How many knowledge points are there in mathematics for three years in senior high school?
Sorting out the knowledge points of senior three mathematics
Sequence is an important content of high school mathematics and the basis of learning advanced mathematics. The examination of the College Entrance Examination is comprehensive, and the examination of arithmetic progression and geometric progression will not be missed every year. The test questions about series are often comprehensive questions, which often combine the knowledge of series with the knowledge of exponential function, logarithmic function and inequality. The test questions often combine arithmetic progression, geometric progression, seeking limit and mathematical induction.
Inquiry problem is a hot spot in the college entrance examination, which often appears when solving the problem of sequence. This chapter also contains a wealth of mathematical ideas. In the subjective questions, important ideas such as function and equation, transformation and reduction, classified discussion, and basic mathematical methods such as collocation method, method of substitution method and undetermined coefficient method are emphasized.
In recent years, the proposition of the college entrance examination on the number sequence mainly has the following three aspects;
(1) series itself, including the concepts, properties, general formulas and summation formulas of arithmetic progression and geometric progression.
(2) Combination of sequence with other knowledge, including combination of sequence with function, equation, inequality, triangle and geometry.
(3) The application of series, in which the growth rate is the main problem. There are three levels of difficulty in the test questions. Most of the small questions are based on basic questions, and most of the answers are based on basic and intermediate questions. It's just that in some places, it's difficult to take the synthesis of sequence and geometry and the synthesis of function and inequality as the last question.
1. On the basis of mastering the definition, nature, general formula, first n items and formulas of arithmetic progression and geometric progression, systematically master the law of solving comprehensive problems of arithmetic progression and geometric progression, deepen the guiding role of mathematical thinking methods in problem-solving practice, and flexibly use the knowledge and methods of sequence to solve related problems in mathematics and real life;
2. In the practice of solving comprehensive and exploratory problems, deepen the understanding of basic knowledge, basic skills and basic mathematical thinking methods, communicate all kinds of knowledge, form a relatively complete knowledge network, and improve the ability to analyze and solve problems.
Further cultivate students' reading comprehension and innovation ability, and comprehensively use mathematical thinking methods to analyze and solve problems.
Sorting out the knowledge points of senior three mathematics
Random sampling
brief introduction
(lottery method, random sample table method) is often used when the total number is small, and its main feature is to extract one by one from the total number;
Advantages: simple operation.
Disadvantages: The overall size is too large to be realized.
way
(1) lottery method
Generally speaking, the lottery method is to number n individuals in the population, write the numbers on the digital labels, put the digital labels into a container, and after stirring evenly, extract one digital label from it every time and extract it for n times continuously to get a sample with a capacity of n.
(The lottery method is simple and easy, suitable for a few people in the crowd. When there are a large number of individuals in the group, it is difficult to "evenly mix the group", which is probably due to the poor representativeness of the samples produced by lottery)
(2) Random number method
In random sampling, another commonly used method is random number method, that is, random number table, random number dice or computer-generated random number are used for sampling.
group sampling
brief introduction
The main feature of stratified sampling is proportional stratified sampling, which is mainly used in the crowd, and there are obvious differences between individuals. * * * Similarity: the probability of each individual being drawn is equal to n/m.
definition
Generally speaking, when sampling, the population is divided into disjoint layers, and then a certain number of individuals are independently extracted from each layer according to a certain proportion, and the individuals extracted from each layer are combined as samples. This sampling method is stratified sampling.
Nested sampling method
definition
What is cluster sampling?
Cluster sampling is also called cluster sampling. It is to combine all the units in the population into several sets that do not cross each other and do not repeat each other, which are called groups; A sampling method in which samples are sampled in groups.
When cluster sampling is applied, each cluster is required to have good representativeness, that is, the differences between units within the cluster are large and the differences between groups are small.
merits and demerits
The advantages of cluster sampling are convenient implementation and saving money;
The disadvantage of cluster sampling is that the sampling error caused by large differences between different groups is often greater than that caused by simple random sampling.
Implementation steps
Firstly, the population is divided into group I, and then several groups are extracted from group I clock to investigate all individuals or units in these groups. The sampling process can be divided into the following steps:
Firstly, the label of clustering is determined.
