AP®MathsCollege BoardCalculus ABStudy GuidesUnit 8: Applications of IntegrationDefinite Integrals in ContextAverage Value of a Function

Average Value of a Function (College Board AP® Calculus AB): Study Guide

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

College Board Calculus ABCollege Board Calculus BC

Average Value of a Function

What is the average value of a function?

  • If f is a continuous function, then the average value of f over the interval open square brackets a comma space b close square brackets is

    • average value of f on open square brackets a comma space b close square bracketsequals fraction numerator 1 over denominator b minus a end fraction integral subscript a superscript b f open parentheses x close parentheses space d x

  • The average value of a function will be a number k

    • where k equals f open parentheses c close parentheses for some c in open square brackets a comma space b close square brackets

      • and such that k times open parentheses b minus a close parentheses equals integral subscript a superscript b f open parentheses x close parentheses space d x

    • This result is referred to as the mean value theorem for integrals

  • This means that the constant function g defined by g open parentheses x close parentheses equals k

    • will represent the same accumulation of change as f between x equals a and x equals b

      • Because integral subscript a superscript b k space d x equals k open square brackets x close square brackets subscript a superscript b equals k open parentheses b minus a close parentheses

  • This can also be interpreted geometrically, as seen in the following diagram

Graph showing the Mean Value Theorem for integrals, with a curve y=f(x) and a horizontal line y=k. Two shaded areas are shown with equal areas. Text explains the theorem.

Examiner Tips and Tricks

Remember that you can't talk about the 'average value of a function' in general

  • The average value is only defined for a particular interval open square brackets a comma space b close square brackets

  • The average value will usually be different for different intervals

Examiner Tips and Tricks

Do not confuse average value (this section) with average rate of change (Unit 2). They are different quantities:

  • Average value of f on left square bracket a comma space b right square bracket is fraction numerator 1 over denominator b minus a end fraction integral subscript a superscript b f left parenthesis x right parenthesis   text d end text x(an integral)

  • Average rate of change of f on left square bracket a comma space b right square bracket is fraction numerator f left parenthesis b right parenthesis minus f left parenthesis a right parenthesis over denominator b minus a end fraction (a difference quotient)

AP FRQs sometimes ask both in the same problem to test that you can tell them apart.

Worked Example

Let f be the function defined by f open parentheses x close parentheses equals sin x.

Calculate the average value of f over the interval open square brackets 0 comma space pi close square brackets.

Answer:

Use average value of f on open square brackets a comma space b close square bracketsequals fraction numerator 1 over denominator b minus a end fraction integral subscript a superscript b f open parentheses x close parentheses space d x

table row cell average space value end cell equals cell fraction numerator 1 over denominator pi minus 0 end fraction integral subscript 0 superscript pi sin x space d x end cell row blank equals cell 1 over pi open square brackets negative cos x close square brackets subscript 0 superscript pi end cell row blank equals cell 1 over pi open parentheses negative cos open parentheses pi close parentheses minus open parentheses negative cos open parentheses 0 close parentheses close parentheses close parentheses end cell row blank equals cell 1 over pi open parentheses negative open parentheses negative 1 close parentheses minus open parentheses negative 1 close parentheses close parentheses end cell row blank equals cell 1 over pi open parentheses 1 plus 1 close parentheses end cell row blank equals cell 2 over pi end cell end table

Average value equals 2 over pi

Examiner Tips and Tricks

Always write the setup in an FRQ, such as table row blank blank cell fraction numerator 1 over denominator pi minus 0 end fraction integral subscript 0 superscript pi sin x space d x end cell end table in the example above.

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Author
Reviewer
Roger B

Author: Roger B

Expertise: Maths Content Creator

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Subject Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.