The national unified examination of some basic courses of online education in pilot universities aims to follow the training goal of applied talents in online education, focus on testing students' mastery and application ability of basic knowledge according to the characteristics of employees' continuing education, and comprehensively improve the teaching quality of modern distance higher education. Advanced Mathematics is one of the basic courses in the national unified examination of online education in modern distance education pilot universities. The examination of this course is a basic level examination, and those who pass the examination should reach the level required by the corresponding higher mathematics course of adult higher education.
Exam object
In the project of "Talent Training Mode Reform and Open Education Pilot" approved by the Ministry of Education, Network Education College of Modern Distance Education Pilot University and Central Radio and TV University, students who have entered undergraduate education since March 1 day, 2004 (including March 1 day) should take the national unified examination for some basic courses of network education.
The syllabus of Advanced Mathematics (B) is suitable for senior high school undergraduate students majoring in science and engineering other than mathematics. Other high school undergraduates who are not majoring in literature, history, law, education and art can also apply for this course.
Examination objectives
Advanced mathematics is one of the compulsory basic courses for students of science and engineering, economic management and other disciplines in colleges and universities. It is a course to cultivate students' computing ability, abstract generalization ability, logical reasoning ability and comprehensive application of learned knowledge to analyze and solve problems. It is also an essential foundation for students to learn further courses and acquire modern scientific and technological knowledge.
The examination goal of this course is to examine students' basic concepts, theories, methods and common operation skills in advanced mathematics, and to test their ability to analyze and solve problems.
The requirements of this outline range from low to high. Concepts and theories are divided into two levels: cognition and understanding, and methods and operations are divided into three levels: understanding, mastering and mastering skillfully.
Examination content and requirements
I. Function, Limit and Continuity
function
1. Examination content
Definition of function, representation of function, piecewise function, inverse function, compound function, implicit function, properties of function (boundedness, parity, periodicity, monotonicity), basic elementary function, elementary function.
2. Examination requirements
(1) Understand the concept of function. Mastering the representation of a function will help you find the domain of the function.
(2) Understand the boundedness, parity, periodicity and monotonicity of functions.
(3) Understand the concepts of piecewise function, inverse function, composite function and implicit function.
(4) Grasp the properties and images of basic elementary functions and understand the concept of elementary functions.
(2) Limit
1. Examination content
Definition and properties of sequence limit, function limit, left limit and right limit of function, concepts and relationships between infinitesimal and infinitesimal, properties and comparison of infinitesimal, four operations of limit, two criteria for the existence of limit (monotone bounded criterion and pinch criterion) and two important limits:
2. Examination requirements
(1) Understand the concepts of sequence and function limit (formal expressions such as ""and ""are not required in the definition of limit).
(2) Will find the limit of the sequence. Will find the limit of the function (including left limit and right limit). The necessary and sufficient conditions for the function to have a limit at one point are understood.
(3) Understand the related properties of limit (uniqueness and boundedness). Four algorithms to master the limit.
(4) Understand the concepts of infinitesimal and infinity. Master the nature of infinitesimal and the relationship between infinitesimal and infinity. Understand the concepts of higher order, same order and equivalent infinitesimal.
(5) Master the method of finding the limit with two important limits.
(3) continuity
1. Examination content
The concept of function continuity: the discontinuous point of left continuous function and right continuous function; Four algorithms of continuous function; Continuity of composite function; Continuity of inverse function; Continuity of elementary function; Properties of continuous functions on closed intervals (maximum theorem, minimum theorem, zero theorem).
2. Examination requirements
(1) Understand the concept of function continuity (including left continuity and right continuity). Discontinuities of the function will be found.
(2) Master four algorithms of continuous function.
(3) Understand the continuity of compound function, inverse function and elementary function.
(4) Understand the properties of continuous functions on closed intervals (maximum theorem, minimum theorem, zero theorem).
Second, the differential calculus of unary function
(a) derivative and differential
1. Examination content
Definition of derivative and differential, left derivative and right derivative, geometric meaning of derivative, relationship between differentiability and continuity of function, four operations of derivative and differential, basic formulas of derivative and differential, derivative of compound function, derivative of implicit function, and higher derivative.
2. Examination requirements
(1) Understand the concept of derivative and its geometric significance. Understand the concepts of left derivative and right derivative.
(2) Understand the relationship between differentiability, differentiability and continuity of functions.
(3) The tangent equation and normal equation of a point on the plane curve will be found.
(4) Master the basic formula of derivative, four algorithms and the derivative method of compound function.
(5) Find the first derivative of the implicit function.
(6) Understand the concept of higher derivative. Will find the second derivative of the function.
