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I read single variable calculus this semester, and the course is using Thomas Calculus as the textbook. But this book is just too huge, a single chapter contains 100 exercise questions! Now I'm looking for a concise and complete textbook. I'm not interested in routine, computational exercises, but rather some challenging problem sets.

I have quite a strong basic knowledge of calculus from high school, but I still have difficulties in solving a few questions from past exam papers. So I'm looking for more challenging exercises. In fact, I'm looking forward to solving Putnam level questions.

Please suggest some textbooks with these features. Thanks in advance.

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See this thread math.stackexchange.com/questions/2395/… –  gaurav Sep 12 '11 at 3:47
Some free online options: Strang ocw.mit.edu/resources/… , Guichard and Koblitz whitman.edu/mathematics/multivariable , Crowell (me) lightandmatter.com/calc , Keisler math.wisc.edu/~keisler/calc.html . Some of these are fairly concise, if that's what you want. –  Ben Crowell Jan 10 '12 at 20:42

4 Answers 4

If you are looking for some challenging calculus problems, I would suggest looking at the HMMT (Harvard-MIT Mathematics Tournament) problems archive. Take a look at the calculus test for the years 1998-2011. I don't think these problems are the routine exercises that you have grown weary of in Thomas' Calculus.

If you want another book that may suit your taste, I recommend "Introduction to Calculus and Analysis" by Richard Courant and Fritz John. I think the problems are a bit harder than those in Thomas' text. But more importantly, Courant and John give a lot of motivation for calculus through discussing its relevance to the physical world (there is even discussion of Fourier Series in the study of a vibrating drum).

Courant and John also introduce some concepts of modern analysis (calculus done more rigorously) such as monotone sequences, a precise discussion of a limit, etc towards the end of the book, if that interests you.

Last but not least, I see that you mentioned the Putnam. I think a great book for doing calculus problems that may prepare you for the Putnam is Titu Andreescu's book- "Problems in Real Analysis: Advanced Calculus on the Real Line".

Andreescu's texts are known to be useful in preparing students for mathematics competitions. In this textbook, Andreescu exposes you to a lot of techniques (especially in dealing with inequalities) that may help you with the Putnam. You may also learn some analysis along the way, but since he is dealing only with the real line, he doesn't "weigh you down" with various topological concepts that may or may not interest you. If you have mastered the concepts of calculus and are looking for a challenge, this book may be for you.

Good luck!

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Thanks! Titu's book is especially interesting! –  george Sep 13 '11 at 1:18
@george: I am glad that you have found Titu's book helpful. I think it is also a good way for you to get introduced to analysis. –  r.g. Sep 13 '11 at 7:39

I was in a similar situation a couple of months ago.So i decided not to read a calculus text untill i watch a video lecture like single variable calculus from MIT. {I'am a self learner by the way}.I would refer to a text only when the situation arises,say for more problem sets i would refer to not one but multiple texts like Ron larson/Edwards calculus.James Stewart's calculus text and many others.

As of now i'am watching Video tutors from Thinkwell,TTC and many others on youtube.I find youtube to be the worlds best resource for learning math and i'am having loads of fun with it.I suggest you do the same.

And not to forget video's from Patrick JMT and khan academy are just amazing.

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You might want to look at Chicago undergraduate mathematics bibliography:

Of course, as we all know, the One True Calculus Book is

Spivak, Calculus

This is a book everyone should read. If you don't know calculus and have the time, read it and do all the exercises. Parts 1 and 2 are where I finally learned what a limit was, after three years of bad-calculus-book “explanations”. The whole thing is the most coherently envisioned and explained treatment of one-variable calculus I've seen (you can see throughout that Spivak has a vision of what he's trying to teach).

The book has flaws, of course. The exercises get a little monotonous because Spivak has a few tricks he likes to use repeatedly, and perhaps too few of them deal with applications (but you can find that kind of exercise in any book). Also, he sometimes avoids sophistication at the expense of clarity, as in the proofs of Three Hard Theorems in chapter 8 (where a lot of epsilon-pushing takes the place of the words “compact” and “connected”). Nevertheless, this is the best calculus book overall, and I've seen it do a wonderful job of brain rectification on many people.

[PC] Yes, it's good, although perhaps more of the affection comes from more advanced students who flip back through it? Most of my exposure to this book comes from tutoring and grading for 161, but I seriously believe that working as many problems as possible (it must be acknowledged that many of them are difficult for first year students, and a few of them are really hard!) is invaluable for developing the mathematical maturity and epsilonic technique that no math major should be without.

Other calculus books worthy of note, and why:

Spivak, The hitchhiker's guide to calculus

Just what the title says. I haven't read it, but a lot of 130s students love it.

Hardy, A course of pure mathematics

Courant, Differential and integral calculus

These two are for “culture”. They are classic treatments of the calculus, from back when a math book was rigorous, period. Hardy focuses more on conceptual elegance and development (beginning by building up R). Courant goes further into applications than is usual (including as much about Fourier analysis as you can do without Lebesgue integration). They're old, and old books are hard to read, but usually worth it. (Remember what Abel said about reading the masters and not the pupils!)

Apostol, Calculus

This is “the other” modern rigorous calculus text. Reads like an upper-level text: lemma-theorem-proof-corollary. Dry but comprehensive (the second volume includes multivariable calculus).

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Spivak's Hitchhiker's Guide is indeed concise, which is what the OP wanted. But it's $18 for 122 pages, which doesn't seem reasonable to me. –  Ben Crowell Jan 10 '12 at 20:47
@Ben Crowell: Do you usually pay for the number of pages? I would argue that paying by the page is the reason many textbooks from american publishers are bad. –  Jørgen Fogh Jun 28 '12 at 10:53

I gathered some online calculus text-books on My blog. Hope these will be useful for you. EDIT: In that list, I mostly liked the Calculus book by Stewart.

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You might want to be careful about posting material that is under copyright. However, I believe it is the policy of this website not to remove anything unless we receive a takedown request. –  Zev Chonoles Sep 11 '11 at 15:37
@Zev Copyright for which? Those books were already there on the web and freely available. Only one book, Stewart's Calculus was sent to me by a reddit user. –  gaurav Sep 11 '11 at 15:41
That's the one I was referring to. I just meant that we would have to remove the link you provided if there was trouble. –  Zev Chonoles Sep 11 '11 at 15:52
@Zev Thanks for pointing it out. I removed the link. Looks fine? –  gaurav Sep 11 '11 at 16:20
Calculus book by Stewart is similar to Thomas' Calculus. Just routine exercises. –  thomas j. Dec 12 '11 at 15:50

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