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Lesson: The Mean Value Theorem and Its Interpretation

In this lesson, we will learn how to use the mean value theorem.

Worksheet: The Mean Value Theorem and Its Interpretation • 4 Questions

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 theorem requires the domain to be an interval, which is not.
  • BThe function is not continuous, and the theorem requires continuity on an interval.
  • CThe theorem requires that the function be differentiable everywhere, which is not.
  • DThe function should be strictly increasing on the interval.
  • EThe function should be strictly decreasing on the 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 differentiable on an interval and not a constant function, then at some .
  • BIf is differentiable on an interval and at each , then is not a constant function.
  • CIf is differentiable on an interval and not a constant function, then at all .
  • DIf is differentiable on an interval and at some , then is not a constant function.
  • EIf is a constant function on an interval , then is differentiable and 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 then and by the mean value theorem, there is a point with
  • Cbecause means that one of or is not zero; we can take as this point
  • Dbecause if on , then would be a constant function
  • Ebecause only constant functions has everywhere

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