Preview: Calculus for Everyone

CALCULUS FOR EV ERYONE

CALCULUS FOR EV ERYONE U N D E R S TA N D I N G T H E

M AT H E M AT I C S O F

C H A N G E

MITCH STOKES W ITH I LLU S TR ATI O N S BY S U M M E R S TO KE S

INHERIT THE HUMANITIES | MOSCOW, IDAHO

Mitch Stokes, Calculus for Everyone: Understanding the Mathematics of Change Text copyright Š2020 by Mitch Stokes.

Illustrations by Summer Stokes, summerstokes.com.

Cover design by Rachel Rosales, orangepealdesign.com.

Published by Roman Roads Press 121 E. 3rd Street, Moscow, Idaho 83843 509-592-4548 | romanroadspress.com Printed in the United States of America.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form by any means, electronic, mechanical, photocopy, recording, or otherwise, without prior permission of the author, except as provided by USA copyright law.

Library of Congress Cataloging-in-Publication Data available on romanroadspress.com.

ISBN-13: 978-1-944482-54-1 ISBN-10: 1-944482-54-7 Version 0.3.0 | June 2020 | Advance Reader Copy

To Morris Kline (1908–92), who changed my view of mathematics . . . and my view of the mathematics of change

CONTENTS ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

HOW TO USE THIS BOOK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii 2 Quizzes and Exams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii 3 Scheduling Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 4 Read the Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 5 Doing the Problems: Study Questions and Exercises. . . . . . . . . . . . . . . . . . . . . xix PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii

1

THE PROBLEM OF CHANGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

2

SPACE, TIME, AND NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

3

Change Is Important. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Order Is Also Important . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Thales Sets a Good Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Pythagoras: A Mathematical Cosmos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Plato the Pythagorean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Plato, Pythagoras, and the Liberal Arts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Plato’s Pythagorean Project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Up Next: Problems with Space, Time, and Numbers. . . . . . . . . . . . . . . . . . . . . 13 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Zeno’s Paradox: Is Change Even Possible?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Problems with Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Representing Physical Objects with Mathematical Ones . . . . . . . . . . . . . . . . . . 24 Problems with Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Quantifying the World: Numbers as Stand-Ins . . . . . . . . . . . . . . . . . . . . . . . . . 27 Problems with Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 The Problem of Motion and User-Friendliness. . . . . . . . . . . . . . . . . . . . . . . . . . 32 Up Next: Is Speed Coherent?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34

THE PARADOX OF SPEED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 3.2 3.3 3.4

Average Speed versus Instantaneous Speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 An Instant: A Point in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Introducing the Delta Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Applying the Delta Notation to Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.5 3.6 3.7

4

Is Our Definition of Speed Coherent? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Up Next: Plato’s Project During Aristotle’s Reign. . . . . . . . . . . . . . . . . . . . . . . 47 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

THE PLATONIC-PYTHAGOREAN PROJECT BEFORE THE SCIENTIFIC REVOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1 The Big Picture (So Far) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 Eudoxus Is the First to Complete His Homework . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Geometry and the Influence of Euclid’s Elements. . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.4 What about Terrestrial Objects?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.5 Aristotle and the Problem of Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.6 Aristotle and the Platonic-Pythagorean Project . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.7 Aristotle’s Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.8 Aristotle’s Physics of Free Fall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.9 Archimedes: Mathematics and Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.10 Up Next: The Scientific Revolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.11 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.12 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5

THE SCIENTIFIC REVOLUTION AND THE NEED FOR CALCULUS . . . . 75

5.1 What Is the Scientific Revolution?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2 Copernicanism, Pythagoreanism, and Cosmology . . . . . . . . . . . . . . . . . . . . . . . 79 5.3 Galileo, Archimedes, and Terrestrial Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.4 Free Fall: Galileo versus the Aristotelians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.5 Updating Galileo’s Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.6 A Way to Calculate Instantaneous Velocity? Alas.. . . . . . . . . . . . . . . . . . . . . . . 92 5.7 Up Next: Newton to the Rescue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.8 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6

NEWTON AND CALCULUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.1 6.2 6.3 6.4 6.5 6.6

7

Newton Unifies the Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 The Enlightenment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 What Is Calculus? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Why Is It Called Calculus?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Up Next: The Essential Toolkit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

THE ESSENTIAL TOOLS: VARIABLES, FUNCTIONS, AND GRAPHS . 111

7.1 7.2 7.3

Variables and the Problem of Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Which Physical Properties Can Numbers Stand For?. . . . . . . . . . . . . . . . . . . . . 115 Interpreting and Predicting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.4 7.5 7.6 7.7 7.8 7.9

8

Functions: Relations between Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Graphs: Pictures of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Descartes and Analytic Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Some Conventions Regarding Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Up Next: Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

A NEW TOOL: THE LIMIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.1 Approaching a Forbidden Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.2 Approaching the Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.3 Remembering a Function’s History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 8.4 Same Limit, Different Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.5 Standing versus Seeing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 8.6 Up Next: Instantaneous Speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.7 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.8 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9

THE METHOD OF APPROXIMATION AND DEFINING INSTANTANEOUS SPEED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 9.1 The Need for Speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 9.2 Free Fall and Average Speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 9.3 The Method of Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9.4 Defining Instantaneous Speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 9.5 Up Next: Calculating Instantaneous Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 9.6 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 9.7 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

10 USING THE METHOD OF INCREMENTS TO CALCULATE INSTANTANEOUS SPEED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.1 Remember: Dividing by 0 is Illegal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 10.2 We Need a Different Formula for Average Velocity. . . . . . . . . . . . . . . . . . . . . . 166 10.3 Finding a Formula for Average Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 10.4 The Big Picture: The Method of Increments. . . . . . . . . . . . . . . . . . . . . . . . . . . 172 10.5 Approaching the Instant from the Opposite Direction. . . . . . . . . . . . . . . . . . . . 174 10.6 Up Next: Finding a Function for Instantaneous Speed. . . . . . . . . . . . . . . . . . . . 176 10.7 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 10.8 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

11 USING THE METHOD OF INCREMENTS TO FIND AN INSTANTANEOUS SPEED FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 11.1 Finding v(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 11.2 Galileo Would Be Proud (or Jealous) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

11.3 Mathemagics? An Example of Mathematics’ Power. . . . . . . . . . . . . . . . . . . . . . 185 11.4 Appreciating v(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 11.5 Up Next: Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 11.6 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 11.7 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

12 THE DERIVATIVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 12.1 Instantaneous Speed and the Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 12.2 Derivative: The Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 12.3 Rate of Change at an Instant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 12.4 The Method of Increments in Terms of Rate of Change. . . . . . . . . . . . . . . . . . . 199 12.5 Example: Using the Method of Increments to Find the Derivative. . . . . . . . . . 200 12.6 Derivatives versus Plain Ol’ Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 12.7 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 12.8 Up Next: Finding the Derivative of y(x) = ax² + bx + c. . . . . . . . . . . . . . . . . . . . 206 12.9 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 12.10 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

13 FINDING MORE DERIVATIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 13.1 The Derivative for Any Function of the Form y(x) = ax². . . . . . . . . . . . . . . . . . 212 13.2 An Infinite Number of Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 13.3 The Derivative for Any Function of the Form y(x) = bx. . . . . . . . . . . . . . . . . . . . 216 13.4 The Derivative for Any Function of the Form y(x) = c. . . . . . . . . . . . . . . . . . . . . 218 13.5 The Derivative for Any Function of the Form y(x) = ax² + bx + c. . . . . . . . . . . . . 220 13.6 The Derivative of a Derivative (of a Derivative). . . . . . . . . . . . . . . . . . . . . . . . 223 13.7 Up Next: The Derivatives of Power Functions. . . . . . . . . . . . . . . . . . . . . . . . . 225 13.8 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 13.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

14 USING THE POWER RULE TO FIND DERIVATIVES . . . . . . . . . . . . . . . . . 229 14.1 The Power Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 14.2 Derivatives for a Sum of Power Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 14.3 Other Kinds of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 14.4 Up Next: Returning to the Problem of Change . . . . . . . . . . . . . . . . . . . . . . . . 238 14.5 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 14.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

15 BACK TO THE PROBLEM OF CHANGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 15.1 15.2 15.3 15.4

Acceleration: How Fast Speed Changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Up Versus Down: Taking Direction into Account. . . . . . . . . . . . . . . . . . . . . . . 245 Dropping an Object. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 Interpreting Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

15.5 Throwing an Object Straight Up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 15.6 Throwing an Object Straight Down. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 15.7 Free Fall in General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 15.8 Why Does Differentiation Work?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 15.9 Up Next: Graphic Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 15.10 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 15.11 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

16 GRAPHS AND SLOPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 16.1 Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 16.2 Slopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 16.3 Start with ∆x and ∆y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 16.4 A Changing Slope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 16.5 The Derivative as Slope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 16.6 Up Next: Graphing the Method of Increments. . . . . . . . . . . . . . . . . . . . . . . . . 279 16.7 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

17 SLOPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 17.1 Finding ∆y/∆x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 17.2 The Derivative: The Instantaneous Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 17.3 Slopes and the Method of Increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 17.4 What about the Power Rule? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 17.5 Up Next:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 17.6 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 17.7 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

18 SLOPES AND THE PROBLEM OF CHANGE . . . . . . . . . . . . . . . . . . . . . . . . . 295 18.1 Mathematical Pictures versus Physical Pictures. . . . . . . . . . . . . . . . . . . . . . . . 296 18.2 Slopes and Free Fall: d(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 18.3 Slopes and Free Fall: v(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 18.4 Slopes and Free Fall: a(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 18.5 Up Next:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 18.6 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 18.7 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

19 MORE INFORMATION FROM DERIVATIVES . . . . . . . . . . . . . . . . . . . . . . . . 307 19.1 Extracting Information from Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 19.2 Slope versus y-Value (Again) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 19.3 Derivatives of Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 19.4 Up Next: Derivatives, Graphs, and Free Falls . . . . . . . . . . . . . . . . . . . . . . . . . . 318 19.5 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 19.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

20 LOOKING CLOSER AT GRAPHS OF FREE FALL . . . . . . . . . . . . . . . . . . . . . 323 20.1 The Problem of Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 20.2 Galileo’s Law of Free Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 20.3 Taking Direction into Account Again. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 20.4 Interpreting the Height Function’s Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 20.5 Interpreting the Velocity’s Function’s Graph . . . . . . . . . . . . . . . . . . . . . . . . . . 329 20.6 Interpreting the Acceleration Function’s Graph. . . . . . . . . . . . . . . . . . . . . . . . . 331 20.7 Combining All Three Functions on a Single Graph. . . . . . . . . . . . . . . . . . . . . 334 20.8 An Object Thrown Vertically Upward. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 20.9 Up Next: Antiderivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 20.10 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 20.11 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

21 THE ANTIDERIVATIVE: UNDOING DERIVATIVES . . . . . . . . . . . . . . . . . . 347 21.1 The Reverse Power Rule: A First Approximation. . . . . . . . . . . . . . . . . . . . . . . 349 21.2 A Problem with the Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 21.3 The Correct Reverse Power Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 21.4 Some New Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 21.5 Antiderivatives for a Sum of Power Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 355 21.6 Antiderivatives: Not Just for Power Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 357 21.7 The Problem of Change and Finding C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 21.8 Notation: Finding Two C’s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 21.9 Finding C’s from a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 21.10 Up Next: Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 21.11 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 21.12 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

22 DEFINING THE INTEGRAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 22.1 22.2 22.3 22.4 22.5 22.6

The Method of Exhaustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 Integrals and Area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Area under a Curve: Integrals and Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 373 Integrals and Circumscribed Rectangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Up Next: The Method of Summation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

23 USING THE METHOD OF SUMMATION TO CALCULATE INTEGRALS 383 23.1 23.2 23.3 23.4 23.5

The Method of Summation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Special Sum Rules and the “+ · · · +” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Inscribed Areas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 Circumscribed Areas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 More (Complicated) Examples: y(x) = x² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

23.6 The Integral Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 23.7 Up Next: The Fundamental Theorem of Calculus. . . . . . . . . . . . . . . . . . . . . . 407 23.8 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 23.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

24 THE FUNDAMENTAL THEOREM OF CALCULUS . . . . . . . . . . . . . . . . . . . . 411 24.1 Derivatives and Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 24.2 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 24.3 The Fundamental Theorem: The Integral Is the Antiderivative . . . . . . . . . . . . 415 24.4 Derivatives and Integrals (Again) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 24.5 Indefinite Integrals (versus Definite Integrals). . . . . . . . . . . . . . . . . . . . . . . . . . 421 24.6 Integrals and the Problem of Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 24.7 Coming to Terms with Calculus: Integrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 24.8 Coming to Terms with Calculus: Differentiate . . . . . . . . . . . . . . . . . . . . . . . . . . 426 24.9 Up Next: More about Areas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 24.10 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 24.11 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

25 INTERPRETING AREAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 25.1 Interpreting the Area of Acceleration Functions. . . . . . . . . . . . . . . . . . . . . . . . 435 25.2 Interpreting the Area of Velocity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 437 25.3 Interpreting the Area of Height Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 25.4 Up Next: Some Final Words about the Problem of Change. . . . . . . . . . . . . . . 444 25.5 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .444 25.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

EPILOGUE: THE PROBLEM OF CHANGE . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

1 Pythagoreanism Today. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 2 (Much) More to Come. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 3 But What about Zeno?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 4 Study Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 5 Exercise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

ACKNOWLEDGMENTS “Show, don’t tell” is such a helpful rule of thumb—for writers, for teachers, for boastful teenage boys. This book is my own attempt to show what I’ve been telling people for fifteen years: that there’s a better way to teach mathematics. And though the book is specifically about calculus, its point is general.

Here are few ways in which I intend Calculus for Everyone to show you a better way to teach mathematics. For one thing, the book majors on the majors (another helpful guideline for nearly everything in life). It spends a lot of time on the high-yield concepts of calculus, those concepts through which all the rest of calculus must been seen. But even more, simply in virtue of understanding these crucial concepts, the student will understand calculus as a whole. It also teaches those high-yield concepts intuitively, and as a coherent whole, as opposed to a hodgepodge of seemingly pedantic and unrelated ideas. Moreover, it puts calculus in its historical and philosophical context, which not only helps explain a surprising amount of Western culture, but actually helps the student understand the workings of calculus itself.

Calculus for Everyone is partly borne out of fifteen years of teaching calculus to liberal arts students, many of whom have had only algebra. During those initial years, I used Morris Kline’s textbook, Calculus: An Intuitive and Physical Approach.

His book is remarkably . . . well . . . intuitive and how I think every calculus student should first approach the subject. But despite the brilliance of Kline’s approach, his book still included far too much material (nearly a thousand pages) and was too uneven in its level of presentation. Calculus for Everyone is intended to avoid those problems while keeping with the spirit of Kline’s general strategy. His influence can

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Acknowledgments

be seen throughout this book, including a few specific homework problems. Indeed, I dedicate the entire work to him.

In addition to seeing the success of my own students, Calculus for Everyone is also the result of my having had seemingly two wildly different careers: one as an engineer, the other, more recently, as an academic philosopher. Seeing firsthand what kind(s) of mathematics engineers learn, what kinds they use most, while also learning what kinds have been philosophically and culturally important, has impressed upon me the importance of calculus to any well-rounded education. I did not go looking for calculus’s significance (or for that of mathematics generally), and if you had told me twenty years ago what I would find, I would have scoffed. But I cannot unsee what I have seen. I hope you too will experience some of the same surprise.

I’m also extremely grateful for Roman Roads Media, not merely for giving me the opportunity to write this book, but for seeing the importance of calculus for a liberal arts education and how calculus is a key component to a larger educational movement that has already been revolutionary. Roman Roads’ president and CEO, Daniel Foucachon, has amazed me with his breadth of competence, his guidance, his support, his energy, and his graciousness. In fact, Roman Roads in general has simply been a great pleasure to work with.

A lot goes into producing a mathematics textbook, more than I ever imagined— and it’s probably best I didn’t know this at the beginning. At the same time, it’s extremely gratifying to see the project come together. Rachel Rosales at Orange Peal Design was just wonderful at giving focus and life to my half-baked cover ideas. With respect to the inside of the book, my wildly talented daughter, Summer Stokes, did all the illustrations as well as designing the internal layout. Having a graphic artist and illustrator of her caliber on this project was the highlight of the whole process, and that this artist is also my little girl (as she always will be) made the whole experience significantly more meaningful.

Others, too, made enormous contributions. Valerie Bost edited the entire manuscript—imagine having to edit a calculus book—and transferred the initial Word document into InDesign, an unenviable task requiring not merely Herculean effort, but also a deftness with subtleties and details. And when it came to the details, my thanks also goes out to my students at New Saint Andrews College throughout the years. In particular, I am grateful for the junior class of 2018–2019, especially Levi Law. These students gave a huge amount of valuable and detailed feedback as

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Chapter  Acknowledgments

they went through a previous version of this book. They were patient, gracious, and encouraging throughout. Jim Nance, too, meticulously read the entire book and I’m pretty sure checked all the examples and problems. And though I have had these folks and others—including Dr. Matthew Abraham—as a safety net, any remaining errors and infelicities are still obviously mine.

As I mentioned, this book required more of my life than I had initially planned and no one knows that better than my wife, Christine. All the many ways she has made this book possible are as invisible to others as they are crucial to me. I thank her most of all.

HOW TO USE THIS BOOK 1 P R E R E Q U I S ITE S

The only prerequisite for this course is a very basic understanding of Algebra 1. The student should, for example, be familiar with polynomials and be able to manipulate such polynomials (e.g., by combining like terms and manipulating exponents). The focus of the book is on the concepts of calculus proper and not on calculus’s further application to increasingly complicated functions, such as rational, trigonometric, exponential, or logarithmic functions. By keeping to polynomials throughout the course (or, more accurately, to power functions), the student will better identify what is the “calculus part” of calculus. Once the concepts of calculus have been mastered, then students can—if they choose—go on to apply those concepts in ever more complicated ways. By better understanding what calculus is not, the student will better understand what it is.

2 Q U I Z Z E S A N D E X A M S

I recommend that you use the study questions and exercises in the back of the book as a pool of potential quiz and exam questions. These should be entirely sufficient, though teachers should modify and supplement them as they see fit. But such supplementation isn’t necessary.

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3 S C H E D U LI N G O P TI O N S

Although the schedule and pace are entirely up to you, the most obvious choice is between either a single-semester or a year-long course. The precise schedule will depend on the specific student(s) and school (which includes homeschools). A teacher may wish to move leisurely through the material over the entire school year to allow the student to digest the material and let it sink in and include time for review. On the other hand, a faster-paced semester-long course may be appropriate for students who have a strong background in mathematics. And in either case, the course would build a strong conceptual foundation for a further, more standard calculus course if you should so choose.

4 R E A D TH E B O O K

Unfortunately, most students do not really read their math textbook. Instead, they merely use it as a kind of reference manual to look up specific concepts and methods when they are stuck on a homework problem. But students should read the text on their own—either before or after the teacher’s presentation of the concepts (or both). One of the goals of this book is to give students practice reading a math book, which is, importantly, unlike reading other types of books.

Of course, on the one hand, a math book should be read like any other book, from front to back, chapter by chapter, line by line. But in other ways, reading math is unique. It will take time for students to adjust to it, and it will take work once they do. A math book should not only be read line by line but word by word and symbol by symbol. Tiny details matter in mathematics, and one small slip—whether it be in writing or reading—can throw everything off course.

Remember: Safety First! Also, even if you have read everything, there will be times when you don’t understand a particular sentence or step. When this happens, don’t go on to the next sentence or step. Read it again, slowly turning it over and over in your mind. Indeed, you may have to read the same sentence over and over. Or you may have to return to the beginning of the section or chapter and begin again. In fact, the great Sir Isaac Newton himself sometimes had to do this. For example, when reading through Rene Descartes’ Geometry,

How to Use This Book

Newton found himself getting stymied almost immediately. He then returned to the beginning of the book and reread until he got stuck again, this time a few pages farther on. He repeated this process until he finished the book. You won’t find yourself in such an extreme situation, but if the top mathematician in the world—the man who invented calculus—had to do this, you shouldn’t be surprised if you have to reread sentences, sections, or chapters. And you should probably be surprised if you don’t.

Mathematics obviously has an additional component beyond ordinary English: mathematical symbols. And when it comes to reading mathematical symbols, there are times when you simply have to stare at the formula or step until you recognize what is happening. And of course, there will be times when no amount of staring or reading helps (whether because the author didn’t present it clearly enough, or he made a mistake that the editors didn’t catch, or you simply don’t have the ability to deal with it on your own). This, again, will happen to everyone at some point.

This all may sound horrible, but it’s not nearly so bad if you know ahead of time that you’ll have to do this. And the benefits of reading this closely and carefully are general and go far beyond the math course itself. In addition to careful thinking, it improves overall attention to detail and observational skills, both of which are important human virtues that apply to nearly all of life. Also, once you realize that reading mathematics takes this much work just about everyone, you’ll be less likely to pigeonhole yourself as someone who is “not a math person.”

5 D O I N G TH E P R O B LE M S : S T U DY Q U E S TI O N S A N D E XE R C I S E S

In this book, there are two types of problems for nearly all the chapters: study questions and exercises. The study questions are easy to answer and are intended to help you identify and memorize important facts and concepts. The exercises, on the other hand, are traditional math problems and typically require much more work. The answers to the study questions are in the chapter text itself, while the Solutions to the exercises are provided in the Solutions manual, which is separate from the main text.

