|
|
Applications of a simple differential equation:
Posted:
Sep 16, 1997 12:28 PM
|
|
Dear fellow list subscribers,
I am one of those who thinks that the changes in the AP syllabus are
significant. Recently, I have been writing problems related to the
"simple" differential equation form, " y'=ky." It is certainly no big
deal to teach students how to "solve" this equation by the so called
separation technique to get a representation of a family of vertically
scaled exponential function solutions. But my reading of the new
syllabi is that beyond the symbol manipulation, we pay attention to
applying this equation to cases of growth of decay. I also assume that
this means paying to the significance of the constant represented by "k"
and to unit designations for all the quantities related by this
equation.
I have given this some thought and perhaps it might be interesting to
some of you to focus on these issues as well.
To be specific, lets say that one of the solutions of the differential
equation is a model for some (time in years, mass in grams of remaining
radioactive carbon 14) data points. Here are some questions:
One calculus book in which I looked refered to the "k" constant as the
"rate of decay." Is this correct?
The "y'" quantities would have unit designation "grams per year." What
then are the unit designations for the "k" constant and the "y"
quantities?
If you find that answering these questions takes some thought, perhaps
you will see why I think the new emphasis on applications and multiple
ways of understanding challenges us as AP calculus teachers.
Sincerely,
Richard Sisley
keckcalc@earthlink.net
|
|
