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The **Mean Value Theorem** is one of the most important tools in Calculus. It states that if the function //f//(//x//) is defined and continuous on the closed interval [//a//,//b//] and differentiable on the open interval (//a//,//b//), then there is at least one number //c// in the interval (//a//,//b//) (that is //a// < //c// < //b//) such that.

The Mean Value Theorem is behind many of the significant rules in calculus. The following statements, in which we assume //f// is differentiable on an open interval //I//, are attributed to the Mean Value Theorem: -//f// ( //x// )=0 everywhere on //I// if and only if //f// is constant on //I//. -If //f// ( //x// )= g ( //x// ) for all //x// on //I//, then //f// and //g// differ at most by a constant on //I//. -If //f// ( //x// )>0 for all //x// on //I//, then //f// is //increasing// on //I//. -If //f// ( //x// )<0 for all //x// on //I//, then //f// is //decreasing// on //I//.

Click here to launch an applet which helps you understand these rules of the Mean Value Theorem. Or here for the Khan Academy lesson on the Mean Value Theorem.


 * Basic Applications of the Mean Value Theorem **

media type="custom" key="9395586" align="left"//As part of the Mean Value Theorem, the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c.// =Rolle's Theorem = The mean value theorem is a generalization of Rolle's theorem, which assumes //f//(//a//) = //f//(//b//), so that the slope of f(c) is zero. In other words, there exists a point in the interval (//a//,//b//) which has a horizontal tangent. In fact, the Mean Value Theorem can be stated also in terms of slopes. is the slope of the line passing through (//a//,//f//(//a//)) and (//b//,//f//(//b//)). So the conclusion of the Mean Value Theorem states that there exists a point such that the tangent line is parallel to the line passing through (//a//,//f//(//a//)) and (//b//,//f//(//b//)).

To see a proof of Rolle's Theorem, click here

media type="custom" key="9395690"
 * Video For Understanding Rolle's Theorem[[image:rolle.gif align="right" caption="A graph that proves Rolle's Theorem"]] **

= Cauchy's Mean Value Theorem = Cauchy's mean-value theorem is a generalization of the usual Mean Value Theorem. Also known as the extended value theorem, it states that if g(x) and f(x) are continuous on the closed interval [a, b], if g(a) ≠ g(b), and if both functions are differentiable on the open interval (a, b), then there exists at least one //c// (a, b) such that

A proof of this extended value theorem can be found at PlanetMath!

= History of the Mean Value Theorem and Taylor's Theorem =

The history of such a rule is difficult to trace, but the basic rule was first discovered by Rolle. He proved this special case when a = b but it was only applicable for polynomials using algebraic == methods. Later, Cauchy helped to form a more modern version while today we have a generalized version called Taylor's theorem.

Taylor stated that if a real-valued function //f// is differentiable at the point //a// then it has a linear approximation at the point //a//. This means that there exists a function //h//1 such that

=﻿ Want Some Practice? =

Our friends at Darmouth have plenty of practice problems you can try out.

At 7 p.m., a car is traveling at 50 miles per hour. Ten minutes later, the car has slowed to 30 miles per hour. Show that at some time between 7 and 7:10 the car's acceleration is exactly 120, in units of miles per hours squared.
 * Or try our own examples!**

At a particular horse race, two horses start at the same time, and finish in a tie. Show that at some time during the race, the horses were running at the same speed.

Suppose f is a differentiable function such that f(1) = 20 f(x) is greater than or equal to 3 x is on the closed interval [1, 6] What is the smallest possible value for f(6)?

If you learned something, you should totally buy these hip, cool Mean Value Theorem accessories!

=MVT Song=

Lyrics: media type="file" key="MVT Song_0001_converted.mp3" width="240" height="20"

To find the slope of lines in Mathematics We often use the phrase rise over run. Divide the change in y over the x’s Carefully, remember how it’s done?

If f ’s continuous and differentiable On a closed interval from a to b. Slope of the secant line equals f prime sometime, This the Mean Value Theorem guarantees

<span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">Calculate the slope, first, of the secant <span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">Also called the average rate of change. <span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">Set f prime at c equal to this number, <span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">Then find all the values c can be.

<span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">If f ’s continuous and differentiable <span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">On a closed interval from a to b. <span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">Slope of the secant line equals f prime sometime, <span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">This the Mean Value Theorem guarantees

<span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">When your points have got the same y value <span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">Zero is the slope you will agree. <span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">Set f prime at c equal to that zero <span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">Max or min’s are possibilities.

<span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">When you need to find the average value <span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">When integrating f from a to b. <span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">You take the integral and just divide it <span style="font-family: Calibri,sans-serif; font-size: 14.6667px;">By the length of b minus the a.

= ﻿﻿**References/Sources** = Husch, Lawrence. "Mean Value Theorem." //Visual Calculus//. University of Tennessee, Knoxville, 2001. Web. 18 May 2011. [].

"Mean Value Theorem." //Wikipedia The Free Encyclopedia//. N.p., 23 Apr 2011. Web. 18 May 2011. [].

Khamsi, Mohamed, and Helmut Knaust. "The Mean-Value Theorem." //S.O.S. Math//. N.p., 2011. Web. 18 May 2011. [].

Dawkins, Paul. "The Mean Value Theorem." //Paul's Online Notes//. Paul Dawkins, 2011. Web. 18 May 2011. [].

Weisstein, Eric. "Cauchy's Mean-Value Theorem." //Wolfram MathWorld//. Mathworld, 1964. Web. 18 May 2011. [].

"Mean Value Theorem." //Cafe Press//. N.p., n.d. Web. 18 May 2011. <http://shop.cafepress.com/mean-value-theorem>.

"Taylor's Theorem." //Wikipedia The Free Encyclopedia//. N.p., 23 Apr 2011. Web. 18 May 2011. http://en.wikipedia.org/wiki/Taylor%27s_theorem

"Mean Value Theorem." //Wikipedia The Free Encyclopedia//. N.p., 23 Apr 2011. Web. 18 May 2011. [].
 * Images**

"Mean Value Theorem." //Brown Sharpie//. Web. 18 May 2011. <http://brownsharpie.courtneygibbons.org/?p=728>.

//Teacher Tube//. Web. 18 May 2011. [].
 * Videos**