Question 1

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 C

Question 2

Use polar coordinates to find the limit   . [Hint: Let   and   , and note that   implies   .]

Accepted Answer

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

Question 3

Discuss the continuity of the function.

Accepted Answer

A)    is continuous everywhere except at those points   such that   . 
B)    is continuous everywhere except at those points   such that   . 
C)    is continuous at those points   such that   . 
D)    is continuous only when   . 
E)    is continuous everywhere in the xy-plane. 
A)    is continuous everywhere except at those points   such that   . 
B)    is continuous everywhere except at those points   such that   . 
C)    is continuous at those points   such that   . 
D)    is continuous only when   . 
E)    is continuous everywhere in the xy-plane. E

Question 4

Find the limit .

Accepted Answer

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

Question 5

Find the limit.

Accepted Answer

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

Question 6

Find the limit.

Accepted Answer

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

Question 7

Find   by using the limits   and   .

Accepted Answer

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

Question 8

Find the limit   for the function   .

Accepted Answer

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

Question 9

Determine the continuity of the composite function   where   and   .

Accepted Answer

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

Question 10

Determine the continuity of the function   .

Accepted Answer

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

Question 11

Find the limit   for the function   .

Accepted Answer

A)    
B)    
C)    
D)    
E)  -5y 
A)    
B)    
C)    
D)    
E)  -5y

Question 12

Find   by using the limits   and   .

Accepted Answer

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

Question 13

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 14

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

Accepted Answer

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

Question 15

Determine the continuity of the function   .

Accepted Answer

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

Question 16

Find the limit   for the function   .

Accepted Answer

A)  -5y 
B)    
C)    
D)  5y 
E)  8x 
A)  -5y 
B)    
C)    
D)  5y 
E)  8x

Question 17

Find the limit.

Accepted Answer

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

Question 18

Use spherical coordinates to find the limit   [Hint: Let   and   , and note that   implies   .]

Accepted Answer

A)  6 
B)  9 
C)  1 
D)  7 
E)    
A)  6 
B)  9 
C)  1 
D)  7 
E)

Question 19

Discuss the continuity of the function at the origin.

Accepted Answer

A)  f is discontinuous at (0,0).
The discontinuity can be removed by setting   instead of 9. 
B)  f is discontinuous at   .
The discontinuity can be removed by setting   instead of 9. 
C)  f is discontinuous at   .
The discontinuity can be removed by setting   = 0 instead of 9. 
D)  f is discontinuous at   .
The discontinuity cannot be removed by changing the value of f at   . 
E)  f is continuous at   . 
A)  f is discontinuous at (0,0).
The discontinuity can be removed by setting   instead of 9. 
B)  f is discontinuous at   .
The discontinuity can be removed by setting   instead of 9. 
C)  f is discontinuous at   .
The discontinuity can be removed by setting   = 0 instead of 9. 
D)  f is discontinuous at   .
The discontinuity cannot be removed by changing the value of f at   . 
E)  f is continuous at   .

Question 20

Find the limit.

Accepted Answer

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

Exam 13: Section 2: Functions of Several Variables

Determine the continuity of the function .

Use polar coordinates to find the limit . [Hint: Let and , and note that implies .]

Discuss the continuity of the function.

Find the limit .

Find the limit.

Find the limit.

Find by using the limits and .

Find the limit for the function .

Determine the continuity of the composite function where and .

Determine the continuity of the function .

Find the limit for the function .

Find by using the limits and .

Determine the continuity of the function .

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

Determine the continuity of the function .

Find the limit for the function .

Find the limit.

Use spherical coordinates to find the limit [Hint: Let and , and note that implies .]

Discuss the continuity of the function at the origin.

Find the limit.

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