8. Original: a < b ⇒ a 2 < b 2 Converse: a 2 < b 2 ⇒ a < b Contrapositive: a 2 ≥ b 2 ⇒ a ≥ b Review and Preview Problems 1. Original: If x > 0 , then x 2 > 0 (AT) Converse: If x 2 > 0 , then x > 0 9. 1 + 1 1 1 1 1 + + + + = 2 4 8 16 32 32 16 8 4 2 1 63 + + + + + = 32 32 32 32 32 32 32 10. 1 + Contrapositive: If x 2 ≤ 0 , then x ≤ 0 (AT) 2. Original: If x 2 > 0 , then x > 0 Converse: If x > 0 , then x 2 > 0 (AT) 4 11. Contrapositive: If x ≤ 0 , then x 2 ≤ 0 3. Original: f differentiable at c ⇒ f continuous at c (AT) Converse: f continuous at c ⇒ f differentiable at c Contrapositive: f discontinuous at c ⇒ f non-differentiable at c (AT) Instructor’s Resource Manual 1 1 4 2 1 7 + = + + = 2 4 4 4 4 4 1 1 1 1 1 12 + 6 + 4 + 3 25 = + + + = = 12 12 i =1 i 1 2 3 4 4 12. (−1) k k = −1 1 −1 1 + + + = 2 4 8 16 2 −8 + 4 − 2 + 1 −5 = 16 16 k =1 ⎛∞⎞ 13. By L’Hopital’s Rule ⎜ ⎟ : ⎝∞⎠ x 1 1 lim = lim = x →∞ 2 x + 1 x →∞ 2 2 Review and Preview 511 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ⎛∞⎞ 14. By L’Hopital’s Rule ⎜ ⎟ twice: ⎝∞⎠ 20. x ∞ 1 2 x +1 t →∞ 1 n2 2n 2 1 lim = lim = = 2 4 2 n →∞ 2n + 1 n→∞ 4n 1 lim 2 t →∞ ⎛∞⎞ 15. By L’Hopital’s Rule ⎜ ⎟ twice: ⎝∞⎠ lim x →∞ x 2 e x = lim x →∞ 2x e = lim x x →∞ 2 ex =0 lim en n →∞ 17. ∞1 1 x = lim n →∞ 2n = lim en t dx = lim n →∞ 1 2 en =0 lim [ ln x ] dx = 21. ∞ 1 1 x2 t 1 t →∞ 1 x2 dx = u = x 2 +1 du = 2 x dx du = ∞ t dx = lim x dx = t →∞ 1 x 2 + 1 x2 + 1 Integral does not converge. 1 1 ∞ 2 x (ln x) 2 dx = lim t t →∞ 2 ( 1 2 ln x +1 2 1 x(ln x) 2 ) ∞ =∞ 1 dx = ln t ⎡ 1⎤ du = lim ⎢ − ⎥ = t →∞ ln 2 u 2 t →∞ ⎣ u ⎦ ln 2 1 ⎤ 1 ⎡ 1 lim ⎢ − = ≈ 1.443 ⎥ t →∞ ⎣ ln 2 ln t ⎦ ln 2 [ ]=∞ dx = lim 22. x ∞ lim Integral does not converge. 18. u x +1 u = ln x du = 1 x dx t →∞ 1 x t = lim ln t 1 t →∞ t →∞ 2 2 Integral does not converge (see problem 17). ⎛∞⎞ 16. By L’Hopital’s Rule ⎜ ⎟ twice: ⎝∞⎠ n2 t 2 +1 1 x t dx = lim ln t 1 Integral converges. dx = t ⎡ 1⎤ ⎡ 1⎤ lim ⎢ − ⎥ = lim ⎢1 − ⎥ = 1 t⎦ t →∞ ⎣ x ⎦1 t →∞ ⎣ Integral converges. 19. 1 ∞ 1 1.001 x dx = lim t 1 t →∞ 1 x1.001 dx = t 1000 ⎤ ⎡ 1000 ⎤ ⎡ lim ⎢ − = lim ⎢1000 − ⎥ ⎥ = 1000 0.001 t →∞ ⎣ x t 0.001 ⎦ ⎦1 t →∞ ⎣ Integral converges. 512 Review and Preview Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 9 CHAPTER Infinite Series 7 26 63 124 215 5. a1 = , a2 = , a3 = , a4 = , a5 = 8 27 64 125 216 9.1 Concepts Review 1. a sequence lim n3 + 3n 2 + 3n (n + 1)3 n →∞ 2. lim an exists (finite sense) n →∞ n n→∞ 1 + 3 n 4. –1; 1 6. a1 = Problem Set 9.1 1 2 3 4 5 1. a1 = , a2 = , a3 = , a4 = , a5 = 2 5 8 11 14 n 1 1 lim = lim = ; 3 n →∞ 3n –1 n →∞ 3 – 1 n converges 5 8 11 14 17 2. a1 = , a2 = , a3 = , a4 = , a5 = 2 3 4 5 6 3 + n2 3n + 2 = lim = 3; n →∞ n + 1 n →∞ 1 + 1 lim n converges a4 = n3 + 3n 2 + 3n n →∞ n3 1 + n3 + 32 = lim 3. bounded above = lim =1 + 32 + 13 n + 3n 2 + 3n + 1 n 5 14 29 , a2 = , a3 = , 3 5 7 50 5 2 77 = , a5 = 9 9 11 3 + 22 3n 2 + 2 3 n = lim = ; 2 n →∞ 2n + 1 n→∞ 2 + 1 n converges lim 1 2 1 3 4 2 7. a1 = – , a2 = = , a3 = – , a4 = = , 3 4 2 5 6 3 5 a5 = – 7 n 1 = lim = 1, but since it alternates lim n →∞ n + 2 n→∞ 1 + 2 n 6 18 38 3. a1 = = 2, a2 = = 2, a3 = , 3 9 17 66 22 102 34 a4 = = , a5 = = 27 9 39 13 lim 4n 2 + 2 n →∞ n 2 + 3n – 1 = lim 4+ n →∞ 1 + 3 n 2 n2 – 1 n2 between positive and negative, the sequence diverges. = 4; converges 4. a1 = 5, a2 = 2 3 4 5 8. a1 = –1, a2 = , a3 = – , a4 = , a5 = – 3 5 7 9 ⎧−1 for n odd cos(nπ) = ⎨ ⎩ 1 for n even lim 14 29 50 77 , a3 = , a4 = , a5 = 3 5 7 9 2 3n + n 3n 2 + 2 = lim = ∞; n →∞ 2n –1 n→∞ 2 – 1 n n →∞ 2n – 1 = lim 1 n→∞ 2 – 1 n = 1 , but since cos(n π ) 2 alternates between 1 and –1, the sequence diverges. lim n diverges Instructor’s Resource Manual Section 9.1 513 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 1 1 1 9. a1 = –1, a2 = , a3 = – , a4 = , a5 = – 2 3 4 5 n 1 cos(nπ) 1 cos ( nπ ) = ( −1) , so – ≤ ≤ . n n n 1 1 lim – = lim = 0, so by the Squeeze n →∞ n n→∞ n Theorem, the sequence converges to 0. 