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lec10_derivatives2

# lec10_derivatives2 - Computing derivatives of functions...

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Computing derivatives of functions Forward & Backward Difference Higher order derivatives Derivatives on a mesh Richardson extrapolation to improve accuracy HW#4 due Thurs –make sure to get addendum (#3) on web. You should be reading in Appendix A.1

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Cost: Here error is of order h (not h 2 ), so approximation is not as good. Benefit: self starting (don’t need values before AND after the point at which you are interested). f ( x 0 ) = f ( x 0 + h ) f ( x 0 ) h O ( h ) f ( x 0 ) = f ( x 0 ) f ( x 0 h ) h + O ( h ) Forward Difference Approximation Backward Difference Approximation
Second derivative is a derivative of the derivative of a function. It is a measure of the CURVATURE (rate of change of the rate of change) of the function at that point. Second order derivatives can be found by ADDING the two Taylor expansions and rearranging for f’’(x) : f ( x 0 ) f ( x 0 h ) 2 f ( x 0 ) + f ( x 0 + h ) h 2 + O ( h 2 )

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Theory suggests that the smaller h is, the better the approximation.
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