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Lesson Worksheet: The Mean Value Theorem Mathematics • Higher Education

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

Q1:

Mason is not convinced that the mean value theorem is true because, he says, the function ๐‘“(๐‘ฅ)=|๐‘ฅ| has the property that if we take ๐‘Ž=โˆ’2 and ๐‘=2, we have ๐‘“(๐‘)โˆ’๐‘“(๐‘Ž)๐‘โˆ’๐‘Ž=0, and yet there is no point ๐‘ฅ where ๐‘“โ€ฒ(๐‘ฅ)=0. What is his error?

  • AThe function is not differentiable at ๐‘ฅ=0. The theorem requires diferentiability on an interval.
  • BThe theorem requires the domain to be an interval, which โ„ 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 ๐‘“(๐‘ฅ)=|๐‘ฅ| is certainly differentiable on โ„โˆ’{0}. But if we take ๐‘Ž=โˆ’1 and ๐‘=1, we have ๐‘“(๐‘)โˆ’๐‘“(๐‘Ž)๐‘โˆ’๐‘Ž=0, and yet there is no point ๐‘ฅ where ๐‘“โ€ฒ(๐‘ฅ)=0. What is her error?

  • AThe function should be continuous on the interval.
  • BThe theorem requires the domain to be an interval, which โ„โˆ’{0} 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 ๐‘ฆ=3+|2๐‘ฅ| over the interval [โˆ’2,2]?

  • AYes
  • BNo

Q4:

Does the mean value theorem apply for the function ๐‘ฆ=3๐‘ฅ2๐œ‹๐‘ฅsin over the interval [0,2]?

  • ANo
  • BYes

Q5:

Does the mean value theorem apply for the function ๐‘ฆ=๐‘ฅ๐‘ฅ+1cos๏Ž„๏Šจ over the interval [0,1]?

  • AYes
  • BNo

Q6:

Does the mean value theorem apply for the function ๐‘ฆ=2๐‘ฅโˆ’4๐‘ฅ+7๏Šฉ over the interval [0,5]?

  • AYes
  • BNo

Q7:

Does the mean value theorem apply for the function ๐‘ฆ=2โˆš๐‘ฅโˆ’4๏Šจ over the interval [โˆ’2,2]?

  • ANo
  • BYes

Q8:

Does the mean value theorem apply for the function ๐‘ฆ=23๐œ‹๐‘ฅtan over the interval [1,3]?

  • AYes
  • BNo

Q9:

For the function ๐‘“(๐‘ฅ)=(๐‘ฅโˆ’1)๏Šฎ, find all the possible values of ๐‘ that satisfy the mean value theorem over the interval [0,2].

Q10:

For the function ๐‘“(๐‘ฅ)=โˆš๐‘ฅ, find all the possible values of ๐‘ that satisfy the mean value theorem over the interval [4,9].

  • A๏„ž52
  • B254
  • C15
  • D0
  • E52

This lesson includes 11 additional questions for subscribers.

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