Examination content: one-dimensional calculus, ordinary differential equation

I. Function, Limit and Continuity

Examination content: the concept and properties of function, the properties and graphics of composite function, inverse function and implicit function piecewise function.

The definitions and properties of sequence limit and function limit, the concepts and relations of infinitesimal and infinitesimal of left limit and right limit of function, the properties and comparison of infinitesimal, four operations of limit, two criteria for the existence of limit: monotone bounded criterion and pinch 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.

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.

3. Understand the concepts of compound function, inverse function, implicit function and piecewise function.

4, master the nature and graphics of basic elementary functions, and understand the concept of elementary functions.

5. Understand the concepts of sequence limit and function limit (including sitting limit and right limit).

6. Understand the concept and basic properties of infinitesimal, master the comparison method of infinitesimal, and understand the concept of infinitesimal and its relationship.

7. To understand the nature of limit and its two criteria, and master the four algorithms of limit, we must skillfully use two important limits.

8. Understanding the concept of function continuity (including left continuity and right continuity) will distinguish the types of function discontinuity points.

9. Understand the properties of continuous function and continuity of elementary function, and understand the properties of continuous function on closed interval (boundedness, maximum theorem, mean value theorem) and its simple application.

Second, the differential calculus of unary function

Examination contents: the concept of derivative, the geometric meaning of derivative, the relationship between function derivability and continuity, the four operations of derivative, the derivative of basic elementary function, the derivative of compound function, inverse function and implicit function, higher-order derivative, the concept and operation rule of differential, and the invariance of first-order differential form.

Rolle theorem and Lagrange mean value theorem and their applications L'H?pital's law, extreme value of function, monotonicity of function, concavity and convexity of function graph, inflection point and asymptote, description of function graph, maximum and minimum value of function.

Examination requirements:

1, understand the concept of derivative and the relationship between derivability and continuity, and understand the geometric meaning of derivative.

2. Master the derivation formula of basic elementary functions, the four arithmetic rules of derivatives and the derivation rules of compound functions, and master the derivation methods of inverse functions and implicit functions and logarithmic derivation methods.

3. By understanding the concept of higher derivative, we can find the higher derivative of simple function.

4. Understand the concept of differential, the relationship between derivative and differential, and the invariance of first-order differential form, and you will find the differential of function.

5. Understand Rolle's theorem, Lagrange's mean value theorem and Cauchy's mean value theorem, and master the simple applications of these three theorems.

6, will use L'H?pital's law to find the limit.

7. Master the method of judging monotonicity of function and its application, and master the solution of extreme value, maximum value and minimum value of function.

8. We will judge the concavity and convexity of the function graph through the derivative, and find the inflection point and oblique asymptote of the function graph.

9, master the basic steps and methods of drawing function, can make simple function graphics.

3. Integral calculus of unary function

Examination contents: the concepts of primitive function and indefinite integral, the basic properties of indefinite integral, the basic integral formula, the concept and basic properties of definite integral, and the integral of rational function; Mean value theorem of definite integral, function defined by variable upper bound definite integral and its derivative, Newton-Leibniz formula, substitution integral method of indefinite integral and definite integral and application of partial integral, generalized integral and definite integral.

Examination requirements:

1. Understand the concepts of original function and indefinite integral, master the basic properties and basic integral formula of indefinite integral, and master the substitution integral method and integration by parts of indefinite integral.

2. Understand the concept and basic properties of definite integral, understand the mean value theorem of definite integral, understand the function defined by variable upper limit definite integral and find its derivative, master Newton-Leibniz formula, and the substitution integral method and partial integral of definite integral.

Fourth, ordinary differential equations

Examination contents: basic concepts of ordinary differential equations, differential equations of separable variables, homogeneous equations, first-order linear differential equations, fully differential equations, higher-order linear differential equations, homogeneous linear differential equations with constant coefficients, and non-homogeneous linear differential equations with constant coefficients.

Examination requirements:

1. Understand differential equations and their concepts such as solutions, orders, general solutions, initial conditions and special solutions.

2. Master the solutions of differential equations, homogeneous differential equations and first-order linear differential equations with separable variables.

Test roll structure

(1) Test scores and test time: the full score of the test paper is 150, and the test time is 180 minutes.

(2) Content ratio

Function, limit, continuity and unary calculus are about 80%; Ordinary differential equation is about 20%.

(3) the proportion of questions

Fill in the blanks and multiple-choice questions about 30%;

Answer questions (including proof) about 70%.

Assigned textbook: Advanced Mathematics (Tongji University 5th Edition)