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Test Bank for Applied Calculus Brief 6th Edition by
Berresford
Chapter 2 Derivatives And Their Uses
1. Complete the table and use it to predict the limit, if it exists.
6x7
f ( x)
1

2
5

x

lim f (x) ?

x

0.5

x
0.51

f (x)

0.501
0.5001
0.5

?

0.4999

0.499
0.49
A) –160.0
B) 80.0
C) –80.0
D)
0.5
E) does not
exist
Ans: C
2.

Use properties of limits and algebraic methods to find the limit, if it exists. lim (8x
13x 2 3x 13)
x

3

A)
B)
C)
D)
E)

–121
121
141
–141
does not exist

Ans: B
3.

2

Find lim
x 5

xx
2x 5

without using a graphing calculator or making tables.

3

A) 2
B) –5
C) 0 D) 4
E)
Ans: D
Berresford/Rockett, Brief Applied Calculus, 6e

4.

Use properties of limits and algebraic methods to find the limit, if it exists. lim

–7

8x

x 14

A) 9
14
B) 1
14
C)

141

D)

149

E) does not
5.
lim
x –5

A)

x 2 9x 14
x 2 2x
2

does
exist Ans: B
6.

not

exist Ans: D
Use properties of limits and algebraic methods to find the
limit, if it exists.
2
E
5 )

2

B)
5
C) 5

Use properties of
find the

2
D)

5

limit, if
it exists.
x 13

x29x8
A)

1712

B) 17
12
C) 12

lim x 2 4 x 32

limits and algebraic methods to

D)

1712

E) does not

exist Ans: B

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Page 38
Berresford/Rockett, Brief Applied Calculus, 6e

7.

Use properties of limits and algebraic methods to find the limit, if it exists.

lim 9
h0

xh2

9x2

h

A) 0
B)
2x
C)
9x
D) 18x
E) does not
exist Ans: D
8.

A graph of y f ( x) is shown and a c-value is given. For this problem, use the graph to
find lim f ( x) .

x

1

c
A) 0
B)
C) –6
D)

5
41

41
44
45

E) does not
exist Ans: A
9.

Use properties of limits and algebraic methods to find the limit, if it exists.
16 7x
for x 3

lim f (x ), where f (x)

x2

x 3

5x
A) 5
B) 6

for x 3

C) –6
D) –5
E) does not
exist Ans: E

10.
Find lim

f ( x) for

x –6+

A)

6

B)

–1

C)

0

D)

1

E)

–6

f ( x) x + 6 .
x+6

Ans: D

©2013 Cengage Learning. All Rights Reserved.
Page 39
Berresford/Rockett, Brief Applied Calculus, 6e

11. Find

lim f ( x) for the graph of f ( x) given below.
x

3+

A)
B) 0
C) -3
D) inf
E) 3
Ans: A

12.
Find lim

1 .

–
x –1

A)
B)

C)
D)

1
0
–1

x +1

E)
Ans: C

13.
Find lim

x 6+

A)
B)
C)
D)
E)
Ans:

–1

.

x– 62

6
0
–6
E

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Page 40

Berresford/Rockett, Brief Applied Calculus, 6e

14. For the given x-value, use the figure to determine whether the function is continuous or
discontinuous at that x-value.

x 5
A) discontinuous
B) continuous
Ans: A
15. Determine whether the function is continuous or discontinuous at the given x-value. x 2
5 if x –4
92
x

f (x )

if x –4

x –4

123

A) discontinuous
B) continuous
Ans: B
16. Determine whether the given function is continuous. If it is not, identify where it
discontinuous.

is

2

y 3x
A)
B)
C)
D)

4x 7
discontinuous at
discontinuous at
discontinuous at
discontinuous at

x 5
x 0
x5
x 10 E)

continuous everywhere

Ans: E
17. Determine whether the function is continuous or discontinuous at the given x-value.
y

x2 5

,x –7

x4
A) continuous
B) discontinuous
Ans: A
Berresford/Rockett, Brief Applied Calculus, 6e

18. Determine whether the given function is continuous. If it is not, identify where it is
discontinuous. You can verify your conclusions by graphing the function with a
graphing utility, if one is available.

y

8x 12 32x 7 x

A) discontinuous at x
1
B) discontinuous at x1
C) discontinuous at x 1
D) discontinuous at x1 2

E) continuous everywhere
Ans: D

2

Berresford/Rockett, Brief Applied Calculus, 6e
P , P , and P ,

19.

By imagining tangent lines at points
1

2

3

positive, zero, or negative at these

points.

A)

At P : positive slope
1

state whether the slopes are

At P

2

B)

: negative slope

At P3 : positive slope
At P : zero slope
1

At P
2

C)

: negative slope

At P3 : positive slope
At P : zero slope
1

At P
2

D)

: positive slope

At P3 : negative slope
At P : positive slope

1

At P
2

E)

: positive slope

At P3 : positive slope
At P : positive slope
1

At P
2

: negative slope

At P3 : negative slope
Ans: C

20. Which graph represents f ( x) if the graph of f ( x) is displayed below?

A)

B)

C)

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Page 44

D)

E)

Ans: C
21. For the given function, find the average rate of change over the specified interval.
f (x ) 5 5x 4x2 over –2, 4
A)
0
B)
–19
C)
19
D)
13
E)
–13
Ans: E
22. Find the average rate of change of f x
A)
B)

8
7 C)

8 x 7 between x

3 and x

8.

