All AP Calculus AB Resources
Example Question #1 : Mean Value Theorem
Find the area under the curve between .
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
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
Example Question #4 : Mean Value Theorem
Consider the region bounded by the functions
between and . What is the area of this region?
The area of this region is given by the following integral:
Taking the antiderivative gives
, evaluated from to .
Thus, the area is given by:
Example Question #5 : Mean Value Theorem
True or false: As a consequence of Rolle's Theorem, has a zero on the interval .
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:
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 #7 : 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 cannot be such that .
There must be such that .
must have a zero on the interval ,
cannot have a zero on the interval ,
None of the statements in the other choices follows from Rolle's Theorem.
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 .
To find the mean value of a function over some interval , one mus use the formula: .
One must then use the inverse Sine function to find the value c: