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The concept of the best answer function is one of the most important concepts in all mathematical concepts. Throughout the development of the concept of function in the past 300 years, many mathematicians have constantly given new ideas to the concept of function from the angles of set, algebra, correspondence and set, thus promoting the development of the whole mathematics. This paper discusses the teaching of function concept through the development and comparative study of function concept. 1, the vertical development of the concept of function 1. 1 The early concept of function-the function under the concept of geometry (G.Galileo, meaning 1564- 1642) is almost covered from beginning to end in the book Two New Sciences. Descartes (France, 1596- 1650) noticed the dependence of one variable on another in his analytic geometry around 1673, but he didn't realize the need to refine the general concept of function at that time, so Newton and Leibniz didn't establish calculus until the late17th century. 1.218th century function concept-function under algebraic concept1718bernoulli Johann (Switzerland, 1667- 1748) was defined on the basis of Leibniz function concept. 1in the middle of the 8th century, L. Euler (Switzerland, 1707- 1783) gave a very vivid function symbol, which has been used ever since. Euler's definition is that the function of a variable is an analytical expression composed of this variable and some numbers, that is, constants, in any way. He called johann bernoulli's definition of function analytic function, and further divided it into algebraic function (only referring to the algebraic operation between independent variables) and transcendental function (trigonometric function, logarithmic function and unreasonable power of variables), and also considered "arbitrary function" (representing the function of drawing any curve). It is not difficult to see that Euler's definition of function is more universal and broader than johann bernoulli's definition. 1.319th century function concept-function under corresponding relation 1822 Fourier (method, 1768- 1830) found that some functions can be represented by curves, one formula or multiple formulas, thus In 1823, Cauchy (method, 1789- 1857) gave the definition of function from the definition of variable, and pointed out that although infinite series is an effective method to specify a function, it is not necessary for a function to have analytical expressions, but he still thinks that the function relationship can be expressed by multiple analytical expressions, which is a big one. In 1837, Dirichlet (Germany, 1805- 1859) thinks that how to establish the relationship between x and y is irrelevant. He expanded the concept of function and pointed out: "For every definite value of X in a certain interval, Y has one or more definite values, so Y is called a function of X." Dirichlet's definition of function avoids all descriptions of dependence in the previous definition of function, which is concise and accurate, and is unconditionally accepted by all mathematicians in a completely clear way. At this point, it can be said that the concept of function and the essential definition of function have been formed, that is, the classical function definition that people often say. After the set theory founded by Cantor (Cantor, German, 1845- 19 18) played an important role in mathematics, Van Buren (United States, 1880- 1960) used "set" and ". 1.4 modern function concept-function under set theory 19 14 F Hausdorf defined function with "ordered couple" in the Outline of Set Theory. Its advantage is that it avoids the fuzzy concepts of "variable" and "correspondence", and its disadvantage is that it introduces the fuzzy concept of "ordered pair". In 192 1, Kuratowski defined "ordered pair" with the concept of set, that is, ordered pair (a, b) is a set,}, thus making Hausdorff's definition very strict. In 1930, the new modern function is defined as: if there is always an element y determined by set n corresponding to any element x of set m, it is said that a function is defined on set m, and it is denoted as y=f(x). Element x is called independent variable and element y is called dependent variable. The definition of function has been tempered and changed for more than 300 years, forming a modern definition form of function, but this does not mean the historical end of the development of function concept. In the 1940s, for the need of physics research, a function called dirac-δ function was discovered. It is not zero at all, but its integral on the whole line is equal to 1, which is inconceivable under the original definitions of function and integral. However, because the concept of function is broad, with the development of other disciplines based on mathematics, the concept of function will continue to expand.

