Ngā Herenga o ngā Mahi Ārai: Kupu Whakataki, Ngā Ariā Taketake me ngā Whakamahinga
He ariā taketake te rohe i roto i te tātaitai e āhei ai tātou ki te tātari i te whanonga o tētahi mahi ina tata atu tana tautohe ki tētahi uara. Ahakoa te āhua whānui o tēnei ariā, he whānui ngā tono o ngā rohe i roto i te oranga o ia rā me ngā momo mara pūtaiao, tae atu ki te pāngarau, te ahupūngao, te ōhanga, me te miihini.
1. Pengantar
Ko te mahi taurangi he mahi i hangaia e ngā pūrinōmā me ngā mahi taurangi taketake pēnei i te tāpiri, te tango, te whakarea, te wehewehe, me te taupū. Hei tauira, ko te mahi \( f(x) = 2x^3 – 5x + 1 \) he mahi taurangi. Ko te rohe o te mahi taurangi, i te kīanga māmā, ko te uara e whakatata atu ana te mahi i te mea e whakatata atu ana tōna taurangi whakauru ki tētahi tau.
2. Whakamāramatanga Whaimana
I roto i te tikanga, ka taea te tuhi i te rohe o tētahi mahi \( f(x) \) i te whakatata atu o \( x \) ki tētahi uara \( c \) penei:
\[ \lim_{{x \to c}} f(x) = L \]
arā, ka whakatata a \( f(x) \) ki \( L \) i te whakatata a \( x \) ki \( c \).
3. Ngā Āhuatanga o ngā Here
Ko ētahi o ngā āhuatanga taketake o ngā rohe e whakamahia whānuitia ana ko:
1. Tepe Pūmau:
Mena ko \( f(x) = k \) te pūmau o \( k \), kāti:
\[ \lim_{{x \to c}} k = k \]
2. Te Rohe o te Tāpiri:
Mena ko \( \lim_{{x \to c}} f(x) = L \) me \( \lim_{{x \to c}} g(x) = M \), kātahi:
\[ \lim_{{x \to c}} [f(x) + g(x)] = L + M \]
3. Tepe Whakarea:
\[ \lim_{{x \to c}} [f(x) \cdot g(x)] = L \cdot M \]
4. Tepe Tohatoha:
Mena \( M \neq 0 \):
\[ \lim_{{x \to c}} \left(\frac{f(x)}{g(x)}\right) = \frac{L}{M} \]
5. Te Rohe o te Hanganga Mahi:
Mena ko \( \lim_{{x \to c}} g(x) = L \) me \( \lim_{{t \to L}} f(t) = M \), kātahi:
\[ \lim_{{x \to c}} f(g(x)) = M \]
4. Ngā Here Mutunga Kore me ngā Here Mutunga Kore
Haunga ngā rohe e tata ana ki tētahi uara, ka taea hoki e ngā rohe te tata ki te mutunga kore. Hei tauira, mō tētahi mahi \( f(x) \), ki te piki haere tonu a \( f(x) \) me te kore here i te whakatata atu o \( x \) ki \( c \), ka tuhia e mātou:
\[ \lim_{{x \to c}} f(x) = \infty \]
I tetahi atu taha, ki te heke te \( f(x) \) me te kore herenga i te whakatata atu o \( x \) ki \( c \), ka tuhia e mātou:
\[ \lim_{{x \to c}} f(x) = -\infty \]
5. Te Ture Hanawiti
He taputapu nui te Sandwich Theorem mō te aromatawai rohe, inā koa he uaua ki te aromatawai tika i te rohe. E kī ana tēnei ariā mēnā ko \( f(x) \leq g(x) \leq h(x) \) mō ngā \( x \) katoa i te taha o \( c \) engari ko te mea pea i \( c \) tonu, ā, mēnā:
\[ \lim_{{x \to c}} f(x) = L = \lim_{{x \to c}} h(x) \]
nā reira:
\[ \lim_{{x \to c}} g(x) = L \]
6. Te Whakamahinga o ngā Herenga o ngā Mahi Ārai
6.1. Ngā Hua Whakaputa
Ko ngā rohe te pūtake o ngā tātaitanga. Ko te tātaitanga o tētahi mahi i tētahi pūwāhi ka homai te tere o te huringa o te mahi i taua pūwāhi. Mena he mahi a \( f(x) \) , ko tōna tātaitanga i \( x = a \) ka homai e:
\[ f'(a) = \lim_{{h \to 0}} \frac{f(a+h) – f(a)}{h} \]
6.2. Whakaurunga
Ka taea hoki te kite i ngā taupū hei rohenga o ngā tapeke mutunga kore. Ko te taupū o \( f(x) \) mai i \( a \) ki \( b \) ka whakaaturia penei:
\[ \int_{a}^{b} f(x) \, dx = \lim_{{n \to \infty}} \sum_{i=1}^{n} f(x_i) \Delta x \]
ko \( x_i \) he pūwāhi i roto i te wā wehewehe, ā, ko \( \Delta x \) te whānui o te wehewehe.
