H¨ogskolan i Sk¨ovde Tentamen i matematik
Kurs: MA152G Matematisk analys
MA123G Matematisk analys f¨or ingenj¨orer Tentamensdag: 2016-03-12 kl 14.30-19.30
Hj¨alpmedel : Inga hj¨alpmedel ut¨over bifogat formelblad. Ej r¨aknedosa.
English version follows after the Swedish.
Tentamen bed¨oms med betyg 3, 4, 5 eller underk¨and, d¨ar 5 ¨ar h¨ogsta betyg. F¨or godk¨ant betyg (3) kr¨avs minst 17 po¨ang fr˚an uppgifterna 1–10, varav minst 3 po¨ang fr˚an uppgifterna 8–10.
Var och en av dessa nio uppgifter kan ge maximalt 3 po¨ang. F¨or var och en av uppgifterna 1–6 kan man v¨alja att i st¨allet f¨or att l¨amna svar utnyttja sitt resultat fr˚an motsvarande dugga fr˚an kurstillf¨allet vt 2016 (duggaresultatlista bifogas). Markera detta genom att skriva ett D ist¨allet f¨or ett kryss i uppgiftsrutan p˚a omslaget. Uppgift 7 kan ers¨attas med aktivt deltagande vid r¨akne¨ovningar under kursens g˚ang. F¨or betyg 4 kr¨avs ut¨over godk¨ant resultat fr˚an 1–10 minst 50% (12 po¨ang) fr˚an uppgift 11–14, f¨or betyg 5 minst 75% (18 po¨ang).
L¨amna fullst¨andiga l¨osningar till alla uppgifter, om inte annat anges. Skriv inte mer
¨an en uppgift p˚a varje blad. Numeriska v¨arden kan anges som uttryck d¨ar faktorer som π och logaritmer ing˚ar ut¨over ”rena siffror” om s˚a beh¨ovs.
Del I. Uppgift 1–10 r¨aknas f¨or godk¨ant betyg. Varje uppgift kan ge upp till 3 po¨ang. F¨or godk¨ant (betyg 3–5) kr¨avs minst 17 po¨ang, varav minst 3 po¨ang p˚a uppgift 8–10. Uppgift 1–6 kan en och en ers¨attas av duggapo¨ang, uppgift 7 genom att ha deltagit i r¨akne¨ovningar.
1. L˚at f (x) =√ x3− 1
(a) Vilken ¨ar (den st¨orsta m¨ojliga) definitionsm¨angden Df f¨or f ? (Vi kr¨aver reella tal som v¨arden.)
(b) f ¨ar str¨angt v¨axande i sin definitionsm¨angd och d¨arf¨or inverterbar. Ange ett uttryck f¨or f−1(x).
(c) Vad har f−1 f¨or definitionsm¨angd, Df−1? 2. L˚at
f (x) = sin π2(x − 2) (4x − 8)(x − 1).
Best¨am f¨oljande gr¨ansv¨arden, om de existerar. Gr¨ansv¨ardet kan vara ett tal, +∞ eller
−∞. Om ett gr¨ansv¨arde inte skulle finnas, ange och motivera detta. (Tips: Se formelbla- det f¨or gr¨ansv¨ardet av sin xx d˚a x → 0.)
(a) lim
x→2f (x) (b) lim
x→1+f (x) (c) lim
x→+∞f (x)
3. Ekvationen
y2− 2xy + ln(x2) = 0
definierar en kurva i xy-planet och inneh˚aller punkten (x, y) = (1, 2). Best¨am kurvans tangentlutning dydx i denna punkt.
4. Betrakta funktionen f (x) = xe(4−x2)/8, definierad p˚a (−∞, ∞) (a) Best¨am eventuella lokala extremv¨arden till f (x), f¨or vilka x de antas, om de ¨ar minima eller maxima. (b) Utred ifall f (x) har n˚agot absolut maximum eller minimum, dvs ett st¨orsta och/eller minsta v¨arde globalt, och vad de i s˚a fall ¨ar.
5. Best¨am v¨ardet av integralen
Z e2 1
(ln x)3 x dx.
6. Best¨am en primitiv funktion F (x) (dvs F0(x) = f (x)) till funktionen f (x) = (2x + 1)ex/2
s˚adan att F (0) = 0.
7. En rektangul¨ar l˚ada (r¨atblock) utan lock ska ha volymen 10 m3. F¨or l˚adans botten g¨aller att den ena sidan ska vara dubbelt s˚a l˚ang som den andra. Materialet f¨or l˚adans botten kostar 10 kronor/m2, medan materialet f¨or l˚adans sidor kostar 6 kronor/m2. Hur mycket kostar materialet till en s˚adan l˚ada om man v¨aljer m˚atten (under ovan angivna villkor) s˚a att kostnaden blir minimal?
8. Best¨am en l¨osning y = f (x) (x > 0) till differentialekvationen
xdy
dx = x2+ x y som uppfyller villkoret f (1) = 3.
9. Best¨am den allm¨anna l¨osningen till differentialekvationen dy
dx+ 4xy = x.
10. (a) [1p] Best¨am den allm¨anna l¨osningen y = y(x) till den homogena differentialekvationen y00+ 4y0+ 29y = 0.
(b) [2p] Best¨am l¨osningen y = y(x) till begynnelsev¨ardesproblemet y00+ 4y0+ 29y = 29x, y(0) = 1, y0(0) = 0.
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Del II. F¨oljande uppgifter r¨aknas f¨or betyg 4 och 5. Varje uppgift kan ge upp till 6 po¨ang, totalt 24. ¨Aven presentationen bed¨oms.
