I try to solve the following task. It is given a finite difference: $$g(x)=f'(x)\approx\frac{f(x+h)-f(x)}{h}$$ $$g'(x)\approx\frac{g(x)-g(x-h)}{h}\approx\frac{\frac{f(x+h)-f(x)}{h}\;-\;\frac{f(x)-f(x-h)}{h}}{h}$$ $$\iff g'(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2}\approx f''(x)$$ I shall determine the order of approximation of $f''$ depending on Big-O.
The order of approximation should say something about the quality of the approximation (size of the possible error?). There wasn't much explanation in the lecture so I am not sure. But they want us to give the order of approximation depending on $\mathcal(O)$ (Landau symbol). (Definition of Big-O is clear)
On our frames is only a barely explained example in which something with Taylor Expansions is done. Still without further explanations it is hard for me to understand what I have to do...
I really hope for your help - please explain very detailed because I am a computer scientist and not a mathematician and could probably missing some more in detail information about certain subjects.
Using the Taylor expansion around $h=0$ $$f(x+n h)=f(x)+h n f'(x)+\frac{1}{2} h^2 n^2 f''(x)+\frac{1}{6} h^3 n^3 f'''(x)+O\left(h^4\right)$$ then $$\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=\frac{h^2 f''(x)+O\left(h^4\right)}{h^2}=f''(x)+O\left(h^2\right)$$