Question 1

Locate all critical points and analyze each graphically.

Accepted Answer

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

Question 2

Locate all critical points and classify them.

Accepted Answer

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

Question 3

Ammonia (NH3) is a gas used for manufacturing fertilizers as well as urethane foams and coatings. A particular gas cylinder contains 17.03 grams of ammonia. The pressure P (in atmospheres (atm)), volume V (in liters (L)), and temperature T (in Kelvin (K)) of the sample are related by the equation   . For the function   , compute   .

Accepted Answer

A)  0.0 atm 
B)  44.2 atm 
C)  14.3 atm 
D)  15.5 atm 
A)  0.0 atm 
B)  44.2 atm 
C)  14.3 atm 
D)  15.5 atm

Question 4

Describe and sketch the domain of the function.

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

Use the contour plot to estimate   and   at the origin.

Accepted Answer

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

Question 7

Compute the indicated limit.

Accepted Answer

A)    
B)    
C)  1 
D)  undefined 
A)    
B)    
C)  1 
D)  undefined

Question 8

Find all first-order partial derivatives.

Accepted Answer

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

Question 9

Write out the third-order Taylor polynomial for   about (0,0).

Accepted Answer

A)    
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

Which of the following is the contour plot of   ?

Accepted Answer

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

Question 12

Match the surface to its contour plot.

Accepted Answer

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

Question 13

Locate all critical points and classify them.

Accepted Answer

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

Question 14

Determine all points at which the given function is continuous.

Accepted Answer

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

Question 15

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 16

Compute the indicated function value.

Accepted Answer

A)    
B)    
C)  -2 
D)  undefined 
A)    
B)    
C)  -2 
D)  undefined

Question 17

Find the total differential of   .

Accepted Answer

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

Question 18

Match the surface to its density plot.

Accepted Answer

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

Question 19

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

Accepted Answer

@#IMG-DLM& (minimum)

Question 20

Find the absolute extrema of   on the region bounded by   and

Accepted Answer

A)    (absolute maximum);   (absolute minimum) 
B)    (absolute maximum);   (absolute minimum) 
C)    (absolute maximum);   (absolute minimum) 
D)    (absolute maximum);   (absolute minimum) 
A)    (absolute maximum);   (absolute minimum) 
B)    (absolute maximum);   (absolute minimum) 
C)    (absolute maximum);   (absolute minimum) 
D)    (absolute maximum);   (absolute minimum)

Locate all critical points and analyze each graphically.

Locate all critical points and classify them.

Describe and sketch the domain of the function.

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

Use the contour plot to estimate and at the origin.

Compute the indicated limit.

Find all first-order partial derivatives.

Write out the third-order Taylor polynomial for about (0,0).

Find the absolute extrema of the function on the region.

Which of the following is the contour plot of ?

Match the surface to its contour plot.

Locate all critical points and classify them.

Determine all points at which the given function is continuous.

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

Compute the indicated function value.

Find the total differential of .

Match the surface to its density plot.

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

Find the absolute extrema of on the region bounded by and

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

Locate all critical points and analyze each graphically.

Locate all critical points and classify them.

Describe and sketch the domain of the function.

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

Use the contour plot to estimate and at the origin.

Compute the indicated limit.

Find all first-order partial derivatives.

Write out the third-order Taylor polynomial for about (0,0).

Find the absolute extrema of the function on the region.

Which of the following is the contour plot of ?

Match the surface to its contour plot.

Locate all critical points and classify them.

Determine all points at which the given function is continuous.

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

Compute the indicated function value.

Find the total differential of .

Match the surface to its density plot.

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

Find the absolute extrema of on the region bounded by and

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