數(shù)學專業(yè)英語課后答案

1、2.1 數(shù)學、方程與比例 （1）數(shù)學來源于人類的社會實踐，包括工農(nóng)業(yè)的勞動，商業(yè)、軍事和科學技術(shù)研究等活動。 Mathematics comes from mans social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. （2）如果沒有運用數(shù)學，任何一個科學技術(shù)分支都不可能正常地發(fā)展。 No modern scientific and t

2、echnological branches could be regularly developed without the application of mathematics. （3）符號在數(shù)學中起著非常重要的作用，它常用于表示概念和命題。 Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. （4）17 世紀之前，人們局限于初等數(shù)學，即幾何、三角和代數(shù)，那時只考慮常數(shù)。 Before 17th cen

3、tury, man confined himself to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered. （5）方程與算數(shù)的等式不同在于它含有可以參加運算的未知量。 Equation is different from arithmetic identity in that it contains unknown quantity which can join operations. （6）方程又稱為條件等式

4、，因為其中的未知量通常只允許取某些特定的值。 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，可以用它來解決許多實際應用問題。Equations are of very great use. We can use equations in many mathematical problems. （8）解方程時要進行一系列移項和同解變形，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 幾何與三角 （1）許多專家都認為數(shù)學是學習其他科學技術(shù)的必備基礎和先決條件。 Many experts recognize that mathematics is the necessary foundation and prerequisite of stu

6、dying other science technology. （2）西方國家的專家認為幾何起源于巴比倫和埃及人的土地測量技術(shù)，其實中國古代的數(shù)學 家對幾何做了許多出色的研究。 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、幾何的學習使學生在思考問題時更周密和審慎，他們將不會盲目接受任何結(jié)論。 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）數(shù)學培養(yǎng)學生的分析問題的能力，使他們能應用毅力、創(chuàng)造性和邏輯推理來解決問題。 Studying mathematics can develop the students ability to analyze prob

8、lems and utilizing perseverance, originality, and logical reasoning in solving the problem. （5）幾何主要不是研究數(shù)，而是形，例如三角形，平行四邊形和圓，雖然它也與數(shù)有關。 Geometry mainly studies hot numbers but figures such as triangles, parallelograms and circles, though it is related with numbers. （6）一個立體（圖形）有長、寬和高；面（曲面或平面）有長和寬，但沒有厚度；

9、線（直線 或曲線）有長度，但既沒有寬度，也沒有厚度；點只有位置，卻沒有大小。 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）射線從某個點出發(fā)無

10、限延伸；兩條從同一點出發(fā)的射線構(gòu)成了角。這兩條射線稱為這個 角的兩邊，當這兩邊位于同一直線上且方向相反時，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）平面上的閉曲線當其

11、中每一點到一個固定點的距離均相等時叫做圓。這個固定點稱為圓 心，經(jīng)過圓心且其兩個端點在圓周上的線段稱為這個圓的直徑，直徑的一半叫做半徑，這條 曲線的長度叫做周長。 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 e

12、ndpoints on the circle. Half of the diameter is called radius. The length of the circle is called circumference.2.3 集合論的基本概念 （1）由小于 10 且能被 3 整除的正整數(shù)組成的集是整數(shù)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，我們通過在括號中

13、列舉元素的辦法來表示集。 When convenient, we shall designate sets by displaying the elements in braces. （3）用符號 表示集的包含關系，也就是說，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）命題 A B 并不排除 B A 的可能性。 The statement AB does not rule out the possibility that BA. （5

14、）基礎集可根據(jù)使用場合不同而改變。 The underlying set may vary from one application to another according to using occasions. （6）為了避免邏輯上的困難，我們必須把元素 x 與僅含有元素 x 的集x區(qū)別開來。 To avoid logical difficulties, we must distinguish between the element x and the set x whose only element is x. （7）圖解法有助于將集合之間的關系形象化。 Diagrams often h

15、elp using visualize relationship between sets. （8）定理的證明僅僅依賴于概念和已知的結(jié)論，而不依賴于圖形。 The proofs of theorems rely only on the definitions of the concepts and known result, not on the diagrams. 2.4 整數(shù)、有理數(shù)與實數(shù) 整數(shù) （1）嚴格說，這樣描述整數(shù)是不完整的，因為我們并沒有說明“依此類推”或“反復加 1” 的含義是什么。 Strictly speaking, this description of the posi

