1. No category

AP Calculus
Sample Examination I
Calculators NOT allowed. You have 45 minutes to complete questions 1-27.
 x 2 + bx,

1. If f is a continuous function defined by f ( x ) = 
π
5sin  2


(A) -6
(B) -5
(C) -4
(D) 4
(E) 5
x≤5

x,

x>5
then b =
2. The graph of y = 3 x 2 − x3 has a relative maximum at
(A)
(B)
(C)
(D)
(E)
(0, 0) only
(1, 2) only
(2, 4) only
(4, -16) only
(0, 0) and (2, 4)
3. A line through the origin rotates around the origin in such a way that the angle, θ , between the line
dθ
for time t ≥ 0 . Which expression give the rate at
and the positive x-axis changes at the rate of
dt
which the slope of the line is changing?
(A)
dθ
dt
(B) cos θ
dθ
dt
(D)
1 dθ
cos 2 θ dt
(E) tan θ
dθ
dt
4. If x + y = xy , then
(A)
dy
is
dx
1
x −1
(D) x + y − 1
(B)
y −1
x −1
(E)
2 − xy
y
(C) − sin θ
(C)
1− y
x −1
(D)
1
10
dθ
dt
108 x 5 + 106 x 4 + 104 x 2
=
x →∞ 109 x 6 + 107 x 5 + 105 x 3
5. lim
(A) 0
(B) 1
(C) -1
(E) −
1
10
6.
The graph of h ( x ) is shown above. Which of the following could be the graph of y = h′ ( x ) ?
(A)
(B)
(D)
(E)
(C)
7. If f ( x ) = 4sin x + 2 , then f ′ ( 0 ) =
(A) -2
(B) 0
(C) 1
(D)
2
2
(E)
2
8. The equation of the tangent line to the curve x 2 + y 2 = 169 at the point (5, -12) is
(A)
(B)
(C)
(D)
(E)
5 y − 12 x = −120
5 x − 12 y = 119
5 x − 12 y = 169
12 x + 5 y = 0
12 x + 5 y = 169
9.
The figure above shows the graph of a function f ( x ) which has horizontal asymptotes of
y = 3 and y = −3 . Which of the following statements are true?
I. f ′ ( x ) < 0 for all x ≥ 0
II. lim f ′ ( x ) = 0
x →+∞
III. lim f ′ ( x ) = 2
x →−∞
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III
10.
The figure above shows the graph of the velocity of a moving object as a function of time. At which of
the marked points is the speed the greatest?
(A) A
(B) B
(C) C
(D) D
(E) E
11.
x
f ( x)
0
0
1
2
3
4
5
0.25 0.48 0.68 0.84 0.95
6
1
6
For the function whose values are given in the table above,
∫ f ( x ) dx is approximated by a Riemann
0
Sum using the value at the midpoint of each of three intervals of width 2. The approximation is
(A)
(B)
(C)
(D)
(E)
2.64
3.64
3.72
3.76
4.64
12. The lim
h →0
(A)
(B)
(C)
(D)
(E)
x+h − x
at x = 3 is
h
-1
0
1
3
Nonexistent
13. If the graph of f ( x ) = 2 x 2 +
(A)
(B)
(C)
(D)
(E)
14.
k
has a point of inflection at x = −1 , then the value of k is
x
-2
-1
0
1
2
∫ sin ( 3x + 4 ) dx =
1
(A) − cos ( 3 x + 4 ) + C
3
(B) − cos ( 3 x + 4 ) + C
(D) cos ( 3 x + 4 ) + C
(E)
1
cos ( 3 x + 4 ) + C
3
15. What are all values of x for which the graph of y =
(A)
(B)
(C)
(D)
(E)
(C) −3cos ( 3 x + 4 ) + C
2
is concave downward?
4− x
No values of x
x<4
x > −4
x < −4
x>4
16. A particle moves along the x-axis in such a way that its position at time t is given by x ( t ) =
What is the acceleration of the particle at time t = 0?
(A)
(B)
(C)
(D)
(E)
-4
-2
-0.6
2
4
1− t
.
1+ t
