Ⅳ. Inspection scope and requirements

I. Required Contents and Requirements

(1) set

The meaning and representation of a set.

(1) Understand the meaning of set and the relationship between elements and set.

(2) Natural language, graphic language and set language (enumeration or description) can be used to describe different specific problems.

2. The basic relationship between sets

(1) Understanding the meaning of inclusion and equality between sets can identify subsets of a given set.

(2) Understand the meaning of complete works and empty sets in specific situations.

3. Basic operations of sets

(1) To understand the meaning of union and intersection of two sets, we need union and intersection of two simple sets.

(2) To understand the meaning of the complement of a subset in a given set, we need the complement of a given subset.

(3) venn diagram can be used to express the basic relations between sets and the basic operations of sets.

(B) the concept of function and basic elementary function Ⅰ

1. function

(1) Knowing the elements that make up a function, we can find the domain and value of some simple functions; Understand the concept of mapping.

(2) In actual situations, appropriate methods (such as image method, list method, analysis method, etc.) will be selected according to different needs.

(3) Understand the simple piecewise function and simply apply it (the function is divided into no more than three segments).

(4) Understand the monotonicity, maximum (minimum) value and geometric significance of the function; Understand the meaning of functional parity.

(5) Using basic elementary functions to analyze the properties of functions.

2. Exponential function

(1) Understand the actual background of the exponential function model.

(2) Understand the meaning of rational exponential power, understand the meaning of real exponential power, and master the operation of power.

(3) Understand the concept and monotonicity of exponential function, grasp the special points that the exponential function image passes through, and draw the exponential function image with 2,3, 10, 1/2, 1/3 as the base.

(4) Exponential function is an important function model.

3. Logarithmic function

(1) Understand the concept of logarithms and their operational properties, and know how to convert general logarithms into natural logarithms or ordinary logarithms by changing the radix formula; Understand the role of logarithm in simplifying operation.

(2) Understand the concept and monotonicity of logarithmic function, master the special points that the logarithmic function image passes through, and draw the logarithmic function images with cardinality of 2, 10 and 1/2.

(3) Understand that logarithmic function is an important function model;

(4) Understand that exponential function and logarithmic function are reciprocal functions.

4. Power function

(1) Understand the concept of power function.

(2) Combination function

Image, understand their changes.

5. Functions and equations

Combined with the image of quadratic function, understand the relationship between function zero and equation root, and judge the existence and number of quadratic equation roots in one variable.

6. Function model and its application

(1) Understand the growth characteristics of exponential function, logarithmic function and power function, and realize the significance of growth of different function types such as linear rise, exponential growth and logarithmic growth with concrete examples.

(2) Understand the wide application of function models (such as exponential function, logarithmic function, power function, piecewise function, etc.).

(3) Preliminary study of solid geometry.

1. Space geometry

(1) Understand the structural features of cylinders, cones, platforms, spheres and their simple combinations, and use these features to describe the structure of simple objects in real life.

(2) Can draw three views of simple space graphics (simple combination of cuboid, sphere, cylinder, cone, prism, etc.). ), can identify the three-dimensional model represented by the above three views, and draw its vertical view by oblique two-sided method.

(3) The parallel projection method will be used to draw three views and straight views of simple space graphics, so as to understand the different representations of space graphics.

(4) Understand the formulas for calculating the surface area and volume of spheres, prisms, pyramids and platforms (no need to memorize formulas).

2. The positional relationship between points, lines and surfaces.

(1) Understand the definition of the positional relationship between a spatial line and a plane, and understand the following axioms and theorems that can be used as the basis of reasoning.

Axiom 1: If two points on a straight line are in a plane, then all points on this straight line are in this plane.

Axiom 2: When three points that are not on a straight line intersect, there is one and only one plane.

Axiom 3: If two non-coincident planes have a common point, then they have one and only one common straight line passing through the point.

Axiom 4: Two lines parallel to the same line are parallel to each other.

Theorem: If two sides of an angle in space are parallel to two sides of another angle, then the two angles are equal or complementary.

(2) Based on the definition, axioms and theorems of solid geometry mentioned above, we should know and understand the nature and judgment of parallelism and verticality of straight lines and planes in space.

