When recently teaching Calculus II to college students, I instructed my students to read and be ready to work through the first 8 or so questions of James Walsh's climate modeling differential equations packet: http://www2.oberlin.edu/math/faculty/walsh/UMAPclimate.pdf

I enjoy this text, because:

1) these problems are well within the reach of students in any Calc II class that addresses examples of the simplest kinds of ODEs (separable, linear first-order, etc.),

2) the questions attack the ODE from different angles, allowing the students to understand the ODE from different perspectives and to integrate new and old mathematical skills, and

3) the familiarity that the long-form nature of this richly contextual problem engenders in the students allows them to translate their mathematical observations and deductions into scientific interpretations. Most of my students are science majors, so this is a win-win.

I capped off this unit in my Calc II class with a speculative question on a take-home quiz that had the students qualitatively examining the the energy-balance ODE for the atmosphere of Mars and making a predictive scientific interpretation for the dust storm phenomena observed by the Mariner 9 probe in 1971. Admittedly, I did perhaps prepare them a little heavy-handedly for answering that question on the quiz, but otherwise, I was fairly satisfied with how well this unit went over with the class.

I would like to utilize more long-form, multi-step, skills-integrating applied mathematics problem packets like this in my calculus classes, and was wondering what you all had uncovered on the internet or had written yourselves that is in the vein of the Walsh packet? I am especially interested in Calc III (multi-variable calculus) applications, but all calculus topics are fine.

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    $\begingroup$ Please tell us what Calc I, II, and III means at your school. There is no standardized definition for these terms. $\endgroup$ – Ben Crowell Dec 31 '18 at 2:00
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    $\begingroup$ Ben, are you just irked by the USism or did you seriously not understand? Even if you are non US, would think you would have been exposed enough on the net to know standard US structure (so I suspect it is the former). Both the terminology and various clues in question make it obvious US sequence. If it is actually the latter, than calc 1: differentiation, calc 2 integration along with simple amount of ODEs and chapter on sequence/series. Calc 3 is divgradcurl, partial differentiation (but not PDEs), line and surface integrals, Stokes/Green theorems. $\endgroup$ – guest Dec 31 '18 at 4:40
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    $\begingroup$ @guest: I meant what I said. It's not standardized. The terms could refer to years, semesters, or quarters. And the curriculum is different at different schools. For example, at my school ODEs are not in the 3rd semester as they seem to be at your school. They're in the 4th semester. $\endgroup$ – Ben Crowell Dec 31 '18 at 16:23
  • $\begingroup$ So, the former. ;-) Yes, I agree there are many different setups and no rigid standard. It is clear to me, that he is referring to stereotypical American standard though. P.s. A full course in ODE is stereotypically standard in 4th semester. A short module on ODE is common at end of second semester though. (For instance many softer science students don't progress to 4th semester calc, so this is what they get.) You can look at the AP curriculum for instance. Or many books like Swokowski, Thomas, even back to Granville. $\endgroup$ – guest Dec 31 '18 at 17:19
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    $\begingroup$ @guest: The meanings of Calculus I, II, III, etc. are not standard within the US, and certainly are not well known outside the US. Generally very few people have much idea about the structure of undergraduate degree programs in countries with which they do not have direct experience, and generally these things are less uniform than one might initially suppose. This sort of question always benefits from making clear that Calculus I is directed to students of such and such a sort and covers such and such topics, that Calculus II is directed to etc. $\endgroup$ – Dan Fox Dec 31 '18 at 18:14

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