To be just a little more succinct than

Jurph, the mean value theorem says (in our speeding metaphor) "You can't average a speed without going that speed for at least a moment".

Remember **rise over run**? That's exactly what the right side of the function is. **f(a) - f(b)** is the rise of the function (the difference between the top and bottom) and **a - b** is the run (the length of the interval of the function. So that's the slope of the straight line connecting **f(a)** and **f(b)**. However, our function f(x) might have lots of curves; it can pretty much go all over the place. If we take a point on the curve of f(x), and find the slope of the curve at that point, we call that f'(x) (we say "f prime of x"). So for a point **c** that is between **a** and **b**, we can find a slope for that point, and we call it **f'(c)**. In a way

In our example, **a**, **b**, and **c** are times - the time we lease the first toll booth, the time we arrive at the second, and some time inbetween, and **f(x)** is the distance we have gon at that time. The right side of the equation above is average speed, and **f'(x)** turns out to be the speed we're going at at that moment. So **f'(c) = (f(a) - f(b))/(a - b)** says that we must have been going our average speed at some point during our trip.

(P.S.: do they really do that with toll booths? That's kind of evil...)