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© Mana Tohu Mātauranga o Aotearoa, 2013. Pūmau te mana.Kia kaua rawa he wāhi o tēnei tuhinga e tāruatia ki te kore te whakaaetanga a te Mana Tohu Mātauranga o Aotearoa.
MĀ TE KAIMĀKA ANAKE
TAPEKE
Te Pāngarau me te Tauanga, Kaupae 2, 201391262M Te whakahāngai tikanga tuanaki hei whakaoti
rapanga
2.00 i te ahiahi Rāhina 18 Whiringa-ā-rangi 2013 Whiwhinga: Rima
Paetae Paetae Kaiaka Paetae KairangiTe whakahāngai tikanga tuanaki hei whakaoti rapanga.
Te whakahāngai tikanga tuanaki mā te whakaaro whaipānga hei whakaoti rapanga.
Te whakahāngai tikanga tuanaki mā te whakaaro waitara hōhonu hei whakaoti rapanga.
Tirohia mehemea e ōrite ana te Tau Ākonga ā-Motu (NSN) kei tō pepa whakauru ki te tau kei runga ake nei.
Me whakautu e koe ngā pātai KATOA kei roto i te pukapuka nei.
Whakaaturia ngā mahinga KATOA.
Ki te hiahia koe ki ētahi atu wāhi hei tuhituhi whakautu, whakamahia te (ngā) whārangi kei muri i te pukapuka nei, ka āta tohu ai i ngā tau pātai.
Tirohia mehemea kei roto nei ngā whārangi 2 – 29 e raupapa tika ana, ā, kāore hoki he whārangi wātea.
HOATU TE PUKAPUKA NEI KI TE KAIWHAKAHAERE HEI TE MUTUNGA O TE WHAKAMĀTAUTAU.
See back cover for an English translation of this cover
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Kia 60 meneti hei whakautu i ngā pātai o tēnei pukapuka.
PĀTAI TUATAHI
(a) Katohuahepāngafmātef (x) = 4x2 – 5x + 2.
Tātaihiaterōnakiotekauwhataofitepūwāhix = 3.
(b) Mōtētahipāngag,
g′(x) = 6x2 – 5.
Kawhakawhititekauwhataogmātepūwāhi(1,4).
Kimihiatepāngag(x).
You are advised to spend 60 minutes answering the questions in this booklet.
QUESTION ONE
(a) Afunctionfisgivenbyf (x) = 4x2 – 5x + 2.
Findthegradientofthegraphoffatthepointwherex = 3.
(b) Forafunctiong,
g′(x) = 6x2 – 5.
Thegraphofgpassesthroughthepoint(1,4).
Findthefunctiong(x).
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(c) Katukunatētahimuraahotea1maiitētahipoti.
Kotōnateitei,hehmitairungaakeitemataotewai,katohuamāte
h = 90t – 5t2 + 2
kottewāā-hēkonamaiitetukutangaotemura.
Heahateteiteimōrahirawaitaeaetemura?
(d) Kotetawhitiāwhio(tepaengaporowhita)otētahirākauhegmita,hettauteroaotewāmaiitōnawhakatōnga,kawhakatauiratiamātepānga
g = – 0.005t 2+0.15t+0.30≤t≤15.
Āheakaeketepāpātangaotetipuotepaengaporowhitaoterākaukite0.04mitaitetau?
1 ohotata
(c) Anemergencyflareisfiredfromaboat.
Itsheight,hmetresabovethesurfaceofthewater,isgivenby
h = 90t – 5t2 + 2
wheretisthetimeinsecondssincetheflarewasfired.
Whatisthemaximumheightreachedbytheflare?
(d) Thedistancearoundatree(itsgirth)gmetres,atatimetyearsafteritisplanted,ismodelledbythefunction
g = – 0.005t 2+0.15t+0.30≤t≤15.
Whenwilltherateofgrowthofthetree’sgirthbe0.04metresperyear?
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(e) g(x) = – x3 + 3x + 2
Mōēheauaraoxkogtepāngahekehaere?
Memātuawhakaatu e koe ngā whakamahinga tuanakiirotoiōmahinga.
(f) Kotepāngarōnakiotētahikōpikokotef ′ (x) = mx+2.Kapātekōpikokingāpūwāhi(2,10)and(–1,– 8).
Kimihiaaf (x),tewhāriteotekōpiko.
(e) g(x) = – x3 + 3x + 2
Forwhatvaluesofxisgadecreasingfunction?
