数学专业英语[1]－The Real Number System

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Exercise ( f) ~: S& r3 C ! j& c8 i; C, k8 j# U

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ⅰ. Turn the following mathematical expressions in English:

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ⅰ)x∈A∪B ⅱ)A∩B=φ

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ⅲ)A={Φ} ⅳ)A={X: a<x<b}

ⅱ.Let A ={2,5,8,11,14} B={2,8,14} C={2,8}

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D={5,11} E={2,8,11}

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ⅰ)B,C,D and E are ____________of A.

ⅱ)C is the ______________of B and E.

ⅲ)A is the ______________of B and D.

ⅳ)The intersection of B and D is ____________

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Read the text carefully and then insert the insert the correct mathematical term in each of the blanks.

ⅲ)Give the definition of each of the following:

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1.A two_ element set.

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2.The difference set of A and B, where A and B are sets.

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ⅳ.Four statements are given below. Among them, there is one and only one statement that cannot be used to express the meaning of A∩B=ф.Point it out and give your reason.

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a) The intersection of A and B is zero.

b) Set A does not intersect set B.

c) The intersection of A and B is zero.

d) Set A and set B are B are disjoint.

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ⅴ.Translate the following passage into Chinese:

It was G.. Cantor who introduced the concept the concept of the set as an object of mathematical study. Cantor stated: “A set is a collection of definite, well_ distinguished Objects of out intuition or thought. These objects are called the elements of the set. “cantor introduced the notions of cardinal and ordinal number and developed what is now known as Set Theory.

ⅵ Translate the following sentences into English:

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1． 若集A 与集B均是集C的子集，则集A与集B的并集仍是集C的子集。

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2． 集A的补（余）集的补集是A 。

[此贴子已经被作者于2004-11-27 12:29:10编辑过]

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数学专业英语－Continuous Functions of One Real Variable / n9 p/ _! V4 L0 c5 e7 }! z# F

This lesson deals with the concept of continuity, one of the most important and also one of the most fascinating ideas in all of mathematics. Before we give a preeise technical definition of continuity, we shall briefly discuss the concept in an informal and intuitive way to give the reader a feeling for its meaning.

Roughly speaking the situation is this: Suppose a function f has the value f ( p ) at a certain point p. Then f is said to be continuous at p if at every nearby point x the function value f ( x ) is close to f ( p ). Another way of putting it is as follows: If we let x move toward p, we want the corresponding function value f ( x ) to become arbitrarily close to f ( p ), regardless of the manner in which x approaches p. We do not want sudden jumps in the values of a continuous function.

Consider the graph of the function f defined by the equation f ( x ) = x –[ x ], where [ x ] denotes the greatest integer < x . At each integer we have what is known ad a jump discontinuity. For example, f ( 2 ) = 0 ,but as x approaches 2 from the left, f ( x ) approaches the value 1, which is not equal to f ( 2 ).Therefore we have a discontinuity at 2. Note that f ( x ) does approach f ( 2 ) if we let x approach 2 from the right, but this by itself is not enough to establish continuity at 2. In case like this, the function is called continuous from the right at 2 and discontinuous from the left at 2. Continuity at a point requires both continuity from the left and from the right.

In the early development of calculus almost all functions that were dealt with were continuous and there was no real need at that time for a penetrating look into the exact meaning of continuity. It was not until late in the 18^th century that discontinuous functions began appearing in connection with various kinds of physical problems. In particular, the work of J.B.J. Fourier(1758-1830) on the theory of heat forced mathematicians the early 19^th century to examine more carefully the exact meaning of the word “continuity”.

A satisfactory mathematical definition of continuity, expressed entirely in terms of properties of the real number system, was first formulated in 1821 by the French mathematician, Augustin-Louis Cauchy (1789-1857). His definition, which is still used today, is most easily explained in terms of the limit concept to which we turn now.

The definition of the limit of a function.

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Let f be a function defined in some open interval containing a point p, although we do not insist that f be defined at the point p itself. Let A be a real number.

