UNIVERSITI TEKNOLOGI PETRONAS FINAL EXAMINATION MAY 2012 SEMESTER COURSE DATE TIME FAMOO3S . CALCULUS 5. 30 PM (3 hours) 01’t SEPTEMBER 2012 (SATURDAY) 2. 30 PM – INSTRUCTIONS TO CANDIDATES 1. Answer ALL questions in the Answer Booklet. Begin EACH answer on a new page. lndicate clearly answers that are cancelled, if any. Where applicable, show clearly steps taken in arriving at the solutions and indicate ALL assumptions, if any. Do not open this Question Booklet until instructed. 2. 3. 4. 5. Note : There are SEVEN (7) pages in this Question Booklet including the cover page.

Universiti Teknologi PETRONAS FAMOO3s 1. a. Evaluate the following limit. €x -l,. …….. lllll ‘r-+0 x2 x [3 marks] b. The highway department is asked to construct a road between point A to point B. Point A lies on an abandoned road that runs on the west of point P. Point B is 3km north from the abandoned road and is 5km east of A as shown in FIGURE 01(b). The engineering division proposes that the road to be constructed by restoring a section of the old road from point A to some point P and constructing a new road from point P to point B.

Given the cost of restoring the old road is RM 2,000,000 per km and the cost of a new road is RM 4,000,000 per km. What is the length of the old road to be restored to minimize the cost of the project? 5km FrcuRE 01(b) [8 marks] FAMOO3S c. Find # in the followings: (? cos e’) (“t'”‘) [5 marks] ytanx- lt x2 :3xy+l [4 marks] 3 a. ii. 4 FAMOO3s 3. Find the equation of the tangent line to the curve x2+4*y+y2=-2 ? t the point (3,-1). [4 marks] b. A canonical paper cup of 8cm diameter and 6cm height is full of water.

The cup springs a leak at the bottom and losses water at the rate of 2cm3 per minute. How fast is the water dropping when the water level is exactly 3cm deep? Evaluate the series [4 marks] d. Determine whether the following series converges or diverges. Justify your answer. il iii, ? k:7 I 5 FAMOO3S 4. a. Find the radius and interval of convergence of the series n=ll s lt-u’ 3? (x-l)’ ) Ll——-. ——,-|. n’ ) [8 marks] b. Find the volume of the solid generated by revolving the region bounded y y=Ji and y: *2 for 0 ( ? ;lt; l aboutx = -). [6 marks] c. Findarclengthof thecurve ! =3×312-l betweeri x:0 and x:1. [6 marks] 6 FAMOO3s 5. Given that (t ^=(? –:) u=l [-, o o 4 r il [2 marks] Determine A B. b. Find the value(s) of x that satisfy the equation -0. [4 marks] Find the inverse of the matrix lt -8 5l .? =l-q 5 -3 l. l_ ‘ -r ‘J [7 marks] Solve the following system of equations using any suitable technique. 2x-y+32=12 )c+ y-z =-3 x +2y -32 = -I0 [7 marks] -END OF PAPER- 7