浅谈改微求切法在数学分析中的应用

Discussion on Effective Method of "Replacing Curve by Straight" in Mathematical Analysis Calculating Tangent Line by Differentiation

直与曲是数学分析的一对基本矛盾,以直代曲是数学分析的一个基本方法,微分学的几何应用正是以直代曲的一个重要体现.微分学的几何应用通常是指:求曲线的切线和法线(或法平面);求曲面的切平面和法线.罗列起来有8种类型共16个公式.虽然这些公式并不难记,但时间长了却容易忘或混淆.本文向大家浅谈一种简便方法——改微求切法.这个方法简便易行、程序机械,是以直代曲的一个典型方法.它可以驾驭这里的全部问题.有了此,就不必记忆公式而直接得到所要求的直线和平面方程.

Straight and bend are a pair of basic contradiction. Replacing Curve by Straight is one of the basic method in mathematical analysis. Geometric application of differential calculus is an important reflect for replacing Curve by Straight. Geometric application of differential calculus usually refers to calculate tangent line and normal（or normal plane） of curve, tangent plane and normal of curved surface. There are eight types totally with sixteen formulas. Although these formulas are not hard to remember, it is easy to forget or confuse. The paper introduces a kind of simple method calculating tangent line by differentia- tion. The method is a typical method of replacing curve by straight. It is easy to operate and the program is mechanical. It can harness all problems here, and can also obtain linear equation and normal equation of plane directly without having to remember formula.