AP Calculus AB : Asymptotic and Unbounded Behavior

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

Example Question #3 : Finding Definite Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can't use the power rule. Instead we end up with: 

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

Example Question #31 : Finding Integrals

Possible Answers:

Correct answer:

Explanation:

The integral of  is .  The constant 3 is simply multiplied by the integral.  

Example Question #31 : Integrals

Possible Answers:

Correct answer:

Explanation:

To integrate , we need to get the two equations in terms of each other. We are going to use "u-substitution" to create a new variable, , which will equal .

Now, if , then 

 

Multiply both sides by  to get the more familiar: 

Note that our , and our original equation was asking for a positive .

That means if we want  in terms of , it looks like this:

Bring the negative sign to the outside:

.

We can use the power rule to find the integral of :

Since we said that , we can plug that back into the equation to get our answer:

Example Question #61 : Asymptotic And Unbounded Behavior

Evaluate the integral below:

 

Possible Answers:

1

Correct answer:

Explanation:

In this case we have a rational function as , where

and

 

can be written as a product of linear factors:

 

 

It is assumed that A and B are certain constants to be evaluated. Denominators can be cleared by multiplying both sides by (x - 4)(x + 4). So we get:

 

First we substitute x = -4 into the produced equation:

Then we substitute x = 4 into the equation:

Thus:

 

Hence:

 

 

Example Question #1 : Finding Integrals By Substitution

Determine the indefinite integral:

Possible Answers:

Correct answer:

Explanation:

, so this can be rewritten as 

 

Set . Then

and 

Substitute:

The outer factor can be absorbed into the constant, and we can substitute back:

Example Question #1 : Finding Integrals By Substitution

Evaluate:

Possible Answers:

Correct answer:

Explanation:

Set . Then

and 

Also, since , the limits of integration change to  and .

Substitute:

 

 

Example Question #1 : Finding Definite Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can use the reverse power rule to find the indefinite integral or anti-derivative of our function:

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

Example Question #5 : Finding Definite Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

As it turns out, since our , the power rule really doesn't help us.  has a special anti derivative: .

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

Example Question #6 : Finding Definite Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

As it turns out, since our , the power rule really doesn't help us.  is the only function that is it's OWN anti-derivative. That means we're still going to be working with .

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

Because  is so small in comparison to the value we got for , our answer will end up being 

Example Question #62 : Calculus Ii — Integrals

What is the indefinite integral of ?

Possible Answers:

Correct answer:

Explanation:

To solve for the indefinite integral, we can use the reverse power rule. We raise the power of the exponents by one and divide by that new exponent. For this problem, that would look like:

Remember, when taking an integral, definite or indefinite, we always add , as there could be a constant involved.

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