Unformatted text preview: ity 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. 20. An equation for a tangent to the graph of y = arcsin x
at the origin is
2 (A) x − 2y = 0 (B) x− y =0 (D) y=0 (E) π x − 2y = 0 (C) x=0 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) f has a relative maximum. 22. If f ( x) = ∫ x 1 0 3 t +2 dt , which of the following is FALSE? (A) f (0) = 0 (B) f is continuous at x for all x ≥ 0 . (C) f (1) > 0 (D) f ′(1) = (E) 1 3
f (−1) > 0 AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 14 1969 AP Calculus BC: Section I
23. If the graph of y = f ( x) contains the point ( 0, 2 ) ,
(A) 3+e − x 2 3+e − x (D) (B) − tan x (B) 3 + e− x (E) 2 24. If sin x = e y , 0 < x < π, what is
(A) dy
−x
=
and f ( x) > 0 for all x, then f ( x) =
dx ye x 2 3+e x (C) 1 + e− x 2 dy
in terms of x ?
dx − cot x (C) cot x 25. A region in the plane is bounded by the graph of y =
x = 2m , m > 0 . The area of this region (D) tan x (E) csc x 1
, the xaxis, 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) AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 15 1969 AP Calculus BC: Section I
dy
= tan x , then y =
dx 27. If (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 e2 x − 1
?
x→0 tan x 28. What is lim
(A) –1 29. ∫0 (
1 (A) 30. (B) 0 3
2 −2
4− x ) 2− 3
3 (C) 1 (D) 2 (E) The limit does not exist. dx = (B) 2 3 −3
4 (C) 3
12 (D) 3
3 (E) 3
2 ∞ (−1) n x n
∑ n ! is the Taylor series about zero for which of the following functions?
n =0 (A) sin x (B) cos x (C) ex (D) e− x (E) ln(1 + x) e1− x (D) e− x (E) −e x 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) 32. For what values of x does the series 1 + 2 x + 3x + 4 x +
(B) x < −1 (A) No values of x + nx + (C) x ≥ −1 converge?
(D) x > −1 (E) All values of x 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 MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 33
4 (E) 16 16 1969 AP Calculus BC: 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
(A) y = − x
(B) y = − x 2
(C) y = − x3
(D) y = x3
(E) y = ln x
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. Of the following choices of δ , which is the largest that could be used successfully with an
arbitrary ε in an epsilondelta proof of lim (1 − 3x ) = −5?
x →2 (A) δ = 3ε ( ) 38. If f ( x) = x 2 + 1
(A) 1
− ln(8e)
2 δ=ε (B)
(2−3 x ) (C) δ= ε
2 (D) δ= (C) 3
− ln(2)
2 (D) − ε
4 (E) δ= (E) ε
5 1
8 , then f ′(1) = (B) − ln(8e) 1
2 1
dy
at x = e ?
39. If y = tan u , u = v − , and v = ln x , what is the value of
v
dx
(A) 0 (B) 1
e (C) 1 AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 2
e (E) sec 2 e 17 1969 AP Calculus BC: Section I
40. If n is a nonnegative integer, then
(A) no n
(D) nonzero n, only 1 ∫0 x n 42. If ∫x 2 (B) 1
0 (1 − x )n dx (B) n even, only
(E) all n ⎧ f ( x) = 8 − x 2 for − 2 ≤ x ≤ 2,
⎪
then
41. If ⎨
2
elsewhere ,
⎪ f ( x) = x
⎩
(A) 0 and 8 dx = ∫ 8 and 16 for
(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 cos x dx = f ( x) − ∫ 2 x sin x dx, then f ( x) = (A) 2 sin x + 2 x cos x + C (B) x 2 sin x + C (C) 2 x cos x − x 2 sin x + C (D) 4 cos x − 2 x sin x + C (E) ( 2 − x2 ) cos x − 4sin x + C 43. Which of the following integrals gives the length of the graph of y = tan x between x = a and
π
x = b , where 0 < a < b < ?
2
b (A) ∫a (B) ∫a (C) ∫a...
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