CALCULUSMADEEASY

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**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

- R. S.

*SECOND EDITION, ENLARGED *

MACMILLAN AND CO.

, LIMITED

- MARTIN

’

S STREET, LONDON

***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 con

siderable 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.

Chapter Page

Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

- To deliver you from the Preliminary Terrors 1
- On Different Degrees of Smallness . . . . . . . . . . . 3

III. On Relative Growings . . . . . . . . . . . . . . . . . . . . . . . . . . 9

- Simplest Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
- Next Stage. What to do with Constants . . . . . . 25
- Sums, Differences, Products and Quotients . . . 34

VII. Successive Differentiation . . . . . . . . . . . . . . . . . . . . . 48

VIII. When Time Varies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

- Introducing a Useful Dodge . . . . . . . . . . . . . . . . . . . 66
- Geometrical Meaning of Differentiation . . . . . . 75
- 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**CALCULUS MADE EASY **viii

Chapter Page

- 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

- Dodges, Pitfalls, and Triumphs . . . . . . . . . . . . . . . . 224

XXI. Finding some Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 232

**Table of Standard Forms . . . . . . . . . . . . . . . . . . . . . . . . **249

Answers to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 252PROLOGUE.

Considering how many fools can calculate, it is surprising that it

should be thought either a diffiffifficult 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 diffiffifficult.

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 diffiffifficult way

.

Being myself a remarkably stupid fellow, I have had to unteach

myself the diffiffifficulties, 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 offff most fififth

–

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,

”

instead of

“

a little bit of.

”

Just as you please. But you will fifind that

these little bits (or elements) may be considered to be indefifinitely small.

(2)

Z

which is merely a long *S*, and may be called (if you like)

“

the

sum of.

”

Thus

Z

*dx *means the sum of all the little bits of *x*; or

Z

*dt *means

the sum of all the little bits of *t*. Ordinary mathematicians call this

symbol

“

the integral of.

”

Now any fool can see that if *x *is considered

as made up of a lot of little bits, each of which is called *dx*, if you

add them all up together you get the sum of all the *dx*

’

s, (which is the**CALCULUS MADE EASY **2

same thing as the whole of *x*)

. The word

“

integral

”

simply means

“

the

whole.

”

If you think of the duration of time for one hour, you may (if

you like) think of it as cut up into 3600 little bits called seconds. The

whole of the 3600 little bits added up together make one hour.

When you see an expression that begins with this terrifying sym

bol, you will henceforth know that it is put there merely to give you

instructions that you are now to perform the operation (if you can) of

totalling up all the little bits that are indicated by the symbols that

follow.

That

’

s all.CHAPTER II.

ON DIFFERENT DEGREES OF SMALLNESS.

We shall fifind that in our processes of calculation we have to deal with

small quantities of various degrees of smallness.

We shall have also to learn under what circumstances we may con

sider small quantities to be so minute that we may omit them from

consideration. Everything depends upon relative minuteness.

Before we fifix any rules let us think of some familiar cases. There

are 60 minutes in the hour, 24 hours in the day, 7 days in the week.

There are therefore 1440 minutes in the day and 10080 minutes in the

week.

Obviously 1 minute is a very small quantity of time compared with

a whole week. Indeed, our forefathers considered it small as com

pared with an hour, and called it

“

one min

`

ute,

”

meaning a minute

fraction

—

namely one sixtieth

—

of an hour. When they came to re

quire still smaller subdivisions of time, they divided each minute into

60 still smaller parts, which, in Queen Elizabeth

’

s days, they called

“

second min

`

utes

”

(*i.e. *small quantities of the second order of minute

ness)

. Nowadays we call these small quantities of the second order of

smallness

“

seconds.

”

But few people know *why *they are so called.

Now if one minute is so small as compared with a whole day, how**CALCULUS MADE EASY **4

much smaller by comparison is one second!

Again, think of a farthing as compared with a sovereign: it is barely

worth more than

1

1000 part. A farthing more or less is of precious little

importance compared with a sovereign: it may certainly be regarded

as a *small *quantity

. But compare a farthing with £1000: relatively to

this greater sum, the farthing is of no more importance than

1

1000

of a

farthing would be to a sovereign. Even a golden sovereign is relatively

a negligible quantity in the wealth of a millionaire.

Now if we fifix upon any numerical fraction as constituting the pro

portion which for any purpose we call relatively small, we can easily

state other fractions of a higher degree of smallness. Thus if, for the

purpose of time,

1

60 be called a *small *fraction, then

1

60

of

1

60 (being a

*small *fraction of a *small *fraction) may be regarded as a *small quantity *

*of the second order *of smallness.

*∗ *

Or, if for any purpose we were to take 1 per cent.

(*i.e. *1

100 ) as a

*small *fraction, then 1 per cent. of 1 per cent.

(*i.e. *1

10*,*000 ) would be a

small fraction of the second order of smallness; and

1

1*,*000*,*000

would be

a small fraction of the third order of smallness, being 1 per cent. of

1 per cent. of 1 per cent.

Lastly, suppose that for some very precise purpose we should regard

1

1*,*000*,*000

as

“

small.