10. a1 = e –1 sin1 ≈ 0.3096, a2 = e –2 sin 2 ≈ 0.1231, a3 = e –3 sin 3 ≈ 0.0070, a4 = e –4 sin 4 ≈ –0.0139, a5 = e –5 sin 5 ≈ –0.0065 –1 ≤ sin n ≤ 1 for all n, so – e – n ≤ e – n sin n ≤ e – n . lim – e – n = lim e – n = 0, so by the Squeeze n →∞ n →∞ Theorem, the sequence converges to 0. 11. a1 = a3 = e2 e4 ≈ 2.4630, a2 = ≈ 6.0665, 3 9 e6 e8 ≈ 23.7311, a4 = ≈ 110.4059, 17 27 e10 ≈ 564.7812 39 Consider e2 x 2e2 x 4e2 x = lim = lim =∞ lim x →∞ x 2 + 3 x –1 x →∞ 2 x + 3 x →∞ 2 by using l’Hôpital’s Rule twice. The sequence diverges. a5 = e2 e4 12. a1 = ≈ 1.8473, a2 = ≈ 3.4124, 4 16 6 8 e e a3 = ≈ 6.3036, a4 = ≈ 11.6444, 64 256 a5 = e 2n 4n 10 e ≈ 21.510 1024 ⎛ e2 =⎜ ⎜ 4 ⎝ n ⎞ e2 > 1 so the sequence diverges. ⎟ , ⎟ 4 ⎠ π π2 13. a1 = – ≈ –0.6283, a2 = ≈ 0.3948, 5 25 a3 = – π3 π4 ≈ –0.2481, a4 = ≈ 0.1559, 125 625 a5 = – π5 ≈ –0.0979 3125 (– π)n n 1 1 + 3 ≈ 1.9821, a2 = + 3 = 3.0625, 4 16 1 1 a3 = + 3 3 ≈ 5.2118, a4 = + 9 ≈ 9.0039, 64 256 1 a5 = + 9 3 ≈ 15.589 1024 14. a1 = n 1 ⎛1⎞ ⎜ ⎟ converges to 0 since –1 < < 1. 4 4 ⎝ ⎠ 3n / 2 = ( 3) n 3 ≈ 1.732 > 1 . diverges since Thus, the sum diverges. 15. a1 = 2.99, a2 = 2.9801, a3 ≈ 2.9703, a4 ≈ 2.9606, a5 ≈ 2.9510 (0.99)n converges to 0 since –1 < 0.99 < 1, thus 2 + (0.99)n converges to 2. 16. a1 = a3 = a5 = 1 2100 ≈ 0.3679, a2 = ≈ 1.72 × 1029 , 2 e e 3100 ≈ 2.57 × 1046 , a4 = 3 e 5100 4100 e 4 ≈ 2.94 × 1058 , ≈ 5.32 × 1067 e5 Consider lim x →∞ x100 ex . By Example 2 of x100 Section 8.2, lim ex x →∞ = 0 . Thus, lim n100 en n →∞ = 0; converges 17. a1 = a3 = a5 = ln1 1 ln 3 3 ln 5 5 = 0, a2 = ln 2 2 ≈ 0.4901, ≈ 0.6343, a4 = ln 4 ≈ 0.6931 , 2 ≈ 0.7198 Consider lim x →∞ ln x x = lim x →∞ 1 x 1 = lim 2 x x →∞ using l’Hôpital’s Rule. Thus, lim n →∞ ln n n 2 x = 0 by = 0; converges. π ⎛ π⎞ = ⎜ – ⎟ , – 1 < – < 1, thus the sequence n 5 ⎝ 5⎠ 5 converges to 0. 514 Section 9.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 18. a1 = a3 = a5 = ln1 2 = 0, a2 = ln 13 6 ln 15 10 lim ln 1n n →∞ 2n 2 ln 14 2 2 a3 = 61/ 6 ≈ 1.3480, a4 = 81/ 8 = 23 / 8 ≈ 1.2968, ≈ –0.4901, a5 = 101/10 ≈ 1.2589 Consider lim (2 x)1/ 2 x . This limit is of the form x →∞ ≈ –0.5089 ln 1x x →∞ x →∞ 20. a1 = 21/ 2 ≈ 1.4142, a2 = 41/ 4 = 21/ 2 ≈ 1.4142, ≈ –0.3466, ≈ –0.4485, a4 = Consider lim = lim − ln 12 2 2x ∞ 0 . Let y = (2 x)1/ 2 x , then ln y = = lim − ln x x →∞ 2x = lim x →∞ − 1x 1 2x = 0 by using l’Hôpital’s Rule. Thus, x ln 2 x 2x ∞ This limit is of the form . ∞ lim 1 1 ln 2 x. 2x x →∞ 2 x ln 2 x = lim x →∞ 1 ln 2 x 1 = lim x = lim =0 x →∞ 2 x x →∞ 2 x →∞ 2 x lim = 0; converges lim (2 x)1/ 2 x = lim eln y = 1 x →∞ 1/ 2 ⎛ 2⎞ 19. a1 = ⎜1 + ⎟ ⎝ 1⎠ = 3 ≈ 1.7321, ⎛ 2⎞ a2 = ⎜ 1 + ⎟ ⎝ 2⎠ ⎛ 2⎞ a3 = ⎜ 1 + ⎟ ⎝ 3⎠ 2/ 2 = 2, 3/ 2 ⎛ 2⎞ a4 = ⎜1 + ⎟ ⎝ 4⎠ 4/2 5/ 2 ⎛5⎞ =⎜ ⎟ ⎝3⎠ x →∞ 1/ 2 n = 1; converges Thus lim (2n) n →∞ n 1 or an = 1 − ; n +1 n +1 1 ⎞ 1 ⎛ lim ⎜1 − = 1; converges ⎟ = 1 − lim n +1⎠ n →∞ ⎝ n→∞ n + 1 21. an = 3/ 2 ≈ 2.1517, 2 9 ⎛3⎞ =⎜ ⎟ = , 2 4 ⎝ ⎠ 22. an = 5/ 2 ⎛ 2⎞ ⎛7⎞ a5 = ⎜ 1 + ⎟ =⎜ ⎟ ≈ 2.3191 5 ⎝ ⎠ ⎝5⎠ 2 Let = h, then as n → ∞, h → 0 and n n/2 ⎛ 2⎞ lim ⎜1 + ⎟ = lim (1 + h)1/ h = e by n⎠ h→0 Theorem 6.5A; converges n →∞ ⎝ n 2 n +1 Consider x 2 x . Now, lim x = lim x →∞ 2 x 1 x →∞ 2 x by l’Hôpital’s Rule. Thus, lim n n →∞ 2n +1 ln 2 =0 = 0; converges n n ; lim 2n − 1 n→∞ 2n − 1 1 1 = lim = , but due to (−1)n , the terms of 2 n →∞ 2 − 1 23. an = (−1)n n the sequence alternate between positive and negative, so the sequence diverges. 24. an = 1 1 − nn−1 = n; lim n = ∞ ; diverges n →∞ 25. an = n 2 2 = n 2 2 = n – (n –1) n – (n – 2n + 1) n 1 1 lim = lim = ; converges 2 n →∞ 2n –1 n→∞ 2 – 1 n ; 2n –1 n Instructor’s Resource Manual Section 9.1 515 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 26. an = n n(n + 1) n2 + n = = ; (n + 1) − n1+1 (n + 1)2 − 1 n 2 + 2n 2 lim n +n n →∞ n 2 + 2n = lim 1 + 1n n→∞ 1 + 2 n = 1; converges 1 n sin x = 1; converges lim x →0 x n2 n2 3n ; n →∞ 3n 29. an = lim 2n n2 2n n →∞ n 2 = lim 2n n →∞ 3n = lim 2 =0 ln 3 by using l’Hôpital’s Rule twice; converges lim n→∞ 3n (ln 3) 2 ; 2n ln 2 2n (ln 2) 2 = lim = ∞; 2 n →∞ 2n n→∞ = lim diverges 30. an = an = 2n 2 − 1 n2 + n 1 1 n +1 – n 1 – ; = = n n + 1 n(n + 1) n(n + 1) to a limit L ≤ 2. 