Te Kaupapa Taketake o te Tātaitai
He maha ngā wā ka mārama te Tātaitai hei "reo" mō te whakamārama i te huringa me te kohikohinga. I tētahi taha, ka ako mātou i ngā pānga hei ine i te tere o te huringa o tētahi mahi. I tētahi atu taha, ka ako mātou i ngā taunga whakauru hei tatau i te kohikohinga, pērā i te horahanga i raro i tētahi kōpiko, i te "tōpū tonu" rānei o tētahi rahinga. Ko te Kaupapa Taketake o te Tātaitai (FTC) he piriti nui e hono ana i ēnei whakaaro e rua: e kitea ana ehara te rerekētanga me te whakaurunga i ngā kaupapa motuhake e rua, engari he mahi tauutuutu e rua. Ko tēnei kaupapa te mea e tino kaha ai te tātaitai i roto i te pūtaiao, te hangarau, te ōhanga, me te maha atu o ngā mara.
Tirohanga Whānui: ngā huringa me te kohikohinga
Whakaarohia he motuka e neke ana i runga i te rori. Ko te tere o te motuka ko te tere o te huringa o te tūranga i roto i te wā, ko te tawhiti i haerea ko te kohikohinga o te "tere" i roto i te wā. I roto i ngā kupu pāngarau, ki te mea ko te \(v(t)\) te tere, ka taea te whakaatu i te tawhiti i haerea mai i te wā \(a\) ki \(b\) mā te tauwehenga.
\[
\int_a^bv(t)\, dt.
\]
I taua wā, mēnā ko te tūnga te \(s(t)\), ko te tere te pānga:
\[
v(t) = s'(t).
\]
E ai ki te Kaupapa Taketake o te Tātaitai he tata te hononga o ēnei mahi e rua: ko te taupū o te pānga ka whakahoki mai i te huringa kupenga o te mahi, ā, ko te pānga o te taupū tuturu ka whakahoki mai i te mahi taketake. Mā tēnei whanaungatanga ka nui ake te pūnaha o te tatau i te horahanga, te tawhiti, te papatipu, te pūngao, me te maha atu o ngā mea.
He whakaritenga tere: he aha ngā taupūnga tuturu me ngā taupūnga pārōnaki?
I mua i te urunga atu ki te whakapuakitanga o te ariā, e rua ngā ariā nui:
1. Te Pānga \(f'(x)\): ka ine i te pikinga o te kauwhata, i te tere rānei o te huringa o \(f(x)\) ina paku rerekē te \(x\). Mā te whakaaro noa, ki te whakaahuahia e \(f(x)\) te tūranga, ka whakaahuahia e \(f'(x)\) te tere.
2. Taupūnga tuturu \(\int_a^bf(x)\,dx\): e ine ana i te kohikohinga o \(f\) i runga i te āputa \([a,b]\). Mā te āhuahanga, ka tautuhia pinepinetia ko te horahanga kua tohua (horahanga pai i runga ake i te tuaka-\(x\), horahanga kino i raro i te tuaka-\(x\)) i raro i te kōpiko \(y=f(x)\) mai i \(x=a\) ki \(x=b\).
Ko te taupū tuturu ka tautuhia ā-turetia e te rohe o te tapeke Riemann, arā, te whakatata i te horahanga mā ngā tapawhā iti, kātahi ka tangohia te rohe ina heke te whānui o ngā tapawhā ki te kore.
Te Whakapuakitanga o te Kaupapa Taketake o te Tātaitai (Wāhanga 1)
E mea ana te Wāhanga 1 o TFK: mēnā he tonu te \(f\) i runga i te \([a,b]\), kātahi ka tautuhia e tātou he mahi hou.
\[
F(x)=\int_a^xf(t)\,dt,
\]
kātahi ka taea te whakaputa i te \(F\) i runga i te \((a,b)\) me
\[
F'(x)=f(x).
\]
He mea tino nui te tikanga: ko te taupū "i hangaia" mai i te \(f\) ka puta te mahi ārai-whakaputa o te \(f\). Arā, ko te tukanga kohikohinga tae noa ki te pūwāhi \(x\) i te wā i wehewehea ai ka hoki ki te tere kohikohinga i taua pūwāhi.
Wāhanga Matatau 1
Mātakitakihia te panoni iti o \(F(x)\) ina whakanuia te \(x\) mā te iti \(\Delta x\):
\[
F(x+\Delta x)-F(x)=\int_a^{x+\Delta x} f(t)\,dt – \int_a^xf(t)\,dt = \int_x^{x+\Delta x} f(t)\,dt.
\]
Mena he iti te \(\Delta x\), he rite tonu tēnei taupū ki te \(f(x)\Delta x\). Nō reira,
\[
\frac{F(x+\Delta x)-F(x)}{\Delta x}\approx f(x).
\]
Ina eke te \(\Delta x\ki te 0\), ka tino tika te whakatata, kia \(F'(x)=f(x)\).
Tauira māmā
Me kī \(f(t)=2t\). Tautuhia
\[
F(x)=\int_0^x 2t\,dt.
\]
E mōhio ana tātou ko \(\int 2t\,dt = t^2\), nō reira \(F(x)=x^2\). Ko te tātaitanga ko \(F'(x)=2x\), e hoki ana ki \(f(x)\). E whakaatu pono ana tēnei i te Wāhanga 1.
