Given a function f(x), use Taylor approximations to derive a second order approximation to f(x0) is given by

f(x0)=af(x0 h)+bf(x0 +h)+cf(x0 +2h)+O(h2). What is the precise form of the error term? Using the formula approximate f(0) where

f(x) = sinx for h = 1/(2p) for p = 1:15. Form a table with columns giving h, the 2

approximation, absolute error and absolute error divided by h . For each indicate to which values they are converging. Finally, verify that the last column appears to be converging to a value derived using the error term.

1. Given a function f(x), use Taylor approximations to derive a second order approximation to f'(x0) is given by f'(x) = a f (xo h) + bf (20 + h) + cf (20 + 2h) + O(h?). What is the precise form of the error term? Using the formula approximate f'(0) where f(x) = sin x for h = 1/(2P) for p = 1:15. Form a table with columns giving h, the approximation, absolute error and absolute error divided by h2. For each indicate to which values they are converging. Finally, verify that the last column appears to be converging to a value derived using the error term