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1、New Words&Expressions:alphabet 字母表 prime 素数，质数displacement 位移 proportional 成比例的domain 定义域 the real-valued function实值函数edge 棱，边 spring constant 弹性系数 graph 图，图形 limit 极限 stretch 拉伸 volume 体积，容积，卷,2.6 函数的概念与函数思想Function concept and function idea,New Words&Expressions(二)P59:admit 准许 mapping 映射forerunner
2、 先行者 presume 假定incorporate 并入，结合 trace 追踪,Various fields of human have to do with relationships that exist between one collection of objects and another.,6-A Informal description of functions,各行各业的人们都必须处理一类事物与另一类事物之间存在的关系。,Graphs,charts,curves,tables,formulas,and Gallup polls are familiar to everyon
3、e who reads the newspapers.,几乎每个人都熟悉图形，图表，曲线，公式和盖洛普民意测验。,These are merely devices for describing special relations in a quantitative fashion.Mathematicians refer to certain types of these relations as functions.,这些只是以定量的方式描述特定关系的方法。数学家将这些关系中的某些类型视作函数。,In this section,we give an informal description
4、of the function concept.A formal definition is given in Section 3.,在本节中，我们给出一个非正式的描述函数的概念。在第3节给出一个正式的定义。,EXAMPLE 1.The force F necessary to stretch a steel spring a distance x beyond its natural length is proportional to x.That is,F=cx,where c is a number independent of x called the spring constant.
5、,把一个钢制的弹簧拉伸到超过其自然长度的距离为x时所需要的力F与x成正比。即，F=cx,这里c是不依赖与x的数，叫做弹性系数。,This formula,discovered by Robert Hooke in the mid-17th century,is called Hookes law,and it is said to express the force as a function of the displacement.,这个公式是在17世纪中叶被胡克发现的，叫做胡克定律，它用来表示力关于位移的函数。,EXAMPLE 2.The volume of a cube is a fun
6、ction of its edge-length.If the edges have length x,the volume V is given by the formula V=x3.,立方体的体积是它棱长的函数。如果棱长为x，那么体积的公式为：V=x3。,EXAMPLE 3.A prime is any integer n1 that cannot be expressed in the form n=ab,where a and b are positive integers,both less than n.The first few primes are 2,3,5,7,11,13
7、,17,19.,素数是大于1且不能表示成n=ab形式的整数，这里a和b都是小于n的正整数。开始的几个素数是2,3,5,7,11,13,17,19.,For a given real number x0,it is possible to count the number of primes less than or equal to x.This number is said to be a function of x even though no simple algebraic formula is known for computing it(without counting)when
8、x is known.,对于一个给定的实数x0，数出小于或者等于x的素数的个数是有可能的。这个数称为x的函数，尽管还没有一个简单代数式可以由已知的x计算(不通过计数求)出它的值。,The word“function”was introduced into mathematics by Leibniz,who used the term primarily to refer to certain kinds of mathematical formulas.,“函数”这个词是由莱布尼茨引入到数学中的，他主要使用这个术语来指代某种数学公式。,It was later realized that L
9、eibnizs idea of function was much too limited in its scope,and the meaning of the word has since undergone many stages of generalization.,后来人们才认识到，莱布尼茨的函数思想适用的范围太过局限了，这个术语的含义从那时起已经过了多次推广。,Today,the meaning of function is essentially this:Given two sets,say X and Y,a function is a correspondence whic
10、h associates with each element of X one and one only element of Y.,如今，从本质上讲，函数的定义如下：给定两个集合X 和Y，函数是X中元素与Y中元素的一一对应。,The set X is called the domain of the function.Those elements of Y associated with the elements in X form a set called the range of the function.(This may be all of Y,but it need not be)
11、,集合X叫做函数的定义域，与X中的元素相对应的Y中的元素的集合叫做函数的值域。(值域可能是整个集合Y,也可能不是。),Letters of the English and Greek alphabets are often used to denote functions.The particular letters f,g,h,F,G,H,and are frequently used for this purpose.,英语字母和希腊字母表通常用于表示函数。为此，一些特定的字母如：f,g,h,频繁使用。,If f is a given function and if x is an obj
12、ect of its domain,the notation f(x)is used to designate that object in the range which is associated to x by the function f;and it is called the value of f at x or the image of x under f.The symbol f(x)is read as“f of x.”,如果f是一个给定的函数，x是它定义域中的一个点，符号f(x)表示值域中按照f对应于x的点，它叫做f在x点的值或者x在f下的像。符号f(x)读作“f of x
13、.”,Seldom has a single concept played so important a role in mathematics as has the concept of function.It is desirable to know how the concept has developed.,6-C The concept of function,在数学中，很少有个概念象函数的概念那样，起那么重要的作用的。因此，需要知道这个概念是如何发展起来的。,This concept,like many others,originates in physics.,这个概念像许多其他
14、概念一样，起源于物理学。,The physical quantities were the forerunners of mathematical variables,and relation among them was called a function relation in the later 16th century.,物理量是数学变量的先驱，他们之间的关系在16世纪后期称为函数关系。,For example,the formula s=16t2 for the number of feet s a body falls in any number of seconds t is a
15、 function relation between s and t,it describes the way s varies with t.例如,代表一物体在若干秒t中下落若干英尺s的公式s=16t2 就是s和t之间的函数关系,它描述了s随t 变化的公式。,The study of such relations led people in the 18th century to think of a function relation as nothing but a formula.对这种关系的研究导致了18世纪的人们认为函数关系只不过是一个公式罢了。,Not specified by
16、this definition is the manner of setting up the correspondence.至于如何建立这种对应关系，这个定义并未详细规定。,It may be done by a formula as the 18th century mathematics presumed but it can equally well be done by a tabulation such as a statistical chart,or by some other form of description.可以如18世纪的数学所假定的那样，用公式来建立，但同样也可以
17、用统计表那样的表格或用某种其他的描述方式来建立。,A typical example is the room temperature,which obviously is a function of time.But this function admits of no formula representation,although it can be recorded in a tabular form or traced out graphically by an automatic device.典型的例子是室温，这显然是一个时间的函数。但是这个函数不能用公式来表示，不过可以用表格的形式
18、来记录或者用一种自动装置以图标形式来追踪。,The modern definition of a function y of x is simply a mapping from a space X to another space Y.A mapping is defined when every point x of X has a definite image y,a point of Y.现代给x的一个函数y所下的定义只是从一个空间X到另一个空间Y的映射。当X空间的每一个点x有一个确定的像点y，即Y空间的一点，那么映射就确定了。,The mapping concept is close
19、 to intuition,and therefore desirable to serve as a basis of the function concept.Moreover,as the space concept is incorporated in this modern definition,its generality contributes much to the generality of the function concept.这个映射概念接近于直观，因此，值得作为函数概念的一个基础。此外，由于这个现代的定义中体现了空间的概念，所以，它的普及性对函数概念的普及性有很大的贡献。,作业：P57 2:(2),(8)3:(1),谢 谢！,
