Question 1

Describe and sketch the domain of the function.

Accepted Answer

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

Question 2

Two electrical resistors with resistances R1 and R2 (in ohms) are wired into an electronics circuit as shown in the figure below. The pair of resistors behaves as a single resistor RT whose resistance is given by   . Once the circuit is energized, the temperatures of the resistors increase, causing the resistances to increase according to:   ,   , where t is in seconds. Find   .

Accepted Answer

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

Question 3

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 4

Compute the directional derivative of   at   in the direction of

Accepted Answer

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

Question 5

Compute the indicated limit.

Accepted Answer

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

Question 6

Locate all critical points and classify them.

Accepted Answer

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

Question 7

Describe the range of the function.

Accepted Answer

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

Question 8

Find the indicated partial derivatives.

Accepted Answer

The answer of Find the indicated partial derivatives.  ...

Question 9

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

Accepted Answer

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

Question 10

Find the indicated partial derivative.

Accepted Answer

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

Question 11

Find the total differential of   .

Accepted Answer

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

Question 12

Find the gradient of the given function.

Accepted Answer

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

Question 13

Find the gradient of the given function.

Accepted Answer

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

Question 14

Find the indicated partial derivative.

Accepted Answer

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

Question 15

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)

Question 16

Locate all critical points and classify them.

Accepted Answer

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

Question 17

Use the chain rule twice to find   for the general composite function involving

Accepted Answer

The answer of Use the chain rule twice to find...

Question 18

Show that the limit does not exist.

Accepted Answer

Answers may vary.
Approach (0,

Question 19

Describe the range of the function.

Accepted Answer

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

Question 20

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...

Describe and sketch the domain of the function.

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

Compute the directional derivative of at in the direction of

Compute the indicated limit.

Locate all critical points and classify them.

Describe the range of the function.

Find the indicated partial derivatives.

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

Find the indicated partial derivative.

Find the total differential of .

Find the gradient of the given function.

Find the gradient of the given function.

Find the indicated partial derivative.

Find the absolute extrema of on the region bounded by and

Locate all critical points and classify them.

Use the chain rule twice to find for the general composite function involving

Show that the limit does not exist.

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.

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

Describe and sketch the domain of the function.

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

Compute the directional derivative of at in the direction of

Compute the indicated limit.

Locate all critical points and classify them.

Describe the range of the function.

Find the indicated partial derivatives.

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

Find the indicated partial derivative.

Find the total differential of .

Find the gradient of the given function.

Find the gradient of the given function.

Find the indicated partial derivative.

Find the absolute extrema of on the region bounded by and

Locate all critical points and classify them.

Use the chain rule twice to find for the general composite function involving

Show that the limit does not exist.

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.

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