泰勒公式在考研题中的妙用

The Magical Use of Talor’s Formula in Postgraduate Examination

摘要: 高等数学是众多专业考研时的必考科目，泰勒公式在高等数学中至关重要。在考研数学中，泰勒公式频繁出现，主要集中在求极限和求高阶导数的题型中。求极限和求高阶导数的方法有多种，但如果能使用泰勒公式求解，往往都可以很大程度降低计算量。本文基于考研数学真题，系统地讨论了带有佩亚诺余项的麦克劳林公式在求极限中，以及泰勒级数在求高阶导数中的灵活应用。并对解题时泰勒公式展开的次数和余项选取办法作出了详细研究，通过对比泰勒公式和其它方法，展现了泰勒公式的优势，为学生理解掌握其应用奠定了基础。

Abstract: Advanced mathematics is a required subject for many majors in postgraduate examinations, and Taylor’s formula is crucial in advanced mathematics. In postgraduate mathematics examination, Taylor’s formula appears frequently, mainly in questions about finding limits and finding high- order derivatives. There are many ways to find limits and higher-order derivatives, but if Taylor’s formula can be used to solve it, the amount of calculation can often be markedly reduced. Based on the postgraduate mathematics examination questions, this article systematically discusses the flexible application of Maclaurin’s formula with Peano remainder in finding limits, and the flexible application of Taylor series in finding higher-order derivatives. A detailed study was made on the number of expansions of Taylor’s formula and the method of selecting remainders when solving problems. By comparing Taylor’s formula with other methods, the advantages of Taylor’s formula were demonstrated, laying a foundation for students to understand and master its application.