Deck 6: Applications of Differentiation

Question 1

Determine whether Rolle's Theorem can be applied to on the closed interval If Rolle's Theorem can be applied, find all values of c in the open interval such that

A)Rolle's Theorem applies; c = 6, c = 18
B)Rolle's Theorem applies; c = 18
C)Rolle's Theorem applies; c = 8, c = 6
D)Rolle's Theorem applies; c = 8
E)Rolle's Theorem does not apply

Answer 1

Rolle's Theorem applies; c = 18

Question 2

Determine whether Rolle's Theorem can be applied to on the closed interval If Rolle's Theorem can be applied, find all values of c in the open interval such that

A)c = 3, c = 8
B)c = 8
C)c = 7, c = 3
D)c = 7
E)Rolle's Theorem does not apply

Answer 2

Rolle's Theorem does not apply

Question 3

Use a graphing utility to graph the function and locate the absolute extrema of the function on the interval .

A)absolute minimum: no absolute maximum
B)absolute minimum: no absolute maximum
C)absolute maximum: absolute minimum:
D)absolute minimum: absolute maximum:
E)absolute minimum: no absolute maximum

Answer 3

absolute minimum: no absolute maximum

Question 4

Locate the absolute extrema of the function on the closed interval [0,6].

A)absolute max: (6, 144) ; absolute min: (2, -16)
B)absolute max: (2, -16) ; absolute min: (6, 144)
C)absolute max: (6, 144) ; no absolute min
D)no absolute max; Absolute min: (6, 144)
E)no absolute max or min

Question 5

Determine whether Rolle's Theorem can be applied to the function on the closed interval . If Rolle's Theorem can be applied, find all numbers c in the open interval such that .

A)Rolle's Theorem applies; c =
B)Rolle's Theorem applies; c = ,
C)Rolle's Theorem applies;
D)Rolle's Theorem does not apply
E)Rolle's Theorem does apply;

Question 6

Locate the absolute extrema of the function on the closed interval .

A)no absolute max; absolute min: (-3, 14)
B)absolute max: (6, -148) ; absolute min: (-3, 14)
C)absolute max: (-3, 14) ; no absolute min
D)absolute max: (-3, 14) ; absolute min: (6, -148)
E)no absolute max or min

Question 7

The ordering and transportation cost C for components used in a manufacturing process is approximated by where C is measured in thousands of dollars and x is the order size in hundreds. According to Rolle's Theorem, the rate of change of the cost must be 0 for some order size in the interval Find this order size. Round your answer to three decimal places.

A)5.174 components
B)5.598 components
C)4.098 components
D)6.674 components
E)8.196 components

Question 8

Find the value of the derivative (if it exists) of at the indicated extremum.

A)
B)
C)
D)
E) is undefined.

Question 9

The formula for the power output of battery is where V is the electromotive force in volts, R is the resistance, and I is the current. Find the current (measured in amperes) that corresponds to a maximum value of P in a battery for which volts and ohm. Assume that a 20-ampere fuse bounds the output in the interval . Round your answer to two decimal places.

A)81.67 amperes
B)13,338.89 amperes
C)40.00 amperes
D)268.00 amperes
E)2.00 amperes

Question 10

Sketch the graph of the function and locate the absolute extrema of the function on the interval .

A)left endpoint: absolute minimumright endpoint: absolute maximum
B)left endpoint: absolute minimumright endpoint: absolute maximum
C)left endpoint: absolute minimumright endpoint: absolute maximum
D)left endpoint: absolute minimumright endpoint: absolute maximum
E)left endpoint: absolute minimumright endpoint: absolute maximum

Question 11

Locate the absolute extrema of the function on the closed interval .

A)absolute maximum: absolute minimum:
B)absolute maximum: absolute minimum:
C)absolute maximum: absolute minimum:
D)absolute maximum: absolute minimum:
E)absolute maximum: absolute minimum:

Question 12

Find any critical numbers of the function , t < 4.

A)0
B)
C)
D)
E)

Question 13

Determine whether Rolle's Theorem can be applied to the function on the closed interval [-1,5]. If Rolle's Theorem can be applied, find all values of c in the open interval (-1,5) such that .

A)Rolle's Theorem applies; c = -2
B)Rolle's Theorem applies; c = 0.5
C)Rolle's Theorem does not apply
D)Rolle's Theorem applies; c = 2
E)Rolle's Theorem applies; c = 3.5

Question 14

Find all critical numbers of the function .

