CORK INSTITUTE OF TECHNOLOGY

INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ

Semester 2 Examinations 2009/10

Module Title: Engineering Mathematics 211

Module Code: MATH7006

School: Building & Civil Engineering

Programme Title: Bachelor of Engineering (Honours) in Civil Engineering-Stage 2

Programme Code: CSTRU-8-Y2

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: Autumn 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) (i) Find a Taylor series expansion of the function

V=f(x,y)=ln(x2-4y)

about the values x=3, y=2. The series is to contain terms deduced from the first and

the second order partial derivatives of f(x,y).

(ii) Bt using differentials (partial derivatives) estimate the value of V where the values

of x and y were estimated to be 4±0.01 and 2±0.02, respectively. (13 marks)

(b) Write down the relationships between Cartesian coordinates x and y and polar

coordinates r and . If stress T=f(x,y) is an arbitrary function in x and y find the

relationships between the partial derivatives of T with respect to x and y and those

with respect to r and . Also show that

2

22

2

2

y

T

x

T

θ

T

r

1

r

T

(5 marks)

(c) Find the maximum/minimum values of V=4x2-8xy+y3+4y+8 (7 marks)

2. (a) By using partial fractions and by completing the square find the Inverse Laplace

Transform of the expression

5s6s

84s

2

(11 marks)

(b) By using Laplace Transformations solve the differential equations

(i)

0(0)yy(0)6ey2

dt

dy

3

dt

yd t

2

2

(ii)

0(0)yy(0)60sin2ty2

dt

dy

3

dt

yd

2

2

(14 marks)

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