Calculus book recommendations (for complete beginner)

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:

Calculus (ELEMENTARY)

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

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.