Tauira o ngā Pātai Kōrero mō te Riemann Sum
Pendahuluan
Ko te tapeke Riemann he ariā taketake i roto i te tātaitai e whakamahia ana hei tautuhi i te tauwehenga tuturu o tētahi mahi. Ka whakamahia e tēnei tikanga te wehenga āputa me te tapeke o ngā horahanga o ngā tapawhā rite hei whakatata i te tauwehenga. Ka matapakihia e tēnei tuhinga te ariā o te tapeke Riemann, tae atu ki ngā tauira me ngā kōrero hei āwhina i te mārama.
Te Ariā Taketake o te Riemannian Sum
I mua i tā tātou kōrero mō ngā tauira, he mea nui kia mārama ki te ariā taketake o ngā tapeke Riemannian. Ka taea te wehewehe i ngā tapeke Riemannian ki ngā momo matua e toru:
1. Te tapeke Riemann maui
2. Te tapeke Riemann matau
3. Te tapeke o te Riemann i waenganui
Mā tēnei tikanga ka wāwāhia te āputa o te mahi hei whakauru ki ngā āputa iti ake he rite te roa. Kātahi ka whakamahia ia āputa hei hanga i tētahi tapawhā rite, ko tōna teitei ka whakatauhia e te uara o te mahi i tētahi pūwāhi motuhake i roto i te āputa iti (mauī, matau, pokapū rānei).
Tātai Whānui mō te Riemann Sum
Me kī, me whakauru te mahi \( f(x) \) mai i \( a \) ki \( b \). Ka wehea e tātou te wā \( [a, b] \) ki \( n \) ngā wā iti rite o te roa \( \Delta x = \frac{ba}{n} \). Ka taea te tuhi i ngā tapeke Riemann mō ngā momo e toru kua whakahuatia i runga ake nei penei:
1. Riemann ki te taha maui:
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x \]
2. Riemann Matau:
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x \]
3. Riemann Waenga:
\[ M_n = \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\right) \Delta x \]
Kei hea:
– Ko te whānui o ia āputa-iti ko \( \Delta x \).
– Ko \( x_i \) te tīmatanga o te wā-iti mō te tapeke Riemann maui.
– Ko \( x_i \) te pito o te wā-iti mō te tapeke Riemann matau.
– Ko \( \frac{x_i + x_{i+1}}{2} \) te pūwāhi o te wā-iti mō te tapeke Riemann waenga.
Ngā Pātai Tauira me te Kōrero
Me matapakihia ngā tauira rapanga mō ia momo Riemann Sum hei hōhonu ake i tō tātou māramatanga.
Tauira 1: Te Tapeke Riemann Mauī
Tātaihia te tapeke Riemann maui mō \( f(x) = x^2 \) i te wā \([0, 2]\) me \( n = 4 \).
Kōrero:
1. Whānuitanga o raro i te wā (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Pūwāhi Wehewehenga Wā (maui):
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5 \]
3. Uara Mahi i te Pūwāhi Wehewehe:
\[ f(x_0) = f(0) = 0^2 = 0 \]
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
4. Te Tapeke Riemann Mauī (Rn):
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x = (0) \cdot 0.5 + (0.25) \cdot 0.5 + (1) \cdot 0.5 + (2.25) \cdot 0.5 \]
\[ L_n = 0 + 0.125 + 0.5 + 1.125 \]
\[ R_n = 1.75 \]
Tauira 2: Te Tapeke Riemann Matau
Tātaihia te tapeke Riemann tika mō \( f(x) = x^2 \) i te wā \([0, 2]\) me \( n = 4 \).
Kōrero:
1. Whānuitanga o raro i te wā (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Pūwāhi Wehewehe Wā (matau):
\[ x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, x_4 = 2.0 \]
3. Uara Mahi i te Pūwāhi Wehewehe:
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
\[ f(x_4) = f(2.0) = (2.0)^2 = 4 \]
4. Te Tapeke Riemann Matau (Rn):
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x = (0.25) \cdot 0.5 + (1) \cdot 0.5 + (2.25) \cdot 0.5 + (4) \cdot 0.5 \]
\[ R_n = 0.125 + 0.5 + 1.125 + 2 \]
\[ R_n = 3.75 \]
Tauira 3: Te Tapeke Riemann Waenga
Tātaihia te tapeke Riemann waenga mō \( f(x) = x^2 \) i te wā \([0, 2]\) me \( n = 4 \).
Kōrero:
1. Whānuitanga o raro i te wā (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Waenganui o te Waenga-iti:
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, \text{ me } x_{n-1}=2.0 \]
Waenganui o te wā poto:
\[tm_0 = \left(\frac{0 + 0.5}{2}\right)=0.25 \]
\[tm_1 = \left(\frac{0.5 + 1.0}{2}\right)=0.75 \]
\[tm_2 = \left(\frac{1.0 + 1.5}{2}\right)=1.25 \]
\[tm_3 = \left(\frac{1.5 + 2.0}{2}\right)=1.75 \]
3. Uara Mahi i Waenganui:
\[ f(0.25) = (0.25)^2 = 0.0625 \]
\[ f(0.75) = (0.75)^2 = 0.5625 \]
\[ f(1.25) = (1.25)^2 = 1.5625 \]
\[ f(1.75) = (1.75)^2 = 3.0625 \]
4. Te Tapeke Riemann Waenga (Mn):
\[ M_n = \sum_{i=0}^{n-1} f(tm_i) \Delta x = (0.0625) \cdot 0.5 + (0.5625) \cdot 0.5 + (1.5625) \cdot 0.5 + (3.0625) \cdot 0.5 \]
\[ M_n = 0.03125 + 0.28125 + 0.78125 + 1.53125 \]
\[ M_n = 2.625 \]
Whakamutunga
Kua matapakihia e tēnei tuhinga te huarahi ki te tatau i ngā tapeke Riemann maui, matau, me waenganui, me ngā tauira taipitopito. Mā te tikanga tapeke Riemann ka whai hua te huarahi ki te whakatau tata i te tauwehenga o tētahi mahi mā te wehewehe i tōna wā ki ngā wā iti, me te tatau i te horahanga katoa o ia wā iti. He mea nui te māramatanga pai ki te tapeke Riemann mō te hunga e ako ana i te tātaitai, e mahi ana rānei me ngā mahi uaua i roto i ngā momo mara pūtaiao.
