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

Find the indicated partial derivative.

Accepted Answer

A)  0 
B)  6w2z 
C)  12wz 
D)  3w2 
A)  0 
B)  6w2z 
C)  12wz 
D)  3w2

Question 3

Use the contour plot to estimate   and   at the origin.

Accepted Answer

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

Question 4

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

Accepted Answer

@#IMG-DLM& (minimum)

Question 5

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

Accepted Answer

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

Question 6

Describe and sketch the domain of the function.

Accepted Answer

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

Question 7

Determine all points at which the given function is continuous.

Accepted Answer

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

Question 8

Find all points at which   .

Accepted Answer

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

Question 9

Determine all points at which the given function is continuous.

Accepted Answer

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

Question 10

A box is to be constructed out of 196 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 11

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 12

Match the surface to its density plot.

Accepted Answer

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

Question 13

Match the surface to its density plot.

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

@#IMG-DLM& (minimum)

Question 17

Suppose that on a fixed budget of $300 you buy x units of product A purchased at $10 apiece and y units of product B purchased at $15 apiece. For the utility function   find x and y that maximize the utility function.

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

Describe the range of the function.

Accepted Answer

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

Question 20

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

Accepted Answer

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

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

Find the indicated partial derivative.

Use the contour plot to estimate and at the origin.

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

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

Describe and sketch the domain of the function.

Determine all points at which the given function is continuous.

Find all points at which .

Determine all points at which the given function is continuous.

A box is to be constructed out of 196 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 direction of maximum change of f at the given point.

Match the surface to its density plot.

Match the surface to its density plot.

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

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

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

Suppose that on a fixed budget of $300 you buy x units of product A purchased at $10 apiece and y units of product B purchased at $15 apiece. For the utility function find x and y that maximize the utility function.

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

Describe the range of the function.

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

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

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

Find the indicated partial derivative.

Use the contour plot to estimate and at the origin.

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

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

Describe and sketch the domain of the function.

Determine all points at which the given function is continuous.

Find all points at which .

Determine all points at which the given function is continuous.

A box is to be constructed out of 196 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 direction of maximum change of f at the given point.

Match the surface to its density plot.

Match the surface to its density plot.

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

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

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

Suppose that on a fixed budget of $300 you buy x units of product A purchased at $10 apiece and y units of product B purchased at $15 apiece. For the utility function find x and y that maximize the utility function.

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

Describe the range of the function.

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

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