Deck 2: Limits and the Derivative

Question 1

Find the limit, if it exists.
Find:

A) - 24
B) 8
C) -8
D) 16

Answer 1

-8

Question 2

Find the limit, if it exists.
Let

A) 5
B) -7
C) -2
D) Does not exist

Answer 2

-7

Question 3

Find the limit, if it exists.
Let

A) -∞
B) -4
C) 0
D) Does not exist

Answer 3

Does not exist

Question 4

Find the limit, if it exists.
Given

A)
B)
C)
D)

Question 5

The graph of a function f is given. Use the graph to answer the question.
Use the graph of f given below to find f(8).

A) 8
B) 12
C) 0
D) -16

Question 6

Find the limit, if it exists.
Find:

A)
B)
C) 0
D) Does not exist

Question 7

Find the limit, if it exists.
Let Find

A)
B) 4
C) -4
D) Does not exist

Question 8

Find the limit, if it exists.
Let

A) -4
B) 0
C) 4
D) Does not exist

Question 9

Find the limit, if it exists.
Find:

A) 1
B) 0
C) -1
D) Does not exist

Question 10

Find the limit, if it exists.
Find:

A) 0
B) -1
C) 1
D) Does not exist

Question 11

Use the graph to evaluate the indicated limit and function value or state that it does not exist.
Find

A) 6; 0
B) Does not exist; 6
C) 0; does not exist
D) 0; 6

Question 12

Find the limit, if it exists.
Find:

A)
B) -11
C) 1
D)

Question 13

Find the limit, if it exists.
Find:

A) 3
B) 10
C) 2
D) Does not exist

Question 14

Find the limit, if it exists.
Evaluate the following limit

A) 2
B)
C)
D) Does not exist

Question 15

Find the limit, if it exists.
Evaluate the following limit.

A) 2
B)
C)
D) Does not exist

Question 16

Find the limit, if it exists.
Given

A)
B)
C)
D)

Question 17

Sketch a possible graph of a function that satisfies the given conditions.
f(0) = 6;

A)
B)
C)
D)

Question 18

Sketch a possible graph of a function that satisfies the given conditions.
f(1) = 4;

A)
B)
C)
D)

Question 19

Provide an appropriate response.
Given that Express the answer as a simplified fraction.

A)
B)
C)
D)

Question 20

Use the graph to evaluate the indicated limit and function value or state that it does not exist.
Find

A) -1; 5
B) 5; -1
C) 5; Does not exist
D) Does not exist; does not exist

Question 21

Find the limit.
Determine the limit.

A)
B)
C) -2
D) 5

Question 22

Solve the problem.
Suppose that the value V of a certain product decreases, or depreciates, with time t, in months, where

A) 16
B) 23
C) 7
D) 19

Question 23

Use -∞ or ∞ where appropriate to describe the behavior at each zero of the denominator and identify all vertical
asymptotes.

A) No zeros of denominator; no vertical asymptotes
B) is a vertical asymptote
C) is a vertical asymptote
D) is a vertical asymptote

Question 24

Use the given graph to find the indicated limit.

A)
B) 3
C) 4
D)

Question 25

Solve the problem.
A company training program determines that, on average, a new employee can do P(x) pieces of work per day after s days of on-the-job training, where

A) 42
B) 105
C) 30
D) Does not exist

Question 26

Use the given graph to find the indicated limit.

A)
B)
C) 4
D) 3

Question 27

Provide an appropriate response.
If the limit at infinity exists, find the limit.

A)
B) 1
C)
D) 0

Question 28

Provide an appropriate response.
Find the vertical asymptote(s) of the graph of the given function.

A) y = -3
B) x = -8
C) x = -6
D) y = 8

Question 29

Provide an appropriate response.
If the limit at infinity exists, find the limit.

A)
B)
C)
D) 0

Question 30

Use the given graph to find the indicated limit.

A) 0
B)
C)
D) 5

Question 31

Find the limit.
Determine the limit.

