Limit continuity of (1) function?

1. Understanding the concept and representation of functions will establish the functional relationship of application problems. 2. Understand the boundedness, monotonicity, periodicity and parity of functions. 3. Understand the concepts of compound function and piecewise function, inverse function and implicit function. 5. Understand the concept of limit, the concepts of left limit and right limit of function and the relationship between the existence of function limit and left limit and right limit. 6. Master the nature of limit and four algorithms. 7. Master two criteria for the existence of limit, and use them to find the limit, and master the method of using two important limits to find the limit. 8. Understand the concepts of infinitesimal and infinitesimal, master the comparison method of infinitesimal, and find the limit with equivalent infinitesimal. 9. Understanding the concept of function continuity (including left continuity and right continuity) will distinguish the types of function discontinuity. 10. Understand the properties of continuous function and continuity of elementary function, understand the properties of continuous function on closed interval (boundedness, maximum theorem, mean value theorem), and apply these properties.

(2) Differential calculus of unary function?

1. Understand the concepts of derivative and differential, understand the relationship between derivative and differential, understand the geometric meaning of derivative, understand the tangent equation and normal equation of plane curve, understand the physical meaning of derivative, describe some physical quantities with derivative, and understand the relationship between function derivability and continuity. 2. Master the four algorithms of derivative and the derivation rules of compound function. Master the derivation formula of basic elementary functions. Understand the four algorithms of differential and the invariance of first-order differential form, and you will find the differential of function. 3. Understand the concept of higher derivative, and you will find the higher derivative of simple function. 4. You will find the derivative of piecewise function. I will find the derivative of implicit function, the function determined by parametric equation and the inverse function. 5. I will understand and apply Rolle theorem, Lagrange mean value theorem and Taylor theorem, and I will understand and apply Cauchy mean value theorem. 6. I will master the method of finding the limit of infinitive with Robida's law. 7. I will understand the concept of function extremum. Master the method of judging monotonicity of function and finding extreme value of function by derivative, and master the method of finding maximum and minimum value of function and its application. 8. Use derivative to judge the concavity and convexity of the function graph (Note: In the interval, let the function have the second derivative. When f'' (x) >; 0, the graph of f(x) is concave; When f "(x) < 0, the graph of f(x) is convex), the inflection point and horizontal, vertical and oblique asymptotes of the function graph will be found, and the function graph will be described.

(3) Integral of unary function

The test requirement is 1. Understand the concepts of original function, indefinite integral and definite integral. 2. Master the basic formula of indefinite integral, the properties of indefinite integral and definite integral and the mean value theorem of definite integral, and master the methods of substitution integral and partial integral. 3. Know how to find the integrals of rational functions, rational trigonometric functions and simple irrational functions. 4. Understand the function of the upper limit of integral and know how to find its derivative. Master Newton-Leibniz formula. 5. Understand the concept of generalized integral, and you will calculate generalized integral. 6. Master the expression and calculation of some geometric physical quantities (the area of plane figure, the arc length of plane curve, the volume and lateral area of rotating body, and the area of parallel section is known solid volume, work, gravity, pressure, center of mass, center of mass, etc.). ) and the average value of the function.

(d) Vector Algebra and Spatial Analytic Geometry

Examination requirements? 1. Understand the spatial rectangular coordinate system and the concept and representation of vectors. 2. Master the operation of vectors (linear operation, quantitative product, cross product, mixed product) and understand the conditions for two vectors to be vertically parallel. 3. Understand the coordinate expressions of unit vector, direction number, direction cosine and vector. Master the method of vector operation with coordinate expressions. 4. Principal plane equation and straight line equation and their solutions. 5. Know the included angles between planes, planes and straight lines, and straight lines. And the relationship between plane and straight line (parallelism, verticality, intersection, etc.) will be used. ) to solve related problems. 6. Will find the distance from a point to a straight line and a point to a plane. 7. Understand the concepts of surface equation and space curve equation. 8. Understand the common equations of quadric surface and its figure, and you will find the equations of simple cylinder and rotating surface. 9. Understand the parametric equation and general equation of space curve. Understand the projection of space curve on the coordinate plane, and you will find it.

(5) Differential calculus of multivariate functions?

