Pārōnaki - he aha te mea tenei? Me pēhea te ki te kitea e te pārōnaki o te mahi?

Me ki pärönaki ratou mahi Pārōnaki - reira etahi o nga ariā taketake o te tuanaki pārōnaki, te wāhanga matua o te tātari pāngarau. Ka rite ki te tühonohono, e rua o ratou tenetere e rave rahi whānui e whakamahia ana i roto i te whakaoti tata katoa raruraru i whakatika i te akoranga o te mahi pūtaiao, me te hangarau.

Ko te putanga o te ariā o te pārōnaki

Hoki te wā tuatahi i hanga reira mārama e te pārōnaki taua, tetahi o nga kaiwhakatū (me ki Isaakom Nyutonom) pārōnaki tuanaki rongonui mathematician German Gotfrid Vilgelm Leybnits. I mua e Mathematicians rau tau 17. whakamahia whakaaro rawa mārama, me te nuinga ki o etahi ore "rorona" o tetahi mahi mohiotia, e tohu ana i te uara tamau iti rawa engari e kore e rite ki te kore, haafaufaa i raro e kore e taea e te mahi e noa. No reira ko reira kotahi anake taahiraa ki te whakataki o fakakaukau o āpiti ore o tohenga mahi me o ratou āpiti tēnā o nga mahi e taea te whakahuatia i roto i ngā o pärönaki o te muri. Na tangohia tenei taahiraa i tata te wā kotahi i te rua pūtaiao nui i runga.

I runga i te hiahia ki te whakatutuki i akiaki rehe mahi raruraru e tu pūtaiao tere whakawhanake ahumahi me te hangarau, hanga Newton ko Leibniz nga ara noa o te kimi i te mahi o te pāpātanga o te huringa (rawa ki te whakaaro ki te tere pūkaha o te tinana o te trajectory mohiotia), i arahina ki te whakataki o taua ariā, rite te mahi pārōnaki me te pārōnaki, ka kitea hoki nga rongoā hātepe kōaro raruraru rite mohiotia ia se (tāupe) tere traversed ki te kitea te ara i arahina e ki te ariā o wāhi Ala.

I roto i nga mahi o Leibniz me Newton o whakaaro tuatahi ka puta reira e nga Pārōnaki - he rite ki te nuku o te tohenga taketake Δh āpiti mahi Δu e taea te pai te tono ki te tātai i te uara o te muri. I roto i te mau parau te tahi atu, kua kitea e ratou e kia he mahi te nuku haere i tetahi wāhi (i roto i tona rohe o whakamāramatanga) Kei te faaite i roto i ona pārōnaki e rua Δu = y '(x) Δh + αΔh wahi α Δh - toenga, tiaki ki te kore rite Δh → 0, nui tere atu i te tūturu Δh.

E ai ki nga kaiwhakatū o tātari pāngarau, te Pārōnaki - ko rite tenei te wā tuatahi i roto i ngā āpiti o tetahi mahi. Ahakoa kahore he e matau he raupapa tepe ariā mārama karu e te uara pārōnaki o te pārōnaki whangai ana ki te mahi, no te Δh → 0 - Δu / Δh → y '(x).

Rerekē Newton, ko wai i matua he ahupūngao me te taputapu pāngarau whakaaro hei taputapu pŭpŭ tauturu mo te ako o ngā raruraru tinana, utua Leibniz atu whakarongo ki tenei kete, tae atu i te pūnaha o tohu ataata, me te mahuki uara pāngarau. Ko ia te tangata e whakaarohia ana te momotuhi paerewa o Pārōnaki mahi dy = y '(x) dx, dx, me te pärönaki o te mahi tautohe rite ratou hononga y' (x) = dy / dx.

Ko te whakamāramatanga hou

He aha te mea te pārōnaki i roto i ngā o te pāngarau hou? he whanaunga tata te reira ki te ariā o te nuku tāupe. Ki te tango te tāupe y he uara tuatahi o y y = _1, ka y = y _2, ka karanga te rerekētanga y ₂ ─ y ₁ ko te uara nuku y. Ka taea e te nuku e pai. kino, me te kore. Ko te kupu "te nuku haere" whakaritea te Δ, Δu tuhi (te pānui i 'raorao y') tohu te uara o te nuku y. na Δu = y ₂ ─ y _1.

Ki te kia te māngai te uara Δu mahi te noho y = f (x) rite Δu = He Δh + α, te wahi A he kahore fakafalala i runga i Δh, t. E. He = const mo te x hoatu, me te α wā, ina whakapaia e Δh → 0 ki ko reira ara tere atu i te Δh tūturu, na te tuatahi ( "ariki") he wā hautanga Δh, a he hoki y = f (x) pārōnaki, denoted dy DF ranei (x) (lau "y de", "de enap i X"). Na reira Pārōnaki - he "matua" rārangi ki te faatura ki te wāhanga o ngā āpiti mahi Δh.

whakamārama aunoa

Kia s = f (t) - te tawhiti i roto i te rārangi tika neke wāhi rauemi i te tūranga tuatahi (t - wā haere). Nuku Δs - ko te wāhi ara i roto i te wā wā Δt, me nga DS pārōnaki = f '(t) Δt - tenei ara, e pai kia puritia wāhi mo te wa taua Δt, ki te mau te reira i te f tere' (t), tae i te wā t . A, no te rerekē te Δt DS ara pohewa ore i te Δs tūturu infinitesimally he tikanga teitei ki te whakapai kanohi ki Δt. Ki te kore e rite ki te kore te tere i te t wa, nga DS uara āwhiwhiwhi homai wāhi rītaha iti.