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Other answers *** 1 History shows that important mathematical concepts have an inestimable effect on the development of mathematics, and the influence of function concepts on the development of mathematics can be said to be eternal and lasting. Looking back at the historical development of the concept of function and seeing the historical process in which the concept of function is continuously refined, deepened and enriched is very beneficial, which not only helps us to improve the clarity of the context of the concept of function. Marx once thought that the concept of function originated from the study of indefinite equations in algebra. Because Diophantine in Roman times had studied indefinite equations, the concept of function had sprouted at least then. Since Copernicus's astronomical revolution, sports have become a common problem for Renaissance scientists. People are thinking: since the earth is not the center of the universe, it has its own rotation and revolution, why does the falling body fall vertically to the earth instead of deflecting? The orbit of this planet is elliptical. What is the principle? In addition, it is not only a problem that scientists try to solve, but also a problem that military strategists ask to solve to study the route, range and height that projectiles can reach on the surface of the earth, as well as the influence of projectile speed on height and range. The concept of function is a mathematical concept derived from the study of motion, and motion is the mechanical source of the concept of function. Long before the concept of function was clearly put forward, mathematicians had contacted and studied many specific functions, such as logarithmic function, trigonometric function, hyperbolic function and so on. Descartes noticed the dependence of one variable on another in Analytic Geometry around 1673, but he didn't realize the need to refine the general concept of function at that time, so Newton and Leibniz didn't establish calculus until 17 century. 1673, Leibniz first used the word "function" to represent "power", and later he used this word to represent the geometric quantities of each point on the curve, such as abscissa, ordinate and tangent length. It can be seen that the original mathematical meaning of the word "function" is quite extensive and vague. Almost at the same time, Newton used another term "flow" to express the relationship between variables in the discussion of calculus until. Swiss mathematician johann bernoulli clearly defined the concept of function on the basis of Leibniz's concept of function. Bernoulli called the quantity formed by variable x and constant in any way "the function of x" and expressed it as yx. Since the operations of connecting variables and constants at that time were mainly arithmetic operations, trigonometric operations, exponential operations and logarithmic operations, Euler later simply named the formula formed by connecting variables X and constants C with these operations as analytic functions and divided them into "algebraic functions" and "transcendental functions". /kloc-in the middle of the 8th century, D'Alembert and Euler introduced the concept of "arbitrary function" successively because of the study of string vibration. When D'Alembert explained the concept of "arbitrary function", he said that it meant "arbitrary analytical formula", while Euler thought it was "arbitrarily drawn curve". Now, it seems that these are all expressions of functions and an extension of the concept of functions. The lack of scientific definition of the concept of function has caused a sharp contradiction between theory and practice. For example, partial differential equations are widely used in engineering technology, but the lack of scientific definition of functions greatly limits the establishment of partial differential equation theory. From 1833 to 1834, Gauss began to turn his attention to physics. In the process of inventing the telegraph with W Wilbur, he did a lot of magnetic experiments. The important theory that force is inversely proportional to the square of distance makes function appear as an independent branch of mathematics, which actually needs to prompt people to further study the definition of function. Later, people gave the definition that if one quantity depends on another quantity, when the latter quantity changes, the former quantity also changes, then the first quantity is called the function of the second quantity. Although this definition has not revealed the essence of function, it is gratifying progress to inject changes and movements into the definition of function? In the history of function concept development, the work of French mathematician Fourier has the greatest influence. Fourier profoundly reveals the essence of functions, and thinks that functions need not be limited to analytic expressions. 