ბიზნესის მათემატიკა by Goga

m. mumlaZe, s. cotniaSvili, k. cercvaZe

biznesis maTematika

@`teqnikuri universitetiâ&#x20AC;?@

saqarTvelos teqnikuri universiteti

m. mumlaZe, s. cotniaSvili, k. cercvaZe

biznesis maTematika

damtkicebulia stu-s saredaqcio-sagamomcemlo sabWos mier

Tbilisi 2009

maTematikuri analizis mniSvnelovan nawilebs warmoadgens integraluri aRricxva da mwkrivTa Teoria. swored am nawilebis da maTi praqtikuli gamoyenebis elementebia Setanili winamdebare saxelmZRvaneloSi. saxelmZRvaneloSi Setanilia agreTve albaTobis Teoriis da maTematikuri statistikis elementebi. saxelmZRvanelo gankuTvnilia umaRlesi profesiuli skolis, aseve ekonomikuri da sainJinro specialobebis bakalavriatis

studentebisaTvis.

© sagamomcemlo saxli ,,teqnikuri universiteti’’, 2009 ISBN 978-9941-14-631-2 http://www.gtu.ge/publishinghouse/ yvela ufleba daculia. am wignis arc erTi nawili (iqneba es teqsti, foto, ilustracia Tu sxva) aranairi formiT da saSualebiT (iqneba es eleqtronuli Tu meqanikuri), ar SeiZleba gamoyenebul iqnas gamomcemlis werilobiTi nebarTvis gareSe. saavtoro uflebebis darRveva isjeba kanoniT.

3 Sinaarsi Sesavali .............................................................................................................................................................................. 6 Tavi 1 gansazRvruli integrali 1. farTobis amocana ..............................................................................................................................................7 2. gansazRvruli integralis Tvisebebi ...........................................................................................9 3. integraluri aRricxvis ZiriTadi formula ..................................................................... 10 4. arasakuTrivi integralebi .................................................................................................................... 14 5. gansazRvruli integralis gamoyeneba ....................................................................................... 16 5.1. farTobis gamoTvla sxvadasxva geometriuli figurisaTvis ........................ 16 5.2 wiris rkalis sigrZis gamoTvla .................................................................................................20 5.3. brunviTi sxeulis moculoba ......................................................................................................... 22 5.4. brunviTi zedapiris farTobi ......................................................................................................... 25 5.5. meqanikuri da fizikuri sidideebis gamoTvla ............................................................. 28 5.5.1. wiris statikuri momenti da simZimis centri ............................................................ 28 5.5..2. brtyeli figuris statikuri momenti da simZimis centri ........................... 30 5.5.3. cvladi siCqariT moZravi materialuri wertilis mier gavlili manZili .................................................................................................................................................................. 33 5.5.4 gansazRvruli integralis gamoyeneba ekonomikur amocanebSi .................. 34 savarjiSoebi 1-li TavisaTvis ......................................................................................................... 35

1. 2. 3. 4. 5. 6.

Tavi 2 ricxviTi mwkrivebi ZiriTadi cnebebi ............................................................................................................................................. 36 krebadi mwkrivis Tvisebebi .................................................................................................................. 37 dadebiT wevrebiani ricxviTi mwkrivis krebadobis niSnebi ......................................................................................................................................... 39 niSancvladi mwkrivebi ..............................................................................................................................42 mwkrivTa absoluturi da pirobiTi krebadoba ............................................................... 43 ricxviTi mwkrivebis gamoyeneba .......................................................................................................44 savarjiSoebi me-2 TaviaTvis .................................................................................................................46

Tavi 3 xarisxovani mwkrivebi 1. ZiriTadi cnebebi .............................................................................................................................................47 2. xarisxovani mwkrivebi. abelis Teorema ...................................................................................47 3 xarisxovani mwkrivis Tvisebebi ......................................................................................................... 50 4. teiloris formula ..................................................................................................................................... 50 5. teiloris mwkrivi .......................................................................................................................................... 53 savarjiSoebi me-3 TavisaTvis ................................................................................................................ 56

1. 2. 3. 4. 5. 6. 7. 8.

Tavi 4 mravali cvladis funqciebi ZiriTadi cnebebi ............................................................................................................................................. 58 ori cvladis funqciis kerZo warmoebulebi .....................................................................60 ori cvladis funqciis sruli diferenciali ................................................................. 62 zedapiris mxebi sibrtye da normali .....................................................................................64 rTuli funqciis warmoebuli ............................................................................................................ 65 aracxadi funqcia da misi warmoebuli .................................................................................66 maRali rigis warmoebulebi da diferencialebi .......................................................... 68 ori cvladis funqciis eqstremumi ..............................................................................................70 savarjiSoebi me-4 TavisaTvis ..............................................................................................................74

4 Tavi 5 orjeradi integralebi 1. cilindruli sxeulis moculoba da orjeradi integralis cneba ............................................................................................................................................ 76 2. ganmeorebiTi integralebi da orjeradi integralis gamoTvla ................................................................................................................................. 77 3. orjeradi integralis gamoTvla marTkuTxa da polarul koordinatTa sistemaSi ................................................................................................. 80 4. orjeradi integralis gamoyeneba .................................................................................................. 84 4.1. brtyeli figuris farTobi ............................................................................................................... 84 4.2. zedapiris farTobi ................................................................................................................................... 84 4.3. brtyeli figuris masa, simZimis centri da statikuri momenti ....................................................................................................................................... 87 4.4. orjeradi integralis sxva gamoyenebebi ............................................................................ 89 savarjiSoebi me-5 TavisaTvis ..............................................................................................................90 Tavi 6 mrudwiruli integralebi 1. pirveli gvaris mrudwiruli integrali ................................................................................ 91 2. meore gvaris mrudwiruli integrali ...................................................................................... 93 3. meore gvaris mrudwiruli integralis meqanikuri mniSvneloba ........................................................................................................................................................... 97 savarjiSoebi me-6 TavisaTvis ................................................................................................................ 98 Tavi 7 diferencialuri gantolebebi 1. ZiriTadi cnebebi ...........................................................................................................................................100 2. umartivesi pirveli rigis Cveulebrivi diferencialuri gantolebebi .............................................................................................................102 2.1. diferencialuri gantoleba gancalebadi cvladebiT .......................................102 2..2. pirveli rigis erTgvarovani diferencialuri gantoleba ............................................................................................................................................................. 103 2.3. wrfivi, pirveli rigis diferencialuri gantoleba ............................................ 105 2.4. bernulis diferencialuri gantoleba ............................................................................ 105 2.5. gantoleba srul diferencialebSi .......................................................................................106 3. wrfivi, meore rigis diferencialuri gantolebani ................................................ 108 3.1. wrfivi, erTgvarovani, meore rigis diferencialuri gantoleba ........................................................................................................................................................... 108 3.2. wrfivi, araerTgvarovani meore rigis diferencialuri gantoleba ............................................................................................................................................................111 4. wrfivi mudmiv koeficientebiani meore rigis diferencialuri gantolebebi ..........................................................................................................113 5. diferencialuri gantolebebis gamoyeneba .........................................................................115 5.1. nivTierebis warmoqmnis da daSlis gantolebebi ......................................................115 5.2. harmoniuli rxevebi ................................................................................................................................. 117 savarjiSoebi me-7 TavisaTvis ..............................................................................................................120 Tavi 8 xdomiloba da misi albaToba 1. xdomiloba. elementarul xdomilobaTa sivrce ........................................................... 121 2. moqmedebebi xdomilobebze ..................................................................................................................... 122 3. albaTobis aqsiomuri ganmarteba ......................................................................................................126

5 4. albaTobis klasikuri gansazRvreba ........................................................................................... 128 5. albaTobis geometriuli gansazRvreba ..................................................................................... 132 6. pirobiTi albaToba. xdomilobaTa namravlis albaToba ...................................... 134 7. xdomilobaTa damoukidebloba ......................................................................................................... 135 8. sruli albaTobis formula. baiesis formula ............................................................... 137 9. damoukidebel cdaTa mimdevroba. bernulis sqema ........................................................140 10. polinaruli sqema. ualbaTesi ricxvi ..................................................................................... 143 11. muavr-laplasis lokaluri da integraluri Teoremebi ......................................146 12. puasonis formula .......................................................................................................................................147 Tavi 9 SemTxveviTi sidideebi 1. SemTxveviTi sidide da misi ganawilebis kanoni .............................................................149 2. SemTxveviTi sididis ricxviTi maxasiaTeblebi ............................................................... 152

1. 2. 3. 4. 5.

Tavi 10 ganawilebis kanonTa ZiriTadi saxeebi binomuri ganawileba ..................................................................................................................................... 158 puasonis ganawileba ..................................................................................................................................... 159 geometriuli ganawileba ......................................................................................................................... 159 Tanabari ganawilebis kanoni ...............................................................................................................160 normaluri ganawilebis kanoni ........................................................................................................160

Tavi 11 did ricxvTa kanoni 1. CebiSevis utoloba .........................................................................................................................................164 2. did ricxvTa kanoni ......................................................................................................................................166

1. 2. 3. 4. 5. 6. 7. 8. 9.

Tavi 12 maTematikuri statistikis elementebi maTematikuri statistikis sagani da ZiriTadi amocanebi ..................................... 168 SerCeviTi meTodi ............................................................................................................................................169 ganawilebis parametrebis statistikuri Sefaseba ....................................................... 171 momentTa meTodi ..............................................................................................................................................174 maqsimaluri dasajerobis meTodi ................................................................................................. 175 empiriuli ganawilebis funqcia ...................................................................................................... 178 ndobis intervalebi ...................................................................................................................................... 179 statistikuri hipoTezebi ........................................................................................................................ 182 parametrul hipoTezaTa Semowmeba ................................................................................................ 186

danarTi1. danarTi 2.

funqciis mniSvnelobaTa cxrili ........................................ 191 funqciis mniSvnelobaTa cxrili ................ 192

danarTi 3. ganawilebis kritikuli wertilebi ............................................................. 193 literatura ................................................................................................................................................................194

6 Sesavali maTematikuri analizi aris maTematikis nawili, romelSic funqciebi Seiswavleba zRvarTa meTodebiT. zRvris cneba mWidrodar aris dakavSirebuli usasrulod mcire sididis cnebasTan. SeiZleba iTqvas, rom maTematikuri analizi Seiswavlis funqciebs usasrulod mcireTa meTodebiT. saxelwodeba ,,maTematikuri analizi’’ aris saxelwodebis ,,usasrulod mcireTa analizis” Semoklebuli saxecvlileba; magram isic Semoklebulia da amdenad sagans ufro zustad aRwers saxelwodeba ,,analizi usasrulod mcireTa saSualebiT”. bunebasa da teqnikaSi, yvelgan gvxvdeba moZraoba, procesebi, romlebic aRiwereba funqciebis meSveobiT; funqciis meSveobiT, aseve SeiZleba aRiweros bunebis kanonebi. aqedan gamomdinare, maTematikuri analizi aris funqciis Seswavlis saSualeba. maTematikur analizs safuZveli Cauyares da ganaviTares iseTma cnobilma bunebismetyvelebma, rogorebic iyvnen niutoni, laibnici, ferma, eileri, abeli da sxva. maTematikuri analizis mniSvnelovan nawilebs warmoadgens integraluri aRricxva da mwkrivTa Teoria. swored am nawilebis da maTi praqtikuli gamoyenebis elementebia Setanili winamdebare saxelmZRvaneloSi. saxelmZRvaneloSi Setanilia agreTve albaTobis Teoriis da maTematikuri statistikis elementebi. albaTobis Teoria, rogorc mecniereba aRmocenda meCvidmete saukunis Sua wlebSi. Tavdapirvelad igi dakavSirebuli iyo azartuli TamaSebis analizTan. am analizisas wamoWrilma problemebma da amocanebma, romlebic ver ixsnebod nen maSin arsebuli maTematikuri meTodebis saSualebiT, miiyvana mecnierebiaxali ideebis da cnebebis warmoqmnamde. aseTi ideebi da cnebebi gvxvdeba cnobili maTematikosebis fermas, paskalis, hiugensis, bernulis da sxvaTa naSromebSi. miuxedavad imisa, rom albaTobis Teoria ganviTarebis sam saukuneze mets iTvlis, maTematikis sruluflebiani dargis statusi moipova meoce saukunis 30-ian wlebSi, rusi maTematikosis a.n.kolmogorovis Sromebis gamoqveynebis Semdeg. man Camoayaliba albaTobis Teoriis aqsiomebi, romelTa safuZvelze albaTobis Teoria warmoadgens mkacr maTematikur disciplinas. albaTobis Teorias efuZneba maTematikuri statistika. garkveuli azriT, ro gorc vnaxavT, maTematikuri statistikis amocanebi warmoadgenen albaTobis Teoriis amocanebis Sebrunebul amocanebs. saxelmZRvanelos im nawilis, romelic moicavs integraluri aRricxvis da mwkrivTa Teoriis elementebs, avtorebi arian goris universitetis profesorebi: malxaz mumlaZe da soso cotniaSvili. im nawilis ki, romelic moicavs albaTobis Teoriis da maTematikuri statistikis elementebs, amave universitetis profesori karlo cercvaZe. avtorebi did madlobas uxdian profesor jemal qirias saxelmZRvanelos redaqtirebisTvis da Rirebuli SeniSvnebisaTvis. saxelmZRvanelo gankuTvnilia umaRlesi profesiuli skolis, aseve ekonomikuri da sainJinro specialobebis bakalavriatis studentebisaTvis.

7 Tavi1 gansazRvruli integrali 1. farTobis amocana vTqvaT, mocemulia sasrul [a, b] segmentze gansazRvruli y = f (x ) funqcia, f ( x ) ≥ 0 , yoveli x ∈ [ a, b] wertilisaTvis. figuras, romelic SemosazRvrulia zemodan y = f (x ) funqciis grafikiT, qvemodan OX ricxviT RerZze mdebare [a, b] monakveTiT, gverdebidan ki x = a da x = b wrfeebiT, mrudwirul trapecias uwodeben (nax.1). SeiZleba adgili hqondes tolobas f ( a ) = 0 an f ( a ) = 0 . aseT SemTTxvevebSi mrudwiruli trapeciis gverdi moiWimeba wertilSi. yoveli brtyeli figura SeiZleba davanawiloT ramdenime mrudwirul trapeciad, romelTa farTobebis jami toli iqneba mocemuli figuris farTobisa. Tu movaxerxebT mrudwiruli trapeciis farTobis gamoTvlas, maSin SevZlebT yovelgvari brtyeli figuris farTobis gamoTvlas. Y y = f (x)

a O

nax.1 X

mrudwiruli trapeciis D farTobis gamosaTvlelad [a, b] segmenti nebismierad davyoT n nawilad. vTqvaT, dayofis wertilebi ganlagebuli arian [a, b] segmentze aseTnairad: a = x0 < x1 < x 2 < ... < x k < x k +1 < ... < x n −1 < x n = b . dayofis wertilebidan x k , k = 1,2,3,..., n − 1 aRvmarToT marTobebi y = f ( x) funqciis grafikis gadakveTamde. TiToeuli am gadakveTis wertilis koordinatebi iqneba ( x k , f ( x k )) . avagoT figura Sedgenili marTkuTxedebisgan, romelTa fuZeebia monakveTebi [ x k , x k +1 ] , k = 0,1,2,..., n − 1 , xolo simaRleebi TiToeuli am monakveTis romelime ξ k ∈ [ x k , x k +1 ] wertilSi aRmarTuli marTobebi grafikis gadakveTamde. TiToeuli marTobis sigrZe f (ξ k ) ricxvis tolia. Y

y = f ( x)

. . .…

ξ2 O ξ0 X ξ3 ξ n −1 x1 x2 b = xn x0 = a x3 x n −1 nax.2 aRvniSnoT agebuli figuris (nax.2) farTobi simboloTi, maSin S n = f (ξ 0 )( x1 − x0 ) + f (ξ1 )( x 2 − x1 ) + ... + f (ξ n −1 )( x n − x n −1 ) anu

8 n −1

S n = ∑ f (ξ k )Δx k ,

(1)

k =0

sadac Δx k = x k +1 − x k warmoadgens [ x k , x k +1 ] monakveTis sigrZes. advilia mixvedra, rom mrudwirili trapeciis farTobi miaxloebiT tolia agebuli figuris farTobisa da SegviZlia davweroT: n −1

D ≈ S n = ∑ f (ξ k )Δx k .

(2)

k =0

Tu (1) gamosaxulebaSi TandaTan gavzrdiT [a, b] segmentis damyofi wertilebis ricxvs ise, rom dayofiT miRebuli monakveTebis max Δx k miiswrafvodes nulisaken, geometriuli maqsimalurisigrZe mosazrebiT, SeiZleba davaskvnaT, rom (2) miaxloebiTi tolobis sizuste TandaTan gaizrdeba da dayofis wertilebis usasrulo gazrdiT zusti gaxdeba. (1) jams [a, b] segmentze y = f ( x ) funqciis rimanis integraluri jami ewodeba. vTqvaT, y = f ( x ) funqcia gansazRvrulia [a, b] segmentze da Rebulobs rogorc arauaryofiT, aseve uaryofiT mniSvnelobebs. gansazRvreba 1.1. Tu [a, b] segmentze gansazRvrul y = f ( x) funqciis rimanis jamebis mimdevrobas: n −1

S n = ∑ f (ξ k )Δx k , n = 1,2,3,..., n,... k =0

gaaCnia sasruli zRvari, rodesac segmentis damyofi wertilebis raodenoba usasrulod izrdeba ise, rom dayofiT miRebuli monakveTebis maqsimaluri sigrZe miiswrafvis nulisken da es zRvari ar aris damokidebili ξ k ∈ [a, b] wertilis amorCevaze, maSin amboben, rom y = f ( x ) funqcia integrebadia [a, b] segme-ntze, xolo zRvars:

lim

n −1

∑ f (ξ

n →∞ max Δxk →0 k = 0

)Δx k

uwodeben y = f ( x) funqciis gansazRvrul integrals [a, b] segmentze. b

gansazRvrul integrals aRniSnaven ase:

∫ f ( x)dx . a

maSasadame b

∫

f ( x)dx =

lim

n −1

∑ f (ξ

n →∞ max Δxk →0 k = 0

)Δx k .

gansazRvruli integralis

∫ f ( x)dx

gamosaxulebaSi f ( x ) funqcias

integralqveSa funqcia ewodeba, x cvlads- saintegracio cvladi, a da b ricxvebs- Sesabamisad, integrirebis qveda da zeda sazRvrebi. SeniSvna: gansazRvrul integralTan dakavSirebiT unda aRvniSnoT, rom arseboben segmentze gansazRvruli wyvetadi funqciebi, [a, b] romlebisTvisac rimanis jamebis mimdevroba krebadia da, Sesabamisad, arsebobs maTi integralebi. aseve arsebobs funqciebi, romlebic ara integrebadia mocemul segmentze. adgili aqvs Teoremebs:

9 integrebadi y = f ( x ) funqcia

Teorema 1.1. yoveli [a, b] segmentze SemosazRvrulia am segmentze. Teorema 1.2. yoveli [a, b] segmentze uwyveti y = f ( x) funqcia integrebadia am segmentze.

2. gansazRvruli integralis Tvisebebi gansazRvruli integralis gamoTvla pirdapir, misi gansazRvrebidan Zalian Znelia martiv SemTxvaSic ki. praqtikaSi gamoiyeneba zogadi da ufro ioli meTodi, romelic gamomdinareobs gansazRvruli integralis Tvisebebidan, romlebsac moviyvanT damtkicebis gareSe. Tviseba 1. Tu f ( x ) da ϕ ( x ) integrebadi funqciebia [a, b] segmentze, maSin funqcia f ( x ) + ϕ ( x ) integrebadia am segmentze da adgili aqvs tolobas: b

∫ [ f ( x) ± ϕ ( x)]dx = ∫ f ( x)dx ± ∫ ϕ ( x)dx . Tviseba 2. Tu kf ( x ) , sadac k tolobas:

f ( x ) integrebadi funqciaa [a, b] segmentze, maSin funqcia raime ricxvia, integrebadia am segmentze da adgili aqvs b

∫ kf ( x)dx = k ∫ f ( x)dx . integralis gansazRvrisas vgulisxmobdiT, rom a < b . Tu a > b , maSin SeTanxmebis safuZvelze iTvleba, rom b

f ( x)dx = − ∫ f ( x)dx .

∫ a

aqedan gamomdinareobs toloba: a

∫ f ( x)dx = 0 . a

Tviseba 3. Tu f ( x ) integrebadi funqciaa [a, b] segmentze da integrirebis segmenti [a, b] gayofilia nawilebad [a, c ] , [c, b ] , maSin arseboben integralebi: c

∫

f ( x)dx ,

∫ f ( x)dx c

da adgili aqvs tolobas: b

∫ f ( x)dx = ∫ f ( x)dx + ∫ f ( x)dx . Tviseba 4. Tu f ( x ) integrebadi funqciaa [a, b] segmentze segmentze f ( x ) ≥ 0 , maSin

da am

∫ f ( x)dx ≥ 0 . a

ukanaskneli Tvisebidan gamomdinareobs, rom Tu funqcia inarCunebs niSans b

∫ f ( x)dx = 0 , a

[a, b] segmentze f ( x )

mxolod maSin, rodesac f ( x ) = 0 . am Tvisebidan aseve gamomdinareobs: Tu f ( x ) ≤ ϕ ( x ) , maSin b

∫ a

f ( x)dx ≤ ∫ ϕ ( x)dx . a

Tviseba 5. Tu f ( x ) integrebadi funqciaa [a, b] segmentze da funqciis mniSvnelobebi am segmentze moTavsebulia m da M ricxvebs Soris, maSin b

m(b − a ) ≤ ∫ f ( x)dx ≤ M (b − a) . a

ukanaskneli Tviseba niSnavs, rom f ( x ) funqciis grafikiT gansazRvruli mrudwiruli trapeciis farTobi moTavsebulia fuZisa da, [a, b] Sesabamisad, m da M simaRleebis mqone marTkuTxedebis farTobebs Soris. es Tviseba SeiZleba gamoyenebuli iqnas integralis miaxloebiTi gamoTvlis dros. Tviseba 6. Tu f ( x ) funqcia uwyvetia [a, b] segmentze, maSin arsebobs iseTi c ∈ [ a, b] wertili, rom adgili aqvs tolobas: b

∫ f ( x)dx = f (c)(b − a) . a

3. integraluri aRricxvis ZiriTadi formula gansazRvruli integralis mniSvneloba ar aris damokidebuli imaze Tu ra simboloTi iqneba aRniSnuli saintegracio cvladi. ase, rom b

∫ f ( x)dx = ∫ f )(t )dt = ∫ f ( z )dz = ... ganvixiloT axla integrali, romlis integrirebis cvladi x sididea. aseTi integrali x cvladis funqciaa:

zeda

sazRvari

I ( x) = ∫ f (t )dt . a

am funqcias integrali cvladi zeda sazRvriT ewodeba. Teorema 1.3. integrali zeda sazRvriT, x

I ( x) = ∫ f (t )dt , a

x cvladis diferencirebadi funqciaa [a, b] segmentze da adgili aqvs tolobas: x ⎤ dI ( x) d ⎡ = ⎢ ∫ f (t )dt ⎥ = f ( x) . dx dx ⎣ a ⎦ es Tviseba gviCvenebs, rom integrali zeda sazRvriT warmoadgens funqciis erT-erT pirvelyofil funqcias: ∫ f ( x)dx = I ( x) + C , sadac C nebismieri mudmivia. radgan x

I ( x) = ∫ f (t )dt , a

f ( x)

11 f ( x ) funqciis erT-erT pirvelyofili funqciaa, amitom x

∫ f (t )dt = F ( x) + C

∗

sadac F ( x ) warmoadgens f ( x ) funqciis romelime pirvelyofil funqcias, C ∗ -konkretuli ricxvia. vipovoT C ∗ . davuSvaT x = a , maSin a

0 = ∫ f (t )dt = F (a) + C ∗ , a

aqedan C = − F (a ) da x

∫ f (t )dt = F ( x) − F (a) . a

Tu axla x = b , miviRebT: x

∫ f (t )dt = F (b) − F (a) .

(3)

es formula warmoadgens integraluri aRricxvis ZiriTad formulas, meornairad mas niuton- laibnicis formulasac uwodeben. sxvaoba F (b) − F ( a ) aRiniSneba ase: F ( x ) |ba . sabolood gveqneba: b

F ( x) |ba = ∫ f ( x)dx , a

sadac F ′( x ) = f ( x ) .

magaliTi. 1. gamovTvaloT integrali

∫x

dx .

amoxsna: 1

∫x

dx = 2 x |10 = 2 ⋅ 1 − 2 ⋅ 0 = 2 .

π 2

magaliTi 2. gamovTvaloT integrali

∫ sin xdx . 0

amoxsna: π 2

∫ sin xdx = − cos x |

π 2 0

= − cos

magaliTi 3. gamovTvaloT integrali

∫ 1

π 2

+ cos 0 = 0 + 1 = 1.

dx . x

amoxsna: 2

dx = ln | x | 12 = ln 2 − ln 1 = ln 2 − 0 = ln 2 . x 1 ganvixiloT gansazRvruli integralis gamoTvlis ramdenime meTodi: a) nawilobiTi integrebis meTodi. rogorc viciT, ganusazRvreli integralis gamosaTvlelad viyenebT nawilobiTi integrebis formulas

∫

∫ udv = uv − ∫ vdu ,

sadac u = u ( x ), v = v ( x ) .

∫ vdu

vTqvaT G ( x ) warmoadgens

simravlis erT-erT elements, maSin funqcia

uv − G ( x ) iqneba ∫ udv simravlis elementi. aqedan gamomdinare: b

∫ udv = uv |

b a

− G ( x) |ba ,

maSasadame b

b ∫ udv = uv |a − ∫ vdu . a

bolo tolobas integralisTvis.

(4)

ewodeba

nawilobiTi

integrebis

wesi

gansazRvruli

π 2

∫ x cos xdx .

magaliTi 4. gamovTvaloT integrali

amoxsna: π

∫ x cos xdx = ∫ xd sin x = x sin x |02 − ∫ sin xdx = =

π 2

+ cos

π 2

− cos 0 =

π 2

⋅ 1 − 0 ⋅ sin

π 2

+ cos x |02 =

− 1. e

∫ x ln xdx .

magaliTi 5. gamovTvaloT integrali

amoxsna:

1 x2 e x2 x2 x2 x2 x2 1 x2 e e e ln ln ln | ln ln | ln | |1 = x xdx = x d = x − d x = x − dx = x − 1 1 1 ∫1 ∫1 ∫1 2 ∫1 2 x 2 2 2 2 2 2 e

e2 e 2 12 e 2 e2 1 e2 + 1 12 1 + = ⋅1 − ⋅ 0 − + = ln e − ln 1 − . 2 2 4 4 2 2 4 4 4 π

∫e

magaliTi 6. gamovTvaloT integrali

cos xdx .

amoxsna: orjer gamoviyenoT nawilobiTi integrebis xerxi, gveqneba: π

∫e 0

sin xdx = − ∫ e d cos x = −e cos x 0 + ∫ cos xde = − e cos x 0 + ∫ e x cos xdx = x

2e π + ∫ e x d sin x = 2e x + e x sin x π0 − ∫ sin xde x = 2e x − ∫ e x sinxdx. 0

2 ∫ e x sin xdx = 2e x anu 0

∫e

sin xdx = e x .

b) Casmis meTodi. zogjer saintegracio cvladis gardaqmna aiolebs gansazRvruli integralis gamoTvlas. adgili aqvs Teoremas:

aqedan

13 b

Teorema 1.4. vTqvaT mocemulia

integrali

∫ f ( x)dx ,

sadac f ( x )

uwyveti funqciaa [a, b] segmentze. davuSvaT, x = ϕ (t ) raime diferencirebadi funqciaa [α , β ] segmentze, amasTan, ϕ (α ) = a, ϕ ( β ) = b da ϕ (t ) ∈ [ a, b] , rodesac t ∈ [α , β ] . maSin adgili aqvs tolobas: β

∫ f ( x)dx = α∫ f [ϕ (t )]ϕ ′(t )dt .

(5)

am Teoremis safuZvelze Cven SegviZlia gardavqmnaT saintegracio cvladi, gamovTvaloT gansazRvruli integrali integrirebis axal sazRvrebSi. 3

∫

magaliTi 7. gamovTvaloT integrali

3 2 − x 2 dx .

amoxsna: 3

∫

3 2 − x 2 dx

integralis gamosaTvlelad movaxdinoT cvladis gardaqmna.

davuSv- aT, x = 3 sin t . rodesac x Rebulobs mniSvnelobas 0 -dan 3-mde, maSin π t icvleba [0, ] SualedSi. (5) formulis gamoyeneba gvaZlevs: 2 3

∫ 0

3 − x dx = ∫ 3 − 3 sin t 3 cos tdt = 3 2

∫

1 − sin t cos tdt = 3 2

∫ cos

tdt =

2 1 + cos 2t 32 2 32 2 2 32 π 2 = 32 ∫ dt = dt + cos 2 tdt = t | + cos 2 tdt = + ∫ cos 2tdt. 0 ∫ ∫ ∫ 2 2 2 4 0 0 0 0 0 bolo integralSi aseve movaxdinoT cvladis gardaqmna: y = 2t , maSin y cvladi miiRebs mniSvnelobas 0 − dan π -mde da sabolood miviRebT: 2

∫

3 2 − x 2 dx =

32 1 32 π 1 32 π + ∫ cos ydy = + sin y |π0 = . 4 20 4 2 4 1

magaliTi 8. gamovTvaloT integrali

∫1+

amoxsna: gamoviyenoT cvladis gardaqmna, x = t 2 , gveqneba: dx = 2dt , t icvleba [0,1] SualedSi, maSasadame: 1

gardaqmnis

dros

2tdt . x 0 1+ t 0 kidev gamoviyenoT cvladis gardaqmna, 1 + t = z , gveqneba: dt = dz , z icvleba [1,2] SualedSi, maSasadame dx

∫1+

=∫

tdt z −1 dz 2 2 ∫0 1 + x = 2∫0 1 + t = 2∫1 z dz = 2∫1 dz − 2∫1 z = 2 z 1 −2 ln z 1 = 4 − 2 − 2 ln 2 = 2 − 2 ln 2 . dx

14 π 4

dx . sin 2 x 0 1 amoxsna: movaxdinoT cvladis gardaqmna, tgx = t , gveqneba dx = dt da t cos 2 x icvleba [0,1] SualedSi. e.i. magaliTi 9. gamovTvaloT

∫1+ a

integrali

π 1

dx cos 2 xdt dt dt dt = ∫0 1 + a 2 sin 2 x ∫0 1 + a 2 sin 2 x = ∫0 1 + tg 2 x + a 2 tg 2 x = ∫0 1 + t 2 + a 2 t 2 = ∫0 1 + (a 2 + 1)t 2 . 4

Tu axla gamoviyenebT cvladis gardaqmnas z = t a 2 + 1 , gveqneba da z icvleba [0, a 2 + 1] SualedSi.

dz a2 +1

= dt

sabolood miviRebT:

π 4

dx ∫0 1 + a 2 sin 2 x =

a 2 +1

1 a +1 2

∫ 0

dz = 1+ z2

a +1 2

arctgz

a 2 +1

a +1 2

arctg a 2 + 1 . a

magaliTi 10. ganvixiloT integrali simetriul [ − a, a ] segmentze: ∫ f ( x)dx , −a

sadac integralqveSa funqcia luwia anu f ( x) = f (− x) . amoxsna: 0

∫

f ( x)dx =

−a

∫

−a

f ( x)dx + ∫ f ( x)dx . 0

pirvel SesakrebSi movaxdinoT cvladis gardaqmna x = −t , gveqneba dx = −dt , integrirebis sazRvarebi gaxdeba Sesabamisad 0 da a . gveqneba: a

∫

−a

f ( x)dx = − ∫ f (−t )dt + ∫ f ( x)dx = ∫ f (t )dt + ∫ f ( x)dx = 2 ∫ f ( x)dx . a

magaliTi 11. ganvixiloT integrali

∫ f ( x)dx ,

simetriul [ − a, a ] sgmentze,

−a

sadac integralqveSa funqcia kentia anu f ( x ) = − f ( − x ) . amoxsna: Tu igive msjelobas gavimeorebT miviRebT: a

−a

∫ f ( x)dx = − ∫ f ( x)dx + ∫ f ( x)dx = 0 .

4. arasakuTrivi integralebi gansazRvruli integralis cnebis Semotanisas vgulisxmobdiT, rom integralqveSa funqcia gansazRvruli iyo integrirebis [a, b] sasrul segmentze. zogjer saWiroa uari vTqvaT am SezRudvebze da ganvixiloT funqciis integralebi usasrulo Sualedze an integralebi sasrul Sualedze, SemousazRvreli funqciis SemTxvevaSi. am SemTxvevebSi ver visargeblebT integralis 1-li gansazRvrebiT. moviyvanoT integralis gansazRvreba aseTi SemTxvevebisTvis.

15 gansazRvreba 1.2. Tu y = f ( x ) integrebadia nebismier [a, b] segmentze, sadac b ∈ [ a, ∞ ) da arsebobs zRvari: b

lim ∫ f ( x)dx ,

b →∞

maSin mas uwodeben f ( x ) funqciis arasakuTriv integrals usasrulo ∞

Sualedze da aRniSnaven ase:

∫ f ( x)dx . a

analogiurad ganisazRvreba arasakuTrivi integralebi: b

∫

−∞

∞

−∞

f ( x)dx = lim ∫ f ( x)dx , a →∞

∫

f ( x)dx = lim ∫ f ( x)dx . a →∞ b →b a

∞

∫x

magaliTi 1. gamovTvaloT arasakuTrivi integrali

,a > 0.

amoxsna: ∞

1 1 1 1 dx dx = lim[− ba ] = lim[ − ] = . 2 ∫a x 2 = lim ∫ b →∞ x b →∞ b →∞ a x b a a ∞

8a 2 dx . magaliTi 2. gamovTvaloT arasakuTrivi integrali ∫ 2 2 0 x + 4a amoxsna: b ∞ 8a 2 8a 2 1 x b dx = lim dx = 8a 2 lim[ arctg 0 = ∫0 x 2 + 4a 2 b →∞ ∫ x 2 + 4a 2 b →∞ 2a 2a 0 = 4a 2 lim[arctg b →∞

b π − arctg 0] = 4a 2 = 2πa 2 . 2a 2 ∞

magaliTi 3. gamovTvaloT arasakuTrivi integrali

∫ 1

dx . x

amoxsna: ∞

dx dx = lim[ln x 1b ] = lim[ln b − 0] = lim ln b = ∞ . ∫1 x = blim ∫ →∞ b →∞ b →∞ b →∞ x 1 es ki niSnavs, rom arasakuTrivi integrali ar arsebobs. gansazRvreba 1.3. vTqvaT, y = f ( x) funqcia integrebadia [a, b] segmentis SigniT, anu integrebadia nebismier [a, b − ε ] segmentze, sadac 0 < ε ≤ b − a , rodesac x → b , amasTan, arsebobs sasruli zRvari magram f ( x ) → ∞ , b −ε

lim ε →0

∫ f ( x)dx ,

maSin

zRvars

ewodeba

arasakuTrivi

integrali

SemousazRvreli funqciidan da aRiniSneba ase:

∫ f ( x)dx . a

analogiurad ganisazRvreba arasakuTrivi integrali, Tu f ( x ) → ∞ , rodesac x → a : b

∫ a

f ( x)dx = lim ε →0

∫ε f ( x)dx .

16 a

∫

magaliTi 4. gamovTvaloT arasakuTrivi integrali

a2 − x2

amoxsna: a

∫ 0

a −ε

dx a2 − x2

= lim ε →0

∫

= lim[arcsin ε →0

a2 − x2

x a

a −ε 0

] = lim arcsin ε →0

a −ε π = . a 2

dx . x 0

∫

magaliTi 5. gamovTvaloT arasakuTrivi integrali amoxsna: 1

dx dx = lim[ln x 1ε ] = lim[ln 1 − ln ε ] = ∞ . ∫0 x = lim ∫ ε →0 ε →0 ε →0 ε x maSasadame arasakuTrivi integrali ar arsebobs. vTqvaT, Tu f ( x ) → ∞ , rodesac x → c , sadac c ∈ (a, b) . Tu erTdroulad arsebob en zRvrebi: c −ε

lim ε →0

∫

f ( x)dx da lim δ →0

∫δ f ( x)dx ,

sadac ε > 0, δ > 0 , maSin arasakuTrivi integrali

f ( x ) SemousazRvreli funqciidan [a, b]

segmentze ewodeba am zRvarTa

jams da aRiniSneba ase: ∫ f ( x)dx . maSasadame a

c −ε

∫

f ( x)dx = lim ε →0

∫

f ( x)dx + lim δ →0

∫δ f ( x)dx .

magaliTi 6. gamovTvaloT arasakuTrivi integrali

∫

dx 3

8 0 +δ

−1

amoxsna: 8

∫

−1

dx 3

0 −ε

= lim ε →ε

∫

−1

dx 3

+ lim δ →0

∫δ

3 = lim x 3 →0 2 ε x

dx 3

0 −ε −1

3 + lim x 2 δ →0 2 2

3 3 3 3 3 9 = lim( (0 − ε ) 3 − (−1) 3 ) + lim( 8 3 − (0 + δ ) 3 ) = − + 6 = . ε →0 2 δ →0 2 2 2 2 2 a dx magaliTi. 7. gamovTvaloT arasakuTrivi integrali ∫ 2 . −a x amoxsna: a

0 −ε

dx dx dx 1 + lim ∫ 2 = lim[− 2 ∫−a x 2 = lim ∫ b →0 ε →0 δ →0 x 0 +ε x −a x

0 −ε −a

] + lim[− δ →0

1 x

a 0 +δ

1 1 1 1 = lim[ − ] + lim[ + ] = ∞. ε →0 ε a δ →0 − a δ

5. gansazRvruli integralis gamoyeneba 5.1. farTobis gamoTvla sxvadasxva geometriuli figurisaTvis gansazRvruli integrali [.a, b] rogorc zemoT aRvniSneT, y = f ( x ) segmentze warmoadgens im figuris farTobs, romelic SemosazRvrulia zemodan f ( x ) funqciis grafikiT, qvemodan OX RerZiT, gverdebidan x = a da x = b wrfeebiT(nax.3). magram zemoT vgulisxmobdiT, rom am SemTxvevaSi adgili hqonda pirobas f ( x ) ≥ 0 . Tu f ( x ) funqciis grafiki gadahkveTs 0 X RerZs an mTlianad mdebareobs am RerZis qvemoT anu [.a, b] segmentis

17 farglebSi f ( x ) ≤ 0 , am dros saqme gveqneba figurasTan, romlis nawili zemodan iqneba SemosazRvruli funqciis grafikiT, nawili- qvemodan. aseT dros rimanis jamis zogierTi an yvela Sesakrebi iqneba uaryofiTi. magram farTobi dadebiTi sididea, amitom aRwerili figuris, romelsac aseve mrudwiruli trapecia ewodeba, farTobi D SeiZleba gamovsaxoT formuliT b

D = ∫ | f ( x) | dx .

(1)

y1 = f1 ( x) D y 2 = f 2 ( x)

y = f ( x)

nax. 3 nax. 4 axla ganvixiloT figura, romelic SemosazRvrulia x = a da x = b y1 = f1 ( x) funqciis grafikiT, qvemodan y 2 = f 2 ( x) wrfeebiT, zemodan funqciis grafikiT. aseTi figura gamosaxulia naxazze (nax.4) misi farTobi warmoadgens ori mrudwiruli trapeciis farTobTa sxvaobas da gamoisaxeba ase: b

D = ∫ | f1 ( x) − f 2 ( x) | dx .

(2)

analogiurad gamoisaxeba im figuris farTobi, romelic SemosazRvrulia mxolod ori wiriT, romlebic y1 = f1 ( x) da y 2 = f 2 ( x) funqciebis grafikebs warmoadgenen (nax.5).

y 2 = f 2 ( x)

y1 = f 2 ( x)

b X

nax.5

magaliTi 1. gmovTvaloT y = sin x , 0 ≤ x ≤ 2π funqciis grafikiT da OX RerZiT Se mosazRvruli figuris farTobi. amoxsna: rodesac 0 ≤ x ≤ π , funqcia sin x ≥ 0 , rodesac π ≤ x ≤ 2π , funqcia sin x ≤ 0 , amitom

18 2π

2π

D = ∫ | sin x | dx = ∫ sin xdx + ∫ − sin xdx = − cos x π0 + cos x π2π = 1 + 1 + 1 + 1 = 4 . magaliTi 2. gamovTvaloT figuris farTobi, romelic SemosazRvrulia parabolebiT: y1 = x 2 da y 2 = x . amoxsna: vipovoT am parabolaTa urTierTgadakveTis wertilebi. amisaTvis amovxsnaT gantoleba x 2 = x . miviRebT x1 = 0, x 2 = 1 . figura, romlis farTobsac veZebT mocemulia naxazze (nax.6).

y=x

−a

X III

O nax.6 misi farTobi iqneba

−b

X nax.7 2

2 1 2 1 1 D = ∫ ( x − x 2 )dx = x 3 − x 3 10 = − = . 3 3 3 3 3 0 vTqvaT mrudwiruli trapeciis zemodan SemomsazRvreli wiri mocemulia parametruli formiT: x = x (t ), y = y (t ), sadac t 0 ≤ t ≤ T . maSin (1) formula x cvladis x = x (t ) gardaqmnis Semdeg miiRebs saxes: T

D = ∫ y (t ) x ′(t )dt . t0

magaliTi

vipovoT

x y + 2 =1 2 a b

elifsiT

SemosazRvruli

figuris

farTobi. amoxsna: CavweroT elifsis gantoleba parametrulad, gveqneba: 0 ≤ α ≤ 2t . sadac radganac elifsi simetriuli x = a cos t , y = b sin t , figuraa(nax.7), jer gamovTvaloT elifsis im nawiliT SemosazRvruli figuris farTobi, romelic imyofeba sakoordinato sibrtyis I meoTxedSii 0

1 1 − cos 2t D = ∫ ydx = ∫ b sin t (− a sin t )dt = − ab ∫ sin 2 tdt = −ab ∫ dt = 4 2 π π π π 2

2 0

= −ab

1 − ab ab ab cos 2tdt = dt + t + sin 2t ∫ ∫ 2π 2 π 2 4 2

π 2

πab 4

sabolood elifsiT SemosazRvruli figuris farTobi D = πab .

19 magaliTi 4. ganvixiloT figura, romelic SemosazRvrulia erTi wertilidan gamosuli ori sxiviT da am sxivebis TiTojer gadamkveTi wiriT. aseT figuras mrudwiruli seqtori ewodeba (nax.8 a). amoxsna: vTqvaT, sxivebi, romlebic SemosazRvraven mrudwirul seqtors gamodian koordinatTa saTavidan da Seadgenen OX RerZTan, Sesabamisad, kuTxeebs: α da β , α < β . gamoviyenoT koordinatTa polaruli sistema da seqtoris SemomsazRvreli wiris gantoleba gamovsaxoT polarul koordinatebSi: r = r (ϕ ), α ≤ ϕ ≤ β .

….….

rn −1

ϕ n −1

ϕk X

β O

ϕ1

nax.8.a

nax.8.b

gamovTvaloT miaxloebiT seqtoris farTobi. amisaTvis davyoT seqtori n nawilad sxivebiT. vTqvaT, ϕ k warmoadgens kuTxes k nomris sxivsa da OX RerZs Soris, xolo Δϕ k = ϕ k +1 − ϕ k , k = 0,1,2,..., n − 1, ϕ 0 = α , ϕ n = β . amgvarad, miviRebT mcire mrudwirul seqtorebs, romlebic SeiZleba CavTvaloT wriul seqtorebad. TiToeuli am wriuli seqtoris farTobi tolia: 1 2 1 rk Δϕ k = [r (ϕ k )] 2 Δϕ k . 2 2 aqedan gamomdinare, mTeli mrudwiruli seqtoris farTobi miaxloebiT toli iqneba sididis: n −1 1 D = ∑ [r (ϕ k )] 2 Δϕ k , k =0 2 1 romelic warmoadgens rimanis jams r (ϕ ) funqciisaTvis [α , β ] segmentze. 2 cxadia, rodesac n → ∞ , ukanaskneli tolobis sizuste izrdeba, amitom sabolood gveqneba: β

1 D = ∫ r 2 (ϕ )dϕ . (3) 2α magaliTi 5. vipovoT figuris farTobi, romelic SemosazRvrulia kardioidiT anu wiriT, romlis gantoleba polarul koordinatebSi aris: r = a (1 + cos ϕ ) (nax. 8 b). amoxsna: aseTi farTobis gamosaTvlelad unda gamoviyenoT (3)

20 formula, gveqneba: 2π

1 a 2 (1 + cos ϕ ) 2 dϕ = 2 ∫0 2π

2π

1 2 1 a ( ∫ dϕ + 2 ∫ cos ϕdϕ + ∫ cos 2 ϕdϕ ) = πa 2 + 0 + a 2 ∫ (1 + cos 2ϕ )dϕ = 2 2 0 0 0 0

= πa 2 +

2π

1 1 3 1 2πa 2 + ∫ cos 2ϕd 2ϕ = πa 2 + sin 2ϕ 4 4 0 2 4

2π 0

3 = πa 2 . 2

5.2 wiris rkalis sigrZis gamoTvla vTqvaT, wiri mocemulia gantolebiT y = f ( x) , sadac f ( x ) funqcia gansazRvrulia da warmoebadia [a, b] segmentze. davyoT [a, b] segmenti n nawilad wertilebiT: a = x0 , x1 , x 2 ,...x k ,..., x n = b . aRvniSnoT x k abscisis Sesabamisi wertili wirze M k -iT, M k ( x k , f ( x k )) (nax.9). Y

M3 M n −1 M n … ...

O x0 = a x1

x3 x n −1 x n = b X

nax.9

amgvarad, avageT texili M 0 M 1 ...M n . misi TiToeuli rgolis sigrZe

Δx k = x k +1 − x k , Δy k = f ( x k +1 ) − f ( x k ) .

gveqneba: 2

| M k M k +1 |= Δx k + Δy k . aqedan

| M k M k +1 |= 1 + (

Δy k 2 ) Δx k . Δx k

Tu gamoviyenebT lagranJis Teoremas, sasruli nazrdis Sesaxeb [ x k , x k +1 ] monakveTze, gveqneba: Δy k = f ′(ξ n )Δx k , sadac x k ≤ ξ k ≤ x k + Δx k = x n +1 . aqedan vRebulobT Semdeg gamosaxulebas:

| M k M k +1 |= 1 + [ f ′(ξ k )]2 Δx k . mTeli texilis sigrZe n −1

n −1

k =0

l n = ∑ | M k M k +1 |= ∑ 1 + [ f (ξ k )] 2 Δx k .

(1)

(1) formulis marjvena mxare warmoadgens rimanis integralur jams 1 + [ f ′( x)] 2 funqciisTvis [a, b] segmentze.

Tu n → ∞ da Δx k → 0 , rodesac k = 0,1,2,..., n − 1 . maSin ricxvs L = lim l n , n →∞ Δx k → 0

ewodeba y = f ( x ) gantolebiT gansazRvruli wiris M 0 M n rkalis sigrZe. Tu (1) tolobaSi gadavalT zRvarze miviRebT: b

L = ∫ 1 + [ f ′( x)]2 dx .

(2)

Tu wiri mocemulia parametruli gantolebebiT x = x (t ), y = y (t ) , sadac ( x(t 0 ), y (t 0 )) koordinatebis mqone wertili Seesabameba M 0 wertils, ( x(T ), y (T )) koordinatebis mqone wertili ki M n wertils, aseT SemTxvevaSi rkalis sigrZe T

L = ∫ [ x ′(t )]2 + [ y ′(t )]2 dt .

(3)

Tu wiri mocemulia polarul koordinatebSi, gantolebiT r = r (ϕ ) , sadac ϕ = α Seesabameba M 0 wertils, ϕ = β ki Seesabameba M n wertils,MmaSin Tu t parametris nacvlad gamoviyenebT ϕ parametrs, gveqneba: x(ϕ ) = r (ϕ ) cos ϕ , y (ϕ ) = r (ϕ ) sin ϕ . aqedan [ x ′(ϕ )]2 + [ y ′(ϕ )]2 = [r ′(ϕ ) cos ϕ ] 2 − 2r ′(ϕ ) cos ϕ ⋅ r (ϕ ) sin ϕ + [r (ϕ ) sin ϕ ] 2 =

= [r ′(ϕ ) sin ϕ ] 2 + 2r ′(ϕ ) sin ϕ ⋅ r (ϕ ) cos ϕ + [r (ϕ ) cos ϕ ] 2 = [r (ϕ )]2 + [r ′(ϕ )]2 . maSasadame β

L = ∫ [r (ϕ )]2 + [r ′(ϕ )]2 dϕ .

(4)

magaliTi 1. vipovoT wiris sigrZe, romelic mocemulia gantolebiT 3 2

y = x , Tu x icvleba koordinatTa saTavidan x = 5 wertilamde. amoxsna: y ′ =

9 1 3 2 4 + 9 x . (3) formulidan gveqneba: x , maSin 1 + [ y ′]2 = 1 + x = 2 4 2 5

1 121 1 335 . (4 + 9 x) 2 50 = (49 2 − 4 2 ) = L = ∫ 4 + 9 x dx = 20 239 27 27

22 magaliTi 2. vipovoT avtomobilis a radiusis mqone borblis raime wertilis mier, borblis erTjer Semotrialebisas, Semowerili rkalis sigrZe. amoxsna: borblis trialis dros misi raime wertili moZraobs wirze, romelsac cikloida ewodeba(nax.10). misi parametruli gantolebaa: x = a (t − sin t ), y = a (1 − cos t ), sadac t parametri icvleba [0.2π ] SualedSi. Y

πa

2π a

nax. 10

x ′(t ) = a (1 − cos t ), y ′ = a ( − sin t ) , amitom saZiebeli rkalis sigrZe tolia 2π

∫

[ x ′(t )]2 + [ y ′(t )]2 dt =

2π

∫

a 2 (1 − cos t ) 2 + a 2 (− sin t ) 2 dt =

. t t 2π = 2a ∫ sin dt = 2a(−2 cos ) 0 = 8a 2 2 0 magaliTi 3. vipovoT kardioidis r = a (1 + cos ϕ ),0 ≤ ϕ ≤ 2π rkalis sigrZe(nax.8 b) amoxsna: gamoviyenoT (4) formula 2π

2π

∫

a 2 (1 + cos ϕ ) 2 + a 2 (− sin ϕ ) 2 dϕ =

∫

a 2 (1 + 2 cos ϕ + cos 2 ϕ + sin 2 ϕ 2 dϕ =

2π

= a ∫ 2(1 + cos ϕ ) dϕ = a ∫ 4 | cos 0

4a sin

2ϕ

ϕ 2

π 0

−4a sin

ϕ 2

2π

ϕ 2

= 4 ∫ cos 0

ϕ ϕ 2

2π

+ 4 ∫ − cos π

ϕ ϕ 2

=8a.

5.3 brunviTi sxeulis moculoba vTqvaT, wiri mocemuli gantolebiT y = f ( x ) , sadac x cvladi Rebulobs mniSvnelobebs [a, b] segmentidan, brunavs OX RerZis garSemo. gamovTvaloT moculoba sxeulisa, romelic SemosazRvrulia wiris brunviT miRebuli zedapirisa da x = a, y = b sibrtyeebiT. davyoT [a, b] segmenti n nawilad wertilebiT: a = x0 , x1 , x 2 ,...x k ,..., x n = b . am abscisebis Sesabamisi ordinatebi aRvniSnoT y k , y k = f ( x k ) . OX RerZis dayofis yovel wertilSi gavavloT am RerZis perpendikularuli sibrtyeebi. es sibrtyeebi dayofen sxeuls fenebad(nax.11). TiToeuli es fena SeiZleba miaxloebiT CavTvaloT wriul cilindrad. am cilindris simaRlea: Δx k = x k +1 − x k , xolo fuZeTa radiuisi- y k . cilindris moculoba iqneba: Δv k = πy k2 Δx k .

23 Z

y k +1

x k +1

nax.11

mTeli sxeulis V moculobisaTvis ki adgili eqneba mixloebiT tolobas: n −1

V ≈ ∑ πy k2 Δx k . 0

bunebrivia sxeulis moculoba iqneba ukanaskneli miaxloebiTi tolobis marjvena mxaris zRvari, rodesac dayofis wertilTa raodenoba miiswrafvis usasrulobisken ise, rom dayofiT miRebuli monakveTebis maqsimaluri sigrZe Δx k miswrafvis nulisaken. maSasadame

V = lim

n −1

∑ πy

n →∞ Δxk →0 0

2 k

Δx k .

aqedan b

V = π ∫ y dx anu V = π ∫ [ f ( x)]2 dx . 2

(1)

Tu wiri mocemulia parametruli gantolebebiT x = x (t ), y = y (t ) , sadac t0 ≤ t ≤ T . aseT SemTxvevaSi T

V = π ∫ [ y (t )]2 x ′(t )dt .

(2)

x2 y2 + = 1 elifsis OX RerZis garSemo brunviT a2 b2 miRebuli elifsoidiT SemosazRvruli sxeulis moculoba. b2 amoxsna: elifsis gantolebidan y 2 = 2 (a 2 − x 2 ) , rodesac − a ≤ x ≤ a . (1) a formuliT miviRebT: magaliTi 1. vipovoT

b2 2 πb 2 πb 2 1 3 2 2 2 2 − = − = π − ( a x ) dx ( a dx x dx ) 2 a b x 2 2 2 ∫ ∫ 3 a a a −a −a −a a

V =π ∫

a −a

4 = πab 2 . 3

magaliTi 2. vipovoT igive elifsis mier OY RerZis garSemo brunviT miRebuli elifsoidiT SemosazRvruli sxeulis moculoba.