Understand the following decision theorem.

◆ If a straight line out of plane is parallel to a straight line in this plane, then this straight line is parallel to this plane.

If two intersecting lines on one plane are parallel to the other plane, then the two planes are parallel.

If a straight line is perpendicular to two intersecting straight lines on a plane, then the straight line is perpendicular to the plane.

If one plane passes through the perpendicular of the other plane, then the two planes are perpendicular to each other.

Understand and prove the following property theorems.

If a straight line is parallel to a plane, then the intersection of any plane passing through this straight line and this plane is parallel to this straight line.

If two parallel planes intersect the third plane at the same time, their intersection lines are parallel to each other.

◆ Two straight lines perpendicular to the same plane are parallel.

If two planes are perpendicular, a straight line perpendicular to their intersection on one plane is perpendicular to the other plane.

(3) Simple propositions that can prove the position relationship of some spatial graphs by using axioms, theorems and conclusions.

(4) Preliminary analysis of plane analytic geometry.

1. Lines and equations

(1) In the plane rectangular coordinate system, grasp the geometric characteristics of determining the position of a straight line by combining specific figures.

(2) Understand the concepts of inclination angle and slope of a straight line, and master the calculation formula of slope of a straight line passing through two points.

(3) According to the slopes of two straight lines, we can judge whether they are parallel or vertical.

(4) Master the geometric characteristics of determining the position of a straight line, master several forms of linear equation (point oblique, two points, general), and understand the relationship between oblique section line and linear function.

(5) The intersection coordinates of two intersecting lines can be obtained by solving the equation.

(6) Master the distance formula between two points and the distance formula from point to straight line, and you will find the distance between two parallel straight lines.

2. Circle sum equation

(1) Grasp the geometric characteristics of the circle, and master the standard equation and general equation of the circle.

(2) According to the given equation of straight line and circle, the positional relationship between straight line and circle can be judged; Can judge the positional relationship between two given circles according to their equations.

(3) Some simple problems can be solved by equations of straight lines and circles.

(4) Understand the idea of dealing with geometric problems by algebraic methods.

3. Spatial Cartesian coordinate system

(1) Understand the spatial rectangular coordinate system, and use the spatial rectangular coordinate system to represent the position of points.

(2) The distance formula between two points in space can be simply applied.

(5) Preliminary algorithm

The significance of 1. algorithm and program block diagram

(1) Understand the significance and ideas of the algorithm.

(2) Understand three basic logical structures of program block diagram: sequence, conditional branch and loop.

2. Basic algorithm statements

Understand the meaning of several basic algorithm statements-input statement, output statement, assignment statement, conditional statement and loop statement.

(6) Statistics

1. Random sampling

(1) Understand the necessity and importance of random sampling.

(2) A simple random sampling method will be used to extract samples from the population; Understand stratified sampling and systematic sampling methods.

2. Estimate the population with samples

(1) Understand the significance and function of distribution, and draw frequency distribution histogram, frequency line graph and stem leaf graph according to the frequency distribution table to realize their own characteristics.

(2) Understand the significance and function of standard deviation of sample data, and calculate the standard deviation of data (without memorizing formulas).

(3) Basic numerical features (such as mean and standard deviation) can be extracted from the sample data, and reasonable explanations are given.

(4) We will use the frequency distribution of samples to estimate the population distribution, and we will use the basic digital characteristics of samples to estimate the basic digital characteristics of the population, so as to understand the idea of using samples to estimate the population.

(5) We will use the basic method of random sampling and the idea of sample estimation to solve some simple practical problems.

3. Correlation of variables

(1) will make a scatter plot of the data of two related variables, and use the scatter plot to understand the correlation between the variables.

(2) Knowing the idea of least square method, we can establish a linear regression equation according to the coefficient formula of the given linear regression equation (the coefficient formula of the linear regression equation does not need to be memorized).

(7) Probability

1. Events and probabilities

(1) Understand the uncertainty and frequency stability of random events, and understand the meaning of probability and the difference between frequency and probability.

(2) Understand mutually exclusive events's two probability addition formulas.

2. Classical probability

(1) Understand classical probability and its probability calculation formula.

