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1969 AP Calculus AB: Section I 90 Minutes—No Calculator Note: In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e). 1. 2. 3. Which of the following defines a function f for which f (− x) = − f ( x) ? (A) f ( x) = x 2 (B) f ( x) = sin x (D) f ( x) = log x (E) f ( x) = e x ln ( x − 2 ) < 0 if and only if (A) x<3 (B) 0< x<3 (D) x>2 (E) x>3 ⎧ 2x + 5 − x + 7 , for x ≠ 2, ⎪ f ( x) = If ⎨ x−2 ⎪ f (2) = k ⎩ (A) 0 4. 8 ∫0 dx 1+ x (A) 1 5. f ( x) = cos x (C) 2< x<3 and if f is continuous at x = 2 , then k = (B) 1 6 (C) 1 3 (D) 1 (E) 7 5 (B) 3 2 (C) 2 (D) 4 (E) 6 (D) 4 (E) not defined = If 3 x 2 + 2 xy + y 2 = 2, then the value of (A) –2 (C) (B) 0 dy at x = 1 is dx (C) 2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 1 1969 AP Calculus AB: Section I 8 6. (A) 0 (E) 7. 1 2 For what value of k will x + (C) 1 (D) The limit does not exist. k have a relative maximum at x = −2? x (B) –2 (C) 2 (D) 4 (E) None of these If p ( x) = ( x + 2 )( x + k ) and if the remainder is 12 when p( x) is divided by x − 1, then k = (A) 2 9. (B) It cannot be determined from the information given. (A) –4 8. 8 ⎛1 ⎞ ⎛1⎞ 8⎜ + h ⎟ − 8⎜ ⎟ 2 ⎠ ⎝2⎠ ? What is lim ⎝ h →0 h (B) 3 (C) 6 (D) 11 (E) 13 When the area in square units of an expanding circle is increasing twice as fast as its radius in linear units, the radius is (A) 1 4π (B) 1 4 (C) 1 π (D) 1 (E) π (E) ln x 10. The set of all points (et , t ) , where t is a real number, is the graph of y = (A) 1 ex (B) 1 ex (C) 1 xex (D) 1 ln x 1⎞ ⎛ 11. The point on the curve x 2 + 2 y = 0 that is nearest the point ⎜ 0, − ⎟ occurs where y is 2⎠ ⎝ 1 1 (B) 0 (C) − (D) −1 (E) none of the above (A) 2 2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 2 1969 AP Calculus AB: Section I 12. If f ( x) = (A) 4 and g ( x) = 2 x, then the solution set of f ( g ( x) ) = g ( f ( x) ) is x −1 ⎧1 ⎫ ⎨ ⎬ ⎩3⎭ (B) {2} (C) {3} (D) {−1, 2} (E) ⎧1 ⎫ ⎨ , 2⎬ ⎩3 ⎭ 13. The region bounded by the x-axis and the part of the graph of y = cos x between x = − π and 2 π π is separated into two regions by the line x = k . If the area of the region for − ≤ x ≤ k is 2 2 π three times the area of the region for k ≤ x ≤ , then k = 2 x= ⎛1⎞ (A) arcsin ⎜ ⎟ ⎝4⎠ (D) π 4 (B) ⎛1⎞ arcsin ⎜ ⎟ ⎝3⎠ (E) π 3 (C) π 6 14. If the function f is defined by f ( x) = x5 − 1, then f −1 , the inverse function of f , is defined by f −1 ( x) = (A) (D) 1 5 x +1 5 x −1 (B) (E) 1 5 x +1 5 x +1 (C) 5 x −1 15. If f ′( x) and g ′( x) exist and f ′( x) > g ′( x) for all real x, then the graph of y = f ( x) and the graph of y = g ( x) (A) intersect exactly once. (B) intersect no more than once. (C) do not intersect. (D) could intersect more than once. (E) have a common tangent at each point of intersection. AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 3 1969 AP Calculus AB: Section I 16. If y is a function of x such that y′ > 0 for all x and y′′ < 0 for all x, which of the following could be part of the graph of y = f ( x) ? 17. The graph of y = 5 x 4 − x5 has a point of inflection at (A) (0, 0) only (B) (3,162) only (D) (0,0) and (3,162 ) (E) (0, 0) and (4, 256) (C) (4, 256) only 18. If f ( x) = 2 + x − 3 for all x, then the value of the derivative f ′( x) at x = 3 is (A) −1 (B) 0 (C) 1 (D) 2 (E) nonexistent 19. A point moves on the x-axis in such a way that its velocity at time t ( t > 0 ) is given by v = ln t . t At what value of t does v attain its maximum? (A) 1 (E) (B) 1 e2 (C) e (D) 3 e2 There is no maximum value for v. AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 4 1969 AP Calculus AB: Section I 20. An equation for a tangent to the graph of y = arcsin (A) x − 2y = 0 (B) x− y =0 (C) x at the origin is 2 x=0 y=0 (D) π x − 2y = 0 (E) 21. At x = 0 , which of the following is true of the function f defined by f ( x) = x 2 + e −2 x ? (A) f is increasing. (B) f is decreasing. (C) f is discontinuous. (D) f has a relative minimum. (E) 22. f has a relative maximum. ( ) d ln e 2 x = dx (A) 1 e 2x (B) 2 e2 x (C) 2x (D) 1 (E) 2 23. The area of the region bounded by the curve y = e2x , the x-axis, the y-axis, and the line x = 2 is equal to (A) e4 −e 2 (B) e4 −1 2 (D) 2e4 − e (E) 2e4 − 2 24. If sin x = e y , 0 < x < π, what is (A) − tan x (B) − cot x (C) e4 1 − 2 2 (E) csc x dy in terms of x ? dx (C) cot x AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) tan x 5 1969 AP Calculus AB: Section I 25. A region in the plane is bounded by the graph of y = x = 2m, m > 0 . The area of this region 1 , the x-axis, the line x = m, and the line x (A) is independent of m . (B) increases as m increases. (C) decreases as m increases. 1 1 ; increases as m increases when m > . 2 2 1 1 increases as m increases when m < ; decreases as m increases when m > . 2 2 (D) decreases as m increases when m < (E) 26. 1 ∫0 x 2 − 2 x + 1 dx is (A) −1 (B) − 1 2 1 2 (D) 1 (E) none of the above (C) 27. If dy = tan x , then y = dx (A) 1 tan 2 x + C 2 (B) sec 2 x + C (D) ln cos x + C (E) sec x tan x + C (C) ln sec x + C (E) 3 3 28. The function defined by f ( x) = 3 cos x + 3sin x has an amplitude of (A) 3− 3 (B) 3 (C) 2 3 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 3+ 3 6 1969 AP Calculus AB: Section I 29. ∫π 4 cos x dx = sin x (A) ln 2 π 2 (B) ln π 4 (C) ln 3 (D) 3 2 ln (E) ln e 30. If a function f is continuous for all x and if f has a relative maximum at (−1, 4) and a relative minimum at (3, − 2) , which of the following statements must be true? (A) The graph of f has a point of inflection somewhere between x = −1 and x = 3. (B) f ′(−1) = 0 (C) The graph of f has a horizontal asymptote. (D) The graph of f has a horizontal tangent line at x = 3 . (E) The graph of f intersects both axes. 31. If f ′( x) = − f ( x) and f (1) = 1, then f ( x) = (A) 1 −2 x + 2 e 2 (B) e − x −1 (C) e1− x (D) e− x (E) −e x 32. If a, b, c, d , and e are real numbers and a ≠ 0 , then the polynomial equation ax 7 + bx5 + cx3 + dx + e = 0 has (A) (B) (C) (D) (E) only one real root. at least one real root. an odd number of nonreal roots. no real roots. no positive real roots. 33. What is the average (mean) value of 3t 3 − t 2 over the interval −1 ≤ t ≤ 2 ? (A) 11 4 (B) 7 2 (C) 8 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 33 4 (E) 16 7 1969 AP Calculus AB: Section I 34. Which of the following is an equation of a curve that intersects at right angles every curve of the 1 family y = + k (where k takes all real values)? x 1 1 (D) y = x3 (E) y = ln x (C) y = − x3 (A) y = − x (B) y = − x 2 3 3 35. At t = 0 a particle starts at rest and moves along a line in such a way that at time t its acceleration is 24t 2 feet per second per second. Through how many feet does the particle move during the first 2 seconds? (A) 32 (B) 48 (C) 64 (D) 96 (E) 192 36. The approximate value of y = 4 + sin x at x = 0.12 , obtained from the tangent to the graph at x = 0, is (A) 2.00 (B) 2.03 (C) 2.06 (D) 2.12 (E) 2.24 37. Which is the best of the following polynomial approximations to cos 2 x near x = 0 ? (A) 1 + 38. x2 ∫ ex 3 x 2 (B) 1 + x (C) 1 − x2 2 (D) 1 − 2x 2 (E) 1 − 2x + x 2 (C) − (E) sec2 e dx = 3 (A) 3 1 − ln e x + C 3 (B) (D) 3 1 ln e x + C 3 (E) ex − +C 3 x3 3e x3 1 3e x 3 +C +C 1 dy 39. If y = tan u , u = v − , and v = ln x , what is the value of at x = e ? v dx (A) 0 (B) 1 e (C) 1 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 2 e 8 1969 AP Calculus AB: Section I 40. If n is a non-negative integer, then 1 ∫0 x (A) no n (D) nonzero n, only n 1 0 (1 − x )n dx for (B) n even, only (E) all n ⎧⎪ f ( x) = 8 − x 2 for − 2 ≤ x ≤ 2, 41. If ⎨ then 2 elsewhere , ⎪⎩ f ( x) = x (A) 0 and 8 dx = ∫ (B) 8 and 16 (C) n odd, only 3 ∫ −1 f ( x) dx is a number between (C) 16 and 24 (D) 24 and 32 (E) 32 and 40 42. What are all values of k for which the graph of y = x3 − 3 x 2 + k will have three distinct x-intercepts? (A) All k > 0 43. (B) All k < 4 (C) k = 0, 4 (D) 0 < k < 4 (E) All k ∫ sin ( 2 x + 3) dx = (A) 1 cos ( 2 x + 3) + C 2 (B) cos ( 2 x + 3) + C (D) 1 − cos ( 2 x + 3) + C 2 (E) 1 − cos ( 2 x + 3) + C 5 (C) 44. The fundamental period of the function defined by f ( x) = 3 − 2 cos 2 (A) 1 (B) 2 (C) 3 − cos ( 2 x + 3) + C πx is 3 (D) 5 (E) 6 (C) 3x 2 g x3 d d d2 2 45. If ( f ( x) ) = g ( x) and ( g ( x) ) = f ( x ) , then 2 f ( x3 ) = dx dx dx ( ( ) (A) f x6 (D) 9 x 4 f x 6 + 6 x g x3 ( ) ( ) ) ( ) (B) g x3 (E) f x 6 + g x3 ( ) ( ) ( ) AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 9 1969 Calculus AB Solutions 1. B Sine is the only odd function listed. sin(− x) = − sin( x) . 2. C ln t < 0 for 0 < t < 1 ⇒ ln ( x − 2 ) < 0 for 2 < x < 3 . 3. B Need to have lim f ( x) = f (2) = k . x →2 k = lim x →2 2x + 5 − x + 7 2x + 5 − x + 7 2x + 5 + x + 7 = lim ⋅ x→2 x−2 x−2 2x + 5 + x + 7 2x + 5 − ( x + 7) 1 1 1 ⋅ = lim = x →2 x−2 2 x + 5 + x + 7 x→2 2 x + 5 + x + 7 6 = lim 8 dx = 2 1+ x 1+ x 8 = 2 ( 3 − 1) = 4 4. D ∫0 5. E Using implicit differentiation, 6 x + 2 xy′ + 2 y + 2 y ⋅ y′ = 0 . Therefore y′ = 0 −2 y − 6 x . 2x + 2 y When x = 1 , 3 + 2 y + y 2 = 2 ⇒ 0 = y 2 + 2 y + 1 = ( y + 1) 2 ⇒ y = −1 dy Therefore 2 x + 2 y = 0 and so is not defined at x = 1 . dx 6. B This is the derivative of f ( x) = 8 x8 at x = 1 2 7 1 ⎛1⎞ ⎛1⎞ f ′ ⎜ ⎟ = 64 ⎜ ⎟ = 2 ⎝2⎠ ⎝2⎠ k k , we need 0 = f ′(−2) = 1 − and so k = 4. Since f ′′(−2) < 0 for k = 4, f x 4 does have a relative maximum at x = −2 . With f ( x) = x + 7. D 8. B p ( x) = q ( x)( x − 1) + 12 for some polynomial q ( x) and so 12 = p (1) = (1 + 2 )(1 + k ) ⇒ k = 3 9. C A = π r2, So, 2 dA dr dA dr and from the given information in the problem = 2π r ⋅ =2 . dt dt dt dt dr dr 1 = 2π r ⋅ ⇒ r = dt dt π AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 160 1969 Calculus AB Solutions 10. E x = e y ⇒ y = ln x 11. B ⎛ x2 ⎞ 1⎞ ⎛ Let L be the distance from ⎜ x , − ⎟ and ⎜ 0, − ⎟ . ⎜ ⎟ 2 ⎠ 2⎠ ⎝ ⎝ 2 ⎛ x2 1 ⎞ L = ( x − 0) + ⎜ − ⎟ ⎜ 2 2⎟ ⎝ ⎠ ⎛ x2 1 ⎞ dL 2L ⋅ = 2x + 2 ⎜ − ⎟ ( x) ⎜ 2 2⎟ dx ⎝ ⎠ 2 2 ⎛ x2 1 ⎞ 2x + 2 ⎜ − ⎟ ( x) 2 ⎜ 2 2⎟ 2 x + x3 − x x3 + x x x + 1 dL ⎝ ⎠ = = = = 2L 2L 2L 2L dx ( ) dL dL < 0 for all x < 0 and > 0 for all x > 0 , so the minimum distance occurs at x = 0 . dx dx The nearest point is the origin. 12. A 13. C 4 1 ⎛ 4 ⎞ = 2⎜ ⎟ ⇒ x − 1 = 4 x − 2; x = 2x −1 3 ⎝ x −1 ⎠ π ⎛ π⎞ ⎛ ⎞ cos x dx ; sin k − sin ⎜ − ⎟ = 3 ⎜ sin − sin k ⎟ 2 ⎝ 2⎠ ⎝ ⎠ π sin k + 1 = 3 − 3sin k ; 4sin k = 2 ⇒ k = 6 k π 2 ∫ −π 2 cos x dx = 3∫ k 14. E y = x5 − 1 has an inverse x = y 5 − 1 ⇒ y = 5 x + 1 15. B The graphs do not need to intersect (eg. f ( x) = −e − x and g ( x) = e − x ) . The graphs could intersect (e.g. f ( x) = 2 x and g ( x) = x ). However, if they do intersect, they will intersect no more than once because f ( x) grows faster than g ( x) . 16. B y′ > 0 ⇒ y is increasing; y′′ < 0 ⇒ the graph is concave down . Only B meets these conditions. 17. B y′ = 20 x3 − 5 x 4 , y′′ = 60 x 2 − 20 x3 = 20 x 2 ( 3 − x ) . The only sign change in y′′ is at x = 3 . The only point of inflection is (3,162). AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 161 1969 Calculus AB Solutions 18. E There is no derivative at the vertex which is located at x = 3 . 19. C dv 1 − ln t dv = 2 > 0 for 0 < t < e and < 0 for t > e , thus v has its maximum at t = e . dt dt t 20. A y (0) = 0 and y′(0) = y= 1 2 x2 1− 4 x =0 1 = 4 − x2 x =0 = 1 . The tangent line is 2 1 x ⇒ x − 2y = 0 . 2 21. B f ′ ( x ) = 2 x − 2e−2 x , f ′ ( 0 ) = −2 , so f is decreasing 22. E ln e2 x = 2 x ⇒ 2 2x e 0 ( 2 1 2x e 2 23. C ∫ 24. C y = ln sin x , y′ = 25. A 26. C 27. C 2m 1 ∫m x dx = ) d d ln e2 x = ( 2 x ) = 2 dx dx dx = ln x 0 = ( ) 1 4 e −1 2 cos x = cot x sin x 2m m = ln ( 2m ) − ln ( m ) = ln 2 so the area is independent of m. 1 1 1 2 x − 1 = ( ) ∫0 0 0 0 2 2 Alternatively, the graph of the region is a right triangle with vertices at (0,0), (0,1), and (1,0). 1 The area is . 2 1 x 2 − 2 x + 1 dx = ∫ 1 1 x − 1 dx = ∫ − ( x − 1) dx = − sin x ∫ tan x dx = ∫ cos x dx = − ln cos x + C = ln sec x + C AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 162 1969 Calculus AB Solutions 28. C 3 cos x + 3sin x can be thought of as the expansion of sin ( x + y ) . Since 3 and 3 are too large for values of sin y and cos y , multiply and divide by the result of the Pythagorean Theorem used on those values, i.e. 2 3 . Then ⎛ 3 ⎞ ⎛1 ⎞ 3 3 3 cos x + 3sin x = 2 3 ⎜⎜ cos x + sin x ⎟⎟ = 2 3 ⎜⎜ cos x + sin x ⎟⎟ 2 2 3 ⎝2 3 ⎠ ⎝2 ⎠ = 2 3 ( sin y cos x + cos y sin x ) = 2 3 sin ( y + x ) ⎛1⎞ where y = sin −1 ⎜ ⎟ . The amplitude is 2 3 . ⎝2⎠ Alternatively, the function f ( x) is periodic with period 2π . f ′( x) = − 3 sin x + 3cos x = 0 π 4π ⎛π⎞ when tan x = 3 . The solutions over one period are x = , . Then f ⎜ ⎟ = 2 3 and 3 3 ⎝3⎠ ⎛ 4π ⎞ f ⎜ ⎟ = −2 3 . So the amplitude is 2 3 . ⎝ 3 ⎠ π2 cos x dx = ln ( sin x ) sin x π2 1 = ln 2 2 29. A ∫π 4 30. E Because f is continuous for all x, the Intermediate Value Theorem implies that the graph of f must intersect the x-axis. The graph must also intersect the y-axis since f is defined for all x, in particular, at x = 0. π4 = ln1 − ln 31. C dy = − y ⇒ y = ce − x and 1 = ce−1 ⇒ c = e ; y = e ⋅ e− x = e1− x dx 32. B If a < 0 then lim y = ∞ and lim y = −∞ which would mean that there is at least one root. x→−∞ x→∞ If a > 0 then lim y = −∞ and lim y = ∞ which would mean that there is at least one root. x→−∞ x→∞ In both cases the equation has at least one root. 33. A 1 2 3 2 1⎛ 3 1 ⎞ 3t − t dt = ⎜ t 4 − t 3 ⎟ ∫ − 1 3 3⎝ 4 3 ⎠ 34. D y′ = − 1 x 2 1 ⎛⎛ 8 ⎞ ⎛ 3 1 ⎞ ⎞ 11 = ⎜ ⎜12 − ⎟ − ⎜ + ⎟ ⎟ = −1 3 ⎝ ⎝ 3 ⎠ ⎝ 4 3 ⎠⎠ 4 2 , so the desired curve satisfies y′ = x 2 ⇒ y = AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 1 3 x +C 3 163 1969 Calculus AB Solutions 35. A a ( t ) = 24t 2 , v(t ) = 8t 3 + C and v(0) = 0 ⇒ C = 0. The particle is always moving to the right, so distance = 36. B ∫0 8t 3dt = 2t 4 y = 4 + sin x , y (0) = 2, y′(0) = L( x ) = 2 + 37. D 2 2 0 = 32 . cos 0 1 = . The linear approximation to y is 2 4 + sin 0 4 1 1 x . L(1.2) = 2 + (1.2) = 2.03 4 4 All options have the same value at x = 0 . We want the one that has the same first and second derivatives at x = 0 as y = cos 2 x : y′(0) = −2sin 2 x For y = 1 − 2 x 2 , y′(0) = −4 x x2 x =0 x =0 = 0 and y′′(0) = −4 cos 2 x 1 − x3 1 3 1 +C e (−3 x 2 dx) = − e − x + C = − 3 ∫ 3 3 3e− x ∫ ex 39. D x = e ⇒ v = 1, u = 0, y = 0; 40. E One solution technique is to evaluate each integral and note that the value is dx = − ( dy dy du dv = ⋅ ⋅ = sec 2 u dx du dv dx Another technique is to use the substitution u = 1 − x ; ) ⎛⎜⎝1 + v12 ⎞⎟⎠ ⎜⎝⎛ 1x ⎟⎠⎞ = (1)( 2) ( e−1 ) = 2e 1 ∫ 0 (1 − x ) Integrals do not depend on the variable that is used and so 3 = −4. = 0 and y′′(0) = −4 and no other option works. 38. C 3 x =0 f ( x ) dx = ∫ 2 ( ) 3 1 ⎞ ⎛ 8 − x 2 dx + ∫ x 2 dx = ⎜ 8 x − x3 ⎟ 2 3 ⎠ ⎝ 2 n 1 n u 0 ∫ 1 3 x 3 1 for each. n +1 0 1 1 0 dx = ∫ u n ( − du ) = ∫ u n du . du is the same as 3 1 n x 0 ∫ 41. D ∫ −1 42. D y = x3 − 3 x 2 + k , y′ = 3 x 2 − 6 x = 3 x( x − 2) . So f has a relative maximum at (0, k ) and a relative minimum at (2, k − 4) . There will be 3 distinct x-intercepts if the maximum and minimum are on the opposite sides of the x-axis. We want k − 4 < 0 < k ⇒ 0 < k < 4 . 43. D ∫ sin ( 2 x + 3) dx = − 2 cos ( 2 x + 3) + C −1 −1 + 2 = 27 1 dx . 3 1 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 164 1969 Calculus AB Solutions 44. C Since cos 2 A = 2 cos 2 A − 1 , we have 3 − 2 cos 2 2π =3 ⎛ 2π ⎞ ⎜ ⎟ ⎝ 3 ⎠ expression has period 45. D πx 2π x = 3 − (1 + cos ) and the latter 3 3 Let y = f ( x3 ) . We want y′′ where f ′( x) = g ( x) and f ′′( x) = g ′( x) = f ( x 2 ) y = f ( x3 ) y′ = f ′( x3 ) ⋅ 3x 2 ( ) y′′ = 3 x 2 f ′′( x3 ) ⋅ 3x 2 + f ′( x3 ) ⋅ 6 x = 9 x 4 f ′′( x3 ) + 6 x f ′( x3 ) = 9 x 4 f (( x3 ) 2 ) + 6 x g ( x3 ) = 9 x 4 f ( x6 ) + 6 x g ( x3 ) AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 165 1973 AP Calculus AB: Section I 90 Minutes—No Calculator Note: In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e). 1. 2. ∫ (x 3 ) − 3 x dx = (A) 3x 2 − 3 + C (D) x4 − 3x + C 4 5 x 2 + 15 x + 25 (D) 225 (B) 5 x3 + 15 x 2 + 20 x + 25 (E) 5 (C) 1125 1 e (B) 2 2 e 2 (C) 4 e 2 (D) 1 e 4 (E) 4 e4 If f ( x) = x + sin x , then f ′( x) = (D) sin x − x cos x (B) 1 − cos x (E) sin x + x cos x (C) cos x y =1 If f ( x) = e x , which of the following lines is an asymptote to the graph of f ? (A) 6. x4 − 3x 2 + C 3 ( ) (A) 1 + cos x 5. (E) x 4 3x 2 − +C 4 2 (C) The slope of the line tangent to the graph of y = ln x 2 at x = e 2 is (A) 4. 4 x4 − 6 x2 + C If f ( x) = x3 + 3 x 2 + 4 x + 5 and g ( x) = 5, then g ( f ( x) ) = (A) 3. (B) y=0 If f ( x) = (A) –1 (B) x=0 (C) y=x (D) y = −x (E) 0 (D) 1 2 (E) 1 x −1 for all x ≠ −1, then f ′(1) = x +1 (B) − 1 2 (C) AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 20 1973 AP Calculus AB: Section I 7. 8. Which of the following equations has a graph that is symmetric with respect to the origin? y= (D) y = ( x − 1) + 1 3 (B) y = − x5 + 3 x (E) y = x2 + 1 − 1 ( ) (C) y = x4 − 2 x2 + 6 2 A particle moves in a straight line with velocity v(t ) = t 2 . How far does the particle move between times t = 1 and t = 2? (A) 9. x +1 x (A) 1 3 (B) If y = cos 2 3 x , then 7 3 (C) 3 (D) 7 (E) 8 dy = dx (A) −6sin 3 x cos 3 x (B) −2 cos 3x (D) 6 cos 3x (E) 2sin 3 x cos 3 x x 4 x5 − attains its maximum value at x = 10. The derivative of f ( x) = 3 5 4 (A) –1 (B) 0 (C) 1 (D) 3 (C) 2 cos 3x (E) 5 3 11. If the line 3x − 4 y = 0 is tangent in the first quadrant to the curve y = x3 + k , then k is (A) 1 2 (B) 1 4 (C) 0 (D) − 1 8 (E) − 1 2 12. If f ( x) = 2 x3 + Ax 2 + Bx − 5 and if f (2) = 3 and f (−2) = −37 , what is the value of A + B ? (A) –6 (E) (B) –3 (C) –1 (D) 2 It cannot be determined from the information given. AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 21 1973 AP Calculus AB: Section I 13. The acceleration α of a body moving in a straight line is given in terms of time t by α = 8 − 6t . If the velocity of the body is 25 at t = 1 and if s (t ) is the distance of the body from the origin at time t, what is s (4) − s (2) ? (A) 20 (B) 24 14. If f ( x) = x 1 3 ( x − 2) 2 3 (C) 28 (D) 32 (E) 42 for all x, then the domain of f ′ is (A) {x x ≠ 0} (B) {x x > 0} (D) {x x ≠ 0 and x ≠ 2} (E) {x x is a real number} (C) {x 0 ≤ x ≤ 2} x 2 15. The area of the region bounded by the lines x = 0, x = 2, and y = 0 and the curve y = e is (A) e −1 2 (B) e −1 (C) 2 ( e − 1) (D) 2e − 1 (E) 2e 2t 3000e 5 16. The number of bacteria in a culture is growing at a rate of per unit of time t. At t = 0 , the number of bacteria present was 7,500. Find the number present at t = 5 . (A) 1, 200e 2 (B) 3, 000e 2 (C) 7,500e 2 (D) 7,500e5 (E) 15, 000 7 e 7 17. What is the area of the region completely bounded by the curve y = − x 2 + x + 6 and the line y =4? (A) 18. 3 2 (B) 7 3 (C) 9 2 (D) 31 6 (E) 33 2 d ( arcsin 2 x ) = dx (A) (D) −1 2 1 − 4x 2 2 1 − 4x 2 (B) (E) −2 4 x2 −1 (C) 1 2 1 − 4x 2 2 4 x2 −1 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 22 1973 AP Calculus AB: Section I 19. Suppose that f is a function that is defined for all real numbers. Which of the following conditions assures that f has an inverse function? (A) The function f is periodic. (B) The graph of f is symmetric with respect to the y-axis. (C) The graph of f is concave up. (D) The function f is a strictly increasing function. (E) The function f is continuous. 20. If F and f are continuous functions such that F ′( x) = f ( x) for all x, then 21. (A) F ′(a ) − F ′(b) (B) F ′(b) − F ′(a ) (C) F (a) − F (b) (D) F (b) − F (a) (E) none of the above 1 ∫ 0 ( x + 1) e (A) x2 +2 x e3 2 b ∫a f ( x) dx is dx = (B) e3 − 1 2 (C) e4 − e 2 (D) e3 − 1 (E) e4 − e 22. Given the function defined by f ( x) = 3 x5 − 20 x3 , find all values of x for which the graph of f is concave up. (A) x>0 (B) − 2 < x < 0 or x > 2 (C) −2 < x < 0 or x > 2 (D) x> 2 (E) −2 < x < 2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 23 1973 AP Calculus AB: Section I 23. 1 ⎛ 2+h⎞ ln ⎜ ⎟ is h→0 h ⎝ 2 ⎠ lim (A) e2 (B) 1 (C) 1 2 (D) 0 (E) nonexistent 24. Let f ( x) = cos ( arctan x ) . What is the range of f ? 25. (A) ⎧ π ⎨x − < x < 2 ⎩ (D) {x π 4 ∫0 (A) π⎫ ⎬ 2⎭ − 1 < x < 1} (B) {x 0 < x ≤ 1} (E) {x − 1 ≤ x ≤ 1} (C) 1 3 (C) {x 0 ≤ x ≤ 1} (E) π +1 4 tan 2 x dx = π −1 4 (B) 1 − π 4 2 −1 (D) 26. The radius r of a sphere is increasing at the uniform rate of 0.3 inches per second. At the instant when the surface area S becomes 100π square inches, what is the rate of increase, in cubic inches 4 ⎛ ⎞ per second, in the volume V ? ⎜ S = 4π r 2 and V = π r 3 ⎟ 3 ⎝ ⎠ (A) 10π 27. 2x 12 ∫0 (B) 12π 1− x (A) 1 − 2 3 2 (C) 22.5 π (D) 25 π (E) 30 π (C) π 6 (D) π −1 6 (E) 2− 3 dx = (B) 1 3 ln 2 4 28. A point moves in a straight line so that its distance at time t from a fixed point of the line is 8t − 3t 2 . What is the total distance covered by the point between t = 1 and t = 2? (A) 1 (B) 4 3 (C) 5 3 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 2 (E) 5 24 1973 AP Calculus AB: Section I 1 . The maximum value attained by f is 2 29. Let f ( x) = sin x − 1 2 (A) 30. (B) 1 ∫1 x−4 (A) − 2 x2 3 2 (D) π 2 (E) 3π 2 (C) ln 2 (D) 2 (E) ln 2 + 2 (C) 8 (D) 16 (E) 32 ) 5 (C) 5x − + C x dx = 1 2 (B) ( ) ln 2 − 2 a , then a = 4 31. If log a 2a = (A) 2 32. (C) (B) 4 5 ∫ 1 + x 2 dx = −10 x ( (A) (1 + x2 ) +C (B) 5 ln 1 + x 2 + C 2x (D) 5arctan x + C (E) 5ln 1 + x 2 + C 2 ( ) 33. Suppose that f is an odd function; i.e., f (− x) = − f ( x) for all x. Suppose that f ′ ( x0 ) exists. Which of the following must necessarily be equal to f ′ ( − x0 ) ? (A) f ′ ( x0 ) (B) − f ′ ( x0 ) (C) 1 f ′ ( x0 ) (D) −1 f ′ ( x0 ) (E) None of the above AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 25 1973 AP Calculus AB: Section I x over the interval 0 ≤ x ≤ 2 is 34. The average value of 1 2 3 (A) (B) 1 2 2 (C) 2 2 3 (D) 1 35. The region in the first quadrant bounded by the graph of y = sec x, x = (E) 4 2 3 π , and the axes is rotated 4 about the x-axis. What is the volume of the solid generated? π2 4 (A) 36. If y = enx , then dx n π −1 (C) π (D) 2π (E) 8π 3 n !e nx (C) n e nx (D) nn e x (E) n !e x 3 + e4 x (D) 4 + e4 x (E) 2 x2 + 4 (E) –5 ) (E) ( 4,8) (C) cos 2 ( xy ) = (B) dy = 4 y and if y = 4 when x = 0, then y = dx 4e4 x (A) 38. If dny n n enx (A) 37. If (B) 2 ∫1 (A) (B) e4 x (C) f ( x − c ) dx = 5 where c is a constant, then 5+c (B) 5 (C) 2−c ∫ 1−c f ( x ) dx = 5−c (D) c − 5 39. The point on the curve 2 y = x 2 nearest to ( 4,1) is (A) ( 0, 0 ) 40. If tan( xy ) = x , then (B) ( 2, 2 ) (C) ( ) 2,1 (D) (2 2, 4 dy = dx (A) 1 − y tan( xy ) sec( xy ) x tan( xy ) sec( xy ) (D) cos 2 ( xy ) x (B) sec 2 ( xy ) − y x (E) cos 2 ( xy ) − y x AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 26 1973 AP Calculus AB: Section I ⎧ x + 1 for x < 0, 41. Given f ( x) = ⎨ ⎩cos π x for x ≥ 0, (A) 1 1 + 2 π (B) − 1 ∫ −1 f ( x) dx = 1 2 (C) 1 1 − 2 π (D) 1 2 (E) 1 − +π 2 42. Calculate the approximate area of the shaded region in the figure by the trapezoidal rule, using 4 5 divisions at x = and x = . 3 3 (A) 50 27 (B) 251 108 (C) 7 3 (D) 127 54 (E) 77 27 (C) − ⎛ x⎞ 43. If the solutions of f ( x) = 0 are –1 and 2, then the solutions of f ⎜ ⎟ = 0 are ⎝2⎠ (A) −1 and 2 (D) − 1 and 1 2 44. For small values of h, the function 4 1 5 and 2 2 (B) − (E) −2 and 4 3 3 and 2 2 16 + h is best approximated by which of the following? (A) 4+ h 32 (B) 2+ h 32 (D) 4− h 32 (E) 2− h 32 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (C) h 32 27 1973 AP Calculus AB: Section I 45. If f is a continuous function on [ a, b ] , which of the following is necessarily true? (A) f ′ exists on ( a , b ) . (B) If f ( x0 ) is a maximum of f, then f ′ ( x0 ) = 0 . (C) ⎛ ⎞ lim f ( x) = f ⎜ lim x ⎟ for x0 ∈ ( a , b ) x→ x0 ⎝ x→ x0 ⎠ (D) f ′( x) = 0 for some x ∈ [ a , b ] (E) The graph of f ′ is a straight line. AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 28 1973 Calculus AB Solutions 1 4 3 2 x − x +C 4 2 1. E ∫ (x 2. E g ( x) = 5 ⇒ g ( f ( x) ) = 5 3. B y = ln x 2 ; y′ = 4. A f ( x) = x + sin x ; 5. A lim e x = 0 ⇒ y = 0 is a horizontal asymptote 6. D 7. B 3 − 3x) dx = 2x x 2 = 2 2 . At x = e 2 , y′ = 2 . x e f ′( x) = 1 − cos x x→−∞ f ′( x) = (1)( x + 1) − ( x − 1)(1) ( x + 1) 2 , f ′(1) = 2 1 = 4 2 Replace x with (− x) and see if the result is the opposite of the original. This is true for B. −(− x)5 + 3(− x) = x5 − 3 x = −(− x5 + 3 x) . 8. B 9. A 10. C 11. B Distance = ∫ 2 1 2 1 t 2 dx = ∫ t 2 dt = t 3 1 3 1 7 = (23 − 13 ) = 1 3 3 d d ( cos 3x ) = 2 cos 3x ⋅ ( − sin 3x ) ⋅ ( 3x ) = 2 cos 3x ⋅ ( − sin 3x ) ⋅ ( 3) dx dx y′ = −6sin 3x cos 3x y′ = 2 cos 3x ⋅ x 4 x5 4 x3 − ; f ′( x) = − x 4 ; f ′′ ( x ) = 4 x 2 − 4 x3 = 4 x 2 (1 − x ) 3 5 3 f ′′ > 0 for x < 1 and f ′′ < 0 for x > 1 ⇒ f ′ has its maximum at x = 1 . f ( x) = Curve and line have the same slope when 3x 2 = ⎛1 3 tangency is ⎜ , ⎝ 2 8 12. C 2 3 1 ⇒ x = . Using the line, the point of 4 2 3 3 ⎛1⎞ 1 ⎞ ⎟ . Since the point is also on the curve, = ⎜ ⎟ + k ⇒ k = . 8 ⎝2⎠ 4 ⎠ Substitute the points into the equation and solve the resulting linear system. 3 = 16 + 4 A + 2 B − 5 and − 37 = −16 + 4 A − 2 B − 5 ; A = −3, B = 2 ⇒ A + B = −1 . AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 172 1973 Calculus AB Solutions 13. D v(t ) = 8t − 3t 2 + C and v(1) = 25 ⇒ C = 20 so v(t ) = 8t − 3t 2 + 20 . 4 4 2 2 s (4) − s (2) = ∫ v ( t ) dt = (4t 2 − t 3 + 20t ) 14. D f ( x) = x1 3 ( x − 2 ) = 32 23 2 1 1 ( x − 2 )−1 3 + ( x − 2 )2 3 ⋅ x −2 3 = x −2 3 ( x − 2 )−1 3 ( 3x − 2 ) 3 3 3 ′ f is not defined at x = 0 and at x = 2 . f ′ ( x ) = x1 3 ⋅ 2 x 2 e x 2 2 2e 0 = 2 ( e − 1) 15. C Area = ∫ 16. C t t dN = 3000e 5 , N = 7500e 5 + C and N (0) = 7500 ⇒ C = 0 dt 0 dx = 2 N 17. C 2 t 5 = 7500e 2 , N ( 5 ) = 7500e2 Determine where the curves intersect. − x 2 + x + 6 = 4 ⇒ x 2 − x − 2 = 0 ( x − 2)( x + 1) = 0 ⇒ x = −1, x = 2 . Between these two x values the parabola lies above the line y = 4. 2 1 ⎛ 1 ⎞ 2 9 Area = ∫ (− x 2 + x + 6) − 4 dx = ⎜ − x3 + x 2 + 2 x ⎟ = −1 2 ⎝ 3 ⎠ −1 2 ( ) 18. D d 1 d 2 2 ⋅ ( 2x) = = ( arcsin 2 x ) = 2 2 dx 1 − 4 x2 1 − ( 2 x ) dx 1− ( 2x) 19. D If f is strictly increasing then it must be one to one and therefore have an inverse. 20. D By the Fundamental Theorem of Calculus, 1 2 b ∫a f ( x) dx = F (b) − F ( a) where F ′( x) = f ( x) . ( 1 1 x2 +2 x 1 2 e (2 x + 2) dx ) = e x + 2 x ( ∫ 2 0 2 ) 1 1 3 0 e3 − 1 e −e = 2 2 ( ) 21. B x ∫ 0 ( x + 1) e 22. B f ( x) = 3 x5 − 20 x3 ; f ′( x) = 15 x 4 − 60 x 2 ; f ′′( x) = 60 x3 − 120 x = 60 x x 2 − 2 +2 x dx = 0 = ( ) The graph of f is concave up where f ′′ > 0 : f ′′ > 0 for x > 2 and for − 2 < x < 0 . AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 173 1973 Calculus AB Solutions 23. C ln ( 2 + h ) − ln 2 1 1 = f ′ ( 2 ) where f ( x ) = ln x ; f ′ ( x ) = ⇒ f ′ ( 2 ) = h →0 h x 2 24. B f ( x) = cos ( arctan x ) ; − lim π π < arctan x < and the cosine in this domain takes on all values in 2 2 the interval (0,1]. π 4 0 ∫ 26. E dV dr dr = 4πr 2 ⋅ = S ⋅ = 100π ( 0.3) = 30π dt dt dt 27. E 28. C tan x dx = ∫ π 4 0 25. B 2 1 2x ∫0 2 1 − x2 (sec 2 x − 1) dx = (tan x − x) dx = − ∫ 1 0 ) ( −2 x dx ) = − 2 ( v ( t ) = 8 − 6t changes sign at t = ) = 2− 3 4 ⎛4⎞ ⎛4⎞ 5 . Distance = x(1) − x ⎜ ⎟ + x(2) − x ⎜ ⎟ = . 3 ⎝3⎠ ⎝3⎠ 3 2 1 30. B π 4 0 Alternative Solution: Distance = ∫ 29. C = 1− 1 1 2 1 − x2 2 1 − 2 1 − x2 2 ( π4 0 v ( t ) dt = ∫ 2 1 8 − 6t dt = 5 3 3 1 1 1 3 is . −1 ≤ sin x ≤ 1 ⇒ − ≤ sin x − ≤ ; The maximum for sin x − 2 2 2 2 2 2 ∫1 x−4 x2 2 ⎛1 4⎞ ⎞ ⎛ dx = ∫ ⎜ − 4 x −2 ⎟ dx = ⎜ ln x + ⎟ 1 ⎝x x⎠ ⎠ ⎝ ( ) a 2 1 = ( ln 2 + 2 ) − ( ln1 + 4 ) = ln 2 − 2 1 a 1 = ⇒ log a 2 = ⇒ 2 = a 4 ; a = 16 4 4 31. D log a 2 32. D ∫ 1 + x2 dx = 5 ∫ 1 + x 2 dx = 5 tan 33. A f (− x) = − f ( x) ⇒ f ′(− x) ⋅ (−1) = − f ′( x) ⇒ f ′(− x) = − f ′( x) thus f ′(− x0 ) = − f ′( x0 ) . 5 1 −1 ( x) + C AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 174 1973 Calculus AB Solutions 3 2 34. C 35. C 1 2 1 2 x dx = ⋅ x 2 ∫ 2 0 2 3 Washers: ∑πr Volume = π∫ 36. A 37. A 38. B 39. B 40. E 41. D π 4 0 2 0 3 1 2 = ⋅ 22 = 2 3 3 ∆x where r = y = sec x . sec 2 x dx = π tan x π4 0 = π(tan π − tan 0) = π 4 y = e nx , y′ = ne nx , y′′ = n 2 enx ," , y ( n) = n n e nx dy = 4 y , y (0) = 4 . This is exponential growth. The general solution is y = Ce 4 x . Since dx y (0) = 4 , C = 4 and so the solution is y = 4e4 x . Let z = x − c . Then 5 = ∫ 2 1 f ( x − c ) dx = ∫ 2−c 1−c f ( z ) dz 1 2 x ) on 2 2 ⎛1 ⎞ the curve to the point (4,1) . The distance L satisfies the equation L2 = ( x − 4 ) + ⎜ x 2 − 1⎟ . ⎝2 ⎠ Determine where L is a maximum by examining critical points. Differentiating with respect dL dL ⎛1 ⎞ to x, 2 L ⋅ changes sign from positive to negative at = 2( x − 4) + 2 ⎜ x 2 − 1⎟ x = x3 − 8 . dx dx ⎝2 ⎠ x = 2 only. The point on the curve has coordinates (2, 2) . Use the distance formula to determine the distance, L, from any point ( x, y ) = ( x , 2 sec ( xy ) ⋅ ( xy′ + y ) = 1, xy′ sec2 ( xy ) + y sec2 ( xy ) = 1 , y′ = 1 ∫ −1 0 1 −1 0 f ( x) dx = ∫ ( x + 1) dx + ∫ cos(π x) dx = = 1 ( x + 1) 2 2 0 1 − y sec2 ( xy ) cos 2 ( xy ) − y = x x sec2 ( xy ) 1 + sin(π x) −1 π 1 0 1 1 1 + ( sin π − sin 0 ) = 2 π 2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 175 1973 Calculus AB Solutions 42. D 2 2 ⎞ 127 1 1 1⎛ ⎛4⎞ ⎛5⎞ ∆x = ; T = ⋅ ⎜12 + 2 ⎜ ⎟ + 2 ⎜ ⎟ + 22 ⎟ = ⎟ 54 3 2 3 ⎜⎝ ⎝3⎠ ⎝3⎠ ⎠ 43. E Solve x x = −1 and = 2; x = −2, 4 2 2 3 44. B 45. C 1 − 1 Use the linearization of f ( x) = x at x = 16 . f ′( x) = x 4 , f ′(16) = 4 32 1 h L( x) = 2 + ( x − 16); f (16 + h) ≈ L(16 + h) = 2 + 32 32 4 This uses the definition of continuity of f at x = x0 . AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 176 1985 AP Calculus AB: Section I 90 Minutes—No Calculator Notes: (1) In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e). (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number. 1. 2 ∫1 x −3 dx = − (A) 2. 7 8 5. (C) 15 64 (D) 3 8 (E) 15 16 4 If y = (A) 4. 3 4 If f ( x) = ( 2 x + 1) , then the 4th derivative of f ( x) at x = 0 is (A) 0 3. − (B) If (B) 24 3 4+ x 2 ( 4 + x2 ) 2 48 (D) 240 (E) 384 dy = dx , then −6 x (C) (B) 3x ( 4 + x2 ) 2 (C) 6x ( 4 + x2 ) 2 (D) −3 ( 4 + x2 ) 2 (E) 3 2x dy = cos ( 2 x ) , then y = dx (A) 1 − cos ( 2 x ) + C 2 1 (B) − cos 2 ( 2 x ) + C 2 (D) 1 2 sin ( 2 x ) + C 2 (E) lim n→∞ 4n 2 n 2 + 10, 000n (A) 0 (C) 1 sin ( 2 x ) + C 2 1 − sin ( 2 x ) + C 2 is (B) 1 2,500 (C) 1 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 4 (E) nonexistent 38 1985 AP Calculus AB: Section I 6. If f ( x) = x, then f ′(5) = (A) 0 7. ln 3 + ln1 1 (D) 5 (E) 25 2 ln 8 ln 2 (B) (C) 4 ∫1 et dt (D) 4 ∫1 ln x dx (E) 4 ∫1 1 dt t ⎛ x⎞ The slope of the line tangent to the graph of y = ln ⎜ ⎟ at x = 4 is ⎝2⎠ 1 8 (A) 9. (C) Which of the following is equal to ln 4 ? (A) 8. 1 5 (B) If 1 ∫ −1 (A) 1 4 (B) 2 0 e − x dx = k , then ∫ −1 −2k −k (B) ( x −1) , then 2 10. If y = 10 (A) ( ln10 )10( (D) 2 x ( ln10 )10 (C) 1 2 (C) − (D) 1 (E) 4 2 e − x dx = k 2 (D) k 2 (E) 2k dy = dx ) (B) ( 2 x )10( ( x −1) (E) x 2 ( ln10 )10 x 2 −1 ) x 2 −1 (C) ( ) ( x −2) x 2 − 1 10 2 ( x −1) 2 2 11. The position of a particle moving along a straight line at any time t is given by s (t ) = t 2 + 4t + 4 . What is the acceleration of the particle when t = 4 ? (A) 0 (B) 2 ( (C) ) 4 (D) 8 (E) 12 ( ) 12. If f ( g ( x) ) = ln x 2 + 4 , f ( x) = ln x 2 , and g ( x) > 0 for all real x, then g (x) = (A) 1 2 x +4 (B) 1 2 x +4 (C) x2 + 4 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) x2 + 4 (E) x+2 39 1985 AP Calculus AB: Section I 13. If x 2 + xy + y 3 = 0 , then, in terms of x and y, (A) − 2x + y x + 3y 2 (B) x + 3y2 − 2x + y (C) dy = dx −2 x 1+ 3y (D) 2 14. The velocity of a particle moving on a line at time t is v meters did the particle travel from t = 0 to t = 4? (A) 32 (B) 40 (C) 1 = 3t 2 64 −2 x x + 3y 3 2 + 5t 2 − 2x + y x + 3 y2 −1 meters per second. How many (D) 80 ( (E) (E) 184 ) 15. The domain of the function defined by f ( x) = ln x 2 − 4 is the set of all real numbers x such that (A) x <2 x ≤2 (B) x >2 (C) (D) x ≥2 (E) x is a real number 16. The function defined by f ( x) = x3 − 3 x 2 for all real numbers x has a relative maximum at x = (A) 17. 1 ∫ 0 xe −2 −x (B) 0 (C) 1 (D) 2 (E) 4 (C) 1 − 2e −1 (D) 1 (E) dx = (A) 1 − 2e (B) −1 2e − 1 18. If y = cos 2 x − sin 2 x , then y′ = (A) −1 (B) (C) −2sin ( 2x ) 0 −2 ( cos x + sin x ) (D) (E) 2 ( cos x − sin x ) 19. If f ( x1 ) + f ( x2 ) = f ( x1 + x2 ) for all real numbers x1 and x2 , which of the following could define f ? (A) f ( x) = x + 1 (B) f ( x) = 2 x (C) f ( x) = 1 x AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) f ( x) = e x (E) f ( x) = x 2 40 1985 AP Calculus AB: Section I 20. If y = arctan ( cos x ) , then dy = dx − sin x (A) (B) − ( arcsec ( cos x ) ) sin x 2 2 1 + cos x 1 (D) ( arccos x ) 2 (E) +1 22. 1 + cos 2 x 1 − x2 is { x : x > 1} , what is the range of f ? { x : −∞ < x < −1} (B) { x : −∞ < x < 0} (D) { x : −1 < x < ∞} (E) { x : 0 < x < ∞} (C) { x : −∞ < x < 1} x2 −1 dx = x +1 1 2 (A) 23. 1 (A) ∫1 ( arcsec ( cos x ) )2 1 21. If the domain of the function f given by f ( x) = 2 (C) (B) 1 5 2 (C) 2 (D) (E) (C) 0 (D) 2 (E) 6 (C) 0 (D) 4 (E) 12 ln 3 d ⎛ 1 1 ⎞ − + x 2 ⎟ at x = −1 is ⎜ 3 dx ⎝ x x ⎠ −6 (A) 24. If ∫ −2 ( x (A) 2 (B) 7 −4 ) + k dx = 16, then k = −12 (B) −4 25. If f ( x) = e x , which of the following is equal to f ′(e)? (A) lim e x+h h →0 h (B) lim (D) e x+h − 1 h →0 h (E) lim lim e x + h − ee h →0 h ee + h − e h →0 h (C) lim ee + h − ee h →0 h AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 41 1985 AP Calculus AB: Section I 26. The graph of y 2 = x 2 + 9 is symmetric to which of the following? I. II. III. The x-axis The y-axis The origin (A) I only 27. 