A)critical numbers:
B)critical numbers:
C)critical numbers:
D)critical numbers:
E)no critical numbers

Question 15

Find the value of the derivative (if it exists) of the function at the extremum point (0,9).

A)0
B)does not exist
C)9
D)-9
E)10

Question 16

Determine whether Rolle's Theorem can be applied to the function on the closed interval . If Rolle's Theorem can be applied, find all numbers c in the open interval such that .

A)Rolle's Theorem applies;
B)Rolle's Theorem applies;
C)Rolle's Theorem applies;
D)Rolle's Theorem applies;
E)Rolle's Theorem does not apply

Question 17

Use a computer algebra system to graph the function and determine all absolute extrema on the interval .

A)absolute maximum: absolute minimum:
B)absolute maximum: absolute minimum:
C)absolute maximum: absolute minimum:
D)absolute maximum: absolute minimum:
E)absolute maximum: absolute minimum:

Question 18

Find the value of the derivative (if it exists) of the function at the extremum point (0,0).

A)0
B)1
C)-1
D)
E)

Question 19

Locate the absolute extrema of the given function on the closed interval [-4,4].

A)absolute max: (2, 1); absolute min (-2, -1)
B)no absolute max; absolute min: (-2, -1)
C)absolute max: (2, 1); absolute min (0, 0)
D)absolute max: (2, 1); no absolute min
E)absolute max: (0, 0); absolute min (-2, -1)

Question 20

Locate the absolute extrema of the function on the closed interval .

A)The absolute maximum is , and it occurs at either endpoint .The absolute minimum is 0, and it occurs at the critical number x = 0.
B)The absolute maximum is , and it occurs only at the right endpoint .The absolute minimum is 0 and it occurs at the critical number x = 0.
C)The absolute maximum is , and it occurs only at the left endpoint .The absolute minimum is 0 and it occurs at the critical number x = 0.
D)The absolute maximum is , and it occurs at the critical number x = 0.The absolute minimum is , and it occurs at the right endpoint .
E)The absolute maximum is , and it occurs at the critical number x = 0.The absolute minimum is , and it occurs at the left endpoint .

Question 21

Which of the following functions passes through the point and satisfies ?

A)
B)
C)
D)
E)

Question 22

A company introduces a new product for which the number of units sold S is where t is the time in months since the product was introduced. Find the average value of during the first year.

A)
B)
C)
D)
E)

Question 23

The height of an object t seconds after it is dropped from a height of 700 meters is Find the time during the first 9 seconds of fall at which the instantaneous velocity equals the average velocity.

A)40.5 seconds
B)15.9 seconds
C)4.5 seconds
D)22.1 seconds
E)2.4 seconds

Question 24

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

A)
B)
C)
D)
E)

Question 25

A plane begins its takeoff at 2:00 P.M. on a 3000-mile flight. After 11.5 hours, the plane arrives at its destination. Explain why there are at least two times during the flight when the speed of the plane is 200 miles per hour.

A)By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 130.4 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 130.4 mi/hr and decelerating from 130.4 mi/hr.
B)By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 173.9 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 173.9 mi/hr and decelerating from 173.9 mi/hr.
C)By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 260.9 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 260.9 mi/hr and decelerating from 260.9 mi/hr.
D)By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 222.2 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 222.2 mi/hr and decelerating from 222.2 mi/hr.
E)By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 444.4 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 444.4 mi/hr and decelerating from 444.4 mi/hr.

Question 26

A company introduces a new product for which the number of units sold S is where t is the time in months since the product was introduced. During what month does equal the average value of S(t) during the first year?

A)March
B)June
C)April
D)May
E)September

Question 27

Identify the open intervals where the function is increasing or decreasing.

A)decreasing: ; increasing:
B)increasing: ; decreasing:
C)increasing: ; decreasing:
D)increasing: ; decreasing:
E)decreasing on

Question 28

Find the critical number of the function

A)
B)
C)
D)
E)

Question 29

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

A)
B)The derivative of f does not exist.
C)
D)
E)

Question 30

Find the open interval(s) on which is increasing or decreasing.

A)increasing on decreasing on
B)increasing on decreasing on
C)increasing on decreasing on
D)increasing on decreasing on
E)increasing on decreasing on

Question 31

For the function :(a) Find the critical numbers of f (if any);(b) Find the open intervals where the function is increasing or decreasing; and(c) Apply the First Derivative Test to identify all relative extrema.Use a graphing utility to confirm your results.

