"Graphing elementary real functions: What can and cannot be done with a CAS"
E.H.A. Gerbracht (*) and W. Struckmann
Institut fuer Netzwerktheorie und Schaltungstechnik
Technische Universitaet Braunschweig
Langer Kamp 19c
D-38106 Braunschweig
Germany
E-mail:e.gerbracht@tu-bs.de
To draw a graph of an elementary real function, one uses analytical
methods to determine zeroes, local maxima or minima, points of
inflections, symmetries, intervals where the function is not defined,
asymptotic behavior, intervals on which the function is increasing or
decreasing and intervals on which the function is concave upward or
downward. Then, these data are used to plot the function. This method
is called ``graphing a function'' and is a common subject of lectures
on calculus or in high-school mathematics education.
In this talk we will give an overview on the theoretical and practical
aspects of the graphing of elementary functions by symbolic means: on
one hand a number of tasks that arise are undecidable for the whole
set of elementary real functions. On the other hand we are able to name
rather large subsets of functions which can be handled completely by an
algorithm. This algorithm has led to a prototype implementation in
MAPLE which is able to graph symbolically most functions occurring in
an undergraduate course on calculus.