# Essential Pre-Calculus

Some more "why you Twitter" evidence here. Our gracious host for TMC14 is teaching Pre-Cal and every so often wants to know connections to Calculus. Maybe you do to?

I've taught Pre-Cal for a while now and Pre-Cal/Calculus concurrently for several years. I have learned so much about how to teach Pre-Cal by observing what really happens the next year (assuming AP Calc specifically). Sidenote: throw your on campus Stats teacher a bone and carve out some time for a Stats primer too, yeah?

In no particular order, here's what I have found to actually matter in Pre-Calculus:

### Rational Functions

The concept of discontinuity comes up here. The key points are horizontal/vertical asymptotes and removable discontinuities (holes). You can talk about continuity in general here. I describe holes as glitches that can be compensated for (there's a problem at x=1, but is that true at x=0.99999?).

### Piecewise Functions

Strictly for the notation and the idea that functions can have multiple behaviors. Piecewise functions aren't particularly practical things on their own other than giving you an opportunity to talk about increasing, decreasing, constant, and undefined behaviors over intervals. Piecewise functions pop up when determining if a function is continuous/differentiable at a point. It's also quite common to use piecewise functions to demonstrate the additive properties of integrals.

Trig is not the focus of Calculus, but it is an assumption that a student has a working knowledge of trig functions. They should know that sin/cos/tan/sec are things with a unique class of behavior. They should be able to spot the graphs of those four easily. They should also be aware that sin(π/4) and other trig values of special angles can be simplified. This is about all the unit circle you need to know:

While students are in Pre-Cal with me they have free use of a full unit circle diagram. In Calc that's not the case and I have them learn this particular subset. It covers 95% of the random trig values you need to apply along the way. Knowing how to find sin(4π) or cos(-π) is a smart move too.

### Exponential/Logarithmic Functions

Exponentials and natural logs are real late-game Calculus, the curriculum has an affinity for them when it comes to separable differential equations. Also a fan favorite with volume problems where it may be necessary to solve something like y = e^(2x) for x. The general use case is a need to know how to integrate/derive functions that make use of them. I use these functions as a part of some algebra review, really just as a "hey, remember these things?" Basic knowledge of their graphs is useful.

### Polynomial Sketching

Function behavior and the connections between f, f', and f'' are the bulk of Calc AB. While it is not necessary to try and introduce the idea of a derivative, it is helpful if a student can roughly sketch f(x) = x (x - 3) (x + 5)^2, note the type of zeros present, and see how the function being even or odd helps them with the sketch. Solving some kooky polynomial for all its real/imaginary roots is not a thing that will happen.

### Limits

If you want to save your local Calc teacher some time, covering Limits is a great way to spend a week and a half at the end of the year. It's a good time to bring back continuity from Rational Functions and discuss how Limits are concerned with the "neighborhood" around a function. Removable discontinuities as merely a "glitch" gets justified here since we can now talk about how right-side and left-side behavior agree, letting us "remove" the offending point from the discussion. There is not need for an exhaustive study. Finding limits graphically, algebraically, one-sided, and at infinity (end behavior! horizontal asymptotes! same thing!) does the job.

### Stuff That Only Matters to Calc BC

If you cover all of the previous topics, you will still have a lot of time during the year. And you might be wondering "what about [blank] which so and so SWEARS we have to teach?" Well, it's probably in the list of things that only matter to students taking Calc BC. Roughly:

Vectors, polar coordinates, parametric functions, summations, converging/diverging series, partial fractions, and inverse trig functions.

Before you freak out, I am not saying you are free to ignore those topics completely. Among them are some of my favorite things to teach in Pre-Cal. Depending on the size of your potential Calc BC population, you aren't doing any harm in condensing or focusing these topics into something you like to teach. If partial fractions leave a bad taste in your mouth and someone's trying to tell you students will suffer if we don't spend three weeks on it, kindly tell them to chill out.

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