~THE MEAN VALUE THEOREM~

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~Also called the Law of the Mean.

~First, let's discuss Rolle's Theorem (it is used to prove the Mean Value Theorem.

~In simple terms, Rolle's Theorem states that if a function is continuous over a closed interval, [a;b] and has a derivative over the open interval, (a;b) and the values of the function at the endpoints, f(a)=f(b)=0, then we can conclude that there must be at least one place between a and b for which the derivative is zero. Think about it...makes sense, yes?

~We will prove Rolle's Theorem in class.

~The Mean Value Theorem is even simpler. It states that, if a function is continuous over a closed interval [a;b] and differentiable over the open interval (a;b), then we can conclude that there will exist at least one value of c between a and b, for which the derivative at c (slope of the tangent line at x=c) has the same value as the slope of the secant line connecting the points a,f(a) and b,f(b). Putting it another
way, we can say that there will be at least one value between a and b where the instantaneous rate of change is equal to the average rate of change.

~This knowledge could save you money and points on your license

~Let's see how.

~Ex: Say you get on the NYS thruway at 9 am in New Paltz and travel (nonstop) to Buffalo. You arrive at Buffalo's toll plaza at exactly 2 pm. A state trooper looks at your card and decides to give you a ticket for speeding. You, of course, are in denial. The trooper takes out a pad and pencil & explains the Mean Value Theorem to you. He says that if you left New paltz at 9 am and arrived at Buffalo at 2 pm, that's 5 hours covering the exact mileage of 400 miles. That means you averaged 80 mph. The speed limit is 70 mph. So, somewhere you had to be going 80 if you averaged 80. In essence, that's what the Mean Value Theorem states. The figures I'm using are for the sake of argument and the actual mileage and times could vary. However, since you listened carefully to the Trooper & was quite polite, he did give you a break..

~We will prove the Mean Value Theorem in class

~Why is this Theorem considered very important?

~Well, it is used to prove many other important properties of functions that we have been assuming from day one.

~Here are just a few:

1) If a function has a zero derivative over an interval, then it is constant
      over that interval.

2) If a function has a positive derivative over an interval, then it is
      increasing over that interval.

3) If a function has a negative derivative over an interval, then it is
      decreasing over that interval.

4) If two functions have equal derivatives over an interval, then they
      must differ by a constant.

~So, know the Mean Value Theorem well, it can save you some grief.

~Many other intertesting properties of functions can be established by the use of this theorem.
It is sometimes referred to, at mathematical tea parties, as a "tool theorem".

~Go to the following link to see an example of finding the prescribed value(s) of c given in the conclusion of the Mean Value Theorem

Example of the Mean Value Theorem