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xx  How to Use This Book Study Questions

Again, the study questions will draw your attention to some of the facts (or “grammar”) of the chapter. They are to ensure that you notice the highlights in each chapter. Nearly all the answers can be found explicitly in the chapter’s text, and the questions are usually asked in the order the text presents the answers. These are entirely open-book questions (though, teachers and parents, see below). Moreover, the questions sometimes overlap. This overlap is designed to help you understand the same idea from different angles.

Let me point out that we won’t begin doing math problems in the traditional sense until later chapters. (Again, the traditional-style problems are called exercises.) That said, the study questions are, strictly speaking, also math problems since they are ultimately about facts and concepts of mathematics. And if you understand these questions, you’ll understand the material far better than if you had done only the exercises; indeed, they will help you understand the exercises. The distinction between study questions and exercises is entirely artificial.

You should neatly write out (or type) your answers to the study questions and save these answers in a dedicated folder or notebook—a central, easy-to-access location for you to readily review throughout the course.

To teachers and parents: these study questions should also be used as quiz and exam questions. You should also orally ask the student(s) these questions as a means of evaluating their understanding of the chapter and to instigate class discussion.

It would also be a great idea to use them as review. You might also want to have students turn in their answers just to keep them accountable, but without carefully checking their answers, since the answers are right there in the text.

Exercises

Again, neatly write out all your work for these exercises and keep them in your course folder. Having them all in one place will to make it easier to study for quizzes and exams.

By the way, it is entirely appropriate—indeed, necessary—to do the exact same problems more than once, even repeatedly (and even in a single sitting).

Repetition is one of the keys to success in mathematics. Review, which is just repetition over a longer period of time, is also crucial, so teachers should assign

5 Doing the Problems: Study Questions and Exercises

previous exercises (and study questions) to keep important material fresh in the student’s mind.

Another key to success in mathematics is neatness and organization. As a general rule, you should carefully write out all of the steps. Admittedly, this is a hassle, but doing the steps merely in your head is dangerous. Again, in mathematics our motto is “Safety First!” Also, do not write too small—and don’t cram your work together. There are no points for saving space, and such savings will likely cost you in the form of simple mistakes. And speaking of simple mistakes, everyone makes them in mathematics, even the experts.

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The calculus has proved to be the richest lode that mathematicians have ever struck. —Morris Kline 1908–1982

P R E FA C E 1

Mathematics and a Classical Christian Education

2

Calculus Isn’t a Luxury

3

The Right Kind of Less

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P R E FA C E 1 M ATH E M ATI C S A N D A C L A S S I C A L C H R I S TI A N E D U C ATI O N

Classical Christian education has come a long way. Countless folks—well, I can’t count them, anyway—have worked extremely hard over the last generation to “repair the ruins.” There’s a lot to be grateful for. Christian educators have made great strides in integrating the Bible with various academic disciplines, recognizing that all truth is God’s truth and is therefore, somehow, related. God’s creation, for example, is a universe, a unified whole, despite its diverse contents. We now have genuine hope for building a Christian worldview in the coming generations.

There is, however, one important area where we’ve not quite been able to put things together, and that’s mathematics. For one thing, we still teach it like the rest of America (but not necessarily like the rest of the world). I’ll say a bit about this in a moment. Perhaps more importantly, we’ve not yet figured out how to integrate mathematics with the rest of the curriculum—with history, philosophy, and theology. Sure, we know that mathematics reflects God’s orderly nature, and good for us—it’s a crucial truth, despite how difficult it is to flesh out the details.

But we no longer see the real importance of mathematics to the broader Western culture, particularly to philosophy. Nor do we see the strong influence of philosophy on mathematics, or the influence of both of these on science (and vice versa). Math, philosophy, and science grew up together, and only recently have we tried to separate them. This is particularly unfortunate given the intellectual and cultural battles that we must deal with today. It would be no exaggeration that science is one of the major

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Preface

cultural forces today and subtly but powerfully shapes our view of reality, including the kinds of things that we believe are possible and impossible. Of course, this is why atheists today have more apparent force than ever: they offer plausible-sounding arguments for the claim that science has shown there is no God—or at least that science hasn’t found God, and that we therefore have no evidence for His existence.

Christians have good reason to be skeptical about these claims, but often their rebuttals are misplaced. We often try to beat secular scientists at their own game, playing science by their rules. Even many creation scientists fall into this trap.

One of the main Solutions to all this is a course or two in the philosophy of science and, in particular, the integrated history of philosophy, science, and mathematics. Only then will we be able to root out the main fallacies of atheistic arguments.

In fact, learning this integrated history—along with its timeless lessons—is necessary for an adequate understanding of philosophy itself. And without understanding philosophy, there is no classical education, Christian or otherwise.

Of course, without knowing how to do math, we’re not going to get very far in understanding its significant place in the history of ideas. This isn’t an either/or; it’s a both/and. We need to learn how to do math and learn how it fits into the broader Western intellectual tradition.

You might find yourself thinking at this point, “Look, we already spend a lot of time on mathematics; there’s no way we can add to the workload.” My response is twofold. First, we need not add to the workload, but need instead to teach math more effectively. Second, learning the historical, philosophical, and theological significance of mathematics will take place mostly at the high school level and above, when students are able to grasp more abstract concepts and fit them together into a coherent picture. (That’s not to say that there’s nothing that can be done in the earlier years, but for now our attention is on students your age.) Also, and perhaps this is a third response, we can’t afford to ask how we could possibly cover anything more than doing problems, as natural as that question is. To say this is akin to saying that we’re too busy driving to get gas.

One of the goals of this book—in addition to teaching calculus classically—is to give both teachers and students an example of how mathematics can be taught from a classical and Christian perspective. That is, I want to show how I think mathematics education can be improved rather than merely tell you. First, I’ll say a few things about calculus in particular.

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xxvi  Preface

2 C A LC U LU S I S N ’ T A LUXU RY

Anyone of average intellectual horsepower—and a very basic understanding of algebra—can learn the fundamentals of calculus. So why is it that only about fourteen percent of our high school students study calculus? Why is calculus seen as a luxury, a delicacy appreciated only by mathematical connoisseurs?

For two reasons, mainly. One, we’re not motivated enough to teach it. We have no idea how central calculus is to Western intellectual culture; as one math teacher helpfully put it, calculus is the mathematical counterpart to the works of Shakespeare. Having a basic grasp of calculus—including why it was invented— puts a lot of western philosophy and science in perspective. In fact, calculus is an ideal place for those involved in classical education to begin teaching mathematics classically, which is why I think this book will be so helpful for us at this stage in the game. The only other bit of mathematics that has had as much impact on the humanities or liberal arts is Euclid’s Elements. If calculus is akin to Hamlet, Euclid is like unto Homer’s Odyssey. Until our students learn the fundamentals of calculus and Euclid’s Elements, they’ll never integrate mathematics with the rest of their studies, and therefore they’ll never really understand the whole.

The second reason calculus is considered a luxury is that it just seems too hard for students who aren’t exceptional at math. But it’s not too hard. I’ve taken liberal arts students who visibly blanch at the mention of mathematics, students who are by no means “math people” and, in eight weeks, taught them real calculus: derivatives, integrals, limits, and the Fundamental Theorem of Calculus. Moreover, they understand the central meanings and purposes of these concepts. And they’re delighted; never did they dream that they’d be doing calculus, much less in so short a time.

The problem isn’t that the fundamentals of calculus are too difficult for the average student. Rather, the problem is that we make it difficult. Not that calculus doesn’t require work, but we make it far harder than it has to be by covering too many topics too quickly, many topics which are unnecessary at that stage of the students’ learning. More isn’t always better, and in the case of mathematics it can cripple the student. The National Science Foundation sees this more-is-better approach as the primary reason for America’s current math debacle. In testimony before the House Committee on Science, NSF director Dr. Rita Colwell said, “U.S. textbooks contain many more topics than those in other countries. For example, the

Preface

science textbooks we give to our eighth graders cover some 67 topics. In Germany, they cover 9 topics. As the saying goes, we are learning less and less about more and more.”1 As a result, the slower students feel like failures, and the students who are hardwired for math never fully flourish but are all the while congratulating themselves on their mere ability to follow mathematical recipes.

An avalanche of concepts makes calculus unnecessarily complicated, masking its overall structure and even its very point.

But simply spending more time on fewer concepts isn’t enough of a change; too much, too fast isn’t the only cause of difficulty. Another cause is covering the right concepts at the wrong time. In fact, there are concepts that should be left out altogether until the core of the discipline is mastered. All concepts aren’t created equal.

This is a general principle for teaching anything. Until a basketball player can dribble the ball in his or her sleep, he or she won’t be ready for complicated plays.

3 TH E R I G HT KI N D O F LE S S

The alternative to all this, of course, is to teach only the right concepts at just the right time, but deciding which concepts to include—and when—isn’t at all easy. It requires familiarity with how mathematics is actually used in the advanced sciences and in industry as well as knowledge of the subject’s historical development, including the overall importance of the individual concepts to that development.

It also takes a healthy knowledge of philosophy—and of how science, math, and philosophy are a cord of three strands. We shouldn’t be surprised at this difficulty.

Again, we’re in the “repairing the ruins” stage of academic reform, which means there’s a lot of work to do. Yet it also means that we’re on the steep part of the learning curve and so there’s potential to reap significant rewards relatively quickly.

This is the one sense in which we may have it easier than subsequent generations.

Our yield promises to be extremely high.

This book, as I said, is an example of what I have in mind.

1 NSF’s FY2000 Budget, Second Hearing on U.S. Math and Science Education—Programs Aimed at Grades K-12, Before the House Science Committee Basic Research Subcommittee, 106th Cong. (1999) (Statement of Rita Colwell, Director of the National Science Foundation), https://www.nsf.gov/about/ congress/106/rc90428k_12edu.jsp.

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We live in a universe that is always changing, full of matter that is always moving.

—Lee Smolin, Time Reborn: From the Crisis in Physics to the Future of the Universe Change is never complete, and change never ceases.

—C.S. Lewis, “De Descriptione Tempore”

CHAPTER 1 THE PROBLEM OF CHANGE

1.1

Change Is Important

1.2

Order Is Also Important

1.3

Thales Sets a Good Example

1.4

Pythagoras: A Mathematical Cosmos

1.5

Plato the Pythagorean

1.6

Plato, Pythagoras, and the Liberal Arts

1.7

Plato’s Pythagorean Project

1.8

Up Next: Problems with Space, Time, and Numbers

1.9

Study Questions

1

1 THE PROBLEM OF CHANGE The goal of this book is to help you understand calculus. But understanding calculus is not the same as merely learning how to solve calculus problems. Obviously you need to be able to solve problems, but you need more. To really get calculus, you also need to know why it was invented—why it was needed in the first place. Moreover, you need to know where it fits in the history of Western thought.

Let’s get started by turning to the beginning of Western classical civilization and a very brief retelling of the birth of Western philosophy and science. Beginning this way is no accident: one of the most important aspects of a classical education is learning various subjects as an integrated whole. After all, disciplines are never really separate from one another. This is particularly true in the story of calculus, where the history of philosophy, science, and mathematics come together in fascinating and crucial ways.

1.1 C H A N G E I S I M P O R TA NT

Around 600 BC, in ancient Greece, Western philosophy began with the problem of change. In a nutshell, the problem can be summarized with a simple question: why do things change? Yet despite the question’s simplicity, the problem of change is one of the single most important problems in philosophy.

Now perceptive people have always noticed that there are lots of different kinds of things in the world. Such people have also noticed that most of these things change: rivers flow, rain falls, trees grow, and even objects in the heavens move daily—albeit silently—while also slowly migrating throughout the year. One of the more obvious ways things change is that they move, sometimes in dangerous 2

1.2 Order Is Also Important

ways: rocks fall, tigers attack, elephants charge, lightning strikes, fire consumes, tornadoes twist.

In fact, the ancient Greeks used the word motion for any kind of change whatsoever, and so the problem of change can also be called the problem of motion. When water freezes or melts, it “moves” from one state to another. When an acorn changes into an oak tree, the acorn moves into a tree. When a child eats a sandwich, some of that sandwich turns into more child, and the child grows, moving to a larger size. And of course, when you drop a pencil, it moves downward, changing location, moving from place to place.

Each of the foregoing changes is an example of the world in motion. In fact, everything in the world is constantly moving. Even a stationary rock moves through time—it is growing older. (Also, remember that a rock is really made up of an enormous number of electrons, protons, and neutrons, which are all swarming around at alarming speeds; we just can’t see this motion.)

The world, then, is a teeming mass of change and motion. Now this change is sometimes predictable, sometimes utterly unpredictable. In either case, humans have always wanted to understand it, and this desire to understand change kickstarted Western philosophy. We’ll also see that the problem of change soon turns into a mathematical problem, a problem that won’t be solved until the invention of calculus. In fact, physics is now the main way we try to solve the problem of change, and the language of physics is calculus. Without calculus, there would be no physics as we know it.

1. 2 O R D E R I S A L S O I M P O R TA NT

The more orderly and well-behaved the change is, the easier it is for us to understand it. Thankfully, the world can be surprisingly well-behaved. This order is surprising because a lot of the world isn’t at all well-behaved. Imagine dropping a box of Legos and watching them scatter across the floor. This sort of mess is a fairly normal occurrence. What wouldn’t be normal is if, when those dropped Legos hit the ground, they spontaneously formed into a Lego Batmobile or AT-AT Walker.

Even though this kind of magic doesn’t happen with Legos, it happens all the time in nature. All around us we see mysterious order-out-of-chaos. Particles

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Chapter 1 The Problem of Change

of matter—atoms, electrons, quarks, or maybe even strings—like tiny Legos, naturally combine to form astoundingly intricate objects, like pine trees, African elephants, viruses, and all manner of beetles. They also compose other miracles, like you and me.

The question that thoughtful people have asked is this: how do we explain the order in the world when it seems as if the world is so often prone to disorder (recall the dropped Legos)? Why on earth are there beetles? Try making one of those out of Legos. Or out of anything, for that matter.

So then, although things change, they often don’t change haphazardly or randomly. Things are often orderly, organized, or otherwise seem to conform to some plan. Moreover, acorns never grow into skunks, and skunks never give birth to snakes or stones. Why? Well this question is just another part of the problem of change, that is, why is change frequently so tidy and organized?1 Again, this was one of the main challenges of early science, which we usually refer to as philosophy.

In fact, until the late 1800s what we call science was called natural philosophy, the philosophy of nature. The birth of Western philosophy was also the birth of science.

In any case, the problem of change—trying to understand motion-in-general— became central in our efforts to understand the universe and our place in it.

1. 3 TH A LE S S E T S A G O O D E X A M P LE

The first Greek philosopher (and scientist) was Thales of Miletus (who lived circa 500s bc). Thales’s answer to the problem of change—a problem which he himself perhaps first posed—was not what you’d expect. His explanation for why there is both change and order in the world is that everything is made of water. From rocks, to trees, to donkeys, to the earth itself, all of it is made of water.

We don’t know exactly why he chose water, but maybe one reason was that water pretty obviously moves (in rivers, at the beach, from the sky). Water can also radically change its form: from solid to liquid to gas. Moreover, living things need water to survive—and of course living things are the most apparent kinds of things that can move or change.

1 Moreover, acorns look a lot different than even the trees they become. Is the acorn the same thing as the resulting tree? Or are they different objects? Is the child the same thing as the adult she grows into?

1.3 Thales Sets a Good Example

Whatever Thales’s reasons, he believed that there was one single type of stuff out which everything is composed. This, he felt, could explain why all the different kinds of objects could change—sometimes drastically—while also explaining the relative order of all this change: all the vast variety of things were ultimately made out of the same stuff. According to Thales, then, there’s a unity (water) to all the diversity of things we see in the world. And unifying things made the world a bit easier to understand. Indeed, unification is one of the main goals of any scientific theory. Scientists, after all, assume we’re living in a universe, that there is unity in all this diversity.

Thales’s Solution to the problem of change was a big step forward in science and philosophy—not because he was correct (he obviously wasn’t), but because he put the problem of change on the table and gave the world a sample Solution. Often, in both science and philosophy, the hardest thing is asking the right question. Once the right kind of question is asked, it puts other folks hot on the trail of an answer.

(When studying philosophy or science, always ask yourself, “What question was this person trying to answer?”)

Now what does any of this have to do with calculus? Or mathematics, for that matter?

Well, for one thing, Thales set the agenda for Western philosophy with his attempt to explain why the world around us changes while maintaining so much order. This problem of change would eventually result in modern science, which would also require the invention of calculus. This will become clearer as the story unfolds.

There’s another reason for Thales’s importance. In addition to being the first philosopher and scientist, he was also the first mathematician, at least as we would use the word mathematician. That is, he was the first person to try to systematically prove mathematical statements (statements like the Pythagorean Theorem, although it wasn’t yet called that).2 Thales may have learned some of his mathematics during his travels to Egypt and Babylonia, but neither of these two cultures did actual mathematical proofs; they just used whatever math happened to work 2 By the way, theorems are just mathematical statements that are proven from other mathematical statements. Ultimately, there must be statements of foundational truths that require no proof, or else you could never prove anything. These special mathematical statements, the ones that require no proof, are called axioms.

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Chapter 1 The Problem of Change

for their practical purposes, whether building or accounting or commerce. In any case, when Thales returned to Greece, he began Western mathematics (and, as we saw, Western science and philosophy—all of these were really the same thing to the Greeks). Not bad for single person. Or a married one.

1. 4 PY TH AG O R A S : A M ATH E M ATI C A L CO S M O S

Speaking of the Pythagorean Theorem, Thales is thought to have been one of Pythagoras’s (ca. 500s bc) teachers, and a very influential one. This is where Thales becomes even more important. Not only did he set Pythagoras on the quest to understand change and motion, but he may also have encouraged Pythagoras’s mathematical leanings by directing him to travel to Egypt and Babylonia for further training.

Of course, today we think of Pythagoras as merely a mathematician. This is a serious misunderstanding. Pythagoras is even more important as a philosopher and scientist. (And the Pythagorean theorem was known long before he was born.)

In fact, Pythagoras may be the most underappreciated thinker in the history of Western thought.

Pythagoras was the first person to view the world as fundamentally mathematical. His view wasn’t merely that mathematics could be used to add and subtract groups of physical objects—people already knew that—but rather that the underlying physical structure of the cosmos was itself mathematical, even the parts we can’t see. So whereas Thales believed that everything is made of water, Pythagoras believed that everything is (somehow) made of numbers. In fact, the Pythagorean motto was “All is number.”

Pythagoras’s view started with a simple but profound discovery. He found that musical intervals could be represented by whole numbers. Consider a string pinned to a wooden board at both ends (or think of a violin, which is just a more complicated version of this.)

Now suppose that the string plays a C note when you pluck it (fig. 1.4).

Now, you can change the note that the string plays by pressing your finger down firmly on the string anywhere between the two ends. And when I say anywhere, I mean just that: there are an enormous number—an infinite number in fact—of

1.4 Pythagoras: A Mathematical Cosmos

different places to hold down the string.

But Pythagoras noticed that if you hold down the string exactly halfway between the two ends, you end up with another C note, only this one an octave higher. To put it in terms of numbers (and this is the important point), the ratio between the two notes, in terms of the length of the string is 1:2 (pronounced one to two). Other notes, for example what are called the fifth and the fourth, (don’t worry about the details), can also be represented by simple ratios of whole numbers. The important thing here is the whole number part.

To us, this isn’t all that impressive. In fact, it’s revolutionary. Notice how userfigure 1.4 friendly the mathematics of musical intervals is. Yet there was no apparent reason for the math to work out so nicely when it came to music and the vibration of strings. Again, there are an infinite number of places we can hold down the string along its length (in addition to the two end points), but it “just so happens” that the special points—where the string makes pleasing sounds—occur where corresponding number ratios are nice and simple. The ratios that correspond to the melodious notes aren’t cumbersome, like 1.09879874523:2.3751324 or 5.8903857403:7.098709873452. Rather, they’re composed of easy-to-deal-with numbers like 1, 2, 3, and 4. There’s nothing particularly special about the easy-to-deal with numbers except that they’re easy for humans to deal with. After all, the universe wouldn’t have to work any harder if it required difficultto-deal-with numbers like 1.09879874523 instead of easy-to-deal-with numbers like 3. It only makes a difference to us. The universe seems to be awfully accommodating to our relatively feeble minds.

Here’s the point: simple mathematics can be used to describe something as seemingly nonmathematical as music. This impressed Pythagoras so much that it eventually led him and his followers (cleverly called Pythagoreans) to believe that all is number, that everything could be described or explained in terms of numbers.

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Chapter 1 The Problem of Change

The great twentieth-century mathematician Morris Kline pointed out that we can unpack the Pythagoreans’ phrase all is number in the following way. There are, he said, really three ideas underlying their motto.3

It would be difficult to overemphasize the impact of these three ideas. Indeed, as we’ll see when we study the Scientific Revolution later in the book, the Pythagoreans’ mathematical philosophy made modern science possible. Without the notion that the world has been mathematically designed by God, modern science wouldn’t have gotten off the ground.

