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

3 a 5 2 b 0 c 10 3 d 5 e 10 the graph

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Unformatted text preview: 3 , then the Intermediate Value Theorem guarantees that (A) f (0) = 0 (B) f ′(c) = 4 for at least one c between –3 and 6 9 (C) −1 ≤ f ( x) ≤ 3 for all x between –3 and 6 (D) f (c) = 1 for at least one c between –3 and 6 (E) f (c) = 0 for at least one c between –1 and 3 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 121 1997 AP Calculus BC: Section I, Part B 82. If 0 ≤ x ≤ 4 , of the following, which is the greatest value of x such that (A) 1.35 (C) 1.41 (D) 1.48 2 x − 2t ) dt ≥ ∫ t dt ? 2 (E) 1.59 dy = (1 + ln x ) y and if y = 1 when x = 1, then y = dx 83. If x 2 −1 x2 (A) e (B) 1 + ln x (C) ln x (D) e 2 x + x ln x −2 (E) 84. (B) 1.38 x ∫ 0 (t e x ln x ∫x 2 sin x dx = (A) − x 2 cos x − 2 x sin x − 2 cos x + C (B) − x 2 cos x + 2 x sin x − 2 cos x + C (C) − x 2 cos x + 2 x sin x + 2 cos x + C (D) − x3 cos x + C 3 2 x cos x + C (E) 85. Let f be a twice differentiable function such that f (1) = 2 and f (3) = 7. Which of the following must be true for the function f on the interval 1 ≤ x ≤ 3 ? I. II. III. (A) (B) (C) (D) (E) The average rate of change of f is 5 . 2 9 . 2 5 The average value of f ′ is . 2 The average value of f is None I only III only I and III only II and III only AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 122 1997 AP Calculus BC: Section I, Part B 86. dx ∫ ( x − 1)( x + 3) = (A) 1 x −1 ln +C 4 x+3 (B) 1 x+3 ln +C 4 x −1 (C) 1 ln ( x − 1)( x + 3) + C 2 (D) 1 ln 2 (E) ln ( x − 1)( x + 3) + C 2x + 2 +C ( x − 1)( x + 3) 87. The base of a solid is the region in the first quadrant enclosed by the graph of y = 2 − x 2 and the coordinate axes. If every cross section of the solid perpendicular to the y-axis is a square, the volume of the solid is given by (A) π ∫ (B) 2 0 ( 2 − y )2 dy 2 ∫ 0 ( 2 − y ) dy (C) π ∫ 2 0 ( 2 − x2 ) (D) 2 ∫ 0 (2 − x ) (E) ∫0 2 2 2 2 dx dx ( 2 − x2 ) dx AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 123 1997 AP Calculus BC: Section I, Part B 88. Let f ( x) = ∫ x2 sin t dt . At how many points in the closed interval ⎡ 0, π ⎤ does the instantaneous ⎣ ⎦ rate of change of f equal the average rate of change of f on that interval? (A) (B) (C) (D) (E) 0 Zero One Two Three Four 89. If f is the antiderivative of (A) − 0.012 x2 1 + x5 (B) 0 such that f (1) = 0 , then f ( 4 ) = (C) 0.016 (D) 0.376 (E) 0.629 90. A force of 10 pounds is required to stretch a spring 4 inches beyond its natural length. Assuming Hooke’s law applies, how much work is done in stretching the spring from its natural length to 6 inches beyond its natural length? (A) (B) (C) (D) (E) 60.0 inch-pounds 45.0 inch-pounds 40.0 inch-pounds 15.0 inch-pounds 7.2 inch-pounds AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 124 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. 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) (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) 5. f ′(c) = ∫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 − 22 (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...
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