I. Function, Limit and Continuity
Examination content
The concept of function and its representation of boundedness, monotonicity, periodicity and parity of function, the properties of composite function, inverse function, piecewise function and implicit function, the establishment of functional relationship of graphic elementary function, the definitions of sequence limit and function limit, the concepts of left limit and right limit of property function, the properties of infinitesimal and their relationship, four operational limits of infinitesimal comparison limit and two important limits: monotone boundedness criterion and pinch criterion; Concept of Function Continuity Types of Discontinuous Points of Functions Continuity of Elementary Functions Properties of Continuous Functions on Closed Interval
Examination requirements
1, understand the concept of function, master the representation of function, and 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 basic elementary function and its graphic properties, and understand the concept of elementary function.
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 limit and right limit.
6. Master the nature of limit and four algorithms.
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 content
The relationship between the geometric meaning of derivative and differential concepts and the derivability and continuity of physical meaning function; Four operations of tangent, normal derivative and differential of plane curve: derivative of basic elementary function; Differential method of higher derivative of function determined by inverse function, implicit function and parametric equation; Invariant differential mean value theorem in first-order differential form; Hospital rules; Concave-convex, inflection point and asymptotic curve of extreme function graph; Maximum and minimum arc differential curvatures of function graphs.
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 understand 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 higher derivative of simple function.
4. Derivative of piecewise function, derivative of implicit function, function determined by parametric 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. 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 depicted.
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 content
The concept of original function and indefinite integral, the basic properties of indefinite integral, the concept of basic integral formula and the basic properties of the mean value theorem of definite integral, the upper limit of integral and the function of 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.
Examination requirements
1, understand the concept of original function, and understand the concepts of 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, and master the integration methods of method of substitution and integration by parts.
3. Can find the integral of rational function, rational formula of trigonometric function and simple unreasonable function.
4. Understand the function of the upper limit of integral, 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 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.
Four, multivariate function calculus
Examination content
Concept of multivariate function, geometric meaning of bivariate function, concept of limit and continuity of bivariate function, properties of bivariate continuous function in bounded closed region, derivative method of partial derivative of multivariate function and fully differential multivariate composite function and implicit function, concepts, basic properties and calculation of extreme value and conditional extreme value of second-order partial derivative multivariate function.
Examination requirements
1, understand the concept of multivariate function and understand the geometric meaning of binary function.
2. Understand the concepts of limit and continuity of binary function and the properties of binary continuous function in bounded closed region.
3. Knowing the concepts of partial derivative and total differential of multivariate function, we can find the first and second partial derivatives of multivariate composite function, total differential, existence theorem of implicit function and partial derivative of multivariate implicit function.
4. 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.
5. Understand the concept and basic properties of double integral, and master the calculation methods of double integral (rectangular coordinates and polar coordinates).
Verb (abbreviation of verb) ordinary differential equation
Examination content
The basic concepts of ordinary differential equations are separated from variable differential equations, homogeneous differential equations, properties and structural theorems of solutions of first-order linear differential equations, which can be reduced to higher order. Some simple applications of second-order homogeneous linear differential equations with constant coefficients are higher than second-order homogeneous linear differential equations with constant coefficients.
Examination requirements
1. Understand differential equations and their concepts such as order, solution, general solution, initial condition and special solution.
2. Mastering the solutions of differential equations with separable variables and first-order linear differential equations can solve homogeneous differential equations.
3. We will solve the following differential equation by order reduction method.
4. Understand the properties of the solution of the second-order linear differential equation and the structure theorem of the solution.
5. 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.
6. Know how to use polynomials, exponential functions, sine functions, cosine functions and their sum and product to solve second-order non-homogeneous linear differential equations with constant coefficients.
7, can use differential equations to solve some simple application problems.