3 ∫0 (B) II only (C) III only (D) I and II only (E) I, II, and III x − 1 dx = (A) 0 (B) 3 2 (C) 2 (D) 5 2 (E) 6 28. If the position of a particle on the x-axis at time t is −5t 2 , then the average velocity of the particle for 0 ≤ t ≤ 3 is (A) −45 (B) −30 (C) −15 (D) −10 (E) −5 29. Which of the following functions are continuous for all real numbers x ? I. II. III. y= 2 x3 y = ex y = tan x (A) None 30. (B) I only (C) II only (D) I and II (E) I and III ∫ tan ( 2x ) dx = (A) −2 ln cos(2 x) + C (B) 1 − ln cos(2 x) + C 2 (D) 2 ln cos(2 x) + C (E) 1 sec(2 x) tan(2 x) + C 2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (C) 1 ln cos(2 x) + C 2 42 1985 AP Calculus AB: Section I 1 31. The volume of a cone of radius r and height h is given by V = π r 2 h . If the radius and the height 3 1 both increase at a constant rate of centimeter per second, at what rate, in cubic centimeters per 2 second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters? (A) 32. ∫ π 3 0 (A) 1 π 2 (B) 10 π (C) 24 π (D) 54 π (E) 108 π sin ( 3x ) dx = −2 (B) − 2 3 (C) 0 (D) 2 3 (E) 2 33. The graph of the derivative of f is shown in the figure above. Which of the following could be the graph of f ? AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 43 1985 AP Calculus AB: Section I 34. The area of the region in the first quadrant that is enclosed by the graphs of y = x3 + 8 and y = x + 8 is (A) 1 4 (B) 1 2 (C) 3 4 (D) 1 (E) 65 4 35. The figure above shows the graph of a sine function for one complete period. Which of the following is an equation for the graph? (A) ⎛π ⎞ y = 2sin ⎜ x ⎟ ⎝2 ⎠ (B) y = sin ( π x ) (D) y = 2sin ( π x ) (E) y = sin ( 2 x ) (C) y = 2sin ( 2 x ) 36. If f is a continuous function defined for all real numbers x and if the maximum value of f ( x) is 5 and the minimum value of f ( x) is −7 , then which of the following must be true? I. The maximum value of f ( x ) is 5. II. The maximum value of f ( x) is 7. III. The minimum value of f ( x ) is 0. (A) I only 37. (B) II only (C) I and II only (D) II and III only (E) I, II, and III (D) 1 ∞ lim ( x csc x ) is x →0 (A) −∞ (B) –1 (C) 0 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (E) 44 1985 AP Calculus AB: Section I 38. Let f and g have continuous first and second derivatives everywhere. If f ( x ) ≤ g ( x ) for all real x, which of the following must be true? f ′( x) ≤ g ′( x) for all real x f ′′( x) ≤ g ′′( x) for all real x I. II. 1 ∫0 III. f ( x) dx ≤ (A) None 39. If f ( x) = (A) (B) (C) (D) (E) f f f f f 1 ∫ 0 g ( x) dx (B) I only (C) III only (D) I and II only (E) I, II, and III ln x , for all x > 0, which of the following is true? x is increasing for all x greater than 0. is increasing for all x greater than 1. is decreasing for all x between 0 and 1. is decreasing for all x between 1 and e. is decreasing for all x greater than e. 40. Let f be a continuous function on the closed interval [ 0, 2] . If 2 ≤ f ( x) ≤ 4, then the greatest possible value of (A) 0 2 ∫0 f ( x) dx is (B) 2 (C) 4 (D) 8 (E) 16 41. If lim f ( x) = L, where L is a real number, which of the following must be true? x →a (A) f ′(a ) exists. (B) f ( x) is continuous at x = a. (C) f ( x) is defined at x = a. (D) f (a) = L (E) None of the above AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 45 1985 AP Calculus AB: Section I 42. d dx (A) (D) x ∫2 1 + t 2 dt = x 1+ x (B) 2 x 1 + x2 − 1 5 (E) 1 + x2 − 5 1 2 1 + x2 − (C) 1 + x2 1 2 5 43. An equation of the line tangent to y = x3 + 3 x 2 + 2 at its point of inflection is (A) y = −6 x − 6 (B) y = −3x + 1 (D) y = 3x − 1 (E) y = 4 x + 1 (C) y = 2 x + 10 44. The average value of f ( x) = x 2 x3 + 1 on the closed interval [ 0, 2] is (A) 26 9 (B) 13 3 (C) 26 3 (D) 13 (E) 26 45. The region enclosed by the graph of y = x 2 , the line x = 2, and the x-axis is revolved about the y -axis. The volume of the solid generated is (A) 8π (B) 32 π 5 (C) 16 π 3 (D) AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 4π (E) 8 π 3 46 1985 Calculus AB Solutions 2 −3 x dx 1 1 = − x −2 2 2 1⎛1 ⎞ 3 = − ⎜ − 1⎟ = . 1 2⎝4 ⎠ 8 1. D ∫ 2. E f ′( x) = 4(2 x + 1)3 ⋅ 2, f ′′(1) = 4 ⋅ 3(2 x + 1) 2 ⋅ 22 , f ′′′(1) = 4 ⋅ 3 ⋅ 2(2 x + 1)1 ⋅ 23 , f (4) (1) = 4!⋅ 24 = 384 3. A y = 3(4 + x 2 ) −1 so y′ = −3(4 + x 2 ) −2 (2 x) = −6 x (4 + x 2 ) 2 ( 4 + x ) (0) − 3(2 x) = −6 x Or using the quotient rule directly gives y′ = (4 + x ) (4 + x ) 2 2 1 C ∫ cos(2 x) dx = 2 ∫ cos(2 x) (2 dx) = 2 sin(2 x) + C 5. D lim 6. C f ′( x) = 1 ⇒ f ′(5) = 1 7. E ∫1 8. B 1 1 ⎛ x⎞ y = ln ⎜ ⎟ = ln x − ln 2, y′ = , y′(4) = 4 x ⎝2⎠ 9. D Since e − x is even, n→∞ 4 2 n + 10000n 2 2 1 4. 4n 2 2 4 =4 10000 1+ n = lim n→∞ 1 4 dt = ln t 1 = ln 4 − ln1 = ln 4 t 2 2 −1) 0 ∫ −1 e − x2 ( dx = 1 1 − x2 1 e dx = k ∫ 2 −1 2 ) 2 d ( x 2 − 1) = 2 x ⋅10( x −1) ⋅ ln(10) dx 10. D y′ = 10( x 11. B v(t ) = 2t + 4 ⇒ a(t ) = 2 ∴ a(4) = 2 12. C f ( g ( x) ) = ln g ( x) 2 = ln x 2 + 4 ⇒ g ( x) = x 2 + 4 ⋅ ln(10) ⋅ ( ) ( ) AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 183 1985 Calculus AB Solutions 13. A 2 x + x ⋅ y′ + y + 3 y 2 ⋅ y′ = 0 ⇒ y′ = − 4 ∫0 2x + y x + 3y2 v ( t ) dt = ∫ 4 3⎞ 5⎞ ⎛ 1 ⎛ 3 2 2 2 ⎜ 3t + 5t ⎟ dt = ⎜ 2t + 2t 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 4 14. D Since v(t ) ≥ 0, distance = 15. C x2 − 4 > 0 ⇒ x > 2 16. B f ′( x) = 3 x 2 − 6 x = 3 x( x − 2) changes sign from positive to negative only at x = 0. 17. C 0 0 = 80 Use the technique of antiderivatives by parts: u=x dv = e− x dx du = dx v = −e − x ( − xe − x + ∫ e − x dx = − xe− x − e− x ) 0 = 1 − 2e − 1 1 18. C y = cos 2 x − sin 2 x = cos 2 x , y′ = −2sin 2 x 19. B Quick solution: lines through the origin have this property. Or, f ( x 1 ) + f ( x 2 ) = 2 x 1 + 2 x 2 = 2( x 1 + x 2 ) = f ( x 1 + x 2 ) 20. A dy 1 d − sin x = ⋅ ( cos x ) = 2 dx 1 + cos x dx 1 + cos 2 x 21. B x > 1 ⇒ x 2 > 1 ⇒ f ( x) < 0 for all x in the domain. lim f ( x) = 0 . lim f ( x) = −∞ . The only x →∞ x →1 option that is consistent with these statements is (B). 2 2 ( x + 1)( x − 1) 2 x2 − 1 1 dx = ∫ dx = ∫ ( x − 1) dx = ( x − 1) 2 1 1 x +1 x +1 2 22. A ∫1 23. B d −3 x − x −1 + x 2 dx 24. D 16 = ∫ ( 2 −2 ) x =−1 ( x 7 + k ) dx = ∫ ( = − 3 x −4 + x − 2 + 2 x 2 −2 x 7 dx + ∫ 2 −2 ) x =−1 2 1 = 1 2 = − 3 + 1 − 2 = −4 k dx = 0 + ( 2 − (−2) ) k = 4k ⇒ k = 4 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 184 1985 Calculus AB Solutions 25. E 26. E f ′(e) = lim h →0 f (e + h ) − f (e ) ee + h − ee = lim h →0 h h I: Replace y with (− y ) : (− y ) 2 = x 2 + 9 ⇒ y 2 = x 2 + 9 , no change, so yes. II: Replace x with (− x) : y 2 = (− x) 2 + 9 ⇒ y 2 = x 2 + 9 , no change, so yes. III: Since there is symmetry with respect to both axes there is origin symmetry. 27. D The graph is a V with vertex at x = 1 . The integral gives the sum of the areas of the two triangles that the V forms with the horizontal axis for x from 0 to 3. These triangles have areas of 1/2 and 2 respectively. 28. C Let x(t ) = −5t 2 be the position at time t. Average velocity = 29. D The tangent function is not defined at x = π 2 so it cannot be continuous for all real numbers. Option E is the only one that includes item III. In fact, the functions in I and II are a power and an exponential function that are known to be continuous for all real numbers x. 30. B ∫ tan(2 x) dx = − 2 ∫ 31. C 1 dV 1 ⎛ dr dh ⎞ 1 ⎛ ⎛1⎞ ⎛ 1 ⎞⎞ V = π r 2h , = π ⎜ 2rh + r 2 ⎟ = π ⎜ 2(6)(9) ⎜ ⎟ + 62 ⎜ ⎟ ⎟ = 24π 3 dt 3 ⎝ dt dt ⎠ 3 ⎝ ⎝2⎠ ⎝ 2 ⎠⎠ 32. D ∫0 33. B f ′ changes sign from positive to negative at x = –1 and therefore f changes from increasing to decreasing at x = –1. 1 π3 x(3) − x(0) −45 − 0 = = −15 3−0 3 −2sin(2 x) 1 dx = − ln cos(2 x) + C cos(2 x) 2 1 sin(3x) dx = − cos(3x) 3 π3 = 0 − 1 2 ( cos π − cos 0 ) = 3 3 Or f ′ changes sign from positive to negative at x = −1 and from negative to positive at x = 1 . Therefore f has a local maximum at x = −1 and a local minimum at x = 1 . 34. A ⎛ 3 3 ∫ 0 ( ( x + 8) − ( x + 8) ) dx = ∫ 0 ( x − x ) dx = ⎜ 1 1 1 2 1 4⎞ x − x ⎟ 4 ⎠ ⎝2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 1 0 = 1 4 185 1985 Calculus AB Solutions 35. D The amplitude is 2 and the period is 2. y = A sin Bx where A = amplitude = 2 and B = 36. B 2π 2π = =π period 2 II is true since − 7 = 7 will be the maximum value of f ( x) . To see why I and III do not ⎧ 5 if ⎪ have to be true, consider the following: f ( x ) = ⎨− x if ⎪ −7 if ⎩ For f 37. D 38. C x ≤ −5 −5 < x < 7 x≥7 ( x ) , the maximum is 0 and the minimum is –7. x =1 x→0 sin x lim x csc x = lim x →0 To see why I and II do not have to be true consider f ( x) = sin x and g ( x) = 1 + e x . Then f ( x) ≤ g ( x) but neither f ′( x) ≤ g ′( x) nor f ′′( x) < g ′′( x) is true for all real values of x. III is true, since f ( x) ≤ g ( x) ⇒ g ( x) − f ( x) ≥ 0 ⇒ 39. E f ′( x) = 2 1 1 1 ∫ 0 ( g ( x) − f ( x) ) dx ≥ 0 ⇒ ∫ 0 f ( x) dx ≤ ∫ 0 g ( x) dx 1 1 1 1 ⋅ − 2 ln x = 2 (1 − ln x) < 0 for x > e . Hence f is decreasing. for x > e . x x x x 2 40. D f ( x) dx ≤ ∫ 4 dx = 8 ∫0 41. E Consider the function whose graph is the horizontal line y = 2 with a hole at x = a . For this function lim f ( x) = 2 and none of the given statements are true. 0 x →a 42. C This is a direct application of the Fundamental Theorem of Calculus: f ′( x) = 1 + x 2 43. B y′ = 3 x 2 + 6 x , y′′ = 6 x + 6 = 0 for x = −1. y′(−1) = −3 . Only option B has a slope of –3. 44. A 1 2 2 3 x x +1 2 ∫0 ( ) 1 2 ( ) ( 1 1 2 dx = ⋅ ∫ x3 + 1 2 3 0 1 2 ) ( 3 x 2 dx = ) 1 3 x +1 6 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 3 2 ⋅ 2 3 2 0 = 26 9 186 1985 Calculus AB Solutions 45. A Washers: ∑ π ( R 2 − r 2 ) ∆y Volume = π ∫ 4 0 ( where R = 2, r = x ) 4 1 ⎞ ⎛ 22 − x 2 dy = π ∫ (4 − y ) dy = π ⎜ 4 y − y 2 ⎟ 0 2 ⎠ ⎝ AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 4 0 = 8π 187 1988 AP Calculus AB: Section I 90 Minutes—No Calculator Notes: (1) In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e). (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number. 1. 2. 3. If y = x 2e x , then 2 xe x (B) x x + 2e x (D) 2x + ex (E) 2x + e What is the domain of the function f given by f ( x) = (A) {x : (D) {x : x ≠ 3} x ≥ 2 and x ≠ 3} (B) {x : (E) {x : ) (C) xe x ( x + 2 ) (C) {x : x2 − 4 ? x−3 x ≤ 2} x ≥ 2} x ≥ 2 and x ≠ 3} A particle with velocity at any time t given by v(t ) = et moves in a straight line. How far does the particle move from t = 0 to t = 2 ? e2 − 1 The graph of y = (A) 5. ( (A) (A) 4. dy = dx ∫ sec x<0 2 (B) e −1 (C) 2e (D) e2 (E) e3 3 (E) x>2 (C) cos 2 x + C −5 is concave downward for all values of x such that x−2 (B) x<2 (C) x<5 (D) x>0 x dx = (A) tan x + C (B) csc 2 x + C (D) sec3 x +C 3 (E) 2sec2 x tan x + C AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 57 1988 AP Calculus AB: Section I 6. 7. 8. If y = ln x dy , then = x dx (A) 1 x (B) x dx ∫ 3x 2 + 5 x 2 (C) ln x − 1 x (D) 2 1 − ln x x 2 1 + ln x (E) x2 = ) 3 2 +C (B) 1 3x 2 + 5 4 ) 1 2 +C (E) 3 3x 2 + 5 2 ( (A) 1 3x 2 + 5 9 (D) 1 3x 2 + 5 3 ( 1 ( ( ) 3 2 +C ) 1 2 +C ( 1 3x 2 + 5 (C) 12 ) 1 2 +C The graph of y = f ( x) is shown in the figure above. On which of the following intervals are dy d2y > 0 and <0? dx dx 2 I. II. III. a< x<b b<x<c c<x<d (A) I only (B) II only (C) III only AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) I and II (E) II and III 58 1988 AP Calculus AB: Section I 9. If x + 2 xy − y 2 = 2, then at the point (1,1) , 3 2 (A) 10. If k ∫0 (B) 1 2 dy is dx − (C) 0 (D) (C) 3 (D) 9 3 2 (E) nonexistent ( 2kx − x2 ) dx = 18, then k = (A) –9 (B) –3 (E) 18 11. An equation of the line tangent to the graph of f ( x) = x(1 − 2 x)3 at the point (1, − 1) is (A) y = −7 x + 6 (B) y = −6 x + 5 (D) y = 2x − 3 (E) y = 7x − 8 (C) 2 2 (C) y = −2 x + 1 (E) 3 ⎛π⎞ 12. If f ( x) = sin x , then f ′ ⎜ ⎟ = ⎝3⎠ (A) − 1 2 (B) 1 2 (D) 13. If the function f has a continuous derivative on [ 0, c ] , then (A) f (c) − f (0) 14. ∫ π 2 0 (B) f (c) − f (0) (C) f (c ) c ∫0 3 2 f ′( x) dx = (D) f ( x) + c (E) f ′′(c) − f ′′(0) cos θ dθ = 1 + sin θ (A) −2 (D) 2 ( ( ) 2 −1 ) 2 −1 (B) −2 2 (E) 2 ( (C) 2 2 ) 2 +1 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 59 1988 AP Calculus AB: Section I 15. If f ( x) = 2 x , then f ′(2) = (A) 1 4 (B) 1 2 (C) 2 2 (D) 1 2 (E) 16. A particle moves along the x-axis so that at any time t ≥ 0 its position is given by x(t ) = t 3 − 3t 2 − 9t + 1 . For what values of t is the particle at rest? (A) No values 17. 1 ∫ 0 ( 3x − 2 ) (A) − 2 (B) 1 only (C) 3 only (D) 5 only (E) 1 and 3 dx = 7 3 (B) − 7 9 1 9 (D) 1 (E) 3 ⎛ x⎞ (C) − sin ⎜ ⎟ ⎝2⎠ ⎛x⎞ (D) − cos ⎜ ⎟ ⎝2⎠ 1 ⎛ x⎞ (E) − cos ⎜ ⎟ 2 ⎝2⎠ (C) ln 2 (D) 2 ln 2 (E) (C) d2y ⎛x⎞ 18. If y = 2 cos ⎜ ⎟ , then = dx 2 ⎝2⎠ ⎛ x⎞ (A) −8cos ⎜ ⎟ ⎝2⎠ 19. 3 ∫2 (A) x 2 x +1 ⎛ x⎞ (B) −2 cos ⎜ ⎟ ⎝2⎠ dx = 1 3 ln 2 2 (B) 1 ln 2 2 1 ln 5 2 20. Let f be a polynomial function with degree greater than 2. If a ≠ b and f (a) = f (b) = 1 , which of the following must be true for at least one value of x between a and b? I. II. III. f ( x) = 0 f ′( x) = 0 f ′′( x) = 0 (A) None (B) I only (C) II only (D) I and II only AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (E) I, II, and III 60 1988 AP Calculus AB: Section I 21. The area of the region enclosed by the graphs of y = x and y = x 2 − 3 x + 3 is (A) 2 3 (B) 1 (C) 4 3 (C) e (D) 2 (E) 14 3 (E) e2 ⎛1⎞ 22. If ln x − ln ⎜ ⎟ = 2, then x = ⎝ x⎠ (A) 1 e 1 e (B) 2 (D) 2e f ( x) is x→0 g ( x ) 23. If f ′( x) = cos x and g ′( x) = 1 for all x, and if f (0) = g (0) = 0 , then lim (A) 24. π 2 (B) 1 (D) −1 (E) nonexistent ( ) d ln x x = dx (A) x ln x (B) ( ln x ) x 25. For all x > 1, if f ( x) = ∫ (A) 1 26. (C) 0 ∫ π 2 0 (A) (B) (C) x 1 ( ) 2 ( ln x ) xln x x (D) ( ln x ) ( xln x−1 ) ( ) (E) 2 ( ln x ) x ln x 1 dt , then f ′( x) = t 1 x (C) ln x − 1 (D) ln x (E) ex (D) 1 (E) π −1 2 x cos x dx = − π 2 (B) –1 (C) 1 − π 2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 61 1988 AP Calculus AB: Section I ⎧⎪ x 2 , x < 3 27. At x = 3 , the function given by f ( x ) = ⎨ is ⎪⎩6 x − 9, x ≥ 3 (A) (B) (C) (D) (E) 28. 4 ∫1 (A) undefined. continuous but not differentiable. differentiable but not continuous. neither continuous nor differentiable. both continuous and differentiable. x − 3 dx = − 3 2 (B) 3 2 (C) 5 2 (D) 9 2 (E) 5 tan 3( x + h ) − tan 3x is h →0 h 29. The lim (A) 0 (B) 3sec 2 (3x) (C) sec2 (3 x) (D) 3cot(3x) (E) nonexistent 30. A region in the first quadrant is enclosed by the graphs of y = e 2 x , x = 1, and the coordinate axes. If the region is rotated about the y -axis , the volume of the solid that is generated is represented by which of the following integrals? 1 (A) 2π ∫ xe2 x dx (B) 2π ∫ e2 x dx (C) π ∫ e 4 x dx (D) π∫ (E) π e 2 ln y dy 4 ∫0 0 1 0 1 0 e 0 y ln y dy AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 62 1988 AP Calculus AB: Section I 31. If f ( x) = (A) x , then the inverse function, f −1 , is given by f −1 ( x) = x +1 x −1 x (B) x +1 x (C) x 1− x (D) x x +1 (E) x 32. Which of the following does NOT have a period of π ? (A) ⎛1 ⎞ f ( x) = sin ⎜ x ⎟ ⎝2 ⎠ (B) f ( x) = sin x (D) f ( x) = tan x (E) f ( x) = tan 2 x (C) f ( x) = sin 2 x 33. The absolute maximum value of f ( x) = x3 − 3 x 2 + 12 on the closed interval [ −2, 4] occurs at x = (A) 4 (B) 2 (C) 1 (D) 0 (E) –2 34. The area of the shaded region in the figure above is represented by which of the following integrals? c (A) ∫ a ( f ( x) (B) ∫b (C) ∫ a ( g ( x) − f ( x) ) dx (D) ∫ a ( f ( x) − g ( x) ) dx (E) c − g ( x) ) dx c f ( x) dx − ∫ g ( x) dx a c c b c ∫ a ( g ( x) − f ( x) ) dx + ∫ b ( f ( x) − g ( x) ) dx AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 63 1988 AP Calculus AB: Section I 35. π⎞ ⎛ 4 cos ⎜ x + ⎟ = 3⎠ ⎝ (A) 2 3 cos x − 2sin x (B) 2 cos x − 2 3 sin x (D) 2 3 cos x + 2sin x (E) 4 cos x + 2 (C) 2 cos x + 2 3 sin x 36. What is the average value of y for the part of the curve y = 3 x − x 2 which is in the first quadrant ? (A) –6 (B) –2 (C) 3 2 (D) 9 4 (E) 9 2 37. If f ( x) = e x sin x , then the number of zeros of f on the closed interval [ 0, 2π] is (A) 0 (B) 1 38. For x > 0, 1 (A) x3 +C (B) ( ) +C (E) 2 10 ∫1 f ( x) dx = 4 and (A) –3 2 (D) 3 (E) 4 ⎛ 1 x du ⎞ ⎜ ∫1 ⎟ dx = u ⎠ ⎝x ln x 2 (D) 39. If ∫ (C) 3 ∫ 10 f ( x) dx = 7, then (B) 0 8 x4 − 2 x2 +C (C) ln ( ln x ) + C (E) 11 ( ln x )2 + C 2 3 ∫1 f ( x) dx = (C) 3 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 10 64 1988 AP Calculus AB: Section I z y x 40. The sides of the rectangle above increase in such a way that when x = 4 and y = 3 , what is the value of (A) 1 3 (B) 1 dz dx dy = 1 and = 3 . At the instant dt dt dt dx ? dt (C) 2 (D) 5 (E) 5 41. If lim f ( x) = 7 , which of the following must be true? x→3 I. II. III. f is continuous at x = 3 . f is differentiable at x = 3 . f (3) = 7 (A) None (B) II only (D) I and III only (E) I, II, and III (C) III only 42. The graph of which of the following equations has y = 1 as an asymptote? (A) y = ln x (B) y = sin x (C) y= x x +1 (D) y= x2 x −1 (E) y = e− x 43. The volume of the solid obtained by revolving the region enclosed by the ellipse x 2 + 9 y 2 = 9 about the x-axis is (A) 2π (B) 4π (C) 6π AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 9π (E) 12π 65 1988 AP Calculus AB: Section I 44. Let f and g be odd functions. If p, r, and s are nonzero functions defined as follows, which must be odd? I. II. III. p ( x) = f ( g ( x) ) r ( x) = f ( x) + g ( x) s ( x) = f ( x) g ( x) (A) I only (B) II only (D) II and III only (E) I, II, and III (C) I and II only 45. The volume of a cylindrical tin can with a top and a bottom is to be 16π cubic inches. If a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can? (A) 3 2 2 (B) 2 2 (C) 3 2 4 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 4 (E) 8 66 1988 Calculus AB Solutions 1. C dy d d = x 2 ⋅ (e x ) + e x ⋅ ( x 2 ) = x 2e x + 2 xe x = xe x ( x + 2) dx dx dx 2. D x 2 − 4 ≥ 0 and x ≠ 3 ⇒ x ≥ 2 and x ≠ 3 3. A Distance = ∫ 4. E Students should know what the graph looks like without a calculator and choose option E. 2 0 2 v ( t ) dt = ∫ et dt = et 0 −1 Or y = −5 ( x − 2 ) ; y ′ = 5 ( x − 2 ) 5. A ∫ sec 2 −2 2 0 = e 2 − e0 = e 2 − 1 −3 ; y ′′ = −10 ( x − 2 ) . y ′′ < 0 for x > 2 . x dx = ∫ d ( tan x ) = tan x + C x⋅ ⎛1⎞ d d (ln x) − ln x ⋅ ( x) x ⋅ ⎜ ⎟ − ln x ⋅ (1) 1 − ln x ⎝ x⎠ dx dx = = x2 x2 x2 D dy = dx 7. D ∫ x(3x 8. B dy d2y > 0 ⇒ y is increasing; < 0 ⇒ graph is concave down . This is only on b < x < c . dx dx 2 9. E 1 + ( 2 x ⋅ y ′ + 2 y ) − 2 y ⋅ y ′ = 0; y ′ = 6. 2 + 5) − 1 2 1 1 1 − 1 1 1 dx = ∫ (3x 2 + 5) 2 ( 6 x dx ) = ⋅ 2(3x 2 + 5) 2 + C = (3x 2 + 5) 2 + C 6 6 3 1+ 2y . This cannot be evaluated at (1,1) and so y ′ does 2 y − 2x not exist at (1,1) . 10. C 1 ⎞ ⎛ 18 = ⎜ kx 2 − x3 ⎟ 3 ⎠ ⎝ k 0 = 2 3 k ⇒ k 3 = 27, so k = 3 3 11. A f ′( x) = x ⋅ 3(1 − 2 x) 2 (−2) + (1 − 2 x)3 ; f ′(1) = −7 . Only option A has a slope of –7. 12. B ⎛π⎞ ⎛π⎞ 1 f ′ ⎜ ⎟ = cos ⎜ ⎟ = ⎝3⎠ ⎝3⎠ 2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 194 1988 Calculus AB Solutions 13. A By the Fundamental Theorem of Calculus π 2 0 −1 2 ∫0 π 12 2 14. D ∫ (1 + sin θ ) 15. B f ( x) = 2 x = 2 ⋅ x ; f ′( x) = 2 ⋅ 16. C c ( cos θ d θ ) = 2 (1 + sin θ ) 1 2 x 0 f ′( x) dx = f ( x) =2 ( c 0 = f (c) − f (0) ) 2 −1 ; f ′(2) = 2 ⋅ 1 2 2 ( = 1 2 ) At rest when 0 = v(t ) = x′(t ) = 3t 2 − 6t − 9 = 3 t 2 − 2t − 3 = 3(t − 3)(t + 1) t = −1, 3 and t ≥ 0 ⇒ t = 3 1 ( 3x − 2 ) dx = 2 1 1 1 1 ( 3x − 2 )2 ( 3 dx ) = ⋅ ( 3x − 2 )3 ∫ 0 3 3 3 1 1 (1 − ( −8) ) = 1 9 17. D ∫0 18. E ⎛ ⎛ 1 ⎛ x⎞ 1⎞ ⎛ x⎞ ⎛ x ⎞ ⎛ 1 ⎞⎞ ⎛ x⎞ y ′ = 2 ⋅ ⎜ − sin ⎜ ⎟ ⋅ ⎟ = − sin ⎜ ⎟ ; y ′′ = − ⎜ cos ⎜ ⎟ ⋅ ⎜ ⎟ ⎟ = − cos ⎜ ⎟ 2 ⎝ 2⎠ 2⎠ ⎝ 2⎠ ⎝ 2 ⎠ ⎝ 2 ⎠⎠ ⎝2⎠ ⎝ ⎝ 3 x ∫ 2 x2 + 1 20. C Consider the cases: I. false if f ( x ) = 1 dx = ( ) 2 = 12 ( ln10 − ln 5) = 12 ln 2 1 3 2 x dx 1 = ln x 2 + 1 2 ∫ 2 x2 + 1 2 19. B 0 = 3 II. This is true by the Mean Value Theorem III. false if the graph of f is a parabola with vertex at x = a+b . 2 Only II must be true. 21. C x = x 2 − 3x + 3 at x = 1 and at x = 3. Area = 22. C ∫1 ( ( 3 )) 4 ⎛ 1 ⎞ − x 2 + 4 x − 3) dx = ⎜ − x3 + 2 x 2 − 3 x ⎟ = ( 1 ⎝ 3 ⎠1 3 x − x 2 − 3 x + 3 dx = ∫ 2 = ln x − ln 3 3 1 = ln x + ln x ⇒ ln x = 1 ⇒ x = e x AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 195 1988 Calculus AB Solutions 23. B By L’Hôpital’s rule (which is no longer part of the AB Course Description), f ( x) f ′( x) f ′(0) cos 0 1 = lim = = = =1 lim x→0 g ( x ) x→0 g ′( x ) g ′(0) 1 1 Alternatively, f ′( x) = cos x and f (0) = 0 ⇒ f ( x) = sin x . Also g ′( x) = 1 and f ( x) sin x g (0) = 0 ⇒ g ( x) = x . Hence lim = lim = 1. x→0 g ( x ) x →0 x 24. C Let y = x ln x and take the ln of each side. ln y = ln x ln x = ln x ⋅ ln x . Take the derivative of y′ 1 1 each side with respect to x. = 2 ln x ⋅ ⇒ y ′ = 2 ln x ⋅ ⋅ x ln x y x x 25. B Use the Fundamental Theorem of Calculus. f ′( x) = 26. E Use the technique of antiderivatives by parts: Let u = x and dv = cos x dx . ∫ 27. E π 2 0 ( x cos x dx = x sin x − ∫ sin x dx ) π 2 0 1 x = ( x sin x + cos x ) π 2 0 = π −1 2 The function is continuous at x = 3 since lim− f ( x) = lim+ f ( x) = 9 = f (3) . Also, the x→3 x→3 derivative as you approach x = 3 from the left is 6 and the derivative as you approach x = 3 from the right is also 6. These two facts imply that f is differentiable at x = 3. The function is clearly continuous and differentiable at all other values of x. 28. C The graph is a V with vertex at x = 3 . The integral gives the sum of the areas of the two triangles that the V forms with the horizontal axis for x from 1 to 4. These triangles have areas of 2 and 0.5 respectively. 29. B This limit gives the derivative of the function f ( x) = tan(3x) . f ′( x) = 3sec 2 (3 x) AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 196 1988 Calculus AB Solutions 30. A Shells (which is no longer part of the AB Course Description) ∑ 2πrh∆x , where r = x, h = e2 x 1 Volume = 2π ∫ xe 2 x dx 0 31. C 32. A 33. A Let y = f ( x) and solve for x. x y x ; xy + y = x ; x( y − 1) = − y ; x = ⇒ f −1 ( x) = y= x +1 1− y 1− x ⎛ x⎞ The period for sin ⎜ ⎟ is ⎝2⎠ 2π = 4π . 1 2 Check the critical points and the endpoints. f ′( x) = 3 x 2 − 6 x = 3 x( x − 2) so the critical points are 0 and 2. x −2 0 2 4 f ( x ) −8 12 8 28 Absolute maximum is at x = 4. 34. D 35. B The interval is x = a to x = c. The height of a rectangular slice is the top curve, f ( x) , minus the bottom curve, g ( x) . The area of the rectangular slice is therefore ( f ( x) − g ( x))∆x . Set up a Riemann sum and take the limit as ∆x goes to 0 to get a definite integral. π⎞ ⎛ ⎛ ⎛π⎞ ⎛ π ⎞⎞ 4 cos ⎜ x + ⎟ = 4 ⎜ cos x ⋅ cos ⎜ ⎟ − sin x ⋅ sin ⎜ ⎟ ⎟ 3⎠ ⎝ ⎝3⎠ ⎝ 3 ⎠⎠ ⎝ ⎛ 1 3⎞ = 4 ⎜⎜ cos x ⋅ − sin x ⋅ ⎟ = 2 cos x − 2 3 sin x 2 2 ⎟⎠ ⎝ 36. C 3x − x 2 = x ( 3 − x ) > 0 for 0 < x < 3 Average value = ( ) 1 3 1⎛ 3 1 ⎞ 3 x − x 2 dx = ⎜ x 2 − x3 ⎟ ∫ 3 0 3⎝ 2 3 ⎠ 3 0 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. = 3 2 197 1988 Calculus AB Solutions 37. D Since e x > 0 for all x, the zeros of f ( x) are the zeros of sin x , so x = 0, π , 2π . 38. E ∫ ⎜⎝ x ∫1 ⎛1 ( ln x )2 2 10 x du ⎞ 1 ⎟ dx = ∫ ln x dx = u ⎠ x ⎛ dx ⎞ ⎟ .This is ∫ u du with u = ln x , so the value is x ⎠ ∫ ln x ⎜⎝ +C f ( x) dx = − ∫ 3 3 40. B x 2 + y 2 = z 2 , take the derivative of both sides with respect to t. 2 x ⋅ 10 Divide by 2 and substitute: 4 ⋅ 41. A 42. C 43. B 1 f ( x) dx − ∫ 3 f ( x) dx = 4 − (−7) = 11 dx dy dz + 2y ⋅ = 2z ⋅ dt dt dt dx 1 dx dx + 3⋅ = 5 ⋅1 ⇒ =1 3 dt dt dt The statement makes no claim as to the behavior of f at x = 3 , only the value of f for input arbitrarily close to x = 3 . None of the statements are true. x x 1 x = lim = lim = 1. 1 x→∞ 1 x→∞ x + 1 x→∞ x 1+ + x x x None of the other functions have a limit of 1 as x → ∞ lim The cross-sections are disks with radius r = y where y = Volume = π ∫ 44. C f ( x) dx = ∫ 10 ∫3 f ( x) dx ; ∫1 10 39. E 3 −3 y 2 dx = 2π ∫ 3 0 ( ) 1 9 − x2 . 3 1 2π ⎛ 1 3⎞ 9 − x 2 dx = ⎜ 9x − x ⎟ 9 9 ⎝ 3 ⎠ 3 0 = 4π For I: p (− x) = f ( g (− x) ) = f ( − g ( x) ) = − f ( g ( x) ) = − p( x) ⇒ p is odd. For II: r (− x) = f (− x) + g (− x) = − f ( x) − g ( x) = − ( f ( x) + g ( x) ) = − r ( x) ⇒ r is odd. For III: s (− x) = f (− x) ⋅ g (− x) = ( − f ( x) ) ⋅ ( − g ( x) ) = f ( x) ⋅ g ( x) = s( x) ⇒ s is not odd. AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 198 1988 Calculus AB Solutions 45. D ( Volume = π r 2 h = 16π ⇒ h = 16r −2 . A = 2π rh + 2π r 2 = 2π 16 r −1 + r 2 ( ) ( ) ) dA dA dA = 2π −16 r −2 + 2r = 4π r −2 −8 + r 3 ; < 0 for 0 < r < 2 and > 0 for r > 2 dr dr dr The minimum surface area of the can is when r = 2 ⇒ h = 4 . AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 199 1993 AP Calculus AB: Section I 90 Minutes—Scientific Calculator Notes: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number. 1. If f ( x) = 3 x2 , then f ′(4) = (A) –6 2. 3. (B) –3 (C) 3 (D) 6 (E) 8 Which of the following represents the area of the shaded region in the figure above? d (A) ∫c (D) (b − a ) [ f (b) − f (a) ] lim f ( y )dy 3n3 − 5n n→∞ n3 − 2n 2 + 1 (A) –5 b (B) ∫ a ( d − f ( x) ) dx (E) (d − c) [ f (b) − f (a) ] (C) 1 (C) f ′(b) − f ′(a) is (B) –2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 3 (E) nonexistent 78 1993 AP Calculus AB: Section I 4. If x3 + 3 xy + 2 y 3 = 17 , then in terms of x and y, (A) (B) − (C) − (D) − (E) 5. − dy = dx x2 + y x + 2 y2 x2 + y x + y2 x2 + y x + 2y x2 + y 2 y2 − x2 1+ 2 y2 If the function f is continuous for all real numbers and if f ( x) = then f (−2) = (A) –4 6. (C) –1 The area of the region enclosed by the curve y = (A) 7. (B) –2 5 36 (B) ln 2 3 (C) ln (D) 0 (D) x + 13 y = 66 4 3 (D) ln 2 3 2 (E) ln 6 2x + 3 at the point (1,5 ) is 3x − 2 (B) 13x + y = 18 (E) (E) 1 , the x-axis, and the lines x = 3 and x = 4 is x −1 An equation of the line tangent to the graph of y = (A) 13 x − y = 8 x2 − 4 when x ≠ −2 , x+2 (C) x − 13 y = 64 −2 x + 3 y = 13 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 79 1993 AP Calculus AB: Section I 8. If y = tan x − cot x, then (A) sec x csc x 9. dy = dx (B) sec x − csc x (C) sec x + csc x (D) sec2 x − csc2 x (E) sec2 x + csc2 x If h is the function given by h( x) = f ( g ( x)), where f ( x) = 3 x 2 − 1 and g ( x) = x , then h( x) = (A) 3x3 − x (B) 3x 2 − 1 (C) 3x 2 x − 1 (D) 3 x − 1 (E) 3x 2 − 1 (D) 1 (E) 2 10. If f ( x) = ( x − 1) 2 sin x, then f ′(0) = (A) –2 (B) –1 (C) 0 11. The acceleration of a particle moving along the x-axis at time t is given by a (t ) = 6t − 2 . If the velocity is 25 when t = 3 and the position is 10 when t = 1 , then the position x(t ) = (A) 9t 2 + 1 (B) 3t 2 − 2t + 4 (C) t 3 − t 2 + 4t + 6 (D) t 3 − t 2 + 9t − 20 (E) 36t 3 − 4t 2 − 77t + 55 12. If f and g are continuous functions, and if f ( x) ≥ 0 for all real numbers x , which of the following must be true? I. II. III. b ∫a b b f ( x) g ( x)dx = ⎛⎜ ∫ f ( x)dx ⎞⎟ ⎛⎜ ∫ g ( x)dx ⎞⎟ ⎝ a ⎠⎝ a ⎠ b b ∫ a ( f ( x) + g ( x) ) dx = ∫ a b ∫a f ( x) dx = (A) I only b ∫a b f ( x)dx + ∫ g ( x)dx a f ( x)dx (B) II only (C) III only (D) II and III only AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (E) I, II, and III 80 1993 AP Calculus AB: Section I 13. The fundamental period of 2 cos(3x) is (A) 14. ∫ 2π 3 3x 2 x3 + 1 (B) 2π (C) 6π (D) 2 (E) 3 dx = (A) 2 x3 + 1 + C (B) 3 3 x +1 + C 2 (C) x3 + 1 + C (D) ln x3 + 1 + C (E) ln( x3 + 1) + C 15. For what value of x does the function f ( x) = ( x − 2)( x − 3) 2 have a relative maximum? (A) –3 (B) − 7 3 (C) − 5 2 (D) 16. The slope of the line normal to the graph of y = 2 ln(sec x) at x = (A) (E) 5 2 π is 4 −2 (B) − 1 2 (C) 1 2 (D) 2 (E) 7 3 nonexistent AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 81 1993 AP Calculus AB: Section I 17. ∫ (x 2 + 1) 2 dx = (A) ( x 2 + 1)3 +C 3 (B) ( x 2 + 1)3 +C 6x (C) ⎛ x3 ⎞ ⎜⎜ + x ⎟⎟ + C ⎝ 3 ⎠ (D) 2 x( x 2 + 1)3 +C 3 (E) x5 2 x3 + + x+C 5 3 2 π 3π ⎛ x⎞ that satisfies the 18. If f ( x) = sin ⎜ ⎟ , then there exists a number c in the interval < x < 2 2 ⎝2⎠ conclusion of the Mean Value Theorem. Which of the following could be c ? (A) 2π 3 (B) 3π 4 (C) ⎪⎧ x3 19. Let f be the function defined by f ( x) = ⎨ ⎪⎩ x about f is true? (A) f is an odd function. (B) f is discontinuous at x = 0 . (C) f has a relative maximum. (D) f ′(0) = 0 (E) f ′( x) > 0 for x ≠ 0 5π 6 (D) π (E) 3π 2 for x ≤ 0, Which of the following statements for x > 0. AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 82 1993 AP Calculus AB: Section I 1 = ( x + 1) 3 20. Let R be the region in the first quadrant enclosed by the graph of y , the line x = 7 , the x-axis, and the y-axis. The volume of the solid generated when R is revolved about the y -axis is given by (A) π ∫ 7 0 (D) 2π ∫ 2 ( x + 1) 3 dx 2 0 (B) 2π ∫ 1 x( x + 1) 3 dx (B) ln( x 2 − 2 x + 2) (C) ln (D) arcsec( x − 1) (E) arctan( x − 1) 1 x2 − 1 x3 have a point of inflection? (C) 2 x2 − 2x + 2 −( x 2 − 2 x + 2) −2 0 2 ( x + 1) 3 dx 0 1 (A) (C) π ∫ 2 7 (B) 1 22. An antiderivative for 0 1 x( x + 1) 3 dx (E) π ∫ ( y 3 − 1) 2 dy 21. At what value of x does the graph of y = (A) 0 7 (D) 3 (E) At no value of x is x−2 x +1 23. How many critical points does the function f ( x) = ( x + 2)5 ( x − 3) 4 have? (A) One (B) Two (C) Three (D) Five (E) Nine (E) –2 2 24. If f ( x) = ( x 2 − 2 x − 1) 3 , then f ′(0) is (A) 4 3 (B) 0 (C) − 2 3 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) − 4 3 83 1993 AP Calculus AB: Section I 25. ( ) d x 2 = dx 2 x−1 (A) (B) (2 x −1 ) x (C) (2 x ) ln 2 (D) (2 x−1 ) ln 2 (E) 2x ln 2 26. A particle moves along a line so that at time t, where 0 ≤ t ≤ π , its position is given by t2 s (t ) = −4 cos t − + 10 . What is the velocity of the particle when its acceleration is zero? 2 (A) –5.19 (B) 0.74 (C) 1.32 (D) 2.55 (E) 8.13 (D) 46.000 (E) 136.364 (D) 1 (E) nonexistent 27. The function f given by f ( x) = x3 + 12 x − 24 is (A) increasing for x < −2, decreasing for −2 < x < 2, increasing for x > 2 (B) decreasing for x < 0, increasing for x > 0 (C) increasing for all x (D) decreasing for all x (E) decreasing for x < −2, increasing for −2 < x < 2, decreasing for x > 2 28. 500 ∫1 (13x − 11x ) dx + ∫ 2500 (11x − 13x ) dx = (A) 0.000 29. lim θ→0 1 − cos θ 2sin 2 θ (A) 0 (B) 14.946 (C) 34.415 is (B) 1 8 (C) 1 4 30. The region enclosed by the x-axis, the line x = 3 , and the curve y = x is rotated about the x-axis. What is the volume of the solid generated? (A) 3π (B) 2 3π (C) 9 π 2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 9 π (E) 36 3 π 5 84 1993 AP Calculus AB: Section I 2 31. If f ( x) = e3ln( x ) , then f ′( x) = (A) 32. 2 ) dx 3 ∫0 4 − x2 (A) 33. If e3ln( x (B) 3 x e3ln( x 2 2 ) (C) 6(ln x) e3ln( x (C) π 6 2 ) (D) 5x 4 (E) 6x5 (D) 1 ln 2 2 (E) − ln 2 (D) 1 3 (E) 2 3 = π 3 (B) π 4 dy = 2 y 2 and if y = −1 when x = 1, then when x = 2, y = dx (A) − 2 3 (B) − 1 3 (C) 0 34. The top of a 25-foot ladder is sliding down a vertical wall at a constant rate of 3 feet per minute. When the top of the ladder is 7 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall? (A) − 7 feet per minute 8 (B) − 7 feet per minute 24 (C) 7 feet per minute 24 (D) 7 feet per minute 8 (E) 21 feet per minute 25 35. If the graph of y = then a + c = (A) –5 ax + b has a horizontal asymptote y = 2 and a vertical asymptote x = −3 , x+c (B) –1 (C) 0 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 1 (E) 5 85 1993 AP Calculus AB: Section I 36. If the definite integral 2 x2 ∫0 e dx is first approximated by using two inscribed rectangles of equal width and then approximated by using the trapezoidal rule with n = 2 , the difference between the two approximations is (A) 53.60 (B) 30.51 (C) 27.80 (D) 26.80 (E) 12.78 37. If f is a differentiable function, then f ′(a ) is given by which of the following? I. II. III. lim f ( a + h) − f ( a ) h lim f ( x) − f (a) x−a lim f ( x + h) − f ( x ) h h →0 x →a x →a (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III 38. If the second derivative of f is given by f ′′( x) = 2 x − cos x , which of the following could be f ( x) ? (A) x3 + cos x − x + 1 3 (B) x3 − cos x − x + 1 3 (C) x3 + cos x − x + 1 (D) x 2 − sin x + 1 (E) x 2 + sin x + 1 39. The radius of a circle is increasing at a nonzero rate, and at a certain instant, the rate of increase in the area of the circle is numerically equal to the rate of increase in its circumference. At this instant, the radius of the circle is (A) 1 π (B) 1 2 (C) 2 π AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 1 (E) 2 86 1993 AP Calculus AB: Section I 40. The graph of y = f ( x) is shown in the figure above. Which of the following could be the graph of y = f 41. ( x )? d x cos(2π u ) du is dx ∫ 0 (A) 0 (B) 1 sin x 2π (C) 1 cos(2πx) 2π (D) cos(2πx) (E) 2π cos(2πx) 42. A puppy weighs 2.0 pounds at birth and 3.5 pounds two months later. If the weight of the puppy during its first 6 months is increasing at a rate proportional to its weight, then how much will the puppy weigh when it is 3 months old? (A) 4.2 pounds (B) 4.6 pounds (C) 4.8 pounds AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 5.6 pounds (E) 6.5 pounds 87 1993 AP Calculus AB: Section I 43. ∫ x f ( x) dx = (A) x f ( x) − ∫ x f ′( x) dx (B) x2 x2 f ( x) − ∫ f ′( x)dx 2 2 (C) x f ( x) − (D) x f ( x) − ∫ f ′( x) dx (E) x2 2 ∫ x2 f ( x) + C 2 f ( x) dx 44. What is the minimum value of f ( x) = x ln x ? (A) −e (B) −1 (C) − 1 e (D) 0 (E) f ( x) has no minimum value. 