Te Whakapuakitanga o te Kaupapa Taketake o te Tātaitai (Wāhanga 2)
E mea ana te Wāhanga 2 o TFK: mēnā he pumau tonu a \(f\) i runga i a \([a,b]\) ā, ko \(F\) he ātete-whakaputa o \(f\) (arā, \(F'(x)=f(x)\)), kātahi
\[
\int_a^bf(x)\,dx = F(b)-F(a).
\]
Koinei te momo ariā e whakamahia whānuitia ana i roto i te tatau taupū. E kī ana, hei tatau i tētahi taupū tino, kāore e hiahiatia kia whakamahia tika te rohe o te tapeke Riemann; kimihia noa te antiderivative o \(F\), kātahi ka aromatawaihia i ngā rohe o runga me raro.
Tauira tātaitanga
Tatau:
\[
\int_1^3 (x^2+1)\,dx.
\]
Ko te antiderivative ko
\[
F(x)=\frac{x^3}{3}+x.
\]
Nā reira:
\[
\int_1^3 (x^2+1)\,dx = \left(\frac{3^3}{3}+3\right)-\left(\frac{1^3}{3}+1\right)
= \left(9+3\right)-\left(\frac{1}{3}+1\right)
=12-\frac{4}{3}=\frac{32}{3}.
\]
Ki te kore a TFK, me tautuhi e tātou te tauwehe hei te rohe o te tapeke o ngā horahanga o ngā tapawhā hāngai, ā, me tatau te rohe—he roa rawa atu.
He aha i kiia ai ko "taketake"?
He mea nui tēnei ariā nā te mea:
1. Honoa ngā ariā matua e rua o te tātaitai: te pānga (huringa) me te taupū (te kohikohi).
2. E whakarato ana i tētahi tikanga mahi: ka taea te tatau i ngā taupūnga tino mā te whakamahi i ngā taupūnga ārai-whakaputa.
3. He maha ngā whakamahinga kei raro: te ahupūngao (te mahi me te pūngao), te tatauranga (te tohatoha me te whai wāhitanga), te ōhanga (te utu katoa me te utu taha), te koiora (te tipu o te taupori), me ētahi atu.
I roto i te ariā, ka noho te tātaitai hei taputapu mārama: ka taea e tātou te whakawhiti i waenga i te tauira "reiti" me te "katoa" me te ngāwari.
Ngā tono e puta pinepine ana
1. Te tawhiti mai i te tere
Mena ko \(v(t)\) te tere, ko te nekehanga kupenga ko:
\[
s(b)-s(a)=\int_a^bv(t)\,dt.
\]
Nō te wāhanga 2 o TFK tēnei mēnā \(v(t)=s'(t)\). Mēnā he kino te \(v(t)\) i ētahi wā, ka homai e te tauwehenga te nekehanga kupenga; mō te tawhiti katoa ka tatauhia i te nuinga o te wā ko \(\int_a^b |v(t)|\,dt\).
2. Te kohikohinga o te tere o te huringa
Mena ka whakakīia he tāke i te tere o te \(r(t)\) rita/meneti, ko te rōrahi e tomo mai ana i te wā \([a,b]\) ko \(\int_a^br(t)\, dt\). Mena he tere rerenga mai, he tere rerenga atu hoki, ko te huringa kupenga o te rōrahi ko te taupū o (rerenga mai − rerenga atu).
3. Te ariā uara toharite mō ngā tauwehenga
Mai i TFK, ka puta ake ngā hua rerekē pēnei i te uara toharite o te mahi:
\[
f_{\text{avg}}=\frac{1}{ba}\int_a^bf(x)\,dx.
\]
He mea nui tēnei i roto i te tātari raraunga me te whakatauira.
Ngā kōrero nui: ngā tikanga me ngā herenga
Ko te tikanga, me whai tonu te mahi \(f\) i te wā e pā ana ki te TFK kia maeneene ai tōna whakapuakitanga. I roto i ngā rangahau anō, ka taea te whakawhānui ake i tēnei ariā ki ngā mahi kāore e tino pumau tonu (hei tauira, ngā mahi ka taea te whakauru ki te Riemannian, ki te Lebesgue rānei i raro i ētahi tikanga), engari mō te tātaitai taketake, ko te whakaaro pumau te paerewa.
Hei tāpiritanga, ka puta mai i ngā taunga whakapūmau ngā horahanga kua tohua, ehara i te mea ko ngā "horahanga āhuahanga parakore" tonu. Mena kei raro iho te kauwhata i te tuaka-x, he kino te taunga. Mō ngā horahanga āhuahanga, ko ngā uara tino, ko ngā wehenga āputa rānei te tikanga e whakamahia ana.
Te Katinga
Ko te Kaupapa Taketake o te Tātaitai te uho e hono ana i te pāngarau me te taupū. I whakaatuhia e te Wāhanga 1 ko te kohikohinga o tētahi mahi tonu, ina wehewehea, ka hoki ki te mahi taketake. I whakaatuhia e te Wāhanga 2 tētahi huarahi tere ki te tatau i ngā pāngarau tino: kimihia noa te pāngarau ā-kore me te aromatawai i te rerekētanga i ngā rohe. Mā tēnei ariā, ehara te tātaitai i te kohinga noa o ngā tikanga pāngarau, engari he anga huatau mō te mārama ki te ao: me pēhea te hurihanga o ngā mea i roto i te wā, me pēhea te kohikohinga o aua huringa ki te tapeke.
Ki te ako koe i ngā tikanga whakauru, i ngā whārite rerekētanga, i ngā tauira ā-tinana rānei ā muri ake nei, ka kite tonu koe i a TFK e mahi ana i muri i ngā whakaaturanga—hei "piriti" e tino whai mana ai te tātaitai.
