Complement of top and number two

Three outlines of mathematics

The third outline of mathematics in 2009 (2008-1-1120: 45: 45) Label: Random talk

Three Outline of Mathematics in 2009

calculus

I. Function, Limit and Continuity

Examination content

The concepts of boundedness, monotonicity, periodicity and parity of functions and the representation of compound function, inverse function and division

Properties of basic elementary functions of piecewise function and implicit function and the establishment of functional relationship of graphic elementary function

The definitions and properties of sequence limit and function limit: left limit and right limit of function; Infinitesimal and infinitesimal.

The concept of infinitesimal and its relationship with infinitesimal properties and the existence of four operational limits of infinitesimal comparison limits.

Two criteria: monotone bounded criterion and pinch criterion.

Two important limitations:

sin 1

lim 1，lim( 1 )x

x x

x

e

x x

?

Concept of functional continuity Types of discontinuous points of functional continuity of elementary functions of continuous functions on closed intervals

nature

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 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 two important things.

Limit the method of finding the limit.

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 quantity.

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, and understand the properties of continuous functions on closed intervals (boundedness, the most

Maximum and minimum theorem, intermediate value theorem), and will apply these properties.

Two. Differential calculus of univariate function

Examination content

The concepts of derivative and differential: the relationship between the geometric meaning of derivative and the derivability and continuity of economic significance function

Four operations of tangent derivative, normal derivative and plane curve derivative of basic elementary function

Differential method for invariance of composite function, inverse function and implicit function in the first differential form of higher derivative

Differential mean value theorem; Monotonicity Criterion of L'H?pital's Law Function: Extreme Function Diagram of Function

The concavity and convexity, inflection point and maximum and minimum value of the drawing function of 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 edge).

The concepts of internal dimension and elasticity), we will find the tangent equation and normal equation of plane curve.

2. Master the four arithmetic rules of derivative and the derivative rule of compound function, master the derivative formula of basic elementary function, and know how to solve.

The derivative of piecewise function will find the derivative of inverse function and implicit function.

If you understand the concept of higher derivative, you will find the higher derivative of a 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's theorem, Lagrange's mean value theorem and Taylor's theorem,

Cauchy mean value theorem, master the simple application of these four theorems.

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

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

Solution and its application.

8. Will judge the concavity and convexity of the function graph by derivative (note: in the interval (a, b), let the function f(x) 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), you will find the function graph.

The inflection point and asymptote of.

9. Can describe the graphics of simple functions.

Three. Integral of unary function

Examination content

The concept of primitive function and indefinite integral The basic properties of indefinite integral Basic integral formula The probability of definite integral

The function of the upper limit of integral and its derivative in the mean value theorem of definite integral

Substitution integral method and partial inverse integral of indefinite integral and definite integral of Newton-Leibniz formula

Application of constant (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 indefinite product.

Variational integral and partial integral.

2. Understand the concept and basic properties of definite integral and the mean value theorem of definite integral, understand the function of the upper limit of integral and find its derivative.

Master Newton-Leibniz formula, substitution integral method of definite integral and integration by parts.

3. I will use definite integral to calculate the area of plane figure, the volume of rotating body and the average value of function, and I will use definite integral to solve it.

Simple economic application problem.

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

Four. Multivariate function calculus

Examination content

The concept of multivariate function, the geometric meaning of binary function, the limit of binary function and the concept of continuity are bounded.

Properties of Binary Continuous Functions on Closed Fields Concept of Partial Derivatives of Multivariate Functions and Calculation of Solutions of Multivariate Composite Functions

Derivation method of second-order partial derivative fully differential multivariate function and derivation method of implicit function extreme value and conditional extreme value,

Concept, basic properties and calculation of simple abnormal double integral on unbounded domain of maximum and minimum double integral

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-order and second-order partial derivative and total differential of multivariate composite function.

Division, will find the total differential, will find the partial derivative of multivariate implicit function.

4. Understand the concepts of multivariate function extreme value and conditional extreme value, master the necessary conditions for the existence of multivariate function extreme value, and understand binary function.

We will find a sufficient condition for the existence of an extreme value of a number, an extreme value of a binary function, and a conditional extreme value by Lagrange multiplier method.

The maximum and minimum of simple multivariate function, and will 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).

Solve the simple abnormal double integral on unbounded domain and calculate it.

5. The original mathematics 4 of infinite series is added, and the original mathematics 3 remains unchanged.

Examination content

Conceptual convergence series and the basic properties and convergence of conceptual series

Geometric series and p series and their necessary conditions for convergence: a judgment method of convergence and divergence of positive series: any term level.

