### All AP Calculus AB Resources

## Example Questions

### Example Question #1 : Use Of The Fundamental Theorem To Represent A Particular Antiderivative, And The Analytical And Graphical Analysis Of Functions So Defined

Evaluate .

**Possible Answers:**

Does not exist

**Correct answer:**

Even though an antideritvative of does not exist, we can still use the Fundamental Theorem of Calculus to "cancel out" the integral sign in this expression.

. Start

. You can "cancel out" the integral sign with the derivative by making sure the lower bound of the integral is a constant, the upper bound is a differentiable function of , , and then substituting in the integrand. Lastly the Theorem states you must multiply your result by (similar to the directions in using the chain rule).

.

### Example Question #2 : Use Of The Fundamental Theorem To Represent A Particular Antiderivative, And The Analytical And Graphical Analysis Of Functions So Defined

The graph of a function is drawn below. Select the best answers to the following:

What is the best interpretation of the function?

Which plot shows the derivative of the function ?

**Possible Answers:**

**Correct answer:**

The function represents the area under the curve from to some value of .

Do not be confused by the use of in the integrand. The reason we use is because are writing the area as a function of , which requires that we treat the upper limit of integration as a variable . So we replace the independent variable of with a dummy index when we write down the integral. It does not change the fundamental behavior of the function or .

The graph of the derivative of is the same as the graph for . This follows directly from the Second Fundamental Theorem of Calculus.

If the function is continuous on an interval containing , then the function defined by:

has for its' derivative .

### Example Question #3 : Use Of The Fundamental Theorem To Represent A Particular Antiderivative, And The Analytical And Graphical Analysis Of Functions So Defined

Evaluate

**Possible Answers:**

**Correct answer:**

Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.

Instead, we make a clever observation of the graph of

Namely, that

This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that