Taylor's theorem

Abstract

Two problems have to be considered when introducing Taylor's formula into a calculus course: motivation for the use of the Taylor polynomial as an approximate function and the choice among different proofs of Taylor's theorem. A number of solutions found in the literature are discussed. Concerning the first problem, we think that the best solution is to find a proof of Taylor's theorem which generates the Taylor polynomial, so that we can dispense with any independent motivation for the polynomial. Such a proof is given at the end of the paper. Concerning the second problem, it is shown that the most common type of proof of Taylor's theorem presents a significant psychological difficulty. A proof which avoids the difficulty is presented, but I nevertheless think that the proof at the end of the paper is still the best choice.