B using interpolating polynomials since we can

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b) Using Interpolating polynomials; Since we can estimate the p th derivative of f(x) simply by differentiating: . We can apply any of our methods to derive the interpolating func- tion and its error. We then differentiate and evaluate at the left most point in order to derive a forward differ- ence approximation, at the right most point to derive a backward difference approximation and at the center point in order to derive a central difference approximation. c) We can apply difference operators (forward, central, backward); we can average forward and backward difference operators, or apply combined forward and backward operators. The order of the leading n th order error term (if it in fact dominates and we are in the so called asymptotic range) controls how fast the solution converges as space or time resolution is reduced. On a log error vs. log grid spacing plot, the error curve will have a slope equal to n . Higher order methods can be applied to help reduce the cost of the computational solutions, since higher order methods can apply larger grid spacing and therefore fewer computational points for equivalent accuracy as compared to lower order methods. Problem 1 Use the method of undetermined coefficients to derive a fourth order accurate central difference approximation
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