Taitara: Ngā Pātai Tauira me te Kōrero mō ngā Whārite Raina Pānga ki ngā Piko
Pendahuluan
He ariā nui te whārite o te pātata ki te kōpiko i roto i te pāngarau, inā koa te tātaitai me te āhuahanga tātari. Ko te pātata ki te kōpiko he rārangi e pā ana ki te kōpiko i tētahi pūwāhi motuhake, engari kāore e whakawhiti. He rite te pari o tēnei rārangi ki te kōpiko i taua pūwāhi. Ko te whāinga o tēnei tuhinga he whakarato tauira me ngā matapakinga mō te whārite o te pātata ki te kōpiko, kia mārama ai ngā kaipānui ki tēnei ariā.
Te Whakamāramatanga me ngā Ariā Taketake
I mua i te urunga atu ki ngā tauira rapanga, he mea pai kia arotakehia ētahi ariā taketake. Ka taea te whakatau i te whārite o te rārangi pātapa ki tētahi kōpiko \( y = f(x) \) i te pūwāhi \((a, f(a))\) mā te whakamahi i te pānga tuatahi o te mahi. Ko ngā mahi whānui e whai ake nei:
1. Te Whakatau i te Pānga o tētahi Mahi: Kimihia te pānga tuatahi \( f'(x) \) o te mahi \( f(x) \).
2. Te Whakatau i te Rōrahi (Pīnakitanga): Tāpirihia te uara \( x = a \) ki te taupū tuatahi \( f'(x) \) hei tiki i te pīnakitanga o te rārangi pātapa i te pūwāhi \( a \).
3. Whakatauhia te Whārite o te Raina Pātata: Whakamahia te pūwāhi \((a, f(a))\) me te pari kua kitea e koe hei hanga i te whārite o te raina pātata. Ka taea te tuhi i te whārite o te rārangi ki te āhua \( y – y_1 = m(x – x_1) \), ko \((x_1, y_1)\) he pūwāhi kei runga i te rārangi, ā, ko \( m \) te pari.
Ngā Pātai Tauira me te Kōrero
Me matapaki tātou i ētahi tauira pātai hei whakamārama i tō tātou māramatanga.
Tauira Pātai 1
I te mea kua hoatu te kōpiko \( y = x^2 + 2x + 1 \). Whakatauhia te whārite o te rārangi pātapa ki te kōpiko i te pūwāhi me te abscissa \( x = 1 \).
Kōrero:
1. Whakatauhia te Pānga Tuatahi:
\[
y = x^2 + 2x + 1
\]
Ko te tuatahi o ngā pānga o \( y \) ko:
\[
\frac{dy}{dx} = 2x + 2
\]
2. Whakatauhia te pikinga i \( x = 1 \):
Whakakapia te \( x = 1 \) ki te tātaitanga tuatahi:
\[
m = 2(1) + 2 = 4
\]
3. Te Whakatau i te Pūwāhi o te Pānga:
Kimihia te uara o \( y \) i \( x = 1 \):
\[
y = (1)^2 + 2(1) + 1 = 4
\]
Nō reira, ko te pūwāhi o te pātata ko \( (1, 4) \).
4. Te Whakatau i te Whārite o te Raina Pānga:
Whakamahia te whārite o te rārangi \( y – y_1 = m(x – x_1) \):
\[
y – 4 = 4(x – 1)
\]
Whakangāwaritia:
\[
y – 4 = 4x – 4
\]
\[
y = 4x – 4 + 4
\]
\[
y = 4x
\]
Nō reira, ko te whārite o te rārangi pātapa ko \( y = 4x \).
Tauira Pātai 2
I te mea kua hoatu te kōpiko \( y = \cos(x) \). Whakatauhia te whārite o te rārangi pātapa ki te kōpiko i te pūwāhi me te abscissa \( x = \frac{\pi}{2} \).
Kōrero:
1. Whakatauhia te Pānga Tuatahi:
\[
y = \cos(x)
\]
Ko te tuatahi o ngā pānga o \( y \) ko:
\[
\frac{dy}{dx} = -\sin(x)
\]
2. Whakatauhia te rōnaki i \( x = \frac{\pi}{2} \):
Whakakapia \( x = \frac{\pi}{2} \) ki te tātaitanga tuatahi:
\[
m = -\sin\left(\frac{\pi}{2}\right) = -1
\]
3. Te Whakatau i te Pūwāhi o te Pānga:
Kimihia te uara o \( y \) i roto i \( x = \frac{\pi}{2} \):
\[
y = \cos\left(\frac{\pi}{2}\right) = 0
\]
Nō reira, ko te pūwāhi pātata ko \( \left(\frac{\pi}{2}, 0\right) \).
4. Te Whakatau i te Whārite o te Raina Pānga:
Whakamahia te whārite o te rārangi \( y – y_1 = m(x – x_1) \):
\[
y – 0 = -1\left(x – \frac{\pi}{2}\right)
\]
\[
y = -x + \frac{\pi}{2}
\]
Nō reira, ko te whārite o te rārangi pātata ko \( y = -x + \frac{\pi}{2} \).
Tauira Pātai 3
I te mea kua hoatu te kōpiko \( y = e^x \). Whakatauhia te whārite o te rārangi pātapa ki te kōpiko i te pūwāhi me te abscissa \( x = 0 \).
Kōrero:
1. Whakatauhia te Pānga Tuatahi:
\[
y = e^x
\]
Ko te tuatahi o ngā pānga o \( y \) ko:
\[
\frac{dy}{dx} = e^x
\]
2. Whakatauhia te pikinga i \( x = 0 \):
Whakakapia te \( x = 0 \) ki te tātaitanga tuatahi:
\[
m = e^0 = 1
\]
3. Te Whakatau i te Pūwāhi o te Pānga:
Kimihia te uara o \( y \) i \( x = 0 \):
\[
y = e^0 = 1
\]
Nō reira, ko te pūwāhi o te pātata ko \( (0, 1) \).
4. Te Whakatau i te Whārite o te Raina Pānga:
Whakamahia te whārite o te rārangi \( y – y_1 = m(x – x_1) \):
\[
y – 1 = 1(x – 0)
\]
\[
y – 1 = x
\]
\[
y = x + 1
\]
Nō reira, ko te whārite o te rārangi pātapa ko \( y = x + 1 \).
Te Katinga
Mā te mārama ki te whakatau i te whārite o te rārangi pātata ki te kōpiko, ka taea e tātou te mārama ake ki te mahi tahi a te āhuahanga me te taurangi ki te tūhura i ngā āhuatanga o ngā kōpiko. Ko ngā mahi whānui ko te kimi i te pānga, te whakatau i te pari, me te hanga i te whārite o te rārangi pātata i runga i ngā pūwāhi me te pari i kitea. Mā te mahi tonu, ka hohonu ake tō tātou mārama ki tēnei kaupapa. Kia pai te ako!
