泰勒公式在若干数学分支中的应用.doc

PAGE 本科学生毕业论文 泰勒公式在若干数学分支中的应用 黑 龙 江 工 程 学 院 The Graduation Thesis for Bachelors Degree Taylor formula in several branches of mathematics Heilongjiang Institute of Technology 黑龙江工程学院本科生毕业论文 PAGE II 摘 要 泰勒公式是数学分析中重要的公式，在解题中有着重要的作用。它将一些复杂函数近似地表示为简单的多项式函数，使问题简单化。它集中体现了微积分“逼近法”的精髓，在微积分学、线性代数、概率等数学学科中都有着重要的应用。 本论文概述了泰勒公式的产生及发展现状；介绍了一元函数泰勒公式的定义及其几个常见函数的展开式；归纳总结了一元函数泰勒公式五个方面的应用，即利用泰勒公式求极限、泰勒公式在证明等式及不等式中的应用、泰勒公式求不定积分、泰勒公式在近似计算中的应用及利用泰勒公式计算行列式。介绍了二元函数泰勒公式的定义，并利用泰勒公式证明极值的充分条件、判定二元函数极限的存在条件、利用泰勒公式求近似值、判定级数和广义积分的敛散性及概率当中严森不等式的证明。 关键词：泰勒公式；佩亚诺余项；拉格朗日余项 ABSTRACT Taylor Formula is an important formula in mathematical analysis, and it plays a necessary partr in problem-solving. It simplifies the problems via making some complicated functions into simple polynomial functions approximately. It embodies the essence of approximation in calculus, and possesses important applications in Mathematics subject such as calculus, linear algebra, the probability, etc. This paper outlines the produce and the development status of Taylor Formula. It introduces the definition of Unary Function Taylor Formula and expanded form of several common functions. It summarizes the applications of Unary Function Taylor Formula in five aspects, that is, getting the limit value, proving the equation and inequalities, evaluating the indefinite integral, applying the approximate calculation and calculating the determinant by Taylor Formula. Moreover, it introduces the definition of Binary Function Taylor Formula, and applies the sufficient basis of proving the limit value by Taylor Formula to determine the existence conditions of Binary Function’ extremum, to get the approximate value, to Judge the progression and the divergence of improper integral and to prove the Yan Sen Inequality in probability. Key words: Taylor formula；Peano remainder；Lagrange Remainder 目 录 TOC \o 1-3 \h \z \u 摘要 I Abstract II 第1章 绪 论 1 1.1 泰勒