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The equation

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f ( x ) = A 3 y# H& Z# y7 k0 `: ^ n2 e! s , a- b# e' G" ?4 P6 F

is read “The limit of f ( x ) , as x approached p, is equal to A”, or “f ( x ) approached A as x approached p.” It is also written without the limit symbol, as follows:

f ( x )→ A as x → p

This symbolism is intended to convey the idea that we can make f ( x ) as close to A as we please, provided we choose x sufficiently close to p.

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Our first task is to explain the meaning of these symbols entirely in terms of real numbers. We shall do this in two stages. First we introduce the concept of a neighborhood of a point, the we define limits in terms of neighborhoods.

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Definition of neighborhood of a point.

Any open interval containing a point p as its midpoint is called a neighborhood of p.

NOTATION. We denote neighborhoods by N ( p ), N₁ ( p ), N₂ ( p ) etc. Since a neighborhood N ( p ) is an open interval symmetric about p, it consists of all real x satisfying p-r < x < p+r for some r > 0. The positive number r is called the radius of the neighborhood. We designate N ( p ) by N ( p; r ) if we wish to specify its radius. The inequalities p-r < x < p+r are equivalent to –r<x-p<r, and to ∣x-p∣< r. Thus N ( p; r ) consists of all points x whose distance from p is less than r.

In the next definition, we assume that A is a real number and that f is a function defined on some neighborhood of a point p (except possibly at p ) . The function may also be defined at p but this is irrelevant in the definition.

Definition of limit of a function.

The symbolism

f ( x ) = A or [ f ( x ) → A as x→ p ] , H- U3 |3 Z3 u% j: F7 ^8 |

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means that for every neighborhood N₁ ( A ) there is some neighborhood N₂ ( p) such that

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f ( x ) ∈ N₁ ( A ) whenever x ∈ N₂ ( p ) and x ≠ p (*)

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The first thing to note about this definition is that it involves two neighborhoods, N₁ ( A) and

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N₂ ( p) . The neighborhood N₁ ( A) is specified first; it tells us how close we wish f ( x ) to be to the limit A. The second neighborhood, N₂ ( p ), tells us how close x should be to p so that f ( x ) will be within the first neighborhood N₁ ( A). The essential part of the definition is that, for every N₁ ( A), no matter how small, there is some neighborhood N₂ (p) to satisfy (*). In general, the neighborhood N₂ ( p) will depend on the choice of N₁ ( A). A neighborhood N₂ ( p ) that works for one particular N₁ ( A) will also work, of course, for every larger N₁ ( A), but it may not be suitable for any smaller N₁ ( A).

The definition of limit can also be formulated in terms of the radii of the neighborhoods

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N₁ ( A) and N₂ ( p ). It is customary to denote the radius of N₁ ( A) byεand the radius of N₂ ( p) by δ.The statement f ( x ) ∈ N₁ ( A ) is equivalent to the inequality ∣f ( x ) – A∣<ε,and the statement x ∈ N₁ ( A) ,x ≠ p ,is equivalent to the inequalities 0 <∣ x-p∣<δ. Therefore, the definition of limit can also be expressed as follows:

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The symbol f ( x ) = A means that for everyε> 0, there is aδ> 0 such that

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∣f ( x ) – A∣<ε whenever 0 <∣x – p∣<δ

“One-sided” limits may be defined in a similar way. For example, if f ( x ) →A as x→ p through values greater than p, we say that A is right-hand limit of f at p, and we indicate this by writing

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f ( x ) = A

In neighborhood terminology this means that for every neighborhood N₁ ( A) ,there is some neighborhood N₂( p) such that

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f ( x ) ∈ N₁ ( A) whenever x ∈ N₁ ( A) and x > p

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Left-hand limits, denoted by writing x→ p^-, are similarly defined by restricting x to values less than p.

If f has a limit A at p, then it also has a right-hand limit and a left-hand limit at p, both of these being equal to A. But a function can have a right-hand limit at p different from the left-hand limit.

The definition of continuity of a function.

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In the definition of limit we made no assertion about the behaviour of f at the point p itself. Moreover, even if f is defined at p, its value there need not be equal to the limit A. However, if it happens that f is defined at p and if it also happens that f ( p ) = A, then we say the function f is continuous at p. In other words, we have the following definition.

Definition of continuity of a function at a point.

