2(x + 1) x f(x) = 3. Find the area of the surface generated by rotating the curve. y = x 3, 0 x 1,

Transkript

1 MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson EXAMINATION IN MATHEMATICS MAA5 Single Variable Calculus, TEN Date: Write time: hours Aid: Writing materials, ruler This examination is intended for the examination part TEN. The examination consists of five randomly ordered problems each of which is worth at maximum 4 points. The pass-marks, 4 and 5 require a minimum of 9, and 7 points respectively. The minimum points for the ECTS-marks E, D, C, B and A are 9, 0,, 6 and 0 respectively. If the obtained sum of points is denoted S, and that obtained at examination TEN S, the mark for a completed course is according to the following: S, S 9 and S + S 4 S, S 9 and 4 S + S S + S 5 S, S 9 and S + S E S, S 9 and 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 justifications and clear answers. All sheets of solutions must be sorted in the order the problems are given in.. Evaluate the integral 7/6 5 + x(4 x) dx, and write the result in as simple form as possible. /. Sketch the graph of the function f defined by f(x) = Also, specify the range of the function. (x + ) x +.. Find the area of the surface generated by rotating the curve about the x-axis. y = x, 0 x, 4. rove that the series n= π n ( 9) n (n)! is convergent. Then, find the sum of the series. 5. Find an equation for the tangent line τ to the curve at the point : (, ). γ : xy arctan(xy) + ln() = π 4 + ln( + x y ) Om du föredrar uppgifterna formulerade på svenska, var god vänd på bladet.

2 MÄLARDALENS HÖGSKOLA Akademin för utbildning, kultur och kommunikation Avdelningen för tillämpad matematik Examinator: Lars-Göran Larsson TENTAMEN I MATEMATIK MAA5 Envariabelkalkyl, TEN Datum: Skrivtid: timmar Hjälpmedel: Skrivdon, linjal Denna tentamen är avsedd för examinationsmomentet TEN. rovet består av fem stycken om varannat slumpmässigt ordnade uppgifter som vardera kan ge maximalt 4 poäng. För godkänd-betygen, 4 och 5 krävs erhållna poängsummor om minst 9, respektive 7 poäng. Om den erhållna poängen benämns S, och den vid tentamen TEN erhållna S, bestäms graden av sammanfattningsbetyg på en slutförd kurs enligt följande: S, S 9 och S + S 4 S, S 9 och 4 S + S S + S 5 Lösningar förutsätts innefatta ordentliga motiveringar och tydliga svar. Samtliga lösningsblad skall vid inlämning vara sorterade i den ordning som uppgifterna är givna i.. Beräkna integralen 7/6 5 + x(4 x) dx, och skriv resultatet på en så enkel form som möjligt. /. Skissa grafen till funktionen f definierad genom f(x) = Specificera även funktionens värdemängd. (x + ) x +.. Bestäm arean av den yta som genereras genom att rotera kurvan kring x-axeln. y = x, 0 x, 4. Bevisa att serien n= π n ( 9) n (n)! är konvergent. Bestäm därefter summan av serien. 5. Bestäm en ekvation för tangenten τ till kurvan i punkten : (, ). γ : xy arctan(xy) + ln() = π 4 + ln( + x y ) If you prefer the problems formulated in English, please turn the page.

3

4

5 MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson Examination TEN EXAMINATION IN MATHEMATICS MAA5 Single Variable Calculus EVALUATION RINCILES with OINT RANGES Academic Year: 06/7 Maximum points for subparts of the problems in the final examination. π + 8 p: Correctly rewrited the integrand into 9 (x ), and correctly by the substitution x = sin( θ ) translated 9 (x ) dx into cos ( θ ) dθ p: Correctly translated the limits of the integral in connection with the substitution x = sin( θ ) p: Correctly rewrited the integrand according to cos ( θ ) = ( + cos( θ )), and correctly found an antiderivative p: Correctly evaluated the antiderivative at the limits and by that correctly found the value of the integral. The graph has a global minimum at : (, ) and global maximum at : (,). The only asymptote is y = 0, and the range is,]. [ p: Correctly found the stationary points and the asymptote p: Correctly classified the stationary points p: Correctly sketched the graph according to how the graph relates to the extreme points and the asymptote p: Correctlys found the range π. (0 0 ) 7 a.u. p: Correctly formulated an explicit integral expression for the area of the surface generated by the curve rotated about the x-axis 4 p: Correctly by e.g. a substitution + 9x = u translated the integral into a standard integral with the integrand u p: Correctly found an antiderivative of the integrand, and correctly evaluated the antiderivative at the limits 4. The series is convergent, and the π sum of the series is [( ) ] p: Correctly, by the ratio test, found that the series is (absolutely) convergent p: Correctly identified the series as, except for the first two terms, the Maclaurin series for the cosine function at the point π p: Correctly found the sum of the series 5. τ : x + y = p: Correctly, with the purpose of finding the slope at the point, implicitly differentiated the equation with respect to x p: Correctly determined the slope dy dx at the point p: Correctly found an equation for the tangent line τ ()