! 1. Gegee / Given! 1. f#x$dx! 12 en / and!4. f#x$dx is. Die waarde van. f#x$dx is / The value of!1. 1 a " 9 1 b 9 1 c 3 1 d 15.

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1 Anwoorde vir WTW68 Semeseroes 4 / Answers for WTW68 Semeser es 4 Beanwoord vrae o op die MERKLESERVORM se KANT Toaal " Answer quesions o on he OPTIC READER FORM on SIDE Toal " Vraag/Quesion Gegee / Given f#$d en / and 4 f#$d. Die waarde van 4 f#$d is / The value of 4 f#$d is a " 9 b 9 c d Ans: b wan / since f#$d f#$d " 4 f#$d f#$d " # 4 f#$d " 9 Vraag/Quesion Die grafiek van y f#$ saamgesel ui n halfsirkel en wee reguilyne, word in die ondersaande skes geoon. The graph of y f#$ consiss of a semi-circle and wo sraigh lines as shown in he skech below. Y f X f#$d a.89 b.8 c.7 d. 7 Ans: d. 7 wan / since f#$d f#$d " f#$d " f#$d #$#$ " " #$#$ ". 7 8 Vrae&4/Quesions &4 Die grafiek van y f#$ word in die ondersaande skes geoon. The graph of y f#$ is shown in he skech below. ww68 4 semeser es p of 8

2 Y f As g#$ f#$d dan he g lokale maksima If g#$ f#$d hen g has local maima a a in en / a and b slegs in / only a c in en 6 / a and 6 d in, 4 en 7 / a, 4 and 7 Ans: c in en 6 / a and 6 wan g$#$ en g$ verander van posiief na negaief in en 6 since g$#$ and g$ changes from posiive o negaive a and 6 As g#$ f#$d dan he g n globale (absolue) maksimum in If g#$ f#$d hen g has an absolue maimum a 4 a 4 b 4 c 4 d 7 Ans: 4 b wan / since g# $ f#$d % 6 f#$d g#6 $ Vraag/Quesion e e d ln a ln " ln b " c e d e geen van hierdie nie / none of hese Ans: b " e wan / since e d ln #$ " #$ " e e #ln$ " # $d #ln $ e e #lne $ " #lne $ Vraag6/Quesion 6 an "d" ww68 4 semeser es p of 8

3 6 a sec " " " C 6 b an" " " " C 6 c an" " " " C 6 d 9 an " " C Ans: 6 c an" " " " C wan / since an "d" #sec " " $d" #sec "$d" " d" an" " " " C Vraag 7 / Quesion 7 u "u 4 du 7 a ln " u 4 " C 7 b arc " C 7 c arc " C 7 d ln " u4 " C (Noasie / Noaion: arcan an " bgan$ Ans: 7 c arc " C u wan / since du u du "u 4 " u arc " C Vraag8/Quesion 8 Die grafiek van y f#$ word in die ondersaande skes geoon. The graph of y f#$ is shown in he skech below. f As g#$ f#$d waer een van die skese is g se grafiek? If g#$ f#$d,hen which one of he skeches is he graph of g? ww68 4 semeser es p of 8

4 8a 8b g g 8c 8d g g Ans: 8c" wan / since g#$ f#$d g#$ f#$d &, #grafiek moe deur oorsprong gaan /herefore he graph mus inersec wih he origin g#$ f#$d f#$d " f#$d f#$d " f#$d g#$ f#$d f#$d " f#$d " f#$d % Vraag 9 / Quesion 9 sin cosd " 6 " 9 a "6 9 b 9 c " 9 d " 9 e geen van hierdie nie / none of hese Ans: 9 a "6 sin#a " b$ sin acosb " cosasin b...#i$ wan / since sin#a " b$ sin acosb " cosasin b...#ii$ # sin#a " b$ " sin#a " b$ sinacosb ui / from..#i$ " #ii$ # sin cosd " " sin# " $ " sin# " $"d " " sin#$ " sin#$"d " cos #$ " cos#$ " " " " " " "6 cos " cos cos #"$ " cos#"$ " ww68 4 semeser es p 4 of 8

