Mathematics three postgraduate entrance examination outline (full version)
First of all, calculus
I. Function, Limit and Continuity
Examination content
Concept and expression of function: boundedness, monotonicity, periodicity and parity of function, inverse function, composite function, implicit function and piecewise function, and the concepts of sequence limit and left and right limit of function limit of graphic elementary function, the concepts of infinitesimal and infinitesimal, the basic properties of infinitesimal relationship and the comparison limit of order, four operations, two important limit functions, the concepts of continuity and discontinuity, and the properties of continuous function on the closed interval of elementary function.
Examination requirements
1. Understand the concept of function and master the representation of function. Deeply understand the boundedness, monotonicity, periodicity and parity of functions.
3. Understand the concepts of compound function, inverse function, implicit function and piecewise function.
4。 Master the nature and graphics of basic elementary functions and understand the concept of elementary functions.
5. The functional relationship in simple application problems will be established.
6. Understand the concepts of sequence limit and function limit (including left and right limits).
7. Understand the concept and basic properties of infinitesimal and master the comparison method of infinitesimal order. Understand the concept of infinity and its relationship with infinitesimal.
8. Understand the nature of limit and two criteria for the existence of limit (monotone bounded sequence has limit and pinch theorem), and master four algorithms of limit, and two important limits will be applied.
9. Understand the concept of function continuity (including left continuity and right continuity).
10. Understand the properties of continuous functions and the continuity of elementary functions, and understand the properties of continuous functions on closed intervals (boundedness, maximum theorem and mean value theorem) and their simple applications.
Second, the differential calculus of unary function
Examination content
The relationship between derivability and continuity of derivative concept function; Four operations of derivative; The concept and algorithm of derivative differentiation of higher derivative of basic elementary function: Hospital Law; The concavity and convexity of the extremum function graph of monotone function: inflection point; And the maximum and minimum values of the asymptote function graph.
Examination requirements
1。 Understand the concept of derivative and the relationship between derivability and continuity, and understand the geometric and economic significance of derivative (including the concepts of margin and elasticity).
2. Master the derivation formula of basic elementary functions, the four operation rules of derivatives and the derivation rules of compound functions; Master the derivative method and logarithmic derivative method of inverse function and implicit function.
3. In order to understand the concept of higher derivative, we can find the second and third derivatives and n-order derivatives of simpler functions.
4. Understand the concept of differential, the relationship between derivative and differential, and the invariance of first-order differential form: master differential method.
5. Understand the conditions and conclusions of Rolle Theorem (ROl 1e), Lagrange Mean Value Theorem (kgrange) and Oluc Mean Value Theorem, and master the simple applications of these three theorems.
6. Will use the Lobida rule to find the limit.
7. Master the method of judging monotonicity of function and its application, and master the solution of extreme value, maximum value and minimum value (including solving simple application problems).
8. Master the judgment method of curve convexity and inflection point and the solution method of curve asymptote.
9. 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
The concept of original function and indefinite integral The basic properties of indefinite integral The concept and basic properties of indefinite integral The integral mean value theorem of partial definite integral The function defined by variable upper limit definite integral and its derivative Newton-Leibniz formula The concept of partial generalized integral and the application of calculating 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; Master the substitution integral method and integration by parts for calculating indefinite integral.
2. Understand the concept and basic properties of definite integral. Master Newton-Leibniz formula, substitution integral method of definite integral and partial integral. Will find the derivative of variable upper bound definite integral.
3. I will use definite integral to calculate the area of plane figure and the volume of rotator, and I will use definite integral to solve some simple economic application problems.
4. Understand the concept of convergence and divergence of generalized integral, master the basic method of calculating generalized integral, and understand the conditions of convergence and divergence of generalized integral.
Four, multivariate function calculus
Examination content
The concept of multivariate function, the geometric meaning of binary function, the limit and continuity of binary function, the properties of binary continuous function in bounded closed region (maximum theorem), the concept and calculation of partial derivative of multivariate composite function, the basic properties and calculation of simple double integral of high-order partial derivative fully differential multivariate function
Examination requirements
1. Understand the concept of multivariate function, and understand the representation and geometric meaning of binary function.
