Deck 4: Applications of Differentiation

Question 1

A rectangular page is to contain 36 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used. ​

A) 10,10
B) 7,7
C) 9,9
D) 13,13
E) 8,8

Answer 1

E

Question 2

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)

Answer 2

B

Question 3

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:

Answer 3

C

Question 4

A plane begins its takeoff at 2:00 P.M. on a 3400-mile flight. After 7.5 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 400 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 716 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 716 mi/hr and decelerating from 716 mi/hr.
B) By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 358 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 358 mi/hr and decelerating from 358 mi/hr.
C) By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 227 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 227 mi/hr and decelerating from 227 mi/hr.
D) By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 302 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 302 mi/hr and decelerating from 302 mi/hr.
E) By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 453 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 453 mi/hr and decelerating from 453 mi/hr.

Question 5

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

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

Question 6

A sector with central angle is cut from a circle of radius 11 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of such that the volume of the cone is a maximum. ​

A)
B)
C)
D)
E)

Question 7

Find all points of inflection on the graph of the function . ​

A)
B)
C)
D)
E)

Question 8

Find the differential dy of the function . ​

A)
B)
C)
D)
E)

Question 9

Find two positive numbers such that the sum of the first and twice the second is 88 and whose product is a maximum. ​

A)
B)
C)
D)
E)

Question 10

On a given day, the flow rate F (cars per hour) on a congested roadway is given by , where v is the speed of the traffic in miles per hour. What speed will maximize the flow rate on the road? Round your answer to the nearest mile per hour. ​

A) 20 miles per hour
B) 18 miles per hour
C) 19 miles per hour
D) 11 miles per hour
E) 21 miles per hour

Question 11

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 minimum:
No absolute maximum
D) absolute minimum:
Absolute maximum:
E) absolute maximum:
Absolute minimum:

Question 12

The radius of a spherical balloon is measured to be 14 inches, with a possible error of 0.05 inch. Use differentials to approximate the maximum possible error in calculating the surface area of the sphere. Round your answer to two decimal places. ​

A) ±21.99 square inches
B) ±61.58 square inches
C) ±184.73 square inches
D) ±123.15 square inches
E) ±17.59 square inches

Question 13

Find all points of inflection on the graph of the function . ​

A)
B)
C)
D)
E)

Question 14

Assume that the amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on this money. Furthermore, the bank can reinvest this money at 27%. Find the interest rate the bank should pay to maximize profit. (Use the simple interest formula.) ​

A)
B)
C)
D)
E)

Question 15

Find the limit. ​ ​

A)
B)
C)
D)
E)

Question 16

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) 437.5
B)
C) 700
D)
E) 1.6

Question 17

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

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

Question 18

The measurement of the side of a square floor tile is 14 inches, with a possible error of inch. Use differentials to approximate the possible propagated error in computing the area of the square. ​

A) square inches
B) square inches
C) square inches
D) square inches
E) square inches

Question 19

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

A)
B)
C)
D)
E)

Question 20

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

A)
B)
C)
D)
E)

Question 21

Analyze and sketch a graph of the function . ​

A)
B)
C)
D)
E)

Question 22

Use differentials to approximate the value of . Round your answer to four decimal places. ​

A) 1.9150
B) 1.9250
C) 1.9550
D) 1.9350
E) 1.9450

Question 23

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

A)
B)
C)
D)
E)

Question 24

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

A) concave downward on ; concave upward 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 25

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 upward on ; concave downward on
D) inflection point at ; concave upward on ; concave downward on
E) inflection point at ; concave downward on ; concave upward on

Question 26

Determine the slant asymptote of the graph of . ​

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

Question 27

Find the point on the graph of the function that is closest to the point . Round all numerical values in your answer to four decimal places. ​

A)
B)
C)
D)
E)

Question 28

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

A)
B)
C)
D)
E)

Question 29

Sketch a graph of the function over the interval . ​

A)
B)
C)
D)
E)

Question 30

Find the point on the graph of the function that is closest to the point . Round all numerical values in your answer to four decimal places. ​

A)
B)
C)
D)
E)

Question 31

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

A) left endpoint: absolute minimum
Right endpoint: absolute maximum
B) left endpoint: absolute minimum
Right endpoint: absolute maximum
C) left endpoint: absolute minimum
Right endpoint: absolute maximum
D) left endpoint: absolute minimum
Right endpoint: absolute maximum
E) left endpoint: absolute minimum
Right endpoint: absolute maximum

Question 32

Match the function with one of the following graphs. ​

A)
B)
C)
D)
E)

Question 33

Find all critical numbers of the function . ​

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

Question 34

Analyze and sketch a graph of the function . ​

A)
B)
C)
D)
E)

Question 35

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

A) The absolute is , and it occurs at the left endpoint .
The absolute is , and it occurs at the right endpoint .
B) The absolute is , and it occurs at the right endpoint .
The absolute is , and it occurs at the left endpoint .
C) The absolute is , and it occurs at the left endpoint .
The absolute is , and it occurs at the right endpoint .
D) The absolute is , and it occurs at the right endpoint .
The absolute is and it occurs at the left endpoint .
E) The absolute is , and it occurs at the left endpoint .
The absolute is , and it occurs at the right endpoint .

