《高等数学(上)》(higher mathematics(1))教学大纲(《高等数学(上)》(高等数学(1))教学大纲)

《高等数学(上)》(higher mathematics(1))教学大纲（《高等数学（上）》（高等数学（1））教学大纲）《高等数学（上）》（高等数学（1））教学大纲一课程编号：：040401。

二课程类型：必修课。

课程学时：80 / 5学分学时适用专业：除信科、强化班外的理、工科各专业先修课程：初等数学三。

课程性质与任务高等数学是我校理工科各专业的一门重要基础课理论课程，是各专业学生一门必修的重要课程。

通过本课程的学习，使学生系统地获得一元函数微积分等基本知识和基本理论；重点介绍极限、导数、积分（不定积分、定积分），并注重培养学生熟练的运算能力和较强的抽象思维能力﹑逻辑推理能力﹑几何直观和空间想象能力，从而使学生学会利用数学知识去分析和解决一些几何﹑力学和物理等方面的实际问题，为学习后续课程和进一步扩大数学知识奠定必要的数学基础。

四。

教学主要内容及学时分配序号主要内容学时一函数、极限与连续十八二导数与微分十五三中值定理及导数的应用十五四不定积分十二五定积分十六定积分的应用八五。

基本要求和基本内容（一）函数与极限1、理解一元函数、反函数、复合函数的定义；2、了解函数的表示和函数的简单性态--有界性、单调性、奇偶性、周期性；3、熟悉基本初等函数与初等函数（包含其定义区间、简单性态和图形）；4、理解数列极限的概念（对定义不作过高要求）；5、熟悉收敛数列的性质-有界性、唯一性；6、了解数列极限的存在准则-单调有界准则、夹逼准则；7、理解函数的极限的定义（包括当和时，函数极限的定义及左、右极限的定义）8、了解函数极限的性质--唯一性、保号性、局部有界性；9、熟练掌握极限的四则运算法则（包括数列极限与函数极限）10、掌握两个重要极限：11、熟悉无穷小量的概念及其运算性质、无穷小量的比较；12、了解无穷大量的概念及其与无穷小量的关系；13、函数极限与无穷小量的关系；14、理解函数的连续性的概念、了解函数的间断点的分类；15、熟悉连续函数的和、差、积、商及复合函数的连续性；16、了解初等函数的连续性，掌握闭区间上连续函数的性质。

（二）导数与微分1、理解函数导数与微分的定义，了解导数与微分的几何意义；2、熟悉函数可导与连续的关系，会用导数来描述一些物理量；3、掌握可导函数的和、差、积、商的求导运算法则；4、掌握复合函数的求导法则和反函数的求导法则；5、熟悉基本初等函数的求导公式及初等函数的求导问题；6、了解高阶导数的概念，会求一些简单函数的高阶导数；7、熟悉隐函数求导法、对数求导法和由参数方程所确定的函数的求导法；8、熟悉微分的基本公式、运算法则和一阶微分形式不变性；（三）中值定理与导数的应用1、理解微分中值定理--罗尔定理、拉格朗日定理、柯西定理；2、掌握罗必塔法则，会利用此法则求函数极限；3、理解函数的极值的概念，会利用导数判别函数的单调性、求函数极值；4. Grasp the maximum and minimum value of function and its application;5, understand the concept of convexity and inflection of the curve, judge the concave and convex interval and inflection point of the function curve;6. Describe the graph of function;7. Understand the concepts of arc differential, curvature and radius of curvature;(four) indefinite integral1, understand the definition of the original function and indefinite integral, understand the geometric meaning of indefinite integral;2. Master the basic properties and basic integral formula ofindefinite integral;3, master indefinite integral variable integral method (the first kind, the second kind of substitution method) and the partial integration method;4. The integral of rational function and some integral which can be rationalized function;(five) definite integral and its application1. Understanding the definition of definite integral and the existence theorem of definite integral;2. Familiar with the basic properties of definite integral - additivity, linear property, comparison property andMean value theorem of definite integral (including integral mean value);3. Understand the integral property and derivative of function of integral upper limit, be familiar with the basic theorem of calculus;4, familiar with the Newton Leibniz formula, grasp the integral element integral method and integral method;5. Understand two kinds of generalized integrals (generalized integrals of unbounded functions and generalized integrals on infinite intervals)Concept and convergence definition of accounting calculation, generalized integral of accounting calculation;6. Understand the approximate calculation method of definite integral (trapezoid method and parabola method);(six) the application of definite integral1. Grasp the element method of definite integral2. Be familiar with the application of definite integral in geometry (seeking the area of plane figure, special volumetric volume and the arc length of plane curve);3. Be familiar with definite integral in physics application (water pressure, change work, object gravitation, etc.);Six. The focus and depth of the course content requirementsBasic task of "higher mathematics" course summary, is to teach calculus (including differential equation etc.) basic knowledge, cultivate students' abstract thinking, logical reasoning, knowledge acquisition, application of mathematical knowledge to solve practical problems, to improve mathematics literacy. In the process of teaching, through the division, induction, analogy, association, geometry and other methods and modern means of education, gradually improve students' mathematical understanding and explore the spirit of innovation. At the same time, to the very important application of mathematical thinking methods, such as the idea of mapping, coordinate the idea of method, limit thought, local linearthoughts, ideas, transform approximation theory, and the idea of optimization, to give enough attention, so that students in the end of this course, have a certain understanding of these thinking methods.Seven. Requirements for students' homeworkAble to finish homework on time and independently.Eight. The relationship between this course and subsequent coursesTo provide the necessary basic mathematical knowledge for the follow-up courses.Nine. Requirements for the cultivation of students' abilityThrough the study of various knowledge points in this course, students' abstract thinking ability and logical thinking ability are cultivated.Ten. Teaching materials and main reference books1. Advanced Mathematics (Fourth Edition). Edited by Tongji University. Higher education press.2. Advanced Mathematics (Fifth Edition). Edited by Tongji University. Higher education press.3. mathematical analysis of engineering foundation. Ma Zhien, edited by Wang Miansheng, higher education press.Eleven. The use of teaching methods and mediaIn the teaching of this course, we can combine the electronic teaching plan demonstration and blackboard writing to improve the teaching efficiency and teaching effect.Twelve. Learning methods and suggestionsIn the course of study, we should pay attention to the study and understanding of the basic concepts, pay attention to the understanding of related nature and problem-solving skills.One。