Chapter 2
1
is
t2
2
2
1
B)  3 C)
D) 3
t
t
t
Difficulty: moderate Section: 2.1

1. The derivative of f (t ) 

2
t
Ans: B
A) 

1
is
t8
–8
–8
8
1
A) 9
B) 7
C) 7 D) 9

–3
A)
B)
C)
D) –3 x
3
3
2 x
2 x
2 x
Ans: A Difficulty: moderate Section: 2.1

4. The derivative of f ( x) 

5. True or False: The derivative of f ( x)  x 2  5 is 2x + 5.
Ans: False Difficulty: easy Section: 2.1
6. True or False: The derivative of f ( x)  x 2  5 is 2x + 5.
Ans: False Difficulty: easy Section: 2.1
7. The equation of the line tangent to the graph of f ( x)  x 2  3x at x = 2 is
A) y = 7x – 4 B) y = 7x – 422 C) y = 7x – 2 D) y = 7x – 144
Ans: A Difficulty: moderate Section: 2.1
8. The equation of the line tangent to the graph of f ( x)  x 2  8 x at x = 6 is
A) y = 20x – 36 B) y = 20x – 1728 C) y = 20x – 6 D) y = 20x – 216
Ans: A Difficulty: moderate Section: 2.1

Page 21


Chapter 2


1
1
1
3
3
3
A) y  x 
B) y  x 
C) y  x 
D) y  x  1
2
2
2
2
2
2
2
Ans: C Difficulty: moderate Section: 2.1
12. True or False: The tangent to the graph of f ( x)  x  3 at x = 2 has slope
Ans: False

Difficulty: moderate

1
.
2

Section: 2.1

13. True or False: The tangent to the graph of f ( x)  x  5 at x = 2 has slope of


f ( x0  h)  f ( x0 )
?
h 0
h
Ans: The average rate of change of oil price with respect to time on the time interval [x0,
x0 + h]; the instantaneous rate of change of oil price with respect to time at time x0.
Difficulty: easy Section: 2.1

represent? What about lim

Page 22


Chapter 2

17. A spherical balloon is being filled with air in such a way that its radius is increasing at the
constant rate of 2 cm/sec. At what rate is the volume of the balloon increasing at the
instant when its surface has area 4 cm2 ? (Note: A sphere of radius r has volume
4
V   r 3 and surface area S  4 r 2 .)
3
Ans: 8 cm 3 /sec
Difficulty: hard Section: 2.1
18. True or False: Differentiating f ( x)  x3  3x  1 gives 3x 2 .
Ans: False Difficulty: easy Section: 2.2
19. True or False: Differentiating f ( x)  x 2  6 x  5 gives 2x1 .
Ans: False Difficulty: easy Section: 2.2
20. Differentiate f ( x)  x8  2
A) 8 x 7  2 B) 8 x9  2 x C) 8x 7 D) 7x 7


Page 23


Chapter 2

1
, differentiate f(x).
x
1
1
Ans: f ( x)  2  3
3
3x
2x 2
Difficulty: moderate Section: 2.2

26. If f ( x)  3 x 

27. Differentiate f ( x)  x 
A) 0

1

B) x C)

Ans: D





1
2 x3

1
x

B) 0 C) 1 D)

2 x 2 x
Ans: A Difficulty: easy



1

2 x
Section: 2.2



1
2 x3

3
.
x

Ans:



31. Differentiate: f ( x) 

32. Find the equation of the tangent line to the curve f ( x)  x3  x 2  8 at the point (1, 8)
Ans: y = x + 7.
Difficulty: moderate Section: 2.2

