Tauira o ngā Pātai Kōrero Whakauru
He ariā taketake te taupū i roto i te tātaitai e whānuitia ana te whakamahinga i roto i ngā momo mara, tae atu ki te ahupūngao, te hangarau, me te ōhanga. Ka tūhuratia e tēnei tuhinga ngā tauira maha o ngā raruraru taupū me ō rātou otinga hei whakarato i tētahi māramatanga hōhonu ake.
1. Te Māramatanga Taketake ki ngā Whakakotahitanga
I roto i ngā kupu māmā noa iho, ko te taupū te mahi whakahuri o tētahi taupū. E rua ngā momo taupū e kōrerohia whānuitia ana, arā:
– Taupūnga Kore-Mutu: he āhua taupūnga tēnei kāore he rohe o runga, o raro hoki, ā, e tohuhia ana e ∫ f(x) dx.
– Taupū Tūturu: he āhua taupū tēnei he rohenga o runga, o raro hoki, ā, ko te tohu ko ∫[a,b] f(x) dx.
Ko te taupū mutunga kore e kiia ana ko te anti-derivative, ā, ko te hua ka uru atu te pūmau C nā te mea ko te āhuatanga o te taupū pumau he kore.
2. Ngā Tauira o ngā Raru Taupū Kore-Mutu
Tauira 1: Taurite Kore Mutunga Māmā
Tātaihia ∫ x^2 dx.
Kōrero:
E mōhio ana tātou ko te ture whakauru taketake mō ∫ x^n dx ko (x^(n+1))/(n+1) + C, ko C te pūmau o te whakauru.
Mō te taupū i runga ake nei, n = 2:
∫ x^2 dx = (x^(2+1))/(2+1) + C
= (x^3)/3 + C.
Nō reira, ko te hua o ∫ x^2 dx ko (x^3)/3 + C.
Tauira 2: Te Whakapūtanga o ngā Mahi Taupūnga
Tātaihia ∫ e^x dx.
Kōrero:
Ko te ture taketake mō te taupū taupū ∫ e^x dx ko e^x + C.
Nō reira, ko te hua o ∫ e^x dx ko e^x + C.
3. Ngā Tauira o ngā Raru Taupū Tūturu
Tauira 1: Taurite Māmā
Tātaihia te ∫[1,3] x^2 dx.
Kōrero:
Tuatahi, ka kitea e tātou te ārai-tāpiri o x^2, arā, (x^3)/3.
Inaianei ka whakakapia ngā here:
∫[1,3] x^2 dx = [(3^3)/3 – (1^3)/3]
= [27/3 – 1/3]
= [9 – 1/3]
= 8 + 2/3, 8.6667 rānei.
Nō reira, ko te hua o ∫[1,3] x^2 dx ko 26/3, ko 8.6667 rānei.
Tauira 2: Taunga Whakakotahi mā te Whakakapinga
Tātaihia ∫[0,2] (2x + 1) dx.
Kōrero:
Tuatahi, ka kitea e tātou te ārai-tāpiri o 2x + 1, arā, ko x^2 + x. Inaianei ka whakakapia ngā herenga:
∫[0,2] (2x+1) dx = [(2^2 + 2) – (0^2 + 0)]
= [(4 + 2) – 0]
= 6.
Nō reira, ko te hua o ∫[0,2] (2x + 1) dx ko te 6.
4. Tauira o ngā Raru Whakauru me te Tikanga Wāhanga
Ko te whakauru ā-wāhanga he tikanga e whakamahia ana ina uaua te tatau tika i te whakauru o te hua o ngā mahi e rua. Ko te tātai mō te whakauru ā-wāhanga ko:
∫ u dv = uv – ∫ v du
Tauira: Ngā Whakapūtanga Wāhanga Pātoru
Tātaihia ∫ xe^x dx.
Kōrero:
I konei ka whakamahia e mātou te tikanga ā-wāhanga. Me kī ko u = x me dv = e^x dx. Kātahi ko du = dx me v = e^x.
I runga i te tātai tauwehe wāhanga:
∫ xe^x dx = xe^x – ∫ e^x dx
= xe^x – e^x + C
= e^x(x – 1) + C.
Nō reira, ko te hua o ∫ xe^x dx ko e^x(x – 1) + C.
5. Ngā Tauira o ngā Raru Taurite Pānga-toru
Tauira: Te Whakapūtanga o ngā Mahi Pānga-toru Taketake
Tātaihia ∫ cos(x) dx.
Kōrero:
Ko te ture matua mō te whakauru o te cos(x) ko sin(x) + C.
Nō reira, ko te hua o ∫ cos(x) dx ko sin(x) + C.
Tauira: Te Whakapūtanga o ngā Mahi Pānga-toru me ngā Herenga
Tātaihia ∫[0,π/2] sin(x) dx.
Kōrero:
Tuatahi, ka kitea e tātou te ātete-whakaputa o te sin(x), arā, ko -cos(x).
Inaianei, whakakapia ngā here:
∫[0,π/2] hara(x) dx = [ -cos(π/2) – (-cos(0)) ]
= [ -0 – (-1) ]
= 1.
Nō reira, ko te hua o ∫[0,π/2] sin(x) dx ko 1.
6. Tauira o te Raru Whakakapinga Taupū
Tauira: Te Whakakapinga Taupū
Tātaihia ∫ 2x sqrt(1-x^2) dx.
Kōrero:
Whakamahia te whakakapinga u = 1-x^2, kātahi ka du = -2x dx.
Kātahi ka huri te taupū ki:
∫ tapawhā (u) (-1/2 du)
= -1/2 ∫ u^(1/2) du
= -1/2 [ (2/3) u^(3/2) ] + C
= -1/3 (1-x^2)^(3/2) + C.
Nō reira, ko te hua o ∫ 2x sqrt(1-x^2) dx ko -1/3 (1-x^2)^(3/2) + C.
7. Whakamutunga
He taputapu tino whai hua ngā taunga whakauru i roto i te pāngarau hei kimi i te horahanga i raro i te kōpiko, te rōrahi, me te maha atu o ngā whakamahinga. He mea nui te mārama ki ngā tikanga whakauru maha, pērā i te whakakapinga, ngā taunga ā-wāhanga, me ngā kaupapa matua o ngā taunga whakauru. Ko te tumanako ka āwhina ngā tauira kua kōrerohia i runga ake nei ki te whakapakari i tō māramatanga ki te ariā o ngā taunga whakauru.
Ko te whakaharatau auau me te mārama ki ngā ariā he mea nui kia matatau ai koe ki ngā taurangi. Me whakaharatau tonu ki ngā taurangi rerekē me ngā āhua mahi rerekē hei whakawhānui ake i tō mōhiotanga ki tēnei mara.
Ko te tumanako ka whai hua tēnei tuhinga māu i roto i te ako i ngā taupūtanga.