Second, divide the whole (n) into several non-overlapping parts, and each part is a group.
Third, according to the sample size, determine the number of groups to be extracted.
Fourthly, by using simple random sampling or systematic sampling method, the determined group number is extracted from the I group.
For example, investigate the situation of middle school students suffering from myopia and make statistics in the last class; Conduct product inspection; All products produced by 1h are sampled and inspected every 8 hours, etc.
Difference from stratified sampling
Cluster sampling and stratified sampling are similar in form, but quite different in fact.
Stratified sampling requires large differences between layers, small differences between individuals or units within layers, small differences between groups and large differences between individuals or units within groups;
The sample of stratified sampling consists of several units or individuals extracted from each layer, while cluster sampling is either cluster sampling or cluster sampling is not.
systematic sampling
definition
When there are many individuals in the group, it is more troublesome to adopt simple random sampling. At this time, the population can be divided into several balanced parts, and then an individual can be extracted from each part according to predetermined rules to get the required samples. This kind of sampling is called systematic sampling.
step
Generally speaking, if you want to extract a sample with a capacity of n from a population with a capacity of n, you can carry out systematic sampling according to the following steps:
(1) First number the n individuals in the population. Sometimes you can directly use your own number, such as student number, admission ticket number, house number, etc.
(2) Determine the segment interval k and the number of segments. When N/n(n is the sample size) is an integer, take k = n/n;
(3) determining the first individual number L (L ≤ K) by simple random sampling in the first paragraph;
(4) Sampling according to certain rules. Usually, the interval k is added with L to get the second number of individuals (l+k), and then K is added to get the third number of individuals (l+2k), and so on until the whole sample is obtained.
Sorting out the knowledge points of senior three mathematics
(A) the first definition of derivative
Let the function y=f(x) be defined in a domain of point x0. When the independent variable x has increment △ x at x0 (x0+△ x is also in the neighborhood), the corresponding function gets increment △ y = f (x0+△ x)-f (x0); If the ratio of △y to △x has a limit when △x→0, the function y=f(x) can be derived at point x0, and this limit value is called the derivative of function y=f(x) at point x0, which is also called f'(x0), which is the first definition of derivative.
(2) The second definition of derivative
Let the function y=f(x) be defined in a domain of point x0. When the independent variable x changes △ x at x0 (x-x0 is also in the neighborhood), the function changes △y=f(x)-f(x0) accordingly. If the ratio of △y to △x is limited when △x→0, then the function y=f(x) is derivable at point x0. This limit value is called that the derivative of function y=f(x) at point x0 is f'(x0), which is the second definition of derivative.
(3) Derivative function and derivative
If the function y=f(x) is differentiable at every point in the open interval I, it is said that the function f(x) is differentiable in the interval I. At this time, the function y=f(x) corresponds to a certain derivative of each certain value of x in the interval I, and forms a new function, which is called the derivative function of the original function y=f(x), and is denoted as y' and f'. Derivative function is called derivative for short.
Monotonicity and its application
1. General steps of studying monotonicity of polynomial function with derivative
(1) Find f \u( x)
(2) Make sure that f￠(x) is in (a, b). Symbol (3) If F ￠ (x) >: 0 is a constant on (a, b), then f(x) is a increasing function on (a, b); If f \u( x)
2. The general steps of finding monotone interval of polynomial function by derivative.
(1) Find f \u( x)
(2) The interval corresponding to the intersection of the solution set of f \ u (x) > 0 and the domain is an increasing interval; f \u( x)& lt； The interval corresponding to the intersection of the solution set of 0 and the domain is the subtraction interval.
Sorting out the knowledge points of senior three mathematics
Definition of 1. sequence
A series of numbers arranged in a certain order is called a series, and each number in the series is called an item of the series.
(1) As can be seen from the definition of series, the numbers of series are arranged in a certain order. If the numbers that make up a series are the same but in different order, then they are not the same series. For example, the series 1, 2,3,4,5 is different from the series 5,4,3,2, 1.
(2) The definition of series does not stipulate that the numbers in the series must be different. Therefore, multiple identical numbers can appear in the same series, such as:-1, 1, 2, 3, 4, … to form a series:-1,-1.
