THE INTERNATIONAL UNIVERSITY IU VIETNAM NATIONAL UNIVERSITY HCMC FINAL EXAMINATION Duration 120 minutes 26 June 2009 SUBJECT CALCULUS I Head of Dept of Mathematics Lecturers Prof DrSc Phan Quoc Khanh N V Thu N Dinh N N Hai J C Harris M D Thanh INSTRUCTIONS Answer ALL questions in Section A and any TWO questions in Section B Calculators and dictionaries allowed All other documents prohibited SECTION A Z A1 10 marks Evaluate the following integrals a x sin xdx b x 5 x 1 x 2 dx A2 4 marks If f0 is continuous on a b show that f x f0 x dx f b 2 f a 2 Z Z b a 2 A3 8 marks State whether the following statements are true or false If false give an explanation or a counterexample If true no explanation is required a If f is continuous and positive on a b thenR b b If f is continuous on a b andR b a f x dx 0 then f x 0 for all x on a b a f x dx 0 A4 10 marks Find the limit R x3 lim x 1 1 cos t2 dt x 1 Hint Do not try to evaluate the integral A5 10 marks Use Newton s Method to nd the root of the equation x3 3x 1 0 in the range 1 x 0 correct to 5 decimal places Show full details of your calculations A6 10 marks Determine the area of the region bounded by y 2x2 10 and y 4x 16 8 marks Find the volume of the solid generated by rotating about the A7 x axis the region under the curve y ex between x 0 and x 1 PLEASE TURN OVER SECTION B Choose 2 out of 3 questions B1 On a certain day in Ho Chi Minh City the temperature T in oC could be modeled by the curve T 0 0004t4 0 12t2 33 12 t 12 where t is the time in hours after noon On this day 12 t 12 nd a 10 marks the average temperature b 10 marks the maximum and minimum temperatures and the times at which they occurred B2 a In a certain factory the cost of producing the rst 10 units is 200 xed After that the marginal cost C0 x measured in dollars per unit at an output level of x units is C0 x 1000 x2 i 6 marks Find the total cost of producing the 11th to 100th units ii 4 marks Find the total cost of producing the rst 100 units b 10 marks An athlete starts to run a race The table below gives her speed during the rst three seconds of the race measured at half second intervals Es timate as accurately as possible the total distance that she traveled during these rst three seconds Explain clearly the method that you use time s speed m s 0 0 0 0 0 5 1 9 1 0 3 3 1 5 4 5 2 0 5 5 2 5 5 9 3 0 6 2 B3 a 10 marks The pressure P and volume V of a sample of gas may both vary with time However under certain conditions they always satisfy the equation P V C where C is a constant Suppose that at a certain instant the volume is 600 cm3 the pressure is 150 kPa and the pressure is increasing at a rate of 20 kPa min At what rate is the volume changing at this instant Is it increasing or decreasing b 10 marks A wooden children s toy is made by taking a solid right circular cone of height 10 cm and base radius 5 cm and drilling a hole of radius 1 cm down its central axis as shown in the gure Calculate the volume of the remaining solid END OF QUESTION PAPER