Second Order Partial Derivatives - Engineering Mathematics - Past Paper, Exams for Engineering Mathematics. Jaypee University of Engineering & Technology

Engineering Mathematics

Description: Main points of this exam paper are: Second Order Partial Derivatives, Taylor Series Expansion, Cartesian Coordinates, Arbitrary Function, Maximum Values, Inverse Laplace Transform, Laplace Transformations, Differential Equations
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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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