Mean Value Theorem Motivation I am currently practicing presenting mathematics to various audiences and am considering the example of the mean value theorem. I was wondering how would I be able to motivate this theorem to a mathematician, to a physicist and to someone who does not have any experience with theoretical mathematics. 
 A: A friend of mine saw a video in school over 40 years ago that went something like this:  A young man was stopped by a police office for speeding.  The office told the young man that he timed his speed over the last few miles or so, and his average speed was over the speed limit.  The young man said, "sure, but how can you be sure that I actually ever exceeded the speed limit?  All you have is an average."  The officer said "Son, haven't you ever heard of the Mean Value Theorem?"
Anyway, I though that was pretty good.
A: Rolle's Theorem is the "What Goes Up Must Come Down" Theorem and work from there.
Well, rather, what goes up and comes down again must stop in the middle.
A: The Mean Value Theorem formalizes the idea that there is always such thing as an average speed. If you're traveling along a straight line (in a train, say), with varying speeds over a certain time interval, then there is some particular  constant speed that would have accomplished the same distance in the same amount of time.
