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So I'm taking Calculus I in college. However I'm not taking the grades I want to and I have sort of difficulties using my teacher material (theoretical and exercises). I'm looking for a book that has a good explanation of the content and also solved exercises (which is a very important thing that I'm missing). So here is a list of books my college has:

Calculus: T. M. Apostol 1994 Vol. I. Reverté

A First Course in Real Analysis: Murrey H. Protter and Charles B. Morrey 1993 Springer-Verlag

Introduction to Real Analysis: R. G. Bartle e D. Sherbert 1991 2nd ed. John Wiley

Advanced Calculus: A. E. Taylor and W. R. Mann 1972 John Wiley

M. Spivak, Calculus, 3rd Ed., Cambridge University Press, 2006.

What is in your opinion the best book for self study (I'm going to repeat the examinations next semester but I'll be studying on my own). If there is a better book than the ones on this list please tell me. Thanks!!

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  • $\begingroup$ Ok, thanks for the advice ! $\endgroup$ – Granger Obliviate Jan 5 '16 at 1:02
  • $\begingroup$ Stay away from books with "advanced calculus" or "real analysis" in the title unless you know that's what you're looking for. These ones give you a much more rigorous, proof-heavy exposition of calculus that would be very difficult for most beginning calc students. $\endgroup$ – Jair Taylor Jan 5 '16 at 1:06
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    $\begingroup$ All of those are "rigorous" calculus books. Is that what you want? You may be better off with a book that tries to take a more intuition-based approach. Here are a few that have some positive reviews: Calculus, Better Explained by Azad, Calculus: An Intuitive and Physical Approach by Kline, ... $\endgroup$ – user137731 Jan 5 '16 at 1:15
  • $\begingroup$ Calculus Made Easy by Thompson, or even James Stewart's Calculus. And of course, there is always Khan Academy for when you can't understand some specific topic in your textbook/ class. $\endgroup$ – user137731 Jan 5 '16 at 1:21
  • $\begingroup$ You should look for Leithold's or Stewart's calculus. They are very intuitive and they provide a good introduction to the topic. Apostol is quite advanced for a beginner. $\endgroup$ – cgo Jan 5 '16 at 8:41
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Some of the books you mentioned deal with real analysis, which go well beyond the level of a first-semester calculus course. You will in fact need three or four semesters of calculus (depending on your institution), linear algebra, and proof-writing skills before you can go near those.

When I was self-studying Calculus I and II over the summer, I used two texts: A First Course in Calculus, Third Edition by Serge Lang and Calculus and Analytic Geometry, Fifth Edition by Thomas and Finney. Both are older editions that I inherited, so you may only be able to get newer editions. I found that the Lang was great for providing a strong theoretical foundation to calculus; his proofs are quite clear and offer deeper insights to the subject than you would simply get from a minimal amount of technique. The Thomas and Finney was suitable for filling in some of the gaps not covered in the Lang such as implicit differentiation and some physical applications of integration such as the shell/washer methods, Theorems of Pappus, moments, center of mass, etc.

I felt that the Lang was deficient when it came to series and sequences; in this respect the Thomas and Finney was far superior in articulating the concepts with plenty of examples.

Both texts had the answers to both the odd- and even-numbered exercises, although I am not sure if this is still true for newer editions.

Aside from book materials, Paul's Online Notes for Calculus are a fine supplementary resource if you need some extra clarification.

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Apostol is a very rigorous book. I studied both Apostol and Stewart. If you have some solid background already, I'd recommend you to get involved in Apostol. However, for a beginner, it is somewhat difficult. Firstly, you can consider a book like Stewart, Thomas and solve some problems to improve your thinking. Then, you can make a transition to rigorous proofs.

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I have read over the Apostol's Calculus Volume 1 line by line after finishing the course I had on the calculus! It was a grief for me that I didn't do this when I had the course.

$1.$ The author is really really good in writing. It always provides motivations for the subjects via historical remarks or intuition.

$2.$ Furthermore, this is best fitted for those who have a nice background in elementary mathematics and want to go through a rigorous theoretical approach to calculus.

$3.$ The proofs are nice and clear and sometimes they challenge you to do some parts by yourself.

$4.$ The book contains final answers to the exercises that does not require a proof.

$5.$ It also makes you ready for a good transition to real analysis in future.

In conclusion, I honestly enjoyed reading this book and suggest this to anyone who may look for a book for a first course in calculus.

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