"Uchungechunge Lwamandla: Izibonelo Nokuzivocavoca" yincwadi enikeza indlela esebenzayo nenamandla yokusebenza ngochungechunge lwamandla. Ngezibonelo ezicacile kanye nokuzivocavoca kwesinyathelo ngesinyathelo, le ncwadi isiza kokubili abafundi nochwepheshe ukuthi baqonde futhi basebenzise imiqondo eyisisekelo yochungechunge lwamandla, okwenza ukufunda kufinyeleleke futhi kuphumelele. Ubhalwe ngolimi olulula, olunenhloso, lo msebenzi uyithuluzi elibalulekile kulabo abafisa ukujulisa ulwazi lwabo kulo mkhakha wezibalo.
Ukuboniswa kwegunya kanye nomthelela ezimweni ezehlukene zezenhlalo, zamasiko nezepolitiki.
Ukuboniswa kwegunya negunya kuvamile ezimweni ezihlukahlukene zezenhlalo, zamasiko, nezombusazwe. Ochungechungeni oluqhutshwa amandla, isibonelo, singabona ngokucacile ukuthi abalingisi basebenzisa kanjani ithonya labo ukuze bafinyelele imigomo yabo.
Esimeni senhlalo, igunya lingabonakaliswa ngokuthinta komzimba, ulimi lomzimba, ngisho nendlela umuntu agqoka ngayo. Esikweni elithile, izimpawu ezithile zamandla zingaziswa kakhulu kunezinye, okuthonya ngokuqondile indlela igunya elibhekwa ngayo.
Emkhakheni wezombangazwe, igunya namandla kubonakala nakakhulu. Abaholi bezombangazwe basebenzisa izinkulumo ezihehayo, izivumelwano zamasu, ngisho nempoqo ukuze balondoloze izikhundla zabo. Kwezinye izimo, igunya livunyelwa ngezinqubo zentando yeningi, kuyilapho kweminye imibuso yezombusazwe, ithonya lisetshenziswa ngendlela enegunya.
Kubalulekile ukuqonda ukuthi lezi zakhi zibonakala kanjani ezimeni ezahlukene ukuze uqonde kangcono ukuguquguquka kwamandla emphakathini wethu.
Ukubonakaliswa okuhlukahlukene kwamandla emiphakathini yesimanje.
Emiphakathini yamanje, singabona ukubonakaliswa kwamandla okuhlukahlukene okugcwele ubudlelwano bomphakathi nezepolitiki. Amandla angazibonakalisa ngezindlela ezahlukene, kungaba ngezikhungo zikahulumeni, izinkampani zamazwe ngamazwe, amaqembu omphakathi ahleliwe, noma abantu abanethonya.
Isibonelo esicacile sokubonakaliswa kwamandla ukulawula okwenziwa izinkampani ezinkulu emnothweni wezwe nepolitiki. Izinkampani abezizwe ngezizwe Bavame ukuba nomthelela omkhulu kunohulumeni basekhaya, bakwazi ukusho imigomo nezinqumo ezithinta ukuphila kwabantu ngokuqondile. Lolu hlobo lwamandla ezomnotho lungobunye bobuso obubonakalayo bamandla emphakathini wanamuhla.
Ngaphezu kwalokho, amandla angakwazi futhi ukuzibonakalisa ngamaqembu ahlelekile omphakathi, njengezinhlangano zomphakathi, izinyunyana, nezinhlangano ezingekho ngaphansi kukahulumeni. Lawa maqembu avame ukuphatha ukuhlanganisa inqwaba yabantu ngemuva kwezizathu ezithile, ecindezela ohulumeni nezikhungo ukuthi zithathe izinyathelo ezizuzisa amaqembu athile emphakathini.
Okokugcina, amandla angaphinde abe khona ezingeni lomuntu ngamunye, ngokusebenzisa abantu abanezikhundla zobuholi emiphakathini noma ezinhlanganweni zabo. Laba bantu abanethonya bangenza izinqumo ezithinta ngokuqondile isiphetho sabantu abaningi, ngaleyo ndlela basebenzise uhlobo oluthile lwamandla phezu kwabo.