Youmustshow the use of calculusinyourworking.
(f) Acurvehasgradientfunctionf ′ (x) = mx+2.Thecurvepassesthroughthepoints(2,10)and(–1,–8).
Findf (x),theequationofthecurve.
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PĀTAI TUARUA
(a) Tuhiatepāngarōnakif ′ (x)mōtepāngaf (x)iraro:
x
f (x)
x
f '(x)
Ki te hiahia koe ki te tā anō i tēnei kauwhata,
whakamahia te tukutuku i te whārangi 26.
QUESTION TWO
(a) Sketchthegradientfunctionf ′ (x)forthefunctionf (x)below:
x
f (x)
x
f '(x)
If you need to redraw this graph, use the
grid on page 27
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(b) Keitewhakakīahetaika2kitemiraka.Kotehōhonuotemirakahedcm,itetmenetiimuriiterututangakatohuamāte
t t td( )4
2
= +
Kimihiatepāpātangaererekēaitehōhonuotemirakaite5menetiimuriitetīmatangaoterututanga.
(c) Kawhakatakahiahekōhatukirotoitētahihōpuawai.
Kawhakaputaingāpōkareporohitahitaitemataotewai.
KotehorahangaA otētahipōkareporohitahita,ā-mitapūrua,katohuamāte
A=π r2
inakotepūtorohermita.
Kimihiatepāpātangaotewhitiotehorahangaotepōkare,epāanakitepūtoro,inakotehorahangahe49πm2.
2kura
(b) Atankisbeingfilledwithmilk.Thedepthofthemilkdcm,atatimetminutesafterpouringstartedisgivenby
t t td( )4
2
= +
Findtherateatwhichthedepthofthemilkischanging5minutesafterthepouringstarted.
(c) Astoneisdroppedintoapool.
Thismakescircularripplesonthesurfaceofthewater.
TheareaA ofacircularripple,insquaremetres,isgivenby
A=πr2
wheretheradiusisrmetres.
Findtherateofchangeoftheareaoftheripple,withrespecttotheradius,whentheareais 49πm2.
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(d) Tuhiatepāngah(x)mōtepāngarōnakih′ (x)iraro,inakoteuaramōrahioh he 5.
Ātawhakaaturiateakitu.
x
h'(x)
5
5–5
–5
x
h(x)
5
5–5
–5
Ki te hiahia koe ki te tā anō i tēnei kauwhata,
whakamahia te tukutuku i te whārangi 26.
(d) Sketchthefunctionh(x)forthegradientfunctionh′ (x)below,giventhatthemaximumvalueofhis5.
Showthevertexclearly.
x
h'(x)
5
5–5
–5
x
h(x)
5
5–5
–5
If you need to redraw this graph, use the
grid on page 27
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ANAKE(e) Katohuaterōnakiotētahikōpikoko y
xx xd
d6 122= − .
Kotetaunga-yotepūwāhihuringamōkitootekōpikohe10.
Kimihiatewhāriteotekōpiko.
(e) Thegradientofacurveisgivenby yx
x xdd
6 122= − .
The y-coordinateoftheminimumturningpointofthecurveis10.
Findtheequationofthecurve.
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(f) Ehaereanatētahimotokākitētahitereaumoukiawhakamahiarāanōngāperekiotemotokā.
Kahuritetereotemotokākitepāpātangaote–0.08tmitahēkona–2imuriitewhakamahingaongāpereki,inakothēkonatewāmaiitewhakamahingaongāpereki.
E3hēkonaimuriitewhakamahingaongāpereki,kotetereotemotokāhe5mitahēkona–1.
Ehiatetawhitiotehaereotemotokāimuaitetūngainawhakamahiangāpereki?
(f) Acaristravellingataconstantspeeduntilthecar’sbrakesareapplied.
Thecar’sspeedchangesatarategivenby–0.08tmetressec–2afterthebrakesareapplied,wheretsecisthetimesincethebrakeswereapplied.
3secondsafterthebrakesareapplied,thespeedofthecaris5metressec–1.
Howfarwillthecartravelwiththebrakesappliedbeforeitstops?
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PĀTAI TUATORU
(a) Kotētahikōpikoy = f (x)kawhitimāte(0,0),ā,kotanapāngarōnakihe
f ′(x) = 4x + 3.
Kimihiangātaungaotepūwāhiotekōpikokox = –3.