1 5 9 13 31. a1 = , a2 = , a3 = , a4 = 2 4 8 16 an is positive for all n, and an +1 < an for all 4n − 7 converges to a limit L ≥ 0. 2n +1 , so {an } n + 2n , so {an } converges ⎛ 1 ⎞ 1 < 1, so an +1 = an ⎜ 1 − and 1 − ⎜ (n + 1) 2 ⎟⎟ (n + 1)2 ⎝ ⎠ {an } converges to a limit L ≥ 0. 3 5 41 34. a1 = 1; a2 = ; a3 = ; a4 = 2 3 24 an < 2 for all n since 1 1 1 1 1 + + + 1+ + + ≤ 0 1 n n! 2 2! 2 2 +1 < 1 = 0; converges n →∞ n( n + 1) 2 2 3 ⎛ 3 ⎞⎛ 8 ⎞ 2 33. a2 = ; a3 = ⎜ ⎟ ⎜ ⎟ = ; 4 ⎝ 4 ⎠⎝ 9 ⎠ 3 ⎛ 3 ⎞ ⎛ 8 ⎞⎛ 15 ⎞ 5 a4 = ⎜ ⎟ ⎜ ⎟⎜ ⎟ = ; ⎝ 4 ⎠ ⎝ 9 ⎠⎝ 16 ⎠ 8 ⎛ 3 ⎞⎛ 8 ⎞ ⎛ 15 ⎞ ⎛ 24 ⎞ 3 a5 = ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎝ 4 ⎠⎝ 9 ⎠ ⎝ 16 ⎠ ⎝ 25 ⎠ 5 an > 0 for all n and an +1 < an since lim n ≥ 2 since an +1 − an = − < 2 for all n, and an < an +1 for all n since an +1 − an = sin n 1 1 27. an = n sin ; lim n sin = lim = 1 since n n→∞ n n→∞ 1 28. an = (–1)n 1 7 17 31 32. a1 = ; a2 = ; a3 = ; a4 = 2 6 12 20 ∞ k ⎛1⎞ ⎜ ⎟ =2 k =0 ⎝ 2 ⎠ the sum never reaches 2. an < an +1 since each term is the previous term plus a positive quantity, so {an } converges to a limit L ≤ 2. 1 3 1⎛3⎞ 7 35. a1 = 1, a2 = 1 + (1) = , a3 = 1 + ⎜ ⎟ = , 2 2 2⎝2⎠ 4 1 ⎛ 7 ⎞ 15 a4 = 1 + ⎜ ⎟ = 2⎝4⎠ 8 1 1 Suppose that 1 < an < 2, then < an < 1, so 2 2 3 1 3 < 1 + an < 2, or < an +1 < 2. Thus, since 2 2 2 1 < a2 < 2, every subsequent term is between 3 2 and 2. 1 1 an < 1, so an < 1 + an = an +1 2 2 and the sequence is nondecreasing, so {an } converges to a limit L ≤ 2. an < 2 thus 516 Section 9.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1⎛ 2⎞ ⎜2+ ⎟ = 2⎝ 2⎠ 1 ⎛ 3 4 ⎞ 17 a3 = ⎜ + ⎟ = , a4 2 ⎝ 2 3 ⎠ 12 36. a1 = 2, a2 = un +1 = 3 + un > un if 3 + un > un 2 or 3 , 2 1 ⎛ 17 24 ⎞ 577 = ⎜ + ⎟= 2 ⎝ 12 17 ⎠ 408 un 2 – un – 3 < 0. un 2 – un – 3 = 0 when Suppose an > 2 , and consider ( 1⎛ 2 ⎞ 2 >2 2⇔ ⎜ an + ⎟ > 2 ⇔ an + an ⎠ an 2⎝ 2 > 0 , which is always true. Hence, an > 2 for all n. Also, n 1 2 3 4 5 6 7 8 9 10 11 lim un ≈ 2.3028 u= 41. ( ) 1 1 + 13 , then 2 ( ( ) ) ( ) Instructor’s Resource Manual ) ) ) ) ( n 1 2 3 4 5 6 7 8 lim un ≈ 1.1118 un 0 1 1.1 1.11053 1.11165 1.11177 1.11178 1.11178 n →∞ ) ( ( ( 1 1 ± 13 so 2 1⎛ 2 ⎞ 40. If a = lim an where an +1 = ⎜ an + ⎟ , then an ⎠ 2⎝ n →∞ 1⎛ 2⎞ a = ⎜ a + ⎟ or 2a 2 = a 2 + 2; a 2 = 2 when a⎠ 2⎝ a = ± 2, so a = 2, since a > 0. 1 3 < 3 + un < 7 + 13 and 2 1 1 3 < 3 + un = un +1 < 7 + 13 = 1 + 13 2 2 ⎛ 1 1 7 + 13 = 1 + 13 can be seen by ⎜⎜ 2 ⎝ 2 squaring both sides of the equality and noting that both sides are positive.) Hence, since 1 0 < u1 = 3 ≈ 1.73 < 1 + 13 ≈ 2.3028, 2 1 3 < un < 1 + 13 for all n; {un } is bounded 2 above. ) ) 1 1 + 13 ≈ 2.3028 since u > 0 and 2 ( n →∞ ( ) 1 1 – 13 < 0. 2 un 1.73205 2.17533 2.27493 2.29672 2.30146 2.30249 2.30271 2.30276 2.30277 2.30278 2.30278 ( ) ( u 2 – u – 3 = 0 when u = 2 < an +1 ≤ an and the 38. Suppose that 0 < un < ( n →∞ series converges to a limit L ≥ 2. 37. ) 39. If u = lim un , then u = 3 + u or u 2 = 3 + u; 1⎛ 2 ⎞ an +1 ≤ an ⇔ ⎜ an + ⎟ ≤ an ⎜ 2⎝ an ⎟⎠ 1 1 ⇔ ≤ an ⇔ 2 ≤ an an 2 which is true. Hence, ) ( an 2 + 2 > 2 2an ⇔ an 2 − 2 2an + 2 > 0 ⇔ ( an − 2 ) ( 1 1 ± 13 , thus un +1 > un if 2 1 1 1 1 – 13 < un < 1 + 13 , 1 – 13 < 0 2 2 2 1 and 0 < un < 1 + 13 for all n, as shown 2 above, so {un } is increasing. Hence, by Theorem D, {un } converges. un = ) 42. Since 1.1 > 1, 1.1a > 1.1b if a > b. Thus, since u3 = 1.1 > 1 = u2 , u4 = 1.11.1 > 1.11 = u3 . Suppose that un < un +1 for all n ≤ N. Then u N +1 = 1.1u N > 1.1u N –1 = u N , since u N > u N –1 by the induction hypothesis. Thus, un is increasing. 