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

The length of the curve y x3 from x 0 to x 2 is given

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Unformatted text preview: base of a solid is the region in the first quadrant enclosed by the parabola y = 4 x 2 , the line x = 1 , and the x-axis. Each plane section of the solid perpendicular to the x-axis is a square. The volume of the solid is (A) 4π 3 16π 5 (B) 4 3 (C) (D) 16 5 (E) 64 5 26. If f is a function such that f ′( x) exists for all x and f ( x) > 0 for all x, which of the following is NOT necessarily true? 1 (A) ∫ −1 f ( x) dx > 0 (B) ∫ −1 2 f ( x) dx = 2∫ −1 f ( x) dx (C) ∫ −1 f ( x) dx = 2∫ 0 f ( x) dx (D) ∫ −1 f ( x) dx = − ∫ 1 (E) ∫ −1 f ( x) dx = ∫ −1 f ( x) dx + ∫ 0 f ( x) dx 1 1 1 1 1 −1 1 f ( x) dx 0 1 27. If the graph of y = x3 + ax 2 + bx − 4 has a point of inflection at (1, − 6 ) , what is the value of b? (A) –3 (E) (B) 0 (C) 1 (D) 3 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. 73 1988 AP Calculus BC: Section I 28. d ⎛π⎞ ln cos ⎜ ⎟ is dx ⎝x⎠ (A) (D) −π ⎛π⎞ x 2 cos ⎜ ⎟ ⎝ x⎠ (B) π ⎛π⎞ tan ⎜ ⎟ x ⎝x⎠ ⎛π⎞ − tan ⎜ ⎟ ⎝x⎠ (C) 1 ⎛π⎞ cos ⎜ ⎟ ⎝x⎠ π ⎛π⎞ tan ⎜ ⎟ x2 ⎝ x⎠ (E) 29. The region R in the first quadrant is enclosed by the lines x = 0 and y = 5 and the graph of y = x 2 + 1 . The volume of the solid generated when R is revolved about the y -axis is (A) ∞ 30. 6π 3 ⎡ ⎛1⎞ ⎢1 − ⎜ ⎟ 2 ⎢ ⎝3⎠ ⎣ (E) 2⎛1⎞ ⎜⎟ 3⎝3⎠ (C) π (E) 544π 15 (C) (D) 16π 3⎛1⎞ ⎜⎟ 2⎝3⎠ (E) 4π ⎛1⎞ ∑ ⎜3⎟ = i=n ⎝ ⎠ (A) 3 ⎛1⎞ −⎜ ⎟ 2 ⎝3⎠ (D) 2⎛1⎞ ⎜⎟ 3⎝3⎠ 2 n n ∫0 8 3 n⎤ ⎥ ⎥ ⎦ n n +1 4 − x 2 dx = (A) 31. 34π 3 (B) 8π (C) i (B) (B) 16 3 (D) 2π 32. The general solution of the differential equation y′ = y + x 2 is y = (A) Ce x (B) Ce x + x 2 (D) e x − x2 − 2 x − 2 + C (E) Ce x − x 2 − 2 x − 2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (C) − x 2 − 2 x − 2 + C 74 1988 AP Calculus BC: Section I 33. The length of the curve y = x3 from x = 0 to x = 2 is given by 2 (A) ∫0 1 + x 6 dx (D) 2π ∫ 2 ∫0 1 + 3x 2 dx (E) 1 + 9 x 4 dx 0 2 (B) ∫0 2 (C) π ∫ 2 1 + 9 x 4 dx 0 1 + 9 x 4 dx 34. A curve in the plane is defined parametrically by the equations x = t 3 + t and y = t 4 + 2t 2 . An equation of the line tangent to the curve at t = 1 is (A) y = 2x (B) y = 8x (D) y = 4x − 5 (E) y = 8 x + 13 35. If k is a positive integer, then lim x→+∞ (A) 0 (B) 1 xk ex (C) y = 2x −1 is (D) k ! (C) e (E) nonexistent 36. Let R be the region between the graphs of y = 1 and y = sin x from x = 0 to x = π . The volume of 2 the solid obtained by revolving R about the x-axis is given by (A) 2π ∫ (D) π ∫ π 2 0 π 2 0 (B) 2π ∫ x sin x dx (E) π ∫ sin 2 x dx π 2 0 π 2 0 (C) π ∫ x cos x dx π 2 0 (1 − sin x )2 dx (1 − sin 2 x ) dx 37. A person 2 meters tall walks directly away from a streetlight that is 8 meters above the ground. If 4 the person is walking at a constant rate and the person’s shadow is lengthening at the rate of 9 meter per second, at what rate, in meters per second, is the person walking? (A) 4 27 (B) 4 9 (C) 3 4 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 4 3 (E) 16 9 75 1988 AP Calculus BC: Section I ∞ xn ∑ n converges? n =1 38. What are all values of x for which the series (A) −1 ≤ x ≤ 1 (B) (D) −1 < x < 1 (E) All real x 39. If −1 < x ≤ 1 (C) −1 ≤ x < 1 (C) 5e tan x dy = y sec 2 x and y = 5 when x = 0, then y = dx (A) e tan x + 4 (B) e tan x + 5 (D) tan x + 5 (E) tan x + 5e x 40. Let f and g be functions that are differentiable everywhere. If g is the inverse function of f and 1 if g (−2) = 5 and f ′(5) = − , then g ′(−2) = 2 (A) 2 41. lim n→∞ 1 2 (D) ∫ 1 x dx 2 ∫1 ∫0 (E) 1 11 dx 2 ∫0 x 4 1 5 1 5 2 ∫ x x dx − (E) −2 (C) (D) ∫ 0 x dx (E) −6 1⎡ 1 2 n⎤ + +… + ⎢ ⎥= n⎣ n n n⎦ (A) 42. If (C) (B) (B) f ( x) dx = 6, what is the value of (A) 6 (B) 3 4 ∫1 1 x dx 1 2 1 f (5 − x) dx ? (C) 0 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) −1 76 1988 AP Calculus BC: Section I 43. Bacteria in a certain culture increase at a rate proportional to the number present. If the number of bacteria doubles in three hours, in how many hours will the number of bacteria triple? (A) 3 ln 3 ln 2 (B) 2 ln 3 ln 2 (C) ln 3 ln 2 ⎛ 27 ⎞ (D) ln ⎜ ⎟ ⎝2⎠ (E) ⎛9⎞ ln ⎜ ⎟ ⎝2⎠ 44. Which of the following series converge? I. ∞ 1 ∑ (−1)n+1 2n + 1 n =1 ∞ II. ∑ 1⎛3⎞ ⎜⎟ n⎝2⎠ ∞ 1 n ln n n =1 III. ∑ n=2 (A) (B) (C) (D) (E) n I only II only III only I and III only I, II, and III 45. What is the area of the largest rectangle that can be inscribed in the ellipse 4 x 2 + 9 y 2 = 36 ? (A) 62 (B) 12 (C) 24 AP Calculus Multiple-Choice Que...
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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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