How I Teach Calculus: A Comedy (The Tragedy of Optimization)

Oh, optimization. The self-proclaimed application of differential calculus. You know, finding the most or least of something, very useful I should say, very useful.

Want to know how many employees to employ? Optimization!

Want to know how much area some arbitrary amount of fence (cough, budget) can enclose? Optimization!

Want to know the dimensions of the most areaed Norman window?

No.

Shhhh… Optimization!

What is optimization? [find the biggest or smallest when you have a (some) knob(s) to fiddle with]

  1. Find a function for the thing thing you want to optimize, and make sure it’s a function of the thing you have control over. This may take some algebra, and is generally where students get stuck. (i.e. A=x^2. Area is what you want to optimize, and x is the thing you can control, in this case the side length of a square.)
  2. Take the derivative of that function. (in this case. dA/dx = 2x)
  3. Set the derivative equal to zero, which is pretty much the only thing you do in Caclulus I for about a month. (0 = 2x) Why? Because if the slope (read: derivative) is zero, the function must be doing something interesting, like turning around.
  4. Solve for x. (x = 0, or we have a minimum area when we don’t have a square, duh, and there’s no such thing as the biggest square).

But now, honestly, if I want to know any of those things I would just plot a graph and point to the hill or valley. Sure, I might make some domain errors, but there are ways to avoid those errors that don’t require the algebra of the Norman-window problem.

In fact. Let’s play a game. I’m going to solve the Norman-window problem two ways and time myself; analytically and graphically.

It only took me three hours by hand! Oh, minutes, but still!

45 second of algebra and plugging into Grapher. 15 of those seconds were used sipping sweet tea.

All I’m getting at here is that if we want kids to respect math, we need to stop making it so silly. If there’s a good way to do something, let’s use it.

My ever-burning quest is for the motivation of new techniques is thwarted during optimization. I often force students to solve problems analytically that would be much easier using another method. This is, of course, fine if you’ve already motivated the necessity of a strategy and are in the let’s-get-fast-at-it stage, but no one needs help doing that anymore (thanks, Sal!).

I’m interested in the part that the websites, remediation software, and all the other blighted silliness can’t help you with: The sell, and right now my sell for analytical optimization is crap.

Here’s where my mind goes: if I don’t get the idea, how can I know how to use Grapher? Isn’t that showing proficiency with optimization? Maybe. But maybe I’ve just memorized the process. Often this is the only thing that students get out of our repeated examples.

What’s worse is that the graphical solution doesn’t even require calculus; I can just point at the highest bit with my hammy finger, and because I know it’s a parabola, I know there’s only one maximum.

I guess what I’m looking for is a situation where analytical methods are better than graphical methods. I.e. Where does Grapher fail?

“Do it this way, because that’s where we are,” isn’t good enough. It’s not good enough because all it does is prepare out students for the ACT. Honestly, I’m done with that. I’m beyond prepping my students for that garbage.

I want extensibility.

I want fearlessness.

I want shoulders thoroughly brushed off.

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