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u
...
danelia, T
...
nadibaiZe
k a l k u l u s i
programa MAPLE-is gamoyenebiT
saleqcio kursi
Tbilisi 2007
1
...
ricxviTi simravleebi
simravlis cneba
...
sailustraciod moviyvanoT maTematikis, rogorc mecnierebis
erT-erTi ganmarteba: maTematika aris mecniereba, romelic Seiswavlis
garkveul struqturebs simravleebze
...
simravle aris garkveul
obieqtTa erToblioba
...
Oobieqtebs, romlebic mocemul simravleSi
Sedian, simravlis elementebi ewodeba
...
, xolo simravlis elementebi
_ patara laTinuri asoebiT: a, b, c, d ,
...
Canaweri “ a ∉ A ” an “ a ∈ A ” niSnavs, rom
a elementi ar ekuTvnis A simravles
...
magaliTi 1
...
vTqvaT, N aris naturalur ricxvTa simravle
...
3
A
simravle Sedgeba mxolod
Tu
a, b, c, d ,
...
magaliTad, Tu 1, 5, 9 aris A simravlis elementebi da is
1
sxva elements ar Seicavs,
maSin SegviZlia CavweroT A = { ,5,9}
...
1
Canaweri { ,2,{2}} cxadia simravles warmoadgens, romelic Seicavs sam
elements
...
)
kalkulusis
kursSi
ZiriTadad
saqme
gveqneba
ricxviT
simravleebTan
...
xSirad gamoviyenebT
simravlis mocemis aseT wess: davasaxelebT raime Tvisebas, romelsac
akmayofilebs aRsaweri erTobliobis yvela wevri da mxolod isini
...
2
simravles
...
3 {x x 2 − 1 = 0 } = {− 1;1}
}
yvela
dadebiT
ricxvTa
simravles, romelic arcerT elements ar Seicavs carieli simravle
ewodeba
...
vTqvaT, mocemulia ori simravle A da B
...
Aam SemTxvevaSi weren A ⊂ B an B ⊃ A
...
carieli simravle yoveli A
simravlis qvesimravled iTvleba
...
)
MmagaliTi 1
...
simravleebi ∅ , {a } ,
{b }, {c }, {a, b }, {a, c } , {b, c }, {a, b, c } A simravlis qvesimravleebia
...
vityviT, rom A da B toli simravleebia da vwerT: A = B
...
5
1
vTqvaT, A = { ,5,9
}
maSin
da B = {5,1,9 }
...
{
magaliTi
1
...
cxadia, rom A = B = C
...
Mmag
...
Aaracariel simravles ewodeba usasrulo, Tu igi sasruli araa
...
moqmedebebi simravleebze
...
A da B simravleebis gaerTianeba ewodeba yvela im
elementebis simravles, romlebic A da B simravleebidan erT-erTs
mainc ekuTvnian
...
Aamgvarad, A ∪ B = {x x ∈ A an x ∈ B }
...
7
A = { ,7,9 } , B = { ,7,10,11
1
1
}
1
A ∪ B = { ,7,9,10,11 }
...
simboloTi
...
A∩ B
mag
...
8
A = { ,7,9 }, B = { ,7,10,11
1
1
} A ∩ B = {1,7}
A da B simravleebis sxvaoba ewodeba A simravlis yvela im
elementebisagan Semdgar simravles, romlebic
B
simravles ar
ekuTvnian
...
A \ B = {x x ∈ A da x ∉ B }
...
9
...
1
...
10
...
B = { ,10,11}
...
vTqvaT, gvaqvs
ramdenime simravle
...
aRniSvna: A1 ∪ A2 ∪
...
k =1
A1 , A2 ,
...
n
aRniSvna: A1 ∩ A2 ∩
...
k =1
A da B simravleebis simetriuli sxvaoba ewodeba ( A \ B ) ∪ ( B \ A)
simravles
...
ganmartebiT A∆B = ( A \ B ) ∪ ( B \ A)
...
11
A = {1,7,9}, B = {1,7,10,11}
A \ B = {9}, B \ A = {10,11}
A∆B = ( A \ B ) ∪ ( B \ A) = {9,10,11}
M raime aracarieli simravlea da
A ⊂ M
...
