7 n + 8 n 3 2n. converges or not. Irrespective whether the answer is yes or no, give an explanation

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1 MÄLARDALEN UNIVERSITY School of Educatio, Culture ad Commuicatio Departmet of Applied Mathematics Examier: Lars-Göra Larsso EXAMINATION IN MATHEMATICS MAA151 Sigle Variable Calculus, TEN2 Date: Write time: 3 hours Aid: Writig materials, ruler This examiatio is iteded for the examiatio part TEN2. The examiatio cosists of five radomly ordered problems each of which is worth at maximum 4 poits. The pass-marks 3, 4 ad 5 require a miimum of 9, 13 ad 17 poits respectively. The miimum poits for the ECTS-marks E, D, C, B ad A are 9, 10, 13, 16 ad 20 respectively. If the obtaied sum of poits is deoted S 2, ad that obtaied at examiatio TEN1 S 1, the mark for a completed course is accordig to the followig: S 1 11, S 2 9 ad S 1 + 2S S 1 11, S 2 9 ad 42 S 1 + 2S S 1 + 2S 2 5 S 1 11, S 2 9 ad S 1 + 2S 2 32 E S 1 11, S 2 9 ad 33 S 1 + 2S 2 41 D S 1 11, S 2 9 ad 42 S 1 + 2S 2 51 C 52 S 1 + 2S 2 60 B 61 S 1 + 2S 2 A Solutios are supposed to iclude rigorous justificatios ad clear aswers. All sheets of solutios must be sorted i the order the problems are give i. 1. Fid the solutio of the iitial value problem y + 4y = 5e 4x, y(0) = 1, y (0) = Determie whether the series =1 coverges or ot. Irrespective whether the aswer is yes or o, give a explaatio of why! 3. Fid the area of the surface geerated by rotatig about the y-axis the triagle which has its vertices at the poits P : (1, 1), Q : (2, 2) ad R : (1, 2). 4. Evaluate the itegral e x l 2 (x) dx, ad write the result i as simple form as possible Sketch the graph of the fuctio f, defied by f(x) = 1 x arcta(x), by utilizig the guidace give by asymptotes ad statioary poits. Om du föredrar uppgiftera formulerade på sveska, var god väd på bladet.

2 MÄLARDALENS HÖGSKOLA Akademi för utbildig, kultur och kommuikatio Avdelige för tillämpad matematik Examiator: Lars-Göra Larsso TENTAMEN I MATEMATIK MAA151 Evariabelkalkyl, TEN2 Datum: Skrivtid: 3 timmar Hjälpmedel: Skrivdo, lijal Dea tetame är avsedd för examiatiosmometet TEN2. Provet består av fem stycke om varaat slumpmässigt ordade uppgifter som vardera ka ge maximalt 4 poäg. För godkäd-betyge 3, 4 och 5 krävs erhålla poägsummor om mist 9, 13 respektive 17 poäg. Om de erhålla poäge beäms S 2, och de vid tetame TEN1 erhålla S 1, bestäms grade av sammafattigsbetyg på e slutförd kurs eligt följade: S 1 11, S 2 9 och S 1 + 2S S 1 11, S 2 9 och 42 S 1 + 2S S 1 + 2S 2 5 Lösigar förutsätts iefatta ordetliga motiverigar och tydliga svar. Samtliga lösigsblad skall vid ilämig vara sorterade i de ordig som uppgiftera är giva i. 1. Bestäm lösige till begyelsevärdesproblemet y + 4y = 5e 4x, y(0) = 1, y (0) = Avgör om serie =1 är koverget eller ej. Oavsett om svaret är ja eller ej, ge e förklarig till varför! Bestäm area av de yta som geereras geom att krig y-axel rotera de triagel som har sia hör i puktera P : (1, 1), Q : (2, 2) och R : (1, 2). 4. Beräka itegrale e x l 2 (x) dx, och skriv resultatet på e så ekel form som möjligt Skissa grafe till fuktioe f, defiierad eligt f(x) = 1 x arcta(x), geom att aväda de vägledig som fås frå asymptoter och statioära pukter. If you prefer the problems formulated i Eglish, please tur the page.

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5 MÄLARDALEN UNIVERSITY School of Educatio, Culture ad Commuicatio Departmet of Applied Mathematics Examier: Lars-Göra Larsso EXAMINATION IN MATHEMATICS MAA151 Sigle Variable Calculus EVALUATION PRINCIPLES with POINT RANGES Academic Year: 2015/16 Examiatio TEN x 1. y ( x x e ) 1 3cos(2 ) 2si(2 ) + + Maximum poits for subparts of the problems i the fial examiatio = 1p: Correctly idetified the differetial equatio as a o- 4 homogeeous liear DE of secod order, ad correctly foud the geeral solutio y h of the corr. homog. DE 1p: Correctly proposed a formula for a part. sol. of the DE 1p: Correctly foud a particular solutio of the DE, ad correctly summarized the geeral solutio of the DE 1p: Correctly adapted the geeral solutio of the DE to the IV 2. The series coverges Oe sceario for the other three poits p: Correctly idetified that the terms of the series are domiated by expoetials 8 i the umerator ad by the expoetials 9 i the the deomiator Aother sceario for the other three poits p: Correctly foud that the limit of the 1p: Correctly oted that the terms of the series are less tha test quatity i the ratio test equals 8 9 e.g. 2 (8 9) 2p: Correctly, from the fact that 8 9 < 1, 2p: Correctly cocluded that the series coverges accordig cocluded that the series coverges to the compariso test (with a compario with the accordig to the ratio test coverget geometric series 2(8 9) ) =1 3. ( ) π a.u. 1p: Correctly realized that the surface cosidered cosists of three parts which most easily are described separately cocerig matters of fidig the total area, ad correctly foud the area of the circular cylider surface of radius 2 1p: Correctly foud the area of the aulus part of the surface 2p: Correctly foud the area of the coic part of the surface, ad correctly added the three cotributios to the area 2 4. (5e e 8) The graph has a local miimum at P : (0,1), ad has the asymptotes x = 1, y = π 2 ad y = π Oe sceario p: Correctly worked out the first step of a partial itegratio 1p: Correctly worked out the secod step of a partial itegratio 1p: Correctly worked out the third step of a partial itegratio Aother sceario p: Correctly, by the substitutio l( x ) = u, traslated the itegral ito ito 1 2 3u 2 u e du 0 3p: Correctly worked out the three steps of a partial itegratio i u 1p: Correctly foud the asymptotes of the graph 1p: Correctly classified the local extreme poit of the graph 1p: Correctly sketched the graph accordig to how the graph relates to the asymptotes o their both sides respectively 1p: Correctly completed the sketch of the graph 1 (1)