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Outline of advanced mathematics examination
First, the quality of the examination.
Advanced Mathematics Examination for Postgraduates of Yangtze University is a level examination with selection function to recruit non-mathematics majors in science. Its main purpose is to test students' mathematical quality, including mastering various contents of advanced mathematics and applying relevant knowledge to solve problems. The examination object is the candidates who take the advanced mathematics examination for postgraduate entrance of Yangtze University.
Second, the basic requirements of the exam
Candidates are required to systematically understand the basic concepts and theories of advanced mathematics and master the basic methods of advanced mathematics. Candidates are required to have abstract thinking ability, logical reasoning ability, spatial imagination ability, calculation ability and the ability to comprehensively apply what they have learned to analyze and solve problems.
Third, the examination method and time
The advanced mathematics examination is in the form of closed-book written examination, with a full score of 150 and an examination time of 180 minutes.
Four. Examination content and requirements
(A), function, limit, continuity
Examination content
The concept of function and its representation are boundedness, monotonicity, periodicity and parity of function, the properties of composite function, inverse function, piecewise function and implicit function, the concepts of graph sequence limit and function limit, the concepts of infinitesimal and infinitesimal and their relations, and the existence of four operational limits of infinitesimal comparison limit. There are two important limits: the concept of function continuity, the type of discontinuity, the concept of elementary function continuity, and the concept of continuous function uniform continuity in 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. Master the method of judging these properties of function.
3. Understand the concept of composite function and the concepts of inverse function and implicit function. Will find the composite function and inverse function of a given function.
4. Master the nature and graphics of basic elementary functions.
5. Understand the concept of limit, the concepts of left limit 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, and use them to make some basic judgments and calculations.
7. Master two criteria for the existence of limit and use them to find the limit. Master the method of using two important limits to find the limit.
8. Understand the concepts of infinitesimal and infinity, 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. Grasp the operational properties of continuous functions and the continuity of elementary functions, and be familiar with the properties of continuous functions on closed intervals (boundedness, maximum theorem, mean value theorem, etc.). ) and apply these properties.
1 1. Understand the concept of uniform continuity of functions.
(2) Differential calculus of unary function
Examination content
The concept of derivative The relationship between geometric meaning and physical meaning of derivative The derivability and continuity of function Four operations of derivatives of tangent and normal Basic elementary function of plane curve Compound function, inverse function, Derivation of implicit function Derivation of function determined by parametric equation Concept of higher-order derivation and geometric meaning of differentiation Differential relation between differentiability and differentiability Algorithm of differential function and application of invariant differential of first-order differential form in approximate calculation L'H?pital's law Taylor formula maximum and minimum value of function, concavity and convexity of monotonicity function graph, inflection point and asymptote function graph, arc differential and curvature calculation.
Examination requirements
1. Understand the concepts of derivative and differential, understand the relationship between derivative and differential, understand the geometric meaning of derivative, find the tangent equation and normal equation of plane curve, understand the physical meaning of derivative, describe some physical quantities with derivative, and master the relationship between function derivability and continuity.
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.
3. If you understand the concept of higher derivative, you will find the n derivative of a simple function.
4. Find the first and second derivatives of piecewise function.
5. Find the first and second derivatives of the implicit function and the function determined by the parametric equation.
6. Find the derivative of the inverse function.
7. Understand and apply Rolle theorem, Lagrange mean value theorem, Cauchy mean value theorem and Taylor theorem.
8. Understand the concept of extreme value of function, master the method of judging monotonicity of function and finding extreme value of function with derivative, master the method of finding maximum and minimum value of function and its simple application.
9. We can judge the concavity and convexity of the function graph by derivative, find the inflection point and horizontal, vertical and oblique asymptotes of the function graph, and describe the function graph.
10. Master the method of finding the limit of indefinite form with L'H?pital's law.
1 1. Understand the concepts of curvature and radius of curvature, and calculate curvature and radius of curvature.
(3) Integral calculus of unary function
Examination content
The concept of original function and indefinite integral The concept of indefinite integral formula and the mean value theorem of definite integral The function defined by variable upper bound definite integral and its derivative Newton-Leibniz formula The substitution integration method of indefinite integral and definite integral and the rational formula of partial integral rational function and trigonometric function and the application of integral The definite integral of generalized integral (infinite integral, deficient integral) simple and unreasonable function.
Examination requirements
1. Understand the concepts of original function and indefinite integral and definite integral.
2. Master the basic formula of indefinite integral, the properties of indefinite integral and definite integral, and the mean value theorem of definite integral. Master Newton-Leibniz formula. Proficient in the substitution integral method of indefinite integral, definite integral and partial integral.
3. Can find the integral of rational function, rational formula of trigonometric function and simple unreasonable function.
4. Understand the function defined by variable upper bound definite integral and find its derivative.
5. Understand the concept of generalized integral (infinite integral, loss integral), master the convergence judgment method of infinite integral and loss integral, and calculate some simple generalized integrals.
6. Master some geometric quantities 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, and the cross-sectional areas of three-dimensional volume, work, gravity and pressure are known) and the average values of functions are expressed and calculated by definite integrals.
(4) Vector Algebra and Spatial Analytic Geometry
Examination content
The condition that the scalar product, cross product and mixed product of the linear operation vector of the concept vector of the vector are vertically parallel; The included angle vector of two vectors and the number of directions and directions of its operation unit vector, the conceptual plane equation of cosine surface equation and space curve equation, the included angle and parallelism of plane to plane, plane to straight line, straight line to straight line, and the distance between vertical condition point and plane and straight line; Equation of surface of revolution with spherical generatrix parallel to coordinate axis and cylindrical rotation axis as coordinate axis; Commonly used quadratic equation and parametric equation of its graphic space curve and projection curve equation of general equation space curve on coordinate plane.
Examination requirements
1. Be familiar with the spatial rectangular coordinate system and understand the concepts of vectors and their modules.
2. Master the operation of vectors (linear operation, scalar product, cross product) and understand the conditions for two vectors to be vertical and parallel.
3. Understand the projection of the vector on the axis, and understand the projection theorem and projection operation. Understand the coordinate expressions of direction number, direction cosine and vector, and master the method of vector operation with coordinate expressions.
4. Principal plane equation and spatial linear equation and their solutions.
5. Will find the plane, the angle between the plane and the straight line, and use the relationship between the plane and the straight line (parallel, vertical, intersecting, etc.). ) to solve related problems.
6. Will find the distance between two points in space, the distance from point to straight line and the distance from point to plane.
7. Understand the concepts of space curve equation and surface equation.
8. Understand the parametric equation and general equation of space curve. Understand the projection of space curve on the coordinate plane and find its equation.
9. Understand the equation, figure and section of quadric surface in common use, and you can find the cylindrical equation of the rotating surface with the coordinate axis as the rotation axis and the generatrix parallel to the coordinate axis.
(5) Differential calculus of multivariate functions
Examination content
The concept of multivariate function, the geometric meaning of binary function, the limit of binary function and the properties of multivariate continuous function in continuous bounded closed region, the concepts of partial derivative and total differential of multivariate function, the necessary and sufficient conditions for the existence of total differential of multivariate composite function, and the derivation of implicit function; Solution of higher order partial derivative; Tangent derivative and normal derivative of space curve and normal plane surface; Taylor formula of gradient binary function; Extreme value and conditional extreme value of multivariate function; Lagrange multiplier method; Maximum and minimum of multivariate function and its simple application: the application of total differential in approximate calculation.
Examination requirements
1. Understand the concept of multivariate function and the geometric meaning of bivariate function.
2. Understanding the concept of limit and continuity of binary function and its basic operation properties, and the relationship between repeated limit and limit of binary function will judge the existence and continuity of limit of binary function at known points and understand the properties of continuous function in bounded closed region.
3. Understand the concepts of partial derivative and total differential of multivariate function, understand the relationship between differentiability, existence and continuity of binary function, find partial derivative and total differential, understand the condition that two mixed partial derivatives of binary function are equal, understand the necessary and sufficient conditions for the existence of total differential, and understand the invariance of total differential form.
4. Master the solution of partial derivative of multivariate composite function.
5. Master the derivation rules of implicit function.
6. Understand the concepts of directional derivative and gradient, and master their calculation methods.
7. Understand the concepts of tangent and normal plane of curves and tangent and normal plane of surfaces, and work out their equations.
8. Understand the second-order Taylor formula of binary function.
9. 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.
10. Understand the application of total differential in approximate calculation.
(6), multivariate function integral calculus
Examination content
The concepts and properties of double integral and triple integral, the calculation and application of double integral and triple integral, the concepts, properties and calculation of two kinds of curve integral relations, Green's formula, the condition that plane curve integral has nothing to do with path, the concepts and properties of two kinds of surface integral relations, and the calculation of two kinds of surface integral relations are all known, as are Gauss formula, Stokes formula, the concepts of divergence and curl, and the application of calculating curve integral and surface integral.
Examination requirements
1. Understand the concepts of double integral and triple integral, and master the properties of double integral.
2. Familiar with the calculation method of double integral (rectangular coordinates, polar coordinates), able to calculate triple integral (rectangular coordinates, cylindrical coordinates, spherical coordinates), and master the method of substitution of double integral.
