"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.