A First Course in Calculus

by Serge Lang

The purpose of a first course in calculus is to teach the student the basic notions of derivative and integral, and the basic techniques and applica­ tions which accompany them. The very talented students, with an ob­ vious aptitude for mathematics, will rapidly require a course in functions of one real variable, more or less as it is understood by professional is not primarily addressed to them (although mathematicians. This book I hope they will be able to acquire from it a good introduction at an early age). I have not written this course in the style I would use for an advanced monograph, on sophisticated topics. One writes an advanced monograph for oneself, because one wants to give permanent form to one's vision of some beautiful part of mathematics, not otherwise ac­ cessible, somewhat in the manner of a composer setting down his sym­ phony in musical notation. This book is written for the students to give them an immediate, and pleasant, access to the subject. I hope that I have struck a proper com­ promise, between dwelling too much on special details and not giving enough technical exercises, necessary to acquire the desired familiarity with the subject. In any case, certain routine habits of sophisticated mathematicians are unsuitable for a first course. Rigor. This does not mean that so-called rigor has to be abandoned.

Genres: Mathematics, Calculus, Science, Nonfiction

746 pages, Hardcover

First published December 1, 1964

About the author

Serge Lang was an influential mathematician in the field of number theory. Algebra is his most famous book.

Librarian Note: There is more than one author in the GoodReads database with this name. See this thread for more information.

Reviews

[Arun · Rating 5 out of 5]

4th edition.
Perfect book to review all the basic concepts of single variable calculus on field R.
If you need epsilon-delta based rigorous proofs on the notion of limit/least upper bound property etc, it is given in the appendix.
It serves as a good textbook companion to MIT's single variable calculus videos by Prof. Jerison. of which I am a big fan. Most chapters had right amount of appeal to geometry before proving the theorems. But the last chapter on power series unabashedly approaches epsilon-delta formalism therefore unsuitable for light reading/skimming.

Side note: This is a calculus book written by an algebraist! I hope to read an Algebra book written by a confirmed Analyst (It looks like "Linear algebra done wrong" might fit the bill).

[Sasha · Rating 4 out of 5]

This was an excellent text for self-studying the beginnings of calculus. It makes proofs accessible, and even taught me how to prove concepts that seemed obvious but were easy to forget when not used for a while. Learning how to derive and prove concepts like this is essential for someone who wants to study physics, which is the case for me.

I have reservations for recommending this book for Calculus II and beyond. Seeing second- and third-semester calculus topics listed in the table of contents raised my hopes for learning Calc II and III from Lang. Beginning with Taylor polynomials, however, Lang's calculus skipped explanations, became unnecessarily abstract when explaining relatively simple ideas, and yet did not cover the operations I needed in enough depth. This text was not sufficient for learning the calculus of vectors and volume integration.

If you understand pre-calculus and trigonometry, go with this text to learn derivative and integral calculus. Go elsewhere for series, vectors, and multivariable functions.

[B2p0n · Rating 5 out of 5]

This is a very good book for Calculus beginners. The new edition of the book has fully solved examples.

[Muhammed · Rating 5 out of 5]

This is a very good book for learning calculus. I intended to use it as a refresher for my knowledge, but it went well beyond being a refresher. That was because Lang is an algebraist; hence, he has a more structural approach to the subject. That approach, combined with his brevity, makes a really good book for a second pass on the subject. He derives many foundational concepts from first principles; the construction is coherent, and his philosophical remarks in certain chapters —such as in Chapter XII— can change your perspective on calculus. It might prove slightly difficult for the mathematically not-so-mature student, but it's definitely worth it.

[Anke Human · Rating 5 out of 5]

Even as a sixteen year old, I feel as if this was a good foundation in calculus for me. The size of the book is not at all intimidating, nor are the Exercises. I also appreciate answers to the exercises at the back of the book. Certainly a lot different than standard high-school texts. I recommend for any high-school or college student.

[Lime Street Labrador · Rating 5 out of 5]

Nice and easy on basic topics from numbers and differentiation to Taylor polynomial and series, with easy-to-follow explanations. Good stepping stone for further studies.

[Petitjean Louis-Frédéric · Rating 4 out of 5]

Trivialement trivial