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

Is y 5 2 x 3 how many points of inflection does

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Unformatted text preview: he 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 22 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 AP Calculus BC: 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. 1 x ( x + 1) dx = ∫0 (A) 0 2. 4e cos(2t) 7 5 (E) 2 sin(2t) (C) 2e (D) 2t cos(2t) 2e 2t (E) cos(2t) e 2t ( 5 5 (B) − (B) x + x2 5 5 (C) 0 (D) (E) 1 (C) 3x 2 (D) x3 (E) x 2 + x3 (C) 2 (D) 7 2 (E) 3+ e 2 (E) −8 ) 2 d xeln x = dx If f ( x) (A) 1 6. (D) The function f given by f ( x) = 3 x5 − 4 x3 − 3 x has a relative maximum at x = (A) 1 + 2 x 5. 16 15 dy = dx e 2t (B) cos(2t) 2t (A) –1 4. (C) If x = e 2t and y = sin(2t ) , then (A) 3. (B) 1 3 = ( x − 1) 2 + e x −2 , then f ′(2) = 2 (B) 3 2 The line normal to the curve y = 16 − x at the point ( 0, 4 ) has slope (A) 8 (B) 4 (C) 1 8 (D) AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. − 1 8 113 1997 AP Calculus BC: Section I, Part A Questions 7-9 refer to the graph and the information below. The function f is defined on the closed interval [ 0,8] . The graph of its derivative f ′ is shown above. 7. The point ( 3,5 ) is on the graph of y = f ( x) . An equation of the line tangent to the graph of f at ( 3,5) (A) y=2 (B) y =5 (C) y − 5 = 2 ( x − 3) (D) y + 5 = 2 ( x − 3) (E) 8. is y + 5 = 2 ( x + 3) How many points of inflection does the graph of f have? (A) (B) (C) (D) (E) Two Three Four Five Six AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 114 1997 AP Calculus BC: Section I, Part A 9. At what value of x does the absolute minimum of f occur? (A) (B) (C) (D) (E) 0 2 4 6 8 10. If y = xy + x 2 + 1 , then when x = −1, (A) 11. 1 2 ∞ ∫1 (A) (B) x (1 + x 2 ) 2 − dy is dx 1 2 − 1 2 (C) –1 (D) –2 (B) − 1 4 (C) 1 4 (D) (E) nonexistent dx is 1 2 (E) divergent 12. The graph of f ′ , the derivative of f , is shown in the figure above. Which of the following describes all relative extrema of f on the open interval ( a, b ) ? (A) (B) (C) (D) (E) One relative maximum and two relative minima Two relative maxima and one relative minimum Three relative maxima and one relative minimum One relative maximum and three relative minima Three relative maxima and two relative minima AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 115 1997 AP Calculus BC: Section I, Part A 13. A particle moves along the x-axis so that its acceleration at any time t is a (t ) = 2t − 7 . If the initial velocity of the particle is 6, at what time t during the interval 0 ≤ t ≤ 4 is the particle farthest to the right? (A) 0 (B) 1 (C) 2 14. The sum of the infinite geometric series (A) 1.60 (B) 2.35 (D) 3 (E) 4 3 9 27 81 ++ + + … is 2 16 128 1, 024 (C) 2.40 (D) 2.45 (E) 2.50 15. The length of the path described by the parametric equations x = cos3 t and y = sin 3 t , for π 0 ≤ t ≤ , is given by 2 π 2 0 (A) ∫ (B) π 2 0 ∫ (C) (D) (E) 16. 3cos 2 t + 3sin 2 t dt −3cos 2t sin t + 3sin 2t cos t dt ∫ π 2 0 9 cos 4 t + 9sin 4 t dt ∫ π 2 0 9 cos 4t sin 2 t + 9sin 4t cos 2 t dt ∫ π 2 0 cos6 t + sin 6 t dt eh − 1 is h →0 2 h lim (A) 0 (B) 1 2 (C) 1 AP Calculus Multiple-Choice Que...
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