1969, 1973, 1985, 1993, 1997, 1998 AP Multiple Choice Sections, AB and BC, Solutions (620)

What is the average value of y for the part of the

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Unformatted text preview: 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) πe2 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 (C) III only (E) I, II, and III 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) 12 π 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 (C) I and II only (E) I, II, and III 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 22 (B) 22 (C) 3 24 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 AP Calculus BC: 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. The area of the region in the first quadrant enclosed by the graph of y = x (1 − x ) and the x-axis is 1 6 (A) 2. ) 2 − 2 3 (D) 5 6 (E) 1 19 3 (C) 9 2 (D) 19 6 (E) (C) − (D) − ( x ) , then 2 x (C) dx = 19 2 If f ( x) = ln (A) 4. x x2 + 2 1 3 (B) ∫0 ( 1 (A) 3. (B) (B) 2 f ′′( x) = − 1 2x 2 1 2x 1 (D) uv′ + u′v w′ (B) u ′vw + uv′w + uvw′ w 2 (E) u ′v′w − uvw′ w 2 (E) 3 2x 2 If u, v, and w are nonzero differentiable functions, then the derivative of (A) 1 6 2 x2 uv is w (C) uvw′ − uv′w − u ′vw w2 uv′w + u ′vw − uvw′ w2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 67 1988 AP Calculus BC: Section I 5. Let f be the function defined by the following. ⎧ sin x, ⎪2 ⎪ x, f ( x) = ⎨ ⎪ 2 − x, ⎪ x − 3, ⎩ x<0 0 ≤ x <1 1≤ x < 2 x≥2 For what val...
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This note was uploaded on 12/29/2010 for the course MATH 214 taught by Professor Smith during the Fall '10 term at Oregon Tech.

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