A) 0
B)
C) -1
D)

Question 32

Describe the end behavior of the function.

A)
B)
C)
D)

Question 33

Solve the problem.
The cost of manufacturing a particular videotape is where x is the number of tapes produced. The average cost per tape, denoted by is found by dividing C(x) by x.

A) 14
B) 6
C) 10
D) Does not exist

Question 34

Use the given graph to find the indicated limit.

A)
B) 3
C)
D) 0

Question 35

Provide an appropriate response.
Find the horizontal asymptote, if any, of the given function.

A) y = 1
B) x = 2, x = -2
C) y = 3, y = -4
D) None

Question 36

Solve the problem.
Suppose that the value V of a certain product decreases, or depreciates, with time t, in months, where

A) 100
B) 30
C) 70
D) 85

Question 37

Solve the problem.
Suppose that the cost C of removing p% of the pollutants from a chemical dumping site is given by Can a company afford to remove 100% of the pollutants? Explain.

A) Yes, the cost of removing p% of the pollutants is $400, which is certainly affordable.
B) No, the cost of removing p% of the pollutants is $400, which is a prohibitive amount of money.
C) Yes, the cost of removing p% of the pollutants is $40,000, which is certainly affordable.
D) No, the cost of removing p% of the pollutants increases without bound as p approaches 100.

Question 38

Provide an appropriate response.
Find the horizontal asymptote, if any, of the given function.

A) y = 0
B)
C)
D) None

Question 39

Use -∞ or ∞ where appropriate to describe the behavior at each zero of the denominator and identify all vertical
asymptotes.

A) is a vertical asymptote
B) 6 is a vertical asymptote
C) is a vertical asymptote
D) is a vertical asymptote

Question 40

Provide an appropriate response.
Find the vertical asymptote(s) of the graph of the given function.

A) x = 9, x = -3
B) x = -9
C) x = 10, x = -10
D) y = 9, y = -3

Question 41

Provide an appropriate response.
Use a sign chart to solve the inequality. Express answers in interval notation.

A)
B)
C)
D)

Question 42

Provide an appropriate response.
Use a graphing utility to find the discontinuities of the given rational function.

A) 3
B) 1
C) -1
D) Continuous at all values of x

Question 43

Sketch a possible graph of a function that satisfies the given conditions.

A)
B)
C)
D)

Question 44

Sketch a possible graph of a function that satisfies the given conditions.

A)
B)
C)
D)

Question 45

Provide an appropriate response.
Use a graphing utility to find the discontinuities of the given rational function.

A) 1
B) 3
C) -1
D) Continuous at all values of x

Question 46

Provide an appropriate response.
Use a sign chart to solve the inequality. Express answers in interval notation.

A)
B)
C)
D)

Question 47

Solve the problem.
The cost of renting a snowblower is $20 for the first hour (or any fraction thereof) and $5 for each additional hour (or fraction thereof) up to a maximum rental time of 5 hours. Write a piecewise definition of the cost C(x)
Of renting a snowblower for x hours. Is C(x) continuous at x = 2.5?

A)
B)
C)
D)

Question 48

Find average rate of change for the function over the given interval.
between x = 4 and x = 8

A) 9
B) 18
C) 14
D) 28

Question 49

Provide an appropriate response.
Use a graphing utility to find the discontinuities of the given rational function.

A) 3
B) -1
C) 1
D) Continuous at all values of x

Question 50

Provide an appropriate response.
Determine where the function is continuous.

A)
B)
C)
D)

Question 51

Provide an appropriate response.
Solve the inequality and express the answer in interval notation:

A)
B)
C)
D)

Question 52

The graph of y = f(x) is shown. Use the graph to answer the question.

A) Yes
B) No

Question 53

The graph of y = f(x) is shown. Use the graph to answer the question.

A) No
B) Yes

Question 54

The graph of y = f(x) is shown. Use the graph to answer the question.

A) Yes
B) No

Question 55

Provide an appropriate response.
Determine where the function is continuous.

