Problem 1: (10 pts) In lab, we showed that if interpolated over the data (xj, fj), (xj+1, fj+1), (xj+2, fj+2) where our nodes are equally spaced so that 8x=xj+1xj = xjxj-1 then using f(x) P2(x), we have 1 f'(xj) (- 3 1 fj + 2 fj + 1 (5 pts) Using the same interpolation information, find an approximation to "(xj). (5 pts) Using f(x) = cos(x), determine the accuracy of your centered difference approximations with respect to the magnitude of 8x for the second derivative. Clearly explain your methodology. Note, in order to measure the error in our approximation, we look at points x; [0,2] where dx = 2/N so that xj = 2j N " j = 0,..., N. We plot the maximum of the absolute value of the error in our approximation over our chosen interval for increasing choices of N.