Calculus mahluko lwemisebenzi enye variables ezininzi

calculus umahluko kukuba nesebe ukuxhomekeka kolungiso, ejongisisa ukuba ithe, ukohluka kunye nokusetyenziswa kwazo nasekufundweni kwemisebenzi.

Indaba

calculus umahluko abonakala isiyalo ezizimeleyo kwisiqingatha sesibini kwenkulungwane ye-17, sibulela umsebenzi Newton Leibniz, ngubani yaqulunqa izibonelelo ezisisiseko ukubala nomahluko wabona uqhagamshelwano phakathi uhlanganiso kunye umahluko. Ekubeni uqeqesho waba kunye ukubalwa integrals, ngokwenjenjalo ukudala isiseko zohlalutyo zemathematika. Imbonakalo ezi calculi wavula ixesha mihla elitsha kwihlabathi zemathematika ukuba ukuvela lwamacandelo amatsha kwisayensi. Wkunyeiselwa amathuba nokusebenzisa imathematika zenzululwazi zendalo kunye zobunjineli.

nezimaphambili ezingundoqo

calculus umahluko isekelwe iingqiqo esisiseko lemathematika. Zezi: inani lokwenyani, inkqubela kunye umda umsebenzi. Emva kwexesha, bathabatha ujongo mihla, kubulelwe calculus ezifunekayo kunye umehluko.

Inkqubo yokudala

Formation of the umahluko calculus ngokohlobo isicelo, kwaye ngoko ke indlela yenzululwazi kwenzeka ngaphambi kokuba ukuvela kwethiyori bulumko, leyo wadalwa yi uNikolay Kuzansky. Umsebenzi wakhe uthathwa ukuba abe kwemvelo ukusuka inzululwazi yamandulo umgwebo. Ngaphandle kwento yokuba sobulumko naye nje kwezezibalo, igalelo lakhe kuphuhliso lenzululwazi kwemathematika ayinakuphikiswa. Kuwo, omnye ngaphandle yokuqala ukuqwalaselwa izibalo njengoko inzululwazi lichaneke, izibalo ukubeka ixesha ibe umbuzo.

Xa zezibalo yamandulo njengenqobo jikelele yaba iyunithi, lo gama i-sobulumko olucetywayo njengenyathelo Infinity entsha ukubuyisela inani ngqo. Malunga nalo ukumelwa omfusa okuchaneka kwinzululwazi zezibalo. ulwazi lwezenzululwazi, ngokwembono yakhe, yahlulahlulwe esinengqiqo ezikrelekrele. Eyesibini luchaneke kakhulu, ngokutsho nzulu, ukususela yangaphambili unika iziphumo luqikelelo nje kuphela.

umbono

Ingcamango esisiseko kunye ingqikelelo calculus umahluko ezinxulumene nomsebenzi kwindawo encinane iingongoma ezithile. Kuba oku kuyimfuneko ukwenza isixhobo zezibalo ukuze isebenze izifundo oziphatha kwindawo encinane amanqaku efakiweyo kufutshane yokuziphatha komsebenzi yomgama okanye polynomial. Ngokusekelwe kule nkcazelo ye esukela kunye umehluko.

Ukuvela nomba ithe kwabangelwa inani elikhulu iingxaki zenzululwazi zendalo kunye nezibalo, nto leyo eyakhokelela ekubeni nokuzimisela lwamaxabiso umda zohlobo olufanayo.

Omnye wemisebenzi eziziintloko banikwa umzekelo, kuqalwa kwiiklasi endala esikolweni, kukuba ukumisela isantya sesindululo yencopho umgca othe ngqo kunye nokwakhiwa komgca tanjenti ye kule igophe. Umahluko edityaniswe oku, ekubeni kusenokwenzeka ukuba esondele umsebenzi kwindawo encinane kwinqanaba komsebenzi yomgama.