16、tive 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）兩個整數(shù)的和、差或積是一個整數(shù)，但是兩個整數(shù)的商未必是一個整數(shù)。 The sum, difference, or product of two integers is an integer, but the quotient of two integers need not be an

17、integer. （3） 這種用幾何來表示實數(shù)的辦法對于幫助我們更好地發(fā)現(xiàn)與理解實數(shù)的性質(zhì)是非常有價值 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）幾何經(jīng)常為一些特定的定理提供證明思路（建議） ，而且，有時幾何的論證比純分析的 （完全依賴于實數(shù)公理的）證明更清晰。 The geometry often

18、 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）一個由實數(shù)組成的集若滿足如下條件則稱為開區(qū)間（open interval） 。 If a set consisting of real numbers satisfies the follow

19、ing conditions we call it an open interval. （6）實數(shù) a 是-a 的相反數(shù)，它們的絕對值相等，且當 a 0 時，其符號不同。 The real number a is the negative number of a and their absolute values are equal. When a 0, their notations are different. （7）每個實數(shù)剛好對應著實軸上的一點，反之，對實軸上的每一點，有且只有一個實數(shù)與之 對應。 Each real number corresponds to exactly one

20、point on this line and, conversely, each point on the line corresponds to one and only one real number. （8）在幾何上，實數(shù)之間的次序關系可以在數(shù)軸上清楚地表示出來。 In geometry, the ordering relation among the real numbers can be expressed clearly in real axis. 2.5 笛卡兒幾何學的基本概念 （1）計算圖形的面積是積分的一種重要應用。 The calculation of figure are

21、a is the important application of the integral. （2）在 x-軸上 O 點右邊選定一個適當?shù)狞c，并把它到 O 點的距離稱為單位長度。 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）對 xy-平面上的每一個點都指定了一個數(shù)對，稱為它的坐標。 Each point in the xy-plane is assigned a pair of numbers, call

22、ed its coordinates. （4）選取兩條互相垂直的直線，其中一條是水平的，另一條是豎立的，把它們的交點記作 O， 稱為原點。 Two perpendicular reference lines are chosen, one horizontal, the other vertical. Their point of intersection, denoted by O, is called the origin. （5）當我們用一對數(shù)（a, b）來表示平面的點時，商定要把橫坐標寫在第一個位置上。 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）微積分與解析幾何在它們的發(fā)展史上已經(jīng)互相融合在一起了。 Throughout their historical development, calculus and analytic geometry have been intimately intertwined. （7）如果想拓展微積分的范圍與應用，需要進一步研究解析幾何，而這種研究需用到向量的 方法。 A deeper study of analyt

24、ic geometry is needed to extend the scope and applications of calculus, and this study will be carried out using vector methods. （8）今后我們要對三維解析幾何做詳細研究，但目前只限于考慮平面解析幾何。 We shall discuss three-dimensional Cartesian geometry in more detail later on; for the present we confine our attention to plane analy

25、tic geometry. 2.6 函數(shù)的概念與函數(shù)思想 （1）常用英語字母和希臘字母來表示函數(shù)。 Letters of the English and Greek alphabets are often used to denote functions. （2）若 f 是一個給定的函數(shù)，x 是定義域里的一個元素，那么記號 f(x)用來表示由 f 確定的 對應于 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 objec

26、t in the range which is associated to x by the function f. （3）該射線將兩個坐標軸的夾角分成兩個相等的角。 The ray makes equal angles with the coordinates axes. （4）可以用許多方式給出函數(shù)思想的圖解說明。 The function idea may be illustrated schematically in many ways. （5）容易證明，絕對值函數(shù)滿足三角不等式。 It is easy to proof that the absolute-value function

27、 satisfies the triangle inequality. （6）對于實數(shù) x0，函數(shù) g(x)表示不超過 x 的素數(shù)的個數(shù)。 For a given real number x0, the function g(x) is defined by the number of primes less than or equal to x. （7）函數(shù)是一種對應，它未必可以表示成一個簡單的代數(shù)公式。 A function is a correspondence. It is not necessary to be expressed by a simple algebraic form

