UNIVERSITY OF MALTAFACULTY OF ENGINEERING and

FACULTY FOR THE BUILT ENVIRONMENTB.Eng.(Hons.)/B.E.& A.(Hons.) Year I

January/February 2009 Assessment Session

MAT1801 Mathematics For Engineers I 21st January 2009

9.15 a.m. - 11.15 a.m.

Calculators and mathematical booklets will be provided. No other calcula-tors are allowed.

Answer THREE questions

1. (a) If z = xy2 cos(y

x

), show that

x∂z

∂x+ y

∂z

∂y= 3z.

11 marks(b) If u = ln(x2 + y2), show that

∂2u

∂x2+

∂2u

∂y2= 0.

11 marks(c) If w =

x

y2z2, find the approximate percentage error in w resulting

from the following errors in x, y and z respectively: 0.3% too large,0.1% too small and 0.2% too large.

13 marks

2. Solve the following differential equations:

(a)dy

dx+ x−1y = x4y3; 18 marks

(b)dy

dx=

x − y

x + y. 17 marks

3. (a) By changing the order of integration, evaluate∫ 1

0

∫ arccos y

0

esin x dxdy.

17 marks(b) Find the volume of the tetrahedron bounded by the planes x = 0,

y = 0, z = 0 and 2x + y + 3z = 6.18 marks

1

4. If y is a function of x, and x = ez, show that

xdy

dx=

dy

dzand x2 d2y

dx2=

d2y

dz2− dy

dz.

7 marksHence, or otherwise, solve the differential equation

x2 d2y

dx2− x

dy

dx+ 2y = 5 sin(ln x),

given that y = 0 anddy

dx= 3 when x = 1. 28 marks

5. (a) Find the Fourier series of the function f defined by

f(x) =

{1 + x if − 1 6 x 6 0,

1 − x if 0 6 x 6 1,

given that f is periodic with period 2.

(b) State Parseval’s identity, and use it to show that

1 +1

34+

1

54+

1

74+ · · · =

π4

96.

25, 10 marks

Page 2 of 3