I Got Something Cool Out Of My Calculus Class

I've long understood the concept of composite functions, but never really had a good practical application of the idea that didn't involve mere multiplication of numbers.  This past weekend, however, I completed my first test in calculus class, and there was a problem on it to state the surface area of a cube as a function of the cube's volume.

The problem was written just that way, but since we'd covered composite functions for one of our lessons, I recognized immediately that that was a great problem that could be solved using composite functions.  Since I covered composite functions in my pre-calculus class, and my students are taking a test later this week that includes a problem about composite functions, I thought the problem from my calculus class would be a great one for my pre-calc test.

I couldn't just say to use composite functions to state the surface area of a cube as a function of its volume.  No, I had to break it up into bite-sized chunks and lead my students towards the solution.  (I consider this to be part of what's called "good teaching".)  My calc instructor can assume that his students have all learned about composite functions before and can rush through the review and assessment; since my students are probably learning the material for the first time, I need to be a bit more "guided" both in my instruction and in the assessment.  I wrote the test today and included a simple f(g(x)) and g(f(x)) composition on it, but thought I'd use the surface area/volume problem as a bonus problem.

I broke the problem up into 3 parts:  surface area as a function of side length, side length as a function of volume, and then the composite function that provides surface area as a function of volume.  The next two parts guided the students to check their work, to see if their formula for SA(V) actually gave the correct answer in the case of a cube with very simple dimensions.

Nifty applications like this is one of the reasons I'm taking the calculus course.