1.1un < 2 if and only if un ln1.1 < ln 2; un < ln 2 ≈ 7.3. Thus, unless un > 7.3, ln1.1 un +1 = 1.1un < 2. This means that {un } is bounded above by 2, since u1 = 0. Section 9.1 517 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 k → 0; using Δx = , an equivalent n n definite integral is 43. As n → ∞, 1 0 sin x dx = [– cos x]10 = – cos1 + cos 0 = 1 – cos1 ≈ 0.4597 1 k → 0; using Δx = , an equivalent n n definite integral is 1 1 π dx = [tan –1 x]10 = tan –1 1 – tan –1 0 = 0 1 + x2 4 44. As n → ∞, 45. n n − (n + 1) −1 1 −1 = = = ; n +1 n +1 n +1 n +1 1 1 < ε is the same as < n + 1. For any given ε n +1 1 ε > 0 , choose N > − 1 then ε n≥N ⇒ 46. For n > 0, n n +1 = n 2 . n 2 n +1 n +1 < ε is the ε n n +1 and L is a lower bound for {an }. Then {– an } is a nondecreasing sequence and –L is an upper bound for {– an }. By what was just proven, {– an } converges to a limit A ≤ –L, so {an } converges to a limit B = –A ≥ L. 49. If {bn } is bounded, there are numbers N and M with N ≤ bn ≤ M for all n. Then an N ≤ an bn ≤ an M . lim an N = N lim an = 0 and n→∞ n→∞ n →∞ by the Squeeze Theorem, and by Theorem C, lim an bn = 0. n →∞ 50. Suppose {an + bn } converges. Then, by Theorem A lim [(an + bn ) – an ] = lim (an + bn ) – lim an . n →∞ n→∞ n→∞ But since (an + bn ) – an = bn , this would mean that {bn } converges. Thus {an + bn } diverges. {an } and {bn } 48. Suppose that {an } is a nondecreasing sequence, and U is an upper bound for {an }, so S = {an : n ∈ } is bounded above. By the completeness property, S has a least upper bound, which we call A. Then A ≤ U by definition and an ≤ A for all n. Suppose that lim an ≠ A, i.e., n →∞ that {an } either does not converge, or does not converge to A. Then there is some ε > 0 such that Section 9.1 < A for all n, 2 which contradicts A being the least upper bound for the set S. For the second part of Theorem D, suppose that {an } is a nonincreasing sequence, 51. No. Consider an = (−1)n and bn = (−1)n +1 . Both < ε. 47. Recall that every rational number can be written as either a terminating or a repeating decimal. Thus if the sequence 1, 1.4, 1.41, 1.414, … has a limit within the rational numbers, the terms of the sequence would eventually either repeat or terminate, which they do not since they are the decimal approximations to 2, which is irrational. Within the real numbers, the least upper bound is 2. 518 ε lim an M = M lim an = 0, so lim an bn = 0 n2 + 1 1 1 = n+ > . n n ε 1 1 Since n + > n , it suffices to take n > . So for n ε 1 any given ε > 0 , choose N > , then 2 A – an > ε for all n, an < A – n →∞ same as n≥N ⇒ A – an ≤ ε for n ≥ N since {an } is nondecreasing and an ≤ A for all n. However, if n →∞ n −1 < ε. n +1 2 A – an > ε for all n, since if A – a N ≤ ε , diverge, but an + bn = (−1) + (−1)n +1 = (−1) n (1 + (−1)) = 0 so n {an + bn } 52. a. converges. f3 = 2, f 4 = 3, f5 = 5, f 6 = 8, f 7 = 13, f8 = 21, f9 = 34, f10 = 55 b. Using the formula, 1 ⎡1 + 5 1 − 5 ⎤ 1 ⎡ 2 5 ⎤ f1 = − ⎢ ⎥= ⎢ ⎥ =1 2 ⎦ 5⎣ 2 5⎣ 2 ⎦ 2 2⎤ ⎡ 1 ⎢⎛ 1 + 5 ⎞ ⎛ 1 − 5 ⎞ ⎥ f2 = ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ 5 ⎢⎝ 2 ⎠ ⎝ 2 ⎠ ⎥ ⎣ ⎦ ⎡ ⎤ 1 1 + 2 5 + 5 − (1 − 2 5 + 5) = ⎢ ⎥ 4 5⎣ ⎦ 1 ⎡4 5 ⎤ = ⎢ ⎥ = 1. 5⎣ 4 ⎦ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. f φ n +1 − (−1)n +1φ − n −1 lim n +1 = lim n →∞ f n n→∞ φ n − (−1) n φ − n n +1 = lim φ n +1 − ( −1)n +1 φ n→∞ φn − φ φ − ( −1) 2 n +1 φ n→∞ n ( x n +1 = lim ( −1)n x ⎛ 1⎞ 54. Let f ( x) = ⎜1 + ⎟ . ⎝ x⎠ ) 1− 2 ( −1) n φ =φ n ⎛ 1⎞ lim ⎜1 + ⎟ = e . n⎠ n →∞ ⎝ 2n ( ) 1 ⎡1 ⎤ 1+ 5 ⎥ − 1+ 5 −1 2 ⎣2 ⎦ ⎛3 5⎞ ⎛1 5⎞ = ⎜⎜ + ⎟⎟ − ⎜⎜ + ⎟⎟ − 1 = 0 ⎝2 2 ⎠ ⎝2 2 ⎠ φ 2 − φ −1 = ⎢ c. ⎛ 1⎞ lim ⎜1 + ⎟ = lim (1 + x )1/ x = e, so x⎠ x →∞ ⎝ x →0 + Therefore φ satisfies x 2 − x − 1 = 0 . Using the Quadratic Formula on x 2 − x − 1 = 0 yields 1± 1+ 4 1± 5 = . 2 2 1+ 5 ; φ= 2 1 2 2(1 − 5) 1 − 5 − =− =− = 1− 5 2 φ 1+ 5 x= 53. x 1 ⎞ ⎛ 55. Let f ( x) = ⎜1 + ⎟ . ⎝ 2x ⎠ x 1/ x 1 ⎞ ⎛ ⎛ x⎞ lim ⎜1 + ⎟ = lim ⎜ 1 + ⎟ + 2x ⎠ 2⎠ x →∞ ⎝ x →0 ⎝ 1/ 2 ⎡⎛ x ⎞ 2 / x ⎤ = lim ⎢⎜ 1 + ⎟ ⎥ 2⎠ ⎥ x →0+ ⎢⎝ ⎣ ⎦ = e1/ 2 , so n 1 ⎞ ⎛ lim ⎜ 1 + ⎟ = e1/ 2 . 