Mathematics three

subject of examination

Calculus, linear algebra, probability theory and mathematical statistics

Test paper structure

I. Total score

The full mark of the test paper is 150, and the test time is 180 minutes.

Second, the content ratio

Calculus is about 56%

Linear algebra accounts for about 22%

Probability theory and mathematical statistics account for about 22%

Third, the problem structure.

8 multiple-choice questions, each with 4 points and ***32 points.

Fill in the blanks with 6 small questions, with 4 points for each small question and 24 points for * *.

Answer 9 small questions (including proof questions), ***94 points.

calculus

I. Function, Limit and Continuity

Examination content

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.

Examination requirements

1. Understand the concept of function and master the expression of function, and you will 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 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 left limit and right limit).

6. Understand the nature of limit and two criteria for the existence of limit, master four algorithms of limit, and master the method of finding limit by using two important limits.

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

8. Understanding the concept of function continuity (including left continuity and right continuity) can determine the type of function discontinuity.

9. 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

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.

Examination requirements

1. Understand the concept of derivative and the relationship between derivability and continuity, understand the geometric and economic significance of derivative (including the concepts of allowance and elasticity), and find the tangent equation and normal equation of plane curve.

2. Master the derivative formula of basic elementary function, the four operation rules of derivative and the derivative rule of compound function, and you can find the derivative of piecewise function, inverse function and implicit function.

If you understand the concept of higher derivative, you will find the higher derivative of a 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 theorem and Lagrange mean value theorem, Taylor theorem and Cauchy mean value theorem, and master the simple application of these four theorems.

6. Will use the Lobida rule to find the limit.

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

8. The concavity and convexity of the function graph can be judged by the derivative (note: in the interval (a, b), let the function f(x) have the second derivative, when the graph of f(x) is concave; When the graph of f(x) is convex, the inflection point and asymptote of the function graph will be found.

3. Integral calculus of unary function

Examination content

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.

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 of upper limit of integral and find its derivative, and master Newton-Leibniz formula, method of substitution and integration by parts of definite integral.

3. I will use definite integral to calculate the area of plane figure, the volume of rotating body and the average value of functions, and I will use definite integral to solve simple economic application problems.

4. Understand the concept of generalized integral and calculate generalized integral.

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, 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.

Examination requirements

1. Understand the concept of multivariate function and the geometric meaning of bivariate 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 and the total differential 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 the simple application problem.

5. Understand the concept and basic properties of double integral, master the calculation method of double integral (rectangular coordinates, polar coordinates), understand and calculate the simple abnormal double integral in unbounded area.

Five, infinite series

Examination content

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.

Examination requirements

1. Understand the concepts of convergence and divergence of series and sum of convergent series.

2. Understand the basic properties of series and the necessary conditions of series convergence, master the convergence and divergence conditions of geometric series and P series, and master the comparison and ratio discrimination methods of positive series convergence.

3. Understand the concepts of absolute convergence and conditional convergence of arbitrary series and the relationship between absolute convergence and convergence, and understand Leibniz discriminant method of staggered series.

4. Will find the convergence radius, convergence interval and convergence domain of power series.

5. Knowing the basic properties of power series in its convergence interval (continuity of sum function, item-by-item derivation, item-by-item integration), we find the sum function of simple power series in its convergence interval, and then we find the sum of several terms of some series.

6. Understand the Maclaurin expansions of,, and.

Six, ordinary differential equations and difference equations

Examination content

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.

Examination requirements

1. Understand differential equations and their concepts such as order, solution, general solution, initial condition and special solution.

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

3. Second-order homogeneous linear differential equations with constant coefficients can be solved.

4. Knowing the properties and structural theorems of solutions of linear differential equations, we can use polynomials, exponential functions, sine functions and cosine functions to solve second-order non-homogeneous linear differential equations with constant coefficients.

5. Understand the concepts of difference and difference equation and their general and special solutions.

6. Understand the solution method of the first-order linear difference equation with constant coefficients.

7. Can use differential equations to solve simple economic application problems.