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Chapter
Page
CONTENTS.
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..................................
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
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
PROLOGUE.
Considering how many fools can calculate, it is surprising that it should be
thought either a difficult or a tedious task for any other fool to learn how to master
the same tricks.
Some calculus-tricks are quite easy. Some are enormously difficult. The fools
who write the textbooks of advanced mathematics—and they are mostly clever
fools—seldom take the trouble to show you how easy the easy calculations are.
On the contrary, they seem to desire to impress you with their tremendous
cleverness by going about it in the most difficult way.
Being myself a remarkably stupid fellow, I have had to unteach myself the
difficulties, and now beg to present to my fellow fools the parts that are not hard.
Master these thoroughly, and the rest will follow. What one fool can do, another
can.
CHAPTER I.
TO DELIVER YOU FROM THE PRELIMINARY TERRORS.
The preliminary terror, which chokes off most fifth-form boys from even
attempting to learn how to calculate, can be abolished once for all by simply
stating what is the meaning—in common-sense terms—of the two principal
symbols that are used in calculating.
These dreadful symbols are: (1) d which merely means “a little bit of.” Thus dx
means a little bit of x; or du means a little bit of u. Or-
dinary mathematicians think it more polite to say “an element of,”
17 V.