Again, we’re no longer surprised at this, but I need to reiterate that there was really no reason for anyone to think that music and numbers should fit together so nicely. In fact, we’re now so used to using numbers to describe things in the world—music, velocity, weight, density, etc.—we entirely expect the world to help us out like this. We think that this relationship between numbers and nonnumbers is something obvious, but there’s a real puzzle here: why should we be able to describe nonmathematical things with mathematics, which seems entirely different from the things it describes? Why, in other words, is mathematics applicable to the nonmathematical (e.g., physical) world?

This is a question that you should keep in mind whenever you study math and particularly as you go through this book. Why on earth does math work? Surprisingly, this question is today often seen as unanswerable. The Nobel prize-winning physicist Eugene Wigner said that “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”4 Wigner, as far as I know, didn’t think that the universe was designed by an intelligent mind, a mind that is importantly mathematical. But anyone who believes in God has a ready explanation for why mathematics so powerfully applies to the physical world: God is a mathematician.

3 Morris Kline, Mathematics in Western Culture (Oxford: Oxford University Press, 1953), 78.

4 Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Symmetries and Reflections: Scientific Essays of Eugene P. Wigner, (Bloomington: Indiana University Press, 1967), 237.

1.5 Plato the Pythagorean

The question—why is the world mathematical?—relates to our original problem of change, as we’ll see. For now, it’s enough for you to know that because the world is mathematical, as the Pythagoreans discovered, any Solution to the problem of change will ultimately include mathematics and, in particular, calculus.

1. 5 P L ATO TH E PY TH AG O R E A N

Today we know Pythagoras’s name because of his famous theorem.5 But, as I said, Pythagoras’s real significance lies in his view of a mathematical cosmos, a view he bequeathed to modern science. This, however, isn’t the full story; it was actually the great Plato who made the Pythagorean worldview world famous. Not only is Plato (perhaps arguably) the most important philosopher of all time—it has been said that the history of philosophy is nothing but a mere footnote to Plato—he was also the most famous Pythagorean. In any case, after Plato was converted to the study of philosophy by Socrates, he traveled beyond Athens to learn from the top Pythagoreans of his day. During this period, Plato was introduced to two crucial Pythagorean ideas, ideas that would eventually form the core of his own philosophy: 1) the human soul is immortal, and 2) the physical world is fundamentally mathematical.

So then, the first of these Pythagorean ideas is that the human soul is immortal, that the soul survives the death of the body. According to Plato and the Pythagoreans, humans are composed of two parts: a physical body and a nonphysical soul or mind.

In fact, Plato divided all of reality into a physical realm and a nonphysical realm.

The physical part is just the ordinary universe we live in. The nonphysical part is a world of Ideas or Forms. These Ideas, according to Plato, are actual nonphysical objects that literally exist outside the physical world, independent of any minds, in what we might call Plato’s heaven, since the world of the Forms is a mysterious, nonphysical realm. Moreover, Plato believed that these Ideas or Forms are “more real” than the physical world; they are absolutely perfect, they exist eternally, and they never, ever change. In fact, the physical universe is nothing more than a shadowy copy of these perfect, immutable Forms. Circles in the physical world, for example, 5 The Pythagorean theorem, you’ll recall, can be put in our modern notation like this: for any rightangled triangle with sides of lengths a, b, and c (c being the longest), a2 + b2 = c2. And though we know it as Pythagoras’s, it was actually discovered long before his time.

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are never perfectly circular; rather they are imperfect copies of the perfect Idea of Circularity (also known as the Form of Circularity). The Forms are ultimate reality for Plato, more real than physical reality.

All of this is connected with the second Pythagorean idea that Plato took to heart—the idea that the physical world is mathematical. As I said, according to Plato, all the mathematical Ideas or Forms exist in the World of the Forms, his heaven.

Moreover, he believed that God (not the Christian God, mind you; Plato called it the demiurge or craftsman) made the physical world in accord with these mathematical Forms. That is, God used the world of the Forms as mathematical blueprints for constructing our universe. Which is why, according to Plato, the world is mathematical: because God made it that way. This, of course, is just what all Pythagoreans believed.

So Plato had a great answer to the question Why does mathematics work? His answer is twofold: first, mathematics works because the world is mathematical, and second, the world is mathematical because God made it that way. We know that this is the right answer. And it’s really the only way, I think, to make sense of Wigner’s perplexity regarding the applicability of mathematics to the physical world.

1.6 P L ATO, PY TH AG O R A S , A N D TH E LI B E R A L A R T S

For Plato and the Pythagoreans, our souls (or minds, which are the same things) are far more important than our bodies. After all, our souls are eternal; our bodies obviously aren’t. It’s important, then, according to Pythagoreans everywhere, to take care of our souls, to purify them. And for Plato, the soul is purified by becoming more like the Forms, more like the eternal, perfect Ideas that inhabit Plato’s heaven. To put the same thing differently, our souls must become more like ultimate reality. The way to do this, said Plato, is to study mathematics.

Since, for Plato, the mind and the soul were pretty much the same thing, when he says we need to take care of our souls, he essentially means that we need to take care of our minds—that we purify our souls or minds by being properly educated.

In Plato’s most famous work, the Republic, he outlines the ideal education, the curriculum that kings must undergo to become philosophers (that is, to become philosopher-kings). This rigorous curriculum spanned decades, and a key to this soul-purifying education is a full ten years of intense mathematical training.

1.6 Plato, Pythagoras, and the Liberal Arts

Plato believed that the study of mathematics improves our souls by training us to take our minds off physical things—the concrete things of our universe—and put them on the more abstract ideas in the world of the Forms. By thinking more abstractly, our minds become more aligned with that perfect world of Ideas.

So Plato believed that a good dose of mathematics was necessary preparation for thinking and leading well. So what? Well, first of all this reiterates how important mathematics was to a great philosopher like Plato, a philosopher who almost single-handedly set the agenda for Western intellectual history. In fact, above the doors of the Academy was posted the following warning: “Let no one ignorant of geometry enter here.”

Second, the education that Plato sketches in his book the Republic is where the West gets the notion of the liberal arts. The first three of the seven liberal arts— grammar, logic, and rhetoric—build the foundation and focus on language and general reasoning skills. These three arts, the trivium, or “three ways,” prepare the student for all subsequent learning. But there are four more liberal arts, not surprisingly called the quadrivium, “four ways”, and these four are (and this might actually surprise you) arithmetic, geometry, astronomy, and music. Plato learned from the Pythagoreans to divide mathematics into these four disciplines. More than half of the liberal arts (four-sevenths, if we’re doing the math) are Pythagorean mathematics. For Plato, the ultimate goal for humans is to become mathematical—to become like reason itself—by studying these arts.

Now it’s pretty clear to us today why the first two disciplines of the quadrivium—arithmetic and geometry—were included in mathematics; but it’s not nearly as clear why the second two were. Well, in the case of astronomy, Plato and the Pythagoreans believed that the motion of the sun, moon, planets, and stars is mathematical, so it can be accurately and precisely mapped using mathematics. By studying astronomy, then, we are actually studying mathematics. Similarly with music: we can study mathematics by studying music or harmony, and we saw a simple case of this earlier with music intervals. As we saw, the physical world, including the objects of astronomy and music, were copied from the world of the Forms by the divine craftsman. Therefore, we study the world of the Forms by studying astronomy and music, just as we do when we study arithmetic and geometry.

Interestingly, the Pythagoreans connected astronomy and music in a strange but historically important way. They believed that the motion of the heavenly bodies produced actual music, their notes combining in harmony to create the “music of

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the spheres.” This Pythagorean idea will be famously pursued during the Scientific Revolution by the scientist Johannes Kepler. Composing music isn’t something that typically happens by accident, so belief in celestial music requires belief in a divine Musician. (Actually, all of the major scientists in the Scientific Revolution were Christians, including Copernicus, Kepler, Galileo and Sir Isaac Newton, and this belief in a divine designer would motivate their studies.)

Again, it is difficult to overemphasize the importance of mathematics for Plato’s philosophy, and therefore it’s difficult to overemphasize math’s importance to the Western intellectual tradition. You will continue to see this throughout the book and therefore continue to see that mathematics involves much more than learning how to perform calculations. To understand the intellectual history of the West, we need to understand the philosophical problems that the main thinkers were attempting to solve along with their motivation for solving them.

This leads us to the other big-ticket item in Plato’s philosophy, where his dedication to Pythagoreanism set the agenda for Western science.

1.7 P L ATO ’S PY TH AG O R E A N P R OJ E C T

We’ve seen that Plato’s ideal for humans—the leaders, anyways—is to become mathematical. Also recall that, according to Plato, the physical world already is mathematical. It was created this way by a divine craftsman according to mathematical principles. This is why, as we saw in the case of musical intervals, mathematics can be used to describe things in the physical world that don’t appear at first (or second) glance to be mathematical. In other words, because the physical world is mathematical, we can expect to understand it using mathematics.

It’s one thing to describe, say, a taut string using numbers, which is difficult enough, and another to describe the myriad phenomena in the world that are far more complex than a string.6 Yet Plato set his sights high. He was confident that, because humans and the world were created by the same mathematically inclined deity, we can hope to eventually describe the entire universe mathematically.

6 That said, physicists are currently working on a potential “theory of everything” called string theory.

According to string theory, all ordinary matter is ultimately composed of tiny, vibrating strings. If string theory turns out to be the final word in physics, this whole mathematical attack on the problem of change begins and ends with strings.

1.8 Up Next: Problems with Space, Time, and Numbers

This was one of Plato’s main philosophical goals. Plato wanted the universe as a whole to be described mathematically—the motion of the sun, moon, stars, planets . . . the entire cosmos. This enormously difficult task is made more difficult by the fact that the heavenly bodies don’t really move in an orderly fashion. For example, if you study the motion of, say, Mars over the course of months you’ll see it reverse direction for a while, only to change its mind, making a U-turn and return along its previous path. This reversal of direction is called retrograde motion.

Such motion is just the beginning of the complexities of celestial motion. It’s an enormously complicated system. In fact, there seems to be little system to it.

But Plato was absolutely sure that behind all this disorder and complexity was, in fact, mathematical order and (relative) simplicity. So he set out a research program for the members of his famous Academy in Athens: create a mathematical system that accurately describes and predicts the motion of the heavens.

Which brings us back to the problem of change—this time the motion of the sun, moon, planets, and stars in all their bewildering meanderings. Plato, too, tackled the problem of change, but with a decidedly Pythagorean twist. This new version of the problem—describing the moving physical world with mathematics—became so important that science is still working on it. Today, whenever scientists use mathematics to uncover the order of nature that hides beneath the appearances, they are helping make Plato’s Pythagorean dream come true. The mathematical sciences are founded on Pythagorean-Platonic foundations, and their goal is to solve the problem of change.

From this point on in Western history, the close interplay between philosophy, mathematics, and science will set the agenda for all three disciplines, not merely science. It is only within the last century that we have forgotten these close connections. Through this book, you’ll begin to recover some of what was lost.

1. 8 U P N E X T: P R O B LE M S W ITH S PAC E , TI M E , A N D NUMBERS

Before we follow the story of change and motion that will eventually lead to the scientific revolution and to the invention of calculus, let’s turn to some conceptual problems with motion—problems that make things very difficult for science,

13

14

Chapter 1 The Problem of Change

mathematics, and philosophy. This will help us understand just why we need calculus, how badly we need it, and why knowledge of calculus is important for our appreciation of Western culture.

1. 9 S T U DY Q U E S TI O N S

1. What is the problem of change?

2. Briefly explain how the problem of change inaugurated Western philosophy.

3. What is the problem of motion?

4. Briefly explain why order is important to the problem of change.

5. Give your own example(s) of motion or change that is not orderly.

6. Give your own example(s) of motion or change that is orderly.

7. What is “natural philosophy”?

8. What was Thales’ answer to the problem of change? How was this answer a step forward in answering the problem?

9. When did Thales live (very roughly)?

10. There were two main ways in which Thales was important. What are they?

11. What was the Pythagorean motto? Explain what this motto means. (Hint: How did Morris Kline unpack the motto?)

12. When did Pythagoras live (very roughly)?

13. What Pythagorean discovery led to the Pythagorean motto?

14. What is the music of the spheres? Why were the Pythagoreans impressed by the connection between music and mathematics? Why do you think that we aren’t usually impressed by this?

15. When we say that mathematics applies to the world, what do we mean exactly?

(Hint: what two kinds of things are similar?)

16. What did Wigner say about the applicability of mathematics to the physical world? Why couldn’t he make sense of why math “works”? How did Plato make sense of it?

1.9 Study Questions

17. What two important views did Plato inherit from the Pythagoreans?

18. According to Plato, humans are divided into two very different kinds of things.

Explain.

19. Plato divided reality into two main realms. Explain.

20. How were Plato’s two “worlds” similar to the two parts of humans?

21. What is “Plato’s Heaven”?

22. What are Forms?

23. Explain how, according to Plato, the physical world is a mere copy. Be sure to give an example.

24. What is Plato’s answer for why the world is mathematical?

25. According to Plato, why was the study of mathematics important?

26. Explain how Plato’s Republic is important in the history of the liberal arts.

What does this have to do with Pythagoras?

27. What is the “music of the spheres”?

28. What was the homework that Plato assigned the Academy? Why is this assignment historically important?

29. How did Plato attack the problem of change? That is, what was his version of the problem?

30. Why is Plato’s problem of change so difficult to solve?

15

CHAPTER 2 S PA C E , T I M E , A N D N U M B E R S

2.1

Zeno’s Paradox: Is Change Even Possible?

2.2

Problems with Space

2.3

Representing Physical Objects with Mathematical Ones

2.4

Problems with Numbers

2.5

Quantifying the World: Numbers as Stand-Ins

2.6

Problems with Time

2.7

The Problem of Motion and User-Friendliness

2.8

Up Next: Is Speed Coherent?

2.9

Study Questions

17

2 S PA C E , T I M E , A N D N U M B E R S In the last chapter, we left off with Plato’s challenge, to mathematically describe the apparently chaotic motion of the heavens. Describing motion with mathematics will occupy science from Plato’s time on. It will really take off during the Scientific Revolution of the 1600s, when Newton and Leibniz invent calculus. After all, calculus allows us to describe the behavior of everything from orbiting planets to gliding hockey pucks to falling anvils. So, naturally enough, using mathematics to describe objects in motion will occupy a lot of our time in this book.

Describing motion with mathematics is difficult for at least two reasons. One is the sheer difficulty of discovering the exact mathematics to describe the universe’s mathematical order. After all, the universe doesn’t come with a built-in science or mathematics textbook. (As I’ll point out later, unveiling the mathematical order isn’t nearly as difficult as it could be.) The second is the deep conceptual problems surrounding the very idea of motion. These problems have caused serious headaches for scientists as well as for mathematicians and philosophers that would be solved only with the invention of calculus. Looking at some of these conceptual problems will help us to understand just why calculus was needed in the first place. It will also help us understand science, philosophy, and their close relation with mathematics.

We will also discover that there is an additional difficulty with the basic concepts related to motion: space, time, numbers, and—as we’ll see in the next chapter— speed. Each of these concepts is central to describing motion with mathematics and is therefore central to calculus.

18

2.1 Zeno’s Paradox: Is Change Even Possible?

2 .1 Z E N O ’S PA R A D OX : I S C H A N G E E V E N P O S S I B LE?

The initial discovery that motion is more complicated than it first seems was made in the context of a fifth century BC philosophical controversy between Parmenides and Heraclitus. Heraclitus, on the one hand, believed that everything changes. He’s responsible for the famous saying, “You can never step into the same river twice.”

After all, he explained, the next time you step into the river, the very stuff that makes up the river—the water—is entirely different water, so the river is an entirely different river. Moreover, you’ve grown older, so it’s not even the same you.

On the other hand, Parmenides believed that nothing whatsoever changes and that the very concept of change is incoherent, like a square circle or married bachelor.

And since physical motion is a kind of change, Parmenides argued that such motion is impossible. One of these arguments was made famous by his student and ardent defender, Zeno of Elea (400s BC). Zeno proposed a thought experiment that attempted to show that motion simply can’t occur, despite appearances to the contrary.

Of course, this is an outrageous suggestion, but you can do the experiment in your head right now. Imagine that you’re about to run a mile. In order to run the entire mile, Zeno pointed out, you’ll obviously have to first run half a mile. No surprise there. But, Zeno continued, before you finish the half mile, you’ll have to first finish half of the half mile, that is, a quarter mile (see fig. 2.1).

figure 2.1

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Chapter 2 Space, Time, and Numbers

Before you finish that distance, though, you’ll have to first finish half of it. And before you finish that distance, you’ll have to first finish half of it. Now, says Zeno, you can keep dividing distances in half indefinitely, so any initial distance can be divided in half an infinite number of times. Of course, as we keep dividing distances in half, the distances become incredibly small. But regardless how small they become, each one can be divided in half. And obviously there’s no way to traverse an infinite number of distances, no matter how small they are. After all, even the tiniest distance would take at least some time to cover, even if it’s an infinitesimal amount of time, and if we add up an infinite number of infinitesimal times, we get an infinite amount of time—an eternity. Therefore, Zeno concluded, motion is impossible.

His conclusion applies to more than just change in location or locomotion: it applies to any change whatsoever, which was Zeno’s point.

Of course, Zeno’s paradox isn’t going to convince you that motion or change is impossible. Instead, you’re probably thinking that there’s something wrong with Zeno’s little thought experiment. And you’re right. Something is wrong with it.

But it turns out that it’s remarkably difficult to say exactly where Zeno goes wrong.

In fact, it wouldn’t be clear how to solve Zeno’s paradox until after calculus was developed two millennia after Zeno (we’ll see this in Chapter 25). Even then, some puzzles remained. Throughout the centuries, many, including the great Aristotle, tried in vain to solve the paradox. In any case, all the initial and obvious attempts to tame Zeno’s paradox were unsuccessful.

The point now is that Zeno puts a real damper on the problem of motion—not because motion is impossible, but because the concept of motion is difficult to analyze or define. But if we’re going to precisely describe motion using mathematics then we’d better be able to precisely define motion. And if we can’t, our desire to understand the natural world will be thwarted. After all, as Aristotle put it, ignorance of motion is ignorance of nature, so our failure to understand motion would be a significant roadblock for science.

There’s one thing to reiterate before moving on. You know that the ancient Greeks used the word motion to refer to any kind of change whatsoever. And while we’ve been discussing Zeno’s paradox, we’ve talked about a specific kind of change, namely, local motion or locomotion, that is, change in location or place. This is no coincidence, since by the time we get to the Scientific Revolution and the birth of modern science, the problem of change will have focused on locomotion.

2.2 Problems with Space

From this point forward, we too will focus on locomotion. But keep in mind that calculus applies to all manner of change.

2 . 2 P R O B LE M S W ITH S PAC E

Zeno’s paradox raises other important conceptual problems. One of the things that Zeno assumes (and that we assume along with him) is that distance can be divided in half indefinitely. That is, we assumed that we could, at least theoretically, divide any distance into infinitely many halves.

And there’s a good reason for assuming this. Let’s return to the previous figure (fig. 2.1). Imagine that you’ve been dividing the halves in half for some time and now the halves are so small you can’t see them without a microscope. But looking at the latest half with a microscope you see that the line can indeed be divided in half again, and that half into another half. Suppose you do this until you can no longer see the line you’re dividing. No problem. Just increase the magnification of the microscope and get back to cutting the line in halves. In the real world, there’s a limit to how many times we can do this, since there’s a limit to what we can see, what we can touch, what we can cut. But that’s only a practical limitation. In theory, this halving process could go on forever. No matter how small a distance is, in theory it can be halved. To put it differently, God, at least, could keep dividing a line in half for eternity, so it seems to be theoretically possible.

The reason a line can be divided in half forever is that lines are very weird objects.

For one thing, a mathematical line—a genuine line—has length but no breadth or thickness. Another way of saying this is that lines have only one dimension. To be sure, any line we draw (or even imagine) will have some thickness. But that is simply because we’re not drawing or imagining genuine, mathematical lines. We can’t represent or imagine mathematical lines without using physical lines, so when we draw a graph of a line on the board or in this book, we’re really not technically drawing a mathematical line. But we’ll pretend that we are. Once we begin to talk about the existence of mathematical lines—or, more generally, just what kind of thing is math?—we get into deep philosophical waters. We can’t always avoid it. And we probably shouldn’t.

I’m going to pause a moment to point out something important about this distinction between perfect, idealized mathematical lines and the imperfect lines that

21

22

Chapter 2 Space, Time, and Numbers

we humans have to settle for. Remember that Plato believed that there really are genuine, mathematical lines—lines with no thickness whatsoever. It would be better to say he believed that there is a single perfect Line, the Form of the Line, or Lineness. This Form—since it’s a Form—doesn’t exist in the physical world but in the ideal (and very mysterious) world of the Forms. According to Plato, every line that we see or draw or imagine is just a shadowy, imperfect copy of the perfect standard in Plato’s heaven. In fact, it was this distinction between ideal mathematical objects and our imperfect copies that led Plato to his theory of the Forms in the first place. It would be difficult to overstate mathematics’ influence on Plato.