2n ⎠ n →∞ ⎝ x 1 ⎞ ⎛ 56. Let f ( x) = ⎜1 + ⎟ . ⎝ x2 ⎠ 1 x ⎛ ⎛ 1 ⎞ 2 ⎞1/ x 1 ⎞ ⎛ lim ⎜ 1 + 2 ⎟ = lim ⎜ 1 + ⎜ ⎟ ⎟ x →∞ ⎝ x →∞ ⎜ x ⎟ x ⎠ ⎝ ⎝ ⎠ ⎠ ⎛1⎞ Using the fact that lim f ( x ) = lim+ f ⎜ ⎟ , we x →∞ x →0 ⎝ x⎠ can write 1 ⎛ ⎛ 1 ⎞ 2 ⎞1/ x lim ⎜ 1 + ⎜ ⎟ ⎟ = lim+ 1 + x 2 x →∞ ⎜ x →0 x ⎟ ⎝ ⎝ ⎠ ⎠ ( From the figure shown, the sides of the triangle have length n – 1 + 2x. The small right triangles 3 marked are 30-60-90 right triangles, so x = ; 2 thus the sides of the large triangle have lengths 2 3 n − 1 + 3 and Bn = n −1+ 3 4 3 2 = n + 2 3n − 2n − 2 3 + 4 while 4 ( ) n(n + 1) ⎛ 1 ⎞ π π ⎜ ⎟ = ( n 2 + n) 2 8 ⎝2⎠ An = lim n →∞ Bn n →∞ lim = lim n→∞ 2 ( π (n2 8 + 2 3n – 2n – 2 3 + 4) π 1+ 1 n 3 1 + 2 n 3 – n2 – 2 3 n2 + 4 n2 1/ x ) = π 2 3 which leads . Then, ( ln y = ln 1 + x 2 ln y = ) 1/ x ( 1 ln 1 + x 2 x lim+ ln y = lim+ ) ( ln 1 + x 2 x x→0 x→0 + n) ) ) = lim+ 3 2 (n 4 ( ( Let y = 1 + x 2 x →0 2 An = 1/ x to the indeterminate form 1∞ . ) ( ) 2x 1 + x2 ) = lim 1+2xx 2 x →0 1 + =0 This gives us lim+ ln y = 0 x →0 ln ⎛⎜ lim+ y ⎞⎟ = 0 ⎝ x →0 ⎠ lim+ y = e0 = 1 or x →0 ( lim+ 1 + x 2 x →0 ) 1/ x =1 n 1 ⎞ ⎛ Thus, lim ⎜1 + 2 ⎟ = 1 . n →∞ ⎝ n ⎠ Instructor’s Resource Manual Section 9.1 519 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. x x ⎛ x −1 ⎞ 57. Let f ( x ) = ⎜ ⎟ . ⎝ x +1⎠ ⎛1⎞ Using the fact that lim f ( x ) = lim+ f ⎜ ⎟ , we x →∞ x →0 ⎝x⎠ can write 1/ x x ⎛ 1x − 1 ⎞ ⎛ x −1 ⎞ = lim ⎜ lim ⎜ ⎟ ⎟ x →∞ ⎝ x + 1 ⎠ x → 0+ ⎜ 1 + 1 ⎟ ⎝x ⎠ 1/ x ⎛ 1− x ⎞ = lim+ ⎜ 1+xx ⎟ ⎟ x →0 ⎜ ⎝ x ⎠ 1/ x ⎛ 1− x ⎞ = lim+ ⎜ ⎟ which leads to the x →0 ⎝ 1 + x ⎠ indeterminate form 1∞ . 1/ x ⎛1− x ⎞ Let y = ⎜ ⎟ ⎝ 1+ x ⎠ lim+ ln y = lim+ x →0 y ⎤⎥ = lim+ ⎦ x →0 = lim+ x →0 1 ⎛1− x ⎞ ln ⎜ ⎟ x ⎝ 1+ x ⎠ ln lim+ ln y = lim+ x →0 1− x 1+ x x −2 1 − x2 n . Then, ⎛ n −1 ⎞ −2 Thus, lim ⎜ ⎟ =e . n →∞ ⎝ n + 1 ⎠ x→0 1 ⎛ 2 x2 + 1 ⎞ ln ⎜ ⎟ x ⎜⎝ 3 x 2 + 1 ⎟⎠ 2 (l'Hopital's Rule) ⎛ 1− x ⎞ lim+ ⎜ ⎟ x →0 ⎝ 1 + x ⎠ or Section 9.1 which leads ( ) ( ) 1/ x 520 1/ x ⎛ 2 x2 + 1 ⎞ = lim+ ⎜ 2 ⎟ x →0 ⎜ 3 x + 1 ⎟ ⎝ ⎠ 1/ x −2 This gives us, ln ⎡⎢ lim+ y ⎤⎥ = −2 ⎣ x →0 ⎦ x →0 1/ x ⎛ 2 x22+1 ⎞ = lim+ ⎜ x2 ⎟ x →0 ⎜ 3 x +1 ⎟ ⎝ x2 ⎠ 1/ x ⎞ ⎟ ⎟ ⎠ ⎛ 2 x2 + 1 ⎞ ln y = ln ⎜ 2 ⎜ 3 x + 1 ⎟⎟ ⎝ ⎠ 1 ⎛1− x ⎞ ln ⎜ ⎟ x ⎝ 1+ x ⎠ lim+ y = e −2 x ⎛ 2 + 12 ⎞ x ⎟ = lim+ ⎜ ⎟ ⎜ 3 + 12 0 x → ⎠ x ⎝ 1/ x 1/ x ln ⎡⎢ lim+ ⎣ x →0 ⎛ 2 + x2 lim ⎜ x →∞ ⎜⎝ 3 + x 2 ⎛ 2 x2 + 1 ⎞ Let y = ⎜ 2 ⎜ 3 x + 1 ⎟⎟ ⎝ ⎠ ⎛ 1− x ⎞ ln y = ln ⎜ ⎟ ⎝ 1+ x ⎠ x →0 ⎛1⎞ Using the fact that lim f ( x ) = lim+ f ⎜ ⎟ , we x →∞ x →0 ⎝x⎠ can write to the indeterminate form 1∞ . . Then, 1/ x ln y = ⎛ 2 + x2 ⎞ 58. Let f ( x) = ⎜ ⎟ . ⎜ 3 + x2 ⎟ ⎝ ⎠ = e−2 ln 2 x2 +1 3 x +1 ⎡ ⎤ ln ⎢ lim+ y ⎥ = lim+ x ⎣ x →0 ⎦ x →0 6x ⎤ ⎡ 4x = lim+ ⎢ 2 − ⎥ (l'Hopital's Rule) x →0 ⎣ 2 x + 1 3 x 2 + 1 ⎦ =0 This gives us, ln ⎡⎢ lim+ y ⎤⎥ = 0 ⎣ x →0 ⎦ 1/ x lim+ y = e0 = 1 or x →0 ⎛ 1− x ⎞ lim+ ⎜ ⎟ x →0 ⎝ 1 + x ⎠ =1 Thus, ⎛ 2 + n2 lim ⎜ n →∞ ⎜ 3 + n 2 ⎝ n ⎞ ⎟⎟ = 1 . ⎠ Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ⎛ 2 + x2 59. Let f ( x ) = ⎜ ⎜ 3 + x2 ⎝ Problem Set 9.2 x2 ⎞ ⎟⎟ ⎠ ∞ ⎛1⎞ Using the fact that lim f ( x ) = lim+ f ⎜ ⎟ , we x →∞ x →0 ⎝ x⎠ can write ⎛ 2 + x2 lim ⎜ x →∞ ⎜⎝ 3 + x 2 ⎛ = lim+ ⎜ x →0 ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ 2 x 2 +1 x2 3 x 2 +1 x2 x2 1/ x ⎞ ⎟ ⎟ ⎠ 2 ∞ 1/ x 2 3. 1/ x 2 ) ⎤ ⎥ (l'Hopital's Rule) ⎥ ⎦⎥ ⎛ 1− x ⎞ lim ⎜ ⎟ x → 0+ ⎝ 1 + x ⎠ Thus, ⎛ 2 + n2 lim ⎜ n →∞ ⎜ 3 + n 2 ⎝ ⎞ ⎟⎟ ⎠ n2 = e −1 . 9.2 Concepts Review 1. an infinite series 2. a1 + a2 + 3. r < 1; 1 6 ⎛ 1⎞ +⎜– ⎟ ⎝ 4⎠ –5 + ; a geometric series 2 k ; a geometric + an a 1– r 2 k ∞ 4. 1/ x 2 x →0 –4 = = e −1 ; a geometric 1 series with a = 3, r = – ; 5 3 3 5 S= = = ⎛ 1⎞ 6 2 1– ⎜ – ⎟ 5 ⎝ 5⎠ Thus, by Theorem B, k k ∞ ⎡ ⎛1⎞ ⎛ 1 ⎞ ⎤ 8 5 31 ⎢2 ⎜ ⎟ + 3 ⎜ – ⎟ ⎥ = + = ⎝ 5 ⎠ ⎥⎦ 3 2 6 k =0 ⎢⎣ ⎝ 4 ⎠ 2 x 2 +1 3 x 2 +1 2 = −1 This gives us, ln ⎡⎢ lim+ y ⎤⎥ = −1 ⎣ x →0 ⎦ or ⎛ 1⎞ +⎜– ⎟ ⎝ 4⎠ 1 ⎛1⎞ ⎛ 1⎞ 3⎜ – ⎟ = 3 – 3 ⋅ + 3⎜ ⎟ − 5 5 ⎝5⎠ ⎝ ⎠ k =0 ( ) x ⎡ −1 = lim+ ⎢ ⎢ 2 x →0 2 x + 1 3x 2 + 1 ⎣⎢ lim+ y = e −1 –3 1 ⎛1⎞ ⎛1⎞ 2⎜ ⎟ = 2 + 2⋅ + 2⎜ ⎟ + 4 4 ⎝4⎠ k =0 ⎝ ⎠ ∞ ⎛ 2 x2 + 1 ⎞ lim+ ln y = lim+ 2 ln ⎜ 2 ⎜ 3x + 1 ⎟⎟ x →0 x →0 x ⎝ ⎠ )( ⎛ 1⎞ =⎜– ⎟ ⎝ 4⎠ 1 7 6 7 1 2 2 8 = = . series with a = 2, r = ; S = 4 1 – 14 43 3 1 ( – k –2 k ∞ . Then, ⎛ 2 x2 + 1 ⎞ ln y = ln ⎜ 2 ⎜ 3 x + 1 ⎟⎟ ⎝ ⎠ y ⎤⎥ = lim+ ⎦ x →0 ; a geometric a = (–4)3 , r = –4; r = 4 > 1 so the series diverges. 