The Calculus Story: A Mathematical Adventure

209 Pages·2018·12.45 MB·English

by  David Acheson

The Calculus Story

The Calculus Story
A Mathematical Adventure
DAVID ACHESON
1
1
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© David Acheson 2017
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First Edition published in 2017
Impression: 1
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British Library Cataloguing in Publication Data
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Library of Congress Control Number: 2017935884
ISBN 978-0-19-880454-3
Printed in Great Britain by
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In memory of
Dr Janet Mills
(1954–2007)
who once claimed that she never quite understood calculus

CONTENTS
1. Introduction 1
2. The Spirit of Mathematics 6
3. Infinity 13
4. How Steep is a Curve? 20
5. Differentiation 25
6. Greatest and Least 32
7. Playing with Infinity 38
8. Area and Volume 45
9. Infinite Series 54
10. ‘Too Much Delight’ 60
11. Dynamics 64
12. Newton and Planetary Motion 70
13. Leibniz’s Paper of 1684 78
14. ‘An Enigma’ 87
15. Who Invented Calculus? 93
16. Round in Circles 100
17. Pi and the Odd Numbers 106
18. Calculus under Attack 112
19. Differential Equations 118
20. Calculus and the Electric Guitar 125
21. The Best of all Possible Worlds? 132
22. The Mysterious Number e 139
23. How to Make a Series 144
viii contents
24. Calculus with Imaginary Numbers 149
25. Infinity Bites Back 154
26. What is a Limit, Exactly? 162
27. The Equations of Nature 167
28. From Calculus to Chaos 174
Further Reading 183
References for Quotations 185
Picture Credits 187
Index 189
1
Introduction
In the summer of 1666, Isaac Newton saw an apple fall in his
garden, and promptly invented the theory of gravity.
That, at least, is the story.
And, however oversimplified this version of events may be,
it makes as good a starting point as any for an introduction
to calculus.
Because the apple speeds up as it falls.
1. Newton and the apple.