Mean Value Theorem Problems
Problems related to the mean value theorem, with detailed solutions, are presented.
Mean Value Theorem.
f '(c) = [f(b) - f(a)] / (b - a).
If f is a function continuous on the interval [ a , b ] and differentiable on (a , b ), then at least one real number c exists in the interval (a , b) such that
The mean value theorem expresses the relatonship between the slope of the tangent to the curve at x = c and the slope of the secant to the curve through the points (a , f(a)) and (b , f(b)).
Problem 1:Find a value of c such that the conclusion of the mean value theorem is satisfied for
f(x) = -2x 3 + 6x - 2
on the interval [-2 , 2]
Solution to Problem 1
f(x) is a polynomial function and is continuous and differentiable for all real numbers. Let us evalute f(x) at x = -2 and x = 2
f(-2) = -2(-2) 3 + 6(-2) - 2 = 2
f(2) = -2(2) 3 + 6(2) - 2 = - 6
Evaluate [f(b) - f(a)] / (b - a)
[f(b) - f(a)] / (b - a) = [ -6 - 2 ] / (2 - -2) = -2
Let us now find f '(x).
f '(x) = -6x 2 + 6
We now construct an equation based on f '(c) = [f(b) - f(a)] / (b - a)
-6c 2 + 6 = -2
Solve for c to obtain 2 solutions
c = 2 sqrt(1/3) and c = - 2 sqrt(1/3)
Below is shown the graph of f, a secant and the two tangent corresponding to the two solutions found. The secant and the two tangents are parallel since their slopes are equal according to the mean value theorem.
Problem 2:Use the mean value theorem to prove that for any two real numbers a and b,
| cos a - cos b| <= | a - b|
Solution to Problem 2
Function cos x is continuous and differentiable for all real numbers. Use the mean value theorem, using 2 real numbers a and b to write
(cos x) ' = [cos a - cos b] / [a - b]
Take the absolute value of both sides
| (cos x) ' | = | [cos a - cos b] / [a - b] |
(cos x)' = - sin x, hence.
| (cos x) ' | < = 1
| [cos a - cos b] / [a - b] | <= 1
| [cos a - cos b] / [a - b] | = |cos a - cos b| / |a - b|
When combined with the above gives
|cos a - cos b| / |a - b| <= 1
Multiply both sides by |a - b| to obtain
|cos a - cos b| <= |a - b|
More references on
Step by Step Math Worksheets SolversNew !
Linear ProgrammingNew !
Online Step by Step Calculus Calculators and SolversNew !
Factor Quadratic Expressions - Step by Step CalculatorNew !
Step by Step Calculator to Find Domain of a Function New !
Free Trigonometry Questions with Answers
Interactive HTML5 Math Web Apps for Mobile LearningNew !
Free Online Graph Plotter for All Devices
Home Page --
HTML5 Math Applets for Mobile Learning --
Math Formulas for Mobile Learning --
Algebra Questions -- Math Worksheets
Free Compass Math tests Practice
Free Practice for SAT, ACT Math tests
Precalculus Tutorials --
Precalculus Questions and Problems
Precalculus Applets --
Equations, Systems and Inequalities
Online Calculators --
Geometry Tutorials --
Geometry Calculators --
Calculus Tutorials --
Calculus Questions --
Applied Math --
Math Software --
High School Math --
Middle School Math --
Math Videos From Analyzemath
Updated: February 2015
Copyright © 2003 - 2016 - All rights reserved