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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 necessiate 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?

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    $\begingroup$ I think this should be on matheducators.stackexchange.com $\endgroup$ – null Nov 5 '15 at 20:43
  • $\begingroup$ This has not been a problem for me, since at every place I've taught such classes in calculus (6 different colleges/universities) there has been either a syllabus to follow or I've been given specific information about what was expected to be covered. Of course, carrying this out is another matter, but for what it's worth, the studiousness of the student body was more significant for me than the percentage of liberal arts majors I had in class. $\endgroup$ – Dave L. Renfro Nov 5 '15 at 20:58
  • $\begingroup$ @DaveL.Renfro But it seems that here, the norm is liberal arts students, not a few mathematically inclined or ambitious ones. Moreover, it sounds like there is no syllabus to "do", one needs to be invented. @ Michael, yes? $\endgroup$ – BrianO Dec 24 '15 at 3:22
  • $\begingroup$ @BrianO: See this 2 February 2012 math-teach post archived at Math Forum. Although I tried to word my comment here to not make any judgment about a correlation between "studiousness" and "liberal arts", my personal experience has actually been that liberal arts majors tend to be quite a bit more studious than the majors one typically sees (such as those I listed in that 2012 post). Of course, physical science and math majors would be great, but very few people (in the U.S., at least) get calculus classes with a lot of these students. $\endgroup$ – Dave L. Renfro Dec 29 '15 at 13:25
  • $\begingroup$ @BrianO: I just realized that the post I linked to in my previous comment doesn't really apply here, because we're talking about calculus courses here and in that 2012 post the math courses (not stated, but certainly implied) were below calculus level (and mostly below precalculus level as well), such as intermediate algebra, college algebra (without trig.), math appreciation, etc. $\endgroup$ – Dave L. Renfro Dec 29 '15 at 13:53
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I would include the formulation of the Riemann integral, and include many applications such as calculating area, volume, center of mass, work, etc.. Approximating and taking a limit is, in my opinion, a beautiful idea that also turns out to be very fruitful.

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  • $\begingroup$ Excellent ideas, all, and +1 on the idea of highlighting the concept of limit, it is beautiful as well as powerful. It's also beautiful that the reals are first countable, and that the rationals are dense, which makes possible actual computation to within known errors. But I wouldn't start with the integral (few do, I think), assuming it will be first one then the other. $\endgroup$ – BrianO Dec 24 '15 at 3:34
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Some material on waves and periodic functions would go over well, I think. Mention, at least, of Fourier's theorem and its consequences; its use in acoustics, & in audio software via the FFT; Chladni patterns.

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