An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

152

1TRACING OF CURVES.

From this table it is immediately evident that fig. 22
represents the curve. For if we take AX, AY as axes,
AB = a, AC = - a; then the table shews that at the point
C the curve passes from the positive to the negative side of
the axis of ax at an angle tan- (- -), as is represented in
the figure; that it touches the axis of x at A, but does not
cross it; that it goes off to an infinite distance when m = AB;
and when x becomes greater than AB it appears on the positive side of the axis of v, evidently in the form FPG, for
y is infinite when i = AB, becomes and continues positive
when xR is greater than AB, and becomes infinite again when
*i is infinite; y therefore must decrease as x gets greater than
AB as far as a certain point after which it must increase
again in order that it may go off to infinity when x becomes
infinite without becoming negative; from which it is evident
that the curve must be in the form FPG. Since y and
dy
- are both infinite when x = co, they must be very large
dx
when x is either a very large positive or a very large negative quantity, and therefore the curve must run off to infinity
on the positive and negative side of the axis of y in the manner
represented in the figure.
Represen- 219. In the plates marked M the Student will find some
tation by
figures of of the different cases which may occur in tracing a curve
differenth represented by figures, to which he will find it useful to refer.
cases which
may occur. Under each figure is set down the corresponding line in the
table from which the figure is deduced. By means of this
plate the use of the table will appear evident.
To deter- 220. Thus we are able to trace the general form of the
form ofthe curve; but this method does not always shew us the points
cureselyre of contrary flexure of the curve; to determine which we must
we must resort to the method given in (180). Thus in the present
examine
whether it case, differentiating y twice, we get
has any
points of d2y (i - a) + 2 a
contrary - 2 -
flexure. dx' ( -a)3
the only real values of ix which make this =0 or o are iz = a,
= a - 2Aa or -a (o, -): so that there can be only two