Cork Institute of Technology
Bachelor of Engineering (Honours) in Structural Engineering – Stage 2
(Bachelor of Engineering in Structural Engineering – Stage 2)
(NFQ – Level 8)
Autumn 2005
Mathematics
(Time: 3 Hours)
Answer FIVE questions.
All questions carry equal marks.
Examiners: Mr. T. Corcoran
Prof. P. O’Donoghue
Mr. T. O Leary
1. (a) (i) Solve the differential equation below by using two different methods
3y(0)01y2
dx
dy ==−
(ii) By using the Tree Term Taylor Method or the Improved Euler Method with
a step of 0.1 estimate the value of y at x=0.1. (12 marks)
(b) By evaluating a double integral locate the centroid of the region in the first and
fourth quadrants bounded by the lines y=2x, y=-2x and the circle x2+y2=20. (8 marks)
2. (a) Show that the Taylor Series expansion of f(x,y)=xln(3x-y) about the values
x=1,y=2 is given by
...2)(y
2
1
2)1)(y2(x1)(x
2
3
2)(y1)3(xy)f(x, 22 +−−−−+−−−−−= (8 marks)
(b) Cartesian coordinates x and y are related to polar coordinates r and θ by the formulae
θ=
−
x
y
tan 1 r= 22 yx +
If stress T=g(r,θ ) is an arbitrary function in r and θ write down the relationships
between the partial derivatives of T with respect to x and y and those with r and θ .
Hence simplify as much as possible the expression
∂
∂
+
∂
∂
y
T
y
x
T
x (7 marks)
(c) Find the maximum and minimum values of V=x3-12xy+6y2+48. (5 marks)
