Outline of advanced mathematics examination
I. Examination requirements
Applicable majors: "2+2" enrollment arts and sciences majors.
The syllabus of advanced mathematics includes calculus, linear algebra and probability theory.
The specific requirements of the examination are in turn three levels: understanding, understanding and mastering, flexible application and comprehensive application.
1. Understanding: It is required to have a basic understanding of the meaning of the listed knowledge, know what this knowledge content is, and identify it in related issues.
2. Understanding and mastering: It requires a profound theoretical understanding of the listed knowledge content and the ability to use knowledge to solve related problems.
3. Flexible and comprehensive application: It requires the system to master the internal relations of knowledge and be able to use the listed knowledge to analyze and solve more complex or comprehensive problems.
Two. Outline content
Calculus part
I. Function, Limit and Continuity
Examination content:
The concept of function and its representation/boundedness, monotonicity, periodicity and parity of function/inverse function, properties of compound function, implicit function, piecewise function/basic elementary function and establishment of functional relationship between graphs/elementary function/application problem/concepts of left limit and right limit of sequence limit and function/concepts of infinitesimal and its relationship/basic properties of infinitesimal and comparison of infinitesimal/four operations of limit/II
Examination requirements:
1. Understanding the concept of function and mastering the expression of function will establish the function relationship in application problems.
2. Understand the boundedness, monotonicity, periodicity and parity of functions.
3. Understand the concepts of compound function, inverse function, implicit function and piecewise function.
4. Grasp the nature and graphics of basic elementary functions and understand the concept of elementary functions.
5. Understand the concepts of sequence limit and function limit (including left and right limit) and the relationship between function limit and left and right limit.
6. Master the nature of the function and the four operations and compound operations of the function limit when the limit exists. Master the method of using two important limits to find the limit.
7. Understand the concepts and basic properties of infinitesimal and infinity, and master the comparison method of infinitesimal order.
8. Understanding the concept of function continuity (including left continuity and right continuity) will distinguish the types of function discontinuity points.
9. Understand the properties of continuous functions and the continuity of elementary functions, understand the properties of continuous functions on closed intervals (boundedness, maximum theorem and mean value theorem) and master the methods of applying these properties to prove them.
Second, the differential calculus of unary function
Examination content
Concept of derivative and differential/Geometric meaning of derivative/Relationship between derivability and continuity of function/Derivative of basic elementary function/Derivative of compound function/Derivative of inverse function and implicit function/Higher derivative of some simple functions /n derivative/Differential mean value theorem and its application/L'H?pital's law/monotonicity of function/concavity and convexity of function graph, inflection point/oblique asymptote of function.
Examination requirements
1. Understand the concept of derivative and the relationship between derivability and continuity, and understand the geometric meaning of derivative, and you will find the tangent equation of plane curve.
2. Master the definition method of finding the derivative value of function; Master the derivation formula of basic elementary function, the four operation rules of derivative and the derivation rules of compound function; Familiar with the derivative rules and logarithmic derivative rules of inverse function and implicit function.
3. To understand the concept of higher-order derivative, we can find the second, third and n-order derivatives of univariate function.
4. Will find the first derivative value of the piecewise function at the piecewise point.
5. Understand the concept of differential and the relationship between derivative and differential.
6. Understand the conditions and conclusions of Rolle's mean value theorem, Lagrange's mean value theorem and Cauchy's mean value theorem, and master the application of these three theorems and the proof methods of related problems.
8. Master the method of finding the limit of infinitive with Robida's law.
9. Master the method of judging monotonicity of function and its application, and master the solution of extreme value, maximum value and minimum value of function (including application problems).
10. Be familiar with the method of judging concave-convex and inflection point of function curve, and the solution of oblique asymptote and vertical asymptote of function curve.
1 1. Master the basic steps and methods of drawing functions, and be able to draw some simple functions.
3. Integral calculus of unary function
Examination content
Concepts of original function and indefinite integral/Basic properties of indefinite integral/Basic integral formula/Substitution integral method of indefinite integral and partial integral/Concept and basic properties of definite integral/Mean value theorem of integral/Variable upper limit integral function of integral and its derivative/Newton-Leibniz formula/Substitution integral method of definite integral and the concept and calculation of partial integral/Application of generalized integral/definite integral.
Examination requirements
1. Understand the concepts of original function and indefinite integral, and master the basic properties and basic integral formula of indefinite integral; Proficient in calculation of indefinite integral and partial integral.
2. Understand the concept and basic properties of definite integral. Master Newton-Leibniz formula, substitution integral method of definite integral and integration by parts. Master the derivative formula of variable upper bound integral function and the compound derivative formula containing this kind of function.
