Question 1

Evaluate the limits using the Limit Laws:
A)   
B)     
C)

Accepted Answer

A)   B)   C)

Question 2

Evaluate the limit or state that it does not exist
A)   
B)   
C)

Accepted Answer

A) 7
B)   C) Limit does not exist

Question 3

Determine the points where the function is not continuous and state the type of the discontinuity: removable, jump, infinite, or none of these.
A)   
B)   
C)

Accepted Answer

A)   , jump
B)   , jump
C)   , removable

Question 4

Find   and use the formal definition of the limit to rigorously prove your result.

Accepted Answer

The answer of Find   and use the formal...

Question 5

Determine the points where the function is not continuous and state the type of discontinuity: removable, jump, infinite, or none of these:
A)   
B)   
C)   
D)

Accepted Answer

A) @#IMG-DLM& , removable
B) @#IMG-DLM&

Question 6

The greatest integer function is defined by   , where   is the unique integer such that   .
The graph of   is shown in the figure. 
A) For which values of   does   exist? 
B) For which values of   does   exist? 
C) For which values of   does   exist?

Accepted Answer

A) All @#IMG-DLM& B) All @#IMG-DLM& C) E

Question 7

The polynomial   surely has a root in the following interval:

Accepted Answer

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

Question 8

The position of a particle is given by   . Compute the average velocity over the time interval   . Estimate the instantaneous velocity at   .

Accepted Answer

Average velocity ove

Question 9

Let   denote the slope of the line segment connecting the origin to the point   on the graph of the equation   . Calculate the average rate of change of   for

Accepted Answer

The answer of Let   denote the slope of...

Question 10

The following function is a counterexample for the converse of the Intermediate Value Theorem, which states:
If   assumes all the values between   and   in the interval   , then   is continuous on   :

Accepted Answer

A)    for   ,   ,   
B)    on   
C)    on   
D)    on   
E)  A and C are correct. 
A)    for   ,   ,   
B)    on   
C)    on   
D)    on   
E)  A and C are correct.

Question 11

Determine whether the following statement is correct. If yes, prove it; otherwise give a counterexample
If   , then   exists.

Accepted Answer

A) True 
 B)False

Question 12

Show that   for all   . Use the Squeeze Theorem to evaluate   .

Accepted Answer

The answer of Show that   for all ...

Question 13

Compute the following one-sided limits:
A)   
B)   
C)

Accepted Answer

The answer of Compute the following one-sided limits:
A)  ...

Question 14

Find   .

Accepted Answer

The answer of Find   ....

Question 15

Evaluate the limit

Accepted Answer

The answer of Evaluate the limit  ...

Question 16

Evaluate the limits
A)   
B)   
C)

Accepted Answer

The answer of Evaluate the limits
A)   
B) ...

Question 17

Determine the one-sided limits at   of the function   shown in the figure and state whether the limit exists at these points.

Accepted Answer

@#IMG-DLM& , limit exists @#IMG-DLM& , l

Question 18

Consider the equation   for   . 
A) Verify that   is a solution. 
B) Show that   for   . Hint: Check the minimum value of the function on the left-hand side. 
C) How many solutions does the equation have?

Accepted Answer

The answer of Consider the equation   for ...

Question 19

The potential energy   of a pendulum of length 1 and mass 2, relative to its rest position is   .
Compute the average rate of change of the potential energy over the angle interval   .

Accepted Answer

The answer of The potential energy   of a...

Question 20

Determine   and   for the function shown in the figure.

Accepted Answer

The answer of Determine   and   for...

Evaluate the limits using the Limit Laws: A) B) C)

Evaluate the limit or state that it does not exist A) B) C)

Determine the points where the function is not continuous and state the type of the discontinuity: removable, jump, infinite, or none of these. A) B) C)

Find and use the formal definition of the limit to rigorously prove your result.

