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:
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.
- 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