(4) The term of a series is different from its number. The term of a series refers to a certain number in this series, which is a function value, equivalent to f(n), while the term of a number refers to the position serial number of this number in the series, which is the value of an independent variable, equivalent to n in f(n).
(5) Order is very important for a series. There are several identical figures. Because of their different arrangement order, the series is also different. Obviously, there is an essential difference between a series and a group of numbers. For example, if the five numbers 2, 3, 4, 5 and 6 are arranged in different order, different series will be obtained, while {2, 3, 4, 5 and 6}.
2. Series classification
(1) According to the number of items in the series, the series can be divided into finite series and infinite series. When writing a series, the last item should be written for a finite series, for example, the series 1, 3, 5, 7, 9, …, 2n- 1 indicates a finite series. If the sequence is written as 1,
(2) According to the relationship between items or the increase or decrease of series, it can be divided into the following categories: increasing series, decreasing series, swinging series and constant series.
3. General term formula of sequence
A sequence is a series of numbers arranged in a certain order, and its essential attribute is to determine the law of this number, which is usually expressed by formula f(n).
Although these two general formulas are different in form, they represent the same series, just as not every functional relationship can be expressed by analytical formula, and not every series can write its general formula. Although some series have general formulas, they may not be formally established. They only know the finite term in front of a series, and there is no other explanation. The series cannot be determined, and the general formula does not exist. For example, the series 1, 2, 3, 4, ...
The items written after the formula are different. Therefore, the induction of general term formula depends not only on its first few terms, but also on the synthesis law of series, and more observation and analysis are needed to truly find the internal law of series. There is no general way to write its general term formula from the first few terms of a series.
In order to understand the general formula of series, emphasize the following points again:
The general term formula of (1) series is actually that the domain is a set of positive integers N_ its finite set {1, 2, ..., n}.
(2) If we know the general term formula of the series, then we can use 1, 2, 3, … instead of N in the formula in turn to find out the terms of this series; At the same time, we can also use the general term formula of series to judge whether a number is an item in the series, and if so, what item it is.
(3) Just as all functional relationships do not necessarily have analytical formulas, not all series have general formulas.
If the approximation is less than 2, the sequence shall be accurate to 1, 0. 1, 0.0 1, 0.00 1, 0.000 1, ... 1, 1.
(4) Some general formulas of series are not necessarily formal, such as:
(5) Some series only give the first few terms, but do not give their composition rules, so the general term formula of series derived from the first few terms is not.
4. Series of images
For series 4, 5, 6, 7, 8, 9, 10, the corresponding relationship between the serial number of each item and this item is as follows:
Serial number: 1234567
Item code: 456789 10
In other words, the above can be regarded as a mapping from one set of serial numbers to another. Therefore, from the perspective of mapping and function, a sequence can be regarded as a positive integer set N_ or its finite set {1, 2,3, ..., n}). When the independent variable takes the value from small to large, the corresponding function value list. The function here is
Because the term of the series is a function value and the serial number is an independent variable, the general term formula of the series is the corresponding function and analytical formula.
Sequence is a special function, which can be expressed intuitively by images.
The sequence is represented by an image. With the serial number as the abscissa and the corresponding item as the ordinate, you can draw a picture to represent an order. When drawing, for convenience, the unit length taken on the two coordinate axes of the plane rectangular coordinate system can be different. From the image representation of the sequence, we can directly see the change of the sequence, but it is not accurate.
Compared with function, sequence is a special function, which is a group of positive integers or a group of finite continuous positive integers headed by 1, and its image is infinite or finite isolated points.
5. Recursive series
A pile of steel pipes is stacked in seven layers, and the number of steel pipes in each layer forms a sequence from top to bottom: 4, 5, 6, 7, 8, 9, 10.
The order ① can also be given by the following method: the number of steel pipes in the first floor from top to bottom is 4, and the number of steel pipes in each floor below is more than that in the previous floor 1 root.
Different from senior one and senior two, reviewing some knowledge of mechanics at this time is aimed at better combining with the college entrance examination outline, especially for students with medium or below average level. At this time, it is necessary to check the leaks and fill the gaps, but at the same time, we must improve our ability and fill the gaps in knowledge and skills. Next is the knowledge points of senior three mathematics compiled by Bian Xiao. I hope you like it!
