In a few days I’ll be setting out to visit some more of the Gulf Islands. One of them doesn’t allow cars, and is powered entirely by generators. At least two lack bank machines. One is rumoured to have “honesty stands” where local artists leave their work unattended and trust visitors to pay for what they take. Only one has more than a thousand full-time residents. I may at some point pop into a cafe for a few minutes on a rainy day to avail myself of the dial-up connection on the island’s IBM 486 , but don’t hold your breath. I will be making notes on postcards, so holler if you want one.
In the meantime, some of you may be able to help me with a project I’m thinking about working on one of these days book I have no excuse not to work on now that I don’t have a pesky job to worry about. A bit of background: for the last five years, I have worked at an academic summer camp for mathematically gifted high school students. One of my favourite classes to teach – and one of the most popular among the campers – has been one that I call “Calculus Without Calculus”.
Those of my readers who know me in real life are well aware that I…well, I don’t hate calculus, so much as I think that calculus doesn’t need me to love it. Calculus gets more than enough attention in the thousands of high schools and universities that inflict it upon every other student that passes through their doors, the overwhelming majority of whom don’t learn it properly and wouldn’t ever use it again even if they did.
Nevertheless, calculus is a natural choice for students who have not learned to think mathematically: for all the terror it strikes in students’ hearts, it’s one of the easier branches of mathematics to reduce to mindless algorithms in a low-level course. Need to maximize some quantity? Set a derivative to zero, and solve. What were we trying to do again?
Calculus Without Calculus is a collection of methods of solving typical calculus problems without taking a single derivative. The two main methods involve inequalities, or exploiting geometric properties of figures, particularly symmetry. The old “maximize area with given cost of rectangular fence” problem? You can complete the square, take a derivative – or you can apply the AM-GM inequality.
The question about getting the best view of the statue on a pedestal, that appears in the chapter on inverse trig functions in every single calculus book? Solvable using elementary circle geometry that could be found in every grade ten math text before it was decreed that geometry should no longer be taught. There are tons of these. I know of three calculus-free methods of finding tangents to ellipses: one using transformations of circles, another using the Cauchy-Schwartz inequality, and one using projective geometry.
A few months ago, I was reading some fiction on the Vietnam War – One To Count Cadence, by James Crumley – and one of the characters mentions in passing that he was able to solve the ladder-around-a-corner problem (you know the one) without calculus. I struggled with that for a long time before a camper provided the key insight. A quick application of Holder’s Inequality, and the result falls right out. (Except that now I’m trying to recreate it. Damn; this is going to keep me up again.)
I’m sure there’s a lot more to say on the topic, and I’d like to write a (short) book on the topic. What I’m looking for: book recs. In particular, recommendations for good books on problem-solving, which tend to spend a lot of time on funky inequalities. I’m especially interested in geometric and otherwise intuitive proofs for the old standbys (AM-GM, Holder’s, C-S…) , and lots of examples – both of single variable problems that one would see in calculus texts, as well as multivariable ones that are a lot easier to solve without calculus.
One of my favourite books of this sort is Problem Solving Through Problems, by Loren Larson, the talented mathematician and educator who first introduced me to this stuff. If you’ve got any recs, or anything even slightly pertinent, I’d love to hear them – post them below so I’ll have something to check out when I return to this big island.
My other reason for posting this, of course, is that now I’ll feel horribly guilty if I don’t actually have anything to show for this in a few weeks.