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 relativ
1997 AP Calculus AB:
Section I, Part B
79. Let f be a function such that lim
h 0
I.
f (2 + h) f (2)
= 5 . Which of the following must be true?
h
f is continuous at x = 2.
II. f is differentiable at x = 2.
III. The derivative of f is continuous at x = 2 .
1993 AP Calculus BC: Section I
x
37. If f ( x) = 1
x
for x 1
then
for x > 1,
(A) 0
(B)
e
0 f ( x)dx =
3
2
(C) 2
(D) e
(E)
e+
1
2
38. During a certain epidemic, the number of people that are infected at any time increases at a rate
proportional to the num
1993 AP Calculus BC: Section I
26. If y = arctan(e 2 x ), then
2e2 x
(A)
1 e4 x
(B)
dy
=
dx
2e 2 x
1+ e
27. The interval of convergence of
(C)
4x
n =0
( x 1) n
3n
e2 x
1+ e
4x
(D)
1
(E)
1 e4 x
1
1 + e4 x
is
(A) 3 < x 3
(B) 3 x 3
(D) 2 x < 4
(C) 2 < x < 4
1993 AP Calculus AB: Section I
8.
If y = tan x cot x, then
(A) sec x csc x
9.
dy
=
dx
(B) sec x csc x
(C) sec x + csc x
(D) sec2 x csc2 x
(E) sec2 x + csc2 x
If h is the function given by h( x) = f ( g ( x), where f ( x) = 3 x 2 1 and g ( x) = x , then h(
1988 AP Calculus BC: Section I
33. The length of the curve y = x3 from x = 0 to x = 2 is given by
2
(A)
0
1 + x 6 dx
(D)
2
2
0
1 + 3x 2 dx
(E)
1 + 9x 4 dx
0
2
(B)
0
2
(C)
2
1 + 9x 4 dx
0
1 + 9x 4 dx
34. A curve in the plane is defined parametrically by
1988 AP Calculus BC: Section I
24. If c is the number that satisfies the conclusion of the Mean Value Theorem for f ( x) = x3 2 x 2 on
the interval 0 x 2, then c =
(A) 0
1
2
(B)
(C) 1
(D)
4
3
(E) 2
25. The base of a solid is the region in the first quadra
1988 AP Calculus BC: Section I
16.
xe2x dx =
xe 2 x e 2 x
+C
(A)
2
4
(B)
(E)
xe 2 x e2 x
+
+C
(D)
2
2
17.
3
2
(A)
xe 2 x e 2 x
+C
2
2
x 2e2 x
+C
4
(C)
xe 2 x e 2 x
+
+C
2
4
3
dx =
( x 1)( x + 2)
33
20
(B)
9
20
(C)
5
ln
2
8
(D) ln
5
(E)
2
ln
5
18. If th
1988 AP Calculus AB: Section I
15. If f ( x) = 2 x , then f (2) =
(A)
1
4
(B)
1
2
(C)
2
2
(D) 1
2
(E)
16. A particle moves along the x-axis so that at any time t 0 its position is given by
x(t ) = t 3 3t 2 9t + 1 . For what values of t is the particle at
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.
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,
1985 AP Calculus BC: Section I
21. If
(A)
f ( x) sin x dx = f ( x) cos x + 3 x 2 cos x dx , then f ( x) could be
3x 2
(B)
x3
(C)
x3
(D) sin x
(E)
cos x
22. The area of a circular region is increasing at a rate of 96 square meters per second. When the are
1985 AP Calculus BC: Section I
16. Which of the following functions shows that the statement If a function is continuous at x = 0 ,
then it is differentiable at x = 0 is false?
(A)
f ( x) = x
4
3
(B)
1
3
(C)
f ( x) =
ln x 2 + 2
(C)
ln x 2 +
f ( x) = x
1
x
1985 AP Calculus BC: Section I
11.
d 1
ln
=
dx 1 x
(A)
12.