1822, he said in his masterpiece Analytical Theory of Heat, "Usually, a function represents a set of connected values or vertical coordinates, each of which is arbitrary ... and we don't assume that these vertical coordinates obey a * * *. They are close in every way. In this book, he expressed a function given by discontinuous "lines" in the form of sum of trigonometric series. More precisely, any periodic function of 2π can be used? Show that among them? Fourier's research fundamentally shook the old traditional thought about the concept of function, which caused great shock in the mathematics field at that time. There is no insurmountable gap between analytical formula and curve, and analytical formula and curve are connected in series. The view that functions are analytical forms eventually becomes a huge obstacle to revealing the relationship between functions. Through an argument, the function definitions of Lobachevsky and Dirichlet came into being. 1834, Russian mathematician Lobachevsky put forward the definition of function: "The function of x is such a number that it has a definite value for each x and changes with it. The function value can be given by an analytical formula or a condition, which provides a method to find all the corresponding values. This dependence of function can exist, but it is not known yet. "This definition establishes the correspondence between variables and functions. 1837, the German mathematician Dirichlet thought that how to establish the relationship between X and Y was not important, so his definition was: "If for every value of X, Y always has a completely certain value corresponding to it, then Y is a function of X"? According to this definition, even if it is expressed as follows, it is still said to be a function (Dirichlet function): f(x)= 1(x is a rational number), 0(x is an irrational number). In this function, if the value of x increases gradually from 0, then f(x) will suddenly change from 0 to 1. In any cell, f(x) will suddenly change from 0 to 1 infinitely. Therefore, it is difficult to express it with one or several formulas, and even whether an expression can be found is a problem. But whether it can be expressed by expression is stated in Dirichlet. Dirichlet's definition of function avoids all the descriptions of dependence in the previous definition of function, and is unconditionally accepted by all mathematicians in a completely clear way. At this point, it can be said that the concept of function and the essential definition of function have been formed, which is the classical definition of function that people often say. The further development of production practice and scientific experiments has caused new sharp contradictions in the concept of function. In the 1920s, people began to study microphysical phenomena. 1930 quantum mechanics came out, and a new function-δ function was needed in quantum mechanics. ρ (x) = 0，x≠0，∞，x = 0。 And then what? The appearance of δ-function has caused a heated debate. According to the original definition of function, only the correspondence between numbers is allowed, and "∞" is not regarded as a number. In addition, it is incredible that there is only one function whose point is not zero for the independent variable, but its integer value is not equal to zero. However, the δ-function is really an abstraction of the actual model. For example, cars and trains cross the bridge, which is natural. P(0)= pressure/contact surface = 1/0 = ∞. At the rest point x≠0, there is no pressure because there is no pressure, that is? P(x)=0。 In addition, we know that the integral of the pressure function is equal to the pressure, that is? Under such historical conditions, the concept of function develops actively, resulting in a new modern function definition: if any element X of set M always has an element Y determined by set N corresponding to it, a function is defined on set M, which is denoted as y=f(x). Element x is called independent variable and element y is called dependent variable. Although there is only one word difference between the modern definition of function and the classical definition in form, it is a major development in concept and a major turning point in the development of mathematics. Modern functional analysis can be used as a sign of this turning point. It studies the functional relations on general sets. The definition of function has been tempered and changed for more than 200 years, forming a modern definition of function, which should be said to be quite perfect. However, the development of mathematics is endless, and the formation of modern definition of function does not mean the historical end of the development of function concept. In the past twenty years, mathematicians have simplified functions into a broader concept-"relationship". Let's set x and y, and define the product set X×Y of x and y as? X×Y={(x，y)|x∈X，y∈Y}。 ? A subset of the product set X×Y, r is called the relationship between x and y, and if (x, y)∈R, it is said that x has a relationship with y, which is denoted as xRy. If (x, y)R, that means x and y have nothing to do? Let F be the relationship between X and Y, that is, FX× Y. If (x, y), (x, z)∈f must have y=z, then F is called a function from X to Y. In this definition, the term "correspondence" has been avoided formally, and all the languages of set theory have been used. From the whole process of the development of the above-mentioned function concepts, we realize how important it is to study, explore and broaden the connotation of mathematical concepts in connection with reality and a large number of mathematical data.

