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

30

29. By Maclaurin's Theorem expand sin (a+b4+cx2+&c.)
into a series of the form A + B + 2 + &c.
30. Having given 1 + x sin y=tan y, find y in terms
of x by Maclaurin's Theorem.
31. If x = m tan ( - nx), prove by Maclaurin's Theorem
Z. m sin 2z
that when - is very small, x -- very nearly.
z 2 mn + cos2
32. Expand ea+b+x +&c. by Maclaurin's Theorem.
33. Prove the following formula:
G,aG3 2 5 XI '
cot x.= - - - -2 - - &c.
3 3. 5 32. 5.7 32. 5. 7
1 a? 14x a
34. Shew that cosec x = - 4 -- --- + &c.
x 1.2.3 1.2.3.4.5.6
35. By Maclaurin's Theorem find the developement of
in descending powers of x.
a3- 1
36. Apply Maclaurin's Theorem to express the arc of a
circle in terms of the secant.
O^ QG3,GX4
37. If my -xy=m, then y=1 + --- + -&c
3m 3 4m 3'm4
38. If my3 X3y- x3 = 0, then will
m4 3m7 12m10
y= - - - &c.
39. If y3 - 2 y2 + x2y _ a3 = 0, then will
a3 2a6 7 a9
Y -+ 5 — +- +&c.
cy" XX X0X
40. If y_-3y+ _x=o, then y == -+ + a &c.,
3 34 36
also, y = V/3/s- - W- &c.
6 81
1 2 4
and y = - - c' - -- &c.
- l 1.2.$ e 3