Incazelo yamandla kufilosofi: ingqikithi yawo, imiqondo kanye nokuzindla ngobunjalo bawo.
Amandla umqondo oyisisekelo kufilosofi, okuxoxwa ngawo kabanzi kuwo wonke umlando. Ingqikithi yayo ihlobene nekhono lokuthonya nokulawula abanye abantu, amaqembu, noma izimo. Amandla angasetshenziswa ngezindlela ezihlukahlukene, kungakhathaliseki ukuthi ukuphoqa, ukukholisa, noma ukuvunyelwa.
Kufilosofi, amandla avame ukuhlaziya maqondana nezakhiwo zokubusa nokuzithoba okukhona emphakathini. Izazi zefilosofi ezinjengoMichel Foucault kanye noFriedrich Nietzsche zahlola uhlobo lwamandla, zigqamisa ubudlelwano bawo nolwazi, ukuziphatha, kanye nobudlelwano bamandla.
Kunemibono ehlukene yamandla, njengamandla epolitiki, amandla ezomnotho, namandla angokomfanekiso. Ngayinye yalezi zinhlobo zamandla inezici zayo kanye nemithelela, ethonya ubudlelwano bomphakathi kanye nokuguquguquka kwamandla emphakathini.
Uchungechunge lwamandla luyizibonelo eziphathekayo zendlela amandla azibonakalisa ngayo ezimweni ezahlukene. Isibonelo sakudala sochungechunge lwamandla isigaba samasosha, lapho abantu bephethe amazinga ahlukahlukene egunya nomthelela. Esinye isibonelo kungaba amandla enkampani, lapho abaphathi besebenzisa amandla phezu kwabasebenzi.
Ukuze uqonde kangcono uhlobo lwamandla, kubalulekile ukwenza izivivinyo ezingokoqobo ezihlola ubudlelwano bamandla ezimeni ezahlukene. Lokhu kungase kuhlanganise ukuhlaziya ukuthi ubani ophethe amandla, ukuthi asetshenziswa kanjani, kanye nemiphumela yalobu budlelwano bamandla kulabo abathintekayo.
Ngokucabanga ngemvelo yamandla nokuhlola uchungechunge lwamandla ezimweni ezehlukene, singanweba ukuqonda kwethu ubudlelwano bamandla emphakathini kanye nemithelela yawo empilweni yomphakathi.
Izinhlobo ezahlukene zethonya negunya ezimweni ezahlukene kanye nobudlelwano phakathi kwabantu.
Ezimweni ezahlukene kanye nobudlelwano phakathi kwabantu, singabona izinhlobo ezahlukene zethonya negunya elinamandla phezu kwabantu abahililekile. Kungakhathaliseki ukuthi kuyinhlangano, emndenini, noma eqenjini labangane, amandla dynamics ahlala ekhona futhi angazibonakalisa ngezindlela ezihlukahlukene.
Isibonelo esicacile sokusebenzisa amandla ukubusa okukhona enkampanini. Umphathi unegunya phezu kwabangaphansi kwakhe futhi angathonya izinqumo zabo, ukuziphatha, nokusebenza komsebenzi. Ngemivuzo, izijeziso, kanye nempendulo, usebenzisa ithonya lakhe futhi ugcina igunya lakhe phezu kweqembu.
Olunye uhlobo lwethonya lungabonwa eqenjini labangane, lapho umuntu onomoya omnene futhi okholisayo ekwazi ukusebenzisa amandla phezu kwamanye amalungu. Imibono kanye nokukhetha kwabo kungaba nomthelela ezinqumweni zeqembu futhi kulolonge ukusebenzisana kwabo kanye nemisebenzi ndawonye.
