本课程利用多媒体与板书相结合进行重难点内容讲授。对基本概念主要采用讲授法和研讨式教学，相关计算方法和技巧主要采用讲授法和启发式教学，并定期利用学习通平台进行章节小测，及时掌握学生阶段性学习情况。 

   针对较抽象的教学内容，组织学生分组进行实例分析研讨，引导学生尝试把抽象问题具体化，提高学生的学习热情。

   本课程团队还特别于2019年创办了“数你行”微信公众号，并挖掘了相当数量的课程思政案例，拍摄了相关的视频节目，大家可以通过搜索“数你行”微信公众号去观看相关案例视频，也可以扫描下面的二维码关注微信公众号了解相关内容。

《Advanced Mathematics (I)》课程教学大纲

课程名称（英文）：Advanced Mathematics (I)

课程性质：（必修课✓、学位课、选修课）

学    分：4.5

总 学 时：72            理论学时：72        实验（或上机）学时：0     

先修课程：

一、课程性质及目的

Being one of the basic courses, Advanced Mathematics aims at imparting mathematical knowledge, cultivating students' ability of logical and abstract thinking, solving practical problems. It could offers necessary theoretical foundation for further subsequent courses and practices. Through the study of this course, students should systematically acquire basic properties of one variable function. At the same time, with exercises and discussions, students are guided to study the textbooks carefully, complete their homework independently, and gradually developed their abilities of abstract thinking, logical reasoning, spatial imagination, analysis and solving practical problems.

二、课程目标

知识：Through the study of this course, students will master the basic concepts and methods of calculus, and be able to apply the learned methods to solve practical problems, gradually improving their learning and innovation abilities.

能力：Advanced mathematics courses can cultivate students' abstract thinking ability, logical reasoning ability, computational ability, spatial imagination ability, innovation ability, self-learning ability, team collaboration ability, and communication ability.

素养：The study of advanced mathematics courses helps to enhance students' comprehensive quality, including abstract thinking ability, problem-solving ability, teamwork ability, etc. Moreover, advanced mathematics requires strict proof and validation, and this rigorous attitude can be extended to other scientific fields, which has a positive impact on students forming a scientific worldview and methodology.

三、课程内容及学时分配

1 Chapter 1 Functions and Continuous （12 Class hour）

1.1 Mappings and functions

1.2 Limits of sequences

1.3 The limit of a function

1.4 Infinitesimal and infinite quantities

1.5 Continuous functions

[重点]：

The method of finding the limit of a function;

Equivalent infinitesimals and using the properties of infinitesimals to find limits;

Continuity of a function and discontinuity points;

Zero point Theorem, Intermediate-value Theorem, and Extremum Theorem.

[难点]：

The proof of the Limit of a Function;

Two Important Limits;

Classification of discontinuity points.

[思政元素]：

By learning the concept of limits, students can establish a sense of accumulation. Not only does learning knowledge require accumulation, but personal growth also requires accumulation. Accumulation is a long-term process, and to achieve success, it is necessary to continuously accumulate professional knowledge and skills.

2 Derivative and Differential （14 Class hour）

2.1 Concept of derivatives

2.2 Rules of finding derivatives

2.3 Higher order derivatives

2.4 Derivation of Implicit Functions and Parametric Equations

2.5 Differential of the Function

[重点]：

Derivative rule and derivative formulas;

Higher order derivatives of commonly used functions;

The derivative of a function determined by a parametric equation;

Finding the differentiation of a function.

[难点]：

The rule for taking derivatives of inverse functions.

[思政元素]：

The concept of derivative emphasizes understanding the overall trend of a function by accumulating small changes, which reflects the importance of accumulation and precipitation, and encourages students to improve themselves by continuously accumulating knowledge and experience.

3 The Mean Value Theorem and Applications of Derivatives （16 Class hour）

3.1 The Mean-value Theorem
3.2 L'Hospital's Rule
3.3 Taylor's Theorem
3.4 Monotonicity, extreme values, global maxima and minima of functions
3.5 Convexity of functions, inflections
3.6 Asymptotes and gaphing of functions

[重点]：

The mean value theorem; L'hospital rule;

Monotonicity of functions, concavity, convexity of curves, and inflection points; Extreme values of functions, maximum and minimum values of functions.

[难点]：

The rule for taking derivatives of inverse functions;

Using the Lopital's Law to find the limit of a function;

The concavity and convexity of curves.

