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First Semester Review
Name____________________
AP Calculus AB
Questions 1-6 are calculator active questions. All other questions are non-calculator.
1. An object moves along the y-axis with coordinate position y(t) and velocity
v(t)=
a.
b.
c.
d.
e.
t − cos(et ) for t ≥ 0 . At time t = 1, the object is
moving downward with negative acceleration
moving upward with negative acceleration
moving downward with positive acceleration
moving upward with positive acceleration
at rest
2. The volume of a gas-filled spherical balloon increases 6 cubic inches for each degree (Celsius)
increase in temperature. If the temperature increases at a constant rate of 2 degrees per minute,
then at what rate (in inches per minute) is the radius of the balloon changing at the instant when
the volume is 36π cubic inches? The volume of a sphere of radius r is
a.
1
π
1
b.
2π
1
c.
3π
4 3
πr .
3
1
6π
2
e.
3π
d.
3. The number of inflection points for the graph of f ( x) = x + cos( x 2 ) in the interval 0 ≤ x ≤ 5 is
a. 6
d. 9
b. 7
e. 10
c. 8
\
4. The position of a weight attached to a spring is given by x(t ) = e−t sin(4t ) . The total distance
traveled by the weight from time t = 0 to t = 1 is nearest to
a. 1.3
d. 1.6
b. 1.4
e. 1.7
c. 1.5
5. A function f is continuous for 0 ≤ x ≤ 5 and differentiable for 0 < x < 5 . Given that
f(0) = -2 and f(5)= 3, which of the following statements must be true?
I . f ′(c) = 1 for some c such that 0 < c < 5
II. f (c) = 0 for some c such that 0 < c < 5
III. f (c) = −1 for some c such that 0 < c < 5
a. I only
d. II and III
b. II only
e. I, II, and III
c. I and II
6. If f ′( x) = tan −1 ( x3 − x) , at how many points is the tangent line to the graph of y = f(x) parallel
to the line y = 2x?
a. None
d. Three
b. One
e. Infinitely many
c. Two
7. If f ( x) = ln(e2 x ) , then f ′( x) =
a. 1
b. 2
c. 2x
d. e −2 x
e. 2e −2 x
8. Which of the following functions are continuous at x = 1 ?
I. ln x
II. e x
III. ln(e x − 1)
a. I only
b. II only
c. I and II only
d. II and III only
e. I, II, and III
9. The position of a particle moving along the x -axis is x(t ) = sin(2t ) − cos(3t ) for time t ≥ 0 .
When t = π , the acceleration of the particle is
a. 9
d. −1/ 9
b. 1/ 9
e. −9
c. 0
10. The derivative of f is x 4 ( x − 2)( x + 3) . At how many points will the graph of f have a
relative maximum?
a. None
d. Three
b. One
e. Four
c. Two
11. If f ( x) = etan x , then f ′( x) =
2
2
d. 2 tan x sec2 x etan
a.
e tan
x
b.
sec2 x etan
c.
tan 2 x etan
2
x
2
x −1
e. 2 tan x e tan
2
2
x
x
12. The slope of the line tangent to the graph of ln( xy ) = x at the point where x = 1 is
a. 0
b. 1
c. e
d. e 2
e. 1 − e
13. If e f ( x ) = 1 + x 2 , then f ′( x) =
1
1 + x2
2x
b.
1 + x2
c. 2 x(1 + x 2 )
a.
2
d. 2 x(e1+ x )
e. 2 x ln(1 + x 2 )
14. The Mean Value Theorem guarantees the existence of a special point on the graph of y =
between ( 0, 0 ) and ( 4, 2 ) . What are the coordinates of this point?
⎛1
⎝2
1 ⎞
⎟
2⎠
a.
( 2, 1)
d. ⎜ ,
b.
(1, 1)
e. None of the above
c.
x
( 2, 2 )
15. A point moves on the x -axis in such a way that its velocity at time t ( t > 0 ) is given by v =
At what value of t does v attain its maximum?
a. 1
d. e 3/ 2
b. e1/ 2
e. There is no maximum value for v .
c. e
ln t
.
t
16. At x = 0 , which of the following is true of the function f defined by f ( x) = x 2 + e−2 x ?
a. f is increasing.
d. f is has a relative minimum.
b. f is decreasing.
e. f is has a relative maximum.
c. f is discontinuous.
dy
in terms of x ?
dx
d. tan x
e. csc x
17. If sin x = e y , 0 < x < π , what is
a.
b.
c.
− tan x
− cot x
cot x
18. If f ( x) = e1/ x , then f ′( x) =
e1/ x
x2
a.
−
b.
−e1/ x
c.
e1/ x
x
e1/ x
x2
1 (1/ x ) −1
e.
e
x
d.
1
, then the set of values for which f increases is
x
a. ( −∞, − 1] ∪ [1, + ∞ ) d. ( 0, ∞ )
19. If f ( x) = x +
b.
c.
[−1,1]
( −∞, ∞)
e.
( −∞, 0) ∪ (0, ∞ )
x −1
for all x ≠ −1 , then f ′(1) =
x +1
a. −1
d. 1/ 2
b. −1/ 2
e. 1
c. 0
20. If f ( x) =
21. If y = sin x and y ( n ) means “the n th derivative with respect to x ,” then the smallest positive
integer
a.
b.
c.
n for which y ( n ) = y is
2
d. 6
4
e. 8
5
22. If f ( x) = cos 2 3x , then
a.
b.
c.
