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.