The examination requires candidates to master the basic concepts, theories and methods of function, limit and continuity, differential calculus of unary function, integral calculus of unary function, infinite series, ordinary differential equations, vector algebra and spatial analytic geometry in "Advanced Mathematics" according to the requirements of this syllabus. Candidates should pay attention to the structure of each part of knowledge and the connection of knowledge; Have certain abstract thinking ability, logical reasoning ability, operational ability and spatial imagination ability; Able to use basic concepts, basic theories and basic methods for reasoning, proof and calculation; Be able to use what you have learned to analyze and solve some simple practical problems.

Examination content

1. Function, Limit and Continuity?

(1) function?

1. Understand the concept of function, find the definition domain, expression and function value of function, and make some simple piecewise function images. ?

2. Master monotonicity, parity, boundedness and periodicity of functions. ?

3. understand the function y? =? (x) and its inverse function y? =? -1(x) (domain, range, image) will find the inverse function of monotone function. ?

4. Master the four operations and compound operations of functions; Master the synthesis process of compound function. ? 5. Master the properties and images of basic elementary functions. ? 6. Understand the concept of elementary function. ?

7. Simple functional relationships of some practical problems will be established. ?

(2) limit?

1. Understand the concept of limit (only the descriptive definition of limit is needed), and can describe the changing trend of function according to the concept of limit. To understand the necessary and sufficient conditions for the existence of the limit of a function at one point, we need the left limit and the right limit of the function at one point. ?

2. Understand the uniqueness, boundedness and sign preservation of limit, and master four algorithms of limit. ?

3. Understand the concepts of infinitesimal and infinitesimal, master the nature of infinitesimal and the relationship between infinitesimal and infinitesimal. Will be infinitely small order (high order, low order, same order, equivalent). Will replace the limit with the equivalent infinitesimal. ?

4. Understand two convergence criteria (pinch criterion and monotone bounded criterion) for the existence of limit, and master two important limits:?

We can use these two important limits to find the limit of the function.

(3) continuous?

1. Understand the concept of function continuity at one point and the relationship between function continuity at one point and function limit at that point. The continuity of the segmentation function at the segmentation point will be judged. ?

2. By understanding the concept of function discontinuity, we can find the discontinuity of function and judge the type of discontinuity. ? 3. Understand that "all elementary functions are continuous within the defined interval", and use the continuity of elementary functions to find the limit of functions. ?

4. Grasp the properties of continuous functions on closed intervals: maximum theorem (boundedness theorem) and intermediate value theorem (existence theorem of zero point). Will use the intermediate value theorem to deduce some simple propositions.

Second, the differential calculus of unary function? (1) derivative and differential?

1. Understand the concept of derivative and its geometric meaning, understand the definitions of left derivative and right derivative, understand the relationship between derivability and continuity of function, and use the definition to find the derivative of function at one point. ? 2. Find the tangent equation and normal equation of a point on the curve. ?

3. Memorize the basic formula of derivative, and use the four operational derivation rules of function, the derivative rule of compound function and the derivative rule of inverse function to find derivative. Will find the derivative of piecewise function. ?

4. Find the derivative of implicit function. Master logarithmic derivative method and derivative method of parameter equation. ? 5. To understand the concept of higher derivative, we can find the N derivative of some simple functions. ?

6. Understand the concept of function differentiation, master the invariance and first-order differential form of differential operation, understand the relationship between differentiability and derivability, and find the first-order differential of function. ? (2) The application of the mean value theorem and derivative?

1. Understand Rolle's mean value theorem, Lagrange's mean value theorem and their geometric significance, and understand

Cauchy mean value theorem, Taylor mean value theorem. Rolle's mean value theorem will be used to prove the existence of the root of the equation. Will use lagrange mean value theorem to prove some simple inequalities. ? 2. Master Robida Law, and you can use Robida Law.

The limit of infinitive. ?

3. We will use the derivative to judge the monotonicity of the function, find the monotonicity interval of the function, and use the monotonicity of the function to prove some simple inequalities. ?

4. Understanding the concept of extreme value of function helps us to find the extreme value and maximum value of function and solve some simple application problems. ? 5. Will judge the convexity of the curve and find the inflection point of the curve. ?

6. Will find the asymptote of the curve (horizontal asymptote, vertical asymptote, oblique asymptote). ? 7. Will describe some simple function graphics.

3. Integral calculus of unary function? (1) indefinite integral?

1. Understand the concepts of primitive function and indefinite integral and their relationship, understand the existence theorem of primitive function, and master the properties of indefinite integral. ?

2. Remember the basic indefinite integral formula. ?

3. Master the method of substitution of the first kind (the "combined" differential method) and method of substitution of the second kind (limited to triangular method of substitution and some simple radical method of substitution) of indefinite integral. ?

4. Master the partial integral of indefinite integral. ?

5. I can find some indefinite integrals of simple rational functions. ? (2) definite integral?

1. Understand the concept and geometric meaning of definite integral. Master the basic properties of definite integral. ? 2. Understand the concept of variable limit integral function and master the method of derivative of variable limit integral function. ? 3. Master Newton-Leibniz formula. ? 4. Master the substitution integral method of definite integral and partial integral. ?

5. Understand the concepts of generalized integral of bounded function on infinite interval and loss integral of unbounded function on finite interval, and master their calculation methods. ?

6. The area of the plane figure and the volume of the rotator obtained by the rotation of the plane figure around the coordinate axis will be calculated by definite integral.

Fourth, infinite series? (1) How many series?

1. Understand the concept of convergence and divergence of series and the basic properties of series, and master the necessary conditions of convergence and divergence of series. ?

2. The convergence and divergence of memory geometric series, harmonic series and P series. You can use positive series

The convergence and divergence of positive series are judged by comparing convergence method and specific convergence method. ?

3. Understand the concepts of absolute convergence and conditional convergence of arbitrary series. Can I use Leibniz? Discrimination of convergence and divergence of staggered series. ? (2) Power series?

1. Understand the concepts of power series, convergence of power series and sum function. Will find the convergence radius and convergence interval of power series. ? 2. Master the sum, difference and product operations of power series. ?

3. Grasp the basic properties of power series in its convergence interval: sum function is continuous, sum function can be derived item by item, and sum function can be integrated item by item. ?

4. Memorizing Maclaurin series of ex, sinx, cosx, ln( 1+x) and1(1-x) will expand some simple elementary functions into power series of x-x0.

5. Ordinary differential equations? (1) first order ordinary differential equation?

1. Understand the concept of ordinary differential equation, and understand the concepts of order, solution, general solution, initial condition and special solution of ordinary differential equation. ?

2. Master the solutions of differential equations and homogeneous equations of separable variables. ? 3. Can solve the first order linear differential equation. ? (2) Second-order linear differential equation with constant coefficients?

1. Understand the structure of solutions of second-order linear differential equations with constant coefficients. ?

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

3. Can solve the second-order non-homogeneous linear differential equation with constant coefficients (the non-homogeneous term is limited to: (Ⅰ)? f(x)=pn(x)e

λx

6. Vector Algebra and Spatial Analytic Geometry? (1) Vector Algebra?

1. Understand the concept of vector, master the representation of vector, find the modulus of vector, the direction cosine of non-zero vector, and the projection of non-zero vector on the axis. ?

2. Master the linear operations of vectors (addition and quantity multiplication) and find the quantity product and cross product of vectors. ? 3. Find the included angle between two non-zero vectors and master the necessary and sufficient conditions for two non-zero vectors to be parallel and vertical. ? (2) Plane and straight line?

1. Find the point equation and general equation of the plane. Will determine the positional relationship between the two planes. ? 2. Will find the distance from a point to a plane. ?

3. Point equation, general equation and parameter equation that can find a straight line. Will determine the positional relationship between two straight lines. ? 4. Will find the distance from a point to a straight line and the distance between two straight lines in different planes. ? 5. Will determine the positional relationship between the straight line and the plane.

Test paper structure?

Total score of test paper: 150? Examination time: 150 minutes? Content ratio of test paper:?

Function, limit and continuity about 20%? The differential calculus of unary function is about 30%? The integral of unary function is about 30%? Infinite series and ordinary differential equations are about 15%? Vector algebra and spatial analytic geometry about 5%? Test paper type score distribution:?

Multiple choice questions * * *? Five questions, each little question? 4? Points, with a total score of 20 points; ? Fill in the blanks *** 10, each small question? 4? A score of 40 points; ?

Calculation problem * * *? 8 questions, with a total score of 60 points; ? Comprehensive questions * * *? 3 questions, each small question 10, with a total score of 30 points.