Ddebuleba*
...
samarTliania tolobebi
C M ( A ∪ B) = C M A ∩ C M B ,
C M ( A ∩ B) = C M A ∪ C M B
...
maSin
(x ∈ C M ( A ∪ B)) ⇒ ( x ∈ M
da
x ∉ A ∪ B ) ⇒ (x ∈ M , x ∉ A
da
x ∈ M , x ∉ B) ⇒ ( x ∈ C M A da x ∈ C M B ) ⇒ x ∈ (C M A ∩ C M B )
...
axla aviRoT nebismieri x elementi C M A ∩ C M B simravlidan
...
miviReT:
C M A ∩ C M B ⊂ C M ( A ∪ B)
...
analogiurad damtkicdeba meore toloba
...
vTqvaT a da b raime obieqtebia
...
vityviT, rom ori ( a , b) da (c, d ) dalagebuli
wyvili tolia mxolod maSin roca a = c da b = d
...
( a , b) da {a, b} sxvadasxva obieqtebia
...
Yyvela
SesaZlebeli
A
da
dalagebuli
( a , b) wyvilebis simravles, sadac
a ∈ A, b ∈ B
B simravleTa
dekartuli
namravli
ewodeba
da
A × B simboloTi
aRiniSneba
...
Tu A = B
maSin A × A
vuwodebT A simravlis dekartul kvadrats da mas avRniSnavT A 2
simboloTi
...
12 A = { ,7,9 }, B = { ,7,10,11 }
A × B = {(1,1), (1,7), (1,10), (1,11), (7,1), (7,7), (7,10), (7,11), (9,1), (9,7), (9,10), (9,11)}
...
13 A = [0,2] , B = [1,2] , maSin am simravleebis dekartuli
namravls SegviZlia mouZebnoT martivi interpretacia sakordinato
sibrtyeze, Tu gavixsenebT, rom ricxvTa yovel dalagebul ( a , b ) wyvils
sibrtyeze Seesabameba erTaderTi wertili, romlis pirveli kordinatia
a , xolo meore b
...
2
...
marTlac, vTqvaT A = [0,2] , B = [1,2] ,
maSin gveqneba
SevniSnoT, rom simravleTa dekartuli namravlis ganmartebis ZaliT Tu
mocemuli simravleebidan erT-erTi simravle carielia maSin namravlic
carieli simravlea
...
qvemoT Cven visargeblebT Semdegi aRniSvnebiT:
N = {1,2,3,
...
} -naturalur ricxvTa simravle
...
, − n,
...
, n,
...
m
m ∈ Z , n ∈ N } -racionalur ricxvTa simravle
...
marTlac, sakmarisia aviRoT
q + q2
, maSin samarTliania q1 < q < q2 utoloba
...
maTematikis saskolo kursidan cnobilia, rom
yovel racionalur ricxvs ricxviT RerZze Seesabameba garkveuli
wertili
...
es
m
wertili Seesabameba iracionalur ricxvs_ 2 , romlis Cawera
n
wiladis saSualebiT SeuZlebelia
...
ricxviTi RerZis yvela
wertilis Sesabamisi ricxvebis erToblioba warmoadgens namdvil
ricxvTa simravles
...
amasTan, sasruli
an usasrulo perioduli aTwiladi warmoadgens racionalur ricxvs,
xolo usasrulo araperioduli aTwiladi warmoadgens iracionalur
ricxvs
...
N ⊂ Z ⊂ Q, Q ∩ I = ∅
...
R = Q ∪ I
...
qvemoT Cven visargeblebT Semdegi saxis SualedebiT:
[a, b] = {x x ∈ R, a ≤ x ≤ b}- Caketili monakveTi (segmenti)
...
[a, b) = {x x ∈ R, a ≤ x < b} -
...
[a,+∞) = {x x ∈ R, a ≤ x}
(a,+∞) = {x x ∈ R, a < x}
(−∞, a] = {x x ∈ R, x ≤ a}
(−∞, a) = {x x ∈ R, x < a}
(−∞,+∞) = R
mocemul kursSi visargeblebT cnobili ricxviTi
utolobebiT:
debuleba:
yoveli
a, b
namdvili
ricxvebisaTvis
utolobebs:
adgili
aqvs
1) a + b ≤ a + b ;
2) | a | − | b | ≤ a − b
...
