Essential Calculus with Applications (Dover Books on Mathematics) by Richard A. Silverman (PDF)

Description

Calculus is an extremely powerful tool for solving a host of practical problems in fields as diverse as physics, biology, and economics, to mention just a few. In this rigorous but accessible text, a noted mathematician introduces undergraduate-level students to the problem-solving techniques that make a working knowledge of calculus indispensable for any mathematician.The author first applies the necessary mathematical background, including sets, inequalities, absolute value, mathematical induction, and other “precalculus” material. Chapter Two begins the actual study of differential calculus with a discussion of the key concept of function, and a thorough treatment of derivatives and limits. In Chapter Three differentiation is used as a tool; among the topics covered here are velocity, continuous and differentiable functions, the indefinite integral, local extrema, and concrete optimization problems. Chapter Four treats integral calculus, employing the standard definition of the Riemann integral, and deals with the mean value theorem for integrals, the main techniques of integration, and improper integrals. Chapter Five offers a brief introduction to differential equations and their applications, including problems of growth, decay, and motion. The final chapter is devoted to the differential calculus of functions of several variables.Numerous problems and answers, and a newly added section of “Supplementary Hints and Answers,” enable the student to test his grasp of the material before going on. Concise and well written, this text is ideal as a primary text or as a refresher for anyone wishing to review the fundamentals of this crucial discipline.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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

⭐, 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.

⭐I first encountered this book more than a decade ago, when I used it as a supplement for self-study in first-year calculus (basically in the manner of a Schaum’s Outlines — just another portable resource with problems and answers). I was happy to find it again! In terms of coverage, it is reminiscent of business-oriented calculus books (for instance, it is very strong on optimization and differentiation techniques but is somewhat light on integration and has little coverage of trigonometric functions), but its level of rigor is easily on par with “standard” undergraduate calculus texts (e.g. Stewart). Since I was especially interested in business/economic applications, which are less abundant in the standard texts, Silverman was perfect. As I discovered later, there are several excellent books in elementary mathematical economics (Chiang/Wainwright and Sydsaeter/Hammond come to mind), but these were expensive then and are now in the neighborhood of $200, and Silverman is still a great read at a great price. Dover is a treasure trove of math classics, but most are fairly advanced books of little use to college math minors like myself. Silverman is an exception: more rigorous than business calculus books, but with fantastic applied problems in finance and basic economics, and good all-around explanations of core calculus ideas (it even includes a brief but pretty good discussion of the epsilon-delta technique in limit problems, something absent even from most math-for-economists books, let alone business calculus textbooks). Bravo to Dover; I hope this one stays in print for years to come!

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