First, the content of "set and function" is cross-complementary, and there are exponential pairs of functions. Parity and increase and decrease are the most obvious observation images.

When the compound function appears, the law of property multiplication is distinguished. To prove it in detail, we must grasp the definition.

Exponential function and logarithmic function are reciprocal functions. Cardinality is not a positive number of 1, and 1 increases or decreases on both sides.

The domain of the function is easy to find. Denominator cannot be equal to 0, even roots must be non-negative, and zero and negative numbers have no logarithm.

The tangent function angle is not straight, and the cotangent function angle is uneven; The real number sets of other functions have intersection in many cases.

Two mutually inverse function have that same monotone property; The images are symmetrical with Y=X as the symmetry axis.

Solve the very regular inverse solution of substitution domain; The domain of inverse function, the domain of original function.

The nature of power function is easy to remember, and the index reduces the score; Keywords exponential function, odd mother and odd son odd function,

Even function with odd mother and even son, even mother non-parity function; In the first quadrant of the image, the function is increased or decreased to see the positive and negative.

Second, trigonometric functions

Trigonometric functions are functions, and quadrant symbols are labeled. Function image unit circle, periodic parity increase and decrease.

The same angle relation is very important, and both simplification and proof are needed. At the vertex of a regular hexagon, the chord is cut from top to bottom.

The numb 1 records that triangle connecte the vertices in the center; The sum of the squares of the downward triangle, the reciprocal relationship is diagonal,

Any function of a vertex is equal to the division of the last two. The inductive formula is good, negative is positive and then big and small,

It is easy to look up the table when it becomes a tax corner, and it is essential to simplify the proof. Half of the integer multiple of two, odd complementary pairs remain unchanged,

The latter is regarded as an acute angle, and the sign is judged as the original function. The cosine of the sum of two angles is converted into a single angle, which is convenient for evaluation.

Cosine product minus sine product, angular deformation formula. Sum and difference products must have the same name, and the complementary angle must be renamed.

The calculation proves that the angle is the first, pay attention to the name of the structural function, the basic quantity remains unchanged, and it changes from complexity to simplicity.

Guided by the principle of reverse order, the product of rising power and falling power and difference. The proof of conditional equality, the idea of equation points out the direction.

Universal formula is unusual, rational formula is ahead. The formula is used before and after, and the deformation is skillfully used.

1 add cosine to think of cosine, 1 subtract cosine to think of sine, power-on angle is halved, and power-on and power-off is a norm.

The inverse function of trigonometric function, in essence, is to find the angle, first find the value of trigonometric function, and then determine the range of angle value.

Using right triangle, the image is intuitive and easy to rename. The equation of a simple triangle is reduced to the simplest solution set.

Three. inequality

The solution to inequality is to use the properties of functions. The unreasonable inequality of the opposite side is transformed into a rational inequality.

From high order to low order, step-by-step transformation should be equivalent. The mutual transformation between numbers and shapes helps to solve problems.

The method of proving inequality is powerful in real number property. Difference is compared with 0, and quotient is compared with 1.

Comprehensive method with good direct difficulty analysis and clear thinking. Non-negative common basic expressions, positive difficulties are reduced to absurdity.

There are also important inequalities and mathematical induction. Graphic function help, drawing modeling construction method.

Fourth, "series"

Arithmetic ratio two series, the sum of n terms in the general formula. Two finites seek the limit, and four operations are the other way around.

The problem of sequence is changeable, and the equation is simplified as a whole calculation. It is difficult to sum series, but it is skillful to eliminate dislocation and transform.

Learn from each other's strong points and calculate the sum formula of split terms. Inductive thinking is very good, just do a program to think about it:

Counting two and seeing three associations, guessing proves indispensable. There is also mathematical induction to prove that the steps are programmed:

First verify and then assume, from k to k plus 1, the reasoning process must be detailed and affirmed by the principle of induction.

Verb (abbreviation of verb) plural

As soon as the imaginary unit I came out, the number set was expanded into a complex number. A complex number and a logarithm, the real and imaginary parts of horizontal and vertical coordinates.

Corresponding to a point on the complex plane, the origin is connected with it in the form of an arrow. The axis of the arrow is opposite to the X axis, and the resulting angle is a radial angle.

