Izvo zvisingaverengeki. Kuverengwa kwekusingaperi kusingaperi

Chimwe chemashoko makuru emasvomhu yekuongorora ndiyo inokosha calculus. Inobatanidza munda wakakura kwazvo wezvinhu, uko yekutanga isinganzwisisiki. Kuisa mamiriro acho akaita sekiyi, kuti kunyange kuchikoro chesekondari inoratidza kuwedzera kwenhamba yemikana uye mikana, inotsanangurwa nepamusoro masvomhu.

Kuonekwa

Pakutanga, kubatanidzwa kunoratidzika sekusingaperi, kwakakosha, asi pakuita zvinoratidza kuti yakaonekwa muna 1800 BC. Nyika yepamutemo inofungidzirwa kuva Ijipiti, sezvo isu tisina kuwana humwe humwe uchapupu hwokuti huripo. Iye, nekuda kwekushaiwa kweruzivo, nguva ino yose yakatarisa sechinhu chinoshamisa. Akambosimbisa zvakare hutano hwekugadzirwa kwesayenzi pakati pevanhu venguva iyo. Pakupedzisira, mabasa ekare echiGiriki emasvomhu anotarisana nezana remakore rechina BC akawanikwa. Vakatsanangura nzira iyo kushandiswa kusingagumi kwakashandiswa, icho chaicho chaiva chekuwana ivhu kana nzvimbo yechipfuva chemazuva (matatu-dimensional uye maviri-dimensional planes, maererano). Nheyo yekuverenga yakanga yakabva pakuparadzanisa chimiro chepakutanga muzvikamu zvisingasviki, kunze kwekuti ivhu (nzvimbo) yavo rave razivikanwa kare. Nokufamba kwenguva, nzira yacho yakakura, Archimedes akashandisa iyo kuwana nzvimbo yemufananidzo. Nhamba dzekuenzanisa panguva imwechete dzakaitwa nevasayendisiti muChina yekare, uyezve vakange vakasiyiwa zvachose nehama dzechiGiriki mune sayenzi.

Development

Kubudirira kwakatevera muzana remakore rechi11 AD raiva basa reArabia "rose" Abuja al-Basri, uyo akawedzera miganhu yezvakanga zvatozivikanwa, kuburikidza nekutsvaga pamusana pekubatanidzwa kwemazwi ekuenzanisa zvikwereti zvezvikamu uye zvikamu zvemasimba kubva kubva kutanga kusvika kunechina, Method of mathematical induction.
Mazuva ano pfungwa dzinoyemura kuti vaIjipita vekare vakagadzira sei zvigadzirwa zvinoshamisa zvekuvakwa, pasina zvigadziriswe zvakasiyana-siyana, kunze kwekuda mavoko avo, asi haisi simba re pfungwa dzevasayendisiti venguva iyo zvisingaiti zvishamiso zvishoma? Mukuenzanisa nenguva ino, upenyu hwavo hunoita sehupfupi, asi mhinduro yekusingaperi isina kukwana yakatorwa kose-kose uye yakashandiswa pakuita kuti hufambire mberi.

Nhanho inotevera yakaitika muzana remakore rechi16, apo nyanzvi yemasvomhu Cavalieri yakabvisa nzira yekusaridzika, yakatorwa naPeter Fermat. Ndivo vanhu vaviri vakateya nheyo dzemazuva ano hutano hunokosha hunozivikanwa panguva ino. Vakabatanidza pfungwa dzekusiyanisa nekubatanidzwa, izvo zvakamboonekwa sezvikwata zvakasimba. Kakawanda, masvomhu ezvenguva izvozvo ainge akaparadzana, zvikamu zvekupedzisira zvakange zviripo pachavo, zvine nzvimbo shoma yekushandiswa. Nzira yekubatana uye kutsvaga mamiriro akafanana ndiyo chete yakarurama panguva iyoyo, nokuda kwake iye zvino nyanzvi yekuongorora masvomhu yakakwanisa kukura uye kukura.

Nenguva yekufamba kwenguva, zvinhu zvose zvakachinja, uye kudanwa kwekubatanidzwa pamwe chete. Kakawanda, yakarongedzwa nevasayendisiti vanoti, zvakadini, somuenzaniso, Newton akashandisa kanda yekanda iyo yaakaisa basa rekushongedza kana kuisa parutivi rwayo. Uku kusabvumirana kwakapfuurira kusvikira muzana ramakore rechi17, apo nyanzvi yezvigadzirwa Gottfried Leibniz akaunza chiratidzo chinowanzozivikanwa kwatiri kwechidzidzo chose chekuongorora masvomhu. Iyo yakatambanudzwa "S" inonyanya kubva pane iyi tsamba yemutauro wechiLatini, sezvo inoreva chiyero chemaspodhi. Zita racho rakapiwa kubatanidzwa naJames Bernoulli mushure memakore 15.

