# AP Calculus AB : Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined

## 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 .

Does not exist

Explanation:

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 ?

Explanation:

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

Explanation:

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