24 gantolebidan x 2 =

amoxsna: aseT SemTxvevaSi, elifsis

rodesac − b ≤ x ≤ b . sxeulis moculoba iqneba:

a2 2 (b − y 2 ) , 2 b

a2 2 4 (b − x 2 )dy = πa 2b . 2 b 3 −b b

V =π ∫

magaliTi 3. vipovoT cikloidis pirveli TaRis OY RerZis garSemo brunviT miRebuli brunviTi zedapiriT SemosazRvruli sxeulis moculoba. amoxsna: cikloidis parametruli gantolebaa x = a (t − sin t ), y = a (1 − cos t ). cikloidis pirveli TaRi aRiwereba t parametris cvlilebiT 0 -dan 2π mde. amitom saZiebeli moculoba, (2) formulis Tanaxmad, tolia 2π

2π

V = π ∫ a 2 (1 − cos t ) 2 a (1 − cos t )dt = πa 3 ∫ (1 − cos t ) 3 dt = πa 3 [ ∫ (1 − 3 cos t )dt + 0

2π

∫ 2 (1 + cos 2t )dt − ∫ (1 − sin 0

3 t )d sin t ] = πa 3 [t − 3 sin t + t + 2

1 3 + sin 2t − sin t + sin 3 t ] 02π = 5π 2 a 3 . 3 4

magaliTi 4. vipovoT kardioidis polaruli RerZis garSemo brunviT miRebuli sxeulis moculoba(nax.8 a). amoxsna: kardioidis gantoleba polarul koordinatebSi aris: r = a (1 + cos ϕ ) . misi parametruli gantoleba iqneba:

⎧ x = a(1 + cos ϕ ) cos ϕ , ⎨ ⎩ y = a (1 + cos ϕ ) sin ϕ . gamoviyenoT (2) formula, gveqneba: 2π

V = π ∫ {a 2 (1 + cos ϕ ) 2 sin 2 ϕ[a(− sin ϕ ) cos ϕ − a(1 + cos ϕ ) sin ϕ ]}dϕ = 0

= −πa

2π 3

∫ (1 + cos ϕ ) 0

sin ϕdϕ − πa 3

2π 3

∫ (1 + cos ϕ )

sin 2 ϕ sin ϕ cos ϕdϕ =

2π

= πa 3 ∫ (1 + cos ϕ ) 4 (1 − cos ϕ )d cos ϕ + 0

+ πa

2π 3

∫ (1 + cos ϕ ) 0

2π

(1 − cos ϕ ) cos ϕd cos ϕ = πa

2π 3

∫ (1 + cos ϕ )

(2 − (1 + cos ϕ ))d (1 + cos ϕ ) +

+ πa 3 ∫ (1 + cos ϕ ) 3 (2 − (1 + cos ϕ ))(−1 + (1 + cos ϕ ))d (1 + cos ϕ ). 0

25 movaxdinoT cvladis gardaqmna t = 1 + cos ϕ da gaviTvaliswinoT, rom cvladi Rebulobs am gardaqmnis dros, rodesac ϕ cvladi izrdeba t mniSvnelobas 2-dan 0-mde. amitom, Tu gaviTvaliswinebT, rom (2) formulaSi t parametri icvleba t 0 ≤ t ≤ T SualedSi, gveqneba: 0

V = −πa 3 ∫ t 4 (2 − t )dt − πa 3 ∫ t 3 (2 − t )(−1 + t )dt = −πa 3 ( ∫ 2t 4 dt − ∫ t 5 dt − 2

− ∫ 2t 3 dt + ∫ 2t 4 dt + ∫ t 4 dt − ∫ t 5 dt ) = 2

− πa 3 (2

t 5

0 2

= −πa 3 (−2 ⋅

−

t 6

0 2

−2

t 4

0 2

t5 5

0 2

t5 5

0 2

−

t6 0 2) = 6

32 64 16 32 32 64 8 + + 2⋅ − 2⋅ − + ) = πa 3 . 5 6 4 5 5 6 3

5.4. brunviTi zedapiris farTobi vTqvaT, wiri mocemuli gantolebiT y = f ( x ) , sadac x cvladi Rebulobs mniSv- nelobebs [a, b] segmentidan da f ( x ) uwyvetad warmoebadia, brunavs OX RerZis garSemo. gamovTvaloT wiris brunviT miRebuli zedapiris farTobi, romelic SemosazRvrulia x = a, y = b sibrtyeebiT. davyoT [a, b] segmenti n nawilad wertilebiT: a = x 0 , x1 , x 2 ,...x k ,..., x n = b . am abscisebis Sesabamisi ordinatebi aRvniSnoT y k , y k = f ( x k ) , xolo ( x k , y k ) koordinatebis mqone wertili M k -iT i = 0,1,2,..., n . Tu yovel rkals M k M k +1 SevcvliT M k M k +1 monakveTiT maSin M 0 M 1 M 2 ...M n texilis brunviT miviRebT zedapirs, romelic Sedgeba wakveTili konusebis gverdiTi zedapirebisgan(nax.12). es gverdiTi zedapirebi gamoiTvleba formuliT: y + y k +1 2π k | M k M k +1 | , 2 sadac | M k M k +1 | texilis erTi M k M k +1 monakveTis sigrZea, magram 5..2 qveTavidan viciT, rom adgili aqvs tolobas:

| M k M k +1 |= 1 + [ f ′( x k )]Δxk = 1 + [ y k′ ]2 Δxk

M0 yk

M k M k +1 y k +1 xk x k +1

nax.12

26 y k +1 = y k + Δy k , mgram f ( x ) uwyvetia, amitom, rodesac Δx k → 0 aseve gvaqvs, rom Δy k → 0 . aqedan gamomdinare, rodesac Δx k sakmaod mcirea, SegviZlia CavTvaloT, rom y k ≈ y k +1 . am faqtis gaTvaliswinebiT:

y k + y k +1 | M k M k +1 |≈ 2πy k 1 + [ y k′ ]2 Δx k . 2 aqedan, brunviTi zedapiris saZiebeli D farTobisTvis gveqneba miaxloebiTi toloba: 2π

n −1

D ≈ ∑ 2πy k 1 + [ y ′k ] 2 Δx k ,

(1)

k =0

romlis marjvena mxarec warmoadgens 2πy 1 + [ y ′] 2 funqciis integralur jams. [a, b] segmentze. Tu [a, b] segmentis dayofis wertilebis raodenobas gavzrdiT ise, rom miRebuli mcire segmentebis maqsimaluri sigrZe Semcirdes, (1) miaxloebiTi tolobis sizuste gaizrdeba. aqedan gamomdinare Tu gadavalT zRvarze, miviRebT:

D = lim

n −1

∑ 2πy

n →∞ Δxk →0 k = 0

1 + [ y k′ ]2 Δx k ,

sabolood ki SegviZlia davweroT: b

D = 2π ∫ y 1 + [ y ′]2 dx ,

(2)

sadac y = f ( x ) . vTqvaT axla wiri mocemulia parametrulad x = x (t ), y = y (t ) , sadac ( x(t 0 ), y (t 0 )) koordinatebis mqone wertili Seesabameba M 0 wertils, ( x(T ), y (T )) koordinatebis mqone wertili ki M n wertils. 5.2 qveTavidan gavixsenoT formula: 2

| M k M k +1 |= Δx k + Δy k . gavyoT tolobis orive mxare t parametris Δt k nazrdze, miviRebT: [ M k M k +1 ] = Δt k

Δx k2 Δy k2 . + Δt k2 Δt k2

gadavideT am tolobis orive mxares zRvarze rodesac Δt k → 0 miviRebT [ M k M k +1 ] Δx k2 Δy k2 . = lim + Δt k →0 Δt k → 0 Δt k Δt k2 Δt k2 lim

sidides

[ M k M k +1 ] )Δt k Δt k →0 Δt k uwodeben wiris rkalis diferencials M k ( lim

wertilSi da aRniSnaven ds

simboloTi. cxadia, aqedan gamomdinare, ds = [ x ′(t )] 2 + [ y ′(t )] 2 dt , dt = Δt . parametrulad mocemuli wiris rkalis sigrZis, CvenTvis kargad cnobili, formula asedac Caiwereba:

27 T

L = ∫ ds . t0

wiris brunviT miRebuli zedapiris farTobis gamosaTvleli formula ki ase: T

D = 2π ∫ y (t ) ds .

(3)

(3) formula igivea, rac T

D = 2π ∫ y (t ) [ x ′(t )]2 + [ y ′(t )]2 dt .

(4)

x2 y2 magaliTi 1. vipovoT 2 + 2 = 1, a > b elifsis OX RerZis garSemo brunviT a b miRebuli zedapiris farTobi. amoxsna: elifsis gantolebidan vRebulobT: b2 y 2 = b2 − 2 x2 . a miRebuli tolobis gawarmoebiT b2 b2 ′ ′ 2 yy = −2 2 x , yy = − x . a2 a aqedan y 1 + [ y ′] = 2

b2 2 b4 2 a2 − b2 2 2 y + [ yy ′] = b − 2 x − 4 x = a − x . a a a2 2

c elifsis eqscentrisiteti ε = = a saZiebeli farTobi iqneba D = 2π

a2 − b2 . a2

b b b 1 a2 εx 2 2 2 2 2 2 2 2 a − x dx = 4 a − x dx = 4 ( x a − x + arcsin ) 0a = ε π ε π ε 2 ∫ ∫ a −a a0 a 2 2ε a a

b (a a 2 − ε 2 a 2 + a 2 arcsin ε ) = 2πab( 1 − ε 2 + arcsin ε ). a magaliTi 2. gamovTvaloT parametrulad mocemuli x = a (t − sin t ), y = a (1 − cos t ), RerZis garSemo brunviT miRebuli cikloidis pirveli TaRis OX zedapiris farTobi. amoxsna: cikloidis(nax.9) pirveli TaRi aRiwereba t parametris cvlilebiT, segmentze [0,2π ]. t y = a (1 − cos t ) = 2a sin 2 x ′(t ) = a − a cos t , y ′(t ) = a sin t , 2 2 2 2 2 2 ds = a − 2a cos t + a cos t + a sin 2 t dt = 2a 2 − 2a 2 cos t dt = 2π

t t t )dt = 4a 2 sin 2 dt = 2a sin dt. 2 2 2 (3) formuliT saZiebeli farTobi: = 2a 2 − 2a 2 (1 − 2 sin 2

2π

D = 2π ∫ 2a sin 2 0

2π

t t t 2a sin dt = 2π ∫ 4a 2 sin 3 dt = 16πa 2 ∫ sin 3 udu = 2 2 2 0 0

cos 3 u 64 = 16πa ( − cos u ) π0 = πa 2 , 3 3 t u= . 2 2

5.5. meqanikuri da fizikuri sidideebis gamoTvla 5.5.1. wiris statikuri momenti da simZimis centri rogorc cnobilia, m masis materialuri wertilis statikuri momenti I R raime R RerZis mimarT ewodeba am wertilis masisa da wertilidan RerZamde d manZilis namravls I L = md . materialur wertilTa sistemis statikuri momenti ki am sistemaSi Semavali yvela wertilis statikuri momentebis jams. vTqvaT, y = f ( x) gantolebiT mocemulia materialuri wiri AB , romelzec masa gadanawilebulia ρ ( x ) simkvrviT wiris yvela wertilSi. davyoT AB wiri n nawilad wertilebiT: A = M 0 , M 1 , M 2 , M 3 ,..., M n −1 , M n = B , dayofiT miRebuli TiToeuli rkalis sigrZe aRvniSnoT s k simboloTi k = 0,1,2,...n − 1 . rogorc 5.2. qveTavidan viciT

s k ≈ Δxk2 + Δy k2 , Δx k = x k +1 − x k , Δy k = y k +1 − y k , sadac ( x k , y k ) warmoadgens M k k = 0,1,2,..., n wertilis koordinatebs. Tu CavTvliT, rom wiris dayofis wertilTa raodenoba sakmaod didia, wiris TiToeuli M k M k +1 rkali SeiZleba CaiTvalos materialur wertilad. aqedan gamomdinare, misi masa miaxloebiT iqneba ρ ( x k ) s k , xolo misi statikuri momenti OX RerZis mimarT y k ρ ( x k ) s k . mTeli wiris

I X , OX RerZis mimarT, miaxloebiTi gamoiTvleba, statikuri momenti rogorc, misi Semadgeneli elementaruli materialuri rkalebis sistemis, statikuri momentebis jami: n −1

I X ≈ ∑ y k ρ ( xk )sk . k =0

toloba zusti iqneba Tu gadavalT integralze, miviRebT: b

I X = ∫ y ( x) ρ ( x) 1 + [ y ( x)]2 dx ,

(1)

sadac

a = x0 , b = x n .

Tu wirs CavwerT parametruli gantolebiT, sadac s parametri icvleba [0, L] SualedSi, sadac L wiris sigrZea, maSin x = x( s ), y = y ( s ), ρ simkvrivec s parametris funqcia, gveqneba:

29 L

I X = ∫ y ( s)ρ ( s)ds .

(2)

analogiurad AB wiris statikuri momomenti OY RerZis mimarT, iqneba: f (b )

∫

Iy =

x( y ) ρ ( y ) 1 + [ x( y )]2 dy ,

(3)

f (a)

I y = ∫ x( s)ρ ( s)ds .

(4)

materialur wertilTa sistemis simZimis centri ewodeba wertils, romelSic sistemis mTeli jamuri masis moTavsebiT igive statikuri momenti eqneba raime R RerZis mimarT, ra statikuri momentic aqvs sistemas am RerZis mimarT. aqedan gamomdinare, materialuri wiris simZimis centris C (ξ ,η ) koordinatebis sapovnelad SegviZlia davweroT gantolebebi: ξM = I Y ,ηM = I X , (5) am gantolebebidan gveqneba I I (6) ξ = Y ,η = X . M M sabolood L

ξ=

∫ x(s) ρ (s)ds 0

, η=

∫ y(s) ρ (s)ds 0

∫ ρ (s)ds

(7)

∫ ρ (s)ds

sadac

∫ ρ (s)ds

integraliT gamoTvlilia materialuri wiris M masa.

CavTvaloT, rom materialur wirze masa Tanabradaa gadanawilebuli da simkvrivis funqcia ρ ( s ) = 1 . (6) tolobebidan meore gavamravloT 2π sidideze, miviRebT: L

2πηM = 2π ∫ yds . 0

am tolobis marjvena mxare warmoadgens wiris OX RerZis garSemo brunviT miRebuli zedapiris farTobs, marcxena mxareze M = L . aqedan gamomdinare wiris OX RerZis garSemo brunviT miRebuli zedapiris farTobi tolia am wiris sigrZisa da misi simZimis centriT Semowerili wrewiris sigrZis namravlis (8) D = 2πηL. am formulas guldenis pirveli formula ewodeba.

30 magaliTi 1. vipovoT x = a cos 3 t , y = a sin 3 t parametruli gantolebiT mocemuli wiris, romelsac astroida ewodeba, im nawilis simZimis centri, π romelic ganisazRvreba t parametris cvlilebiT [0, ] SualedSi(nax.13 a). 2 D moxsna: (8) formulidan gveqneba: η = . 2πL π

D = 2π ∫ y (t )ds = 2π ∫ a sin 3 t [−3a cos 2 t sin t ] 2 + [3a sin 2 t cos t ] 2 dt = π

2 1 6 2π ∫ 3a sin 3 t sin 2 t cos 2 t dt = 6πa 2 ∫ sin 4 t cos tdt = 6πa 2 ( sin 5 t ) 02 = πa 2 . 5 5 0 0 2

L = ∫ ds = ∫ [−3a cos 2 t sin t ] 2 + [3a sin 2 t cos t ] 2 dt = ∫ 3a sin 2 t cos 2 t dt = 0

π π

3 2 3 3 3 = 3a ∫ sin t cos tdt = a ∫ sin 2tdt = a ∫ sin udu = a (− cos u ) π0 = a. 4 2 2 0 4 0 0 .

maSasadame

η=

2 a. 5

Cˆ (ξˆ,ηˆ )

O X nax.13

x x + Δx

X nax. 14

rogorc naxazidan Cans, astroidis mier OY RerZis garSemo brunvisas igive farTobis zedapiri miiReba, amitom simZimis centris abscisac igive 2 unda iyos, gveqneba: ξ = a. 5 5.5.2. brtyeli figuris statikuri momenti da simZimis centri vTqvaT mocemulia mrudwiruli trapecia(nax.13), romelic SemosazRvrulia zemodan f ( x ) funqciis grafikiT, qvemodan OX RerZiT, gverdebidan x = a da x = b wrfeebiT. vTqvaT, masa am figuraze Tanabrad aris gadanawilebuli simkvriviT ρ = 1 , aseT SemTxvevaSi mrudwiruli trapeciis yoveli ΔD elementis masa toli iqneba misi farTobis.

Δx . radgan Δx mcirea, CavTvaliT es amovWraT trapeciidan zoli fuZiT zoli marTkuTxedad(nax.14). vTqvaT, zolis farTobia ΔD . misi simZimis centri mdebareobs marTkuTxedis diagonalebis gadakveTis Cˆ (ξˆ,ηˆ ) wertilSi. zolis simaRles aRvniSnoT y -iT, simZimis centris ordinati1 ηˆ = y , abscisa2 Δx . zolis statikuri momenti OX RerZis mimarT iyos ΔI X , xolo ξˆ = x + 2 OY Re- rZis mimarT- ΔI Y . zolis masa iqneba ΔD = yΔx . aqedan y 1 ΔI X ≈ yΔx, ΔI Y ≈ yΔx ( x + Δx ) = xyΔx + y[Δx] 2 . 2 2 Tu gaviTvaliswinebT, rom Δx mcirea, kidev ufro mcire iqneba sidide 1 [ Δx ] 2 . 2 am faqtis gaTvaliswinebiT mTeli trapeciis statikuri momentebsaTvis gveqneba mixloebiTi tolobebi: y2 (9) I X ≈ ∑ Δ, I Y ≈ ∑ xyΔx. 2 Tu am tolobebSi gadavalT zRvarze, rodesac Δx → 0 , mviRebT ukve zust gamosaxulebebs statikuri momentebisaTvis: b

1 2 y dx,I Y = ∫ xydx. (10) 2 ∫a a vTqvaT mrudwiruli trapeciis simZimis centria C (ξ ,η ) . simZimis centris gansazRvrebidan gamomdinare, adgili eqneba tolobebs: ξM = I Y ,ηM = I X . M trapeciis masaa. aqedan, sabolood, simZimis centris sadac koordinatebisaTvis gveqneba: IX =

ξ=

∫ xydx a b

, η=

∫ ydx

1 2 y dx 2 ∫a b

(11)

∫ ydx

a b

sadac, Cven SemTxvevaSi, M = ∫ ydx anu trapeciis masa misi farTobis tolia. a

ukanaskneli tolobebidan meoris orive mxare gavamravloT 2πD sidideze, miviRebT: b

2πDη = π ∫ y 2 dx . a

Tu gaviTvaliswinebT, rom π ∫ y 2 dx tolia mrudwiruli trapeciis OX RerZis a

garSemo brunviT miRebuli sxeulis moculobisa, miviRebT: (12) V = 2πDη . miRebul formulas guldenis meore formula ewodeba. es formula gviCvenebs, rom brtyeli figuris masTan TanaukveTi RerZis garSemo brunviT miRebuli sxeulis moculoba tolia am figuris

32 farTobisa da misi simZimis centris mier RerZis garSemo Semowerili wrewiris sigrZis namravlis. magaliTi 1. vipovoT cikloidis(nax.9) pirveli TaRiT SemosazRvruli figuris simZimis centri. amoxsna: cikloidis parametruli gantolebaa x = a (t − sin t ), y = a (1 − cos t ). misi pirveli TaRi aRiwereba t parametris [0,2π ] SualedSi cvlilebiT. jer vipovoT farTobi: 2π

2π

∫ ydx = ∫ a 0

2π

2πa 2 +

(1 − cos t ) tdt = ∫ (1 − 2 cos t + cos t )dt =a (1 − sin t ) 2

2π 0

2π

1 + cos 2t dt = 2 0

+∫

a 1 2πa 2 (t + sin 2t ) 02π = 2πa 2 + = 3πa 2 . 2 2 2 2

Semdeg moculoba: 2π

V = π ∫ a (1 − cos t ) a (1 − cos t )dt = πa 2

2π 3

2π

∫ 2 (1 + cos 2t )dt − ∫ (1 − sin 0

∫ (1 − cos t ) 0

2π

dt = πa [ ∫ (1 − 3 cos t )dt + 3

3 t )d sin t ] = πa 3 [t − 3 sin t + t + 2

3 1 + sin 2t − sin t + sin 3 t ] 02π = 5π 2 a 3 . 4 3 Tu gamoviyenebT guldenis meore formulas, miviRebT: 5π 2 a 3 = 2π 3πa 2η . aqedan simZimis centris ordinati V 5π 2 a 3 5 η= = = a. 2 2πD 2π 3πa 6 cikloidis pirveli TaRiT SemosazRvruli mrudwiruli trapecia simetriulia x = πa wrfis mimarT, amitom simZimis centri am wrfeze mdebareobs. aqedan gamomdinare, misi abscisa ξ = πa . magaliTi 2. vipovoT im brtyeli figuris simZimis centri, romelic SemosazRvrulia wirebiT y = x 2 , y = x (nax.7). amoxsna: statikuri momentebis gansazRvrebidan da (9), (10) formulebidan gamomdinareobs Semdegi: 1

ξ=

∫ x(

x − x )dx

∫(

x − x )dx

,η =

0 1

∫(

x − x dx 2)

2 2 ∫ ( x − x )dx = ∫ xdx − ∫ x dx = 0

1 ( x − x 2 ) 2 dx ∫ 20

1 +1 2

x x 2+1 1 + 0 1 2 +1 +1 2

1 0

2 1 1 − = 3 3 3

33 1

3 2

3 +1 2

x 3 +1 2

2 3 ∫ x( x − x )dx = ∫ x dx − ∫ x dx = 0

3 2

1 0

x 3+1 3 +1

1 0

2 1 3 − = 5 4 20 3

x2 2 2 4 ( ) 2 − = − + = x x dx xdx x dx x dx ∫0 ∫0 ∫0 ∫0 2

x 2 1 x5 1 1 2 1 3 1 − − + = . 0 0 + 0 = 3 5 2 5 5 10 +1 2 3 3 am gamoTvlebidan gamomdinare ξ = . ,η = 20 20 5.5.3. cvladi siCqariT moZravi materialuri wertilis mier gavlili manZili vTqvaT materialuri wertili moZraobs siCqariT, romelic warmoadgens drois uwyvet funqcias v = v (t ) da saWiroa gavigoT mis mier drois T0 momentidan T1 momentamde gavlili manZili S . davyoT [T0 , T1 ] Sualedi n nawilad drois momentebiT T0 = t 0 , t1 , t 2 ,...t k ,..., t n = T1 . materialuri wertilis mier drois mcire [t1 , t t +1 ] monakveTSi gavlili manZili miaxloebiT toli iqneba sididis, sadac Δ i = t i +1 − t i . aqedan gamomdinare, mTel [T0 , T1 ] monakveTSi gavlili manZilisaTvis gveqneba miaxloebiTi toloba: S ≈

∑ (t i=0

i +1

− t i ) v (t i ) .

(13)

rac ufro met nawilad davyofT [T0 , T1 ] Sualeds, ise rom dayofiT miRebuli monakveTebis maqsimaluri sigrZe max Δ i miiswrafodes nulisaken, (13) tolobis sizuste TandaTan ufro gaizrdeba. amitom rogorc zemoT ganxilul SemTxvevebSia, adgili eqneba zust tolobas: T1

S = ∫ v(t )dt .

(14)

(14) formulidan SeiZleba ganvsazRvroT saSualo siCqare: T1

∫ v(t )dt

. T1 − T0 analogiuri formula gveqneba drois T0 momentidan T1 momentamde cvladi aCqarebiT moZravi materialuri wertilis siCqaris nazrdis gamosaTvlelad: T1

Δv = ∫ a(t )dt , T0

sadac a (t ) droSi cvladi aCqarebaa. rac Seexeba materialuri wertilis siCqares T1 momentSi igi ganisazRvreba tolobiT:

34 T1

v(T1 ) = v(T0 ) + Δv = v(T0 ) + ∫ a(t )dt .

(15)

5.5.4 gansazRvruli integralis gamoyeneba ekonomikur amocanebSi (13) formulis analogiur formulebTan gveqneba saqme, rodesac gvinda gamovTvaloT sawarmos mogeba an gamoSvebuli produqciis raodenoba drois T0 momentidan T1 momentamde: T1

P = ∫ M (t )dt , T0

sadac M (t ) drois t momentSi miRebuli mogebaa an sawarmos mier drois t momentSi gamoSvebuli produqciis raodenoba(simZlavre). (15) formulis analogiur formulasTan gveqneba saqme Tu gvinda gamovTvaloT sawarmos Semosavali an sawarmoo xarjebis raodenoba drois T1 momentamde: T1

R(T1 ) = R(T0 ) + ∫ N (t )dt , T0

sadac N (t ) Semosavlis an sawarmoo xarjebis zrdis siCqarea drois t momentSi, R(T0 ) ki- drois T0 momentamde miRebuli Semosavali an gaweuli xarjebi. vTqvaT, axla N1 (t ) sawarmos Semosavlis zrdis siCqarea, N 2 (t ) ki- xarjebis zr-dis siCqarea, maSin, sanam N1 (t ) ≥ N 2 (t ) , sawarmo imuSavebs mogebaze, drois es intervali iqneba [0, T ∗ ] , sadac T ∗ drois is momentia, rodesac N1 (t ) = N 2 (t ) . aqedan gamomdinare, sawarmos maqsimaluri mogeba T∗

L = ∫ [ N 1 (t ) − N 2 (t )]dt . 0

funqcias p = D ( x) , sadac p raime produqciis sacalo fasia, xolo x im momx-marebelTa raodenoba, romelTac surT am fasad SeiZinon produqciis erTeulili. am funqcias moTxovnis funqcia ewodeba, xolo funqcias p = S (x ) , sadac x aRniSnuli produqciis im mwarmoeblTa raodenobaa, romelTac surT p fasad gayidon produqciis erTeuli, miwodebis funqcia. wertils koordinatebiT ( x ∗ , p ∗ ) wonasworobis bazris wertili ewodeba (nax.15). cxadia, bazarze wonasworobis miRwevamde, Warbi moTxovnis raodenoba PD da Warbi produqciis raodenoba PS Sesabamisad, gamoiTvleba Semdegi formulebiT: P D(x) PD

S ( x)

∗

p PS

x∗

nax.15

35 x∗

x∗

PD = ∫ ( D( x) − p )dx, PS = ∫ ( p ∗ − S ( x))dx ∗

6. savarjiSoebi 1-li TavisaTvis π 2

π 4

4 1 I gamoTvaleT integralebi: 1) ∫ x dx , 2) ∫ ( x + 2 )dx , 3) ∫ x dx , 4) ∫ sin 4 xdx , x 1 1 0 0 2

1 1 2 ⎧⎪ x 2 ,0 ≤ x < 1, dx cos x 5) ∫ sin 2 xdx ,6) ∫ f ( x)dx , Tu f ( x) = ⎨ , 7) ∫ dx , 8) ∫ , 9) ∫ e 2 x dx , ⎪⎩ x ,1 ≤ x ≤ 2. π sin x 4 − x2 o 0 0 0

10) ∫ 1

ex −1

2 xdx , 14) 4 − x 2 dx , 2 2 ∫ 0 ( x + 1) − 2

dx , 11) ∫ e x (e x − 1)dx , 12) ∫ x 2 a 2 − x 2 dx , 13) ∫ 0

π 2

15) ∫ 1

dx x x2 −1

4 ln x dx , 18) ∫ xe x dx , 19) ∫ x 2 sin xdx , 20) ∫ sin x dx , x 1 0 0 0 e

, 16) ∫ sin 2 x cos xdx , 17) ∫ 0

π 1

21) ∫ x ln(x 2 + 1)dx , 22) ∫

xdx . II.gamovTvaloT Semdegi wirebiTSemosazRvrul sin 2 x

brtyel figuraTa farTobebi: a) y = 6 x − x 2 , y = 0 ,b) y = x 2 + 1, y = 0, x = 0, x = 2; g) y = 8 + 2 x − x 2 , y = 2 x + 4 , d) y = x 2 − 4 x, y = 0 , e) y = ln x, y = 0, x = 2, x = 8 , 2 1 v) y = sin x, y = x ,z) y = arcsin x, x = , y = 0 . III. ipoveT rkalebis sigrZeebi 2 π x wertilebs Soris a) y = e , x = 0, x = e ; b) y = ln x , x = 3 , x = 8 ; g) y = arcsin(e − x ) , x = 0, x = 1 . IV. gamoTvaleT Semdegi wirebiT SemosazRvruli figuris brunviT miRebuli sxeulebis moculoba: a) y = x − x 2 , x = 0 , OX RerZis garSemo; b) y = x 3 , x = 0, y = 1 , OY RerZis garSemo; g) y = sin 2 x, x = 0, x = π , OX RerZis garSemo. V. gamoTvaleT Semdegi wirebis rkaliT brunviT π miRebuli zedapiris farTobi: a) y = sin x , x = 0, x = , OX RerZis garSemo; 2 −x b) y = e , x = 0, x = ∞ , OX RerZis garSemo; g) y = ln x , y = 0, y = −∞ , OY RerZis garSemo.

36 Tavi 2 ricxviTi mwkrivebi 1. ZiriTadi cnebebi vTqvaT mocemulia sidideTa raime mimdevroba u1 , u 2 , u 3 ,..., u n ,... . usasrulo jams: u1 + u 2 + u 3 + ... + u n + ... , (1) romlis Sesakrebebic mocemuli mimdevrobis wevrebia mwkrivi ewodeba. Tu sidideebi u1 , u 2 , u 3 ,..., u n ,... ricxvebia, mwkrivs ricxviTi mwkrivi ewodeba, Tu funqciebia, funqcionaluri mwkrivi. am paragrafSi Cven ganvixilavT ricxviT mwkrivebs. u n sidides mwkrivis zogadi wevri ewodeba. ganvixiloT jamebi: s1 = u1 , s 2 = u1 + u 2 , (2) s 3 = u1 + u 2 + u 3 , s 4 = u1 + u 2 + u 3 + u 4 , .................................., s n = u1 + u 2 + u 3 + ... + u n . am jamebs (1) mwkrivis kerZo jamebi ewodeba. gansazRvreba 2.1. Tu (1) mwkrivis kerZo jamebis mimdevrobas s1 , s 2 , s 3 ,..., s n ,... gaaCnia sasruli zRvari, maSin mwkrivs ewodeba krebadi, xolo zRvars lim s n ewodeba mwkrivis jami. n→∞

Tu kerZo jamebis mimdevrobas sasruli zRvari ar gaaCnia, maSin mwkrivs ewodeba ganSladi. magaliTi 1. gamovikvlioT CvenTvis kargad cnobili mwkrivis, geometriuli progresiis, 1 + q + q 2 + q 3 + ... + q n + ... (3) krebadobis sakiTxi. skolis maTematikis kursidan viciT, rom progresiis pirveli n wevris jami, rodesac q ≠ 1 , gamoisaxeba formuliT:

1− qn s n = 1 + q + q + q + ... + q = . 1− q 2

aqedan gamomdinare 1 − lim q n 1− qn n →∞ . = lim s = lim n →∞ n →∞ 1 − q 1− q 1 Tu | q |< 1 , maSin lim q n = 0 da lim s n = . maSasadame, Tu | q |< 1 , (3) mwkrivi n →∞ n →∞ 1− q krebadia. davuSvaT, | q |> 1 , maSin lim q n = ∞ da, aqedan gamomdinare, lim s n = ∞ , n

n →∞

n→∞

maSasadame, am SemTxvevaSi (3) mwkrivi ganSladia. davuSvaT, exla q = 1 , maSin (3) mwkrivs eqneba saxe 1 + 1 + 1 + ... .

37 am mwkrivis kerZo jamia s n = 1 + 1 + ... + 1 = n , s n = lim n = ∞ , amgvarad, aseT n →∞

SemTxveSi, (3) mwkrivi ganSladia. axla ganvixiloT SemTxveva q = −1 , (3) mwkrivs eqneba saxe 1 − 1 + 1 − 1 + ... . am mwkrivis kerZo jamebi, Tu n luwia, tolia nulis; xolo, Tu n kentia, maSin es kerZo jamebi tolia erTis. aseT SemTxvevaSi lim s n ar arsebobs, n→∞

maSasadame (3) mwkrivi ganSladia. mwkrivis aRniSvnisTvis gamoiyeneba, agreTve Semoklebuli Cawera: ∞

u1 + u 2 + u 3 + ... + u n + ... = ∑ u n . n =1

2. krebadi mwkrivis Tvisebebi ricxviTi mwkrivebis Tvisebebi CamovayaliboT Teoremebis saxiT. Teorema 2.1. Tu mocemuli ∞

u1 + u 2 + u 3 + ... + u n + ... = ∑ u n

(1)

n =1

mwkrividan amoviRebT pirvel k wevrs, maSin miRebuli mwkrivi u k +1 + u k + 2 + u k + 3 + ... + u n + ... =

∞

∑u

n = k +1

(2)

krebadia maSin da mxolod maSin, Tu krebadia mocemuli mwkrivi. (2) mwkrivs (1) mwkrivis naSTi ewodeba. Teorema 2.2. Tu mwkrivebi u1 + u 2 + u 3 + ... + u n + ... , v1 + v 2 + v3 + ... + v n + ... krebadia da maTi jamebi tolia Sesabamisad A da B , maSin mwkrivi, romelic miiReba mocemuli mwkrivebis wevr-wevrad SekrebiT an gamoklebiT: u1 ± u 2 ± u 3 ± ... ± u n ± ... , aseve krebadia da misi jami tolia A ± B . Teorema 2.3. Tu mwkrivi u1 + u 2 + u 3 + ... + u n + ... krebadia da misi jamia A , maSin mwkrivi Cu1 + Cu 2 + Cu 3 + ... + Cu n + ... , sadac C raime ricxvia, aseve krebadia da misi jami tolia CA . Semdegi Tviseba warmoadgens mwkrivis krebadobis aucilebel pirobas. Teorema 2.4. Tu mwkrivi ∞

u1 + u 2 + u 3 + ... + u n + ... = ∑ u n n =1

krebadia, maSin misi zogadi wevri u n miiswrafvis nulisken n indeqsis usasrulo zrdis dros anu lim u n = 0 . n →∞

am Tvisebidan gamomdinareobs Semdegi faqti, Tu mwkrivis zogadi wevri ar miiswrafis nulisken, n indeqsis usasrulo zrdis dros, maSin mwkrivi ganSladia. magram, mwkrivis zogadi wevris nulisken miswrafeba anu piroba lim u n = 0 ar aris sakmarisi mwkrivis krebadobisTvis. marTlac n→∞

ganvixiloT mwkrivi:

38 1 1 1 (3) + + ... + + ... , 2 3 n romelsac harmoniul mwkrivs uwodeben. cxadia, misi zogadi wevrisaTvis 1 1 sruldeba piroba lim u n = lim = 0 . un = n→∞ n→∞ n n davajgufoT (3) mwkrivis Sesakrebebi Semdegnairad: 1+

1 ⎛1 1⎞ ⎛1 1 1 1⎞ ⎛1 1 1⎞ ⎛1 1 1 ⎞ (4) + ⎜ + ⎟ + ⎜ + + + ⎟ + ⎜ + + ... + ⎟ + ⎜ + + ... + ⎟ + ... . 2 ⎝ 3 4 ⎠ ⎝ 5 6 7 8 ⎠ ⎝ 9 10 16 ⎠ ⎝ 17 18 32 ⎠ rogorc vxedavT SesakrebTa TiToeuli jgufis bolo Sesakrebs aqvs saxe: 1 da mezobeli jgufebis bolo Sesakrebebs Soris aseTi saxis ricxvebi 2n ar gvxvdeba, amasTan, rac ufro marcxniv mdebareobs SesakrebTa jgufi, miT metia mis bolo SesakrebSi mniSvnelis xarisxi. ganvixiloT mwkrivi v1 + v 2 + v3 + ... + v n + ... , (5) sadac 1 ⎛1 1⎞ ⎛1 1 1 1⎞ v1 = 1, v 2 = , v3 = ⎜ + ⎟, v 4 = ⎜ + + + ⎟,... 2 ⎝3 4⎠ ⎝5 6 7 8⎠ advilia mixvedra, rom am mwkrivis TiToeuli wevri vn pirvelis garda, 1+

Sedgba 2 n − 2 Sesakrebisgan. ganvixiloT aseve mwkrivi (6) w1 + w2 + w3 + ... + wn + ... , sadac 1⎞ 1 1 ⎛1 ⎛1 1 1 1⎞ ⎛1 1⎞ w1 = 1, w2 = , w3 = ⎜ + ⎟, w4 = ⎜ + + + ⎟, w5 = ⎜ + + ... + ⎟, 16 ⎠ 2 ⎝ 16 16 ⎝8 8 8 8⎠ ⎝4 4⎠ 1 ⎞ 1 ⎛ 1 + ... + ⎟,.... w6 = ⎜ + 32 ⎠ ⎝ 32 32 1 am mwkrivis TiToeuli wevrisaTvis, garda pirvelisa, gvaqvs wn = . 2 Tu n ≥ 3 , gvaqvs v n > wn . aqedan gamomdinare, (5) da (6) mwkrivebis kerZo jamebisTvis gveqneba: s n′ > s n′′ . n −1 n +1 1 1 1 . s n′′ = 1 + + + ... + = 1 + = 2 2 2 2 2

n −1

n +1 = ∞ . radgan s n′ > s n′′ , aseve gveqneba n →∞ 2 n →∞ lim s n′ = ∞ . maSasadame (5) mwkrivi ganSladia.

aqedan gamomdinare, gveqneba lim s n′′ = lim n→∞

arsebobs iseTi Tu s k , k = 1,2,3,... (3) mwkrivis kerZo jamebia, maSin k1 , k 2 , k 3 ,...k n ,... , rom s kn = s ′n . amasTan, Tu n → ∞ , maSin k n → ∞ . mimdevrobis zRvris erT-erTi Tvisebidan gamomdinareobs, rom krebadi mimdevrobis yoveli qvemimdevroba krebadia. amitom, Tu mimdevroba s1 , s 2 ,..., s k ,... krebadia, krebadi unda iyos misi qvemimdevrobac: s k1 , s k 2 ,..., s k n ,... , magram s kn = s ′n da

lim s n′ = ∞ . n→∞

miviReT

39 winaaRmdegoba, maSasdame (3) mwkrivis kerZo jamebis

mimdevroba ganSladia. miuxedavad imisa, rom (3) mwkrivis zogadi wevri miiswrafvis nulisaken, es mwkrivi ar aris krebadi. Ese-igi mwkrivis krebadobis aucilebeli piroba (Teorema 2.4) ar yofila sakmarisi misi krebadobisTvis. 3. dadebiT wevrebiani ricxviTi mwkrivis krebadobis niSnebi mwkrivs u1 + u 2 + u 3 + ... + u n + ... ewodeba dadebiTwevrebiani, Tu igi akmayofilebs pirobas u n ≥ 0, n = 1,2,3,..., . moviyvanoT aseTi mwkrivebis krebadobis niSnebi. 1. mwkrivis krebadobis Sedarebis niSani. Tu mocemulia ori dadebiTwevrebiani mwkrivi u1 + u 2 + u 3 + ... + u n + ... , (1) v1 + v 2 + v3 + ... + v n + ... , (2) romlebisTvisac adgili aqvs pirobas v n ≤ u n , n = 1,2,3,... maSin, Tu krebadia (1) mwkrivi, krebadi iqneba (2) mwkrivic. aseve, Tu ganSladia (2) mwkrivi, ganSladi iqneba (1) mwkrivic. magaliTi 1. gamovikvlioT krebadobaze Semdegi mwkrivi: π π π (3) sin + sin + ... + sin n + ... 2 4 2 amoxsna: gamoviyenoT Sedarebis niSani rogorc cnobilia sin x ≤ x , rodesac π π π 0 ≤ x ≤ . aqedan gamomdinare, radgan 0 ≤ n ≤ , n = 1,2,3,... , gveqneba 2 2 2 π π sin n ≤ n , n = 1,2,... . 2 2 mwkrivi π π π (4) + + ... + n + ... 2 4 2 1 warmoadgens geometriul progresias, q = < 1 , amitom igi krebadia. 2 Sedarebis niSnidan gamomdinare krebadi iqneba (3) mwkrivic. 2. mwkrivis krebadobis dalamberis niSani. Tu dadebiTwevrebiani u1 + u 2 + u 3 + ... + u n + ... mwkrivisTvis, dawyebuli n indeqsis(nomris) romeliRac mniSvnelobidan, adgi-li aqvs utolobas: y n +1 ≤ q < 1, un sadac q damokidebuli araa n -is mniSvnelobaze, maSin mocemuli mwkrivi krebadia. aseve, Tu dawyebuli n indeqsis romeliRac mniSvnelobidan, adgili aqvs utolobas: y n +1 >1, un

40 maSin mwkrivi ganSladia.

dalamberis niSani SeiZleba zRvruli formiTac iqnes warmodgenili: Tu u lim n +1 = q , maSin n →∞ u n u1 + u 2 + u 3 + ... + u n + ... mwkrivi krebadia, rodesac q < 1 da ganSladi, rodesac q > 1 . Tu q = 1 , maSin mwkrivi SeiZleba krebadi iyos, SeiZleba ganSladi, aseT SemTxvevaSi mwkrivis krebadobis sxva niSani unda gamoviyenoT. magaliTi 2. gamovikvlioT krebadobaze Semdegi mwkrivi: 1 22 33 44 nn + + + + ... + n + ... . 3 3 2 2! 333! 3 4 4! 3 n! amoxsna: gamoviyenoT dalamberis niSani, u (n + 1) n +1 3 n n! (n + 1) n +1 1 (n + 1) n 1 1 1 lim n +1 = lim n n +1 lim = lim = = lim(1 + ) n = e < 1 . n n n→∞ u n →∞ n 3 3 n →∞ n 3 n →∞ n 3 (b + 1)! n→∞ 3(n + 1)n n aqedan davaskvniT, rom mwkrivi krebadia. magaliTi 3. gamovikvlioT krebadobaze Semdegi mwkrivi: ( 2)n 2 2 2 + + ... + + ... . 2n − 1 3 5 amoxsna: rogorc wina magaliTSi u 2n − 1 ( 2 ) n +1 (2n − 1) = 2 ⋅1 = 2 > 1. = 2 lim lim n +1 = lim n →∞ u n →∞ n →∞ 2n + 1 (2n + 1)( 2 ) n n maSasadame mwkrivi ganSladia. magaliTi 4. gamovikvlioT krebadobaze Semdegi mwkrivi: 1 1 1 1 + 2 + 2 + ... + 2 + ... 2 3 n amoxsna: am SemTxvevaSic dalamberis niSani gvaZlevs u n +1 n2 1 = lim = =1. lim n →∞ u n ( n + 1) 2 1 2 n lim (1 + ) n →∞ n maSasadame mwkrivis krebadobaze verafers vityviT. 3. mwkrivis krebadobis koSis niSani. Tu dadebiTwevrebiani u1 + u 2 + u 3 + ... + u n + ... mwkrivisTvis, dawyebuli n indeqsis(nomris) romeliRac mniSvnelobidan, adgili aqvs utolobas: n u < q < 1, n sadac q damokidebuli araa n -is mniSvnelobaze, maSin mocemuli mwkrivi krebadia. aseve, Tu dawyebuli n indeqsis romeliRac mniSvnelobidan, adgili aqvs utolobas: n u . > 1, n maSin mwkrivi ganSladia. koSis niSani SeiZleba zRvruli formiTac iqnes warmodgenili: 2+

41 Tu lim n u n = q , n →∞

maSin

u1 + u 2 + u 3 + ... + u n + ... mwkrivi krebadia, rodesac q < 1 da ganSladi, rodesac q > 1 .

maSin rogorc dalamberis niSnis SemTxvevaSi, aqac gvaqvs: Tu q = 1, mwkrivi SeiZleba krebadi iyos, SeiZleba ganSladi. aseT SemTxvevaSic mwkrivis krebadobis sxva niSani unda gamoviyenoT. magaliTi 5. gamovikvlioT krebadobaze Semdegi mwkrivi: ∞ n . ∑ n n =1 2 amoxsna: gamoviyenoT mwkrivis krebadobis koSis niSnis zRvruli forma. 1 ln n n →∞ n

1 lim n1 n→∞ n

n n 1 lim ln n 1 1 1 1 1 1 n = e = e = e0 = < 1. lim n = lim n n = lim e ln n = e n →∞ n n →∞ 2 2 2 2 2 2 n →∞ 2 n →∞ 2 2 maSasadame, mwkrivi krebadia. aq zRvris gamosaTvlelad gamoyenebul iqna ln n lopitalis wesi saxis wiladisTvis. n magaliTi 6. gamovikvlioT krebadobaze Semdegi mwkrivi n

un =

lim

⎛ n +1 ⎞ ⎜ ⎟ . ∑ n =1 ⎝ 2 n − 1 ⎠ amoxsna: gamoviyenoT koSis niSani n

∞

⎛ n +1 ⎞ un = ⎜ ⎟ , ⎝ 2n − 1 ⎠ n

n +1 ⎛ n +1 ⎞ , advili misaxvedria, rom, rodesac n > 5 , maSin ⎟ = ⎜ 2n − 1 ⎝ 2n − 1 ⎠ n +1 2 < < 1 . maSasadame mwkrivi krebadia. 2n − 1 3 4. mwkrivis krebadobis koSis integraluri niSani. Tu dadebiT wevrebiani u1 + u 2 + u 3 + ... + u n + ... mwkrivis zogadi wevri miiswrafvis nulisken da naturaluri argumentis funqcia u n = f (n) , iseTia, rom f ( x ) gansazRvrulia [1, ∞) Sualedze; amasTan, arsebobs arasakuTrivi integrali n

∞

∫ f ( x)dx , 1

maSin mwkrivi iqneba krebadi, winaaRmdeg SemTxvevaSi ganSladi. magaliTi 7. gamovikvlioT dadebiT wevrebiani ∞ 1 ∑ n = 2 n ln n mwkrivis krebadoba. amoxsna: gamoviyenoT koSis integraluri niSani

42 1 . n ln n aqedan gamomdinare gveqneba arasakuTrivi integrali f ( n) = ∞

∫ x ln x dx . 2

d ln x 1 = lim ln(ln x) 2a = lim[ln(ln a ) − ln(ln 2)] = ∞ , dx = lim ∫ a →∞ a → ∞ a →∞ a →∞ ln x x ln x 2 2 maSasadame mwkrivi ganSladia. magaliTi 8. gamovikvlioT dadebiTwevrebiani ∞ 1 ∑ α n =1 n mwkrivis krebadoba. amoxsna: aqac gamoviyenoT koSis integraluri niSani 1 f ( x) = α . x gveqneba arasakuTrivi integrali lim ∫

∞

(5)

∫ x α dx . 2

Tu α > 1 , maSin

x 1−α a a 1−α 1 1 1 . = = − dx lim [ lim [ ]= 1 α a →∞ a →∞ 1 − α a →∞ 1 − α α −1 1−α 2 x maSasadame arasakuTrivi integrali arsebobs da mwkrivi krebadia. Tu α < 1 , maSin a

lim ∫

1 x 1−α lim ∫ α dx = lim[ a →∞ a →∞ 1 − α 2 x am SemTxvevaSi mwkrivi ganSladia. Tu α = 1 , maSin gveqneba a

a 1

a 1−α 1 = lim[ − ] = ∞. a →∞ 1 − α 1−α

1 lim ∫ dx = lim ln x 1a = ∞ . a →∞ a →∞ x 2 maSasadame mwkrivi ganSladia. am magaliTidan gamomdinare mwkrivi ∞ 1 ∑ 2 n =1 n krebadia. (5) mwkrivs ganzogadebuli harmoniuli mwkrivi ewodeba. igi xSirad gamoiyeneba mwkrivTa krebadobis Sedarebis niSnis gamoyenebis dros.

4. niSancvladi mwkrivebi Tu mwkrivis u1 + u 2 + u 3 + ... + u n + ... wevrebs Soris gvxvdeba rogorc dadebiTi, aseve uaryofiTi wevrebi, maSin mwkrivs niSancvladi ewodeba. niSancvlad mwkrivebs Soris, xSirad gvxvdeba iseTebi, romlebSic icvleba yoveli momdevno wevris niSani. aseT mwkrivebs aqvT saxe u1 − u 2 + u 3 − u 4 + ... + (−1) n −1 u n + ... , an saxe − u1 + u 2 − u 3 + u 4 − ... + (−1) n u n + ... ,

43 aseT mwkrivebs sadac sidideebi u n n = 1,2,3,... dadebiTia. niSanmonacvle ewodeba. niSanmonacvle mwkrivebisTvis arsebobs krebadobis sakmarisi niSani: niSanmonacvle mwkrivis krebadobis laibnicis niSani. Tu niSanmonacvle u1 − u 2 + u 3 − u 4 + ... + (−1) n −1 u n + ... , − u1 + u 2 − u 3 + u 4 − ... + (−1) n u n + ... mwkrivebis wevrebis absoluturi mniSvnelobebi klebadia da zogadi wevrebi miiswrafvian nulisken, rodesac n miiswrafvis usasrulobisken, maSin es mwkrivebi krebadia.

magaliTi 1.

gamovikvlioT Semdegi mwkrivi krebadobaze: 1 1 1 1 1 − + − + ... + ( −1) n −1 + ... . n 2 3 4 amoxsna: laibnicis niSnidan gamomdinare, krebadia. magaliTi 2. gamovikvlioT Semdegi mwkrivi krebadobaze: 3 5 7 2n + 1 − + − + ... + (−1) n −1 + ... . 1⋅ 2 2 ⋅ 3 3 ⋅ 4 n(n + 1) amoxsna: mwkrivi krebadia radgan | u n +1 |<| u n | da lim u n = 0 . n →→

laibnicis niSani ar warmoadgens niSanmonacvle mwkrivis krebadobis aucilebel pirobas. es kargad Cans Semdegi magaliTidan. magaliTi 3. gamovikvlioT mwkrivi: 1−

1 1 1 1 1 + 3 − 2 + ... + − + ... 2 3 ( 2n − 1) ( 2n) 2 2 3 4

(1)

krebadobaze. amoxsna: am mwkrivisTvis laibnicis piroba | u n +1 |<| u n | ar sruldeba, magram mwkrivi krebadia. marTlac, es mwkrivi SeiZleba warmovadginoT 1 1 1 (2) + ... 1 + 3 + 3 + ... + ( 2n − 1) 3 3 5 da 1 1 1 1 (3) + 2 + 2 ... + + ... 2 ( 2n) 2 2 4 6 mwkrivebis sxvaobis saxiT. Tu SevadarebT (2) mwkrivs krebad 1 1 1 1 + 3 + 3 + ... + 3 + ... 2 3 n mwkrivTan da (3) mwkrivs krebad 1 1 1 1 + 2 + 2 + ... + 2 + ... 2 3 n mwkrivTan, dadebiT wevrebianmwkrivebis Sedarebis niSnidan davinaxavT, rom es mwkrivebi krebadia. maSasadame, krebadi iqneba maTi sxvaobac, rac (1) mwkrivis tolia. 5. mwkrivTa absolituri da pirobiTi krebadoba ganvixiloT mwkrivi u1 + u 2 + u 3 + ... + u n + ... , romlis wevrebic nebismieri niSnisaa.

(1)

44 SevadginoT mwkrivi

| u1 | + | u 2 | + | u 3 | +...+ | u n | +... , (2) romlis wevrebic warmoadgens (1) mwkrivis Sesabamisi wevrebis absolutur mniSvnelobebs. gansazRvreba 2.2. (1) mwkrivs ewodeba absoluturad krebadi, Tu (2) mwkrivi krebadia. (1) mwkrivs ewodeba pirobiTad krebadi, Tu igi krebadia, magram (2) mwkrivi ganSladia. magaliTi 1. mwkrivi 1 1 1 1 + − + ... + (−1) n −1 + ... 2 3 4 n pirobiTad krebadia, radgan misi wevrebis absolitur mniSvnelobebisgan Sedgenili, CvenTvis kargad cnobili harmoniuli mwkrivi, 1 1 1 1 + + + ... + + ... , n 2 3 ganSladia. mwkrivis absolitur krebadobasa da Cveulebriv krebadobas Soris arsebobs kavSiri, romelic Semdegi TeoremiT ganisazRvreba. Teorema 2.5. Tu (1) mwkrivi krebadia absoluturad, maSin igi Cveulebrivadac krebadia. mwkrivis absoluturad krebadobis dasadgenad SeiZleba gamoviyenoT krebadobis yvela is niSani, romelsac viyenebT dadebiTwevrebiani mwkrivebisTvis. Teorema 2.6 (koSis Teorema). Tu absoluturad krebad mwkrivebSi wevrebis mimdevrobas nebismierad SevcvliT, maSin miRebuli mwkrivi absoluturad krebadi iqneba da misi jami mocemuli mwkrivis jamis tolia. aseT faqts adgili ar aqvs pirobiTad krebadi mwkrivebisTvis. marTlac, ganvixiloT mwkrivi 1 1 1 1 (3) 1 − + − + ... + (−1) n −1 + ... , n 2 3 4 rogorc viciT, es mwkrivi pirobiTad krebadia. vTqvaT, misi jamia A . gadavanacvloT am mwkrivSi wevrebi ise, rom yoveli dadebiTniSniani wevris Semdeg modiodes ori momdevno uaryofiTi wevri: 1 1 1 1 1 1 1 1 1 (4) 1 − − + − − + − − + − ... 2 4 3 6 8 5 10 12 7 davajgufoT am mwkrivSi wevrebi Semdegnairad: 1 1 1 1 1 1 1 1 1 − − + − − + − − + ... . 3

6 8 5 10 N2 4

12 1−

miviRebT: 1⎛ 1 1 1 1 1 1 1 1 1 1 ⎞ 1 − + − + − + ... = ⎜1 − + − + − ... ⎟ = A . 2⎝ 2 3 4 5 2 4 6 8 10 12 ⎠ 2 maSasadame, (3) mwkrivis wevrebis gadanacvlebiT miRebuli (4) mwkrivis jami orjer naklebia (3) mwkrivis jamze.

6. ricxviTi mwkrivebis gamoyeneba magaliTi 1. ras udris kreditis Rirebuleba L , Tu

misi sididea

45 A, yovelTviuri saprocento ganakveTi n% , kreditis dafarvis vada m weli, kreditis dafarva xdeba yovelTviurad erTi da igive sididis TanxiT. A amoxsna: yovelTviurad kreditis dasafaravad Sesatani Tanxaa B = . m × 12 n , meore TveSipirvel TveSi Sesatani Tanxaa a1 = B + A × 100 n n , mesame TveSi Sesatani a3 = B + ( A − 2 B ) × da ase a 2 = B + ( A − B) × 100 100

Semdeg a12 m = B + [ A − (12m − 1) B ] ×

n . 100

aqedan gamomdinare, kreditis

Rirebulebaa n n n n A× + ( A − B) × ( A − 2 B) × + ... + ( A − (12m − 1) B ] × = 100 100 100 100 n { A + ( A − B) + ( A − 2 B ) + [ A − (12m − 1) B]} = = 100 n {12mA − [ B + 2 B + ... + (12m − 1) B ]} = 100 Tu gavixsenebT ariTmetikuli progresiis jamis formulas, gveqneba: 12 m − 1 B + 2 B + ... + (12m − 1) B = 12 mB = (72m 2 − 6m) B . 2 aqedan n A n 1 L= [12mA − (72m 2 − 6m) ]= (6m + ) A . 2 100 12m 100 magaliTi 2. saxelmwifom gadawyvita ekonomikis stimulirebis mizniT SemoiRos sagadasaxado SeRavaTebi. vTqvaT, adamianma miiRo sagadaxado Se RavaTi 600 lari da daxarja Tanxis 80%. vTqvaT, adamianTa jgufi, romelic am daxarjul Tanxas iRebs, jamurad, rogorc xelfasis nawils, xarjavs mis 80%, aseve, adamianTa sxva jgufi romelic, iRebs, jamurad, rogorc xelfasis nawils, wina daxarjul jamur Tanxas, xarjavs am Tanxis 80% da ase grZeldeba usasrulod. gamovTvaloT erTi 600_lariani sagadasaxado SeRavTis ekonomikuri efeqturoba. amoxsna: yvela im adamianis daxarjul jamuri Sromis Rirebuleba, romlebic dasaqmdnen 600- lariani arapirdapiri investiciiT, SeiZleba warmovadginoT usasrulod klebadi a1 + a 2 + a 3 + ... + a n + ... geometriuli progresiis jamis saxiT, romlis pirveli wevria a1 = 0,8 ⋅ 600 = 480 , xolo mniSvneli- q = 0,8 . Tu gavixsenebT usasrulod klebad geometriuli progresiis jamis formulas, gveqneba: a 480 a1 + a 2 + a3 + ... + a n + ... = 1 = = 2400 . 1 − q 0,2 maSasadame, erTi 600- lariani sagadasaxado SeRavTi iwvevs 2400 lari Rirebulebis daxarjuli Sromis raodenobis anazRaurebas da Sesabamisi namati produqciis Seqmnas. magaliTi 3. vTqvaT, rezervuari icleba mis fskerSi arsebuli naxvretidan siCqariT, romelic warmoadgens diskretul drois funqcias

46 da

aqvs

saxe:

v (n) =

c , c > 0, k > 0 , e kn

vipovoT

rezervuarSi

arsebuli

nivTierebis raodenoba. amoxsna: rezervuarSi drois sawyis momentSi arsebuli nivTierebis raodenoba tolia rezervuaris mTlianad daclis, misgan gadmoRvrili mTeli nivTierebis raodenobisa. es raodenoba ki warmoadgens Semdegi mwkrivis: c c c c + 2 k + 3k + ... + nk + ... k e e e e jams, romelic tolia:

1 k 1 c e =c k . 1 e −1 1− k e

7. savarjiSoebi me-2 TavisaTvis 1.ipoveT Semdegi mwkrivebis jami: ∞ ∞ ∞ ∞ (−1) n 2 50 1 1 1 a) ∑ ( + ) , b) ∑ [ ] , g) ∑ n , d) ∑ ( 2 n + 2 n +1 ) , 2n − 1 10 n =1 2 n + 1 n =1 ( 2 n + 1)( 2 n − 1) n =1 7 n =1 10 ∞ 5 e) ∑ n . n =1 10 2. krebadia Tu ganSladi Semdegi mwkrivebi: ∞ ∞ ∞ 1 1 n a) ∑ , b) ∑ (1 + ) n , g) ∑ ln(1 + ) . n n n =1 1000 n + 1 n =1 n =1 3. Sedarebis niSnis gamoyenebiT gamoikvlieT Semdegi mwkrivebis krebadoba: ∞ ∞ ∞ ∞ ∞ 1 1 1 1 1 , b) ∑ 3 , g) ∑ , d) ∑ 2 n sin n , e) ∑ , v) a) ∑ 2 3 n −1 n =1 n =1 n + 3 n =1 n =1 ln n n =1 n(n 2 + 2) ∞

∑

. n( n + 1) 4. dalamberis niSnis gamoyenebiT gamoikvlieT Semdegi mwkrivebis krebadoba: ∞ ∞ ∞ ∞ ∞ n!2 n 3n 2n − 1 n n a) ∑ , b) ∑ , g) ∑ n , e) ∑ n , d) ∑ n . 2 n =1 ( 2n − 1)! n =1 n! n =1 3 n =1 n =1 n 5. gamoikvlieT absolutur da pirobiT krebadobaze Semdegi mwkrivebi: ∞ ∞ ∞ ∞ ∞ ∞ (−1) n +1 (−1) n (−1) n +1 n (−1) n (−1) n cos n a) ∑ ,B b) ∑ , g) ∑ , d) ∑ , e) ∑ , v) ∑ n! n n =1 n =1 2n − 1 n =1 2n − 1 n =1 n =1 ln n n =1 n n + 1 . n =1

47 Tavi 3 xarisxovani mwkrivebi 1. ZiriTadi cnebebi vTqvaT, mocemulia erT da igive Sualedze gansazRvrul funqciaTa mimdevroba u1 ( x), u 2 ( x), u 3 ( x),..., u n ( x),... . funqcionaluri mwkrivi ewodeba usasrulo jam ∞

u1 ( x) + u 2 ( x ) + u 3 ( x) + ... + u n ( x ) + ... = ∑ u n ( x ) .

