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

If 3 1 d 189 e 233 f x dx 23 and f x f

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Unformatted text preview: m f ( x) x →a x→b lim f ( x) = 2 x →a lim f ( x) = 2 x→b lim f ( x) = 1 x→b lim f ( x) does not exist. x →a AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 104 1997 AP Calculus AB: Section I, Part A 16. The area of the region enclosed by the graph of y = x 2 + 1 and the line y = 5 is (A) 14 3 (B) 16 3 17. If x 2 + y 2 = 25 , what is the value of (A) 18. ∫ π 4 0 − 25 27 e tan x cos 2 x (B) − 7 27 (C) d2y dx 2 28 3 (D) 32 3 (E) 8π at the point ( 4,3) ? (C) 7 27 (D) 3 4 (E) 25 27 (C) e −1 (D) e (E) e +1 dx is (A) 0 (B) 1 19. If f ( x) = ln x 2 − 1 , then f ′( x) = (A) (B) (C) (D) (E) 2x x2 − 1 2x 2 x −1 2x x2 − 1 2x 2 x −1 1 2 x −1 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 105 1997 AP Calculus AB: Section I, Part A 20. The average value of cos x on the interval [ −3,5] is (A) (B) sin 5 − sin 3 2 (C) sin 3 − sin 5 2 (D) sin 3 + sin 5 2 (E) 21. sin 5 − sin 3 8 sin 3 + sin 5 8 x is x→1 ln x lim (A) 0 (B) 1 e (C) 1 (D) e (E) nonexistent 22. What are all values of x for which the function f defined by f ( x) = ( x 2 − 3)e − x is increasing? (A) There are no such values of x . (B) x < −1 and x > 3 (C) −3 < x < 1 (D) −1 < x < 3 (E) All values of x 23. If the region enclosed by the y-axis, the line y = 2 , and the curve y = x is revolved about the y-axis, the volume of the solid generated is (A) 32π 5 (B) 16π 3 (C) 16π 5 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 8π 3 (E) π 106 1997 AP Calculus AB: Section I, Part A 24. The expression 1 1⎛ 1 2 3 50 ⎞ + + + ⋅⋅⋅ + ⎜ ⎟ is a Riemann sum approximation for 50 ⎜ 50 50 50 50 ⎟ ⎝ ⎠ x dx 50 (A) (B) ∫0 (C) 11x dx 50 ∫ 0 50 (D) 11 x dx 50 ∫ 0 (E) 25. ∫0 1 50 x dx 50 ∫ 0 1 x dx ∫ x sin(2 x) dx = (A) 1 x − cos(2 x) + sin(2 x) + C 2 4 (B) 1 x − cos(2 x) − sin(2 x) + C 2 4 (C) 1 x cos(2 x) − sin(2 x) + C 2 4 (D) −2 x cos(2 x) + sin(2 x) + C (E) −2 x cos(2 x) − 4sin(2 x) + C AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 107 1997 AP Calculus AB: Section I, Part B 40 Minutes—Graphing Calculator Required Notes: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (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. e2 x 76. If f ( x) = , then f ′( x) = 2x (A) 1 (B) (C) (D) (E) e 2 x (1 − 2 x) 2 x2 e2 x e 2 x (2 x + 1) x2 e 2 x (2 x − 1) 2 x2 77. The graph of the function y = x3 + 6 x 2 + 7 x − 2 cos x changes concavity at x = (A) –1.58 (B) –1.63 (C) –1.67 78. The graph of f is shown in the figure above. If 3 ∫1 (D) –1.89 (E) –2.33 f ( x) dx = 2.3 and F ′( x) = f ( x), then F (3) − F (0) = (A) 0.3 (B) 1.3 (C) 3.3 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 4.3 (E) 5.3 108 1997 AP Calculus AB: Section I, Part B 79. Let f be a function such that lim h →0 I. f (2 + h) − f (2) = 5 . Which of the following must be true? h f is continuous at x = 2. II. f is differentiable at x = 2. III. The derivative of f is continuous at x = 2 . (A) I only (B) II only (C) I and II only (D) I and III only (E) II and III only 2 80. Let f be the function given by f ( x) = 2e 4 x . For what value of x is the slope of the line tangent to the graph of f at ( x, f ( x) ) equal to 3? (A) 0.168 (B) 0.276 (C) 0.318 (D) 0.342 (E) 0.551 81. A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and watches an eastbound train traveling at 60 meters per second. At how many meters per second is the train moving away from the observer 4 seconds after it passes through the intersection? (A) 57.60 (B) 57.88 (C) 59.20 (D) 60.00 (E) 67.40 82. If y = 2 x − 8 , what is the minimum value of the product xy ? (A) –16 (B) –8 (C) –4 (D) 0 (E) 2 83. What is the area of the region in the first quadrant enclosed by the graphs of y = cos x, y = x, and the y-axis? (A) 0.127 (B) 0.385 (C) 0.400 (D) 0.600 (E) 0.947 84. The base of a solid S is the region enclosed by the graph of y = ln x , the line x = e, and the x-axis. If the cross sections of S perpendicular to the x-axis are squares, then the volume of S is (A) 1 2 (B) 2 3 (C) 1 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 2 (E) 13 (e − 1) 3 109 1997 AP Calculus AB: Section I, Part B 85. If t...
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