Tikanga Wehewehe i te Kimi Pūtake
Ko te tikanga wehewehe he tikanga tau e whakamahia ana hei kimi i ngā pūtake o tētahi whārite kore-raina. E mōhiotia ana hoki tēnei tikanga ko te tikanga tapahi āputa nā te mea e uru ana ki te wehewehe i tētahi āputa kia tutuki rā anō te tika e hiahiatia ana. Ka matapakihia e tēnei tuhinga ngā mātāpono taketake, ngā mahi, ngā painga, ngā ngoikoretanga, me ngā tauira whakatinanatanga o te tikanga wehewehe.
Ngā Mātāpono Taketake o te Tikanga Wehewehe
Ko te tikanga wehewehe i ahu mai i te Ariā a Bolzano, e kī ana mēnā he uara tohu rerekē kei tētahi mahi tonu \(f(x)\) i ngā pūwāhi e rua \(a\) me \(b\), arā, \(f(a)\cdot f(b) < 0\), kāti kotahi te pūtake kei roto i te wā \([a, b]\). Ko tēnei mātāpono te pūtake matua o te tikanga wehewehe, e whakawhāitihia haeretia ana te wā \([a, b]\) kia tae rā anō ki te pūtake e hiahiatia ana.
Ngā Hipanga o te Tikanga Wehewehe
Ka taea te whakamārama i te tukanga tikanga wehewehe mā roto i ngā mahi e whai ake nei:
1. Whakatauhia te Wā Tīmatanga:
Tīpakohia kia rua ngā pūwāhi \(a\) me \(b\) kia < 0\ te \(f(a)\cdot f(b)\). Me whai i tēnei wā \([a, b]\) te pūtake e rapuhia ana e koe.
2. Te Tatau i te Waenganui:
Tātaihia te pūwāhi o te wā \[ c = \frac{a + b}{2} \].
3. Aromatawai Mahi:
Tātaihia te uara o \(f(c)\).
4. Whakawhāitihia te Wā:
a. Mena ko te \(f(a)\cdot f(c) < 0\), kei roto te pūtake i te wā \([a, c]\). Whakakapia te \(b\) ki te \(c\).
b. Mena ko te \(f(b)\cdot f(c) < 0\), kei roto te pūtake i te wā \([c, b]\). Whakakapia te \(a\) ki te \(c\).
5. Te Whakahoki:
Whakahokia ngā taahiraa 2-4 kia iti rawa te wā \([a, b]\) kia tata rānei te \(f(c)\) ki te kore me te manawanui kua tohua.
Tauira Whakatinanatanga
Hei whakamārama ake i te āhua, me titiro tātou ki tētahi tauira o te whakamahi i te tikanga wehewehe ki te whārite \(f(x) = x^2 – 4\).
1. Whakatauhia te Wā Tīmatanga:
Kōwhiria \(a = 0\) me \(b = 3\). Ka tirohia e mātou ngā uara \(f(0)\) me \(f(3)\):
\[
f(0) = 0^2 – 4 = -4
f(3) = 3^2 – 4 = 5
\]
Nā te mea ko te \(f(0) \cdot f(3) < 0\), he tika tēnei wā.
2. Te Whakahoutanga Tuatahi:
\[
c = \frac{0 + 3}{2} = 1.5 \\
f(1.5) = (1.5)^2 – 4 = -1.75
\]
Nā te mea ko te \(f(0) \cdot f(1.5) < 0\), ka whakawhāitihia e mātou te mokowā ki te \([0, 1.5]\).
3. Te Whakahoutanga Tuarua:
\[
c = \frac{0 + 1.5}{2} = 0.75 \\
f(0.75) = (0.75)^2 – 4 = -3.4375
\]
Nā te mea ko te \(f(0) \cdot f(0.75) < 0\), ka whakawhāitihia e mātou te mokowā ki te \([0, 0.75]\).
