Farqiga calaamadaha - waxa uu yahay waxan? Sida loo helo kala duwan ee shaqada?

Oo ay la socdaan taagayo hawlaha ay Farqiga calaamadaha - waxaa qaar ka mid ah fikradaha aasaasiga ah ee kalkulaska kala duwan ee, qaybta ugu muhiimsan ee falanqaynta xisaabta. Sida saari Karin xidhiidh, oo labadoodaba waxaa ka qarniyo dhawr ah si weyn loo isticmaalaa in lagu xaliyo dhamaan dhibaatooyinka kacay ee koorsada of hawlaha sayniska iyo farsamada.

ah ee fikirka ah kala duwan

Waayo, markii ugu horeysay ka dhigtay in ay kala duwan cad in noocan oo kale ah, mid ka mid ah Aasaasayaasha ah (oo ay la socdaan Isaakom Nyutonom) kalkulas kala duwan xisaab Jarmal caanka Gotfrid Vilgelm Leybnits. Ka hor inta in xisaabyahannada qarnigii 17aad. loo isticmaalo fikrad aad u caddeyn iyo caddayn ah ee qaar ka mid ah infinitesimal "Ciddi leh" ee shaqo kasta oo la yaqaan, oo wakiil ka ah qiimaha aad u yar si joogto ah, laakiin aan loo siman yahay eber, hoose oo la qiimeeyo shaqada si fudud ma noqon karaan. Sidaa awgeed ay ahayd oo kaliya hal talaabo inay hordhaca ah ee fikradaha isabdal infinitesimal ee doodaha shaqo iyo isabdal ah ay ka soo jeedaan oo ah hawlaha oo la sheegi karaa in loo eego taagayo doonana. Oo tallaabo waxaa lagu qabtay ku dhowaad isku mar kor ku xusan laba cilmibaadhe oo weyn.

Iyada oo ku saleysan baahida loo qabo in wax laga qabto dhibaatooyinka farsamo wax ku ool ah oo degdeg ah in ay waajahaan sayniska si degdeg ah horumarinta warshadaha iyo technology, Newton iyo Leibniz (gaar ahaan marka la eego xawaaraha farsamo jidhka dhabbihii loo yaqaan), taas oo keentay in hordhaca ah ee fikradaha sida abuuray siyaabo badan ee loo helo hawlaha heerka isbedelka, sida shaqada derivative iyo kala duwan, iyo sidoo kale laga helaa xal dhibaatada bedelka geynta sida loo yaqaan per se (variable) xawaare ku safray si aad u ogaato jidka ayaa keentay in fikradda ah muhiim Ala.

In shuqullada Leibniz iyo Newton ayaa fikrad marka hore u muuqday in oo kala ah - saami la inremantiga oo ka mid ah dood aasaasiga ah Δh isabdal hawlaha Δu in si guul dabaqi karo si loo xisaabiyo qiimaha dambaysta ah. In si kale loo dhigo, waxay ogaadeen in shaqo inremantiga laga yaabaa in mar kasta (gudahood ay domain ee qeexidda) waxaa lagu qeexay iyada oo ay derivative labada Δu = y '(x) Δh + αΔh meesha α Δh - ka dhiman, in lasugo eber sida Δh → 0, ka dhakhso badan Δh dhabta ah.

Sida laga soo xigtay aasaasayaasha falanqaynta xisaabeed, oo kala ah - this dhab waa muddo xileedka koowaad ee isabdal ah hawlaha kasta. Xataa isagoo aan isku xigxig xadka fikrad cad oo qeexan ayaa la fahamsan yahay aad maskax ahaan in qiimaha kala duwan ee derivative ah waxay u egtahay in ay u shaqeeyaan marka Δh → 0 - Δu / Δh → y '(x).

Si ka duwan Newton, kuwaas oo ugu horayn ahaa jirka ah iyo qalab xisaabeed loo tixgeliyaa sida qalab kaabayaal waxbarasho ee dhibaatooyin jidheed, Leibniz bixisay fiiro gaar ah in la hirgaliyo qalabkan, oo ay ku jiraan nidaamka calaamadaha muuqaalka iyo la fahmi karo qiimaha xisaabta. Waxa uu ahaa kii jeediyay sal caadiga ah haysashada xun Farqiga calaamadaha function = y '(x) DX, DX, iyo derivative of function muran sida ay y xiriir' (x) = haysashada / DX.

Qeexidda casri ah

Waa maxay faraqa u eego xisaabta casriga ah? Waxaa si dhow la xiriira fikirka ah inremantiga ah variable. Haddii variable ah y qaadataa qiimaha ugu horeysay ee y y = _1, ka dibna y = y _2, y farqiga ₂ ─ y ₁ waxaa lagu magacaabaa y qiimaha inremantiga ah. inremantiga waxay noqon kartaa positive. negative iyo eber. ereyga "inremantiga" waxaa loo qoondeeyey Δ, Δu qoro (u akhri 'y Delta') oo tusinaya qiimaha y inremantiga ah. sidaas Δu = y ₂ ─ y _1.

Haddii qiimaha Δu shaqo aan loo aabo yeelin y = f (x) matali karaa sida Δu = A Δh + α, halkaas oo waa ku tiirsanaanta ma on Δh, t. E. A = const for x la siiyay, iyo α dheer marka Δh → 0 socotaa in waana ka dhaqso badan Δh dhabta ah, ka dibna marka hore ( "sayidkayga") xilli saami Δh, waxaana loogu talagalay y = f (x) kala duwan, muujinaysaa kale haysashada ama DF (x) (akhri "y de", "de eff ka X"). Sidaa darteed xun Farqiga - a toosan "ugu weyn" marka la eego qaybaha ee isabdal hawlaha Δh.

sharaxaad farsamo

Ha s = f (t) - fog ee xariiq toosan dhaqaaqin dhibic wax laga qabto, hore (t - waqtiga safarka). Inremantiga Δs - waa dhibic jidka lagu jiro bareeg waqti Δt ah, iyo DS kala duwan ee = f '(t) Δt - jidkaa, oo barta lagu qaban lahaa waqti isku mid ah Δt, haddii ay ceshan f xawaaraha' (t), soo gaartay xilli t . Marka Δt DS jidka ah infinitesimal khayaali ka duwan Δs dhabta infinitesimally isagoo si sare marka la eego Δt. Haddii xawaaraha mar t ma qummana in eber, ee DS qiimo qiyaas siinayaa dhibic eexda yar.

fasiraadda joomateri

Ha L line waa garaaf ee y = f (x). Markaas Δ x = MQ, Δu = aad afkiisa '(eeg. Hoos Jaantuska). Taabte MN jebiyo Δu jarjar laba qaybood, QN iyo nm '. First iyo Δh saami QN = MQ ∙ tg (QMN xagal) = Δh f '(x), t. E QN waa kala duwan haysid.

Qaybta labaad ee farqiga u Δu NM'daet ─ haysid, marka Δh dhererka → 0 nm 'hoos xitaa si ka dhaqso badan inremantiga ee doodda, sida uu leeyahay amarka smallness ka badan Δh. Xaaladdan oo kale, haddii f (x) ≠ 0 (Taabte non-isku midka ah OX) qaybaha QM'i QN u dhigma; in si kale loo dhigo nm 'si degdeg ah hoos (Si of smallness ay ka sareeya) ka badan total inremantiga Δu = aad afkiisa ku ah'. Tani waa caddayn in Jaantuska (qeybta dhowaanin M'k M NM'sostavlyaet oo dhan yar boqolkiiba aad afkiisa 'qeybta).

Sidaas daraaddeed, garaaf ahaan kala duwan shaqo aan loo aabo yeelin waa loo siman yahay si inremantiga ee duwo ee Taabte ee.

Derivative iyo kala duwanaanshaha

factor A ee muddo xileedka koowaad ee shaqo inremantiga hadal la siman yahay qiimaha ay f derivative '(x). Sidaas darteed, la xiriirta ka dib markii - haysashada = f '(x) Δh ama DF (x) = f' (x) Δh.

Waxaa la og yahay in inremantiga ah ee dood madax bannaan oo waa loo siman yahay si ay faraqa Δh = DX. Iyadoo la raacayo, waxaynu u qori karnaa: f '(x) DX = haysid.