The test requirement is 1. Understand the concept of multivariate function and the geometric meaning of bivariate 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, find total differential, understand the necessary and sufficient conditions for the existence of total differential, and understand the invariance of total differential form. 4. Understand the concepts of directional derivative and gradient. And master its calculation method. 5. Master the solution of the first and second partial derivatives of multivariate composite functions. 6. Understand the existence theorem of implicit function, and you will find the partial derivative of multivariate implicit function. 7. Understand the concepts of tangent plane and normal plane of space curve and tangent plane and normal plane of surface, and you will find their equations. 8. Understand the second-order Taylor formula of binary function. 9. Understand the concepts of multivariate function extremum and conditional extremum, and master the necessary conditions for the existence of multivariate function extremum. Knowing the sufficient conditions for the existence of extreme value of binary function, we can find the extreme value of binary function, find the conditional extreme value by Lagrange multiplier method, find the maximum and minimum value of simple multivariate function, and solve some simple application problems.

(6) Integral calculus of multivariate functions

Test requires 1. Understand the concepts of double integral and triple integral, understand the properties of double integral and understand the mean value theorem of double integral. 2. Master the calculation method of double integrals (rectangular coordinates and polar coordinates) and calculate triple integrals (rectangular coordinates, cylindrical coordinates and spherical coordinates). 3. Understand the concepts, properties and relationships of two kinds of curve integrals. 4. Master the calculation methods of two kinds of curve integrals. 5. Master Green's formula, and use the condition that the plane curve integral has nothing to do with the path to find the original function of the total differential of binary function. 6. Understand the concepts, properties and relations of two kinds of surface integrals, master the calculation methods of two kinds of surface integrals, and master the method of calculating surface integrals with Gaussian formula. Stokes formula will be used to calculate the curve integral. 7. The concepts of dissolution and curling will be introduced. Some geometric and physical quantities (area, volume, surface area, arc length, mass, center of mass, centroid, moment of inertia, gravity, work and flow, etc. ) will be calculated by multiple integral, curve integral and surface integral.

(7) infinite series

The test requirement is 1. Understand the concepts of convergence and divergence and sum of convergent constant series, and master the basic properties of series and the necessary conditions for convergence. 2. Master the conditions of geometric series and convergence and divergence of series. 3. Master the comparison of convergence of positive series and the ratio discrimination method. I can use the root value discrimination method. 4. Master the Leibniz discriminant method of staggered series. 5. Understand 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 series and the concept of function. 7. Understand the concept of convergence radius of power series and master the convergence radius of power series. Solution of convergence interval and convergence domain. 8. Knowing the basic properties of power series in its convergence interval (continuity of sum function, item-by-item derivation, item-by-item integration), we will find the sum function of some power series in its convergence interval, and then find the sum of some polynomial series. 9. Know the necessary and sufficient conditions for the function to expand into Taylor series. 10. Master the maclaurin expansion of,, and. They will be used to indirectly expand some simple functions into power series. 1 1. Knowing the concept of Fourier series and Dirichlet's convergence theorem, we expand the functions defined on the ground into Fourier series, and expand the functions defined on the ground into sine series and cosine series, and write the expressions of Fourier series and functions.

(8) Ordinary differential equations

The test requirement is 1. Understand the concept of differential equation and its order, solution, general solution, initial condition and special solution. 2. Master the solutions of differential equations with separable variables and first-order linear differential equations. 3. Know how to solve homogeneous differential equations, Bernoulli equations and fully differential equations. Can solve some differential equations with simple variables. 4. The order reduction method will be used to solve the following differential equations: 5. Understand the properties and structure of the solutions of linear differential equations. 6. Mastering the solution of second-order homogeneous linear differential equations with constant coefficients will solve some homogeneous linear differential equations with constant coefficients higher than the second order. 7. Know how to solve free terms such as polynomial, exponential function, sine function, cosine function and the second-order non-constant coefficient of their sum and product.

Second, linear algebra.

(1) determinant?

Examination content: the concept and basic properties of determinant? Theorem of expansion of determinant by row (column)?

Examination requirements: 1. Understand the concept and properties of determinant. 2. Applying the properties of determinant and determinant, the determinant is calculated according to the row (column) expansion theorem.

(2) Matrix?

Examination contents: the concept of matrix, linear operation of matrix, the concept and properties of transposed inverse matrix of determinant matrix of multiplication matrix, necessary and sufficient conditions for matrix reversibility, elementary transformation of adjoint matrix, rank matrix equivalent block matrix of elementary matrix and its operation?