A function f is said to be continuous at a point p if

( a ) f is defined at p, and ( b ) f ( x ) = f ( p )

This definition can also be formulated in term of neighborhoods. A function f is continuous at p if for every neighborhood N₁ ( f(p)) there is a neighborhood N₂ (p) such that

f ( x ) ∈ N₁ ( f (p)) whenever x ∈ N₂ ( p).

In theε-δterminology , where we specify the radii of the neighborhoods, the definition of continuity can be restated ad follows:

Function f is continuous at p if for every ε> 0 ,there is aδ> 0 such that

∣f ( x ) – f ( p )∣< ε whenever ∣x – p∣< δ

In the rest of this lesson we shall list certain special properties of continuous functions that are used quite frequently. Most of these properties appear obvious when interpreted geometrically ; consequently many people are inclined to accept them ad self-evident. However, it is important to realize that these statements are no more self-evident than the definition of continuity itself, and therefore they require proof if they are to be used with any degree of generality. The proofs of most of these properties make use of the least-upper bound axiom for the real number system.

THEOREM 1. (Bolzano’s theorem) Let f be continuous at each point of a closed interval [a, b] and assume that f ( a ) an f ( b ) have opposite signs. Then there is at least one c in the open interval (a ,b) such that f ( c ) = 0.

THEOREM 2. Sign-preserving property of continuous functions. Let f be continuious at c and suppose that f ( c ) ≠ 0. Then there is an interval (c-δ,c +δ) about c in which f has the same sign as f ( c ).

THEOREM 3. Let f be continuous at each point of a closed interval [a, b]. Choose two arbitrary points x₁ < x₂ in [a, b] such that f ( x₁ ) ≠ f ( x₂ ) . Then f takes every value between f ( x₁ ) and f (x₂ ) somewhere in the interval ( x_1, x₂ ).

THEOREM 4. Boundedness theorem for continuous functions. Let f be continuous on a closed interval [a, b]. Then f is bounded on [a, b]. That is , there is a number M > 0, such that∣f ( x )∣≤ M for all x in [a, b].

THEOREM 5. (extreme value theorem) Assume f is continuous on a closed interval [a, b]. Then there exist points c and d in [a, b] such that f ( c ) = sup f and f ( d ) = inf f .

Note. This theorem shows that if f is continuous on [a, b], then sup f is its absolute maximum, and inf f is its absolute minimum

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Vocabulary . o) ~; Z" }8 r- K- H

continuity 连续性 assume 假定,取

continuous 连续的 specify 指定, 详细说明

continuous function 连续函数 statement 陈述,语句

intuitive 直观的 right-hand limit 右极限

corresponding 对应的 left-hand limit 左极限

correspondence 对应 restrict 限制于

graph 图形 assertion 断定

approach 趋近,探索,入门 consequently 因而,所以

tend to 趋向 prove 证明

regardless 不管,不顾 proof 证明

discontinuous 不连续的 bound 限界

jump discontinuity 限跳跃不连续 least upper bound 上确界

mathematician 科学家 greatest lower bound 下确界

formulate 用公式表示,阐述 boundedness 有界性

limit 极限 maximum 最大值

Interval 区间 minimum 最小值

open interval 开区间 extreme value 极值

equation 方程 extremum 极值

neighborhood 邻域 increasing function 增函数

midpoint 中点 decreasing function 减函数

symmetric 对称的 strict 严格的

radius 半径(单数) uniformly continuous 一致连续

radii 半径(复数) monotonic 单调的

inequality 不等式 monotonic function 单调函数

equivalent 等价的

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Notes

1. It wad not until late in the 18th century that discontinuous functions began appearing in connection with various kinds of physical problems.

意思是:直到十八世纪末,不连续函数才开始出现于与物理学有关的各类问题中.

这里It was not until …that译为“直到……才”

2. The symbol f ( x ) = A means that for every ε> 0 ,there is a δ> 0, such that

|f( x ) - A|<ε whenever 0 <| x – p |<δ

注意此种句型.凡涉及极限的其它定义,如本课中定义函数在点P连续及往后出现的关于收敛的定义等,都有完全类似的句型,参看附录IV.

有时句中there is可换为there exists; such that可换为satisfying; whenever换成if或for.