Research on the Development Trend and Countermeasures of Population Aging in China: Population aging is the trend of population development in the world today. This change of population age structure is widely and profoundly affecting all aspects of human social life. Population aging has increasingly become a major population issue of concern to all countries in the world. As the largest developing country in the world, China is facing an aging population under the background of "getting old before getting rich", which will surely bring profound changes to economic and social development. I. Present situation of the elderly population in China China is the country with the largest and fastest growing elderly population in the world. Since 1980, the elderly population over 60 years old in China has been growing at an average annual rate of 3%. The results of the "Fifth General Plan" show that the population aged 65 and above has reached 88 1 10000, accounting for 6.96% of the total population. China has entered an aging society. According to the prediction of relevant experts (pictured in the middle), by 2050, the number of elderly people over 65 in China will reach more than 320 million, accounting for about 1/5 of China's total population and 1/4(2) of the world's elderly population. At the same time, the aging trend of the elderly population in China is becoming increasingly obvious. It is estimated that by 2040, the population over 80 will reach 56 million. It is worth noting that although the average fertility level in rural areas is higher than that in cities, due to a large number of young and middle-aged people flowing from rural areas to cities, the current aging rate in rural areas is 7.35% and that in cities and towns is 6.30%. The aging trend of the elderly population in China is also very noticeable. At present, the elderly population in China is growing at an annual rate of 5.4%. According to experts' estimation, it will reach 27.8 million by 2020. The rapid aging of the population in China is mainly the result of the excessive number of people born in the 1950s and 1960s, and the life expectancy of people has greatly increased. When developed countries enter an aging society, it is generally $654.38+more than 0.000 million, while our current per capita GNP is less than $654.38 +0.000. Compared with western developed countries, China's aging is fast and sudden, ahead of industrialization and modernization, which will bring severe challenges to China's economic and social development. At present, China's life expectancy and mortality rate are close to the level of developed countries, and the fertility rate has reached below the replacement level. With the baby boomers entering the old age in the middle of the 20th century, it can be predicted that the first half of the 20th century will be the fastest period of population aging in China. 2. The main difficulties faced by the elderly population (1) The urban elderly have a growing desire to be adopted by social welfare institutions, but the number of welfare institutions is still difficult to meet the needs of the elderly. A survey released by the Ministry of Civil Affairs ⑤: At present, there are 28 16 social welfare institutions in China, with 220,000 beds and more than 70,000 adoptions; There are 3,700 collective welfare institutions with more than 870,000 beds and more than 660,000 adoptions. In total, there are more than 40,000 public welfare organizations, 1.09 million beds and more than 830,000 adoptions. Compared with hundreds of millions of welfare clients in China (the elderly population exceeds1.300 million, orphans and disabled children number millions), the number of beds for centralized adoption is only a little over 0.7%, while the number of people for centralized adoption is less than 0.6%, which is far from the ratio of 5-7% in developed countries. (2) The health status of the elderly is not optimistic. Compared with non-elderly population, the elderly population will have some age-related characteristics in physiology. With the continuous growth of age, physiological function, resistance and health status decline. According to the statistics of the national health department from 65438 to 0994, "the prevalence rate of the elderly over 65 years old in cities in China is 60.2%, and the prevalence rate of the total urban population is 23.7%, which is 1.54 times higher than that of the total population. The prevalence rate of the elderly in rural areas is 22.6%, and that of the rural population is 7.4%, which is 2. 1 times higher than the total population. ⑥ "All these show that the more elderly people in a society, the greater the possibility of getting sick, and the greater the demand for medical services. Especially in rural areas, the proportion of elderly people enjoying free medical care is quite low, and the huge rural elderly population in China will exert great pressure on the relatively scarce rural health resources. (3) The living environment of the elderly needs to be improved. For most old people, it's time for them to spend their old age in peace and enjoy family happiness. However, for those who can't spend their old age safely, the old age is the last period of life. Some old people face four kinds of violations: first, physical violations. Some elderly people have a very difficult life because of mobility difficulties and lack of care from their children. Sometimes there will be cases of violent beating and scolding, abandonment and even killing of the elderly. The second is spiritual infringement. Some young people in society are impolite, disrespectful and even bully the elderly, causing mental harm to the elderly. The third is social infringement. With the