Question 36

Locate the absolute extrema of the function on the closed interval . Round your answers to four decimal places. ​

A) left endpoint:
Right endpoint: absolute maximum
Critical points: absolute minimum
B) left endpoint: absolute minimum
Right endpoint: absolute maximum
Critical points: none
C) left endpoint: absolute minimum
Right endpoint: absolute maximum
Critical points:
D) left endpoint: absolute minimum
Right endpoint: absolute maximum
Critical points: none
E) left endpoint:
Right endpoint: absolute maximum
Critical points: absolute minimum

Question 37

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

Question 38

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

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

Question 39

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 the critical number of the function. Round numerical values in your answer to the nearest whole number. ​

A)
B)
C)
D)
E)

Question 40

The measurements of the base and altitude of a triangle are found to be 27 and 26 centimeters. The possible error in each measurement is 0.1 centimeter. Use differentials to estimate the propagated error in computing the area of the triangle. Round your answer to four decimal places. ​

A) ±2.915 square centimeters
B) ±2.385 square centimeters
C) ±2.65 square centimeters
D) ±3.0475 square centimeters
E) ±3.18 square centimeters

Question 41

A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing process. The flow rate R (in liters per hour) at time t (in hours) is given by the linear model . Write the linear model in exponential form. ​

A)
B)
C)
D)
E)

Question 42

Find any critical numbers of the function , . ​

A)
B)
C)
D)
E)

Question 43

Sketch a graph of the function over the interval . ​

A)
B)
C)
D)
E)

Question 44

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 during the first year? ​

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

Question 45

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

A)
B)
C)
D)
E)

Question 46

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)
B) ,
C) ,
D)
E) Rolle's Theorem does not apply

Question 47

Analyze and sketch a graph of the function . ​

A)
B)
C)
D)
E)

Question 48

Find the cubic function of the form where and the coefficients are real numbers, which satisfies the conditions given below. ​
Relative maximum: Relative minimum: Inflection point: ​

A)
B)
C)
D)
E)

Question 49

Analyze and sketch a graph of the function . ​

A)
B)
C)
D)
E)

Question 50

For the function , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x f(x)
​

A) -0.7500
B) 0.5000
C) 1.6800
D) -1.4000
E) does not exist

Question 51

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 52

Analyze and sketch a graph of the function . ​

A)
B)
C)
D)
E) none of these

Question 53

Find the differential of the function . ​

A)
B)
C)
D)
E)

Question 54

Find the relative minima of on the interval by applying the First Derivative Test. Round numerical values in your answer to three decimal places. ​

A) relative minima:
B) relative minima:
C) relative minima:
D) relative minima:
E) relative minima:

Question 55

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 23 cubic centimeters. Find the radius, r, of the cylinder that produces the minimum surface area. Round your answer to two decimal places. ​

A)
B)
C)
D)
E)

Question 56

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 57

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 58

Find the limit. ​ ​

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

Question 59

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) units
B) units
C) units
D) units
E) units

Question 60

The measurement of the radius of the end of a log is found to be 10 inches, with a possible error of inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log. ​

A) ±24π square inches
B) ±16π square inches
C) ±10π square inches
D) ±20π square inches
E) ±14π square inches

Question 61

Find the open interval(s) on which the function is decreasing in the interval . Round numerical values in your answer to three decimal places. ​

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

Question 62

The graph of f is shown below. On what interval is an increasing function? ​

A)
B)
C)
D)
E)

Question 63

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

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

Question 64

Find the point on the graph of the function that is closest to the point . ​

A)
B)
C)
D)
E)

Question 65

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

A) left endpoint: absolute minimum
Right endpoint: absolute maximum
Critical points: none
B) left endpoint: absolute minimum
Right endpoint: absolute maximum
Critical points: none
C) left endpoint: absolute minimum
Right endpoint:
Critical points: absolute maximum
D) left endpoint: absolute maximum
Right endpoint: absolute minimum
Critical points: none
E) left endpoint: absolute minimum
Right endpoint: absolute maximum
Critical points:

Question 66

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

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

Question 67

For the function , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x f(x)
​

A) 3
B) -5
C) -1
D)
E) None of these answers

Question 68

The total stopping distance T of a vehicle is where T is in feet and x is the speed in miles per hour. Use differentials to approximate the percent change in total stopping distance as speed changes from to miles per hour. Round your answer to one decimal place. ​

A) 33.3%
B) 14.7%
C) 29.1%
D) 332.0%
E) 114.7%

Question 69

The graph of f is shown below. For which value of x is zero? ​ ​

A)
B)
C)
D)
E)

Question 70

Find the relative maxima of on the interval by applying the First Derivative Test. Round numerical values in your answer to three decimal places. ​

A) relative maxima:
B) relative maxima:
C) relative maxima:
D) relative maxima:
E) relative maxima:

Question 71

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

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

Question 72

Find the open interval(s) on which the function is increasing in the interval . Round numerical values in your answer to three decimal places. ​

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

Question 73

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

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

Question 74

Find all points of inflection, if any exist, of the graph of the function . Round your answers to two decimal places. ​

A)
B)
C)
D)
E) no points of inflection

Question 75

A meteorologist measures the atmospheric pressure P (in kilograms per square meter) at altitude h (in kilometers). The data are shown below. Use the regression capabilities of the graphing utility to find a linear model for the revised data points obtained by plotting the points . ​ 0
5
10
15
20 ​
​

A)
B)
C)
D)
E)

Question 76

Match the function with one of the following graphs. ​

A)
B)
C)
D)
E)

Question 77

The graph of f is shown below. For which value of x is minimum? ​ ​

A)
B)
C)
D)
E)

Question 78

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

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

Question 79

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)
(b) increasing: ; decreasing:
(c) relative max: ; relative min:
B) (a)
(b) decreasing: ; increasing:
(c) relative min: ; relative max:
C) (a)
(b) increasing: ; decreasing:
(c) relative max: ; no relative min.
D) (a)
(b) increasing: ; decreasing:
(c) relative max: ; relative min:
E) (a)
(b) decreasing: ; increasing:
(c) relative min: ; relative max:

Question 80

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

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