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 2010 Assessment Session

MAT1801 Mathematics For Engineers I 4th February 2010

9.15 a.m. - 11.15 a.m.

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

Answer THREE questions

1. (a) Let u be a twice-differentiable function of x and y, where x = r cos θ andy = r sin θ.

(i) Find∂u

∂rand

∂u

∂θ.

(ii) Show that

∂2u

∂θ2= −r cos θ

∂u

∂x− r sin θ

∂u

∂y+ r2 sin2 θ

∂2u

∂x2

− 2r2 sin θ cos θ∂2u

∂x∂y+ r2 cos2 θ

∂2u

∂y2.

(iii) Find∂2u

∂r2, and show that

∂2u

∂x2+

∂2u

∂y2=

∂2u

∂r2+

1

r

∂u

∂r+

1

r2

∂2u

∂θ2.

Note: You may assume that∂2u

∂x∂y=

∂2u

∂y∂x.

25 marks

(b) If w =x3y

z4, find the approximate percentage error in w resulting from the

following errors in x, y and z respectively: 0.2% too large, 0.4% too smalland 0.1% too large.

10 marks

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2. Solve the following differential equations:

(a)dy

dx+ y = xy3; 18 marks

(b)dy

dx=

x2 + y2

x(x + y). 17 marks

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

0

∫ 2

x

3y4 cos(xy2

)dy dx.

17 marks

(b) Find the volume of the region bounded by the planes x = 0, y = 0, z = 0,2y + z = 4 and the surface x = 4 − y2.

18 marks

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

dy

dx= cos x

dy

dzand

d2y

dx2= cos2 x

d2y

dz2− sin x

dy

dz.

7 marks

Hence solve the differential equation

d2y

dx2+ tan x

dy

dx− y cos2 x = 2esin x cos2 x,

given that y = 1 anddy

dx= 0 when x = 0.

28 marks

5. The function f is defined by f(x) = 1 − x for 0 < x < 1.

(a) Expand f in a Fourier sine series and show that

1 − 1

3+

1

5− 1

7+ · · · =

π

4.

(b) Expand f in a Fourier cosine series and show that

1 +1

9+

1

25+

1

49+ · · · =

π2

8.

12, 5, 14, 4 marks

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