The post has a fairly serious real world, war story example of where some calculus did something significant in business and the calculus was actually quite eas

The post has a fairly serious real world, war story example of where some calculus did something significant in business and the calculus was actually quite easy, no course in differential equations needed, and basically a routine exercise in first calculus. Students might like to see that; calculus teachers might like to show it to them. That is, just first calculus can already make some significant contact with the real world. For the girl whose mother wants her daughter to go to Princeton, I mentioned some more in calculus that might be impressive for Princeton and make moot the student's A-. Indeed, there was long Harvard's Math 55 where the three texts were Halmos, Finite Dimensional Vector Spaces, Rudin, Principles of Mathematical Analysis, and Spivak, Calculus on Manifolds. The course was something of a math boot camp and aimed at freshmen. There was an article on the course at http://www.american.com/archive/2008/march-april-magazine-contents/why-can2019t-a-woman-be-more-like-a-man/?searchterm=Sommers http://www.american.com/archive/2008/march-april-magazine-co... maybe no longer available. For a high school student to have done well in that material might be impressive at Princeton. Linear algebra is perfectly accessible to high school students, basically easier than calculus, although the classic by Halmos may be regarded as better for a second course. The Rudin text I mentioned. And the third edition of that Rudin text contains the core of what is in the Spivak text. If that girl and her mother want to pursue such things, then I'd recommend that the two of them chat with a good college math prof once each three months or so, maybe once a month. Working in a small group that also chats with a college math prof occasionally would also be good. I also mentioned some introductions to stochastic optimal control that are accessible to high school students. For the more advanced material, the students might regard that as a destination on the horizon. I did the high school calculus teachers and their students and that mother and her daughter a favor, possibly valuable for them and not easy to find elsewhere. Moreover, I know what I'm talking about: I studied calculus, advanced calculus, measure theory, and probability theory based on measure theory, and much more, hold a Ph.D. in pure and applied math from a world class research university, and have published peer-reviewed original research using advanced calculus, measure theory, and probability theory based on measure theory. My Ph.D. dissertation was in stochastic optimal control. Did I mention I know what I'm talking about? I did them a favor.