数学专业英语

,第二级,第三级,第四级,第五级,New Words&Expressions:,acceleration,加速度,interval,区间,altitude,高度,numerator,分子,approach,趋于,rectilinear motion,直线运动,bound,界，限,slope,斜率,derivative,导数,tangent,正切，切线,fraction,分数，分式,velocity,速度,2.8,函数的导数和它的几何意义,The Derivative of a Function and Its Geometric interpretation,Key points:,the definition of the derivative of a function,Difficult points:,s,ome relevant terms,The example described in the foregoing section points the way to the,introduction,of the concept of derivative.,8-A,The derivative of a function,上一节描述的例子指出了介绍导数概念的方法。,We begin with a function,f,defined at least on some,open interval,(,a,b,)on the,x,-axis.,我们从一个至少定义在,x,轴的开区间,(,a,b,),上的函数入手。,Then we choose a,fixed point,x,in this,interval,and introduce the,difference quotient,where the number,h,which may be positive or negative(but not zero),is such that,x,+,h,also lie in(,a,b,).,接下来在这个区间中选择一个固定点，并且引进差商，这里数,h,可正可负,(,但是不能为,0),，且使得,x,+,h,也在,(,a,b,),中。,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 the change of,f,in the,interval,joining,x,to,x,+,h,.,这个商的分子度量了当,x,由,x,变到,x,+,h,时函数的改变量。差商本身表示函数,f,在连接,x,与,x,+,h,的区间上的平均变化率。,Now we let,h,approach,zero and see what happens to this quotient.,现在令,h,趋于,0,，看这个商如何变化。,If the quotient approaches some definite value 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,”).,如果差商以某个确定的值为极限,(,这蕴含着不论,h,取正的值趋于,0,还是取负的值趋于,0,，其极限一样,),，那么这个极限称为,f,在,x,点,的导数，记作,f,(,x,)(,读成“,f,一撇,x,”),。,Thus,the formal definition of,f,(,x,)may be stated as follows:,这样，,f,(,x,),的正式定义可以叙述如下：,DEFINITION.The,derivative,f,(,x,)is defined by the equation,provided the limit exists.The number,f,(,x,)is also called the rate of change of,f,at,x,.,如果这个极限存在，则这个等式定义了导数,f,(,x,),，它也称为,f,在,x,处的变化率。,By comparing(8.2)with(7.3)in the foregoing section,we see that the concept of,instantaneous velocity,is merely an example of the concept of derivative.,通过对比,(8.2),和上节的,(7.3),式，可以看出瞬时速度的概念只是导数概念的一个例子。,The velocity,v,(,t,)is equal to the,derivative,f,(,t,),where,f,is the function which measures position.This is often described by saying that,velocity,is the rate of change of position,with respect to,time.,速度,v,(,t,),是位置函数,f,的导数。这通常说成速度是位移关于时间的变化率。,In general,the limit process which produces,f,(,x,)from,f,(,x,)gives us 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,.,一般地,由,f,(,x,),产生,f,(,x,),的极限的过程提供了一种方法,从一个给定的函数,f,得到一个新的函数,f,。这个过程叫做微分法。,f,叫做,f,的一阶导数。,If,f,in turn,is defined on an,open interval,we can try to compute its,first derivative,denoted by,f,and called,the second derivative of,f,.,反过来，如果,f,定义在一个开区间上，我们也可以尝试计算它的一阶导数，记为,f,，叫做,f,的二阶导数。,For,rectilinear motion,the first derivative of,velocity,(second derivative of position)is called,acceleration,.(P67,第四段第一句话,),在直线运动中，速度的一阶导数称为加速度。,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.,8-B,Geometric interpretation of the derivative as a slope,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的思想。,A portion of the graph of a function,f,is shown in Figure 2-8-1.Two of the points,P,and,Q,are shown with respective coordinates(,x,f,(,x,)and(,x,+,h,f,(,x,+,h,).,函数,f,图象的一部分如图,2-8-1,所示。,P,，,Q,两点有各自的坐标。,Consider the,right triangle,with,hypotenuse,PQ,;its,altitude,f,(,x,+,h,)-,f,(,x,),represents the difference of the,ordinate,s of the two points,Q,and,P,.,考虑斜边为,PQ,的直角三角形，它的高度,f,(,x,+,h,)-,f,(,x,),表示,P,、,Q,两点纵坐标的差。,Therefore,the,difference quotient,represents the,trigonometric tangent,of the angle,a,that,PQ,makes with the,horizontal,.,因此，差商表示,PQ,与水平线的夹角,a,的正切。,The real number,tan,a,is called the,slope,of the line through,P,and,Q,and it provides a way of measuring the“steepness”of the line.,实数,tan,a,叫做直线,PQ,的斜率，它提供了一种度量直线陡峭程度的方法。,For example,if,f,is a linear function,say,f,(,x,)=,mx,+,b,the difference quotient has the value,m,so,m,is the slope of the line.,例如，如果,f,是一个线性函数，那么差商的值为,m,那么,m,就是直线的斜率。,Suppose now that,f,has a derivative at,x,.,This means that the difference quotient approaches a certain limit,f,(,x,)as,h,approaches 0.,现在假设,f,在,x,有导数。这意味着当,h,趋于,0,时，差商趋于某一个极限,f,(,x,),。,When this is interpreted geometrically it tells us that,as,h,gets nearer to 0,the point,P,remains fixed,Q,moves along the curve toward,P,and the line through,PQ,changes its direction in such a way that its slope approaches the number,f,(,x,)as a limit,.,其几何意义为,当,h,趋于,0,时,点,P,保持不动,而点,Q,沿曲线趋近,P,同时,经过,PQ,的直线以这样的方式改变方向,即其斜率趋于数值,f,(,x,),并以它为极限,.,For this reason,it seems that natural to,define,the,slope,of the curve at,P,to,be the number,f,(,x,).The line through,P,having this slope is called the,tangent line,at,P,.,正因如此，把曲线在,P,点的斜率定义成数,f,(,x,),看起来是自然的。通过,P,且具有这个斜率的直线叫做在,P,点的切线。,本小节重点掌握