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A First Course in Calculus (Undergraduate Texts in Mathematics) [Hardcover]

Serge Lang
5.0 out of 5 stars  See all reviews (1 customer review)
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Book Description

1 Mar 1998 0387962018 978-0387962016 1986. 5th Corr.
This fifth edition of Lang's book covers all the topics traditionally taught in the first-year calculus sequence. Divided into five parts, each section of A FIRST COURSE IN CALCULUS contains examples and applications relating to the topic covered. In addition, the rear of the book contains detailed solutions to a large number of the exercises, allowing them to be used as worked-out examples -- one of the main improvements over previous editions.

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Product details

  • Hardcover: 752 pages
  • Publisher: Springer; 1986. 5th Corr. edition (1 Mar 1998)
  • Language: English
  • ISBN-10: 0387962018
  • ISBN-13: 978-0387962016
  • Product Dimensions: 4.8 x 16 x 23.3 cm
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Bestsellers Rank: 324,739 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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Most Helpful Customer Reviews
2 of 2 people found the following review helpful
5.0 out of 5 stars Excellent first calculus text 3 July 2009
Format:Hardcover|Verified Purchase
This edition is similar in content to his book "Short Calculus", only about 3 times as big. Like "Short Calculus", it is intended as an introductory course - first year at University and possibly for good A' level students.

The material is presented in a very clear and easy to understand. It starts right at the begining. The first two chapters contains preliminary material essential to understand Calculus. The first 4 parts of the book has the same chapters as "Short Calculus", but containing extra material. If you have done both A'level and Further A' level mathematics, then you will have seen most of the topics in these first 4 parts of the book. The 5th part concerns functions of several variables. You will typically only see this at University.

Serge Lang was one of the main contributors to Nicholas Boubaki, and is both an eminent mathematician and teacher. So there is no surprise that the material contains rigour, even though the concepts are expressed so clearly and simply. Besides the clear explanations, there are some excellent proofs that are so much simpler than those I have seen in A'level texts. A good example is for addition formulae - cos(A+B)that only requires pythagoras theorem and a single identity that can itself be derived from pythagoras theorem for the proof.

Each chapter contains numerous exercises. These start of very easy and gradually get more difficult. In the appendix at the end of the book appears the answers to many of the questions in these exercises. It is the perfect book for self-study.

This has become one of my favourate introductory calculus texts. I highly recommend this book to all those readers interested in mathematics.
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Amazon.com: 4.5 out of 5 stars  8 reviews
23 of 24 people found the following review helpful
4.0 out of 5 stars Effectively conveys key concepts and skills. 1 Aug 2008
By N. F. Taussig - Published on Amazon.com
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.
13 of 13 people found the following review helpful
5.0 out of 5 stars shines without all the bling and flash 24 Jun 2007
By tech book guy - Published on Amazon.com
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.
27 of 31 people found the following review helpful
4.0 out of 5 stars Calculus for beginning college students 28 Aug 2002
By A Customer - Published on Amazon.com
Format:Hardcover|Verified Purchase
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.
12 of 12 people found the following review helpful
5.0 out of 5 stars Promotes real understanding of calculus 26 Mar 2008
By Coleman Nee - Published on Amazon.com
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.
5 of 5 people found the following review helpful
3.0 out of 5 stars Good book, not great 14 Feb 2010
By W. Ghost - Published on Amazon.com
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.
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