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90 Minutes—No Calculator
Note: In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e).
1. 2. 3. Which of the following defines a function f for which f (− x) = − f ( x) ?
(A) f ( x) = x 2 (B) f ( x) = sin x (D) f ( x) = log x (E) f ( x) = e x ln ( x − 2 ) < 0 if and only if (A) x<3 (B) 0< x<3 (D) x>2 (E) x>3 ⎧
2x + 5 − x + 7
, for x ≠ 2,
⎪ f ( x) =
If ⎨
x−2
⎪ f (2) = k
⎩ (A) 0 4. 8 ∫0 dx
1+ x (A) 1 5. f ( x) = cos x (C) 2< x<3 and if f is continuous at x = 2 , then k = (B) 1
6 (C) 1
3 (D) 1 (E) 7
5 (B) 3
2 (C) 2 (D) 4 (E) 6 (D) 4 (E) not defined = If 3 x 2 + 2 xy + y 2 = 2, then the value of (A) –2 (C) (B) 0 dy
at x = 1 is
dx (C) 2 AP Calculus Multiple-Choice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 1 1969 AP Calculus AB: Section I
8 6. (A) 0
(E)
7. 1
2 For what value of k will x + (C) 1 (D) The limit does not exist. k
have a relative maximum at x = −2?
x (B) –2 (C) 2 (D) 4 (E) None of these If p ( x) = ( x + 2 )( x + k ) and if the remainder is 12 when p( x) is divided by x − 1, then k =
(A) 2 9. (B) It cannot be determined from the information given. (A) –4
8. 8 ⎛1
⎞
⎛1⎞
8⎜ + h ⎟ − 8⎜ ⎟
2
⎠
⎝2⎠ ?
What is lim ⎝
h →0
h (B) 3 (C) 6 (D) 11 (E) 13 When the area in square units of an expanding circle is increasing twice as fast as its radius in
linear units, the radius is
(A) 1
4π (B) 1
4 (C) 1
π (D) 1 (E) π (E) ln x 10. The set of all points (et , t ) , where t is a real number, is the graph of y = (A) 1
ex (B) 1
ex (C) 1
xex (D) 1
ln x 1⎞
⎛
11. The point on the curve x 2 + 2 y = 0 that is nearest the point ⎜ 0, − ⎟ occurs where y is
2⎠
⎝
1
1
(B) 0
(C) −
(D) −1
(E) none of the above
(A)
2
2 AP Calculus Multiple-Choice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 2 1969 AP Calculus AB: Section I
12. If f ( x) = (A) 4
and g ( x) = 2 x, then the solution set of f ( g ( x) ) = g ( f ( x) ) is
x −1 ⎧1 ⎫
⎨ ⎬
⎩3⎭ (B) {2} (C) {3} (D) {−1, 2} (E) ⎧1 ⎫
⎨ , 2⎬
⎩3 ⎭ 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 ⎜ ⎟
⎝4⎠
(D) π
4 (B) ⎛1⎞
arcsin ⎜ ⎟
⎝3⎠ (E) π
3 (C) π
6 14. If the function f is defined by f ( x) = x5 − 1, then f −1 , the inverse function of f , is defined by
f −1 ( x) = (A) (D) 1
5 x +1 5 x −1 (B) (E) 1
5 x +1 5 x +1 (C) 5 x −1 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. AP Calculus Multiple-Choice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 3 1969 AP Calculus AB: Section I
16. If y is a function of 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) (0, 0) only (B) (3,162) only (D) (0,0) and (3,162 ) (E) (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 (D) 2 (E) nonexistent 19. A point moves on the x-axis in such a way that its velocity at time t ( t > 0 ) is given by v = ln t
.
t At what value of t does v attain its maximum?
(A) 1
(E) (B) 1
e2 (C) e (D) 3
e2 There is no maximum value for v. AP Calculus Multiple-Choice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 4 1969 AP Calculus AB: Section I
20. An equation for a tangent to the graph of y = arcsin
(A) x − 2y = 0 (B) x− y =0 (C) x
at the origin is
2 x=0 y=0 (D) π x − 2y = 0 (E) 21. At x = 0 , which of the following is true of the function f defined by f ( x) = x 2 + e −2 x ?
(A) f is increasing.
(B) f is decreasing. (C) f is discontinuous. (D) f has a relative minimum.
(E)
22. f has a relative maximum. ( ) d
ln e 2 x =
dx
(A) 1
e 2x (B) 2
e2 x (C) 2x (D) 1 (E) 2 23. The area of the region bounded by the curve y = e2x , the x-axis, the y-axis, and the line x = 2 is
equal to
(A) e4
−e
2 (B) e4
−1
2 (D) 2e4 − e (E) 2e4 − 2 24. If sin x = e y , 0 < x < π, what is
(A) − tan x (B) − cot x (C) e4 1
−
2 2 (E) csc x dy
in terms of x ?
dx
(C) cot x AP Calculus Multiple-Choice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) tan x 5 1969 AP Calculus AB: Section I
25. A region in the plane is bounded by the graph of y = x = 2m, m > 0 . The area of this region 1
, the x-axis, the line x = m, and the line
x (A) is independent of m .
(B) increases as m increases. (C) decreases as m increases.
1
1
; increases as m increases when m > .
