te pāngarau me te tauanga, kaupae 2, 2013 - nzqa.govt.nz · 2 te pāngarau me te tauanga 91262m,...

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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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Te Pāngarau me te Tauanga 91262M, 2013

MĀ TE KAIMĀKA

ANAKE

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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Mathematics and Statistics 91262, 2013

ASSESSOR’S USE ONLY

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MĀ TE KAIMĀKA

ANAKE

(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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MĀ TE KAIMĀKA

ANAKE

(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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MĀ TE KAIMĀKA

ANAKE

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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MĀ TE KAIMĀKA

ANAKE

(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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MĀ TE KAIMĀKA

ANAKE

(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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MĀ TE KAIMĀKA

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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MĀ TE KAIMĀKA

ANAKE

(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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MĀ TE KAIMĀKA

ANAKE

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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MĀ TE KAIMĀKA

ANAKE

(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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MĀ TE KAIMĀKA

ANAKE

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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MĀ TE KAIMĀKA

ANAKE

(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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MĀ TE KAIMĀKA

ANAKE

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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MĀ TE KAIMĀKA

ANAKETAU PĀTAI

He puka anō mēnā ka hiahiatia.Tuhia te (ngā) tau pātai mēnā e hāngai ana.

29



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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te pāngarau me te tauanga, kaupae 2, 2013 - nzqa.govt.nz · 2 te pāngarau me te tauanga 91262m,...

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