(2) Calculate the number of basic events and the probability of some random events.

3. Random Numbers and Geometric Probability

(1) Understand the meaning of random numbers and be able to estimate the probability through simulation.

(2) Understand the meaning of geometric probability.

(VIII) Basic elementary function II (trigonometric function)

1. The concept of arbitrary angle and arc system

(1) Understand the concepts of arbitrary angles and arc systems.

(2) The radian and angle can be changed.

2. Trigonometric function

(1) Understand the definition of trigonometric functions (sine, cosine and tangent).

(2) It can be deduced from the trigonometric function line in the unit circle.

The inductive formulas of sine, cosine and tangent of α and π α can be drawn.

Understand the periodicity of trigonometric functions.

(3) Understand the properties of sine function and cosine function in the interval [0,2π] (such as monotonicity, maximum and minimum, intersection with X axis, etc.). ). Understand that the tangent function is in the interval.

Monotonicity in).

(4) Understand the basic relationship of trigonometric functions with the same angle:

(5) Understand the function

The physical meaning of; Can draw.

To understand these parameters.

Influence on the change of function image.

(6) Understanding trigonometric function is an important function model to describe the phenomenon of periodic change, and trigonometric function can be used to solve some simple practical problems.

(9) Plane vector

The Practical Background and Basic Concepts of 1. Plane Vector

(1) Understand the actual background of the vector.

(2) Understand the concept of plane vector and the meaning that two vectors are equal.

(3) Understand the geometric representation of vectors.

2. Linear operation of vectors

(1) Master the operation of vector addition and subtraction and understand its geometric meaning.

(2) Master the operation of vector multiplication and its geometric meaning, and understand the meaning of two vector lines.

(3) Understand the nature and geometric significance of vector linear operation.

3. The basic theorem and coordinate representation of plane vector.

(1) Understand the basic theorem of plane vector and its significance.

(2) Master the orthogonal decomposition of plane vector and its coordinate representation.

(3) Coordinates are used to represent the addition, subtraction and multiplication of plane vectors.

(4) Understand the condition that plane vector lines are represented by coordinates.

4. The product of plane vectors

(1) Understand the meaning of plane vector quantity product and its physical meaning.

(2) Understand the relationship between the product of plane vectors and vector projection.

(3) Having mastered the coordinate expression of the scalar product, we can calculate the scalar product of the plane vector.

(4) The product of available quantities indicates the included angle between two vectors, and the product of available quantities determines the vertical relationship between two plane vectors.

5. The application of vectors

(1) will use vector method to solve some simple plane geometry problems.

(2) Using vector method to solve simple mechanical problems and other practical problems.

(10) trigonometric identity transformation

1. formulas of trigonometric functions of sum and difference of two angles.

(1) will use the product of vectors to derive the cosine formula of the difference between two angles.

(2) Sine formula and tangent formula of two-angle difference are derived from cosine formula of two-angle difference.

(3) The sine, cosine and tangent formulas of the sum of two angles and the sine, cosine and tangent formulas of two angles will be derived from the cosine formula of the difference between two angles, so as to understand their internal relations.

2. Simple trigonometric identity transformation

You can use the above formula to carry out simple identity transformation (including derivation of sum-difference product, sum-difference product and half-angle formula, but you don't need to remember these three formulas).

(1 1) Solving Triangle

1. Sine theorem and cosine theorem

Master sine theorem and cosine theorem and solve some simple triangle measurement problems.

2. Application

Can use knowledge and methods such as sine theorem and cosine theorem to solve some practical problems related to measurement and geometric calculation.

(XII) Series

The Concept and Simple Representation of 1. Sequence

(1) Understand the concept of sequence and several simple representations (list, image, general formula).

(2) Understand that sequence is a special function whose independent variable is a positive integer.

2. Arithmetic series, geometric series

(1) Understand the concepts of arithmetic progression and geometric progression.

(2) Master the general formula of arithmetic progression and geometric progression and the sum formula of the first n items.

(3) Be able to identify the arithmetic relationship or proportional relationship of sequence in specific problem situations, and use relevant knowledge to solve corresponding problems.

(4) Understand the relationship between arithmetic progression and linear function, geometric series and exponential function.

