Punte: Intern Marks: Internal WTW 168 : CALCULUS. EKSAMEN / EXAMINATION Eksterne eksaminator / External examiner: Me / Ms R Möller

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1 Outeursreg voorbehou UNIVERSITEIT VAN PRETORIA Departement Wiskunde en Toegepaste Wiskunde Copright reserved UNIVERSITY OF PRETORIA Department of Mathematics and Applied Maths November 005 Punte / Marks:35 Td / Time: 90 min Punte: Intern Marks: Internal Veranderinge Changes Ekstern Eternal WTW 68 : CALCULUS EKSAMEN / EXAMINATION Eksterne eksaminator / Eternal eaminer: Me / Ms R Möller Van / Surname: VOORNAME / FIRST NAMES: STUDENTNOMMER: STUDENT NUMBER: HANDTEKENING SIGNATURE: Studierigting / Field of Stud B C E L M N P R S Z Ander / Other Telefoonnommer gedurende eksamenperiode / Telephone number during eam period Interne eksaminatore / Internal eaminers Me / Ms AE du Preez Prof MJ Schoeman Mnr / Mr HW van Wk Lees eers die volgende instruksies. Slegs nie-programmeerbare sakrekenaars mag gebruik word.. Die vraestel bestaan uit bladse tot (vrae -6) ) asook n merkleesvorm. Kontroleer of u vraestel volledig is. 3. Doen alle krapwerk op die vraestel. Dit word nie nagesien nie.. As u meer as die beskikbare ruimte vir n antwoord nodig het, gebruik dan enige blanko spasie en dui dit duidelik aan deur n raam daarom te trek. 5. Geen potloodwerk word nagesien nie. 6. Korreksievloeistof (bv.tipp-e ) mag nie gebruik word nie. 7. Geen antwoordboek mag uit die lokaal geneem word nie 8. Toon alle bewerkings duidelik Interne eksaminatore / Internal eaminers Dr SM Maepa Me / Ms E Smith First read the following instructions. Onl non-programmable calculators ma be used.. The question paper consists of pages to (questions -6). as well as an optic reader form Check if our paper is complete. 3. Scribbling can be done on the script. This will not be marked.. If ou need more than the available space for an answer, ou ma also use an empt space. Indicate it clearl b framing it. 5. Work in pencil will not be marked. 6. Use of correction fluid (e.g.tipp-e) is not allowed. 7. No test scripts ma be removed from the venues 8. Show all computations clearl

2 WTW68 eksamen 005 / WTW68 eam 005 Beantwoord vrae tot op die MERKLESERVORM se KANT Answer questions to on the OPTIC READER FORM on SIDE Vraag / Question sin 3 d a b 0 c sin tan sin tan d e Kan nie gevind word nie / Can not be found Notasie / Notation: tan arctan bgtan. Vraag / Question Om 3 d te bepaal, sal die volgende substitusie geskik wees To evaluate 3 d the following substitution will be appropriate a 3 tan,, d 3 sin,, b 3 c 3 tan,, e Vrae 3 tot / Questions 3 to Laat / Let f. Vraag 3 / Question 3 Die grafiek van f is The graph of f is 3a - 3b 3c 3d e Vraag / Question Die area van die omwentelingsliggaam wanneer f, 0, om die as wentel, word gegee deur The area of the surface of revolution when f, 0, revolves about the ais is given b a 0 d b 0 5 d c 0 5 d d 0 5 d e wtw68 eam 005 p of

3 Vraag 5 / Question 5 Beskou die kromme met parametriese vergelkings e t, e t, t R. Die grafiek van hierdie kromme word gegee deur Consider the curve with parametric equations e t, e t, t R. The graph of this curve is given in 5a 5b 5c 5d 5e Vraag 6 / Question 6 Die gebied in die Cartesiese vlak gedefinieer deur die pool ongelkhede 0, r word gegee in The region defined b the polar inequalities 0, r is given in 6a 6b 6c 6d e Vraag 7 / Question 7 Die definisieversameling van die vektorwaardige funksie r t cost i sint j tk is The domain of the vector valued function r t cost i sint j tk is 7a 0, 7b, 7c, 7e 7d 0, Vraag 8 / Question 8 Gegee / Given f,. Die koördinate van die punt waar f, f, 0 is, is The coordinates of the point where f, f, 0 are 8a 0,0,0 8b,0,0 8c,0, 8e 8d 0,, wtw68 eam 005 p of

