Calculus With Applications

Abstract

Our purpose in writing a calculus text has been to help students learn at first hand that mathematics is the language in which scientific ideas can be precisely formulated, that science is a source of mathematical ideas that profoundly shape the development of mathematics, and that mathematics can furnish brilliant answers to important scientific problems. This book is a thorough revision of the text Calculus with Applications and Computing by Lax, Burstein, and Lax. The original text was predicated on a number of innovative ideas, and it included some new and nontraditional material. This revision is written in the same spirit. It is fair to ask what new subject matter or new ideas could possibly be introduced into so old a topic as calculus. The answer is that science and mathematics are growing by leaps and bounds on the research frontier, so what we teach in high school, college, and graduate school must not be allowed to fall too far behind. As mathematicians and educators, our goal must be to simplify the teaching of old topics to make room for new ones. To achieve that goal, we present the language of mathematics as natural and comprehensible, a language students can learn to use. Throughout the text we offer proofs of all the important theorems to help students understand their meaning; our aim is to foster understanding, not “rigor.”We have greatly increased the number of worked examples and homework problems.We have made some significant changes in the organization of the material; the familiar transcendental functions are introduced before the derivative and the integral. The word “computing” was dropped from the title because today, in contrast to 1976, it is generally agreed that computing is an integral part of calculus and that it poses interesting challenges. These are illustrated in this text in Sects. 4.4, 5.3, and 10.4, and by all of Chap. 8. But the mathematics that enables us to discuss issues that arise in computing when we round off inputs or approximate a function by a sequence of functions, i.e., uniform continuity and uniform convergence, remains.We have worked hard in this revision to show that uniform convergence and continuity are more natural and useful than pointwise convergence and continuity. The initial feedback from students who have used the text is that they “get it.” This text is intended for a two-semester course in the calcu