Question 1

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.Then...

Question 2

Find the second-order partial derivatives of the given function.Show that the mixed partial derivatives  and  are equal.

Accepted Answer

A)

B)

C)

D)

Question 3

Find the first partial derivatives of the function.

Accepted Answer

A)   
 
B)   
 
C)   
 
D)

Question 4

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) (5, 4); relative maximum value: f(5, 4) = 34 
B) (4, 5); relative minimum value: f(4, 5) = 25 
C) (4, 5); saddle point: f(4, 5) = 34

Question 5

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

Accepted Answer

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

Question 6

Evaluate the double integral  for the given function f(x, y) and the region R.  ; R is the rectangle defined by

Accepted Answer

A)   
B)   
C)   
D)

Question 7

Maximize the function  subject to the constraints  .

Accepted Answer

A) Max.of  at
 
B) Max.of  at
 
C) Max.of  at
 
D) Max.of  at
 
E) Max.of  at

Question 8

Find the second-order partial derivatives of the given function.Show that the mixed partial derivatives  and  are equal.

Accepted Answer

A)

B)

C)

D)

Question 9

Maximize the function  subject to the constraint  .

Accepted Answer

The answer of Maximize the function  subject to the...

Question 10

Find the first partial derivatives of the function.

Accepted Answer

A)    
B)    
C)    
D)

Question 11

Evaluate the double integral  for the function  and the region  .  and  is bounded by  ,  ,  and  .

Accepted Answer

The answer of Evaluate the double integral  for the...

Question 12

Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than  inches.Find the diamension of a cylendrical package of greatest volume that can be sent throught the mail.What is the volume of such a package? Hint: The length plus the grith is  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 13

Maximize the function  subject to the constraint  .

Accepted Answer

A)   
B)   
C)   
D)

Question 14

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

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 15

Find the domain of the function.
f(m, n) = 2m + 4n

Accepted Answer

The answer of Find the domain of the function.
f(m, n)...

Question 16

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

Accepted Answer

A)   
B)   
C)   
D)

Question 17

Sketch the level curves of the function corresponding to the given values of z.

Accepted Answer

A)   
B)   
C)   
D)

Question 18

Evaluate the double integral  for the given function f(x, y) and the region R.

Accepted Answer

A) 9 
B) 4 
C) 72 
D) 63 
E) 8

Question 19

Let  .Compute  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 20

Minimize the function  subject to the constraint  .

Accepted Answer

The answer of Minimize the function  subject to the...

Exam 8: Calculus of Several Variables

Find the second-order partial derivatives of the given function.Show that the mixed partial derivatives and are equal.

Find the first partial derivatives of the function.

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.

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

Evaluate the double integral for the given function f(x, y) and the region R. ; R is the rectangle defined by

Maximize the function subject to the constraints .

Find the second-order partial derivatives of the given function.Show that the mixed partial derivatives and are equal.

Maximize the function subject to the constraint .

Find the first partial derivatives of the function.

Evaluate the double integral for the function and the region . and is bounded by , , and .

Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than inches.Find the diamension of a cylendrical package of greatest volume that can be sent throught the mail.What is the volume of such a package? Hint: The length plus the grith is .

Maximize the function subject to the constraint .

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

Find the domain of the function. f(m, n) = 2m + 4n

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

Sketch the level curves of the function corresponding to the given values of z.

Evaluate the double integral for the given function f(x, y) and the region R.

Let .Compute .

Minimize the function subject to the constraint .