3. Let…and assume (suppose)…Then…

这一句型是定理叙述的一种最常见的形式;参看附录IV.一般而语文课 Let假设条件的大前提,assume (suppose)是小前提(即进一步的假设条件),而if是对具体而关键的条件的使用语.

4. Approach在这里是“趋于”,“趋近”的意思,是及物动词.如:

f ( x ) approaches A as x approaches p. Approach有时可代以tend to. 如f ( x ) tends to A as x tends to p.值得留意的是approach后不加to而tend之后应加to.

5. as close to A as we please = arbitrarily close to A..

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Exercise 0 r8 [, x: [+ T4 X8 E; q' U

I. Fill in each blank with a suitable word to be chosen from the words given below:

independent domain correspondence

associates variable range

(a) Let y = f ( x ) be a function defined on [a, b]. Then

(i) x is called the ____________variable.

(ii) y is called the dependent ___________.

(iii) The interval [a, b] is called the ___________ of the function.

(b) In set terminology, the definition of a function may be given as follows:

Given two sets X and Y, a function f : X → Y is a __________which ___________with each element of X one and only one element of Y.

II. a) Which function, the exponential function or the logarithmic function, has the property that it satisfies the functional equation

f ( xy ) = f ( x ) + f ( v )

b) Give the functional equation which will be satisfied by the function which you do not choose in (a).

III. Let f be a real-valued function defined on a set S of real numbers. Then we have the following two definitions:

i) f is said to be increasing on the set S if f ( x ) < f ( y ) for every pair of points x and y with x < y.

ii) f is said to have and absolute maximum on the set S if there is a point c in S such that f ( x ) < f ( c ) for all x∈ S.

Now define

a) a strictly increasing function;

b) a monotonic function;

c) the relative (or local ) minimum of f .

IV. Translate theorems 1-3 into Chinese.

V. Translate the following definition into English:

定义:设E 是定义在实数集 E 上的函数,那么, 当且仅当对应于每一ε>0(ε不依赖于E上的点)存在一个正数δ使得当 p 和 q 属于E且|p –q| <δ时有|f ( p ) – f ( q )|<ε,则称f在E上一致连续.

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Historical Introduction

Newton and Leibniz,quite independently of one another,were largely responsible for developing the ideas of integral calculus to the point where hitherto insurmountable problems could be solved by more or less routine methods.The successful accomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch of calculus,differential calculus.

The central idea of differential calculus is the notion of derivative.Like the integral,the derivative originated from a problem in geometry—the problem of finding the tangent line at a point of a curve.Unlile the integral,however,the derivative evolved very late in the history of mathematics.The concept was not formulated until early in the 17^th century when the French mathematician Pierre de Fermat,attempted to determine the maxima and minima of certain special functions.

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Fermat’s idea,basically very simple,can be understood if we refer to a curve and assume that at each of its points this curve has a definite direction that can be described by a tangent line.Fermat noticed that at certain points where the curve has a maximum or minimum,the tangent line must be horizontal.Thus the problem of locating such extreme values is seen to depend on the solution of another problem,that of locating the horizontal tangents.

This raises the more general question of determining the direction of the tangent line at an arbitrary point of the curve.It was the attempt to solve this general problem that led Fermat to discover some of the rudimentary ideas underlying the notion of derivative.

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At first sight there seems to be no connection whatever between the problem of finding the area of a region lying under a curve and the problem of finding the tangent line at a point of a curve.The first person to realize that these two seemingly remote ideas are,in fact, rather intimately related appears to have been Newton’s teacher,Isaac Barrow(1630-1677).However,Newton and Leibniz were the first to understand the real importance of this relation and they exploited it to the fullest,thus inaugurating an unprecedented era in the development of mathematics.

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Although the derivative was originally formulated to study the problem of tangents,it was soon found that it also provides a way to calculate velocity and,more generally,the rate of change of a function.In the next section we shall consider a special problem involving the calculation of a velocity.The solution of this problem contains all the essential fcatures of the derivative concept and may help to motivate the general definition of derivative which is given below.

