Date of writing: August 2004

Compilation teaching and research section: the first mathematics teaching and research section

Applicable majors: undergraduate and junior college majors with a single semester of 70-84 hours.

First, the nature and tasks of the course.

Advanced Mathematics (Class C) is an important basic theory course required by students of related majors in colleges and universities, which serves to cultivate high-quality professionals needed by China's socialist modernization.

Through the study of this course, students should get:

1. function, limit and continuity;

2. One-variable function calculus and its application:

3. Infinite series

Basic concepts, basic theories and basic operational skills. To lay the necessary mathematical foundation for studying the follow-up courses and further acquiring mathematical knowledge.

Second, the basic content and requirements of the course

(1) Function, Limit and Continuity

1. Understand the concept of function.

2. Understand the monotonicity, periodicity and parity of functions.

3. Understand the concepts of inverse function and composite function.

4. Be familiar with the nature and graphics of basic elementary functions.

5. Be able to enumerate the functional relationships in simple practical problems.

6. Understand the descriptive definition of limit.

7. Four algorithms to master the limit.

8. Understand two limit existence criteria (pinch criterion and monotone bounded criterion). Will use two important limits to find the limit.

9. Understand the concepts of infinitesimal and infinity. Master the comparison of infinitesimal.

10. Knowing the concept of function continuity at one point, we can determine the type of discontinuity.

1 1. Understand the continuity of elementary functions. Know the properties of continuous functions on closed intervals (mean value theorem and maximum-minimum theorem).

(b) Differential calculus of univariate function

1. Understand the concepts of derivative and differential. Understand the geometric meaning of derivative and the relationship between differentiability and continuity of function. Some physical quantities can be described by derivatives.

2. Understand the algorithm of derivative and differential (including the invariance of differential form) and the basic formula of derivative. Understand the concept of higher derivative. Can skillfully find the first and second derivatives of elementary functions.

3. Understand the solution of the first and second derivatives of functions determined by implicit functions and parameter expressions.

4. Understand Rolle Theorem and Lagrange Theorem.

5. Understand the concept of function extremum. Master the methods of finding the extreme value of function, judging the increase and decrease of function and the concavity and convexity of function graph, and finding the inflection point of function graph. , and can describe the graph of the function (including horizontal asymptote and vertical asymptote). It can solve the simple application problem of maximum and minimum.

6. I will use the Lobida rule to find the limit of the type.

(3) Integral of unary function

1. Understand the concepts and properties of indefinite integral and definite integral.

2. Familiar with the basic formula of indefinite integral. Proficient in parts substitution integration method of indefinite integral and definite integral. Master the integral of simple rational function.

3. Understand the function of definite integral with variable upper limit and its derivative theorem. Familiar with Newton)-Leibniz formula.

4. Understand the concept of generalized integral.

5. Some geometric quantities (such as area, volume, etc.) will be expressed by definite integral.

(4) Infinite series

1. Understand the concepts of convergence and divergence of infinite series and summation. Necessary conditions for understanding the convergence of infinite series. Know the basic properties of infinite series.

2. Understand the convergence of geometric series and P series.

3. Master the comparative convergence method of positive series. Familiar with the ratio convergence method of positive series.

4. Master Leibniz theorem of staggered series.

5. Understand the concepts of absolute convergence and conditional convergence of infinite series, and the relationship between absolute convergence and convergence.

6. Know the convergence domain of function term series and the concept of function.

7. Mastering the solution of convergence domain of simple power series (endpoint convergence can be ignored).

8. Know some basic properties of power series in its convergence interval.

Master Maclaurin expansions, and use these expansions to expand some simple functions into power series.

Three. Requirements and measures for strengthening quality education in an all-round way and improving hands-on ability and innovation ability.

At the same time of imparting knowledge, through various teaching links, students' ability to abstract and summarize problems, logical reasoning ability, spatial imagination ability and self-study ability are gradually cultivated, especially to cultivate students' more skilled computing ability and comprehensive application of learned knowledge to analyze and solve problems.

Through the cultivation of the above abilities, we can improve our hands-on ability and innovation ability and strengthen quality education.

Four. Necessary explanations and requirements

[1], the classification scheme is mainly based on class hours, students' level, semester and professional characteristics.

(2) Use different words to distinguish the level of basic requirements. Concepts and theories are divided into three levels from high to low: understanding, understanding and knowing, and operation methods are divided into three levels from high to low: mastering, mastering, knowing or being able. The word "familiarity" is equivalent to "understanding" and "mastering".

(3) Different teaching methods can be adopted in the same class according to the characteristics of students in various aspects. The teaching purpose can be divided into "basic theory", "method application" and "exploration" from low to high. On the premise of ensuring the teaching content, teachers can master the teaching methods flexibly.

[4], the same course should be unified teaching materials, unified teaching content, unified examination.

Five, class allocation table (see table), the reference class range is 70-84 hours.

6. Credits: 5 points.

Seven, recommended textbooks and reference books

Textbook: three sets of advanced mathematics