Nā Laʻana o nā Nīnau e Kūkākūkā ana i ka Hoʻohana ʻana o nā Derivatives ma nā ʻAno ʻEpekema like ʻole
Pendahuluan
He manaʻo nui ka derivative i ka calculus me nā noi he nui ma nā ʻano ʻepekema like ʻole a me ka ʻenekinia. Hōʻike ia i ka wikiwiki o ka loli o kahi hana a hiki ke hoʻohana ʻia e ʻike i ka maxima a me ka minima, hoʻoponopono i nā pilikia optimization, a kālailai i ka ulu ʻana a me ka emi ʻana i loko o kahi ʻōnaehana. Ma kēia ʻatikala, e kūkākūkā mākou i kekahi mau pilikia hoʻohālike a kūkākūkā i nā noi o nā derivatives ma nā ʻano ʻepekema like ʻole, e like me ke kino, ka hoʻokele waiwai, ka biology, a me ka ʻenekinia.
1. ʻEpekema: Ka Hoʻolalelale ʻana ma ke ʻano he hopena o ka wikiwiki
I loko o ke kinoea, ʻo ka wikiwiki ka hopena o ka wikiwiki e pili ana i ka manawa. ʻO kahi pilikia hoʻohālike ma kēia ʻano penei:
Laʻana pilikia:
Neʻe kekahi mea ma kahi laina pololei me ka hana kūlana \(s(t) = 4t^3 – 3t^2 + 2t – 1\) mika, kahi ʻo \(t\) i kekona. E hoʻoholo i ka wikiwiki a me ka wikiwiki o ka mea ma \(t = 2\) kekona.
Kūkākūkā:
ʻO ka wikiwiki ka hopena mua o ke kūlana e pili ana i ka manawa:
\[ v(t) = s'(t) = \frac{d}{dt}(4t^3 – 3t^2 + 2t – 1) \]
\[ v(t) = 12t^2 – 6t + 2 \]
ʻO ka hoʻolalelale ka hopena mua o ka wikiwiki e pili ana i ka manawa:
\[ a(t) = v'(t) = \frac{d}{dt}(12t^2 – 6t + 2) \]
\[ a(t) = 24t – 6 \]
No laila, ʻo ka wikiwiki ma \(t = 2\) kekona penei:
\[ v(2) = 12(2)^2 – 6(2) + 2 = 48 – 12 + 2 = 38 \, \kikokikona{m/s} \]
ʻO ka hoʻolalelale ʻana ma \(t = 2\) kekona penei:
\[ a(2) = 24(2) – 6 = 48 – 6 = 42 \, \kikokikona{m/s}^2 \]
2. Hoʻokele waiwai: Hoʻonui i ka loaʻa kālā
I ka hoʻokele waiwai, hoʻohana pinepine ʻia nā derivatives e hoʻoholo i nā helu kiʻekiʻe a i ʻole ka palena iki o kahi hana waiwai a i ʻole ke kumukūʻai. ʻO kahi pilikia laʻana ma kēia pōʻaiapili:
Laʻana pilikia:
Hana kahi ʻoihana i nā waiwai me kahi hana waiwai \(P(x) = -2x^2 + 12x – 20\), kahi \(x\) ka helu o nā ʻāpana i hana ʻia a kūʻai ʻia aku. Ehia mau ʻāpana e pono e hana ʻia e hoʻonui i ka loaʻa kālā, a he aha ka loaʻa kālā nui loa?
Kūkākūkā:
No ka hoʻonui ʻana i ka loaʻa kālā, pono mākou e ʻimi i ka derivative mua o \(P(x)\) a ʻimi i kona mau kiko koʻikoʻi.
P'(x) = \frac{d}{dt}(-2x^2 + 12x – 20)
\[ P'(x) = -4x + 12 \]
E huli i ke kiko koʻikoʻi ma ka hoʻoponopono ʻana i ka \(P'(x) = 0\):
\[ -4x + 12 = 0 \]
\[ x = 3 \]
Pono mākou e hōʻoia i ka \(x = 3\) he kiko kiʻekiʻe loa ma ka hoʻohana ʻana i ka lua o ka derivative.
