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AP Calculus – Final Review Sheet When you see the words …. 1. Find the zeros This is what you think of doing 2. Find equation of the line tangent to f (x ) at (a, b ) 3. Find equation of the line normal to f (x ) at (a, b ) 4. Show that f (x ) is even 5. Show that f (x ) is odd 6. Find the interval where f (x ) is increasing 7. Find interval where the slope of f (x ) is increasing 8. Find the minimum value of a function 9. Find the minimum slope of a function 10. Find critical values 11. Find inflection points 12. Show that lim f (x ) exists x! a 13. Show that f (x ) is continuous 14. Find vertical asymptotes of f (x ) 15. Find horizontal asymptotes of f (x ) www.MasterMathMentor.com Stu Schwartz 16. Find the average rate of change of f (x ) on [a, b] 17. Find instantaneous rate of change of f (x ) at a 18. Find the average value of f (x ) on [a, b] 19. Find the absolute maximum of f (x ) on [a, b] 20. Show that a piecewise function is differentiable at the point a where the function rule splits 21. Given s (t ) (position function), find v(t ) 22. Given v(t ), find how far a particle travels on [a, b] 23. Find the average velocity of a particle on [a, b] 24. Given v(t ), determine if a particle is speeding up at t = k 25. Given v(t ) and s (0 ), find s (t ) ! 26. Show that Rolle’s Theorem holds on [a, b] 27. Show that Mean Value Theorem holds on [a, b] 28. Find domain of f (x ) 29. Find range of f (x ) on [a, b] 30. Find range of f (x ) on (" !, ! ) 31. Find f !(x ) by definition www.MasterMathMentor.com Stu Schwartz 32. Find derivative of inverse to f (x ) at x = a 33. y is increasing proportionally to y 34. Find the line x = c that divides the area under f (x ) on [a, b] to two equal areas x d 35. f (t )dt = dx !a d u 36. " f (t ) dt dx a 37. The rate of change of population is … ! 38. The line y = mx + b is tangent to f (x ) at (a, b ) 39. Find area using left Riemann sums 40. Find area using right Riemann sums 41. Find area using midpoint rectangles 42. Find area using trapezoids 43. Solve the differential equation … x 44. Meaning of ! f (t )dt a 45. Given a base, cross sections perpendicular to the x-axis are squares 46. Find where the tangent line to f (x ) is horizontal 47. Find where the tangent line to f (x ) is vertical 48. Find the minimum acceleration given v(t ) www.MasterMathMentor.com Stu Schwartz 49. Approximate the value of f (0.1) by using the tangent line to f at x = 0 50. Given the value of f (a ) and the fact that the antiderivative of f is F, find F (b ) 51. Find the derivative of f (g (x )) b 52. Given b ! f (x )dx , find ! [f (x )+ k ]dx a a 53. Given a picture of f !(x ), find where f (x ) is increasing 54. Given v(t ) and s (0 ), find the greatest distance from the origin of a particle on [a, b] 55. Given a water tank with g gallons initially being filled at the rate of F (t ) gallons/min and emptied at the rate of E (t ) gallons/min on [t1 ,t 2 ], find a) the amount of water in the tank at m minutes 56. b) the rate the water amount is changing at m 57. c) the time when the water is at a minimum 58. Given a chart of x and f (x ) on selected values between a and b, estimate f !(c ) where c is between a and b. dy 59. Given , draw a slope field dx 60. Find the area between curves f (x ), g (x ) on [a, b] 61. Find the volume if the area between f (x ), g (x ) is rotated about the x-axis www.MasterMathMentor.com Stu Schwartz BC Problems 62. Find lim x "# 63. Find ! # f (x) if lim f (x) = lim g(x) = 0 x "# x "# g(x) " 0 f (x) dx ! # dP P& dP k = kP%1" ( 64. = P(M " P) or $ M' dt dt M ! dx where x 2 + ax + b factors + ax + b ! 66. The position vector of a particle moving in the ! ), y(t ) plane is r(t ) = x(t ! a) Find the velocity. 67. b) Find the acceleration. 65. Find ! "x 2 ! 68. c) Find the speed. 69. a) Given the velocity vector v(t ) = x(t ), y(t ) and position at time 0, find the position vector. 