Chapter 3:

Applications of Derivatives

The Mean Value Theorem for Derivatives (often called MVT):

Suppose you take 5 10 point quizzes in a math class.  Does the fact that your quiz average = 8 imply that you scored exactly 8 on one of the quizzes?

(If you're in doubt:  Think of scoring 10, 10, 10, 10 and 0.)

The Mean Value Theorem for Derivatives essentially says that this cannot happen for continuous averages.  That is, if you travel from point A to point B and average 60 mph, then at some point on the trip, you were traveling exactly 60 mph (instantaneous velocity.)

Example:  Suppose between mile marker 30 and mile marker 70 on the Interstate, your position can be given by s(t) = 0.1t² - 3t + 30, where t is given in minutes.

Notice, first, that you aren't really going that slowly.  The statement "yeah, well I only averaged 60 mph" doesn't really tell the whole story, does it?  In fact, to begin with, you're going the other direction.  Then, at some point you reverse yourself, and head back toward mile post 70.

A)  How fast are you going at time = 0 (in miles per hour?)  What's with the negative????

B)  How fast are you going at time = 40 minutes (when you finally reach mile post 70), again in miles per hour?

C)  When and where did you switch directions?

D)  Verify that your average rate of speed from time t = 0 to time t = 40 minutes is 60 mph.

E)  The Mean Value Theorem asserts that at some point, you were traveling exactly 1 mile/minute = 60 mph.

Try to identify that point on the graph, above.

Use a bit of calculus and Algebra to find the point on the graph, above.

Answers:

Here's the picture, with the point for part (E) drawn in:

So, graphically, we're showing a tangent line that is parallel to the secant line.  The existence of such a line somewhere in the interior is what is guaranteed by the Mean Value Theorem for Derivatives.

The steps in the proof of the MVT are just as beautiful and important as the proof of the Max-Min Theorem, and, in fact, the Max-Min Theorem is a cornerstone of the proof.  Once again, study the steps carefully.

The first step is to prove a special case of the MVT, called "Rolle's Theorem."  Rolle's Theorem usually insists that f(b) = f(a) = 0, but it's not really necessary that the values at the end points are zero, just that they're equal to each other:

Now that Rolle's Theorem is established, we proceed to the result we actually wanted, called the Mean Value Theorem for Derivatives (there will be another one for Integrals):

Here's a GeoGebra Applet allowing you to explore the conclusion of the MVT for Derivatives:

Applet:

To prove the MVT, we employ a common trick in proving Calculus results: we "lift" the result to the case where the values at the end points are not the same, by subtracting the general case function down to the case where the values of f(b) and f(a) are equal (and, in fact, = 0.)  We take the function we want, and subtract off the secant line:

g(x) = f(x) - secant line.

Here's an applet demonstrating the "lifting" by subtraction (notice that it may lift f(x) up or down or twist it....)

Applet:

Then, we apply Rolle's Theorem to g(x), and finally see what this implies for f(x).

This applet demonstrates that any point, C, that satisfies the conclusion of Rolle's Theorem for g(x) automatically also satisfies the Mean Value Theorem for f(x):

Applet:

Examples like the homework:

Next Time, A Few Last Results and Observations: