## Bad Optimization Problems

I thought that Jack M made an interesting comment about this question:

There aren't any. There may be situations where it's possible to apply optimization to solve a problem you've encountered, but in none of these cases is it honestly worth the effort of solving the problem analytically. I optimize path lengths every day when I walk across the grass on my way to classes, but I'm not going to get out a notebook and calculate an optimal route just to save myself twelve seconds of walking every morning. Mathematics beyond basic arithmetic is simply not useful in ordinary life. But I'm not sure if that's exactly what you mean. – JackM

To some extent, I agree with this comment. With few exceptions, mathematics beyond basic arithmetic is simply not useful in everyday life. Students know this, and you'll have trouble convincing them otherwise.

Because of this, I've always found "everyday"-style calculus problems a little artificial. Consider the following problem from Stewart's *Calculus: Concepts and Contexts*.

A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

The proper response to this question is: who cares? Is there any reason to calculate this length precisely? Why would anyone ever use calculus to compute this? If you have an actual building and an actual ladder, you could just try it and see if the ladder fits. If you don't have a specific ladder in mind (e.g. you are buying a ladder), the thing to do would be to *draw* the situation on paper and then use a ruler to estimate the minimum length. Of course, it's neat that you can use calculus to solve this problem precisely, but this is more of a curiosity than a legitimate application.

Chris specifically mentions the farmer fence problem, the wire-cutting problem, and the Norman window problem as not relevant to the students' lives. I agree—none of these problems are relevant. But it's not because the students aren't farmers, or wire-cutters, or architects. Even in a class full of future farmers, the fence problem would still be bad, because *farmers don't use calculus to plan their fences*.

## Good Optimization Problems

What calculus *is* useful for is science, economics, engineering, industrial operations, finance, and so forth. That is, it's useful for *all the things that make our society run*. Most students who take calculus at a university are planning to go into one of these fields, so calculus will be relevant in their lives—specifically in their future studies and in their professions.

Here's something that's closer to a real-life optimization problem:

When a critically damped RLC circuit is connected to a voltage source, the current $I$ in the circuit varies with time according to the equation
$$
I \;=\; \biggl(\frac{V}{L}\biggr)te^{-Rt/(2L)}
$$
where $V$ is the applied voltage, $L$ is the inductance, and $R$ is the resistance (all of which are constant).

Suppose an RLC circuit with a resistance of 30.0 volt/amp and an inductance of 0.400 volt·sec/amp is attached to a 12.0-volt voltage source. Find the maximum current that will occur in the circuit.

This is at least close to something that a physics or engineering student might actually come across in their future studies. It's real in a way that the farmer fence problem isn't, and even students who don't plan to study physics can sense that this is a legitimate application. (By the way, if you have good students, you might even ask them to come up with a formula for the maximum current, without giving them specific numbers for $V$, $L$ and $R$. This has the advantage that it can't simply be solved using a graphing calculator.)

Of course, this isn't actually a *constrained* optimization problem—it's just an optimization problem. I'm not actually aware of any place in science that simple constrained optimization problems arise, although there are examples from economics (maximizing utility), finance (optimal portfolios), and industrial design (e.g. shape of a can type problems). When I cover constrained optimization in calculus, I usually stick to industrial-type problems (best cans, best shipping crates/boxes, best pipeline across a river, etc.), but that's probably just because I don't know enough about economics or finance to make up problems that involve them.

Finally, I should mention that I've never found the optimization portion of Calculus I particularly compelling. It's good to introduce the idea of optimization, but setting the derivative equal to zero isn't actually a very useful optimization technique by itself. It only really works for simple formulas—for anything complicated it just replaces one essentially numerical problem (finding the maximum of a function) with another (finding roots of a function). I agree that it should be covered, but it's far from the most important application of calculus.

