WTW 158 : 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 Copyright reserved UNIVERSITY OF PRETORIA Department of Mathematics and Applied Maths Junie / June 004 Punte / Marks:50 Tyd / Time: 0 min Punte : Intern Marks : Internal Veranderinge Changes Punte : Ekstern Marks :External WTW 58 : CALCULUS EKSAMEN / EXAMINATION Eksterne eksaminator / External examiner: Me/Ms R Möller 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 0 (vrae -8) asook n merkleesvorm. 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 0 (questions -8) and an optic reader form. 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 removed from the venues 9. Show all computations clearly
2 Beantwoord vrae tot op die MERKLESERVORM se KANT Answer questions to on the OPTIC READER FORM on SIDE Vraag / Question Die waardeversameling van f x sin x is The range of f x sin x is (Notasie / Notation: sin x arcsinx bgsinx a, b 6, 6 c,0 d, Vraag / Question Bepaal / Find: lim x x x x x a lim x lim b lim x x c d x x x x x x Vraag / Question lim lnx e x a b e c d geen van hierdie nie / none of these Vraag 4 / Question 4 As / If f x x dan / then f x 4 a x 4 b x ln 4 c x x 4 d ln x Vraag 5 / Question 5 D x cos x 5 a sin x 5 c cos x sin x x 5 b cos x sin x 5 d geen van hierdie nie / none of these x Vraag 6 / Question 6 D x tan e x 6 a e x e 4x 6 b e x e 4x 6 c D x tan e x D x cote x e x cosec e x 6 d e x tan e x sec e x (Notasie / Notation: tan x arctanx bgtanx wtw exam p of 0
3 Vraag 7 / Question 7 e y e y dy 7 a e y e y 7 b e y 7 c e y y 7 d geen van hierdie nie / none of these Vraag 8 / Question 8 t sec t dt 8 a ln t tant 8 b t tant 8 c t 8 d geen van hierdie nie / none of these sec ttant Vraag 9 / Question 9 Watter van die volgende is korrek? / Which of the following is / are correct? i sinxcosx dx sin x ii sinxcosx dx cos x iii sinxcosx dx cosx 9 a slegs i is korrek / only i is correct 9 b i en ii is korrek / i and ii are correct 9 c al die antwoorde is korrek / all the answers are correct 9 d geen van hierdie nie / none of these Vraag 0 / Question 0 u u du 0 a u ln u 0 b u u ln u 0 c 0 d geen van hierdie nie / none of these u Vraag / Question ln y 4 y dy a y ln y 4 4 b ln y 4 4 d geen van hierdie nie / none of these c ln y ln y 4 4 wtw exam p of 0
4 Vraag / Question sin x dx a 6 cos x b c x sinx x sinx d geen van hierdie nie / none of these Vraag / Question Laat / Let f x coshx en / and g x sinhx. Vir watter waarde(s) van x sal f en g sny? For which value(s) of x will f and g intersect? a x.5 b f en g sny oneindig veel kere / f and g intersect infinitely many times c f en g sny mekaar nooit / f and g never intersect d geen van hierdie nie / none of these Vraag 4 / Question 4 Bepaal dy dx as cos xy y. / Find dy dx if cos xy y. 4 a cos xy 4 b y sin xy x sin xy 4 d geen van hierdie nie / none of these 4 c y sin xy x sin xy Vraag 5 / Question 5 Vind die funksie f wat aan die volgende voorwaardes voldoen: Find the function f that satisfies the following conditions: f e, f 0, f a e 4 5 b 4 e 4 5 d nie een van hierdie nie / none of these 5 c 4 e Vraag 6 / Question 6 Laat a,, en b 0,, vektore wees, dan is a b Let a,, and b 0,, be vectors then a b 6 a,4, 6 b 6 c i j k 6 d, 4, wtw exam p of 0
