# Worksheet: The Mean Value Theorem

In this worksheet, we will practice interpreting and using the mean value theorem and Rolle’s theorem.

Q1:

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

Q2:

Madison 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 should be continuous on the interval.
• BThe theorem requires the domain to be an interval, which is not.
• CThe function should be strictly increasing on the interval.
• DThe theorem requires that the function be differentiable on its domain.
• EThe function should be strictly decreasing on the interval.

Q3:

Does the mean value theorem apply for the function over the interval ?

• ANo
• BYes

Q4:

Does the mean value theorem apply to the function over the interval ?

• ANo
• BYes

Q5:

Does the mean value theorem apply for the function over the interval ?

• AYes
• BNo

Q6:

Does the mean value theorem apply for the function over the interval ?

• AYes
• BNo

Q7:

For the function , find all the possible values of that satisfy the mean value theorem over the interval .

• A
• B
• C
• D0
• E

Q8:

For the function , find all the values of that satisfy the mean value theorem over the interval .

Q9:

Does the mean value theorem apply for the function over the interval ?

• ANo
• BYes

Q10:

For the function , find all the possible values of that satisfy the mean value theorem over the interval .

• A2
• B
• C12
• D
• E0

Q11:

For the function , find all the possible values of that satisfy the mean value theorem over the interval .

• A,
• B,
• C,
• D
• E

Q12:

For the function , find all the possible values of that satisfy the mean value theorem over the interval .

Q13:

A rock is dropped from a height of 81 ft. Its position seconds after it is dropped until it hits the ground is given by the function .

Determine how long it will take for the rock to hit the ground.

• A0 s
• B s
• C s
• D s
• E s

Find the average velocity, , of the rock from the point of release until it hits the ground.

Find the time according to the mean value theorem when the instantaneous velocity of the rock is .

• A s
• B s
• C s
• D0 s
• E s

Q14:

Does the mean value theorem apply for the function over the interval ?

• AYes
• BNo

Q15:

Does the mean value theorem apply for the function over the interval ?

• AYes
• BNo

Q16:

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

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

• AThere are points with but .
• BThere are points with but .
• CThere is a point where .
• DWhenever satisfy , then .
• EThere are points with but .

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 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 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.

Q17:

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 a constant function on an interval , then is differentiable and at all .
• CIf is differentiable on an interval and at some , then is not a constant function.
• DIf is differentiable on an interval and not a constant function, then at some .
• 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 means that one of or is not zero; we can take as this point
• Cbecause only constant functions has everywhere
• Dbecause then and by the mean value theorem, there is a point with
• Ebecause if on , then would be a constant function