calculus

I. Function, Limit and Continuity

Examination content

The concept and representation of function, the boundedness, monotonicity, periodicity and parity of function, inverse function, piecewise function and implicit function, the properties of basic elementary function and the establishment of functional relationship of graphic elementary function.

Definitions and properties of sequence limit and function limit, left limit and right limit of function, concepts and relationships of infinitesimal and infinitesimal, properties of infinitesimal and four operational limits of infinitesimal, two important limits: monotone bounded criterion and pinch criterion;

Concept of Function Continuity Types of Discontinuous Points of Functions Continuity of Elementary Functions Properties of Continuous Functions on Closed Interval

Examination requirements

1. Understand the concept of function and master the expression of function, and you 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.

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 limit and right limit).

6. Understand the nature of limit and two criteria for the existence of limit, master four algorithms of limit, and master the method of finding limit by using two important limits.

7. Understand the concept and basic properties of infinitesimal, master the comparison method of infinitesimal, and understand the concept of infinitesimal and its relationship with infinitesimal.

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 function and continuity of elementary function, understand the properties of continuous function on closed interval (boundedness, maximum theorem and mean value theorem), and apply these properties.

Second, the differential calculus of unary function

Examination content

The relationship between the geometric meaning of derivative and differential concepts and the derivability and continuity of economic functions; Four operations of tangent, normal derivative and differential of plane curve: derivative compound function of basic elementary function; Differential method of inverse function and implicit function; Invariant differential mean value theorem in first-order differential form; Hospital rules; The concavity and convexity of extreme function graph: inflection point and asymptote describe the maximum and minimum value of function graph.

Examination requirements

1. Understand the concept of derivative and the relationship between derivability and continuity, understand the geometric and economic significance of derivative (including the concepts of allowance and elasticity), and find the tangent equation and normal equation of plane curve.

2. Mastering the derivation formula of basic elementary function, four arithmetic rules of derivation and the derivation rule of compound function, we can obtain the derivation of piecewise function, inverse function and implicit function.

3. If you understand the concept of higher derivative, you will find the higher derivative of simple function.

4. Understand the concept of differential, the relationship between derivative and differential, and the invariance of first-order differential form, and you will find the differential of function.

5. Understand Rolle theorem, Lagrange mean value theorem, Taylor theorem and Cauchy mean value theorem, and master the simple application of these four theorems.

6. Will use the Lobida rule to find the limit.

7. Master the method of judging monotonicity of function, understand the concept of function extreme value, and master the solution and application of function extreme value, maximum value and minimum value.

8. The concavity and convexity of the function graph can be judged by the derivative (note: in the interval, let the function have the second derivative. When, the figure is concave; When the graph is convex, you will find the inflection point and asymptote of the function graph.

9. Graphics that can describe 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 of basic integral formula and the basic properties of the mean value theorem of definite integral The function of upper limit of integral and its derivative Newton-Leibniz formula replaces the integral method of indefinite integral and definite integral and the application of partial integral Abnormal (generalized) integral definite integral.

Examination requirements

1. Understand the concepts of original function and indefinite integral, master the basic properties and basic integral formula of indefinite integral, and master the substitution integral method and integration by parts of indefinite integral.

2. Understand the concept and basic properties of definite integral, understand the mean value theorem of definite integral, understand the function of upper limit of integral and find its derivative, and master Newton-Leibniz formula, method of substitution and integration by parts of definite integral.

3. Will use definite integral to calculate the area of plane figure, the volume of rotating body and the average value of function, and will use definite integral to solve simple economic application problems.

4. Understand the concept of generalized integral and calculate generalized integral.

Four, multivariate function calculus

Examination content

Concept of multivariate function, geometric meaning of binary function, concept of limit and continuity of binary function, concept and calculation of partial derivative of multivariate function in bounded closed region, derivative method of multivariate composite function and derivative method of implicit function, extreme value and conditional extreme value of second-order partial derivative fully differential multivariate function, concept of max-min double integral, basic properties and calculation of simple abnormal double integral in unbounded region.

Examination requirements

1. Understand the concept of multivariate function and the geometric meaning of bivariate function.

2. Understand the concepts of limit and continuity of binary function and the properties of binary continuous function in bounded closed region.

3. Knowing the concepts of partial derivative and total differential of multivariate function, we can find the first and second partial derivatives of multivariate composite function and the total differential and partial derivative of multivariate implicit function.

