Differentiation Derive the central-differences O(Deltax 2) accurate scheme for the second derivative using the Taylor series for f(x ± Delta x) given by f(x ± Delta x) = f(x) ± Delta xf'(x) + (Delta x2/2)f"(x) ± (Delta x3/3!)f"'(x) + (Delta c4/4!) f"'(c). Be sure to keep the truncation error. Construct the differentiation matrix A which takes the first derivative of a vector f of length n, using the following central-difference scheme: f'(x) = f(x + Delta x) - f(x - Delta x)/2Delta x + O(Delta x2) Assume periodic boundary conditions, so that f(n + 1) = f(1).