A)(a) (b) decreasing: ; increasing: (c) relative min:
B)(a) (b) increasing: ; decreasing: (c) relative max:
C)(a) (b) decreasing: ; increasing: (c) relative min: ; relative max:
D)(a) (b) decreasing: ; increasing: (c) relative min:
E)(a) (b) increasing: ; decreasing: (c) relative max:

Question 32

Determine whether the Mean Value Theorem can be applied to the function on the closed interval [0,14]. If the Mean Value Theorem can be applied, find all numbers c in the open interval (0,14) such that .

A)MVT applies; 3.5
B)MVT applies;
C)MVT applies; 7
D)MVT applies;
E)MVT does not apply

Question 33

For the function : (a) Find the critical numbers of f (if any);
(b) Find the open intervals where the function is increasing or decreasing; and
(c) Apply the First Derivative Test to identify all relative extrema.
Then use a graphing utility to confirm your results.

A)(a) x = 0 , 4 (b) increasing: ; decreasing: (c) relative max: ; relative min:
B)(a) x = 0 , 4 (b) decreasing: ; increasing: (c) relative min: ; relative max:
C)(a) x = 0 , 1 (b) increasing: ; decreasing: (c) relative max: ; relative min:
D)(a) x = 0 , 1 (b) decreasing: ; increasing: (c) relative min: ; relative max:
E)(a) x = 0 , 1 (b) increasing: ; decreasing: (c) relative max: ; no relative min.

Question 34

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

A)
B)
C)
D)
E)

Question 35

Find the relative extremum of by applying the First Derivative Test.

A)relative maximum:
B)relative minimum:
C)relative maximum:
D)relative minimum:
E)relative minimum:

Question 36

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

A)
B)
C)
D)
E)

Question 37

Determine whether the Mean Value Theorem can be applied to the function on the closed interval [5,15]. If the Mean Value Theorem can be applied, find all numbers c in the open interval (5,15) such that .

A)MVT applies; c = 9
B)MVT applies; c = 11
C)MVT applies; c = 10
D)MVT applies; c = 12
E)MVT applies; c = 8

Question 38

Identify the open intervals where the function is increasing or decreasing.

A)decreasing: ; increasing:
B)increasing: ; decreasing:
C)increasing on
D)decreasing on
E)decreasing on ; increasing on

Question 39

Use the graph of the function given below to estimate the open intervals on which the function is increasing or decreasing.

A)increasing on ; decreasing on
B)increasing on ; decreasing on
C)increasing on ; decreasing on
D)increasing on ; decreasing on
E)increasing on ; decreasing on

Question 40

The height of an object t seconds after it is dropped from a height of 250 meters is Find the average velocity of the object during the first 10 seconds.

A)-100.0 m/sec
B)-5.1 m/sec
C)100.0 m/sec
D)49.0 m/sec
E)-49.0 m/sec

Question 41

Match the function with one of the following graphs.

A)
B)
C)
D)
E)

Question 42

Suppose the deflection D of a beam of length L is where x is the distance from one end of the beam. Find the value of x that yields the maximum deflection .

A)
B)
C)
D)
E)

Question 43

The resistance R of a certain type of resistor is where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to determine the minimum resistance for this type of resistor. Round your answer to two decimal places.

A)
B)
C)
D)
E)

Question 44

The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes.

A)
B)
C)
D)
E)none of the above

Question 45

Find all relative extrema of the function . Use the Second Derivative Test where applicable.

A)
B)
C)
D)
E)

Question 46

Find the cubic function of the form are real numbers, which satisfies the conditions given below.

A)
B)
C)
D)
E)

Question 47

Find the limit.

A)-3
B)
C)
D)7
E)3

Question 48

The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes.

A)
B)
C)
D)
E)none of the above

Question 49

Suppose the annual sales S of a new product is given by where t is time in years. Find the exact time when the annual sales are increasing at the greatest rate. Round your answer to three decimal places.

A)
B)
C)
D)
E)

Question 50

Determine the open intervals on which the graph of is concave downward or concave upward.

A)concave downward on ; concave upward on
B)concave upward on
C)concave downward on
D)concave upward on ; concave downward on
E)concave downward on ; concave upward on

Question 51

Match the function with one of the following graphs.

A)
B)
C)
D)
E)

Question 52

Find all relative extrema of the function . Use the Second Derivative Test where applicable.