Xa kuthelekiswa ne ingqikelelo yemvelaphi ngumsebenzi okanye umahluko wokwenene, inkcazo nomahluko idlule kwi umsebenzi ngokubanzi, ingakumbi umfanekiselo isithuba Euclidean kwenye.

uvumelane

Makhe lihamba ndawana kwicala y-axis, kuba ixesha sithatha x, apho kulinganiswa ukususela ekuqaleni umzuzwana. Chaza intshukumo onjalo kunokwenzeka ngenxa umsebenzi y = f (x), nto leyo ehambisana ukusa kwixesha ngalinye yokudada x nokulungelelanisa ngongoma displaceable. Le umnxeba umsebenzi ukukhanda ukuba umthetho motion. Uphawu iphambili isindululo, ingakumbi obungalinganiyo, nguye lo Velocity nje ngephanyazo. Xa ingongoma isusiwe ecaleni y-axis ngokomthetho ka ubucukubhede, ixesha random Lake uzuza nokulungelelanisa x f (x). Ekuhambeni kwexesha yokudada x + Δh, apho Δh imela ukwenyuselwa kwexesha, kuya kordinaty f (x + Δh). ifomula Ngaloo wabumba Δy = f (x + Δh) - f (x), nto leyo ibizwa ngokuba umsebenzi okukodwa. Kuyinto indawo kwindlela onqamlezwe ebudeni bexesha ukusuka x kwi x + Δh.

Ngokunxulumene elivela kuzo uqina ngexesha ithe ilawulwa. Le livela nawuphi na umsebenzi kwindawo esisigxina ebizwa ngokuba umda (kuqikelelwa ukuba ikhona). It bangathunyelwa kubalinganiswa ezithile:

f '(x), y', Ewe, iDF / DX, Dy / DX, DF (x).

Inkqubo yokubala yemvelaphi call umahluko.

calculus mahluko kwemisebenzi nesebenzayo ezininzi

Le ndlela lisetyenziswa xa kubalwa umsebenzi sifundo, variables eziliqela. Xa kukho yezahlukileyo ezimbini ux no y, ukuba ithe buso ngokumalunga x kwindawo A ibizwa ngokuba yemvelaphi lo msebenzi x kunye y eliqingqiweyo.

Ngamana kuboniswa simboli zilandelayo:

f '(x) (x, y), u' (x), ∂u / ∂x kunye ∂f (x, y) '/ ∂x.

izakhono ezifunekayo

Ukuze bafunde ngempumelelo yaye bakwazi ukucombulula diffury izakhono ezifunekayo uhlanganiso kunye umahluko. Ukwenza kube lula ukuqonda umehluko zibalo, kufuneka iqondwe isihloko ithe kunye elingenammiselo ephambili. Kwakhona ayikho buhlungu ukufunda ukukhangela ukuba yemvelaphi umsebenzi lokukwekwa. Oku kungenxa yokuba inkqubo yokufunda badla uya kusebenzisa integrals nokuzahlula.

Types of umehluko zibalo

Phantse wonke umsebenzi ulawulo ezinxulumene ne yokuqala-order umehluko zibalo, kukho 3 iintlobo kwizibalo: bomzi, kunye variables separable, inhomogeneous yomgama.

Kukho kwakhona ngakumbi ezinqabileyo iintlobo zibalo kunye nomahluko zizonke, quadratic Bernoulli kaThixo, kunye nabanye.

solutions imithetho

Okokuqala, sifanele sikhumbule ukuba equation aljibra kwekhosi yesikolo. Aqulathe ezitshintshayo kunye namanani. Ukuze ukusombulula inxaki eqhelekileyo kufuneka bafumane amanani ukwanelisa imeko ethile. Ngokuqhelekileyo, ezi zibalo kuba iingcambu enye, yaye ukuze lingqinisiswe kufuneka kufakwe kuphela eli xabiso endaweni engaziwayo.