WeSmaritia Tu Mmcdari Semdegi
1
1
1
...
1
2
...
vTqvaT, mocemulia simravleebi:
A = { ,2} ,
1
B = {{ }, {2}} ,
1
C = {{ }, { ,2}} da D = {{ }, {2}, { ,2}}
...
4
...
mocemulia simravleebi: A = { 1;
ipoveT:
a) ( A ∪ B ) ∩ C ;
b)
d)
g) ( A ∩ B ) ∪ C ;
6
...
mocemulia simravleebi: A = { 2;
ipoveT:
a) ( A ∩ B ) ∪ C ;
g) ( A ∪ B ) ∩ C ;
3; 4;
},
B = { 3; 7; 9
},
C = { 3; 8; 9
}
...
5; 8; } , B = { 8; 9; 11
},
C = { 8; 10;
}
...
4; 7; } , B = { 7; 8; 9
},
C = { 7; 9;
}
...
mocemulia simravleebi: A = { 5; 7; 8; } , B = { 7; 13 } , C = { 7; 12; 13
ipoveT:
b) ( A \ B ) × ( B \ C ) ;
a) ( A ∩ B ) ∪ C ;
d)
g)
( A ∪ B) ∩ C ;
( A ∩ B) × ( B ∪ C )
...
daStrixeT A × B , Tu
a) A = [1;3] , B = [− 1;4] ;
g) A = (2;4) , B = (− 2;1)
}
...
d) A = [0;4] ;
B = (0;2)
...
daStrixeT A ∪ B, A ∩ B, A \ B, B \ A, A∆B ,Tu
a)
b)
11
...
g) A ∪ ∅ = A ;
d) ( A \ B ) ∪ B = A ∪ B
...
SeamowmeT tolobebi:
b) A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C ;
a) A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C ;
g) A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) ;
d) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) ;
e) ( A ∩ C ) ∪ ( B ∩ D) ⊂ ( A ∪ B) ∩ (C ∪ D) ;
13*
...
14*
...
SeamowmeT tolobebi
...
A \ ( B ∪ C ) = ( A \ B) \ C ;
b)
16*
...
g) samarTliani aris Tu ara toloba
( A × B) ∪ ( X × Y ) = ( A ∪ X ) × ( B × Y ) ?
17
...
2
...
aRniSnuli cnebis gasaazreblad SevniSnoT, rom
fizikuri da
geometriuli kanonzomierebebi gamoisaxeba rogorc ricxvebs Soris
damokidebulebebi
...
magaliTad, Tu viciT, rom kvadratis gverdis
sigrZea x, maSin mis farTobs gamovTvliT calsaxad, formuliT
S = x 2
...
maSasadame kanonzomierebebi,
romlis drosac erTi sididis mniSvneloba calsaxad gansazRvravs
meore sididis mniSvnelobas, aRiwerebian ricxviTi funqciebiT
...
vityviT, rom E simravleze mocemulia (gansazRvrulia) funqcia mniSvnelobebiT R
simravleSi, Tu mocemulia f wesi, romlis saSualebiTac E
simravlis yovel elements Seesabameba R simralis erTaderTi
elementi
...
E simravles funqciis gansazRvris are ewodeba, f -s ki
⎯→
_ funqciis mocemis wesi
...
Tu argumentis
konkretul x0 mniSvnelobas mocemuli f wesiT eTanadeba y 0 ∈ R
ricxvi, maSin y0 -s vuwodebT f funqciis mniSvnelobas x0 wertilSi
x argumentisaTvis ki vwerT
da CavwerT y0 = f ( x0 )
...
Aam SemTxvevaSi y -s damokidebuli cvladi ewodeba
...
Gganmartebis Tanaxmad, gansazRvris aris nebismier
ricxvs
f wesiT Seesabameba erTaderTi y ricxvi, romelic unda
SevniSnoT agreTve, rom y = x 2
daviTvaloT formuliT y = x 2
...
am
SemTxvevaSi, bunebrivia, am wesiT mocemuli funqciis gansazRvris
ares warmoadgens (0,+∞) Sualedi
...
t da z
...