4. Understand the concepts of extreme value and conditional extreme value of multivariate function, master the necessary conditions of extreme value of multivariate function, understand the sufficient conditions of extreme value of binary function, find the extreme value of binary function, find the conditional extreme value by Lagrange multiplier method, find the maximum value and minimum value of simple multivariate function, and solve the simple application problem.

5. Understand the concept and basic properties of double integral, and master the calculation method of double integral (rectangular coordinates. Polar coordinates), understand the simple abnormal double integral of unbounded region and calculate it.

Five, infinite series

Examination content

Concept of Convergence and Divergence of Constant Term Series Basic properties and necessary conditions of geometric series and series convergence and their convergence criteria Absolute convergence and conditional convergence of positive term series Power series of arbitrary term series and its convergence radius. The basic properties of power series sum function in convergence interval (referring to open interval) and convergence domain; the solution of simple power series sum function; the power series expansion of elementary function.

Examination requirements

1. Understand the convergence and divergence of series. The concept of the sum of convergent series.

2. Understand the basic properties of series and the necessary conditions of convergence and divergence of series, master the conditions of convergence and divergence of geometric series and series, and master the comparative judgment method and ratio judgment method of convergence and divergence of positive series.

3. Understand the concepts of absolute convergence and conditional convergence of arbitrary series and the relationship between absolute convergence and convergence, and understand Leibniz discriminant method of staggered series.

4. Will find the convergence radius, convergence interval and convergence domain of power series.

5. Knowing the basic properties of power series in its convergence interval (continuity of sum function, item-by-item derivation, item-by-item integration), we can find the sum function of simple power series in its convergence interval.

6. understand; Understand ... and maclaurin inflation.

Six, ordinary differential equations and difference equations

Examination content

Basic concepts of ordinary differential equations The properties and structural theorems of solutions of first-order homogeneous linear differential equations with separable variables The concepts of difference equations of second-order homogeneous linear differential equations with constant coefficients and simple non-homogeneous linear differential equations The general solutions and special solutions of first-order linear differential equations are simply applied.

Examination requirements

1. Understand differential equations and their concepts such as order, solution, general solution, initial condition and special solution.

2. Master the solutions of differential equations, homogeneous differential equations and first-order linear differential equations with separable variables.

3. Second-order homogeneous linear differential equations with constant coefficients can be solved.

4. Knowing the properties and structure theorems of solutions of linear differential equations, you can solve second-order non-homogeneous linear differential equations with constant coefficients with polynomial, exponential function, sine function and cosine function as free terms.

5. Understand the concepts of difference and difference equation and their general and special solutions.

6. Understand the solution method of the first-order linear difference equation with constant coefficients.

7. Can use differential equations to solve simple economic application problems.

linear algebra

I. Determinants

Examination content

The concept and basic properties of determinant The 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, concept and properties of transposed inverse matrix of determinant matrix, necessary and sufficient conditions for matrix reversibility, equivalent block matrix of elementary transformation of matrix and rank matrix of elementary matrix and its operation.

Examination requirements

1. Understand the concept of matrix, the definitions and properties of identity matrix, quantitative matrix, diagonal matrix and triangular matrix, and the definitions and properties of symmetric matrix, antisymmetric matrix and orthogonal 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, 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. Understand the concepts of elementary transformation of matrix and elementary matrix and matrix equivalence, understand the concept of matrix rank, and master the method of finding the inverse matrix and rank of matrix by elementary transformation.

5. Understand the concept of block matrix and master the algorithm of block matrix.

Third, the vector

Examination content

Linear combination of concept vectors of vectors and linear representation of vector groups Linear correlation and maximum linear independent equivalent vector group of linear independent vector group Orthogonal normalization method of linear independent vector group Rank between rank of vector group and rank of matrix.

Examination requirements

1. Understand the concept of vectors and master the operations of vector addition and multiplication.

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. By understanding the concept of maximal linearly independent group of vector group, we can 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 matrix and the rank of its row (column) vector group.

5. Understand the concept of inner product and master the Schmidt method of orthogonal normalization of linear independent vector groups.

Fourth, linear equations.

Examination content

Cramer's law for linear equations: Determination of existence and nonexistence of 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. will use Cramer's rule to solve linear equations.

2. Master the method of judging the existence and non-existence of non-homogeneous linear equations.

3. 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.

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.

