What kind of materials are needed to build the structure of calculus?

The present article looks at another approach for presenting the ideas of calculus, instead of the traditional ‘limit based calculus’. Building the ideas of calculus from real applications, the first elements that should be studied are the ‘differential forms’. From these, the process of integration and the derivative appear naturally and are given meaning. Furthermore, the techniques of numerical and graphical integration fit very nicely within this approach. This paper discusses first, the different interpretations that can be associated with the three main elements of calculus: the differential, the derivative and the integral, and compares the approach suggested in this article with the conventional one. In doing this, it is shown how the Fundamental Theorem of Calculus can be interpreted as a ‘continuous sum’ of changes and it is shown why the geometric model of the integral as an area under the curve is a restricted view in many ways. The article then turns to examining the constructive materials of calculus. Five main components are identified: (1) variation of differences and sums; (2) successive approximations; (3) linearization; (4) behaviour and orders of magnitude; and (5) infinite processes.