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Unformatted text preview: 3 , then
the Intermediate Value Theorem guarantees that
(A) f (0) = 0
(B) f ′(c) = 4
for at least one c between –3 and 6
9 (C) −1 ≤ f ( x) ≤ 3 for all x between –3 and 6
(D) f (c) = 1 for at least one c between –3 and 6
(E) f (c) = 0 for at least one c between –1 and 3
AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 121 1997 AP Calculus BC:
Section I, Part B
82. If 0 ≤ x ≤ 4 , of the following, which is the greatest value of x such that
(A) 1.35 (C) 1.41 (D) 1.48 2 x − 2t ) dt ≥ ∫ t dt ?
2 (E) 1.59 dy
= (1 + ln x ) y and if y = 1 when x = 1, then y =
dx 83. If x 2 −1
x2 (A) e (B) 1 + ln x (C) ln x (D) e 2 x + x ln x −2 (E)
84. (B) 1.38 x ∫ 0 (t e x ln x ∫x 2 sin x dx = (A) − x 2 cos x − 2 x sin x − 2 cos x + C (B) − x 2 cos x + 2 x sin x − 2 cos x + C (C) − x 2 cos x + 2 x sin x + 2 cos x + C (D) − x3
cos x + C
3
2 x cos x + C (E) 85. Let f be a twice differentiable function such that f (1) = 2 and f (3) = 7. Which of the following
must be true for the function f on the interval 1 ≤ x ≤ 3 ?
I.
II.
III.
(A)
(B)
(C)
(D)
(E) The average rate of change of f is 5
.
2 9
.
2
5
The average value of f ′ is .
2
The average value of f is None
I only
III only
I and III only
II and III only AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 122 1997 AP Calculus BC:
Section I, Part B
86. dx ∫ ( x − 1)( x + 3) =
(A) 1
x −1
ln
+C
4
x+3 (B) 1
x+3
ln
+C
4
x −1 (C) 1
ln ( x − 1)( x + 3) + C
2 (D) 1
ln
2 (E) ln ( x − 1)( x + 3) + C 2x + 2
+C
( x − 1)( x + 3) 87. The base of a solid is the region in the first quadrant enclosed by the graph of y = 2 − x 2 and the
coordinate axes. If every cross section of the solid perpendicular to the yaxis is a square, the
volume of the solid is given by
(A) π ∫
(B) 2
0 ( 2 − y )2 dy 2 ∫ 0 ( 2 − y ) dy (C) π ∫ 2
0 ( 2 − x2 ) (D) 2
∫ 0 (2 − x ) (E) ∫0 2 2 2 2 dx dx ( 2 − x2 ) dx AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 123 1997 AP Calculus BC:
Section I, Part B
88. Let f ( x) = ∫ x2 sin t dt . At how many points in the closed interval ⎡ 0, π ⎤ does the instantaneous
⎣
⎦
rate of change of f equal the average rate of change of f on that interval? (A)
(B)
(C)
(D)
(E) 0 Zero
One
Two
Three
Four 89. If f is the antiderivative of
(A) − 0.012 x2
1 + x5 (B) 0 such that f (1) = 0 , then f ( 4 ) =
(C) 0.016 (D) 0.376 (E) 0.629 90. A force of 10 pounds is required to stretch a spring 4 inches beyond its natural length. Assuming
Hooke’s law applies, how much work is done in stretching the spring from its natural length to 6
inches beyond its natural length?
(A)
(B)
(C)
(D)
(E) 60.0 inchpounds
45.0 inchpounds
40.0 inchpounds
15.0 inchpounds
7.2 inchpounds AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 124 1998 AP Calculus AB:
Section I, Part A
55 Minutes—No Calculator Note: 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. 1
What is the xcoordinate of the point of inflection on the graph of y = x3 + 5 x 2 + 24 ?
3
(A) 5 2. (B) 0 (C) − 10
3 (D) –5 (E) − 10 The graph of a piecewiselinear function f , for −1 ≤ x ≤ 4 , is shown above. What is the value of
4 ∫ −1 f ( x) dx ?
(A) 1
3. 2 ∫1 (A) 1
x2 (B) 2.5 (C) 4 (D) 5.5 (E) 8 7
24 (C) 1
2 (D) 1 (E) dx = − 1
2 (B) AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 2 ln 2 125 1998 AP Calculus AB:
Section I, Part A
4. If f is continuous for a ≤ x ≤ b and differentiable for a < x < b , which of the following could be
false?
f (b) − f (a)
for some c such that a < c < b.
b−a (A)
(B) f ′(c) = 0 for some c such that a < c < b. (C) f has a minimum value on a ≤ x ≤ b. (D) f has a maximum value on a ≤ x ≤ b. (E) 5. f ′(c) = ∫a b f ( x) dx exists. x ∫ 0 sin t dt =
(A) sin x 6. If x 2 + xy = 10, then when x = 2, (A) 7. e ∫1 (A) 8. (B) − cos x − 7
2 (B) –2 (C) cos x (D) cos x − 1 (E) 1 − cos x dy
=
dx
(C) 2
7 (C) e2
1
−e+
2
2 (D) 3
2 (E) 7
2 (E) e2 3
−
22 (E) 1 ⎛ x2 − 1 ⎞
dx =
⎜
⎜x⎟
⎟
⎝
⎠
1
e−
e (B) 2 e −e 2 e −2 (D) Let f and g be differentiable functions with the following properties:
(i)
(ii) g ( x) > 0 for all x
f (0) = 1 If h( x) = f ( x) g ( x) and h′( x) = f ( x) g ′( x), then f ( x) =
(A) f ′( x) (B) g ( x) (C) ex AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 0 126 1998 AP Calculus AB:
Section I, Part A 9. The...
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