上海第二大学专升本考试大纲《高等数学》（一）（AdvancedMathem..

上海第二大学专升本考试大纲《高等数学》(一)（Advanced Mathematics for higher education in Shanghai Second University
(1)）
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Shanghai second university upgraded examination syllabus
Advanced Mathematics (1)
First, the nature of the examination
The syllabus of advanced mathematics is formulated by Shanghai Second Polytechnic University
Two, examination objectives
The entrance examination of advanced mathematics emphasizes the investigation of students' basic knowledge, basic skills and thinking ability, computing ability, and the ability to analyze and solve problems
Three, the content and basic requirements of the examination
Function, limit and continuity
(1) examination contents
The concept of function and basic characteristics; sequence, function limit; rules of limits; two important limits;
comparison of infinitesimal and order; the continuity and discontinuity of function; properties of continuous functions on closed interval
(two) examination requirements
1. understand the concept of function
Understanding the parity, monotonicity, periodicity and boundedness of functions
Understand the concept of inverse function and understand the concept of composite function
Understanding the concept of elementary functions
A functional relationship between simple and practical problems is established
2. understand the concept of sequence limit and function limit (not required to give)
The two criterion of existence of the limit property (uniqueness, boundedness, number preserving) and limit (the forcing criterion and the monotone bounded criterion)
3. grasp the operation of the limit of function; master the limit calculation method
Grasp two important limits
And use two important limits to find the limit
4. understand the concepts of infinitesimal, infinity, higher order infinitesimal and equivalent infinitesimal
The limit of equivalent infinitesimal
5. understand the concept of function continuity; understand the concept of function discontinuity point
The types of discontinuity points (the first kind of removable, jump discontinuity points and second kinds of breakpoints) are distinguished
6. understand the continuity of elementary functions; understand the properties of continuous functions on closed intervals
Some simple conclusions can be proved by the properties
Two. Derivative and differential
(1) examination contents
The concept of derivative and the rule of derivation; the derivative of function determined by implicit function and parametric equation; the derivative of higher order; the concept and arithmetic of differential
(two) examination requirements
1. understand the concept of derivative and its geometric meaning
Understanding the relation between differentiable and continuous functions
The tangent normal equation of plane curve;
2. master the four operation rules of derivative and the derivation rule of compound function; master the derivation formula of basic elementary function
The derivative of function is skilled
3. master the implicit function and parameter equation of the derivative method (first order); grasp the logarithmic derivative method
4. understand the concept of higher order derivatives
Grasp the first and two order derivatives of elementary functions
The n order derivatives of simple functions
5. understand the concept of differentiation
Understanding the operation of differential calculus and the invariance of first order differential form
Differentiation of functions
Three, mean value theorem and derivative application
(1) examination contents
The Rolle theorem and Lagrange theorem; L'Hospital Rule; function monotonicity and convexity and inflection point, extreme value curve
(two) examination requirements
1. understand the Rolle mean value theorem and Lagrange mean value theorem (the analysis and proof of the theorem is not required), and prove some simple conclusions with the mean value theorem
2. master used l'Hopital's rule for
The method of equal infinitive limit
3. understand the concept of function extremum
The method of judging the monotonicity of the function and the extremum of the function by derivative is mastered. The monotonicity of the function is used to prove the inequality, and the application problem of the simpler maximum and minimum is obtained
4. judge the concavity and convexity of curves by derivative
The inflection point of the curve
Four, indefinite integral
(1) examination contents
Primitive function and indefinite integral concept
Indefinite integral substitution method
Indefinite Integration by parts
(two) examination requirements
1. understand the concepts and properties of primitive functions and indefinite integrals
2. master the basic formula of indefinite integral, change element integration method and parts integration method (desalination special integration skills training)
The general method of integral of rational function does not require
Some simple rational functions can be properly trained as examples of two kinds of integral methods
Five, definite integral and its application
(1) examination contents
The concept and properties of definite integral
Integral variable upper limit function
newton-leibniz formula
Integral integration method and partial integration method for definite integral
The generalized integral on infinite interval and the application of definite integral -- calculating the area of plane figure and volume of revolving body
(two) examination requirements
1. understand the concept of definite integral
Understanding the properties of definite integral and the mean value theorem of integrals
2. understand the concept and property of integral variable upper limit function
Master Newton Leibniz formula
Correct calculation of definite integral by using this formula correctly
3. master definite integral change element method and partial integration method
4. understand the element method of definite integral
The area of an accounting plane figure and the volume of a revolving body
5. understand the concept of generalized integrals on infinite intervals
And the generalized integral on infinite interval is obtained
Six. Differential equations
(1) examination contents
Basic concepts of differential equations
Separable variable differential equations and homogeneous equations
Linear differential equation of first order
Two order linear differential equation with constant coefficients
(two) examination requirements
1. understand the differential equations and the order, solution, general solution, initial conditions and particular solutions of differential equations