28、ula. （8）在函數(shù)的定義中，關于定義域和值域中的對象，沒對其性質(zhì)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 序列及其極限 序列及其極限 （1）序列各項對 n 的相關性常利用下標來表示，寫成如下形式： a n , x n 等。 The dependence of every team of sequence on n is denoted by using subscript, and we write

29、a n , x n and so on. （2）以正整數(shù)集為定義域的函數(shù)稱為序列。 A function whose domain is the set of all positive integers is called an infinite sequence. （3）一個復值序列收斂當且僅當它的實部和虛部分別收斂。 A complex-valued sequence converges if and only if both the real part and the imaginary part converge separately. （4） 一個序列 a n 若滿足： 對任意正數(shù)

30、， 存在另一個正數(shù) N （N可能與 有關） 使得 a n - L 對所有 n N 成立，就稱 a n 收斂于 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， 該集的每一個成員都用一個正整數(shù)標上

31、記號。 這樣一來， 就可以談論第一項、 第二項和一般項，即第 n 項。 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）若無另加申明，本章研究的序列都假定具有實的項或復的項。 Unless otherwise specified, all sequences in this chapter are a

32、ssumed to have real or complex terms. （7）作為日常用語，sequence 和 series 是同義詞；但作為數(shù)學術(shù)語，它們表示不同的概念。 In everyday usage of the English language, the words “sequence” and “series” are synonyms, but in mathematics these words have special technical meanings. （8）術(shù)語“收斂序列”指的是具有有限極限的序列，因此，極限為無限的序列不是收斂的， 而是發(fā)散的。 The ph

33、rase “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 函數(shù)的導數(shù)和它的幾何意義 （1）差商表示函數(shù) f 在連接 x 與 x+h 的區(qū)間上的平均變化率。 The different quotient is referred to as the average rate of the change of f in the interval joinin

34、g x to x+h. （2）速度等于位置函數(shù)的導數(shù)。 Velocity is equal to the derivative of positing. （3）由定義導數(shù)的過程所提供的幾何解釋以一種自然的方式導出了關于曲線的切線思想。 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）差商表示直線 PQ 與水平線的夾角的正切。 The differe

35、nce quotient represents the trigonometric tangent of the angle that PQ makes with the horizontal. （5）在直線運動中，速度的一階導數(shù)稱為加速度。 For rectilinear motion, the first derivative of velocity is called acceleration. （6）我們約定 f（0）=f，即函數(shù) f 的零階導數(shù)就等于它本身。 We make the convention that f（0）=f, that is the zeroth derivate

36、 is the function itself. （7）在運動的 9 秒鐘內(nèi)，物體的速度由 v (0) = -144 變成了 v (9) =144，也就是說，速度總共 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）當 從 0 增加到/2 時，tan 所對應的直線趨于豎直位置。As increases from 0

37、 to /2 , tan approach a vertical position. 2.9 微分方程簡介（1）此時，微分方程就有無窮多個解，C的每個值對應一個解。The differential equation has infinitely many solutions, one for each value of C.（2）微分方程的階指的是方程中最高階導數(shù)的階。By the order of an equation is meant the order of the highest derivative which appears.（3）我們可以由已知的粒子運動速度或者加速度計算出粒子

38、的位置。We could try to compute the position of a moving particle from a knowledge of its velocity or acceleration.（4）如果一個微分方程的未知函數(shù)是多元函數(shù)，則稱為偏微分方程。Ordinary and partial, depend on whether the unknown is a function of just one variable or of two or more variables.（5）微分方程的研究直接受到力學、天文學和數(shù)學物理的推動。The study of d

39、ifferential equations has been directly inspired by mechanics, astronomy, and mathematical physics.（6）許多應用問題要求我們從方程的解集中選出一個在某個點具有指定值的解。In many problems it is necessary to select from the collection of all solutions one having a prescribed value at some point.（7）確定滿足邊界條件的解的問題稱為邊值問題。The problem of determining such a solution that satisfies boundary condition is called a boundary-value problem.（8）人們設計許多高速運行的計算機來對各