1/ x 2 ln ⎡⎢ lim+ ⎣ x →0 ⎛ 1⎞ ⎜– ⎟ k =1 ⎝ 4 ⎠ = (–4)3 + (–4)4 + (–4)5 + with which leads to the indeterminate form 1∞ . ln 2 1 2. ⎛ 2x2 + 1 ⎞ = lim+ ⎜ 2 ⎟ x →0 ⎜ 3 x + 1 ⎟ ⎝ ⎠ ⎛ 2 x2 + 1 ⎞ Let y = ⎜ 2 ⎜ 3x + 1 ⎟⎟ ⎝ ⎠ k 1 1 1 1⎛1⎞ ⎛1⎞ ⎜ ⎟ = + ⋅ + ⎜ ⎟ + 7 7 7 7⎝7⎠ k =1 ⎝ 7 ⎠ 1 1 series with a = , r = ; S = 7 = 7 7 1 – 17 1/ x 2 ⎛ 2 + 12 x = lim ⎜ x →0+ ⎜ 3 + 12 x ⎝ ⎞ ⎟ ⎟ ⎠ 1. 2 5 5 1 5⎛1⎞ ⎛1⎞ 5⎜ ⎟ = + ⋅ + ⎜ ⎟ + 2 2 2 2⎝2⎠ k =1 ⎝ 2 ⎠ ; a geometric 5 5 1 series with a = , r = ; S = 2 = 2 2 1 – 12 ∞ ⎛1⎞ 3⎜ ⎟ k =1 ⎝ 7 ⎠ k +1 3 49 1 – 17 = = 5. 2 = 3 3 1 3 ⎛1⎞ + ⋅ + ⎜ ⎟ + 49 49 7 49 ⎝ 7 ⎠ geometric series with a = S= 5 2 1 2 3 49 6 7 = ; a 3 1 ,r = ; 49 7 1 14 Thus, by Theorem B, k k +1 ∞ ⎡ 1 69 ⎛1⎞ ⎛1⎞ ⎤ . ⎢2 ⎜ ⎟ – 3 ⎜ ⎟ ⎥ = 5 – = 4 7 14 14 ⎝ ⎠ ⎝ ⎠ k =1 ⎣⎢ ⎦⎥ 4. diverges Instructor’s Resource Manual Section 9.2 521 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ∞ 5. k –5 4 3 2 1 1 2 = – – – – +0+ + + k + 2 3 4 5 6 8 9 k =1 ∞ ; 6. 1 – k5 k –5 = lim = 1 ≠ 0; the series k →∞ k + 2 k →∞ 1 + 2 k 2 9 9 9 9⎛9⎞ ⎛9⎞ ⎜ ⎟ = + ⋅ + ⎜ ⎟ + 8 8 8 8 8⎝8⎠ k =1 ⎝ ⎠ ; a geometric 9 9 9 series with a = , r = ; > 1, so the series 8 8 8 diverges. lim k diverges. ∞ 7. 1 ⎞ ⎛ 1 1⎞ ⎛ 1 1 ⎞ ⎛ 1 1 ⎞ ⎛1 ⎜ – ⎟ = ⎜ – ⎟+⎜ – ⎟+⎜ – ⎟+ –1⎠ ⎝ 2 1⎠ ⎝ 3 2 ⎠ ⎝ 4 3 ⎠ k k k =2 ⎝ ⎛1 ⎞ ⎛1 1⎞ S n = ⎜ – 1⎟ + ⎜ – ⎟ + ⎝2 ⎠ ⎝3 2⎠ lim Sn = lim –1 + n →∞ n→∞ ; 1 ⎞ ⎛1 1 ⎞ 1 ⎛ 1 +⎜ – ⎟+⎜ – ⎟ = –1 + ; n – 1 n – 2 n n – 1 n ⎝ ⎠ ⎝ ⎠ ∞ 1 1 ⎞ ⎛1 = –1, so ⎜ – ⎟ = –1 n k =2 ⎝ k k – 1 ⎠ ∞ 8. ∞ ∞ 3 1 1 =3 which diverges since diverges. k =1 k k =1 k k =1 k ∞ 9. k! k =1 100 1 2 6 + + + 100 10, 000 1, 000, 000 = k n +1 1 an , a1 = . an > 0 for all n, and for n >99, an +1 > an , so the 100 100 sequence is eventually an increasing sequence, hence lim an ≠ 0. The sequence can also be described by Consider {an }, where an +1 = n →∞ an = n! 100 n ∞ , hence k! k =1 100 k ∞ 10. 2 2 2 2 = + + + 3 8 15 k =1 ( k + 2) k diverges. = ∞ 1 ⎞ ⎛1 1 ⎞ ⎛ 1 1 ⎞ ⎛ 1 1 ⎞ ⎛ 1 1 ⎞ ⎛1 ⎜ – ⎟ = ⎜ – ⎟+⎜ – ⎟+⎜ – ⎟+⎜ – ⎟+ ⎝1 3 ⎠ ⎝ 2 4 ⎠ ⎝ 3 5 ⎠ ⎝ 4 6 ⎠ k =1 ⎝ k k + 2 ⎠ 1 ⎞ ⎛1 1 ⎞ ⎛ 1⎞ ⎛1 1⎞ ⎛1 1⎞ ⎛ 1 Sn = ⎜1 – ⎟ + ⎜ – ⎟ + ⎜ – ⎟ + + ⎜ – ⎟+⎜ – ⎟ 3 2 4 3 5 n – 1 n + 1 n n + 2⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ 1 1 1 3 2n + 3 3 2n + 3 – = 1+ – = – = – 2 2 n + 1 n + 2 2 (n + 1)(n + 2) 2 n + 3n + 2 2+ 3 ∞ 2 3 3 2n + 3 3 3 n n2 = – lim = , so lim Sn = – lim = . 2 3 2 2 n→∞ n + 3n + 2 2 n→∞ 1 + + 2 2 2 n →∞ k =1 ( k + 2) k n n ∞ 11. ⎛e⎞ ⎜ ⎟ k =1 ⎝ π ⎠ k +1 2 e e ( ( π) π) S= = 2 ∞ 12. 522 4k +1 k =1 7 k –1 2 π–e π 1 – πe 2 2 2 ⎛e⎞ ⎛e⎞ e ⎛e⎞ ⎛e⎞ = ⎜ ⎟ +⎜ ⎟ ⋅ +⎜ ⎟ ⎜ ⎟ + ⎝π⎠ ⎝π⎠ π ⎝π⎠ ⎝π⎠ = e2 ≈ 5.5562 π(π – e) 2 = 16 4 ⎛4⎞ + 16 ⋅ + 16 ⎜ ⎟ + 1 7 ⎝7⎠ Section 9.2 2 e ⎛e⎞ ; a geometric series with a = ⎜ ⎟ , r = < 1; π ⎝π⎠ ; a geometric series with a = 16, r = 4 16 16 112 < 1; S = = = 3 7 3 1 – 74 7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 13. ∞ ⎛ 3 3 – ⎜⎜ 2 k2 k = 2 ⎝ ( k – 1) ⎞ ⎛3 3⎞ ⎛ 3 3⎞ ⎛3 3 ⎞ ⎟⎟ = ⎜ – ⎟ + ⎜ – ⎟ + ⎜ – ⎟ + ; ⎠ ⎝ 1 4 ⎠ ⎝ 4 9 ⎠ ⎝ 9 16 ⎠ ⎛ 3⎞ ⎛ 3 1⎞ ⎛1 3 ⎞ 3 3 ⎛ Sn = ⎜ 3 – ⎟ + ⎜ – ⎟ + ⎜ – ⎟ + + ⎜ – ⎜ (n – 2)2 (n – 1)2 4 ⎠ ⎝ 4 3 ⎠ ⎝ 3 16 ⎠ ⎝ ⎝ 3 3 = 3– ; lim S = 3 – lim = 3, so 2 n →∞ n →∞ n n n2 ∞ ⎛ 3 3 ⎞ – ⎜⎜ ⎟ = 3. 2 k 2 ⎟⎠ k = 2 ⎝ (k – 1) ∞ 14. 2 2 2 2 2 = + + + + k – 5 1 2 3 4 k =6 =2 ∞ 1 1 which diverges since diverges. k k =1 k =1 k = 2 10 1 1 – 10 ∞ 2⎛1⎞ ⎜ ⎟ 10 ⎝ 10 ⎠ k =1 21 = 100 1 1 – 100 100 21. Let s = 1 – r, so r = 1 – s. Since 0 < r < 2, –1 < 1 – r < 1, so ∞ 21 ⎛ 1 ⎞ ⎜ ⎟ k =1 100 ⎝ 100 ⎠ ∞ = k –1 ∞ k =0 (1 − s ) s k k =0 (1 – s ) s k –1 = (−1) k x k = 22. ∞ r (1 − r )k = k =1 21 7 = = 99 33 13 = 1000 1 1 − 1000 ∞ k =0 = k –1 36 10,000 727 + = 100 1 − 1 1980 s < 1, and 17. 0.013013013... = ∞ 1− s =1 1− s (− x) k = k =0 ∞ (– x)k –1; k =1 if –1 < x < 1 then –1 < –x < 1 so x < 1 ; ∞ 13 ⎛ 1 ⎞ ⎜ ⎟ k =1 1000 ⎝ 1000 ⎠ k –1 ∞ k =1 (– x) k –1 = 1 1 = 1 − (− x) 1 + x 13 = 999 18. 