Common problems and knowledge points of each part of mathematics in senior one for postgraduate entrance examination.

I. Function, Limit and Continuity

1 Find the composite function of piecewise function;

2 Find the limit or known limit to determine the constant in the original formula;

3 Discuss the continuity of function and judge the type of discontinuity;

4 comparison of infinitesimal order;

Discuss the number of zeros of continuous function in a given interval, or judge whether the equation has real roots in a given interval.

Two. Differential calculus of univariate function

1 Find the derivative and differential of a given function (including higher-order derivative), the derivative of implicit function and the function determined by parametric equation, especially the discussion on the differentiability of piecewise function and function with absolute value;

2. Use Robida's law to find the limit of infinitive;

3 discuss the extreme value of function, the root of equation and prove the inequality of function;

4 Use Rolle's theorem, Lagrange's mean value theorem, Cauchy's mean value theorem and Taylor's mean value theorem to prove related propositions, such as "Prove that there is at least one satisfaction in the open interval ……", and proving such problems often requires the construction of auxiliary functions;

5. The application of maximum and minimum in geometry, physics and economy. To solve this kind of problems, it is mainly to determine the objective function and constraint conditions, and to determine the discussion interval;

6. Use derivative to study the behavior of function and describe the graph of function, and find the asymptote of curve.

Three. Integral of unary function

1 calculation problem: calculate indefinite integral, definite integral and generalized integral;

2. Questions about variable upper bound integral: such as derivation, limit, etc.

3. The proof of mean value theorem and integral property:

4 definite integral application problems: calculation area, volume of rotating body, arc length of plane curve, area of rotating surface,

Pressure, gravity, variable force work, etc.

Comprehensive examination questions.

Four. Vector Algebra and Spatial Analytic Geometry

1 calculation problem: find the quantity product, cross product and mixed product of vectors;

2. Find the linear equation and plane equation;

3. Determine the parallel and vertical relationship between the plane and the straight line, and find the included angle;

4. Establish the equation of revolution surface;

Five topics related to the application of differential calculus of multivariate functions in geometry or linear algebra.

Verb (abbreviation of verb) Differential calculus of multivariate functions

1 Determine whether the binary function is continuous at one point, whether the partial derivative exists, whether it is differentiable, and whether the partial derivative is continuous;

② Find the first and second partial derivatives of multivariate functions (especially those with abstract functions) and the first and second partial derivatives of implicit functions;

3. Find the directional derivatives and gradients of binary and ternary functions;

Find the tangent plane and normal direction of the surface and the tangent plane and normal direction of the space curve. This kind of problem is a comprehensive problem of multivariate function differential calculus, vector algebra and spatial analytic geometry, which should be reviewed together;

The application of extreme value or conditional extreme value of multivariate function in geometry, physics and economy; Find the maximum and minimum of binary continuous function in bounded plane region. This part of the application problem needs knowledge from other fields, so candidates should pay attention to it when reviewing.

Integrals of Multivariate Functions of intransitive Verbs

1 Calculation of double and triple integrals in various coordinates, and exchange order of repeated integrals;

2. Calculation of the first kind of curve integral and surface integral;

3. Calculate the second kind (coordinate) curve integral, Green formula, Stokes formula and their applications;

4. Calculation of the second kind (coordinate) surface integral, Gauss formula and its application;

5. Comprehensively calculate gradient, divergence and curl;

60% off integration, line and surface integration application; Find the area, volume, weight, center of gravity, gravity, variable force work, etc. Mathematics candidates should pay enough attention to this part of the content and questions.

Seven. infinite series

1 determines the convergence, divergence, absolute convergence and conditional convergence of series terms;

2. Find the convergence radius and convergence domain of power series;

3 Find the sum function of power series or the sum of several series;

4. Expand the function into a power series (including writing the convergence domain);

5 Expand the function into Fourier series, or a given Fourier series, and determine its sum at a certain point (usually using Dirichlet theorem);

6 comprehensive proof questions.

Eight. differential equation

1 Find the general solution or special solution of a typical first-order differential equation: This kind of problem first discriminates the type of the equation. Of course, some equations do not directly belong to the type we have learned. At this time, the common method is to switch x and y or make appropriate variable substitution to turn the original equation into the type we have learned;

2. Solve the reducible equation;

3. Find the special solution or general solution of homogeneous and inhomogeneous linear equations with constant coefficients;

4. Establish differential equations and solve them according to actual problems or given conditions;

5 Comprehensive questions, the common ones are the synthesis of the following contents: variable upper bound definite integral, multiple integral in variable integral domain, line integral has nothing to do with path, necessary and sufficient conditions of total differential, partial derivative and so on.