Absolute convergence and conditional convergence, convergence radius and convergence of staggered series of sequence and power series of Leibniz theorem

Convergence interval (open interval) and function power series and the basis of convergence domain power series on convergence interval

The solution of the sum function of this simple power series; Power series expansion of elementary functions.

Examination requirements

1. Understand the concepts of convergence and divergence of term series and sum of convergent series.

2. Understand the basic properties of series and the necessary conditions for series convergence, and master the convergence and divergence of geometric series and P series.

◆ Master the comparison and comparison of convergence 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 staggered series.

Leibniz discriminant method.

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

5. Understand the basic properties of power series in its convergence interval (continuity of sum function, derivative item by item, integral item by item), and it will

Find the sum function of simple power series in its convergence interval.

6. Understand ex, sinx, cosx, ln( 1? X) and (1? x)？ Maclaurin expansion.

Ordinary differential equation of intransitive verbs

Examination content

Basic concepts of ordinary differential equations First-order linearity of separable variable differential equations and homogeneous differential equations

Properties of solutions of linear differential equations and structural theorems of solutions: original mathematics 4, new addition, original number

Learn the new content of the original mathematics 4 of the second order homogeneous linear differential equation with constant coefficients, and the original mathematics 3 remains unchanged.

And the new content of the original mathematics, 4 simple non-homogeneous linear differential equations, and 3 invariant differences and differences of the original mathematics.

The concept of equation, the new content of primitive mathematics, the special solution of primitive mathematics, and the general solution of primitive mathematics.

4 new content, original mathematics 3 invariant first-order linear difference square original mathematics 4 new content,

Simple application of the original mathematics 3. The new content of the original mathematics 4. Invariant range differential equation 4.

Examination requirements

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

2. Master the methods of solving 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. Understand the properties and structural theorems of solutions of linear differential equations, and know that the free terms of solutions are polynomials, exponential functions and sine functions.

Second-order non-homogeneous linear differential equations with constant coefficients for digital and cosine functions.

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

Two. Matrix

Examination content

The line of the product of the power matrix of the linear operation matrix of the concept matrix

The concept and properties of transposed inverse matrix of matrix: necessary and sufficient conditions for matrix invertibility

Elementary transformation of adjoint matrix equivalence of rank matrix of elementary matrix

Block matrix and its operation

Examination requirements

1. Understand the concept of matrix, the definition and properties of identity matrix, quantitative matrix, diagonal matrix and triangular matrix, and understand

Definition and properties of symmetric matrix, antisymmetric matrix and orthogonal matrix.

2. Master the linear operation, multiplication, transposition and its operation rules of matrices, and understand the rows of the power of a square matrix and the product of a square matrix.

Properties of the formula.

3. Understand the concept of inverse matrix, master the properties of inverse matrix and the necessary and sufficient conditions for matrix reversibility, and understand adjoint matrix.

Concept, will use adjoint matrix to find the inverse matrix.

4. Understand the concepts of matrix elementary transformation and elementary matrix and matrix equivalence, understand the concept of matrix rank, and master the use of elementary transformation.

The method of finding the rank and inverse matrix of a matrix.

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

Three. vectors

Examination content

The linear combination of concept vectors of vectors and the linear representation of vector groups are linearly related and linearly independent.

The vector group rank of the equivalent vector group matrix of the largest linearly independent group of quantities

Orthogonal normalization method of inner product linear independent vector group of relation vector

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 linearity of vector groups.

Correlation properties and discrimination methods of correlation and linear independence.

3. Understand the concept of maximal linearly independent group 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 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.

Four. linear equation

Examination content

Cramer's law for linear equations: Determination of homogeneous linearity of linear equations with and without solutions

Basic solution system of equations and general solution (derivation group) of nonhomogeneous linear equations and corresponding homogeneous linear equations

General solution of nonhomogeneous linear equations.

Examination requirements

1. will use Cramer's rule to solve linear equations.

2. Master the judgment method of non-homogeneous linear equations with and without solutions.

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.

Law.

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. Eigenvalues and eigenvectors of matrices

Examination content

Concepts of eigenvalues and eigenvectors of matrices, property similarity transformation, concepts and property moments of similar matrices.

Necessary and Sufficient Conditions for Similar Diagonalization of Matrices and the Sum of Eigenvalues, Eigenvectors and Real Symmetric Matrices of Similar Diagonal Matrices

Its similar diagonal matrix

Examination requirements

1. Understand the concepts of matrix eigenvalues and eigenvectors, master the properties of matrix eigenvalues, and master the sum of matrix eigenvalues.

Eigenvector method.

2. Understand the concept of matrix similarity, master the properties of similar matrices, and understand the necessary and sufficient conditions for matrix similarity diagonalization.

Master the method of transforming matrix into similar diagonal matrix.

