The Mean Value Theorem

    Given a function that is differentiable on an open interval and continuous at the endpoints the Mean Value Theorem states there exists a number in the open interval where the slope of the tangent line at this point on the graph is the same as the slope of the line through the two points on the graph determined by the endpoints of the interval. The "mean" in the Mean Value Theorem is referring to the mean (average) rate of change of in the interval.  

Proposition (Mean Value Theorem) Let be a function that is continuous on and differentiable on Then there exists at least one number in such that



    Proof. The equation of the secant line through and is



Let be the difference between and Then



We can see that Because is continuous on and differentiable on so is By Rolle's Theorem, there exists a number in such that which means



and so as desired.

Example (Mean Value Theorem) Find all numbers in the interval such that for the following functions on the given interval.

on
    
    Solution. We determine,



Therefore, since is not in

Example (Mean Value Theorem) Find all numbers in the interval such that for the following functions on the given interval.

on

    Solution. We determine,



Therefore, since is not in

Example (Mean Value Theorem) Find all numbers in the interval such that for the following functions on the given interval.

on

    Solution. We determine,



We find, using a sketch:

Example (Mean Value Theorem) Find all numbers in the interval such that for the following functions on the given interval.  

on

    Solution. We determine,



Since is continuous on and differentiable on we know this must exist. In fact as we see that because the Mean Value Theorem says that is in the open interval Thus we can compute the limit



by the squeeze theorem.

    If an object moves in a straight line with position function then the average velocity between and is and the velocity at is Thus the Mean Value Theorem tells us that at some time between and the instantaneous velocity is equal to that average velocity.

Example (Application of the Mean Value Theorem) Two stationary patrol cars equipped with radar are 1.2 miles apart on a street. As a truck passes the first patrol car, its speed is clocked at 35 miles per hour. One and half minutes later, when the truck passes the second patrol car, its speed is clocked at 30 miles per hour. Prove that the truck must have exceeded the speed limit (of 35 miles per hour) at some time during the one and half minutes.

    Solution. Let be the time when the truck passes the first patrol car. The time when it passes the second patrol car is hour.  By letting represent the distance (in miles) travelled by the truck, we have and So the average velocity is
    


Assuming that the position function is differentiable, we can apply the Mean Value Theorem to conclude that the truck must have been travelling at a rate of 48 miles per hour sometime during the one and half minutes.     

• print this page
• pages linking here
• related pages

Cite this as:
Mean Value Theorem
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/mean-value-theorem.html