MAT455eee

CONFIDENTIAL CS/JAN 2012/MAT455

UNIVERSITI TEKNOLOGI MARA FINAL EXAMINATION

COURSE COURSE CODE EXAMINATION TIME

FURTHER CALCULUS FOR ENGINEERS MAT455

JANUARY 2012 3 HOURS

INSTRUCTIONS TO CANDIDATES

1. This question paper consists of five (5) questions. Answer ALL questions in the Answer Booklet. Start each answer on a new page. 2.

3. Do not bring any material into the examination room unless permission is given by the invigilator.

Please check to make sure that this examination pack consists of:

i) the Question Paper ii) an Answer Booklet - provided by the Faculty

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO This examination paper consists of 4 printed pages

Hak Cipta Universiti Teknologi MARA CON FIDENTIAL

CONFIDENTIAL 2 CS/JAN 2012/MAT4S5

QUESTION 1

a) Let an = '-, determine the convergence or divergence of n(n + 2f

0 {*n}?

ii) E a" n=l

(3 marks)

b) Use an appropriate test to determine whether the following series converges or diverges.

3 5 ^ ) V ntUV^-l)+4"

") I 00 3nn

... x- i COS Mi

n=o n +1 (10 marks)

c) Find the value(s) of x where the series V (- 1)"(X + 3)" ^ ' converges.

(7 marks) =1 3"4"!

Hak Cipta Universiti Teknologi MARA CONFIDENTIAL

CONFIDENTIAL 3 CS/JAN 2012/MAT455

QUESTION 2

00 (_ j \ / ( + l

/c=i k7 - 6

(5 marks)

a) Determine whether the series Y ^'- converges absolutely, converges

conditionally or diverges.

2 2y+1

b) Use the transformations u = x - 2y and v = y to evaluate I I *Jx-2y dxdy . 0 2y

(7 marks)

c) A thin plate of constant density k bounded by the parabola y = 3x2 and the liney = 3 has a mass of 4k. Find the center of mass of the plate.

(8 marks)

QUESTION 3

a) Use the integral test to determine whether V * -, c-, -^ converges or diverges. La (n-2)ln(n- 2) n=A

(4 marks)

b) Sketch the region of integration of [ [ yex dxdy and evaluate the integral by reversing Jo Jo

the order of integration. (9 marks)

c) Evaluate [ i U y 2 +z2 dV where Q is the solid region that lies inside the cylinder Q

y2+z2 =16 between the planes x = 0 and x = 3.

(7 marks)


CONFIDENTIAL 4 CS/JAN 2012/MAT455

QUESTION 4

a) Given the vector field F( x,y,z) = (3x2y2z)J + (2x3yz-2j] + (x3y2 -e~z)k .

i) Show that F is conservative.

ii) Find the potential function #>such that cp = VF.

iii) Hence, find the work done in moving a particle under this field of force from point A(1,1,0) to point B( 3, 2,1).

(10 marks)

b) Use the Green's Theorem to evaluate to y2dx + (3x3 +2xy)dy where C is the C

semicircle x2 +y2 = 9 from point (0, -3) to point (0, 3) and followed by the line segment from point (0, 3) to point (0, -3), oriented counterclockwise.

(10 marks)

QUESTION 5

a) Evaluate & Fdr using Stokes' Theorem where F(x,y,z) = y3J + xz] + x2k and C is c

the boundary of the parabolic surface z = 1-x2 in the first octant cut off by the plane y = 4.

(10 marks)

b) Use the Divergence Theorem to find the outward flux $FndS where s

F(x,y,z) = xzJ + y2k and S is the surface of solid bounded by the cone z = -\x2 +y2 and the plane z = 2 in the first octant.

(10 marks)

END OF QUESTION PAPER


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