and, similarly, if (a+h)3 be a certain function, we may write (2) thus

function(function) + (dc)h,

and these two results are precisely the same, and it is found that whatever the function be, the result is the

same.

and then

88 There appears to be a slight difficulty here which we will not pass over. We have said let h=0, (dc)=2a, and immediately afterwards we multiply (dc) by h, and one might be led to suppose that this product, viz., (dc)h, would naturally be 0 also. Not so, however. We only say, what would be the value of de1 of the function, supposing h were 0, and we obtain a certain result a certain quantity. Then, quite apart from that operation, we multiply another quantity (h) by this quantity.

89. We have said that it is found that of whatever form the function be, we always have, as an approximation, when h is small,

function(function) + (dc1)h ; we will give a simple example in support of this. Let the function be

3(a + h)2+4(a+h) +1

Therefore, omitting the term involving h2, we have function = (3a2+4a+1)+h(6a+4). But since function=3(a+h)2+4(a+h)+1 (function)=3a2+4a+1 (dc)=2x 3a +4 = 6a+ 4.

and

Therefore we have again

function (function)+(de1)h.

(This form is a deduction from Taylor's theorem.) 90. We will now show how this result may be practically utilized in approximating to the roots of an equation,

Now, by trial, 15 is found to be near one of the roots. Let h be the difference between 15 and the root; that is, let x=1·5+h, which is of the form (a+h).

Therefore,

and

(function) a3-3a+1

=(1·5)3-3×15+1
= −.125,
(dc1)=3a2 - 3

=3×(1·5)2 - 3
=6.75-3,

We can now take this as an approximation, as we did 1.5, and so may get a result, by proceeding in this manner, as near to one of the roots as we please.

91. If we wish to find the limit of a fraction, as the variable gradually approaches a certain limit, in the case where the fraction becomes of the form o

or

∞

we may employ the process of differentiation, and by this means get rid of all artifice in arriving at the correct result.

F

The method of Bernouilli is to differentiate the numerator and denominator separately, until they do not both vanish, for the value of the limit of the variable.

In No. 2 of "Examples worked out" we found the 2x+5 value of the fraction when x was infinite, by an 4x+6'

artifice. We shall get the same result by the method of differentiating.

92. Again, find the real value of the fraction

XIX. Maxima and Minima.

93. The value of a function is said to be a maximum or a minimum according as the particular value is greater or less than the values which both immediately precede and immediately succeed it.

94. If, then, a function continually increases or continually decreases, it cannot have a maximum or a minimum.

95. If a function increase at a diminishing rate, like a stone thrown straight up in the air, until at a certain point it ceases to increase and begins to diminish (ie., in the case of the stone, to diminish its height from the ground), then, at the turning point, it has its greatest value, and the values which immediately precede and immediately succeed this value are less than this value, and therefore it is a maximum.

96. Again, if the function decrease until, at a certain point, it ceases to diminish and begins to increase, then the values on either side of it are greater than it, and consequently it is a minimum. Such, for instance, would be the case, if a cork were forced into a vessel filled with water, it would attain its minimum distance from the bottom of the vessel at the turning point, when it began to rise. Take the stone thrown straight up into the air as another instance: it decreases in velocity until at the turning point it is a minimum, and then begins to increase.

97. Let APB be a circle, and AB its diameter, and

let a straight line move from A so as to be always perpendicular to AB and have its other extremity in the circumference of the circle; it will increase until it reaches the position CP, and then diminish_until it reaches B; and in the position CP it will have its maximum value.

Again, a straight line drawn so as to have one extremity in MN, and its other extremity on the circumference, will first have such a position as MA, and will gradually diminish until it reaches the opsition TP, and then it will increase until it reaches the position NB.

Therefore, at the turning point, in the position TP it has its minimum value.

98. A function may have more than one maximum or minimum; in fact may have an endless number of both, for a function may increase until it has reached a maximum, and then diminish until it reaches a minimum, and then increase again to a maximum, and so on. From the nature of the case the maximum and minimum values must alternate-that is, there cannot be two maximum values succeeding each other without a minimum value intervening, and vice versa. The troughs and crests of the waves of the sea give minima and maxima with regard to a horizontal line. The tide furnishes another example of maxima and minima.

99. The sine of an angle-i.e., the semi-chord-as the angle varies from 0 to 360°, is a minimum at 0 and 180°, and a maximum at 90° and 270°; the values of the sine at any angles on either side of a maximum being smaller, and on either side of a minimum being larger than the maximum and minimum values-viz. (in a circle of radius 1), 1 and 0.

100. Now if, as the variable increases, the function increases, its rate of variation must be positive; but if, as the variable increases, the function diminishes, its rate of variation must be negative-that is to say, in