The length of the arrow shaft is a model, and the numbers are often combined. Algebraic geometric triangles, try to transform each other.

The essence of algebraic operation is I polynomial operation. The positive integer of I is the second time, and there are four numerical periods.

Some important conclusions, cleverly remember the results. The ability of mutual transformation between reality and reality is great, and complex number equals transformation.

Solve with equations and pay attention to the whole substitution. On the geometric operation diagram, the addition parallelogram,

Subtractive triangle rule judgment; Multiplication and division operations, reverse and forward rotation, expansion and contraction of annual module length.

In triangular operation, it is necessary to distinguish between radiation angle and mode. It is very convenient to take a square and make a square by using Demofo formula.

The radial angle operation is very strange, and the product quotient is used to sum the difference. These four properties are inseparable, such as equality module and yoke,

Two will not be real numbers, and the comparison size is not allowed. Complex numbers are very close to real numbers, so we should pay attention to the essential differences.

Six, "permutation, combination and binomial theorem"

The two principles of addition and multiplication are the laws that run through. What has nothing to do with order is combination, but what needs order is arrangement.

Two formulas, two properties, two ideas and methods. Arrange and combine the summary, and the application questions must be transformed.

It is common sense to arrange and combine together and choose the back row first. Special elements and positions should be considered first.

Don't worry too much, and don't miss too much. Punching is a skill. Arrange combinatorial identities and define proof modeling tests.

On binomial theorem, China Yang Hui Triangle. Two properties, two formulas, function assignment transformation.

Seven. solid geometry

The trinity of point, line and surface is represented by cone billiards. All distances start from points, and all angles are made of lines.

High school solid geometry

Vertical parallelism is the key point, and the concept must be clear in the proof. Line, line, plane, plane, triple cycle.

When the whole idea of the equation is solved, it becomes consciousness. Before calculation, it is necessary to prove and draw the removed figure.

Auxiliary lines of solid geometry, usually vertical lines and planes. The concept of projection is very important and the key to solving problems.

The dihedral angle and volume projection formulas of lines with different planes are vivid. Axioms are naturally three vertical lines, which solve many problems.

Eight, "plane analytic geometry"

Directed line segments, straight circles, elliptic hyperbolic parabolas, polar coordinates of parametric equations and the combination of numbers and shapes are called normal forms.

Descartes' viewpoint pair, point and ordered real number pair correspond to each other, which opens up a new way of geometry.

The two ideas reflect each other and turn into ideas to fight the front line; The undetermined coefficient method is actually the idea of equations.

Summarize three types, draw a curve to solve the equation, and give the curve of the equation and the relationship between the curves.

Four tools are magic weapons with good coordinate parameters; Plane geometry can not be lost, find the complex number of rotation transformation.

Analytic geometry is geometry, and you can't get carried away. Graphics are intuitive and detailed, and mathematics is mathematics.

Edit this required mathematics course 1

1.set

(about 4 class hours)

The Meaning and Representation of (1) Set

Senior high school mathematics (15 sheets)

① Understand the meaning of set and the "subordinate" relationship between elements and set through examples.

② We can choose natural language, graphic language and assembly language (enumeration or description) to describe different specific problems and feel the significance and function of assembly language.

(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.

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 require 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 express the relations and operations of sets, and the role of intuitive graphs in understanding abstract concepts can be realized. 2. The concept of function and basic elementary function

(about 32 class hours)

(1) function

① Further understand that function is an important mathematical model to describe the dependence between variables, and on this basis, learn to describe functions with sets and corresponding languages, and understand the role of correspondence in describing the concept of functions; Knowing the elements that make up a function, we can find the definition and value range of some simple functions; Understand the concept of mapping.

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

③ Understand the simple piecewise function and apply it simply.

④ Understand the monotonicity, maximum (minimum) value and its geometric significance of the function through the learned function, especially the quadratic function; Understand the meaning of parity with specific functions.

⑤ Learn to use function images to understand and study the properties of functions (see example 1).

(2) Exponential function

(1) (cell division, the decay of archaeological C, the change of drug residues in human body, etc. ), and understand the actual background of exponential function model.

② Understand the meaning of rational exponential power, understand the meaning of real exponential power through concrete examples, and master the operation of power.

