Ngā Tauira Pātai me te Kōrero mō ngā Pānga Taurite o ngā Mahi Taurite
He ariā taketake te pāngarau i roto i te tātaitai, e whakamahia ana hei whakaahua i te tere o te huringa o tētahi mahi. Mō ngā mahi pāngarau, ka āwhina te pāngarau i a tātou ki te mārama me pēhea te pānga o ngā huringa o ngā koki ki te uara o te mahi. I roto i tēnei tuhinga, ka matapakihia e tātou ētahi tauira raruraru me ngā otinga e pā ana ki ngā pāngarau o ngā mahi pāngarau.
He Kupu Whakataki ki ngā Mahi Pāngatoru
Ko ngā mahi matua o te pātoru e whakamahia whānuitia ana ko te sine (sin), te cosine (cos), te tangent (tan), te secant (sec), te cosecant (cosec), me te cotangent (cot). He pānga motuhake tō ia mahi:
1. \( \frac{d}{dx} \sin(x) = \cos(x) \)
2. \( \frac{d}{dx} \cos(x) = -\sin(x) \)
3. \( \frac{d}{dx} \tan(x) = \sec^2(x) \)
4. \( \frac{d}{dx} \sec(x) = \sec(x) \tan(x) \)
5. \( \frac{d}{dx} \csc(x) = -\csc(x) \cot(x) \)
6. \( \frac{d}{dx} \cot(x) = -\csc^2(x) \)
Mā tēnei māramatanga taketake, ka taea e tātou te neke atu ki ngā tauira rapanga me ngā otinga hōhonu ake.
Tauira Pātai 1: Te Pūtake o te Mahi Sine
Pātai
Kimihia te pānga o te mahi \( f(x) = 3\sin(x) \).
Te Whakatau
Hei kimi i te pānga o te mahi \( f(x) = 3\sin(x) \), ka taea e tātou te whakamahi i ngā ture taketake o ngā pānga me ngā pūmau i roto i te tātaitai. Ko te pānga o \( \sin(x) \) ko \( \cos(x) \).
\[
f'(x) = 3 \cdot \frac{d}{dx} \sin(x) = 3\cos(x)
\]
Nō reira, ko te pānga o \( f(x) = 3\sin(x) \) ko \( 3\cos(x) \).
Tauira 2: Te Huinga o ngā Mahi Sine me Cosine
Pātai
Kimihia te pānga o te mahi \( g(x) = 2\sin(x) + 4\cos(x) \).
Te Whakatau
Hei kimi i te pānga o te mahi \( g(x) = 2\sin(x) + 4\cos(x) \), ka taea e tātou te whakamahi i ngā ture pānga taketake me te tautuhi i ia pānga o \( \sin(x) \) me \( \cos(x) \).
\[
g'(x) = 2 \cdot \frac{d}{dx} \sin(x) + 4 \cdot \frac{d}{dx} \cos(x)
\]
E mōhio ana mātou:
\[
\frac{d}{dx} \sin(x) = \cos(x)
\]
\[
\frac{d}{dx} \cos(x) = -\sin(x)
\]
Nō reira:
\[
g'(x) = 2 \cos(x) + 4(-\sin(x)) = 2\cos(x) – 4\sin(x)
\]
Nō reira, ko te tauwehenga o \( g(x) = 2\sin(x) + 4\cos(x) \) ko \( 2\cos(x) – 4\sin(x) \).
Tauira 3: Te Mahi Tapawhā o te Sine
Pātai
Kimihia te pānga o te mahi \( h(x) = (\sin(x))^2 \).
Te Whakatau
Hei kimi i te pānga o te mahi \( h(x) = (\sin(x))^2 \), ka taea e tātou te whakamahi i te ture mekameka.
Tuatahi, ka whakatakotoria e tātou \( u = \sin(x) \), kia \( h(x) = u^2 \).
E mōhio ana tātou ko te taupū o \( u^2 \) e pā ana ki \( u \) ko \( 2u \), ā, ko te taupū o \( u \) e pā ana ki \( x \) ko \( \cos(x) \).
Nō reira,
\[
\frac{d}{dx} (\sin(x))^2 = 2 (\sin(x)) \cdot \cos(x)
\]
Nō reira, ko te pānga o \( h(x) = (\sin(x))^2 \) ko \( 2\sin(x)\cos(x) \).
Tauira Pātai 4: Pānga Tātai
Pātai
Kimihia te pānga o te mahi \( f(x) = \tan(x) \).
Te Whakatau
Hei kimi i te pānga o \( f(x) = \tan(x) \), ka whakamahia e mātou te whakamāramatanga o te pānga o te pānihi.
\[
\frac{d}{dx} \tan(x) = \sec^2(x)
\]
Nō reira, ko te pānga o \( f(x) = \tan(x) \) ko \( \sec^2(x) \).
Tauira 5: Te Huinga o ngā Mahi Tāngā me ngā Mahi Momo
Pātai
Kimihia te pānga o te mahi \( p(x) = \tan(x)\sec(x) \).
Te Whakatau
Hei kimi i te pānga o te hua o ngā taumahi e rua, me whakamahi tātou i te ture hua.
\[
(fg)' = f'g + fg'
\]
Ko te wāhi \( f(x) = \tan(x) \) me \( g(x) = \sec(x) \).
E mōhio ana mātou:
\[
f'(x) = \hēkona^2(x)
\]
\[
g'(x) = \h(x)\tan(x)
\]
Nō reira:
\[
p'(x) = \tan(x) \cdot \sec(x) \tan(x) + \sec(x) \cdot \sec^2(x)
\]
\[
p'(x) = \hc^2(x) \t^2(x) + \hc^3(x)
\]
Nō reira, ko te pānga o \( p(x) = \tan(x)\sec(x) \) ko \( \sec^2(x) \tan^2(x) + \sec^3(x) \).
Tauira Pātai 6: Ngā Mahi Taurite me Ngā Mahi Taurite
Pātai
Kimihia te pānga o te mahi \( q(x) = \csc(x) – \cot(x) \).
Te Whakatau
Hei kimi i te pānga o \( q(x) = \csc(x) – \cot(x) \), ka whakamahia e mātou ngā whakamāramatanga o te pānga o te cosecant me te cotangent.
\[
\frac{d}{dx} \csc(x) = -\csc(x)\cot(x)
\]
\[
\frac{d}{dx} \cot(x) = -\csc^2(x)
\]
Nō reira:
\[
q'(x) = -\csc(x)\cot(x) – (-\csc^2(x))
\]
\[
q'(x) = -\csc(x)\cot(x) + \csc^2(x)
\]
Nō reira, ko te tauwehenga o \( q(x) = \csc(x) – \cot(x) \) ko \( -\csc(x)\cot(x) + \csc^2(x) \).
Whakamutunga
I roto i tēnei tuhinga, kua matapakihia e mātou ngā tauira me ngā otinga e pā ana ki ngā pānga o ngā mahi whārite. Mai i ngā mahi taketake pēnei i te sine me te cosine, ki ngā huinga uaua ake pēnei i te hua o te tangent me te secant, me ngā pānga o te cosecant me te cotangent. Ehara i te mea he whai hua anake te mārama ki ngā pānga o ngā mahi whārite i roto i te pāngarau parakore engari he whānui hoki ngā tono i roto i te ahupūngao, te miihini, me ētahi atu mara e whakamahi ana i te huringa mahi me ngā tere o te huringa.
Mā te whakaharatau i ētahi atu rapanga, ka pai ake tō tātou māramatanga ki ngā pānga o ngā mahi pākoki. Ko te tumanako, ka āwhina tēnei tuhinga i a koe ki te mārama ki te ariā me ngā whakamahinga o ngā pānga i roto i ngā mahi pākoki!