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

The intransitive verb square

Examination content

Quadratic form and its matrix represent contract transformation and rank inertia theorem of quadratic form of contract matrix. Standard sum of quadratic form

By using orthogonal transformation and collocation method and the positive definiteness of its matrix, the canonical form is transformed into the quadratic form of canonical form.

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 canonical form and canonical form of quadratic form, and inertia theorem, and use positive.

The quadratic form is transformed into the standard form by the intersection transformation matching 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 sample space events and the basic properties of complete event group probability concept probability

The basic formula of classical 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, and calculate classical probability and geometric probability.

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 testing,

Master the calculation method of related event probability.

Second, random variables and their distribution

Examination content

Concept and properties of random variable random variable distribution function Probability distribution of discrete random variable and continuous random variable

Probability density of variables Common random variable distribution Random variable function distribution

Examination requirements

1. Understand the concept of random variables and the distribution function F (x)? P{X？ x }(？ x？ The concept and nature of),

Calculate the probability of an event associated with a random variable.

2. Understand the concept and probability distribution of discrete random variables, and master 0- 1 distribution, binomial distribution B(n, p), geometry.

Distribution, hypergeometric distribution, Poisson distribution P (? ) and its application.

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 the uniform distribution U (a, b) and normal distribution.

n(？ ,? 2) Exponential distribution and its application, where the parameter is? (0) e(？ Probability density of)

for

, 0,

( )

0, 0.

e x

f x

x

If x>

if

5. Find the distribution of random variable function.

Three, multidimensional random variables and their distribution

Examination content

Probability distribution, edge distribution and conditional distribution of multidimensional random variables and their distribution functions II

Independence and irrelevance of probability density, marginal probability density and conditional density of multidimensional continuous random variables

The distribution of common two-dimensional random variables The distribution of functions 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 two-dimensional randomness.

Edge distribution and conditional distribution of variables.

3. Understand the concepts of independence and irrelevance of random variables, master the conditions of independence of random variables, and understand random variables.

The relationship between irrelevance and quantity independence.

4. Grasp the two-dimensional uniform distribution and two-dimensional normal distribution n (2 2, understand the probability meaning of parameters.

1 2 1 2 ? ,? ; ? ,? ; ? )

5. The distribution of its function will be found according to the joint distribution of two random variables, and the distribution of its function will be found according to the joint distribution of several independent random variables.

Distribution Find the distribution of its function.

Fourth, the numerical characteristics of random variables

Examination content

Mathematical expectation (mean), variance and standard deviation of random variables and their properties The mathematical expectation of random variable function is tangent to snow.

Moment, covariance, correlation coefficient and their properties of Markov inequality

Examination requirements

1. Understand the concept of digital characteristics of random variables (mathematical expectation, variance, standard deviation, moment, covariance, correlation coefficient),

Will use the basic properties of digital characteristics to master the digital characteristics of common distribution.

2. Know the mathematical expectation of random variable function.

3. Understand Chebyshev inequality

Law of Large Numbers and Central Limit Theorem

Examination content

Chebyshev's law of large numbers Bernoulli's law of large numbers Qinqin's law of large numbers

Morville's Law-Laplace Theorem

theorem

Examination requirements

1. Understand Chebyshev's law of large numbers, Bernoulli's law of large numbers and Hinchin's law of large numbers (independent and identically distributed random variable sequence

Law of large numbers)

2. Understand the central limit theorem of de moivre-Laplace (binomial distribution takes normal distribution as the limit distribution) and Levi-Linde.

Berg's central limit theorem (the central limit theorem of independent and identically distributed random variable sequences), and will be approximated by related theorems.

Calculate the probability of random events.

Six, the basic concept of mathematical statistics, the original mathematics 4 new content, the original mathematics 3 unchanged.

Examination content

Sample Mean, Sample Variance and Sample Moment of Empirical Distribution Function of Statistic of Simple Random Sample.

2 distribution t distribution f distribution quantile normal population commonly used sampling distribution

Examination requirements

1. Understand the concepts of population, simple random sample, statistics, sample mean, sample variance and sample moment, where sample square

The definition of difference is

2 2

1

1 ( )

1

n

I

I

S X X

n？

2. Understanding is produced? Typical patterns of 2 variables, t variables and f variables; Understand the standard normal distribution. 2 distribution, t distribution and

The upper side of the F distribution? Quantile, will look up the corresponding numerical table.

3. Grasp the sampling distribution of sample mean, sample variance and sample moment of normal population.

4. Understand the concept and properties of empirical distribution function.

7. Add original mathematics 4 to parameter estimation, reduce original mathematics 3, and completely remove hypothesis testing.

Examination content

Concept estimator and moment estimator of point estimation maximum likelihood estimation method

Examination requirements

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.