③ To understand the concept and significance of exponential function, we can draw the image of specific exponential function with the help of calculator or computer, and explore and understand the monotonicity and special points of exponential function.

④ In the process of solving simple practical problems, I realized that exponential function is an important function model (see Example 2).

(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; By reading the materials, we can understand the history of logarithm and its role in simplifying operations.

② Through concrete examples, we can intuitively understand the quantitative relationship described by the logarithmic function model, preliminarily understand the concept of logarithmic function, and realize that logarithmic function is an important function model; With the help of calculator or computer, we can draw images of specific logarithmic functions and explore and understand the monotonicity and special points of logarithmic functions.

③ Know that exponential function and logarithmic function are reciprocal functions (A >；); 0，a≠ 1)。

(4) Power function

Understand the concept of power function through examples; Combine the images of functions to understand their changes.

(5) Functions and equations

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

(2) According to the image of a specific function, it is a common method to find the approximate solution of the corresponding equation by dichotomy with the help of a calculator.

(6) Function model and its application

① Compare the growth differences of exponential function, logarithmic function and power function with calculation tools; Combined with examples, we can understand the meaning of growth of different function types such as linear rise, exponential explosion and logarithmic growth.

② Collect some examples of function models (exponential function, logarithmic function, power function, piecewise function, etc. ) It is often used in social life to understand the wide application of functional models.

(7) Practice homework

According to a certain theme, collect some historical events and figures (Kepler, Galileo, Descartes, Newton, Leibniz, Euler, etc. )/kloc-For those who have played an important role in the development of mathematics around the 0/7th century, or some examples of functions in real life, write an article about the formation, development or application of the concept of functions in the form of group cooperation and communicate in class. See the requirements of mathematical culture for specific requirements.

Edit this math required course 2 1. A preliminary study on solid geometry

(about 18 class hours)

(1) space geometry

① Using physical models and computer software to observe a large number of spatial graphics, we can understand the structural characteristics of columns, cones, platforms, balls and their simple combinations, and can use these characteristics 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, can make models with materials (such as cardboard), and can draw their own front views by oblique double-sided method.

③ By observing the views and straight views drawn by two methods (parallel projection and central projection), we can understand the different representations of spatial graphics.

(4) Complete the internship, such as drawing some views and front views of 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

(1) With the help of the cuboid model, on the basis of intuitive knowledge and understanding of the positional relationship between points, lines and surfaces in space, the definition of the positional relationship between lines and surfaces in space is abstracted, and the following axioms and theorems that can be used as the basis of reasoning are understood.

Axiom 1: If two points on a straight line are on a plane, then the straight line is on 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.

Theorem: If two sides of two angles in space are parallel to each other, then the two angles are equal or complementary.

② Based on the above-mentioned definitions, axioms and theorems of solid geometry, we can know and understand the related properties and judgments of parallelism and verticality of straight lines and planes in space through intuitive perception, operational confirmation and speculative argumentation.

Operation confirmation, summed up the following judgment theorem.

◆ If the straight line out of the plane is parallel to the straight line in the plane, the straight line is parallel to the plane.

◆ Two intersecting straight lines in one plane are parallel to another plane, so the two planes are parallel.

◆ If a straight line is perpendicular to two intersecting straight lines in the plane, the straight line is perpendicular to the plane.

◆ If one plane intersects the perpendicular of another plane, the two planes are perpendicular.

The operation is confirmed, and the following property theorems are summarized and proved.

◆ If a straight line is parallel to a plane, the intersection line between any plane passing through the straight line and the plane is parallel to the straight line.

◆ If two planes are parallel, the intersection lines obtained by the intersection of any plane and these two planes are parallel to each other.

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

◆ If two planes are perpendicular, the straight line perpendicular to the intersection line in one plane is perpendicular to the other plane.

③ We can use the conclusions to prove some simple propositions of spatial relationship.

2. Analysis of Plane Analytic Geometry

(about 18 class hours)

(1) row sum equation

(1) In the plane rectangular coordinate system, combined with specific graphics, the geometric characteristics of determining the position of a straight line are explored.

② Understand the concepts of inclination angle and slope of a straight line, experience the process of describing the slope of a straight line by algebraic method, and master the calculation formula of the 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) According to the geometric characteristics of determining the position of a straight line, explore and master several forms of linear equation (point oblique, two points, general), and understand the relationship between oblique line and linear function.

