Analysis of advanced mathematics test sites and summary of frequently asked questions 1. Function, Limit and Continuity
Examination requirements
1. Understand the concept of function and master the expression of function, and you will establish the functional relationship of 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 concept of limit, the concept of left and right limit of function and the relationship between the existence of function limit and left and right limit.
6. Master the nature of limit and four algorithms of limit.
7. Master two criteria for the existence of limit, and use them to find the limit, and master the method of using two important limits to find the limit.
8. Understand the concepts of infinitesimal and infinitesimal, master the comparison method of infinitesimal, and find the limit with equivalent infinitesimal.
9. Understanding the concept of function continuity (including left continuity and right continuity) will distinguish the types of function discontinuity points.
10. 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.
Second, the differential calculus of unary function
Examination requirements
1. Understand the concepts of derivative and differential, understand the relationship between derivative and differential, and understand the relationship between differentiability and continuity of functions.
2. Master the four algorithms of derivative and the derivative rule of compound function, and master the derivative formula of basic elementary function. Knowing the four algorithms of differential and the invariance of first-order differential form, we can find the differential of function.
If you understand the concept of higher derivative, you will find the higher derivative of a simple function.
4. The derivative of piecewise function, implicit function, function determined by parameter equation and inverse function can be obtained.
5. Understand and apply Rolle theorem, Lagrange mean value theorem, Taylor theorem, and Cauchy mean value theorem.
6. Master the method of using L'H?pital's law to find the limit of indefinite form.
7. Understand the concept of extreme value of function, master the method of judging monotonicity of function and finding extreme value of function with derivative, and master the method of finding maximum and minimum value of function and its application.
8. The derivative will be used to judge the concavity and convexity of the function graph (Note: in the interval, let the function have the second derivative. At that time, the figure was concave; At that time, the graph was convex), the inflection point and horizontal, vertical and oblique asymptotes of the function graph were found, and the function graph was portrayed.
9. Understand the concepts of curvature, circle of curvature and radius of curvature, and calculate curvature and radius of curvature.
3. Integral calculus of unary function
Examination requirements
1. Understand the concept of original function and the concepts of indefinite integral and definite integral.
2. Master the basic formula of indefinite integral, the properties of indefinite integral and definite integral, the mean value theorem of definite integral, and the substitution integral method and integration by parts.
3. Can find the integral of rational function, rational formula of trigonometric function and simple unreasonable function.
4. If you understand the function of the upper limit of integral, you will find its derivative and master Newton-Leibniz formula.
5. Understand the concept of generalized integral and calculate generalized integral.
6. Master the expression and calculation of some geometric and physical quantities (the area of a plane figure, the arc length of a plane curve, the volume and lateral area of a rotating body, the area of a parallel section, the volume, work, gravity, pressure, center of mass, centroid, etc. of a known solid. ) and definite integral to find the average value of the function.
Six frequently asked questions
Problem 1: Find the limit.
Finding the limit is the basic requirement of higher mathematics, so it is also the content that must be tested every year. No matter Math 1, Math 2 and Math 3, every year's exam questions will be involved, but sometimes they will appear in the form of 4-point small questions with simple topics; Sometimes it appears as a big problem and the methods to be used are comprehensive. For example, a big problem may need several methods, such as equivalent infinitesimal substitution, Taylor expansion, Robida rule, separation factor, important limit and so on. Sometimes candidates need to choose a variety of methods to complete the questions comprehensively. In addition, the study of derivative of piecewise function at individual points, asymptotic line of function graph, continuity and derivability of function defined by limit form also needs to use limit means to achieve the goal.
Question 2: Prove equality or inequality with the mean value theorem, and prove inequality with the monotonicity of the function.
Although you can't say that you have to take the exam every year, it will basically be involved in nine years out of ten years. The proof of equality includes the use of four common differential mean value theorems (namely Rolle mean value theorem, Lagrange mean value theorem, Cauchy mean value theorem and Taylor mean value theorem) and a definite integral mean value theorem. Inequalities can sometimes be proved by the mean value theorem and monotonicity of functions. There is a difficulty in using Taylor's mean value theorem, but the probability of examination is not great.
Question 3: One-variable function takes derivative, and multivariate function takes partial derivative.
The problem of derivative mainly examines the basic formula and operational ability, of course, including the ability to deal with functional relations. The derivation of univariate function may be based on parametric equation, variable limit integral and even higher derivative in application problem. The partial derivatives of multivariate functions (mainly binary functions) are tested almost every year. The given functions may be complex explicit functions or implicit functions (including those determined by equations).
In addition, the extreme value and conditional extreme value of binary function are closely related to practical problems and are a key point of investigation. The necessary and sufficient conditions of extremum involve partial derivatives of binary functions.
Question 4: Sequence problem
The discrimination of convergence and divergence of constant series (especially positive series and staggered series), the essential significance of conditional convergence and divergence, and absolute convergence are the key points of examination, but they often appear in the form of small questions. The convergence radius, convergence interval, convergence domain and sum function of function series (power series, Fourier series is used for logarithmic candidates, but the frequency of examination is not high) and the power series expansion of functions at one point often occupy higher scores in the examination.
Question 5: Calculation of integral
The calculation of integral includes indefinite integral, definite integral and generalized integral, and the calculation of double integral. For freshmen, it is often mainly the calculation of triple integral, curve integral and surface integral. This is mainly based on the examination of operation ability and problem-solving skills, supplemented by the examination of familiarity with formulas and spatial imagination. In the review, we should pay attention to the flexible handling of some problems, such as the application of geometric meaning of definite integral, the application of formulas of center of gravity and center of mass, and the application of symmetry.
Question 6: Differential equation
The method of solving ordinary differential equations is fixed. Whether it is a first-order linear equation, a separable variable equation, a homogeneous equation or a higher-order homogeneous and heterogeneous equation with constant coefficients, as long as we remember the common forms and pay attention to the accuracy of operation, there is no problem in correct operation in the examination room. However, it should be noted here that there is often a way to test differential equations in the postgraduate entrance examination, that is, usually give the equation a general solution or a special solution, and now give the equation a general solution or a special solution. This requires candidates to master the equation and the relationship between its general solution and special solution.