Chapter 3 2 Use the properties of limits to help decide whether

Solve the problem.

62)

The force F (in N) exerted by a cam on a lever is given by F =x4–13x3+45x2–67x +30, where x (1



x  5) is the distance (in cm) from the center of rotation of the cam to the edge of the cam in contact

with the lever. Find the instantaneous rate of change of F with respect to x when x =4 cm.

62)

A)

–31 N/cm

B)

–139 N/cm

C)

–75 N/cm

D)

–94 N/cm

Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value.

63)

lim

x

1

x2+9x –10

x –1

63)

A)

9

B)

Does not exist

C)

11

D)

0

Use a graphing utility to find the limit, if it exists.

64)

lim

x

49x2+6x +8

2x

64)

A)



B)

3.5

C)

Does not exist

D)

–3.5

Sketch the derivative of the graph.

65)

21

A)

B)

C)

D)

Find all values x = a where the function is discontinuous.

66)

k(x) =ex +2

66)

A)

Nowhere

B)

a < – 2

C)

a >2

D)

a > – 2

Find the equation of the tangent line to the curve when x has the given value.

67)

f(x) =x2+2 ; x = – 3

67)

A)

y = – 3x – 7

B)

y = – 6x – 16

C)

y = – 6x – 14

D)

y = – 6x – 7

22

Find all points where the function is discontinuous.

68)

A)

x = 4, x = 2

B)

x = 4

C)

x = 2

D)

None

Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value.

69)

lim

x

5x2+6x –3

–4x2+5

69)

A)

–5

3

B)



C)

–5

4

D)

0

70)

lim

h

0

(x + h)3– x3

h

70)

A)

Does not exist

B)

3x2+ 3xh + h2

C)

0

D)

3x2

Sketch the derivative of the graph.

71)

23

A)

B)

C)

D)

Estimate the slope of the tangent line to the curve at the given point.

72)

A)

3

B)

1

C)

undefined

D)

0

Find the equation of the secant line through the points where x has the given values.

73)

f(x) =x2+3x; x =3, x =4

73)

A)

y =10x

B)

y =10x +12

C)

y =12x –10

D)

y =10x –12

Solve the problem.

74)

In order to boost business, a ski resort in Vermont is offering rooms for $125 per night with every

fourth night free. Let C(x) represent the total cost of renting a room for x days. Sketch a graph of

C(x) on the interval (0, 6] and determine the cost for staying 4 1

2 days.

74)

A)

C 4 1

2= $500

B)

C 4 1

2= $375

25

C)

C 4 1

2= $625

D)

C 4 1

2= $500

Decide whether the limit exists. If it exists, find its value.

75)

lim

x–1/2 f(x)

75)

A)

–2

B)

0

C)

Does not exist

D)

–1

Find the equation of the tangent line to the curve when x has the given value.

76)

f(x) =x2

4 ; x = – 4

76)

A)

y = – 2x –4

B)

y = – 2x +4

C)

y = – 8x –4

D)

y = – 2x –8

Find the value of the constant k that makes the function continuous.

77)

g(x) =x2–7 if x <6

5kx if x 6

77)

A)

k =29

30

B)

k =29

C)

k =6

5

D)

k =13

Use the properties of limits to help decide whether each limit exits. If a limit exists, find its value.

78)

Let f(x) =x2+ 1 if x < 0

2if x 

0 . Find lim

x–4f(x).

78)

A)

2

B)

1

C)

17

D)

Does not exist

Find the average rate of change for the function over the given interval.

79)

y = 5x + 7 between x = – 1 and x = 0

79)

A)

–1

6

B)

5

C)

1

2

D)

–28

27

Decide whether the limit exists. If it exists, find its value.

80)

lim

x

f(x)

80)

A)



B)

0

C)

Does not exist

D)

–2

Find the equation of the tangent line to the curve when x has the given value.

81)

f(x) =x3

2 ; x =2

81)

A)

y =2x +8

B)

y =8x +6

C)

y =2x –8

D)

y =6x –8

Decide whether the limit exists. If it exists, find its value.

82)

lim

x

1 f(x)

82)

A)

0

B)

Does not exist

C)

2

D)

1

Sketch the derivative of the graph.

