Question 1

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

Accepted Answer

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 
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 2

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.
​

Accepted Answer

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:   
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 3

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

Accepted Answer

A)  concave down on   ; no points of inflection 
B)  concave downward on   ; concave upward on   ; inflection points at   and   
C)  concave upward on   ; concave downward on   ; inflection points at   and   
D)  concave up on   ; no points of inflection 
E)  none of the above 
A)  concave down on   ; no points of inflection 
B)  concave downward on   ; concave upward on   ; inflection points at   and   
C)  concave upward on   ; concave downward on   ; inflection points at   and   
D)  concave up on   ; no points of inflection 
E)  none of the above

Question 4

Find the differential dy of the function   . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 5

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 6

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   . ​

Accepted Answer

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

Question 7

The radius of a spherical balloon is measured to be 8 inches, with a possible error of 0.08 inch. Use differentials to approximate the maximum possible error in calculating the volume of the sphere. Round your answer to two decimal places. ​

Accepted Answer

A)  ±32.17 cubic inches 
B)  ±20.11 cubic inches 
C)  ±96.51 cubic inches 
D)  ±64.34 cubic inches 
E)  ±16.08 cubic inches 
A)  ±32.17 cubic inches 
B)  ±20.11 cubic inches 
C)  ±96.51 cubic inches 
D)  ±64.34 cubic inches 
E)  ±16.08 cubic inches

Question 8

Find the limit. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 9

Determine the x-coordinate(s) of any relative extrema and inflection points of the function   . ​

Accepted Answer

A)  relative minimum:   ; inflection point:   
B)  no relative extrema; inflection point:   
C)  relative minimum:   ; no inflection points 
D)  no relative extrema; inflection point:   
E)  relative maximum:   inflection point:   
A)  relative minimum:   ; inflection point:   
B)  no relative extrema; inflection point:   
C)  relative minimum:   ; no inflection points 
D)  no relative extrema; inflection point:   
E)  relative maximum:   inflection point:

Question 10

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 11

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. ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 12

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. ​

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

Analyze and sketch a graph of the function   . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 15

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. ​

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

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

Accepted Answer

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:   
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 18

Analyze and sketch a graph of the function   . ​

Accepted Answer

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

Question 19

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 100 square meters. ​

Accepted Answer

A)  Dimensions:   
B)  Dimensions:   
C)  Dimensions:   
D)  Dimensions:   
E)  Dimensions:   
A)  Dimensions:   
B)  Dimensions:   
C)  Dimensions:   
D)  Dimensions:   
E)  Dimensions:

Question 20

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

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

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.

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

Find the differential dy of the function .

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

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 .

The radius of a spherical balloon is measured to be 8 inches, with a possible error of 0.08 inch. Use differentials to approximate the maximum possible error in calculating the volume of the sphere. Round your answer to two decimal places.

Find the limit.

Determine the x-coordinate(s) of any relative extrema and inflection points of the function .

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

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.

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.

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

Analyze and sketch a graph of the function .

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.

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

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

Analyze and sketch a graph of the function .

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 100 square meters.

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

Preparation for Calculus

Limits and Their Properties

Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Anal

Filters

Exam 4: Applications of Differentiation

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

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. ​

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

Find the differential dy of the function . ​

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

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 . ​

The radius of a spherical balloon is measured to be 8 inches, with a possible error of 0.08 inch. Use differentials to approximate the maximum possible error in calculating the volume of the sphere. Round your answer to two decimal places. ​

Find the limit. ​ ​

Determine the x-coordinate(s) of any relative extrema and inflection points of the function . ​

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

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. ​

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. ​

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

Analyze and sketch a graph of the function . ​

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. ​

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

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

Analyze and sketch a graph of the function . ​

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 100 square meters. ​

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

Preparation for Calculus

Limits and Their Properties

Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Anal

Filters

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

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.

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

Find the differential dy of the function .

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

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 .

The radius of a spherical balloon is measured to be 8 inches, with a possible error of 0.08 inch. Use differentials to approximate the maximum possible error in calculating the volume of the sphere. Round your answer to two decimal places.

Find the limit.

Determine the x-coordinate(s) of any relative extrema and inflection points of the function .

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

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.

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.

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

Analyze and sketch a graph of the function .

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.

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

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

Analyze and sketch a graph of the function .

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 100 square meters.

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