5 Vraag 7 / Question 7 Die afstand vanaf die punt P,,5 The distance from the point P,,5 na die YZ vlak is to the YZ plane is 7 a 7 b 7 c 0 7 d 9 Vraag 8 / Question 8 n Lyn L het vergelykings x, A line L has equations x, y y z. n Punt op die lyn L is z. A point on the line L is 8 a 0,, 8 b,, 8 c,, 8 d 0,, Vraag 9 / Question 9 n Lyn L het vergelykings x A line L has equations x y y z 4. n Vektor ewewydig aan die lyn L z 4. A vector parallel to the line L is is 9 a,,4 9 b,, 4 9 c,,4 9 d,,0 Vraag 0 / Question 0 Laat f n funksie wees waarvan ons die volgende weet. / Let f be a function with the properties lim x f x, lim x f x, lim f x en / and lim f x. x Watter van die volgende is korrek? / Which of the following is / are correct? 0 a f het n horisontale asimptoot / f has a horizontal asymptote 0 b f het n vertikale asimptoot x / f has a vertical asymptote x 0 c f het n vertikale asimptoot y / f has a vertical asymptote y 0 d Niks kan van f gesê word nie / Nothing can be said of f x Vraag & / Question & Laat / Let f x lim x sin x x 0 x sin x, x 0 a, x 0. a b 0 c die limiet bestaan nie / the limit does not exist d f is kontinu op R as a / f is continuous on R if a a b 0 c a kan nie bereken word nie / a cannot be calculated d Totaal / Total wtw exam p 4 of 0
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 clearly. Vraag / Question Laat / Let f x e x x met / with f x x e x x en / and f x 8x 4x e x x.. Bepaal (indien moontlik) die lokale maksimumwaarde van f. Find (if possible) the local maximum value of f.. Vind die interval(le), indien enige, waarop f konkaaf na bo is. Find the interval(s), if any, on which f is concave upward. wtw exam p 5 of 0
7 Vraag 4 / Question 4 Skets n funksie f wat kontinu is op R en die volgende voorwaardes bevredig. Sketch a function f that is continuous on R and satisfies the following conditions. f is onewe / f is odd f x 0 as / if 0 x f x 0 as / if x f x 0 as / if 0 x f x 0 as / if x lim x f x y Rofwerk / Rough work y x x 4 Vraag 5 / Question 5 Bepaal / Find (5.) t t dt. wtw exam p 6 of 0
8 (5.) x x dx. wtw exam p 7 of 0
9 Vraag 6 / Question 6 Laat / Let y x x. (6.) Bepaal / Find dy dx. (6.) Bepaal / Find lim x y. wtw exam p 8 of 0
10 Vraag 7 / Question 7 Stel vas of die lyne L en L mekaar sny of nie. Indien L en L mekaar sny, gee ook die koördinate van die snypunt. Determine whether the lines L and L intersect or not. If L and L intersect, give the co-ordinates of the point of intersection. L : x 4 t y,t R L : x s y s,s R z t z 4s 4 wtw exam p 9 of 0
11 Vraag 8 / Question 8 Dit kan aangetoon word dat die straal water wat uit n brandslang spuit, n hoogte van y m x mx bokant n punt x meter vanaf die spuitkop bereik. 80 m is die helling van die spuitkop en neem aan dat die spuitkop op grondvlak gehou word. It can be shown that the stream of water emitted by a fire hose will have height y 80 m x mx above a point located x metres from the nozzle. m is the slope of the nozzle and the nozzle is assumed to be on ground level. Helling m / Slope m Water Hoogte y / Height y Grondvlak / Ground level x meter / metres (8.) As m, bepaal die afstand x waar n maksimum hoogte bereik sal word. If m, determine the distance x that will result in maximum height (8.) As n brandweerman x 40 meter weg van n gebou moet staan, teen watter helling moet hy die spuitkop hou om die maksimum hoogte te bereik. If a firefighter must stand x 40 metres away from a building, at what slope must he aim to reach the maximum height. wtw exam p 0 of 0
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