| Te Taurangi Taumata 8
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| Ngā Whāinga Paetae
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| Te torotoro tauira, pānga
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| I roto i ngā horopaki whai tikanga, me mōhio te ākonga :
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| 1. ki te whakamahi raupapa, tapeke raupapa hei whakatauira pūāhua tuturu, whakatau rānei , ka whakamaori ai i ngā otinga;
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series; |
| 2. ki te hopara, ki te whakamaori i te putahitanga o ngā raupapa me ngā tapeke raupapa;
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convergence; sequence |
| 3. ki te kōwhiri , ki te whakatutuki whawhenga, whakaaritanga kauwhata e tika ana mo ngā tau hiato;
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manipulation
complex numbers |
| 4. ki te whakatauira pūāhua tūturu,
pūāhua whakatau me te whakamahi anō i ngā tikanga tuhi
kauwhata rārangi , ki te kimi, ki te whakamāori whakaotinga
tino tika;
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optimal solutions |
5. ki te whakamahi tikanga kauwhata hei torotoro, hei whakaahua i ngā pānga e pēnei ana te āhua : y = xa, y =  i ngā pānga manunu hoki;
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graphical technique; function
piece-wise |
| 6. ki te kōwhiri tauira e tika ana
mo ngā raraunga tuturu, pēnei i te tikanga tau pū
kōaro, i te tau pū huri kōaro rānei , ka
tātari, ka whakamāori ai i ngā otinga;
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real data; log-log
semi-log |
| 7. ki te ta kauwhata pānga koaro, huri koaro rānei , ka whakamarama ai i ngā hononga i waenganui i ēnei me ngā pānga ake.
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sketch; inverse function; reciprocal function; original |
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Te torotoro whārite , kīanga
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| I roto i ngā horopaki whai tikanga, me mōhio te ākonga :
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| 8. ki te hōpara, ki te kimi huinga,
kōwhiringa rānei mai i tētahi kohinga taonga;
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| 9. ki te whakawhānui , ki te whakamahi
kīanga huarua i ngā wāhi he tau tōpū iti
te taupū;
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expand; binomial
exponent |
| 10. ki te whakamahi whārite tukutahi
hei whakatauira pūāhua tūturu, pūāhua whakatau
rānei , ki te whakamaori hoki i ngā otinga i roto i tētahi
horopaki kua whakaritea atu;
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simulataneous equations; real situation |
| 11. ki te whakamahi tikanga tau, hangarau rānei hei whakaoti whārite rārangi kore;
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numerical methods; non-linear equations |
| 12. ki te whakamahi, ki te hapono i te ture tauwehe, ture toenga hoki;
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prove; factor theorem; remainder theorem |
| 13. ki te whakaoti whārite pūrua , me ngā whārite e pēnei ana te āhua : zn = a (n = he tau tōrunga ), e pēnei ana rānei : z = rcisø;
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quadratic equation
positive integer |
| 14. ki te whakatutuki whawhenga e taea
ai ngā kīanga pākoki te whakamahi ki ētahi atu wāhanga
o te mātauranga pāngarau;
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trigonometrical expression |
| 15. ki te kimi otinga mai i ngā whārite pākoki , tae atu hoki ki te otinga whānui .
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He Tauira Horopaki
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- TE MANAAKI MANUHIRI&;Mehemea
tokowaru ngā tangata hei ata whakanoho ki te tepu, kai ai, e hia
ngā kōwhiringa whakanoho ka taea, e rerekē ai te āhua
o tā rātou noho?
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| He Tauira Mahi
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| Te torotoro tauira, pānga
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| Anei ētahi tauira mahi e taea ai ngā whāinga paetae o tenei taumata te whakatutuki:
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| 1. Me mōhio te ākonga ki te whakamahi raupapa, tapeke raupapa hei whakatauira pūāhua tuturu, whakatau rānei , ka whakamaori ai i ngā otinga;
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| 2. Me mōhio te ākonga ki te hopara, ki te whakamaori i te putahitanga o ngā raupapa me ngā tapeke raupapa;
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- he whakamahi tikanga tuhi kauwhata, whakatauira, tātaitai hei
torotoro raupapa, tapeke raupapa tihoi, pūtahi, piupiu rānei
(tae atu hoki ki ngā tapeke taupū, taupū kōaro);
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- he torotoro tuhinga, kauwhata, rorohiko, tātaitai rānei hei hōpara
i ngā tikanga kauwhata.
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| 3. Me mōhio te ākonga ki te whakatauira pūāhua
tuturu, pūāhua whakatau me te whakamahi anō i ngā tikanga
tuhi kauwhata rārangi ki te kimi, ki te whakamāori whakaotinga
tino tika;
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- he whakatauira pūāhua whaitake mai i ngā horopaki putaiao, tauhokohoko, mātauranga , papori;
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4. Me mōhio te ākonga ki te whakamahi tikanga kauwhata hei torotoro, hei whakaahua i ngā pānga e pēnei ana te āhua : y = xa, y =  i ngā pānga manunu hoki;
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- he whakamahi pānga manunu e ara noa ake ana i ngā horopaki whaitake.