4. Master the method of using definite integral to calculate the area of plane figure and the volume of rotating body around X axis and Y axis, and use definite integral to calculate the average value of function.
5. Understand the concept and conditions of convergence and divergence of generalized integrals, and master the substitution integral method and integration by parts for calculating generalized integrals.
Four, multivariate function calculus
Examination content
Concept of multivariate function/geometric meaning of bivariate function/concept of limit and continuity of bivariate function/necessary and sufficient conditions for partial derivative and total differential of multivariate function/derivative of multivariate composite function and implicit function/extreme value and conditional extreme value of second-order Taylor formula of bivariate function/Lagrange multiplier method/maximum and minimum value of multivariate function and its simple application/concept and properties of double integral/calculation of double integral
Examination requirements
1, understand the concept of multivariate function and understand the geometric meaning of binary function.
2. Understand the concepts of limit and continuity of binary functions and the properties of continuous functions in bounded closed regions.
3. Understand the concepts of partial derivative and total differential of multivariate function, and you will find total differential.
4. Master the solution of the first and second partial derivatives of multivariate composite functions.
5. Master the derivation rules of binary implicit function.
6. Understand the second-order Taylor formula of binary function.
7. Understand the concepts of multivariate function extremum and conditional extremum, master the necessary and sufficient conditions for the existence of multivariate function extremum, find binary function extremum, use Lagrange multiplier method to find conditional extremum, find the maximum and minimum of simple binary function, and master the solution method of unconditional extremum or conditional extremum application problem skillfully.
8. Understand the concept and properties of double integral.
9, master the calculation method of double integral (rectangular coordinates, polar coordinates).
Five, infinite series
Examination content
Concept of convergence and divergence of constant series/concept of convergence series/concept of series sum/basic properties and necessary conditions of convergence/discrimination of geometric series and P series and their convergence/convergence of positive series/absolute convergence and conditional convergence of staggered series, concept/function of convergence domain and function of Leibniz theorem/function series and its convergence radius, convergence interval (referring to open interval) and convergence domain/sum function of power series.
Examination requirements
1. Understand the concepts of convergence and sum of convergent constant series, and master the basic properties of series and the necessary conditions for convergence.
2. Master the conditions of convergence and divergence of geometric series and P series.
3. Master the comparison discrimination method and ratio discrimination method of convergence of positive series.
4. Master the Leibniz discriminant method of staggered series.
5. Master the concepts of absolute convergence and conditional convergence of arbitrary series, and the relationship between absolute convergence and convergence.
6. Understand the convergence domain of function term series and the concept of function.
7. Understand the concept of convergence radius of power series and master the solution of convergence radius, convergence interval and convergence domain of power series.
8. Knowing some basic properties of power series in its convergence interval (continuity of sum function, item-by-item differentiation, item-by-item integration), we can find the sum function of simple power series in its convergence interval, and then find the sum of constant series.
9. Understand the necessary conditions for the function to expand into Taylor series.
10, master the maclaurin expansion of α. They will be used to indirectly expand some simple functions into power series.
Sixth, ordinary differential equations
Examination content
Basic concept of ordinary differential equation/Differential equation with separable variables/First-order linear differential equation/Bernoulli equation/Structure theorem of solution/Second-order homogeneous linear differential equation with constant coefficients/Simple application of second-order inhomogeneous linear differential equation with constant coefficients/Differential equation.
Examination requirements
1. Understand differential equations and their concepts such as solutions, orders, general solutions, initial conditions and special solutions.
2. Master the solutions of differential equations with separable variables and first-order linear differential equations.
3. Master the solution of homogeneous differential equation and Bernoulli equation.
4. Understand the properties of solutions of linear differential equations and the structure theorem of solutions.
5. Master the solution of second-order homogeneous linear differential equation with constant coefficients.
6. Can use polynomials, exponential functions, sine functions and cosine functions to solve second-order non-homogeneous linear differential equations with constant coefficients.
Linear algebraic part
I. Determinants
Examination content
Concept and basic properties of determinant/expansion theorem of determinant by row (column)
Examination requirements
1. Understand the concept of determinant and master its properties.
2. The properties of determinant and determinant expansion theorem will be applied to calculate determinant.
Second, the matrix
Examination content
Concept of matrix/linear operation of matrix/multiplication of matrix/power of determinant of square matrix/product of square matrix/concept and property of matrix inversion/necessary and sufficient condition of matrix invertibility/elementary transformation of adjoint matrix/rank of matrix/matrix equivalence/block matrix and its operation.
