Question 1

Use the given contour plot to estimate the linear approximation of   at   .

Accepted Answer

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

Question 2

Describe the range of the function.

Accepted Answer

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

Question 3

Find the total differential of   .

Accepted Answer

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

Question 4

Show that the indicated limit exists.

Accepted Answer

Answers may vary.
Find the lim

Question 5

Which of the following is the contour plot of   ?

Accepted Answer

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

Question 6

Show that the limit does not exist.

Accepted Answer

Answers may vary.
Approach (0,

Question 7

Compute the linear approximation of the function at the given point.

Accepted Answer

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

Question 8

Find the indicated partial derivative.

Accepted Answer

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

Question 9

Compute the indicated function value.

Accepted Answer

A)  -53 
B)  -29 
C)  -45 
D)  -98 
A)  -53 
B)  -29 
C)  -45 
D)  -98

Question 10

Find the gradient of the given function.

Accepted Answer

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

Question 11

Use a graphing utility to sketch the graph of   . Use the viewpoint   and   (see figure).

Accepted Answer

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

Question 12

Find a function with the given properties.

Accepted Answer

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

Question 13

A box is to be constructed out of 16 square feet of material. Suppose the bottom of the box must be reinforced by doubling up the material (essentially, there are two bottoms of the box). Find the dimensions l, w, and h that maximizes the volume of the box.

Accepted Answer

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

Question 14

Find the gradient of the given function.

Accepted Answer

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

Question 15

Use the chain rule to find the indicated derivative.   , where

Accepted Answer

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

Question 16

Find an equation of the tangent plane to the parametric surface at the indicated point. S is defined by   ; at u = -4 and v = -1

Accepted Answer

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

Question 17

Find the total differential of   .

Accepted Answer

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

Question 18

Show that the function   satisfies the diffusion equation   .

Accepted Answer

Answer det

Question 19

Use the given contour plot to estimate the linear approximation of   at   . Assume each contour line represents a 1 unit increase from left to right with the left-most contour line corresponding to a value of 1.

Accepted Answer

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

Question 20

Determine all points at which the given function is continuous.

Accepted Answer

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

Use the given contour plot to estimate the linear approximation of at .

Describe the range of the function.

Find the total differential of .

Show that the indicated limit exists.

Which of the following is the contour plot of ?

Show that the limit does not exist.

Compute the linear approximation of the function at the given point.

Find the indicated partial derivative.

Compute the indicated function value.

Find the gradient of the given function.

Use a graphing utility to sketch the graph of . Use the viewpoint and (see figure).

Find a function with the given properties.

A box is to be constructed out of 16 square feet of material. Suppose the bottom of the box must be reinforced by doubling up the material (essentially, there are two bottoms of the box). Find the dimensions l, w, and h that maximizes the volume of the box.

Find the gradient of the given function.

Use the chain rule to find the indicated derivative. , where

Find an equation of the tangent plane to the parametric surface at the indicated point. S is defined by ; at u = -4 and v = -1

Find the total differential of .

Show that the function satisfies the diffusion equation .

Use the given contour plot to estimate the linear approximation of at . Assume each contour line represents a 1 unit increase from left to right with the left-most contour line corresponding to a value of 1.

Determine all points at which the given function is continuous.

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

Use the given contour plot to estimate the linear approximation of at .

Describe the range of the function.

Find the total differential of .

Show that the indicated limit exists.

Which of the following is the contour plot of ?

Show that the limit does not exist.

Compute the linear approximation of the function at the given point.

Find the indicated partial derivative.

Compute the indicated function value.

Find the gradient of the given function.

Use a graphing utility to sketch the graph of . Use the viewpoint and (see figure).

Find a function with the given properties.

A box is to be constructed out of 16 square feet of material. Suppose the bottom of the box must be reinforced by doubling up the material (essentially, there are two bottoms of the box). Find the dimensions l, w, and h that maximizes the volume of the box.

Find the gradient of the given function.

Use the chain rule to find the indicated derivative. , where

Find an equation of the tangent plane to the parametric surface at the indicated point. S is defined by ; at u = -4 and v = -1

Find the total differential of .

Show that the function satisfies the diffusion equation .

Use the given contour plot to estimate the linear approximation of at . Assume each contour line represents a 1 unit increase from left to right with the left-most contour line corresponding to a value of 1.

Determine all points at which the given function is continuous.

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