01:198:323 Lecture Notes - Lecture 2: Constant Function
29 Feb 2016
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Let f (x) be a given function, and assume it has deriv- atives around some point x = a (with as many deriva- tives as we nd necessary). For the error in the taylor polynomial pn(x), we have the formulas f (x) pn(x) = 1 n!z x (x a)n+1f (n+1)(cx) (x t)nf (n+1)(t) dt a. The point cx is restricted to the interval bounded by x and a, and otherwise cx is unknown. Rst form of this error formula, although the second is more precise in that you do not need to deal with the unknown point cx. Consider the special case of n = 0. Then the taylor polynomial is the constant function: f (x) p0(x) = f (a) The rst form of the error formula becomes f (x) p0(x) = f (x) f (a) = (x a) f0(cx) with cx between a and x.
