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Calculus Made Easy Books by Martin Gardner CALCULUS MADE EASY BOOKS BY MARTIN GARDNER Fads and Fallacies in the Name Order and Surprise of Science The Whys of a Philosophical Mathematics, Magic, and Mystery Scrivener Great Essays in Science (ed.) Puzzles from Other Worlds Logic Machines and Diagrams The Magic Numbers of Dr. The Scientific American Book of Matrix Mathematical Puzzles and Knotted Doughnuts and Other Diversions Mathematical Entertainments The Annotated Alice The Wreck of the Titanic Foretold The Second Scientific American Riddles of the Sphinx Book of Mathematical Puzzles The Annotated Innocence of and Diversions Father Brown Relativity for the Million The No-Sided Professor (short The Annotated Snark stories) The Ambidextrous Universe Time Travel and Other The Annotated Ancient Mariner Mathematical Bewilderments New Mathematical Diversions The New Age: Notes of a Fringe from Scientific American Watcher The Annotated Casey at the Bat Gardner's Whys and Wherefores Perplexing Puzzles and Penrose Tiles to Trapdoor Ciphers Tantalizing Teasers How Not to Test a Psychic The Unexpected Hanging and The New Ambidextrous Universe Other Mathematical Diversions More Annotated Alice Never Make Fun of a Turtle, My The Annotated Night Before Son (verse) Christmas The Sixth Book of Mathematical Best Remembered Poems (ed.) Games from Scientific American Fractal Music, Hypercards, and Codes, Ciphers, and Secret Writing More Space Puzzles The Healing Revelations of Mary The Snark Puzzle Book Baker Eddy The Flight of Peter Fromm (novel) Martin Gardner Presents Mathematical Magic Show My Best Mathematical and Logic More Perplexing Puzzles and Puzzles Tantalizing Teasers Classic Brainteasers The Encyclopedia of Impromptu Famous Poems of Bygone Days Magic (ed.) Aha! Insight Urantia: The Great Cult Mathematical Carnival Mystery Science: Good, Bad, and Bogus The Universe Inside a Science Fiction Puzzle Tales Handkerchief Aha! Gotcha The Night Is Large Wheels, Life, and Other Last Recreations Mathematical Amusements Visitors from Oz CALCULUS MADE EASY BEING A VERY-SIMPLEST INTRODUCTION TO THOSE BEAUTIFUL METHODS OF RECKONING WHICH ARE GENERALLY CALLED BY THE TERRIFYING NAMES OF THE DIFFERENTIAL CALCULUS AND THE INTEGRAL CALCULUS Silvanus P. Thompson, F.R.S. AND Martin Gardner Newly Revised, Updated, Expanded, and Annotated for its 1998 edition. palgrave Cl Martin Gardner 1998 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with * the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London WH 4LP. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988. Original edition written by Silvanus P. Thompson and published 1910 Subsequent editions 1914, 1946 This edition first published in North America 1998 by St Martin's Press Published by PALGRAVE Houndmills, Basingstoke, Hampshire RG21 6XS and 175 Fifth Avenue, New York, N. Y. 10010 Companies and representatives throughout the world PALGRAVE is the new global academic imprint of St. Martin's Press LLC Scholarly and Reference Division and Palgrave Publishers Ltd (formerly Macmillan Press Ltd). ISBN 978-0-333-77243-0 ISBN 978-1-349-15058-8 (eBook) DOI 10.1007/978-1-349-15058-8 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. A catalogue record for this book is available from the British Library. 10 9 8 7 6 5 07 06 OS 04 03 02 CONTENTS Preface to the 1998 Edition 1 Preliminary Chapters by Martin Gardner 1. What Is a Function? 10 2. What Is a Limit? 18 3. What Is a Derivative? 30 Calculus Made Easy by Silvanus P. Thompson Publisher's Note on the Third Edition 36 Prologue 38 I. To Deliver You from the Preliminary Terrors 39 II. On Different Degrees of Smallness 41 III. On Relative Growings 45 IV. Simplest Cases 51 V. Next Stage. What to Do with Constants 59 VI. Sums, Differences, Products, and Quotients 66 VII. Successive Differentiation 79 VIII. When Time Varies 83 IX. Introducing a Useful Dodge 94 X. Geometrical Meaning of Differentiation 103 XI. Maxima and Minima 116 XII. Curvature of Curves 132 XIII. Partial Fractions and Inverse Functions 139 XlV. On True Compound Interest and the Law of Organic Growth 150 Xv. How to Deal with Sines and Cosines 175 XVI. Partial Differentiation 184 v vi Contents XVII. Integration 191 XVIII. Integrating as the Reverse of Differentiating 198 XIX. On Finding Areas by Integrating 210 XX. Dodges, Pitfalls, and Triumphs 227 XXI. Finding Solutions 235 XXII. A Little More about Curvature of Curves 249 XXIII. How to Find the Length of an Arc on a Curve 263 Table of Standard Forms 276 Epilogue and Apologue 279 Answers to Exercises 281 Appendix: Some Recreational Problems Relating to Calculus, by Martin Gardner 296 Index 326 About the Authors 330 PREFACE TO THE 1998 EDITION