VAN / SURNAME: VOORNAME / FIRST NAMES: STUDENTENOMMER / STUDENT NUMBER: HANDTEKENING / SIGNATURE: SEL NR / CELL NO:

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1 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 ANALISE / ANALYSIS EXAM / EKSAMEN 11 NOVEMBER 2013 OM / AT 16:00 TYD / TIME: 120 min PUNTE / MARKS: 42 VAN / SURNAME: VOORNAME / FIRST NAMES: STUDENTENOMMER / STUDENT NUMBER: HANDTEKENING / SIGNATURE: SEL NR / CELL NO: Eksterne eksaminator / External examiner: Prof NF J van Rensburg Interne eksaminatore / Internal examiners: Prof. I Broere, Prof. M Sango, Mr. WS Lee, Dr. AJ van Zyl PUNTE / MARKS V1/Q1 V2/Q2 V3/Q3 V4/Q4 V5/Q5 TOTAAL / Afdeling A / Afdeling B TOTAL Punt behaal / Section A Section B Mark obtained MAKS / MAX 11 MAKS / MAX INSTRUKSIES 1. Hierdie vraestel bestaan uit hierdie voorblad en nog 7 bladsye wat vrae in TWEE AFDEL- INGS bevat. Kontroleer of jou vraestel volledig is. 2. Geen elektroniese toerusting (bv. sakrekenaar, ipad, ens.) mag gebruik word nie. 3. Doen alle rofwerk 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 enige iets wat in rooi ink gedoen is, word nagesien nie. 6. As jy korrigeerink (Tipp-Ex of soortgelyk) gebruik, verbeur jy die reg om werk wat nagesien is te bevraagteken of te beweer dat werk nie nagesien is nie. Outeursreg voorbehou INSTRUCTIONS 1. This paper consists of this cover page and 7 more pages containing questions in TWO SEC- TIONS. Check whether your paper is complete. 2. No electronic equipment (e.g. calculator, ipad, etc.) may be used. 3. Do all scribbling 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. If you use correcting fluid (Tipp-Ex or similar), you lose the right to question the marking or claim that work has not been marked. Copyright reserved

2 1 SECTION A / AFDELING A: Use a soft pencil for filling in the optic reader form. Fill in your personal information in pencil on the optic reader form on Side 1. Fill in your student number from top to bottom and then code it. If you make a mistake when coding your student number, we will not be able to take your mark into account. Hint: First circle your answers on this question paper, then transfer your answers to the optic reader form when you are certain of your choices. You may not erase wrong answers on the optical reader form. For each question there is only one correct answer. Gebruik n sagte potlood om die optiese leservorm in te vul. Vul jou persoonlike inligting in potlood in op die optiese leservorm op Kant 1. Vul jou studentenommer van bo tot onder in en kodeer dit dan. As jy n fout maak met die kodering van jou studentenommer, sal ons nie jou punt in berekening kan bring nie. Wenk: Omkring eers jou antwoorde op hierdie vraestel en dra dan die antwoorde oor na die optiese leservorm as jy seker is van jou keuses. Jy mag nie verkeerde antwoorde uitvee op die optiese leservorm nie. Vir elke vraag is daar net een korrekte keuse. 1. [1 mark] Read the following statements carefully to determine which one is not a property that holds for all real numbers a, b and c. (a) a + (b + c) = (a + b) + c. (b) There exists a unique real number 0 such that a+0 = a. (c) a (b + c) = a b + a c (d) a b or b a (e) If a b then a c b c. 1. [1 punt] Lees die volgende bewerings versigtig en bepaal watter een nie n eienskap is wat geld vir alle reële getalle a, b en c nie. (a) a + (b + c) = (a + b) + c. (b) Daar n unieke reële getal 0 só dat a + 0 = a. (c) a (b + c) = a b + a c (d) a b of b a (e) As a b dan is a c b c. 2. [2 marks] Consider the set A = { m n : m, n N} of positive rational numbers. Which of the following statements are true? 1. The set A has an upper bound. 2. The set A has a least upper bound. 3. For every ɛ R, ɛ > 0 there is a number a A such that a < ɛ. 2. [2 punte] Beskou die versameling A = { m n : m, n N} van positiewe rasionale getalle. Watter van die volgende uitsprake is waar? 1. Die versameling A het n bogrens. 2. Die versameling A het n kleinste bogrens. 3. Vir elke ɛ R, ɛ > 0 is daar n getal a A só dat a < ɛ. (a) Only 1. / Slegs 1. (b) Each of 1. and 2. / Elk van 1. en 2. (c) Only 3. / Slegs 3. (d) Each of 1., 2. and 3. / Elk van en 3. (e) None of 1., 2. and 3. / Nie een van 1., 2. en 3. nie. 3. [2 marks] Which one of the following is the definition of the sequence (a n ) tends to infinity as n tends to infinity? (a) for every positive real number K and integer N we have n > N = a n > K. (b) for every positive real number K there exists an integer N such that n > N = a n > K. (c) for every integer N there exists a positive real number K such that n > N = a n > K. (d) for every positive real number K there exists an integer N such that a n > K = n > N. (e) for every positive real number K there exists a δ > 0 such that 0 < n < δ = a n > K. 3. [2 punte] Watter een van die volgende is die definisie van die ry (a n ) neig na oneindig as n neig na oneindig? (a) vir elke positiewe reële getal K en heelgetal N geld dat n > N = a n > K. (b) vir elke positiewe reële getal K bestaan daar n heelgetal N só dat n > N = a n > K. (c) vir elke heelgetal N bestaan daar n positiewe reële getal K só dat n > N = a n > K. (d) vir elke positiewe reële getal K bestaan daar n heelgetal N só dat a n > K = n > N. (e) vir elke positiewe reële getal K bestaan daar n δ > 0 só dat 0 < n < δ = a n > K.

