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Since I'll be working ("I" being the original poster, Andrew L) as either a high school math teacher or adjunct at a university as well as private tutoring, to make ends meet for the next year or so (hopefully entering a PHD program after that, finally), I suspect I'll be teaching calculus a great deal and at several different levels. I wanted to ask what everyone's favorite calculus text is. What worries me about this question is the same problem the board seems to have whenever this kind of question is asked: Since many mathematicians begin as gifted students, they have—to put it kindly—unrealistic expectations of what level textbook most students can handle. For example, when asked about undergraduate topology texts for beginners, more then a few posters put Robert Switzer and Peter May's books. And in the same thread, Hatcher was listed as an undergraduate text. I suppose that'd be reasonable at Harvard or in Germany, where students actually have a reasonable education coming in. Unfortunately, this is America and whether or not students can read who enter college isn't a foregone conclusion.

Another problem that this question has is a lot of mathematicians choose a purely theoretical calculus course devoid of physical applications. The applications of calculus are not only incredibly important in their own right, but without them, a lot of the concepts of calculus become downright mysterious in not only their relevance, but how anyone ever thought of them. As beautiful as Michael Spivak's Calculus is—and it is quite wonderful—the fact that it only has one toy example involving celestial mechanics to me is a serious defect that would force me to supplement it extensively.

Here are my choices: For an honors course consisting of the very best students, I would use either Spivak supplemented with an extensive set of composed notes focusing on history and applications or the very unusual and wonderful book, Practical Analysis in One Variable by Donald Estep. Estep's book is clearly a calculus text which is being sold as an analysis text-this is what Estep thinks a general calculus course SHOULD look like. The book uses a rigorous approach to calculus through numerical approximation of physical models such as Verhulst population models, chemical equilibrium, Newtonian and Einsteinian mechanics and a lot more. It's a fascinating read and I highly recommend it to all.

For "regular" students, the choice is clear: Gilbert Strang's Calculus. Beautifully written, clearly motivated with lots of examples, applications and diagrams and nothing is thrown out without clear explanation. It's also more advanced then the usual plug-and-chug books in its choice of topics and applications—very careful without being rigorous. And lots of good conceptual exercises that will force students to think, rather then calculate mindlessly.

Ok, that's my 2 picks. What about everyone else? What calculus books would you use—and make clear what kinds of students you have. It matters!

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The question/rant ratio is abnormally low. –  Victor Protsak Jun 3 '10 at 0:40
    
I think it feels less like a "rant" now that it's divided into paragraphs. –  Charles Staats Jun 3 '10 at 3:42
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There is something Very wrong here with how the software apportions authorship on this question! At first sight, it looks as though Harald Hanche-Olsen is the one who'll be working as a teacher! I guess I ought to inform our Head of Department about this ... (in case it's not clear: the last sentence is a joke. Also, for those who don't know, Harald and I are (usually) at the same university.) –  Andrew Stacey Jun 3 '10 at 13:33
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Andrew is more than welcome to try to get me in trouble. I'll have his head off in no time. But seriously, using “I” in a community wiki question is asking for trouble. And how is the software doing percentages? There is no way I wrote 54% of that post. –  Harald Hanche-Olsen Jun 3 '10 at 22:24
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For students NOT at the top of the heap, maybe we need not only (1) "calculus with motivational examples from physics" but also similar books (2) "calculus with motivational examples from biology" and (3) "calculus with motivational examples from finance" ... Students who have not taken (and do not intend to take) physics are baffled by books (1), but maybe books (2) or (3) is what they need. –  Gerald Edgar Sep 27 '12 at 17:50
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6 Answers

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I wonder whether an English translation exists, but the greatest source of teaching ideas for me was always the Collection of problems in analysis by B.P. Demidovich (the Moscow students all know this book as "Demidovich"). The collection covers the related theoretic material as well (without proofs).

As for textbooks, Whittaker and Watson's course in analysis is still one of the most contemporary treatises.

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I think an english translation does exist: amazon.com/Problems-Mathematical-Analysis-Boris-Demidovich/dp/… –  alex Jun 3 '10 at 0:32
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I used Demidovich's book when I studied calculus a few years ago: a truly useful resource! Russian, English and Spanish editions can be found at gigapedia.com. –  Alberto García-Raboso Jun 4 '10 at 2:35
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SHHHHHHHHH,Alberto! Gigapedia does NOT exist!!! –  Andrew L Jun 4 '10 at 8:21
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Obi-Wan: These aren't the droids you're looking for. Stormtrooper: These aren't the droids we're looking for. Obi-Wan: He can go about his business. Stormtrooper: You can go about your business. Obi-Wan: Move along. Stormtrooper: Move along... move along. –  Alberto García-Raboso Jun 4 '10 at 12:08
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I find your lack of faith disturbing... –  Alberto García-Raboso Jun 5 '10 at 15:47
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It has been some years since I was in academia, but I see that Ostebee and Zorn's Calculus from Graphical, Numerical, and Symbolic Points of View is still available, and I thought it was a good text for the "standard" college calculus course.

http://www.stolaf.edu/people/zorn/ozcalc/index.html

This is a product of the so-called "calculus reform" movement, but it is a relatively "moderate" example of a reform textbook and I think will not offend those with a more traditional mindset. A glance at the website shows that it seems to have resisted (to some extent) the market pressure to create a constant stream of new editions, each bigger and more expensive than the last.

