Can someone explain the mean value theorem to me?

Question

psymon · Answer

So, Im sure you have a formula for this in a textbook somewhere or in notes. Ill write it down and then see if we can explain what it is telling us:

$$f'(c) = \frac{ f(b)-f(a) }{ b-a }   $$

So let's explain the f'(c) part first.

psymon · Answer

So first off, f'(x) just means derivative. C is usually always meant to mean a constant, just some number. So f'(c) means we take the derivative and then plug in a point. Now just saying that gives us no context, though. 
So what a derivative does is give us an equation for slope. When you take the derivative of a function, you can plug in any x-coordinate and get the slope at that x-coordinate. So for a quick example, if I have a function 3x^2 and I want to know the slope at the point (2,12) I do this. I take the derivative to get 6x. Then I plug in the x-coordinate of 2. Doing that gives me 12. 12 is the slope of the graph of 3x^2 when x = 2. So basically, f'(c) means the slope at some point on a graph. That make sense so far?

anonymous · Answer

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