2. Understand the intuitive meaning of limit and continuity of binary function.
3. Understand the concepts of partial derivative and total differential of multivariate function, master the solution of partial derivative and total differential of compound function, and use the derivative rule of implicit function.
4. Understand the concepts of multivariate function extremum and conditional extremum/master the necessary conditions for the existence of multivariate function extremum, and understand the sufficient conditions for the existence of binary function extremum. Will find the extreme value of binary function. Lagrange multiplier method will be used to find conditional extremum. Can find the maximum and minimum of simple multivariate function, and can solve some simple application problems.
5. Understand the concept and basic properties of double integral, and master the calculation methods of double integral (rectangular coordinates and polar coordinates). Will calculate simple double integrals on unbounded regions.
Five, infinite series
Examination content
The concept of convergence and divergence of constant series, basic properties and necessary conditions for convergence, the concept of convergence of geometric series and convergence of positive series, the absolute convergence and conditional convergence of arbitrary series, the Leibniz theorem, the concept of convergence radius, convergence region (referring to open interval) and the basic properties of power series sum function in convergence interval, the solution of simple power series sum function and the power series expansion of elementary function
Examination requirements
1. Understand the concepts of convergence and divergence of series and sum of convergent series.
2. Master the necessary conditions of series convergence and the basic properties of convergent series. Master the conditions of convergence and divergence of geometric series and p series. Master the comparison discrimination method and D'Alembert (ratio) discrimination method of positive series.
3. Understand the concepts of absolute convergence and conditional convergence of arbitrary series, master Leibniz discriminant method of staggered series, and master the discriminant method of absolute convergence and conditional convergence.
4. Will find the convergence radius and convergence domain of power series.
5. Understand the basic properties of power series in convergence domain (continuity of sum function, item-by-item differentiation, item-by-item integration), and we will find some simple sum functions of power series.
6. Master the expansions of (abbreviated) power rank numbers, and use these expansions to indirectly expand some simple functions into power series.
Six, ordinary differential equation and envy equation
Examination content
Concept of differential equation solution, general solution, initial condition and special solution of differential equation with separable variables, second-order homogeneous linear equation with constant coefficient and simple non-homogeneous linear equation, general solution and simple application of first-order linear difference equation with constant coefficient.
Examination requirements
1. Understand the concepts of order, general solution, initial condition and special solution of differential equations.
2. Master the solutions of equations with separable variables, homogeneous equations and first-order linear equations.
3. Polynomial, exponential function, sine function, cosine function and their sum and product can be used to solve the second-order homogeneous linear equation with constant coefficients and the second-order inhomogeneous linear differential equation with constant coefficients.
4. Understand the concepts of difference and difference equation and their general and special solutions.
5. Master the solution method of the first-order linear difference equation with constant coefficients.
6. Will apply differential equations and difference equations to solve some simple economic application problems.
Second, linear algebra.
I. Determinants
Examination content
The concept and basic properties of determinant Using the determinant expansion theorem of row (column) Clem rule
Examination requirements
1. Understand the concept of threshold determinant.
2. Mastering the properties of determinant will apply the properties of determinant and the expansion theorem of determinant line by line (column) to calculate determinant.
3. Will use Cramer's rule to solve linear equations.
Second, the matrix
Examination content
Concepts of matrix identity matrix, diagonal matrix, quantized matrix, triangular matrix, symmetric matrix and orthogonal matrix, concepts of transposed inverse matrix of matrix product matrix and adjoint matrix of property matrix, elementary transformation of elementary matrix block matrix and rank of its operation matrix.
Examination requirements
1. Understand the concept of matrix, and understand the definitions and properties of several special matrices.
2. Master the addition, multiplication and multiplication of matrices and their algorithms; Master the properties of matrix transposition; Master the properties of determinant of square matrix product.
3. Understand the concept of inverse matrix and master the properties of inverse matrix. Will use the adjoint matrix to find the inverse of the matrix.
4. Understand the elementary transformation of matrix and the concept of elementary matrix; In order to understand the concept of rank of matrix, we will use elementary transformation to find the inverse sum rank of matrix.