(1)

n =1

Tu CavsvamT am mwkrivis TiToeul wevrSi x cvladis fiqsirebul mniSvnelobas funqciaTa saerTo gansazRvris aridan, miviRebT ricxviT mwkrivs, romelic SeiZleba krebadi iyos, SeiZleba ganSladi. vTqvaT, D im wertilTa simravlea funqciaTa saerTo gansazRvris aridan, romelTa CasmiTac x cvladis adgilze, miiReba krebadi mwkrivi. ganvixiloT D simravleze gansazRvruli f ( x ) funqcia, romelic simravlis TiToeul wertils Seusabamebs im ricxviTi mwkrivis jams, romelic miiReba (1) mwkrivisagan x cvladis magivrad am wertilis CasmiT. aseT funqcias, Tu D carieli simravle ar aris, uwodeben (1) funqcionaluri mwkrivis jams, xolo D simravles (1) mwkrivis krebadobis ares. funqcionaluri mwkrivebidan Cven ganvixilavT xarisxovan mwkrivebs. 2. xarisxovani mwkrivebi. abelis Teorema gansazRvreba 3.1. xarisxovani mwkrivi ewodeba Semdegi saxis mwkrivs: ∞

a 0 + a1 ( x − x 0 ) + a 2 ( x − x 0 ) 2 + ... + a n ( x − x 0 ) n + ... = ∑ a n ( x − x 0 ) 2 ,

(2)

n =1

sadac x 0 mocemuli ricxvia, a 0 , a1 , a 2 ,..., a n ,... - cnobili ricxviTi koeficientebi. Tu x0 = 0 , maSin (2) mwkrivi miiRebs saxes: ∞

a 0 + a1 x + a 2 x 2 + ... + a n x n + ... = ∑ a n x n . n =1

cxadia, yoveli xarisxovani mwkrivi ikribeba x = 0 wertilSi da misi jamia a0 . arseboben xarisxovani mwkrivebi, romlebic ikribebian mxolod x = 0 ∞

wertilSi, magaliTad, mwkrivi

∑ (nx)

. marTlac, rodesac n → ∞ , n − is

n −1

romeliRac mniSvnelobisTvis nx 0 > 2 . aqedan gamomdinare, (nx 0 ) n > 2 n da

lim(nx0 ) 2 > lim 2 n = ∞ . n →∞

n →∞

maSasadame, ar sruldeba mwkrivis krebadobis aucilebeli piroba arc erTi x ≠ 0 wertilisTvis. arseboben xarisxovani mwkrivebi, romlebic krebadia mTel (−∞, ∞) Sualedze. marTlac mwkrivi ∞ x (3) ( )n ∑ n −1 n

48 krebadia yoveli x0 ∈ (−∞, ∞) wertilisTvis, radganac rodesac n → ∞ , n − is n 1 rome- liRac mniSvnelobisTvis | |< n da Tu SevadarebT (4) mwkrivs x0 2 ∞ 1 krebad ∑ 2 n =1 2 mwkrivs, mwkrivTa Sedarebis niSnidan davadgenT, rom is krebadia. arseboben mwkrivebi, romlebic krebadia namdvil ricxvTa RerZis wertilebis nawilze da ganSladia danarCen wertilebze. marTlac mwkrivi ∞ x (4) ( )n ∑ n −1 3 krebadia rodesac − 3 < x < 3 , radgan am SemTxvevaSi igi warmoadgens x 1 x geometriul progresias, romlis mniSvnelicaa q = , | q |=| |< < 1 . Tu 3 3 3 x ∈ (−∞,−3] ∪ [3, ∞ ) , maSin (5) mwkrivi ganSladia, radgan progresiis x mniSvnelis moduli | q |=| |> 1 . 3 adgili aqvs Semdeg mniSvnelovan Teoremas: Teorema 3.1 (abelis Teorema). 1) Tu xarisxovani mwkrivi ∞

a 0 + a1 x + a 2 x 2 + ... + a n x n + ... = ∑ a n x n

(5)

n =1

krebadia x = x0 wertilSi, maSin igi absoluturad krebadia nebismier x wertilSi, romelic akmayofilebs utolobas: | x |<| x0 | . 2) Tu xarisxovani mwkrivi ganSladia x = x0 , maSin igi ganSladia nebismier x wertilSi, romelic akmayofilebs utolobas: | x |>| x0 | . abelis Teoremidan gamomdinareobs, rom arsebobs zRvruli ricxvi R > 0 , romelsac aqvs Semdegi Tvisebebi: (5) mwkrivi krebadia x cvladis yvela im mniSvnelobisTvis, romlebic akmayofileben utolobas: | x |< R da mwkrivi ganSladia x cvladis yvela im mniSvnelobisTvis, romlebic akmayofileben utolobas: | x |> R . am R ricxvs (5) mwkrivis krebadobis radiuss uwodeben. Tu mwkrivi krebadia mxolod erT x = 0 wertilSi, maSin krebadobis radiusi R = 0 . Tu mwkrivi krebadia ricxviTi RerZis nebismier wertilSi, maSin R = ∞ . ( − R, R ) intervali (1) mwkrivis krebadobis ares warmoadgens. abelis Teorema arafers gveubneba ( − R, R ) intervalis sazRvrebze: x = − R, x = R , mwkrivis krebadobaze, amitom am wertilebSi mwkrivis krebadoba unda Semowmdes yovel calkeul SemTxvevaSi. gamovTvaloT (5) mwkrivis krebadobis radiusi R . amisTvis gamoviyenoT ∞

mwkrivis krebadobis dalamberis niSani

∑| a n =1

x n | mwkrivisTvis

|a | a xn lim | n +1 n |=| x | lim n +1 , n →0 n→0 | a | an x n sadac

a n +1 . n →∞ a n

r = lim

r | x |< 1 ,maSin (5) mwkrivi absoluturad krebadia, Tu

ganSladi. maSasadame (5) mwkrivi krebadia, rodesac

a 1 = lim n . aqedan n a r n +1 krebadobis radiusi gamoiTvleba tolobiT: ganSladi,

rodesac

| x |>

R = lim n →∞

| x |<

gamomdinare

r | x |> 1 , maSin

a 1 = lim n n → ∞ r a n +1 (5)

an . a n +1

mwkrivis

(6)

∞

xn mwkrivis krebadobis radiusi. ∑ n = 0 n! amoxsna: (6) formuliT gveqneba a (n + 1)! R = lim n = lim = lim(n + 1) = ∞ . n →∞ a n →∞ n →∞ n! n +1

magaliTi 1. gamovTvaloT

x n −1 ∑ n =1 n ∞

magaliTi 2.

gamovTvaloT

mwkrivis krebadobis radiusi da

krebadobis are. amoxsna: (6) formuliT gveqneba: a n +1 = 1. R = lim n = lim n →∞ a n →∞ n n +1 maSasadame R = 1, D = ( −1,1) . gamovikvlioT mwkrivis krebadobis sakiTxi krebadobis aris boloebze. rodesac x = 1 , maSin miiReba ∞ 1 1 1 x n −1 = 1 + + + ... + + ... ∑ 2 3 n n =1 n harmoniuli mwkrivi, romelic ganSladia. rodesac x = −1 , maSin saqme gvaqvs ∞ 1 1 1 1 x n −1 = 1 − + − + ... + (−1) n −1 + ... ∑ 2 3 4 n n =1 n mwkrivTan, romelic, rogorc viciT, pirobiTad krebadia. Tu movaxdenT cvladis gardaqmnas x = y − x 0 , maSin (5) mwkrivi gardaiqmneba (2) mwkrivad. aqedan gamomdinare, Tu x cvladi Rebulobs mniSvnelobas (5) mwkrivis krebadobis aridan- ( − R, R ) , maSin y icvleba intervalSi ( x 0 − R, x0 + R ) , amitom (2) mwkrivis krebadobis radiusi igive iqneba, krebadobis are ki Seicvl- eba da gaxdeba ( x 0 − R, x0 + R ) . rogorc ricxviTi mwkrivebi, xarisxovani mwkrivebic SeiZleba SevkriboT an misi yoveli wevri gavamravloT raime mudmiv ricxvze.

50 ∞

∑ an x n

∞

∑b x

mwkrivis krebadobis radiusia R1 ,

n=0

n =0

∞

krebadobis radiusi ki- R2 ,

maSin

∑ (a n =0

+ bn ) x n mwkrivis krebadobis radiusi

akmayofilebs utolobas: R ≥ min{R1 , R2 } . ∞

mwkrivis

∑ an x n mwkrivis krebadobis radiusia R1 ,

∞

∑ Ca

maSin

n =1

krebadobis radiusi R

n =1

x n mwkrivis

daakmayofilebs utolobas: R ≥ R1 .

3 xarisxovani mwkrivis Tvisebebi ∞

Teorema 3.2. Tu xarisxovani mwkrivi

∑a n =0

krebadia ( − R, R )

Sualedze,

∞

maSin

misi jami

f ( x ) = ∑ a n x n warmoadgens uwyvet funqcias am Sualedis n =0

SigniT. Teorema 3.3. krebadi xarisxovani mwkrivi SeiZleba wevr-wevrad gavawarmovoT krebadobis ( − R, R ) Sualedis SigniT. ∞

es niSnavs, rom Tu f ( x) = ∑ a n x n ,

maSin

n =0

∞

f ′( x) = ∑ nan x n−1 = a + 2a1 x + 3a2 x 2 + ...., n =1

∞

f ′′( x) = ∑ n(n − 1)a n x n − 2 = 2a 2 + 6a3 x 2 + ....

n =1

................................................................................... gawarmoebiT miRebuli mwkrivebis krebadobis radiusebi igive iqneba, rac iyo mocemuli mwkrivis krebadobis radiusi. Teorema 3.4. krebadi xarisxovani mwkrivi SeiZleba wevr-wevrad gavaintegroT yovel Sualedze [a, b] , romelic mdebareobs krebadobis ( − R, R ) Sualedis SigniT. magaliTad b

∫ a

f ( x)dx = ∫ a 0 dx + ∫ a1 xdx + ∫ a 2 x 2 dx + ... + ∫ a n x n + = a 0 x +

a x n +1 a1 x 2 a 2 x 3 + + ... + n + ... . 2 3 n +1

amasTan, krebadobis radiusi darCeba igive (− R, R ) . 4. teiloris formula vTqvaT, funqcia f ( x ) gansazRvrulia da aqvs uwyveti warmoebuli f ′( x ) raime x 0 wertilis midamoSi. maSin niuton-laibnicis formuliT gvaqvs x

f ( x) = f ( x0 ) + ∫ f ′(t )dt . x0

Tu f ′(x ) warmoebadia x 0 wertilis midamoSi nawilobiTi integrirebiT miviRebT:

51 x

∫

f ′(t )dt = − ∫ f ′(t )d ( x − t ) = − f (t )( x − t ) x0

x xo

+ ∫ f ′′(t )( x − t )dt = x0

f ( x0 )( x − x0 ) +

∫ f ′′(t )( x − t )dt.

aqedan gamomdinare

f ( x) = f ( x0 ) + f ′( x0 )( x − x) 0 +

∫ f ′(t )( x − t )dt .

f ′′( x) warmoebadia x 0 wertilis midamoSi, analogiurad miviRebT:

f ( x) = f ( x0 ) + f ′( x0 )( x − x) 0 +

1 1 f ′′( x)( x − x0 ) 2 + ∫ f ′′′(t )( x − t ) 2 dt . 2 2 x0

Tu f ( x ) funqcia warmoebadia meoTxe rigamde, asev gveqneba:

f ( x) = f ( x 0 ) + f ′( x 0 )( x − x) 0 +

1 1 1 f ′′( x)( x − x0 ) 2 + f ′′′( x0 )( x − x0 ) + ∫ f 2! 3! 3! x0

( 4)

(t )( x − t ) 3 dt .

am process gavagrZelebT da davrwmundebiT, rom Tu f ( x ) warmoebadia n + 1 rigamde adgili eqneba formulas: 1 1 f ( x) = f ( x0 ) + f ′( x0 )( x − x 0 ) + f ′′( x 0 )( x − x 0 ) 2 + .. + f ( n ) ( x0 )( x − x0 ) + ... + 2! n! (1) x 1 ( n +1) n + ∫f (t )( x − t ) . n! x0 am formulas ewodeba f (x ) funqciis teiloris formula x 0 wertilSi, naSTis integraluri formiT. funqcias x

Rn ( x ) =

1 f ( n +1) (t )( x − x 0 ) n dt n! x∫0

(2)

ki naSTiTi wevri ewodeba. Tu gamoviyenebT saSualo mniSvnelobis formulas integralisTvis, (2) funqcia Caiwereba ase

Rn ( x ) = f

( n +1)

(ξ )

1 f ( n +1) (ξ ) n ( x x ) dt − = ( x − x0 ) n +1 , 0 ∫ (n)! x0 (n + 1)! x

(3)

sadac x 0 ≤ ξ ≤ x . naSTiTi wevris am formas lagranJis forma ewodeba. Tu CavTvliT, rom f ( x) = f ( 0 ) ( x) da gaviTvaliswinebT (3) formulas teiloris formula SeiZleba ufro kompaqturad CavweroT: n f n ( x0 ) f ( n +1) (ξ ) f ( x) = ∑ ( x − x0 ) n + ( x − x0 ) n +1 . (4) n! (n + 1)! n =1 (1) an (5) formulis marjvena mxares uwodeben funqciis gaSlas ( x − x 0 ) orwevris xarisxebad x 0 wertilSi, n rigamde. teiloris formulis naSTiTi wevrisTvis gvaqvs: n f n ( x0 ) f ( n +1) (ξ ) max | f ( n +1) ( x) | n +1 n n +1 | f ( x) − ∑ ( x − x 0 ) | =| ( x − x 0 ) |≤ δ , (5) n! (n + 1)! (n + 1)! n =1

52 sadac δ warmoadgens x 0 wertilis ganxiluli midamos sigrZes. ganvixiloT teiloris formula zogierTi elementaruli funqciisaTvis x0 = 0 wertilSi: 1. f ( x) = e x . rogorc viciT, am funqciis warmoebulebi nebismier rigamde erTi da igivea: (e x ) ( k ) = e x . amitom, teiloris formulas x0 = 0 wertilSi naSTiTi wevris lagranJis formiT, eqneba saxe: x2 x3 xn eξ ex = 1+ x + x n +1 . + + ... + + n! (n + 1)! 2! 3! 2. f ( x) = sin x . (sin x ) ′ = cos x = sin(

π 2

+ x ), (sin x ) ′′ = (sin(

π 2

+ x )) ′ = sin( 2

π 2

+ x ),..., (sin x) ( n ) = sin( n

π 2

+ x) .

nπ . rodesac n luwia f ( n ) (0) = 0 , rodesac n 2 kentia, anu rodesac n = 2k + 1 , adgili aqvs tolobas f ( 2 k +1) (0) = (−1) k . amgvarad, teiloris formulas sin x funqciisTvis, x 0 = 0 wertilSi, naSTis lagranJis formiT, aqvs saxe: x3 x5 x 2 n +1 (−1) n +1 2 n +3 sin x = x − x cos ξ , + + ... + (−1) n + 3! 5! (2n + 1)! (2n + 3)! sadac 0 ≤ ξ ≤ x . 3. f ( x ) = cos x . iseve, rogorc sin x funqciisTvis

aqedan gamomdinare, f ( n ) (0) = sin

(cos x) ( n ) = cos(n luwi n = 2k ricxvebisTvis

(cos x) ( 2 k ) = cos( 2k

kenti n = 2k + 1 ricxvebisTvis

+ x) ,

+ x ) = cos( kπ + x ) = (−1) k cos x ,

(cos x ) ( 2 k +1) = cos( kπ +

+ x ) = ( −1) k +1 sin x .

2 aqedan gamomdinare, teiloris formulas cos x funqciisTvis, x0 = 0 wertilSi, naSTis lagranJis formiT, aqvs saxe: x2 x4 x 2n (−1) n +1 2 n + 2 cos x = 1 − x cos ξ , + + ... + (−1) n + 2! 4! 2n! (2n + 2)! sadac 0 ≤ ξ ≤ x . 4. f ( x ) = ln(1 + x ) . −1 2 1 f ′( x) = ,.... . , f ′′′( x) = , f ′′( x) = 2 1+ x (1 + x ) 3 (1 + x) induqciiT miviRebT (−1) n −1 (n − 1)! ( n) f ( x) = (1 + x) n

53 da f ( n ) (0) = (−1) n −1 (n − 1)! . amgvarad, teiloris formulas ln(1 + x) funqciisTvis, x 0 = 0 wertilSi, naSTis lagranJis formiT, aqvs saxe: 1 x2 x3 x4 xn x n +1 ln(1 + x) = x − + − + ... + (−1) n −1 + (−1) n ⋅ , 2! 3! 4! n! n + 1 (1 + ξ ) n +1 sadac 0 ≤ ξ ≤ x . magaliTi 1. gavSaloT funqcia f ( x) = e x teiloris formuliT x0 = 0 wertilSi mesame rigamde. amoxsna: f ′( x ) = e x , f ′′( x ) = e x , f ′′′( x ) = e x , f ( x ) = e x da f (0) = 1, f ′(0) = 1, f ′′(0) = 1, f ′′′(0) = 1, f ( 4 ) (0) . aqedan gamomdinare, yoveli x ∈ R namdvili ricxvisTvis x 2 x 3 eξ 4 ex = 1+ x x , + + 2! 3! 4! sadac 0 < ξ ≤ x . magaliTi 2. gavSaloT funqcia f ( x) = sin x teiloris formuliT x 0 wertilSi mesame rigamde. amoxsna: (sin x)′ = cos x, (sin x)′′ = − sin x, (sin x)′′′ = − cos x, (sin x) ( 4 ) = sin x . aqedan gamomdinare − sin x0 − cos x0 sin x = sin x0 + cos x 0 ⋅ ( x − x 0 ) + ( x − x0 ) 2 + ( x − x0 ) 2 + 2 3! sin ξ + ( x − x0 ) 4 . 4! (5) formula saSualebas gvaZlevs gamovTvaloT funqciaTa miaxloebiTi mni- Svnelobani mocemuli sizustiT. magaliTi 3. gamovTvaloT sin 1 ε = 0,0001 sizustiT. amoxsna: 13 15 12 n +1 (−1) n +1 2 n +3 n | sin 1 − 1 − + + ... + (−1) |≤| 1 | ⋅ max | cos ξ |≤ ε = 0,0001 3! 5! (2n + 1)! (2n + 3)! anu

1 1 1 1 + + ... + (−1) n |≤| |≤ ε = 0,0001 . 3! 5! (2n + 1)! (2n + 3)! aqedan gamomdinareobs, ( 2n + 3)!≥ 10000 . am utolobidan miviRebT n ≥ 4 . maSasadame sin 1 -is ε = 0,0001 sizustiT gamosaTvlelad sakmarisia gavSaloT funqcia sin x teiloris formuliT, meoTxe rigamde, x0 = 0 wertilSi. | sin 1 − 1 −

5. teiloris mwkrivi vTqvaT, funqcia f (x ) gansazRvrulia da aqvs uwyveti warmoebulebi nebismier rigamde, raime x 0 wertilis midamoSi. xarisxovan mwkrivs

f ( n ) ( x0 ) ( x − x0 ) n n!

∞

∑ n =0

ewodeba f (x ) funqciis teiloris mwkrivi x 0 wertilSi. teiloris mwkrivs, rodesac x0 = 0 , maklorenis mwkrivi ewodeba. ganvixiloT teiloris mwkrivi zogierTi elementaruli funqciisaTvis: 1. f ( x) = e x . vTqvaT, x0 = 0 , maSin am wertilSi funqciis teiloris mwkrivs eqneba saxe: n

∞

x . ∑ n = 0 n!

(1)

|u | |x| xn = 0 , amitom mwkrivis . lim n +1 = lim n! n→∞ | u n | n→∞ n + 1 krebadobis dalamberis niSnidan gamomdinare, (1) mwkrivi absoluturad krebadia, nebismieri x ∈ R ricxvisTvis. mwkrivis krebadobis aucilebeli | x |n pirobidan gamomdinare lim = 0 . (1) mwkrivis kerZo jami n → ∞ n! k k xn x k +1 ξ x x Sk = ∑ x . = e − Rk = e − ∑ n = 0 n! n = 0 ( k + 1)! mwkrivis zogadi wevria u n =

xn x k +1 ξ = e x − lim x = ex − 0 = ex . k →∞ k →∞ k → ∞ (k + 1)! n = 0 n! sabolood SegviZlia davweroT k

lim(e x − Rk ) = lim ∑

∞

x e =∑ . n = 0 n! tolia Tavisi teiloris mwkrivis jamis. x

maSasadame f ( x) = e x 2. f ( x) = sin x . funqciis teiloris mwkrivs x0 = 0 wertilSi eqneba saxe:

x 2 n +1 (−1) . ∑ (2n + 1)! n =0 ∞

(2)

x 2 n +1 misi zogadi wevri u n = (−1) , igi dalamberis niSniT krebadia: (2n + 1)! |u | | x |2 lim n +1 = lim = 0. n → ∞ ( 2 n + 2)(2n + 3) n →∞ | u | n n

| x | 2 n +1 = 0 . (2) mwkrivis kerZo jami n →∞ (2n + 1)! k k x 2 n +1 x 2 k +3 cos ξ . S k = ∑ (−1) k = sin x − Rk = sin x − ∑ (−1) k +1 (2n + 1)! (k + 1)! n =0 n =0

amitom lim(−1) n

lim(sin x − Rk ) = lim ∑ (−1) k

k →∞

sabolood

k →∞

n =0

x 2 n +1 (−1) k +1 2 n +3 = sin x − lim x cos ξ = sin x − 0 = sin x . k →∞ ( 2k + 3)! (2n + 1)!

55 x 2 n +1 . sin x = ∑ (−1) (2n + 1)! n=0 ∞

3. f ( x ) = cos x . sin x funqciis analogiurad, cos x funqciis teiloris mwkrivisTvis x0 = 0 wer- tilSi gveqneba: ∞ (−1) n 2 n x . cos x = ∑ (3) n! n =0 4. f ( x ) = ln(1 + x ) . funqciis teiloris mwkrivs x0 = 0 wertilSi eqneba saxe: ∞

∑ n =1

(−1) n −1 n x . n

(4)

(4) mwkrivis zogadi wevria

un =

(−1) n −1 n x . n

(4) mwkrivis krebadobis radiusi

(−1) n − 2 |a | R = lim n −1 = lim n −n1−1 = 1 . n →∞ | a | n →∞ ( −1) n n maSasadame mwkrivi krebadia yvela x ∈ (−1,1) wertilisTvis. amitom lim u n = n→∞

(−1) n −1 n x = 0. n

rogorc wina SemTxvevaSi: k

lim (ln(1 + x) − Rk ) = lim ∑ (−1) n −1 k →∞

k →∞

n =1

xn x k +1 1 = ln(1 + x) − lim (−1) k ⋅ = ln(1 + x) − 0 = ln(1 + x) k → ∞ n k + 1 (1 + ξ ) k +1

. maSasadame ∞

ln(1 + x) = ∑ n =1

(−1) n −1 n x . n

(5)

5. f ( x) = (1 + x) α funqciis teiloris mwkrivs x0 = 0 wertilSi eqneba saxe: ∞

α (α − 1)...(α − (n − 1))

n =1

1+ ∑

xn .

(6)

rogorc zemoT ganxilul SemTxvevaSi, SegviZlia vaCvenoT, rom es mwkrivi krebadia, krebadobis radiusiT R = 1 , misi jami ki udris (1 + x) α . 1 6. f ( x) = . 1+ x2 analogiurad SeiZleba vaCvenoT, rom am funqciis teiloris mwkrivi x0 = 0 wertilSi aris

56 ∞

∑ (−1)

n −1

x 2 ( n −1) ,

(7)

n =1

romlis krebadobis radiusia R = 1 . magaliTi 1. gamovTvaloT 2 . amoxsna: (6) formuliT gveqneba: 1 1 1 1 1 1 1 1 ( − 1) ( − 1)( − 2) ( − 1)...( − (n − 1)) 1 1 2 2 1 + x = (1 + x) 2 = 1 + x + 2 2 x2 + 2 2 x 3 + ... + 2 2 x n + ... 2 2! 3! n! aqedan gamomdinare 1 1 1 ( − 1) − ( − (n − 1)) 1 1 1 3 1 15 1 2 2 = 1+1 = 1+ − + − + .. + 2 2 + ... . 2 4 2! 8 3! 16 4! n! magaliTi 2. gamovTvaloT ricxvi π . amoxsna: (7) mwkrivis wevr-wevrad integrebiT miviRebT:. x

1 1 1 1 dt = x − x 3 + x 5 + ... + (−1) n −1 x 2 n −1 + ... . 2 3 5 2 n − 1 0 1+ t 1 1 1 1 1 1 π = 4arctg1 = 4(1 − + − + − + ... + (−1) n −1 + ...) . 2n − 1 3 5 7 9 11

arctgx = ∫

∫e

magaliTi 3. gamovTvaloT integrali

− x2

amoxsna:

am integralis gamoTvla niuton-laibnicis formuliT

SeuZlebelia, vinaidan e − x

funqciis pirvelyofili ar gamoisaxeba

elementaruli funqciis saSualebiT. amitom gavSaloT funqcia e − x teiloris mwkrivad, gveqneba: 2 x 2n − x2 x4 − x6 e−x = 1 + + + + ... + (−1) n + ... . n! 1! 2! 3! 3.4 Teoremis ZaliT 1

∫ 0

e − x dx = ∫ dx + ∫ 2

− x2 x4 − x6 x 2n dx + ∫ dx + ∫ dx + ... + ∫ (−1) n dx + ... 1! 2! 3! n! 0 0 0

anu 1

2 1 1 1 1 + ... + (−1) n + ... . e − x dx = 1 − + − 3 10 42 (2n + 1)n! 0 miRebuli toloba saSualebas gvaZlevs gamovTvaloT integralis miaxloebiTi mniSvneloba.

∫

6. savarjiSoebi me-3 TavisaTvis I. ipoveT Semdegi xarisxovani mwkrivebis krebadobis are: ∞ ∞ ∞ ∞ ∞ ∞ ∞ xn xn x 2 n −1 n x n n 2 n −1 n 2 n2 , 3) , 4) , 4) , 5) 1) ∑ x n , 2) ∑ n , 3) ∑ ( ) x 3 x ( ) , ∑ ∑ ∑ ∑ n n =0 n =1 n n =1 n.2 n =1 2 n + 1 n =1 2n − 1 n =0 n =1 n + 1 2 3

∞ n! x n n!( x + 3) n ( x + 2) n 6) ∑ n , 7) ∑ , 8) . ∑ nn nn n =1 n n =1 n =1 II. eteiloris mwkrivad da ipoveT krebadobis radiusebi Semdegi

∞

funqciebisTvis: 1) a x , a > 0 , 2) sin 2 x , 3) ln(2 + x) , 4) cos 2 x , 5) xe −2 x , 6) cos 2 x , 7) e x ,

sin t ln(1 + t ) x dt , 11) ∫ dt , , 10) ∫ 2 t t 9+ x 0 0 1 15) sin 2 x cos 2 x , 16) . 4 − x4 x

12) (1 + e x ) 3 ,13) ∫ 0

dt 1− t

, 14) ln( x 2 + 3 x + 2) ,

58 Tavi 4 M mravali cvladis funqciebi 1. ZiriTadi cnebebi bunebaSi da praqtikuli saqmianobis dros xSirad gvxvdeba iseTi sidideebi, romelTa cvlilebac sxva ramodenime sididis cvlilebazea damokidebuli. amis safuZvelze bunebrivia, funqcionaluri damokidebulebis cneba unda gafarTovdes da Semovides mravali cvladis funqciis cneba. X 1 , X 2 ,..., X n , Y ricxviTi simravlis gansazRvreba 4.1. vTqvaT qvesimravleebia. Sesabamisobas, romelic X 1 × X 2 × ... × X n dekartis namravlis yovel elements Seusabamebs Y simravlis erT, sruliad gansazRvrul elements mravali cvladis funqcia ewodeba. mravali cvladis funqcias aRniSnaven ase: f : X 1 × X 2 × ... × X n → Y . ( x1 , x 2 ,...x n ) ∈ X 1 × X 2 × ... × X n elementis Sesabamisi y ∈ Y elementi aRiniSneba ase: y = f ( x1 , x 2 ,...x n ) . ukanaskneli Canaweri, xSirad funqciis aRsaniSnadac gamoiyeneba. X 1 × X 2 × ... × X n simravles funqciis gansazRvris are ewodeba, Y simravles funqciis cvlilebis are. n = 1 , saqme gveqneba CvenTvis kargad cnobil erTi cvladis Tu funqciasTan, Tu n = 2 saqme gveqneba ori cvladis funqciasTan, Tu n = 3 , sami cvladis funqciasTan da ase Semdeg. Cven ganvixiloT ori cvladis funqcia z = f ( x, y ) . yvelaferi rac naTqvami iqneba qvemoT, aseTi funqciebisaTvis marTebuli iqneba nebismieri raodenobis cvladis funqciebisTvisac. yovel ricxviT ( x, y ) wyvils calsaxad Seesabameba iseTi wertili sakoordinato sibrtyeze, romlis koordinatebic warmodgenilia mocemuli wyviliT. amitom nebismieri ori cvladis funqcia warmoadgens sibrtyis wertilebis Sesabamisobas ricxviTi simravlis elementebTan. aseve, sami cvladis funqcia warmoadgens sivrcis wertilebis Sesabamisobas ricxviTi simravlis elementebTan. Sesabamisad, ori an sami cvladis funqciis gansazRvris ared SeiZleba CaiTvalos wertilTa simravle sibrtyeze an sivrceSi. x , y cvladebs z = f ( x, y ) funqciis argumentebs uwodeben. magaliTebi: 1. z = x 2 + 2 y + 1 warmoadgens ori cvladis funqcias, x da y cvladebs SeuZliaT miiRon nebismieri, namdvili ricxviT mniSvneloba,, amitom misi gansazRvris area R 2 = R × R simravle. aseve z cvladsac SeuZlia miiRos nebismieri namdvili ricxviTi mniSvnelobani, amitom misi cvlilebis area R simravle. 2. u = x 2 + y 2 + z 2 warmoadgens sami cvladis funqcias. x, y , z cvladebs SuZliaT miiRon nebismieri, namdvili ricxviT mniSvneloba, amitom funqciis gansazRvris area R 3 = R × R × R simravle. u cvladi Rebulobs mxolod arauaryofiT namdvil mniSvnelobebs, amitom funqciis cvlilebis area R + = {x ∈ R | x ≥ 0} simravle. 3. z = 1 − x 2 − y 2 funqciis gansazRvris ares warmoadgens iseTi wyvilebis qvesim-ravle R 2 = R × R simravleSi, romelTa komponentebic akmayofileben utolobas x 2 + y 2 ≤ 1 . cvlilebis area R + . x 2 + y 2 ≤ 1 utoloba gansazRvravs

59 im wertilTa simravles, romelic funqciis gansazRvris ares warmoadgens. mocemul kerZo SemTxvevaSi es simravlea wre, centriT koordinatTa saTaveSi da radiusiT 1. iseve, rogorc erTi cvladis funqciisaTvis, arsebobs ori cvladis funqciis grafikis cnebac. grafiki ewodeba z = f ( x, y ) funqciis samganzomilebiani sivrcis iseTi wertilebis simravles, romelTa koordinatebia ( x, y , f ( x, y )) (nax.16). martiv SemTxvevaSi, es simravle warmoadgens zedapirs, romlis gantolebacaa z = f ( x, y ) . M ( x, y , f ( x, y ))

z = f ( x, y )

Y y0 + δ N 0 ( x0, y 0 )

y0 − δ

O x0 − δ x0 + δ

x X

N ( x, y )

nax.16 nax.17 SemovitanoT ramdenime mniSvnelovani cneba. vTqvaT δ > 0 raime arauaryofiTi ricxvia, N 0 ( x0 , y 0 ) raime wertilia sibrtyeze. N ( x, y ) wertilTa simravles, romelTa koordinatebic sibrtyis im akmayofileben utolobebs 0 <| x − x 0 |< δ , 0 <| y − y 0 |< δ , N 0 ( x0 , y 0 ) wertilis δ midamo ewodeba. vityviT, rom N ( x, y ) wertili miiswrafvis N 0 ( x0 , y 0 ) wertilisken, Tu manZili am wertilebs Soris miiswrafvis nulisaken anu

lim | NN 0 |= lim ( x − x0 ) 2 + ( y − y 0 ) 2 = 0 . ukanasknel tolobas adgili aqvs maSin da mxolod maSin, rodesac x → x0 da y → y 0 . N 0 ( x0 , y 0 ) wertils ewodeba sibrtyeze moTavsebuli D aris zRvruli wertili, Tu nebismieri ε > 0 ricxvisaTvis, D areSi arsebobs iseTi N ( x, y ) wertili, rom 0 <| x − x0 |< ε , 0 <| y − y 0 |< ε . N 0 ( x0 , y 0 ) wertilis δ midamo geometriulad warmoadgens im kvadratis Siga wertilebis simravles, romlis centria N 0 ( x0 , y 0 ) wertili, xolo gverdebi 2δ sigrZis monakveTebia(nax.17).

60 gansazRvreba 4..2. A ricxvs ewodeba z = f ( x, y ) ori cvladis funqciis zRvari N 0 ( x0 , y 0 ) wertilSi, Tu nebismieri ε > 0 ricxvisTvis arsebobs ricxvi δ > 0 , rom rodesac 0 <| x − x 0 |< δ , 0 <| y − y 0 |< δ , maSin | f ( x, y ) − A |< ε . ori cvladis funqciis zRvari N 0 ( x0 , y 0 ) wertilSi aRiniSneba ase:

A = lim f ( x, y ) x → x0 y → y0

lim

N ( x , y ) → N 0 ( x0 , y 0 )

f ( x, y ) .

gansazRvreba 4.3. raime D areze gansazRvrul z = f ( x, y ) funqcias ewodeba uwyveti N 0 ( x 0 , y 0 ) ∈ D wertilSi, Tu am funqciis zRvari mocemul wertilSi emTxveva mis mniSvnelobas am wertilSi anu lim f ( x, y ) = f ( x0 , y 0 ) . x → x0 y → y0

Tu funqcia uwyvetia D aris TiToeul wertilSi, maSin amboben, rom funqcia uwyvetia am areze. sidideebs Δx = x − x0 , Δy = y − y 0 uwodeben Sesabamis argumentTa nazrdebs, xolo sidides Δz = f ( x, y ) − f ( x0 , y 0 ) funqciis nazrds. funqciis z = f ( x, y ) uwyvetoba N 0 ( x0 , y 0 ) wertilSi niSnavs, rom funqciis nazrdi Δz → 0 rodesac Δx → 0 da Δy → 0 anu argumentTa usasrulod mcire nazrdebs Seesabameba usasrulod mcire funqciis nazrdi. sibrtyis im wertils sadac funqciis uwyvetobis piroba irRveva, funqciis wyvetis wertili ewodeba. 1 magaliTebi: 1 funqciis erTaderTi wyvetis wertilia z= 2 x + y2 koordinatebiT x = 0, y = 0 . ⎧ 1 ,x ≠ y ⎪ z = ⎨x − y , 2. ⎪1, x = y ⎩ funqcis wyvetis wertilia yvela is wertili, romlisTvisac x = y .

2. ori cvladis funqciis kerZo warmoebulebi vTqvaT, mocemulia funqcia z = f ( x, y ) gansazRvruli raime D areze da N 0 ( x0 , y0 ) ∈ D . ganvixiloT sidide Δz x = f ( x 0 + Δx , y 0 ) − f ( x 0 , y 0 ) , am sidides mocemuli funqciis kerZo nazrdi ewodeba x argumentis mimarT. analogiurad ganisazRvreba funqciis kerZo nazrdi y argumentis mimarT: Δz y = f ( x 0 , y 0 + Δy ) − f ( x 0 , y 0 ) . gansazRvreba 4.4. Tu arsebobs sasruli zRvari Δz f ( x0 + Δx, y 0 ) − f ( x0 , y 0 ) lim x = lim , Δx →0 Δx Δx →0 Δx

61 maSin amboben, rom funqcias kerZo warmoebuli z = f ( x, y ) gaaCnia N 0 ( x0 , y0 ) ∈ D wertilSi x argumentis mimarT, zRvars ki uwodeben funqciis kerZo warmoebuls x argumentis mimarT, am wertilSi. am kerZo warmoebulis aRsaniSnad gamoiyeneba erT-erTi Semdegi ∂z ∂f simboloebidan: , , z ′x , f x′ . Tu saWiroa mivuTiToT, romel wertilSa ∂x ∂x funqciis kerZo warmoebuli, maSin viyenebT aRniSvnebs: ∂z ∂f | x = x0 , | x = x0 , f x′ ( x0 , y 0 ) . ∂x y = y0 ∂x y = y0 analogiurad, Tu arsebobs sasruli zRvari Δz y f ( x0 , y 0 + Δy ) − f ( x0 , y 0 ) = lim lim , Δy →0 Δy Δy →0 Δy maSin amboben, rom funqcias kerZo warmoebuli z = f ( x, y ) gaaCnia N 0 ( x 0 , y 0 ) ∈ D wertilSi y argumentis mimarT, zRvars ki uwodeben funqciis kerZo warmoebuls y argumentis mimarT, am wertilSi. am kerZo warmoebulis aRniSvnebi wina SemTxvevis analogiuria. magaliTi 1. vipovoT z = ax 2 + sin 3 ( x + 2 y 2 ) funqciis kerZo warmoebulebi z ′x da z ′y . amoxsna: rogorc kerZo warmoebulebis gansazRvrebidan Cans, rom gamovTvaloT kerZo warmoebuli erT erTi argumentis mimarT, saWiroa meore argumenti CavTvaloT mudmiv sidided da funqcia CavTvaloT erTi cvladis funqcid, Semdeg gamovTvaloT am erTi cvladis funqciis warmoebuli z ′x = 2ax + 3 sin 2 ( x + 2 y 2 ) , z ′y = 6 y sin 2 ( x + 2 y 2 ) . Z

α C

O K l1 P

X nax.18

N 0 ( x0 , y 0 ) γ 1

l2 γ 2 P

62 gavarkvioT, ra geometriuli azri aqvT z = f ( x, y ) funqciis kerZo warmoebulebs N 0 ( x 0 , y 0 ) ∈ D wertilSi(nax. 18). ganvixiloT gantolebiT gansazRvruli zedapiri z = f ( x, y ) samganzomilebian Oxyz sivrceSi. N 0 ( x 0 , y 0 ) ∈ D wertils am zedapirze Seesabameba wertili M 0 ( x0 , y 0 , z 0 ) , sadac z 0 = f ( x0 , y 0 ) . N 0 ( x0 , y 0 ) wertilze gavataroT sibrtye α , romelic paraleluria YOZ sibrtyis. misi gantoleba iqneba x = x0 . cxadia es sibrtye gadakveTs zedapirs raRac CK wirze da gaivlis M 0 ( x0 , y 0 , z 0 ) wertilze. wiri, romelic miiReba sibrtyis da zedapiris gadakveTiT, SeiZleba warmovadginoT gantolebaTa sistemiT ⎧ z = f ( x, y ) , ⎨ ⎩ x = x0 an erTi gantolebiT z = f ( x 0 , y ) , romelic miiReba am sistemidan. funqcia z = f ( x 0 , y ) erTi cvladis funqciaa. z = f ( x, y ) funqciis kerZo warmoebuli z ′y = f y′ ( x 0 , y 0 ) iqneba am erTi cvladis funqciis Cveulebrivi warmoebuli y = y 0 wertilSi. aqedan gamomdinare, z ′y = f y′ ( x 0 , y 0 ) warmoadgens

CK wiris l1 mxebis sakuTxo koeficients anu l1 mxebis XOY sibrtyesTan daxris kuTxis tangenss. axla N 0 ( x0 , y 0 ) wertilze gavataroT sibrtye α , romelic paraleluria XOZ sibrtyis. misi gantoleba iqneba: y = y 0 . am sibrtyis zedapirTan kveTiT miRebuli PQ wiris gantoleba iqneba z = f ( x 0 , y ) . kerZo warmoebuli z ′y = f y′ ( x 0 , y 0 ) warmoadgens PQ wiris l 2 mxebis sakuTxo koeficients anu l 2 mxebis XOY

sibrtyesTan daxris kuTxis tangenss.

3. ori cvladis funqciis sruli diferenciali vTqvaT, mocemulia funqcia z = f ( x, y ) gansazRvruli raime D areze, amasTan D areze arsebobs z ′x = f x′ ( x0 , y 0 ) , z ′y = f y′ ( x 0 , y 0 ) kerZo warmoebulebi. vTqvaT N 0 ( x 0 , y 0 ) ∈ D , ganvixiloT sidide Δz = f ( x0 + Δx, y 0 + Δy ) − f ( x0 , y 0 ) , am sidides z = f ( x, y ) funqciis sruli nazrdi ewodeba. gansazRvreba 4..5. z = f ( x, y ) funqcias ewodeba diferencirebadi funqcia N 0 ( x0 , y 0 ) wertilSi, Tu misi sruli nazrdi warmoidgineba Semdegi saxiT: (1) Δ z = A Δ x + B Δ y + α ( Δ x ) Δ x + β ( Δy ) Δ y , sadac A da B raime ricxvebia, α ( Δx), β ( Δy ) funqciebi akmayofileben pirobebs: lim α (Δx) = 0, lim β (Δy) = 0 . Δx →0

Δy →0

(1) tolobis marjvena mxaris wrfiv nawils AΔx + BΔy ewodeba z = f ( x, y ) funqcis sruli diferenciali N 0 ( x0 , y 0 ) wertilSi da aRiniSneba ase:

x = x0 y = y0

= AΔx + BΔy .

63 rogorc vxedavT, sruli diferenciali warmoadgens Δx da Δy argumentebis wrfiv funqcias. davadginoT A da B ricxvebis mniSvnelobani. amisaTvis (1) tolobis orive mxare gavyoT jer Δx -ze, Semdeg miRebuli tolobis orive mxares gadavideT zRvarze, rodesac Δx → 0 , miviRebT: f x′ ( x0 , y 0 ) = A . analogiurad, (1) tolobis orive mxare gavyoT jer Δy -ze, Semdeg miRebuli tolobis orive mxares gadavideT zRvarze, rodesac Δy → 0 , miviRebT: f y′ ( x 0 , y 0 ) = B . am dazustebebis Semdeg, z = f ( x, y ) funqcis sruli diferenciali Δx da N 0 ( x0 , y 0 ) wertilSi, SeiZleba warmovadginoT, rogorc Δy argumentebis konkretuli wrfivi funqcia

x = x0 y = y0

′ = f x′ ( x0 , y 0 )Δx + f y′ ( x 0 , y 0 )Δy .

x da y cvladebis diferencialebi dx, dy Tu CavTvliT, rom rac am cvladebis nazrdebi Δx da Δy , SeiZleba davweroT dz

x = x0 y = y0

igivea,

′ = f x′ ( x0 , y 0 )dx + f y′ ( x0 , y 0 )dy .

Tu funqcia diferencirebadia D aris nebismier wertilSi, maSin mas uwodeben diferencirebads D areze. magaliTi 1. vipovoT π z = ax 2 + sin 3 ( x + 2 y 2 ) funqciis sruli diferenciali N (π , ) wertilSi. 2 amoxsna:

x =π y=

π 2

= (2ax + 3 sin 2 ( x + 2 y 2 ))′x

x =π y=

π 0

dx + (6 y sin 2 ( x + 2 y 2 ))′y

x =π y=

dy = (2aπ + 3)dx + 3πdy . π 2

sruli diferenciali saSualebas gvaZlevs gamovTvaloT funqciis miaxloebiTi mniSvneloba, marTlac f ( x 0 + Δx, y 0 + Δy ) − f ( x0 , y 0 ) = f x′ ( x0 , y 0 )Δx + f x′ ( x0 , y 0 )Δy + α (Δx)Δx + β (Δy )Δy . Δx da Δy nazrdebis sakmaod mcire mniSvnelobebisTvis gveqneba miaxloebiTi toloba: f ( x0 + Δx, y 0 + Δy ) − f ( x0 , y 0 ) ≈ f x′ ( x0 , y 0 )Δx + f x′ ( x0 , y 0 )Δy . am miaxloebiTi tolobidan SegviZlia davweroT: f ( x0 + Δx, y 0 + Δy ) ≈ f ( x0 , y 0 ) + f x′ ( x0 , y 0 )Δx + f x′ ( x0 , y 0 )Δy . Tu x0 ≤ x ≤ x0 + Δx da y 0 ≤ y ≤ y 0 + Δy , sakmaod mcire Δx da Δy nazrdebis mniSvnelo- bebisTvis gveqneba: f ( x0 , y ) ≈ f ( x 0 , y 0 ) + f x′ ( x 0 , y 0 )Δx + f x′ ( x0 , y 0 )Δy . (2) magaliTi 2. gamovTvaloT z = e xy funqciis mniSvneloba, rodesac x = 0,35, y = −0,5 . amoxsna: (2) formuliT: z = e 0,35⋅( −0,5) = e 0⋅0 + (−0,5)e 0 0,35 + 0,35e 0 (−0,5) = 1 − 0,35 = 0,65 .

64 4. zedapiris mxebi sibrtye da normali rogorc cnobilia, D areze diferencirebadi z = f ( x, y ) funqcis grafiki warmoadgens zedapirs. amovirCioT wertili N 0 ( x0 , y 0 ) D areSi. am wertilis amorCeviT ganisazRvreba wertili zedapirze M 0 ( x0 , y 0 , f ( x0 , y 0 )) . rogorc geometriidan aris cnobili, zedapiris mxebi sibrtye mocemul wertilSi warmoadgens sibrtyes, romelic gadis zedapiris am wertilze gamaval mis nebismier or mxeb wrfeze da es sibrtye erTaderTia. aseTi sibrtyis gantoleba, zogadad, unda iyosSemdegi saxis: z − z 0 = A( x − x0 ) + B( y − y 0 ) .

l1 l2 zedapiris da mxebi wrfeebi (nax.18), cxadia, M 0 ( x0 , y 0 , f ( x0 , y 0 )) , wertilSi gamaval mxeb sibrtyeze.

mdebareoben

l1 wrfis wertilTa koordinatebi ( x0 , y, z ) saxisaa, radganac es wrfe mxeb sibrtyeze mdebareobs, amitom l1 wrfis wertilebisTvis adgili unda hqondes tolobas: z − z 0 = B( y − y 0 ) . meore mxriv, l1 wrfis gantolebaa: z − z 0 = f y′ ( x 0 , y 0 )( y − y 0 ) . maSasadame B = f y′ ( x 0 , y 0 ) . analogiurad SeiZleba gamovTvaloT A sididis mniSvnelobac, gveqneba: A = f x′ ( x0 , y 0 ) . am gamoTvlebis Semdeg mxebi sibrtyis gantoleba miiRebs saxes: (1) z − z 0 = f x′ ( x 0 , y 0 )( x − x 0 ) + f y′ ( x 0 , y 0 )( y − y 0 ) . gansazRvreba 4..6. zedapiris M0 wertilSi mxebi sibrtyis marTobul wrfes, romelic amave wertilze gadis, M 0 wertilSi zedapiris normali ewodeba. rogorc analizuri geometriidan aris cnobili (1) saxis gantolebiT gansazRvruli sibrtyis M 0 ( x0 , y 0 , z 0 ) wertilSi gamavali marTobuli wrfis gantolebas unda hqondes saxe: z − z0 x − x0 y − y0 = = . (2) −1 f x′ ( x0 , y 0 ) f y′ ( x 0 , y 0 ) aqedan gamomdinare (2) normalis gantolebaa. magaliTi 1. vipovoT z = 2 x 2 + 3 y 2 paraboloidis M M 0 (1,1,5) wertilSi mxebi sibrtyis da normalis gantolebani. amoxsna: (2 x 2 + 3 y 2 )′x x =1 = 4 x x =1 = 4 , (2 x 2 + 3 y 2 )′y y =1 = 6 y y =1 = 6 . aqedan gamomdinare, mxebi sibrtyis gantoleba iqneba: z − 5 = 4( x − 1) + 6( y − 1) . normalis gantoleba ki z − 5 x −1 y −1 . = = −1 4 6

65 5. rTuli funqciis warmoebuli vTqvaT, mocemulia funqcia z = f (u , v) gansazRvruli da diferencirebadi raime U areze. vTqvaT, u da v , Tavis mxriv warmoadgens x da y cvladebis funqc-iebs: u = u ( x, y ) , v = v ( x, y ) , romlebic gansazRvruli arian da diferencirebadi raime D areze da romelTa mniSvnelobisgan Sedgenili wyvilebi (u , v) Sedis U areSi. aseT SemTxvevaSi z warmoadgens rTul funqcias z = f (u ( x, y ), v( x, y )) = F ( x, y ) . davafiqsiroT y cvladi da x cvlads mivceT nazrdi Δx . rogorc me-3 qveTavidan dan viciT: Δz = f u′Δu x + f v′Δv x + α (Δu )Δu x + β (Δv)Δv x . Tu am tolobis orive mxares gavyofT Δx -ze miviRebT:

Δu Δv Δu Δv Δz = f u′ x + f v′ x + α (Δu x ) x + β (Δv x ) x . (1) Δx Δx Δx Δx Δx u = u ( x, y ) da v = v ( x, y ) uwyveti funqciebia, amitom Tu Δx → 0 , maSin Δu → 0 , Δv → 0 . Δx → 0 Tu (1) tolobaSi gadavalT zRvarze, rodesac da lim α ( Δ x ) = 0 , lim β ( Δ y ) = 0 gaviTvaliswinebT, rom , miviRebT: Δx →0

Δy →0

∂z ∂u ∂v (2) = f u′ + f v′ . ∂x ∂x ∂x analogiurad, Tu davafiqsirebT x cvlads da y cvlads mivcemT nazrds, miviRebT: ∂u ∂v ∂z = f u′ + f v′ . (3) ∂y ∂y ∂y ∂z magaliTi 1. vipovoT z = sin((ln( xy )e x + y ) funqciis kerZo warmoebulebi ∂x ∂z da . ∂y amoxsna: u = ln xy , v = e x + y , z = sin uv . ∂z ∂ sin uv ∂u ∂ sin uv ∂v 1 = + = cos(ln( xy )e x + y )e x + y + cos(ln( xy )e x + y ) ln( xy )e x , ∂x ∂u ∂x ∂v ∂x x

∂z ∂ sin uv ∂u ∂ sin uv ∂v 1 = + = cos(ln( xy )e x + y )e x + y + cos(ln( xy)e x + y ) ln( xy)e y . ∂y ∂u ∂y ∂v ∂y y Tu mocemulia funqcia z = f ( x, y ) da x = x (t ), y = y (t ) , maSin z = f ( x (t ), y (t )) ∂z ∂f dx ∂f dy . warmoadgens t argumentis rTul funqcias da = + ∂t ∂x dt ∂y dt z = f (u ( x, y ), v( x, y )) = F ( x, y ) rTuli funqciis diferenciali dz = Fx′dx + Fy′ rogorc (2) da (3) formulebidan Cans, Semdegi saxis unda iyos:

66 ∂u ∂v ∂v ∂u + f v′ )dy . + f v′ )dx + ( f u′ ∂y ∂y ∂x ∂x es toloba gardavqmnaT Semdegnairad ∂u ∂v ∂v ∂u dz = f u′ ( dx + dy ) + f v′( dx + dy ) . ∂y ∂x ∂y ∂x

dz = ( f u′

Tu gaviTvaliswinebT, rom

du =

(4)

∂u ∂v ∂v ∂u dx + dy , dv = dx + dy , ∂x ∂y ∂x ∂y

(4) toloba miiRebs saxes:

dz = f u′du + f v′dv . rogorc vxedavT, imis miuxedavad iqneba u da v cvladebi damoukidebeli, Tu TiToeuli maTgani damokidebuli iqneba sxva x da y cvladebze, adgili aqvs sruli diferencialis formis invariantulobas anu ucvlelobas. sruli diferencialis formis invariantulobas cxadia aqvs formaluri da ara Sinaarsobrivi xasiaTi. rodesac u da v cvladebi damoukidebelni arian, du da dv sidideebi warmoadgenen am cvladebis mudmiv nazrdebs, maSin, rodesac u = u ( x, y ) , v = v ( x, y ) , du da dv sidideebi warmoadgenen am funqciebis diferencialebs. sruli diferencialis formis invariantuloba saSualebas gvaZlevs vaCvenoT Semdegi formulebis samarTlianoba: f gdf − fdg . d ( f ± g ) = df ± dg , d ( fg ) = fdg + gdf , d ( ) = g g2

magaliTi 2. vipovoT z = ln( x 2 + y 2 ) funqciis sruli diferenciali. amoxsna: ∂ ln u 1 z = ln u ,sadac u = x 2 + y 2 . aqedan dz = = du , du = 2 xdx + 2 ydy . u ∂u sabolood: 2 xdx + 2 ydy . dz = x2 + y2 6. aracxadi funqcia da misi warmoebuli vTqvaT funqcia z = F ( x, y ) gansazRvrulia raime D areze. ganvixiloT gantoleba (1) F ( x, y ) = 0 . am gantolebiT cvladebi x da y erTmaneTTan garkveul urTierTkavSirSi arian. Tu x cvladis yoveli mniSvnelobisaTvis, raime ricxviTi Sualedidan, arsebobs y cvladis iseTi mniSvneloba, romelic x cvladis am mniSvnelobasTan erTad akmayofilebs (1) gantolebas, maSin saqme gvaqvs aracxad funqciasTan y = f ( x ) , romelic ganisazRvreba (1) gantolebiT. SeiZleba sxvanairadac iyos, y cvladis yoveli mniSvnelobisaTvis, raime ricxviTi Sualedidan, arsebobdes x cvladis iseTi mniSvneloba, romelic y cvladis am mniSvnelobasTan erTad akmayofilebs (1) gantolebas maSin (1) gantoleba gansazRvravs aracxad funqcis x = ϕ ( y ) . rodesac vlaparakobT aracxad funqciaze, mxedvelobaSi gvaqvs funqciis mocemis erT-erTi xerxi da ara misi raime Tviseba. zogierT SemTxvevaSi

67 SesaZlebelia (1) gantolebis amoxsna erT-erTi cvladis mimarT, advilad gadavdivarT aracxadi funqciidan Cveulebriv funqciaze. magaliTad F ( x, y ) = x 3 + y 2 ( x − 1) = 0 aracxadi funqciidan advilad vRebulobT Cveulebriv funqcias:

maSin

x3 . x −1 magram es yovelTvis ar xerxdeba. magaliTad, x y = y x gantolebis amoxsna SeuZlebelia romelime cvladis mimarT. anu ar SegviZlia cxadad wamovadginoT Sesabamisoba y = f (x ) an x = ϕ ( y ) . aracxadad mocemuli funqcia realuri funqciaa, aseTi funqciebi mravali amocanis gadaWris procesSi SeiZleba Segvxvdes, amitom wamoiWreba maTi diferencrebadobis sakiTxic. vTqvaT, F ( x, y ) = 0 gantoleba gansazRvravs aracxad funqcias. am aracxadi y = f (x ) funqciisTvis, cxadia, adgili aqvs igivur tolobas: F ( x, y ( x )) = 0 . gavawarmooT am tolobis orive mxare x cvladiT, rTuli funqciis gawarmoebis wesiT, miviRebT: ∂F ∂F ∂F dy = + =0. ∂x ∂x ∂y dx ∂F ≠ 0 , gveqneba: am ukanasknel tolobaSi, Tu CavTvliT, rom ∂y ∂F dy (2) = − ∂x . ∂F dx ∂y 3 magaliTi 1. vipovoT y + 3 y = x gantolebiT mocemuli y = f ( x) aracxadi dy . funqciis warmoebuli dx ∂F ∂F amoxsna: F ( x, y ) = y 3 + 3 y − x = 0 , = 3 y 2 + 3 . (2) formulis = −1 , ∂x ∂y gamoyenebiT gveqneba: dy −1 1 . =− 2 = 2 dx 3 y + 3 3( y + 1) magaliTi 2. vTqvaT, aracxadi funqcia mocemulia gantolebiT x+ y= a. dy vipovoT . dx amoxsna: F ( x, y) = x + y − a = 0 , y=

68 ∂F 1 ∂F 1 = = , . (2) formulis ∂y 2 y ∂x 2 x magaliTi 3. vipovoT

gamoyenebiT, gveqneba:

dy y =− . dx x

x2 y2 − = 1 hiperbolis mxebi, romelic gadis 32 4 2

M (1,5) wertilze. amoxsna:

x2 y2 − − 1 = 0 gantolebiT mocemuli, y = f ( x ) aracxadi 32 4 2 funqciis grafikis mxebis povnis amocanis ekvivalenturia. aseTi funqciis grafikis, M (1,5) wertilze gamavali, mxebi wrfis gantolebas eqneba saxe: dy y − y0 = ( x − x0 ) . dx funqciis grafikis mxebis sakuTxo koeficienti tolia am funqciis warmoebulisa, Sesabamis wertilSi 2 y dy 4 2 x 42 dy ∂F 2x ∂F =− 2 , = 2 . aqedan gamomdinare, = da mxebis = 2 , x =1 dx 3 y dx y =5 3 2 ⋅ 5 ∂y ∂x 3 4 gantoleba iqneba: 16 y −5 = ( x − 1) . 45 es amocana, F ( x, y ) =

7. maRali rigis warmoebulebi da diferencialebi vTqvaT, funqcia z = f ( x, y ) gansazRvrulia da diferencirebadi raime D ∂f ∂f areze. misi kerZo warmoebulebi , , romlebic aseve ori cvladis ∂x ∂y funqciebia, SesaZlebelia aRmoCndnen diferencirebadi raime areze. Tu ∂f ∂f ∂f ∂ ∂ ∂ ∂y movaxdenT maTi gawar- moebas x da y cvladebiT, gveqneba ∂x , ∂x , , ∂x ∂x ∂y ∂f ∂ ∂y kerZo warmoebulebi. maT uwodeben z = f ( x, y ) funqciis meore rigis ∂y kerZo warmoebulebs da aRniSnaven, Sesabamisad,

∂2 f ∂2 f ∂2 f ∂2 f , , , ∂x∂y ∂y∂x ∂y 2 ∂x 2

f xx′′ , f xy′′ , f yx′′ , f yy′′ , simboloebiT.

