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© Mana Tohu Mātauranga o Aotearoa, 2018. Pūmau te mana. Kia kaua rawa he wāhi o tēnei tuhinga e whakahuatia ki te kore te whakaaetanga tuatahi a te Mana Tohu Mātauranga o Aotearoa.

MĀ TE KAIMĀKA ANAKE

TAPEKE

Tuanaki, Kaupae 3, 201891579M Te whakahāngai i ngā tikanga pāwhaitua hei

whakaoti rapanga

9.30 i te ata Rātū 13 Whiringa-ā-rangi 2018 Whiwhinga: Ono

Paetae Kaiaka KairangiTe whakahāngai i ngā tikanga pāwhaitua hei whakaoti rapanga.

Te whakahāngai i ngā tikanga pāwhaitua mā te whakaaro whaipānga hei whakaoti rapanga.

Te whakahāngai i ngā tikanga pāwhaitua mā te whakaaro waitara hōhonu hei whakaoti rapanga.

Tirohia mēnā e rite ana te Tau Ākonga ā-Motu (NSN) kei runga i tō puka whakauru ki te tau kei runga i tēnei whārangi.

Me whakamātau koe i ngā tūmahi KATOA kei roto i tēnei pukapuka.

Tuhia ō mahinga KATOA.

Tirohia mēnā kei a koe te Pukapuka Tikanga Tātai me ngā Tūtohi L3–CALCMF.

Mēnā ka hiahia whārangi atu anō koe mō ō tuhinga, whakamahia te (ngā) whārangi wātea kei muri o tēnei pukapuka, ka āta tohu ai i te tau tūmahi.

Tirohia mēnā e tika ana te raupapatanga o ngā whārangi 2 –27 kei roto i tēnei pukapuka, ka mutu, kāore tētahi o aua whārangi i te takoto kau.

ME HOATU RAWA KOE I TĒNEI PUKAPUKA KI TE KAIWHAKAHAERE Ā TE MUTUNGA O TE WHAKAMĀTAUTAU.

Tuanaki 91579M, 2018

MĀ TE KAIMĀKA

ANAKE

TŪMAHI TUATAHI

(a) Whiriwhiria 6x 8x3 dx

(b) Whakaotihiatewhāritepārōnakidydx

= e2x + 1x,inakox=1,kātikoy = 2.

(c) Whiriwhiria 2x 7x 5

6

8

dx

Mewhakamahirawaitetuanakikawhakaatuingāotingaotemahipāwhaituakahiahiatiaheiwhakaotiiterapanga.

2

QUESTION ONE

(a) Find 6x 8x3 dx

(b) Solvethedifferentialequationdydx

= e2x + 1x,giventhatwhenx = 1, y = 2.

(c) Find 2x 7x 5

6

8

dx

Youmustusecalculusandshowtheresultsofanyintegrationneededtosolvetheproblem.

3

Calculus 91579, 2018

ASSESSOR’S USE ONLY


MĀ TE KAIMĀKA

ANAKE(d) WhakaotihiatewhāritepārōnakiSolve the differential equationdydx

= cos2xe y

given that y = 0 when x = π4.inakoSolve the differential equation

dydx

= cos2xe y

given that y = 0 when x = π4.mēnāSolve the differential equation

dydx

= cos2xe y

given that y = 0 when x = π4..


4

(d) Solve the differential equationdydx

= cos2xe y

given that y = 0 when x = π4.


5




MĀ TE KAIMĀKA

ANAKE

(e) Ewhakaatuanatehoahoaoraroneiitekauwhataotepānga f (x) =12ex −1( ).

Q (k,k)

f (x)

x

EtakotoanatepūwāhiQ(k,k)kiteānau. Keiteroheatewāhangakaurukuitehoahoaneieteānau,etetuaka-xmeterārangix = k.

Mewhakaatukotehorahangaoterohekuakaurukutiahef (x) = 12ex −1( )

k.


6

(e) Thediagrambelowshowsthegraphofthefunction f (x) =12ex −1( ).

Q (k,k)

f (x)

x

ThepointQ(k,k)liesonthecurve. Theshadedregioninthediagramisboundedbythecurve,thex-axisandthelinex = k.

Showthattheshadedregionhasanareaoff (x) = 12ex −1( )

k.


7




MĀ TE KAIMĀKA

ANAKE

TŪMAHI TUARUA

(a) Whiriwhiria sec2 x + sec2x tan2x( )dx∫

(b) Whiriwhiriateuaraok,inako x1

k

∫ dx = 523

.


8

QUESTION TWO

(a) Find sec2 x + sec2x tan2x( )dx∫

(b) Findthevalueofk,giventhat x1

k

∫ dx = 523

.


9




MĀ TE KAIMĀKA

ANAKE

(c) Ewhakaatuanatehoahoairaroingākauwhataongāpāngay=cos2x me y=sin2x.

y = cos2x

y

x k

y = sin2x

Whiriwhiriateuaraok kia x x xcos sin d 1

2

k

2 2

0∫( )− = .


10

(c) Thediagrambelowshowsthegraphsofthefunctionsy=cos2xandy=sin2x.

y = cos2x

y

x k

y = sin2x

Findthevalueofksuchthat x x xcos sin d 1

2

k

2 2

0∫( )− = .