(XIII) Inequality

1. inequality relation

Understand the unequal relationship between the real world and daily life and the actual background (group) of inequality.

2. Unary quadratic inequality

(1) will abstract a quadratic inequality model from the actual situation.

(2) Understand the relationship between the unary quadratic inequality and the corresponding quadratic function and unary quadratic equation through the function image.

(3) Can solve the quadratic inequality of one variable and design the program block diagram for the given quadratic inequality of one variable.

3. Binary linear inequalities and simple linear programming problems

(1) will abstract a set of binary linear inequalities from the actual situation.

(2) Knowing the geometric meaning of binary linear inequality, we can use plane region to represent binary linear inequality.

(3) Some simple binary linear programming problems are abstracted from the actual situation and solved.

4. Basic inequality:

(1) Understand the process of proving basic inequalities.

(2) Basic inequalities can be used to solve simple maximum (minimum) problems.

(14) Common logical terms

(1) Understand the concept of proposition.

(2) Understanding the proposition in the form of "If P, then Q" and its inverse proposition, negative proposition and negative proposition will analyze the relationship among the four propositions.

(3) Understand the meaning of necessary conditions, sufficient conditions and necessary and sufficient conditions.

(4) Understand the meanings of logical conjunctions "or", "and" and ".

(5) Understand the meanings of universal quantifiers and existential quantifiers.

(6) Can correctly deny the proposition containing quantifiers.

(15) conic curve and equation

(1) Understand the actual background of conic section and its role in depicting the real world and solving practical problems.

(2) Master the definition, geometry, standard equation and simple properties of ellipse and parabola (range, symmetry, fixed point, eccentricity).

(3) Understand the definition, geometric figure and standard equation of hyperbola, and know its simple geometric properties (range, symmetry, fixed point, eccentricity, asymptote).

(4) Understand the corresponding relationship between curves and equations.

(5) Understand the idea of combining numbers with shapes.

(6) Understand the simple application of conic curve.

(16) space vector and solid geometry

(1) Understand the concept, basic theorem and significance of space vector, and master the orthogonal decomposition and coordinate representation of space vector.

(2) Master the linear operation of space vector and its coordinate representation.

(3) Grasp the quantity product of space vector and its coordinate representation, and use the quantity product of vector to judge the * * * line and vertical line of vector.

(4) Find the direction vector of the straight line and the normal vector of the plane.

(5) The parallel and vertical relations among lines, lines and planes can be expressed by vector language.

(6) Some theorems (including triple perpendicularity theorem) about the positional relationship between a straight line and a plane can be proved by vector method.

(7) We can use vector method to calculate the included angles of straight lines, straight lines and planes, and planes and planes, and understand the application of vector method in studying geometric problems.

(XVII) Derivatives and their applications

(1) Understand the practical background of the concept of derivative.

(2) Understand the geometric meaning of the derivative intuitively through the function image.

(3) Find the function according to the definition of derivative.

(c is a constant).

(4) We can use the derivative formula of basic elementary function and the four operations given below to find the derivative of simple function, and we can find the derivative of simple composite function (only the composite function with the shape of f(ax+b)).

Derivative formula and derivative operation formula of common basic elementary functions;

(c is a constant);

n∈N+

;

(a>0, while a ≠1);

(a>0 and a ≠ 1).

Commonly used derivative algorithm:

Rule 1

.

Rule 2

.

Rule 3

(5) Understand the relationship between monotonicity and derivative of function; The monotonicity of functions can be studied by using derivatives, and the monotone interval of functions can be found (in which polynomial functions are generally not more than three times).

(6) Understand the necessary and sufficient conditions for the function to obtain the extreme value at a certain point; Will use derivatives to find the maximum and minimum values of functions (in which polynomial functions generally do not exceed three times); Will find the maximum and minimum value of the function in the closed interval (where the polynomial function generally does not exceed three times).

(7) Using derivatives to solve some practical problems. ..

(8) Understand the actual background, basic ideas and concepts of definite integral.

(9) Understand the meaning of the basic theorem of calculus.

