Discussion:  All Topics 
Topic:  "Graphing Calculator" Software 
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Subject:  Family of curves 
Author:  Paul Muckerheide 
Date:  Nov 10 2003 
stimulate deeper understanding. Plotting data for average temperatures in say 10
cities from Seattle to New York to Minneapolis to San Diego for 12 months for 3
years has been tried here in Math Analysis (Pre Calc). Questions about
predictions of temperatures on certain days in the future or instantaneous rate
of temperature change are asked to encourage the visualization of the plot to
have depth.
I want students to notice a,b, c, and d in terms of the period, phase
shift , amplitude and vertical shifter before words are used and before
associated with the alpha symbols, sometimes some students make the
connections. The start is helped by having done similar approach for polynomial
curves and I intentionally emphasize the number of relative max and mins (Aleph
null?) to set students up for Taylor's polynomial approximations.
The discussion revolves around the concept that the fundamental theorem
of algebra and the number of zeroes for the instaneous rate of change function
imply that the model for the data would have to be a polynomial with a countably
infinite number of terms. Perhaps f(x) = asinb(xc) +d might be more
convenient.
 
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