2012 - 8803A SEMESTER 2 20:2 THE UNIVERSITY OF SYDNEY...

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Unformatted text preview: 8803A SEMESTER 2 20:2 THE UNIVERSITY OF SYDNEY SCHOOL OF MATHEMATICS AND STATISTICS MATH1003 INTEGRAL CALCULUS ANn MODELLING November 2012 LECTURERS: S. Santra, N. Saunders, M. Wechselberger, Z. Zhang TIME ALLOWED: One and a half hours Famiiy Name: ............................................................ Other Names: This examination has two sections: Multiple Choice and Extended Answer. The Multiple Choice Section is worth 50% of the total examination; there are 25 questions; the questions are of equal value; all questions may be attempted. Answers to the Multiple Choice questions must be entered on the Multiple Choice Answer Sheet. The Extended Answer Section is worth 50% of the total examination; there are 3 questions; the questions are of equal value; all questions may be attempted; working must be shown. There is a table of integrals after the last question in this booklet. Approved non-programmable calculators may be used. THE QUESTION PAPER MUST NOT BE REMOVED FROM THE EXAMINATION ROOM. PAGE 1 OF 30 MARKER’S use ONLY 8003A SEMESTER 2 2012 PAGE 14 0F 30 Extended Answer Section Answer these questions in the answer 500k(3) provided. Ask for extra books if you need them. 1. (a) (2’) Determine the general solution of the differential equation 012?; dy mm 4% m 21 = . 053:3 + dm 3; U {2 marks] 8003A SEMESTER 2 2012 PAGE 15 OF 30 (ii) Find the particular solution which satisfies the initial conditions 5 dy m 5 [2 marks} 8003A SEMESTER 2 2012 PAGE 16 OF 30 (iii) What is the long—time behaviour of this particular solution, 1.8. What value does Mt) take in the limit as t —+ 00? [1 mark] 8003A SEMESTER 2 2012 PAGE 17 OF 30 (b) Determine the general solution of the following system of equations M : w2xm5y, 823+2y. [3 marks] 8003A SEMESTER 2 2012 PAGE :8 OF 30 (c) Fermentation breaks down sugar according to the law d2: Where m is the amount of sugar at time t, f is the amount of yeast that changes 10 1 _ _ . and a 2 — 13 a. proportzonahty constant that 1 + t2 5 describes the rate of fermentation. with time according to f (t) = (2') Find the amount of sugar as a, function of time. [3 marks] 8003A SEMESTER ‘2 2012 PAGE 19 OF 30 (ii) Suppose that there were initially 80 units of sugar present. What; is the precise iimiting amount of sugar as t -> 00? [1 mark] 8003A SEMESTER 2 2012 PAGE 20 OF 30 2. (a) Consider the definite integra} 64 f 232/3 dzc. 0 (2') What is the minimum number N of equaliy spaced subintervals needed such that the maximum error of the upper or lower Riemann sum estimates for this definite integral is less than 10? [2 marks] 8003A SEMESTER 2 2012 PAGE 21 OF 30 (ii) Write down expressions for the upper Riemann sum UN and lower Riemann sums L N for this definite integrai using N = 64 equally spaced subintervals. [2 marks] 8003A SEMESTER 2 2012 PAGE 22 OF 30 64 (121') Find the precise value of the integrai f 531/3 dm. 0 Combine this answer with resuits from the preview; part (ii) to Show that 64 192 < Elsi/3 < 196. Show all necessary working. [2 marks] Ic=1 8003A SEMESTER 2 2012 PAGE 23 OF 30 (1)) Use integration by substitution to find the precise value of the definite integral 4 mw2 md. fgx2~4m+5 a; [3 marks] 8003A SEMESTER 2 2012 PAGE 24 OF 30 (C) Use integration by parts to find the indefinite integral Ik m/tk(1nt)dt, keN [3 marks] 8003A SEMESTER 2 2012 PAGE 25 OF 30 71' 3. (a) Compute the area enclosed by the curve y 2 Siam, the lines y m :13, :1: "m; —§ and [3 marks] 8003A SEMESTER 2 2012 PAGE 26 or 30 (b) Consider the solid generated by rotating this area in (at) around sac—axis. (2) Using the disk method (Le. slicing the solid perpendicuieriy to xii—axis), set up the expression which represents the volume of this soiid and compute the volume. You are required to write down the expression for the volume of a general slice. [3 marks] 8003A SEMESTER 2 2812 PAGE 27 OF 30 (ii) Using the shell method (is. slicing the solié using area slices parallel to a:— axis)7 set up the expression which represents the volume of this solid. It is NOT required to compute this expression. You are required to write down the expression for the volume of a. general slice. [2 marks} 8003A SEMESTER 2 2012 PAGE 28 OF 30 (0) Consider In : /(sin'1 :13)“ dm. For n 2 2, prove the foliowing reductiofi formula. In m Main“: 3:)” + m/ 1 —— 51:2(sixf31 :z:)""‘l -»~ n02 w 1)In_2. [4 marks] 8003A SEMESTER 2 2012 PAGE 29 OF 30 8003A SEMESTER 2 2012 PAGE; 38 OF 38 Tabie of Standard Integrals $37,4- E n+1 xndx: +0 (7175—1) 9./secgasdm=tana:+0 d3: m mm + C 10. COSGC2$d$ = —— cotm + C" 23 3. 6md$mew+o 11. secmdlenisecx+tanzcj +0 sinxda: = m (2031: + C 12. cosecwda: 21n|cosecaz “ (:0th + C coshxdar: = sinhm + C 6. tanmda: : min|c0823| + C 14. t“ \\\\\\\ cosmdxmsinx+0 13. /sinhxdxmcoshx+0 7. cotxdx21n|sinx§+0 15. tanhxd$=lncoshx+0 dx 1 "1 cc d2: , “1(53) . w— —~ 16. mmsm ——+C :c<a 8 [694-332 atan (a)+0 x/a2—x2 a (I I ) 17. Lusinh*é E) +Cflln(m~é—Va¢2+ag) +C” V562+a2 a 18. "flww=coshmi(£)+0m1n<x+xfcc2—a2)+0’ (33>a) Vm2ma2 a +0 ($75dza) End of Extended Answer Section THIS ES THE LAST PAGE OF THE QUESTION PAPER. ...
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