Taylor's Theorem: The Elusive c is Not So Elusive

Abstract

SummaryFor a suitably nice, real-valued function f defined on an open interval containing [a, b], f (b) can be expressed as pn(b) (the nth Taylor polynomial of f centered at a) plus an error term of the (Lagrange) form f(n+1)(c)/(n + 1)! for some c in (a, b). This article is for those who think that not much more can be said about where in (a, b) c is located. Supplementary materials, including further numerical illustrations and sketches of some proofs, are available at http://faculty.colostate-pueblo.edu/rick.kreminski.