Back to our discussion of lines. Lines are, of course, made up of points. But points themselves have no width (just as lines have no width); neither do points have height or depth. They have no dimensions at all, no extension; they are pure location. So, even though we represent points with small dots—using pencil, pen, printer ink, or pixels—dots have dimensions to them, no matter how small they are.

Points do not. They have no size at all. In fact, Euclid (not very helpfully) defines a point as “that which has no part.”

But how can something that has length—a line—be made up of things that have no size, not even length? After all, zero plus zero plus zero plus . . . is still zero.

Unfortunately, we’ll have to ignore this question for now. But keep it in the back of your mind. In any case, let’s assume that we can construct lines out of points.

After all, mathematics is largely based on this assumption. Notice something else that’s strange about lines being composed of points. Because points have no size whatsoever, a line of any length will be made out of an infinite number of points (infinity isn’t technically a number but, again, ignore this for now). After all, points take up no space at all and so we can fit as many as we like into any size line segment. Imagine a line segment that’s 12 inches long. How many points will fit on this line? An infinite number of them. Now divide that line in half. How many points will fit on this 6-inch line? Again, an infinite number. In fact, no matter how small a line is, it will be composed of an infinitely large set of points. So then, any line—no matter how long or short—will be made out of the same number of points, namely, an infinite amount.

Another odd thing to notice about the structure of lines is that there are no gaps in a line, no empty places, no place where there’s no point. To put this another way, lines are continuous, that is, without gaps. They are packed full of points. There’s

23

2.2 Problems with Space

an infinity of points in any line segment, no matter how small the segment. Lines are infinitely “dense.”

And because a line is continuous, there is always at least one other point between any two points. So, even if you choose two points as close together as you can possibly imagine (and even closer), there will be a third point between them (fig. 2.2).

figure 2.2

In fact, there will be an infinite number of points between them. The implication here is that no point is next to any other point; there are always points between any two points. No point has an immediate neighbor.

Although the importance of continuity will become clearer when we talk about numbers a little later, suffice it to say for now that these strange properties of lines and points allow us to see that even very simple mathematical objects give us more trouble than we first bargained for. Or, to put it more positively, they’re far more interesting than they first seem. But we’re not looking at the continuity and infinite divisibility of points and lines out of mere curiosity. Rather, these notions are crucial for our understanding calculus—indeed, for understanding why we need calculus at all.

24

Chapter 2 Space, Time, and Numbers

2 . 3 R E P R E S E NTI N G P H YS I C A L O B J E C T S W ITH M ATH E M ATI C A L O N E S

The objects we’ve been discussing are points and lines—the fundamental entities of geometry. But notice that we began our discussion by talking about space, which is physical. That is, points and lines are mathematical objects, not physical ones. Maybe this difference can help us solve Zeno’s paradox of motion: perhaps the problems associated with continuity and infinite divisibility are limited to mathematics, and so don’t apply to physical distances in the real world. If so, then Zeno’s paradox is only a mathematical problem.

Unfortunately, this proposed Solution won’t work. Geometry is the science of space; the reason we can use geometry to mathematically describe physical space is that lines and space are importantly alike. And they’re alike in ways that make both mathematical and physical objects susceptible to problems of continuity and infinity. We’re now so used to using mathematics to talk about the physical world that we did it automatically in figure 2.1, where we represented distance with a line.

Without much thought, we used a mathematical line to stand for a physical length in space. Similarly, we used mathematical points to stand for physical points in space.

In fact, representing physical objects with mathematical objects is the key to solving the problem of change, and the problem of motion in particular. Without mathematics, we can barely get started on the problem of change because, as the Pythagoreans discovered, the physical world is mathematical. Without mathematics, we can’t adequately describe motion—at least not in any significant detail.

And this will become more apparent throughout the book. For now, however, the important point is that the similarity between mathematical objects and the physical world is a double-edged sword. On the one hand, without this similarity, we couldn’t use mathematics in science and therefore couldn’t do science as we know it. On the other hand, this similarity poses real-life problems for anyone who takes up Plato’s Pythagorean challenge, including scientists who need mathematics to describe some aspect of nature.

The continuity of space and lines—and the corresponding concept of infinity— has always forced mathematicians, philosophers, and scientists to lose sleep. Notice that the continuity of lines results in the infinity of points between any two points.

If lines weren’t continuous, they wouldn’t be as interesting. And, as I alluded to,

25

2.4 Problems with Numbers

it also led Newton to invent calculus. But there are other important mathematical objects that suffer from continuity and infinite divisibility; perhaps you’ve heard of these objects: numbers. Numbers also have the same strangeness as their geometrical siblings. And since numbers are required for physics and calculus, we ought to take a look at them.

2 . 4 P R O B LE M S W ITH N U M B E R S

To explain how the problems associated with continuity and infinity are found in numbers, let’s look at one of the most helpful, insightful inventions of all time: the humble number line (fig. 2.4a).

figure 2.4a

As its name suggests, the number line is a melding of two entirely different realms: the realm of numbers and the realm of points (and, in this case, a line)—it unites arithmetic and geometry in wedded bliss. This marriage will, in the 1600s, make possible our wildly useful coordinate system, the x and y graph (called Cartesian coordinates after Rene Descartes). The Cartesian coordinate system will help make calculus possible, and we’ll talk about it more in a later chapter.

For now, notice that, on the number line, each point is paired with its own personal number. That is, every point on the line has a unique numerical address. And just as we can speak of points between points, this pairing of points and numbers will allow us to speak of numbers between numbers. For example, 1.5 is between 1 and 2. And there’s more space between the numbers 1 and 2, space to fit additional

26

Chapter 2 Space, Time, and Numbers

numbers, like 1.125, 1.25, and 1.75. But if we simply had a crowd of numbers instead of an orderly line (fig. 2.4b) we couldn’t see this relationship of betweenness.

figure 2.4b

Putting numbers on a line, as it turns out, is an enormously fruitful idea.

Now, consider the numbers 1 and 1.125. Between them, there’s still room for more numbers: 1.001 is one of these numbers. (You can see where this is going.)

And of course there are numbers between 1 and 1.001, like 1.0005 (fig. 2.4c). Like Captain America, we can do this all day long. Longer, in fact. Forever. There’s even room between 1 and 1.00000000000000001. For any two numbers you choose, no matter how close they are, there’s room for a third number between them (notice that close and room are spatial metaphors). For any two numbers there is always another number between them. As we saw with points, no number is ever next to any other number.

figure 2.4c

2.5 Quantifying the World: Numbers as Stand-Ins

Recall that, in the case of a line, there are infinitely many points between any two points, no matter how close together those two points are to each other. Now, because we’ve assigned a number to each point, the same is true of numbers: for any two numbers you choose, there are infinitely more numbers between them. And, just as with points, this is true no matter how close the original two numbers are.

So, numbers and lines share the strangeness associated with continuity and infinity. This is because numbers and mathematical points have a very similar structure, a structure that can be visually displayed on the number line. In other words, the structure that points and numbers share allows us to “see” numbers and how they’re related to one another. For example, we can more easily see the greater than relationship between two numbers because the larger number is located to the right of the smaller number (fig. 2.4d). This may seem trivial but it’s actually one of the most important ideas in mathematics, which we’ll see when we discuss the invention of Cartesian coordinates in a later chapter.

figure 2.4d

2 . 5 QUANTIF YING THE WORLD: NUMBERS AS STAND -INS

Let’s pause here a moment to consider something that’s central to calculus and physics—and to science in general. We saw earlier in this chapter that we can represent physical space with mathematical points and lines (we do this any time we draw a number line or Cartesian coordinate graph). That is, we were able to represent physical objects (points in space) with mathematical objects (mathematical points).

And we just saw in the previous section (2.4) that points and numbers also have a similar structure. Therefore, physical space and numbers will also share a similar

27

28

Chapter 2 Space, Time, and Numbers

structure.1 So then, we have the following three-way structural similarity between space, points, and numbers (keep in mind that space is physical while points and numbers are mathematical objects):

figure 2.5

It would be difficult to overstate the importance of this additional connection between the physical realm and the mathematical realm, that is, between numbers and physical space. The similar structure between physical space and the set of all numbers is what makes physics possible. If we want to use numbers in physics—and we do—numbers must have a structure similar to physical objects and properties.

It just so turns out that this similarity is found all over nature. For example, in addition to numbers standing for space (as in the number line and Cartesian coordinates), we can use numbers to stand for mass, speed, electrical charge, density, and time (we’ll talk about time in the next section). To put it differently, we can quantify these things, that is, we can represent them with numbers, with specific quantities. Quantifying the world is what physics is all about, and quantifying allows us to work on the problem of change. In particular, when we quantify nature, 1 Notice that we just applied a rule similar to the following: If a = b and b = c then a = c. But, in this case, we replaced the “=” with “has a similar structure to.” This is known as a transitive property.

2.6 Problems with Time

we can talk about the physical world by talking about numbers. This way, we can discover important truths about the world by calculating numbers—numbers that stand for physical properties. When we say we’re calculating the mass or speed of an object, technically speaking, we’re only calculating numbers. But these numbers are stunt doubles for physical objects, so by watching the numbers we can learn about the physical world.

2 .6 P R O B LE M S W ITH TI M E

One of the central problems in philosophy is the problem of change and, in particular, the problem of motion—which will be, for us, the problem of change in location.

Change in location obviously occurs in space. When you walk, you move from one location in space to another. So understanding space is important for understanding motion. But notice something else: change takes time. Moving from one place to another therefore takes time. In order to understand many common forms of motion or change, then, we need to also understand time.

Or at least understand it better than we do. We’ll ignore the fact that we don’t know what time is. Is time a thing? Does it exist? Or when we say that things change in time, we’re merely speaking metaphorically (again, using a spatial metaphor)? Is time just a concept we use to compare the motion of two or more objects? For example, when we time how long it takes for a race car to complete a quarter mile, are we doing anything more than merely comparing the movement of the stopwatch’s hand with the movement of the car? These are real and serious questions that physicists and philosophers struggle to answer. But we’ll largely ignore these interesting questions.

In any case, we think of time as being similar to space in important ways (hence the spatial metaphor in the previous paragraph or when we talk about a point in time or about someone traveling through time). In fact, we think of time in spatial terms very naturally. For example, we use timelines when studying history. We represent time (history) with a mathematical line, and by putting dates on that line we can see relations between times (dates). It’s much easier to see that there’s significantly more time separating the time of Jesus and the present, than the time separating Jesus and King David (fig. 2.6a). We can even quantify this relation by saying that there is roughly twice as much time.

29

30

Chapter 2 Space, Time, and Numbers

figure 2.6a

We represent physical time with mathematical lines in physics, too. Take the following example, one that we’ll encounter in a later chapter (fig. 2.6b). Don’t worry about all the details of the graph for now. Just notice that we’re plotting space (distance in feet) versus time (in seconds). That is, distance is represented by the vertical axis and time by the horizontal axis.

figure 2.6b

This graph is a mathematical picture of an object being dropped from a height of 80 feet. The t-axis (the horizontal axis) is a timeline, in this case a timeline of a small part of the dropped object’s history: from the time it was dropped to the time it hit the ground. But instead of marking the line with dates, we mark it with

31

2.6 Problems with Time

seconds (the time elapsed from the moment it was dropped). In any case, this kind of timeline is frequently used in science, and we’ll be dealing with them quite a bit in this book. (Again, we’ll talk about the power and beauty of graphs when we discuss Cartesian coordinates.)

So again, we’re representing physical time with a mathematical line, 2 just as we did with physical space (which, you’ll note, we’ve also done here on the vertical axis). Moreover, on the horizontal axis we’re representing time mathematically by using both a mathematical line and numbers, again, just as we’ve done with physical space. All these concepts are so connected, so similar, that it’s easy to overlook yet difficult to see their connections.

The similarities between physical time and physical space are important. One of the main benefits of their similarities is that space is easier to understand than time and, therefore, if we can think of time in spatial terms, we can deal with time a bit better. But there’s a downside to the similarities between space and time: some of the mysteries of space carry over into the concept of time. In particular, continuity and infinity show up again.

Imagine a timeline, and on it we’ve marked off an hour (fig. 2.6c).

figure 2.6c

On this line, we can identify the half-hour mark by dividing the line exactly in half. This is just how Zeno’s paradox of motion began, which is no coincidence. We 2 Or more precisely (you’ll remember), we’re representing physical time with a physical line (on the page) which itself is a representation of a mathematical line.

32

Chapter 2 Space, Time, and Numbers

can present Zeno’s paradox in terms of the time during which motion takes place (instead of in terms of the distance traveled, as we did in the original presentation of the paradox). Not too surprisingly, we get almost exactly the same problems.

Time is continuous; there are no gaps in time. This means—as you now know—that we can forever divide time intervals in half. No matter how small an interval of time is, it can always be divided into two smaller time intervals. This will probably be the only situation in life where we have more than enough time.

So then, because time has a structure similar to space, the strangeness of space due to continuity and infinity infects time as well. But things aren’t all bad. In fact, this similar structure is what allows us to work on the ancient problem of motion using mathematics. We saw this double edge earlier.

2 .7 TH E P R O B LE M O F M OTI O N A N D U S E RF R I E N D LI N E S S

In a mere two chapters, we’ve learned quite a lot about the problem of change, its origins, some of its importance for Western philosophy and science, and a few of the conceptual difficulties that will require calculus to overcome. In the next chapter we’ll look at problems associated with the concept of speed, but for a moment let’s stop and focus on some encouraging news.

As we’ve seen, the physical world is often structured like the mathematical world. That is, there are important similarities between physical structures and mathematical ones despite the fact that physical objects are not mathematical objects. In this chapter we’ve focused on two main features of the physical world: space and time (see fig. 2.7, which is the same as fig. 2.5, but with physical time added to the diagram).

By looking at just these two features we’ve covered a lot of ground. After all, every physical object exists in space and time; space and time make up the entire arena or stage in which science is played out. If these two physical features didn’t have a structure that we could link to mathematics, we wouldn’t be able to do any physics, and the problem of motion wouldn’t have made it much past Thales.

2.7 The Problem of Motion and User-Friendliness

figure 2.7

So then, space and time are the two most fundamental features of the physical world. And we’ve learned that they both have a structure similar to the mathematical world and can be represented by the two most fundamental objects of the mathematical world, namely, points and numbers. Notice, too, that points and numbers are the most basic objects of geometry and arithmetic, respectively. In any case, the similarity between the basic objects of both the physical and mathematical realms is what makes physics possible—and what makes calculus necessary.

Notice something else. The tight and orderly mathematical-physical connection makes working on the problem of motion much, much simpler than it would have been without this connection. The world could have been much more difficult to figure out—impossible, in fact. But it’s remarkably user-friendly, seemingly fit for humans to grasp it. Mathematics is something orderly and convenient, something humans can get their minds around. Is it likely that the universe is accidentally structured in a way that is itself mathematical? That seems like a stretch. Rather, many people—myself included—think that the best explanation for this user-friendliness is that there’s someone who wants us to know about His creation. As Newton and other founders of modern science believed, by working

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on the problem of motion and discovering more about the universe, we are actually doing theology.

2 . 8 U P N E X T: I S S P E E D CO H E R E NT ?

We’ve seen, then, that there is both good and bad news when it comes to representing the world with mathematics. The good news is that the physical world has a similar structure to mathematical objects like lines and numbers. And it’s this similarity that makes it possible to use mathematics to tackle the problem of change. The bad news, however, is that this similarity means that the conceptual problems associated with lines and numbers will infect the physical world, making the universe’s user-friendliness a bit less friendly.

Moreover, this problem shows up in another important concept: our concept of speed. This isn’t surprising, really; after all, our notion of speed relies on the more fundamental notions of distance (i.e., space) and time. Therefore, in order to apply mathematics to the problem of change, we’ll need to also address the problem of speed. So let’s turn to the paradox that speed presents.

2 . 9 S T U DY Q U E S TI O N S

1. What is Plato’s challenge?

2. What are two of the difficulties with describing motion using mathematics?

There is another difficulty. What is it?

3. What was the philosophical controversy that sparked Zeno’s paradox?

4. Very roughly, when did Zeno live?

5. What was Zeno’s outrageous suggestion? What was his argument for this?

6. Of course, you don’t believe Zeno’s suggestion. So what is the real problem that Zeno’s paradox poses for us? In other words, what does Zeno’s paradox rightly point out to us?

7. What is Aristotle’s little motto about motion and nature?

2.9 Study Questions

8. In what way did the ancient Greeks use the word “motion” differently from our ordinary usage?

9. What is one of the assumptions about distances that Zeno (very naturally) makes in his argument?

10. What is the difference between a genuine mathematical line and any line that we draw or imagine? How is this difference related to Plato’s theory of the Forms?

11. Why is it strange to say that mathematical lines are made out of points?

12. Are lines of different lengths made up of different numbers of points? Explain.

13. What does “continuous” mean when we’re talking about lines?

14. Explain why no point is immediately next to any other point on a line.

15. What are the fundamental or most basic entities/objects of geometry?

16. What part of mathematics can be called the “science of space”?

17. How are mathematical lines and physical distance in space different? How are they alike? Why are their similarities important?

18. Give some example of mathematical objects from geometry. Give some examples of physical objects. Is a square a physical or a mathematical object?

19. How is the similarity between mathematical objects and the physical world a “double-edged sword”?

20. Why is Plato’s challenge also called “Plato’s Pythagorean challenge”?

21. What two realms does the number line unite?

22. What do we mean by “numerical address” in the case of the number line?

23. Why can we say that 1.5 is between 1 and 2?

24. Why is no number immediately next to any other number?

25. Numbers and lines share the strangeness of what two important (and related) concepts?

26. In what way can we “see” numbers?

27. Draw the triangle diagram that shows the relation between numbers, shapes, (lines & points), and space. Can you explain it to someone who hasn’t read about this? Why are these relations crucial for science?

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28. How can the behavior of numbers tell us about the behavior of the physical world? Can you give specific examples? (We haven’t yet covered examples in this book, so you’ll have to pull from previous classes.)

29. We often represent time spatially. Explain and give two examples (the examples from the book).

30. Draw the updated version of the “similar structure” triangle.

31. The similarities between the structures of mathematics and those of the physical universe can cause real headaches. But there are more positive than negative implications. How might these positive implications provide evidence for a divine designer?

4 T H E P L AT O N I C - PY T H A G O R E A N P R OJ E C T B E F O R E T H E S C I E N T I F I C R E VO LU T I O N We’ve looked at some of the difficulties that confronted philosophers, scientists, and mathematicians as they worked on the problem of change before the invention of calculus. Despite these difficulties, neither scientists nor philosophers nor mathematicians stopped working on the Platonic-Pythagorean project (the general project of uncovering the mathematical order of nature that hides behind the chaotic appearances). Between Pythagoras and the Scientific Revolution of the 1600s were centuries filled with efforts to mathematically describe the physical world. In this chapter, we’ll highlight the most important efforts, keeping a close eye on how they relate to the problem of change and the need for calculus.

Before we do that, we should take a moment to stop and catch our breath and review how far we’ve come since the beginning of the book.

4 .1 TH E B I G P I C T U R E (S O FA R)

Your goal here is to learn calculus, to really understand its core fundamentals.

But in order to genuinely understand calculus, you must first understand why it’s needed in the first place. That’s why we began at the beginning, at the very start of Western science—which is also the beginning of philosophy and mathematics (since all three disciplines were inseparable at the time, as they should be). We learned that the problem of change was one of the main topics of conversation for Greek 50

4.1 The Big Picture (So Far)

philosophers, with Pythagoras and Plato giving it a decidedly mathematical twist.

And because these first philosophers were also the first scientists, we discovered why motion and change are still such an enormous part of contemporary science.

One of the main goals of science, especially since Plato (given his unparalleled influence), has been to understand change in mathematical terms.

Change is everywhere, and it’s been one of the main interests of scientists since the beginning of science. Not only that, but change has been considered a type of motion, and motion—locomotion in particular—brings with it its own interesting problems, as we’ve seen. One thing that helps us to understand physical change or motion is by attaching numbers to it, by quantifying it. Which, of course, takes mathematics. Yet so far we’ve seen that motion can be pretty stubborn when it comes to allowing us to use mathematics.

This will be why we need calculus: to help us with the ongoing problem of change, a problem that has been at the forefront of the Western intellectual tradition since its beginning. Without calculus, we’d have no science, and one of the main problems of philosophy would remain almost entirely unaddressed.

Recall that there were two main difficulties with the project of describing nature mathematically. The first difficulty was that mathematically characterizing real-life physical phenomena is just plain hard. To use mathematics to talk about simple numbers and geometrical shapes is one thing; to apply mathematics to the complex and seemingly chaotic physical is world quite another.

The second problem is really a whole set of problems—problems associated with the very concept of change, especially locomotion, or the change in location. We saw that these problems had their source in the more basic concepts of space, time, and numbers. We also saw that the problem with these three concepts arose from two additional concepts: continuity and infinity.

These problems associated with space, time, and numbers then seeped into the related concept of speed. For example, just as the idea of a spatial points caused concern—they have no dimensions and all the weirdness associated with that—temporal points (i.e., instants) gave rise to the question of how to characterize velocity at an instant, or instantaneous velocity. Surely an object in motion is moving at each instant it is moving. But no distance is traversed in an instant. And zero distance was only one problem associated with temporal instants. Our ordinary definition of speed collapses into meaninglessness if we try to use it for instantaneous speed,

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because the denominator becomes 0. The very concept of instantaneous speed seems incoherent.

