Question 1

Find all first-order partial derivatives.

Accepted Answer

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

Question 2

Match the surface to its density plot.

Accepted Answer

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

Question 3

Compute the directional derivative of   at   in the direction of

Accepted Answer

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

Question 4

Determine all points at which the given function is continuous.

Accepted Answer

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

Question 5

Show that the function   has exactly one critical point, which is an absolute minimum.

Accepted Answer

Solution details may vary. @#IMG-DLM& @#IMG-DLM& [

Question 6

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

Accepted Answer

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

Question 7

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

Question 8

Find all first-order partial derivatives.

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

Which of the following is the contour plot of   ?

Accepted Answer

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

Question 11

Suppose that on a fixed budget of $400 you buy x units of product A purchased at $20 apiece and y units of product B purchased at $20 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 12

Which of the following is the contour plot of   ?

Accepted Answer

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

Question 13

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

Accepted Answer

@#IMG-DLM& (minimum)

Question 14

Find the indicated partial derivative.

Accepted Answer

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

Question 15

Compute the indicated limit.

Accepted Answer

A)  undefined 
B)    
C)  0 
D)    
A)  undefined 
B)    
C)  0 
D)

Question 16

Describe the range of the function.

Accepted Answer

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

Question 17

In the contour plot, the locations of two local extrema and one saddle point are visible. Identify these critical points.

Accepted Answer

The answer of In the contour plot, the locations of...

Question 18

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 19

Use implicit differentiation to find   and   Assume that the equation defines z as a differentiable function near each

Accepted Answer

The answer of Use implicit differentiation to find  ...

Question 20

Numerically approximate all critical points. Classify each.

Accepted Answer

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

Find all first-order partial derivatives.

Match the surface to its density plot.

Compute the directional derivative of at in the direction of

Determine all points at which the given function is continuous.

Show that the function has exactly one critical point, which is an absolute minimum.

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

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

Find all first-order partial derivatives.

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

Which of the following is the contour plot of ?

Suppose that on a fixed budget of $400 you buy x units of product A purchased at $20 apiece and y units of product B purchased at $20 apiece. For the utility function find x and y that maximize the utility function.

Which of the following is the contour plot of ?

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

Find the indicated partial derivative.

Compute the indicated limit.

Describe the range of the function.

In the contour plot, the locations of two local extrema and one saddle point are visible. Identify these critical points.

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

Use implicit differentiation to find and Assume that the equation defines z as a differentiable function near each

Numerically approximate all critical points. Classify each.

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

Find all first-order partial derivatives.

Match the surface to its density plot.

Compute the directional derivative of at in the direction of

Determine all points at which the given function is continuous.

Show that the function has exactly one critical point, which is an absolute minimum.

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

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

Find all first-order partial derivatives.

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

Which of the following is the contour plot of ?

Suppose that on a fixed budget of $400 you buy x units of product A purchased at $20 apiece and y units of product B purchased at $20 apiece. For the utility function find x and y that maximize the utility function.

Which of the following is the contour plot of ?

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

Find the indicated partial derivative.

Compute the indicated limit.

Describe the range of the function.

In the contour plot, the locations of two local extrema and one saddle point are visible. Identify these critical points.

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

Use implicit differentiation to find and Assume that the equation defines z as a differentiable function near each

Numerically approximate all critical points. Classify each.

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