I have one question which asks to derive a finite difference formula to approximate $f''(x)$ in the form of $$f''(x)\approx Af(x+2h)+Bf(x+h)+Cf(x)$$ with the method of undetermined coefficients.
Unfortunately, I don't understand what this question actually means. I'd appreciate if someone could clarify what exactly has to be done here.
Taylor expand $f(x+2h)$ and $f(x+h)$ about the point $x$. Choose $A,B,C$ so that the coefficient of $f(x)$ is zero, the coefficient of $f'(x)$ is zero, and the coefficient of $f''(x)$ is $1$. If possible you would also want to make the coefficients of higher derivatives be zero as well.