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

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Unformatted text preview: ailable at apcentral.collegeboard.com. x ( sin x ) x −1 ( sin x ) x ( x cot x + ln ( sin x ) ) (C) ( cos x ) 52 1985 AP Calculus BC: Section I 27. If f is the continuous, strictly increasing function on the interval a ≤ x ≤ b as shown above, which of the following must be true? b I. ∫a II. ∫a III. ∫a f ( x) dx < f (b)(b − a) b f ( x) dx > f (a)(b − a ) b f ( x) dx = f (c)(b − a) for some number c such that a < c < b (A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III x 28. An antiderivative of f ( x) = e x +e is (A) 29. e x+e x 1+ e x (B) (1 + e x ) e x+e x (C) e1+e (C) x (D) e x + e π 4 x (E) ee x π⎞ ⎛ sin ⎜ x − ⎟ 4⎠ ⎝ lim is π π x→ x− 4 4 (A) 0 (B) 30. If x = t 3 − t and y = (A) 1 8 (B) 1 2 3t + 1, then (D) 1 dy at t = 1 is dx 3 8 (C) 3 4 31. What are all values of x for which the series (D) ∞ ∑ n =1 (A) −1 ≤ x < 1 (B) (E) nonexistent −1 ≤ x ≤ 1 (C) ( x − 1)n n 8 3 (E) 8 converges? 0< x<2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 0 ≤ x < 2 (E) 0≤ x≤2 53 1985 AP Calculus BC: Section I 32. An equation of the line normal to the graph of y = x3 + 3 x 2 + 7 x − 1 at the point where x = −1 is (A) 33. If 4 x + y = −10 (B) x − 4 y = 23 (C) 4 x − y = 2 (D) x + 4 y = 25 (E) x + 4 y = −25 1 dy = −2 y and if y = 1 when t = 0, what is the value of t for which y = ? 2 dt (A) − ln 2 2 − (B) 1 4 (C) ln 2 2 (D) 2 2 (E) ln 2 34. Which of the following gives the area of the surface generated by revolving about the y-axis the arc of x = y 3 from y = 0 to y = 1? 1 (A) 2π ∫ y 3 1 + 9 y 4 dy 0 1 (B) 2π ∫ y 3 1 + y 6 dy 0 1 (C) 2π ∫ y 3 1 + 3 y 2 dy 0 1 (D) 2π ∫ y 1 + 9 y 4 dy 0 1 (E) 2π ∫ y 1 + y 6 dy 0 35. The region in the first quadrant between the x-axis and the graph of y = 6 x − x 2 is rotated around the y-axis. The volume of the resulting solid of revolution is given by (A) ∫0 ( ) (B) ∫0 (C) ∫0 (D) ∫ 0 π (3 + 9− y ) 2 (E) ∫ 0 π (3 + 9− y ) 2 2 6 π 6 x − x2 6 2πx 6 x − x 2 dx 6 πx 6 x − x 2 ( ( 6 9 dx ) ) 2 dx dy dy AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 54 1985 AP Calculus BC: Section I 36. 1 3 ∫ −1 x 2 dx is (A) –6 (B) –3 37. The general solution for the equation (C) 0 (D) 6 nonexistent dy + y = xe− x is dx (A) x2 − x y= e + Ce− x 2 x2 − x − x e +e +C (B) y = 2 (D) 38. (E) y = x e − x + Ce − x (E) y = C1e x + C2 x e− x lim x→∞ 1 xx 1 + 5e ( ) (C) y = − e− x + C 1+ x is (A) 0 (B) 1 (D) e5 (C) e (E) nonexistent 39. The base of a solid is the region enclosed by the graph of y = e − x , the coordinate axes, and the line x = 3 . If all plane cross sections perpendicular to the x-axis are squares, then its volume is (A) (1 − e−6 ) 2 (B) 1 −6 e 2 (C) e −6 (D) e −3 (A) (D) 1− u2 du u (B) 2 1− u2 du 4u (E) ∫1 ∫1 ⎛ x⎞ 1− ⎜ ⎟ ⎝ 2 ⎠ dx = x 4 1− u2 du u 4 1− u2 du 2u ⌠ x is made, the integral ⎮ ⎮ 2 ⌡2 2 1 − e−3 2 4 40. If the substitution u = (E) ∫2 ∫2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (C) 2 ∫1 1− u2 du 2u 55 1985 AP Calculus BC: Section I 3 2 41. What is the length of the arc of y = x 2 from x = 0 to x = 3? 3 (A) 8 3 (B) 4 (C) 14 3 (D) 16 3 (E) 7 3 2 (E) 9 2 42. The coefficient of x3 in the Taylor series for e3 x about x = 0 is (A) 1 6 (B) 1 3 (C) 1 2 (D) 43. Let f be a function that is continuous on the closed interval [ −2,3] such that f ′(0) does not exist, f ′(2) = 0, and f ′′( x) < 0 for all x except x = 0. Which of the following could be the graph of f ? 44. At each point ( x , y ) on a certain curve, the slope of the curve is 3 x 2 y . If the curve contains the point ( 0,8 ) , then its equation is 3 (A) y = 8e x (D) y = ln ( x + 1) + 8 (B) y = x3 + 8 (E) 1 ⎡⎛ 1 ⎞ ⎛ 2 ⎞ ⎛ 3n ⎞ 45. If n is a positive integer, then lim ⎢⎜ ⎟ + ⎜ ⎟ + … + ⎜ ⎟ n→∞ n ⎢⎝ n ⎠ ⎝n⎠ ⎝n⎠ ⎣ 2 1 (A) ∫0 (D) ∫0 3 1 2 dx (B) 3∫ x 2 dx (E) 3∫ x2 3 y = ex + 7 y 2 = x3 + 8 1 0 3 0 (C) 2⎤ ⎥ can be expressed as ⎥ ⎦ 2 ⎛1⎞ ⎜ ⎟ dx ⎝x⎠ (C) 3 ∫0 2 ⎛1⎞ ⎜ ⎟ dx ⎝ x⎠ x 2 dx AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 56 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. If y = x 2e x , then dy = dx ( (A) (B) x x + 2e x (D) 2. 2 xe x 2x + ex (E) What is the domain of the function f given by f ( x) = {x : (D) {x : (B) x ≠ 3} x ≥ 2 and x ≠ 3} e2 − 1 The graph of...
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