17. At what value(s) of x does f ( x ) = x 4 − 8 x 2 have a relative minimum?
(A)
(B)
(C)
(D)
(E)
0 and -2 only
0 and 2 only
0 only
-2 and 2 only
-2, 0, and 2
18. The edge of a cube is increasing at the rate of 0.05 centimeters per second. In terms of the edge of
the cube, s, what is the rate of change of the volume of the cube, in cubic centimeters per second?
19. If
(A)
(B)
(C)
(D)
(E)
0.053
0.05s 2
0.05s 3
0.15s 2
3s 2
6
− 2 x + 2 ) dx is approximated by a left endpoint Reimann Sum (n = 3) on the x-axis, then the
∫(x
2
0
approximation is
(A)
(B)
(C)
(D)
(E)
20. If
24
26
28
48
76
dy
d2y
=
= x 2 y 2 , then
dx
dx 2
(C) 2 x + 2 x 2 y 3
(A) 2 xy 2
(B) 4 x3 y 3
(D) 2 x 2 y + 2 xy 2
(E) 2 x 4 y 3 + 2 xy 2
21. Let f be the function give by f ( x ) = x3 . What are all values of c that satisfy the conclusion of the
Mean Value Theorem on the closed interval [-1, 2]?
(A) 0 only
(B) 1 only
(D) -1 and 1
(E) − 3 and
(C)
3
3 only
22.
x
Let g ( x ) = ∫ f ( t ) dt . The figure above shows the graph of g. Which of the following could be the
a
graph of y =
d
f ( x ) ]?
dx
(A)
(B)
(D)
(E)
(C)
23. If for all real numbers x, f ′ ( x ) < 0 and f ′′ ( x ) > 0 , which of the following curves could be part of
the graph of f ?
(A)
(B)
(D)
(E)
(C)
24.
The functions f and g are piecewise linear functions whose graph are shown above. If
h ( x ) = f ( x ) g ( x ) , then h′ ( 3) =
(A) −
2
25. If
∫ ( 2x
3
8
3
(B) −
1
3
2
3
(C) 0
(D)
(C) 1
(D) 2
(E)
8
3
− kx 2 + 2k ) dx = 12 , the k must be
0
(A) -3
(B) -2
(E) 3
x≤0
 x,
26. Let f ( x ) be the function defined by f ( x ) = 
. The value of
 x + 1, x > 0
3
5
7
(A)
(B)
(C) 3
(D)
2
2
2
1
∫ xf ( x ) dx =
−2
(E)
11
2
1 
27. The average value of the function f ( x ) = cos  x  on the closed interval [-4, 0] is
2 
1
1
1
(A) − sin ( 2 )
(B) − sin ( 2 )
(C) cos ( 2 )
2
4
2
(D)
1
sin ( 2 )
4
(E)
1
sin ( 2 )
2
STOP WHEN FINISHED
AP Calculus
Sample Examination I
Calculators ALLOWED. You have 45 minutes to complete questions 28-43.
28.
The graph above shows the distance s ( t ) from a reference point of a particle moving on a number line,
as a function of time. Which of the points marked is closest to the point where the acceleration first
becomes negative?
(A) A
(B) B
(C) C
(D) D
(E) E
29. Let f be the function given by f ( x ) = tan x and let g be the function given by g ( x ) = x 2 . At what
value of x in the interval 0 ≤ x ≤ π do the graphs of f and g have parallel tangent lines?
(A)
(B)
(C)
(D)
(E)
0
0.660
2.083
2.194
2.207
30. Let f ( t ) =
1
for t > 0. For what value of t is f ′ ( t ) equal to the average rate of change of f on [a, b]?
t
(A) − ab
(D)
1
ab
ab
(E)
11 1
 − 
2b a
31. Let f and g be differentiable functions such that
(B) 15
f (1) = 4,
g (1) = 3,
1
ab
f ′ ( 3) = −5
f ′ (1) = −4, g ′ (1) = −3, g ′ ( 3) = 2
If h ( x ) = f ( g ( x ) ) , then h′ (1) =
(A) -9
(C) −
(B)
(C) 0
(D) -5
(E) -12
32.
x
0
1
2
3
4
5
6
7
8
9
