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

CHAPTER XV.
ON THE METHOD OF TRACING THE GENERAL FORM OF A CURVE
FROM ITS EQUATIONS.
IT is often necessary to make out and trace the general
form of a curve from its equation, without actually calculating its exact dimensions, or ascertaining the precise positions
of its remarkable points: we now proceed to state how
this may be done in most cases.
217. If we can find y in terms of a, it is easy in general How to
to trace the general form of the curve by determining the trache n
curve when
values of v which make y=0 or co, the corresponding values wecanCfind
y min terms
dy of x.
of d-, and the signs which y has between its zero and indx
finite values. We may do this in the following manner, viz.
In one column write in order the values of ix which make
y = 0 or co, and in addition to these, the value xv =o:
opposite to these values, in another column, write down the
corresponding values of y, and between each two put the
sign which y has between them (y will always have the same
sign between each two of these values, since it can only
change its sign in passing through 0 or co): and in a third
d 7y
column put the corresponding values of Y. By means of
div
such a table it will be easy, in most cases, to trace the general
form of the curve, as the following example will shew.
218. To trace the general form of the curve whose Example.
equation is
y --;
a, - a,
here y is zero when v = 0 or - a, and infinite when v = a:
hence the values of,' to be written down are -a, 0, a, C;