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1、-2.1 数学、方程与比例 （1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。 Mathematics comes from mans social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. （2）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。 No modern scientific and
2、technological branches could be regularly developed without the application of mathematics. （3）符号在数学中起着非常重要的作用，它常用于表示概念和命题。 Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. （4）17 世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常数。 Before 17th ce
3、ntury, man confined himself to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered. （5）方程与算数的等式不同在于它含有可以参加运算的未知量。 Equation is different from arithmetic identity in that it contains unknown quantity which can join operations. （6）方程又称为条件等
4、式，因为其中的未知量通常只允许取某些特定的值。 Equipment is called an equation of condition in that it is true only for certain values of unknown quantities in it. （7）方程很有用，可以用它来解决许多实际应用问题。Equations are of very great use. We can use equations in many mathematical problems. （8）解方程时要进行一系列移项和同解变形，最后求出它的根，即未知量的值。 To solve the
5、 equation means to move and change the terms about without making the equation untrue, until the root of the equation is obtained, which is the value of unknown term. 2.2 几何与三角 （1）许多专家都认为数学是学习其他科学技术的必备基础和先决条件。 Many experts recognize that mathematics is the necessary foundation and prerequisite of st
6、udying other science technology. （2）西方国家的专家认为几何起源于巴比伦和埃及人的土地测量技术，其实中国古代的数学 家对几何做了许多出色的研究。 The western experts think that geometry had its origin in the measurements by the Babylonians and Egyptians of their lands. Infect, the ancient Chinese mathematicians made much remarkable study for geometry. （3
7、）几何的学习使学生在思考问题时更周密和审慎，他们将不会盲目接受任何结论。 In studying geometry, the student is taught to think clearly and critically and he is led away from the practice of blind acceptance of any conclusions. （4）数学培养学生的分析问题的能力，使他们能应用毅力、创造性和逻辑推理来解决问题。 Studying mathematics can develop the students ability to analyze pro
8、blems and utilizing perseverance, originality, and logical reasoning in solving the problem. （5）几何主要不是研究数，而是形，例如三角形，平行四边形和圆，虽然它也与数有关。 Geometry mainly studies hot numbers but figures such as triangles, parallelograms and circles, though it is related with numbers. （6）一个立体（图形）有长、宽和高；面（曲面或平面）有长和宽，但没有厚度
9、；线（直线 或曲线）有长度，但既没有宽度，也没有厚度；点只有位置，却没有大小。 A solid (figure) has length, width and height. A surface (curved surface or plane surface) has length and width, but no thickness. A line (straight line or curved line) has length, but no width and thickness. A point has position, but no dimension. （7）射线从某个点出发
10、无限延伸；两条从同一点出发的射线构成了角。这两条射线称为这个 角的两边，当这两边位于同一直线上且方向相反时，所得的角是平角。 A ray starts from a point and extends infinitely far. Two rays starting from one point form an angle, which are called two edges of the angle. When two edges lie in the same line and have opposite direction named plane angle. （8）平面上的闭曲线当
11、其中每一点到一个固定点的距离均相等时叫做圆。这个固定点称为圆 心，经过圆心且其两个端点在圆周上的线段称为这个圆的直径，直径的一半叫做半径，这条 曲线的长度叫做周长。 A circle is a closed curve lying in one plane, all points of which are equidistant from a fixed point. The fixed point called the center. A diameter of a circle is a line segment through the center of the circle with
12、endpoints on the circle. Half of the diameter is called radius. The length of the circle is called circumference. 2.3 集合论的基本概念 （1）由小于 10 且能被 3 整除的正整数组成的集是整数集的子集。 The set consisting of those positive integers less than 10 which are divisible by 3 is a subset of the set of all integers. （2）如果方便，我们通过在括
13、号中列举元素的办法来表示集。 When convenient, we shall designate sets by displaying the elements in braces. （3）用符号 表示集的包含关系，也就是说，式子 A B 表示 A 包含于 B。 The relation is referred to as set inclusion; AB means that A is contained in B. （4）命题 A B 并不排除 B A 的可能性。 The statement AB does not rule out the possibility that BA.
