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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, TEN1 Date: 2015-08-10 Write time: 3 hours Aid: Writing materials This examination is intended for the examination part TEN1. The examination consists of eight randomly ordered problems each of which is worth at maximum 3 points. The pass-marks 3, 4 and 5 require a minimum of 11, 16 and 21 points respectively. The minimum points for the ECTS-marks E, D, C, B and A are 11, 13, 16, 20 and 23 respectively. If the obtained sum of points is denoted S1 , and that obtained at examination TEN2 S2 , 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. Determine whether lim n→∞ √ √ n2 + 6n − n2 + 1 exists or not. If the answer is no: Give an explanation of why! If the answer is yes: Give an explanation of why and find the limit! 2. Explain why 3a2 − 2a + 4/3 + . . . is a geometric series for a 6= 0 . Also, find those a for which the series converges, and determine the corresponding sums. 3. Find to the function x y f (x) = 2 the antiderivative F that have (x + 2)(x + 4) the value 1 at the point −3. 4. Determine to the differential equation y 00 − 6y 0 + 9y = 0 the solution that satisfies the initial conditions y(0) = 1 , y 0 (0) = 5. 5. Let f (x) = e−4( x−2) . Find an equation for the tangent line to the curve y = f (x) at the point P whose x-coordinate is equal to 4. 6. Let f (x) = 7. The product of two positive real numbers a and b equals 8. Which are the numbers if the weighted sum 2a + 9b of them is a minimum? Prove your conclusion! √ x+1 . Find the function expression, the domain and the range of x−1 the composition f ◦ f . Z 8. π | sin(2x)| dx . Evaluate the integral 0 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, TEN1 Datum: 2015-08-10 Skrivtid: 3 timmar Hjälpmedel: Skrivdon Denna tentamen är avsedd för examinationsmomentet TEN1. Provet består av åtta stycken om varannat slumpmässigt ordnade uppgifter som vardera kan ge maximalt 3 poäng. För godkänd-betygen 3, 4 och 5 krävs erhållna poängsummor om minst 11, 16 respektive 21 poäng. Om den erhållna poängen benämns S1 , och den vid tentamen TEN2 erhållna S2 , bestäms graden av sammanfattningsbetyg på en slutförd kurs enligt 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. Avgör om lim n→∞ √ √ n2 + 6n − n2 + 1 existerar eller ej. Om svaret är nej: Ge en förklaring till varför! Om svaret är ja: Ge en förklaring till varför och bestäm gränsvärdet! 2. Förklara varför 3a2 − 2a + 4/3 + . . . är en geometrisk serie för a 6= 0 . Bestäm även de a för vilka serien konvergerar, och bestäm motsvarande summor. 3. Bestäm till funktionen x y f (x) = 2 den primitiv F som har (x + 2)(x + 4) värdet 1 i punkten −3 . 4. Bestäm till differentialekvationen y 00 − 6y 0 + 9y = 0 den lösning som satisfierar begynnelsevillkoren y(0) = 1 , y 0 (0) = 5. 5. Låt f (x) = e−4( x−2) . Bestäm en ekvation för tangenten till funktionskurvan y = f (x) i den punkt P vars x-koordinat är lika med 4. 6. Låt f (x) = 7. Produkten av två positiva reella tal a och b är lika med 8. Vilka är talen om den viktade summan 2a + 9b av dem är minimal? Bevisa din slutsats! √ x+1 . Bestäm funktionsuttrycket, definitionsmängden och värdex−1 mängden för sammansättningen f ◦ f . Z 8. π | sin(2x)| dx . Beräkna integralen 0 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 IN MATHEMATICS MAA151 Single Variable Calculus EVALUATION PRINCIPLES with POINT RANGES Academic Year: 2014/15 Examination TEN1 – 2015-08-10 Maximum points for subparts of the problems in the final examination 1. The limit exists and is equal to 3 1p: Correctly extended the difference of square roots with the conjugate, and correctly simplified the numerator 1p: Correctly algebraically prepared for determining the limit 1p: Correctly determined the limit 2. It is a geometric series since the each quotient of two terms following each other has the common value 2 (3a) . The series converges for a 23 , i.e. for (a 23 ) (a 23 ) . 3 The sum of the series equals 9a 1p: Correctly explained why the series is a geometric series 1p: Correctly determined the a for which the series converges 1p: Correctly determined the sum of the series 2 3a x2 x4 3. F ( x) 1 ln 4. y (1 2 x) e 3 x Note: The student who has stated that y Ae 3x Be 3x is the general solution of the differential equation, and who has not found any explanation to the impossible conditions occuring when adapting to the initial values, can not obtain any other sum of points than 0p. 1p: Correctly found the partial fractions of f (x) 1p: Correctly found the general antiderivative of f 1p: Correctly adapted the antiderivative to the value at 3 1p: Correctly found one solution of the DE 1p: Correctly found the general solution of the DE 1p: Correctly adapted the general solution to the initial values, and correctly summarized the solution of the IVP 5. x y 5 1p: Correctly found the derivative of the the function f , all with the purpose of determining the slope at the point P 1p: Correctly evaluated the function f and the derivative f at the point P 1p: Correctly found an equation for the tangent line to the curve y f (x) at the point P 6. f f ( x) x 1p: Correctly found the expression for f f (x) 1p: Correctly found the domain of the composition f f 1p: Correctly found the range of the composition f f where D f f V f f ( ,1) (1, ) 7. a 6 and b 8. 2 4 3 1p: Correctly for the optimization problem formulated a function of one variable, and correctly found the derivative of the function 1p: Correctly worked out a first derivative test 1p: Correctly found the numbers a and b which give the minimum value of the weighted sum 2a 9b 1p: Correctly divided the integral in two integrals, each in which the absolute value bars can be removed 1p: Correctly determined an antiderivative for each integrand 1p: Correctly determined the value of the integral 1 (1)