Average value of a function worksheet

Flatten the function until it is a straight line that shuts in the same area. That height is its average value, and it is the integral divided by the width of the interval.

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Pick more than one for a sheet that mixes them.

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Sigma Prep · sigmaprep.io/worksheets Name Average Value of a Function Date Period Find each average value, or the c asked for. Find the average value of a function Find the average value of $f (x) = - 3 x^{2} + 5 x$ on $[2, 3]$ Find the average value of $f (x) = 2 x^{2} - x$ on $[1, 2]$ Find the average value of $f (x) = - x^{2} + 4 x - 1$ on $[0, 1]$ Find the average value of $f (x) = \frac{9}{x^{2}}$ on $[1, 4]$ Find the average value of $f (x) = 2 x^{2} - x + 6$ on $[3, 7]$ Find the average value of $f (x) = {(3 x - 1)}^{2}$ on $[0, 1]$ Find the average value of $f (x) = 3 x^{2} - 5 x + 5$ on $[2, 4]$ Find the average value of $f (x) = \sin x$ on $[0, π]$ Find the average value of $f (x) = - 2 x^{2} - 5 x + 3$ on $[0, 1]$ Find the average value of $f (x) = 3 \cos (2 x)$ on $[0, \frac{π}{4}]$ Find average values beyond polynomials Find the average value of $f (x) = 5 \cos x$ on $[0, \frac{π}{2}]$ Find the average value of $f (x) = 4 \sqrt{x}$ on $[0, 16]$ Find the average value of $f (x) = \frac{5}{x}$ on $[1, e]$ Find the average value of $f (x) = 2 \sin x$ on $[0, π]$ Find the average value of $f (x) = \sqrt{x}$ on $[0, 9]$ Find where $f$ equals its average value Find $c$ in $[0, \ln 3]$ given by the Mean Value Theorem for Integrals for $f (x) = e^{x}$ . Find $c$ in $[1, 4]$ where $f (x) = x^{2}$ equals its average value. Find $c$ in $[0, 2]$ such that $f (c)$ is the average value of $f (x) = \sqrt[3]{x}$ on $[0, 2]$ . Find $c$ in $[2, 5]$ at which $f (x) = x^{2}$ takes on its average value over the interval. Find $c$ in $[0, 6]$ at which $f (x) = \sqrt{x}$ takes on its average value over the interval. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Average Value of a Function

Date Period

Find each average value, or the c asked for.

Find the average value of a function
Find the average value of $f (x) = - 3 x^{2} + 5 x$ on $[2, 3]$
Find the average value of $f (x) = 2 x^{2} - x$ on $[1, 2]$
Find the average value of $f (x) = - x^{2} + 4 x - 1$ on $[0, 1]$
Find the average value of $f (x) = \frac{9}{x^{2}}$ on $[1, 4]$
Find the average value of $f (x) = 2 x^{2} - x + 6$ on $[3, 7]$
Find the average value of $f (x) = {(3 x - 1)}^{2}$ on $[0, 1]$
Find the average value of $f (x) = 3 x^{2} - 5 x + 5$ on $[2, 4]$
Find the average value of $f (x) = \sin x$ on $[0, π]$
Find the average value of $f (x) = - 2 x^{2} - 5 x + 3$ on $[0, 1]$
Find the average value of $f (x) = 3 \cos (2 x)$ on $[0, \frac{π}{4}]$

Find average values beyond polynomials
Find the average value of $f (x) = 5 \cos x$ on $[0, \frac{π}{2}]$
Find the average value of $f (x) = 4 \sqrt{x}$ on $[0, 16]$
Find the average value of $f (x) = \frac{5}{x}$ on $[1, e]$
Find the average value of $f (x) = 2 \sin x$ on $[0, π]$
Find the average value of $f (x) = \sqrt{x}$ on $[0, 9]$

Find where $f$ equals its average value
Find $c$ in $[0, \ln 3]$ given by the Mean Value Theorem for Integrals for $f (x) = e^{x}$ .
Find $c$ in $[1, 4]$ where $f (x) = x^{2}$ equals its average value.
Find $c$ in $[0, 2]$ such that $f (c)$ is the average value of $f (x) = \sqrt[3]{x}$ on $[0, 2]$ .
Find $c$ in $[2, 5]$ at which $f (x) = x^{2}$ takes on its average value over the interval.
Find $c$ in $[0, 6]$ at which $f (x) = \sqrt{x}$ takes on its average value over the interval.



