Worksheet: The Mean Value Theorem and Its Interpretation

The mean value theorem is one of the most important theoretical tools in calculus. It is used if the function is defined and continuous on a known interval and differentiable on that interval. This worksheet assesses your understanding of the mean value theorem and its uses.

Q1:

Stephanie is not convinced that the mean value theorem is true because, she says, the function is certainly differentiable on . But if we take and , we have , and yet there is no point where . What is her error?

AThe function is not continuous, and the theorem requires continuity on an interval.
BThe function should be strictly increasing on the interval.
CThe theorem requires the domain to be an interval, which is not.
DThe function should be strictly decreasing on the interval.
EThe theorem requires that the function be differentiable everywhere, which is not.

Q2:

Consider the result: If is differentiable on an interval and , then , a constant, for all .

Which of the following statements says exactly the same thing as the constant function result?

AIf is differentiable on an interval and at each , then is not a constant function.
BIf is differentiable on an interval and at some , then is not a constant function.
CIf is differentiable on an interval and not a constant function, then at some .
DIf is a constant function on an interval , then is differentiable and at all .
EIf is differentiable on an interval and not a constant function, then at all .

If is differentiable on an interval and not constant, we get points with . How does this show that at some point ?

Abecause then and by the mean value theorem, there is a point with
Bbecause if on , then would be a constant function
Cbecause then and by the mean value theorem, there is a point with
Dbecause only constant functions has everywhere
Ebecause means that one of or is not zero; we can take as this point

Q3:

Gabriel is not convinced that the mean value theorem is true because, he says, the function has the property that if we take and , we have , and yet there is no point where . What is his error?

AThe function is not continuous. The theorem requires continuity on an interval.
BThe function should be strictly increasing on the interval.
CThe function is not differentiable at . The theorem requires diferentiability on an interval.
DThe function should be strictly decreasing on the interval.
EThe theorem requires the domain to be an interval, which is not.

Q4:

Consider the statement that if is a differentiable function on an interval and there, then is strictly increasing on .

Which of the following statements is equivalent to the above?

AIf is differentiable on an interval and strictly increasing there, then at some point .
BIf is differentiable on an interval and strictly increasing there, then at all points .
CIf is differentiable on an interval and not strictly increasing there, then at some point .
DIf is differentiable on an interval and there, then is not strictly increasing on .
EIf is differentiable on an interval and not strictly increasing there, then at all points .

What does it mean for a function to not be strictly increasing on the interval ?

AWhenever satisfy , then .
BThere are points with but .
CThere are points with but .
DThere are points with but .
EThere is a point where .

Using the equivalent statement to the main result, how can you use the mean value theorem to prove the equivalent statement?

AIf is differentiable on and not strictly increasing, then take with . By the mean value theorem, we get between and where , so .
BIf is differentiable on and not strictly increasing, then take with . By the mean value theorem, we get between and where , so .
CIf is differentiable on and not strictly increasing, then take with . By the mean value theorem, we get between and and where , so .
DIf is differentiable on and not strictly increasing, then take with . By the mean value theorem, we get between and where , so .
EIt is not possible to prove the statement using the mean value theorem.