6.3. Ngā Whārite Whakarerekētanga
Ka whakamahia ngā rohenga hei kimi otinga mō ngā whārite rerekētanga. Ko ngā whārite rerekētanga he whārite e whai wāhi ana ngā mahi me ō rātou pānga, ā, ka whakamahia hei whakatauira i ngā āhuatanga taiao, pērā i te nekehanga, te tipu o te taupori, me ngā huringa o ngā kukū matū.
6.4. Ahupūngao
I roto i te ahupūngao, ka whakamahia ngā rohe i roto i ngā ariā maha pēnei i te tere inamata, te whakaterenga, me ngā ture nekehanga a Newton. Hei tauira, ko te tere inamata te rohe o te tere toharite ina tata te wā ki te kore.
7. Ngā Tauira Pātai me te Kōrero
Tauira 1: Te rohenga o tētahi mahi pūrau
Kimihia \( \lim_{{x \to 3}} (2x^2 + 5x – 4) \).
Kōrero:
Whakakapia tika te \( x = 3 \) ki roto i te mahi:
\[ 2(3)^2 + 5(3) – 4 = 2(9) + 15 – 4 = 18 + 15 – 4 = 29 \]
Nō reira, \( \lim_{{x \to 3}} (2x^2 + 5x – 4) = 29 \).
Tauira 2: Te Herenga o ngā Mahi Whaitake
Kimihia \( \lim_{{x \to 2}} \frac{x^2 – 4}{x – 2} \).
Kōrero:
Ka puta te āhua kore-taurite i tēnei mahi \(\frac{0}{0}\). Mā te tauwehe i te taunga:
\[ \frac{x^2 – 4}{x – 2} = \frac{(x-2)(x+2)}{x-2} \]
I muri i te whakangawari:
\[ \frac{(x-2)(x+2)}{x-2} = x+2 \quad (x \neq 2) \]
Nā reira:
\[ \lim_{{x \to 2}} \frac{x^2 – 4}{x – 2} = \lim_{{x \to 2}} (x+2) = 2 + 2 = 4 \]
Whakamutunga
Ko te rohenga o tētahi mahi taurangi he ariā taketake i roto i te tātaitai e whakarato ana i te māramatanga ki te whanonga o tētahi mahi i te mea ka whakatata atu tētahi taurangi ki tētahi uara. He mea nui te mārama ki ngā rohenga hei mārama ki ngā ariā matatau ake o te tātaitai, pērā i te wehewehenga me te whakauru. He whānuitia ngā tono o ngā rohenga, e kapi ana i ngā mara ako me te oranga o ia rā. Mā te mārama pai ki ngā rohenga, ka taea e tātou te tūhura me te whakatau i ngā raruraru uaua i roto i te pāngarau me te pūtaiao.