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A Problem Involving Velocity

Suppose a projectile is fired straight up from the ground with initial velocity of 144 feet persecond.Neglect friction,and assume the projectile is influenced only by gravity so that it moves up and back along a straight line.Let f(t) denote the height in feet that the projectile attains t seconds after firing.If the force of gravity were not acting on it,the projectile would continue to move upward with a constant velocity,traveling a distance of 144 feet every second,and at time t we woule have f(t)=144 t.In actual practice,gravity causes the projectile to slow down until its velocity decreases to zero and then it drops back to earth.Physical experiments suggest that as the projectile is aloft,its height f(t) is given by the formula

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(1) f(t)=144t –16 t²

The term –16t² is due to the influence of gravity.Note that f(t)=0 when t=0 and when t=9.This means that the projectile returns to earth after 9 seconds and it is to be understood that formula (1) is valid only for 0<t<9.

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The problem we wish to consider is this:To determine the velocity of the projectile at each instant of its motion.Before we can understand this problem,we must decide on what is meant by the velocity at each instant.To do this,we introduce first the notion of average velocity during a time interval,say from time t to time t+h.This is defined to be the quotient.

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Change in distance during time interval =f(t+h)-f(t)/h

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- z1 a. z3 D3 [3 i& s: c, j2 D0 l$ A$ X Length of time interval

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This quotient,called a difference quotient,is a number which may be calculated whenever both t and t+h are in the interval[0,9].The number h may be positive or negative,but not zero.We shall keep t fixed and see what happens to the difference quotient as we take values of h with smaller and smaller absolute value.

The limit process by which v(t) is obtained from the difference quotient is written symbolically as follows:

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V(t)=lim_(h→0) [f(t+h)-f(t)]/h

The equation is used to define velocity not only for this particular example but,more generally,for any particle moving along a straight line,provided the position function f is such that the differerce quotient tends to a definite limit as h approaches zero.

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The example describe in the foregoing section points the way to the introduction of the concept of derivative.We begin with a function f defined at least on some open interval(a,b) on the x axis.Then we choose a fixed point in this interval and introduce the difference quotient

[f(x+h)-f(x)]/h

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where the number h,which may be positive or negative(but not zero),is such that x+h also lies in(a,b).The numerator of this quotient measures the change in the function when x changes from x to x+h.The quotient itself is referred to as the average rate of change of f in the interval joining x to x+h.

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Now we let h approach zero and see what happens to this quotient.If the quotient.If the quotient approaches some definite values as a limit(which implies that the limit is the same whether h approaches zero through positive values or through negative values),then this limit is called the derivative of f at x and is denoted by the symbol f’(x) (read as “f prime of x”).Thus the formal definition of f’(x) may be stated as follows:

Definition of derivative.The derivative f’(x)is defined by the equation

f’(x)=lim_(h→o)[f(x+h)-f(x)]/h

provided the limit exists.The number f’(x) is also called the rate of change of f at x.

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In general,the limit process which produces f’(x) from f(x) gives a way of obtaining a new function f’ from a given function f.This process is called differentiation,and f’ is called the first derivative of f.If f’,in turn,is defined on an interval,we can try to compute its first derivative,denoted by f’’,and is called the second derivative of f.Similarly,the nth derivative of f denoted by f^(n),is defined to be the first derivative of f^(n-1).We make the convention that f^(0)=f,that is,the zeroth derivative is the function itself.

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Vocabulary

differential calculus微积分 differentiable可微的

intergral calculus 积分学 differentiate 求微分

hither to 迄今 integration 积分法

insurmountable 不能超越 integral 积分

routine 惯常的 integrable 可积的

fuse 融合 integrate 求积分

originate 起源于 sign-preserving保号

evolve 发展，引出 axis 轴（单数）

tangent line 切线 axes 轴（复数）

direction 方向 contradict 矛盾

horizontal 水平的 contradiction 矛盾

vertical 垂直的 contrary 相反的

rudimentary 初步的，未成熟的 composite function 合成函数，复合函数

area 面积 composition 复合函数

intimately 紧密地 interior 内部

exploit 开拓，开发 interior point 内点

inaugurate 开始 imply 推出，蕴含

projectile 弹丸 aloft 高入云霄

friction摩擦 initial 初始的

gravity 引力 instant 瞬时

rate of change 变化率 integration by parts分部积分

attain 达到 definite integral 定积分

defferential 微分 indefinite integral 不定积分

differentiation 微分法 average 平均

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Notes - Z9 m+ N- r3 W$ O5 j

1. Newton and Leibniz,quite independently of one another,were largely responsible for developing…by more or less routine methods.

