Come up with the following two formulae:
- $$f'(x) ≈ \frac 1 {4h}( f (x + 2h) − f (x − 2h) )$$
- $$f''(x) ≈ \frac 1 {4h^2} ( f (x + 2h) − 2 f (x) + f (x − 2h) )$$
and establish the error formula for using these two.
The answer from textbook for a is $−\dfrac{2}{3h^2}f'''(ξ)$.
I know to use the fact that for one-sided formula $$f(x)≈ \frac 1 h ( f (x + h) − f (x)),$$ the error term is $-\dfrac {1} 2 h f''(ξ )$ but in order to find the formula, does it mean to find f(x)? the original function?
Basically, they ask you to use Taylor series using $$f(x+n h)=f(x)+h n f'(x)+\frac{1}{2} h^2 n^2 f''(x)+\frac{1}{6} h^3 n^3 f^{(3)}(x)+\frac{1}{24} h^4 n^4 f^{(4)}(x)+O\left(h^5\right)$$
Apply it to the rhs for each question with the proper values of $n$.