谈微积分中的近似与精确

Approximation and Accuracy in Calculus

为揭示微积分中所蕴含的哲学思想,使用案例法、分析法和应用法,对微积分中几个最重要的概念进行分析研究,发现都蕴含着"近似与精确"这一对矛盾的对立统一与转化,同时,也蕴含着其他一些哲学思想,比如,量变与质变、否定之否定以及直与曲等。充分利用"近似与精确"这对矛盾的转化,在解决实际问题时,把无规则的总量微元后就可以看成是规则的,从而解决这个问题。现实中很多数据,只要求出达到某个精确度的近似值即可。

To reveal the philosophy contained in calculus, the case-law, analysis method,and application method are used. Analysis of several of the most important concepts in calculus research reveal that all contain ＂approximate and exact＂ unity of opposites of the contradiction and transformation at the same time. Also, a number of other philosophical thoughts are contained in calculus, such as quantitative and qualitative changes, the negation as well as straight and curve, and so on. When facing practical issue, the full use of ＂approximate and exact＂ conversion of this contradiction and transformation of rules and no rules help to solve the problem. Besides, for real-world data, just a certain precision of approximate value is enough.