2. master the solution of differential equations with separable variables
3. solving homogeneous equations (which can be transformed into separable variable differential equations)
4. understand the constant variation method of first order linear differential equation
Grasp the solution of first order linear differential equation
5. understand the structure of solutions of two order linear differential equations
Grasp the two order constant coefficient homogeneous linear differential equation solution method
6. the special solution method of the two order constant coefficient non-homogeneous linear differential equation with the free term as the simple function by the method of undetermined coefficients
Seven 、 space analytic geometry vector algebra
(1) examination contents
Space rectangular coordinate system, vector and its operation, space plane and its equation, space straight line and its equation, two times curved surface
(two) examination requirements
1. understand the concept of space Cartesian coordinate system
Understand the concept of vector and its representation; find the distance between two points in space
2. grasp the operation of vector (linear operation, scalar product, vector product)
Understand the condition of two vectors vertical and parallel
3. will seek plane equation, straight line equation
4. master the plane and plane, straight line and plane, straight line and line parallel and vertical conditions
The distance from point to plane will be calculated
5. understand the concept of surface equation
Understanding the equation and its figure of two quadric surfaces
Eight. Multivariate functional differential calculus
(1) examination contents
The concept of two variable function, the limit and continuity of two variables function
Derivation rules of partial derivative, total differential and multivariate function
Implicit function derivation formula
Geometric applications of multivariate functional differential calculus
Extreme value of multivariate function
(two) examination requirements
1. understand the concept of two variable function
Understanding the concept of multivariate functions
2. to understand the concept of limit and continuity of two variables function
The limit of some simple functions of two variables
3. understanding the concepts of partial derivatives and total differential functions of two variables
Necessary conditions and sufficient conditions for the existence of total differential
Grasp the calculation method of partial derivative and total differential of multivariate function
4. grasp the first derivative of the multivariate composite function
5. solving the first order partial derivative of implicit
function
6. understand the tangent of the curve and the normal plane, tangent plane and normal of the surface
And they will find their equations;
7. understanding the concept of extreme value and conditional extreme value of two variables function
The extremum of a simple function of two variables
Understanding the Lagrange multiplier method
The application of some simple maximum and minimum values will be discussed
Nine. Multivariate function calculus
(1) examination contents
The concept and properties of double integral and three integral, double integral and calculation of three integral
Curvilinear integral and Green formula
(two) examination requirements
1. understand the concept and nature of double integral
2. master the calculation method of double integral (Cartesian
coordinates, polar coordinates)
3. understand the concept of three integral
Three simple integrals (Cartesian coordinates, cylindrical coordinates) that can be calculated simply
4. understand the concept of two types of curvilinear integrals
Understanding the properties of two kinds of curvilinear integrals and the relation between two kinds of curve integrals
Grasp the calculation method of two kinds of curve integral
6. master Green formula
Mastering the condition and application of plane curve integral and path independent
Ten, infinite series
(1) examination contents
The concept and properties of series of constant terms
The discrimination of convergence and divergence of constant term series and the concept and property of power series
Power series expansion of function
(two) examination requirements
1. understand infinite series and the concepts of convergence, divergence, sum
Understanding the basic properties of infinite series and the necessary conditions for convergence
2. grasp the convergence of geometric series and series
3. to grasp the ratio of positive series of convergence method
Understanding the comparison and convergence method of positive series
4. master Leibniz's theorem of alternating series
Understanding the concept of absolute convergence and conditional convergence
The absolute convergence and conditional convergence of alternating series
5. understand the concept of power series
Grasp the convergence radius, convergence interval, convergence domain and the solution of sum function of power series
6., the McLaughlin expansion uses some simple functions to expand into power series
Four, teaching materials
A series of textbooks for advanced application talents training in the new century
Higher mathematics (upper and lower)
Chief editor, Department of Applied Mathematics, Tongji University
Higher Education Press
Five. Reference books
Advanced Mathematics (Sixth Edition)
Upper and lower volumes)
Tongji University Applied Mathematics Department editor in chief, Tongji University press
Guide to the complete solution of advanced mathematics exercises
Editor in chief of Applied Mathematics Department, Shanghai Second Polytechnic University
Six, examination rules
The proportion of each part of the higher mathematics in the test paper is about one yuan function calculus, about 50%
Space analytic geometry and multivariate function calculus about 30%
The differential equation is about 10%
Series 10% or so
The test paper includes three types of questions:
multiple-choice questions, filling in the blanks and answering questions
Multiple-choice questions and cloze tests accounted for about 40% of the total score
Answer questions accounted for about 60% of the total score
According to the relative difficulty, the test questions are divided into easy questions, middle questions and difficult questions
These three difficulty questions accounted for 40%, 40% and 20% of the total score respectively
The questions of all types are sorted according to the principle of "easy to difficult"
Calculators are not allowed in exams
The examination form is written in closed form
The exam time is 120 minutes
The full score of the test paper is 150 One。