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Unformatted text preview: 4/10/2011 1 • Derivative, what does it mean physically • Finite differences using Taylor series • Forward differencing Lecture 17: Numerical differentiation • Backward differencing • Central differencing • Interpretation of finite differences • Higher order derivatives • Example: NGSIM data set: computation of velocity • Be careful of noise... Differentiation Definition of the derivative Most basic way to define the derivative: as the “slope of the tangent”, defined by the limit below: f x x f x f x 4/10/2011 2 Idea of finite differencing: discretization Create a grid along which the function will be discretized, and the limit of the ratio in the previous slide will be approximated. Illustration: numerical approx of ratio (1) If h is small, this should provide a good approximation of the derivative 4/10/2011 3 Illustration: numerical approx of ratio (2) If h is small, this should provide a good approximation of the derivative Illustration: numerical approx of ratio (3) If h is small, this should provide a good approximation of the derivative 4/10/2011 4 Formal expression: Taylor series Remember, approximations of functions with small variations can be expressed with Taylor series. We use the following notation Remember, approximations of functions with small variations can be expressed with Taylor series. We use the following notation Formal expression: Taylor series Remember, approximations of functions with small variations can be expressed with Taylor series. We use the following notationexpressed with Taylor series....
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 Spring '08
 HOROWITZ
 Numerical Analysis, Derivative, Taylor Series, Mathematical analysis, Numerical Differentiation

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