Inxaki umahluko iyafana oku. Ngokubanzi, inxaki lo umyalelo wokuqala siqulathe:

• variable Independent.
• A livela umsebenzi wokuqala.
• Umsebenzi okanye oxhomekeke umahluko.

Kwezinye iimeko, inokuba akukho mntu ayaziwa, x okanye y, kodwa ibalulekile njengoko kuyimfuneko ukuba uvumelane yokuqala, kungekho avela eliphezulu ukuya isisombululo kunye calculus umahluko ziyinyaniso.

Ukusombulula inxaki umahluko - oko kuthetha ukufumana isethi yonke imisebenzi ukuba ibinzana efanelekileyo elinikiweyo. iiseti ezinjalo kwemisebenzi idla ngokuba ulawulo isisombululo jikelele.

calculus ebalulekile

calculus ezifunekayo yenye amacandelo ukuxhomekeka kolungiso, ejongisisa ingqiqo efunekayo, iimpawu kunye neendlela izibalo zalo.

Ngokufuthi ekubaleni elipheleleyo lwenzeka xa kubalwa kummandla imilo yegophe. Xa oku kuthetha indawo umda, ekuyeni apho indawo kwangaphambili shape polygon sibhalwe ukunyuka okuthe chu ngesandla sakhe, kwaye kwicala data ingenziwa ngaphantsi kwe naliphi na ixabiso elincinane engenasizathu ekhankanyiweyo ngaphambili.

Ingcamango iphambili ekubaleni indawo nayiphi imilo zejiyometri into yokubala ummandla uxande, ngoko kukho ubungqina bokuba indawo yayo lilingana imveliso ubude nobubanzi. Xa kuziwa kwageometry, koko wenza zonke izakhiwo usebenzisa irula kunye ikhampasi, uze umlinganiselo ubude ukuya ububanzi lixabiso nenengqiqo. Xa kubalwa indawo kanxantathu lungelo kumiselwa ukuba ufake unxantathu elandelayo, uxande kwakheka. Kule ndawo ye parallelogram sibalwa indlela efanayo kodwa kancinane ezinobunzima, ngaphakathi uxande kunye unxantathu. Xa kummandla polygon ingqalelo yi triangles ifakiwe kuyo.

Ekumiseleni inceba ngendlela engaqondakaliyo, le ndlela okungalungelaniyo ijika. Ukuba uwaphule elisebenzileyo ngokwezikweri ngamnye, oko kuya kuhlala iindawo zihlala. Kulo mzekelo, zama ukusebenzisa zangaphantsi ezimbini, kunye neengxande ngasentla ngezantsi, ngenxa yabo ziquka igrafu umsebenzi kwaye ayibandakanyi. Kubalulekile apha indlela ukwahlukana nale noxande. Kwakhona, ukuba singathatha ikhefu ngakumbi nangakumbi yancitshiswa, ummandla phezulu nasezantsi kufuneka idibane kwi ixabiso ethile.

Kufuneka ubuyele unendlela eyahlula zibe iingxande. Kukho iindlela ezimbini ethandwayo.

Riemann waye ngokusesikweni ingcaciso efunekayo, wadalwa yi Leibniz kunye Newton, njengoko indawo subgraph. Kulo mzekelo, siye saxubusha isafobe eziquka inani elithile iingxande nkqo ezifunyenwe ngokwahlula lekhefu. Xa ngokwaphula ukuhla kukho umda apho kwindawo yehlise isafobe esinjalo, lo mda ubizwa ngokuba Riemann nengxam komsebenzi kwindawo lethuba elifutshane elicaziweyo.

