Question 1

Calculate the first two steps of the steepest ascent algorithm from the given starting point.

Accepted Answer

A)  Step 1:   Step 2:   
B)  Step 1:   Step 2:   
C)  Step 1:   Step 2:   
D)  Step 1:   Step 2:   
A)  Step 1:   Step 2:   
B)  Step 1:   Step 2:   
C)  Step 1:   Step 2:   
D)  Step 1:   Step 2:

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

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 4

Use implicit differentiation to find   .

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

Find the value of the maximum rate of change of f at the given point.

Accepted Answer

A)  0 
B)  12 
C)    
D)  13 
A)  0 
B)  12 
C)    
D)  13

Question 7

Find all points at which   .

Accepted Answer

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

Question 8

Describe and sketch the domain of the function.

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

Find an equation of the tangent plane to the surface at the given point.

Accepted Answer

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

Question 12

Show that the limit does not exist.

Accepted Answer

Answers may vary.
Approach (-1

Calculate the first two steps of the steepest ascent algorithm from the given starting point.

Find the indicated partial derivative.

Use implicit differentiation to find .

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

Find the value of the maximum rate of change of f at the given point.

Find all points at which .

Describe and sketch the domain of the function.

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 the constraint .

Find an equation of the tangent plane to the surface at the given point.

Show that the limit does not exist.

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

Calculate the first two steps of the steepest ascent algorithm from the given starting point.

Find the indicated partial derivative.

Use implicit differentiation to find .

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

Find the value of the maximum rate of change of f at the given point.

Find all points at which .

Describe and sketch the domain of the function.

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 the constraint .

Find an equation of the tangent plane to the surface at the given point.

Show that the limit does not exist.

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