--J, Universit Malaysia PAHANG Engineering Technology Creativity

FACULTY OF INDUSTRIAL SCIENCES & TECHNOLOGY FINAL EXAMINATION

COURSE : APPLIED CALCULUS

COURSE CODE : BUM2123

LECTURER TAN LIT KEN RAHIMAH BINTI JUSOH@AWANG SITI FATIMAH BINTI AHMAD ZABIDI EZZATUL FARHAIN BINTI AZMI

DATE : 10 JUNE 2014

DURATION : 3 HOURS

SESSION/SEMESTER : SESSION 2013/2014 SEMESTER II

PROGRAMME CODE : BAA/BEE/BECIBEP/BFF/BFM/BHAIBHM/BKB/ BKC/BKG/BMA/BMM/BPS/BSB/BSK/BSP/BTC

INSTRUCTIONS TO CANDIDATES

1. This question paper consists of NINE (9) questions. Answer ALL questions.

2. All answers to a new question should start on a new page. 3. All calculations and assumptions must be clearly stated. 4. Candidates are not allowed to bring any material other than those allowed by

the invigilator into the examination room.

EXAMINATION REQUIREMENTS:

1. APPENDIX - Table of Formulas

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO

This examination paper consists of EIGHT (8) printed pages including the front page.

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QUESTION 1

(a) Find the center and the radius of the sphere

2x2 +2y2 +2z 2 —6x+2y+z+6=O

(4 Marks)

(b) Sketch the surface of

z = x 2 +4

(2 Marks)

QUESTION 2

Consider a plane that passes through the points A(O, - 1,2), B(- 2,1,1) and C(- 3,2, - 2).

(i) Find the equation of the plane.

(ii) Determine the intersection point between the plane with the line

x=3—t, y=2t-8, z=1O

(11 Marks)

QUESTION 3

Sketch and find the area of the region that lies inside the cardioid r = 2 +2 sin 0 and

outside the circle r = I

(10 Marks)

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QUESTION 4

A vector equation of a curve is given by

r(t) = (sin t - t cost) I + (cost + t sin t) j

Find

(i) The arc length in the interval [0, 27r].

(5 Marks)

(ii) The curvature and the radius of the curvature.

(6 Marks)

QUESTION 5

(a) The position vector of a moving particle is given by

r(t) = ---cosrt i+ --sin ntj+3tk

Find its velocity and speed at t =

(5 Marks)

(b) Find the tangential component, a7 , and normal component, a, for the vector

equation

r(t) = e 2' i + 4t j + e 2 ' k

(10 Marks)

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VA

z

CONFIDENTIAL BAA/BEE/BEC/BEP/BFF/BFM/BHA[BHF/BKB/BKC/ BKG/BMA/BMM/BPS/BSBIBSKJBSP/BTC/1 31 4IIIBUM2I 23

QUESTION 6

The lengths x,y, and z of the edges of a rectangular box change with time. At the

instant when x =2, y = 4, z = 4,

dxdy - = = 2m / sec dt dt

and

dz - = —4m / sec. dt

Find the rate of change of the interior diagonal D. Hence, determine whether it is

increasing or decreasing in length.

X

(14 Marks)

QUESTION 7

(a) Suppose

/ y,z) \ y—x

f = z+w

find

(4 Marks)

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(b) Given

3e' +z cos (xyz)_-4x+3y

find at (0 1, 1) 3x

(6 Marks)

QUESTION 8

A rectangular lamina with vertices (0,0), (0,1), (1,1), and (1,0) has density function

y+l

Find its centre of gravity.

(13 Marks)

QUESTION 9

Consider

4 Ji6.v J32-x--v

JJJJ(x,y,z)dv J0 J0 J2 +y2+z2dzdydx

(i) Sketch the region G

(2 Marks)

(ii) Evaluate the triple integral by changing to spherical coordinates.

(8 Marks)

END OF QUESTION PAPER

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CONFIDENTIAL BAA/BEE/BEC/BEP/BFFIBFM!BHA/BHF/BKB/BKC/ BKG/BMA/BMM/BPS/BSBIBSK/BSP/BTC/131 411/BUM2 123

APPENDIX - TABLE OF FORMULAS

TRIGONOMETRIC IDENTITIES

cos 2 9 + sin' 0 = 1

cos 2 9='(1+ cos 20)

sin e = (1— cos 20)

POLAR COORDINATES

x=rcos9 y=rsin0

Area: A-_Jir2d0

7 7 y

x+y 7 =r tan0=— x

SECOND PARTIALS TEST

D(a, b) = f (a, b)j, (a, b) - (f, (a, b))2

VECTORS AND GEOMETRY OF SPACE

Equation of a Sphere (x—h)2 +(y—k) 2 +(z _l)2 = r2

Vector Equation of Lines r = r0 + tv

Parametric Equation of Lines x=x0 +at y=y0 +bt z=z0+ct

EQUATION OF PLANES

Scalar Equation of the Plane

Distance between a Point and Plane

a(x—x0)+b(y— y o ) + c(z—z0)O

- ax0 + by0 + cz0 + d

- / 7 7 7 a +b +c

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VECTOR FUNCTIONSb b f(2 (2 (dz 2

- Arc Length L = Jr'(t)t =dt) dt) dt) di' a a

Unit Tangent Vector T(t) = r'(t)

Unit Normal Vector N(t) = T'(t)

Binormal Vector B(t) = T(t) x N(t)

T'(/) r'(t) x r"(t)j - v x Curvature =

r '(1 ) r'(t )M 3 -

Radius of Curvature p = I / ic

Tangential Component v a

=

vxa ll Normal Component ON=

MULTIPLE INTEGRALS

Area A = ft dA = J J rdrd9

Volume (Rectangular & Polar Coordinates)

V = JJf(x,y)dA = JJf(r cos 8,r sin 6)rdrd9

Cylindrical Coordinates

x=r cos O y=rsinO z=z

JJJ f(x, y, z)dV = JJJ f(r, 0, z)rdzdrd0

VA

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Spherical Coordinates

x=psincosO y=psinq,sin9 z=p Cos p

JJJf(x, y, z)dV = JJJf(p 0, (p) p 2 sin çodpd0dço

hf3z\2 +

( ,,Z 2

Su rface Area s=JJ_J - +1 dA ax

I?

Mass M=JJS(x,y)dA

M),1 Center of gravity X= -- = —JJxS(x,y)dA

MM/? M 1 j=_L=—jJy6(x,y)dA

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