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

What are all values of k for which the graph of y x3

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Unformatted text preview: odd, only 3 ∫ −1 f ( x) dx is a number between (C) 16 and 24 (D) 24 and 32 (E) 32 and 40 42. What are all values of k for which the graph of y = x3 − 3 x 2 + k will have three distinct x-intercepts? (A) All k > 0 43. (B) All k < 4 (C) k = 0, 4 (D) 0 < k < 4 (E) All k ∫ sin ( 2 x + 3) dx = (A) 1 cos ( 2 x + 3) + C 2 (B) cos ( 2 x + 3) + C (D) 1 − cos ( 2 x + 3) + C 2 (E) 1 − cos ( 2 x + 3) + C 5 (C) 44. The fundamental period of the function defined by f ( x) = 3 − 2 cos 2 (A) 1 (B) 2 (C) 3 − cos ( 2 x + 3) + C πx is 3 (D) 5 (E) 6 (C) 3 x 2 g x3 d d d2 2 45. If ( f ( x) ) = g ( x) and ( g ( x) ) = f ( x ) , then 2 f ( x3 ) = dx dx dx ( () (A) f x6 (D) 9 x 4 f x 6 + 6 x g x3 () () (B) () ) g x3 (E) () f x 6 + g x3 () () AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 9 1969 AP Calculus BC: 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. 1 t are The asymptotes of the graph of the parametric equations x = , y = t t +1 (A) (D) 2. x = 0 only x = 0, y = 1 (B) (C) (D) (E) ( −1, 0 ) x = −1, y = 0 (B) ( 0, 0 ) (C) ( 0,1) (D) ⎛ π⎞ ⎜ 1, ⎟ ⎝ 4⎠ (E) ⎛ π⎞ ⎜ 1, ⎟ ⎝ 2⎠ 8 ∫0 ( 2,1) (1,1) ( 2, 2 ) ⎛1 1 ⎞ ⎜2, ⎟ 2⎠ ⎝ None of the above dx 1+ x (A) 1 = (B) 3 2 (C) 2 If 3 x 2 + 2 xy + y 2 = 2, then the value of (C) (D) 4 (E) 6 (D) 4 (E) not defined dy a t x = 1 is dx (A) –2 5. (C) The Mean Value Theorem guarantees the existence of a special point on the graph of y = x between ( 0, 0 ) and ( 4, 2 ) . What are the coordinates of this point? (A) 4. (B) (E) What are the coordinates of the inflection point on the graph of y = ( x + 1) arctan x ? (A) 3. x = 0, y = 0 x = −1 only (B) 0 2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 10 1969 AP Calculus BC: Section I 8 6. 1 (C) 1 (D) The limit does not exist. 2 It cannot be determined from the information given. (A) 0 (E) 7. (B) For what value of k will x + (A) –4 8. 8 ⎛1 ⎞ ⎛1⎞ 8⎜ + h ⎟ − 8⎜ ⎟ 2 ⎠ ⎝2⎠ ? What is lim ⎝ h →0 h k have a relative maximum at x = −2? x (B) –2 (C) (D) (B) (E) None of these 1 (E) ( − g ( x) )2 − ( f ( x) )2 (C) −4 f ( x ) g ( x ) −2 ( − g ( x ) + f ( x ) ) The area of the closed region bounded by the polar graph of r = 3 + cos θ is given by the integral 2π (A) ∫0 (D) 10. (D) 4 If h( x) = f 2 ( x) − g 2 ( x) , f ′( x) = − g ( x) , and g ′( x) = f ( x), then h′( x) = (A) 0 9. 2 ∫ 0 ( 3 + cos θ ) d θ 1 ∫0 (A) x2 + 1 ∫0 (E) π x2 π (B) 3 + cos θ d θ 2∫ 3 + cos θ d θ π2 0 (C) 2∫ π2 0 ( 3 + cos θ ) d θ 3 + cos θ d θ dx = 4−π 4 (B) ln 2 (C) 0 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 1 ln 2 2 (E) 4+π 4 11 1969 AP Calculus BC: Section I 1⎞ ⎛ 11. The point on the curve x 2 + 2 y = 0 that is nearest the point ⎜ 0, − ⎟ occurs where y is 2⎠ ⎝ 1 (A) 2 (B) 0 1 (C) − 2 (D) −1 (E) none of the above 12. If F ( x) = ∫ x 0 2 e −t dt , then F ′( x) = − x2 (A) 2 xe (D) (B) (E) e− x − 1 2 −2 xe e− x − x2 (C) e− x 2 +1 − x2 + 1 −e 2 13. The region bounded by the x-axis and the part of the graph of y = cos x between x = − π and 2 π π is separated into two regions by the line x = k . If the area of the region for − ≤ x ≤ k is 2 2 π three times the area of the region for k ≤ x ≤ , then k = 2 x= ⎛1⎞ (A) arcsin ⎜ ⎟ (B) ⎝4⎠ ⎛1⎞ arcsin ⎜ ⎟ ⎝3⎠ 14. If y = x 2 + 2 and u = 2 x − 1, then (A) (D) 2 x2 − 2 x + 4 ( 2 x − 1) 2 x (C) π 6 (D) π 4 (E) π 3 (C) x2 dy = du (B) 6 x2 − 2 x + 4 (E) 1 x AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 12 1969 AP Calculus BC: Section I 15. If f ′( x) and g ′( x) exist and f ′( x) > g ′( x) for all real x, then the graph of y = f ( x) and the graph of y = g ( x) (A) intersect exactly once. (B) intersect no more than once. (C) do not intersect. (D) could intersect more than once. (E) have a common tangent at each point of intersection. 16. If y is a function x such that y′ > 0 for all x and y′′ < 0 for all x, which of the following could be part of the graph of y = f ( x) ? 17. The graph of y = 5 x 4 − x5 has a point of inflection at (A) (D) ( 0, 0 ) only ( 0, 0 ) and ( 3,162 ) (B) (E) ( 3,162 ) only ( 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 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 2 (E) nonexistent 13 1969 AP Calculus BC: Section I 19. A point moves on the x-axis in such a way that its veloc...
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