45. If Newton’s method is used to approximate the real root of x3 + x − 1 = 0 , then a first approximation x1 = 1 would lead to a third approximation of x3 = (A) 0.682 (B) 0.686 (C) 0.694 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 0.750 (E) 1.637 88 1993 Calculus AB Solutions 1 1 1. C 2. B 3 3 3 f ′( x) = x 2 ; f ′(4) = ⋅ 4 2 = ⋅ 2 = 3 2 2 2 Summing pieces of the form: (vertical) ⋅ (small width) , vertical = ( d − f ( x) ) , width = ∆x Area = b ∫ a ( d − f ( x) ) dx 3n3 − 5n 3− 5 n2 3. D Divide each term by n3 . lim 4. A 3x 2 + 3 ( y + x ⋅ y′ ) + 6 y 2 ⋅ y′ = 0; y′(3 x + 6 y 2 ) = −(3 x 2 + 3 y ) n→∞ n3 y′ = − 3x2 + 3 y 3x + 6 y 2 =− − 2n 2 + 1 = lim n→∞ 2 1 1− + 3 n n =3 x2 + y x + 2 y2 5. A x2 − 4 ( x + 2)( x − 2) lim = lim = lim ( x − 2) = −4. For continuity f (−2) must be –4. x→−2 x + 2 x→−2 x→−2 x+2 6. D Area = ∫ 7. B y′ = 8. E y′ = sec 2 x + csc2 x 9. E h( x ) = f 4 3 1 dx = ( ln x − 1 x −1 2 ⋅ (3 x − 2) − (2 x + 3) ⋅ 3 (3x − 2) 2 ( x )=3 x 2 ) 3 = ln 3 − ln 2 = ln 32 4 ; y′(1) = −13 . Tangent line: y − 5 = −13( x − 1) ⇒ 13x + y = 18 − 1 = 3x2 − 1 10. D f ′( x) = 2( x − 1) ⋅ sin x + ( x − 1) 2 cos x ; f ′(0) = (−2) ⋅ 0 + 1⋅1 = 1 11. C a (t ) = 6t − 2; v(t ) = 3t 2 − 2t + C and v(3) = 25 ⇒ 25 = 27 − 6 + C ; v(t ) = 3t 2 − 2t + 4 x(t ) = t 3 − t 2 + 4t + K ; Since x(1) = 10, K = 6; x(t ) = t 3 − t 2 + 4t + 6 . AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 206 1993 Calculus AB Solutions 12. B The only one that is true is II. The others can easily been seen as false by examples. For example, let f ( x) = 1 and g ( x) = 1 with a = 0 and b = 2. Then I gives 2 = 4 and III gives 2 = 2 , both false statements. 2π 2π = B 3 13. A period = 14. A Let u = x3 + 1. Then 15. D f ′( x) = ( x − 3) 2 + 2( x − 2)( x − 3) = ( x − 3)(3 x − 7); f ′( x) changes from positive to negative at 7 x= . 3 16. B 17. E 18. D 19. E 3x 2 ∫ 3 x +1 dx = ∫ u −1/ 2 du = 2u1/ 2 + C = 2 x3 + 1 + C sec x tan x = 2 tan x; y′(π 4) = 2 tan(π 4) = 2 . The slope of the normal line sec x 1 1 − =− y′(π 4) 2 y′ = 2 Expand the integrand. ∫ (x 2 + 1) 2 dx = ∫ ( x 4 + 2 x 2 + 1) dx = 1 5 2 3 x + x + x+C 5 3 ⎛ 3π ⎞ ⎛π⎞ ⎛ 3π ⎞ ⎛ π⎞ f ⎜ ⎟ − f ⎜ ⎟ sin ⎜ ⎟ − sin ⎜ ⎟ 2 ⎝2⎠ = ⎝ 4 ⎠ ⎝4⎠ = 0 . Want c so that f ′(c) = ⎝ ⎠ 3π π π π − 2 2 1 ⎛c⎞ f ′(c) = cos ⎜ ⎟ = 0 ⇒ c = π 2 ⎝2⎠ The only one that is true is E. A consideration of the graph of y = f ( x) , which is a standard cubic to the left of 0 and a line with slope 1 to the right of 0, shows the other options to be false. AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 207 1993 Calculus AB Solutions 20. B Use Cylindrical Shells which is no part of the AP Course Description. The volume of each 7 1 3 shell is of the form (2π rh) ∆x with r = x and h = y. Volume = 2π ∫ x ( x + 1) dx . 0 21. C y = x −2 − x −3 ; y′ = −2 x −3 + 3 x −4 ; y′′ = 6 x −4 − 12 x −5 = 6 x −5 ( x − 2) . The only domain value at which there is a sign change in y′′ is x = 2 . Inflection point at x = 2 . 22. E ∫ 23. C A quick way to do this problem is to use the effect of the multiplicity of the zeros of f on the graph of y = f ( x) . There is point of inflection and a horizontal tangent at x = −2 . There is a horizontal tangent and turning point at x = 3 . There is a horizontal tangent on the interval (−2,3) . Thus, there must be 3 critical points. Also, f ′( x) = ( x − 3)3 ( x + 2) 4 (9 x − 7) . 1 2 x − 2x + 2 dx = ∫ ( 1 2 ( x − 2 x + 1) + 1 ) − 1 3 1 dx = ∫ ( x − 1) 2 +1 dx = tan −1 ( x − 1) + C 24. A 2 f ′( x) = x 2 − 2 x − 1 3 25. C d x (2 ) = 2 x ⋅ ln 2 dx 26. D v(t ) = 4sin t − t ; a (t ) = 4 cos t − 1 = 0 at t = cos −1 (1 4) = 1.31812; v(1.31812) = 2.55487 27. C f ′( x) = 3 x 2 + 12 > 0 . Thus f is increasing for all x. 28. B ∫1 500 (13x − 11x ) dx + ∫ 500 2 ( 2x − 2) , 2 4 f ′(0) = ⋅ (−1) ⋅ (−2) = 3 3 (11x − 13x ) dx = ∫ ⎛ 13x 11x ⎞ 2 = ∫ (13x − 11x ) dx = ⎜ − ⎜ ln13 ln11 ⎟⎟ 1 ⎝ ⎠ 2 1 = 500 1 (13x − 11x ) dx − ∫ 500 2 (13x − 11x ) dx 132 − 13 112 − 11 − = 14.946 ln13 ln11 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 208 1993 Calculus AB Solutions 29. C Use L’Hôpital’s Rule (which is no longer part of the AB Course Description). lim θ→0 1 − cos θ 2 2sin θ sin θ 1 1 = lim = θ→0 4sin θ cos θ θ→0 4 cos θ 4 = lim A way to do this without L’Hôpital’s rule is the following lim θ→0 30. C 1 − cos θ 2sin 2 θ = lim 1 − cos θ θ→0 2(1 − cos 2 θ) 1 − cos θ 1 1 = lim = θ→0 2(1 − cos θ)(1 + cos θ) θ→0 2(1 + cos θ) 4 = lim Each slice is a disk whose volume is given by π r 2 ∆x , where r = x . 3 3 0 0 Volume = π∫ ( x ) 2 dx = π ∫ x dx = 2 π 2 x 2 f ( x) = e3ln( x ) = eln( x ) = x6 ; f ′( x) = 6 x5 32. A ∫ 33. B 0 = 9 π. 2 6 31. E ∫0 3 ⎛u⎞ = sin −1 ⎜ ⎟ + C , a > 0 ⎝a⎠ a2 − u 2 du 3 ⎛ x⎞ = sin −1 ⎜ ⎟ 2 ⎝2⎠ 4− x dx 3 0 ⎛ 3⎞ π −1 = sin −1 ⎜⎜ ⎟⎟ − sin (0) = 3 ⎝ 2 ⎠ 1 −1 = 2x + C ; y = . Substitute the point (1, −1) y 2x + C −1 1 1 ⇒ C = −1, so y = . When x = 2, y = − . to find the value of C. Then −1 = 2+C 1− 2x 3 Separate the variables. y −2 dy = 2dx ; − 34. D Let x and y represent the horizontal and vertical sides of the triangle formed by the ladder, the wall, and the ground. dx dy dx dx 7 x 2 + y 2 = 25; 2 x + 2 y = 0; 2(24) + 2(7)(−3) = 0; = . dt dt dt dt 8 35. E For there to be a vertical asymptote at x = −3 , the value of c must be 3. For y = 2 to be a horizontal asymptote, the value of a must be 2. Thus a + c = 5 . 36. D Rectangle approximation = e0 + e1 = 1 + e ( ) Trapezoid approximation. = 1 + 2e + e 4 / 2 . Difference = (e 4 − 1) / 2 = 26.799 . AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 209 1993 Calculus AB Solutions 37. C I and II both give the derivative at a. In III the denominator is fixed. This is not the derivative of f at x = a . This gives the slope of the secant line from ( a , f (a ) ) to ( a + h , f (a + h) ) . 38. A 1 f ′( x) = x 2 − sin x + C , f ( x) = x3 + cos x + Cx + K . Option A is the only one with this form. 3 39. D A = π r 2 and C =2π r ; 40. C The graph of y = f x > 0 , x and x dA dr dC dr dA dC = 2π r and = 2π . For = , r = 1. dt dt dt dt dt dt ( x ) is symmetric to the y-axis. This leaves only options C and E. For are the same, so the graphs of f ( x ) and f ( x ) must be the same. This is option C. 41. D Answer follows from the Fundamental Theorem of Calculus. t 42. B ⎛ 3.5 ⎞ 2 This is an example of exponential growth. We know from pre-calculus that w = 2 ⎜ ⎟ is ⎝ 2 ⎠ an exponential function that meets the two given conditions. When t = 3 , w = 4.630 . Using calculus the student may translate the statement “increasing at a rate proportional to its weight” to mean exponential growth and write the equation w = 2e kt . Using the given conditions, 3.5 = 2e 43. B t⋅ ln(1.75) ; w = 2e ; ln(1.75) = 2k ; k = 2 ln(1.75) 2 . When t = 3 , w = 4.630 . Use the technique of antiderivative by parts, which is no longer in the AB Course Description. The formula is ∫ u dv = uv − ∫ v du . Let u = f ( x) and dv = x dx. This leads to 1 ∫ x f ( x) dx = 2 x 44. C 2k 2 f ( x) − 1 x 2 f ′( x) dx . 2∫ 1 f ′( x) = ln x + x ⋅ ; f ′( x) changes sign from negative to positive only at x = e −1 . x 1 f (e−1 ) = −e−1 = − . e AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 210 1993 Calculus AB Solutions 45. B Let f ( x) = x3 + x − 1 . Then Newton’s method (which is no longer part of the AP Course Description) gives xn+1 = xn − x2 = 1 − f ( xn ) x 3 + x −1 = xn − n 2 n f ′( xn ) 3xn + 1 1+1−1 3 = 3 +1 4 3 ⎛3⎞ 3 + −1 3 ⎜⎝ 4 ⎟⎠ 4 59 x3 = − = = 0.686 2 4 86 ⎛3⎞ 3⎜ ⎟ +1 ⎝4⎠ AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 211 1997 AP Calculus AB: Section I, Part A 50 Minutes—No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number. 1. 2. 2 3 (A) (B) (C) (D) (E) 2 4 6 36 42 ∫ 1 (4 x If f ( x) = x 2 x − 3, then f ′( x) = 3x − 3 (A) 2x − 3 x (B) 2x − 3 1 (C) 2x − 3 −x + 3 (D) 2x − 3 5x − 6 (E) 3. If 2 2x − 3 b ∫a (A) 4. − 6 x) dx = f ( x) dx = a + 2b, then a + 2b + 5 (B) b ∫ a ( f ( x) + 5) dx = 5b − 5a (C) 7b − 4 a (D) 7b − 5a (C) –1 (D) –3 (E) 7b − 6 a 1 If f ( x) = − x3 + x + , then f ′(−1) = x (A) 3 (B) 1 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (E) –5 100 1997 AP Calculus AB: Section I, Part A 5. The graph of y = 3 x 4 − 16 x3 + 24 x 2 + 48 is concave down for (A) x<0 (B) x>0 (C) x < −2 or x > − (D) x< (E) 2 <x<2 3 2 3 2 or x > 2 3 t 6. 1 2 e dt = 2∫ (A) 7. −t e +C (B) e − t 2 +C (C) t 2 e +C (D) t 2 2e +C (E) et + C d cos 2 ( x3 ) = dx (A) 6 x 2 sin( x3 ) cos( x3 ) (B) 6 x 2 cos( x3 ) (C) sin 2 ( x3 ) (D) −6 x 2 sin( x3 ) cos( x3 ) (E) −2sin( x3 ) cos( x3 ) AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 101 1997 AP Calculus AB: Section I, Part A Questions 8-9 refer to the following situation. A bug begins to crawl up a vertical wire at time t = 0 . The velocity v of the bug at time t, 0 ≤ t ≤ 8 , is given by the function whose graph is shown above. 8. At what value of t does the bug change direction? (A) 2 9. (B) 4 (C) 6 (D) 7 (E) 8 What is the total distance the bug traveled from t = 0 to t = 8 ? (A) 14 (B) 13 (C) 11 (D) 8 10. An equation of the line tangent to the graph of y = cos(2 x) at x = (A) π⎞ ⎛ y −1 = − ⎜ x − ⎟ 4⎠ ⎝ (B) π⎞ ⎛ y − 1 = −2 ⎜ x − ⎟ 4⎠ ⎝ (C) π⎞ ⎛ y = 2⎜ x − ⎟ 4⎠ ⎝ (D) π⎞ ⎛ y = −⎜ x − ⎟ 4⎠ ⎝ (E) π⎞ ⎛ y = −2 ⎜ x − ⎟ 4⎠ ⎝ AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (E) 6 π is 4 102 1997 AP Calculus AB: Section I, Part A 11. The graph of the derivative of f is shown in the figure above. Which of the following could be the graph of f ? 12. At what point on the graph of y = (A) ⎛1 1⎞ ⎜ ,− ⎟ ⎝2 2⎠ ⎛1 1⎞ (B) ⎜ , ⎟ ⎝ 2 8⎠ 1 2 x is the tangent line parallel to the line 2 x − 4 y = 3 ? 2 (C) 1⎞ ⎛ ⎜1, − ⎟ 4⎠ ⎝ AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) ⎛ 1⎞ ⎜1, ⎟ ⎝ 2⎠ (E) ( 2, 2 ) 103 1997 AP Calculus AB: Section I, Part A 13. Let f be a function defined for all real numbers x. If f ′( x) = 4 − x2 x−2 , then f is decreasing on the interval (A) ( −∞, 2 ) (B) ( −∞, ∞ ) (C) ( −2, 4 ) (D) ( −2, ∞ ) (E) ( 2, ∞ ) 14. Let f be a differentiable function such that f (3) = 2 and f ′(3) = 5 . If the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f, that approximation is (A) 0.4 (B) 0.5 (C) 2.6 (D) 3.4 (E) 5.5 15. The graph of the function f is shown in the figure above. Which of the following statements about f is true? (A) (B) (C) (D) (E) lim f ( x) = lim f ( x) x →a x→b lim f ( x) = 2 x →a lim f ( x) = 2 x→b lim f ( x) = 1 x→b lim f ( x) does not exist. x →a AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 104 1997 AP Calculus AB: Section I, Part A 16. The area of the region enclosed by the graph of y = x 2 + 1 and the line y = 5 is (A) 14 3 (B) 16 3 17. If x 2 + y 2 = 25 , what is the value of (A) 18. ∫ π 4 0 − 25 27 e tan x cos 2 x (B) − 7 27 (C) d2y dx 2 28 3 (D) 32 3 (E) 8π at the point ( 4,3) ? (C) 7 27 (D) 3 4 (E) 25 27 (C) e −1 (D) e (E) e +1 dx is (A) 0 (B) 1 19. If f ( x) = ln x 2 − 1 , then f ′( x) = (A) (B) (C) (D) (E) 2x x2 − 1 2x 2 x −1 2 x x2 − 1 2x 2 x −1 1 2 x −1 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 105 1997 AP Calculus AB: Section I, Part A 20. The average value of cos x on the interval [ −3,5] is 21. (A) sin 5 − sin 3 8 (B) sin 5 − sin 3 2 (C) sin 3 − sin 5 2 (D) sin 3 + sin 5 2 (E) sin 3 + sin 5 8 x is x→1 ln x lim (A) 0 (B) 1 e (C) 1 (D) e (E) nonexistent 22. What are all values of x for which the function f defined by f ( x) = ( x 2 − 3)e − x is increasing? (A) There are no such values of x . (B) x < −1 and x > 3 (C) −3 < x < 1 (D) −1 < x < 3 (E) All values of x 23. If the region enclosed by the y-axis, the line y = 2 , and the curve y = x is revolved about the y-axis, the volume of the solid generated is (A) 32π 5 (B) 16π 3 (C) 16π 5 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 8π 3 (E) π 106 1997 AP Calculus AB: Section I, Part A 24. The expression 25. 1 1 ⎛ 1 2 3 50 ⎞ + + + ⋅⋅⋅ + ⎜⎜ ⎟ is a Riemann sum approximation for 50 ⎝ 50 50 50 50 ⎟⎠ x dx 50 (A) ∫0 (B) ∫0 (C) 1 1 x dx 50 ∫ 0 50 (D) 1 1 x dx 50 ∫ 0 (E) 1 50 x dx 50 ∫ 0 1 x dx ∫ x sin(2 x) dx = (A) 1 x − cos(2 x) + sin(2 x) + C 2 4 (B) 1 x − cos(2 x) − sin(2 x) + C 2 4 (C) 1 x cos(2 x) − sin(2 x) + C 2 4 (D) −2 x cos(2 x) + sin(2 x) + C (E) −2 x cos(2 x) − 4sin(2 x) + C AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 107 1997 AP Calculus AB: Section I, Part B 40 Minutes—Graphing Calculator Required Notes: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number. e2 x 76. If f ( x) = , then f ′( x) = 2x (A) 1 (B) (C) (D) (E) e 2 x (1 − 2 x) 2 x2 e 2x e 2 x (2 x + 1) x2 e 2 x (2 x − 1) 2 x2 77. The graph of the function y = x3 + 6 x 2 + 7 x − 2 cos x changes concavity at x = (A) –1.58 (B) –1.63 (C) –1.67 78. The graph of f is shown in the figure above. If 3 ∫1 (D) –1.89 (E) –2.33 f ( x) dx = 2.3 and F ′( x) = f ( x), then F (3) − F (0) = (A) 0.3 (B) 1.3 (C) 3.3 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 4.3 (E) 5.3 108 1997 AP Calculus AB: Section I, Part B 79. Let f be a function such that lim h →0 I. f (2 + h) − f (2) = 5 . Which of the following must be true? h f is continuous at x = 2. II. f is differentiable at x = 2. III. The derivative of f is continuous at x = 2 . (A) I only (B) II only (C) I and II only (D) I and III only (E) II and III only 2 80. Let f be the function given by f ( x) = 2e 4 x . For what value of x is the slope of the line tangent to the graph of f at ( x, f ( x) ) equal to 3? (A) 0.168 (B) 0.276 (C) 0.318 (D) 0.342 (E) 0.551 81. A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and watches an eastbound train traveling at 60 meters per second. At how many meters per second is the train moving away from the observer 4 seconds after it passes through the intersection? (A) 57.60 (B) 57.88 (C) 59.20 (D) 60.00 (E) 67.40 82. If y = 2 x − 8 , what is the minimum value of the product xy ? (A) –16 (B) –8 (C) –4 (D) 0 (E) 2 83. What is the area of the region in the first quadrant enclosed by the graphs of y = cos x, y = x, and the y-axis? (A) 0.127 (B) 0.385 (C) 0.400 (D) 0.600 (E) 0.947 84. The base of a solid S is the region enclosed by the graph of y = ln x , the line x = e, and the x-axis. If the cross sections of S perpendicular to the x-axis are squares, then the volume of S is (A) 1 2 (B) 2 3 (C) 1 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 2 (E) 1 3 (e − 1) 3 109 1997 AP Calculus AB: Section I, Part B 85. If the derivative of f is given by f ′( x) = e x − 3 x 2 , at which of the following values of x does f have a relative maximum value? (A) –0.46 (B) 0.20 (C) 0.91 (D) 0.95 (E) 3.73 86. Let f ( x) = x . If the rate of change of f at x = c is twice its rate of change at x = 1 , then c = (A) 1 4 (B) 1 (C) 4 (D) 1 2 (E) 1 2 2 87. At time t ≥ 0 , the acceleration of a particle moving on the x-axis is a (t ) = t + sin t . At t = 0 , the velocity of the particle is –2. For what value t will the velocity of the particle be zero? (A) 1.02 (B) 1.48 (C) 1.85 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 2.81 (E) 3.14 110 1997 AP Calculus AB: Section I, Part B 88. Let f ( x) = ∫ x a h(t ) dt , where h has the graph shown above. Which of the following could be the graph of f ? AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 111 1997 AP Calculus AB: Section I, Part B x 0 0.5 1.0 1.5 2.0 f ( x) 3 3 5 8 13 89. A table of values for a continuous function f is shown above. If four equal subintervals of [ 0, 2] are used, which of the following is the trapezoidal approximation of (A) 8 (B) 12 (C) 16 (D) 24 2 ∫0 f ( x) dx ? (E) 32 90. Which of the following are antiderivatives of f ( x) = sin x cos x ? I. F ( x) = sin 2 x 2 II. F ( x) = cos 2 x 2 III. F ( x) = − cos(2 x) 4 (A) I only (B) II only (C) III only (D) I and III only (E) II and III only AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 112 1997 Calculus AB Solutions: Part A 1. C 2 ∫1 (4 x3 − 6 x) dx = ( x 4 − 3 x 2 ) f ( x) = 1 x(2 x − 3) 2 ; 2 1 = (16 − 12) − (1 − 3) = 6 1 f ′( x) = (2 x − 3) 2 + x(2 x − 3) − 1 2 = (2 x − 3) − 1 2 (3 x − 3) = (3x − 3) 2x − 3 2. A 3. C ∫a 4. D 1 1 1 = −3 + 1 − 1 = −3 f ( x) = − x3 + x + ; f ′( x) = −3 x 2 + 1 − 2 ; f ′(−1) = −3(−1) 2 + 1 − x x (−1) 2 5. E b b b a a ( f ( x) + 5) dx = ∫ f ( x)dx + 5∫ 1 dx = a + 2b + 5(b − a) = 7b − 4a y = 3 x 4 − 16 x3 + 24 x 2 + 48; y′ = 12 x3 − 48 x 2 + 48 x; y′′ = 36 x 2 − 96 x + 48 = 12(3x − 2)( x − 2) 2 2 y′′ < 0 for < x < 2, therefore the graph is concave down for < x < 2 3 3 t t 6. C 1 2 e dt = e 2 + C 2∫ 7. D d ⎛d ⎞ ⎛d ⎞ cos 2 ( x3 ) = 2 cos( x3 ) ⎜ (cos( x3 ) ⎟ = 2 cos( x3 )(− sin( x3 ) ⎜ ( x3 ) ⎟ dx ⎝ dx ⎠ ⎝ dx ⎠ = 2 cos( x3 )(− sin( x3 )(3 x 2 ) 8. C The bug change direction when v changes sign. This happens at t = 6 . 9. B Let A1 be the area between the graph and t-axis for 0 ≤ t ≤ 6 , and let A 2 be the area between the graph and the t-axis for 6 ≤ t ≤ 8 Then A1 = 12 and A 2 = 1 . The total distance is A1 + A 2 = 13 . 10. E π⎞ ⎛π⎞ ⎛π⎞ ⎛ y = cos(2 x); y′ = −2sin(2 x); y′ ⎜ ⎟ = −2 and y ⎜ ⎟ = 0; y = −2 ⎜ x − ⎟ 4⎠ ⎝4⎠ ⎝4⎠ ⎝ 11. E Since f ′ is positive for −2 < x < 2 and negative for x < −2 and for x > 2, we are looking for a graph that is increasing for −2 < x < 2 and decreasing otherwise. Only option E. 12. B y= 1 2 1 1 ⎛1 1⎞ x ; y′ = x; We want y′ = ⇒ x = . So the point is ⎜ , ⎟ . 2 2 2 ⎝ 2 8⎠ AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 217 1997 Calculus AB Solutions: Part A 4 − x2 ; f is decreasing when f ′ < 0 . Since the numerator is non-negative, this is x−2 only when the denominator is negative. Only when x < 2 . 13. A f ′( x) = 14. C f ( x) ≈ L( x) = 2 + 5( x − 3); L( x) = 0 if 0 = 5 x − 13 ⇒ x = 2.6 15. B Statement B is true because lim− f ( x) = 2 = lim+ f ( x) . Also, lim f ( x) does not exist x →a x→b x→a because the left- and right-sided limits are not equal, so neither (A), (C), nor (D) are true. 16. D 17. A 18. C 19. D 20. E 1 3 2 ∫ −2 (5 − ( x + 1))dx = 2(4 x − 3 x ) 2 The area of the region is given by 2 ⎛ 8 ⎞ 32 = 2⎜8 − ⎟ = 0 ⎝ 3⎠ 3 4 x 2 + y 2 = 25; 2 x + 2 y ⋅ y′ = 0; x + y ⋅ y′ = 0; y′(4,3) = − ; 3 25 ⎛ 4⎞ ⎛ 4⎞ x + y ⋅ y′ = 0 ⇒ 1+y ⋅ y′′ + y′ ⋅ y′ = 0; 1 + (3) y′′ + ⎜ − ⎟ ⋅ ⎜ − ⎟ = 0; y′′ = − 27 ⎝ 3⎠ ⎝ 3⎠ ∫ π 4 0 ∫ π 4 0 e tan x 2 cos x e tan x 2 cos x dx is of the form dx = e tan x f ( x) = ln x 2 − 1 ; π 4 0 ∫e u du where u = tan x. . = e1 − e0 = e − 1 f ′( x) = 1 d 2 2x ( x − 1) = 2 x − 1 dx x −1 2 ⋅ 1 5 1 1 cos x dx = (sin 5 − sin(−3)) = (sin 5 + sin 3) ; Note: Since the sine is an odd function, ∫ 8 −3 8 8 sin(−3) = − sin(3) . x is nonexistent since lim ln x = 0 and lim x ≠ 0 . x→1 ln x x→1 x→1 21. E lim 22. D f ( x) = ( x 2 − 3)e − x ; f ′( x) = e− x (− x 2 + 2 x + 3) = −e− x ( x − 3)( x + 1); f ′( x) > 0 for − 1 < x < 3 23. A 2 2 0 0 Disks where r = x . V = π ∫ x 2 dy = π ∫ y 4 dy = π 5 y 5 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 2 0 = 32π 5 218 1997 Calculus AB Solutions: Part A 24. B Let [ 0,1] be divided into 50 subintervals. ∆x = Using f ( x) = x , the right Riemann sum 1 1 2 3 ; x1 = , x2 = , x3 = , ⋅⋅⋅, x50 = 1 50 50 50 50 50 ∑ f ( xi )∆x is an approximation for i =1 25. A 1 ∫0 x dx . Use the technique of antiderivatives by parts, which was removed from the AB Course Description in 1998. u=x du = dx dv = sin 2 x dx 1 v = − cos 2 x 2 1 1 1 1 ∫ x sin(2 x) dx = − 2 x cos(2 x) + ∫ 2 cos(2 x) dx = − 2 x cos(2 x) + 4 sin(2 x) + C AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 219 1997 Calculus AB Solutions: Part B 76. E f ( x) = e2 x 2e2 x ⋅ 2 x − 2e2 x e 2 x (2 x − 1) ; f ′( x) = = 2x 4x2 2 x2 77. D y = x3 + 6 x 2 + 7 x − 2 cos x . Look at the graph of y′′ = 6 x + 12 + 2 cos x in the window [–3,–1] since that domain contains all the option values. y′′ changes sign at x = −1.89 . 78. D F (3) − F (0) = ∫ f ( x)dx = ∫ f ( x)dx + ∫ f ( x)dx = 2 + 2.3 = 4.3 3 1 3 0 0 1 ( Count squares for 1 ∫ 0 f ( x)dx ) 79. C The stem of the questions means f ′(2) = 5 . Thus f is differentiable at x = 2 and therefore continuous at x = 2. We know nothing of the continuity of f ′ . I and II only. 80. A f ( x) = 2e4 x ; f ′( x) = 16 xe4 x ; We want 16 xe4 x = 3. Graph the derivative function and the function y = 3, then find the intersection to get x = 0.168 . 81. A Let x be the distance of the train from the dx crossing. Then = 60 . dt dS dx dS x dx S 2 = x 2 + 702 ⇒ 2 S = 2x ⇒ = . dt dt dt S dt After 4 seconds, x = 240 and so S = 250 . dS 240 = (60) = 57.6 Therefore dt 250 82. B P ( x) = 2 x 2 − 8 x; P′( x) = 4 x − 8; P′ changes from negative to positive at x = 2. P (2) = −8 83. C cos x = x at x = 0.739085. Store this in A. 84. C Cross sections are squares with sides of length y. 2 2 e e 1 1 2 Volume = ∫ y 2 dx = ∫ ln x dx = (x ln x − x) 85. C 86. A A ∫ 0 (cos x − x) dx = 0.400 e 1 = (e ln e − e) − (0 − 1) = 1 Look at the graph of f ′ and locate where the y changes from positive to negative. x = 0.91 f ( x) = x ; f ′( x) = 1 2 x ; 1 2 c = 2⋅ 1 2 1 ⇒ c= 1 4 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 220 1997 Calculus AB Solutions: Part B 87. B 1 a(t ) = t + sin t and v(0) = −2 ⇒ v(t ) = t 2 − cos t − 1; v(t ) = 0 at t = 1.48 2 88. E f ( x) = ∫ h( x)dx ⇒ f (a) = 0, therefore only (A) or (E) are possible. But f ′( x) = h( x) and x a therefore f is differentiable at x = b. This is true for the graph in option (E) but not in option (A) where there appears to be a corner in the graph at x = b. Also, Since h is increasing at first, the graph of f must start out concave up. This is also true in (E) but not (A). 89. B 90. D 1 1 T = ⋅ (3 + 2 ⋅ 3 + 2 ⋅ 5 + 2 ⋅ 8 + 13) = 12 2 2 1 F ( x) = sin 2 x 2 1 F ( x) = cos 2 x 2 1 F ( x) = − cos(2 x) 4 F ′( x) = sin x cos x Yes F ′( x) = − cos x sin x No 1 F ′( x) = sin(2 x) = sin x cos x 2 Yes AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 221 1998 AP Calculus AB: Section I, Part A 55 Minutes—No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number. 1. 1 What is the x-coordinate of the point of inflection on the graph of y = x3 + 5 x 2 + 24 ? 3 (A) 5 2. (B) 0 (C) − 10 3 (D) –5 (E) −10 The graph of a piecewise-linear function f , for −1 ≤ x ≤ 4 , is shown above. What is the value of 4 ∫ −1 f ( x) dx ? (A) 1 3. 2 ∫1 (A) 1 x2 (B) 2.5 (C) 4 (D) 5.5 (E) 8 7 24 (C) 1 2 (D) 1 (E) dx = − 1 2 (B) AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 2 ln 2 125 1998 AP Calculus AB: Section I, Part A 4. 5. If f is continuous for a ≤ x ≤ b and differentiable for a < x < b , which of the following could be false? f (b) − f (a) for some c such that a < c < b. b−a (A) f ′(c) = (B) f ′(c) = 0 for some c such that a < c < b. (C) f has a minimum value on a ≤ x ≤ b. (D) f has a maximum value on a ≤ x ≤ b. (E) ∫a b f ( x) dx exists. x ∫ 0 sin t dt = (A) sin x 6. If x 2 + xy = 10, then when x = 2, (A) 7. e ∫1 (A) 8. (B) − cos x − 7 2 (B) –2 (C) cos x (D) cos x − 1 (E) 1 − cos x dy = dx (C) 2 7 (C) e2 1 −e+ 2 2 (D) 3 2 (E) 7 2 (E) e2 3 − 2 2 (E) 1 ⎛ x2 − 1 ⎞ ⎜⎜ ⎟⎟ dx = x ⎝ ⎠ 1 e− e (B) 2 e −e 2 e −2 (D) Let f and g be differentiable functions with the following properties: (i) (ii) g ( x) > 0 for all x f (0) = 1 If h( x) = f ( x) g ( x) and h′( x) = f ( x) g ′( x), then f ( x) = (A) f ′( x) (B) g ( x) (C) ex AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 0 126 1998 AP Calculus AB: Section I, Part A 9. The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph shown above. Of the following, which best approximates the total number of barrels of oil that passed through the pipeline that day? (A) 500 (B) 600 (C) 2, 400 (D) 3, 000 (E) 10. What is the instantaneous rate of change at x = 2 of the function f given by f ( x) = (A) −2 (B) 1 6 11. If f is a linear function and 0 < a < b, then (A) 0 (B) 1 1 2 (C) (C) b ∫a (D) 2 4,800 x2 − 2 ? x −1 (E) 6 f ′′( x) dx = ab 2 (D) b−a (E) b2 − a 2 2 ⎪⎧ ln x for 0 < x ≤ 2 then lim f ( x) is 12. If f ( x) = ⎨ 2 x →2 ⎪⎩ x ln 2 for 2 < x ≤ 4, (A) ln 2 (B) ln 8 (C) ln16 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 4 (E) nonexistent 127 1998 AP Calculus AB: Section I, Part A 13. The graph of the function f shown in the figure above has a vertical tangent at the point ( 2, 0 ) and horizontal tangents at the points (1, − 1) and ( 3,1) . For what values of x, −2 < x < 4 , is f not differentiable? (A) 0 only (B) 0 and 2 only (C) 1 and 3 only (D) 0, 1, and 3 only (E) 0, 1, 2, and 3 14. A particle moves along the x-axis so that its position at time t is given by x(t ) = t 2 − 6t + 5 . For what value of t is the velocity of the particle zero? (A) 1 15. If F ( x) = ∫ (A) (B) 2 x 0 (C) 3 (D) 4 (E) 5 (C) 2 (D) 3 (E) 18 t 3 + 1 dt , then F ′(2) = −3 (B) −2 ( ) 16. If f ( x) = sin e − x , then f ′(x) = (A) − cos(e − x ) (B) cos(e − x ) + e − x (C) cos(e − x ) − e − x (D) e − x cos(e− x ) (E) −e − x cos(e− x ) AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 128 1998 AP Calculus AB: Section I, Part A 17. The graph of a twice-differentiable function f is shown in the figure above. Which of the following is true? (A) f (1) < f ′ (1) < f ′′ (1) (B) f (1) < f ′′ (1) < f ′ (1) (C) f ′ (1) < f (1) < f ′′ (1) (D) f ′′ (1) < f (1) < f ′ (1) (E) f ′′ (1) < f ′ (1) < f (1) 18. An equation of the line tangent to the graph of y = x + cos x at the point ( 0,1) is (A) y = 2x +1 y = x +1 (B) (C) y=x (D) y = x −1 (E) y=0 19. If f ′′( x) = x ( x + 1)( x − 2 ) , then the graph of f has inflection points when x = 2 (A) –1 only (B) 2 only (C) –1 and 0 only 20. What are all values of k for which (A) –3 21. If (B) 0 k ∫ −3 x 2 (D) –1 and 2 only (E) –1, 0, and 2 only dx = 0 ? (C) 3 (D) –3 and 3 (E) –3, 0, and 3 dy = ky and k is a nonzero constant, then y could be dt (A) 2e kty (B) 2e kt (C) e kt + 3 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) kty + 5 (E) 1 2 1 ky + 2 2 129 1998 AP Calculus AB: Section I, Part A 22. The function f is given by f ( x) = x 4 + x 2 − 2 . On which of the following intervals is f increasing? (A) ⎛ 1 ⎞ , ∞⎟ ⎜− 2 ⎝ ⎠ (B) 1 ⎞ ⎛ 1 , ⎜− ⎟ 2 2⎠ ⎝ (C) ( 0, ∞ ) (D) ( −∞, 0 ) (E) 1 ⎞ ⎛ ⎜ −∞, − ⎟ 2⎠ ⎝ AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 130 1998 AP Calculus AB: Section I, Part A 23. The graph of f is shown in the figure above. Which of the following could be the graph of the derivative of f ? AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 131 1998 AP Calculus AB: Section I, Part A 24. The maximum acceleration attained on the interval 0 ≤ t ≤ 3 by the particle whose velocity is given by v(t ) = t 3 − 3t 2 + 12t + 4 is (A) 9 (B) 12 (C) 14 (D) 21 (E) 40 25. What is the area of the region between the graphs of y = x 2 and y = − x from x = 0 to x = 2? (A) 2 3 (B) 8 3 (C) 4 (D) x 0 1 2 f ( x) 1 k 2 14 3 (E) 16 3 26. The function f is continuous on the closed interval [ 0, 2] and has values that are given in the table above. The equation f ( x) = (A) (B) 0 1 must have at least two solutions in the interval [ 0, 2] if k = 2 1 2 (C) 1 (D) 2 (E) 3 27. What is the average value of y = x 2 x3 + 1 on the interval [ 0, 2] ? (A) 26 9 (B) 52 9 (C) 26 3 (D) 52 3 (E) 24 (C) 4 (D) 4 3 (E) 8 ⎛π⎞ 28. If f ( x) = tan(2 x), then f ′ ⎜ ⎟ = ⎝6⎠ (A) 3 (B) 2 3 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 132 1998 AP Calculus AB: Section I, Part B 50 Minutes—Graphing Calculator Required Notes: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number. 76. The graph of a function f is shown above. Which of the following statements about f is false? (A) f is continuous at x = a . (B) f has a relative maximum at x = a . (C) x = a is in the domain of f. (D) (E) lim f ( x) is equal to lim− f ( x) . x →a + x →a lim f ( x) exists . x →a 77. Let f be the function given by f ( x) = 3e 2 x and let g be the function given by g ( x) = 6 x3 . At what value of x do the graphs of f and g have parallel tangent lines? (A) (B) (C) (D) (E) −0.701 −0.567 −0.391 −0.302 −0.258 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 133 1998 AP Calculus AB: Section I, Part B 78. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the circumference C, what is the rate of change of the area of the circle, in square centimeters per second? (A) − ( 0.2 ) π C (B) − ( 0.1) C (C) − ( 0.1) C 2π (D) ( 0.1)2 C (E) ( 0.1)2 π C 79. The graphs of the derivatives of the functions f, g, and h are shown above. Which of the functions f, g, or h have a relative maximum on the open interval a < x < b ? (A) (B) (C) (D) (E) f only g only h only f and g only f, g, and h 80. The first derivative of the function f is given by f ′( x) = does f have on the open interval ( 0,10 ) ? (A) (B) (C) (D) (E) cos 2 x 1 − . How many critical values x 5 One Three Four Five Seven AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 134 1998 AP Calculus AB: Section I, Part B 81. Let f be the function given by f ( x) = x . Which of the following statements about f are true? f is continuous at x = 0 . f is differentiable at x = 0 . f has an absolute minimum at x = 0 . I. II. III. (A) I only (B) II only (C) III only (D) I and III only (E) II and III only 82. If f is a continuous function and if F ′( x) = f ( x) for all real numbers x, then (A) 2 F (3) − 2 F (1) (B) 1 1 F (3) − F (1) 2 2 (C) 2 F (6) − 2 F (2) (D) F (6) − F (2) (E) 1 1 F (6) − F (2) 2 2 83. If a ≠ 0, then lim x →a (A) 1 a2 x2 − a2 x4 − a4 (B) 3 ∫ 1 f ( 2 x ) dx = is 1 2a 2 (C) 1 6a 2 (D) 0 (E) nonexistent dy = ky , where k is a constant and t is measured in dt years. If the population doubles every 10 years, then the value of k is 84. Population y grows according to the equation (A) 0.069 (B) 0.200 (C) 0.301 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 3.322 (E) 5.000 135 1998 AP Calculus AB: Section I, Part B 2 x f ( x) 5 7 8 10 30 40 20 85. The function f is continuous on the closed interval [ 2,8] and has values that are given in the table above. Using the subintervals [ 2,5] , [5, 7 ] , and [ 7,8] , what is the trapezoidal approximation of 8 ∫ 2 f ( x) dx ? (A) 110 (B) 130 (C) 160 (D) 190 (E) 210 86. The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line x + 2 y = 8 , as shown in the figure above. If cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid? (A) 12.566 (B) 14.661 (C) 16.755 (D) 67.021 (E) 134.041 87. Which of the following is an equation of the line tangent to the graph of f ( x) = x 4 + 2 x 2 at the point where f ′( x) = 1? (A) (B) (C) (D) (E) y = 8x − 5 y = x+7 y = x + 0.763 y = x − 0.122 y = x − 2.146 88. Let F ( x) be an antiderivative of (A) 0.048 (B) 0.144 ( ln x )3 . If x (C) F (1) = 0, then F (9) = 5.827 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 23.308 (E) 1,640.250 136 1998 AP Calculus AB: Section I, Part B 89. If g is a differentiable function such that g ( x) < 0 for all real numbers x and if ( ) f ′( x) = x 2 − 4 g ( x) , which of the following is true? (A) (B) (C) (D) (E) f has a relative maximum at x = −2 and a relative minimum at x = 2 . f has a relative minimum at x = −2 and a relative maximum at x = 2 . f has relative minima at x = −2 and at x = 2 . f has relative maxima at x = −2 and at x = 2 . It cannot be determined if f has any relative extrema. 90. If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is decreasing at a rate of 3 inches per minute, which of the following must be true about the area A of the triangle? (A) (B) (C) (D) (E) A is always increasing. A is always decreasing. A is decreasing only when b < h . A is decreasing only when b > h . A remains constant. 91. Let f be a function that is differentiable on the open interval (1,10 ) . If f (2) = −5, f (5) = 5, and f (9) = −5 , which of the following must be true? I. II. III. (A) (B) (C) (D) (E) f has at least 2 zeros. The graph of f has at least one horizontal tangent. For some c, 2 < c < 5, f (c) = 3 . None I only I and II only I and III only I, II, and III 92. If 0 ≤ k < π π and the area under the curve y = cos x from x = k to x = is 0.1, then k = 2 2 (A) 1.471 (B) 1.414 (C) 1.277 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 1.120 (E) 0.436 137 1998 Calculus AB Solutions: Part A 1. D y′ = x 2 + 10 x ; y′′ = 2 x + 10; y′′ changes sign at x = −5 2. B ∫ −1 f ( x)dx = ∫ −1 f ( x)dx + ∫ 2 4 2 4 f ( x)dx = Area of trapezoid(1) – Area of trapezoid(2) = 4 − 1.5 = 2.5 2 1 2 dx = ∫ x −2 dx = − x −1 2 1 2 3. C ∫1 4. B This would be false if f was a linear function with non-zero slope. 5. E ∫ 0 sin t dt = − cos t 0 = − cos x − (− cos 0) = − cos x + 1 = 1 − cos x 6. A Substitute x = 2 into the equation to find y = 3. Taking the derivative implicitly gives d 2 x + xy = 2 x + xy′ + y = 0 . Substitute for x and y and solve for y′ . dx 7 4 + 2 y′ + 3 = 0; y′ = − 2 x 2 1 x x ( e 1 = ) e e x2 − 1 1 3 ⎛1 ⎞ ⎛1 ⎞ ⎛1 ⎞ 1 dx = ∫ x − dx = ⎜ x 2 − ln x ⎟ = ⎜ e 2 − 1⎟ − ⎜ − 0 ⎟ = e 2 − 1 2 x x ⎝2 ⎠1 ⎝2 ⎠ ⎝2 ⎠ 2 7. E ∫1 8. E h( x) = f ( x) g ( x) so, h′( x) = f ′( x) g ( x) + f ( x) g ′( x) . It is given that h′( x) = f ( x) g ′( x) . Thus, f ′( x) g ( x) = 0 . Since g ( x) > 0 for all x, f ′( x) = 0 . This means that f is constant. It is given that f (0) = 1 , therefore f ( x) = 1 . 9. D Let r (t ) be the rate of oil flow as given by the graph, where t is measured in hours. The total number of barrels is given by 24 ∫0 r (t )dt . This can be approximated by counting the squares below the curve and above the horizontal axis. There are approximately five squares with area 600 barrels. Thus the total is about 3, 000 barrels. 