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

# Let f x cos arctan x what is the range of f a d 25 x

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Unformatted text preview: y f ( x) = 3 x5 − 20 x3 , find all values of x for which the graph of f is concave up. (A) x>0 (B) − 2 < x < 0 or x > 2 (C) −2 < x < 0 or x > 2 (D) x> 2 (E) −2 < x < 2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 23 1973 AP Calculus AB: Section I 23. 1 ⎛ 2+h⎞ ln ⎜ ⎟ is h→0 h ⎝ 2 ⎠ lim (A) e2 (B) 1 (C) 1 2 (D) 0 (E) nonexistent 24. Let f ( x) = cos ( arctan x ) . What is the range of f ? (A) (D) 25. ⎧ π ⎨x − < x < 2 ⎩ {x π4 ∫0 (A) π⎫ ⎬ 2⎭ (B) 0 < x ≤ 1} (E) − 1 < x < 1} {x (C) {x − 1 ≤ x ≤ 1} (C) 1 3 {x 0 ≤ x ≤ 1} (E) π +1 4 tan 2 x dx = π −1 4 (B) 1 − π 4 2 −1 (D) 26. The radius r of a sphere is increasing at the uniform rate of 0.3 inches per second. At the instant when the surface area S becomes 100π square inches, what is the rate of increase, in cubic inches 4 ⎛ ⎞ per second, in the volume V ? ⎜ S = 4π r 2 and V = π r 3 ⎟ 3 ⎝ ⎠ (A) 10π 27. 2x 12 ∫0 (B) 12π 1− x (A) 1 − 2 3 2 (C) 22.5 π (D) 25 π (E) 30 π (C) π 6 (D) π −1 6 (E) 2− 3 dx = (B) 13 ln 24 28. A point moves in a straight line so that its distance at time t from a fixed point of the line is 8t − 3t 2 . What is the total distance covered by the point between t = 1 and t = 2? (A) 1 (B) 4 3 (C) 5 3 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 2 (E) 5 24 1973 AP Calculus AB: Section I 1 . The maximum value attained by f is 2 29. Let f ( x) = sin x − 1 2 (A) (B) 1 ∫1 x−4 (A) 30. − 2 x2 3 2 (D) π 2 (E) 3π 2 (C) ln 2 (D) 2 (E) ln 2 + 2 (C) 8 (D) 16 (E) 32 ) 5 (C) 5 x − + C x dx = 1 2 (B) () ln 2 − 2 a , then a = 4 31. If log a 2a = (A) 2 32. (C) (B) 4 5 ∫ 1 + x 2 dx = −10 x ( (A) (1 + x2 ) +C (B) 5 ln 1 + x 2 + C 2x (D) 5arctan x + C (E) 5ln 1 + x 2 + C 2 ( ) 33. Suppose that f is an odd function; i.e., f (− x) = − f ( x) for all x. Suppose that f ′ ( x0 ) exists. Which of the following must necessarily be equal to f ′ ( − x0 ) ? (A) f ′ ( x0 ) (B) − f ′ ( x0 ) (C) 1 f ′ ( x0 ) (D) −1 f ′ ( x0 ) (E) None of the above AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 25 1973 AP Calculus AB: Section I x over the interval 0 ≤ x ≤ 2 is 34. The average value of 1 2 3 (A) (B) 1 2 2 (C) 2 2 3 (D) 1 35. The region in the first quadrant bounded by the graph of y = sec x, x = (E) 4 2 3 π , and the axes is rotated 4 about the x-axis. What is the volume of the solid generated? π2 4 (A) 36. If y = enx , then dx n π −1 (C) π (D) 2π (E) 8π 3 n !e nx (C) n e nx (D) nn e x (E) n !e x 3 + e4 x (D) 4 + e4 x (E) 2 x2 + 4 (E) –5 ) (E) ( 4,8) (C) cos 2 ( xy ) = (B) dy = 4 y and if y = 4 when x = 0, then y = dx 4e4 x (A) 38. If dny n n enx (A) 37. If (B) 2 ∫1 (A) (B) e4 x (C) f ( x − c ) dx = 5 where c is a constant, then 5+c (B) 5 (C) 2−c ∫ 1−c f ( x ) dx = 5−c (D) c − 5 39. The point on the curve 2 y = x 2 nearest to ( 4,1) is (A) ( 0, 0 ) 40. If tan( xy ) = x , then (B) ( 2, 2 ) (C) ( ) 2,1 (D) (2 2, 4 dy = dx (A) 1 − y tan( xy ) sec( xy ) x tan( xy ) sec( xy ) (D) cos 2 ( xy ) x (B) sec 2 ( xy ) − y x (E) cos 2 ( xy ) − y x AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 26 1973 AP Calculus AB: Section I ⎧ x + 1 for x < 0, 41. Given f ( x) = ⎨ ⎩cos π x for x ≥ 0, (A) 11 + 2π (B) − 1 ∫ −1 f ( x) dx = 1 2 (C) 11 − 2π (D) 1 2 (E) 1 − +π 2 42. Calculate the approximate area of the shaded region in the figure by the trapezoidal rule, using 4 5 divisions at x = and x = . 3 3 (A) 50 27 (B) 251 108 (C) 7 3 (D) 127 54 (E) 77 27 (C) − ⎛ x⎞ 43. If the solutions of f ( x) = 0 are –1 and 2, then the solutions of f ⎜ ⎟ = 0 are ⎝2⎠ (A) (D) − 4 − (E) 1 and 1 2 44. For small values of h, the function 1 5 and 2 2 (B) −1 and 2 3 3 and 2 2 −2 and 4 16 + h is best approximated by which of the following? (A) 4+ h 32 (B) 2+ h 32 (D) 4− h 32 (E) 2− h 32 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (C) h 32 27 1973 AP Calculus AB: Section I 45. If f is a continuous function on [ a, b ] , which of the following is necessarily true? (A) f ′ exists on ( a , b ) . (B) If f ( x0 ) is a maximum of f, then f ′ ( x0 ) = 0 . (C) ⎛ ⎞ lim f ( x) = f ⎜ lim x ⎟ for x0 ∈ ( a , b ) x→ x0 ⎝ x→ x0 ⎠ (D) f ′( x) = 0 for some x ∈ [ a , b ] (E) The graph of f ′ is a straight line. AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 28 1973 AP Calculus BC: Section I 90 Minutes—No Calculator N...
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