0.125125125... = = k –1 36 ∞ 71 ⎛ 1 ⎞ + ⎜ ⎟ 100 k =1 10, 000 ⎝ 100 ⎠ 71 = 2 = 9 16. 0.21212121 ⎞ ⎟⎟ ⎠ 20. 0.36717171... = ∞ 15. 0.22222 = ⎞ ⎛ 3 3 – ⎟⎟ + ⎜⎜ 2 n2 ⎠ ⎝ (n – 1) 125 1000 1 1 − 1000 = 19. 0.4999... = ∞ 125 ⎛ 1 ⎞ ⎜ ⎟ 1000 ⎝ 1000 ⎠ k =1 k –1 125 999 4 ∞ 9 ⎛1⎞ + ⎜ ⎟ 10 k =1 100 ⎝ 10 ⎠ k –1 9 = 4 1 + 100 = 10 1 − 1 2 10 Instructor’s Resource Manual Section 9.2 523 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. k = ln k – ln(k + 1) k +1 Sn = (ln1 – ln 2) + (ln 2 – ln 3) + (ln 3 – ln 4) + 23. ln lim Sn = lim – ln(n + 1) = – ∞, thus n →∞ n →∞ ∞ ln k =1 + (ln( n – 1) – ln n) + (ln n – ln( n + 1)) = ln1 – ln(n + 1) = – ln(n + 1) k diverges. k +1 ⎛ 1 ⎞ k2 –1 24. ln ⎜ 1 – = ln = ln(k 2 – 1) – ln k 2 = ln[(k + 1)(k – 1)] – ln k 2 = ln(k + 1) + ln(k – 1) – 2 ln k ⎟ 2 2 k ⎝ k ⎠ Sn = (ln 3 + ln1 – 2 ln 2) + (ln 4 + ln 2 – 2 ln 3) + (ln 5 + ln 3 – 2 ln 4) + +(ln n + ln(n – 2) – 2 ln(n – 1)) + (ln(n + 1) + ln(n – 1) – 2 ln n) = –ln 2 + ln(n + 1) – ln n = – ln 2 + ln lim Sn = – ln 2 + lim ln n →∞ n →∞ n +1 n n +1 n +1⎞ ⎛ = – ln 2 + ln ⎜ lim ⎟ = – ln 2 + ln1 = – ln 2 n ⎝ n→∞ n ⎠ 2 ⎛2⎞ ⎛2⎞ ⎛2⎞ 25. The ball drops 100 feet, rebounds up 100 ⎜ ⎟ feet, drops 100 ⎜ ⎟ feet, rebounds up 100 ⎜ ⎟ feet, drops ⎝3⎠ ⎝3⎠ ⎝3⎠ 2 ⎛2⎞ 100 ⎜ ⎟ , etc. The total distance it travels is ⎝3⎠ 2 3 2 3 ⎛2⎞ ⎛2⎞ ⎛2⎞ ⎛2⎞ ⎛2⎞ ⎛2⎞ 100 + 200 ⎜ ⎟ + 200 ⎜ ⎟ + 200 ⎜ ⎟ + ... = −100 + 200 + 200 ⎜ ⎟ + 200 ⎜ ⎟ + 200 ⎜ ⎟ + ... ⎝3⎠ ⎝3⎠ ⎝3⎠ ⎝3⎠ ⎝3⎠ ⎝3⎠ = −100 + 26. Each gets ∞ ⎛2⎞ 200 ⎜ ⎟ ⎝3⎠ k =1 k –1 = −100 + 200 = 500 feet 1 − 23 ∞ 1 1 1 1⎛1 1⎞ 1⎛1⎞ + ⋅ + ⎜ ⋅ ⎟ + ... = ⎜ ⎟ 4 4 4 4⎝4 4⎠ k =1 4 ⎝ 4 ⎠ 1 4 k –1 = 1− 1 4 = 1 3 (This can be seen intuitively, since the size of the leftover piece is approaching 0, and each person gets the same amount.) ∞ 27. $1 billion + 75% of $1 billion + 75% of 75% of $1 billion + ... = k =1 ∞ 28. $1 billion (0.90) k –1 = k =1 ($1 billion)0.75k –1 = $1 billion = $4 billion 1 − 0.75 $1 billion = $10 billion 1 − 0.90 29. As the midpoints of the sides of a square are connected, a new square is formed. The new square has sides 1 2 1 times the sides of the old square. Thus, the new square has area the area of the old square. Then in the next step, 2 1 of each new square is shaded. 8 Area = ∞ 1 1 1 1 1 1⎛1⎞ ⋅1 + ⋅ + ⋅ + ... = ⎜ ⎟ 8 8 2 8 4 k =1 8 ⎝ 2 ⎠ The area will be 524 Section 9.2 k –1 = 1 8 1− 1 2 = 1 4 1 . 4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. ∞ 1 1⎛8⎞ 1⎛8 8⎞ + ⎜ ⎟+ ⎜ ⋅ ⎟+ 30. 9 9⎝9⎠ 9⎝9 9⎠ 31. 1⎛8⎞ = ⎜ ⎟ k =1 9 ⎝ 9 ⎠ k –1 = 1 9 = 1; the whole square will be painted. 1 − 89 ∞ 3 3 ⎛ 1 1 ⎞ 3 ⎛ 1 1 ⎞⎛ 1 1 ⎞ 3⎛ 1 ⎞ + ⎜ ⋅ ⎟ + ⎜ ⋅ ⎟⎜ ⋅ ⎟ + ... = ⎜ ⎟ 4 4 ⎝ 4 4 ⎠ 4 ⎝ 4 4 ⎠⎝ 4 4 ⎠ k =1 4 ⎝ 16 ⎠ k –1 = 3 4 1 1 − 16 = 4 5 The original does not need to be equilateral since each smaller triangle will have 1 area of the previous larger 4 triangle. ∞ π 32. Ratio of inscribed circle to triangle is 3 3 , so π 3⎛1⎞ ⋅ ⎜ ⎟ k =1 3 3 4 ⎝ 4 ⎠ k –1 = ( )= π 4 3 1 − 14 π 3 3 (This can be seen intuitively, since every small triangle has a circle inscribed in it.) 33. a. We first note that, at each stage, the number of sides is four times the number in the previous stage and the length of each side is one-third the length in the previous stage. Summarizing: Stage # of sides length/ side (in.) perimeter 0 3 9 27 pn () 36 ( ) ⎛4⎞ 27 ⎜ ⎟ ⎝3⎠ 1 3(4) 9 n 3(4n ) 9 1 3 1 3n n n 4 ⎛4⎞ The perimeter of the Koch snowflake is lim pn = lim 27 ⎜ ⎟ which is infinite since > 1 . 