70. b) When does the particle stop? ! 71. c) Find the slope of the tangent line to the vector at t1. 72. Find the area inside the polar curve r = f (" ) . ! 73. Find the slope of the tangent line to the polar ! curve r = f (" ) . ! 74. Use Euler’s method to approximate f (1.2) given dy , ( x 0 , y 0 ) = (1,1) , and "x = 0.1 ! dx 75. Is the Euler’s approximation an underestimate or ! an overestimate? ! ! n ax 76. Find " x e dx where a, n are integers 77. Write a series for x n cos x where n is an integer ! www.MasterMathMentor.com ! Stu Schwartz 78. Write a series for ln(1+ x) centered at x = 0 . " 79. 1 #n p converges if….. ! ! n=1 # 1& 80. If f (x) = 2 + 6x + 18x 2 + 54 x 3 + ... , find f % " ( $ 2' ! 81. Find the interval of convergence of a series. ! ! 82. Let S4 be the sum of the first 4 terms of an alternating series for f (x). Approximate f (x) " S4 !83. Suppose f (n ) (x) = (n + 1) n! . Write the first four 2n ! terms and the general term of a series for f (x) ! centered at x = c 84. Given a Taylor series, find the Lagrange form of ! the remainder for the 4th term. ! ! x2 x3 85. f ( x ) = 1+ x + + + ... 2! 3! n ! "1) x 2n+1 ( x3 x5 x 7 86. f ( x ) = x " + " + ... + + ... 3! 5! 7! (2n + 1) ! n ! "1) x 2n ( x2 x 4 x6 87. f ( x ) = 1" + " + ... + + ... 2! 4! 6! (2n) ! 88. Find m " (sin x ) (cos x ) n dx where m and n are integers ! dy 89. Given x = f (t ), y = g (t ), find dx ! d2y 90. Given x = f (t ), y = g (t ), find dx 2 ! 91. Given f ( x ) , find arc length on [ a, b] ! 92. x = f (t ), y = g (t ), find arc length on [t1 ,t2 ] ! ! ! 93. Find horizontal tangents to a polar curve r = f (" ) www.MasterMathMentor.com ! Stu Schwartz 94. Find vertical tangents to a polar curve r = f (" ) 95. Find the volume when the area between ! y = f ( x ), x = 0, y = 0 is rotated about the y-axis. 96. Given a set of points, estimate the volume under the curve using Simpson’s rule. 97. Find the dot product: u1 , u2 " v1 ,v2 98. Multiply two vectors: ! www.MasterMathMentor.com Stu Schwartz AP Calculus – Final Review Sheet When you see the words …. 1. Find the zeros 4. Show that f (x ) is even This is what you think of doing Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator Take derivative - f !(a ) = m and use y ! y1 = m(x ! x1 ) "1 Same as above but m = f !(a ) Show that f (! x ) = f (x ) - symmetric to y-axis 5. Show that f (x ) is odd Show that f (! x ) = ! f (x ) - symmetric to origin 6. Find the interval where f (x ) is increasing 12. Show that lim f (x ) exists Find f !(x ), set both numerator and denominator to zero to find critical points, make sign chart of f !(x ) and determine where it is positive. Find the derivative of f !(x ) = f !!(x ), set both numerator and denominator to zero to find critical points, make sign chart of f !!(x ) and determine where it is positive. Make a sign chart of f !(x ), find all relative minimums and plug those values back into f (x ) and choose the smallest. Make a sign chart of the derivative of f !(x ) = f !!(x ), find all relative minimums and plug those values back into f !(x ) and choose the smallest. Express f !(x ) as a fraction and set both numerator and denominator equal to zero. Express f !!(x ) as a fraction and set both numerator and denominator equal to zero. Make sign chart of f !!(x ) to find where f !!(x ) changes sign. (+ to – or – to +) Show that lim# f ( x ) = lim+ f ( x ) 13. Show that f (x ) is continuous Show that 1) lim f (x ) exists ( lim# f ( x ) = lim+ f ( x ) ) 2. Find equation of the line tangent to f (x ) on [a, b] 3. Find equation of the line normal to f (x ) on [a, b] 7. Find interval where the slope of f (x ) is increasing 8. Find the minimum value of a function 9. Find the minimum slope of a function 10. Find critical values 11. Find inflection points x! a x "a x "a x! a ! 