I am a student who would like to become a teacher, so I am currently following courses in education. One of the things I learned, is that students like authentic, realistic problems. However, much of the extremum problems (in one variable) appear very 'fabricated'. I only know a few 'realistic extremum problems':

  1. minimizing some cost function
  2. maximizing some return function
  3. maximizing the volume of a box.

A realistic example of the third problem can be found here, where the volume of a box is maximized. This box is the kind of box you receive a package in: it is closed, with overlapping flaps.

With respect to multivariable functions, I can think of one extra example: determining a regression line.

However, this list seems rather short. Can anyone give extra examples of real-life situations where we wish to maximize/minimize some sort of function? (For my class, single variable functions are the most interesting, but I equally like multivariable examples.)

  • $\begingroup$ Could you summarize the "realistic box problem" that you link to? That link requires Java in the browser, which many people (including myself) disable for security reasons. $\endgroup$ – Rory Daulton Oct 27 '18 at 10:15
  • $\begingroup$ @RoryDaulton: Here is that problem: "A sheet of cardboard is rectangular, 14 inches long and 8.5 inches wide. If six congruent cuts (denoted in black) are to be made into the cardboard and five folds (denoted by dotted lines) made to adjoin the cuts so that the resulting piece of cardboard is to be folded to form a closeable rectangular box (see figures below). How should this be done to get a box of largest possible volume?" $\endgroup$ – Joseph O'Rourke Oct 27 '18 at 10:51
  • $\begingroup$ @JosephO'Rourke: Thanks! Is there any way to show the figures with the cuts and folds and final box? $\endgroup$ – Rory Daulton Oct 27 '18 at 11:23
  • $\begingroup$ 1 covers a lot of ground: number of shale wells in a section, number of salespeople, hours to keep a store open, etc. $\endgroup$ – guest Oct 27 '18 at 13:22
  • $\begingroup$ There is an entire scientific discipline called Operations Research that grew out of military logistics in WW I. It sits somewhere between Applied Maths, Statistics, Logistics, and more recently, Computer Science. It is essentially concerned with Optimization, which is basically finding Extrema, i.e. the "best" or "cheapest " solution Basically, every OR problem is an extrema problem. $\endgroup$ – Jörg W Mittag Oct 27 '18 at 21:33

Here are some optimization problems that were harder than a simple homework problem:

Notice that in these examples, more effort went into designing the functions to be optimized, and in choosing the parameter to be varied, than went into performing the calculus.

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  • 2
    $\begingroup$ Perhaps one could invent a two-variable function $g(x,y)$ of math-effort $x$ vs. optimization set-up effort $y$ then analyze $g$ subject the to the set of all word problems. If this set was compact then we could optimize $g$. Perhaps the choice of $g$ ought to depend on the course of major study for the student ? $\endgroup$ – James S. Cook Oct 27 '18 at 16:01

Some example types:

  1. Minimizing potential energy of any realistic physical system. Examples:
    • 0D: The point where a rolling ball/flowing water might* come to rest (*might not, if momentum carries it past the low point).
    • 1D: The curve described by a hanging chain/flexible rope (in equilibrium).
    • 2D: The surface of a soap film (in equilibrium).
    • Generally, anything that doesn't move or change: look at the world and wonder. [For instance, how hard do you have to push or jump to move your classroom building out of its potential well?)
  2. Maximizing a utility function subject to a budget constraint.
  3. Fermat's principle: Light takes the path of least time.
  4. Machine learning: minimize the loss function of a particular learning task. (It and the subfield, data mining, are hot fields right now.)
  5. In a network of resistors connecting voltage $V_A$ to voltage $V_B$, the voltages at the nodes of the network minimize power dissipation.
  6. Space trajectories that minimize fuel use (Mission Design).
  7. When using the central difference formula $[f(x+h)-f(x-h)]/(2h)$ to approximate the derivative $f'(x)$ using floating-point numbers with a machine epsilon of $\epsilon$, what choice of $h$ minimizes the relative error of the approximation? [Note: This is a somewhat ill-defined problem, since it may depend on $f$ and $x$.]
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Some physics examples: --

Given that the range of a projectile is $R=(v^2/2g)\sin\theta\cos\theta$, prove that the maximum range is achieved for $\theta=45$ degrees.

An electrical transmission line of resistance $x$ is in series with a load of resistance $y$. For a fixed voltage $V$, the useful power dissipated in the load is $P= V^2y(x+y)^{-2}$. Show that this is maximized, for fixed $x$, when $y=x$.

Two atoms will interact through electrical forces between their protons and electrons. One fairly good approximation to the total electrical energy $U$ is the Lennard-Jones formula, $U(r) = k\left[\left(\frac{a}{r}\right)^{12}-2\left(\frac{a}{r}\right)^{6}\right]$. Infer the units of $a$. Find the equilibrium value of $r$, which occurs at the minimum of $U$, and show that your answer has units that make sense.

