1.set

The Meaning and Representation of (1) Set

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

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 By understanding the meaning of inclusion and equality between sets, we can identify a subset of a given set.

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

(3) Basic operations of sets

In order to understand the meaning of union and intersection of two sets, we will find the union and intersection of two simple sets.

② Understanding the meaning of the complement set of a subset in a given set will lead to the complement set of a given subset.

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

2. Function concept and basic elementary function Ⅰ (exponential function, logarithmic function, power function)

(1) function

① Knowing the elements that make up a function, we can find the definition domain and value domain 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 apply it simply.

④ Understand the monotonicity, maximum (minimum) value and geometric significance of the function; Combined with specific functions, understand the meaning of function parity.

⑤ Understand and study the properties of functions by using function images.

(2) Exponential function

① Understand the actual background of exponential function model.

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

③ Understand the concept of exponential function, understand the monotonicity of exponential function, and grasp the special points of exponential function images.

④ Know that exponential function is an important function model.

(3) Logarithmic function

(1) Understand the concept of logarithm and its operational properties, and know that general logarithm can be converted into natural logarithm or ordinary logarithm by changing the base formula; Understand the role of logarithm in simplifying operation.

② Understand the concept of logarithmic function; Understand the monotonicity of logarithmic function and master the special points that the function image passes through.

③ Knowing 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 Combine the image of the function to understand its changes.

(5) Functions and equations

① With the image of quadratic function, we can understand the relationship between the zero point of function and the roots of equation, and judge the existence and number of roots of quadratic equation in one variable.

② According to the image of a specific function, the approximate solution of the corresponding equation is obtained by dichotomy.

(6) Function model and its application

① Understand the growth characteristics of exponential function, logarithmic function and power function, and know the significance of growth of different function types such as linear rise, exponential growth and logarithmic growth.

② Understand the wide application of function model (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 their direct views by oblique two-sided method.

③ Two methods, parallel projection and central projection, 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) Engineering drawings and front views of some buildings (the requirements for size and lines are not strict without affecting the graphic characteristics).

⑤ 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

Understand the definition of the relationship between spatial straight line and plane position, 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, axiom and theorem of solid geometry mentioned above, we should know and understand the nature and judgment of parallelism and perpendicularity 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 and any plane passing through the straight line intersects the plane, the straight line is parallel to the intersection 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 Some simple propositions that can prove the spatial position relationship by using axioms, theorems and conclusions.

4. Analysis of Plane Analytic Geometry

(1) row sum equation

(1) in the plane rectangular coordinate system, combined with specific graphics to determine the geometric characteristics of the straight line position.

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 Two straight lines can be judged to be parallel or vertical according to their slopes.

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

⑤ The coordinates of the intersection of two straight lines can be obtained by solving the equation.

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

② We can judge the positional relationship between the straight line and the circle according to the given equation of the straight line and the circle; 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.

Understand the idea of dealing with geometric problems by algebraic method.

(3) Spatial Cartesian coordinate system

(1) know the space rectangular coordinate system, you will use the space rectangular coordinate system to represent the position of the point.

② Derive the distance formula between two points in space.

5. Preliminary algorithm

The meaning of (1) algorithm, program block diagram.

① Understand the meaning and idea of the algorithm.

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 Using simple random sampling method to extract samples from the population; Understand stratified sampling and systematic sampling methods.

(2) Overall estimation

① Understand the significance and function of distribution, list the frequency distribution table, draw the frequency distribution histogram, frequency line diagram and stem leaf diagram, and understand their respective characteristics.

Understand the significance and function of standard deviation of sample data and calculate the standard deviation of data.

③ Basic numerical features (such as mean and standard deviation) can be extracted from the sample data and explained reasonably.

④ The frequency distribution of samples will be used to estimate the population distribution, and the basic digital characteristics of samples will be used to estimate the basic digital characteristics of the population, so as to understand the idea of estimating the population with samples.

⑤ The basic method of random sampling and the idea of sample estimation will solve some simple practical problems.

