Mathematics 1 Exam Outline

Exam Subjects: Advanced mathematics, linear algebra, probability theory and mathematical statistics

Examination form and examination paper structure

I. Full score and test time

The examination paper is divided into 150 points, the test time is 180 minutes

Second, answer the way

The answer is closed book, written test

Third, the content structure of the test paper

About 56% Higher Education

Linear algebra approx. 22%

Probability theory and mathematical statistics about 22%

IV. structure of test paper types

Single topic 8 small questions, 4 points for each small topic, total 32 points

Fill in the blanks 6 small questions, 4 points for each small question, total 24 points

Answer question (including proof) 9 small questions, total 94 points

Advanced mathematics

I. Function, limit, continuous

Exam Content

The concept of function and the nature of the bounded, monotonic, periodic and odd-even complex functions, inverse functions, piecewise functions and basic elementary functions of the implicit function and the establishment of the function relation of the graph elementary function

The definition of sequence limit and function limit and the concept of left limit and right limit infinitesimal and infinite mass of the function and the relationship between the properties of infinitesimal quantity and the arithmetic limit of the limit of infinite small quantity two criteria exist: monotone bounded criterion and clamping criterion two important limits:

function continuous conceptual function discontinuity type the nature of continuity function on continuous closed interval of elementary function

Exam Requirements

1. Understanding the concept of functions and mastering the representation of functions will establish the functional relationship of application problems.

2. Understanding the boundedness, monotonicity, periodicity, and parity of functions

3. Understanding the concepts of complex functions and piecewise functions, and understanding the concepts of inverse functions and implicit functions

4. Grasp the nature of basic elementary functions and their graphs to understand the concept of elementary functions

5. Understanding the concept of limits, understanding the concept of the left and right limits of functions, and the relationship between the limit existence and the left limit and the right limit

6. Mastering the nature of limit and arithmetic law

7. Grasp the two criteria of the limit, and will use them to limit, master the use of two important limits to limit the method

8. Understanding the concept of infinite quantity and infinite quantity, mastering the comparison method of infinitesimal quantity, will use the equivalent infinite quantity to find the limit

9. Understanding the concept of function continuity, including left and right continuous, will discriminate the type of function discontinuity.

10. To understand the nature of continuous functions and the continuity of elementary functions, to understand the properties of continuous functions (boundedness, maximums and minima theorems, and intermediate value theorems) on closed intervals, and to apply these properties

Two or one meta-functional calculus

Exam Content

The relationship between the geometric meaning of derivative and differential concept derivative and the relation between the arithmetic and continuity of the physical meaning function the tangent and normal derivative of the plane curve and the derivative of the derivative of the basic elementary function, the inverse function, implicit function and parametric equation of the function determined by the differential method of the higher order derivative of the first order differential form of the invariance differential mean value theorem indeterminate form (L ' Hospital) Law function monotonicity of the discriminant function of the extremum function graph of the concave and convex properties, Inflection point and asymptote function graph description function maximum and minimum value arc differential curvature concept curvature circle and radius of curvature

Exam Requirements

1. Understanding the concept of derivative and differential, understanding the relationship between derivative and differential, understanding the geometric meaning of the derivative, the tangent equation and the normal equation of the plane curve, understanding the physical meaning of the derivative, using the derivative to describe some physical quantities, understanding the relationship between the conductivity and continuity of the function

2. Master the Arithmetic Law of derivative and the derivation rule of compound function, master the derivative formula of basic elementary function. Understanding the arithmetic law of differential and invariance of First order differential form will find the differential of function

2. Mastering the basic formula of indefinite integral, mastering the properties of indefinite integral and definite integral and the mean value theorem of definite integral, mastering the exchange-element integral method and the partial integral method

3. The integrals of rational function, triangular function and simple irrational function will be obtained.

4. A function that understands the upper limit of the integral will seek its derivative and master the Newton-Leibniz formula

5. Understand the concept of anomalous integrals and calculate anomalous integrals

6. Master the use of definite integral to express and calculate some geometrical and physical quantities (area of plane graph, arc length of plane curve, volume and side area of rotating body, parallel section area is known three-dimensional volume, work, gravity, pressure, centroid, shape heart, etc.) and average of function

Four, vector algebra and spatial analytic geometry

Exam Content

Vector concept vector of the linear operation vector of the quantity product and the vector product vector of the mixed product two vector vertical, parallel conditions two vectors of the angle vector coordinate expression and its operation unit vector direction number and direction cosine surface equation and space curve equation of the concept plane equation linear equation plane and plane, plane and line, The angle between the straight line and the straight line, and the distance between the parallel and perpendicular conditions to the plane and the point to the straight line. The two-dimensional surface equation and the parametric equation of the graph space curve and the projection curve equation of the general equation space curve on the coordinate plane of the spherical cylindrical surface

Exam Requirements

1. Understanding spatial Cartesian coordinates, understanding the concept of vectors and their representations

2. Mastering the operation of vectors (linear operation, quantity product, vector product, mixed product), understanding the conditions of two vectors perpendicular and parallel

3. Understand the coordinate expressions of unit vector, direction number and direction cosine and vector, and master the method of vector operation with coordinate expression.