Examination requirements: 1. Understand the concepts and properties of matrix, identity matrix, quantitative matrix, diagonal matrix, triangular matrix, symmetric matrix and antisymmetric matrix. 2. Master the linear operation, multiplication, transposition and its operation rules of matrix, and understand the determinant properties of square matrix power and square matrix product. 3. Understand the concept of inverse matrix, master the properties of inverse matrix and the necessary and sufficient conditions for matrix reversibility. In order to understand the concept of adjoint matrix, we will use adjoint matrix to find the inverse matrix. 4. Understand the concept of elementary transformation of matrix, the properties of elementary matrix, the concept of matrix equivalence and the concept of matrix rank, and master the method of finding the rank and inverse matrix of matrix by elementary transformation. 5. Understand the block matrix and its operation.

(3) Vector?

Exam content:? The linear combination of concept vectors of vectors and the relationship between linear correlation of linear representation vector group and the largest linear independent group of linear independent vector group being equivalent to the rank of rank vector group of vector group and the rank of matrix, as well as the orthogonal normalization method of inner product linear independent vector group of related concepts.

Examination requirements:? 1. Understand the concepts of n-dimensional vectors, linear combinations of vectors and linear representations. 2. Understand the concepts of 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 concepts of maximum linearly independent group and rank of vector group, and you will find the maximum linearly independent group and rank of vector group. 4. Understand the concept of vector group equivalence and the relationship between the rank of a matrix and the rank of its row (column) vector group. 5. Understand the concepts of N-dimensional vector space, subspace, basis, dimension and coordinates. 6. Understand the formulas of base transformation and coordinate transformation, and you will find the transfer matrix. 7. Understand the concept of inner product and master the Schmidt method of orthogonal normalization of linear independent vector groups. 8. Understand the concepts and properties of normalized orthogonal basis and orthogonal matrix.

(4) Linear equations?

Exam 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 Structures of Solutions of Homogeneous Linear Equations and General Solutions of Non-homogeneous Linear Equations in General Solution Space?

Examination requirements? L you can use clem's law. 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, general solution and solution space of homogeneous linear equations, and master the solution of basic solution system and general solution of homogeneous linear equations. 4. Understand the structure of solutions of nonhomogeneous linear equations and the concept of general solutions. 5. Master the method of solving linear equations with elementary line transformation.

(5) What are the eigenvalues and eigenvectors of a matrix?

Exam content:? Concepts of eigenvalues and eigenvectors of matrices, property similarity transformation, concepts of similar matrices and necessary and sufficient conditions for similar diagonalization of property matrices, eigenvalues, eigenvectors and similar diagonal matrices of real symmetric matrices of similar diagonal matrices?

Examination requirements:? 1. Understand the concepts and properties of eigenvalues and eigenvectors of a matrix, and you will find the eigenvalues and eigenvectors of the matrix. 2. Understand the concept and properties of similar matrix and the necessary and sufficient conditions for similar diagonalization of matrix, and master the method of transforming matrix into similar diagonal matrix. 3. Master the properties of eigenvalues and eigenvectors of real symmetric matrices.

(6) quadratic?

Exam content:? Quadratic form and its matrix represent contract transformation and rank inertia theorem of quadratic form of contract matrix. The canonical form and canonical form of quadratic form are transformed into canonical quadratic form and the positive definiteness of its matrix by orthogonal transformation and matching method.

Examination requirements:? 1. Master quadratic form and its matrix representation, understand the concept of quadratic form rank, understand the concepts of contract change and contract matrix, and understand the concepts of standard form and standard form of quadratic form and inertia theorem. 2. Master the method of transforming quadratic form into standard form by orthogonal transformation, and will transform quadratic form into standard form by matching method. 3. Understand the concepts of positive definite quadratic form and positive definite matrix, and master their discrimination methods.

Third, probability theory and mathematical statistics

(a) random events and probability?

Exam content:? The relationship between random events and sample space events and the basic properties of concept probability of complete operation event group probability; Classical probability, geometric probability, basic formula of conditional probability; Independent repeated testing of events?

Examination requirements:? 1. Understand the concept of sample space (basic event space), understand the concept of random events, and master the relationship and operation of 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 of 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.

(2) Random variables and their distribution?

