Question 1

Let   represent the temperature at each point on the sphere   . Find the maximum temperature on the curve formed by the intersection of the sphere and the plane   .

Accepted Answer

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

Question 2

Sketch the surface given by the function.

Accepted Answer

A)    
B)    
C)    
D)    
E)  none of these 
A)    
B)    
C)    
D)    
E)  none of these

Question 3

Find symmetric equations of the tangent line to the curve of intersection of the surfaces   ,   at the point   . ​

Accepted Answer

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

Question 4

Use polar coordinates and L'Hopital's Rule to find the limit.

Accepted Answer

A)    
B)  13 
C)    
D)    
E)  0 
A)    
B)  13 
C)    
D)    
E)  0

Question 5

Find and simplify the function   at the given value   .

Accepted Answer

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D)    
E)    
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E)

Question 6

The least squares regression line for the points shown in the graph is   . Calculate S, the sum of the squared errors. ​   ​

Accepted Answer

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

Question 7

Find symmetric equations of the normal line to the surface   at the point   .

Accepted Answer

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

Question 8

Identify the point(s) on the surface   where the tangent plane is horizontal. ​

Accepted Answer

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B)    ​ 
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E)    ​ 
A)    ​ 
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Question 9

Find the partial derivative   for the function   .

Accepted Answer

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

Question 10

The average length of time that a customer waits in line for service is   where y is the average arrival rate, written as the number of customers per unit of time, and x is the average service rate, written in the same units. Evaluate   . Note: x and y are given as the number of customers per hour. ​

Accepted Answer

A)  13 min 
B)  10 min 
C)  7 min 
D)  8 min 
E)  12 min 
A)  13 min 
B)  10 min 
C)  7 min 
D)  8 min 
E)  12 min

Question 11

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 12

The radius of a right circular cylinder is increasing at a rate of 9 inches per minute, and the height is decreasing at a rate of 7 inches per minute. What is the rate of change of the volume when the radius is 13 inches and the height is 36 inches?

Accepted Answer

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

Question 13

Find the limit   for the function   . ​

Accepted Answer

A)  -4y 
B)  0 
C)    
D)  4y 
E)  4x 
A)  -4y 
B)  0 
C)    
D)  4y 
E)  4x

Question 14

Determine the continuity of the function   .

Accepted Answer

A)  continuous except at   
B)  continuous except at   
C)  continuous except at   
D)  continuous except at   
E)  continuous everywhere 
A)  continuous except at   
B)  continuous except at   
C)  continuous except at   
D)  continuous except at   
E)  continuous everywhere

Question 15

Find three positive numbers x, y, and z whose sum is 32 and   is a maximum. ​

Accepted Answer

A)  x = 11, y = 6, z = 15 
B)  x = 6, y = 15, z = 11 
C)  x = 15, y = 11, z = 6 
D)  x = 8, y = 16, z = 8 
E)  x = 8, y = 8, z = 16 
A)  x = 11, y = 6, z = 15 
B)  x = 6, y = 15, z = 11 
C)  x = 15, y = 11, z = 6 
D)  x = 8, y = 16, z = 8 
E)  x = 8, y = 8, z = 16

Question 16

Sketch the level curves of the function   for the given c-values   .

Accepted Answer

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

Question 17

Describe the domain of the function   .

Accepted Answer

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

Question 18

For   , evaluate   at the point   . ​

Accepted Answer

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

Question 19

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)

Question 20

For   , evaluate   at the point   . ​

Accepted Answer

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

Let represent the temperature at each point on the sphere . Find the maximum temperature on the curve formed by the intersection of the sphere and the plane .

Sketch the surface given by the function.

Find symmetric equations of the tangent line to the curve of intersection of the surfaces , at the point .

Use polar coordinates and L'Hopital's Rule to find the limit.

Find and simplify the function at the given value .

The least squares regression line for the points shown in the graph is . Calculate S, the sum of the squared errors.

Find symmetric equations of the normal line to the surface at the point .

Identify the point(s) on the surface where the tangent plane is horizontal.

Find the partial derivative for the function .

The average length of time that a customer waits in line for service is where y is the average arrival rate, written as the number of customers per unit of time, and x is the average service rate, written in the same units. Evaluate . Note: x and y are given as the number of customers per hour.

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

The radius of a right circular cylinder is increasing at a rate of 9 inches per minute, and the height is decreasing at a rate of 7 inches per minute. What is the rate of change of the volume when the radius is 13 inches and the height is 36 inches?

Find the limit for the function .

Determine the continuity of the function .

Find three positive numbers x, y, and z whose sum is 32 and is a maximum.

Sketch the level curves of the function for the given c-values .

Describe the domain of the function .

For , evaluate at the point .

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 , evaluate at the point .

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integration

Vector Anal

Filters

Exam 13: Functions of Several Variables

Let represent the temperature at each point on the sphere . Find the maximum temperature on the curve formed by the intersection of the sphere and the plane .

Sketch the surface given by the function.

Find symmetric equations of the tangent line to the curve of intersection of the surfaces , at the point . ​

Use polar coordinates and L'Hopital's Rule to find the limit.

Find and simplify the function at the given value .

The least squares regression line for the points shown in the graph is . Calculate S, the sum of the squared errors. ​ ​

Find symmetric equations of the normal line to the surface at the point .

Identify the point(s) on the surface where the tangent plane is horizontal. ​

Find the partial derivative for the function .

The average length of time that a customer waits in line for service is where y is the average arrival rate, written as the number of customers per unit of time, and x is the average service rate, written in the same units. Evaluate . Note: x and y are given as the number of customers per hour. ​

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

The radius of a right circular cylinder is increasing at a rate of 9 inches per minute, and the height is decreasing at a rate of 7 inches per minute. What is the rate of change of the volume when the radius is 13 inches and the height is 36 inches?

Find the limit for the function . ​

Determine the continuity of the function .

Find three positive numbers x, y, and z whose sum is 32 and is a maximum. ​

Sketch the level curves of the function for the given c-values .

Describe the domain of the function .

For , evaluate at the point . ​

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 , evaluate at the point . ​

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integration

Vector Anal

Filters

Find symmetric equations of the tangent line to the curve of intersection of the surfaces , at the point .

The least squares regression line for the points shown in the graph is . Calculate S, the sum of the squared errors.

Identify the point(s) on the surface where the tangent plane is horizontal.

The average length of time that a customer waits in line for service is where y is the average arrival rate, written as the number of customers per unit of time, and x is the average service rate, written in the same units. Evaluate . Note: x and y are given as the number of customers per hour.

Find the limit for the function .

Find three positive numbers x, y, and z whose sum is 32 and is a maximum.

For , evaluate at the point .

For , evaluate at the point .