āhuahanga tikanga

Kia te L rārangi ko te kauwhata o y = f (x). Na Δ x = MQ, Δu = QM '(kite. Figure raro). Pātapa MN wawahi tapahia Δu kia rua nga wahi, QN ko NM '. Tuatahi ko Δh he hautanga QN = MQ ∙ TN (koki QMN) = Δh f '(x), t. E QN he pārōnaki dy.

Ko te wāhanga tuarua o te rerekētanga Δu NM'daet ─ dy, ina heke Δh roa → 0 NM 'ara tere atu i te nuku o te tautohe, arā, te mea i te tikanga o te itinga runga ake i Δh. I roto i tenei take, ki te f '(x) ≠ 0 (pātapa kore-whakarara OX) wāhanga QM'i QN ōrite; i roto i te kupu atu NM 'heke tere (tikanga o itinga o tona teitei) atu i te nuku katoa Δu = QM'. Ka kitea i roto i te Whakaahua (tata wāhanga M'k M NM'sostavlyaet katoa ōrau QM 'wāhanga iti) tenei.

Na, kauwhata anō he rite ki te nuku o te whakarite o te pātapa mahi te noho.

Pārōnaki me te pārōnaki

He take i roto i te wā tuatahi o te faaiteraa nuku mahi he rite ki te uara o ona f pārōnaki '(x). Ko te kupu, ko te whai ake e pā ana - dy = f '(x) Δh DF ranei (x) = f' (x) Δh.

Kei te mohiotia te reira e te nuku o te tautohe motuhake he rite ki tona pārōnaki Δh = dx. Runga i, Ka taea e tatou te tuhituhi: f '(x) dx = dy.

Te kimi (i ētahi wā ka mea ki hei te "whakatau") Pārōnaki whakamana e te taua tikanga rite mo nga pärönaki. homai he rārangi o ratou te raro nei.

He aha te ake ao: ko te nuku o te tautohe tona pārōnaki ranei

Tenei te mea e tika ana ki te hanga i te tahi mau haamaramaramaraa. Māngai uara f '(x) pārōnaki Δh taea, ka whakaaro x rite te tautohe. Ko taea te mahi e he matatini, i roto i nei e taea x kia he mahi o te t tautohe. Katahi te māngai o te faaiteraa pārōnaki o f '(x) Δh, rite ki te tikanga, e kore e taea; anake i roto i te take o te ti'aturiraa rārangi x = i + b.

Ka rite ki ki te tātai f '(x) dx = dy, ka i roto i te take o motuhake tautohe x (ka dx = Δh) i roto i te take o te fakafalala tawhā o x t, he reira pārōnaki.

Hei tauira, te faaiteraa 2 x Δh he hoki y = x ² tona pārōnaki, ka x ko te tautohe. Tatou i teie nei x = t ² me te amo t tohenga. Na y = x ² = t ^4.

aru ana tēnei e (t + Δt) ² = t ² + 2tΔt + Δt ^2. No reira Δh = 2tΔt + Δt ^2. No reira: 2xΔh = 2t ² (2tΔt + Δt ^2).

E kore te mea tenei faaiteraa hautanga ki Δt, a na reira ko inaianei e kore te 2xΔh pārōnaki. Ka taea te kitea te reira i te whārite y = x ² = t ^4. Ko reira rite dy = 4t ³ Δt.

Ki te tangohia e tatou i te 2xdx kīanga, ko reira te pārōnaki y = x ² mo tetahi tautohe t. Ko e mo'oni, ka x = t ² whiwhi dx = 2tΔt.

Na 2xdx = 2t ² 2tΔt = 4t ³ .DELTA.t, t. E. Ko te Pārōnaki faaiteraa tuhia e taurangi e rua rerekē hāngai.

Whakakapi āpiti Pārōnaki

Ki te f '(x) ≠ 0, ka Δu me dy ōrite (ka Δh → 0); ki te f '(x) = 0 (tikanga, me te dy = 0), e kore e ōrite ratou.

Hei tauira, ki te y = x ^2, ka Δu = (x + Δh) ² ─ x ² = 2xΔh + Δh ² me dy = 2xΔh. Ki te x = 3, ka to tatou Δu = 6Δh + Δh ² me dy = 6Δh e he ōrite tika Δh ² → 0, ka kore e ōrite x = 0 uara Δu = Δh ² me dy = 0.