Emkhayeni, igunya labazali phezu kwezingane liyisibonelo esivelele sokusebenzisa amandla. Ngemithetho, imingcele, nezindinganiso zokuziphatha, abazali baba nomthelela ekuziphatheni nasekukhuleni kwezingane zabo, beziqondisa ekwakhiweni kobuntu bazo nezindinganiso.
Ukubona nokuqonda lezi zinhlobo zamandla kuyisisekelo sokuhlalisana okunempilo nokulinganayo ezimweni ezehlukene zomphakathi.
Uchungechunge Lwamandla: Izibonelo Nokuzivocavoca
Uma uchungechunge lwamandla iqukethe isamba samatemu ngesimo samandla okuguquguquka x , noma ngaphezulu ngokuvamile, ka xc , lapho c iyinombolo yangempela engaguquki. Ngokuphawula okufinyeziwe, uchungechunge lwamandla ivezwa ngale ndlela elandelayo:
Na n (x -c) n = a o + a 1 (x – c) + a 2 (x – c) 2 + a 3 (x – c) 3 +... + a n (x – c) n
Lapho ama-coefficients a o , a 1 , a 2 … izinombolo zangempela futhi uchungechunge luqala ku-n = 0.
Lolu chungechunge lugxile enanini c okuyinto engaguquki, kodwa ungakhetha lokho c ilingana no-0; kulokhu, uchungechunge lwamandla lwenziwa lula ukuze:
Na n x n = a o + a 1 x + a 2 x 2 + a 3 x 3 + ... + a n x n
Uchungechunge luqala ngokuthi um o (xc) 0 e a ou x 0, ngokulandelana. Kodwa siyazi ukuthi:
(xc) 0 =x 0 = 1
Ngakho-ke, um o (xc) 0 = um ou x 0 = um o (itemu elizimele)
Into enhle ngochungechunge lwamandla ukuthi ungakwazi ukuveza imisebenzi ngayo, futhi lokhu kunezinzuzo eziningi, ikakhulukazi uma ufuna ukusebenza ngomsebenzi oyinkimbinkimbi.
Kulesi simo, esikhundleni sokusebenzisa umsebenzi ngokuqondile, ukuthuthukiswa kwawo ochungechungeni lwamandla kusetshenziswa, okungaba lula ukutholakala, ukuhlanganisa noma ukusebenza ngenombolo.
Yiqiniso, konke kuncike ekuhlanganeni kochungechunge. Uchungechunge luyahlangana lapho inani elikhulu lamagama lengezwa, okuholela enanini elingashintshi. Futhi uma sengeza imigomo eyengeziwe, sizoqhubeka nokuthola lelo nani.
Isebenza njengochungechunge lwamandla
Njengesibonelo somsebenzi ovezwe njengochungechunge lwamandla, ake sithathe f (x) = e x .
Lo msebenzi ungavezwa ngokochungechunge lwamandla ngale ndlela elandelayo:
e x ≈ 1 + x + (x 2 /2!) + (x 3 /3!) + (x 4 /4!) + (x 5 / 5!) +…
Kuphi! = n. (n-1). (n-2). (n-3) ... futhi uthola u-0! = 1.
Masisebenzise umshini wokubala ukuze siqinisekise ukuthi uchungechunge lufana ngempela nomsebenzi ocaciswe ngokusobala. Isibonelo, ake siqale ngokusetha x = 0.
Siyazi ukuthi futhi 0 = 1. Ake sibone ukuthi uchungechunge lwenzani:
e 0 ≈ 1 + 0 + (0 2 /2!) + (0 3 /3!) + (0 4 /4!) + (0 5 / aba-5!) + … = 1
Futhi manje ake sizame x = i-1 . Isibali siyakubonisa lokho e 1 = 2,71828 bese siyiqhathanisa nochungechunge:
e eyodwa ≈ 1 + 1 + (1 2 /2!) + (1 3 /3!) + (1 4 /4!) + (1 5 / the 5!) + … = 2 + 0.5000 + 0.1667 + 0.0417 + 0,0083 + … ≈ 2.7167
Ngamatemu angu-5 kuphela, sesivele sinokufana nse e-2.71 . Uchungechunge lwethu luntula okwengeziwe kancane, kodwa njengoba amagama engeziwe engezwa, ngokuqinisekile aguqukela enanini eliqondile le e . Isethulo sinembile uma n → ∞ .