14、（5）基础集可根据使用场合不同而改变。 The underlying set may vary from one application to another according to using occasions. （6）为了避免逻辑上的困难，我们必须把元素 x 与仅含有元素 x 的集x区别开来。 To avoid logical difficulties, we must distinguish between the element x and the set x whose only element is x. （7）图解法有助于将集合之间的关系形象化。 Diagrams often
15、 help using visualize relationship between sets. （8）定理的证明仅仅依赖于概念和已知的结论，而不依赖于图形。 The proofs of theorems rely only on the definitions of the concepts and known result, not on the diagrams. 2.4 整数、有理数与实数 整数 （1）严格说，这样描述整数是不完整的，因为我们并没有说明“依此类推”或“反复加 1” 的含义是什么。 Strictly speaking, this description of the po
16、sitive integers is not entirely complete because we have not explained in detail what we mean by the expressions “and so on”, or “repeated addition of 1”. （2）两个整数的和、差或积是一个整数，但是两个整数的商未必是一个整数。 The sum, difference, or product of two integers is an integer, but the quotient of two integers need not be a
17、n integer. （3） 这种用几何来表示实数的办法对于帮助我们更好地发现与理解实数的性质是非常有价值 的。 This device for representing real numbers geometrically is a very worthwhile aid that helps us to discover and understand better certain properties of real numbers. （4）几何经常为一些特定的定理提供证明思路（建议） ，而且，有时几何的论证比纯分析的 （完全依赖于实数公理的）证明更清晰。 The geometry oft
18、en suggests the method of proof of a particular theorem, and sometimes a geometric argument is more illuminating than a purely analytic proof (one depending entirely on the axioms for the real numbers). （5）一个由实数组成的集若满足如下条件则称为开区间（open interval） 。 If a set consisting of real numbers satisfies the foll
19、owing conditions we call it an open interval. （6）实数 a 是-a 的相反数，它们的绝对值相等，且当 a 0 时，其符号不同。 The real number a is the negative number of a and their absolute values are equal. When a 0, their notations are different. （7）每个实数刚好对应着实轴上的一点，反之，对实轴上的每一点，有且只有一个实数与之 对应。 Each real number corresponds to exactly on
20、e point on this line and, conversely, each point on the line corresponds to one and only one real number. （8）在几何上，实数之间的次序关系可以在数轴上清楚地表示出来。 In geometry, the ordering relation among the real numbers can be expressed clearly in real axis. 2.5 笛卡儿几何学的基本概念 （1）计算图形的面积是积分的一种重要应用。 The calculation of figure a
21、rea is the important application of the integral. （2）在 x-轴上 O 点右边选定一个适当的点，并把它到 O 点的距离称为单位长度。 On the x-axis a convenient point is chosen to the right of O and its distance from O is called the unit distance. （3）对 xy-平面上的每一个点都指定了一个数对，称为它的坐标。 Each point in the xy-plane is assigned a pair of numbers, ca
22、lled its coordinates. （4）选取两条互相垂直的直线，其中一条是水平的，另一条是竖立的，把它们的交点记作 O， 称为原点。 Two perpendicular reference lines are chosen, one horizontal, the other vertical. Their point of intersection, denoted by O, is called the origin. （5）当我们用一对数（a, b）来表示平面的点时，商定要把横坐标写在第一个位置上。 When we write a pair of numbers such as
23、 (a, b) to represent a point, we agree that the abscissa or x-coordinate, a, is written first. （6）微积分与解析几何在它们的发展史上已经互相融合在一起了。 Throughout their historical development, calculus and analytic geometry have been intimately intertwined. （7）如果想拓展微积分的范围与应用，需要进一步研究解析几何，而这种研究需用到向量的 方法。 A deeper study of anal
24、ytic geometry is needed to extend the scope and applications of calculus, and this study will be carried out using vector methods. （8）今后我们要对三维解析几何做详细研究，但目前只限于考虑平面解析几何。 We shall discuss three-dimensional Cartesian geometry in more detail later on; for the present we confine our attention to plane ana
25、lytic geometry. 2.6 函数的概念与函数思想 （1）常用英语字母和希腊字母来表示函数。 Letters of the English and Greek alphabets are often used to denote functions. （2）若 f 是一个给定的函数，x 是定义域里的一个元素，那么记号 f(x)用来表示由 f 确定的 对应于 x 的值。 If f is a given function and if x is an object of its domain, the notation f(x) is used to designate that obj
