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

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Unformatted text preview: y = (A) 5. xe x ( x + 2 ) {x : (C) {x : (E) {x : x2 − 4 ? x−3 x ≤ 2} x ≥ 2} x ≥ 2 and x ≠ 3} A particle with velocity at any time t given by v(t ) = et moves in a straight line. How far does the particle move from t = 0 to t = 2 ? (A) 4. (C) 2x + e (A) 3. ) ∫ sec x<0 2 (B) e −1 (C) 2e (D) e2 (E) e3 3 (E) x>2 (C) cos 2 x + C −5 is concave downward for all values of x such that x−2 (B) x<2 (C) x<5 (D) x>0 x dx = (A) tan x + C (B) csc 2 x + C (D) sec3 x +C 3 (E) 2sec2 x tan x + C AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 57 1988 AP Calculus AB: Section I 7. If y = ln x dy , then = x dx (A) 6. 1 x (B) x dx ∫ 3x 2 + 5 (C) ln x − 1 x (D) 2 ) 3 2 +C (B) 1 3x 2 + 5 4 ) 1 2 +C (E) 3 3x 2 + 5 2 ( (A) (D) 8. x 2 1 − ln x x 2 1 + ln x (E) x2 = 1 3x 2 + 5 9 1 3x 2 + 5 3 ( 1 ( ( ) 3 2 +C ) 1 2 +C ( 1 3x 2 + 5 (C) 12 ) 1 2 +C The graph of y = f ( x) is shown in the figure above. On which of the following intervals are dy d2y > 0 and <0? dx dx 2 I. II. III. a< x<b b<x<c c<x<d (A) I only (B) II only (C) III only AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) I and II (E) II and III 58 1988 AP Calculus AB: Section I 9. If x + 2 xy − y 2 = 2, then at the point (1,1) , 3 2 (A) 10. If k ∫0 (B) 1 2 dy is dx − (C) 0 (D) (C) 3 3 2 (D) 9 (E) nonexistent ( 2kx − x2 ) dx = 18, then k = (A) –9 (B) –3 (E) 18 11. An equation of the line tangent to the graph of f ( x) = x(1 − 2 x)3 at the point (1, − 1) is (A) y = −7 x + 6 (B) y = −6 x + 5 (D) y = 2x − 3 (E) y = 7x − 8 (C) 2 2 (C) y = −2 x + 1 (E) 3 ⎛π⎞ 12. If f ( x) = sin x , then f ′ ⎜ ⎟ = ⎝3⎠ (A) − 1 2 (B) 1 2 (D) 13. If the function f has a continuous derivative on [ 0, c ] , then (A) f (c) − f (0) 14. ∫ π 2 0 (B) f (c) − f (0) (C) f (c ) c ∫0 3 2 f ′( x) dx = (D) f ( x) + c (E) f ′′(c) − f ′′(0) cos θ dθ = 1 + sin θ (A) −2 (D) 2 ( ( ) 2 −1 ) 2 −1 (B) −2 2 (E) 2 ( (C) 2 2 ) 2 +1 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 59 1988 AP Calculus AB: Section I 15. If f ( x) = 2 x , then f ′(2) = (A) 1 4 (B) 1 2 (C) 2 2 (D) 1 2 (E) 16. A particle moves along the x-axis so that at any time t ≥ 0 its position is given by x(t ) = t 3 − 3t 2 − 9t + 1 . For what values of t is the particle at rest? (A) No values 17. 1 ∫ 0 ( 3x − 2 ) (A) − 2 (B) 1 only (C) 3 only (D) 5 only (E) 1 and 3 dx = 7 3 (B) − 7 9 1 9 (D) 1 (E) 3 ⎛ x⎞ (C) − sin ⎜ ⎟ ⎝2⎠ ⎛x⎞ (D) − cos ⎜ ⎟ ⎝2⎠ 1 ⎛ x⎞ (E) − cos ⎜ ⎟ 2 ⎝2⎠ (C) ln 2 (D) 2 ln 2 (E) (C) d2y ⎛x⎞ 18. If y = 2 cos ⎜ ⎟ , then = dx 2 ⎝2⎠ ⎛ x⎞ (A) −8 cos ⎜ ⎟ ⎝2⎠ 19. 3 ∫2 (A) x 2 x +1 ⎛ x⎞ (B) −2 cos ⎜ ⎟ ⎝2⎠ dx = 13 ln 22 (B) 1 ln 2 2 1 ln 5 2 20. Let f be a polynomial function with degree greater than 2. If a ≠ b and f (a) = f (b) = 1 , which of the following must be true for at least one value of x between a and b? I. II. III. f ( x) = 0 f ′( x) = 0 f ′′( x) = 0 (A) None (B) I only (C) II only (D) I and II only AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (E) I, II, and III 60 1988 AP Calculus AB: Section I 21. The area of the region enclosed by the graphs of y = x and y = x 2 − 3 x + 3 is (A) 2 3 (C) 4 3 (C) (B) 1 e (E) 14 3 (E) (D) 2 e2 ⎛1⎞ 22. If ln x − ln ⎜ ⎟ = 2, then x = ⎝ x⎠ (A) 1 e 1 e (B) 2 (D) 2e f ( x) is x→0 g ( x ) 23. If f ′( x) = cos x and g ′( x) = 1 for all x, and if f (0) = g (0) = 0 , then lim (A) 24. π 2 (B) 1 (D) −1 (E) nonexistent () d ln x x = dx (A) x ln x (B) ( ln x ) x 25. For all x > 1, if f ( x) = ∫ (A) 1 26. (C) 0 ∫ π 2 0 (A) (B) (C) x 1 () 2 ( ln x ) xln x x (D) ( ln x ) ( xln x−1 ) () (E) 2 ( ln x ) x ln x 1 dt , then f ′( x) = t 1 x (C) ln x − 1 (D) ln x (E) ex (D) 1 (E) π −1 2 x cos x dx = − π 2 (B) –1 (C) 1 − π 2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 61 1988 AP Calculus AB: Section I ⎧ x2 , x < 3 ⎪ 27. At x = 3 , the function given by f ( x ) = ⎨ is ⎪6 x − 9, x ≥ 3 ⎩ (A) (B) (C) (D) (E) 28. 4 ∫1 (A) undefined. continuous but not differentiable. differentiable but not continuous. neither continuous nor differentiable. both continuous and differentiable. x − 3 dx = − 3 2 (B) 3 2 (C) 5 2 (D) 9 2 (E) 5 tan 3( x + h ) − tan 3x is h →0 h 29. The lim (A) 0 (B) 3sec 2 (3x) (C) sec2 (3 x) (D) 3cot(3x) (E) nonexistent 30. A region in the first quadrant is enclosed by the graphs of y = e 2 x , x = 1, and the coordinate axes. If the region is rotated about the y -axis , the volume of the solid that is...
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