4. Mastering plane equation and linear equation and its seeking method

5. The angle between plane and plane, plane and line, straight line and straight line will be asked, and the relation between plane and line will be solved (parallel, perpendicular, intersect, etc.).

6. The distance to the line and the point to the plane will be asked.

7. Understanding the concepts of surface equations and spatial curve equations

8. Understanding the equations and graphs of commonly used two-times surfaces, the equations for simple cylinder and rotational surfaces are obtained.

9. Understand the parametric equations and general equations of spatial curves. To understand the projection of the spatial curve on the coordinate plane, and to find the equation of the projection curve

Five, multivariate function differentiation

Exam Content

The concept of multivariate function two the geometrical meaning of the function two the limit of the function and the concept of continuity the necessary condition and sufficient conditions for the existence of multivariate functions with the properties of multivariate continuous functions in bounded closed regions

The derivation of multivariate complex function and implicit function Fah Derivative Party Wizard number and tangent of the gradient space curve and the tangent plane of the normal plane surface and the Taylor formula of the normals two-ary function the extremum of the multivariate function and the maximum value and the minimum value of the multivariate function of the extremum and its simple application

Exam Requirements

1. Understanding the concept of multivariate functions and understanding the geometrical meaning of two-ary functions

2. Understanding the concept of limit and continuity of two-tuple functions and the properties of continuous functions on bounded closed areas

3. Understanding the concept of multiple function partial derivative and total differential will perfection the differential, understand the necessary conditions and sufficient condition of the existence of the full differential, and understand the invariance of the whole differential form.

4. Understand the concept of the number of side guides and gradients, and master their calculation methods

5. Mastering the method of first order and second derivative of multivariate compound function

6. Understanding the existence theorem of implicit function and finding the partial derivative of multiple implicit function

7. To understand the tangent of the spatial curve and the concept of tangent plane and normals of the plane and surface, the equations will be obtained.

8. Understanding the Taylor formula for a two-tuple function

9. Understanding the concept of multivariate function extremum and conditional extremum, mastering the necessary condition of the existence of multivariate function extremum, understanding sufficient condition of the existence of the extremum of the two-yuan function, finding the extremum of the two-yuan function, using Lagrange multiplier method to find the maximum and minimum value of the simple multivariate function, and solving some simple application problems

The integration of multivariate functions

Exam Content

The concept, properties, calculation and application of two-integral and Sanchong integrals the concept, properties and calculation of two kinds of curve integrals Green's formula plane curve integration and path-independent condition binary function The original function of the total differential of two kinds of curved area of the concept, The relation between the properties and the calculation of the two types of curved surface area Gaussian (Gauss) formula Stokes (Stokes) formula divergence, Curl concept and application of calculation curve integral and curved area Division

Exam Requirements

1. Understand the concept of double integral and Sanchong integral, understand the nature of the re-integral, and understand the mean value theorem of two integrals

2. Master the calculation method of double integral (rectangular coordinates, polar coordinate), will calculate the triple integral (rectangular coordinate, cylinder coordinates, spherical coordinates)

3. Understanding the concepts of two kinds of curve integrals, understanding the properties of two kinds of curve integrals and the relationship between the two kinds of curve integrals

4. Mastering the method of calculating two kinds of curve integrals

5. Mastering the green formula and applying the plane curve integral to the path-independent condition will find the original function of the total differential of the two-element function

6. Understanding the concept of two kinds of curved area, properties and the relationship between two types of curved area, master the method of calculating two kinds of curved area, master the method of calculating curved area by Gauss formula, and calculate the curve integral by Stokes formula

7. Understand the concept of divergence and curl and calculate

8. Some geometrical and physical quantities (area, volume, surface area, arc length, mass, center of mass, centroid, moment of inertia, gravity, work and flow) of the plane graph are calculated by using the integral, curvilinear and curved areas.

