WTW 158 : CALCULUS EKSAMEN / EXAMINATION Eksterne eksaminator / External examiner: Prof NFJ van Rensburg
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1 Outeursreg voorbehou UNIVERSITEIT VAN PRETORIA Departement Wiskunde en Toegepaste Wiskunde Copyright reserved UNIVERSITY OF PRETORIA Department of Mathematics and Applied Maths 4 Junie / June 00 Punte / Marks:50 Tyd / Time: 50 min Punte : Intern Marks : Internal Veranderinge Changes Punte : Ekstern Marks :External WTW 58 : CALCULUS EKSAMEN / EXAMINATION Eksterne eksaminator / External examiner: Prof NFJ van Rensburg Van / Surname : Voorname / First names: Studentenommer / Handtekening / Student number: Signature: Studierigting / Field of Study: B C E L M N P R S Z Ander / Other Telefoonnommer gedurende eksamenperiode / Telephone number during exam period Interne eksaminatore / Internal examiners Mev / Mrs A E du Preez Prof JC Engelbrecht Dr S M Maepa Prof M J Schoeman Interne eksaminatore / Internal examiners Lees eers die volgende instruksies. Slegs nie -programmeerbare sakrekenaars mag gebruik word.. Die vraestel bestaan uit bladsye tot 9 (vrae -7) Kontroleer of u vraestel volledig is.. Doen alle krapwerk op die blanko bladsy(e). Dit word nie nagesien nie. 4. As u meer as die beskikbare ruimte vir n antwoord nodig het, gebruik dan ook die blanko bladsy(e) en dui dit duidelik aan deur n raam daarom te trek. 5. Geen potloodwerk word nagesien nie. 6. Korreksievloeistof (bv.tipp-ex ) mag nie gebruik word nie. 7. Geen antwoordboek mag uit die lokaal geneem word nie 8. Toon alle bewerkings duidelik First read the following instructions. Only non-programmable calculators may be used.. The question paper consists of pages to 9 (questions -7). Check if your paper is complete.. Scribbling can be done on the empty page(s). This will not be marked. 4. If you need more than the available space for an answer, you may also use the empty page(s). Indicate it clearly by framing it. 5. Work in pencil will not be marked. 6. Use of correction fluid (e.g.tipp-ex) is not allowed. 7. No test scripts may be re moved from the venues 9. Show all computations clearly
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3 Vraag / Question Skets die grafiek van f x bgtan x. Sketch the graph of f x arctan x. Y - 0 X - Vraag / Question Bepaal dy dx as / Determine dy dx if i y x x ii y x iii x y xy wtw58 exam p of 9
4 Vraag / Question Bepaal / Evaluate i x x 4 dx ii x cosx dx iii cosh x dx iv sin x dx x wtw58 exam p of 9
5 Vraag 4 / Question 4 Bepaal die vergelyking van die raaklyn aan die grafiek van f x bgsinx by x. Find the equation of the tangent line to the graph of f x arcsinx at x. Vraag 5 / Question 5 Bepaal n formule vir f n x as f x ln x, n N. Find a formula for f n x if f x ln x, n N. Vraag 6 / Question 6 Laat / Let f x x x, f x x en / and f x x x x. 4 Toon alle berekeninge en bepaal die Show all computations and determine the i vertikale asimptoot van f. vertical asymptote of f. wtw58 exam p of 9
6 ii horisontale asimptoot van f. horizontal asymptote of f. iii lokale minimumwaarde van f. local minimum value of f. iv interval(le) waarop f konkaaf na bo is. interval(s) where f is concave upward. wtw58 exam p4 of 9
7 Vraag 7 / Question 7 Skets n funksie f met die volgende eienskappe : Sketch a function f with the following properties: f 0, f 0 0 f x 0 as / if 0 x f x 0 as / if x 0 en / and x f x 0 as / if x lim x f x Y Rofwerk / Rough work Y - X - X Vraag 8 / Question 8 Gegee / Given f x x a x a, as / if x a 4, as / if x a i Bepaal die waarde van a waarvoor f kontinu is op R. Determine the value of a such that f is continuous on R. wtw58 exam p5 of 9
8 ii Skets die grafiek van f vir hierdie a. Sketch the graph of f for this a. Y X Vraag 9 / Question 9 Bepaal / Evaluate sin xcos xdx Vraag 0 / Question 0 Die grafiek van y f x gaan deur,. As f x x, bepaal f x. The graph of y f x passes through,. If f x x, determine f x. wtw58 exam p6 of 9
9 Vraag / Question Bepaal die afmetings van die reghoek met maksimum oppervlakte wat in n reghoekige driehoek met afmetings soos in die gegewe skets, ingeskryf kan word. Determine the dimensions of the rectangle with maximum area that can be inscribed in a right angled triangle with dimensions as in the given sketch. 6 y x 4 Vraag / Question Bepaal n eenheidsvektor loodreg (ortogonaal) op beide a i j en b i k. Determine a unit vector perpendicular (orthogonal) to both a i j and b i k. wtw58 exam p7 of 9
10 Vraag / Question Bepaal die vektorprojeksie van a i k op b i j. Determine the vector projection of a i k on b i j. Vraag 4 / Question 4 Bepaal simmetriese vergelykings van die lyn deur die oorsprong en wat ewewydig is aan die lyn met parametriese vergelykings x t, y t, z t, t R. Determine symmetric equations for the line through the origin and parallel to the line with parametric equations x t, y t, z t, t R. Vraag 5 / Question 5 Bepaal n vergelyking van die vlak deur die punt,, en loodreg op die lyn met vergelykings x, y z. Determine an equation for the plane through the point,, and perpendicular to the line with equations x, y z. wtw58 exam p8 of 9
11 Vraag 6 / Question 6 Bepaal n vergelyking van die vlak deur die punt,6,5 en ewewydig aan die vlak x y z 4. Determine an equation for the plane through the point,6,5 and parallel to the plane x y z 4. Vraag 7 / Question 7 Beskryf (of skets) die gebied in R wat voorgestel word deur x y 4. Describe (or sketch) the region in R that is represented by x y 4. wtw58 exam p9 of 9
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