1. ## Showing Differentiation

Let $I\subset \mathbb{R}$ be an open interval, let $f:I \rightarrow \mathbb{R}$ be differentiable on $I$, and suppose $f''(a)$ exists at $a\in I$. Show that
$f''(a) = \lim_{h\to 0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}$
Give an example where this limit exists, but the function does not have a second derivative at $a$ .

2. Originally Posted by CrazyCat87
Let $I\subset \mathbb{R}$ be an open interval, let [tex]f:I \rightarrow \mathbb{R} be differentiable on $I$, and suppose $f''(a)$ exists at $a\in I$. Show that
$f''(a) = \lim_{h\to 0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}$
Give an example where this limit exists, but the function does not have a second derivative at $a$ .

Use Taylor polynomials of order 2 around $a\,\,\,for\,\,\,f(a+h)\,\,\,and\,\,\,f(a-h)$ :

$f(a+h)=f(a)+f'(a)h+\frac{f''(a)h^2}{2!}+O(h^3)$

$f(a-h)=f(a)+f'(a)(-h)+\frac{f''(a)h^2}{2!}+O(h^3)$

Now add both eq's above and solve for $f''(a)$ and let $h\rightarrow 0$

Tonio

3. Does $O(h^3)$ represent the remainder term of the function? I'm confused as to why it disappears..

4. Originally Posted by CrazyCat87
Does $O(h^3)$ represent the remainder term of the function? I'm confused as to why it disappears..

It doesn't disappear: when we add the corresponding eq's we get $2O(h^3)$ , and when $h\rightarrow 0$ this, of course, vanishes in the limit...

Tonio

5. ah yes