Ngā Mahi Whakarea me te Wehewehe
He maha ngā mahi pāngarau e whai wāhi ana ki ngā momo mara pūtaiao, tae atu ki te ōhanga, te miihini, te ahupūngao, me ētahi atu. Ko ngā mahi matua e rua e whakamahia whānuitia ana i roto i te whakahaere mahi ko te whakarea me te wehewehe. He ariā me ngā tono motuhake tō ēnei mahi e rua, he mea nui kia mārama. Ka matapakihia e tēnei tuhinga ngā mahi whakarea me te wehewehe: ō rātou whakamāramatanga, ō rātou āhuatanga, ngā ture, me ngā tauira.
Te Whakarea Mahi
Whakamāramatanga
He mahi rua te whakarea mahi e tango ana i ngā mahi e rua, ā, ka puta he mahi hou. Mehemea he mahi e rua ā tātou \( f \) me \( g \), kātahi ka tuhia te whakarea o ēnei mahi e rua hei \( f(x) \cdot g(x) \) me \( (fg)(x) \).
Ngā Āhuatanga o te Whakarea Mahi
1. Whakawhitiwhiti: He whakawhitiwhiti te whakarea o ngā mahi, arā, \( f(x) \cdot g(x) = g(x) \cdot f(x) \).
2. Honohono: He hononga anō hoki te whakarea o ngā mahi, arā, \( (f(x) \cdot g(x)) \cdot h(x) = f(x) \cdot (g(x) \cdot h(x)) \).
3. Tohatoha: Ka tohatohahia te whakarea o ngā mahi ki runga i te tāpiri o ngā mahi, arā, \( f(x) \cdot (g(x) + h(x)) = f(x) \cdot g(x) + f(x) \cdot h(x) \).
Tauira
Mehemea ko \( f(x) = 2x + 3 \) me \( g(x) = x^2 \), ko te hua o ngā mahi e rua ko:
\[ (fg)(x) = f(x) \cdot g(x) = (2x + 3) \cdot x^2 = 2x^3 + 3x^2 \].
E whakaatu ana me pēhea te whakakotahi i ngā mahi e rua mā te whakarea hei whakaputa i tētahi mahi hou he rerekē ōna āhuatanga i te mahi taketake.
Te Wehenga o ngā Mahi
Whakamāramatanga
Ko te wehewehenga mahi, i roto i te hinengaro, ko te mahi o te tango i ngā mahi e rua, ka whakaputa i tētahi mahi hou, koia te haurua o ngā mahi e rua. Mehemea kei a tātou ngā mahi \( f \) me \( g \), kātahi ka tuhia te wehewehenga \( f \) ki \( g \) hei \( \frac{f(x)}{g(x)} \) me \( \left(\frac{f}{g}\right)(x) \), mena ko \( g(x) \neq 0 \).
Ngā Āhuatanga o te Wehenga Mahi
1. Kāore i te Whakawhitiwhiti: Ehara te wehenga o ngā mahi i te whakawhitiwhiti, arā, \( \frac{f(x)}{g(x)} \neq \frac{g(x)}{f(x)} \).
2. Kāore i te Honohono: Kāore hoki te wehenga o ngā mahi i te honohono, arā, \( \frac{f(x)}{g(x)/h(x)} \neq \left(\frac{f(x)}{g(x)}\right)/h(x) \).
3. Tohatoha: He tohatoha te mahi wehewehe ki te wehewehenga o ngā huānga, arā, \( f(x)/g(x) = f(x) \cdot \frac{1}{g(x)} \).
Tauira
Mehemea ko \( f(x) = x^2 + 2x \) me \( g(x) = x \), ko te wehenga o ngā mahi e rua ko:
\[ \left(\frac{f}{g}\right)(x) = \frac{x^2 + 2x}{x} = x + 2 \].
E whakaatu ana me pēhea te whakakotahi i ngā mahi e rua mā te wehewehe hei whakaputa i tētahi mahi hou he rerekē ōna āhuatanga i te mahi taketake.