2
2
1
1
increases as m increases when m < ; decreases as m increases when m > .
2
2 (D) decreases as m increases when m <
(E) 26. 1 ∫0 x 2 − 2 x + 1 dx is (A) −1 (B) − 1
2 1
2
(D) 1
(E) none of the above
(C) 27. If dy
= tan x , then y =
dx (A) 1
tan 2 x + C
2 (B) sec 2 x + C (D) ln cos x + C (E) sec x tan x + C (C) ln sec x + C (E) 3 3 28. The function defined by f ( x) = 3 cos x + 3sin x has an amplitude of
(A) 3− 3 (B) 3 (C) 2 3 AP Calculus Multiple-Choice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 3+ 3 6 1969 AP Calculus AB: Section I
29. ∫π 4 cos x
dx =
sin x (A) ln 2 π 2 (B) ln π
4 (C) ln 3 (D) 3
2 ln (E) ln e 30. If a function f is continuous for all x and if f has a relative maximum at ( −1, 4) and a relative
minimum at (3, − 2) , which of the following statements must be true?
(A) The graph of f has a point of inflection somewhere between x = −1 and x = 3.
(B) f ′(−1) = 0
(C) The graph of f has a horizontal asymptote.
(D) The graph of f has a horizontal tangent line at x = 3 .
(E) The graph of f intersects both axes. 31. If f ′( x) = − f ( x) and f (1) = 1, then f ( x) =
(A) 1 −2 x + 2
e
2 (B) e − x −1 (C) e1− x (D) e− x (E) −e x 32. If a, b, c, d , and e are real numbers and a ≠ 0 , then the polynomial equation
ax 7 + bx5 + cx3 + dx + e = 0 has
(A)
(B)
(C)
(D)
(E) only one real root.
at least one real root.
an odd number of nonreal roots.
no real roots.
no positive real roots. 33. What is the average (mean) value of 3t 3 − t 2 over the interval −1 ≤ t ≤ 2 ?
(A) 11
4 (B) 7
2 (C) 8 AP Calculus Multiple-Choice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 33
4 (E) 16 7 1969 AP Calculus AB: Section I
34. Which of the following is an equation of a curve that intersects at right angles every curve of the
1
family y = + k (where k takes all real values)?
x
1
1
(D) y = x3
(E) y = ln x
(C) y = − x3
(A) y = − x
(B) y = − x 2
3
3
35. At t = 0 a particle starts at rest and moves along a line in such a way that at time t its acceleration
is 24t 2 feet per second per second. Through how many feet does the particle move during the first
2 seconds?
(A) 32 (B) 48 (C) 64 (D) 96 (E) 192 36. The approximate value of y = 4 + sin x at x = 0.12 , obtained from the tangent to the graph at
x = 0, is (A) 2.00 (B) 2.03 (C) 2.06 (D) 2.12 (E) 2.24 37. Which is the best of the following polynomial approximations to cos 2 x near x = 0 ?
(A) 1 + 38. x2 ∫ ex 3 x
2 (B) 1 + x (C) 1 − x2
2 (D) 1 − 2x 2 (E) 1 − 2x + x 2 (C) − (E) sec2 e dx =
3 (A) 3
1
− ln e x + C
3 (B) (D) 3
1
ln e x + C
3 (E) ex
−
+C
3
x3
3e x3 1
3e x 3 +C +C 1
dy
39. If y = tan u , u = v − , and v = ln x , what is the value of
at x = e ?
v
dx
(A) 0 (B) 1
e (C) 1 AP Calculus Multiple-Choice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 2
e 8 1969 AP Calculus AB: Section I
40. If n is a non-negative integer, then 1 ∫0 x (A) no n
(D) nonzero n, only n 1
0 (1 − x )n dx for (B) n even, only
(E) all n ⎧⎪ f ( x) = 8 − x 2 for − 2 ≤ x ≤ 2,
41. If ⎨
then
2
elsewhere ,
⎪⎩ f ( x) = x
(A) 0 and 8 dx = ∫ (B) 8 and 16 (C) n 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) 3x 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. (B)
(C)
(D)
(E) ( −1, 0 ) (C) 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 5. x = 0 only
x = 0, y = 1 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) 3
2 (C) 2 If 3 x 2 + 2 xy + y 2 = 2, then the value of dy
at x = 1 is
dx (A) –2 (C) (B) 0 2 AP Calculus Multiple-Choice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 4 (E) 6 (D) 4 (E) not defined 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) 10. (D) 4 (E) None of these 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 ( − g ( x) )2 − ( f ( x) )2 (B) 1 (C) (E) −2 ( − g ( x ) + f ( x ) ) −4 f ( x) g ( 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) ∫ 0 ( 3 + cos θ) d θ 1 ∫0 (A) 3 + cos θ d θ π x2
x2 + 1 π (B) ∫0 (E) 2∫ 3 + cos θ d θ
π 2 0 (C) 2∫ π 2 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) e− x − 1 2 (B) −2 xe (E) 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 )
( 0, 0 ) only (B) and ( 3,162 ) (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 velocity at time t ( t > 0 ) is given by v = ln t
.
t At what value of t does v attain its maximum?
(A) 1
(E) (B) 1
e2 (C) e (D) 3
e2 There is no ...

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