Math Post: Pre-Calculus

The idea of the first two weeks in this school year of teaching calculus is to spend some time doign pre-calculus. 

 

In previous years I didn't bother with that and lunged right into the calculus content.  One of my colleagues at that time went against the grain and actually did pre-calculus at first.  As a result, he got behind the rest of us for a substantial time, but his argument was that he work would pay off later.  Actually he was spot on and I'm now applying that wisdom.

 

One of the key pre-calculus topics I want to cover is that of function compositions.  In so doing, I figured out that I can help students deal with the dreaded *chain rule* that they will encounter later on.  The chain rule is a real hurdle for stuents in an intro calculus class, and they tend to hit the wall once introduced to it.

 

However, it just occured to me that doing function compositions where each function has a *linear* slope, or a constant slope, can introduce the major conceptual thrust of the chain rule without actually doing any calculus.

 

It is then a simple matter of being able to show that when you compose the two linear functions and look at the slope of the composition, the effect is to multiply the slopes of each function.  That is the thrust behind chain rule, except in calculus of course the functions are not necessarily linear. 

 

Why this isn't found in any pre-calculus textbooks is a mystery to me, as it seems a perfect thing to include.