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: 05-06-08 Write time: 3 hours Aid: Writing materials 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, 4 and 5 require a minimum o, 6 and points respectively. The minimum points or the ECTS-marks E, D, C, B and A are, 3, 6, 0 and 3 respectively. I the obtained sum o points is denoted S, and that obtained at examination TEN S, the mark or a completed course is according to the ollowing S, S 9 and S + S 4 3 S, S 9 and 4 S + S 53 4 54 S + S 5 S, S 9 and S + S 3 E S, S 9 and 33 S + S 4 D S, S 9 and 4 S + S 5 C 5 S + S 60 B 6 S + S 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.. Find the general antiderivative o x (x) = xe x.. Find the area o the bounded region precisely enclosed by the curves y = 3 x and y = x + x. 3. Let be the unction deined by (x) = x +. In what intervals is (x + ) the unction convex? 4. Let (x) = arcsin(x). State the domain and the range o and respectively, and sketch in separate igures the graphs o the unctions. 5. Solve the initial value problem xy y = 3 x, y() = 3. 6. Find the coeicients o the power series in x representing determine the interval o convergence o the power series. x +. Also, 7. Prove that the unction deined by (x) = x 5 + x 3 + x is invertible, and ind the derivative o at the point 3. 8. Determine whether lim x 0 cos(x) e x x 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! 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: 05-06-08 Skrivtid: 3 timmar Hjälpmedel: Skrivdon 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, 4 och 5 krävs erhållna poängsummor om minst, 6 respektive poäng. Om den erhållna poängen benämns S, och den vid tentamen TEN erhållna S, bestäms graden av sammanattningsbetyg på en slutörd kurs enligt S, S 9 och S + S 4 3 S, S 9 och 4 S + S 53 4 54 S + S 5 Lösningar örutsätts inneatta ordentliga motiveringar och tydliga svar. Samtliga lösningsblad skall vid inlämning vara sorterade i den ordning som uppgiterna är givna i.. Bestäm den generella primitiva unktionen till x (x) = xe x.. Bestäm arean av det begränsade område som precis innesluts av kurvorna y = 3 x och y = x + x. 3. Låt vara unktionen deinierad enligt (x) = x +. I vilka intervall (x + ) är unktionen konvex? 4. Låt (x) = arcsin(x). Ange deinitionsmängden och värdemängden ör respektive, och skissa i separata igurer graerna till unktionerna. 5. Lös begynnelsevärdesproblemet xy y = 3 x, y() = 3. 6. Bestäm koeicienterna i den potensserie i x som representerar även konvergensintervallet ör potensserien. x +. Bestäm 7. Bevisa att unktionen deinierad enligt (x) = x 5 + x 3 + x är inverterbar, och bestäm derivatan till i punkten 3. 8. Avgör om lim x 0 cos(x) e x x existerar eller ej. Om svaret är nej: Ge en örklaring till varör! Om svaret är ja: Ge en örklaring till varör och bestäm gränsvärdet! I you preer the problems ormulated in English, please turn the sheet.
MÄLARDALEN UNIVERSITY School o Education, Culture and Communication Department o Applied Mathematics Examiner: Lars-Göran Larsson Examination TEN 05-06-08. F( x) x ) e x 4 ( C where C is a constant EXAMINATION IN MATHEMATICS MAA5 Single Variable Calculus EVALUATION PRINCIPLES with POINT RANGES Academic Year: 04/5 Maximum points or subparts o the problems in the inal examination p: Correctly worked out the irst progressive step in inding the antiderivative by parts p: Correctly worked out the second progressive step in inding the antiderivative by parts p: Correctly included a constant in an otherwise correctly ound antiderivative. ( 5 3 3) a.u. p: Correctly ound the intersection o the two enclosing curves, and correctly ormulated an integral or the area p: Correctly determined the needed antiderivatives p: Correctly ound the limits in the integral and the area 3. is convex in the interval [, ) p: Correctly ound the second derivative o p: Correctly actorized the second derivative o, and correctly worked out a test or convexity o p: Correctly determined the interval where is convex 4. D [,] and V, ] [ [ D, ] and V,] [ p: Correctly stated the domains and the ranges o the unctions and p: Correctly sketched the graph o the unction p: Correctly sketched the graph o the unction 5. y 4x x p: Correctly written the DE in standard orm, correctly determined an integrating actor, and correctly reormulated the let-hand-side o the DE as an exact derivative p: Correctly ound the general solution o the DE p: Correctly adapted the general solution to the initial value, and correctly summarized the solution o the IVP 6. c x k n k where c ( k ( ) ) x k0 The interval o convergence is (,) n p: Correctly expanded ( x ) in a power series in x p: Correctly identiied the coeicients o the power series p: Correctly determined the interval o convergence 7. is invertible since ( x) 0 on D ( ) (3) 9 p: Correctly proved that is invertible p: Correctly ound that (3) p: Correctly determined the value o ( ) (3) 8. The limit exists and is equal to 4 p: Correctly expanded cos(x ) and x e in their Maclaurin series p: Correctly algebraically prepared or determining the limit p: Correctly determined the limit ()