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MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson EXAMINATION IN MATHEMATICS MAA151 Single Variable Calculus, TEN2 Date: 2017-03-24 Write time: 3 hours Aid: Writing materials, ruler This examination is intended for the examination part TEN2. The examination consists of five randomly ordered problems each of which is worth at maximum 4 points. The pass-marks 3, 4 and 5 require a minimum of 9, 13 and 17 points respectively. The minimum points for the ECTS-marks E, D, C, B and A are 9, 10, 13, 16 and 20 respectively. If the obtained sum of points is denoted S2 , and that obtained at examination TEN1 S1 , the mark for a completed course is according to the following: S1 ≥ 11, S2 ≥ 9 and S1 + 2S2 ≤ 41 → 3 S1 ≥ 11, S2 ≥ 9 and 42 ≤ S1 + 2S2 ≤ 53 → 4 54 ≤ S1 + 2S2 → 5 S1 ≥ 11, S2 ≥ 9 and S1 + 2S2 ≤ 32 → E S1 ≥ 11, S2 ≥ 9 and 33 ≤ S1 + 2S2 ≤ 41 → D S1 ≥ 11, S2 ≥ 9 and 42 ≤ S1 + 2S2 ≤ 51 → C 52 ≤ S1 + 2S2 ≤ 60 → B 61 ≤ S1 + 2S2 → A Solutions are supposed to include rigorous justifications and clear answers. All sheets of solutions must be sorted in the order the problems are given in. 1. Find the real numbers x for which the power series ∞ X (4x − 12)n n=1 n . is convergent. Are there any of those x for which the series is not absolutely convergent, i.e. is (only) conditionally convergent? ( y 0 = (xy)3 , 2. Solve the initial-value problem 3. A closed (with bottom and lock) cylindrical tin is to hold π liters. Find the radius and height of the can so that the consumption of material (the number of square decimeter thin sheet) becomes a minimum. ( y(1) = − 21 . x = 12 cos2 (θ) , 4. Find the length of the curve 5. Evaluate the generalized integral Z ∞ y = 21 sin2 (θ) , 9/2 0 ≤ θ ≤ π. dx , 4x(x − 6) + 45 and write the result in as simple form as possible. Om du föredrar uppgifterna formulerade på svenska, var god vänd på bladet. MÄLARDALENS HÖGSKOLA Akademin för utbildning, kultur och kommunikation Avdelningen för tillämpad matematik Examinator: Lars-Göran Larsson TENTAMEN I MATEMATIK MAA151 Envariabelkalkyl, TEN2 Datum: 2017-03-24 Skrivtid: 3 timmar Hjälpmedel: Skrivdon, linjal Denna tentamen är avsedd för examinationsmomentet TEN2. Provet består av fem stycken om varannat slumpmässigt ordnade uppgifter som vardera kan ge maximalt 4 poäng. För godkänd-betygen 3, 4 och 5 krävs erhållna poängsummor om minst 9, 13 respektive 17 poäng. Om den erhållna poängen benämns S2 , och den vid tentamen TEN1 erhållna S1 , bestäms graden av sammanfattningsbetyg på en slutförd kurs enligt följande: S1 ≥ 11, S2 ≥ 9 och S1 + 2S2 ≤ 41 → 3 S1 ≥ 11, S2 ≥ 9 och 42 ≤ S1 + 2S2 ≤ 53 → 4 54 ≤ S1 + 2S2 → 5 Lösningar förutsätts innefatta ordentliga motiveringar och tydliga svar. Samtliga lösningsblad skall vid inlämning vara sorterade i den ordning som uppgifterna är givna i. 1. Bestäm de reella tal x för vilka potensserien ∞ X (4x − 12)n n n=1 . är konvergent. Är det några av dessa x för vilka serien inte är absolutkonvergent, dvs. är (endast) betingat konvergent? ( y 0 = (xy)3 , 2. Lös begynnelsevärdesproblemet 3. En sluten (med botten och lock) cylinderformad konservburk ska rymma π liter. Bestäm burkens radie och höjd så att åtgången av material (antalet kvadratdecimeter tunn plåt) blir minimal. ( y(1) = − 21 . x = 21 cos2 (θ) , 4. Bestäm längden av kurvan 5. Beräkna den generaliserade integralen Z ∞ y = 21 sin2 (θ) , 9/2 0 ≤ θ ≤ π. dx , 4x(x − 6) + 45 och skriv resultatet på en så enkel form som möjligt. If you prefer the problems formulated in English, please turn the page. MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson Examination TEN2 – 2017-03-24 EXAMINATION IN MATHEMATICS MAA151 Single Variable Calculus EVALUATION PRINCIPLES with POINT RANGES Academic Year: 2016/17 Maximum points for subparts of the problems in the final examination 1. The series is convergent for 114 ≤ x < 134 . 1p: Correctly, by e.g. the ratio test, found that the series is Of those x , it is only conditionally absolutely convergent for | 4( x − 3) | < 1 , i.e. for 11 13 11 convergent for x = 4 . , and hopefully correctly mentioned that the 4 < x< 4 series definitely is divergent for | x − 3 | > 14 1p: Correctly found that the series is convergent for x = 114 1p: Correctly found that the series is divergent for x = 134 1p: Correctly stressed that the series is only conditionally convergent for x = 114 2. y=− 3. 4. 2 9 − x4 1 dm 2 2 height = 3 dm = 3 4 dm 2 radius = 3 1p: Correctly for the optimization problem formulated a function of one variable 2p: Correctly found and concluded about the local extreme points of the function 1p: Correctly stated the radius and height which minimizes the consumption of material ------------------------------------- One scenario ------------------------------------------------l.u. if the curve has been interpreted as the path back and forth between the 1p: Correctly formulated an integral for the length of the curve (an integral with explicit expressions for the points P : ( 12 ,0) and Q : (0, 12 ) derivatives dx dθ and dy dθ ) 1p: Correctly rewrited the integrand into 2 cos(θ ) sin(θ ) 1 2 l.u. if the curve has been inter1p: Correctly treated the absolute values when integrating preted as the path between the points 1 1 1p: Correctly found an antiderivative of the integrand, and P : ( 2 ,0) and Q : (0, 2 ) correctly found the length of the curve 2 Notice: Both interpretations/answers are accepted provided they have been justified. 5. 1p: Correctly identified the differential equation as nonlinear and separable, and correctly found general antiderivatives of both sides of the separated differential equation 1p: Correctly adapted the integrated equation to the IV 1p: Correctly solved for y 2 1p: Correctly solved for y (the IV had to be used once more) π 24 ---------------------------------- Another scenario ---------------------------------------------- 2p: Correctly noticed that the curve is the straight line x + y = 12 back and forth between two points, P and Q 1p: Correctly found the length of the straight line between the points P and Q 1p: Correctly found the length of the path back and forth between the points P and Q 1p: Correctly rewrited the denominator of the integrand into (2 x − 6) 2 + 9 , and correctly by the substitution 2( x − 3) = 3u translated the integrand into 1 (6(1 + u 2 )) 1p: Correctly translated the limits of the integral in connection with the substitution 2( x − 3) = 3u 1p: Correctly found an antiderivative of the integrand 1p: Correctly evaluated the antiderivative at the limits, and by that also correctly found the value of the integral 1 (1)