Question 1

Find the domain of the function. ​   ​

Accepted Answer

A)  All real values of   and   
B)  All real values of   and   satisfying   
C)  All real values of   and   satisfying   
A)  All real values of   and   
B)  All real values of   and   satisfying   
C)  All real values of   and   satisfying

Question 2

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function.
​   ​
Find the critical point(s) of the function.
​
Find the point(s) of maximum.
​
Find the point(s) of minimum.
​
Find the relative extrema of the function.

Accepted Answer

The answer of Find the critical point(s) of the function....

Question 3

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. ​
If   and   , then f must have a critical point at (a, b).
​

Accepted Answer

A)  False. In relative extremum   and   . 
B)  True. It follows from definition. 
A)  False. In relative extremum   and   . 
B)  True. It follows from definition.

Question 4

Find the volume of the solid bounded above by the surface   and below by the plane region R.   and ​R is the region bounded by the graphs of   and   .

Accepted Answer

The answer of Find the volume of the solid bounded...

Question 5

Let   . Compute the following.
​   __________
​   __________
​   __________
​   __________
​   __________

Accepted Answer

The answer of Let   . Compute the following.
​...

Question 6

Find the maximum and minimum values of the function ​   ​
Subject to the constraint   .
​

Accepted Answer

A)  Maxima :   Minima :   
B)  Maxima :   Minima :   
C)  Maxima :   Minima :   
D)  Maxima :   Minima :   
A)  Maxima :   Minima :   
B)  Maxima :   Minima :   
C)  Maxima :   Minima :   
D)  Maxima :   Minima :

Question 7

Find the total differential of the function. ​   ​

Accepted Answer

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

Question 8

Find the first partial derivatives of the function. ​   ​

Accepted Answer

A)    ​ 
B)  ​   ​ 
C)  ​   ​ 
D)  ​   ​ 
A)    ​ 
B)  ​   ​ 
C)  ​   ​ 
D)  ​   ​

Question 9

The flow of blood through an arteriole in cubic centimeters per second is given by
​   ​
where l is the length (in cm) of the arteriole, r is its radius (in cm), p is the difference in pressure between the two ends of the arteriole (in dyne/cm2), and k is the viscosity of blood (in dyne-sec/cm2). Find the approximate percentage change in the flow of blood if an error of 2% is made in measuring the length of the arteriole and an error of 1% is made in measuring its radius. Assume that p and k are constant.
​
__________ %

Accepted Answer

The answer of The flow of blood through an arteriole...

Question 10

Find the total differential of the function.
​

Accepted Answer

The answer of Find the total differential of the function.
​...

Question 11

Use a double integral to find the volume of the solid shown in the figure.
​   ​
__________ cu units.

Accepted Answer

The answer of Use a double integral to find the...

Question 12

Use a double integral to find the volume of the solid shown in the figure. ​   ​

Accepted Answer

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

Question 13

Find the total differential of the function. ​
​   ​

Accepted Answer

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

Question 14

Find the total differential of the function. ​   ​

Accepted Answer

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

Question 15

Maximize the function ​   ​
Subject to the constraint   .
​

Accepted Answer

A)    
B)  ​   
C)  ​   
D)  ​   
A)    
B)  ​   
C)  ​   
D)  ​

Question 16

Find the first partial derivatives of the function. ​   ​

Accepted Answer

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

Question 17

The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function
​   ​
where ​x stands for the number of finished units and ​y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
​
P = $__________ per week.

Accepted Answer

The answer of The Country Workshop's total weekly profit (in...

Question 18

Find the maximum and minimum values of the function ​   ​
Subject to the constraint   .
​

Accepted Answer

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

Question 19

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. ​   ​

Accepted Answer

A)    ; relative maximum value:   
B)    ; relative minimum value:   
C)    ; saddle point:   
D)  there are no critical points 
A)    ; relative maximum value:   
B)    ; relative minimum value:   
C)    ; saddle point:   
D)  there are no critical points

Find the domain of the function.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If and , then f must have a critical point at (a, b).

Find the volume of the solid bounded above by the surface and below by the plane region R. and R is the region bounded by the graphs of and .

Let . Compute the following. __

Find the maximum and minimum values of the function Subject to the constraint .

Find the total differential of the function.

Find the first partial derivatives of the function.

Find the total differential of the function.

Use a double integral to find the volume of the solid shown in the figure. __________ cu units.

Use a double integral to find the volume of the solid shown in the figure.

Find the total differential of the function.

Find the total differential of the function.

Maximize the function Subject to the constraint .

Find the first partial derivatives of the function.

Find the maximum and minimum values of the function Subject to the constraint .

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function.

Preliminaries

Functions, Limits, and the Derivative

Differentiation

Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Differential Equations

Probability and Calculus

Taylor Polynomials and Infinite Series

Trigonometric Functions

Filters

Exam 8: Calculus of Several Variables

Find the domain of the function. ​ ​

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. ​ If and , then f must have a critical point at (a, b). ​

Find the volume of the solid bounded above by the surface and below by the plane region R. and ​R is the region bounded by the graphs of and .

Let . Compute the following. ​ __________ ​ __________ ​ __________ ​ __________ ​ __________

Find the maximum and minimum values of the function ​ ​ Subject to the constraint . ​

Find the total differential of the function. ​ ​

Find the first partial derivatives of the function. ​ ​

Find the total differential of the function. ​

Use a double integral to find the volume of the solid shown in the figure. ​ ​ __________ cu units.

Use a double integral to find the volume of the solid shown in the figure. ​ ​

Find the total differential of the function. ​ ​ ​

Find the total differential of the function. ​ ​

Maximize the function ​ ​ Subject to the constraint . ​

Find the first partial derivatives of the function. ​ ​

Find the maximum and minimum values of the function ​ ​ Subject to the constraint . ​

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. ​ ​

Preliminaries

Functions, Limits, and the Derivative

Differentiation

Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Differential Equations

Probability and Calculus

Taylor Polynomials and Infinite Series

Trigonometric Functions

Filters

Find the domain of the function.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If and , then f must have a critical point at (a, b).

Find the volume of the solid bounded above by the surface and below by the plane region R. and R is the region bounded by the graphs of and .

Let . Compute the following. __

Find the maximum and minimum values of the function Subject to the constraint .

Find the total differential of the function.

Find the first partial derivatives of the function.

Find the total differential of the function.

Use a double integral to find the volume of the solid shown in the figure. __________ cu units.

Use a double integral to find the volume of the solid shown in the figure.

Find the total differential of the function.

Find the total differential of the function.

Maximize the function Subject to the constraint .

Find the first partial derivatives of the function.

Find the maximum and minimum values of the function Subject to the constraint .

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function.