This is where we currently stand in our story. Let’s continue it, looking in this chapter at a few of the efforts to mathematically describe nature prior to calculus, efforts that will highlight the importance and power of calculus.

On with the story.

4 . 2 E U D OXU S I S TH E F I R S T TO CO M P LE TE H I S H O M E WO R K

Let’s reconvene our historical story of math, science, and philosophy, returning to where we left off at the end of Chapter 1. There our story stopped at Plato (in Chapters 2 and 3 we paused to look at the conceptual weirdness related to change and motion). Plato’s Pythagoras-inspired project—to mathematically describe the heavenly motions—is exceptionally difficult, even without the conceptual problems that arise from continuity and infinity, which made things doubly difficult. But if scientists could bracket or set aside the problems of infinity and continuity, then perhaps they could make some progress on mathematically describing nature.

Is it possible set aside these important issues?

Yes—at least for a time—which is exactly what happened. Even though the sun, moon, planets, and stars are in motion, they move so slowly that we can easily think of each night as a snapshot of their relative positions. Because their movement is almost imperceptible, we can get by without having to consider their instantaneous speed. Plotting a location for a planet or star each night is relatively easy. And by thinking of Plato’s project in terms of a series of photographs rather than as one continuous motion, the issues of continuity and infinity could be sidestepped.

But even if that made the task easier, it was by no means easy. The planets, stars, moon, and sun all move in fairly complicated ways. But the mathematical tools available at the time were pretty limited compared to what we have today. Plato had charged his Academy to explain this unruly motion using only different sized spheres that spin at constant speeds. But the motion of the planet Mercury was not at all circular or spherical. Mercury generally moves eastward but will also reverse directions from time to time (fig. 4.2a).

4.2Â Â Â Eudoxus Is the First to Complete His Homework

figure 4.2a

And this is just the beginning of the irregularities, so mathematicians had to be extremely clever to account for this motion using only spheres. It was as if the apparently random stumblings of a drunk man had to be described with nothing but perfect circles (fig. 4.2b).

figure 4.2b

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Although the motion of all the astronomical phenomena is more complicated than this, this sketch gives you an idea of the general strategy: taking motion that seems chaotic and finding the mathematical order behind it. The circles are the mathematical order behind the meandering path.

The first mathematician to turn in his math homework was Plato’s student Eudoxus (d. 355 bc). Eudoxus’s system of spheres was extremely complicated. To model Mercury’s motion, for example, he used a mini-system of four concentric spheres, all centered on the earth, with each sphere rotating on different axes. Note that each axis was aligned in different directions and attached to the next sphere outward (fig. 4.2c).

figure 4.2c

Like a complicated gyroscope, Mercury’s system of four spheres rotated about each other, with Mercury attached to one of those spheres, moving in what was

4.3 Geometry and the Influence of Euclid’s Elements

decidedly not circular motion. Arriving at this system was an extremely difficult task, so Eudoxus’s accomplishment is unbelievably impressive. And that’s not the entirety of it; he constructed a mini-system like this for each planet as well as additional systems for the sun and moon—for a total of twenty-seven spheres.1 Eudoxus’s system didn’t perfectly match the heavens’ actual motion (no system ever does, not even today’s versions), but it came closer than it had a right to. And as astronomers (who were mathematicians) made more accurate observations, they added spheres to match these new observations. And on it went for centuries, greater accuracy requiring increased mathematical complexity.

It is interesting that neither Plato nor Eudoxus believed that these spheres actually exist. That is, they didn’t think that Mercury was literally attached to rotating physical spheres. They were merely interested in mapping the motion of Mercury (and the other celestial objects). Think of it this way: suppose you wanted to describe the motion of a bowling ball dropped from the top of a two-hundred-foot tower (this kind of example will become very important for us as we proceed). And let’s say you had the mathematics to determine how fast the object travels at each point. This would be quite a feat, but it wouldn’t tell you what is making the object fall. To be sure, something is causing the object to fall, but what exactly is difficult to say. In fact, it’s so difficult to say that finding the unseen cause of motion will be a major part of the story of science, mathematics, and philosophy.

So Plato and Eudoxus were interested in mathematically describing the motion that they could see, while being content to remain ignorant about what they couldn’t, that is, about what is causing the motion. This distinction between describing and finding the cause is crucial to keep in mind. Yet most of the time it’s ignored, even by scientists.

4 . 3 G E O M E TRY A N D TH E I N F LU E N C E O F E U C LI D ’S E LE M E NT S

Did Eudoxus really do what Plato had asked of the Academy? Is his system really a mathematical system? Where are the numbers? This is a good question, and to answer it we need to remember that spheres are geometrical objects. In fact, the 1 (Lloyd 1970), 90.

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Greeks saw geometry as the main discipline of mathematics, with the numerical calculations of arithmetic being largely a matter for the slave class, who used them for commerce and other practical tasks. So when Plato charged the Academy to find the mathematical order of the heavens, he was asking them to find geometrical order. Remember the sign above the doors of the Academy: “Let no one ignorant of geometry enter here.” Geometry just was mathematics for the ancient Greeks.

This monumental switch in emphasis, from arithmetic to geometry, had taken place almost immediately after Pythagoras discovered the mathematical order of vibrating strings.2 In fact, the canon of ancient Greek mathematics is Euclid’s Elements of Geometry (fig. 4.3 shows a fragment of an ancient copy), essentially a summary of the mathematics of Plato’s Academy. The Elements contains no numbers, only theorems and proofs about shapes. Even the Elements’ theorems or propositions on number theory were all proven using geometry. For example, operations on numbers or magnitudes are discussed by Euclid as operations on

figure 4.3 2 This switch from arithmetic to geometry is an important and crucial part our intellectual heritage. The short version is that the Pythagoreans discovered what we would now call irrational numbers. The longer version is much more interesting. (Legend says that the Pythagorean who discovered irrational numbers was unceremoniously dumped over the side of a boat and drowned for his efforts.)

4.4 What about Terrestrial Objects?

lengths of lines. For example, Proposition 7 in Book Five of the Elements is “Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.” The proof of this is done entirely with lengths of lines, rather than numbers, as stand-ins for the magnitudes. As it turns out, the Elements would be the West’s main mathematical textbook for the next 2,500 years. Indeed, perhaps only the Bible surpasses the Elements’ influence on humanity (which is a story for another time, but it’s also the reason I said in the preface that the Elements is the Odyssey of mathematics.)

4 . 4 W H AT A B O U T TE R R E S TR I A L O B J E C T S?

Notice that Plato’s homework assignment was to mathematically capture the motion of the heavenly, astronomical realm and not the motion of ordinary objects down here on earth. This distinction between celestial and terrestrial objects is an important one. The task of mapping the heavens—as difficult as that was—was significantly easier than trying to use mathematics to accurately describe the motion of most earthly objects. This is because many of the terrestrial or mundane (literally earthly) objects that interest us also move very quickly. Just think of how fast objects fall when they’re dropped. They move so quickly that it’s very difficult to say much more about them than that they fall downward and that they gain speed as they fall. And the mathematical relationships that characterize their speed are not at all obvious, to put it mildly.3 So then, in general, a planet’s change in location is much easier to mathematically tame than the motion of a falling rock. This is why, for centuries—at least when it came to mathematically studying the terrestrial realm, scientists focused on well-behaved stationary objects. You have to learn to crawl before you walk or run.

3 To be sure, the Pythagoreans were successful in describing the ratios of string lengths in a musical scale, but this these ratios of lengths didn’t change with time—they weren’t moving. Of course, the strings themselves were vibrating, and therefore in motion, but this motion wasn’t the phenomenon that the Pythagoreans were describing. Instead, they were describing stationary distances of nodes that produced specific notes. To describe the actual vibration of strings would require more sophisticated mathematics, mathematics that the ancients didn’t have. And, by the way, the mathematics of waves will become incredibly important for physics. Indeed, contemporary quantum mechanics says that particles like electrons actually behave like waves. Moreover, the present candidate for a theory of everything is called string theory.

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4 . 5 A R I S TOTLE A N D TH E P R O B LE M O F C H A N G E

While Eudoxus focused on astronomy, another of Plato’s students was more interested in the terrestrial realm. Aristotle (d. 323 bc) departed from his teacher Plato in an important way: he denied that there was a world of the Forms. Aristotle still believed in Forms, but he believed that they were present within individual ordinary objects here in our physical universe. For example, each physical table has within it (somehow . . . don’t ask) its own Form of Tableness, and that this Form is what makes it a table. That is, each table was composed of both matter and form.

Matter is the stuff the table is made of, whereas form is what defines the stuff as a table (as opposed to, say, a turtle).

It’s not clear exactly how Aristotle meant us to understand him on this, but the only thing that need worry us here is that, for him, there is no extra world outside of the cosmos. For this reason, Aristotle directed his attention on the physical cosmos, what Plato would have called the World of Change. Contrary to Plato, who believed that we could have genuine knowledge only of the eternal unchanging World of the Forms, Aristotle believed that we can have knowledge about the ever-changing physical world.

Plato and Aristotle’s different emphases—the former focusing on the otherworldly realm of the Forms, the latter on the physical world down here—is nicely pictured in Raphael’s School of Athens (fig. 4.5).

Raphael depicts Plato pointing upward to the world of the Forms and Aristotle gesturing toward the earth.

Or perhaps they’re pretending to play basketball, Plato spinning a ball on his fingertip while Aristotle is dribbling another on the ground.

The point of all this is that Aristotle’s interest in the physical world of change figure 4.5 (rather than on unchanging Forms) made

4.6 Aristotle and the Platonic-Pythagorean Project

the problem of change even more central to philosophy and science, which were really same discipline (remember that what we call science was called natural philosophy until the late 1800s).

4 .6 A R I S TOTLE A N D TH E P L ATO N I C- PY TH AG O R E A N P R OJ E C T

Now what does any of this have to do with mathematics, much less with calculus?

Well, for one thing, Aristotle’s emphasis on earthly phenomena will affect his ability to deal with the problem of change mathematically. Because the motion of terrestrial objects is so much more difficult to mathematically describe, there was little Aristotle could do within the constraints of the mathematics of his time.4 That’s not to say that Aristotle didn’t think mathematics was important. On the contrary, he believed that mathematics was the most secure kind of knowledge we can have and that all our reasoning should copy mathematical reasoning. In fact, it was Aristotle who essentially invented the discipline of logic, a discipline he developed by copying the mathematical method we now find in Euclid’s Elements.

Moreover, Aristotle and his students were still interested in Plato’s project of mathematically describing the heavens. To that end, they modified Eudoxus’s celestial system of spheres, making it more accurate as well as more complicated, increasing the number of spheres to more than fifty.

As for terrestrial objects, Aristotle approached the problem of change without mathematics. Indeed, he believed that merely describing how objects move wasn’t the main goal of natural philosophy. Recall that Plato and Eudoxus believed that describing or mapping the motion of the planets was the main goal, rather than finding the cause of the planets’ motion. Aristotle, on the other hand, believed that the primary goal of natural philosophy (i.e., the primary goal of science) is to discover what causes objects to move the way they do, not merely to describe their motion.

Indeed, when it came to his system of celestial spheres, Aristotle didn’t consider these spheres to be merely theoretical mathematical tools to describe the motions.

4 It would be more accurate to say that there was little his mathematical students could do, since, like Plato, Aristotle’s own expertise wasn’t in doing the actual mathematics.

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Rather, he believed that they were physical. According to him, they were actual transparent crystalline spheres composed of ether or quintessence (quintessence means “fifth essence”; the other four essences, or elements, were earth, air, fire, and water, and were what terrestrial objects were presumed to be made of). So, in the case of the celestial motion, his system of spheres was both mathematical and physical.

Though his system of spheres was nearly the extent of his mathematical attack on the problem of change, Aristotle devoted an enormous amount of effort to determining physical causes of earthly motion and change in general. If you’ve studied Aristotle at all, you probably know that his doctrine of the four causes— material, efficient, formal, and final—is central to his philosophy. But this doctrine is really his Solution to the problem change. This simply highlights the importance of this problem: one of the most famous philosophical doctrines in history is actually a result of Aristotle’s attempts to get to the bottom of change. And this effort—despite its lack of mathematics—is part of the story of calculus, as you’ll see when we come to the Scientific Revolution. After all, this revolution was an overthrowing of Aristotle’s largely nonmathematical science, and to understand the revolution you’ll need to understand what was being overthrown. Let’s briefly look at some key features of Aristotle’s natural philosophy of change.

4 .7 A R I S TOTLE ’S P H YS I C S

Let’s return to the case of a falling object, a baseball, say. What causes the ball to fall? Our answer would be gravity (which is actually not as clear an answer as you might think). That is, we would say that the earth exerts an external force on the ball, pulling the ball to toward its center. But Aristotle believed that the cause of the ball’s motion toward the earth is something inside the ball, something internal to it. This something is the ball’s nature, and it’s this nature that causes the ball to move the way it does, just as a cat’s nature causes it to chase a mouse, moving the cat toward its goal.

Let’s unpack this a bit. Aristotle believed that everything in the terrestrial realm (a realm which stretched from the center of the earth to just inside the moon’s orbit) was made of some combination of the four elements, earth, air, fire, and water. Each

4.7 Aristotle’s Physics

element has a natural place of rest in the terrestrial realm. Earth’s natural place, for example, is the center of the cosmos, whereas fire’s natural place is the outermost area of the terrestrial realm, just inside the moon’s orbit (the natural resting places for water and air were between these two extremes). Aristotle believed that each element is always “trying” to get to its natural place of rest. They were trying to reach a goal. This explains what causes objects to fall: they are composed mostly of the element earth, and this earthy element naturally moves toward the center of earth. Earth (the element) has an internal nature that causes it to seek its natural place of rest. Similarly, it is fire’s internal nature that makes it move upward in an attempt to reach its resting place just beneath the moon.

So then, Aristotle believed that the cause of free fall is an internal nature that causes an object to behave the way it does. (Free fall simply means that the falling object is free from any force acting on it, other than gravity.) In fact, all objects, from rocks to trees to alligators had their own unique nature that caused them to be what they are and to do what they do. Finding natures, then, was the goal of Aristotle’s natural philosophy. The Greek word for nature is physis, the source of our word physics. Aristotle’s physics, then, was a search an object’s nature, the cause of its motion and change. An object’s nature is, Aristotle said, its principle of change.

In reference to the meaning of the word motion, Aristotle says that “if it were unknown, the meaning of ‘nature’ too would be unknown.” (Physics 200b14) The medieval Scholastics put his point this way: “Ignorance of motion is ignorance of nature.” Again, until the late 1800s, science was called natural philosophy. It would be difficult to underestimate the West’s obsession with motion and change.

Notice that I used words like trying and seeking when explaining the natural motion of objects. This seems to imply that objects have intentions—or at least that they are living beings. But Aristotle didn’t think that objects were, strictly speaking, alive. Nevertheless, he did think of the cosmos as more like a gigantic organism than like a giant clock, as we might. This could be because he was an enthusiastic biologist (his father was a physician, perhaps the court physician to Alexander the Great’s father, King Philip of Macedonia). In any case, Aristotle looked at the universe as kind of a living organism rather than as a mathematically designed machine. This would influence the way he thought about science and probably helped subsequent scientists to be satisfied with their limited mathematical tools.

Mathematics simply wouldn’t be as important to them.

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So then, Aristotle’s analysis of motion had more in common with biology than with what we now think of as physics, thus his talk of natures and seeking and trying. This analysis will have a great effect on the West’s understanding of gravity, falling objects, and other related phenomena. We’ll see this later when we look at Galileo’s revolutionary study of motion.

4 . 8 A R I S TOTLE ’S P H YS I C S O F F R E E FA LL

One last thing before we leave Aristotle. Despite his search for physical causes, he actually attempted something like a mathematical description for falling objects. And this law of free fall—this law of falling objects—would be a main source of conflict during the Scientific Revolution. Aristotle puts the law this way in his book On the Heavens: A given weight moves a given distance in a given time; a weight which is as great and more [i.e., greater] moves the same distance in a less time, the times being in inverse proportion to the weights. For instance, if one weight is twice another, it will take half as long over a given movement. Further, a finite weight traverses any finite distance in a finite time. Cael 1.6 273b30ff

Of course, this doesn’t look at all like a mathematical statement, but remember that until the 1600s, with Descartes’s invention of algebraic notation, mathematics was done with ordinary words. This made mathematics doubly difficult. Every problem was a word problem! It would be difficult to overstate how important a simple and efficient mathematical notation is. In fact, the notation is often just as important as the concepts it stands for.

Aristotle’s law of free fall is actually pretty straightforward.

First, his law is stated in terms of the ratio of two different instances of free fall.

That is, his law is put in terms of a comparison between the free fall of two different objects (fig. 4.8). We can use the delta notation or not. Let’s do both, just for practice. We’ll use the subscript 1 to denote that we’re talking about ball 1, and we’ll use 2 to denote ball 2. In this diagram, ∆h1 and d1, both stand for the distance ball 1 falls. Again, we’re just doing this so you can have practice going back and forth between notations. ∆h1 just denotes the change in the ball’s height, which is why

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4.8 Aristotle’s Physics of Free Fall

we’re using the h. ∆t2 and t2 both stand for the time it takes ball 1 to fall a distance of ∆h1 (which we can also denote by d1). We’ll use w1 to represent the weight of ball 1. Something similar to all of this can be said for ball 2.

figure 4.8

Given this diagram, we can express Aristotle’s law in modern algebra this way in delta notation (you don’t need to necessarily see how I got this from Aristotle’s word problem above; I’ll unpack its meaning as we proceed):

Again, where w1 and w2 are the individual weights of the two objects, ∆h1 and ∆h2 are the distances each object travels (written here as a change in height), and ∆t1 and ∆t2 are the times it takes each object to fall. Look closely at the formula, noticing which subscripts are where.

So Aristotle does have a mathematical description for the motion of falling bodies. And notice, too, that this mathematical description applies to the terrestrial

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realm and not the celestial realm. This is extremely important: Aristotle is trying to tackle the much more difficult problem of change for objects that change very quickly. Notice, too, that this formula doesn’t say what causes the objects to fall but simply describes their falling behavior. And don’t forget, Aristotle had a theory for what caused the objects to behave this way, namely, their natures.

We can also write it without delta notation (again for practice), which looks like this:

Now, since average velocity (vave) is equal to distance traveled divided by time traveled (d/t) we can rewrite the law as follows. Pay attention to the subscript numbers.

We can see from this last version of the formula that Aristotle believed that heavier objects fall faster than lighter ones (make sure you see this). For example, an object ten times heavier will fall ten times faster. Suppose ball 1 weighs 10 pounds, and ball 2 weighs 5 pounds (maybe they’re bowling balls). We know therefore that

and if we rearrange this formula we have

In other words, ball 1, which is twice as heavy as ball 2, falls twice as fast as ball 2.

Unfortunately, this isn’t true in real life, as Galileo would take great pains to point out. But it’s harder than you might think to prove it wrong (and harder still to find the correct law). In any case, it’s Aristotle’s attempt at a mathematical description of free fall, his stab at the Platonic-Pythagorean project. And it gives you an idea of how hard it is to tackle the problem of change mathematically.

4.9 Archimedes: Mathematics and Statics

4 . 9 A R C H I M E D E S : M ATH E M ATI C S A N D S TATI C S

Applying mathematics to terrestrial objects in motion was largely unsuccessful until the Scientific Revolution. But mathematics still had a part to play until then.

Astronomers continued to construct ever more accurate and complicated mathematical descriptions of the heavens. Not only that, but progress was made in mathematically characterizing stationary earthly objects. We now call the discipline that studies forces acting on stationary objects statics (static just means “stationary.”)

This next step in mathematical science—the inauguration of statics—was made by Archimedes of Syracuse (d. 212 bc, fig. 4.9a).

figure 4.9a

Archimedes is most famous for . . . um . . . running naked through the streets shouting “I found it!” (“Eureka!” in Greek). But he’s also known for his mathematical description of the forces acting on a lever. A seesaw is a type of lever. So is a pair of pliers. In any case, Archimedes found a mathematical formula for a balanced (and therefore stationary) lever. To put it differently, the formula describes a lever that is in equilibrium, that is, the forces acting on the lever cancel each other out. We call Archimedes’s mathematical law regarding levers, not very creatively but very descriptively, the Law of the Lever. He expressed it this way in On the Equilibrium of Planes:

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Two magnitudes [weights] . . . balance at distances reciprocally proportional to the magnitudes.

Again, taken as-is, in mere words, this isn’t very clear, so let’s first consider a picture.

figure 4.9b

Now add a little algebra (don’t worry how I got this; just trust me):

We can also rearrange the equation:

Notice that in the diagram, the lever is perfectly balanced. And notice in the formula we have an equals sign. This equals sign tells us that each side of the formula—each side of the equation—is balanced. This is why, when we manipulate equations, whatever we do to one side, we will have to do to the other to keep it balanced. So if you subtracted 3 from one side of the equation you would need to subtract 3 from the other side.

