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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
¸ñÂ÷
9791190399746
CONTENTS.
Chapter Page
Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
I. To deliver you from the Preliminary Terrors 1
II. On Different Degrees of Smallness . . . . . . . . . . . 3
III. On Relative Growings . . . . . . . . . . . . . . . . . . . . . . . . . . 9
IV. Simplest Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
V. Next Stage. What to do with Constants . . . . . . 25
VI. Sums, Differences, Products and Quotients . . . 34
VII. Successive Differentiation . . . . . . . . . . . . . . . . . . . . . 48
VIII. When Time Varies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
IX. Introducing a Useful Dodge . . . . . . . . . . . . . . . . . . . 66
X. Geometrical Meaning of Differentiation . . . . . . 75
XI. Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
XII. Curvature of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
XIII. Other Useful Dodges . . . . . . . . . . . . . . . . . . . . . . . . . . 118
XIV. On true Compound Interest and the Law of Or£¿ganic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
vii
Chapter Page
XV. How to deal with Sines and Cosines . . . . . . . . . . . 162
XVI. Partial Differentiation . . . . . . . . . . . . . . . . . . . . . . . . 172
XVII. Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
XVIII. Integrating as the Reverse of Differentiating 189
XIX. On Finding Areas by Integrating . . . . . . . . . . . . . . 204
XX. Dodges, Pitfalls, and Triumphs . . . . . . . . . . . . . . . . 224
XXI. Finding some Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 232
Table of Standard Forms . . . . . . . . . . . . . . . . . . . . . . . . 249
Answers to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 252