The Mean Value Theorem states that:
Given f is a function which is continuous and differentiable on the closed interval between a and b. Then there exists a point c in (a, b) such that:
f`(c) = f(b) - f(a)
b - a
That is... if you have a continuous function f between two points x=a and x=b, there will always be a point x=c on the function where the derivative f`(c) is equal to the gradient of the chord joining f(a) and f(b).
Here is my poor attempt at illustrating this:
| ,' ,'
| f(c) ,' ,'
| .x^`. ,'
| ,'/ .\f(b)
| ,' / ,' \ /
| / ,' `_/
| a c b
Oh dear, that is quite poor...
The 'curve' represents our function f(x). The points where f(a) and f(b) are joined by a dotted line, and f(c) is indicated by the x. The tangent to the curve at x=c is also represented by a dotted line. The point of the theorem is that it says there will always be a point x=c in the interval between x=a and x=b where the tangent to the curve at x=c is parallel to that of the line joining the points f(a) and f(b).
This theorem is useful for Convergence Testing and Proving a function has only one root in a given interval, amongst other things. (feel fre to /msg :)
Compare: Rolle's theorem; Intermediate Value Theorem
Props to buo for teaching me this shit so I could node it.