Mean Value Theorem

Wow, so much more work for a blog post. Not used to that...

1. To understand what the mean value theorem means, you need to be able to translate it into English. f '(c) refers to the derivative at x=c and [f(b)-f(a)]/(b-a) refers to the slope of the secant line through the interval [a,b]. The mean value theorem, therefore, means that if there is a differentiable function on the interval [a,b], which implies continuity, then there is a guarenteed point at which the slope of the tangent line equals the slope of the secant line for that interval. For my example, I used the equation f(x)=4cos x.

If we consider the interval to be [0,4pi], then there are numerous points in which the slope of the tangent line equals the slope of the tangent line. The slope of the secant line is 0, and so all max and min in the interval have the same slope of 0 (the calculations are easy enough, so I leave them all to you).

2. I'm not 100% sure why it only works for differentiable functions, but my reasoning is this. If the function is both continuous and differentiable, then you can get any number as a derivative, either large or small (assuming the function is a curve). If the function is a line, then there's only one possible derivative, so by all points would have the same slope as the secant line because the function is the secant line. If the function is either not differentiable (cusp, corner, discontinuity, vertical tangent) or discontinuous, it misses that crucial charecteristic that curves have, it being that the derivative can be wither large or small. Yes, my explanation isn't the best, so lets look at a case of each and see why they fail.


This is continuous but not differentiable: f(x)=abs(x). If we pretend the interval is [-5,5], then the slope of the secant line is 0. However, there are only two possibilities for derivatives in the actual graph: -1 and 1. Relating back to what I said before, it doesn't have the charecteristic that curves have that the derivative can be large or small.



The graph is f(x)=abs(x)/x (yes, I know the graph looks strange, but it's the calculator). If we take the interval to be [-1,1], on which the function isn't continuous, then the slope of the secant line is 1. However, the only option for the derivative is 0. Again, the graph lacks the charecteristic of all curves, in that the derivative can be anything, large or small.

(Note: Sorry if the pictures aren't clear, I somewhat experimented on their size.)