Average Value of a Function

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 6.5 – Average Value of a Function

Expected Educational Results

• Objective 13–1: I can compute the average value of a function on $[a,b]$$[a,b]$.
• Objective 13–2: I can use the Mean Value Theorem for Integrals to find a value $c\in [a,b]$$c\in [a,b]$ such that $f(c)=f_{\text{average}}$$f(c)=f_{\text{average}}$.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Average Value of a Function

Definition: Average Value of a Function

The average value of $f(x)$$f(x)$ on $[a,b]$$[a,b]$ is $f_{\text{average}}={\displaystyle \frac{1}{b-a}}{\int }_a^{b}f(x)dx$$f_{\text{average}}=\dfrac{1}{b-a}\int_a^b{f(x)\,dx}$

Example 01: Find the average value of $f(x)=x^2+3x+6$$f(x)=x^2+3x+6$ on $[-2,2]$$[-2,2]$.

Solution 01:

$f_{\text{ave}}={\displaystyle \frac{1}{2-(-2)}}{\int }_{-2}^2(x^2+3x+6)dx={\displaystyle \frac{1}{4}}{({\displaystyle \frac{1}{3}}{x}^3+{\displaystyle \frac{3}{2}}{x}^2+6x)}_{-2}^2={\displaystyle \frac{1}{4}}\left({\displaystyle \frac{88}{3}}\right)={\displaystyle \frac{22}{3}}$$f_{\text{ave}}=\dfrac{1}{2-(-2)}\int_{-2}^2{(x^2+3x+6)\,dx}=\dfrac{1}{4}\left(\dfrac{1}{3}x^3+\dfrac{3}{2}x^2+6x\right)_{-2}^2=\dfrac{1}{4}\left(\dfrac{88}{3}\right)=\dfrac{22}{3}$

Check Your Work

Use Technology to Verify Average Value

NOTE: The use of technology in this activity is to verify, i.e., check, your computation of average value. You must show work on assessments to receive credit for all Calculus II work.

Mathematica

To check the answer in Example 01:

1
(* Find the average value of x^2+3x+6 from x=-2 to x=2 *)
2
Integrate[x^2 + 3x + 6,{x, -2, 2}]/(2-(-2))

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. Integrate[ ] has two arguments:

   1. The function in the integrand of the average value integral;

   2. $\{x,a,b\}$$\{x,a,b\}$, where

      1. $x$$x$ is the independent variable
      2. $a$$a$ is the lower limit of integration, i.e., the left-hand endpoint of the interval;
      3. $b$$b$ is the upper limit of integration, i.e., the right-hand endpoint of the interval.

3. For help on using the Integrate[ ] function:

   1. In Mathematica, execute the code: $\text{?Integrate}$$\text{?Integrate}$
   2. Click on $\vee$$\or$ near the bottom-left of output
   3. Click on local
   4. Read how to use the Integrate[ ] function – you will be able to copy-paste code.

4. You may need parens, $($$(\,$ and $)$$\,)$, to group multiple terms in the numerator and denominator.

5. Remember, correct Mathematica code will be all black except for variables.

6. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Investigation 01

1. Find the average value of $y=\frac{1}{x}$$y=\frac{1}{x}$ on $[1,4]$$[1,4]$.
2. Find the average value of $y=\sin (2x)$$y=\sin{(2x)}$ on $[0,\frac{\pi }{4}]$$[0,\frac{\pi}{4}]$.
3. Find the average value of $y=2^x$$y=2^x$ on $[0,2]$$[0,2]$.
4. Find the average value of $y=(x-3{)}^2$$y=(x-3)^2$ on $[2,5]$$[2,5]$.
5. Find the average value of $y=x\cos (3x)$$y=x\cos{(3x)}$ on $[0,2\pi ]$$[0,2\pi]$.
6. Find the average value of $y=\ln (x)$$y=\ln{(x)}$ on $[1,10]$$[1,10]$.
7. Find the average value of $y=\frac{1}{1+x^2}$$y=\frac{1}{1+x^2}$ on $[-1,1]$$[-1,1]$.
8. Find the average value of $y={\tan }^{-1}(x)$$y=\tan^{-1}{(x)}$ on $[0,5]$$[0,5]$.

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Last Modified: Thursday, 8 October 2020 7:40 EDT