Ngā Tauira Pātai e Matapaki ana i te Tuhituhi i ngā Pānga Mahi
Ko te pānga o tētahi mahi he ariā taketake i roto i te tātaitai, e whakamahia whānuitia ana i roto i ngā momo mara pūtaiao, pērā i te ahupūngao, te ōhanga, te koiora, me te hangarau. Ka ine te pānga o tētahi mahi i te tere o te huringa o tōna uara e pā ana ki ngā huringa o ōna taurangi motuhake. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira rapanga e pā ana ki te tuhi i te pānga o tētahi mahi, me ngā whakamārama.
Tauira Pātai 1: Te Pūtake o ngā Mahi Māmā
Pātai: Kimihia te pānga tuatahi o te mahi \( f(x) = 3x^2 + 5x + 7 \).
Kōrero:
Hei whakatau i te pānga tuatahi o tētahi mahi \( f(x) \), ka whakamahia e mātou ngā ture taketake o te wehewehenga, arā:
\[
\frac{d}{dx}(ax^n) = anx^{n-1}
\]
Nō reira, ka taea e tātou te tatau i te taupū o ia kupu i roto i te mahi penei:
\[
f'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(5x) + \frac{d}{dx}(7)
\]
\[
f'(x) = 3 \cdot 2x^{2-1} + 5 \cdot 1x^{1-1} + 0
\]
\[
f'(x) = 6x + 5
\]
Nō reira, ko te pānga tuatahi o te mahi \( f(x) = 3x^2 + 5x + 7 \) ko \( f'(x) = 6x + 5 \).
Tauira Pātai 2: Ngā Pānga o ngā Mahi Pāngatoru
Pātai: Kimihia te pānga tuatahi o te mahi \( g(x) = \sin(x) + \cos(x) \).
Kōrero:
Ka whakamahia e mātou ngā ture taketake mō ngā pānga pākoki:
\[
\frac{d}{dx}(\sin(x)) = \cos(x)
\]
\[
\frac{d}{dx}(\cos(x)) = -\sin(x)
\]
Nā reira:
\[
g'(x) = \frac{d}{dx}(\sin(x)) + \frac{d}{dx}(\cos(x))
\]
\[
g'(x) = \cos(x) – \sin(x)
\]
Nō reira, ko te pānga tuatahi o te mahi \( g(x) = \sin(x) + \cos(x) \) ko \( g'(x) = \cos(x) – \sin(x) \).
Tauira Pātai 3: Te Pūtake o te Mahi Whakarea
Pātai: Kimihia te pānga tuatahi o te mahi \( h(x) = x^2 \sin(x) \).
Kōrero:
Mō ngā mahi he hua o ngā mahi e rua, ka whakamahia e mātou te ture whakarea:
\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]
Me kī ko \( u(x) = x^2 \) me \( v(x) = \sin(x) \). Kātahi:
\[
u'(x) = \frac{d}{dx}(x^2) = 2x
\]
\[
v'(x) = \frac{d}{dx}(\sin(x)) = \cos(x)
\]
Mā te whakamahi i te ture whakarea, ka taea e tātou te tuhi:
\[
h'(x) = [x^2]' \sin(x) + x^2 [\sin(x)]'
\]
\[
h'(x) = 2x \sin(x) + x^2 \cos(x)
\]
Nō reira, ko te pānga tuatahi o te mahi \( h(x) = x^2 \sin(x) \) ko \( h'(x) = 2x \sin(x) + x^2 \cos(x) \).
Tauira Pātai 4: Te Pūtake o tētahi Mahi Whakatakotoranga
Pātai: Kimihia te pānga tuatahi o te mahi \( k(x) = \sin(x^2) \).
Kōrero:
Mō ngā mahi e tito ana i ngā mahi e rua, ka whakamahia e mātou te ture mekameka:
\[
\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
\]
Me \( f(u) = \sin(u) \) me \( u = x^2 \). Kātahi ka \( f'(u) = \cos(u) \) me \( g'(x) = \frac{d}{dx}(x^2) = 2x \).
Mā te whakamahi i te ture mekameka, ka taea e tātou te tuhi:
\[
k'(x) = \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x
\]
Nō reira, ko te pānga tuatahi o te mahi \( k(x) = \sin(x^2) \) ko \( k'(x) = 2x \cos(x^2) \).
Tauira Pātai 5: Te Whakaputa o ngā Mahi Whaitake
Raru: Kimihia te pānga tuatahi o te mahi \( m(x) = \frac{2x}{x^2 + 1} \).
Kōrero:
Mō ngā mahi ko te haurua o ngā mahi e rua, ka whakamahia e mātou te ture haurua:
\[
\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2}
\]
Me kī ko \( u(x) = 2x \) me \( v(x) = x^2 + 1 \). Kātahi:
\[
u'(x) = 2
\]
\[
v'(x) = \frac{d}{dx}(x^2 + 1) = 2x
\]
Mā te whakamahi i te ture quotient, ka taea e tātou te tuhi:
\[
m'(x) = \frac{[2x]'(x^2 + 1) – 2x[x^2 + 1]'}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2(x^2 + 1) – 2x \cdot 2x}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2x^2 + 2 – 4x^2}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{-(2x^2 – 2)}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2}
\]
Nō reira, ko te pānga tuatahi o te mahi \( m(x) = \frac{2x}{x^2 + 1} \) ko \( m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2} \).
Whakamutunga
I roto i tēnei tuhinga, kua matapakihia e mātou ētahi tauira o ngā raruraru e pā ana ki ngā pānga o ngā mahi, mai i ngā mahi māmā, ngā mahi pākoki, te whakarea, te hanganga, me ngā mahi whaitake. E whakaatu ana ia tauira i te whakamahinga tika o ngā ture pānga, pērā i te ture taketake, te ture mekameka, te ture whakarea, me te ture wāhanga. He mea nui te mārama ki te whakamahi i ēnei ture hei whakaoti rapanga tātaitai uaua ake puta noa i ngā momo marautanga. Mā te mahi me te whakangungu tonu ka āwhina i te whakapakari i tō māramatanga me ō pūkenga ki te wehewehe i ngā mahi.
