高等数学中泰勒中值定理及拓展应用
Taylor’s Mean Value Theorem in Advanced Mathematics and Its Extended Applications

泰勒中值定理及其对应的泰勒公式是微分学的重点内容，也是教学难点。在高等数学课程中，泰勒公式主要用于求解未定式的极限，近似计算以及证明某些与中值问题有关的结论。本文主要探讨泰勒公式的思想方法，利用泰勒中值定理证明曲线的凹凸性判定定理，中值问题，以及求解微分方程等拓展应用。
The Taylor mean value theorem and its extended applications are key contents of calculus, and it is also a difficult point in the course. In the advanced mathematics courses, Taylor’s formula is mainly used to get the limits of undetermined forms, calculate, and prove certain mean value problems. This article mainly explores the idea of Taylor’s formula, and uses it to discuss the concavity and convexity of curves, prove some mean value problems, and solve differential equations.