Kuyini okubalulekile, futhi kuyini incazelo yalo ngokomzimba

Ukuvela komqondo wokubaluleka kwakubangelwa isidingo sokuthola umsebenzi wokutholwa yi-derivative, kanye nokunquma ubukhulu bomsebenzi, indawo yezibalo eziyinkimbinkimbi, ibanga elihambahambayo, emigqeni echazwe ngamakherubi achazwa ngamagama angabonisi.

Kusukela eqinisweni Futhi izazi ze-physics ziyazi ukuthi umsebenzi ulingana nomkhiqizo wamandla ngaphezu kwebanga. Uma konke ukunyakaza kwenzeka ngejubane elivamile noma ibanga linqotshwa nokusetshenziswa kwamandla afanayo, ngakho konke kucacile, usanda kubandisa. Kuyini okubalulekile kokuhlala njalo? Lena umsebenzi ohambisanayo wefomu y = kx + c.

Kodwa amandla angashintsha ngokuhamba kwesikhathi komsebenzi, futhi ngenye indlela yokuthembela kwemvelo. Isimo esifanayo sibangelwa ukubalwa kwebanga elihamba uma ijubane lingavamile.

Ngakho-ke, kucacile lokho okubalulekile. Ukuqaphela njengenani lemikhiqizo yamagugu omsebenzi ngokulinganisa okungenamkhawulo kwempikiswano ichaza ngokugcwele incazelo ebalulekile yalo mqondo njengoba indawo yesibalo ihlolwe kusuka phezulu ngenhla yomsebenzi, kanye nasemaphethelweni ngemingcele yencazelo.

UJean Gaston Darboux, isazi sezibalo saseFrance, engxenyeni yesibili yekhulu le-19, uchaze ngokucacile lokho okubalulekile. Wakwenza ngokucacile ukuthi, akukona nzima, ngisho nangomfundi osemncane esikoleni esiphakeme ukuqonda lo mbuzo.

Ake sithi kukhona umsebenzi wanoma yiluphi uhlobo oluyinkimbinkimbi. I-axis ye-ordinates, lapho izimiso zokuphikisana zihlelwe khona, zihlukaniswe ngezikhathi ezincane, ngokuqinisekile zilinganiselwe, kepha njengoba umqondo wokungapheli komzimba kunengqondo, kunokwanele ukucabanga nje ngamagatsha amancane, okubaluleka ngokuvamile okuvezwa yileligama lesiGreki elithi Δ (delta).

Lo msebenzi "wawunqunywa" zibe izitini ezincane.

Inani ngalinye lempikiswano lapho lihambisana nephuzu ku-axis i-ordinate lapho amanani ahambisana nalo msebenzi ahlelwe khona. Kodwa njengoba imingcele yendawo ekhethiwe yilezi ezimbili, khona-ke izindinganiso zomsebenzi zizoba ezimbili, ezinkulu nezincane.

Isibalo semikhiqizo yamanani amakhulu nge-increment Δ ibizwa ngokuthi isibalo esikhulu saseDarboux, futhi siboniswe ngu S. Ngakho-ke, amanani amancane esifundeni esinqunyiwe aphindwe yi-A ndawonye enza ifomu elincane laseDarboux. Isigaba ngokwalo sifana nomgwaqo we-trapezoid, ngoba ukuvinjelwa komsebenzi womsebenzi kunganakwa nge-incrementimal increment. Indlela elula yokuthola indawo yalesi sibalo se-geometric ukwengeza imikhiqizo yexabiso elikhudlwana nelincane kune-Δ-increment futhi lihlukanise ngamabili, okungukuthi, ukuchaza ukuthi i-arithmetic isho ukuthini.

Nansi iDarboux ebalulekile:

S = Σf (x) Δ isamba esincane;

S = Σf (x + Δ) Δ isamba esikhulu.

Ngakho, yini ebalulekile? Indawo engaphansi komsebenzi wokusebenza kanye nemingcele yencazelo kuyoba:

∫f (x) dx = {(S + s) / 2} + c

Okungukuthi, isilinganiso samanani se-arithmetic samanani amakhulu nezincane zaseDarboux yinto engaguquki, engabonakali ngokuhlukaniswa.

Ukuqhubekela phambili ekuboniseni kwejometri yalo mqondo, incazelo yenyama yomzimba ibuye icacile. Indawo yalesi sibalo, ehlukaniswa umsebenzi we-velocity, futhi ehlanganiswe yisikhathi sesikhathi eduze kwe-abscissa axis, kuyoba ubude bendlela ehamba ngayo.

L = ∫f (x) dx esikhathini esivela ku-t1 kuya ku-t2,

Kuphi

I-F (x) umsebenzi we-velocity, okungukuthi, ifomula lapho ihluka khona nesikhathi;

L ubude bomzila;

I-T1 - isikhathi sokuqala kwendlela;

I-T2 yisikhathi sokuphela kwendlela.

Ngokuvumelana nomgomo ofanayo, ubukhulu bomsebenzi bunqunywe, kuphela eceleni kwe-abscissa ibanga lizobekwa, futhi ngokuqondisa ubukhulu bamandla asetshenziselwa iphuzu ngalinye.