3 2 4. [1 mark] Which one of the following sequences is not a null sequence? 4. [1 punt] Watter een van die volgende rye is nie n nulry nie? (a) ( 1 n ) (b) ((n 3 2 n ) (c) ( 17n n! ) (d) ( 2n n 2 ) (e) (( 0.19) n ) 5. [2 marks] Consider the following statements about a continuous function f : A B, where A is a bounded interval. I. The inverse f 1 of f is continuous. II. If f is strictly increasing, then the inverse f 1 of f is also strictly increasing. III. B is necessarily a bounded interval. Which of these statements are true? 5. [2 punte] Beskou die volgende bewerings oor n kontinue funksie f : A B, waar A n begrensde interval is. I. Die inverse van f 1 van f is kontinu. II. As f streng stygend is, dan is die inverse f 1 van f ook streng stygend. III. B is noodwendig n begrensde interval. Watter van hierdie bewerings is waar? (a) None of (I), (II) or (III). / Nie een van (I), (II) of (III) nie. (b) Only (I). / Slegs (I). (c) Only (I) and (II). / Slegs (I) en (II). (d) Only (I) and (III). / Slegs (I) en (III). (e) (I), (II) and (III). / (I), (II) en (III). 6. [2 marks] Let L and U have their usual meaning in integration theory, so that L is the supremum of the set of lower sums, and U the infimum of the set of upper sums. Consider the following functions mapping the interval { [0, 1] to R: 1 if x is rational f(x) = 0 if x is irrational g(x) = sin( x { 2 ) 0 if 0 x 1 h(x) = 2 x + 1 if 1 2 < x 1 For which of these functions is it true that L = U? (a) Only f. (b) Only g. (c) Only h. (d) f and g. (e) g and h. 6. [2 punte] Veronderstel L en U het hul gewone betekenisse in integrasieteorie: naamlik L is die supremum van die versameling laersomme, en U is die infimum van die versameling bosomme. Beskou die volgende funksies { van die interval [0, 1] na R: 1 as x rasionaal is f(x) = 0 as x irrasionaal is g(x) = sin( x { 2 ) 0 as 0 x 1 h(x) = 2 x + 1 as 1 2 < x 1 Vir watter van hierdie funksies is dit waar dat L = U? (a) Slegs f. (b) Slegs g. (c) Slegs h. (d) f en g. (e) g en h. 7. [2 marks] Consider the following three series: 7. [2 punte] Beskou die volgende drie reekse: Which of these three series is (or are) convergent? I. r=1 1 2 r II. r=1 1 r III. r=1 1 r 2 Watter van hierdie reekse is konvergent? (a) Only (I). / Slegs (I). (b) Only (II). / Slegs (II). (c) Only (III). / Slegs (III). (d) (I) and (II). / (I) en (II). (e) (I), (II) and (III). / (I), (II) en (III).

4 3 SECTION B / AFDELING B: QUESTION 1 / VRAAG 1 (a) Define what it means to say that the number l is the infimum of the non-empty bounded set E. / Definieer die betekenis van die uitspraak die getal l is die infimum van die nie-leë begrensde versameling E. [1] (b) Find the infimum of the set / Vind die infimum van die versameling E = { 2 n2 n 2 : n N}, and motivate briefly why your value for the infimum satisfies the properties stated in your answer to part (a). / en motiveer kortliks hoekom jou waarde vir die infimum die eienskappe, gelys in jou antwoord tot deel (a), bevredig. [2] The goal of parts (c) and (d) of this question is to prove the density property of the set of rational numbers. / Die doel van dele (c) en (d) van hierdie vraag is om die digtheidseienskap van die versameling rasionale getalle te bewys. (c) Let x and y be any two real numbers with x < y. Use the fact that N is not bounded above, to show that there is a natural number n such that 1 n < y x. / Laat x en y enige twee reële getalle wees met x < y. Gebruik die feit, dat N nie van bo begrens is nie, om te bewys daar n natuurlike getal n is só dat 1 n < y x. [2] (d) Let m be the smallest integer which has the property that x < m n. Show the steps that are needed to deduce that x < m n < y. (Hint: m = m ) / Laat m die kleinste heelgetal wees met die eienskap dat x < m n. Toon die stappe wat nodig is om af te lei dat x < m n < y. (Wenk: m = m ) [2]

