I have been told to derive "the central difference approximation for f''(x) accurate to O(h^4) by applying Richardson extrapolation to the central difference approximation of O(h^2).
I have done this and got:
$$f''(x_i)=\frac{-f(x_{i+2})+16f(x_{i+1})-30f{x_i}+16f(x_{i-1})-f(x_{i-2})}{12h^2}.$$ For error: O(h^4)
using this I have then been given some data points, and computed a f''(0.2) where 0.2 is part of the data set. This is all fine, I am vaguely understanding how this works. However, it is now asking me to estimate the second derivative of $$f(x)=sin(x)+e^{-x}$$ at x=0, using h=0.1 and h=0.5, I literally have no idea how to do this, am i just replacing my $x_{i}, x_{i-1}... $ with 0 and then substituting each value of h? it makes no sense to me. If someone could point me in the right direction that'd be great.