Chapter 1: Function, Limit and Continuity

Examination content

The concept and representation of function, boundedness, monotonicity, periodicity and parity of function, the properties of basic elementary functions of inverse function, piecewise function and implicit 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 relations 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, master the expression 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, 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. Understand the concept of infinity 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, mean value theorem), and apply these properties.

Chapter 2: 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 significance function; Four operations of tangent, normal derivative and differential of plane curve; Differential method of derivative of basic elementary function: differential method of higher derivative of inverse function and implicit function; Invariant differential mean value theorem in first-order differential form; The regular function of the hospital; Monotonicity of extreme value function; Discriminating the concavity and convexity of function graph: inflection point; And the maximum and minimum values of the asymptote 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. Master the derivative formula of basic elementary function, the four operation rules of derivative and the derivative rule of compound function, and you can find the derivative of piecewise function, 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 theorem, Lagrange mean value theorem, Taylor theorem and Cauchy mean value theorem, and master the simple application of these four theorems. Men?

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. The concavity and convexity of the function graph can be judged by the derivative (note: in the interval (a, b), let the function f(x) have the second derivative. At that time, the graph of f(x) was concave; When the graph of f(x) is convex), you will find the inflection point and asymptote of the function graph.

9. Graphics that describe simple functions.

Comparison: "Understanding Taylor Theorem" and "(Note: In the interval (a, b), let the function f(x) have the second derivative. At that time, the graph of f(x) was concave; At that time, the graph of f(x) was convex) "

Analysis: 1, Taylor's theorem in previous years is unnecessary for students in Grade Three, but in view of the importance and conciseness of Taylor's formula in approximate expressions of some complex functions, it is necessary for candidates to understand it; Secondly, although Taylor theorem was not required in previous years, some students often used Taylor theorem in the process of solving problems in exams, so whether it is a super-rigid solution has always been controversial, so it is necessary to make it clear.

2. Comments on Article 8 are specially noted in the syllabus for the sake of unification, because there are many versions of teaching materials and the nature of judgment is different.

Suggestion: 1. Since it is new content, candidates must strengthen this practice, master the basic ideas and solutions, and clarify concepts and formulas during the review process. However, don't have any psychological burden, thinking that the new content may be more difficult to test. In fact, you will know from the requirements of the outline that the requirements of this knowledge point are relatively low and belong to the understanding content. So as long as you review and master the basic content, basic questions and solutions.

2. Try to use some symbols and definitions consistent with the outline in the review process.

Chapter 3: Integral 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 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 substitution integral method and integration by parts of definite integral.

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

4. Understand the concept of generalized integral and be able to calculate generalized integral.

Chapter 4: Calculus of Multivariate Functions

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, concept, basic properties and calculation of simple abnormal double integral of second-order partial derivative fully differential multivariate function.

Examination requirements

1, understand the concept of multivariate function and understand the geometric meaning of binary 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 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 abnormal double integral on unbounded domain and calculate it.

Chapter 5: Infinite Series

Examination content

Convergence and Divergence of Constant Term Series Basic Properties and Necessary Conditions of Conceptual Series Convergence Absolute Convergence and Conditional Convergence of Arbitrary Term Series and Leibniz Theorem Power Series and Its Convergence Radius, Convergence Interval (refers to the open interval) and Convergence Domain Basic Properties of Sum Function of Simple Power Series in its Convergence Interval Solution of Power Series Expansion of Sum Function 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 of convergence and divergence of series, master the conditions of convergence and divergence of geometric series and P series, master the comparison of convergence and divergence of positive series and the ratio discrimination method, and use the root value discrimination method.

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

4. 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 find the sum function of simple power series in its convergence interval, and then we find the sum of several terms of some series.

6. Master maclaurin expansions of sum, and use them to indirectly expand simple functions into power series.

Chapter 6: Ordinary differential equations and difference equations.

Examination content

The basic concept of ordinary differential equation, separable variable differential equation, homogeneous differential equation, the nature and structure theorem of the solution of first-order linear differential equation, and the second-order constant coefficient homogeneous line? ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ ⒎ 935

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. Understand the properties and structure theorems of solutions of linear differential equations, and 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, general solution and special solution.

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

7. Will apply differential equations and difference equations to solve simple economic application problems.

linear algebra

Chapter 1: Determinant

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.

Chapter 2: Matrix

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.

Chapter 3: 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 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. Master the correlation properties and discrimination methods of linear correlation and linear independence of vector groups.

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.

Chapter four: 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 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.

Chapter 5: Eigenvalues and eigenvectors of matrices.

Examination content

The concepts of eigenvalues and eigenvectors of matrices, the concepts of property similarity matrices and the necessary and sufficient conditions for the similarity diagonalization of property matrices? Alas, the poorer, the worse.

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.

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.

Chapter 1: Random Events and Probability

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.

Chapter 2: 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 concept of random variables and the concept and properties of distribution function; 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 (), 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, where the parameter is λ (λ >; The density function of the exponential distribution of 0) is.

5. Find the distribution of random variable function.

Contrast: the new syllabus gives the standard letter representation of distribution, which may mean that candidates should memorize and master this standard writing.

Chapter 3: 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. Master the edge distribution and conditional distribution of two-dimensional random variables.

3. Understand the concepts of independence and irrelevance of random variables, and master the conditions of mutual independence of random variables; Understand the relationship between independence 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 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.

Contrast: the new syllabus gives the standard letter representation of distribution, which may mean that candidates should memorize and master this standard writing.