摘要导数是反映函数中因变量随自变量变化的快慢程度的数学量，是高等数学的基本概念之一，是其核心部分。导数的概念在中学数学教学中应用广泛，贯穿于中学阶段所学知识。在中学数学教学中，导数作用于各个板块，并在解题方面起到重要作用。71278

在调查分析了导数在中学数学教学中的应用的研究现状、中学数学教材以及各类教材参考书籍之后，结合学习数学以来自身对导数相关知识的见解，综合分析导数在中学数学的各个板块中的应用，选取相对重要的两个模块——几何和代数，对导数在这两方面的中学数学教学中的应用进行了分析整理。借助例题的分析与解答，归纳整合出导数在中学教学中的几类作用，介绍导数在起各类作用时的具体应用方法及教师教授时的注意点。有利于教师的教学和学生对导数相关知识的框架的建构。

The derivative is a mathematical quantity which reflects the speed of the change of the independent variable in the function with the change of the independent variable. It is one of the basic concepts of advanced mathematics, and it’s the core part of it. Similarly, the concept of derivative is widely used in middle school mathematics teaching, throughout the knowledge which we learn in middle school. In the math teaching of middle school, the derivative is used in the several kinds of aspects, and plays an important role in problem solving.

After the investigation and analysis of the research status in the application of the derivative in the middle school mathematics teaching, the high school textbooks and reference books, with the own opinion of the knowledge of derivative, analyzed the application of derivative in the middle school mathematics synthetically and finally choose two key sections, one is algebraic and another is geometry. Analyzed and sorted out the application of derivative in these two sections. With the help of the analysis and explain of the examples, integrated and induced several effects of derivative in high school teaching. Classify those effects and introduce the methods of application of derivative in each kind with examples and the lime light which teachers should notice in teaching. It is conducive to the teachers' teaching and the construction of students' knowledge framework.

毕业论文关键词：导数； 中学教学； 几何； 代数； 应用方法

Keyword: Derivative; The teaching of middle school; Geometry; Algebra; Method of application

目    录

1.教材的剖析 5

2.导数在几何方面的应用 6

2.1与曲线切线有关题型 6

2.2与导数定义有关题型 9

2.3与函数单调性、极值、最值有关题型 11

3.导数在代数方面的应用 14

3.1与不等式相关题型 14

3.1.1证明不等式 14

3.1.2不等式恒成立、任意性、存在性问题 16

3.1.3参数取值问题 17

3.2与数列有关题型 19

4.结论与意义 20

4.1结论 20

4.2意义 20

引言

从八年级上册第五章一次函数开始，学生接触到导数相关知识内容。自此，导数相关知识贯穿整个中学的数学学习过程。有不少学者对导数在中学数学中的应用已经有了一定的研究，如：李孝杰主要研究了运用导数解决数列相关问题[1]，李长明主要在初等数学范畴内对导数在不等式方面的应用做了研究[2]。本文运用内容分析法，参考教材以及前人的研究成果，结合己见，在文中对导数应用进行分类总结，并结合例题对各类问题中导数的应用方法做了介绍。对导数在中学数学教学中应用的研究，有利于教师讲授条理更清晰，促使学生自主建立学习框架，主动分类归纳整理，更快更好的接受知识，理解知识并学会举一反三，能灵活应用知识。