2. We will use Taylors theorem to derive the central difference approxi- mation.

(a) (2 points) Using Taylor's theorem (Chapter 1 slides, slide 29/40), find an approximation for f'(x0) using: f(xo + h) f(xo h) (1) 2h This is similar to example 1.2 in the text (AG), where the forward difference approximation is computed. Show that the discretization error satisfies () f(xo + h) f(xo h) h2 f'(xo) -f' (20) (2) 2h 6 for small h. You may treat matrix inversion as an elementary operation - you do not need to compute it yourself. (b) (2 points) Working off the given script for example 1.3 (page 8) of the text, write a MATLAB script using the above approximation. Show both round-off and discretization errors. Compare the forward difference from the text and the central difference that we just computed. You may plot both types of difference on the same figure for added clarity