Uma ukuhlaziywa kwangaphambilini kuphindaphindiwe kwe n = 2 , kutholwa imiphumela efana kakhulu.
Ngale ndlela, siyaqiniseka ukuthi umsebenzi we-exponential f (x) = e x ingamelwa yilolu chungechunge lwamandla:
Uchungechunge lwamandla eJiyomethri
Umsebenzi f (x) = e x akuwona kuphela umsebenzi osekela ukumelwa kochungechunge lwamandla. Ngokwesibonelo, umsebenzi f ( x) = 1/1 – x ibukeka ifana kakhulu neyaziwayo uchungechunge lwejometri oluguqukayo :
emhlophe n = a / 1 – r
Vele usethe okuthi = 1 kanye no-r = x ukuze uthole uchungechunge olufanele lwalo msebenzi, olugxile ku-c = 0:
Kodwa-ke, kuyaziwa ukuthi lolu chungechunge luguqukela ku-│r│ <1, ngakho-ke, ukumelwa kuvumeleke kuphela ku-interval (-1,1), noma ngabe umsebenzi uvumeleke kubo bonke x ngaphandle kuka-x = 1.
Uma ufuna ukuchaza lo msebenzi kolunye uhla, vele ugxile enanini elifanele futhi usuqedile.
Ungakuthola kanjani ukuthuthukiswa kwe-serial kwamandla omsebenzi
Noma yimuphi umsebenzi ungathuthukiswa ube uchungechunge lwamandla olugxile kokuthi c, inqobo nje uma unokuphuma kokuphuma kwawo wonke ama-oda kokuthi x = c. Inqubo isebenzisa ithiyori elandelayo, ebizwa ngokuthi Ithiyori kaTaylor:
Ake u-f abe umsebenzi (x) onokuphuma kokunye kokuhleleka n , kukhonjiswe ngokuthi f (n) , esekela ukuthuthukiswa kochungechunge lwamandla ebangeni le I . Ukuthuthukiswa kwayo kwe Taylor series ed:
Ukuze:
f (x) = f (c) + f '(c), (xc) + f' '(c) (XC) 2 /2 + f ”' (c) (XC) 3 /6 + ... R n
Lapho u-R n , okuyitemu lesishiyagalolunye lochungechunge, libizwa ngokuthi i ukuphumula :
Uma c = 0, uchungechunge lubizwa Uchungechunge lwe-Maclaurin .
Lolu chungechunge olwethulwa lapha lufana nochungechunge oluthulwe ekuqaleni, kodwa manje sinendlela yokuthola ngokusobala ama-coefficients wethemu ngayinye, enikezwe ngu:
Nokho, kufanele kuqinisekiswe ukuthi uchungechunge luhlangana nomsebenzi ozomelwa. Kuvele ukuthi akuwona wonke uchungechunge luka-Taylor oluguqukela ku-f(x), okucatshangelwe ekubalweni kwama-coefficients. a n .
Lokhu kwenzeka ngoba mhlawumbe okuphuma kokunye komsebenzi, okuhlolwe ngo x = c, kuhambisana nenani elifanayo nokuphuma kokunye, futhi ku x = c . Kulokhu, ama-coefficients azofana, kodwa ukuthuthukiswa bekuyoba okungaqondakali, njengoba bekungekho isiqiniseko sokuthi yimuphi umsebenzi ohambisana nawo.
Ngenhlanhla, kukhona indlela yokuthola:
Imibandela yokuhlangana
Ukuze ugweme ukungaqondakali, uma u-R n → 0 uma n → ∞ kukho konke x esikhaleni I, uchungechunge luguqulela ku-f (x).