(b) (i) Kimihiatetaunga-xotepūwāhiotekauwhataog(x) = 0.5x2 – 5xkoterōnakieriteanakite2.
(ii) Kimihiatewhāriteotepātapakitekōpikog(x) = 0.5x2 – 5xitepūwāhi(8,–8).
QUESTION THREE
(a) Acurvey = f (x)passesthrough(0,0)andhasgradientfunction
f ′(x) = 4x + 3.
Findthecoordinatesofthepointonthecurvewherex = –3.
(b) (i) Findthex-coordinateofthepointonthegraphofg(x) = 0.5x2 – 5xwherethegradientisequalto2.
(ii) Findtheequationofthetangenttothecurveg(x) = 0.5x2 – 5xatthepoint(8,–8).
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(c) Etohuatahitiaanatekauwhataotepāngah(x)metanapāngarōnakiiraro.
x
h(x)
2
4
6
8
2 4 6 8 10–2
–2
x
h'(x)
2
4
–4
2 4 6 8 10–2
–2
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Kimihiatewhāriteoh(x).
Memātuawhakamahi koe i ngā tikanga tuanakikiawhiwhiaiitōwhakautu.
(d) Hepūwāhihuringatōtekōpikoof (x) = Px2 + Qx+2inako x 23= .
Kawhititekōpikomātepūwāhi(1,9).
Kimihiangātaungaotepūwāhiotekōpikoinakox = 3.
(c) Thegraphofthefunctionh(x)togetherwiththatofitsgradientfunctionaregivenbelow.
x
h(x)
2
4
6
8
2 4 6 8 10–2
–2
x
h'(x)
2
4
–4
2 4 6 8 10–2
–2
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Findtheequationofh(x).
Youmustuse calculus methodstoobtainyouranswer.
(d) Thecurveoff (x) = Px2 + Qx+2hasaturningpointwhen x 23= .
Thecurvepassesthroughthepoint(1,9).
Findthecoordinatesofthepointonthecurvewherex = 3.
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(e) E12ngātokomaitaioteangaotētahikereti,ā,koteroatapekeongātokoheLcm.
Koteroaotekeretieruawhakareangaakeitewhānui.
Whakaaturiakoteroaotekeretihe L9 cminakoterōrahihemōrahi.
length 2xwidth x
height hteiteih
roa2xwhānuix
(e) Theframeofacrateismadeupof12steelrodsthathaveatotallengthofLcm.
Thelengthofthecrateistwicethewidth.
Showthatthelengthofthecratewillbe L9 cmwhenthevolumeisamaximum.
length 2xwidth x
height h
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KitehiahiakoekitetuhianōitekauwhatamōtePātaiTuarua(a),tuhiakitetukutukuiraro. Kiamāramatetohukotēheatekauwhatakahiahiakoekiamākahia.
x
f '(x)
KitehiahiakoekitetuhianōitekauwhatamōtePātaiTuarua(d),tuhiakitetukutukuiraro. Kiamāramatetohukotēheatekauwhatakahiahiakoekiamākahia.
x
h(x)
5
5–5
–5
IfyouneedtoredrawyourgraphfromQuestionTwo(a),drawitonthegridbelow.Makesureitisclearwhichgraphyouwantmarked.
x
f '(x)
IfyouneedtoredrawyourgraphfromQuestionTwo(d),drawitonthegridbelow.Makesureitisclearwhichgraphyouwantmarked.
x
h(x)
5
5–5
–5
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ANAKETAU PĀTAI
He puka anō mēnā ka hiahiatia.Tuhia te (ngā) tau pātai mēnā e hāngai ana.
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QUESTION NUMBER
Extra paper if required.Write the question number(s) if applicable.
© New Zealand Qualifications Authority, 2013. All rights reserved.No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.
Level 2 Mathematics and Statistics, 201391262 Apply calculus methods in solving problems
2.00 pm Monday 18 November 2013 Credits: Five
Achievement Achievement with Merit Achievement with ExcellenceApply calculus methods in solving problems.
Apply calculus methods, using relational thinking, in solving problems.
Apply calculus methods, using extended abstract thinking, in solving problems.
Check that the National Student Number (NSN) on your admission slip is the same as the number at the top of this page.
You should attempt ALL the questions in this booklet.
Show ALL working.
If you need more space for any answer, use the page(s) provided at the back of this booklet and clearly number the question.
Check that this booklet has pages 2 – 29 in the correct order and that none of these pages is blank.
YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.
English translation of the wording on the front cover
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