Customer Reviews

Essential Calculus with Applications (Dover Books on Mathematics)

5 star:		(7)
4 star:		(2)
3 star:		(2)
2 star:		(0)
1 star:		(0)

 

 

I am a graduate student in computer science, and I've forgotten quite a bit of math from college. I picked up this book and made a point to read it over the summer and do as many problems as I could. It is a very "tight" book, in the sense that there is not a lot of fluff (hence the "essential" in the title). The book seemed to maintain a nice balance between too hard and too easy so that I was always challenged. It is a very good book for me, and has a sufficient level of rigor so that I feel like I get some practice in that area as well. I would highly recommend it to someone wishing to learn calculus. In addition, it's always nice to find an author you like, because it usually means other books by that author will be good. As a bonus, it has solutions (or hints) to all the exercises so you can check your progress.

Silverman's book is much clearer than other books on calculus that I have seen recently. Everything is concise and there is no extra "fluff". Some may wonder about some seemengly strange orderings in the book such as introducing the differential before introducing limits but this serves to help create an intuitive understanding of limits before explaining their mathematical meaning. Otherwise, the problems range from extremely simple to challenging. All of the problems are reachable by just having read the previous section and are not out of the reach of a careful reader. Most of the problems have solutions and / or hints at the back of the book. The book's title includes the word "applications" and the book indeed does contain them. After each chapter on a new topic, the following chapter contains applications. In addition, the last two chapters on differential equations and multi variable calculus are better than in most introductory books. Essential Calculus is suited for class study or for self study and should be accessable to all with high school mathematics under their belt.

It is awesome that I can find most of the important calculus theorems and their proofs in only 250 pages. The author really shows all the proof clearly. This book is much better than the calculus book that I am using in my class. It is definitely worth more than 10 bucks!

This is a very neat little introduction to basic single variable calculus that also includes a final chapter on partial differentiation. It's fairly rigorous and would lead naturally into a first course in real analysis. The problems are all fairly easy and many can be `done in your head' (so saving paper) but they are typically not as boring as the usual `follow the method' type problems you see in many of the big introductory calculus texts. Some problems even ask you to construct proofs of simple theorems. This is all good stuff and for the money, who can argue? Given the aims, coverage and price, I don't see how it could be much better - five stars from me.

Richard Silverman's text carefully develops and clearly presents the aspects of differential and integral calculus that it covers. The examples are illuminating. Most of the theorems are proved, so the text is largely self-contained. The exercises, which range from routine calculations to difficult problems, derivations, or proofs that require insight and ingenuity to solve, are interesting. Answers or hints are provided for virtually every exercise in the text in two answer keys, one of which was adapted for this edition from an instructor's manual for an earlier edition of the text. Consequently, this text is suitable for self-study.

However, using this text could be problematic for a student new to the subject. The text is dense. A novice would benefit from a more leisurely treatment that includes more examples and a greater number of routine problems at the start of the exercise sets so that she or he could obtain more practice with the techniques of differentiation and integration treated here before being confronted with the challenging problems at the end of the exercise sets. The reader must contend with occasional errors, which, alas, is not unusual in a mathematics text. More importantly, this text omits important topics, which also makes it problematic for those who wish to review calculus.

The calculus of trigonometric functions is omitted entirely, with consequent implications for the applications of differentiation and integration that can be discussed or the integration techniques that can be introduced. While Silverman assumes that the reader is familiar with the partial fraction decomposition of a rational expression with non-repeated linear factors in the denominator, he does not cover integration by partial fractions. Applications of integration are limited to those that involve the solution of simple differential equations, so there is no coverage of the calculation of volumes, surface areas, or arc lengths. There is minimal coverage of sequences. The treatment of infinite series is limited to geometric series, so Taylor series are omitted entirely. These omissions severely limit the usefulness of this text, when used in isolation, for students of mathematics, the physical sciences, or engineering.

If you can live with these limitations, you will benefit from working through this text because of Silverman's clear exposition, interesting examples and problems, and his provision of answers to those problems. If not, you may wish to track down a copy of Silverman's Modern Calculus and Analytic Geometry, a self-contained text that treats single- and multi-variable calculus. I have found that text to be a useful reference.

Silverman begins the book with a review of the foundations you will need in order to understand the differential and integral calculus covered in the text. The topics covered include sets, number systems, proofs by mathematical induction, inequalities, absolute value, intervals and neighborhoods, lines, functions, and graphs. A weakness of the text is that there are several proofs that should be handled by mathematical induction for which Silverman proves the result for the cases n = 1, 2, and 3, only to assert that the method used can be generalized to higher values of n.