(7) Understand the concept of differential. You can find the differential of a function.
(2) The application of differential mean value theorem and derivative.
1. Examination content
Differential mean value theorem (Rolle's theorem, Lagrange's mean value theorem), L'H?pital's law, discrimination of monotonicity of function, extreme value of function, convexity and inflection point of function graph of maximum and minimum value of function.
2. Examination requirements
(1) Understand Rolle Theorem and Lagrange Mean Value Theorem.
(2) To master the method of finding the limit of infinitives with Robida's law.
(3) Master the method of judging monotonicity of function by derivative.
(4) Understand the concept of function extremum. Master the method of finding the extreme value, maximum value and minimum value of a function and solve simple application problems.
(5) Will judge the concavity and convexity of the plane curve. You will find the inflection point of the plane curve.
3. Integral calculus of unary function
indefinite integral
1. Examination content
The concepts of original function and indefinite integral, the basic properties of indefinite integral, the basic formula of indefinite integral, the substitution integral method of indefinite integral and partial integral.
2. Examination requirements
(1) Understand the concepts of primitive function and indefinite integral. Master the basic properties of indefinite integral.
(2) Master the basic formula of indefinite integral.
(3) method of substitution, who is familiar with indefinite integral, method of substitution, who is familiar with indefinite integral (triangle method of substitution and simple radical method of substitution only).
(4) Mastering the partial integral of indefinite integral.
(2) definite integral
1. Examination content
Concept and basic properties of definite integral, geometric meaning of definite integral, function defined by variable upper limit integral and its derivative, Newton-Leibniz formula, integral method of partial substitution of definite integral, application of definite integral (area of plane figure, volume of rotating body).
2. Examination requirements
(1) Understand the concept of definite integral. Understand the geometric meaning of definite integral. Master the basic properties of definite integral.
(2) Understanding the significance of the upper-bound integral of variables as a function of its upper limit, we will find the derivative of this kind of function.
(3) Master Newton-Leibniz formula.
(4) Proficient in the part substitution integral method of definite integral.
(5) The definite integral will be used to calculate the area of the plane figure and the volume of the rotator.
Four, multivariate function calculus
Differential calculus of multivariate functions
1. Examination content
Concept of multivariate function, limit and continuity of binary function, first-order partial derivative and total differential, derivative method of compound function and implicit function, second-order partial derivative, extreme value of binary function.
2. Examination requirements
(1) Understand the concept of multivariate function. Understand the concepts of limit and continuity of binary functions.
(2) Understand the concept of partial derivative. Understand the concept of total differential.
(3) The first and second partial derivatives of binary function can be obtained, and the total differential of binary function can be obtained.
(4) Mastering the solution of the first-order partial derivative of composite function.
(5) Find the first partial derivative of the implicit function determined by the equation.
(6) Understand the necessary and sufficient conditions for the existence of extreme value of binary function. Will find the extreme value of binary function.
(2) Double integral
1. Examination content
The concept and properties of double integral, the calculation method of double integral.
2. Examination requirements
(1) Understand the concept and properties of double integral.
(2) Master the calculation method of double integral in rectangular coordinate system, and exchange the order of integral.
(3) Double integral will be calculated by polar coordinate system.
Verb (abbreviation of verb) ordinary differential equation
1. Examination content
Basic concepts of ordinary differential equations, differential equations with separable variables, homogeneous differential equations, first-order linear differential equations.
2. Examination requirements
(1) Understand differential equations and their concepts such as order, solution, general solution, initial condition and special solution.
(2) Master the solution methods of differential equations with separable variables and first-order linear differential equations.
(3) It can solve homogeneous differential equations.
Test paper structure and question type
First, test scores.
100.
Second, the type of test questions
Multiple choice questions, fill-in-the-blank questions and solutions.
The form of multiple-choice questions is one of four choices, that is, choose a correct answer from four alternative answers to each question.
Fill in the blanks only need to fill in the results directly, without writing out the calculation process and reasoning process.
Solution questions include calculation questions, application questions, proof questions, etc. Solving problems requires writing instructions, calculation steps or derivation process.
Third, the proportion of questions.
About 20% of multiple-choice questions, about 30% of fill-in-the-blank questions, and about 50% of answers (including no more than 5% of proof questions).
(A) the difficulty of the test questions
According to the degree of difficulty, the test questions are divided into easy questions, medium questions and difficult questions, and the score ratio is about 4∶4∶2.
(B) the proportion of test paper content
Univariate function calculus (including functions and limits) is about 65%, multivariate function calculus is about 25%, and ordinary differential equation is about 10%.
Examination method and time
Examination method: closed-book written examination (calculators are not allowed).
Examination time: 120 minutes.