3. Understand the concepts, properties and relationships of two kinds of curve integrals.
4. Master the calculation methods of two kinds of curve integrals.
5. Master Green's formula, master the condition that plane curve integral has nothing to do with path, and find the original function of total differential.
6. Understand the concepts, properties and relations of two kinds of surface integrals, master the calculation methods of two kinds of surface integrals, and use Gaussian formula and Stokes formula to calculate surface integrals and curve integrals.
7. The concepts of dissolution and rotation are introduced and calculated.
8. Understand the integral with parametric variables and Leibniz formula.
9. We can use multiple integrals, curve integrals and surface integrals to find some geometric physical quantities (area of plane figure, area of surface, volume of object, arc length of curve, mass, center of gravity, moment of inertia, gravity, work and flow of object, etc.). ).
(7), infinite series
Examination content
The basic properties and necessary conditions of constant series and its convergence and convergence of conceptual series and conceptual series; Convergence of geometric series and P series and discrimination of positive series: absolute convergence and conditional convergence of staggered series and Leibniz theorem: convergence domain of function series; The concept power series of sum function and its convergence radius: the basic properties of convergence interval (referring to open interval) and the power series of convergence domain in convergence interval; Solution of simple power series sum function: application of power series expansion of Taylor series elementary function in approximate calculation: Fourier coefficient of function and Dirichlet theorem of Fourier series: sine series and cosine series of Fourier series function on [-l, l]. Uniform convergence of series of function terms.
Examination requirements
1. Understand the concepts of convergence and sum of convergent constant series, and master the basic properties of series and the necessary conditions for convergence.
2. Master the conditions of convergence and divergence of geometric series and P series.
3. Master the comparison of convergence of positive series and the ratio discrimination method, and use the root value discrimination method.
4. Master the Leibniz discriminant method of staggered series.
5. Understand the concepts of absolute convergence and conditional convergence of arbitrary series, and the relationship between absolute convergence and conditional convergence.
6. Understand the convergence domain of function term series and the concept of function.
7. Understand the concept of convergence radius of power series and master the solution of convergence radius, convergence interval and convergence domain of power series.
8. Knowing some basic properties of power series in its convergence interval (continuity of sum function, item-by-item differentiation, item-by-item integration), we will find the sum function of some power series in its convergence interval, and then find the sum of some series.
9. Understand the necessary and sufficient conditions for the function to expand into Taylor series.
10. Master maclaurin expansions of some common functions such as ex, sin x, cos x, ln( 1+x) and (1+x)α, and use them to indirectly expand some simple functions into power series.
1 1. Approximate calculation will be made by power series expansion of the function.
12. Understand the concept of Fourier series and Dirichlet theorem, expand the function defined on [-l, l] into Fourier series, the function defined on [0, l] into sine series and cosine series, and the function with a period of 2l into Fourier series.
13. If we know the uniform convergence of the series of function terms and its properties, we can judge the uniform convergence of the series of function terms.
(8) Ordinary differential equations
Examination content
The basic concept of ordinary differential equation The separable variable homogeneous differential equation First-order linear differential equation Bermoulli equation Total differential equation Some differential equations which can be solved by substitution of simple variables The properties and structure theorems of solutions of second-order homogeneous linear differential equation with constant coefficients The second-order homogeneous linear differential equation with constant coefficients is higher than that of non-homogeneous linear differential equation with constant coefficients. Some second-order homogeneous linear differential equations with constant coefficients Euler equation power series solution differential equation simple solution constant coefficient linear differential equation simple application differential equation
Examination requirements
1. Master the concepts of differential equation and its order, solution, general solution, initial condition and special solution.
2. Master the solutions of differential equations with separable variables and first-order linear differential equations.
3. I can solve homogeneous differential equations, Bernoulli equations and total differential equations, and I will replace some differential equations with simple variables.
4. The following equations will be solved by order reduction method: y(n)=f(x), y "= f (x, y'), y" = f (y, y').
5. Understand the properties of solutions of linear differential equations and the structure theorem of solutions. Understand the constant variation method for solving second-order inhomogeneous linear differential equations.
6. Master the solution of second-order homogeneous linear differential equations with constant coefficients, and be able to solve some homogeneous linear differential equations with constant coefficients higher than the second order.
7. Polynomials, exponential functions, sine functions, cosine functions and their sum and product can be used to solve second-order non-homogeneous linear differential equations with constant coefficients.
8. Euler equation can be solved.
9. Understand the power series solution of differential equation.
10. Understand the solution of simple linear differential equations with constant coefficients.
1 1 can solve some simple application problems with differential equations.
Verb (abbreviation of verb) main references
Advanced Mathematics (Volume I and Volume II) (fourth edition), edited by Mathematics Teaching and Research Section of Tongji University, Higher Education Press, 1996.
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