11




MĀ TE KAIMĀKA

ANAKE

(d) Kataeatewhakaterengaotētahiahanoatewhakatauiramāte

whāritea(t) = 2t +1

, where t ≥ 0. inakoa(t) = 2

t +1, where t ≥ 0.

inakoatewhakaterengaoteahanoaitems–2

ko ttewāā-hēkonamaiitetīmatangaoteinewā. He9ms–1teterengaoteahanoainakot = 3. Ehiatetawhitiihaereaiteahanoaingāhēkonae8tuatahiitewāeineaanatenekehanga?


12

(d) Anobject’saccelerationcanbemodelledbytheequationa(t) = 2t +1

, where t ≥ 0.

where aistheaccelerationoftheobjectinms–2

andtisthetimeinsecondsfromthestartoftiming.

Theobjecthasavelocityof9ms–1whent = 3. Howfardidtheobjecttravelinthefirst8secondsofitstimedmotion?


13




MĀ TE KAIMĀKA

ANAKE

(e) Kotepapatipu,mkaramu,otētahikānaraemuraanaitethaoraimuriitewhakakātangatuatahikataeatewhakatauiramātewhāritepārōnaki

dmdt

= −k(m−10) where k > 0 and m ≥10.inakodmdt

= −k(m−10) where k > 0 and m ≥10. me dmdt

= −k(m−10) where k > 0 and m ≥10.

Kotepapatiputīmataotekānarahe140karamu.

Ingāhaorae3imurimaikuahauruakētepapatipuotekānara.

Whiriwhiriateroaotewāehekeaitepapatipuotekānarakite50karamu.


14

(e) Themass,mgrams,ofaburningcandlethoursafteritwasfirstlitcanbemodelledbythedifferentialequation

dmdt

= −k(m−10) where k > 0 and m ≥10.

Theinitialmassofthecandlewas140grams.

3hourslaterthemassofthecandlehadhalved.

Findthelengthoftimeitwilltakeforthemassofthecandletoreduceto50grams.


15




MĀ TE KAIMĀKA

ANAKE

TŪMAHI TUATORU

(a) Whiriwhiria 4x( )2+ 4x + 4

xdx

(b) Whakamahiangāuarairaroheiwhiriwhiriitētahiāwhiwhitangaki f (x)dx0

3

∫ ,mātewhakamahiiteTureaSimpson.

x 0 0.5 1 1.5 2 2.5 3f (x) 0.3 0.75 1.1 1.35 1.6 1.15 0.5

(c) Whiriwhiriateuaraokinako

3e0.5x0

k

∫ dx = 75

.

Ka haere tonu te Tūmahi Tuatoru i te whārangi 18.

16

QUESTION THREE

(a) Find 4x( )2+ 4x + 4

xdx

(b) Usethevaluesgiveninthetablebelowtofindanapproximationto f (x)dx0

3

∫ ,usingSimpson’sRule.

x 0 0.5 1 1.5 2 2.5 3f (x) 0.3 0.75 1.1 1.35 1.6 1.15 0.5

(c) Findthevalueofkgiventhat 3e0.5x0

k

∫ dx = 75 .

17



Question Three continues on page 19.


MĀ TE KAIMĀKA

ANAKE(d) Ewhakaatuanatehoahoaoraroingākauwhataongāpānga y = x2 and y = x3 me te y = x2 and y = x3 .

y

x

y = x2

y = 3 x

Whiriwhiriatehorahangaoteroheiwaengaingākauwhata(kuakaurukutiairotoitehoahoa).


18

(d) Thediagrambelowshowsthegraphsofthefunctions y = x2 and y = x3 .

y

x

y = x2

y = 3 x

Findtheareaoftheregionbetweenthegraphs(shownshadedinthediagram).


19




MĀ TE KAIMĀKA

ANAKE

(e) Ewhakaatuanatehoahoaoraroneiitekauwhataotepānga f (x) = (2x – 1)4.

Q (1,1)

y

xP

f (x) = (2x – 1)4

Katūtakiteānauitetuaka-xiPmeterārangiotekauwhatahepātapakiteānauitepūwāhiQ(1,1).

Whiriwhiriatehorahangakuaroheaeteānau,tetuaka-x,metepātapakiteānauiQ(kuakaurukutiairotoitehoahoa).


20

(e) Thediagrambelowshowsthegraphofthefunctionf(x) = (2x – 1)4.

Q (1,1)

y

xP

f (x) = (2x – 1)4

Thecurvemeetsthex-axisatPandthelineonthegraphisatangenttothecurveatthepointQ(1,1).

Findtheareaoftheregionboundedbythecurve,thex-axis,andthetangenttothecurveatQ(shownshadedinthediagram).


21




MĀ TE KAIMĀKA

ANAKETAU TŪMAHI

He whārangi anō ki te hiahiatia.Tuhia te (ngā) tau tūmahi mēnā e tika ana.

22

23



QUESTION NUMBER

Extra paper if required.Write the question number(s) if applicable.


MĀ TE KAIMĀKA

ANAKETAU TŪMAHI


24

25



QUESTION NUMBER



MĀ TE KAIMĀKA

ANAKETAU TŪMAHI


26

27



QUESTION NUMBER


Level 3 Calculus, 201891579 Apply integration methods in solving problems

9.30 a.m. Tuesday 13 November 2018 Credits: Six

Achievement Achievement with Merit Achievement with ExcellenceApply integration methods in solving problems.

Apply integration methods, using relational thinking, in solving problems.

Apply integration 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.

Make sure that you have the Formulae and Tables Booklet L3–CALCF.

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 – 27 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.

91

57

9M

English translation of the wording on the front cover

tuanaki, kaupae 3, 2018 - nzqa.govt.nz · tuanaki 91579m, 2018 mĀ t kaimĀka anak (e) ko te...

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