Indlela yesibini kukulungisa le Lebesgue efunekayo, owakhiwa yokuba endaweni yokuzahlula elijongene ndawo phezu yinxalenye integrand lokudidiyela ngoko ke inani elipheleleyo amaxabiso afunyenwe kule mimandla, ngokwamakhefu wahlula uluhlu yayo wamaxabiso, uze ushwankathelwa kunye namanyathelo ahambelanayo imifanekiso isichasi ezi integrals.

zixhobo ze

Enye inzuzo ukuba ukufunda umehluko kunye calculus ebalulekileyo Fikhtengol'ts wabhala - "ze umehluko kunye calculus ezifunekayo." incwadi yakhe sisixhobo esingundoqo isifundo uhlalutyo zezibalo, nto leyo ndamelana iintlelo ezininzi neenguqulelo kwezinye iilwimi. Owadala kubafundi kunye ixesha elide esetyenziswa ezahlukeneyo zamaziko emfundo njengenye izibonelelo ezingundoqo sifundo. Inika mfundo kunye lwezakhono zeprakhtikhali. Okokuqala yapapashwa ngo-1948.

umsebenzi wophando algorithm

Ukuhlola iindlela umsebenzi umahluko calculus, kufuneka ulandele sele unike algorithm:

1. Fumana domain lo msebenzi.
2. Fumana esineengcambu alenxaki elinikiweyo.
3. Bala ukuzibaxa izinto. Ukuze wenze oku, siya ukubala i yemvelaphi kwaye kwindawo apho ilingana zero.
4. We shintsha ixabiso afunyenweyo EQ.

Zezinto umehluko zibalo

Ulawulo komyalelo lokuqala (ngenye indlela, ezohlukileyo calculus enye bume) kunye neentlobo zazo:

• Kunye variables separable equation: f (y) dy = g (x) DX.
• Esilula equation okanye ezohlukileyo calculus umsebenzi omnye variable, ukuba ifomula: y '= f (x).
• I-linear lokuqala-order ulawulo nonuniform: y '+ P (x) y = Q (x).
• Bernoulli zahlula equation: y '+ P (x) y = Q (x) y a.
• Equation ukwehluka lilonke: P (x, y) DX + Q (x, y) dy = 0.

Umehluko zibalo ucwangco yesibini kunye neentlobo zazo:

• Eyohlobo olulodwa yomgama ukuze umahluko yesibini equation kunye ^{okuza rhoqo: y n + ngu- bani '+ qy} = 0 p, q fanele R.
• Inhomogeneous yomgama ukuze yesibini ezohlukileyo equation kunye ^nexabiso okuza ^{rhoqo: y n + ngu- bani '+ qy} = f (x).
• Eyohlobo olulodwa yomgama zahlula ^{equation: y n + p (x)} y '+ q (x) y = 0, kwaye inhomogeneous yesibini umyalelo ^{equation: (x) y n + p} y' + q (x) y = f (x).

umehluko zibalo kwemiyalelo ephezulu kunye neentlobo zazo:

• Umahluko equation, evumela ukuncitshiswa ^{komyalelo: F (x, y (k} ), y (k + ^{1), .., y (n)} = 0.
• A equation linear ucwangco eliphezulu ^{_{yohlobo: y (n) + f (}} not translate the ₁₎ y ^(n-1) + ... + f ₁ y '+ f ₀ y = 0, kwaye ^{_{inhomogeneous: y (n) + f (}} n _-1) y ^(n-1) + ... + f ₁ y '+ f ₀ y = f (x).

Amanqanaba zokusombulula ingxaki kunye quadratic umahluko

Ngoncedo kolawulo kude ukuba isonjululwe kuphela yemathematika okanye iingxaki emzimbeni, kodwa ke iingxaki ezahlukeneyo eziphilayo, ezoqoqosho, yokuhlalisana kunye nabanye. Nakuba iintlobo ezahlukeneyo izihloko, kufuneka lulandele ukulandelelana logic omnye zokusombulula ezi ngxaki;

1. Ukuzoba up ukulawula. Enye yamanqanaba anzima, efuna ukuchaneka eliphezulu, ngenxa yokuba nayiphi na impazamo kuya kukhokelela kwiziphumo engalunganga ngokupheleleyo. Kuyimfuneko ukuba athathele ingqalelo yonke imibandela echaphazela inkqubo ze imisele imiqathango ekuqaleni. Kufuneka kwakhona isekelwe kwizibakala nezigqibo ubhalo.
2. Kuba ekusombululeni zibalo. Le nkqubo kulula ukuya kwindawo yokuqala, ekubeni kufuna ukuphunyezwa engqongqo kuphela izibalo.
3. Uhlahlelo kunye nokuhlolwa kweziphumo. isisombululo ethathwe kufuneka ihlolwe ukuba ukufakelwa ukubaluleka kunye nemiqathango isiphumo.

Umzekelo ukusetyenziswa umehluko zibalo amayeza

Ukusebenzisa ulawulo kude kummandla wamayeza ifumaneka ekwakhiweni imodeli epidemiological zezibalo. Asimele silibale ukuba ezi zibalo nazo zifumaneka eziphilayo kunye chemistry, akufutshane amayeza, kuba kudlala indima ebalulekileyo ekufundeni kwamanani ezahlukeneyo yebhayoloji kunye neenkqubo kwemichiza emzimbeni womntu.

Kulo mzekelo, ukusasazeka bhubhani usulelo sinokunyangwa xa uluntu ekwanti. Abemi ziye zohlulwa ngokwendidi ezintathu:

• Abosulelekileyo, inani x (t), olu sapho lwaluquka abantu, abathwali ezasulelayo, nganye na esosulelayo (okuthatha bufutshane).
• Uhlobo lwesibini luquka sifo nabantu y (t), unako nokosuleleka ngokudibana ezosulelekileyo.
• Uhlobo lwesithathu kuquka nabantu eliyinjubaqa z (t), nto leyo omzimba okanye ezilahlekileyo ngenxa yokugula.

Inani abantu rhoqo, ukugcina ngokuzalwa, ukufa zendalo yokufudukela akujongwa. Ngundoqo iya kuba nantoni na awayefike sele ezimbini.

sifo Percent ngexesha elithile kwindawo ulingana no'x '(t) y (t) (vekeleke ngokusekelwe ingcamango ukuba inani lezehlo ngokomlinganiselo inani engenelayo phakathi abaguli kunye namalungu ephendulayo, nto isilinganiselo yokuqala umlinganiselo ukuba x (t) y (t)), ngo ngoko ke inani lamatyala liyenyuka, kwaye inani ucutho singangena ngeqondo ebalwayo yi izembe ifomula (t) y (t) (a> 0).

Inani non-ekuthethwe izilwanyana wafa okanye lufundiwe sifo, ukwanda ngeqondo apho ngokomlinganiselo inani leemeko, 'bx' (t) (b> 0).

Ngenxa yoko, uyakwazi ungazimiseli simiso selitye kwizibalo nazo zonke izibonisi zontathu ngokusekelwe izigqibo zayo.

ukusebenzisa UMZEKELO economics

calculus umahluko isoloko isetyenziswa Uhlalutyo lwezoqoqosho. Umsebenzi ophambili kweli Uhlalutyo lwezoqoqosho ithathwa ukuba isifundo amaxabiso lwezoqoqosho, nanto leyo ibhalwe ngohlobo lo msebenzi. Oku kusetyenziswa ekucombululeni iingxaki ezifana notshintsho nokunyuka kwerhafu yengeniso ngoko nangoko emva, imirhumo entry, utshintsho ngongeniselo xa ukutshintsha ixabiso lemveliso, ziziphi umlinganiselo ukuba kufakwe abasebenzi umhlala kunye izixhobo ezitsha. Ukusombulula iingxaki ezinjalo, kufuneka ukwakha umsebenzi wonxibelelwano yezahlukileyo engenayo, oko, emva kokuba zifundwa calculus umahluko.