Page 24


Chapter 2

33. Find the equation of the tangent line to the curve f ( x)  x3  x 2  1 at the point (1, 1).
Ans: y = x
Difficulty: moderate Section: 2.2
34. Find the equation of the tangent to the graph of f ( x)  x 2  9 x  16 at the point (1, 8).
Ans: y = –7x + 15
Difficulty: moderate Section: 2.2
35. Find the equation of the tangent to the graph of f ( x)  x 2  2 x  9 at the point (1, 12).
Ans: y = 2x + 9
Difficulty: moderate Section: 2.2
36. Find the equation of the tangent line to the graph of f ( x)  x 2  1 at (1, 2).
A) Not defined B) y = 2 C) x = 1 D) y = 2x
Ans: D Difficulty: moderate Section: 2.2
37. Find the equation of the tangent line to the graph of f ( x)  x 2  4 at the point (4, 20).
A) y = 8x – 12 B) Not defined C) y = 20 D) x = 4
Ans: A Difficulty: moderate Section: 2.2
38. Find the equation of the line that is tangent to the curve f ( x)  5  3x 2  x5 at the point
(1, 7).
Ans: y = x + 6

1
2
D) y  x 
7
7

41. Find the equation of the tangent line to the graph of f ( x) 
1
2
1
2
1
B) y   x 
C) y 
x
x
49
7
7
7
49
Ans: A Difficulty: moderate Section: 2.2

A) y  

Page 25

 1
 7,  .
 7

years from now the average level of carbon monoxide in the air will be
Q(t )  0.07t 2  0.2t  2.8 ppm. The rate that the carbon monoxide level will change with
respect to time 2 years from now will be 0.048 ppm/yr.
Ans: False Difficulty: hard Section: 2.2
46. True or False: The gross annual earnings of a certain company were
E (t )  0.2t 2  9t  30 thousand dollars where t is the number of years since its formation
in 1990. The gross annual earnings with respect to t in 1995 are growing at 13.75%.
Ans: True Difficulty: hard Section: 2.2
47. True or False: An environmental study of a certain suburban community suggests that t
years from now the average level of carbon monoxide in the air will be
Q(t )  0.05t 2  0.3t  3.2 parts per million (ppm). The rate that the carbon monoxide
level will change with respect to time 4 years from now will be 0.4 ppm/yr.
Ans: False Difficulty: hard Section: 2.2
48. An appliance store manager estimates that for x television ads run per day,
R( x)  0.01x3  x 2  3x  200 refrigerators will be sold per month. Find R(4) and
interpret what it tells us about sales.
R(4)  203.36; they'll sell about 203 refrigerators if they run 4 ads per day.
A)
R(4)  4.52; they'll sell about 5 refrigerators if they run 4 ads per day.
B)
R(4)  4.52; sales will be increasing at about 5 refrigerators per month per ad
C)
when they're running 4 ads.
R(4)  203.36; the cost of refrigerators will be rising by $203.36 if they're selling
D)
4 per day.
Ans: C Difficulty: easy Section: 2.2

Page 26


A) 2 B) 2t + 2 C) 2t D) t
Ans: A Difficulty: moderate Section: 2.2
54. The displacement function of a moving object is described by s(t )  t 3  2t  1 . What is
the velocity of the object as a function of t?
A) 3t 2 B) 3t 2  2 C) 3 D) 2
Ans: B Difficulty: easy Section: 2.2
55. An object moves along a line in such a way that its position at time t is
s(t )  t 3  27t 2  195t  4 . Find the velocity and acceleration of the object at time t.
When is the object stationary?
A)
v(t )  3t 2  54t  195 ; a(t) = 6t – 54; t = 5 and 13
B)
v(t )  3t 2  54t  195 ; a(t) = 6t – 54; t = 9
C)
v(t )  3t 2  18t  195 ; a(t) = 6t – 18; t = 5
D)
v(t )  3t 2  54t  195 ; a(t) = 6t – 54; t = 5
Ans: A Difficulty: moderate Section: 2.2