Limits are introduced via the derivative. Silverman emphasizes the geometric meaning of the derivative as the slope of the tangent line to a point by introducing the Delta notation. He continues to use the Delta notation in his algebraic proofs, which I sometimes found unhelpful. After discussing properties of limits and continuity, Silverman discusses properties of the derivative and techniques of differentiation. These include the sum, difference, product, and quotient rules and the Chain Rule. He explains velocity and acceleration in terms of the derivative and demonstrates how to solve related rate problems. Silverman discusses properties of continuous and differentiable functions, including the Intermediate Value Theorem and Mean-Value Theorem. He also explains how to find local and global extrema, determine concavity, and solve optimization problems.

Silverman introduces the antiderivative and the definite integral and discusses their properties. The natural logarithm of x, where x > 0, is introduced as the area under the curve 1/x between 1 and x. Properties of the logarithm are derived from the integral used to calculate this area. The exponential function with base e is introduced as the inverse of the natural logarithm function. With this knowledge in hand, logarithmic differentiation is introduced. Silverman's treatment of logarithmic and exponential functions is a particular strength of this book.

Silverman demonstrates the techniques of integration by substitution and integration by parts and how to deal with improper integrals. He then illustrates how to solve some simple ordinary differential equations, particularly by the method of separation of variables. The problem sets become noticeably harder at this point. The methods for solving differential equations are applied to growth and decay problems and motion problems.

The final chapter introduces multi-variable calculus. Silverman discusses limits, the Chain Rule, and extrema problems for functions of several variables, while relating these topics to their single-variable analogues. The treatment of extrema problems includes a discussion of Lagrange multipliers. Silverman's treatment of this material is less rigorous than his treatment of single-variable calculus. More results are stated without proof. However, there is still an increase in difficulty. The text concludes with the most challenging problem set in the book.

While I do not recommend using this book in isolation, it is useful as a review of the topics it covers and as a source of interesting problems.

Addendum: (25 August 2009) I call your attention to problem 16 in section 3.8 (Optimization Problems). It reads:

"Given a point P inside an acute angle, let L be the line segment through P cutting off the triangle of least area from the angle. Show that P bisects the part of L inside the angle. Show that this property also characterizes the point P in problem 10."

Problem 10 reads:

"Given a point P = (a, b) in the first quadrant, find the line through P which cuts off the angle of least area from the quadrant."

In the second answer key at the back of the text, Silverman outlines a solution for problem 16 in terms of the slope of line L in which he chooses a coordinate system in which the angle's vertex coincides with the origin and its sides coincide with the positive x-axis and the line y = mx, where m, x > 0. This choice of coordinates does not work.

In attempting to solve the problem using the same coordinate system with a different method (expressing the x-intercept of line L in terms of the coordinates of point P), I discovered that I could not account for the possibility that line L was vertical, a possibility that Silverman's solution does not take into account. My friend, Christopher T. Allen, pointed out that the possibility that line L was vertical was simply an artifact of the choice of coordinates that Silverman and I had used. Chris chose his coordinates so that the vertex of the angle coincides with the origin and its sides coincide with the positive y-axis and the line y = mx, where m, x > 0 (the reciprocal of the value of m above), thereby eliminating the possibility that line L was vertical. With this choice of coordinates, he adapted my method of solution (with the y-intercept replacing the x-intercept) in order to complete the proof. Using the same coordinate system that Chris had used, I was able to prove the result using the method that Silverman outlined (albeit with the reciprocal slope).

The book is nicely, and concisely, organized. Based on the topics covered and the explanations presented, I would place it somewhere between typical calculus service course texts and introductory texts on real analysis. As a result it can be used for self-study in preparation for a more applications-oriented calculus course, a prelude to a real analysis course, or a short refresher for those who've had calculus in the past.

There is a brief introductory chapter (35 pages) before the chapter on Differential Calculus begins. Subsequent chapters are Differentiation as a Tool, Integral Calculus, and Integration as a Tool. The text concludes with a chapter on Functions of Several Variables.

Although this Dover edition includes additional answers not available in the original, the solution section was not retyped. The additional answers are not integrated into the original answer section; they are included, in a different font, as a separate section after the original. This is only a minor issue, but it does require some extra 'page flipping' as answers are checked. However, the additional answer section is of real value for self-study, probably the most likely reason this book will be purchased.