Helitaanka (oo mararka qaarkood la sheegay in ay "go'aan") Farqiga calaamadaha lagu sameeyaa by xeerarka la mid tahay sida taagayo ah. Liiska A, iyaga waa la hoos ku siin.

Waa maxay more universal: inremantiga doodda ama ay kala duwan

Halkan waxaa lagama maarmaan ah in la sameeyo qaar ka mid ah xujooyin. qiimaha Matalaadda f '(x) faraqa Δh suurto gal ah markii la tixgelinayo x sida wax xuja ah. Laakiin shaqo wuxuu noqon karaa mid adag oo ah, taas oo x noqon kartaa shaqo ah t doodda. Markaas wakiil ka ah ee ra'yi kala duwan ee f '(x) Δh ah, sida caadiga ah, ma suurtowdo, marka laga reebo in ay dhacdo tiirsanaanta toosan x = at + b.

Sida ay u f caanaha '(x) DX = haysid, ka dibna ay dhacdo muran madax banaan x (markaas DX = Δh) in ay dhacdo tiirsanaanta parametric of x t, waa kala duwan.

Tusaale ahaan, hadal 2 x Δh waxaa y = x ² ay kala duwan marka x waa muran. Waxaan hadda x = t ² iyo u qaadan muran t. = X ² = t ⁴ Markaasaa ^y.

Tani waxaa la socda (t + Δt) ² = t ² + 2tΔt + Δt ^2. Sidaa Δh = 2tΔt + Δt ^2. Sidaa awgeed, 2xΔh = 2t ² (2tΔt + Δt ^2).

hadal Tani ma aha saamiyeed si ay Δt, oo sidaas daraaddeed haatan 2xΔh lama kala duwan yahay. Waxaa laga heli karaa ka shaqee isla'egta y = x ² = t ^4. Waxaa kale haysashada siman = 4t ³ Δt.

Haddii aan soo qaadanno 2xdx muujinta, waa y ku kala duwan = x ² wixii t muran. Indeed, marka x = t ² heli DX = 2tΔt.

Sidaas 2xdx = 2t ² 2tΔt = 4t ³ .DELTA.t, t. E. xun Farqiga calaamadaha The hadal duubay by laba doorsoome kala duwan ku beeganto.

Bedelaadda xun Farqiga calaamadaha isabdal

Haddii f '(x) ≠ 0, ka dibna Δu oo u dhiganta haysashada (marka Δh → 0); haddii f (x) = 0 (macnaha iyo haysashada = 0), ma ay joogaan u dhigma.

Tusaale ahaan, haddii y = x ^2, ka dibna Δu = (x + Δh) ² ─ x ² = 2xΔh + Δh ² iyo haysashada = 2xΔh. Haddii x = 3, ka dibna waxaan ku leeyihiin Δu = 6Δh + Δh ² iyo haysashada = 6Δh in ay yihiin wax u dhigma ugu wacan Δh ² → 0, marka x = 0 qiimaha Δu = Δh ² iyo haysashada = 0 ma yihiin wax u dhigma.

Dhab ahaantii Tani, si wadajir ah ula qaabka fudud ee kala duwan ee (m. E. Linearity marka la eego Δh), waxaa badanaa loo isticmaalaa xisaabinta qiyaas, on male ah in Δu ≈ haysashada for yar Δh. Raadi function kala duwan sida caadiga ah waxaa ka hawl yar ka badan si loo xisaabiyo qiimaha saxda ah ee inremantiga ah.

Tusaale ahaan, waxaan leenahay saaraayo macdan ah ku laayeen x = 10.00 cm. On kulaylinta ku laayeen dheeraado on Δh = 0,001 cm. Sidee kordhay mugga sedjibeke V? Waxaan leenahay V = x ^2, si DV = 3x ² = Δh 3 ∙ ∙ ⁰ 10 2/01 = 3 (cm ^3). Kordhinta kala duwan u dhigma ΔV DV, si ΔV = 3 cm ^3. xisaabinta Full siin lahaa ³ ΔV = 10,01 ─ March ¹⁰ = 3.003001. Laakiin natiijada oo dhan lambar aan lagu kalsoonaan karin hore mooyee, Sidaa darteed, waxaa weli loo baahan yahay si ay u duubaan ilaa 3 cm ^3.