increase of age, the discrimination ability of the elderly is also declining. Some people in society cheat them, sell inferior products to the elderly, and cheat the elderly in law and civil affairs. The fourth is material infringement. Cases in which children and relatives seize houses and compete for property often occur in various places. Due to the development of social economy and the implementation of subsistence allowance, the problem of material infringement on the elderly has been alleviated, but it can not be ignored. Pension funds are seriously inadequate. Due to the increase of retirees and the improvement of treatment level, the expenditure of China's endowment insurance fund is increasing every year, which leads to the increasing gap between income and expenditure in some areas. In 2000, only 37.907 million people in China received retirement pension, and in 2006, only 33.865438+0 million people received old-age insurance. At present, the gap between the current income and expenditure of pensions is mainly solved by the transfer payment of the central government. However, experts pointed out that with the increasing elderly population in China, the task of endowment insurance will be more arduous. To continue to ensure that pensions are paid in full and on time, it is not enough to rely solely on fund collection and financial subsidies. It is necessary to further broaden the financing channels of pension funds. (5) China's aging population has brought certain social and economic impacts: First, the growth of the elderly population will change the dependency ratio of the population, and the increase of the dependent population will definitely increase the burden on the working population. Second, the aging of the age structure of the labor force is accompanied by the aging of the population, which will definitely affect the improvement of labor productivity. Third, the aging population has greatly increased the social security cost of the elderly and brought a heavy burden to the government. Fourth, the aging population requires adjusting the existing industrial structure to meet the special needs of the elderly population for material and spiritual culture. Fifth, the aging of the population has changed the family size and structure, weakening the family's pension function. Three. Countermeasures and Suggestions Family pension, collective pension and social assistance are the three main forms of rural pension in China. At present, family pension is still the main form of rural old-age security. In fact, family pension is a kind of "feedback" pension based on personal lifelong labor accumulation and intergenerational exchange within the family; Collective support is mainly a kind of "five guarantees" with relief nature for the lonely and widowed elderly, including decentralized support and centralized support. Social pension includes rural social pension insurance, family planning pension insurance and farmers' pension insurance system implemented in a few places. At present, a series of major social changes closely related to farmers' pension in China have made the traditional pension model face severe challenges, and the increasing elderly population has put forward higher requirements for China's existing social security system. To solve the problem of population aging in China, the key is to seize the favorable opportunity of light population support burden and abundant labor resources in the next 20 years, promote all-round economic and social development, establish and improve the social security system covering the whole society, and create a good population environment for building a well-off society in an all-round way. (a) attach great importance to and pay close attention to the problem of population aging, and will solve the problem of aging into the overall strategy of national economic and social development. 1g-7 summit held in Denver in June, 1997 put forward the idea of active aging. 1in may, 1999, the European union adopted an aging policy communique, namely, "building an eu for all ages? Promote prosperity and intergenerational harmony. " The basic content of active aging strategy is to create conditions for the elderly to participate in society in various ways, so that the elderly can better adapt to the development and changes of an aging society. Advocating the strategy of active aging requires governments at all levels and coordinating agencies for aging work to implement the Law on the Protection of the Rights and Interests of the Elderly and local laws and regulations on aging, formulate development plans and related policies for the cause of aging in different periods, and take practical measures to protect the rights and interests of the elderly in accordance with the development policy of government-led, social participation and universal care and the goal of "providing old people with a sense of security, medical care, education, learning and happiness". (two) widely mobilize social forces to set up socialized old-age care. Socialized pension mainly includes: first, diversification of investment subjects. It is necessary to change the single investment channel that relied mainly on state and collective investment in the past, and form a situation in which the state, collective, enterprises and individuals invest through multiple channels and pension institutions with various ownership systems develop together. The second is to disclose customers. In the past, welfare institutions only faced the elderly with "three noes" and "five guarantees", and in the future they will face the elderly in the whole society. The third is the diversification of service forms. In the past, it was basically to support the "three