Sometimes doors are built with mechanisms that automatically close them after they have been opened. The designer can set both the strength of the spring and the amount of friction. If there is too much friction in relation to the strength of the spring, the door takes too long to close, but if there is too little, the door will oscillate. For an optimal design, we get motion of the form $x=ct e^{-bt}$, where $x$ is the position of some point on the door, and $c$ and $b$ are positive constants. (Similar systems are used for other mechanical devices, such as stereo speakers and the recoil mechanisms of guns.) In this example, the door moves in the positive direction up until a certain time, then stops and settles back in the negative direction, eventually approaching $x=0$. This would be the type of motion we would get if someone flung a door open and the door closer then brought it back closed again. (a) Infer the units of the constants $b$ and $c$. (b) Find the door's maximum speed (i.e., the greatest absolute value of its velocity) as it comes back to the closed position. (c) Show that your answer has units that make sense.

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  • $\begingroup$ The range example can be done without calculus, using $\sin \theta \cos \theta = \dfrac{1}{2} \sin (2 \theta)$. $\endgroup$ – BobaFret Oct 27 '18 at 22:08
  • $\begingroup$ It's not hard to make the range problem much harder. Just make the initial and final altitudes differ. Then the double-angle formula will be no help... but I like the given problem. $\endgroup$ – James S. Cook Nov 26 '18 at 1:14

There are many volume-of-a-box questions. I like this one, simpler than what the OP cites:

Given a rectangle, cut out squares from the corners so you can fold it up to a box, without a top, of maximal volume.

The rectangle might be specialized to a square, as below. See also The Math Forum.

          (Image from patrickJMT YouTube video.)

In response to @RoryDaulton, here is the box problem to which the OP @Student points: BoxFlaps

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  • $\begingroup$ Thanks for showing me the box problem. I'll try using it in my calculus class--it is more realistic than the box problem I have been using. $\endgroup$ – Rory Daulton Nov 18 '18 at 0:16

Many important machine learning algorithms, such as training a neural network, require solving a large optimization problem (essentially tuning the neural network weights to minimize the classification error on a training dataset). The whole deep learning revolution is powered by optimization.

In radiation therapy (for cancer treatment), an optimization problem is solved to find optimal beam intensities that target the tumor most effectively while sparing healthy tissue.

Magnetic resonance imaging (MRI) works by solving an optimization problem that finds an image of the human body which is most consistent with the measurements obtained by the MRI machine (and also consistent with prior knowledge we have of what images of the human body should look like). Other medical imaging techniques also use optimization for the image reconstruction step.

In Statistics, many important optimization problems arise as special cases of Maximum Likelihood Estimation. Linear regression is a classic example with a ton of applications.

The book Convex Optimization by Boyd and Vandenberghe discusses many applications of convex optimization.

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Here is a real life example with millions of dollars (maybe billions) depending on the answer:

http://phx.corporate-ir.net/phoenix.zhtml?c=197380&p=irol-presentations (MAY18, IR presentation, page 20).

The key insight is to maximize unit net present value (middle row), not per well performance. You have a tradeoff when adding extra wells to a section. Each well gives you more total oil from the land, BUT the per well performance goes down (because there is a mix of cannibalization with old wells versus accessing new rock by the extra fracturing) and as the spacing gets tighter, it becomes more and more cannibalization. And each well has a capital cost (there is a small capital economy of scale from running on the same pad [and even smaller OPEX scale savings from pad operations] but most of the capital cost remains.)

This is almost the exact same problem as adding salaried business development (sales) resources to a territory.

I'm NOT saying to present this problem to the kids with all the modeling involved (taking business insights and converting into algebra and Excel). I would actually opt more for giving them idealizing problems with formula shown where they just do the derivatives and check for extrema. But maybe when they are encountering these issues in the work world (and tradeoffs like this exist in almost every business), they have some "feel" for the issue from having done it in calculus class and it just resonates off a remote memory. And I don't even care that this problem is discrete (can't have half a well) versus continuous. The concepts convey.

And note, this is a "return" problem. So it doesn't meet your restrictions of some other category. But I just want you to see how rich, rich, rich that area is. I'm not even saying it for the kids benefits but for yours. So that you feel like the concept is worthwhile.

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I also like the example of regression and least squares as a nice application of optimization a differentiation with respect to something other than $x$.

In my class, I also discuss partial derivatives and give some examples of envelope problems. Suppose a 13 foot ladder slides down a wall, what is the curve that it traces out?

Describe the ladder as an equation determined by its $x$ an $y$ intercepts and some parameter $a$, and consider a specific value $x$, say $x = 4$. Then, we have $y = 17 -a -52/a$.

If we look at what happens as we run through different values of $a$ we find that the $y$'s increase then decrease, moving first towards and then away from the envelope, as we see below with the little red dots. Thus, we have the curve at the maximum value when we differentiate our expressions with respect to $a$.


Now, more generally, we have

$$y = \frac{a - 13}{a}(x - a) = 13 + x - a - \frac{13x}{a}$$

$$\frac{\partial y}{\partial a} = -1 + 13x/a^2$$

$$a = \sqrt{13x}$$

and we can substitute this value for $a$ back in our original equation, yielding the description for the curve. This example is from Analysis by its History by Hairer and Wanner.

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