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Extrapolation formula for numerical derivative

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Derive the extrapolation formula and show its error converges to zero more rapidly than does the error of (*).

Please see the attachment for the formula.

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Solution Summary

We give a rigorous derivation of the extrapolation formula.

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If f(x) is twice differentiable, we have:

f(x+h) = f(x) + h f'(x) + h^2/2 f''(x) + o(h^2) (1)

where the "little o" error term o(h^2) denotes a function that goes to zero for h to zero, faster than h^2: For every epsilon > 0 there exists a delta>0 such that:

|o(h^2)| < epsilon h^2 when |h| < ...

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