Sigma Prep · sigmaprep.io/worksheets Name Average Value of a FunctionAnswer keyVersion 1 Date Period Find each average value, or the c asked for. Find the average value of a function Find the average value of $f (x) = - 3 x^{2} + 5 x$ on $[2, 3]$ $- \frac{13}{2}$ Find the average value of $f (x) = 2 x^{2} - x$ on $[1, 2]$ $\frac{19}{6}$ Find the average value of $f (x) = - x^{2} + 4 x - 1$ on $[0, 1]$ $\frac{2}{3}$ Find the average value of $f (x) = \frac{9}{x^{2}}$ on $[1, 4]$ $\frac{9}{4}$ Find the average value of $f (x) = 2 x^{2} - x + 6$ on $[3, 7]$ $\frac{161}{3}$ Find the average value of $f (x) = {(3 x - 1)}^{2}$ on $[0, 1]$ $1$ Find the average value of $f (x) = 3 x^{2} - 5 x + 5$ on $[2, 4]$ $18$ Find the average value of $f (x) = \sin x$ on $[0, π]$ $\frac{2}{π}$ Find the average value of $f (x) = - 2 x^{2} - 5 x + 3$ on $[0, 1]$ $- \frac{1}{6}$ Find the average value of $f (x) = 3 \cos (2 x)$ on $[0, \frac{π}{4}]$ $\frac{6}{π}$ Find average values beyond polynomials Find the average value of $f (x) = 5 \cos x$ on $[0, \frac{π}{2}]$ $\frac{10}{π}$ Find the average value of $f (x) = 4 \sqrt{x}$ on $[0, 16]$ $\frac{32}{3}$ Find the average value of $f (x) = \frac{5}{x}$ on $[1, e]$ $\frac{5}{e - 1}$ Find the average value of $f (x) = 2 \sin x$ on $[0, π]$ $\frac{4}{π}$ Find the average value of $f (x) = \sqrt{x}$ on $[0, 9]$ $2$ Find where $f$ equals its average value Find $c$ in $[0, \ln 3]$ given by the Mean Value Theorem for Integrals for $f (x) = e^{x}$ . $c = \ln ⁣ (\frac{2}{\ln 3})$ Find $c$ in $[1, 4]$ where $f (x) = x^{2}$ equals its average value. $c = \sqrt{7}$ Find $c$ in $[0, 2]$ such that $f (c)$ is the average value of $f (x) = \sqrt[3]{x}$ on $[0, 2]$ . $c = \frac{27}{32}$ Find $c$ in $[2, 5]$ at which $f (x) = x^{2}$ takes on its average value over the interval. $c = \sqrt{13}$ Find $c$ in $[0, 6]$ at which $f (x) = \sqrt{x}$ takes on its average value over the interval. $c = \frac{8}{3}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Average Value of a FunctionAnswer keyVersion 1

Date Period

Find each average value, or the c asked for.

Find the average value of a function
Find the average value of $f (x) = - 3 x^{2} + 5 x$ on $[2, 3]$ $- \frac{13}{2}$
Find the average value of $f (x) = 2 x^{2} - x$ on $[1, 2]$ $\frac{19}{6}$
Find the average value of $f (x) = - x^{2} + 4 x - 1$ on $[0, 1]$ $\frac{2}{3}$
Find the average value of $f (x) = \frac{9}{x^{2}}$ on $[1, 4]$ $\frac{9}{4}$
Find the average value of $f (x) = 2 x^{2} - x + 6$ on $[3, 7]$ $\frac{161}{3}$
Find the average value of $f (x) = {(3 x - 1)}^{2}$ on $[0, 1]$ $1$
Find the average value of $f (x) = 3 x^{2} - 5 x + 5$ on $[2, 4]$ $18$
Find the average value of $f (x) = \sin x$ on $[0, π]$ $\frac{2}{π}$
Find the average value of $f (x) = - 2 x^{2} - 5 x + 3$ on $[0, 1]$ $- \frac{1}{6}$
Find the average value of $f (x) = 3 \cos (2 x)$ on $[0, \frac{π}{4}]$ $\frac{6}{π}$

