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: 208-0-0 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 2 53 5 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 2. 6. Find to the dierential equation y 8y + 6y = 0 the solution that satisies the initial conditions y(0) =, y (0) = 7. 7. 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.
MÄLARDALENS HÖGSKOLA Akademin ör utbildning, kultur och kommunikation Avdelningen ör tillämpad matematik Examinator: Lars-Göran Larsson TENTAMEN I MATEMATIK MAA5 Envariabelkalkyl, TEN Datum: 208-0-0 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 2 53 5 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 2. 6. Bestäm till dierentialekvationen y 8y +6y = 0 den lösning som satisierar begynnelsevillkoren y(0) =, y (0) = 7. 7. 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.
MÄLARDALEN UNIVERSITY School o Education, Culture and Communication Department o Applied Mathematics Examiner: Lars-Göran Larsson Examination TEN 208-0-0 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 5. 20 3 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 ) ()