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| 5. Me mōhio te ākonga ki te ta kauwhata pānga
koaro, huri kōaro rānei , ka whakamarama ai i ngā hononga
i waenganui i ēnei me ngā pānga ake;
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- he whakamahi tikanga kauwhata, taurangi rānei hei torotoro, hei
whakaahua i ngā pānga kōaro me ngā pānga huri
kōaro; he torotoro anō i ngā pānga i waenganui i
ēnei, tae atu hoki ki ngā pānga taupū, taupū
kōaro, pākoki .
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| Te torotoro whārite , kīanga
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| 6. Me mōhio te ākonga ki te hopara, ki te kimi huinga, kowhiringa rānei mai i tētahi kohinga taonga;
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| 7. Me mōhio te ākonga ki te whakawhānui ,
ki te whakamahi kīanga huarua i ngā wahi he tau tōpū
iti te taupu;
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- he whakatutuki whiriwhiringa, whakaritenga e puta mai ai ētahi
kōwhiringa, kowhiringa raupapa rānei , te ture huarua rānei
mō ngā tau tōpū iti tōrunga .
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| 8. Me mōhio te ākonga ki te whakamahi whārite
tukutahi hei whakatauira pūāhuatūturu , pūāhua
whakatau rānei , ki te whakamāori hoki i ngā otinga i roto
i tētahi horopaki kua whakaritea atu;
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| 9. Me mōhio te ākonga ki te whakamahi tikanga tau, hangarau rānei hei whakaoti whārite rārangi kore;
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- he hōpara, he whakatauira pūāhua tūturu, whakatau
rānei , me te whakamahi anō i ngā tikanga tuhi kauwhata,
taurangi, āhuahanga, hangarau.
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| 10. Me mōhio te ākonga ki te whakaoti whārite pūrua , me ngā whārite e pēnei ana teahua: zn = a ( n = he tau tōrunga ), e pēnei ana rānei : z = rcisø;
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- he torotoro i te roanga atu o te punaha tau, tae atu hoki ki ngā tau hiato;
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- he whawhe tau hiato e pēnei ana te āhua : a + ib, kua tuhia rānei hei tau pākoki hei whakaoti whārite pūrua , whārite aha kē rānei , tae atu hoki ki era e pēnei ana te āhua : zn = a (n = he tau topu tōrunga ) , e pēnei kē ana rānei : z = rcisø, me te whakamahi anō i ta De Moivre ture.
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| 11. Me mōhio te ākonga ki te whakatutuki whawhenga e taea ai ngā kīanga pākoki te whakamahi ki ētahi atu wāhanga o te mātauranga pāngarau ;
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- he torotoro i ngā pānga i waenganui i ngā kīanga pākoki me ngā pūāhua i puta mai ai aua kīanga .
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| He Tauira Aro Matawai
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| He tauira noa ēnei hei āwhina i te kaiwhakaako:
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- He whakamahi tātaitai (pumanawa rorohiko rānei ) hei hopara i te putahitanga o ngā raupapa pēnei i te f(n) =1 me te g(n) = ( 1 + 1 )n ka whakatau tata i te uara o ngā tepe, mehemea ra he tepe. Anei anō ētahi raupapa e tika ana kia whiriwhirihia:
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mā tenei tauira ka mōhio te kaiwhakaako e pēhea ana te matatau o ngā ākonga ki te:
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- He tatai whirwhiringa raupapa, whiriwhiringa raupapa-kore, e puta noa ake ana i ngā horopaki whai tikanga. Hei whakatauira, anei te āhua o ngā tauwaka i: Ingarangi&;A 657HDC, Poipiripi&;MPQ 049, Aotearoa&;PQ5674. E hia katoa ngā tauwaka ka taea te whakaputa i ēnei whenua e toru?
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mā tenei tauira ka mōhio te kaiwhakaako e pēhea ana te matatau o ngā ākonga ki te:
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- He whakaoti whārite tukutahi, e toru ngā taurangi o roto, ki te tikanga whakakore. Hei whakatauira, me whakaoti ēnei ki te tikanga whakakore:
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mā tenei tauira ka mōhio te kaiwhakaako e pēhea ana te matatau o ngā ākonga ki te:
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- He tuhi pūrongo e whakaatu ana i te kaha o te hiahiatia o ngā
tikanga tau (te tikanga "wāhanga-rua" me tā Newton
rāua ko Raphson tikanga) hei whakaoti whārite . Me whakamārama
hoki te āhua o tēnā , o tēnā , me te hua o
te huri i te uara timata. Me whakaoti te ākonga i tētahi whārite
kua whakaritea hei tirotiro ki te tikanga tau, ka torotoro ai i tētahi
pānga nana anō i kōwhiri .
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mā tenei tauira ka mōhio te kaiwhakaako e pēhea ana te matatau o ngā ākonga ki te:
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- He whakamahi tahi i te tātaitai (pumanawa tuhi kauwhata rānei ) me te ture toenga, hei tautuhi tauwehe, aputa rānei kei reira e mau ana ngā pūtake o tētahi whārite .
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mā tenei tauira ka mōhio te kaiwhakaako e pēhea ana te matatau o ngā ākonga ki te:
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- He kimi otinga tuturu, pohewa rānei o ngā whārite e pēnei ana te āhua :
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mā tenei tauira ka mōhio te kaiwhakaako e pēhea ana te matatau o ngā ākonga ki te:
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