榆林学院本科毕业论文
摘 要
泰勒公式是微积分中一个重点和难点内容，它能将某些复杂函数近似地表示成简单的多项式函数，体现了微积分“逼近法”的思想精髓，成为解决数学问题的有力工具.基于此，本论文探讨了泰勒公式在诸多数学问题中的应用.为此，首先阐述了泰勒公式的基本思想理论；其次，重点总结介绍了泰勒公式在求解极限、证明不等式与等式、计算行列式，研究函数性态和近似计算与误差估计等方面的应用；同时,每种应用下都给予了相应的典型例题加以具体说明.通过本文的探讨介绍，充分显示出泰勒公式在解题中所发挥出的重要作用.
关键词：泰勒公式，高阶导数，行列式，误差估计
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榆林学院本科毕业论文
ABSTRACT
Taylor formula is an important and difficult content in the Calculus, it can take some complex functions approximately expressed as the simple polynomial functions, and it reflects the essence of \the calculus, so it is a powerful tool to solve many mathematical problems. Based on this, this paper discussed the application of Taylor formula in solving many mathematical problems. For this reason, firstly , described the basic idea and theory of Taylor formula; Secondly, summaried and described the application of Taylor formula in solving problems like limit calculation, proving the inequalities and the equations, studying the state of function, approximate calculation and error estimation and so on; At the same time, the corresponding typical examples are given under each application for the specific descriptions. Through the introduction of this paper, demonstrated fully the significant role of Taylor formula in solving a lot of mathematical problems.
Key words：Taylor formula，Higher derivative，Determinant，Error estimation
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榆林学院本科毕业论文
目 录
摘 要 .................................................................................................................................................. I ABSTRACT ............................................................................................................................................ II 引 言 .................................................................................................................................................. 1 1 泰勒公式 ........................................................................................................................................ 2
1.1 泰勒公式的引入 ................................................................................................................. 2 1.2 常见函数的泰勒展开式 ..................................................................................................... 4 2 泰勒公式的应用 ............................................................................................................................ 6
2.1 证明不等式与等式 ............................................................................................................. 6 2.2 求解极限 ............................................................................................................................ 9 2.3 计算有理函数的不定积分 ............................................................................................... 11 2.4 判别级数敛散性 ............................................................................................................... 12 2.5 研究函数性态................................................................................................................... 15
2.5.1 判断函数的极值 .................................................................................................. 15 2.5.2 求函数的拐点 ...................................................................................................... 17 2.5.3 判别函数的凹凸性 .............................................................................................. 19 2.6 求高阶导数....................................................................................................................... 20 2.7 计算行列式....................................................................................................................... 21 2.8 证明中值定理................................................................................................................... 23 2.9 近似计算与误差估计 ....................................................................................................... 27 3 结束语 .......................................................................................................................................... 31 参考文献 ............................................................................................................................................ 32 致 谢 ................................................................................................................................................ 33
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榆林学院本科毕业论文
引 言
多项式函数是各类函数中最简单的一种，对于一些比较复杂的函数，为了便于数值计算和理论分析，可以用多项式逼近函数，为了更好更方便地研究一些函数，则需要寻求更广泛，更高精度的近似公式来表示函数，这就引入了泰勒公式.泰勒公式集中体现了微积分“逼近法”的精髓.泰勒公式在微分学中占有很重要的位置，尤其在解决一些计算和证明问题中有十分重要的作用.泰勒公式适用于函数具有二阶或二阶以上连续导数的命题，它的一般应用多见于近似计算.而在诸多数学分析教材中，对泰勒公式的介绍仅局限在如何进行函数的泰勒展开，并未对其展开深入的阐述说明.
实际上，泰勒公式除应用于近似计算外，还在其他方面有着广泛且重要的应用.如用泰勒公式来求函数极限，用泰勒公式证明不等式，讨论级数收敛等等.本文详细介绍几种泰勒公式的应用，以典型例题进行具体说明，以期举一反三，拓宽思路，从而加深对泰勒公式的认识和理解.
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