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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering- Stage 2
(EMECH_8_Y2)
Summer 2008
Mathematics
(Time: 3 Hours)
Answer FIVE questions.
All questions carry equal marks.
Examiners: Mr. P. Clarke
Prof. M. Gilchrist
Mr. T O Leary
- (a) Solve the differential equation
dy +3y 12 y(0) 1 dx
By using the Three Term Taylor Method with a step of 0.1 estimate the value of y
at x=0.1. Calculate the error in this approximation. (8 marks)
(b) Find the general solution of the differential equation
2
2
d i di +4 +4i=50cos3t dt dt
Express the steady state part of the solution as a single periodic function of the form
Rsin(3t-α). Write down the maximum and minimum values of this function and find
the smallest positive values of t for which these extreme values hold. (8 marks)
(c) Show that the expression
(6x
2 -6y)dx-(6x-8y)dy
is the total derivative of a function. Find this function. Hence solve the exact
differential equation
(6x
2 -6y)dx-(6x-8y)dy=0 y(1)=2. (4 marks)
- (a) Find a Taylor Series expansion of the function f(x,y)=xln(x-3y) about the values
x=4,y=1. The series is to contain terms deduced from second order partial derivatives.
(7 marks)
(b) Find the partial derivatives of u and v with respect to x and y where
-1 2y u=sin x
v=
2 2 x -4y.
(i) If T=f(u) is an arbitrary function in u show that
T T
x y 0 x y
(ii) Estimate the value of v if the values of x and y were estimated to be 5 and 2
with maximum errors of 0.06 and 0.03, respectively. (8 marks)
(c) Find the minimum value of V=x
2 +2xy where y=6+x. Eliminate one of the variables
and use a Lagrangian Multiplier. (5 marks)
- (a) Find the Inverse Laplace transform of the expressions
(i) 2
(s+1)(s +4)
(ii) 2
4s+
(s-1)(s-3)
(10 marks)
(b) By using Laplace Transforms solve the differential equation
2 t 2
d y dy +2 5y 16e y(0)=y (0)= dt dt
(c) Find the zero and the poles of the transfer function L[f(t)]
L[y] where
2 t
(^2 )
d y dy 7 40y 34 ydt f(t) y(0) y (0) 0 dt dt
(4 marks)
- (a) A force F =6x
2 i +24xy j moves a mass the perimeter of the semi-elliptical region
2 2 x y
x≥ 0
Find the work done by the force.
By evaluating a double integral locate the centroid of this region. (9 marks)
(b) The region R is the sector of the circle x
2 +y
2 =5 lying in the first and fourth quadrants
with vertices (0,0), (1,2) and (1,-2).
(i) Evaluate the line integral
C
2ydx 4xdy
where C is the perimeter of this region. The direction of C is anticlockwise.
(ii) If V is the volume with a constant cross section that is described by the region R
above and if 0≤z≤2 then evaluate the triple integral
V
6xzdV
(11 marks)
- (a) A variate can only assume values between x=0 and x=2. The probability density
function is given by
p(x)=A(3x
2 +1).
Find the value of A, the mean value of the distribution and the median value of the
distribution correct to two places of decimal. This value is close to x=1.5. (8 marks)
(b) A particular chemical is packed into bags with a mean content of 5kg and a standard
deviation of 0.015kg. What percentage of bags contain (i) more than 5.02kg and
(ii)between 4.98 and 4.99kg of chemical? EU regulations demand that a minimum
content be printed on each bag and not more than 0.2% of bags are to weigh less than
this minimum content. Find this minimum content. (6 marks)
(c) In a manufacturing process items are packed into batches of 50 and for a number of
these batches the number of defective items were counted.
Number of defectives 0 1 2 3 ≥ 4
Number of batches 70 21 8 1 0
Calculate the average number of defective items per batch. Calculate the probability
that a batch of 200 of these items contains at most two defective items. Use both the
Binomial and the Poisson distributions. (6 marks)
f(x) f ′ (x) a=constant
x
n nx
n-
lnx
x
e
ax ae
ax
sinx cosx
cosx -sinx
sin
2
1-x
1 x sin a
− ^
2 2
a -x
uv
dx
du v dx
dv u +
v
u
2 v
dx
dv u dx
du v −
f(x)
f(x)dx a=constant
sinx -cosx
cosx sinx
e
ax
a
e
ax