Seven, infinite series

Exam Content

Convergence and divergence of constant term series-concept of convergence series and the basic properties of the concept series and the necessary conditions for the convergence of the progression and the convergence of the convergent progression and the convergence of the absolute convergence of the arbitrary term of the series and the convergence of the convergent function and the convergent radius , Convergence interval (the opening interval) and the fundamental properties of the function power series in the convergence interval of the series of simple power series and functions of the function of the sum of the Fourier (Fourier) coefficients of the power series expansion function and the Fourier series Dirichlet (Dirichlet) theorem functions on the Fourier series function on the sine series and the remainder Chord Series

Exam Requirements

1. Understanding the concepts of convergence, divergence, and convergence of the series of constants, mastering the basic properties of the series and the necessary conditions for convergence

2. Mastering the conditions of convergence and divergence of geometric progression and progression

3. The comparative discriminant method and the ratio discriminant method for mastering the convergence of the positive series will be used to discriminate the root value.

4. Leibniz discriminant method for mastering staggered series

5. Understanding the concept of absolute convergence and conditional convergence and the relationship between absolute convergence and convergence of arbitrary term series

6. Understanding the convergent domain of the series of function items and the concept of function

7. Understand the concept of convergent radius of power series, and grasp the convergence radius, convergence interval and the method of convergence region of power series

8. Understanding the basic properties of a power series in its convergent interval (and the continuity of the function, the derivative and the itemized integrals), the sum function of some power series in the convergent interval is obtained, and the sum of some number of series is obtained.

9. Understanding the sufficient and necessary conditions for function expansion to Taylor series

10. Master,,, and McLoughlin (Maclaurin), will use them to indirectly expand some simple functions into power series

11. Understanding the concept of Fourier series and the Dirichlet convergence theorem, the functions defined on the above are expanded into Fourier series, the functions defined on the above are expanded into sine series and cosine series, and the expressions of Fourier series and functions are written out.

Eight, ordinary differential equation

Exam Content

Basic concept variables of ordinary differential equations separable differential equation homogeneous differential equation First order linear differential equation Bernoulli (Bernoulli) equation full differential equation can be solved by simple variable substitution some differential equations the properties of the solutions of the linear differential equations of the higher order differential equations and the structure theorem of the solution Simple application of differential equation of Euler (Euler) equation of second order constant coefficient nonhomogeneous linear differential equation with simple coefficients homogeneous linear differential equation

Exam Requirements

1. Understanding the concepts of differential equations and their order, solution, general solution, initial conditions and special solutions

2. Mastering the differential equation with separable variables and solving the first order linear differential equation

3. Will solve the homogeneous differential equation, Bernoulli equation and the whole differential equation, and will solve some differential equations with simple variable substitution.

4. The following forms of differential equations are solved by descending order method

5. Understanding the properties of solutions of linear differential equations and the structure of their solutions

6. Mastering the solution of the second order homogeneous linear differential equation with constant coefficients, and solving some homogeneous linear differential equations with constant coefficients higher than the second time

7. The second order constant coefficients non-homogeneous linear differential equations with polynomial, exponential function, sine function, cosine function and their and product are solved by the free term.

8. Will solve the Euler equation.

9. Will solve some simple application problems with differential equations.

Linear algebra

First, determinant

Exam Content

Determinant of the concept and basic nature of determinant by row (column) expansion theorem

Exam Requirements
1. Understanding the concept of determinant, mastering the nature of determinant

2. The determinant is computed by the nature of the determinant and the determinant by row (column) expansion theorem

Second, The Matrix

Exam Content

Matrix of concepts matrix of linear operations matrix multiplication matrices of the exponential matrix of the determinant matrices of the polynomial of the transformation of the concept and nature of the matrix reversible sufficient and necessary conditions adjoint matrix matrix Elementary Transform Elementary matrix matrix of the rank matrix of the equivalent block matrix and its operation

Exam Requirements

1. Understanding the concept of matrix, understanding the unit matrix, the quantity matrix, the diagonal matrix, the triangular matrix, the symmetry matrix and the anti-symmetric matrix, and their properties

2. Master the linear operation, multiplication, transpose and their operation rules of matrices, and understand the properties of the determinant of the power of the square and the product of the square matrix.