A)
B)
C)
D)

Question 56

Find average rate of change for the function over the given interval.
Find the average rate of change for if x changes from 2 to 8.

A) 7
B)
C)
D) 2

Question 57

Provide an appropriate response.
Use a sign chart to solve the inequality. Express answers in interval notation.

A)
B)
C)
D)

Question 58

Provide an appropriate response.
Use a graphing utility to approximate the partition numbers of the function to four decimal places:

A)
B)
C)
D)

Question 59

Find average rate of change for the function over the given interval.
between x = -9 and x = -4

A)
B) 730
C)
D)

Question 60

Provide an appropriate response.
Determine the x-values, if any, at which the function is discontinuous.

A) 1
B) -1, 0, 1
C) -1, 1
D) None

Question 61

Find the instantaneous rate of change for the function at the value given.
Find the instantaneous rate of change for the function

A) 6
B) -39
C) -14
D) -41

Question 62

Provide an appropriate response.
Use the four step p

A) 10x + 5h - 3
B)
C) 5h - 3
D) 10x - 3

Question 63

List the x-values in the graph at which the function is not differentiable.

A) x = -3, x = 0, x = 3
B) x = -2, x = 0, x = 2
C) x = -2, x = 2
D) x = -3, x = 3

Question 64

Provide an appropriate response.
Use the four step process to find for the function

A)
B)
C)
D)

Question 65

Use the definition to find the derivative at x.

A)
B)
C)
D)

Question 66

Provide an appropriate response.
Find the slope of the line tangent to the graph of the function at the given value of x.

A) 65
B) -52
C) -50
D) 67

Question 67

Find average rate of change for the function over the given interval.
Find the average rate of change of y with respect to x if x changes from 3 to 5 in the function

A) 22
B) 11
C) 4
D) 9

Question 68

Solve the problem.
Suppose an object moves along the y-axis so that its location is at time x (y is in meters and x is in seconds). Find the average velocity for x changing from 3 to 3 + h seconds.

A)
B) 7 - h m/s
C) 12 + h m/s
D) 7 + h m/s

Question 69

List the x-values in the graph at which the function is not differentiable.

A) x = 0
B) x = 1
C) x = -1
D) x = 2

Question 70

Use the definition to find the derivative at x.

A)
B) 6 - 12x
C) -12x
D) 6 - 6x

Question 71

Provide an appropriate response.
Find the slope of the secant line joining (2, f(2)) and (3, f(3)) for

A) 55
B) 15
C) -55
D) -15

Question 72

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

A)
B)
C) y = - 39x - 80
D) y = 13x - 16

Question 73

Provide an appropriate response.
Use the four step process to find f'(x) for the function

A)
B)
C)
D)

Question 74

Provide an appropriate response.
Find the slope of the graph at the point (1, 2).

A) -1
B) -2
C) 2
D) 1

Question 75

Find the instantaneous rate of change for the function at the value given.
Find the instantaneous rate of change for the function

A) 120
B) 15
C) 23
D) 16

Question 76

Solve the problem.
Suppose an object moves along the y-axis so that its location is at time x (y is in meters and x is in seconds). Find the instantaneous velocity at x = 4 seconds.

A) 9 m/s
B) 10 m/s
C) 8 m/s
D) 20 m/s

Question 77

Provide an appropriate response.
, find the slope of the tangent line at x = 4.

A) 20
B) 16
C) 8
D) 22

Question 78

Solve the problem.
Suppose an object moves along the y-axis so that its location is at time x (y is in meters and x is in seconds). Find the average velocity (the average rate of change of y with respect to x) for x changing from 2 to
9 seconds.

A) 12 m/s
B) 3 m/s
C) 15 m/s
D) 84 m/s

Question 79

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

A) y = 14x - 46
B) y = -14x + 46
C) y = 7x + 46
D) y = -2x

Question 80

Use the definition to find the derivative at x.
f(x) = 13x - 12

A) 13x
B) 1
C) 13
D) -13