Question 1

Find the maximum value of the directional derivative at the point   of the function   . Round your answer to two decimal places.

Accepted Answer

A)  0.55 
B)  0.14 
C)  0.56 
D)  0.10 
E)  0.06 
A)  0.55 
B)  0.14 
C)  0.56 
D)  0.10 
E)  0.06 D

Question 2

Use the gradient to find the directional derivative of the function at P in the direction of Q.

Accepted Answer

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

Question 3

The temperature at the point   on a metal plate is   . Find the direction of greatest increase in heat from the point   . Round all numerical values in your answer to three decimal places.

Accepted Answer

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

Question 4

For function   , find the maximum value of the directional derivative at (3,2).

Accepted Answer

A)    /8 
B)    /56 
C)    /224 
D)    /112 
E)    /168 
A)    /8 
B)    /56 
C)    /224 
D)    /112 
E)    /168

Question 5

Use the gradient to find a normal vector to the graph of the equation at the given point.

Accepted Answer

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

Question 6

The temperature at the point   on a metal plate is modeled by   ,   . Find the directions of no change in heat on the plate from the point   .

Accepted Answer

A)  There will be no change in directions perpendicular to the gradient   . 
B)  There will be no change in directions parallel to the gradient   . 
C)  There will be no change in directions parallel to the gradient   . 
D)  There will be no change in directions perpendicular to the gradient   . 
E)  There will be no change in directions perpendicular to the gradient   . 
A)  There will be no change in directions perpendicular to the gradient   . 
B)  There will be no change in directions parallel to the gradient   . 
C)  There will be no change in directions parallel to the gradient   . 
D)  There will be no change in directions perpendicular to the gradient   . 
E)  There will be no change in directions perpendicular to the gradient   .

Question 7

Use the gradient to find the directional derivative of the function at P in the direction of Q.

Accepted Answer

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

Question 8

Find the directional derivative of the function at P in the direction of   .

Accepted Answer

A)  4/   
B)  -8/   
C)  12/   
D)  24/   
E)  -24/   
A)  4/   
B)  -8/   
C)  12/   
D)  24/   
E)  -24/

Question 9

Find the path of a heat-seeking particle placed at point   on a metal plate with a temperature field   .

Accepted Answer

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

Question 10

Find the maximum value of the directional derivative at the point   of the function   . Round your answer to two decimal places.

Accepted Answer

A)  899.28 
B)  814.59 
C)  922.05 
D)  933.23 
E)  538.80 
A)  899.28 
B)  814.59 
C)  922.05 
D)  933.23 
E)  538.80

Question 11

Find the directional derivative of the function at P in the direction of   .

Accepted Answer

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

Question 12

Find the gradient of the function at the given point.

Accepted Answer

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

Question 13

The surface of a mountain is modeled by the equation   . A mountain climber is at the point   . In what direction should the climber move in order to ascend at the greatest rate? Round all numerical values in your answer to one decimal place.

Accepted Answer

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

Question 14

The temperature at the point   on a metal plate is modeled by   ,   . Find the direction of greatest increase in heat from the point   .

Accepted Answer

A)  The greatest increase is in the direction of the gradient   . 
B)  The greatest increase is in the direction of the gradient   . 
C)  The greatest increase is in the direction of the gradient   . 
D)  The greatest increase is in the direction of the gradient   . 
E)  The greatest increase is in the direction of the gradient   . 
A)  The greatest increase is in the direction of the gradient   . 
B)  The greatest increase is in the direction of the gradient   . 
C)  The greatest increase is in the direction of the gradient   . 
D)  The greatest increase is in the direction of the gradient   . 
E)  The greatest increase is in the direction of the gradient   .

Question 15

Find   for function   .

Accepted Answer

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

Question 16

Find the directional derivative of the function at P in the direction of   .

Accepted Answer

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

Question 17

Find the gradient of the function at the given point.

Accepted Answer

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

Question 18

Find the gradient of the function at the given point.

Accepted Answer

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

Question 19

Find the directional derivative of the function   at   in the direction of   . Round your answer to two decimal places.

Accepted Answer

A)  -11.64 
B)  -1.87 
C)  -5.86 
D)  -0.70 
E)  -5.62 
A)  -11.64 
B)  -1.87 
C)  -5.86 
D)  -0.70 
E)  -5.62

Question 20

Find the directional derivative of the function in the direction of   .

Accepted Answer

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

Exam 13: Section 7: Functions of Several Variables

Find the maximum value of the directional derivative at the point of the function . Round your answer to two decimal places.

Use the gradient to find the directional derivative of the function at P in the direction of Q.

The temperature at the point on a metal plate is . Find the direction of greatest increase in heat from the point . Round all numerical values in your answer to three decimal places.

For function , find the maximum value of the directional derivative at (3,2).

Use the gradient to find a normal vector to the graph of the equation at the given point.

The temperature at the point on a metal plate is modeled by , . Find the directions of no change in heat on the plate from the point .

Use the gradient to find the directional derivative of the function at P in the direction of Q.

Find the directional derivative of the function at P in the direction of .

Find the path of a heat-seeking particle placed at point on a metal plate with a temperature field .

Find the maximum value of the directional derivative at the point of the function . Round your answer to two decimal places.

Find the directional derivative of the function at P in the direction of .

Find the gradient of the function at the given point.

The surface of a mountain is modeled by the equation . A mountain climber is at the point . In what direction should the climber move in order to ascend at the greatest rate? Round all numerical values in your answer to one decimal place.

The temperature at the point on a metal plate is modeled by , . Find the direction of greatest increase in heat from the point .

Find for function .

Find the directional derivative of the function at P in the direction of .

Find the gradient of the function at the given point.

Find the gradient of the function at the given point.

Find the directional derivative of the function at in the direction of . Round your answer to two decimal places.

Find the directional derivative of the function in the direction of .

Section 1: Preparation for Calculus

Section 2: Preparation for Calculus

Section 3: Preparation for Calculus

Section 4: Preparation for Calculus

Section 5: Preparation for Calculus

Section 6: Preparation for Calculus

Section 1: Limits and Their Properties

Section 2: Limits and Their Properties

Section 3: Limits and Their Properties

Section 4: Limits and Their Properties

Section 5 : Limits and Their Properties

Section 1 : Differentiation

Section 2: Differentiation

Section 3: Differentiation

Section 4 : Differentiation

Section 5: Differentiation

Section 6: Differentiation

Section 7: Differentiation

Section 8: Differentiation

Section 1 : Applications of Differentiation

Section 2: Applications of Differentiation

Section 3: Applications of Differentiation

Section 4: Applications of Differentiation

Section 5: Applications of Differentiation

Section 6: Applications of Differentiation

Section 7: Applications of Differentiation

Section 8: Applications of Differentiation

Section 1: Integration

Section 2: Integration

Section 3: Integration

Section 4: Integration

Section 5: Integration

Section 6: Integration

Section 7: Integration

Section 8: Integration

Section 9: Integration

Section 1: Differential Equations

Section 2: Differential Equations

Section 3: Differential Equations

Section 4: Differential Equations

Section 5: Differential Equations

Section 1: Applications of Integration

Section 2: Applications of Integration

Section 3: Applications of Integration

Section 4: Applications of Integration

Section 5: Applications of Integration

Section 6: Applications of Integration

Section 7: Applications of Integration

Section 1: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 2: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 3: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 4: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 5: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 6: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 7: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 8: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 1: Infinite Series

Section 2: Infinite Series

Section 3: Infinite Series