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

⑥ Explore and master the distance formula between two points and the distance formula from point to straight line, and find the distance between two parallel straight lines.

(2) Circle sum equation

(1) review and determine the geometric characteristics of the circle, explore and master the standard equation and general equation of the circle in the plane rectangular coordinate system.

② According to the given equation of straight line and circle, we can judge the positional relationship between straight line and circle and between them.

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

(3) During the initial study of plane analytic geometry, I realized the idea of using algebraic method to deal with geometric problems.

(4) Spatial Cartesian coordinate system

(1) Through specific situations, feel the necessity of establishing a spatial rectangular coordinate system, understand the spatial rectangular coordinate system, and describe the position of points by using the spatial rectangular coordinate system.

② By representing the coordinates of the vertices of a special cuboid (each side is parallel to the coordinate axis), the distance formula between two points in space is explored.

Edit this math required course 3 1. Preliminary algorithm

(about 12 class hours)

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

(1) by analyzing the process and steps to solve specific problems (such as solving binary linear equations, etc.). ), we can understand the idea and significance of the algorithm.

② Through imitation, operation and exploration, experience the process of expressing and solving problems by designing program block diagram. In the process of solving specific problems (such as solving ternary linear equations, etc. ), understand the three basic logical structures of program block diagram: sequence, conditional branch and loop.

(2) Basic algorithm statements: Through the process of transforming the program block diagram of specific problems into program statements, we can understand several basic algorithm statements-input statements, output statements, assignment statements, conditional statements and loop statements, and further understand the basic idea of the algorithm.

(3) By reading the algorithm cases in ancient mathematics in China, we can understand the contribution of ancient mathematics in China to the development of mathematics in the world.

2. Statistics

(about 16 class hours)

(1) random sampling

(1) can raise some valuable statistical questions from real life or other disciplines.

② Understand the necessity and importance of random sampling in combination with specific practical problem situations.

③ In the process of solving statistical problems, learn to use simple random sampling method to extract samples from the population; Through case study, we can understand the methods of stratified sampling and systematic sampling.

④ Data can be collected through experiments, consulting materials and designing questionnaires.

(2) estimate the population with samples

① Understand the significance and function of distribution through examples. In the process of representing sample data, learn to list the frequency distribution table, draw the frequency distribution histogram, frequency line diagram and stem leaf diagram (see example 1), and understand their respective characteristics.

② Understand the significance and function of standard deviation of sample data through examples, and learn to calculate the standard deviation of data.

③ We can reasonably select samples according to the needs of practical problems, extract basic numerical features (such as mean and standard deviation) from sample data, and make reasonable explanations.

④ In the process of solving statistical problems, we will further understand the idea of estimating the population with samples. We will estimate the population distribution with the frequency distribution of samples and estimate the basic digital characteristics of the population with the basic digital characteristics of samples. Understand the randomness and numerical characteristics of sample frequency distribution.

⑤ We will use the basic method of random sampling and the idea of sample estimation to solve some simple practical problems; Through the analysis of data, we can provide some basis for rational decision-making, understand the role of statistics and understand the difference between statistical thinking and deterministic thinking.

⑥ Form a preliminary evaluation consciousness of data processing.

(3) Correlation of variables

① Make a scatter plot by collecting the data of two related variables in the real question, and use the scatter plot to intuitively understand the correlation between variables.

② Experiencing the process of describing the linear correlation of two variables with different estimation methods. Knowing the idea of least square method, we can establish a linear regression equation according to the given coefficient formula of linear regression equation (see Example 2).

3. Possibility

(about 8 class hours)

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

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

(3) Through examples, we can understand the classical probability and its probability calculation formula, and use enumeration method to calculate the number of basic events and the probability of some random events.

(4) Knowing the meaning of random numbers, we can use simulation methods (including random numbers generated by calculators for simulation) to estimate the probability and get a preliminary understanding of the meaning of geometric probability (see Example 3).

(5) By reading the materials, we can understand the cognitive process of human beings to random phenomena.

Edit this math required course 4 1. Trigonometric function.

(about 16 class hours)

(1) Any angle and radian

Understand the concept of arbitrary angle and radian system, and realize the conversion between radian and angle.