A)relative max: ; no relative min
B)relative min: ; no relative max
C)no relative min; no relative max
D)relative max: ; relative min:
E)relative min: ; relative max:

Question 53

Find the points of inflection and discuss the concavity of the function.

A)inflection point at ; concave downward on ; concave upward on
B)inflection point at ; concave upward on ; concave downward on
C)inflection point at ; concave downward on ; concave upward on
D)inflection point at ; concave upward on ; concave downward on
E)inflection point at ; concare downward on ; concave upward on

Question 54

Determine the open intervals on which the graph of the function is concave upward or concave downward.

A)
B)
C)
D)
E)

Question 55

Suppose the total cost C for ordering and storing x units is What order size will produce a minimum cost? Round your answer to the nearest unit produced.

A)
B)
C)
D)
E)

Question 56

The resistance R of a certain type of resistor is where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to find

A)
B)
C)
D)
E)

Question 57

Suppose a manufacturer has determined that the total cost C of operating a factory is where x is the number of units produced. At what level of production will the average cost per unit be minimized? (The average cost per unit is C/x.)

A)
B)
C)
D)
E)

Question 58

Determine the open intervals on which the graph of is concave downward or concave upward.

A)concave downward on
B)concave downward on ; concave upward on
C)concave upward on ; concave downward on
D)concave downward on ; concave upward on
E)concave upward on ; concave downward on

Question 59

Find the points of inflection and discuss the concavity of the function .

A)inflection point at x = 6; concave down on
B)no inflection points; concave down on
C)inflection point at x = 6; concave up on
D)no inflection points; concave up on
E)inflection point at x = 0; concave up on ; concave down on

Question 60

Find all relative extrema of the function Use the Second Derivative Test where applicable.

A)relative max: ; no relative min
B)relative max: ; no relative min
C)no relative max or min
D)relative min: ; no relative max
E)relative min: ; no relative max

Question 61

The graph of a function f is is shown below. Sketch the graph of the derivative .

A)
B)
C)
D)
E)

Question 62

Find the limit.

A)
B)
C)0
D)
E)-4

Question 63

Analyze and sketch a graph of the function .

A)
B)
C)
D)
E)

Question 64

A heat probe is attached to the heat exchanger of a heating system. The temperature T (in degrees Celsius) is recorded t seconds after the furnace is started. A model for the data recorded for the first two minutes is given by Find

A)1501
B)82
C)1583
D)51
E)29

Question 65

Analyze and sketch a graph of the function .

A)
B)
C)
D)
E)none of the above

Question 66

The graph shows the temperature T, in degrees Fahrenheit, of molten glass t seconds after it is removed from a kiln. Find

A)
B)
C)
D)
E)

Question 67

Find the limit.

A)
B)
C)1
D)0
E)does not exist

Question 68

A model for the average typing speeds S (words per minute) of a typing student after t weeks of lessons is given by Find

A)96 words per minute
B)24 words per minute
C)52 words per minute
D)104 words per minute
E)16 words per minute

Question 69

A business has a cost of for producing x units. The average cost per unit is Find the limit of as x approaches infinity.

A)250.0
B)300.0
C)0
D)1.0
E)1.2

Question 70

The graph of a function f is is shown below. Sketch the graph of the derivative .

A)
B)
C)
D)
E)

Question 71

The graph of a function f is is shown below. Sketch the graph of the derivative .

A)
B)
C)
D)
E)

Question 72

Determine the slant asymptote of the graph of .

A)
B)
C)
D)
E)no slant asymptote

Question 73

Sketch the graph of the function using any extrema, intercepts, symmetry, and asymptotes.

A)
B)
C)
D)
E)

Question 74

Analyze and sketch a graph of the function .

A)
B)
C)
D)
E)

Question 75

Analyze and sketch a graph of the function .

A)
B)
C)
D)
E)

Question 76

Sketch the graph of the relation using any extrema, intercepts, symmetry, and asymptotes.

A)
B)
C)
D)
E)

Question 77

Find the limit.

A)
B)
C)7
D)1

Question 78

Find the limit.

A)8
B)
C)
D)1
E)

Question 79

Sketch the graph of the function using any extrema, intercepts, symmetry, and asymptotes.

A)
B)
C)
D)
E)

Question 80

Sketch the graph of the relation using any extrema, intercepts, symmetry, and asymptotes.

A)
B)
C)
D)
E)