3. Understanding the concept of inverse matrices, mastering the properties of inverse matrices and the sufficient and necessary conditions for invertible matrices, understanding the concept of adjoint matrices, and using adjoint matrices to find inverse matrices

4. Understanding the concept of elementary transformation of matrices, understanding the nature of elementary matrices and the concept of matrix equivalence, understanding the concept of rank of matrices, mastering the method of finding the rank and inverse matrix of matrices by elementary transformation

5. Understanding the chunking matrix and its operations

Three, Vector

Exam Content

The linear combination of the conceptual vectors of vectors and the linear correlation of the linear representations of vector groups and the linear relationship of the maximal linear independent vector groups of the vector group of the rank vector groups of the ranks to the rank of the matrix, the relation between the rank of the vector space and the ranking of the matrices, and the orthogonal normalization method of the inner product linear independent vector Group of Matrix and its properties

Exam Requirements

1. Understanding the concept of linear combinations and linear representations of dimensional vectors and vectors

2. Understanding the concept of linear correlation and linear independence of Vector group, and mastering the related properties and discriminant methods of linear correlation and linear independence of Vector Group

3. Understanding the concept of the rank of the maximal linear independent group and the vector group of the vector group, the maximal linear independent group and the rank of the vector group are obtained.

4. Understanding the concept of vector group equivalence, understanding the relationship between the rank of a matrix and its row (column) vector Group

5. Understand the concepts of dimensional vector space, subspace, base, dimension, coordinates, etc.

6. Understanding the base transformation and coordinate transformation formulas, the transition matrix is obtained

7. Understanding the concept of inner product and mastering the orthogonal normalization of the linear independent vector Group Schmitt (Schmidt) method

8. Understand the concepts of canonical orthogonal and orthogonal matrices and their properties

Iv. Systems of linear equations

Exam Content

Kramer (Cramer) Law of linear equations group with the sufficient and necessary conditions of a non-0 solution for a homogeneous linear equation group with a sufficient and necessary solution to the conditions of a system of linear equations the general solution of the solution of a homogeneous linear equation group

Exam Requirements

L will use the Kramer rule.

2. The sufficient and necessary conditions for understanding the non-0 solutions of homogeneous linear equations and the sufficient and necessary conditions for the solution of nonhomogeneous linear systems

3. Understanding the basic Solution system, general solution and the concept of solution space of homogeneous linear equations, grasping the basic solution system and solution of the homogeneous linear equation group

4. Understanding the structure of the solution of nonhomogeneous linear equations and the concept of general solution

5. Mastering the method of solving linear equations with elementary line transformation

V. Eigenvalues and eigenvectors of matrices

Exam Content

The concept of the eigenvalues and eigenvectors of matrices, the concept of similarity transformation, the concepts of similarity matrices and the sufficient and necessary conditions for the similarity diagonalization of matrices and the eigenvalues, eigenvectors and similar diagonal matrices of real symmetric matrices of similar diagonal matrices

Exam Requirements

1. To understand the concept and properties of the eigenvalues and eigenvectors of matrices, the eigenvalues and eigenvectors of matrices are obtained.

2. To understand the sufficient and necessary conditions for the concept and properties of the similarity matrix and the similarity diagonalization of the Matrix, to master the method of the matrix as a similar diagonal matrix

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

六、二次 type

Exam Content

Two-time type and its matrix representation contract transformation and contract matrix two order of rank inertia theorem for two times standard shape and canonical form with orthogonal transformation and matching method two times type is standard shape two sub-type and its matrix positive qualitative

Exam Requirements

1. Master the two-time type and its matrix representation, understand the concept of two-order rank, understand the concept of contract transformation and contract matrix, understand the standard form of two-type, the concept of canonical form and the inertia theorem

2. Mastering the method of using orthogonal transformation two-order form as standard form, the method of matching two-order type to standard form

3. Understanding the concept of positive definite two-and positive-definite matrices and mastering its discriminant method

Probability theory and Mathematical statistics

I. Random events and probabilities

Exam Content

The relationship between random events and sample space events and the basic properties of the concept probability of the probability of complete event group the basic formula of probability probability of probabilistic condition of classical probabilistic geometry event independence independent repetition test

Exam Requirements

1. Understanding Sample Space

(Basic event space) concept, understanding the concept of random events, mastering the relationship and operation of events

2. Understanding the concept of probability, conditional probability, mastering the basic nature of probability, will calculate the classical probability and geometry probability, grasp the probability of addition formula, subtraction formula, multiplication formula, full probability formula and Bayesian (Bayes) formula

3. Understanding the concept of event independence, mastering the probability calculation by using event independence, understanding the concept of independent repetition test, and mastering the method of calculating the probability of event

Ii. random variables and their distributions

Exam Content

The concept of random variable random variable distribution function and its properties probability distribution of discrete random variables distribution of random variable distribution of stochastic variable with probability density of random variables

Exam Requirements

1. Understanding the concept of random variables, understanding the concept and nature of distribution functions, and calculating the probability of events associated with random variables

2. Understanding the concept of discrete random variables and their probability distributions, mastering 0-1 Distributions, two distributions, geometric distributions, hypergeometric distributions, Poisson (Poisson) distributions and their applications

3. Understanding the conclusion and application conditions of Poisson's theorem, the Poisson distribution is used to approximate the two-item distribution.