oko kuyimfuneko rhoqo ukufumana umsebenzi kakhulu kangangoko kwinkalo yezoqoqosho: kwemveliso eliphezulu, imivuzo ephezulu, iindleko ubuncinane njalo njalo. Icandelo ngalinye elinjalo umsebenzi lweempikiswano omnye okanye ngaphezulu. Umzekelo, imveliso libe nokuthathelwa ingqalelo njengomsebenzi zabasebenzi kunye eyinkunzi. Kule meko, ukufumana ixabiso elifanelekileyo kungancitshiswa ukuze sifumane ubuninzi okanye ubuncinane umsebenzi kweziganga omnye okanye ngaphezulu.

iingxaki ezinjalo ukudala iklasi iingxaki extremal kwintsimi kwezoqoqosho, apho ufuna calculus umahluko. Xa kufuneka isalathisi yezoqoqosho ukunciphisa okanye wenze nkulu njengomsebenzi nabanye parameters, lo ukwenyuselwa ratio ubuninzi indawo umsebenzi kwingxoxo ziya kuthanda zero ukuba ukwenyuselwa Kwempikiswano uyabaleka zero. Kungenjalo, xa isimo sengqondo esinjalo kuvame kwixabiso ethile entle okanye embi, ingongoma elikhankanyiweyo ezifanelekileyo, ngenxa yokuba ngokwandisa okanye ukunciphisa ingxabano zingatshintshwa ixabiso abaxhomekeke ulwalathiso olufunayo. Xa isigama umahluko calculus, oku kuya kuthetha ukuba iimeko eziyimfuneko umsebenzi likhulu lixabiso zero of ithe yayo.

Uqoqosho ayikho ingxaki akuqhelekanga lokufumana extremum komsebenzi eziguquguqukayo eziliqela, ngenxa yokuba izikhombisi zoqoqosho zenziwe izinto ezininzi. Imiba efana ziqondwa kakuhle kwithiyori kwemisebenzi kweziganga eziliqela, indlela yokubala umahluko. ngxaki ziquka nje kuphela nkulu yaye kuncitshiswe umsebenzi, kodwa imida. Le mibuzo zinxulumene iinkqubo zemathematika, yaye kusonjululwa noko kolu ngoncedo iindlela ephuhliswe ngokukhethekileyo nayo isekelwe phezu kweli sebe yezenzululwazi.

Phakathi iindlela ezohlukileyo calculus esetyenziswa kuqoqosho, kukho icandelo ibalulekileyo vavanyo lukhulu. Kwinkalo yezoqoqosho, eli gama libhekisela iseti iindlela zophando yokusebenza bume kwaye iziphumo xa utshintsha ivolyum indalo, ngesifo sephepha, olusekelwe kucazululo lwamaxabiso zabo mda. Ukucutha sesalathiso ingqalelo ithe okanye yezabelo akakhethi variables eziliqela.

calculus mahluko nesebenzayo ezininzi - nesihloko ebalulekileyo uhlalutyo zezibalo. Ukuze isifundo eneenkcukacha, ungasebenzisa iindlela ezahlukeneyo zokuncedisa ekufundiseni kumaziko emfundo ephakamileyo. Enye Fikhtengol'ts abadumileyo wadala - "ze umehluko kunye calculus ezifunekayo." Yimalini lwegama ngesisombululo umehluko zibalo ubaluleke kakhulu ukuba izakhono ukusebenza integrals. Xa kukho calculus mahluko nemisebenzi omnye variable, isigqibo kuba lula. Nangona, kufanele kuqatshelwe, kulandela le migaqo efanayo emayisetyenziswe. Ukuziqhelisa, ukuphanda umsebenzi calculus umahluko, landela nje algorithm sele ekhoyo, nto leyo enikwe kwisikolo samabanga aphakamileyo, kwaye inzima kuphela intwana encinane kunye nokuqaliswa variables ezintsha.