数学专业英语（课堂PPT）

1、New Words & Expressions:acceleration 加速度加速度 interval 区间区间altitude 高度高度 numerator 分子分子approach 趋于趋于 rectilinear motion 直线运动直线运动 bound 界，限界，限 slope 斜率斜率derivative 导数导数 tangent 正切，切线正切，切线 fraction 分数，分式分数，分式 velocity 速度速度2.8 函数的导数和它的几何意义函数的导数和它的几何意义The Derivative of a Function and Its Geometric int

2、erpretationKey points: 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。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

3、少定义在我们从一个至少定义在x轴的开区间轴的开区间(a,b)上的函数入上的函数入手。手。Then we choose a fixed point x in this interval and introduce the difference quotientwhere 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，接下来在这个区间中选择一个固定点，并且引进差商，这里数这里数h可正可负可正可负(但是

4、不能为但是不能为0)，且使得，且使得x+h也在也在(a,b)中。中。()( )f xhf xhThe 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这个商的分子度量了当x由由x变到变到x+h时函数的改变量。时函数的改变量。

5、差商本身表示函数差商本身表示函数 f 在连接在连接 x 与与 x + h 的区间上的平的区间上的平均变化率。均变化率。 Now we let h approach zero and see what happens to this quotient.现在令现在令h趋于趋于0，看这个商如何变化。，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

6、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如果差商以某个确定的值为极限(这蕴含着不论这蕴含着不论 h 取取正的值趋于正的值趋于 0 还是取负的值趋于还是取负的值趋于 0，其极限一样，其极限一样)，那，那么这个极限称为么这个极限称为 f 在在 x点点 的导数，记作的导数，记作f (x) (读成读成“f一撇一撇x”)。Th

7、us, the formal definition of f(x) may be stated as follows:这样，这样， f(x)的正式定义可以叙述如下：的正式定义可以叙述如下：DEFINITION. The derivative f(x) is defined by the equationprovided 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如果这个极限存在，则这个等式定义了导数f(x) ，它，它也称为也称为f在在x处的变化

8、率。处的变化率。0()( )( )lim(8.2)hf xhf xfxhBy 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.通过对比通过对比(8.2)和上节的和上节的(7.3)式，可以看出瞬时速度的式，可以看出瞬时速度的概念只是导数概念的一个例子。概念只是导数概念的一个例子。The velocity v(t) is equal to th

9、e 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.速度速度v(t)是位置函数是位置函数f的导数。这通常说成速度是位移的导数。这通常说成速度是位移关于时间的变化率。关于时间的变化率。In general, the limit process which produces f(x) from f(x) giv

10、es 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.一般地一般地, 由由f(x)产生产生f(x)的极限的过程提供了一种方法的极限的过程提供了一种方法, 从一个给定的函数从一个给定的函数 f 得到一个新的函数得到一个新的函数f。这个过程。这个过程叫做微分法。叫做微分法。 f叫做叫做f的一阶导数。的一阶导数。If f , in turn, is defin

11、ed 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，如果反过来，如果f定义在一个开区间上，我们也可以尝定义在一个开区间上，我们也可以尝试计算它的一阶导数，记为试计算它的一阶导数，记为f ，叫做，叫做f的二阶导数。的二阶导数。For rectilinear motion, the first derivative of velocity (second derivative of position) is

12、 called acceleration. (P67 第四段第一句话第四段第一句话)在直线运动中，速度的一阶导数称为加速度。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

13、以一种自然的方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。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).函数函数f图象的一部分如图图象的一部分如图2-8-1所示。所示。P，Q两点有各自两点有各自的坐标。的坐标。 Consider the right triangle with

14、hypotenuse PQ; its altitude, f(x+h)-f(x), represents the difference of the ordinates of the two points Q and P.考虑斜边为考虑斜边为PQ的直角三角形，它的高度的直角三角形，它的高度f(x+h)-f(x)表表示示P、Q两点纵坐标的差。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.

15、因此，差商表示因此，差商表示PQ与水平线的夹角与水平线的夹角a的正切。的正切。The real number tana is called the slope of the line through P and Q and it provides a way of measuring the “steepness” of the line.实数实数tana叫做直线叫做直线PQ的斜率，它提供了一种度量直线的斜率，它提供了一种度量直线陡峭程度的方法。陡峭程度的方法。For example, if f is a linear function, say f(x)=mx+b, the differen

16、ce quotient has the value m, so m is the slope of the line.例如，如果例如，如果f是一个线性函数，那么差商的值为是一个线性函数，那么差商的值为m, 那那么么m就是直线的斜率。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.现在假设现在假设f在在x有导数。这意味着当有导数。这意味着当h趋于趋于0时，差商趋时，

17、差商趋于某一个极限于某一个极限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.其几何意义为其几何意义为, 当当 h 趋于趋于

18、0时时, 点点 P 保持不动保持不动, 而点而点Q 沿沿曲线趋近曲线趋近P, 同时同时, 经过经过 PQ 的直线以这样的方式改变的直线以这样的方式改变方向方向, 即其斜率趋于数值即其斜率趋于数值 f (x), 并以它为极限并以它为极限.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，把曲线在正因如此，把曲线在P点的斜率定义成数点的斜率定义成数f (x)看起来看起来是自然的。通过是自然的。通过P且具有这个斜率的直线叫做在且具有这个斜率的直线叫做在P点点的切线。的切线。本小节重点掌握本小节重点掌握1. 如果差商以某一个确定的值为极限，则该极限叫做如果差商以某一个确定的值为极限，则该极限叫做f