I'm trying to think of topics which motivate study of calculus and at the same time necessitate exposure to a healthy cross-section of its core ideas, i.e. the exposure isn't too narrow.
An example I've encountered a few times recently is Fourier Analysis. I've read two excursions into that topic recently, one for language students , another for music students .
Both get into a derivation of e, a discussion of logs and In, of derivatives as limits, of Taylor and Maclaurin series, of Eulers e^j(pi) = -1 and CIS. Plus there's adding functions, integral notation, derivative of trig functions, background on complex numbers etc. etc.
My interest is in curricula which start with some overview challenge or topic which drives or motivates delving into X, where X is some technical domain, e.g. calculus. A parallel example would be using cryptography to drive a study of group theory. You start with an application, and use that as a guiding outline for fishing out what you need from a broader, more generic theory.
Aside from Fourier Analysis, which other topic areas motivate coverage of a lot of interconnected calculus topics? We usually see the classical mechanics stuff a lot, starting with velocity and acceleration. Maybe it's used too much? Electromagnetism is another such topic, and gives us more of a workout with vectors, gets to the div, grad, curl stuff, plus partial differentiation. The problem here is fewer majors seem to need a "semi-casual" understanding of Maxwell's Equations (they're either science and engineering majors, and therefore have a lot of calculus) or they don't study them at all. So maybe that's why it's harder to find these kinds of "from the ground up" approaches to electromagnetism, although I'm not saying they don't exist (recommendations?).
Not that any of these topics are mutually exclusive of course -- I'm just wondering what's a good motivating topic that might be used with students who presumably start with not a lot of background, and therefore need several puzzle pieces explained in order to fit a coherent system together. It seems that Fourier Analysis is already proving itself a winner in this role, but I was wondering what other topics might be leading contenders.