Tenei meka, tahi ki te hanganga ohie o te pārōnaki (m. E. Linearity ki te faatura ki Δh), Kei te maha whakamahia i roto i te tātaitanga āwhiwhiwhi, i runga i te whakaaro e Δu ≈ dy mō iti Δh. Kimihia te mahi pārōnaki ko te tikanga māmā atu ki te tātai i te uara tangohia o te nuku.

Hei tauira, to tatou konganuku kupiki ki mata x = 10.00 cm. I te whakawera i te mata roa i runga i Δh = 0,001 cm. Pehea nui haere rōrahi mataono V? E tatou V = x ^2, kia e AM = 3x ² = Δh 3 ∙ ∙ February ¹⁰ 0/01 = 3 (cm ^3). Nui ake ΔV pārōnaki ōrite AM, kia e ΔV = 3 cm ^3. e hoatu tātaitanga tonu ³ ΔV = 10,01 ─ March ¹⁰ = 3.003001. Ko te hua o mati katoa anake te hārakiraki tuatahi; Na, ko reira tika tonu ki a tawhio ake ki te 3 cm ^3.

Oia mau, he pai tenei huarahi anake, ki te he e taea ki te whakatau tata i te uara homai ki hapa.

mahi pārōnaki: tauira

Kia tamata a ki te kitea te pārōnaki o te pānga y = x ^3, te kimi i te pārōnaki. Kia tatou e hoatu i te tautohe nuku Δu me te tautuhi.

Δu = (Δh + x) ³ ─ x ³ = 3x ² + Δh (Δh 3xΔh ² + ^3).

Here, e kore e te tau whakarea A = 3x ² whakawhirinaki i runga i Δh, kia e te wā tuatahi he hautanga Δh, te tahi atu melo o te 3xΔh Δh ² + ³ ka heke tere atu i te nuku o te tautohe Δh → 0. Nā tēnei, he mema o te 3x ² Δh ko te pārōnaki o y = x ^3:

dy = 3x ² Δh = 3x ² dx d ranei (x ³⁾ = 3x ² dx.

Ai d (x ³⁾ / dx = 3x ^2.

Dy inaianei kitea tatou i te pānga y = 1 / x i te pārōnaki. Na d (1 / x) / dx = ─1 / x ^2. Na dy = ─ Δh / x ^2.

Pārōnaki e hoatu taketake mahi taurangi i raro.

tātaitanga āwhiwhiwhi whakamahi pārōnaki

Hei arotake i te f pānga (x), me ona pārōnaki f '(x) i te x = te he maha uaua, engari ki te mahi i te taua i roto i te takiwā o x = te kore he ngāwari. Na ka haere mai ki te āwhina o te faaiteraa āwhiwhi

f (he + Δh) ≈ f '(he) Δh + f (he).

homai tenei te uara āwhiwhiwhi o te mahi i ngā āpiti iti i roto i tona pārōnaki Δh f '(he) Δh.

Na reira, tenei tātai homai he kīanga āwhiwhiwhi mo te mahi i te wāhi mutunga o te wahi o te roa Δh rite te moni o tona uara i te tīmatanga o te wahi (x = a) me te pārōnaki i roto i te wāhi tīmata taua. Tika o te tikanga mo te whakatau i nga uara o te mahi whakaatu i raro i te tātuhi.

Heoi matau me te whakapuaki tangohia mo te uara o te pānga x = te + Δh homai e tātai ngā āpiti tahuti (ranei, i ētahi wā, tātai o Lagrange)

f (he + Δh) ≈ f '(ξ) Δh + f (he),

te wahi i te pūwāhi x = te + ξ kei roto i te wā i x = te ki x = te + Δh, ahakoa tona tūranga tangohia he unknown. Ko te tātai tangohia taea ki te aromātai i te hapa o te tātai āwhiwhi. Ki te hoatu e matou i roto i te Lagrange tātai ξ = Δh / 2, ahakoa mutu reira ki te kia tika, engari homai, ka rite ki te tikanga, he huarahi pai nui atu i te faaiteraa taketake i roto i ngā o te pārōnaki.

hapa tātai aro mātai mā te tono pārōnaki

Ine mea , i roto i te parau tumu, hē, ka kawea ki te raraunga ine hāngai ki te hapa. E āhuatanga ratou na roto i te whāiti te hapa tino, ranei, i roto i te poto, te hapa rohe - pai, mārama rawa te hapa i roto i te uara pū (ranei i tino rite ki reira). Whāiti te hapa whanaunga i huaina ko te huawehe whiwhi e wehewehe i te reira e te uara pū o te uara whanganga.

Kia tātai tangohia y = f (x) mahi e whakamahia ana ki te vychislyaeniya y, engari te uara o x ko te hua ine, me reira e hopoi mai i te hapa y. Na, ki te kitea te whāiti hapa pū │Δu│funktsii y, te whakamahi i te tātai

│Δu│≈│dy│ = │ f '(x) ││Δh│,

i reira │Δh│yavlyaetsya hapa haurokuroku tautohe. Me häunga │Δu│ rahinga whakarunga, rite tātai hē iho ko te whakakapinga o te nuku i runga i te tātaitanga pārōnaki.