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

The complete interval of convergence of the series k

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Unformatted text preview: b 1 + sec 2 x dx (D) ∫a b 1 + tan 2 x dx (E) ∫a b 1 + sec 4 x dx b x 2 + tan 2 x dx x + tan x dx AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 18 1969 AP Calculus BC: Section I 44. If f ′′( x) − f ′( x) − 2 f ( x) = 0, f ′(0) = −2, and f (0) = 2, then f (1) = (A) e 2 + e −1 (B) 1 C) (D) e 2 0 45. The complete interval of convergence of the series ∞ ∑ k =1 ( x + 1)k k2 (A) 0< x<2 (B) 0≤ x≤2 (D) −2 ≤ x < 0 (E) (E) 2e −1 (C) −2 < x ≤ 0 is −2 ≤ x ≤ 0 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 19 1973 AP Calculus AB: Section I 90 Minutes—No Calculator Note: In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e). 1. ∫ (x 3 ) − 3 x dx = (A) (D) 2. 3x 2 − 3 + C (B) x4 − 3x + C 4 (E) 5 x 2 + 15 x + 25 (B) () 1 e (B) 2 2 e 2 (C) 4 e 2 (D) 1 e 4 (E) 4 e4 If f ( x) = x + sin x , then f ′( x) = (A) 1 + cos x (D) (B) (E) sin x − x cos x 1 − cos x sin x + x cos x (C) cos x y =1 If f ( x) = e x , which of the following lines is an asymptote to the graph of f ? (A) 6. 5 (C) 1125 The slope of the line tangent to the graph of y = ln x 2 at x = e 2 is (A) 5. 5 x3 + 15 x 2 + 20 x + 25 (E) (D) 225 4. x4 − 3x 2 + C 3 x 4 3x 2 − +C 4 2 (C) If f ( x) = x3 + 3 x 2 + 4 x + 5 and g ( x) = 5, then g ( f ( x) ) = (A) 3. 4 x4 − 6 x2 + C y=0 If f ( x) = (A) –1 (B) x=0 (C) y=x (D) y = −x (E) 0 (D) 1 2 (E) 1 x −1 for all x ≠ −1, then f ′(1) = x +1 (B) − 1 2 (C) AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 20 1973 AP Calculus AB: Section I 7. Which of the following equations has a graph that is symmetric with respect to the origin? x +1 x (B) (A) (D) 8. y= 3 y = x2 + 1 − 1 ( ) (C) y = x4 − 2 x2 + 6 2 A particle moves in a straight line with velocity v(t ) = t 2 . How far does the particle move between times t = 1 and t = 2 ? (A) 9. (E) y = ( x − 1) + 1 y = − x5 + 3 x 1 3 (B) If y = cos 2 3 x , then 7 3 (C) 3 (D) 7 (E) 8 dy = dx (A) −6 sin 3 x cos 3 x (B) −2 cos 3 x (D) 6 cos 3 x (E) 2 sin 3 x cos 3 x (C) (E) x 4 x5 − attains its maximum value at x = 10. The derivative of f ( x) = 3 5 4 (A) –1 (B) 0 (C) 1 (D) 3 2 cos 3 x 5 3 11. If the line 3x − 4 y = 0 is tangent in the first quadrant to the curve y = x3 + k , then k is (A) 1 2 (B) 1 4 (C) 0 (D) − 1 8 (E) − 1 2 12. If f ( x) = 2 x3 + Ax 2 + Bx − 5 and if f (2) = 3 and f (−2) = −37 , what is the value of A + B ? (A) –6 (E) (B) –3 (C) –1 (D) 2 It cannot be determined from the information given. AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 21 1973 AP Calculus AB: Section I 13. The acceleration α of a body moving in a straight line is given in terms of time t by α = 8 − 6t . If the velocity of the body is 25 at t = 1 and if s (t ) is the distance of the body from the origin at time t, what is s (4) − s (2) ? (A) 20 (B) 24 14. If f ( x) = x 1 3 ( x − 2) 2 3 (C) 28 (D) 32 (E) 42 for all x, then the domain of f ′ is (A) {x x ≠ 0} (B) {x x > 0} (D) {x x ≠ 0 and x ≠ 2} (E) {x x is a real number} (C) {x 0 ≤ x ≤ 2} x 2 15. The area of the region bounded by the lines x = 0 , x = 2, and y = 0 and the curve y = e is (A) e −1 2 (B) e −1 (C) 2 ( e − 1) (D) 2e − 1 (E) 2e 2t 3000e 5 16. The number of bacteria in a culture is growing at a rate of per unit of time t. At t = 0 , the number of bacteria present was 7,500. Find the number present at t = 5 . (A) 1, 200e 2 (B) 3, 000e 2 (C) 7,500e 2 (D) 7,500e5 (E) 15, 000 7 e 7 17. What is the area of the region completely bounded by the curve y = − x 2 + x + 6 and the line y =4? (A) 18. 3 2 (B) 7 3 (C) 9 2 (D) 31 6 (E) 33 2 d ( arcsin 2 x ) = dx (A) (D) −1 2 1 − 4 x2 2 1 − 4 x2 (B) (E) −2 4 x2 −1 (C) 1 2 1 − 4 x2 2 4 x2 −1 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 22 1973 AP Calculus AB: Section I 19. Suppose that f is a function that is defined for all real numbers. Which of the following conditions assures that f has an inverse function? (A) The function f is periodic. (B) The graph of f is symmetric with respect to the y-axis. (C) The graph of f is concave up. (D) The function f is a strictly increasing function. (E) The function f is continuous. 20. If F and f are continuous functions such that F ′( x) = f ( x) for all x, then (A) F (a) − F (b) (D) F (b) − F (a) (E) 21. F ′(b) − F ′(a ) (C) f ( x) dx is F ′(a ) − F ′(b) (B) b ∫a none of the above 1 ∫ 0 ( x + 1) e (A) x2 +2 x e3 2 dx = (B) e3 − 1 2 (C) e4 − e 2 (D) e3 − 1 (E) e4 − e 22. Given the function defined b...
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