11. Tv˚a punkter A och B ¨ar sk¨arningspunkterna mellan parabelkurvan y = x2 och en linje y = kx + m. Best¨am den punkt P p˚a parabelns b˚age mellan A och B som g¨or att arean av triangeln ABP blir maximal. Vilken denna punkt ¨ar kommer f¨orst˚as att bero p˚a punkterna A och B (eller linjeparametrarna k och m).
12. Best¨am om integralen
Z 1 0
ln x + 1
√1 − x
dx
¨ar konvergent, och i s˚a fall, dess v¨arde.
13. En kropp i ett xyz-koordinatsystem begr¨ansas av planen x = 0 och x = b > 0 samt av ytan (y2+ z2)(1 + x)3 = 1. (a) Visa att den ¨ar rotationssymmetrisk kring x-axeln.
Best¨am kroppens volym, i koordinatsystemets volymenheter, som en funktion av b, dels (b) genom skivmetoden, dels (c) genom metoden med cylindriska skal.
(d) Har volymen n˚agot gr¨ansv¨arde d˚a b → ∞?
14. Best¨am den allm¨anna l¨osningen (f¨or x > −1/3) till differentialekvationen (1 + 4x2)(3x + 1)dy
dx = (2x + 3)y.
Lycka till! /SK
The exam is graded 5, 4, 3 or U, where 5 is the highest grade and U is fail. For passed result (grade 3) at least 17 points are needed from problems 1–10 (Part I), among these at least 3 points from problems 8–10. Each of these 10 problems may yield 3 points. From each of problems 1–6 you may choose to use the results from the pre-tests (dugga) instead of giving a solution to the exam problems. (The results from the pre-tests are found appended.) In case the pre-test result is used no solution shall be given to the exam problem, and you shall write a D instead of an X in the corresponding square on the envelope. If you have been active in the tutorials during the course, the points for problem seven will be awarded for free. For grade 4 the requirements for grade 3 shall be met, and further at least 50% (12 points) in part II (problems 11–14). For grade 5 at least 75% (18 points) in part II is required.
Give full solutions to all problems. Don’t write more than one problem at each page, use only one side of the sheet. Numerical values may be given as expressions including factors like π and logarithms, if needed.
Part I. Problems 1–10 is for passing. Each problem can give up to 3 points. To pass the course (grade 3–5) at least 17 points are required, whereof at least 3 points from problems 8–10. Problems 1–6 may one by one be substituted for by pre-test grades, problem 7 by participating in tutorials during the course.
1. Let f (x) =√ x3− 1
(a) Which is the (largest possible) domain Df of f ? (We require real values.)
(b) f is strictly increasing in its domain, and is therefore invertible. Find an expression for f−1(x).
(c) Which is the domain Df−1 of f−1? 2. Let
f (x) = sin π2(x − 2) (4x − 8)(x − 1).
Find the following limits, if they exist. The limit may be a number, +∞ or −∞. If a limit should not exist, state and motivate this. (Hint: See cheat sheet for the limit of sin xx as x → 0.)
(a) lim
x→2f (x) (b) lim
x→1+f (x) (c) lim
x→+∞f (x)
3. The equation
y2− 2xy + ln(x2) = 0
defines a curve in the xy plane which contains the point (x, y) = (1, 2). Find the tangent slope dydx of the curve in this point.
4. The function f (x) = xe(4−x2)/8 is defined in (−∞, ∞) (a) Find the local extreme values of f (x), if they exist, and for which x they are taken, and for each, find if it is a local minimum or maximum. (b) Examine if f (x) has any absolute maximum or minimum, that is, a greatest and/or least value globally, and in that case, their values.
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5. Find the value of the integral
Z e2 1
(ln x)3 x dx.
6. Find a primitive function (anti-derivative) F (x) (i.e. F0(x) = f (x)) to the function f (x) = (2x + 1)ex/2
such that F (0) = 0.
7. A rectangular box without a top lid shall have a volume of 10 m3. For the base of the box one side shall have twice the length of the other. The material for the rectangular bottom costs 10 kronor/m2, while the material for the sides costs 6 kronor/m2. How much does the material for a box cost, if we choose the dimensions (under the conditions given) so that the cost is minimized=?
8. Find a solution y = f (x) (x > 0) of the differential equation
xdy
dx = x2+ x y which fulfills the condition f (1) = 3.
9. Find the general solution of the differential equation dy
dx+ 4xy = x.
10. (a) [1p] Find the general solution y = y(x) of the homogeneous differential equation y00+ 4y0+ 29y = 0.
(b) [2p] Find the solution y = y(x) of the initial value problem y00+ 4y0+ 29y = 29x, y(0) = 1, y0(0) = 0.
Part II. The following problems are for grades 4 and 5. Each problem yields up to 6 points.
The assessment also includes the presentation.
11. Two points A and B are the intersection between the parabola y = x2 and a line y = kx + m. Find the point P on the arc of the parabola between A and B which maximizes the area of the triangle ABP . This point will of course depend on the points A and B.
12. Find whether the integral
Z 1 0
ln x + 1
√1 − x
dx is convergent or not. If so, find its value.
13. A solid in the xyz coordinate system is bounded by the planes x = 0 and x = b > 0 the surface (y2+ z2)(1 + x)3 = 1. (a) Show that the solid is a solid of revolution around the x axis.
Find the volume, in the volume units of the coordinate system, as a function of b, (b) by cross sections, and (c) by cylindrical shells.
(d) Does the volume have a limit as b → ∞? Which?
14. Find the general solution (for x > −1/3) of the differential equationen (1 + 4x2)(3x + 1)dy
dx = (2x + 3)y.
Good luck! /SK
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