∂2 f ∂2 f meore rigis kerZo warmoebulebs: , uwodeben Sereul kerZo ∂x∂y ∂y∂x warmoebulebs. Tu funqciis meore rigis warmoebulebic aseve diferencirebadi funqciebia raime areze, SeiZleba ganvixiloT maTi kerZo warmoebulebic, romlebsac mocemul z = f ( x, y ) funqciis mesame rigis kerZo warmoebulebs

69 uwodeben.

maT

aRniSnaven

Semdegnairad:

∂3 f ∂x 3

′′′ , f xxx

∂3 f ∂x 2 ∂y

∂3 f ∂3 f ′ ′ ′ ′′′ . an , an f yyy f xyy ∂x∂y 2 ∂y 3 analogiurad ganisazRvreba meoTxe rigis, mexuTe rigis da ase Semdeg kerZo warmoebulebi. magaliTi 1. vipovoT z = x 3 + x 2 y + y 3 funqciis yvela meore rigis kerZo warmoe-buli. amoxsna: pirveli rigis kerZo warmoebulebia: ′′′ , f xxy

∂z ∂z = x 2 + 3y 2 . = 3 x 2 + 2 xy , ∂y ∂x meore rigis kerZo warmoebulebia: ∂2 f ∂2 ∂2z ∂2 f = 2 x = 2 x 6 2 = x + y , , , = 6y . ∂x∂y ∂y∂x ∂x 2 ∂y 2 y magaliTi 2. vipovoT z = arctg funqciis yvela meore rigis x warmoebuli. amoxsna: pirveli rigis kerZo warmoebulebia: 1 1 x 1 y −y =− 2 = 2 z ′x = , z ′y = . 2 2 y x x + y2 y x x +y 1 + ( )2 1 + ( )2 x x meore rigis kerZo warmoebulebia: − (x 2 + y 2 ) + 2 y 2 y2 − x2 2 xy ′ ′ = z = , , z ′xx′ = 2 xy (x + y 2 )2 (x2 + y 2 )2 (x2 + y 2 )2

kerZo

x 2 + y 2 − 2x 2 y2 − x2 = , (x 2 + y 2 )2 (x 2 − y 2 )2 − 2 xy . z ′yy′ = 2 (x + y 2 )2 am orive magaliTSi, meore rigis Sereuli kerZo warmoebulebi warmoebulebi erTmaneTis toli aRmoCnda. garkveul pirobebSi es zogadi faqtia. adgili aqvs Semdeg Teoremas: Teorema 4.1(Svarcis Teorema). Tu z = f ( x, y ) funqcias N 0 ( x0 , y 0 ) wertilSi da mis raime midamoSi gaaCnia uwyveti meore rigis Sereuli kerZo warmoebulebi f xy′′ ( x, y ) da f yx′′ ( x, y ) , maSin gawarmoebis Tanamimdevrobas mniSvneloba ar aqvs anu f xy′′ ( x, y ) = f yx′′ ( x, y ) . rogorc viciT, z = f ( x, y ) funqciis sruli diferenciali N 0 ( x0 , y 0 ) wertilSi tolia: dz = f x′dx + f y′dy = f x′Δx + f y′Δy . es sidide damokidebulia x, y , Δx, Δy cvladebze. vipovoT misi, rogorc x, y cvladebze damokidebuli funqciis sruli diferenciali z ′yx′ =

70 d ( dz ) = d ( f x′dx + f y′dy ) = d ( f x′Δx + f y′Δy ) = f xx′′ Δxδx + f xy′′ Δxδy + f yx′′ Δxδx + f yy′′ Δyδy ,

(1)

sadac δx da δy warmoadgens x, y argumentebis nazrdebs, romlebic gansxvavdebian amave argumentebis Δx, Δy nazrdebisagan. rodesac δx = Δx da δy = Δy , (1) sidides uwodeben z = f ( x, y ) funqciis meore rigis srul diferencials da aRniSnaven ase: d 2 z an d 2 f ( x, y ) . Svarcis Teoremis Tanaxmad f xy′′ ( x, y ) = f yx′′ ( x, y ) . amitom funqciis meore rigis sruli diferenciali, rodesac δx = Δx = dx , δy = Δy = dy , gamoisaxeba ase: d 2 z = f xx′′ dx 2 + 2 f xy′′ dxdy + f yy′′ dy 2 . funqciis meore rigis sruli diferencialis forma ukve aRar aris invariantuli, radganac u da v cvladebis x, y cvladebze damokidebulebis SemTxvevaSi Δu , Δv nazrdebi ukve aRar iqnebian mudmivi sidideebi, isini damokidebuli iqnebian x, y cvladebze da (1) tolobaSi gaCndeba maTi kerZo warmoebulebi am cvladebis mimarT. analogiurad ganisazRvreba funqciis ufro maRali rigis sruli diferencialic. 8. ori cvladis funqciis eqstremumi gansazRvreba 4.7. N 0 ( x0 , y 0 ) wertils ewodeba z = f ( x, y ) funqciis maqsimumis (minimumis) wertili, Tu arsebobs am wertilis iseTi midamo, romelSic f ( x0 , y 0 ) warmoadgens funqciis udides (umcires) mniSvnelobas anu am midamoSi adgili aqvs utolobas: f ( x0 , y 0 ) > f ( x, y ) ( f ( x0 , y0 ) < f ( x, y )) . funqciis maqsimumis da minimumis wertilebs eqstremumis wertilebi ewodeba. grafikulad eqstremumis wertilebs aqvT saxe:

f ( x0 , y 0 )

X nax.19

N 0 ( x0 , y 0 )

nax.20

71 da aRwevs vTqvaT z = f ( x, y ) funqcia diferencirebadia maqsimums N 0 ( x0 , y 0 ) wertilSi(nax.19. davafiqsiroT y = y 0 da vcvaloT x . funqcia f ( x, y 0 ) erTi cvladis funqciaa, igi Tavis maqsimums aRwevs, rodesac x = x0 da f ( x 0 , y 0 ) > f ( x, y 0 ) , x cvladis yoveli mniSvnelobisaTvis, romelic sakmarisad axloa x 0 -Tan. rogorc erTi cvladis funqciebisTvis, am SemTxvevaSi df ( x, y 0 ) ∂f ( x, y 0 ) x = x0 = x = x0 = 0 . dx ∂x exla davafiqsiroT x = x0 da vcvaloT y . erTi cvladis funqcia f ( x0 , y ) maqsimums aRwevs, rodesac y = y 0 , am SemTxvevaSic df ( x0 , y ) ∂f ( x0 , y ) y = y0 = y = y0 = 0 . dy ∂y maSasadame, sabolood gvaqvs: ∂f ( x, y ) ∂f ( x, y ) (1) x = x0 = 0 , x = x0 = 0 . ∂x ∂y y = y0 y = y0 analogiur Sedegs miviRebT, im SemTxvevaSic, Tu N 0 ( x0 , y 0 ) minimumis wertili iqneba. amgvarad, (1) toloba diferencirebadi z = f ( x, y ) funqciisaTvis warmoadgens aucilebel pirobas, rom N 0 ( x0 , y 0 ) iyos eqstremumis wertili. magaliTi 1. ganvixiloT z = x 2 − y 2 funqcia. geometriulad am funqciiT gansazRvruli zedapiri warmoadgens hiperbolur paraboloids, misi kerZo warmoebulebia es warmoebulebi N 0 (0,0) wertilSi z ′x = 2 x, z ′y = −2 y . Rebuloben nulovan mniSvnelobas. N 0 (0,0) wertilSi funqciis mniSvneloba z = 0 . Tu gavixsenebT, rom wertili koordinatebiT (0,0,0) hiperbolur paraboloidze ar warmoadgens arc maqsimumis arc minimumis wertils, vinaidan arseboben wertilebi am zedapirze, romlebic imyofebian rogorc XOY OY sibrtyis zemoT, aseve mis qvemoT. am magaliTidan Cans, rom (1) tolobebi ar warmoadgenen imis sakmaris pirobas, rom N 0 (0,0) iyos maqsimumis an minimumis wertili. wertilebs, sadac funqciis kerZo warmoebulebi nulis tolia funqciis kritikuli wertilebi ewodeba. rogorc me-4 qveTavidandan viciT, z = f ( x, y ) funqciiT gansazRvruli zedapiris N 0 ( x0 , y 0 ) wertilze gamavali mxebi sibrtyis gantolebaa:

∂f ∂f x = x0 ( x − x 0 ) + x= x ( y − y0 ) . ∂x y = y0 ∂y y = y00 Tu N 0 ( x0 , y 0 ) wertili funqciis kritikuli wertilia, maSin z − z0 =

∂f ( x, y ) ∂f ( x, y ) x = x0 = 0 , x = x0 = 0 . ∂x ∂y y = y0 y = y0 aseT SemTxvevaSi mxebi sibrtyis gantolebas eqneba saxe: z − z0 = 0 . es ki warmoadgens N 0 ( x0 , y 0 ) wertilze gamavali XOY OY sibrtyis paraleluri sibrtyis, gantolebas.

72 maSasadame, kritikul wertilebSi, mxebi sibrtye XOY OY sakoordinato sibrtyis paraleluria. exla moviyvanoT funqciis eqstremumis sakmarisi pirobebi. vTqvaT, N 0 ( x0 , y 0 ) wertili z = f ( x, y ) funqciis kritikuli wertilia da funqcias am wertilSi gaaCnia meore rigis kerZo warmoebulebi f xx′′

x = x0 y = y0

= A, f xy′′

x = x0 y = y0

= B, f yy′′

x = x0 y = y0

=C.

SevadginoT gamosaxuleba Δ = AC − B 2 .

MmaSin: 1) Tu Δ > 0 , maSin eqstremumi arsebobs. a) N 0 ( x0 , y 0 ) wertili minimumis wertilia, Tu A > 0 an C > 0 . b) N 0 ( x0 , y 0 ) wertili maqsimumis wertilia, Tu A < 0 an C < 0 . 2) Tu Δ < 0 eqstremumi ar arsebobs. 3) Tu Δ > 0 arafris Tqma ar SeiZleba, kvleva unda gagrZeldes. aqedan gamomdinare z = f ( x, y ) funqciis gamokvleva eqstremumze unda moxdes Semdegi praqtikuli xerxiT: 1) viTvliT funqciis kerZo warmoebulebs: f x′ ( x, y ) da f y′ ( x, y ) . 2) vpoulobT funqciis kritikul wertilebs, amisaTvis vxsniT gantolebaTa sistemas: ⎧ f x′ ( x, y ), ⎨ ′ ⎩ f y ( x, y ). 3) viTvliT meore rigis kerZo warmoebulebs kritikul wertilebSi:

f xx′′

x = x0 y = y0

= A, f xy′′

x = x0 y = y0

= B, f yy′′

x = x0 y = y0

=C.

4) vadgenT Δ = AC − B 2 gamosaxulebas da vaxdenT mis analizs. magaliTi 2. gamovikvlioT funqcia z = x 3 + y 3 − 3axy, a > 0 eqstremumze. amoxsna: gamovTvaloT pirveli rigis kerZo warmoebulebi f x′ ( x, y ) = 3 x 2 − 3ay , f y′ ( x, y ) = 3 y 2 − 3ax . vipovoT funqciis kritikuli wertilebi 2 ⎪⎧3x − 3ay = 0, ⎨ 3 ⎪⎩3 y − 3ay = 0. pirveli gantolebidan y =

x2 , am tolobidan da meore gantolebidan a

x4 miviRebT: 2 = ax anu x( x 3 − a 3 ) = 0 . aqedan ki x1 = 0, x 2 = a , Sesabamisad a y = 0, y = a . gamovTvaloT funqciis meore kerZo warmoebulebi: f xx′ = 6 x, f xy′ = −3a, f yy′ = 6 y .

N1 (0,0) da N 2 (a, a) kritikul wertilebSi maTi mniSvneloba iqneba: N1 (0,0) wertilisaTvis A = 0, B = −3a, C = 0 . N 2 (a, a) wertilisTvis

A = 6a, B = −3a, C = 6a .

N1 (0,0) wertilisaTvis Δ = AC − B 2 = 0 − 3a ,

radgan a > 0 , amitom

Δ < 0. es ki niSnavs, rom N1 (0,0) wertilSi eqstremumi ar arsebobs. N 2 (a, a) wertilisTvis Δ = AC − B 2 = 36 a 2 − 9a 2 = 27 a 2 > 0 . amitom N 2 (a, a) wertilSi eqstremumi arsebobs. amasTan, radgan A = 6a > 0 , saqme gvaqvs minimumis wertilTan. magaliTi 3. vTqvaT, unda damzaddes yuTi, romlis moculoba iqneba 48 kub. decimetri da Sedgenili iqneba ori ganyofilebisagan, romlebic gayofilia erTmaneTisagan tixriT. vipovoT im masalis minimaluri raodenoba, romelic saWiroa yuTis dasamzadeblad(nax.21).

x y

nax.21

amoxsna: masalis raodenoba, romelic saWiroa yuTis dasamzadeblad, tolia sididis: M = xy + 2 xz + 3 yz . yuTis moculoba V = xyz . ukanaskneli 48 tolobidan ganvsazRvroT z , gveqneba z = . aqedan xy 48 48 96 144 M = xy + 2 x + 3 y = xy + + . xy xy y x gamovTvaloT M ( x, y ) funqciis kritikuli wertilebi. amisaTvis amovxsnaT gantolebaTa sistema: ∂M ( x, y ) 144 = y − 2 = 0, ⎧ ∂x ∂x ⎪ ⎨ ∂M ( x, y ) 96 ⎪ x = − = 0. ⎩ ∂y ∂y 2 miviRebT: x = 6, y = 4 ,

∂ 2 M ( x, y ) ∂x 2

∂ 2 M ( x, y ) ∂y 2

x =6

288 x3

x =6

y =4

192 y3

y =4

= =

∂ 2 M ( x, y ) 288 4 = ,B = ∂x∂y 216 3 192 =3 64

x =6 y =4

= 1,

74 48 4 = 2 , A = > 0, Δ = AC − B 2 = 4 − 1 = 3 > 0 . es ki xy 3 niSnavs, rom saqme gvaqvs minimumis wertilTan. amgvarad, yuTi rom davamzadoT minimaluri raodenobis masaliT saWiroa misi zomebi iyos: sigrZe x = 6 dm, sigane y = 4 dm, simaRle z = 2 dm. magaliTi 4. firma uSvebs x aTas A tipis produqts da y aTas B tipis produqts. produqciis realizaciiT miRebuli Semosavlis funqciaa: R ( x, y ) = 2 x + 3 y , xolo xarjebis funqcia: C ( x, y ) = x 2 − 2 xy + 2 y 2 + 6 x − 9 y + 5 . ra raodenobiT unda damzaddes TiToeuli tipis produqti, rom firmam miiRos maqsimaluri mogeba. amoxsna: firmis mogebis funqcia tolia: P ( x, y ) = R ( x, y ) − C ( x, y ) = −4 x − x 2 + 2 xy − 2 y 2 + 12 y − 5 . misi kritikuli wertilebis mosaZebnad amovxsnaT gantolebaTa sistema: da aqedan gamomdinare: z =

∂P( x, y ) ⎧ = −4 − 2 x + 2 y = 0, ∂x ⎪ ⎨ ∂P( x, y ) ⎪ = 2 x − 4 y + 12 = 0. ⎩ ∂y miviRebT: x = 2, y = 4 . aqedan A = −2, B = 2, C = −4 , Δ = AC − B 2 = 8 − 4 > 4, A < 0 . maSasadame, saqme gvaqvs maqsimumis wertilTan. amgvarad maqsimaluri mogebis misaRebad firmam unda daamzados 2000 cali A tipis da 4000 cali B tipis produqti.

savarjiSoebi me-4 TavisaTvis I. ipoveT Semdegi funqciebis kerZo warmoebulebi.: 1) z = 4 x + 5 y − 6 , 2), z = x 3 − y 2 + 7 x + 3 y + 1 3) z = x 2 ( y + x) , 4) z = e xy , 5) z = x 2 y − xy 2 , 6) z = 2 x 2 − xy + y 2 − x − 5 y + 8 , 7) z = y 3 + 2 x 2 y 2 − 3 x − 2 y + 8 , 8) z = x ln y + x 2 − 4 x − 5 y + 3 . II. ipoveT ipoveT Semdegi rTuli funqciebis kerZo warmoebulebi: 1) z = sin( x + y ) ,2) z = ln( x + y ) ,3) z = x + y ,4) z = e 2

sin

y x

,5) z =

x x −y 2

, 6) z = arctg

x dz dz dz z = , x = e t , y = ln t ,8 )ipoveT ,Tu y dt dt dt x , x = 3t 2 , y = t 2 + 1 . Tu z = ln sin y III. ipoveT Semdegi funqciebis meore rigis kerZo warmoebulebi: x+ y 1) z = 2 xy + y 2 , 2) z = ln( x 2 + y 2 ) , 3) z = sin( xy ) , 4) z = arctg , 5) z = x y . 1 − xy IV. ipoveT Semdegi aracxadi funqciebis Cveulebrivi warmoebulebi:

7)ipoveT

1) x 4 + 2 xy 5 = 0 , 2) xy + e x = 0 , 3) sin xy + x 2 = 0 , 5) ln sin( x + y ) − tg ( xy ) = 0 , x 6) +1 = 0. 2 x +y V. ipoveT Semdeg funqciaTa sruli diferencialebi:

y . x

75 3) z = yx y , 4) z = ln( x 2 + y 2 ) , 5) z = ln tg

1) z = x 2 + y 2 − 3 xy , 2) z = sin 2 x + cos 2 y ,

6) z = x 2 y 2 . VI. gamoikvlieT eqstremumze Semdegi funqciebi: 1) z = ( x − 1) 2 + 2 y 2 , 2) z = x 2 + xy + y 2 2 x − y , 3) z = x 3 y − 2 y 2 , 4) z = ( x − 1) 2 − 2 y 2 , 5) z = ( x + y )e 2

−( x2 + y 2 )

2 3

, 6) z = 1 − ( x + y ) . 2

y , x

76 Tavi 5. orjeradi integralebi 1. cilindruli sxeulis moculoba da orjeradi integralis cneba vTqvaT, XOY OY sakoordinato sibrtyis D areze gansazRvrulia z = f ( x, y ) arauaryofiTi funqcia. ganvixiloT samganzomilebiani sxeuli, romelic zemodan SemosazRvrulia mocemuli funqciis grafikiT, qvemodan SemosazRvrulia XOY OsibrtyiT, gverdebidan cilindruli zedapiriT, romlis msaxvelic Z RerZis paraleluria, mimmarTveli wiri ki- D aris SemomsazRvreli konturi. aseT sxeuls cilindruli sxeuli ewodeba(nax.22)

Y DD D D

ΔD k

nax.22

D ares ki am cilundruli sxeulis fuZe. ganxiluli cilindruli sxeulis moculoba aRvniSnoT V simboloTi. davyoT D are n nawilad TiToeuli es nawili aRvniSnoT ΔDk

simboloTi ΔS k k = 1,2,..., n. . am mcire areTa farTobebi aRvniSnoT simboloTi. ganvixiloT cilindruli sxeulebi, romlebsac fuZed aqvT dayofiT miRebuli mcire ΔDk areebi da zemodan SemosazRvruli arian z = f ( x, y ) gantolebiT gansazRvruli zedapiris im nawilebiT, romlebic proeqcirdebian am mcire areebze. amovirCioT TiToeul ΔDk areSi nebismieri wertili N k (ξ k ,η k ) ,. SevcvaloT TiToeuli cilindruli sxeuli cilindrebiT, romelTa simaRleebi tolia f (ξ k ,η k ) , k = 1,2,3,... . am cilindrebis moculobaTa jami n

Rn = ∑ f (ξ k ,η k )ΔS k .

(1)

k =1

cxadia, raRac miaxloebiT, tolia cilindruli sxeulis moculobisa. amitom SegviZlia davweroT: n

V ≈ ∑ f (ξ k ,η k )ΔS k . k =1

(2)

77 dD XOY OY sibrtyis raime D aris diametri ewodeba {d ( M , N )} simravlis zust zeda sazRvars, d = sup {d ( M , N )} , sadac d ( M , N ) manZilia M , N ∈D

D aris nebismier or, M da N wertils Soris. TandaTan ufro met nawilad davyoT D are ise, rom dayofiT miRebuli ΔDk nawilebis maqsimaluri diametri d Dk miiswrafodes nulisken. aseT SemTxvevaSi mosalodnelia (2) miaxloebiTi toloba TandaTan ufro zusti gaxdes da usasrulobaSi am miaxloebiTi tolobis marjvena mxare daemTxvas marcxena mxares. maSin cilindruli sxeulis moculobisTvis gveqneba:

V =

lim

∑ f (ξ

n →∞ max d D K →0 k =1

,η k )ΔS k .

(3)

(3) jams, z = f ( x, y ) funqciis rimanis jami ewodeba, D aris TiToeul dayofas Tavisi rimanis jami Seesabameba. amgvarad. saqme gvaqvs z = f ( x, y ) funqciis rimanis jamebis mimdevrobasTan D areze. gansazRvreba 5.1. Tu z = f ( x, y ) funqciis rimanis jamebis mimdevrobas D areze gaaCnia zRvari, rodesac dayofis nawilebis ricxvi usasrulod izrdeba ise, rom am nawilebis maqsimaluri diametri miiswrafis nulisken. amasTan es zRvari damokidebuli ar aris ΔDk areSi N k (ξ k ,η k ) wertilis SerCevaze, maSin amboben, rom z = f ( x, y ) funqcia integrebadia D areze. xolo zRvars: n

∑ f (ξ

lim

n →∞ max d D K →0 k =1

,η k )ΔS k

uwodeben am funqciis orjerad integrals D areze da aRniSnaven ase: ∫∫ f ( x, y)ds . D

f ( x, y ) funqcias integralqveSa funqcia ewodeba, D ares- saintegacio are, ds sidides- farTobis elementi. rogorc davinaxeT, orjeradi integrali geometriulad warmoadgens z = f ( x, y ) funqciT gansazRvruli cilindruli sxeulis moculobas anu gvaqvs

V = ∫∫ f ( x, y )ds . D

mtkicdeba Teorema: Teorema 5.1. nebismieri D areze gansazRvruli uwyveti funqcia integrebadia am areze. rogorc davinaxeT, integrali ori cvladis funqciisaTvis ganisazRvreba zustad erTi cvladis funqciis integralis analogiurad. amitom erTi cvladis funqciis integralis yvela Tviseba, axasiaTebs orjerad integralsac. 2. ganmeorebiTi integralebi da orjeradi integralis gamoTvla rogorc wina paragrafSi vnaxeT, cilindruli sxeulis, romelic zemodan SemosazRvrulia z = f ( x, y ) funqciis grafikiT, qvemodan D - ariT, moculoba gamoisaxeba formuliT:

V = ∫∫ f ( x, y )ds D

gamovTvaloT es moculoba sxvanairad. davuSvT, rom D are iseTia, rom misi SemomsazRvreli konturi XOY O sakoordinato sibrtyis RerZebis paralelur wrfeebs gadahkveTs araumetes or wertilSi(nax.23). D( y k )

Z z = f ( x, y k )

y0 = a

z = f ( x, y )

yn = b Y

P x1 = ϕ1 ( y ) q

x2 = ϕ 2 ( y) Q

nax.23 nax..24 gavavloT XOY sakoordinato sibrtyeSi OX RerZis paraleluri wrfeebi, romlebic exebian D aris SemomsazRvrel konturs. Sexebis A da B wertilebi am konturs yofen or nawilad, es nawilebi wirebs warmoadgenen. vTqvaT, maTi gantolebebia: x1 = ϕ1 ( y ), x2 = ϕ 2 ( y ) . OY RerZze segmenti [a, b] , sadac a da b warmoadgenen A da B wertilebis abscisebs, Sesabamisad, davyoT n nawilad, wertilebiT: a = y 0 , y1 , y 2 ,..., y n −1 , y n = b . yovel y k wertilze, k = 1,2,..., n − 1 , gavataroT XOZ sibrtyis paraleluri sibrtyeebi. es sibrtyeebi, cilindrul sxeulTan TanakveTaSi, gvaZleven mrudwirul trapeciebs. ganvixiloT mrudwiruli trapecia, romelic y = y k sibrtyeze mdebareobs. es trapecia zemodan SemosazRvrulia wiriT, romlis gantoleba z = f ( x, y k ) , qvemodan [ϕ1 ( y k ), ϕ 2 ( y k )] monakveTiT. rogorc erTjeradi integralis gansazRvrebidan viciT, trapeciis farTobi

S ( yk ) =

ϕ 2 ( yk )

∫ f ( x, y

)dx .

ϕ1 ( y k )

aqedan gamomdinare, yoveli y ∈ [ a, b] wertilisTvis, gveqneba:

S ( y) =

ϕ2 ( y )

∫ f ( x, y )dx .

ϕ1 ( y )

79 maSin cilindruli sxeulis moculobisTvis

gveqneba miaxloebiT toloba:

n −1

V ≈ ∑ S ( y k )Δy k . k =0

Tu TandaTan gavzrdiT [a, b] segmentis damyofi wertilebis ricxvs ise, rom Δy k segmentebis maqsimaluri sigrZec Semcirdes, ukanaskneli miaxloebiTi toloba ufro zusti gaxdeba. aqedan gamomdinare, adgili eqneba zust tolobas:

lim

n −1

∑ S(y

n →∞ max Δy k →0 k = 0

)Δy k ,

anu b

V = ∫ S ( y )dy . a

sabolood vRebulobT ganmeorebiT integrals b ⎛ ϕ2 ( y ) ⎞ (1) V = ∫ ⎜ ∫ f ( x, y )dx ⎟dy . ⎟ ⎜ a ⎝ ϕ1 ( y ) ⎠ analogiurad SeiZleba vaCvenoT, rom q ⎛ φ2 ( x ) ⎞ V = ∫ ⎜ ∫ f ( x, y )dy ⎟dx , (2) ⎜ ⎟ p ⎝ φ11 ( x ) ⎠ sadac x1 = ϕ1 ( y ), x2 = ϕ 2 ( y ) im wirebs warmoadgenen, romlebadac yofen D aris SemomsazRvrel konturs XOY sakoordinato sibrtyeSi OY RerZis paralelur mxeb wrfeebze mdebare Sexebis P da Q wertilebi. aqedAgamomdinare orjeradi integralisTvis, gveqneba: q ⎛ φ2 ( x ) b ⎛ ϕ2 ( y ) ⎞ ⎞ ⎜ f ( x, y )dx ⎟dy = ⎜ f ( x, y )dy ⎟dx . f ( x , y ) ds (3) = ∫∫D ∫p ⎜ φ ∫( x) ∫a ⎜ ϕ ∫( y ) ⎟ ⎟ ⎠ ⎝ 1 ⎝ 11 ⎠ f ( x, y ) ≥ 0 . orjeradi integralis gansazRvrisas vgulisxmobdiT, rom vTqvaT, es piroba ar sruldeba, maSin davyoT D are iseT D1 , D2 nawilebad, sadac funqcia niSans inarCunebs. vTqvaT, D2 areze funqcia uaryofiTia, am areebze aviRoT | f ( x, y ) | funqciis orjeradi integrali

∫∫ | f ( x, y ) | ds D2

da CavTvaloT, rom

∫∫ f ( x, y)ds = − ∫∫ | f ( x, y ) | ds . D

sabolood gveqneba:

∫∫ f ( x, y)ds = ∫∫ f ( x, y)ds + ∫∫ f ( x, y )ds . D

orjeradi integralis gansazRvrisas aseve vgulisxmobdiT, rom D aris SemomsazRvreli konturi, sakoordinato RerZebis paraleluri wrfeebiT, gadaikveTeboda ara umetes or wertilSi. vTqvaT gadakveTis wertilebis ricxvi metia, maSin yovelTvis SegviZlia are sakoordinato RerZebis paraleluri wrfeebiT davyoT iseT nawilebad, rom am nawilebis

80 konturebma wrfeebi gadakveTos ara umetes or

SemomsazRvrelma wertilSi (nax.24). aseT SemTxvevaSi integrals gamoviTvliT dayofiT miRebul TiToeul areze. adgili eqneba tolobas: ∫∫ f ( x, y)ds = ∫∫ f ( x, y)ds + ∫∫ f ( x, y )ds + ∫∫ f ( x, y)ds . D

magaliTi 1. gamovTvaloT z = x − y − 4 zeapiriT, XOY da XOZ sakoordinato sibrtyeebiT SemosazRvruli cilindruli sxeulis moculoba. amoxsna: aseTi cilindruli sxeulis fuZeaparaboliT da y = x 2 − 4 da X X RerZiT SemosazRvruli brtyeli figura, RerZi am parabolas gadahkveTs x = −2 da x = 2 wertilebSi. fuZis konturi Sedgeba y == x 2 − 4 parabolis nawilisagan da [-2,2] monakveTisagan X RerZze. aqedan gamomdinare, Tu gamoviyenebT (2) formulas, gveqneba: 2 2 0 0 0 ⎛ 0 ⎞ ⎛ ⎞ V = ∫ ⎜ ∫ ( x 2 − y − 4)dy ⎟dx = ∫ ⎜ x 2 ∫ dy − ∫ ydy − 4 ∫ dy ⎟dx = ⎜ ⎟ ⎜ 2 ⎟ − 2⎝ x 2 − 4 − 2⎝ x −4 x2 −4 x2 −4 ⎠ ⎠ 2

1 ( x 2 (0 − x 2 + 4) − (0 − ( x 2 − 4) 2 ) − 2 x2 −4

∫

1 − 4(0 − x + 4)))dx = − ∫ x 4 dx + 4 ∫ x 2 dx = 2 −2 −2 2

−−

1 5 x 10

2 −2

4 3 x 3

2 −2

=−

32 32 24 24 − + + = 16 − 10,4 = 5,6. 10 10 3 3

3. orjeradi integralis gamoTvla marTkuTxa da polarul koordinatTa sistemaSi vTqvaT vTqvaT XOY sakoordinato sibrtyis D areze gansazRvrulia z = f ( x, y ) arauaryofiTi funqcia. davyoT D are X da YRerZebis paraleluri wrfeebiT Sedgenili badiT(nax.25). Y Dk q y k +1 yk

ηk p

x k x k +1

b X

nax. 25

ξ2 TiToeuli danayofis Dk farTobi, cxadia iqneba ΔS k = Δx k Δy k sadac Δx k = x k +1 − x k , Δy k = y k +1 − y k , k = 0,1,2,..., n − 1 . TiToeuli danayofidan aviRoT nebismieri (ξ k ,η k ) wertili, Tu z = f ( x, y ) funqcia integrebadia, gveqneba:

∫∫

f ( x, y )ds =

n −1

∑ f (ξ

lim

n →∞ max Δxk → 0 k = 0 max Δy k →0

,η k )Δx k Δy k .

Dk danayofis farTobis gamosaxuleba ΔS k = Δx k Δy k iZleva safuZvels davweroT toloba ds = dxdy , maSasadame

∫∫ f ( x, y)ds = ∫∫ f ( x, y)dxdy . D

wina paragrafSi gamoyvanili formulebis saSualebiT SegviZlia davweroT:

∫∫ D

q ⎛ φ2 ( x ) ⎞ ⎞ ⎛ ϕ2 ( y ) ⎟ ⎜ f ( x, y )dxdy = ∫ ∫ f ( x, y )dx dy = ∫ ⎜ ∫ f ( x, y )dy ⎟dx . ⎟ ⎜ ⎟ ⎜ p ⎝ φ11 ( x ) a ⎝ ϕ1 ( y ) ⎠ ⎠ b

es formula ZalaSia maSinac, rodesac z = f ( x, y ) niSans D areze.

funqcia ar inarCunebs

magaliTi.1. gamovTvaloT orjeradi integrali

∫∫ ( x

+ y 2 )dxdy

areze, romelic SemosazRvrulia parabolebiT: y = x 2 , x = y 2 . amoxsna: es parabolebi erTmaneTs kveTen wertilebSi, romelTa abscisebia 0 da 1 (nax.26). y = x2

M (r , ϕ , z )

x y

O ϕ O

Y nax.26.

N (r , ϕ )

nax.27

⎛ x 2 ⎞ ⎜ ( x + y 2 )dy ⎟dx = + = x y dxdy ( ) ∫∫D ∫0 ⎜ ∫2 ⎟ ⎝x ⎠ 1

1 1 1 = ∫ x ( x − x )dx + ∫ ( x 2 − x 6 )dx = ∫ x 5 dx − ∫ x 4 dx + ∫ x 2 dx − ∫ x 6 dx = 30 30 30 0 0 0 2

11 2 1 2 1 6 1 1 1 + − = − + − = . 5 3 7 7 5 15 21 35 5 33 +1 +1 2 2 marTkuTxa koordinatTa sistemasTan erTad, sivrceSi gamoiyeneba polarul koordinatTa sistema. polarul koordinatTa sistemaSic wertilis mdebareoba sivrceSi, calsaxad ganisazRvreba ricxvTa sameuliT: r , ϕ , , z . es sameuli miiReba Semdegnairad(nax.26): vTqvaT, sivrceSi ukve gvaqvs marTkuTxa koordinatTa sistema OXYZ ,sivrceSi arsebul M wertilidan davuSvaT marTobi OXY sibrtyeze. am marTobis fuZe N SevaerToT koordinatTa O saTavesTan. am operaciebis Sedegad miviRebT sam sidides: r =| ON | sadac tolobis marjvena mxares gvaqvs ON monakveTis ON monakveTs Soris, z ki_ warmoadgens sigrZe, ϕ -kuTxe OX RerZsa da M wertilis aplikats. kuTxe ϕ aiTvleba dadebiTi mimarTulebiT anu saaTis isris sawinaaRmdego mimarTulebiT da icvleba [0,2π ] SualedSi, r ≥ 0, xolo z ∈ ( −∞, ∞ ) . cxadia ricxvTa es sameuli calsaxad gansazRvravs M wertils sivrceSi. polarul koordinatTa sistema marTkuTxa koordinatTa sistemasTan dakavSirebulia Semdegnairad: =

−

x = r cos ϕ , y = r sin ϕ , z = z ,

(1)

an Semdegnairad:

y , z = z. x Tu mocemulia funqcia z = f ( x, y ) , polarul sakoordinato cvladebze gadasvliT gveqneba: z = f ( r cos ϕ , r sin ϕ ) = g (r , ϕ ) . vTqvaT, OXY sibrtyeze mocemulia ΔD mcire zomis are, romlic ori mxridan SemosazRvrulia koordinatTa saTavidan gamomavali ori l1 ,l 2 sxiviT, ori mxare ki r1 , r2 - radiusiani rkalebiT(nax.28). ΔD aris farTobi: ΔS = rΔϕΔr . vTqaT, z = f ( r , ϕ ) funqcia gansazRvrulia da integrebadi D areze. Tu D ares davyofT saTavidan gamosuli sxivebiT da radialuri rkalebiT Sedgenili badiT, dayofis TiToeuli are iseTive iqneba rogoric iyo zemoT ganxiluli ΔD are. radgan z = f ( r , ϕ ) funqcia integrebadia, D areis aseTnairi dayofebiT gansazRvruli misi rimanis jamebis mimdevroba krebadi iqneba da gveqneba: r = x 2 + y 2 , ϕ = Arctg

83 Y

ΔD

r1 Δϕ

Δr rΔϕ

r=2

r2 D

ϕ2

ϕ1

r =1

nax.28

X nax.29

∫∫ f (r ,ϕ )ds = lim ∑ f (r, ϕ )rΔϕΔr ,

(2)

ΔS →0 n→∞

sadac n dayofis elementTa raodenobaa, ΔS - dayofis elementTa maqsilaluri farTobi. Tu gaviTvaliswinebT, rom dr = Δr , dϕ = Δϕ , gveqneba ds = rΔϕΔr . sabolood

∫∫ f (r, ϕ )ds =∫∫ f (r,ϕ )rdϕdr . D

(3)

Tu D are SemosazRvrulia wirebiT: r = ρ1 (ϕ ), r = ρ 2 (ϕ ), ϕ ∈ [α , β } , romelTa gantolebebic mocemulia polarul koordinatTa sistemaSi, maSin β ⎛ ρ 2 (ϕ ) ⎞ ⎜ f (r , ϕ )r ⎟dϕ . (4) f ( r , ϕ ) rd ϕ dr = ∫∫D ∫⎜ ∫ ⎟ α ⎝ ρ1 (ϕ ) ⎠ Tu O koordinatTa saTave D aris SigniTaa moTavsebuli, maSin kuTxe ϕ icvleba [0,2π ] SualedSi da (3) formulas aqvs saxe: 2 π ρ (ϕ ) ⎛ ⎞ (5) = f ( r , ϕ ) rd ϕ dr ∫∫D ∫0 ⎜⎜ ∫0 f (r , ϕ )r ⎟⎟dϕ . ⎝ ⎠ zogjer integralis gamoTvla polarul koordinatebSi ufro moxerxebulia, amitom Tu mocemulia D areze integrebadi funqcia z = f ( x, y ) da gvinda gamovTvaloT integrali

∫∫ f ( x, y)dxdy , D

(1) formulebis gamoyenebiT SegviZlia gadavideT polarul koordinatebze, gveqneba (6) ∫∫ f ( x, y)dxdy = ∫∫ f (r cos ϕ , sin ϕ )rdϕdr D

magaliTi 2. gamovTvaloT integrali

∫∫ ( x + y)dxdy ,sadac

D 2

are

warmoadgens OX , OY RerZebiTa da x 2 + y = 1, x 2 + y 2 = 4 wrewirTa rkalebiT SemosazRvrul figuras(nax.29). amoxsna: gadavideT polarul koordinatebze, aris SemomsazRvreli rkalebis gantolebebi iqneba 1 = ρ1 (ϕ ),2 = ρ 2 (ϕ ) , aqedan gamomdinare,

84 miviRebT: π

2 ⎛2 ⎞ 7 ⎜ ⎟ ∫∫D ( x + y)dxdy = ∫∫D (r cos ϕ + sin)rdϕdr = ∫0 ⎜⎝ ∫1 (r cos ϕ + r sin ϕ )rdr ⎟⎠dϕ = ∫0 3 (cos ϕ + sin ϕ )dϕ = 2

7 14 = (sin ϕ − cos ϕ ) 02 = . 3 3

4. orjeradi integralis gamoyeneba 4.1. brtyeli figuris farTobi integrali ∫∫ ds ,Tu gavixsenebT orjeradi

ganvixiloT

integralis

gansazRvras, es integrali tolia iqneba D aris farTobis. aqedan gamomdinare orjeradi integrali SeZleba gamoviyenoT brtyeli figureaTa farTobebis saangariSod. magaliTi 1. gamovTvaloT im figuris farTobi, romelic SemosazRvrulia OX RerZiT y 2 = 4ax paraboliT da x + y = 3a wrfiT, a > 0 (nax.30). Y y 2 = 4ax x + y = 3a

O a X nax.30 amoxsna: vipovoT parabolis da wrfis gadakveTis wertili, amisTvis amovxsnaT sistema ⎧ x + y = 3a, ⎨ 2 ⎩ y = 4ax. y1 = −2a + 4a = 2a, y 2 − 2a − 4a = −6a . radgan miviRebT y 2 + 4ay + 12a 2 = 0 , aqedan Cven gvainteresebs y cvladis arauaryofiTi mniSvnelobebi, amitom aviRebT mxolod y1 = 2a mniSvnelobas. maSasadame y icvleba 0- dan 2a - mde, am dros x icvleba x1 =

y2 − dan x2 = 3a − y - mde. 4a

amgvarad gveqneba: 2 a 3a − y

∫∫ ds = ∫ ( D

∫ dx)dy = ∫ (3a − y

y2 4a

6a 2 − 2a 2 −

y2 y2 y3 )dy = 3ay − − 4a 2 12a

y =2a y =0

8 2 10 2 a = a . 12 3

4.2. zedapiris farTobi vTqvaT mocemulia z = f ( x, y ) gantolebiT gansazRvruli S zedapiri. davafiqsiroT raime M 0 ( x0 , y 0 , z 0 ) wertili am zedapirze. am wertilSi da

∂f ∂y

85 kerZo warmoebulebi aRvniSnoT, Sesabamisad, p da q simboloebiT. ∂f ∂f CavTvaloT, rom f ( x, y ), , uwyveti funqciebia. ∂x ∂y rogorc viciT zedapiris M 0 ( x0 , y 0 , z 0 ) wertilSi mxebi sibrtyis gantolebaa: z − z 0 = p ( x − x0 ) + q ( y − y 0 ) . amave wertilSi normalis gantoleba iqneba: z − z 0 x − x0 y − y 0 . = = −1 p q am normalis mimmarTveli kosinusebia: ±p ±q , cos β = , cos α = 1+ p2 + q2 1+ p2 + q2 ±1 . cos γ = 1+ p2 + q2 zedapiris proeqcia OXY sibrtyeze aRvniSnoT D simboloTi. davyoT D are nawilebad ΔDk , k = 1,2,..., n . TiToeuli am nawilidan aRvmarToT cilindri zedapiris gadakveTamde. gadakveTaSi miviRebT zedapiris ΔS k nawils. aviRoT am nawilze M k ( x k , y k , z k ) wertili da gavavloT am wertilSi mxebi sibrtye. agebuli cilindri gavagrZeloT mxebi sibrtyis gadakveTamde. cilindri mxeb sibrtyes amokveTs nawils, romelic aRvniSnoT ΔS k -iT. TiToeuli agebuli cilindrisTvis gveqneba Sesabamisi mxebi sibrtyis nawili ΔS k , k = 1,2,..., n . mxeb sibrtyeTa es nawilebi qmnian mravalkuTxeds, romelSic Cawerilia S zedapiri. gansazRvreba 5.2. S zedapiris farTobi ewodeba zRvar lim

∑ ΔS

n →∞ max d ΔDk → 0 k =1

(1)

sadac d ΔDk warmoadgens D aris ΔDk nawilis diametrs.

Z n

ΔS k

M ( xk , y k , z k ) ΔS k

Y ΔDk

Δy k nax.31

Δx k

86 gamovTvaloT axla mocemuli zedapiris farTobi.

M k ( x k , y k , z k ) wertilze

gamavali mxebi sibrtyis ΔS k nawilis mier OXY sibrtyesTan Sedgenili kuTxe tolia mocemul wertilSi zedapiris n normalsa da Z RerZs Soris kuTxis, romelic zemoT aRniSnuli gvaqvs γ simboloTi(nax. 31). ~ ~ vTqvaT ΔDk aris farTobi tolia ΔS k , maSin ΔS k = ΔS k cos γ . aqedan ~ ΔS , ΔS k = cos γ ±1 magram radganac cos γ = , amitom ΔS k = ± 1 + p 2 + q 2 . 2 2 1+ p + q π normalis mimarTuleba avirCioT ise, rom 0 < γ < , maSin gveqneba cos γ > 0 2 da ΔS k = + 1 + p 2 + q 2 . zedapiris farTobis gansazRvrebis Tanaxmad: n

S = lim ∑ ΔS k = k =1 n →∞ max d ΔDk →0

lim

∑

n →∞ max d ΔDk →0 k =1

1+ p2 + q2 .

(2) formulis marjvena mxare tolia orjeradi integralis

(2)

∫∫

1 + p 2 + q 2 ds ,

maSasadame Cveni zedapiris farTobi S = ∫∫ 1 + p 2 + q 2 ds , D

∂f ∂f sadac p = da q = . ∂y ∂x aq ganvixileT is SemTxveva, rodesac zedapiri gadakveTs z RerZis paralelur wrfeebs mxolod erT wertilSi, winaaRmdeg SemTxvevaSi zedapirs davyofT aseT nawilebad, gamovTvliT TiToeuli nawilis farTobs da Sedegebs SevkrebT. imis mixedviT, Tu ra mdebareoba aqvs zedapirs OXYZ sivrceSi, zogjer xelsayrelia ganvixiloT misi proeqcia YOZ an XOZ sibrtyeSi. magaliTi 1. vipovoT sferuli zedapiris im nawilis farTobi, romelsac amoWris sferos centrze gamavali cilindri, Tu sferos gantolebaa a a2 x 2 + y 2 + z 2 = a , cilindris gantoleba ki ( x − ) 2 + y 2 = (nax.32). 2 4 amoxsna: Z

a Y

87 nax.32

cilindris fuZis SemomsazRvreli wriuli konturis gantoleba iqneba a a2 (x − )2 + y 2 = . gamovTvaloT p da q sidideebi 2 4 x y ∂z ∂z p= = = ,q = . 2 2 2 2 ∂y ∂x a −x −y a − x2 − y2 1+ p2 + q2 =

aqedan

a a2 − x2 − y2

. CvenTvis saintereso zedapiris farTobi

S = ∫∫ D

a a2 − x2 − y2

ds ,

a a2 utolobiT gansazRvrul wres. sadac D warmoadgens ( x − ) 2 + y 2 ≤ 2 4 Cveni zedapiri OXZ sibrtyis mimarT simetriulia, amitom zedapiris farTobis meoTxedisTTvis gveqneba: ⎛ a 2 −( x − a ) 2 ⎞ ⎟ a⎜ 4 2 S a a ⎜ dy ⎟dx = ds = ∫ = ∫ 2 2 2 2 2 2 ⎜ ⎟ 4 ∫∫ a −x −y a −x −y D 0 0 ⎜ ⎟ ⎝ ⎠ a

= a∫ 0

a2 a −( x − ) 2 4 2

1 a2 − x2

= a ∫ arcsin u 0

dx = a ∫ arcsin 0

π 4

a2 (

π 4

2 π

1 a2 − x2

∫

[ a2 − x2

1 1− u2

du =

4 x 1 2 2 dx = 2a 2 ∫ t ⋅tgt dt = a t ⋅ tgtdtgt = a t ⋅ dtg 2 t = 2 2 ∫ ∫ a+x cos t 0 0 0

= a 2 t ⋅ tg 2 t 04 − a 2 ∫ tg 2 tdt = a 0

dy ]dx = a ∫

y2 1− 2 a − x2

x a+ x

∫

[

a2 a −( x − ) 2 4 2 a2 − x2

− a2 ∫ ( 0

π π 1 − 1)dt = a 2 − a 2 tg 04 + a 2 = 2 4 4 cos t

− 1),

sadac t = arcsin

x π 1 da dx = tgt dt . maSasadame S = 4a 2 ( − 1) . 2 a+x 4 cos t

4.3. brtyeli figuris masa, simZimis centri da statikuri momenti vTqvaT, mocemulia materialuri sxeuli D , romelic warmoadgens brtyel figuras. vTqvaT, aseve, m masa gadanawilebulia am figuraze ρ = ρ ( N ) zedapiruli simkvriviT,sadac N figuris cvladi wertilia. rom gavigoT figuris masa, igi unda davyoT mcire ΔDk , k = 1,2,..., n. nawilebad, Nk romelTa simkvrive SeiZleba CavTvaloT mudmiv sidided. Tu

88 warmoadgens figuris ΔDk nawilis wertils. am nawilis masa Δmk miaxloebiT tolia sididis ρ ( N k )ΔS k , sadac ΔS k warmoadgens ΔDk nawilis

farTobs. aqedan gamomdinare, mTeli figurisaTvis SeiZleba davweroT miaxloebiTi toloba: n

m ≈ ∑ ρ ( N k ) ΔS k . k =1

Tu figuras davyofT ufro mcire diametris nawilebad, miaxloebiTi tolobis sizuste gaizrdeba, amitom SegviZlia davweroT:

m = lim

∑ ρ (N

n→∞ d ΔD K → 0 k =1

)ΔS k

anu

m = ∫∫ ρ ( N )ds . D

rogorc viciT, figuris ΔDk nawilis statikuri momenti OX RerZis mimarT, miaxloebiT SeiZleba gamoisaxos formuliT: I xΔDk = y k Δm k = y k ρ ( N k ) ΔS k , sadac yk warmoadgens N k wertilis ordinats, romelic SegviZlia miaxloebiT CavTvaloT, figuris ΔDk nawilidan OX RerZamde manZilad. aqedan gamomdinare, mTeli D brtyeli figuris statikuri momenti OX RerZis mimarT toli iqneba sididis:

I x = lim

∑y

n→∞ d ΔD K → 0 k =1

ρ ( N k )ΔS k ,

anu

I x = ∫∫ yρ ( N )ds . D

analogiurad gamoiTvleba statikuri momenti OY RerZis mimarTac: I y = ∫∫ xρ ( N )ds . D

brtyeli figuris simZimis centris C (ξ ,η ) koordinatebi, rogorc viciT, ganisazRvreba tolobebiT: Iy I ξ = ,η = x . m m aqedan gamomdinare, sabolood ∫∫D xρ ( N )ds ∫∫D yρ ( N )ds ,η = . ξ= ρ ( N ) ds ρ ( N ) ds ∫∫ ∫∫ D

Tu brtyeli figura erTgvarovania anu ρ ( n) = C , sadac C mudmivi sididea, wina formulebs eqnebaT saxe: ∫∫D yds ∫∫D xds , η= . ξ= ds ds ∫∫ ∫∫ D

89 magaliTi 1. vipovoT erTgvarovani, x 2 + y 2 ≤ a 2 , y ≥ 0 utolobiT gansazRvruli D naxevarwris simZimis centri.

amoxsna: vTqvaT, naxevarwris simZimis centria C (ξ ,η ) . CavTvaloT, rom masis ganawilebis simkvrive ρ ( N ) = 1 . naxevarwre D SemosazRvrulia naxevarwrewiriT: x2 + y2 = a2 , y ≥ 0 , aqedan y = a 2 − x 2 . jer gamovTvaloT statikuri momenti I x ⎛ I x = ∫∫ yds = ∫ ⎜ ⎜ − a⎝ D a

a ⎞ ⎟dx = 1 (a 2 − x 2 )dx = 1 ( 2 ax − 1 x 3 ) ydy ∫0 ⎟ −∫a 2 2 3 ⎠

a2 − x2

a −a

2 3 a . 3

naxevarwris masa, roca ρ ( N ) = 1 , toli iqneba misi farTobis:

πa 2 2

am gamoTvlebidan gveqneba

2 3 a 4a ξ= 3 2 = . 3π πa 2 rac Seexeba figuris simZimis centris ordinats, igi SeiZleba gamovTvaloT uSualod. marTlac, Cveni naxevarwre warmoadgens erTgvarovan brtyel materialur sxeuls, romelic simetriulia OY RerZis mimarT. aseT SemTxvevaSi simZimis centri OY RerZze mdebareobs. maSasadame 4a naxevarwris simZimis centri yofila wertili C (0, ) . 3π 4.4. orjeradi integralis sxva gamoyenebebi vTqvaT funqcia z = f ( x, y ) funqcia gansazRvrulia da integrebadi D areze. funqciis saSualo mniSvneloba am areze ewodeba sidides

∫∫ f ( x, y)dxdy D

∫∫ dxdy

sadac mniSvnelSi myofi integrali warmoadgens D aris farTobs. magaliTi 1. vTqvaT, mTavrobam gadawyvita ekonomikis stimulirebis mizniT gaakeTos erTjeradi 10% sagadaxado SeRavaTi, ris Sedegadac mosalodnelia ekonomikaSi damatebiT y miliardi dolaris investicia, sadac 5 ≤ y ≤ 7 . Tu yoveli fizikuri piri an korporacia daxarjavs x aRniSnuli iribi investiciiT miRebuli Semosavlebis nawils, sadac 0,6 ≤ x ≤ 0,8 . ekonomikaSi cnobili mravlobiTobis principis Tanaxmad, mTliani ekonomikuri efeqti (fulad erTeulebSi) gamoisaxeba formuliT: y . vipovoT sagadaxado SeRavaTis saSualo ekonomikuri efeqt S ( x, y ) = 1− x

90 y

amoxsna:

saSualo

ekonomikuri efeqti

warmoadgens marTkuTxa 0,6 ≤ x ≤ 0.8;5 ≤ y ≤ 7 . 0 ,8

ares,

romelic

0 ,8

y y y2 dxdy = dx ( dy ) = ( ∫∫D 1 − x ∫ ∫5 1 − x ∫ 2(1 − x) 0,6 0, 6 = −12 ln z

z =0, 2 z =0, 4

y =7 y =5

S =

∫∫ 1 − x dxdy D

0,4 ganisazRvreba

0, 2

)dx =

sadac

utolobebiT

0, 2

12 12 ∫0,61 − x dx = −0∫,4 z dz =

= −12 ln 0,2 + 12 ln 0,4 = 12 ln 2 = 12 ln 2.

12 ln 2 = 30 ln 2 ≈ 20,8 miliard dolars. 0,4 magaliTi 2. kob-duglasis warmoebis funqcia. Tu warmoebaSi movaxdenT x 10 ≤ x ≤ 20 , aTasi, samuSaosaaTis da dolaris, y milioni 1≤ y ≤ 2 investicias, maSin warmoebuli raime detalis aTas erTeulTa raodenoba N ( x, y ) = x 0, 75 y 0,5 . vipovoT warmoebuli detalebis saSualo raodenoba amoxsna: 20 2 1 x 0,75 x = 20 y 1, 25 = N = ∫∫ x 0,75 y 0, 25 dxdy = ∫ x 0, 75 ∫ y 0, 25 dy = x =10 10 D 1 , 75 1 , 25 10 1 aqedan S =

8 (21, 25 − 1)(201, 75 − 101, 75 ) = 8,375 175 aTas erTeuls. magaliTi 3. avtomobilis damuxruWebis manZili iangariSeba formuliT: L = 0,0000133 xy 2 , sadac x avtomobilis wonaa, y -avtomobilis siCqare. gavigoT saSualo damuxruWebis manZili avtomobilebisTvis, romelTa wona 2000 da 3000 kilograms Sorisaa, siCqare ki 75 da 90 kilometr/saaTs Soris. amoxsna: 1 0,0000133 x 2 x =3000 y 3 y =95 2 0,0000133xy dxdy = L= x = 2000 y = 75 = 2,512 km. 15000 ∫∫ 15000 2 3 D savarjiSoebi Mme-5 TavisaTvis I. gamoTvaleT orjeradi integralebi: x2 xdxdy , sadac D warmoadgens paraboliT da y = x wrfiT 1) ∫∫ 2 y = 2 2 D x + y SemosazRvrul ares; 2) gadadiT polarul koordinatebze da gamoTvaleT 2 2 2 2 ∫∫ ( x + y )dxdy , sadac D warmoadgens x + y = 5 wrewiriT SemosazRvrul D

wres; 3)

∫∫

x 2 − y 2 dxdy , sadac D warmoadgens samkuTxeds

wveroebiT O (0,0), A(1,−1), B (1,1) ; dxdy 4) ∫∫ , sadac D warmoadgens wres radiusiT a , romelic 2a − x D sakoordinato RerZebs da mdebareobs pirvel meoTxedSi. II. gamoTvalT areTa farTobebi, romlebic SemosazRvrulia:

exeba

91 1) y = 10 x + 25, y = −6 x + 9 parabolebiT;2) r = a (1 + cos ϕ ), r = a cos ϕ , a > 0 wirebiT; 3) x = y , x = 2 y , x + y = a, x + 3 y = a, a > 0 wrfeebiT. III. gamoTvaleT moculobebi sxeulebisa, romlebic SemosazRvrulia zedapirebiT: x2 y2 z2 2, 2 2 2 1) az = y , x + y = r , z = 0 ;2) y = x , y = 2 x , x + z = 6, z = 0 ;3) 2 + 2 + 2 = 1 . a b c 2

Tavi 6 mrudwiruli integralebi 1. pirveli gvaris mrudwiruli integrali vTqvaT sibrtyeze mdebare araCaketili l wiris rkalze, romlis boloebia wertilebi A, B , gansazRvrulia z = f (M ) funqcia, sadac M ( x, y ) rkalis cvladi wertilia. f ( M ) = f ( x, y ) , davyoT AB rkali n nawilad, nebismierad, wertilebiT M 1 ( x1 , y1 ), M 2 ( x 2 , y 3 ),..., M n −1 ( x n −1 , y n −1 ) (nax.33).

Y y k +1

ηk yk A

x k ξ k x k +1

nax. 33

davuSvaT A = M 0 ( x0 , y 0 ), B = M n ( x n , y n ) da aviRoT nebismierad M k (ξ k ,η k ) wertili M k M k +1 rkalidan, Δl k iyos am rkalis sigrZe, max Δl k iyos AB rkalis n nawilad dayofiT miRebuli rkalebis sigrZeTa maqsimumi. ganvixiloT jami: n −1

∑ f (ξ k =0

,η k )Δl k ,

(1)

gansazRvreba 6.1. Tu arsebobs (1) jamis zRvari:

lim

n −1

∑ f (ξ

n→∞ max Δlk →0 k = 0

,η k )Δl k ,

romelic damokidebuli araa M k (ξ k ,η k ) wertilis arCevaze, maSin am zRvars uwodeben z = f ( M ) funqciis pirveli gvaris mrudwiruli integrals AB rkalze da aRniSnaven ase: ∫ f ( x, y)dl . AB

92 l vTqvaT wiri mocemulia parametrulad, uwyvetad warmoebadi x = x (t ), y = y (t ) funqciebiT, sadac t parametri icvleba [T1 , T2 ] SualedSi da AB rkalis boloebis koordinatebia, Sesabamisad: A( x(t 0 ), y (t 0 )), B ( x(T ), y (T )); t 0 , T ∈ [T1 , T2 ] . davyoT [t 0 , T ] segmenti n nawilad wertilebiT t1 , t 2 ,..., t n −1 , Sesabamisad, AB rkalic daiyofa n nawilad wertilebiT M 1 ( x(t1 ), y (t1 )), M 2 ( x(t 2 ), y (t 3 )),..., M n −1 ( x(t n −1 ), y (t n −1 )) . f ( M ) = f ( x, y ) funqcia warmoadgens rTul funqcias: f ( M ) = f ( x, y ) = f ( x(t ), y (t )) , f ( M k ) = f ( x(t k ), y (t k )) .

rogorc viciT, M k M k +1 rkalis sigrZisaTvis gvaqvs miaxloebiTi toloba:

Δl k ≈ Δxk2 + Δy k2 ≈ ( x ′(t k )) 2 + ( y ′(t k )) 2 ⋅ Δt k . aseve viciT, rom rkalis dl diferencialisTvis adgili aqvs tolobas: dl = ( x ′(t )) 2 + ( y ′(t )) 2 ⋅ dt .

yovelive aqedan T

∫ f ( x, y)dl = ∫ f ( x(t ), y(t ))

( x ′(t )) 2 + ( y ′(t )) 2 dt .