Determine the points where the function is not continuous and state the type of discontinuity: removable, jump, infinite, or none of these: A) B) C) D)

The greatest integer function is defined by , where is the unique integer such that . The graph of is shown in the figure. A) For which values of does exist? B) For which values of does exist? C) For which values of does exist?

The polynomial surely has a root in the following interval:

The position of a particle is given by . Compute the average velocity over the time interval . Estimate the instantaneous velocity at .

Let denote the slope of the line segment connecting the origin to the point on the graph of the equation . Calculate the average rate of change of for

The following function is a counterexample for the converse of the Intermediate Value Theorem, which states: If assumes all the values between and in the interval , then is continuous on :

Determine whether the following statement is correct. If yes, prove it; otherwise give a counterexample If , then exists.

Show that for all . Use the Squeeze Theorem to evaluate .

Compute the following one-sided limits: A) B) C)

Find .

Evaluate the limit

Evaluate the limits A) B) C)

Determine the one-sided limits at of the function shown in the figure and state whether the limit exists at these points.

Consider the equation for . A) Verify that is a solution. B) Show that for . Hint: Check the minimum value of the function on the left-hand side. C) How many solutions does the equation have?

The potential energy of a pendulum of length 1 and mass 2, relative to its rest position is . Compute the average rate of change of the potential energy over the angle interval .

Determine and for the function shown in the figure.

Precalculus Review

Differentiation

Applications of the Derivative

The Integral

Applications of the Integral

Exponential Functions

Techniques of Integration

Further Applications of the Integral and Taylor Polynomials

Introduction to Differential Equations

Infinite Series

Parametric Equations, Polar Coordinates, and Conic Sections

Vector Geometry

Calculus of Vector-Valued Functions

Differentiation in Several Variables

Multiple Integration

Line and Surface Integrals

Fundamental Theorems of Vector Analysis

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Exam 2: Limits

Evaluate the limits using the Limit Laws: A) B) C)

Evaluate the limit or state that it does not exist A) B) C)

Determine the points where the function is not continuous and state the type of the discontinuity: removable, jump, infinite, or none of these. A) B) C)

Find and use the formal definition of the limit to rigorously prove your result.

Determine the points where the function is not continuous and state the type of discontinuity: removable, jump, infinite, or none of these: A) B) C) D)

The greatest integer function is defined by , where is the unique integer such that . The graph of is shown in the figure. A) For which values of does exist? B) For which values of does exist? C) For which values of does exist?

The polynomial surely has a root in the following interval:

The position of a particle is given by . Compute the average velocity over the time interval . Estimate the instantaneous velocity at .

Let denote the slope of the line segment connecting the origin to the point on the graph of the equation . Calculate the average rate of change of for

The following function is a counterexample for the converse of the Intermediate Value Theorem, which states: If assumes all the values between and in the interval , then is continuous on :

Determine whether the following statement is correct. If yes, prove it; otherwise give a counterexample If , then exists.

Show that for all . Use the Squeeze Theorem to evaluate .

Compute the following one-sided limits: A) B) C)

Find .

Evaluate the limit

Evaluate the limits A) B) C)

Determine the one-sided limits at of the function shown in the figure and state whether the limit exists at these points.

Consider the equation for . A) Verify that is a solution. B) Show that for . Hint: Check the minimum value of the function on the left-hand side. C) How many solutions does the equation have?

The potential energy of a pendulum of length 1 and mass 2, relative to its rest position is . Compute the average rate of change of the potential energy over the angle interval .

Determine and for the function shown in the figure.

Precalculus Review

Differentiation

Applications of the Derivative

The Integral

Applications of the Integral

Exponential Functions

Techniques of Integration

Further Applications of the Integral and Taylor Polynomials

Introduction to Differential Equations

Infinite Series

Parametric Equations, Polar Coordinates, and Conic Sections

Vector Geometry

Calculus of Vector-Valued Functions

Differentiation in Several Variables

Multiple Integration

Line and Surface Integrals

Fundamental Theorems of Vector Analysis

Filters