Calculus book recommendations (for complete beginner)

Well I have not started calculus yet but I am really keen to.
I would love if you suggest some books.

Points to be noted:

  1. I really don’t like the way textbooks are written so please no “textbooks”
  2. I am COMPLETELY beginner in calculus. I know a little bit of trigonometry and what functions are but not really in depth.The book should start from the base, I mean really from the base. I may require a precalculus book too if the books don’t cover that.
  3. I would really love if the book shows how calculus was developed, why it was developed and things like that.

Well,thanks in advance!

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If you want to learn calculus, you should ensure you have mastered material typically covered in a Precalculus course. And if you want to learn calculus, you’re going to have to have some sort of “textbook.” And some are better than others.

That said, a very nice supplement to a textbook is Michael Spivak’s
A Hitchhiker’s Guide to Calculus. It won’t replace a calculus textbook, but it really is great reading to understand calculus a bit more intuitively. And it outlines the development of Calculus, and the motivation for its development to some degree. You might enjoy this site that gives timeline of the history of calculus.

I’ll also provide a link to the Khan Academy, where you can review pre-requisite material, and supplement your journey through Calculus with video lectures, practice problems, etc.

Finally, here is a link to Paul’s Online Math Notes. The link will take you to the Calculus I notes, but there’s a menu at the top of the page where you can select notes for algebra/precalculus. Paul’s Notes are really an instructive tutorial that allows you to proceed at your own pace, provides exercises, organizes the material into “modules” so you can work through and digest sub-sections/topics progressively.

Quoted from here:


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).

Janusz, Calculus

The worst calculus book ever written. This was the 150s text in
1994–95; it tries to give a Spivak-style rigorous presentation in
colorful mainstream-calculus-book format and reading level. Horrible.
Take a look at it to see how badly written a mathematics book can be.

See more recommendations here:
Chicago undergraduate mathematics bibliography

Below are two well known books for what you seem to be looking for.

W. W. Sawyer, What is Calculus About? (1962).

David Berlinski, A Tour of the Calculus (1997).

I love Spivak, Courant and Hardy the most, and the previous posters have mentioned them. But there is one basic foundations book which made me end up LOVING calculus (and WANT to read the above books): LV Tarasov’s Calculus: Concepts for High School. It’s a Soviet book available for download online.

  • Very small book, written in the form of a conversation between the student and the teacher.
  • Builds from the very basics, and covers a wide expanse (even DEs) in a short number of pages.
  • Every concept is first explained intuitively, and then rigorously formulated.
  • You won’t ever “forget” anything you learn, because you will be capable of building all the “rules” from scratch quickly. It will also help you figure when analysis doesn’t work.
  • So if you think sequentially, and want to know the “Why?” wherever possible, this book’s brilliant for the beginner.

Not all textbooks are written the same…

I’d look at William Chen’s lecture notes, and at the books by (and recommended by) the Trillia Group. Look around for lecture notes, there are sets of nice ones (and homework, exams with solutions, etc) at OCW. On Coursera they also carry lectures as videos.

There are many, many more excellent resources. Wikipedia covers details, the Wikiversity has textbooks in different stages of completion, and you might want to rummage in Wikibooks.

I recommend Precalculus: Mathematics for Calculus by Stewart and after that Calculus Early Transcendentals by Stewart. I really like Stewart’s style of writing as he provides many examples, there are a lot of good exercises (some with online hints), and the structure of his books is very good.

For pre-calculus I would recommend Pre-Calculus For Dummies.

For calculus I would recommend Calculus For Dummies & Calculus II For Dummies.

You can also buy these three books if you are a really absolute beginner:

  1. Algebra I For Dummies
  2. Algebra II For Dummies
  3. Geometry For Dummies

You can see for more details about the books.

You could try H. Jerome Keisler’s Elementary Calculus which is a high school introductory text using infinitesimals (as opposed to limits), freely downloadable from the author’s website as a PDF.

For a textbook in calculus, I would recommend Schaum’s outline series for calculus:

It starts off with the absolute basics (about lines and circles) and ends up on advanced multivariable calculus. It has huge numbers of solved and unsolved problems so I would definitely recommend that. They are also available in other maths topics, such as differential geometry and probability.

I would also consider using the website Brilliant if you require a more basic foundation:

This has plenty of unsolved problems from precalculus and algebra to more advanced topics.

For the history of calculus, as well as the book by Spivak mentioned above, I would recommend the history of the calculus and its conceptual development by Carl Boyer:

It gives a comprehensive overview of the history and development of calculus. I hope this helps. I can give more recommendations of you would like.