Page 27


Chapter 2

56. The displacement function of a moving object is described by s(t )  t 3  2t  3 . What is
the velocity of the object as a function of time?
A) 3t 2  2 B) 3t 2 C) 3 D) 2
Ans: A Difficulty: easy Section: 2.2
57. True or False: If the displacement of a moving object is s(t )  t 3 , the acceleration is 6t.
Ans: True Difficulty: easy Section: 2.2


2x 1
, what is f ( x) ?
7x  6
19
Ans: f ( x ) 
(7 x  6) 2
Difficulty: moderate Section: 2.3

63. If f ( x) 

Page 28


Chapter 2

x2
x2
2
2
x  4x
x  4x
A)
B)
C) 2x D) –x
2
( x  2)
( x  2) 2
Ans: A Difficulty: moderate Section: 2.3


Ans: True Difficulty: hard Section: 2.3
69. If f ( x) 
Ans:

3x  1
, what is f ( x) ?
x 1
4

 x  1

2

Difficulty: moderate

Section: 2.3

70. Find the equation of the line that is tangent to the curve f ( x) 
(1, –1).
Ans: y = –9x + 8
Difficulty: hard Section: 2.3

Page 29

5x2  7 x  1
at the point
5  4 x3


Chapter 2


2t  3
with respect to t when t = 5?
t 5

17
7
C) 10 D)
10
10
Difficulty: hard Section: 2.3

13
100
Ans: A

A)

B)

74. What is the rate of change of f (t ) 

6t  3
with respect to t when t = 18?
t 3

1
1
B) 
C) 21 D) –21


3
.
1  x3

1  x 

3 3

Difficulty: hard

Section: 2.3

Page 30


Chapter 2

78. Find f ( x) , where f ( x)  x3  4 .
Ans: 6x
Difficulty: easy Section: 2.3
79. The temperature in degrees Fahrenheit inside an oven t minutes after turning it on can be
modeled with the function
400t  70
. Find F (5) and interpret what it tells us about the temperature.
F (t ) 
t 1
Round your answer to 2 decimal places.
Ans: F (5)  9.17 ; After 5 minutes, the temperature is increasing at the rate of 9.17
degrees per minute.

2
f ( x )   3
 3
A)
C)
8x 6x x
15
120
f ( x)  
 6
B)
D)
3
16 x 6 x x
Ans: C Difficulty: moderate Section: 2.3

83. Find f ( x ) if f ( x) 

84. Find
Ans:

dy
if y  3 u and u  x 4  3x3  7 .
dx
4 x3  9 x 2

3 3  x 4  3x3  7 

Difficulty: hard


Ans: 6 x5  15 x 4  8 x3  3 x 2  4 x  1
Difficulty: hard Section: 2.4

85. Find

86. Find

dy
if y  u 3  4u 2  5 and u  x 2  x  7
dx

Ans: 3  2 x  1  x 2  x  7   16 x 3  24 x 2  104 x  56
2

Difficulty: hard

Section: 2.4

dy
if y  3 u and u  x 4  3x3  2
dx
4 x3  9 x 2
Ans:
3( x 4  3x3  2)2 / 3
Difficulty: hard Section: 2.4

87. Find

88. Find
Ans:

8

 x  2

2

Difficulty: hard

Section: 2.4

(3  5 x)3
, then f ( x)  5(2 x  1) .
( x 2  x  1)2
Difficulty: moderate Section: 2.4

90. True or False: If f ( x) 
Ans: False

2x  3
x 2  3x  5
, then f ( x) 
.
1  3x
1  3x
Difficulty: moderate Section: 2.4

91. True or False: If f ( x) 
Ans: False

92. True or False: An equation for the tangent line to the curve f ( x)  3x 2  5 x at the

Ans: y = 0.11x + 1.42
Difficulty: hard Section: 2.4
99. Find all points on the graph of the function f ( x)  x 3  4 x  16  where the tangent line is
horizontal.
Ans: (0, 0) and (–3, –108)
Difficulty: moderate Section: 2.4
100. Find all points on the graph of the function f ( x) 

x2
where the tangent line is
x2

horizontal.
A) There are none. B) (2, 1) C) (0, 0) and (–4, –8)
Ans: C Difficulty: moderate Section: 2.4