)
zemoTTqmulidan
gamomdinare,
funqciis
mocema
niSnavs,
rogorc gansazRvris E aris miTiTebas, aseve f Sesabamisobis
mocemas
...
aseT
SemTxvevaSi funqciis gansazRvris aris qveS gulisxmoben yvela im
x ricxvTa erTobliobas, romelTaTvisac f (x) gamosaxulebas azri
1
aqvs
...
kalkulusis mocemul kursSi Cven
am midgomiT visargeblebT
...
vityviT,
rom es funqciebi tolia Tu maTi gansazRvris areebi toli
simravleebia; E = F da yoveli x ∈ E ricxvisaTvis f ( x ) = g ( x)
...
Ees funqciebi toli funqciebi ar aris, vinaidan maTi gansazRvris
areebi erTmaneTs ar emTxveva, Tumca f ( x ) = g ( x ) = x 2 , roca x ∈ [0, ∞ )
...
vTqvaT mocemulia ori funqcia f : E → R da
g:F → R
...
(aseT
g funqcia aris
f
F
funqcia
SemTxvevaSi SeiZleba ubralod ase vTqvaT:
f
simravleze emTxveva g funqcias
...
sailustraciod moviyvanoT kidev erTi magaliTi:
x2 −1
ganvixiloT funqciebi f ( x ) =
da g ( x ) = x − 1
...
rogor vipovoT mocemuli y = f (x) funqciis mniSvneloba
gansazRvris aris ama Tu im
wertilSi? ganvixiloT magaliTi:
2
vTqvaT, f ( x) = x + 2 x
...
f (2) = 2 2 + 2 ⋅ 2 = 8 ;
f ( x + 5) = ( x + 5) 2 + 2( x + 5) = x 2 + 12 x + 35
...
moviyvanoT funqciis grafikis ganmarteba
...
Aam funqciis grafiki ewodeba yvela SesaZllebeli
wyvilisagan Semdgar
simravles, sadac x ∈ E
...
Aamgvarad,
Γ f = {( x, y ) x ∈ E , y = f ( x)} = {( x, f ( x )) x ∈ E}
...
es saSualebas gvaZlevs
funqciis
grafiki TvalsaCinod warmovidginoT, rogorc sakoordinato xoy
sibrtyis qvesimravle
...
f :E →R
funqcia
...
maSasadame: f ( X ) = {y ∈ Y | ∃x, x ∈ X da y = f (x)}
...
moyvanili ganmartebidan SegviZlia CamovayaliboT algoriTmi,
imisa Tu rogor davadginoT fiqsirebuli y ∈ R ricxvi ekuTvnis Tu
ara funqciis mniSvnelobaTa ares
...
magaliTi 2
...
f : R → R , f ( x) = x 2 funqciis grafikis, Γ f = {( x, x 2 ) x ∈ R}
eskizs, rogorc skolis kursidan viciT, aqvs saxe (ix
...
TvalsaCinoebisTvis, mocemuli funqciisaTvis ganvixiloT zemoT
moyvanili cnebebTan dakavSirebuli, ramdenime sakiTxi
...
2)
funqciis
mniSvnelobaTa
f ( R ) = [0,+∞ )
...
maSin f ( x) = y 0 gantolebas anu x 2 = y 0 gantolebas aqvs erTi mainc
amonaxsni funqciis gansazRvris aridan
...
Yyovelive
zemoTTqmuls
geometriuli interpretacia
...
magaliTi 2
...
vTqvaT funqcia, mocemulia Semdegi saxiT
⎧ x, x ≤ 0,
f ( x) = ⎨
...
i
...
(im faqts, rom funqciis mniSvneloba x = 0
wertilSi aris 0 da ara -1 naxazze avRniSnavT isriT)
...
3
...
⎪2, x = 0
⎩
maSin f (0) = 2
...
4
...
maSin advili dasanaxia, rom f (−1) = 1, f (0) = 0, f (1) = 2
...
5
Tu funqciis grafiks aqvs Semdegi saxe,
maSin f (1) = 2
...