Introductory courses in calculus are now routinely taught to high school students and college freshmen. For students who hope to become mathematicians or to enter professions that require a knowledge of calculus, such courses are the highest hurdle they have to jump. Studies show that almost half of college freshmen who take a course in calculus fail to pass. Those who fail almost always abandon plans to major in mathematics, physics, or engi­ neering-three fields where advanced calculus is essential. They may even decide against entering such professions as architecture, the behavioral sciences, or the social sciences (especially econom­ ics) where calculus can be useful. They exit what they fear will be too difficult a road to consider careers where entrance roads are easier. One reason for such a high dropout rate is that introductory calculus is so poorly taught. Classes tend to be so boring that stu­ dents sometimes fall asleep. Calculus textbooks get fatter and fat­ ter every year, with more multi color overlays, computer graphics, and photographs of eminent mathematicians (starting with New­ ton and Leibniz), yet they never seem easier to comprehend. You look through them in vain for simple, clear exposition and for problems that will hook a student's interest. Their exercises have, as one mathematician recently put it, "the dignity of solving crossword puzzles." Modern calculus textbooks often contain more than a thousand pages-heavy enough to make excellent doorstops-and more than a thousand frightening exercises! Their prices are rapidly approaching $100. "Why do calculus books weigh so much?" Lynn Arthur Steen 1 2 Preface to the 1998 Edition asked in a paper on "Twenty Questions for Calculus Reformers" that is reprinted in Toward a Lean and Lively Calculus (Math­ ematical Association of America, 1986), edited by Ronald Doug­ las. Because, he answers, "the economics of publishing compels authors. to add every topic that anyone might want so that no one can reject the book just because some particular item is omitted. The result is an encyclopaedic compendium of tech­ niques, examples, exercises and problems that more resemble an overgrown workbook than an intellectually stimulating introduc­ tion to a magnificent subject." "The teaching of calculus is a national disgrace," Steen, a math­ ematician at St. Olaf College, later declared. "Too often calculus is taught by inexperienced instructors to ill-prepared students in an environment with insufficient feedback." Leonard Gillman, writing on "The College Teaching Scandal" (Focus, Vol. 8, 1988, page 5), said: "The calculus scene has been execrable for many years, and given the inertia of our profession is quite capable of continuing that way for many more." Calculus has been called the topic mathematicians most love to hate. One hopes this is true only of teachers who do not appreci­ ate its enormous power and beauty. Howard Eves is a retired mathematician who actually enjoyed teaching calculus. In his book Great Moments in Mathematics I found this paragraph: Surely no subject in early college mathematics is more excit­ ing or more fun to teach than the calculus. It is like being the ringmaster of a great three-ring circus. It has been said that one can recognize the students on a college campus who have studied the calculus-they are the students with no eyebrows. In utter astonishment at the incredible applica­ bility of the subject, the eyebrows of the calculus students have receded higher and higher and finally vanished over the backs of their heads. Recent years have seen a great hue and cry in mathematical circles over ways to improve calculus teaching. Endless confer­ ences have been held, many funded by the federal government. Dozens of experimental programs are underway here and there. Some leaders of reform argue that while traditional textbooks Preface to the 1998 Edition 3 get weightier, the need for advanced calculus is actually dimin­ ishing. In his popular Introduction to the History of Mathematics. Eves sadly writes: "Today the larger part of mathematics has no, or very little connection with calculus or its extensions." Why is this? One reason is obvious. Computers! Today's digi­ tal computers have become incredibly fast and powerful. Con­ tinuous functions which once could be handled only by slow ana­ log machines can now be turned into discrete functions which digital computers handle efficiently with s~ep-by-step algo­ rithms. Hand-held calculators called "graphers" will instantly graph a function much too complex to draw with a pencil on graph paper. The trend now is away from continuous math to what used to be called finite math, but now is more often called discrete math.
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