The difference between a clear and lucid math book and one that is, to
coin a politically correct phrase, organizationally challenged can be
the difference between success and failure for the math student.
I can't tell you that I have read through even a sizeable fraction
of the calculus texts and study guides that are on the market, but I have looked at
a bunch of them. If you are taking a college calculus course,
you probably have already been assigned a required text. You will,
of course, have to buy that text, whether it is any good or not,
just so that you can do the homework assignments. But it doesn't
hurt anything besides your pocket book to have other references
at your fingertips. So here I shall make my recommendations from
among the books I am familiar with that are still in print.
If you click on a book, it will take you to the page of
where you can order that title. The prices listed here are those posted
by Amazon.com at the time I prepared this page.
Here is a book I highly recommend for anybody who tends to fall asleep while reading math books. The style in How to Ace Calculus: The Streetwise Guide by Joel Hass, Abigail Thompson, Colin Conrad Adams is so lively that I had to put it down in order avoid succumbing to the urge to plagiarize it. In the actual calculus material, you don't have to wade through a lot of stuff to get to the critical points. This book specializes in critical points -- that is the stuff you must know to pull down a good grade. And it describes them language so plain it will have you chuckling from time to time. It also has some great practical advice on how to select a good teacher, how to ask questions, how to study, etc. The examples worked in the text are not the really difficult ones that sometimes might show up on an exam, but you have to learn the not-so-hard ones before you can do the hard ones. To those of you who thought you could never understand math, this book is worth many times its price.
An outfit called Schaum's Outline Series has been publishing study aids in a wide variety of subjects for decades now. I once had a coworker who was fond of saying that he got his degree from Schaum's Institute of Technology. There are several calculus volumes that they publish. One of them I am quite familiar with, and that is the one by Frank Ayres and Elliot Mendelson. There are brief, clear explanations of the concepts and methods. But the emphasis is upon worked examples. There are over 1000 of them. For what it's worth, this book has sold over a million copies in the 30 years it's been in print.
Here's another from the Schaum's series. It has fewer worked problems, covers fewer topics, but has much more explanatory text. And the explanations are organized in a consistent way. Each section is divided into the concepts of Approximation, Refinement, and Limit. The author, Eli Passow, shows how these three stages are effective at attacking a wide variety of problems.
Having a different viewpoint of a topic always helps. Silvanus P. Thompson believed that calculus did not have to be difficult. In the first decade of the 20th century, he wrote a little book called Calculus Made Easy in which he appealed to concepts that most of us already have to explain how calculus works. His approach is different from the one commonly used in college calculus courses today (no epsilons and no deltas). He uses a concept he calls "orders of smallness." But when married to elementary algebra it leads to the very same conclusions you would learn in any calculus course. More recently Martin Gardner (author of Scientific American's Mathematical Games column) has annotated Thompson's work, and this edition is now available from St. Martin's Press. Some of Gardner's comments explain how Thompson's approach to a topic differs from the approaches that are popular today. The important point, though, is that Thompson's methods and his elegantly simple explanations lead you to an understanding that is every bit as useful and valid as the one you will get in class. And once you have traveled Thompson's road, your instructor's road won't seem nearly as rocky.
Of all the beginning calculus texts that have been foist onto students, there is one that, in my opinion, stands head and shoulders above the rest. Apparently I am not the only one to think so. Enough others feel this way to have kept this book in print for nearly 40 years. This is Calculus and Analytic Geometry by George B. Thomas Jr. at M.I.T. It is now in its 9th edition. Later editions have some contributions from Thomas' associate, Ross L. Finney. The development is logical, there are loads of worked examples, the explanations are clear, and the authors have had 9 editions and several decades to bring this work very near perfection.
For anyone planning a career in science or engineering, I recommend that you eventually get Mathematical Methods in the Physical Sciences by Mary Boas. It is a broad survey of theory and methods that you will be using to solve problems that involve physics or engineering. It is quite accessible to anybody who has had two semesters of calculus, and the explanations of the methods are clear and concise. When you take your advanced undergrad courses, this book may prove to be invaluable.
If your second or third semester calculus course covers differential equations, you will want this book. It begins by explaining concepts in linear algebra, which is a skill you will need if you plan to continue your math education. It then goes on to the basics of differential equations, with emphasis on linear differential equations. There is plenty of explanatory text and worked examples.
Here are two very useful reference books for those of you who will be using math in your future studies or careers:
More books will be appearing on this list soon, including titles that I recommend for those of you who plan to continue your math studies beyond first year calculus.
email me at hahn@netsrq.com