Ukuzivocavoca umzimba
- Ukuzivocavoca okuxazululiwe 1
Thola uchungechunge lwamandla ejiyomethri lomsebenzi f (x) = 1/2 – x igxile ku-c = 0.
Isixazululo
Umsebenzi onikeziwe kufanele uvezwe ngendlela yokuthi ufane ngokuseduze ngangokunokwenzeka 1/1 x, uchungechunge lwawo lwaziwa. Ngakho-ke, masiphinde sibhale inombolo nedinominetha, ngaphandle kokushintsha isisho sangempela:
1/2 – x = (1/2) / [1 – (x / 2)]
Njengoba u-½ engaguquguquki, ushiya ukufinyezwa futhi ubhalwe ngokuvumelana nokuguquguquka okusha x / 2:
Qaphela ukuthi u-x = 2 akayona ingxenye yesizinda somsebenzi futhi, ngokombandela wokuhlanganisa onikezwe esigabeni. I-Geometric Power Series , ukuthuthukiswa kuvumeleke kokuthi │x / 2│ <1 noma ngokulinganayo -2
- Ukuzivocavoca okuxazululiwe 2
Thola imigomo yokuqala emi-5 yokuthuthukiswa kochungechunge lwe-Maclaurin lomsebenzi f (x) = sin x.
Isixazululo
Isinyathelo 1
Okokuqala, sithola ama-derivatives:
-Okususelwe kuhlelo 0: kuwumsebenzi ofanayo f (x) = isono x
-Okuphuma kokukuqala: (sin x) ´ = cos x
-Okuphuma kokunye okwesibili: (sin x) ´´ = (cos x) ´ = – isono x
-Okuphuma kokuthathu: (isono x) ´´´ = (-sen x) ´ = – cos x
-Okuphuma kokuhlanu: (isono x) ´´´´ = (- cos x) ´ = isono x
Isinyathelo 2
Bese okuphuma kokunye kuhlolwa kokuthi x = c, njengokuthuthuka kwe-Maclaurin, c = 0:
isono 0 = 0; i-cos 0 = 1; - isono 0 = 0; -cos 0 = -1; isono 0 = 0
isigaba 3
Ama-coefficients a n zakhiwe ;
a o = 0/0! = 0; a 1 = 1/1! = 1; a 2 = 0/2! = 0; a 3 = -1/3! a 4 = 0/4! = 0
Isinyathelo 4
Ekugcineni, uchungechunge luhlanganiswa ngokusho:
isono x ≈ 0.x 0 + 1. x 1 + 0 .x 2 – (1/3!) x 3 + 0 x 4 … = x – (1/3!)) x 3 +...
Ingabe umfundi udinga amagama engeziwe? Uma kukhona okuningi, uchungechunge lusondela kakhulu kumsebenzi.
Qaphela ukuthi kunephethini kuma-coefficients, igama elilandelayo elingelona uziro lithi 5 futhi zonke izinombolo eziyinqaba nazo zihlukile ku-0, izimpawu ezishintshanayo, njenge:
isono x ≈ x – (1/3!)) x 3 + (1/5!)) x 5 – (1/7!)) x 7 +….
Ishiywe njengomsebenzi wokuhlola ukuthi iyahlangana, i umbandela do i-quotient ingasetshenziselwa ukuhlangana kochungechunge.
Izinkomba
- Isisekelo se-CK-12. Power Series: emele imisebenzi kanye nokusebenza. Ithathwe ku: ck12.org.
- Engler, A. 2019. I-Integral Calculus. National University of the Coast.
- Larson, R. 2010. I-One-Variable Calculus. Uhlelo lwesi-9. UMcGraw Hill.
- Imibhalo Yezibalo Yamahhala. Uchungechunge lwamandla. Ithathwe ku: math.liibretexts.org.
- I-Wikipedia. Uchungechunge lwamandla. Ithathwe ku: es.wikipedia.org.