(XVIII) Reasoning and proof

(1) Understand the meaning of sensible reasoning, make simple reasoning through induction and analogy, and understand the role of sensible reasoning in mathematical discovery.

(2) Understand the meaning of deductive reasoning and the relationship and difference between perceptual reasoning and deductive reasoning; Mastering the syllogism of deductive reasoning can make some simple deductive reasoning.

(3) Understand two basic methods of direct proof: analysis and synthesis; Understand the thinking process and characteristics of analytical methods and comprehensive methods.

(4) Understand the thinking process and characteristics of reduction to absurdity.

(5) Understand the principle of mathematical induction, and use mathematical induction to prove some simple mathematical propositions.

(nineteen) the expansion of the number system and the introduction of complex numbers.

(1) Understand the basic concepts of complex numbers and the necessary and sufficient conditions for the equality of complex numbers.

(2) Understand the algebraic representation of complex numbers and their geometric significance; Algebraic complex numbers can be represented by points or vectors on the complex plane, and the complex numbers corresponding to points or vectors on the complex plane can be represented in algebraic form.

(3) Be able to perform four operations in the form of complex algebra and understand the geometric meaning of the addition and subtraction operations of two specific complex numbers.

(20) Counting principle

(1) Knowing the principle of classified addition counting and the principle of step-by-step multiplication counting can correctly distinguish "class" from "step" and solve some simple practical problems by using the two principles.

(2) Understand the concept of permutation and the formula of permutation number, and use the formula to solve some simple practical problems.

(3) Understand the concept of combination and the formula of combination number, and use the formula to solve some simple practical problems.

(4) Using binomial theorem to solve simple problems related to binomial expansion.

(21) Probability statistics

(1) Understand the concept of finite-valued discrete random variables and their distribution tables, and understand the importance of distribution tables to the description of random phenomena, so we will find some distribution tables of finite-valued discrete random variables.

(2) Understand the hypergeometric distribution and its derivation process, and simply apply it.

(3) Understand the concept of conditional probability, understand the concept that two events are independent of each other, understand the model and binomial distribution of n independent repeated tests, and solve some simple practical problems.

(4) Understanding the concepts of mean and variance of finite discrete random variables will help us to find the mean and variance of simple discrete random variables and solve some simple problems by using the concepts of mean and variance of discrete random variables.

(5) Understand the characteristics and significance of normal distribution curve with the help of intuitive histogram.

(6) Understand the basic ideas, methods and simple applications of regression.

(7) Understand the idea, method and preliminary application of independence test.

Second, the content and requirements of the exam

(a) Seminar on Geometric Proof

(1) Understand the definition and properties of similar triangles and the parallel cutting theorem.

(2) The following theorems will be proved and applied: ① right triangle projective theorem; ② Theorem of circle angle; ③ The tangent judgment theorem and property theorem of the circle; ④ Intersecting chord theorem; ⑤ The property theorem and judgement theorem of the quadrilateral inscribed in a circle; ⑥ Cutting line theorem.

(2) Coordinate system and parameter equation

(1) Understand the function of coordinate system and the change of plane figure under the telescopic transformation of plane rectangular coordinate system.

(2) Knowing the basic concept of polar coordinates, we can describe the position of points in polar coordinates and realize the mutual conversion between polar coordinates and rectangular coordinates.

(3) In the polar coordinate system, we can give the polar coordinate equation expressed by simple figures (such as a straight line crossing the pole, a circle crossing the pole or a circle with the center at the pole).

(4) Understand the parameter equation and the meaning of parameters.

(5) Can choose appropriate parameters to write the parametric equations of straight lines, circles and conic curves.

(3) Special lectures on inequality

(1) Understand the geometric meaning of absolute value, and prove the following inequality by using the geometric meaning of inequality with absolute value:

∣a+b∣≤∣a∣+∣b∣；

∣a-b∣≤∣a-c∣+∣c-b∣；

(2) We will use the geometric meaning of absolute value to solve the following kinds of inequalities:

∣ax+b∣≤c；

∣ax+b∣≥c；

∣x-c+∣x-b∣≥a

(3) Through some simple questions, understand the basic methods of proving inequality: comparison, synthesis and analysis.

Magician Tang hopes it will be useful to you! ! !