Question

数学专业英语翻译谢谢了

摘 要 泰勒公式是数学分析中非常重要的基本内容,它在函数估计和近似计算方面等广为应用．也是函数的极值、幂级数理论等众多数学理论的基础.泰勒公式的基本思想是用多项式来逼近一个已知函数，而这个多项式的系数由给定函数的各阶导数确定，并对这种逼近给出有效地估计.因此全面认识泰勒公式尤为重要.本文首先介绍了一元和多元泰勒公式的定义,然后分别从证明中值公式，证明不等式，导数的中值估计,关于界的估计,求无穷远处的极限，中值点的极限，函数的极限，近似计算,函数方程中的应用等方面对泰勒公式的应用做了归纳.关键词：泰勒公式；皮亚诺型余项；拉格朗日余项；麦克劳林公式

Answer 1

Abstract
Taylor's formula is the mathematical analysis of the basic content is very important, it is similar in function estimation and calculation of such widely used. Is also a function of extreme value theory, and many other mathematical theory of power series basis. Taylor formula for the basic idea is to use polynomials to approximate a known function, and the coefficients of the polynomial by the given function to determine the derivative , and this approach gives an effective estimate. Therefore, comprehensive understanding of Taylor's formula is particularly important. This paper introduces the Taylor formula of one dollar and multi-definition, then the proof of the value from the formula, respectively, that inequality, the median estimate derivatives, on industry estimates, demand infinity limit, the median point of the limit function limits, approximation, function equations and so on Taylor formula to do the induction.

【Key words】: Taylor's formula; Peano-type remainder; Lagrange remainder; Maclaurin formula
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Answer 2

Taylor formula is very important in the mathematical analysis, it is the basic content in function and widely applied in approximate calculation. Also, the function extreme value theory of power series, etc. Of the basic theory of mathematics is the basic thought of Taylor formula with polynomial function approximation to a known, but the coefficients of polynomial function by a derivative of this approach, and are effectively. Therefore a comprehensive understanding of Taylor formula is particularly important. This paper introduces a pluralistic Taylor formula and the definition, then respectively from the proof of formula, to testify inequality, the median estimate, derivative boundary estimate about the limits of infinity, and the limit value point limit function, approximate calculation, the function equation, the application of Taylor formula aspects of application.Keywords: Taylor formula, More than skin jarno type, Lagrange more, Ms McLaughlin formula