# Natural, rich, calculus questions

We have the good fortune of having "lab sections" here at my college. I'm interested in conducting some activities in the spirit of this talk. However, even in my stash of inquiry-based learning resources I can't easily find a large number of natural questions outside the usual suspects (falling bodies and other physics examples) that aren't somehow artificial-looking. I'm mostly interested in questions that lead into topics before related rates and optimization...since those topics have been around so long in calculus courses that the relevant ideas can be found "canned" easily.

Question: What are some very good inquiry-based activities for calculus that lead students to the discovery of basic definitions in the subject via natural examples?

One more point of clarification: Ideally, I'd like to see problems that lead to calculus topics through a "Martin Gardner" feel...if this helps...rather than some contrived artifice meant to jam the definition of derivative down the throats of unwilling students...

Remark: Perhaps this will be helpful: http://www.iblcalculus.com/. The bottle graph activities are nice (things like this can be found in Hughes-Hallett) and the traffic camera activity is pretty good, too. You can imagine a student wondering, outside of a mathematics class, how traffic cameras can measure a car's speed. That is another litmus test for the quality of activity...would a student come up with it on his or her own under minimally invasive circumstances?

Here is a "lowbrow" example that seemed to work well today: I drew a circle on the board, then the tangent line to the circle at a point and a radius of the circle connecting the origin to that point. Then I asked the students to use the definition of derivative to verify that the angle formed by any radius and the tangent line to the circle at the terminus of that radius is a right angle. This question was very simple to state, as well as straightforward and natural, but incorporates and reinforces several concepts from the course.

• Take a look at comap.com for some ideas. Commented Oct 22, 2015 at 13:54
• These slides of mine be useful: lightandmatter.com/alr/alr_fund.pdf . They're about 50% active-learning activities and 50% presentation of material. They're meant to work with this book: lightandmatter.com/fund .
– user507
Commented Oct 22, 2015 at 15:03
• (A comment, which you may have explored/broached in class, but which is rather un-related to your question.) The upper-semicircle has a relative [well, absolute...] max at $x = 0$; so the tangent line there has slope $0$. The radius associated to this point aims due north, so they form a right angle. For any other point on the circle, draw in the relevant constructions and rotate to the aforementioned scenario; the angle between radius and tangent line doesn't change, and so it's right (as desired). Commented Oct 25, 2015 at 0:19
• @Benjamin: This will be nice to reintroduce when the students are learning about extrema. Thanks! Commented Oct 26, 2015 at 12:10
• I think profit maximization is a powerful way to introduce (or at least run applied word problems) maximization problems Something like adding salespeople to a territory (more sales, but declining benefit/person added because of competition) versus cost (say linear per salesman). Money is important. Unfortunately, most of the examples on YouTube are kind of clunky. Commented Apr 1, 2023 at 0:45

There are quite a few good resources that could be adapted, from various "projects". I haven't read all of them but some of these could work.

As to topics, here are a few econ/business ones that could be interesting for you - though finding complete stuff ready-made may be tough thus far, there are many texts nowadays that have fairly realistic versions of this that could be discussed.

• Revenue optimization derived solely from a demand function and theory
• Deriving elasticity - I find this quite well-suited to IBL
• Storage costs for inventory - could start with VERY simple cost functions and go to more and more realistic ones to motivate integration
• Degree-days! But you have to live in a cold-weather state for this to gain traction.

I was just now working on my presentation for the Joint Mathematics Meetings in January, and put together links to some blogs that have great resources.

Check out:

• Thanks. I fixed it again, though, so it points to the posts tagged as calculus. Commented Oct 30, 2015 at 23:27

I recall that Mount Holyoke College built a fascinating unit around witchcraft, but I cannot find curricular details at the moment. Perhaps these references might indicate the potential.

(1) Schwartz, Robert M. History and Statistics: The Case of Witchcraft in Early Modern Europe and New England. Research Foundation of State University of New York, 1992. (Mount Holyoke web excerpt.)

(2) There is a detailed list in Wikipedia of those executed for witchcraft by date and nationality.

(3) Here is a chart specific to Salem:

(Image from the Salem Witchcraft Site.)

An example that I once started a calculus class with was how to build a reflecting telescope. Imagine the light from a distant object arrives as parallel rays, and you want to build a mirror that will focus all of them at a single point. What shape should the mirror be? Obviously it will have to be curved; but then how do we figure out at what angle light will be reflected from a curved mirror? We know how light bounces off of a straight mirror — incidence equals refraction — so maybe we can pretend that a curved mirror is approximately straight. Etc. Once you can find the derivative of $x^2$, checking that a parabolic mirror really does focus parallel rays at a point is not too hard of a trigonometry problem (having the additional advantage of forcing them to remember trigonometry and its geometric meaning).

I find the calculus based explanation for why we can't go into space just by bundling together a bunch of low power rockets (as it turns out ISP matters a lot!). I found it especially neat because it was the first bit of calculus based physics that really clicked as "oh, this is why we need calculus," because the chain rule shows up when calculating accelerations as the mass changes.