Question 1

Find the gradient of   at

Accepted Answer

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

Question 2

Use implicit differentiation to find   .

Accepted Answer

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

Question 3

State the chain rule for finding   for the general composite function involving

Accepted Answer

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

Question 4

Find all first-order partial derivatives.

Accepted Answer

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

Question 5

Find all first-order partial derivatives.

Accepted Answer

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

Question 6

Use implicit differentiation to find   .

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

Match the surface to its contour plot.

Accepted Answer

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

Question 9

Match the surface to its density plot.

Accepted Answer

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

Question 10

Find the absolute extrema of the function on the region.

Accepted Answer

A)  minimum:  
Maximum:   
B)  minimum:  
Maximum:   
C)  minimum:  
Maximum:   
D)  minimum:  
Maximum:   
A)  minimum:  
Maximum:   
B)  minimum:  
Maximum:   
C)  minimum:  
Maximum:   
D)  minimum:  
Maximum:

Question 11

Show that the limit does not exist.

Accepted Answer

Answers may vary.
Approach (0,

Question 12

Find the total differential of   .

Accepted Answer

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

Find the gradient of at

Use implicit differentiation to find .

State the chain rule for finding for the general composite function involving

Find all first-order partial derivatives.

Find all first-order partial derivatives.

Use implicit differentiation to find .

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

Match the surface to its contour plot.

Match the surface to its density plot.

Find the absolute extrema of the function on the region.

Show that the limit does not exist.

Find the total differential of .

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

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 13: Functions of Several Variables and Partial Differentiation

Find the gradient of at

Use implicit differentiation to find .

State the chain rule for finding for the general composite function involving

Find all first-order partial derivatives.

Find all first-order partial derivatives.

Use implicit differentiation to find .

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

Match the surface to its contour plot.

Match the surface to its density plot.

Find the absolute extrema of the function on the region.

Show that the limit does not exist.

Find the total differential of .

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

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