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Average Value of a Function Practice Problems

Integrate the function over the interval, then divide by the interval's length b − a. For the mean value theorem problems, set f(c) equal to the average value and solve for c in the interval. Exact answers like 2/pi or e - 1 are accepted, as are decimals.

  1. 1.

    Find the average value of on .

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    . Divide by : the average value is .

  2. 2.

    Find the average value of on .

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    . Divide by 6: the average value is .

  3. 3.

    Find the average value of on .

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    . Divide by : the average value is .

  4. 4.

    Find the average value of on .

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    . Divide by 4: .

  5. 5.

    Find the average value of on .

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    . Divide by 4: .

  6. 6.

    Find the average value of on .

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    . Divide by : .

  7. 7.

    Find the average value of on .

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    . The length is 1, so the average value is .

  8. 8.

    Find the average value of on .

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    , since is odd and the interval is symmetric. The average value is .

  9. 9.

    Find the average value of on .

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    . Divide by : .

  10. 10.

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

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    , so . Solve : .

  11. 11.

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

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    From problem 2, . Solve : . Only lies in .

  12. 12.

    The temperature hours after 6 a.m. is degrees. Find the average temperature from 6 a.m. to 6 p.m., to two decimal places.

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    The interval is . , so degrees.
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Average Value of a Function Practice Problems

These problems follow the average value of a function lesson. Problems 1–9 ask for an average value; 10 and 11 ask for the from the mean value theorem for integrals; 12 is a word problem.

The method

  1. Evaluate .
  2. Divide by the length of the interval, .
  3. For the mean value theorem for integrals, set equal to that average and keep the solutions in .

Common mistakes to watch for

  • Not dividing by : the integral alone is an area.
  • Dividing by when .
  • Averaging the endpoint values , which only works for straight lines.

Frequently Asked Questions

Do I have to give an exact answer?+

No. Exact forms such as 2/pi, e - 1 or 2sqrt(3) are accepted, and so are decimals rounded to three places.

Why is my answer too big by a factor?+

The most common slip is forgetting to divide the integral by b - a, or dividing by b instead of b - a. On [1, 3] the length is 2, not 3.

More practice problems

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