(3) Correlation of variables

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

② Knowing the idea of least square method, we can establish a linear regression equation according to the given coefficient formula of linear regression equation.

7. possibility

(1) Events and Probability

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

② Understand the probability addition formula of two mutually exclusive events.

(2) Classical probability

Understand classical probability and its probability calculation formula.

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

(3) Random number and geometric probability

Understand the meaning of random number, and estimate the probability through simulation.

② Understand the meaning of geometric probability.

8. Basic elementary function Ⅱ (trigonometric function)

(1) The concept of arbitrary angle and arc system

① Understand the concept of any angle.

Understand the concept of radian system and be able to convert radian and angle.

(2) Trigonometric function

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

② Inductive formulas of sine, cosine and tangent of α and π α can be derived from trigonometric function lines in the unit circle, and images can be drawn to understand the periodicity of trigonometric functions.

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

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

⑤ Understand the physical meaning of the function; Can draw an image and understand the influence of parameters 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 will 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.

Understand the concept of plane vector and the meaning of equality of two vectors.

③ Understand the geometric representation of vectors.

(2) Linear operation of vectors

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

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.

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

③ Coordinates can be 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

Understand the meaning and physical meaning of plane vector product.

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

Master the coordinate expression of scalar product, and can calculate the scalar product of plane vector.

④ The included angle between two vectors can be expressed by the product of quantities, and the vertical relationship between two plane vectors can be judged by the product of quantities.

(5) Application of carrier

① Some simple plane geometry problems can be solved by vector method.

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

10. trigonometric identity transformation

The sum and difference formulas of (1) trigonometric functions

① The cosine formula of the difference between two angles will be derived by using the product of vectors.

② Sine and tangent formulas of two-angle difference can be derived from cosine formula of two-angle difference.

③ The sine, cosine and tangent formulas of the sum of the two angles can be derived by using the cosine formula of the difference between the two angles, and the sine, cosine and tangent formulas of the two angles can be derived 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 triangles

(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.

12. Order

The Concept and Simple Representation of (1) Sequence

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

② Understanding sequence is a kind of function whose independent variable is a positive integer.

(2) arithmetic progression and geometric progression

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

(2) to master the general formula of arithmetic progression and geometric progression and the summation formula of the first n items.

③ We can identify the arithmetical relationship or proportional relationship of sequence in specific problem situations, and solve corresponding problems with relevant knowledge.

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

13. Inequality

(1) inequality relation

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

(2) One-dimensional quadratic inequality

① A quadratic inequality model will be abstracted 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 unary quadratic inequality and design the program block diagram for the given unary quadratic inequality.

(3) Binary linear inequalities and simple linear programming problems.

① A set of binary linear inequalities is abstracted from the actual situation.

② Understand the geometric meaning of binary linear inequality, and express binary linear inequality by grouping in plane areas.

③ Some simple binary linear programming problems will be abstracted from the actual situation and solved.

(4) Basic inequality:

① Understand the proof process of basic inequality.

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

14. Common logical terms

(1) proposition and its relationship

Understand the concept of proposition.

② Understanding the inverse proposition, negative proposition and negative proposition of a proposition in the form of "if P, then Q" will analyze the relationship among the four propositions.

③ Understand the meanings of necessary conditions, sufficient conditions and necessary and sufficient conditions.

(2) Simple logical connectives

Understand the meaning of logical conjunctions "or" and "and".

(3) Full name quantifiers and existential quantifiers

Understand the meaning of universal quantifiers and existential quantifiers.

(2) Can correctly deny the proposition containing quantifiers.

15. Conic curves and equations

(1) conic curve

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

Master the definition, geometry, standard equation and simple properties of ellipse and parabola.

Understand the definition, geometry and standard equation of hyperbola, and know its simple geometric properties.

④ Understand the simple application of conic curve.

⑤ Understand the idea of combining numbers with shapes.

(2) Curves and equations

Understand the relationship between the curve of the equation and the equation of the curve.

16. Space vector and solid geometry

(1) space vector and its operation

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

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

(3) master 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.

