Question 1

Compute the directional derivative of   at   in the direction of

Accepted Answer

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

Question 2

Match the surface to its contour plot.

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

Question 3

Find all first-order partial derivatives.

Accepted Answer

The answer of Find all first-order partial derivatives.  ...

Question 4

Find the direction of maximum change of f at the given point.

Accepted Answer

A)  in the direction of   
B)  in the direction of   
C)  in the direction of   
D)  in the direction of   
A)  in the direction of   
B)  in the direction of   
C)  in the direction of   
D)  in the direction of

Question 5

Use Lagrange multipliers to find the closest point on the given curve to the indicated point.

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D)

Question 6

Use implicit differentiation to find   .

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D)    
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D)

Question 7

Locate all critical points and classify them.

Accepted Answer

A)    local minimum 
B)    local maximum 
C)    local minimum;   local maximum 
D)    local maximum;   local minimum 
A)    local minimum 
B)    local maximum 
C)    local minimum;   local maximum 
D)    local maximum;   local minimum

Question 8

Show that the indicated limit exists.

Accepted Answer

Answers may vary.
Find the lim

Question 9

Find an equation for the tangent plane at the given point.

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

Question 10

The number n of gas particles in a leak-tight container is approximately described by the equation   , where T is the temperature of the gas in kelvins (K), P is the pressure of the gas in atmospheres (atm), and V is the volume of the container in liters (L). For a particular sample, an experimenter finds T = 296.6 $\pm$ 0.01 K, P = 1.05 $\pm$ 0.01 atm, and V = 1.302 $\pm$ 0.002 L. Use a linear approximation to estimate the range of the computed value of n.

Accepted Answer

A)  0 to   
B)    to   
C)    to   
D)    to   
A)  0 to   
B)    to   
C)    to   
D)    to

Question 11

Match the surface to its contour plot.

Accepted Answer

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

Question 12

Match the surface to its density plot.

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

Question 13

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. At   and a wind speed of 30 mph, how much change in wind chill can be expected if the wind speed increases by 10 mph? Temperature (degrees Fahrenheit)
Wind speed
(mph)

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

Question 14

Use the contour plot to estimate   and   at the origin.

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

Question 15

Sketch the indicated traces of   .

Accepted Answer

The answer of Sketch the indicated traces of  ...

Question 16

Maximize   , subject to the constraints   and   .

Accepted Answer

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

Question 17

Find an equation for the tangent plane at the given point.

Accepted Answer

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

Question 18

Find the indicated partial derivatives.

Accepted Answer

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

Question 19

Describe the range of the function.

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

Question 20

For   find   such that   .

Accepted Answer

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

Compute the directional derivative of at in the direction of

Match the surface to its contour plot.

Find all first-order partial derivatives.

Find the direction of maximum change of f at the given point.

Use Lagrange multipliers to find the closest point on the given curve to the indicated point.

Use implicit differentiation to find .

Locate all critical points and classify them.

Show that the indicated limit exists.

Find an equation for the tangent plane at the given point.

Match the surface to its contour plot.

Match the surface to its density plot.

Use the contour plot to estimate and at the origin.

Sketch the indicated traces of .

Maximize , subject to the constraints and .

Find an equation for the tangent plane at the given point.

Find the indicated partial derivatives.

Describe the range of the function.

For find such that .

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters

Exam 14: Functions of Several Variables and Partial Differentiation

Compute the directional derivative of at in the direction of

Match the surface to its contour plot.

Find all first-order partial derivatives.

Find the direction of maximum change of f at the given point.

Use Lagrange multipliers to find the closest point on the given curve to the indicated point.

Use implicit differentiation to find .

Locate all critical points and classify them.

Show that the indicated limit exists.

Find an equation for the tangent plane at the given point.

Match the surface to its contour plot.

Match the surface to its density plot.

Use the contour plot to estimate and at the origin.

Sketch the indicated traces of .

Maximize , subject to the constraints and .

Find an equation for the tangent plane at the given point.

Find the indicated partial derivatives.

Describe the range of the function.

For find such that .

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters