Ngā Huanga o ngā Mahi Pāngatoru
I roto i ngā pāngarau matatau, inā koa ko te tātaitai, he maha ngā wā ka tūtaki tātou ki ngā mahi pāngarau pēnei i te sine (sin), te cosine (cos), te secant (sec), te cosecant (csc), te tangent (tan), me te cotangent (cot). I roto i tēnei horopaki, he mea nui te mōhio ki ngā pānga o ēnei mahi, inā koa mō ngā tono i roto i te ahupūngao, te hangarau, me te pūtaiao rorohiko. Ka whakamāramahia e tēnei tuhinga me pēhea te whakatau i ngā pānga o ēnei mahi pāngarau.
Kupu Whakataki ki ngā Hua Whakaputa
I mua i te matapaki i ngā pānga o ngā mahi pākoki, me arotake poto te ariā o tētahi pānga. Mā te pānga o tētahi mahi ka homai te tere o te huringa o taua mahi e pā ana ki tōna taurangi motuhake. I roto i ngā kupu āhuahanga, mā te pānga o tētahi mahi f(x) i tētahi pūwāhi x ka homai te rōnaki, te pari rānei, o te rārangi pātata ki te kōpiko f(x) i taua pūwāhi.
Mā te pāngarau, ko te whakamāramatanga tuatahi o te mahi f(x) e whai ake nei:
\[ f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) – f(x)}{\Delta x} \]
He rite tonu tēnei whakamāramatanga mō ngā mahi pākoki, engari ka māmā ake mēnā ka mōhio tātou ki ētahi pārōnaki taketake o ngā mahi pākoki.
Ngā Huarahi o ngā Mahi Pāngatoru Taketake
1. Te pānga sine (sin x)
Ko te mahi sine tētahi o ngā mahi trigonometric tino taketake. Ko te pānga o te sin x ko cos x. I ahu mai tēnei i ētahi rohe me te arapūrei rerekētanga.
\[ \frac{d}{dx}(\sin x) = \cos x \]
Arā, ki te mea ko f(x) = sin x, ko f'(x) = cos x.
2. Te Whakapūtanga o te Kosinī (cos x)
Ko te kōsina tētahi atu mahi whārite taketake. Ko te pānga o te kōsina x ko -sin x.
\[ \frac{d}{dx}(\cos x) = -\sin x \]
Arā, mēnā ko f(x) = cos x, ko f'(x) = -sin x.
3. Te Whakapūtanga o te Pānga (tan x)
Ko te mahi tānga ko te ōwehenga o te sine me te cosine. Ko te pānga o te tan x ko te sec^2 x. Ka taea tēnei te whiwhi mā te whakamahi i te ture pānga mō ngā mahi hiato (mekameka).
\[ \frac{d}{dx}(\tan x) = \sec^2 x \]
Arā, mēnā ko te f(x) = te parauri x, ko te f'(x) = te hēkona² x.
4. Te Whakapūtanga o te Cotangent (cot x)
Ko te cotangent te whakahuri o te tangent. Ko te tauwehenga o te cot x ko -csc² x.
\[ \frac{d}{dx}(\cot x) = -\csc^2 x \]
Arā, mēnā ko f(x) = cot x, ko f'(x) = -csc² x.
5. Te Whakapūtanga Momotu (sekona x)
Ko te mahi secant te whakahuri o te cosine. Ko te pānga o te sec x ko te sec x tan x.
\[ \frac{d}{dx}(\hēkona x) = \hēkona x \tan x \]
Arā, mēnā ko te f(x) = te hēkona x, ko te f'(x) = te hēkona x te parauri x.
6. Te Whakapūtanga Taurite (csc x)
Ko te mahi cosecant te whakahuri o te sine. Ko te pārōnaki o csc x ko -csc x cot x.
\[ \frac{d}{dx}(\csc x) = -\csc x \cot x \]
Arā, mēnā ko f(x) = csc x, ko f'(x) = -csc x cot x.
Te Whakamahinga o ngā Ture Whakaputa ki ngā Mahi Pāngatoru
Kia mōhio tātou ki ngā pānga taketake o ngā mahi pākoki, ka taea e tātou te whakawhānui atu ki ngā tono uaua ake mā te whakamahi i ngā ture pānga pērā i te ture mekameka, te ture hua, me te ture tapeke.
1. Ture mekameka
Ka whakamahia te ture mekameka ina he mahi kei roto i ngā mahi e rua, neke atu rānei. Ngā tauira o tōna whakamahinga:
Mena he mahi tā tātou \( g(x) = \sin(3x^2) \), ka taea e tātou te whakamahi i te ture mekameka hei kimi i tōna pānga:
\[ g'(x) = \frac{d}{dx}[\sin(3x^2)] \]
\[ = \cos(3x^2) \cdot \frac{d}{dx}[3x^2] \]
\[ = \cos(3x^2) \cdot 6x \]
\[ = 6x \cos(3x^2) \]
2. Ngā Ture Hua
Ka whakamahia te ture hua ina he mahi tā tātou e hua ana i ngā mahi e rua, neke atu rānei. Ngā tauira o tōna whakamahinga:
Mena ko te \( h(x) = x^2 \sin(x) \), mā te ture hua:
\[ h'(x) = \frac{d}{dx}[x^2 \cdot \sin(x)] \]
\[ = x^2 \cdot \cos(x) + \sin(x) \cdot \frac{d}{dx}[x^2] \]
\[ = x^2 \cos(x) + \sin(x) \cdot 2x \]
\[ = x^2 \cos(x) + 2x \sin(x) \]
3. Ngā Ture Tau
Ka whakamahia te ture tāpeke ina he mahi tā tātou e rite ana ki te tāpeke o ngā mahi e rua, neke atu rānei. Ngā tauira o tōna whakamahinga:
Mena ko te f(x) = sin(x) + cos(x):
\[ f'(x) = \frac{d}{dx}[\sin(x) + \cos(x)] \]
\[ = \frac{d}{dx}[\sin(x)] + \frac{d}{dx}[\cos(x)] \]
\[ = \cos(x) + (-\sin(x)) \]
\[ = \cos(x) – \sin(x) \]
Ngā Mahi Pānga Taurite Whakamuri me ō Rātou Pānga
Haunga ngā mahi whārite taketake, kei a tātou anō ngā mahi whārite whakamuri pēnei i te sin^-1 x (arcsin x), cos^-1 x (arccos x), me te tan^-1 x (arctan x). He mea nui anō hoki ngā pānga o ēnei mahi i roto i ngā tono o te tātaitai.
Hei tauira:
– Te pūtake o te arcsin x:
\[ \frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1 – x^2}} \]
– Te pūtake o te arccos x:
\[ \frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1 – x^2}} \]
– Te pūtake o te arctan x:
\[ \frac{d}{dx}(\arctan x) = \frac{1}{1 + x^2} \]
Whakamutunga
He taahiraa taketake te ako i ngā pānga o ngā mahi pākoki ine. Ko ngā pānga o ngā mahi taketake pēnei i te sin, cos, tan, cot, sec, me te csc he turanga pakari mō te tātari me te whakaoti rapanga uaua ake i roto i ngā momo kaupapa ako. Hei tāpiri, mā te mārama ki te ture mekameka, te ture hua, me te ture tapeke ka āwhina i a tātou ki te whakahaere i ngā pānga o ngā mahi uaua ake. He mea tino nui tēnei mōhiotanga i roto i ngā tono mahi me ngā tono ariā maha, tae atu ki te ahupūngao, te hangarau, me te pūtaiao rorohiko.