Tsanangudzo yemhando

Kubatana kusingagumi kunoenderana zvakananga pane tsanangudzo yekusagadzirisa, saka funga nezvayo kutanga.

Icho chinotanga kuita basa chinopesana nezvinobva, zvichiita zvinonziwo chepakutanga. Zvimwe zvisingaiti: kusagadzirisa kwebasa d ibasa D iro rinobva kune v == V V = = v. Kutsvaga kwekusagadzirisa ndiko kuverengwa kwekusingaperi kusingagumi, uye iyo purogiramu pachayo inonzi kushandiswa.

Muenzaniso:

Basa s (y) = y ³ , uye rinopesana naro S (y) = (y 4/4).

Izvo zvakasarudzwa zvese zvinoshandiswa pasi pekufungisisa ndezvisingagadziriswi zvakakwana; zvinonzi zvinotevera: ∫v (x) dx.

Sezvo V (x) inongova imwe yepakutanga yemabasa ekutanga, tine mazwi: ∫v (x) dx = V (x) + C, apo C inogara iripo. Izvo zvinongoramba zvisinganzwisisiki zvinonzwisiswa sechinhu chipi zvacho chinogara chiripo, sezvo chibereko chacho chiri zero.

Properties

Izvo zvinhu zvisingagumi zvakakosha zviripo zvinobva pane tsanangudzo yekutanga uye nzvimbo yezviwanikwa.
Chimbofunga pfungwa dzinokosha:

• Iko kunokosha kwechirwere chisingaiti zvachose icho chaicho chinopesana uye pamwe neC constant constant) ∫V '(x) dx = V (x) + C;
• Izvo zvinobva mukubatanidzwa kwebasa ndiro basa rekutanga <=> (∫v (x) dx) '= v (x);
• Izvo zvinogara zvichitorwa kubva kuchiratidzo chekukosha <=> ∫kv (x) dx = k∫v (x) dx, kana k isingaiti;
• Iyo inokosha inotorwa kubva pachiyero inofananidzwa zvakaenzana nemari yezvibatanidzwa <=> ∫ (v (y) + w (y)) dy = ∫v (y) dy + ∫w (y) dy.

Kubva pazvinhu zviviri zvekupedzisira zvinogona kugumiswa kuti kusingagumi kusinganzwisisi kwakafanana. Nokuda kweizvi, tine: ∫ (kv (y) dy + ∫ lw (y)) dy = k∫v (y) dy + l∫w (y) dy.

Pakugadzirisa, tinotarisa mienzaniso yezvigadziriswe zvekusingaperi kusinganisi.

Zvakakosha kuwana hutano ∫ (3sinx + 4cosx) dx:

• ∫ (3sinx + 4cosx) dx = ∫3sinxdx + ∫4cosxdx = 3∫sinxdx + 4∫cosxdx = 3 (-cosx) + 4sinx + C = 4sinx - 3cosx + C.

Kubva pamuenzaniso watinogona kugumisa: hauzive kuti ungagadzirisa sei zvisingaverengeki zvinyorwa? Ingoona zvose zvinopesana! Uye heino maitiro ekutsvaga pasi apa.

Nzira uye mienzaniso

Kuti tive nekugadzirisa iyo inokosha, tinogona kushandisa nzira dzinotevera:

• Shandisa tafura yakagadzirwa;
• Gadzirisa nezvikamu;
• Gadzirisa nekuchinja shanduko;
• Kupfupisa pasi pechiratidzo chekusiyanisa.

Tables

Nzira iri nyore uye inofadza zvikuru. Parizvino, mashematical analysis inogona kuzvirumbidza nematafura akawandisa, mune izvo zvinyorwa zvinyorwa zvinogadziriswa zvinongororwa. Mune mamwe mazwi, pane matanho, anotorwa pamberi pako uye kwauri, anoramba achingovashandisa. Heino mutsara weiyo tafura huru iyo iyo inenge mimwe mienzaniso yose ine mhinduro inogona kuwanikwa:

• ∫0dy = C, apo C inogara iripo;
• ∫dy = y + C, apo C inogara iripo;
• ∫y ⁿ dy = (y ^{n + 1} ) / (n + 1) + C, apo C inogara iripo, uye n isiri nhamba;
• ∫ (1 / y) dy = ln | y | + C, apo C inoramba iripo;
• ∫e ^y dy = e ^y + C, apo C inogara iripo;
• ∫k ^y dy = (k ^y / ln k) + C, apo C inoramba iripo;
• ∫cosydy = siny + C, apo C inogara iripo;
• ∫sinydy = -cosy + C, apo C inogara iripo;
• ∫dy / cos ² y = tgy + C, apo C inogara iripo;
• ∫dy / sin ² y = -ctgy + C, apo C inogara iripo;
• ∫dy / (1 + y ² ) = arctgy + C, apo C inoramba iripo;
• ∫chydy = kunyara + C, uko C inoramba iripo;
• ∫shydy = chy + C, apo C inogara iripo.

