# 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.
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
.
(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)