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

If f x 2 x 3 for all x then the value of the

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Unformatted text preview: (B) (3,162) only (D) (0,0) and (3,162 ) (E) (0, 0) and (4, 256) (C) (4, 256) only 18. If f ( x) = 2 + x − 3 for all x, then the value of the derivative f ′( x) at x = 3 is (A) −1 (B) 0 (C) 1 (D) 2 (E) nonexistent 19. A point moves on the x-axis in such a way that its velocity at time t ( t > 0 ) is given by v = ln t . t At what value of t does v attain its maximum? (A) 1 (E) (B) 1 e2 (C) e (D) 3 e2 There is no maximum value for v. AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 4 1969 AP Calculus AB: Section I 20. An equation for a tangent to the graph of y = arcsin (A) x − 2y = 0 (B) x− y =0 (C) x at the origin is 2 x=0 y=0 (D) π x − 2y = 0 (E) 21. At x = 0 , which of the following is true of the function f defined by f ( x) = x 2 + e −2 x ? (A) f is increasing. (B) f is decreasing. (C) f is discontinuous. (D) f has a relative minimum. (E) 22. f has a relative maximum. ( ) d ln e 2 x = dx (A) 1 e 2x (B) 2 e2 x (C) 2x (D) 1 (E) 2 23. The area of the region bounded by the curve y = e2 x , the x-axis, the y-axis, and the line x = 2 is equal to (A) e4 −e 2 (B) e4 −1 2 (D) 2e4 − e (E) 2e4 − 2 24. If sin x = e y , 0 < x < π, what is (A) − tan x (B) − cot x (C) e4 1 − 22 (E) csc x dy in terms of x ? dx (C) cot x AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) tan x 5 1969 AP Calculus AB: Section I 25. A region in the plane is bounded by the graph of y = x = 2m, m > 0 . The area of this region 1 , the x-axis, the line x = m, and the line x (A) is independent of m . (B) increases as m increases. (C) decreases as m increases. 1 1 ; increases as m increases when m > . 2 2 1 1 increases as m increases when m < ; decreases as m increases when m > . 2 2 (D) decreases as m increases when m < (E) 26. 1 ∫0 x 2 − 2 x + 1 dx is (A) −1 (B) − 1 2 1 2 (D) 1 (E) none of the above (C) 27. If dy = tan x , then y = dx (A) 1 tan 2 x + C 2 (B) sec 2 x + C (D) ln cos x + C (E) sec x tan x + C (C) ln sec x + C (E) 33 28. The function defined by f ( x) = 3 cos x + 3sin x has an amplitude of (A) 3− 3 (B) 3 (C) 23 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 3+ 3 6 1969 AP Calculus AB: Section I ∫π 4 cos x dx = sin x (A) 29. ln 2 π2 (B) ln π 4 (C) ln 3 (D) 3 2 ln (E) ln e 30. If a function f is continuous for all x and if f has a relative maximum at (−1, 4) and a relative minimum at (3, − 2) , which of the following statements must be true? (A) The graph of f has a point of inflection somewhere between x = −1 and x = 3. (B) f ′(−1) = 0 (C) The graph of f has a horizontal asymptote. (D) The graph of f has a horizontal tangent line at x = 3 . (E) The graph of f intersects both axes. 31. If f ′( x) = − f ( x) and f (1) = 1, then f ( x) = (A) 1 −2 x + 2 e 2 (B) e − x −1 (C) e1− x (D) e− x (E) −e x 32. If a, b, c, d , and e are real numbers and a ≠ 0 , then the polynomial equation ax 7 + bx5 + cx3 + dx + e = 0 has (A) (B) (C) (D) (E) only one real root. at least one real root. an odd number of nonreal roots. no real roots. no positive real roots. 33. What is the average (mean) value of 3t 3 − t 2 over the interval −1 ≤ t ≤ 2 ? (A) 11 4 (B) 7 2 (C) 8 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 33 4 (E) 16 7 1969 AP Calculus AB: Section I 34. Which of the following is an equation of a curve that intersects at right angles every curve of the 1 family y = + k (where k takes all real values)? x 1 1 (D) y = x3 (E) y = ln x (C) y = − x3 (A) y = − x (B) y = − x 2 3 3 35. At t = 0 a particle starts at rest and moves along a line in such a way that at time t its acceleration is 24t 2 feet per second per second. Through how many feet does the particle move during the first 2 seconds? (A) 32 (B) 48 (C) 64 (D) 96 (E) 192 36. The approximate value of y = 4 + sin x at x = 0.12 , obtained from the tangent to the graph at x = 0, is (A) 2.00 (B) 2.03 (C) 2.06 (D) 2.12 (E) 2.24 37. Which is the best of the following polynomial approximations to cos 2 x near x = 0 ? (A) 1 + 38. x2 ∫ ex 3 x 2 (B) 1 + x (C) 1 − x2 2 (D) 1 − 2 x 2 (E) 1 − 2 x + x2 (C) − (E) sec2 e dx = 3 (A) 3 1 − ln e x + C 3 (B) (D) 3 1 ln e x + C 3 (E) ex − +C 3 x3 3e x3 1 3e x 3 +C +C 1 dy 39. If y = tan u , u = v − , and v = ln x , what is the value of at x = e ? v dx (A) 0 (B) 1 e (C) 1 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 2 e 8 1969 AP Calculus AB: Section I 40. If n is a non-negative integer, then 1 ∫0 x (A) no n (D) nonzero n, only n 1 0 (1 − x )n dx for (B) n even, only (E) all n ⎧ f ( x) = 8 − x 2 for − 2 ≤ x ≤ 2, ⎪ 41. If ⎨ then 2 elsewhere , ⎪ f ( x) = x ⎩ (A) 0 and 8 dx = ∫ (B) 8 and 16 (C) n...
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