83)

A)

B)

29

C)

D)

Solve the problem.

84)

The graph shows the amount of potential energy V(x) (in arbitrary energy units) stored in a large

rubber band that is stretched a distance of x inches beyond its normal length.

The magnitude of the force required to hold the rubber band at the position x = a is the derivative

of the potential energy with respect to x, evaluated at the point x = a. Sketch a graph of the

magnitude of the force versus x.

84)

A)

B)

30

C)

D)

Use a graphing utility to find the limit, if it exists.

85)

lim

x

6

x2+2x –48

x2–36

85)

A)

–1

6

B)

7

6

C)

0

D)

Does not exist

Find the value of the constant k that makes the function continuous.

86)

h(x) =

7x2+25x –12

x +4 if x –4

3x + k if x = – 4

86)

A)

k = – 19

B)

k =52

C)

k =4

D)

k = 0

Use a graphing utility to find the limit, if it exists.

87)

lim

t

25t2–125

t –5

87)

A)

25

B)

5

C)

Does not exist

D)

125

Find f'(x) at the given value of x.

88)



f(x) =x2– 9x – 1; Find f (–5).

88)

A)

–19

B)

–20

C)

–10

D)

69

Estimate the slope of the tangent line to the curve at the given point.

89)

A)

–1

2

B)

–3

2

C)

–1

D)

1

Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value.

90)

lim

x

9

x–3

x –9

90)

A)

0

B)

1

6

C)

3

D)

1

3

Solve the problem.

91)

A particular strain of influenza is known to spread according to the function p(t) =1

4(t2+ t), where

t is the number of days after the first appearance of the strain and p(t) is the percentage of the

population that is infected. Find the instantaneous rate of change of p with respect to t at t =3.

91)

A)

2% per day

B)

3

2% per day

C)

7

4% per day

D)

3% per day

Find the average rate of change for the function over the given interval.

92)

y =3

x + 2 between x = 1 and x = 4

92)

A)

–1

6

B)

1

2

C)

–28

D)

–2

Complete the table and use the result to find the indicated limit.

93)

If f(x) =x– 2, find lim

x 

4 f(x).

x 3.9 3.99 3.999 4.001 4.01 4.1

f(x)

93)

A)

x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) –0.02516 –0.00250 –0.00025 0.00025 0.00250 0.02485 ; limit = 0

B)

x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 1.47736 1.49775 1.49977 1.50022 1.50225 1.52236 ; limit = 1.50

C)

x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 3.9000 2.9000 1.9000 2.0000 3.0000 4.0000 ; limit =

D)

x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 3.9000 2.9000 1.9000 2.0000 3.0000 4.0000 ; limit = 1.95

Give an appropriate answer.

94)

Let lim

x –4f(x) =6 and lim

x –4g(x) = – 5. Find lim

x –4[f(x) – g(x)].

94)

A)

1

B)

11

C)

–4

D)

6

Estimate the slope of the tangent line to the curve at the given point.

95)

A)

1

B)

–1

C)

2

D)

1

2

Find f'(x) at the given value of x.

96)



f(x) = – 7x2+2x +6; Find f (3).

96)

A)

44

B)

–36

C)

–34

D)

–40

Solve the problem.

97)

The current value of an annuity per period is given by P =R

i–R

i(1 + i)n , where n is the number of

periods, i is the interest rate, and R is the amount of the periodic payment. Find the limit of the

current value equation as n approaches infinity to derive an expression for the current value for an

annuity that makes payments in perpetuity.

97)

A)

P =R

i

B)

P = 0

C)

P =

D)

P =R

i– R

Sketch the derivative of the graph.

35

98)

A)

B)

C)

D)

Solve the problem.

99)

The profit from the expenditure of x thousand dollars on advertising is given by

P(x) =1040 + 25x – 3x2. Find the marginal profit when the expenditure is x =9.

99)

A)

1040 thousand dollars

B)

225 thousand dollars

C)

171 thousand dollars

D)

–29 thousand dollars

Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value.

100)

lim

x

3x + 1

11x – 7

100)

A)

0

B)

3

11

C)

–1

7

D)



Complete the table and use the result to find the indicated limit.

101)

If f(x) =x + 1

x + 1 , find lim

x 

1 f(x).