### All AP Calculus AB Resources

## Example Questions

### Example Question #1 : Mean Value Theorem

Find the area under the curve between .

**Possible Answers:**

**Correct answer:**

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

### Example Question #1 : Mean Value Theorem

Find the area bounded by

**Possible Answers:**

**Correct answer:**

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 .

### Example Question #3 : Mean Value Theorem

**Possible Answers:**

**Correct answer:**

### Example Question #2 : Mean Value Theorem

Consider the region bounded by the functions

and

between and . What is the area of this region?

**Possible Answers:**

**Correct answer:**

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 #3 : Mean Value Theorem

Let .

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

**Possible Answers:**

True

False

**Correct answer:**

False

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 #6 : Mean Value Theorem

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

**Possible Answers:**

**Correct answer:**

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?

**Possible Answers:**

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 must be such that .

There cannot be such that .

**Correct answer:**

There must be such that .

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 #8 : Mean Value Theorem

Find the mean value of the function over the interval .

**Possible Answers:**

**Correct answer:**

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: