Matematika Ekonomistebisatvis
revaz danelia,
nana CxaiZe, Tamar kvaracxeliadadada
maTematika ekonomistebisaTvis
s a x e l m Z R v a n e l o
bakalavris mosamzadeblad saqarTvelos saxelmwifo
sasoflo-sameurneo universitetis
ekonomikur-humanitaruli fakultetis
studentebisaTvis
Tbilisi
2008w.
umaRlesi maTematikis winamdebare saxelmZRvanelo Sedgenilia saqarTvelos
saxelmwifo sasoflo-sameureneo universitetis ekonomikur-humanitaruli fakultetis
bakalavrebisaTvis amJamad moqmedi programis mixedviT. naSromSi umaRlesi maTematikis
amocanebisa da savarjiSoebis gverdiT mocemulia martivi amocanebi ekonomikis
sferodan da yoveli maTgani dawvrilebiTaa amoxsnili. amasTan dakavSirebiT wignSi
gadmocemulia im ekonomikuri cnebebis gansazRvrebebi, romelic saWiroa ekonomikuri
amocanebis gamosakvlevad.
recenzentebi:fizika-maTematikis mecnierebaTa doqtori,
profesori roland omanaZe
ekonomikis akademiuri doqtori,
sruli profesori nugzar iosebaSvili
redaqtori: fizika-maTematikis mecnierebaTa doqtori,
sruli profesori jemal rogava
2
winasityvaoba
Tanamedrove ekonomistebisaTvis da agreTve sxva dargebis specialistebisaTvis
aucilebelia maTematikis zogierTi dargebis arazedapiruli codna. sabazro
ekonomikis pirobebSi ekonomikur specialobebze maTematikis swavleba axlebur
midgomas moiTxovs. ukan unda movitovoT am specialobebze maTematikis swavlebis
abstraqtuloba, romlis drosac srulad ar aris gaTvaliswinebuli ekonomikuri
mecnierebis Taviseburebani, radgan igi xSirad Rebulobda Sablonuri Sinaarsis
amocanebis amoxsnaSi erTferovani varjiSis xasiaTs, rasac SesaZloa raRac formaluri
Cvevebis ganviTarebisaken ki mivyavdiT, magram xels ar uwyobda Tavisufali azrovnebis
ganviTarebas. saTanado yuradReba ar eqceoda maTematikaSi miRebuli Sedegebis
gamoyenebas. arada ekonomikaSi aseTi SesaZleblobani uamravia.
saqarTvelos saxelmwifo sasoflo-sameurneo universitetis ekonomikis
fakultetis studentebi dRemde moklebulni arian umaRlesi maTematikis
saxelmZRvanelos mSobliur enaze. aseTi saxelmZRvanelos saWiroeba ki didi xania
momwifda. saxelmZRvanelos Sedgenisas vxelmZRvanelobdiT ekonomikur fakultetze
(bakalavriati) amJamad moqmedi programiT, sadac umaRlesi maTematikis kurss
(praqtikulis CaTvliT) eTmoba 140 saaTi. aqedan gamomdinare saxelmZRvaneloSi
mocemuli masalis gadmocemisas avtorebi umTavresad eyrdnobian geometriul mxares
(TvalsaCinoebas) da mkiTxvelis intuicias. sakiTxebis mkacr logikur dasabuTebas,
rogorc es tradiciul sauniversiteto kursebSia mocemuli Cven ar mivmarTavT.
wignSi mocemulia amoxsnili amocanebi da savarjiSoebi. Cveni rCevaa:
studentma pirveli dakvirvebuli wakiTxvis Semdeg damoukideblad amoxsnas igive
amocana Tu savarjiSo, ris Sedegadac igi daeufleba praqtikul saqmianobaSi
wamoWril ama Tu im sakiTxis damoukideblad gadawyvetis unar-Cvevebs.
avtorebi did madlierebis grZnobiT miiReben yvela im SeniSvnas, romelic
waadgeba naSromis daxvewas.
3
Tavi I. simravleTa Teoria. maTematikuri induqciis meTodi.
kombinatorika. finansuri maTematikis elemntebi
$1. elementaruli cnebebi simravleTa Teoriidan
saleqcio kursis programidan gamomdinare, Cvens mizans ar warmoad-
gens simravleTa Teoriis gadmocema. vTvliT, rom mkiTxveli icnobs
namdvil ricxvTa simravles da am simravleSi moqmedebebis ZiriTad
kanonebs. amitom gakvriT SevexebiT im elementarul cnebebs simravleTa
Teoriidan, romelTa gareSe gadmosacemi kursis ageba SeuZlebelia.
1. simravlis cneba maTematikis erTerTi ZiriTadi sawyisi cnebaa. misi
ganmarteba ufro martivi cnebebis saSualebiT TiTqmis SeuZlebelia. amitom
Cven am cnebis aRweriT unda davkmayofildeT. ase magaliTad, SeiZleba
vilaparakoT sibrtyis yvela wertilTa simravleze, an raime
mravalkuTxedis yvela gverdis simravleze da sxva. sityva simravlis
nacvlad SeiZleba vixmaroT erToblioba, klasi, sistema da sxva. im
obieqtebs, romlebic am simravleSi arian gaerTianebuli, am simravlis
elementebsUuwodeben. savldebulo araa, rom simravle maTematikuri
obieqtebis erTobliobas warmoadgendes. ase magaliTad, SesaZlebelia
ganvixiloT aRebul momentSi Tbilisis mcxovrebTa simravle. yoveli piri,
romelic aRniSnul momentSi TbilisSi cxovrobs, Cveni simravlis elementi
iqneba. simravle gansazRrulia, roca cnobilia am simravlis yvela
elementi. simravlis mocema SeiZleba an masSi Semavali yvela elementis
ubralo CamoTvliT (roca es SesaZlebelia), an im Tvisebebis dasaxelebiT,
romelic axasiaTebs simravleSi Semaval yvela elements.
simravleebs avRniSnavT didi laTinuri asoebiT: ase magaliTad:
A, B, E, M. xolo mis elementebs patara asoebiT:
4
Tu M simravlis yvela elementis saerTo saxelia e, maSin weren:
M ={ }
e .
Tu M simravle Sedgeba a, b, c, d...l elementebisagan, maSin weren:
M ={a,b,c,d,...}
l .
Tu a obieqti Sedis M simravleSi, am garemoebas ase avRniSnavT:
a ∈ M . rac ikiTxeba Semdegnairad: `a ekuTvnis M simravles~ an `a
warmoadgens M simravlis elements~
Tu a obieqti araa M simravlis elementi, maSin davwerT: a ∉ M da
vityviT `a ar ekuTvnis M simravles~.
zogierTi gamoTqmis gamartivebis mizniT, SemoaqvT agreTe e.w.
carieli simravlis cneba, ase ewodeba simravles, romelsac arc erTi
elementi ar gaaCnia. mas aRniSnaven ∅ simboloTi. masaSadame rogori a
obieqti ar unda aviRoT, yovelTvis gveqneba a ∉ ∅ .
yvela im x elementTa simravle, romlebsac gaaCniaT raime p Tviseba,
aRiniSneba Semdegnairad: {x : }
p . magaliTad, yvela im naturalur ricxvTa
simravle, romlebic naklebia 100-ze ase aRiniSneba {x : x ∈ N,
x <
}
100
sadac N-naturalur ricxvTa simravlea.
A simravles ewodeba B simravlis qvesimravle, Tu A simravlis yoveli
elementi ekuTvnis B simravles da weren: Α⊂Β (ikiTxeba `A Sedis B-Si). Tu
A simravle ar aris Tu B simravlis qvesimravle , maSin weren: Α⊄Β.
miRebulia, rom carieli simravle nebismieri A simravlis qvesim-
ravlea. e.i. ∅⊂Α.
cxadia, nebismieri A simravle Tavis Tavis qvesimravles warmoadgens,
e.i Α⊂Α.
Tu A simravle B simravlis qvesimravlea, xolo A-Si arsebobs erTi
elementi mainc, romelic B-s ar ekuTvnis, maSin A-s ewodeba B-s sakuTrivi
qvesimravle.
Tu Β⊃Α da Β⊂Α, maSin A da B simravleebs toli ewodeba da weren Β=Α.
ori A da B simravleebis gaerTianeba (jami) ewodeba yvela im
elementis simravles, romlebic A da B simravleebidan erTerTs mainc
ekuTvnis da Α∪Β simboloTi aRiniSneba. cxadia, rom Α∪Α=Α.
5
Tu gvaqvs ramdenime simravle A1, A2, A3, . . .,An, maSin am simravleTa
gaerTianeba ewodeba yvela im elementis simravles, romlebic ekuTvnis
erT-erTs mainc mocemuli simravleebidan. simravleTa gaerTianeba
n
aRiniSneba ase: A ∪A ∪
∪ A
A
1
2
L
n an ∪ k .
k =1
ori A da B simravlis TanakveTa (namravli) ewodeba yvela im elementis
simravles, romlebic erTdroulad ekuTvnis, rogorc A ise B simravles da
Α∩Β simboloTi aRiniSneba.
axla vTqvaT, mocemulia ramdenime simravle A1, A2, A3, . . .,An, maSin am
simravleTa TanakveTa (namravli) ewodeba yvela im elementis simravles,
romlebic erTdroulad ekuTvnis ekuTvnis yvela mocemul simravles. am
SemTxvevaSi simravleTa TanakveTa ase aRiniSneba:
n
A ∩A ∩ ∩ A
A
1
2
L
n an ∩ k .
k =1
A da B simravlis Usxvaoba ewodeba A simravlis yvela im elementis
simravles, romlebic B simravles ar ekuTvnian da A\B simboloTi
aRiniSneba.
cxadia, rom Tu Α∩Β=∅, maSin A\B=A.
advilia Cveneba, rom moqmedebebs simravleebze gaaCniaT Semdegi
Tvisebebi:
1.
A ∪ B = B ∪ ,
A A ∩ B = B ∩ ,
A (komutaciuroba);
2. A ∪ (B ∪ C) = (A ∪ B) ∪ C, A ∩ (B ∩ C) = (A ∩ B) ∩ C (asociaciuroba);
3. A ∪ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), (distribuciuloba).
ganvixiloT raime bunebis elementTa M simravle da vigulisxmoT,
rom masSi Semavali elementebisaTvis raime niSnis mixedviT dadgenilia
garkveuli rigi ise, rom Sesrulebulia Semdegi pirobebi:
1. Tu a da b warmoadgenen M simravlis sxavdasxva elementebis raime
wyvils, maSin erT-erT maTgans ufro dabali rigi aqvs, vidre meores,
(amasTan Tu magaliTad a-s ufro dabali rigi aqvs, vidre b-s, maSin b-s ar
SeiZleba ufro dabali rigi hqondes vidre a-s. aseT SemTxvevaSi Cven
vityviT, rom M simravleSi dawesebuli rigis mixedviT a win uswrebs b-s,
an b aris a-s momdevno).
2. Tu a-s ufro dabali rigi aqvs vidre b-s, xolo b-s ufro dabali
rigi aqvs vidre c-s, maSin a-s ufro dabali rigi aqvs vidre c-s.
6
Tu simravleSi SemoRebulia rigis cneba, ise rom igi zemoaRniSnul
or pirobas akmayofilebs, maSin mas dalagebuli simravle ewodeba, xolo
im niSans, romelic am pirobebs akmayofilebs da garkveul rigs amyarebs M
simravlis elementebs, Soris simravlis dalagebis wesi hqvia.
Tu a-s ufro dabali rigi aqvs vidre b-s Cven davwerT a p b
(ikiTxeba: a win uswrebs b-s). am simbolos gamoyenebiT zemoTaRniSnuli ori
piroba, romelsac unda akmayofilebdes simravlis dalagebis wesi SeiZleba
CavweroT ase:
1) Tu a ≠ b da a,b ∈ M maSin a
b
p an a b
f amasTan, roca a b
p ,
SeuZlebelia gvqondes b
a
p .
2) Tu a
b
p da b c
p maSin a c
p .
magaliTad, vTqvaT M aRniSnavs sxvadasxva asakis adamianTa jgufs.
ganvixiloT am simravlis ori elementi, e.i. ori pirovneba Cveni jgufidan
da mas, vinc ufro axalgazrdaa, dabali rigi mivaniWoT, meoresTan
SedarebiT; es wesi M simravles dalagebul simravled aqcevs da simravlis
dalegebis wess gansazRvravs asaki.
2. namdvil ricxvaTa simravle. radgan namdvil ricxvTa simravlis
ZiriTadi Tvisebebi da ariTmetikuli moqmedebebi namdvil ricxvebze mkiT-
xvelisaTvis cnobilia maTematikis saskolo kursidan, amitom Cven maTze ar
SevCerdebiT. avRniSnoT mxilod Semdegi ori Tviseba:
1. dalagebuloba. Tu x ≠ y maSin an x<y , an x>y .
2. uwyvetoba. X da Y namdvil ricxvTa ori simravlea, Tu nebismieri
x ∈ X da y ∈Y ricxvebisaTvis marTebulia utoloba x ≤ y ,
arsebobs erTi mainc iseTi C risxvi, rom yoveli x ∈ X da y ∈Y
ricxvebisaTvis sruldeba utoloba x ≤ c ≤ y .
SevniSnoT, rom uwyvetobis Tviseba, romelic namdvil ricxvTa
simravles gaaCnia, ar gaaCnia racionalur ricxvTa simravles. magaliTad,
vTqvaT X = {x ∈ Q : x < 2 } da Y = {y ∈ Q : y > 2} cxadia, rom Tu x ∈ X da y ∈Y
maSin x ≤ y magram ar arsebobs iseTi racionaluri C ricxvi, rom yoveli
x ∈ X da y ∈Y marTebuli iyos utoloba x ≤ c ≤ y .
3. ricxviTi ReZi. ricxvTa Sualedebi. wrfes, romelzedac
fiqsirebulia raime O wertili (saTave), arCeulia dadebiTi mimarTuleba
7
da sigrZis erTeuli (masStabi) ricxviTi RerZi anu ricxviTi wrfe
ewodeba (nax.1).
0 1 M(x)
nax. 1
ricxviTi RerZis yovel M wertils SeiZleba Sevusabamod erTaderTi
namdvili x ricxvi da piriqiT. es urTierTcalsaxa Sesabamisoba SeiZleba
damyardes Semdegi wesiT: x = OM (OM monakveTis sigrZes), Tu 0-dan M-saken
emTxveva RerZis mimaTulebas da winaaRmdeg SemTxvevaSi x = − OM .
am x ricxvs M wertilis koordinati ewodeba. es garemoeba ase Caiwereba:
M(x). cxadia, rom O-saTvis koordinatia 0. mocemulia wertili RerZze
niSnavs, rom mocemulia wertilis koordinati (SemdgomSi namdvil ricxvsa
da mis Sesabamis wertils RerZze gavaigivebT).
ganvixiloT namdvil ricxvTa simravlis zogierTi qvesimravle,
romlebTac ricxviTi Sualedebi ewodeba.
vTqvaT
a da b namdvili ircxvebia da a<b.
gansazRvrebebi: yvela im namdvil x ricxvTa simravles, romlebic
akmayofileben utolobas a ≤ x ≤ b , Caketili Sualedi (monakveTi)
ewodeba da [a;b] simboloTi aRiniSneba, e.i.
[a;b] = {x ∈ R : a ≤ x ≤ }
b
analogiurad
(a;b)= {x∈R :a < x < }
b .
simravlis Ria Sualedi (intervali) ewodeba, xolo
[a;b) = {x ∈ R : a ≤ x < }
b ,
(a;b]= {x∈R :a ≤ x ≤ }
b ,
Sesabamisad naxevrad Ria Sualedebi marjvnidan da marcxnidan ewodebaT. a
da b ricxvebs ganxiluli Sualedebis sazRvrebi an boloebi ewodeba, xolo
b-a ricxvs Sualedis sigrZe.
yvela im namdvil x ricxvTa simravles, romlebic akmayofileben
utolobas x ≥ a usasrulo Sualedi ewodeba da [a;+∞) simboloTi aRiniSneba
e.i.
[a;+∞) = {x∈ R : x ≥ }
a .
8
usasrulo Sualedebs warmoadgenen agreTve Semdegi simravleebi:
(a;+∞)= {x∈R: x > }
a
(−∞;b]= {x∈R : x ≤ }
b
(a;b) = {x ∈ R : a < x < }
b
namdvil ricxvTa R simravlec usasrulo Sualeds warmoadgens da
igi ase aRiniSneba R = (− ∞;+∞).
a ∈ R ricxvis midamo U(a) ewodeba am ricxvis Semcvel yovel
intervals.
a ∈ R ricxvis ε midamo U(a; ε) ewodeba (a-ε; a+ε) intervals:
U(a; ε)=(a-ε; a+ε).
SeniSvna. simbolo +∞ ar aris ricxvi. es pirobiTi niSania da aRni-
Snavs im faqts, rom x cvlad sidides SeuZlia miiRos meti mniSvnelobebi,
vidre winaswar aRebuli nebismierad didi dadebiTi ricxvia.
- ∞ pirobiTi niSania da aRniSnavs im faqts, rom x cvlad sidides
SeuZlia miiRos naklebi mniSvnelobebi, vidre winaswar aRebuli
nebismierad mcire uaryofiTi ricxvia.
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ras ewodeba simravlis elementebi? simravlis nawili? carieli
simravle?
2. ras ras niSnavs, rom A da B simravleebs Soris damyrebulia
urTierTcalsaxa Tanadoba?
3. rogor simravleebs ewodeba ekvivalenturi?
4. rogor simravles ewodeba usasrulo simravle? sasruli simravle?
5. ras niSnavs, rom c ricxvi moTavsebulia a da b ricxvebs Soris?
6. rogor ricxvebs ewodeba namdvili ricxvebi?
7. ras ewodeba intervali? segmenti? naxevarintervali? naxevarsegmenti?
8. rogor ricxvebs ewodeba racionaluri?
9
9. wrfis ra wertils ewodeba racionaluri wertili?
II. praqtikuli savarjiSoebi
1. mocemulia simravleebi: A = {a,b, c, d , }
e da B = {c, d,e, f , g}. ipoveT
A B, A B, A \ B
I
U
.
2. SejibrebaSi monawileobda 9 moTxilemure da 20 mocigurave.
4 moTxilamure mociguravec iyo. ramdeni sporcmeni monawileobda
SejibrebaSi?
3. daamyareT urTierTcalsaxa Sesabamisoba [0,1] segmentsa da [a,b] segments
Soris.
4. daamyareT urTierTcalsaxa Sesabamisoba ]a,b[ intervalsa da ricxvTa
RerZs Soris.
5. daamyareT urTierTcalsaxa Sesabamisoba [0 [
1
, da [0,+∞[ simravleebs
Soris.
6. daamyareT urTierTcalsaxa Sesabamisoba wrewirsa da wrfes Soris.
$2 maTematikuri induqciis meTodi da
kombinatorikis elementebi
1. maTematikuri induqcia (an sruli induqcia) ewodeba naturaluri
ricxvebis Sesaxeb Teoremebis damtkicebis meTods. arsebobs Teoremebi,
romlebic WeSmaritia, zogierTi erTmaneTis momdevno naturalur
ricxvTaTvis, da aseve arsebobs Teoremebi, romlebic WeSmaritia
sazogadod, nebismieri naturaluri ricxvisaTvis. maTematikuri induqciis
(an sruli induqciis) meTodi xSirad gamoiyeneba imis dasamtkiceblad, rom
Teorema WeSmaritia zogadi SemTxvevisaTvis e.i. nebismieri naturaluri
ricxvisaTvis.
10
Teorema. vTqvaT T(n) aris Teorema naturaluri n ricxvis Sesaxeb.
dauSvaT, rom:
1. Teorema T WeSmaritia n0 naturaluri ricxvisaTvis;
2. Tu T Teorema WeSmaritia naturaluri k≥n0 ricxvisaTvis maSin
igi WeSmaritia momdevno naturaluri k+1 ricxvisaTvisac.
Tu es ori piroba Sesrulda maSin T Teorema WeSmaritia nebismieri
n≥n0 ricxvisaTvis. Tu n0=1 gvaqvs sruli induqcia, xolo n0>1 arasruli
induqcia.
damtkiceba. dauSvaT, rom T(n) Teorema ar aris WeSmariti, romelime
N>n0 naturaluri ricxvisaTvis, maSin igi ar unda iyos WeSmariti N-1
naturaluri ricxvisaTvisac, vinaidan winaaRmdeg SemTxvevaSi, e.i. Tu T(n)
Teorema WeSmaritia N-1 naturaluri ricxvisaTvis, maSin meore pirobis
ZaliT igi WeSmariti iqneboda N naturaluri ricxvisaTvisac. analogiurad
miviRevT, rom Tu T(n) Teorema ar aris WeSmariti N-1 naturaluri
ricxvisaTvis, maSin igi ar iqneba WeSmariti N-2 naturaluri
ricxvisaTvisac da a.S.
amrigad, rogori didi ar unda iyos N naturaluri ricxvi,
TanmimdevrobiT TiTo erTeulis gamoklebiT mivalT im daskvnamde, rom T(n)
Teorema ar aris WeSmariti ar aris n0 naturaluri ricxvisaTvis, rac
pirvel pirobas ewinaaRmdegeba. maSasadame daSveba imisa, rom Teorema ar
aris WeSmariti N ≥ n0 ricxvisaTvis mcdaria.
n n +
+
2
2
2
(
)(
1 2
)
1
magaliTi 1. davamtkicoT 1 +
n
2 +
+ n =
L
tolobis
6
WeSmariteba.
amoxsana: roca n0=1 dasamtkicebeli toloba WeSmaritia, marTlac
1
(
1 + )(
1 2 ⋅1 + )
1
1⋅ 2 ⋅ 3
12 =
=
= 1
6
6
1=1
dauSvaT toloba samarTliania, roca n=k e.i.WeSmaritia toloba:
k k +
+
2
2
2
(
)(
1 2
)
1
1 +
k
2 +
+ k =
L
6
elementaruli gardaqmnebiT miviRebT:
11
2
k k +
k +
k k +
k +
k +
12 + 22
2
+
+ k +
L
(k + )2
(
)(
1 2
)
1
1 =
+ (k + )2
(
)(
1 2
)
1
6(
)
1
1 =
+
=
6
6
6
(k + )
1
k +
2
(
)
1
(2k + k + 6k + 6) =
(2 2
k + 7k + 6)
6
6
frCxilebSi moTavsebuli wevri kvadratuli samwevria, romlis
3
fesvebia k = −2
k = −
(2 2
k + 7k + 6) = (k + 2)(2k +
1
da 2
, amitom
)
3 es ukanaskneli
2
SevitanoT dasamtkicebel tolobaSi, miviRebT
k +
k +
k +
k +
k + +
+ +
12 +
k
22
2
+
+ k +
L
(k + )2 (
)(
1
2)(2
)
3
(
)
1 [(
)
1
][
1 2(
)
1
]1
1 =
=
.
6
6
es niSnavs, rom dasamtkicebeli Teorema WeSmaritia. aqedan, ki
gamomdinareobs, rom toloba marTebulia nebismieri naturaluri n
ricxvisaTvis.
magaliTi 2. davamtkicoT, rom nebismieri n ricxvisaTvis adgili aqvs
n
n−
q −
2
1
1
tolobas: a + a q + a q +
+ a q
= a
1
1
1
L
1
1
(geometriuli progresiis n wevris
q −1
jamis formula).
amoxsna: roca n=2 gvaqvs
2
q −1
(q − )(
1 q + )
1
a + a q = a
= a
= a (q + )
1 = a q + a
1
1
1
1
1
1
1
q −1
q −1
amrigad dasamtkicebeli toloba marTebulia roca n=2 .
axla davamtkicoT, rom Tu toloba marTebulia naturaluri k≥2
ricxvisaTvis, maSin igi marTebuli iqneba momdevno k+1 ricxvisaTvisac,
k
k −
q −
2
1
1
pirobidan a + a q + a q +
+ a q
= a
,
1
1
1
L
1
1
gamomdinareobs, rom
q −1
k 1
+
k −
k
q
−
2
1
1
a + a q + a q +
+ a q
+ a q = a
,
1
1
1
L 1
1
1
q −1
marTlac,
k
k
k −
k
q −
⎛
k
q −
⎞
2
1
1
1
a + a q + a q +
+ a q
+ a q = a
+ a q = a
+ qk =
1
1
1
L
1
1
1 q −1
1
1 ⎜⎜ q −1
⎟
⎟
⎝
⎠
qk −1
k 1
+
k
k 1
+ q
−
+
q
q
−1
= a
= a
1
q −1
1
q −1
maSasadame Teorema marTebulia nebismieri n ricxvisaTvis.
12
2. kombinatorikis elementebi. xSirad Teoriuli da praqtikuli
amocanebis gadasawyvetad saWiroa sasruli simravlis elementebisagan
sxvadasxva kombinaciebis Sedgena da raime wesiT Sedgenili yvela SesaZlo
kombinaciis raodenobis gamoangariSeba. aseT amocanebs, kombinatoruls
uwodeben, xolo maTematikis nawils, romelic dasaxelebul amocanebis
amoxsnas Seiswavlis - kombinatorikas.
erTi da igive sasruli simravlis elementebi SeiZleba davalagoT
sxvadasxva rigiT imisa da mixedviT, Tu simravlis romel elements
avarCevT pirvel elementad, romels meored da a.S.
gansazRvreba: sasrul simravleSi dadgenil rigs misi gadanacvleba
ewodeba.
n-elementiani simravlis yvela SesaZlo gadanacvlebaTa ricxvi Pn-iT
aRiniSneba. cxadia, igi damokidebulia mxolod simravlis elementTa
raodenobaze.
Tu simravle Sedgeba erTi elementisagan, maSin SesaZlebelia
erTaderTi gadancvleba e.i. Pn=1. Tu simravle Sedgeba ori elementisagan
{a,b}, maSin SesaZlebelia ori gadanacvleba (a,b) da (b,a) e.i. P2=2.
Tu simravle Sedgeba sami elementisagan – {a,b,c}, maSin SesaZlebelia eqvsi
gadanacvleba: (a,b,c), (a,c,b), (c,a,b), (c,b,a) (b,a,c) (b,c,a) e.i. P3=6. advili
SesamCnevia, rom P1=1, P2=1.2, P3=1.2.3. maTematikuri induqciis principis
gamoyenebiT davamtkicod, rom
Pn=1⋅ 2⋅
n
L
3
(1)
rogorc zemoT vnaxeT, rodesac n=1 es formula marTebulia, dauSvaT,
rom formula marTebulia n=k –saTvis, e.i. Pk=1⋅ 2 ⋅
k
L
3
vaCvenoT, formula
samarTliani iqneba n=k+1, e.i.
Pk+1=1⋅ 2⋅3 k ⋅(k + )
1
L
.
marTlac, imisaTvis, rom miviRoT k+1 elementisagan yvela SesaZlo
gadancvleba, saWiroa TiToeul k elementian adrindel gadanacvlebas
mivuerToT axali (k+1) elementi, romelic SeiZleba davayenoT 1-el, me-2,
me_3, . . . k –ur da (k+1)-e adgilze. amrigad, yoveli k elementiani gadanac-
vleba warmoqmnis (k+1)-axal gadanacvlebas, amitom (k+1) elementiani
simravlis yvela SesaZlo gadanacvlebaTa ricxvi iqneba Pk=(k+1), e.i.
P
= P (k + )
1 = 1⋅ 2 ⋅ 3 k ⋅ (k + )
1
k 1
+
k
L
.
13
maTematikuri induqciis principis Tanaxmad (1) formula samarTliania
nebismieri
n-saTvis.
1⋅ 2 ⋅
n
L
3
namravli aRiniSneba n!
(ikiTxeba
n faqtoriali). miRebulia, rom 0!=1 da 1!=1 maSasadame
P = !
n
n
.
gansazRvreba: n-elementiani simravlis nebismier m-elementian dala-
gebul qvesimravles (m ≤ n) ewodeba wyoba n elementisagan m-ad. igi
aRiniSneba m
An simboloTi. maSasadame n elementisagan k-elementiani wyoba
aris simravle, romelic Seicavs k-elementians, aRebuls n elementisagan da
dalagebuls garkveuli mimdevrobiT. amasTanave, erTi da imave k-elemen-
tisagan Semdgari ori wyoba iTvleba sxvadasxvagvarad, Tu isini
gansxvavdebian elementTa ganlagebiT. cxadia, rom A1 = n
n
. marTlac, n-dan
erTi elementi n xerxiT SeiZleba avarCioT, xolo am elementisagan miiReba
erTaderTi dalagebuli simravle.
axla davamtkicoT, rom roca 1 ≤ m ≤ n , maSin m 1
+
m
A
= (n − m)A .
n
n
imisaTvis,
rom n elementisagan SevadginoT yvela SesaZlo (m+1) elementiani wyoba,
SeiZleba moviqceT Semdegnairad: jer avarCioT raime m raodenobis elemen-
ti da movaTavsoT isini pirvel m adgilze, amis Sesruleba SeiZleba m
An
xerxiT. (m+1) adgilze SeiZleba movaTavsod nebismieri elementi darCe-
nili (n-m) elementidan. amrigad yoveli m elementiani wyoba warmoqmnis
(n-m) raodenobis m+1 elementian wyobas, maSasadame, sul gveqneba
m
(n − m)An
wyoba, e.i. m 1
+
m
A
= (n − m)A
A1 =
n
n . Tu gaviTvaliwinebT, rom
n da visargeblebT
n
damtkicebuli formuliT, gveqneba:
A1 = n
n
,
2
A = (n − )
1 1
A = n(n − )
1
n
n
,
3
A = (n − )
2 2
A = n(n − )(
1 n − )
2
n
n
. . . . . . . . . . . . . . . . . . . . . . . . .
m 1
+
A
= (n − m + )
1 Am = n(n − )(
1 n − )
2
(n − m + )
1
n
n
L
magram, n-dan (n-m+1)-mde naturalur ricxvTa namravli faqtorialis
saSualebiT SeiZleba Semdegnairad gamovsaxoT:
n(n − )
1
(n − m − )
1
3⋅ 2 ⋅1
!
n
n(n − )
1
(n − m + )
1 =
L
L
=
L
1⋅ 2⋅3 (n − m − )(
1 n − m)
(n − m)!
L
14
!
n
e.i. Am =
n
(n − m)!
SevniSnoT, rom An = P = !
n
n
n
.
gansazRvreba: n elemntiani simravlis nebismier m elementian
qvesimravles ewodeba jufTeba n elementisagan m-ad.
cxadia, rom mocemuli simravlis ori m-elemntiani jufTeba
sxvadasxvaa. mxolod maSin, rodesac isini gansxvavdebian erTi elementiT
mainc, xolo m elemntiani wyobebi rom gansxvavdebodnen, sakmarisia isini
gansxvavdebodnen dalagebiT.
n elementiani simravlis yvela SesaZlo m elementian jufTebaTa
m
A
ricxvi m
C
m
n
C =
n simboloTi aRiniSneba. davamtkicoT, rom:
n
..
Pn
imisaTvis, rom mocemuli n-elementisagan SevadginoT yvela
SesaZlebeli m-elemntiani wyoba, SeiZleba moviqceT Semdegnairad:
jer n-elementidan gamovyoT romelime m-elementi, rac m
Cn xerxiT SeiZleba
ganxorcieldes. gamoyofili m-elementi davalagoT sxvadasxvanairad, rac
Pm xerxiT SeiZleba Sesruldes. amgvarad, miviRebT m
C P
n
m dalagebul
m
A
simravles. e.i. m
m
A = C P
m
n
C =
n
n
n , saidanac
n
. Tu gaviTvaliswinebT,
Pn
!
n
rom Am =
n
da P
(n − m)!
n= !
n , miviRebT
!
n
C m =
n
.
!
m (n − m)!
daumtkiceblad moviyvanoT jufTebaTa ricxvis Semdegi Tvisebebi:
1.
m
n−m
C = C
n
n
;
2.
m
m 1
+
m 1
+
C + C
= C
n
n
n 1
+ ;
3.
0
1
n
n
C + C +
+ C = 2
n
n
L
n
.
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ra aris maTematikuri induqcia?
15
2. ras Seiswavlis kombinatorika?
3. ras ewodeba wyoba? jufTeba ? gadanacvleba ?
4. moiyvaneT jufTebaTa ricxvis Tvisebebi.
II. praqtikuli savarjiSoebi
1.
daamtkiceT maTematikuri induqciis meTodiT Semdegi formulebi
( n ∈ N ):
1.1. a = a + d (n − )
1
n
1
– ariTmetikuli progresiis zogadi wevris formula.
a + a
1.2. S
1
n
=
⋅ n
n
– ariTmetikuli progresiis wevrTa jamis formula.
2
1.3.
1
−
b =
n
b q
n
1
– geometriuli progresiis zogadi wevris formula.
2. daamtkiceT Semdegi tolobebi:
n( + )
1
2.1 1 +
n
2 + 3 +
+ n =
L
, n ∈ N
2
2.2
2
1 + 3 + 5 +
+ (2n − )
1 = n
L
, n ∈ N
3. gamoTvaleT:
3
A
4
5
A
+ A
12
A
(n + )!
3
15
a)
13
15
14
3
3
; g)
; d)
A
+
; b)
A
3
A
3
A
⋅ 12 !
(n + (
)!
1
2
n − )
4
15
14
15
16
4. amoxseniT gantoleba:
4
15 2
A
4
x
2
1
2
1
a) A ⋅ P
= 42 p
C =
A + C =
A
− C =
x
x−4
x−4 ; b)
x
4
; g)
256
x
x
; d)
79
x −1
x
$3. procentebis gamoTvla
1. dagroveba. Tanxas, romelsac ixdian fulad saSualebaTa sargeb-
lobisaTvis procenti ewodeba. procentis Sefardebas fulad saSualebaTa
k sididesTan (raodenobasTan) p procentuli ganakveTi ewodeba.
P
p =
100
k
16
ekonomikuri amocanebis gadawyvetisas Zalze moxerxebulia e.w.
xvedriTi procentuli ganakveTiT sargebloba.
i – xvedriTi procentuli ganakveTi _ aris is, Tanxa, romelic moaqvs
p
P ⋅100
P
1 lars erT weliwadSi. gvaqvs i =
=
=
% , sadac n welTa ricxvia.
n
kn
nk
Tu procenti Tavdapirvel Tanxas emateba, maSin xdeba Tanxis
dagroveba. Tu damatebul procents ar daericxeba procenti, maSin gvaqvs
martivi procenti. Tu procenti emateba kapitals, romelsac is ericxeboda
da momdevno periodSi procenti ericxeba aseTnairad gazrdil kapitals,
maSin amboben, rom gvaqvs rTuli procenti.
ganvixiloT martivi procentis cneba. vTqvaT k – sawyisi Tanxaa, i –
xvedriTi procentuli ganakveTi, maSin n – wlis Semdeg martivi procenti
iqneba kni. n – wlis Semdeg martivi procentiT miRebuli Kn – dagroveba
tolia
K = k + kni = k 1
( + ni)
n
(1)
miRebul formulaSi monawileobs oTxi sidide: k, kn, i, da n. (1) for-
mulidan gamovTvliT yovel maTgans, Tu sami danarCeni cnobili sidideebia.
miviRebT
K
k
n
=
(2)
1+ ni
K − k
n
n
=
(3)
ik
K − k
i
n
=
(4)
nk
magaliTi 1. 5000 lari gacemulia 4 procentad. gamoiangariSeT 3 wlis
ganmavlobaSi miRebuli dagroveba.
amoxsna. pirobis Tanaxmad k=5000, p=4, n=3. visargeblod (1) formuliT:
4
k3=5000(1+3i), magram = p
i
=
= 04
,
0
SevitanoT i=0,04 k
100 100
3-is saangariSo
formulaSi miviRebT:
k3=5000(1+3⋅0,04)=5000⋅1,12=5600 lari.
magaliTi 2. ramdeni lari mogvcems or weliwadSi 784 000 lars, Tu
igi gacemulia yovelwliurad martiv 6 procentad.
17
amoxsna. mocemulia p=6, n=2 Kn=784 000. unda gamovTvaloT sawyisi
Tanxa. gamoviyenoT (2) formula, gveqneba:
784 000
784 000
k =
=
= 700 000 lari.
1+ 2 ⋅ ,
0 06
12
,
1
magaliTi 3. yovelwliurad 2000 lari gacemulia martiv 5 procentad.
ramdeni wlis Semdeg mogvcems igi 2500 lar mogebas?
amoxsna: mocemulia p=5, k=2000, Kn=2500. unda gamovTvaloT n, (3)
formula gvaZlevs:
2500 − 2000
500
n =
=
= 5 weli.
2000⋅ ,
0 05
100
ganvixiloT rTuli procentis cneba. vTqvaT k sawyis fulad
saSualebas n wlis ganmavlobaSi ericxeba rTuli procenti i – xvedriTi
procentuli ganakveTiT;
k – fuladi Tanxis sidide pirveli wlis Semdeg Seadgens
k = k + ik = k 1
( + i)
1
meore wlis Semdeg
2
k = k + ik = k 1
( + i) = k 1
( + i)
2
1
1
1
mesame wlis Semdeg
3
k = k + ik = k 1
( + i)
3
2
2
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
n wlis Semdeg
n
k = k
+ ik
= k 1
( + i)
n
n 1
−
n 1
−
e.i.
K = k 1
( + i)n .
n
(5)
gamosaxulebas 1+i=r ewodeba rTuli procentis koeficienti. amrigad
n
K = kr
n
(6)
maSasadame n wlis ganmavlobaSi i – xvedriTi procentis daricxviT
fulad saSualebaTa saboloo raodenoba, romlis sawyisi raodenoba iyo k
romelic tolia sawyisi Tanxa gamravlebuli rTuli procentis
koeficientis xarisxze, romlis maCvenebelia welTa ricxvi.
magaliTi 4. Semnaxvel salaroSi aTi wliT Setanilia anabari 100000
lari, ra Tanxas gadaixdis Semnaxveli salaro am periodis bolos Tu P=3.
18
amoxsna: mocemulia n=10, k=100 000, p=3; unda gamovTvaloT Kn.
visargebloT (6) formuliT.
n
⎛
⎞
10
5
3 10
K = kr = 100000 1
( + i) = 10 1
( + K )
n
n ⎜1 +
⎟ = 105 ⋅ ,
1 0310 =
134 392 lari
⎝
100 ⎠
r sididis n xarisxis gamoTvla sakmarisad Sromatevadi procesia. misi
gamartiveba SegviZlia logariTmebis cxrilis gamoyenebiT.
me-(6) formulaSi monawileobs 4 sidide Kn, k, n da xvedriTi
procentuli ganakveTi – i. yoveli maTganis gamoTvla SesaZlebelia Tu
cnobilia danarCeni sami. magaliTad, Tu cnobilia Kn, p, n, maSin
K
(6) formulidan gveqneba:
n
k =
, sadac
n
r
P
r = 1+ i = 1+
(7)
100
Tu cnobilia Kn , k da n, i–maSin xvedriTi procentuli ganakveTi
Semdegnairad gamoiTvleba: gavalogariTmoT (6) formulis orive mxare,
miviRebT:
lg K = lg k + nlg r
n
, saidanac:
lg K − lg k
n lg r = lg K − lg k
lg r
n
=
n
da
.
n
logariTmebis cxrilis gamoyenebiT gamoviTvliT r-is mniSvnelobas
p
da misi saSualebiT vpoulobT p-s. radgan r = 1+ i = 1+
, gveqneba
100
p = 100r −100 (8)
miRebuli formuliT ekonomikur analizSi dgindeba zrdis tempebis
saSualo mniSvneloba.
magaliTi 5. xuTwliani gegmis Sesasrulebel periodSi produqciis
moculoba unda gaizardos 85 %-iT. rogori unda iyos zrdis saSualo
tempi?
amoxsna: vTqvaT, sagegmo wlis dasawyisSi gvaqvs produqciis k
moculoba. xuTi wlis Semdeg gveqneba k5=kr5, magram k5=1,85k e.i. 1,85k=kr5.
5
r =
;
85
,
1
5
r =
85
,
1
= 131
,
1
radgan r=1+i, amitom 1+i=1,131, saidanac i=0,131, magram
P
i =
; p = 100⋅ i = 100⋅ 131
,
0
=
%
1
,
13
100
me-(6) formulidan gveqneba, rom
19
log K − log k
n
n
=
(9)
log r
Tanxis saboloo sididis gamosaTvlelad procenti SeiZleba
daematos sawyis Tanxas weliwadSi erTxel wlis bolos, agreTve
weliwadSi m – jer, magaliTad yovel TveSi. am SemTxvevaSi procentebi 1
1
i
larze
wlis Semdeg Seadgens
, xolo 1 lari Tanxa Tavisi namatiT
m
m
1
i
wlis Semdeg iqneba: 1 +
m
m
2
2
⎛
i ⎞
wlis Semdeg iqneba: ⎜1 +
⎟ da a.S. ...
m
⎝
m ⎠
m
⎛
i ⎞
1 wlis Semdeg iqneba: ⎜1+
⎟ da a.S. ...
⎝
m ⎠
mn
⎛
i ⎞
n wlis Semdeg iqneba: ⎜1 +
⎟
⎝
m ⎠
amgvarad, Tu i – procentuli ganakveTia, xolo procentuli daricxva
weliwadSi xdeba m – jer, sawyisi k Tanxis saboloo sidide n wlis Semdeg
iqneba
mn
⎛
i ⎞
K = k⎜1+
⎟
n
(10)
⎝
m ⎠
Tu procentebis daricxva xdeba uwyvetad, e.i. m→∞, maSin
in
m
⎡
⎤
i
⎢⎛
⎞ ⎥
mn
mn
⎛
i ⎞
⎛
i
⎜
⎟
⎞
⎢
1
⎥
in
K = lim k⎜1+ ⎟ = k lim⎜1+ ⎟ = k lim ⎜1+
⎟
= ke
n
.
m→∞
m→∞
m
⎝
m ⎠
⎝
m
→∞⎢
⎠
⎜
m
⎥
⎟
⎢⎜
⎟ ⎥
⎝
i
⎢
⎠ ⎥
⎣
⎦
(zRvarsa da zRvris gamoTvlas gavecnobiT SemdegSi).
maSasadame, uwyveti daricxvis SemTxvevaSi
ni
K = ke
n
(11)
yvelaze arsebiTi da moulodneli momenti, rac did yuradRebas
imsaxurebs uwyveti daricxvis dros aris Semdegi: miuxedavad imisa, rom
daricxvebis m raodenoba usasrulobisaken miiswrfvis, rac iwvevs
Sesabamisi saboloo Tanxebis zrdas, zRvruli saboloo Tanxa rCeba
SemosazRvruli da gamoiTvleba (11) formuliT.
20
magaliTi 6. Tanxa – k =100 000 lari dabandebulia n=3 wliT, yovl-
wliurad p=6 rTul procentad. gamoTvleT saboloo Tanxa, Tu
procentebis daricxva xdeba yoveli Tvis bolos.
amoxsna: amocanis pirobebis gaTvaliswinebiT (10) formulidan
miviRebT
⎛
0,06 12 3
⎞ ⋅
K =
⎜ +
⎟
=
+
=
3
100 000 1
100 000 1
(
0,005)36
119 620 lari
⎝
12 ⎠
magaliTi 7. gamoTvaleT k=100 000 laris saboloo sidide, romelic
dabandebulia p=6 rTul procentad, Tu procentebis daricxva xdeba
uwyvetad 3 wlis ganmavlobaSi.
amoxsna: (11) formulaSi SevitanoT amocanis pirobebi, gveqneba
K = 100 000 30⋅,06
e
= 119 720
3
2. perioduli Senatani. Tu sxvadasxva periodis ganmavlobaSi yoveli
maTganis dasawyisSi bankSi Sedis p rTuli procentuli ganakveTiT mudmivi
k Tanxa, maSin k – s ewodeba perioduli Senatani. gamovTvaloT is Tanxa,
romelic am gziT dagrovdeba n wlis Semdeg.
pirveli Senatani k – dabandebulia n wliT. amrigad, misgan miiReba
p
Tanxa
n
k = kr ,
r = 1 +
1
;
100
meore Senatani k – dabndebulia (n-1) wliT. amrigad, misgan miiReba
Tanxa
1
−
k =
n
kr
2
;
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
ukanaskneli Senatani dabandebulia mxolod 1 wliT. amrigad, misgan
miiReba Tanxa k = kr
n
; am Tanxebis jami aris saZiebeli A Tanxa. amrigad
n
n 1
−
A = kr + kr
+ + kr = kr 1
(
2
n 1
−
+ r + r + + r ).
L
L
frCxilebSi moTavsebuli gamosaxuleba warmoadgens geometriul
n
n−
r −
2
1
progresias mniSvneliT r. amitom 1
1
+ r + r +
+ r
=
L
. maSasadame
r −1
kr(r n − )
1
A =
(12)
r −1
3. martivi da rTuli ganakveTebiT sargeblobisas Tanxis dagrovebis
procesebis Sefaseba. dasmuli amocanis Sesaswavlad saWiroa erTmaneTs
SevadaroT (1) da (5) formulebiT miRebuli sidideebi. dauSvaT, rom am
21
formulebSi tolia sawyisi Tanxac, sargeblobis ganakveTic da daricxvis
periodebis n raodenobac.
gvaqvs
K '
k 1
(
ni)
1 ni
K
n
+
+
⎧ ′n
ganakveTia
martivi
sargeblis
=
=
'
n
n
⎨
sadac,
K
k 1
( + i)
1
( + i)
K
n
⎩ ′n
ganakveTia
rTuli
sargeblis
ganvixiloT Semdegi sami SemTxveva:
'
K
1 + i
1. roca n=1, miviRebT n =
= 1, maSasadame dagrovili Tanxebi tolia.
'
K
1 + i
n
'
K
2. roca 0
1
p n p da i 0
f , maSin mtkicdeba, rom 1+ ni f (1+ )n
i , e.i. n > 1.
'
Kn
es niSnavs, rom sargeblis martivi ganakveTi ufro met mogebas
iZleva.
3. roca n>1 da i>0, maSin mtkicdeba, rom 1 + ni p (1+ )n
i , maSasadame
'
Kn <1.
'
Kn
es niSnavs, rom rTuli ganakveTi ufro met mogebas iZleva. cxadia
n –is zrdasTan erTad gansxvaveba dagrovil Tanxebs Soris sul ufro
izrdeba.
magaliTi 8. banki sTavazobs meanabreebs erTi wlis ganmavlobaSi sar-
geblis rTul Tviur 2%-ian daricxvas, xolo meore banki – sargeblis
martiv 2,2%-ian daricxvas. roml bankSi ufro xelsayrelia meanabrisaTvis
fulis Setana?
rogorc viciT
,
2 2
'
1 +
12
K
k 1
( + ni)
,
1 264
n
100
=
=
≈
< 1
'
K
k 1
( + i)n
1
( + ,
0 02)12
,
1 268
n
amitom meanabrisaTvis Tanxis Setana xelsayrelia pirvel bankSi.
4. sargeblis p wliur da p* Tviur rTul ganakveTebs Soris
kavSiri. (erTi da imave sawyis da saboloo Tanxebis SemTxveva).
vTqvaT Tanxa (K lari) dabandebulia sargebli p* Tviuri ganakveTiT.
maSin n wlis (anu 12-n Tvis) Semdeg sabolood dagrovili Tanxa
*
K
(m = 12n)
m
gamoiTvleba formuliT
*
* m
K = K 1
( + i )
m
(13)
22
davsvaT Semdegi amocana: sargeblis rogori Tviuri p ganakveTiT unda
davabandoT sawyisi K Tanxa, rom n wlis Semdeg saboloo Kn Tanxa (13)
formuliT mocemuli Tanxis toli iyos?
pirobis Tanaxmad K * = K ,
m = 12n
* 12n
K 1
( + i )
= K 1
( + i)
m
n
, e.i.
n
anu
1
( + i*)12 = 1+ i . aqedan miviRebT Semdeg or tolobas
i = 1
(
*
+ i )12 −1 (14)
da
*
12
i = 1+ i −1 (15)
amrigad Tanxis dabandeba sargeblis p rTuli ganakveTiT eqvivalen-
turia Tanxis dabandebisa sargeblis rTuli p* ganakveTiT, sadac i da i*
p
dakavSirebulia (14) da (15) formulebiT. cxadia, rom p* ≠
. ufro metic me-
12
p
(15) tolobis gamoyenebiT martivad vaCvenebT, rom p* <
.
12
marTlac, radgan (1+x)m<1+mx, Tu 0<m<1 da x>0 amitom (15) tolobidan
1
1
i
miviRebT 121 + i −1 = (i + )
1 12 −1 < 1 +
i −1 =
, maSasadame *
i
i <
;
12
12
12
aqedan vaskvniT, rom Tu sargeblis wliuri rTuli ganakveTia P, maSin
p
Tanxis moTavseba bankSi rTuli Tviuri
ganakveTiT momgebiania
12
meanbresaTvis da wamgebiania bankisaTvis.
analogiurad SegviZlia davamyaroT kavSirebi, Tu sawyisi da saboloo
Tanxebi erTi da igivea
1. sargebelis kvartalur da wliur rTul ganakveTebs Soris;
2. sargebelis Tviur da wliur rTul ganakveTebs Soris;
3. sargeblis dRiur da wliur rTul ganakveTebs Soris.
studentebs vurCevT TviTon Caataros msjelobani da miRebuli
Sedegebidan gaakeTos daskvnebi.
magaliTi 9. ganvsazRvroT sargeblis Tviuri rTuli 2 %-iani ganakve-
Tis eqvivalenturi rTuli ganakveTi.
mocemulia, rom p*=2%; saZiebelia p. (14) formulis Tanaxmad
i = 1
(
*
+ i )12 −1 = 1
( + 02
,
0
)12 −1 = ,
1 2682413 −1 ≈ ,
0 2682
radgan = p
i
≈ ,
0
;
2682 p=26,82%
100
23
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ras ewodeba procenti? ricxvis procenti?
2. ras ewodeba martivi procenti? rTuli procenti?
3. ra aris daricxvis periodi?
4. ras ewodeba sargebeli?
5. ra formuliT gamoiTvleba saboloo dagrovili Tanxa?
II. praqtikuli savarjiSoebi:
1. bankSi dabandebulia 5000 lari 2 wlis vadiT sargeblis wliuri
9 % -iani (martivi) ganakveTiT. gamoTvaleT dagrovili Tanxa.
2. firmam Semnaxvel bankSi daabanda 25 000 lari (oTxi TviT) dawyebuli
18 ivlisisdan. wliuri saprocento ganakveTia i=0,09 (martivi).
gamoTvaleT dagrovili Tanxa 18 noembrisaTvis.
3. moqalaqem daabanda 45 000 lari 03 ivlisidan 01 seqtembramde wliuri
i=0,08 saprocento ganakveTiT (martivi). ra Tanxas miiRebs igi
01 seqtembers?
4. ra Tanxa unda davabandoT bankSi martivi wliuri 8 % saprocento
ganakveTiT, rom 9 Tvis Semdeg miviRoT 60 000 lari?
5. moqalaqem bankSi daabanda 5000 lari 5 wliT. ra Tanxas miiRebs igi am
periodis gasvlis Semdeg, Tu banki iZleva sargebels 9 % wliuri
saprocento ganakveTiT (rTuli).
6. 200 000 lari dabandebulia 3 wliT yovelwliuri i=0,06 saprocento
ganakveTiT (rTuli). gamoTvaleT saboloo Tanxa, Tu procentebis
daricxva xdeba yoveli Tvis bolos.
7. mamam gadawyviT Tavisi danazogi 30 000 lari gaunawilos sam Svils,
pirobiTad A, B da C - s da daubandos bankSi im angariSiT, rom
TiToeuls 18 wlis asakSi daugrovdes Tanabari Tanxa. gamoTvaleT ra
Tanxebs daubandebda Svilebs, Tu im momentSi Svilebis wlovaneba iyo
24
12, 13 da 16 weli Sesabamisad da banki iZleoda sargebels i=7,5 %
saprocento ganakveTiT (rTuli).
8. ra Tanxa dagrovdeba 5 wlis Semdeg, Tu bankSi yovelwliuri
perioduli Senatani aris 5
000 lari, xolo yovelwliuri
saprocento ganakveTia 9 % (rTuli)?
$ 4 dikontireba. jamuri (dagrovili) sargebeli. grZelvadiani
kreditebis dafarva. dafarvis koeficienti. anuiteti.
1. diskonti. sawyisi Tanxis gamoTvlas misi saboloo sididis
mixedviT diskontireba ewodeba. saboloo diskontirebul Kn Tanxis da
sawyis sadiskontirebo K Tanxis sxvaobas diskonti ewodeba. amrigad gvaqvs
D = K − K
n
(1)
SemovitanoT diskontis ganmarteba ise, rogorc es aris ganmartebuli
Tanamedrove ekonomikur leqsikonSi: diskonti _(ingl.) discount– fasdakleba
- da niSnavs:
1.
gansxvavebas
fasiani
qaRaldebis
mimdinare
sabirJo
Rirebulebasa da mis nominals (dafarvis fasi) Soris.
2. gansxvavebas valutis forvardul kurssa (kursi, romelic
dafiqsirebulia garigebis dadebis momentisaTvis, magram gada-
saxdelia momavalSi) da dauyoneblivi gadaxdis kurss Soris.
3. Tamasuqebis aRricxvas gadaxdis dromde daTqmuli
procentebis gamoklebiT.
4. erTi da imave saqonelze fasTa Soris gansxvavebas misi
mowodebis sxvadasxva drois gaTvaliswinebiT;
5. saqonelze fasdaklebas.
dikonturi politika niSnavs centraluri bankebis mier gatarebul
saaRricxvo, fulad sakredito politikas, romelic kreditze saaRricxvo
saprocento ganakveTis gazrdaSi an SemcirebaSi gamoixateba anu gamoiyeneba
sasesxo kapitalis miReba-gacemis regulirebis mizniT. ase, magaliTad,
25
saaRricxvo ganakveTis gazrdiT banki xels uwyobs kreditze moTxovnis
Semcirebas, xolo procentis SemcirebiT masze moTxovnis gaaqtiurebas.
diskontirebis problemas vxvdebiT kapitaldabandebaTa ekonomikuri
efeqtianobis gamoTvlis dros. cxadia, rom martivi procentis SemTxvevaSi
K
K
n
n – diskontirebuli fuladi saSualeba Seadgens K = (
; rTuli
1+ i)n
K
procentis SemTxvevaSi, radgan
n
K =
, amitom fulad saSualebaTa K
n
r
n
Tanxis diskontirebuli mniSvnelobaa
n
K = Kr
n
.
sidides
1
v =
(2)
n
r
ewodeba diskontis koeficienti.
maSasadame rTuli procentis SemTxvevaSi
n
K = K v
n
.
e.i. Kn
fulad saSualebaTa diskontirebuli mniSvnelobis
gamosaTvlelad am mniSvnelobaTa Tanxa mravldeba diskontis koeficientze
im xarisxSi, romelic welTa raodenobis tolia. gamoTvlebis gamartivebis
mizniT praqtikaSi sargebloben cxrilebiT, sadac mocemulia diskontis
koeficientebi, Sesabamisi xvedriTi procentuli ganakveTebisa da
wlebisaTvis.
magaliTi 1. gamovTvaloT ra Tanxa unda Seitanon Semnaxvel
salaroSi, rom yovelwliurad 3 %-is daricxviT aTi wlis Semdeg miviRoT
200 000 lari.
cxadia, K=K10v10=200 000v10. cxrilSi vpoulobT, rom v10=0,74409,
maSasadame diskontirebuli Tanxa aris K= 200 000⋅ 0,74409=148 918 lari.
rodesac ar gagvaCnia diskontis koeficientebis cxrili, maSin
vsargeblobT logariTmebis cxriliT, kerZod K= 200 000v10 – valogariTmebT,
gveqneba
log K = log 20000 −10log ,
1 03 saidanac gamoTvlebiT vRebulobT
K= 148 918 lars.
magaliTi 2. erTi da igive warmoebisaTvis arsebobs manqanis SeZenis
ori varianti. pirveli manqanis eqspluataciis vada sami welia, meore
manqanis eqvsi weli. manqanis SeZenisa da eqspluataciis danaxarjebi
mocemulia Semdegi cxriliT:
26
weli 1 2
3
4
5
6
I manqana
1000
200
400
II
manqana 1700 100 200 300 400 500
gamoTvaleT, romeli ufro xelsayrelia: 2-jer SeviZinoT pirveli
saxis manqana, Tu erTxel meore saxis manqana (vigulisxmoT, rom xvedriTi
procentuli ganakveTi 10-is tolia).
pirveli manqanis SeZenisa da eqspluataciis diskontirebuli
danaxarjebi Seadgens
200 400 1000 200 400
100 +
+
+
+
+
= 2647
1
,
1
1
,
1 2
1
,
1 3
1
,
1 4
1
,
1 5
meore manqanis SeZenisa da eqspluataciis diskontirebuli
danaxarjebi iqneba:
100 200 300 400 500
1700 +
+
+
+
+
= 2765
1
,
1
1
,
1 2
1
,
1 3
1
,
1 4
1
,
1 5
miRebuli Sedegebidan vaskvniT, rom ufro xelsayrelia pirveli
varianti.
2. jamuri (dagrovili) sargebeli. vTqvaT bankidan sesxad aRebulia
K lari, n wlis vadiT sargeblis martivi wliuri P%-iani ganakveTiT.
SevadginoT valis gadaxdis gegma, Tu movale banks yovelwliurad
K
ubrunebs
lars da mimdinare wlis valis P%. gamovTvaloT bankis mogeba.
n
amovweroT cxrilis saxiT wlebis mixedviT sesxebisa da Sesabamisi
sargeblis gadasaxadebis monacemebi:
wlebis
wlis
erTi wlis
ori wlis
...
n wlis Semdeg
raodenoba
dasawyisi
Semdeg
Semdeg
K
K
K
sesxi
K
K −
K − 2
...
K − n
= 0
n
n
n
wliuri
KP
⎛
K ⎞ KP
⎡
K ⎤ KP
_
⎜ K −
⎟
...
K − (n − )
1
⎢
⎥
sargebeli
100
⎝
n ⎠100
⎣
n ⎦ 100
am cxrilidan Cans, Tu ra Tanxa unda gadaixados yovelwliurad
debitorma (debitori anu movale, fulis msesxebeli, kreditori,U anu
27
mevale - fulis gamxsesxebeli). yovlwliuri valebi (meore sveti) Seadgenen
K
n wevrian ariTmetikul progresias mniSvneliT -
.
n
SevniSnoT, rom bankis mogebas warmoadgens yovelwliurad darCenili
valebis Sesabamisi sargeblis P %-iani gadasaxadebi. gamoviTvaloT bankis
mogeba n wlis Semdeg amisaTvis SevniSnoT, rom gadasaxdeli sargeblis
mimdevroba (mesame sveti) warmoadgens ariTmetikul progresias, romlis
KP
⎡
K ⎤ KP
KP
pirveli wevri a =
K − (n − )
1
d =
1
, bolo wevria
, sxvaoba
,
100
⎢
⎥
⎣
n ⎦ 100
100n
xolo wevrTa ricxvia n. ariTmetikuli progresiis wevrTa jamis formulis
Tanaxmad gveqneba:
1 ⎧ kp
⎡
k ⎤ p ⎫
S = ⎨
+ k − (n
⎢
− )
1 ⎥
⎬n
n
2 ⎩100 ⎣
n ⎦100⎭
elementaruli gardaqmnebiT miviRebT
KP( + )
1
S =
n
n
200
Sn - Tanxas uwodeben jamur (dagrovil) sargebels n wlis
ganmavlobaSi. es aris Tanxa, romelsac mogebis saxiT miiRebs banki n wlis
ganmavlobaSi gasesxebul K kapitalTan erTad.
magaliTi 3. 1000 lari aRebulia sesxaT aTi wlis vadiT sargeblis
wliuri martivi 5%-iani ganakveTiT. vipovoT jamuri (dagrovili) sargebeli
aTi wlis Semdeg, Tu movale banks yovelwliurad ubrunebs 100 lars da
mimdinare wlis valis 5%-s. gamovTvaloT bankis mogeba.
K
1000
radgan
=
= 100 lari da igi ixdis mimdinare wlis vadis 5%-s,
n
10
amitom kmayofildeba magaliTi 3-is pirobebi. SegviZlia gamoviyenoT jamuri
sargeblis Sn -is formula, miviRebT:
1000⋅5⋅11
S =
= 25⋅11 = 275
10
lari.
200
maSasadame jamuri sargebeli aTi wlisSemdeg Seadgens 275 lars.
magaliTi 4. 2000 lari gacemulia sesxad sargeblis 5%-iani
ganakveTiT, saboloo Tanxaa 3000 lari. gansazRvreT sesxis xangrZlivoba.
K − K
1⎛ K
⎞
1 ⎛ K
⎞ ⎛ K
⎞100
rogorc viciT n
n
n
n
n
=
= ⎜
−1⎟ =
⎜
−1⎟ = ⎜
−1⎟
ik
i ⎝ K
p
⎠
⎝ K
⎠ ⎝ K
⎠ p
100
28
Cven SemTxvevaSi Kn =3000, K =2000=5, amitom
⎛ 3000
⎞100 ⎛ 3
⎞
1
n = ⎜
−1⎟
= ⎜ −1⎟20 = ⋅20 = .
10
⎝ 2000
⎠ 5
⎝ 2
⎠
2
maSasadame sexis xangrZlivoba aTi welia.
3. grZelvadiani kreditebis dafarva. sakredito dawesebulebebis
mier grZelvadiani kreditebis gacemisas damuSavebuli unda iyos
kreditebis gadaxdis gegma, e.i. garkveuli wlebis ganmavlobaSi movalisagan
kreditebis dafarvis gegma, amasTan, movale ixdis yovelwliur gadasaxads
ise, rom garkveuli wlebis Semdeg dafaros mTeli krediti aRebuli TanxiT
sargeblobis procentis CaTvliT. Cveulebrivi kreditis dasafarav Seatans
ixdian yoveli wlis bolos.
vTqvaT, K aris krediti, gacemuli n wliT, xolo R kreditis
dasafaravi Sesatani, romelsac ixdian yoveli wlis bolos.
pirvel Sesatans ixdian wlis bolos da maSasadame misi
diskontirebuli sidide aris Rv,
meore Sesatans ixdian meore wlis bolos da maSasadame misi
diskontirebuli sidide aris Rv2
- - - - - -
ukanasknel Sesatans ixdian n wlis bolos, da maSasadame misi
diskontirebuli sidide aris Rvn.
krediti dafaruli iqneba, Tu yvela gadaxdili Sesatanebis sawyisi
Rirebuleba kreditiT warmodgenili fuladi saSualebebis tolia e.i. Tu
2
K = Rv + Rv +
+
n
Rv = Rv
L
( 2
n 1
−
v + v +
+ v
L
)
sadac, frCxilebSi moTavsebuli gamosaxuleba geometriul
n 1
−
n−
v
−
2
1
1
progresias warmoadgens mniSvneliT v , amitom v + v +
+ v
=
L
,
v −1
v(vn − )
1
maSasadame K = R
, saidanac
v −1
K (v − )
1
R =
(3)
v( n
v − )
1
radgan v = 1/ r, gveqneba
29
⎛ 1
⎞
K⎜ −1⎟
⎝ r
⎠
(1− r) n
n
r
r (r − )
1
R =
= k
= K
; viciT, rom r = 1+ i ,
1 ⎛ 1
⎞
1 − n
n
r
r −1
⎜
−1⎟
r ⎝ n
r
⎠
amitom
(1+i)ni
R = K (
(4)
1+ i)n −1
gamosaxulebas
(1+i)ni
(
(5)
1+ i)n −1
ewodeba dafarvis koeficienti
sxvadsxva procentuli ganakveTebisa da vadebisaTvis arsebobs am
koeficientebis gamosaTvleli cxrilebi (ix. cxrili 3)
magaliTi 5. cnobilia, rom K=5 000 000, i=0,05; n=12 unda gavigoT R .
mesame cxrilidan vpoulobT, rom dafarvis koeficientia 0,11283 e.i.
kreditis dasafaravad saWiro Sesatania R = 5000000 ⋅ 0 11283
,
=
lari
564150
miRebuli krediti, rom daifaros 12 wlis ganmavlobaSi movalem
yovelwliurad unda gadaixados 564 150 lari. igi mTlianad Seitans 6769800
lars. kreditoris mogebaa 6769800-5000000=1769800 lari.
magaliTi 6. ganvsazRvroT yovelTviuri gadasaxadi Tu 100 000 lari
aRebulia valad sargeblis rTuli 8%-iani ganakveTiT 25 wlis vadiT.
mTlianad ra Tanxas gadaixdis movale.
P
8
pirobis Tanaxmad, K=100 000, i=
=
= 08
,
0
; 1+i=1,08 n=25. mocemul pi-
100 100
robebSi dafarvis koeficientia
,
1 0825 ⋅ ,
0 08
8484726
,
6
⋅ ,
0 08
5478778
,
0
=
=
= ,
0 0936787
,
1 0825 −1
8484726
,
6
−1
8484726
,
5
wliuri Sesatani gamoiTvleba (4) farmuliT.
R = 100 000 ⋅ 0936787
,
0
=
87
,
9367
lari.
R
e.i. movale Tviurad gadaixdis
= 9367 87
,
:12 = 780,65583 ≈ 780,
lars
66
,
12
xolo mTlianad gadaixdis 25К = 25 ⋅ 9367687 =
lars
234196675
.saintereso, Tu
ramdenad daiklebs vali erTi wlis Semdeg?
30
amocanis pirobis Tanaxmad movale ixdis fiqsirebul gadasaxads
yovelTviurad, maSin rodesac 8%-iani daricxva valze xdeba yovel
wliurad. vali pirveli wlis gasvlis Semdeg aRvniSnoT K1, cxadia, rom
K = K + K ⋅ 0,08 − R = ,
1 08K − R = 108000 − 9367 87
,
= 98632
1
lari
13
,
. e.i. valma daik-
lo 100000-98632,13=1367,87 lariT, miuxedavad imisa, rom gadaxdilia
9367,87 lari. (!)
4. anuiteti. zemoT Cven miviReT
n
K = K v
n
formula erTi saboloo
Kn Tanxis diskontirebisaTvis. axla gavecnoT, rogor xdeba sasruli
raodenobis saboloo Tanxebis dikontireba sxvadasxva drois periodis
mixedviT. aseTi tipis amocanebTan maSin gvaqvs saqme, rodesac xdeba raime
Tanxis dabandeba im pirobiT, rom drois yoveli fiqsirebuli periodis
(magaliTad, yoveli wlis) Semdeg xdeba erTi da imave fiqsirebuli Tanxis
moxsna am anabridan, aseT process ekonomikaSi anuiteti ewodeba. e.i.
anuiteti ewodeba fuladi nakadebis mimdevrobas.
drois raime intervalis, magaliTad n wlis Sesabamisi anuitetis
sawyisi Tanxa ewodeba im Tanxas, romelic uzrunvelyofs yovelwliur
fiqsirebul K0 Tanxis nakads, (romelic ixsneba angariSidan).
n wlis Semdeg saanabro angariSi ixureba e.i. saanabro angariSze
darCenili Tanxa 0-is tolia. maSasadame, anuiteti yovelwliuri
gadasaxadia. igi grZelvadiani sesxis saxeobaa, romlis drosac kontraqtiT
dadgenili sasesxo periodis gavlis Semdeg kreditori ZiriTad TanxasTan
erTad yovelwliurad iRebs saprocento ganakveTiT dadgenil procentis
Tanxas.
magaliTi 7. vipovoT sawyisi Tanxa im anuitetisa, romelic yoveli
wlis bolos iZleva 10 000 lars Semosavals 10 wlis ganmavlobaSi, Tu
sargeblis wliuri rTuli ganakveTia 7%.
cnobilia, rom yovelwliuri gamosatani Tanxa R = 10000 lari, n = 10
P
da 1+ i = 1+
=1+ 07
,
0
= 07
,
1
; unda gamovTvaloT K -sawyisi Tanxa. viciT, rom
100
(1+i)n
(1+i)n −1
R =
i
K (
- formula (4), saidanac K = R
; maSasadame, anuiteti
1+ i)n −1
i(1+ i)n
warmoadgens grZelvadiani kreditis dafarvis Sebrunebul amocanas.
miRebul formulaSi CavsvaT ricxviTi mniSvnelobebi, miviRebT
31
,
1 0710 −1
,
1 9671511 −1
0,9671511
K = 10000
= 1000
= 10000
=
0,07 ⋅ ,
1 0710
0,07 ⋅ ,
1 9671511
0 1377005
,
= 10000 ⋅ 7,0235845 =
lari
845
,
70235
amrigad, amocanaSi aRniSnuli anuitetis sawyisi Tanxaa 70235,845 lari.
igi uzrunvelyofs pirobas: meanabrem aT weliwadSi miiRos 100 000 lari.
rogorc, vnaxeT anuitetis Tanxa n wlis manZilze, romelic yoveli
wlis bolos iZleva K lar Semosavals, sargeblis wliuri rTuli
procentiani ganakveTiT, gamoiTvleba formuliT
(1+i)n −1
K = R
(6)
i( + i) ,
1
n
sadac n – welTa ricxvi fiqsirebulia, R - yovelwliurad kreditis
dasafaravi Sesatania. saintereso SemTxvevaa, rodesac welTa ricxvi ragind
didia, e.i. n → ∞ . am SemTxvevis gansaxilvelad saWiroa zemoTmoyvanil
formulaSi gadavideT zRvarze roca n → ∞ , gveqneba
(1+i)n −1
lim K = lim R
, (7)
n→∞
n→∞
i(1+ i)n
R
(1+i)n −1 R ⎡
1
⎤ R ⎡
1
⎤
K =
lim R
lim 1
1 lim
(8)
n→∞
i
i(1+ i) =
n
⎢ −
n→∞
i
⎣
(1+i)n ⎥ = ⎢ −
⎦
i ⎣ n→∞ (1+ i)n ⎥⎦
radganac i > 0 , amitom 1 + i > 1 e.i. lim(1+ i)n = ∞ . miRebuli Sedegi
n→∞
gaviTvaliswinoT ukanasknel tolobaSi, miviRebT
R
K =
(9)
i
maSasadame, anuitetis Tanxa, rodesac welTa ricxvi sakmarisad didia,
tolia yovelwliurad kreditis dasafaravad Tanxa gayofili xvedriT
procentul ganakveTze.
magaliTi 8. Tu magaliT 7-Si SevcvliT pirobas `aTi wlis
ganmavlobaSi~ pirobiT `welTa ricxvi sakmarisad didia~ ( n → ∞ ), miRebuli
10 000
formulis Tanaxmad miviRebT: K =
=142857 14
, lari.
,
0 07
maSasadame Tu dabandebulia 142857,14 lari, maSin anuiteti
yovelwliurad iZleva 10 000 lar Semosavals.
32
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ra aris diskonti? diskontireba?
2. ra aris sargeblia martivi ganakveTi? rTuli ganakveTi?
3. rogor finansur operacias ewodeba anuiteti?
4. ras ewodeba anuitetis sawyis Tanxa?
5. risi tolia yovelwliurad kreditis dasafaravad Tanxa gayofili
xvedriT procentul ganakveTze?
II. praqtikuli savarjiSoebi
1. ra Tanxa unda davabandoT bankSi, rom 20 wlis Semdeg dagrovdes
100 000 lari, Tu banki iZleva i =5% (rTul) yovelwliur sargebels?
2. gamoTvaleT bankSi dabandebuli sawyisi Tanxa, Tu cnobilia , rom
yovelwliuri saprocento ganakveTi i =5% (rTul) da 4 wlis Semdeg
saboloo Tanxam Seadgina 18 232,5 lari.
3. ra Tanxa unda davabandoT 3% (rTuli) wliuri saprocento
ganakveTiT, rom 4 wlis Semdeg dagrovdes 60 775 lari?
4. ramdeni wliT unda davabandoT 1 000 lari i =0,05 (rTul) wliuri
saprocento ganakveTiT, rom dagvigrovdes 13 200 lari?
5. krediti, romelic Seadgens erT milion lars, unda daifaros 20
weliwadSi yoveli wlis bolos valis erTnairi Tanxis gadaxdiT. ra Tanxa
unda ixados movalem (debitorma) yovelwliurad, rom krediti dafaros, Tu
i =0,05 (rTuli).
6. movalem (debitorma) aiRo krediti 120 000 laris raodenobiT 8
wliT. gamoTvaleT kreditis dafarvis yovelwliuri R Senatani da jamuri
davalianeba, Tu yovelwliuri saprocento ganakveTia i =0,05 (rTuli).
7. kompaniam 5 wlis ganmavlobaSi unda miiRos yovelwliurad
Tanabari R=10
000 lari. ipoveT aseTi fuladi nakadebis mimdevrobis
(anuitetis) mimdinare (sawyisi) Rirebuleba, Tu yovelwliuri saprocento
ganakveTia i =0,10 (rTuli).
33
8. ipoveT mimdinare Rirebuleba im anuitetisa, romelic yoveli wlis
bolos iZleva 5 000 lar Semosavals 5 wlis manZilze, Tu yovelwliuri
saprocento ganakveTia i =5 % (rTuli).
$ 5 investiciebis Sefaseba da Sedareba
Cveni mizania SeviswavloT gansxvavebul parametriani sainvesticio
proeqtebi finansuri momgebianobis TvalsazrisiT, romelic saSualebas
mogvcems ramdenime proeqtidan SevarCioT CvenTvis yvelaze xelsayreli
winadadeba.
sailustraciod amovxsnaT Semdegi konkretuli amocana: vTqvaT
sainvesticio proeqti iTxovs 15 000 laris investirebas da garantias
iZleva, rom sam weliwadSi igi daubrunebs investors 20 000 lars. amasTan
cnobilia, rom safinanso bazarze dominanturi wliuri rTuli ganakveTi
5%-ia.
gamovTvaloT:
(1) 15 000 laris Sesabamisi saboloo Tanxa 3 wlis Semdeg;
(2) 20 000 laris Sesabamisi diskontirebuli Tanxa, Tu drois
intervali 3 weliwadia;
(3) sargeblis ra wliuri rTuli ganakveTi Seesabameba Tanxis zrdas
3 wlis intervalSi 15 000-dan 20 000 laramde?
(4) sasurvelia Tu ara finansuri TvalsazrisiT aseTi investicia?
(5) Seicvleboda Tu ara (d) punqtis rekomendacia, sargeblis
dominanturi wliuri rTuli ganakveTi, rom 12% yofiliyo.
(1) K = 15000, n
= ,
3 P = 5 . unda gamovTvaloT K3 .
rogorc viciT
34
K = K + i =
+
⋅
=
⋅
=
3
(1 )3 15
(
000 1
,
0 005)315 000 ,
1 053 15 000 157625
,
1
375
,
17364
lari.
(2) unda gamovTvaloT 20 000 laris diskontirebuli Tanxa, e.i. misi
Sesabamisi sawyisi Tanxa.
K
K =
=
=
3
(
lari.
1 + i)
20000 : 157625
,
1
17276,751
3
(3) sami wlis ganmavlobaSi 15 000 – dan 20 000 laramde zrdis
Sesabamisi sargeblis wliuri ganakveTi, rogorc viciT gamoiTvleba
formuliT:
3
3
K
K
20 000
2
K = K + i
+ i
=
i =
− =
− =
− ≈
3
(1 1) ; (1 1)
3 ;
3
1
1 3
3
1
1
,
0
1
;
K
K
15 000
5
P
miviReT, rom i ≈ 1
,
0
1
= i
P = i 100 = 1
,
0 ⋅100 =
1
magram
1 ;
10 , maSasadame
100
1
1
saZiebeli sargeblis ganakveTia P = 10%
1
.
(4) gavaanalizoT. sasurvelia, Tu ara amocanaSi aRwerili
investiciis ganxorcieleba. amocanis pirobis Tanaxmad investori sawyisi
15 000 laris nacvlad ibrunebs 20 000 lars, e.i. igebs 5 000 lars. mas, rom
sawyisi fuli daebandebina bazarze, maSin miiRebda 17 364 lars, da mogeba
darCeboda 2 364 lari, sanacvlod 5 000 larisa. maSasadame sainvesticio
proeqtSi Tanxis dabandeba finansurad ufro momgebiania. Tu
gavaalalizebT (2) punqtSi miRebul Sedegs, kvlav igive daskvnamde mivalT.
marTlac, imisaTvis, rom safinanso bazarze sami wlis Semdeg miviRoT
sainvesticio proeqtiT SemoTavazebuli 20 000 lari, amisaTvis bazadze
unda dabanddes sawyisi Tanxa 17 276 lari, rac 2 276 lariT aRemateba
sainvesticio proeqtiT moTxovnil Tanxas.
sainvesticio proeqtiT gaTvaliswinebul saboloo Tanxis Sesabamis
diskontirebul sididesa da sainvesticio proeqtiT moTxovnil sawyisi
Tanxis sidides Soris sxvaobas wminda sawyisi sidide ewodeba. Cven
SemTxvevaSi wminda sawyisi sididea
17 276,751- 15 000=2276,751 lari.
cxadia, Tu wminda sawyisi sidide dadebiTia, maSin sainvesticio
proeqtSi monawileoba, finansurad momgebiania. es ukanaskneli aris
35
erTerTi ZiriTadi kriteriumi sainvesticio proeqtis momgebianobis
Sesafaseblad.
meores mxriv (3) punqtSi vaCveneT, rom sainvesticio proeqtSi
monawileoba tolfasia sargeblis wliuri, rTuli 10%-iani ganakveTiT
sawyisi Tanxis dabandebisa, rac aRemateba sabazro 5%-ian ganakveTs,
maSasadame sainvesticio proeqtSi monawileoba finansurad ufro
momgebiania.
sargeblis im wliur rTul ganakveTs, romelic sainvesticio drois
periodSi uzrunvelyofs sainvesticio Tanxis zrdas, sainvesticio
proeqtiT gansazRvrul saboloo Tanxamde, ewodeba sargeblis Siga
ganakveTi.
Cven SemTxvevaSi sargeblis Siga ganakveTia 10%. cxadia, Tu sargeblis
Siga ganakveTi metia safinonso bazris sargeblis dominatur ganakveTze,
maSin sainvesticio proeqti momgebiania. es pirobac erTerTi kriteriumia
sainvesticio proeqtebis momgebianobis Sesafaseblad. cxadia, igi srul
TanxmobaSia zemoT Camoyalibebul kriteriumTan, romelic proeqtis
momgebianobas afasebs wminda sawyisi sididis saSualebiT.
(5) Tu safinanso bazris sargeblis ganakveTi gaxdeba 12 %, maSin igi
gadaaWarbebs sainvesticio proeqtis Siga ganakveTs. amitom am SemTxvevaSi
proeqtSi monawileoba ara sasurvelia finansurad, radgan safinanso
bazarze dabandeba ufro met mogebas moutans investors, vidre
sainvesticio proeqtSi monawileoba. marTlac, 15 000 lari dabandebuli
12%-s mogvcems.
K = 15 000 +
=
⋅
=
⋅
=
3
(1 12
,
0
)3 15 000 12
,
1 3 15 000 ,
1 404928
92
,
21073
lari, rac 1073,92
lariT metia sainvesticio proeqtiT gaTvaliswinebul saboloo Tanxaze.
gavecnoT wminda sawyisi sididis da sargeblis Siga ganakveTis
gamosaTvlel formulebs. SemovitanoT aRniSvnebi:
wminda sawyisi sidide aRvniSnoT NPV (Net present value) simboloTi;
sargeblis Siga ganakveTi aRvniSnoT IRR (Internal rate of returne)
simboloTi.
vTqvaT:
36
K1 - proeqtiT gaTvaliswinebuli sawyisi sainvesticio Tanxaa;
K2 -proeqtiT gaTvaliswinebuli saboloo Tanxaa, romelic ubrundeba
investors K < K
1
2 ;
Pm -safinanso bazris (dominanturi) sargeblis wliuri rTuli
ganakveTi;
n -sainvesticio periodis xangrZlivoba;
NPV -ganmartebis Tanaxmad aris K2 Tanxis Sesabamisi diskontirebuli
Tanxisa da sawyisi K
K
1 Tanxis sxvaoba, magram
2 -is diskontirebuli
K
⎛
Pm ⎞
sididea
2
−n
= K 1
( + i ) , sadac i -iT ⎜i =
m
⎟ aRniSnulia safinanso
n
2
m
1
( + i )
m
100
m
⎝
⎠
bazris (dominanturi) sargeblis wliuri xvedriTi rTuli ganakveTi.
maSasadame
⎛
P
−n
⎞
NPV = K 1
( + i )−n − K
NPV = K 1
m
+
− K
2
m
1 anu
2 ⎜⎜
⎟
⎟
1 (1)
⎝
100 ⎠
gamovTvaloT, ra rTuli ganakveTi Seesabameba sawyis K1 Tanxis
zrdas saboloo K2 Tanxamde n wlis ganmavlobaSi. Tu ix avRniSnavT
sargebelis saZiebel xvedriT ganakveTs, gveqneba
n
K
K
K = K
+ i
+ i = n
i = n
−
2
1 (1
x ) ;
1
1 ;
1
;
1
x
x
K
K
2
2
e.i.
K
1
i = n
−1
x
(2)
K2
P
IRR funqciis asagebad gaviTvaliswinoT, rom
x
i =
x
maSin
100
(2) formulidan miviRebT
⎡ K
⎤
P = IRR
x
(
)
2
= ⎢n
−1⎥
%;
100
(3)
⎣ K1
⎦
Tu saqme exeba sxvadasxva sainvesticio proeqtebis Sedarebas, saWiroa
gamoviyenoT orive (1), (2) an (3) kriteriumi ufro momgebiani proeqtis Se-
sarCevad. rodesac saqme exeba sxvadasxva Tanxebs, maSin IRR kriteriumi
yovelTvis ver iZleva finansur mogebis TvalsazrisiT saukeTeso proeqtis
arCevis saSualebas. aseT SemTxvevaSi unda mivmarToT NPV kriteriums. am
mizniT ganvixiloT
37
magaliTi 1. vTqvaT iuridiul pirs aqvs 30 000 lari da Tanxis
investireba SesaZlebelia or A da B proeqtebidan mxolod erTSi.
A proeqti iTxovs sawyis sainvesticio Tanxas (A)
K
=1000
1
lars da n=4 wlis
Semdeg abrunebs (A)
K
= 1200
2
lars. B proeqti iTxovs sawyis sainvesticio
Tanxas
(B)
K
= 30 000
(B)
K
= 35
1
lars da n=4 wlis Semdeg abrunebs
000
2
lars.
cnobilia, rom safinanso bazris sargeblis wliuri rTuli ganakveTia
P = 3%
m
. romel proeqtSi ufro momgebiani fulis dabandeba?
zemoT moyvanili daskvnebis Tanaxmad, unda gamovTvaloT NPV da IRR
orive proeqtisaTvis, miRebuli Sedegebi SevadaroT erTmaneTs A da B mis
safuZvelze miviRoT Sesabamisi gadawyvetileba.
gvaqvs
(NPV )
( A)
= K
1
( + i )−n − K = 1200 ⋅ ,
1 03 4
− −1000 = 1066 1844
,
−1000 ≈ 66
A
2
m
i
lari
18
,
.
(NPV )
( B )
= K
1
( + i )−n − K = 35000 ⋅ ,
1 03−4 − 30000 =
B
2
m
i
= 35000 : 1255088
,
1
− 30000 ≈ 1097,
lari
046
radgan, orive SemTxvevaSi NPV dadebiTia, amitom bazarTan SedarebiT
orive proeqti momgebiania: amovarCioT optimaluri proeqti. gvaqvs fulis
dabandebis Semgedgi ori varianti:
1. investori aTas lars daabandebs A proeqtSi; darCenil 29 000 lars –
bazarze. bazridan ki is miiRebs 29 000 ⋅ ,
1 034 =
,
32639 76 laris. am varian-
tidan investori miiRebs jamSi 1 200+32639,76=33839,76 lars.
2. investori ocdaaTi aTas lars mTlianad daabandebs B proeqtSi. maSin
igi 4 wlis Semdeg miiRebs 35 000 lars. cxadia, rom B proeqtSi
monawileoba ukeTesia, radgan igi 35 000-33839,76=1160,24 lariT met
Semosavals iZleva.
gamovTvaloT sargeblis Siga ganakveTi –IRR orive proeqtisaTvis
IRR =
A
[4 ,12 − ]
1100 ≈ ,
4 7(%)
⎡ 35000 ⎤
IRR = 4
B
⎢
−1 100
⎥
≈
(%)
9
,
3
30000
⎣
⎦
rogorc vxedavT ufro maRali sargeblis Siga ganakveTi aqvs A
proeqts, Tumca rogorc vnaxeT B proeqtSi monawileoba ufro momgebiania,
vidre A proeqtSi. amrigad, IRR kriteriumi gansxvavebuli Tanxebis
SemTxvevaSi araa sando, radganac patara Tanxis didi procenti SeiZleba
38
ufro mcire aRmoCndes, vidre didi Tanxis patara procenti (rogorc es
aRmoCnda ganxilul amocanaSi).
axla ganvixiloT Semdegi tipis amocana, romelic exeba sainvesticio
proeqtebis Sedarebas, rodesac investori iRebs mravaljerad Semosavals
garkveuli periodebis Semdeg.
vTqvaT investors aqvs 20 000lari, romelic SeiZleba dabanddes ori
A da B sainvesticio proeqtebidan mxolod erTSi. dabandebuli 20 000 laris
SemTxvevaSi TiToeuli proeqti iRebs garantias, rom 4 wlis ganmavlobaSi
yovelwliurad daubrunebs investors garkveul
Tanxebs romlebic
mocemulia Semdegi cxrilis saxiT:
weliwadi 1
2
3
4
sul
investoris A proeqti
a1 =6 000
a2 =6 000
a3 =10 000
a4 =8000 27
000
Semosavali B proeqti b1 =10 000 b2 =6 000
b3 =9 000
b4 =1 000
26 000
SevarCioT, romel proeqtSia umjobesi Tanxis dabandeba, Tu bazarze
sargeblis wliuri rTuli ganakveTia 11 %.
garegnulad A proeqti ufro momgebiani Cans vidre B proeqti, radgan
igi TiTqos da 4 weliwadSi 1 000 lariT met Semosavals iZleva.
gamovikvliT marTlac asea Tu ara.
cxrilidan, Cans, rom erTidaigive 10 000 lar gadasaxads A proeqtiT
investori iRebs mesame wlis bolos, xolo B proeqtiT pirveli wlis
bolos. radgan es miRebuli 10 000 lari investors SeuZlia kvlav
daabandos bazarze, amitom A da B proeqtebis aRniSnuli gadasaxadi ar
aris erTmaneTis eqvivalenturi. aSkaraa, rom B proeqtis mier pirveli
wlis bolos gadaxdili 10 000 lars ufro meti `fasi~ aqvs, vidre A
proeqtis mier gadaxdil igive Tanxas mesame wlis bolos. imisaTvis, rom
gavarkvioT romeli proeqti jobia investorisaTvis, daviTvaloT
proeqtebiT gaTvaliswinebuli gadasaxadebis Sesabamisi diskontirebuli
(sawyisi) Tanxebis jami.
Tu gaviTvaliswinebT, rom bazris sargeblis wliuri rTuli
ganakveTia 11 %, diskontirebuli *
a
b
k da
*
k TanxebisaTvis miviRebT
39
*
6000
*
3000
a =
=
,
5405 405 ≈
,
5405 41 a =
=
8672
,
2434
≈
1
87
,
2434
da a.S.
1
,
1
2
1
,
1 2
diskontirebuli Tanxebi
weliwadi
A proeqti
B proeqti
1
*
a =
,
5405 41
*
b =
,
9000
1
01
1
2
*
a =
87
,
2434
*
b =
,
4869
2
73
2
3
*
a =
91
,
7311
*
b =
3
72
,
6580
3
4
*
a =
85
,
5269
*
b =
,
658
4
73
4
sul 20422,04 21109,14
SevadginoT cxrili miRebuli cxrilidan aSkaraa, rom A proeqtSi
monawileoba tolfasia mimdinare momentSi bazarze 20422,04 laris
dabandebisa, xolo B proeqtSi monawileoba – 21109,14 laris dabandebisa.
amitom B proeqtSi monawileoba ufro momgebiania, e.i. Cveni warmodgena A
proeqtis ufro momgebianobis Sesaxeb mcdari aRmoCnda.
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ras ewodeba wminda sawyis sidide? rogor gamoiTvleba wminda sawyis
sidide?
2. ras ewodeba sargeblis Siga ganakveTi? rogor gamoiTvleba
sargeblis Siga ganakveTi?
3. rogor aRiniSneba wminda sawyis sidide? sargeblis Siga ganakveTi?
4. ra kriteriumebiTaa SesaZlebeli sainvensticio proeqtebis Sefaseba?
II. praqtikuli savarjiSoebi
40
1. iuridiul pirs aqvs 30
000 lari da Tanxis invenstireba
SesaZlebelia ori A da B proeqtebidan mxolod erTSi. A proeqti iTxovs
sawyis sainvensticio Tanxas 1 000 lars da 4 wlis Semdeg abrunebs 1 200
lars. B proeqti iTxovs sawyis sainvensticio Tanxas 30 000 lars da
4 wlis Semdeg abrunebs 35 000 lars. cnobilia, rom safinanso bazris
sargeblis wliuri rTuli ganakveTia 3%. romel proeqtSi ufro
momgebiania fulis dabandeba?
2. sainvensticio proeqti invenstors hpirdeba 20
000 laris
invensticiiT 80 000 laris mogebas 5 weliwadSi. gamoTvaleT sargeblis
Sida ganakveTi da gansazRvreT, momgebiania Tu ara es proeqti, Tu
sargeblis sabazro wliuri rTuli ganakveTia 6 %.
3. vTqvaT, invenstors aqvs saSualeba miiRos monawileoba sami A , B
da C proeqtebidan mxolod erTSi. A proeqti moiTxovs sawyis 20 000
lars, B proeqti - 30 000 lars, xolo C proeqti - 100 000 lars. amasTan,
A proeqti garantias iZleva, rom invenstors daubrunebs 25 000 lars,
B proeqti garantias iZleva, rom invenstors daubrunebs 37 000 lars,
xolo C proeqti - 117 000 lars (sami wlis Semdeg). romeli proeqtia
ufro momgebiani, Tu sabazro wliuri ganakveTia 5 %?
4. firmas aqvs arCevani daabandos 10 000 lari ori A da B proeqtebidan
erTSi. am proeqtebidan Semosavali wlebis mixedviT mocemulia cxriliT
(qvemoT). cnobilia, rom sargeblis sabazro wliuri rTuli ganakveTia 15 %.
romeli proeqtia ufro momgebiani?
weliwadi
1 2 3 4 5
sul
firmis
A proeqti a1 =2 000
a2 =2 000
a3 =3 000
a4 =3 000
a5 =3 000
13 000
Semosavali B proeqti b1 =1 000
b2 =1 000
b3 =2 000
b4 =6 000
b5 =4 000
14 000
41
Tavi II. wrfivi algebris elementebi.
wrfivi algebris elementebis zogierTi gamoyeneba
ekonomikur kvlevebSi
$ 1. matricTa Teoriis elementebi
1. matricis cneba. imisaTvis, rom warmoebulma biznesma mogvitanos
mogeba amitom, warmoebis dagegmva unda emyarebodes saTanadod
mowesrigebul sistemas, romlis daxmarebiTac martivad da mokled
aRiwereba damokidebulebani, romelsac adgili aqvs warmoebaSi. aseTi
mowesrigebuli sistema SeiZleba TvalnaTliv warmovadginoT Sesabamisi
cxriliT.
magaliTad, ganvixiloT materialuri warmoebis sxvadasxva dargebis
mier urTierTmiwodebis informaciis sistema. Tu i=1, 2, 3, 4 saTanadod
aRniSnavs Sesabamisi dargis nomers, maSin produqciis urTierTmiwodebis
cxrils aqvs Semdegi saxe
dargi
I
II
III
IV
I
ϖ
ϖ
ϖ
ϖ
11
12
13
14
II
ϖ
ϖ
ϖ
ϖ
21
22
23
24 (1)
III
ϖ
ϖ
ϖ
ϖ
31
32
33
34
IV
ϖ
ϖ
ϖ
ϖ
41
42
43
44
am cxrilSi ϖ (i, j = ,
1
,
3
,
2
)
4
ij
aRniSnavs produqciis im moculobas,
romelsac i-uri dargi awvdis j-uri dargs. ase magaliTad, ϖ ,ϖ ,ϖ ,ϖ
11
12
13
14
42
Sesabamisad aRniSnaven produqciis moculobebs, romelsac I dargi awvdis
sxva dargebs da a.S.
Tu normebs miviRebT, rogorc informaciis sistemas, maSin
analogiurad SeiZleba aRvweroT warmoebis dagegmva. magaliTad, Tu
warmoeba amzadebs oTxi saxis produqcias: 1,2,3,4 da maTi warmoebisaTvis
saWiroa sami saxis nedleuli 1,2,3, maSin materialur danaxarjTa normebi,
romelic waroadgens momaragebis gegmis safuZvels SeiZleba mocemuli iyos
cxriliT:
a
a
a
a
11
12
13
14
a
a
a
a
21
22
23
24 (2)
a
a
a
a
31
32
33
34
sadac a (i = ,
1
;
3
,
2
j = ,
1
,
3
,
2
)
4
i, j
aRniSnavs i-uri nedleulis danaxarjis
normas j-uri produqtis erTeulis sawarmoeblad. ase magaliTad,
I nedleulis danaxarjis normebi 1,2,3,4 produqciis erTeulze aris
a , a , a , a
11
12
13
14 .
(1) da (2) cxrilebi warmoadgenen e.w. matricis kerZo SemTxvevebs,
romelic ganixileba maTematikaSi. ekonomikur kvlevebSi isini farTo
gamoyenebas pouloben, gansakuTrebiT, ki warmoebis dagegmvaSi.
m× n zomis matrica ewodeba maTkuTxa cxrils, romelic Seicavs m
striqons da n svets. TviTon ricxvebs, romlebic Seadgenen matricas,
matricis elementebi ewodebaT. matricas aRniSnaven A, B, C, ... laTinuri
didi asoebiT, xolo elementebs ormagi indeqsis mqone patara asoebiT ai, j ,
sadac i aRniSnavs striqonis nomers j ki svetis nomers. agreTve gamoiyeneba
(ai,j)n aRniSvna. e.i.
m
⎛ a
a
a
11
12
L
1n ⎞
⎜
⎟
n
a
a
a
A = (a
21
22
L
2
i, j )
= ⎜
n ⎟
m
⎜
⎜ K
K
K
K ⎟⎟
⎝ a
a
a
m1
m 2
L
mn ⎠
imisaTvis, rom A da B matrica erTmaneTs SevadaroT, pirvl rigSi
isini erTnairi zomisa unda iyvnen. Tu A da B matrica erTairi zomisaa:
A = (a
B = b
i, j )n da
( i,j)n , maSin vityviT, rom A da B matricebi tolia Tuki
m
m
a = b
i, j
i, j nebismieri i=1,2,...m da j=1,2,...n-sTvis. aseT SemTxvevaSi A=B.
43
(cxadia, rom matricebis sxvagvar Sedarebaze uazrobaa, igulisxmeba, rom
SeuZlebelia A>B cnebis ramdenadme azriani ganmatreba).
roca matrica mxolod erTo striqonisagan Sedgeba, maSin mas veqtor-
striqons uwodeben, xolo roca – mxolod erTi svetisagan, maSin mas
veqtor-svets uwodeben. maSasadame
⎛b11 ⎞
⎜
⎟
⎜b21 ⎟
A = (a
a
a )
B =
11
12
L
1n
- veqtor-striqonia, xolo,
⎜
⎟ ki -
M
⎜
⎜
⎟
⎟
⎝b 1n ⎠
veqtor-sveti.
Tu matricaSi m=n maSin matricas kvadratuli ewodeba.
matricis elementebs, romelTa striqonisa da svetis nomrebi emTxveva
diagonaluri elementebi ewodebaT da isini adgenen matricis diagonals.
magaliTad, kvadratuli matricis diagonals adgenen a , a ,
, a
11
22 K
nn
elementebi.
Tu matricis yvela elementi nulis tolia, maSin matricas
nulovaniUBewodeba, e.i.
⎛ 0
0
0
L
⎞
n
⎜
⎟
0
0
0
O = (0) =
m
⎜
L
⎟
⎜K K K K⎟
⎝ 0
0
0
L
⎠
Tu matricis yvela elementi, garda diagonalurisa, tolia nulis,
maSin matricas diagonaluri ewodeba. magaliTad diagonaluria matrica
⎛ 1
0
0
L 0 ⎞
⎜
⎟
⎜ 0
2
0
0 ⎟
L
⎜
⎟
A = ⎜ 0
0
3
L 0 ⎟
⎜
⎟
⎜K K K K K⎟
⎜
⎟
⎝ 0
0
0
n
L
⎠
Tu diagonaluri matricis yvela diagonaluri elementi 1-is tolia,
maSin matricas erTeulovani ewodeba:
⎛ 1
0
0
0
L
⎞
⎜
⎟
0
1
0
0
⎜
L
⎟
E =
0
0
1
0
⎜
L
⎟
⎜K K K K K⎟
⎝ 0
0
0
1
L
⎠
44
2. moqmedebani matricebze. A matricis namravli λ ricxvze ewodeba B
matricas (B=λA), Tu bi,j=λai,j (i=1,2,3,....m; j=1,2,3,...n). am ganmartebidan
uSualod gamomdinareobs, rom saerTo Tanmamravli SeiZleba marticis
gareT gamovitanoT.
Tu A da B orive m × n zomis matrica, maSin maTi jami ewodeba eseT C
matricas (C=A+B), romlis elementebic A da B matricis Sesabamisi
elementebis jamis tolia:
c = a + b ,
= ,
1 2, ,
= ,
1 2, ,
ij
ij
ij i
K
m
j
K
n ,
cxadia, rom A+O=A. advili SesamCnevia, rom A-B=A+(-1)B.
rogorc ukve vnaxeT nebismieri ori matricis arc Sedareba SeiZleba
da arc Sekreba. intuiciurad cxadia, garkveuli SezRudvebi iarsebebs
matricebis gamravlebis drosac.
A = (a
B = b
m ×
i, j )n da
( i,j)p , matricebis namravli ewodeba p zomis
m
n
C = (c
c
i,k )p matricas, sadac
elementi aris A matricis i-uri striqonis
m
i,k
elementebis B matricis k-uri svetis Sesabamis elementebze namravlTa jami
e.i.
n
i = ,
1 ,
2
, m
K
c
= a b + a b + + a b = ∑a b
k
i,
i1 1k
i2 2k
L
in nk
ij
jk
(1)
j 1
=
j = ,
1 ,
2
, p
K
SevniSnoT, rom matricebis namravli ganimarteba mxolod im SemTxve-
vaSi, roca pirveli matricis svetebis ricxvi meore matricis striqonebis
ricxvis tolia. magaliTad, rom vipovoT C namravli matricis c23 elementi,
amisaTvis saWiroa A matricis me-2 striqonis elementebi gavamravloT B
matricis me-3 svetis Sesabamis elementebze.
⎛ b ⎞
13
⎜
⎟
⎜b23 ⎟
c = (a
a
a
a ) ⎜
⋅ b ⎟ = a b + a b + a b + a b
23
21
22
23
L
2n
33
21 13
22 23
23 33
2n n3
⎜
⎟
⎜ M ⎟
⎜
⎟
b
⎝ n3 ⎠
magaliTi 1. vipovoT Α⋅Β Tu
⎛1 0 3⎞
⎛1 2 − 3⎞
⎜
⎟
A = ⎜⎜
⎟
⎟ , B = ⎜2 0 0⎟
⎝2 0 − 2⎠
⎜
⎟
⎝3 2 4⎠
45
mocemul magaliTSi A matricis zomebia 2 × 3 da B matricisa 3 × 3 .
namravlis elementebia
⎛1⎞
⎛0⎞
⎜ ⎟
⎜ ⎟
c =
− ⋅⎜ ⎟ = + − = −
c = 1 2 − 3 ⋅ ⎜0⎟ = −
11
(1 2
3) 2
1 4 9
;
4 12 (
)
;
6
⎜3⎟
⎜ ⎟
⎝ ⎠
2
⎝ ⎠
⎛3⎞
⎛1⎞
⎜ ⎟
⎜ ⎟
c =
− ⋅ ⎜ ⎟ = + − = −
c = 2 0 − 2 ⋅ ⎜2⎟ = 2 + 0 − 6 = −
13
(1 2
3) 0
3 0 12
;
9 21 (
)
;
4
⎜4⎟
⎜ ⎟
⎝ ⎠
3
⎝ ⎠
⎛0⎞
⎛3⎞
⎜ ⎟
⎜ ⎟
c =
− ⋅⎜ ⎟ = + − = − c =
− ⋅⎜ ⎟ = + − = −
21
(2 0
2) 0
6 0 8
;
2
22
(2 0
2) 0
0 0 4
;
4
⎜2⎟
⎜ ⎟
⎝ ⎠
4
⎝ ⎠
maSasadame
⎛− 4 − 6 − 9⎞
⎛4 6 9⎞
A ⋅ B =
= (−
⎜
⎜
⎟
⎟
⎜
⎜
)
1
⎟
⎟
⎝− 4 − 4 − 2⎠
⎝4 4 2⎠
Tu ganvixilavT B ⋅ A namravls is saerTod ar iarsebebs, radgam am
SemTxvevaSi B matricis svetTa raodenoba gansxavavebulia A matricis
striqonTa raodenobaze (3≠2).
⎛1 0⎞
⎛−1 2⎞
magaliTi 2. vTqvaT A = ⎜⎜
⎟
⎟ , B = ⎜⎜
⎟
⎟ ,
⎝2 3⎠
⎝ 3 4⎠
⎛−1 2 ⎞
⎛ 3
6 ⎞
maSin A⋅ B =
,
⎜
⎜
B ⋅ A = ⎜⎜
⎟
⎟
⎝ 7 16⎟⎟⎠
⎝11 12⎠
maSasadame A⋅ B ≠ B ⋅ A
aqedan SeiZleba gavakeToT Semdegi daskvna: sazogadod A⋅ B ≠ B ⋅ A ; e.i.
matricTa namravli arakomuraciuria. e.i. SeiZleba arsebobdes A ⋅ B
namravli, magram ar arsebobdes B ⋅ A namravli.
martivad mowmdeba, rom Sekrebisa da gamravlebis operaciebi
matricTa simravleSi xasiaTdeba Semdegi TvisebebiT:
1. A + B = B +
A
2. λ(δA) = (λλδ)
3. A − A =
O
4.
A(B + C) = AB + AC
5. A + O =
A
(A
6.
+ B)C = AC + BC
7. λ(A + B) = λA + λB
8.
A(BC) = (AB)C
46
im SemTxvevaSi roca A ⋅ B = B ⋅ A , amboben, rom A da B matricebi
komutireben. magaliTad E erTeulovani matrica komutirebs nebismier
kvadratul matricasTan: EA=AE. marTlac E-erTeulovani matrica iseTive
rols TamaSobs, rogorsac ricxvebis gamravlebis dros ricxvi 1-iani.
namdvil ricxvTa simravleSi Tu ab = 0 , maSin am ricxvTagan erTi
mainc nulis tolia. matricebSi es ase ar aris.
⎛1 1⎞
⎛ 1
1 ⎞
magaliTad, a) Tu A =
,
⎜
⎜
B =
,
⎜
⎜
⎟
⎟ maSin A⋅ B = 0
⎝1 1⎟⎟⎠
⎝−1 −1⎠
⎛0 1⎞
b) Tu A =
,
2
⎜
⎜
maSin A = A⋅ A = 0
⎝0 0⎟⎟⎠
3. transponirebuli matrica.UBvTqvaT mocemulia kvadratuli A= (ai, j )n
n
matrica. misi transponirebuli matrica ewodeba B matricas (B=AT), Tu
b = a
i, j
i, j e.i. A matricis striqonebma da svetebma adgilebi Seicvales.
matricis transponireba SeiZleba ganimartos m × n zomis A matricis-
aTvisac. maSin AT iqneba n × m zomis matrica e.i. am SemTxvevaSic matricis
striqonebi Caiwereba svetebad da piriqiT svetebi Caiwereba striqonebad.
magaliTad: Tu
⎛1 2⎞
⎜
⎟
⎛1 2 3 4⎞
T
⎜2 3⎟
A = ⎜⎜
⎟
⎟ maSin A =
.
⎝2 3 4 5⎠
⎜
⎟
3 4
⎜
⎜
⎟
⎟
⎝4 5⎠
advili dasamtkicebelia Semdegi Tvisebebi:
(A )T
T
T
T
T
T
T
T
T
T
= A;
( A
λ ) = A
λ
(A + B) = A + B
(AB) = A B .
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ras ewodeba matrica? matricis rigi?
47
2. ras ewodeba kvadratuli matrica? erTeulovani matrica? nulovani
matrica? diagonaluri matrica?
3. rogor matricebs ewodebaT toli?
4. rogor SevkriboT ori matrica? ras ewodeba matricis ricxvze
namravli?
5. rogor gavamravloT A da B matrica?
6. aris Tu ara A da B matricebis namravli komutaciuri?
II. praqtikuli savarjiSoebi
1. ipoveT A ⋅ B , Tu:
⎛ −1⎞
⎛1 2⎞
⎜
⎟
⎜
⎟
⎜
⎟
⎛1 0
3 ⎞
⎜ 2 3⎟
⎛1 2⎞
1.1 A = ⎜2 ⎟ , B = ⎜⎜
⎟
⎟ ; 1.2 A = ⎜
⎟ , B =
;
1
2
−
⎜
⎜
⎟
⎟
⎜
⎟
⎝
1⎠
⎜ 3 4⎟
⎝0 1⎠
⎝1 ⎠
⎜
⎟
⎝ 4 5⎠
⎛ 1 3 −1⎞
⎛ 2
3
−1⎞
⎜
⎟
⎜
⎟
1.3 A = ⎜ 2
5
1 ⎟ , B = ⎜ 0
− 5 1 ⎟
⎜
⎟
⎜
⎟
⎝−1 3 1 ⎠
⎝− 3
1
0 ⎠
⎛−1 3 −1⎞
⎛ 4
− 3 − 7⎞
⎜
⎟
⎜
⎟
1.4 A = ⎜ 2
0
3 ⎟ , B = ⎜ 50 − 52 37 ⎟
⎜
⎟
⎜
⎟
⎝−1 30 10 ⎠
⎝− 3
41
0 ⎠
⎛1 3⎞
⎜
⎟
⎛ 2
5
− 4⎞
1.5 A = ⎜ 2 1⎟ , B = ⎜⎜
⎟
⎟
⎜
⎟
⎝−1 − 3
0 ⎠
⎝3 2⎠
⎛a b c ⎞
⎛ 2a
b
3
c
3 ⎞
⎜
⎟
⎜
⎟
1.6 A = ⎜ s d
f ⎟ , B = ⎜ 3s
4d
− 7 f ⎟
⎜
⎟
⎜
⎟
⎝ x v m⎠
⎝− x − 2v − m ⎠
2. ipoveT k ⋅ A , Tu:
48
⎛ 12 ⎞
⎜
⎟
⎛ − 27
3 ⎞
⎜
−1
⎟
3
⎜ 13 ⎟
1
2.1. k =
da A =
4 ; 2.2. k = , xolo
⎜ 4
2 ⎟
A =
;
4
⎜
⎟
⎜
⎟
3
⎜ 1
⎟
3 − 5
⎜ 1 ⎟
⎜
⎟
⎝
⎠
⎝ 3
⎠
3. ipoveT A+B da A_B Tu:
⎛− 5⎞
⎛ 1 ⎞
⎛ 1 3 −1⎞
⎛ 2
3
−1⎞
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
3.1 A = ⎜ 3 ⎟ , B = ⎜ − 7⎟ ; 3.2 A = ⎜ 2
5
1 ⎟ , B = ⎜ 0
− 5 1 ⎟ ;
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎝ 0 ⎠
⎝− 4⎠
⎝−1 3 1 ⎠
⎝− 3
1
0 ⎠
⎛−1 3 −1⎞
⎛ 4
− 3 − 7⎞
⎛ −
2
sin α ⎞
⎛ −
2
cos α ⎞
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
3.3 A = ⎜ 2
0
3 ⎟ , B = ⎜ 50 − 52 37 ⎟ ; 3.4 A = ⎜
2
sin β ⎟ , B = ⎜
2
cos β ⎟
⎜
⎟
⎜
⎟
⎜
2
⎟
⎜
2
⎟
⎝−1 30 10 ⎠
⎝− 3
41
0 ⎠
⎝− 2sin γ ⎠
⎝− 2cos γ ⎠
A
$ 2. magaliTebi informaciis matriculi saxiT warmodgenaze.
ganvixiloT konkretuli
magaliTi 1. vTqvaT, warmoeba iyenebs oTxi N1, N2, N3 da N4 nedleuls
da amzadebs sami tipis P1, P2, P3 da P4 produqts. CavweroT matriculi
saxiT warmoebisaTvis saWiro nedleuls danaxarjebis normebi da gavaana-
lizoT miRebuli cxrili.
a11- iT avRinSnoT N1 saxis nedleulis danaxarjTa norma P1 produq-
tis erTeulis warmoebisas, a -
12 iT - P2 produqtis erTeulis warmoebisas,
xolo a -
a
13 iT – P3 produqtis erTeulis warmoebisas. analogiurad,
21 - iT
avRinSnoT N2 saxis nedleulis danaxarjTa norma P1 produqtis erTeulis
warmoebisas da a.S. sazogadod a -
ij iT avRinSnoT Ni saxis nedleulis
danaxarjTa norma Pj
produqtis erTeulis warmoebisas, cxadia
i = ,
1 2, ,
3 ;
4 j = ,
1 2,3 . Tu CavwerT aRniSnul informacias cxrilis saxiT,
miviRebT 4× 3 zomis matricas.
⎛ a
a
a
11
12
13 ⎞
⎜
⎟
⎜a
a
a
21
22
23 ⎟
⎜
⎟
a
a
a
31
32
33
⎜
⎜
⎟
⎟
⎝a
a
a
41
42
43 ⎠
49
SevniSnoT, rom i–ur striqonSi mdgomi a , a
a
i1
i 2
da i3 elementebis
jami gviCvenebs Ni saxis nedleulis mTlian danaxarjs samive saxis pro-
duqtis TiTo erTeulis sawarmoeblad, xolo j–ur svetSi mdgomi
a , a
a
,
a
1 j
2 j
3 j
da
4 j elementebis jami gviCvenebs Pj produqtis erTeulis
warmoebisaTvis saWiro oTxive nedleulis danaxarjebs. am matricaSi
zogierTi elementis nulTan toloba gamoricxuli ar aris. magaliTad, Tu
a = 0
12
, es niSnavs, rom N1 saxis nedleuli P2 saxis produqtis
sawarmoeblad ar gamoiyeneba.
magaliTi 2. ori qarxnis mier warmoebuli produqcia igzavneba sam
sawyobSi. TiToeul sawyobSi pirveli qarxnidan produqtis erTeulis
gadazidvaze ixarjeba, Sesabamisad, 2 lari, 3 lari, 4 lari, xolo meore
qarxnidan – 1 lari, 5 lari, 2 lari. CavweroT orive qarxnis satransporto
xarjebi matriculi saxiT.
radgan cnobilia TiToeuli qarxnis mier erTeuli produqtis
gadazidvis xarjebi, amitom satransporto xarjebi SegviZlia CavweroT
Semdegi matriciT:
⎛2 3 4⎞
⎜⎜
⎟
⎟ .
⎝1 5 2⎠
aq pirveli striqoni miuTiTebs pirveli qarxnis mier warmoebuli
erTeuli produqtis gadazidvis xarjebs samive sawyobSi, xolo meore
striqoni – meore qarxnis mier warmoebuli erTeuli produqtis gadazidvis
xarjebs.
magaliTi 3. orma momxmarebelma maRaziaSi SeiZina sami dasaxelebis
produqti: Saqari, yveli da karaqi. pirvelma SeiZina 1 kg Saqari, 2 kg yveli
da 1kg karaqi. meorem ki – 2 kg Saqari, 3 kg yveli da 2kg karaqi.
ganvsazRvroT TiToeuli momxmareblis danaxarji, Tu 1 kg Saqari Rirs 1
lari, 1 kg yveli – 3 lari, xolo 1 kg karaqi – 5 lari.
orive momxmareblis mier SeZenili produqtebis raodenoba (kg-obiT)
⎛1 2 1⎞
xasiaTdeba matriciT - ⎜⎜
⎟
⎟ , sadac pirveli striqoni Seesabameba
⎝2 3 2⎠
pirveli momxmareblis navaWrs, xolo meore – meore momxmareblis navaWrs.
50
⎛1⎞
⎜ ⎟
produqtebis Rirebuleba (larebSi) xasiaTdeba matriciT - ⎜3⎟
⎜ ⎟
⎝5⎠
martivi dasanaxia, rom am matricebis gadamravlebiT miviRebT orive
momxmareblis danaxarjebis cxrils.
⎛1⎞
⎛1 2 1⎞ ⎜ ⎟ ⎛ 1⋅1+ 2 ⋅ 3 +1⋅ 5 ⎞ ⎛12⎞
⎜⎜
⎟
⎟ ⎜3⎟=
=
⎜
⎜
⎟
⎟ ⎜⎜ ⎟⎟
⎝2 3 2⎠ ⎜ ⎟ ⎝2 ⋅1+ 3⋅ 3 + 2 ⋅ 5⎠ ⎝21⎠
⎝5⎠
aqedan Cans, rom pirvelma momxmarebelma daxarja 12 lari, xolo
meorem ki 21 lari.
magaliTi 4. firma uSvebs sami P1, P2, da P3 saxis produqts, romlebsac
yidis or C1 da C
2 momxmarebelze. am momxmareblebis mier nayidi produqtebis
raodenobebi gamosaxulia matriciT
P
P
P
1
2
3
⎛ 6 7 9⎞ (c )
1
A =
.
⎜
⎜
⎝ 2 1 2⎟⎟ (c )
⎠ 2
TiToeuli produqtis erTeulis fasi mocemulia matriciT
(P )
(P )
(P )
1
2
3
= (100 500 200)T
B
.
samive saxis produqtis gamosaSvebad firma iyenebs oTxi N1, N2, N3 da N4
saxis nedleuls. am nedleulis danaxarjis normebi (tonobiT) TiToeuli
saxis produqtis erTeulis sawarmoeblad gamosaxulia matriciT:
(N ) (N ) (N ) (N )
1
2
3
4
⎛1
0
0
1⎞ (P1)
⎜
⎟
C =
.
⎜1
1
2
1⎟ (P2 )
⎜0
0
1
1⎟
⎝
⎠ (P3 )
oTxive nedleulis TiToeuli tonis Rirebuleba (larebSi) aRwerilia
matriciT
(N ) (N ) (N ) (N )
= ( 1
2
3
4
20
10
15
15)T
D
.
51
damatebiT SemoviRoT erTstriqoniani matrica E(1 1).
vipovoT qvemoT miTiTebuli yvela matrica da aRvweroT maTi
ekonomikuri Senaarsi:
(a) AB, (b) AC, (g) CD, (d) ACD, (e) EAB, (v) EACD, (z) EAB – EACD.
⎛6 ⋅100 + 7 ⋅ 500 + 9 ⋅ 200⎞ ⎛5900⎞
(a) AB =
=
⎜
⎜
⎟
⎟ ⎜⎜
⎟
⎟
⎝ 2 ⋅100 +1⋅ 500 + 2 ⋅ 200 ⎠ ⎝1100 ⎠
miRebuli matrica gamosaxavs TiToeuli mamxmareblis mier nayidi
saqonlis mTlian fass;
⎛ 6⋅1+ 7 ⋅1+ 9⋅0 6⋅0 + 7 ⋅1+ 9⋅0 6⋅0 + 7 ⋅ 2 + 9⋅1 6⋅1+ 7 ⋅1+ 9⋅1 ⎞
(b)
AC=
=
⎜
⎜
⎟
⎟
⎝ 2⋅1+1⋅1+ 2⋅0 2⋅0 +1⋅1+ 2⋅0 2⋅0 +1⋅ 2 + 2⋅1 2⋅1+1⋅1+ 2⋅1 ⎠
⎛13 7 23 22⎞
= ⎜⎜
⎟
⎟ .
⎝ 3 1 4
5 ⎠
miRebuli matriciT gamoisaxeba nedleulis raodenoba (tonobiT),
romelic ixarjeba TiToeuli momxmareblis mier nayidi produqtis
warmoebisaTvis;
⎛1⋅ 20 + 0⋅10 + 0⋅15 +1⋅15⎞ ⎛35⎞
⎜
⎟ ⎜ ⎟
(g) CD= ⎜1⋅ 20 +1⋅10 + 2⋅15 +1⋅15 ⎟ = ⎜75⎟ .
⎜
⎟ ⎜ ⎟
⎝0⋅ 20 + 0⋅10 +1⋅15 +1⋅15⎠ ⎝30⎠
miRebuli matrica gvaZlevs nedleulis mTlian Rirebulibas
(larebSi), romelic ixarjeba samive saxis produqtis erTi erTeulis
warmoebisaTvis;
⎛13 7 23 22⎞
T
⎛1105⎞
(d) ACD = ( AC)D =
(20 10 15 15) =
⎜
⎜
⎟
⎟
⎜
⎜
⎟
⎟ .
⎝ 3 1 4
5 ⎠
⎝ 205 ⎠
miRebuli matriciT ganisazRvreba daxarjuli nedleulis mTliani
fasi, romelic Seesabameba TiToeuli momxmareblis mier nayid produqts;
EAB = (EA) = (1⋅ 6 +1⋅ 2 1⋅ 7 +1⋅1+1⋅ 9 +1⋅ 2)(100 500 200)T
B
=
(e)
(8 8
)
11 ⋅ (100 500 200)T = (8 ⋅100 + 8 ⋅ 500 +11⋅ 200) = (7000).
miRebuli matriciT gamoisaxeba mTliani Semosavali, romelsac firma
momxmareblisagan Rebulobs.
52
T
(v) EACD = (8 8
)
11 (35 75 30) = (8⋅35 + 8⋅75 +11⋅30) =
)
1210
(
.
miRebuli matrica gvaZlevs daxarjuli nedleulis mTlian fass;
(z) EAB-EACD=(7000)-(1210)=(5790).
miRebuli matrica gamosaxavs mogebas gadasaxadebisa da xelfasis
gadaxdamde.
davaleba:
CawereT damoukideblad informacia matriculi saxiT da amoxseniT
1. orma diasaxlisma bazarSi SeiZina 4 sasursaTo produqti:
kartofili, yveli, xorci da Tafli. erTma diasaxlisma SeiZina 2 kg.
kartofili, 1 kg yveli, 2 kg xorci da 1 kg Tafli. xolo meorem _ 3 kg.
kartofili, 2 kg yveli, 1,5 kg xorci da 1 kg Tafli. gamoTvaleT TiToeuli
diasaxlisis danaxarji, Tu kartofilis fasia 0,5 lari, yvelis – 5 lari,
xorcis _ 3 lati da Taflis – 8 lari.
2. ori gamyidveli bazarSi erTnairi produqtebiT vaWrobda. erTma
gayida 50 kg vaSli da 30 kg atami, xolo meorem _ 60 kg vaSli da 20 kg
atami. gamoTvaleT TiToeuli gamyidvelis amonagebi, Tu 1 kg vaSlis fasia
0,8 lari, xolo 1 kg atmis – 1,2 lari.
$ 3 determinantebi. Sebrunebuli matrica.
matricis rangi
53
1. Sebrunebuli matrica. mocemulia n rigis kvadratuli matrica.
vTqvaT, I imave rigis erTeulovani matricaa. iseT kvadratul X matricas,
romlisTvisac
XA = AX = I (1)
ewodeba A matricis Sebrunebuli matrica da aRiniSneba
1
−
A simboloTi.
Teorema 1. A matricas Sebrunebuli matricis Sebrunebuli matrica
−1
−1
aris A matrica, e.i. (A−1 ) = A . A−1 ⋅ (A−1 ) = I
marTlac, Sebrunebuli matricis ganmartebidan gamomdinareobs, rom
A−1 ⋅ (A )−
−
1
1
= I . Tu am tolobis orive mxares gavamravlebT marcxnidan A
matricaze, miviRebT
A(A−1 ⋅ A −
−
1
1
)
( ) = AI
radganac
−1
AA = 1, gvaqvs
−1
(A−1 ) = A
Teorema 2. ori erTi da igive zomis matricebis namravlis Sebrune-
buli matrica tolia Sebrunebuli matricebis namravlisa Sebrunebuli
mimdevrobiT, e.i.
(AB) 1−
−1
1
−
= B A
marTlac, gvaqvs:
AB(B−1A−1 ) = A(BB−1 )A−1 = A⋅ I ⋅ A−1 = AA−1 = I
(B−1A−1)AB = B−1(A−1A)B = B−1 ⋅I ⋅B = BB−1 = I
saidanac
AB(B 1−A−1 ) = (B 1−A−1 )AB maSasadame Sebrunebuli matricis
ganmartebis Tanaxmad
1
−
−1
B A warmoadgens AB matricis Sebrunebuls, e.i.
1
−
(AB)
−1
−1
= B A
54
Sebrunebuli matricis gamoTvlis xerxi vuCvenoT magaliTze:
⎛2 3 ⎞
gamovTvaloT A = ⎜⎜
⎟
⎟ matricis Sebrunebuli martica.
⎝5 −1⎠
c
c
−
⎛
1
11
12 ⎞
davuSvaT, rom A matricis Sebrunebulia A = ⎜⎜
⎟
⎟ matrica.
⎝c
c
21
22 ⎠
c
c
c
c
c
c
−
⎛
⎛
⎞ 2 3 ⎞ ⎛ 2 + 5
3
−
⎞ ⎛1 0⎞
gveqneba
1
11
12
11
12
11
12
A A =
=
=
;
⎜
⎜
⎜
⎜
⎟
⎟
⎝c c ⎝5 −1⎟⎟ ⎜⎜2c + 5c 3
⎟
⎟ ⎜⎜
⎠
⎠ ⎝
c − c ⎠ ⎝0 1⎟⎟
21
22
21
22
21
22
⎠
2c + 5c = ,
1
2c + 5c = ,
0
maSasadame,
11
12
21
22
⎩
⎨
⎧3c −c = 0.
c
c
11
12
⎩
⎨
⎧3 − =1.
21
22
Tu amovxsniT TiToeul sistemas, miviRebT:
1
3
5
2
c =
c =
c =
c = −
11
;
;
;
.
17
12
17
21
17
22
17
e.i.
⎛ 1
3 ⎞
⎜
⎟
−1
A = ⎜17
17 ⎟
5
2
⎜
⎜
−
⎟
⎟
⎝17
17 ⎠
SeniSvna. Sebrunebuli matricis gamoTvlis zemoTmoyvanili xerxi
rTuldeba marticis rigis zrdasTan erTad. matricis Sebrunebuli
matricis povnis ufro moxerxebul xerxs aRvwerT Semdgom paragrafSi.
2. determinanti. determinantis cnebis SemoReba sxvadasxvagvarad
SeiZleba. Cven iseT gzas avirCevT, romelic saSualebas mogvcems SemdgomSi
SedarebiT martivad gamovTvaloT da gamoviyenoT determinantebi.
Tu A kvadratuli matricaa, maSin mis determinants aRniSnaven A an Δ
simboloTi. daviwyoT n=1-dan da TandaTan ganvsazRvroT n × n zomis
kvadratuli matricis determinanti.
n=1. maSin (A) = a
A = a
11 da
11
⎛a
a
11
12 ⎞
n=2. maSin A = ⎜⎜
⎟
⎟ da ganvmartoT
⎝a
a
21
22 ⎠
55
a
a
11
12
A =
= a a − a a
11 22
12 21 .
a
a
21
22
⎛a
a
a
11
12
13 ⎞
⎜
⎟
n=3. maSin, A = ⎜a
a
a
21
22
23 ⎟ .
⎜
⎟
⎝ a
a
a
31
32
33 ⎠
misi Sesabamisi mesame rigis determinanti ganvsazRvroT Semdegi
wesiT:
a
a
a
a
a
a
22
23
12
13
12
13
A = a
− a
+ a
.
11
21
31
a
a
a
a
a
a
32
33
32
33
22
23
vTqvaT, Cven ukve gansazRvruli gvaqvs n × n kvadratuli matricis
determinanti da gvinda ganvszRvroT (n + )
1 × (n + )
1 zomis determinanti. maSin
viqceviT wina punqtis msgavsad: viRebT
⎛ a
a
a
a
11
12
L
1n
1,n+1 ⎞
⎜
⎟
⎜ a
a
a
a
21
22
L
2n
2,n+1 ⎟
A = ⎜
⎟
K
K
K
K
K
⎜
⎟
a
a
a
a
n1
n 2
L
nn
n,n+
⎜
⎜
1 ⎟⎟
⎝ a
a
a
a
n+ ,
1 n
n+1,2
L
n+1n
n+ ,
1 n+1 ⎠
matricis pirvel svets da TandaTan viwyebT A determinantis
gaSlas. viRebT a11 elements, matricidan amovSliT pirvel striqonsa da
pirvel svets, vamravlebT a11 – darCenili matricis determinantze. Semdeg
viRebT a21 elements, matricidan amovSliT meore striqonsa da pirvel
svets, vamravlebT a21 – darCenili matricis determinantze da a.S
zogadad viRebT aj1 elements, matricidan amovSliT j-ur striqonsa da
pirvel svets, vamravlebT (-1)j+1j1 – darCenili matricis determinantze.
sabolood yvela aseTi namravls (sul aseTi n+1- ia) SevkribavT da
miviRebT A matricis determinants.
SeniSvna. determinantis aseTi ganmarteba emyareba determinantis Tvi-
sebebs. magram Cven mizanSewonilad CavTvaleT mainc aseTi saxis
gansazRvrebis SemoReba, radgan Cveni azriT es garkveulad aadvilebs
determinantis gamoTvlebs.
56
3. minori da algebruli damateba. vTqvaT A n × n zomis kvadratuli
matricaa. ganvixiloT misi aij elementi. Tu matricidan amovSliT j-ur
striqonsa da i-ur svets, maSin miReba (n − )
1 × (n − )
1 zomis kvadratuli
matrica. am matricis determinants ewodeba aij elementis minori da
aRiniSneba Mij simboloTi. Tu Mij minors gavamravlebT (-1)i+j – ze, miReba aij
elementis algebruli damateba. igi aRiniSneba Aij simboloTi, e.i.
Aij=(-1)i+jMij .
algebruli damatebebis gamoTvlebs Zalian didi mniSvneloba aqvT
determinantebisa da Sebrunebuli matricebis gamoTvlisas, amitom cota
meti yuradRebiT movekidebiT.
laplasis Teorema. kvadratuli matricis determinanti tolia
matricis nebismieri striqonis (an svetis) elementebisa da maTi algebruli
damatebebis namravlebis jamisa:
A = a A + a A +
+ a A
i1 i1
i2+
i2
L
in
in
A = a A + a
A +
+ a A
1 j
1 j
2 j+
2 j
L
nj
nj
maSasadame, determinantis gamoTvla SeiZleba misi gaSliT nebismieri
striqonis
an svetis mimarT. amitom praqtikuli gamoTvlebisas
mizanSewonilia im striqonis an svetis amorCeva, sadac bevri nulebia.
determinantis Tvisebebi:
I. Tu determinantis romelime striqonis an romelime svetis yvela
elementi nulis tolia, maSin determinantic nulis tolia;
II. Tu determinantis ori striqoni an sveti erTnairia, maSin
determinanti nulis tolia;
III. Tu determinantSi adgilebs SeucvliT or striqonsa an or svets,
maSin determinanti mxolod niSans Seicvlis;
IV. Tu determinantis romelime striqonis an svetis elementebi
Seicaven saerTo Tanamamravls, is SeiZleba gamovitanoT determinantis
niSnis gareT;
SeniSvna. matricis ricxvze gamravleba gansxvavdeba determinantis
ricxvze gamravlebisagan. matricis ricxvze gamravlebisas am matricis
yvela elementi unda gavamravloT ricxvze. im dros, rodesac
57
determinantis ricxvze gamravlebisas saWiroa erTi romelime striqoni an
sveti gavamravloT mocemul ricxvze.
V.
Tu determinantis romelime striqonis an svetis elementebs
davumatebT sxva striqonis an svetis elementebs gamravlebul erTi da
igive ricxvze, amiT determinantis mniSvneloba ar Seicvleba;
VI.
Tu determinantis romelime striqonis an svetis elementebs
gavamravlebT sxva striqonis an svetis algebrul damatabebze da am
namravlebs SevkribavT, miRebuli jami nulis toli iqneba;
VII.
Tu determinantis ori striqonis an svetis elementebi
proporciulia, maSin determinanti nulis tolia;
VIII. transponirebuli matricis determinanti matricis
determinantis tolia;
IX.
ori kvadratuli matricis namravlis determinanti matricTa
determinantebis namravlis tolia, e.i.
AB = A ⋅ B .
SevniSnoT, rom es toloba samarTliania roca B ⋅ A ≠ A⋅ B .
4. Sebrunebuli matricis gamoTvla determinantebis gamoyenebiT.
ganvixiloT kvadratuli matrica
⎛ a
a
a
11
12
L
1n ⎞
⎜
⎟
a
a
a
A = ⎜ 21
22
L
2n ⎟ .
⎜
⎜ K
K K K ⎟⎟
⎝ a
a
a
n1
n 2
L
nn ⎠
radganac Sebrunebuli matrica Cven ganvmarteT kvadratuli matrici-
saTvis, amitom mas SeiZleba hqondes Sebrunebuli matrica. bunebrivad ismis
amocana – rogor matricas aqvs Sebrunebuli. am sakiTxze pasuxs iZleva
Semdegi Teorema.
Teorema. kvadratul matricas maSin da mxolod maSin aqvs
Sebrunebuli matrica, roca misi determinanti gansxvavdeba nulisagan.
(kvadratul A matricas ewodeba gansakuTrebuli, Tu A = 0 , xolo Tu
A ≠ 0 , maSin aragansakuTrebuli).
damtkiceba. a) vTqvaT A matricas aqvs Sebrunebuli A-1 matrica. maSin
58
AA−1 = I da amitom
1
−
1
AA
= I
⇒
−
A A
= 1.
1
−
1
maSasadame, A ≠ 0 da A
=
;
A
b) vTqvaT A ≠ 0 da vaCvenoT, rom arsebobs matrica. axla Cven ufro
mets vaCvenebT, faqtiurad aq miuTiTebT, Tu rogor unda avagoT A-1 .
ganvixiloT A* matrica, romelsac A matricis mikavSirebul matricas
uwodeben da romlis elementebic Semdegi wesiT aigeba. A*=( * n
a ) ,
a
ij n sadac
*
ij
aris AT matricis algebruli damateba. maSasadame *
T
a = A = A
ij
ij
ji .
vTqvaT B=A*A=(
n
b ) ,
ij n
maSin
⎧ A ,
i
Tu
= j,
n
n
*
*
⎪
b =
ij
∑ A A =
is
sj
∑ A a =
is
sj
⎨
s=1
s=1
⎪
⎩0,
i
Tu
≠ j.
amitom
⎛ A
0
0 ⎞
⎜
⎟
⎜ 0
A
0 ⎟
B = ⎜
⎟
K
K
K
⎜
⎜
⎟
⎟
⎝ 0
0
A ⎠
analogiurad miiReba, rom
⎛ A
0
0 ⎞
⎜
⎟
*
⎜ 0
A
0 ⎟
A ⋅ A = ⎜
⎟
K
K
K
⎜
⎜
⎟
⎟
⎝ 0
0
A ⎠
1
maSin cxadia, rom Tu aviRebT matricas
*
C =
A , maSin A-1=C.
A
vaCvenoT, rom A-1 matrica calsaxadaa gansazRvruli.
vTqvaT arsebobs X da Y romlebic toli araa A-1-is da AX=I, YA=I.
maSin A-1AX=A-1 da YAA-1 =A-1 saidanac X=Y=A-1.
59
Teoremis damtkicebidan gamomdinare CamovayaliboT Sebrunebuli
matricis agebis algoriTmi:
1)
unda vipovoT mocemuli matricis determinanti A Tu A ≠ 0 ,
maSin A-1 arsebobs, Tu A =0, maSin A-1 ar arsebobs;
2)
vipovoT transponirebuli matrica AT.
3)
unda gamovTvaloT AT matricis elementebis algebruli
damatebebi da vipovoT mierTebuli matrica A* da gamovTvaloT
Sebrunebuli matrica
1
−
1
*
A =
A formuliT.
A
5. matricis rangi. ganvixiloT A = (a)nm matrica SevarCioT am
matricaSi nebismieri r striqoni da r sveti. cxadia, r ≤ min{m, }
n arCeuli
striqonebisa da svetebis gadakveTaze mdgomi elementebisagan Sedgenil
determinants matricis r rigis minori ewodeba.
mTel r ricxvs ewodeba matricis rangi, Tu misi r rigis minorTa
Soris erTi mainc gansxvavebulia nulisagan, xolo yvela r maRali rigis
minori nulis tolia.
A matricis rangi aRiniSneba simboloTi rangA. am ganmartebidan
gamomdinareobs, rom Tu r rigis minorTa Soris erTi mainc gansxvavebulia
nulisagan, xolo yvela (r+1) rigis minori (Tu aseTi arsebobs) nulis
tolia, maSin A matricis rangi r ricxvis tolia.
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ras ewodeba A matricis Sebrunebuli matrica?
2. ras ewodeba minori da algebruli damateba?
3. CamoayalibeT determinantis Tvisebebi.
4. ras ewodeba gansakuTrebuli matrica? aragansakuTrebuli
matrica?
5. ras ewodeba matricis rangi?
60
II. praqtikuli savarjiSoebi
1. ipoveT A-1, Tu:
⎛ 2 1 −1⎞
⎛1 5⎞
⎛3 1⎞
⎜
⎟
1.1 A= ⎜⎜
⎟
⎟ ; 1.2 ⎜⎜
⎟
⎟ ; 1.3 A = ⎜5 2
0 ⎟
⎝3 4⎠
⎝5 2⎠
⎜
⎜
⎟
⎟
⎝ 6 5
3 ⎠
⎛ 1 3 −1⎞
⎛ 2
3
−1⎞
⎜
⎟
⎜
⎟
1.4. A = ⎜ 2
5
1 ⎟ ; 1.5 A = ⎜ 0
− 5
1 ⎟
⎜
⎟
⎝−1 3 1 ⎠
⎜
⎜
⎟
⎟
⎝ − 3
1
0 ⎠
2. gamoTvaleT:
2
1 0 0
2
1
−1
1
1
1
0
1 3 2
2.1. 5
2
0 ; 2.2. x
y
z ; 2.3.
;
0
0 0 5
6
5
3
2
2
2
x
y
z
−1 2 0 0
−
2
x
1
x
1 − t
2 t
3 7
2
2
+
+
2.4.
; 2.5. 0
− x −1 ; 2.6. 1
t
1
t
;
2 0
2
− 2 t
1 − t
x
1
− x
2
2
1 + t
1 + t
3. amoxseniT gantoleba:
cos8x − sin 5x
a +1
b − c
4sin x
1
3.1.
= 0 ; 3.2
= 0 ; 3.3.
= 0 ;
sin 8x
cos 5x
2
a + a
ab − ac
1
cos x
4. gamoTvaleT Semdegi matricebis rangi:
⎛ 1
2
−1 0⎞
⎛ 1 2
1
− 2⎞
⎜
⎟
⎜
⎟
4.1. A = ⎜ 1
2
4
2⎟ ; 4.2. A = ⎜ 2 1
−1
0 ⎟ ;
⎜
⎜
⎟
⎟
⎜
⎜
⎟
⎟
⎝ − 3 − 2
6
2⎠
⎝ 1 1 −1
0 ⎠
⎛ 2
3
−1⎞
⎛1 2 1 1⎞
⎜
⎟
⎜
⎟
4.3. A = ⎜ 0
− 5
1 ⎟ ; 4.4. A = ⎜ 0 1 1 1⎟ ;
⎜
⎜
⎟
⎟
⎜
⎜
⎟
⎟
⎝ − 3
1
0 ⎠
⎝1 1 0 0⎠
61
$ 4 wrfiv gantolebaTa sistema
1. zogadi cnebebi. ganvixiloT m wrfiv gantolebaTa sistema n
ucnobiT:
⎧a x + a x +
+ a x = b ,
L
⎪ 11 1
12 2
1n n
1
⎪
⎪a x + a x + + a x = b ,
21 1
22 2
L
2n n
2
⎨
(1)
⎪
L
L
L
L
L
L
L
L
L
L
L
L
⎪
⎪
⎩a x + a x +
+ a x = b .
m1 1
m 2 2
L
mn n
m
sadac,
aij da bij (i=1,2,...,m; j=1,2,3,...n) nebismieri namdvili ricxvebia, xolo
x , x ,
x
1
2 K n ucnobi sidideebia. aij ricxvebs sistemis koeficientebi
ewodebaT, xolo bij ricxvebs ki gantolebis Tavisufali wevrebi. Tu m ≠ n
maSin sistemas marTkuTxa sistemas, xolo Tu m = n , maSin kvadratuli
sistema ewodeba.
Semoklebuli formiT sistema Caiwereba semdegi saxiT:
n
∑ a x = b ,
(i = ,
1
,...,
3
,
2
n .)
ij
j
i
j 1
=
sistemis amonaxsni ewodeba x , x ,..., x
1
2
n cvladebis iseT mniSvnelobebs,
romelic sistemis yovel gantolebas WeSmarit ricxviT tolobad aqcevs.
62
Tu sistemas erTi amonaxseni mainc aqvs, maSin mas Tavsebadi ewodeba.
Tu sistemas ara aqvs arcerTi amonaxseni, maSin mas ara Tavsebadi
ewodeba.
⎧x + x =10
magaliTebi: I. sistema ⎨ 1
2
⎩x − 3x = 2
1
2
Tavsebadia da mas erTaderTi amonaxseni (x = ,
8
x = 2)
1
2
aqvs;
⎧x + x = 10
II. sistema ⎨ 1
2
araTavsebadia.
⎩2x + 2x = 5
1
2
wrfiv algebrul gantolebaTa sistemasTan dakavSirebiT ismeba ori
ZiriTadi sakiTxi:
I.
sistemis Tavsebadobis gamokvleva;
II.
Tavsebadi sistemis yvela amonaxsenis povna.
advili dasanaxia, rom Tu SemoviRebT Semdegi saxis aRniSvnebs:
⎛ a
a
a
x
b
L
n ⎞
⎛ ⎞
⎛ ⎞
11
12
1
1
1
⎜
⎟
⎜ ⎟
⎜ ⎟
a
a
a
x
b
21
22
L
2
A = ⎜
n ⎟,
2
X = ⎜
⎟,
2
B = ⎜
.
⎟
⎜
⎜ K
K
K
K ⎟⎟
⎜
⎜ M ⎟⎟
⎜
⎜ M ⎟⎟
⎝ a
a
a
x
b
m1
m 2
L
mn ⎠
⎝ n ⎠
⎝ n ⎠
maSin (1) gantolebaTa sistema SeiZleba Caiweros
AX = B (2)
matriculi formiT.
2. krameris wesi. ganvixiloT kvadratuli sistema (m=n)
⎧a x + a x +
+ a x = b ,
L
⎪ 11 1
12 2
1n n
1
⎪
⎪a x + a x + + a x = b ,
21 1
22 2
L
2n n
2
⎨
(3)
⎪
L
L
L
L
L
L
L
L
L
L
L
L
⎪
⎪
⎩a x + a x +
+ a x = b .
n1 1
n 2 2
L
nn n
n
da am sistemis koeficientebisagan Sedgenili determinanti avRniSnoT Δ-Ti
e.i.
a
a
a
11
12
L
1n
a
a
a
21
22
L
2n
Δ =
K
K K K
a
a
a
n1
n 2
L
nn
63
ganvixiloT agreTve n rigis egreTwodebuli damxmare determinantebi:
Δ1,Δ2,..,Δn, sadac Δj (j=1,2,3...,n) miiReba sistemis Δ determinantisagan misi
j-uri svetis SecvliT Tavisufali wevrebisagan Sedgenili svetiT. e.i.
a
a
b
a
a
11 L
,
1 j 1
−
1
1, j 1
+ L
1n
a
a
b
a
a
21 L 2, j 1
−
2
2, j 1
+ L
2n
A =
K
K
K
K
a
a
b
a
a
n1 L n, j 1
−
n
n, j 1
+ L
nn
mtkicdeba, rom Tu Δ≠0, maSin (3) sistema Tavsebadia da mas erTaderTi
amonaxseni aqvs, romelic moicema formulebiT:
Δ
x
j
=
, j = ,
1 2, ,...,
3
n
j
(4)
Δ
miRebul (4) formulebs krameris formulebi ewodeba.
cxadia, rom krameris formulebis gamoyeneba SeiZleba mxolod maSin,
am roca Δ≠0, SemTxvevaSi sistemas erTaderTi amonaxsni aqvs. agreTve
mtkicdeba, rom Tu Δ=0 da Δj –ebidan erTi mainc gansxvavebulia nulisagan,
maSin sistemas amonaxseni ara aqvs. Tu Δ=0 da Δj= 0, maSin sistemas uamravi
amonaxseni aqvs.
3. sistemis amoxsna matriculi xerxiT. ganvixiloT (2) matriculi
formiT Cawerili gantolebaTa sistema
AX = B .
davuSvaT, rom A matrica aragadagvarebulia e.i. Δ= A ≠ 0 . maSin
rogorc, viciT arsebobs A matricis Sebrunebuli A-1 matrica. gavamravloT
(2) – matriculi gantolebis orive mxare marcxnidan A-1 matricaze
1
A− (AX )
1
= A− B ⇒ [( 1
A− )
1
A X = A− B]⇒ [
1
IX = A− B]
1
⇒ X = A− .
B
e.i.
1
X
A−
=
B. (5)
rac warmoadgens (2) gantolebis amonaxsns.
axla ganvixiloT gantolebaTa marTkuTxovani sistema m ≠ n . am
sistemis amonaxsenis arseboba mWidrodaa dakavSirebuli Semdeg or
64
matricasTan
⎛ a
a
a
a
a
a
b
11
12
L
1n ⎞
⎛ 11
12
L
1n
1 ⎞
⎜
⎟
⎜
⎟
⎜ a
a
a
⎟
~
⎜ a
a
a
b ⎟
21
22
L
2 n
21
22
L
2 n
2
A = ⎜
⎟, A = ⎜
⎟
⎜ K
K
K
K ⎟
⎜ K
K
K
K
K ⎟
⎜
⎟
⎜
⎟
⎝ a
a
a
a
a
a
b
m1
m 2
L
mn ⎠
⎝ n1
n 2
L
nn
n ⎠
~
A matricas sistemis gafarToebuli matrica ewodeba. mtkicdeba
Semdegi
Teorema (kroneker-kapeli). wrfiv algebrul gantolebaTa sistemis
TavsebadobisaTvis aucilebeli da sakmarisi, rom sistemis matricis rangi
~
udrides gafarToebuli matricis rangs, e.i. rang A = rangA ; amasTan
~
1.
Tu
rangA = rang A = n , maSin sistemas aqvs mxolod erTi
amonaxseni;
~
2.
Tu rangA = rang A < n , maSin sistemas aqvs amonaxsenTa usasrulo
simravle;
~
3.
Tu rangA ≠ rang A , maSin sistema araTavsebadia.
4. erTgvarovan gantolebaTa sistema. wrfiv gantolebaTa sistemas
ewodeba erTgvarovani, Tu (1) sistemis yvela Tavisufali wevri nulis
tolia:
⎧a x + a x +
+ a x = 0,
L
⎪ 11 1
12 2
1n n
⎪
⎪a x + a x + + a x = 0,
21 1
22 2
L
2n n
⎨
⎪
L
L
L
L
L
L
L
L
L
L
L
L
⎪
⎪
⎩a x + a x +
+ a x = .
0
m1 1
m 2 2
L
mn n
cxadia, rom erTgvarovani sistema yovelTvis Tavsebadia, radgan e.w.
trivialuri (nulovani) amonaxseni mas yovelTvis aqvs.
advili dasanaxia, rom Tu m=n da matricis rangia n, maSin sistemis
determinanti ar udris nuls da amitom sistemas mxolod nulovani
amonaxseni aqvs. maSasadame, arativialuri (aranulovani) amonaxsnebi aqvs
mxolod iseT sistemebs, romelSic gantolebaTa raodenoba naklebia
cvladebis raodenobaze an, Tuki isini tolia, maSin determinantia nulis
toli. sxavgvarad: erTgvarovan sistemas aratrivialuri amonaxsnebi aqvs
65
maSin, da mxolod maSin roca rangi naklebia sistemis ucnobebis
raodenobaze.
avRniSnoT erTgvarovani sistemis amonaxseni x = k ,
x = k ,..., x = k ,
1
1
2
2
n
n
striqonis saxiT e = (k , k ,..., k )
1
1
2
n .
advili dasanaxia, rom Tu e1 erTgvarovani sistemis amonaxsenia, maSin
e
λ = ( k
λ , k
λ ,...,λk )
e
e
1
1
2
n -ic sistemis amonaxsenia. aseve, Tu 1 da 2 sistemis
amonaxsnebia, maSin ( e
e
1 + 2 )-ic amonaxsenia.
maSasadame, erTgvarovani sistemis amonaxsenTa nebismieri wrfivi kom-
binacia isev sistemis amonaxsenia.
magaliTi 1. ganvsazRvroT bazarze Setanili sami
urTierTdamokidebuli saqonlis wonasworobis P1, P2 da P3 fasebi, Tu
isini akmayofileben Semdeg pirobebs:
⎧2P + 4P + P = 77,
1
2
3
⎪
⎪
⎨4P + 3P + 7P = 114,
1
2
3
⎪
⎪
⎩2P + P + 3P = 48.
1
2
3
jer gamovTvaloT sistemis Δ determinanti:
2 4 1
3 7
4 7
4 3
Δ = 4 3 7 = 2
− 4
+
= 2 9
( − 7) − 4 12
( −14) + (4 − 6) = 10 .
1 3
2 3
2 1
2 1 3
e.i. Δ=10≠0, amitom sistemas gaaCnia erTaderTi amonaxseni. am
amonaxsenis sapovnelad gamovTvaloT, Δ1, Δ2, Δ3 – damxmare determinantebi.
77 4 1
2 77 1
2 4 77
Δ = 114 3 7 =
,
100
Δ = 4 114 7 =
,
130
Δ = 4 3 114 = 50.
1
2
3
48 1 3
2 48 3
2 1 48
krameris formulebiT davadgenT, rom
Δ
Δ
Δ
1
P =
= ,
10
2
P =
= ,
13
3
P =
= .
5
1
2
3
Δ
Δ
Δ
amrigad, saZiebeli wonasworobis fasebia P =
,
10 P =
,
13 P = 5
1
2
3
magaliTi 2. amovxsnaT matriculi xerxiT Semdegi sistema
⎪
⎧x + y + 2z = − ,
1
⎨2x − y + 2z = − ,
4
⎪
⎩4x + y + 4z = −2.
SemoviRoT aRniSvnebi:
66
⎛1 1 2⎞
⎛ x ⎞
⎛ −1⎞
⎜
⎟
⎜ ⎟
⎜
⎟
A = ⎜ 2 −1 2 ,
⎟
X = ⎜ y ,⎟
B = ⎜− 4 .⎟ .
⎜4 1 4⎟
⎜ ⎟
⎜
z
− 2⎟
⎝
⎠
⎝ ⎠
⎝
⎠
mocemuli sistema, matriculi saxiT ase gadaiwereba
AX = B .
radgan
1
1
2
A = 2 −1 2 = 6 ≠ 0 ,
4
1
4
amitom
⎛ A
A
A
11
12
13 ⎞ ⎛ − 1 ⎞
⎜
⎟ ⎜
⎟
1
1
X = −
A B =
⎜ A
A
A ⎟ ⋅ ⎜ − 4⎟
21
22
23
,
A ⎜⎜
⎟
⎟ ⎜⎜ ⎟⎟
⎝ A
A
A
31
32
33 ⎠ ⎝ − 2 ⎠
sadac
−1 1
1
1
1
−1
A = (− )
1 2
= 6
− , A = (− )
1 3
= 2
− , A = (− )
1 4
= −4,
11
2
4
21
2
4
31
2
2
2
4
1
2
1
2
A = (− )
1 3
= 0, A = (− )
1 4
= −4, A = (− )
1 5
= 2,
12
2
4
22
4
4
31
2
2
2
4
1
4
1
2
A = (− )
1 4
= 6, A = (− )
1 5
= ,
3 A = (− )
1 6
= 3
− .
13
−1 1
23
1 1
31
1
−1
sabolood
⎛
1
2 ⎞
⎜−1 −
⎟
⎛ x ⎞ ⎜
3
3 ⎛
⎟ −1⎞ ⎛ 1 ⎞
⎜ ⎟
2
1 ⎜
⎟ ⎜
⎟
⎜
⎟
⎜ y ⎟ = 0
−
⎜− 4⎟ = ⎜ 2 ⎟
⎜
⎜ ⎟
3
3 ⎜
⎟
⎟ ⎜
⎟
⎝ z ⎠ ⎜
1
1 ⎝
⎟ − 2⎠ ⎝− 2⎠
⎜ 1
− ⎟
⎝
2
2 ⎠
SevniSnoT, rom krameris wesiT, aseve matriculi xerxiT SeiZleba
amoixsnas mxolod iseTi wrfiv gantolebaTa sistemebi, romlebSic ucnobTa
ricxvi gantolebaTa ricxvis tolia da sistemis determinanti
gansxvavebulia nulisagan.
magaliTi 3. gamovikvlioT sistema
⎪
⎧x + 4y + 5z = ,
10
⎨2x + 8y + 10z =
,
20
⎪
⎩x − y + z = 1.
67
pirvel rigSi, amovweroT sistemis matrica da gafarToebuli
matrica
⎛1 4
5 ⎞
⎛1 4
5 10 ⎞
⎜
⎟
~
⎜
⎟
A = ⎜ 2
8
10⎟,
A = ⎜2 8 10 20⎟ .
⎜
⎟
⎜
⎟
⎝1 −1 1 ⎠
⎝1 −1 1
1 ⎠
Tu gamoviTvliT, miviRebT, rom A = 0 e.i. A matrica gadagvarebulia.
amitom krameris xerxs ver gamoviyenebT. martivad SevamowmebT, rom rangA=2,
radgan misi meore rigis minori
2
8
= 2
− − 8 = 10
− ≠ 0
1 −
.
1
~
aseve martivad vaCvenebT, rom gafarToebul A matricis yvela mesame
~
rigis minori nulis tolia da rang A =2. amitom kroneker-kapelis Teoremis
Tanaxmad mocemul sistemas gaaCnia amonaxsenTa usasrulo simravle.
vipovoT es amonaxsnebi. amisaTvis mocemuli sistema gadavweroT Semdegi
saxiT:
⎪
⎧x + 4y = 10 − 5z,
⎨x − y = 1 − z,
⎪
⎩2x + 8y +10z = 20.
pirveli ori gantolebidan miviRebT:
14 − 9z
9 − 4z
x =
; y =
.
5
5
uSualo SemowmebiT martivad vaCvenebT, rom sistemis mesame
gantoleba avtomaturad kmayofildeba z-s nebismieri mniSvnelobisaTvis,
⎛
14 − 9z
9 − 4z
⎞
amitom Semdegi sameuli ⎜ x =
; y =
z ⎟ warmoadgens mocemuli
⎝
5
5
⎠
sistemis amonaxsens nebismieri z-saTvis.
magaliTi 4. amovxsnaT erTgvarovani sistema
⎪
⎧x + 2x + x = ,
0
1
2
3
⎨3x − 3x − 3x = ,
0
⎪ 1
2
3
⎩5x + x − x = 0.
1
2
3
1
2
2
1
2
A = 3 − 3 − 3 = 0 , xolo meore rigis minori
= 9
− ≠ 0
3 −
. amitom sis-
3
5
1
−1
temis matricis rangi _ rangA=2, amitom sistemas aqvs aratrivialuri
68
amonaxsnebi. vipovoT es amonaxsnebi. amisaTvis aviRoT gamoTvlili minoris
x + 2x + x = 0
warmomqmneli pirveli ori gantoleba
1
2
3
⎩
⎨
⎧3x −3x −3x =0, x3 –s mivaniWoT
1
2
3
nebismieri mniSvneloba c.
x + 2x = −c,
miviRebT:
1
2
⎩
⎨
⎧x − x = .c
1
2
1
2
saidanac x = c,
x = − c,
1
maSasadame sistemis amonaxsenia
3
2
3
1
2
x = c, x = − c, x = c,
1
sadac c nebismieri mudmivia.
3
2
3
3
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. amowereT n ucnobiani m wrfiv gantolebaTa sistema. ra
SemTxvevaSi uwodeben am sistemas erTgvarovans? araerTgvarovans?
2. ras uwodeben wrfiv gantolebaTa sistemis amonaxsns? ras ewodeba
sistemis amoxsna?
3. rogor sistems uwodeben Tavsebads? araTavsebads?
4. CamoayalibeT krameris wesi.
5. CawereT n ucnobiani kvadratuli wrfiv gantolebaTa sistema
matriculi saxiT.
6. CawereT gantolebaTa sistemis amonaxsni matriculi saxiT.
7. ra kavSiri arsebobs matricis rangsa da aranulovan minorTa
maqsimalur rigs Soris?
8. rogor wrfiv gantolebaTa sistemas uwodeben marTkuTxovans?
9. CawereT wrfiv gantolebaTa sistemis matrica da gafarToebuli
matrica.
10. CamoayalibeT kroneker-kapelis Teorema.
69
II. praqtikuli savarjiSoebi
1. amoxseniT krameris wesiT:
⎧ 4x − y = 5
⎧ 2x − y = 3
⎧3x + 2 y = 12
1.1. ⎨
; 1.2. ⎨
; 1.3. ⎨
⎩3x + 2y = 1
⎩3x + 2y = 1
⎩2x + 5y = 19
⎧ 2x − x + x = 0
⎧ 3x − 2x + x = 2
⎪
1
2
3
⎪ 1
2
3
1.4. ⎨3x + 2x − 5x = 7
4x + x + 2x = 7
1
2
3
; 1.5. ⎨
1
2
3
⎪
⎪
⎩ x + 3x − 2x = 7
6x − 4x − 2x = 1
1
2
3
⎩ 1
2
3
⎧ 2x + x + x = 4
⎧ 2x + x + x = 4
⎪
1
2
3
⎪
1
2
3
1.6. ⎨ 4x + 2x + 2x = 5
4x + 2x + 2x = 8
1
2
3
; 1.7. ⎨
1
2
3
⎪
⎪
⎩6x + 3x + 3x = 10
6x + 3x + 3x = 12
1
2
3
⎩ 1
2
3
2. m parametris ra mniSvnelobisaTvis aqvs gantolebaTa sistemas:
⎧(3 + m)x + 4 y = 5 − 3m
⎨
⎩ 2x + (5 + m) y = 8
uamravi amonaxsni?
3. kroneker-kapelis Teoremis gamoyenebiT daadgineT Tavsebadia Tu
ara gantolebaTa sistema:
⎧ 5x − x + 2x + x = 7
⎧ 7x + 3x = 2
⎪ 1
2
3
4
⎪ 1
2
3.1. ⎨2x + x + 4x − 2x = 1
x − 2x = −3
1
2
3
4
; 3.2. ⎨ 1
2
⎪
⎪
⎩x − 3x − 6x + 5x = 0
4x + 9x = 11
1
2
3
4
⎩ 1
2
4. amoxseniT Semdegi matriculi gantolebebi
⎛1 2⎞
⎛3 5⎞
⎛3 − 2⎞ ⎛ −1 2⎞
4.1.
⋅ x =
⎜
⎜
⎟
⎟
⎜
⎜
⎟
⎟ ; 4.2. x ⋅
=
⎜
⎜
⎟
⎟ ⎜⎜
⎟
⎟ ;
⎝3 4⎠
⎝5 9⎠
⎝5 − 4⎠ ⎝ − 5 6⎠
⎛ 2
1
− 4⎞
⎛ 6 ⎞
⎛1 1⎞
⎜
⎟
⎜
⎟
4.3. x ⋅
= (2 0)
⎜
⎜
; 4.4 ⎜ − 3
2
−1⎟ ⋅ x = ⎜ − 2⎟
⎝1 1⎟⎟⎠
⎜
⎜
⎟
⎟
⎜
⎜
⎟
⎟
⎝ 1
2
1 ⎠
⎝ 0 ⎠
70
$ 5 matricTa algebris gamoyeneba ekonomikaSi
1. warmoebis dargSoris balansi. vTqvaT warmoeba Sedgeba sxvadasxva
n dargisagan. drois erTeulSi (magaliTad weliwadSi) am dargebis warmo-
ebis moculobebia Q ,Q , ,Q
1
2 L
n . TiToeuli dargis produqcia gamoiyeneba sxva
dargebis mier da agreTve nawilobriv – dargis SegniT.
i-dargis produqciis raodenoba, romelic ixarjeba j-dargis saWi-
roebisaTvis avRniSnoT qij (i,j=1,2,3,...,n). ase rom qij iqneba i-iuri dargis
produqciis is raodenoba, romelic moixmareba, imave i-uri dargis SigniT.
rogorc, wesi i-iuri dargis produqciis mxolod nawili moixmareba
warmoebaSi. am produqciis nawili ixarjeba iseTi miznebisaTvis, romelic
dakavSirebuli araa warmoebasTan; magaliTad: eqsporti, kapitaluri
dabandeba, maragis gazrda da sxv.
i-iuri dargis produqts, romelic ar gamoiyeneba sawarmoo danaxarje-
bisaTvis ewodeba i-iuri dargis saboloo produqti da aRiniSneba qj.
yoveli dargis produqciis ganawileba SeiZleba aRvweroT e.w.
dargTaSoriso kavSirebis cxriliT:
materialuri
dargTaSorisi nakadebi dargebSi
warmoebis
saboloo
warmoebis
moculoba
produqti
dargi
1 2
...
n
1
a1
q11
q12
.. q1n
q1
2
a2
q21
q22
... q2n
q2
.... ...
... ...
...
...
...
n an
qn1
qn2
... qnn
qn
dargTaSoris kavSirebis cxrilSi SeiZleba SevajamoT, mxolod
striqonebis elementebi. svetebis elementebis Sekreba ar SeiZleba, radgan
71
isini warmoadgenen sxvadasxva dargebis produqciebs, romlebic sxvadasxva
erTeulebiT izomeba.
cxadia, rom
⎧Q = q + q +
+ q + q ,
L
⎪ 1
11
12
1n
1
⎪
⎪Q = q + q + + q + q ,
2
21
22
L
2n
2
⎨
(1)
⎪
L
L
L
L
L
L
L
L
L
L
L
⎪
⎪
⎩Q = q + q +
+ q + q .
n
n1
n 2
L
nn
n
miviReT gantolebaTa sistema, romelsac warmoebis sabalanso ganto-
lebebi ewodeba.
isini gviCveneben, rom mocemuli dargis warmoebis mTliani moculoba
tolia produqciaTa im nakadebis jamisa, romlic am dargebidan midis sxva
dargebSi, mimatebuli mocemuli dargis SigniT, moxmarebuli produqciisa
da saboloo produqtis jami.
SemoviRoT aRniSvna:
q
a
ij
=
, i, j = ,
1
,...,
3
,
2
n
ij
. (2)
Qj
cxadia, rom aij -iT aRniSnulia i-iuri dargis produqciis is
moculoba, romelic saWiroa j-iuri dargis produqciis erTi erTeulis
warmoebisaTvis.
aij -ricxvebs warmoebis teqnologiuri koefeicientebi
ewodeba. am koeficientebis saSualebiT SevadginoT n rigis kvadratuli
matrica:
⎛ a
a
a
11
12
L
1n ⎞
⎜
⎟
⎜ a
a
a ⎟
21
22
L
2n
A = ⎜
⎟ . (3)
⎜ K
K K K ⎟
⎜
⎟
⎝ a
a
a
n1
n 2
L
nn ⎠
es matrica axasiaTebs warmoebis teqnikur pirobebs sagegmo periodSi.
amitom mas warmoebis teqnologiuri matricas uwodeben.
vTqvaT, drois gansaxilvel periodSi cnobilia aij -sidideebi, e.i.
(3) matrica. (2)-dan gamomdinareobs, rom
q = a Q
ij
ij
j
ukanaskneli tolobis gaTvaliswinebiT (1) – sabalanso
gantolebebidan miviRebT:
72
⎧Q = a Q + a Q +
+ a Q + q
L
⎪ 1
11
1
12
2
1n
n
1
,
⎪
⎪Q = a Q + a Q +
+ a Q + q ,
2
21
1
22
2
L
2n
n
2
⎨
(4)
⎪
L
L
L
L
L
L
L
L
L
L
L
L
L
L
⎪
⎪
⎩Q = a Q + a Q +
+ a Q + q .
n
n1
1
n 2
2
L
nn
2
n
miRebul sistemaSi igulisxmeba, rom a
q
Q
ij da
i cnobilia, xolo
1
ucnobi. (4) gantolebaTa sistema CavweroT matriculi formiT
Q = AQ + q (5)
sadac
⎛ Q
⎛ q1 ⎞
1 ⎞
⎜
⎟
⎜ ⎟
⎜Q
q
2 ⎟
⎜ 2 ⎟
Q = ⎜ ⎟ , q =
,
M
⎜ ⎟
M
⎜
⎜
⎟
⎟
⎜
⎜ ⎟⎟
⎝Q
q
n ⎠
⎝ n ⎠
xolo A – warmoebis teqnologiuri matricaa. saidanac miviRebT
Q − AQ = q;
(I − A Q
) = q .
miRebuli matriculi gantoleba gavamravloT marcxnidan (I
) 1−
− A -ze
gveqneba
(I − A) 1−(I − A)Q = (I − A)q;
IQ = (I − A) 1
− q .
maSasadame
1
−
Q = (I − A) q (6)
cxadia, am procesSi igulisxmeboda, rom determinanti I − A ≠ 0
me-(6) formula iZlva saSualebas gamovTvaloT warmoebis dagebis
moculebebi ( Qj sidideebi) Tu cnobilia warmoebis teqnologiuri A
matrica da qj – saboloo produqtebis raodenobebi. Qj – sidideebis moZebnas
uwodeben warmoebis gegmis Sedgenas.
vTqvaT gamoviTvaleT, (I
) 1−
− A da miviReT
⎛ A
A
A
11
12
L
1n ⎞
⎜
⎟
−1
A
A
A
(I − A) = ⎜ 21
22
L
2n ⎟ .
⎜
⎜ K
K
K
K ⎟⎟
⎝ A
A
A
n1
n 2
L
nn ⎠
MmaSin (6) gantolebidan miviRebT
73
⎛ Q
A q
A q
A q
1 ⎞
⎛
+
+
+
,
11 1
12 2
L
1n
n ⎞
⎜
⎟ ⎜
⎟
⎜Q ⎟ ⎜ A q + A q +
+ A q ,⎟
2
21 1
22 2
L
2n
n
⎜
⎟ = ⎜
⎟ . (7)
⎜ M ⎟ ⎜
L
L
L
L
L
L
L
L
L
L
⎟
⎜
⎟ ⎜
⎟
⎝Q
A q
A q
A q
n ⎠
⎝
+
+
+
.
n1 1
n 2 2
L
nn
n ⎠
saidanac
⎧
n
⎪ Q = A q + A q +
+ A q =
A q
1
11 1
12 2
L
1n n
∑
,
1 j
j
⎪
j=1
n
⎪
⎪Q = A q + A q + + A q = A q
2
21 1
22 2
L
2n n
∑
,
⎨
2 j
j (8)
⎪
j=1
⎪
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
n
⎪Q = A q + A q + + A q =
A q
n
n1 1
n 2 2
L
nn
n
∑
.
⎪
nj
j
⎩
j=1
(8) gantolebebi gviCveneben, rom (I
) 1−
− A matricis elementebi warmoad-
genen sidideebs, romlebic gansazRvraven raodenobriv damokidebulebebs
yvela dargebis saboloo produqtebs Soris; amasTanave dargebis warmoebis
moculobebi wrfivadaa damokidebuli dargebis saboloo produqtebze.
(I
) 1−
− A matricis elemntebs aqvT garkveuli ekonomikuri Sinaarsi.
aviRoT, magaliTad, k-uri dargis saboloo produqti, Tu vigulisxmebT,
rom Seicvala mxolod am dargis saboloo produqti, xolo sxva dargebis
saboloo produqtebi ucvleli darCa, maSin (8) sistemis yoveli gantoleba
daamyarebs wrfiv damokidebulebas am dargis produqciis saerTo
raodenobasa da k-uri dargis saboloo produqts Soris. am SemTxvevaSi
damokidebuli cvladis nazrdis Sefardeba damoukidebeli cvladis
nazrdTan, toli iqneba am cvladis koeficientisa
Q
Δ
i = A
(i = ,
1 2, ,...,
3
n) (9)
q
ik
Δ k
am formulidan Cans, rom Aik koeficienti gansazRvravs i-iuri dargis
warmoebis im moculobas, romelic saWiroa k-iuri dargis saboloo pro-
duqtis erTi erTeuliT gazrdisaTvis. Aik koeficientebs uwodeben erTobli-
vi moxmarebis koeficientebs, xolo (I
) 1−
− A matricis – erToblivi
moxmarebis koeficientTa matricas.
es koeficientebi asruleben Zalian mniSvnelovan rols warmoebis
dagegmvaSi. Tu viciT (I
) 1−
− A matrica, SegviZlia ganvsazRvroT
74
materialuri warmoebis gegmis sxvadasxva varianti, romlebic Seesabamebian
saboloo produqtis sxvadasxva variantebs.
magaliTi 1. vTqvaT, warmoebaSi 3 dargia da mocemulia warmoebis
teqnologiuri matrica:
⎛ 3
,
0
1
,
0
,
0 0⎞
⎜
⎟
A = ⎜ ,
0 2
3
,
0
,
0 2⎟ .
⎜
⎟
⎝ 1
,
0
,
0 0
1
,
0 ⎠
gegmis winaswari damuSavebis dros miRebulia momxmarebis
produqtebis ori varianti:
1 .0
)
1
(
q = 100
,
000
)
1
(
q = 300
,
000
)
1
(
q = 200
;
000
1
2
3
20.
(2)
q
= 200
,
000
(2)
q
= 300
,
000
)
1
(
q = 100 000.
1
2
3
saWiroa ganisazRvros warmoebis gegmis Sesabamisi variantebi.
⎛ ,
0 7
− 1
,
0
,
0 0 ⎞
⎜
⎟
I − A = ⎜ − ,
0 2
,
0 7
− ,
0 2⎟ .
⎜
⎟
⎝ − 9
,
0
,
0 0
9
,
0
⎠
misi determinanti I − A = ,
0 439 ≠ 0 e.i. aragadagvarebulia.
vipovoT Sebrunebuli (I
) 1−
− A matrica. gvaqvs
⎛ ,
1 49643
,
0 21378
,
0 04751⎞
(I − ) ⎜
⎟
−1
A
= ⎜ ,
0 47506
,
1 49643
33253
,
0
⎟ .
⎜
⎟
⎝ 16627
,
0
,
0 02375
11639
,
1
⎠
pirveli variantisTvis gveqneba
⎛
)
1
(
Q ⎞
1
⎛100 000 ⎞ ⎛ 223 279⎞
⎜
⎟
⎜
)
1
(
Q
I A
2
⎟ = ( − ) ⎜
⎟ ⎜
⎟
−1⎜300 000⎟ = ⎜ 562 941⎟ ,
⎜
)
1
(
⎟
⎜
⎟ ⎜
⎟
⎝Q3 ⎠
⎝200 000⎠ ⎝247 030⎠
saidanac
)
1
(
Q = 223
,
279
)
1
(
Q = 562
,
941
)
1
(
Q = 247 030
1
2
3
.
analogiurad, meore variantisaTvis miviRebT
(2)
Q
= 368
,
171
(2)
Q
= 577
,
194
(2)
Q
= 152 018
1
2
3
.
miRebuli sidideebi gansazRvraven warmoebis gegmis Sesabamis
variantebs.
Tu visargeblebT tolobiT q = a Q
ij
ij
j SeiZleba ganvsarRvroT dargTa
Soris nakadis sidideebi da maTi orive SesaZlo varianti.
75
2. warmoebaSi dasaqmebulobis gansazRvra. warmoebis mier
produqciis gamoSveba moiTxovs ara marto sxvadasxva masalisa da
warmoebis sagnebis (Sromis iaraRebis) danaxarjebs, aramed cocxali Sromis
sxvadasxva danaxarjebsac.
davuSvaT, rom:
1) warmoeba iyofa n dargad da warmoebis teqnologiuri matrica aris
A = (Aij )n ;
n
2) cocxali Sromis danaxarjebi (romelic izomeba samuSao saaTebiT
an momuSaveTa raodenobiT) i-uri dargis produqciis erTeulze
aris zi(i=1,2,..,n).
3) sagegmo periodSi i-uri dargis saboloo produqtia qi(i=1,2,..,n)
saWiroa ganisazRvros cocxali Sromis aucilebeli danaxarjebis
sidideebi sagegmo periodSi.
rogorc viciT, mocemul saboloo sazogadoebriv produqts
Seesabameba calkeuli dargebis produqciaTa gansazRvruli moculobebi,
sagegmo periodSi maT unda Seadginon
⎧
n
⎪ Q = A q + A q +
+ A q =
A q
1
11 1
12 2
L
1n n
∑
,
1 j
j
⎪
j=1
n
⎪
⎪Q = A q + A q + + A q = A q
2
21 1
22 2
L
2n n
∑
,
⎨
2 j
j (10)
⎪
j=1
⎪
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
n
⎪Q = A q + A q + + A q =
A q
n
n1 1
n 2 2
L
nn n
∑
.
⎪
nj
j
⎩
j=1
sadac Aij i-uri dargis produqciis danaxarjTa j-uri dargis saboloo
produqcia erTeulze. zemoT moyvanili meore punqtidan cxadia, rom
cocxali Sromis aucilebeli sruli danaxarjebi sagegmo periodSi
ganisazRvreba formuliT
n
z = Q z + Q z +
+ Q z =
Q z
1 1
2 2
L
n n
∑ i i . (11)
i=1
aqedan, Tu mxedvelobaSi miviRebT gantolebaTa (10) sistemas, gveqneba:
n
n
n
z = z1∑ A q + z
A q
z z
A q
1 j
j
2 ∑
+ +
2 j
j
L
n 1 ∑ nj j . (12)
j=1
j=1
j=1
miRebuli (12) formulidan gamomdinareobs, rom saboloo produqtis
yovel variants mocemuli warmoebis teqnologiuri matricisa da erTeul
produqtze cocxali Sromis cnobili danaxarjebis pirobebSi, cocxali
76
Sromis aucilebeli danaxarjebis srulad garkveuli gegma Seesabameba. z
sididis moZebnas ekonomikaSi Sromis gegmis Sedgenas uwodeben.
magaliTi 2. warmoeba dayofilia 3 dargad; mocemulia warmoebis
teqnologiuri matrica
⎛ 3
,
0
1
,
0
,
0 0⎞
⎜
⎟
A = ⎜ ,
0 2
3
,
0
,
0 2⎟ .
⎜
⎟
⎝ 1
,
0
,
0 0
1
,
0 ⎠
cocxali Sromis danaxarji erTeul produqtze dargebis mixedviT
Seadgens:
z = ,
10
z = ,
15
z = 20
1
2
3
.
gegmis SemuSavebis winaswar etapze miRebuli saboloo produqtze
moTxovnilebis ori varianti:
1 .0
)
1
(
q = 100
,
000
)
1
(
q = 300
,
000
)
1
(
q = 200
;
000
1
2
3
2 .0
(2)
q
= 200
,
000
(2)
q
= 300
,
000
)
1
(
q = 100 000.
1
2
3
saZiebelia Sromis gegmis Sesabamisi variantebi.
produqciis moculobebi: pirvel variantSi (ix. magaliTi 1)
)
1
(
Q = 223
,
279
)
1
(
Q = 562
)
1
(
Q = 247 030
1
,
941
2
3
, meore variantSi miviRebT
(2)
Q
= 368
,
171
(2)
Q
=152
(2)
Q
= 577
1
018
3
,
,
194
2
amgvarad, Sromis saerTo danaxarjebi,
romlebic aucilebelia saboloo sazogadoebrivi produqtis dagegmili
moculobis sawarmoeblad, pirveli variantis mixedviT Seadgens
)
1
(
z = 223279⋅10 + 562941⋅15 + 247030⋅ 20 = 15617305
xolo meore variantis mixedviT
)
1
(
z = 358171⋅10 + 577194⋅15 +152018⋅ 20 =15319980
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ras ewodeba i –uri dargis saboloo produqti?
2. romeli elementebis Sekreba SeiZleba dargTaSoriso kavSirebis
cxrilSi?
3. ras ewodeba warmoebis sabalanso gantoleba?
4. ras uwodeben warmoebis teqnologiur koeficientebs?
77
5. ras ewodeba warmoebis teqnologiuri matrica?
6. ras ewodeba warmoebis gegemis Sedgena?
7. rogor koeficientebs uwodeben erToblivi moxmarebis
koeficientebs?
8. ras uwodeben erToblivi moxmarebis koeficientTa matricas?
II. praqtikuli savarjiSoebi
1. sami urTierT SeuRlebuli dargis teqnologiur koeficientTa
matrica aris
⎛ 0,3
0 1
,
0 ⎞
⎜
⎟
A = ⎜ 0,2
0,3
0,2⎟
⎜
⎜
⎟
⎟
⎝ 0 1
,
0
0 1
, ⎠
cocxali Sromis danaxarji erTeul produqciis erTeulze Seadgens:
z = ,
3
z = ,
5
z = 10
1
2
3
.
cnobilia saboloo produqtze moTxovnilebis ori varianti:
.
1
(1)
q
= 100 000,
(1)
q
= 300 000,
(1)
q
= 200
;
000
1
2
3
.
2
( 2)
q
= 200 000,
( 2)
q
= 300 000,
(1)
q
= 100
.
000
1
2
3
gansazRvreT Sromis sruli cocxali danaxarji yoveli dargis
erTeul produqciaze.
2. warmoeba iyofa sam dargad. mocemulia warmoebis teqnologiuri
matrica
⎛ 0,3
0 1
,
0 ⎞
⎜
⎟
A = ⎜ 0,2
0,3
0,2⎟
⎜
⎜
⎟
⎟
⎝ 0 1
,
0
0 1
, ⎠
cnobilia saboloo produqtze moTxovnilebis ori varianti:
.
1
(1)
q
= 1000,
(1)
q
=
,
3000
(1)
q
=
;
2000
1
2
3
.
2
( 2)
q
=
,
3000
( 2)
q
=
,
2000
(1)
q
= 1000
1
2
3
gansazRvreT warmoebis gegmis Sesabamisi varianti.
78
Tavi III. analizuri geometriis elementebi. analizuri
geometriis elementebis zogierTi gamoyeneba
ekonomikur
amocanebSi
$ 1 wrfe sibrtyeze
1. or wertils Soris manZili. mocemulia or wertili
A(x1,x2) da B(y1,y2;)
vipovoT maT Soris manZili.
nax. 1–dan ABC marTkuTxaa, amitom piTagoras Teoremis Tanaxmad
2
2
2
AB = AC + CB . (1)
magram AC = x − x
BC = y − y
1
2 ,
1
2 ,
miRebuli mniSvnelobebi SevitanoT (1) gantolebaSi, miviRebT:
2
2
AB = (x − x
+ y − y
2
1 )
( 2
)2
1
;
saidanac miviRebT or wertils Soris manZilis saZiebel formulas:
2
d = (x − x
+ y − y
2
1 )
( 2
)2
1
(2)
79
y
y2 B
d y2 - y1
y1 C
A x2 - x1
0 x1 x2 x
nax. 1
2. wiris gantoleba sibrtyeze. vTqvaT sibrtyeze mocemulia raRac
wiri, romlis koordinatebi x da y raRac wesiT arian dakavSirebuli.
(nax. 2) gantolebas, romelic wiris koordinatebs erTmaneTTan akavSirebs,
mocemuli wiris gantoleba ewodeba.
zogadi saxiT wiris gantolebaa F(x,y)=0 (Tu ki es SesaZlebelia
y=f(x)).
A
y y
M(x,y)
M (x,y) A
0 x x
B
nax. 2 nax. 3
magaliTi 1. vipovoT iseTi wiris gantoleba, romlis wertilebi
Tanabari manZilebiTaa daSorebuli A(-2;1) da B(-3;-4) wertilebisagan. (nax. 3).
radgan
d ( ;
A M ) = d( ;
B M ) (sadac d( ;
A M ) aRniSnavs manZils A-dan M-mde)
amitom
80
2
2
2
(x + 2) + (y − )
1 = (x + 3) + (y + 4)2 .
aqedan x + 5y = 10
−
faqtiurad yvela wiris gantoleba arsebobs (Tumca xSirad am
gantolebis moZebna Zalian rTuli saqmea). meore mxriv, yvela gantoleba
rodi aRwers raime wirs. magaliTad 2
2
x + y + 4 = 0 gantoleba ar aRwers
araviTar wirs.
y
M(x,y)
B(0,b) α
N(x,b)
α
0 x
nax. 4
3. wrfis gantoleba sibrtyeze. vTqvaT, raRac wrfe kveTs Oy RerZs
π
B(0;b) wertilSi da Ox RerZTan adgens α kuTxes, sadac 0 < α <
(nax. 4)
2
aviRoT wrfeze nebismieri M ( ;
x y) wertili (mimdinare wertili),
MN
y − b
maSin BMN marTkuTxa samkuTxedidan miiReba, rom tgα =
=
. Tu
BN
x
SemovitanT aRniSvnas k = tgα , maSin miviRebT, rom
y = kx + b (3)
miRebuli gantolebas ewodeba wrfis gantoleba kuTxuri
koeficientiT. qvemoT moyvanilia sxvadasxva kerZo SemTxveva:
y y y
k<0
b
k>0
a
0
x x x
y=kx y=b y=a
nax. 5
81
1. Tu b=0, maSin miiReba y=kx da esaa wrfe, romelic gadis
koordinatTa saTaveSi. amasTan, Tu k>0 maSin wrfe Ox RerZis
dadebiT mimarTulebasTan adgens maxvil kuTxes, Tu k<0 maSin
adgens blagv kuTxes;
2. Tu a=0, maSin k = α
tg = 0 da wrfe Ox RerZis paraleluria;
π
3. Tu α =
, maSin tgα ar arsebobs da wrfe Oy RerZis paraleluria.
2
4. mocemul wertilze gamavali wrfis gantoleba. vTqvaT M (x , y )
1
1
π
wertilze gadis raRac wrfe (nax. 6), romelic Ox RerZTan adgens α ≠
2
kuTxes. radgan M (x , y )
y = kx +
1
1 wertili wrfeze mdebareobs, amitom
b
1
1
. maSin
wrfis gantolebaa:
y − y = k(x − x )
1
1
. (4)
sadac k nebismieri ricxvia da miiRebuli formula warmoadgens
M (x , y )
1
1 wertilze gamaval wrfeTa konis gantolebas (nax. 7).
y y M(x,y)
M(x,y)
Y
α
0 x 0 x
nax. 6 nax. 7
5. or wertilze gamavali wrfis gantoleba. vTqvaT M (x , y )
1
1
1 da
M (x , y )
2
2
2 ori wertilia da gvinda vipovoT wrfe, romelic gadis am or
wertilze.
y
M2
M1
0 x
82
nax. 8
amovweroT M (x , y )
1
1
1 wertilze gamavali wrfeTa konis gantoleba:
y − y = k(x − x )
M (x , y
1
1 . radgan
)
2
2
2 wertili ekuTvnis am wrfes, amitom
y − y
y − y = k(x − x )
2
1
y − y =
(x − x
2
1
2
1 , maSin
)
1
1
x − x
2
1
y − y
y − y
1
2
1
=
(5)
x − x
x − x
1
2
1
miRebuli warmoadgens or wertilze gamaval wrfis gantolebas.
6. wrfis gantoleba RerZTa monakveTebSi. vTqvaT, gvinda vipovoT im
wrfis gantoleba, romelic gadis (
A a )
0
, da B( ,
0 b) wertilebze. or
wertilze gamavali wrfis (5) gantolebidan miviRebT:
y − 0
x − a
=
.
b − 0
0 − a
x
y
saidanac
+
= 1 (6)
a
b
7. wrfis zogadi gantoleba. wrfis zogadi saxis gantolebaa
Ax + By + C = 0 (7)
sadac A da B erTdroulad nulis toli ar aris.
A
C
1.
Tu B ≠ 0 , maSin y = −
x − ; miviReT wrfis gantoleba sakuTxo
B
B
koeficientiT.
C
2.
Tu B = 0 da A ≠ 0 , maSin wrfis gantolebaa x = − .
A
8. kuTxe or wrfes Soris. vTqvaT mocemulia ori wrfe Semdegi
gantolebebiT y = k x + b ;
y = k x + b
1
1 da
;
2
2 da saWiroa maT Soris ϕ
kuTxis moZebna. (nax. 9) -dan Cans, rom ϕ = α + α ;
1
2
y
Y
ϕ α1
α1 α2
O
0 x
83
nax. 9
tgα − tgα
k − k
maSin
2
1
2
1
tgϕ = tg(α −α ) =
=
2
1
,
1+ tgα tgα
1+ k k
1
2
1 2
maSasadame kuTxe or wrfes Soris gamoiTvleba formuliT:
k − k
2
1
tgϕ =
(8)
1 + k k
1
2
saidanac martivad miiReba
9. wrfeTa perpendikularobisa da paralelobis pirobebi. cxadia,
rom Tu 1+ k k = 0
1 2
, maSin wrfeebi urTierT perpendikularulia, saidanac
1
k = −
2
(9)
k1
xolo roca k − k = 0
2
1
, maSin wrfeebi paraleluria
k = k
1
2 (10)
Tu wrfeebi mocemulia zogadi saxis gantolebebiT
A x + B y + C = ;
0
1
1
1
A x + B y + C = ,
0
2
2
2
maSin Sesabamisad miviRebT paralelurobisa (11) da marTobulobis (12)
pirobebs:
A
B
1
1
=
; (11)
A
B
2
2
A A + B B y + C C = 0
1
2
1
2
1
2
(12)
10. wrfeTa gadakveTis wertili. vTqvaT mocemulia ori wrfe
A x + B y + C = 0
A x + B y + C =
1
1
1
da
,
0
2
2
2
romlebic paralelurni ar arian. maSin
imisaTvis, rom vipovoT am ori wrfis gadakveTis wertili. unda amovxsnaT
wrfiv gantolebaTa sistema:
⎧A x + B y + C = ;
0
⎨ 1
1
1
⎩A x + B y + C = ,
0
2
2
2
A
B
radganac, 1
1
≠
, amitom gantolebaTa sistemas aqvs amonaxsni
A
B
2
2
84
− C
B
A
− C
1
1
1
1
− C
B
A
− C
2
2
2
2
x =
, y =
(13)
A
B
A
B
1
1
1
1
A
B
A
B
2
2
2
2
da es amonaxsni warmoadgens am ori wrfis gadakveTis wertilis
koordinatebs.
11. manZili wertilidan wrfemde. vTqvaT mocemulia M (x , y )
0
0 wertili
da wrfe Ax + By + C = 0 .
y
M
N
L
0 x
nax. 10
M wertilidan L wrfemde manZili ewodeba MN perpendikularis
sigrZes. amrigad, M wertilze unda gavataroT L wrfis perpendikularuli
wrfe. am wrfis gantolebaa Bx − Ay + D = 0 da radgan es wrfe gadis M (x , y )
0
0
wertilze, amitom Bx − Ay + D = ,
0
0
0
maSin saZiebeli wrfis gantolebaa
Bx − Ay + (Ay − Bx ) = 0
0
0
.
saZiebeli N wertilis koordinatebis sapovnelad unda amovxsnaT
sistema:
Ax + By + C = ,
0
⎩
⎨
⎧Bx − Ay +(Ay − Bx ) = 0.
0
0
romelic mogvcems N wertilis koordinatebs; amis Semdeg vpoulobT
MN monakveTis sigrZes. sabolood miviRebT
Ax + By + C
0
0
d =
(14)
2
2
A + B
aqve SevniSnoT _ imisaTvis, rom vipovoT manZili or paralelur
wrfes Soris, saWiroa, romelime wrfeze aviRoT nebismieri wertili da
85
Semdeg zemoT miRebuli formuliT vingariSoT manZili aRebuli
wertilidan meore wrfemde.
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. dawereT or wertils Soris manZilis gamosaTvleli formula.
2. rogori saxe aqvs wrfis zogadi saxis gantolebas?
3. rogori saxe aqvs wrfis gantolebas kuTxuri koeficientiT?
4. dawereT wrfeTa konis gantoleba.
5. rogori saxe aqvs or wertilze gamaval wrfis gantolebas?
6. rogori saxe aqvs wrfis gantolebas RerZTa monakveTebSi.
7. dawereT or wrfes Soris kuTxis gamosaTvleli formula.
8. dawereT wrfeT paralelobisa da marTobulobis pirobebi.
9. rogor gamoiTvleba manZili wertilidan wrfemde?
10. rogor vipovoT ori wrfis gadakvetis wertilis koordinatebi?
II. praqtikuli savarjiSoebi
1. ipoveT wertili, romelic 5 erTeuliTaa daSorebuli rogorc
A(2;1) wertlidan ise Oy RerZidan;
2. ipoveT samkuTxedis gverdebis Suawertilebi, Tu misi wveroebia:
A(2;-1), B(4;3), C(-2;1);
3. dawereT im wrfis gantoleba, romelic paraleluria 12x+5y=52
wrfisa da misgan daSorebulia d=2 manZiliT;
4. gaatareT M(-1;2) wertilze wrfe, romelic 5x-2y-5=0 wrfis
marTobia;
5. gaatareT M(-1;2) wertilze wrfe, romelic 5x-2y-5=0 wrfis
marTobia;
6. ipoveT y –is mniSvneloba, Tu manZili M(10;y) wertilidan N(2:-7)
wertilamde udris 17 erTeuls;
7. mocemulia tolgverda ABC samkuTxedis ori A(-1;1) da B(2;3) wvero.
moZebneT mesame C wvero;
86
8. x + 2 y − 5 = 0 da 3x − y −1 = 0 wrfeebis gadakveTis wertilze gaatareT
wrfe, romelic x − y + 3 = 0 wrfesTan adgens
0
45 kuTxes;
9. SeadgineT im wrfis gantoleba, romelic gavlebulia 3x − y − 3 = 0 ,
2
11
y = − x +
wrfeTa gadakveTis wertilze pirveli wrfis
3
2
marTobulad;
10. ipoveT wrfe, romelic 4x + 3y + 1 = 0 wrfis paraleluria da misgan
daSorebulia 3 erTeuliT.;
11. dawereT im wrfis gantoleba, romelic gadis koordinatTa
saTaveze da ordinatTa RerZTan adgens
0
150 -ian kuTxes;
12. ipoveT monakveTis Sua wertilis koordinatebi, Tu bolo
wertilebis koordinatebia: (9;-2) da (1:5);
13. ipoveT sakuTxo koeficientebi da monakveTebi, romlebsac mokveTs
koordinatTa RerZebze 2x − y + 3 = 0 wrfe;
14. ipoveT
manZili
Semdeg
paralelur
wrfeebs
Soris:
x − 2y + 5 = 0 da 2x − 4y +1 = 0 ;
15. ipoveT manZili Semdeg wertilebs Soris: L(10;-3) da F(4;5).
$ 2 meore rigis wirebi
1. wrewiri. skolis kursidan cnobilia, rom wrewiri im wertilTa
simravlea, romlebic toli manZiliT arian daSorebuli raRac O(x0,y0)
wertilidan. Tu M(x,y) wrewiris nebismieri wertilia, maSin d(Μ,0)=R.
maSasadame wrewiris gantolebaa
2
2
2
(x − x ) + (y − y ) = R
0
0
(1)
Tu wrewiris centri koordinatTa saTaveSia, maSin wrewiris gantolebaa
x 2 + y 2 = R2 . (2)
ganvixiloT meore rigis or ucnobiani zogadi saxis gantoleba
2
2
Ax + Bxy + Cy + Dx + Ey + F = 0 .
A,B, da C erTdroulad nulis toli ar xdebian. gamovikvlioT rodis
aris es gantoleba wrewiris gantoleba. amisaTvis amovweroT wrewiris
gantoleba gaSlili saxiT:
2
x − 2
2
2
x x + x + y − 2
2
2
y y + y − R = ,
0
0
0
0
0
87
A C
saidanac Cans, rom B=0 da
=
. maSasadame, wrewiris zogadi gantolebaa:
1
1
2
2
Ax + Ay + Dx + Ey + F = 0 (3)
aqedan, Tu gamovyofT srul kvadratebs
2
2
2
2
⎛
D ⎞
⎛
E ⎞
D + E − 4AE
⎜ x +
⎟ + ⎜ y +
⎟ =
2
⎝
2A ⎠
⎝
2A ⎠
4A
cxadia, rom saWiroa Sesruldes piroba:
2
2
D + E − 4AE
> ,
0
4 2
A
⎛
8
F ⎞
maSin wrewiris centria ⎜
0 −
,−
⎟ wertili da radiusia:
⎝ 2A 2A ⎠
2
2
D + E − 4AE
R =
.
4 A
B2.Uelifsi. ganvixiloT isev meore rigis orcvladiani gantoleba,
romelSic B = ,
0 A ≠ 0 C ≠ .
0 gamovyoT sruli kvadratebi, miviRebT
2
2
2
2
⎛
D ⎞
⎛
E ⎞
D
E
A⎜ x +
⎟ + C⎜ y +
⎟ =
+
− F.
⎝
2A ⎠
⎝
2C ⎠
4A 4C
2
2
D
E
D
E
avRniSnoT x =
, y = −
, ρ =
+
− F.
0
2
0
A
2C
4A 4C
miviRebT
(
A x − x )2 + C y − y
= ρ
0
(
0 )2
2
simartivisTvis jer dauSvaT, rom x = y = 0
0
0
, maSin wiris gantolebaa
2
Ax +
2
Cy = ρ
am wirs elifsi ewodeba, Tuki AC>0. e.i. elifsi ewodeba sibrtyis im
wertilTa simravles, romlis yoveli wertilidan or mocemul
wertilebamde manZilebis jami mudmivi sididea.
vigulisxmoT, rom A>0, C>0. cxadia Cven gvainteresebs SemTxveva, roca
ρ > 0 . am SemTxvevaSi miviRebT gantolebas:
x 2
y 2
+
= 1 (4)
a 2
b2
88
romelsac elifsis kanonikuri gantoleba ewodeba. am gantolebaSi
ρ
ρ
a =
da b =
sidideebi warmoadgenen elifsis naxevar RerZebs. roca
A
C
a = b miiReba elifsis kerZo SemTxveva – wrewiri.
y
O
b
(x;y)
d2
d1
x
-a
-c
c
a
-b
nax. 1
(a ),
0
;
(−a ),
0
;
( ;
0 b), ( ;
0 b
− ) wertilebs elifsis wveroebi ewodebaT.
(c ),
0
;
(−c )
0
; wertilebs, sadac
2
2
c = a − b , elifsis fokusebi ewodebaT.
aviRoT elifsis nebismieri ( ;
x y) wertili da gamovTvaloT jami
d + d
1
2 . marTlac
⎛
2
b
⎞
d = (x − 2
c) + 2
y =
2
x − 2cx + 2
c + 2
y =
2
x − 2cx +
2
(a − 2
b ) +
2
b −
2
x
=
1
⎜
⎜
2
⎟
⎟
⎝
a
⎠
2
2
2 2
⎛
⎞
2
b
c x
⎛ cx
⎞
cx
= x 1−
− 2
2
cx + a ==
− 2
2
cx + a = ⎜
− a⎟ = a −
= a − ε .
x
⎜
⎜
2 ⎟⎟
2
a
a
⎝
⎠
⎝ a
⎠
a
c
sadac ε =
elifsis eqscentrisitetia.
a
d2 -is analogiuri gamoTvla gvaZlevs
d = a + x
ε
2
maSasadame
d + d = 2a
1
2
e.i. manZilebis jami elefsis nebismieri wertilidan elifsis
fokusebamde mudmivia.
89
2. hiperbola. zemoT Cven meore rigis orcvladiani gantoleba
miviyvaneT
(
A x − x
C y y
0 )2 +
( − 0)2 = ρ
saxeze da vnaxeT roca AC > 0 , maSin es gantoleba gvaZlevs elifss. roca
am gantolebaSi AC < 0 maSin gantoleba aRwers wirs, romelsac hiperbola
ewodeba.
vTqvaT
A > 0 da C < 0 , maSin SesaZlebelia sami SemTxveva
ρ > ,
0 ρ = ,
0 ρ < 0
1. ρ > ,
0 maSin miiReba hiperbolas kanonikuri gantoleba
x 2
y 2
−
= 1 (5)
a 2
b2
ρ
ρ
sadac a =
da b =
- Sesabamisad hiperbolas namdvili da
A
− C
warmosaxviTi naxevarRerZebia. e.i. hiperbola ewodeba sibrtyis im
wertilTa simravles, romlis yoveli wertilidan or mocemul
wertilebamde (romelsac fokusebs vuwodebT) manZilebis sxvaobis moduli
mudmivi sididea.
hiperbolas fokusebia (c )
0
; da (−c )
0
; wertilebi. sadac
2
2
c = a + b ,
xolo misi wveroebia A (−a 0
; )
A (−a 0
;
2
da
)
1
.
Tu am SemTxvevaSic gamoviTvliT d
d
1 da
2 , maSin mudmivi gamova
d − d = 2a
1
2
sxvaoba. gadavweroT hiperbolas gantoleba Semdegi saxiT:
b
2
2
y = ±
x − a
a
roca x sakmaod didia, maSin x2 − a2 ≈ x da aqdan vaskvniT, rom
b
hiperbola uaxlovdeba y = ± x wrfeebs, roca x → ∞ . am wrfeebs
a
hiperbolis asimptotebi ewodebaT.
Yy
d1
d2
x
-c
A
-a
a
2
A1
c
90
nax. 2
2. ρ = ,
0 maSin hiperbola gadagvardeba da miiReba ori wrfe
b
b
y =
x , y = − x .
a
a
3. ρ < ,
0 maSin miiReba hiperbola
2
2
y
x
−
=1.
2
2
b
a
cxadia, rom ubralod (5) SemTxvevaSi miRebuli hiperbola
`amotrialdeba~ (sakoordinato RerZebi Seicvlebian).
3. parabola. kvlav ganvixiloT meore rigis orcvladiani zogadi
gantoleba, romelSic B = 0 , A = 0 da C ≠ 0 maSasadame
2
Cy + Dx + Ey + F = 0 .
Tu D=0, maSin miiReba ori horizontaluri wrfe. amitom es saintereso
SemTxveva ar aris. D ≠ 0 maSin srul kvadratebamde Sevsebis Semdeg
⎛
E 2
⎞
E 2
miviRebT: C⎜ y +
⎟ + Dx + F =
.
⎝
C
2 ⎠
C
4
F
E 2
E
D
avRniSnoT: x = −
+
, y =
2 p = −
0
, maSin miiReba parabolas
D 4DC
0
C
2
C
kanonikuri gantoleba.
2
( y − y
= 2p x − x
0 )
(
0 ) (6)
Yy
p<0
y
0
x
x0
0
91
nax. 3
⎛ p ⎞
Tu 2
y = 2 px, maSin ⎜
⎟ wertils parabolas fokusi ewodeba, xolo
⎝ 0
;
2 ⎠
p
x = − wrfes, ki parabolas direqtrisa.
2
y
M(x;y)
d1
p
−
2
d2
O
x
nax. 4
(x0;y0) wertils parabolas saTave ewodeba, xolo p ricxvs parabolis
parametri. roca p>0, maSin parabolas Stoebi marjvnivaa mimarTuli, xolo
roca p<0, maSin marcxniv. y=y0 wrfes parabolas simetriis RerZi ewodeba.
roca parabola gadis koordinatTa saTaveze, maSin misi gantolebaa
y 2 = 2px (7)
p
advili dasanaxia, rom d = x +
1
da
2
⎛
p ⎞2
2
2
p
⎛
p ⎞
d =
x
y
2
p
x − px +
+ 2 px = ⎜ x + ⎟ = x +
2
⎜ − ⎟ + 2 =
⎝
2 ⎠
4
⎝
2 ⎠
2
miviReT parabolas erTerTi ZiriTadi damaxasiaTebeli Tviseba d1=d2.
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
92
1. ras ewodeba elifsi? hiperbola?
2. rogori saxe aqvs elifsis kanonikur gantolebas?
3. rogori saxe aqvs hiperbolas kanonikur gantolebas?
4. ras ewodeba elifsis eqscentrisiteti?
5. rogor wrfeebs ewodebaT hiperbolas asimptotebi?
6. rogori saxe aqvs parabolas kanonikur gantolebas?
7. ras ewodeba parabolas fokusi?
II. praqtikuli savarjiSoebi
1. ipoveT Semdegi wrewiris radiusi da centris koordinatebi:
2
2
x + y − 4x = 0
2. ipoveT Semdegi wrewiris radiusi da centris koordinatebi:
2
2
x + y + 6y − 7 = 0
3. ipoveT Semdegi wrewiris radiusi da centris koordinatebi:
2
2
x + y + 2x −10y +1 = 0
4. ipoveT Semdegi wrewiris radiusi da centris koordinatebi:
3 2
x + 3 2
y − 4x − 6y −15 = 0
5. ipoveT Semdegi wrewiris radiusi da centris koordinatebi:
2
2
x + y − 4x + 2y +1 = 0
6. gansazRvreT elifsis eqscentrisiteti, Tu misi did RerZi 3-jer
aRemateba mcire RerZs.
7. gansazRvreT elifsis eqscentrisiteti, Tu misi RerZebis fardobaa 5:3
8. SeadgineT elifsis gantoleba, Tu fokusebs Soris manZili udris
6-s, xolo didi naxevarRerZia 5.
9. SeadgineT elifsis gantoleba, Tu eqscentrisiteti e=0,8, xolo didi
naxevarRerZia 10.
10. SeadgineT elifsis gantoleba, Tu mcire naxevarRerZia 3, xolo
2
eqscentrisiteti e =
2
11. ipoveT RerZTa sigrZeebi, fokusebi da eqscentrisiteti Semdegi
elifsis: 9 2
x + 25 2
y = 225
93
12. ipoveT RerZTa sigrZeebi, fokusebi da eqscentrisiteti Semdegi
elifsis: 16 2
2
x + y = 16
13. SeadgineT hiperbolis kanonikuri gantoleba, Tu wveroebs Soris
manZilia 8, fokusTa Soris manZili ki 10.
14. ipoveT RerZTa sigrZeebi, fokusebi da eqscentrisiteti Semdegi
elifsis: 25 2
x +169 2
y = 4225
15. gamoTvaleT hiperbolis naxevarRerZebi, Tu fokusuri manZilia 8,
xolo direqtrisebs Soris manZili ki 6.
18
16. ipoveT hiperboli, Tu direqtrisebs Soris manZilia
, xolo
5
warmosaxviTi naxevarRerZi ki 4.
17. ipoveT RerZebi, fokusebi da eqscentrisiteti Semdegi hiperbolis:
7 2
x − 2 2
y = 14
18. ipoveT RerZebi, fokusebi da eqscentrisiteti Semdegi hiperbolis:
4 2
x − 9 2
y = 25
$ 3 wrfivi funqciebi ekonomikur amocanebSi
am paragrafSi Cven ganvixilavT wrfis gantolebis cnebis gamoyenebas
ekonomikaSi. rogorc vnaxeT sibrtyeze wrfis gantolebaa:
y = ax + b . (1)
amitom bunebrivia, rom y = ax + b -s vuwodoT wrfivi funqcia. am
funqcias didi gamoyeneba aqvs ekonomikaSi. amis sailustraciod
ganvixiloT ramdenime ekonomikuri amocana. SevniSnoT, rom TiTqmis yvela
amocanis amoxsnisas mogviwevs garkveuli cvladebis Semoyvana. wminda
maTematikuri Teoriisagan gansxvavebiT, aq cvladebi SezRudulia da isini
SeiZleba icvlebodnen mxolod amocanis pirobebiT gansazRvrul
simravleebze. amitom yovel konkretul amocanaSi aucilebelia miuTiToT
cvladebis Sesabamisi dasaSvebi simravleebi, rom Tavidan aviciloT mcdari
pasuxebi da gavakeToT swori daskvnebi.
amocana 1. tvirTis gadatanis xarjebis gansazRvra. vTqvaT, erTi da
imave tvirTis gadatana mocemuli qalaqidan 20 km-iT daSorebul punqtamde
94
Rirs 25 lari, xolo 40 km-iT daSorebul punqtamde – 45 lari. ra eRireba
imave tvirTis gadatana x manZilze mocemuli qalaqidan, Tu cnobilia, rom
damokidebulebma manZilsa da tvirTis gadatanis xarjebs Soris wrfivia?
y
45
25
5
0
20 40 x
nax. 1
manZili qalaqidan daniSnul punqtamde x km-ia. tvirTis gadatanis
xarji avRniSnoT y lariT. amocanis pirobis Tanaxmad, saZiebel
damokidebulebas gansazRvravs wrfivi funqcia – wrfe. amasTan, roca x=20,
maSin y=25; xolo roca x=40, maSin y=45; e.i. aRniSnuli wrfe gaivlis (20;25)
da (40;45) wertilebze (nax. 1 ) misi Sesabamisi gantoleba iqneba:
x − 20
y − 25
=
40 − 20
45 − 25
aqedan miviRebT saZiebeli wrfis gantolebas y=x+5, romelic
gansazRvravs gadatanis xarjebs, e.i. x km-ze gadatanis xarji iqneba (x+5)
lari. aq x>0; y>0
amocana 2. warmoebis simZlavris gansazRvra. qarxnis sawarmoo
simZlavrea 200 litri ludi saaTSi an 600 litri limonaTi. qarxanas
SeuZlia erTdroulad awarmoos ludica da limonaTic. SevadginoT
gantoleba, romelic daaxasiaTebs qarxnis sawarmoosimZlavres Tu
damokidebuleba warmoebuli limonaTisa da luds Soris wrfivia.
Ox RerZze gadavzomoT qarxnis mier erT saaTSi warmoebuli ludis
raodenoba, xolo Oy RerZze erT saaTSi warmoebuli limonaTis raodenoba.
amocanis pirobidan gamomdinare Sesabamisi wrfivi damokidebulebis
grafiki gadis (200;0) da (0;600) wertilebze, amitom saZiebeli wrfis
gantoleba RerZTa monakveTebSi iqneba:
95
x
+ y = 1.
200
600
y
y
50
600
0 200 x
0 8 x
nax. 2 nax. 3
O
aqedan 3x+y=600. am gantolebiT ganisazRvreba qarxnis sawaroo
simZlavre, romelic akavSirebs 1 saaTSi warmoebuli ludisa da limonaTis
raodenobebs. cxadia, rom 0 ≤ x ≤ 200 da 0 ≤ y ≤ 600 (nax.2).
amocana 3. produqciis danaxarjis gansazRvra drois mixedviT.
vTqvaT, raime warmoebas aqvs 50 tona sawvavi, romelic unda daixarjos 8
dRis ganmavlobaSi. vipovoT kavSiri dasaxarji sawvavisa da gasuli
dReebis raodenobebs Soris, Tu es damokidebuleba wrfivia.
amocanis pirobis Tanaxmad, roca dReTa raodenoba 0 tolia
dasaxarjia 50 t. sawvavi,xolo 8 dRis Semdeg sawvavis raodenobaa 0,
maSasadame, Tu Ox RerZze gadavzomavT dReebis raodenobas, xolo Oy RerZze
sawvavis raodenobas (nax.3), maSin saZiebeli grafiki (wrfe) gaivlis (0;50)
da (8;0) wertilebze. RerZTa monakveTebSi misi gantolebaa:
x
+ y = .
1
8 50
aqedan 25x+4y=200. swored esaa saZiebeli kavSiri. cxadia, rom 0 ≤ x ≤ 8
da 0 ≤ y ≤ 50 .
amocana 4.UtvirTis gadatanis optimaluri variantis arCeva. vTqvaT,
tvirTis gadatanis xarjebi (larebSi) x km-ze pirveli saxis transportiT
gamoisaxeba y=0,1x+15 formuliT, meore saxis transportiT ki y=0,06x+25
formuliT. ra saxis transportiTaa ufro xelsayreli tvirTis gadatana.
amocanis gadaswyvetad avagoT amocanaSi miTiTebuli wrfivi
funqciebis Sesabamisi L1 da L2 wrfeebi (nax. 4). cxadia, rom x>0, amitom
96
ganvixilavT mxolod mis Sesabamis naxevarwrfeebs. maTi gadakveTis
wertili miZebneba:
y = 1
,
0 x + ,
15
sistemis ⎩⎨⎧y = ,006x + 25.
amonaxsniT. miviRebT A=A(250;40).
Y
y N1
A(250.40)
40 M2
L
2
25
M1
15 L
1
M
0 x0 250 N x
nax. 4
naxazidan vaskvniT, rom Tu 0 < x < 250
0
, maSin x0-is Sesabamisi xarjebi
iqneba pirveli saxis transportiT
)
1
(
y = MM
0
1 meore saxis transportiT
(2)
y
= MM
MM < MM
0 < x <
0
2 . radgan
1
2 amitom roca
250
0
tvirTis gadazidva
ufro xelsayrelia pirveli saxis transportiT. analogiurad roca
x > 250
0
, maSin tvirTis gadazidva xelsayrelia meore saxis transportiT.
Tu manZili 250 –is tolia, maSin orive transportis SemTxvevaSi erTi da
igivea da 40 laris tolia.
amocana 5, kompleqsuri sawarmos optimaluri muSaobis programis
Sedgena. meqanikur saamqros erTi Tvis ganmavlobaSi SeuZlia daamzados A
saxis manqanebisaTvis nawilebis 60 kompleqti. an B saxis manqanebisaTvis
nawilebis 120 kompleqti. amwyob saamqros erTi Tvis ganmavlobaSi SeuZlia
aawyos A saxis 120 manqana an B saxis 80 manqana. SevadginoT orive saamqros
muSaobis yvelaze optimaluri da ekonomikuri programa, Tu TiToeuli
saamqros simZlavre iseTia, rom A da B saxis gamoSvebul produqtebs Soris
damokidebuleba wrfivia. (yvelaze optimaluri da ekonomikuri programa
aris iseTi sasurveli da idealuri reJimi, rodesac meqanikuri saamqro erT
TveSi awarmoebs n raodenobis, A saxisa da m raodenobis B saxis
97
kompleqtebs da amwyobi saamqroc erT TveSi aawyobs n raodenobis, A saxisa
da m raodenobis B saxis manqanebs. es faqtobrivad niSnavs, rom
simZlavreTa Seucvlelad am saamqros kopleqsi unaSTod da mocdenis
gareSe muSaobs).
Ox RerZze gadavzomoT A saxis produqciis raodenoba, xolo Oy-ze B
saxisas. pirobis Tanaxmad, meqanikuri saamqros mier gamoSvebuli A da B
saxis kompleqtebs, Soris damokidebulebis wrfe (60;0) (0;120) wertilebze
gadis. misi gantoleba iqneba
x
+ y =1,
60 120
anu 2x + y = 120 . es gantoleba niSnavs Semdegs: meqanikuri saamqros
simZlavre iseTia, rom mas SeuZlia erT TveSi awarmoos x kompleqti A
saxis manqanisaTvis da y=120-2x kompleqti B saxis manqanisaTvis. aseve,
amwyobi saamqros mier gamoSvebuli A da B saxis manqanebis raodenobebs
Soris damokidebuleba moicema (120;0) da (0;80) wertilebze gamavali wrfiT.
misi gantolebaa
x
+ y =1.
120 80
Nanu 2x+3y=240. amragad, amwyobi saamqros simZlavre iseTia, rom Tu
erT TveSi igi gamouSvebs A saxis manqanas, maSin man unda gamouSvas
1
y = (240 − 2x) raodenobis B saxis manqana. avagoT miRebuli wrfeebis
3
grafiki da gavanalizoT igi (nax.5).
y
Y
120
80
M(30,60)
60
0
30 60 120 x
nax. 5
98
2x + y =
,
120
es ori wrfe ikveTeba M wertilSi, romlis koordinatebia ⎩⎨⎧2x + 3y = 240.
sistemis amonaxseni: x =
;
30 y = 60 e.i. M(30;60). radgan M wertili orive
grfiks ekuTvnis, es niSnavs, rom meqanikuri saamqros Tavisi simZlavris
pirobebSi SeuZlia awarmoos 30 kompleqti A saxis manqanisaTvis da 60
komtleqti B saxis manqanisaTvis, xolo amwyob saamqros SeuZlia gamouSvas
30 cali A saxis manqana da 60 cali B saxis manqana. maSasadame, M (30;60)
wertili iZleva yvelaze optimaluri da ekonomikuri muSaobis programas,
romlis moZebnac warmoadgenda Cvens mizans.
amocana 6. warmoebis maqsimaluri mogebis gansazRvra. vTqvaT,
fabrika awarmoebs ori A da B saxis produqcias. A saxis erTi erTeulis
warmoebas sWirdeba erTi samuSao saaTi da 2kvt/sT eleqtro energia. B
saxis erTi erTeulis warmoebas 2sT da 1kvt/sT eleqtroenergia. dRis
ganmavlobaSi orive saxis produqciis warmoebisaTvis gankuTvnilia
araumetes 8 saaTisa da 7kvt/sT eleqtroenergiisa. mogeba A saxis pro-
duqciis erTeulze 30 laria, xolo B saxis erTeulze-20 lari. rogor war-
vmarToT warmoeba, rom mogeba iyos maqsimaluri?
vTqvaT, fabrikam awarmoa x raodenobis A saxis produqcia da y rao-
denobis B saxis produqcia. SevniSnoT, rom x ≥ 0
y ≥ 0 . pirobis Tanaxmad
mogeba gamoiTvleba formuliT 30x+2y. aRniSnuli raodenobis produqciis
warmoeba amoiTxovs x+2y saaTs da (2x+y) kvt/sT eleqtroenergias. amitom
amocanis pirobis Tanaxmad miviRebT sistemas:
⎧
1
⎧x + 2 y ≤ ;
8
⎪y ≤ − x + 4,
⎪
⎪2x + y ≤ 7;
⎪
2
⎨
⎨
anu
y ≤ −2x + 7,
⎪x ≥ ;
0
⎪x ≥
⎪
;
0
⎩y ≥ 0.
⎪
⎩y ≥ 0.
amovxsanaT igi grafikuli xerxiT. naxazze (nax. 6) daStrixuli are
warmoadgens saZiebel amonaxsns. SevniSnoT, rom Tu Tavisi SinaarsiT x da y
mTeli ricxvebia, maSin utolobaTa sitemis amonaxseni iqneba mxolod mTel
koordinatebiani wertilebi, romlebic ekuTvnian daSrtixul ares (isini
naxazze gamuqebulia). Cveni mizania, vnaxoT x da y parametrebis rogori
mniSvnelobisaTvis iqneba mogebis 30x+20y=c. funqcia maqsimaluri.
99
y
7
6
4
M(2;3)
0 4 Lc 8 x
nax. 6
SevniSnoT, rom Oxy sakoordinato sibrtyeze avagebT am wrfivi
funqciis grafiks (c>0 ganvixiloT rogorc mudmivi parametri), maSin
Sesabamis wrfes gaaCnia Semdegi ekonomikuri interpretacia:
Tu A da B saxis warmoebuli produqciebis raodenobebia x0
y
da
0
amasTan (x y
0, 0 )∈ Lc , maSin mogeba tolia c-si e.i. mogebis 30x+20y funqcia Lc
wiris gaswvriv mudmivia da c-s tolia. cxadia, rom 30x+20y=c funqciis
grafiki 30x+20y=0 gantolebiT mocemuli wrfis paraleluria.
c parametris zrdas mosdevs grafikis paraleluri gadatana marjvniv.
c-s zrda eqvivalenturia mogebis zrdisa.
imisaTvis rom vipovoT maqsimaluri mogeba xsenebul pirobebSi, unda
movZebnoT Lc wrfis iseTi mdgomareoba, rodesac Lc wrfe Seicavs
daStrixuli aris erT wertils mainc da amasTan mis marjvniv daStrixuli
aris arcerTi weritili ar Zevs. amas mivaRwevT Lc wrfis TandaTanobiT
paraleluri gadataniT marjvniv. Cvens SemTxvevaSi Lc wrfis aseTi zRvruli
mdebareoba miiRweva maSin, rodesac is gaivlis M(2;3) wertilze, maSin
c=30⋅2+20⋅3=120 da amitom Sesabamisi wrfis gantolebaa
x
30x+20y=120 anu + y = 1.
4 6
es wrfe Ox daOy RerZebs hkveTs Sesabamisad (4;0) da (0;6) wertilebSi
da amitom mdebareobs x+2y=8 da 2x+y=7 wrfeebs Soris. amrigad, moZebnilia
Lc wrfis ukiduresad zRvruli marjvena mdebareoba, rodesac is Seicavs
erTerT wertils daStrixuli aredan, kerZod M(2;3) wertils da mis
100
marjvniv daStrixuli aris sxva wertilebi aRar mdebareoben. aqdan daskvna:
mogeba maqmaluri iqneba rodesac iwrmoeba A saxis ori erTeuli da B saxis
sami erTeuli produqciebi. xolo maqsimaluri mogeba tolia 30⋅2+20⋅3=120
laris.
aseTi tipis amocanebi ganekuTvnebian wrfivi programirebis amocanaTa
klass.
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. aCveneT magaliTze raSi mdgomareobs tvirTvis gadatanis xarjebis
gansazRvra.
2. ganixileT warmoebis simZlavris gansazRvra konkretul magaliTze.
3. ganixileT amocana produqtis danaxarjis gansazRvraze drois
mixedviT.
4. aCveneT konkretul amocanaze raSi mdgomareobs tvirTvis gadatanis
optimaluri variantis arCeva?
5. ganixileT amocana kompleqsuri sawarmos optimaluri muSaobis
programis SedgenasTan dakavSirebiT.
II. praqtikuli savarjiSoebi
1. firmis danaxarjebi 200 cali saqonlis dasamzadeblad Seadgens 250
lars, xolo 700 erTeulis dasamzadeblad -560 lars. SeadgineT
danaxarjebis funqcia (im pirobiT, rom danaxarjebis funqcia wrfivia) da
gamoTvaleT danaxarjebi, Tu firmam daamzada:
1.
150
erTeuli;
2. 500 erTeuli; 3. 1000 erTeuli;
2.
warmoebis danaxarji garkveuli saqonlis 100 erTeulis
sawarmoeblad Seadgens 300 lars, xolo 500 erTeulis sawarmoeblad – 600
lars. gansazRvreT warmoebis danaxarjebi, Tu vigulisxmebT, rom
danaxarjebis funqcia wrfivia.
101
3. ori saxis transportiT gadazidvis xarjebi gamoisaxeba funqciebiT
y=50x+150 da y=25x+250. sadac x manZilia, xolo y satransporto xarjebi.
rodisaa ufro ekonomiuri meore saxis transportis gamoyeneba.
4. firmis TanamSromeli valdebulia imuSaos kviraSi 40 saaTi. Tu misi
samuSaos saaTebis raodenob akviarSi gadaaWarbebs 40 saaTs, maSin mas
yovel zenormatiul saaTSi uxdian 2-jer met Tanxas. rogoria
TanamSromlis saaTobrivi anazRaureba, Tu kvirasi 55 saaTis muSaobisaTvis
man miiRo 420 lari?
Tavi IV. maTemtikuri analizis Sesavali. funqciis cnebis
gamoyeneba ekonomikur amocanebSi
$ 1 funqcia
funqciis cneba maTematikuri analizis erT-erTi ZiriTadi cnebaa.
bunebis movlenebi, procesebi, kerZod, teqnikuri da ekonomikuri procesebi
xSirad aRiwereba da Seiswavleba funqciis saSualebiT.
1. cvladi da mudmivi sidideebi. cvladi ewodeba iseT sidides,
romelic mocemuli sakiTxis pirobebSi Rebulobs sxvadasxva ricxviT
mniSvnelobas. mudmivi ewodeba iseT sidides, romelsac mocemuli sakiTxis
pirobebSi aqvs erTi da igive ricxviTi mniSvneloba. Tu raime sidides
avRniSnavT x an a asoTi, es ar niSnavs imas, rom cvladia es sidide, Tu
mudmivi, amitom sididis cvlilebis xasiaTi yovelTvis gansakuTrebiT unda
iyos aRniSnuli. erTi da igive sidide erT SemTxvevaSi SeiZleba iyos
mudmivi, meoreSi ki cvladi. aseT sidides parametri ewodeba. magaliTad,
Tu r radiusian wres misi radiusisSeucvlelad vagoravebT wrfeze, an
centris mdebareoba Seicvleba, xolo farTobi
2
π ⋅ r mudmivi darCeba. Tuki
102
wris centrs uZravad davtovebT da r radiuss SevcvliT, wris
2
π ⋅ r
farTobi Seicvleba. amrigad wris farTobi erT SemTxvevaSi mudmivia,
meoreSi ki – cvladi, da radgan orive SenTxvevaSi wris farTobi aris
2
π ⋅ r , amitom r aris parametri.
arsebobs iseTi mudmivi sidideebi, romlebic Tavis mniSvnelobas
nebismier pirobebSi inarCuneben. aseT sidideebs absoluturi mudmivi
sidideebi ewodeba. magaliTad, samkuTxedis Siga kuTxeebis jami da
wrewiris sigrZis fardoba diametrTan absoluturad mudmivis sidideebia.
cvladi sidideebi Cveulebriv aRiniSneba laTinuri anbanis
ukanaskneli asoebiT x, y, z, u, w, t, xolo mudmivi sidideebi ki pirveli
asoebiT a,b,c,...
2. funqciis cneba. sxvadasxva sidides Soris warmoSobil
damokidebulebaTa Seswavlas mivyavarT funqciis cnebamde. SesaZlebelia,
rom garkveul pirobebSi erTi sididis yovel mniSvnelobas Seesabamebodes
meore sididis garkveuli mniSvneloba. Tu gvaqvs aseTi Sesabamisoba, maSin
amboben rom meore sidide pirveli sididis funqciaa. magaliTad, haeris
temperatura dRis sxvadasxva droisaTvis sxvadasxvaa, ris gamoc haeris
temperatura drois funqciaa, an sxvadasxva radiusian sferos sxvadasxva
moculoba aqvs, amitom sferos moculoba aris radiusis funqcia. axla
moviyvanoT funqciis zusti gansazRvra, romelic ekuTvnis dirixles.
Tu namdvil ricxvTa E simravlis yovel x ricxvs Seesabameba raime
wesiT erTi y namdvili ricxvi, maSin vityviT, rom y aris x –is funqcia.
TviT E simravles funqciis gansazRvris are ewodeba.
is faqti, rom y aris x –is funqcia, ase Caiwereba: y = f (x) , da
ikiTxeba igreki udris ef iqss. am SemTxvevaSi ambobnen, rom x da y
cvladebs Soris arsebobs funqcionaluri damokidebuleba. x –s ewodeba
damoukidebeli cvladi, anu argumenti, y –s ki damokidebuli cvladi.
aq f aRniSnavs ara sidides, aramed damoukidebel da damokidebul
cvladebs Soris Sesabamisobis kanons. e.i. f SeiZleba aRniSnavdes
maTematikur im operaciaTa erTobliobas, romlebic unda vawarmooT x –ze,
rom miviRoT y. magaliTad,
2
y = x . aq f aRniSnavs Semdegs: x unda
avaxarisxoT kvadratSi, rom miviRoT y.
103
Tu f (x) gansazRvrulia E simravleze da x0 aris E simravlis
raime elementi, maSin f (x )
x
0
warmoadgens 0 is Sesabamis ricxvs da mas
ewodeba f (x) funqciis mniSvneloba x
f (x
0 wertilSi.
) funqciis yvela
mniSvnelobaTa simravles am funqciis cvlilebis are ewodeba.
3. calsaxa da mravalsaxa funqcia. funqciis grafiki. funqciis
mocemis xerxebi. funqciis cnebis gansazRvris mixedviT, y aris x –is
funqcia, Tu x –is yovel mniSvnelobas mocemuli aredan Seesabameba y –is
erTi garkveuli mniSvneloba. magram zogjer SeiZleba funqciis cnebis ise
ganzogadeba, rom argumentis mocemul mniSvnelobas Seesabamebodes ara
erTi, aramed ramodenime mniSvneloba (usasrulod bevric ki). aseT
SemTxvevaSi y = f (x) warmoadgens x –is mravalsaxa funqcias, xolo rodesac
argumentis yovel mniSvnelobas funqciis gansazRvris aredan Seesabameba
funqciis erTi garkveuli mniSvneloba, amboben, rom mocemuli funqcia
calsaxaa. (SeniSvna: SemdgomSi Cven ganvixilavT calsaxa funqcias, Tu
sawinaaRmdego ar iqneba naTqvami. da aseve Cven ganvixilavT mxolod
ricxviT funqcias, e.i. iseT funqcias, romlis gansazRvrisa da
mniSvnelobaTa areebi ricxviTi simravleebia. e.i. E( f ) ⊂ R ).
vTqvaT, D areze gansazRvrulia y = f (x) funqcia. x –is yovel
mniSvnelobas mocemuli D aredan Seesabameba y –is garkveuli mniSvneloba.
am gziT miviRebT (x; y) wyvilebis simravles. aviRoT sibrtyeze marTkuTxa
koordinatTa Oxy sistema (nax. 1).
y
M
y
0 x x
nax. 1
104
zemoT aRniSnul namdvil ricxvTa yovel (x; y) wyvils Seesabameba
sibrtyis garkveuli M wertili, romlis abscisaa x, ordinati ki y.
amgvarad sibrtyeze miviRebT wertilTa simravles, romelsac f (x)
funqciis grafiki ewodeba. e.i. f (x) funqciis grafiki ewodeba im wertilTa
geometriul adgils, romelTa koordinatebi aRebul funqcionalur
damokidebulebas akmayofileben, anu f funqciis Γ grafiki warmoadgens
sibrtyis wertilTa Semdeg simravles Γ = {(x, y)
2
∈ R ; x ∈ D( f ), y = f (x }
) .
f (x) funqciis grafiks, Cveulebriv wiri ewodeba, xolo y = f (x)
gantolebas ki wiris gantoleba. (SeniSvna: wiris cneba maTematikis erT-
erTi urTulesi cnebaa, amitom ar mogvyavs misi gansazRvra).
funqcia mocemulia es niSnavs, rom dadgenilia wesi, romlis ZaliT
argumentis mniSvnelobaTa mixedviT moiZebneba funqciis Sesabamisi
mniSvneloba. arsebobs funqciis mocemis sami xerxi:
1. analizuri (formuliT) xerxi.
2. cxriluri xerxi – amoweren damoukidebeli cvladis mTel rig
mniSvnelobebsa da funqciis Sesabamis mniSvnelobebs. amiT Sedgeba
garkveuli cxrili. mag. logariTmebis cxrili, trigonometriuli
funqciebis cxrili da sxv.
3. grafikuli xerxi – (aRwerili iyo zemoT). es xerxi mizanSewonilia
maSin, rodesac funqciis mocema analizurad sakmaod Znelia. igi
xSirad gamoiyeneba maTematikaSi funqciis ama Tu im Tvisebis
sailustraciod.
maTematikur analizSi upiratesobas aniWeben funqciis mocemis
analizur xerxs. analizurad mocemuli funqciis gansazRvris are ewodeba
im mniSvnelobis simravles, romelTaTvisac Sesabamis formulas azri aqvs.
2x − 4
magaliTi 1. ipoveT funqciis y =
gansazRvris are.
3 − 6x
kvadratuli fesvi gansazRvrulia mxolod arauaryofiTi
2x − 4
ricxvebisaTvis. amitom
≥ 0 . am utolobis amoxsna gvaZlevs mocemuli
3 − 6x
funqciis gansazRvris areTa mniSvnelobebs, e.i. im namdvil ricxvTa
1
erTobliobas, romlebic akmayofileben utolobas:
x ≤ 2
p
.
2
105
⎛ 2x
⎞
magaliTi 2. ipoveT funqciis y = lg⎜
−1⎟ gansazRvris are.
⎝ x +1
⎠
aTobiTi logariTmi gansazRvrulia mxolod dadebiTi
ricxvebisaTvis, amitom x argumentis yvela mniSvneloba mocemuli funqciis
2x
gansazRvris aredan unda akmayofilebdes
−1 0
f utolobas. Tu am
x +1
x −1
utolobis marcxena mxareSi SevasrulebT gamoklebas, miviRebT:
0
f . am
x +1
wiladis mricxveli dadebiTia, roca x 1
f , da uaryofiTi, roca x 1
p . xolo
mniSvneli dadebiTia, roca x
1
−
f
, da uaryofiTi, roca x
−1
p
. mTlianad
wiladi dadebiTia, roca x
−1
p
da x
1
f . x –is yvela es mniSvneloba
SeiZleba Caiweros x
1
f erTi utolobis saxiT.
3. racionaluri funqciebi. cxadi da aracxadi algebruli funqciebi.
transedenturi funqciebi. funqciaTa umartives klass mravalwevrebi
Seadgenen. mravalwevri ewodeba
2
n
y = a + a x + a x +
+ a x
0
1
2
L
n
(1)
saxis funqcias, sadac x damoukidebeli cvladia, a , a , a ,
, a
0
1
2 K
n warmoadgenen
namdvil ricxvebs (mravalwevris koeficientebs), xolo n aris nebismieri
arauaryofiTi mTeli ricxvi. am n ricxvs ewodeba mravalwevris xarisxi.
xSirad, mravalwevrs mTel racionalur funqcias an polinoms
uwodeben. mravalwevris kerZo saxeebia, magaliTad, wrfivi funqcia
y = ax + b , (a ≠ 0). kvadratuli funqcia y = ax2 + bx + c , (a ≠ 0). funqciaTa
Semdegi ufro farTo klasia racionalur funqciaTa klasi. ori
mravalwevris fardobas ewodeba racionaluri funqcia:
2
n
a + a x + a x +
+ a x
0
1
2
L
n
y =
(2)
2
n
b + b x + b x +
+ b x
0
1
2
L
n
aq igulisxmeba, rom mricxvelsa da mniSvnels saerTo fesvi ara aqvT.
aseTi funqciis gansazRvris ares Seadgens x –is yvela mniSvneloba,
romlebic mniSvnels nulad ar aqceven.
axla ganvixiloT ori x da y cvladis mravalwevri. x da y
cvladebis F (x, y) mravalwevri ewodeba
i
k
a x y
ik
saxis wevrTa jams, sadac
aik namdvili ricxvebia, xolo i da k maCveneblebi – arauryofiTi mTeli
106
ricxvebi. (i + k) ricxvs ewodeba
i
k
a x y
ik
wevris xarisxi an rigi. e.i.
F (x, y) mravalwevris xarisxi ewodeba misi wevrebis xarisxebs Soris
udidess. magaliTad,
7
4
3
7
11
F (x, y) = 3x y + 5x y − 6x y aris me-18 xarisxis
mravalwevri. ori cvaldis racionaluri funqcia ewodeba ori cvladis
ori mravawevris fardobas.
racionalur funqcias
ax + b
y =
(3)
cx + d
⎛ a
b ⎞
sadac a, b, c, d mudmivebi akmayofileben pirobebs (c ≠ 0) da ⎜
≠ ⎟ ,
⎝ c
d ⎠
ewodeba
wiladur-wrfivi funqcia. am funqciis gansazRvris area
⎧ d ⎫
(−∞;+∞) \ ⎨− ⎬ simravle. racionaluri funqciisaTvis damaxasiaTebelia is,
⎩ c ⎭
rom am funqciis mniSvnelobis gamosaTvlelad sakmarisia argumentze
SevasruloT sasruli raodenoba oTxi ariTmetikuli moqmedebisa,
rogoricaa Sekreba, gamokleba, gamravleba da gayofa. SemovitaniT Semdegi
ganmartebebi:
Sekrebas, gamoklebas, gamravlebas, gayofasa da fesvis amoRebas
algebruli operaciebi ewodeba.
y = f (x) funqcias ewodeba cxadi alagebruli funqcia, Tu yoveli
x–isaTvis, funqciis gansazRvris aredan, misi Sesabamisi y–is
gamosaTvlelad saWiroa x –ze vawarmooT mxolod sasruli ricxvi
algebruli operaciebisa.
y –s ewodeba x –is aracxadi alagebruli funqcia, Tu is F (x, y) = 0
gantolebis amonaxsnia, sadac F (x, y) mravalwevria x da y cvladebis
miarT. sazogadod aracxad alagebrul funqciaTa klasi ufro farToa,
vidre cxad alagebrul funqcia klasi. algebrul funqcias, romelic
racionaluri ar aris iracionaluri funqcia ewodeba. funqcias, romelic
algebruli ar aris transedenturi funqcia ewodeba. magaliTad,
sin x
, cos x, tgx
, ctgx
transedenturi funqciebia.
4. zrdadi da klebadi, SemosazRvruli da SemousazRvreli, luwi
da kenti, perioduli funqciebi. raime D simravleze gansazRvrul f (x)
funqcias ewodeba zrdadi am simravleze, Tu am simravlis nebismieri
107
x
da x
f (x ) ≤ f (x
x
x
1
2 ricxvisaTvis
)
1
2
, rodesac 1 p 2 (nax. 2). xolo
gansazRvrul f (x) funqcias ewodeba klebadi am simravleze, Tu am
simravlis nebismieri x
da
x
f (x ) ≤ f (x
1
2
ricxvisaTvis
)
1
2
, rodesac
x
x
1 p
2 (nax. 3).
y y A
B
A B
0 a b x 0 a b x
nax. 2 nax. 3
D simravleze gansazRvrul f (x) funqcias ewodeba monotonuri am
simravleze, Tu igi zrdadia an klebadi.
D simravleze gansazRvrul f (x) funqcias ewodeba SemosazRvruli am
simravleze, Tu arsebobs iseTi dadebiTi M ricxvi, rom am simravlis
yoveli x wertilisaTvis marTebulia utoloba: f (x) ≤ M , an
− M ≤ f (x) ≤ M .
f (x) funqcias ewodeba SemousazRvreli D simravleze, Tu yoveli
ragind didi dadebiTi A ricxvisaTvis moiZebneba am simravlis iseTi x0
wertili, rom f (x )
A
0
f .
f (x) funqcias ewodeba luwi, Tu nebismieri x –isaTvis funqciis
gansazRvris aredan marTebulia toloba f (−x) = f (x) . luwi funqciis
magaliTebia y = x2 , y = cos x, y
2
= sin x .
f (x) funqcias ewodeba kenti, Tu nebismieri x –isaTvis funqciis
gansazRvris aredan marTebulia toloba f (−x) = − f (x) . kenti funqciis
magaliTebia y = x3
,
y = sin x
,
y = tgx .
raime areSi gansazRvrul f (x) funqcias ewodeba perioduli w
periodiT, an w – perioduli funqcia, sadac w
0
f , Tu am ares yoveli
x wertilisaTvis x ± w wertilebi ekuTvnis funqciis gansazRvris ares da
108
marTebulia toloba f (x + w) = f (x) . magaliTad, cos x
sin
da
x perioduli
funqciebia da maTi periodia π
2 . aseve, tgx
ctgx
da
-c perioduli
funqciebia da maTi periodia π . Cveulebriv, funqciis periods uwodeben
yvela dadebiT periods Soris umciress. aqve SevniSnavT, rom funqciis
periodTa Soris SeiZleba ar arsebobdes umciresi. mag. f (x) = 1 funqcia
periodulia da misi periodia nulisagan gansxvavebuli nebismieri namdvili
ricxvi α , vinaidan f (x + α ) = 1 = f (x) .
5. Seqceuli funqcia. rTuli funqcia. y = f (x) toloba x cvladi
sididis yovel dasaSveb mniSvnelobas Seusabamebs y cvladi sididis
sruliad gansazRvrul mniSvnelobas. magram zog SemTxvevaSi y = f (x)
Tanafardoba SeiZleba ganvixiloT, rogorc iseTi toloba, romelic y
cvladi sididis yovel mniSvnelobas Seusabamebs x cvladi sididis
sruliad gansazRvrul mniSvnelobas. es garemoeba naTelvyoT konkretul
magaliTebze:
magaliTi 3.
x
y = 2 toloba y –is yovel dadebiT mniSvnelobas
Seusabamebs x –is Semdeg mniSvnelobas: x = log y
y =
2
. Tu
1, maSin
x = log 1 = 0
y =
x = log 2 =
y =
x = log
2
; Tu
2 , maSin
1
2
; Tu
3 , maSin
3
2
da a. S.
maSasadame,
x
y = 2 toloba gansazRvravs x –s, rogorc y cvladi sididis
raRac funqcias. cxadi saxiT es funqcia ase Caiwereba x = log y
2
.
π
π
magaliTi 4. Tu −
≤ x ≤ , maSin y = sin x toloba y –is yovel
2
2
mniSvnelobas [−
]1
;
1 Sualedidan Seusabamebs x ricxvs, romelic arcsin y -is
π
toli. roca y = 1
− , maSin x = arcsin(− )
1 = − ; roca y = 0 , maSin x = arcsin 0 = 0 ;
2
1
1
π
roca y =
, maSin x = arcsin
=
da a.S. maSasadame, y = sin x
2
2
4
π
π
toloba −
≤ x ≤ damatebiTi pirobisas gansazRvravs x -s, rogorc
2
2
y cvladi sididis raRac funqcias. cxadi saxiT es funqcia Semdegnairad
Caiwereba x = arcsin y .
zogadad, vTqvaT, rom y = f (x) tolobis mixedviT y cvladi sididis
yoveli dasaSvebi mniSvnelobisaTvis SesaZlebelia x cvladi sididis erTi
da mxolod erTi mniSvnelobis aRdgena. maSin es toloba gansazRvravs x-s,
109
rogorc y –is raRac funqcias. avRniSnoT es funqcia ϕ asoTi x = ϕ ( y) .
am formulaSi y asrulebs argumentis rols, xolo x – funqciis.
Cveulebrib
x asos iyeneben argumentis aRsaniSnavad. amitom im
funqcionalur damokidebulebas, romelic ϕ asoTia aRniSnuli, Cven ase
gadavwerT: y = ϕ (x) .
aseTnairad gansazRvrul y = ϕ (x) funqcias y = f (x) funqciis
Seqceuli funqcia ewodeba.
SevniSnoT, rom y = f (x) funqciisa da misi Seqceuli y = ϕ (x)
funqciis gansazRvrisa da cvlilebis areebi, ase vTqvaT, rolebs icvlian.
e.i. is, rac f (x) funqciisaTvis gansazRvris are iyo Seqceuli y = ϕ (x)
funqciisaTvis cvlilebis ared iqceva da is, rac f (x) funqciisaTvis
cvlilebis are iyo Seqceuli y = ϕ (x) funqciisaTvis gansazRvris ared
iqceva. magaliTad,
x
y = 2 funqciisaTvis gansazRvris area yvela namdvil
ricxvTa erToblioba, xolo cvlilebis are – yvela dadebiTi ricxvis
erToblioba. misi Seqceuli x = log y
2
funqciisaTvis ki gansazRvris area
yvela dadebiTi ricxvis erToblioba, xolo yvela namdvil ricxvTa
erToblioba cvlilebis are.
vTqvaT, U areze gansazRvrulia u cvladis funqcia y = f (u) , sadac u
cvladi Tavis mxriv warmoadgens D areze gansazRvrul x cvladis ϕ(x)
funqcias. vigulisxmoT, rom ϕ(x) funqciis mniSvnelobebi moTavsebulia U
areze. maSin am ori funqciis saSualebiT SegviZlia axali funqcia ase:
Tu x nebismieri ricxvia D aredan, maSin mas u = ϕ (x) funqcia Seusabamebs
garkveul u namdvil ricxvs, romelsac Tavis mxriv y = f (u) funqcia
Seusabamebs garkveul y ricxvs. amrigad, y aris x -is funqcia y = F (x) .
aseTnairad gansazRvrul y = F (x) funqcias ewodeba y = f (u) , u = ϕ (x)
funqciebiT gansazRvruli rTuli funqcia. mas agreTve ase aRniSnaven
y = f [ϕ(x)] da amboben, rom y funqcia agebulia f (u) da ϕ(x) funqciebis
superpoziciiT.
magaliTi 5.
5
y = u da
6
u = 1+ 25x funqciebis suprpoziciiT
5
miviRebT
6
y = (1+ 25x) .
110
magaliTi 6. y = tgu ,
5
u = v da v = x +1 funqciebis
superpoziciiT miviRebT
5
y = tg x +1 .
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. rogor sidideebs ewodeba cvladi da mudmivi?
2. rogor sidides ewodeba parametri? absoluturad mudmivi sidide?
3. ras ewodeba funqcia?
4. ras ewodeba funqciis gansazRvris are? funqciis cvlilebis are?
5. funqciis warmodgenis ramdeni ZiriTadi wesi arsebobs?
6. rogor funqcias ewodeba zrdadi? klebadi? monotonuri?
7. rogor funqcias ewodeba SemosazRvruli? ras niSnavs, rom funqcia
SemousazRvrelia?
8. rogor funqcias ewodeba luwi? kenti? perioduli?
9. ras ewodeba funqciis periodi?
10. rogor aris gansazRvruli Seqceuli funqcia?
11. moiyvaneT rTuli funqciis gansazRvra.
II. praqtikuli savrjiSoebi
1. moZebneT Semdegi funqciis gansazRvris areebi:
2 + x
1.1. y = x −1 ; 1.2.
3
y = x +1 ; 1.3.
3
y = x − x ; 1.4 y = lg
;
2 − x
2
x − 3x + 2
1.5. y = lg
; 1.6. y = sin 2x
x +1
2. gamoarkvieT, romeli funqciaa kenti da romeli luwi:
2.1.
x
− x
f (x) = a + a ; 2.2.
x
− x
f (x) = a − a ;
2.3.
3
2
3
2
f (x) = 1− x + x − 1+ x + x ;
1+ x
f (x) = lg(
2
x + 1+ x )
2.4. f (x) = lg
; 2.5.
1− x
3. mocemulia funqcia:
x − 2
1
3.1. f (x) =
, ipoveT f (0), f (−2), f (− ), f ( 2 )
x +1
2
111
x − 2
3.2. ϕ(x) =
, ipoveT ϕ(0),
ϕ
),
1
(
ϕ
(2),
ϕ
( 2
− ),
ϕ
(4)
x +1
4. moZebneT y = f [ϕ (x)], Tu
4.1.
2
f (x) = x ,
x
ϕ(x) = 2 ;
4.2.
3
f (x) = x ,
3
ϕ(x) = 5x −1 ;
$ 2 ricxviTa mimdevroba. mimdevrobis zRvari.
ricxvaTa mwkrivi.
1. ganmarteba. Tu mocemulia wesi, romlis mixedviTac yovel
naturalur n ricxvs Seesabameba garkveuli an ricxvi, maSin vityviT, rom
mocemulia ricxviTi mimdevroba
a ,a ,
a
1
2 L,
,
n L
ricxviT mimdevrobas aRniSnaven {a
a
n }∞
n=1 an { n } simboloebiT.
maSasadame, Tu mocemuli gvaqvs raRac ricxviTi simravle da
movaxerxeT am simravleSi Semavali ricxvebis gadanomra e.i. yovel ricxvs
mivaniWeT nomeri ise, rom arcerTi nomeri or sxvadasxva ricxvs ar
ekuTvnis, maSin aseTi saxiT gadanomrili simravle iqneba mimdevroba.
a
a
1 - s ewodeba mimdevrobis pirveli wevri,
2 -s meore da a.S.
an -s ewodeba n-uri anu zogadi wevri. zogadi wevris mocema niSnavs mTeli
mimdevrobis mocemas. magaliTad,
1
1 1
1
Tu a =
,
1 , , ,
n
, maSin gveqneba mimdevroba
L ,L
n
2 3
n
n
1 2 3
n
Tu a =
, , , ,
n
, maSin gveqneba mimdevroba
L
,L
n + 1
2 3 2
n +1
Tu a = (− )n
1
,
1 − ,
1
,
1 −
n
, maSin gveqneba mimdevroba
L
1
mimdevrobebis mocema Ziridad xdeba analizuri an rekurentuli
xerxiT vityviT, rom mimdevroba mocemulia analizuri xerxiT, Tu
mocemulia formula, romelic gviCvenebs, Tu ra moqmedebani unda
SevasruloT n-ze rom miviRoT mimdevrobis nebismieri wevri.
112
3
magaliTad, Tu a =
n
, maSin SegviZlia amovweroT mimdevrobis
3n + 1
3
3
3
nebismieri wevri. kerZod a = , a =
, a =
1
da a.S.
4
5
16
20
61
vityviT, rom mimdevroba mocemulia rekurentuli xerxiT, Tu
mocemulia pirveli an ramdenime sawyisi wevri da formula, romlis
saSualebiTac gamoiTvleba mimdevrobis danarCeni wevrebi.
rekurentuli xerxiT mocemuli mimdevrobebis magaliTebia:
1. a = 2
2
a
= a , n ≥
2
2
2
a = a = 2 ; a = a = 4
1
da
1
n 1
+
n
cxadia, rom
2
2
1
3
da a.S.
2
2. a = a
a
= qa , n ≥
1
da
1
n 1
+
n
(rekurentulad mocemulia usasrulo
geometriuli progresia) a = a, a = a q,
2
a = q a,
1
2
1
3
L da a.S.
3. a = a da a
= a + d, n ≥1
1
n 1
+
n
(rekurentulad mocemulia usasrulo
ariTmetikuli progresia) a = a, a = a + d, a = a + 2d,
1
2
1
3
1
L
2. monotonuri mimdevroba. mimdevrobas {2n − }
1 e.i. 1,3,5,... saxis
mimdevrobas gaaCnia Tviseba: misi yoveli wevri metia wina wevrze, xolo
⎧1 ⎫
1 1
⎨ ⎬ mimdevrobis (e.i. ,
1 , ,L) yoveli wevri naklebia mis wina wevrze.
⎩n⎭
2 3
{a
a < a <
< a < a
n } mimdevrobas ewodeba zrdadi, Tu 1
2
L
<
n
n+1
Lda
ewodeba klebadi, Tu a > a >
a
a
a
1
2
L >
>
>
n
n+1
L. { n} mimdevrobas ewodeba
araklebadi Tu a ≥ a ≥
a
a
1
2
L ≥
≥
≥
n
n+1
L da ewodeba arazrdadi, Tu
a ≤ a ≤
a
a
1
2
L ≤
≤
≤
n
n+1
L
zrdad, klebad, araklebad da arzrdad mimdevrobebs monotonuri
mimdevrobebi ewodeba. cxadia, rom Tu
a − a
> 0
n
n 1
+
maSin {an} mimdevroba klebadia. Tu
a − a
≥ 0
n
n 1
+
maSin {an} mimdevroba arazrdadia.
magaliTi 1. vaCvenoT, rom mimdevroba, romlis zogadi wevri
n
a =
n
zrdadia.
n + 1
amisaTvis unda davamtkicoT utoloba a
− a > 0
n 1
+
n
marTlac,
113
n + 1
n
n + 1
2
n
n + 2n + 1
2
− n − 2n
1
a
− a =
−
=
−
=
=
n 1
+
n
n + 1 + 1 n + 1
n + 2
n + 1
(n + )(
1 n + 2)
(n + )(
1 n + 2)
1
wiladi
n-nebismieri mniSvnelobisaTvis dadebiTia e.i. a
− a > 0
(n + )(
1 n + 2)
n 1
+
n
mimdevroba zrdadia.
arseboben mimdevrobebi, romlebic arc klebadia da arc zrdadi, e.i.
aramonotonuri mimdevrobebi. magaliTad, mimdevroba romlis zogadi wevria
n
a = (− )
1
n
aramonotonuria.
{an} mimdevrobas ewodeba zemodan SemosazRvruli, Tu arsebobs
iseTi M ricxvi, rom nebismieri n ∈ N sruldeba utoloba a ≤ M
n
, xolo
qvemodan SemosazRvruli, Tu a ≥ M
n
.
{an} mimdevrobas ewodeba SemosazRvruli, Tu is erTdroulad
SemosazRvrulia qvemodanac da zemodanac. e.i. nebismieri n naturaluri
ricxvisaTvis samarTliania
m < a < M
n
,
sadac M da m ricxvebi n-sagan damoukidebeli mudmivebia. am
utolobaSi ¸niSani SeiZleba Seicvalos ≤ niSniT.
⎧1 ⎫
1
magaliTi 2. mimdevroba ⎨ ⎬ SemosazRvrulia, radgan 0 <
< 1;
⎩n⎭
n
⎧ 1 ⎫
1
n
magaliTi 3. mimdevroba ⎨
⎬SemosazRvrulia, radgan ≤
<1;
⎩n +1⎭
2
n +1
1+ (− )
1 n
magaliTi 4. mimdevroba, romlis zogadi wevria a =
n
Semo-
2
sazRvrulia.
marTlac, am mimdevrobis yoveli wevri tolia 0-is an 1-is.
a = ,
0
a = 1 a = ,
0
a = ,
1
0 ≤ a ≤
1
2
3
4
L amitom
1
n
.
magaliTi 5. mimdevroba {2n + }
1 SemousazRvrelia zemodan.
radgan a = 2n + 1
a
a
n
, maSin n >0 da n-is zrdasTan erTad n usasrulod
izrdeba, rac imas niSnavs, rom nebismieri fiqsirebuli M > 0 ricsvisaTvis
miZebneba iseTi k nomeri, rom a
a >
k gadaaWarbebs M ricxvs e.i.
M
k
3. mimdevrobis dagrovebis wertili. ganvixiloT mimdevroba {an},
n
1 2 3
100
sadac a =
, , , ,
n
maSin es mimdevrobaa
L
,L
n +1
2 3 2
101
114
ganvixiloT wertili 1 da misi nebismieri ε midamo (1-ε;1+ε). vnaxoT
am mimdevrobis ramdeni wertili moxvdeba (1-ε;1+ε). SualedSi.
n
1
1
1− ε <
<1+ ε ⇒ −ε <
< ε ⇒ n > −1.
n +1
n +1
ε
aqedan, cxadia, rom aRniSnul midamoSi moxdeba mimdevrobis yvela wevri
⎡1
⎤
⎡1
⎤
1
dawyebuli a
N =
−1 +
−1
−
N -dan, sadac
;
1
⎢
⎥
⎣ε
⎦
⎢
⎥
⎣ε
iT aRniSnulia
1
⎦
ε
ricxvis
mTeli nawili, xolo am midamos gareT darCeba a ,a , , a
1
2 L
N mimdevrobis
wevrTa sasruli raodenoba. aseT wertilebs mimdevrobis dagrovebis
wertilebi ewodeba.
ganxilul magaliTSi mimdevrobas hqonda mxolod erTi dagrovebis
wertili. dagrovebis wertilebis raodenoba SeiZleba metic iyos. ase,
1
magaliTad, mimdevrobas, romlis zogadi wevria a
n
= (− )
1 +
n
ori dagrovebis
n
wertili aqvs. ganvixiloT magaliTi, romelSic mimdevrobis zogadi wevri
gamoiTvleba formuliT
⎧
1
⎪1+ , Tu n = 3k
⎪
k
⎪
a =
1
n
⎨2 + , Tu n = 3k −1
⎪
k
⎪
1
⎪3 + , Tu n = 3k − 2
⎩
k
maSin am mimdevrobas eqneba sami dagrovebis wertili 1, 2 da3.
4. mimdevrobis zRvari. ganvixiloT {an} mimdevroba. vityviT, rom a
wertili aris am mimdevrobis zRvari, Tu a ricxvis nebismieri ε midamos
gareT aris {an} mimdevrobis wevrTa sasruli raodenoba. es garemoeba ase
Caiwereba:
a = lim an .
n→∞
Tu mimdevrobas zRvari aqvs, maSin mas krebadi mimdevroba ewodeba;
winaaRmdeg SemTxvevaSi ganSladi.
cxadia, Tu a aris {an} mimdevrobis zRvari, maSin (a-ε; a+ε) midamoSi
moxvdeba wevrTa usasrulo raodenoba, e.i. a dagrovebis wertilia. aqve
SevniSnoT, rom mimdevrobas, rogorc zemoT vnaxeT SeiZleba hqondes
araerTi dagrovebis wertili, amitom bunebrivia SekiTxva: `mimdevrobis
115
dagrovebis wertili aris Tu ara misi zRvari?~ am SekiTxvaze pasuxs
iZleva Semdegi
Teorema 1: Tu mimdevrobas zRvari aqvs, is erTaderTia.
damtkiceba. dauSvaT sawinaaRmdegoi. vTqvaT {an} mimdevrobas aqvs
ori zRvari a = lim a
b = lim a
b ≠
n da
n , sadac
a .
n→∞
n→∞
b − a
vigulisxmoT, rom b > a ganvixiloT ε =
da aviRoT a da b
3
ricxvebis ε midamoebi: (a − ε; a + ε ), (b − ε;b + ε ). cxadia, rom es midamoebi ar
gadaikveTebian
(//////////////) (//////////////)
a b
nax.1
pirobiT a = lim a
a − ε; a + ε
n , amitom (
) midamos gareT darCeba am
n→∞
midevrobis wevrTa sasruli raodenoba, maSasadame b ricxvi ar warmoadgens
mocemuli mimdevrobis zRvars, rac Cvens daSvebebs ewinaaRmdegeba. (ufro
metic, b ricxvi ar SeiZleba iyos am midevrobis dagrovebis wertilic ki)
Teorema damtkicebulia.
Teorema 2: Tu mimdevrobas zRvari aqvs, maSin is SemosazRvrulia.
damtkiceba: mocemulia a = lim an , maSin a ricxvis nebismieri
n→∞
ε - midamos gareT SesaZloa moxvdes am midevrobis wevrTa mxolod
sasruli raodenoba. aviRoT ε=1 da amovweroT (a-1;a+1) midamos gareT
moxvedrili wevrebi: vTqvaT maT Soris udidesis modulia M1. maSin aviRoT
M = max(a +1; a − ;
1 M
a ≤
1 ) da cxadia, rom nebismieri n-saTvis
M
n
Teorema
damtkicebulia. sazogadod Sebrunebuli Teorema ar aris samarTliani.
magaliTad
n
a = (− )
1
n
SemosazRvrulia, magram mas zRvari ar gaaCnia (aqvs
ori dagrovebis wertili 1 da –1).
5. mimdevrobis zRvris arsebobis niSani. daumtkiceblad Camovayali-
bebT mimdevrobis arsebobis aucilebel da sakmaris niSans, romelsac
xSirad zRvris ganmartebadac xmaroben.
116
UTeorema 3: imisaTvis, rom a ricxvi iyos {an} mimdevrobis zRvari,
aucilebeli da sakmarisia, rom nebismieri ε>0 ricxvisTvis arsebobdes
iseTi naturaluri N(ε) ricxvi, rom adgili hqondes utolobas
a − a < ε,
roca
n > N (ε )
n
.
magaliTi 1. davamtkicoT, rom
1
lim = 0 .
n→∞ n
gamoviyenoT zemoT Camoyalibebuli Teorema da aviRoT nebismieri ε>0.
ganvixiloT (−ε;ε) Sualedi. maSin saaWiroa, rom sruldebodes piroba
− ε < 1 < ε
n
1
cxadia, rom es utolobebi sruldeba maSin, roca n > ε . aviRoT
⎡1⎤
1
1
N (ε ) =
+1
⎢ ⎥
. cxadia, rom Tu n > N (ε ) , maSin
< ε da −ε < < ε e.i.
⎣ε ⎦
n
n
sruldeba zRvris arsebobis aucilebeli da sakmarisi piroba, maSasadame
1
lim = 0 .
n→∞ n
6. mimdevrobis usasrulo (arasakuTrivi) zRvari. vTqvaT, {an}
ganSladi mimdevroba araa SemosazRvruli da mis wevrebs gaaCniaT Tviseba:
nebismieri M>0 ricxvisaTvis moiZebneba iseTi N(M) ricxvi, rom
a > M roca n > N(M ),
n
an
a < −M roca n > N(M ),
n
maSin vityviT, rom Sesabamisad {an} mimdevroba miiswrafvis +∞-ken an
- ∞ _ken da formalurad CavwerT
lim a = +∞
an
lim a = −∞ .
→∞ n
n
→∞ n
n
sxva sityvebiT, rom vTqvaT pirvel SemTxvevaSi n- is usasrulod
zrdas mosdevs an ricxvebis SemousazRvrelad zrda, amitom isini gada-
aWarbebn nebismier dadebiT ricxvs, xolo meore SemTxvevaSi an ricxvebis
SemousazRvrelad mcirdebian da naklebi gaxdebian nebismierad didi
modulis mqone uaryofiT ricxvze. am SemTxvevaSi amboben, rom an ganSladi
mimdevroba krebadia Sesabamisad +∞-ken an -∞-ken.
117
cxadia, rom ariTmetikuli an=a1+d(n-1) progresiisaTvis gveqneba
lim
a = +∞
d
roca
> 0.
→∞ n
n
lim
an
a = −∞
d
roca
< 0.
→∞ n
n
SevniSnoT, rom Tu a ≠ 0 da a → +∞
a
an
→ −∞
n
n
n
, maSin orive
1
SemTxvevaSi _ lim
= 0.
n→∞ an
cxadia, rom arsebobs ganSladi SemousazRvreli mimdevrobebi,
romlebic araa krebadi arc +∞-ken da arc -∞-ken. magaliTad an=(-1)nn2 araa
krebadi arc +∞-ken da arc -∞-ken, radgan am mimdevrobaSi nebismieri nomris
Semdeg gvxvdeba, rogorc Zalian didi dadebiTi ricxvi (roca n luwia) ise
moduliT Zalian didi uaryofiTi ricxvi (roca n kentia).
Teorema 4: Tu {a
da
b
lim a =
n }
{ n} mimdevrobebi krebadia da
a
n
,
n→∞
⎧a ⎫
limb = b
a + b ,
a ⋅
n
b ,
n
, maSin krebadia { n
n }
{ n n} ⎨ ⎬ (am ukanasknelSi b≠0)
n→∞
⎩bn ⎭
mimdevrobebi da adgili aqvs
an
a
1.lim(a + b = + ; 2
.lim
=
;
3
.lim
=
n
n )
a b
a b
ab
n n
tolobebs.
n→∞
n→∞
n→∞ b
b
n
damtkiceba. zRvris arsebobis niSnidan gamomdinareobs, rom
ε
ε
a − a < , roca n > N (′ε ) da b − b < ,roca n > N (′ε )
n
.
2
n
2
Tu
n > N (ε ), sadac N(ε ) =
{
max N (
′ ε),n > N (′ε }
) maSin erTdroulad
sruldeba
ε
a − a < , da b − b < ε roca n > N(ε )
n
2
n
utolobebi. amitom
(a +b
roca n > N (ε )
n
n ) − a + b = (a − a
n
)−(b −b
n
)
ε ε
≤ a − a + b − b < + = ε,
n
n
,
2 2
e.i.
lim(a + b = a + b = lima + limb
n
n )
n
n .
n→∞
n→∞
n→∞
analogiurad mtkicdeba danarCeni ori toloba.
rogorc zemoT vnaxeT, Tu mimdevrobas zRvari aqvs SemosazRvrulia,
magram yvela SemosazRvrul mimdevrobas zRvari ar gaaCnia. sainteresoa
118
romel SemosazRvrul mimdevrobas ar gaaCnia zRvari. am sakiTxze pasuxs
iZleva
vaierStrasis Teorema: nebismier SemosazRvrul monotonur
mimdevrobas aqvs zRvari.
Zalian xSirad saWiroa utolobaSi zRvarze gadasvla. am piroblemas
exeba Semdegi
Teorema 5. {a
da
b
n }
{ n} krebadi mimdevrobebia da yoveli n-saTvis
a < b an a ≤ b
n
n
n
n , maSin
lim a ≤ limb
n
n .
n→∞
n→∞
Teorema 6. Tu lim a = limb = c
n
n
da yoveli n-saTvis
n→∞
n→∞
a ≤ c ≤ b
n
n
n ,
maSin {cn} mimdevroba krebadia da
lim c = c
n
.
n→∞
6-s uwodeben ori policielis Teoremas.
7. ricxvi e. umaRles maTematikaSi didi mniSvneloba aqvs ricxvs,
romelsac e simboloTi aRniSnaven. es ricxvi ganisazRvreba, rogorc {an}
midevrobis zRvari, romlis zogadi wevria
n
⎛
⎞
a =
1
⎜1+ ⎟
n
,
⎝
n ⎠
e.i.
n
⎛
1 ⎞
lim a =
⎜
lim 1 + ⎟ = e
n
. (1)
n→∞
n→∞⎝
n ⎠
e ricxvi iracionaluri ricxvia da misi miaxloebiTi mniSvnelobaa
e ≈
59045
7182818284
,
2
L
TxuTmeti niSnis sizustiT. e ricxvs ewodeba neperis ricxvi.
(1) toloba xSirad gamoiyeneba konkretuli tipis zRvrebis gamoTvlis
dros. kerZod, (1) tolobis safuZvelze mtkicdeba, rom mimdevrobas
∞
⎪
⎧⎛ a ⎞mb ⎪⎫
⎨⎜1+
⎟ ⎬
⎪⎝
m ⎠ ⎪
⎩
⎭m=1
zRvari gaaCnia da
119
mb
⎛
a ⎞
ab
lim⎜1+
⎟ = e (2)
m→∞⎝
m ⎠
sadac a da b nebismieri ricxvebia.
Tu logariTimis fuZe e ricxvia, maSin mas naturaluri logariTmi
ewodeba da aRiniSneba
b
log = lnb
e
. davamtkicod (2) formula. gveqneba:
ab
m
m
ab
⎡
⎤
a
a
⎛
⎞
⎢⎛
⎞ ⎥
mb
⎛
a
⎜
⎟
⎜
⎟
⎞
1
⎢
1
⎥
lim⎜1 +
⎟ = lim⎜1+
⎟
= lim ⎜1+
⎟
m→∞
m
⎝
m
∞
→
m
m→∞⎢
m
⎥
⎠
⎜
⎜
⎟
⎟
⎢⎜⎜
⎟
⎟ ⎥
⎝
a ⎠
⎝
a
⎢
⎠ ⎥
⎣
⎦
m
SemovtanoT aRniSvna
= N . cxadia, rom roca m → ∞ , maSin N → ∞ .
a
maSasadame
ab
m
⎡
⎤
a
⎢⎛
⎞ ⎥
mb
⎛
a ⎞
⎢⎜
⎟
1
⎥
ab
lim⎜1 +
⎟ = lim ⎜1+
⎟
= e .
→∞
→∞ ⎢
⎥
m
m
⎝
m ⎠
⎜
m
⎢
⎟
⎜
⎟ ⎥
⎢⎝
a ⎠ ⎥
⎣
⎦
8. ricxviTi mwkrivi. mwkrivis jami. ricxviTi mimdevrobis cnebasTan
mWidrodaa dakavSirebuli ricxviTi mwkrivisa da mwkrivis jamis cnebebi.
ganvixiloT raime {an}∞n=1 ricxviTi mimdevroba da SevadginoT Semdegi
gamosaxuleba:
a + a +
1
2
L+ a +
n
L (3)
SevniSnoT, rom es gamosaxuleba aris formaluri jami, radgan masSi
Sesakrebebis ricxvi usasruloa da Sekrebis ganxorcieleba elementarul
maTematikaSi cnobili algoriTmebiT SeuZlebelia, (3) tipis usasrulo
raodenobis SesakrebTa formalur jams ricxviTi mwkrivi ewodeba. xolo an
ricxvebs – mwkrivis wevrebi. an -s uwodeben mwkrivis zogad wevrs.
ismeba kiTxva: SeiZleba, Tu ara (3) gamosaxulebas mivaniWoT raime
azri? am kiTxvaze pasuxis gasacemad SevadginoT Semdegi ricxvebis
mimdevroba:
120
S = a
1
1
S = a + a
2
1
2
S = a + a + a
3
1
2
3
L
L
L
L
L
L
L
L
S = a + a +
+ a
n
1
2
L
n
L
L
L
L
L
L
L
L
miRebul {Sn}∞n=1 mimdevrobas (3) mwkrivis kerZo jamebis mimdevroba
ewodeba, xolo Sn ricxvs – n-uri kerZo jami. intuicia gvkarnaxobs, rom
rodesac n → ∞ maSin Sn jamSi Sesakrebad Seva (3) gamosaxulebis `TiTqmis
yvela wevri~. am mosazrebidan gamomdinare SemoviRoT Semdegi
gansazRvreba: Tu (3) mwkrivis kerZo jamebis {Sn} mimdevrobas gaaCnia S
zRvari, rodesac n → ∞ , maSin (3) mwkrivs ewodeba krebadi, xolo lim S = S
n
n→∞
zRvars – (3) mwkrivis jami ewodeba da igi ase Caiwereba
a + a +
+ a +
=
a
S
1
2
L
n
L ∑
∞
=
n
.
n=1
Tu {S
lim S =
n } mimdevrobas ar gaaCnia zRvari, an
±∞ , maSin (3) mwkrivs
→∞
n
n
ewodeba ganSladi. cxadia, rom Tu lim S = S
lim S
=
n
, maSin
S
n−1
amitom
n→∞
n→∞
lim a = lim
n
(S − S =
S −
S
= S − S =
n
n 1
− )
lim
lim
0
n
n 1
−
.
n→∞
n→∞
n→∞
n→∞
amrigad, Tu mwkrivi krebadia, maSin misi zogadi wevri a → ,
0
n
roca
n → ∞ . es aris mwkrivis krebadobis aucilebeli piroba. maSasadame, Tu
lim a ≠ 0 mwkrivi ganSladia. magram es piroba ar aris sakmarisi mwkrivis
→∞ n
n
krebadobisaTvis. magaliTad mtkicdeba, rom e.w. harmoniuli mwkrivi
+ 1
1
1
+ 1 +
1
L+ +L ganSladia, miuxedavad imisa, rom lima = lim = 0 .
2 3
n
→∞ n
n
n→∞ n
CamovayaliboT mwkrivis krebadobis ramdenime sakmarisi piroba:
121
Sedarebis niSani: vTqvaT ∑
∞
a
b
n da ∑
∞
n dadebiT wevrebiani mwkrivebia
n=1
n=1
(e.i. a > ,
0 b > 0
n
n
).
1. Tu ∑
∞ b
a ≤ b
n mwkrivi krebadia da yoveli n-saTvis
n
n , maSin
n=1
∑∞an mwkrivi krebadia.
n=1
2. Tu ∑
∞
b
a ≥ b
n mwkrivi ganSladia da yoveli n-saTvis
n
n , maSin
n=1
∑∞an mwkrivic ganSladia.
n=1
a
dalamberis niSani: vTqvaT, mocemulia ∑
∞
a
n+1
lim
=
n mwkrivi da
q .
n→∞
n=1
an
1. Tu q<1, maSin mwkrivi krebadia;
2. Tu q >1, maSin mwkrivi ganSladia
koSis niSani: vTqvaT mocemulia ∑
∞ a
n
lim a =
n mwkrivi da
q
n
n→∞
n=1
1. Tu q<1, maSin mwkrivi krebadia;
2. Tu q >1, maSin mwkrivi ganSladia
SevniSnoT, rom dalamberisa da koSis zemoT Camoyalibebuli
Teoremebi q=1 SemTxvevaSi ver iZlevian pasuxs mwkrivis krebadobaze. aseT
SemTxvevaSi mwkrivis krebadobis dasadgenad saWiroa damatebiTi gamokvleva
(am sakiTxs aq ar SevexebiT).
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ras ewodeba obieqtTa mimdevroba? ricxvTa mimdevrobis mocemis ra
wesebi iciT?
122
2. rogor mimdevrobas ewodeba zemodan SemosaRvruli? qvemodan
SemosaRvruli?
3. rogor mimdevrobas ewodeba zrdadi mimdevroba? klebadi
mimdevroba? arazrdadi mimdevroba? araklebadi mimdevroba?
monotonuri?
4. ras niSnavs (xn )n 1
≥ mimdevrobis zRvaria a ricxvi?
5. SemosazRvrulia Tu ara krebadi mimdevroba?
6. krebadia Tu ara yoveli SemosazRvruli mimdevroba?
7. SeiZleba Tu ara, rom krebad mimdevrobas ori zRvari hqondes?
8. raSi mdgomareobs vaierStrasis Teorema?
9. ras ewodeba neperis e ricxvi?
10. ras ewodeba mwkrivi? mwkrivis zogadi wevri? mwkrivis jami?
11. raSi mdgomareobs mwkrivis krebadobis aucilebeli piroba?
12. raSi mdgomareobs koSisa da dalamberis niSani?
II. praqtikuli savarjiSoebi
1. dawereT mimdevroba maTi zogadi wevris mixedviT:
n
⎛ 1 ⎞
n
1.1 x = ⎜ −
⎟
x = −1 2n +
x = cos n
x = n + −1
n
; 1.2 n
( ) (
)1; 1.3
π ; 1.4 n
( )n ;
⎝ 2 ⎠
n
nπ
2nsin
3 + (− )n
1
2
1
⎛ π ⎞
1.5. x =
x =
a =
a = sin
n
⎜ n
n
; 1.6.
; 1.7.
; 1.8
⎟ ;
n
n
n +1
n
−n
2
⎝ 3 ⎠
(− )1n
2n −1
π
1.9. a = 2 + 5n
a =
a =
a = tg
n
; 1.10. n
; 1.11.
; 1.12.
;
2
n
3 + n
5
n
n + 2
( )n 1
1 +
−
( )n
1 1
+
−
n
⎛
3 ⎞
1.13. a =
a =
x = ⎜−
⎟
n
; 1.14.
; 1.15.
n
n
n
3
n
n
⎝
+1
2
⎠
2n +1
2. aCveneT, rom x =
n →
n
mimdevrobis zRvari, roca
∞ , aris 2.
n +1
ipoveT N , Tu ε = 1
,
0 ; ε = 01
,
0
;
123
1
3. aCveneT, rom x = 1 +
n →
n
mimdevrobis zRvari, roca
∞ , aris 1.
n
ipoveT N , Tu ε = 01
,
0
; ε = 1
,
0 ;
n
4. aCveneT, rom x =
n →
n
mimdevrobis zRvari, roca
∞ , aris 1.
n +1
ipoveT N , Tu ε = 01
,
0
;
1
5. aCveneT, rom x =
n →
n
mimdevrobis zRvari, roca
∞ , aris 0.
2
n
1
rogori n–isaTvis Sesruldeba
ε utoloba, Tu ε = 01
,
0
;
2 p
n
ε = ,
0 001 ; ε = 0001
,
0
;
$ 3 funqciis zRvari da uwyvetoba
1. funqciis zRvris cneba. ZiriTadi Teoremebi funqciis zRvris
Sesaxeb. vidre funqciis zRvris mkacr maTematikur ganmartebas moviyvandeT,
ganvixiloT magaliTi.
vTqvaT,
2
f (x) = x . Tu x argumenti gairbens rig mniSvnelobebs,
krebads 2 ricxvisaken, maSin f (x) funqcia gairbens gairbens rig
mniSvnelobebs, krebads 4 ricxvisaken. amas SevniSnavT, Tu ganvixilavT
2
x − 4 funqciis miaxloebiT mniSvnelobaTa Semdeg cxrils:
x
1,96 1,97 1,98 1,99 2,00
2,01 2,02
2
x (miaxloebiT)
3,84 3,88 3,92 3,96 4,00 4,04 4,08
2
x − 4 (miaxloebiT) 0,16 0,12 0,08 0,04 0 0,04 0,08
124
rac ufro axloa x argumentis mniSvneloba 2- Tan, imdenad mcirea
2
x − 4 sxvaobis absoluturi mniSvneloba. amaSi SeiZleba davrwmundeT
mkacri maTematikuri msjelobiTac, ganxiluli cxrilis gareSe.
davamtkicoT, rom ragind mcire dadebiTi ε ricxvic ar unda aviRoT,
yovelTvis SeiZleba x=2 wertilis Semcveli iseTi Sualedi gamoiyos, rom
am Sualedis yoveli wertilisaTvis Sesruldes utoloba:
2
x − 4
ε
p (1)
marTlac, (1) utoloba tolfasia ormagi utolobisa:
− ε
2
x −
ε
p
4 p (2)
saidanac miviRebT:
4 − ε
2
4 + ε
p x p
4 − ε
4 + ε
p x p
(3)
(gaviTvaliswineT x –is mxolod dadebiTi mniSvnelobebi, radganac
2
y = x funqciis yofaqceva am SemTxvevaSi gvainteresebs mxolod x=2
wertilis maxloblobaSi). amgvarad, (1) utoloba sruldeba (3)
SualedSi, romelic x=2 wertils moicavs. magaliTad, 2
x − 4
0 1
,
p
utoloba (ε = 0 )
1
, sruldeba
,
3 9
4 1
,
p x p
SualedSi, anu ,
1 98
2,02
p x p
,
xolo
2
x − 4
0,01
p
utoloba (ε = 0, )
01 sruldeba
,
3 99
4,01
p x p
SualedSi,
anu ,
1 998
2,002
p x p
.
(3) Sualedi aigeba geometriuladac. vTqvaT, A aris
2
y = x funqciis
grafikis wertili, romlis abscisaa x=2 (nax. 1).
y
C
y
ε
A
A+ε
B ε
A
A-ε
0 x
0 B
0-η x0 x0+η x
1 2 C1 x
125
nax. 1 nax. 2
am wertilis orive mxares gavavloT ori horizontaluri wrfe,
romlebic A wertilidan daSorebulia ε manZiliT. es wrfeebi gadakveTen
2
y = x parabolis marjvena nawils B da C wertilebSi. Tu maTgan marTobebs
davuSvebT abscisTa RerZze, miviRebT B C
1
1 monakveTs. swored es monakveTi
warmoadgens ( 4 − ε ; 4 + ε ) Sualeds, romelic adre miviReT algebruli
gziT.
e.i. Tu x argumentisaTvis 2-is sakmaod axlo mniSvnelobebs SevarCevT,
maSin
2
y = x funqciis mniSvnelobani ragind mcired iqnebian gansxvavebuli
4 –sagan. magaliTad, SeiZleba mivaRwioT imas, rom Sesruldes utolobani:
2
x − 4
0,001
p
, 2
x − 4
0,0001
p
da a. S. aseT SemTxvevaSi bunebrivia ricxv
4 –s ewodos
2
y = x funqciis zRvari, roca x miiswrafvis 2 –isaken.
SegaxsenebT, rom wrfeze romelime x0 wertilis midamo ewodeba
yovel Ria Sualeds, romelic aRniSnul x0 wertils Seicavs.
A ricxvs ewodeba f (x) funqciis zRvari x0 wertilSi, Tu
nebismieri dadebiTi ε ricxvisaTvis arsebobs iseTi dadebiTi η ricxvi,
rom
f (x) − A
ε
p , rodesac 0
x − x
η
p
0 p
, (4)
simbolurad es aRiniSneba ase:
lim f (x) = A (5)
x→x0
an f (x) → A , roca x → x0 . SevniSnoT, rom η ricxvi
damokidebulia ε -ze, amasTanave, sazogadod, igi mcirdeba ε -Tan erTad.
gamovarkvioT funqciis zRvris cnebis geometriuli Sinaarsi. rogorc,
viciT
f (x) gantoleba dekartes Oxy koordinatTa sistemis mimarT
gamosaxavs garkveul wirs (nax. 2). Oy RerZze aviRoT A wertilis
]A −ε; A + ε[ midamo, sadac ε nebismieri dadebiTi ricxvia. im SemTxvevaSi,
roca adgili aqvs (4) utolobas Ox RerZze arsebobs x0 wertilis iseTi
]x −η;x +η
0
0
[ midamo, rom am midamos yoveli x wertilisaTvis, romelic
126
gansxvavebulia x
f (x
0 wertilisagan,
) funqciis mniSvneloba A wertilis
]A −ε; A + ε[ midamoSi. maSasadame, y = f (x) fuqciis grafiki, romelic
Seesabameba ]x −η; x +η
0
0
[ intervals, ar gamova daStrixuli ares gareT
(nax. 2), garda SesaZlebelia, grafikis erTi wertilisa, romelic
Seesabamebodes x0 wertils.
daumtkiceblad moviyvanoT ZiriTadi Teoremebi funqciaTa zRvrebis
Sesaxeb.
1. mudmivi sididis zRvari TviT am mudmivi sididis tolia:
lim C = C
x→x0
2. mudmivi mamravli SeiZleba zRvris niSnis gareT gamovitanoT:
lim [k ⋅ f (x)] = k ⋅ lim f (x)
x→ 0
x
x→ 0
x
3. funqciaTa jamis (sxvaobis) zRvari am funqciebis zRvarTa jamis
(sxvaobis) tolia:
lim [ f (x) ± ϕ (x)] = lim f (x) ± lim ϕ (x)
x→ 0
x
x→ 0
x
x→ 0
x
4. funqciaTa namravlis zRvari am funqciebis zRvarTa namravlis
tolia:
lim [ f (x) ⋅ϕ (x)] = lim f (x) ⋅ lim ϕ (x)
x→ 0
x
x→ 0
x
x→ 0
x
5. ori funqciis Sefardebis zRvari am funqciebis zRvarTa
Sefardebis tolia:
lim f (x)
f (x)
x→ 0
x
⎛
⎞
lim
=
,
⎜
lim ϕ(x) ≠ 0⎟ .
x→ 0
x ϕ
(x)
lim ϕ(x)
⎝ x→ 0x
⎠
x→ 0
x
ganvixiloT magaliTebi funqciaTa zRvrebis sapovnelad.
3x − 2
magaliTi 1. ipoveT lim
x→4 x + 2
ganayofis zRvris Tvisebis ZaliT gveqneba:
lim(3x − 2)
lim 3x − lim 2
3lim x − 2
3x − 2
x→
x→
x→
x→
⋅ −
4
4
4
4
3 4 2
5
lim
=
=
=
=
=
x→4 x + 2
lim(x + 2)
lim x + lim 2
lim x + 2
4 + 2
3
x→4
x→4
x→4
x→4
3
x − 8
magaliTi 2. ipoveT lim
;
x→2 x − 2
127
roca x → 2 , mocemuli wiladis mricxvelica da mniSvnelic → 0 . amitom
ganayofis zRvris Sesaxeb Teoremis uSualo gamoyeneba ar SeiZleba, magram
SesaZlebelia, rom Sevkveco mocemuli wiladi:
3
x − 8
(x − 2)( 2
x + 2x + 4)
2
=
= x + 2x + 4
x − 2
x − 2
amis Semdeg zRvari advilad gamoiTvleba.
3
x − 8
lim
= lim( 2
x + 2x + 4) = 22 + 2 ⋅ 2 + 4 = 12
x→2 x − 2
x→2
2. funqciis uwyvetoba da wyveta. funqciis calmxrivi zRvrebi.
funqciis uwyvetoba marjvnidan da marcxnidan. funqciis nazrdi.
ganvixiloT f (x) funqcia gansazRvruli x0 wertilis midamoSi.
x
f (x
0 wertilis midamoSi gansazRvrul
) funqcias ewodeba uwyveti
x
f (x
x
0 wertilSi, Tu
)
0
sasrulia da 0 wertilSi funqciis zRvari da
funqciis mniSvneloba Tanatolia:
lim f (x) = f (x )
0
x→ 0
x
e.i. yoveli dadebiTi ε ricxvisaTvis ( ε
0
f ) moiZebneba iseTi
dadebiTi η ricxvi, rom f (x) − f (x )
ε
x − x
0
p , rodesac
η
0 p
.
vTqvaT,
f (x) funqcia gansazRvrulia [a,b] segmentze, garda
SesaZlebelia am segmentis raime x0 wertilisa. samarTliania Semdegi
gansazRvrebebi:
x
f (x
0 wertils ewodeba
) funqciis wyvetis wertili, Tu
Sesrulebulia erT-erTi Semdegi pirobebidan:
1. x
f (x
0 wertilSi
) funqcia ar aris gansazRvruli;
2. lim f (x) ar arsebobs;
x→ 0
x
3. x
f (x
lim f (x
0 wertilSi
) funqcia gansazRvrulia, arsebobs
) , magram
x→ 0
x
lim f (x) ≠ f (x )
x
0 . e.i. funqciis zRvari ar udris funqciis mniSvnelobas
x→
0
0
x
wertilSi.
funqcias, romelsac [a,b] segmentze aqvs wyvetis raime x0 wertili, am
segmentze wyvetili funqcia ewodeba.
128
1
magaliTi 3. vTqvaT, mocemulia f (x) =
funqcia. es funqcia
x
gansazRvrulia x –is yvela mniSvnelobisaTvis, garda x=0 mniSvnelobisa.
x=0 wertilSi funqcia gansazRvruli ar aris. maSasadame, x=0 wertili
1
warmoadgens mocemuli f (x) =
funqciis wyvetis wertils.
x
A ricxvs ewodeba ]x ;b
f (x
0
[ intervalSi gansazRvruli
) funqciis
marjvena zRvari x0 wertilSi, Tu yoveli dadebiTi ε ricxvisaTvis
arsebobs iseTi η
0
f ricxvi, rom
f (x) − A
ε
p , rodesac 0 x − x
η
p
0 p
,
am SemTxvevaSi weren ase:
lim f (x) = A .
x→x +
0
f (x) funqciis marjvena zRvari x
f (x +
0 wertilSi aRniSnaven
)
0
simboloTi. Tu f (x) funqcia gansazRvrulia x0 wertilSic da
f (x +) = f (x )
x
0
0
, e.i. funqciis marjvena zRvari 0 wertilSi da funqciis
mniSvneloba am wertilSi Tanatolia, maSin f (x) funqcias ewodeba
marjvnidan uwyveti x0 wertilSi.
A ricxvs ewodeba ]a; x
f (x
0 [ intervalSi gansazRvruli
) funqciis
marcxena zRvari x0 wertilSi, Tu yoveli dadebiTi ε ricxvisaTvis
arsebobs iseTi dadebiTi η
0
f ricxvi, rom
f (x) − A
ε
p , rodesac 0 x −
η
p
x
0
p ,
am SemTxvevaSi weren ase:
lim f (x) = A .
x→x −
0
f (x) funqciis marcxena zRvari x
f (x −
0 wertilSi aRniSnaven
)
0
simboloTi. Tu f (x) funqcia gansazRvrulia x0 wertilSic da
f (x −) = f (x )
x
0
0
, e.i. funqciis marcxena zRvari 0 wertilSi da funqciis
mniSvneloba am wertilSi Tanatolia, maSin f (x) funqcias ewodeba
marcxnidan uwyveti x0 wertilSi.
funqciis marjvena da marcxena zRvars funqciis calmxrivi zRvrebi
ewodeba. zRvris ganmartebidan gamomdinareobs, rom Tu f (x) funqcias aqvs
129
zRvari x0 wertilSi, maSin arsebobs funqciis marjvena da marcxena
zRvrebi x
f (x
0 wertilSi da es zRvrebi
) funqciis zRvris tolia. e.i. Tu
f (x) funqcias x0 wertilSi ara aqvs romelime calmxrivi zRvari, an orive
calmxrivi zRvari arsebobs, magram isini erTmaneTisagan gansxvavdebian,
maSin x0 wertilSi funqcias ar eqneba zRvari. aqedan gamomdinareobs
Teorema 6. f (x) funqcias aqvs marjvena da marcxena zRvari x0
wertilSi da isini Tanatolia, maSin funqcias eqneba zRvari x0 wertilSi
da igi calmxrivi zRvrebis saerTo mniSvnelobis tolia.
damtkiceba: vTqvaT, f (x +) = f (x −) = A
0
0
. am tolobebis Tanaxmad,
yoveli dadebiTi ε ricxvisaTvis arsebobs iseTi dadebiTi η
0
f ricxvi,
rom f (x) − A
ε
p , rodesac 0 x − x
η
p
0 p
,
f (x) − A
ε
p , rodesac 0 x −
η
p
x
0
p .
aqedan gamomdinareobs, rom
f (x) − A
ε
p , rodesac 0
x − x
η
p
0 p
.
maSasadame, A ricxvi aris f (x) funqciis zRvari x0 wertilSi.
amrigad,
f (x) funqciis zRvris arsebobisaTvis x0 wertilSi
aucilebelia da sakmarisi, rom f (x) funqcias hqondes Tanatoli marjvena
da marcxena zRvari.
analogiurad,
f (x) funqciis uwyvetobisaTvis x0 wertilSi
aucilebelia da sakmarisi, rom adgili hqondes tolobebs
f (x +) = f (x −) = f (x )
f (x
a,b
0
0
0
.
) funqcias ewodeba uwyveti ]
[ intervalSi,
Tu igi uwyvetia am intervalis yovel wertilSi. xolo f (x) funqcias
ewodeba uwyveti [a,b] segmentze, Tu igi uwyvetia ]a,b[ intervalSi da,
garda amisa a wertilSi uwyvetia marjvnidan, b wertilSi ki – marcxnidan.
SemovitanoT funqciis nazrdis cneba. vTqvaT, h warmoadgens ]a,b[
intervalSi argumentis ori mniSvnelobis sxvaobas x − x = h
0
. am sxvaobas
aRniSnaven x
Δ simboloTi (ikiTxeba `delta iqsi~) da ewodeba x argumentis
nazrdi. maSin f (x )
0
funqciis Sesabamis mniSvnelobaTa sxvaoba iqneba
130
f (x + x
Δ ) − f (x )
0
0
romelsac y = f (x) funqciis nazrdi ewodeba da aRiniSneba y
Δ simboloTi
(ikiTxeba `delta igreki~). e.i.
y
Δ = f (x + x
Δ ) − f (x )
0
0 (6)
Tu x
a,b
0 -is nacvlad aviRebT ]
[ intervalSi nebismier x wertils, maSin
f (x) funqciis nazrdi x wertilze iqneba
y
Δ = f (x + x
Δ ) − f (x) (7)
SevniSnoT, rom x cvladis win simbolo Δ aRniSnavs am cvladis
nazrds, amitom x
Δ warmoadgens erT mTlian aRniSvnas da ara Δ -s x –ze
namravls. vnaxoT ras warmoadgens argumentisa da funqciis nazrdi
geometriulad. vTqvaT, sibrtyeze agebulia y = f (x) funqciis grafiki
(nax. 3).
ganvixiloT argumentis raime x mniSvneloba, funqciis Sesabamisi
mniSvneloba iqneba f (x) . naxazze, Tu x=Om, maSin funqciis Sesabamisi
mniSvneloba gamoisaxeba Mm ordinatiT.
axla gadavideT axal x + x
Δ = On mniSvnelobaze. maSin funqciis
mniSvneloba am wertilSi iqneba f (x + x
Δ ) , romelic naxazze nN ordinatiT
gamoisaxeba.
y N
Δy
M K
Δx
f(x) f(x+Δx)
x
0 m n x
x+Δx
nax. 3
131
gavataroT M wertilze KM monakveti Ox RerZis paralelurad Nn
wrfis gadakveTamde. maSin (nax. 3) -ze argumentisa da funqciis nazrdi
iqneba: x
Δ = On − Om = mn = MK
y
Δ = f (x + x
Δ ) − f (x) = nN − mM = nN − nK = KN .
3. zogierTi trigonometriuli utoloba da maTi gamoyeneba
zRvrebis gamoTvlisas. SesaniSnavi zRvrebi. davamtkicoT Semdegi
π
lema 1: nebismieri x maxvili 0 p x p
kuTxisaTvis
2
sin x
x tgx
p p
. (8)
y
C
B
x A
0
D
0 π/2 x
nax. 4 nax. 5
damtkiceba: vTqvaT, (nax. 4) –ze OA = OB =
,
1
AOB
∠
= x radians. BD
da AC marTobebia OA –sadmi. cxadia, OAB samkuTxedis farTobi OAB
wriuli seqtoris farTobze naklebia, xolo OAB wriuli seqtoris
farTobi, Tavis mxriv OAC samkuTxedis farTobze naklebia, magram
1
1
2
π ⋅OA
x
1
1
S
= BD ⋅ OA = BD
Δ
S
=
⋅ x =
S
= OA ⋅ AC =
OAB
;
;
AC .
2
2
seqt.OAB
2π
2
ΔOAC
2
2
Tu gaviTvaliswinebT, rom BD=sinx,
AC=tgx, miviRebT
1
x
1
sin x
tgx
p p
. aqedan gamomdinareobs rom sin x
x tgx
p p
.
2
2
2
⎛ π ⎞
(8) utolobis grafikul ilustracias warmoadgens ( nax. 5). ⎜ ;
0
⎟
⎝ 2 ⎠
SualedSi y=x funqciis grafiki y=sinx funqciis grafikis zemoT da amave
132
dros y=tgx funqciis grafikis qvemoT mdebareobs. SevCerdeT sin x
x
p
utolobaze ufro dawvrilebiT. Cven igi davamtkiceT im daSvebiT, rom
π
0 p x p . magram aseT SemTxvevaSi x da sinx dadebiTia. amitom x = x ,
2
sin x = sin x da sin x x
p utoloba SeiZleba Semdegi saxiT gadaiweros:
sin x
x
p
(9)
es utoloba marTebulia agreTve x –is uaryofiTi mniSvnelobebisTvisac,
⎛ π ⎞
romlebic ⎜ −
;0⎟ SualedSia moTavsebuli, radgan sin(−x) = − sin x = sin x
⎝ 2 ⎠
da − x = x .
(9) utoloba ase gadavweroT sin x − 0
x
p
. Tu x nulisaken
miiswrafvis, maSin ramdenadac sin x − 0 naklebia x -ze, isic nulisaken
miisawrafvis. kerZod, sin x − 0 SeiZleba naklebi gaxdes, vidre 0,1; 0,01;
0,001 da a.S. sazogadod, ragind mcire dadebiTi ε ricxvic ar unda aviRoT,
yovelTvis SeiZleba imas mivaRwioT, rom Sesruldes sin x − 0
ε
p utoloba.
amisaTvis saWiroa x –is SerCeva (− ε ;ε ) SualedSi. es ki niSnavs, rom
limsin x = 0 (10)
x→0
e.i. sinus x –is zRvari, roca x → 0 , nulis tolia. vinaidan, 0=sin0,
amitom miRebuli Sedegi arsebiTad imas niSnavs, rom y=sinx funqcia
uwyvetia, roca x=0.
lema 2: nebismieri x maxvili kuTxisaTvis (gamosaxuli radianebiT)
1 − cos x
x
p . (11)
x
damtkiceba: saskolo kursidan cnobilia, rom 1 − cos x = 2sin 2
, magram
2
x
x
x
x
x
radgan 0
sin
1
p
p , amitom sin2
sin
p
. lema 1-is Tanaxmad, sin
p .
2
2
2
2
2
e.i.
2 x
x
x
1 − cos x = 2sin
2sin
2 = x
p
p
2
2
2
133
Tu x kuTxe maxvilia, maSin x da 1_cosx dadebiTia, amitom x = x ,
1 − cos x = 1 − cos x . maSin (11) ase SeiZleba gadavweroT:
1 − cos x
x
p
(12)
es utoloba, cxadia, marTebulia maSinac, rodesac x
0
p , vinaidan
1 − cos(−x) = 1 − cos x da − x = x . (12) –dan gamomdinareobs, rom
lim cos x = 1 (13)
x→0
e.i. kosinus x –is zRvari, roca x → 0 , erTis tolia. vinaidan, 1=sin0,
amitom miRebuli Sedegi arsebiTad imas niSnavs, rom y=cosx funqcia
uwyvetia, roca x=0. exla davamtkicoT Semdegi Teorema:
sin x
Teorema 7.
funqciis zRvari aris erTi, rodesac x → 0 . e.i.
x
sin x
lim
= 1 (14)
x→0
x
damtkiceba: vTqvaT, x → 0 da amave dros dadebiTi rCeva. maSin
π
SeiZleba CavTvaloT, rom 0 p x p
da amitom lema 1-is Tanaxmad
2
sin x
x tgx
p p
. amasTan am utolobaSi Semavali yvela gamosaxuleba
dadebiTia. ganvixiloT sami wiladi
sin x
sin x
sin x
,
da
.
sin x
x
tgx
rogorc viciT. im wiladebs Soris romlebsac toli mricxveli aqvT, is
iqneba naklebi, romlis mniSvnelic metia. e.i.
sin x
sin x
sin x
sin x
f
f
an 1
cos x
f
f
.
sin x
x
tgx
x
gavamravloT es utoloba wevr-wevrad (_1) –ze, maSin utolobis niSnebi
mopirdapired Seicvleba
sin x
−1
−
−cos x
p
p
.
x
Tu am utolobis TiToeul wevrs mivumatebT 1 –s miviRebT:
sin x
sin x
0 1 −
1 − cos x
p
p
, lema 2-is ZaliT gveqneba 0 1 −
x
p
p .
x
x
134
sin x
sin x
radgan x
0
f da 1−
0
f , amitom 1−
da x sidideebi ukanasknel
x
x
utolobaSi SeiZleba SevcvaloT maTi absoluturi mniSvnelobebiT. amis
sin x
Sedegad miviRebT, rom 1 −
x
p
. rac cxadia, SeiZleba Caiweros Semdegi
x
saxiT:
sin x
−1
x
p
(15)
x
(15) utoloba miviReT im pirobiT, rom x
0
f , magram igi marTebulia maSinac,
sin x
roca x
0
p , radgan
funqcia luwia da maSasadame
x
sin(−x)
sin x
−1 =
−1 .
(−x)
x
sin x
Tu x nulisaken miiswrafvis, maSin, rogorc es (15) –dan Cans
−1 miT
x
ufro miiswrafvis nulisaken. ragind mcirec unda iyos dadebiTi ε
ricxvi, yovelTvis SeiZleba mivaRwioT imas, rom Sesruldes utoloba
sin x
−1 ε
p
x
amisaTvis saWiroa x –is SerCeva (− ε ;ε ) SualedSi. es ki niSnavs, rom
sin x
lim
= 1
x→0
x
ganvixiloT magaliTebi
sin 3x
magaliTi 4. vipovoT lim
x→0
x
mocemuli wiladis mricxveli da mniSvneli gavamravloT 3-ze.
sin 3x
3sin 3x
=
x
3x
SemoviRoT aRniSvna 3x = y . cxadia x → 0 pirobidan gamomdinareobs, rom
agreTve y → 0 . amitom
sin 3x
3sin 3x
3sin y
sin y
lim
= lim
= lim
= 3lim
= 3⋅1 = 3
x→0
x→0
x
3
y→0
y→0
x
y
y
135
1 − cos x
magaliTi 5. vipovoT lim
2
x→0
x
2 x
2sin
x
1 − cos x
2
Tu gamoviyenebT 1 − cos x = 2sin 2
igiveobas, miviRebT: lim
= lim
.
2
2
2
x→0
x→0
x
x
x
SemovitanoT axali cvladi y =
⇒
x = 2 y , maSin
2
1 − cos x
2sin 2 y
1
sin 2 y
1
sin y
sin y
1
1
lim
= lim
= lim
= lim
⋅ lim
= ⋅1⋅1 =
2
x→0
y→0
x
(2 y)2
2
2
y→0
y
2 y→0
y→0
y
y
2
2
(14) –s uwodeben SesaniSnav zRvars. $2.7 –Si ganvixileT neperis ricxvi,
romelsac agreTve uwodeben SesaniSnav zRvars. e.i. gvaqvs SesaniSnavi
sin x
x
⎛
1 ⎞
zRvrebi: 1. lim
= 1 da 2. lim⎜1+ ⎟ = e .
x→0
x
x→∞⎝
x ⎠
bolo tolobidan gvaqvs, rom
x
⎛
1 ⎞
1
3. lim ⎜1 + ⎟ = e da 4. lim (1 + x)x = e .
x→−∞⎝
x ⎠
x→−∞
maTac agreTve uwodeben SesaniSnav zRvrebs.
Semdeg magaliTebSi ganvixiloT zRvrebi, romlebsac pirobiTad
SeiZleba mniSvnelovani zRvrebi vuwodoT:
tgx
magaliTi 6. gamovTvaloT lim
x→0
x
tgx
sin x
sin x
1
vinaidan
=
=
⋅
, amitom miviRebT
x
x cos x
x
cos x
tgx
⎛ sin x
1 ⎞
sin x
1
lim
= lim⎜
⋅
⎟ = lim
⋅ lim
= 1
x→0
x→0
x
⎝ x
cos
x→0
x→0
x ⎠
x
cos x
tgx
e.i. lim
= 1.
x→0
x
tgax
magaliTi 7. gamovTvaloT lim
,
a
sadac
≠ 0 ;
x→0
x
avRniSnoT ax = y , maSin gveqneba
tgax
tgy
⎛
tgy ⎞
tgy
lim
= lim
= lim a ⋅
= a lim
= a ⋅1 = a
→0
→0
→0⎜
⎜
⎟
⎟
x
x
y
y
y
⎝
y
y
⎠
→0 y
a
tgax
e.i. lim
= a .
x→0
x
136
sin bx
magaliTi 8. gamovTvaloT lim
b
sadac
≠ 0 ;
x→0
x
sin bx
analogiurad miviRebT: lim
= b
b
sadac
≠ 0 .
x→0
x
tgax
magaliTi 9. gamovTvaloT lim
sadac
a ≠ 0
, b ≠
0
x→0 sin bx
mricxveli da mniSvneli gavyoT x –ze, sadac x ≠ 0 , miviRebT
tgax
lim
tgax
x→0
x
a
tgax
a
lim
=
= e.i. lim
=
x→0 sin bx
sin bx
b
x→0
lim
sin bx
b
x→0
x
x
⎛
k ⎞
magaliTi 10. gamovTvaloT lim⎜1 + ⎟
x→∞⎝
x ⎠
k
1
SemoviRoT aRniSvna
= , maSin
x
y
k
x
ky
y
⎛
k
⎡
⎤
⎞
⎛
1 ⎞
⎛
1 ⎞
k
lim⎜1+ ⎟ = lim 1+
= ⎢lim 1+
⎥ = e
→∞
→∞⎜
⎜
⎟
⎟
→0⎜
⎜
⎟
⎟
x
x
y
⎝
x ⎠
⎝
y ⎠
⎢
⎝
y ⎠ ⎥
⎣
⎦
x
⎛
k ⎞
e.i.
k
lim⎜1+ ⎟ = e
x→∞⎝
x ⎠
1
ln( + x)
magaliTi 11. gamovTvaloT lim
x→0
x
Tu visargeblebT logariTmis TvisebiT gveqneba
1
ln( + x)
1
1
⎡
⎤
lim
= lim 1
ln( + x) x = ln⎢lim 1
( + x) x ⎥ = ln e = 1
x→0
x→0
x→0
x
⎣
⎦
1
ln( + x)
e.i. lim
= 1
x 0
→
x
e.w. mniSvnelovan zRvrebs miekuTvneba, agreTve, Semdegi zRvruli
tolobebi, romelTa damtkicebac mkiTxvelisaTvis migvindia.
k 1+ x −1
1
a x −1
ex −1
1. lim
= ; 2. lim
= ln a ; 3. lim
= 1;
x→0
x
k
x→0
x
x→0
x
a x −1
1
( + x)n −1
log 1
( + x)
4. lim
= 1; 5. lim
= 1; 6.
a
lim
= log e
a
;
x→0 x ln a
x→0
nx
x→0
x
arcsin x
arctgx
7. lim
= 1; 8. lim
= 1
x→0
x
x→0
x
137
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ras niSnavs, rom f (x) funqciis zRvari x0 wertilSi aris A ricxvi?
2. moiyvaneT gansazRvra f (x) funqciis calmxrivi zRvrebisa x0
wertilSi.
3. ras niSnavs, rom f (x) funqcia uwyvetia x0 wertilSi? uwyvetia
marjvnidan? uwyvetia marcxnidan?
4. ras niSnavs, rom f (x) funqcia uwyvetia [a,b] intervalSi? ]a,b[
segmentze?
5. ras niSnavs, rom f (x) funqcia wyvetilia x0 wertilSi?
6. SeiZleba Tu ara, rom funqcia gansazRvruli iyos x0 wertilSi,
magram iyos uwyveti? iyos wyvetili?
II. praqtikuli savarjiSoebi
2n +1
1. aCveneT, rom x =
n →
n
mimdevrobis zRvari, roca
∞ , aris 2.
n +1
ipoveT N , Tu ε = 1
,
0 ; ε = 01
,
0
;
1
2. aCveneT, rom x = 1 +
n →
n
mimdevrobis zRvari, roca
∞ , aris 1. ipoveT
n
N , Tu ε = 1
,
0 ; ε = 01
,
0
;
n
3. aCveneT, rom x =
n →
n
mimdevrobis zRvari, roca
∞ , aris 1. ipoveT
n +1
N , Tu; ε = 01
,
0
;
1
4. aCveneT, rom x =
n →
n
mimdevrobis zRvari, roca
∞ , aris 0. rogori
2
n
1
n–isaTvis Sesruldeba
ε utoloba, Tu ε = 1,
0 ; ε = 01
,
0
;
2 p
n
5. gamoTvaleT Semdegi zRvrebi:
138
3 2
x + 4 x − 5
2
x − x +
lim (
5
1
x 2 + x + 1 − x )
5.1. lim
; 5.2.
5.3. lim
;
x → ∞
x +
x → ∞
6 2
x + 2 x + 1
x → ∞
3
7
2 4
x − x + 3
3
x − 2 2
x − 4
⎛ x 3
⎞
5.4. lim
; 5.5
; 5.6.
− x ;
3
lim
4
lim ⎜⎜ 2
⎟
⎟
x → ∞
x +
x → ∞
x +
x −
x → ∞
x − 8 x + 5
7
1
⎝
1
⎠
2 2
x − x + 3
⎛
2
x
2 3
x
⎞
2
x + x − 12
5.7. lim
; 5.8.
−
; 5.9.
;
3
lim ⎜⎜
2
⎟
⎟
lim 2
x →
x − x +
x → ∞
4 x + 1
8 x −
x → ∞
x − 8 x + 5
⎝
1 ⎠
3
5
6
2
x − 1
x − 1
2 2
x − 9 x + 4
5.10. lim
; 5.11.
; 5.12
;
2
lim
lim
3
2
x →
x + x −
x →
x − x +
2
3
2
x → 1
x − 1
4
20
3
x + 4 2
x
3
x − 3 x + 2
4
x − 16
5.13.
lim
; 5.14.
; 5.15.
2
lim 4
lim
x →
x − x +
x → −
x + x −
4
12
1
4
3
x → 2
x −
2
3 4
x − 4 3
x + 1
2 x + 7 − 5
3 x
5.16. lim
; 5.17.
; 5.18.
3
2
lim
lim
x →
x − x − x +
1
1
x → 9
3 −
x
x → 0
5 + x −
5 − x
x
sin 3
tgbx
3 1 + 2 x − 1
5.19. lim
; 5.20. lim
; 5.21. lim
;
x → 0
x
x →
sin(
2
0
ax + x )
x → 0
x
3
⎛
2k ⎞x
⎛
x ⎞ x
x
3 −1
5.22. lim⎜1 +
⎟ = 1; 5.23. lim⎜1+ ⎟ = 1; 5.24. lim
;
x→0 ⎝
x ⎠
x→0 ⎝
2 ⎠
x→0
5x
e3x −1
2
ex − cos x
5.25 lim
; 5.26. lim
x→0
6x
2
x→0
x
$4. elementarul funqciaTa klasi
funqciaTa im simravlidan, romelsac maTematikur analizSi
ganixilaven, gamoyofen e.w. elementarul funqciebs. ZiriTadi
elementaruli funqciebs warmoadgens Semdegi:
1. y = C , sadac C mudmivia.
2.
α
y = x , sadac α nebismieri namdvili ricxvia.
139
3. y = a x
(
a
0, a ≠
f
)1.
4. trigonometriuli funqciebi y = sin x, y = cos x, y = tgx, y = ctgx, y = sec x ,
y = cos ecx .
5. Seqceuli trigonometriuli funqciebi y = arcsin x,
y = arccos x,
y = arctgx, y = arcctgx, y = arc sec x , y = arccosecx .
gansazRvreba: y = f (x) funqcias ewodeba elementaruli funqcia, Tu
igi warmoidgineba erTi formuliT, romelic Sedgenilia ZiriTadi
elementaruli funqciebisagan maTze ariTmetikuli operaciebisa da
superpoziciaTa sasrul ricxvjer gamoyenebiT. amrigad, elementaruli
funqciebi yovelTvis mocemulia analizurad.
x
elementaruli funqciebia, magaliTad y = ln cos x, y = ctgx ⋅ arctgx ,
2x
y =
.
araelementarulia funqcia y = E(x) , sadac E(x) aRniSnavs udides mTel
ricxvs, romelic ar aRemateba x -s. y = x funqcia elementarulia,
vinaidan igi ase warmoidgineba
2
y = x . mtkicdeba Semdegi
Teorema: nebismieri elementaruli funqcia uwyvetia Tavis
gansazRvris areSi.
Cven davamtkicebT zogierTi elementaruli funqciebis uwyvetobas
Tavis gansazRvris areSi.
1. ganvixiloT funqciebi y = sin x, y = cos x, rogorc viciT, es
funqciebi gansazRvrulia x –is yvela namdvili ricxviTi
mniSvnelobebisaTvis, anu (− ∞;+∞) SualedSi. aviRoT nebismieri namdvili
ricxvi x0 . wina paragrafSi moyvanili mtkicebis Tanaxmad, adgili eqneba
utolobebs
lim sin x = sin x , lim cos x = cos x .
0
0
x→x
→
0
x
x0
aqedan gamomdinareobs, rom y = sin x
da
y = cos x uwyveti funqciebia
x
x
0 wertilze da vinaidan
0 iyo nebismieri namdvili ricxvi, amitom
y = sin x
da
y = cos x uwyveti funqciebia Tavis gansazRvris (− ∞;+∞)
SualedSi.
sin x
2. axla aviRoT y = tgx funqcia. vinaidan tgx =
da
cos x
y = sin x
da
y = cos x uwyveti funqciebia (− ∞;+∞) SualedSi, amitom y = tgx
140
funqcia, rogorc ori uwyveti funqciis fardoba uwyvetia yvalgan, sadac
mniSvneli ar udris nuls, anu igi uwyvetia yvelgan Tavis gansazRvris
π
areSi, xolo x =
k = 0 ,
1
,
,
2
k
(2k + )1 ,
K wertilebSi ar aris gansazRvruli.
2
analogiurad, y = ctgx funqcia uwyvetia yvelgan Tavisi gansazRvris areSi,
xolo x = π
k
k = 0 ,
1
,
,
2
k
,
K wertilebSi ar aris gansazRvruli.
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ras ewodeba elementaruli funqcia?
2. romelia ZiriTadi elementaruli funqciebi?
3. moiyvaneT damtkiceba zogierTi elementaruli funqciebis uwyvetobisa
Tavis gansazRvris areSi.
II. praqtikuli savarjiSoebi
gamoikvlieT Semdegi funqciebidan romeli da sad ganicdis wyvetas
(aageT garfikebi):
1
1.
2
y = x ; 2. y =
;
x + 3
1
3. y = ctgx ; 4. y = sin .
x
$5. usasrulod mcire da usasrulod didi sidideebi.
usasrulod mcireTa Sedareba. ekvivalenturi usasrulod mcireeebi
141
1. usasrulod mcire da usasrulod didi sidideebi. x0 wertilis
midamoSi gansazRvrul f (x) funqcias ewodeba usasrulod mcire x0
wertilSi, Tu misi zRvari nulis tolia
lim f (x) = 0 .
x→ 0
x
es imas niSnavs, rom usasrulod mcire sididis absoluturi
mniSvneloba, garkveuli droidan dawyebuli, naklebia yovel, winaswar
aRebul, nebismierad mcire ε ricxvze da SemdegSiac aseTive rCeba. e.i.
rogoric ar unda iyos nebismierad mcire dadebiTi ε ricxvi, garkveuli
droidan dawyebuli, usasrulod mcire x-sidide daakmayofilebs utolobas:
x
ε
p .
x
+ ∞ an
−
x =
0 SeiZleba iyos
∞ . Tu
+∞
0
, am arasakuTrivi wertilis
midamoa yoveli ]a,+∞[ Sualedi, xolo Tu x = −∞
0
, maSin midamoa yoveli
]− ∞,a[ Sualedi. SevniSnoT, rom Zalian mcire sidide ar aris usasrulod
mcire sidide, vinaidan usasrulod mcire sidide cvladi sididea, Zalian
mcire sidide ki mudmivia.
funqciisa zRvris gansazRvrebidan gamomdinreobs, rom Tu
lim f (x) = A , maSin f (x) − A sxvaoba usasrulod mcirea x wertilSi. da
x→x
0
0
piriqiT, Tu f (x) = A + α (x) , sadac α (x) usasrulos mcirea x0 wertilSi,
maSin A iqneba f (x) funqciis zRvari x0 wertilSi (zRvris gamosaxva
tolobis saSualebiT).
f (x) funqcias ewodeba usasrulod didi x0 wertilSi, Tu yoveli
dadebiTi A ricxvisaTvis arsebobs iseTi dadebiTi η ricxvi, rom
f (x)
A
f , rodesac 0
x − x
η
p
0 p
.
am SemTxvevaSi davwerT
lim f (x) = ∞ .
x→ 0
x
amboben, rom cvladi sidide usasrulod didia, Tu misi absoluturi
mniSvneloba, garkveuli droidan dawyebuli metia yovel winaswar aRebul
nebismierad did dadebiT N ricxvze da SemdegSiac aseTive rCeba. e.i.
rogoric ar unda iyos nebismierad didi dadebiTi N ricxvi, garkveuli
droidan dawyebuli usasrulod didi x-sidide daakmayofilebs utolobas:
142
x
N
f
.
Teorema 1: usasrulod didi sididis Sebrunebuli sidide,
usasrulod mcirea.
damtkiceba: vTqvaT, f (x) usasrulod didi sididea x0 wertilSi.
1
davamtkicoT, rom
usasrulod mcirea amave wertilSi. aviRoT
f (x)
nebismieri dadebiTi ε ricxvi, radgan lim f (x) = +∞ , amitom moiZebneba
x→ 0
x
1
iseTi dadebiTi η ricxvi, rom f (x) f
0
x − x
ε , rodesac
η
p
0 p
. aqedan
1
1
vRebulobT
ε
p , rodesac 0
x − x
η
p
lim
=
0 p
. e.i.
0 .
f (x)
x→ 0
x f (x)
avRniSnoT usasrulod mcire sidideTa zogierTi Tviseba.
Teorema 2. usasrulod mcireTa jami usasrulod mcirea (igulisxmeba,
rom SesakrebTa ricxvi sasrulia).
damtkiceba: vTqvaT, mocemulia ramdenime usasrulod mcire ricxviT n.
x , x , , x
1
2 K
n . rogoric ar unda iyos nebismierad mcire dadebiTi ε ricxvi,
garkveuli droidan dawyebuli, yveal mocemuli usasrulod mcire xi
ε
cvladi x
(i = ,
1 2, , n
i p
)
K
. da SemdegSic aseTive darCeba. magram meores
n
mxriv gvaqvs x + x +
+ x ≤ x + x +
+ x
x + x +
+ x ≤
1
2
L
n
1
2
L
n . e.i.
ε
1
2
L
n
. es ki
amtkiceb, rom x + x +
+ x
1
2
L
n jami usasrulod mcirea.
aqedan uSualod gamomdinreobs, rom ori usasrulod mcires sxvaoba
usasrulod mcire iqneba an nuli. marTlac, vinaidan x − x = x + (−x )
1
2
1
2
,amitom
jamisaTvis debuleba damtkicebulia. da Tu orve usasrulod mcire tolia,
maSin maTi sxvaoba nulis toli iqneba.
Teorema 3. mudmivi sididisa da usasrulod mcires namravli
usasrulod mcirea.
143
damtkiceba: vTqvaT x usasrulod mcirea, xolo a nebismieri mudmivi
ε
sidide. garkveuli droidan dawyebuli x p , sadac ε nebismierad mcire
a
dadebiTi ricxvia. magram radgan xa = x ⋅ a , amitom miviRebT xa
ε
p . e.i.
Teorema 4. usasrulod mcireTa namravli usasrulod mcirea.
damtkiceba: jerjerobiT ganvixiloT ori x
da
x
1
2
usasrulod
mcire. cxadia, rom garkveuli droidan dawyebuli x1 gaxdeba romelime
mudmiv dadebiT sidideze naklebi, amasTanave x2 -is cvlilebis dros
ε
dadgeba iseTi momenti, rodesac adgili eqneba utolobas x2 p , sadac ε
a
nebismierad mcire dadebiTi ricxvia.
ε
axla ganvixiloT x ⋅ x = x ⋅ x
a ⋅ = ε
x ⋅ x
1
2
1
2 p
. amrigad,
ε
1
2 p
.
a
advili dasamtkicebelia, rom ramdenic ar unda iyos usasrulod
mcire maTi namravli yovelTvis usasrulod mcire iqneba. kerZod. Tu yvela
mamravli Tanatolia, e. i. Tu x usasrulod mcirea, maSin n
x -ic
usasrulod mcire iqneba, sadac n mTeli dadebiTi ricxvia.
Teorema 5. usasrulod mcire sidideTa Sefardeba SeiZleba iyos
rogorc usasrulod mcire, ise sasruli da usasrulod didic.
damtkiceba: es gamomdinareobs Semdegi magaliTebidan. Tu x
usasrulod mcirea. maSin Teorema 4 –is Tanaxmad, 2
x -ic usasrulod mcire
x2
iqneba. Sefardeba
usasrulod mcirea, vinaidan igi x –is tolia. axla
x
aviRoT romelime sasruli a sidide. Teorema 3 –is Tanaxmad ax usasrulod
ax
mcirea,
fardoba sasrulia, vinaidan igi a -s tolia. dasasrul aviRoT
x
x
1
fardoba; es usasrulod didia, vinaidan igi -is tolia da Tu x
2
x
x
1
usasrulod mcirea, maSin rogorc viciT usasrulod didi iqneba.
x
144
SevniSnoT, rom garda aq CamoTvlili SemTxvevbisa, SesaZlebelia
iseTi SemTxveva, rodesac usasrulod mcire sidideTa Sefardebas ara aqvs
1
zRvari. magaliTad, Tu Tu x usasrulod mcirea, maSin x sin namravli
x
1
x sin x
1
agteRve usasrulod mcirea, magram
Sefardebas, romelic sin
-is
x
x
tolia ara aqvs zRvari. amrigad, rogorc gamoirkva ori usasrulod mcire
sidideTa Sefardeba ganuzRvrelobas warmoadgens.
Teorema 6. ori usasrulod didi sidideTa Sefardeba SeiZleba iyos
rogorc usasrulod didi, ise sasruli da usasrulod mcirec.
damtkiceba: es gamomdinareobs iqidan, rom ori usasrulod didi
y
1
y
sidideTa Sefardeba 1 SeiZleba ase warmovadginoT 1 , magram es ukve
y
1
2
y2
ori usasrulod mcireTa fardobaa da amgvarad Teorema damtkicebulia.
aseve mtkicdeba Semdegi ori Tvisebac
Teorema 7. usasrulod mciresa da usasrulod didis namravli
SeiZleba iyos, rogorc usasrulod mcire, ise sasruli da usasrulod
didic.
Teorema 8. ori usasrulod didis sxvaoba SeiZleba iyos, rogorc
usasrulod didi, ise sasruli da usasrulod mcirec.
amrigad, usasrulod mcireTa fardoba, ori usasrulod didis
fardoba, usasrulod mcires usasrulod didze namravli da ori
usasrulod didis sxvaoba ganuzRvrelobebs warmoadgens.
2. usasrulod mcireTa Sedareba. ekvivalenturi usasrulod
α
mcireeebi. Tu α
da
β usasrulod mcireTa β fardoba usasrulod mcirea,
maSin α -s ewodeba maRali rigis usasrulod mcire β -s mimarT.
α
Tu α
da
β usasrulod mcireTa β fardoba usasrulod didia, maSin
α -s ewodeba dabali rigis usasrulod mcire β -s mimarT.
145
α
Tu α
da
β usasrulod mcireTa β fardobis zRvari nulisagan
gansxvavebuli sasruli ricxvia, maSin α
da β
-s
ewodeba
erTi da imave
rigis usasrulod mcireebi.
zogjer saWiroa zustad daxasiaTeba erTi usasrulod mciris
nulisagan miswrafebisa meoresTan SedarebiT. amisaTvis SemoviRoT
ganmarteba: β usasrulod mcires ewodeba k –rigis usasrulod mcire α
usasrulod mcires mimarT, Tu α k
β
da
erTi da imave rigis usasrulod
mcireebia.
magaliTad, Tu α usasrulod mcirea, maSin 2
3
4
α , α
, α arian
1
Sesabamisad, meore, mesame da rigis usasrulod mcireebi α -s mimarT.
4
α
α
da
β usasrulod mcireebs ewodeba ekvivalenturi, Tu lim = 1
β
. Tu
α
da
β ekvivalenturi usasrulod mcireebia, maSin weren α ~ β . cxadia,
rom Tu α ~ β , maSin β ~ α . xolo Tu α ~ β da β ~ α , maSin α ~ γ .
magaliTad, sinx da x ekvivalenturi usasrulod mcireebia, rodesac
sin x
x → 0 , radgan lim
= 1.
x→0
x
davamtkicoT Teorema, romelsac gamoyeneba aqvs ori usasrulod
mciris fardobis zRvris gamoTvlis dros.
Teorema 9. ori usasrulod mciris fardobis zRvari ar Seicvleba,
Tu am usasrulod mcireebs SevcvliT maTi ekvivalenturi usasrulod
mcireebiT.
damtkiceba: vTqvaT, α
da
β usasrulod mcireebia da α ~ α′ , β ~ β ′ .
α
α ′
dasamatkicebelia, rom Tu arsebobs lim
lim
β , maSin iarsebebs
β ′ da
α
α ′
α ′
marTebulia toloba lim
= lim
β
β ′ ; β ′ Sefardeba ase warmovadginoT
α ′ α ′ β α
α ′
α ′
β
α
α
=
⋅
⋅
lim
= lim
⋅ lim
⋅ lim
= lim
β ′ α β ′ β ; aqedan
β ′
α
β ′
β
β
146
axla moviyvanoT zogierTi ekvivalenturi usasrulod mciris
magaliTebi:
sin x
1. sin x ~ x , rodesac x → 0 , vinaidan lim
= 1;
x→0
x
tgx
2. tgx ~ x , rodesac x → 0 , vinaidan lim
= 1;
x→0
x
λ
3. (1 + x) −1 ~ λx , rodesac x → 0 , sadac λ namdvili ricxvia, vinaidan
(1+ x)λ −1
lim
= 1;
x→0
λx
a x −1
4. a x −1 ~ x ln a , rodesac x → 0 , vinaidan lim
= 1;
x→0 x ln a
ex −1
5. e x −1 ~ x , rodesac x → 0 , vinaidan lim
= 1;(4-s kerZo SemTxvevaa).
x→0
x
1
ln( + α )
6.
1
ln( + α ) ~ α , rodesac α → 0 , vinaidan lim
= 1;
α →0
α
arcsin x
7. arcsin x ~ x , rodesac x → 0 , vinaidan lim
= 1;
x→0
x
arctgx
8. arctgx ~ x , rodesac x → 0 , vinaidan lim
= 1;
x→0
x
x
k 1+ x −1
9. k 1 + x −1 ~
, rodesac x → 0 , vinaidan lim
= 1.
k
x→0
x
k
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ras ewodeba usasrulod mcire sidide x0 wertilSi?
2. ra damokidebuleba arsebobs usasrulod mcire da usasrulod did
sidideebs Soris?
3. Tu lim f (x) = A , maSin f (x) − A sxvaoba rogori sididea x wertilSi?
x→x
0
0
4. vTqvaT, f (x) = A + α (x) , sadac α (x) usasrulos mcirea x0 wertilSi,
xolo A mudmivia. ras warmoadgens maSin f (x) funqciisaTvis A
wertili?
147
5. ras niSnavs, rom α usasrulod mcire maRali rigis usasrulod
mcirea β usasrulod mcires mimarT?
6. ras niSnavs, rom α
da
β usasrulod mcireebi erTi da imave rigis
usasrulod mcireebia?
7. moiyvaneT ekvivalenturi ori usasrulod mcires gansazRvra.
8. SesaZlebelia Tu ara , rom ori usasrulod mcire yovelTvis
SevadaroT?
9. rodis ewodeba α -s k –rigis usasrulod mcire β usasrulod mcires
mimarT?
10.
SeiZleba Tu ara ori usasrulod mcires fardobis zRvris
gamoTvlisas mricxveli da mniSvneli SevcvaloT maTi ekvivalenturi
usasruld mcireebiT?
II. praqtikuli savarjiSoebi
1. SeiswavleT x cvladis mimrT Semdeg usasrulod mcireTa rigi:
1.1. 2x; 1.2.
3
3x ; 1.3. 3 x ;
2. Tu x → 0 , maSin uCveneT qvemoTmoyvanili funqciebis cvalebadobis
xasiaTi:
a
π
2.1 a − x ; 2.2. ctg
; 2.3. tgx − lg a ;
2x
1
1
x
2.4
; 2.5.
; 2.6. arcsin .
2
2
4x − 3a
tgx − lg a
a
3. Tu x cvladi usasrulod didia, maSin uCveneT romelia usasrulod
mcire, sasruli ricxvi da usasrulod didi qvemo magaliTebidan:
1
3.1. 2
x ; 3.2. 2
−
x ; 3.3 1+ ; 3.4. x
e− ;
x
1
1
1
3.5. ctg
; 3.6. sin
; 3.7 1 + 6x ; 3.8. 4 +
.
x
x − a
x
4. gansazRvreT x usasrulod mciris mimarT Semdegi gamosaxulebebis
rigi:
1
4.1. tgx − sin x ; 4.2. 1− cos x ; 4.3. 2
3
x − 3x ; 4.4. sin 3x ; 4.5. tg3x
2
148
$ 6 moTxovnis, miwodebisa da danaxarjis
funqciebi. wonasworobis fasi
sanam ZiriTad sakiTxze gadavidodeT mokled gavmartoT, ras
Seiswavlis mikroekonomika. SemovitanoT mikroekonomikis ganmarteba ise,
rogorc es aris ganmartebuli Tanamedrove ekonomikur leqsikonSi.
mikroekonomika – (berZ. micros - mcire), es aris ekonomikuri
mecnierebis sfero, romelic dakavSirebulia SedarebiT mciremasStabiani
ekonomikuri procesebis, subieqtebis, movlenebis, ZiriTadSi sawarmos,
firmis, mewarmeTa da maT Soris sameurneo saqmianobasTan, ekonomikur
urTierTobebTan. mikroekonomikis yuradRebis centrSia mwarmoeblebi da
momxmareblebi, maT mier miRebuli gadawyvetilebebi warmoebis moculobis,
gayidvis, yidvis, moxmarebis fasebis, danaxarjebis, mogebis
gaTvaliswinebiT. mikroekonomika aseve Seiswavlis subieqtebis sabazro
moqmedebas, warmoebis procesSi maT urTierTobas, ganawilebas, gacvlas,
moxmarebas. mikroekonomikis Seswavlis obieqtis agreTve urTierTobani
mwarmoeblebs, mewarmeebsa da saxelmwifos Soris.
Cven ganvixilavT mikroekonomikis or ZiriTad sferos romelTac ewo-
debaT miwodeba da moTxovna. Cveni mizania gavaanalizoT sabazro
ekonomikis metad mniSvnelovani sakiTxi, romlesac uwodeben miwodebisa da
moTxovnis wonasworobas.
SemoviRoT moTxovnisa da miwodebis funqciebis cneba.
vTqvaT, bazris moTxovna raime fiqsirebul nawarmze aris Q. cxadia, Q
ricxvi, romelic aRniSnavs moTxovnili nawarmis raodenobas da izomeba
(garkveuli produqciis Sesabamisi) erTeulebiT. avRniSnoT erTeuli
produqciis sabazro fasi P simboloTi. sabazro ekonomikis pirobebSi
moTxovna Q damokidebulia sabazro P fassze.
Q = f (P) . (1)
149
f funqciis konkretuli saxe dgindeba an ekonomikuri Teriidan an
sabazro monacemebidan. (1) tipis damokidebulebas uwodeben moTxovnis
funqcias da amis misaTiTeblad f -s indeqsad miaweren D asos:
Q = f (P)
D
(2)
sazogadod fD SeiZleba Zalian rTuli funqcia iyos, SevniSnoT, rom
fD sazogadod damokidebulia P fasze, momxmareblis Y Semosavalze
alternatiuli produqciis PS fassze, damatebiTi saqonlis Pc fassze,
reklamis A danaxrjze da momxmarebelTa T gemovnebaze. Cven ganvixilavT, im
SemTxvevas, rodesac fD wrfivi funqciaa P-s mimarT, kerZod
Q = f (P) = a P + b
D
1
1 , (3)
sadac a1 da b1 raime konkretuli mudmivebia, romlebsac ekonomikaSi
parametrebs uwodeben. realur cxovrebaSi raime produqciaze fasis zrda
iwvevs am produqciaze moTxovnis Semcirebas e.i. (3) funqcia unda iyos
klebadi, es ki niSnavs, rom a1 <0. amitom Sesabamisi grafiki OP RerZis
dadebiT mimarTulebasTan blagv kuTxes Seadgens. tradiciulad,
ekonomistebi moTxovnis funqcias weren ara (2) formiT, aramed
P = g (Q)
D
saxiT. e.i. fass gamosaxaven, rogorc moTxovnis funqcias. es tolfasia, imis
rom (2) gantolebidan vipovoT P cvladi Q cvladis saSualebiT. amis
Sesabamisad grafikis agebis dros vertikalur RerZze gadazomaven P fass,
hirizontalur RerZze ki- Q moTxovnas. (3) wrfivi damokidebulebidan
martivad miviRebT
P = g (Q) = aQ + b,
D
(4)
1
1
sadac a =
,
b =
SevniSnoT, rom radgan a < 0 amitom a < 0 e.i. (4)
a
b
1
1
1
funqcia klebadia. maSasadame misi grafiki OQ RerZTan Seadgens blagv
kuTxes. mas uwodeben moTxovnis wirs (wrfes). ekonomikuri Sinaarsidan
gamomdinare, cxadia, rom P ≥ 0 daQQQ ≥ 0 amasTan P=0 niSnavs, rom
faqtobrivad produqcia ufasod eZleva yvela msurvels, xolo Q=0 niSnavs,
rom moTxovna gansaxilvel produqciaze ar arsebobs. gavarkvioT (4)
gantolebaSi b parametris ekonomikuri Sinaarsi. cxadia, roca P=b maSin
Q=0 e.i., rodesac fasi aris b toli, maSin produqciaze moTxovna ar
150
aresebobs (raRac mizezis gamo, magaliTad maRali fasis gamo) amrigad b>0
da produqciis fasi bazarze SemosazRvrulia am b ricxviT. aseve cxadia,
b
b
rom rodesac P=0 maSin Q = −
> 0 . amitom − ricxvi miuTiTebs bazris
a
a
maqsimalur moTxovnas.
amocana 1. avagoT moTxovnis wiri, Tu moTxovnis funqciaa P = 3
− Q + 75 .
1. ras udris fasi, Tu moTxovnaa 23?
2. ras udris moTxovna, rodesac fasia 18?
3. rogor icvleba fasi moTxovnis erTi erTeuliT Semcirebisas?
moTxovnis wiris (wrfis) asagebad movZebnoT misi OQ da OP RerZebTan
gadakveTis wertilebi: rodesac P=0 maSin Q=25; xolo Q=0, maSin P=75; e.i.
wrfe gadis (25;0) da (0;75) wertilebze (nax.1)
P
P
75
d
0 25 Q
0 Q
nax. 1 nax. 2
1. mocemuli moTxovnis funqcidan miviRebT: Tu Q=23; maSin
P= _3⋅23+75=6 e.i. am SemTxvevaSi produqciis fasia P=6.
2. rodesac P=18; maSin Q=19, e.i. rodesac fasia P=18, maSin moTxovnaa
Q=19;
3. rodesac moTxovnaa Q, maSin fasia
P= _ 3Q+75
amitom rodesac moTxovna iqneba Q-1, maSin fasi iqneba P1= _3(Q-1)+75=_3Q+78.
aqedan miviRebT P1_ P= (-3Q+78)-(-3Q+75)=3 anu P1=P+3. amrigad, Tu moTxovna
Semcirda erTi erTeuliT, maSin fasi izrdeba sami erTeuliT.
axla ganvixiloT miwodebis funqcia. igi amyarebs Sesabamisobas raime
produqciis erTeulis P fassa da amave produqciis Q raodenobas Soris,
151
romlis bazarze Setanasac gegmavs mwarmoebeli. ekonomikuri Teoria da
realuri cxovreba uCvenebs, rom fasis zrdas mosdevs miwodebis zrda.
amitom mowodebis funqcia
P = g (Q)
S
(5)
zrdadi funqciaa. Cven ganvixilavT konkretul SemTxvevas, roca igi
wrfivia, e.i.
P = g (Q) = cQ + d ,
S
(6)
sadac c da d mudmivebia, radganac miwodebis funqcia zrdadia, amitom
kuTxuri koeficienti c>0. radgan fasi yovelTvis dadebiTia, amitom d
parametric dadebiTia, e.i. c>0, d>0. aqedan vaskvniT, rom misi Sesabamisi
grafiki OQ RerZTan adgens maxvil kuTxes da OP RerZs kveTs d wertilSi
(nax.2) am grafiks miwodebis wiri (wrfe) ewodeba. zemoT moyvanili
msjelobidan (ix. nax.3) gamomdinareobs, rom mwarmoebeli dadgmavs produq-
ciis Setanas bazarze mxolod maSin Tu fasi gadaaWarbebs d sidides. avagoT
axla erTsa da imave OPQ sibrtyeze moTxovnis LD da miwodebis LS wirebi
(nax.3.)
P
P1
LLS – P=cQ+d
B C
A
P0
L
E F
P
LD _ P=aQ+b
2
0 Q (2) Q
(1)
(1)
(1)
S
0 QD
QS
QD Q
nax. 3
CavataroT nax.3 -is analizi. ganvixiloT P1 fasis Sesabamisi B da C
wertilebi LD da LS wrfeebze. radgan B ∈ L
C
da
∈ L
p
s amitom P1 fass
Seesabameba
)
1
(
Q
Q
Q
Q
D moTxovna da
)
1
(
S miwodeba, amasTan
)
1
(
D <
)
1
(
S e.i. moTxovna
CamorCeba miwodebas. es ki niSnavs rom bazari gajerebuli miwodebuli
produqciiT da amitom es produqcia mTlianad ar gaiyideba. amrigad, am
SemTxvevaSi bazarze gvaqvs Warbi produqcia.
152
axla ganvixiloT P2
E ∈ L
da F ∈ L
fasis Sesabamisi
S
D wertilebi, cxa-
dia, rom P
(2
(2
2
Q
Q
fass Seesabameba
)
D moTxovna da
)
S
miwodeba, amasTan
(2)
Q
(2
Q
D >
)
S
, e i. moTxovna Warbobs mowodebas, es ki niSnavs, rom moTxovna
mTlianad ver kmayofildeba da saqme gvaqvs produqciis deficitTan. es
orive varianti arasasurveli, sabazro ekonomikisaTvis.
ganvixiloT P0
L
da L
fasis Sesabamisi situacia, mas Seesabameba S
D
wrfeebis saerTo A wertili, romlis abscisaa Q0. bunebrivia, rom P0 fasis
Seesabamisi Q0 moTxovna da Q0 miwodeba erTmaneTis tolia. e.i. moTxovna
emTxveva mowodebas. am SemTxvevaSi amboben, rom bazari gawonasworebulia.
(iyideba im raodenobis produqcia, ra raodenobac miewodeba bazars).
P0 fass ewodeba wonasworobis fasi, Sesabamis Q0 – wonasworobis
sidide (moculoba).
cxadia gawonasworebuli bazari warmoadgens idealur variants. igi
ganisazRvreba Semdegi sistemiT:
P = aQ +
⎪
⎧
b,
⎨
⎪
⎩P = cQ + d.
miRebul sistemis (Q0;P0) gansazRvravs moTxovnis L
da L
D
S wrfeebis
saerTo A wertilis koordinatebs.
amocana 1. moTxovnisa da miwodebis funqciebi Sesabamisad mocemulia
tolobebiT
P = g (Q) = 2
− Q + 50
D
(7)
1
P = g (Q) = Q +
,
25
S
(8)
2
sadac (7) gantolebaSi Q aris moTxovna, xolo (8) gantolebaSi – miwodeba.
1. ganvsazRvroT wonasworobis fasi da wonasworobis sidide.
2. mTavrobam gadawyvita daawesos fiqsirebuli gadasaxadi 5 laris
odenobiT produqciis yovel gayodul erTeulze. vipovoT am
RonisZiebis gavlena bazris wonasworobaze.
1. avagoT
moTxovnis
P
L
L
D
da miwodebis
S
wirebi (nax. 4). maTi
50
gadakveTis A wertilis LS
koordinatebiT
30 A
davadgenT wonasworobis
fassa
25
153
LD
0 10 25 Q
da wonasworobis sidides.
nax. 4
⎪
⎧ p = −2Q + ,
50
⎨
1
⎪ p = Q + .
25
⎩
2
sistemis amoxsniT miviRebT P0=30; Q0=10. amrigad, wonasworobis fasia,
P0=30; wonasworobis sidide Q0=10.
2. Tu mTavroba daawesebs 5 lar fiqsirebul gadasaxads produqciis
yovel gayidul erTeulze, maSin moTxovnis funqcia (7) ucvleli darCeba,
miwodebis funqcia (8) ki Seicvleba. marTlac, Tu produqciis erTeuli P
larad iyideboda, rasac momxmarebeli firmas uxdida, axla ukve momxmareb-
lis mier gadaxrili P laridan 5 lari saxelmwifos `miaqvs~, xolo P-5
lari rCeba firmas. amitom miwodebis funqciaSi P-s magier unda aviRoT P-5.
amis Sedeged miviRebT axali situaciis Sesabamis miwodebis funqcias
1
1
P − 5 =
Q + 25
anu
P = Q + 30 . (9)
2
2
am funqciis Sesabamisi grafiki (nax. 4)-zea agebuli wyvetili xaziT. wonas-
worobis axali 1
P
Q
0 -sa fasisa da wonasworobis
1
0 axali sididis misaRebad
unda amovxsnaT sistema:
⎪
⎧P = −2Q + ,
50
⎨
1
⎪P = Q + .
30
⎩
2
miviRebT 1
P
Q
0 =34,
1
0 =8. eseni wonasworobis axali B wertilis
koordinatebia. (nax. 4). gavaanalizoT miRebuli efeqti, rac mTavrobis
aRniSnulma gadawyvetilebam gamoiwvia. jer erTi moTxovnis wiri ucvleli
154
darCa, xolo mowodebis wirma aiwia zemoT 5 erTeuliT, aman gamoiwvia
wonsworobis fasis gazrda P0=30 - dan 1
P0 =34, amrigad, momxmareblisaTvis
4 lariT gaZvirda, garda amisa, firmam erTeuli produqciis gayidviT
miRebuli 34 laridan, 5 lari mTavrobas unda gadauxados, ris gamoc mas
29 lari, anu 1 lariT naklebi rCeba, vidre mTavrobis gadawyvetilebande
rCeboda. e.i. Catarebuli analizi gviCvenebs, rom mTavrobis gadawyvetilebam
Secvala bazris wonasworobis mdgomareoba. bazris axali wonasworobis
damyareba iwvevs produqciis 4 lariT gaZvirebas da miwodebas 10 dan 8
erTeulamde Semcirebas. garda amisa, 5 lariani gadasaxadi ase nawildeba:
4 lars ixdis momxmarebeli (radgan misTvis 4 lariT moxda gaZvireba),
xolo erT lars firma (radgan firmis Semosavalama produqciis
erTeulze gaangariSebiT daiklo erTi lariT).
zemoT gadawyvetili amocanebi Seeesabameboda erTsaqonlian bazars,
e.i. gansaxilveli gansxvavebuli produqciis
maxasiaTeblebi
(fasi, moTxovna, miwodeba) damokidebulia bazris sxva produqtebisagan.
ganvixiloT SemTxveva, rodesac gvaqvs ori produqti, romlebic
urTierT damoukidebulia, e.i. maTi moTxovnis raodenobebi da fasebi
gavlenas axdenen erTmaneTze e.w. ori saqonliani bazari.
amocana 2. bazarze Semodis ori urTierTdamokidebuli produqti. pir-
veli saxis produqtis moTxovnis da miwodebis funqciebia Sesabamisad
Q = 10 − 2P + P
1
1
2 (10)
Q = 3
− + 2P
1
1 (11)
xolo meore saxis produqtisa ki aris
Q = 5 + 2P − 2P
2
1
2 (12)
Q = −2 + 3P2
2
(13)
P1 da P2 Sesabamisad pirveli da meore saxis produqtis erTeulis fasebia.
ganvsazRvroT am pirobebSi or saqonliani bazris wonasworobis fasebi da
wonaswobis moculobebi.
rogorc viciT wonasworoba niSnavs, rom moTxovna orive produqtze
emTxveva maT mowodebas. radgan (10) tolobaSi Q1 moTxovnaa, (11) tolobaSi
ki miwodeba, maTi gatoleba gvaZlevs
10 − 2P + P = 3
− + 2P
4
anu
P − P = 13.
1
2
1
1
2
analogiurad (12) da (13)-dan miviRebT
155
5 + 2
P − P = − + P,,
anu
P − P = −
1
2 2
2 3 2
2 1 5 2
7.
miRebuli orive gantoleba unda Sesruldes erTdroulad. amitom miviRebT
sistemas:
4P − P = ,
13
1
2
⎩
⎨
⎧2P −5P = −7.
1
2
misi amonaxsnebia P = ,
4
P = 3
1
2
romlebic wonasworobis saZiebeli fasebia,
maSin (11) da (13) tolobebidan miviRebT Q = ,
5
Q = 7
1
2
. isini wonasworobis
saZiebeli moculobebia, Sesabamisad pirveli da meore produqtisaTvis.
davaleba:
I. SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ras ewodeba moTxovnis funqcia da rogori saxe aqvs mas?
2. raze SeiZleba iyos damokidebuli moTxovnis funqcia? rogoria
moTxovnis funqciaSi parametrebis ekonomikuri Sinaarsi?
3. ras ewodeba miwodebis funqcia da rogori saxe aqvs mas? rogoria
miwodebis funqciaSi parametrebis ekonomikuri Sinaarsi?
4. moTxovnisa da miwodebis wirebis daxmarebiT axseniT ras niSnavs
bazarze Warbi produqcia? produqciis deficiti? wonasworoba?
II. praqtikuli savarjiSoebi
100
1. moTxovnis funqcia gamoisaxeba formuliT Q =
,
sadac
P + 6
Q moTxovnis funqciaa, P saqonlis fasia. gamoTvaleT moTxovna,
rodesac P fasi aris: P = 1; P = ,
1 5 ; P = 2 ; P = 2,5. aageT moTxovnis
funqciis grafiki. fasis zrda ra gavlenas axdens moTxovnaze?
400
2. moTxovnis fasis funqcia mocemulia formuliT P =
.
Q + 5
gamoTvaleT Semosavali (amonagebi), romelic miiReba Q = 5
raodenobis gayidviT.
156
3. miwodebis funqcias aqvs saxe Q = 2P , sadac P saqonlis fasia, Q
miwodebis sididea. gamoTvaleT miwodeba, roca P fasi aris: P = 2 ;
P = 4,5; P = 6 . aageT miwodebis funqciis grafiki. fasis zrda ra
gavlenas axdens miwodebaze?
4. mocemulia moTxovnis funqcia Q = −3P + 15 da miwodebis funqcia
Q = 2P + 3. garfikulad warmoadgineT orive funqcia da ipoveT
wonasworobis fasi P
Q
0 da raodenoba
0 .
5. italielma ekonomistma vilfredo paretom gamoikvlia kapitalistur
a
sazogadoebaSi Semosavlebis ganawileba da miiRo formula y =
,
n
x
sadac y aRniSnavs im pirTa raodenobas, romelTac gaaCniaT
Semosavali aranakleb x -isa. x Semosavalia, xolo a da
n
mudmivebia. vTqvaT:
5.1. a = 2 000
n
da
000
000
= ,
1 5 . ipoveT: a) im pirTa raodenoba, romelTa
Semosavali metia an toli 10 000 –is. b) umciresi Semosavali 100
yvelaze mdidar pirovnebas Soris.
5.2 a = 4 000
n
da
000
000
= ,
1 2 . ipoveT: a) im pirTa raodenoba, romelTa
Semosavali metia an tolia 1 000 000 –is. b) umciresi Semosavali
1 000 yvelaze mdidar pirovnebas Soris.
6. sawarmos mTliani danaxarjebis funqciaa C (Q) = 0
− Q
1
,
3 + 300Q
T
, sadac
Q saqonlis raodenobaa, CT - mTliani danaxarjebia fulad
erTeulebSi. gamoTvaleT mTliani da saSualo danaxarjebi, roca
warmoebuli saqonlis raodenoba Q udris: Q = 1; Q = 2 ; Q = 3.
C
miTiTeba: saSualo danaxarjebi gamoiTvleba SefardebiT T .
Q
Tavi V. diferencialuri aRricxvis elementebi da maTi
zogierTi gamoyeneba ekonomikur amocanebSi
$ 1. funqciis warmoebuli. misi geometriuli da
ekonomikuri interpretacia.
157
funqciis SeswavlisaTvis mTavari mniSvneloba eniWeba imas, rom Cven
gvqondes iseTi aparati, romelic daaxasiaTebs funqciis cvalebadobis
siCqares argumentis cvalebadobasTan dakavSirebiT, romelsac didi
mniSvneloba aqvs funqcionaluri damokidebulebis bunebis Sesaswavlad.
1. funqciis warmoebuli. vTqvaT, raime SualedSi mocemulia y = f (x)
funqcia. am SualedSi aviRoT romelime x0 wertili da mivaniWoT mas x
Δ
nazrdi, romelsac argumentis nazrdi ewodeba. vigulisxmoT, rom Δx ≠ 0 da
amasTan, (x + x
Δ
x
0
) wertili Zevs 0 wertilis midamoSi. argumentis x
Δ
nazrdiT Secvlisas funqciac miiRebs Sesabamis nazrds, romelic
aRiniSneba
y
Δ simboloTi da gamoiTvleba formuliT
y
Δ = f (x + x
Δ ) − f (x )
0
0
. funqciis saSualo nazrdi aris
y
Δ
f (x + x
Δ ) − f (x )
=
0
0
(1)
x
Δ
x
Δ
rac axasiaTebs funqciis cvlilebis siCqares, rodesac argumenti icvleba
x
x + x
Δ
0 _ dan ( 0
) _ mde. rac ufro mcirea x
Δ (anu rac ufro axlosaa x
Δ
nulTan), miT ufro zustad aRwers es Sefardeba funqciis cvlilebis
siCqares x0 wertilis uSualo maxloblobaSi. Tu arsebobs sasruli an
y
Δ
usasrulo zRvari lim
, anu rac igivea
x
Δ →0
x
Δ
y
Δ
f (x + x
Δ ) − f (x )
lim
= lim
0
0
(2)
x
Δ →0
x
x
Δ
Δ →0
x
Δ
maSin am zRvars ewodeba y = f (x) funqciis warmoebuli x0 wertilSi da
aRiniSneba y′ an f ′(x )
y′
0
simboloTi. zogjer y′ -s nacvalad weren - x , amiT
xazgasmulia, rom warmoebuli aRebulia x cvladiT.
maSasadame, funqciis warmoebuli ewodeba funqciis nazrdisa da
argumentis nazrdis Sefardebis zRvars, rodesac argumentis nazrdi
nebismierad miiswarafis nulisaken.
Tu (2) zRvari sasrulia, maSin warmoebuls sasruli ewodeba, xolo
Tu igi usasruloa warmoebuls usasrulo ewodeba. amrigad, mocemul
wertilSi warmoebuli warmoadgens ricxvs. Tu raime Sualedis yovel
wertilSi arsebobs sasruli warmoebuli, maSin warmoebuli warmoadgens
158
x –is funqcias mocemul SualedSi. funqciis warmoebulis gamoTvlis
operacias gawarmoeba ewodeba.
funqciis marjvena da marcxena warmoebuli x0 wertilSi ase
ganisazRvreba:
zRvrebs
f (x + x
Δ ) − f (x )
f (x + x
Δ ) − f (x )
lim
0
0
da lim
0
0
x
Δ →0+
x
Δ
x
Δ →0−
x
Δ
ewodeba Sesabamisad marjvena da marcxena warmoebulebi x0 wertilSi. es
warmoebulebi aRiniSneba f ′(x +)
f ′(x −
0
da
)
0
simboloebiT.
advili SesamCnevia, rom f (x) funqciis warmoebadobisaTvis x0
wertilSi aucilebelia da sakmarisi x0 wertilSi marjvena da marcxena
warmoebulebis Tanatoloba.
funqcias ewodeba warmoebadi [a,b] segmentze, Tu igi warmoebadia am
segmentis yvela Siga wertilSi da a wertilSi funqcias aqvs marjvena
warmoebuli, b wertilSi _ marcxena.
2. warmoebadi funqciebi. y = f (x) funqciis warmoebuli x wertilSi
ganisazRvreba, rogorc zRvari
f (x + x
Δ ) − f (x)
f ′(x) = lim
(3)
x
Δ →0
x
Δ
magram, rogorc cnobilia zRvrebi yovelTvis ar arseboben. swored amitom
arc warmoebuli arsebobs yovelTvis.
magaliTis saxiT ganvixiloT Semdegi funqcia f (x) = x . vaCvenoT, rom
am funqciis warmoebuli x=0 wertilSi f ′(0) gansazRvruli ar aris.
marTlac warmoebulis gansazRvris Tanaxmad
f (0 + x
Δ ) − f (0)
0 + x
Δ − 0
x
Δ
f ′(0) = lim
= lim
= lim
(4)
x
Δ →0
x
x
Δ
Δ →0
x
x
Δ
Δ →0
x
Δ
Tu x
Δ ise miiswrafvis nulisaken, rom rCeba dadebiTi, maSin
Δx
x
Δ = x
Δ da lim
= 1.
Δx→0
Δx
( Δx 0)
f
Tu x
Δ ise miiswrafvis nulisaken, rom rCeba uaryofiTi, maSin
Δx
x
Δ = − x
Δ da lim
= 1
− .
Δx→0
Δx
( Δx 0)
p
159
Tu (4) tolobaSi zRvari iarsebebda, maSin igi ar iqneboda
damokidebuli imaze, Tu rogor miiswarfis x
Δ nulisaken. sinamdvileSi ki
es ase ar aris. magram aqedan SesaZlebelia mxolod erTi daskvnis gakeTeba,
rom (4) tolobaSi zRvari ar arsebobs. e.i. f (x) = x funqciisaTvis x=0
wertilSi warmoebuli ar arsebobs.
siZneles ar warmoadgens imis Cveneba, rom yvela danarCen wertilSi
f (x) = x funqciis warmoebuli arsebobs (damtkiceba mkiTxvelisaTvis
migvindia) da udris
⎧ ,
1
Tu
x
0
f
f ′(x) = ⎨
⎩− ,
1
Tu
x
0
f
f (x) funqcias uwodeben warmoebads x wertilSi, Tu arsebobs
sasruli an usasrulo f ′(x) . Tu funqcia warmoebadia, romelime Sualedis
TiToeul wertilSi maSin amboben, rom igi warmoebadia mTels am
SualedSi. magaliTad, zemoT ganxiluli funqcia warmoebadia yovel
SualedSi, romelic ar Seicavs x=0 wertils; y=x funqcia yvelgan
warmoebadia.
aqve SevniSnavT, rom SesaZlebelia funqcia, romelic wyvetilia x=a
wertilSi, ar SeiZleba iyos warmoebadi am wertilSi. e.i. warmoebadi
SeiZleba iyos mxolod da mxolod uwyveti funqciebi. magram ise ar unda
vifiqroT, rom x=a wertilSi yoveli uwyveti funqcia warmoebadia am
wertilSi. magaliTad y = x funqcia uwyvetia x=0 wertilSi. magram
rogorc zemoT vaCveneT warmoebadi ar aris am wertilSi. arsebobs ufro
damajerebeli magaliTebic, rom funqcia SesaZloa yvelgan uwyveti iyos,
magram ar iyos warmoebadi (Cven aseT magaliTebs ar ganxilavT, radgan is
scildeba Cveni programis sazRvrebs).
3. warmoebulis geometriuli mniSvneloba. jer SemoviRoT mxebi
wrfis ganmarteba. Cven aqamde saqme gvqonda mxolod wrewiris mxebTan,
mocemul M wertilSi wrewiris mxebs vuwodebdiT wrfes, romelsac
wrewirTan erTi da mxolod erTi saerTo M wertili hqonda. magram aseTi
gansazRvra yvela mrudisaTvis ar aris marTebuli.
160
N
C
M2
M3 M1
A B
M P
N
M
D
nax. 1 nax. 2
magaliTad, bunebrivia CavTvaloT, rom AB wrfe Seexeba MNP mruds
M wertilSi (nax. 1). Tumca am mrudTan mas ara aqvs erTi, aramed ori
saerTo M da P wertili. CD wrfes, piriqiT, mxolod erTi saerTo
wertili aqvs MNP mrudTan _ N wertili, magram sruliadac ar iqneba
bunebrivi, rom igi CavTvaloT mxebad MNP mrudisadmi.
imisaTvis, rom ganvsazRvroT nebismieri mrudis mxebi mis romelime M
wertilSi (nax. 2). aviRoT am mrudze kidev erTi wertili M1 da gavavloT
mkveTi MM1 .
M
M /
M ///
M // M /
nax. 3
Tu M1 wertils vamoZravebT mocemul mrudze ise, rom igi usazRvrod
miuaxlovdes M wertils, maSin mkveTi ibrunebs M wertilis garSemo da
mimdevrobiT daikavebs MM
MM
MM
1 ,
2 ,
3 da a. S. mdebareobebs. MN mkveTis
zRvruli mdebareoba mogvcems mrudis mxebs M wertilSi.
161
gansazRvreba: mrudis mxebi M wertilSi ewodeba MM1 mkveTis
zRvrul mdebareobas, roca mrudze moZravi M
1
wertili usazRvrod
uaxlovdeba M wertils.
mrudze moZravi M
1
wertili SeiZleba usazRvrod uaxlovdebodes
M wertils sxvadasxva mxridan. magaliTad, nax. 3-ze M ′ wertili
uaxlovdeba M wertils ara zemodan, rogorc es nax. 2-zea naCvenebi,
aramed qvemodan. am SemTxvevaSi saqme gvaqvs sxva mkveTebTan M
M ′ ,
M
M
′,
M
M
′′ da a. S., Tumca maTi zRvruli mdebareoba igive MN mxebia.
y N
M1(x+Δx,y+Δy)
α L
P
M(x,y)
K
β
Δx
x x+Δx x
nax. 4
vTqvaT, nax. 4-ze warmodgenili KL mrudi aris y = f (x) funqciis
grafiki. aviRoT masze ori wertili: erTi M koordinatebiT (x,y) da
M
x + Δx, y + Δy
1 _ (
). gavavloT abscisTa RerZis paraleluri MP monakveTi.
Δ
Δ
x
MM P
MP = x
Δ
M P = Δy
1
-Si
,
.
1
amitom
im α kuTxis tangensis tolia,
y
Δ
romelsac MM
Δx →
1 mkveTi adgens abscisTa RerZTan. roca
0 , M wertili
adgilze rCeva, xolo M1 mrudis gaswvriv usazRvrod uaxlovdeba M –s.
MM1 mkveTi amasobaSi icvlis mimarTulebas. amasTan erTad icvleba α
x
Δ
kuTxec da adgili aqvs tolobas
= tgα . zRvarze gadasvlisas MM
y
Δ
1
qorda daikavebs MN mxebis mdebareobas, Seadgens, ra abscisTa RerZTan
162
x
Δ
romeliRac β kuTxes. cxadia, am dros tgβ = lim tgα . magram tgα =
,
x
Δ →0
y
Δ
x
Δ
amitom tgβ = lim
= y′ .
x
Δ →0
y
Δ
amgvarad, y = f (x) funqciis warmoebuli x wertilSi warmoadgens im
mxebis daxris kuTxis tangenss, romelic gavlebulia funqciis
grafikisadmi x abscisas mqone wertilze.
4. warmoebulis ekonomikuri azri. vTqvaT, Y aris ekonomikuri
sidide (danaxarjebi, amonagebi, mogeba,...), romelic x cvldis Y = f (x)
funqciaa da igi warmoebadia. am funqciis warmoebuls marJinaluri anu
zRvruli funqcia ewodeba da aRiniSneba Y
= f ′(x) .
zR.
magaliTad, danaxarjebis funqciis warmoebuli aris zRvruli
danaxarjebi, amonagebis funqciis warmoebuli aris zRvruli amonagebi da
a. S. ganvixiloT dawvrilebiT martivi ekonomikuri amocana marJinaluri
(zRvruli) amonagebis Sesaxeb.
mTliani amonagebis funqciis (TR) = f (Q) grafiki aris wiri,
romelic gamosaxulia (nax. 5) _ze. naxazidan Cans, rom f (Q) zrdadi
funqciaa ( ;
0 b) SualedSi.
(TR)
f(Q2+h) B
f(Q2)
A
f(Q1+h)
f(Q
2)
0 Q1 Q1+h a Q2 Q2+h b Q
nax. 5
amasTan, igi ufro swrafad izrdeba ( ;
0 a) SualedSi (oA wiri), vidre (a;b)
SualedSi (AB wiri), am funqcias gaaCnia Semdegi ekonomikuri Sinaarsi.
163
Tu SevadarebT erTmaneTs gayiduli saqonlis erTi da imave
raodenobiT zrdas Q
Q +
( ;
0 a
Q
1 raodenobidan
h
1
_mde
) SualedSi da 2
raodenobidan Q + h
(a;b
2
_mde
) SualedSi, davrwmundebiT, rom pirvel
SemTxvevaSi amonagebis cvlileba (namati) saqonlis erTeulze
gaangariSebiT aris
f (Q + h −
1
) f (Q1)
,
h
xolo meore SemTxvevaSi ki
f (Q + h −
2
) f (Q2 )
.
h
naxazidan isic Cans, rom
f (Q + h − f Q
f Q + h − f Q
1
)
( 1)f ( 2 ) ( 2 ),
amitom
f (Q + h −
+
−
1
) f (Q1) f (Q h
2
) f (Q2 )
f
h
h
aqedan ki gamomdinareobs Semdegi: gayiduli produqciis h raodenobiT
gazrdisas mTliani amonagebis saSualo cvlileba (namati) produqciis
erTeulze gaangariSebiT ufro metia ( ;
0 a) intervalis Q1 wertilisaTvis,
vidre (a;b) intervalis Q2 wertilisaTvis. amrigad Sefardeba
f (Q + h) − f (Q)
(5)
h
aRwers mTliani amonagebis `cvlilebis siCqares~ produqciis erTeulze
gaangariSebiT, rodesac gayiduli produqciis raodenoba izrdeba Q –Ddan
Q+h –Dmde. rac ufro mcirea h –ricxvi, miT ufro zustad axasiaTebs
aRniSnuli Sefardeba f(Q) funqciis `cvlilebis siCqares~ Q
-s
maxloblobaSi. idealurad zusti maxasiaTebeli ki iqneba (5)
gamosaxulebis zRvari, rodesac h → 0 .
Tu es zRvari arsebobs, maSin mas rogorc zemoT ganvmarteT
marJinaluri anu zRvruli amonagebi ewodeba da aRiniSneba (MK)
simboloTi.
f Q
( + h) − f Q
( )
(MK ) = lim
(6)
h→0
h
gavecnoT me-(5) da me- (6) saxis gamosaxulebebis geometriul Sinaarss.
164
(TR) T
B
f(Q+h) (TR)=f(Q)
A ϕ
f(Q) C
α
0 Q Q+h Q
nax. 6
vTqvaT, mTliani Semosavlis (TR)=f(Q) funqciis grafiki (nax. 6) _ze
gamosaxuli wiria. maSin, cxadia, rom ΔABC _dan
f (Q + h)− f (Q)
BC
=
= tgϕ (7)
h
AC
amrigad, (7) gamosaxuleba warmoadgens funqciis grafikis A(Q; f (Q)) da
B(Q + h; f (Q + h)) wertilebze gavlebuli mkveTis mier OQ RerZis dadebiT
mimarTulebasTan Sedgenili kuTxis tangenss (anu AB wrfis kuTxur
koeficients). rodesac h → 0 , maSin cxadia, rom B wertili miiswarfvis A
wertilisaken, xolo AB mkveTi wrfe _ AT wrfisaken, romelic
warmoadgens grafikis mxebs A wertilSi. aseve, cxadia, rom am SemTxvevaSi
ϕ = BAC
∠
miiswrafis α = TAC
∠
kuTxisaken. amitom, tgϕ → tgα . SevniSnoT,
rom tgα warmoadgens AT wrfis kuTxur koeficients.
amrigad, Tu arsebobs (6) zRvari, maSin (MK) marJinaluri amonagebi,
argumentis Q mniSvnelobisaTvis, ricxobrivad mTliani amonagebis
funqciis grafikis (Q; f (Q)) wertilze gavlebuli mxebis kuTxuri
koeficientis tolia.
davaleba:
165
SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
10. ras ewodeba funqciis warmoebuli x0 wertilSi?
11. ras ewodeba funqciis sasruli warmoebuli? usasrulo warmoebuli?
marjvena warmoebuli? marcxena warmoebuli? warmoebadi [a,b]
segmentze?
12. raSi mdgomareobs warmoebulis geometriuli Sinaarsi?
13. raSi mdgomareobs warmoebulis ekonomikuri Sinaarsi?
praqtikuli savarjiSoebi:
1. ipoveT y=f(x) funqciis y
Δ = f
Δ (x) nazrdi x0 wertilSi, Tu
a)
2
y = x , x = ,
1
Δ
x = 0 1
,
y = 4x , x = 2, Δ
x = 0
− ,
0
b)
5
0
π
g) y = lg x, x = ,
1
Δ
x = 9
y = sin x, x = 0, Δ
x = −
0
d)
0
6
2. ipoveT y=f(x) funqciis argumentis x
Δ nazrdis Sesabamisi y
Δ nazrdi
x wertilSi, Tu:
y = ax + b; y = ax2 + bx + c; y = a x ; y = ln x; y = cos x; y = sin x
$ 2 gawarmoebis ZiriTadi wesebi. elementarul
166
funqciaTa warmoebulebis cxrili
vTqvaT, u(x) da v(x) warmoebadi funqciebia, xolo C mudmivi ricxvia.
moviyvanoT gawarmoebis ZiriTadi wesebis damtkicebebi:
Teorema 1. ori funqciis namravlis warmoebuli tolia pirveli
funqciis warmoebulisa da meore funqciis namravls plus meore funqciis
warmoebulisa da pirveli funqciis namravli.
damtkiceba. vTqvT, w(x) funqcia ori u(x) da v(x) funqciis namravlia
w(x) = u(x) ⋅ v(x) . CavweroT igive ufro mokled w = u ⋅ v . vTqvaT, u(x) da v(x)
funqciebi warmoebadia. iqneba Tu ara warmoebadi maTi namravli?
gvaqvs w
Δ = w(x + x
Δ ) − w(x) = u(x + x
Δ ) ⋅ v(x + x
Δ ) − u(x)v(x) ,
radgan
u(x + x
Δ ) − u(x) = u
Δ , v(x + x
Δ ) − v(x) = v
Δ
aqedan u(x + x
Δ ) = u + u
Δ , v(x + x
Δ ) = v + v
Δ ,
maSasadame
w
Δ = (u + u
Δ )(v + v
Δ ) − uv = uv + u v
Δ + v u
Δ + u
Δ v
Δ − uv = u v
Δ + v u
Δ + u
Δ v
Δ
w
Δ
v
Δ
u
Δ
u
Δ
amis gamo
= u ⋅
+ v ⋅
+
⋅ v
Δ .
x
Δ
x
Δ
x
Δ
x
Δ
u
Δ
v
Δ
rodesac, Δx → 0 , miviRebT: u → u,
v
→ v
,
→ u′
,
→ v′ .
x
Δ
x
Δ
vaCvenoT, rom Tu Δx → 0 , maSin Δv → 0 . marTlac
Δv
Δv =
⋅ Δx → v′ ⋅ 0 = 0 ,
Δx
amgvarad,
w
Δ
lim
= u ⋅ v′ + u′ ⋅ v + u′ ⋅ 0 = uv′ + u v
′ .
x
Δ →0
x
Δ
maSasadame, gansaxilvel SemTxvevaSi namravlis warmoebuli arsebobs
da adgili aqvs tolobas:
′
(u ⋅ v) = u v
′ + v u
′ (8)
Teorema 2. mudmivi mamravli SeiZleba gavitanoT warmoebulis niSnis
gareT.
167
damtkiceba. ganvixiloT g(x) = af (x) funqcia, sadac a raime ricxvia,
xolo f(x) nebismieri warmoebadi funqcia. vaCvenoT, rom g(x) funqcia
warmoebadia da
g′(x) = a f′ (x) (9)
marTlac,
g(x + x
Δ ) − g(x)
af (x + x
Δ ) − af (x)
f (x + x
Δ ) − f (x)
=
= a ⋅
.
x
Δ
x
Δ
x
Δ
f (x + x
Δ ) − f (x)
vinaidan f(x) nebismieri warmoebadi funqciaa, amitom
x
Δ
Sefardebis zRvari, rodesac Δx → 0 arsebobs da udris f ′(x) . amitom
agreTve arsebobs zRvari
g(x + x
Δ ) − g(x)
lim
x
Δ →0
x
Δ
da udris a f′ (x) . e.i. (9) toloba damtkicebulia.
Teorema 3. Tu u(x) da v(x) funqciebi warmoebadia, maSin maTi jamic
w(x) = u(x) + v(x) warmoebadi iqneba, amasTan Sesruldeba toloba:
w′(x) = u′(x) + v′(x) .
e.i. ori funqciis jamis warmoebuli am funqciebis warmoebulTa jamis
tolia.
damtkiceba. gvaqvs
w
Δ (x) = w(x + x
Δ ) − w(x) = [u(x + x
Δ ) + v(x + x
Δ )]− [u(x) + v(x)] =
= [u(x + x
Δ ) − u(x)]+ [v(x + x
Δ ) − v(x)]
amitom
w
Δ
u(x + x
Δ ) − u(x) v(x + x
Δ ) − v(x)
=
+
x
Δ
x
Δ
x
Δ
vinaidan u(x) da v(x) funqciebi warmoebadia, amitom arsebobs maTi zRvrebi:
u(x + x
Δ ) − u(x)
v(x + x
Δ ) − v(x)
lim
= u′(x) da lim
= v′(x) .
x
Δ →0
x
Δ
x
Δ →0
x
Δ
maSasadame arsebobs zRvaric
w
Δ
lim
= w′(x) .
x
Δ →0
x
Δ
amrigad,
w′(x) = u′(x) + v′(x) (10)
168
miviReT ori funqciis jamis warmoebulis formula. amgvaradve
vRebulobT ori funqciis sxvaobis warmoebulis formulas:
′
[u(x) − v(x)] = u′(x) − v′(x) (11)
Tumca is SeiZleba jamis warmoebulis formulaze daviyvanoT, Tu
u(x) − v(x) gamosaxulebas ganvixilavT, rogorc jams u(x) + (− v(x)) da
gamoviyenebT Teorema 2-s.
SevniSnoT, rom Teorema-3 marTebulia SesakrebTa nebismieri
ricxvisaTvis:
′
(u + u +
+ u
= u′ + u′ +
+ u′
1
2
L
n )
1
2
L
n
vTqvaT, u da v aris x argumentis raime funqciebi da cnobilia am
funqciebis u′
v
da
′ warmoebulebi. ibadeba kiTxva, SeiZleba Tu ara
u
vipovoT Sefardebis warmoebuli im wertilebSi, sadac v noli ar
v
gaxdeba? am amocanis amosaxsnelad SevasruloT Semdegi gardaqmnebi.
u(x + x
Δ ) u(x) u + u
Δ
u
uv + v u
Δ − uv − u v
Δ
v u
Δ − u v
Δ
−
=
− =
=
v(x + x
Δ ) v(x)
v + v
Δ
v
v(v + v
Δ
v2
)
+ v ⋅ v
Δ
amis gamo
u(x + Δx) − u(x)
v(x + Δx)
v(x)
1
⎡ Δu
Δv ⎤
=
.
2
⎢
⋅ v − u ⋅
⎥
Δx
v + v ⋅ Δv ⎣ Δx
Δx ⎦
ramdenadac u da v funqciebi warmoebadi arian, maSin arsebobs zRvrebi:
u
Δ
v
Δ
lim
= u′ da lim
= v′ . garda amisa,
x
Δ →0
x
Δ
x
Δ →0
x
Δ
⎡
v
Δ
⎤
lim [ 2
v + v ⋅ v
Δ ]
2
2
2
= lim v + v ⋅
⋅ x
Δ = v + v
v ′ ⋅ 0 = v
Δ →
Δ → ⎢
⎥
x 0
x 0 ⎣
x
Δ
⎦
amitom arsebobs zRvaric
u(x + Δx) − u(x)
′
v(x + Δx)
v(x)
⎛ u ⎞
lim
= ⎜ ⎟ , romelic
Δx→0
Δx
⎝ v ⎠
u v
′ − v u
′
- is tolia.
2
v
amgvarad Cven davamtkiceT Semdegi
169
Teorema 4. Tu u da v funqciebi warmoebadi arian, maSin im
u
wertilebSi, sadac v gansxvavdeba nulisagan, Sefardeba agreTve
v
warmoebadia da
′
⎛ u ⎞
u v
′ − v u
′
⎜ ⎟ =
2
⎝ v ⎠
v
im funqciebis gasawarmoeblad, romlebsac vxvdebiT praqtikaSi,
sargebloben ubralo, magram mniSvnelovani formulebiT, romelTa zepirad
codna aucilebelia. axla swored gadavdivarT am formulebis gamoyvanaze.
1. mudmivi sidide funqciis kerZo saxea, rodesac argumentis yovel
mniSvnelobas funqciis erTi da igive ricxviTi mniSvneloba Seesabameba.
maSasadame, roca arguments mivcemT nazrds, mudmivi C sidide ucvleli
darCeba. amgvarad, misi nazrdi nulis tolia. nulis Δx –Tan fardoba
yovelTvis nulia rogoric ar unda iyos Δx da amgvarad, zRvrisaken
gadasvlisas nulis mniSvnelobas miviRebT. formulebiT es ase Caiwereba:
vTqvaT, raime X SualedSi mocemulia funqcia y = f (x) = C , sadac C mudmivia.
vipovoT am funqciis warmoebuli. aviRoT X Sualedis Siga x wertili, Δx
nazrdi imdenad mcire SegviZlia aviRoT, rom x+Δx miekuTvnos agreTve X
Sualeds. amitom
Δy
Δy
Δy = f (x + Δx) − f (x) = 0 da maSasadame,
= 0 . aqedan lim
= 0.
Δx
Δx→0 Δx
e.i. X Sualedis yovel Siga x wertilSi f ′(x) = 0 . amrigad, mudmivi
sididis warmoebuli nulis tolia.
2. vipovoT f (x) = x funqciis warmoebuli. gvaqvs: f (x) = x ,
f (x + x
Δ ) = x + x
Δ . amitom, y
Δ = f (x + x
Δ ) − f (x) = (x + x
Δ ) − x = x
Δ , maSasadame,
Δy
Δ
Δ
= x =
y
1. aqedan gamomdinareobs, rom y′ = lim
= lim 1 = 1. e.i. im funqciis
Δx
Δx
Δx→0
Δx→0
Δx
warmoebuli, romelic damoukidebeli cvladis tolia udris erTs.
3. vipovoT
2
f (x) = x funqciis warmoebuli. gvaqvs:
2
f (x) = x ,
2
f (x + x
Δ ) = (x + x
Δ ) . amitom,
2
2
y
Δ = f (x + x
Δ ) − f (x) = (x + x
Δ ) − x = [ 2
2
x + 2x x
Δ + ( x
Δ ) ]
2
2
− x = 2x x
Δ + ( x
Δ ) ,
y
Δ
maSasadame,
= 2x + x
Δ . aqedan gamomdinareobs, rom
x
Δ
170
y
Δ
y′ = lim
= lim (2x + x
Δ ) = lim 2x + lim ( x
Δ ) = 2x + 0 = 2x . e.i.
2
f (x) = x
x
Δ →0
x
x
Δ →0
x
Δ →0
x
Δ →0
Δ
funqciis warmoebuli 2x –is tolia (pirveli ori magaliTisagan
gansxvavebiT, aq warmoebuli x –zea damokidebuli).
4. xarisxovani funqciis warmoebuli. zemoT davamtkiceT, rom (x)′ = 1,
(x2 )′ = 2x . Tu gamoviyenebT Teoremas ori funqciis namravlis warmoebulis
Sesaxeb advilad miviRebT x –is sxva nebismieri sxva naturaluri xarisxis
warmoebuls. magaliTad
( ′
′
′
3
x ) = ( 2
x ⋅ x) = ( 2
x )
2
2
2
x + x (x)′ = 2x ⋅ x + x ⋅1 = 3x ;
( ′
′
′
4
x ) = ( 3
x ⋅ x) = ( 3
x )
3
2
3
3
x + x (x)′ = 3x ⋅ x + x ⋅1 = 4x ;
( ′
′
′
5
x ) = ( 4
x ⋅ x) = ( 4
x )
4
3
4
4
x + x (x)′ = 4x ⋅ x + x ⋅1 = 5x .
imisaTvis, rom vipovoT
n
y = x funqciis warmoebuli, saWiroa n
maCvenebeli koeficientad aviRoT, xolo x –is maCvenebli erTiT davwioT.
′
e.i. ( n
x )
n−1
= n ⋅ x
. xarisxovani funqciis warmoebulis es wesi marTebulia ara
mxolod naturaluri, aramed nebismieri namdvili α maCveneblis
α ′
SemTxvevaSi: (x )
α −1
= α ⋅ x
, (x
0
f ).
5. sinusis da kosinusis warmoebuli. vTqvaT, y = sin x . vipovoT y′ .
mivceT x –s nazrdi x
Δ maSin y + y
Δ = sin(x + x
Δ ). aqedan, y
Δ = sin(x + x
Δ )− sin x
A − B
A + B
da cnobili formulis _ sin A − sin B = 2sin
cos
_is Tanaxmad,
2
2
Δx
Δ
sin
x
⎛
Δx ⎞
Δy
2
⎛
Δx ⎞
miviRebT Δy = 2sin
cos⎜ x +
⎟ , maSasadame
=
cos⎜ x +
⎟ , saidanac
2
⎝
2 ⎠
Δx
Δx
⎝
2 ⎠
2
⎡
x
Δ
⎤
x
Δ
sin
sin
⎢
2
⎛
x
Δ ⎞⎥
2
⎛
x
Δ ⎞
y′ = lim ⎢
cos⎜ x +
⎟⎥ = lim
⋅ lim cos⎜ x +
⎟ = 1⋅ cos x = cos x .
x
Δ →0
x
Δ
x
⎝
2
Δ →0
⎠
x
Δ
x
Δ →0
⎢
⎥
⎝
2 ⎠
⎢
⎣ 2
⎥
⎦
2
amgvarad,
′
(sin x) = cos x
sruliad analogiurad, kosinusebis sxvaobis formulis
A + B
A − B
cos A − cos B = 2
− sin
sin
gamoyenebiT, miviRebT
2
2
171
′
cos x = − sin x
6. tangensisa da kotangensis warmoebulebi. vTqvaT, y = tgx , maSin
sin x
y =
da wiladis gawarmoebis wesis Tanaxmad
cos x
′
⎛ sin x ⎞
cos x cos x − sin x(− sin x)
2
cos x
2
+ sin x
1
y′ = ⎜
⎟ =
=
=
.
⎝ cos x
2
⎠
cos x
2
cos x
2
cos x
e.i.
′
1
(tgx) =
2
cos x
analogiuri msjelobiT miiReba, rom
′
1
(ctgx) = −
2
sin x
7. maCvenebliani funqciis warmoebuli. ganvixiloT y = a x
(a 0,a ≠
f
)1
maCvenebliani funqcia. am funqciis warmoebulis sapovnelad vatarebT
Cveulebriv gardaqmnebs
y
Δ
Δ −1
x a x
x
x+ x
Δ
x
x
y = a , y + y
Δ = a
= a ⋅ aΔ , Δy = x Δx
a a − x
a = x
a ⋅ ( Δx
a − )
1 ,
= a
x
Δ
x
Δ
gadavideT zRvarze, roca Δx → 0 da mxedvelobaSi miviRoT SesaniSnavi
au −1
Δ −1
x
a x
zRvari lim
= ln a , miviRebT y′ = a lim
= a x ln a . e.i.
u→0
u
x
Δ →0
x
Δ
′
(a x ) = a x ln a
roca a=e, maSin lne=1, amitom (e neperis ricxvia)
′
( x
e )
x
= e
8. logariTmuli funqciis warmoebuli. vTqvaT, y = ln x . aqac vatarebT
Cveulebriv gardaqmnebs y + y
Δ = ln(x + x
Δ ),
⎛
x
Δ ⎞
ln⎜1+
⎟
⎛ x + Δx ⎞
⎛
Δx ⎞
y
Δ
⎝
x ⎠
Δy = ln(x + Δx) − ln x = ln⎜
⎟ = ln⎜1+
⎟ ,
=
.
⎝
x
⎠
⎝
x ⎠
x
Δ
x
Δ
miRebul tolobaSi gadavideT zRvarze da mxedvelobaSi miviRoT
a(1+ z)
SesaniSnavi zRvari lim
= 1, gveqneba
z→0
z
172
⎛
x
Δ ⎞
⎡
⎛
x
Δ ⎞⎤
ln⎜1+
⎟
ln⎜1+
⎟
⎝
x
⎢
⎠
1
⎝
x ⎥
⎠
1
y′ = lim
= lim ⎢ ⋅
⎥ = . e.i.
x
Δ →0
x
x
Δ →0
Δ
⎢ x
x
Δ
⎥
x
⎢
x
⎥
⎣
⎦
′
1
(ln x) =
.
x
elementarul funqciaTa warmoebulebis cxrils aqvs Semdegi saxe
′
′
C
1.
=
C
0 =
const;
(
2. x n ) = n ⋅ xn−1 n
, ∈ R
( ′
′
3. a x ) = a x
lna;
(
4.
e x ) = e x ;
(
′
′
5. log
=
=
a
)
1
x
;
(
6.
) 1
lnx
;
xlna
x
′
′
(
7. sinx) =
cosx;
(
8.
cosx) = −sinx
(
′
′
9.
)
1
tgx =
;
(
10.
)
1
ctgx = −
;
cos2 x
sin2 x
(
′
′
11.
)
1
arcsinx =
(
12.
;
)
1
arccosx = −
;
1 − x 2
1 − x 2
(
′
′
13.
)
1
arctgx =
;
(
14.
)
1
arcctgx = −
;
1 + x 2
1 + x 2
davaleba:
SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. moiyvaneT funqciis jamis, sxvaobis, namravlisa da fardobis
warmoebulTa gamosaTvleli formulebi.
2. amowereT ZiriTadi elementaruli funqciebis warmoebulebis cxrili.
praqtikuli savarjiSoebi:
173
1. ipoveT y=f(x) funqciis y
Δ = f
Δ (x) nazrdi x0 wertilSi, Tu
a)
2
y = x , x = ,
1
Δ
x = 0 1
,
y = 4x , x = 2, Δ
x = −0,
0
b)
5
0
π
g) y = lg x, x = ,
1
Δ
x = 9
y = sin x, x = 0, Δ
x = −
0
d)
0
6
2. ipoveT y=f(x) funqciis argumentis x
Δ nazrdis Sesabamisi y
Δ nazrdi
x wertilSi, Tu:
y = ax + b; y = ax2 + bx + c; y = a x ; y = ln x; y = cos x; y = sin x
3. gamoTvaleT Semdegi funqciebis warmoebulebi:
1
1. a) y = 7 b) f ′ 16
( ) , Tu f (x) = x x g)
2
f (x) = x −
2
2x
π
cos x
2. a)
8
y = x b) y (′ ) , Tu y( x) =
g) f (x
x
) = 4 ⋅ cos x
6
1 − sin x
1
4x4 3 x2
4
3. a)
15
y = x b) y = 16 ⋅ ( x − ) g) y =
− + 3x
x
x2
x
arctgx
tgx
4. a)
7
y = 6x b) y(x) =
g) f ( x ) =
2
1 + x
2
1 − x
1
2
1
5. a)
q
y = ax b)
2
f (x) = x −
g)
9
3
2
f (x) =
x −
− x
2
2x
2
9
2x
1− sin x
x
3
6. a)
3
y = 5 x b) f (x) =
g) y =
1 + cos x
ln 3 ⋅ arctgx
4 3
2
4x
x
14. a) y =
b) f ( x) = e x ⋅ log x g) y=x3-9x2+15x-3
2
5
x
5 6
x
1
3
x
2
2
15. a) y = 6x +
−
+ x − 4 b) y( x) = tgx ⋅ ln x g) y =
3
x
x
2
2
1 − x
1 − x
cos x
16. a) y′ )
3
( , Tu y =
b) y( x) =
g) y=x3-6x2+9x-3
1 + x
1 − sin x
1
17. a) y′ )
8
( , Tu
3
y =
x b)
2
5
4
f (x) = x − + 7x + 5x
x
$ 3 funqciis diferenciali. diferencialis umartivesi
174
Tvisebebi. diferencialis geometriuli mniSvneloba.
1. diferencialis gansazRvra. warmoebulis cnebasTan mWidros aris
dakavSirebuli diferencialis cneba (`diferenciali~ laTinuri sityvaa da
niSnavs sxvaobas).
vTqvaT, y = f (x) aris raime funqcia, romelsac garkveul x wertilSi
aqvs f ′(x) warmoebuli, warmoebulis gansazRvrebis Tanaxmad
y
Δ
lim
= f ′(x) = y′ .
x
Δ →0
x
Δ
radgan cvladsa da mis zRvars Soris sxvaoba usasrulod mcirea,
y
Δ
Δy
amitom
− y′ usasrulod mcirea, roca Δx → 0 . davuSvaT, rom
− y′ = α ,
x
Δ
Δx
y
Δ
roca Δx → 0 da
= α + y′ saidanac
x
Δ
y
Δ = α x
Δ + y x
Δ
′ (1)
ρ = α x
Δ sidide ufro maRali rigis usasrulod mcirea, vidre misi
mamravlebi (kerZod, vidre x
Δ ), rogorc ori usasrulod mciris namravli.
amgvarad,
Δy = y Δ
′ x + ρ (2)
Tu y′ ≠ 0 , maSin marjvena nawilSi pirveli Sesakrebi imave rigis
usasrulod mcirea, rogorc x
Δ , xolo meore Sesakrebi ρ ufro maRali
rigisaa. amitom mcire Δx nazrdis SemTxvevaSi meore Sesakrebi naklebi
mniSvnelobisaa, vidre pirveli. am pirvel Sesakrebs (miuxedavad imisa y′ ≠ 0
Tu ara) uwodeben funqciis diferencials, ufro zustad, am cnebas aqvs
aseTi
gansazRvreba: y = f (x) funqciis diferenciali x wertilSi ewodeba
am x wertilSi funqciis y′ = f ′(x) warmoebulisa da argumentis nebismieri
Δx nazrdis namravls. diferenciali aRiniSneba dy
df
an
(x) (ikiTxeba
`de igrek~, `de ef iqs~) simboloTi. e.i.
dy = y x
Δ
′ (3)
175
amgvarad, dy diferenciali damokidebulia or sidideze: x wertilsa
da Δx nazrdze. kerZod, Tu y = f (x) = x , gvaqvs f ′(x) = 1. ris gamoc
bunebrivia, rom dx = x
Δ , amitom
dy = y dx
′ = f ′(x)dx (4)
e.i. funqciis diferenciali tolia funqciis warmoebulisa da
argumentis diferencialis namravlis.
′
vTqvaT, y = sin x, maSin dy = d (sin x) = (sin x) dx da dy = cos xdx . Tu
x
y = e ,
′
maSin dy = (ex ) dx = e xdx da a.S.
(4) tolobidan gvaqvs
dy
f ′(x) = y′ =
(5)
dx
maSasadame, funqciis warmoebuli udris funqciis diferencialisa da
argumentis diferencialis fardobas.
dy
dy
simbolo
ikiTxeba ase: `de igrek de iqsiT~.
-s uwodeben
dx
dx
laibnicis simbolos, xolo y′ -s lagranJis simbolos.
mtkicdeba Semdegi Teoremebi:
Teorema 1: Tu y = f (x) funqcias x wertilSi aqvs nulisagan
gansxvavebuli sasruli warmoebuli, maSin f (x) funqciis y
Δ nazrdi
eqvivalenturia dy = f ′(x) x
Δ diferencialisa, rodesac Δx → 0 .
SeniSvna: Tu f ′(x) = 0 , maSin dy = 0 , magaram y
Δ nazrdi rogorc wesi,
nulisagan gansxvavebulia. maSasadame, y
Δ da dy sidideTa eqvivalenturoba
irRveva, rodesac f ′(x) = 0 .
Tu funqciis nazrdi SeiZleba warmovidginoT (1) formuliT, maSin x
wertilSi y = f (x) funqcias diferencirebadi ewodeba.
Teorema 2: imisaTvis, rom y = f (x) funqcia diferencirebadi iyos x
wertilSi, aucilebelia da sakmarisi, is am wertilSi iyos warmoebadi.
amrigad, erTi cvladis funqciis warmoebadoba da diferencirebadoba
erTmaneTis eqvivalenturia.
Tu y = f (x) funqcia diferencirebadia x wertilSi, maSin igi uwyvetia
amve wertilSi. Sebrunebuli winadadeba samarTliani ar aris.
176
vinaidan Δy = dy + ρ , amitom funqciis dy diferenciali da misi nazrdi
y
Δ erTmaneTisagan gansxvavdeba ρ usasrulod mcireTi, romelic ufro
maRali rigisaa, vidre x
Δ . misi ugulvebelyofiT miviRebT Semdeg
miaxloebiT tolobas y
Δ ~ dy . miRebul tolobas didi gamoyeneba aqvs
miaxloebiT gamoTvlebSi.
Tu u ,v da w warmoebadi funqciebia, maSin adgili aqvs tolobebs:
1. dC = 0 2. d (u + v + w) = du + dv + dw
3. d (uv) = udv + vdu 4. d (Cu ) = Cdu
⎛ u ⎞ udv − vdu
5. d ⎜ ⎟ =
6. df (u) = f ′(u)du, u
= ϕ (x)
2
⎝ v ⎠
dv
2. funqciis diferencialis geometriuli mniSvneloba. vTqvaT
y = f (x) funqcia diferencirebadia x wertilSi. gamovarkvioT ras
warmoadgens geometriulad mocemuli funqciis diferenciali. avagoT
y = f (x) wiri (nax. 1).
y N
Q
M P
α
Δx
0 a b x
nax. 1
naxazidan Cans, rom
y = aM ,
x
Δ = ab = MP , y
Δ = bN − bP = PN, f ′(x) = tgα .
funqciis diferenciali ase warmogvidgeba
dy = f ′(x) x
Δ = tgα ⋅ MP .
177
magram MPQ samkuTxedidan gvaqvs tgα ⋅ MP = PQ , maSasadame dy = PQ . amrigad
y = f (x) funqciis diferenciali geometriulad warmoadgens mxebis
wertilis ordinatis nazrds.
magaliTad, vipovoT f (x) = 3 3
x − 5 2
x +15 funqciis diferenciali:
′
dy = (3x3 − 5x2 + 15) dx = (3x2 − 5)dx
davaleba:
SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ras ewodeba funqciis diferenciali?
2. CamoayalibeT funqciis diferencialis geometriuli arsi.
praqtikuli savarjiSoebi:
1. ipoveT Semdegi funqciebis diferenciali:
arctgx
tgx
1.
7
y = 6x 2. y(x) =
3. f ( x) =
2
1 + x
2
1 − x
1
2
1
4.
q
y = ax 5.
2
f (x) = x −
6.
9
3
2
f (x) =
x −
− x
2
2x
2
9
2x
1− sin x
7.
3
y = 5 x 8. f (x) =
1 + cos x
178
$4. Seqceuli funqciis warmoebuli. rTuli funqciis
warmoebuli. maRali rigis warmoebulebi. laibnicis formula.
1. Seqceuli funqciis warmoebuli. vTqvaT, mocemulia y = f (x)
dy
funqciis
warmoebuli da saWiroa amis mixedviT Sebrunebuli x = ϕ( y)
dx
funqciis warmoebulis gamoTvla. vinaidan am ukanasknel SemTxvevaSi
damoukidebeli cvladis rols y asrulebs, amitom x funqciis warmoebuli
dx
unda iyos
; magram rogorc Cveulebrivi wiladi, Cven SegviZlia es
dy
warmoebuli ase warmovadginoT:
dx
1
=
(1)
dy
dy
dx
dy
amgvarad, Tu cnobilia pirdapiri funqciis
warmoebuli, maSin
dx
dx
Sebrunebuli funqciis
warmoebuli gamoiTvleba (1) formuliT.
dy
mtkicdeba Semdegi
Teorema: vTqvaT, [a; b] segmentze arsebiTad zrdadi da uwyveti
y = f (x) funqcia warmoebadia segmentis Siga x
x = ϕ( y
0 wertilSi. Tu
) aris
f (x) funqciis Seqceuli funqcia, maSin igi warmoebadia Sesabamis y0
wertilSi da marTebulia toloba
1
ϕ′( y ) =
1
( ′
0
)
f ′(x )
0
1. arcsinx da arccosx funqciebis gawarmoeba: vTqvaT,
y = arcsin x , maSin x = sin y .
(1) –is Tanaxmad
179
dy
1
1
1
=
=
=
;
dx
dx
d sin y
cos y
dy
dy
⎛ π π ⎞
vinaidan, funqciis Sebrunebis cnebis Tanaxmad, y icvleba ⎜ −
, ⎟
⎝ 2 2 ⎠
SualedSi, amitom cos y
0
f da maSasadame, cos y
2
= 1− sin y . xolo sin y ,
Tanaxmad aRebuli gantolebisa, aris x, amitom:
dy
1
1
=
=
;
2
2
dx
1 − sin y
1 − x
maSasadame,
d arcsin x
1
=
2
dx
1 − x
dx
da d (arcsin x) =
.
2
1 − x
axla vTqvaT,
y = arccos x , maSin x = cos y .
(1) –is Tanaxmad
dy
1
1
1
=
=
= −
;
dx
dx
− sin y
sin y
dy
vinaidan, funqciis Sebrunebis cnebis Tanaxmad, y icvleba (0,π ) SualedSi,
amitom sin y
0
f da maSasadame, sin y
2
= 1− cos y . xolo cos y , Tanaxmad
aRebuli gantolebisa, aris x, amitom:
d arccos x
1
= −
2
dx
1 − x
dx
da d (arccos x) = −
.
2
1− x
2. arctgx da arcctgx funqciebis gawarmoeba: vTqvaT,
y = arctgx , maSin x = tgy .
maSin, cxadia
dy
1
1
=
=
;
dx
dx
1
dy
2
cos y
magram
180
2
1
cos y
2
+ sin y
=
= 1+ tg 2 y
2
cos y
2
cos y
xolo tgy , Tanaxmad aRebuli gantolebisa, aris x, amitom:
darctgx
1
=
2
dx
1 + x
dx
da d (arctgx) =
.
2
1 + x
axla vTqvaT,
y = arcctgx , maSin x = ctgy .
Tanaxmad zogadi wesebisa:
dy
1
1
=
=
;
dx
dx
1
−
dy
2
sin y
magram
2
1
sin y
2
+ cos y
=
= 1+ ctg 2 y
2
sin y
2
sin y
vinaidan tgy , Tanaxmad aRebuli gantolebisa, aris x, amitom:
darcctgx
1
= −
2
dx
1 + x
dx
da d (arctgx) = −
.
2
1+ x
2. rTuli funqciis warmoebuli. vTqvaT, mocemulia funqcia f (u) ,
sadac u Tavis mxriv aris x-is funqcia: u = g(x) . maSin, rogorc CvenTvis
cnobilia, f (g(x))funqcias ewodeba x argumentis rTuli funqcia. Tu
argumentis raime x mniSvnelobebisaTvis u = g(x) funqcia warmoebadia, xolo
u-s saTanado mniSvnelobisaTvis warmoebadia f (u) funqciac, maSin arsebobs
F (x) = f (g(x)) rTuli funqciis warmoebuli da adgili aqvs tolobas.
′
F (x) = [ f (g(x))] = f ′(u)
⋅ g′(x) = f ′(g(x))⋅ g′(x)
u=g ( x)
(2)
mtkicdeba Semdegi
Teorema: vTqvaT, y = f (u) , sadac u = g(x) . Tu argumentis raime x
mniSvnelobebisaTvis u = g(x) funqcia diferencirebadia, xolo u-s saTanado
mnisvnelobisaTvis y = f (u) funqciac diferencirebadia, maSin arsebobs
rTuli y = f [g (x)] funqciis warmoebuli da marTebulia toloba
181
dy
dy du
=
⋅
(2* )
dx
du dx
e.i. rTuli funqciis warmoebuli udris mocemuli funqciis warmoebuls
damxmare cvladiT, gamravlebuls damxmare cvladis warmoebulze
damoukidebeli cvladiT.
SeniSvna 1: (2) formula SeiZleba gavavrceloT im SemTxvevazec,
rodesac funqcia warmodgenilia ara ori, aramed ramdenime funqciis
erTobliobis saSualebiT. magaliTad, Tu
y = f (u) , u = ϕ (v) , v = ψ (x)
maSin
dy
dy du dv
=
⋅
⋅
(
dx
du dv dx
*)
rTuli funqciisaTvis ZiriTad elementarul funqciaTa
warmoebulebis cxrili ase gadaiwereba
( ′
1 un ) = n ⋅ un−1 ⋅ u′ ,
n
∈ R
( ′
′
2. a u ) = aulna ⋅ u′
;
(
3. e u ) = eu ⋅ u ;′
(
′
′
4. log
=
⋅ ′
= ⋅ ′
a
)
1
u
u
;
(
5.
) 1
lnu
u ;
ulna
u
(
′
′
6. sinu) = cosu ⋅ u′
;
(
7.
cosu) = −sinu ⋅ u′
(
′
′
8.
)
1
tgu =
⋅ u′
;
(
9.
)
1
ctgu = −
⋅ u ;′
cos2u
sin2u
(
′
′
10.
)
1
arcsinu =
⋅ u′
(
11.
;
)
1
arccosu = −
⋅ u′ ;
1 − u2
1 − u2
(
′
′
12.
)
1
arctgu =
⋅ u′
;
(
13.
)
1
arcctgu = −
⋅ u ;′
1 + u2
1 + u2
magaliTi 1. vipovoT f (x)
3
= x +1 funqciis warmoebuli.
amoxsna: SemoviRoT aRniSvnebi:
f (u) = u da u = g(x)
3
= x +1,
maSin (2) formulis gamoyenebiT miviRebT
′
′
x
3
1
2
3 2
F (x) = f ′(u)
⋅ g′(x) =
u=
( u)
⋅ (x + )
1 =
⋅ 3x =
g ( x)
3
u=x +
2
3
1
u u=x 1
+
2
3
x +1
magaliTi 2. vipovoT y = cos3 1
(
4
+ x ) funqciis warmoebuli.
182
amoxsna:
3
y = u , u = cos v ,
4
v = 1+ x
rTuli funqciis gawarmoebis (*) wesis Tanaxmad
dy
du
dv
2
3
= 3u ,
= −sin v,
= 4x
du
dv
dx
maSasadame,
dy
2
= 3u ⋅ (− sin v)
3
2
4
4
3
⋅ 4x = 12
− cos x 1
( + x )
1
sin( + x )x .
dx
SeniSvna 2: warmoebulis gamoTvlaSi ZiriTadi roli miekuTvneba
rTuli funqciis gawarmoebis wess. rTuli funqciis gasawarmoeblad
Semogvaqvs damxmare cvladebi. magaliTad,
y = tg5x
rTuli funqcia u da v damxmare cvladebis saSualebiT ase warmovadginoT:
1
2
y = u , u = tgv , v = 5x
rac saSualebas gvaZlevs gamoviyenoT rTuli funqciis gawarmoebis (*)
wesi. studentma unda miaRwios am wesis gamoyenebis srul gagebas. amis
Semdeg damxmare cvladebis SemoReba saWiro ar aris da am cvladebis
SemouReblad unda SevZloT funqciis warmoebulis moZebna. axla y = tg5x
funqcia ase gavawarmooT:
′
dy
1
⎡
⎤
1
1 1
−
′
1
1
′
5
= (tg5x)2 = (tg5x)2 ⋅ (tg5x) =
⋅
(5x) =
dx
⎢
⎥
⎣
⎦
2
2 tg5x cos2 5x
2 tg5x ⋅ cos2 5x
3. maRali rigis warmoebuli. cxadia, y=f(x) funqciis f (
′ x)
warmoebuli isev x cvladis funqciaa. Tu f(x) funqcias aqvs warmoebuli,
maSin am warmoebuls ewodeba f(x) funqciis meore rigis warmoebuli da
aRiniSneba f ′(x
an
)
y ′ simboloTi. Tu meore rigis warmoebuls gaaCnia
warmoebuli, maSin mas uwodeben mesame rigis warmoebuls da aRniSnaven
f ′′(x
an
)
y ′′ simboloTi. sazogadod, (n_1) rigis warmoebulis warmoebuls
ewodeba n-uri rigis warmoebuli da aRiniSneba n
n
f (x
an
)
y simboloTi.
SevniSnoT, rom meoTxe rigis warmoebulidan dawyebuli miRebuli
aRniSvnebi (4)
( 4)
y
= f
(x),
(5)
(5)
y
= f
(x), , (n)
( n)
y
= f
(x)
K
. frCxilebi iwereba
imisaTvis, rom ganvasxvaoT (n)
y warmoebuli n
y xarisxisagan.
183
Tu y = f (x) da x
Δ = dx x argumentis nazrdia, maSin (aq ( )n
dx nacvlad
weren
n
dx ) (n
y ) (dx)n namravls ewodeba y = f (x) funqciis n rigis
diferenciali da aRniSneba d n y
d n
an
f (x) simboloTi
n
( n)
n
d y = y dx (3)
n
( )
d y
saidanac n
n
f (x) = y
=
.
n
dx
magaliTad, vipovoT f (x) = 3 3
x + 2 2
x − 3x +1 funqciis mesame rigis
warmoebuli. TanamimdevrobiTi gawarmoebiT miviRebT
f (
′ x) = 9 2
x + 4x −
,
3
f (
′ x) = 18x +
,
4
f (
′′ x) =
18 .
4. rTuli funqciis umaRlesi rigis diferenciali. aviRoT y = f (u)
funqcia, sadac u damoukidebeli cvladia. maSin
2
2
d y = f ′(u)du (4)
axla vTqvaT, rom u aris x –is funqcia u = ϕ(x) , maSin du = ϕ′(x)dx .
movZebnoT mocemuli rTuli funqciis meore rigis diferenciali x
cvladiT. gvaqvs:
d 2 y = d (dy) = d [ f (u)du] = [ f ′(u)du]du + f ′(u)d (du) = f ′(u)du 2 + f ′(u)d 2u , e.i.
d 2 y = f ′(u)du 2 + f ′(u)d 2u (5)
aqedan Cans, rom ganxilul or SemTxvevas Soris gansxvaveba aris.
roca u aris damoukidebeli cvaldi, maSin gvaqvs (4) toloba, Tuki u
damokidebulia x-ze, gveqneba (5) toloba.
5. laibnicis formula. cxadia, algebruli jamis gawarmoebis wesi
ucvlelad gadaitaneba nebismieri rigis warmoebulebze, magram
sayuradReboa ori funqciis namravlis mimdevrobiTi gawarmoeba. mtkicdeba
Semdegi Teorema:
Teorema: Tu raime X SualedSi u = u(x)
v
da
= v(x) funqciebs aqvs
yvela rigis warmoebuli n rigamde CaTvliT, maSin y = uv namravlsac aqvs
yvela rigis warmoebuli n rigamde CaTvliT da marTebulia toloba
−
n n −
−
n n −
n − k −
n
n
n
(
)
1
n
(
)
1
(
)
1
L
( )
( )
(
1)
(
2)
( n−k ) ( k )
( n)
y
= u v + nu
v′ +
u
v ′ +
+
u
v
+
+ uv
L
L
(6)
!
2
k!
(6) formulas ewodeba laibnicis formula.
magaliTi 3. gamovTvaloT y = x sin ax n –uri rigis warmoebuli.
184
amoxsna: laibnicis formulaSi aviRoT u = sin ax
v
da
= x . maSin
π
n
n
n
( )
⎛
⎞
u
= a sin⎜ ax +
⎟ , v′ = 1,
( n)
v ′ = v ′′ =
= v
= 0
L
. amitom (6) formulis Tanaxmad
⎝
2 ⎠
⎡
n
n
nπ
n
π
( )
1
⎛
⎞
⎛
( − )
1
⎞⎤
y
= −
a
⎢ax sin⎜ ax +
⎟ + nsin⎜ ax +
⎟⎥ .
⎣
⎝
2 ⎠
⎝
2
⎠⎦
davaleba:
SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. rogor gamoiTvleba Seqceuli funqciis warmoebuli?
2. rogor gamoiTvleba rTuli funqciis warmoebuli?
3. amowereT rTuli funqciis warmoebulis formula.
4. ras ewodeba funqciis me-2 rigis warmoebuli? me-3 rigis
warmoebuli? n –uri rigis warmoebuli?
5. ras ewodeba funqciis n –uri rigis diferenciali?
6. rogor gamoiTvleba rTuli funqciis umaRlesi rigis
diferenciali?
7. amowereT (zepirad) laibnicis formula.
praqtikuli savarjiSoebi:
I. gamoTvaleT Semdegi funqciebis warmoebulebi:
2x + 3
2
1. y
2
= ln sin
+ sin x 2.
2
2 2
y = (a − x ) 3. y = ln(ln tgx ) − arcsin(
x
e
)
x +1
1
x
1 cos x
4.
4
y = 1
( + tgx) 5. y = x ln x − x 6. y = ln tg
−
2
2
2
2 sin x
x
tg 2
1
7. y = ln
3
2
2
2
y = tg x +
y = 1 + x arctgx − ln( x + x + 1
1 + sin
x 8.
tgx 9.
)
3
x
2
ctg x
10. y = arctg
11. y = x ln( 2
x
x + )
1 − 2 x + 2arctg
12. y
2
= 3
a
2
185
2
x + 1 + x
1
x
1
x − a
13. y = ln
+ ln(1 + 2 x ) 14. y = arctg 15. y =
ln
2
x + 1 − x
a
a
2a
x + a
16. y = ln( x 2 +
x 4 + 1) + sin 3 5x cos 5 3x 17.
2
y = 1 + x
x 2 − 1
arctg
1 2x(3sin3x+2cos3x)
x
1 x 2
x
y =
e
+ arctg
18. y =
−
5
+ e
19.
2
13
1− x
3x + 2
3
20. y = x ln x +
1
ln( − x) 21. y =
x − 1
2
+ arctg x − 1
4 2
x
4
arctg x
x −
2
2
1
22.
− x
y = arcsin 2x + a 23. y = 1
( + tg x)e
arcsin
2
x + 1
2
24.
5 arctgx
y =
25. y = ecos x sin x
x
− x
e − e
ln cos arctg
6
2
y =
2
5
−
−
+
x
x
x
2
26.
1
3
x 3 + ln x
27.
2
y = 3
ln
sin 2 x
5
x + 2
x
x
+
ln
tg
5
4
1
x4 1
+
2
29. y = ln cos x 30. y = e
arctg ln
3
x
x2
sin
3
x − 2 cos x
31. y = ln
32. y =
x
x + 14 − 4
II. ipoveT:
πx
33. f ′ )
1
( , Tu f (x
x
) = 7 ⋅ ln x 34. y (′ )
2 , Tu y = −3 sin
4
1 − x
π
35. y′ )
3
( , Tu y =
36. y (
′ ) , Tu y = ln sin x
1 + x
4
37. y′ )
8
( , Tu
3
y =
x
III. gamoTvaleT Semdegi funqciebis II rigis warmoebulebi:
5 6
x
1
3
x
2
2
38. y = 6x +
−
+ x − 4 39 y( x) = tgx ⋅ ln x 40. y =
3
x
x
2
2
1 − x
1 − x
cos x
41. y′ )
3
( , Tu y =
42. y( x) =
43. y=x3-6x2+9x-3
1 + x
1 − sin x
1
44. y′ )
8
( , Tu
3
y =
x 45.
2
5
4
f (x) = x − + 7x + 5x
x
arctgx
46. f ′ )
1
( , Tu f (x
x
) = 7 ⋅ ln x 47. y =
48. y=2x3-15x2+36x
2
1 + x
186
$ 5. warmoebulis umartivesi gamoyenebani
gawarmoebis ZiriTadi wesebis gacnobis Semdeg SesaZlebloba gvaqvs
davamtkicoT diferencialuri aRricxvis ZiriTadi Teoremebi, romelTac
didi gamoyeneba aqvT. SemovitanoT Semdegi ganmartebebi:
f (x) funqcias aqvs lokaluri maqsimumi x0 wertilSi, Tu arsebobs
x
x ≠ x
0 wertilis iseTi midamo, rom yoveli
0 -saTvis am midamodan
sruldeba utoloba
f (x )
f (x)
0
f
.
xolo, x0 wertils ewodeba lokaluri maqsimumis wertili.
analogiurad, ityvian, rom f (x) funqcias aqvs fardobiTi minimumi
x
x
x ≠ x
0 wertilSi, Tu arsebobs
0 wertilis iseTi midamo, rom yoveli
0 -
saTvis am midamodan sruldeba utoloba
f (x )
f (x)
0
p
.
xolo, x0 wertils ewodeba lokaluri minimumis wertili.
maqsimumisa da minimumis wertilebs uwodeben eqstremumis wertilebs,
xolo funqciis mniSvnelobebs am wertilebSi – funqciis eqstremumebs.
1. fermas Teorema (eqstremumis aucilebeli piroba):Tu f (x)
funqcia warmoebadia [a;b] SualedSi da amasTanave, Sualedis romelime
Siga ξ wertilze igi aRwevs Tavis udides an umcires mniSvnelobas, maSin
aRniSnul wertilze funqciis warmoebuli nulis tolia. e.i.
f ′(ξ ) = 0 (1)
damtkiceba: marTlac, simartivisaTvis SemoviRoT, rom f (ξ )
warmoadgens funqciis udides mniSvnelobas [a;b] SualedSi. vinaidan ξ
Sualedis Siga wertilia, amitom sakmaod mcire h sididisaTvis adgili
eqneba pirobas:
f (ξ + h) − f (ξ ) ≤ 0 .
e.i. rogoric ar unda iyos mcire h sidide, f (ξ + h) − f (ξ ) sxvaoba
aradadebiTia. maSasadame,
f (ξ + h) − f (ξ )
≤ 0 , roca h 0
f (2)
h
187
da
f (ξ + h) − f (ξ )
≥ 0 , roca h 0
p (3)
h
axla, radgan funqcia SualedSi warmoebadia, amitom Tu gadavalT zRvarze,
rodesac h → 0 , miviRebT:
f (ξ + h) − f (ξ )
f (ξ + h) − f (ξ )
f ′(ξ ) = lim
≤ 0 da f ′(ξ ) = lim
≥ 0
h→0
h
h→0
h
es ki moxdeba mxolod maSin, roca f ′(ξ ) = 0 . e.i.
2. rolis Teorema: Tu f (x) funqcia uwyvetia daxurul [a;b]
SualedSi, amasTan warmoebadia SigniT da garda amisa, f (a) = f (b) = 0 , maSin
am Sualedis SigniT moiZebneba x -is erTi mainc iseTi ξ mniSvneloba, sadac
f (x) funqciis warmoebulis mniSvneloba iqneba noli. e.i.
f ′(ξ ) = 0 (4)
damtkiceba: Tu f (x) funqcia mudmiv mniSvnelobas inarCunebs [a;b]
SualedSi, maSin misi warmoebuli Sualedis yovel wertilze iqneba nuli,
e.i. f ′(x) = 0 da rolis Teorema TavisTavad cxadia. axla Tu f (x) funqcia
[a;b] SualedSi ar aris mudmivi, maSin funqciis an M udidesi an m
umciresi mniSvneloba uTuod unda gansxvavdebodes f (a) = f (b)
mniSvnelobisagan. maSasadame, f (x) funqcia Sualedis romelime Siga ξ
wertilze miaRwevs an M udides an m umcires mniSvnelobas, magram
fermas Teoremis Tanaxmad, aRniSnul ξ wertilze f ′(ξ ) = 0 . e.i.
geometriulad rolis Teorema ase gamoiTqmeba: Tu funqcia
warmoebadia [a;b] SualedSi da saTanado grafikis ordinatebi a da b
wertilebze tolia, maSin [a;b] Sualedis Sig iarsebebda erTi mainc iseTi
ξ wertili, sadac mrudis mxebi Ox RerZis paraleluria.
3. koSis Teorema: Tu f (x) da ϕ(x) ori uwyveti funqciaa daxurul
[a;b] SualedSi da warmoebadi yvela Siga wertilze, garda amisa, ϕ′(x)
warmoebuli Sualedis Sig arsad ar xdeba nuli, maSin adgili aqvs Semdeg
tolobas:
188
f (b) − f (a)
f ′(ξ )
=
ϕ
(5)
(b) − ϕ(a)
ϕ′(ξ )
sadac, ξ erT-erTi mniSvnelobaa [a;b]-s SigniT.
damtkiceba: marTlac, ganvixiloT damxmare funqcia:
F (x) ≡ f (x) − f (a) − P[ϕ (x) − ϕ (a)],
sadac, P jerjerobiT ucnobi mudmivia. es funqciebi warmoebadia da, garda
amisa, F (a) = 0 ; axla SevarCioT P mudmivi ise, rom adgili hqondes agreTve
pirobas: F (b) = 0 ; e.i.
f (b) − f (a) − P[ϕ (b) − ϕ (a)] = 0
saidanac
f (b) − f (a)
P = ϕ
(6)
(b) − ϕ(a)
radgan F (x) funqcia warmoebadia da akmayofilebs pirobebs:
F (a) = 0 da F (b) = 0 ,
amitom rolis Teoremis ZaliT SegviZlia vTqvaT, rom [a;b] Sualedis
SigniT moiZebneba erTi mainc iseTi ξ mniSvneloba, sadac F ′(ξ ) = 0 ; Tu
gamovTvliT F ′(x) warmoebuls, miviRebT
F ′(x) ≡ f ′(x) − Pϕ′(x),
da, maSasadame, unda arsebobdes iseTi ξ , rom
f ′(ξ ) − ϕ
P ′(ξ ) = 0,
′(ξ )
saidanac,
= f
P
ϕ′
. Tu am tolobaSi CavsvamT (6)-s, gveqneba
(ξ )
f (b) − f (a)
f ′(ξ )
=
ϕ
(b) − ϕ(a)
ϕ′(ξ )
sadac a
b
p ξ p .
SeniSvna: is garemoeba, rom ξ moTavsebulia [a;b] Sualedis SigniT,
SeiZleba agreTve Caiweros ase: ξ = a + Θ(b − a) , sadac Θ erTze naklebi
dadebiTi ricxvia (marTlac, rodesac Θ = 0 , maSin cxadia, ξ = a ; xolo Tu
Θ = 1, maSin ξ = b ; yvela sxva SemTxvevaSi ξ moTavsdeba a -sa da b –s Soris).
e.i. 0
1
p Θ p .
189
4. lagranJis Teorema: Tu f (x) funqcia uwyvetia [a;b] SualedSi
da warmoebadia yvela Siga wertilze, maSin adgili aqvs tolobas
f (b) − f (a) = (b − a) f ′(ξ ) , (7)
sadac a
b
p ξ p .
damtkiceba: marTlac, koSis TeoremaSi miviRoT ϕ(x) ≡ x , maSin
ϕ′(x) = 1. maSasadame, ϕ′(x) ≠ 0 piroba [a;b] SualedSi Sesrulebuli iqneba da
Cven gveqneba:
f (b) − f (a)
f ′(ξ )
=
, a
b
p ξ p
b − a
1
aqedan, f (b) − f (a) = (b − a) f ′(ξ ) , a
b
p ξ p . e.i.
Sedegi 1: vuCvenoT, rom Tu [a;b] segmentis yovel wertilze igivurad
f ′(x) ≡ 0 , maSin f (x) funqcia aris mudmivi. marTlac, aseT SemTxvevaSi
Sualedis yoveli ori x
da
x
1
2 mniSvnelobisaTvis gveqneba
f (x ) − f (x ) = x −x f ′ ξ =
2
1
( 2 1) ( ) 0
aqedan f (x ) = f (x
2
1 ) ,
rac amtkicebs SualedSi f (x) funqciis mudmivobas. e.i.
Sedegi 2: Tu [a;b] segmentis yovel wertilze igivurad f ′(x) ≡ ϕ′(x) ,
maSin
f (x) da ϕ(x) funqciebi erTmaneTisagan mxolod mudmiviT
gansxvavdeba. marTlac, Tu avRniSnavT F (x) ≡ f (x) − ϕ(x) , maSin, cxadia, F ′(x)
igivurad nuli iqneba segmentis yovel wertilze, da wina Sedegis ZaliT
miviRebT F (x) ≡ f (x) − ϕ(x) ≡ const , saidanac f (x) = ϕ (x) + const .
lagranJis Teoremas martivi geometriuli interpretacia aqvs.
marTlac, ganvixiloT mrudi:
y = f (x) .
aviRoT masze ori M da N wertili, romlebic a da b abscisebs
Seesabameba. MNP samkuTxedidan miviRebT (nax.1)
NP
= tgNMP ,
MP
190
f (b) − f (a)
e.i.
= tgNMP (8)
b − a
lagranJis Teoremis Tanaxmad, me- (8) toloba f ′(ξ ) -is toli unda
iyos. magram es [a;b] SualedSi mrudis erT-erT wertilze gavlebuli
mxebis sakuTxo koeficientia. maSasadame, lagranJis Teoremas is
geometriuli Sinaarsi aqvs, rom [a;b] SualedSi moiZebneba erTi mainc iseTi
Q wertili y = f (x) mrudze, sadac mxebi MN qordis paraleluria.
y
Q N
M P
f(a) f(b)
0 a ξ b x
nax. 1
dasasrul, SevniSnoT, rom vinaidan me-(7) formulaSi Sedis f (b) − f (a)
sxvaoba, romelic f (x) funqciis nazrds (sasrulo) warmoadgens, amitom
lagranJis Teoremas xSirad sasrulo nazrdis Teoremas uwodeben.
5. funqciis zrdadoba da klebadoba. warmoebulis saSualebiT
advilia nebismieri warmoebadi funqciis zrdadobisa da klebadobis
Sualedebis moZebna. mtkicdeba Semdegi
Teorema 5: Tu f (x) funqciis f ′(x) warmoebuli dadebiTia [a;b]
SualedSi, maSin f (x) funqcia monotonurad zrdadia am SualedSi, xolo
Tu warmoebuli uaryofiTia am SualedSi, maSin f (x) funqcia monotonurad
klebadi am SualedSi.
moviyvanoT am Teoremis geometriuli interpretacia: y = f (x) funqciis
warmoebuli, roca x = x0 , im mxebis kuTxuri koeficientis tolia, romelic
gavlebulia y = f (x) funqciis grafikis im wertilze, romlis abscisa x0 -ia.
piroba f ′(x)
0
f aRniSnavs, rom gansaxilvel SualedSi mxebTa kuTxuri
191
koeficientebi dadebiTia. magram es SesaZlebelia mxolod imSemTxvevaSi,
roca mxebTa mier Ox RerZis dadebiT mimarTulebasTan Sedgenili kuTxeebi
maxvilia (nax. 2). maSin y = f (x) funqciis grafiki x –is zrdisas sul
maRla miemarTeba. es ki niSnavs, rom funqcia y = f (x) monotonurad
izrdeba.
y y
0 x 0 x
nax. 2 nax. 3
y
y
max max
y=x2-4x+3
min min
0 1 2 3 x
0 x1 x2 x3 x4 x
-1
nax. 4 nax. 5
SemTxveva, roca [a,b] SualedSi f ′(x)
0
p , analogiurad ganixileba.
piroba f ′(x)
0
p niSnavs, rom y = f (x) funqciis grafikisadmi gavlebul
mxebTa kuTxuri koeficientebi uaryofiTia. magram es SesaZlebelia im
SemTxvevaSi, roca mxebTa mier Ox RerZis dadebiT mimarTulebasTan
Sedgenili kuTxeebi blagvia (nax. 3). maSin y = f (x) funqciis grafiki x –is
192
zrdisas sul dabla da dabla miemarTeba. es ki niSnavs, rom funqcia
y = f (x) monotonurad klebulobs.
magaliTad, ganvsazRvroT f (x)
2
= x − 4x + 3 funqciis zrdadobisa da
klebadobis Sualedebi. gvaqvs: f ′(x) = 2x − 4 . rodesac x
2
f , f ′(x) 0
f , xolo
rodesac x
2
p , f ′(x) 0
p . maSasadame f (x)
2
= x − 4x + 3 funqcia zrdadia, roca
x
2
f da klebadia, roca x 2
p (ix. nax. 4).
6. warmoebulis gamoyeneba funqciis lokaluri eqstremumis
sapovnelad. funqcias raime SualedSi SeiZleba hqondes erTi an ramodenime
maqsimumi da minimumi (nax. 5), an ar hqondes eqstremumi.
rogor davadginoT, aqvs Tu ara funqcias eqstremumi da Tu aqvs,
rogor vipovoT igi? amisaTvis, cxadia unda visargebloT fermas TeoremiT:
Tu wertili aris y = f (x) warmoebadi funqciis lokaluri eqstremumis
wertili, maSin f ′(x) warmoebuli am wertilSi xdeba nulis toli:
f ′(a) = 0 . rogorc zemoT avRniSneT, funqciis warmoebulis nulTan toloba
eqstremumis arsebobis mxolod aucilebeli pirobaa da ara sakmarisi.
SeiZleba funqciis warmoebuli raime wertilSi nulis toli iyos, magram
am wertilSi funqcias eqstremumi ar hqondes. mivceT am naTqvams
geometriuli interpretacia.
y y
0 a x 0 a x
nax. 6 nax. 7
vTqvaT, x = a wertili aris y = f (x) funqciis lokaluri maqsimumis
wertili (nax. 6). maSin am funqciis grafikis im wertilze gavlebuli mxebi,
romlis abscisaa a, Ox RerZis paraleluri iqneb. am mxebis kuTxuri
koeficienti nulis tolia, magram, rogorc cnobilia, es kuTxuri
193
koeficienti f ′(a) -s toli unda iyos. maSasadame, f ′(a) = 0 . analogiuri axsna
aqvs im SemTxvevas, roca x = a wertili aris y = f (x) funqciis lokaluri
minimumis wertili (nax. 7). xazi unda gaesvas im garemoebas, rom f ′(a) = 0
miRebuli piroba Seexeba mxolod x = a wertilSi warmoebad funqciebs.
SevniSnoT, rom funqcias lokaluri eqstremumi SeiZleba hqondes im
wertilzec, romelzec warmoebuli ar arsebobs. wertilebs, romlebzec
funqcias SeiZleba hqondes eqstremumi, kritikuli wertili ewodeba.
magaliTad, f (x = ax2
)
+ bx + c funqcias gaaCnia erTaderTi lokaluri
b
eqstremumi x = −
wertilSi. amaSi vrwmundebodiT mocemuli kvadratuli
2a
⎛
b 2
⎞
−
2
b2 4ac
samwevridan sruli kvadratis gamoyofiT ax + bx + c = a⎜ x +
⎟ −
.
⎝
2a ⎠
4a
⎛
b ⎞
advili Sesamowmebeli, rom f ⎜
′ −
⎟ = 0 . marTlac, vinaidan f (x = ax2
)
+ bx + c ,
⎝ 2a ⎠
⎛
b ⎞
amitom f ′(x) = 2ax + b da amitom f ⎜
′ −
⎟ = 0 .
⎝ 2a ⎠
bunebrivad ismis kiTxva: Tu viciT, rom x = a aris y = f (x) funqciis
lokaluri eqstremumis wertili, rogor ganvsazRvroT, ra saxis eqstremums
iZleva igi _ maqsimums Tu minimums?
vTqvaT, f ′(x)
0
p , roca x a
p , xolo f ′(x) 0
f , roca x a
f . maSin x = a
wertilis maxloblobaSi f (x) funqcia klebadia a -s marcxniv mdebare
wertilebSi, xolo zrdadia a -s marjvniv mdebare wertilebSi (nax. 7). am
SemTxvevaSi x = a wertili aris lokaluri minimumis wertili. Tu f ′(x)
0
f ,
roca x
a
p da f ′(x) 0
p roca x a
f , maSin piriqiT, f (x) funqcia a -s
marcxniv mdebare wertilebSi zrdadi iqneba, xolo a -s marjvniv _
klebadi. am SemTxvevaSi x = a wertili iqneba lokaluri maqsimumis
wertili iqneba (sur. 6).
e.i. f ′(x) warmoebuli x = a wertilSi nulad iqceva da, amasTan, am
wertilze gadasvlisas igi niSans icvlis `_~ - dan `+~ -ze, maSin a wertili
am funqciis lokaluri minimumis wertili iqneba. Tu f ′(x) warmoebuli
x = a wertilSi nulad iqceva, xolo am wertilze gadasvlisas igi niSans
icvlis `+~- dan `_~ -ze, maSin a wertili f (x) funqciis lokaluri
maqsimumis wertili iqneba.
194
magaliTad, f (x = ax2
)
+ bx + c funqciis warmoebuli f ′(x) = 2ax + b
b
b
nulad iqceva, roca x = −
. vTqvaT, a
0
f , maSin, Tu x −
p
, gveqneba:
2a
2a
b
2ax
−b
p
, 2ax + b
0
p . xolo Tu x −
f
, miviRebT 2ax
−b
f
, 2ax + b
0
f .
2a
b
amgvarad, Tu a
0
f , maSin x = −
wertilze gadasvlisas
2a
f (x = ax2
)
+ bx + c funqciis warmoebuli niSans icvlis `_~ - s `+~ -ze. amitom
b
x = −
wertili am funqciis lokaluri minimumis wertilia. studentebs
2a
vandobT damoukideblad ganixilon SemTxveva, rodesac a
0
p da
b
warmoebulis saSualebiT darwmundnen, rom x = −
wertili am SemTxvevaSi
2a
aris f (x = ax2
)
+ bx + c funqciis lokaluri maqsimumis wertili.
y
y=x3
0 x
nax. 8
ar unda vifiqroT, rom Tu f ′(a) = 0 , x = a wertili, rogorc zemoT
avRniSneT, uTuod lokaluri eqstremumis wertilia. magaliTad,
3
f (x) = x
funqciisaTvis gveqneba:
2
f ′(x) = 3x da amitom f ′(0) = 0 . magram, rogorc es am
funqciis grafikidan Cans (nax. 8). x = 0 wertili ar aris arc lokaluri
minimumis wertili da arc lokaluri maqsimumis wertili. es imiT aixsneba,
rom x = 0 wertilze warmoebuli
2
f ′(x) = 3x nulad iqceva ise, rom niSans ar
icvlis. rogorc x
0
p -is, ise x 0
f - isaTvis f ′(x) 0
f . SeiZleba imis
damtkiceba, rom Tu f ′(a) = 0 , xolo x = a wertilze gadasvlisas f ′(x)
195
warmoebuli niSans ar icvlis, maSin x = a wertili arc lokaluri
minimumis wertilia da arc lokaluri maqsimumis. wertilebs, sadac f (x)
funqciis
f ′(x) warmoebuli nulad iqceva, stacionaruli wertilebi
ewodeba, xolo funqciis mniSvnelobebs am wertilze _ am funqciis
stacionaruli mniSvnelobebi.
e.i. arsebobs eqstremumis sakmarisi piroba ori saxiT:
I piroba. Tu x
f ′(x
0 kritikul wertlze gadasvlisas warmoebulma
) -
ma niSani Seicvala plusidan minusze, maSin am wertilze funqcias aqvs
maqsimumi. Tu x
f ′(x
0 kritikul wertlze gadasvlisas warmoebulma
) -ma
niSani Seicvala minusidan plusze, maSin am wertilze funqcias aqvs
minimumi. Tu warmoebuli niSans ar icvlis, maSin funqcias eqstremumi ara
aqvs.
II piroba. Tu x0 kritikul wertlze meore rigis warmoebuli
uaryofiTia, e.i. Tu f ′(x )
0
0
p , maSin am wertilze funqcias aqvs maqsimumi da
y
= f (x )
x
max
0
. Tu 0 kritikul wertlze meore rigis warmoebuli dadebiTia,
e.i. Tu f ′(x )
0
y
= f (x
0
f , maSin am wertilze funqcias aqvs minimumi da
)
min
0
.
zemoTqmulidan gamomdinare, eqstremumze gamokvlevis algoriTmi ase
Camoyalibdeba:
1. vipovoT y = f (x) funqciis gansazRvris are D( f ).
2. vipovoT f ′(x) da amovxsnaT gantoleba f ′(x) = 0 da davadginoT sad ar
arsebobs f ′(x) . amiT miviRebT kritikul wertilebs.
3. gamovikvlioT kritikuli wertilebi I da II pirobis saSualebiT. Tu
x
y
= f (x
x
0 maqsimumis wertilia, maSin
)
max
0
, Tu 0 minimumis wertilia,
maSin y
= f (x )
min
0
.
7. funqciis grafikis amozneqiloba da Cazneqiloba. gadaRunvis
wertili. funqciis grafikis asimptotebi. SemovitanoT Semdegi
gansazRvreba: funqciis grafiks ewodeba amozneqili (Cazneqili) raime
SualedSi, Tu misi yvela wertili mdebareobs grafikis nebismieri mxebis
qvemoT (zemoT) (nax. 9., nax. 10). mtkicdeba Semdegi
Teorema 6 (grafikis amozneqilobisa da Cazneqilobis sakmarisi
pirobebi): Tu y = f (x) funqcias aqvs raime SualedSi meore rigis
196
warmoebuli da f ′(x ) ≤ 0
( f ′ x ≥ 0
0
)
0
( )
am Sualedis yvela wertilSi, maSin
funqciis grafiki amozneqilia (Cazneqilia) am SualedSi.
gansazRvreba: gadaRunvis wertili ewodeba grafikis iseT wertils,
romelic amozneqilobas yofs Cazneqilobisagan. mtkiceba Semdegi
Teorema 7 (gadaRunvis wertilis arsebobis sakmarisi piroba): Tu x0
wertilSi funqcias aqvs pirveli rigis warmoebuli, meore rigis
warmoebuli amave wertilSi nulia, e.i. f ′(x ) = 0
x
0
da 0 wertilze
gadasvlisas f ′(x
M x , f (x )
y = f (x
0 ) niSans icvlis, maSin
0 ( 0
0 ) aris
) funqciis
grafikis gadaRunvis wertili (nax. 11, nax. 12).
y y
Cazneqili
amozneqili
0 x 0 x
nax. 9 nax. 10
nax. 11 nax. 12
vTqvaT, M aris y = f (x) funqciis grafikis wertili. Tu am
wertilis erTi koordinati mainc absoluturi sididiT usasrulobisaken
miiswrafis, maSin vityviT, rom M wertili miiswrfis usasrulobisaken. am
197
SemTxvevaSi Zneli warmosadgenia grafikis saxe, magram mainc xerxdeba am
siZnelis daZleva grafikis SedarebiT romelime cnobil grafikTan.
umartivesi Sesadarebeli grafikia wrfe. grafikis im Stos formis
Seswavla, romlis wertilis erTi koordinati mainc usasrulod izrdeba,
xdeba asimptotis saSualebiT.
gansazRvreba: f (x) funqcis grafikis asimptoti ewodeba wrfes,
romelsac is Tviseba aqvs, rom manZili grafikis M wertilidan am
wrfemde miiswrafis nulisaken, roca M wertili usasrulobisaken
miiswrafis, anu usasrulod Sordeba koordinatTa saTaves. arsebobs sami
saxis asimptoti:
o vertikaluri, romlis gantolebaa x = a (nax. 13),
o horizontaluri, romlis gantolebaa y = b (nax. 14).
o daxrili, romlis gantolebaa y = kx + b , sadac
f (x)
k = lim
(
b = lim f (x) − kx (
x→∞
x
*) da
[
]
x→∞
**) (nax. 15).
SeniSvna: Tu (*) an (**) zRvrebidan erT-erTi mainc ar arsebobs, maSin
y = f (x) funqcias daxrili asimptoti ara aqvs.
y
y horizontaluri
asimptoti
vertikaluri
b
asimptoti
a x
0 x
nax. 14
y
M
nax. 13
0 daxrili x
asimptoti
sur. 15
198
davaleba:
SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. CamoayalibeT diferencialuri aRricxvis ZiriTadi Teoremebi.
2. ras ewodeba lokaluri maqsimumis, minimumis wertilebi?
3. ras ewodeba fardobiTi maqsimumis, minimumis wertilebi?
4. ras ewodeba eqstremumis wertlebi? kritikuli wertilebi?
5. CamoayalibeT eqstremumis sakmarisi piroba.
6. CamoayalibeT grafikis amozneqilobisa da Cazneqilobis sakmarisi
pirobebi.
7. ras ewodeba funqcis grafikis asimptoti?
8. ras ewodeba gadaRunvis wertili? CamoayalibeT gadaRunvis wertilis
arsebobis sakmarisi piroba.
praqtikuli magaliTebi:
1. gamoikvlieT funqcia eqstremumze:
1
2
a) y =
x3 − x2 − 3x b) y =
x
e −1
3
1
1
g) y = x + 1 +
d) y = ln x +
x −1
x
2. ipoveT funqciis grafikis amozneqilobis, Cazneqilobis Sualedebi
da gadaRunvis wertilebi:
a) f (x)
4
= x −12 3
x + 48 2
x − 50 b)
x
f (x) = e
g) f (x) = 3 5
x − 5 4
x + 3x − 2 d)
4
2
x
f (x) = x + x + e
3. ipoveT funqciis grafikis asimptotebi
2
x
2x −1
a) y =
b) y =
2
x + 8
(x − )2
1
199
$ 6. ganuzRvrelobaTa gaxsna. lopitalis wesi
vTqvaT, F (x) funqcia gansazRvruli ar aris x = a wertilSi, magram
zRvari aqvs, rodesac x → a . aseTi zRvris moZebnas ganuzRvrelobaTa
gaxsna ewodeba.
0
1. ganuzRvrelobebi. davamtkicoT Semdegi
0
Teorema (lopitalis): vTqvaT, f (x) da g(x) funqciebi gansazRvrulia
a wertilis raime modamoSi. Tu
1. f (a) = g(a) = 0 ,
2. f ′(a) da g′(a) arsebobs, amasTan g′(a) ≠ 0 ,
f (x)
maSin arsebobs lim
da adgili aqvs tolobas
x→a g( x)
f (x)
f ′(x)
lim
=
(1)
x→a g(x)
g′(x)
damtkiceba: radganac, lim f (x) = 0 da lim g(x) = 0 , amitom SegviZlia
x→a
x→a
vigulisxmoT, rom f (a) = 0, g(a) = 0 . am pirobebSi f (x) da g(x) funqciebi
uwyvetia a wertilSi. a wertilis aRniSnul midamoSi aviRoT raime x
wertili. koSis formulis Tanaxmad
f (x) − f (a)
f ′(c)
=
, sadac a
c
x
p p .
g(x) − g(a)
g′(c)
radgan f (a) = 0, g(a) = 0 , amitom gveqneba
f (x)
f ′(c)
=
(2)
g(x)
g′(c)
f ′(x)
cxadia, Tu x → a , maSin c → a . pirobis Tanaxmad arsebobs lim
,
x→a g′(x)
f ′(c)
f ′(x)
amitom lim
= lim
-is da (2) tolobis Tanaxmad marTebulia (1).
c→a g′(c)
x→a g′( x)
ln(1+ x)
magaliTi 1: vipovoT lim
;
x→0
sin x
200
0
amoxsna: aq gvaqvs
saxis ganuzRvreloba. lopitalis Teoremis
0
1
ln(1+ x)
1+ x
1
Tanaxmad lim
= lim
= = 1.
x→0
sin
x→0
x
cos x
1
2
x − 7x +10
2x − 7
3
magaliTi 2: lim
= lim
= − ;
2
x→2
x − 4
x→2
2x
4
∞
2.
saxis ganuzRvreloba.
∞
∞
SeniSvna: SeiZleba davamtkicoT, rom amave xerxiT SeiZleba
saxis
∞
ganuzRvrelobis gaxsna:
x
x
x
x
e
e
e
e
magaliTi 3: lim
= lim
= lim
= lim
= +∞ ;
x→+∞
3
x→+∞
2
x→+∞
x→+∞
x
3x
6x
6
magaliTi 4:
′
ln sin x
⎛ ln sin x ⎞
⎛ cos x 2cos 2x ⎞
cos x sin 2x
lim
= lim⎜
⎟ = lim⎜
:
⎟ = lim
=
x→0 ln sin 2
x→0
x
⎝ ln sin 2
x→0
x ⎠
⎝ sin x
sin 2
x→0
x ⎠
2 cos 2x sin x
sin 2x
2
=
x
lim
= lim
= 1.
x→0 2sin
x→0
x
2x
3. 0 ⋅ ∞ saxis ganuzRvreloba. vTqvaT, lim f (x) = 0 da lim g(x) = ∞ ,
x→a
x→a
maSin
f (x) ⋅ g(x) namravli x=a
wertilSi mogvcems 0 ⋅ ∞ saxis
ganuzRvrelobas. igi kvlav SeiZleba gavxsnaT lopitalis wesiT. marTlac,
f (x)
lim f (x) ⋅ g(x) = lim
.
x→a
x→a
1
g(x)
∞
es ukanaskneli ki warmoadgens
saxis ganuzRvrelobas. es ki gaixsneba
∞
lopitalis wesiT.
⎡
π
a
x
x ⎤
magaliTi 5: vipovoT lim e
e tg
.
x→a ⎢(
− )
⎥
⎣
2a ⎦
amoxsna:
201
a
x
x
⎡
x
π ⎤
e − e
− e
lim
→ ⎢( a
x
e − e )tg
= lim
= lim
=
⎥
x a ⎣
2
1
x→a
a ⎦
x
π
x a π ⎛
x −
→
ctg
π ⎞
⎜sin2
⎟
2a
2a ⎝
2a ⎠
2a
x
π
a
π
a
x
2
2
a
2
2
=
lim e ⋅ sin
=
e ⋅ sin
a
=
e .
x→a
π
2a
π
2
π
4. ∞ − ∞ saxis ganuzRvreloba. Tu lim f (x) = ∞ da lim g(x) = ∞ ,
x→a
x→a
maSin lim[ f (x) − g(x)] ∞ − ∞ saxis ganuzRvrelobas. am saxis ganuzRvrelobis
x→a
0
gaxsna xdeba saxis ganuzRvrelobamde dayvaniT. marTlac
0
1
1
−
g(x)
f (x)
lim[ f (x) − g(x)] = lim
;
x→a
x→a
1
f (x)g(x)
⎛
1 ⎞
magaliTi 6: vipovoT lim⎜ ctgx − ⎟
x→0 ⎝
x ⎠
1 −
1
⎛
1 ⎞
x − tgx
2
cos x
2
cos x −1
amoxsna: lim⎜ ctgx − ⎟ = lim
= lim
= lim
=
x→0 ⎝
x ⎠ x→0
x→
xtgx
0
x
x→0 x +
2
tgx +
cos xtgx
2
cos x
sin 2 x
2 cos x sin x
0
= − lim
= − lim
= = 0.
x→0 x + cos x sin
x→0
x
1+ cos 2x
2
5. ∞ 0
0
1 ,0 ,∞ saxis ganuzRvrelobebi. am tipis ganuzRvrelobebis
gaxsna daiyvaneba 0 ⋅ ∞ saxis ganuzRvrelobis gaxsnamde Semdegi formulis
gamoyenebiT
g ( x )
[ f (x)]
g ( x) ln f ( x)
= e
, ( f (x)
0
f ).
x
⎛
2a ⎞
magaliTi 7: gamovTvaloT lim cos
;
x→∞⎜
⎜
x ⎟⎟
⎝
⎠
x
⎛
a ⎞
amoxsna: SemoviRoT aRniSvna y =
2
−
⎜
⎜cos
, maSin
1
ln y = x ln cos 2ax .
x ⎟⎟
⎝
⎠
saidanac
202
1
−
ax−
a
x
lim ln y = lim(x ln cos 2
1
ax )
ln cos 2
= lim
= −
lim
=
1
x→∞
x→∞
x→∞
x−
2 x→∞ ctg 2
1
ax−
2
1
sin 2 2
1
ax−
⎛ sin2 2 1
ax− ⎞
= − lim
= −a lim⎜
⎟ = −a.
2
1
x→∞
x−
x→∞⎜
⎝
2
1
ax−
⎟
⎠
davaleba:
SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. ras ewodeba ganuzRvrelobis gaxsnis lopitalis wesi?
2. rogor SeiZleba gavxsnaT 0 ⋅ ∞ saxis ganuzRvreloba? ∞ − ∞ saxis
ganuzRvreloba?
3. rogor SeiZleba gavxsnaT ∞ 0
0
1 ,0 ,∞ saxis ganuzRvrelobebi?
praqtikuli magaliTebi:
gamoTvaleT zRvari lopitalis wesiT:
0
∞
1. da
saxis ganuzRvrelobis gaxsna
0
∞
x − 2
ex −1
a) lim
b) lim
3
x→2 x − 8
x→0 sin x
ex −1
ln x
g) lim
d) lim
x→0 cos x −1
x→0 ln sin x
2. 0 ⋅ ∞ , ∞ − ∞ , ∞ 0
0
1 ,0 ,∞ saxis ganuzRvrelobis gaxsna
2
a) lim x ln x b)
2
lim
− x
x e
g)
x
lim x
x→0
x→∞
x→0
⎛
1 ⎞
d) lim(sin x)tgx e) lim⎜ tgx − ⎟
x→0
x→0 ⎝
x ⎠
203
$ 7. zRvruli (marJinaluri) amonagebi. danaxarji da mogeba.
zRvruli (marJinaluri) midrekileba dazogvisa
da moxmarebisadmi
1. zRvruli (marJinaluri) amonagebi. marJinaluri amonagebi (MR)
gansazRvrulia tolobiT
f (Q + Q
Δ ) − f (Q)
(MR) = lim
Q
Δ →0
Q
Δ
sadac, (TR)=f(Q) aris mTliani amonagebis funqcia. aq Q aris moTxovna,
anu gayiduli produqciis raodenoba. Tu es zRvari arsebobs, cxadia, rom
igi aris f(Q) funqciis warmoebuli. amrigad moviReT, rom
d (TR)
(MR) =
= (TR)′ (1)
dQ
e.i. marJinaluri amonagebi aris mTliani amonagebis funqciis warmoebuli
moTxovnis Q cvladiT. xSirad marJinalur amonagebs gansazRvraven,
rogorc mTliani amonagebis cvlilebas, rodesac moTxovna Q izrdeba erTi
erTeuliT. aseT midgomas vuwodoT marJinaluri funqciis miaxloebiTi
gamoTvla argumentis (moTxovnis) erTi erTeuliT gazrdis meTodiT da
avRniSnoT igi
∗
(MR) simboloTi. ganvixiloT mTliani amonagebis (TR)=f(Q)
funqciis grafiki (nax. 1)
(TR)
T
B
A
C
0 Q Q+1 Q
nax. 1
rogorc ukve viciT
(MR) = f (Q) = tg TAC
∠
= TC
xolo
204
(MR ∗
) = f (Q + )
1 − f (Q) = BC = tg BAC
∠
aqedan Cans, rom (MR) da
∗
(MR) sideeebs Soris sxvaoba TB
monakveTis sigrZe, garkveul SemTxvevaSi SeiZleba sulac ar iyos Zalian
mcire. amitom sizustis TvalsazrisiT, umjobesia visargebloT
marJinaluri amonagebis (1) ganmartebiT.
magaliTi 1: moTxovnis funqciaa P = 60 − Q
1. vipovoT mTliani amonagebis (TR)=f(Q) funqciis gamosaxuleba da misi
marJinaluri funqcia;
2. gamovTvaloT marJinaluri amonagebis funqciis mniSvneloba, roca Q=50
3. gamovTvaloT mTliani amonagebis funqcia argumentis Q=50 da Q=51
mniSvnelobebisaTvis. SevadaroT (MR)∗ = f ( )
51 − f (50) da (MR)
marJinaluri amonagebi Q=50 mniSvnelobebisaTvis.
amoxsna: 1 . gamoviyenoT mTliani amonagebis formula
(TR)=QP (2)
miviRebT
2
(TR) = QP = (60 − Q) ⋅ Q = 60 ⋅ Q − Q , misi Sesabamisi marJinaluri
funqcia iqneba
d TR
(
)
(MR) =
= f ′(Q) = 60 − 2Q
dQ
2. marJinaluri funqciis mniSvneloba, rodesac Q=50 gamoiTvleba
Semdegi tolobiT:
(MR) = f ′(50) = −40
3. rodesac Q=50 da Q=51, maSin mTliani amonagebisaTvis Sesabamisad
miviRebT f(50)=500 da f(51)=459. amitom mTliani amonagebis cvlileba,
rac mosdevs Q raodenobis erTi erTeuliT gazrdas, gamoiTvleba sxvaobiT
(MR)∗ = f ( )
51 − f (50) = −41
amrigad, miviReT, rom moTxovnis cvladis Q=50 mniSvnelobebisaTvis
marJinaluri funqciis zusti sidide (-40) da marJinaluri funqciis
miaxloebiTi mniSvneloba (- 41) ar emTxveva erTmaneTs.
2. danaxarji da mogeba. saSualo amonagebi (AR) moTxovnis Q doneze,
anu gayiduli saqonlis Q raodenobisaTvis ganisazRvreba tolobiT
TR
(
)
( AR) =
(3)
Q
205
(2) formulis gamoyenebiT, sadac P = g (Q)
D
aris moTxovnis funqcia,
miviRebT
P ⋅ Q
( AR) =
= P = g (Q) (4)
Q
D
e.i. saSualo amonagebi emTxveva moTxovnis funqcias.
mTliani danaxarjis (TC) funqcia, romelic warmodgindeba
fiqsirebuli (mudmivi) (FC) danaxarjisa da mTliani cvalebadi
danazarjis jamis saxiT, avRniSnoT K(Q) _Ti. e.i.
TC
(
) = K (Q) = F (C) + V (C) ⋅ Q
sadac, (VC) aris produqciis erTeulis warmoebisaTvis saWiro cvalebadi
danaxarji, xolo Q warmoebuli produqciis raodenoba.
Cveni mizania davaxasiaToT danaxarjis K funqciis cvlilebis siCqare
warmoebuli produqciis Q raodenobis cvlasTan dakavSirebiT. vTqvaT,
warmoebuli produqciis raodenoba gaizarda Q
Δ raodenobiT, maSin
Sesabamisi danaxarji iqneba K (Q + ΔQ) . e. i. warmoebuli produqciis Q
Δ
nazrds Seesabameba warmoebis danaxarjis nazrdi: ΔK (Q) = K (Q + ΔQ) − K (Q) .
amitom warmoebis danaxarjis saSualo nazrdi warmoebuli produqciis
K
Δ (Q)
erTeulze gaangariSebiT gamoiTvleba SefardebiT
.
Q
Δ
K
Δ (Q) Sefardebis zRvars, rodesac ΔQ → 0, ewodeba warmoebis
Q
Δ
marJinaluri anu zRvruli (MC) danaxarji:
K
Δ (Q)
d TC
(
)
(MC) = lim
= K ′(Q) =
(5)
Q
Δ →0
Q
Δ
dQ
amrigad, marJinaluri danaxarji K ′(Q) , romelic gamoTvlilia Q
raodenobis produqciisaTvis, gamosaxavs warmoebis danaxarjis cvlilebas
sawarmoebeli produqciis erTeulze gaangariSebiT
K
Δ ≈ K ′(Q) ⋅ Q
Δ = dK (Q) (6)
aqedan gamomdinareobs, rom warmoebis xarjebis nazrdi miaxloebiT udris
warmoebis marJinaluri danaxarjisa da produqciis nazrdis namravls, anu
danaxarjis K funqciis diferencials. aqac SegviZlia SemoviRoT
marJinaluri funqciis miaxloebiTi mniSvnelobis gamosaTvlelad formula,
206
romlelic Seesabameba argumentis (warmoebuli produqciis raodenobis)
erTi erTeuliT gazrdisas miRebuli cvlilebis dadgenas:
(MC)∗ = Δ(TC) = ΔK (Q) = K (Q + )
1 − K (Q) ≈ K ′(Q) ⋅1 (7)
es Tanafardoba miiReba (6) tolobidan, Tu davuSvebT, rom ΔQ = 1 .
ekonomikaSi
∗
(MC) sidides uwodeben danaxarjs, romelic saWiroa
produqciis (Q+1) _e erTeulis warmoebisaTvis. rogorc vxedavT, es sidide
garkveuli miaxloebiT udris marJinalur danaxarjs.
magaliTi 2: vTqvaT, warmoebis saSualo danaxarjis (AC) funqcia
13
mocemulia Semdegi tolobiT ( AC) = 2Q + 6 +
Q
a. vipovoT marJinaluri danaxarjis (MC) funqcia.
b. rogori iqneba marJinaluri danaxarjis saSualebiT gamoTvlili
sruli danaxrjis cvlileba, Tu warmoebuliproduqciis raodenoba
15-dan 12 erTeulamde mcirdeba? rogoria AaRniSnul situaciaSi
mTliani danaxarjis zusti cvlileba?
amoxsna: a. pirvel rigSi unda vipovoT mTliani danaxarjis
TC
(
)
(TC) funqcia ( AC) =
, amitom TC
(
) = ( AC) ⋅ Q . magaliTis pirobis
Q
gaTvaliswinebiT martivad davaskvniT (TC) = K (Q) = 2 2
Q + 6Q + 3.
(5) tolobis Tanaxmad, marJinaluri danaxarjis funqciisaTvis miviRebT
d (TC)
(MC) =
= K ′(Q) = 4Q + 6
dQ
b. radgan warmoebuli produqciis raodenoba Q=15 erTeulidan
klebulobs Q + ΔQ = 12 erTeulamde, amitom ΔQ = 12 − 5 = −3 . argumentis am
cvlilebis Sesabamisi warmoebis mTliani danaxarjis cvlilebis
gamosaTvlelad gamoviyenoT (6) formula. miviRebT
Δ(TC) ≈ K ′ 15
( ) ⋅ (− )
3 = (4 ⋅15 + 6) ⋅ (− )
3 = 198
−
amrigad, aRniSnuli meTodiT miviRebT, rom danaxarji daiklebs 198
erTeuliT. gamovTvaloT exla mTliani danaxarjis zusti cvlileba igive
pirobebSi:
Δ(TC) = K 12
( ) − k 15
( ) = 2 ⋅122 + 6 ⋅12 +13 − (2 ⋅152 + 6 ⋅15 + )
13 =
= 2 12
( 2 −152 ) + 6 12
(
−15) = 2 ⋅ (− )
81 − 6 ⋅ (− )
3 = −162 −18 = 180
−
207
miRebuli Sedegebis Sedareba gviCvenebs, rom mTliani danaxarjis zust da
miaxloebiT mniSvnelobebs Soris sxvaoba 18 erTeulia.
mogebis funqcia π gamoiTvleba Semdegi formuliT
π (Q) = (TR) − (TC) ,
sadac (TR) aris mTliani amonagebi, xolo (TC) mTliani danaxarji. aq Q
aris relizebuli (gayiduli) produqciis raodenoba. produqciis
realizaciisaTvis Q
Δ raodenobiT gazrdas Seesabameba mogebis nazrdi
π
Δ (Q) = π (Q + Q
Δ ) − π (Q) ,
amitom, mogebis saSualo nazrdi sarealizacio produqtis erTeulze
Δπ (Q)
gaangariSebiT gamoiTvleba SefardebiT
.
Q
Δ
Δπ (Q) Sefardebis zRvars, rodesac ΔQ → 0, ewodeba zRvruli an
Q
Δ
marJinaluri mogeba. Tu π funqcia warmoebadia, maSin
π
Δ (Q)
lim
= π ′(Q) (8)
ΔQ→0
Q
Δ
amrigad, marJinaluri mogeba, romelic gamoiTvleba Q raodenobis
sarealizacio produqciisaTvis, gamosaxavs mogebis myisier cvlilebas
(nazrds) sarealizacio produqciis erTeulze gaangariSebiT. (8) _dan
gamomdinareobs, rom mogebis nazrdi miaxloebiT gamoiTvleba formuliT
π
Δ (Q) ≈ π ′(Q) Q
Δ = dπ (Q) (9)
e.i. mogebis nazrdi MmiaxloebiT udris marJinaluri mogebisa
sarealizacio produqciis nazrdis namravls, anu mogebis π funqciis
diferencials. Tu ΔQ = 1 , maSin
Δπ = π (Q + )
1 − π (Q) ≈ π ′(Q) ⋅1 (10)
e.i. mogebis nazrdi, romelic Seesabameba realizebuli produqciis erTi
erTeuliT gazrdas garkveuli miaxloebiT, SegviZlia CavTvaloT
marJinaluri mogebis tolad.
magaliTi 3: produqciis warmoebis fiqsirebuli (mudmivi) danaxarjia
(FC)=100 lari, xolo cvalebadi danaxarji produqciis erTeulze
Seadgens (VC)=3 lars. moTxovnis funqcia P=200-Q
a. vipovoT marJinaluri mogebis funqcia;
b. daaxloebiT, rogor Seicvleba mogeba relizebuli produqciis erTi
erTeuliT gazrdisas, Tu aRebul momentSi produqciis ralizaciis
208
donea 80 erTeuli. gamovTvaloT mogebis zusti cvlileba da
SevadaroT miRebul miaxloebiT mniSvnelobas.
amoxsna: a. radgan cnobilia mudmivi danaxarji da erTeuli
produqciis warmoebis cvalebadi danaxarji, mTliani danaxarjis funqcia
iqneba TC
(
) = (FC) + VC
(
) ⋅ Q = 100 + 3⋅ Q . amocanis pirobidan gamomdunareobs,
rom mTliani amonagebi gamoiTvleba formuliT
2
(TR) = P ⋅ Q = (200 − Q) ⋅ Q = 200 ⋅ Q − Q
amitom mogebis funqciisaTvis miviRebT
π (Q) = (TR) − (TC) = 200
2
⋅ Q − Q − 100
(
+ 3Q)
2
= −Q +197Q −100
viciT, rom marJinaluri mogeba aris mogebis funqciis warmoebuli. amitom
marJinaluri mogeba iqneba π ′(Q) = 2
− Q +197
b. amocanis meore kiTxvaze pasuxis gasacemad SevniSnoT, rom am
SemTxvevaSi Q = 80
Δ
da
Q = 1 mogebis miaxloebiTi cvlilebis mosaZebnad
gamoviyenoT (9) toloba: Δπ (80) ≈ π ′(80) ⋅1 = 2
− ⋅80 +197 = 37 .
amrigad, Tu aRebul momentSi iyideba 80 erTeuli, masin erti
erTeuliT metis gayidva mogebas gazrdis daaxloebiT 37 erTeuliT
(lariT). gamovTvaloT mogebis zusti nazrdi, Tu
Δ
Q = 1 .
Δπ (80) = π ( )
81 − π (80) = 812
−
+197 ⋅81− ( 802
−
+197 ⋅80 −100) = 36
vxedavT, rom mogebis cvlilebis miaxloebiT da zust mniSvnelobebs Soris
sxvaobaa erTi erTeuli.
3. zRvruli (marJinaluri) midrekileba dazogvisa da
moxmarebisadmi. exla gamovikvlioT, Tu rogoria moxmarebis xarjebisa da
danazogis cvlilebaTa tendencia erovnuli Semosavlis cvlilebasTan
dakavSirebiT, risTvisac gamoviyenoT warmoebulis cneba.
moxmarebisa da danazogis funqciebi avRniSnoT, Sesabamisad, C(Y) da
S(Y) –iT. vigulisxmoT, rom vixilavT martiv models da Y erovnuli
Semosavli gamoiyeneba mxolod moxmarebis xarjebisa da danazogisaTvis. e.i.
Y= C(Y)+ S(Y) (11)
cxadia, rom erovnuli Semosavlis ΔY cvlilebas mosdevs
moxmarebis funqciisa da danazogis funqciis cvlilebebi, romlebic
erovnuli Semosavlis erT erTeulze gadaangariSebiT gamoiTvleba
SefardebebiT:
209
C
Δ Y
( )
S
Δ Y
( )
da
Y
Δ
Y
Δ
es Sefardebebi gviCveneben C da S `cvlilebis saSualo siCqares~
(Y,Y + Y
Δ ) intervalSi, anu maTi cvlilebis tendenciebs. am SefardebaTa
zRvrebs, rodesac ΔY → 0 , ewodebaT, Sesabamisad, marJinaluri anu
zRvruli midrekileba (miswrafeba) momxmareblebisadmi _ MPC da
marJinaluri anu zRvruli midrekileba (miswrafeba) dazogvisadmi _ MPS.
C
Δ (Y )
MPC = lim
= C′(Y
)
ΔY →0
Y
Δ
(12)
S
Δ (Y )
MPS = lim
= S′(Y
)
ΔY →0
Y
Δ
Tu gavawarmoebT (11) tolobis orive mxares Y cvaldiT da gamoviyenebT (12)
tolobebs, miviRebT
MPC +
MPS = C′(Y ) + S′(Y ) =
1 (13)
magaliTi 4: moxmarebis funqcia mocemulia tolobiT:
C(Y ) = 0,01
2
⋅Y + 0,2 ⋅ y + 50
vipovoT MPC da MPS, rodesac Y=30. gavaanalizoT miRebuli Sedegi.
amoxsna: moxmarebis funqciis warmoebuli gvaZlevs MPC marJinalur
midrekilebas momxmareblisadmi. MPC = C′(Y ) = 0,02Y + 0,2 , Tu Y=30. maSin
MPC = 0,02 ⋅ 30 + 0,2 = 0 8
, marJinaluri midrekileba dazogvisadmi MPS
gamovTvaloT (13) formulis gamoyenebiT MPS = 1- MPC = 0,2 . miRebuli
Sedegebi gviCvenebs, rom Tu erovnuli Semosavlis done 30 erTeulia, maSin
misi gazrda erTi erTeuliT gamoiwvevs moxmarebis gazrdas daaxloebiT 0,8
erTeuliT, xolo dazogvis gazrdas _ daaxloebiT 0,2 erTeuliT. amrigad,
erovnuli Semosavlis aRniSnul doneze midrekileba momxmareblisadmi
ufro didia, vidre dazogvisadmi.
davaleba:
SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. gansazRvreT marJinaluri amonagebi. moiyvaneT marJinaluri
amonagebis meorenairi ganmarteba.
210
2. geometriulad aCveneT sxvaoba (MR) da(MR)* sideebs Soris.
3. ganmarteT saSualo amonagebis sidide moTxovnis Q donisaTvis.
4. gansazRvreT warmoebis marJinaluri anu zRvruli (MC) danaxarji.
5. amowereT marJinaluri funqciis miaxloebiTi mniSvnelobis
gamosaTvleli formula.
6. ganmarteT zRvruli anu marJinaluri mogeba.
7. amowereT mogebis nazrdis miaxloebiTi gamosaTvleli formula.
8. amowereT formula, romelic amyarebs kavSirs moxmarebis zRvrul
midrekilebasa da dazogvis zRvrul midrekilebas Soris.
praqtikuli savarjiSoebi:
1. mocemulia mTliani amonagebis funqcia =100Q-Q2
a) vipovoT marJinaluri amonagebis funqcia;
b) vTqvaT, aRebul momentSi produqciaze moTxovnaa Q=60. ra
cvlilebebs ganicdis (TR) mTliani amonagebi, Tu moTxovna
gaizrdeba 2 erTeuliT?
2. warmoebis yoveldRiuri danaxarji, romelic damokidebulia
warmoebuli produqciis moculobaze, ganisazRvreba formuliT
2
K (Q) = 100Q + 0 1
, Q , 0 ≤ Q ≤ 100 . vipovoT warmoebis zRvruli (marJinaluri)
danaxarji, Tu warmoebis moculoba tolia:
a) 3 erTeulis;
b) 5,5 erTeulis.
miaxloebiT ramdeniT gaizrdeba warmoebis xarjebi, rodesac
produqcia izrdeba:
g) 200 erTeulidan 200,5 erTeulamde?
d) 800 erTeulidan 800,5 erTeulamde?
SeadareT g) da d) punqtebis pasuxebi warmoebis xarjebis zust
nazrdebs.
211
$ 8. funqciis elastikuroba. moTxovnilebis elastikuroba
fasis mixedviT. miwodeba da miwodebis elastikuroba. sruli da
saSualo danaxarjebis elastikuroba
1. funqciis elastikuroba. warmoebulis daxmarebiT SeiZleba
gamovTvaloT damokidebuli cvladis nazrdi, romelic Seesabameba
damoukidebeli cvladis nazrds. bevr amocanaSi moxerxebulia
gamovTvaloT damokidebuli cvladis namatis procenti (SefardebiTi
nazrdi), romelic Seesabameba damoukidebeli cvladis namatis procents,
amaa mivyavarT funqciis elastikurobis cnebamde (zogjer mas uwodeben
fardobiT warmoebuls).
mocemulia y = f (x) funqcia. davuSvaT, rom damoukidebeli cvladis
x
Δ
nazrdia
x
Δ , xolo damoukidebeli x cvladis SefardebiTi nazrdi
.
x
damokidebuli cvladis Sesabamisi nazrdi aris y
Δ = f (x + x
Δ ) − f (x) , xolo
y
Δ
damokidebeli cvladis SefardebiTi nazrdia
. funqciis (damokidebeli
y
cvladis) SefardebiTi nazrdis Sefardeba damoukidebeli cvladis
y
Δ
x
Δ
SefardebiT nazrdTan aris
:
.
y
x
es Sefardeba gviCvenebs, ramdenjer didia funqciis SefardebiTi
nazrdi argumentis SefardebiT nazrdze. es SeiZleba, agreTve, CavweroT
Semdegi saxiT:
y
Δ
x
Δ
y
Δ
x
:
=
⋅ (1)
y
x
x
Δ
y
Tu arsebobs y = f (x) funqciis warmoebuli, maSin
⎛ y
Δ
x
Δ ⎞
⎛ y
Δ
x ⎞
x
y
Δ
x
x dx
lim
:
= lim
⋅
=
lim
=
f ′(x) =
⋅
(2)
Δx→0⎜⎜⎝ y
x ⎟⎟
x
Δ →0⎜⎜
⎠
⎝ x
Δ
y ⎟⎟⎠ y x
Δ →0
x
Δ
y
y dy
(2) zRvars, e.i. y = f (x) funqciis SefardebiTi nazrdisa da
damoukidebeli cvladis SefardebiTi nazrdis Sefardebis zRvars,
rodesac damoukidebeli cvladis nazrdi Δx → 0 , ewodeba y = f (x)
funqciis elastikuroba x cvladis mimarT.
212
y = f (x) funqciis elastikuroba aRiniSneba E ( y)
x
simboloTi.
maSasadame,
x dx
E ( y) =
⋅
x
(3)
y dy
elastikuroba x cvladis mimarT aris funqciis miaxloebiTi procentuli
nazrdi, romelic Seesabameba damoukidebeli cvladis 1% -iT gazrdas.
magaliTi 1: gamoTvaleT funqciis elastikuroba y=3x-6
amoxsna: elastikurobis ganmartebis ZaliT gvaqvs:
x dx
x
3x
x
E ( y) =
⋅
=
⋅ 3 =
=
x
y dy
(
3 x − 2)
3x − 6
x − 2
Tu, magaliTad, x=10, maSin funqciis elastikuroba aris
10
10
5
=
=
10 − 2
8
4
5
es niSnavs. rom Tu x cvladi gaizarda 1% -iT, maSin y gaizrdeba %-iT.
4
magaliTi 2: gamoTvaleT funqciis elastikuroba
2
y = 1+ 2x − x
amoxsna: elastikurobis ganmartebis ZaliT gvaqvs:
2
x dx
x
2x − 2x
E ( y) =
⋅
=
(2 − 2x) =
x
2
2
y dy
1 + 2x − x
1 + 2x − x
Tu, magaliTad, x=1, maSin funqciis elastikuroba aris
2 ⋅1 − 2 ⋅12
= 0
1 + 2 ⋅1 −1
es niSnavs. rom Tu x cvladi gaizarda 1% -iT (1-da, 1,01 –mde), maSin
damoukidebeli cvladis mniSvneloba ar Sicvleba (daaxloebiT).
2. moTxovnilebis elastikuroba fasis mixedviT. mocemul saqonelsa
da mis fass Soris arsebuli funqcionaluri damokidebuleba (im pirobiT,
rom sxva saqonlis fasi, myidvelTa Semosavali da moTxovnilebaTa
struqtura mudmivi sidideebia) saSualebas gvaZlevs fasi ganvsazRvroT
saTanadod gansazRvruli moTxovnilebis mixedviT. magram, mTel rig
ekonomikur gamokvlevebSi, aucilebelia ganvsazRvroT ara moTxovnilebis
sidide, aramed moTxovnilebis cvlileba, gamowveuli fasebis
gansazRvruli cvlilebiT. sxvanairad rom vTqvaT, saWiroa ganvsazRvroT
moTxovnilebis elastikuroba fasis mimarT.
vigulisxmoT, rom
moTxovnilebs q damokidebulia P fasisagan
213
q = f (P) (1)
vTqvaT, P
Δ fasis nazrdia, xolo q
Δ moTxovnilebis Sesabamisi
P
Δ
nazrdi. fasis fardobiTi cvlileba aris
, xolo moTxovnilebis
P
q
Δ
q
Δ
P
Δ
fardobiTi cvlilebaa
. Sefardeba
:
gamosaxavs moTxovnilebis
q
q
P
fardobiT cvlilebas, roca saqonlis fasi izrdeba erTi procentiT.
moTxovnilebis elastikuroba fasis mimarT ewodeba zRvars
⎛ q
Δ
P
Δ ⎞
⎛ q
Δ
P ⎞
P
q
Δ
P dq
E (q) = E = lim
:
= lim
⋅
=
lim
=
⋅
P
c
(2)
P
Δ →0⎜⎜⎝ q
P ⎟⎟
P
Δ →0⎜⎜
⎠
⎝ P
Δ
q ⎟⎟⎠ q P
Δ →0
P
Δ
q dP
da, maSasadame,
P dq
E =
⋅
c
. (3)
q dP
moTxovnilebis elastikuroba fasis mimarT, daaxloebiT
gansazRvravs, Tu rogor icvleba moTxovnileba saqonelze, roca misi fasi
izrdeba 1% -iT. umravles SemTxvevaSi moTxovnilebis funqcia aris
klebadi, radganac saqonlis fasis gazrdiT moTxovnileba masze mcirdeba.
maSasadame, aseT SemTxvevaSi
dq
0
p
dP
imisaTvis, rom avicdinoT uaryofiTi ricxvebi, moTxovnilebis
elastkurobis Seswavlis dros moRebulia, rom
P dq
E = −
⋅
c
(4)
q dP
Tu E
1
c f , e.i. Tu fasebis 1% -iT gazrda iwvevs moTxovnilebis
Semcirebas, maSin amboben, rom moTxovnileba elastikuria.
Tu E = 1
c
, e.i. Tu fasebis 1% -iT gazrda iwvevs moTxovnilebis 1% -iT
Semcirebas, maSin amboben, rom moTxovnileba neitraluria.
Tu 0
E
1
p c p , e.i. Tu fasebis 1% -iT gazrda iwvevs moTxovnilebis 1%
-ze metiT Semcirebas, maSin amboben, rom moTxovnileba araelastikuria.
magaliTi 3: Tu moTxovnilebis funqcia aris q=10-P, maSin
moTxovnilebis elastikuroba etoleba
P dq
P
P
E = −
⋅
= −
⋅ (− )
1 =
c
q dP
10 − P
10 − P
214
Tu P=2, maSin
2
1
E =
=
c
8
4
es niSnavs, rom fasi 2-ia, maSin fasis 1% -iT gazrda gamoiwvevs
1
moTxovnilebis % -iT Semcirebas.
4
C
magaliTi 4: Tu moTxovnilebis funqcia aris q =
(C mudmivia),
P
maSin moTxovnilebis elastikuroba etoleba
P dq
P ⎛ C ⎞
2
−2
E = −
⋅
= −
⋅
c
⎜ −
⎟ = P ⋅ P = 1
2
q dP
C
⎝ P ⎠
P
maSasadame, es moTxovnileba fasis ukuproporciulia, maSin nebismieri
fasisaTvis, moTxovnilebis elastikuroba erTis tolia.
3. miwodeba da miwodebis elastikuroba. miwodebaSi igulisxmeba
romelime saqonelis raodenoba, romelic gankuTvnilia erTeul droSi
gasayidad. rogorc wesi, romelime saqonlis miwodeba mocemul periodSi
aris fasis zrdadi funqcia. yvela sxva danarCen erTnair pirobebSi
miwodeba mocemuli fasis dros ufro metia, vidre miwodeba naklebi fasis
dros. magram aris SemTxvevebi, roca miwodeba izrdeba fasebis
SemcirebasTan erTad. magaliTad xorblis fasis dacema xSirad aizulebs
glexs, gazardos miwodeba, Tu mas surs gansazRvruli Semosavlis miReba.
vTqvaT, S = S (P) aris miwodebis funqcia, ΔP fasis nazrdia, xolo
Δ
P
Δ
S Sesabamisi miwodebis nazrdi. fasis SefardebiTi nazrdi aris
,
P
S
Δ
xolo miwodebis fardobiTi nazrdi
.
S
zRvars
⎛ S
Δ
P
Δ ⎞
⎛ S
Δ
P ⎞
P
S
Δ
P dS
P
E (S ) = E = lim ⎜
:
⎟ = lim ⎜
⋅ ⎟ =
lim
=
⋅
=
S ′
P
P
(5)
P
Δ →0⎝ S
P
ΔP
⎠
→0⎝ P
Δ
S ⎠ S P
Δ →0
P
Δ
q dP
q
ewodeba miwodebis elastikuroba fasis mimarT. maSasadame, miwodebis
elastikuroba fasis mimarT daxloebiT gansazrvravs miwodebis naxrds
procentebSi, 1% -iT gazrdis dros.
215
4. sruli da saSualo danaxarjebis elastikuroba. Tu sawarmo
awarmoebs romeliRac saqonlis x erTeuls da gansazRvrulia srul
danaxarjTa funqcia, maSin srul danaxarjTa elastikuroba aris:
dK x
dK K
E (K ) = E =
⋅
=
⋅
x
K
(6)
dx K
dx x
da, maSasadame, srul danaxarjTa elastikuroba aris zRvruli
danaxarjebis fardoba.
K
π =
_ saSualo danaxarjis elastikuroba Seadgens:
x
dK
x
−
dK
K
2
x
− K
x dπ
x
dx
x
dx
x dK
E (π ) = E =
⋅
=
⋅
=
⋅
=
−1 = E −1
x
π
2
2
K
π dx
K
x
k
x
K dx
x
da, maSasadame, saSualo danaxarjTa elastikuroba erTeuliT mcire srul
danaxarjTa elastikurobaze.
Tu E = 1
K
, maSin saSualo danaxarjTa elastikuroba tolia nulis.
e.i. E = 0
π
, rac imas niSnavs, rom saSualo danaxarjebi mudmivia. aqedan
gamomdinareobs, rom
dK
dK
K
x
− K =
an
0
=
(7)
dx
dx
x
maSasadame, Tu srul danaxarjTa elastikuroba erTis tolia, maSin sruli
zRvruli danaxarjebi tolia sruli saSualo danaxarjebisa.
davaleba:
SeamowmeT Teoriuli masalis codna Semdegi kiTxvebis saSualebiT
1. raSi mdgomareobs warmoebulis ekonomikuri azri?
2. ras ewodeba warmoebis zRvruli danaxarji?
3. ganmarteT ras ewodeba zRvruli mogeba.
4. CamoayalibeT funqciis elastikurobis cneba.
5. CamoayalibeT moTxovnilebis elastikuroba fasis mimarT.
6. ra SemTxvevaSia moTxovnileba neitraluri?
7. rodisaa moTxovnileba araelastikuri?
216
8. ganmarteT miwodebiselastikuroba fasis mimarT.
9. ganmarteT srul danaxarjTa elastikuroba.
10. CamoayalibeT saSualo danaxarjTa elastikuroba.
praqtikuli savarjiSoebi:
1.
srul danaxarjTa mruds aqvs saxe K=6log(1+3x). gansazRvreT
zRvrul danaxarjTa mrudi.
2.
daamtkiceT, rom Tu E [ f (x)]
x
aris f(x) funqciis elastikuroba,
maSin xf(x) funqciis elastikuroba, aris E [ f (x)] + 1
x
3.
gamoTvaleT y=x3-1 funqciis elastikuroba. ipoveT funqciis
elastikurobis maCvenebeli, roca: a) x=1 b) x=5
4.
gamoTvaleT
5 x
y = e funqciis elastikuroba. ipoveT funqciis
elastikurobis maCvenebeli, roca: a) x=1 b) x=0 g) x=2
5.
gamoTvaleT y = 5log x funqciis elastikuroba. ipoveT funqciis
elastikurobis maCvenebeli, roca: a) x=10 b) x=e g) x=e4
6.
gamoTvaleT y = ax + b funqciis elastikuroba, sadac a da b
mudmivebia.
7.
gamoTvaleT
m
y = ax funqciis elastikuroba, sadac a da b
mudmivebia.
20 + 4P2
8.
romeliRac saqonlis miwodebis funqcia aris S =
, xolo
1 +10P
25 − P4P2
moTxovnilebis funqciaa q =
. gansazRvreT wonasworobis
1 +10P
fasi. e.i. fasi, romlis drosac moTxovnileba da miwodeba
wonaswordeba, agreTve moTxovnilebis elastikuroba da miwodeba
am fasisaTvis.
217
$ 9. ekonomikuri amocanebi funqciis eqstremumis gamokvlevaze
magaliTi 1: dawesebuleba TveSi amzadebs produqciis x erTeuls.
warmoebis finansuri dagroveba dakavSirebulia gamoSvebuli produqciis x
moculobasTan formuliT:
A = 0
− ,01 3
x + 300x − 500
vipovoT warmoebuli:
A′ = 0
− ,03 2
x + 300
aqedan gamomdinare, A′
0
p , Tu − 0,03 2
x + 300 0
p , anu 3 2
x − 30000 0
f , saidanac
2
x −10000 0
f utoloba sruldeba, roca x −100
p
an x
100
f
.
maSasadame, Tu produqciis gamoSveba gadaaWarbebs 100 erTeuls, maSin
warmoebis finansuri dagroveba iklebs.
magaliTi 2: vigulisxmoT, rom romelime saqonlis moTxovnileba
ganisazRvreba am saqonlis fasiT Semdegi formuliT
q = f ( p)
sadac, p –saqonlis fasia, xolo q- Sesabamisi moTxovnileba. mosaxleobis
saerTo danaxarji am saqonelze (amonagebi saqonlis gayidviT) Seadgens _
u = pq , xolo zRvruli mogeba aris
p
u′ = q + pq′ = q 1
( +
q′) .
q
vinaidan moTxovnilebis alstikuroba fasis mimarT aris
p
E = −
q′
c
q
amitom vRebulobT
u′ = q 1
( − E )
c
(1)
es gantoleba gansazRvravs damok