Te Taupānga Mahi Whakarea me te Wehewehe
1. Ōhanga
I roto i te ōhanga, he maha ngā wā ka whakamahia te whakarea me te wehewehe i ngā mahi i roto i te tātari utu me te moni whiwhi. Hei tauira, mēnā ko \( R(x) \) te mahi moni whiwhi, ā, ko \( C(x) \) te mahi utu, ka taea te tatau i te hua penei \( P(x) = R(x) – C(x) \). Mēnā he mahi te moni whiwhi o te maha o ngā waeine i hokona me te utu mō ia waeine, ka taea te tatau i te mahi \( R(x) \) mā te whakarea i te mahi o te maha o ngā waeine me te utu mō ia waeine.
2. Tikanga
He maha ngā whakamahinga a ngā kaihangarau i ngā mahi whakarea me te wehewehe i roto i te tātari pūnaha. Hei tauira, i roto i te tātari ara iahiko, ka taea te tatau i te aukati whakakotahi o ngā wāhanga e rua e honoa ana i roto i te raupapa mā te whakarea i ngā mahi aukati o ia wāhanga. Waihoki, ka whakamahia ngā mahi wehewehe i roto i te whakahaere pūnaha hei whakatau i te urupare a te pūnaha ki tētahi whakaurunga motuhake.
3. Ahupūngao
I roto i te ahupūngao, he maha ngā ariā e whakamahi ana i te whakarea me te wehewehe i ngā mahi. Hei tauira, ko te mahi i mahia e te kaha ki runga i tētahi mea e neke ana ka taea te tatau hei tauwehenga o te mahi kaha ki runga i te mahi tawhiti. I tetahi atu taha, ko te ariā o te tere toharite i roto i te nekehanga ka taea te tātari mā te wehewehe i te mahi tawhiti katoa ki te mahi wā katoa.
Ngā Ture Whakapūtātanga mō te Whakarea me te Wehewehenga o ngā Mahi
I roto i te tātaitai, he mea tino nui ngā ture pārōnaki mō te whakarea me te wehewehe i ngā mahi.
Ture Hua
Mena he rerekētanga ka taea te wehewehe i a \( f(x) \) me \( g(x) \), ko te pānga o \( f(x) \cdot g(x) \) ko:
\[ (fg)'(x) = f'(x)g(x) + f(x)g'(x) \].
Ture Whakarea
Mena he rerekētanga ka taea te wehewehe i a \( f(x) \) me \( g(x) \), ko te pānga o \( \frac{f(x)}{g(x)} \) ko:
\[ \left(\frac{f}{g}\right)'(x) = \frac{f'(x)g(x) – f(x)g'(x)}{g(x)^2} \],
me te tikanga \( g(x) \neq 0 \).
Tauira
Mehemea \( f(x) = x^2 \) me \( g(x) = x + 1 \), ka tatauhia e tātou te pānga o \( f(x) \) whakareatia ki \( g(x) \).
1. \( f'(x) = 2x \)
2. \( g'(x) = 1 \)
3. E ai ki ngā ture whakarea:
\[ (fg)'(x) = 2x(x + 1) + x^2(1) = 2x^2 + 2x + x^2 = 3x^2 + 2x \].
Nā, tatauhia te pānga o \( \frac{f(x)}{g(x)} \).
1. E ai ki ngā ture wehewehe:
\[ \left(\frac{f}{g}\right)'(x) = \frac{(2x)(x + 1) – (x^2)(1)}{(x + 1)^2} = \frac{2x^2 + 2x – x^2}{(x + 1)^2} = \frac{x^2 + 2x}{(x + 1)^2} \].
Whakamutunga
He ariā taketake te whakarea me te wehewehe i ngā mahi i roto i te arapūrei me te tātaitai, me ngā tono maha puta noa i te whānuitanga o ngā marautanga. He mea nui te mārama ki ngā āhuatanga, ngā ture o te wehewehe, me ngā tono mahi o ēnei mahi mō te tātari tika me te whai hua. Ahakoa he tohunga pāngarau koe, he miihini, he tohunga ōhanga rānei, he pūkenga tino whai hua te pūkenga ki te mahi me te whakarea me te wehewehe i ngā mahi.