4 Vraag 9 / Question 9 Die definisieversameling van die funksie f, ln word gegee deur The domain of the function f, ln is given b 9a 9b 9c 9d = e 0, Vraag 0 / Question 0 Die grafiek van f, is The graph of f, is 0a z 0b z 0c z 0d z 0e Vraag / Question Die 3de graad Maclaurin-polinoom (Talor-polinoom om 0) vir f sin word gegee deur The Maclaurin polnomial (Talor polnomial about 0) of degree 3 for f sin is given b a 3 6 b 3 e 6 c 3 8 d 3 3 wtw68 eam 005 p 3 of

5 Ans: b Vrae tot 3 / Questions to 3 Gegee / Given z 6. Vraag / Question Die gegewe oppervlak(liggaam) sn die vlak in n The given surface intersects the plane in a a parabool / parabola b sirkel / circle c ellips / ellipse d reguit ln / straight line e Vraag 3 / Question 3 Die vergelking van die raakln aan die kromme in Vraag, in die punt,, is The equation of the tangent line to the curvein Question, at the point,, is 3a z, 3b 3c z, 3d 3e Vraag / Question Die parsiële breuke van The partial fractions of is are a e b c d Totaal / Total wtw68 eam 005 p of

6 BEANTWOORD ALLE VERDERE VRAE OP HIERDIE VRAESTEL en TOON alle bewerkings duidelik aan. ANSWER ALL THE FOLLOWING QUESTIONS ON THE SCRIPT and SHOW all computations clearl. Vraag 5 / Question Beskou die skets van f sin. Consider the sketch of f sin Benader sin d deur gebruik te maak van die Middelpuntreël met n 5. (Gee jou antwoord 0 tot desimale akkuraat). Verduidelik ook watter oppervlaktehierdie benadering voorstel deur dit op die skets hierbo aan tetoon. Approimate sin d b using the Midpoint Rule with n 5. (Give our answer correct to 0 decimals). Also, eplain which area this approimation represents b indicating it on the sketch above. wtw68 eam 005 p 5 of

7 Vraag 6 / Question 6 Gegee / Given e d e e C. Bepaal of e d konvergeer of divergeer.toon alle berekeninge. 0 Determine whether e d converges or diverges. Show all computations. 0 3 Vraag 7 / Question 7 Bepaal / Evaluate i arcsin 3 d (Notasie / Notation: arcsin sin bgsin). 3 wtw68 eam 005 p 6 of

8 ii d 3 iii tan 3 d Vraag 8 / Question 8 Vind as d dt Find if d dt t 3t t 3t en 3 0. and 3 0. wtw68 eam 005 p 7 of

9 Vraag 9 / Question 9 Vind parametriesevergelkings vir die raaklnaan die vektorwaardige funksie r t sint i t cost j e t k in die punt waar t 0. Find parametric equations for the tangent line to the vector valued function r t sint i t cost j e t k at the point where t 0. Vraag 0 / Question 0 Laat / Let u 5. Is u u? Verduidelik / Eplain.. wtw68 eam 005 p 8 of

10 Vraag / Question Laat / Let z f, Skets die kontoerkrommes vir f, 6 6 vir k 0,,3 en. Dui die k waardes op die skets aan. Sketch the level curves for f, 6 6 for k 0,,3 and. Indicate the k values on the sketch Vind die vergelking van die raakvlak aan z f, in die punt,,. Find an equation of the tangent plane to z f, at the point,,. 5 wtw68 eam 005 p 9 of

11 Vraag / Question n Metaalplaat is geleë in die vlak en beslaan die reghoek 0 0, 0 8, waar en in meters gemeet word. Die temperatuur in die punt, op die plaat is T,, waar T gemeet word in grade Celsius. Temperature is gemeet in puntewat ewever van mekaar af is en getabuleer. A metal plate is situated in the plane and occupies the rectangle 0 0, 0 8, where and are measured in meters. The temperature at the point, on the plate is T,, where T is measured in degrees Celsius. Temperatures at equall spaced points were measured and recorded in the table. \ Benader die waarde van T 6,. Estimate the value of T 6,.. Benader die waarde van T 6,. Estimate the value of T 6,. wtw68 eam 005 p 0 of

12 Vraag 3 / Question 3 Twee partikels beweeg langs die ruimtekrommes r t t,7t,t en r t t 3,t,5t 6 vir t 0. Sal die partikels bots? Verduidelik. Two particles travel along the space curves r t t,7t,t and r t t 3,t,5t 6 for t 0. Do the particles collide? Eplain. 3 wtw68 eam 005 p of