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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 2 Examinations 2009/
Module Title: : Engineering Mathematics 211
Module Code: MATH
School: Building & Civil Engineering
Programme Title: Bachelor of Engineering (Honours) in Civil Engineering-Stage 2
Programme Code: CSTRU-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: Summer 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.
- (a) Given that
x
u f(x,y) arctan 2y v=
2y
ln 3x
find the first order partial derivatives of u and v with respect to x and y.
(i) Find a Taylor Series expansion of f(x,y) about the values x=2,y=1. The series is to contain terms deduced from first and second order partial derivatives.
(ii) If T=xf(v) where f(v) is an arbitrary function in v show that
T
y
y T x
x T = ∂
(iii) Estimate the value of u where the values of x and y were estimated to be 2±0. and 1±0.004, respectively (16 marks)
(b) .An open rectangular box is to contain 9 m^3 of liquid. In order to reduce heat loss the base is insulated at a cost of €4 per m^2 and all other sides at a cost of €6 per m^2. Find the dimensions of the box so that the cost is at a minimum value. Calculate the minimum cost. (9 marks)
- (a) Find the Inverse Laplace Transform of the expressions
(i) s 8 s 19 s 12
(^3) + 2 + + (ii)^ s 4 s 13
6s 12 (^2) − +
− (^) (11 marks)
(b) By using Laplace Transformations solve the differential equations
(i) 2 y 20sin4t y(0) 0 dt
dy (^) + = =
(ii) y 20e y(0) y(0) 0 dt
2 dy dt
dy (^) + + = 2t (^) = ′ = (14 marks)
- (a) R is the triangular region with vertices (-1,0), (1,0) and (0,2).
(i) If C is the perimeter of this region evaluate the line integral
C
12x^2 dx 12y^2 dy
(ii) By evaluating appropriate double integrals find the second moment of area of this region about the x-axis. Sum vertically and sum horizontally. (13 marks)
(b) A sector of a circle lies in the first quadrant and this sector is enclosed by the line y=x, the x-axis and the circle x 2 +y^2 =4. The vertices of this region are the points (0,0), (2,0) and ( 2 , 2 ).
(i) If C is the perimeter of this region evaluate the line integral
C
2xdx 4ydy
(ii) A volume V is of uniform cross section. This cross sectional area is the sector of the circle described above and the height is described by 0≤z≤4. For this volume V evaluate the triple integral
V
x^2 zdV
Note: cos^2 A=^1 2
(1+cos2A) sin^2 A=^1 2
(1-cos2A) (12 marks)
- (a) By using the Three Term Taylor Method with a step of 0.1 estimate the value of y at
x=0.2 where
2 xy y(1) 5 dx
dy (^) = =
Solve this differential equation and calculate the error in the estimate above. (8 marks)
(b) Select any two of the following:
(i) By using the Method of Variation of Parameters find the general solution of the differential equation x 2
2 2y 16xe dt
3 dy dt
d y− + =
Note: ∫ = − 2
ax ax ax a
e a
xe dx xe where “a” is a constant. (9 marks)
(ii) Solve for x and for y where
x 2 y y(0) 0 dt
dy
4 x y x(0) 4 dt
dx
(9 marks)
(iii) By using two different methods find the minimum value of V=3x^2 +y^2 +z 2 where x+y+z=7. (9 marks)
DERIVATIVES
f(x) a=constant f ′ (x) x n nx n−^1 lnx x
eax a eax sinx cosx cosx -sinx uv dx
vdu dx
u dv+
tan-1^
a
x 2 2
a x +a tan-1^ x 2
x +
v
u
v^2
dx
udv dx
v du−
