Worksheets · Calculus

Mean value theorem worksheet

Draw the straight line from one end of the curve to the other. Somewhere between the ends the curve runs parallel to it, and the theorem says so for every function that is smooth enough.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Mean Value Theorem

Date Period

Find each value of c, or say why the theorem does not apply.

  1. Apply the Mean Value Theorem

    f(x)=2x3−6x2−12x+69 on [−3,3]. Find the value of c guaranteed by the mean value theorem.
  2. Find the c guaranteed by the mean value theorem for f(x)=1x on [2,8].
  3. Find every c that the mean value theorem promises for f(x)=x+2x on [4,9].
  4. f(x)=−2x3−6x2−3x−13 on [−3,0]. Find the value of c guaranteed by the mean value theorem.
  5. Find the c guaranteed by the mean value theorem for f(x)=x on [0,9].
  6. Find every c that the mean value theorem promises for f(x)=x3+2x2−4x on [0,3].
  7. f(x)=x3−3x2+5x+7 on [−1,2]. Find the value of c guaranteed by the mean value theorem.
  8. f is differentiable, f(1)=−8 and f(4)=−20. Must there be a c in (1,4) with f′(c)=−4? Why?
  9. Find every c that the mean value theorem promises for f(x)=x3−x2+2x on [−2,0].
  10. f(x)=−x3+15x2−59x+40 on [1,7]. Find the value of c guaranteed by the mean value theorem.
  11. f is differentiable, f(3)=9 and f(8)=−16. Must there be a c in (3,8) with f′(c)=−5? Why?
  12. Find every c that the mean value theorem promises for f(x)=x3−2x on [−3,−1].
  13. f(x)=−2x3+18x2−29x−47 on [−1,5]. Find the value of c guaranteed by the mean value theorem.
  14. f is differentiable, f(1)=6 and f(4)=18. Must there be a c in (1,4) with f′(c)=4? Why?
  15. Find every c that the mean value theorem promises for f(x)=x3+x on [−2,2].

  1. Decide if the Mean Value Theorem applies, and if it does, find cc

    Does the Mean Value Theorem apply to f(x)=1x−1 on [−1,4]? If so, find every c.
  2. Does the Mean Value Theorem apply to f(x)=x2−x−2 on [0,6]? If so, find every c.
  3. Does the Mean Value Theorem apply to f(x)=∣x+1∣ on [−2,2]? If so, find every c.
  4. Does the Mean Value Theorem apply to f(x)=1x−2 on [1,3]? If so, find every c.
  5. Does the Mean Value Theorem apply to f(x)=∣x∣ on [−1,3]? If so, find every c.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Mean Value TheoremAnswer keyVersion 1

Date Period

Find each value of c, or say why the theorem does not apply.

  1. Apply the Mean Value Theorem

    f(x)=2x3−6x2−12x+69 on [−3,3]. Find the value of c guaranteed by the mean value theorem.c=−1
  2. Find the c guaranteed by the mean value theorem for f(x)=1x on [2,8].c=4
  3. Find every c that the mean value theorem promises for f(x)=x+2x on [4,9].c=6
  4. f(x)=−2x3−6x2−3x−13 on [−3,0]. Find the value of c guaranteed by the mean value theorem.c=−2
  5. Find the c guaranteed by the mean value theorem for f(x)=x on [0,9].c=94
  6. Find every c that the mean value theorem promises for f(x)=x3+2x2−4x on [0,3].c=53
  7. f(x)=x3−3x2+5x+7 on [−1,2]. Find the value of c guaranteed by the mean value theorem.c=0
  8. f is differentiable, f(1)=−8 and f(4)=−20. Must there be a c in (1,4) with f′(c)=−4? Why?Yes, by the mean value theorem
  9. Find every c that the mean value theorem promises for f(x)=x3−x2+2x on [−2,0].c=1−193
  10. f(x)=−x3+15x2−59x+40 on [1,7]. Find the value of c guaranteed by the mean value theorem.c=3
  11. f is differentiable, f(3)=9 and f(8)=−16. Must there be a c in (3,8) with f′(c)=−5? Why?Yes, by the mean value theorem
  12. Find every c that the mean value theorem promises for f(x)=x3−2x on [−3,−1].c=−393
  13. f(x)=−2x3+18x2−29x−47 on [−1,5]. Find the value of c guaranteed by the mean value theorem.c=1
  14. f is differentiable, f(1)=6 and f(4)=18. Must there be a c in (1,4) with f′(c)=4? Why?Yes, by the mean value theorem
  15. Find every c that the mean value theorem promises for f(x)=x3+x on [−2,2].c=−233 or c=233

  1. Decide if the Mean Value Theorem applies, and if it does, find cc

    Does the Mean Value Theorem apply to f(x)=1x−1 on [−1,4]? If so, find every c.No: f is not continuous at x=1
  2. Does the Mean Value Theorem apply to f(x)=x2−x−2 on [0,6]? If so, find every c.Yes: c=3
  3. Does the Mean Value Theorem apply to f(x)=∣x+1∣ on [−2,2]? If so, find every c.No: f is not differentiable at x=−1
  4. Does the Mean Value Theorem apply to f(x)=1x−2 on [1,3]? If so, find every c.No: f is not continuous at x=2
  5. Does the Mean Value Theorem apply to f(x)=∣x∣ on [−1,3]? If so, find every c.No: f is not differentiable at x=0

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Find the value of c for f(x)=x2 on [0,4]

  1. The line across goes from 0 to 16 over a width of 4, so its slope is 4.
  2. The derivative is 2x, which is 4 at x equals 2.
  3. So c is 2, which is the midpoint, as it always is for a parabola.

The answer is c=2.

Where students go wrong

Setting the derivative to zero. That is what Rolle asks for, and Rolle is the special case where the line across happens to be flat; in general the line has a slope and the derivative has to match it.

Also called MVT, find c or average rate of change equals instantaneous.

What you can put on this worksheet

The mean value theorem
f(x)=3x2+1 on [−6,0]. Find the value of c guaranteed by the mean value theorem. → c=−3
Does it apply? If so, find c
Does the Mean Value Theorem apply to f(x)=∣x∣ on [−3,1]? If so, find every c. → No: f is not differentiable at x=0
Roots, quotients, trig, and a table
f is differentiable, f(3)=−8 and f(8)=−23. Must there be a c in (3,8) with f′(c)=−3? Why? → Yes, by the mean value theorem
Two values of c, quotients and roots
Find every c that the mean value theorem promises for f(x)=x3+3x on [−2,2]. → c=−233 or c=233

Questions about these worksheets

Yes. Every mean value theorem sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the mean value theorem questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 4: the mean value theorem, does it apply? if so, find c, roots, quotients, trig, and a table and two values of c, quotients and roots. Tick as many as you want and set how many of each, or let it spread them evenly.

Setting the derivative to zero. That is what Rolle asks for, and Rolle is the special case where the line across happens to be flat; in general the line has a slope and the derivative has to match it.