subject of examination
Calculus, linear algebra, probability theory and mathematical statistics
calculus
I. Function, Limit and Continuity
Examination content
Concept and representation of function, boundedness, monotonicity, periodicity and parity of function, properties and graphs of composite function, inverse function, piecewise function, implicit function and basic elementary function.
The Establishment of Function Relation of Elementary Function
The definitions and properties of sequence limit and function limit: left limit and right limit of function; The concepts of infinitesimal and infinity and their relationship; The nature of infinitesimal and four operational limits of infinitesimal: and two criteria for the existence of operational limits (monotone bounded criterion and pinch criterion).
Two important limitations,
Concept of Function Continuity Types of Discontinuous Points of Functions Continuity of Elementary Functions Properties of Continuous Functions on Closed Interval
Adjust knowledge points: adjust "the establishment of function relationship of simple application problems" to "the establishment of function relationship"
Examination requirements
1。 Understand the concept of function, master the representation of function, and establish the function relationship in simple application problems.
2。 Understand the boundedness, monotonicity, periodicity and parity of functions.
3。 Understand the concepts of compound function and piecewise function. Understand the concepts of inverse function and implicit function.
4。 Master 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 concept and basic properties of infinitesimal. Master the comparison method of infinitesimal. Understand the concept of infinity and its relationship with infinitesimal.
7。 In order to understand the nature of limit and the two criteria of existence, and master the four algorithms of limit, it is necessary to apply two important limits.
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, mean value theorem), and apply these properties. "
Examination requirements change: 2005-"9. Understand the properties of continuous functions and the continuity of elementary functions, and understand the properties of continuous functions on closed intervals (boundedness, maximum theorem, mean value theorem) and their simple applications. "
Second, the differential calculus of unary function
Examination content
The concept of derivative, the geometric meaning of derivative and the relationship between derivability and continuity of economic significance function; Four operations of tangent derivative and normal derivative of plane curve: the concept and operation rule of derivative of higher derivative of basic elementary function; Invariant differential mean value theorem in first-order differential form; Hospital rules; Monotonicity of extreme value function; The concavity and convexity of function graph; Inflection point; And the maximum and minimum values of the drawing function of the asymptote function graph.
Adjust knowledge points: integrate the concepts and algorithms of derivative with those of differential.
Examination requirements
1。 Understand the concept of derivative and the relationship between derivability and continuity, and understand the geometric and economic significance of derivative (including the concepts of margin and elasticity). Can find the tangent equation and normal equation of plane curve.
2. Master the derivation formula of basic elementary functions, the four operation rules of derivatives and the derivation rules of compound functions; Can find the derivative of piecewise function, can find the derivative of inverse function and implicit function. "
Examination requirements change: "2. Master the derivation formula of basic elementary function, the four operation rules of derivative and the derivation rule of compound function; Master the derivative method of inverse function and implicit function, and understand the logarithmic derivative method. "
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 and Cauchy mean value theorem, and master the simple application of these three theorems.
6。 Will use L'H?pital's law 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。 Derivative will be used to judge the concavity and convexity of function graph, and the inflection point and asymptote of function graph will be found.
9。 A graph describing a simple function.
Third, the integral calculus of unary function
Examination content
Concepts of primitive function and indefinite integral Basic properties of indefinite integral Basic integral formula Concept and basic properties of definite integral
The function of the upper limit of the mean value theorem of definite integral and its derivative Newton-Leibniz formula, the substitution integration method of indefinite integral and definite integral and the application of partial integral 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 for calculating 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, master Newton-Leibniz formula and the integral method of partial substitution 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, concepts, basic properties and calculation of extreme value and conditional extreme value of second-order partial derivative fully differential multivariate function, and simple generalized 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 understand 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 multivariate function extreme value and conditional extreme value, grasp the necessary conditions for multivariate function extreme value, understand the sufficient conditions for binary function extreme value, find binary function extreme value, and use Lagrange multiplier method to find conditional extreme value. Can find the maximum and minimum of simple multivariate function, and can solve some simple application problems.
5。 Understand the concept and basic properties of double integral, and master the calculation methods of double integral (rectangular coordinates and polar coordinates). Understand the simple generalized double integral on unbounded domain and its calculation.
Five, infinite series
Examination content
Convergence and Divergence of Constant Term Series Basic Properties and Necessary Conditions of the Concept of Constant Term Series Sum Convergence and Divergence of Geometric Series and P Series Discrimination of Convergence and Divergence of Positive Term Series Absolute Convergence and Conditional Convergence and Divergence of Arbitrary Term Series Intersecting with Leibniz Theorem Power Series and Its Convergence Radius, Convergence Interval and Convergence Region
The basic properties of power series sum function in its convergence interval: the solution of simple power series sum function: the power series expansion of elementary function.
Examination requirements
1。 Understand the concepts of convergence and divergence of series and sum of convergent series.
2。 Master the basic properties of series and the necessary conditions for series convergence. Master the conditions of convergence and divergence of geometric series and p series. To master the comparison and ratio methods of convergence of positive series, the root value method will be used.
3。 Understand the concepts of absolute convergence and conditional convergence of arbitrary series and the relationship between absolute convergence and convergence. Master 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, and then we can find the sum of some series with several terms.
6。 Master Maclaurin expansions of,, and, and use them to indirectly expand simple functions into power series.
Six, ordinary differential equations and difference equations
Examination content
Basic concept of ordinary differential equation Variable separable variable differential equation Homogeneous differential equation First-order linear differential equation Second-order homogeneous linear differential equation with constant coefficient and simple nonhomogeneous linear differential equation Simple application of general solution and special solution of first-order linear differential equation with constant coefficient.
New knowledge points: properties of solutions of linear differential equations and structural theorems of solutions
Examination requirements
1。 Understand the concept of differential equation and its order, solution, general solution, initial condition and special solution.
2。 Master the solution methods 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 structural theorems of solutions of linear differential equations, we can use polynomials, exponential functions, sine functions, cosine functions and their sum and product to solve second-order non-homogeneous linear differential equations with constant coefficients.
5。 Understand the concepts of difference and difference equation and their general and special solutions.
6。 Master the solution method of first-order linear difference equation with constant coefficients.
7。 Can use differential equation and difference equation 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 the properties of determinant.
2。 Will apply the properties of determinant and determinant expansion theorem to calculate determinant by row (column).
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, master the properties of inverse matrix, 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
The linear combination of concept vectors of vectors and the linear representation of vector groups are linearly related to the largest linear independent group of linear independent vector groups. Orthogonal normalization method of inner product linear independent vector group of relation vector between rank of vector group and rank of matrix.
Examination requirements
1。 Understand the concept of vector and master the addition and multiplication of vector.
2。 Understand the concepts of linear combination and linear representation of vectors, linear correlation of vector groups, linear independence and so on. Master the correlation 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 will find 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.
Changes in examination requirements: 2005 "4. Understand the concept of vector group equivalence and the relationship between the rank of a matrix and its row (column) vector group. "
5。 Understand the concept of inner product and master Schmidt's 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 law 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.
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, grasp 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 will use orthogonal transformation and collocation method to transform quadratic form into standard form.
3. Understand the concepts of positive definite quadratic form and positive definite matrix, and master their discrimination methods. "