I. Function, Limit and Continuity

Examination content

Concept and representation of function, boundedness, monotonicity, periodicity and parity of function, properties and graphs of basic elementary functions of inverse function, piecewise function and implicit function.

The concept of sequence limit and function limit The concept of infinitesimal and its relationship The nature of infinitesimal and the four operational limits of infinitesimal comparison Two important limits are monotone bounded criterion and pinching criterion:

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The concept of functional continuity The discontinuous point type of functional continuity of elementary functions The concept of uniform continuity of continuous functions on closed intervals.

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, and master the methods to judge these properties of functions.

3. Understand the concepts of composite function, inverse function and implicit function, and 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.

Second, the differential calculus of unary function

Examination content

The relationship between the geometric meaning of conceptual derivative and the derivability and continuity of physical meaning function; Four operations of derivatives of tangent and normal basic elementary functions of plane curve: derivative of compound function, inverse function and implicit function; derivative of higher derivative of function determined by parameter equation; concept of derivative of higher derivative; geometric meaning of differential; relationship between differentiability and differentiability; algorithm of differential and solution of functional differential; application of invariant differential in the form of first-order differential in approximate calculation; L'H?pital's differential mean value theorem (L'Hospital) Taylor's law; maximum and minimum functions; monotonicity functions; graphic concavity and convexity.

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 differentiation and the invariance of the first-order differential form, we can differentiate the function.

3. Understand the concept of higher-order derivative and find the n-order derivative of simple function.

4. The first and second derivatives of piecewise function can be obtained.

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, the mean value theorem of definite integral, Newton-Leibniz formula, method of substitution and partial integral of indefinite integral and definite integral.

3. Know the integral of rational function, rational 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. Grasp the representation and calculation of some geometric 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 area is a known three-dimensional volume, work, gravity and pressure) and the average value of functions through definite integral.

4. Vector Algebra and Spatial Analytic Geometry

Examination content

The condition that the number product, cross product and mixed product of the linear operation vector of the concept vector of the vector are vertical and parallel, the coordinate expression of the included angle vector of the two vectors and its operation unit vector direction number and direction cosine surface equation and the concept plane equation of space curve equation, straight line equation plane and plane, plane and straight line, straight line and straight line included angle and parallel, 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 condition that two vectors are vertical and parallel.

3. Understand the projection of vectors on the axis, the projection theorem and the operation of projection. Understand the coordinate expressions of direction number and direction cosine, 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. 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. By understanding the equations, graphs and sections of quadric surfaces in common use, we can find the cylindrical equation of the surface of revolution with the rotation axis as the center and the generatrix parallel to the rotation axis.

5. Differential calculus of multivariate functions

Examination content

Concept of multivariate function, geometric meaning of bivariate function, limit of bivariate function and properties of multivariate continuous function in continuous bounded closed region, concepts and solutions of partial derivative and total differential of multivariate function, necessary and sufficient conditions for existence of total differential, derivative of higher-order partial derivative of implicit function, tangent derivative and normal derivative of space curve and Taylor formula of gradient bivariate function, extreme value and conditional extreme value of multivariate function, maximum and minimum value of Lagrange multiplier method and its simple application 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 understanding 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 partial derivative 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.

Six, 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 Gaussian of plane curve integral has nothing to do with path, the concepts and properties of two kinds of surface integral and the calculation of two kinds of surface integral relations are all known.

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, flow, etc.). ).

Seven, infinite series

Examination content

The basic properties and necessary conditions of the convergence of the concept series of the sum of constant series and its convergence and divergence, the judgment method of the convergence of geometric series and P series and its convergence positive series, the absolute convergence and conditional convergence of staggered series and Leibniz theorem, the 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 within the 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]; Function term series.

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 convergence and divergence conditions of geometric series and P series.

3. To master the comparison method and ratio method of positive series convergence, the root value method will be used.

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.

Eight. ordinary differential equation

Examination content

The basic concept of ordinary differential equations can be separated into variable differential equations, homogeneous differential equations, first-order linear differential equations, Bermoulli, and all differential equations can be solved by simple variable substitution. Some differential equations can reduce the price. Properties and structure theorems of solutions of higher order linear differential equations. Second order homogeneous linear differential equation with constant coefficients. Second order non-homogeneous linear differential equation with constant coefficients. Some Euler equations with constant coefficients. Power series solutions of differential equations. Simple application of solving differential equations with constant coefficients.

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. Homogeneous differential equations, Bernoulli equations and total differential equations can be solved, and some differential equations can be replaced by 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 and structure theorems of solutions of linear differential equations, and solve the second-order nonhomogeneous linear differential equations by constant variation method.

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.

The main references of intransitive verbs

Advanced Mathematics (Volume I and Volume II) (fourth edition), edited by Mathematics Teaching and Research Section of Tongji University, Higher Education Press, 1996.