3

D) 11
E) 5
Ans: A
23. Find the instantaneous rate of change of the function f x
A)

6x 2 5x at x

30

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Page 45

2.

B)

26

C)

41

D)

42

E)

29

Ans: E
24. For the function in this problem, find the instantaneous rate of change of the function
at the given value.
f (x ) 9x 2 5x 5; x 4

A)
B)
C)
D)
E)
Ans:

0
41
31
67
77
D

25. For the function in this problem, find the slope of the tangent line at the given value. f
(x ) 5x 2 9x 9; x 1
A)
B)
C)
D)
E)
Ans:

1
14
–4
0
19
A

26. Find the slope of the tangent at x –1. f
(x ) 6x 2 2x
A)
–14
B)
–4
C)
–10
D)
4
E)
0
Ans: C
27. For the function in this problem, find the derivative, by using the

2

5x 3x 9
A)

5x2

B)
C)

2

5x
10x

3x 9
3x

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Page 46

definition. f (x )

D)
5x 3
E)
10x 3
Ans: E

28. Find the slope of the tangent to the graph of f (x) at any
point. f (x) 9x2 6x
A)
B)
C)
D)
E)
Ans:

18x 6
18x 6
9x 6
9 x 2 6x
3x
A

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Page 47

29. Find f ' x

of f x –7 x 8 by using the definition of the derivative.

A)

f' x 8

B)

f ' x–7

f ' x7x

C)

D) f ' x7 E) f ' x –7 x
Ans: B
30. Write the equation of the line tangent to the graph of f (x) at x –1. f
(x) 5x2 8x
A)
y –2 x 2
B)

y –2 x 2

C)

y –2 x

D)

y –2 x 5

E)

y –2 x 5

Ans: D

31. The population of a town is f x
0x

20 ). Find f ' x

3 x 2 15 x 200 people after x weeks (for

to find the instantaneous rate of change of the population after

8 weeks.
A)
48
B)

64

C)

33

D)

31

E)

49

Ans: C

32. An automobile dealership finds that the number of cars that it sells on day x of an
advertising campaign is S x x 2 18x (for 0 x 7 ). Find S ' x to find the instantaneous
rate of change on day x 2 .
A) 14
B) 18
C) 16
D) 22
E)
21
Ans: A
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Page 48

Berresford/Rockett, Brief Applied Calculus, 6e

33. Differentiate the given function.

96x6

y
A)
B)

6x5

C)

9x7

D)

54x5

9x6

E)
9 x5
Ans: E

34.
Find the derivative of g w
A)

5

4

20 4 w .

g w

3

w
3

B)

4

g

w20

w
4

4

w

3

C)
g w
D)
g w

54 w3

E)
g w 20 4 w3
Ans: A
35.

Find the derivative of the
function. y 5x 1 9x 2 13

A)

–5 x

2

18x

B)

–5 x 2

18x 3

C)

–5 18x 1

D)

–5 x

2

9x

3

3

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Page 49

E)

–5 x

1

9x

2

Ans: B

36.

For the function given, find f '(x).

f (x ) x 4

13x 8

A)

x3

13

B)

4x3

8

C)

4x3

13

D)

4x4 4

E)

x 13x 8

13x

Ans: C

©2013 Cengage Learning. All Rights Reserved.
Berresford/Rockett, Brief Applied Calculus, 6e

37.
A)

Find the derivative of the function. 8 / 3
–24 x 11/ 3
C)
D)

30x

–24 x 11/ 3
–24 x 5 / 3

–24 x 5 / 3

B)

13 / 3

10 / 3

30x

f (x ) 9x

7/3

30x 13 / 3
30x 7 / 3

–72 x 11/ 3
Ans: C

90x

E)

13 / 3

38. 8

Find the derivative of f x
A)
2
f x
3

4

4

B)
f x
f

x

x5

4
4

D)

x

2
4

C)

x.

x3

4
f x

5

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Page 50

9x

x

4

E)

f x

2
4

x5

Ans: B
39.

Find the derivative of the

function. y 7x 4 2x 2 6x

7
A)

28 x 4

4x2

B)

28 x 3

4x 6

C)

7x3

2x 6

D)

28 x 3

4x

E)

7x4
2x
Ans: B
40.

6x 7

6x 7

2

Find the derivative of the function.
21

h(x ) 11x

11

8

19x

7x

14x 6

A)

220 x 20

190 x10

B)

231x 21

209 x11

C)

11x 20

D)

231x 20

E)

220 x 21 190 x11 49 x 8 14x
Ans: D

19 x10

49 x7

14

56 x 8 14x
7 x7

209 x10

14

56 x7 14

©2013 Cengage Learning. All Rights Reserved.

