Unformatted text preview: ailable at apcentral.collegeboard.com. x ( sin x ) x −1 ( sin x ) x ( x cot x + ln ( sin x ) ) (C) ( cos x ) 52 1985 AP Calculus BC: Section I 27. If f is the continuous, strictly increasing function on the interval a ≤ x ≤ b as shown above, which
of the following must be true?
b I. ∫a II. ∫a III. ∫a f ( x) dx < f (b)(b − a) b f ( x) dx > f (a)(b − a ) b f ( x) dx = f (c)(b − a) for some number c such that a < c < b (A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III x 28. An antiderivative of f ( x) = e x +e is
(A) 29. e x+e x 1+ e x (B) (1 + e x ) e x+e x (C) e1+e (C) x (D) e x + e π
4 x (E) ee x π⎞
⎛
sin ⎜ x − ⎟
4⎠
⎝
lim
is
π
π
x→
x−
4
4 (A) 0 (B) 30. If x = t 3 − t and y =
(A) 1
8 (B) 1
2
3t + 1, then (D) 1 dy
at t = 1 is
dx 3
8 (C) 3
4 31. What are all values of x for which the series (D)
∞ ∑ n =1 (A) −1 ≤ x < 1 (B) (E) nonexistent −1 ≤ x ≤ 1 (C) ( x − 1)n
n 8
3 (E) 8 converges? 0< x<2 AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 0 ≤ x < 2 (E) 0≤ x≤2
53 1985 AP Calculus BC: Section I
32. An equation of the line normal to the graph of y = x3 + 3 x 2 + 7 x − 1 at the point where x = −1 is
(A)
33. If 4 x + y = −10 (B) x − 4 y = 23 (C) 4 x − y = 2 (D) x + 4 y = 25 (E) x + 4 y = −25 1
dy
= −2 y and if y = 1 when t = 0, what is the value of t for which y = ?
2
dt (A) − ln 2
2 − (B) 1
4 (C) ln 2
2 (D) 2
2 (E) ln 2 34. Which of the following gives the area of the surface generated by revolving about the yaxis the arc
of x = y 3 from y = 0 to y = 1?
1 (A) 2π ∫ y 3 1 + 9 y 4 dy
0 1 (B) 2π ∫ y 3 1 + y 6 dy
0 1 (C) 2π ∫ y 3 1 + 3 y 2 dy
0 1 (D) 2π ∫ y 1 + 9 y 4 dy
0 1 (E) 2π ∫ y 1 + y 6 dy
0 35. The region in the first quadrant between the xaxis and the graph of y = 6 x − x 2 is rotated around
the yaxis. The volume of the resulting solid of revolution is given by
(A) ∫0 ( ) (B) ∫0 (C) ∫0 (D) ∫ 0 π (3 + 9− y ) 2 (E) ∫ 0 π (3 + 9− y ) 2 2 6 π 6 x − x2 6 2πx 6 x − x 2 dx 6 πx 6 x − x 2 ( ( 6 9 dx ) ) 2 dx
dy
dy AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 54 1985 AP Calculus BC: Section I
36. 1 3 ∫ −1 x 2 dx is
(A) –6 (B) –3 37. The general solution for the equation (C) 0 (D) 6 nonexistent dy
+ y = xe− x is
dx (A) x2 − x
y=
e + Ce− x
2 x2 − x − x
e +e +C
(B) y =
2 (D) 38. (E) y = x e − x + Ce − x (E) y = C1e x + C2 x e− x lim x→∞ 1
xx
1 + 5e ( ) (C) y = − e− x + C
1+ x is (A) 0 (B) 1 (D) e5 (C) e (E) nonexistent 39. The base of a solid is the region enclosed by the graph of y = e − x , the coordinate axes, and the line
x = 3 . If all plane cross sections perpendicular to the xaxis are squares, then its volume is (A) (1 − e−6 )
2 (B) 1 −6
e
2 (C) e −6 (D) e −3 (A) (D) 1− u2
du
u (B) 2 1− u2
du
4u (E) ∫1
∫1 ⎛ x⎞
1− ⎜ ⎟
⎝ 2 ⎠ dx =
x 4 1− u2
du
u 4 1− u2
du
2u ⌠
x
is made, the integral ⎮
⎮
2
⌡2 2 1 − e−3 2 4 40. If the substitution u = (E) ∫2
∫2 AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (C) 2 ∫1 1− u2
du
2u 55 1985 AP Calculus BC: Section I
3 2
41. What is the length of the arc of y = x 2 from x = 0 to x = 3?
3
(A) 8
3 (B) 4 (C) 14
3 (D) 16
3 (E) 7 3
2 (E) 9
2 42. The coefficient of x3 in the Taylor series for e3 x about x = 0 is
(A) 1
6 (B) 1
3 (C) 1
2 (D) 43. Let f be a function that is continuous on the closed interval [ −2,3] such that f ′(0) does not exist,
f ′(2) = 0, and f ′′( x) < 0 for all x except x = 0. Which of the following could be the graph of f ? 44. At each point ( x , y ) on a certain curve, the slope of the curve is 3 x 2 y . If the curve contains the
point ( 0,8 ) , then its equation is
3 (A) y = 8e x (D) y = ln ( x + 1) + 8 (B) y = x3 + 8 (E) 1 ⎡⎛ 1 ⎞ ⎛ 2 ⎞
⎛ 3n ⎞
45. If n is a positive integer, then lim ⎢⎜ ⎟ + ⎜ ⎟ + … + ⎜ ⎟
n→∞ n ⎢⎝ n ⎠
⎝n⎠
⎝n⎠
⎣
2 1 (A) ∫0 (D) ∫0 3 1 2 dx (B) 3∫ x 2 dx (E) 3∫ x2 3 y = ex + 7 y 2 = x3 + 8 1
0 3
0 (C) 2⎤ ⎥ can be expressed as
⎥
⎦ 2 ⎛1⎞
⎜ ⎟ dx
⎝x⎠ (C) 3 ∫0 2 ⎛1⎞
⎜ ⎟ dx
⎝ x⎠ x 2 dx AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 56 1988 AP Calculus AB: Section I
90 Minutes—No Calculator Notes: (1) In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e).
(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 y = x 2e x , then dy
=
dx ( (A) (B) x x + 2e x (D) 2. 2 xe x
2x + ex (E) What is the domain of the function f given by f ( x) = {x : (D) {x : (B) x ≠ 3} x ≥ 2 and x ≠ 3} e2 − 1 The graph of...
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This note was uploaded on 12/29/2010 for the course MATH 214 taught by Professor Smith during the Fall '10 term at Oregon Tech.
 Fall '10
 smith
 Calculus

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