揭示微积分的基本思想方法——用以指导微积分的教学

Revealing the Basic Idea and Method of Calculus—To Guide the Teaching of Calculus

摘要: 遵循“用‘已知’认识‘未知’、用‘已知’研究‘未知’、用‘已知’解决‘未知’”这一基本认知准则，通过分析变速直线运动的瞬时速度(引出导数)和变速直线运动的路程(引出定积分)问题，揭示微积分的基本思想方法。与导数、定积分、重积分、曲线积分及曲面积分等相关的问题的解决都遵循微积分的基本思想方法。这一思想方法的揭示统一了微分学与积分学的基本思想方法，对减轻微积分教学难度，提高教学质量，使学生学到微积分的真谛产生积极影响。

Abstract: “Using known knowledge to learn unknown facts, to explore unknown world, and to solve un-known problems” is a basic cognitive principle. We follow this principle to analyze the instanta-neous speed and the distance of variable rectilinear motion, which extract the concepts of derivative and integral respectively, and then to reveal the basic idea and method of calculus. From the analysis of the practical examples, we can see that the resolving of the problems related to derivative, definite integral, double integral, triple integral, curvilinear integrals and surface integral all follow the basic idea and method of calculus. The basic ideas and methods of differential calculus and integral calculus are unified. The study of this methodology can reduce the calculus teaching difficulty, improve the teaching quality and bring students a thorough understanding of core knowledge behind calculus.