10
f ( x)
20
19.5
18
15.5
12
7.5
2
-4.5
-12
-20.5
-30
Some values of a continuous function are given in the table above. Using 10 subintervals of equal
10
length, the Trapezoidal Rule approximation for
∫ f ( x ) dx is
0
(A)
(B)
(C)
(D)
(E)
7.500
32.500
33.325
33.333
57.500
33. Let R ( t ) represent the rate at which water is leaking out of a tank, where t is measured in hours.
Which of the following expressions represents the total amount of water in gallons that leaks out in
the first three hours?
(A) R ( 3) − R ( 0 )
3
(D)
(B)
R ( 3) − R ( 0 )
3
(C)
3−0
∫ R ( t ) dt
0
3
1
(E) ∫ R ( t ) dt
30
∫ R′ ( t ) dt
0
x3
d
34.
sin ( t 2 ) dt =
dx ∫x
(A) sin ( x 6 ) − sin ( x 2 )
(B) 6 x 2 sin ( x 3 ) − 2sin x
(C) 3x 2 sin ( x 6 ) − sin ( x 2 )
(D) 6 x5 sin ( x 6 ) − 2sin ( x 2 ) (E) 2 x 3 cos ( x 6 ) − 2 x cos ( x 2 )
35.
t (sec)
a (t )
( ft / sec )
2
0
2
4
6
8
2
3
4
3
2
The table for the acceleration of a particle from 0 to 8 is given in the table above. If the velocity at t = 0
is 4 feet per second, the approximation value of the velocity, in feet per second, at t = 8, computed using
the Right Riemann Sum with four subdivisions of equal length is
(A) 4
(B) 12
(C) 16
(D) 24
(E) 28
36. Suppose that f ( x ) is an even function and let
1
7
0
0
∫ f ( x ) dx = 5 and ∫ f ( x ) dx = 1 .
−1
What is
∫ f ( x ) dx ?
−7
(A) -5
(B) -4
37.
(C) 0
(D) 4
(E) 5
f (t )
x
If g ( x ) = ∫ f ( t ) dt and f ( t ) is shown above, how many points of inflection does g ( x ) have?
0
(A) None
38.
(B) One
(C) Two
(D) Three
(E) Four
∫ ( sec x )( tan x ) dx =
2
2
(A)
1 3
tan x + C
3
(B) tan 3 x + C
(D)
1 3
sec x + C
3
(E) tan 2 x + C
(C)
1
tan 2 x + C
2
39. A tank is being filled with water at the rate of 300 t gallons per hour with t > 0 measured in hours.
If the tank is originally empty, how many gallons of water are in the tank after 4 hours?
(A) 600
40.
∫
(B) 900
(C) 1200
(D) 1600
(E) 2400
x ( x + 2 ) dx =
(A)
x +2 x +C
3
2 52 4 23
(D) x + x + C
5
3
2 32 4 12
(B) x + x + C
5
3

x 2  2 32
(E)
 x + 2x  + C
2 3

(C)
3
1
x+
+C
2
x
41.
As shown in the figure above the function f ( x ) consists of a line segment from (0, 4) to (8, 4) and onequarter of a circle with radius of 4. What is the average (mean) value of this function on the interval [0,
12]?
(A) 2
(B) 3.714
(C) 3.9
(D) 22.283
(E) 41.144
42. The amount A ( t ) of a certain item produced in a factory is given by
A ( t ) = 4000 + 48 ( t − 3) − 4 ( t − 3)
3
where t is the number of hours of production since the beginning of the workday at 8:00 a.m. At
what time is the rate of production increasing most rapidly?
(A) 8:00 a.m
(B) 10:00 a.m
(D) 12:00 noon
(E) 1:00 p.m.
(C) 11:00 a.m.
43.
f ′( x)
Let f be a differentiable function for all x. The graph of f ′ ( x ) is shown above. If f ( 2 ) = 10 , which of
the following best approximates the maximum value of f ( x ) ?
(A) 30
B) 50
C) 70
D) 80
E) 110