Sorting out the knowledge points of senior three mathematics
Sequence is an important content of high school mathematics and the basis of learning advanced mathematics. The examination of the College Entrance Examination is comprehensive, and the examination of arithmetic progression and geometric progression will not be missed every year. The test questions about series are often comprehensive questions, which often combine the knowledge of series with the knowledge of exponential function, logarithmic function and inequality. The test questions often combine arithmetic progression, geometric progression, seeking limit and mathematical induction.
Inquiry problem is a hot spot in the college entrance examination, which often appears when solving the problem of sequence. This chapter also contains a wealth of mathematical ideas. In the subjective questions, important ideas such as function and equation, transformation and reduction, classified discussion, and basic mathematical methods such as collocation method, method of substitution method and undetermined coefficient method are emphasized.
In recent years, the proposition of the college entrance examination on the number sequence mainly has the following three aspects;
(1) series itself, including the concepts, properties, general formulas and summation formulas of arithmetic progression and geometric progression.
(2) Combination of sequence with other knowledge, including combination of sequence with function, equation, inequality, triangle and geometry.
(3) The application of series, in which the growth rate is the main problem. There are three levels of difficulty in the test questions. Most of the small questions are based on basic questions, and most of the answers are based on basic and intermediate questions. It's just that in some places, it's difficult to take the synthesis of sequence and geometry and the synthesis of function and inequality as the last question.
1. On the basis of mastering the definition, nature, general formula, first n items and formulas of arithmetic progression and geometric progression, systematically master the law of solving comprehensive problems of arithmetic progression and geometric progression, deepen the guiding role of mathematical thinking methods in problem-solving practice, and flexibly use the knowledge and methods of sequence to solve related problems in mathematics and real life;
2. In the practice of solving comprehensive and exploratory problems, deepen the understanding of basic knowledge, basic skills and basic mathematical thinking methods, communicate the links of all kinds of knowledge, form a relatively complete knowledge network, and improve the ability to analyze and solve problems.
Further cultivate students' reading comprehension and innovation ability, and comprehensively use mathematical thinking methods to analyze and solve problems.
Sorting out the knowledge points of senior three mathematics
Random sampling
brief introduction
(lottery method, random sample table method) is often used when the total number is small, and its main feature is to extract one by one from the total number;
Advantages: simple operation.
Disadvantages: The overall size is too large to be realized.
way
(1) lottery method
Generally speaking, the lottery method is to number n individuals in the population, write the numbers on the digital labels, put the digital labels into a container, and after stirring evenly, extract one digital label from it every time and extract it for n times continuously to get a sample with a capacity of n.
(The lottery method is simple and easy, suitable for a few people in the crowd. When there are a large number of individuals in the group, it is difficult to "evenly mix the group", which is probably due to the poor representativeness of the samples produced by lottery)
(2) Random number method
In random sampling, another commonly used method is random number method, that is, random number table, random number dice or computer-generated random number are used for sampling.
group sampling
brief introduction
The main feature of stratified sampling is proportional stratified sampling, which is mainly used in the crowd, and there are obvious differences between individuals. * * * Similarity: the probability of each individual being drawn is equal to n/m.
definition
Generally speaking, when sampling, the population is divided into disjoint layers, and then a certain number of individuals are independently extracted from each layer according to a certain proportion, and the individuals extracted from each layer are combined as samples. This sampling method is stratified sampling.
Nested sampling method
definition
What is cluster sampling?
Cluster sampling is also called cluster sampling. It is to combine all the units in the population into several sets that do not cross each other and do not repeat each other, which are called groups; A sampling method in which samples are sampled in groups.
When cluster sampling is applied, each cluster is required to have good representativeness, that is, the differences between units within the cluster are large and the differences between groups are small.
merits and demerits
The advantages of cluster sampling are convenient implementation and saving money;
The disadvantage of cluster sampling is that the sampling error caused by large differences between different groups is often greater than that caused by simple random sampling.