Examination requirements
1. Understand the concepts of matrix, identity matrix, quantitative matrix, diagonal matrix, triangular matrix, symmetric matrix and antisymmetric matrix, and their properties.
2. Master the linear operation, multiplication, transposition and its operation rules of matrix, and understand the determinant of the power of square matrix and the product of square matrix.
3. Understand the concept of inverse matrix, grasp the properties of inverse matrix and the necessary and sufficient conditions of matrix reversibility, understand the concept of adjoint matrix, and use adjoint matrix to find inverse matrix.
4. Master the elementary transformation of matrix, understand the properties of elementary matrix and the concept of matrix equivalence, understand the concept of matrix rank, and master the method of finding matrix rank and inverse matrix by elementary transformation.
5. Understand the block matrix and its operation.
Third, the vector
Examination content
The concept of vector/linear combination of vectors is linearly independent of the linear representation of vector group/linear correlation of vector group/relationship between the rank of maximal linearly independent group of vector group/rank of equivalent vector group/rank of matrix/orthogonal normalization method/normalized orthogonal basis/orthogonal matrix of linearly independent vector group and its properties
Examination requirements
1. Understand the concepts of n-dimensional vectors, linear combinations of vectors and linear representations.
2. Understand the definitions and properties of linear correlation and linear independence of vector groups, and judge the linear correlation and linear independence of vector groups.
3. Understand the concepts of maximal linearly independent group and rank of vector group, and find the maximal linearly independent group and rank of vector group.
4. Understand the concept of vector group equivalence and the relationship between the rank of vector group and the rank of matrix.
5. Master the Schmidt method of orthogonal normalization of linear independent vector groups.
6. Understand the concept and properties of orthogonal matrix.
Fourth, linear equations.
Examination content
Cramer's Law of Linear Equations/Necessary and Sufficient Conditions for Homogeneous Linear Equations to Have Non-zero Solutions/Necessary and Sufficient Conditions for Non-homogeneous Linear Equations to Have Solutions/Properties and Structure of Solutions of Linear Equations/Basic Solution Systems and General Solutions of Homogeneous Linear Equations/General Solutions of Non-homogeneous Linear Equations.
Examination requirements
1. Cramer's law can be used.
2. Understand the necessary and sufficient conditions for homogeneous linear equations to have nonzero solutions and nonhomogeneous linear equations to have solutions.
3. Understand the concepts of basic solution system and general solution of homogeneous linear equations, and master the solution of basic solution system and general solution of homogeneous linear equations skillfully.
4. Understand the structure and concept of general solution of nonhomogeneous linear equations, and master the solution of general solution of nonhomogeneous linear equations skillfully.
5. Master the method of solving linear equations with elementary line transformation.
Eigenvalues and eigenvectors of verb (abbreviation of verb) matrix
Examination content
Concepts, properties/similarity transformation of eigenvalues and eigenvectors of matrices, concepts and necessary and sufficient conditions of similar diagonalization of properties/matrices, eigenvalues, eigenvectors and similar diagonal matrices of similar diagonal matrices/real symmetric matrices.
Examination requirements
1. Understand the concepts and properties of eigenvalues and eigenvectors of matrices, and master the methods of finding eigenvalues and eigenvectors of matrices.
2. Understand the concept and properties of similar matrix and the necessary and sufficient conditions of matrix similarity diagonalization, and master the method of transforming matrix into similar diagonal matrix.
3. Understand the properties of eigenvalues and eigenvectors of real symmetric matrices.
Sixth, quadratic form
Examination content
Quadratic form and its matrix representation/contract transformation and contract matrix/rank/inertia theorem of quadratic form/standard form and standard form of quadratic form/transforming quadratic form into standard form/quadratic form and positive definiteness of its matrix by orthogonal transformation and collocation method
Examination requirements
1. Master quadratic form and its matrix representation, understand the concepts of quadratic form rank, contract transformation and contract matrix, and understand the concepts of canonical form and canonical form of quadratic form and inertia theorem.
2. Master the method of transforming quadratic form into standard form by orthogonal transformation and transforming quadratic form into standard form by matching method.
Understand the positive definiteness of quadratic form and corresponding matrix and its discrimination method.
Part of probability theory
I. Random events and probabilities
Examination content
The relationship between random events and sample space and operations/events/complete event groups/concepts of probability/basic properties of probability/basic formulas of classical probability/geometric probability/conditional probability/independence of probability/independent repetition test.
Examination requirements
1. Understand the concept of sample space (basic event space), understand the concept of random events, and master the relationship and operation between events.