(2) Application of space vector

Understand the direction vector of a straight line and the normal vector of a plane.

② The vertical and parallel relationships between straight lines, straight lines and planes can be expressed in vector language.

(3) Some theorems about the positional relationship between a straight line and a plane (including the theorem of three perpendicular lines) can be proved by vector method.

④ The calculation of included angles between straight lines, between straight lines and planes, and between planes can be solved by vector method, so as to understand the application of vector method in studying geometric problems.

17. Derivative and its application

The Concept of (1) Derivative and Its Geometric Significance

① Understand the actual background of the concept of derivative.

② Understand the geometric meaning of derivative.

(2) the operation of derivative

① According to the definition of derivative, we can find the derivative of function (c is a constant).

② We can use the derivative formula of basic elementary function given in table 1 and the four algorithms of derivative to find the derivative of simple function, and we can find the derivative of simple compound function (only the compound 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).

Rule 1

Rule number two

Rule three.

(3) The application of derivative in function research.

① 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).

(2) 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).

(4) Optimization in life.

Will use derivatives to solve some practical problems. ..

(5) Basic theorems of definite integral and calculus

① Understand the actual background, basic ideas and concepts of definite integral.

The significance of understanding the basic theorem of calculus.

18. Reasoning and proof

(1) Rational reasoning and deductive reasoning

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

Understand the importance of deductive reasoning, master the basic models of deductive reasoning, and use them for some simple reasoning.

Understand the connection and difference between perceptual reasoning and deductive reasoning.

(2) Direct proof and indirect proof

① Understand the two basic methods of direct proof-analysis and synthesis; Understand the thinking process and characteristics of analytical methods and comprehensive methods.

② Understand a basic method of indirect proof-reduction to absurdity; Understand the thinking process and characteristics of reduction to absurdity.

(3) Mathematical induction

Knowing the principle of mathematical induction, we can prove some simple mathematical propositions by mathematical induction.

19. Extension of number system and introduction of complex number

The concept of (1) complex number

Understand the basic concepts of complex numbers.

Understanding the necessary and sufficient conditions for the equality of complex numbers.

Understand the algebraic representation of complex numbers and its geometric significance.

(2) Four operations of complex numbers

① Four operations can be performed in the form of complex algebra.

Understand the geometric meaning of addition and subtraction in complex algebraic form.

20. Counting principle

(1) classification addition counting principle and step-by-step multiplication counting principle

① Understand the principles of classified addition counting and classified multiplication counting;

② We can analyze and solve some simple practical problems by using the principle of classified addition counting or step-by-step multiplication counting.

(2) permutation and combination

Understand the concepts of permutation and combination.

② Formulas of permutation number and combination number can be deduced by counting principle.

③ It can solve simple practical problems.

(3) binomial theorem

① The binomial theorem can be proved by counting principle.

② Using binomial theorem to solve simple problems related to binomial expansion.

2 1. Probability and statistics

(1) probability

① Understand the concept of finite discrete random variable and its distribution table, and understand the importance of distribution table in describing random phenomena.

Understand hypergeometric distribution and its derivation process, and can be simply applied.

③ Understand the concepts of conditional probability and independence of two events, 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 can calculate the mean and variance of simple discrete random variables and solve some practical problems.

⑤ Use the histogram of practical problems to understand the characteristics and significance of normal distribution curve.

(2) Statistical cases

Understand the following common statistical methods and apply them to solve some practical problems.

(1) independence test

Understand the basic idea, method and simple application of independence test (only 2×2 contingency table is needed).

(2) Regression analysis

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

(two) the content and requirements of the examination.

1. Symposium on Geometric Proof

(1) Understanding the parallel wire cutting theorem will prove and apply the right triangle projective theorem.

(2) Will prove and apply the theorem of circle angle, the judgment theorem of circle tangent and the property theorem.

(3) We will prove and apply the intersection theorem, the property theorem and judgment theorem of quadrilateral inscribed in a circle, and the tangent line theorem.

