Calculus Made Easy

by Silvanus Phillips Thompson

Layout 2

CALCULUS MADE EASY:

 

BEING A VERY-SIMPLEST INTRODUCTION TO
THOSE BEAUTIFUL METHODS OF RECKONING
WHICH ARE GENERALLY CALLED BY THE
TERRIFYING NAMES OF THE

 

DIFFERENTIAL CALCULUS

AND THE

INTEGRAL CALCULUS.

 

BY

F. R. S.

 

SECOND EDITION, ENLARGED

 

MACMILLAN AND CO., LIMITED
ST. MARTIN’S STREET, LONDON
1914

MACMILLAN AND CO., Limited

LONDON • BOMBAY • CALCUTTA
MELBOURNE

THE MACMILLAN COMPANY

NEW YORK • BOSTON • CHICAGO
DALLAS • SAN FRANCISCO

THE MACMILLAN CO. OF CANADA, Ltd.

TORONTO

COPYRIGHT.

First Edition 1910.
Reprinted 1911 (twice), 1912, 1913.
Second Edition 1914.

 

What one fool can do, another can.
(Ancient Simian Proverb.) 

PREFACE TO THE SECOND EDITION.

The surprising success of this work has led the author to add a considerable number of worked examples and exercises. Advantage has also been taken to enlarge certain parts where experience showed that further explanations would be useful.

The author acknowledges with gratitude many valuable suggestions and letters received from teachers, students, and—critics.

October, 1914.

CONTENTS.

1. CHAPTER

   PAGE

2. Prologue ................................................................................................................................................................................................................................................................................................................................................................................................

   xi

3. I.

   To Deliver You From The Preliminary Terrors ................................................................................................................................................................................................................................................................................................................................................................................................

   1

4. II.

   On Different Degrees of Smallness ................................................................................................................................................................................................................................................................................................................................................................................................

   3

5. III.

   On Relative Growings ................................................................................................................................................................................................................................................................................................................................................................................................

   9

6. IV.

   Simplest Cases ................................................................................................................................................................................................................................................................................................................................................................................................

   18

7. V.

   Next Stage. What to do with Constants ................................................................................................................................................................................................................................................................................................................................................................................................

   26

8. VI.

   Sums, Differences, Products, and Quotients ................................................................................................................................................................................................................................................................................................................................................................................................

   35

9. VII.

   Successive Differentiation ................................................................................................................................................................................................................................................................................................................................................................................................

   49

10. VIII.

    When Time Varies ................................................................................................................................................................................................................................................................................................................................................................................................

    52

11. IX.

    Introducing a Useful Dodge ................................................................................................................................................................................................................................................................................................................................................................................................

    67

12. X.

    Geometrical Meaning of Differentiation ................................................................................................................................................................................................................................................................................................................................................................................................

    76

13. XI.

    Maxima and Minima ................................................................................................................................................................................................................................................................................................................................................................................................

    93

14. XII.

    Curvature of Curves ................................................................................................................................................................................................................................................................................................................................................................................................

    112

15. XIII.

    Other Useful Dodges ................................................................................................................................................................................................................................................................................................................................................................................................

    121

16. XIV.

    On true Compound Interest and the Law of Organic Growth ................................................................................................................................................................................................................................................................................................................................................................................................

    134

17. XV.

    How To Deal With Sines And Cosines ................................................................................................................................................................................................................................................................................................................................................................................................

    165

18. XVI.

    Partial Differentiation ................................................................................................................................................................................................................................................................................................................................................................................................

    175

19. XVII.

    Integration ................................................................................................................................................................................................................................................................................................................................................................................................

    182

1. XVIII.

   Integrating as the Reverse of Differentiating ................................................................................................................................................................................................................................................................................................................................................................................................

   191

2. XIX.

   On Finding Areas by Integrating ................................................................................................................................................................................................................................................................................................................................................................................................

   206

3. XX.

   Dodges, Pitfalls, and Triumphs ................................................................................................................................................................................................................................................................................................................................................................................................

   226

4. XXI.

   Finding some Solutions ................................................................................................................................................................................................................................................................................................................................................................................................

   234

5. XXII.

   Epilogue and Apologue ................................................................................................................................................................................................................................................................................................................................................................................................

   249

6. Table of Standard Forms ................................................................................................................................................................................................................................................................................................................................................................................................

   252

7. Answers to Exercises ................................................................................................................................................................................................................................................................................................................................................................................................

   254