数学专业英语

1、New Words 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 re

2、al 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tana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

3、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 fi

4、xed, 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

5、 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,1. 如果差商以某一个确定的值为极限，则该极限叫做f在x点的导数。 2. 通过P且具有斜率f (x)的直线叫做在P点的切线。,If the difference quotient approaches some definite value as a limit，then this limit is called the derivative of f at x.,The line through