Second Derivative as an average displacement of neighbouring points? Ian Stewart suggested in one of his books that 'the second derivative of a point is the average displacement of its surrounding points from that point.' Can someone interpret this and explain how this is true? I've tried integrating to +/- h and taking h to zero, tried epsilon delta definition but hardly anything worked. All I could show was that the sign of second derivative was related to the sign of the displacement.
 A: Take $h$ normally distributed with mean $x$ and variance $ε$. Then the expectation of the Taylor series looks like
$$\mathbb E_h(f(x+h))=\mathbb E_h(f(x)+f'(x)h+\tfrac12f''(x)h^2+...)=f(x)+\tfrac12f''(x)ε^2+O(ε^4)$$
So the average of the displacement $f(x+h)-f(x)$ is dominated by the term with the second derivative. 
This is one interpretation, but if it is the correct one I cannot decide without the source.
A: One possible interpretation involves inverting the roles of "average" and "displacement".  Namely, the second derivative can be viewed as the displacement of the average, in the following sense:
Consider the average of the nearby values $f(c+h)$ and $f(c-h)$ - clearly, it is $\frac{f(c+h)+f(c-h)}{2}$. This is in general different from $f(c)$. By what amount is the average displaced compared to $f(c)$? by $$\frac{f(c+h)+f(c-h)}{2}-f(c).$$ Now divide by $h^2$, and let $h$ tend to zero.  What you get (up to factor of 1/2) is the second derivative $$f''(c)=\lim_{h \to 0} \frac{f(c+h)+f(c-h)-2f(c)}{h^2}.$$