The concept of balance, or equilibrium, will become a surprisingly fruitful metaphor in mathematics. In an equation, we have a mathematical equilibrium, a

4.9 Archimedes: Mathematics and Statics

balance of concepts. Thinking of equations as a balanced lever or balanced scales will become important for the further application of mathematics to the physical world. It will be no coincidence that Galileo, the man who breathes new life into the Platonic-Pythagorean project of mathematizing the physical world—motion in particular—will look to Archimedes as his hero and inspiration for this project.5 Thinking of the world in terms of levers in equilibrium will be part of the key to unlocking the mathematical secrets of the universe.

In any case, Archimedes’s formula helps explains why a lever makes a useful tool. Suppose w1 is 1,000 pounds, and w2 is the force you need to apply to lift w1.

If d1 is short enough and d2 long enough, you need apply relatively little force to lift the 1,000 pounds. For example, if d1 is a foot and d2 is 10 feet, you would need apply only 100 pounds, so as long as you weigh more than 100 pounds, you can just lean on the lever. Archimedes picturesquely expressed the force-multiplying principle of levers by saying something like, “Give me a place to stand and a lever long enough, and I can move the world” (fig. 4.9c).

figure 4.9c

Archimedes also discovered mathematical principles for objects floating in water (or in any liquid, for that matter), and presented this work in On Floating Bodies.

He discovered that the upward force keeping the object afloat is equal to the weight of the liquid that the floating object displaces (fig. 4.9d).

5 For those who’d like to dig a bit deeper into Galileo’s inspiration, life, work, character, and conflicts.

I’ve written a brief biography, titled Galileo, that was published by Thomas Nelson in 2011.

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figure 4.9d

Or as he put it, Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced. On Floating Bodies I.5

This, too, can be understood in terms of a lever, balance, or scale (the reason for putting it in these terms will become a little clearer later) (fig. 4.9e).

figure 4.9e

4.10 Up Next: The Scientific Revolution

Notice that the lever is balanced. Keeping the previous picture of a floating ball in mind, the weight of ball (wb) on the right is equal to the weight of the fluid displaced (wf) by the ball. The buoyant force of the water pushing up on the ball is equal to the weight of the displaced water. We can put it in terms of algebra this way:

And since wb = wf, (we said that the weights are the same), then wb/wf will be 1.

From this we therefore know that df /db will also be 1, so df = db. And this is exactly what we have drawn. Of course the key to all this is discovering that

which is called Archimedes Principle. This discovery is what excited Archimedes so much that he ran naked through the streets shouting, “I found it!” In that case, it seems that he was the floating body.

In any case, Archimedes’s Principle is a mathematical description of stationary objects, objects whose forces are in equilibrium, so it’s a principle of statics. But in this case, it is also a principle of hydrostatics, since it describes phenomena in water or some other fluid.

So then, Archimedes successfully applied mathematics to a number of important terrestrial phenomena, which was an enormous step forward. Before Archimedes, mathematicians had applied mathematics to the heavens but not to objects on earth, at least not to an appreciable extent (but remember the Pythagoreans and the application of mathematics to strings and musical intervals). The ability to mathematically describe the motion of earthly objects was still centuries away, awaiting Newton’s calculus.

4 .10 U P N E X T: TH E S C I E NTI F I C R E VO LU TI O N

After Archimedes, statics (including hydrostatics) continued but made few advances, at least compared to the innovative work of Archimedes.

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And as far as the study of the heavens was concerned, the high-water mark will be the book Almagest, (which means “the greatest”) by the great Egyptian astronomer and mathematician, Claudius Ptolemy (d. AD 168). In it, Ptolemy not only presents his accurate and complicated mathematical model of the heavens, but also introduces trigonometry (which means, literally, the measurement of triangles) into Western science.

In the next chapter, we will pick up our story with Copernicus’s innovations in the 1500s, many centuries after Aristotle, Archimedes, and Ptolemy. By this time, Aristotle had become the main source of scientific understanding His teaching would eventually be combined with Christian doctrine to create the somewhat schizophrenic system of Scholasticism. This combination of Aristotle with Christianity lent a lot of credibility to Aristotle’s natural philosophy in the Christian West. Moreover, Aristotle’s entire corpus of work—including not only his physics but also his metaphysics, logic, politics, aesthetics, ethics, and biology—was so comprehensive, elaborate, and internally consistent that it was hard to deny his genius.

It was easy to be impressed and difficult not to believe that he was on to something.

In the meantime, Plato’s more Pythagorean approach to science had been overshadowed, partly due to the difficulty of applying mathematics to the physical world.

All this will change in the Scientific Revolution, a revolt in which Pythagoras and Plato will be vindicated. Aristotle’s dictum, “Ignorance of motion is ignorance of nature” will remain in play. But Aristotle himself will be dethroned. When the dust settles, Newton will have become king.

4 .11 S T U DY Q U E S TI O N S

1. What is the “Platonic-Pythagorean project”?

2. What do I say is your main goal here with respect to calculus?

3. Who gave the problem of change a mathematical twist?

4. What is one of the main goals of science?

5. What do we mean by “quantifying” motion?

6. What were two main difficulties with the project of describing nature mathematically? For the “second” difficulty, make sure you can name the concepts that give rise to the difficulties.

4.11 Study Questions

7. What are the difficulties specifically related to speed?

8. What is Plato’s “Pythagoras-inspired” project?

9. Explain how it was possible to temporarily set aside the conceptual problems associated with motion. Why was the Pythagorean problem—when it came to heavens—still difficult?

10. Who was the first to answer Plato’s challenge? What geometrical (i.e., mathematical) shape did he use? How many of these did it take to describe the motion of Mercury?

11. How many spheres did Eudoxus use for the entire cosmos? Was his system exact?

12. What did Plato and Eudoxus think caused the planets’ motion? Was it the spheres? What is the important distinction related to this; that is, cause as opposed to what?

13. The sign above the entrance to Plato’s Academy said, “Let no one ignorant of geometry enter here” and not “Let no one ignorant of mathematics enter here.” Why? 14. What is Euclid’s Elements? Why is it so important? What is so surprising about it?

15. Why is the distinction between celestial and terrestrial objects important for the problem of change?

16. How did Aristotle differ from Plato regarding Forms?

17. What was science called for most of Western history?

18. Did Aristotle think that mathematics was important? Explain?

19. Aristotle’s view of the Forms made the problem of change a bit more difficult to solve mathematically. How so?

20. Was Aristotle interested in Plato’s project? Explain.

21. An answer to a previous question was the distinction between descriptions and causes. How was this distinction related to the difference between Plato and Aristotle’s goal of science or natural philosophy?

22. What were the celestial spheres made of, according to Aristotle?

23. What did the ancient Greeks (including Aristotle) think composed terrestrial objects?

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24. Was Aristotle’s system of spheres a mathematical or physical system?

25. What are the four causes and why are they important?

26. What do we say causes a dropped ball to fall? What does Aristotle say the cause is? In each case, where are these causes?

27. What is a “nature” according to Aristotle? How does an object’s nature cause it to fall?

28. According to Aristotle, what causes fire to rise?

29. What does the Greek word physis mean? What word do we derive from it?

30. What is the “principle of change”?

31. In your own words, what is the goal of Aristotle’s physics?

32. The medieval Scholastics paraphrased Aristotle’s goal of physics with a famous saying. What was that saying?

33. Did Aristotle think that objects like rocks were alive since they “sought” their natural place when dropped?

34. Why did Aristotle state his law of free fall with only words, and not with mathematical symbols?

35. From memory, draw and label the diagram of the two free falling balls from figure 4.8 36. What is Aristotle’s law of free fall in terms of a formula (without the delta notation).

37. Write Aristotle’s law of free fall using the delta notation.

38. Aristotle’s law of fall is actually false. What do you think is wrong with it?

39. Does Aristotle’s law of free fall say what causes objects to fall? Explain.

40. What is the discipline that studies forces acting on stationary objects?

41. When did Archimedes die? When did Aristotle die?

42. What does “Eureka” mean?

43. What does it mean to say that a lever is in equilibrium?

44. Write out, in words, Archimedes’ Law of the Lever. Draw a diagram for this but without labeling.

4.12 Exercises

45. Label the Law of the Lever diagram and write out as a formula.

46. How might a lever be an analogy for an equation?

47. What is Archimedes’ Principle?

48. Draw a diagram showing how Archimedes’ Principle can be thought of as a lever problem.

49. What is hydrostatics?

50. How was Archimedes’ work a step forward in mankind’s work on the problem of change?

51. Who wrote the Almagest? When did he die? What does “Almagest” mean?

52. What does “trigonometry” mean?’

53. What is Scholasticism?

54. Why were Scholastic philosophers and theologians so impressed with Aristotle?

55. Why didn’t Plato’s Pythagorean approach to science advance very much between Aristotle and the time of the Scholastics?

56. Who will eventually replace Aristotle as the ruler of science after the Scientific Revolution?

4 .12 E XE R C I S E S

1. Suppose two objects are dropped from the same height and one of the objects is 100 pounds and the other is 10 pounds. According to Aristotle’s law of free fall, how much more time will it take for the smaller object to hit the ground than the larger one?

2. Suppose a man weighing 200 pounds is sitting on a see-saw perfectly balancing a 500-pound bear on the other end. If the distance from the pivot point (that is, the fulcrum) to the bear is 10 ft, what is the distance from the fulcrum to the man?

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12 T H E D E R I VAT I V E We’ve solved the paradox of speed, at least to the degree that we have a coherent definition of instantaneous velocity,

and a way to calculate it (by using the Method of Increments). That’s a big deal in the grand scheme of the problem of change.

Remember that there are three main concepts in calculus—the limit, the derivative, and the integral—all related to one another in a single theorem—the Fundamental Theorem of Calculus. In this chapter, we’re going to discuss the derivative, focusing on its definition and showing how it is related to the limit and the Method of Increments. In subsequent chapters, we’ll focus on taking the derivatives of various functions.

12 .1 I N S TA NTA N E O U S S P E E D A N D TH E D E R I VATI V E

We have all the tools we need to define the derivative; indeed you’re already familiar with the derivative. The limit of the average speed ∆d/∆t as ∆t approaches zero,

is what we call “the derivative of d with respect to t.” That is,

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12.1 Instantaneous Speed and the Derivative

This doesn’t specify just what the function d is, although because we’re taking the derivative with respect to t, we know that d is a function of t, that is, it’s d(t). The definition of instantaneous speed only requires that distance traveled be a function of time.

That said, in the previous chapters, we’ve been working closely with the function expressed by the formula

But we could have taken the derivative of a different distance function, a different d(t). For example (and we’ll do this later), we could take the derivative of

This function applies to a different physical situation than simple free fall of an object dropped from a stationary position, although the physical situation is similar, as we’ll see. In any case, for this function, we would have to go through our Method of Increments for d(t) just as before, first finding a formula for ∆d/∆t for some particular instant, even if we just call that instant t rather than some number, and then taking the limit of ∆d/∆t as ∆t approached 0. And though d(t) = −16t2 + 100t + 3 might require a bit more algebra to find a formula for ∆d/∆t, the steps of the Method of Increments are exactly the same.

So then, it turns out that the Method of Increments is simply a way of finding or calculating the derivative of a function. The derivative was originally called the derivative because in finding the derivative of a function we derive one function from another. But we also say that we differentiate a function to find its derivative and call this process differentiation. So far, we’ve been deriving v(t) from d(t). In any case, the Method of Increments is not only about speed, but about derivatives in general. We’ll see that we can use the Method to calculate the derivative for functions of many different types.

But focusing on instantaneous speed for a moment, notice that we can also express its definition this way:

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The qualifier “with respect to time” tells us that t is the independent variable in this function. The notation d(t) tells us the same thing. This qualifier provides a bit more detail than we need; after all, what other independent variable is there in d(t)? Time is the only other property represented in our distance function. But there are many functions with more than one independent variable—it’s just that we won’t consider them in this course. That being the case, we can simplify our definition as follows:

which is just to say

Putting all this together we can better see that

Again this definition is true for any d(t) function.

Even more important, if we change the notation a bit, we also have the definition of a derivative for any function whatsoever.

12 . 2 D E R I VATI V E : TH E D E F I N ITI O N

Let’s return to the distinction between dependent and independent variables.

Remember that the independent variable is the input of the function, and the dependent variable is the output. The dependent variable is dependent because its value depends upon the input.

Again, for our d(t), d is the dependent variable and t is the independent variable.

So then, we can say this:

12.2 Derivative: The Definition

Therefore, in the above case, the derivative of d(t) can be defined as follows:

Next, remember that d(t) is the name of a function of distance versus time. Yet the definition of a derivative applies to functions in general, not just those about distance and time. Suppose we let y(x) stand for any, generic function. Taking this into account and just rewriting things we have the general definition of a derivative:

Notice that “change in independent variable” again shows up twice: under the limit symbol and in the denominator. The fact that the change in the independent variable is both going to 0 and is in the denominator is one of the things that makes the derivative both dangerous and powerful.

Also notice that this definition doesn’t depend on what kind of things the variables stand for. In particular, there is no mention of speed or time. This definition applies to all kinds of functions, whether they have a physical meaning or not. So the derivative of a function won’t necessarily be instantaneous speed; that will depend on whatever y(x) is. The derivative of a function—which itself will be a function—will depend on that original function. But of course, in the case of a function

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representing distance as a function of time, the derivative will be instantaneous speed. Yet there are many other uses of the derivative besides finding instantaneous speed. As I said, the world is remarkably user-friendly.

Now that we have the general definition of the derivative, we can see that, to use the Method of Increments, all we need to find is a formula for ∆y/∆x and then take its limit as ∆x approaches 0. That is,

At least that’s the overall strategy. In practice it may be really difficult to get a decent formula for ∆y/∆x, but the strategy is simple, even if in practice it requires some heavy lifting. For now, we’re only worried about the definition of the derivative and the general strategy of deriving it, and in this course we’ll use only simple functions so that we can focus on the main concepts of calculus.

12 . 3 R ATE O F C H A N G E AT A N I N S TA NT

Let’s remind ourselves that calculus is part of the Solution to the problem of change.

The specific type of change we were interested in when looking at Galileo’s law of free fall, d(t) = 16t2, was what we call motion or change in location, or what the ancients called locomotion (remember, they often referred to any kind of change whatsoever as motion).

Another way of putting this is that, when it came to free fall, we were interested in how fast the distance or location changed—i.e., we were interested in speed. Or we might put it, as mathematicians do, the rate of change in distance with respect to time or rate of change in location with respect to time. (We will speak in terms of either distance or location.) Though speaking of speed in terms of “rate of change in location with respect to time” is clunky, it’s precise; and sometimes the trade-offs are worth it, at least when we’re trying to understand calculus.

Moreover, the real prize is instantaneous speed: how much an object’s location is changing at an instant. (Again, putting it this way highlights the paradox of

12.3 Rate of Change at an Instant

instantaneous speed: surely there is a speed at an instant, but of course no distance or time changes at all at an instant.) That is, we’re looking for the instantaneous rate of change in location with respect to time.

In fact, some mathematicians say that the single most important idea of calculus is the instantaneous rate of change of one variable with respect to another. Given the centrality of instantaneous rate of change, we could, then, define calculus as follows:

Again, in the case of speed, we have a specific kind of rate of change in mind: the rate of change in location or distance with respect to time. But we can also have rates of change in other things. For example, when you boil water on the stove, the temperature of the water changes, and so we have the rate of change in temperature with respect to time.

Strangely, rates of change don’t always have to change with respect to time, even though they’re called rates, which can be confusing at first. For example, the air temperature might change the higher in the atmosphere you go. In this case, we would speak of the rate of change in temperature with respect to altitude. Even though we’re talking about the rate of change, we’re not interested in how fast the air temperature changes over time, say, over the course of a day. Even at a given instant in time (imagine a kind of snapshot), the temperature changes with elevation. We need to think of rate a bit more broadly here. Instead of change in some property with respect to time, it is the change in some property with respect to some other property.

For a function—let’s return to calling it y(x)—the rate of change in y with respect to x is really how much (not necessarily how fast) y is changing as x is changing.

When we say that it’s the rate of change at an instant or the instantaneous rate of change, we don’t necessarily mean that the change is occurring in time, though we might. The word instant or instantaneous will sometimes mean an instant of time but will generally mean when the interval of the independent variable approaches 0, when the independent variable’s interval becomes infinitesimally small. So when speak of the instantaneous rate of change in temperature with respect to altitude, we are referring to how much the temperature is varying at a specific height—a point—above the earth.

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Notice too that, in a derivative, we have two variables changing simultaneously; each changes while the other changes (at least generally speaking; there may be special cases where the values of one or both variables don’t change, but those are fairly boring cases.) And the derivative tells us how much the dependent variable is changing Returning to our definition of the derivative, we can express it in terms of the phrase “instantaneous rate of change.”

Notice something else: whereas the derivative is the instantaneous rate of change, the average rate of change over some interval is given simply by ∆y/∆x. That is,

Let’s take the example of speed. Average speed is the average rate of change in distance with respect to time. That is,

Instantaneous speed, on the other hand, is instantaneous rate of change in distance with respect to time:

Of course, we’re thinking of a distance function as d(t) since we’re used to working with Galileo’s law of free fall. But we can speak of a change in location

12.4 The Method of Increments in Terms of Rate of Change

generally, perhaps calling it x. In that case, we’d have essentially the same thing but with slightly different notation.

Perhaps you’re thinking, “Why even mention such a trivial change, the change from d to x?” Because one of the keys to success in mathematics is learning to generalize or handle different variables for the same (or similar) concepts without thinking you’re dealing with entirely different kinds of mathematics. Changing notation is one of those things that frequently trips us up when learning math.

Moreover, we often need the notation to be more descriptive of what the variable is standing for, whether it’s referring to distance travelled, location, or height. These can have subtle but important differences, as you’ll see.

12 . 4 TH E M E TH O D O F I N C R E M E NT S I N TE R M S O F R ATE O F C H A N G E

We can modify the wording for the Method of Increments, in terms of rate of change:

Much of the work in calculus courses comes not from trying to understand what a derivative is, but from trying to calculate derivatives for the gazillion different kinds of functions. And because many of these functions are idiosyncratic and ornery, students can get distracted from the meaning of calculus, spending most of their blood, sweat, and tears on the tricks specific to each particular kind

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of function. Not that the details aren’t important; they’re just not as important for those who are first trying to learn what calculus is, to understand the big picture.

Indeed, after you understand the fundamentals of calculus, the variation can help you understand calculus even more. But only after.

Now let’s do an example.

12 . 5 E X A M P LE : U S I N G TH E M E TH O D O F I N C R E M E NT S TO F I N D TH E D E R I VATI V E

Using the Method of Increments, let’s find the derivative of y(x) = 3x2, and let’s approach the point x from values greater than x. Now, notice this function y(x) as nothing to do with speed, as far as we can tell; it’s just a generic function. But other than that difference, we proceed just as we did in the last chapter. As in that chapter, we’re looking for a function of x rather than a specific number. Given that we’ll approach x from values greater than x, here’s a way to draw a diagram that will help guide us in setting up the problem.

figure 12.5

In this case, x is the name of the generic point in which we’re interested (instead of 1 or 5 or whatever); we want a value that can stand for all points. In other words, our point of interest is x, and this will allow us—once we’ve found the derivative function—to find the instantaneous rate of change at any x-value we want.

12.5 Example: Using the Method of Increments to Find the Derivative

Step 1: Find a formula for ∆y/∆x.

Since y(x) = 3x2, ∆y is the difference between value of y at the two endpoints of the interval ∆x, that is, the difference between y evaluated at x + ∆x and y evaluated at x.

Let’s just call those two corresponding y-values y2 and y1, respectively (see fig. 12.5).

Remember, for the delta notation, we subtract the smaller (y1) from the larger (y2)

And so for ∆y/∆x we have

Step 2: Take the limit of ∆y/∆x as ∆x approaches 0.

To get the derivative of y(x) = 3x2, we take the limit as follows:

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12 .6 D E R I VATI V E S V E R S U S P L A I N O L’ LI M IT S

Obviously, limits are an important part of calculus. But there’s often a difference between the derivative of a function and the limit of that same function. This is important and is often missed. It’s important because distinguishing between the two will help you further understand the definition of a derivative.

For any function, y(x),

The two sides of the equation are not the same even though they both include the concept of a limit. The left-hand side is the derivative of y(x) with respect to x.

Derivatives are a special use of a limit. The right-hand side is not the derivative, but simply the limit of y(x) as x approaches 0 (and not as ∆x approaches 0). Of course, there are two main reasons for the difference. The first is that the limit is being performed on two different functions, namely ∆y/∆x, which will be a function of ∆x, and y(x), which is of course a function of x. To put it simply,

The second difference is that the independent variables that approach 0 are different, namely ∆x and x. That is,

Again, the derivative of y(x) is just one use of the limit, a tool that can be used on all sorts of functions and for all sorts of purposes. Using the generic function y(x),

is not the same as

12.7 Notation

Let’s take a specific case of this difference, going back to Galileo’s law of free fall:

We saw that the derivative of d(t) at 3 seconds (approaching the instant of interest from time values less than 3 seconds) is

which is the instantaneous speed at 3 seconds. The final answer is in feet per second because ∆d/∆t is in feet per second—distance on top, time on the bottom. The derivative of d(t) is very different from the following limit of d(t) (and we don’t have a special name for it):

This is expressed in feet because d(t) is distance and is in feet. There may be cases where you would want to take the limit of Galileo’s function this way, but right now it’s not clear what such a case might be. It basically says that, as the time gets closer to the moment the ball was dropped (the initial time, ti), the distance traveled approaches 0. Entirely true, but not very interesting.