3 n →∞ n →∞ ⎝3⎠ b. We note the following: 3 2 s 4 2. The number of new triangles added at each stage is equal to the number of sides the figure had at the previous stage and 1. The area of an equilateral triangle of side s is 3. the area of each new triangle at a given stage is 3 (side length at that stage)2 . Using results from part a. we 4 can summarize: Stage Additional triangles Area of each new Δ Additional area, An (col 2, part a.) (see col 3, part a.) ( ) 3 2 9 4 ( ) 36 0 original 3 2 9 4 1 3 3 2 3 4 n 3 4n −1 ( ) Instructor’s Resource Manual 3⎛ 9 ⎞ ⎜ ⎟ 4 ⎝ 3n ⎠ 2 ( ) ⎛4⎞ 3 3⎜ ⎟ ⎝9⎠ n−2 Section 9.2 525 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Thus the area of the Koch snowflake is ∞ n=0 An = 81 3 27 3 ∞ ⎛4⎞ + + 3 3⎜ ⎟ 4 4 ⎝9⎠ n =1 n −1 = 81 3 27 3 ⎛ 3 3 ⎞ ⎟ + +⎜ ⎜ (1 − 4 ) ⎟ 4 4 9 ⎠ ⎝ = 81 3 1 ⎛ 81 3 ⎞ 4 ⎛ 81 3 ⎞ 8 ⎛ 81 3 ⎞ + ⎜⎜ ⎟+ ⎜ ⎟= ⎜ ⎟ 4 3 ⎝ 4 ⎟⎠ 15 ⎜⎝ 4 ⎟⎠ 5 ⎜⎝ 4 ⎟⎠ Note: By generalizing the above argument it can be shown that, no matter what the size of the original equilateral triangle, the area of the Koch snowflake constructed from it will be 8 5 times the area of the original triangle. 34. We note the following: 1. Each triangle contains the angles 90, ,90 − succeeding triangle. Summarizing: # triangle base height 1 h cos h sin 2 h sin cos h sin 2 n h (sin n −1 ) cos h sin n 2. The height of each triangle will be the hypotenuse of the area An 1 2 h sin cos 2 1 2 3 h sin cos 2 1 2 2 n −1 cos h sin 2 ∞ Thus the total area of the small triangles is A = n =1 ∞ Now consider the infinite geometric series S = An = h 2 ⎛ cos ⎞ ∞ 2 n −1 ⎜ ⎟ (sin ) 2 ⎝ sin ⎠n = 2 (sin 2 )n −1 = n =1 ∞ then: (sin 2 )n −1 = S − 1 = n=2 In 1 cos 2 −1 = sin 2 cos 2 1 1 − sin = 1 cos 2 A= Therefore: ABC , height = h and base = h tan ; thus the area of 2 ABC = h 2 ⎛ cos ⎞ ⎛ sin 2 ⎜ ⎟⎜ 2 ⎝ sin ⎠ ⎜⎝ cos 2 ⎞ h2 tan ⎟= ⎟ 2 ⎠ 1 h2 (h tan )h = tan , the same as A . 2 2 35. Both Achilles and the tortoise will have moved. 100 + 10 + 1 + = ∞ 1 1 ⎛1⎞ + + ... = 100 ⎜ ⎟ 10 100 ⎝ 10 ⎠ k =1 k –1 100 1 = 111 yards 1 9 1 − 10 Also, one can see this by the following reasoning. In the time it takes the tortoise to run run d yards. Solve d = 100 + 526 Section 9.2 d yards, Achilles will 10 d 1000 1 .d = = 111 yards 10 9 9 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 36. a. Say Trot and Tom start from the left, Joel from the right. Trot and Joel run towards each other at 30 mph. Since they are 60 miles apart they will meet in 2 hours. Trot will have run 40 miles and Tom will have run 20 miles, so they will be 20 miles apart. Trot and Tom will now be approaching each other at 30 mph, so they will meet after 2/3 hour. Trot will have run another 40/3 miles and will be 80/3 miles from the left. Joel will have run another 20/3 miles and will be at 100/3 miles from the left, so they will be 20/3 miles apart. They will meet after 2/9 hour, during which Trot will have run 40/9 miles, etc. So Trot runs 40 + 40 40 + + 3 9 = ∞ ⎛1⎞ 40 ⎜ ⎟ k =1 ⎝ 3 ⎠ k –1 = 40 1 − 13 39. Let X = number of rolls needed to get first 6 For X to equal n , two things must occur: 1. Mary must get a non-6 (probability = 2. Mary rolls a 6 (probability = ⎛5⎞ Pr( X = n) = ⎜ ⎟ ⎝6⎠ ) on the nth n −1 ⎛1⎞ ⎜ ⎟ and ⎝6⎠ 1 n − 1 ∞ ⎛5⎞ ⎛1⎞ 6 =1 = ⎜ ⎟ ⎜ ⎟ 6 6 1 − ⎝ ⎠ ( 56 ) n =1 ⎝ ⎠ ∞ 40. EV ( X ) = n ⋅ Pr ( X = n ) = n =1 ∞ 1 ⎛5⎞ n⋅ ⋅⎜ ⎟ 6 ⎝6⎠ n =1 n −1 1⎛ 1 ⎞ ∞ 1 ⎛ 6 ⎞⎛ 5/ 6 = ⎜ ⎟ n ⋅ pn = ⎜ ⎟ ⎜ 6 ⎝ p ⎠ n =1 6 ⎝ 5 ⎠ ⎜ (1 − 5 / 6 )2 ⎝ 1 ⎛ 6 ⎞ ⎛ 5 ⎞ ⎛ 36 ⎞ = ⎜ ⎟⎜ ⎟⎜ ⎟ = 6 6 ⎝ 5 ⎠⎝ 6 ⎠⎝ 1 ⎠ 37. Note that: ⎞ ⎟ ⎟ ⎠ ∞ 1. If we let tn be the probability that Peter wins on his nth flip, then the total probability that Peter wins is T = n −1 and T = ∞ ⎛ 1 ⎞⎛ 4 ⎞ ⎜ ⎟⎜ ⎟ n =1 ⎝ 3 ⎠ ⎝ 9 ⎠ ⎛ 13 ⎞ 1 9 3 ⎜ ⎟= ⋅ = ⎜ (1 − 4 ) ⎟ 3 5 5 9 ⎝ ⎠ In this case (see problem 37), ∞ n −1 2 p ⎡(1 − p ) ⎤ ⎣ ⎦ n =1 p (2 p − p ) 2 = converges, and c ≠ 0. Then ∞ 3. The probability that Peter wins on his nth flip requires that (i) he gets a head on the nth flip, and (ii) neither he nor Paul gets a head on their previous n-1 flips. Thus: 2 tn = p ⎡(1 − p ) ⎤ ⎣ ⎦ k =1 tn 2. The probability that neither man wins in their k k ⎛2 2⎞ ⎛4 ⎞ first k flips is ⎜ ⋅ ⎟ = ⎜ ⎟ . ⎝3 3⎠ ⎝9 ⎠ ⎛ 1 ⎞⎛ 4 ⎞ tn = ⎜ ⎟ ⎜ ⎟ ⎝ 3 ⎠⎝ 9 ⎠ 41. (Proof by contradiction) Assume ∞ n =1 = 1 6 roll. Thus, b. Tom and Joel are approaching each other at 20 mph. They are 60 miles apart, so they will meet in 3 hours. Trot is running at 20 mph during that entire time,