FAKULTEIT INGENIEURSWESE FACULTY OF ENGINEERING Vakkursus / Subject course: Toegepaste Wiskunde B264 Applied Mathematics B264 Tweede Eksamen: Desember 2010 Second Examination: December 2010 Volpunte / Full marks: 75 Tydsduur / Duration: 2.5 h Tyd / Time: 9:00 Datum / Date: 3 Desember 2010 Kantoorgebruik / Office Use Eksaminatore / Examiners: H Coetzer, SJ van der Walt, BM Herbst Vraag/Question Punte/Marks Nasiener/Examiner Student / Student: Van in blokletters / Surname in block letters: Volle voorname / Full first names: Handtekening / Signature: Studentenommer / Student number: TOTAAL/TOTAL 1. Beantwoord al die vrae. Answer all the questions. Instruksies / Instructions 2. Sakrekenaars soos voorgeskryf vir die eerste twee jaar van die BIng mag gebruik word. Calculators as prescribed for the first two years of the BEng may be used. 3. Eksamenreëls en -voorskrifte verskyn op die agterblad. Examination rules and instructions appear on the rear cover.
Afdeling 1 (20 punte) VRAAG 1 (10 punte) Beskou die golfvergelyking QUESTION 1 (10 marks) Consider the wave equation 2 u t 2 = 2 u a2, x [0, L], t [0, ) x2 wat die vibrasie van n snaar modelleer. Sien die figure hieronder: Snaarsegment s / String segment s which models the vibration of a string. See the figures below: Vergroting van s / Enlargement of s (a) Herlei die golfvergelyking vanuit eerste beginsels. Alle aannames moet duidelik gestel word en simbole moet gedefinieer word. Gee onder meer die fisiese interpretasie van die konstante a. (b) Verduidelik waarom die volgende twee randvoorwaardes van toepassing is op hierdie probleem, u(0, t) = 0 en/and u(l, t) = 0, t [0, ). (a) Derive the wave equation from first principles. All assumptions have to be stated clearly and symbols must be defined. Give, amongst others, the physical interpretation of the constant a. (b) Explain why the following two boundary conditions are applicable to this problem, (c) Interpreteer die volgende twee aanvangsvoorwaardes deur te verwys na die fisiese probleem, u(x, 0) = f(x) en/and u (x, 0) = g(x), t (c) Interpret the following two initial conditions by referring to the physical problem, x [0, L]. 1
VRAAG 2 (10 punte) Beskou Laplace se vergelyking, QUESTION 2 (10 marks) Consider Laplace s equation, 2 u x + 2 u = 0, x [0, 1], y [0, 1], (1) 2 y2 onderhewig aan vier randvoorwaardes, soos aangedui in die figuur hieronder. subject to four boundary conditions, as indicated in the figure below. (a) Toon volledig aan dat die produkoplossings wat Laplace se vergelyking (vergelyking (1)), sowel as die volgende twee randvoorwaardes, (a) Show in detail that the product solutions that satisfy Laplace s equation (equation (1)), as well as the following two boundary conditions, u(0, y) = 0 en/and u(1, y) = 0, y [0, 1], (2) bevredig, gegee word deur are given by u n (x, y) = {a n cosh(nπy) + b n sinh(nπy)} sin(nπx), n = 1, 2,... (b) Vind dan die spesifieke oplossing u(x, y) wat, afgesien van Laplace se vergelyking (vergelyking (1)) en die randvoorwaardes (voorwaardes (2)) in (a), ook die volgende twee randvoorwaardes bevredig, (b) Then find the specific solution u(x, y) that, in addition to Laplace s equation (equation (1)) and the boundary conditions (conditions (2)) in (a), also satisfies the following two boundary conditions, u(x, 0) = 0 en/and u(x, 1) = sin(πx), x [0, 1]. 2