Implementation steps
Firstly, the population is divided into group I, and then several groups are extracted from group I clock to investigate all individuals or units in these groups. The sampling process can be divided into the following steps:
Firstly, the label of clustering is determined.
Second, divide the whole (n) into several non-overlapping parts, and each part is a group.
Third, according to the sample size, determine the number of groups to be extracted.
Fourthly, by using simple random sampling or systematic sampling method, the determined group number is extracted from the I group.
For example, investigate the situation of middle school students suffering from myopia and make statistics in the last class; Conduct product inspection; All products produced by 1h are sampled and inspected every 8 hours, etc.
Difference from stratified sampling
Cluster sampling and stratified sampling are similar in form, but quite different in fact.
Stratified sampling requires large differences between layers, small differences between individuals or units within layers, small differences between groups and large differences between individuals or units within groups;
The sample of stratified sampling consists of several units or individuals extracted from each layer, while cluster sampling is either cluster sampling or cluster sampling is not.
systematic sampling
definition
When there are many individuals in the group, it is more troublesome to adopt simple random sampling. At this time, the population can be divided into several balanced parts, and then an individual can be extracted from each part according to predetermined rules to get the required samples. This kind of sampling is called systematic sampling.
step
Generally speaking, if you want to extract a sample with a capacity of n from a population with a capacity of n, you can carry out systematic sampling according to the following steps:
(1) First number the n individuals in the population. Sometimes you can directly use your own number, such as student number, admission ticket number, house number, etc.
(2) Determine the segment interval k and the number of segments. When N/n(n is the sample size) is an integer, take k = n/n;
(3) determining the first individual number L (L ≤ K) by simple random sampling in the first paragraph;
(4) Sampling according to certain rules. Usually, the interval k is added with L to get the second number of individuals (l+k), and then K is added to get the third number of individuals (l+2k), and so on until the whole sample is obtained.
Sorting out the knowledge points of senior three mathematics
(A) the first definition of derivative
Let the function y=f(x) be defined in a domain of point x0. When the independent variable x has increment △ x at x0 (x0+△ x is also in the neighborhood), the corresponding function gets increment △ y = f (x0+△ x)-f (x0); If the ratio of △y to △x has a limit when △x→0, the function y=f(x) can be derived at point x0, and this limit value is called the derivative of function y=f(x) at point x0, which is also called f'(x0), which is the first definition of derivative.
(2) The second definition of derivative
Let the function y=f(x) be defined in a domain of point x0. When the independent variable x changes △ x at x0 (x-x0 is also in the neighborhood), the function changes △y=f(x)-f(x0) accordingly. If the ratio of △y to △x is limited when △x→0, then the function y=f(x) is derivable at point x0. This limit value is called that the derivative of function y=f(x) at point x0 is f'(x0), which is the second definition of derivative.
(3) Derivative function and derivative
If the function y=f(x) is differentiable at every point in the open interval I, it is said that the function f(x) is differentiable in the interval I. At this time, the function y=f(x) corresponds to a certain derivative of each certain value of x in the interval I, and forms a new function, which is called the derivative function of the original function y=f(x), and is denoted as y' and f'. Derivative function is called derivative for short.
Monotonicity and its application
1. General steps of studying monotonicity of polynomial function with derivative
(1) Find f \u( x)
(2) Make sure that f￠(x) is in (a, b). Symbol (3) If F ￠ (x) >: 0 is a constant on (a, b), then f(x) is a increasing function on (a, b); If f \u( x)
2. The general steps of finding monotone interval of polynomial function by derivative.
(1) Find f \u( x)
(2) The interval corresponding to the intersection of the solution set of f \ u (x) > 0 and the domain is an increasing interval; f \u( x)& lt； The interval corresponding to the intersection of the solution set of 0 and the domain is the subtraction interval.
Sorting out the knowledge points of senior three mathematics
Definition of 1. sequence
A series of numbers arranged in a certain order is called a series, and each number in the series is called an item of the series.
(1) As can be seen from the definition of series, the numbers of series are arranged in a certain order. If the numbers that make up a series are the same but in different order, then they are not the same series. For example, the series 1, 2,3,4,5 is different from the series 5,4,3,2, 1.
