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Worksheet: The Mean Value Theorem and Its Interpretation

Q1:

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

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 a constant function on an interval , then is differentiable and at all .
  • BIf is differentiable on an interval and not a constant function, then 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 at each , then is not a constant function.

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 only constant functions has everywhere
  • Dbecause means that one of or is not zero; we can take as this point
  • Ebecause then and by the mean value theorem, there is a point with

Q3:

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 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.

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 there, then is not strictly increasing on .
  • 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 not strictly increasing there, then at some point .
  • 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 are points with but .
  • DThere is a point where .
  • EWhenever satisfy , then .

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 .
  • BIt is not possible to prove the statement using the mean value theorem.
  • 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 .
  • EIf is differentiable on and not strictly increasing, then take with . By the mean value theorem, we get between and where , so .