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 Subject: RE: The exponential function Author: Alan Cooper Date: Apr 27 2004
Hi Gene,

I agree that by just zooming in on the point (0,1) it takes good eyes to go
beyond one dp. But if you look closely at the case with b=2.7 you may note that
at w=h=.125 the red graph "lifts off" from the blue asymmetrically and that the
symmetry is best at b=2.72. Also, at b=2.7, if we zoom in further to
w=h<.003, then by checking the bottom left and top right corners we may be
able to see that there really is a crossing - and again this can be eliminated
by going to b=2.72

The key to getting more dp is to move the viewpoint slightly. e.g. at x=0.0001
y=1.0001, b=2.7 produces a significant separation at w=h=10^-5 (with red below
blue which indicates a crossing - there's lots of opportunity for getting
students to think about why that is true, and to come up with their own
suggestions about what choices are most useful here). Coincidence of the lines
in this window is restored at b=2.718, and then for more figures we can go back
closer to (0,1), crank up the magnification, re-adjust b, and repeat.

The best I have been able to convince myself of is b=2.71828 with x=10^-6 and
w=5*10^-13

For me the graph breaks up at around 10^-14 (all calculations are done with
java 'doubles' so it has to happen eventually) and the applet sometimes freezes
at this point (which is probably my fault). I haven't tried Safari, but use both
MSIE and Netscape7 with Java installed directly from java.com (this is much more
user friendly than Sun's old site used to be, so I don't mind asking students to
upgrade if they have older versions installed)

Thanks for checking it out.
Alan