(2)

am formulis marjvena mxares gvaqvs gansazRvruli integrali, romlis gamoTvlis meTodi CvenTvis ukve nacnobia. maSasadame, pirveli gvaris mrudwiruli integralis gamoTvla daiyvaneba gansazRvruli integralis gamoTvlaze. vTqvaT, exla wiri mocemulia gantolebiT y = y ( x ) . aseT SemTxvevaSi SeiZleba pirdapir davweroT wiris parametruli gantoleba: x = x, y = y ( x ) , sadac parametris rols asrulebs x cvladi. davuSvaT A wertilis koordinatebia ( a, y ( a )) , B wertilis ki (b, y (b)) . maSin rkalis wertilebisTvis x cvladi Rebulobs mniSvnelobebs [a, b] Sualedidan, amitom, (2) formulaSi Tu SevcvliT t parametrs x parametriT, gveqneba: b

∫ f ( x, y)dl = ∫ f ( x, y( x))

(1 + ( y ′( x)) 2 dx .

magaliTi. 1. gamovTvaloT pirveli gvaris mrudwiruli integrali ∫ xydl , AB

AB rkalze, romelic warmoadgens x 2 + y 2 = a 2 wrewiris, sakoordinato sibrtyis pirvel meoTxedSi mdebare nawils. amoxsna: gadavideT wrewiris parametrul gantolebaze x = a cos t , y = a sin t , sadac sakoordinato sibrtyeSi mdebare wrewiris nawilisTvis t parametri

π icvleba SualedSi: [0, ] . aseT SemTxvevaSi (2) formuliT gveqneba: 2

93 π 2

2 2 2 2 ∫ xydl = ∫ a cos t ⋅ a sin t a cos t + a sin t dt = 0

a 3 ∫ sin t cos tdt = 0

1 − a 3 cos θ 4

π 0

π π

1 3 1 2 1 a ∫ sin 2tdt = a 3 ∫ sin 2td 2t = a 3 ∫ sin θdθ = 4 0 2 0 4 0 2

1 3 1 3 a3 a + a = . 4 4 2

2. meore gvaris mrudwiruli integrali vTqvaT sibrtyeze mdebare araCaketili l wiris rkalze, romlis boloebia wertilebi A, B , gansazRvrulia ori funqcia P = P ( x, y ) da Q = Q ( x, y ) . davyoT AB rkali n nawilad, nebismierad: M 1 ( x1 , y1 ), M 2 ( x 2 , y 3 ),..., M n −1 ( x n −1 , y n −1 ) .

davuSvaT, A = M 0 ( x0 , y 0 ), B = M n ( x n , y n ) , aviRoT nebismierad wertili M k M k +1 rkalidan da gamovTvaloT am wertilSi mniSvnelobebi P (ξ k ,η k ) da Q(ξ k ,η k ) . ganvixiloT jami: n −1

∑ P(ξ k =0

,η k )Δx k + Q(ξ k ,η k )Δy k ,

M k (ξ k ,η k )

funqciaTa

(1)

sadac Δx k = x k +1 − x k , Δy k = y k +1 − y k . gansazRvreba 6.2. Tu arsebobs (1) jamis zRvari: lim

n −1

∑ P(ξ

n→∞ max Δxk →0 k = 0 max Δy k →0

,η k )Δx k + Q(ξ k ,η k )Δy k ,

romelic damokidebuli araa M k (ξ k ,η k ) wertilis amorCevaze, maSin am zRvars uwodeben meore gvaris mrudwirul integrals P = P ( x, y ) da Q = Q ( x, y ) funqciebisaTvis, AB rkalze da aRniSnaven ase: ∫ P( x, y)dx + Q( x, y)dy . AB

Tu dayofis wertilebis aTvlas daviwyebT B wertilidan, maSin Δx k = x k +1 − x k , Δy k = y k +1 − y k sidideebi niSans Seicvlian, gansxvavebiT Δl k sididisgan, romelic rkalis sigrZes warmoadgens da yovelTvis dadebiTia. aqedan gamomdinare, meore gvaris mrudwiruli integrali, gansxvavebiT pirveli gvaris mrudwiruli integralisagan, integrebis mimarTulebis SecvliT, niSans Seicvlis. maSasadame adgili eqneba tolobas: (2) ∫ P( x, y)dx + Q( x, y)dy = − ∫ P( x, y)dx + Q( x, y)dy . AB

vTqvaT β k warmoadgens kuTxes M k M k +1 qordasa da OX RerZs Soris(nax.34). naxazidan Cans, rom Δx k ≈ Δl k cos β k , Δy k ≈ Δl k sin β k ,sadac Δl k warmoadgens M k M k +1 rkalis sigrZes. Tu gadavalT diferencialze gveqneba:

94 dx = cos β ( x, y ) dl , dy = sin( x, y ) β dl .

(3)

ukanasknel tolobebSi β ( x, y ) kuTxea wiris cvlad M wertilSi gamaval L mxebsa da OX RerZs Soris, Tu gaviTvaliswinebT (3) tolobebs, gveqneba: (4) ∫ P( x, y)dx + Q( x, y)dy = ∫ [ P( x, y) cos β ( x, y) + Q( x, y) sin β ( x, y)]dl . AB

(4) toloba gviCvenebs integralebs Soris.

kavSirs

pirveli

meore

gvaris

mrudwirul

Y M k +1

Δy k

β ( x, y )

M ( x, y )

M k βk

Δx k

nax.34

l wiri mocemulia parametrulad, uwyvetad warmoebadi vTqvaT, x = x (t ), y = y (t ) funqciebiT, sadac t parametri icvleba [T1 , T2 ] SualedSi da AB rkalis boloebis koordinatebia, Sesabamisad: A( x(t 0 ), y (t 0 )), B ( x(T ), y (T )); t 0 , T ∈ [T1 , T2 ] . n nawilad davyoT [t 0 , T ] segmenti wertilebiT t1 , t 2 ,..., t n −1 , Sesabamisad, n AB rkalic daiyofa nawilad wertilebiT M 1 ( x(t1 ), y (t1 )), M 2 ( x(t 2 ), y (t 3 )),..., M n −1 ( x(t n −1 ), y (t n −1 )) . am dros P = P ( x, y ) da Q = Q ( x, y ) funqciebi warmoadgenen rTul funqciebs: P = P ( x, y ) = P ( x (t ), y (t )) , Q = Q ( x, y ) + Q ( X (t ), y (t )) , amasTan dx = x ′(t ) dt , dy = y ′(t ) dt . Tu gaviTvaliswinebT ukanasknel mimarTebebs. gveqneba: t

∫ P( x, y)dx + Q( x, y)dy = ∫ [ P( x(t )) x′(t ) + Q( x(t ), y(t )) y ′(t )]dt .

(5)

amgvarad, meore gvaris mrudwiruli integralis gamoTvla daviyvaneT gansazRvruli integralis gamoTvlaze. xSirad saqme gvaqvs iseT meore gvaris mrudwiruli integralTan, rodesac integralqveSa gamosaxuleba Seicavs mxolod erT Sesakrebs:

∫

f ( x, y )dx = ∫ f ( x(t ), y (t )) x ′(t )dt , 0

∫ f ( x, y)dy = ∫ f ( x(t ), y(t )) y ′(t )dt .

axla ganvixiloT SemTxveva, rodesac wiri mocemulia gantolebiT y = y ( x) aseT SemTxvevaSi SeiZleba pirdapir davweroT wiris parametruli gantoleba:

95 x asrulebs cvladi. sadac parametris rols x = x, y = y ( x ) , davuSvaT A wertilis koordinatebia ( a, y ( a )) , B wertilis ki- (b, y (b)) . maSin rkalis wertilebisTvis x cvladi Rebulobs mniSvnelobebs [a, b] Sualedidan, amitom, (2) formulaSi, Tu SevcvliT t parametrs x parametriT, gveqneba: b

∫ P( x, y)dx + Q( x, y)dy = ∫ [ P( x, y( x)) + Q( x, y( x)) y ′( x)dx .

(6)

SeniSvna: Tu AB rkali l wirze Sedgeba AC , CD ,..., FB nawilebisgan, maSin mrudwiruli integralis gansazRvridan gamomdinare

∫ P( x, y)dx + Q( x, y)dy = ∫ P( x, y)dx + Q( x, y)dy + ∫ P( x, y)dx + Q( x, y)dy + ...

(7)

... + ∫ P( x, y )dx + Q( x, y )dy. FB

mrudwiruli integralis gansazRvrisas vgulisxmobdiT, rom AB rkali l wirze ar warmoadgenda Sekrul konturs. Tu C = AB rkali Sekruli konturia anu A = B , maSin am konturze aviRebT raime D wertils, mxedvelobaSi miviRebT ukanasknel formulas, ris Sedegadac gveqneba:

∫ P( x, y)dx + Q( x, y)dy = ∫ P( x, y)dx + Q( x, y)dy + ∫ P( x, y)dx + Q( x, y)dy =

∫

P( x, y )dx + Q( x, y )dy +

(8)

∫ P( x,y)dx + Q( x, y)dy.

integreba AD da DA rkalebze xdeba erTi da igive mimarTulebiT anu A wertilidan D wertilisken, pirvel SesakrebSi; D wertilidan A wertilisken, meore SesakrebSi. amasTan, dadebiT mimarTulebad iTvleba is mimarTuleba, romlis drosac konturze moZraobisas mis mier SemosazRvruli are rCeba Cvengan marcxniv. mrudwiruli integrali Caketil C konturze aRiniSneba ase: ∫ P( x, y)dx + Q( x, y)dy C . magaliTi 1. gamovTvaloT ∫ xydx + ( y − x)dy integralis mniSvneloba AB

A(0,0) da B (1,1) wertilebis SemaerTebel rkalze, Tu: 1) AB warmoadgens monakveTs wrfeze y = x ,

2) AB warmoadgens rkals parabolaze y = x 2 , 3) AB warmoadgens rkals kubur parabolaze y = x 3 . amoxsna: 1) am SemTxvevaSi wiris parametrul gantolebas aqvs saxe: x = x, y = x , sadac x parametri icvleba SualedSi [0,1] . gveqneba: 1

2 ∫ xydx + ( y − x)dy = ∫ [ x − ( x − x)]dx = ∫ x dx = 2

x3 3

1 0

1 . 3

2) am SemTxvevaSi wiris parametruli gantolebaa x = x, y = x 2 , sadac x parametri icvleba SualedSi [0,1] . gveqneba: 1

2 ∫ xydx + ( y − x)dy = ∫ [ x − ( x − x)]2dx = 3∫ x dx − 2∫ x = 3

3x 3 2 x 2 − 4 3

1 0

1 . 12

96 3) am SemTxvevaSi wiris parametruli gantolebaa x = x, y = x 3 , sadac x parametri icvleba SualedSi [0,1] . gveqneba: 1

4 3 2 5 4 3 ∫ xydx + ( y − x)dy = ∫ [ x − ( x − x)]3x dx = 3∫ x dx + ∫ x dx − 3∫ x dx

3x 1 1 x x + − . 0 = 2 5 4 20 magaliTi 2. gamovTvaloT

∫ P( x + y)dx + 2 xdy

integralis mniSvneloba, sadac

C warmoadgens y = x2 da funqciebis grafikTa nawilebisgan y= x warmoqmnil Sekrul konturs. amoxsna: C konturi Sedgeba ori sxvadasxva wirze mdebare, OA da OA rkalebisagan, romelTagan pirveli SeiZleba warmovadginoT parametruli gantolebiT: x = x, y = x 2 , x ∈ [0,1] , meore Semdegi parametruli gantolebiT: x = x, y =

x , x ∈ [0,1] . gveqneba:

∫ ( x + y)dx + 2 xdy = ∫ ( x + y)dx + 2 xdy + ∫ ( x + y)dx + 2 xdy ,

sadac tolobis marjvena mxaris integrireba xdeba pirvel SesakrebSi meore wirze mdebare OA rkalze; meore SesakrebSi ki- pirvel wirze mdebare AO rkalze − ∫ ( x + y )dx + 2 xdy = ∫ ( x + y )dx + 2 xdy , OA

amitom 1

∫ ( x + y)dx + 2 xdy = ∫ ( x + y)dx + 2 xdy − ∫ ( x + y)dx + 2 xdy = ∫ ( x + OA

x + 2x

1 2 x

)dx −

1 x2 1 5 − ∫ ( x + x 2 + 2 x 2 x)dx = ∫ ( x + 2 x )dx + ∫ ( x + 5 x 2 )dx = ( x 2 + 2 ) 10 −(( x 2 + x 3 ) 10 ) = 3 2 2 3 0 0 0 2 1 4 1 5 = + − + =3 2 3 2 3

vTqvaT, XOY sibrtyeze mocemulia D are, romelic SemosazRvrulia erTi mTliani, Caketili konturiT. aseT areebs srulad bmul areebs uwodeben(nax.33). aviRoT ori A(ξ 0 ,η 0 ), B(ξ1 ,η1 ) wertili D aris SigniT. vTqvaT, funqciebi P ( x, y ), Q ( x, y ) gansazRvrulia am areze. SevaerToT A da B wertilebi ori nebismieri AnB da AmB rkalebiT da

• B (ξ 1 ,η )

m D

• A(ξ 0 ,η 0 ) )

n X

ganvixiloT mrudwiruli integralebi:

nax. 35

∫ P( x, y)dx + Q( x, y)dy , ∫ P( x, y)dx + Q( x, y)dy , ∫ P( x, y)dx + Q( x, y)dy .

AnB

AmB

(9)

bolo integrali gansazRvrulia C = AnB + BmA Caketil konturze, igi nulis toli iqneba maSin da mxolod maSin, rodesac adgili eqneba tolobas: (10) ∫ P( x, y)dx + Q( x, y)dy = ∫ P( x, y)dx + Q( x, y)dy . AnB

AmB

radganac AnB da AmB rkalebi nebismierad aris aRebuli, amitom (10) toloba niSnavs, rom mrudwiruli integrali P ( x, y ), Q ( x, y ) funqciebisagan, damokidebuli ar aris A da B wertilebis SemaerTebeli wiris SerCevaze anu damokidebuli ar aris integrirebis gzis SerCevaze. mtkicdeba Teorema: Teorema 6.1. srulad bmul D areze gansazRvruli P ( x, y ), Q ( x, y ) diferencirebadi funqciebisTvis mrudwiruli integrali ∫ P( x, y )dx + Q( x, y )dy ar aris damokideuli A da B wertilebis AB

SemaerTebeli wiris SerCevaze

maSin da mxolod maSin, rodesac

∂P( x, y ) ∂Q( x, y ) = . ∂y ∂x am

Teoremidan

gamomdinareobs,

rom

(11) Tu

integrali ∫ P( x, y )dx + Q( x, y )dy AB

damokidebuli ar aris integrirebis gzis SerCevaze, maSin integrali ∫ P( x, y)dx + Q( x, y)dy = 0 yovel Caketil D areSi moTavsebul C konturze. C

3. meore gvaris mrudwiruli integralis meqanikuri mniSvneloba vTqvaT, OYX sibrtyeze mocemulia D are, romlis TiToeul wertilSi G modebulia Zala F ( x, y ) . davuSvaT, materialuri wertili moZraobs am egreT wodebuli Zaluri velis moqmedebiT raime l wirze. gamovTvaloT muSaoba E , romelic sruldeba am velis mier materialuri wertilis wiris A wertilidan B wertilamde gadaadgilebisas. davyoT AB rkali n nawilad, nebismierad, wertilebiT: M 1 ( x1 , y1 ), M 2 ( x 2 , y 3 ),..., M n −1 ( x n −1 , y n −1 ) . G davuSvaT, A = M 0 ( x0 , y 0 ), B = M n ( x n , y n ) . vTqvaT, F ( x, y ) Zalis gegmilebi OX da OY RerZebze, Sesabamisad, tolia f1 ( x, y) da f 2 ( x, y ) sidideebis. Tu AB rkalis dayofiT miRebuli rkalebis maqsimaluri sigrZe Δl sakmaod mcirea, maSin SegviZlia CavTvaloT, rom M k M k +1 rkalis wertilebze G modebuli Zalebi miaxloebiT tolia F (ξ k ,η k ) sididis, sadac (ξ k ,η k ) warmoadgens am rkalze moTavsebul, nebismierad arCeul wertilis koordinatebs. aqedan gamomdinare, im muSaoa E k , romelic sruldeba, velis

mier, materialuri wertilis gadaadgilebisas

M k M k +1

rkalis gaswvriv

98 tolia:

daaxloebiT E k ≈ f1 (ξ kη k )Δx k + f 2 (ξ k ,η k )Δy k ,Esadac Δx k = x k +1 − x k , Δy k = y k +1 − y k . wiris A wertilidan B wertilamde gadaadgilebisas gveqneba miaxloebiTi toloba n −1

E ≈ ∑ f 1 (ξ k ,η k )Δx k + f 2 (ξ k ,η k )Δy k , k =0

rac ufro met nawilebad davyofT AB rkals, ise rom Δl sidide sul ufro da ufro Semcirdes, zRvarze gadasvliT miviRebT zust tolobas: n −1

E = lim ∑ f1 (ξ k ,η k )Δx k + f 2 (ξ k ,η k )Δy k , n →∞ Δl →0 k = 0

maSasadame

∫ f ( x, y)dx + f 1

( x, y )dy .

es ki niSnavs, rom meore gvaris mrudwiruli integralis mniSvneloba warmoadgens Zaluri velis mier materialuri wertilis raime wirze moZraobisas Sesrulebul muSaobas. magaliTi 1. x = a cos t , y = a sin t , t ∈ [0,2π ] parametruli gantolebiT G gansazRvrul wrewirze modebulia F ( P, Q ) Zaluri velis mier Sesrulebuli muSaoba, sadac P = x + y , Q = 2 x , materialuri wertilis am wrewirze moZraobisas. amoxsna: rogorc aRvniSneT, es muSaoba gamoiTvleba formuliT: E = ∫ ( x + y )dx + 2 xdy , C

sadac C mocemuli wrewiria. P = a (cos t + sin t ); q = 2a sin t , dx = − a sin tdt , dy = a cos tdt . am tolobebis gaTvaliswinebiT 2x

E = ∫ ( x + y )dx + 2 xdy = ∫ [−a 2 (cos t + sin t ) sin t + 2a 2 cos 2 t ]dt = 0

1 1 = ∫ [2 cos 2 t − sin 2 t − cos t sin t ]dt = a 2 ∫ [(1 + cos 2 x) − (1 − cos 2t ) − sin 2t ]dt = 2 2 0 0 2x

a2 a2 = a ∫ dt + a ∫ cos 2tdt − dt + 2 ∫0 2 0 0 2

−

a2 t 2

2x 0

a2 sin 2t 4

2x 0

a2 cos 2t 4

2x 0

a2 ∫0 cos 2tdt − 2

∫ sin 2tdt = a t 2

2x 0

a2 + sin 2t 2

2x 0

−

= 2πa 2 − πa 2 = πa 2 .

savarjiSoebi me-6 TavisaTvis I gamoTvaleT Semdegi pirveli gvaris mrudwiruli integralebi: 1) ∫ C

ds x2 + y2 + 4

,sadac

C wrfis

O (0,0), A(1,2) ; 2) ∫ xyds , sadac C

monakveTia,

C warmoadgens

romelic

aerTebs

wertilebs

x2 y2 + = 1 elifsis meoTxeds, a2 b2

99 romelic mdebareobs sakoordinato sibrtyis pirvel meoTxedSi; 2 3t , z = t 3 ,0 ≤ t ≤ 1 wiris rkals. 3) ∫ ( x + z )ds , sadac C warmoadgens x = t , y = 2 C II. gamoTvaleT Semdegi meore gvaris mrudwiruli integralebi: 1) ∫ ( x 2 − 2 xy)dx + (2 xy + y 2 )dy , sadac AB warmoadgens y = x 2 parabolis rkals AB

romelic moqceulia A(1,1) da B ( 2,4) wertilebs Soris; 2) ∫ y 2 dx + y 2 dy , sadac C

warmoadgens x = a cos t , y = b sin t

elifsis zeda naxevars da integrireba

xdeba saaTis isris sawinaaRmdego mimarTulebiT; 3) ∫ cos ydx − sin xdy , sadac AB

warmoadgens meore sakoordinato kuTxis abscisia 2, B wertilis ordinati- 2.

biseqtrisas.

wertilis

100 Tavi 7 diferencialuri gantolebebi 1. ZiriTadi cnebebi Cveulebrivi n -uri rigis diferencialuri gantoleba ewodeba gantolebas: ⎛ dy d 2 y dny⎞ F ⎜⎜ x, y, , 2 ,..., n ⎟⎟ = 0 , (1) dx dx dx ⎠ ⎝ sadac F Tavisi argumentebis uwyveti funqciaa. (1) gantolebis amonaxsns warmoadgens n rigamde warmoebadi funqcia y = ϕ (x ) , romlis Casma gantolebaSi iwvevs mis igivur tolobad gadaqcevas. gansazRvridan gamomdinare pirveli da meore rigis Cveulebriv diferencialur gantolebebs eqnebaT saxe: ⎛ dy d 2 y ⎞ dy ⎞ ⎛ F ⎜ x, y, ⎟ = 0; F ⎜⎜ x, y, , 2 ⎟⎟ = 0 . dx ⎠ dx dx ⎠ ⎝ ⎝ xSirad n -uri rigis diferencialuri gantoleba micemulia aseTi saxiT: dny (2) = f ( x, y, y ′, y ′′,..., y ( n −1) ) . n dx Cveulebrivi diferencialuri gantoleba rom iyos n -uri rigis, F dny funqcia aucileblad damokidebuli unda iyos argumentze, miuxedavad dx 2 imisa, igi damokidebuli iqneba Tu ara sxva argumentebze. dy ganvixiloT pirveli rigis diferencialuri gantoleba: = f ( x ) . (3) am dx gantolebis marcxena mxares dgas misi y = ϕ (x ) amonaxsnis warmoebuli, f (x ) funqciisa, amitom adgili eqneba tolobas romelic tolia dy ∫ dx = ∫ f (x )dx aqedan gamomdinare gveqneba tolobac y = ∫ f (x )dx . maSasadame (3) gantolebis amonaxsni erTaderTi ki ar aris, aramed warmoadgens namdvil ricxvTa simravliT parametrizebul funqciaTa simravles, vinaidan funqciis ganusazRvreli integrali swored aseT simravles warmoadgens. maSasadame (3)gantolebis amonaxsni SeiZleba warmovadginoT ase: (4) y = ϕ ( x, C ) , sada C parametria, romelic Rebulobs mniSvnelobebs namdvil ricxvTa simravleSi. exla ganvixiloT diferencialuri gantoleba: d2y = f (x ) . (5) dx 2 am gantolebis marcxena mxares dgas misi y = ϕ (x ) amonaxsnis meore rigis warmoebuli, amitom adgili eqneba tolobebs: ⎛ d2y ⎞ d2y ( ) dx = f x dx , ∫ ⎜⎜⎝ ∫ dx 2 dx ⎟⎟⎠dx = ∫ ∫ f (x )dx dx . ∫ dx 2 ∫ sabolood ki miviRebT: y = ∫ ∫ f ( x )dx dx .

(

)

maSasadame (3) gantolebis amonaxsni erTaderTi ki ar aris, aramed warmoadgens namdvil ricxvTa wyvilebiT parametrizebul funqciaTa

101 simravles, radganac funqciis orjer mimdevrobiT aRebuli ganusazRvreli integrali warmoadgens aseT simravles. y = ϕ ( x; C1 , C 2 ) , (6) C1 ,C 2 parametrebia, romlebic namdvil ricxvTa simravlidan sadac Rebuloben mniSvnelobebs. (3) da (5) gantolebebis (4) da (6) tipis amonaxsnebs Sesabamisad uwodeben am gantolebebis zogad amonaxsnebs. Tu parametrebs C , C1 ,C 2 mivcemT raime konkretul mniSvnelobebs maSin miviRebT C = C 0 C1 = C10 , C 2 = C 20 , egreTwodebul kerZo amonaxsnebs Sesabamisad: y = ϕ ( x, C 0 ) , y = ϕ (x; C10 , C 20 ) . (7) zogadi amonaxsnis cneba gvaqvs (1) saxis gantolebebisTvisac. mtkicdeba, rom (1) saxis gantolebis zogad amonaxsns aqvs saxe: y = ϕ ( x; C1 , C 2 ,...., C n ) . Tu C1 , C 2 ,..., C n

parametrebs

mivcemT

miviRebT

gantolebis

kerZo

mniSvnelobebs

C10 , C 20 ,..., C n0 ,

amonaxsns- y = ( x, C10 , C 20 ,..., C n0 ) .

maSasadame

konkretul

SeiZleba vilaparakoT Cveulebrivi diferencialuri gantolebis zogad da kerZo amonaxsnebze, gantolebis nebismieri rigis SemTxvevaSic. CavTvaloT x da y dekartes marTkuTxa koordinatebad sibrtyeze. vTqvaT, funqcia y = ϕ (x ) (2) gantolebis raime amonaxsnia, am funqciis grafiks (2) gantolebis integraluri wiri ewodeba. diferencialuri gantolebis amonaxsnTa Zebnis dros, xSirad ismeba egreTwodebuli koSis amocana: vipovoT (1) an (2) gantolebis iseTi amonaxsni y = ϕ (x) , rom (8) ϕ ( x0 ) = y 00 , ϕ ′( x 0 ) = y 01 , ϕ ′′( x0 ),..., ϕ ( n −1) ( x0 ) = y 0n −1 sadac x 0 , y 00 , y 01 , y 02 ,..., y 0n −1 winaswar mocemuli ricxvebia. (8) pirobebs koSis amocanis sawyisi pirobebi ewodeba. gansazRvreba7.1 vityviT, rom D = {( x, y1 , y 2 ,..., y n ) | x0 − a ≤ x ≤ x 0 + a , y 0i − bi ≤ y i ≤ y 0i + bi , i = 12,..., n} areze gansazRvruli funqcia f ( x, y1 , y 2 ,..., y n ) akmayofilebs lifSicis pirobas, Tu arsebobs iseTi N > 0 namdvili ricxvi, rom yoveli x -isaTvis, romelic akmayo filebs utolobas x − x0 ≤ a da y i , i = 1,2,..., n cvladebis yoveli

ori

romlebic

y i1 , i = 1,2,..., n

akmayofileben

y i2 , i = 1,2,..., n

mniSvnelobaTa

sistemisTvis,

yi1 − y0i ≤ bi , yi2 − y0i ≤ bi , i = 1,2,..., n

utolobebs

sruldeba piroba:

(

) (

)

f x, yi1 , yi1 ,..., yi1 − f x, yi2 , yi2 ,..., yi2 ≤ ∑ N yi1 − yi2 . i =1

n rigis (2) saxis gantolebisaTvis, koSis amocanis amonaxsnis arseboba da erTaderToba gamomdinareobs Semdegi Teoremidan: Teorema 7.1.(koSis Teorema): vTqvaT, D = {( x, y, y ′, y ′′,..., y ( n −1) ) | x0 − a ≤ x ≤ x0 + a, y 0 − b ≤ y ≤ y 0 + b, y 0( i ) − bi ≤ y ( i ) ≤ y 0(i ) + bi ; i = 1,2,...n − 1} marTkuTxedze gansazRvruli funqcia f ( x, y, y ′, y ′′,..., y ( n −1) ) akmayofilebs lifSicis pirobas da SemosazRvrulia- | f ( x, y, y ′, y ′′,..., y ( n −1) ) |< M . maSin arsebobs

102

dny = f ( x, y, y ′, y ′′,..., y ( n −1) ) n dx gantolebis erTaderTi amonaxsni y = ϕ (x ) , romelic gansazRvrulia ⎧ b⎫ x0 − h ≤ x ≤ x0 + h Sualedze, sadac h = min ⎨a, ⎬ da akmayofilebs pirobebs: ⎩ M⎭ 0 1 ϕ ( x0 ) = y 0 , ϕ ′( x 0 ) = y 0 , ϕ ′′( x0 ),..., ϕ ( n −1) ( x0 ) = y 0n −1 , sadac x 0 , y 00 , y 01 , y 02 ,..., y 0n −1 winaswar mocemuli ricxvebia. rogorc koSis Teorema gviCvenebs, rodesac (2) gantolebis marjvena mxare akmayofilebs garkveul pirobebs, misi amonaxsni arsebobs, erTaderTia da akmayofilebs pirobebs 0 1 ( n −1) n −1 ϕ ( x0 ) = y 0 , ϕ ′( x 0 ) = y 0 , ϕ ′′( x0 ),..., ϕ ( x 0 ) = y 0 , sadac x 0 da y 0 winaswar mocemuli ricxvebia. XOY sakoordinato sibrtyis nebismierad geometriulad es niSnavs, rom dafiqsirebul ( x0 , y 0 ) wertilze gaivlis (2) gantolebis mxolod erTi integraluri wiri. Tu vcvliT koSis amocanis sawyis pirobebs, miviRebT (2) diferencialuri gantolebis sxvadasxva amonaxsns, amitom am gantolebis zogadi amonaxsni SeiZleba warmovadginoT ase: sadac x0 y = ϕ (x; x 0 , y 00 , y 01 , y 02 ,...., y 0n −1 ) , dafiqsirebuli ricxvia, y 00 , y 01 , y 02 ,..., y 0n −1 - cvladi sidideebi, Rebuloben mniSvnelobas namdvil ricxvTa simravlidan.

romlebic

2. umartivesi pirveli rigis Cveulebrivi diferencialuri gantolebebi 2.1. diferencialuri gantoleba gancalebadi cvladebiT diferencialuri gantoleba gancalebadi cvladebiT Semdegi saxisaa: dy (1) = f ( x )ϕ ( y ) , dx sadac marjvena mxare warmoadgens or sxvadasxva argumentze damokidebul funqciis namravls. es gantoleba gardavqmnaT Semdegnairad: gavyoT orive mxare ϕ ( y ) - ze da Semdeg gavamravloT dx -ze, miviRebT: dy = f ( x )dx . ϕ(y) miviReT toloba, romlis marcxena mxare damokidebulia y argumentze, marjvena ki x argumentze, amasTan tolobis orive mxares gvaqvs raRac funqciebis diferencialebi. Tu aviRebT orive mxaris ganusazRvrel integralebs, miviRebT tolobas: dy (2) ∫ ϕ ( y ) = ∫ f (x )dx . miRebuli toloba warmoadgens x da y cvlad sidideebs Soris aracxadi saxis damokidebulebas: Φ ( x, y , C ) = 0 , (3)

103 romlis saSualebiTac y cvladi, zog SemTxvevaSi, SeiZleba gamovsaxoT x da C cvladebis saSualebiT. aseT SemTxvevaSi miviRebT (1) gantolebis zogad amonaxsns:

y = φ (x, C ) , Tu aseTi ram ar moxerxda anu Tu y cvladi ar gamisaxeba (3) tolobidan danarCeni cvladebis saSualebiT, maSin (1) gantoleba (3) tolobis miRebiT, mainc iTvleba amoxsnilad da (3) tolobas uwodeben am gantolebis zogad integrals. magaliTi . amovxsnaT diferencialuri gantoleba gancalebadi cvladebiT dy (4) = y sin x . dx amoxsna: am gantolebis gardaqmniT vRebulobT: dy (5) ∫ y = ∫ sin xdx . Tu movaxdenT tolobis orive mxaris integrirebas, miviRebT ln y = − cos x + C . maSasadame Cveni gantolebis zogadi amonaxsna iqneba: y = e − cos x = eC e − cos x = Ce − cos x . Tu C = C0 , gveqneba kerZo amonaxsni: y = C0e − cos x .

Tu (x0 , y 0 ) (4) gantolebis marjvena mxaris gansazRvris aris iseTi wertilia, romelic mdebareobs am gantolebis integralur wirze, am SemTxvevaSi y0 ≥ 0 , maSin (5) toloba SeiZleba ase gadavweroT: y

dt ∫y t + C0 = x∫ sin tdt + C1 . 0 0

(6) x

∫ sin tdt

gveqneba: ln y = ∫ sin tdt + C , sadac C = C1 + ln y0 − C0 , aqedan y = eC e x 0

∫ sin tdt

= Ce x 0

rodesac x = x0 , maSin y = y0 , amitom y0 = C . maSasadame im integraluri wiris gantoleba anu (4) gantolebis kerZo amonaxsni iqneba: x

∫ sin tdt

y = y0e x 0

2.2. pirveli rigis erTgvarovani diferencialuri gantoleba ori cvladis funqcias y = f ( x, y ) ewodeba erTgvarovani, Tu nebismieri t ∈ R namdvili ricxvisaTvis adgili aqvs tolobas f (x, y ) = f (tx, ty ) . diferencialur gantolebas: dy (1) = f ( x, y ) dx

ewodeba erTgvarovani Tu misi erTgvarovan funqcias. radgan

f (x, y )

erTgvarovani

104 marjvena mxare f (x, y ) funqciaa,

amitom

warmoadgens

⎛ y⎞ f (x, y ) = f ⎜1, ⎟ . ⎝ x⎠

aqedan

gamomdinare, (1) gantolebas eqneba Semdeg saxe: dy ⎛ y⎞ (2) = f ⎜1, ⎟ . dx ⎝ x⎠ y dy du SemovitanoT cvladTa gardaqmna u = , maSin y = ux da =u+ x . amis x dx dx Semdeg (2) gantolebis nacvlad gveqneba: du du 1 u+ = ( f (1, u ) − u ) . x = f (1, u ) , anu dx dx x

bolo gantolebis marjvena mxare warmoadgens x da u cvladebze damokidebul funqciaTa namravls. amgvarad (1) gantoleba daviyvaneT gancalebadcvladebian pirveli rigis diferencialur gantolebamde, romlis amoxsnac ukve viciT. magaliTi. amovxsnaT diferencialuri gantoleba dy 2 xy . = 2 dx x − y 2 amoxsna: gantolebis marjvena mxare erTgvarovani funqciaa. marTlac: x 2 xy 2 f ( x, y ) = 2 = = f (1, ) . 2 x y y x −y − y x dy du SemovitanoT aRniSvna y = ux , maSin da gantoleba miiRebs saxes: =u+x dx dx du u + u 3 du 2u x = . am gantolebidan miviRebT . es ki warmoadgens u+x = dx 1 − u 2 dx 1 − u 2 gantolebas gancalebadi cvladebiT: dx u2 −1 = du . x u (u 2 + 1) am gantolebisTvis SegviZlia davweroT zogadi integrali dx u2 −1 ∫ x = ∫ u (u 2 + 1) du . Tu gamovTvliT tolobis orive mxares mdgom ganusazRvrel integralebs, x(u 2 + 1) = C . CavsvaT am mivi- RebT ln x = ln(u 2 + 1) − ln u + ln C . aqedan gveqneba u y tolobaSi u cvladic magier , miviRebT gantolebis zogad integrals x x 2 + y 2 = yC . unda aRvniSnoT, rom gantolebis amonaxsnia aseve funqcia y = 0 , rac mowmdeba uSualod.

105 2.3. wrfivi pirveli rigis diferencialri gantoleba wrfiv pirveli rigis diferencialur gantolebas aqvs saxe: dy + P(x ) y = Q(x ) , dx sadac P(x ) da Q( x ) saerTo gansazRvris aris mqone funqcebia. Tu Q(x ) = 0 , maSin (1) gantolebas ewodeba wrfivi pirveli rigis erTgvarovani diferencialuri gantoleba da aqvs saxe:

dy + P(x ) y = 0 . dx igi warmoadgens gancalebadcvladebian gantolebas da misi amonaxsnia − P ( x )dx y = Ce ∫ .

(1)

zogadi

Tu Q( x ) ≠ 0 , maSin (1) gantolebas ewodeba wrfivi, araerTgvarovani diferencialuri gantoleba. mis amosaxsnelad viyenebT egreTwodebul mudmivTa variaciis meTods, romelic SemdegSi mdgomareobs: (1) gantolebis amonaxsns veZebT aseTi funqciis saxiT: − P ( x )dx y = C ( x )e ∫ , (2)

romlis CasmiTac (1) gantolebaSi miviRebT: − P ( x )dx − P ( x )dx dC ( x ) ∫ − P ( x )dx e − C ( x )e ∫ P ( x ) + P ( x )C ( x )e ∫ = Q(x ) , dx aqedan vRebulobT: P ( x )dx C ( x ) = Q(x )e ∫ +C.

Tu CavsvamT C ( x ) -is gamoTvlil mniSvnelobas (2) tolobaSi miviRebT (1) gantolebis zogad amonaxsns: − P ( x )dx ⎛ ∫ P ( x )dx dx + C ⎞ . y = e∫ (3) ⎟ ⎜ ∫ Q( x )e ⎠ ⎝ magaliTi. vipovoT diferencialuri gantolebis dy + y = e −x dx zogadi amonaxsni. amoxsna: gamoviyenoT (3) formula, miviRebT zogad amonaxsns: − dx dx y = e ∫ ⎛⎜ ∫ e − x e ∫ dx + C ⎞⎟ , ⎝ ⎠ −x sabolood y = e ( x + C ) . 2.4. bernulis diferencialuri gantoleba bernulis diferencialuri gantoleba ewodeba gantolebas: dy (1) + P(x ) y = Q(x ) y n . dx Tu n = 0 , cxadia, bernulis gantoleba warmoadgens pirveli rigis wrfiv araerTgvarovan gantolebas, xolo Tu n = 1 - pirveli rigis erTgvarovan diferencialur gantolebas. vTqvaT, n > 1 , maSin Tu (1) gantolebis orive mxares gavyofT y n -ze miviRebT:

106 dy + P ( x ) y − n +1 = Q (x ) . dx dz dy . aqedan SemovitanoT axali cvladi z = y − n +1 , gveqneba = (− n + 1) y − n dx dx vRebulobT: dz + (− n + 1)P ( x )z = (− n + 1)Q ( x ) . dx es gantoleba wrfivi, pirveli rigis diferencialuri gantolebaa, romlis amoxsnis meTodic zemoT ganvixileT. y −n

Tu n > 0 (1) gantolebas aqvs agreTve trivialuri amonaxsnic y = 0 . magaliTi. vipovoT dy + xy = x 3 y 3 dx gantolebis zogadi amonaxsni. 1 1 2 dy dz amoxsna: aq n = 3 , gamoviyenoT gardaqmna: z = 3−1 = 2 , gveqneba . =− 3 dx y y y dx bolo tolobis gaTvaliswinebiT da gantolebis y 3 -ze gayofiT miviRebT 1 dy 1 dz + x 2 = x 3 . aqedan gveqneba: − 2 xz = −2 x 3 . 3 dx y y dx miRebuli gantoleba warmoadgens wrfiv, pirveli rigis diferencialur gantolebas, misi zogadi amonaxsnia: 2 xdx − 2 xdx z = e ∫ ⎛⎜ − 2∫ x 3e ∫ dx + C ⎞⎟ ⎝ ⎠ anu z = e x ( ∫ − 2 x 3 e − x dx + C ) = e x ( ∫ x 2 de − x + C ) = e x (e − x x 2 + e − x + C ) = x 2 + 1 + Ce x . 2

radgan z =

1 , amitom sabolood gveqneba. y2 1 y= . 2 x 2 + 1 + Ce x

2.5.. gantoleba srul diferencialebSi ganvixiloT diferencialuri gantoleba: M ( x, y )dx + N ( x, y )dy = 0 . (1) u (x; y ) Tu (1) gantolebis marcxena mxare warmoadgens romeliRac funqciis srul diferencials anu: ∂u ∂u du ( x, y ) = dx + dy = M ( x, y )dx + N ( x, y )dy , ∂y ∂x maSin (1) diferencialur gantolebas ewodeba gantoleba srul diferencialebSi. cxadia es gantoleba pirveli rigisaa. Teorema 7.2. (eileris Teorema). M ( x, y )dx + N ( x, y )dy gamosaxuleba maSin da mxol-od maSin warmoadgens raime u (x; y ) funqciis srul diferencials, rodesac ∂M ( x, y ) ∂N ( x, y ) = . ∂y ∂x

107 am faqtis gaTvaliswinebiT avagoT (1) gantolebis amonaxsni. davuSvaT, (1) gantolebaa srul diferencialebSi, maSin arsebobs iseTi funqcia u (x, y ) , ∂u (x, y ) ∂M ( x, y ) ∂N ( x, y ) ∂u (x; y ) = N ( x, y ) da = rom . am to-lobis x = M ( x, y ) , ∂y ∂y ∂x ∂x cvladiT integrirebiT miviRebT: x

u (x, y ) = ∫ M (t , y )dt + ϕ ( y ) . x0

miRebuli tolobis y cvladiT gawarmoebiT miviRebT:

∂u ( x, y ) ∂M (t , y ) =∫ dt + ϕ ′ ( y ) , ∂y y ∂ x0 x

∂M ( x, y ) ∂N ( x, y ) ∂u (x, y ) = N ( x, y ) tolobebis = da ∂y ∂y ∂x gaTvaliswinebiT gveqneba: bolo tolobidan

∂N (t , y ) dt + ϕ ′( y ) , ∂t x0 x

N ( x, y ) = ∫ anu

N ( x, y ) = N ( x, y ) − N ( x0 , y ) + ϕ ′( y ) . aqedan gamomdinare ϕ ′( y ) = N ( x0 , y ) da y

ϕ ( y ) = ∫ N ( x0 ;τ )dτ + C . y0

maSasadame x

u ( x, y ) = ∫ M (t , y )dt + ∫ N ( x0 ,τ )dτ + C . u ( x, y ) funqciis sruli diferenciali du = M ( x, y )dx + N ( x, y )dy = 0 .

radganac

du = 0 , amitom

u (x, y ) = const

da toloba ∫ M (t , y )dt + ∫ N ( x0 ,τ )dτ = C

warmoadgens (1) gantolebis zogad integrals. aq yvelgan igulisxmeba, rom wertili ( x0 , y 0 ) Sedis uwyvetad warmoebadi

M ( x, y ) da N ( x; y ) funqciebis saerTo gansazRvris areSi. magaliTi. vipovoT (3 x 2 + 6 xy 2 )dx + (6 x 2 y + 4 y 3 )dy = 0 gantolebis zogadi amonaxsni. ∂M ( x, y ) ∂N ( x, y ) = 12 xy , maSasadame saqme gvaqvs gantolebasTan = amoxsna: ∂x ∂y ∂u srul diferencialebSi. = 3 x 2 + 6 xy 2 , aqedan u ( x, y ) = x 3 + 3 x 2 y 2 + ϕ ( y ) . ∂x ∂u − 6 x 2 y = N − 6 x 2 y = 4 y 3 , aqedan ϕ ( y ) = y 4 + C , gamovTvaloT ϕ ′( y ) . ϕ ′( y ) = ∂y maSasadame u ( x, y ) = x 3 + 3 x 2 y 2 + y 4 = C warmoadgens gantolebis zogad integrals

108 3. wrfivi, meore rigis diferencialuri gantolebani wrfivi, meore rigis diferencialuri gantoleba ewodeba gantolebas: (1) y ′′ + a1 ( x ) y ′ + a 2 ( x ) y = f ( x ) , sadac f ( x), ai ( x ), i = 1,2 saerTo (a, b) gansazRvris aris mqone uwyveti funqciebia. Tu f ( x ) = 0 , maSin gantolebas ewodeba erTgvarovani, sxva SemTxvevaSi araerTgvarovani. funqcia (2) − (a1 ( x ) y ′ + a 2 ( x ) y ) + f ( x ) gansazRvrulia areze, romelic ganisazRvreba utolobebiT: x 0 − a ≤ x ≤ x 0 + a, y 0 − b ≤ y ≤ y 0 + b, y 01 − b ≤ y ′ ≤ y 10 + b. SemosazRvrulia da rogorc x , y , y ′ cvladebis funqcia, akmayofilebs lifSicis pirobas, amitom rogorc am Tavis 1-l qveTavSi vaCveneT, ( y ′)′ = −(a1 ( x) y ′ + a 2 ( x) y ) gantolebiisaTvis koSis amocanas eqneba erTaderTi amonaxsni y = ϕ (x) romelic gansazRvrulia [ x0 − h; x0 + h] segmentze, sadac h ≤ a . a, b ricxvebi SeiZleba iyos usasrulod didic. mtkicdeba, rom (1) gantolebisTvis anu meore rigis wrfivi diferencialuri gantolebisaTvis koSis amocanis amonaxsni gansazRvrulia f ( x), ai ( x ), i = 1,2 funqciebis saerTo gansazRvris areze, Cven SemTxvevaSi (a; b) intervalze. ganvixiloT Sesabamisoba L , romelic meore rigamde warmoebad y funqcias Seusabamebs sidides y ′′ + a1 ( x ) y ′ + a 2 ( x ) y , cxadia, rom es Sesabamisoba wrfivia anu L[ y1 + y 2 ] = L[ y1 ] + L[ y 2 ], L[ky] = kL[ y] . L Sesabamisobas wrfivi diferencialuri operatori ewodeba.

3.1. wrfivi, erTgvarovani, meore rigis diferencialuri gantoleba ganvixiloT meore rigis wrfivi, erTgvarovani diferencialuri gantoleba: (3) y ′′ + a1 ( x ) y ′ + a 2 ( x ) y = 0 es gantoleba asedac SeiZleba CavweroT: L[ y ] = 0 . Tu y1 , y 2 funqciebi L[ y ] = 0 gantolebis kerZo amonaxsnebia, maSin αy1 + βy 2 , sadac α , β nebismieri mudmivebia, aseve misi kerZo amonaxsni iqneba. marTlac, radganac L[ y1 ] = 0, L[ y 2 ] = 0 , amitom L operatoris wrfivobis gamo gveqneba L[αy1 βy 2 ] = αL[ y1 ] + βL[ y 2 ] = 0 . (4)

y1 (x ), y 2 ( x ) (3)gantolebisor amonaxsns, romelebic ewodeba wrfivad damokidebuli, Tu gansazRvrulia (a, b) Sualedze, arseboben namdvili ricxvebi: α1 , α 2 ,romelTagan erTi mainc gansxvavebulia

109 nulisagan da adgili aqvs tolobas: α 1 y1 + α 2 y 2 = 0 . Tu es amonaxsnebi ar arian wrfivad damokidebuli, maSin maT uwodeben wrfivad damoukidebels. vTqvaT (3) gantolebis amonaxsnebi y1 (x ), y 2 ( x ) pirvel rigamde warmoebadi funqciebia, ganvixiloT determinanti: yy W ( y1 , y 2 ) = W ( x ) = 1 2 = y1 y 2′ − y1′ y 2 . (5) y1′ y 2′ am determinants (3) gantolebis y1 (x ), y 2 ( x ) amonaxsnTa sistemis vronskis determinanti ewodeba. vronskis determinants aqvs Semdegi Tvisebebi, romlebic Teoremebis saxiTaa Camoyalibebuli: Teorema 7.3. Tu funqciaTa sistema wrfivad damokidebulia, maSin misi vronskis determinanti W ( x ) nulis tolia.

Teorema 7.24 Tu y1 , y 2 (3) gantolebis kerZo amonaxsnebisagan Sedgenili wrfivad damoukidebeli sistemaa, maSin misi vronskis determinanti W ( x ) ≠ 0 yvela x ∈ (a; b) wertilisaTvis. (3) gantolebis y1 , y 2 , romlebic Seadgenen wrfivad damoukidebel sistemas am gantolebis amonaxsnTa fundamenturi sistema ewodeba. (3) gantolebisaTvis arsebobs amonaxsnTa fundamenturi sistema. marTlac, ganvixiloT ricxviTi determinanti, romelic gansxvavebulia nulisagan: a11a12 ≠ 0. a 21a 22 aseTi determinantis ageba yovelTvis aris SesaZlebeli. marTlac, vTqvaT, y1 , y 2 (3) gantolebis iseTi amonaxsnebia, romlebic akmayofilebens pirobebs: y1 ( x 0 ) = a11 , y1(1) ( x 0 ) = a 21 . y 2 ( x 0 ) = a12 , y 2(1) ( x 0 ) = a 22 .

Tu ganvixilavT amonaxsnTa sistemas determinantis mniSvneloba x0 wertilSi W (x0 ) ≠ 0 .

y1 , y 2 ,

cxadia

misi

vronskis

aqedan gamomdinare, y1 , y 2 sistema wrfivad damoukidebelia. (3) gantolebis zogadi amonaxsni y Caiwereba ase: y = C1 y1 + C 2 y 2 , sadac y1 , y 2 am gantolebis amonaxsnTa fundamenturi sistemaa, C i , i = 1,2 nebismieri mudmivebia. marTlac, vTqvaT, y amonaxsni akmayofilebs pirobas: y ( x 0 ) = y 00 , y (1) ( x 0 ) = y 10 . ganvixiloT wrfiv gantolebaTa sistema: C1 y 10 + C 2 y 02 = y 0 , (6) 2 C1 y 101 + C 2 y 01 = y 10 1 2 sadac y 01 = y1′ ( x 0 ), y 02 = y 2′ ( x 0 ) . gantolebaTa

sistemis

determinantia

W ( x0 ) ,

igi

udris

nuls,

radgan sistema y1 , y 2 wrfivad damoukidebelia, amitom mas aqvs erTaderTi y = C1 y1 + C 2 y 2 , cxadia es funqcia, (6) amonaxsni. ganvixiloT funqcia

110 tolobebidan

gamomdinare,

akmayofilebs

pirobebs:

y (x 0 ) = y 00 , y (1) (x 0 ) = y 01 ,

amitim koSis amocanis amonaxsnis erTaderTobis gamo y = y = C10 y1 + C 20 y 2 . es ki niSnavs imas, rom Cveni diferencialuri gantolebis zogadi amonaxsni warmodgindeba ase: y = C1 y1 + C 2 y 2 . Tu y1 , y 2 , y 3 (3) gantolebis amonaxsnebia, maSin isini Seadgenen wrfivad damokidebul sistemas. marTlac Tu y1 , y 2 wrfivad damokidebuli sistemaa, maSin misi momcveli y1 , y 2 wrfivad aseve wrfivad damokidebulia. Tu sistema y1 , y 2 , y 3 damoukidebuli sistemaa, maSin y 3 = C1 y1 + C 2 y 2 . amitom C1 y1 + C 2 y 2 + (− y 3 ) = 0 . es niSnavs, rom sistema y1 , y 2 wrfivad damokidebulia. YTu gantolebebs y ′′ + a1′ ( x ) y ′ + a 2′ ( x ) y = 0, y ′ + a1′′( x ) y ′ + a 2′′ ( x ) y = 0 aqvT saerTo amonaxsnTa fundamenturi sistema, maSin isini erTmaneTs emTxvevian. marTlac, davuSvaT a i′ (x ) ≠ a i′′( x ), i = 1,2. , maSin Tu ganvixilavT am ori gantolebis sxvaobas: ( a1′ ( x ) − a1′′( x )) y ′ + ( a 2′ ( x ) − a ′2′ ( x )) y = 0 igi iqneba pirveli rigis wrfivi erTgvarovani diferencialuri gantoleba, mas eqneba zustad erTi funqciis Semcveli amonaxsnTa fundamenturi sistema. mocemuli gantolebebis yvela amonaxsni aseve maTi sxvaobis amonaxsnicaa. gantolebebis saerTo fundamenturi amonaxsni ki Seicavs zustad or funqcias amgvarad miviReT winaaRmdegoba, maSasadame Cveni daSveba ar yofila swori, amitom ai′ ( x ) = ai′′( x ) . exla avagoT iseTi wrfivi erTgvarovani gantoleba, romlis amonaxsnTa fundamenturi sistema winaswar mocemuli: y1 , y 2 wrfivad damokidebuli sistemaa. ganvixiloT determinanti: y1 y 2 y y1(1) y 2(1) y (1) , y1( 2 ) y 2( 2 ) y ( 2 ) romlis bolo sveti Sedgeba saZiebeli funqciisa da misi warmoebulebisagan. Tu am determinants gavutolebT nuls da gavSliT bolo svetis mimarT miviRebT meore rigis gantolebas: A0 y ′′ + A1 y ′ + A2 y = 0 , (7)

sadac Ai = ( −1) i + 2 +1 M i ,3 , sadac M i ,3 i -uri, i = 1,2,3 striqonis da mesame svetis

amoRebiT miRebuli minoria, A0 = W [ y1 , y 2 ] = W ( x ) . Tu gavyofT (7) gantolebis orive mxares A0 -ze miviRebT gantolebas: (8) y ′′ + a1 y ′ + ... + a 2 y = 0 , sadac y1 y 2 W ′( x ) , (9) W (x ) W (x ) sadac W ′(x ) vronskis determinantis warmoebulia. ganvixiloT pirveli rigis wrfivi erTgvarovani diferencialuri gantoleba a1 = −

y1(2 ) y 2(2 ) .

=−

111

W ′( x ) + a1W ( x ) = 0

(10)

rogorc viciT, misi zogadi amonaxsnia W ( x ) = e ∫ , rac Seexeba kerZo amonaxsns, Tu davsvamT (10) gantolebisTvis koSis amocanas sawyisi pirobebiT: x = x0 , W0 = W ( x0 ) , gveqneba kerZo amonaxsnic: − a1dx

∫

− a1dt

W (x ) = W ( x0 )e x0 , (11) (10) da (11) tolobebs liulvilis formulebi ewodeba. magaliTi. gamoviyenoT liulvilis (11) formula y′′ + a1 y + a2 y = 0 gantolebis zogadi amonaxsnis mosaZebnad, rodesac cnobilia misi erTi kerZo amonaxsni - y1 . amoxsna: SevadginoT mocemuli gantolebisaTvis vronskis determinanti da gamoviyenoT liulvilis formula: y1 y − a1dx = Ce ∫ , y1′ y ′

− a1dx aqedan y1 y ′ − y1′ y = Ce ∫ . miviRebT:

tolobis

orive

mxares

gavyofT

y12 -ze

− a dx d ⎛ y′ ⎞ 1 ⎜⎜ ⎟⎟ = 2 Ce ∫ 1 . dx ⎝ y1 ⎠ y1

⎞ ⎛ 1 − a1dx aqedan ki sabolood gveqneba: y = y1 ⎜⎜ ∫ 2 Ce ∫ dx ⎟⎟ . ⎠ ⎝ y1

3.2.wrfivi, araerTgvarovani meore rigis diferencialuri gantoleba wrfivi, araerTgvarovani meore rigis diferencialuri gantoleba ewodeba gantolebas: (1) L[ y ] = y ′′ + a1 ( x ) y ′ + a 2 ( x ) y = f ( x ) , sadac marjvena mxare f (x ) igivurad ar udris nuls. wrfiv, erTgvarovan n -uri rigis diferencialuri gantolebas, romelsac igive koeficientebi aqvs, L[ y ] = y + a1 ( x ) y ′ + a 2 ( x ) y = 0 , (2) uwodeben (1) araerTgvarovani gantolebis Sesabamis erTgvarovan gantolebas. vTqvaT (1) gantolebis kerZo amonaxsnia Y ( x ) , maSin gveqneba L[Y ] = f ( x ) , SemovitanoT axali saZiebeli funqcia z (3) y =Y + z. CavsvaT es gamosaxuleba(1) gantolebaSi, miviRebT L[ y ] = L[Y ] + L[ z ] = f ( x) , radgan Y (x ) (1) gantolebis amonaxsnia, amitom L[ z ] = 0 . maSasadame z funqcia warmoadgens (2) gantolebis amonaxsns. vTqvaT exla (2) gantolebis amonaxsnTa fundamenturi sistemaa y1 , y 2 , maSin mis zogad amonaxsns rogorc wina paragrafidan aris cnobili, aqvs saxe: C1 y1 + C 2 y 2 . CavsvaT es

gamosaxuleba z -is zogad amonaxsns

112 (3) formulaSi, miviRebT (1) gantolebis

magivrad

y = C1 y1 + C 2 y 2 + Y . (4) maSasadame Tu cnobilia (1) gantolebis erTi kerZo amonaxsni, maSin am gantolebis amonaxsni warmoadgens aRniSnuli kerZo amonaxsnisa da Sesabamisi (2) erTgvarovani gantolebis zogadi amonaxsnis jams. amgvarad wrfivi araerTgvarovani meore rigis diferencialuri gantolebis amosaxsnelad, sakmarisia vicodeT am gantolebis raime kerZo amonaxsni da Sesabamisi erTgvarovani diferencialuri gantolebis amonaxsnTa fundamenturi sistema. egrewodebuli lagranJis mudmivTa variaciis meTodiT vipovoT (1) gantolebis kerZo amonaxsni. vTqvaT, y1 , y 2 (2) erTgvarovani diferencialuri gantolebis amonaxsnTa fundamenturi sistemaa. (2) gantolebis zogad amonaxsns aqvs saxe: y = C1 y1 + C 2 y 2 , sadac C1 ,C 2 mudmivi sidideebia. veZeboT (1) araerTgvarovani diferencialuri gantolebis kerZo amonaxsni aseTi saxiT: y = C1 ( x) y1 + C 2 ( x) y 2 , (5) sadac C1 ( x), C 2 ( x) warmoadgenen x cvladze damokidebul ucnob funqciebs. vipovoT es funqciebi. amisaTvis gavawarmooT (5) toloba, miviRebT: dC ( x ) dC 2 ( x ) . (6) y ′ = C1 ( x ) y1′ + C 2 ( x) y 2′ + y1 1 + y2 dx dx radgan (1) gantolebis amonaxsns veZebT (5) saxiT, bunebrivia CavTvaloT, rom mis warmoebulsac erTgvarovani gantolebis SemTxvevis analogiurad, unda hqondes saxe: y ′ = C1 ( x) y1′ + C 2 ( x) y 2′ . (7) amitom gavutoloT nuls (6)tolobis marjvena mxaris SesakrebTa im nawilis jami, romlebic Seicaven C i ( x), i = 1,2 funqciebis warmoebulebs. dC ( x ) dC 2 ( x ) y1 1 + y2 = 0 . (8) dx dx gavawarmooT exla (8) toloba, miviRebT dC n ( x ) dC ( x ) dC 2 ( x) . (9) y ′′ = C1 ( x ) y1′′ + C 2 ( x ) y ′2′ + ... + C n ( x ) y n′′ + y1′ 1 + y 2′ + ... + y n′ dx dx dx SevitanoT (1) gantolebaSi (5),(9) tolobebis marjvena nawilebi da agreTve saZiebeli amonaxsnis warmoebulis (7) gamosaxuleba, miviRebT: 2

∑ C ( y ′′ + a ( x) y ′ + ... + a i =1

dC1 ( x) dC 2 ( x ) + y 2′ = f ( x) . dx dx funqciebTan mdgomi Tanamamravlebi

( x) y i + y1′

am tolobis marjvena nawilSi C i nulis tolia, radgan TiToeuli y i , i = 1,2 warmoadgens (2) gantolebis amonaxsns. aqedan gamomdinare: dC ( x) dC 2 ( x ) y1′ 1 + y 2′ = f ( x) . dx dx dC1 ( x ) dC 2 ( x) amgvarad, miviReT wrfiv algebrul gantolebaTa sistema , dx dx ucnobebis mimarT

113 dC n ( x ) dC1 ( x ) dC 2 ( x ) + y2 + ... + y n = 0, dx dx dx dC n ( x ) dC ( x ) dC 2 ( x ) y1′ 1 + y 2′ + ... + y n′ = 0. dx dx dx am gantolebaTa sistemis determinanti warmoadgens (2) erTgvarovani diferencialuri gantolebis amonaxsnTa fundamenturi sistemis vronskis determinants, amitom gansxvavebuli iqneba nulisgan. maSasadame algebrul sistemas amonaxsni aqvs da erTaderTia. am amonaxsnis TiToeuli dC i ( x ) komponenti = ϕ i ( x ), i = 1,2 , sadac TiToeuli ϕ i ( x) warmoadgens x dx cvladis uwyvet funqcias. maTi integrirebiT vRebulobT: C i ( x) = ∫ ϕ i ( x)dx, i = 1,2 . y1

maSasadame

(1) gantolebis kerZo amonaxsnia: 2

y = ∑ y i ∫ ϕ i ( x)dx . i =1

sabolood (1) wrfivi araerTgvarovani gantolebis zogadi amonaxsni iqneba: 2

y = C1 y1 + C 2 y 2 + ∑ y i ∫ ϕ i ( x)dx , i =1

sadac C1 ,C 2 nebismieri mudmivebia. magaliTi. amovxsnaT araerTgvarovani diferencialuri gantoleba xy ′′ − y ′ = x 2 . amoxsna: Sesabamisi erTgvarovani gantoleba iqneba: xy ′′ − y ′ = 0 . igi advilad y ′′ 1 dy ′ dx C = , = ixsneba Semdegnairad: , ln y ′ = ln x + C , y ′ = Cx, y = x 2 + D . aqedan y′ x y′ x 2 gamomdinare, amonaxsnTa fundamenturi sistema iqneba 1, x 2 . veZeboT mocemuli araerTgvarovani gantolebis kerZo amonaxsni aseTi saxiT y = C1 + C 2 x 2 . Sesabamisad gveqneba:

dC 2 1 dC x x2 x3 = , C 2 = + γ 2 , 1 = − , C1 = − + γ 1 . dx 2 2 2 6 dx sabolood saZiebeli zogadi amonaxsni x3 y = γ1 + γ 2 x2 + , 3 sadac γ 1 , γ 2 nebismieri mudmivebia. 4. wrfivi mudmiv koeficientebiani meore rigis diferencialuri gantolebebi ganvixiloT wrfivi mudmiv koeficientebiani erTgvarovani diferencialuri gantoleba: (1) L[ y ] = y ′ + ... + a1 y ′ + a 2 y = 0 . am gantolebaSi vgulisxmobT, rom a1 , a 2 namdvili ricxvebia. vipovoT am gantolebis amonaxsnTa fundamenturi sistema. amisaTvis moviqceT Semdegnairad: veZeboT misi kerZo amonaxsnebi aseTi saxiT: y = e kx . maSin y ′ = ke kx , y ′′ = k 2 e kx . Tu CavsvamT am monacemebs (1) gantolebaSi, miviRebT:

114

[ ]

(

)

L e kx = e kx k 2 + a1 k + a 2 = 0 .

