Calculate derivative using Richaradson's deferred approach to limit. This is a preferred method for numeric differentiation since it's most precise.
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A three point formula can be constructed which uses the difference in results of the forward and backward two point difference schemes, and computes a three point derivative of that to get the second derivative. |
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This formula can be obtained by writing out the four Taylor series of ƒ(x ± h) and ƒ(x ± 2h) up to terms of h 3 (or up to terms of h 5 to get an error estimation as well). |
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An online calculator for approximating a definite integral using the midpoint (mid-ordinate) rule, with steps shown. |