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)

3. In answering the following you are required to use the Method of Undetermined

Coefficients. No marks will be awarded if any other method is used.

Select any three parts of the following:

(a) In the theory Beam Struts the differential equation below arises

2EI

Wx

- yω

dx

yd 2

2

2

where

EI

P

ω2

Solve this differential equation where y=0 at x=0 and at x=L to show that

L

x

ωLsin

ωxsin

2P

WL

y

Show that the strut fails if the load P reaches the critical value

2

2

2

L

EI

ω

.

(9 marks)

(b) Solve the differential equation

0(0)yy(0)244y

dx

dy

4

dx

yd

2

2

(8 marks)

(c) Find the general solution of the differential equation

2tsin20

dt

dy

2

dt

yd

2

2

(8 marks)

(d) Find the general solution of the differential equation

t

2

26e2y

dt

dy

3

dt

yd

(9 marks)

4. (a) R is the triangular region with vertices (-1,0), (1,0) and (1,4).

(i) If C is the perimeter of this region evaluate the line integral

C

22 dy12ydx6x

(ii) Evaluate the double integrals

R

3dA80y

(13 marks)

(b) A volume V is in the form of a right circular cylinder and is described by

V: x2+y2≤4 0≤z≤3

(i) Evaluate the line integral

C

24xydydx3x

where C is the perimeter of the base.

(ii) For this volume V evaluate the triple integral

V

2zdV4x

Note: cos2A=

1

2

(1+cos2A) sin2A=

1

2

(1-cos2A) (12 marks)

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University:
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