Gganmarteba: funqcias f : E → R uwodeben mudmiv funqcias Tu misi
mniSvnelobaTa simravle erTelementiania
...
funqciaTa kompozicia
...
f da g funqciebis kompozicia anu rTuli funqcia
ganimarteba Semdegnairad
( f g )( x) = f ( g ( x )) ,
g ( x) ∈ D( f )
...
1 sazogadod, f g da g f sxvadasxvaa
...
vTqvaT, f ( x) = x 2 + 2 x + 3, g ( x) = x + 3
( f g )( x) = f ( g ( x)) = f ( x + 3) = ( x + 3) 2 + 2( x + 3) + 3 = x 2 + 8 x + 18
( g f )( x) = g ( f ( x)) = g ( x 2 + 2 x + 3) = x 2 + 2 x + 3 + 3 = x 2 + 2 x + 6
cxadia, rom am SemTxvevaSi ( f g )( x ) ≠ ( g f )( x )
...
2);
> h:=x->f(g(x));
> h(x);
> r:=x->g(f(x));
> r(x);
h := x → f( g( x ) )
( x + 3 )2 + 2 x + 9
r := x → g( f( x ) )
x2 + 2 x + 6
> plot([h(x),g(x)],x=-2
...
magaliTi 1-dan
Cans, rom aseT tolobas yovelTvis ara aqvs adgili
...
2
...
rogorc zemoT vnaxeT, Tu gvaqvs f ( x ) = 3 x − 2 da
1
2
funqciebi, maSin ( f g )( x ) = ( g f )( x ) = x
...
SevniSnoT, rom Cven
faqtiurad orive SemTxvevaSi ganvixileT funqciaTa kompozicia
...
3
3
3
rogorc vxedavT, f da g funqciebi urTierTSeqceul moqmedebebs
axorcieleben
...
vidre Seqceuli funqciis mkacr ganmartebas moviyvandeT,
moviyvanoT urTierTcalsaxa funqciis ganmarteba
...
f -s ewodeba urTierT
calsaxa funqcia, Tu yoveli ori x1 ≠ x2 -Tvis gansazRvris aridan
gvaqvs f ( x1 ) ≠ f ( x2 )
...
es funqcia ar
aris urTierTcalsaxa funqcia, radgan x –is or gansxvavebul
mniSvnelobas y –is erTidaigive mniSvneloba Seesabameba
...
magram, Tu SevzRudavT
gansazRvris ares [0,+∞ )
Sualedamde, miviRebT
urTierTcalsaxa funqcias
...
vityviT, rom f da g
Seqceuli funqciebia, Tu isini
urTierTcalsaxa funqciebia da
sruldeba toloba
( f g )( x ) = ( g f )( x ) = x
...
analogiurad, f aris
g -s Seqceuli funqcia da CavwerT f ( x) = g −1 ( x)
...
i Cvens mier zemoT ganxiluli f ( x ) = 3 x − 2 da g ( x) = x +
3
3
funqciebisTvis gvaqvs:
1
2
f ( x) = 3x − 2
f −1 ( x) = x +
3
3
1
2
g −1 ( x ) = 3 x − 2
g ( x) = x +
3
3
SeniSvna 2
...
f ( x)
praqtikulad rogor vipovoT Seqceuli funqcia
...
vTqvaT, f ( x ) = 5 x + 2
...
Semdeg x da
y -s adgilebs vucvliT: x = 5 y + 2 da vxsniT gantolebas y -is
1
2
mimarT: y = x −
...
bolos vamowmebT ( f f −1 )( x) = ( f −1 f )( x) = x tolobis
5
5
marTebulobas
...
aviRoT
1
2
funqciebi
...
2,y=-2
...
es wesi
yovelTvis marTebulia mocemuli
funqciisa da misi Seqceuli funqciis
grafikebisTvis
...
vTqvaT, mocemulia
funqcia f : E → R
...
magaliTi 2
...
ganvixiloT funqcia
⎧x 3 ,
f ( x) = ⎨
⎩2 ,
x ∈ (−∞, 0]
x ∈ (0 ,+∞)
E rogorc vxedavT, es funqcia zrdadia Tavis gansazRvris
areze
...
magaliTi 2
...
ganvixiloT funqcia
⎧− x ,
f ( x) = ⎨
⎩− 2 ,
x ∈ (−∞, 0]
x ∈ (0 ,+∞)
rogorc vxedavT, Ees funqcia klebadia Tavis
gansazRvris areze
...