意思是：在很大程度上是牛顿和莱伯尼，他们相互独立地把积分学的思想发展到这样一种程度，使得迄今一些难于超越的问题可以或多或少地用通常的方法加以解决。

这里responsible for的基本意义是：“对…负责”，但也可作“应归功于”解，这里应理解为“归功于”。

2．The example described in the foregoing section points the way to the introduction of the concept of derivative.

意思是：前面一节所描述的例子指出了引进导数概念的方法。

这里described是过去分词，foregoing是现在分词，两者都用作定语，切不可误认described为过去式谓语。类似句子如：

We begin with a function f defined on some interval(a,b);

3. The quotient itself is referred to as the average of change of f in the interval joining x to x+h.

意思是：商本身是指区间x到x+h上f的平均变化率。这里be referred to as意思是：“把…

认为是“

4．We make the convention that f⁰=f

意思是：我们约定（按惯例）f⁰=f

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Exercise

I.1.Fill in the missing words in column B such that the word in column B corresponds to the word in column A in the same sense as “integration”corresponds to “differentiation”

A B

Differentiation integration

Differential

Differentiate

Differentiable

1. Then choose the correct word from either column A or column B and insert it in each of the blanks.

(i)The process of finding the derivative of a function is called( )

(ii) If f(x) has a derivative at the point x₀ then f(x) is said to be ( )at x_0.

(iii) ∫f(x)dx is called the indefinite( ) of f(x).

II Translate the following two examples into Chinese(pay attention to the phrases used):

Example 1 Find the derivative of

f(x)=(3x+1)⁴/(x²+2)³

solution:Taking the logarithms of both sides,we have

ln f(x)=4ln(3x+1)-3ln(x²+2)

Differentiating both sides of the above equation,we obtain

f’(x)/f(x)=12/(3x+1)-6/(x²+2)

(1) and (2)together yield

f’(x)=[(3x+1)⁴/(x2+2)³][12/(3x+1)-6/(x²+2)]

Example 2 Integrate ∫ x² cosxdx

Solutionet u=x²,v=sin x;

Then du=2xdx,dv=cos x dx

So we have I=∫x² cos x dx =∫u dv

By applying integration by parts,we have

I=∫udv=uv-∫vdu=x² sinx-2∫x sinx dx (1)

Applying integration by parts once again to the indefinite integral∫xsinxdx,we get

∫x sinx dx = -x cos x +sin x +c (2)

Substituting (2)into (1)yields

∫ x² cos x dx=x² sin x+2x cos x-2 sin x +c

where c is an arbitrary constant.

III. Translate the following example into English:

求y=ln(2x²-4)的导数

【解】令y=ln u,u=(2x2-4),则dy/du=1/u. du/dx=4x (1)

据复合函数求导数的公式，我们有

dy/du=dy/du-du/dx (2)

把（1）代入（2）式得dy/dx=(1/u)4x

把u换为2x²-4,最后得dy/dx=4x/(2x²-4)

IV. Theorem Let f be defined on an open interval I,and assume that f has a relative maximum or a relative minimum at an interior point c of I.If the derivative f’(c )exists,then f’(c )=0

The proof of this theorem is given in Chinese as follows.Turn it into English:

证明：

（1） 在I上定义函数Q（x）

Q(x)=[f(x)-f(c )]/(x-c) x≠c

f’(c ) x=c

(2) 因为f’(c)存在，故当x趋向c时，Q（x）趋向Q（c），也即Q（x）在点x＝c连续

（2） 我们将证明：若f’（c）＝Q（c）≠0，则导致矛盾；

（3） 设Q（c）>0，根据连续函数的保号性质，存在c点的一个领域，在此领域里，Q（x）是正的；

（4） 因此在此领域内，对所有x≠c，Q（x）的分子和分母同号；

（5） 即是说，当x>c时，f（x）>f(c ),而当x<c时，f（x）<f（c）这与f在x＝c处有一极值相矛盾；

（6） 因此Q（c）>0不可能，同理可证Q（c）<0也不真。