”

Thus, if a fifirst

–

rate chronometer is not to lose

or gain more than half a minute in a year, it must keep time with

an accuracy of 1 part in 1*, *051*, *200. Now if, for such a purpose, we

*∗*

The mathematicians talk about the second order of

“

magnitude

”

(*i.e. *

great

ness) when they really mean second order of *smallness*. This is very confusing to

beginners.**DIFFERENT DEGREES OF SMALLNESS **5

regard

1

1*,*000*,*000 (or one millionth) as a small quantity, then

1

1*,*000*,*000

of

1

1*,*000*,*000 , that is

1

1*,*000*,*000*,*000*,*000 (or one billionth) will be a small quantity

of the second order of smallness, and may be utterly disregarded, by

comparison.

Then we see that the smaller a small quantity itself is, the more

negligible does the corresponding small quantity of the second order

become. Hence we know that *in all cases we are justifified in neglecting *

*the small quantities of the second*

*—*

*or third *(or higher)

*—*

*orders*, if only

we take the small quantity of the fifirst order small enough in itself.

But, it must be remembered, that small quantities if they occur in

our expressions as factors multiplied by some other factor, may become

important if the other factor is itself large. Even a farthing becomes

important if only it is multiplied by a few hundred.

Now in the calculus we write *dx *for a little bit of *x*. These things

such as *dx*, and *du*, and *dy*, are called

“

difffferentials,

”

the difffferential

of *x*, or of *u*, or of *y*, as the case may be.

[You *read *them as *dee*

*–*

*eks*,

or *dee*

*–*

*you*, or *dee*

*–*

*wy*

.

] If *dx *be a small bit of *x*, and relatively small of

itself, it does not follow that such quantities as *x *

*dx*, or *x*

2

*dx*, or *a*

*x *

*dx *

are negligible. But *dx **× **dx *would be negligible, being a small quantity

of the second order.

A very simple example will serve as illustration.

Let us think of *x *as a quantity that can grow by a small amount so

as to become *x*+*dx*, where *dx *is the small increment added by growth.

The square of this is *x*

2

+ 2*x *

*dx *+ (*dx*)

2 . The second term is not

negligible because it is a fifirst

–

order quantity; while the third term is of

the second order of smallness, being a bit of, a bit of *x*

2 . Thus if we*x *

*x *

Fig. 1.

**CALCULUS MADE EASY **6

took *dx *to mean numerically, say,

1

60

of *x*, then the second term would

be

2

60

of *x*

2

, whereas the third term would be

1

3600

of *x*

2 . This last term

is clearly less important than the second. But if we go further and take

*dx *to mean only

1

1000

of *x*, then the second term will be

2

1000

of *x*

2

, while

the third term will be only

1

1*,*000*,*000

of *x*

- 2.

Geometrically this may be depicted as follows: Draw a square

(Fig

. 1) the side of which we will take to represent *x*. Now suppose

the square to grow by having a bit *dx *added to its size each way

.

The enlarged square is made up of the original square *x*

2

, the two

rectangles at the top and on the right, each of which is of area *x *

*dx *

(or together 2*x *

*dx*), and the little square at the top right

–

hand corner

which is (*dx*)

2 . In Fig

. 2 we have taken *dx *as quite a big fraction

of *x*

—

about

1

5

. But suppose we had taken it only

1

100

—

about the

thickness of an inked line drawn with a fifine pen. Then the little corner

square will have an area of only

1

10*,*000

of *x*

2

, and be practically invisible.

Clearly (*dx*)

2

is negligible if only we consider the increment *dx *to be

itself small enough.

Let us consider a simile.*x *

*x *

*x *

*x *

*dx *

*dx *

*dx *

*dx *

Fig. 2.

*x *

*dx *

*x *

*dx *(*dx*)

2

*x*

2

Fig. 3.

**DIFFERENT DEGREES OF SMALLNESS **7

Suppose a millionaire were to say to his secretary: next week I will

give you a small fraction of any money that comes in to me. Suppose

that the secretary were to say to his boy: I will give you a small fraction

of what I get. Suppose the fraction in each case to be

1

100 part. Now

if Mr. Millionaire received during the next week £1000, the secretary

would receive £10 and the boy 2 shillings. Ten pounds would be a

small quantity compared with £1000; but two shillings is a small small

quantity indeed, of a very secondary order. But what would be the

disproportion if the fraction, instead of being

1

100 , had been settled at

1

1000 part? Then, while Mr. Millionaire got his £1000, Mr. Secretary

would get only £1, and the boy less than one farthing!

The witty Dean Swift

*∗ *

once wrote:

“

So, Nat

’

ralists observe, a Flea

“

Hath smaller Fleas that on him prey

.

“

And these have smaller Fleas to bite

’

em,

“

And so proceed *ad infifinitum*.

”

*∗*

*On Poetry: a Rhapsody *(p

. 20), printed 1733

—

usually misquoted.**CALCULUS MADE EASY **8

An ox might worry about a flflea of ordinary size

—

a small creature of

the fifirst order of smallness. But he would probably not trouble himself

about a flflea

’

s flflea; being of the second order of smallness, it would be

negligible. Even a gross of flfleas

’

flfleas would not be of much account to

the ox.CHAPTER III.

ON RELATIVE GROWINGS.

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