泰勒公式的探究式教学新设计

A New Design of Inquiry Teaching of Taylor Formula

本文基于波利亚的合情推理,给出泰勒公式教学的一种新设计.首先通过超越数的近似小数表示,类比出超越函数的近似多项式表示,接着探究确定多项式系数需要的条件,归纳出带佩亚诺余项的n阶泰勒公式;然后,类比其零阶情形与有限增量公式,得到带拉格朗日余项的泰勒公式,并通过构造辅助函数,给出其简单证明.最后,讨论了泰勒多项式的唯一性,函数及其导函数的泰勒公式的关系,余项对函数逼近范围的影响,及泰勒公式的常见应用.

This paper presents a new instructional design for Taylor s formula based on Polya s plausible reasoning.Firstly,the approximate polynomial representation of the transcendental function is derived by comparing it with the approximate decimal representation of the transcendental number.The conditions for determining the approximate polynomial coefficients are then discussed,leading to the derivation of the n-th order Taylor formula with the Peano remainder term.Subsequently,the Taylor formula with the Lagrange remainder is obtained by comparing the zero order case with the finite increment formula,and a simple proof is provided by constructing auxiliary functions.Finally,the paper discusses the uniqueness of Taylor polynomials,the relationship between the expansion of functions and their derivatives,the influence of remainder terms on the approximation range of functions,and some applications of Taylor s formula.