An easier way to understand this (and, in my limited experience, teach it) is to explain it with a physical metpahor:
You leave point A heading for point B. Point A is the beginning of a turnpike--a highway with no exits except the tollbooths--and Point B is your exit, where your lovely vacation and some tawdry sex await you. (Don't leave out the tawdry sex! It makes the lesson easier to remember) Needless to say, you're in a hurry. Well, there are 65 miles to cover, and you cover them in less than an hour--actually 50 minutes. The cops--if there were any--didn't pull you, so you're safe, right? Wrong.
As you hand your toll ticket to the lady in the booth, she smiles, and with your receipt, hands you a ticket for speeding: 13 miles over the limit. How can this be?
Well, dummy, here it is: Let "a" and "b" be the times you passed through points A and B, respectively. And let f(t) be your position at time t. Your average velocity for the trip is given by
f(a) - f(b)
a - b
...or, Hmmmm... 78 mph. The Mean Value Theorem demands the existence of a time "c" at which your velocity (the derivative of f(t), or f'(t))was equal to 78mph. QED. Damn you, Joseph-Louis Lagrange!
Some points you need to go over with the class, once they stop giggling:
- The function f(t) is differentiable in the open interval - I should hope so! You always have a velocity, which is the derivative of your position. Condition one, check.
- The function f(t) is continuous in the closed interval - this means that when your car is sitting at the tollbooth and you're taking the first ticket, or when you're parked at point B getting your speeding ticket, you and your car and your position must all be defined. More to the point, you won't be getting on or off of the turnpike except at places able to measure your displacement and time; hence, for the purposes of this metaphor, both are always defined. Condition two, check.