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

If f x x 1 2 sin x then f 0 a 2 b 1 c

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Unformatted text preview: stion Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 24 2 (E) 36 77 1993 AP Calculus AB: Section I 90 Minutes—Scientific Calculator 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. 1. If f ( x) = 3 x2 , then f ′(4) = (A) –6 2. (B) –3 3 (D) 6 (E) 8 Which of the following represents the area of the shaded region in the figure above? d ∫c (D) (b − a ) [ f (b) − f (a) ] lim f ( y )dy 3n3 − 5n n→∞ n3 − 2n 2 + 1 (A) –5 b (B) (A) 3. (C) ∫ a ( d − f ( x) ) dx (E) (d − c) [ f (b) − f (a) ] (C) 1 (C) f ′(b) − f ′(a) is (B) –2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 3 (E) nonexistent 78 1993 AP Calculus AB: Section I 4. If x3 + 3 xy + 2 y 3 = 17 , then in terms of x and y, (A) − (B) − (C) − (D) − (E) 5. dy = dx x2 + y x + 2 y2 x2 + y x + y2 x2 + y x + 2y x2 + y 2 y2 − x2 1+ 2 y2 If the function f is continuous for all real numbers and if f ( x) = then f (−2) = (A) –4 6. (C) –1 The area of the region enclosed by the curve y = (A) 7. (B) –2 5 36 (B) ln 2 3 (C) ln (D) 0 (D) x + 13 y = 66 4 3 (D) ln 2 3 2 (E) ln 6 2x + 3 at the point (1,5 ) is 3x − 2 (B) 13x + y = 18 (E) (E) 1 , the x-axis, and the lines x = 3 and x = 4 is x −1 An equation of the line tangent to the graph of y = (A) 13 x − y = 8 x2 − 4 when x ≠ −2 , x+2 (C) x − 13 y = 64 −2 x + 3 y = 13 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 79 1993 AP Calculus AB: Section I 8. If y = tan x − cot x, then (A) sec x csc x 9. dy = dx (B) sec x − csc x (C) sec x + csc x (D) sec2 x − csc2 x (E) sec2 x + csc2 x If h is the function given by h( x) = f ( g ( x)), where f ( x) = 3 x 2 − 1 and g ( x) = x , then h( x) = (A) 3x3 − x (B) 3x 2 − 1 (C) (D) 3 x − 1 (E) 3x 2 − 1 (D) 1 3x 2 x − 1 (E) 2 10. If f ( x) = ( x − 1) 2 sin x, then f ′(0) = (A) –2 (B) –1 (C) 0 11. The acceleration of a particle moving along the x-axis at time t is given by a (t ) = 6t − 2 . If the velocity is 25 when t = 3 and the position is 10 when t = 1 , then the position x(t ) = (A) 9t 2 + 1 (B) 3t 2 − 2t + 4 (C) t 3 − t 2 + 4t + 6 (D) t 3 − t 2 + 9t − 20 (E) 36t 3 − 4t 2 − 77t + 55 12. If f and g are continuous functions, and if f ( x) ≥ 0 for all real numbers x , which of the following must be true? I. II. III. b ∫a b b f ( x) g ( x)dx = ⎛ ∫ f ( x)dx ⎞ ⎛ ∫ g ( x)dx ⎞ ⎜a ⎟⎜ a ⎟ ⎝ ⎠⎝ ⎠ b b ∫ a ( f ( x) + g ( x) ) dx = ∫ a b ∫a f ( x) dx = (A) I only b ∫a b f ( x)dx + ∫ g ( x)dx a f ( x)dx (B) II only (C) III only (D) II and III 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 80 1993 AP Calculus AB: Section I 13. The fundamental period of 2 cos(3x) is (A) 14. ∫ 2π 3 3x 2 x3 + 1 (B) 2π (C) 6π (D) 2 (E) 3 dx = (A) 2 x3 + 1 + C (B) 33 x +1 + C 2 (C) x3 + 1 + C (D) ln x3 + 1 + C (E) ln( x3 + 1) + C 15. For what value of x does the function f ( x) = ( x − 2)( x − 3) 2 have a relative maximum? (A) –3 (B) − 7 3 (C) − 5 2 (D) 16. The slope of the line normal to the graph of y = 2 ln(sec x) at x = (A) 7 3 (E) 5 2 π is 4 −2 (B) − 1 2 (C) 1 2 (D) 2 (E) nonexistent AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 81 1993 AP Calculus AB: Section I 17. ∫ (x 2 + 1) 2 dx = (A) ( x 2 + 1)3 +C 3 (B) ( x 2 + 1)3 +C 6x (C) ⎛ x3 ⎞ ⎜ + x⎟ + C ⎜3 ⎟ ⎝ ⎠ (D) 2 x( x 2 + 1)3 +C 3 (E) x5 2 x3 + + x+C 5 3 2 π 3π ⎛ x⎞ that satisfies the 18. If f ( x) = sin ⎜ ⎟ , then there exists a number c in the interval < x < 2 2 ⎝2⎠ conclusion of the Mean Value Theorem. Which of the following could be c ? (A) 2π 3 (B) 3π 4 (C) ⎧ ⎪ x3 19. Let f be the function defined by f ( x) = ⎨ ⎪x ⎩ about f is true? (A) 3π 2 for x ≤ 0, Which of the following statements for x > 0. f ′(0) = 0 (E) (E) f has a relative maximum. (D) π f is discontinuous at x = 0 . (C) (D) f is an odd function. (B) 5π 6 f ′( x) > 0 for x ≠ 0 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 82 1993 AP Calculus AB: Section I 1 = ( x + 1) 3 20. Let R be the region in the first quadrant enclosed by the graph of y , the line x = 7 , the x-axis, and the y-axis. The volume of the solid generated when R is revolv...
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