Find the composite function of piecewise function; Find the limit or known limit and determine the constant in the original formula; Discuss the continuity of function and judge the type of discontinuity; Comparison of infinitesimal orders; 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
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 derivability of piecewise function and function with absolute value; Using Robida's law to find the limit of infinitive; Discuss the extreme value of function, the root of equation, and prove the inequality of function; Using 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 …", it is often necessary to construct auxiliary functions to prove such problems; 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; 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
Calculation problems: calculate indefinite integral, definite integral and generalized integral; Problems about variable upper bound integral: such as derivative, limit, etc. The proof of integral mean value theorem and integral property: the application of definite integral: 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
Calculation problems: find the quantity product, cross product and mixed product of vectors; Find linear equation and plane equation; Determine the parallel and vertical relationship between plane and straight line, and find the included angle; Establishing an equation of a rotating surface; 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
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; 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; Find the directional derivative and gradient of binary and ternary functions; Find the tangent plane and normal of the surface and the tangent plane and normal 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
The calculation of double integral and triple integral in various coordinates, and the exchange order of repeated integral; Calculation of curve integral and surface integral of the first kind; Calculation of the second kind (coordinate) curve integral, Green formula, Stokes formula and their applications; Calculation of the second kind (coordinate) surface integral, Gaussian formula and its application; Comprehensive calculation of gradient, divergence and curl; Double 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
Determine the convergence, absolute convergence and conditional convergence of series; Find the convergence radius and convergence domain of power series; Find the sum function of power series or the sum of several series; Expand the function into a power series (including writing the convergence domain); When a function is expanded into a Fourier series, or a Fourier series has been given, it is necessary to determine its sum at a certain point (usually by Dirichlet theorem); Comprehensive proof questions.
Eight. differential equation
Find the general solution or special solution of a typical first-order differential equation: this kind of problem is to distinguish the types of equations first. Of course, some equations do not directly belong to the type we are studying. 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 studied; Solve the reducible equation; Find the special solution or general solution of homogeneous and inhomogeneous equations with linear constant coefficients; According to practical problems or given conditions, the differential equation is established and solved; In the comprehensive problem, the following contents are commonly synthesized: definite integral with variable upper limit, 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.