Ngā Tauira Pātai me te Kōrero mō te Ture Mekameka i roto i ngā Pānga
Ko te ture mekameka tētahi o ngā ariā tino taketake o te tātaitanga rerekētanga, e whakamahia ana hei tatau i te pānga o tētahi mahi e titoa ana e ngā mahi e rua, neke atu rānei. I roto i tēnei tuhinga, ka matapakihia e mātou te ariā taketake o te ture mekameka, me pēhea te whakamahi, me ngā tauira o tōna whakamahinga i roto i ngā raruraru pānga e puta pinepine ana i te kura tuarua me te whare wānanga.
1. Whakataki ki te Ture Mekameka
I mua i te urunga atu ki te tauira raruraru, me mārama tuatahi tātou ki te tikanga o te ture mekameka. E mea ana te ture mekameka mēnā e rua ā tātou mahi rerekētanga \( f \) me \( g \), ā, e hiahia ana tātou ki te kimi i te pānga o te hanganga o ngā mahi \( h = f(g(x)) \), ko te pānga o \( h \) ko:
\[ h'(x) = f'(g(x)) \cdot g'(x) \]
I ngā kupu māmā noa iho, ka tatauhia e tātou te pānga o te mahi o waho i runga i te g(x), kātahi ka whakareatia te hua ki te pānga o te mahi o roto \( g(x) \).
2. Te Mārama ki te Mahi a te Hanganga
I mua i tā tātou urunga atu ki ngā tauira raruraru, he mea nui kia mārama ki ngā mahi hanganga. Ko te mahi hanganga he mahi e whiwhihia ana mā te whakauru i tētahi mahi ki tētahi atu. Hei tauira, mēnā kei a tātou \( f(x) = \sin(x) \) me \( g(x) = x^2 \), ko te hanganga o ngā mahi e rua ko \( h(x) = f(g(x)) = \sin(x^2) \).
I roto i ngā mahi tito, he maha ngā wā ka whakaarohia ko \( g(x) \) te "mahinga o roto" me \( f(x) \) te "mahinga o waho". I tēnei tauira, ko te mahi o roto ko \( x^2 \) ā, ko te mahi o waho ko sine.
3. Ngā Tauira Pātai me te Kōrero
Me titiro tātou ki ētahi tauira rapanga e whakamahi ana i te ture mekameka hei whakaoti rapanga.
Tauira 1:
I runga i te mahi \( y = \cos(3x^2) \), kimihia te pānga tuatahi o y e pā ana ki a x.
Kōrero:
Tuatahi, ka tautuhia e tātou ngā mahi ā-roto me ngā mahi ā-waho. I konei, ko te mahi ā-roto ko \( g(x) = 3x^2 \) ā, ko te mahi ā-waho ko \( f(g) = \cos(g) \).
E mōhio ana mātou:
1. \( g'(x) = 6x \)
2. \( f'(g) = -\sin(g) \)
Mā te ture mekameka, ka whiwhi tātou:
\[ y' = f'(g(x)) \cdot g'(x) = -\sin(3x^2) \cdot 6x \]
Nō reira, ko te tauwehenga o \( y = \cos(3x^2) \) ko:
\[ y' = -6x \sin(3x^2) \]
Tauira 2:
Kimihia te taupū tuatahi o \( h(x) = e^{5x^3 + 2x} \).
Kōrero:
Ko te mahi ā-roto i konei ko \( g(x) = 5x^3 + 2x \) ā, ko te mahi ā-waho ko \( f(g) = e^g \).
E mōhio ana mātou:
1. \( g'(x) = 15x^2 + 2 \)
2. \( f'(g) = e^g \)
Mā te ture mekameka, ka whiwhi tātou:
\[ h'(x) = f'(g(x)) \cdot g'(x) = e^{5x^3 + 2x} \cdot (15x^2 + 2) \]
Nō reira, ko te tauwehenga o \( h(x) = e^{5x^3 + 2x} \) ko:
\[ h'(x) = (15x^2 + 2)e^{5x^3 + 2x} \]
Tauira 3:
Kimihia te taupū tuatahi o \( y = \ln(4x^2 – 5) \).
