UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS WTW 220 - ANALISE / ANALYSIS EKSAMEN / EXAM 12 November 2012 TYD/TIME: 120 min PUNTE / MARKS: 40 Interne eksaminatore / Internal examiners: Prof I Broere, Dr AJ van Zyl Eksterne eksaminator / External examiner: Prof NF J van Rensburg VAN / SURNAME: VOORNAME / FIRST NAMES: STUDENTENOMMER / STUDENT NUMBER: FOONNO. GEDURENDE EKSAMENPERIODE / PHONE NO. DURING EXAM PERIOD: Totaal / Total 1. The paper consists of pages 1 to 7 (questions 1 to 5). Check whether your paper is complete. / Die vraestel bestaan uit bladsye 1 tot 7 (vrae 1 tot 5). Kontroleer of jou vraestel volledig is. 2. If you need more than the available space for an answer, use the facing page and indicate it clearly. / As jy meer as die beskikbare ruimte vir n antwoord nodig het, gebruik dan ook die teenblad en dui dit duidelik aan. 3. Only non-programmable calculators may be used. / Slegs nie-programmeerbare sakrekenaars mag gebruik word.
Question 1 Vraag 1 (a) Let S be a subset of the set R of real numbers. Define the following concepts: / Laat S n deelversameling van die versameling R van reële getalle wees. Definieer die volgende begrippe: (i) The set S is bounded above. / Die versameling S is van bo begrens. (ii) The number s is the supremum of S (written as sups) / Die getal s is die supremum van S (geskryf as sups). (b) Remember the Completeness Axiom which states that every bounded non-empty set of real numbers has a supremum. Now use it to prove the Archimedean Postulate which claims that the set N of natural numbers is not bounded above. / Onthou dat die Volledigheidsaksioma ons verseker dat daar vir elke begrensde nie-leë versameling reële getalle n supremum is. Gebruik nou hierdie feit om die Archimediese Postulaat te bewys wat sê dat die versameling N van natuurlike getalle nie van bo begrens is nie. 1
(c) Is the set { 4+n : n N} bounded above? Motivate your answer. / n 2 Is die versameling { 4+n : n N} van bo begrens? Motiveer jou antwoord. n 2 Question 2 Vraag 2 (a) Define what it means that the sequence (a n ) is convergent. / Definieer wat dit beteken dat die ry (a n ) konvergent is. (b) Use known results on the convergence of sequences to prove that the sequence (a n ) is a null sequence, with / Gebruik bekende resultate oor die konvergensie van rye om te bewys dat (a n ) n nulry is, met a n = 3n + 17n 5. n! (c) Define what it means to say that a sequence (a n ) is monotone / Definieer wat dit beteken om te sê dat die ry (a n ) monotoon is. (d) Give an example of a sequence which is monotone but not convergent / Gee n voorbeeld van n ry wat monotoon is maar nie konvergent is nie. 2
(e) Let (a n ) be a bounded sequence of real numbers and define, for every N N, the set S N = {a n : n > N}. Now prove the following part of the Bolzano-Weierstrass Theorem: If every S N has a maximum element, then (a n ) has a convergent subsequence. / Laat (a n ) n begrensde ry reële getalle wees en definieer, vir elke N N, die versameling S N = {a n : n > N}. Bewys nou die volgende deel van die Bolzano-Weierstrass Stelling: As elke S N n maksimale element het, dan het (a n ) n konvergente deelry. Question 3 Vraag 3 (a) Remember that every continuous function on an interval of the form [a, b] is bounded on [a, b]. Use this result to prove that every continuous function on [a, b] attains a minimum value somewhere on [a, b]. / Onthou dat elke kontinue funksie op n interval van die vorm [a, b] begrens is op [a, b]. Gebruik hierdie resultaat om te bewys dat elke kontinue funksie op [a, b] n minimumwaarde iewers op [a, b] bereik. [4] 3
(b) Now give an example of a function w : [0, 1] R which is not bounded on [0, 1] and which does not attain a minimum value somewhere on [0, 1]. / Gee nou n voorbeeld van n funksie w : [0, 1] R wat nie begrens op [0, 1] is nie en wat nie n minimumwaarde iewers op [0, 1] bereik nie. (c) Suppose that f is a continuous function on [a, b] and let K = f(x 1 ) + f(x 2 ) for some (fixed) values x 1 and x 2 with a < x 1 < x 2 < b. Use the Intermediate Value Property to prove that there exists a real number c in (a, b) such that f(c) = K. / Veronderstel f is n kontinue funksie op [a, b] 2 en laat K = f(x 1 ) + f(x 2 ) vir twee (vaste) waardes x 1 en x 2 met a < x 1 < x 2 < b. Gebruik die Tussenwaardestelling om te bewys dat daar n reële getal c in (a, b) is só dat f(c) = K. 2 4
Question 4 Vraag 4 (a) Use the concepts of partition P, lower sum L(P ) and upper sum U(P ) to define what it means for a bounded function f to be Riemann-integrable on the interval [a, b]. / Gebruik die begrippe van partisie P, laersom L(P ) en bosom U(P ) om te definieer wat dit beteken vir n begrensde funksie f om Riemann-integreerbaar op die interval [a, b] te wees. (b) Formulate Riemann s condition that relates Riemann integrability to the difference between upper and lower sums. / Formuleer die Riemannvereiste wat Riemann integreerbaarheid in verband bring met die verskil tussen bo- en laersomme. (c) Prove that if f is a bounded decreasing function on the interval [a, b], then f is Riemann integrable on [a, b]. / Bewys dat as f n begrensde dalende funksie is op die interval [a, b], dan is f Riemann integreerbaar op [a, b]. 5
(d) Recall the following property of Riemann sums: If f is Riemann integrable on [a, b] and (P n ) is any sequence of partitions with corresponding maximum subinterval lengths (h n ) and corresponding sequence of sample point sets (Q n ), then the sequence of Riemann sums (S(P n, Q n )) converges to the b a f(x)dx if (h n) converges to zero as n. / Onthou die volgende eienskap van Riemannsomme: As f Riemann integreerbaar is op [a, b] en (P n ) is enige ry partisies met ooreenstemmende maksimum deelintervallengtes (h n ) en ooreenstemmende ry versamelings van keusepunte (evaluasiepunte) (Q n ), dan sal die ry Riemannsomme konvergeer na b a f(x)dx as (h n) neig na nul soos n. Use this, or another method, to prove that if f(x) g(x) on [a, b], then / Gebruik hierdie, of n ander metode, om te bewys dat as f(x) g(x) op [a, b], dan sal b a f(x)dx b a g(x)dx. Question 5 Vraag 5 (a) Define the n th partial sum of a series and use this definition to state what it means for the series r=1 a r to be convergent. / Definieer die n de parsiële som van n reeks en gebruik die definisie om te beskryf wat dit beteken vir die reeks r=1 a r om te konvergeer. 6
(b) Prove the following part of the Second Comparison Test. Let r=1 a r and r=1 b r be positiveterm series such that / Bewys die volgende deel van die Tweede Vergelykingstoets. Laat r=1 a r en r=1 b r positiewe-term reekse wees waarvoor a n lim = L 0. n b n Suppose that r=1 b r converges. Prove that r=1 a r converges. / Veronderstel dat r=1 b r konvergeer. Bewys dat r=1 a r konvergeer. (c) Determine all values of the real numbers k and p for which Bepaal al die waardes van die reële getalle k en p waarvoor r=1 r=1 1 r p k is a convergent series. n konvergente reeks is. 1 r p k 7