Customer Reviews

A First Course in Calculus (Undergraduate Texts in Mathematics)

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As a high school teacher, I used this text with great success several times for both AP Calculus BC and AP Calculus AB courses. It is my favorite calculus text to teach from, because it is very user-friendly and the material is presented in such an eloquent way. There are no gratuitous color pictures of people parachuting out of airplanes here. Opening this book is like entering a temple: all is quiet and serene. Epsilon-delta is banished to an appendix, where (in my opinion) it belongs, but all of the proofs are there, and they're presented in a simple (but not unsophisticated) way, with a minimum of unnecessary jargon or obtuse notation. He doesn't belabor the concept of "limit"; most calculus books beat this intuitively obvious concept into the ground. Even though it doesn't cover all of the topics on the AP syllabus, I would rather supplement and use this text rather than any other.

Serge Lang's text does an effective job of teaching you the skills you need to solve challenging calculus problems, while teaching you to think mathematically. The text is principally concerned with how to solve calculus problems. Key concepts are explained clearly. Methods of solution are effectively demonstrated through examples. The challenging exercises reinforce the concepts, while enabling you to develop the skills required to solve hard problems. Answers to the majority of exercises (not just the odd-numbered ones) are provided in a hundred page appendix, making this text suitable for self-study. In some sections, such as related rates and max-min problems, Lang provides many fully worked out solutions.

As effectively as Lang conveys the key concepts and teaches you how to solve problems, he does not neglect the subject's logical development. Topics are introduced only after their logical foundations have been laid. Results are derived. Theorems are proved when Lang feels that they will add to the reader's understanding. Through his exposition and his grouping of logically related exercises, Lang teaches the reader how a mathematician thinks about the subject.

The book is divided into five sections: review of basic material, differentiation and elementary functions, integration, Taylor's formula and series, and functions of several variables. The heart of the course is the middle three sections.

Most of the topics covered in the review of basic material should be familiar to most readers. However, it is still worth reading since there are challenging problems, properties of the absolute value function are derived from defining the absolute of a number as the square root of the square of the number, conic sections and dilations may be unfamiliar to some readers, and Lang views the material through the prism of a mathematician who knows what concepts are important for understanding higher mathematics.

Lang introduces the derivative as the slope of a curve in order to motivate the introduction of the idea of a limit. Next, Lang teaches you techniques of differentiation and shows you how to use them solve applications such as related rate problems. After a detailed discussion of the sine and cosine functions, Lang introduces the Mean Value Theorem and illustrates how it can be used for curve sketching and solving for maxima or minima. Lang covers properties of inverse functions before concluding the section by defining the natural logarithm of x as the area under the curve y = 1/x between 1 and x and defining the exponential function f(x) = e^x as its inverse.

The integral is introduced as the area under a curve, with the natural logarithm taken as the motivating example. Lang explains the relationship between integration and differentiation before introducing techniques of integration and their applications. Integration with respect to polar and parametric coordinates is introduced to expand the range of applications. The exercises introduce additional tricks that enable you to solve integrals that do not succumb to the basic techniques. A table of integrals is included on the inside of the book's front and back covers.

Lang's demonstrates the power of differential and integral calculus through his discussion of approximation of functions through their Taylor polynomials. This chapter should also give you an idea of how your calculator calculates square roots and the values of trigonometric, exponential, and logarithmic functions. The behavior of series, including convergence and divergence tests, concludes the material on single variable calculus.

The material on functions of several variables in the final section of the book is covered in somewhat greater detail in Lang's Calculus of Several Variables (Undergraduate Texts in Mathematics). Since the corresponding chapters in that text include additional sections on the cross product, repeated partial derivatives, and further techniques in partial differentiation and an expanded section on functions depending only on their distance from the origin, I chose to read these chapters in Lang's multi-variable calculus text. The material that is included here, on vectors, differentiation of vectors, and partial differentiation, should give the reader a solid foundation for a course in multi-variable calculus.