12 .7 N OTATI O N

Mathematicians are always looking for good, simple ways to express more complicated ideas. Indeed, mathematical and scientific breakthroughs may depend on the notation: how clear it is, how much it might spark new ideas, etc. Let’s spend a few moments on the various kinds of notations associated with derivatives.

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We’ve made a step toward simplifying how we write the derivative of y(x).

Remember that the really complicated version is something like “the limit of the rate of change in y with respect to x as the change in x approaches 0.” We can now express this much more simply as

and we can refer to all this as “the derivative of y(x).”

But we can do even better. Mathematician’s today use Leibniz’s notation (Newton wasn’t at all interested in making calculus user-friendly; Leibniz worked hard at it) and denote the derivative of y(x) this way:

This dy/dx notation makes sense if you recall that the average rate of change in y with respect to x can be written as

and that the ∆ is an uppercase delta, the Greek equivalent of an uppercase D. Recall also that ∆ is our symbol for “change in.” So we can think of the ∆ as a relatively big change and a lowercase d to remind us that the intervals referred to by the ∆ are getting smaller and smaller as they approach zero. (NOTE: d in this case isn’t standing for the function d = 15t2; it’s purely a coincidence that we’re using d for both.) So then, we can write the derivative using this notation as follows:

This may not seem like much, but it’s an amazingly simple and densely packed statement of the derivative. Not only that, but this notation reminds us visually that the derivative is the instantaneous rate of change—again, the lowercase d that

12.7 Notation

evokes the idea of a tiny or infinitesimal interval. It simultaneously reminds us that the derivative is also associated with the average rate of change over some interval (∆y/∆x).

But there’s an even simpler shorthand for the derivative, though it doesn’t give us as much information about what the derivative is:

or, dropping the “(x)”

Where the ′ is called a prime, and we call y′, y prime. This notation is helpful because y′ is faster to write than dy/dx. But speed isn’t always of the essence in mathematics; in fact, it almost always takes a backseat to safety. Again, much of the success in learning or practicing mathematics is the ability to pay attention to detail and to proceed carefully. Remember, safety first.

In any case, to summarize our intermission on notation, we have, for the derivative

Keep in mind that we’re using y(x) as a kind of generic name for all functions. The derivative notation applies to other names of functions as well. For example,

Also, the independent variable doesn’t have to be x, or even t. We could have

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Here’s another bit of notation: when we’ve taken the derivative, we sometimes evaluate it at some specific point of interest (although we don’t have to). There’s a notation that we can use to remind us when a derivative is at such a specific point.

Suppose we are interested in an object’s velocity one second after it was dropped.

Suppose also that the distance function is given by y(t). We’re interested in the derivative y(t) evaluated at t = 1 s, that is, we want to find

Some of these notations we’ll use more than others, but hopefully you’ll become comfortable with them all. Familiarity with different notation for the same idea helps us get a handle on the idea.

12 . 8 U P N E X T: F I N D I N G TH E D E R I VATI V E O F y ( x) = ax 2 + bx + c

Now that we have a definition of the derivative, it’s time to use it. In the next chapter we’ll use the Method of Increments to find the derivative of any function that has the form

We’ll also look at the idea of taking the derivative of derivatives. And though it will be nice to have in hand a ready-made formula for the derivative of such functions, the process by which we get the formula is just as important. We want to practice the Method of Increments to establish it firmly in our minds to better understand one of the most important concepts in mathematics (the derivative, that is).

12 . 9 S T U DY Q U E S TI O N S

1. In symbols, what is the definition of instantaneous speed, v(t)? What is the process for calculating it?

12.9 Study Questions

2. What are the three central concepts of calculus? What theorem ties them all together?

3. Why do we use the word “derivative”? What other related terms will we use to talk about the process of finding derivatives?

4. What is the definition of instantaneous speed in terms of the derivative of a function?

5. In terms of “input” and “output,” what is the difference between independent and dependent variables? In what sense is the dependent variable dependent?

6. In d(t) what is the independent variable?

7. In terms of dependent and independent variables what is the definition of a derivative? (This was the “ugly” version.) Now write it in symbols, using y(x).

8. What are the two main steps for finding the derivative of a function y(x)?

9. In terms of “rate of change” what is instantaneous speed?

10. In terms of “rate of change” what is calculus?

11. Is the rate in “rate of change” always about a change in time? Explain.

12. In terms of “independent variable” what does an “instant” mean generally (and not just for time)?

13. Express, “The instantaneous rate of change in y with respect to x” using symbols.

14. Express, “The average rate of change in y with respect to x” using symbols.

15. Write out the steps of the Method of Increments using the phrase, “rate of change.”

16. Explain the following:

17. Are all derivatives limits? Are all limits derivatives?

18. What are some ways to express (in symbols) the derivative of y(x)?

19. What will we do in the next chapter?

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12 .10 E XE R C I S E S

1. Using the Method of Increments, find the derivative of y(x)=5x2. Approach x from values greater than x.

2. Using your results from the previous question, find

for y(x)=5x2.

3. For y =3x, find y using the Method of Increments. Approach x from values less than x.

4. For y(x)=5x2, find the limit of y(x) as x approaches 3.

5. Using the Method of Increments, find the function for the instantaneous rate of change for f(x) = 2x3.

22 DEFINING THE INTEGRAL We’re now ready to study the third fundamental concept in calculus: the integral.

In this chapter we’ll look at what an integral is, that is, how we define the integral.

In the next chapter, we’ll look at a method for calculating the integral before finally seeing how it’s directly relevant to the problem of change.

22 .1 TH E M E TH O D O F E XH AU S TI O N

Another important problem calculus addressed was that of finding areas of complicated shapes—including the area under the curve of a function’s graph. This might not seem very significant, and it’s certainly not clear how it’s related to the problem of change, but remember that something as seemingly insignificant as finding the slope of a curve gave enormous benefits for the study of change. But for now, let’s just start with areas.

It’s relatively easy to calculate areas the areas of polygons—shapes that are bounded by straight lines. For example, the area of a square or triangle doesn’t present much of a problem and, even in the case of a complicated polygon, we can often divide the area into a number of triangles and add up the area of these triangles to get the total area. For example, figure 22.1a the shape in figure 22.1a.

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22.1 The Method of Exhaustion

It’s much more difficult to calculate the areas of shapes with curved borders, especially irregularly curved borders, since triangles, with their straight edges, don’t fit curves exactly. But Archimedes had remarkable success when he extended the strategy of adding up smaller, more manageable areas, applying it to shapes with curved sides. Although his method gave only approximations of the areas, he could make the approximation as close to the actual area as he wanted. Let’s look at this strategy.

For starters, we could conceivably approximate the area of a circle with a single triangle (see fig. 22.1b), but this obviously won’t give us a very good approximation.

We could improve on this pretty sad approximation by using, say, a square—which itself can be easily divided into triangles. This will give us a better approximation of a circle’s area (it couldn’t be much worse), but it’s still pretty far from the mark.

Yet—as figure 22.1b shows—as we choose polygons with more and more sides, we can get better and better approximations. In fact, this process of using polygons with increasing numbers of sides can continue as long as you wish, assuming you have the time and motivation.

figure 22.1b

Now notice that you can have two kinds of approximations: approximations that are larger than the actual area and those that are smaller. That is, you can either overestimate or underestimate the actual area. It depends on whether the approximating shape is inscribed inside the circle or circumscribed outside it. But in either case, the more sides your polygon has, the better it will approximate the actual area of the circle, and it usually doesn’t matter whether you underestimate or overestimate.

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figure 22.1c

This process is known as the method of exhaustion. Although it doesn’t really mean “continue until you’re exhausted” thinking of it this way will help you remember it. In this context, exhaust means, roughly, “try out and throw out” the answers that are either too large or too small, until you get the answer you want. We use it this way in the familiar phrase exhaust the possibilities.

Let’s call all the little individual areas A1, A 2, A 3, . . . (remember, the ellipsis “ . . . ” means “and so on”). Suppose, for example, we use a 12-sided regular polygon to approximate a circle.

figure 22.1d

The area of the polygon is

22.2 Integrals and Area

Each of the individual areas or mini-areas is the area of a triangle, which is relatively easy to calculate. We also know—since we constructed it this way—that the area of the polygon approximates the area of the circle. That is,

(where the symbol ≅ means “approximately equal to”). Therefore,

As the number of mini-areas increases, the size of each mini-area gets smaller.

And it’s pretty clear that the more mini-areas you use to approximate the circle, the closer you’ll get to the circle’s actual area. Again, you can get as close to the correct answer as you want: just keep increasing the number of mini-areas. So, if the number of mini-areas were 100, your approximation would be much better than, say, if the number of mini-areas were 12.

But why stop at 100? Why not a million mini-areas? Or 10100 mini-areas (that is, a 1 with 100 zeros)? To be sure, if we had to calculate each mini-area by hand we wouldn’t do this, but surely we could make a simple computer program to do it for us. Just call the number of mini-areas n, that is,

and then choose n = 1,000,000 to plug into this simple program, go get some coffee, and let the computer do it for you (in fact, today, computers would probably calculate it so quickly you might only have time to blink or take a breath).

You might see where we’re going with this. As n gets larger, the approximated area approaches the exact or actual area. This language of approach will no doubt remind you of limits. And this isn’t a coincidence—we’ll use limits to define (and to calculate) integrals.

22 . 2 I NTE G R A L S A N D A R E A

The strategy, then, of finding the areas of complicated shapes is to divide the shape into mini-areas whose areas are easy to calculate, and then add up these mini-areas.

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This is a pretty simple strategy, and if you understand it, you already understand the strategy behind the next key concept in calculus: the integral.

The first thing to notice is that an integral is an area. The second thing is that an integral is a sum of mini-areas. The third thing—and I’ll have to do some more explaining—is that, for an integral, the number of mini-areas approaches infinity. Let’s summarize this before moving on.

As I said before, the key to the integral is this last point: it allows the number of mini-areas to approach infinity. Though approach is too strong, because no matter how high you count, you’re never any closer to infinity than when you began (remember some of the paradoxes of infinity). It would be better to say that the number of mini-areas tends toward infinity, since you’re at least heading in the direction of infinity. In any event, we’ll use the word approach.

A summation of n mini-areas—let’s call this sum Sn—would be written as

Again, the number of mini-areas, n, could be any number whatsoever, 1 or 15 or a million, but we want it to approach infinity in order to get Sn as close to the actual area as possible. Again, this is a job for limits, and so we take the limit of this summation as n approaches ∞. This, in fact, is just what an integral is. The total area, A, is given by this formula:

As n approaches infinity, the size of the mini-areas approaches zero. Think of it this way: in order to fit more and more mini-areas into the circle (or whatever shape you have) the mini-areas must get smaller and smaller. If you continue this line of thinking, then as the number of mini-areas approaches infinity, their size must approach zero. Again, this would raise weird issues for mathematicians over the first 150 years after calculus was invented. We, however, will just ignore these issues.

Let’s now transfer what we know about integrals and areas to functions and their graphs.

22.3 Area under a Curve: Integrals and Functions

22 . 3 A R E A U N D E R A C U RV E : I NTE G R A L S A N D F U N C TI O N S

You now know that the integral is an area calculated by adding together an unimaginably large number of unimaginably tiny mini-areas. The next thing we need is the fact that we can graph shapes in Cartesian coordinates. After all, even though we’ll be working with functions, we’re talking about shapes here, and the graph of a function gives us a function’s “shape.”

Let’s begin by looking at the graph of y(x) = x2, but keep in mind that what we say here will be applicable to any function that we can integrate. In any case, this function will give us a complicated enough shape. Let’s avoid any unnecessary creativity and simply call the total area of the shape A.

figure 22.3a

The shape A is the area under the curve y above the x-axis and bounded on either side by vertical lines at x = a and x = b. We’ll often just call this the area under the curve for short. This shape is complicated enough to introduce how integrals can help us to calculate the area of nearly any shape.

The next thing we do is divide A into rectangular mini-areas. We’re using rectangles rather than triangles because the area of a rectangle is even easier to calculate than the area of a triangle: it’s simply base times height.

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figure 22.3b

So then, let’s divide the area A into n rectangles:

figure 22.3c

The ellipsis ( . . . ) between the fourth rectangle and the nth rectangle reminds us that there are other rectangles between them, but we just don’t know how many, since we don’t know exactly how large n is. We’re assuming n can be any number. So n is simply the total number of rectangles. And each of the mini-areas is A1, A2, A3, . . . , An. Notice that the rectangles are inscribed, and the sum of the mini-areas is smaller than the total area, the sum of mini-areas must grow toward the total area A as n increases. That is, we’re underestimating the total area in the figure 22.3c. But we could have circumscribed the rectangles, and so could have begun with an overestimation of the area, which we’ll do later. Either way, defining the integral will be the same. There

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22.3 Area under a Curve: Integrals and Functions

will be a slight difference, however, when it comes to calculating integrals. Not much, but enough that we’ll need to pay attention. Again, we’ll do some examples later.

Of course, the fewer mini-areas we have, the further away our sum of the areas will be from the actual area. Consider a case where n is small:

figure 22.3d

The gaps are the problem here, and shrinking the size of each mini-area will shrink the size of the gaps. Notice the difference in gap size as we increase n in the next two graphs.

figure 22.3e

figure 22.3f

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Since each of the rectangles has an area of base × height, let’s figure out what the base and height of each rectangle will be in terms of Cartesian coordinates. Let’s call the base of each rectangle ∆x. Now notice that the top of each rectangle touches the graph of y(x) = x2 at a single point. The height, then, is the value of y(x) at the rectangle’s point of contact with y(x). That is, we can calculate the height of each rectangle by plugging in the appropriate x-value into y(x) = x2. We’ll stipulate that the width of each mini-rectangle is the same and refer to this width as Δx. (You’ll see why in a bit.) Let’s label what we have so far:

figure 22.3g

A lot is going on in figure 22.3g. First, notice that the details inside the circle on the graph are magnified in the circle to the left. Second, the height of each miniarea is the y-value of the curve where the curve and mini-area touch. Each mini-area’s height is evaluated at the corresponding x-value. In other words, to get y1, you plug x1 into the function y(x) = x2. Let’s put this another way (fig. 22.3h): the point where A1’s rectangle touches the curve y(x) = x2 is given by the coordinates (x1, y1).

figure 22.3h

22.3 Area under a Curve: Integrals and Functions

Again, ∆x is the width of each rectangle’s base, so as n, the number of mini-areas, approaches ∞, the width of each mini-area, ∆x, approaches 0, and therefore so does the area of each rectangle.

We’re now ready to write the formula for the mini-areas in terms of x and y. For the area A1, the base is ∆x and the height is y1.

figure 22.3i

In general, the area of each Ai is yi × ∆x and so the sum of n mini-areas—remember that we called this sum Sn—is given by

We want n to be as large as possible, to go toward infinity. So we still need to take the limit of S as n approaches ∞. This next step is simple enough—to write, at least.

Remember that A is the total area under the curve y(x) = x2 that we’re ultimately after. Sn by itself for some finite n is really just an approximation and not equal to A itself. But to precisely calculate A, not merely approximate it, we take the limit of Sn as n approaches ∞:

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From what we know about Sn, we can put things as follows:

Again, this just is the definition of an integral:

where

figure 22.3j

Let’s take stock of what we’ve done. We’ve taken the ancient method of exhaustion and combined it with limits to get the definition of an integral. And analytic geometry and its Cartesian coordinates allowed us to talk about integrals in terms of areas and formulas. Notice that, although we used a specific function y(x) = x2 throughout our discussion here, our definition of an integral is entirely general—it applies to functions in general.

We still haven’t seen an example of how we calculate the integral (we’ll do that in the next chapter). Right now we want to make sure we understand the idea of an integral. Of course, calculating integrals will help us understand even more.

22.4 Integrals and Circumscribed Rectangles

Another thing. Remember that during our entire discussion, we’ve used an example of inscribed rectangles, rectangles that are inside the area we’re interested in.

But we would get the same exact definition if we had considered a circumscribed example. Whether we approach the actual area from an underestimation or an overestimation, the calculation that we end up with is the same. Although we won’t go into all the detail we did for the inscribed case, I want to make sure we look at what’s going on in circumscribed case. The reason is that when it comes to actually calculating or evaluating integrals, there’s enough of a difference between the two cases to matter.

22 . 4 I NTE G R A L S A N D C I R C U M S C R I B E D R E C TA N G LE S

In figure 22.4a, we’ve kept everything the same as in the earlier inscribed case but made the rectangles larger, so that parts of them are outside the area we’re interested in.

figure 22.4a

As before, we’re interested first in adding up the area of n rectangles and then, second, taking the limit of this sum as n approaches ∞. And the area for a given

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rectangle—for example, A1—a is y1∙∆x. We even calcluate the height yi the same way.

The difference is the where the y-value is. Now that the rectangles rise above the function’s curve, the y-value that we need for the height of the rectangle is located where the rectangle’s upper right corner touches the function’s curve.

But in both cases—both the inscribed and the circumscribed—the height is given by the y-value where the function’s curve touches the top of the rectangle.

In the inscribed case, this point occurs at the upper left corners of the rectangles, in the circumscribed case it occurs in the upper right-hand corners. And notice, because we’re interesting in where yi touches the curve, and in this case, the upper right-hand corner is touching the curve, the x-value will also be moved accordingly.

figure 22.4b

So again, even though we’re calculating the height of the rectangles at a different location, the area for each rectangle is still Ai = yi ∙∆x, it’s just that the values of yi and xi will be different. And the sum of the area of n rectangles is the same as before:

22.5 Up Next: The Method of Summation

22 . 5 U P N E X T: TH E M E TH O D O F S U M M ATI O N

Now that we’re clear about the definition of an integral—that it’s the area underneath a curve and that it’s a summation of mini-areas—we’re ready to use this definition to calculate the areas of complicated shapes. So then, in the next chapter we’ll use the Method of Summation for integrals, a method which is analogous to the Method of Increments we used for derivatives.

22 .6 S T U DY Q U E S TI O N S

1. What is the method of exhaustion?

2. From the figures in the text, draw what the method of exhaustion might look like for approximating the area of a circle for both underestimating and overestimating the area.

3. What do we call the case of overestimating? Of underestimating?

4. What does “exhaustion” mean in the context of the method of exhaustion?

5. What happens to the approximation of the area as the number of mini-areas increases?

6. What letter did we use to represent the number of mini-areas?

7. In the text we gave three characteristics of the integral. Write them out.

8. What concept or tool of calculus—a concept that is related to the word “approach”—was added to the method of exhaustion to get the notion of an integral?

9. Why is it odd to say that “n approaches ∞”?

10. In words, what is Sn? 11. In Sn and the limit notation, what is the formula for the total area?

12. In formula form, what is the definition of an integral?

13. What happens to the size of the mini-areas as n approaches ∞?

14. Why do Cartesian coordinates help us when calculating the area of shapes?

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15. For the function y(x)=x2, sketch the drawing that shows the area under the curve in Figure 22.3a.

16. What letters did we use to represent the two endpoints (where the sides run into the x-axis) of the area?

17. What is the area formula for a rectangle?

18. Draw the graph in Figure 22.3c. (Eventually make sure you can draw it from memory.) Are the rectangles inscribed or circumscribed?

19. Draw the graph in Figure 22.3d. What is important to notice about the gaps?

20. As we go from Figure 22.3d, to Figure 22.3e, to Figure 22.3f, what happens to the gaps? What happens to the mini-areas?

21. Draw the graph in Figure 22.3g. Make it large enough to draw the details.

22. In Figure 22.3g, what is important about where the rectangles touch the curve of y(x)=x2? What symbol are we using to refer to the width (i.e., base) of each mini-rectangle? What symbol for the height? How would we calculate that height?

23. Draw Figure 22.3h. What are the coordinates of the point where the rectangle touches the curve y(x)=x2? What is the formula for the height of the rectangle?

Where did we get that formula?

24. Looking at mini-area 1 in Figure 22.3i, what is the formula for its area?

25. In terms of Ai ’s what is the formula for Sn? 26. In terms of yi ’s and ∆x’s, what is the formula for Sn? 27. In terms of Sn and the limit notation, give the definition of the integral.

28. In terms of Ai’s and the limit notation, give the definition of the integral.

29. In terms of y’s and ∆x’s and the limit notation, give the definition of the integral.

30. Draw Figure 22.4, the circumscribed case. What important details change from the inscribed case? Are any of the formulas or symbols different?

31. Is the definition of the integral any different if we begin with the circumscribed case?

32. What will we do in the following chapter? What method will we use to do this?

CHAPTER 23 U S I N G T H E M E T H O D O F S U M M AT I O N T O C A LC U L AT E I N T E G R A LS

23.1

The Method of Summation

23.2

Special Sum Rules and the “+ · · · +”

23.3

Inscribed Areas

23.4 Circumscribed Areas 23.5 More (Complicated) Examples: y(x) = x² 23.6 The Integral Notation 23.7

Up Next: The Fundamental Theorem of Calculus

23.8 Study Questions 23.9 Exercises

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23 U S I N G T H E M E T H O D O F S U M M AT I O N T O C A LC U L AT E I N T E G R A LS We now know what an integral is:

When we first stated it this way, we didn’t call it the definition of an integral but that’s what it is. After all, it’s just a restatement of the symbolic version of our definition:

But we have yet to use this definition it to calculate an area of an actual shape.