5 Vraag / Quesion d d co# " $d " a ln sin#"99$ " ln sin# " $ b co# " $ c co# " $ d geen van hierdie nie / none of hese Ans: b co# " $ d wan / since d co# " $d d " " co# " $d " d " co# " #" $$"#"$ co# " $ BEANTWOORD ALLE VERDERE VRAE OP HIERDIE VRAESTEL en TOON alle bewerkings duidelik aan ANSWER ALL THE FOLLOWING QUESTIONS ON THE SCRIPT and SHOW all compuaions clearly. Vraag / Quesion #a$ Gebruik die middelpunreël me n 4 om bgsin d e benader. Gee jou anwoord o desimale akkuraa. Use he midpoin rule wih n 4 o esimae arcsin d. Give your answer correc o decimals places. Ans: % " 4 4 Parisie / Pariion: &,,,,' " Middelpune / Midpoins: 4 ; 4 " 4 ; 4 " 4 ; arcsin d arcsin " arcsin " arcsin " arcsin arcsin " arcsin " arcsin " arcsin #. ".844 ".67 ".64$ 4 #.$ " 7 8 4" #b$ Konroleer jou anwoord in #a$ deur bgsin d e bepaal. (m.a.w. inegreer) Check your answer in #a$ by evaluaing arcsind. (i.e. inegrae) Ans: bgsin d arcsin d arcsin( " d " arcsin" arcsin" " # " $ " #"$ d " " " # " $ " # " $ " # " $ ".78.7 Of / Or Laa / Le arcsin u # sin u en / and cosudu d ww68 4 semeser es p of 8

6 As / If dan is / hen u arcsin As / If dan is / hen u arcsin # arcsin d ucosudu #sin u$u( " sin udu sin " #sin $ " cosu( " cos " cos ".78.7 " Vraag / Quesion Diegrafiekevan y " " en y " word in die ondersaande skes geoon. The graphs of y " " and y " are shown in he skech below. y Bepaal die oppervlake van die ingekleurde gebied. Find he area of he shaded region. Ans: Snypune / Poins of inersecion: " " " # " # # " $ # of / or " Area " #" " $ " # " $"d " " " "d " " "#$ " #$ "#"$ " " #"$ " 6 " Vraag / Quesion Bepaal / Evaluae " # " $d. # " $d "# " #"$$d.# " #$$d Ans: " d " "d " #$ " #"$ " " " " "#$ " "#$ " " 9 "6 Of / Or: Diegrafiekvan/The graph of f#$ " ww68 4 semeser es p 6 of 8

7 Area #$#"$ " #$#"$ "6 " Vraag 4 / Quesion 4 Die gemiddelde waarde van n funksie f op n inerval a, b" word gedefinieer deur b b"a f#$d. Bepaal die gemiddelde waarde van f#$ ln op,". a The average value of a funcion f on an inerval a, b" is defined by b f#$d. Find he average value of a funcion f#$ ln on,". b"a a Ans: b b"a f#$d a " lnd lnd ln " d ln " ln " d #9ln$ " 9 #9ln$ " 9 " 9 9 ln " " Vraag / Quesion Bepaal / Evaluae " d. Ans: Laa / Le # d sec udu " # d "an u sec udu sec u du cos u of / or ' du sinu cos u secusec u du sinucos u du secu an u" du secuan u secuan u cos " u sinu cos u sinu sec udu...sien / see $$ " secu du sinu # " an u$du du sinu " sin u cos u secuan u " du " sinu du sinu sinu cos u sec u " cos ecu"du cos ecu " sin ucos " u"du sec udu " cos ecu cos ecu"co u cos ecu"co u du du cos ecu"co u of / or cos ecu cos ecu"co u du " sin ucos" udu sec u " ln cos ecu " co u " C ln cosecu " co u " cos " u " c sec u " ln cos ecu " co u " C ln cosecu " co u " sec u " c ww68 4 semeser es p 7 of 8

8 " " ln " " " C ln " " " " " c (As / If dan is / hen co u, sec u " # sec u " en / and cosec u co u " # $ " # cosecu # $ " " of gebruik $ n driehoek / or use a riangle.) $$ sinu sec udu cosecusec udu cosecu " #" cosecuco u$du cosecu " cosecu# $du cosecu " cosecudu cosecu " ln cosecu " co u " C {Ne erloops / by he way: ln cosecu " co u ln cos ecu"co u ln cos ecu"co u ln cosecu " co u " " ln cosecu " co u ' cos ecu"co u cos ecu"co u ln cos ec u"co u cos ecu"co u 4" ww68 4 semeser es p 8 of 8