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There has long been debate about whether a first year undergraduate course in discrete mathematics would be better for students than the traditional calculus sequence. The purpose of this question is not to further that debate, but to inquire whether any text has tried to teach calculus by emphasizing probability as a primary motivating example.

It seems that such a text would introduce integrals before derivatives, which I know many authors have tried.

I am asking this question because it seems industry has use for people proficient with probability, and pure mathematics makes great use of these ideas as well (think of probabilistic methods in number theory, for example). It seems that a good course of the above type would be cosmopolitan enough to attract students with tastes in either pure or applied mathematics to major in mathematics.

EDIT: In the absence of a perfect text for this, it may be worth a collaborative effort by mathematicians to write such a text. Is there a way to "crowdsource" writing a text like this as a wiki?

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    $\begingroup$ Sans answers thus far, might I ask whether you've an interest in writing or even just outlining such a text? $\endgroup$
    – user614
    Apr 4, 2014 at 4:42
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    $\begingroup$ I don't think that "crowdsourcing" such a text is a good idea. Good textbooks are written by a very small set of people (often a singleton) with a clear vision. As someone said, "there's a reason 'designed by committee' is not a compliment"... $\endgroup$
    – mbork
    Apr 9, 2014 at 18:05
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    $\begingroup$ @mbork: You are probably right. Perhaps an outline would be interesting, though? $\endgroup$
    – Jon Bannon
    Apr 9, 2014 at 21:24
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    $\begingroup$ @JonBannon: I don't think so, either. An outline is a result of vision of how and what to present. My proposal would be: take one-two knowledgeable people, gather some proposals (a few dozen, maybe), and then make them sit down for a few days (or weeks) to produce a vision (including a rough outline). Only then should they write the book. $\endgroup$
    – mbork
    Apr 9, 2014 at 21:26
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    $\begingroup$ BTW, I like this question a lot (though, alas, I have little knowledge of probability). $\endgroup$
    – mbork
    Apr 9, 2014 at 21:27

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I think this one may suit your needs, at least partially: Richard W. Hamming, Methods of Mathematics Applied to Calculus, Probability, and Statistics, http://store.doverpublications.com/0486439453.html

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This isn't exactly a full answer for you, but Approximately Calculus has a large bit on the prime number theorem as a motivation, which was motivated by Gauss' thinking of prime distribution probabilistically and where using integrals doesn't make sense unless you are thinking of integrating a pdf.

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