CONFIDENTIAL CS/APR 2011/MAT235/241/243

UNIVERSITI TEKNOLOGI MARA FINAL EXAMINATION

COURSE

COURSE CODE

EXAMINATION

TIME

CALCULUS II FOR ENGINEERS / MATHEMATICS IIIA / MATHEMATICS III

MAT235/241/243

APRIL 2011

3 HOURS

INSTRUCTIONS TO CANDIDATES

1. This question paper consists of four (4) questions.

2. Answer ALL questions in the Answer Booklet. Start each answer on a new page.

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

4. 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 3 printed pages © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL

CONFIDENTIAL 2 CS/APR 2011/MAT235/241/243

QUESTION 1

6x + 2

(x + 3)(x-l) '

6x + 2

a) Express — —^ in the form of partial fractions.

r ox -t- z , Hence, solve -ax.

J ( x + 3 ) ( x - l ) 2

(6 marks)

•*• - .2 , / : . . , . „ . , 1 ..2 b) Given that f(x,y) = 3x +6x + xy + — y -6y, find the critical points and

determine the relative maximum, relative minimum, and saddle point(s) if any.

(9 marks)

QUESTION 2

CO

a) Find \xe~xdx using integration by parts. Hence, evaluate Jxe~xdx. 0

(5 marks)

b) Find f cos4(2x)sin3(2x) dx.

(5 marks)

f X

c) Find , dx by using trigonometric substitution x = 3 tan 6 . J V9 + x2

(6 marks)

© Hak Cipta Universiti Teknologi MARA CONFIDENTIAL

CONFIDENTIAL 3 CS/APR 2011/MAT235/241/243

QUESTION 3

1 a) Let f(x,y,z) = -zy2+sm(7rx). Use total differential to approximate the change in

f(x,y, z) from x = 0,y = 2,z = 2 to x = — ,y = l.9, z- 2.1. Give your answer correct to

4 decimal places.

/ j2 \ / o 2 \ /

b) Given that V = sin(2x + 3y). Prove that — - - 3 ^ - — = 3V. dy 9x

c) Solve the following first order linear equation: dy lnx

x— + y = dx x

(4 marks)

(5 marks)

(5 marks)

QUESTION 4

a) A heated object is allowed to cool in a room which has a constant temperature of 20 °C. It cools from 90°C to 55°C in 30 minutes. Find

i) the time it will take for the object to cool to 30 °C. ii) the temperature of the object after 8 minutes.

(7 marks)

T2 I

b) Solve the equation — — - 4 — + 3y = e dx dx

, 3 J

(8 marks)

END OF QUESTION PAPER

© Hak Cipta Universiti Teknologi MARA CONFIDENTIAL