One thing to look out for when selecting a textbook is that if it is very popular, then solutions to all the exercises may be easy to find electronically. I don't know if this is true of Ostebee and Zorn.

Another thing to consider is whether you want to make a symbolic algebra package like Maple or Mathematica an essential part of the course experience. I taught a "Calculus with Maple" course once and while there were pros and cons, overall I thought it was a good idea. The course materials were put together from a variety of sources, but nowadays I think there are a number of commercial "Calculus with Mathematica" packages to choose from.

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I had calculus with this book as a high school student. While I don't recall having any special feelings about the book, I came out of that experience with a good grasp on the basic ideas of differentiation and integration, and also with a good knowledge of how to grasp functions. (My experience teaching suggests this is not common.) On the other hand, at some point in college I saw an epsilon-delta proof and was totally unaware that other people remembered these from calculus classes. I don't know that there weren't such proofs for us, though; I was not so interested in math at the time. –  Allison Smith Jun 7 '10 at 20:52
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For an honors intro to analysis (and rigorous mathematics) course:

Baby Rudin supplemented with extensive course notes. I really like Spivak, but he completely neglects the topological aspects of analysis that motivate the epsilon-delta formalism. He also doesn't really talk about things like compactness, connectedness, etc, which are extremely important to learn before you get to a real course in general topology.

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I think it's great that Spivak doesn't talk about compactness or connectedness. Or, rather, he doesn't use those words. This kind of thing goes a long way towards making sure courses are not just about absorbing piles of definitions but have some content. –  Mike Benfield Jun 4 '10 at 4:49
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For CALCULUS,Harry?!? Ladies and Gentlemen,the prosecution rests......... –  Andrew L Jun 4 '10 at 5:45
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That was funny like a brain tumor without health insurance,Harry....... –  Andrew L Jun 4 '10 at 8:21
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I wasn't aware that brain tumors could buy health insurance, Andrew. –  Harry Gindi Jun 4 '10 at 8:39
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I don't get it. Can lampshades buy health insurance in civilized countries as well? How about tea kettles? –  Harry Gindi Jun 4 '10 at 22:02
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When I was in high school I used "integrated physics and calculus" to learn basic physics and multivariable calculus (on my own, not for a class), along with one of the standard calculus texts (I don't remember specifically which). I really liked the physical motivation used in defining concepts, and really think it would be a great book for an applications-oriented course. And I think it did a much better job in most cases than the standard calculus text.

http://www.amazon.com/Integrated-Physics-Calculus-Andrew-Rex/dp/0201473968

I remember later, taking a college level calculus course and being really irritated that the only application of calculus they could think of was "suppose we have a tank of brine..." A whole quarter of solving that problem over and over is not anyone's idea of fun, or applications!

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It also doesn't give students a whole lot of added insight after the second time around,Jeremy. If Spivak had a lot of physical applications in addition to the beautiful theoretical presentation,it'd be my text of choice for math and serious physics majors hands down. –  Andrew L Jun 3 '10 at 0:46
    
I can definitely agree with that. Although Spivak would obviously not work so well at the High School level like the question mentioned, while I think the book I used could work there. –  jeremy Jun 3 '10 at 1:08
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Apostol's two volumes in Calculus supplemented with Spivak Calculus & Calculus on Manifolds could be good and enough for introductory in Calculus before serious Mathematical Analysis, e.g. three books by Rudin, Zorich, Amann, Stein...

But I still cannot avoid great classics by Hardy - A Course of Pure Mathematics and Courant - An Introduction to Calculus and Analysis.

These five books in basic calculus is even more than enough, I cannot see any reason to read more, since time's limited, move on to more advanced topics could be much more beneficial even though some basic skills (e.g. computational skills) not trained well, fortunately it can always be trained again in higher topic.

Finally, personally, I feel non-smooth study could be much healthier, read deep and hard books even if we're only beginners, put yourself in the 'ocean' of high point of view, just enjoy, we'll understand it silently when we are moving on and on.

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I think that "Calculus of one variable" by Joseph W. Kitchen is the best. The perfect equilibrium between, rigourosity, motivating words (not only propositions and exercises) and applications.

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