1. The sum of two non-negative numbers x and y equals 4. Which is the smallest interval that surely contains the number x 3 + 3y 2?

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1 MÄLARDALEN UNIVERSITY School o Education, Culture and Communication Department o Applied Mathematics Examiner: Lars-Göran Larsson EXAMINATION IN MATHEMATICS MAA5 Single Variable Calculus, TEN Date: Write time: 3 hours Aid: Writing materials, ruler This examination is intended or the examination part TEN. The examination consists o eight randomly ordered problems each o which is worth at maximum 3 points. The pass-marks 3, and 5 require a minimum o, 6 and 2 points respectively. The minimum points or the ECTS-marks E, D, C, B and A are, 3, 6, 20 and 23 respectively. I the obtained sum o points is denoted S, and that obtained at examination TEN2 S 2, the mark or a completed course is according to the ollowing S, S 2 9 and S + 2S 2 3 S, S 2 9 and 2 S + 2S S + 2S 2 5 S, S 2 9 and S + 2S 2 32 E S, S 2 9 and 33 S + 2S 2 D S, S 2 9 and 2 S + 2S 2 5 C 52 S + 2S 2 60 B 6 S + 2S 2 A Solutions are supposed to include rigorous justiications and clear answers. All sheets o solutions must be sorted in the order the problems are given in.. The sum o two non-negative numbers x and y equals. Which is the smallest interval that surely contains the number x 3 + 3y 2? 2. For which x is the series ln(x) + ln 2 (x) + ln 3 (x) +... convergent? Find the sum o the series or these x. 3. Find the inverse o the unction deined by (x) = speciy the domain and the range o the inverse. x + 2. Especially,. Find out whether [ ] x 3 lim x + (x ) x3 2 (x + ) 2 exists or not. I the answer is no: Give an explanation o why! I the answer is yes: Give an explanation o why and ind the limit! 5. Find the area o the bounded region which is precisely enclosed by the curves γ : y = x and γ 2 : y = 2 x Find to the dierential equation y 8y + 6y = 0 the solution that satisies the initial conditions y(0) =, y (0) = Find to the unction x (x) = value 0 at the point 0. 6 x 2 9 the antiderivative F that have the 8. Prove that the unction deined by (x) = x ln(x) is invertible, and ind the derivative o the inverse at the point 2e. Om du öredrar uppgiterna ormulerade på svenska, var god vänd på bladet.

2 MÄLARDALENS HÖGSKOLA Akademin ör utbildning, kultur och kommunikation Avdelningen ör tillämpad matematik Examinator: Lars-Göran Larsson TENTAMEN I MATEMATIK MAA5 Envariabelkalkyl, TEN Datum: Skrivtid: 3 timmar Hjälpmedel: Skrivdon, linjal Denna tentamen är avsedd ör examinationsmomentet TEN. Provet består av åtta stycken om varannat slumpmässigt ordnade uppgiter som vardera kan ge maximalt 3 poäng. För godkänd-betygen 3, och 5 krävs erhållna poängsummor om minst, 6 respektive 2 poäng. Om den erhållna poängen benämns S, och den vid tentamen TEN2 erhållna S 2, bestäms graden av sammanattningsbetyg på en slutörd kurs enligt S, S 2 9 och S + 2S 2 3 S, S 2 9 och 2 S + 2S S + 2S 2 5 Lösningar örutsätts inneatta ordentliga motiveringar och tydliga svar. Samtliga lösningsblad skall vid inlämning vara sorterade i den ordning som uppgiterna är givna i.. Summan av två icke-negativa tal x och y är lika med. Vilket är det minsta intervall som garanterat innehåller talet x 3 + 3y 2? 2. För vilka x är serien ln(x) + ln 2 (x) + ln 3 (x) +... konvergent? Bestäm seriens summa ör dessa x. 3. Bestäm inversen till unktionen deinierad enligt (x) = särskilt inversens deinitionsmängd och värdemängd.. Utred om [ ] x 3 lim x + (x ) x3 2 (x + ) 2 x + 2. Speciicera existerar eller ej. Om svaret är nej: Ge en örklaring till varör! Om svaret är ja: Ge en örklaring till varör och bestäm gränsvärdet! 5. Bestäm arean av det begränsade område som precis innesluts av kurvorna γ : y = x och γ 2 : y = 2 x Bestäm till dierentialekvationen y 8y +6y = 0 den lösning som satisierar begynnelsevillkoren y(0) =, y (0) = Bestäm till unktionen x (x) = 6 x 2 9 punkten 0. den primitiv F som har värdet 0 i 8. Bevisa att unktionen deinierad enligt (x) = x ln(x) är inverterbar, och bestäm derivatan till inversen i punkten 2e. I you preer the problems ormulated in English, please turn the page.

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5 MÄLARDALEN UNIVERSITY School o Education, Culture and Communication Department o Applied Mathematics Examiner: Lars-Göran Larsson Examination TEN EXAMINATION IN MATHEMATICS MAA5 Single Variable Calculus EVALUATION PRINCIPLES with POINT RANGES Academic Year: 207/8 Maximum points or subparts o the problems in the inal examination. x 3 + 3y 2 [20,6] p: Correctly or the problem ormulated a unction o one variable including the speciication o its domain p: Correctly ound and concluded about the local extreme points o the unction p: Correctly ound the range o the unction, and thereby the 3 2 interval in which the number x + 3y lies e 2. The series converges i x ( e, ). ln( x) The sum o the series equals ln( x) p: Correctly noted that the series is geometric, and correctly ound the upper (non-included) limit o the interval o convergence p: Correctly ound the lower (non-included) limit o the interval o convergence p: Correctly ound the sum o the series 3. 2 ( x) = 2 + x D = (0, ), R ( 2, = ) p: Correctly ound the expression o ( x) p: Correctly ound the domain o p: Correctly ound the the range o. The limit exists and is equal to p: Correctly brought the terms together with a least common denominator and correctly simpliied the numerator p: Correctly identiied the dominating actors as x + p: Correctly ound the limit a.u. p: Correctly ound the intersection o the two enclosing curves, and correctly ormulated an integral or the area p: Correctly ound the needed antiderivatives p: Correctly ound the limits in the integral and the area 6. y = ( 3x + ) e x Note: The student who has stated that y = Ae x + Be x is the general solution o the dierential equation, and who has not ound any explanation to the impossible conditions occuring when adapting to the initial values, can not obtain any other sum o points than 0p. p: Correctly ound one solution o the DE p: Correctly ound the general solution o the DE p: Correctly adapted the general solution to the initial values, and correctly summarized the solution o the IVP 7. 3 x F( x) = ln 3 + x 8. is invertible since ( x) > 0 on D ( ) (2e ) = '( e = 9 ) p: Correctly ound the partial ractions o (x) p: Correctly ound the general antiderivative o p: Correctly adapted the antiderivative to the value at 0 p: Correctly proved that is invertible p: Correctly ound that ( 2e ) = e p: Correctly determined the value o ( ) (2e ) ()