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Mean Value Theorem Practice Problems

Find the average rate of change on the interval, set f′(c) equal to it, and keep only the solutions strictly between the endpoints. Each problem uses a function that meets the theorem's conditions.

  1. 1.

    Find the value of guaranteed by the mean value theorem for on .

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    The average rate of change is . Then , so .

  2. 2.

    Find for on .

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    and , so the average rate is . Then gives .

  3. 3.

    For on , the mean value theorem says equals some number for a in . What is that number?

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    equals the average rate of change: .

  4. 4.

    Find for on . A fraction or a decimal is fine.

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    and , so the average rate is . Then gives .

  5. 5.

    Find for on . You can type sqrt( ).

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    The average rate is . Then , so . Only is in .

  6. 6.

    Find for on .

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    is continuous on and differentiable on . The average rate is . Then , so and .

  7. 7.

    Find for on .

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    The average rate is . Then , so and (the root is outside the interval).

  8. 8.

    Find every for on . Separate them with commas; you can type sqrt( ).

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    and , so the average rate is . Then , so and . Both are in .

  9. 9.

    Find for on . You can type e.

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    The average rate is . Then , so .

  10. 10.

    Find for on . Give a decimal rounded to 3 places.

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    The average rate is . Then , so .

  11. 11.

    Suppose and for all . What is the largest possible value of ?

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    By the mean value theorem, for some in . Since , .

  12. 12.

    Find for on . Give a decimal rounded to 3 places.

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    The average rate is . Then , so .
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Mean Value Theorem Practice Problems

These problems follow the mean value theorem lesson. Most ask for the value of where the tangent slope equals the average slope; the last few use the theorem to bound a function.

The method

  1. Check that is continuous on and differentiable on .
  2. Compute the average rate of change .
  3. Set equal to that number and solve for .
  4. Keep only the solutions with .

Common mistakes to watch for

  • Setting instead of equal to the average rate.
  • Keeping a solution outside the interval, such as on .
  • Flipping the fraction: the change in goes on top.

Frequently Asked Questions

How do I find c in a mean value theorem problem?+

Compute the average rate of change (f(b) − f(a)) / (b − a), set f′(c) equal to it, and solve for c. Keep only the solutions strictly between a and b.

How should I type my answers?+

Exact forms like sqrt(3), 2/sqrt(3), 1/2 and e - 1 are accepted, and so are decimals rounded as the question asks. For more than one value, separate them with commas in any order.

More practice problems

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