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Cork Institute of Technology
Bachelor of Engineering (Honours) in Structural Engineering – Stage 2
(Bachelor of Engineering in Structural Engineering - Stage 2)
(NFQ – Level 8)
Summer 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
- (a) (i) By separating the variables or by using an integrating factor solve the differential
equation
2 xy 6x y(0) 1 dx
dy − = =
(ii) By using the Three Taylor Method or the Improved Euler Method with a step
of 0.1 estimate the value of y at x=0.1. (8 marks)
(b) Write down three terms of a Taylor Series expansion of f(x) about x=a and
include a remainder term. Hence show that
f(a h) f(a - h)
2h
f (a) O(h )
Estimate the value of f ′(1) where f(0.95)=2.22, f(1.00)=2.10 and f(1.05)=2.08.
(5 marks)
(c) By using double integrals find the second moment of area of the triangular region with
vertices (-1,0), (1,0) and (0,2) about the x-axis. Sum vertically and sum horizontally.
(7 marks)
- (a) Variables u and v are related to variables x and y by the formulae
−
2x
y u f(x,y) tan
1 v=
2 2 x − y.
(i) Find a Taylor Series expansion for the function f(x,y) about the values x=1,y=2.
The series is to contain terms obtained from second order partial derivatives.
(ii) Estimate the value of v where the values of x and y were estimated to be
5±0.04 and 3±0.02, repectively.
(iii) If P=f(u,v) is an arbitrary functions in u and v show that
v
P
v y
P
y x
P
x ∂
(13 marks)
(b) Find the minimum value of V=2x
2 +y
2 +z
2 where 2x+y+z=8. Eliminate one of
the variables and use a Lagrangian Multiplier. (7 marks)
- (a) Find the Inverse Laplace Transform of the expressions
(i) s 4s -4s 16
24s 72
3 2
(ii) s 4s 20
10s 8
2
(7 marks)
(b) By using Laplace Transforms solve the differential equations
(i) 2y 40sin2t y(0) y(0) 0 dt
dy 3 dt
d y
2
2
(ii) 6 y=4e y(0)=y(0) 0 dt
dy 5 dt
d y 2t
2
2
− + ′ = (13 marks)
- (a) Evaluate the line integral
C
zdx ydy zdz
where C is the line passing from (1,1,1) to (2,0,3). (4 marks)
(b) (i) Evaluate the line integral
C
2 2 (6x 6y )dx 24 xydy
where C is the perimeter of the sector of the circle x
2 +y
2 =8 in the first and fourth
quadrants with vertices (0,0), (2,2) and (2,-2).
(ii) By calculating an appropriate double integral locate the centroid of the sector above.
(11 marks)
(c) A volume V is of constant cross sectional area and is described by
y
x
2 2
For this volume V evaluate the triple integral
V
4xzdV (5 marks)
- (a) By using the Method of Variation of Parameters find the general solution of
the differential equation:
x
10e y dx
dy 2 dx
d y
x
2
2
− + = (7 marks)
(b) Solve for x where
4 x y y(0) 0 dt
dy
5 x y x(0) 3 dt
dx
(6 marks)
(c) Show that the function f(x)=x
4 +2x
2 satisfies the criteria of the Mean Value
Theorem for derivatives over the interval [0,2]. By using the Newton Raphson
Method find correct to two places of decimal the value of x that satisfies the
conclusion of the theorem. This value is close to x=1.2. (7marks)
- (a) A variate x can only assume values between x=-1 and x=1and its probability
density function is given by p(x)=k(1-x
2 ). Find (i) the value of k, (ii) the mean value,
(iii) the modal value and (iv) the median value of this distribution. (8marks)
(b) If the weights of goods packed by a certain machine are Normally Distributed
with a mean weight of 5kg and a standard deviation of 0.035kg. What percentage
of packages will weigh (i) less than 5.08 and (ii) between 4.98 and 5.09 kg?
(iii) Find the value of λ if 95% of packages weigh between 5±λ. (5 marks)
(c) After production of items by a manufacturer the items are packed into batches
of 20 and in a number of these batches the number of defectives were counted
No. of defectives (^0 1 2 3) ≥ 4
No. of batches 66 29 4 1 0
Calculate the average defective rate. Using the Binomial and the Poisson
Distributions calculate the probability that a random sample of 100 items contains
at least 3 defectives. (7 marks)
LAPLACE TRANSFORMS
For a function f(t) the Laplace Transform of f(t) is a function in s defined by
F(s) e f(t)dt
st
0
−
∞
∫ where s>0.
f(t) F(s)
A=constant A
s
t
N N!
s
N + 1
e
at 1
s −a
sinhkt k
s k
2 2 −
coshkt s
s k
2 2 −
sin ωt ω
s ω
2 2
cos ωt s
s
2 2
e f(t)
at F(s-a)
f (t) ′ sF(s)-f(0)
f (t) ′′ (^) s F(s)^2 − sf(0) − f (o)′
f(u)du
0
t
F(s)
s
f(u)g(t u)du
0
t
F(s)G(s)
U(t-a) (^) e
s
-as
f(t-a)U(t-a) (^) e F(s)
−as
δ ( t − a) e
-as
Note: coshA
e e
sinhA
e e
A A A A
=
− −
