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

**Q1: **

Nader 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 should be strictly decreasing 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 function is not differentiable at . The theorem requires diferentiability on an interval.
- EThe function is not continuous. The theorem requires continuity on an interval.

**Q2: **

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

**Q3: **

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

- AYes
- BNo

**Q4: **

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

- AYes
- BNo

**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 ?

- ANo
- BYes

**Q7: **

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

- A
- B0
- C
- D
- 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 ?

- AYes
- BNo

**Q10: **

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

- A2
- B12
- C
- 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: **

**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.

- A s
- B 0 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
- D 0 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 ?

- ANo
- BYes