Eigenvalues and eigenvectors of verb (abbreviation of verb) matrix

Examination content

The concepts of eigenvalues and eigenvectors of matrices, the concepts of property similarity matrices and the necessary and sufficient conditions for similar diagonalization of property matrices, and the eigenvalues and eigenvectors of real symmetric matrices of similar diagonal matrices and similar diagonal 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 matrix, understand the necessary and sufficient conditions for matrix similarity to diagonal, and master the method of transforming matrix into similar diagonal matrix.

3. Master the properties of eigenvalues and eigenvectors of real symmetric matrices.

Sixth, quadratic form

Examination 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. Understand the concept of quadratic form, express quadratic form in matrix form, and understand the concepts of contract transformation and contract matrix.

2. Understand the concept of rank of quadratic form, the concepts of standard form and standard form of quadratic form, and inertia theorem, and transform quadratic form into standard form by orthogonal transformation and collocation method.

3. Understand the concepts of positive definite quadratic form and positive definite matrix, and master their discrimination methods.

Probability and mathematical statistics

I. Random events and probabilities

Examination content

The relationship between random events and events in sample space and the basic properties of complete operation concept probability Basic formula of classical probability of event group probability Geometric probability Conditional independent repetition test of probability 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.

Second, random variables and their distribution

Examination 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 Distribution of random variable functions

Examination requirements

1. Understand the concepts of random variables and distribution functions.

The concept and properties of will 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, geometric 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, and master uniform distribution, normal distribution, exponential distribution and their applications, in which the probability density of exponential distribution with parameters is

5. Find the distribution of random variable function.

Thirdly, the distribution of multidimensional random variables.

Examination content

Probability distribution, edge distribution and conditional distribution of multidimensional random variables and their distribution functions Probability density, marginal probability density and conditional density of two-dimensional continuous random variables The independence and irrelevance of common two-dimensional random variables The function distribution of two or more random variables.

Examination requirements

1. Understand the concept and basic properties of the distribution function of multidimensional random variables.

2. Understand the probability distribution of two-dimensional discrete random variables and the probability density of two-dimensional continuous random variables, and master the edge distribution and conditional distribution of two-dimensional random variables.

3. Understand the concepts of independence and irrelevance of random variables, master the conditions of mutual independence of random variables, and understand the relationship between irrelevance 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. The distribution of function will be found according to the joint distribution of two random variables, and the distribution of function will be found according to the joint distribution of several independent random variables.

Fourth, the numerical characteristics of random variables

Examination content

Mathematical expectation (mean), variance and standard deviation of random variables and their properties; Mathematical expectation of random variable function; Moment, covariance, correlation coefficient and their properties of Chebyshev inequality.

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.

3. Understand Chebyshev inequality.

Law of Large Numbers and Central Limit Theorem

Contrast: No change.

Basic concepts of mathematical statistics of intransitive verbs

Contrast:

1. Modify the concept of "population, simple random sample, statistics, sample mean, sample variance and sample moment" to "population, simple random sample, statistics, sample mean, sample variance and sample moment".

2. The understanding of "upper quantile of standard normal distribution, distribution, distribution" in test requirement 2 is changed to "upper quantile of standard normal distribution, distribution, distribution".

3. Delete the sample mean difference and sample variance ratio of the normal population from test requirement 3.

4. In test requirement 4, understand "the concept and properties of empirical distribution function" instead of "the concept and properties of empirical distribution function".

5. Delete "the empirical distribution function will be obtained according to the sample value" from test requirement 4.

Seven. parameter estimation

Contrast:

The content of 1. test is deleted from the concept of "selection criteria of estimators, interval estimation of mean of a single normal population, interval estimation of population variance and standard deviation of a single normal population, and interval estimation of mean difference and variance ratio of two normal populations".

2. In the test requirement 1, the concept of "point estimation, estimator and estimated value of parameters" is changed to "point estimation, estimator and estimated value of parameters".

3. "Understanding the concepts of unbiased estimator, validity (minimum variance) and consistency (consistency) is deleted from the inspection requirement 1, and unbiased estimator will be verified".

4. Test requirement 3 deleted "Master the general method of establishing confidence intervals of unknown parameters (bilateral and unilateral); Master the solution of the confidence interval of the mean, variance, standard deviation and moment of the normal population and the numerical characteristics related to it. "

5. Test requirement 4 deleted "the solution to master the confidence interval of the mean difference and variance ratio of two normal populations and related numerical characteristics".

Eight, hypothesis testing

Contrast: delete the whole chapter

There may be a small adjustment every year, so pay attention to it in August and September.