Kana zvichidikanwa, tora matanho maviri, tauya kuiswa mutafura yekuona uye unakidzwe kukunda. Muenzaniso: ∫cos (5x -2) dx = 1 / 5∫cos (5x-2) d (5x-2) = 1/5 x chivi (5x-2) + C.

Nechisarudzo zvakajeka kuti pamuenzaniso wekutafura mubatanidzwa hauna huwandu hwehuwandu hwe 5. Tinohuwedzera, kuwedzera ne 1/5 muhumwe, kuitira kuti mazwi ose haashanduki.

Kubatana nezvikamu

Funga mabasa maviri - z (y) uye x (y). Dzinofanirwa kuramba dzichinyatsosiyaniswa pane zvese zveshoko retsanangudzo. Nechimwe chekusiyanisa kwezvinhu tine: d (xz) = xdz + zdx. Kubatanidza mativi maviri ehuenzana, tinowana: ∫d (xz) = ∫ (xdz + zdx) => zx = ∫zdx + ∫xdz.

Kudzokorora kuenzanisa kwakaguma, tinowana chirevo chinotsanangura nzira yekubatanidzwa nezvikamu: ∫zdx = zx - ∫xdz.

Nei zvichidikanwa? Ichokwadi ndechokuti mimwe mimwe mienzaniso ine mukana wekuita nyore, kutaura zvishoma, kuderedza ∫zdx ku ∫xdz, kana iyo yekupedzisira iri pedyo nepepanular form. Uyewo, iri shanduro inogona kushandiswa kanopfuura kamwechete, kubudirira kwechigumisiro chakanaka.

Nzira yekugadzirisa kusingagumi inobatanidza nenzira iyi:

• Zvakakosha kuverenga ∫ (s + 1) e ^2s ds

∫ (x + 1) e ^2s ds = {z = s + 1, dz = ds, y = 1 / 2e ^2s , dy = e ^2x ds} = ((s + 1) e ^2s ) / 2-1 / 2 ∫e ^2s dx = ((s + 1) e ^2s ) / 2-e ^2s / 4 + C;

• Iwe unoda kuverenga ∫lnsds

Ll = C.

Variable replacement

Iyi nheyo yekugadzirisa kusawirirana kusingagumi isingasviki pakudiwa kudarika mbiri yapfuura, kunyange yakaoma kunzwisisa. Iyo nzira inowanikwa mune zvinotevera: regai V (x) ive inokosha yeimwe basa v (x). Muchiitiko icho chinobatanidzwa pachavo mumuenzaniso chakaoma, pane mukana mukuru wekuvhiringidzika uye kuenda nenzira isina kururama. Kuti udzivise izvi, kuchinja kubva pane zvinoshandiswa x kusvika kuZ kunoshandiswa, umo mazwi ose anonyatsojekesa kana kuvimba kwe z pa x kuchichengetedzwa.

Mumathematical language, inoratidzika seiyi: ∫v (x) dx = ∫v (y (z)) y '(z) dz = V (z) = V (y ^-1 (x)), x = y ( Z) is permutation. Uye, zvechokwadi, basa rinopesana z = y ^-1 (x) rinotsanangura zvizere kuderera uye kuwirirana kwezvimwe zvinoshandiswa. Nyaya inokosha ndeyokuti kukasiyana dx kunotsanangurika nekushandurwa patsva dz, sezvo kushandurwa kwechitsinhanho muchibvumirano chisingaverengeki kunoreva kushandiswa kwayo kwose kwose, uye kwete chete mubatanidzwa.

Muenzaniso:

• Zvakakosha kuwana ∫ (s + 1) / (s ² + 2s - 5) ds

Tinoshandisa kushandiswa z = (s + 1) / (s ² + 2s-5). Zvadaro zv = 2sds = 2 + 2 (s + 1) ds <=> (s + 1) ds = dz / 2. Somugumisiro, tinowana mazwi anotevera, izvo zviri nyore kwazvo kuverenga:

∫ (s + 1) / (s ² + 2s-5) ds = ∫ (dz / 2) / z = 1 / 2ln | z | + C = 1 / 2ln | s ² + 2s-5 | + C;

• Zvakakosha kuwana hutano huno ∫2 ^s e ^s dx

Nokuda kwekugadzirisa, tinonyora zvakare mazwi muchimiro chinotevera:

∫2 ^{s s} ds = ∫ (2e) ^s ds.