2) f (a ) exists 3) lim f (x ) = f (a ) x "a x "a x!a 14. Find vertical asymptotes of f (x ) Do all factor/cancel of! f (x ) and set denominator = 0 15. Find horizontal asymptotes of f (x ) Find 16. Find the average rate of change of f (x ) on [a, b] Find www.MasterMathMentor.com lim f (x ) and lim f (x ) x "! x # !" f (b )! f (a ) b!a Stu Schwartz 17. Find instantaneous rate of change of f (x ) at a 18. Find the average value of f (x ) on [a, b] Find f !(a ) b Find ! f (x )dx a b-a Make a sign chart of f !(x ), find all relative maximums and plug those values back into f (x ) as well as finding f (a )and f (b ) and choose the largest. First, be sure that the function is continuous at x = a . Take the derivative of each piece and show that lim# f !(x ) = lim f !(x ) 19. Find the absolute maximum of f (x ) on [a, b] 20. Show that a piecewise function is differentiable at the point a where the function rule splits x"a + x"a 21. Given s (t ) (position function), find v(t ) 22. Given v(t ), find how far a particle travels on [a, b] Find v(t ) = s !(t ) b Find ! v(t )dt a 23. Find the average velocity of a particle on [a, b] b Find 24. Given v(t ), determine if a particle is speeding up at t = k ! 25. Given v(t ) and s (0 ), find s (t ) ! 26. Show that Rolle’s Theorem holds on [a, b] a = s ( b) # s ( a ) b#a b#a Find v(k )and a(k ). Examine their signs. If both positive, the particle is speeding up, if different signs, then the particle is slowing down. s(t ) = " v (t ) dt + C Plug in t = 0 to find C Show that f is continuous and differentiable on the interval. If f ( a ) = f ( b) , then find some c in [ a, b] ! 27. Show that Mean Value Theorem holds on [a, b] 28. Find domain of f (x ) 29. Find range of f (x ) on [a, b] " v (t ) dt ! such that f "(c) = 0. Show that f is continuous and differentiable on the interval. Then find some c such that! ! f b # f ( a) !f "(c) = ( ) . b#a Assume domain is ("#,#) . Restrictable domains: denominators ≠ 0, square roots of only non negative numbers, log or ln of only positive numbers. Use max/min techniques to rind relative max/mins. ! Then examine f ( a ), f ( b) Use max/min techniques to rind relative max/mins. Then examine lim f ( x ) . 30. Find range of f (x ) on (" !, ! ) x "±# 31. Find f !(x ) by definition ! f ( x + h) $ f ( x) f "( x ) = lim or h#0 h ! f ( x ) $ f ( a) f "( a ) = lim x #a x$a 32. Find derivative of inverse to f (x ) at x = a ! www.MasterMathMentor.com dy implicitly (in terms dx of y). Plug your x value into the inverse relation and Interchange x with y. Solve for Stu Schwartz solve for y. Finally, plug that y into your 33. y is increasing proportionally to y dy = ky translating to y = Ce kt dt 34. Find the line x = c that divides the area under f (x ) on [a, b] to two equal areas c dy . dx b ! f (x )dx = ! f (x )dx a c x 35. d f (t )dt = dx !a 36. d dx u " f (t ) dt a 37. The rate of change of population is … ! 2nd FTC: Answer is f (x ) 2nd FTC: Answer is f (u ) du dx 39. Find area using left Riemann sums dP = ... dt Two relationships are true. The two functions share the same slope ( m = f !(x )) and share the same y value at x1 . A = base[x0 + x1 + x 2 + ... + x n !1 ] 40. Find area using right Riemann sums A = base[x1 + x 2 + x3 + ... + x n ] 41. Find area using midpoint rectangles Typically done with a table of values. Be sure to use only values that are given. If you are given 6 sets of points, you can only do 3 midpoint rectangles. base [x0 + 2 x1 + 2 x2 + ... + 2 xn!1 + xn ] A= 2 This formula only works when the base is the same. If not, you have to do individual trapezoids. Separate the variables – x on one side, y on the other. The dx and dy must all be upstairs. The accumulation function – accumulated area under the function f (x ) starting at some constant a and ending at x. The area between the curves typically is the base of 38. The line y = mx + b is tangent to f (x ) at (x1 , y1 ) 42. Find area using trapezoids 43. Solve the differential equation … x 44. Meaning of ! f (t )dt a 45. Given a base, cross sections perpendicular to the x-axis are squares b your square. So the volume is ! (base )dx 2 a 46. Find where the tangent line to f (x ) is horizontal Write f !