Tuanaki, Kaupae 3, 2012

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1 See ck cove fo Eglish tsltio of this cove LCALCMF 9908 Tuki, Kupe, 0.00 Rāhi 6 Whiig-ā-gi 0 TE PUKAITI O NGĀ TIKANGA TĀTAI ME NGĀ TŪTOHI mō 9065M, 9066M, 9068M me 9069M Tiohi tēei pukiti hei whkutu i gā pāti o ō Pukpuk Whkutu, Pāti hoki. Tiohi meheme kei oto ei gā whāgi 7 e upp tik, ā, kāoe hoki he whāgi wāte. KA TAEA TĒNEI PUKAITI TE PUPURI HEI TE MUTUNGA O TE WHAKAMĀTAUTAU. M Tohu Mātug o Aoteo, 0. Pūmu te m. Ki ku w he wāhi o tēei tuhig e tāuti ki te koe te whketg te M Tohu Mātug o Aoteo.

2 TE TUANAKI ĒTAHI TURE WHAI HUA TE TAURANGI Ngā Whāite Pūu Mēā kāti + 0 c ± Ngā Tupū Kōo log log ( ) log + log Ngā Tu Hito z + i cisθ (cosθ + isi θ) z i cis ( θ) (cosθ isi θ) z zz ( + ) θ g z TE ĀHUAHANGA TAUNGA Te Rāgi Tootik Whāite m( ) Te Poohit ( ) + ( ) ko te (,) te pū, ko te te pūtoo Te Uhi log log log i cos θ ā, siθ, (t,t) āei Aothi (,0) Rāgi whkite log log ( ) log log log Te Tue De Moive Mēā he tu tōpū, kāti, ( cis θ) cis (θ) Te Poop āei +, (cosθ,siθ ) Aothi (c,0) (c,0) i c Tue Tohuu ( + ) C 0 +! ( )!! Pūkētg: e c Te Pūweewee, (secθ,tθ ) āgi pātt ± āei Ku homi ēthi u o i te tūtohi i o ei. Aothi (c,0) (c,0) i c Pūkētg: e c

3 CALCULUS USEFUL FORMULAE ALGEBRA Qudtics If + 0 the ± Logithms c log log ( ) log + log log log log log ( ) log log log log Comple umes z + i cis! (cos! + isi!) z! i cis (!!) (cos!! isi!) z zz ( + )! g z whee cos! d si! De Moive s Theoem If is itege, the ( cis!) cis (!) COORDINATE GEOMETRY Stight Lie Equtio m( ) Cicle ( ) + ( ) hs cete (,) d dius Pol o (t,t) Focus (,0) Diecti Ellipse + o (cosθ,siθ) Foci (c,0) (c,0) whee c Ecceticit: e c Biomil Theoem ( + ) ! C ( )!! Hpeol o (secθ,tθ) smptotes ± Foci (c,0) ( c,0) whee c Some vlues of e give i the tle elow. Ecceticit: e c

4 TE TUANAKI Kimi Pāōki f( ) f ( ) l e si cos t sec e cos si sec sec t cosec cosec cot cot cosec Ngā Tikg Pāwhitu f( ) f( ) f ( ) f( ) + + ( ) l l f( ) Ngā Mātāpoo Tuthi f ( + h) f( ) f ( ) lim h 0 h Te Pāg Twhā. d d d. Te Tue mō te Otig Whku ( f.g) f.g + g. f Te Tue mō te Otig Wehe f g Te Tue Pāg Hito, te Tue Mekmek āei ( f ( g) ) f g. f f.g g ( g). g Te Rōhi o te Huihg NGĀ TIKANGA TAU Te Tue Tp Te Tue Simpso mēā āei mēā āei u v mēā āei f (u) ā u g() kāti f () i weg me k huihi i te tuk- Rōhi π f () i h, ā, f ( ) f () uv kāti kāti du.du h ( ) i h, ā, he tuu te., f ( ) u dv + v du du v v u dv h ( ) + ( ) whke

5 5 CALCULUS Diffeetitio f( ) f ( ) l e e si cos cos si t sec sec sec t cosec cosec cot cot cosec Itegtio f( ) f( ) + + ( ) l f ( ) l f( ) f( ) Fist piciples f ( + h) f( ) f ( ) lim h 0 h Pmetic Fuctio. Poduct Rule ( f.g) f. g + g. f o if uv the u dv + v du Quotiet Rule f g g. f f. g g o if u v Composite Fuctio o Chi Rule f ( g) f ( g). g ( ) du v the u dv o if f (u) d u g() the du.du Volume of Revolutio f () etwee d otted out the -is Volume π NUMERICAL METHODS Tpezium Rule f () h + + ( ) 0 whee h Simpso s Rule d f ( ) f () h + + ( ) + ( ) 0 whee h, f ( ) d is eve. v d d d.

