Unformatted text preview: umed to be the set of all real
numbers x for which f (x) is a real number. 76. The graph of a function f is shown above. Which of the following statements about f is false?
(A) f is continuous at x = a . (B) f has a relative maximum at x = a . (C) x = a is in the domain of f. (D)
(E) lim f ( x) is equal to lim− f ( x) . x →a + x →a lim f ( x) exists . x →a 77. Let f be the function given by f ( x) = 3e 2 x and let g be the function given by g ( x) = 6 x3 . At what
value of x do the graphs of f and g have parallel tangent lines?
(A)
(B)
(C)
(D)
(E) −0.701
−0.567
−0.391
−0.302
−0.258 AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 133 1998 AP Calculus AB:
Section I, Part B
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) − ( 0.1) C
2π (D) ( 0.1)2 C (E) ( 0.1)2 π C 79. The graphs of the derivatives of the functions f, g, and h are shown above. Which of the functions
f, g, or h have a relative maximum on the open interval a < x < b ?
(A)
(B)
(C)
(D)
(E) f only
g only
h only
f and g only
f, g, and h 80. The first derivative of the function f is given by f ′( x) =
does f have on the open interval ( 0,10 ) ?
(A)
(B)
(C)
(D)
(E) cos 2 x 1
− . How many critical values
x
5 One
Three
Four
Five
Seven AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 134 1998 AP Calculus AB:
Section I, Part B
81. Let f be the function given by f ( x) = x . Which of the following statements about f are true?
f is continuous at x = 0 .
f is differentiable at x = 0 .
f has an absolute minimum at x = 0 . I.
II.
III. (A) I only (B) II only (C) III only (D) I and III only (E) II and III only 82. If f is a continuous function and if F ′( x) = f ( x) for all real numbers x, then
(A) 2 F (3) − 2 F (1) (B) 1
1
F (3) − F (1)
2
2 (C) 2 F (6) − 2 F (2) (D) F (6) − F (2) (E) 3 ∫ 1 f ( 2 x ) dx = 1
1
F (6) − F (2)
2
2 83. If a ≠ 0, then lim x →a (A) 1
a2 x2 − a2
x4 − a4 (B) is
1
2a 2 (C) 1
6a 2 (D) 0 (E) nonexistent dy
= ky , where k is a constant and t is measured in
dt
years. If the population doubles every 10 years, then the value of k is 84. Population y grows according to the equation (A) 0.069 (B) 0.200 (C) 0.301 AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 3.322 (E) 5.000 135 1998 AP Calculus AB:
Section I, Part B
2 x
f ( x) 5 7 8 10 30 40 20 85. The function f is continuous on the closed interval [ 2,8] and has values that are given in the table
above. Using the subintervals [ 2,5] , [5, 7 ] , and [ 7,8] , what is the trapezoidal approximation of
8 ∫ 2 f ( x) dx ?
(A) 110 (B) 130 (C) 160 (D) 190 (E) 210 86. The base of a solid is a region in the first quadrant bounded by the xaxis, the yaxis, and the line
x + 2 y = 8 , as shown in the figure above. If cross sections of the solid perpendicular to the xaxis
are semicircles, what is the volume of the solid?
(A) 12.566 (B) 14.661 (C) 16.755 (D) 67.021 (E) 134.041 87. Which of the following is an equation of the line tangent to the graph of f ( x) = x 4 + 2 x 2 at the
point where f ′( x) = 1?
(A)
(B)
(C)
(D)
(E) y = 8x − 5
y = x+7
y = x + 0.763
y = x − 0.122
y = x − 2.146 88. Let F ( x) be an antiderivative of
(A) 0.048 (B) 0.144 ( ln x )3 . If
x (C) F (1) = 0, then F (9) =
5.827 AP Calculus MultipleChoice Question Collection
Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 23.308 (E) 1,640.250 136 1998 AP Calculus AB:
Section I, Part B
89. If g is a differentiable function such that g ( x) < 0 for all real numbers x and if ( ) f ′( x) = x 2 − 4 g ( x) , which of the following is true? (A)
(B)
(C)
(D)
(E) f has a relative maximum at x = −2 and a relative minimum at x = 2 .
f has a relative minimum at x = −2 and a relative maximum at x = 2 .
f has relative minima at x = −2 and at x = 2 .
f has relative maxima at x = −2 and at x = 2 .
It cannot be determined if f has any relative extrema. 90. If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is
decreasing at a rate of 3 inches per minute, which of the following must be true about the area A of
the triangle?
(A)
(B)
(C)
(D)
(E) A is always increasing.
A is always decreasing.
A is decreasing only when b < h .
A is decreasing only when b > h .
A remains constant. 91. Let f be a function that is differentiable on the open interval (1,10 ) . If f (2) = −5, f (5) = 5, and
f (9) = −5 , which of the following must be true?
I.
II.
III.
(A)
(B)
(C)
(D)
(E) f has at least 2 zeros.
The graph of f has at least one horizontal tangent.
For some c, 2 < c &l...
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