I have some caveats. There are numerous errors, including some in the answer key. Some terminology is nonstandard, notably the use of bending up (down) for concave up (down), or missing, limiting the text's usefulness as a reference. In the chapter on Taylor polynomials, when Lang requests an answer accurate to n decimal places, what he really means is that the error in the answer should be less than 1/10^n, which is not the same thing. At one point, Lang claims that the Extreme Value Theorem, which he leaves unnamed, is obvious. I turned to the more rigorous texts Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (Second Edition) by Tom M. Apostol and Calculus by Michael Spivak, where I discovered proofs covering one and half pages of text of the Extreme Value Theorem and a preliminary result on which it depends that Lang does not state until an appendix much later in the book. Perhaps Lang meant the Extreme Value Theorem is intuitive. While I found much of the text to be clear, I sometimes found myself turning to Apostol's text for clarification when I read Lang's proofs.

Despite my reservations, I think this text is well worth reading. Reading the text and working through the exercises gives you a good understanding of the key concepts and techniques in calculus, enables you to develop strong problem solving skills, prepares you well for more advanced mathematics courses, and gives you a sense of how mathematicians think about the subject.

I needed to bring my high school calculus up to speed for first year physics studies and found this to be the only book which covered the necessary ground. The material is presented in a thorough manner with the great majority of topics shown with proofs. The book is very well organized and there are abundant worked examples. Some problems are offered which deal with matters not covered in the text, but usually there is a worked example given among the answers. Lang deals with the material in a clear fashion so that the subject matter is usually not difficult to follow.On the negative side I can say that there is no human touch between the covers. His sole attempt at humor is an item following a list of problems in which he notes "relax". In the foreword he exhibits his firm belief that many freshmen arrive unprepared for college calculus, which may be true. But nowhere in the book is there a note of encouragement, so it cannot be described as reader friendly. Finally the index is pathetic--just three pages for a book of 624 pages, so that finding things can be frustrating.

I had to take a refresher calculus course as a prerequisite to get into graduate school, but the assigned text (Edwards and Penney) was horrible. Like every other mass market calculus, it was filled with colorful diagrams and digressions on how to use calculators, but little in the way of explanation. Fortunately I found Lang's calculus in the university book store and it cured all of my problems. Unlike the bloated E&P, Lang's book is clear and concise. E&P covers more material to be sure, but for the essentials nothing beats Lang. After reading this book calculus became easy for me again. Which is as it should be, since calculus is a surprisingly simple subject if expalined well.

This book by the late Prof. Lang covers calculus in a clear and concise manner. I own more than a few
calculus books and this book is one of my favorites. The book looks like a math
book in that it is not a 1200 page glossy coloring book with multi-colored inserts on every
page. I think that the style of this book is a hugh improvement over most of the books on the market. I think a student who
buys this book along with a good calculus study guide would be very well set.

The book is OK, but I wouldn't say it's great. There are lots of exercises that
ask you to do simple symbolic manipulation so you'll remember rules -- but there are
too few exercises that require the reader to actually think harder and be creative. The explanations are often shallow and not as stimulating as they could be, in my opinion.

Some examples of sections that I think are not well written are the one about implicit differentiation (the discussion is too short and not clear, and there are less exercises in this section than in others); the one about rate of change (some examples are boring, like "find the rate of change of the area of a circle given the rate of change of its diameter"; he does not make it clear that he's always derives with relation to time and that, for example, the radius and height of a cylinder should be understood as functions of time, so there's a feeling of sloppiness).

It's a good book,anyway. Now, it becomes a really great book when compared to the
colorful, flashy books available today.

As earlier reviewers noted, there's nothing fancy about this book, just a clear explanation of single variable calculus.

a book that focuses on the foundation without trying to do too much and it does that very well. self-contained and easy-to-follow, this book promotes understanding of the basics. very good