Find average values beyond polynomials
Find the average value of $f (x) = 5 \cos x$ on $[0, \frac{π}{2}]$ $\frac{10}{π}$
Find the average value of $f (x) = 4 \sqrt{x}$ on $[0, 16]$ $\frac{32}{3}$
Find the average value of $f (x) = \frac{5}{x}$ on $[1, e]$ $\frac{5}{e - 1}$
Find the average value of $f (x) = 2 \sin x$ on $[0, π]$ $\frac{4}{π}$
Find the average value of $f (x) = \sqrt{x}$ on $[0, 9]$ $2$

Find where $f$ equals its average value
Find $c$ in $[0, \ln 3]$ given by the Mean Value Theorem for Integrals for $f (x) = e^{x}$ . $c = \ln ⁣ (\frac{2}{\ln 3})$
Find $c$ in $[1, 4]$ where $f (x) = x^{2}$ equals its average value. $c = \sqrt{7}$
Find $c$ in $[0, 2]$ such that $f (c)$ is the average value of $f (x) = \sqrt[3]{x}$ on $[0, 2]$ . $c = \frac{27}{32}$
Find $c$ in $[2, 5]$ at which $f (x) = x^{2}$ takes on its average value over the interval. $c = \sqrt{13}$
Find $c$ in $[0, 6]$ at which $f (x) = \sqrt{x}$ takes on its average value over the interval. $c = \frac{8}{3}$

How to do these

Find the average value of $f (x) = x^{2}$ on $[0, 3]$

The integral of x squared from 0 to 3 is 9.
The interval is 3 wide.
So the average value is 9 over 3, which is 3.

The answer is $3$ .

Where students go wrong

Forgetting to divide by the width. The integral is the area, not the height, and it only becomes an average once it is spread across the interval it came from.

Also called mean value theorem for integrals, average value or mean value of a function.

What you can put on this worksheet

Average value of a function: Find the average value of $f (x) = - 3 x + 3$ on $[2, 5]$ → $- \frac{15}{2}$
Trig, exponential and roots: Find the average value of $f (x) = 4 \sin x$ on $[0, π]$ → $\frac{8}{π}$
Find the c: Find $c$ in $[0, 4]$ at which $f (x) = 9 - 2 x$ takes on its average value over the interval. → $c = 2$
Quartics, negative ends, and a substitution: Find the average value of $f (x) = 3 \sqrt{x}$ on $[0, 9]$ → $6$

Questions about these worksheets

Yes. Every average value of a function 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 average value of a function 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: average value of a function, trig, exponential and roots, find the c and quartics, negative ends, and a substitution. Tick as many as you want and set how many of each, or let it spread them evenly.

Forgetting to divide by the width. The integral is the area, not the height, and it only becomes an average once it is spread across the interval it came from.

Average value of a function worksheet

Select your difficulty

Find the average value of a function

Find average values beyond polynomials

Find where fff equals its average value

Find the average value of a function

Find average values beyond polynomials

Find where fff equals its average value

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these average value of a function worksheets free?+

Do the average value of a function worksheets come with an answer key?+

Are these average value of a function worksheets printable?+

What grade are these average value of a function worksheets for?+

Can I get a different version of the same average value of a function worksheet?+

Is this a free alternative to Kuta Software for average value of a function?+

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What kinds of question are on a average value of a function worksheet?+

What do students get wrong with average value of a function?+

Find where $f$ equals its average value

Find where $f$ equals its average value