(2) frCxilebSi moTavsebul mravalwevrs (1) gantolebis maxasiaTebel mravalwevrs uwodeben. (2) tolobis marcxena mxare warmoadgens ori Tanamamravlis namravls, romelTagan pirveli - e kx arasodes nuli ar aris, meore Tanamamravli: F (k ) = k 2 + a1 k + a 2 (3) nulis tolia k cvladis im mniSvnelobebisTvis, romlebic warmoadgenen (3) mravalwevris fesvebs. amis gamo, y = e k ′x iqneba (1) gantolebis amonaxsni, mxolod maSin, rodesac k ′ warmoadgens (1) gantolebis maxasiaTebeli mravalwevris fesvs. SemTxveva 1. es is SemTxvevaa, rodesac (2) maxasiaTebel mravalwevrs gaaCnia 2 gansxvavebuli fesvi: k1 , k 2 , maSin gveqneba amonaxsnTa sistema: y1 = e k1 x , y 2 = e k 2 x . (4) vaCvenoT, rom (4) sistema warmoadgens (1) gantolebis amonaxsnTa fundamentur sistemas. amisaTvis ganvixiloT vronskis determinanti: e k1 x e k 2 x ( k1 + k 2 ) 11 . (5) W (x ) = e = k1 k 2 k1e k1x k 2 e k 2 x . am tolobis marjvena mxares mdgomi determinanti nuli ar aris, Tu ricxvebi: k1 , k 2 gansxvavebulia erTmaneTisagan, radganac Cven SemTxvevaSi k1 , k 2 warmoadgens (3) mravalwevris gansxvavebul fesvebs, amitom W (x ) ≠ 0 . Sedegad (4) sistema wrfivad damoukidebelia da warmoadgens (1) gantolebis amonaxsnTa fundamentur sistemas. ganvixiloT SemTxveva, rodesac (3) mravalwevrs aqvs 2 gansxvavebuli kompleqsuri fesvi. vTqvaT, (1) gantolebis maxasiaTebeli mravalwevris fesvi k1 = u + iv , maSin aseve iqneba am maxasiaTebeli misi SeuRlebuli ricxvi k 2 = u − iv mravalwevris fesvi. k1 fesvisTvis gveqneba:

[

] [ = L[e cos vx] + L[e

] [

]

0 = L e (u +iv ) x = L e ux (cos vx + ivx) = L e ux cos vx + ie ux sin vc = ux

] [

i sin vx = L e ux cos vx + iL e ux sin vx

miRebuli tolobidan gvaqvs:

[

]

[

]

L e ux cos vx = −iL e ux sin vx , niSnavs, rom L e ux cos vx = 0, L e ux sin vx = 0 . maSasadame y1 = e ux cos vx, y 2 = e ux sin vx funqciebi warmoadgenen (1) gantolebis namdvil amonaxsnebs da, cxadia, Seadgenen amonaxsnTa fundamentur sistemas.

[

]

[

]

SemTxveva 2. es is SemTxvevaa, rodesac maxasiaTebel mravalwevrs gaaCnia jeradi fesvebi. am SemTxvevaSi gansxvavebul fesvTa raodenoba (4) tolia erTis, k ′ = k1 = k 2 . cxadia erTi kerZo amonaxsnia y = e k ′x . rom miviRoT meore amonaxsni, romelic am amonaxsnTan Seadgens fundamentur sistemas, SeviswavloT L diferencialuri operatoris moqmedeba uv namravlze. rogorc viciT

115

(uv )″ = u ′′v + 2u ′v′ + uv′′ (uv )′ = u ′v + uv′ uv = uv Tu gavamravlebT pirvel striqons 1-ze, meore striqons a1 -ze, mesame striqons a 2 -ze da Semdeg SevkrebT, miviRebT: v′ v ′′ L[uv] = vL[u ] + L1 [u ] + L2 [u ] , (6) 1! 2! sadac: L1 [u ] = 2u ′ + a1u , L2 [u ] = 2u . TiToeuli Lr , r = 1,2 warmoadgens wrfiv diferencialur operators. cxadia, maTi maxasiaTebeli mravalwevri Fr (k ) = F (r ) (k ), r = 1,2 , sadac F (k ) (1) gantolebis maxasiaTebeli mravalwevria. gamovTvaloT exla (6), Tu u = e kx da v = x L xe kx = xL e kx + L1 (e kx ) . radgan L1 e kx = e kx F ′(k ) , amitom gveqneba: L xe kx = e kx ( xF (k ) + F ′(k )) . (7) k ′ aris (1) gantolebis maxasiaTebeli mravalwevris oris jeradobis fesvi,, maSin F (k ′) = 0, F ′(k ′) = 0 . Tu (7) tolobaSi k -s magivrad SevitanT k ′ , miviRebT L[ xe k ′x ] = 0 , maSasadame

[ ]

y = xe k ′x warmoadgens kerZo amonaxsns. cxadia, funqciaTa funqcia k ′x k ′x sistema e , xe wrfivad damoukidebelia da fundamenturi (1) gantolebisTvis. rac iTqva am paragrafSi, Seexeboda wrfivi, erTgvarovani, mudmiv koeficientebiani diferencialuri gantolebis amonaxsnTa fundamenturi sistemis agebas. rac Seexeba araerTgvarovan: (8) L[ y ] = y ′′ + a1 y ′ + a 2 y = f ( x ) diferencialur gantolebas, Sesabamisi erTgvarovani gantolebis amonaxsnTa fundamenturi sistemis safuZvelze, mudmivTa variaciis meTodiT, SegviZlia movZebnoT misi kerZo amonaxsni da Semdeg, viciT ra Sesabamisi erTgvarovani gantolebis zogadi amonaxsni, rogorc zemoT aRvwereT, SegviZlia gamovsaxoT am araerTgvarovani gantolebis zogadi amonaxsnic.

5. diferencialuri gantolebebis gamoyeneba 5.1. nivTierebis warmoqmnis da daSlis gantolebebi nivTierebaTa warmoqmnis da daSlis bevri procesi akmayofilebs Semdeg pirobebs: nivTierebis raodenobis cvlilebis siCqare proporciulia mocemul momentSi nivTierebis raodenobaze damokidebuli raRac funqciis da aseve am nivTierebis xasiaTze, garemo pirobebze(temperatura, wneva ganaTebuloba da sxva). vTqvaT, nivTierebis raodenoba drois t momentSi aris x(t ) . misi dx (t ) cvlilebis siCqare iqneba . zemoT Tqmulidan gamomdinare, gveqneba: dt

116 dx (t ) (1) = k (t ) f ( x ) . dt rodesac nivTierebis maxasiaTeblebi da garemo pirobebi mudmivia k (t ) koeficientic iqneba mudmivi. (1) warmoadgens diferencialur gantolebas gancalebadi cvladebiT. misi amoxsnis meTodebi CvenTvis ukve cnobilia. magaliTi 1. baqteriaTa gamravlebis modeli. vTqvaT konkretuli saxis baqteriaTa jamuri masa drois t momentSi aris x(t ) da maTi gamravleba xdeba mudmiv garemoSi anu proporciulobis koeficienti k mudmivia. aseT SemTxvevaSi baqteriaTa gamravlebis procesi aRiwereba diferencialuri gantolebiT dx (t ) (2) = kx . dt pirobis Tanaxmad x(t ) da x ′(t ) arauaryofiTi sidideebia amitom k k > 0, koeficientic arauaryofiTia. sainteresoa SemTxveva, rodesac radgan Tu k = 0 , araviTari gamravleba ar xdeba. (2) gantoleba wrfivi, pirveli rigis gantolebaa, misi zogadi amonaxsnia x = Ce kt , sadac C nebismieri mudmivia. Tu viciT k koeficientis mniSvneloba da baqteriaTa masa m0 drois fiqsirebul t 0 momentSi anu

x(t 0 ) = m0 , maSin m 0 = Ce kt , C = m0 e − kt0 ,

aqedan gamomdinare: x (t ) = m0 e k ( t −t0 ) . magaliTi 2. radioaqtiuri daSla. eqsperimentebiT dadgenilia, rom radioaqtiuri nivTierebis daSlis siCqare proporciulia drois mocemul momentamde dauSleli nivTierebis raodenobisa. Tu x(t ) warmoadgens drois t momentamde darCenili dauSleli nivTierebis dx (t ) masas, maSin daSlis siCqare akmayofilebs gantolebas: dt dx(t ) (3) = − kx(t ) . dt niSani minusi, gantolebis marjvena mxares, niSnavs, rom xdeba nivTierebis raodenobis Semcireba da ara gazrda. (3) gantolebis zogadi amonaxsnia x(t ) = Ce − kt , sadac C nebismieri mudmivia. misi mniSvneloba, iseve rogorc wina magaliTSi, gamoiTvleba formuliT: C = m0 e kt0 , sadac m0 nivTierebis masaa drois fiqsirebul t 0 momentSi. maSasadame radioaqtiuri da- Slis procesi aRiwereba funqciiT: (4) x (t ) = m 0 e − k ( t − t 0 ) . praqtikaSi radioaqtiuri daSlis siCqare xasiaTdeba egreTwodebuli naxevar-daSlis periodiT anu im droiT, romelSic xdeba arsebuli nivTierebis naxevris daSla. aRvniSnoT naxevardaSlis periodi T simboloTi. gamovsaxoT k naxevarda-lis T periodis saSualebiT.

117 (4) formulidan, rodesac t = t 0 + T , kT = ln 2, k =

gveqneba

− ln 2 . amgvarad, x (t ) = m0 e T

t 0 = 0 , maSin x(t ) = m0 2

−

t T

t −t0 ln 2 T

m0 = m0 e − kT , amitom 2

anu x (t ) = m0 2

−

t −t0 T

. kerZod, rodesac

5.2. harmoniuli rxevebi ganvixiloT gantoleba (1) x ′′ + ω 2 x = 0 . 2 2 am gantolebis maxasiaTebeli mravalwevria k + ω . kompleqsuri fesvebia: k1 = iω , k 2 = −iω . aqedan gamomdinare, (1) gantolebis amonaxsnTa fundamenturi sistemaa y1 = cos ωt , y 2 = sin ωt , zogadi amonaxsni x = C1 cos ωt + C 2 sin ωt . (2) vaCvenoT, rom arseboben ricxvebi A, α , rom C1 = A cos α , C 2 = − A sin α . Tu ganvixilavT am tolobebs rogorc orucnobian algebrul gantolebebs davinaxavT, rom maT amonaxsni aqvT. amitom (2) amonaxsni SeiZleba ase warmovadginoT: x = A cosα cos ωt − A sin α sin ωt anu (3) x = A cos(ωt + α ) , sadac A, α nebismieri namdvili ricxvebia. Tu CavTvliT, rom t cvladi warmoadgens droiT parametrs, (2) funqcia aRwers egreTwodebul harmoniul rxeviT process. | A | ricxvs uwodeben rxevis amplitudas, α ricxvs rxevis sawyis fazas, (1) gantolebas ki harmoniuli rxevis gantolebas. advili misaxvedria, rom drois erTeulSi rxevaTa ricxvi

ν =

ω , 2π

am ricxvs rxevis sixSire ewodeba. vTqvaT (1) gantolebisTvis mocemuli gvaqvs pirobebi: x(0) = x 0 , x ′(0) = v 0 . (2) zogadi amonaxsnidan miviRebT C1 = x0 , C 2 =

koSis

amocanis

sawyisi (4)

. aqedan gamomdinare, kerZo

amonaxsni, romelic Seesabameba (4) sawyis pirobebs iqneba: v x = x0 cos ωt + 0 sin ωt .

am funqciiT aRwerili rxevis amplitudis gamosaTvlelad unda amovxsnaT gantolebaTa sistema: ⎧ A cos α = x 0 , ⎪ :⎨ ν0 ⎪− A sin α = ω . ⎩ am sistemidan miviRebT

118 A =x + 2

2 0

v02

ω2

saidanac

x 02 +

v 02

ω2

(5)

rac Seexeba sawyis fazas

α = arccos

x0 x02 +

v02

(6)

ω2

magaliTi 1. wertilis rxeva F drekadi Zalis moqmedebiT. vTqvaT m masis mqone P wertilze F ZaliT moqmedebs drekadi zambara(nax.36). x

nax. 36 niutonis kanonisTanaxmad, P wertilis moZraobis gantolebaa mx ′′ = F . hukis kanonis mixedviT, drekadi Zala pirdapirproporciulia P wertilis O wonasworobis mdgomareobidan gadaxris sididis da mimarTulia wertilis moZraobis sawinaaRmdegod. amitom SegviZlia davweroT F = −kx , sadac k > 0 . amgvarad, gveqneba gantoleba mx′′ + kx = 0 . (7) es gantoleba harmoniuli rxevis gantolebaa sixSiriT ω =

k . misi zogadi m

amonaxsni iqneba:

k k t + C 2 sin t. m m Sesabamisad koSis amocanis amonaxsni (7) gantolebisaTvis, x(0) = x 0 , x ′(0) = v 0 sawyisi pirobebiT, iqneba: x = C1 cos

k m k t + v0 sin t. m k m aqedan gamomdinare harmoniuli rxevis amplituda x = x0 cos

m 2 v0 . k rogorc vxedavT, P wertilis rxevis sixSire ω damokidebuli ar aris sawyis pirobebze, igi damokidebulia mxolod wertilis masaze da zambaris drekadobaze. rac Seexeba amplitudas da sawyis fazas, maTi gamosaTvleli formulebidan gamomdinare, isini arsebiTad arian damokidebuli sawyis pirobebze. magaliTi. 2. maTematikuri saqanis (qanqaris) mcire rxevebi. maTematikuri saqani warmoadgens m masis P wertils, romelic simZimis Zalis A = x02 +

119 meSveobiT moZraobs vertikalur sibrtyeSi moTavsebul wrewirze. aRniSnuli wrewiris radiuss saqanis sigrZe ewodeba(nax. 37). A ϕ L L P F2 • F1 • O Fϕ nax.37 saqanis moZraobis wrewirze SemovitanoT sakuTxo koordinati ϕ , wrewirze yvelaze dabla myofi wertili aRvniSnoT O simboloTi, dadebiTad CavTvaloT mimarTuleba marcxnidan marjvniv. P wertilze moqmedebs simZimis Zala F = mg , romelic mimarTulia vertikalurad qvemoT. aRniSnuli Zala iyofa mdgenelebad F1 , F2 . pirveli Zala F1 mimarTulia wrewiris radiusis gaswvriv da ar iwvevs moZraobas, igi gawonasworebulia sxva saxis ZalebiT. meore F2 Zala mimarTulia wrewiris mxebis gaswvriv da iw-vevs P wertilis moZraobas. F2 Zalis sidide tolia mg sin ϕ . es Zala ewinaaR- mdegeba ϕ kuTxis gazrdas, amitom saqanis moZraobis gantoleba unda iyos: mLϕ ′′ = − mg sin ϕ , sadac ϕ ′′ kuTxuri aCqarebaa, L -wrewiris radiusi anu saqanis sigrZe, Lϕ ′′ xazovani aCqareba. Tu miRebuli gantolebis orive mxares gavyofT m masaze, miviRebT Lϕ ′′ = − g sin ϕ , romelsac maTematikuri saqanis gantolebaewodeba. am gantolebis amoxsna sakmaod rTulia, amitom Cven ganvixilavT SemTxvvevas, rodesac adgili aqvs egreTwodebul mcire rxevebs anu, rodesac ϕ kuTxis sidide icvleba sakmaod mcire intervalSi. rogorc viciT mcire

kuTxeebisaTvis adgili miviRebT ganto-lebas:

aqvs

tolobas:

sin ϕ ≈ ϕ .

pirobis

Sedegad

Lϕ ′′ + gϕ = 0 , romelsac maTematikuri saqanis mcire rxevebis gantoleba ewodeba. cxadia, es gantoleba warmoadgens harmoniuli rxevis gantolebas, misi zogadi amonaxsni iqneba:

ϕ = A(

g t +α) L

g g t + C 2 sin t. L L rogorc wina SemTxvevaSi, SeiZleba ganvixiloT koSis amocana Sesabamisi sawyisi pirobebiT da amovxsnaT igi. unda aRvniSnoT, rom saqanis mcire rxevebis sixSire ω damokidebulia mxolod saqanis sigrZeze da mcirdeba misi gazrdis SemTxvevaSi. sixSirisTvis gvaqvs formula:

ϕ = C1 cos

120

1 g , 2π L sadac g ≈ 9,8 m/wm 2 , L L saqanis sigrZe. L ^ savarjiSoebi me-7 TavisaTvis I amoxseniT gantolebani gancalebadi cvladebiT:

ω=

1) xy ′ − y = y 2 , 2) y ′tgx = y , 3) xyy ′ = 1 − x 2 , 4) y ′ sin x = y ln x; y = 1 , rodesac x =

π 2

II. amoxseniT erTgvarovani gantolebani: y 1) y ′ = − 1 , 2) ydx + (2 xy − y )dy , 3) xdy − ydx = x 2 + y 2 dx , x 4) ( x 2 − 3 y 2 )dx + 2 xydy = 0 ; y = 1 , rodesac x = 2 . III. amoxseniT pirveli rigis wrfivi diferencialuri gantolebani: y 2y 1) y ′ − = x, 2) y ′ + = x 2 , 3) y 2 dx − (2 xy + 3)dy = 0 , 4) xy ′ + y − e x = 0 ; y = b , rodesac x x x = a. IV. amoxseniT gantolebani srul diferencialebSi: xdy − ydx 1) ( x + y ) dx + ( x + 2 y ) dy = 0 , 2) ( x 2 + y 2 + 2 x)dx + 2 xydy = 0 , 3) xdx + ydy = 2 . x + y2 V. amoxseniT meore rigis mudmiv koeficientebiani diferencialuri gantolebani: 1) y ′′ − 5 y ′ + 6 y = 0 , 2) y ′′ − 9 y = 0 , 3) y ′′ + 2 y ′ + y = 0 , 4) y ′′ − 5 y ′ + 4 y = 0 ; y = 5, y ′ = 8 , rodesac x = 0 . VI. amoxseniT mudmivTa variaciis meTodiT: 2 ex 1 1) y ′′ + y = tgx , 2) y ′′ + y = , 3) y ′′ − 2 y ′ + y = , 4) y ′′ − 2 y = 4 x 2 e x , 5) y ′′ + y = ctgx . x cos x

121 Tavi 8 1.xdomiloba. elementarul xdomilobaTa

sivrce

albaTobis Teoriis EerT-erTi ZiriTadi cnebaa “xdomiloba”, romelic Tavis mxriv ganimarteba cdis (eqsperimentis) daxmarebiT. (cda) eqsperimenti ewodeba pirobaTa garkveuli kompleqsis ganxorcielebas.

magaliTad: 1.

lilvis diametris gazomva; 2. detalis Semowmeba vargisiobaze; 3.kamaTlis gagoreba; 4. monetis agdeba da a.S. igulisxmeba, rom: 1)Eeqsperimentis

Catarebamde

SesaZlebelia

yvela

SesaZlo

Sedegis

miTiTeba; 2)

eqsperimentis

Catarebamde

SeuZlebelia

misi

Sedegis

calsaxad

gansazRvra; 3) SesaZlebelia eqsperimentis mravaljeradi ganmeoreba. eqsperimenti

aRiwereba

misi

urTierTgamomricxavi

Sedegebis

CamonaTvaliT. am Sedegebs elementaruli xdomilobebi ewodeba, xolo maT srul

erTobliobas

elementarul

xdomilobaTa

sivrce,

aRvniSnavT Ω-asoTi, xolo elementarul xdomilobas

romelsac

asoTi indeqsiT

an uindeqsoT. magaliTi 1. vTqvaT, eqsperimenti niSnavs simetriuli monetis agdebas. vakvirdebiT,

,,modis’’

davardnil

monetaze.

eqsperimentis

Sedegebi:,,gerbis mosvla’’ (g) da ,,safasuris mosvla’’ (s). elementarul xdomilobaTa sivrce iqneba:

Ω={g,s}.

magaliTi 2. agoreben erT kamaTels. cdas eqvsi Sedegi SeiZleba hqondes (zeda waxnagze wertilebis raodenoba). elementarul xdomilobaTa sivrce iqneba: Ω={1,2,3,4,5,6}. magaliTi 3. agdeben or sxvadasxva monetas. aq oTxi SesaZlo Sedegia mosalodneli

(vasxvavebT,

xdomilobaTa sivrce iqneba: magaliTi

monetas

romel

monetaze

movida).

elementarul

Ω={gg,gs,sg,ss}. agdeben

gerbis

pirvel

gamoCenamde.

elementarul xdomilobaTa sivrce iqneba:

Ω={g,sg,ssg,sssg,...}. magaliTi 5. msroleli esvris wriuli formis samiznes, romelsac yoveli gasrolisas axvedrebs. Tu wris radiusia R da samiznis sibrtyeSi SemoRebulia marTkuTxa koordinatTa sistema, romlis saTave samiznis centrSia da cdis TiToeul Sedegs (samiznis gansazRvrul wertilSi

moxvedras)

122 am wertilis

SevusabamebT

koordinatebs,

maSin

elementarul xdomilobaTa sivrce iqneba yvela iseTi ricxviTi wyvilis erToblioba, romlebic Semdegnairad Caiwereba:

Ω={

pirvel sam magaliTSi elementarul xdomilobaTa simravle sasrulia, meoTxe

magaliTSi_Tvladi,

xolo

mexuTe

magaliTSi

elementarul

raime

erTobliobas

xdomilobaTa simravle araa Tvladi. elementaruli xdomilobas,

xdomilobebis ubralod

xdomilobas

uwodeben.

amrigad,

Sedgenil xdomiloba,

elementarul xdomilobaTa sivrcis qvesimravlea. vityviT, rom cdis Sedegad xdeba A xdomiloba, Tu cdas mohyva iseTi elementarul xdomiloba, romelic A xdomilobas ekuTvnis. ganixileba

sami

saxis

xdomiloba:

aucilebeli,

SeuZlebeli

SemTxveviTi. gansazRvreba. adgili

xdomilobas, romelsac yoveli cdis Sedegad aqvs

aucilebeli

aRvniSnavT

xdomiloba

ewodeba.

Aaucilebel

xdomilobas

asoTi.

gansazRvreba. xdomilobas, romelsac adgili ar SeiZleba hqondes erTi

cdis

Sedegad

SeuZlebeli

xdomilobas aRvniSnavT

xdomiloba

ewodeba.

arc

SeuZlebel

asoTi.

gansazRvreba. xdomilobas, romelsac cdis Sedegad SeiZleba hqondes an ar

hqondes

adgili

SemTxveviTi

xdomiloba

ewodeba.

SemTxveviT

xdomilobebs aRvniSnavT laTinuri anbanis didi asoebiT, indeqsiT an uindeqsod: AA, B, C,

da a.S.

2. moqmedebebi xdomilobebze gansazRvreba. xdomilobas

vityviT, rom A xdomiloba iwvevs B xdomilobas , Tu B

adgili

aqvs

yovelTvis,

roca

xdeba

xdomiloba.

simbolurad CavwerT: A B, ikiTxeba ,, A xdomiloba iwvevs B xdomilobas’’. magaliTi.

vTqvaT, A

aris

xdomiloba,

naturaluri ricxvi martivia’’ xolo dasaxelebuli

naturaluri

ricxvi

,,SemTxveviT

dasaxelebuli

B iyos xdomiloba ,,SemTxveviT kentia’’.

nebismieri martivi ricxvi aucileblad kenticaa.

cxadia,

A B,

radganac

SevniSnoT,

rom

123 nebismieri xdomiloba

iwvevs

aucilebel

xdomilobas, A â&#x201E;Ś. aseve, nebismieri xdomiloba iwvevs Tavis Tavs, A A. am

gansazRvrebidan

gamomdinare,

elementaruli

xdomiloba

elementarul xdomilobaTa sivrce SeiZleba ganisazRvros Semdegnairad.

B xdomilobas vuwodebT elementarul xdomilobas, Tu

gansazRvreba.

ar arsebobs iseTi A xdomiloba, romelic iwvevs B xdomilobas da gansxvavebulia BB-sgan. gansazRvreba.

eqsperimentTan

dakavSirebul

yvelaEelementarul

xdomilebaTa erTobliobas elementarul xdomilobaTa sivrce ewodeba da aRvniSnavT

asoTi, romelsac aucilebel xdomilobasTan gavaigivebT.

gansazRvreba. ori A da B xdomilobis gaerTianeba ewodeba iseT C xdomilobas romelsac adgili aqvs maSin, rodesac xdeba an A xdomiloba an B xdomiliba, an orTave. im faqts, rom A xdomilobis gaerTianeba B xdomilobasTan aris C xdomiloba simbolurad aRvniSnavT: A B=C. magaliTi

vTqvaT,

eqsperimenti

mdgomareobs

yuTidan

burTulis

amoRebaSi, romelSic 20 erTnairi burTulaa, gadanomrili 1-dan 20-mde. ATi aRvniSnoT xdomiloba imisa, rom SemTxveviT amoRebuli burTulis nomeri luwi ricxvia, xolo B-Ti SemTxveviT amoRebuli burTulis nomeri metia

9-ze.

A={2;4;6;8;10;12;14;16;18;20};

Ee.i.

B={10;11;12;13;14;15;16;17;18;19;20}. ganmartebis Tanaxmad: A B={2;4;6;8;10;11;12;13;14;15;16;17;18;19;20}. xdomilobebis gaerTianebis operacias aqvs Semdegi Tvisebebi: 1. nebismieri A da BBB xdomilobisaTis A B= B A (komutatiuroba); 2. nebismieriA,BBB

xdomilobisaTis

(A B)

(B C)

(asociaciuroba); 3. Tu A

maSin A B= B; A

gansazRvreba. ori A da B xdomilobis TanakveTa ewodeba iseT C xdomilobas, romelsac adgili aqvs maSin da mxolod maSin, rodesac xdeba A da TanakveTa

B xdomiloba

erTdroulad. im faqts, rom A xdomilobis

B xdomilobasTan aris C xdomiloba simbolurad aRvniSnavT:

A B=C.G Gganxilul me-6 magaliTSi ganmartebis Tanaxmad: A B={10;12;14;16;18;20}. xdomilobebis kveTas aqvs Semdegi Tvisebebi:

nebismieri

124 BBB xdomilobisaTis

(komutatiuroba); 2.

nebismieri

BBB

xdomilobisaTis(A

;

(asociaciuroba); 3. Tu A

maSin A B= A;

xdomilobaTa

gaerTianebisa

gamoisaxeba Semdegi

kveTis

operaciebs

kavSiri

TanafardobiT:

A (B C)=(A

gansazRvreba.

Soris

ori

A da

(distribuciuloba). B xdomilobis

sxvaoba

ewodeba

iseT

xdomilobas, romelsac adgili aqvs maSin da mxolod maSin rodesac xdeba

A da ar xdeba

B xdomiloba. im faqts, rom A xdomilobis sxvaoba B

xdomilodasTan

aris C xdomiloba

simbolurad

aRvniSnavT:

A B=C.

Gganxilul me-6 magaliTSi ganmartebis Tanaxmad:

A B={2;4;6;8}; B-A={11;13;15;17;19}. gansazRvreba.Oori A da B xdomilobis simetriuli sxvaoba ewodeba iseT

C xdomilobas, romelsac adgili aqvs maSin da mxolod maSin, rodesac xdeba an A xdomiloba an B xdomiliba, magram ara orTave erTdroulad. im faqts, rom A xdomilobis simetriuli sxvaoba xdomiloba simbolurad avRniSnavT:

B xdomilobasTan aris C

A B=C. Gganxilul me-6 magaliTSi

ganmartebis Tanaxmad:

A B={2;4;6;8;11;13;15;17;19}. gansazRvreba. aRiniSneba

A xdomilobis uaryofa

simboloTi:

ewodeba â&#x201E;Ś

xdomilobas da

A. Gganxilul me-6 magaliTSi ganmartebis

Tanaxmad: ={1;3;5;7;9;11;13;15;17;19}; xdomilobis uaryofis uaryofa TviT es xdomilobaa, xdomilobaTa

gaerTianebisa

TanakveTis

=A.

operaciebi

SeiZleba

ganisazRvros maSinac rodesac gvaqvs Tvladi raodenobis xdomilobebi: maSin =C aris xdomiloba, romelic xdeba maSin da mxolod maSin rodesac xdeba xdomilobebidan erTi mainc;

125 xolo =C aris xdomiloba, romelic xdeba maSin da mxolod maSin rodesac xdeba yoveli

xdomiloba. or A da B xdomilobas ewodeba uTavsebadi xdomilobebi

gansazRvreba.

Tu maTi erTdroulad moxdena SeuZlebelia, e.i. gansazRvreba. .

xdomilobaTa erTobliobas ewodeba wyvil

wyvilad UuTavsebadi xdomilobebi Tu nebismieri ori maTgani uTavsebadi

i j i,j=1,2,3,â&#x20AC;Ś,n.

xdomilobebia. gansazRvreba. xdomilobaTa

xdomilobaTa

sruli

xdomilobebia

jgufi,

maTi

isini

gaerTianeba

erTobliobas

wyvil-wyvilad

aucilebeli

ewodeba

uTavsebadi

xdomilobaa,

e.i.

Sesrulebulia Semdegi ori piroba: 1.

magaliTi.

i j i,j=1,2,3,â&#x20AC;Ś,n.

amwyobma saamqrom miiRo sam sxvadasxva meqanikur

damzadebuli erTi dasaxelebis n detali, maT Soris pirvel saamqroSi,

saamqroSi

damzadebulia

meore saamqroSi, xolo danarCeni

mesame saamqroSi. _ iyos xdomiloba imisa, rom alalbedze aRebuli detali detali damzadebulia

pirvel

saamqroSi,

damzadebulia meore saamqroSi, xdomilobebi

_xdomiloba

imisa,

rom

detali

_ detali damzadebulia mesame saamqroSi.

qmnian xdomilobaTa srul jgufs, cxadia: ;

savarjiSo magaliTebi 1) aris xdomiloba_kamaTelze movida luwi ricxvi, _kamaTelze movida 3-iani,

aris xdomiloba

aris xdomiloba _kamaTelze movida 5-iani. ras

niSnavs xdomilobebi:

2) xaratis mier damzadebuli detali SeiZleba iyos pirveli xarisxis (xdomioloba ), (xdomiloba ) .

meore raSi

xarisxis

mdgomareobs

(xdomiloba ),

xdomilobebi:

mesame ?;

xarisxis

126 3) [0, 1] intervalze alalbedze virCevT

wertils.

xdomiloba_ alalbedze arCeuli wertili ekuTvnis [0,

aris [ intervals,

aris xdomiloba_ alalbedze arCeuli wertili ekuTvnis ]0, ras niSnavs

4) gansazRvreT 5)

ipoveT

yvela

[ intervals.

xdomilobebi?

xdomilobebi, Tu xdomiloba,

romelic

akmayofilebs

pirobas:

3. albaTobis aqsiomuri ganmarteba davuSvaT, Ω nebismieri nebismieri elementarul xdomilebaTa sivrcea rogorc viciT, xdomiloba warmoadgens Ω sivrcis qvesimravles. vTqvaT, mocemuli gvaqvs xdomilobaTa F klasi (xdomilobaTa erToblioba). gansazRvreba. Sesrulebulia

xdomilobaTa

klass

ewodeba

algebra,

Semdegi pirobebi:

1.SeuZlebeli

aucilebeli

xdomilobebi

ekuTvnis

Fklass,

θ F;Ω F.

2. Tu xdomiloba ekuTvnis Fklass, maSin misi uaryofac ekuTvnis

Fklass, e.i. A(A

3. Tu xdomilobebi A da B ekuTvnian Fklass, maSin maTi gaerTianebac ekuTvnis F klass, e. i. (A F, B F gansazRvreba.

xdomilobaTa

A F

F). klass

ewodeba

algebra,

Sesrulebulia Semdegi pirobebi: 1.SeuZlebeli

θ F;Ω F.

aucilebeli

xdomilobebi

ekuTvnis

Fklass,

127 2.

xdomiloba

ekuTvnis Fklass, e.i. A(A

Fklass,

ekuTvnis

maSin

misi

uaryofac

xdomilobebi

ekuTvnian

Fklass,

gaerTianebac da TanakveTac ekuTvnis F klass, e.i.

maSin

gansazRvreba. P ricxviT funqcias, romelic gansazRvrulia algebris

elementebisaTvis

Eewodeba

albaToba,

maTi

igi

akmayofilebs

Semdeg sam aqsiomas:L 1. arauaryofiTobis aqsioma. nebismieri A xomilobisaTvis Fklasidan,

P(A) 0; 2.normirebis aqsioma. aucilebeli xdomilobis albaToba erTis tolia,

P(Ω)=1; 3.Tvladi aditiurobis aqsioma. Tu

wyvil-wyvilad

uTavsebadi xdomilobebia, maSin maTi gaerTianebis albaToba Sesabamisi albaTobebis jamis tolia

P( albaTobis maTematikoss a.

zemomoyvanili

gansazaRvreba

ekuTvnis

gamoCenil

rus

kolmogorovs, romlis sapativcemulod mas kolmogorovis

aqsiomebs uwodeben. albaTobis aqsiomuri ganmartebidan gamomdinareobs Semdegi ZiriTadi Tvisebebi: Tviseba1. SeuZlebeli xdomilobis albaToba nulis tolia,

P( )=0.

damtkiceba: SeuZlebeli xdomiloba warmovadginoT, rogorc Tvladi raodenobis SeuZlebeli xdomilobebis gaerTianeba:

maSin mesame aqsioms Tanaxmad:

P( )=P( )+P( )+P( )+…+P( )+… pirveli

aqsioms

Tanaxmad

P( ) 0,

romlis

(1) gaTvaliswinebiTac

tolobas SeiZleba adgili hqondes maSin da mxolod maSin, roca P( )=0.

(1)

128 sasruli raodenobis wyvil-wyvilad

Tviseba 2. Tu

uTavsebadi xdomilobebia, maSin maTi gaerTianebis albaToba Sesabamisi albaTobebis jamis tolia

damtkiceba:

xdomilobaTa

gaerTianeba

SeiZleba

warmovadginoT Semdegi saxiT: (2) (2) tolobis marjvena mxareSi Semavali xdomilobebi wyvil-wyvilad uTavsebadi xdomilobebia, Tu gaviTvaliswinebT me-3 aqsiomas da pirvel Sedegs miviRebT: )=

=P(

)+P(

)+…P( )+P( )+P( )+..+P( )+….=P(

)+P(

)+…P( )

Tviseba 3. Tu A xdomiloba iwvevs B xdomilobas, maSin A xdomilobis albaToba naklebia an toli B

xdomilobis albaTobaze

da P(B-A)=P(B)-P(A). damtkiceba:

radgan

A iwvevs B- s

davweroT: B=A+(B-A),

SeiZleba

amasTanave A da (B-A) xdomilobebi uTavsebadi xdomilobebia, maSin me-2 Tvisebis Tanaxmad

P(B)=P(A)+P(B-A)B

(3)

1-li aqsiomis Tanaxmad P(B-A) 0 (3)-dan davaskvniT:

P(A) P(B); Tviseba

xdomilobebis

ori

xdomilobis

Sesabamisi

P(B-A)=P(B)-P(A). gaerTianebis

albaTobebis

jams

albaToba

tolia

gamoklebuli

am maTi

erTdroulad moxdenis albaToba:

P(A B)= P(A)+P(B)-P(A B). damtkiceba:

ori

xdomilobebis

gaerTianeba

SeiZleba

warmovadginoT Semdegi saxiT:

A B= (A-A B) (B-A B)

A B) .

(4)

advilad davrwmundebiT, romMme-4 tolobis marjvena mxares mdgomi xdomilobebi uTavsebadi xdmilobebia. me-2 da me-3 Tvisebis

gamoyenebiT

SegviZlia davweroT:

P(A B)=P (A-A B)

(B-A B)

A B) = P(A)-P(AB)+P(B)-P(AB)+P(AB))=

= P(A)+P(B)-P(AB).

129 me-4 Tvisebidan gamomdinareobs, rom PP(A B) P(A)+P(B). Tviseba

sawinaaRmdego

xdomilobis

albaToba

tolia

erTs

gamoklebuli mocemuli xdomilobis albaToba, e.i. P( )=1-P(A). damtkiceba: sawinaaRmdego

xdomilobis ganmartebis Tanaxmad: P =Ω-A,

maSin me-2 Tvisebis da normirebis aqsiomis Tanaxmad:

P( )=P(Ω-A)=P( )-P(A)=1-P(A). gansazRvreba.

sameuls

sadac: Ω elementarul

}

{

xdomilobaTa

sivrcea; F- algebra; P-albaToba, albaTuri sivrce ewodeba.

4. albaTobis klasikuri gansazRvreba davuSvaT,

elementarul

xdomilobaTa

sivrce

Seicavs

sasruli

raodenobis elementarul xdomilobebs . elementarul

xdomilobas

SevusabamoT

ricxvi,

i=1,2,3,…,n,

romlebic akmayofileben Sedeg or pirobas: 1.

cxadia, rom aseTi elementarul xdomilobaTa sivrcis

nebismieri

qvesimravle warmoadgens xdomilobas. vTqvaT,

nebismieri

sivrcidan

romelic

AA={

,…,

qvemoT

xdomilobaa

Ω

Seicavs

elementarul

xdomilobaTa

elementarul

xdomilobas:

mocemul

gansazRvrebas

albaTobis

klasikuri

gansazRvreba

ewodeba. gansazRvreba.

A xdomilobis albaToba ganvsazRvroT Semdegnairad: P(A) =

advilad

vaCvenebT,

rom

aseTnairad

. gansazRvruli

albaToba

akmayofilebs Semdeg Tvisebebs: 1. nebismieri A xdomilobis albaToba arauaryofiTi ricxvia, P(A) 0.

2. P(Ω)=1, marTlac, gansazRvrebis Tanaxmad, P( )=

TuUA

130 xdomilobebia, uTavsebadi

P(AUB)=P(A)+P(B). marTlac, vTqvaT: A={ AUB={

maSin

,…,

,…, ,…,

maSin

}; B={

,…,

gansazRvrebis Tanaxmad,

P(AUB)=

=P(A)+P(B).

zemoT naCvenebi sami pirobis SesrulebiT

faqtiurad SevamowmeT, rom

aseTnairad gansazRvruli albaToba akmayofilebs kolmogorovis samive aqsiomas.

aqedan

gamomdinare,

adgili

eqneba

aqsiomebidan

gamomdinare

ZiriTad Sedegebs. im SemTxvevaSi, tolia

anu

mosalodneli,

rodesac

SemTxveviTi

elementarul xdomilobaTa

eqsperimentis

yvela

(am

e.i.

Sedegi

SemTxvevas

albaTobebi erTnairadaa

klasikuri

sqema

ewodeba). maSin me-2 Tvisebis Tanaxmad: . Tu Axdomiloba Seicavs k raodenobis elementarul xdomilobas maSin

P(A) =

. e.i. xdomilobis albaToba tolia misi xelSemwyob Sedegebis

raodenoba Sefardebuli yvela SesaZlo Sedegebis raodenobasTan. ganvixiloT SemTxveva, rodesac elementarul xdomilobaTa sivrce Ω Seicavs Tvladi raodenobis elementarul xdomilobebs:

aseTi elementarul xdomilobaTa sivrcis warmoadgens

xdomilobas,

romelic

nebismieri qvesimravle A

SeiZleba

Seicavdes

Tvladi

raodenobis elementebs: AA={ elementarul

xdomilobas

,…, SevusabamoT

}. ricxvi,

romlebic akmayofileben Sedeg or pirobas: 1.

A xdomilobis albaToba ganvsazRvreba Semdegnairad: P(A)=

i=1,2,3,…,n,…

131 elementaruli xdomilobaTa sivrcisaTvis

sasruli naCveneb

Tvisebebs

daemateba

e.w.

Tvladi

aditiurobis

zemoT

Tviseba:

agdeben

erT

wyvil-wyvilad uTavsebadi xdomilobebia, maSin . magaliTi.

agdeben

garCevad

kamaTels

orjer

kamaTels. vipovoT albaToba imisa, rom: 1) mosul qulaTa jami tolia 8-is; 2) mosul qulaTa jami naklebia an toli 5-ze. 3) mosul qulaTa jami metia 10-ze. amoxsna. i iyos pirvel kamaTelze mosuli qulaTa ricxvi, xolo j meore kamaTelze

mosul

qulaTa

ricxvi.

elementarul

xdomilobaTa

sivrce

iqneba:

â&#x201E;Ś={ (2, 4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3), (5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

A-Ti aRvniSnoT xdomiloba imisa, rom mosul qulaTa jami tolia 8-is; B-Ti aRvniSnoT xdomiloba imisa, rom mosul qulaTa jami naklebia an toli 5-ze; C-Ti aRvniSnoT xdomiloba imisa, rom mosul qulaTa jami metia 10-ze, maSin A={

={(2,6),(3,5),(4,4),(5,3),(6,2)};

B={

={(1,1),(1,2),(1,3)(1,4),(2,1)(2,2),(2,3)(3,1),(3,2),(4,1)

};

C={

={(5,6),(6,5),(6,6)}.

gvaqvs klasikuri sqema. kamaTelze movida

-Ti aRvniSnoT albaToba imisa, rom pirvel

qula, xolo meoreze

qula (

. analogiurad miviRebT: magaliTi. simetriul monetas agdeben gerbis pirvel gamoCenamde. 1) vipovoT albaToba imisa, rom gerbi pivrelad mova luwnomrian cdaSi. amoxsna. aq elementarul xdomilobaTa sivrce iqneba:

â&#x201E;Ś={(g),(s,g),(s,s,g),(s,s,s,g),...}.

iyos

xdomiloba

imisa,

132 rom gerbi

pirvelad

mova

luwnomrian

cdaSi, maSin

A={(s,g),(s,s,s,g),(s,s,s,s,s,g),...}. aRvniSnoT xdomiloba imisa, rom wina yvela

cdaSi safasuri (

=(g),

}, A={

aRniSvniT SevusabamoT

-ur cdaSi movida gerbi, xolo =(s,g), .

=(s,s,g),...). miRebuli

elementarul xdomilobas

albaToba. advilad mowmdeba normirebis piroba,

marTlac = aq

Cven

gamoviyeneT

usasrulod

wevrTa jamis formula, Tu

= klebadi

geometriuli

pirveli wevria da

progresiis

<1, maSin

= saZiebeli

xdomilobis albaToba =

savarjiSo magaliTebi 1) yuTSi, erTmaneTisagan mxolod feriT gansxvavebuli, 10 TeTri da 20 Savi burTulaa. ras udris albaToba imisa, rom alalbedze amoRebuli burTula TeTri feris iqneba? 2) alalbedze asaxeleben naturalur ricxvs, romelic ar aRemateba 24s. ras udris albaToba imisa, rom es ricxvi warmoadgens 24-is gamyofs? 3) ras udris albaToba imisa, rom alalbedze dasaxelebuli orniSna ricxvis cifrTa jami 12 iqneba? 4) sastumroSi 15 stumaria, maTgan 10 turistia, 5 biznesmeni. alalbedze irCeven 3 stumars. ras udris albaToba imisa, rom samive turistia? 5)

raodenobis

nomerSi.

stumari yovel

alalbedze nomerSi

nawildeba

SeiZleba

sastumros

moTavsdes

raodenobis stumari. ras udria albaToba imisa, rom pirvel nomerSi moTavsdeba nomerSi

raodenobis stumari, meoreSi stumari (

= )?

-raodenobis da a. S.

-ur

students

133 8 friadosania.

Soris

jgufidan

alalbedze airCies 9 studenti. ipoveT albaToba imisa, rom maT Soris 5 friadosania. 7)

xaratis

mier

damzadebuli

standartuli

detalebis

fardobiTi

sixSire mdgradia da yovelTvis 0,9-is tolia. ramdeni detali daamzada xaratma cvlis ganmavlobaSi, Tu arastandartuli detalebis raodenoba 50-is toli aRmoCnda? 8)

xarisxis saxelmwifo inspeqtori sasursaTo maRaziaSi amowmebs rZis

produqtebs. cnobilia, rom rZis 20 paketidan ori amJavebulia. inspeqtori SemTxveviT irCevs 2 pakets 20-dan. rogoria albaToba imisa, rom maT Soris: a) arc erTi ar iqneba amJavebuli; b) mxolod erTi iqneba amJavebuli; g) orive amJavebuli iqneba. 5. albaTobis geometriuli gansazRvreba

albaTobis

klasikuri

elementaruli

xdomilobaTa

raodenobis

elementebs.

gansazRvrebidan

moiTxoveba,

sivrce

Seicavdes

praqtikaSi

xSirad

sasruli gvxvdeba

Ω

rom an

Tvladi

iseTi

saxis

eqsperimentebi romelTa Sesabamisi elementaruli xdomilobaTa sivrce araTvladia

aseT SemTxvevaSi xdomilobebi,

romelTaTvisac arsebobs

albaTobebi, ekuTvnian garkveul klass

(es klasi, rogorc albaTobis

aqsiomuri ganmartebisas aRvniSneT aris

algebra) sailustraciod Cven

SemovifarglebiT e.w. albaTobis geometriuli gansazRvrebiT, sadac Ω elementarul

xdomilobaTa

sivrced

ganixileba

raime

geometriuli

obieqti: wrfis sasruli monakveTi; wrewiri; rkali; wre; sibrtyis sasruli farTobis mqone nawili; sasruli moculobis sivrculi sxeuli da a.S. gansazRvreba. warmoadgens

raime

davuSvaT,

Ω

geometriul

elementarul obieqts:

wrfis

xdomilobaTa sasruli

sivrce

monakveTi;

wrewiri; sibrtyis sasruli farTobis mqone nawili, sasruli moculobis mqone sivrculi sxeuli, da a.S. A xdomilobis albaToba vuwodoT wilads

P(A)= sadac

warmodgens A xdomilobisa da Ω sivrcis zomas,

Sesabamisad. advili

saCvenebelia,

rom

aseTnairad

akmayofilebs kolmogorovis samive aqsiomas.

gansazRvuli

albaToba

magaliTi.

kvadratSi,

134 romlis gverdi

santimetria

Caxazulia

wre. ipoveT albaToba imisa, rom kvadratSi SemTxveviT aRebuli wertili wreSi moxvdeba. amoxsna. â&#x201E;Ś elementarul xdomilobaTa sivrcea kvadratis wertilebi, xdomilobas, romlis albaTobasac

romlis farTobia 100 kv.santimetri. veZebT,

aris

wris,

wertilebi

romlis

farTobia

25 .

albaTobis

geometriuli gansazRvrebis Tanaxmad:

P(A)=

= .

savarjiSo magaliTebi. 1)

-radiusian wreSi moTavsebulia

-radiusiani mcire wre. ipoveT

albaToba imisa, rom did wreSi alalbedze dasmuli wertili mcire wreSic moxvdeba. 2) 1 metris siganisa da 2 metris sigrZis magidaze moTavsebulia 10 sm siganisa da 20 sm sigrZis Ffurceli. ipoveT albaToba imisa, rom magidaze alalbedze dasmuli wertili furcelze ar moxvdeba. 3)

-radiusian wreSi Caxazulia wesieri samkuTxedi. ipoveT albaToba

imisa, rom wreSi alalbedze dasmuli wertili moxvdeba samkuTxedSic. 4) qariSxalma daaziana satelefono xazi 160-e da 290-e kilometrebs Soris.

ras udris albaToba imisa, rom dazianeba moxda xazis me-200-240

kilometrebs Soris.

6.pirobiTi albaToba. xdomilobaTa namravlis albaToba. vTqvaT, {â&#x201E;Ś;F;P} albaTuri sivrcea, xolo A da B nebismieri xdomilobebia

F klasidan

iseTi,

rom

P(B)>0. davsvaT

Semdegi

amocana:

vipovoT

xdomilobis albaToba Tu cnobilia, rom B xdomilobas hqonda adgili? aseTi saxis albaTobas ewodeba A xdomilobis albaToba B pirobiT da aRiniSneba

simboloTi da ikiTxeba Semdegnairad: A xdomilobis

albaToba B pirobiT. magaliTi. vTqvaT, cnobilia, rom erTi kamaTlis gagorebisas

movida

luwi cifri da gvainteresebs albaToba imisa, rom es cifri aris oriani. am

albaTobis

sapovnelad

visargebloT

albaTobis

klasikuri

gansazRvrebiT,

135 SesaZlo SemTxvevaTa

yvela

raodenoba

aris

xolo xelSemwyob SemTxvevaTa raodenoba 1, e.i. saZiebeli albaToba iqneba da ara

rogorc iqneboda iqneboda im SemTxvevaSi, araviTari wina

piroba, rom ar yofiliyo cnobili. maSasadame, sazogadod davaskvniT, rom

P(A)

gansazRvreba. A xdomilobis albaToba

B pirobiT ewodeba wilads

romlis mricxvelia A da B xdomilobebis erTdroulad moxdenis albaToba mniSvneli ki B pirobis albaToba e.i.

, Tu P(B)

(1)

advili saCvenebelia, rom pirobiTi albaToba akmayofilebs albaTobis samive aqsiomas. pirobiTi albaTobis gansazRvrebis Tanaxmad:

, Tu P(A)

(2)

radgan AB=BA, (1) da (2)-dan davwerT: PP(AB)=P(BA)=P(B)P (3)

warmoadgens

formulas

ori

=P(B)P

xdomilobaTa

(3)

namravlis

TanamamravlisaTvis,

albaTobis

romelic

gamosaTvlel

SeiZleba

ganzogaddes

TanamamravlTa nebismieri sasruli raodenobisaTvis: P(

)=P(

magaliTi.

studentma

…P

programiT

(4)

gaTvaliswinebuli

sakiTxidan

moamzada 40. ras udris albaToba imisa, rom students Sexvdeba bileTi, romlis samive sakiTxi momzadebuli aqvs? amoxsna.

aRvniSnoT xdomiloba imisa, rom students Sexvda bileTi

romlis pirveli sakiTxi momzadebuli aqvs,

aRvniSnoT xdomiloba imisa,

rom students Sexvda bileTi, romlis meore sakiTxi momzadebuli aqvs, aRvniSnoT xdomiloba imisa, rom students Sexvda bileTi, romlis mesame sakiTxi

momzadebuli

aqvs.

Cven

namravlis albaToba. (4) tolobis P(

)=P(

gvainteresebs

xdomilobaTa

Tanaxmad: P

savarjiSo magaliTebi

136 1) yuTSi 20 TeTri da 15 Savi burTulaa. rigrigobiT iReben TiTo burTulas. ras udris albaToba imisa, rom meore burTula TeTri iqneba Tu cnobilia, rom pirveli iyo Savi? 2) amwyobma saamqrom miiRo 100 detali, romelTagan 5 arastandartulia. ras udris albaToba imisa, rom alalbedze aRebuli sami detalidan samive arastandartuli iqneba? 3) 25-kacian jgufSi 7 oTxosani da 3 xuTosani studentia. ras udris albaToba imisa, rom alalbedze gamoZaxebuli sami studentidan: a) samive xuTosani iqneba; b) samive oTxosani iqneba; g) arc erTi ar iqneba oTxosani an xuTosani? xdomilobaTa damoukidebloba gansazRvreba. or A da B xdomilobas ewodeba damoukidebeli, Tu maTi erTdroulad

moxdenis

albaToba Sesabamisi

albaTobebis namravlis

tolia:

P(AB)=P(A)P(B).

(1)

dauSvaT P(A)>0,P(B)>0, pirobiTi albaTobis ganmartebis Tanaxmad:

P P

, =

(2)

(3)

vTqvaT, A da B damokidebeli xdomilobebia, maSin Tu gaviTvaliswinebT (1) tolobas (2) da (3) tolobebSi miviRebT:

P anu,

xdomilobebi

P damoukideblebia,

=P(B), maSin

erTi

xdomilobis

moxdena ar cvlis meoris moxdenis albaTobas. xdomilobebs ewodeba wyvil _ wyvilad

gansazRvreba.

damoukidebeli, Tu nebismieri i da k-saTvis i

P( gansazRvreba.

)=P( )P(

). xdomilobebs

(4) ewodeba

erTobliobaSi, Tu nebismieri k xdomilobisaTvis

damoukidebeli

, maTi erTdroulad

moxdenis albaToba Sesabamisi albaTobebis namravlis tolia:

(5)

SevniSnoT, rom Tu xdomilobebi damoukideblebia erToblobaSi, maSin isini wyvil-wyvilad damoukideblebicaa. marTlac (5) tolobaSi, roca k=2,

miviRebT ar

(4)

137 magram wyvil-wyvilad

tolobas.

gamomdinareobs

damoukidebloba

damoukideblobidan

erTobliobaSi.

sailustraciod

moviyvanoT magaliTi. magaliTi.

(bernSteini)

tetraedris sam

erTgvarovani

waxnagze aweria

masalisgan

damzadebuli

cifrebi 1, 2, da 3, xolo

waxnagze es samive cifri erTad.

meoTxe

iyos xdomiloba imisa, rom zemoT

asrolili tetraedri daeca waxnagze, romelzec aweria cifri k (k=1, 2, 3). vaCvenoT,

rom

xdomilobebi

wyvil-wyvilad

damoukidebeli

xdomilobebia. albaTobis klasikuri gansazRvrebis Tanaxmad:

=P(

)=P(

)= ; PP(

da radganac sruldeba pirobebi:

)=P(

xdomilobebi

)P(

); P(

)=P(

)P(

); P(

)=P(

)P(

)

wyvil-wyvilad damoukidebeli xdomilobebia.

axla gamovikvlioT am xdomilobebis erTobliobaSi damoukidebloba. radgan p(

)= , xolo P( P(

e.i.