1
1 x
1
x 1
(B)
(C) 1 x
(D)
x 1
(E)
(1 x )2
dx
( x 1)( x + 2) =
(A)
1
x 1
ln
+C
3
x+2
(B)
1
x+2
ln
+C
3
x 1
(D)
( ln
(E)
ln ( x 1)( x + 2) 2 + C
x 1
)( ln
x+2 )+C
(C)
1
ln ( x 1)( x + 2) + C
3
1985 AP Calculus AB: Section I
20. If y = arctan ( cos x ) , then
dy
=
dx
sin x
(A)
(B) ( arcsec ( cos x ) ) sin x
2
2
1 + cos x
1
(D)
( arccos x )
2
(E)
+1
1 + cos 2 x
1
1 x2
is cfw_ x : x > 1 , what is the range of f ?
(A)
cfw_ x : < x < 1
(B)
cfw_ x :
1985 AP Calculus AB: Section I
13. If x 2 + xy + y 3 = 0 , then, in terms of x and y,
(A)
2x + y
x + 3y
2
(B)
x + 3y2
2x + y
(C)
dy
=
dx
2 x
1+ 3y
(D)
2
14. The velocity of a particle moving on a line at time t is v
meters did the particle travel from t =
1985 AP Calculus AB: Section I
6.
If f ( x) = x, then f (5) =
(A) 0
7.
ln 3 + ln1
1
(D) 5
(E)
25
2
ln 8
ln 2
(B)
(C)
4
1
et dt
(D)
4
1
ln x dx
(E)
4
1
1
dt
t
x
The slope of the line tangent to the graph of y = ln at x = 4 is
2
1
8
(A)
9.
(C)
Which of the
1973 AP Calculus BC: Section I
22. A particle moves on the curve y = ln x so that the x-component has velocity x(t ) = t + 1 for t 0 .
At time t = 0 , the particle is at the point (1, 0 ) . At time t = 1 , the particle is at the point
(A)
(B)
( e2 , 2 )
(
1973 AP Calculus BC: Section I
17. The number of bacteria in a culture is growing at a rate of 3, 000e 2t 5 per unit of time t. At t = 0 ,
the number of bacteria present was 7,500. Find the number present at t = 5 .
(A) 1, 200e 2
(B)
3, 000e 2
(C) 7,500e
1973 AP Calculus AB: Section I
x over the interval 0 x 2 is
34. The average value of
1
2
3
(A)
(B)
1
2
2
(C)
2
2
3
(D) 1
35. The region in the first quadrant bounded by the graph of y = sec x, x =
(E)
4
2
3
, and the axes is rotated
4
about the x-axis. Wh
1973 AP Calculus AB: Section I
23.
1 2+h
ln
is
h0 h 2
lim
(A)
e2
(B) 1
(C)
1
2
(D) 0
(E) nonexistent
24. Let f ( x) = cos ( arctan x ) . What is the range of f ?
(A)
(D)
25.
x < x <
2
cfw_x
4
0
(A)
2
(B)
0 < x 1
(E)
1 < x < 1
cfw_x
(C)
cfw_x
1 x 1
(
1973 AP Calculus AB: Section I
13. The acceleration of a body moving in a straight line is given in terms of time t by = 8 6t . If
the velocity of the body is 25 at t = 1 and if s (t ) is the distance of the body from the origin at time
t, what is s (4) s
1973 AP Calculus AB: Section I
90 MinutesNo Calculator
Note: In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e).
1.
(x
3
)
3 x dx =
(A)
(D)
2.
3x 2 3 + C
(B)
x4
3x + C
4
(E)
5 x 2 + 15 x + 25
(B)
( )
1
e
(B)
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.
1969 AP Calculus AB: Section I
40. If n is a non-negative integer, then
1
0 x
(A) no n
(D) nonzero n, only
n
1
0
(1 x )n dx
for
(B) n even, only
(E) all n
f ( x) = 8 x 2 for 2 x 2,
41. If
then
2
elsewhere ,
f ( x) = x
(A) 0 and 8
dx =
(B) 8 and 16
(C)