Exam content:? Concept and properties of distribution function of random variables Probability distribution of discrete random variables Probability density of continuous random variables Distribution of common random variables Random variable function?

Test requirements: 1. Understand the concept of random variables. Understand the distribution function? The concept and nature of. The probability of events related to random variables will be calculated. 2. Understand the concept and probability distribution of discrete random variables, and master 0- 1 distribution, binomial distribution, geometric distribution, hypergeometric distribution, Poisson distribution and their applications. 3. Understand 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, and master uniform distribution, normal distribution, exponential distribution and their applications, where the parameter is λ (λ >; The probability density of exponential distribution of 0) is 5. The distribution of random variable function will be found.

(3) Multidimensional random variables and their distribution?

What is the content of the exam? Multidimensional random variables and their distributions Probability distribution, edge distribution and conditional distribution of two-dimensional discrete random variables Probability density, marginal probability density and conditional density of two-dimensional continuous random variables? Independence and irrelevance of random variables The distribution of two-dimensional random variables is commonly used. The distribution of simple functions of two or more random variables?

Examination requirements? 1. Understand the concept of multidimensional random variables, understand the concept and properties of multidimensional random variable distribution, understand the probability distribution, edge distribution and conditional distribution of two-dimensional discrete random variables, understand the probability density, edge density and conditional density of two-dimensional continuous random variables, and find the probability of related events of two-dimensional random variables. 2. Understand the concepts of independence and irrelevance of random variables, and master the conditions of mutual independence of random variables. 3. Grasp the two-dimensional uniform distribution, understand the probability density of the two-dimensional normal distribution, and understand the probability meaning of its parameters. 4. Will find the distribution of simple functions of two random variables, and will find the distribution of simple functions of multiple independent random variables.

(4) What are the numerical characteristics of random variables?

What is the content of the exam? Mathematical expectation (mean), variance, standard deviation and their properties of random variables Mathematical expectation moment, covariance, correlation coefficient and their properties of random variable functions?

Examination requirements? 1. Understand the concept of numerical characteristics of random variables (mathematical expectation, variance, standard deviation, moment, covariance, correlation coefficient), and use the basic properties of numerical characteristics to master the numerical characteristics of common distributions? 2. Know the mathematical expectation of random variable function.

(5) Law of large numbers and central limit theorem?

What is the content of the exam? Chebyshev Inequality Chebyshev's Law of Large Numbers Bernoulli's Law of Large Numbers De Morville-Laplace Theorem Levy-Lindbergh Theorem?

Examination requirements? 1. Understanding Chebyshev Inequality. 2. Understand Chebyshev's law of large numbers, Bernoulli's law of large numbers and Sinchin's law of large numbers (the law of large numbers of independent and identically distributed random variable sequences). 3. Understand de moivre-Laplace Theorem (binomial distribution takes normal distribution as the limit distribution) and Levi-Lindbergh Theorem (central limit theorem of independent identically distributed random variable sequence).

(6) Basic concepts of mathematical statistics?

What is the content of the exam? Sample variance and sample moment distribution of population sample mean Simple random sample statistics of quantile normal population common sampling distribution?

Examination requirements? 1. Understand the concepts of population, simple random sample, statistics, sample mean, sample variance and sample moment, where sample variance is defined as:? 2. Understand the concept and nature of distribution, distribution and distribution, understand the concept of upper quantile and look up the table for calculation. 3. Understand the common sampling distribution of normal population.

(7) Parameter estimation?

What is the content of the exam? Concept estimation of point estimation and estimated value Method of moment estimation Maximum likelihood estimation Method of estimation criterion Interval estimation Concept Interval estimation of mean and variance of a single normal population Interval estimation of mean difference and variance ratio of two normal populations?

The test requirement is 1. Understand the concepts of point estimation, estimator and parameter estimation. 2. Master moment estimation method (first-order moment, second-order moment) and maximum likelihood estimation method. 3. Understand the concepts of unbiased estimation, validity (minimum variance) and consistency (consistency) and verify unbiased estimation. 4. Understand the concept of interval estimation.

(8) Hypothesis test?

What is the content of the exam? Two types of false hypotheses in significance test hypothesis test of mean and variance of single and two normal populations?

Examination requirements? 1. Understand the basic idea of significance test, master the basic steps of hypothesis test, and understand two possible errors in hypothesis test. 2. Master the hypothesis test of the mean and variance of single and two normal populations.