Page 51

Berresford/Rockett, Brief Applied Calculus, 6e

41.
Find the derivative of h x 3 3 2
6.

x

3

x
A)

1

h x
B)

1

x

3

3

x4

2

2
h x
C)

2

2

h x
D)

E)

1

x3

x

3

x4

2

x3

1

x

2

2

2

2

3

h x

x

2
h x

2

x

2

x3

Ans: C
42. At the indicated point, find the instantaneous rate of change of the
function. R (x ) 17x 2x 2 , x 3
A) 29

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Page 52

B) 52
C) 19

D) 21
E) 23
Ans: A

43. If f x 60 4 ,

x3

972

find f 81 .

4

x

A)

f 81 14

B)

f 81 15

C)
f 81

D)

21

f 81 16

E)

Ans: D

f 81 26

44. Find the derivative at the given x-value with the appropriate
rule. y 8 24 x at x 9
A) –8
B) –64
C) 8
D) –4
E) 0
Ans: D
Berresford/Rockett, Brief Applied Calculus, 6e

45. 5
If

df
f xx , find

.
dx

x –2

A) df
–32
dx

x –2

B) df
–192

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Page 53

dx

x –2

C) df
320
dx

x –2

D) df
–128
dx

x –2

E) df
80
dx

x –2

Ans: E
46. If f
x

250

30 x ,

find df

.

x

dx

x 25

A) df

2
dx

x 25

B) df

–2
dx

x 25

C) df

10
dx

x 25

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Page 54

D) df

–10
dx

x 25

E) df

4

dx
x
25

Ans: A
47. Suppose the Marginal Cost Businesses can buy multiple licenses for PowerZip data
compression software at a total cost of approximately C x 24x2 3 dollars for x
licenses. Find the derivative of this cost function at x 64 .
A)
C 64 8

B)

C 644

C)

C 642

D)

C 64 12

E)
C 64 6
Ans: B
48. Suppose the number of people newly inflected on day t of a flu epidemic is f t
t 3 (for 0 t 13) . Find the instantaneous rate of change of this number on
day 10.
A)

6 3 x 8 x 1 by using the Product Rule. Simplify
your

answer.
A)

3

f x

3 x2
B)

3

f x

x2

6 32 3 x

C) 6 64 3 x
f x
3

D)

f x

x2
2 64 3 x

3 x2

E)

f x

2 32 3 x
2
3

x

Ans: D

50.

Find ds

if s

t6

8 .

8 t3

dt

A)

6t 8

6t

B)

9t 8

48t 5

5
9t 8
6t
3t
8
5
2
E) 9t 48t 24t
Ans: E

D)

5

24t

2

3t 2 C) 6t 8

48t

5

24t

2

2

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Page 56

1 32

x

Berresford/Rockett, Brief Applied Calculus, 6e
51.

Find the derivative, but do not simplify your answer.
y7x73x39x3x5
8 x 8 9 x9 6
A)

7

3

7x
B)

4

3x

7

15 x

C)

D)

6

4

7

9 x 15 x

81x 49

2

4

7

9x

9 15 x

2

49 x

81x 49

64 x

x6 9x2

9 3 x 5 8 x 8 9 x9 6

8

64 x

49 x

6

8

5

9x

x 6 9 x2 9
64 x

81x8

8

9 3x

9

8x

67

9x

x 7 3 x 3 9 x 15 x 4 64 x 7

81x8

E)
7 x 7 3 x 3 9 x 15 x 4 64 x 7 81x 849 x 6

9x2 9 3x5 8x8

9 x9

6

Ans: A

52.

28

14 15

Find the derivative of f z

z

Simplify your answer.
A)
f z43z 42 z
B)

f z42 z 43

29z 30

z2

C)

f z42z 43

z2

D)

f z43 z 42 30 z29 1

E)

f z43 z42 1

Ans: E

53.
Find the derivative of x16

.

6 x5
B)

6

C)

x7
1

6x

54.

Find the indicated derivative and simplify.

Page 53

C ( x) for C ( x) 2

7x 4x 3 7

A)

14 x 2 2 x4

21

2 x4 7 2

B)

2

x

4

2x

21

72

2 x4
C)
x 2 2 x4

2 x4
D)

21

72

7 x 2 2 x4

2 x4

E)

2

21

72

4

7x 2x

2 x4

21

72

Ans: D

55.

x 5
Find the derivative of f x
A)

4 x2

5 by using Quotient Rule. Simplify your answer.

2

12 x 40 x 5 f x

4 x2

53
B)

f x4 x

40 x 5

4 x2

53

2

2

C)

f x4 x

40 x 5

52

4 x2
2

D)

f x4 x

40 x 5

52

4 x2
2

E)

f x12 x

40 x 5

4 x2

52

Ans: D

©2013 Cengage Learning. All Rights Reserved.
Page 54
Berresford/Rockett, Brief Applied Calculus, 6e

56.

Find the indicated derivative and simplify.
dydx
y

64x x22

A)

4

2