AP Calculus AB : Mean Value Theorem

Example Questions

Example Question #1 : Mean Value Theorem

Find the area under the curve between .

Explanation:

To find the area under the curve, we need to integrate. In this case, it is a definite integral.

Example Question #2 : Mean Value Theorem

Find the area bounded by

Explanation:

The easiest way to look at this is to plot the graphs. The shaded area is the actual area that we want to compute. We can first find area bounded by  and  in the first quadrant and subtract the excessive areas. The area of that rectangle box is 6. The area under the curve  is .

The area of the triangle above the curve  is 2. Therefore, the area bounded is .

Explanation:

Example Question #1 : Mean Value Theorem

Consider the region bounded by the functions

and

between  and .  What is the area of this region?

Explanation:

The area of this region is given by the following integral:

or

Taking the antiderivative gives

, evaluated from  to .

, and

.

Thus, the area is given by:

Example Question #2 : Mean Value Theorem

Let .

True or false: As a consequence of Rolle's Theorem,  has a zero on the interval .

True

False

False

Explanation:

By Rolle's Theorem, if  is continuous on  and differentiable on , and , then there must be  such that . Nothing in the statement of this theorem addresses the location of the zeroes of the function itself. Therefore, the statement is false.

Example Question #3 : Mean Value Theorem

As a consequence of the Mean Value Theorem, there must be a value  such that:

Explanation:

By the Mean Value Theorem (MVT), if a function  is continuous and differentiable on , then there exists at least one value  such that , a polynomial, is continuous and differentiable everywhere; setting , it follows from the MVT that there is  such that

Evaluating  and :

The expression for  is equal to

,

the correct choice.

Example Question #4 : Mean Value Theorem

is continuous and differentiable on .

The values of  for five different values of  are as follows:

Which of the following is a consequence of Rolle's Theorem?

There must be  such that .

None of the statements in the other choices follows from Rolle's Theorem.

cannot have a zero on the interval ,

must have a zero on the interval ,

There cannot be  such that .

There must be  such that .

Explanation:

By Rolle's Theorem, if  is continuous on  and differentiable on , and , then there must be  such that

is given to be continuous. Also, if we set , we note that . This sets up the conditions for Rolle's Theorem to apply. As a consequence, there must be  such that .

Incidentally, it does follow from the given information that  must have a zero on the interval , but this is due to the Intermediate Value Theorem, not Rolle's Theorem.

Example Question #5 : Mean Value Theorem

Find the mean value of the function  over the interval .

Explanation:

To find the mean value of a function over some interval , one mus use the formula: .

Plugging in

Simplifying

One must then use the inverse Sine function to find the value c: