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(This question was proposed during the area51 phase.)

It's common for chemistry/biology/physics majors to be required to take certain calculus courses. At my school, chem/bio students must take up through Calc II (integration, separable diff eqs, infinite sequences/series) and physics students up through Calc III (vector calc). We don't have the resources necessarily to develop a separate "Calc for science students" course. Yet, this semester, I find myself with a section of 25 students entirely from other science majors.

I'd like to appeal to the interests of these students and, hopefully, demonstrate that calculus can be both useful and interesting for them. Alas, many textbook applications are somewhat contrived, or don't represent the way the students will have to handle data/problem-solving in their actual scientific work (i.e. the numbers are too "round", the modeling function is already given with no motivation as to its applicability/accuracy, etc.), while other application problems are just too difficult to assign, given the range of abilities in the class.

Do you have any in-class activities or take-home assignments that demonstrate a calculus concept/technique in the context of a scientific problem and have effectively appealed to the interests of science majors in your course?

I am specifically curious about activities/assignments you have used before, and would like a description of how you think it was effective (and what could be improved, if need be), and to whom it appealed. I don't want this thread to be full of "Try something with topic X, that might work."

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    $\begingroup$ I'd ask their teachers of courses taken in their areas before/concurrently/soon after for examples. Perhaps you can show how to solve some previously hairy problem. In any case, having the other teachers aware of what you are up to might make them use your subject matter in class... $\endgroup$ – vonbrand Apr 2 '14 at 3:50
  • $\begingroup$ @vonbrand: Yes, I thought of that, and asked a biology professor for suggestions, only to hear back that "we almost never use calculus in our work" :-/ ? $\endgroup$ – Brendan W. Sullivan Apr 2 '14 at 3:52
  • $\begingroup$ Population models offer many excuses for differential equations. Chemical reactions kinetics ditto. $\endgroup$ – vonbrand Apr 2 '14 at 4:53
  • $\begingroup$ @vonbrand: Very much agreed! And I went through some such examples in class early this term. I just found it oddly disconcerting that a bio department chair would say such a thing. $\endgroup$ – Brendan W. Sullivan Apr 2 '14 at 4:54
  • $\begingroup$ I was a science major. I would probably go pretty easy on the applied problems. For one thing, word problems are usually harder than algebraic ones. (You are going to make it even more so with messy numbers and such.) The calculus class is better as an idealized tool that is done simpler first. Then in courses they can get into more complicated modeling (but really only the physicists will). Kinematics is probably your safest bet as it is intuitive for anyone. I would avoid anything electrical. $\endgroup$ – guest Oct 24 '18 at 6:09
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I took an excellent set of related rates problems from Jim Belk's webpage and turned it into a nice worksheet. I spun the problems this way, to motivate them:

We have now covered related rates. This means you now have an extra power that works in all your science classes. Anytime anyone gives you an equation -- a fact -- that is true about the world, you have secret knowledge that gives you an extra bonus fact about the world! This is because you can take their equation and take the derivative of both sides with respect to time.

  • $PV = nRT$? Why not differentiate both sides? Your chemistry teacher might not even know that $P\frac{dV}{dt} + V\frac{dP}{dt} = nR\frac{dT}{dt}$! And if the gas is actually gradually being created by a reaction, you can declare n to be a variable instead of a constant, and take the derivative that way to get a different formula.

  • Kinetic Energy is $E = \frac{1}{2}mv^2$? Why not differentiate both sides? $\frac{dE}{dt} = mv\frac{dv}{dt}$. What's that all about? I don't even know. It's Power = something, because I know the units of $\frac{dE}{dt}$. (It doesn't matter that I do know what it means because I looked it up; the effect is greater when I say that I don't know -- it gives the students more of a feeling of power, which is the whole point.)

  • $pH = -\log_{10} [H^+]$? Why not differentiate both sides? $\frac{d(pH)}{dt} = \frac{1}{(\ln 10) [H^+]}\frac{d[H^+]}{dt}$. It... apparently tells you how fast the pH changes depending on how fast the concentration is changing. That could actually be useful! And it shows that a reaction happening at a constant rate doesn't cause the pH to change at a constant rate.

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    $\begingroup$ Well, I'm happy to hear that someone is actually using my problems! See this MathOverflow question for more discussion of the nature of related rates. $\endgroup$ – Jim Belk Apr 1 '14 at 14:40
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    $\begingroup$ @JimBelk: I'm quite intrigued and inspired by these problems! Have you found that these adequately motivate other science majors to invest in the corresponding calculus course? Or do they tend to pay attention to the relevant bits and then tune out? I'm working within this context right now, this semester, so I'm curious. Thanks! $\endgroup$ – Brendan W. Sullivan Apr 2 '14 at 5:09
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    $\begingroup$ @brendansullivan07 I usually include science-based problems throughout Calculus I, and it certainly seems to help the students see the relevance of the course. Really, there's very little material in first-semester calculus that doesn't pertain to science. $\endgroup$ – Jim Belk Apr 2 '14 at 13:31
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Perhaps consult the appropriate parts of Real World Applications of Mathematics

Also try COMAP. Their materials include free materials and materials available to members. Many (at least the members' material) are classified according to the subject area of the application and the math topic to be applied.

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