高等数学中的辅助函数设计

Discussion on auxiliary function in advanced mathematics

函数是描述变量之间关系的重要工具,是微积分学研究的主要对象。因此,微积分中许多问题都离不开函数,适当地构造辅助函数,可以达到事半功倍的效果。在理工科院校高等数学课程教学过程中,洛尔定理、Language中值定理是教学的重点和难点,学生很难理解和掌握利用中值定理解决的证明问题。通过规律性地构造辅助函数,加深了学生对于这个难点问题的理解和应用。另外不等式的证明也是高等数学课程中的常见问题之一,运用单调性及Lagrange中值定理结合辅助函数是解决此类问题比较常用的方法。在利用单调性证明不等式问题中,通常情况下是将不等式两边相减之后的函数作为辅助函数,在利用Lagrange中值定理证明不等式问题中一般采用逆推法,适当选取辅助函数可使问题迎刃而解。

Function is an important tool to describe the relationships between variables, it is the main research object of calculus, therefore, many problems are inseparable from the functions in calculus, so it can achieve a multiplier effect if we construct auxiliary function reasonably. In the teaching process of higher mathematics course in universities of science and technology, Rolle＇s Theorem and Lagrange Mean Value Theorem are emphasis of the teaching process, many students have difficulty in understanding and mastering the use of Mean Value Theorem to solve the proof problems, In this paper we can deepen students＇ understanding and using of this difficult problem by constructing auxiliary function regularly. Moreover, the proof of inequality is one of the frequently asked questions in higher mathematics course, we often use monotonicity and Lagrange Mean Value Theorem which are combined with auxiliary function to solve such problems. In the proofs of inequality problems by using monotonicity, we subtract both sides of the inequality and then we obtain a function which usually regard as auxiliary function; Furthermore, we often apply inverse operation in the proofs of inequality problems by using Lagrange Mean Value Theorem, then the problems are solved very easily by selecting appropriate auxiliary function.