Question 1

Use differentials to estimate the amount of metal in a closed cylindrical can that is 12 cm high and 8 cm in diameter if the metal in the top and bottom is 0.09 cm thick and the metal in the sides is 0.01 cm thick. (rounded to the nearest hundredth.)

Accepted Answer

A) 6.7  
B) 6.91  
C) 6.99  
D) 8.34  
E)

Question 2

Find the second partial derivatives of the function

Accepted Answer

The answer of Find the second partial derivatives of the...

Question 3

Use the definition of partial derivatives as limits to find  if  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 4

Describe the level surfaces of the function  .

Accepted Answer

The answer of Describe the level surfaces of the function...

Question 5

Find all the saddle points of the function.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 6

Find and classify the relative extrema and saddle points of the function  for  and  .

Accepted Answer

A) Saddle point  
B) Relative minimum  
C) Relative maximum  
D) None

Question 7

Find the indicated partial derivative.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 8

Sketch the graph of the function

Accepted Answer

The answer of Sketch the graph of the function ...

Question 9

Evaluate the limit.

Accepted Answer

A)   
B) 0 
C) 2 
D) the limit does not exist 
E) 1

Question 10

Find all the second partial derivatives.

Accepted Answer

The answer of Find all the second partial derivatives. ...

Question 11

Find the equation of the tangent plane to the given surface at the specified point.

Accepted Answer

The answer of Find the equation of the tangent plane...

Question 12

Find the absolute extrema of the function  on the region bounded by the disk defined by  .

Accepted Answer

The answer of Find the absolute extrema of the function...

Question 13

Use Lagrange multipliers to find the maximum value of the function subject to the given constraints.

Accepted Answer

The answer of Use Lagrange multipliers to find the maximum...

Question 14

Use the Chain Rule to find  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 15

Consider evaluating the limit  (a) Evaluate the limit along the path  (b) Evaluate the limit along the path  (c) What can you conclude about the existence of  ?

Accepted Answer

The answer of Consider evaluating the limit  (a) Evaluate...

Question 16

Find the maximum rate of change of  at the point  (2,1). In what direction does it occur?

Accepted Answer

A)   
B)   
C)   
D)   
E) none of these

Question 17

Determine where the function  is continuous.

Accepted Answer

A)   
B)   
C)   
D)

Question 18

Find the limit.

Accepted Answer

The answer of Find the limit. ...

Question 19

Use the linearization L(x, y) of the function.  at  to approximate  .

Accepted Answer

The answer of Use the linearization L(x, y) of the...

Question 20

Find the indicated partial derivative.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Exam 14: Partial Derivatives

Use differentials to estimate the amount of metal in a closed cylindrical can that is 12 cm high and 8 cm in diameter if the metal in the top and bottom is 0.09 cm thick and the metal in the sides is 0.01 cm thick. (rounded to the nearest hundredth.)

Find the second partial derivatives of the function

Use the definition of partial derivatives as limits to find if .

Describe the level surfaces of the function .

Find all the saddle points of the function.

Find and classify the relative extrema and saddle points of the function for and .

Find the indicated partial derivative.

Sketch the graph of the function

Evaluate the limit.

Find all the second partial derivatives.

Find the equation of the tangent plane to the given surface at the specified point.

Find the absolute extrema of the function on the region bounded by the disk defined by .

Use Lagrange multipliers to find the maximum value of the function subject to the given constraints.

Use the Chain Rule to find .

Consider evaluating the limit (a) Evaluate the limit along the path (b) Evaluate the limit along the path (c) What can you conclude about the existence of ?

Find the maximum rate of change of at the point (2,1) . In what direction does it occur?

Determine where the function is continuous.

Find the limit.

Use the linearization L(x, y) of the function. at to approximate .

Find the indicated partial derivative.