Afdeling 2 / Section 2 (55) VRAAG 1 (10) Beskou die volgende lineêre vergelykings: QUESTION 1 (10) Consider the following linear equations: 2x 1 + 4x 2 3x 3 = 1 3x 1 2x 2 + x 3 = 17 4x 2 + 3x 3 = 9 a. Skryf die stelsel in die vorm Ax = b. b. Bereken die LU-faktorisering van A. c. Gebruik hierdie faktorisering om x te vind. a. Write the system in the form Ax = b. b. Compute the LU-factorisation of A. c. Use this factorisation to nd x. VRAAG 2 (15) Beskou die volgende stroombaan. QUESTION 2 (15) Consider the following circuit. 25V 1Ω 1Ω 2Ω I 1 1Ω 3Ω I 2 15V 3Ω 5Ω 15V I 3 3Ω 5Ω 7Ω 4Ω I 4 5V 7Ω 5V 7Ω a. Onthou Kircho se wet (spannings in 'n lus sommeer na nul) en Ohm se wet (V = IR) en skryf die lusvergelykings vir strome I 1, I 2, I 3 en I 4 neer. b. Skryf die vergelykings in die vorm Ax = b and los op vir I 1, I 2, I 3 en I 4 (gebruik 'n metode van jou keuse). c. Bevestig nou jou antwoord deur Ax te bereken. a. Recall Kircho's law (voltages around a loop sum to zero) and Ohm's law (V = IR) and write down the loop equations for currents I 1, I 2, I 3 and I 3. b. Write the equations in the form Ax = b and solve for I 1, I 2, I 3 and I 3 (use any method your prefer). c. Now, verify your answer by computing Ax. 3
Beskou die transformasie VRAAG 3 (5) QUESTION 3 (5) Consider the transformation T : x 1 x 2 x 3 ( ) x 1 + x 2. x 1 + x 2 + x 3 a. Wys dat hierdie transformasie lineêr is, en vind dan die ooreenstemmende standaardmatriks. b. Vind 'n intreevektor wat afbeeld op [6, 7] T. Bestaan daar meer as een so 'n vektor? a. Show that this transformation is linear; then nd its standard matrix. b. Find an input vector of which the image is [6, 7] T. Does there exist more than one such a vector? Beskou die matriks VRAAG 4 (5) Consider the matrix QUESTION 4 (5) A = 1 2 3 4 0 1 2 3 0 0 1 2. a. Bereken die LU-dekomposisie van A. b. Vind 'n basis vir die kolomruimte van A. c. Vind 'n basis vir die nulruimte van A. a. Computer the LU-factorisation of A. b. Find a basis for the column space of A. c. Find a basis for the null space of A. Beskou die matrikse VRAAG 5 (10) Consider the matrices QUESTION 5 (10) A = 2 0 2 1 1 1 0 0 3, C = ( 3 2 1 8 ) ( 4 1, D = 0 1 ). a. Bereken die determinant van A. b. Wys dat det(cd) = det(c) det(d) (dit geld vir enige twee vierkantige matrikse). c. Maak gebruik van (b) en wys dat det(a 1 ) = 1/ det(a). Wenk: AA 1 = I. a. Compute the determinant of A. b. Show that det(cd) = det(c) det(d) (this holds for any two square matrices). c. Using (b), show that det(a 1 ) = 1/ det(a). Hint: AA 1 = I. 4
VRAAG 6 (10) QUESTION 6 (10) a. Toon aan die hand van 'n skets dat die kleinstekwadraat-oplossing van Ax = b, waar A = [a 1, a 2 ] T, met a 1 en a 2 lineêre onafhanklike kolomvektore in R 3 en b R 3, b / KolA, gevind kan word deur die volgende vergelyking op te los: a. Show, using a sketch, that the least squares approximation of Ax = b, where A = [a 1, a 2 ] T, with a 1 and a 2 linearly independent column vectors in R 3 and b R 3, b / ColA, can be found by solving the following equation: A T Aˆx = A T b. b. Vind die vergelyking van die reguitlyn wat die volgende punte die beste pas in terme van die kleinste-kwadraatfout. b. Find the line that best approximates the following points in a least squares sense. (0, 2), (2, 12), (3, 16), (3, 18) 5
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