Sida iska cad, habkan waa mid waxtar leh oo keliya haddii ay suurto gal tahay in la qiyaaso qiimaha siinno qaladka.

shaqo kala duwan: tusaalooyin

Aan isku dayno in la helo kala duwan ee y = x shaqada ^3, helidda derivative ah. Ha na siiso inremantiga muran Δu iyo qeexo.

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

Halkan, Wehliyaha A = 3x ² kuma xirna Δh, si ay muddo xileedka koowaad saami Δh, xubin kale oo ka 3xΔh Δh ² + ³ marka Δh → 0 hoos u dhaqso badan inremantiga ee doodda. Sidaas awgeed, xubin ka mid ah 3x ² Δh waa kala duwan ee y = x ^3:

haysashada = 3x ² Δh = 3x ² DX ama d (x ³⁾ = 3x ² DX.

War sidee d (x ³⁾ / DX = 3x ^2.

Haysashada Waxaan hadda wuxuu heli y shaqada = 1 / x by derivative ah. Markaas d (1 / x) / DX = ─1 / x ^2. Sidaa darteed haysashada = ─ Δh / x ^2.

Farqiga calaamadaha hawlaha aasaasiga ah aljebrada ayaa hoos siiyey.

xisaabaha qiyaas la isticmaalayo kala duwan

Si loo qiimeeyo f function (x), iyo derivative f '(x) ee x = a inta badan waa ay adag tahay, laakiin si la mid ah sameeyo oo goobta joogay ee x = a ma fududa. Markaas waxaa u timid in ay gargaarka ah ee ra'yi dhiibashada ka qiyaas

f (a + Δh) ≈ f '(a) Δh + f (a).

Tani waxay ku siinaysaa qiimo saxda ah ee shaqada ee isabdal yar iyada oo ay faraqa Δh f '(a) Δh.

Sidaa darteed, caanaha this siinayaa weer dhaw ee shaqada barta dhamaadka qayb ka mid ah dherer Δh sida wadarta ah ee ay qiimaha at Bar bilaw qaybtii (x = a) iyo faraqa ee dhibic isku upka. Sax ah habka lagu ogaanayo qiyamka shaqo hoose ayaa muujineysa sawirka.

Si kastaba ha ahaatee yaqaan iyo muujinta dhabta ah qiimaha shaqada x = a + Δh siiyo by formula isabdal uguna (ama, haddii kale, formula Lagrange ee)

f (a + Δh) ≈ f '(ξ) Δh + f (a),

halkaas oo dhibic x = a ah + ξ waa in bareeg ka x = a in x = a + Δh, inkasta oo ay booska saxda ah lama oga. formula dhabta ah u ogolaanaya in ay qiimeeyaan qaladka formula qiyaastii. Haddii aan ku riday Lagrange formula ξ ah = Δh / 2, inkastoo ay jireyso noqon sax ah, laakiin, sida caadiga ah, hab aad uga wanaagsan hadal asalka ah marka la eego faraqa u.

qaaciidooyinka Qiimeynta baadi by codsanaya kala duwan

Cabirka alaabtii , mabda ', aan sax ahayn, oo waxaad soo gelisaa si xogta cabbirka u dhiganta qaladka. Waxay waxaa lagu gartaa yareysey qaladka buuxda, ama, ee gaaban, xadka qaladka - positive, si cad aad iyo aad u baadi qiimaha buuxda (ama ugu badnaan loo siman yahay, waxa uu). Xaddididda qaladka qaraabo waxa loo yaqaan qeybtu waxay heli by it qaybinaya by qiimaha waadaxa ah ee qiimaha qiyaasay.

Ha formula dhabta ah y = f (x) function loo isticmaalaa in lagu vychislyaeniya y, laakiin qiimaha x waa natiijada cabbirka, oo sidaas daraaddeed waxay keentaa baadi y. Markaas, in la helo yareysey baadi buuxda │Δu│funktsii y, iyadoo la isticmaalayo caanaha

│Δu│≈│dy│ = │ f '(x) ││Δh│,

halkaas oo │Δh│yavlyaetsya baadi yarna uu muran. tiro │Δu│ waa in la kor lagu soo koobay, sida xisaabinta sax ahayn laftiisa waa bedelka ee inremantiga ah xisaabinta kala duwan ee.