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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 1 Examinations 2010/
Module Title: MATH7004: Engineering Mathematics 201
Module Code:MATH
School: School of Mechanical & Process Engineering
School of Manufacturing, Biomedical & Facilities Engineering
Programme Title:Bachelor of Engineering (Honours) in Mechanical Engineering-Stage 2
Programme Code: EMECH-8-Y
DBE2-8-Y
CSTRY-8-Y
ECPEN-8_Y
External Examiner(s):Dr.P.Robinson
Internal Examiner(s):Mr. T. O Leary
Instructions: Select any four questions. The questions carry equal marks.
Duration: 2 Hours
Sitting: Winter 2010
Requirements for this examination:
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the
correct examination paper.
If in doubt please contact an Invigilator.
1. (a) Verify that the function f(x)=x
4 +4x
2 +4x satisfies the criteria of the Mean Value Theorem
for derivatives over the interval [0,2]. Find correct to two places of decimal a value of x
that satisfies the conclusion of the theorem. There is a value is close to x=1.80.
(9 marks)
(b) Write down two terms of a Taylor Series expansion of f(x) about x=a. Verify that
the approximation below ( theTrapezoidal Rule) holds.
f(a h) f(a) f(x)dx h
a h
a
(8 marks)
(c) In constructing a closed rectangular tank with a reinforced base it costs €12m
construct the base and €4m
- to construct all other sides. Find the dimensions of the
tank of maximum volume that can be constructed for €192. Also calculate this
maximum volume. You are required to use a Lagrangian Multiplier. No marks will
be awarded if any other method is used. (8 marks)
Note: f(a) .... !
(x-a) f(x) f(a) (x-a)f(a) 2
2
f(x )
f(x ) x x
n
n n n
2. (a) By completing the square and by using partial fractions find the Inverse Laplace
Transform of the expression
s 4s 3
2s 4
2
(8 marks)
(b) By using Laplace Transformations solve the differential equations
(i) x 6 e x(0) x(0) 0 dt
dx 2 dt
d x 2t
2
2
(ii) 2x 20cos2t x(0) x(0) 0 dt
dx 3 dt
d x
2
2
(17 marks)
4. In answering the following question you are required to use the Method of Undetermined
Coefficients. No marks will be awarded if any other method is used.
(a) Solve the differential equation
10 x=100t x(0) x(0)= 0 dt
dx
d x
2
2
^
(9 marks)
(b) Find the general solution of the differential equation
=120cos2t dt
dy
d y
2
2
. (8 marks)
(c) Find the general solution of the differential equation
x 2
2
y= 6e dx
dy 2 dx
d y . (8 marks)
5. (a) Find the Fourier Series for the periodic function below
f(t)=
1 tif- 1 t 0
1 tif 0 t 1 f(t+2)=f(t) (10 marks)
Note:
2 2 2 2
( 1 )sin( ) cos( ) ( 1 )cos( )
( 1 )sin( ) cos( ) ( 1 )cos( )
n
n t
n
t n t t n t dt n
n t
n
t n t t n tdt
2 2 2 2
( 1 )cos( ) sin( ) ( 1 )sin( )
( 1 )cos( ) sin( ) ( 1 )sin( )
n
n t
n
t n t t n t dt n
n t
n
t n t t n t dt
A 0 2
sinA 0 0 0
cosA 1 - 1 1
(b) Solve for x where
3 x 2y y(0) 0 dt
dy
4x y x(0) 8 dt
dx
( 6 marks)
(c) Find the poles of the transfer function L[f(t)]
L[y(t)] for the system described by
21 y 26 ydt f(t) y(0) y(0) 0 dt
dy 6 dt
d y t
(^20)
2
(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 1 s
n!
e
at 1
s a
sinhkt k
s k
2 2
coshkt s
s k
2 2
sin t
s
2 2
cos t 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)
Note: coshA
e e
sinhA
e e
A A A A
