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
xintercepts?
(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 MultipleChoice 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 MultipleChoice 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 MultipleChoice 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 xaxis 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 MultipleChoice 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 MultipleChoice 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 xaxis in such a way that its veloc...
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