## The Mean Value Theorem

If is a differentiable function with a continuous derivative and if are any two points then there is a point between and at which This result is also geometrically obvious' once you draw the picture. Mean Value Theorem: somewhere between a and b , the graph must be parallel to the chord The expression is just the slope of the chord So the theorem is saying that there must be a point between and at which the slope of the graph is equal to the slope of the chord This may seem very simple--it is very simple--but it is also very important. In most of its applications we think of the result in the following way, which is easily derived. (Note that the theorem does not tell us what c is--just that it must be there.)

Here's a simple application of this theorem. Suppose that between and Then must be positive (whatever c is). So must be bigger than So, in an obvious sense, the function must be increasing.

Here's another typical application. Let Then So we must have This result is rubbish. For example, tan( ) = tan(0) = 0. But 0 - 0 is not bigger than - 0. What's gone wrong?

The problem here is that is not a differentiable function between 0 and --it blows up at When using these theorems, no matter how obvious' they are, you really must check that the conditions of the theorem are precisely satisfied. 