)P(

) P(

)P(

)P( )P(

)= , amitom ),

xdomilobebi erTobliobaSi

damokidebeli xdomilobebia,

miuxedavad maTi wyvil-wyvilad damoukideblobisa. savarjiSo magaliTebi 1) albaToba imisa, rom avtomobili SekeTebis gareSe gaivlis 100000 km-s yovelTvis mudmivia da udris 0,8. ras udria albaToba imisa, rom sami miRebuli avtomanqanidan samive SekeTebis gareSe gaivlis 100000 km-s? 2)

ras

udris

albaToba

imisa,

rom

oTxi

kamaTlis

erTdroulad

gagorebisas oTxiveze erTi da igive ricxvi mova? 3) erTi msrolelisTvis mizanSi moxvedris albaTobaa 0,8, meorisaTis0,9.

ras

udria

albaToba

imisa,

rom

erTdroulad

gasrolisas

moaxvedrebs mizanSi, meore ki aacdens?

sruli albaTobis formula. baiesis formula

vTqvaT, {â&#x201E;Ś;F;P} albaTuri sivrcea, xolo

erTi

138 xdomilobaTa

sruli

jgufia,

maSin

A xdomilobis

nebismieri

albaToba F klasidan gamoiTvleba formuliT

P(A)=

(1) marTlac, radgan

xdomilobaTa sruli jgufia =â&#x201E;Ś

(2)

tolobis

orive

(2)

mxare

gavamravloT

xdomilobaze

gaviTvaliswinoT, rom Aâ&#x201E;Ś=A, miviRebT

radgan xdomilobebia,

(3)

xdomilobebi wyvil-wyvilad

uTavsebadi

wyvil-wyvilad

uTavsebadi

iqneba

xdomilobebic. Tu (3) tolobaSi gamoviyenebT sasruli aditiurobis da xdomilobaTa

namravlis

albaTobis

gamosaTvlel

formulas,

miviRebT

P(A)=P( =P

)= +P

+...+P

(4) (4) formulas sruli albaTobis formula ewodeba. magaliTi. amwyobma saamqrom miiRo sam sxvadasxva meqanikur

saamqroSi

damzadebuli erTi dasaxelebis 2000 detali, maT Soris 600 damzadebulia pirvel saamqroSi, 650 _ meore saamqroSi, xolo 750 _mesame saamqroSi. ras udris albaToba imisa, rom alalbedze aRebuli detali arastandartuli aRmoCndeba, Tu cnobilia, rom pirveli saamqro saSualod

95% standartul

produqcias uSvebs, meore saamqro _ 99%-s, xolo mesame saamqro _ 98%-s? amoxsna. A iyos xdomiloba imisa, rom alalbedze aRebuli detali arastandartuli damzadebulia

_xdomiloba

aRmoCndeba.

pirvel

saamqroSi,

damzadebulia meore saamqroSi, xdomilobebi

imisa,

_xdomiloba

imisa,

rom rom

es es

detali detali

_ detali damzadebulia mesame saamqroSi.

qmnian

xdomilobaTa cxadia A

A=A sruli albaTobis formulis Tanxmad

srul

jgufs,

139 P(A)=P( P(

, P(

P(A

)+ P(

P(A

da P(

P(A

)

gamoiTvleba albaTobis klasikuri gansazRvrebis

gamoyenebiT:

P( P(A

=0,3; P(

)=1-0,95=0,005; P(A

=0,325; P(

=0,375;

)=1-0,99=0,01; P(A

)=1-0,98=0.02.

sabolood miviRebT:

P(A)=0,3.0,05+0,325.0,01+0,375.0,02=0,02575. sruli

albaTobis

formulaSi

Semaval

xdomilobebs

hipoTezebs uwodeben. (4) formuliT nebismieri A viTvliT

hipoTezaTa

cnobili

albaTobebisa

xdomilobis albaTobas

xdomilobis

pirobiTi

albaTobebis saSualebiT. praqtikaSi xSirad saWiro xdeba Sebrunebuli amocanis

gadawyveta.

kerZod

mocemulia

hipoTezaTa

, P(

),..., P(

)

albaTobebi da gvainteresebs, Tu rogor Seicvlebian isini, Tu cnobilia, rom A

xdomiloba moxda. e.i. unda vipovoT

pirobiT.

pirobiTi

albaTobis

xdomilobis albaToba A

xdomilobaTa

namravlis

albaTobis

formulas

ganmartebis Tanaxmad: = Tu

ukanasknelSi

gaviTvaliswinebT

sruli

miviRebT: =

(5)

(5) formula warmoadgens baiesis formulas, romelsac hipoTezaTa albaTobis formulasac uwodeben. advili misaxvedria, rom =1. magaliTi. wina magaliTis pirobebSi, davuSvaT, rom alalbedze aRebuli detali aRmoCnda arastandartuli da gvainteresebs, Tu rogori albaTobiT SeiZleba mivakuTvnoT es detali TiToeul saamqros. (i=1,2,3).

amoxsna. unda vipovoT =

savarjiSo magaliTebi

, , .

erT

140 moTavsebulia pirveli xaratis mier damzadebuli

yuTSi

100 detali, meore xaratis mier damzadebuli 80 detali, mesame xaratis mier

damzadebuli

120

detali.

ipoveT

albaToba

imisa,

rom

yuTidan

alalbedze amoRebuli detali standartuli iqneba, Tu pirveli xarati saSualod 95% standartul detals amzadebs, meore xarati_ 98%-s, mesame ki_ 90%-s. 2) sawyobSi moitanes erTi dasaxelebis 1000 detali, maT Soris 300 damzadebulia Ppirvel saamqroSi, 200_ meore saamqroSi, xolo 500_mesame saamqroSi. cnobilia, rom Ppirvel saamqroSi mzaddeba saSualod 95% standartuli detali, meoreSi_90%, mesameSi ki_85%. ras udris albaToba imisa, rom alalbedze amoRebuli detali standartuli iqneba. 3)

amwyobi

saamqrosaTvis

samagrs

amzadebs

sami

avtomati,

pirveli

iZleva saWiro samagrebis 25%-s, meore_35%-s, mesame ki_40%-s. pirveli avtomati saSualod iZleva 0,1% wuns, meore_0,3%, mesame ki_0,2%-s. ipoveT albaToba

imisa,

albaToba imisa,

rom

amwyobi

saamqro

miiRebs

wundebul

samagrs

Tu samagri wundebuli aRmoCnda, is pirveli saamqros

mieraa damzadebuli. 4) albaToba imisa, rom pirveli msroleli mizans daazianebs aris 0,6. xolo meorisaTvis -0,7. orive msrolelma erTmaneTisgan damoukideblad gaisrola, ris Sedegac samizne dazianda. ras udris albaToba imisa, rom mizani pirvelma msrolelma daaziana? 5)

gzaze,

sadac

benzingasamarTi

sadguria,

gamovlil

satvirTo

avtomobilTa ricxvi ise Seefardeba msubuq avtomobilTa ricxvs, rogorc 2/3.

albaToba

imisa,

rom

gamvleli

avtomobili

benziniT

gaimarTeba,

satvirTo avtomobilisTvis aris 0,1, xolo msubuqi avtomobilisaTvis 0,2. gasamarT sadgurTan mivida avtomanqana. ras udris albaToba imisa, rom es avtomanqana satvirToa? 6) pirvel kurselTa 60% vaJia. vaJebis 80%-s da gogonebis 75%-s aqvs fexburTis abonementi. damlagebelma pirvel kurselTa auditoriaSi ipova abonementi. ras udris albaToba imisa, rom es abonementi vaJisaa? 7) jgufSi 20 studentia, romelTa Soris 4 xuTosania, 10 oTxosani da 6 samosani. albaToba imisa, rom dafasTan gamoZaxebuli studenti amocanas amoxsnis

xuTosanisTvis

0,9-is

tolia,

oTxosanisaTvis

_0,7-is,

samosanisaTvis ki_0,5-is. ras udris albaToba imisa, rom: a) dafasTan

141 alalbedze gamoZaxebuli studenti amocanas

amoxsnis;

dafasTan

alalbedze gamoZaxebuli orive studenti amocanas amoxsnis. 8)

avadmyofi

grZnobs

Zlier

statistikurad 50%-is SemTxvevaSi

gulis

SeiZleba gamoiwvios

daavadebam, xolo 20%-is SemTxvevaSi

is SemTxvevaSi

eqimTan mkurnalobis SemTxvevaSi tolia,

tkivilebs

-isa_0,8-is,

xolo

areSi,

rac

daavadebam, 30%daavadebam. ubnis

davadebis gankurnebis albaToba 0,7-is isa

_0,9-is.

avadmyofi

gankurna, ras udris albaToba imisa, rom avadmyofs hqonda

ubnis

eqimma

davadeba?

8 damoukidebel cdaTa mimdevroba. bernulis sqema

vTqvaT, vatarebT raime cdas, romlis ganmeorebac SeiZleba mravaljer da vakvirdebiT A xdomilobis moxdenis faqts. vityviT, rom adgili aqvs warmatebas, Tu A xdomiloba moxda mocemul cdaSi, winaaRmdeg SemTxvevaSi vityviT, adgili aqvs marcxs. mravali praqtikuli amocanis gadawyvetisas sainteresoa ara romelime konkretuli cdis Sedegi, aramed warmatebaTa saerTo raodenoba cdaTa garkveul

seriaSi.

sainteresoa

ara

magaliTad, romelime

msroleli

konkretuli

cdis

isvris

mizanSi

Sedegi,

aramed

50-jer is

ramdenjer daaziana samizne. gansazRvreba. cdaTa iseT mimdevrobas, romlis nebismieri cdis Sedegi gavlenas

axdens

momdevno

cdebis

SesaZlo

SedegTa

albaTobebze,

damoukidebel cdaTa mimdevroba ewodeba. cdas, romelSic vakvirdebiT A

xdomilobis moxdenis an armoxdenis

faqts orSedegiani, an binaruli cda ewodeba.

xdomilobis moxdenis

(warmatebis) albaToba aRvniSnoT p â&#x20AC;&#x201C;Ti, xolo armoxdenis (marcxis)

q-Ti:

P(A)=p; P( )=q; p+q=1. gansazRvreba. orSedegian damoukidebel cdaTa mimdevrobas, warmatebis mudmivi albaTobiT, bernulis sqema ewodeba. davuSvaT vatarebT n damoukidebel cdas, k-Ti aRvniSnoT ricxvi n damoukidebel cdaSi, xolo

warmatebaTa

(k)-Ti albaToba imisa, rom

damoukidebel cdaSi adgili eqneba k warmatebas.

magaliTi.

142 isvrian samjer.

mizanSi

vipovoT

albaToba

imisa,

rom samizne daziandeba: a) arc erTxel, b) erTjer, g) orjer, d) samjer? Tu mizanSi moxvedris albaToba Yyovel cdaSi tolia 0,7.

-uriT aRvniSnoT i-ur cdaSi mizanSi moxvedris (warmatebis)

amoxsna.

iqneba i-ur cdaSi acilebis (marcxis) xdomiloba.

xdomiloba, maSin

avagoT elementarul xdomilobaTa sivrce

â&#x201E;Ś={(

); (

);(

); (

);(

)}.

-Ti aRvniSnoT xdomiloba imisa, rom warmatebas adgili hqonda k-jer: =(

;

)+(

radganac

)+( )+(

)+( );

); ).

damoukidebeli xdomilobebia, damoukidebeli iqneba

xdomilobebic.

gamoviyenebT

damoukidebeli

xdomilobebis

namravlis da uTavsebadi xdomilobaTa jamis albaTobebis gamosaTvlel formulebs, miviRebT: (0)=

(1)=

)=P((

)+(

))=P(

)+P(

)+(

))=P(

)+P(

)=0,7.0,7.0,3+0,7.0,3.0,7+0,3.0,7.0,7=0,441;

+P( (3)=

)+(

)=0,7.0,3.0,3+0,3.0,7.0,3+0,3.0,3.0,7= 0,189;

+P( (2)=

)=0,3.0,3.0,3=0,027;

)=P

)=0,7.0,7.0.7=0,343.

Teorema. bernulis sqemis n damoukidebel cdaSi k

warmatebis

(k)

albaToba gamoiTvleba formuliT: (k)=

. sadac

p warmatebis

q=1-p

albaTobaa,

marcxis,

xolo

_jufdebaTa ricxvi n elementidan k elementad. damtkiceba.

-uriT aRvniSnoT i-ur cdaSi warmatebis xdomiloba, maSin

iqneba i-ur cdaSi marcxis xdomiloba. cdis n-jer ganmeorebis Sedegad warmatebis

k-jer

moxdenis

xdomiloba

aRvniSnoT B-Ti.

igi

SeiZleba

warmovidginoT Semdegnairad:

... +

xdomilobebi

... ...

... +....+

...

+...+

(1) ...

...

...;

(

...

143 wyvil-wyvilad )

+....+

uTavsebadi

xdomilobebia, maTi raodenoba iqneba imdeni, ramden jufdebac SeiZleba SevadginoT n elementidan k elementad. cdaTa damoukideblobis gamo, TiToeuli

xdomilobis

albaToba

masSi

Semavali

xdomilobebis

albaTobebis namravlis tolia. Tu (1) tolobaSi gamoviyenebT daomukidebeli xdomilobebis namravlis da uTavsebadi xdomilobaTa jamis albaTobebis gamosaTvlel formulebs, miviRebT: (k)= P(B)=

+...+P(

...

... +....+

...

... )...P(

...

)...P(

...P(

)+...+

)...P(

+....+

)= .

(3)

amiT Teorema damtkicebulia. (3) formulas bernulis formula ewodeba. bernulis

formulis

gamoyenebiT

SegviZlia

gamoviTvaloT

Semdegi

albaTobebi:

1) albaToba imisa, rom n damoukidebel cdaSi adgili eqneba aranakleb i warmatebas: (k

;

2) albaToba imisa, rom n damoukidebel cdaSi adgili eqneba ara umetes i warmatebas: (k

3) albaToba imisa, rom

;

n damoukidebel cdaSi warmatebaTa ricxvi ar

aRemateba i-s da aranaklebia j-ze: (j

;

aqve SevniSnoT, rom (0

savarjiSo magaliTebi 1) ojaxSi 5

bavSvia. ipoveT albaToba imisa, rom maT Soris 2 vaJia, Tu

cnobilia, rom vaJis dabadebis albaToba 0, 51-is tolia.

144 2) ras udris albaToba imisa, iqneba

standartuli,

rom

cnobilia,

rom

avtomobilidan am

markis

zustad

avtomobilebis

standartulobis albaToba mudmivia da tolia 0,9-is. 3)

vipovoT

albaToba

yovel

imisa,

eqsperimentSi

rom

xdomiloba,

0,9-is

tolia,

romlis

xuTjer

moxdenis Catarebul

eqsperimentSi moxdeba aranakleb oTxisa. 4) mizani dazianebulad CaiTvleba, Tu samiznes moxvdeba aranakleb sami Wurvi. ipoveT samiznis dazianebis albaToba xuTi gasrolis Semdeg, Tu yoveli gasrolisas moxvedris albaToba 0,8-is tolia. 5) albaToba imisa, rom studenti amoxsnis mocemul amocanas, aris 0,8. gamoTvaleT albaToba imisa, rom igi miiRebs CaTvlas Tu cnobilia, rom man 5 mocemuli amocanidan unda amoxsnas aranakleb 4 amocana. polinaruli sqema. ualbaTesi ricxvi praqtikaSi xSirad saqme gvaqvs iseT eqsperimentTan, romlis SesaZlo SedegTa ricxvi metia orze. aseT SemTxvevaSi SesaZlebelia bernuli sqemis ganzogadeba. vTqvaT, raime eqsperimentis SesaZlo Sedegebia Sesabamisi albaTobebi

e.i.

, xolo maTi

. cxadia, rom

. -iT aRvniSnoT albaToba imisa, rom n damoukidebel cdaSi xdomilobas adgili eqneba da a.S.

-jer,

xdomilobas adgili eqneba

SevniSnoT, rom marTebulia

xdomilobas adgili eqneba

-jer

-jer. Tu

, i=1,2,...,m

, maSin

formula: =

-s uwodeben polinarul albaTobas. magaliTi. 1000 detalisagan Semdgari partiidan arCeven 5 detals. vipovoT albaToba imisa, rom maTgan 3 iqneba pirveli xarisxis, erTi_meore xarisxis da erTic_mesame xarisxis, Tu cnobilia, rom MmTel partiaSi aris 700 pirveli xarisxis, 200_meore xarisxis da 100_mesame xarisxis detali.

amoxsna.

145 rom SerCeul detals Semowmebis Semdeg

(igulisxmeba,

ukan abruneben).

-Ti aRvniSnoT albaToba imisa, rom SerCeuli detali

iqneba k_uri xarisxis (k=1,2,3). albaTobis klasikuri ganmartebis Tanaxmad gveqneba: =0,7;

. Tu visargeblebT polinaruli albaTobis formuliT, miviRebT: (3,1,1)=

n damoukidebel cdaSi albaToba

(k)

yvelaze

.0,2.0,1=0,1372. warmatebaTa im ricxvs romlisTvisac binomuri did

mniSvnelobas

iRebs

ualbaTesi

ricxvi

ewodeba. ualbaTesi ricxvis sapovnelad, fiqsrebuli n>1- saTvis binomuri albaToba

(k) ganvixiloT, rogorc k-s (k=0,1,2,...,n) funqcia. is zrdadia (k+1)>

s im mniSvnelobebisaTvis, roca <

(k); xolo klebadi, roca

(k+1)

(k). ganvixiloT fardoba:

saidanac vRebulobT:

k<np-q

>1, roca

(1)

(k), rogorc k-s (k=0,1,2,...,n) funqcia zrdia, roca

e.i.

roca

k<np-q ;

k>np-q ,

(2)

(k), rogorc k-s (k=0,1,2,...,n) funqcia klebadia, roca

e.i.

(2)-dan davaskvniT, rom

k>np-q . (1) da

(k)-s, rogorc k-s (k=0,1,2,...,n) funqcia udides

mniSvnelobas iRebs, roca k=[np-q] (simbolo [np-q] niSnavs (np-q)-s mTel nawils).

ualbaTesi ricxvis sapovnelad miviRebT formulas

[np-q]=[np-(1-p)]=[(n+1)p-1] . Tu ((n+1)p-1) mTeli ricxvia, maSin

(k)-s gaaCnia ori maqsimumi

((n+1)p-1)

da (n+1)p wertilebSi. magaliTi. Tu bernulis sqemaSi cdaTa ricxvi n=19, p=0,36. ualbaTesi ricxvi

[np-q]=[19.0,36-0,64]=[7,2]=7. Tu

cdaTa ricxvi n=19, p=0,15. maSin

np-q= =19.0,15-0,85=2 ,e.i.gveqneba ori ualbaTesi ricxvi 2 da 3.

146 savarjiSo magaliTebi 1)

jgufSi

danarCeni

10 ki

studentia, samosani.

maT

Soris

vipovoT

friadosania,

albaToba

imisa,

rom

kargosani,

SerCeuli

studentidan sami friadosania, ori kargosani, erTi samosani. 2) amwyobma saamqrom erTi da igive dasaxelebis 12 detali miiRo. maTgan 5 damzadebulia pirvel avtomatze, 4-meore avtomatze, 3_ mesameze. vipovoT albaToba imisa, rom 5 SerCeuli detalidan sami damzadebulia pirvel avtomatze, TiTo ki meore da mesame avtomatebze. 3)

albaToba

imisa,

rom

xaratis

mier

damzadebuli

detali

arastandartulia udris 0,1-s. ipoveT arastandartul detalTa ualbaTesi ricxvi, Tu xaratma 30 detali daamzada. 4) meqanikurma saamqrom gamouSva 10000 detali. albaToba imisa, rom detali standartuli iqneba 0,77-is tolia. ipoveT standartul detalTa ualbaTesi ricxvi. 5) xdomilobis albaToba yovel cdaSi 0,3-ia. ramdeni damoukidebeli cdaa saWiro, rom am xdomilobis mosvlis ualbaTesi ricxvi iyos 60? 6)

ipoveT

damoukidebel

xdomilobis cdaSi

moxdenis

xdomlobis

albaToba, moxdenis

cnobilia,

ualbaTesi

rom

ricxvi

25-is

tolia.

10.muavr-laplasis lokaluri da integraluri Teoremebi bernulis formulis gamoyeneba praqtikulad sakmaod Znelia, roca cdaTa ricxvi didia. magaliTad, Tu n=1500, p=0,75, k=850: (850)=

rac sakmaod mouxerxebeli gamosaTvlelia. am

siZnelis

gadasalaxavad

gamoiyeneba

muavr-laplasis

lokalur

Teorema: Teorema. albaToba

bernulis

sqemis

cdaSi

miaxloebiT gamoiTvleba formuliT: ,

warmatebis

147

sadac

, xolo

funqciis mniSvnelobebi

mocemulia cxrilis saxiT (danarTi 1).

funqcia luwia

aqve SevniSnoT, rom

magaliTi. albaToba imisa, rom xaratis mier damzadebuli

detali

standartuli iqneba 0,8-is tolia, vipovoT albaToba imisa, rom xaratis mier damzadebuli 1000 detalidan 8020 standartuli aRmoCndeba. amoxsna. SemoRebul aRniSvnebSi: n=1000; p=0,8; q=1-0,8=0,2; k=8020. saZiebeli albaToba: (8020)=

sadac =0,5 . cxrilSi (danarTi

) vpoulobT, rom

=0,3521, amitom

(8020)=

.0,3521=0,0088.

im SemTxvevaSi, rodesac gvainteresebs albaToba imisa, rom warmatebaTa ricxvi k moTavsebulia raime SualedSi sargebloben

formuliT,

romelic

(

da n didi ricxvia

miiReba

laplasis

integraluri

Teoriebis safuZvelze. Teorema. bernulis sqemis n cdaSi albaToba imisa, rom warmatebaTa raodenoba k moTavsebuli iqneba

ricxvebs Soris, miaxloebiT

gamoiTvleba formuliT: , sadac ,

funqcias albaTobis integrali anu laplasis funqcia ewodeba. igi

gamoisaxeba

elementaruli

funqciebis

saSualebiT,

misi

148 mniSvnelobebi mocemulia cxrilis saxiT (danarTi 2). SevniSnoT, rom kenti funqciaa

= _

magaliTi. albaToba imisa, rom amwyobi saamqros mier miRebuli detali aRmoCndeba arastandartuli 0,1-is tolia. vipovoT albaToba imisa, rom 1600 miRebul detalSi standartul detalTa raodenoba iqneba aranakleb 1400-sa. amoxsna. mocemulobis Tanaxmad: n=1600; p=0,9; q=0,1;

= (1400

= 13,(3); a =

=1400;

=1600;

=3,(3).

)

=0,5+0,4996= 0,9996. puasonis formula rogorc vanxeT, rodesac cdaTa ricxvi sakmaod didia, sargebloben muavr-laplasis

miaxloebiTi

formuliT.

SemTxvevaSi

daSvebuli

cdomileba miT metia, rac ufro mcirea warmatebis albaToba. rodesac warmatebis albaToba p<0,1, mimarTaven puasonis miaxloebiT formulas. amocana ismis ase, vipovoT albaToba imisa, rom n damoukidebel cdaSi A xdomileba moxdeba zustad k-jer, Tu cdaTa ricxvi n Zalian didia, xolo TiToeul cdaSi warmatebis albaToba axlosaa nulTan (aseT xdomilobebs iSviaTi xdomilobebi ewodeba). puasonis

formulis

daSveba: namravli Teorema. albaToba

gamoyenebisas

inarCunebs mudmiv

bernulis

sqemis

gakeTebulia

erTi

mniSvnelovani

mniSvneloba anu P

warmatebis

dazianebis

albaToba

cdaSi

miaxloebiT gamoiTvleba formuliT: .

magaliTi.

transportirebisas

mza

produqciis

aris 0,0002. vipovoT albaToba imisa, rom gagzavnili 5000 erTeulidan daziandeba 3. Aamoxsna. n=5000; p=0,0002;k=3; =np=5000.0,0002=1. puasonis formulis Tanaxmad: (3)=

=0,06.

149 savarjiSo magaliTebi 1) Rvinis qarxanas gamougzavnes 5000 cali muxis kasri. albaToba imisa, rom kasri gzaSi daziandeba 0,0002-is tolia.

ras udris albaToba imisa,

rom qarxanaSi miRebuli kasrebidan sami kasri dazianebuli aRmoCndeba. 2) vaJis dabadebis albaTobaa 0,51. ras udris albaToba imisa, rom 100 axalSobili bavSvidan 50 vaJia. 3) 1000 adamianSi 8 eqimia. gamoTvaleT albaToba imisa, SemTxveviT arCeuli 100 adamianidan arc erTi eqimi ar iqneba. 4) gamoTvaleT albaToba imisa, rom 200 agdebis SemTxvevaSi liTonis monetaze mova gerbi aranakleb 95-jer da ara umetes 105-jer. 5) kamaTlis gagoreba xdeba 12000-jer. ras udris albaToba imisa, rom ,,6’’ mosvlis ricxvi moTavsebuli iqneba (1900; 2100) SualedSi. 6) msrolelis mizanSi moxvedris albaToba erT gasrolaze udris 0,75s.

ipoveT

albaToba

imisa,

rom

100

gasrolidan

msroleli

mizanSi

moaxvedrebs aranakleb 70-jer. 7) albaToba imisa, rom SeZenili iqneba latariis wamgebiani bileTi udris

0,1-s.

ras

udris

albaToba

imisa,

rom

600

nayidi

bileTidan

wamgebiani bileTebis raodenoba iqneba aranakleb 48 da ara umetes 55-sa.

Tavi 9 1. SemTxveviTi sidide da misi ganawilebis kanoni. albaTobis Teoriis erT-erTi ZiriTadi cnebaa SemTxveviTi sididis cneba.

sanam

SemTxveviT

sidides

ganvmartavT

ganvixiloT

ramdenime

magaliTi: kamaTlis gagorebisas cdis Sedegi aris ama Tu im waxnagis zemoT moqceva. radgan kamaTlis waxnagebze dasmulia wertilebi 1-dan 6-is CaTvliT, cdis Sedegs SevusabamoT ricxvi, romelic moqceul

waxnagze

wertilebis

raodenobas.

warmoadgens zemoT

SemTxvevaSi

SegviZlia

SemovitanoT X –sidide, romelsac SeuZlia miiRos 1,2,3,4,5 da 6-is toli mniSvnelobebi

imisda

mixedviT,

romeli

waxnagi

aRmoCnda

zemoT

moqceuli. ori kamaTlis gagorebisas SegviZlia SemovitanoT X –sidide, romelsac SeuZlia miiRos 2,3,4,...,12 -is toli mniSvnelobebi (qulaTa jami) imisda mixedviT, Tu romel da meore Sedegi,

waxnagebi aRmoCndnen zemoT moqceuli pirvel

kamaTlze. SevniSnoT, rom cdebs yovelTvis ar gaaCnia iseTi romlelebic

ricxviT

gamoisaxebian.

magaliTad,

yuTSi

moTavsebulia,

150 zomis, TeTri,

erTnairi

Savi,

wiTeli

burTulebi. alalbedze amoRebuli burTula SeiZleba aRmoCndes

mwvane am oTxi

Fferidan erT-erTi. cxadia, elementaruli xdomilobebi ricxvebiT ar gamoisaxeba, magram, SegviZlia ganvixiloT X â&#x20AC;&#x201C;sidide, romelsac SeuZlia miiRos 1,2,3 da 4-is toli mniSvnelobebi imisda mixedviT, Tu romeli ferisaa alalbedze amoRebuli burTula. ganxiluli neismieri

magaliTebidan

cda

aRiweros

SeiZleba ricviTi

vivaraudoT, sididiT,

rom

SesaZlebelia

romlis

mniSvneloba

damokidebulia cdis Sedegze. cdis Sedegebi ki warmoadgenen elementarul xdoilobaTa

sivrcis

elementarul funqcias. nebismieri

elementebs,

xdomilobaTa

â&#x20AC;&#x201C;sidide

warmoadgens

elementebze

sivrcis

gansazRvrul

-ze misi mniSvneloba aRvniSnoT X( ) simboloTi. davuSvaT x [X<x]

ricxvia.

xdomilobaTa

simravle,

CanaweriT

aRvniSnoiT

X( )

romelTaTvisac

<x,

im e.i.

elementarul [X<x]

aris

elementarul xdomilobaTa sivrcis qvesimravle. gansazRvreba. SemTxveviTi sidide ewodeba sivrceze gansazRvrul X( )

elementarul xdomilobaTa

funqcias, romelTaTvisac [X<x] simravle

yoveli x-ricxvisaTvis warmoadgens xdomilobas, garkveuli albaTobiT. SevniSnoT, rom [X<x] simravlesTan erTad, xdomilobebs warmoadgenen [X>x] da [X=x] simravleebi da maTac garkveuli albaTobebi Seesabameba. ganixileba ori tipis SemTxveviTi sidideebi: diskretuli da uwyveti. gansazRvreba. SemTxveviT sidides ewodeba diskretuli Tu is iRebs sasruli an Tvladi raodenobis mniSvnelobebs. ganvixiloT diskretuli tipis SemTxveviT sidide, (SevTanxmdeT,

romlis SesaZlo mniSvnelobebia

SemTxveviTi

sididis

SesaZlo

mniSvnelobebi

dalagebulia zrdis mixedviT). SemoviRoT aRniSvnebi: ,..., rogorc

viciT,

[X= ]

simravle

yoveli

,... (i=1,2,...,n,...)

xdomilobas, garkveuli albaTobiT: (1)

warmoadgens

gansazRvreba.

151 romlis pirvel

cxrils,

striqonSi

mocemulia

SemTxveviTi sididis mniSvnelobebi, xolo meore striqonSi Sesabamisi albaTobebi SemTxveviTi sididis ganawilebis kanoni ewodeba:

. .

. . .

. .

. . .

rogorc vxedavT ganawilebis kanoni mocemulia (1) tolobiT. SevniSnoT, rom

roca i j ,normirebis aqsiomis da

uTasebadi xdomilobebis jamis albaTobis Tvisebis gamoyenebiT miviRebT: e.i.

magaliTi. SemTxveviTi sidides warmoadgens ori kamaTlis agdebisas mosuli qulaTa jami. SevadginoT am SemTxveviTi sididis ganawilebis kanoni. 2

1 0

1 1

1 2

ganvixiloT X SemTxveviTi sidide da [X<x] xdomiloba. am xdomilobis damokidebulia x-ze da warmoadgens namdvili cvladis

albaToba

namdvil funqcias. am funqcias aRvniSnavT F(x)-iT da ganawilebis vuwodebT:

funqcias

F(x)=

ganawilebis funqcias gaaCnia Semdegi Tvisebebi: 1) ganawilebis funqcia araklebadia namdvil ricxvTa simravleze: Tu

maSin F( ) F

2) F(+ )=1, F(- )=0, rogorc cnobilia F( 3) ganawilebis

funqcia

uwyetia

marcxnidan

; namdvil

ricxvTa

simravleze. SemTxveviTi sididis

ganawilebis funqcia warmoadgens SemTxveviTi

sididis universalur maxasiaTebels. ganawilebis funqciis saSualebiT SeiZleba gamovTvaloT X SemTxveviTi sididis raime [

] intervalSi moxvedris albaToba:

152 F( ) F

)=P(X< )-P(X< )=

gansazRvreba. X SemTxveviTi sidides ewodeba uwveti tipis SemTxveviTi sidide,

F(x)=

misi

ganawilebis

, sadac

funqcia

warmoidgineba

Semdegi

saxiT:

arauaryofiTi funqciaa.

funcias X uwveti tipis SemTxveviTi sididis ganawilebis simkvrive ewodeba.

cxadia

SemTxveviTi

sididis

ganawilebis

albaToba ganawilebis

ganawilebis simkvrives Soris arsebobs Tanafardoba: X uwveti tipis SemTxveviTi sididis raime [

funqciasa

] intervalSi moxvedris

simkvrivis saSualebiT gamoiTvleba Semdegnairad:

)=F( ) F

ukanaskneli Tanafardobidan gamomdinareobs uwveti tipis SemTxveviTi sididis F(

ganawilebis simkvrivis normirebis piroba:

) F

=1, marTlac,

=1.

savarjiSo magaliTebi 1) msroleli mizanSi isvris 3-jer. yoveli gasrolisas moxvedris albaToba udris 0,4-s. yoveli moxvedrisas msrolels eTvleba 5 qula. dawereT miRebuli qulebis ganawilebis kanoni. 2) kamaTels agoreben 3-jer. dawereT 6-ianis mosvlaTa ganawilebis kanoni. 3)

msroleli,

romelsac

oTxi

vazna

aqvs,

mizanSi

isvris

pirvel

moxvedramde. yoveli gasrolisas moxvedris albaToba udris 0,8-s. ipoveT daxajuli vaznebis ganawilebis kanoni. 4) albaToba imisa, rom biblioTekaSi studentisaTis saWiro wigni Tavisufalia, udris 0,4-s. SeadgineT im biblioTekebis ganawilebis kanoni, romlebic unda inaxulos studentma, Tu qalaqSi oTxi biblioTekaa. 5) SemTxveviTi sididis ganawilebis funqcia mocemulia Semdegi saxiT:

vipovoT (0,25; 0,75) intervalSi moxvedris albaToba. 6) SemTxveviTi sididis ganawilebis funqcia mocemulia Semdegi saxiT:

153

ipoveT ganawilebis simkvrive. 2. SemTxveviTi sididis ricxviTi maxasiaTeblebi SemTxveviTi sididis ganawilebis dasaxasiaTeblad xSirad sakmarisia ramdenime

iseTi

ricxviTi

maCveneblis

codna,

romlebic

gamoxataven

SemTxveviTi sididis arsebiT Tvisebebs. aseTi ricxviTi maxasiaTeblebia: SemTxveviTi sididis maTematikuri lodini; SemTxveviTi sididis dispersia; sxvadasxva rigis sawyisi da centraluri momentebi; mediana; moda da sxva. vTqvaT, mocemulia diskretuli tipis SemTxveviTi sididis ganawilebis kanoni:

. .

. . .

. .

. . .

gansazRvreba. lodini

diskretuli

ewodeba,

misi

tipis

SesaZlo

SemTxveviTi

mniSvnelobis

sididis

maTematikuri

Sesabamis

albaTobaze

namravlis jams, Tu es jami arsebobs. X

SemTxveviTi

sididis

maTematikuri

lodini

aRiniSneba

M(X)

simboloTi:

M(X) = e.i.

diskretuli

= tipis

SemTxveviTi

sididis

(1)

maTematikuri

lodini

arsebobs TuU(1) tolobis marjvena mxares mdgari mwkrivi absoluturad krebadia.

SevniSnoT,

rom

SemTxveviTi

sidide

iRebs

sasruli

raodenobis sasrul mniSvnelobebs es jami yovelTvis iarsebebs. vTqvaT

uwveti

tipis

SemTxveviTi

sididea,

romlis

ganawilebis

simkvrivea . gansazRvreba. Tu integrali

absoluturad krebadia, maSin

mas uwyveti tipis X SemTxveviTi sididis maTematikuri lodini ewodeba:

M(X)=

maTematikur lodins gaaCnia Semdegi ZiriTadi Tvisebebi:

154 Tviseba 1. mudmivis maTematikuri lodini TviT am mudmivis tolia. marTlac, raime C

mudmivi SeiZleba ganvixiloT, rogorc SemTxveviTi

sidide, romelic C mniSvnelobas iRebs albaTobiT erTi maTematikuri lodinis ganmartebis Tanaxmad: M(C)=1C=C. Tviseba 2. SemTxveviTi sidideTa jamis maTematikuri lodini SesakrebTA maTematikuri lodinebis jamis tolia:

M(X+Y)=M(X)+M(Y). vTqvaT, X-is SesaZlo mniSvnelobebia albaTobebi

. ,

xolo

SemTxveviTi

analogiurad Y-is

Sesabamisi

...,

,...

, -Ti

mniSvnelobas

mniSvnelobebs:

...,

,...,

aRvniSnoT

SemTxveviTi sidide miiRebs mniSvnelobas miiRebs

mniSvnelobebia maSin (X+Y)

miiRebs

...,

SesaZlo

albaTobebi

sidide

, xolo Sesabamisi

=P[

albaToba

imisa,

rom X

, xolo Y SemTxveviTi sidide

i,j=1,2,3,...

;

maTematikuri

lodinis ganmartebis Tanaxmad gveqneba:

(1)

M(X+Y)= sruli albaTobis formulis Tanaxmad: =

(2)

Tu (2)-s gaviTvaliswinebT (1) tolobaSi miviRebT:

M(X+Y) me-2

Tviseba

= M(X)+M(Y).

samarTliania

SesakrebTa

nebismieri

sasruli

raodenobisaTvis. Tviseba 3. damoukidebel SemTxveviT sidideTa namravlis maTematikuri lodini TanamamravlTaAmaTematikuri lodinebis namravlis tolia:

M(XY)=M(X)M(Y). gamoviyenoT namravlis

me-3

Tvisebis

SesaZlo

=P[

SemTxveviTi sidideebia gveqneba: M(XY)=

Cvenebis

mniSvnelobebia

i,j=1,2,3,... =

dros

miRebuli

aRniSvnebi.

albaTobiT

ramdenadac X daY Y damoukidebeli

maTematikuri lodinis ganmartebis Tanaxmad

155 M(X)M(Y).

zemoT naCvenebi (1), (2) da (3) Tvisebebidan gamomdinareobs Sedegi. Tu a da b nebismieri mudmivebia, xolo X daY Y SemTxveviTi sidideebi, maSin

M(aX+bY)=aM(x)+bM(y). gansazRvreba. (X-a) sxvaobas ewodeba X SemTxveviTi sididis gadaxra a ricxvisgan. roca a=M(X) , maSin (X-M(X))

sxvaobas ubralod gadaxra

ewodeba, xolo (X-M(X)) SemTxveviT sidides ki _ cetrirebul SemTxveviTi sidides uwodeben. Teorema.

centrirebuli

SemTxveviTi

sididis

maTematikuri

lodini

M(X-M(X)) =0.

nulis tolia.

damtkiceba. maTematikuri lodinis Tvisebebis gamoyenebiT miviRebT:

M(X-M(X)) =M(X)-M(M(X))=M(X)-M(X)=0. maTematikuri

lodini

warmoadgens

SemTxveviTi

sididis

erT-erT

ZiriTad ricxviT maxasiaTebels, magram, rogorc Semdgom vnaxavT, mxolod maTematikuri

lodini

kmara

SemTxveviTi

sididis

dasaxasiaTeblad.

kerZod, maTematikuri lodini ar iZleva warmodgenas SemTxveviTi sididis mniSvnelobebze

maTi

gafantulobis

Sesaxeb

maTematikuri

lodinis

irgvliv. SemTxveviTi gafantulobis

sididis

Tavisi

dasaxasiaTeblad

maTematikuri

sargebloben

lodinis

ricxviTi

irgvliv

maxasiaTebliT

romelsac dispersia ewodeba. gansazRvreba.

gadaxris

kvadratis

maTematikur

lodins

SemTxveviTi

sididis dispersia ewodeba da aRiniSneba D(X) simboloTi: D(X)=M

maTematikur lodins ganmartebis Tanaxmad: D(X)= rodesac X aris diskretuli tipis SemTxveviTi sidide. D(X)= Tu

uwveti

tipis

SemTxveviTi

sididea,

romlis

maTematikuri

lodinis

Tvisebebs

ganawilebis

simkvrivea . Tu

gamoviyenebT

gamosaTvlelad miviRebT ufro moxerxebul formulas:

dispersiis

156 +

D(X)=M =M(

= M(

=M(

)-M(2XM(X))+

e.i. SemTxveviTi sididis dispersia udris misi kvadratis maTematikur lodins gamoklebuli maTematikuri lodinis

kvadrati.

SemTxveviTi sididis dispersias gaCnia Semdegi Tvisebebi: Tviseba 1. mudmivis dispersia nulis tolia. marTlac, D(C)=M[C Tviseba

=M[C-

mudmivi

=0.

Tanamamravli

gamodis

dispersiis

niSnis

gareT

kvadratSi axarisxebuli:

D(CX)=

D(X).

marTlac,

D(CX)=M(

)= M[

Tviseba

damoukidebel

SemTxveviT

sidideTa

jamis

dispersia

Sesabamisi dispersiebis jamis tolia

D(X+Y)=D(X)+D(Y). dispersiis ganmartebis Tanaxmad gveqneba:

D(X+Y)=M

+2M[X-M(X)][Y-M(Y)]+ M =D(X)+2M[X-M(X)]M[Y-M(Y)]+D(Y)=D(X)+ D(Y),

radganac

M[X-M(X)]=M[Y-M(Y)]=0. gansazRvreba.

kvadratul

fesvs

dispersiidan

SemTxveviTi

saSualo kvadratuli gadaxra ewodeba da aRiniSneba

sididis

simboloTi.

= SevniSnoT, rom dispersias gaaCnia SemTxveviT sididis ganzomilebis kvadratis

ganzomileba,

ganzomileba

aqvs,

rac

xolo

saSualo

SemTxveviT

kvadratuli sidides,

gadaxras

amasTanave

igive orTave

arauaryofiTi sidideebia da axasiaTebs mis gafantulobas maTematikuri lodinis

irgvliv.

saSualo

kvadratuli

Tvisebebi: Tviseba 1.

sadac C mudmivia.

gadaxras

gaaCnia

Semdegi

Tviseba

157 Y damoukidebeli

SemTxveviTi

sidideebia, maSin . gansazRvreba. SemTxveviT

SemTxveviT

sididis

sidides,

gayofiT

romelic

saSualo

miiReba

centrirebuli

kvadratuli

gadaxraze

normirebuli SemTxveviT sidide ewodeba da aRiniSneba: =

normirebuli SemTxveviTi sididis maTematikuri lodini nulis tolia, dispersia erTis. MM

=M[

M[X-M(X)]=0,

D[(X-M(X)]

)] =

D[X]=1.

gansazRvreba. SemTxveviTi sididis k_uri xarisxis maTematikur lodins

k-rigis sawyisi momenti ewodeba da aRiniSneba =

simboloTi:

k=0,1,2,3,....

gansazRvreba.

centrirebuli

SemTxveviTi

sididis

k-uri

xarisxis

maTematikur lodins k-rigis centraluri momenti ewodeba da aRiniSneba simboloTi:

k=0,1,2,3,....

advili saCvenebelia, rom: 1;

=M(X);

=1;

=0;

=D(X)=

-4

-3

;

gansazRvreba. mocemulia X SemTxveviTi sidide romlis ganawilebis funqciaa F(x) maSin misi mediana ewodeba iseT sruldeba pirobebi: F( ) zRvaria

da F(

)

, sadac F(

ricxvs, romlisTvisac

) , F funqciis marjvena

wertilze.

gansazRvreba. X SemTxveviTi sididis asimetriis koeficienti ewodeba ricxvs da aRiniSneba simboloTi:

gansazRvreba.

158 SemTxveviTi sididis =

sidides da aRiniSneba simboloTi:

eqcesi

ewodeba

-3)

-3).

gansazRvreba. diskretuli tipis X SemTxveviTi sididis moda ewodeba mis im SesaZlo mniSvnelobas, romlis Sesabamisi albaToba udidesia. uwyveti tipis X SemTxveviTi sididis moda ewodeba ganawilebis simkvrivis lokaluri maqsimumis wertils. savarjiSo magaliTebi 1)

SemTxveviTi sididis ganawilebis kanoni mocemulia cxrilis saxiT: -5

0.4

0,3

0,1

0,2

vipovoT maTematikuri lodini da dispersia. 1) vipovoT

dispersia

saSualo

kvadratuli

gadaxra,

SemTxveviTi sididis ganawilebis kanoni mocemulia cxrilis saxiT:

3)vipovoT da

Q131

140

160

180

0.05

0,10

0,25

0,60

SemTxveviTi sididis maTematikuri lodini, Tu cnobilia,

SemTxveviTi sidideTa maTematikuri lodinebi: a) b) 4)

SemTxveviTi SemTxveviTi

sidideebi

sididis

damoukidebeli

dispersia,

arian.

vipovoT

cnobilia,

rom

SemTxveviTi sididis ganawilebis kanoni mocemulia cxrilis saxiT.

-2

0.4

0,3

0,1

0,2

vipovoT pirveli sami rigis sawyisi da centraluri momentebi.

SemTxveviTi sidide ganawilebulia Tanabrad [3, 8] intervalze.

vipovoT maTematikuri lodini da dispersia.

159

Tavi 10 ganawilebis kanonTa ZiriTadi saxeebi

1. binomuri ganawileba ganvixiloT

orSedegiani

damoukidebel

cdaTa

mimdevroba.

warmatebaTa ricxvi sasrul cdaTa mimdevrobaSi aRvniSnoT X â&#x20AC;&#x201C;iT. cxadia, igi warmoadgens SemTxveviT sidides, ganawilebis kanoniT: 0

â&#x20AC;Ś. . . . . .

gansazRvreba.

diskretuli

tipis

SemTxveviTi

sidides

ewodeba

binomurad ganawilebuli Tu misi SesaZlo mniSvnelobebia 0,1,2,3,...,n. xolo Sesabamisi albaTobebi gamoiTvleba

formuliT:

PP(X=k)= vipovT

binomurad

ganawilebuli

(k)=

SemTxveviTi

sididis

maTematikuri

lodini da dispersia. -iT aRvniSnoT i-ur cdaSi warmatebaTa ricxvi (cxadia, igi udris . SevniSnoT, rom

erTs an nuls), maSin

, sadac i=1,2,3,...n ; e.i.

sididis ganawilebis kanons eqneba saxe:

=0(1-p)+1p=p;

)=p;

= M(

SemTxveviTi

)

= p- =p(1-p)=pq. Tu

gamoviyenebT, damoukidebeli SemTxveviTi sidideebis jamis maTematikuri lodinisa da dispersiis gamosaTvlel formulas, miviRebT: =p+p+...+p=np ;

pq+pq+...+pq=npq.

puasonis ganawileba

160 gansazRvreba. diskretuli tipis X

SemTxveviT

sidides

ewodeba

puasonis kanoniT ganawilebuli Tu misi SesaZlo mniSvnelobebia 0, 1, 2, 3, ... ,k,.... xolo Sesabamisi albaTobebi gamoiTvleba formuliT: PP(X=k)= puasonis

kanoniT

ganawilebuli

SemTxveviTi

sididis

lodini da dispersia erTmaneTis tolia da emTxveva

maTematikuri

parametrs:

2. geometriuli ganawileba vTqvaT, bernulis sqemiT cdebs vatarebT pirvel warmatebamde. cdaTa ricxvi, romelic saWiroa pirvel warmatebamde aRvniSnoT X-iT. cxadia, X=k,

(k=1,2,3,...,n,...) niSnavs, rom wina (k-1) cdaSi marcxs hqonda adgili, xolo

â&#x20AC;&#x201C;ur cdaSi warmatebas. Tu gaviTvaliswinebT damoukidebel xdomilobaTa namravlis albaTobis gamosaTlel formulas, miviRebT:

P(X=k)= gansazRvreba. diskretuli tipis X geometriuli

. SemTxveviTi

kanoniT ganawilebuli Tu misi SesaZlo

sidides

ewodeba

mniSvnelobebia 0,

1, 2, 3, ... ,k,.... xolo Sesabamisi albaTobebi gamoiTvleba formuliT:

P(X=k)= geometriulad lodini

ganawilebuli xolo dispersia

SemTxveviTi =

sididis

maTematikuri

3. Tanabari ganawilebis kanoni gansazRvreba. uwyveti tipis X intervalze

SemTxveviTi

sidides

ewodeba [a,b]

Tanabrad ganawilebuli Tu, mis ganawilebis funqciaa:

F(x)= ganawilebis simkvrivea

vipovoT

[

161 intervalze Tanabrad ganawilebuli SemTxveviTi

]

sididis maTematikuri lodini da dispersia: =

;

= =

4. normaluri ganawilebis kanoni gansazRvreba.

uwyveti

tipis

SemTxveviTi

sidides

ewodeba

normaluri kanoniT ganawilebuli, Tu mis ganawilebis simkvrives aqvs Semdegi saxe:

f(x)= a da

sadac

parametrebi

kanoniT

ricxvebia,

ewodeba

maTematikuri

lodinis

ganawilebuli

aRiniSneba N(a ,

, romlebsac

Sesabamisad

dispersiis:

SemTxveviT

normaluri

tolia

SemTxveviTi =

sidide

ganawilebis

sididis

normaluri

parametrebiT

a da

) simboloTi.

praqtikaSi didi gamoyeneba aqvs normalur ganawilebas parametrebiT 0 da 1, mas standartuli normaluri

ganawileba ewodeba.

standartuli normaluri ganawilebis simkvrivea

(x)=

xolo ganawilebis funqcia:

F(x)=

dt .

funqcia dakavSirebulia laplasis

= vTqvaT,

SemTxveviTi

sidide

funqciasTan tolobiT

. ganawilebulia

normaluri

kanoniT, maSin rogorc viciT simkvrivis saSualebiT SemTxveviTi sididis raime SualedSi moxvedris albaToba gamoiTvleba Semdegnairad:

162

cvladTa gardaqmniT es albaToba gamoisaxeba laplasis

funqciis

saSualebiT, kerZod . magaliTi.

vTqvaT,

SemTxveviTi

(1)

sidide

ganawilebulia

normaluri kanoniT, vipovoT albaToba imisa, rom gadaxris absoluturi mniSvneloba naklebi iqneba 3-ze. amoxsna. visargebloT (1) formuliT: = =

ganvixiloT

(1)

=2.0,1179=0,2358. tolobis

kerZo

SemTxveva,

rodesac

e.i.

0,9973

kanoniT

toli

albaTobiT

SeiZleba

ganawilebuli

mniSvnelobebs

0,9973.

CavTvaloT,

rom

normaluri

sidide

miiRebs

SemTxveviTi Sualedidan.

faqts

emyareba

rogorc

Teoriul, ise praqtikul amocanebSi gamoyenebis mqone e.w. sami sigmas wesi, romlis

Tanaxmadac,

normalurad

maTematikuri lodinidan

ganawilebuli

SemTxveviTi

sididis

-ze meti gadaxra praqtikulad SeuZlebeli

xdomilobaa.

gama-ganawileba gansazRvreba. uwyveti tipis X

SemTxveviTi

sidides

ewodeba gama-

kanoniT ganawilebuli, Tu igi Rebulobs mxolod dadebiT mniSvnelobebs da mis ganawilebis simkvrives aqvs Semdegi saxe:

sadac

ganawilebis parametrebia, xolo

163

eileris meore gvaris integrali. gama-kanoniT ganawilebuli SemTxveviTi sididis maTematikuri lodini da dispersia Sesabamisad tolia: ;

beta-ganawileba gansazRvreba. uwyveti tipis X

SemTxveviT

sidides

ewodeba beta-

kanoniT ganawilebuli, Tu igi Rebulobs mniSvnelobebs (0,1) intervalidan da mis ganawilebis simkvrives aqvs Semdegi saxe: , sadac

beta-kanoniT

ganawilebis parametrebia. ganawilebuli

gama-kanoniT

ganawilebuli

SemTxveviTi

sididis maTematikuri lodini da dispersia Sesabamisad tolia: ;

ganawileba vTqvaT

mocemuli

gvaqvs

ganawilebiT

damoukidebeli

SemTxveviTi

ganvsazRvroT axali SemTxveviTi

sidideebi sidide

tolobiT: . gansazRvreba. SemTxveviT ewodeba

(1)

sidides, romelic mocemulia (1) tolobiT

kanoniT ganawilebuli,

Tavisuflebis xarisxiT.

ricxvi gviCvenebs, Tu ramdeni Sesakrebia (1) tolobis marjvena mxares da

warmoadgens

kanoniT ganawilebis erTaderT parametrs. cxadia,

164 Tavisuflebis xarisxis mqone

SemTxveviTi

sididis

ganawilebis simkvrives aqvs Semdegi saxe:

Tavisuflebis xarisxis mqone

SemTxveviTi

sididis maTematikuri

lodini da dispersia Sesabamisad tolia:

stiudentis ganawileba vTqvaT

mocemuli romlebic

SemTxveviTi

damoukidebeli

SemTxveviTi

ganawilebulia

ganvixiloT

sidide Semdegi

sidideebi: ;

SemTxveviTi sididis ganawilebis simkvrives aqvs Semdegi saxe:

SemTxveviT

sidides ewodeba stiudentis kanoniT ganawilebuli

SemTxveviTi sidide, misi maTematikuri lodini da dispersia, Sesabamisad, tolia:

, roca

Tavi 11 did ricxvTa kanoni

1. CebiSevis utoloba SemTxveviT sidideTa didi raodenobis jamebis yofaqceva specialur gamokvlevas saWiroebs. magaliTad, ganvixiloT damoukidebeli, erTnairad ganawilebuli mniSvneloba

SemTxveviTi sidideebis ariTmetikuli saSualo

165 = aRvniSnoT:

M( )=a,

D( )=

maSin

M( )=M(

D( ) =DD(

= na=a,

)

rogorc vxedavT,

(2)

SemTxveviT sidides igive maTematikuri lodini

aqvs, rac TiToeul Sesakrebs, magram dispersia TiToeuli

(1)

Sesakrebis.

e.i.

SesakrebTa

-jer ufro mcirea, vidre

raodenobis

zrdasTan

erTad

mcirdeba gafantuloba Tavisi saSualos mimarT. ganvixiloT

diskretuli

tipis

SemTxveviTi

sidide,

gaaCnia sasruli dispersia da maTematikuri lodini: M(X)=a, iyos

winaswar

albaToba imisa, rom

dasaxelebuli

raime

dadebiTi

romelsac

D(X)=

ricxvi.

SevafasoT

SemTxveviTi sidide miiRebs mniSnelobas

intervalidan. CebiSevis utoloba. albaToba imisa, rom

SemTxveviTi sididis Tavisi

maTematikuri lodinidan gadaxris absoluturi sidide ar aRemateba aranaklebia, vidre ]

(3)

damtkiceba. ganvixiloT xdomiloba:

maSin = da radgan =1, gveqneba: . amitom,

(3)

utolobis

dasamtkiceblad

(4) sakmarisia

albaTobis Sefaseba. ganmartebis Tanaxmad,

SemTxveviTi sididis dispersia

-s

166 . (5) amitom

mwkrivis Tu

marjvena

avjamavT

(5)

mxareSi

TiToeuli

mxolod

romlebisTvisac

maT

Sesakrebi

nawils,

arauaryofiTia,

kerZod,

wevrebs,

, jami ar gaizrdeba. .

jami kidev ufro Semcirdeba, Tu yovel

-s SevcvliT

-iT:

. saidanac ]

(6)

(4) tolobis ZaliT ] magaliTi.

SevafasoT

albaToba

cnobilia,

rom amoxsna. visargebloT (6) formuliT, gveqneba . sazogadod

CebiSevis

utoloba

iZleva

maTematikuri

lodinisgan

SemTxveviTi sididis gadaxris mxolod uxeS Sefasebas, rasac adasturebs Semdegi magaliTi. magaliTi. vTqvaT, M(X)=a,

D(X)=

maSin =

Tu davuSvebT, rom

=0,(1).

SemTxveviTi sidide ganawilebulia normalurad

parametrebiT, maSin rogorc viciT, imave xdomilobis albaToba didi sizustiT tolia 0,0027, rac gacilebiT naklebia vidre 0,(1).

2. did ricxvTa kanoni vTqvaT, SemTxveviT

damoukidebeli, sidideTa

mimdevrobaa,

dispersiiT. ganvixiloT =

erTnairad

maTematikuri

ganawilebuli

lodiniTa

167 sidideTa mimedevroba. (1) da (2) formulis da

SemTxveviT

(6) utolobis ZaliT, nebismieri

dadebiTi ricxvisaTvis gveqneba: ]

Tu gaviTvaliswinebT, rom

, roca

miviRebT ]=0.

ukanaskneli

Tanafardoba

samarTliania

SemTxvevaSic,

roca

SemTxveviT sidideeebs ar gaaCniaT sasruli dispersia. sabolood SeiZleba CamovayaliboT Teorema, romelic cnobilia did ricxvTa kanonis saxeliT. Teorema.

(did

ricxvTa

kanoni)

erTnairad

ganawilebuli SemTxveviTi sidideebia saerTo maSin nebismieri

maTematikuri lodiniT,

dadebiTi ricxvisaTvis samarTliania toloba:

GgansazRvreba. SemTxveviT sidideTa albaTobiT krebadi

mimdevrobas ewodeba

SemTxveviT sididisaken Tu nebismieri

dadebiTi

ricxvisaTvis samarTliania toloba: . Teorema

(CebiSevis

Teorema)

wyvil-wyvilad

damoukidebel SemTxveviT sidideTa mimdevrobaa, romelTa dispersiebis mimdevroba

SemosazRvrulia maSin

erTi

nebismieri

igive

mudmiviT,

ricxvisaTvis

adgili

e.i. aqvs

tolobas: . CebiSevis Teoremis arsi mdgomareobs SemdegSi: miuxedavad imisa, rom calkeul damoukidebel SemTxveviT sidideebs SeuZliaT miiRon Tavisi maTematikuri lodinisgan sagrZnoblad gansxvavebuli mniSvnelobebi, didi raodenobis

SemTxveviT

SemTxveviTi

sidide),

xasiaTs.

sxva

sidideTa

garkveuli

sityvebiT,

rom

saSualo

azriT, vTqvaT,

ariTmetikuli

(rogorc

kargavs

SemTxveviTi

sididis

calkeul

SemTxveviT

sidides

SeiZleba gaaCndes didi gadaxra, magram maTi saSualo ariTmetikulis gadaxra sakmarisad mcirea, roca SesakrebTa ricxvi didia.