5. Understand the concept of block matrix and master the algorithm of block matrix.
Third, the vector
Examination content
The linear combination of the sum of the concept vector of the vector and the product vector of the vector and the linear representation of the vector group; The concept, properties and discrimination of linear elements of vector groups: the rank of vector groups with maximal linear elements.
Examination requirements
1. Understand the concept of vectors and master the addition and multiplication of vectors.
2. Understand the concepts of linear combination and linear representation of vectors, linear correlation and linear independence of vector groups, and master the related properties and discrimination methods of linear correlation and linear independence of vector groups.
3. Understand the concept of maximal independent group of vector group and master the method of finding maximal independent group of vector group.
4. Understand the concept of the rank of vector group, understand the relationship between the rank of matrix and the rank of its row (column) vector group, and find the rank of vector group.
Fourth, linear equations.
Examination content
The solution of linear equations and the determination of solutions and meta-solutions of linear equations; The basic solution system of homogeneous linear equations and the relationship between the solutions of nonhomogeneous linear equations and the corresponding homogeneous linear equations (derivative group); General solution of nonhomogeneous linear equations
Examination requirements
1. Understand the concept of solutions of linear equations, and master the judgment method of solutions and non-solutions of linear equations.
2. Understand the concept of basic solution system of homogeneous linear equations, and master the solution and general solution of basic solution system of homogeneous linear equations.
3. Master the solution of the general solution of non-homogeneous linear equations, and express the general solution of non-homogeneous linear equations with its special solution and the basic solution system of the corresponding derivative group.
Eigenvalues and eigenvectors of verb (abbreviation of verb) matrix
Examination content
Eigenvalues and eigenvectors of matrices Similar diagonal matrices Eigenvalues and eigenvectors of real symmetric matrices
Examination requirements
1. Understand the concepts of matrix eigenvalues and eigenvectors, master the properties of matrix eigenvalues, and master the methods of finding matrix eigenvalues and eigenvectors.
2. Understand the concept of matrix similarity, master the properties of similar matrices, understand the necessary and sufficient conditions of matrix diagonalization, and master the method of transforming matrices into similar diagonal matrices.
3. Master the properties of eigenvalues and eigenvectors of real symmetric matrices.
Sixth, quadratic form
Examination content
Quadratic form and its matrix represent the rank inertia theorem of quadratic form of contract matrix. Quadratic canonical form and orthogonal transformation of canonical form Quadratic form and positive definiteness of its matrix
Examination requirements
1. Understand the concept of quadratic form and express quadratic form in matrix form.
2. Understand the concepts of rank of quadratic form, standard form of quadratic form and standard form (knowing the conditions and conclusions of inertia theorem, we will abandon orthogonal transformation and collocation method to convert quadratic form into standard form. The concepts of positive definite quadratic form and positive definite matrix, and master the properties of positive definite matrix.
Third, probability theory and mathematical statistics
I. Random events and probabilities
Examination content
The relationship between random events and sample space events, the independence of event operation and natural events, the definition of complete event group probability, the basic properties of probability, classical probability conditional probability, normal formula, multiplication formula, total probability formula and Bayesian formula, independent repetition test.
Examination requirements
1. Understand the concept of sample 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 and calculate classical probability; Master the addition and multiplication formulas of probability, as well as the total probability formula and Bayesian formula.
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 the distribution function and its probability distribution of random variables; the probability distribution of discrete random variables and the probability density of continuous random variables; the probability distribution and joint (probability) distribution of common random variables; the joint probability distribution and edge distribution of two-dimensional discrete random variables; the concept of the quantile of the sum of the joint probability density and edge density of two-dimensional continuous random variables.
Examination requirements
1. Understand the concept of random variables and their probability distribution; Understand the concept and properties of distribution function f (x) = p {x ≤ x}; Calculate the probability of events related to random variables.
2. Understand the concept and probability distribution of discrete random variables, and master 0- 1 distribution, binomial distribution, super JLnn distribution, POison distribution and their applications.
3. Understand the concept of continuous random variables and their probability density, and master the relationship between probability density and distribution function; Master uniform distribution, exponential distribution, normal distribution and their applications.
