数学专业英语 微分学论文翻译.doc

信息0801 姬彩云 200801010129Differential Calculus 微分学Historical Introduction历史介绍/ D/ x* p2 R6 r$ $Newton and Leibniz,quite independently of one another,were largely responsible for developing the ideas of integral calculus to the point where hitherto insurmountable problems could be solved by more or less routine methods.The successful accomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch of calculus,differential calculus.+ V0 E$ f! k% w6 l- ( The central idea of differential calculus is the notion of derivative.Like the integral,the derivative originated from a problem in geometrythe 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.Fermats 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 Newtons 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 一个设计速度的问题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.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 0t9.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.ength of time intervalThis quotient,called a difference quotient,is a number which may be calculated whenever both t and t+h are in the interval0,9.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:* O4 ?/ n, # H# y3 m! KThe 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.8 b+ N, j, b; w; VThe 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 quotientf(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牛顿和莱布尼茨，完全相互独立，主要负责开发积分学思想的地步，迄今无法解决的问题可以解决更多或更少的常规方法。这些成功的人主要是由于他们能够将积分学和微分融合在一起的事实，。中心思想是微分学的概念衍生。像积分，衍生起源于一个问题的几何问题找到切线曲线在一点的。然而，衍生发展非常晚在数学的历史。这一概念是没有制定直到早在第十七世纪法国数学家费尔马彼埃尔，试图确定9 B; p t5 U+ ! 费马的想法，基本上是很简单的，可以理解为如果我们指的一个曲线和假设在其每一点这个曲线有一定的方向，可以说是由一个切线。费马发现在某些点在曲线有一个最大或最小，切线必须水平。因此定位问题这样极端的价值观被认为取决于解决另一个问题，即定位水平切线。这就提出了更一般的问题，确定方向的切线在任意一点的曲线。这是试图解决这一问题，使得费马发现一些基本的思想的基本概念衍生。 h3 A P2 R5 ?% s + c+ S; s$ e乍一看似乎没有任何关系之间找到问题的区域面积躺在曲线和发现问题的切线的曲线在一点。第一个人意识到这些看似遥远的想法是，事实上，而不是密切相关的似乎是牛顿的老师，艾萨克巴罗（1630-1677）。然而，牛顿和莱布尼茨是第一个了解真正的重要性，这种关系，他们利用它的最大，从而开创了前所未有的时代，在数学的发展。. K+ b1 U* ) 1 虽然衍生最初制定研究问题的切线，但不久就发现它还提供了一种方法来计算速度，更一般地，该函数变化率。在下一节中我们将考虑的特殊问题涉及计算速度。这个问题的解决方案包含了所有的基本特征的导数的概念和可能有助于激发普通导数的定义。$ A* |. G2 9 v. W! z猜射弹发射是直接从地面与初始速度为144英尺/秒。忽略摩擦，并承担弹只有重力的影响使其移动和沿直线。让f(t)表示高度英尺的弹丸达到秒后发射。如果力重力不身体力行，弹丸会继续向上