Abstract:

Calculus has been developing for almost three hundreds years. It was first introduced by Newton and Leibniz. Without the basis of rigorous theory, it was ambiguous, which is called the first generation of calculus.  Later, it was established on the concept of limit in a rigorous and abstract way, called the second generation of calculus. However, it is difficult to understand for most beginners. In this paper, we introduce two concepts of function A and function B, on which a new calculus is built in a rigorous and simple way. The new calculus is called the third generation of calculus. Due to its basis being elementary mathematics, the difficulty of its teaching and learning is reduced greatly. In addition, it is expected that this new calculus will have a profound influence on mechanization of calculus.  

Key words: elementarization of calculus, limit, mechanization of mathematics