4. Te Tuatoru o ngā Whakahoutanga:
\[
c = \frac{0 + 0.75}{2} = 0.375 \\
f(0.375) = (0.375)^2 – 4 = -3.859375
\]
Nā te mea ko te \(f(0) \cdot f(0.375) < 0\), ka whakawhāitihia e mātou te mokowā ki te \([0, 0.375]\).
Ka haere tonu tēnei tukanga kia tae rā anō ki te tika e hiahiatia ana. I ia taahiraa, ka whakawhāitihia te wā \([a, b]\), ā, ka tatauhia, ka aromatawaihia hoki te pūwāhi \(c\) kia tata rā anō a \(f(c)\) ki te kore.
Ngā Painga o te Tikanga Wehewehe
1. Māmā, ā, he māmā noa iho te mārama:
He tino māmā te tikanga wehewehe, ā, he ngāwari hoki ki te mārama, ahakoa mō te hunga tauhou ki ngā tikanga tau.
2. Te Huihuinga Kua Whakamanahia:
Mena he tonu te mahi e arotakengia ana, ā, he tika te whiriwhiri i te wā tīmatanga, ka ū tonu te tikanga wehe rua ki te pūtake.
3. Kāore e hiahiatia he hua tāpiri:
Kāore e hiahiatia te tatau i ngā tātaitanga o ngā pānga e te tikanga wehewehe, nō reira he pai mō ngā mahi he uaua, he kore rānei e taea te tatau i ngā tātaitanga tuatahi.
Ngā Huakore o te Tikanga Wehewehe
1. Te Huinga Pōturi:
Ahakoa e oatitia ana te huihuinga, he puhoi te tikanga wehewehe i te nuinga o te wā ki te whakataurite ki ētahi atu tikanga pēnei i a Newton-Raphson.
2. Me whai pūtake te wā:
Hei whakamahi i te tikanga wehewehe, me mōhio tātou ki te wā kei roto te pūtake. Ki te kore, kāore e taea te whakamahi i te tikanga.
3. Kāore e whai hua mō ngā mahi uaua:
Mō ngā mahi he maha ngā pūtake, he tino uaua rānei ā rātou whanonga, kāore pea e whai hua te tikanga wehewehe.
Ngā Taupānga o te Ao Tūturu
He whānui te whakamahinga o te tikanga wehewehe i roto i ngā momo mara o te pūtaiao me te hangarau. Ko ētahi o ngā tono o te ao tūturu ko:
1. Hangarau Ā-iwi:
I roto i te tātari hanganga, ka whakamahia te tikanga weherua hei whakatau i ngā pūwāhi ka puta ai te whakarerekētanga mōrahi i runga i tētahi kaha, i tētahi momo nekehanga rānei.
2. Ahupūngao:
I roto i te ahupūngao, ka whakamahia te tikanga wehewehe hei kimi otinga mō ngā whārite pūngao me ngā āhua taurite i roto i ngā pūnaha hihiri.
3. Ōhanga:
I roto i te ōhanga, ka taea te whakamahi i te tikanga wehewehe hei kimi i ngā pūwāhi taurite mākete, i ētahi atu uara nui rānei.
4. Te Hōtaka Rorohiko:
I roto i te hōtaka rorohiko, he maha ngā whakamahinga o ngā rauropi kimi-pūtake pēnei i te tikanga wehewehe i waenga i ngā momo tono tau me te whakatauira.
Whakamutunga
He taputapu māmā engari he tino whai hua te tikanga wehewehe i waenga i ngā tau, hei kimi i ngā pūtake o ngā whārite kore-raina. Nā ōna mātāpono taketake ngāwari ki te mārama me te whakapūmautanga kua whakamanahia, he pai tēnei tikanga mō ngā raruraru tau maha. Ahakoa he ngoikoretanga ōna, pērā i te whakapūmautanga puhoi me te hiahia mō tētahi wā kei roto te pūtake, nā ngā painga o te tikanga wehewehe i whai tikanga ai i roto i ngā tono o te ao tūturu. Mō te hunga e rapu ana ki te mārama ki ngā kaupapa matua o te kimi pūtake, he tīmatanga pai te tikanga wehewehe i waenga i ngā tau.
