摘要：我们在证明某个微积分问题时，经常会通过已有的条件无法直接推导出想要的结论，这时可以尝试运用构造辅助函数的方法。根据问题中的条件将结论变换，从而构造出一个适当的辅助函数，再运用有关的定理、结论推导出命题的结论。构造辅助函数的方法在分析定理和命题的证明中占据着非常重要的地位，并且一般的构造思想和方法也是有规律可循的。本文通过实例剖析构造辅助函数的原则，总结构造辅助函数的几种方法，探讨构造辅助函数在微积分中的重要作用与应用技巧。
关键词：微积分； 辅助函数； 罗尔中值定理； 拉格朗日中值定理； 柯西中值定理
Abstract:When we prove some mathematical problems in calculus, we often can’t prove the conclusion directly through the existing conditions. In this time, we can attempt to use the structural method of auxiliary function. According to the conditions in the proposition, the conclusion can be changed to construct the related auxiliary function. Using the relevant theorem again, the conclusion of the proposition is obtained. It is very important for the method of building auxiliary function in the analysis of the theorem and propositions. The general ideas and methods to build the auxiliary function also follow certain rules. In this paper, we will analyze the principles of the structural method of auxiliary function, summarize several methods of the structural method of auxiliary function and explore the important role and the application skills of auxiliary functions in calculus.
Keywords: Calculus；Auxiliary function；Rolle mean value theorem; Lagrange mean value theorem；Cauchy mean value theorem.
