UNIVERSITY OF MALTAFACULTY OF ENGINEERING, FACULTY of ICT and

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

January/February 2012 Assessment Session

MAT1801 Mathematics For Engineers I 27th January 2012

9.15 a.m. - 11.15 a.m.

Calculators and mathematical booklets will be provided. No other calculatorsare allowed.

Answer THREE questions

1. (a) If u = ln

(x+

1

y

), verify that

∂2u

∂y∂x=

∂2u

∂x∂y.

(b) If w =x2y4

z3, find the approximate percentage error in w resulting from the

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

(c) Solve the following differential equation:

2(1 + x2)dy

dx+ (2x− 1)y = earctanx−xy−1.

10, 10, 15 marks

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

0

∫ arccos(√y)

0

esinx

√y

dxdy.

(b) Find the volume of the region bounded by the planes x = 0, y = 0 andz = 0, and the surfaces z = y2 and y = 1− x2.

17, 18 marks

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

xdy

dx=

dy

dzand x2

d2y

dx2=

d2y

dz2− dy

dz.

7 marks

Hence solve the differential equation

x2d2y

dx2+ x

dy

dx+ 4y = 3 sin(2 ln(x)),

given that y =3

4and

dy

dx= −3

4when x = 1.

28 marks

4. (a) (i) Use the ratio test to show that the series∞∑n=1

5n

n!converges. Hence,

or otherwise, show that the series∞∑n=1

(−1)n5n

n!converges.

(ii) Use the nth root test to show that the series∞∑n=1

5n

n5diverges. (You

may use the fact that limn→∞

n√n = 1.)

(b) (i) Find the Fourier series of the the function given by f(x) = x(π − x)for 0 6 x 6 π, and periodic with period π.

(ii) Show that1

12− 1

22+

1

32− 1

42+ · · · = π2

12.

9, 5, 16, 5 marks

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