10. D 11. A f ′( x) = ( x − 1)(2 x) − ( x 2 − 2)(1) ( x − 1) 2 ; f ′(2) = (2 − 1)(4) − (4 − 2)(1) (2 − 1) 2 =2 Since f is linear, its second derivative is zero. The integral gives the area of a rectangle with zero height and width (b − a) . This area is zero. AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 228 1998 Calculus AB Solutions: Part A 12. E lim f ( x) = ln 2 ≠ 4 ln 2 = lim+ f ( x) . Therefore the limit does not exist. x →2− x →2 13. B At x = 0 and x = 2 only. The graph has a non-vertical tangent line at every other point in the interval and so has a derivative at each of these other x’s. 14. C v(t ) = 2t − 6; v(t ) = 0 for t = 3 15 By the Fundamental Theorem of Calculus, F ′( x) = x3 + 1, thus F ′(2) = 23 + 1 = 9 = 3 . D 16. E f ′( x) = cos(e− x ) ⋅ d −x d ⎛ ⎞ (e ) = cos(e− x ) ⎜ e− x ⋅ (− x) ⎟ = −e− x cos(e− x ) dx dx ⎝ ⎠ 17. D From the graph f (1) = 0 . Since f ′(1) represents the slope of the graph at x = 1 , f ′(1) > 0 . Also, since f ′′(1) represents the concavity of the graph at x = 1 , f ′′(1) <0 . 18. B y′ = 1 − sin x so y′(0) = 1 and the line with slope 1 containing the point (0,1) is y = x + 1 . 19. C Points of inflection occur where f ′′ changes sign. This is only at x = 0 and x = −1 . There is no sign change at x = 2. 20. A ∫ −3 x 21. B The solution to this differential equation is known to be of the form y = y (0) ⋅ ekt . Option (B) is the only one of this form. If you do not remember the form of the solution, then separate the variables and antidifferentiate. dy = k dt ; ln y = kt + c1; y = ekt +c1 = ekt ec1 ; y = cekt . y 22. C f is increasing on any interval where f ′( x) > 0 . f ′( x) = 4 x3 + 2 x = 2 x(2 x 2 + 1) > 0 . k 2 1 dx = x3 3 k −3 = ( ) ( ) 1 3 1 k − (−3)3 = k 3 + 27 = 0 only when k = −3. 3 3 Since ( x 2 + 1) > 0 for all x, f ′( x) > 0 whenever x > 0 . 23. A The graph shows that f is increasing on an interval (a, c) and decreasing on the interval (c, b) , where a < c < b . This means the graph of the derivative of f is positive on the interval (a, c) and negative on the interval (c, b) , so the answer is (A) or (E). The derivative is not (E), however, since then the graph of f would be concave down for the entire interval. AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 229 1998 Calculus AB Solutions: Part A 24. D 25. D 26. A The maximum acceleration will occur when its derivative changes from positive to negative or at an endpoint of the interval. a (t ) = v′(t ) = 3t 2 − 6t + 12 = 3(t 2 − 2t + 4) which is always positive. Thus the acceleration is always increasing. The maximum must occur at t = 3 where a(3) = 21 The area is given by ∫ 2 2 x 0 1 ⎞ ⎛1 − (− x) dx = ⎜ x3 + x 2 ⎟ 2 ⎠ ⎝3 2 0 Any value of k less than 1/2 will require the function to assume the value of 1/2 at least twice because of the Intermediate Value Theorem on the intervals [0,1] and [1,2]. Hence k = 0 is the only option. 1 27. A 28. E 8 14 = +2= . 3 3 3 2 1 2 2 3 1 2 1 1 2 ⎛1 ⎞ x x + 1 dx = ∫ ( x3 + 1) 2 ⎜ ⋅ 3 x 2 ⎟ dx = ⋅ ⋅ ( x3 + 1) 2 ∫ 0 0 2 2 2 3 3 ⎝3 ⎠ f ′( x) = sec2 (2 x) ⋅ 0 3 3 1 ⎛ 2 2 ⎞ 26 ⎜ = 9 −1 ⎟ = ⎟ 9 9⎜ ⎝ ⎠ d ⎛π⎞ ⎛π⎞ (2 x) = 2sec 2 (2 x); f ′ ⎜ ⎟ = 2sec 2 ⎜ ⎟ = 2(4) = 8 dx ⎝6⎠ ⎝3⎠ AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 230 1998 Calculus AB Solutions: Part B 76. A From the graph it is clear that f is not continuous at x = a . All others are true. 77. C Parallel tangents will occur when the slopes of f and g are equal. f ′( x) = 6e 2 x and g ′( x) = 18 x 2 . The graphs of these derivatives reveal that they are equal only at x = −0.391 . dA dr dr dA = 2πr . However, C = 2πr and = −0.1 . Thus = −0.1C . dt dt dt dt 78. B A = πr 2 ⇒ 79. A The graph of the derivative would have to change from positive to negative. This is only true for the graph of f ′ . 80. B Look at the graph of f ′( x) on the interval (0,10) and count the number of x-intercepts in the interval. 81. D Only II is false since the graph of the absolute value function has a sharp corner at x = 0. 82. E Since F is an antiderivative of f, 83. B 84. A lim x →a x2 − a2 4 x −a 4 = lim x→a ( x 3 ∫1 f (2 x) dx = x2 − a2 2 2 2 2 − a )( x + a ) = lim x→a ( x 1 1 3 F (2 x) 1 = ( F (6) − F (2) ) 2 2 1 2 2 +a ) = 1 2a 2 A known solution to this differential equation is y (t ) = y (0)e k t . Use the fact that the population is 2 y (0) when t = 10. Then 2 y (0) = y (0)ek (10) ⇒ e10 k = 2 ⇒ k = (0.1) ln 2 = 0.069 1 1 1 ⋅ 3 ( f (2) + f (5) ) + ⋅ 2 ( f (5) + f (7) ) + ⋅1( f (7) + f (8) ) 2 2 2 85. C There are 3 trapezoids. 86. C Each cross section is a semicircle with a diameter of y. The volume would be given by 8 ∫0 87. D 2 2 1 ⎛ y⎞ π 8 ⎛8− x ⎞ π ⎜ ⎟ dx = ∫ ⎜ ⎟ dx = 16.755 2 ⎝2⎠ 8 0 ⎝ 2 ⎠ Find the x for which f ′( x) = 1 . f ′( x) = 4 x3 + 4 x = 1 only for x = 0.237 . Then f (0.237) = 0.115 . So the equation is y − 0.115 = x − 0.237 . This is equivalent to option (D). AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 231 1998 Calculus AB Solutions: Part B 88. C F (9) − F (1) = ∫ 9 1 (ln t )3 dt = 5.827 using a calculator. Since F (1) = 0 , F (9) = 5.827. t Or solve the differential equation with an initial condition by finding an antiderivative for (ln x)3 1 . This is of the form u 3du where u = ln x . Hence F ( x) = (ln x) 4 + C and since x 4 1 F (1) = 0 , C = 0. Therefore F (9) = (ln 9)4 = 5.827 4 89. B The graph of y = x 2 − 4 is a parabola that changes from positive to negative at x = −2 and from negative to positive at x = 2 . Since g is always negative, f ′ changes sign opposite to the way y = x 2 − 4 does. Thus f has a relative minimum at x = −2 and a relative maximum at x = 2 . 90. D 91. E 1 The area of a triangle is given by A = bh . Taking the derivative with respect to t of both 2 dA 1 ⎛ db dh ⎞ sides of the equation yields = ⎜ ⋅ h + b ⋅ ⎟ . Substitute the given rates to get dt 2 ⎝ dt dt ⎠ dA 3 dA 1 < 0 . This is true = (3h − 3b) = (h − b) . The area will be decreasing whenever dt 2 dt 2 whenever b > h . I. True. Apply the Intermediate Value Theorem to each of the intervals [2,5] and [5,9] . II. True. Apply the Mean Value Theorem to the interval [2,9] . III. True. Apply the Intermediate Value Theorem to the interval [2,5]. 92. D ∫ π 2 cos x dx k ⎛π⎞ = 0.1 ⇒ sin ⎜ ⎟ − sin k = 0.1 ⇒ sin k = 0.9 . Therefore k = sin −1 (0.9) = 1.120 . ⎝2⎠ AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 232 AP Calculus 2003 Calculators ARE NOT Permitted On This Portion Of The Exam 28 Questions - 55 Minutes 1) Give f(g(1)), given that a) -8/9 b) 7/3 c) 2 d) 4/3 e) -2/9 2) Find the slope of the tangent line to the graph of f at x = 4, given that a) -8 b) -10 c) -9 d) -5 e) -7 3) Determine a) ∞ b) 0 c) ½ d) 3/10 e) 1 4) Let f(x) = x 3 A region is bounded between the graphs of y = -1 and y = f(x) for x between -1 and 0, and between the graphs of y = 1 and y = f(x) for x between 0 and 1. Give an integral that corresponds to the area of this region. a) b) c) d) e) 5) Given that Determine the change in y with respect to x. a) b) c) d) e) 6) Compute the derivative of -4 sec (x) + 2 csc (x) a) b) c) d) e) 7) Compute a) - π b) 3/2 π c) ½ π d) π e) 0 8) Determine a) b) c) d) e) 9) Give the equation of the normal line to the graph of at the point ( 0 , 2 ). a) b) c) d) e) 10) Determine the concavity of the graph of at x = π. a) 8 b) -10 c) 4 d) -8 e) -6 11) Compute a) b) c) d) e) 12) Give the value of x where the function has a local maximum. a) 4 b) -2 c) 2 d) -4 e) 3 13) The slope of the tangent line to the graph of at x = 0 is 4. Give the value of c. a) -2 b) 4 c) 8 d) -4 e) -8 14) Compute a) b) c) d) e) 15) What is the average value of the function x = -3 to x = -1? a) 7/3 b) -4 c) 5 d) 14/3 e) 3 16) Compute a) 1 b) ¼ π c) π g (x) = (2x + 3) 2 on the interval from d) 2 e) -1 17) Find the instantaneous rate of change of at t = 0. a) -3 b) -3/4 c) 0 d) -4 e) -5/4 18) Compute a) b) c) d) e) 19) A solid is generated by rotating the region enclosed by the graph of the lines x = 1, x = 2, and y = 1, about the x-axis. Which of the following integrals gives the volume of the solid? a) b) c) d) e) 20) Compute a) ∞ b) 0 c) -5/2 d) -2 e) undefined 21) Given y > 0 and If the point is on the graph relating x and y, then what is y when x = 0? a) 3 b) 2 c) 1 d) 6 e) 10 22) Determine a) ½ π b) 1/3 π c) π d) 1/6 π e) ¼ π 23) Determine a) b) c) d) e) 24) A particle's acceleration for t > 0 is given by a(t) = 12 t + 4. The particle's initial position is 2 and its velocity at t = 1 is 5. What is the position of the particle at t = 2? a) 10 b) 12 c) 16 d) 4 e) 20 25) Determine a) -1 b) 1 c) 0 d) 2/3 e) -2/3 26) Determine the derivative of at x = /b. a) b) c) d) e) 27) Compute the derivative of a) b) c) d) e) 28) Determine a) b) c) d) e) Calculators ARE Permitted On This Portion Of The Exam 17 Questions - 50 Minutes 1) Give a value of c that satisfies the conclusion of the Mean Value Theorem for Derivatives for the function on the interval [1,3]. a) 9/4 b) 3/2 c) ½ d) 2 e) 5/4 2) The function is invertible. Give the derivative of f -1 at x = 2. a) b) c) d) e) 3) The derivative of f is graphed below. Give a value of x where f has a local maximum. a) -4 b) -1 c) -5/2 d) There is no such value of x e) 1 4) Let Which of the following is (are) true? 1) f is continuous at x = -2. 2) f is differentiable at x = 1. 3) f has a local minimum at x = 0. 4) f has an absolute maximum at x = -2. a) 2 and 4 b) 3 only c) 2 only d) 1 and 3 e) 1 and 4 5) Given Determine a) 10 b) -3 c) There is not enough infomration d) -6 e) 5 6) Give the approximate location of a local maximum for the function a) b) c) d) e) 7) Give the approximate average value of the function f(x) = 4 x ln (2x) over the interval [1,4]. a) 19.71 b) 12.54 c) 16.71 d) 18.02182670 e) 18.71 8) The region enclosed by the graphs of is rotated around the y-axis to generate a solid. What is the volume of the solid? a) 0.8380 b) 0.7855 c) 1.676 d) 1.047 e) 2.356 9) What is the approximate instantaneous rate of change of the function at t = /7? a) -.9009 b) -7.207 c) 3.3473 d) 0.4341 e) -1.030 10) What is the error when the integral is approximated by the Trapezoidal rule with n = 3? a) 0.011 b) 0.032 c) 0.109 d) 0.059 e) 0.051 11) The amount of money in a bank account is increasing at the rate of dollars per year, where t is measured in years. If t = 0 corresponds to the year 2005, then what is the approximate total amount of increase from 2005 to 2007. a) $18,350 b) $4,500 c) $21,250 d) $32,560 e) $16,250 12) A particle moves with acceleration and its initial velocity is 0. For how many values of t does the particle change direction? a) 3 b) 2 c) 1 d) 0 e) 4 13) At what approximate rate (in cubic meters per minute) is the volume of a sphere changing at the instant when the surface area is 5 square meters and the radius is increasing at the rate of 1/3 meters per minute? a) 5.271 b) 1.700 c) 1.667 d) 1.080 e) 2.714 14) A rectangle has one side on the x-axis and the upper two vertices on the graph of Give a decimal approximation to the maximum possible area for this rectangle. a) 1.649 b) 1 c) -1 d) 0.5458 e) 0.6065 15) A rough approximation for ln(5) is 1.609. Use this approximation and differentials to approximate ln(128/25). a) 1.633 b) 1.621 c) 1.632 d) 1.585 e) 1.597 16) The function is differentiable everywhere. What is n? a) -9 b) 13 c) -17 d) -11 e) -14 17) Which of the following functions has a vertical asymptote at x = -1 and a horizontal asymptote at y = 2? a) b) c) d) e) Answers Part A 1) d) 2) e) 3) b) 4) b) 5) e) 6) a) 7) c) 8) c) 9) b) 10) d) 11) c) 12) c) 13) c) 14) e) 15) a) 16) d) 17) a) 18) b) 19) a) 20) d) 21) b) 22) b) 23) a) 24) c) 25) d) 26) b) 27) b) 28) c) Part B 1) d) 2) e) 3) b) 4) b) 5) e) 6) a) 7) c) 8) c) 9) b) 10) d) 11) c) 12) c) 13) c) 14) e) 15) a) 16) d) 17) a) Advanced Placement Program AP® Calculus AB Practice Exam The questions contained in this AP® Calculus AB Practice Exam are written to the content specifications of AP Exams for this subject. Taking this practice exam should provide students with an idea of their general areas of strengths and weaknesses in preparing for the actual AP Exam. Because this AP Calculus AB Practice Exam has never been administered as an operational AP Exam, statistical data are not available for calculating potential raw scores or conversions into AP grades. This AP Calculus AB Practice Exam is provided by the College Board for AP Exam preparation. Teachers are permitted to download the materials and make copies to use with their students in a classroom setting only. To maintain the security of this exam, teachers should collect all materials after their administration and keep them in a secure location. Teachers may not redistribute the files electronically for any reason. © 2008 The College Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners. Visit the College Board on the Web: www.collegeboard.com. AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA CALCULUS AB SECTION I, Part A Time—55 minutes Number of questions—28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and place the letter of your choice in the corresponding box on the student answer sheet. Do not spend too much time on any one problem. In this exam: (1) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x ) is a real number. (2) The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the prefix “arc” (e.g., sin -1 x = arcsin x ). GO ON TO THE NEXT PAGE. -2- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Ú cos (3x ) dx = 1. (A) -3sin (3x ) + C 1 (B) - sin (3 x ) + C 3 (C) 1 sin (3 x ) + C 3 (D) sin (3x ) + C (E) 3sin (3x ) + C 2. 2 x6 + 6 x3 is xÆ0 4 x5 + 3x3 lim (A) 0 (B) 1 2 (C) 1 (D) 2 (E) nonexistent GO ON TO THE NEXT PAGE. -3- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Ï x 2 - 3 x + 9 for x £ 2 f ( x) = Ì for x > 2 Ó kx + 1 3. The function f is defined above. For what value of k, if any, is f continuous at x = 2 ? (A) 1 (B) 2 (C) 3 (D) 7 (E) No value of k will make f continuous at x = 2. 4. If f ( x ) = cos3 (4 x ) , then f ¢( x ) = (A) 3cos2 (4x ) (B) -12 cos2 ( 4 x ) sin (4 x ) (C) -3cos2 ( 4 x ) sin (4 x ) (D) 12 cos2 ( 4 x ) sin (4 x ) (E) - 4 sin 3 ( 4x ) GO ON TO THE NEXT PAGE. -4- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 5. The function f given by f ( x ) = 2 x 3 - 3 x 2 - 12 x has a relative minimum at x = (A) -1 (B) 0 (C) 2 (D) 3 - 105 4 (E) 3 + 105 4 6. Let f be the function given by f ( x ) = (2 x - 1) ( x + 1) . Which of the following is an equation for the line tangent to the graph of f at the point where x = 1 ? 5 (A) y = 21x + 2 (B) y = 21x - 19 (C) y = 11x - 9 (D) y = 10 x + 2 (E) y = 10 x - 8 GO ON TO THE NEXT PAGE. -5- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Û e x dx = Ù ı x 7. (A) 2e (B) 1 e 2 (C) e x x x +C +C (D) 2 x e (E) +C x +C 1e x +C 2 x x 0 2 4 6 f ( x) 4 k 8 12 8. The function f is continuous on the closed interval [0, 6] and has the values given in the table above. The trapezoidal approximation for 6 Ú0 f ( x ) dx found with 3 subintervals of equal length is 52. What is the value of k ? (A) 2 (B) 6 (C) 7 (D) 10 (E) 14 GO ON TO THE NEXT PAGE. -6- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 9. A particle moves along the x-axis so that at any time t > 0, its velocity is given by v(t ) = 4 - 6t 2 . If the particle is at position x = 7 at time t = 1, what is the position of the particle at time t = 2 ? (A) -10 (B) -5 (C) -3 (D) 3 (E) 17 ax 2 + 12 . The figure above shows a portion of the graph of f. Which of the x2 + b following could be the values of the constants a and b ? 10. The function f is given by f ( x ) = (A) a = -3, b = 2 (B) a = 2, b = -3 (C) a = 2, b = -2 (D) a = 3, b = - 4 (E) a = 3, b = 4 GO ON TO THE NEXT PAGE. -7- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 11. What is the slope of the line tangent to the graph of y = (A) - 1 e 12. If f ¢( x ) = (A) 2 (B) - 3 4e (C) - 1 4e e- x at x = 1 ? x +1 1 4e (D) (E) 1 e 2 and f ( e ) = 5, then f (e ) = x (B) ln 25 (C) 5 + 2 2 - 2 e e (D) 6 (E) 25 GO ON TO THE NEXT PAGE. -8- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Ú (x 13. 3 ) +1 2 dx = (A) 1 7 x + x+C 7 (B) 1 7 1 4 x + x + x +C 7 2 ( ) (C) 6 x 2 x 3 + 1 + C (D) ( ) 3 1 3 x +1 +C 3 ( x3 + 1) 3 (E) 14. 9x2 +C e( 2 + h ) - e 2 = h hÆ0 lim (A) 0 (B) 1 (C) 2e (D) e2 (E) 2e2 GO ON TO THE NEXT PAGE. -9- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 15. The slope field for a certain differential equation is shown above. Which of the following could be a solution to the differential equation with the initial condition y(0 ) = 1 ? (A) y = cos x (B) y = 1 - x 2 (C) y = e x (D) y = 1 - x 2 (E) y = 1 1 + x2 GO ON TO THE NEXT PAGE. -10- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 16. If f ¢( x ) = x - 2 , which of the following could be the graph of y = f ( x ) ? (A) (B) (C) (D) (E) GO ON TO THE NEXT PAGE. -11- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 17. What is the area of the region enclosed by the graphs of f ( x ) = x - 2 x 2 and g( x ) = -5 x ? (A) 7 3 (B) 16 3 (C) 20 3 (D) 9 (E) 36 18. For the function f, f ¢( x ) = 2 x + 1 and f (1) = 4. What is the approximation for f (1.2 ) found by using the line tangent to the graph of f at x = 1 ? (A) 0.6 (B) 3.4 (C) 4.2 (D) 4.6 (E) 4.64 GO ON TO THE NEXT PAGE. -12- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 19. Let f be the function given by f ( x ) = x 3 - 6 x 2 . The graph of f is concave up when (A) x > 2 (B) x < 2 (C) 0 < x < 4 (D) x < 0 or x > 4 only (E) x > 6 only 20. If g( x ) = x 2 - 3 x + 4 and f ( x ) = g ¢( x ) , then (A) - 14 3 (B) -2 (C) 2 3 Ú1 f ( x ) dx = (D) 4 (E) 14 3 GO ON TO THE NEXT PAGE. -13- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 21. The graph of f ¢, the derivative of the function f, is shown above for 0 £ x £ 10. The areas of the regions between the graph of f ¢ and the x-axis are 20, 6, and 4, respectively. If f (0 ) = 2, what is the maximum value of f on the closed interval 0 £ x £ 10 ? (A) 16 (B) 20 (C) 22 (D) 30 (E) 32 22. If f ¢( x ) = ( x - 2 )( x - 3) ( x - 4 ) , then f has which of the following relative extrema? 2 3 I. A relative maximum at x = 2 II. A relative minimum at x = 3 III. A relative maximum at x = 4 (A) I only (B) III only (C) I and III only (D) II and III only (E) I, II, and III GO ON TO THE NEXT PAGE. -14- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 23. The graph of the even function y = f ( x ) consists of 4 line segments, as shown above. Which of the following statements about f is false? (A) lim ( f ( x ) - f (0 )) = 0 x Æ0 (B) lim f ( x ) - f (0 ) =0 x (C) lim f ( x ) - f (- x ) =0 2x (D) lim f ( x ) - f (2 ) =1 x-2 (E) lim f ( x ) - f (3 ) does not exist. x-3 xÆ0 xÆ0 xÆ2 x Æ3 GO ON TO THE NEXT PAGE. -15- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 24. The radius of a circle is increasing. At a certain instant, the rate of increase in the area of the circle is numerically equal to twice the rate of increase in its circumference. What is the radius of the circle at that instant? (A) 1 2 (B) 1 (C) (D) 2 2 25. If x 2 y - 3 x = y3 - 3, then at the point ( -1, 2 ) , (A) - 7 11 (B) - 7 13 (C) - 1 2 (E) 4 dy = dx (D) - 3 14 (E) 7 GO ON TO THE NEXT PAGE. -16- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 26. For x > 0, f is a function such that f ¢( x ) = ln x 1 - ln x . Which of the following is true? and f ¢¢( x ) = x x2 (A) f is decreasing for x > 1, and the graph of f is concave down for x > e. (B) f is decreasing for x > 1, and the graph of f is concave up for x > e. (C) f is increasing for x > 1, and the graph of f is concave down for x > e. (D) f is increasing for x > 1, and the graph of f is concave up for x > e. (E) f is increasing for 0 < x < e, and the graph of f is concave down for 0 < x < e3 2 . 27. If f is the function given by f ( x ) = (A) 0 (B) 7 2 12 (C) 2x Ú4 2 t 2 - t dt, then f ¢(2 ) = (D) 12 (E) 2 12 GO ON TO THE NEXT PAGE. -17- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 28. If y = sin -1 (5 x ) , then (A) 1 1 + 25x 2 (B) 5 1 + 25x 2 (C) (D) (E) dy = dx -5 1 - 25x 2 1 1 - 25x 2 5 1 - 25x 2 END OF PART A OF SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY. DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO. -18- B B B B B B B B B CALCULUS AB SECTION I, Part B Time—50 minutes Number of questions—17 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and place the letter of your choice in the corresponding box on the student answer sheet. Do not spend too much time on any one problem. In this exam: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x ) is a real number. (3) The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the prefix “arc” (e.g., sin -1 x = arcsin x ). GO ON TO THE NEXT PAGE. -19- B B B B B B B B B 76. A particle moves along the x-axis so that at any time t ≥ 0 its velocity is given by v(t ) = t 2 ln (t + 2 ) . What is the acceleration of the particle at time t = 6 ? (A) 1.500 77. If (B) 20.453 3 5 (C) 29.453 (D) 74.860 (E) 133.417 5 Ú0 f ( x ) dx = 6 and Ú3 f ( x ) dx = 4, then Ú0 (3 + 2 f ( x )) dx = (A) 10 (B) 20 (C) 23 (D) 35 (E) 50 GO ON TO THE NEXT PAGE. -20- B B B B B B B B B 78. For t ≥ 0 hours, H is a differentiable function of t that gives the temperature, in degrees Celsius, at an Arctic weather station. Which of the following is the best interpretation of H ¢(24 ) ? (A) The change in temperature during the first day (B) The change in temperature during the 24th hour (C) The average rate at which the temperature changed during the 24th hour (D) The rate at which the temperature is changing during the first day (E) The rate at which the temperature is changing at the end of the 24th hour 79. A spherical tank contains 81.637 gallons of water at time t = 0 minutes. For the next 6 minutes, water flows out of the tank at the rate of 9sin ( t + 1 ) gallons per minute. How many gallons of water are in the tank at the end of the 6 minutes? (A) 36.606 (B) 45.031 (C) 68.858 (D) 77.355 (E) 126.668 GO ON TO THE NEXT PAGE. -21- B B B B B B B B B 80. A left Riemann sum, a right Riemann sum, and a trapezoidal sum are used to approximate the value of 1 Ú0 f ( x ) dx, each using the same number of subintervals. The graph of the function f is shown in the figure above. Which of the sums give an underestimate of the value of 1 Ú0 f ( x ) dx ? I. Left sum II. Right sum III. Trapezoidal sum (A) I only (B) II only (C) III only (D) I and III only (E) II and III only GO ON TO THE NEXT PAGE. -22- B B B B B B B B B 81. The first derivative of the function f is given by f ¢( x ) = x - 4e - sin(2 x ) . How many points of inflection does the graph of f have on the interval 0 < x < 2 p ? (A) Three (B) Four (C) Five (D) Six (E) Seven 82. If f is a continuous function on the closed interval [a, b], which of the following must be true? (A) There is a number c in the open interval (a, b ) such that f (c ) = 0. (B) There is a number c in the open interval (a, b ) such that f ( a ) < f (c ) < f (b ) . (C) There is a number c in the closed interval [a, b] such that f (c ) ≥ f ( x ) for all x in [a, b]. (D) There is a number c in the open interval (a, b ) such that f ¢(c ) = 0. (E) There is a number c in the open interval (a, b ) such that f ¢(c ) = f (b ) - f ( a ) . b-a GO ON TO THE NEXT PAGE. -23- B B B B B B B x 2.5 2.8 3.0 3.1 f ( x) 31.25 39.20 45 48.05 B B 83. The function f is differentiable and has values as shown in the table above. Both f and f ¢ are strictly increasing on the interval 0 £ x £ 5. Which of the following could be the value of f ¢(3) ? (A) 20 (B) 27.5 (C) 29 (D) 30 (E) 30.5 84. The graph of f ¢, the derivative of the function f, is shown above. On which of the following intervals is f decreasing? (A) [2, 4] only (B) [3, 5] only (C) [0, 1] and [3, 5] (D) [2, 4] and [6, 7] (E) [0, 2] and [4, 6] GO ON TO THE NEXT PAGE. -24- B B B B B B B B B x2 x2 and y = for 1 £ x £ 4, as shown in 10 10 the figure above. For this loudspeaker, the cross sections perpendicular to the x-axis are squares. What is the volume of the loudspeaker, in cubic units? 85. The base of a loudspeaker is determined by the two curves y = (A) 2.046 (B) 4.092 (C) 4.200 (D) 8.184 (E) 25.711 GO ON TO THE NEXT PAGE. -25- B B B B x f ( x) B B B 3 4 5 6 7 20 17 12 16 20 B B 86. The function f is continuous and differentiable on the closed interval [3, 7]. The table above gives selected values of f on this interval. Which of the following statements must be true? I. The minimum value of f on [3, 7] is 12. II. There exists c, for 3 < c < 7, such that f ¢(c ) = 0. III. f ¢( x ) > 0 for 5 < x < 7. (A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III GO ON TO THE NEXT PAGE. -26- B B B B B B B B B 87. The figure above shows the graph of f ¢, the derivative of the function f, on the open interval -7 < x < 7. If f ¢ has four zeros on -7 < x < 7, how many relative maxima does f have on -7 < x < 7 ? (A) One (B) Two (C) Three (D) Four (E) Five 88. The rate at which water is sprayed on a field of vegetables is given by R (t ) = 2 1 + 5t 3 , where t is in minutes and R (t ) is in gallons per minute. During the time interval 0 £ t £ 4, what is the average rate of water flow, in gallons per minute? (A) 8.458 (B) 13.395 (C) 14.691 (D) 18.916 (E) 35.833 GO ON TO THE NEXT PAGE. -27- B B B B B B B x f ( x) f ¢( x ) g( x ) g ¢( x ) 1 3 –2 –3 4 B B 89. The table above gives values of the differentiable functions f and g and their derivatives at x = 1. If h( x ) = (2 f ( x ) + 3) (1 + g ( x )) , then h ¢(1) = (A) -28 (B) -16 (C) 40 (D) 44 (E) 47 90. The functions f and g are differentiable. For all x, f ( g( x )) = x and g( f ( x )) = x. If f (3) = 8 and f ¢(3) = 9, what are the values of g(8) and g¢(8) ? (A) g(8) = 1 1 and g ¢(8) = 3 9 (B) g(8) = 1 1 and g ¢(8) = 3 9 (C) g(8) = 3 and g ¢(8) = -9 (D) g(8) = 3 and g ¢(8) = (E) g(8) = 3 and g ¢(8) = 1 9 1 9 GO ON TO THE NEXT PAGE. -28- B B B B B B B B B 91. A particle moves along the x-axis so that its velocity at any time t ≥ 0 is given by v(t ) = 5te - t - 1. At t = 0, the particle is at position x = 1. What is the total distance traveled by the particle from t = 0 to t = 4 ? (A) 0.366 (B) 0.542 (C) 1.542 (D) 1.821 (E) 2.821 ( ) 92. Let f be the function with first derivative defined by f ¢( x ) = sin x 3 for 0 £ x £ 2. At what value of x does f attain its maximum value on the closed interval 0 £ x £ 2 ? (A) 0 (B) 1.162 (C) 1.465 (D) 1.845 (E) 2 END OF SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART B ONLY. DO NOT GO ON TO SECTION II UNTIL YOU ARE TOLD TO DO SO. ________________________________________________ -29- AP® Calculus Instructions for Section II Free-Response Questions Write clearly and legibly. Cross out any errors you make; erased or crossed-out work will not be graded. Manage your time carefully. During the timed portion for Part A, work only on the questions in Part A. You are permitted to use your calculator to solve an equation, find the derivative of a function at a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your question, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results. During the timed portion for Part B, you may continue to work on the questions in Part A without the use of a calculator. For each part of Section II, you may wish to look over the questions before starting to work on them. It is not expected that everyone will be able to complete all parts of all questions. • Show all of your work. Clearly label any functions, graphs, tables, or other objects that you use. Your work will be graded on the correctness and completeness of your methods as well as your answers. Answers without supporting work may not receive credit. Justifications require that you give mathematical (noncalculator) reasons. • Your work must be expressed in standard mathematical notation rather than calculator syntax. For example, 5 2 Ú1 x dx may not be written as fnInt(X2, X, 1, 5). • Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If you use decimal approximations in calculations, your work will be graded on accuracy. Unless otherwise specified, your final answers should be accurate to three places after the decimal point. • Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x ) is a real number. -31- CALCULUS AB SECTION II, Part A Time—45 minutes Number of problems—3 A graphing calculator is required for some problems or parts of problems. Ê pt2 ˆ gallons per hour 1. The rate at which raw sewage enters a treatment tank is given by E (t ) = 850 + 715cos Á Ë 9 ˜¯ for 0 £ t £ 4 hours. Treated sewage is removed from the tank at the constant rate of 645 gallons per hour. The treatment tank is empty at time t = 0. (a) How many gallons of sewage enter the treatment tank during the time interval 0 £ t £ 4 ? Round your answer to the nearest gallon. (b) For 0 £ t £ 4, at what time t is the amount of sewage in the treatment tank greatest? To the nearest gallon, what is the maximum amount of sewage in the tank? Justify your answers. (c) For 0 £ t £ 4, the cost of treating the raw sewage that enters the tank at time t is (0.15 - 0.02t ) dollars per gallon. To the nearest dollar, what is the total cost of treating all the sewage that enters the tank during the time interval 0 £ t £ 4 ? GO ON TO THE NEXT PAGE. -32- 2. Let R and S in the figure above be defined as follows: R is the region in the first and second quadrants bounded by the graphs of y = 3 - x 2 and y = 2 x. S is the shaded region in the first quadrant bounded by the two graphs, the x-axis, and the y-axis. (a) Find the area of S. (b) Find the volume of the solid generated when R is rotated about the horizontal line y = -1. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an isosceles right triangle with one leg across the base of the solid. Write, but do not evaluate, an integral expression that gives the volume of the solid. t (minutes) 0 4 8 12 16 H (t ) (∞C ) 65 68 73 80 90 3. The temperature, in degrees Celsius (∞C ) , of an oven being heated is modeled by an increasing differentiable function H of time t, where t is measured in minutes. The table above gives the temperature as recorded every 4 minutes over a 16-minute period. (a) Use the data in the table to estimate the instantaneous rate at which the temperature of the oven is changing at time t = 10. Show the computations that lead to your answer. Indicate units of measure. (b) Write an integral expression in terms of H for the average temperature of the oven between time t = 0 and time t = 16. Estimate the average temperature of the oven using a left Riemann sum with four subintervals of equal length. Show the computations that lead to your answer. (c) Is your approximation in part (b) an underestimate or an overestimate of the average temperature? Give a reason for your answer. (d) Are the data in the table consistent with or do they contradict the claim that the temperature of the oven is increasing at an increasing rate? Give a reason for your answer. END OF PART A OF SECTION II -33- CALCULUS AB SECTION II, Part B Time—45 minutes Number of problems—3 No calculator is allowed for these problems. 4. Let f be the function given by f ( x ) = (ln x )(sin x ) . The figure above shows the graph of f for 0 < x £ 2 p . The function g is defined by g( x ) = x Ú1 f (t ) dt for 0 < x £ 2 p . (a) Find g(1) and g¢(1) . (b) On what intervals, if any, is g increasing? Justify your answer. (c) For 0 < x £ 2 p , find the value of x at which g has an absolute minimum. Justify your answer. (d) For 0 < x < 2 p , is there a value of x at which the graph of g is tangent to the x-axis? Explain why or why not. GO ON TO THE NEXT PAGE. -34- 5. Consider the differential equation dy x = , where y π 0. dx y (a) The slope field for the given differential equation is shown below. Sketch the solution curve that passes through the point (3, -1) , and sketch the solution curve that passes through the point (1, 2 ) . (Note: The points (3, -1) and (1, 2 ) are indicated in the figure.) (b) Write an equation for the line tangent to the solution curve that passes through the point (1, 2 ) . (c) Find the particular solution y = f ( x ) to the differential equation with the initial condition f (3) = -1, and state its domain. 6. Let g( x ) = xe - x + be - x , where b is a positive constant. (a) Find lim g ( x ) . x Æ• (b) For what positive value of b does g have an absolute maximum at x = 2 ? Justify your answer. 