Piv txwv ntawm Cov Lus Nug Sib Tham Txog Kev Siv Cov Derivatives hauv Ntau Qhov Chaw Tshawb Fawb
Pendahuluuan
Tus derivative yog ib lub tswv yim tseem ceeb hauv calculus nrog ntau daim ntawv thov hauv ntau qhov chaw ntawm kev tshawb fawb thiab engineering. Nws piav qhia txog tus nqi ntawm kev hloov pauv ntawm ib qho function thiab tuaj yeem siv los txheeb xyuas qhov siab tshaj plaws thiab qhov tsawg tshaj plaws, daws cov teeb meem optimization, thiab tshuaj xyuas kev loj hlob thiab kev poob qis hauv ib lub system. Hauv tsab xov xwm no, peb yuav tham txog ntau qhov teeb meem piv txwv thiab tham txog kev siv cov derivatives hauv ntau qhov chaw ntawm kev tshawb fawb, xws li physics, economics, biology, thiab engineering.
1. Physics: Kev nrawm ua ib qho kev sib txuas ntawm qhov ceev
Hauv physics, qhov kev nrawm yog qhov derivative ntawm qhov ceev piv rau lub sijhawm. Ib qho piv txwv teeb meem hauv cov ntsiab lus no yog:
Piv txwv teeb meem:
Ib yam khoom txav mus raws ib txoj kab ncaj nraim nrog qhov chaw ua haujlwm \(s(t) = 4t^3 – 3t^2 + 2t – 1\) meters, qhov twg \(t\) yog nyob rau hauv vib nas this. Txheeb xyuas qhov ceev thiab kev ua kom nrawm ntawm yam khoom ntawm \(t = 2\) vib nas this.
Kev Sib Tham:
Qhov ceev yog thawj qhov derivative ntawm txoj hauj lwm nrog rau lub sijhawm:
\[ v(t) = s'(t) = \frac{d}{dt}(4t^3 – 3t^2 + 2t – 1) \]
\[ v(t) = 12t^2 – 6t + 2 \]
Kev nrawm yog thawj qhov derivative ntawm qhov ceev piv rau lub sijhawm:
\[ a(t) = v'(t) = \frac{d}{dt}(12t^2 – 6t + 2) \]
\[ ib(t) = 24t – 6 \]
Yog li, qhov ceev ntawm ∑(t = 2) vib nas this yog:
\[ v(2) = 12(2)^2 – 6(2) + 2 = 48 – 12 + 2 = 38 \, \text{m/s} \]
Qhov kev nrawm ntawm ∙ (t = 2) vib nas this yog:
\[ a(2) = 24(2) – 6 = 48 – 6 = 42 \, \text{m/s}^2 \]
2. Kev Lag Luam: Kev Txhim Kho Cov Nyiaj Tau Los
Hauv kev lag luam, cov derivatives feem ntau siv los txiav txim siab qhov siab tshaj plaws lossis qhov tsawg kawg nkaus ntawm qhov txiaj ntsig lossis tus nqi ua haujlwm. Ib qho piv txwv teeb meem hauv cov ntsiab lus no yog:
Piv txwv teeb meem:
Ib lub tuam txhab tsim khoom nrog lub luag haujlwm khwv tau nyiaj \(P(x) = -2x^2 + 12x – 20\), qhov twg \(x\) yog tus naj npawb ntawm cov khoom tsim thiab muag. Yuav tsum tsim pes tsawg cov khoom kom tau txiaj ntsig ntau tshaj plaws, thiab qhov nyiaj tau ntau tshaj plaws yog dab tsi?
Kev Sib Tham:
Yuav kom tau txiaj ntsig ntau tshaj plaws, peb yuav tsum nrhiav thawj qhov derivative ntawm \(P(x)\) thiab nrhiav nws cov ntsiab lus tseem ceeb.
\[ P'(x) = \frac{d}{dt}(-2x^2 + 12x – 20) \]
\[ P'(x) = -4x + 12 \]
Nrhiav qhov chaw tseem ceeb los ntawm kev daws qhov \(P'(x) = 0\):
\[ -4x + 12 = 0 \]
\[ x = 3 \]
Peb yuav tsum xyuas kom meej tias \(x = 3\) yog qhov siab tshaj plaws los ntawm kev siv qhov thib ob derivative.
\[ P”(x) = \frac{d}{dt}(-4x + 12) \]
\[ P”(x) = -4 \]
Vim tias \(P”(3) = -4 < 0\), qhov no qhia tau tias \(x = 3\) yog qhov siab tshaj plaws. Qhov txiaj ntsig siab tshaj plaws yog: \[ P(3) = -2(3)^2 + 12(3) - 20 \] \[ P(3) = -18 + 36 - 20 \] \[ P(3) = -2 \] Yog li, tus lej ntawm cov chav uas yuav tsum tau tsim kom tau txiaj ntsig siab tshaj plaws yog 3 chav, thiab qhov txiaj ntsig siab tshaj plaws yog -2. 3. Biology: Kev Loj Hlob ntawm Cov Neeg Hauv biology, cov khoom siv los tshuaj xyuas cov nqi loj hlob ntawm cov neeg. Ib qho piv txwv teeb meem hauv cov ntsiab lus no yog: Piv txwv Teeb Meem: Xav tias qhov loj ntawm cov neeg ntawm ib hom tsiaj yog muab los ntawm kev ua haujlwm \(P(t) = 100e^{0.05t}\), qhov twg \(t\) yog lub sijhawm hauv xyoo. Nrhiav qhov txiaj ntsig loj hlob ntawm cov neeg ntawm \(t = 10\). Kev daws teeb meem: Qhov txiaj ntsig loj hlob ntawm cov neeg yog qhov khoom siv ntawm \(P(t)\) nrog rau lub sijhawm: \[ P'(t) = \frac{d}{dt}(100e^{0.05t}) \] \[ P'(t) = 100 \cdot 0.05e^{0.05t} \] \[ P'(t) = 5e^{0.05t} \] Tus nqi loj hlob ntawm cov pej xeem ntawm \(t = 10\) yog: \[ P'(10) = 5e^{0.05(10)} \] \[ P'(10) = 5e^{0.5} \] Los ntawm kev xam tus nqi exponential ntawm \(e^{0.5}\) (kwv yees li 1.64872): \[ P'(10) \approx 5 \cdot 1.64872 \] \[ P'(10) \approx 8.2436 \] Yog li, tus nqi loj hlob ntawm cov pej xeem ntawm \(t = 10\) yog kwv yees li 8.24 tus tib neeg ib xyoos. 4. Engineering: Kev Tsim Qauv Zoo Tshaj Plaws ntawm Cov Hluav Taws Xob Circuits Hauv engineering, tshwj xeeb tshaj yog hluav taws xob kev tsim kho vaj tse, cov khoom siv derivatives raug siv los ua kom zoo dua qhov kev tsim qauv circuit. Ib qho piv txwv teeb meem hauv cov ntsiab lus no yog: Piv txwv Teeb Meem: Muab ib qho kev siv hluav taws xob \(P(R) = V^2 / R + I^2 R\), qhov twg \(V\) yog qhov hluav taws xob tas li, \(I\) yog qhov tam sim no tas li, thiab \(R\) yog qhov tsis kam. Txheeb xyuas tus nqi ntawm \(R\) uas txo qis kev siv hluav taws xob. Kev daws teeb meem: Tus thawj derivative ntawm \(P(R)\) nrog rau \(R\) yog: \[ P'(R) = \frac{d}{dR}\left(\frac{V^2}{R} + I^2 R\right) \] \[ P'(R) = -\frac{V^2}{R^2} + I^2 \] Txhawm rau nrhiav tus nqi ntawm \(R\) uas txo qis kev siv hluav taws xob, peb pom \(P'(R) = 0\): \[ -\frac{V^2}{R^2} + I^2 = 0 \] \[ \frac{V^2}{R^2} = Kuv^2 \] \[ R^2 = \frac{V^2}{I^2} \] \[ R = \frac{V}{I} \] Yog li, tus nqi tiv thaiv \(R\) uas txo qis kev siv hluav taws xob yog \(R = \frac{V}{I}\). Xaus Lus Los ntawm cov piv txwv saum toj no, peb tau pom tias lub tswv yim ntawm derivatives siv li cas hauv ntau qhov chaw ntawm kev tshawb fawb xws li physics, kev lag luam, biology, thiab engineering. Kev nkag siab tob txog derivatives thiab lawv cov ntawv thov tso cai rau peb daws ntau yam teeb meem nyuaj thiab ua kom zoo dua cov txheej txheem hauv lub neej tiag tiag. Derivatives yog cov cuab yeej analytical muaj zog heev thiab muaj txiaj ntsig zoo hauv kev nkag siab txog dynamics thiab kev hloov pauv hauv ntau qhov xwm txheej.
