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Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 143 1998 AP Calculus BC:
Section I, Part A
18. Which of the following series converge?
I. ∞ n ∑ n+2 II. n =1 (A)
(B)
(C)
(D)
(E) ∞ ∑ n =1 cos(nπ)
n III. ∞ 1 ∑n
n =1 None
II only
III only
I and II only
I and III only 19. The area of the region inside the polar curve r = 4 sin θ and outside the polar curve r = 2 is given
by
3π 1π
2
(A)
∫ 0 ( 4sin θ − 2 ) d θ
2
5π ( 14
2
(B)
∫ π ( 4sin θ − 2 ) d θ
2
4 ) 16
16sin 2 θ − 4 d θ
(D)
2∫π (E) 6 20. When x = 8 , the rate at which 3 ( 5π 16
2
(C)
∫ π ( 4sin θ − 2 ) d θ
2
6 ) 1π
16sin 2 θ − 4 d θ
2 ∫0 x is increasing is 1
times the rate at which x is increasing. What
k is the value of k ?
(A) 3 (B) 4 (C) 6 (D) 8 (E) 12 1
1
21. The length of the path described by the parametric equations x = t 3 and y = t 2 , where
3
2
0 ≤ t ≤ 1 , is given by
1 ∫0 t 2 + 1 dt 1 t 2 + t dt 1 t 4 + t 2 dt (A)
(B) ∫0 (C) ∫0 (D) 11
4
∫ 0 4 + t dt
2 (E) 1 12
t 4t 2 + 9 dt
6 ∫0 AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 144 1998 AP Calculus BC:
Section I, Part A
b ∫
b→∞ 1 22. If lim (A) ∞ ∑ n =1
∞ (B) ∑ n =1 (C) xp
1
np
1
np ∞ ∑ n =1 (D) dx 1
n ∞ ∑ n =1 1
n p −1 ∞ (E) ∑ n =1 p−2 1
n p +1 is finite, then which of the following must be true? converges diverges converges converges diverges 23. Let f be a function defined and continuous on the closed interval [ a, b ] . If f has a relative
maximum at c and a < c < b , which of the following statements must be true?
I. f ′(c) exists.
II. If f ′(c) exists, then f ′(c) = 0 .
III. If f ′′(c) exists, then f ′′(c) ≤ 0 .
(A) II only (B) III only (C) I and II only (D) I and III only AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (E) II and III only 145 1998 AP Calculus BC:
Section I, Part A 24. Shown above is a slope field for which of the following differential equations? 25. (B) dy
= x2
dx (C) dy
= x+ y
dx (D) (B) dy
= 1+ x
dx (A) dy x
=
dx y 0 (C) 1
3 (D) 1 (E) dy
= ln y
dx ∞ 2 − x3
x e dx is ∫0 − (A) 1
3 (E) divergent dP
P⎞
⎛
= P⎜2−
⎟,
dt
5000 ⎠
⎝
where the initial population P (0) = 3, 000 and t is the time in years. What is lim P(t ) ? 26. The population P (t ) of a species satisfies the logistic differential equation t →∞ (A) 2,500 ∞ 27. If ∑ an x n (B) 3, 000 (C) 4, 200 (D) 5, 000 (E) 10, 000 is a Taylor series that converges to f ( x) for all real x, then f ′(1) = n =0 ∞ (A) x t2 28. lim (B) 0 ∫1 e a1 (C) ∞ ∑ an (D) e
2 (D) n =0 ∞ ∑ nan (E) e (E) nonexistent n =1 ∑ nan n−1 n =1 dt x→1 x2 − 1 (A) 0 is
(B) 1 (C) AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 146 1998 AP Calculus BC:
Section I, Part B
50 Minutes—Graphing Calculator Required Notes: (1) The exact numerical value of the correct answer does not always appear among the choices
given. When this happens, select from among the choices the number that best approximates
the exact numerical value.
(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.
76. For what integer k, k > 1 , will both ∞ ∑ n =1 (A) 6 (B) 5 ( −1)kn
n (C) ∞ n ⎛k⎞
and ∑ ⎜ ⎟ converge?
n =1 ⎝ 4 ⎠
4 (D) 3 ( (E) 2 ) 77. If f is a vectorvalued function defined by f (t ) = e −t , cos t , then f ′′(t ) =
(A) −e−t + sin t (B) e −t − cos t (D) ( e−t , cos t ) (E) ( e−t , − cos t ) (C) ( −e−t , − sin t ) 78. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the
circumference C, what is the rate of change of the area of the circle, in square centimeters per
second?
(A) − ( 0.2 ) π C (B) − ( 0.1) C (C) − (D) ( 0.1)2 C (E) ( 0.1)2 π C ( 0.1) C
2π AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 147 1998 AP Calculus BC:
Section I, Part B
79. Let f be the function given by f ( x) = ( x − 1)( x 2 − 4)
x2 − a continuous for all real numbers x?
(A)
(B)
(C)
(D)
(E) . For what positive values of a is f None
1 only
2 only
4 only
1 and 4 only ( ) 80. Let R be the region enclosed by the graph of y = 1 + ln cos 4 x , the xaxis, and the lines x = −
2
. The closest integer approximation of the area of R is
3 and x =
(A) 2
3 (B) 1 0 (C) 2 (D) 3 (E) 4 (D) (E) dy
d2y
2
81. If
= 1 − y , then
=
dx
dx 2
(A) −2 y (B) −y 82. If f ( x) = g ( x) + 7 for 3 ≤ x ≤ 5, then (A) 2∫ (B) 2∫ (C) 2∫ (D) ∫3 (E) ∫3 5
3
5
3
5
3 5 5 (C) −y 1− y2 y 1
2 5 ∫ 3 [ f ( x) + g ( x)] dx = g ( x) dx + 7
g...
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 Calculus, AP, Natural logarithm, College Entrance Examination Board, Appalachian State Mountaineers football seasons, Calculus MultipleChoice Question

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