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

CHAP. X.

MISCELLANEOUS EXAMPLES.
BY reference to Articles (213)-(251), the solutions of
the following Theorems and Problems may be effected.
1. The equation of a curve being y- 1 = ( -2)3, it is
required to prove that it cuts the axis of v when = 1 and
tan X= 32: and the axis of y when y= -7 and tan Y=-
2. The curve whose equation is ay2= 2 ( -b), meets
the axis of x when x = 0 and x = b, and corresponding to them
tan X-= - and tan X= oo: required a proof.
a
3. If a curve be defined by the equation, (a -) = y (a + y),
it meets the axis of x when x = and =a, so that tan X
= + 1: and it meets the axis of y when y = O and y= -a,
so that tan Y= + 1: required a proof.
4. The curve whose equation is V-y = a + x, meets the
axis of x when x = - a and tan X = 0: required a proof.
5. If the equation of a curve be a3y=XI4-bX3-bb2V2;
then will it meet the axis of x when x = 0 and x = b (1 + 5),
tan X=o and tan X=; and the axis of y when y=o
and tan Y = oo: required a proof.
6. It is required to prove that the ellipse defined by the
equation y2 - 2Xy + 3$2 + 2y- 4 -3 = O, meets the axis of v
when = 9(2 + /13) and tan X= (5 T+ A13): also the axis
of y when the values of y are 1 and -, and tan Y= and 2,