MEANING OF DIFFERENTTATION 85
01.76 Will be 3 times as great as dy; as magnied in
Fig. 21. So, draw the line in Fig. 20 at this slope.
I . .I_._-
FIG. 21.
(3) Now for a slightly harder case.
Let y = aw? + 1).
Again the curve will start on the yaxis at a
THE LAW OF ORGANIC GROWTH 145
and therefore
when m=m, 3/:(2'718281 etc); that is, 31:6,
thus nally demonstrating that
m 902 m3 m4
=1+i+m+m+m+em
[NOTEH ow to read exponentials. For the benet
of those who have no tutor at hand it may be of use
to state that
MEANING OF DIFFERENTIATION 91
Exercises VIII. (See page 291 for Answers.)
(1) Plot the curve 31:19025, using a scale oi
millimetres. Measure at points corresponding to
different values of x, the angle of its slope.
Find, by differentiating the equation, t
ON TRUE COMPOUND INTEREST 135
stocking, or locking it up in his safe. Then, if he
goes on for 10 years, by the end of that time he will
have received 10 increments of 10 each, or 100,
making, with the original 100, a total of 200 in all.
His property will
CHAPTER XIII.
OTHER USEFUL DODGES.
Partial Fractions.
WE have seen that when we differentiate a fraction
we have to perform a rather complicated operation;
and, if the fraction is not itself a simple one, the result
is bound to be a complicated expression
OTHER USEFUL DODGES 129
which gives
3.27 1 =(Aac+B)(a,-+1)
+(0w+D)(a,-+ 1)(2m2 .1.)+E(2x2 1)2.
For as: 1, this gives E: 4. Replacing, trans-
posing, collecting like terms, and dividing by (n+1,
we get
16w3_]5m2+3=2am3+20m2+w(A -0)+(BD)'
Hence 20:16 and 0:
CURVA TURE OF CURVES 115
into a downward or negative slope, so that in this
,dig/.
da/
Go back now to the examples of the last chapter
and verify in this way the conclusions arrived at as to
whether in any particular case there is a maximum
or a minimum.
THE LAW OF ORGANIC GROWTH 149
This gives 2%; = 63 = :1: + a;
hence, reverting to the original function (see p. 131),
we get d3/ 1 1
dw _ 979 = 97+?
012/
Next try 3/ = loglo ac.
First change to natural logarithms by multiplying
by the modulus 04343. This
THE LAW OF ORGANIC GROWTH 141
It is, however, worth while to find another way of
calculating this immensely important gure.
Accordingly, we will avail ourselves of the binomial
theorem, and expand the expression (141) in that
well-known way. u
The binomia
SINES AND COSINES 173
(7)y=J1mn-29; y=(1+3tan?9)*-
Let 3 tan 6 = v.
dy 1
_ . _ _ _ .
CI-+1) d1 2J1+U (seep 68)"
(lv_ 2
d06tan0sec 0
(for, if tan6=u,
d1; du
H 2. _= . 2
1231; , du 6%, de =sec 9;
hence 53 = 6 tan 6 $602 6);
dy_ 6 tan 0sec29
d9 2J1+3tan20
MEANING OF DIFFERENTIATION 89
that is, it must be a solution of the system of simul-
taneous equations formed by coupling together the
equations of the curves. Here the curves meet one
another at points given by the solution of
cfw_y=2.7f-+2,
3/: Jza;+2 o
CURVATURE OF CURVEs' 117
d30_ +cfw_)(20+2P)
dP2_ (c+P)4
which is positive for all the values of P; hence
[7 . .
P = + A; 0 corresponds to a mlnlmum.
Now
(5) The total cost per hour 0 of lighting a building
with N lamps of a certain kind is
_ 0, EPCe
0N<T
PARTIAL DIFFERENTIATION 181
(5) Find the total differential of y=u3 sinv; of
logm,
v .
(6) Verify that the sum of three quantities as, y, 2,
whose product is a constant k, is minimum when
these three quantities are equal.
(7) Find the maximum or minimum o
CHAPTER XI.
MAXIMA AND MINIMA.
