Nā mea loaʻa helu: helu ana kāu kiʻina hana a me nā examples

Paha ka manaʻo o ka derivative mea kama'āina i nā mea a pau o mākou, mai ke kula kiʻekiʻe. IeAUPIIe, IAa IO i nä haumäna i pilikia hoʻomaopopo i kēia mea kanalua he mea nui mea. Ua Ua 'Imi hoʻohana' ia i loko o kekahi mau wahi ola kanaka ka o, a me ka nui 'enekinia, i ka nānā' ana kaʻoi loa ma makemakika omaulia loaa mai ma ka derivative. Akā, ma mua lede i ka Ikepili o ka mea he derivative o ka nui me ka mea e huli a ma kahi e hele mai lākou i loko o mākaukau, delve i kona wahi i loko o mōʻaukala.

moʻolelo

I ka manaʻo o ka derivative, i mea ke kumu o ka makemakika Ka Ikepili, ua hamama (a hiki maikaʻi e olelo "invented" no ka mea, mea, e like me ia, 'aʻole e nei ma ka maoli) Isaakom Nyutonom, ka mea mākou a pau i ike mai i ka loaʻa mai o ke kānāwai o ka umekaumaha. Ua ua ia i ka mea mua hoʻohana i kēia manaʻo i loko o ke kālaikūlohea no ka paa ano o ka māmā a me ka hoʻonui o kino. A me ka nuiʻepekema nō hoolea aku Newton no keia nani hana mea hou i hana muaʻole, no ka mea, ma ka mea, ua invented i ke kumu o ka palena pau a me ka integral calculus, ka factual kumu o ke kula a pau o ka makemakika kapaia "makemakika Ka Ikepili". Ina paha i ka manawa o ka Nobel uku, Newton paha ina ua loaa ia i kekahi mau manawa.

'Aʻole me nā manaʻo nui. Ma waho aʻeo ia Newton ma ka hooulu ana o derivative a me ka integral hana na hookiekieia geniuses o makemakika like Leonhard Euler, Lagrange a me Louis Gotfrid Leybnits. He mahalo ia ia mākou i ka kumumanaʻo o ka palena pau calculus i loko o ka palapala ma i mea ole a hiki i kēia lā. Incidentally, keia mea Leibniz loaa ka geometric ano o ka derivative, i ua mea oi ma mua o ka hene o ka tangent a hiki i ka pakuhi o ke kuleana pili i.

He aha mea he derivative o ka helu? Iki hou mea i wahi i loko o ke kula.

He aha mea he derivative?

Hoakaka i kēia manaʻo i loko o kekahi mauʻano likeʻole. Ke simplest wehewehe: nā mea loaʻa - ka mea, o ka uku o ka loli kuleana pili i. Ho i ka pakuhi o kekahi papa y me ka m. Inā ia mea i pololei, ka mea, ua loaa kekahi curves i loko o ka pakuhi, ka manawa o ka mahuahua ana a me ka emi ana. Inā e lawe kekahi infinitesimal wā mawaena o ka haʻalele 'ana, ka mea e lilo he pololei laina Hoʻohana. No laila, i ka lākiō o ka nui o ka infinitesimal Hoʻohana o ka y me a hiki i ka nui o ka m i kahi kūlike o, a e lilo i derivative no ka hana ma ka hāʻawi wahi. Inā i ka hana me ka a pau, ma mua ma kekahi mau 'wahi mākou noonoo, he hana no ka derivative ua loaa, oa i kekahi kaukaʻi ma ka X y me.

Eia hou kekahi, ke kaawale mai ka physical ano o ka derivative like me ka hana ana o ka uku o ka loli, he mea no hoi he geometric pono. Ma ka mea, ua e kūkākūkā.

