Question 1

Describe and sketch the domain of the function.

Accepted Answer

A)    
B)    
C)    
D)

Question 2

Use Lagrange multipliers to find the maximum and minimum of the function  subject to the constraint  .

Accepted Answer

A)   
B)   
C)   
D)

Question 3

Find the directions of maximum and minimum changes of  at  and the values of the maximum and minimum rates of change.

Accepted Answer

The answer of Find the directions of maximum and minimum...

Question 4

Compute the directional derivative of f at the given point in the direction of the indicated vector.

Accepted Answer

A)   
B)   
C)   
D)

Question 5

Describe and sketch the domain of the function.

Accepted Answer

A)    
B)    
C)    
D)

Question 6

If the temperature at the point  is given by  find the direction from the point  in which the temperature decreases most rapidly.

Accepted Answer

A)   
B)   
C)   
D)

Question 7

Find all points at which  .

Accepted Answer

A)   
B)   
C)   
D)  (0, 0)

Question 8

Use the contour plot to estimate  and  at the origin.

Accepted Answer

A)   
B)   
C)   
D)

Question 9

Sketch the path of steepest ascent from the indicated point.

Accepted Answer

The answer of Sketch the path of steepest ascent from...

Question 10

Find the gradient of  at

Accepted Answer

A)   
B)   
C)   
D)

Question 11

Use the chain rule twice to find  for the general composite function involving

Accepted Answer

The answer of Use the chain rule twice to find...

Question 12

The pressure P of a gas in a leak-tight container can be approximated by  , where V is the volume of the container, T is its absolute (Kelvin) temperature, and k is a constant. Find  and  .

Accepted Answer

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

Question 13

Show that the limit does not exist.

Accepted Answer

The answer of Show that the limit does not exist....

Question 14

Suppose the production of a firm is modeled by  where k measures capital (in millions of dollars) and l measures the labor force (in thousands of workers). Suppose that  and  the labor force is increasing at the rate of 90 and capital is growing at a rate of $200,000 per year. Determine the rate of change of production.

Accepted Answer

A)  Production is increasing at a rate of 13.27 units per year. 
B)  Production is increasing at a rate of 2.13 units per year. 
C)  Production is increasing at a rate of 10.15 units per year. 
D)  Production is increasing at a rate of 21.29 units per year.

Question 15

Solid bodies, such as mechanical flywheels, experience stress when rotating. The stress s on a flywheel with uniform density d (kg/m3) and radius r (meters) is approximated by  where  is the rotational speed of the flywheel in radians per second. Estimate the percent increase in the stress if the rotational speed of a uniform flywheel is increased from 3205 to 3355 rad/s. [Note: you should be able to answer the question with the given information: no additional information is necessary.]

Accepted Answer

A)  9% 
B)  3% 
C)  0.02% 
D)  5%

Question 16

Show that the indicated limit exists.

Accepted Answer

The answer of Show that the indicated limit exists. ...

Question 17

Sketch the indicated traces of  .

Accepted Answer

The answer of Sketch the indicated traces of  ....

Question 18

Use polar coordinates to find the indicated limit, if it exists. Note that  is equivalent to  .

Accepted Answer

A)  0 
B)  1 
C)  -1 
D)  The limit is undefined.

Question 19

Wind chill is a combination of temperature and wind speed that expresses how cold the air feels. The larger the wind speed, the more rapidly heat is lost to the air and thus the colder it feels. The table shows the wind chill for selected temperatures and wind speeds. For the function  , find  . Temperature (degrees Fahrenheit)
Wind speed
(mph)

Accepted Answer

A)   
B)   
C)   
D)

Question 20

Find the gradient of the given function at the indicated point.

Accepted Answer

A)   
B)   
C)   
D)

Exam 14: Functions of Several Variables and Partial Differentiation

Describe and sketch the domain of the function.

Use Lagrange multipliers to find the maximum and minimum of the function subject to the constraint .

Find the directions of maximum and minimum changes of at and the values of the maximum and minimum rates of change.

Compute the directional derivative of f at the given point in the direction of the indicated vector.

Describe and sketch the domain of the function.

If the temperature at the point is given by find the direction from the point in which the temperature decreases most rapidly.

Find all points at which .

Use the contour plot to estimate and at the origin.

Sketch the path of steepest ascent from the indicated point.

Find the gradient of at

Use the chain rule twice to find for the general composite function involving

The pressure P of a gas in a leak-tight container can be approximated by , where V is the volume of the container, T is its absolute (Kelvin) temperature, and k is a constant. Find and .

Show that the limit does not exist.

Show that the indicated limit exists.

Sketch the indicated traces of .

Use polar coordinates to find the indicated limit, if it exists. Note that is equivalent to .

Find the gradient of the given function at the indicated point.