1I wonder how Dr. Pangloss would answer. We live in the best of all possible worlds, but there is not a really good answer to this question. But Calculus of Variations is a really cool subject! – user52817 – 2015-05-11T18:31:07.207

@JackM: re:

I'm not going to get out a notebook and calculate an optimal route just to save myself twelve seconds of walking every morningSounds like someone who's into bounded rationality! (Or at least someone wiser and more cognisant of the finiteness of lifespan than me. You wouldn't believe the number of pennies I've salvaged for the price of a pound (each) or my love of premature optimisation.) Incidentally, if you would save 12 seconds each time as opposed to savingon(i.e., part of) 12 seconds, and if there are enough instances of it, I might think the savings to be worth it. – Vandermonde – 2015-11-25T03:40:13.6601Whoops.... I stopped myself doing a cost-benefit comparison and estimating the minimal number of days required to break even, and subsequently almost did a cost-benefit analysis of cost-benefit analyses. – Vandermonde – 2015-11-25T03:43:11.600

Since you are asking for examples related to calculus, the following discrete examples are only a comment: 1. Planning a trip in a city where there are mulitple ways by bus/subway/train to get from A to B. 2. Almost irrelevant, but something I encountered as a problem as a teen: Given a record tape of 2x45mins and some songs of different lengths (which aren't all required to be on the tape) - what's the best way to put songs on the tape with as few empty space at the end? – Roland – 2014-04-11T15:05:53.520

9There aren't any. There may be situations where it's possible to apply optimization to solve a problem you've encountered, but in none of these cases is it honestly worth the effort of solving the problem analytically. I optimize path lengths every day when I walk across the grass on my way to classes, but I'm not going to get out a notebook and calculate an optimal route just to save myself twelve seconds of walking every morning. Mathematics beyond basic arithmetic is simply

notuseful in ordinary life. But I'm not sure if that's exactly what you mean. – Jack M – 2014-04-11T15:45:16.66714@JackM That's an extremely depressing way to think about mathematics, and I'm sorry you think that way. – Chris Cunningham – 2014-04-11T16:07:39.287

10@ChrisCunningham What's depressing is the notion that mathematics

hasto be relevant to practical life in order to be interesting. – Jack M – 2014-04-11T16:52:56.0335I think you've misjudged my agenda here. Mathematics is beautiful on its own, but optimization problems could additionally be relevant. It sounds like someone once told you that the purpose of mathematics is to apply to the world, and you were very displeased. I don't think anyone here is advocating such a position -- I'm certainly not! – Chris Cunningham – 2014-04-11T19:00:49.853

@JackM, The simple act of catching a thrown object is something we can easily find in our daily lives, and involves a whole bunch of math... but our brains are hard-wired to perform those calculations without thinking about them (and without even necessarily

knowinghow to perform the math formally) – Brian S – 2014-04-11T20:47:08.623I think there are a lot of great points made and it's is great that conversations like these are had regarding math and application. As one user pointed out, a lot of optimization problems have no "real-world" application and it is difficult getting a student on board and excited about math with these problems. However, I think the point with these problems and with math in general is it gives us valuable critical thinking and problem-solving skills. [Mod note: see below for the rest.] – James S. – 2017-09-26T18:28:42.780

[Continuing the comment above converted from an answer] So although the actual problem and topics may not have significance to the "real-world", what is significant is the cognitive development and asssociated skills gained from performing these problems. – quid – 2017-09-27T10:42:14.253

I agree with James. I'm not some super design engineer but have a pretty good applied background (from conversation here, more than most of the forum). And I benefited from simple optimization problems. For one thing just the intuitive drive to THINK about optimzation is helpful. So many people don't have a feel for equilibrium points. FWIW, it's not clear to me if Chris was looking for optimization like calculus of variations, like linear programming, or just a min/max calculus problem. Regardless, I think there are plenty of physical and economic examples. I wouldn't stress over slight simpl – guest – 2017-09-27T04:07:33.507

(1) First of all, people do ring in fences, make windows, and do mechanical things. There is a tendency in academia to think everyone works for Google, but the US has been having a manufacturing renaissance in general and in particular in oil, gas, and chemicals. (2) Something good about these examples is they are intuitive. If your example involves something hard to understand as a topic such as maybe circuits than that is not as helpful if you do something that everyone can easily visualize such as a fence. (P.s. quid's continuation above was of James's comment, not of mine.) – guest – 2017-12-08T20:02:48.060

@ChrisCunningham Ok, done. – user11235 – 2014-04-19T20:22:25.823