导数在高中数学教学的应用研究——几何体的表面积与体积的关系

Research on the Application of Derivatives in High School Mathematics Teaching—The Relationship between Surface Area and Volume of Geometric Bodies

摘要: 通过数形结合的方式，在高三导数专题复习中创造性地构建了一个立体化的知识探究体系：通过对数学本质的深度挖掘发现几何体表面积与体积导数之间的普适关系，揭示导数概念的几何内涵。以几何直观引导代数推导，有机构建知识网络。“数缺形时少直观，形缺数时难入微；数形结合百般好，隔离分家万事休”。不仅提高了复习效率，还通过自主探究让学生在知识再发现的过程中体会到数学的统一美与逻辑严谨性，实现从“学会”到“会学”的质变。

Abstract: Through the combination of numbers and shapes, a three-dimensional knowledge exploration system was creatively constructed in the special review of high school derivatives: by deeply exploring the essence of mathematics, the universal relationship between geometric surface area and volume derivatives was discovered, revealing the geometric connotation of the concept of derivatives. Geometric intuition is used to guide algebraic deduction and organically construct a knowledge network. “When numbers are lacking in form, they are less intuitive, and when shapes are lacking in form, they are difficult to be precise. The combination of numbers and forms is excellent in every way, and isolation and separation make everything go smoothly.” Not only does it improve review efficiency, but it also allows students to experience the unified beauty and logical rigor of mathematics through independent exploration in the process of knowledge rediscovery, achieving a qualitative change from “learning” to “being able to learn”.