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.
TORONTOCALCULUS 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
viiCALCULUS 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 theCALCULUS 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, howCALCULUS 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
+ 2x
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 wex
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 2x
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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