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Mathematics for 3D Game Programming and Computer Graphics, Third Edition by Eric Lengyel

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Appendix D. Taylor Series

Derivation

Let f(x) be a function whose first n derivatives exist on some interval I. Suppose that we wish to approximate f(x) near the value x = c in I using a degree n polynomial pn (x) so that

Equation D.1. 

whenever x is small. The derivatives of pn(x) evaluated at x = c are the following.

Equation D.2. 

We can determine the coefficients ai by requiring that

Equation D.3. 

The polynomial pn (x) is thus given by

Equation D.4. 

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