Discussion:  Roundtable 
Topic:  "Graphing Calculator" Software 
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Subject:  RE: The exponential function 
Author:  Alan Cooper 
Date:  Apr 25 2004 
least one part of what you are asking for  namely the identification of e by
'trial and error'. It allows the first couple of decimal places to be determined
by just zooming in on the yintercept, and by moving just slightly away and
zooming in to a window size around 10^12 at x around 10^7 it is possible
to get several more places.
It's at http://www.langara.bc.ca/~acooper/mathlets/expozoom/index.htm
and I have just submitted it as a 'MathTool'  along with the more powerful
general purpose graphing applet and component library on which it is based.
On Sep 19, 2003, Gene wrote:
I first showed them several exponential functions using Graphing Calculator
(2^x, 3^x, etc.), then used the Derivative and Tangent Line applet on the same
functions. They surmised that the derivative of an exponential function should
be an exponential function. We manipulated the definition of derivative to see
that the derivative of a^x is of the form ca^x, where c is the derivative at
0.
They agreed it would be nice to have a number for which the c turned out to be
one, and by trying various numbers like 2.7.2.71, 2.718 in Derivative and
Tangent Line Applet (and zooming), we got a number "a" for which the derivative
of a^x was pretty close to itself. By then almost everybody was recognizing
e.
This worked well but there may be applets that do it cleaner. Know any?
I passed back the quiz I gave last meeting. Some folks had problems figuring out
the shapes of derivative curves, as in the Derivative Puzzle applets, and I
recommended they go play with these till they can do them. I promised to put
similar problems on midterm and/or final and expect most who need it will
actually do it.
 
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