Nā Hana Hoʻonui a me ka Māhele
Hoʻokani pinepine nā hana makemakika i kahi kuleana ma nā ʻano ʻepekema like ʻole, me ka hoʻokele waiwai, ka ʻenekinia, ka physics, a me nā mea ʻē aʻe. ʻElua mau hana kumu i hoʻohana pinepine ʻia i ka hoʻoponopono hana ʻo ia ka hoʻonui a me ka mahele. Loaʻa i kēia mau hana ʻelua nā manaʻo a me nā noi kūikawā e pono e hoʻomaopopo. E kūkākūkā hohonu kēia ʻatikala i nā hana hoʻonui a me ka mahele: ko lākou wehewehe ʻana, nā waiwai, nā lula, a me nā laʻana.
Hoʻonui Hana
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ʻO ka hoʻonui hana he hana binary e lawe ana i ʻelua mau hana a hana i kahi hana hou. Manaʻo mākou he ʻelua mau hana kā mākou \( f \) a me \( g \), a laila ua kākau ʻia ka hoʻonui ʻana o kēia mau hana ʻelua ʻo \( f(x) \cdot g(x) \) a i ʻole \( (fg)(x) \).
Nā Waiwai o ka Hoʻonui Hana
1. Hoʻololi: He hoʻololi ka hoʻonui ʻana o nā hana, ʻo ia hoʻi \( f(x) \cdot g(x) = g(x) \cdot f(x) \).
2. Hoʻopili: ʻO ka hoʻonui ʻana o nā hana he mea pili nō hoʻi ia, ʻo ia hoʻi \( (f(x) \cdot g(x)) \cdot h(x) = f(x) \cdot (g(x) \cdot h(x)) \).
3. Hoʻokaʻawale: Hoʻokaʻawale ʻia ka hoʻonui ʻana o nā hana ma luna o ka hoʻohui ʻana o nā hana, ʻo ia hoʻi \( f(x) \cdot (g(x) + h(x)) = f(x) \cdot g(x) + f(x) \cdot h(x) \).
Eia kekahi laʻana
Manaʻo ʻia \( f(x) = 2x + 3 \) a me \( g(x) = x^2 \), a laila ʻo ka huahana o nā hana ʻelua:
\[ (fg)(x) = f(x) \cdot g(x) = (2x + 3) \cdot x^2 = 2x^3 + 3x^2 \].
Hōʻike ia pehea e hiki ai ke hoʻohui ʻia ʻelua mau hana ma o ka hoʻonui ʻana e hana i kahi hana hou me nā ʻano ʻokoʻa mai ka hana mua.
Māhele o nā Hana
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ʻO ka mahele hana, ma ke ʻano naʻauao, ʻo ia ka hana o ka lawe ʻana i ʻelua mau hana a hana i kahi hana hou ʻo ia ka quotient o nā hana ʻelua. Manaʻo mākou he mau hana kā mākou \( f \) a me \( g \), a laila ua kākau ʻia ka mahele ʻana o \( f \) e \( g \) ʻo \( \frac{f(x)}{g(x)} \) a i ʻole \( \left(\frac{f}{g}\right)(x) \), inā ʻo \( g(x) \neq 0 \).
Nā Waiwai o ka Māhele Hana
1. ʻAʻole Commutative: ʻAʻole commutative ka mahele ʻana o nā hana, ʻo ia hoʻi \( \frac{f(x)}{g(x)} \neq \frac{g(x)}{f(x)} \).
2. ʻAʻole pili: ʻAʻole pili pū ka mahele ʻana o nā hana, ʻo ia hoʻi \( \frac{f(x)}{g(x)/h(x)} \neq \left(\frac{f(x)}{g(x)}\right)/h(x) \).
3. Hoʻokaʻawale: He hoʻokaʻawale ka hana mahele i ka mahele o nā mea, ʻo ia hoʻi \( f(x)/g(x) = f(x) \cdot \frac{1}{g(x)} \).
Eia kekahi laʻana
Manaʻo ʻia \( f(x) = x^2 + 2x \) a me \( g(x) = x \), a laila ʻo ka mahele ʻana o nā hana ʻelua penei:
\[ \left(\frac{f}{g}\right)(x) = \frac{x^2 + 2x}{x} = x + 2 \].
Hōʻike ia pehea e hiki ai ke hoʻohui ʻia ʻelua mau hana ma o ka puʻunaue ʻana e hana i kahi hana hou me nā ʻano ʻokoʻa mai ka hana mua.
Noi Hana Hoʻonui a me ka Māhele
1. Hoʻokele waiwai
I loko o ka hoʻokele waiwai, hoʻohana pinepine ʻia ka hoʻonui ʻana a me ka mahele ʻana o nā hana i ka loiloi kumukūʻai a me ka loaʻa kālā. No ka laʻana, inā ʻo \( R(x) \) ka hana loaʻa kālā a ʻo \( C(x) \) ka hana kumukūʻai, a laila hiki ke helu ʻia ka loaʻa kālā e like me \( P(x) = R(x) – C(x) \). Inā he hana ka loaʻa kālā o ka helu o nā ʻāpana i kūʻai ʻia me ke kumukūʻai no kēlā me kēia ʻāpana, a laila hiki ke helu ʻia ka hana \( R(x) \) ma ka hoʻonui ʻana i ka hana o ka helu o nā ʻāpana a me ke kumukūʻai no kēlā me kēia ʻāpana.
2. ʻAno hana
Hoʻohana pinepine nā ʻenekinia i nā hana hoʻonui a me ka mahele i ka loiloi ʻōnaehana. No ka laʻana, i ka loiloi kaapuni, hiki ke helu ʻia ka impedance i hui pū ʻia o nā ʻāpana ʻelua i hoʻopili ʻia ma ke ʻano moʻo ma ka hoʻonui ʻana i nā hana impedance o kēlā me kēia ʻāpana. Pēlā nō, hoʻohana ʻia nā hana mahele i ka mana ʻōnaehana e hoʻoholo i ka pane a ka ʻōnaehana i kahi mea hoʻokomo kikoʻī.
3. ʻIke kino
I loko o ke kinoea, hoʻohana nā manaʻo he nui i ka hoʻonui a me ka mahele ʻana o nā hana. No ka laʻana, hiki ke helu ʻia ka hana i hana ʻia e ka ikaika ma luna o kahi mea e neʻe ana ma ke ʻano he hoʻohui o ka hana ikaika ma luna o ka hana mamao. I kekahi ʻaoʻao, hiki ke kālailai ʻia ka manaʻo o ka wikiwiki awelika i ka neʻe ʻana ma ka puʻunaue ʻana i ka hana mamao holoʻokoʻa e ka hana manawa holoʻokoʻa.
Nā Rula Hoʻohui no ka Hoʻonui a me ka Māhele ʻana o nā Hana
I ka calculus, he mea nui loa nā lula derivative o ka hoʻonui a me ka puʻunaue ʻana o nā hana.
Rula Huahana
Inā hiki ke hoʻokaʻawale ʻia ʻo \( f(x) \) a me \( g(x) \), a laila ʻo ka derivative o \( f(x) \cdot g(x) \) penei:
\[ (fg)'(x) = f'(x)g(x) + f(x)g'(x) \].
Rula Quotient
Inā hiki ke hoʻokaʻawale ʻia ʻo \( f(x) \) a me \( g(x) \), a laila ʻo ka derivative o \( \frac{f(x)}{g(x)} \) penei:
\[ \left(\frac{f}{g}\right)'(x) = \frac{f'(x)g(x) – f(x)g'(x)}{g(x)^2} \],
me ke kūlana \( g(x) \neq 0 \).
Eia kekahi laʻana
Manaʻo ʻia \( f(x) = x^2 \) a me \( g(x) = x + 1 \), helu mākou i ka derivative o \( f(x) \) i hoʻonui ʻia e \( g(x) \).
1. \( f'(x) = 2x \)
2. \( g'(x) = 1 \)
3. Wahi a nā lula o ka hoʻonui ʻana:
\[ (fg)'(x) = 2x(x + 1) + x^2(1) = 2x^2 + 2x + x^2 = 3x^2 + 2x \].
I kēia manawa, e helu i ka derivative o \( \frac{f(x)}{g(x)} \).
1. Wahi a nā lula mahele:
\[ \left(\frac{f}{g}\right)'(x) = \frac{(2x)(x + 1) – (x^2)(1)}{(x + 1)^2} = \frac{2x^2 + 2x – x^2}{(x + 1)^2} = \frac{x^2 + 2x}{(x + 1)^2} \].
Ka hopena
ʻO ka hoʻonui ʻana a me ka mahele ʻana o nā hana he mau manaʻo koʻikoʻi ia i ka algebra a me ka calculus, me nā noi he nui ma nā ʻano aʻo like ʻole. ʻO ka hoʻomaopopo ʻana i nā waiwai, nā lula o ka hoʻokaʻawale ʻana, a me nā noi kūpono o kēia mau hana he mea nui ia no ka loiloi pololei a maikaʻi. Inā he makemakika ʻoe, ʻenekinia, a he loea waiwai paha, he mākaukau waiwai nui ka hiki ke hana me ka hoʻonui ʻana a me ka mahele ʻana o nā hana.