(x ) as a fraction. Set the numerator equal to zero. 47. Find where the tangent line to f (x ) is vertical Write f !(x ) as a fraction. Set the denominator equal to zero. 48. Find the minimum acceleration given v(t ) First find the acceleration a(t ) = v !(t ). Then minimize the acceleration by examining a !(t ). Find the equation of the tangent line to f using 49. Approximate the value of f (0.1) by using the www.MasterMathMentor.com Stu Schwartz tangent line to f at x = 0 50. Given the value of F (a ) and the fact that the antiderivative of f is F, find F (b )1 y ! y1 = m(x ! x1 ) where m = f !(0 ) and the point is (0, f (0)). Then plug in 0.1 into this line being sure to use an approximate (! )sign. Usually, this problem contains an antiderivative you cannot take. Utilize the fact that if F (x )is the b antiderivative of f, then " F (x )dx = F (b )! F (a ). So a solve for F (b ) using the calculator to find the definite integral. f !(g (x ))" g !(x ) 51. Find the derivative of f (g (x )) b 52. Given b ! f (x )dx , find ! [f (x )+ k ]dx a b b b ! [f (x )+ k ]dx = ! f (x )dx + ! kdx a a 53. Given a picture of f !(x ), find where f (x ) is increasing 54. Given v(t ) and s (0 ), find the greatest distance from the origin of a particle on [a, b] a a Make a sign chart of f !(x ) and determine where f !(x ) is positive. Generate a sign chart of v(t ) to find turning points. Then integrate v(t ) using s (0 ) to find the constant to find s (t ). Finally, find s(all turning points) which will give you the distance from your starting point. Adjust for the origin. 55. Given a water tank with g gallons initially being filled at the rate of F (t ) gallons/min and emptied at the rate of E (t ) gallons/min on [t1 ,t 2 ], find a) the amount of water in the tank at m minutes 56. b) the rate the water amount is changing at m g + ! (F (t )" E (t ))dt 57. c) the time when the water is at a minimum F (m )! E (m )=0, testing the endpoints as well. 58. Given a chart of x and f (x ) on selected values between a and b, estimate f !(c ) where c is between a and b. Straddle c, using a value k greater than c and a value h f (k )! f (h ) less than c. so f #(c ) " k !h 59. Given dy , draw a slope field dx t2 t m d (F (t )! E (t ))dt = F (m )! E (m ) dt t" dy , drawing dx little lines with the indicated slopes at the points. Use the given points and plug them into 60. Find the area between curves f (x ), g (x ) on [a, b] b A = ! [f (x )" g (x )]dx , assuming that the f curve is a above the g curve. 61. Find the volume if the area between f (x ), g (x ) is rotated about the x-axis b 2 2' $ A = " * & f ( x ) # g ( x ) ) dx assuming that the f curve is % ( a above the g curve. www.MasterMathMentor.com Stu Schwartz ! www.MasterMathMentor.com Stu Schwartz BC Problems f (x) if lim f (x) = lim g(x) = 0 x "# x "# x "# g(x) 62. Find lim 63. Find ! # " 0 h f (x) dx ! lim $ f ( x ) dx h"# ! dP k # dP P& = kP%1" 64. = P(M " P) or ! ( $ M' dt dt M ! 65. Find Use L’Hopital’s Rule. "x 2 dx where x 2 + ax + b ! + ax + b ! factors 66. The position vector of a particle moving ) = x(t ), y(t ) in the plane is r(t ! ! a) Find the velocity. 67. b) Find the acceleration. ! 68. c) Find the speed. 0 Signals logistic growth. dP lim =0$ M =P t "# dt Factor denominator and use Heaviside partial fraction technique. v(t ) = x "(t ), y "(t ) a(t ) = x ""(t ), y ""(t ) ! 2 [ x "(t )] + [ y "(t )] v(t ) = ! 69. a) Given the velocity vector v(t ) = x(t ), y(t ) ! and position at time 0, find the position vector. 70. b) When does the particle stop? ! ! ! 71. c) Find the slope of the tangent line to the vector at t1. ! 72. Find the area inside the polar curve r = f (" ) . ! ! 73. Find the slope of the tangent line to the polar curve r = f (" ) . ! ! 