EXAMINATION PAPEES.
153
11. Define a liquid.
Prove that the surface of a liquid at rest is horizontal.
12. A body A, weighing one pound, is found to float with half its
volume under water. It is then attached to a body B, weighing
two pounds, and the combined mass is found to float with |ths
of its volume under water. Find the specific gravity (1) of A,
(2) of B.
DIFFERENTIAL CALCULUS.
Tuesday, 26th June. 10 a.m. to 12 a.m.
Give your wording in full.
Endeavour rather to answer a moderate number of Questions accu¬
rately and fully than to answer a large number imperfectly.
1. When is one quantity said to be a function of another ?
Define a differential coefficient, and prove that, if y be a func¬
tion of x, and a: be a function of t, x —.
dt dx dt
2. Find-^j^ and in each of the following cases :—
(1.) y = (x+ay, (2.) y — ax, (3.) y=sin x + cos x.
3. Obtain the differential coefficients of the following functions
1—*
y ~ ~vT+P’
3s-i b+a cos x
a+b cos x ’
4. State and prove Leibnitz’s Theorem for the differentiation of a
product of two functions.
Find
d^x2 cos x
daft
5. Enunciate Taylor’s Theorem.
Deduce Maclaurin’s Theorem, and apply it to expand sin x in
ascending powers of x.
6. If be a fraction such that both numerator and denominator
<p(x)
vanish when x = a, find the limit to which the fraction tends as
x approaches the value a.
Find the limiting values of
(1)
2 —7*+8a:2—3a?
3—8a:+7a:2—2a?
when a: = 1.