1 The forward difference formula can be expressed as h xo xo h xo ro 0 h Use Richardson extrapolation to derive an O h formula for f ro In addition obtain a table similar to the one appearing at the end of Lecture Note 15 EXAMPLE 6 5 Extrapolating the Derivative of the Natural Logarithm Let s reconsider the approximation of the derivative of the function f x Inx using the second order central difference approximation formula With xo 2 we ind h 0 1 0 05 0 025 Dh f zo h f zo h Error 2h 0 500417292 0 500104205 0 500026043 h 0 1 0 05 4 17 x 10 4 1 04 x 10 4 2 60 x10 5 Applying the extrapolation formula to these results produces 4Dh 2 Dh 3 Error 0 499999842 1 58 x 10 7 0 499999989 1 10 x10 8 Not only has the approximation error been significantly reduced as a result of extrapolation the cut in error due to the cut in the step size is also larger In par iouler it onnoers on if the extrapolated values are fourth order opproximations