Ka geometricʻano

Nā mea loaʻa helu ana ia lakou iho i ka kekahi helu ole i ka pono me ka naauao, aole ia i lawe aku i kekahi manaʻo ia mea. Ua huli mai i ka derivative ua hōʻike ole wale i ka ulu ana a me ke emi o ka papa, a me ka hene o ka tangent a hiki i ka pakuhi o ka hana ma ia wahi. 'Aʻole loa akaka ho'ākāka' ana. E mai ia ia ia i loko o au mamuli. Ina mākou i ka pakuhi o ka papa (e lawe panee hālawa). Ua loaʻa he mana loa helu o nā kumu, akā, ma laila nā wahi ma kahi e wale he hookahi aʻoi, i ka i kā mākou a palena iki. Ma kekahi mau wahi, e hiki ke unuhi pololei i ka laina, a makemake e p ÷ haku i ka pakuhi o ka hana ma ia wahi. Keia laina e e kapaʻia he tangent. Ina ua paa ia i hiki i ke kuʻina me ka iho (axis) bipi. A loaa ma waena o ka tangent, a me ka iho (axis) bipi, a me ka huina e e hooponoponoia ia mea, ma ka derivative. More manaʻoi, ka tangent o keia huina, e like me ia.

E ka kamailio iki e pili ana i ka mau hihia a me nā mea loaʻa E na helu mai hoʻokolokolo ai.

Special hihia

he derivative waiwai ma kekahi wahi -, nā mea loaʻa o nā helu e like me mākou i ai mamua ae nei. Eia, no ka laʻana, lawe i ka papa y me = m ^2. Ke derivative o ka m - nui, akā, i loko o mau - i kekahi kuleana pili i like i ka 2 * m. Inā mākou pono, e hoʻomaulia i ka derivative, no ka mea hoike, ma ka wahi m ₀ = 1, e hele mākou y me '(1) = 2 * 1 = 2. It Aloha loa noonoo ole. An hana hoihoi hihia o ka derivative o ka luna 'helu. E hele i loko o ka piha ho'ākāka o ka mea he luna 'helu, ua makemake ole. Lawa ia i ka olelo i keia helu a iloko ona i kakauia na ai-kapaia ka mana pa alima - i ka helu o kona hale ma lākou, -1. Ke i ka ho omaulia ana o keia derivative mea hiki wale nō ma lalo o ke ana ma lalo nei:

1) Aia pono e mua mea ka helehelena nā mea loaʻa o ka maoli a me ka mana ae māhele o y me a me X.

2) i nā kūlana o ka Cauchy-Riemann pili i like ka helehelena i ho'ākāka 'ia ma ka paukū mua.

Kekahi hana hoihoi hihia, eia nae,ʻaʻole like huikau like me ka mua kekahi, o ka derivative o ka io ka helu. I ka mea, i kekahi io helu hiki e na poe like me ka 'oluʻolu, hoonuiia e -1. Ua pono, ka derivative, a me ka ikaika kuleana pili i like i ke a eia i hoonuiia e ka derivative o ke kuleana pili i.

It e e hoihoi 'e aʻo e pili ana i ke kūlana o nā mea loaʻa i loko o kā lākou kela la i keia ola ana, a me keia mea kēia manawa, a kamaʻilio aku ai.

palapala noi

Paha kēlā me kēia o ko mākou ma ka liʻiliʻi koke i loko o ka wa e ola ana hopu iaʻu iho me ka manao ana makemakika meaʻaʻole paha ia e pono no ia ia. A ua ka huikau mea like me ka derivative pahaʻaʻohe hana. I mea, ka makemakika - kahua 'epekema, a pau kona hua? Acaeoey?ʻano nui ke kālaikūlohea, kemika, kilohōkū, a hiki i ka hoʻokele waiwai. Derivative ia i ka hoʻomaka o ka makemakika Ka Ikepili, a haawi mai i ka manawa e hoʻoholo i ka hopena, mai ka pakuhi o ka oihana, a mākou i aʻo ai i ke ano o na kanawai o ke 'ano a me ka huli ia i kā lākou mau pono, no ka mea.

ka hopena

O ka holo ana, i kela kanaka keia kanaka hiki e pono i ka derivative ma ka maoli ola. Akā, makemakika e hoʻomōhala ana i ke kūpili mea, e oiaio no, e pono ai. 'Aʻole no ka mea no ka mea, ua kapaʻia makemakika ke aliʻi wahine o sciences: ia komo ana o ke kumu o ka naʻauao o nā mahinaʻai o ka ike.