Show that $f'(x)$ can be approximated by using the following formula. $f'(x) = (4f(x + h) − 3f(x) − f(x − 2h))/6h$ Find the error term and order for the approximation formula.
Confused how to start this question. I know how to prove the Central difference formula for f'' but not this one. Any help would be appreciated.
We have
$f(x+h)=f(x)+hf'(x)$
$+\frac{h^2}{2}f''(c_1)$
and
$f(x-2h)=f(x)-2hf'(x)$
$+2h^2f''(c_2)$
thus
$$6hf'(x)=$$
$$4f(x+h)-3f(x)-f(x-2h)+2h^2(f''c_2)-f''(c_1))$$
the error term is
$$\frac{h}{3}|f''(c_2)-f''(c_1)|\leq\frac{2h}{3}M.$$
where $M=\sup |f''(x)|$.
it's a first order approximation.