(2) Trigonometric function

① Understand the definition of trigonometric functions (sine, cosine and tangent) with the help of the unit circle.

② Derive inductive formulas (sine, cosine and tangent) with the help of trigonometric function lines in the unit circle, and draw pictures to understand the periodicity of trigonometric functions.

③ Understand the properties of sine function, cosine function and tangent function (such as monotonicity, maximum and minimum value, image intersecting with X axis, etc.). ) with the help of images.

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

⑤ Understand the practical significance with concrete examples; With the help of the image drawn by calculator or computer, we can observe the influence of parameters a and ω on the change of function image.

⑥ trigonometric function can be used to solve some simple practical problems, and it is recognized that trigonometric function is an important function model to describe periodic changes.

2. Plane vector

(about 12 class hours)

The Practical Background and Basic Concepts of (1) Plane Vector

Through the analysis of force and other examples, we can understand the actual background of vector, the meaning of plane vector and vector equality, and the geometric representation of vector.

(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 understand its geometric meaning and the meaning of two vector lines.

③ Understand the linear operation properties of vectors and their geometric significance.

(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 will be used to represent the addition, subtraction and multiplication of plane vectors.

(4) understand the condition that the plane vector * * * straight line is represented by coordinates.

(4) the product of plane vectors

① Understand the meaning and physical meaning of the product of plane vectors through examples such as "work" in physics.

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

(3) Grasp the coordinate expression of the product of quantity, and carry out the product operation 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

Through the process of solving some simple plane geometric problems, mechanical problems and other practical problems with vector method, I realize that vector is a tool to deal with geometric and physical problems and cultivate the ability to calculate and solve practical problems.

3. Trigonometric identity transformation

(about 8 class hours)

(1) experienced the process of deriving the cosine formula of the difference between two angles by using the product of vectors, and further realized the function of vector method.

(2) Sine, cosine and tangent formulas of sum and difference of two angles and sine, cosine and tangent formulas of two angles can be derived from cosine formula of difference of two angles, so as to understand their internal relations.

(3) We can use the above formula to carry out simple identity transformation (including guiding and deducing product sum and difference, product sum and difference, and half-angle formula, but we don't need to remember).

Edit this math required course 5 1. Solve triangles.

(about 8 class hours)

(1) By exploring the relationship between the sides and angles of any triangle, we can master the sine theorem and cosine theorem and solve some simple triangle measurement problems.

(2) Be able to use knowledge and methods such as sine theorem and cosine theorem to solve some practical problems related to measurement and geometric calculation.

Step 2: Order

(about 12 class hours)

The Concept and Simple Representation of (1) Sequence

Understand the concept of sequence and several simple expressions (list, image, general formula), and understand that sequence is a special function.

(2) arithmetic progression and geometric progression

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

② Explore and master the general formula of arithmetic progression and geometric progression and the formula of the sum of the first n items.

(3) In specific problem situations, we can find the arithmetic relationship or proportional relationship of the sequence, and use relevant knowledge to solve corresponding problems (see example 1).

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

3. Inequality

(about 16 class hours)

(1) inequality relation

Feel a lot of unequal relations between the real world and daily life, and understand the actual background (group) of inequality.

(2) One-dimensional quadratic inequality

① Experience the process of abstracting a quadratic inequality model from the actual situation.

(2) Understand the relationship between unary quadratic inequality and corresponding functions and equations through function images.

(3) Can solve the unary quadratic inequality, and try to 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) To understand the geometric meaning of binary linear inequality, we can express binary linear inequality by plane region (see Example 2).

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

(4) Basic inequality:

① Explore and understand the process of proving basic inequalities.

② The basic inequality will be used to solve the simple maximum (minimum) problem (see Example 4).

Property Exponent and Logarithm of Function

(1) domain, range, corresponding rules

(2) Monotonicity

For any x 1, x2∈D

If x 1

If x 1

(3) Parity

If f (-x) = f(x), f(x) is called an even function for any X in the domain of function F (x).

If f (-x) =-f(x), then f(x) is called odd function.

(4) periodicity

For any x in the definition domain of function f(x), if there is a constant t that makes f(x+T)=f(x), it is said that f(x) is a fractional exponential power of periodic function (1).

Mathematics elective course