26、ect in the range which is associated to x by the function f. （3）该射线将两个坐标轴的夹角分成两个相等的角。 The ray makes equal angles with the coordinates axes. （4）可以用许多方式给出函数思想的图解说明。 The function idea may be illustrated schematically in many ways. （5）容易证明，绝对值函数满足三角不等式。 It is easy to proof that the absolute-value functi
27、on satisfies the triangle inequality. （6）对于实数 x0，函数 g(x)表示不超过 x 的素数的个数。 For a given real number x0, the function g(x) is defined by the number of primes less than or equal to x. （7）函数是一种对应，它未必可以表示成一个简单的代数公式。 A function is a correspondence. It is not necessary to be expressed by a simple algebraic fo
28、rmula. （8）在函数的定义中，关于定义域和值域中的对象，没对其性质做出任何限制。 The function idea places no restriction on the nature of the objects in the domain X and in the range Y. 2.7 序列及其极限 序列及其极限 （1）序列各项对 n 的相关性常利用下标来表示，写成如下形式： a n , x n 等。 The dependence of every team of sequence on n is denoted by using subscript, and we writ
29、e a n , x n and so on. （2）以正整数集为定义域的函数称为序列。 A function whose domain is the set of all positive integers is called an infinite sequence. （3）一个复值序列收敛当且仅当它的实部和虚部分别收敛。 A complex-valued sequence converges if and only if both the real part and the imaginary part converge separately. （4） 一个序列 a n 若满足： 对任意正
30、数 ， 存在另一个正数 N （N可能与 有关） 使得 a n - L 对所有 n N 成立，就称 a n 收敛于 L。 A sequence a n is said to have a limit L if, for every positive number , there is another positive number N (which may depend on ) such that In this case, we say the sequence a n converges to L. an ? L for all n N. （5） 重要的是， 该集的每一个成员都用一个正整数
31、标上记号。 这样一来， 就可以谈论第一项、 第二项和一般项，即第 n 项。 The important thing is that each member of the set has been labeled with an integer so that we may speak of the first term, the second term and in general, the nth term. （6）若无另加申明，本章研究的序列都假定具有实的项或复的项。 Unless otherwise specified, all sequences in this chapter are
32、 assumed to have real or complex terms. （7）作为日常用语，sequence 和 series 是同义词；但作为数学术语，它们表示不同的概念。 In everyday usage of the English language, the words “sequence” and “series” are synonyms, but in mathematics these words have special technical meanings. （8）术语“收敛序列”指的是具有有限极限的序列，因此，极限为无限的序列不是收敛的， 而是发散的。 The
33、phrase “convergent sequence” is used only for a sequence whose limit is finite. A sequence with an infinite limit is said to diverge not convergence. 2.8 函数的导数和它的几何意义 （1）差商表示函数 f 在连接 x 与 x+h 的区间上的平均变化率。 The different quotient is referred to as the average rate of the change of f in the interval join
34、ing x to x+h. （2）速度等于位置函数的导数。 Velocity is equal to the derivative of positing. （3）由定义导数的过程所提供的几何解释以一种自然的方式导出了关于曲线的切线思想。 The procedure used to define the derivative has a geometric interpretation which leads in a natural way to the idea of a tangent line to a curve. （4）差商表示直线 PQ 与水平线的夹角的正切。 The diffe
35、rence quotient represents the trigonometric tangent of the angle that PQ makes with the horizontal. （5）在直线运动中，速度的一阶导数称为加速度。 For rectilinear motion, the first derivative of velocity is called acceleration. （6）我们约定 f（0）=f，即函数 f 的零阶导数就等于它本身。 We make the convention that f（0）=f, that is the zeroth deriva
36、te is the function itself. （7）在运动的 9 秒钟内，物体的速度由 v (0) = -144 变成了 v (9) =144，也就是说，速度总共 增加了每秒 288 英尺。 During the 9 seconds of motion the velocity changes from v (0) = -144 to v (9) =144, that is, the total increase in velocity is 288 feet per second. （8）当 从 0 增加到/2 时，tan 所对应的直线趋于竖直位置。As increases from 0 to /2 , tan approach a vertical position. -第 6 页-