6 6 cosec θ siθ sec θ cosθ cot θ tθ cot θ cosθ siθ si A TE PĀKOKI Te Tue Aho si B Te Tue Wheu c + c sic cosc Ngā Whāite k Poo Ahko gā U K Whkuu Atu cos θ + si θ t θ + sec θ cot θ + cosec θ Ngā Otig Whāui Mēā si θ si α kāti θ + ( ) α Mēā cos θ cos α kāti θ ± α Mēā t θ t α kāti θ + α ko te, he tu tōpū hko Ngā Koki Hito si(a ± B) si Acos B ± cos Asi B cos( A ± B) cos Acos B t A ± t B t( A ± B) t At B Ngā Koki Reu sia si Acos A t A t A t A cosa cos A ± cos A si A si A ± si Asi B 6 Ngā Otig Whku si A cos B si( A + B) + si( A B) cos A si B si( A + B) si( A B) cos Acos B cos( A + B) os( A B) si Asi B cos( A B) cos( A + B) Ngā Otig Tāpii C + D C D si C + si D si cos C + D C D si C si D cos si C + D C D cosc os D cos cos C + D C D cosc cos D si si TE INE Te Tptou Hohg sic Te Tp Hohg ( + )h Te Pewg Hohg θ Te o o te pew θ Te Rgo Rōhi π h Hohg mt kōpiko πh Te Koeko Rōhi π h Hohg mt kōpiko πl i ko te l te teitei o te tīth Te Poi Rōhi π Hohg mt π

7 7 TRIGONOMETRY cosec θ siθ sec θ cosθ cot θ tθ cot θ cosθ siθ Sie Rule si A si B Cosie Rule c sic 6 Poducts si Acos B si( A+ B) + si( A B) cos Asi B si( A+ B) si( A B) cos Acos B cos( A+ B) os( A B) si Asi B cos( A B) cos( A+ B) Sums sic + si D si C + D cos C D sic si D cos C + D si C D cosc os D cos C + D cos C D cosc cos D si C + D si C D c + cosc Idetities cos θ + si θ MEASUREMENT Tigle Ae sic t θ + sec θ cot θ + cosec θ Tpezium Geel Solutios If si θ si α the θ π + ( ) α If cos θ cos α the θ π ± α If t θ t α the θ π + α whee is itege Compoud Agles si(a ± B) si Acos B ± cos Asi B cos( A ± B) cos Acos B si Asi B t( A ± B) Doule Agles t A ± t B t At B sia si Acos A t A t A t A cosa cos A si A cos A si A Ae ( + )h Secto Ae θ Ac legth θ Clide Volume! h Cuved sufce e!h Coe Volume! h Cuved sufce e! l whee l slt height Sphee Volume! Sufce e!

8 LCALCMF Level Clculus, 0.00 pm Mod 6 Noveme 0 FORMULAE AND TABLES BOOKLET fo 9065, 9066, 9068 d 9069 Refe to this ooklet to swe the questios i ou Questio d Aswe ooklets. Check tht this ooklet hs pges 7 i the coect ode d tht oe of these pges is lk. YOU MAY KEEP THIS BOOKLET AT THE END OF THE EXAMINATION. New Zeld Qulifictios Authoit, 0. All ights eseved. No pt of this pulictio m e epoduced mes without the pio pemissio of the New Zeld Qulifictios Authoit.

Tuanaki, Kaupae 3, 2014

Tuanaki, Kaupae 3, 2014 See ck cover for n English trnsltion of this cover LCALCMF 9908 Tunki, Kupe, 0 9.0 i te t Rātū 8 Whiring-ā-rngi 0 TE PUKAITI O NGĀ TIKANGA TĀTAI ME NGĀ TŪTOHI mō 9577M, 9578M me 9579M Tirohi tēnei pukiti