Question 1

Examine the function   for relative extrema and saddle points. ​

Accepted Answer

A)  saddle point:   ; relative minimum:   
B)  relative minimum:   ; relative maximum:   
C)  saddle points:   ,   
D)  saddle point:   ; relative minimum:   
E)  relative minimum:   ; relative maximum:   
A)  saddle point:   ; relative minimum:   
B)  relative minimum:   ; relative maximum:   
C)  saddle points:   ,   
D)  saddle point:   ; relative minimum:   
E)  relative minimum:   ; relative maximum:

Question 2

Find three positive numbers x,y,and z whose sum is 27 and product is a maximum. ​

Accepted Answer

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

Question 3

Determine whether there is a relative maximum,a relative minimum,a saddle point,or insufficient information to determine the nature of the function   at the critical point   ​   ​

Accepted Answer

A)  insufficient information to determine the nature of the function at   
B)  relative maximum at ​   
C)  saddle point at   
D)  relative minimum at   
A)  insufficient information to determine the nature of the function at   
B)  relative maximum at ​   
C)  saddle point at   
D)  relative minimum at

Question 4

Evaluate the following integral. ​   ​

Accepted Answer

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

Question 5

Sketch the yz-trace of the equation: ​   ​

Accepted Answer

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

Question 6

Sketch the graph of the plane given by   ​

Accepted Answer

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

Question 7

Find the center and radius of the sphere whose equation is   Round your answer to two decimal places,where applicable. ​

Accepted Answer

A)  center:   ; radius: 2.00 
B)  center:   ; radius: 2.00 
C)  center:   ; radius: 8.00 
D)  center:   ; radius: 4.00 
E)  center:   ; radius: 8.00 
A)  center:   ; radius: 2.00 
B)  center:   ; radius: 2.00 
C)  center:   ; radius: 8.00 
D)  center:   ; radius: 4.00 
E)  center:   ; radius: 8.00

Question 8

Evaluate   and   for the function   at the point   Round your answer to two decimal places. ​

Accepted Answer

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

Question 9

A company sells two products whose demand functions are given by   and   So,the total revenue is given by   Estimate the average revenue if the price   varies between $25 and $50 and the price   varies between $25 and $50. ​

Accepted Answer

A)  $37,985 
B)  $40,135 
C)  $37,483 
D)  $39,000 
E)  $40,545 
A)  $37,985 
B)  $40,135 
C)  $37,483 
D)  $39,000 
E)  $40,545

Question 10

Find three positive numbers x,y,and z whose sum is 18 and the sum of the squares is a minimum. ​

Accepted Answer

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

Question 11

Examine the function   for relative extrema. ​

Accepted Answer

A)  relative minimum at   
B)  relative maximum at   
C)  relative minimum at   
D)  relative maximum at   
E)  no relative extrema 
A)  relative minimum at   
B)  relative maximum at   
C)  relative minimum at   
D)  relative maximum at   
E)  no relative extrema

Question 12

Describe the trace of the surface given by the function below in the xy-plane. ​   x2 + y2 + z2 = 1
​

Accepted Answer

A)  parabola 
B)  ellipse (not circle) 
C)  circle 
D)  hyperbola 
E)  line 
A)  parabola 
B)  ellipse (not circle) 
C)  circle 
D)  hyperbola 
E)  line

Question 13

Identify the quadric surface. ​   ​

Accepted Answer

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

Examine the function for relative extrema and saddle points.

Find three positive numbers x,y,and z whose sum is 27 and product is a maximum.

Determine whether there is a relative maximum,a relative minimum,a saddle point,or insufficient information to determine the nature of the function at the critical point

Evaluate the following integral.

Sketch the yz-trace of the equation:

Sketch the graph of the plane given by

Find the center and radius of the sphere whose equation is Round your answer to two decimal places,where applicable.

Evaluate and for the function at the point Round your answer to two decimal places.

A company sells two products whose demand functions are given by and So,the total revenue is given by Estimate the average revenue if the price varies between $25 and $50 and the price varies between $25 and $50.

Find three positive numbers x,y,and z whose sum is 18 and the sum of the squares is a minimum.

Examine the function for relative extrema.

Describe the trace of the surface given by the function below in the xy-plane. x² + y² + z² = 1

Identify the quadric surface.

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Exam 7: Functions of Several Variables

Examine the function for relative extrema and saddle points. ​