\[ P”(x) = \frac{d}{dt}(-4x + 12) \]
\[ P”(x) = -4 \]
ʻOiai ʻo \(P”(3) = -4 < 0\), hōʻike kēia he kiko kiʻekiʻe loa ʻo \(x = 3\). ʻO ka loaʻa kālā kiʻekiʻe loa: \[ P(3) = -2(3)^2 + 12(3) - 20 \] \[ P(3) = -18 + 36 - 20 \] \[ P(3) = -2 \] No laila, ʻo ka helu o nā ʻāpana e pono e hana ʻia e hoʻonui i ka loaʻa kālā he 3 mau ʻāpana, a ʻo ka loaʻa kālā kiʻekiʻe loa he -2. 3. Biology: Ka wikiwiki o ka ulu ʻana o ka heluna kanaka I ka biology, hoʻohana ʻia nā derivatives e kālailai i nā wikiwiki o ka ulu ʻana o ka heluna kanaka. ʻO kahi pilikia hoʻohālike ma kēia ʻano: Ka pilikia hoʻohālike: Manaʻo ʻia ua hāʻawi ʻia ka nui o ka heluna kanaka o kahi ʻano e ka hana \(P(t) = 100e^{0.05t}\), kahi ʻo \(t\) ka manawa ma nā makahiki. E ʻimi i ka wikiwiki o ka ulu ʻana o ka heluna kanaka ma \(t = 10\). Hoʻonā: ʻO ka wikiwiki o ka ulu ʻana o ka heluna kanaka ka derivative o \(P(t)\) e pili ana i ka manawa: \[ P'(t) = \frac{d}{dt}(100e^{0.05t}) \] \[ P'(t) = 100 \cdot 0.05e^{0.05t} \] \[ P'(t) = 5e^{0.05t} \] ʻO ka nui o ka ulu ʻana o ka heluna kanaka ma \(t = 10\) penei: \[ P'(10) = 5e^{0.05(10)} \] \[ P'(10) = 5e^{0.5} \] Ma ka helu ʻana i ka waiwai exponential o \(e^{0.5}\) (ma kahi o 1.64872): \[ P'(10) \approx 5 \cdot 1.64872 \] \[ P'(10) \approx 8.2436 \] No laila, ʻo ka nui o ka ulu ʻana o ka heluna kanaka ma \(t = 10\) ma kahi o 8.24 mau kānaka i kēlā me kēia makahiki. 4. ʻEnekinia: Hoʻolālā Kūpono o nā Kaapuni Uila I ke ʻenekinia, ʻoiai ka ʻenekinia uila, hoʻohana ʻia nā derivatives e hoʻomaikaʻi i ka hoʻolālā kaapuni. ʻO kahi pilikia laʻana ma kēia ʻano: Laʻana Pilikia: Hāʻawi ʻia kahi hana hoʻohana mana \(P(R) = V^2 / R + I^2 R\), kahi \(V\) he voltage mau, \(I\) he au mau, a ʻo \(R\) he kū'ē. E hoʻoholo i ka waiwai o \(R\) e hōʻemi ana i ka hoʻohana mana. Hoʻonā: ʻO ka derivative mua o \(P(R)\) e pili ana iā \(R\) ʻo ia: \[ P'(R) = \frac{d}{dR}\left(\frac{V^2}{R} + I^2 R\right) \] \[ P'(R) = -\frac{V^2}{R^2} + I^2 \] No ka loaʻa ʻana o ka waiwai o \(R\) e hōʻemi ana i ka hoʻohana mana, loaʻa iā mākou \(P'(R) = 0\): \[ -\frac{V^2}{R^2} + I^2 = 0 \] \[ \frac{V^2}{R^2} = I^2 \] \[ R^2 = \frac{V^2}{I^2} \] \[ R = \frac{V}{I} \] No laila, ʻo ka waiwai kū'ē \(R\) e hōʻemi ana i ka hoʻohana ʻana i ka mana ʻo \(R = \frac{V}{I}\). Hopena Mai nā laʻana ma luna, ua ʻike mākou pehea e hoʻopili ʻia ai ke kumumanaʻo o nā derivatives i nā ʻano ʻepekema like ʻole e like me ka physics, economics, biology, a me ka ʻenekinia. ʻO ka hoʻomaopopo hohonu ʻana i nā derivatives a me kā lākou noi e hiki ai iā mākou ke hoʻoponopono i nā pilikia paʻakikī like ʻole a hoʻomaikaʻi i nā ʻōnaehana i ke ola maoli. He mau mea hana loiloi ikaika loa nā derivatives a he mea pono ia i ka hoʻomaopopo ʻana i nā dynamics a me nā loli i nā ʻano like ʻole.