4. Understanding the concept of continuous random variable and its probability density, mastering uniform distribution, normal distribution, exponential distribution and its application

5. Will find the distribution of random variable functions

Three-dimensional random variables and their distributions

Exam Content

The probability distribution, edge distribution and conditional distribution of the two-dimensional stochastic variable with multi-dimensional random variables and their distributions the probability density, the edge probability density and the independent of the conditional density random variable and the independence of the non-relativity common two-dimensional random variable distribution of simple functions of two and more than two random variables

Exam Requirements

1. Understanding the concept of multidimensional random variables, understanding the concept and nature of the distribution of multidimensional random variables, understanding the probability distribution, edge distribution and conditional distribution of two-dimensional discrete random variables, understanding probability density, edge density and conditional density of two-dimensional continuous random variables, and finding the probability of events related to two-dimensional random variables

2. Understanding the independence of random variables and the concept of non-relativity, mastering the conditions of independent random variables

3. Grasp the two-dimensional uniform distribution, understand the probability density of two-dimensional normal distribution, understand the probability significance of the parameters

4. The distribution of simple functions of two random variables will be obtained, and the distribution of simple functions of multiple independent random variables will be obtained.

Iv. numerical characteristics of random variables

Exam Content

Mathematical expectation (mean), variance, standard deviation and its properties of random variables mathematical expectation moment, covariance, correlation coefficient and properties of random variable function

Exam Requirements

1. Understanding the concept of random variable digital features (mathematical expectation, variance, standard deviation, moment, covariance, correlation coefficients) will use the basic properties of digital features and master the digital characteristics of commonly used distributions.

2. Will seek the mathematical expectation of the function of random variables

The law of large numbers and the central limit theorem

Exam Content

Chebyshev (Chebyshev) Inequality Chebyshev large number law Bernoulli (Bernoulli) Large number Law sinzing (khinchine) Large number law di Moivre-Laplace (Demoivre-laplace) theorem Levi-Lindbergh (Levy-lindberg) theorem

Exam Requirements

1. Understanding Chebyshev Inequalities

2. Learn about the law of large numbers of Chebyshev, the law of Bernoulli large numbers and the law of large numbers of sinzing (the law of large numbers of independent and distributed random variable sequences)

3. The Zhei Moivre-Laplace theorem (two distributions with normal distribution as the limit distribution) and the Levi-Lindbergh theorem (the central limit theorem for the sequence of independent and distributed random variables)

Vi. Basic concepts of mathematical statistics

Exam Content

Common sampling distributions of the average sample variance and distribution distribution of sample moment distributions in the total individual simple random sample statistic sample

Exam Requirements

1. Understanding the concepts of general, simple random samples, statistics, sample mean, sample variance, and sample moments

2. Understand the concept and nature of distribution, distribution, and distribution, understand the concept of the upper-side division, and look at the table calculation

3. Understanding common sampling distributions for normal populations

Vii. parameter estimation

Exam Content

Concept estimation of point estimation and estimation of the maximum likelihood estimation method for estimating the estimation of the value of an estimate for a standard interval estimate interval estimation of mean and variance of a single normal population two normal population mean difference and variance ratio

Exam Requirements

1. Understanding the concept of point estimates, estimates, and estimates of parameters

2. Master moment Estimation Method (first order moment, second moment) and maximum likelihood estimation method

3. Understanding the concept of unbiased, effective (minimum variance) and conformance (consistency) of estimators and verifying the unbiased nature of estimators

4, understanding the concept of interval estimation, will find the average value of a single normal population and the confidence interval of variance, will find two normal population mean difference and variance ratio of confidence intervals

Eight, hypothesis test

Exam Content

Hypothesis test of mean and variance of two kinds of errors of single and two normal populations for the hypothesis test of significance test

Exam Requirements

1. Understand the basic idea of the significance test, grasp the basic steps of hypothesis testing, and understand the two kinds of errors that hypothesis testing can produce

2. Hypothesis test for mastering the mean and variance of single and two normal populations

Mathematics 1 Exam Outline