[思政元素]：

Rolle's Theorem requires a function to be continuous within a certain interval, differentiable within a certain subinterval, and the function values at both ends to be equal. This can be likened to the process of socialist construction, which requires us to consider the continuity and stability of the entire process while also paying attention to local development and details on the road to achieving a comprehensive well-off society, ensuring that every link can smoothly transition and lay a solid foundation for the overall harmonious development.

4 Indefinite Integrals （10 Class hour）

4.1 Concepts and properties of indefinite integrals
4.2 Integration by substitution
4.3 Integration by parts
4.4 Integration of rational functions

[重点]：

Two types of integration by substitutios;

Integration by parts;

The area of a planar figure, the volume of a rotating body, and the arc length of a planar curve.

[难点]：

Using the second type of substitution integration method to calculate integration;

Integrating the product of inverse trigonometric functions and power functions;

Integration of Rational Function.

[思政元素]：

The calculation methods of indefinite integrals, such as substitution and partial integration, reflect the strategies and skills of problem-solving. By using this, students can be educated to be good at changing their thinking and finding suitable solutions when encountering problems, which is in line with the concept of socialist innovative development.

5 Definite Integrals （12 Class hour）

5.1 Concepts and properties of definite integrals
5.2 The fundamental theorems of calculus
5.3 Integration by substitution and by parts in definite  integrals
5.4 Improper integrals

5.5 Applications of definite integrals

[重点]：

Integral mean value theorem;

Newton-Leibniz Formula;

Using two types of substitution integration and partial integration methods to obtain definite integrals.

[难点]：

The derivative of integral upper and lower limit functions;

Using the second type of substitution integration method and partial integration to find definite integrals;

Using polar coordinates to calculate the area of a planar figure and the arc length of a planar curve.

[思政元素]：

By combining definite integrals with real-life problems such as calculating the volume and distance of objects, students can realize that mathematics is not abstract, but closely related to real-life situations, thereby enhancing their practical abilities.

6 Differential Equations （8 Class hour）

6.1 Basic concepts of differential equations
6.2 First-order differential equations
6.3 Reducible second-order differential equations

6.4 Higher-order linear differential equations

6.5 Higher-order linear equations with constant coefficients

[重点]：

The solution of first-order linear differential equations;

Reducible second-order differential equations;

General solutions to second-order homogeneous linear differential equations with constant coefficients;

General solutions to second-order non-homogeneous linear differential equations with constant coefficients in the two specific form. 

[难点]：

The method of variation of constant; Rreducible second-order differential equations;

General solutions to second-order non-homogeneous linear differential equations with constant coefficients in the two specific form.

[思政元素]：

Differential equations can be used to study and describe the laws of change in things in nature, which can help students understand the laws of development of things and conform to the philosophical thinking about the laws of development in socialist construction.

四、实验教学内容及要求

This course does not involve experimental operations.

五、课程目标对毕业要求的支撑

六、达成课程目标的途径和措施

1.Based on the nature of the course objectives and the learning characteristics of students, choose appropriate teaching methods, such as teaching, practice, discussion, etc., to promote student learning and achieve the objectives.

2. Combine offline teaching with online teaching, combine knowledge content explanation with ideological and political education in the curriculum, and form a unique three-dimensional blended teaching mode and teaching method.

3. Adopting a teaching method that combines multimedia teaching with classroom discussions, using heuristic teaching, discussion-based teaching, analogical teaching, etc., to cultivate students' scientific thinking ability, discover and analyze problems.

4. Based on the actual situation of the school and college, develop an evaluation method for the achievement of course objectives in this major, which will be implemented by the course leader. All teachers responsible for the course will participate in the evaluation.

七、考核方式、成绩评定及课程目标达成评价依据

1. Assessment method: Advanced mathematics courses are usually assessed through written exams, which include regular assignments, daily tests, and final exams.

2. Score evaluation: The score evaluation of Advanced mathematics courses is usually based on a percentage system, with daily tests accounting for 6%, homework accounting for 14%, and final exams accounting for 80%.

3. Evaluation criteria for achieving course objectives: The evaluation criteria for achieving course objectives in Advanced mathematics courses mainly include course outlines, teaching plans, textbooks, lesson plans, courseware, classroom discussions, exam papers, and other teaching links. The evaluation subjects mainly include students, full-time teachers, teaching supervisors, etc. The evaluation method is mainly based on the course teaching outline, with course assessment results, expert supervision results, peer listening scores, and student evaluation results as the data support basis.

成绩评定细则

Advanced mathematics courses are usually assessed through written exams, which include regular assignments, daily tests, and final exams. The score evaluation of Advanced mathematics courses is usually based on a percentage system, with daily tests accounting for 6%, homework accounting for 14%, and final exams accounting for 80%.

