# Fundamental Theorem of Calculus

Worked example: Finding derivative with fundamental theorem of calculus | Khan Academy
Worked example: Finding derivative with fundamental theorem of calculus | Khan Academy

Table of Contents

## Example Questions

### Example Question #1 : Integrals

Evaluate .

Does not exist

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 : Integrals

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
?

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 #1 : Integrals

Evaluate

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

### Example Question #1 : Fundamental Theorem Of Calculus

Find

Does not exist

Does not exist

The one side limits are not equal: left is 0 and right is 3

### Example Question #2 : Fundamental Theorem Of Calculus

Which of the following is a vertical asymptote?

When approaches 3, approaches .

Vertical asymptotes occur at values. The horizontal asymptote occurs at

.

### Example Question #3 : Fundamental Theorem Of Calculus

What are the horizontal asymptotes of ?

Compute the limits of as approaches infinity.

### Example Question #4 : Fundamental Theorem Of Calculus

Write the domain of the function.

The answer is

The denominator must not equal zero and anything under a radical must be a nonnegative number.

### Example Question #5 : Fundamental Theorem Of Calculus

What is the value of the derivative of at x=1?

First, find the derivative of the function, which is:

Then, plug in 1 for x:

The result is .

### Example Question #1 : Calculus 3

Evaluate the following limit:

Does not exist.

First, let’s multiply the numerator and denominator of the fraction in the limit by .

As becomes increasingly large the and terms will tend to zero. This leaves us with the limit of .

.

The answer is .

### Example Question #1 : Fundamental Theorem Of Calculus

Let and be inverse functions, and let

.

What is the value of ?

Since and are inverse functions, . We can differentiate both sides of the equation with respect to to obtain the following:

We are asked to find , which means that we will need to find such that . The given information tells us that , which means that . Thus, we will substitute 3 into the equation.

The given information tells us that.

The equation then becomes .

We can now solve for .

.

The answer is .

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