168 CebiSevis Teoremis kerZo saxes warmoadgens i. bernulis Teorema, romelic did ricxvTa kanonis umartives formas warmoadgens. Teorema.

vTqvaT,

aris

xdomilobis

moxdenis

damoukidebel cdaSi. vigulisxmoT, rom TiToeul cdaSi moxdenis albaToba aris

, maSin nebismieri

ricxvi xdomilobis

ricxvisaTvis adgili aqvs

tolobas:

SevniSnoT, rom xolo

sidide aris

aris

xdomilobis

xdomilobis fardobiTi sixSire,

moxdenis

albaToba.

bernulis

Teorema

akavSirebs am or sidides: fardobiT sixSires da albaTobas. pirveli empiriuli

sididea,

meore

Teoriuli.

bernulis

Teorema

imaze

migviTiTebs, rom Tu cdaTa ricxvi didia, rogorc Semdgom vnaxavT, maSin fardobiTi sixSire amave xdomilobis ucnobi Teoriuli albaTobis ,, kargiâ&#x20AC;&#x2122;â&#x20AC;&#x2122;

Sefasebaa.

Teorema

saSualebas

iZleva

albaTobis

Teoria,

rogorc SemTxveviT xdomilobaTa kanonzomierebebis maTematikuri modeli, daukavSirdes praqtikas, e.i. gvqondes ama Tu im praqtikuli amocanis gadawyvetis saSualeba. savarjiSo magaliTebi 1)

CebiSevis

utolobis

gamoyenebiT

albaTobebi,

aris

SevafasoT normalurad

ganawilebuli SemTxveviTi sidide. 2)

SemTxveviTi sididis maTematikuri lodini M (X)=1, xol dispersia CebiSevis

utolobis

gamoyenebiT

SevafasoT

utolobis albaToba. 3) visargebloT CebiSevis utolobiT,

SevafasoT albaToba imisa, rom

<0,2. Tu 4) mocemulia

gamoyenebiT qvemodan SevafasoT 5)

CebiSevis utolobis

is mniSvneloba.

SemTxveviTi sididis maTematikuri lodini M (X)=1, xol dispersia SevafasoT qvemodan Semdegi xdomilobebis albaTobebi: , diskretuli

SemTxveviTi sidide mocemulia ganawilebis kanoniT:

0,3

169 0,6

0,2

0,8

CebiSevis utolobis gamoyenebiT SevafasoT

utolobis

albaToba. 7) diskretuli

SemTxveviTi sidide mocemulia ganawilebis kanoniT:

0,1

0,4

0,6

0,2

0,3

0,5

CebiSevis

utolobis

gamoyenebiT

SevafasoT

albaToba

imisa,

rom

Tavi 12 MmaTematikuri statistikis elementebi 1.maTematikuri statistikis sagani da ZiriTadi amocanebi maTematikuri SemTxveviT

statistika,

movlenebs

maTematikuri

rogorc

statistika

albaTobis

masTan

iyenebs

Teoria,

dakavSirebul

albaTobis

AAაmocanebs.

Teoriis

meTodebs,

sargeblobs analogiuri cnebebiT, magram miuxedavaT amisa, igi rogorc

damoukidebeli

garkveuli

azriT,

kerZod, Tu

sagani.

albaTobis

(mizans)

kanonzomierebebis amocanas

Teoriis

statistikis

amocanebis

Teoriis ZiriTadi amocanaa

modelidan sxvadasxva rTuli albaTuri

maTematikuri

Seiswavlis

ganixileba amocanebi,

Sebrunebulia.

mocemuli albaTuri

xdomilobebis albaTobebis gamoTvla da dadgena, maTematikuri statistikis ZiriTad

warmoadgens

SemTxveviT

movlenis

dakvirvebebis

safuZvelze dasabuTebuli statistikuri daskvnebis gamotana Sesaswavli SemTxveviTi movlenis albaTuri modelis Sesaxeb.

170 ganvixiloT eqsperimenti, romelSic vakvirdebiT

magaliTisTvis

raime A xdomilobis moxdena ar moxdenisfaqts. Cveni mizania safuZvelze vipovoT (SevafasoT) saqme gvaqvs

maTematikuri

statistikis tipur

sidide,

p=P(A) albaToba.

ucnobi

amocanasTan. Tu n cdaSi A

adili hqonda m-jer, ucnobi P(A)

xdomilobas aviRoT

A xdomilobis

dakvirvebaTa

albaTobis Sesafaseblad

garkveuli azriT, axlos unda iyos P(A)

romelic,

albaTobasTan, rogorc Semdgom vnaxeT amis garantias iZleva did ricxvTa kanoni. ucnob parametrTa statistikuri Sefaseba warmoadgens maTematikuri statistikis

erT-erT

araparametruli

ZiriTad

Sefasebis

amocanas.

amocanebi

ganixileba

(ganawilebis

kanoni,

agreTve

ganawilebis

funqcia, simkvrive,...), statistikuri hipoTezebi.

3. SerCeviTi meTodi maTematikuri

statistikis

warmoadgens

e.w.

SerCeviTi

ganvixiloT

magaliTi.

kvlevis

meTodi.

vTqvaT

erT-erT

misi

ZiriTad

arsis

gvainteresebs

meTods

ukeT

gasarkvevad

wundebul

detalTa

raodenobis dadgena qarxnis mier gamoSvebuli 1000000 detalidan. cxadia, yvela detalis Semowmeba Zalze Sromatevadi saqmea, zogjer SeuZlebelic (SeiZleba detali Semowmebis Sedegad FgafuWdes), amitom qarxnis

mier

gamoSvebul

wundebul

detalTa

moxerxebulia

raodenobaze

daskvna

gavakeToT mis raRac nawilze dakvirvebis Sedegad. gansazRvreba. erTgvarovan obieqtTa erTobliobis zogadi Tvisebebis kvlevis

statistikur

meTods,

romlis safuZvelia am obieqtTa mxolod

nawilis SerCeva da Seswavla, SerCeviTi meTodi ewodeba. gansazRvreba. yvela im erTgvarovan obieqtTa xdeba

SerCeva,

elementTa

generaluri

erTobliobas,

daskvna generaluri

erToblioba

romelTa

erTobliobis

erTobliobas saidanac

ewodeba,

Seswavlis

xolo

safuZvelze

SerCeuli keTdeba

Sesaxeb, SerCeviTi erToblioba anu

SerCeva ewodeba. imisaTvis, rom SerCevis safuZvelze

gakeTebuli daskvna iyos ‘’swori’’

saWiroa SerCeva iyos warmomadgenlobiTi. did ricxvTa kanonis Tanaxmad, SerCeva iqneba

warmomadgenlobiTi Tu igi SemTxveviTia, e.i.

generaluri

erTobliobis

171 obieqts SerCevaSi

yovel

moxvedris

erTnairi

albaToba aqvs. SerCeva rodesac

SeiZleba yovel

ganxorcieldes

Semowmebul

ornairad:

obieqts

SerCeva

aRar

daubruneblad,

vabrunebT

generalur

erTobliobaSi da SerCeva dabrunebiT, rodesac yovel Semowmebul obieqts vabrunebT generalur erTobliobaSi, SemdgomSi Cven ganvixilavT SerCevas dabrunebiT.

igulisxmeba, rom

mxolod

SerCevis procesi ar cvlis

generaluri erTobliobis ganawilebas. SerCeviT meTods SeiZleba mieces Semdegi interpretaciac, romelic dafuZnebuli iqneba elementarul xdomilobaTa sivrcis da SemTxveviTi sididis

cnebebze.

amiT

saSualeba

mogvecema

statistikuri

amocana

CamovayaliboT da amovxsnaT albaTobaTa TeoriaSi miRebuli meTodebiT. vTqvaT,

vatarebT

raime

cdas,

romlis

Sesabamisi

elementarul

xdomilobaTa sivrcea Ω. ganvixiloT Ω sivrceze X SemTxveviTi sidide, romlis ganawilebis funqciaa F(x) . davuSvaT cdas vimeorebT n-jer , i-uri cdis

Sedegad

aRvniSnoT

SemTxveviTi

sididis

mier

miRebuli

mniSvneloba

–iT. miviRebT mimdevrobas: (1)

aRniSnuli

mimdevroba

SeiZleba

ganvixiloT

Semdegnairadac:

ganvixiloT damoukidebel SemTxveviT sidideTa mimdevroba (2) romlis ganawilebis

yoveli

wevri

kanoniT.

ganawilebulia

iyos

SemTxveviTi

SemTxveviTi sididis

sididis

mier

miRebuli

realizacia (i=1,2,..,n). SemdgomSi (1) da (2) mimdevrobebs gavaigivebT. gansazRvreba. n generaluri

moculobis SerCeva F(x) ganawilebis funqciis mqone

erTobliobidan

ganawilebul

SemTxveviT

TiToeuli wevri ganawilebulia SemTxveviTi

ewodeba

sididis

damoukidebel,

erTnairad

sidideTa mimdevrobas, romlis

F(x) ganawilebis kanoniT. SemdgomSi, X

realizaciis

Sedegad

miRebul

mniSvnelobebs

davalagebT zrdis mixedviT, mimdevrobas (3)

, variaciuli mwkrivi ewodeba.

(3) mimdevrobaSi SesaZlebelia elementTa ganmeoreba. vTqvaT, sul gvaqvs

r gansxvavebuli

mniSvneloba

amasTan

–uri

SerCevaSi ewodeba

gvxdeba

172 (i=1,2,..,r,

-jer

â&#x20AC;&#x201C;uri elementis sixSire, xolo Sefardebas

ricxvs =

fardobiTi

sixSire . SerCevis

Cawera mosaxerxebelia cxrilis saxiT, romlis pirvel

striqonSi Cawerilia SerCevis strionSi

maTi

Sesabamisi

gansxvavebuli mniSvnelobebi, xolo meore fardobiTi

sixSireebi.

aseT

cxrils

SemTxveviTi sididis statistikuri ganawilebis cxrili ewodeba:

Tu X uwyveti

. .

tipis SemTxveviTi sididea, maSin ganawilebis cxrilis

Sesadgenad iqcevian Semdegnairad:

X SemTxveviTi sididis mier miRebul

mniSvnelobaTa simravles vyofT k nawilad wertilebiT ganvixiloT intervalebi: [

]; ]

];...;]

] .

-iT aRvniSnoT dakvirvebaTa is raodenoba (1) SerCevidan, romlebic ] intervalSi (i=1,2,..,k). xolo

moTavsdnen [

]

. . .

= . ]

. . .

4. ganawilebis parametrebis statistikuri Sefaseba ganvixiloT

raime obieqtTa

generaluri

erToblioba da, vTqvaT,

gvainteresebs am obieqtTa raime niSan-Tvisebis Seswavla. davukavSiroT mocemul generalur

erTobliobas SemTxveviTi sidide, romelic am niSan-

Tvisebas Seesabameba. aseTi SemTxveviTi sididis ganawilebis kanoni amave dros warmoadgens ganawilebis

generaluri erTobliobis dasakvirvebel niSan-Tvisebis

kanons. am

ganawilebis

ricxviT

maxasiaTeblebs

ewodeba

generaluri erTobliobis ricxviTi parametrebi. vTqvaT, mocemulia n

moculobis

mqone generaluri erTobliobidan

SerCeva F(x) ganawilebis funqciis

173 sidideebis

gansazRvreba.

S=S(

nebismier

) funqcias statisika ewodeba.

statistikis magaliTebia funqciebi: =

da a.S.

X SemTxveviT sidideze dakvirvebis Sedegad miRebuli statistika X -is ucnobi

parametris

Sefasebas

warmoadgens, amitom SemdgomSi gavaigivebT

statistikis da Sefasebis cnebebs. cxadia, Sefasebis aseTi Zalian

ganmarteba

zogadia da gamoxatavs im mosazrebas, rom Sefasebebi unda

avagoT SerCevis

saSualebiT da arafers gveubneba, Tu ramdenad

axlosaa

igi generaluri erTobliobis Sesabamis ricxviT maxasiaTeblebTan.

ganvixiloT

Sefaseba

vipovoT

sidide romlis ganawilebis funqcia

parametrs F(x, ). n

Seicavs ucnob saSualebiT

SemTxveviTi

rogorc

ucnobi zemoT

nebismier

moculobis

SerCevis

parametris garkveuli azriT ‘’kargi’’

aRvniSneT,

parametris

Sefaseba

funqcias. cxadia,

SemTxveviTi sidideebi,

amitom

ewodeba TviTonaa

aris SemTxveviTi sidide, romlis

ganawilebis kanoni damokidebulia X

SemTxveviTi sididis ganawilebis

kanonze da SerCevis moculobaze. imisaTvis, rom

parametris Sefaseba iyos praqtikulad Rirebuli, igi

unda akmayofilebdes garkveul moTxovnebs. gansazRvreba. statistikur Sefasebas ewodeba wertilovani Sefaseba, Tu igi ganisazRvreba erTi ricxviT. wertilovan Sefasebebs moeTxoveba Zaldebuloba,

Caunacvlebloba

efeqturoba,

romlebsac

qvemoT

Sefasebas

ewodeba

ganvmartavT. gansazRvreba.

parametris

Caunacvlebeli (gadauadgilebadi), Tu misi maTematikuri lodini SerCevis nebismieri

moculobisaTvis

udris

Sesafasebeli

parametris

mniSvnelobas

M( ) = rogoric ar unda iyos

(1)

WeSmarit

174 parametris

gansazRvreba.

Sefasebas

ewodeba Zaldebuli, Tu nebismieri

ricxvisaTvis sruldeba Semdegi

Tanafardoba: PPP[ gansazRvreba. ufro

]

roca

parametris

efeqturi,

Sefasebas

ewodeba

vidre

< M

efeqturobis

sazomad miRebulia Sefardeba . ganvixiloT infM

yvela SesaZlo

-is mimarT.

Sefasebas

romlisTvisac es qveda sazRvari miiRweva ewodeba efeqturi Sefaseba. parametris

gansazRvreba. asimptoturad

efeqturi

Sefasebas

Sefaseba,

romlisTvisac

sruldeba

ewodeba zRvruli

toloba:

parametris

amrigad, rodesac veZebT ucnobi Sefasebas

unda

gaviTvaliswinoT

zemoT

moyvanili

moTxovnebi:

Sesafaseblad

ganvixiloT

. vTqvaT, mocemulia n

moculobis

Zaldebuloba, Caunacvlebloba da efeqturoba. magaliTi. statistika

ucnobi

maTematikuri

lodinis

). SerCevis ganmartebis

SerCeva generaluri erTobliobidan ( Tanaxmad,

damoukidebeli

erTnairad

ganawilebuli

SemTxveviTi sidideebia. aRvniSnoT:

M( )=a,

D( )=

maSin

M( )=M( D( ) =DD(

)= )

= na=a, =

(1) .

CebiSevis utolobis Tanaxmad (2)-dan miviRebT:E

(2)

175 PP[ roca

]

aqedan cxadia, rom PP[

]

roca

(3)

(1) da (3) tolobebi gviCvenebs, rom SerCevis saSualo ariTmetikuli aris

maTematikuri

lodinis

Zaldebuli

Caunacvlebeli

Sefaseba.

analogiurad vaCvenebT, rom ucnobi dispersiis Sefaseba

aris Zaldebuli da Caunacvlebeli Sefaseba.

5. momentTa meTodi X SemTxveviTi

ganvixiloT Seicavs ucnob maTematikuri

sidide,

romlis

ganawilebis

funqcia

parametrs. f(x, ) iyos ganawilebis simkvrive. maSin, cxadia,

M(X)

lodini

empiriuli saSualo

iqneba

parametris

funqcia:

M(X)=

warmoadgens SerCeviTi mniSvnelobebis

funqcias. ganvixilavT ori saxis moments _ Teoriuls da empiriuls. Teoriuls vuwodebT generaluri erTobliobis WeSmarit moments, xolo empiriuls_ SerCevis safuZvelze gamoTvlil moments. momentTa

meTodis

arsi

mdgomareobs

SemdgomSi:

xdeba

ganawilebis

Teoriuli da empiriuli Sesabamisi momentebis gatoleba, ris Sedegadac miiReba

gantoleba

ucnobi

parametris

amonaxseni

= ,

mimarT:

warmoadgens

ucnobi

romlis

parametris

Sefasebas. Tu SerCevis safuZvelze unda Sefasdes ara erTi, aramed ramdenime, vTqvaT, k- ucnobi

parametri

, maSin unda vipovoT generaluri

ganawilebis pirveli, meore, da a. S. k-uri rigis Teoriuli momentebi: (

(

), ... ,

(

Semdeg vipovoT Sesabamisi empiriuli momentebi: , da gavutoloT isini erTmaneTs:

, ... ,

176 (

(i=1,2,..,k).

(4)

(4) sistemis amonaxseni:

warmoadgens

= (

, (i=1,2,..,k)

ucnobi

parametrTa

statitikur

Sefasebebs

momentTa meTodiT. Mmtkicdeba, rom momentTa meTodiT miRebuli Sefasebebi Zaldebulia, magram ar aris efeqturi da arc asimptoturad efeqturi, miuxedavaT amisa, am meTods, simartivis gamo, gamoviyenebT maSin, roca sxva meTodebiT Zneldeba Sefasebebis povna. magaliTi. vTqvaT, mocemulia n ganawilebis kanonis mqone

moculobis

SerCeva

maCvenebliani

generaluri erTobliobidan

Sesafasebelia misi erTaderTi parametri

rogorc viciT, maCvenebliani

ganawilebis pirveli rigis Teoriul moments aqvs saxe: =

SerCevis anu pirveli rigis empiriuli momenti aris

maTi

gatolebiT miviRebT:

= saidanac

Sefaseba iqneba: =

5. maqsimaluri dasajerobis meTodi SerCevis meTodia

saSualebiT

maqsimaluri

Sefasebis

dasajerobis

povnis

erT-erTi

meTodi,

mniSvnelovani

SemuSavebuli

cnobili

statistikosis fiSeris mier. misi arsi SemdegSi mdgomareobs: vTqvaT

) X SemTxveviTi sidididan,

mocemulia n moculobis SerCeva (

parametrs F(x, ).

romlis ganawilebis funqcia Seicavs ucnob misi

ganawilebis

simkvrive.

SevecadoT

Sefaseba,

vipovoT

romlisTvisac

ganxorcielebis albaToba iqneba maqsimaluri.

f(x, ) iyos

parametris mocemuli

iseTi

SerCevis

177 damoukidebeli Tanaxmad,

albaToba

xdomilobebis

albaTobebis

SemTxveviTi

imisa,

rom

Sedegad miviRebT zustad mocemul f(

)= f(

). f(

sidideze

wesis

dakvirvebis

SerCevas aris )... f(

aRvniSnoT es gamosaxuleba L( L(

gamravlebis

). ) simboloTi:

)... f(

(1)

Tu SemTxveviTi sidide diskretulia, maSin gveqneba: L(

)=P(

). P(

)... P(

(2)

sadac )=P(

), (i=1,2,..,n).

(1) da (2) formulebiT mocemul funqciebs maqsimaluri dasajerobis funqciebi ewodeba. maqsimaluri dasajerobis meTodis arsidan gamomdinare, parametris Sesafaseblad unda aviRoT iseTi L(

romlisTvisac funqcia

) aRwevs maqsimums. rogorc

viciT,

funqcia

parametris

mimarT

aRwevs

maqsimums, rodesac =0 .

(3)

gantolebis amonaxsni iqneba maqsimaluri dasajerobis meTodiT parametris Sefaseba. rogorc maTematikuri analizidanaa cnobili, funqciebis amitom

gamoTvlebis

ukeTesia amovxsnaT

maqsimumis

gamartivebis

wertilebi

mizniT

)

L( emTxveva

erTmaneTs,

gantolebis

nacvlad

Semdegi gantoleba:

=0. Tu Sesafasebelia ara erTi, aramed k maSin maqsimaluri dasajerobis meTodiT

(4)

ucnobi Sefasebebi

parametri, miiReba

Semdegi

gantolebaTa sistemis amoxsniT: = 0, (i=1,2,..,k). maqsimaluri dasajerobis meTodiT miRebul Sefasebebs aqvs (garkveuli azriT) kargi Tvisebebi:

178 dasajerobis meTodiT

maqsimaluri

miRebuli

Sefaseba

aris

Zaldebuli. 2.

parametris

Sefaseba

aris

asimptoturad

efeqturi Sefaseba. 3. maqsimaluri dasajerobis meTodiT miRebuli Sefaseba asimptoturad normaluria. rac imas niSnavs, rom ganawileba n-is

zrdasTan

erTad

-is, rogorc SemTxveviTi sididis, miiswrafvis

normaluri

ganawilebis

kanonisaken. 4. Tu arsebobs

parametris efeqturi Sefaseba, maSin

gantolebas

aqvs erTaderTi amonaxsni, romelic emTxveva am Sefasebas. maqsimaluri dasajerobis meTodiT miRebul Sefasebas aqvs uaryofiTi mxareebic, kerZod, am meTodiT miRebuli Sefasebebi yovelTvis ar aris Caunacvlebadi.

garda

amisa,

dasajerobis

gantoleba

zogjer

rTuli

amosaxsnelia. magaliTi. vTqvaT, bernulis cdaTa sqemaSi Sesafasebelia ’’warmatebis’’ ucnobi p albaToba. maSin (2) formulis Tanaxmad L(

(4) gantoleba miiRebs saxes: = sadanac vpovulobT

–

= 0.

magaliTi. ganvixiloT normaluri ganawilebis a da Sefasebis povnis amocana

parametrebis

maqsimaluri dasajerobis meTodiT.

am SemTxvevaSi (1) da (4) formula miiRebs saxes: L(

;

= _

Sefasebis sapovnelad vipovoT , da amovxsnaT gantolebaTa sistema:

miviRebT:

179 ;

rogorc

vxedavT,

maqsimaluri

dasajerobis

meTodiT

miRebuli

maTematikuri lodinis Sefaseba Caunacvlebadia, xolo dispersiis Sefaseba araa Caunacvlebadi Sefaseba.

6. empiriuli ganawilebis funqcia Cven ganvixileT iseTi amocanebi, rodesac cnobili iyo ganawilebis saxe da ucnobi iyo misi zogierTi parametri (maTematikuri lodini, disersia

sxva).

vakeTebdiT ucnobi

SemTxveviT

sidideze

dakvirvebis

saSualebiT

parametris, garkveuli azriT ‘’kargi’’ Sefasebas.

praqtikaSi xSirad gvxvdeba iseTi amocanebi, rodesac Sesafasebelia ara marto ganawilebis parametrebi, aramed ganawilebis zogadi saxec da igi

unda

dadgindes

SerCevis

safuZvelze,

tipis

amocanebs

araparametruli amocanebi ewodeba. vTqvaT, mocemulia n moculobis SerCeva

(

(1)

SemTxveviTi sidididan, romlis ganawilebis

SevecadoT

(1)

SerCevis

avagoT X

safuZvelze

funqcia ucnobia.

SemTxveviTi

sididis

ganawilebis funqcis Sefaseba. rogorc

viciT,

ganawilebis

funqcia

F(x)=P(X<x) warmoadgens

albaTobas, rom SemTxveviTi sidide miiRebs mniSvnelobas

imis

,x[

intervalidan. aRvniSnoT

–iT (1) SerCevis im mniSvnelobaTa raodenoba,

romlebic ekuTvnis

,x[ intervals. did ricxvTa kanoniT sakmaod

didi n-isaTvis gansazRvreba.

P(X

=] _

) (x)

. funqcias,

romelic

nebismieri

namdvili

ricxvisaTvis gansazRvrulia tolobiT (x)= ewodeba

SemTxveviTi

sididis

miRebuli (1) SerCevis safuZvelze.

(2) empiriuli

ganawilebis

funqcia

SevniSnoT,

rom

(1)

180 SerCevis mniSvnelobebi

(x) Ffunqciac SemTxveviTi

sididebia, maTi saSualebiT gansazRvruli (x)

iqneba.

funqcias,

F(x) funqciis

rogorc

SemTxveviTi

Sefasebas aqvs

Semdegi

Tvisebebi: 1. (x) funqcia, ganawilebis F(x) funqciis aragadadgilebadi Sefasebaa M( (x))= F(x); (x) funqcia, aris ganawilebis F(x) funqciis Zaldebuli Sefaseba

]

P[ X

SemTxveviTi

sididis

roca n

empiriuli

ganawilebis

funqcias

aqvs

ganawilebis funqciis yvela Tviseba:

(- )=0,

2. Tu 3.

(+ )=1; maSin

( )

( ) araklebadia;

(x) funqcia marcxnidan uwyvetia.

7. ndobis intervalebi vTqvaT, mocemulia n moculobis SerCeva

)

(

(1)

X SemTxveviTi sidididan, romlis ganawilebis ucnob

parametrs. aqamde ucnobi

funqcia F(x, ) Seicavs

parametris Sefasebas vaxdendiT erTi

garkveuli ricxviT, romelsac wertilovan Sefasebas vuwodebT. Tu cdaTa ricxvi

didia,

xolo

wertilovani

Sefaseba

aragadadgilebadobiT da ZaldebulobiT, maSin Sefaseba Sesafasebel

xasiaTdeba ‘’kargad cvlis’’

parametrs.

rodesac cdaTa ricxvi arcTu ise didia,

parametris SemTxveviTi

xasiaTidan gamomdinare, SeiZleba miviRoT didi cdomiloba. am SemTxvevaSi ufro

mosaxerxebelia

(1)

SerCevis

intervali, rom sakmaod didi moTavsebulia = (

safuZvelze

iseTi

]

albaTobiT SegveZlos vTqvaT, rom

intervalSi,

) ramdenadac

avagoT

= (

)

SemTxveviTi sidideebia, xolo

fiqsirebuli

181 amitom SeiZleba

ricxvia,

xdomilobis albaTobaze xolo

intervals

intervali,

[

-s ewodeba ndobis albaToba,

albaTobis Sesabamisi ndobis intervali.

ganawilebis ucnobi ]- ,+ [

vilaparakoT

parametris yvelaze uxeS Sefasebas warmoadgens 1-is

toli

albaTobiT

SegviZlia

vTqvaT,

rom

]- ,+ [, magram, cxadia, aseTi Sefaseba uvargisia, radgan igi ar iZleva araviTar informacias

parametris WeSmarit mniSvnelobaze. sasurvelia

ndobis intervali ise aigos, rom misi sigrZe iyos rac SeiZleba mcire, xolo

ndobis albaToba, rac SeiZleba didi. rogorc wesi, orTave

amocanis

erTdroulad

gadawyveta

Semdegnairad: winaswar irCeven

SeuZlebelia,

amitom

iqcevian

ndobis albaTobas ise, rom igi axlos

iyos erTTan da eZeben mis Sesabamis umciresi sigrZis ndobis intervals. ndobis intervalis agebis procesi ganvixiloT

konkretul magaliTze.

amasTan davuSvebT, rom X aris normalurad ganawilebuli SemTxveviTi sidide a da

parametrebiT.

vTqvaT, vatarebT raRac fizikuri sididis erTmaneTisgan damoukidbel gazomvebs.

iTvleba,

rom

gazomvis

cdomilobebi

ganawilebulia

normalurad, amitom gazomvis Sedegic ganawilebulia normalurad. Tu adgili ar aqvs sistematur cdomilobas SegviZlia vTqvaT, rom M(X)=a. zemoTqmulidan gamomdinare, gazomvaTa Sedegebis damuSavebis ZiriTadi amocana ’’gasazomi sididis WeSmariti mniSvnelobis dadgena’’ maTematikurad Camoyalibdeba, rogorc normalurad ganawilebuli SemTxveviTi sididis maTematikuri lodinis Sefasebis amocana. am amocanis amoxsnas didi n-saTvis iZleva empiriuli saSualo: =

Tu gazomvaTa ricxvi n arcTu ise didia cdomileba

da a-s Soris

SeiZleba sakmaod didi iyos, amitom saWiroa aigos iseTi ]

[ intervali,

ndobis albaTobiT moTavsebuli iqneba a ricxvi.

romelSic mocemuli ndobis cnobilia.

intervalis 2. roca

1. vTqvaT SemTxveviT

ageba

ganvixiloT

SemTxvevaSi:

roca

ucnobia. cnobilia,

sidideTa

jami

viciT

aseve

rom

normalurad

ganawilebulia

ganawilebuli

normalurad,

radgan

182 sidideebi normaluradaa ganawilebuli a da

SemTxveviTi parametrebiT.

SemTxveviTi sididec ganawilebulia normalurad a da

= parametrebiT.

ganvixiloT normirebuli SemTxveviTi sididide

(1)

romelic ganawilebulia normalurad parametrebiT 0 da 1, amitom mocemuli

-saTvis vipovoT iseTi

, rom Sesruldes toloba

= ,

(2)

rogorc viciT, albaToba

= sadac

= 2

_1,

(3)

(X) aris standartuli normaluri ganawilebis funqcia,

(X)= (-X).

-s mosaZebnad unda amovxsnaT gantoleba 2

+1 anu

(

romelic sakmao miaxloebiT ixsneba

funqciis cxrilis

saSualebiT (ix.danarTi 2 ). gadavweroT (2) Semdegi saxiT:

= anu

P[ --

<a< +

]= .

(4)

es niSnavs imas, rom a maTematikuri lodinis

ndobis albaTobis

Sesabamisi ndobis intervali aris

-rogorc

(5)-dan

gamomdinareobs,

[ . ndobis

(5) intervalis

sigrZe

damokidebulia mxolod SerCevis moculobaze, amasTan, cdaTa ricxvis zrdasTan erTad mcirdeba. intervalis centri moTavsebulia 2. axla ganvixiloT SemTxveva, roca gamosaxulebaSi Sedis ori ucnobi _a da

wertilSi.

ucnobia. am SemTxvevaSi parametri. Tu

nacvlad am gamosaxulebaSi SevitanT mis Sefasebas

(1)

parametris

183 =

miviRebT

U= mtkicdeba, ganawileba

rom

U SemTxveviT

Tavisuflebis

(6)

sidides

xarisxiT,

gaaCnia

romlis

e.w.

stiudentis

ganawilebis

simkvrivea

. imisaTvis, rom avagoT a maTematikuri lodinis

ndobis albaTobis

Sesabamisi ndobis intervali, unda vipovoT iseTi

, rom Sesruldes

toloba:

= anu amovxsnaT gantoleba: =

(7)

ndobis intervali maTematikuri lodinisaTvis

ndobis albaTobiT

iqneba:

[ .

8. statistikuri hipoTezebi vTqvaT, mocemulia erTgvarovan obieqtTa generaluri erToblioba. X iyos SemTxveviTi sidide, romelic gamoxatavs generaluri erTobliobis obieqtTa ucnobi

raime

F(x)

niSan_Tvisebas. ganawilebis

gvainteresebs funqciis

saxis

SemTxveviTi

dadgena.

sididis

generaluri

erTobliobis bunebidan, SeiZleba gvqondes winaswari varaudi (hioTeza), rom

ganawilebis

funqcias

aqvs

raRac

ganawilebis funqcia Seicavs ucnob SeiZleba gamovTqvaT HhipoTeza

F(x)

saxe.

analogiurad,

parametrs, garkveuli mosazrebiT

, sadac

cnobili sididea.

hipoTezas vuwodoT statistikuri, Tu is exeba ucnobi

ganawilebis

funqciis saxes an ucnob parametrs. bunebrivia,

daSvebul

hipoTezasTan

erTad

ganvixiloT

misi

sawinaaRmdego hipoTezac. Tu Semowmebis Sedegad daSvebuli hipo痺容za ar gamarTlda, maSin adgili eqneba mis sawinaaRmdegos.

daSvebul

184 hipoTezas ewodeba nulovani da aRiniSneba

(ZiriTad)

simboloTi.

misgan

gansxvavebul

alternatiuli da aRiniSneba

nebismier

hipoTezas

ewodeba

simboloTi.

magaliTad, Tu nulovani hipoTeza mdgomareobs imaSi, rom normalurad ganawilebuli SemTxveviTi sididis maTematikuri lodini a=0, maSin misi erT-erTi

alternatiuli

Caiwereba:

: a=0;

hipoTeza

iqneba

a 0. es

faqti

mokled

ase

: a 0.

wess, romelic gansazRvravs pirobebs, romlis drosac Sesamowmebel hipoTezas miviRebT an uarvyofT ewodeba statistikuri kriteriumi. cxadia, hipoTezis

Semowmeba

vRebulobT

SerCevidan.

SemTxveviTi SerCevis

xdeba e.i.

monacemebis

statistikuri

safuZvelze,

kriteriumi

romelsac

adgens

wess,

ra monacemebis dros miiReba mocemuli hipoTeza da

ra monacemebis dros ara. statistikuri hipoTezebis Semowmebis dros SeiZleba daSvebul

iqnes

ori tipis Secdoma anu, rogorc maT uwodeben, pirveli an meore gvaris Secdoma. Secdoma pirveli gvarisaa, rodesac WeSmariti xolo meore gvarisaa, rodesac mcdari hipoTezis

Sesamowmeblad

hipoTeza mcdarad,

hipoTeza CaiTvleba WeSmaritad.

avagoT

statistikuri

kriteriumi

Semdegnairad: 1.

SemoviRoT

specialurad

SerCeuli

SemTxveviTi

sidide,

romlis

zusti an zRvruli ganawileba cnobilia. cxadia, es sidide SerCevis monacemebis funqciaa. aRvniSnoT igi K simboloTi (mas Cven SemdgomSi gavaigivebT

statistikur

ganvsazRvroT, miviRebT

kriteriumTan).

misi

saSualebiT

unda

hipoTezas Tu ukuvagdebT mas.

2. davafiqsiroT pirveli gvaris Secdomis albaToba

, romelsac

mniSvnelobis done ewodeba. mniSvnelobis done es is sazRvaria, romlis gadalaxvis Semdeg gansxvaveba SerCevis monacemebsa da nulovan hipoTezas Soris arsebiTia, e.i. monacemebi nulovani hipoTezis winaaRmdegia. 3. SemoviRoT

-s mimarT alternatiuli hipoTeza

4. davadginoT W kritikuli are anu statistikuri kriteriumis im namdvil

mniSvnelobaTa

simravle,

romelTa

miRebisas

hipoTeza

ukuigdeba. kriteriumis im mniSvnelobaTa simravle, romelebzec

hipoTezas

185 vuwodoT dasaSveb

miviRebT,

wertils, romelic

mniSvnelobaTa

are.

yofs kritikul da dasaSveb mniSvnelobaTa areebaá&#x192;&#x201C;,

kritikuli wertili ewodeba. zemoT

moyvanili

oTxi

punqti

saSualebas

gvaZlevs

avagoT

statistikuri kriteriumi. praqtikulad, SeiZleba Tavidanve davafiqsiroT (Cveulebrivad

pirveli gvaris Secdomis albaToba

=0,1; 0,05; 0,02; 0,01).

yvela praqtikulad mniSvnelovani kriteriumisaTvis Sedgenilia cxrilebi da P(K W)=1

tolobidan SeiZleba

kritikuli wertilis dadgena. Cven

vagebT kritikul ares im mosazrebidan, rom nulovani hipoTezis dros kriteriumis kritikul areSi moxvedris albaTobaa

. sasurvelia vicodeT,

ras udris albaToba imisa, rom kriteriumi moxvdeba kritikul areSi, roca samarTliania alternatiuli hipoTeza. kriteriumis

simZlavre,

sxva

sityvebiT,

vuwodoT am albaTobas

kriteriumis

simZlavre

aris

albaToba imisa, rom nulovan hipoTezas ukuvagdebT, Tu samarTliania alternatiuli hipoTeza. vTqvaT, simZlavre

meore iqneba

gvaris 1-

Secdomis

aqedan

Cans,

albaTobaa rom

rac

maSin

kriteriumis

ufro

metia

kriteriumis

simZlavre, miT ufro mcirdeba meore gvaris albaToba

amrigad, albaToba imisa, rom kriteriumi Cavardeba kritikul areSi, roca samarTliania

hipoTeza, udris

-s da amave dros albaToba imisa, hipoTeza,

rom kriteriumi Cavardeba kritikul areSi, roca samarTliania

unda iyos maqsimaluri. am or pirobas ewodeba kriteriumis simZlavris maqsimizaciis postulati, rac analizurad ase Caiwereba:

P(K W statistikur

kriteriums,

)= ; P(K W

romelic

)=max.

akmayofilebs

(1) am

postulats,

ewodeba umZlavresi kriteriumi. rogorc vnaxeT, rac ufro mcirea

, miT ufro kargi kritikuli

are gvaqvs, magram Tu SerCevis n moculoba fiqsirebulia, erTdrouli Semcireba SeuZlebelia, radgan Tu

-s

-s SevamcirebT, maSin

gaizrdeba. da

-s erTdrouli Semcirebis erTaderTi gza SerCevis moculobis

gazrdaa, romelic, Tavis mxriv, siZneleebTan aris dakavSirebuli.

kriteriumis

186 ageba statistikur hipoTezaTa Semowmebis

erT-erTi ZiriTadi amocanaa.

umZlavresi kriteriumebis agebis ararTuli

umZlavresi

meTodebi

dadgenilia

mxolod

martivi

hipoTezebisaTvis.

hipoTezas

ewodeba martivi, Tu igi calsaxad gansazRvravs ganawilebis funqcias, winaaRmdeg SemTxvevaSi hipoTezas ewodeba rTuli. magaliTad, hipoTeza imisa, rom normaluri ganawilebuli SemTxveviTi X sididis maTematikuri lodini udris nuls, xolo dispersia_erTs martivi hipoTezaa, radgan igi calsaxad

gansazRvravs

SemxveviTi

sididis

ganawilebis

funqcias.

hipoTeza, normalurad ganawilebuli SemxveviTi X sididis maTematikuri lodini udris nuls,

dispersia nebismeri

dadebiTi ricxvia, rTuli

hipoTezaa. im SemTxvevaSi, roca

hipoTezebi martivia, adgili aqvs neiman-

pirsonis Teoremas. Teorema. Tu ZiriTadi hipoTeza martivebia,

Sesabamisad,

da alternatiuli

= ;

hipoTeza

) warmoadgenen dasajerobis funqciebs, gamoTvlils

hipoTezebisaTvis,

xolo

SemTxveviTi

sidide,

romlidanac

da aRebulia

SeCeva, uwyvetia, maSin arsebobs kriteriumi, romelic aris umZlavresi hipoTezisaTis

hipoTezis

mimarT.

kritikuli

are

TviTon

kriteriumi ganisazRvreba utolobiT

L( sadac

dadebiTi

mniSvnelovnebis

) ricxvia,

L( romlis

), mniSvneloba

damokidebulia

doneze.

magaliTi. vipovoT umZlavresi kritriumi gulisxmobs,

(1)

rom

normalurad

maTematikuri lodini aris

dispersia cnobilia da udris

ganawilebuli

hipoTezisaTis, romelic SemTxveviTi

alternativa gveubneba, rom

X sididis , xolo

, xolo amokrefis moculobaa n.

amoxsna. SevadginoT dasajerobis funqciebi, roca maTematikuri lodini aris Sesabamisad

da L(

187 CavsvaT es gamosaxulebebi (1)-

Si, miviRebT:

C me-2

utolobis

orive

(2)

mxaris

galogariTmebiT

garkveuli

gardaqmnebiT miviRebT: (

_n(

)

) =n

Tu ukanasknelSi gaviTvaliswinebT, rom

da amovxsniT

mimarT miviRebT:

amgvarad,

neiman_pirsonis

Teoremis

ZaliT,

vipoveT

kriteriumi

ganvsazRvreT kritikuli are: marTlac, kriteriumi unda aviRoT amokrefiT ; Tu

saSualo mniSvneloba,

>0, maSin kritikul areSi moxvdeba

romelic

gadaaWarbebs

kritikul areSi Cavardeba ricxvze.

ricxvs.

-is yvela is

<0,

maSin

-is yvela is mniSvneloba romelic naklebia BB

AA da B ricxvebi unda SevarCioT ise, rom Sesruldes (1)

toloba.

9. parametrul ganvixiloT

hipoTezaTa Semowmeba

kriteriumi,

romelic

miekuTvneba

parametrul

kriteriumebs, e.i. iseT kriteriumebs, romlebic ganixilaven hipoTezebs ganawilebis

ucnobi

parametrebis

Sesaxeb,

roca

ganawilebis

saxe

cnobilia. davuSvaT,

moculobis

SerCeva

normalurad

maTematikuri

lodini

ganawilebuli

SemTxveviTi

sidididan,

dispersia

cnobilia. maSin, rogorc viciT, SerCeviT saSualos =

romlis

xdeba

ucnobia,

(1)

xolo

agreTve

188 normaluri ganawileba

aqvs

vTqvaT, Sesamowmebelia hipoTeza :

parametrebiT

alternatiuli hipoTeziT

. statistikur kriteriumad miviRoT SemTxveviTi sidide: .

(2)

Tu mniSvnelovnebis done aris

, maSin kritikuli are, romelic meore

gvaris Secdomas minimalurs gaxdis, moiZebneba

sididis mniSvnelobis

mixedviT. Tu

, maSin kritikuli are iqneba marjvniv. e. i. mas ekuTvnis (2)

kriteriumis yvela is mniSvneloba, romelic gadaaWarbebs iseTia, rom sruldeba utoloba

wertils. Tavis mxriv,

. aqedan,

kritikul

kritikuli

aris

dadgena

(3) siZneles

warmoadgens

visargeblebT standartuli normaluri ganawilebis (parametrebiT (0,1) ) cxrilebiT. Tu

, maSin kritikuli are iqneba marcxniv, e. i. mas ekuTvnis (2)

kriteriumis

yvela

wertilze, sadac

mniSvneloba,

romelic

naklebia

kritikul

moiZebneba analogiurad.

magaliTi. vTqvaT, generaluri erTobliobis ganawileba normaluria, . SerCevis moculoba alternatiuli

hipoTeziT

. Sesamowmebelia hipoTeza :

pirvel

rigSi

, davafiqsiroT

. vTqvaT, SerCeviTi saSualo

mnSvnelobis done

radgan

, amitom avagoT marjvena kritikuli are. (3) tolobis safuZvelze da normaluri

ganawilebis

kritikul ares ekuTvnis

cxrilebiT

vpoulobT

amitom

kriteriumis yvela is mniSvneloba, romelic

gadaaWarbebs 1,65-s. Cven SemTxvevaSi kriteriumis mniSvneloba

radganac es mniSvneloba kritikuli aris gareTaa, amitom ara gvaqvs safuZveli uarvyoT

hipoTeza.

189 ganvixiloT SemTxveva, rodesac normalurad

ganawilebuli

moculobis

SemTxveviTi

SerCeva

xdeba

romlis

orive

sidididan,

parametri, maTematikuri lodini da dispersia ucnobia. am SemTxvevaSi statistikur kriteriumad miviRoT SemTxveviTi sidide: . sadac

aris

(4)

-s Sefaseba: =

mtkicdeba,

rom

(4)

sidides

aqvs

stiudentis

ganawileba

Tavisuflebis xarisxiT. kritikuli aris arCevis principi igivea, rac wina magaliTSi

gamoviyenebT

gansxvavebiT, stiudentis

rom

kritikuli

ganawilebis

wertilis

cxrils

dasazusteblad Tavisuflebis

xarisxiT. magaliTi.

mocemulia

ganawilebuli maTematikuri

=26

SemTxveviTi lodini

moculobis

sidididan,

davafiqsiroT monacemebiT marcxena

Tavisuflebis

ares.

xarisxi

orive

parametri,

ucnobia.SevamowmoT

done:

. vTqvaT,

kritikul

normalurad

hipoTeza

mniSvnelobis

romlis

dispersia

Tu sawinaaRmdegoa

SerCeva

stiudentis

26-1=25,

gamovTvaloT

. radganac ganawilebis

SerCevis , vagebT

cxrilebiT,

vpoulobT:

roca amitom

kritikul ares ekuTvnis kriteriumis mniSvnelibaTa simravle, romlebic naklebia -2,485-ze. Cven SemTxvevaSi:

e. i. kriteriumis mniSvneloba aRmoCnda kritikul areSi, es niSnavs, rom unda ukuvagdoT

hipoTeza da miviRoT alternatiuli hipoTeza. savarjiSo magaliTebi

SemTxveviT

sidideze

dakvirvebis

Sedegebi

mocemulia

Semdegi

cxrilis saxiT:

N 1

X 10

13 16 12 1

8 9

10 11 12 13 14 15 16 17 18 19 20

15 6

14 7

15 8

16 20 15 10 14 11

190 davyoT (0; 20) Sualedi 4 tol nawilad

SevadginoT

statistikuri ganawilebis cxrili. 2)

SemTxveviT sidideze dakvirvebis Sedegebi mocemulia Semdegi

cxrilis saxiT: N

10 11 12 13 14 15 16 17 18 19 20

10 4

aageT statistikuri ganawilebis cxrili. 3) ipoveT Semdegi statistikuri mwkrivis saSualebiT mocemuli SerCevis empiriuli ganawilebis funqcia: 2

4) SemTxveviT SerCeuli 100 studentis simaRlis gazomvis Sedegebi aRmoCnda Semdegi: simaRle

154158

rao-ba

156-

162-

162 10

166-

166 14

170 26

170-

174-

174 28

178-

178 12

182 8

ipoveT SerCeviTi saSualo da SerCeviTi dispersia. 4) tranzistoris

erT-erTi

parametris

Semowmebam

mogvca

Semdegi

Sedegebi: nomeri 1

mniS-ba 4,40

4,31

4,40

4,65

4,71

4,54

4,34

4,56

ipoveT SerCeviTi saSualo mniSvneloba daSerCeviTi dispersia. SemTxveviTi sidides gaaCnia binomuri ganawileba.

6) cnobilia, rom momentTa

meTodis

saSualebiT

SevafasoT ucnobi warmatebis

(

SerCevaze

dafuZnebiT

albaToba, Tu:

a) b) 7) cnobilia, rom

SemTxveviTi sidide ganawilebulia puasonis kanoniT

, sadac

ucnobi parametria. maqsimaluri dasajerobis

meTodis saSualebiT vipovoT SerCevaze dafuZnebuli Sefaseba, Tu SerCevas aqvs Semdegi saxe:

ucnobi parametris

191 a) 14, 13, 17, 15, 20, 25, 13, 22. b) 12, 14, 9, 8, 15, 7, 11, 8. 8) mocemul nivTierebaSi rkinis Semcvelobaze xangrZlivi dakvirvebis Sedegad

dadginda

albaTobiT,

standartuli

nivTierebaSi

rkinis

gadaxra

0,12%.

Semcvelobis

ipoveT

ndobis

intervali,

0,95 Tu

analizis Sedegad aRmoCnda, rom saSualo Semcvelobaa 32,56%. 9)

naTurebis

didi

partiidan

alalbedze

SearCies

100

naTura.

SerCevidan aRebuli naTurebis naTebis saSualo xangrZlivoba aRmoCnda 1000 sT.

ipoveT

naTurebis

mTeli

partiis

saSualo

naTebis

drois

ndobis

intervali, ndobis 0,95 albaTobiT, Tu cnobilia, rom naTuris naTebis saSualo kvadratuli gadaxra

=40 sT-s.

192 danarTi1.

funqciis mniSvnelobaTa cxrili

0.0

0.3989

3989

3988

3986

3984

3982

3980

3877

3973

0.1

3970

3965

3961

3956

3951

3945

3939

3932

3925

3918

0.2

3910

3902

3894

3885

3876

3867

3857

3847

3836

3825

0.3

3814

3802

3790

3778

3765

3752

3739

3726

3712

3697

0.4

3683

3668

3653

3637

3621

3605

3589

3572

3555

3538

0.5

3521

3503

3485

3467

3448

3429

3410

3391

3372

3352

0.6

3332

3312

3292

3271

3251

3230

3209

3187

3166

3144

0.7

3123

3101

3079

3056

3034

3011

2989

2966

2943

2902

0.8

2897

2874

2850

2827

2803

2780

2756

2732

2709

2685

0.9

2661

2637

2613

2589

2565

2541

2516

2492

2468

2444

1.0

0.2420

2396

2371

2347

2323

2299

2275

2251

2227

2203

1.1

2179

2155

2131

2107

2083

2059

2036

2012

1989

1965

1.2

1942

1919

1895

1872

1849

1826

1804

1781

1758

1736

1.3

1714

1961

1669

1647

1626

1604

1582

1561

1539

1518

1.4

1497

1476

1456

1635

1415

1394

1374

1354

1334

1315

1.5

1295

1276

1257

1238

1219

1200

1182

1163

1145

1127

1.6

1109

1092

1074

1057

1040

1023

1006

0989

0973

0957

1.7

0940

0925

0909

0893

0878

0863

0848

0833

0818

0804

1.8

0790

0775

0761

0748

0734

0721

0707

0694

0681

0669

1.9

0656

0644

0632

0620

0608

0596

0584

0573

0562

0551

2.0

0.0540

0529

0519

0508

0498

0488

0478

0468

0459

0449

2.1

0440

0431

0422

0413

0404

0396

0387

0379

0371

0363

2.2

0355

0347

0339

0332

0325

0317

0310

0303

0297

0290

2.3

0283

0277

0270

0264

0258

0252

0246

0241

0235

0229

2.4

0224

0219

0213

0208

0203

0198

0194

0189

0184

0180

2.5

0175

0171

0167

0163

0158

0154

0151

0147

0143

0139

2.6

0136

0132

0129

0126

0122

0119

0116

0113

0110

0107

2.7

0104

0101

0099

0096

0093

0091

0088

0086

0084

0081

2.8

0079

0077

0075

0073

0071

0069

0067

0065

0063

0061

2.9

0060

0058

0056

0055

0053

0051

0050

0048

0047

0046

3.0

0.0044

0043

0042

0040

0039

0038

0037

0036

0035

0034

3.1

0033

0032

0031

0030

0029

0028

0027

0026

0025

3.2

0024

0023

0022

0021

0020

0019

0018

3.3

0017

0016

0015

0014

0013

3.4

0012

0011

0010

0009

3.5

0009

0008

0007

0006

3.6

0006

0005

0004

3.7

0004

0003

3.8

0003

0002

3.9

0002

0001

193 funqciis mniSvnelobaTa cxrili

danarTi 2. x

0.0

0.0000

0040

0080

0120

0159

0199

0239

0279

0319

0359

0.1

0398

0438

0478

0517

0557

0596

0639

0675

0714

0753

0.2

0793

0832

0871

0909

0948

0987

1026

1103

1064

1141

0.3

1179

1217

1255

1293

1331

1386

1406

1443

1480

1517

0.4

1554

1591

1628

1664

1700

1736

1772

1808

1844

1879

0.5

1915

1950

1985

2019

2054

2088

2123

2157

5190

2224

0.6

2257

2291

2324

2356

2389

2421

2454

2486

2517

2549

0.7

2580

2611

2642

2673

2703

2734

2764

2793

2823

2852

0.8

2881

2910

2939

2967

2995

3023

3051

3078

3106

3133

0.9

3159

3186

3212

3238

3264

3289

3315

3340

3365

3389

1.0

3413

3437

3461

3485

3508

3531

3554

3577

3599

3621

1.1

3643

3665

3686

3708

3728

3749

3770

3790

3810

3830

1.2

3849

3869

3888

3906

3925

3943

3962

3980

3997

4015

1.3

4032

4049

4066

4082

4099

4115

4131

4147

4162

4177

1.4

4192

4207

4222

4236

4251

4265

4279

4292

4306

4319

1.5

4332

4345

4357

4370

4382

4394

4406

4418

4429

4441

1.6

4452

4463

4474

4484

4495

4505

4515

4525

4535

4545

1.7

4554

4564

4573

4582

4591

4599

4608

4616

4625

4633

1.8

4641

4648

4656

4664

4671

4678

4686

4692

4699

4706

1.9

4713

4719

4726

4732

4738

4744

4750

4756

4761

4767

2.0

4772

4778

4783

4788

4793

4798

4803

4808

4812

4817

2.1

4821

4826

4830

4834

4838

4842

4846

4850

4854

4857

2.2

4861

4864

4868

4871

4874

4878

4881

4884

4887

4890

2.3

4893

4896

4898

4901

4904

4906

4909

4911

4913

4916

2.4

4918

4920

4922

4924

4927

4929

4930

4932

4934

4936

2.5

4938

4940

4941

4943

4945

4946

4948

4949

4951

4952

2.6

4953

4955

4956

4957

4958

4960

4961

4962

4963

4964

2.7

4965

4966

4967

4968

4969

4970

4971

4972

4973

4974

2.8

4974

4975

4976

4977

4978

4979

4980

4981

2.9

4981

4982

4983

4984

4985

4986

3.0

0.4986

3.1

4990

3.2

49931

3.3

49952

3.4

49966

3.5

4998

3.6

4998

3.7

49989

3.8

49993

3.9

49995

4.0

499968

4.5

499997

5.0

49999997

194 danarTi 3.

ganawilebis

kritikuli wertilebi

Tavisuflebis

mniSvnelobis done

xarisxi n

00.1

0.025

0.05

0.95

6.6

5.0

3.8

0.0039

0.00098

0.00016

9.2

7.4

6.0

0.103

0.051

0.020

11.3

9.4

7.8

0.352

0.216

0.115

13.3

11.1

9.5

0.711

0.484

0.297

15.1

12.8

11.1

0.15

0.831

0.554

16.8

14.4

12.6

0.64

1.24

0.872

18.5

16.0

14.1

2.17

1.69

1.24

20.1

17.5

15.5

2.73

2.18

1.65

21.7

19.0

16.9

3.33

2.70

2.09

23.2

20.5

18.3

3.94

3.25

2.56

24.7

21.9

19.7

4.57

3.82

3.05

26.2

23.3

21.0

5.23

4.40

3.57

27.7

24.7

22.4

5.89

5.01

4.11

29.1

26.1

23.7

6.57

5.63

4.66

30.6

27.5

25.0

7.26

6.26

5.23

32.0

28.8

26.3

7.96

6.91

5.81

33.4

30.2

27.6

8.67

7.56

6.41

34.8

31.5

28.9

9.39

8.23

7.01

36.2

32.9

30.1

10.1

8.91

7.63

37.6

34.2

31.4

10.9

9.59

8.26

38.9

35.5

32.7

11.6

10.3

8.90

40.3

36.8

33.9

12.3

11.0

8.54

41.6

38.1

35.2

13.1

11.7

10.2

43.0

39.4

36.4

13.8

12.4

10.9

44.3

40.6

37.7

14.6

13.1

11.5

45.6

41.9

38.9

15.4

13.8

12.2

47.0

43.2

40.1

16.2

14.6

12.9

48.3

44.5

41.3

16.9

15.3

13.6

49.6

45.7

42.6

17.7

16.0

16.3

50.9

47.0

43.8

18.5

16.8

15.0

0.975

0.89

195 literatura 1. p. zeragia, umaRlesi maTematikis kursi 1-2 tomi. gam."ganaTleba", Tbilisi 1972w 2. T. Tofuria, umaRlesi maTematikis kursi 1-2 tomi. gam."ganaTleba", Tbilisi 1975w. 3. Слободская В.А., Краткий курс высшей математики. Издательство ,,Высшая школа,, Москва 1962г. 4. Глаголев А.А. , Солнцева Т.В. Курс высшей математики. изд. ,. Высшая школа., Москва 1971г . 5. Richard L. Faber, Marvin I. Fridman, James L. Kaplan., Applied calculus. " West publishing company". New york,1986. 6. Смирнов С.И., Курс высшей математики. т. 1-2.Москва, ,,Наука,, 1974 г. 7. Фихтенгольц Г.М., Курс дифференцифльного и интегрального исчисления. т. 1-2. Москва, ,,Наука,,. 1770 г. 8. Задачи и упражнения по математическому анализу. Под редакцией Б.П. Демидовича. Москва, ,,Наука,,. 1772 г. 9. Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen. Calculus for Business, Economics, Life Sciences & Social Sciences (11th Edition) (2007) 10. b. doWviri, albaTobis Teoria da maTematikuri statistika. Tbilisi: Tsu gamomcemloba, 1984. 11. h. krinski, maTematika ekonomistebisaTis, Tsu gamomcemloba, 1974. 12. n. lazrieva, m. mania, g. mari, a. mosiZe, a. toronjaZe, T. toronjaZe, T. ServaSiZe, albaTobis Teoria da maTematikuri statistika ekonomistebisaTis. Tbilisi: fondi ,,evrazia’’, 2000. 13. g. mania,

albaTobis Teoria da maTematikuri statistika. Tbilisi: Tsu

gamomcemloba, 1976. 14. i.sxirtlaZe, T. tuRuSi, a.osiZe, a. civaZe, m, nadareiSvili, albaTobis Teoria da maTematikuri statistika. Tbilisi: “ganaTleba”, 1990. 15.g.

mania,

anTelava,

ediberiZe,

albaTobis

Teoria

maTematikuri

statistikis amocanaTa krebuli. Tbilisi: Tsu gamomcemloba, 1980. 16. g. mari, a. mosiZe, z. cigroSvili, statistika. Tbilisi: ESM ,1996.

196 17.T. ServaSiZe, albaTobis Teoria. (leqciaTa

kursi)

Tbilisi:

Tsu

gamomcemloba, 1980.

18.Боровков А.А., Теория вероятностей, Москва „Наука“ 1976. 19.Боровков А.А., Математическая статистика, Москва „Наука“ 1984. 20. Гнеденко Б.В. Курс теория вероятностей, Москва „Наука“ 1988.

21. Дубин-Барковский И.В., Смирнов Н. А., Курс теория вероятностей и математической статистики для технических приложений. Москва „Наука“ 1980. 22. Крамер Г., Математические методы статистики. Москва ,,Мир“ 1975. 23. Шириаев А.Н., Вероятность, Москва „Наука“ 1989.

ibeWdeba avtorTa mier warmodgenili saxiT

gadaeca warmoebas 28.05.2009. xelmowerilia dasabeWdad 15.07.2009. qaRaldis zoma 60X84 1/8. pirobiTi nabeWdi Tabaxi 12. tiraJi 100 egz.

sagamomcemlo saxli `teqnikuri universiteti~, Tbilisi, kostavas 77