A LIMIT–FREE APPROACH TO DERIVATIVES: Report on a Classroom Project

Led by the idea that “. . . using a graphing calculator to zoom in functions is one of the best ways of seeing local linearity” (Hughes-Hallet, Gleason et al. 1994), we discuss here the role of new technology in teaching and understanding mathematics and present a complete “route to derivatives”, which is particularly suitable for undergraduate teaching. Starting with a formal definition of the tangent to a graph we are led to the fundamental theorem of calculus. The basic definition (the slope of the tangent) relies on graphic concepts, like screen resolution and pixels’ dimensions, so, essentially, it’s “limit–free”. The fundamental theorem of calculus is presented in a discretized version, showing that the trapezoidal rule for areas and the central difference formula for the tangents’ slopes are, in a sense, inverse one to each other. At an higher level, we can note that the central difference rule can be applied to solve numerically differential equations. Everything has been implemented on an intermediate graphic calculator (actually the TI–82, TI–83 and TI–83 Plus), and carried out in the last three years on a total population of about 750 students of life sciences, pharmacy and chemistry, aged 18–19, from two different universities in Italy. As a result we noticed a substantial improvement of lowranked and middle-ranked students. We present also the project a “teacher bank”, consisting of TI–GraphLink compatible files containing related programs and data, and more advanced teaching tools as Mathematica notebooks and QuickTime movies, especially generated by TI–8∗ screenshots.