Exercise 27 [3 x 5 Points]: a) Consider the backward differencing scheme for computing the first order derivative of a function f. Apply the idea of Richardson extrapolation to derive an estimate with quadratic error term in the step-size h. Compute the exact not just asymptotic) error. b) Apply Richardson extrapolation to the estimate from a) in order to derive an estimate with even better asymptotic error. Compute the exact (not just asymptotic) error. c) Apply the estimates from a) and b) and the backward differencing scheme itself to compute f'(0) for a function f(1) = 100.22 + 1 and step size h = 0.2. For which of the three estimates can we assume that the error is zero? Why