Interesting calculus book

Question

(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.

$:)$

Answer 1

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.

Answer 2

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

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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.

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