(4) Understand the meaning of parallel projection, and understand parallel projection through the positional relationship between cylinder and plane; It is proved that the section of a plane and a cylindrical surface is an ellipse (a circle in special cases).

(5) Understand the following theorems:

Theorem In space, take a straight line as the axis, and the included angle between the straight line and the intersection point O is α, and rotate around it to get a conical surface with O as the vertex and generatrix. Take plane π, and if its included angle with the axis is β (π is parallel, β = 0), then:

① β > α, and the intersection of plane π and cone is ellipse;

② β = α, and the intersection of plane π and cone is parabola;

③ β < α, and the intersection of plane π and cone is hyperbola.

(6) The above theorem ① will be proved by using Danderling's two spherical surfaces (as shown in the figure, these two spherical surfaces are located inside the cone, one above the plane π and the other below the plane, which are tangent to both the plane π and the conical surface, and their tangent points are f and e respectively): when β >; When α, the intersection of plane π and cone is ellipse. (In the figure, the tangent points of the upper and lower spherical surfaces and the conical surface are points B and C respectively, and the line segment BC intersects the plane π at point A)

(7) The following results will be proved:

(1) In (6), the intersection of a dandelion sphere and a conical surface is a circle, which is parallel to the bottom of the cone, and the plane in which this circle lies is π';

(2) If the intersection of plane π and plane π' is m, take any point A on the ellipse in (5) (1) and the tangent point of dandelion ball and plane π is f, then the ratio of the distance from point A to point F to the distance from point A to line M is a constant E (point F is the focus of this ellipse, and line M is the directrix of the ellipse. )

(8) Understand the proof in Theorem (5)③, and understand the limit result of plane π when β approaches α infinitely.

2. Coordinate system and parameter equation

(1) coordinate system

① Understand the function of coordinate system.

(2) to understand the changes of plane graphics under the telescopic transformation of plane rectangular coordinate system.

(3) the position of the point can be expressed in polar coordinates, and the difference between polar coordinates and plane rectangular coordinates can be understood, and polar coordinates and rectangular coordinates can be converted to each other.

④ The equation of a simple figure (such as a straight line crossing the pole, a circle crossing the pole or a circle with its center at the pole) can be given in polar coordinates. By comparing the equations of these figures in polar coordinate system and plane rectangular coordinate system, we can understand the significance of choosing the appropriate coordinate system when using equations to represent plane figures.

⑤ Understand the representation methods of points in space in cylindrical coordinate system and spherical coordinate system, and compare them with the representation methods of points in rectangular coordinate system to understand their differences.

(2) Parameter equation

① Understand the parameter equation and the meaning of parameters.

Can choose appropriate parameters to write parameter equations of straight lines, circles and conic curves.

③ Understand the generation process of cycloid and involute, and derive their parameter equations.

④ Understand the generation process of other cycloids, understand the application of cycloids in practice, and understand the role of cycloids in expressing planetary motion orbits.

3. Lecture 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:

①∣a+b∣≤∣a∣+∣b∣；

②∣a-b∣≤∣a-c∣+∣c-b∣；

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

∣ax+b∣≤c；

∣ax+b∣≥c；

∣x-a∣+∣x-b∣≥c.

(2) Understand several different forms of Cauchy inequality, understand its geometric meaning and prove it.

① Cauchy inequality vector form: | α|? |β|≥|α? β|.

② ≥ .

③ + ≥

(usually called trigonometric inequality).

(3) Discuss the general situation of Cauchy inequality by parameter matching method: ≥.

(4) Discuss rank inequality with vector recursion method.

(5) Understand the principle and application scope of mathematical induction, and prove some simple problems with mathematical induction.

(6) Bernoulli inequality will be proved by mathematical induction:

Is a positive integer greater than 1), and it is known that Bernoulli inequality also holds when n is a real number greater than 1.

(7) We can prove some simple problems with the above inequalities. We can use the mean inequality and Cauchy inequality to find the extreme value of some specific functions.

(8) Understand the basic methods of proving inequality: comparison, synthesis, analysis, reduction to absurdity and scaling.