2. Understand the concepts of probability and conditional probability, master the basic properties of probability, calculate classical probability and geometric probability, and master the addition formula, subtraction formula, multiplication formula, total probability formula and Bayesian formula for calculating probability.
3. Understand the concept of event independence and master the probability calculation with event independence; Understand the concept of independent repeated test and master the calculation method of related event probability.
Second, random variables and their probability distribution
Examination content
The concept and properties of random variables and their probability distribution/distribution function of random variables/probability distribution of discrete random variables/probability density of continuous random variables/probability distribution of common random variables/probability distribution function of random variables.
Examination requirements
1. Understand the concept of random variables and their probability distribution; Understand the probability distribution function of random variable x
The concept and nature of; Master the calculation method of event probability related to random variables.
2. Understand the concept of discrete random variables and their probability distribution, and master 0- 1 distribution, binomial distribution, hypergeometric distribution, Poisson distribution and their applications.
3. Grasp the conclusion and application conditions of Poisson theorem, and use Poisson distribution to approximately represent binomial distribution.
4. Understand the concept of continuous random variables and their probability density, be familiar with the probability density functions of uniform distribution, normal distribution and exponential distribution, and master the application problems of calculating the probability of related events by using the probability density functions of continuous random variables such as uniform distribution, normal distribution and exponential distribution.
6. Master the method of finding the probability distribution of simple function random variables according to the probability distribution of random variables.
Three, two-dimensional random variables and their joint probability distribution
Examination content
Joint distribution function of two-dimensional random variables/joint probability distribution, edge distribution and conditional distribution of discrete two-dimensional random variables/joint probability density, edge density/independence and correlation of continuous two-dimensional random variables/probability distribution of common two-dimensional random variables/probability distribution of functions of two random variables.
Examination requirements
1. Understand the concept and basic properties of joint distribution function of two-dimensional random variables.
2. Understand the concept, properties and two basic expressions of joint distribution of two-dimensional random variables: discrete joint probability distribution of two-dimensional random variables and continuous joint probability density of two-dimensional random variables. Master the method of finding the edge distribution of two random variables when the joint distribution is known.
3. Understand the concepts of independence and correlation of random variables, and master the conditions of independence of random variables; Understand the relationship between independence and independence of random variables.
4. Grasp the two-dimensional uniform distribution and two-dimensional normal distribution, and understand the probability meaning of parameters.
5. Master the method of finding the function probability distribution of two random variables according to their joint probability distribution.
Fourth, the numerical characteristics of random variables
Examination content
Mathematical expectation (mean), variance and standard deviation of random variables and their properties/Mathematical expectation/moment, covariance and correlation coefficient of random variable function and their properties.
Examination requirements
1. Understand the concept of digital characteristics of random variables (mathematical expectation, variance, standard deviation, moment, covariance, correlation coefficient), calculate the digital characteristics of specific distribution by using the basic properties of digital characteristics, and master the digital characteristics of common distribution.
2. Master the method of finding the mathematical expectation of random variables according to their probability distribution; Master the method of finding the mathematical expectation of two random variables according to their joint probability distribution.
Law of Large Numbers and Central Limit Theorem
Examination content
Chebyshev's Law of Large Numbers/Bernoulli's Law of Large Numbers/Sinchin's Law of Large Numbers/De Morville-Laplace Theorem/Levi-Lindbergh Theorem
Examination requirements
1. Understand the conditions and conclusions of Chebyshev's law of large numbers, Bernoulli's law of large numbers and Hinchin's law of large numbers (the law of large numbers of independent and identically distributed random variables).
2. Grasp the conclusions and application conditions of the central limit theorem of de moivre-Laplacian (binomial distribution is normal distribution) and Levi-Lindbergh (central limit theorem of independent identically distributed random variables), and use relevant theorems to approximately calculate the probability of related events.
Three. Form and structure of test paper
The examination paper is in the form of closed book and written test. The full score of the whole paper is 150, and the examination time is 150 minutes.
Test questions are divided into five types: multiple-choice questions, fill-in-the-blank questions, calculation questions, application questions and proof questions.
Multiple choice questions are single choice questions of four choices and one type; As long as the results are directly filled in, there is no need to write out the calculation process or derivation process; Calculation questions, application questions and proof questions must be written in words, calculation steps or derivation process.
The scores of the five types of questions are roughly as follows: multiple-choice questions and fill-in-the-blank questions account for about 30%, calculation questions account for about 45%, application questions account for about 17%, and proof questions account for about 8%.
The proportion of calculus, linear algebra and probability theory in the test paper is roughly: calculus 50%, linear algebra 25% and probability theory 25%.