# What topics should be included in a course matching these specifications?

I posted this question on m.s.e., where I upvoted the two answers, both of which said rather little by comparison to what the question asks. Hence this present posting.

Say you have a calculus classroom full of liberal-arts majors who are not particularly mathematically inclined. Your goal is NOT to teach them everything on a list of topics that will be needed in other courses (either later calculus courses or courses in physics or engineering or statistics or biology or other subjects to which calculus is applied). (As anyone with any common sense would do) you will include only ten percent or less of the topics in the usual lists and perhaps examine each included topic in more depth than it might normally get, but perhaps also proceed more slowly than you would if you needed to complete all the topics in the usual list. Rather, your goal is to impress them with (1) the ways in which calculus has played a role and continues to play a role in the world --- in the sciences and engineering and philosophy, etc. --- and with (2) the fact that calculus is a considerable intellectual and aesthetic achievement. This might necessitate presenting a few applications in the sciences, not usually found in the first-year calculus text, rather than concentrating on techniques.

My question is: What topics would you include in such an (in the present day) unusual course?

• Why calculus? I mean, I heard an interesting talk a few years ago on presenting higher dimensions as a math for liberal arts course. Basically the course was built around Flatland. If you free yourself from making it about calculus in particular then there is much more freedom in the selection of topics. I'd like to see a general education course on the big picture of math these days, a selection of mathematical stories which presents math as an art rather than a process. I think this is a very interestng question. – James S. Cook Dec 24 '15 at 2:01
• How long is this course? One quarter, two quarters, one semester, two semesters? – Jasper Dec 24 '15 at 23:14
• Will students be able to go directly from this class to a class that expects an understanding of calculus? Or is this strictly an "Appreciation of Calculus" class? – Jasper Dec 24 '15 at 23:15
• Are the students entering the class expected to know the basics, like: What is a variable? What does an expression like "5 meters per second" or "50 miles per hour" mean? How to set up a story problem, and label variables? How to make a graph with an x-axis, a y-axis, and an origin? How to find the slope of a line? How to calculate the area of a triangle and a rectangle? – Jasper Dec 24 '15 at 23:19
• @Jasper Please tell me there's nowhere in the states where those things are not covered at high school (or earlier). – Jessica B Dec 25 '15 at 9:31

My first idea would be to aim to include applications at different levels (ones they might be able to calculate, ones they can get maths software to do, high-level maths research) and also across different subjects. Some ideas:

• relationship between area/volume/perimeter/surface-area of circles and spheres, maybe cones
• some fluid dynamics, maybe in relation to aeroplanes
• some modelling, such as populations in ecosystems
• the Numberphile discussion of John Nash's embedding of a torus
• designing a suspension bridge, or similar
• finding an equilibrium point of some physical system, and testing whether it is stable (and therefore safe to build)

Other initial ideas, which might or might not be sensible, would be something in economics/related to financial markets, and designing a route for a spaceship (ie variable acceleration, change of mass... generally a situation where the formulae they have previously met don't work).

I would also want to show them that in some cases approximation does not work, so the full theory does have value. I might also discuss the Fundamental Theorem of Calculus - that it's not obvious why rates-of-change should be related to areas-under-graphs, although I'm not sure whether the students would appreciate that.

I would probably not include all the various calculation rules (chain rule, integration by parts etc) beyond what is directly needed for any calculations you are doing.

Key points for goal #1:

• Professor Krugman's intuitive summary of how a derivative is the limit of a sequence of differentials. Start with the graph of a function you want to differentiate. Find the slope over a modest delta X. Repeat for the next delta X. Optionally repeat again. The shape of the derivative should now be obvious, but also obviously has kinks or steps (because it was computed using finite length steps). "Now we take off our glasses…" and the derivative looks like the true curve. (For this level of class, the example curve is probably smooth.)

Teach students that they can check their work.

• Always do a sanity check.

• If possible, check by substitution.

• Integrals can be used to check derivatives, and vice versa.

How to set up problems

• How to identify boundary conditions and field conditions. If you cannot find enough independent conditions, your solution will give you a range of possibilities instead of a single answer.

• Teach students to keep track of their units. (This also means that students usually should not use calculators until the end of a problem.) Derivatives usually involve dividing by length or time. Integrals usually involve multiplying by length or time.

Common applications

• Position -> Velocity -> Acceleration -> Jerk.

• Energy -> Power

• The importance of conservation laws. Simple examples of the Reynolds Transport Theorem.

Exponentials, and common applications

• The derivative of an exponential curve. The integral of an exponential curve. If profits are proportional to capital invested, and a fixed portion of the profits are reinvested, you get exponential growth. Demonstrate the power of exponential growth. Ask the students to consider real-world limits to exponential growth.

• The integral (from time = now to time = forever in the future) of an exponential decay is inversely proportional to the decay rate.

• The integral (from time = 30 years in the future to time = forever in the future) of the same exponential decay is the previous integral, times the amount of decay in 30 years.

• Applications of the exponential decay integrals to mortgages, radiation exposure, …. Dare the students to come up with other examples where exponential decay might be a good model.

Sine waves, and common applications

• Show how having the second derivative be proportional to the original value, but opposite in sign, results in sine and cosine waves. Show the forms of what became known as Maxwell's Equations, and that they have this property. Point out that this allowed Maxwell to describe light as a wave, and understand its speed.

• Point out that there had to be a negative feedback in there somewhere, or it would have exponentially grown out of control. (And that would violate a conservation law.) Dare the students to come up with other examples of wave phenomena. Dare the students to guess where the negative feedback comes from.

Optionally mention some cool stuff

• Any repeating pattern can be modeled using a sum of sine and cosine waves. Fourier transforms are used to digitize songs, to compress images, to transmit data via radio waves, et cetera.

For a look at the intellectual development of calculus (not a textbook nor a skeleton for a course, but something to peek at for the instructor), I'd recommend Dunham's "The Calculus Gallery" (2005). His books on the history of mathematics are outstanding expositions, accessible with a high-school background (and interest enough to work through the material).

This will also say little in comparison to the breadth of your question.

I would include some of the history of how the notion of limit of a function, $\lim_{x \to a} f(x)$, emerged over time. I was surprised to learn this concept was not present (or at least not clear) in the work of either Newton or Liebnitz. It took another hundred$+$ years for this to reach its modern $\epsilon$-$\delta$ form.

Including the major historical struggles fits in well with your 2nd goal, to convey the "intellectual and aesthetic achievement" of calculus.

• I think it took about 200 years. – Michael Hardy Dec 24 '15 at 4:27
• But I'm not sure limits should be mentioned at all. – Michael Hardy Dec 24 '15 at 4:28
• I wonder if you should write $\varepsilon$-$\delta$, with a hyphen, rather than $\varepsilon{-}\delta$, with a minus sign? ${}\qquad{}$ – Michael Hardy Dec 24 '15 at 4:29
• While it pains me greatly to observe it, the epsilon/delta definition of a limit is already stricken out of many mainline calculus courses and textbooks. So one that expects to cut out 90% of a standard course is highly unlikely to include it (as evidenced by @MichaelHardy's comment above). – Daniel R. Collins Dec 24 '15 at 8:32
• If you had any interest in this idea, then you'd want to include non-standard analysis as an alternative to limits also. – Sue VanHattum Dec 24 '15 at 13:38

Have you checked out the Discovering the Art of Mathematics project? Their 11 books, one of which is Calculus, are free for educational use, and they provide lots of supportive community. If we taught a liberal arts calculus course, this is what I'd want to use, I think.

I see that your question was about topics, not textbooks. They have lots of experience teaching courses like this, and would be good people to ask that question of. They encourage building your own textbook by using chapters from various books in their series, so will have people in their community who have created all sorts of hybrid courses.

One really long answer to your question that I am fond of is the book Calculus in Context. Selected topics from the first four chapters might make for a nice one-semester course, though perhaps it has too much of an emphasis on technique for what you have in mind. But I like the explanation of why calculus is relevant and why successive approximation is important through the concrete SIR model. And I think that some understanding of technique is necessary to truly have a meaningful appreciation of why calculus is necessary and useful in applications.

Most of all, I like their philosophy that with the appropriate use of a calculator or computer, one can get a sense for how computations are carried out and how complex systems are analyzed. If students come away with a sense for how differential equations naturally model the world around us and how basic techniques such as Euler's method can be used to approximate solutions and why computers are naturally suited to carrying out such computations if they are programmed properly, then I think the course has served a significant purpose.

Do hand-waving-in-the-air lectures building up to the calculus of variations. Then get them to build two slopes - one with a straight path and one with a cycloid path that they can actually roll two balls down side-by-side to see the fastest one for real.

• How much experience do you have teaching this? With typical student, who, remember, have never suspected that mathematics is not an entirely dogmatic subject, so they don't know that one knows things by deriving them from simpler things. – Michael Hardy Dec 2 '17 at 16:40

Kinematics is easiest conceptual example of derivative and something everyone has visceral physical experience with (acceleration, speed, position). Optimization (particularly business or manufacturing type problems) is something everyone can understand as a practical use of calculus. And is a KEY concept, use of calculus.

Something I learned about giving academic science talks is to always mention impact of science knowledge on humanity, usually some commercial impact (could cure cancer, make computer chips cheaper, etc.) It is motivation to listen to some of the boring science talk.

Disagree with answer above pushing calculus of variations (for weak students in intro calc course?) This is perfect example of the opposite of "getting it" in terms of the question. I would consider to avoid trig functions also (for semester long course).

Eschew integration by parts and partial fractions and u substitution. Cut related rates, diffyQ survey, and series.

Give the kids the understanding of what you will cover/won't cover at beginning of course and in individual lectures. I find mathematicians tend to trust people to go along with them on a winding road of explication but without knowing where it is heading. This is poor pedagogy.

Also, if you put it in concept "what calculus is about" it is actually useful knowledge even if the kids can't do partial fractions. Even knowing what the typical courses are and what other majors require them. I've never had some courses but I know what they are about and where they are needed. This is a practical man's way of thinking about a topic.

I would mix in a little "history of calculus" (the Newton, Leibniz rivalry; how/when the topic started to be used by Prussian artillerists and Voltaire [I am making that last one up, but you get point]). Humanists want to know what something is about and how it influenced people even if they don't know it in detail. (E.g. famous books you have not read but still know about, etc.)

P.s. Jasper had some important questions (e.g. one semester, no credit for moving on or prereq to physics, would think so?) that you did not answer.