Piv txwv ntawm Riemann Sum Discussion Questions
Pendahuluuan
Tus lej Riemann sum yog ib lub tswv yim tseem ceeb hauv kev xam lej siv los txhais qhov integral ntawm ib qho function. Txoj kev no siv kev faib ntu thiab qhov sib ntxiv ntawm cov cheeb tsam ntawm cov duab plaub fab kom kwv yees qhov integral. Tsab xov xwm no yuav tham txog lub tswv yim ntawm tus lej Riemann sum, suav nrog cov piv txwv thiab kev sib tham kom yooj yim nkag siab.
Lub Tswv Yim Tseem Ceeb ntawm Riemannian Sum
Ua ntej peb tham txog cov piv txwv, nws yog ib qho tseem ceeb kom nkag siab txog lub tswv yim yooj yim ntawm Riemannian sums. Riemannian sums tuaj yeem muab faib ua peb hom tseem ceeb:
1. Sab laug Riemann sum
2. Tus lej Riemann sab xis
3. Nruab nrab ntawm Riemann sum
Txoj kev no rhuav tshem lub sijhawm ntawm lub luag haujlwm kom sib xyaw ua ke rau hauv cov ntu me me uas ntev sib npaug. Txhua qhov ntawm cov ntu no tom qab ntawd siv los tsim ib lub duab plaub fab uas qhov siab yog txiav txim siab los ntawm tus nqi ntawm lub luag haujlwm ntawm ib qho chaw tshwj xeeb hauv lub sijhawm ntu (sab laug, sab xis, lossis nruab nrab).
Tus Qauv Dav Dav rau Riemann Sum
Xav tias peb xav koom ua ke lub luag haujlwm \(f(x)\) los ntawm \(a\) mus rau \(b\). Peb faib lub sijhawm \([a, b]\) rau hauv \(n\) sib npaug zos ntawm qhov ntev \(\Delta x = \frac{ba}{n}\). Cov lej Riemann rau peb hom uas tau hais los saum toj no tuaj yeem sau ua raws li hauv qab no:
1. Sab laug Riemann:
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x \]
2. Riemann sab xis:
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x \]
3. Nruab Nrab Riemann:
\[ M_n = \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\right) \Delta x \]
Qhov twg:
- \( \Delta x \) yog qhov dav ntawm txhua qhov subinterval.
- \( x_i \) yog qhov pib ntawm i-th subinterval rau sab laug Riemann sum.
- \( x_i \) yog qhov kawg ntawm i-th subinterval rau qhov Riemann sum sab xis.
- \( \frac{x_i + x_{i+1}}{2} \) yog qhov nruab nrab ntawm i-th subinterval rau qhov nruab nrab Riemann sum.
Cov Lus Nug Piv Txwv thiab Kev Sib Tham
Cia peb tham txog cov teeb meem piv txwv rau txhua hom Riemann Sum kom peb nkag siab tob dua.
Piv txwv 1: Sab laug Riemann Sum
Xam tus lej sab laug Riemann rau \( f(x) = x^2 \) ntawm qhov sib nrug \([0, 2]\) nrog \( n = 4 \).
Kev Sib Tham:
1. Qhov Dav Ntawm Qhov Sib Txawv (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Lub Chaw Faib Sib Nrug (sab laug):
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5 \]
3. Tus nqi ua haujlwm ntawm qhov chaw faib:
\[ f(x_0) = f(0) = 0^2 = 0 \]
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
4. Sab laug Riemann Sum (Ln):
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x = (0)\cdot 0.5 + (0.25)\cdot 0.5 + (1)\cdot 0.5 + (2.25)\cdot 0.5 \]
\[L_n = 0 + 0.125 + 0.5 + 1.125 \]
\[ L_n = 1.75 \]
Piv txwv 2: Tus lej Riemann sab xis
Xam tus lej Riemann raug rau \( f(x) = x^2 \) ntawm qhov sib nrug \([0, 2]\) nrog \( n = 4 \).
Kev Sib Tham:
1. Qhov Dav Ntawm Qhov Sib Txawv (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Lub Chaw Faib Sib Nrug (sab xis):
\[ x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, x_4 = 2.0 \]
3. Tus nqi ua haujlwm ntawm qhov chaw faib:
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
\[ f(x_4) = f(2.0) = (2.0)^2 = 4 \]
4. Sab xis Riemann Sum (Rn):
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x = (0.25) 0.5 + (1) 0.5 + (2.25) 0.5 + (4) 0.5 \]
\[ R_n = 0.125 + 0.5 + 1.125 + 2 \]
\[ R_n = 3.75 \]
Piv txwv 3: Nruab Nrab Riemann Sum
Xam tus nqi nruab nrab ntawm Riemann rau \( f(x) = x^2 \) ntawm qhov sib nrug \([0, 2]\) nrog \( n = 4 \).
Kev Sib Tham:
1. Qhov Dav Ntawm Qhov Sib Txawv (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Nruab Nrab ntawm Subinterval:
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, \text{ thiab } x_{n-1}=2.0 \]
Nruab nrab ntawm subinterval:
\[tm_0 = \left(\frac{0 + 0.5}{2}\right)=0.25 \]
\[tm_1 = \left(\frac{0.5 + 1.0}{2}\right)=0.75 \]
\[tm_2 = \left(\frac{1.0 + 1.5}{2}\right)=1.25 \]
\[tm_3 = \left(\frac{1.5 + 2.0}{2}\right)=1.75 \]
3. Tus nqi ntawm qhov nruab nrab ntawm kev ua haujlwm:
\[ f(0.25) = (0.25)^2 = 0.0625 \]
\[ f(0.75) = (0.75)^2 = 0.5625 \]
\[ f(1.25) = (1.25)^2 = 1.5625 \]
\[ f(1.75) = (1.75)^2 = 3.0625 \]
4. Nruab Nrab Riemann Sum (Mn):
\[ M_n = \sum_{i=0}^{n-1} f(tm_i) \Delta x = (0.0625) 0.5 + (0.5625) 0.5 + (1.5625) 0.5 + (3.0625) 0.5 \]
\[ M_n = 0.03125 + 0.28125 + 0.78125 + 1.53125 \]
\[ M_n = 2.625 \]
Xaus
Tsab xov xwm no tau tham txog yuav ua li cas xam cov lej Riemann sab laug, sab xis, thiab nruab nrab, nrog rau cov piv txwv ntxaws. Txoj kev suav lej Riemann muab txoj hauv kev zoo los kwv yees qhov sib xyaw ntawm ib qho kev ua haujlwm los ntawm kev faib nws lub sijhawm ua me me thiab xam tag nrho thaj chaw ntawm txhua lub sijhawm. Kev nkag siab zoo txog Riemann sum yog qhov tseem ceeb rau cov neeg kawm calculus lossis ua haujlwm nrog cov haujlwm nyuaj hauv ntau qhov kev tshawb fawb.
