-j Universiti J Malaysia

PAHANG Engineering Technology . Creativity

FACULTY OF INDUSTRIAL SCIENCES & TECHNOLOGY FINAL EXAMINATION

COURSE : APPLIED CALCULUS/CALCULUS! ENGINEERING MATHEMATICS II! INDUSTRIAL CALCULUS/ENGINEERING MATHEMATICS I/ENGINEERING MATHEMATICS III

COURSE CODE : BUM2123/BAM1O13/BKU1O23/BSU1O13/ BMM1 1 13/BET2543

LECTURER : SAMSUDIN BIN ABDULLAH NOR IZZATI BINTI JAINI FARAHANI BINTI SAIMI SITI FATIMAH BINTI HAM AHMAD ZABIDI NOR AZILA BINTI CHE MUSA

DATE : 5JUN 2012

DURATION : 3 HOURS

SESSION/SEMESTER : SESSION 2011/2012 SEMESTER II

PROGRAMME CODE : BAAIBAEIBEE1BEP/ BFF/BFM/ BKB/BKCIBKG/

INSTRUCTIONS TO CANDIDATES 1. This question paper consists of FOUR (4) questions. Answer ALL questions. 2. All answers to a new question should start on a new page. 3. All the 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 FIVE (5) printed pages including the front page.

CONFIDENTIAL BAA/BAE/BEEIBEP/BFF/BFM1BKBiBKC1BKG1BMAIBMB1Ml BMMIBSB/BSK/1 1 12IIIBUM2123IBAM1O13JBKU1O23/BSU1O13/BMM1 1 131BET2543

QUESTION 1

(a) Find the area of the region that is inside of the cardioid r 4+4 cos 0 and outside

the circle r = 6.(8 Marks)

(a) Find the distance between the planes 3x + 2y - z = 6 and— 3x - 2y + z =12.

(7 Marks)

(b) Determine whether the line L and 4 are parallel, skew or intersect. If

intersect, find the point of intersection.

L1 :x=3-4t, y=2-3t, z=1+2t

L2 :x=s, y=1+2s, z=2+3s

(12 Marks)

QUESTION 2

(a) Find the length of the arc of circular helix with vector equation

r(t) = (3 sin t,3 cos t, t)

from the point (0,3,0) to (3,0,7r12).

(5 Marks)

(b) Suppose

r(t) (2 cos 2t,2 sin 2t, t)

Find

(i) the unit tangent vector, T(t) and T(0)

(ii) the unit normal vector, N(t) and N(0)

(iii) the curvature, K(t), and the radius of curvature, p(t).

(13 Marks)

(c) Find the velocity, v(t), and position vectors, r(t).

a(t) = 2costi+3sintj+el, v(0) = i —2j+k, r(0) = j — i

(8 Marks)

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CONFIDENTIAL BAAIBAEIBEE/BEP/BFFIBFMIBKBIBKCIBKGIBMAIBMB/BMFIBMI BMM/BSBIBSKJ1 1121L1BUM2123/BAM1O13/BKU1O23/BSU1O13IBMM1 1 13/BET2543

QUESTION 3

(a) Evaluate the indicated partial derivatives.

z=Jx2 —y; z(2,l)

(4 Marks)

(b) Use chain rule to find -az

u=-1,v=1 au

z=e 2"; x=2u-3v; y=-

(6 Marks)

(c) Determine the relative extrema and saddle point (if any) of the graph

f(x,y)=x2 +xy+y 2 —3x

(12 Marks)

QUESTION 4

(a) Evaluate the double integral by converting to polar coordinates.

o__ J J Jx2+y2dydx

(7 Marks)

(b) Find the surface area of the portion of the paraboloid z = 1— (x2 + y2 ) that is

above the xy-plane.

(8 Marks)

(c) Use cylindrical coordinates to find the volume of the solid within the cylinder

X2 +Y2 =4 and between the planes z=O and y+z=6.

(10 Marks)

END OF QUESTION PAPER

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CONFIDENTIAL BAA[BAE[BEEIBEPIBFFIBFM1BKB[BKC1BKGJBMA/BMBIBMFIBMI BMM/BSB/BSK/1 1 1211/BUM2123IBAM1O13[BKU1 0231BSU1013/BMM1 1 13/BET2543

APPENDIX - TABLE OF FORMULAS

TRIGONOMETRIC IDENTITIES

sin 20=2 sin 0 cos 0

cos 2 0 + sin' 0 =1

cos 2 0=--(1+ cos 20)

sin 0= . (1— cos 20)

POLAR COORDINATES

x=rcosO y=rsin0

rf ,

Arc Length dO Ld0)

tan 0 =x

Area A=1' r2dO 2

SECOND DERIVATIVE TEST D(a, b) = f (a, b)f, (a, b) - (f, (a, b))2

VECTORS AND GEOMETRY OF SPACE

Equation of 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 = + at z = z0 + at

EQUATION OF PLANES

Scalar Equation of the Plane

Distance between a Point and Plane

a(x—x0)+b(y— yo ) +c(z—z0)=O

D= ax0 +by0 +cz0 +dI

+ b2 + C2

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CONFIDENTIAL BAAIBAE/BEEIBEP/BFFIBFM/BKB1BKC/BKG1BMAIBMB1BMFIBMI BMMIBSB/BSK/1 112IIIBUM2123IBAM1O13/BKU1O23JBSU1O131BMM 1 113/BET2543

VECTOR FUNCTIONS

b /( 2(2 (dZ)2

+ dtArc Length L =jlr'(t)Idt = dt) dt

Unit Tangent Vector

Radius of Curvature

T(t)— r(t) IIr'(t)II

N(t)— Y(t) IT'(t)I

B(t) = T(t)xN(t)

lT'(t)l !r'(t) x r"(t) JI =

Vr'(t)I lIr'(t)I

p=1I1c

Unit Normal Vector

Binormal Vector

Curvature

MULTIPLE INTEGRALS

A = JfdA = Jjrdrd0 :

Volume (Rectangular & Polar Coordinates)

V = JJf(x, y)dA cos 0, r sin 0)rdrd9

Cylindrical/Spherical Coordinates

x=rcos0 psinq5cos0 y =r sin 9 = p sin q sin 0 z = z = pcosqi

JJJf(x, y, z)dV = JJJf(r, 9, z)rdzdrd9 JJff(p 9, q)p sin q5dpdWçb

Surface Area s=ifF(!

z+ + dA) y)

- Centroid (x,y,z) x = -- JJfxdV y = z = iJJJydV -V

JJJzdV G VG

Area

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