Kōrero:
Ko te mahi ā-roto ko \( g(x) = 4x^2 – 5 \) ā, ko te mahi ā-waho ko \( f(g) = \ln(g) \).
E mōhio ana mātou:
1. \( g'(x) = 8x \)
2. \( f'(g) = \frac{1}{g} \)
Mā te ture mekameka, ka whiwhi tātou:
\[ y' = f'(g(x)) \cdot g'(x) = \frac{1}{4x^2 – 5} \cdot 8x \]
Nō reira, ko te pānga o \( y = \ln(4x^2 – 5) \) ko:
\[ y' = \frac{8x}{4x^2 – 5} \]
Tauira 4:
I runga i te mahi \( y = (3x^2 + 2x + 1)^4 \), kimihia tōna pānga.
Kōrero:
Ko te mahi ā-roto ko \( g(x) = 3x^2 + 2x + 1 \) ā, ko te mahi ā-waho ko \( f(g) = g^4 \).
E mōhio ana mātou:
1. \( g'(x) = 6x + 2 \)
2. \( f'(g) = 4g^3 \)
Mā te ture mekameka, ka whiwhi tātou:
\[ y' = f'(g(x)) \cdot g'(x) = 4(3x^2 + 2x + 1)^3 \cdot (6x + 2) \]
Nō reira, ko te pānga o \( y = (3x^2 + 2x + 1)^4 \) ko:
\[ y' = 4(3x^2 + 2x + 1)^3 (6x + 2) \]
4. Ngā Take Motuhake me te Whakawhanaketanga o ngā Ture Mekameka
I ētahi wā, kāore te ture mekameka e mutu ki te hanganga o ngā mahi e rua anake. Tērā ētahi wā he hanganga o ngā mahi neke atu i te rua te mahi, hei tauira: \( h(x) = f(g(k(x))) \).
Mō ngā mahi e toru, ka taea te whakamahi i te ture mekameka i roto i ngā paparanga:
\[ h'(x) = f'(g(k(x))) \cdot g'(k(x)) \cdot k'(x) \]
Ka kite tātou i ia paparanga, ka tatauhia e tātou ngā pānga o ngā paparanga o waho i mua i te neke atu ki ngā pānga o ngā paparanga o roto.
Tauira 5:
I te mea kua hoatu \( y = \sqrt{\ln(2x^2 + 1)} \), kimihia tōna taupū.
Kōrero:
Ko te mahi o roto rawa ko \( k = 2x^2 + 1 \), waenganui: \( g = \ln(k) \) me waho: \( f = \sqrt{g} \).
E mōhio ana mātou:
1. \( k'(x) = 4x \)
2. \( g'(k) = \frac{1}{k} \)
3. \( f'(g) = \frac{1}{2\sqrt{g}} \)
Me whakamahi te ture mekameka i roto i ngā paparanga:
\[ y' = f'(g(k(x))) \cdot g'(k(x)) \cdot k'(x) = \frac{1}{2\sqrt{\ln(2x^2 + 1)}} \cdot \frac{1}{2x^2 + 1} \cdot 4x \]
Nō reira, ko te pānga o \( y = \sqrt{\ln(2x^2 + 1)} \) ko:
\[ y' = \frac{4x}{2(2x^2 + 1)\sqrt{\ln(2x^2 + 1)}} \]
5. Whakamutunga
He mea nui te ture mekameka i roto i te tātaitai rerekētanga, inā koa ka pā ki te hanganga o ngā mahi. Mā te mārama me te matatau ki te ture mekameka ka whakaratohia he tūāpapa pakari mō te aro atu ki ngā raruraru uaua ake i roto i te tātaitai. Kua matapakihia e tēnei tuhinga ētahi tauira nui hei whakarato i tētahi māramatanga pakari mō te whakamahinga o te ture mekameka ki ngā taupatupatu. Ko te tumanako ka whai hua tēnei kōrero ki ngā ākonga, ā, ka taea te whakamahi ki ngā momo āhuatanga pāngarau wero.