74. Use Euler’s method to approximate dy , ( x 0 , y 0 ) = (1,1) , and f (1.2) given ! dx "x = 0.1 75. Is the Euler’s approximation an underestimate or an overestimate?! ! ! 2 " x (t ) dt + " y (t ) dt + C Use s(0) to find C, remembering it is a vector. s(t ) = v(t ) = 0 " x (t ) = 0 AND y (t ) = 0 This is the acceleration vector at t1. " A= 2 1 2 f (" ) d" # 2 "1 [ ] ! dy dy d" x = r cos" , y = r sin " # = dx dx d" dy = dy ("x ), ynew = yold + dy dx Look at sign of www.MasterMathMentor.com dy d2y in the interval. This gives you and dx dx 2 Stu Schwartz ! n ax 76. Find "xe dx where a, n are integers 77. Write a series for x n cos x where n is an ! integer ! 78. Write a series for ln(1+ x) centered at ! x = 0. increasing.decreasing/concavity. Draw picture to ascertain answer. Integration by parts, " u dv = uv # " v du + C x2 x 4 x6 cos x = 1" ! + " + ... 2! 4! 6! Multiply each term by x n Find Maclaurin polynomial: ( n) f ""(0) 2 f """(0) 3 f (0) n ! Pn ( x ) = f (0) + f "(0) x + x + x +…+ x 2! 3! n! ! " ! 79. 1 #n p converges if….. ! n=1 ! ! p >1 80. If f (x) = 2 + 6x + 18x 2 + 54 x 3 + ... , find # 1& ! f %" ( $ 2' Plug in and factor. This will be a geometric series: " a # ar n = 1$ r n=0 81. Find the interval of convergence of a series. ! Use a test (usually the ratio) to find the interval and then test convergence at the endpoints. 82. Let S4 be the sum of the first 4 terms of an This is the error for the 4th term of an alternating series which th alternating series for f (x). Approximate is simply the 5 term. It will be positive since you are looking for an absolute value. f (x) " S4 ! (n + 1) n! (n ) 83. Suppose f ! . Write the (x) = 2n ! first four terms and the general term of a series for f (x) centered at x = c 84. ! Given a Taylor series, find the Lagrange form of the remainder for the nth term ! ! at x = c. where n is an integer ! 85. f ( x ) = 1+ x + x2 x3 + + ... 2! 3! ! ! ! 86. f ( x ) = x " n 5 m " (sin x ) (cos x ) www.MasterMathMentor.com ! n 2! ( x # c) 2 +…+ f ( n) (c) ( x # c) ! f ( x ) = sin x n f ( x ) = cos x dx where m and n ! n n! ! You need to determine the largest value of the 5th derivative of f at some value of z. Usually you are told this. Then: ( n+1) f ( z) x " c n+1 Rn ( x ) = ( ) (n + 1) ! 2n+1 "1) x 2n ( x2 x 4 x6 ! + ... 87. f ( x ) = 1" + " + ... + 2! 4! 6! (2n) ! 88. Find f ( x ) = f (c) + f "(c)( x # c) + f ""(c) f ( x) = ex ("1) x !+ ... x x + + ... + 3! 5! (2n + 1) ! 3 You are being given a formula for the derivative of f (x) . If m is odd and positive, save a sine and convert everything else to cosine. The sine will be the du. If n is odd and positive, Stu Schwartz are integers 89. Given x = f (t ), y = g (t ), find dy dx ! 2 ! Given x = f (t ), y = g (t ), find d y 90. dx 2 ! 91. ! Given f ( x ) , find arc length on [ a, b] save a cosine and convert everything else to sine. The cosine will be the du. Otherwise use the fact that: 1" cos2x 1+ cos2x sin 2 x = and cos2 x = 2 2 dy dy dt = dx dx dt d " dy % $ ' d 2 y d " dy % dt # dx & = $ '= dx dx 2 dx # dx & dt b L= # a ! 92. x = f (t ), y = g (t ) , find arc length on ! [t1 ,t2 ] ! ! ! 93. Find horizontal tangents to a polar curve r = f (" ) ! ! 94. Find vertical tangents to a polar curve r = f (" ) ! ! ! ! 95. Find the volume when the area between ! y = f ( x ), x = 0, y = 0 is rotated about the y-axis. 96. Given a set of points, estimate the volume under the curve using Simpson’s rule on [a, b] . 97. Find the dot product: u1 , u2 " v1 ,v2 98. Multiply two vectors: ! [ ] 2 1+ f "( x ) dx " dx %2 " dy %2 L = ( $ ' + $ ' dt # dt & # dt & t1 x = r cos" , y = r sin " t2 Find where r sin " = 0 where r cos" # 0 x = r cos" , y = r sin " Find where r cos " = 0 where r sin " # 0 b Shell method: V = 2" $ radius# height dx where b is the root. 0 A" b#a [ y0 + 4 y1 + 2 y2 + 4 y3 + 2 y4 + ... + 4 yn#1 + yn ] 3n ! u1 , u2 " v1 ,v2 = u1v1 + u2v2 You get a scalar. ! ! www.MasterMathMentor.com ! Stu Schwartz