A QUANTITY which varies continuously is said to
pass by (or through) a maximum or minimum value
when, in the course of its variation, the immediately
preceding and following values are both smaller or
greater, respectively,
MAXIMA AND MINIMA 97
Now, before we go on to any further cases, we have
two remarks to make. When you are told to equate
dy
olw
wits of your own) a kind of resentment, because you
to zero, you feel at rst (that is if you have any
know that 3: has all sort
OTHER USEFUL DODGES 127
This method can always be used; but the method
shown rst will be found the quickest in the case of
factors in m only.
Case III. When among the factors of the denomi-
nator there are some which are raised to some power,
one must all
PARTIAL DIFFERENTIATION 179
triangle is A =JW, where
s is the half perimeter, 15, so that A =J15P, Where
P=<15w><15y)<w+y15>
= avg/2+ 902:1; - 15.202 1 5y2 - 45mg + 450a;- + 4503/ 3375.
Clearly A is maximum when P is maximum.
For a maximum (clearly it w
MAXIMA AND MINIMA 99
Try another simple problem in maxima and minima.
Suppose you were asked to divide any number into
two parts, such that the product was a maximum?
How would you set about it if you did not know
the trick of equating to zero? I suppose
MAXIMA AND MINIMA 95
Plot these values as in Fig. 27.
It will be evident that there will be a maximum
somewhere between cfw_17:1 and 90:2; and the thing
loo/cs as if the maximum value of y ought to be
about 2%. Try some intermediate values. If ac=1i,
3/:2
' THE LAW OF ORGANIC GROWTH 159
The values of 6" and e are continually required
in different branches of physics, and as they are given
in very few sets of mathematical tables, some of the
values are tabulated here for convenience.
a a" 1 6
10000 1 '0000
CURVATURE OF CURVES 119
. (4) Find the maxima and minima of
5
y = 2.7; + 1 + a?
(5) Find the maxima and minima of
3
yx2+m+i
(6) Find the maxima and minima of
_ 5.76
y 2TH?
(7) Find the maxima and minima of
3x3
y=$
(8) Divide a number N into two parts
MAXIMA AND MINIMA 109
keep always in the same proportion to each other; .
that is, at any instant, the cylinder is similar to the
original cylinder. When the radius of the base is
7' feet, the surface area is increasing at the rate of
20 square inches per
THE LAW OF ORGANIC GROWTH 143
But, when n is made indenitely great, this
simplies down to the following:
cfw_Edi-2
e: 1+.av+|2 1.+etc .
+L+ L4
This series is called the exponential series.
' The great reason why 6 is regarded of importance
is that 63 poss
OTHER USEFUL DODGES 133
hence
3 2
3y 1 _<.1_+'!ff 11:32.11: _._3_.
d3 E 62/ . 33 /~3
1
Let us take as another example y=-.-
3/6+5
The inverse function is 6:5353 or 9:3/3-5, and
(l9_ _4_ 3' :Z
017 .33, _3~/(6+o).
(l1/_ 1
It follows that ,as might have
OTHER USEFUL DODGES 12a '
Suppose we wish to go back from 3aff1 to the
. 1 2
components whlch we know are w+1 and 901. If
we did not know what those components were we can
still prepare the way by writing:
3w+1_ 390+]
'wZ 1 (w+1)(a; 1) =m+1
leaving
MAXIMA AND MINIMA 105
You will get
d8 :20 4R 21:2
07$:96X7J4R2w2 J? 4R2
For maximum or minimum we must have
4R2 2.702
J41? 1:2 = O
that is, 4132 2m2=0 and $=RJ.
The other side = J4RZ 2R2: RJ; the two sides
are equal; the gure is a square the side of whic
CUHVATURE OF CURVES 113
other words, whether the curve curves up or down
towards the right
Suppose a slope constant, as in Fig. 31.
Here, 3: is of constant value.
Suppose, however, a case in which. like Fig. 32,
the slope itself is getting greater upwards
SINES AND COSINES 167'
The accompanylng curves, Figs. 44 and 45, show,
plotted to scale, the values of y=sin 0, and Zg=cos 9,
for the corresponding values of 6.
_-_-.2;
9,
FIG. 44. 168 CALCULUS MADE EASY
Take next the cosine.
Let y = cos 6.
Now cos 9 =
CHAPTER XV.
HOW TO DEAL WITH SINES AND COSINES.
GREEK letters being usual to denote angles, we will
take as the usual letter for any variable angle the
letter 9 (theta).
Let us consider the function
y=sin 6.
FIG. 43.
d(sin9).
d9
or, in other words, if