数学专业英语－Differential Calculus

yunjiang · 发表于 2004-5-6 09:29:10

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>Historical Introduction
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>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.
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>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 17th century when the French mathematician Pierre de Fermat,attempted to determine the maxima and minima of certain special functions.
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.
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.
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.
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
(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.
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.
Change in distance during time interval =f(t+h)-f(t)/h
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Length of time interval</TD></TR></TABLE>
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:
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.
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
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.
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.
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.

yunjiang · 发表于 2004-5-6 09:29:26

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0cm 0cm 0pt; TEXT-ALIGN: center" align=center><A>Vocabulary</A><

0cm 0cm 0pt">differential calculus微积分 differentiable可微的<

0cm 0cm 0pt">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

yunjiang · 发表于 2004-5-6 09:29:43

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0cm 0cm 0pt; TEXT-ALIGN: center" align=center>Notes<

0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; tab-stops: list 18.0pt; mso-list: l69 level1 lfo14">1. Newton and Leibniz,quite independently of one another,were largely responsible for developing…by more or less routine methods.<

0cm 0cm 0pt 18pt">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

yunjiang · 发表于 2004-5-6 09:30:02

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0cm 0cm 0pt; TEXT-ALIGN: center" align=center>Exercise<

0cm 0cm 0pt">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”<

0cm 0cm 0pt"> A B Differentiation integration Differential Differentiate Differentiable1. 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 x0. (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 haveln f(x)=4ln(3x+1)-3ln(x2+2) Differentiating both sides of the above equation,we obtainf’(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 Solution

et 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 )=0The 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≠cf’(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。

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