5 4 QUESTION 2 / VRAAG 2 (a) Define what is means to say that a sequence (a n ) is convergent. / Definieer die betekenis van die frase die ry (a n ) is konvergent. [1] (b) Remember the Bolzano-Weierstrass Theorem (BWT) which says that every bounded sequence has a convergent subsequence (and the fact that its proof is given by constructing a monotone subsequence). For (ii) and (iii) below, let a n = 3+( 1) n and b n = ( 1) n 5. / Onthou die Bolzano-Weierstrass Stelling (BWS) wat sê dat elke begrensde ry n konvergente deelry het (en die feit dat die bewys daarvan gegee is deur n monotone deelry te konstrueer). Vir (ii) en (iii) hieronder, laat a n = 3 + ( 1) n en b n = ( 1) n 5. (i) Define what it means to say that a sequence (c n ) is bounded. / Definieer die betekenis van die frase die ry (c n ) is begrens. [1] (ii) Show that (a n + b n ) (with a n and b n as given above) is a bounded sequence. / Bewys dat (a n + b n ) (met a n en b n soos hierbo gegee) n begrensde ry is. [2] (iii) Since (a n +b n ) is bounded, it has a convergent subsequence (by the BWT). Find a convergent subsequence of (a n +b n ). / Die ry (a n +b n ) het, omdat dit begrens is, n konvergente deelry (uit die BWS). Vind n konvergente deelry van (a n + b n ). [2] (iv) Is the sequence (a n + b n ) also convergent? Motivate your answer. / Is die ry (a n + b n ) ook konvergent? Motiveer jou antwoord. [1]

6 5 QUESTION 3 / VRAAG 3 (a) Define what it means to say that a function f is right-continuous at the point a. / Definieer die betekenis van die frase n funksie f is regskontinu by die punt a. [1] (b) Let f : [a, b] R be right-continuous at a. Let B = {x : x [a, b] and f is bounded on [a, x].} Let c = sup B. Explain carefully how one can use the right-continuity to deduce, firstly, that f is bounded on some interval [a, a + δ), and secondly, that c > a. / Laat f : [a, b] R regskontinu in a wees. Laat B = {x : x [a, b] en f is begrens op [a, x]}. Laat c = sup B. Verduidelik in detail hoe mens die regskontinuïteit kan gebruik om te bewys dat, eerstens f begerens is op n interval [a, a + δ), en, tweedens, dat c > a. [3] (c) Let f be continuous on J = [a, b]. The boundedness property states that (1) f is bounded on [a, b], and (2) f attains both a maximum value and a minimum value somewhere on [a, b]. Use the boundedness property and the intermediate value property to show that f(j) is a closed bounded interval. / Laat f kontinu op J = [a, b] wees. Die begrensdheidseienskap sê dat (1) f begrens is op [a, b], en (2) dat f sowel n maksimumwaarde as n minimumwaarde iewers op [a, b] bereik. Gebruik die begrensdheidseienskap en die tussenwaardestelling om te bewys dat f(j) n geslote en begrensde interval is. [3]

7 6 QUESTION 4 / VRAAG 4 (a) Let f be a bounded function, defined on [a, b]. Given a partition P = {x 0, x 1,..., x n } of the interval [a, b], and the notations m i = inf{f(x) : x i 1 x x i }, M i = sup{f(x) : x i 1 x x i }, define the lower and upper sums: / Laat f n begrensde funfsie wees wat begrens op [a, b] is. Vir n gegewe partisie P = {x 0, x 1,..., x n } van die interval [a, b], en gegee dat m i = inf{f(x) : x i 1 x x i }, M i = sup{f(x) : x i 1 x x i }, definieer die laersom en bosom: L(P ) = U(P ) = [1] (b) Define the symbols L and U and define what it means to say that f is integrable over [a, b]. / Definieer die simbole L en U en definieer die betekenis van die frase f is integreerbaar oor [a, b]. [2] (c) The Riemann criterion says that a bounded function is integrable on [a, b] if and only if for each ɛ > 0 there exists a partition P such that U(P ) L(P ) < ɛ. Using the usual notation, the small span property of continuous functions states that for each ɛ > 0 there is a partition P of [a, b] such that M i m i < ɛ for each i = 1,..., n. Use this to prove that every continuous function f : [a, b] R is integrable on [a, b]. / Die Riemann kriterium sê dat n begrensde funksie integreerbaar op [a, b] is as en slegs as vir elke ɛ > 0 daar n partisie P van [a, b] bestaan só dat U(P ) L(P ) < ɛ. In die gewone notasie sê die kleinspanstelling vir kontinue funksies dat daar vir elke ɛ > 0 n partisie P van [a, b] bestaan só dat M i m i < ɛ vir elke i = 1,..., n. Gebruik hierdie gegewens om te bewys dat elke kontinue funksie f : [a, b] R integreerbaar op [a, b] is. [4]

8 7 QUESTION 5 / VRAAG 5 Use one of the convergence tests to determine whether the following series is convergent or divergent. Name the test that you use. / Gebruik een van die konvergensietoetse om te bepaal of die volgende reeks konvergent of divergent is. Noem die toets wat jy gebruik. [3] r=1 1 r 2 r