3 (c) Find all values of b, if any, for which the graph of g has a point of inflection on the interval 0 < x < •. Justify your answer. STOP END OF EXAM -35- AP® Calculus AB Multiple-Choice Answer Key No. 1 Correct Answer C No. 76 Correct Answer C 2 D 77 D 3 C 78 E 4 B 79 A 5 C 80 D 6 B 81 B 7 A 82 C 8 D 83 D 9 C 84 E 10 D 85 D 11 B 86 B 12 D 87 A 13 B 88 C 14 D 89 D 15 E 90 E 16 E 91 D 17 D 92 C 18 D 19 A 20 C 21 C 22 A 23 B 24 D 25 A 26 C 27 E 28 E -37- AP® Calculus AB Free-Response Scoring Guidelines Question 1 Ê pt 2 ˆ The rate at which raw sewage enters a treatment tank is given by E (t ) = 850 + 715cos Á gallons Ë 9 ˜¯ per hour for 0 £ t £ 4 hours. Treated sewage is removed from the tank at the constant rate of 645 gallons per hour. The treatment tank is empty at time t = 0. (a) How many gallons of sewage enter the treatment tank during the time interval 0 £ t £ 4 ? Round your answer to the nearest gallon. (b) For 0 £ t £ 4, at what time t is the amount of sewage in the treatment tank greatest? To the nearest gallon, what is the maximum amount of sewage in the tank? Justify your answers. (c) For 0 £ t £ 4, the cost of treating the raw sewage that enters the tank at time t is (0.15 - 0.02t ) dollars per gallon. To the nearest dollar, what is the total cost of treating all the sewage that enters the tank during the time interval 0 £ t £ 4 ? (a) 4 Ú0 E (t ) dt ª 3981 gallons 2: (b) Let S (t ) be the amount of sewage in the treatment tank at time t. Then S ¢(t ) = E (t ) - 645 and S ¢(t ) = 0 when E (t ) = 645. On the interval 0 £ t £ 4, E (t ) = 645 when t = 2.309 and t = 3.559. t (hours) 0 1 : integral 1 : answer Ï 1 : sets E (t ) = 645 Ô 1 : identifies t = 2.309 as Ô 4: Ì a candidate Ô 1 : amount of sewage at t = 2.309 Ô Ó 1 : conclusion amount of sewage in treatment tank 0 2.309 E (t ) dt - 645 (2.309) = 1637.178 3.559 E (t ) dt - 645 (3.559) = 1228.520 2.309 Ú0 3.559 Ú0 4 { 3981.022 - 645(4) = 1401.022 The amount of sewage in the treatment tank is greatest at t = 2.309 hours. At that time, the amount of sewage in the tank, rounded to the nearest gallon, is 1637 gallons. (c) Total cost = 4 Ú0 (0.15 - 0.02t ) E (t ) dt = 474.320 The total cost of treating the sewage that enters the tank during the time interval 0 £ t £ 4, to the nearest dollar, is $474. -38- Ï 1 : integrand Ô 3 : Ì 1 : limits ÔÓ 1 : answer AP® Calculus AB Free-Response Scoring Guidelines Question 2 Let R and S in the figure above be defined as follows: R is the region in the first and second quadrants bounded by the graphs of y = 3 - x 2 and y = 2 x. S is the shaded region in the first quadrant bounded by the two graphs, the x-axis, and the y-axis. (a) Find the area of S. (b) Find the volume of the solid generated when R is rotated about the horizontal line y = -1. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an isosceles right triangle with one leg across the base of the solid. Write, but do not evaluate, an integral expression that gives the volume of the solid. 3 - x 2 = 2 x when x = -1.63658 and x = 1 Let a = -1.63658 (a) Area of S = 1 x Ú0 2 dx + = 2.240 3-x ı (( Ù (b) Volume = p Û 1 a 2 2 Ú1 (3 - x ) dx 3 Ï 1 : integrands Ô 3 : Ì 1 : limits ÔÓ 1 : answer ) - (2 x + 1) ) dx 2 +1 2 Ï 2 : integrand Ô 4 : Ì 1 : limits and constant ÔÓ 1 : answer = 63.106 or 63.107 (c) Volume = ( 1 1 3 - x2 - 2 x 2 Úa ) 2 dx 2: -39- { 1 : integrand 1 : limits and constant AP® Calculus AB Free-Response Scoring Guidelines Question 3 t (minutes) 0 4 8 12 16 H (t ) (∞C) 65 68 73 80 90 The temperature, in degrees Celsius (∞C) , of an oven being heated is modeled by an increasing differentiable function H of time t, where t is measured in minutes. The table above gives the temperature as recorded every 4 minutes over a 16-minute period. (a) Use the data in the table to estimate the instantaneous rate at which the temperature of the oven is changing at time t = 10. Show the computations that lead to your answer. Indicate units of measure. (b) Write an integral expression in terms of H for the average temperature of the oven between time t = 0 and time t = 16. Estimate the average temperature of the oven using a left Riemann sum with four subintervals of equal length. Show the computations that lead to your answer. (c) Is your approximation in part (b) an underestimate or an overestimate of the average temperature? Give a reason for your answer. (d) Are the data in the table consistent with or do they contradict the claim that the temperature of the oven is increasing at an increasing rate? Give a reason for your answer. (a) H ¢ (10 ) ª H (12 ) - H (8) 80 - 73 7 = = ∞C min 12 - 8 4 4 (b) Average temperature is 16 Ú0 1 16 16 Ú0 { 1 : difference quotient 1 : answer with units Ï 1 : 1 16 H (t ) dt Ô 16 Ú0 3: Ì 1 : left Riemann sum Ô Ó 1 : answer H (t ) dt H (t ) dt ª 4 ◊ (65 + 68 + 73 + 80 ) Average temperature ª 2: 4 ◊ 286 = 71.5∞C 16 (c) The left Riemann sum approximation is an underestimate of the integral because the graph of H is increasing. Dividing by 16 will not change the inequality, so 71.5∞C is an underestimate of the average temperature. 1 : answer with reason (d) If a continuous function is increasing at an increasing rate, then the slopes of the secant lines of the graph of the function are increasing. The slopes of the secant lines for the four intervals in 3 5 7 10 , respectively. the table are , , , and 4 4 4 4 Since the slopes are increasing, the data are consistent with the claim. OR By the Mean Value Theorem, the slopes are also the values of H ¢(ck ) for some times c1 < c2 < c3 < c4 , respectively. Since these derivative values are positive and increasing, the data are consistent with the claim. Ï 1 : considers slopes of Ô four secant lines Ô 3 : Ì 1 : explanation Ô 1 : conclusion consistent Ô with explanation Ó -40- AP® Calculus AB Free-Response Scoring Guidelines Question 4 Let f be the function given by f ( x ) = (ln x )(sin x ) . The figure above shows the graph of f for 0 < x £ 2 p . The function g is defined by g( x ) = x Ú1 f (t ) dt for 0 < x £ 2 p . (a) Find g(1) and g¢(1) . (b) On what intervals, if any, is g increasing? Justify your answer. (c) For 0 < x £ 2 p , find the value of x at which g has an absolute minimum. Justify your answer. (d) For 0 < x < 2 p , is there a value of x at which the graph of g is tangent to the x-axis? Explain why or why not. (a) g(1) = 1 Ú1 f (t ) dt = 0 and g ¢(1) = Ï 1 : g(1) 2: Ì Ó 1 : g ¢(1) f (1) = 0 (b) Since g ¢( x ) = f ( x ) , g is increasing on the interval 1 £ x £ p because f ( x ) > 0 for 1 < x < p . (c) For 0 < x < 2 p , g ¢( x ) = f ( x ) = 0 when x = 1, p . g ¢ = f changes from negative to positive only at x = 1. The absolute minimum must occur at x = 1 or at the right endpoint. Since g(1) = 0 and g (2 p ) = 2p Ú1 f (t ) dt = p Ú1 f (t ) dt + 2p Úp f (t ) dt < 0 by comparison of the two areas, the absolute minimum occurs at x = 2 p . (d) Yes, the graph of g is tangent to the x-axis at x = 1 since g(1) = 0 and g ¢(1) = 0. -41- 2: { 1 : interval 1 : reason Ï 1 : identifies 1 and 2 p as candidates Ô - or Ô indicates that the graph of g Ô 3: Ì decreases, increases, then decreases Ô Ô 1 : justifies g(2 p ) < g(1) Ô 1 : answer Ó 2: { 1 : answer of “yes” with x = 1 1 : explanation AP® Calculus AB Free-Response Scoring Guidelines Question 5 Consider the differential equation dy x = , where y π 0. dx y (a) The slope field for the given differential equation is shown below. Sketch the solution curve that passes through the point (3, -1) , and sketch the solution curve that passes through the point (1, 2 ) . (Note: The points (3, -1) and (1, 2 ) are indicated in the figure.) (b) Write an equation for the line tangent to the solution curve that passes through the point (1, 2 ) . (c) Find the particular solution y = f ( x ) to the differential equation with the initial condition f (3) = -1, and state its domain. Ï 1 : solution curve through (3, -1) 2: Ì Ó 1 : solution curve through (1, 2 ) (a) Curves must go through the indicated points, follow the given slope lines, and extend to the boundary of the slope field or the x-axis. (b) dy 1 = dx (1, 2) 2 1 : equation of tangent line An equation for the line tangent to the solution 1 curve is y - 2 = ( x - 1) . 2 (c) y dy = x dx 1 2 1 2 y = x +A 2 2 y2 = x 2 + C C = -8 Since the particular solution goes through (3, -1) , y must be negative. y = - x 2 - 8 for x > 8 -42- Ï Ï 1 : separates variables Ô 1 : antiderivatives Ô Ô Ô Ô 5 : Ì 1 : constant of integration Ô 1 : uses initial condition ÔÔ Ô 6: Ì Ó 1 : solves for y Ô Ô Note: max 2 5 [1-1-0-0-0] if no Ô constant of integration Ô ÓÔ 1 : domain AP® Calculus AB Free-Response Scoring Guidelines Question 6 Let g( x ) = xe - x + be - x , where b is a positive constant. (a) Find lim g( x ) . xÆ• (b) For what positive value of b does g have an absolute maximum at x = 2 ? Justify your answer. 3 (c) Find all values of b, if any, for which the graph of g has a point of inflection on the interval 0 < x < •. Justify your answer. (a) lim g( x ) = 0 1 : answer xÆ• Ï 2 : g ¢( x ) ÔÔ 2 4 : Ì 1 : solves g ¢ = 0 for b 3 Ô ÔÓ 1 : justification (b) g ¢( x ) = e - x - xe - x - be - x = (1 - x - b ) e - x g¢ ( 23 ) = ( 13 - b) e When b = -2 3 =0ﬁb= ( ) () 1 3 1 2 - x e- x . , g ¢( x ) = 3 3 2 2 , g ¢( x ) > 0 and for x > , g ¢( x ) < 0. 3 3 1 Therefore, when b = , g has an absolute maximum 3 2 at x = . 3 For x < (c) g ¢¢( x ) = -e - x - (1 - x - b ) e - x = ( x - 2 + b ) e - x If 0 < b < 2, then g ¢¢( x ) will change sign at x = 2 - b > 0. Therefore, the graph of g will have a point of inflection on the interval 0 < x < • when 0 < b < 2. -43- Ï 2 : g ¢¢( x ) Ô 4 : Ì 1 : interval for b ÔÓ 1 : justification ® AP Calculus AB Exam SECTION I: Multiple Choice 2012 DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Score 50% Writing Instrument Pencil required Part A Instructions Section I of this exam contains 45 multiple-choice questions and 4 survey questions. For Part A, fill in only the circles for numbers 1 through 28 on page 2 of the answer sheet. For Part B, fill in only the circles for numbers 76 through 92 on page 3 of the answer sheet. The survey questions are numbers 93 through 96. Indicate all of your answers to the multiple-choice questions on the answer sheet. No credit will be given for anything written in this exam booklet, but you may use the booklet for notes or scratch work. After you have decided which of the suggested answers is best, completely fill in the corresponding circle on the answer sheet. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely. Here is a sample question and answer. Number of Questions 28 Time 55 minutes Electronic Device None allowed Part B Number of Questions 17 Time 50 minutes Electronic Device Graphing calculator required Use your time effectively, working as quickly as you can without losing accuracy. Do not spend too much time on any one question. Go on to other questions and come back to the ones you have not answered if you have time. It is not expected that everyone will know the answers to all of the multiple-choice questions. Your total score on the multiple-choice section is based only on the number of questions answered correctly. Points are not deducted for incorrect answers or unanswered questions. Form I Form Code 4IBP4-Q-S Minimum 20% post-consumer waste 66 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA CALCULUS AB SECTION I, Part A Time— 55 minutes Number of questions—28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. In this exam: (1) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x ) is a real number. (2) The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the prefix “arc” (e.g., sin -1 x = arcsin x ). Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -3- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 1. If y = x sin x, then dy = dx (A) sin x + cos x (B) sin x + x cos x (C) sin x - x cos x (D) x (sin x + cos x ) (E) x (sin x - cos x ) 2. Let f be the function given by f ( x ) = 300 x - x 3 . On which of the following intervals is the function f increasing? (A) ( - •, -10] and [10, • ) (B) [ -10, 10] (C) [0, 10] only (D) [0, 10 3 ] only (E) [0, •) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -4- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Ú sec x tan x dx = 3. (A) sec x + C (B) tan x + C (C) sec 2 x +C 2 (D) tan 2 x +C 2 (E) sec 2 x tan 2 x +C 2 4. If f ( x ) = 7 x - 3 + ln x, then f ¢(1) = (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -5- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 5. The graph of the function f is shown above. Which of the following statements is false? (A) lim f ( x ) exists. xÆ2 (B) lim f ( x ) exists. x Æ3 (C) lim f ( x ) exists. xÆ4 (D) lim f ( x ) exists. xÆ5 (E) The function f is continuous at x = 3. 6. A particle moves along the x-axis. The velocity of the particle at time t is 6t - t 2 . What is the total distance traveled by the particle from time t = 0 to t = 3 ? (A) 3 (B) 6 (C) 9 (D) 18 (E) 27 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -6- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA ( ) ( ) 5 7. If y = x 3 - cos x , then y ¢ = (A) 5 x 3 - cos x 4 ( ) 5 (3 x 2 + sin x ) 4 (B) 5 3 x 2 + sin x (C) ( ) ( ) ⴢ (3x 2 + sin x ) (D) 5 3 x 2 + sin x (E) 5 x 3 - cos x 4 ⴢ (6 x + cos x ) 4 t (hours) 4 7 12 15 R (t ) (liters/hour) 6.5 6.2 5.9 5.6 8. A tank contains 50 liters of oil at time t = 4 hours. Oil is being pumped into the tank at a rate R(t ) , where R(t ) is measured in liters per hour, and t is measured in hours. Selected values of R(t ) are given in the table above. Using a right Riemann sum with three subintervals and data from the table, what is the approximation of the number of liters of oil that are in the tank at time t = 15 hours? (A) 64.9 (B) 68.2 (C) 114.9 (D) 116.6 Unauthorized copying or reuse of any part of this page is illegal. (E) 118.2 GO ON TO THE NEXT PAGE. -7- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA ÏÔ (2 x + 1)( x - 2 ) f (x) = Ì x-2 ÔÓ k for x π 2 for x = 2 9. Let f be the function defined above. For what value of k is f continuous at x = 2 ? (A) 0 (B) 1 (C) 2 (D) 3 (E) 5 10. What is the area of the region in the first quadrant bounded by the graph of y = e x (A) 2e - 2 (B) 2e (C) e -1 2 (D) Unauthorized copying or reuse of any part of this page is illegal. e -1 2 2 and the line x = 2 ? (E) e - 1 GO ON TO THE NEXT PAGE. -8- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 11. Let f be the function defined by f ( x ) = x - 2 for all x. Which of the following statements is true? (A) f is continuous but not differentiable at x = 2. (B) f is differentiable at x = 2. (C) f is not continuous at x = 2. (D) lim f ( x ) π 0 xÆ2 (E) x = 2 is a vertical asymptote of the graph of f. 4 e 12. Using the substitution u = x , Û Ù ı1 (A) 2 16 u Ú1 e du (B) 2 4 u Ú1 e du x x dx is equal to which of the following? (C) 2 2 u Ú1 e du Unauthorized copying or reuse of any part of this page is illegal. (D) 1 2 u e du 2 Ú1 (E) 4 u Ú1 e du GO ON TO THE NEXT PAGE. -9- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 13. The function f is defined by f ( x ) = (A) 2 (B) 6 (C) 8 { 2 for x < 3 What is the value of x - 1 for x ≥ 3. (D) 10 5 Ú1 f ( x ) dx ? (E) 12 14. If f ( x ) = x 2 - 4 and g( x ) = 3 x - 2, then the derivative of f ( g( x )) at x = 3 is (A) 7 5 (B) 14 5 (C) 18 5 (D) 15 21 Unauthorized copying or reuse of any part of this page is illegal. (E) 30 21 GO ON TO THE NEXT PAGE. -10- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 15. The graph of a differentiable function f is shown above. If h( x ) = x Ú0 f (t ) dt, which of the following is true? (A) h(6 ) < h ¢(6 ) < h ¢¢(6 ) (B) h(6 ) < h ¢¢(6 ) < h ¢(6 ) (C) h ¢(6 ) < h(6 ) < h ¢¢(6 ) (D) h ¢¢(6 ) < h(6 ) < h ¢(6 ) (E) h ¢¢(6 ) < h ¢(6 ) < h(6 ) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -11- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 16. A particle moves along the x-axis with its position at time t given by x (t ) = (t - a )(t - b ) , where a and b are constants and a π b. For which of the following values of t is the particle at rest? (A) t = ab (B) t = a+b 2 (C) t = a + b (D) t = 2 ( a + b ) (E) t = a and t = b Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -12- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 17. The figure above shows the graph of f. If f ( x ) = of y = g( x ) ? x Ú2 g(t ) dt, which of the following could be the graph (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -13- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA lim 18. hÆ0 (A) 0 ln (4 + h ) - ln ( 4 ) is h (B) 1 4 (C) 1 (D) e 19. The function f is defined by f ( x ) = line tangent to f at ( x, y ) has slope (E) nonexistent x . What points ( x, y ) on the graph of f have the property that the x+2 1 ? 2 (A) (0,0 ) only (B) ( 21 , 15 ) only (C) (0,0 ) and ( -4,2 ) ( 23 ) (D) (0,0 ) and 4, (E) There are no such points. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -14- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 20. Let f ( x ) = (2 x + 1) and let g be the inverse function of f. Given that f (0 ) = 1, what is the value of g¢(1) ? 3 (A) - 2 27 (B) 1 54 (C) 1 27 (D) 1 6 (E) 6 21. The line y = 5 is a horizontal asymptote to the graph of which of the following functions? (A) y = sin (5 x ) x (B) y = 5 x (C) y = 1 x-5 Unauthorized copying or reuse of any part of this page is illegal. (D) y = 5x 1- x (E) y = 20 x 2 - x 1 + 4 x2 GO ON TO THE NEXT PAGE. -15- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 22. Let f be the function defined by f ( x ) = ln x . What is the absolute maximum value of f ? x (A) 1 (B) 1 e (C) 0 (D) -e (E) f does not have an absolute maximum value. 23. If P (t ) is the size of a population at time t, which of the following differential equations describes linear growth in the size of the population? (A) dP = 200 dt (B) dP = 200t dt (C) dP = 100t 2 dt (D) dP = 200 P dt (E) dP = 100 P 2 dt Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -16- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 24. Let g be the function given by g( x ) = x 2e kx , where k is a constant. For what value of k does g have a critical 2 point at x = ? 3 (A) -3 (B) - 3 2 (C) - 1 3 (D) 0 Unauthorized copying or reuse of any part of this page is illegal. (E) There is no such k. GO ON TO THE NEXT PAGE. -17- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 25. Which of the following is the solution to the differential equation condition y( p ) = 1 ? dy = 2sin x with the initial dx (A) y = 2 cos x + 3 (B) y = 2 cos x - 1 (C) y = -2 cos x + 3 (D) y = -2 cos x + 1 (E) y = -2 cos x - 1 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -18- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 26. Let g be a function with first derivative given by g ¢( x ) = interval 0 < x < 2 ? x -t2 Ú0 e dt. Which of the following must be true on the (A) g is increasing, and the graph of g is concave up. (B) g is increasing, and the graph of g is concave down. (C) g is decreasing, and the graph of g is concave up. (D) g is decreasing, and the graph of g is concave down. (E) g is decreasing, and the graph of g has a point of inflection on 0 < x < 2. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -19- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 27. If ( x + 2 y ) ◊ (A) - 10 3 dy d2y at the point (3, 0 ) ? = 2 x - y, what is the value of dx dx 2 (B) 0 (C) 2 (D) 10 3 (E) Undefined Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -20- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 28. For t ≥ 0, the position of a particle moving along the x-axis is given by x (t ) = sin t - cos t. What is the acceleration of the particle at the point where the velocity is first equal to 0 ? (A) - 2 (B) -1 (C) 0 (D) 1 (E) 2 END OF PART A OF SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY. DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO. Unauthorized copying or reuse of any part of this page is illegal. -21- B B B B B B B B B CALCULUS AB SECTION I, Part B Time— 50 minutes Number of questions—17 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO QUESTIONS NUMBERED 76–92. YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET. In this exam: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x ) is a real number. (3) The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the prefix “arc” (e.g., sin -1 x = arcsin x ). Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -24- B B B B B B B B B 76. The graph of the function f is shown in the figure above. For which of the following values of x is f ¢( x ) positive and increasing? (A) a (B) b (C) c (D) d (E) e Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -25- B B B B B B B B B 77. Let f be a function that is continuous on the closed interval [2, 4] with f (2 ) = 10 and f ( 4 ) = 20. Which of the following is guaranteed by the Intermediate Value Theorem? (A) f ( x ) = 13 has at least one solution in the open interval (2, 4 ) . (B) f (3) = 15 (C) f attains a maximum on the open interval (2, 4 ) . (D) f ¢( x ) = 5 has at least one solution in the open interval (2, 4 ) . (E) f ¢( x ) > 0 for all x in the open interval (2, 4 ) . 78. The graph of y = e tan x - 2 crosses the x-axis at one point in the interval [ 0, 1]. What is the slope of the graph at this point? (A) 0.606 (B) 2 (C) 2.242 (D) 2.961 Unauthorized copying or reuse of any part of this page is illegal. (E) 3.747 GO ON TO THE NEXT PAGE. -26- B B B B B B B B B 79. A particle moves along the x-axis. The velocity of the particle at time t is given by v(t ) , and the acceleration of the particle at time t is given by a(t ) . Which of the following gives the average velocity of the particle from time t = 0 to time t = 8 ? (A) a(8) - a(0 ) 8 (B) 1 8 v(t ) dt 8 Ú0 (C) 1 8 v(t ) dt 8 Ú0 (D) 1 8 v(t ) dt 2 Ú0 (E) v(0 ) + v(8) 2 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -27- B B B B B B B B B 80. The graph of f ¢, the derivative of the function f, is shown above. Which of the following statements must be true? I. f has a relative minimum at x = -3. II. The graph of f has a point of inflection at x = -2. III. The graph of f is concave down for 0 < x < 4. (A) I only (B) II only (C) III only (D) I and II only Unauthorized copying or reuse of any part of this page is illegal. (E) I and III only GO ON TO THE NEXT PAGE. -28- B B B B B ( B B B B ) 81. Water is pumped into a tank at a rate of r (t ) = 30 1 - e - 0.16t gallons per minute, where t is the number of minutes since the pump was turned on. If the tank contained 800 gallons of water when the pump was turned on, how much water, to the nearest gallon, is in the tank after 20 minutes? (A) 380 gallons (B) 420 gallons (C) 829 gallons (D) 1220 gallons (E) 1376 gallons 82. If f ¢ ( x ) = x 4 + 1 + x 3 - 3 x, then f has a local maximum at x = (A) -2.314 (B) -1.332 (C) 0.350 (D) 0.829 Unauthorized copying or reuse of any part of this page is illegal. (E) 1.234 GO ON TO THE NEXT PAGE. -29- B B B B B B B B B 83. The graph above gives the velocity, v, in ft/sec, of a car for 0 £ t £ 8, where t is the time in seconds. Of the following, which is the best estimate of the distance traveled by the car from t = 0 until the car comes to a complete stop? (A) 21 ft (B) 26 ft (C) 180 ft (D) 210 ft (E) 260 ft 84. For -1.5 < x < 1.5, let f be a function with first derivative given by f ¢( x ) = e following are all intervals on which the graph of f is concave down? ( x4 - 2 x2 +1) - 2. Which of the (A) ( - 0.418, 0.418) only (B) ( -1, 1) (C) ( -1.354, - 0.409) and (0.409, 1.354) (D) ( -1.5, -1) and (0, 1) (E) ( -1.5, -1.354) , ( - 0.409, 0 ) , and (1.354, 1.5) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -30- B B B B B B B B B 85. The graph of f ¢, the derivative of f, is shown in the figure above. The function f has a local maximum at x = (A) -3 (B) -1 (C) 1 (D) 3 (E) 4 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -31- B B B B 86. If f ¢ ( x ) > 0 for all real numbers x and B B B B B 7 Ú4 f (t ) dt = 0, which of the following could be a table of values for the function f ? (A) x 4 5 7 (B) x 4 5 7 f (x) -4 -3 0 f (x) -4 -2 5 4 5 7 f (x) -4 6 3 (D) x 4 5 7 f (x) 0 0 0 (E) x 4 5 7 f (x) 0 4 6 (C) x Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -32- B B B B B B B B B 87. The graph of f ¢¢, the second derivative of f, is shown above for -2 £ x £ 4. What are all intervals on which the graph of the function f is concave down? (A) -1 < x < 1 (B) 0< x<2 (C) 1 < x < 3 only (D) -2 < x < -1 only (E) -2 < x < -1 and 1 < x < 3 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -33- B B B B B B B B B 88. A person whose height is 6 feet is walking away from the base of a streetlight along a straight path at a rate of 4 feet per second. If the height of the streetlight is 15 feet, what is the rate at which the person’s shadow is lengthening? (A) 1.5 ft/sec (B) 2.667 ft/sec (C) 3.75 ft/sec (D) 6 ft/sec 89. A particle moves along a line so that its acceleration for t ≥ 0 is given by a (t ) = velocity at t = 0 is 5, what is the velocity of the particle at t = 3 ? (A) 0.713 (B) 1.134 (C) 6.134 (D) 6.710 Unauthorized copying or reuse of any part of this page is illegal. (E) 10 ft/sec t+3 t3 + 1 . If the particle’s (E) 11.710 GO ON TO THE NEXT PAGE. -34- B B B 90. Let f be a function such that 24 f (t ) dt = 5 24 f (t ) dt = 20 12 f (t ) dt = 5 12 f (t ) dt = 20 (A) Ú12 (B) Ú12 (C) Ú6 (D) Ú6 (E) B 12 Ú6 B B B B B f (2 x ) dx = 10. Which of the following must be true? 6 Ú3 f (t ) dt = 5 x –2 0 3 5 6 f ¢( x ) 3 1 4 7 5 91. Let f be a polynomial function with values of f ¢( x ) at selected values of x given in the table above. Which of the following must be true for -2 < x < 6 ? (A) The graph of f is concave up. (B) The graph of f has at least two points of inflection. (C) f is increasing. (D) f has no critical points. (E) f has at least two relative extrema. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -35- B B B B B B B B B 92. Let R be the region in the first quadrant bounded below by the graph of y = x 2 and above by the graph of y = x . R is the base of a solid whose cross sections perpendicular to the x-axis are squares. What is the volume of the solid? (A) 0.129 (B) 0.300 (C) 0.333 (D) 0.700 Unauthorized copying or reuse of any part of this page is illegal. (E) 1.271 GO ON TO THE NEXT PAGE. -36- Answer Key for AP Calculus AB Practice Exam, Section I Multiple-Choice Questions Question # Key 1 B 2 B 3 A 4 E 5 C 6 D 7 E 8 C 9 E 10 A 11 A 12 C 13 D 14 A 15 A 16 B 17 A 18 B 19 C 20 D 21 E 22 B 23 24 25 26 27 28 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 A A E A A A E A D B E D C D D C B E B E B B A ® AP Calculus AB Exam 2013 SECTION I: Multiple Choice DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Score 50% Writing Instrument Pencil required Part A Instructions Section I of this exam contains 45 multiple-choice questions and 4 survey questions. For Part A, fill in only the circles for numbers 1 through 28 on page 2 of the answer sheet. For Part B, fill in only the circles for numbers 76 through 92 on page 3 of the answer sheet. The survey questions are numbers 93 through 96. Indicate all of your answers to the multiple-choice questions on the answer sheet. No credit will be given for anything written in this exam booklet, but you may use the booklet for notes or scratch work. After you have decided which of the suggested answers is best, completely fill in the corresponding circle on the answer sheet. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely. Here is a sample question and answer. Number of Questions 28 Time 55 minutes Electronic Device None allowed Part B Number of Questions 17 Time 50 minutes Electronic Device Graphing calculator required Use your time effectively, working as quickly as you can without losing accuracy. Do not spend too much time on any one question. Go on to other questions and come back to the ones you have not answered if you have time. It is not expected that everyone will know the answers to all of the multiple-choice questions. Your total score on the multiple-choice section is based only on the number of questions answered correctly. Points are not deducted for incorrect answers or unanswered questions. Form I Form Code 4JBP4-Q-S 66 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA CALCULUS AB SECTION I, Part A Time—55 minutes Number of questions—28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. In this exam: (1) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which is a real number. (2) The inverse of a trigonometric function f may be indicated using the inverse function notation prefix “arc” (e.g., or with the ). Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -3- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA is 1. (A) (B) 0 (D) (E) nonexistent , then 2. If (A) 10 (C) 1 (B) 9 (C) 7 (D) 5 (E) 3 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -4- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 3. Which of the following definite integrals has the same value as ? (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -5- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 4. Which of the following is an equation of the line tangent to the graph of at the point ? (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -6- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 5. If g is the function given by , on which of the following intervals is g decreasing? (A) (B) (C) (D) (E) 6. (A) (B) (C) (D) Unauthorized copying or reuse of any part of this page is illegal. (E) GO ON TO THE NEXT PAGE. -7- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 7. Let f be the function given by of f at ? (A) (B) . What is the instantaneous rate of change (C) (D) 6 (E) 17 at time t is shown above 8. A particle moves along a straight line. The graph of the particle’s velocity , where j, k, l, and m are constants. The graph intersects the horizontal axis at , , for and and has horizontal tangents at and . For what values of t is the speed of the particle decreasing? (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -8- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA . For which of the following values of x is f not 9. Let f be the function given by continuous? and (A) (B) , only , and 2 (C) only (D) and 2 only (E) 2 only for time 10. A particle moves along the x-axis with velocity given by ? at time , what is the position of the particle at time position (A) 13 (B) 15 (C) 16 (D) 17 . If the particle is at (E) 25 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -9- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 11. Let f be the function defined by concave down? graph of (A) . On which of the following intervals is the only (B) (C) (D) (E) only only Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -10- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 12. For which of the following does ? I. II. III. (A) I only (B) II only (C) III only (D) I and II only (E) I and III only Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -11- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 13. Let f be a differentiable function such that possible value for ? (A) (B) (C) 0 for all x. Of the following, which is not a and (D) 1 (E) 2 14. Let f be the function given above. What are all values of a and b for which f is differentiable at (A) and (B) and (C) and b is any real number (D) ? , where b is any real number (E) There are no such values of a and b. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -12- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 15. The table above gives values for the functions f and g and their derivatives at given by (A) 16. If (A) 0. What is the value of , where (B) , then (B) (C) 2 at (D) 3 . Let k be the function ? (E) 8 is (C) (D) Unauthorized copying or reuse of any part of this page is illegal. (E) GO ON TO THE NEXT PAGE. -13- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA , which of the following could be the value of a ? 17. If (A) (B) (D) (E) 2 , then 18. If (A) (C) (B) (C) (D) Unauthorized copying or reuse of any part of this page is illegal. (E) GO ON TO THE NEXT PAGE. -14- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 19. The figure above shows the graph of the function g and the line tangent to the graph of g at . What is the value of the function given by (A) (B) (C) (D) . Let h be ? (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -15- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 20. For (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -16- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 21. The graph of a function f is shown above. What is the value of (A) 6 (B) 8 (C) 10 (D) 14 ? (E) 18 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -17- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 22. The function f is continuous for all real numbers, and the average rate of change of f on the closed interval is , there is no value of c such that . For . Of the following, which must be true? (A) (B) does not exist. (C) (D) for all x in the open interval . (E) f is not differentiable on the open interval 23. Let f be the function defined by ? of f, what is the value of (A) (B) . for all x and the point . If (C) (D) 3 Unauthorized copying or reuse of any part of this page is illegal. is on the graph (E) GO ON TO THE NEXT PAGE. -18- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 24. The function g is given by interval ? (A) (B) . What is the absolute minimum value of g on the closed (C) 0 (D) 2 (E) 6 25. Which of the following is the solution to the differential equation condition with the initial ? (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -19- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA ? 26. Which of the following is an antiderivative of (A) (B) (C) (D) (E) 27. For time , the height h of an object suspended from a spring is given by the average height of the object from (A) 16 (B) (C) to . What is ? (D) Unauthorized copying or reuse of any part of this page is illegal. (E) GO ON TO THE NEXT PAGE. -20- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 28. The function f is defined by for . What is the x-coordinate of the point of inflection where the graph of f changes from concave down to concave up? (A) (B) (C) (D) (E) END OF PART A OF SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY. DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO. Unauthorized copying or reuse of any part of this page is illegal. -21- B B B B B B B B B CALCULUS AB SECTION I, Part B Time—50 minutes Number of questions—17 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO QUESTIONS NUMBERED 76–92. YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET. In this exam: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which is a real number. (3) The inverse of a trigonometric function f may be indicated using the inverse function notation prefix “arc” (e.g., or with the ). Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -24- B B B B B B B B B 76. The graph of the function f shown above consists of two line segments and a semicircle. Let g be defined by . What is the value of (A) 0 (B) (C) ? (D) Unauthorized copying or reuse of any part of this page is illegal. (E) GO ON TO THE NEXT PAGE. -25- B B B B B B B B B 77. The volume of a sphere is decreasing at a constant rate of 3 cubic centimeters per second. At the instant when the radius of the sphere is decreasing at a rate of 0.25 centimeter per second, what is the radius of the sphere? (The volume V of a sphere with radius r is (A) 0.141 cm (B) 0.244 cm .) (C) 0.250 cm (D) 0.489 cm 78. Let f and g be continuous functions such that , and ? . What is the value of (A) 3 (B) 7 (C) 11 (D) 15 (E) 0.977 cm (E) 19 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -26- B B B 79. The figure above shows the graph of could be the graph of f ? B B B , the derivative of the function f . If (A) (B) (C) (D) B B B , which of the following (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -27- B B B B B B B B B 80. For time , the position of a particle traveling along a line is given by a differentiable function s. If s is and s is decreasing for , which of the following is the total distance the particle increasing for ? travels for (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -28- B B B B B B B B B . If the initial 81. A cup of tea is cooling in a room that has a constant temperature of 70 degrees Fahrenheit minutes, is and the temperature of the tea changes at the rate temperature of the tea, at time degrees Fahrenheit per minute, what is the temperature, to the nearest degree, of the tea after 4 minutes? (A) (B) (C) (D) Unauthorized copying or reuse of any part of this page is illegal. (E) GO ON TO THE NEXT PAGE. -29- B B B B B B 82. The derivative of the function f is given by B . On the interval B B , at which of the following values of x does f have a relative maximum? (A) and 0 (B) and 1.075 (C) , 0.542, and 1.396 (D) and 1.396 only (E) 0.542 only 0 0 0.5 4 1 10 1.5 18 2 28 2.5 40 3 54 83. The table above gives selected values for a continuous function f . If f is increasing over the closed interval , which of the following could be the value of (A) 50 (B) 62 (C) 77 ? (D) 100 (E) 154 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -30- B B B B B B B B B 84. The graph of a function f is shown in the figure above. Which of the following statements is true? (A) (B) f is continuous at . (C) (D) (E) does not exist. its position is given by 85. A particle moves along the x-axis so that at time velocity of the particle at the first instance the particle is at the origin? (A) (B) (C) (D) 0 Unauthorized copying or reuse of any part of this page is illegal. . What is the (E) 0.065 GO ON TO THE NEXT PAGE. -31- B B B B for all x and 86. If (A) B B B B B for all x, which of the following could be a table of values for f ? (B) (D) (C) 4 (E) 4 4 4 4 0 3 0 4 0 5 0 5 0 6 1 1 1 4 1 6 1 7 1 7 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -32- B B B B B B 87. Let f be the function with first derivative given by extrema does f have on the open interval (A) Two (B) Three (C) Four B B B . How many relative ? (D) Five Unauthorized copying or reuse of any part of this page is illegal. (E) Six GO ON TO THE NEXT PAGE. -33- B B B B B B B B B 88. The graph of a twice-differentiable function f is shown in the figure above. Which of the following is true? (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -34- B B B B B B B 89. What is the volume of the solid generated when the region bounded by the graph of and is revolved about the y-axis? (A) 3.464 (B) 4.500 (C) 7.854 (D) 10.883 B B and the lines (E) 14.137 90. The population P of a city grows according to the differential equation , where k is a constant and t is measured in years. If the population of the city doubles every 12 years, what is the value of k ? (A) 0.058 (B) 0.061 (C) 0.167 (D) 0.693 Unauthorized copying or reuse of any part of this page is illegal. (E) 8.318 GO ON TO THE NEXT PAGE. -35- B B B B 91. The function f is continuous and (A) (B) 3 (C) 6 B (B) f is not bounded on (D) 12 B B ? (E) 24 . If f does not attain a maximum value on , . . (C) f does not attain a minimum value on . (D) The graph of f has a vertical asymptote in the interval (E) The equation B . What is the value of 92. The function f is defined for all x in the closed interval which of the following must be true? (A) f is not continuous on B . does not have a solution in the interval Unauthorized copying or reuse of any part of this page is illegal. . GO ON TO THE NEXT PAGE. -36- Answer Key for AP Calculus AB Practice Exam, Section I Question 1: D Question 24: A Question 2: B Question 25: C Question 3: B Question 26: E Question 4: E Question 27: D Question 5: D Question 28: B Question 6: B Question 76: D Question 7: C Question 77: E Question 8: C Question 78: A Question 9: D Question 79: C Question 10: A Question 80: E Question 11: D Question 81: A Question 12: E Question 82: E Question 13: E Question 83: B Question 14: A Question 84: D Question 15: C Question 85: C Question 16: D Question 86: E Question 17: B Question 87: E Question 18: B Question 88: D Question 19: B Question 89: E Question 20: D Question 90: A Question 21: A Question 91: B Question 22: E Question 92: A Question 23: B ® AP Calculus AB Exam 2014 SECTION I: Multiple Choice DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Score 50% Writing Instrument Pencil required Part A Instructions Section I of this exam contains 45 multiple-choice questions and 4 survey questions. For Part A, fill in only the circles for numbers 1 through 28 on page 2 of the answer sheet. For Part B, fill in only the circles for numbers 76 through 92 on page 3 of the answer sheet. The survey questions are numbers 93 through 96. Indicate all of your answers to the multiple-choice questions on the answer sheet. No credit will be given for anything written in this exam booklet, but you may use the booklet for notes or scratch work. After you have decided which of the suggested answers is best, completely fill in the corresponding circle on the answer sheet. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely. Here is a sample question and answer. Number of Questions 28 Time 55 minutes Electronic Device None allowed Part B Number of Questions 17 Time 50 minutes Electronic Device Graphing calculator required Use your time effectively, working as quickly as you can without losing accuracy. Do not spend too much time on any one question. Go on to other questions and come back to the ones you have not answered if you have time. It is not expected that everyone will know the answers to all of the multiple-choice questions. Your total score on the multiple-choice section is based only on the number of questions answered correctly. Points are not deducted for incorrect answers or unanswered questions. Form I Form Code 4JBP6-S 66 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA CALCULUS AB SECTION I, Part A Time— 55 minutes Number of questions—28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. In this exam: (1) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x) is a real number. (2) The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the prefix “arc” (e.g., sin -1 x = arcsin x ). Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -3- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Ú2 (3t x 1. 2 ) - 1 dt = (A) x 3 - x - 6 (B) x 3 - x (C) 3 x 2 - 12 (D) 3 x 2 - 1 (E) 6 x - 12 2. What is the slope of the line tangent to the graph of y = ln (2 x ) at the point where x = 4 ? (A) 1 8 (B) 1 4 (C) 1 2 (D) 3 4 (E) 4 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -4- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 3. If f ( x ) = 4 x -2 + (A) -62 4. 2 1 2 x + 4, then f ¢(2 ) = 4 (B) -58 (C) -3 (D) 0 (E) 1 dx Ú1 2 x + 1 = (A) 2 ln 2 (B) 1 ln 2 2 (C) 2 (ln 5 - ln 3) Unauthorized copying or reuse of any part of this page is illegal. (D) ln 5 - ln 3 (E) 1 (ln 5 - ln 3) 2 GO ON TO THE NEXT PAGE. -5- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 5. The figure above shows the graph of the function f . Which of the following statements are true? I. lim f ( x ) = f (2 ) x Æ 2- II. lim f ( x ) = lim f ( x ) x Æ 6- III. x Æ6+ lim f ( x ) = f (6 ) x Æ6 (A) II only (B) III only (C) I and II only (D) II and III only (E) I, II, and III Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -6- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA ( ( )) = d sin3 x 2 dx 6. ( ) 3sin 2 ( x 2 ) 6 x sin 2 ( x 2 ) 3sin 2 ( x 2 ) cos ( x 2 ) 6 x sin 2 ( x 2 ) cos ( x 2 ) (A) cos3 x 2 (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -7- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA x3 is x Æ • e3 x 7. lim (A) 0 (B) 2 9 (C) 2 3 8. Using the substitution u = sin (2 x ) , (A) -2 1 Ú1 2 u 5 1 1 5 u du 2 Ú1 2 (C) 1 3 2 5 u du 2 Ú0 (D) 1 0 u 5 du 2Ú 3 2 0 Ú 3 2u 5 p 2 Úp 6 (E) infinite sin 5 (2 x ) cos (2 x ) dx is equivalent to du (B) (E) 2 (D) 1 du Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -8- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 9. The function f has a first derivative given by f ¢( x ) = x ( x - 3) ( x + 1) . At what values of x does f have a relative maximum? 2 (A) -1 only (C) -1 and 0 only (B) 0 only (D) -1 and 3 only Ï x 2 - 7 x + 10 Ô f ( x ) = Ì b ( x - 2) ÔÓ b (E) -1, 0, and 3 for x π 2 for x = 2 10. Let f be the function defined above. For what value of b is f continuous at x = 2 ? (A) -3 (B) 2 (C) 3 (D) 5 (E) There is no such value of b. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -9- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 11. For 0 £ x £ 6, the graph of f ¢, the derivative of f, is piecewise linear as shown above. If f (0 ) = 1, what is the maximum value of f on the interval? (A) 1 (B) 1.5 (C) 2 (D) 4 (E) 6 12. Let f be the function given by f ( x ) = 9 x. If four subintervals of equal length are used, what is the value of the right Riemann sum approximation for (A) 20 (B) 40 (C) 60 2 Ú0 f ( x ) dx ? (D) 80 (E) 120 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -10- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA d Ê x +1 ˆ = dx ÁË x 2 + 1 ˜¯ 13. (A) (B) x2 + 2 x - 1 ( x2 + 1) 2 - x2 - 2 x + 1 x2 + 1 (C) - x2 - 2 x + 1 (D) 3x2 + 2 x + 1 (E) ( x2 + 1) ( x2 + 1) 2 2 1 2x Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -11- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 14. The velocity of a particle moving along the x-axis is given by v(t ) = sin (2t ) at time t. If the particle is at x = 4 p when t = 0, what is the position of the particle when t = ? 2 (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -12- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 15. The function y = g( x ) is differentiable and increasing for all real numbers. On what intervals is the function ( ) y = g x 3 - 6 x 2 increasing? (A) ( - •, 0] and [ 4, • ) only (B) [0, 4] only (C) [2, • ) only (D) [6, • ) only (E) 16. ( -•, • ) lim x Æ 3- (A) -3 x-3 is x-3 (B) -1 (C) 1 (D) 3 (E) nonexistent Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -13- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 17. If f ( x ) = ae - ax for a > 0, then f ¢( x ) = (A) e - ax (B) ae - ax (C) a 2 e - ax (D) - ae - ax (E) - a 2e - ax Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -14- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 18. A student attempted to solve the differential equation dy = xy with initial condition y = 2 when x = 0. In dx which step, if any, does an error first appear? 1 Step 1: Û Ù dy = ıy x2 +C 2 Step 2: ln y = Step 3: y = e x Ú x dx 2 2 +C Step 4: Since y = 2 when x = 0, 2 = e0 + C. Step 5: y = e x 2 2 +1 (A) Step 2 (B) Step 3 (C) Step 4 (D) Step 5 (E) There is no error in the solution. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -15- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 19. For what values of x does the graph of y = 3 x 5 + 10x 4 have a point of inflection? (A) x = - 8 only 3 (B) x = -2 only (C) x = 0 only (D) x = 0 and x = - 8 3 (E) x = 0 and x = -2 20. ln ( x + 3) - ln (5) is x-2 xÆ2 lim (A) 0 (B) 1 5 (C) 1 2 (D) 1 (E) nonexistent Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -16- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 21. Functions w, x, and y are differentiable with respect to time and are related by the equation w = x 2 y. If x is decreasing at a constant rate of 1 unit per minute and y is increasing at a constant rate of 4 units per minute, at what rate is w changing with respect to time when x = 6 and y = 20 ? (A) -384 (B) -240 (C) -96 (D) 276 (E) 384 22. Let f be the function defined by f ( x ) = 2 x 3 - 3 x 2 - 12 x + 18. On which of the following intervals is the graph of f both decreasing and concave up? (A) ( -•, -1) (B) (-1, 21 ) (C) ( -1, 2) Unauthorized copying or reuse of any part of this page is illegal. (D) ( 12 , 2) (E) (2, • ) GO ON TO THE NEXT PAGE. -17- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA when x < -1 Ï3 x + 5 f (x) = Ì 2 Ó - x + 3 when x ≥ -1 23. If f is the function defined above, then f ¢( -1) is (A) -3 (B) -2 (C) 2 24. Let f be the function defined by f ( x ) = (D) 3 (E) nonexistent (3 x + 8)(5 - 4 x ) . Which of the following is a horizontal asymptote to (2 x + 1)2 the graph of f ? (A) y = - 6 (B) y = -3 (C) y = - 1 2 (D) y = 0 (E) y = 3 2 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -18- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 25. If y = x 2 - 2 x and u = 2 x + 1, then (A) ( (2 x + 1) 26. For x > 0, (A) ) 2 x2 + x - 1 d dx (B) 6 x 2 - 3 x - 2 2 Ú1 x dy = du (C) 4x (D) x - 1 (E) 1 x -1 1 dt = 1 + t2 1 2 x (1 + x ) (B) 1 2 x (1 + x ) (C) 1 1+ x Unauthorized copying or reuse of any part of this page is illegal. (D) x 1+ x (E) 1 1+ x GO ON TO THE NEXT PAGE. -19- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 27. A particle moves on the x-axis so that at any time t, 0 £ t £ 1, its position is given by x (t ) = sin (2 p t ) + 2 p t. For what value of t is the particle at rest? (A) 0 (B) 1 8 (C) 1 4 (D) 1 2 (E) 1 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -20- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 28. Shown above is a slope field for which of the following differential equations? (A) dy = xy - x dx (B) dy = xy + x dx (C) dy = y - x2 dx (D) dy = ( y - 1) x 2 dx (E) dy 3 = ( y - 1) dx END OF PART A OF SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY. DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO. Unauthorized copying or reuse of any part of this page is illegal. -21- B B B B B B B B B CALCULUS AB SECTION I, Part B Time— 50 minutes Number of questions—17 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO QUESTIONS NUMBERED 76–92. YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET. In this exam: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x) is a real number. (3) The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the prefix “arc” (e.g., sin -1 x = arcsin x ). Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -24- B B B B B B B B B 76. A particle moves along a straight line so that at time t > 0 the position of the particle is given by s(t ) , the velocity is given by v(t ) , and the acceleration is given by a(t ) . Which of the following expressions gives the average velocity of the particle on the interval [2, 8] ? (A) 1 8 a(t ) dt 6 Ú2 (B) 1 8 s(t ) dt 6 Ú2 (C) s(8) - s(2 ) 6 (D) v(8) - v(2 ) 6 (E) v(8) - v(2 ) Ê 1 ˆ 77. If sin Á 2 is an antiderivative for f ( x ) , then Ë x + 1 ˜¯ (A) - 0.281 (B) - 0.102 2 Ú1 f ( x ) dx = (C) 0.102 (D) 0.260 Unauthorized copying or reuse of any part of this page is illegal. (E) 0.282 GO ON TO THE NEXT PAGE. -25- B B B B B B B B B 78. The function f is differentiable and increasing for all real numbers x, and the graph of f has exactly one point of inflection. Of the following, which could be the graph of f ¢, the derivative of f ? (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -26- B B B B B B B B B 79. A vase has the shape obtained by revolving the curve y = 2 + sin x from x = 0 to x = 5 about the x-axis, where x and y are measured in inches. What is the volume, in cubic inches, of the vase? (A) 10.716 (B) 25.501 (C) 33.666 (D) 71.113 x f (x) 1 2.4 3 3.6 5 5.4 (E) 80.115 80. The table above gives selected values of a function f. The function is twice differentiable with f ¢¢( x ) > 0. Which of the following could be the value of f ¢(3) ? (A) 0.6 (B) 0.7 (C) 0.9 (D) 1.2 Unauthorized copying or reuse of any part of this page is illegal. (E) 1.5 GO ON TO THE NEXT PAGE. -27- B B B B B B B B B 81. At time t = 0 years, a forest preserve has a population of 1500 deer. If the rate of growth of the population is modeled by R(t ) = 2000e0.23t deer per year, what is the population at time t = 3 ? (A) 3987 (B) 5487 (C) 8641 (D) 10,141 Unauthorized copying or reuse of any part of this page is illegal. (E) 12,628 GO ON TO THE NEXT PAGE. -28- B B B B B B B B B 82. The figure above shows the graph of f ¢, the derivative of function f, for -6 < x < 8. Of the following, which best describes the graph of f on the same interval? (A) 1 relative minimum, 1 relative maximum, and 3 points of inflection (B) 1 relative minimum, 1 relative maximum, and 4 points of inflection (C) 2 relative minima, 1 relative maximum, and 2 points of inflection (D) 2 relative minima, 1 relative maximum, and 4 points of inflection (E) 2 relative minima, 2 relative maxima, and 3 points of inflection Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -29- B B B B 83. Let f and g be continuous functions such that 1 f ( x ) - 3g ( x )) dx ? ( ı 2 the value of Û Ù B 6 B 6 B B B 0 Ú0 f ( x ) dx = 9, Ú3 f ( x ) dx = 5, and Ú3 g( x ) dx = -7. What is 3 0 (A) -23 (B) -19 (C) - 17 2 (D) 19 Unauthorized copying or reuse of any part of this page is illegal. (E) 23 GO ON TO THE NEXT PAGE. -30- B B B B B B B B B 84. The regions A, B, and C in the figure above are bounded by the graph of the function f and the x-axis. The area of region A is 14, the area of region B is 16, and the area of region C is 50. What is the average value of f on the interval [ 0, 8] ? (A) 6 (B) 10 (C) 40 3 (D) 80 3 (E) 48 85. A particle moves along the x-axis so that its velocity at time t ≥ 0 is given by v(t ) = distance traveled by the particle from t = 0 to t = 2 ? (A) 0.214 (B) 0.320 (C) 0.600 (D) 0.927 Unauthorized copying or reuse of any part of this page is illegal. t2 - 1 . What is the total t2 + 1 (E) 1.600 GO ON TO THE NEXT PAGE. -31- B B B B B ( B B B B ) 86. Line is tangent to the graph of y = e x at the point k , e k . What is the positive value of k for which the y-intercept of is 1 ? 2 (A) 0.405 (B) 0.768 (C) 1.500 (D) 1.560 (E) There is no such value of k. 87. A differentiable function f has the property that f ¢( x ) £ 3 for 1 £ x £ 8 and f (5) = 6. Which of the following could be true? I. f (2 ) = 0 II. f (6 ) = -2 III. f (7) = 13 (A) I only (B) II only (C) I and II only (D) I and III only (E) II and III only Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -32- B B B B B B B B B 88. The graph of the differentiable function f is shown in the figure above. Let h be the function defined by h( x ) = x Ú0 f (t ) dt. Which of the following correctly orders h(2), h¢(2 ) , and h¢¢(2 ) ? (A) h(2 ) < h ¢(2 ) < h¢¢(2 ) (B) h ¢(2 ) < h(2 ) < h¢¢(2 ) (C) h ¢(2 ) < h ¢¢(2 ) < h(2 ) (D) h ¢¢(2 ) < h(2 ) < h ¢(2 ) (E) h ¢¢(2 ) < h ¢(2 ) < h(2 ) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -33- B B B B B B B B B 89. What is the area of the region enclosed by the graphs of y = e x - 2, y = sin x, and x = 0 ? (A) 0.239 (B) 0.506 (C) 0.745 (D) 2.340 (E) 3.472 90. A particle moves along a line so that its velocity is given by v(t ) = -t 3 + 2t 2 + 2 - t for t ≥ 0. For what values of t is the speed of the particle increasing? (A) (0, 0.177) and (1.256, • ) (B) (0, 1.256 ) only (C) (0, 2.057) only (D) (0.177, 1.256 ) only (E) (0.177, 1.256) and (2.057, • ) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -34- B B B B B B B B B 91. Let F be a function defined for all real numbers x such that F ¢( x ) > 0 and F ¢¢( x ) > 0. Which of the following could be a table of values for F ? (A) x 1 2 3 4 F(x) –3 –4 –6 –9 (B) x 1 2 3 4 F(x) –3 –1 3 19 (C) x 1 2 3 4 F(x) –3 0 3 6 (D) x 1 2 3 4 F(x) –3 5 11 13 (E) x 1 2 3 4 F(x) –3 –4 –3 0 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -35- B B B B B B x f (x) g( x ) f ¢( x ) –4 0 –9 5 –2 4 –7 4 0 6 –4 2 2 7 –3 1 4 10 –2 3 B B B 92. The table above gives values of the differentiable functions f and g, and f ¢, the derivative of f, at selected values of x. If g( x ) = f -1 ( x ) , what is the value of g¢( 4 ) ? (A) - 1 3 (B - 1 4 (C) - 3 100 (D Unauthorized copying or reuse of any part of this page is illegal. 1 4 (E) 1 3 GO ON TO THE NEXT PAGE. -36- Question 1: A Question 24: B Question 2: B Question 25: D Question 3: D Question 26: A Question 4: E Question 27: D Question 5: C Question 28: A Question 6: E Question 76: C Question 7: A Question 77: A Question 8: D Question 78: A Question 9: A Question 79: E Question 10: E Question 80: B Question 11: D Question 81: D Question 12: C Question 82: A Question 13: C Question 83: B Question 14: D Question 84: A Question 15: A Question 85: D Question 16: B Question 86: B Question 17: E Question 87: C Question 18: B Question 88: E Question 19: B Question 89: C Question 20: B Question 90: E Question 21: C Question 91: B Question 22: D Question 92: D Question 23: E ® AP Calculus AB Exam 2015 SECTION I: Multiple Choice DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Score 50% Writing Instrument Pencil required Part A Instructions Section I of this exam contains 45 multiple-choice questions and 4 survey questions. For Part A, fill in only the circles for numbers 1 through 28 on page 2 of the answer sheet. For Part B, fill in only the circles for numbers 76 through 92 on page 3 of the answer sheet. The survey questions are numbers 93 through 96. Indicate all of your answers to the multiple-choice questions on the answer sheet. No credit will be given for anything written in this exam booklet, but you may use the booklet for notes or scratch work. After you have decided which of the suggested answers is best, completely fill in the corresponding circle on the answer sheet. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely. Here is a sample question and answer. Number of Questions 28 Time 55 minutes Electronic Device None allowed Part B Number of Questions 17 Time 50 minutes Electronic Device Graphing calculator required Use your time effectively, working as quickly as you can without losing accuracy. Do not spend too much time on any one question. Go on to other questions and come back to the ones you have not answered if you have time. It is not expected that everyone will know the answers to all of the multiple-choice questions. Your total score on the multiple-choice section is based only on the number of questions answered correctly. Points are not deducted for incorrect answers or unanswered questions. Form I Form Code 4KBP6-S 66 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA CALCULUS AB SECTION I, Part A Time— 55 minutes Number of questions—28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. In this exam: (1) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x) is a real number. (2) The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the prefix “arc” (e.g., sin -1 x = arcsin x ). Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -3- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA ( ) 1 Û dx = Ù 5e2 x + x ı 1. (A) 5 2x 2 e + 2 +C 2 x (B) 5 2x e + ln x + C 2 (C) 5e 2 x + 2 +C x2 (D) 5e 2 x + ln x + C (E) 10e 2 x - 1 +C x2 2. If f ( x ) = x + (A) 1 16 3 , then f ¢( 4) = x (B) 5 16 (C) 1 (D) 7 2 (E) Unauthorized copying or reuse of any part of this page is illegal. 49 4 GO ON TO THE NEXT PAGE. -4- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 2 3 Ú x ( x + 5) 6 3. ( ) dx = (A) 1 3 x +5 3 (B) 1 3 1 4 x x + 5x 3 4 (C) 1 3 x +5 7 (D) 3 2 3 x x +5 7 (E) 1 3 x +5 21 6 +C ( ( ) 7 ( ( ) ) 6 +C +C ) 7 7 +C +C Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -5- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA x 0 25 30 50 f (x) 4 6 8 12 4. The values of a continuous function f for selected values of x are given in the table above. What is the value of the left Riemann sum approximation to (A) 290 (B) 360 (C) 380 50 Ú0 f ( x ) dx using the subintervals [0, 25], [25, 30], and [30, 50] ? (D) 390 (E) 430 ÏÔ x 2 sin ( p x ) for x < 2 f (x) = Ì 2 ÔÓ x + cx - 18 for x ≥ 2 5. Let f be the function defined above, where c is a constant. For what value of c, if any, is f continuous at x = 2 ? (A) 2 (B) 7 (C) 9 (D) 4p - 4 (E) There is no such value of c. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -6- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 6. Which of the following is an antiderivative of 3sec 2 x + 2 ? (A) 3 tan x (B) 3tan x + 2 x (C) 3sec x + 2 x (D) sec3 x + 2 x (E) 6sec 2 x tan x 7. If f ( x ) = x 2 - 4 and g is a differentiable function of x, what is the derivative of f ( g ( x )) ? (A) 2 g ( x ) (B) 2 g ¢ ( x ) (C) 2 xg ¢ ( x ) (D) 2 g ( x ) g ¢ ( x ) Unauthorized copying or reuse of any part of this page is illegal. (E) 2 g ( x ) - 4 GO ON TO THE NEXT PAGE. -7- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA ( ) dy = y 2 4 - y 2 . If y = g( x ) is the solution to the dx differential equation with the initial condition g( -2 ) = -1, then lim g( x ) is 8. Shown above is a slope field for the differential equation x Æ• (A) - • (B) -2 (C) 0 (D) 2 (E) 3 9. If f ¢¢ ( x ) = x ( x + 2 ) , then the graph of f is concave up for 2 (A) x < 0 (B) x > 0 (C) -2 < x < 0 (D) x < -2 and x > 0 (E) all real numbers Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -8- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 10. If y = sin x cos x, then at x = (A) - 11. 1 2 (B) - 1 4 p dy , = 3 dx (C) 1 4 (D) 1 2 (E) 1 x2 - 9 is xÆ-3 x 2 - 2x - 15 lim (A) 0 (B) 3 5 (C) 3 4 (D) 1 (E) nonexistent Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -9- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA p 12. What is the average rate of change of y = cos ( 2 x ) on the interval È 0, ˘ ? ÎÍ 2 ˙˚ (A) - 4 p (B) -1 13. If y3 + y = x 2 , then (A) 0 (B) (C) 0 (D) 2 2 (E) 4 p dy = dx x 2 (C) 2x 3y 2 (D) 2x - 3y 2 Unauthorized copying or reuse of any part of this page is illegal. (E) 2x 1 + 3y 2 GO ON TO THE NEXT PAGE. -10- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 14. The graph of y = f ( x ) on the closed interval [0, 4] is shown above. Which of the following could be the graph of y = f ¢( x ) ? (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -11- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA if x < 1 Ï 3x - 2 f (x) = Ì Ó ln (3 x - 2 ) if x ≥ 1 15. Let f be the function defined above. Which of the following statements about f are true? I. lim f ( x ) = lim f ( x ) xÆ1- xÆ1+ II. lim f ¢( x ) = lim f ¢( x ) xÆ1- xÆ1+ III. f is differentiable at x = 1. (A) None (B) I only (C) II only (D) II and III only (E) I, II, and III Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -12- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 16. The function f is defined by f ( x ) = 2x 3 - 4x 2 + 1. The application of the Mean Value Theorem to f on the interval 1 £ x £ 3 guarantees the existence of a value c, where 1 < c < 3, such that f ¢( c ) = (A) 0 (B) 9 (C) 10 (D) 14 (E) 16 17. The velocity v, in meters per second, of a certain type of wave is given by v(h ) = 3 h , where h is the depth, in meters, of the water through which the wave moves. What is the rate of change, in meters per second per meter, of the velocity of the wave with respect to the depth of the water, when the depth is 2 meters? (A) - 3 4 2 (B) - 3 8 2 (C) 3 2 2 (D) Unauthorized copying or reuse of any part of this page is illegal. 3 2 (E) 4 2 GO ON TO THE NEXT PAGE. -13- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 18. If dy = -10e - t dt 2 (A) 20 e - 6 and y (0 ) = 20, what is the value of y (6 ) ? (B) 20 e -3 (C) 20 e -2 (D) 10 e -3 (E) 5e -3 19. Let f be the function with derivative defined by f ¢( x ) = x 3 - 4x. At which of the following values of x does the graph of f have a point of inflection? (A) 0 (B) 2 3 (C) 2 3 (D) 4 3 (E) 2 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -14- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 20. Let f be the function given by f ( x ) = ( x - 4 )(2 x - 3) . If the line y = b is a horizontal asymptote to the graph ( x - 1)2 of f, then b = (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 21. The base of a solid is the region bounded by the x-axis and the graph of y = 1 - x 2 . For the solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid? (A) 2 3 (B) 4 3 (C) 2 (D) 2p 3 (E) Unauthorized copying or reuse of any part of this page is illegal. 4p 3 GO ON TO THE NEXT PAGE. -15- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA kx , where k is a constant. For what values of k, if any, x +1 is f strictly decreasing on the interval ( -1, 1) ? 22. Let f be the function given by f ( x ) = 2 (A) k < 0 (B) k = 0 (C) k > 0 (D) k > 1 only (E) There are no such values of k. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -16- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 23. Which of the following is an equation for the line tangent to the graph of y = 3 where x = -1 ? x Ú-1 e -t3 dt at the point (A) y - 3 = -3e ( x + 1) (B) y - 3 = -e ( x + 1) (C) y - 3 = 0 (D) y - 3 = 1 ( x + 1) e (E) y - 3 = 3e ( x + 1) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -17- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 24. Which of the following is the solution to the differential equation y( 0 ) = 3 ? dy = 5y 2 with the initial condition dx (A) y = 9e 5 x (B) y = 1 5x e 9 (C) y = e 5x + 9 (D) y = 3 1 - 15 x (E) y = 3 1 + 15 x Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -18- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA sin 25. lim hÆ0 (A) 0 ( p3 + h) - sin ( p3 ) is h (B) 1 2 (C) 1 (D) 3 2 (E) nonexistent 26. An object moves along a straight line so that at any time t ≥ 0 its velocity is given by v(t ) = 2 cos (3t ) . What is the distance traveled by the object from t = 0 to the first time that it stops? (A) 0 (B) p 6 (C) 2 3 (D) p 3 (E) Unauthorized copying or reuse of any part of this page is illegal. 4 3 GO ON TO THE NEXT PAGE. -19- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA x f (x) f ¢( x ) 0 49 0 1 2 –8 2 –1 –80 27. The table above gives selected values for a differentiable and decreasing function f and its derivative. If f -1 is ( ) the inverse function of f, what is the value of f -1 ¢ (2 ) ? (A) -80 (B) - 1 8 (C) - 1 80 (D) 1 80 Unauthorized copying or reuse of any part of this page is illegal. (E) 1 8 GO ON TO THE NEXT PAGE. -20- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 28. The top of a 15-foot-long ladder rests against a vertical wall with the bottom of the ladder on level ground, as shown above. The ladder is sliding down the wall at a constant rate of 2 feet per second. At what rate, in radians per second, is the acute angle between the bottom of the ladder and the ground changing at the instant the bottom of the ladder is 9 feet from the base of the wall? (A) - 2 9 (B) - 1 6 (C) - 2 25 (D) 2 25 (E) 1 9 END OF PART A OF SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY. DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO. Unauthorized copying or reuse of any part of this page is illegal. -21- B B B B B B B B B CALCULUS AB SECTION I, Part B Time— 50 minutes Number of questions—17 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO QUESTIONS NUMBERED 76–92. YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET. In this exam: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x) is a real number. (3) The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the prefix “arc” (e.g., sin -1 x = arcsin x ). Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -24- B B B B B B B B B 76. The function P (t ) models the population of the world, in billions of people, where t is the number of years since January 1, 2010. Which of the following is the best interpretation of the statement P¢(1) = 0.076 ? (A) On February 1, 2010, the population of the world was increasing at a rate of 0.076 billion people per year. (B) On January 1, 2011, the population of the world was increasing at a rate of 0.076 billion people per year. (C) On January 1, 2011, the population of the world was 0.076 billion people. (D) From January 1, 2010 to January 1, 2011, the population of the world was increasing at an average rate of 0.076 billion people per year. (E) When the population of the world was 1 billion people, the population of the world was increasing at a rate of 0.076 billion people per year. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -25- B B B B B B B x 0 2 4 6 8 10 f (x) 5 7 8 0 –15 –20 B B 77. Let f be a differentiable function with selected values given in the table above. What is the average rate of change of f over the closed interval 0 £ x £ 10 ? (A) -6 (B) - 5 2 (C) -2 (D) - 2 5 (E) 2 5 78. The rate at which motor oil is leaking from an automobile is modeled by the function L defined by L (t ) = 1 + sin t 2 for time t ≥ 0. L (t ) is measured in liters per hour, and t is measured in hours. How ( ) much oil leaks out of the automobile during the first half hour? (A) 1.998 liters (B) 1.247 liters (C) 0.969 liters (D) 0.541 liters (E) 0.531 liters Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -26- B B B B x 0 B f (x) 3 f ¢( x ) 4 B g( x ) 2 B B B g ¢( x ) p 79. The table above gives values of the differentiable functions f and g and their derivatives at x = 0. f (x) , what is the value of h ¢( 0 ) ? If h( x ) = g( x ) (A) 8 - 3p 4 (B) 3p - 8 4 (C) 4 p (D) Unauthorized copying or reuse of any part of this page is illegal. 2 - 3p 2 (E) 8 + 3p 4 GO ON TO THE NEXT PAGE. -27- B B B B B B B B B 80. The figure above shows the graph of f ¢, the derivative of a function f, for 0 £ x £ 2. What is the value of x at which the absolute minimum of f occurs? (A) 0 (B) 1 2 (C) 1 (D) 3 2 (E) 2 81. What is the area of the region enclosed by the graphs of y = 4 x - x 2 and y = (A) 1.707 (B) 2.829 (C) 5.389 (D) 8.886 Unauthorized copying or reuse of any part of this page is illegal. x ? 2 (E) 21.447 GO ON TO THE NEXT PAGE. -28- B B B B B B B B B 82. The graph of f ¢, the derivative of f, is shown above. The line tangent to the graph of f ¢ at x = 0 is vertical, and f ¢ is not differentiable at x = 2. Which of the following statements is true? (A) f ¢ does not exist at x = 2. (B) f is decreasing on the interval (2, 4 ) . (C) The graph of f has a point of inflection at x = 2. (D) The graph of f has a point of inflection at x = 0. (E) f has a local maximum at x = 0. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -29- B B B B B B B B B 83. A particle moves along the x-axis so that its position at time t > 0 is given by x (t ) and dx = -10t 4 + 9t 2 + 8t. The acceleration of the particle is zero when t = dt (A) 0.387 (B) 0.831 (C) 1.243 (D) 1.647 84. The function f is continuous on the closed interval [1, 7]. If then 7 Ú1 (E) 8.094 f ( x ) dx = 42 and 3 Ú7 f ( x ) dx = -32, 3 Ú1 2 f ( x ) dx = (A) -148 (B) 10 (C) 12 (D) 20 Unauthorized copying or reuse of any part of this page is illegal. (E) 148 GO ON TO THE NEXT PAGE. -30- B B B B B B B B B 85. Let y = f ( x) define a twice-differentiable function and let y = t ( x) be the line tangent to the graph of f at x = 2. If t ( x ) ≥ f ( x) for all real x, which of the following must be true? (A) f (2) ≥ 0 (B) f ¢(2) ≥ 0 (C) f ¢(2) £ 0 (D) f ¢¢(2) ≥ 0 (E) f ¢¢(2) £ 0 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -31- B B B B B B B B B 86. The vertical line x = 2 is an asymptote for the graph of the function f. Which of the following statements must be false? (A) lim f ( x ) = 0 xÆ2 (B) lim f ( x ) = - • xÆ2 (C) lim f ( x ) = • xÆ2 (D) lim f ( x ) = 2 xÆ • (E) lim f ( x ) = • xÆ • Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -32- B B B B B B B B 87. The graph of the piecewise linear function f is shown above. Let h be the function given by h( x ) = B x Ú-1 f (t ) dt. On which of the following intervals is h increasing? (A) [ -1, 3] (B) [0, 5] (C) [2, 5] only (D) [2, 9] (E) [3, 9] only Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -33- B B B B B B B B B ( ) 88. The first derivative of the function f is given by f ¢ ( x ) = sin x 2 . At which of the following values of x does f have a local minimum? (A) 2.507 (B) 2.171 (C) 1.772 (D) 1.253 (E) 0 89. If lim f ( x ) = f (a ) , then which of the following statements about f must be true? xÆa (A) f is continuous at x = a. (B) f is differentiable at x = a. (C) For all values of x, f ( x ) = f (a ) . (D) The line y = f (a ) is tangent to the graph of f at x = a. (E) The line x = a is a vertical asymptote of the graph of f. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -34- B B B B B B B B B 90. The temperature F, in degrees Fahrenheit (∞F ) , of a cup of coffee t minutes after it is poured is given by F (t ) = 72 + 118e -0.093t . To the nearest degree, what is the average temperature of the coffee between t = 0 and t = 10 minutes? (A) 93∞F (B) 119∞F (C) 146∞F (D) 149∞F (E) 154∞F ( ) 91. If f ¢( x ) = cos x 2 and f (3) = 7, then f (2 ) = (A) 0.241 (B) 5.831 (C) 6.416 (D) 6.759 Unauthorized copying or reuse of any part of this page is illegal. (E) 7.241 GO ON TO THE NEXT PAGE. -35- B B B B B B B B B 92. The graph of the function h is shown in the figure above. Of the following, which has the greatest value? (A) Average value of h over [ -3,2] (B) Average rate of change of h over [ -3,2] 2 (C) Ú-3 h ( x ) dx (D) Ú-3 h ( x ) dx 0 (E) h ¢ (0 ) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -36- Answer Key for AP Calculus AB Practice Exam, Section I Question 24: D Question 2: A Question 25: B Question 3: E Question 26: C Question 4: A Question 27: B Question 5: B Question 28: A Question 6: B Question 76: B Question 7: D Question 77: B Question 8: C Question 78: D Question 9: B Question 79: A Question 10: A Question 80: E Question 11: C Question 81: B Question 12: A Question 82: C Question 13: E Question 83: B Question 14: D Question 84: D Question 15: C Question 85: E Question 16: C Question 86: A Question 17: C Question 87: E Question 18: B Question 88: A Question 19: C Question 89: A Question 20: C Question 90: D Question 21: B Question 91: D Question 22: A Question 92: B Question 23: B ® AP Calculus AB Exam 2016 SECTION I: Multiple Choice DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Score 50% Writing Instrument Pencil required Part A Instructions Section I of this exam contains 45 multiple-choice questions and 4 survey questions. For Part A, fill in only the circles for numbers 1 through 28 on page 2 of the answer sheet. For Part B, fill in only the circles for numbers 76 through 92 on page 3 of the answer sheet. The survey questions are numbers 93 through 96. Indicate all of your answers to the multiple-choice questions on the answer sheet. No credit will be given for anything written in this exam booklet, but you may use the booklet for notes or scratch work. After you have decided which of the suggested answers is best, completely fill in the corresponding circle on the answer sheet. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely. Here is a sample question and answer. Number of Questions 28 Time 55 minutes Electronic Device None allowed Part B Number of Questions 17 Time 50 minutes Electronic Device Graphing calculator required Use your time effectively, working as quickly as you can without losing accuracy. Do not spend too much time on any one question. Go on to other questions and come back to the ones you have not answered if you have time. It is not expected that everyone will know the answers to all of the multiple-choice questions. Your total score on the multiple-choice section is based only on the number of questions answered correctly. Points are not deducted for incorrect answers or unanswered questions. Form I Form Code 4LBP6-S 66 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA CALCULUS AB SECTION I, Part A Time— 55 minutes Number of questions—28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. In this exam: (1) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x) is a real number. (2) The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the prefix “arc” (e.g., sin -1 x = arcsin x ). Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -3- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA dy = dx 1. If y = cos 2 x, then (A) -2sin 2x (B) - sin 2x 2 3 Ú x ( x - 1) 10 2. (A) ˆ x3 Ê x 4 - x˜ 3 ÁË 4 ¯ ( x3 - 1) (C) sin 2x (D) 2sin 2x (E) 2sin x dx = 10 +C 11 (B) (C) +C 11 ( ) x 2 x3 - 1 11 11 ( x3 - 1) +C 11 (D) (E) +C 33 ( ) x3 x3 - 1 33 11 +C Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -4- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 9x4 + 1 is x Æ• 4 x 2 + 3 lim 3. (A) 1 3 4. If y = (B) 3 4 (C) 3 2 (D) 9 4 (E) infinite ( ) dy x 5 , then = x +1 dx (A) 5 (1 + x ) 4 (B) x4 ( x + 1)4 (C) 5x 4 ( x + 1)4 Unauthorized copying or reuse of any part of this page is illegal. (D) 5x 4 ( x + 1)6 (E) 5 x 4 (2 x + 1) ( x + 1)6 GO ON TO THE NEXT PAGE. -5- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA t (minutes) r (t ) (gallons per minute) 0 4 7 9 9 6 4 3 5. Water is flowing into a tank at the rate r (t ) , where r (t ) is measured in gallons per minute and t is measured in minutes. The tank contains 15 gallons of water at time t = 0. Values of r (t ) for selected values of t are given in the table above. Using a trapezoidal sum with the three intervals indicated by the table, what is the approximation of the number of gallons of water in the tank at time t = 9 ? (A) 52 (B) 57 (C) 67 (D) 77 (E) 79 6. The slope of the line tangent to the graph of y = ln (1 - x ) at x = -1 is (A) -1 (B) - 1 2 (C) 1 2 (D) ln 2 Unauthorized copying or reuse of any part of this page is illegal. (E) 1 GO ON TO THE NEXT PAGE. -6- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 7. For which of the following pairs of functions f and g is lim x Æ• f (x) infinite? g( x ) (A) f ( x ) = x 2 + 2 x and g( x ) = x 2 + ln x (B) f ( x ) = 3 x 3 and g( x ) = x 4 (C) f ( x ) = 3 x and g( x ) = x 3 (D) f ( x ) = 3e x + x 3 and g( x ) = 2e x + x 2 (E) f ( x ) = ln (3 x ) and g( x ) = ln ( 2 x ) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -7- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Ú 8. 4 0 x x +9 2 (A) -2 dx = (B) - 2 15 (C) 1 (D) 2 (E) 5 9. Let f be the function with derivative given by f ¢( x ) = -2 x (1 + x 2 ) 2 . On what interval is f decreasing? (A) [ 0, •) only (B) ( - •, 0] only 1 1 ˘ , (C) È only ÍÎ 3 3 ˙˚ (D) ( - •, •) (E) There is no such interval. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -8- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 10. Ú (e x ) + e dx = (A) e x + C (B) 2e x + C (C) e x + e + C Unauthorized copying or reuse of any part of this page is illegal. (D) e x +1 + ex + C (E) e x + ex + C GO ON TO THE NEXT PAGE. -9- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 11. The graph of the function f is shown in the figure above. Which of the following could be the graph of f ¢, the derivative of f ? (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -10- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 12. If 0 < c < 1, what is the area of the region enclosed by the graphs of y = 0, y = (A) ln (1 - c ) (B) ln ( ( 1c ) (C) ln c (D) 1 -1 c2 1 , x = c, and x = 1 ? x (E) 1 - 1 c2 ) d tan -1 x + 2 x = dx 13. (A) (B) (C) 1 1 + 2 x sin x 1 1- x 2 1 1- x 2 - 43 x + 1 x (D) 1 - 43 x 2 1+ x (E) 1 1 + 2 x 1+ x Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -11- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 14. If y = f ( x ) is a solution to the differential equation 2 dy = e x with the initial condition f (0 ) = 2, which of the dx following is true? (A) f ( x ) = 1 + e x (B) f ( x ) = 2 xe x 2 2 x t2 Ú1 e dt (D) f ( x ) = 2 + Ú0 e (E) f ( x ) = 2 + Ú2 e (C) f ( x ) = x t2 x t2 dt dt Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -12- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 15. A function f (t ) gives the rate of evaporation of water, in liters per hour, from a pond, where t is measured in hours since 12 noon. Which of the following gives the meaning of 10 Ú4 f (t ) dt in the context described? (A) The total volume of water, in liters, that evaporated from the pond during the first 10 hours after 12 noon (B) The total volume of water, in liters, that evaporated from the pond between 4 P.M. and 10 P.M. (C) The net change in the rate of evaporation, in liters per hour, from the pond between 4 P.M. and 10 P.M. (D) The average rate of evaporation, in liters per hour, from the pond between 4 P.M. and 10 P.M. (E) The average rate of change in the rate of evaporation, in liters per hour per hour, from the pond between 4 P.M. and 10 P.M. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -13- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 16. The first derivative of the function f is given by f ¢( x ) = 3 x 4 - 12 x 3 . What are the x-coordinates of the points of inflection of the graph of f ? (A) x = 3 only (B) x = 4 only (C) x = 0 and x = 2 (D) x = 0 and x = 3 (E) x = 0 and x = 4 17. Let f be the function defined by f ( x ) = (A) - 1 24 (B) 5 24 (C) 1 . What is the average value of f on the interval [ 4, 6] ? x 1 3 ln 2 2 (D) ln Unauthorized copying or reuse of any part of this page is illegal. 3 2 (E) 1 ln 2 2 GO ON TO THE NEXT PAGE. -14- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Ê x, 1 ˆ , and Ê 3, 1 ˆ are the vertices of a rectangle, where x ≥ 3, as shown in the ÁË ˜ ÁË ˜ x2 ¯ x2 ¯ figure above. For what value of x does the rectangle have a maximum area? 18. The points (3, 0 ) , ( x, 0 ) , (A) 3 (B) 4 (C) 6 (D) 9 (E) There is no such value of x. 19. What are all values of x for which (A) -2 only (B) 0 only 2 3 Úx t dt is equal to 0 ? (D) -2 and 2 only (C) 2 only Unauthorized copying or reuse of any part of this page is illegal. (E) -2, 0, and 2 GO ON TO THE NEXT PAGE. -15- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 20. Let h be the function defined by h( x ) = to the graph of h at the point where x = (A) y = x Úp 4 sin 2 t dt. Which of the following is an equation for the line tangent p ? 4 1 2 (B) y = 2 x p 4 (C) y = x (D) y = (E) y = ( 1 p x2 4 ( ) 2 p x2 4 ) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -16- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA x f (x) –1 0 3 5 –30 –2 10 18 21. The table above gives selected values for a twice-differentiable function f. Which of the following must be true? (A) f has no critical points in the interval -1 < x < 5. (B) f ¢( x ) = 8 for some value of x in the interval -1 < x < 5. (C) f ¢( x ) > 0 for all values of x in the interval -1 < x < 5. (D) f ¢¢( x ) < 0 for all values of x in the interval -1 < x < 5. (E) The graph of f has no points of inflection in the interval -1 < x < 5. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -17- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 22. A particle moves along the x-axis so that at time t ≥ 0, the acceleration of the particle is a (t ) = 15 t . The position of the particle is 10 when t = 0, and the position of the particle is 20 when t = 1. What is the velocity of the particle at time t = 0 ? (A) -14 (B) 0 (C) 5 (D) 6 (E) 10 23. Which of the following is the solution to the differential equation the point ( 0, 1) ? (A) y = e x dy 2 xy whose graph contains = 2 dx x +1 2 (B) y = x 2 + 1 ( ) (C) y = ln x 2 + 1 ( ) (D) y = 1 + ln x 2 + 1 ( ) (E) y = 1 + 2 ln x 2 + 1 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -18- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA r3 , where r is the 3 radius of the base, in feet. The circumference of the base is increasing at a constant rate of 5p feet per hour. When the circumference of the base is 8p feet, what is the rate of change of the volume of the pile, in cubic feet per hour? 24. Sand is deposited into a pile with a circular base. The volume V of the pile is given by V = (A) 25. 8 p (B) 16 (C) 40 (D) 40p (E) 80p e -1- h - e -1 is h hÆ 0 lim (A) -1 (B) -1 e (C) 0 (D) 1 e (E) nonexistent Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -19- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 26. Let f be the function given by f ( x ) = x 3 + 5 x. For what value of x in the closed interval [1,3] does the instantaneous rate of change of f equal the average rate of change of f on that interval? 7 3 (A) 13 3 (B) (C) 27. If e xy - y 2 = e - 4, then at x = (A) e 4 (B) e 2 (C) 5 6 (D) (E) 19 3 dy 1 = and y = 2, dx 2 4e 8-e (D) 4e 4-e Unauthorized copying or reuse of any part of this page is illegal. (E) 8 - 4e e GO ON TO THE NEXT PAGE. -20- AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 28. Let f be the function defined by f ( x ) = x 3 + x 2 + x. Let g( x ) = f -1 ( x ) , where g(3) = 1. What is the value of g¢(3) ? (A) 1 39 (B) 1 34 (C) 1 6 (D) 1 3 (E) 39 END OF PART A OF SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY. DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO. Unauthorized copying or reuse of any part of this page is illegal. -21- B B B B B B B B B CALCULUS AB SECTION I, Part B Time— 50 minutes Number of questions—17 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO QUESTIONS NUMBERED 76–92. YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET. In this exam: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x) is a real number. (3) The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the prefix “arc” (e.g., sin -1 x = arcsin x ). Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -24- B B B B B B B B B 76. The graph of a function f is shown above. Which of the following limits does not exist? (A) lim f ( x ) x Æ1- (B) lim f ( x ) x Æ1 (C) lim f ( x ) x Æ 3- Unauthorized copying or reuse of any part of this page is illegal. (D) lim f ( x ) x Æ3 (E) lim f ( x ) xÆ5 GO ON TO THE NEXT PAGE. -25- B B B B B B B B B 77. Let f be a function that is continuous on the closed interval [1, 3] with f (1) = 10 and f (3) = 18. Which of the following statements must be true? (A) 10 £ f (2 ) £ 18 (B) f is increasing on the interval [1, 3]. (C) f ( x ) = 17 has at least one solution in the interval [1, 3]. (D) f ¢( x ) = 8 has at least one solution in the interval (1, 3) . (E) 3 Ú1 f ( x ) dx > 20 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -26- B B B B B B B B B 78. Let R be the region bounded by the graphs of y = e x , y = e3 , and x = 0. Which of the following gives the volume of the solid formed by revolving R about the line y = -1? (A) p Ú0 (e 3 3 - e x + 1 dx (B) p Ú0 (e 3 - e x - 1 dx (C) p Ú0 ÈÎÍ(e 3 - ex ) (D) p Ú0 ÈÍÎ(e 3 - ex ) (E) p Ú0 ÈÍÎ(e 3 +1 3 3 3 3 ) ) 2 2 2 + 1˘ dx ˚˙ 2 - 1˘ dx ˙˚ ) - (e x + 1) 2 2˘ ˙˚ dx 79. The number of people who have entered a museum on a certain day is modeled by a function f (t ) , where t is measured in hours since the museum opened that day. The number of people who have left ( ) the museum since it opened that same day is modeled by a function g(t ) . If f ¢(t ) = 380 1.02t and Ê p (t - 4) ˆ g ¢(t ) = 240 + 240sin Á , at what time t, for 1 £ t £ 11, is the number of people in the Ë 12 ˜¯ museum at a maximum? (A) 1 (B) 7.888 (C) 9.446 (D) 10.974 Unauthorized copying or reuse of any part of this page is illegal. (E) 11 GO ON TO THE NEXT PAGE. -27- B B B B B B B x 0 1 2 3 f (x) 5 2 3 6 f ¢( x ) –3 1 3 4 B B 80. The derivative of the function f is continuous on the closed interval [ 0, 4]. Values of f and f ¢ for selected values of x are given in the table above. If (A) 0 (B) 3 (C) 5 4 Ú0 f ¢(t ) dt = 8, then (D) 10 f ( 4) = (E) 13 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -28- B B B B B B B B B 81. A slope field for a differential equation is shown in the figure above. If y = f ( x ) is the particular solution to the differential equation through the point ( -1, 2 ) and h( x ) = 3 x ⴢ f ( x ) , then h ¢( -1) = (A) - 6 (B) -2 (C) 0 (D) 1 (E) 12 82. If f is a continuous function such that f ( 2 ) = 6, which of the following statements must be true? (A) lim f ( 2 x ) = 3 x Æ1 (B) lim f (2 x ) = 12 xÆ2 (C) lim x Æ2 f ( x ) - f ( 2) =6 x-2 ( ) (D) lim f x 2 = 36 x Æ2 (E) lim ( f ( x )) = 36 2 xÆ2 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -29- B B B B B B B 83. A particle moves along a straight line with velocity given by v(t ) = 5 + et acceleration of the particle at time t = 4 ? (A) 0.422 (B) 0.698 (C) 1.265 84. A home uses fuel oil at the rate r (t ) = 10 + 8sin (D) 8.794 3 B B for time t ≥ 0. What is the (E) 28.381 ( 60t ) gallons per day, where t is the number of days from the beginning of the heating season. To the nearest gallon, what is the total amount of fuel oil used from t = 0 to t = 60 days? (A) 7 gal (B) 14 gal (C) 600 gal (D) 821 gal Unauthorized copying or reuse of any part of this page is illegal. (E) 1004 gal GO ON TO THE NEXT PAGE. -30- B B B B B B B B B 85. The function f is defined on the open interval 0.4 < x < 2.4 and has first derivative f ¢ given by ( ) f ¢ ( x ) = sin x 2 . Which of the following statements are true? I. f has a relative maximum on the interval 0.4 < x < 2.4. II. f has a relative minimum on the interval 0.4 < x < 2.4. III. The graph of f has two points of inflection on the interval 0.4 < x < 2.4. (A) I only (B) II only (C) III only (D) I and III only (E) II and III only Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -31- B B B B B B B B B 86. The graph of the function f, which has a domain of [0, 7], is shown in the figure above. The graph consists of a quarter circle of radius 3 and a segment with slope -1. Let b be a positive number such that b Ú0 f ( x ) dx = 0. What is the value of b ? (A) 3.760 (B) 5.548 (C) 5.659 (D) 6.760 (E) There is no such value of b. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -32- B B B B B B B B B ( ) 87. The first derivative of the function g is given by g ¢ ( x ) = cos p x 2 for - 0.5 < x < 1.5. On which of the following intervals is g decreasing? (A) - 0.5 < x < 0 (B) 0 < x < 1 (C) 0.707 < x < 1.225 (D) 1.225 < x < 1.414 (E) 1.414 < x < 1.5 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -33- B B B B B B B B B 88. The height above the ground of a passenger on a Ferris wheel t minutes after the ride begins is modeled by the differentiable function H, where H (t ) is measured in meters. Which of the following is an interpretation of the statement H ¢(7.5) = 15.708 ? (A) The Ferris wheel is turning at a rate of 15.708 meters per minute when the passenger is 7.5 meters above the ground. (B) The Ferris wheel is turning at a rate of 15.708 meters per minute 7.5 minutes after the ride begins. (C) The passenger’s height above the ground is increasing by 15.708 meters per minute when the passenger is 7.5 meters above the ground. (D) The passenger’s height above the ground is increasing by 15.708 meters per minute 7.5 minutes after the ride begins. (E) The passenger is 15.708 meters above the ground 7.5 minutes after the ride begins. 89. A particle moves along a straight line for 6 seconds so that its velocity, in centimeters per second, is modeled by the graph shown. During the time interval 0 £ t £ 6, what is the total distance the particle travels? (A) 2 cm (B) 3.5 cm (C) 4 cm (D) 6.5 cm Unauthorized copying or reuse of any part of this page is illegal. (E) 8.5 cm GO ON TO THE NEXT PAGE. -34- B B B B B B B B B 90. Let f be a twice-differentiable function on the open interval (a, b ) . If f ¢( x ) > 0 on (a, b ) and f ¢¢( x ) < 0 on (a, b ) , which of the following could be the graph of f ? (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -35- B B B B B B B B B 91. The graphs of f and g are shown above. If h ( x ) = f ( x ) g( x ) , then h ¢(6 ) = (A) -9 (B) -7 (C) 1 (D) 7 (E) 9 92. In the xy-plane, the graph of the twice-differentiable function y = f ( x ) is concave up on the open interval (0, 2 ) and is tangent to the line y = 3 x - 2 at x = 1. Which of the following statements must be true about the derivative of f ? (A) f ¢( x ) £ 3 on the interval (0.9, 1) . (B) f ¢( x ) ≥ 3 on the interval (0.9, 1) . (C) f ¢( x ) < 0 on the interval (0.9, 1.1) . (D) f ¢( x ) > 0 on the interval (0.9, 1.1) . (E) f ¢( x ) is constant on the interval (0.9, 1.1) . Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -36- Answer Key for AP Calculus AB Practice Exam, Section I Question 1: A Question 24: C Question 2: D Question 25: B Question 3: B Question 26: B Question 4: D Question 27: C Question 5: C Question 28: C Question 6: B Question 76: D Question 7: C Question 77: C Question 8: D Question 78: E Question 9: A Question 79: B Question 10: E Question 80: E Question 11: B Question 81: E Question 12: B Question 82: E Question 13: E Question 83: C Question 14: D Question 84: D Question 15: B Question 85: D Question 16: A Question 86: D Question 17: C Question 87: C Question 18: C Question 88: D Question 19: D Question 89: D Question 20: D Question 90: C Question 21: B Question 91: A Question 22: D Question 92: A Question 23: B Answer Key for AP Calculus AB Practice Exam, Section I Question 1: D Question 24: C Question 2: B Question 25: D Question 3: C Question 26: C Question 4: B Question 27: C Question 5: A Question 28: B Question 6: B Question 29: A Question 7: D Question 30: B Question 8: A Question 76: D Question 9: D Question 77: B Question 10: A Question 78: C Question 11: A Question 79: B Question 12: D Question 80: A Question 13: D Question 81: D Question 14: B Question 82: D Question 15: C Question 83: C Question 16: A Question 84: D Question 17: A Question 85: B Question 18: D Question 86: C Question 19: C Question 87: B Question 20: C Question 88: C Question 21: A Question 89: A Question 22: B Question 90: C Question 23: A 2017 AP Calculus AB Question Descriptors and Performance Data Multiple-Choice Questions Question Learning Objective Essential Knowledge 1 2.1C 2.1C4 2 3.3B(a) 3.3B3 3 2.1C 2.1C3 4 3.2B 3.2B2 5 2.1C 2.1C4 6 2.1A 2.1A1 7 3.2C 3.2C2 8 2.3B 2.3B1 9 3.2C 3.2C1 Mathematical Practice for AP Calculus 1 Implementing algebraic/computational processes Implementing algebraic/computational processes Implementing algebraic/computational processes Connecting multiple representations Implementing algebraic/computational processes Implementing algebraic/computational processes Reasoning with definitions and theorems Implementing algebraic/computational processes Connecting multiple representations Implementing algebraic/computational processes Implementing algebraic/computational processes Mathematical Practice for AP Calculus 2 Key % Correct Building notational fluency D 91 Building notational fluency B 63 Building notational fluency C 80 Connecting concepts B 68 Building notational fluency A 76 Connecting concepts B 57 Building notational fluency D 69 Connecting concepts A 68 Connecting concepts D 84 Building notational fluency A 42 Building notational fluency A 61 10 3.3B(b) 3.3B5 11 2.1C 2.1C5 12 2.3D 2.3D1 Connecting concepts Implementing algebraic/computational processes D 80 13 2.3F 2.3F1 Connecting multiple representations Connecting concepts D 52 14 3.5A 3.5A1 Connecting concepts Building notational fluency B 18 15 1.1A(b) 1.1A3 Connecting multiple representations Connecting concepts C 41 Connecting concepts A 60 Connecting concepts A 42 Connecting concepts D 68 Connecting concepts C 57 Connecting concepts C 24 Implementing algebraic/computational processes Implementing algebraic/computational processes Reasoning with definitions and theorems Implementing algebraic/computational processes Reasoning with definitions and theorems 16 2.2A 2.2A1 17 2.3C 2.3C1 18 3.3A 3.3A2 19 2.3B 2.3B2 20 1.2A 1.2A3 21 2.1A 2.1A2 Connecting concepts Building notational fluency A 47 Connecting concepts B 39 Building notational fluency A 42 C 44 D 60 C 30 22 3.5A 3.5A2 Implementing algebraic/computational processes 23 3.4D 3.4D2 Connecting concepts 24 3.4B 3.4B1 Connecting concepts 25 2.1C 2.1C3 Connecting multiple representations 26 1.1C 1.1C3 Implementing algebraic/computational processes Implementing algebraic/computational processes Implementing algebraic/computational processes Reasoning with definitions and theorems 27 2.2A 2.2A3 Connecting multiple representations Connecting concepts C 61 Implementing algebraic/computational processes Connecting concepts B 46 28 2.3C 2.3C2 29 1.1D 1.1D1 Building notational fluency Connecting concepts A 30 3.2A2 Implementing algebraic/computational processes Reasoning with definitions and theorems B 21 30 3.2A(a) 2017 AP Calculus AB Question Descriptors and Performance Data Question Learning Objective Essential Knowledge 76 2.2A 2.2A2 Mathematical Practice for AP Calculus 1 Mathematical Practice for AP Calculus 2 Key % Correct Connecting multiple representations Connecting concepts D 81 Reasoning with definitions and theorems B 55 77 3.3B(b) 3.3B2 Implementing algebraic/computational processes 78 1.1D 1.1D1 Building notational fluency Connecting concepts C 63 79 2.3A 2.3A2 Connecting concepts Building notational fluency B 67 Implementing algebraic/computational processes Connecting concepts A 65 Connecting multiple representations Connecting concepts D 49 Connecting concepts D 42 Connecting concepts C 72 80 2.2A 2.2A1 81 2.2A 2.2A3 Implementing algebraic/computational processes Implementing algebraic/computational processes 82 3.4C 3.4C1 83 2.2A 2.2A1 84 3.4E 3.4E1 Connecting concepts Building notational fluency D 89 85 2.4A 2.4A1 Reasoning with definitions and theorems Connecting concepts B 63 86 2.2A 2.2A1 Connecting concepts Connecting multiple representations C 44 87 2.2A 2.2A2 Connecting multiple representations Connecting concepts B 44 Connecting concepts C 61 Connecting concepts A 37 Connecting concepts C 54 88 2.3B 2.3B1 89 3.4B 3.4B1 90 1.2A 1.2A1 Implementing algebraic/computational processes Reasoning with definitions and theorems Reasoning with definitions and theorems Free-Response Questions Question Learning Objective Essential Knowledge 1 1.2B|2.3C|3.4C 1.2B1|2.3C1|3.4C1 2 2.3D|3.2C|3.4A|3.4D 2.3D1|3.2C2|3.4A2|3.4D1 3 2.1A|2.2A|3.2C|3.3A 2.1A1|2.2A1|3.2C1|3.3A2,3.3A3 4 1.1C|2.1C|3.2B|3.3B(b) 1.1C3|2.1C2|3.2B2|3.3B2 5 3.3B(b)|3.4D 3.3B2,3.3B5|3.4D1,3.4D2 6 2.1C|2.3A|2.3C|3.1A|3.3B(b)|3.4A 2.1C2,2.1C4|2.3A1|2.3C2|3.1A2 |3.3B2|3.4A2 Mathematical Practice for AP Calculus Reasoning with definitions and theorems|Connecting concepts|Implementing algebraic/computational processes|Building notational fluency|Communicating Reasoning with definitions and theorems|Connecting concepts|Implementing algebraic/computational processes|Connecting multiple representations|Building notational fluency|Communicating Reasoning with definitions and theorems|Connecting concepts|Implementing algebraic/computational processes|Connecting multiple representations|Building notational fluency|Communicating Reasoning with definitions and theorems|Connecting concepts|Implementing algebraic/computational processes|Connecting multiple representations|Building notational fluency|Communicating Reasoning with definitions and theorems|Connecting concepts|Implementing algebraic/computational processes|Connecting multiple representations|Building notational fluency|Communicating Reasoning with definitions and theorems|Connecting concepts|Implementing algebraic/computational processes|Building notational fluency|Communicating Mean 3.04 3.01 3.49 2.44 4.46 5.39 Sample Questions for Calculus AB: Section I 1. What is (a) 1 Æ ( + ) - ( ) 2 2 (c) 0 (d) -1 (e) The limit does not exist. (b) 2. At which of the five points on the graph in the figure at the right are both negative? (a) (b) (c) ( d) (e) 3. A B C D E y d2y dy and 2 dx dx A B D C O E x The slope of the tangent to the curve y 3 x + y 2 x 2 = 6 at ( 2, 1) is (a) - 3 2 (b) -1 (c) - 5 14 3 (d) 14 (e) 0 18 © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. Sample Questions for Calculus AB: Section I 4. Let S be the region enclosed by the graphs of y = 2 x and y = 2 x 2 for 0 £ x £ 1. What is the volume of the solid generated when S is revolved about the line y = 3 ? - ) - ( - ) ) ı (( Û (( - ) - ( - ) ) ı (a) Û (b) ( ) (c) Ú - 5. Û (d) Ù ı ÊÊ ˆ Ê - Á ÁË ¯ Ë Ë Û (e) Ù ı ÊÊ Á ÁË Ë ˆ ˆ ˜¯ ˜¯ ˆ Ê - ˆ ˆ Ë ˜¯ ¯ ˜¯ Which of the following statements about the function given by f ( x ) = x 4 - 2 x3 is true? (a) The function has no relative extremum. (b) The graph of the function has one point of inflection and the function has two relative extrema. (c) The graph of the function has two points of inflection and the function has one relative extremum. (d) The graph of the function has two points of inflection and the function has two relative extrema. (e) The graph of the function has two points of inflection and the function has three relative extrema. 6. If f ( x ) = sin 2 (3 - x ) , then f ¢(0) = (a) (b) (c) (d) (e) 7. –2 cos 3 –2 sin 3 cos 3 6 cos 3 2 sin 3 cos 3 6 sin 3 cos 3 Which of the following is the solution to the differential equation (a) (b) (c) where for for for (d) for (e) for © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. 19 Sample Questions for Calculus AB: Section I 8. What is the average rate of change of the function f given by f ( x ) = x 4 - 5 x on the closed interval [0, 3] ? (a) 8.5 (b) 8.7 (c) 22 (d) 33 (e) 66 9. The position of a particle moving along a line is given by s (t ) = 2t 3 - 24t 2 + 90t + 7 for t ≥ 0. For what values of t is the speed of the particle increasing? (a) (b) (c) ( d) (e) 10. 