八、教材及参考文献

教材：Advanced Mathematics (I)，北京：北京邮电大学出版社，2017.

参考文献：同济大学数学系编《高等数学》（上）第七版，北京：高等教育出版社，2014.

大纲制定： 王荣欣        大纲审定：蔡霞

制定日期：2023年12月

河北科技大学教学日历

（2023—2024学年第1学期）                          

课程名称： Advanced Mathematics（1）  

专    业：中澳计算机/软件/信管/电信                    

任课教师：  阎晨光/王荣欣            

                                                   系主任   纪玉德     

注明：如遇学校统一调整，教学进程以教师实际上课安排为准。

                                                         《高等数学》课程学习指南与建议

一、引言

      《高等数学》作为我校理工科的核心基础课程，共份上下两册，涵盖极限、导数、微分、不定积分、定积分及其应用。《高等数学》课程不仅承载着培养学生逻辑思维、抽象思维及数学素养的重要使命，也是后续专业课程学习不可或缺的基石。以下是一份详细的《高等数学》课程学习指南与建议，旨在帮助同学们高效、系统地掌握这门课程。

二、学习方法与建议

        1. 预习与复习

        预习： 在每次课前，提前阅读教材相关章节，标记不理解的概念和例题。预习时，可以尝试自己推导公式、解答例题，这有助于在课堂上更快地跟上老师的节奏，深入理解知识点。

        复习： 课后及时复习当天所学内容，巩固记忆，查漏补缺。可以通过做练习题、总结笔记、回顾课堂讲解等方式进行。复习时，要注重知识点的串联和扩展，形成完整的知识体系。

        2. 课堂参与

        积极听讲： 课堂上保持专注，紧跟老师的思路。对于难点和重点，要特别留意老师的讲解和示范。

        主动提问： 遇到不理解的问题，要勇于提问，不要害羞或拖延。与同学和老师交流讨论，可以加深对问题的理解。

        笔记记录： 记录老师讲课的要点、例题解法及自己的疑惑。笔记要简洁明了，方便日后查阅。

        3. 习题练习

        大量练习： 高等数学是一门需要大量练习的学科。通过做题，可以加深对知识点的理解和记忆，提高解题能力。建议每天安排一定时间进行习题练习，从基础题开始，逐步增加难度。

        错题整理： 将做错的题目整理到错题本上，分析错误原因，总结解题方法和技巧。定期回顾错题本，避免重复犯错。

        模拟联系： 临近考试或阶段测试时，进行模拟训练，熟悉考试题型和难度，提高应试能力。

        4. 资源利用

        教材与参考书： 充分利用教材及配套参考书，深入理解基本概念和定理。参考书可以提供不同的解题思路和方法，有助于拓宽视野。

        网络资源： 利用网络资源如数你行微信公众号中的教学视频等，这些资源通常包含丰富的例题和解析，有助于巩固所学知识。

        学习小组： 与同学组成学习小组，共同讨论问题、分享学习资源和经验。小组学习可以促进思维碰撞和相互启发，提高学习效率。

        5. 心态调整

        保持自信： 高等数学虽然难度较大，但并非不可逾越。要相信自己的能力，保持积极的心态面对挑战。

        持之以恒： 学习高等数学需要耐心和毅力。要制定合理的学习计划并坚持执行，不要轻易放弃。

        适时放松： 学习之余要适当放松身心，避免过度疲劳导致学习效率下降。可以通过运动、听音乐、阅读等方式来放松自己。

三、总结与展望

        《高等数学》作为一门基础而重要的学科，对于培养学生的数学素养和解决问题的能力具有不可替代的作用。希望同学们能够掌握有效的学习方法，积极投入学习，不断提升自己的数学水平和综合素质。同时，也要认识到高等数学的学习是一个长期而持续的过程，需要持之以恒的努力和不断的探索。在未来的学习和工作中，愿同学们能够继续保持对数学的热爱和追求，用数学的力量去创造更加美好的未来。

教材：Advanced Mathematics (I)，北京：北京邮电大学出版社，2017.

参考文献：同济大学数学系编《高等数学》（上）第七版，北京：高等教育出版社，2014.

1. “数你行”微信公众号

   大家可以通过扫描下面的二维码关注“数你行”微信公众号，公众号有学、看、听三个板块，其中”学“板块中包含每一章的知识点串讲、难题讲解。

2. 同济大学高等数学网页，https://www.icourse163.org/course/TONGJI-53004?from=searchPage&outVendor=zw_mooc_pcssjg_