The single most important mathematical tool in computational science is the Taylor series. It is used to derive new methods and also for the analysis of the accuracy of approximations. We will use the series many times in this text. Right here, we just introduce it and present a few applications.
Suppose you know the value of a function f at some point x0, and you are interested in the value of f close to x. More precisely, suppose we know f (x0) and we want an approximation of f (x0 + h) where h is a small number. If the function is smooth and h is really small, our first approximation reads f(xo + h) » f(xo). (A.13)
That approximation is, of course, not very accurate. In order to derive a more accurate approximation, we have to know more about f at x0. Suppose that we know the value of f (x0) and f'(x0), then we can find a better approximation of f (x0 + h) by recalling that
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