Mean Value Theorem - Worksheets

Oct 8, 2026

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Mean Value Theorem worksheet problems from Infinite Calculus, asking students to find values of c that satisfy the Mean Value Theorem for given functions on specified intervals. It includes tasks to decide whether the theorem can be applied by checking continuity and differentiability on the interval or interior. One critical thinking problem uses the theorem to prove an inequality involving sin a and sin b for real numbers a and b with a ≠ b, guiding learners through the MVT framework and bounding with properties of cos.

LanguageEnglish

CategoryExam

FormatPDF

Pages4

Reading Time10 min

AccessFree to download

Mean Value Theorem Problems

Find c values

Determine applicability

Critical thinking: sin a - sin b inequality

Kuta Software-Infinite Calculus Name Mean Value Theorem Date Period For each problem, find the values of c that satisfy the Mean Value Theorem. 1) y = −x2 + 8x − 17; [3, 6] 2) y = x 3 − 9x2 + 24x − 18; [2, 4] 3) y = − ~~ ~~x22~~ ~~ + x − ~~ ~~12~~ ~~ ; [−2, 1] 4) y = x22~~ ~~ − 2x − 1; [−1, 1] 5) y = x 3 + 3x2 − 2; [−2, 0] 6) y = −x3 + 4x2 − 3; [0, 4] x 2 − 9 x 2 7) y = ; [1, 4] 8) y = ; [−4, 1] 3x 2x − 4 1 1 9) y = −(−2x + 6)2 ; [−2, 3] 10) y = −(−5x + 25)2 ; [3, 5] For each problem, determine if the Mean Value Theorem can be applied. If it can, find all values of c that satisfy the theorem. If it cannot, explain why not. 11) y = − ~~ ~~4xx~~ ~~8~~ ~~ ; [−3, −1] 12) y = −x24~~ ~~9~~ ~~ ; [1, 3] 2 2 13) y = −(6x + 24) 3 ; [−4, −1] 14) y = (x − 3) 3 ; [1, 4] Critical thinking question: 15) Use the Mean Value Theorem to prove that sin a − sin b ≤ a − b for all real values of a and b where a ≠ b. Kuta Software-Infinite Calculus Name Mean Value Theorem Date Period For each problem, find the values of c that satisfy the Mean Value Theorem. 1) y = −x2 + 8x − 17; [3, 6] {~~ ~~92~~ ~~ } 2) y = x 3 − 9x2 + 24x − 18; [2, 4] {~~ ~~9~~ ~~+3~~ ~~3~~ ~~ , ~~ ~~9~~ ~~−3~~ ~~3~~ ~~ } 3) y = − ~~ ~~x22~~ ~~ + x − ~~ ~~12~~ ~~ ; [−2, 1] 4) y = x22~~ ~~ − 2x − 1; [−1, 1] {− ~~ ~~12~~ ~~ } {0} 5) y = x 3 + 3x2 − 2; [−2, 0] {~~ ~~−3~~ ~~~~ ~~3~~ ~~ , −3~~ ~~−3~~ ~~3~~ ~~ } 7) y = x~~ ~~23−x~~ ~~9~~ ~~ ; [1, 4] {2} 6) y = −x3 + 4x2 − 3; [0, 4] {~~ ~~83~~ ~~ } 8) y = 2xx~~ ~~4~~ ~~ ; [−4, 1] {2 − 6 } 1 9) y = −(−2x + 6)2 ; [−2, 3] {~~ ~~74~~ ~~ } 1 10) y = −(−5x + 25)2 ; [3, 5] {~~ ~~92~~ ~~ } For each problem, determine if the Mean Value Theorem can be applied. If it can, find all values of c that satisfy the theorem. If it cannot, explain why not. 11) y = − ~~ ~~4xx~~ ~~8~~ ~~ ; [−3, −1] 12) y = −x24~~ ~~9~~ ~~ ; [1, 3] The function is not continuous on [−3, −1] { 3 } 2 2 13) y = −(6x + 24) 3 ; [−4, −1] 14) y = (x − 3) 3 ; [1, 4] {− ~~ ~~289~~ ~~ } The function is not differentiable on (1, 4) Critical thinking question: 15) Use the Mean Value Theorem to prove that sin a − sin b ≤ a − b for all real values of a and b where a ≠ b. Let f (x) = sin x. Use the interval [a,b] . By the MVT, we know that there is at least one c sin b − sin a sin b − sin a such that b − a = cos c. We know cos c ≤ 1 for all c. Therefore, b − a ≤ 1, ~~ ~~~~ ~~sin~~ ~~aasinb~~ ~~b~~ ~~ ≤ 1, and sin a − sin b ≤ a − b . Create your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)

FAQs

What does the worksheet ask you to find for the first set of problems?

For each problem, you find the values of c that satisfy the Mean Value Theorem on the given interval.

How do you decide whether the Mean Value Theorem can be applied?

Check whether the function is continuous on the closed interval and differentiable on the open interval; if not, the worksheet indicates why the theorem cannot be applied.

How does the critical thinking problem use the Mean Value Theorem?

It sets f(x)=sin x on the interval [a,b], applies the MVT to obtain a relationship involving cos c, and then uses cos c ≤ 1 to derive sin a − sin b ≤ a − b for a ≠ b.

Oct 8, 2026

Upload by Angel