Tinoreva ne = 2e (kuburikidza nekugadzirisa nharo iyi chiito, hachiri s), tinopa isu, pakutanga kutarisa, yakaoma kunzwisisa, kune chekutanga chemafomu:

∫ (2e) ^s ds = ∫a ^s ds = ^s / lna + C = (2e) ^s / l (2e) + C = 2 ^{s s} / l (2 + lone) + C = 2 ^{s s s} / (L2 + 1) + C.

Dhirowa pasi pechiratidzo chekusiyanisa

Nenzira yakawanda, iyi nzira yekusagadzikana kusinganzwisisi ihama yechimunana yechinoshandiswa chinoshandiswa pakugadziriswa, asi pane kusiyana kwekugadzira. Ngationgororei zvakadzama.

Kana ∫v (x) dx = V (x) + C uye y = z (x), ipapo ∫v (y) dy = V (y) + C.

Panguva imwecheteyo, munhu haafaniri kukanganwa zvishoma zvishoma kushandurwa, pakati peizvi:

• Dx = d (x + a), kupi pane chero nguva;
• Dx = (1 / a) d (dhi + b), pane zvakare inogara iripo, asi kwete yakaenzana nezero;
• Xdx = 1 / 2d (x2 + b);
• Sinxdx = -d (cosx);
• Cosxdx = d (sinx).

Kana tikafunga nyaya yakawanda kana tichienzanisa kusakosha kusingagumi, mienzaniso inogona kuderedzwa kusvika kumusorosoro mukuru w '(x) dx = dw (x).

Mienzaniso:

• Zvakakosha kuwana ∫ (2s + 3) ² ds, ds = 1 / 2d (2s + 3)

∫ (2s + 3) ² ds = 1 / 2∫ (2s + 3) ² d (2s + 3) = (1/2) x ((2s + 3) ² ) / 3 + C = (1/6) X (2s + 3) ² + C;

∫tgsds = ∫sins / cossds = ∫d (coss) / coss = -ln | coss | + C.

Kubatsira paIndaneti

Mune zvimwe zviitiko, mhaka inogona kuva uremeso kana kudiwa kwekukurumidzira, unogona kushandisa mazano ari paIndaneti, kana kuti, shandisa iyo calculator isina chokwadi. Pasinei nezvose zvinoratidzika zvakaoma kunzwisisa uye kukakavadzana kwezvibatanidza, sarudzo yavo inowanikwa pane imwe sarudzo, iyo yakavakirwa pamutemo "kana kwete ..., ipapo ...".

Zvechokwadi, kunyanya mienzaniso yakajeka yakadai sekugadzirisa zvinhu haigone kuve yakakosha, sezvo pane zviitiko apo sarudzo inowanikwa zvakajeka, "kukanganisa" kuunza zvimwe zvinhu muitiro, nokuti nzira dzakajeka dzemugumisiro haigoni kuwanikwa. Pasinei nekupesana kwemashoko aya, ndezvechokwadi, sezvo masvomhu, mumitemo, isingazivikanwi nesayenzi, uye inofunga basa rayo rekutanga yekuwedzera miganhu yemikana. Zvechokwadi, zvakaoma kwazvo kutama uye nekukudziridza paunenge uchimhanya-mune mazano, saka usafungidzira kuti mienzaniso yekugadzirisa kusinganzwisisiki kwatakakapa ndiyo yakakwirira yemikana. Zvisinei, regai tidzokere kune rumwe rutivi rwehutano hwenyaya yacho. Zvichida kutarisa kuverenga iwe unogona kushandisa mazano ayo zvinhu zvose zvakanyorwa pamberi pedu. Kana pane chido chekutsvaga kuverenga kwekutaura kwakaoma, haigoni kuparidzirwa, iwe unofanirwa kushandura kune yakawanda software. Zvakakosha kubhadhara kutanga kune zvose kuMatLab.

Kushanda

Mhinduro yekusagadzikana kusingagumi pakutarisa kwekutanga inoratidzika yakarambana zvachose kubva pakuitika, sezvo zvakaoma kuona zvionekwe zvirongwa zvekushandisa. Zvechokwadi, hazvigoni kushandiswa zvakananga kupi zvako, asi dzinofungidzirwa kuti inonyanya kukosha pakati pechikamu chekuita zvisarudzo zvinoshandiswa mukuita. Nokudaro, kubatana kunopesana zvakasiyana, nekuda kwekuti inoshingaira kubatanidzwa mukugadzirisa matanho.
Nekudaro, kuenzanisa uku kunobatsira zvakakwana pakugadzirisa kwezvinetso zvemagetsi, kuverenga kwetrajectories uye kupisa conductivity - muchidimbu, zvinhu zvose zvinoumba zvino uye zvinogadzira ramangwana. Kusingagumi kusinganzwisisiki, mienzaniso yatakakurukura pamusoro apa, haina maturo chete pakutanga, sezvo ichi chikonzero chekuita zvimwe zvitsva zvinowanikwa.