I. Function, Limit and Continuity
Concept and representation of function, boundedness, monotonicity, periodicity and parity of function, composite function, inverse function, piecewise function and implicit function, properties and graphs of basic elementary function, and establishment of relationship between elementary function and function.
Definition and properties of sequence limit and function limit, left limit and right limit of function, concepts and relationships between infinitesimal and infinitesimal, properties and comparison of infinitesimal, four operations of limit, two criteria for the existence of limit: monotone bounded criterion and pinching criterion, two important limits:
The concept of function continuity, the types of function discontinuity points, the continuity of elementary functions, and the properties of continuous functions on closed intervals.
Second, the differential calculus of unary function
Concepts of derivative and differential, geometric and economic significance of derivative, relationship between derivability and continuity of function, tangent and normal of plane curve, four operations of derivative and differential, derivative of basic elementary function, differential method of compound function, inverse function and implicit function, higher derivative, invariance of first-order differential form, differential mean value theorem, Lobida rule, discrimination of monotonicity of function, extreme value of function.
3. Integral calculus of unary function
Concepts of primitive function and indefinite integral, basic properties of indefinite integral, basic integral formula, concept and basic properties of definite integral, mean value theorem of definite integral, upper bound function of integral and its derivative, Newton-Leibniz formula, substitution integral method and partial integral of indefinite integral and definite integral, abnormal (generalized) integral and application of definite integral.
Four, multivariate function calculus
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, concept and calculation of partial derivative of multivariate function, derivative method of multivariate composite function and implicit function, second-order partial derivative, total differential, extreme value and conditional extreme value, maximum and minimum value of multivariate function, concept, basic properties and calculation of double integral, simple abnormal double integral in unbounded region.
Five, infinite series
Concept of convergence and divergence of constant series, concept of convergence series sum, basic properties and necessary conditions of convergence, geometric series and P series and their convergence, judgment of convergence of positive series, absolute convergence and conditional convergence of arbitrary series, staggered series and Leibniz theorem, power series and its convergence radius, convergence interval (referring to open interval) and convergence domain, sum function of power series, basic properties of power series in its convergence interval and simple sum of power series.
Six, ordinary differential equations and difference equations
Basic concepts of ordinary differential equations, differential equations with separable variables, homogeneous differential equations, first-order linear differential equations, properties and structure theorems of solutions of linear differential equations, second-order homogeneous linear differential equations with constant coefficients and simple non-homogeneous linear differential equations, concepts of difference and difference equations, general and special solutions of difference equations, first-order linear differential equations with constant coefficients and simple applications of differential equations.