CONFIDENTIAL CS/OCT 2012/MAT235/241/243

UNIVERSITI TEKNOLOGI MARA FINAL EXAMINATION

COURSE

COURSE CODE

EXAMINATION

TIME

CALCULUS II FOR ENGINEERS / MATHEMATICS IIIA / MATHEMATICS III

MAT235/241/243

OCTOBER 2012

2 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 page © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL

CONFIDENTIAL 2 CS/OCT 2012/MAT235/241/243

QUESTION 1

a) Use the integration by parts to find f x4 In x dx.

A _ J _

b) If z = (2x - 3)2ey , x = and y = t + cost, evaluate — when t = 0. v ; t + 1 dt

c) Given x + 3 A Bx + C

+ (x - 1 ) ( X 2 + 1) x - 1 x2 + 1

i) Find the values of A, B and C.

f x + 3 ii) Hence, evaluate I dx.

J (x - 1 ) ( x 2 + 1)

(4 marks)

(5 marks)

(3 marks)

(3 marks)

QUESTION 2

a) Use differentials to approximate (2.01 )3 ^/15.98.

b) 2

Given a function f(x, y) = — y3 - 2xy - 12y + x2

(5 marks)

i) Find all the critical points of the function. (5/4 marks)

ii) Determine the relative maximum, relative minimum and saddle point(s), if any. (2Y2 marks)

© Hak Cipta Universiti Teknologi MARA CONFIDENTIAL

CONFIDENTIAL 3 CS/OCT 2012/MAT235/241/243

QUESTION 3

a) Evaluate — -dx J ( x 2 + 2 ) 3

0 (5 marks)

b) Find the general solution of the first order linear differential equation dy . 2

x ^ L + y = sin x. dx y

(5 marks)

c) Evaluate I , dx by using a trigonometric substitution x = 4 sec 9. J V x 2 - 16

(6 marks)

QUESTION 4

a) The growth of a certain bacteria is increasing at a rate proportional to the number present. If the number of bacteria is 100 at 1 p.m. and 156 at 2 p.m.,

i) find the expected number of bacteria at 3.30 p.m., (5 marks)

ii) at what time will the bacteria double the initial number? (3 marks)

d v dy iv b) Solve the second order differential equation — - + — — 2y = 5e + x.

dx2 d x

(8 marks)

END OF QUESTION PAPER

© Hak Cipta Universiti Teknologi MARA CONFIDENTIAL