That is, we don’t yet know how to integrate a function. Now’s the time to learn the process of integration.

Just as we did with derivatives, we’re going to first use a rather painful method before showing you the easy way out. Recall that, with derivatives, we first used the Method of Increments to show you what is going on when we take the derivative of a function. Only then did I allow you to take the shortcut for differentiating power functions, that is, the Power Rule. We’re going to do something similar for the integral. We’ll call the first, complicated method the Method of Summation. We’ll see the easier method in the next chapter.

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23.1 The Method of Summation

2 3 .1 TH E M E TH O D O F S U M M ATI O N

First, based on our definition of the integral, the Method of Summation will have two main steps:

This is extremely simple, but not necessarily easy (but nearly all the work is in the first step; step 2 is a piece of cake). Nevertheless, keep these two steps in the front of your mind the entire time. The big picture is just as important as the details.

Let me be frank with you: these Method of Summation problems are likely to be the most difficult of the course, largely because there are so many steps. But once you learn the overall strategy, they’re simply a matter of keeping the algebra straight. They’re tedious and require attention to detail, but it’s not hard to understand the strategy. And though there are many steps, each step is usually easy. Just pay attention and don’t rush things.

2 3 . 2 S P E C I A L S U M R U LE S A N D TH E “+ · · · +”

Before we begin, I need to give you a set of tools (consider them gifts—we’re not going to prove them). We’ll need these tools to deal with the fact that we never specify n with an actual number. This is why, you’ll recall, we had to use the ellipsis ( · · · ) in our formula for the summation of n areas:

But we’re just not equipped to handle the “ · · · ” very easily, so we need a way to get rid of it. Here are some rules for summations that will help us in the examples to come. I’ll explain as we go how to use them. For now, just notice that they are handy rules that get rid of the · · · and replace it with a formula in terms of n.1 1 See, for example, Morris Kline Calculus: An Intuitive and Physical Approach, Chapter 9.

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On to the examples.

2 3 . 3 I N S C R I B E D A R E A S Example 1

Find the area under the curve y(x) = 4x between x = 1 and x = 3. That is, find A. Use inscribed rectangles.

figure 23.3a

Of course, we really don’t need an elaborate process like integration to calculate this area. For example, we could easily just divide the area into a rectangle and triangle and add up these two mini-areas (see figure 22.3b). There wouldn’t be any need to consider an infinitely large number of mini-areas.

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23.3 Inscribed Areas

But we’re going to need integration for other, more difficult shapes, and it’s best that we learn how to integrate on simpler shapes. Walking comes before running.

Solution

Begin by drawing the picture, labeling all the relevant details. This takes a bit of work but it’s crucial, since you’ll use your labeled diafigure 23.3b gram to set up your algebra problem. I suggest something like the following order, but as long as you label everything, the exact order is up to you.

(Since this pretty much how every problem will be solved using the Method of Summation, I’m putting a box around these steps.)

1. Draw the axes and graph of the function.

2. Label the endpoints 1 and 3 on the x axis, drawing the two sides of the area, that is, a vertical line from 1 on the x-axis up to the line of y(x) = 4x, and a vertical line from 3 on the x-axis up to the line of y(x) = 4x.

3. Draw the mini-area rectangles and label them A1, A2, A3, A4, . . . , An. Notice how these—according to the way the problem was presented to us—are inscribed rectangles. This will determine where the rectangles touch the graph of y(x) = 4x. This is very important. Also pay attention to the . . . between the A4 rectangle and the final rectangle, An. 4. Label, the heights of the mini-rectangles (y1, y2, y3, . . . , yn) where the rectangles touch the graph, as well as the x-values at which to evaluate them (x1, x2, x3, . . . , xn 5. Label the ∆x’s.

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figure 23.3c

Step 1 of the Method of Summation

Find a formula for Sn in terms of n. That is, find the area of n rectangles:

We’ll eventually want this in terms of n since we’ll ultimately be taking the limit of Sn as n → ∞, but we’ll do this in steps. Also, we’ll introduce a bit of notation for Sn to help us remember that Sn is an underestimation for finite n’s. Let’s put a bar underneath Sn:2

We’ll first find formulas for the heights (all the yi’s). We can say this another way, which turns out to be useful. Find all the yi’s and xi’s, where the i stands for each rectangle’s identifying numbers. For the first rectangle, i = 1 and so its yi and xi will be y1 and x1. Similarly for the second rectangle: i = 2 and its yi and xi will be y2 and x2. And so on, until the nth rectangle, where i = n, which gives us yn and xn.

To do this, we need to find formulas for xi, and then plug each xi into the function y(x) = 4x to get each yi. I’ll first tabulate this information and explain it afterward. To get these, I paid close attention to the diagram. The information in the table is essentially a description of the diagram. The tabulated information would 2 See Kline 1967, 233.

23.3 Inscribed Areas

make no sense without the diagram. The diagram is the centerpiece; you can begin to see why it’s so important to draw and label first.

We only calculated the first three terms before the · · · and the nth term. We could have written out more terms, but you’ll see that we don’t really need to. The table shows that we did things very methodically, which is fitting for mathematics problems, especially when they have a lot of steps. For each i we found the xi, and then plugged this xi into the function y(x) = 4x to get the corresponding yi.

The only tricky thing is figuring out what xn is: in particular, how did we get the (n−1) for the nth term? Let’s go through this slowly.

It’s fairly clear that the value of x2 is simply x1 + ∆x, and because x1 = 1, x2 = 1 + ∆x.

For each xi after that, we’re moving to the right one more ∆x. So x3 = 1 + ∆x + ∆x, or combining the two ∆x’s we get 1 + 2∆x. This much is pretty easy. It’s the xn that will be a bit trickier. The question is, how many ∆x’s have we moved over from 1 to reach xn?

Well, notice that for i = 2, the number in front of the ∆x is 1 (since ∆x = 1∙∆x).

For i = 3, the number in front of the ∆x is 2. And, going back to i = 1, the number in front of the ∆x is 0, since 1 = 1 + 0∙∆x.

We’ve got a pattern here. The number in front of each ∆x is i−1 for each i. That is, in this example, for every i,

and so when i = n, we have

Now let’s go back to the formula for Sn, plugging in all the yi ’s (the heights— notice that the bases or the Δx’s are already there).

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From here on out, it’s mostly a matter of algebra.

As an important aside, in algebra, when we rearrange all the terms, there are usually a variety of ways to do it. As we proceed, you might think, “Why are we doing it this way?” or “I would have never thought of that!” That’s okay. You’re not supposed to come up with these moves; at this point you’re simply supposed to be able to follow them and then copy them. This is often how the great artists began, and it’s an underappreciated key to the study of mathematics. Let the geniuses come up with the clever tricks. We mortals simply learn from them and are better for it.

Besides, even the geniuses had to start somewhere.

So watch closely how we distribute terms and combine them; it’s pretty clever (I certainly didn’t come up with it). You’ll get practice copying this method later.

For now, take it slowly and make sure you understand each step. When reading any math problem, you cannot skim or simply “get a feel” for what’s going on. You must see how each step follows from the previous step. Don’t feel bad if it takes you a while to think about how we get each step from the previous one. And if you see it immediately, even better. But that’s rare.

Back to the algebra.

Let’s move the bases of the mini-rectangles, the ∆x’s, to the left side of each term. We’re eventually going to distribute the ∆x’s, so I usually put them out front.

Notice that there is a 4∆x for each term (keep this in mind for later). We can now go ahead and get rid of most of the parentheses by distributing the 4∆x’s through the parentheses. (In later examples I won’t explain every little detail, but the first time through it’s good to be thorough.)

23.3 Inscribed Areas

Notice that there are a number of 4∆x terms that are on their own:

In fact, as I said earlier, there’s a 4∆x for each term, for each i or each mini-rectangle. We also know that there are n terms total since there are n mini-rectangles. If we had, say, three terms, we could combine 4∆x + 4∆x + 4∆x into 3 · 4∆x. And we can do something similar for n terms: we can combine 4∆x + 4∆x + 4∆x + · · · + 4∆x for n terms into n · 4∆x. This is where Sum Rule 1 comes in.

So we replace all those 4∆x’s with 4n∆x (notice that I put the 4 out front, but you don’t need to):

We’ve simplified things a bit, but we’re also going to have to do something about the “ · · · ”. We’ll need a Sum Rule here, too, but it’s not clear which one.

We’re going to factor out the 4(∆x)2’s from the last three terms to get something really convenient:

The convenient part is the sequence in square brackets, [1 + 2 + · · · + (n − 1)], which matches Sum Rule 3.

Our summation of the mini-areas, Sn now becomes

No more “ · · · ”!

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Next, we’re going to replace the ∆x’s with equivalent values, but in terms of n (this is because we’re going to be taking the limit of Sn as n approaches ∞, not as ∆x approaches ∞). Thankfully, we can easily represent the rectangle’s widths in terms of n. The width of each rectangle is the total length of the all the bases together, divided by the total number of rectangles. Since all the Δx’s are the same size, each Δx is equal to the sum of all the bases divided by n. The sum of all the bases is really just the base of the total area A, which in this case is 3−1 or 2. Notice we don’t have any units of measurement; we haven’t specified whether we’re measuring inches or feet or millimeters. So the width of each mini-area’s base is

Each rectangle, therefore, is 2/n wide. Now replace each ∆x with 2/n and start simplifying things:

This, then, is the area of n rectangles under the curve y(x) = 4x between x = 1 and x = 3. Step 1 is done. And the problem’s almost over. On to Step 2.

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23.4 Circumscribed Areas

Step 2 of the Method of Summation

Allow n to approach ∞ to get the actual area A.

As n gets larger and larger, 8/n gets smaller and smaller. And as n approaches ∞, 8/n approaches 0.

And there we have it, the final answer:

Let’s try another one, this time, instead of inscribed rectangles, we’ll try circumscribed rectangles.

2 3 . 4 C I R C U M S C R I B E D A R E A S Example 2

Find the area under the curve y(x) = 4x between x = 1 and x = 3. That is, find A. Use circumscribed rectangles this time.

figure 23.4a

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Solution

Again, draw and label the graph. The key difference from the previous example is that the rectangles extend beyond the y = 4x line this time around.

figure 23.4b

Step 1 of the Method of Summation

Find a formula for Sn in terms of n.

This time, we’ve put a bar over Sn to indicate that Sn is an overestimation.3 Also, in this case, x1 isn’t at 1, but at 1 + Δx due to the location of y1; this will be the main difference from Example 1, which will make the algebra a little bit different (and therefore one of the sum rules we use will be correspondingly different from the inscribed case).

3 See Kline 1967, 235.

23.4 Circumscribed Areas

Notice that this time xn was easier to come up with since it’s a bit easier to see that, in general,

Now, plugging in the yi’s to Sn and going through the algebra, we get the following:

Again, we’ll use Sum Rule 1 to reduce the number of terms.

Then we’ll factor out a 4(∆x)2.

and use Sum Rule 2 to replace the sequence, [1 + 2 ++ n].

Replacing the · · · series, we get

The next step, recall, is to write the width of each rectangle in terms of n rather than as ∆x. Again, this is just the base of the entire area divided by the number of rectangles, n:

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And Sn becomes

The rest is just a matter of simplifying with algebra:

Step 2 of the Method of Summation

Let n approach ∞ to get the actual area A.

Just as before (and good thing too) we get the following:

We should have expected to get the same answer for both examples. This is because the only thing we changed was the direction from which the approximated area approached the actual area. In the inscribed case it was from below; that is, we began with the sum of mini-areas that was less than the actual area, an underestimation.

23.5 More (Complicated) Examples: y(x) = x²

In the circumscribed case we approached the actual area from above: the sum of the mini-areas was initially larger than the actual area, that is, an overestimation.

We can see this in another way. Look at the difference in the sum of n miniareas in each case:

We now know that the actual area we were looking for is 16. So, in each Sn, the second term is the amount we’re off by, the error in our approximation. The goal was to make that error approach 0, to disappear, which we did by allowing n to run off toward infinity. Notice, though, that in the inscribed case, the error—when it’s not 0—makes Sn less than 16, less than the actual area. Again, in the inscribed case, Sn is an underestimate of the area. In the circumscribed case, it’s just the opposite: the Sn is an overestimate when the error term is anything other than 0:

2 3 . 5 M O R E (CO M P LI C ATE D) E X A M P LE S : y ( x) = x ²

Now that you’ve had some practice with the Method of Summation, let’s look at some slightly more complicated cases. Whereas in the last two examples we integrated functions whose graphs are lines—that is, first-order functions—let’s move up to second-order functions like y(x) = x2. In other words, we’re moving from lines to parabolas. I’ll leave everything else the same. Not much changes, but enough does that it will be helpful to see a couple of examples.

Example 3

Find the area under the curve y(x) = x2 between x = 1 and x = 3. That is, find A. Use inscribed rectangles.

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figure 23.5a

Solution

Draw and label the graph.

figure 23.5b

Step 1 of the Method of Summation

Find a formula for Sn.

23.5 More (Complicated) Examples: y(x) = x²

Plug in the yi’s to Sn and do the algebra. This time the algebra is more complicated because of the squared values. Make sure to write out every step. Don’t take shortcuts or do things only in your head. Remember, safety first, especially at first, when you’re new to the material. And don’t forget F.O.I.L. (first, outside, inside, last). You’ll need it. (And this is going to make you really appreciate the shortcut for integration that I’ll introduce later. Some of the algebra is pretty tiring.)

As usual, we’ll use Sum Rule 1 to reduce the number of terms:

Now, group together all the (∆x)2 terms and all the (∆x)3 terms into two separate series. Each series gets its own “ · · · ”. But don’t add them together to reduce them (you’ll see why in a moment):

Notice that we didn’t combine the 2(∆x)2 with 4(∆x)2 to get 6(∆x)2. Instead, we left them separate. You don’t have to see why right now, just notice that we did. Now,

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factor out a 2(∆x)2 from the first series (in the first set of square brackets). Also factor out a (∆x)3 from the second series. You’ll see that this will give us something to apply our sum rules to.

The first series looks like one for Sum Rule 3, but the second series doesn’t look like any of our rules until we notice that 1 = 12 and 4 = 22. Then the second series is transformed into something familiar and we have

So, we can use Sum Rule 3 on the first series and Sum Rule 5 on the second series.

The next thing to do is rewrite ∆x in terms of n. As before:

Making this substitution, we get

Again, more algebra:

23.5 More (Complicated) Examples: y(x) = x²

Step 2 of the Method of Summation

Let n approach ∞ to get the actual area A.

Similar to what we did in the previous section, let’s work through this same problem, this time with the rectangles circumscribed.

Example 4

Find the area under the curve y(x) = x2 between x = 1 and x = 3. That is, find A. Use circumscribed rectangles.

figure 23.5c

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Solution

Draw and label the graph.

figure 23.5d

Step 1 of the Method of Summation

Find a formula for Sn.

You know the drill.

23.5 More (Complicated) Examples: y(x) = x²

Again, use Sum Rule 1 to combine all the ∆x’s.

Then group together all the (∆x)2 terms and all the (∆x)2 terms into two separate series.

Next, factor out 2(∆x)2 from the first series and (∆x)3 from the second series.

Again, the first series is fine as it is—we can use Sum Rule 2a on it. And like last time, we can modify the second series by remembering that 1 = 12, 4 = 22, and 9 = 32.

So we can use Sum Rule 3 on the first series and Sum Rule 4 on the second series.

Now replace ∆x, since

which gives us

Again, more algebra:

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Step 2 of the Method of Summation

Let n approach ∞ to get the actual area A.

Again, we arrive at the same answer whether we approach the area from above (circumscribed rectangles) or from below (inscribed rectangles). As before the area of n rectangles has an error term associated with it, which is everything after the 26/3 in each case. This error disappears as n approaches ∞.

2 3 .6 TH E I NTE G R A L N OTATI O N

Since the integral can be interpreted naturally as an area when graphed, we’ve mostly been using A to stand for an integral:

23.6 The Integral Notation

But there’s another, perhaps more descriptive notation for integrals, one that also keeps the idea of summation at the forefront of our minds. Here it is. (I’ll explain it after presenting it.) For a function y(x), the area under the curve (and above the x-axis) between x = a and x = b is denoted by

where

This symbol, ∫, is an elongated s standing for sum. It was introduced by Leibniz in the 1600s. The a and b at the bottom and top of the ∫ stand for the x-values of the endpoints of the entire area’s base (1 and 3 in our previous examples).

The ydx stands for the area of each mini-rectangle. You can also think of it as having an invisible multiplication sign in it: y · dx (although it’s not usually written this way). The y is the height of each mini-rectangle and the dx is the base of each mini-rectangle. Just like in the derivative notation, the ∆ becomes d to indicate that the change in x is infinitesimal. In other words, dx is an infinitely small ∆x. Therefore the ydx isn’t merely the area of a mini-rectangle, but the area of an infinitely thin minirectangle. In other words, dx actually includes the notion of a limit. That is,

so

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All together, we have

To put it in terms of the term integral,

So

We’ve been using y(x) to stand for any function. There are of course, many ways to represent a function and you should know a few of the most prevalent. Many calculus books use

to indicate that y is a function of x: that y is the dependent variable and x is the independent variable. These books also tend to use f(x) in the same way we’ve been using y. And this is just fine. Here’s what our definition of the integral would be using this notation:

Of course, it really doesn’t matter what you call the function. We’ve seen that a function by any other name is still a function. Instead of y(x) or f(x), we can use d(t), g(z), or u(v). It doesn’t change the meaning or definition of the integral.

We can now formulate the four examples from this chapter with our new notation. Example 1, you’ll recall, was stated as “Find the area under the curve y(x) = 4x between x = 1 and x = 3.” In our efficient integral notation, it becomes

23.7 Up Next: The Fundamental Theorem of Calculus

Example 2 would be the same thing (I’ve left out the inscribed/circumscribed distinction, since the integral symbol doesn’t account for that, and later, we’ll not need to either). Examples 3 and 4, which both asked us to “Find the area under the curve y(x) = x2 between x = 1 and x = 3” become

2 3 .7 U P N E X T: TH E F U N DA M E NTA L TH E O R E M O F C A LC U LU S

In this chapter you’ve become familiar with integration, that is, the process of taking the integral of a function. Recall the three main concepts of calculus:

You’ve now learned all three. In the next chapter you’ll see how they all fit together. Indeed, you’ll see how they’re virtually different ways of looking at the same thing. Although this was only discovered after the development of the three concepts, the revelation that all three are intimately connected—and expressed in the Fundamental Theorem of Calculus—was one of the most satisfying discoveries in intellectual history. It is in this theorem that we come to realize the essence of calculus, and therefore truly understand it. It is the core of calculus.

2 3 . 8 S T U DY Q U E S TI O N S

1. In words, what is an integral?

2. In symbols, what is an integral?

3. What was the long method of taking the derivative of a function? How did we take a shortcut?

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4. What is the name of the long method for taking the integral of a function?

5. What are the two main steps of the Method of Summation? Which is the hardest step?

6. What is an ellipsis? How will the special summation rules help us with it?

7. In terms of y’s and ∆x’s, what is the formula for Sn? 8. What does the bar under Sn indicate about Sn? 9. In the symbols yi and xi, what does the i stand for?

10. What does the bar over Sn indicate about Sn? 11. For Example 1 the summation of n mini-areas was given by

while for Example 2 the summation of n mini-areas was given by

What do the positive and negative signs tell us?

12. What is

in terms of our elongated “s” integral notation? What does “ydx” as a whole stand for? What does y stand for? What does dx stand for?

13. In terms of the limit notation what is dx?

14. What do the a and b stand for in the formula

15. What does “ f(x)dx” as a whole stand for in the formula

What does f(x) stand for? What does dx stand for?

16. What are the three central concept in calculus? What is the “core” of calculus?

23.9 Exercises

2 3 . 9 E XE R C I S E S

For all of the following use the Method of Summation; you may have a list of the special sum rules in front of you.

1. Find the area under the curve y(x)=4x between x = 1 and x = 3. That is, find A. Use inscribed rectangles. (This is just Example 1 from the chapter.)

2. Find

using circumscribed rectangles. (This is really just the same as Example 2 from the chapter.)

3. Find

using inscribed rectangles. (This is just Example 3 from the chapter.)

4. Find the area under the curve y(x)=x2 between x = 1 and x = 3. That is, find A. Use circumscribed rectangles. (This is Example 4 from the chapter.)

5. Find the area under the curve y(x)=x2 between x = 0 and x = 2. That is, find A. Use circumscribed rectangles.

6. Find

using inscribed rectangles.4 7. Find

using circumscribed rectangles.

4 Exercises 6–9 are from Kline, 240.

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8. Find

beginning with an overestimation of the sum of the mini-rectangles.

9. Find

beginning with an overestimation of the sum of the mini-rectangles.

10. Find the area under the curve y(x)=x2 between x = 1 and x = 5 beginning with an underestimation of Sn.