Calculus books are supposed to start with a review of PreCalculus. I think that’s a Lemma or something.
Here’s the problem with starting with PreCalculus – the people that need it don’t pay attention to it because they don’t know they need it yet. For instance – if you have this fraction:
6 x + 84 ---------- 2
You know you can’t just cancel the 6 and the 2. Well, maybe you don’t know it – but you certainly don’t have the patience in a Calculus class to write out:
2( 3 x + 42) --------------- = 3 x + 42 2
You think you know what you’re doing and then you get to the quotient rule and you substitute in for the appropriate things in the formula
(f/g)' = f'g-g'f ------- g^2
and you go ahead and end up canceling the g’s. Anyone who has graded calc homework or exams has seen that a bunch of times.
Most of what trips people up in Calculus is not the Calc but the PreCalc. But they aren’t ready to listen to the PreCalc the first week of Calculus because they think they know it already. After all, they’re in a Calculus class. They wouldn’t be in a calc class if they needed precalc.
So my preference is JIT PreCalc (Java programmers recognize that as “Just in Time”). When I’ve got your attention because you perceive a need, you will be ready for me to review the PreCalculus in place and not in this separate module that you’ll tune out.