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Sma 2200: Calculus Iii Question Paper

Sma 2200: Calculus Iii

Course:Bachelor Of Science In Acturial Science

Institution: Dedan Kimathi University Of Technology question papers

Exam Year:2011

SUPPLEMENTARY SMA 2200 Calculus III Page 1 of 3
Page 1 of 3
DEDAN KIMATHI UNIVERSITY OF TECHNOLOGY
University Examinations 2010/2011
SECOND YEAR SUPPLIMENTARY/SPECIAL EXAMINATION FOR THE DEGREE OF
BACHELOR OF SCIENCE IN ACTURIAL SCIENCE/TELECOMMUNICATION
& INFORMATION ENGINEERING
SMA 2200: CALCULUS III
DATE: 8TH NOVEMBER 2010 TIME: 2.00 PM – 4.00 PM
INSTRUCTIONS: ANSWER QUESTION ONE (COMPULSORY) AND ANY
OTHER TWO QUESTIONS
QUESTION ONE (30 MARKS)
a) Find the Cartesian form of the polar equation r ?1?2sin? . (3 Marks)
b) Evaluate
0
1 cos 2
lim
x sin
x
? x x
?
(3 Marks)
c) Test for convergence or divergence of the series
? ? 1
1
1 3
!
n n
n n
? ?
?
?
?
(3 Marks)
d) Show that if 4 x and higher powers of x are neglected then
Hence estimate 0.1 e cos0.1 correct to four decimal places. (5 Marks)
e) Apply the integral test to the series 2
1
1
1 n n
?
? ? ? (3 Marks)
f) Given that 2 2 U ? ln x ? y , show that
2 2
2 2 0
U U
x y
? ?
? ?
? ?
(5 Marks)
g) Evaluate ? , ?
R
?? f x y dA for ? ? 2 2 f x, y ? x ? 2y ?1 an R ???x, y? 0 ? x ? 4,1? y ? 2? .
(5 Marks)
h) Find the y-coordinate of the centre of mass of a thin homogenous plate of density
3g/cm3 which is bounded by the lines y ? 2x , y = 0 and x = 1. (3 Marks)
SUPPLEMENTARY SMA 2200 Calculus III Page 2 of 3
Page 2 of 3
QUESTION TWO (20 MARKS)
a) (i). State the Rolle’s Theorem of differential calculus. (2 Marks)
(ii). Verify the MVT of differential calculus for the function ? ? 2 f x ? 2x ?7x ?10 on
the ?2,5? . (5 Marks)
b) Discuss the continuity of the function ? ? 2
2 3, 0 2
3, 2 4
x x
f x
x x
? ? ? ?
? ?
? ? ? ?
(5 Marks)
c) (i) Convert the polar equation r ? ?2?1? cos? ? to the Cartesian form and hence
sketch the graph. (8 Marks)
QUESTION THREE (20 MARKS)
a) (i). Evaluate
3
3 2
3 2
lim
n 4 3 1
n n
?? n n
?
? ?
4 Marks)
(ii). Determine the convergence or divergence of the series 3
1
2n
n n
?
? ?
.(4 Marks)
b) Find the radius of convergence of the power series
? ? ? ? 1 2
1
1 2 1
n n n
n
x
?
?
?
? ? ?
(6 Marks)
c) Prove that 2 3 8
tan 1 2 2
4 3
x x x x
?? ?
? ? ? ? ? ? ? ?
? ?
(6 Marks)
SUPPLEMENTARY SMA 2200 Calculus III Page 3 of 3
Page 3 of 3
QUESTION FOUR (20 MARKS)
(a) For ? ? 3 2 f x, y, z ? xy z ? 4x y , defined for x, y, z ? 0 , compute xy f and xyz f .
(5 Marks)
(b) Find the length of the arc of the curve
3
2 2
1
3
y ? x ? between 0 and 1.
(5 Marks)
(c) Prove that
1 2 1 1
sin sin cos sin n n n n
x dx x x x dx
n n
? ? ?
? ? ? ? ? , where n is a positive integer. Hence
evaluate dx. (10 Marks)
QUESTION FIVE (20 MARKS)
(a) Evaluate the double integral
? ? 2 2
R
?? x ? y dA
Where R is the region bounded by 2 y ? x , x ? 2 and y = 1. (8 Marks)
(b) Determine the volume generated by revolving the area bounded by the curve
2 y ? 4? x and the line y ? 2? x about the x – axis. (8 Marks)
(c) Evaluate
2x e dx
?
??
?
(4 Marks)

Sma 2200: Calculus Iii Question Paper

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