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(Yeah, I've checked all similar questions, and this is not a duplicate)

I'm learning calculus for the first time, and all books I'm seeing is roughly falling into two categories:

a. Contains interesting problems which requires more than half hour to solve, but either doesn't covers the history (so you keep on wondering how on earth one would think of that/what the hell is the point behind this definition), or doesn't covers the conceptual difficulties (eg Understanding what "infinitesemals" of $dx$ actually means, why you can or can't treat $\frac{dx}{dy}$ as a fraction etc).

b. Contains interesting problems and explains motivations and intuition and all that, but is way too hard and requires way too much prerequisites to read.

Is there a good calculus book which doesn't requires you having extra background in calculus, (so develops it from scratch), but contains interesting and hard problems, and also provides the historical background and the intuitions ?

Bonus (but not strictly necessary) if it contains hints to the hard problems.

$:)$

katana_0
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2 Answers2

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My favorite book is the one by Courant. This is part 2.

A modern book that I like is Lax/Terrell.

As for the history, Toeplitz's book is very good, but also very detailed.

Jan Peter Schäfermeyer
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On OP's request, I am converting my comment into an answer.

I would suggest you to grab a copy of G H Hardy's A Course of Pure Mathematics 10th edition. It should be available free of cost online if you search enough. This book is especially suitable for young students (age 15-16 years) who are totally new to calculus. And it is meant for self study.

The distinguishing feature of the book is the focus on rigor. This is exactly how a mathematics textbook should be written. The concepts of calculus are developed from scratch. The book expects a basic knowledge of algebra (complex numbers are developed in the book itself) and a bare minimum geometry to understand some geometric arguments.

The book also contains very challenging problems (mostly from the infamous Mathematical Tripos which Hardy despised) and Hardy provides hints / solutions to the most difficult problems. The simpler problems are designed to help students learn the application of concepts studied so far.

If one really studies this book completely (even without solving all the challenging problems) one will never face any conceptual difficulties in calculus. I believe much of the conceptual difficulties in calculus are a result of crappy calculus textbooks and such crap is published purposely to justify a first course in real analysis at undergraduate level.

Note: The book is an old classic and some notation / terminology may be outdated. You need to map it with the corresponding new stuff.

A review of this book with some personal touch is available here.

Paramanand Singh
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  • Excellent answer, and almost exactly fit for the criterion. Thanks a lot +1 – katana_0 Oct 07 '17 at 13:19
  • @AlexKChen: you may also have a look at other books which I found interesting (though not exactly for calculus). – Paramanand Singh Oct 07 '17 at 13:28
  • I'd like to know which the crappy books are so I can warn people. – AlvinL Oct 11 '17 at 07:10
  • @AlvinLepik: I used to read such books in 1997 and I don't think they are worthwhile to remember. You can try any calculus book in your country which is designed for high school students and there is a good chance that they are crappy. – Paramanand Singh Oct 11 '17 at 07:36
  • @AlvinLepik: if you browse through the questions here on calculus and limits you will find that there is ample confusion in the minds of students. I think these students have good intentions (are not lazy so to speak), but they lack good books/teachers. – Paramanand Singh Oct 11 '17 at 07:39
  • @ParamanandSingh Calculus is a diet coke version of analysis in the uni I go to. The notion of limit is handled rigorously in analysis and in general topology (which are not first year topics). A thing I find quite amazing, even in case of Hardy's book, that they make no mention about propositional calculus. If one is to really pick up a book that seeks to rigorously disect all the problems, one would think they'd suggest to get familiar with propositional calculus.

    ..and if it were so obvious, why would these students have so much confusion about various notions? :/

     – AlvinL Oct 11 '17 at 07:49
  • @AlvinLepik: people may have different opinions, but I think that that was the best decision Hardy made not to include the symbols $\forall, \exists$ in his book. The issue with limit definition is not logical (propositional calculus), but rather that students have utter lack of appreciation of inequalities. Just ask if any student has a problem with the statement: for every positive integer $n>1$ there is a corresponding prime $p$ such that $p$ divides $n$. Cont'd. – Paramanand Singh Oct 11 '17 at 07:58
  • @AlvinLepik: typically when one is asked to prove that $x^{2}\to 4$ as $x\to 2$, students start solving the inequality $|x^2-4|<\epsilon$. Solving inequality is 2 way implication and definition of limit involves one way implication. – Paramanand Singh Oct 11 '17 at 08:01
  • @ParamanandSingh But that is completely logical. We are provided that $|x-2|<\delta$. If we chose the $\delta >0$ suitable, we would get the result we need and verify the implication. I digress.. I was struggling hard with these notions too, now they are plain as day to me and it's not because I some how "got used to these ideas", but because I had an intense practice of propositional calculus (I came to realize the importance of it near the end of my SECOND year. I should have known it was important right from the get go.) – AlvinL Oct 11 '17 at 08:09
  • Let us continue this discussion in chat. – Paramanand Singh Oct 11 '17 at 08:10