4. Understand the concept of two-dimensional random variables and the concept, properties and two basic forms of joint distribution of two-dimensional random variables: discrete joint probability distribution and edge distribution, continuous joint probability density and edge density; Will use two-dimensional probability distribution to find the probability of related events.
5. Understand the concepts of independence and irrelevance of random variables, and master the conditions of independence of discrete and continuous random variables.
6. Grasp the two-dimensional uniform distribution; Understand the density function of two-dimensional normal distribution and the probability significance of its parameters.
7. Master the basic method of finding the probability distribution of its simpler function according to the probability distribution of independent variables; Will find the probability distribution of the sum of two random variables; Understand the typical mode of generating χ2 variable, χ 2 variable and F variable; Understand the quantiles of standard normal distribution: χ2 distribution, t distribution and f distribution, and look up the corresponding numerical table.
Third, the numerical characteristics of random variables
Examination content
Mathematical expectation, variance, standard deviation and their basic properties of random variables; Mathematical expectation of random variable function; Chebyshev inequality; Covariance and properties of two random variables; Correlation coefficient and properties of two random variables.
Examination requirements
1. Understand the concept of digital characteristics of random variables (expectation, variance, standard deviation, covariance, correlation coefficient), use the basic properties of digital characteristics to calculate the digital characteristics of specific distributions, and master the digital characteristics of common distributions.
2. According to the probability distribution of random variable 1, the mathematical expectation eg (x) of its function is obtained; The mathematical expectation eg(x, y) of its function g (x, y) will be obtained according to the joint probability distribution of harmonic y of random variables.
3. Master Chebyshev inequality.
Fourth, the law of large numbers and the central limit theorem
Examination content
Chebyshev's law of large numbers Bemoulli's law of large numbers Khinchine's law of large numbers Poisson's theorem Lemoff-Laplace's theorem (binomial distribution takes normal distribution as the limit distribution) Levi-Lindbergh's theorem (central limit theorem of independent and identical distribution)
Examination requirements
1. Understand the conditions and conclusions of Chebyshev, Bernoulli and Qin Xin's law of large numbers, and understand its intuitive significance.
2. Grasp the conclusion and application conditions of Poisson theorem, and use Poisson distribution to approximately calculate the probability of binomial distribution.
3. Grasp the conclusions and application conditions of Mo Hoff-Laplacian central limit theorem and Levi-Lindbergh central limit theorem, and use relevant theorems to approximately calculate the probability of random events.
Five, the basic concepts of mathematical statistics
Examination content
Sample Mean, Sample Variance, Sample Moment of Empirical Distribution Function of Statistic of Simple Random Sample.
Examination requirements
Understand the concepts of population, simple random sample, statistics, sample mean and sample variance; Understand the empirical distribution function; Grasp the sampling distribution of normal population (standard normal distribution, χ2 distribution, F distribution, T distribution).
Parameter estimation of intransitive verbs
Examination content
Concept estimator of point estimation and estimation value moment estimation method Selection of maximum likelihood estimation Concept of standard estimator interval estimation Square test of interval estimation of single normal population mean and interval estimation of single normal population standard deviation interval estimation of mean difference and variance ratio of two normal populations
Examination requirements
1. Understand the concepts of point estimation, estimator and parameter estimation; Understand the concepts of unbiased estimator, minimum variance (validity) and consistency (consistency), and check unbiased estimator.
2. Master moment estimation method and maximum likelihood estimation method.
3. Master the solution of confidence intervals of mean and variance of a single normal population.
4. Grasp the solution of confidence interval of mean difference and variance ratio of two normal populations.
Seven, hypothesis testing
Examination content
The basic idea and steps of significance test and the hypothesis test of mean difference and variance of single and two normal populations
Examination requirements
1。 Understand the basic idea of significance construction research, master the basic steps of hypothesis testing, and understand two possible mistakes in hypothesis testing.
2. Understand the hypothesis test of the mean and variance of a single and two normal populations.
Test paper structure
(1) content ratio
Calculus is about 50%
Linear algebra accounts for about 25%
Probability theory and mathematical statistics account for about 25%
(B) the proportion of questions
Fill in the blanks and multiple-choice questions about 30%
Answer questions (including proof questions) about 70%