AP Calculus AB : Asymptotic and Unbounded Behavior

Study concepts, example questions & explanations for AP Calculus AB

varsity tutors app store varsity tutors android store

Example Questions

Example Question #24 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the integral:

Possible Answers:

1

Correct answer:

Explanation:

In order to find the antiderivative, add 1 to the exponent and divide by the exponent. 

Example Question #25 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate:

Possible Answers:

Correct answer:

Explanation:

Example Question #26 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate:

Possible Answers:

Correct answer:

Explanation:

You should first know that the derivative of .

Therefore, looking at the equation you can see that the antiderivative should involve something close to: 

Now to figure out what value represents the square take the derivative of  and set it equal to what the original integral contained. 

Since the derivative of  contains a 3 that the integral does not show, we know that the square is equal to . Thus, the answer is .

Example Question #31 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate:

Possible Answers:

Correct answer:

Explanation:

The antiderivative of . The derivative of . However, since there is no 2 in the original integral, we must divide  by 2. Therefore, the answer is

Example Question #32 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the integral:

Possible Answers:

Correct answer:

Explanation:

When taking the antiderivative add one to the exponent and then divide by the exponent. 

Example Question #51 : Asymptotic And Unbounded Behavior

Evaluate the integral:

Possible Answers:

Cannot be evaluated 

Correct answer:

Explanation:

The derivative of . Therefore, the antiderivative of  is equal to itself. 

Example Question #34 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate:

Possible Answers:

Can't be determined from the information given.

Correct answer:

Explanation:

 and

 

Recall that  is an odd function and  is an even function.

Thus, since  is an odd function, the integral of this function from  to  will be zero.

 

 

Example Question #35 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate this indefinite integral:

Possible Answers:

Correct answer:

Explanation:

To approach this problem, first rewrite the integral expression as shown below:

.

Then, recognize that , and substitute this into the integral expression:

Use substitution, letting  and .  The integral can then be rewritten as

  

Evaluating this integral gives

.

Finally, substituting  back into this expression gives the final answer:

(As this is an indefinite integral,  must be included).

Example Question #1 : Finding Definite Integrals

Evaluate:

Possible Answers:

Correct answer:

Explanation:

 

Example Question #1 : Finding Integrals

Find  

Possible Answers:

Correct answer:

Explanation:

This is most easily solved by recognizing that .  

Learning Tools by Varsity Tutors