Cork Institute of Technology
Higher Certificate in Engineering in Mechanical Engineering โ Award
(National Certificate in Engineering in Mechanical Engineering โ Award)
(NFQ โ Level 6)
Summer 2005
Mathematics
(Time: 3 Hours)
Answer FIVE questions.
Examiners: Ms. J. English
Dr. D. Cremin
Mr. J. Connelly
Mr. R. Simpson
Q1. (a) Find dy
dx for the parametric equations
x
te
t
=
2. and yte
t
=..
Find the equation of the tangent to the curve at t = 2.5. [7 marks]
(b) If
42 3 0xxy yโโ=sin( ) , use implicit differentiation to find dy
dx at the point (3,2).
[6 marks]
(c) Show that the function 3
() 3 4
f
tt t
=
โโ
has a root between t=2 and t=3.
Use three iterations of the Newton-Raphson method to find the root correct to two
decimal places. [7 marks]
Q2. (a) Given z = 7x3 + 4xy2-3y3 , find
,
zz
x
y
โโ
โโ
and
2
2
z
x
โ
โ [6 marks]
(b) The resistance of a length of wire is given by
2
kPL
RD
= where k is a constant.
Calculate the approximate percentage change in R when L is increased by 3.5%, P is
increased by 0.2% and D is decreased by 2%. [8 marks]
(c) Locate the turning points on the curve y = -5x3 +2x2 +3x and establish whether they are
maximum or minimum points. [6 marks]
Q3. Determine each of the following integrals:
