On Concavity, Calculus, and Crescents

Concavity characterizes so much of the world — from the distribution of sizes of animal species, to the potential return-on-investment from small vs large companies, to the temperature of a warm body in a cold room, to the results of sports training.

To explain concavity, I’ll talk about training for a race.

Training for the Mile Race

If you’re like me and haven’t slipped into your jogging gear for a few months (ok, maybe a few years), then the first time you hit the track you will be quite slow. My mile time had dropped to 8:31 — and I used to run sub-6! Oh well. The good news is that I was quick to improve. After only a week my time dropped to 7:44. That’s what I call “low-hanging fruit”. Imagine if an Olympic athlete could drop their time by 9% in a week!

I continued to train and here were my times from each week when I tested myself at the end of the week.

The story gets much more boring from here on out because I had left the land of low-hanging fruit and entered the land of diminishing returns. I was still getting faster every week, but not by as much. And so it is with Olympians–months of intense training often lead to speed gains of less than a second.

Calculus!

This whole story can be summarized with just two mathematical statements about the function f which maps from [tex]\{ \mathrm{amount of training} \} \to \{ \mathrm{speed} \} [/tex].

1. f ' > 0 — training makes you faster —
2. f ” < 0 at a decreasing rate.

As well as everything began…so badly did it end

But it’s not the end of my story. I kept up my regimen of running right through the summer, when it was warm. But as the weather worsened, my willpower waned. When I wouldn’t work out, I would get worse – weaker, wimpier. By winter I was worthless.

And so, in just exactly the opposite pattern as my times had leapt up in the spring and gradually improved over the summer, so did they, come fall, gradually start getting worse f ' < 0, and then plummeted back to sloth once November hit. By Black Friday, I was about as fast as a jelly donut.

My up-then-down crescent was exactly, exactly a concave function. Fast up, slow up, slow down, fast down. Unfortunately.

Well, at least I was fast… once. Oh, negative f ”, I’ll best you one of these years.

Conclusion

Concavity is common, and when couched as calculus, can be condensed to a curt comment: f ” < 0.

About

Chris Waggoner is a mathematical psychologist from Indiana, USA. He writes the Human Mathematics blog, which discusses applications of math to human behavior, emotion, thought, and imagination.

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