noes" elderly. There should be various forms of services in the future, especially through various community services, such as door-to-door, nursing and day care. Fourth, the combination of service team professionals and volunteers. (three) to promote medical and health institutions to actively provide health services for the elderly. It is necessary to vigorously develop community health services, and list elderly health services as an important part of community health services and incorporate them into community development planning; Vigorously develop the community health care service system. Through system reform, institutional reform, resource reorganization and other forms, we will guide grassroots medical and health institutions to transform into community service centers, transform wards into various forms of places, and provide life, health care and medical services for the elderly; Broaden financing channels, increase investment in community health, and constantly improve the economic security level of medical and health care for the elderly; Carry out various forms of health care services for the elderly, expand the medical care, nursing, health care, rehabilitation and other services for the elderly, and increase assistance to the elderly who are seriously ill and have difficulties living due to serious illness. (4) Increase investment. In order to alleviate the increasingly severe pressure of pension payment in the future, we must implement the method of "walking on two legs". On the one hand, it is necessary to further expand the coverage of endowment insurance, strengthen fund collection, increase fund income, and play the role of the main channel of endowment insurance funds; On the other hand, we should continue to adjust the structure of fiscal expenditure and gradually increase the proportion of social security expenditure in the total fiscal expenditure. At the same time, we will further study how to enrich the social security fund by issuing lottery tickets. (5) Strengthen the construction of welfare service system for the elderly. It is necessary to strengthen the construction of welfare service facilities for the elderly and create a good home care environment for the elderly. It is necessary to formulate and implement various preferential policies, such as giving longevity subsidies to the elderly over 100 every month; The elderly hold the "senior citizen card" in the city where their household registration is located, and can enjoy various preferential services except public transportation in other parts of the province; Urban elderly people do not bear the burden of social fund-raising and other social services, and rural elderly people do not bear voluntary labor or pay various collection fees; Courts at all levels should give priority to cases involving the elderly. If it is really difficult for the elderly to pay the legal fees when they file a lawsuit because their legitimate rights and interests are infringed, they may postpone or reduce their fees. If they need the help of a lawyer and are unable to pay the fees, they can get legal aid. At the same time, carry out moral education of respecting, supporting and helping the elderly among the whole people, and establish a good social fashion. (Author: Huan Wang, Master of Laws, Zhejiang University Law School, teaching assistant. Zhu Yaogeng: Master of Arts, Chief Clerk, Research Office, General Office of the National Population and Family Planning Commission. )

Experimental report on the harm of correction fluid to human body | [< <; ][& gt； & gt Pen, ink, paper and inkstone, I believe students of any era are familiar with it, even modern students. Students living in modern times no longer have to go through complicated processes such as inkstone washing and ink research. When the right-hand ballpoint pen is pressed, it immediately dances, while the dragon and the phoenix dance, and the font spreads like water. If you are too careless, when you open the bottle cap with your left hand, a modern "debugging agent" called correction fluid will cover it. When the liquid dries, it will be written correctly. I don't know that this popular chemical product has great influence on human body. Today, our group is going to study the problem of "the harm of correction fluid to human body". Correction fluid (also known as correction fluid, Likebo, correction fluid) is a white opaque pigment, which is coated on paper to cover typos, and can be re-written on it after drying. Traditionally, correction fluid is packed in a small bottle with a small broom or triangle tied to the bottle cap. Foam plastics are immersed in correction fluid. Because the correction fluid is highly volatile (that is, it dries quickly), unused correction fluid can easily become too thick and difficult to use if left in the bottle. Therefore, correction fluid manufacturers will generally give diluent for sale. Before the invention of word processor, correction fluid made typing or writing more convenient.

In recent years, pen-type correction fluid began to appear. This pen is equipped with a spring. By pressing the pen tip on the paper, the correction fluid can be drawn out, which is more uniform than scanning, and will not dry as easily as bottled.

In order to know the situation quickly, the students are first divided into two groups. A group of people tried various brands of correction fluids in bookstores and stationery stores and found that they all had a special pungent smell. If they smell it for a long time, some students in poor health will feel dizzy and have headaches. The situation of individual products is particularly prominent. If you choose it carelessly, it will be a potential hazard to your health.