(2) The definition of series does not stipulate that the numbers in the series must be different. Therefore, multiple identical numbers can appear in the same series, such as:-1, 1, 2, 3, 4, … to form a series:-1,-1.
(4) The term of a series is different from its number. The term of a series refers to a certain number in this series, which is a function value, equivalent to f(n), while the term of a number refers to the position serial number of this number in the series, which is the value of an independent variable, equivalent to n in f(n).
(5) Order is very important for a series. There are several identical figures. Because of their different arrangement order, the series is also different. Obviously, there is an essential difference between a series and a group of numbers. For example, if the five numbers 2, 3, 4, 5 and 6 are arranged in different order, different series will be obtained, while {2, 3, 4, 5 and 6}.
2. Series classification
(1) According to the number of items in the series, the series can be divided into finite series and infinite series. When writing a series, the last item should be written for a finite series, for example, the series 1, 3, 5, 7, 9, …, 2n- 1 indicates a finite series. If the sequence is written as 1,
(2) According to the relationship between items or the increase or decrease of series, it can be divided into the following categories: increasing series, decreasing series, swinging series and constant series.
3. General term formula of sequence
A sequence is a series of numbers arranged in a certain order, and its essential attribute is to determine the law of this number, which is usually expressed by formula f(n).
Although these two general formulas are different in form, they represent the same series, just as not every functional relationship can be expressed by analytical formula, and not every series can write its general formula. Although some series have general formulas, they may not be formally established. They only know the finite term in front of a series, and there is no other explanation. The series cannot be determined, and the general formula does not exist. For example, the series 1, 2, 3, 4, ...
The items written after the formula are different. Therefore, the induction of general term formula depends not only on its first few terms, but also on the synthesis law of series, and more observation and analysis are needed to truly find the internal law of series. There is no general way to write its general term formula from the first few terms of a series.
In order to understand the general formula of series, emphasize the following points again:
The general term formula of (1) series is actually that the domain is a set of positive integers N_ its finite set {1, 2, ..., n}.
(2) If we know the general term formula of the series, then we can use 1, 2, 3, … instead of N in the formula in turn to find out the terms of this series; At the same time, we can also use the general term formula of series to judge whether a number is an item in the series, and if so, what item it is.
(3) Just as all functional relationships do not necessarily have analytical formulas, not all series have general formulas.
If the approximation is less than 2, the sequence shall be accurate to 1, 0. 1, 0.0 1, 0.00 1, 0.000 1, ... 1, 1.
(4) Some general formulas of series are not necessarily formal, such as:
(5) Some series only give the first few terms, but do not give their composition rules, so the general term formula of series derived from the first few terms is not.
4. Series of images
For series 4, 5, 6, 7, 8, 9, 10, the corresponding relationship between the serial number of each item and this item is as follows:
Serial number: 1234567
Item code: 456789 10
In other words, the above can be regarded as a mapping from one set of serial numbers to another. Therefore, from the perspective of mapping and function, a sequence can be regarded as a positive integer set N_ or its finite set {1, 2,3, ..., n}). When the independent variable takes the value from small to large, the corresponding function value list. The function here is
Because the term of the series is a function value and the serial number is an independent variable, the general term formula of the series is the corresponding function and analytical formula.
Sequence is a special function, which can be expressed intuitively by images.
The sequence is represented by an image. With the serial number as the abscissa and the corresponding item as the ordinate, you can draw a picture to represent an order. When drawing, for convenience, the unit length taken on the two coordinate axes of the plane rectangular coordinate system can be different. From the image representation of the sequence, we can directly see the change of the sequence, but it is not accurate.
Compared with function, sequence is a special function, which is a group of positive integers or a group of finite continuous positive integers headed by 1, and its image is infinite or finite isolated points.
5. Recursive series
A pile of steel pipes is stacked in seven layers, and the number of steel pipes in each layer forms a sequence from top to bottom: 4, 5, 6, 7, 8, 9, 10.
The order ① can also be given by the following method: the number of steel pipes in the first floor from top to bottom is 4, and the number of steel pipes in each floor below is more than that in the previous floor 1 root.
Different from senior one and senior two, the department of mechanics is reviewed at this time.