−6sin 3 x cos 3 x
−2 cos 3x
2 cos 3x
dy
=
dx
d. 6 cos 3x
e. 2sin 3 x cos 3 x
23. If f ( x) = ln(ln x) , then f ′( x) =
a.
b.
c.
1
x
1
ln x
ln x
x
d. x
e.
1
x ln x
f ( x) − f (2)
= 0 , which of the following must be true?
x →2
x−2
The limit of f ( x) as x approaches 2 does not exist.
f is not defined at x = 2 .
The derivative of f at x = 2 is 0 .
f is continuous at x = 0 .
f (2) = 0
24. If f is a function such that lim
a.
b.
c.
d.
e.
25. 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
d. f ( x) = x 4/ 3
b.
c.
f ( x) = x −1/ 3
f ( x) = x1/ 3
e. f ( x) = x3
26. Let f be the function defined by the following.
For what values of x is f NOT continuous?
a. 0 only
d. 0 and 2 only
b. 1 only
e. 0 , 1 , and 2
c. 2 only
x<0
⎧sin x
⎪ x2
0 ≤ x <1
⎪
f ( x) = ⎨
⎪2 − x 1 ≤ x < 2
⎪⎩ x − 3
x≥2
27. If y 2 − 2 xy = 16 , then
dy
=
dx
x
d.
y−x
y
b.
e.
x− y
y
c.
y−x
28. If f ( x) = e x , then ln ( f ′(2) ) =
a. 2
d.
b. 0
e.
2
c. 1/ e
a.
y
2y − x
2y
x− y
2e
e2
29. Which of the following pairs of graphs could represent the graph of a function and the graph of its
derivative?
a. I only
b. II only
c. III only
30. lim
h →0
sin( x + h) − sin x
=
h
a. 0
b. 1
c. sin x
d. I and III
e. II and III
d. cos x
e. nonexistent
31. If x + 7 y = 29 is an equation of the line normal to the graph of f at the point (1, 4 ) , then
f ′(1) =
a. 7
b. 1/ 7
c. −1/ 7
d. −7 / 29
e. −7
32. A polynomial p ( x) has a relative maximum at ( −2, 4 ) , a relative minimum at (1, 1) , a relative
maximum at ( 5, 7 ) and no other critical points. How many real zeros does p ( x) have?
a. One
b. Two
c. Three
d. Four
e. Five
33. If the graph of y = x3 + ax 2 + bx − 4 has a point of inflection at (1, − 6 ) , then what is the value
of b ?
a. −3
b. 0
c. 1
34.
d
⎛π⎞
ln cos ⎜ ⎟ =
dx
⎝x⎠
−π
a.
⎛π⎞
x 2 cos ⎜ ⎟
⎝x⎠
⎛π⎞
b. − tan ⎜ ⎟
⎝x⎠
1
c.
⎛π⎞
cos ⎜ ⎟
⎝x⎠
d. 3
e. It cannot be determined from the information given.
d.
π
⎛π⎞
tan ⎜ ⎟
x
⎝x⎠
e.
π
⎛π⎞
tan ⎜ ⎟
2
x
⎝x⎠
35. A person 2 meters tall walks directly away from a streetlight that is 8 meters above the ground.
If the person is walking at a constant rate and the person’s shadow is lengthening at a rate of 4 / 9
meter per second, at what rate, in meters per second, is the person walking?
a. 4 / 27
d. 4 / 3
b. 4 / 9
e. 16 / 9
c. 3 / 4
36. If f and g are continuously differentiable functions, each with domain the set of all real
numbers, and if g is the inverse function of f and f (a) = b , then g ′(b) is
a.
b.
c.
1
f ′(a)
1
f (b)
f ′( a )
d. f ′(b)
e. f ′(a) ⋅ g (b)
37. If the line y = 3x − 5 is tangent to the graph of y = f ( x) at the point ( 4, 7 ) , then
lim
h →0
f (4 + h) − f (4)
is
h
a. −5
b. 3
c. 4
d. 7
e. nonexistent
38. Let f be a function whose domain is the open interval ( −3, 4 ) , and let the derivative of f have
the graph shown in the figure above. The function f is increasing on which of the following
intervals.
a. ( −3, − 1) and ( 2, 4 ) d. ( −1, 2 ) only
b.
c.
( −2, 0) and (3, 4 )
( −3, − 2) and (0, 3)
e.
(0, 3) only
( )′
−1
(0) ?
39. If f (x) = x 3 − 7x 2 + 25x − 39 then what is the value of f
d. 1 0
e. 25
a. -0.04
b. 0.04
c. 0.10
Answers:
1. D
2.C
14. B
15. C
27. C
28. A
3. C
16. B
29. D
4. E
17. C
30. D
5. E
18. A
31. A
6. A
19. A
32. B
7. B
20. D
33. B
8. E
21. B
34. E
9. E
22. A
35. D
10. B
23. E
36. A
11. D
24. C
37. B
12. A
25. C
38. B
13. B
26. C
39. C
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