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Mean Value Theorem (College Board AP® Calculus AB)

Study Guide

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Mean value theorem

What is the mean value theorem?

  • The mean value theorem is an important result in calculus

  • It states that:

    • If a function f is continuous over the closed interval open square brackets a comma space b close square brackets

      • and differentiable over the open interval open parentheses a comma space b close parentheses

    • Then there exists a value x equals c in the interval open parentheses a comma space b close parentheses

      • such that f to the power of apostrophe open parentheses c close parentheses equals fraction numerator f open parentheses b close parentheses minus f open parentheses a close parentheses over denominator b minus a end fraction

    • I.e. that there will be a point within that open interval open parentheses a comma space b close parentheses

      • where the instantaneous rate of change f to the power of apostrophe open parentheses c close parentheses

      • is equal to the average rate of change over the interval

A graph of a function with tangent and average rate of change lines illustrating the mean value theorem
An illustration of the mean value theorem

Examiner Tips and Tricks

When using the mean value theorem on the exam

  • Be sure to justify that the theorem is valid

    • I.e. that the function is continuous on open square brackets a comma space b close square brackets

    • and differentiable on open parentheses a comma space b close parentheses

  • Remember that if a function is differentiable on an interval

    • then it is also continuous on that interval

Worked Example

A social sciences researcher is using a function m to model the total mass of all the garden gnomes appearing on lawns in a particular neighborhood at time t. The function m is twice-differentiable, with m open parentheses t close parentheses measured in kilograms and t measured in days.

The table below gives selected values of m open parentheses t close parentheses over the time interval 0 less or equal than t less or equal than 12.

t

(days)

0

3

7

10

12

m open parentheses t close parentheses

(kilograms)

24.9

36.0

70.3

89.7

89.1

Justify why there must be at least one time, t, for 10 less or equal than t less or equal than 12, at which m to the power of apostrophe open parentheses t close parentheses, the rate of change of the mass, equals -0.3 kilograms per day.

Answer:

Showing that a function's derivative has a particular value at an unspecified point is a job for the mean value theorem

But first you have to justify why m open parentheses t close parentheses is continuous; along with being differentiable, that will make the mean value theorem valid

Remember that a differentiable function is automatically also continuous

m open parentheses t close parentheses differentiable rightwards double arrow space m open parentheses t close parentheses continuous

Now calculate the average rate of change of m between x equals 10 and x equals 12 using fraction numerator f open parentheses b close parentheses minus f open parentheses a close parentheses over denominator b minus a end fraction

fraction numerator m open parentheses 12 close parentheses minus m open parentheses 10 close parentheses over denominator 12 minus 10 end fraction equals fraction numerator 89.1 minus 89.7 over denominator 2 end fraction equals fraction numerator negative 0.6 over denominator 2 end fraction equals negative 0.3

Now everything is in place to justify the result using the mean value theorem

m open parentheses t close parentheses is twice-differentiable, which means m open parentheses t close parentheses is differentiable, which means m open parentheses t close parentheses is continuous

The average rate of change of m between t equals 10 and t equals 12 is -0.3 kilograms per hour

Therefore by the mean value theorem there must be at least one time, t, for 10 less or equal than t less or equal than 12, at which m to the power of apostrophe open parentheses t close parentheses equals -0.3 kilograms per hour

Rolle's theorem

What is Rolle's theorem?

  • Rolle's theorem is a special case of the mean value theorem

    • It occurs when f open parentheses a close parentheses equals f open parentheses b close parentheses in the mean value theorem,

      • Which means that f open parentheses b close parentheses minus f open parentheses a close parentheses equals 0

  • Rolle's theorem states that:

    • If a function f is continuous over the closed interval open square brackets a comma space b close square brackets

      • and differentiable over the open interval open parentheses a comma space b close parentheses

    • And if f open parentheses a close parentheses equals f open parentheses b close parentheses

    • Then there exists a value x equals c in the interval open parentheses a comma space b close parentheses

      • such that f to the power of apostrophe open parentheses c close parentheses equals 0

    • I.e. that there will be a point within that open interval open parentheses a comma space b close parentheses

      • where the instantaneous rate of change f to the power of apostrophe open parentheses c close parentheses is equal to zero

    • This means there will be a horizontal tangent at that point

      • and hence a local minimum or maximum point somewhere between x equals a and x equals b

A graph of a function with tangent and average rate of change lines illustrating Rolle's theorem
An illustration of Rolle's theorem

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths

    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

    Author: Dan Finlay

    Expertise: Maths 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.