Oscar E. Fernandez - Everyday Calculus: Discovering the Hidden Math All Around Us

EVERYDAY CALCULUS

EVERYDAY CALCULUS

Discovering the Hidden Math All around Us

OSCAR E. FERNANDEZ

PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD

Copyright 2014 by Princeton University Press

Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540

In the United Kingdom: Princeton University Press,

6 Oxford Street, Woodstock, Oxfordshire OX20 1TW

press.princeton.edu

All Rights Reserved

Library of Congress Cataloging-in-Publication Data

Fernandez, Oscar E. (Oscar Edward)

Everyday calculus: discovering the hidden math all around us / Oscar E. Fernandez.

pages cm

Includes bibliographical references and index.

ISBN 978-0-691-15755-9 (hardcover: acid-free paper)

1. CalculusPopular works. I. Title.

QA303.2.F47 2014 515dc232013033097

British Library Cataloging-in-Publication Data is available

This book has been composed in Minion Pro

Printed on acid-free paper.

Typeset by S R Nova Pvt Ltd, Bangalore, India

Printed in the United States of America

1 3 5 7 9 10 8 6 4 2

Dedicado a Zoraida,
eres la belleza de mi vida
y tambin a nuestra hija
mi nia, t sers mi consentida
y por supuesto a mi mam
sin tu amor aqu no estuviera

CONTENTS

CHAPTER 5 Take a Derivative and Youll Feel Better

PREFACE

SINCE THE LATE 1600S, when calculus was being developed by the greatest mathematical minds of the day, scores of people across the world have asked the same question: When am I ever going to use this?

If youre reading this, youre probably interested in the answer to this question, as I was when I first started learning calculus. There are answers, like Calculus is used by engineers when designing X, but this is more a statement of fact than an answer to the question. The pages that follow answer this question in a very different way, by instead revealing the hidden mathematicscalculus in particularthat describes our world.

To tell this revelatory tale Ill take you through a typical day in my life. You might be thinking: A typical day? Youre a mathematician! How typical can that be? But as youll discover, my day is just as normal as anyone elses. In the morning I sometimes feel groggy; I spend what feel like hours in traffic (even though theyre only minutes) on my way to work; throughout my day I choose what to eat and where to eat it; and at some point I think about money. We dont pay attention to these everyday events, but in this book Ill peel back the facade of daily life and uncover its mathematical DNA.

Calculus will explain why our blood vessels branch off at certain angles ().

includes a refresher on functions and graphs to get you started, and appendices 17 include the calculations mentioned throughout the book, which are indicated by superscripts that look like this.1 (Youll also find footnotes indicated by Roman numerals and endnotes indicated by Arabic numerals.) Finally, on the next page youll find a breakdown of the mathematics discussed in each chapter.

Whether youre new to calculus, youre studying calculus, or its been a few years since youve seen it, youll find a whole new way of looking at the world in the next few chapters. You may not see fancy formulas flashing before your eyes when you finish this book, but Im hopeful that youll achieve an enlightenment akin to what Neo in The Matrix experiences when he learns that a computer code underlies his reality. Although Im not as cool as Morpheus, I look forward to helping you emerge through the other end of the rabbit hole.

Oscar Edward Fernandez
Newton, MA

CALCULUS TOPICS DISCUSSED BY CHAPTER

The chart below details the calculus topics discussed in each chapter.

EVERYDAY CALCULUS

CHAPTER 1

WAKE UP AND SMELL THE FUNCTIONS

ITS FRIDAY MORNING. The alarm clock next to me reads 6:55 a.m. In five minutes itll wake me up, and Ill awake refreshed after sleeping roughly 7.5 hours. Echoing the followers of the ancient mathematician Pythagoraswhose dictum was All is numberI deliberately chose to sleep for 7.5 hours. But truth be told, I didnt have much of a choice. It turns out that a handful of numbers, including 7.5, rule over our lives every day. Allow me to explain.

A long time ago at a university far, far away I was walking up the stairs of my college dorm to my room. I lived on the second floor at the time, just down the hall from my friend Eric Johnsons room. EJ and I were in freshman physics together, and I often stopped by his room to discuss the class. This time, however, he wasnt there. I thought nothing of it and kept walking down the narrow hallway toward my room. Out of nowhere EJ appeared, holding a yellow Post-it note in his hand. These numbers will change your life, he said in a stern voice as he handed me the note. Off in the corner was a sequence of numbers:

Like Hurley from the Lost television series encountering his mystical sequence of numbers for the first time, my gut told me that these numbers meant something, but I didnt know what. Not knowing how to respond, I just said, Huh?

EJ took the note from me and pointed to the number 1.5. One and a half hours; then another one and a half makes three, he said. He explained that the average human sleep cycle is 90 minutes (1.5 hours) long. I started connecting the numbers in the shape of a W. They were all a distance of 1.5 from each otherthe length of the sleep cycle. This was starting to sound like a good explanation for why some days Id wake up feeling like a million bucks, while other days I was just out of it the entire morning. The notion that a simple sequence of numbers could affect me this much was fascinating.