How to Do Well in AP Calculus

Principals, managers, and students love asking this question, lifting up the same rock hoping a better answer is there this time. The natural state of the universe is to use the least energy possible and so we look for efficiency, shortcuts, some magical panacea that will rid us of the ills of our humanity. The reality is simple: baby, you're the problem.

It starts with the foundations. You're going to take the course and honestly you're probably going to learn a fair amount of calculus just fine. You'll understand the concepts, you'll say oh, rate of change, that's a derivative, or area under the curve, integral. You'll go to set up the problem and all will be well and good until you trip over your shoelaces because you can't do simple algebra.

Step 1: Memorize your unit circle. Know your trig rules and log rules. Recognize common function graphs and their domain and range. Knowing these 'tricks' makes the seemingly difficult trivial.

Step 2: actually do step 1. This is the part where you nod along and then don't do it. I know because I'm guilty too. Nobody wants to drill trig identities. It's flossing. Everybody knows they should, nobody does, and then one day the dentist looks at you with that particular disappointment reserved for children and the chronic procrastinators. Every year I watch some kid sail through an optimization problem, draw the box, label the sides, write the volume function, take the derivative, a thing of beauty, I could frame it, and then x³ − 4x = 0 and they divide both sides by x. Just like that. Zero, poof, gone. Sent to the cornfield. (Never divide by the variable. Never. I don't care what your heart tells you. Factor it out. The root you throw away is always the one you needed.) The calculus was fine. It was the eighth grade math that got you.

Euclid, so the story goes, was asked by King Ptolemy whether there wasn't some quicker way to learn geometry than slogging through all thirteen books of the Elements, him being a king and presumably busy with king things. Euclid told him there is no royal road. Twenty-three hundred years, a printing press, the internet, and several trillion dollars of GPUs later, still no road. Lots of people selling maps to it though. (Some of them have very good ads.)

Step 3: do the problems. (Preferably here, my wife would appreciate that) Not watch the problems. Not read the worked solution and go "yep, makes sense," which is the academic equivalent of watching the Food Network and assuming you can now make a soufflé. Do them. With a pencil. On paper. Get stuck. Stay stuck a while. Being stuck is not a sign you're doing it wrong, being stuck is the whole damn point, it's the feeling of your brain actually building something instead of bobbing along like a dashboard hula girl. The answer key is for afterwards. If you're flipping to the back after thirty seconds you're not studying, you're transcribing.

And mix them up. Homework comes in neat little boxes. Section 4.3, related rates, twenty related rates problems, and by number twelve you could do them in your sleep and some of you do. The AP exam is not so polite. It doesn't announce itself. It just sits there, a word problem about a conical tank leaking at some ungodly rate, daring you. Knowing how to do a related rates problem and knowing that this is a related rates problem are two different skills. Dig up September's stuff in February. It's coming back in May.

Step 4. Show your work. I will die on this hill, I've already bought the plot. Oh how I loathed this as a teen but theres nobody zealous like the converted. On the free response the answer is worth a pittance and the reasoning is worth the rest. "Relative max at x = 2" earns you the square root of nothing. "Relative max at x = 2 because f′ changes from positive to negative" earns you the point. The people grading these are called Readers, not Mind Readers, and as someone who has graded a stack of tests at eleven at night I can tell you generosity is not in surplus. Write your units. If they ask for gallons per hour and you write 12, you've told me 12 of something. Bananas, maybe. I'll be honest I'm lazy about this one too, it's 12 somethings and I don't particularly care. Fortunately for me I've already passed the test. Bummer for you. C'est la vie.

Now. The machines. It's a screed, of course this part was coming. Step 5. Learn your calculator cold, because fumbling for the intersect function while the clock runs is its own special circle of hell. But check what it tells you. It doesn't know anything. It does exactly what you told it to, and if you told it wrong it will do the wrong thing very fast and with total confidence. A tank that holds fifty gallons does not drain at three thousand gallons a minute. That's not a leak, that's the Hoover Dam.

As for AI, sure if you've got self control (I'm doubtful, not because I'm doubtful of you but because I'm doubtful of all of us like I said we all want to find the easy path and it's just too damn easy to say just show me the answer, and move along), use it. Ask it why the chain rule works. Ask it for ten more problems like the one that just stole your lunch money. Ask it to explain like you're twelve, then like you're a pirate, or a princess, whatever gets it in your head. What you can't do is have it do the homework, because nobody, and I mean nobody, has ever cared about your homework. I'm not reading it for the plot. The homework is the treadmill. You can pay someone to run on it for you, but then you've got to live with the results.

Step 6: Ask questions. Your teacher wants you to, I promise, even if we look tired (we are tired, but not when you ask questions, then we're juiced). The kid who stays after to ask why the integral of 1/x is ln|x| and not ln x has made my whole week. Gold stars for life. The rat-bastard kid who sits silently confused from October and confesses in March is why I'm balding. It's not like it was a secret, I grade your tests. How many D's does it take before you recognize, 'maybe I'm not doing so hot'?

Step 7: Prepare. Don't cram. Twenty minutes a day beats a five-hour Red Bull vigil the night before, and it isn't close. Sleep. On test day skip what's got you stuck and come back to it, and never leave a multiple choice blank. There's no penalty for guessing. There should be, there used to be, but lucky for you we've gotten soft. Blank is a zero. C is a prayer, and prayer is greater than zero.

None of this is new or particularly unique to math. Your parents say it. Your parents' teachers said it, their teachers said it, Euclid said it to a king who probably wanted to have him flogged for it. We keep waiting for the thing that makes all this unnecessary, the calculator, the internet, the chatbot, the implant they'll stick in your brain (it's a trap to sell you ads), and it keeps not coming, because the thing doing the learning was never the machine.

It's you, baby. It was always you.