Question 1

Evaluate the double integral   Round your answer to two decimal places,where applicable. ​

Accepted Answer

A)  4.50 
B)  4.75 
C)  5.25 
D)  6.00 
E)  5.75 
A)  4.50 
B)  4.75 
C)  5.25 
D)  6.00 
E)  5.75

Question 2

Set up an integral for both orders of integration,and use the more convenient order to evaluate the integral below over the region R. ​   ​
​

Accepted Answer

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

Question 3

Sketch the graph of the plane given by   ​

Accepted Answer

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

Question 4

Use a double integral to find the volume of the solid bounded by the graphs of the equations   ,   ,   ,   ,   ,   ​

Accepted Answer

A)  ​10 
B)  ​   
C)  ​9 
D)  18 
E)  ​2 
A)  ​10 
B)  ​   
C)  ​9 
D)  18 
E)  ​2

Question 5

Evaluate the double integral   Round your answer to one decimal place. ​

Accepted Answer

A)  72.5 
B)  82.5 
C)  92.5 
D)  21.5 
E)  11.0 
A)  72.5 
B)  82.5 
C)  92.5 
D)  21.5 
E)  11.0

Question 6

The population density (in people per square mile)for a coastal town can be modeled by   where x and y are measured in miles.What is the population inside the rectangular area defined by the vertices   ,   ,   ,and   ? Round to the nearest integer. ​

Accepted Answer

A)  83,552 people 
B)  77,552 people 
C)  80,052 people 
D)  82,552 people 
E)  81,552 people 
A)  83,552 people 
B)  77,552 people 
C)  80,052 people 
D)  82,552 people 
E)  81,552 people

Question 7

Use a symbolic integration utility to evaluate the double integral.Round your answer to four decimal places. ​   ​
​

Accepted Answer

A)  8.1747 
B)  9.1211 
C)  6.2031 
D)  7.8522 
E)  9.4362 
A)  8.1747 
B)  9.1211 
C)  6.2031 
D)  7.8522 
E)  9.4362

Question 8

Describe the level curves for the function   for the c-values given by   ​

Accepted Answer

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

Question 9

Use Lagrange multipliers to find the given extremum.Assume that x and y are positive. ​
Maximize   Constraint   ​

Accepted Answer

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

Question 10

Identify the quadric surface. z2 - x2 -   y2 = 1

Accepted Answer

A)  The graph is a hyperboloid of one sheet. 
B)  The graph is an ellipsoid. 
C)  The graph is an elliptic cone. 
D)  The graph is hyperboloid of two sheets. 
E)  The graph is an elliptic paraboloid. 
A)  The graph is a hyperboloid of one sheet. 
B)  The graph is an ellipsoid. 
C)  The graph is an elliptic cone. 
D)  The graph is hyperboloid of two sheets. 
E)  The graph is an elliptic paraboloid.

Question 11

Sketch the level curves for the function below for the given c-values   ​   ​

Accepted Answer

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

Question 12

The two planes   and   are perpendicular? ​

Accepted Answer

A) True 
 B)False

Question 13

Sketch the region of integration   ​

Accepted Answer

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

Question 14

Sketch the trace of the intersection of plane   with the sphere: ​   ​

Accepted Answer

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

Question 15

The utility function   is a measure of utility (or satisfaction)derived by a person from the consumption of two products x and y.Suppose the utility function is   Determine the marginal utility of product x. ​

Accepted Answer

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

Question 16

Use Lagrange multipliers to find the minimum distance from the circle   to the point   Round your answer to the nearest tenth. ​

Accepted Answer

A)  13.2 
B)  ​160.0 
C)  ​173.7 
D)  ​10.8 
E)  ​116.8 
A)  13.2 
B)  ​160.0 
C)  ​173.7 
D)  ​10.8 
E)  ​116.8

Question 17

Sketch the yz-trace of the equation: ​   ​

Accepted Answer

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

Question 18

Test for relative extrema and saddle points. ​   ​

Accepted Answer

A)  saddle point at   
B)  saddle point at   
C)  saddle point at   
D)  relative minimum at   
E)  relative minimum at   
A)  saddle point at   
B)  saddle point at   
C)  saddle point at   
D)  relative minimum at   
E)  relative minimum at

Question 19

Find the coordinates of the point that is located nine units in front of the yz-plane,six units to the left of the xz-plane,and five units above of the xy-plane. ​

Accepted Answer

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

Question 20

Use the function   to find   ​

Accepted Answer

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

Evaluate the double integral Round your answer to two decimal places,where applicable.

Set up an integral for both orders of integration,and use the more convenient order to evaluate the integral below over the region R.

Sketch the graph of the plane given by

Use a double integral to find the volume of the solid bounded by the graphs of the equations , , , , ,

Evaluate the double integral Round your answer to one decimal place.

The population density (in people per square mile)for a coastal town can be modeled by where x and y are measured in miles.What is the population inside the rectangular area defined by the vertices , , ,and ? Round to the nearest integer.

Use a symbolic integration utility to evaluate the double integral.Round your answer to four decimal places.

Describe the level curves for the function for the c-values given by

Use Lagrange multipliers to find the given extremum.Assume that x and y are positive. Maximize Constraint

Identify the quadric surface. z² - x² - y²= 1

Sketch the level curves for the function below for the given c-values

The two planes and are perpendicular?

Sketch the region of integration

Sketch the trace of the intersection of plane with the sphere:

The utility function is a measure of utility (or satisfaction)derived by a person from the consumption of two products x and y.Suppose the utility function is Determine the marginal utility of product x.

Use Lagrange multipliers to find the minimum distance from the circle to the point Round your answer to the nearest tenth.

Sketch the yz-trace of the equation:

Test for relative extrema and saddle points.

Find the coordinates of the point that is located nine units in front of the yz-plane,six units to the left of the xz-plane,and five units above of the xy-plane.

Use the function to find

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Exam 7: Functions of Several Variables

Evaluate the double integral Round your answer to two decimal places,where applicable. ​

Set up an integral for both orders of integration,and use the more convenient order to evaluate the integral below over the region R. ​ ​ ​

Sketch the graph of the plane given by ​

Use a double integral to find the volume of the solid bounded by the graphs of the equations , , , , , ​

Evaluate the double integral Round your answer to one decimal place. ​

The population density (in people per square mile)for a coastal town can be modeled by where x and y are measured in miles.What is the population inside the rectangular area defined by the vertices , , ,and ? Round to the nearest integer. ​

Use a symbolic integration utility to evaluate the double integral.Round your answer to four decimal places. ​ ​ ​

Describe the level curves for the function for the c-values given by ​

Use Lagrange multipliers to find the given extremum.Assume that x and y are positive. ​ Maximize Constraint ​

Identify the quadric surface. z2 - x2 - y2 = 1

Sketch the level curves for the function below for the given c-values ​ ​

The two planes and are perpendicular? ​

Sketch the region of integration ​

Sketch the trace of the intersection of plane with the sphere: ​ ​

The utility function is a measure of utility (or satisfaction)derived by a person from the consumption of two products x and y.Suppose the utility function is Determine the marginal utility of product x. ​

Use Lagrange multipliers to find the minimum distance from the circle to the point Round your answer to the nearest tenth. ​

Sketch the yz-trace of the equation: ​ ​

Test for relative extrema and saddle points. ​ ​