3 < t < 4 only t > 4 only t > 5 only 0 < t < 3 and t > 5 3 < t < 4 and t > 5 Ú ( x - 1) (a) (b) (c) (d) (e) x dx = 3 1 x +C 2 x 2 3 2 1 1 2 x + x +C 3 2 1 2 x -x+C 2 2 5 2 2 3 2 x - x +C 5 3 1 2 x + 2 x3 2 - x + C 2 x2 - 4 ? xÆ• 2 + x - 4 x2 11. What is lim (a) -2 1 (b) 4 1 (c) 2 ( d) 1 (e) The limit does not exist. 20 © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. Sample Questions for Calculus AB: Section I y O x 12. The figure above shows the graph of y = 5 x - x 2 and the graph of the line y = 2 x. What is the area of the shaded region? 25 6 9 (b) 2 (c) 9 (a) (d) 27 2 45 (e) 2 13. If which of the following is true? (a) and (b) and (c) and (d) and (e) and © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. 21 Sample Questions for Calculus AB: Section I 14. Which of the following is a slope field for the differential equation (a) (b) (c) (d) dy x = ? dx y (e) 22 © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. Sample Questions for Calculus AB: Section I Part B Sample Multiple-Choice Questions A graphing calculator is required for some questions on this part of the exam. Part B consists of 17 questions. Following are the directions for Section I, Part B, and a representative set of 10 questions. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. In this exam: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number. (3) The inverse of a trigonometric function f may be indicated using the inverse function notation f –1 or with the prefix “arc” (e.g., sin–1 x = arcsin x). 15. A particle travels along a straight line with a velocity of v(t ) = 3e(- t 2) sin ( 2t ) meters per second. What is the total distance, in meters, traveled by the particle during the time interval 0 £ t £ 2 seconds? ( a) (b) (c) (d) (e) 0.835 1.850 2.055 2.261 7.025 16. A city is built around a circular lake that has a radius of 1 mile. The population density of the city is f ( r ) people per square mile, where r is the distance from the center of the lake, in miles. Which of the following expressions gives the number of people who live within 1 mile of the lake? (a) Ú ( ) (b) Ú ( + ( )) (c) Ú ( + ( )) (d) Ú ( ) (e) Ú ( + ( )) © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. 23 Sample Questions for Calculus AB: Section I y 2 1 O –1 1 2 3 4 x 17. The graph of a function f is shown above. If lim f ( x ) exists and f is not xÆb continuous at b, then b = (a) (b) (c) (d) (e) –1 0 1 2 3 x 1.1 1.2 1.3 1.4 f (x) 4.18 4.38 4.56 4.73 18. Let f be a function such that f ¢¢( x ) < 0 for all x in the closed interval [1, 2]. Selected values of f are shown in the table above. Which of the following must be true about f ¢(1.2) ? (a) (b) (c) (d) (e) f ¢(1.2) < 0 0 < f ¢(1.2) < 1.6 1.6 < f ¢(1.2) < 1.8 1.8 < f ¢(1.2) < 2.0 f ¢(1.2) > 2.0 19. Two particles start at the origin and move along the x-axis. For 0 £ t £ 10, their respective position functions are given by x1 = sin t and x2 = e -2t - 1. For how many values of t do the particles have the same velocity? (a) (b) (c) (d) (e) 24 None One Two Three Four © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. Sample Questions for Calculus AB: Section I y (0, 2) (−1, 0) 0 (2, 0) x (−2, −2) Graph of f 20. The graph of the function f shown above consists of two line segments. If g is the function defined by g ( x ) = (a) (b) (c) (d) (e) –2 –1 0 1 2 x Ú0 f (t ) dt , then g ( -1) = © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. 25 Sample Questions for Calculus AB: Section I 21. The graphs of five functions are shown below. Which function has a nonzero average value over the closed interval [ - ] (a) (b) (c) (d) (e) and 22. A differentiable function f has the property that using the local linear approximation for f at is the estimate for (a) (b) (c) (d) (e) 26 What 2.2 2.8 3.4 3.8 4.6 © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. Sample Questions for Calculus AB: Section I 23. Oil is leaking from a tanker at the rate of R(t ) = 2,000e - 0.2t gallons per hour, where t is measured in hours. How much oil leaks out of the tanker from time t = 0 to t = 10 ? (a) (b) (c) ( d) (e) 54 gallons 271 gallons 865 gallons 8,647 gallons 14,778 gallons Ê ˆ 24. If ¢( ) = Á and f (0) = 1, then f ( 2) = Ë ˜¯ (a) –1.819 (b) –0.843 (c) –0.819 (d) 0.157 (e) 1.157 Answers to Calculus AB Multiple-Choice Questions Part A 1. a 2. b 3. c 4. a 5. c 6. b †7. c 8. c 9. e 10. d 11. b 12. b 13. e 14. e Part B 15.* d 16. d 17. b 18. d 19.* d 20. b 21. e 22. a 23.* d 24.* e *Indicates a graphing calculator-active question. †For resources on differential equations, see the Course Home Pages for Calculus AB and Calculus BC at AP Central. © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. 27 SAMPLE QUESTIONS AP Calculus AB and AP Calculus BC Exam ® ® Originally published in the Fall 2014 AP® Calculus AB and AP® Calculus BC Curriculum Framework Sample Questions AP Calculus AB/BC Exam AP Calculus AB Sample Exam Questions Multiple Choice: Section I, Part A A calculator may not be used on questions on this part of the exam. 1. is (A) (B) (C) 1 (D) nonexistent Learning Objectives Essential Knowledge LO 1.1C: Determine limits of functions. EK 1.1C3: Limits of the indeterminate forms LO 2.1C: Calculate derivatives. EK 2.1C2: Specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric. © 2016 The College Board and may be evaluated using L’Hospital’s Rule. Mathematical Practices for AP Calculus MPAC 1: Reasoning with definitions and theorems MPAC 3: Implementing algebraic/computational processes Return to Table of Contents 1 Sample Questions 2. AP Calculus AB/BC Exam is (A) 1 (B) 3 (C) 9 (D) nonexistent Learning Objectives Essential Knowledge LO 1.1C: Determine limits of functions. EK 1.1C2: The limit of a function may be found by using algebraic manipulation, alternate forms of trigonometric functions, or the squeeze theorem. LO 1.1A(b): Interpret limits expressed symbolically. © 2016 The College Board EK 1.1A2: The concept of a limit can be extended to include one-sided limits, limits at infinity, and infinite limits. Mathematical Practices for AP Calculus MPAC 3: Implementing algebraic/computational processes MPAC 2: Connecting concepts Return to Table of Contents 2 Sample Questions AP Calculus AB/BC Exam 3. The graph of the piecewise-defined function f is shown in the figure above. The graph has a vertical tangent line at and horizontal tangent lines at and What are all values of x, at which f is continuous but not differentiable? (A) (B) and (C) and (D) and Learning Objectives Essential Knowledge Mathematical Practices for AP Calculus LO 2.2B: Recognize the connection between differentiability and continuity. EK 2.2B1: A continuous function may fail to be differentiable at a point in its domain. MPAC 4: Connecting multiple representations LO 1.2A: Analyze functions for intervals of continuity or points of discontinuity. EK 1.2A3: Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes. © 2016 The College Board MPAC 2: Connecting concepts Return to Table of Contents 3 Sample Questions AP Calculus AB/BC Exam 4. An ice sculpture in the form of a sphere melts in such a way that it maintains its spherical cubic meters per hour. At shape. The volume of the sphere is decreasing at a constant rate of what rate, in square meters per hour, is the surface area of the sphere decreasing at the moment and the when the radius is 5 meters? (Note: For a sphere of radius r, the surface area is ) volume is (A) (B) (C) (D) Learning Objectives Essential Knowledge LO 2.3C: Solve problems involving related rates, optimization, rectilinear motion, (BC) and planar motion. EK 2.3C2: The derivative can be used to solve related rates problems, that is, finding a rate at which one quantity is changing by relating it to other quantities whose rates of change are known. LO 2.1C: Calculate derivatives. EK 2.1C5: The chain rule is the basis for implicit differentiation. © 2016 The College Board Mathematical Practices for AP Calculus MPAC 2: Connecting concepts MPAC 3: Implementing algebraic/computational processes Return to Table of Contents 4 Sample Questions AP Calculus AB/BC Exam 5. Shown above is a slope field for which of the following differential equations? (A) (B) (C) (D) Learning Objective Essential Knowledge LO 2.3F: Estimate solutions to differential equations. EK 2.3F1: Slope fields provide visual clues to the behavior of solutions to first order differential equations. © 2016 The College Board Mathematical Practices for AP Calculus MPAC 4: Connecting multiple representations MPAC 2: Connecting concepts Return to Table of Contents 5 Sample Questions AP Calculus AB/BC Exam 6. Let f be the piecewise-linear function defined above. Which of the following statements are true? I. II. III. (A) None (B) II only (C) I and II only (D) I, II, and III Learning Objectives Essential Knowledge Mathematical Practices for AP Calculus LO 2.1A: Identify the derivative of a function as the limit of a difference quotient. EK 2.1A2: The instantaneous rate of change of a function at a point can be expressed by MPAC 2: Connecting concepts MPAC 5: Building notational fluency or provided that the limit exists. These are common forms of the definition of the derivative and are denoted LO 1.1A(b): Interpret limits expressed symbolically. © 2016 The College Board EK 1.1A2: The concept of a limit can be extended to include one-sided limits, limits at infinity, and infinite limits. Return to Table of Contents 6 Sample Questions 7. If AP Calculus AB/BC Exam for then (A) (B) (C) (D) Mathematical Practices for AP Calculus Learning Objectives Essential Knowledge LO 3.3A: Analyze functions defined by an integral. EK 3.3A2: If f is a continuous function on the interval then where x is between a and b. LO 2.1C: Calculate derivatives. © 2016 The College Board EK 2.1C4: The chain rule provides a way to differentiate composite functions. MPAC 1: Reasoning with definitions and theorems MPAC 3: Implementing algebraic/computational processes Return to Table of Contents 7 Sample Questions AP Calculus AB/BC Exam 8. Which of the following limits is equal to (A) (B) (C) (D) Learning Objective Essential Knowledge LO 3.2A(a): Interpret the definite integral as the limit of a Riemann sum. EK 3.2A3: The information in a definite integral can be translated into the limit of a related Riemann sum, and the limit of a Riemann sum can be written as a definite integral. © 2016 The College Board Mathematical Practices for AP Calculus MPAC 1: Reasoning with definitions and theorems MPAC 5: Building notational fluency Return to Table of Contents 8 Sample Questions AP Calculus AB/BC Exam 9. The function f is continuous for The graph of f shown above consists of five line segments. What is the average value of f on the interval (A) (B) (C) (D) Learning Objectives Essential Knowledge LO 3.4B: Apply definite EK 3.4B1: The average value of a function f over integrals to problems involving an interval is the average value of a function. LO 3.2C: Calculate a definite integral using areas and properties of definite integrals. © 2016 The College Board EK 3.2C1: In some cases, a definite integral can be evaluated by using geometry and the connection between the definite integral and area. Mathematical Practices for AP Calculus MPAC 1: Reasoning with definitions and theorems MPAC 4: Connecting multiple representations Return to Table of Contents 9 Sample Questions 10. Let AP Calculus AB/BC Exam be a solution to the differential equation where k is a constant. Values of f for selected values of t are given in the table above. Which of the following is an expression for (A) (B) (C) (D) Learning Objective Essential Knowledge Mathematical Practices for AP Calculus LO 3.5B: Interpret, create and solve differential equations from problems in context. EK 3.5B1: The model for exponential growth and decay that arises from the statement “The rate of change of a quantity is proportional MPAC 3: Implementing algebraic/computational processes to the size of the quantity” is MPAC 4: Connecting multiple representations © 2016 The College Board Return to Table of Contents 10 Sample Questions AP Calculus AB/BC Exam Multiple Choice: Section I, Part B A graphing calculator is required for some questions on this part of the exam. 11. The graph of the derivative of the function f, is shown above. Which of the following could be the graph of f ? (A) (B) (C) (D) © 2016 The College Board Return to Table of Contents 11 Sample Questions AP Calculus AB/BC Exam Learning Objectives Essential Knowledge LO 2.2A: Use derivatives to analyze properties of a function. EK 2.2A3: Key features of the graphs of are related to one another. LO 2.2B: Recognize the connection between differentiability and continuity. EK 2.2B2: If a function is differentiable at a point, then it is continuous at that point. © 2016 The College Board Mathematical Practices for AP Calculus and MPAC 4: Connecting multiple representations MPAC 2: Connecting concepts Return to Table of Contents 12 Sample Questions AP Calculus AB/BC Exam 12. The derivative of a function f is given by intervals is f decreasing? (A) and (B) and (C) and (D) and for Learning Objective Essential Knowledge LO 2.2A: Use derivatives to analyze properties of a function. EK 2.2A1: First and second derivatives of a function can provide information about the function and its graph including intervals of increase or decrease, local (relative) and global (absolute) extrema, intervals of upward or downward concavity, and points of inflection. © 2016 The College Board On what Mathematical Practices for AP Calculus MPAC 4: Connecting multiple representations MPAC 2: Connecting concepts Return to Table of Contents 13 Sample Questions AP Calculus AB/BC Exam 13. The temperature of a room, in degrees Fahrenheit, is modeled by , a differentiable function of the number of minutes after the thermostat is adjusted. Of the following, which is the best interpretation of (A) The temperature of the room is 2 degrees Fahrenheit, 5 minutes after the thermostat is adjusted. (B) The temperature of the room increases by 2 degrees Fahrenheit during the first 5 minutes after the thermostat is adjusted. (C) The temperature of the room is increasing at a constant rate of minute. degree Fahrenheit per (D) The temperature of the room is increasing at a rate of 2 degrees Fahrenheit per minute, 5 minutes after the thermostat is adjusted. Mathematical Practices for AP Calculus Learning Objectives Essential Knowledge LO 2.3A: Interpret the meaning of a derivative within a problem. EK 2.3A1: The unit for divided by the unit for x. LO 2.3D: Solve problems involving rates of change in applied contexts. EK 2.3D1: The derivative can be used to express information about rates of change in applied contexts. © 2016 The College Board is the unit for f MPAC 2: Connecting concepts MPAC 5: Building notational fluency Return to Table of Contents 14 Sample Questions AP Calculus AB/BC Exam 14. A function f is continuous on the closed interval with and Which of the following additional conditions guarantees that there is a number c in the open interval such that (A) No additional conditions are necessary. (B) f has a relative extremum on the open interval (C) f is differentiable on the open interval (D) exists. Learning Objective Essential Knowledge LO 2.4A: Apply the Mean Value Theorem to describe the behavior of a function over an interval. EK 2.4A1: If a function f is continuous over the interval and differentiable over the interval the Mean Value Theorem guarantees a point within that open interval where the instantaneous rate of change equals the average rate of change over the interval. © 2016 The College Board Mathematical Practices for AP Calculus MPAC 1: Reasoning with definitions and theorems MPAC 5: Building notational fluency Return to Table of Contents 15 Sample Questions AP Calculus AB/BC Exam 15. A rain barrel collects water off the roof of a house during three hours of heavy rainfall. The height of the water in the barrel increases at the rate of feet per hour, where t is the time in hours since the rain began. At time hour, the height of the water is foot. What is the height of the water in the barrel at time hours? (A) (B) (C) (D) Learning Objectives Essential Knowledge LO 3.4E: Use the definite integral to solve problems in various contexts. EK 3.4E1: The definite integral can be used to express information about accumulation and net change in many applied contexts. LO 3.3B(b): Evaluate definite integrals. EK 3.3B2: If is continuous on the interval and is an antiderivative Mathematical Practices for AP Calculus MPAC 2: Connecting concepts MPAC 3: Implementing algebraic/computational processes of , then © 2016 The College Board Return to Table of Contents 16 Sample Questions AP Calculus AB/BC Exam 16. A race car is traveling on a straight track at a velocity of 80 meters per second when the brakes seconds. From time to the moment the race car stops, the are applied at time acceleration of the race car is given by meters per second per second. During this time period, how far does the race car travel? (A) (B) (C) (D) Learning Objectives Essential Knowledge LO 3.4C: Apply definite integrals to problems involving motion. EK 3.4C1: For a particle in rectilinear motion over an interval of time, the definite integral of velocity represents the particle’s displacement over the interval of time, and the definite integral of speed represents the particle’s total distance traveled over the interval of time. LO 3.1A: Recognize antiderivatives of basic functions. EK 3.1A2: Differentiation rules provide the foundation for finding antiderivatives. © 2016 The College Board Mathematical Practices for AP Calculus MPAC 2: Connecting concepts MPAC 3: Implementing algebraic/computational processes Return to Table of Contents 17 Sample Questions AP Calculus AB/BC Exam Answers and Rubrics (AB) Answers to Multiple-Choice Questions 1. B 2. B 3. C 4. A 5. A 6. B 7. D 8. D 9. B 10. A 11. A 12. A 13. D 14. C 15. D 16. B © 2016 The College Board Return to Table of Contents 22 AP Calculus AB and AP Calculus BC ® ® Course and Exam Description Effective Fall 2016 New York, NY Sample Exam Questions AP Calculus AB Sample Exam Questions Multiple Choice: Section I, Part A A calculator may not be used on questions on this part of the exam. is AP CALCULUS AB SAMPLE EXAM QUESTIONS 1. The graphs of the functions f and g are shown above. The value of (A) 1 (B) 2 (C) 3 (D) nonexistent Learning Objective Essential Knowledge LO 1.1C: Determine limits of functions. EK 1.1C1: Limits of sums, differences, products, quotients, and composite functions can be found using the basic theorems of limits and algebraic rules. AP Calculus AB/BC Course and Exam Description Mathematical Practice for AP Calculus MPAC 4: Connecting multiple representations MPAC 2: Connecting concepts Return to Table of Contents © 2015 The College Board 47 Sample Exam Questions 2. (A) 6 (B) 2 (C) 1 (D) 0 Learning Objective Essential Knowledge LO 1.1C: Determine limits of functions. EK 1.1C3: Limits of the indeterminate forms and may be evaluated using L’Hospital’s Rule. Mathematical Practice for AP Calculus MPAC 3: Implementing algebraic/computational processes AP CALCULUS AB SAMPLE EXAM QUESTIONS MPAC 5: Building notational fluency 48 AP Calculus AB/BC Course and Exam Description Return to Table of Contents © 2015 The College Board Sample Exam Questions 3. If then (A) (B) (C) (D) Learning Objective Essential Knowledge LO 2.1C: Calculate derivatives. EK 2.1C4: The chain rule provides a way to differentiate composite functions. Mathematical Practice for AP Calculus MPAC 3: Implementing algebraic/computational processes AP Calculus AB/BC Course and Exam Description Return to Table of Contents © 2015 The College Board AP CALCULUS AB SAMPLE EXAM QUESTIONS MPAC 5: Building notational fluency 49 Sample Exam Questions 4. Three graphs labeled I, II, and III are shown above. One is the graph of f, one is the graph of and one is the graph of Which of the following correctly identifies each of the three graphs? AP CALCULUS AB SAMPLE EXAM QUESTIONS f 50 (A) I II III (B) II I III (C) II III I (D) III I II Learning Objective Essential Knowledge Mathematical Practice for AP Calculus LO 2.2A: Use derivatives to analyze properties of a function. EK 2.2A3: Key features of the graphs of f, and are related to one another. MPAC 2: Connecting concepts AP Calculus AB/BC Course and Exam Description MPAC 4: Connecting multiple representations Return to Table of Contents © 2015 The College Board Sample Exam Questions 5. The local linear approximation to the function g at is What is the value of (A) 4 (B) 5 (C) 6 (D) 7 Learning Objective Essential Knowledge LO 2.3B: Solve problems involving the slope of a tangent line. EK 2.3B2: The tangent line is the graph of a locally linear approximation of the function near the point of tangency. Mathematical Practice for AP Calculus MPAC 2: Connecting concepts MPAC 1: Reasoning with definitions and theorems AP CALCULUS AB SAMPLE EXAM QUESTIONS AP Calculus AB/BC Course and Exam Description Return to Table of Contents © 2015 The College Board 51 Sample Exam Questions 6. For time the velocity of a particle moving along the x-axis is given by At what values of t is the acceleration of the particle equal to 0? (A) 2 only (B) 4 only (C) 2 and 4 (D) 2 and 5 Essential Knowledge LO 2.3C: Solve problems involving related rates, optimization, rectilinear motion, (BC) and planar motion. EK 2.3C1: The derivative can be used to solve rectilinear motion problems involving position, speed, velocity, and acceleration. LO 2.1C: Calculate derivatives. EK 2.1C3: Sums, differences, products, and quotients of functions can be differentiated using derivative rules. MPAC 2: Connecting concepts MPAC 3: Implementing algebraic/computational processes AP CALCULUS AB SAMPLE EXAM QUESTIONS Learning Objective Mathematical Practice for AP Calculus 52 AP Calculus AB/BC Course and Exam Description Return to Table of Contents © 2015 The College Board Sample Exam Questions 7. The cost, in dollars, to shred the confidential documents of a company is modeled by C, a differentiable function of the weight of documents in pounds. Of the following, which is the best interpretation of Cʹ(500) = 80? (A) The cost to shred 500 pounds of documents is $80. (B) The average cost to shred documents is dollar per pound. (C) Increasing the weight of documents by 500 pounds will increase the cost to shred the documents by approximately $80. (D) The cost to shred documents is increasing at a rate of $80 per pound when the weight of the documents is 500 pounds. Learning Objective Essential Knowledge LO 2.3D: Solve problems involving rates of change in applied contexts. EK 2.3D1: The derivative can be used to express information about rates of change in applied contexts. Mathematical Practice for AP Calculus MPAC 2: Connecting concepts MPAC 5: Building notational fluency AP CALCULUS AB SAMPLE EXAM QUESTIONS AP Calculus AB/BC Course and Exam Description Return to Table of Contents © 2015 The College Board 53 Sample Exam Questions 8. Which of the following integral expressions is equal to (A) (B) (C) (D) Mathematical Practice for AP Calculus Learning Objective Essential Knowledge LO 3.2A(b): Express the limit of a Riemann sum in integral notation. EK 3.2A2: The definite integral of a continuous denoted by function f over the interval is the limit of Riemann sums as the widths of the subintervals approach 0. That is, AP CALCULUS AB SAMPLE EXAM QUESTIONS where MPAC 1: Reasoning with definitions and theorems MPAC 5: Building notational fluency is a value in the ith subinterval, is the width of the ith subinterval, n is the number of subintervals, and is the width of the largest subinterval. Another form of the definition is and 54 where is a value in the ith subinterval. AP Calculus AB/BC Course and Exam Description Return to Table of Contents © 2015 The College Board Sample Exam Questions 9. If f is the function defined above, then is (A) (B) (C) (D) undefined Essential Knowledge LO 3.2C: Calculate a definite integral using areas and properties of definite integrals. EK 3.2C3: The definition of the definite integral may be extended to functions with removable or jump discontinuities. AP Calculus AB/BC Course and Exam Description MPAC 2: Connecting concepts MPAC 3: Implementing algebraic/computational processes Return to Table of Contents © 2015 The College Board AP CALCULUS AB SAMPLE EXAM QUESTIONS Learning Objective Mathematical Practice for AP Calculus 55 Sample Exam Questions 10. (A) (B) (C) (D) Essential Knowledge LO 3.3B(a): Calculate antiderivatives. EK 3.3B5: Techniques for finding antiderivatives include algebraic manipulation such as long division and completing the square, substitution of variables, (BC) integration by parts, and nonrepeating linear partial fractions. MPAC 3: Implementing algebraic/computational processes MPAC 5: Building notational fluency AP CALCULUS AB SAMPLE EXAM QUESTIONS Learning Objective Mathematical Practice for AP Calculus 56 AP Calculus AB/BC Course and Exam Description Return to Table of Contents © 2015 The College Board Sample Exam Questions 11. At time t, a population of bacteria grows at the rate of grams per day, where t is measured in days. By how many grams has the population grown from time days to days? (A) (B) (C) (D) Learning Objective Essential Knowledge LO 3.4A: Interpret the meaning of a definite integral within a problem. EK 3.4A2: The definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over that interval. Mathematical Practice for AP Calculus MPAC 2: Connecting concepts MPAC 3: Implementing algebraic/computational processes AP CALCULUS AB SAMPLE EXAM QUESTIONS AP Calculus AB/BC Course and Exam Description Return to Table of Contents © 2015 The College Board 57 Sample Exam Questions 12. The right triangle shown in the figure above represents the boundary of a town that is bordered by a highway. The population density of the town at a distance of x miles from the highway is where is measured in thousands of people per square mile. modeled by According to the model, which of the following expressions gives the total population, in thousands, of the town? AP CALCULUS AB SAMPLE EXAM QUESTIONS (A) (B) (C) (D) 58 Learning Objective Essential Knowledge Mathematical Practice for AP Calculus LO 3.4A: Interpret the meaning of a definite integral within a problem. EK 3.4A3: The limit of an approximating Riemann sum can be interpreted as a definite integral. MPAC 2: Connecting concepts AP Calculus AB/BC Course and Exam Description MPAC 5: Building notational fluency Return to Table of Contents © 2015 The College Board Sample Exam Questions 13. Which of the following is the solution to the differential equation p initial condition yÊ ˆ = -1 ? Ë 4¯ with the (A) (B) (C) (D) Learning Objective Essential Knowledge LO 3.5A: Analyze differential equations to obtain general and specific solutions. EK 3.5A2: Some differential equations can be solved by separation of variables. Mathematical Practice for AP Calculus MPAC 3: Implementing algebraic/computational processes MPAC 2: Connecting concepts AP CALCULUS AB SAMPLE EXAM QUESTIONS AP Calculus AB/BC Course and Exam Description Return to Table of Contents © 2015 The College Board 59 Sample Exam Questions 14. The graph of the function f is shown in the figure above. For how many values of x in the open interval is f discontinuous? (A) one AP CALCULUS AB SAMPLE EXAM QUESTIONS (B) two (C) three (D) four 60 Learning Objective Essential Knowledge LO 1.2A: Analyze functions for intervals of continuity or points of discontinuity. EK 1.2A3: Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes. AP Calculus AB/BC Course and Exam Description Mathematical Practice for AP Calculus MPAC 2: Connecting concepts MPAC 4: Connecting multiple representations Return to Table of Contents © 2015 The College Board Sample Exam Questions 15. x 0 1 5 2 2 The table above gives selected values of a differentiable and decreasing function f and its derivative. If g is the inverse function of f, what is the value of (A) (B) (C) (D) 5 Essential Knowledge LO 2.1C: Calculate derivatives. EK 2.1C6: The chain rule can be used to find the derivative of an inverse function, provided the derivative of that function exists. MPAC 3: Implementing algebraic/computational processes AP CALCULUS AB SAMPLE EXAM QUESTIONS Learning Objective Mathematical Practice for AP Calculus MPAC 4: Connecting multiple representations AP Calculus AB/BC Course and Exam Description Return to Table of Contents © 2015 The College Board 61 Sample Exam Questions Multiple Choice: Section I, Part B A graphing calculator is required for some questions on this part of the exam. 16. The derivative of the function f is given by At what values of x does f have a relative minimum on the interval (A) and (B) (C) AP CALCULUS AB SAMPLE EXAM QUESTIONS (D) 62 Learning Objective Essential Knowledge LO 2.2A: Use derivatives to analyze properties of a function. EK 2.2A1: First and second derivatives of a function can provide information about the function and its graph including intervals of increase or decrease, local (relative) and global (absolute) extrema, intervals of upward or downward concavity, and points of inflection. AP Calculus AB/BC Course and Exam Description Mathematical Practice for AP Calculus MPAC 2: Connecting concepts MPAC 3: Implementing algebraic/computational processes Return to Table of Contents © 2015 The College Board Sample Exam Questions 17. The second derivative of a function g is given by on what open intervals is the graph of g concave up? (A) (B) only only (C) (D) For only and Learning Objective Essential Knowledge LO 2.2A: Use derivatives to analyze properties of a function. EK 2.2A1: First and second derivatives of a function can provide information about the function and its graph including intervals of increase or decrease, local (relative) and global (absolute) extrema, intervals of upward or downward concavity, and points of inflection. Mathematical Practice for AP Calculus MPAC 2: Connecting concepts MPAC 3: Implementing algebraic/computational processes AP CALCULUS AB SAMPLE EXAM QUESTIONS AP Calculus AB/BC Course and Exam Description Return to Table of Contents © 2015 The College Board 63 Sample Exam Questions 18. The temperature, in degrees Fahrenheit H given by of water in a pond is modeled by the function where t is the number of days since January 1 What is the instantaneous rate of change of the temperature of the water at time days? (A) (B) (C) (D) Essential Knowledge LO 2.3D: Solve problems involving rates of change in applied contexts. EK 2.3D1: The derivative can be used to express information about rates of change in applied contexts. AP CALCULUS AB SAMPLE EXAM QUESTIONS Learning Objective 64 AP Calculus AB/BC Course and Exam Description Mathematical Practice for AP Calculus MPAC 2: Connecting concepts MPAC 3: Implementing algebraic/ computational processes Return to Table of Contents © 2015 The College Board Sample Exam Questions 19. x 0 2 4 8 3 4 9 13 0 1 1 2 The table above gives values of a differentiable function f and its derivative at selected values of x. If h is the function given by which of the following statements must be true? (I) h is increasing on (II) There exists c, where such that (III) There exists c, where such that (A) II only (B) I and III only (C) II and III only Learning Objective Essential Knowledge LO 2.4A: Apply the Mean Value Theorem to describe the behavior of a function over an interval. EK 2.4A1: If a function f is continuous over and differentiable over the interval the interval the Mean Value Theorem guarantees a point within that open interval where the instantaneous rate of change equals the average rate of change over the interval. LO 1.2B: Determine the applicability of important calculus theorems using continuity. EK 1.2B1: Continuity is an essential condition for theorems such as the Intermediate Value Theorem, the Extreme Value Theorem, and the Mean Value Theorem. AP Calculus AB/BC Course and Exam Description AP CALCULUS AB SAMPLE EXAM QUESTIONS (D) I, II, and III Mathematical Practice for AP Calculus MPAC 1: Reasoning with definitions and theorems MPAC 4: Connecting multiple representations Return to Table of Contents © 2015 The College Board 65 Sample Exam Questions 20. Let h be the function defined by If g is an antiderivative of h and what is the value of (A) (B) (C) (D) Learning Objective Essential Knowledge LO 3.3B(b): Evaluate definite integrals. EK 3.3B2: If f is continuous on the interval and is an antiderivative MPAC 1: Reasoning with definitions and theorems MPAC 2: Connecting concepts AP CALCULUS AB SAMPLE EXAM QUESTIONS of f, then Mathematical Practice for AP Calculus 66 AP Calculus AB/BC Course and Exam Description Return to Table of Contents © 2015 The College Board Sample Exam Questions Answers and Rubrics (AB) Answers to Multiple-Choice Questions 1 C 2 B 3 B 4 C 5 D 6 C 7 D 8 A 9 B 10 A 11 C 12 D 13 B 14 C 15 A 16 C 17 B 18 B 19 C 20 D AP Calculus AB/BC Course and Exam Description Return to Table of Contents © 2015 The College Board 71