Mehlala ea Lipotso tse Buisanang ka Tšebeliso ea Li-Derivatives Mafapheng a Fapaneng a Saense
Pendahuluan
Derivative ke mohopolo oa motheo ka har'a calculus o nang le lits'ebetso tse ngata mafapheng a fapaneng a saense le boenjiniere. E hlalosa sekhahla sa phetoho ea mosebetsi 'me e ka sebelisoa ho khetholla maxima le minima, ho rarolla mathata a ntlafatso, le ho sekaseka kholo le ho fokotseha ha sistimi. Sehloohong sena, re tla buisana ka mathata a 'maloa a mehlala le ho buisana ka ts'ebeliso ea derivatives mafapheng a fapaneng a saense, joalo ka fisiks, moruo, baeloji le boenjiniere.
1. Fisiks: Ho Potlaka e le Lebelo le Tsoang ho Lebelo
Fisiks, ho potlakisa ke derivative ea lebelo mabapi le nako. Mohlala oa bothata moelelong ona ke:
Mohlala oa mathata:
Ntho e tsamaya moleng o otlolohileng ka mosebetsi wa boemo \(s(t) = 4t^3 – 3t^2 + 2t – 1\) metres, moo \(t\) e leng ka metsotsoana. Fumana lebelo le ho potlaka ha ntho ka metsotsoana \(t = 2\).
Puisano:
Lebelo ke ntlha ea pele e tsoang boemong mabapi le nako:
\[ v(t) = s'(t) = \frac{d}{dt}(4t^3 – 3t^2 + 2t – 1) \]
\[ v(t) = 12t^2 – 6t + 2 \]
Ho potlaka ke derivative ea pele ea lebelo mabapi le nako:
\[ a(t) = v'(t) = \frac{d}{dt}(12t^2 – 6t + 2) \]
\[ a(t) = 24t – 6 \]
Kahoo, lebelo la metsotsoana ea \(t = 2\) ke:
\[ v(2) = 12(2)^2 – 6(2) + 2 = 48 – 12 + 2 = 38 \, \text{m/s} \]
Ho potlaka ha metsotsoana ya \(t = 2\) ke:
\[ a(2) = 24(2) – 6 = 48 – 6 = 42 \, \text{m/s}^2 \]
2. Moruo: Ntlafatso ea Phaello
Moruong, hangata di-derivatives di sebediswa ho fumana dintlha tse hodimo kapa tse tlase tsa phaello kapa mosebetsi wa ditjeo. Mohlala wa bothata moelelong ona ke:
Mohlala oa mathata:
Khamphani e hlahisa thepa e nang le mosebetsi oa phaello \(P(x) = -2x^2 + 12x – 20\), moo \(x\) e leng palo ea liyuniti tse hlahisoang le tse rekisoang. Ke liyuniti tse kae tse lokelang ho hlahisoa ho eketsa phaello, 'me phaello e phahameng ka ho fetisisa ke efe?
Puisano:
Ho eketsa phaello, re hloka ho fumana derivative ea pele ea \(P(x)\) le ho fumana lintlha tsa eona tsa bohlokoa.
\[ P'(x) = \frac{d}{dt}(-2x^2 + 12x – 20) \]
\[ P'(x) = -4x + 12 \]
Fumana ntlha ea bohlokoa ka ho rarolla \(P'(x) = 0\):
\[ -4x + 12 = 0 \]
\[x = 3 \]
Re hloka ho netefatsa hore \(x = 3\) ke ntlha e phahameng ka ho fetisisa ka ho sebedisa derivative ya bobedi.
\[ P”(x) = \frac{d}{dt}(-4x + 12) \]
\[ P”(x) = -4 \]
Kaha \(P”(3) = -4 < 0\), sena se bontsha hore \(x = 3\) ke ntlha e phahameng ka ho fetisisa. Phaello e phahameng ka ho fetisisa ke: \[ P(3) = -2(3)^2 + 12(3) - 20 \] \[ P(3) = -18 + 36 - 20 \] \[ P(3) = -2 \] Kahoo, palo ea diyuniti tse lokelang ho hlahiswa ho eketsa phaello ke diyuniti tse 3, mme phaello e phahameng ka ho fetisisa ke -2. 3. Biology: Sekgahla sa Kgolo ya Baahi Thutong ya baeloji, di-derivatives di sebediswa ho sekaseka sekgahla sa kgolo ya baahi. Bothata ba mohlala moelelong ona ke: Mohlala Bothata: A re re boholo ba baahi ba mofuta bo fanwa ke mosebetsi \(P(t) = 100e^{0.05t}\), moo \(t\) e leng nako ka dilemo. Fumana sekgahla sa kgolo ya baahi ho \(t = 10\). Tharollo: Sekgahla sa kgolo ya baahi ke derivative ya \(P(t)\) mabapi le nako: \[ P'(t) = \frac{d}{dt}(100e^{0.05t}) \] \[ P'(t) = 100 \cdot 0.05e^{0.05t} \] \[ P'(t) = 5e^{0.05t} \] Sekgahla sa kgolo ya baahi ho \(t = 10\) ke: \[ P'(10) = 5e^{0.05(10)} \] \[ P'(10) = 5e^{0.5} \] Ka ho bala boleng ba exponential ba \(e^{0.5}\) (hoo e ka bang 1.64872): \[ P'(10) \hoo e ka bang 5 \cdot 1.64872 \] \[ P'(10) \hoo e ka bang 8.2436 \]
Kahoo, sekgahla sa kgolo ya baahi ho \(t = 10\) ke batho ba ka bang 8.24 ka selemo. 4. Boenjiniere: Moralo o Motle wa Dipotoloho tsa Motlakase Boenjiniereng, haholoholo boenjiniere ba motlakase, di-derivatives di sebediswa ho ntlafatsa moralo wa potoloho. Mohlala wa bothata moelelong ona ke: Mohlala wa Bothata: Ha ho fanwa ka mosebetsi wa tshebediso ya matla \(P(R) = V^2 / R + I^2 R\), moo \(V\) e leng motlakase o sa fetoheng, \(I\) ke motlakase o sa fetoheng, mme \(R\) ke kganyetso. Fumana boleng ba \(R\) bo fokotsang tshebediso ya matla. Puisano: Sehlahiswa sa pele sa \(P(R)\) mabapi le \(R\) ke: \[ P'(R) = \frac{d}{dR}\left(\frac{V^2}{R} + I^2 R\right) \] \[ P'(R) = -\frac{V^2}{R^2} + I^2 \] Ho fumana boleng ba \(R\) bo fokotsang tshebediso ya matla, re fumana \(P'(R) = 0\): \[ -\frac{V^2}{R^2} + I^2 = 0 \] \[ \frac{V^2}{R^2} = I^2 \] \[ R^2 = \frac{V^2}{I^2} \] \[ R = \frac{V}{I} \] Kahoo, boleng ba khanyetso \(R\) bo fokotsang tshebediso ya matla ke \(R = \frac{V}{I}\). Qetello Ho tsoa mehlaleng e kaholimo, re bone kamoo mohopolo oa li-derivative o sebelisoang kateng mafapheng a fapaneng a mahlale a kang fisiks, moruo, baeloji le boenjiniere. Kutloisiso e batsi ea li-derivative le ts'ebeliso ea tsona e re lumella ho rarolla mathata a fapaneng a rarahaneng le ho ntlafatsa litsamaiso bophelong ba 'nete. Li-derivative ke sesebelisoa se matla haholo sa tlhahlobo 'me li na le thuso ho utloisiseng matla le phetoho maemong a fapaneng.