Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

A)  4.9967 
B)  4.9767 
C)  4.9667 
D)  4.9867 
E)  5.0067 
A)  4.9967 
B)  4.9767 
C)  4.9667 
D)  4.9867 
E)  5.0067

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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   . 
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 5

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

Accepted Answer

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:   
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 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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. ​

Accepted Answer

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 
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 9

Determine whether the Mean Value Theorem can be applied to the function   on the closed interval   . If the Mean Value Theorem can be applied, find all numbers c in the open interval   such that   . ​

Accepted Answer

A)  MVT applies;   
B)  MVT applies;   
C)  MVT applies;   
D)  MVT applies;   
E)  MVT applies;   
A)  MVT applies;   
B)  MVT applies;   
C)  MVT applies;   
D)  MVT applies;   
E)  MVT applies;

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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   
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 12

Analyze and sketch a graph of the function   . ​

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

Match the function   with one of the following graphs. ​

Accepted Answer

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

Question 15

Determine whether the Mean Value Theorem can be applied to the function   on the closed interval   . If the Mean Value Theorem can be applied, find all numbers c in the open interval   such that   . ​

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

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

Accepted Answer

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   
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 18

Compare dy and   for   at   with   . Give your answers to four decimal places. ​

Accepted Answer

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

Question 19

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. ​

Accepted Answer

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 
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 20

Determine the slant asymptote of the graph of   . ​

Accepted Answer

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

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

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

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

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

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

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

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

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.

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

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

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

Analyze and sketch a graph of the function .

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

Match the function with one of the following graphs.

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

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

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

Compare dy and for at with . Give your answers to four decimal places.

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.

Determine the slant asymptote of the graph of .

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

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

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

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

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

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

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

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

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. ​

Determine whether the Mean Value Theorem can be applied to the function on the closed interval . If the Mean Value Theorem can be applied, find all numbers c in the open interval such that . ​

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

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

Analyze and sketch a graph of the function . ​

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

Match the function with one of the following graphs. ​

Determine whether the Mean Value Theorem can be applied to the function on the closed interval . If the Mean Value Theorem can be applied, find all numbers c in the open interval such that . ​

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

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

Compare dy and for at with . Give your answers to four decimal places. ​

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. ​

Determine the slant asymptote of the graph of . ​

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

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

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

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

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

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

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

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

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.

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

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

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

Analyze and sketch a graph of the function .

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

Match the function with one of the following graphs.

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

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

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

Compare dy and for at with . Give your answers to four decimal places.

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.

Determine the slant asymptote of the graph of .