Questions range from easy to hard, with harder questions marked by asterisks. Asterisked questions ask the reader to solve problems that are only peripherally related to the section's topic, or where no problem solving exemplar had been previously presented. For example, the reader is asked to find the rational number representations for a set of repeating decimal numbers when the procedure to do this had not yet been discussed. Fortunately, in many of these cases problems have been sequenced so that the results of earlier questions can provide insight into solving later ones.

This book was written before the "questions proliferation" juggernaut of recent times. So, although there is the occasional section with thirty or so questions, most sections contain twenty of less well chosen questions reinforcing the section's core concepts, with asterisked questions asking the reader to consider new material. Thus, readers can consider all problems and still complete the book within a reasonable period.

In proofs, the author often takes the interesting approach of returning to basic definitions rather than using theorems previously proven. For example, the section on inequalities presents theorems with algebraic proofs that multiplying both sides of an inequality by the same positive (negative) quantity does not (does) change the sense of the inequality. However, in the solutions section, the author often returns to basic definitions for proofs, e.g., a > b means a - b > 0, even where the use of previously proven theorems would have produced more concise results.

There are some minor additional issues: In the section on sets, Venn diagrams are unfortunately missing. These usually prove helpful to those new to this topic. Occasionally, solutions in the "hints and answers" section do not provide either answers or hints. For example, a problem asks the reader to "verify" a statement. The answers section provides the unhelpful 'solution', "Trivial, but give details anyway". In a similar vein, occasionally a question such as "(why?)" is asked within the text, with the answer posed for the reader but with no further explanation presented.

Although, the back cover states this can be "ideal as a primary text...", it does not cover many topics now typically included in introductory calculus courses. Thus, this will not be its main application. However, its conciseness combined with the excellent style of presentation, means it can be highly recommended as a self-study refresher for those who previously studied calculus, as a supplement to those currently in a calculus course, or as a comparatively short prelude for courses in calculus or real analysis.

This book apparently was written for the the typically one semester course in introductory calculus designed for business, education, and life science majors. For this purpose, it is excellent. It is very clearly written and has good problems.
However, it should be mentioned that the book does not cover trig functions at all. Thus, it is not suitable for courses intended for physical science or engineering majors. It does cover log and exponential functions though, in a way that was new to me. Rather than deriving the derivative of log, it simply defines log in terms of the integral of 1/x. The exponential is then defined as the inverse of log.

I started reading this in a bookstore one day and was hooked in by the author's style. His exposition is clear and succinct. It had been over ten years since I took calculus and I was about to take a course in analysis so I wanted to refresh myself. I was very glad to find out that the text was on proof side in the sense that basic calculus theorems were proven, and delta and epsilons were given their due. An introductory chapter on absolute value was also nice as it suddenly is thrust to the forefront in analysis. In essence, I was able to accomplish a twofold purpose. On the one hand, I was able to review the calculus that I had learned so long ago. On other hand, I learned some basic analysis that I had not seen before. With this text, I felt well prepared to take the analysis class. It's excellent for reviewing calculus while moving forward at the same time. And at just over 200 pages, it's possible to get through the entire text. It's also great for self-study with answers or hints for all problems in the back. Though it should be pointed out that it is more focused on the theoretical side. Sure, applications are present, but no more so than in standard calculus texts that I have seen.

I have completed Calculus in University (Engineering), I used this as a nice review to keep sharp. Picked up on a few things and better understand a few things (perhaps simply because I care about math now more then back then), but the book is useful. Whether you want a quick review or you are studying for the first time this is a cheap way to do so, and the challenging questions are pretty useful. Any slight hesitations about the book are meaningless (like two editions of answers, the different sizes of similar books by dover).... It is a great book that one can get for less than $20. I have no real regrets about buying this book.

I like this book. The author's writing style is excellent, and that is what makes this worthwhile.

The only issue is that Essential=less. The author avoids trig functions compeletely, which simplifies and makes this book a good choice for those who are looking for a nice refresher.

It does have some value to the self-study, but it should not be the only text used. The structure of the book is a bit odd, and at times his presentation could use a few more examples.
Especially when he goes through various differentation techniques.

He does provide enough of a foundation with functions, lines, and preliminaries like intervals and numbers.

The odd part is how he structured the instruction. Basically he is concise with the presentation of theory, and really puts the application into later chapters. The coverage of integration is outstanding. His sections on applications is very helpful.

Overall he covers the main theorems of Calculus, but it is Essential, and not everything a student needs to know. But it is a great intro or refresher.