D) (0, 0)

101. True or False: If f ( x)  x 2  x , then f ( x)  0 at x = 0 and x = 2.
Ans: False Difficulty: hard Section: 2.4

Page 33


Chapter 2

3
.
(1  3 x 2 )3/ 2
Section: 2.4

Ans: 285.7 people/year
Difficulty: hard Section: 2.4
109. True or False: It is estimated that t years from now, the population of a certain suburban
7
community will be p(t )  30 
thousand. An environmental study indicates that
2t  1
the average daily level of carbon monoxide in the air will be C ( p)  0.3 p 2  p  30
parts per million (ppm) when the population is p thousand. The rate at which the level of
pollution is changing with respect to time 3 years from now is about 0.084 ppm per year.
Ans: True Difficulty: hard Section: 2.4
110. It is estimated that t years from now, the population of a certain community will be
6
thousand. An environmental study indicates that the average daily level
p(t )  14 
3t
of carbon monoxide in the air will be C ( p)  0.5 p 2  2 p  30 units when the
population is p thousand. The rate at which the level of carbon monoxide will be
changing 3 years from now is
A) –0.078 ppm per thousand people
C) 1.000 ppm per thousand people
B) 0.078 ppm per thousand people
D) –1.000 ppm per thousand people
Ans: B Difficulty: hard Section: 2.4
111. True or False: The function f ( x) 
decreases from 3 to 2.7.
Ans: False Difficulty: hard

x
 5 will decrease by approximately 0.6 as x

between 11:00 and 11:45 A.M.?
A) Approximately 19 radios
C) Approximately 14 radios
B) Approximately 855 radios
D) Approximately 39 radios
Ans: C Difficulty: moderate Section: 2.5
116. True or False: If x3  y3  x  y , then
Ans: False

Difficulty: moderate

dy 3x 2  1
.

dx 3 y 2  1
Section: 2.6

dy
, where xy 3  3x 2  7 y .
dx
6 x  y3
6x 2
3
A) y 3  6 x  7 B)
C)
D)
y

6
x

3 1
dy
5 .
, where 
x 2y
dx
6 y2
Ans:  2
x
Difficulty: moderate Section: 2.6

119. Find

dy
 2x  3y .
dx
Section: 2.6

120. True or False: If x 2  3xy  y 2  15 , then
Ans: False

Difficulty: moderate

121. True or False: If x 2 y  xy 2  7 , then
Ans: False

Difficulty: moderate

dy
 2 xy  y 2 .

47
47
Ans: C Difficulty: moderate Section: 2.6
127. True or False: The equation for the tangent line to the curve x 2  2 xy  y 3 at the point (1,
–1) is y = –1.
Ans: True Difficulty: hard Section: 2.6

Page 37


Chapter 2

128. Use implicit differentiation to find
A) 80x3
Ans: B
129. Find
Ans:

d2y
for 4 x5  11y  100 .
2
dx

80 3
C) 60 x 2  11 D) 60 x 2  100
x
11
Difficulty: easy Section: 2.6

B) 

C) It cannot be determined
B) A decrease of 1.24 hours
D) No change
Ans: B Difficulty: hard Section: 2.6
132. Suppose the output at a certain factory is Q  4 x 2  5x1 y1  3 y1 units, where x is the
number of hours of skilled labor used and y is the number of hours of unskilled labor. The
current labor force consists of 30 hours of skilled labor and 30 hours of unskilled labor.
Use calculus to estimate the change in unskilled labor y that should be made to offset a 1hour increase in skilled labor x so that output will be maintained at its current level.
A) –2.55 hours B) –1.76 hours C) –0.39 hours D) 0.39 hours
Ans: A Difficulty: moderate Section: 2.6

Page 38