As promised, here comes the worksheet for drugs in waterways. As I mentioned yesterday, this was the worksheet hardest to construct so far.
I did a lot of reading on pharmaceuticals in waterways and I liked the paper "Attenuation of Wastewater-Derived Contaminants in an Effluent-Dominated River," from 2006. It discusses the importance of biodegradation and photolysis in breaking down naproxen (brand name Aleve) in the Trinity River, which is full of water from a wastewater treatment plant much of the time. It's got some great graphs and a good set of supplementary calculations and information, which allowed me to pick out the pre-calculus problem inherent in the analysis.
The worksheet is set up as three pages that might be usable somewhat independently: the first page sets up the problem and asks about domain and range, the second asks about limits, and the third asks about piecewise functions. When used in a classroom, you're going to want to know that students will need calculators and could perhaps profitably use graphing software. (A little hand-graphing is good for them, though, as it's so useful later on.)
- continuity, and
- piecewise functions.
First page: challenge them to think symbolically about domain and range, and physically. Negative depth in the river doesn't make sense. They don't need to know the values of or , though they'd prefer to!
Second page: I've deliberately given a limit (as z goes to zero) that will not be easy using limit rules (unless there's a trick I haven't seen). It is very easy using L'Hopital's rule or a power series expansion of the exponential so tell students they will soon have tools to do this rigorously. For now, they can use numerical calculation and grapple with the idea of "limit." Students also already have the value at zero, given implicitly in problem 3 by the scientists. We have a removable discontinuity at , always a somewhat challenging concept for students. The limit as z goes to infinity can be done with limit rules.
Third page: Piecewise functions! Some students like 'em, others hate 'em, and many mostly understand but don't know how to write 'em. Coach students through correct 'mathematical grammar' for piecewise functions. There's also a chance to practice writing nice coherent sentences explaining why a function is continuous at the end of the page. At every university or college I've taught at, students have wanted more examples of how to write mathematics -- this is a good chance!
Along the path to this worksheet, I learned why water is blue!
Ok. Off to enjoy some sunlight now. I'm going to wear my mineral sunscreen so that I don't have to worry about avobenzone sloughing off of me into my local waterways... Once limits and continuity are covered, we're only a few steps away from the definition of derivative.