UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA FAKULTEIT NATUUR- EN LANDBOUWETENSKAPPE / FACULTY OF NATURAL AND AGRICULTURAL SCIENCES DEPARTEMENT WISKUNDE EN TOEGEPASTE WISKUNDE DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS WTW 263 NUMERIESE METODES / NUMERICAL METHODS VAN/SURNAME: November 2003 TYD/TIME: 90 min PUNTE/MARKS: 40 VOORNAME/FIRST NAMES: STUDENTENOMMER/STUDENT NUMBER: HANDTEKENING/SIGNATURE: Eksterne eksaminator / External examiner: Me/Ms A Labuschagne Interne eksaminatore / Internal examiners : Me/Ms A Griffiths Mev/Mrs L Mostert PUNTE MARKS LEES DIE VOLGENDE INSTRUK- SIES 1. Die vraestel bestaan uit bladsye 1 tot 9. (vrae 1 tot 8 ). Kontroleer of jou vraestel volledig is. 2. Gebruik 4 desimale syfers tensy anders gemeld in n vraag. 3. Doen alle krapwerk op die teenblad. Dit word nie nagesien nie. 4. As jy meer as die beskikbare ruimte vir n antwoord nodig het, gebruik dan ook die teenblad en dui dit asseblief duidelik aan. 5. Geen potloodwerk of enigiets wat in rooi ink gedoen is, word nagesien nie. 6. Slegs nie-programmeerbare sakrekenaars mag gebruik word. READ THE FOLLOWING INSTRUC- TIONS 1. The paper consists of pages 1 to 9. (questions 1 to 8). Check whether your paper is complete. 2. Use 4 decimal digits except stated different in a question. 3. Do all calculations on the facing page. It will not be marked. 4. If you need more than the available space for an answer, use the facing page and please indicate it clearly. 5. No pencil work or any work in red ink will be marked. 6. Only non-programmable pocket calculators may be used. Outeursreg voorbehou Copyright reserved 1
VRAAG 1 QUESTION 1 n Buigbare metaalstrook wat 10 m lank is, word in n boogvorm gebuig deur die twee punte met n 8 m kabel aan mekaar vas te maak. Die radius r van die sirkelboog bevredig die volgende vergelyking: r sin ( ) 5 4 = 0. r A flexible strip of metal 10 m long is bent into the shape of a circular arc by the ends being secured together by means of a cable that is 8 m long. The radius r of the circular arc satisfies the following equation: Gebruik die Newton-Raphson metode met r 0 = 4 en bepaal die eerste twee iterasies r 1 en r 2. Use the Newton-Raphson method with r 0 = 4 and find the first two iterations r 1 and r 2. (5) 1
VRAAG 2 QUESTION 2 Beskou die stelsel lineêre vergelykings Consider the system of linear equations 7x + y z = 8 2x + y + 9z = 12 3x 7y + 2z = 4 2.1 Gee Gauss-Seidel iterasie formules vir hierdie stelsel sodat konvergensie van die ry iterasies na die oplossing vooraf verseker is. 2.1 Give Gauss-Seidel iteration formulae for this system for which convergence of the sequence of iterations will be assured beforehand. x k+1 = y k+1 = z k+1 = (1) 2.2 Begin met P 0 = (0, 0, 0) en los die stelsel op met die Gauss-Seidel metode. Itereer totdat P k+1 P k 1 < 0.1. 2.2 Start with P 0 = (0, 0, 0) and solve the system with the Gauss-Seidel method. Iterate until P k+1 P k 1 < 0.1. k x k y k z k P k+1 P k 1 (4) 2
VRAAG 3 QUESTION 3 Beskou die nie-lineêre stelsel vergelykings Consider the non linear system of equations 2xy 3 = 0 x 2 y 2 = 0. Neem (p 0, q 0 ) = (1.5, 0.9) en bepaal die eerste iterasie (p 1, q 1 ) met Newton se metode. Toon al u berekeninge. Take (p 0, q 0 ) = (1.5, 0.9) and find the first iteration (p 1, q 1 ) with Newton s method. Show all your calculations. (5) 3
VRAAG 4 QUESTION 4 Die traagheidsmoment I van n drielem skroef van n skip, met afmetings soos in die skets, word gegee deur The moment of inertia I of a threebladed ship s propeller whose dimensions are shown in the figure, is given by I = 3(9496)π 2(9.8) + 3(9496) 9.8 4.5 1 r 2 Adr. waar A die deursnitoppervlakte van die skroef op n afstand r vanaf die middelpunt van die skroef is. Gebruik die trapesiumreël en die data in die tabel en bepaal n benadering vir I. Toon alle berekeninge. where A is the area of the cross-section of the propeller at a distance r from the center of the hub. Use the trapezoidal rule and the data in the table to find an approximation for I. Show all calculations. r m 1 1.5 2 2.5 3 3.5 4 4.5 A m 2 0.3 0.5 0.62 0.7 0.6 0.5 0.27 0 (4) 4
VRAAG 5 QUESTION 5 Bepaal n geskikte intervalwydte h sodanig dat die saamgestelde Simpson reël gebruik kan word om 1 3 xe x dx met n akkuraatheid van 10 4 te bepaal. Die fout vir die saamgestelde Simpsonreël word gegee deur Determine a suitable interval width h so that the composite Simpson rule can be used to find 1 3 xe x dx with an accuracy of 10 4. The error for the composite Simpson rule is given by E s (f, h) = (b a)f 4 (c)h 4 180 (4) 5
VRAAG 6 QUESTION 6 Beskou die beginwaardeprobleem Consider the initial value problem dy dt = t2 y, y(0) = 3 Gebruik die Runge-Kutta vierde orde metode met staplengte h = 0.4 en bepaal n benadering vir y(0.4). Wys al jou berekeninge. Use the Runge-Kutta fourth order method with h = 0.4 and find an approximation for y(0.4). Show all your calculations. (4) 6
VRAAG 7 QUESTION 7 Beskou die beginwaardeprobleem Consider the initial value problem x + x + 4x = 10t, x(0) = 0, x (0) = 0 7.1 Herskryf die probleem as n stelsel van eerste-orde differensiaalvergelykings met die toepaslike beginwaardes. 7.1 Rewrite the problem as a system of first order differential equations with the appropriate initial conditions. (2) 7.2 Gebruik die metode van Heun om n benadering vir x(0.4) te bereken. Gebruik twee tydstappe. 7.2 Use Heun s method to compute an approximation for x(0.4). Use two time steps. (4) 7
VRAAG 8 QUESTION 8 Beskou die kromme Consider the curve y = Cxe Dx + K 8.1 Toon aan dat die kromme gelineariseer kan word tot Y = AX + B deur die volgende keuse 8.1 Show that the curve can be linearized as Y = AX + B by choosing ( ) y K Y = ln x, X = x, A = D, B = ln C (2) 8.2 Die volgende data gee rentekoerse vir verskillende termyne. Pas die kromme in 8.1 deur hierdie data. Neem K = 8. 8.2 The following data gives interest rates for different terms. Fit the curve in 8.1 to this data. Let K = 8. Termyn (x jaar) r% Term (x years) (y) 1 10 2 12.5 3 13.5 4 12.75 5 11.5 6 11.3 nog spasie op bl 9 more space on page 9 8
(5) 8.3 Vind die rentekoers vir n termyn van 10 jaar. 8.3 Find the interest rate for a term of 10 years. (1) 9