2
...
f funqcias ewodeba mkacrad klebadi E simravleze, Tu
am simravlis yoveli x1 da x 2 ricxvisaTvis
f ( x1 ) > f ( x 2 ) , roca x1 < x 2
...
8 ganvixiloT funqcia
f ( x) = − x 3
es funqcia mkacrad klebadia R –ze
...
SeniSvna 2
...
SeiZleba mocemuli funqcia ar iyos
monotonuri
...
ganvixiloT funqcia
f ( x) = x 2
es funqcia ar aris monotonuri
...
'
'
rogorc vxedavT, f ( x1' ) > f ( x 2 ) , roca x1' < x 2 ,xolo
'
'
f ( x1'' ) < f ( x 2' ) , roca x1'' < x 2'
...
vTqvaT,
mocemulia f : E → R
...
magaliTi 2
...
ganvixiloT funqcia
f ( x) = − x 2 + 1
rogorc vxedavT, yoveli x –Tvis f ( x) ≤ 1
...
i
f ( x) = − x 2 + 1 funqcia zemodan SemosazRvrulia
...
10
...
e
...
f funqcias ewodeba SemosazRvruli, Tu igi
F
SemosazRvrulia rogorc qvemodan, ise zemodan
...
11
...
e
...
- 1 ≤ cos x ≤ 1
f funqcias ewodeba SemousazRvreli E simravleze,
F
Tu nebismieri dadebiTi A ricxvisTvis moiZebneba am
simravlis iseTi x0 wertili , rom
f ( x0 ) > A
...
12
...
Ee
...
f : (0,1] → R
...
magram f araa SemosazRvruli,
x
radgan igi araa zemodan SemosazRvruli
...
aviRoT
1
x0 =
, x0 ∈ (0,1] maSin
2A
f ( x0 ) =
e
...
f ( x) =
1
= 2A > A
...
x
magaliTi 2
...
ganvixiloT
SualedSi
...
i
...
f ( x) = −
1
x
funqcia (0,1]
es funqcia zemodan SemosazRvrulia, radgan yoveli
1
x -sTvis (0,1]-dan
f ( x) = − < 0
...
marTlac, aviRoT nebismieri ragind didi A0 dadebiTi
1
ricxvi ( A > 1 )
...
x0
e
...
f ( x) = −
L
1
araa SemosazRvruli (0,1] -ze
...
vTqvaT, f funqcia gansazRvrulia
koordinatTa saTavis mimarT simetriul E SualedSi
...
magaliTi 2
...
i
...
luwi funqciis grafiki simetriulia ordinatTa
RerZis mimarT
...
e
...
f ( x) = tan x ;
f : (−
π π
, )→R
...
magaliTi 2
...
SeniSvna 2
...
simetriul Sualedze gansazRvruli
funqcia SeiZleba ar iyos arc luwi da arc kenti
...
marTlac,
f (− x) = (− x) + 3 = − x + 3 ⇒ f (− x) ≠ − f ( x) da
f (− x) ≠ f ( x)
...
vTqvaT, mocemulia f : E → R
...
magaliTi 2
...
f ( x) = sin x periodulia periodiT 2π
...
f ( x − T ) = f (( x − T ) + T ) = f ( x)
...
Cveulebriv, funqciis
periods uwodeben yvela
dadebiT periodebs Soris
umciress
...
SevniSnoT, rom funqciis periodTa
Soris SeiZleba ar arsebobdes umciresi
...
funqciis eqstremumi
...
x0 ∈ E wertils ewodeba f funqciis
lokaluri maqsimumis wertili, Tu moiZebneba x0 -is iseTi
ε − midamo ( x0 − ε , x0 + ε ), rom yoveli x –Tvis ( x0 − ε , x0 + ε ) ∩ E
simravlidan gvaqvs
f ( x) ≤ f ( x0 )
...
x = −2, x = −1, x = 2, x = 4 wertilebi lokaluri maqimumis
wertilebia
x0 ∈ E wertils ewodeba f funqciis lokaluri
minimumis wertili, Tu moiZebneba x0 –is iseTi ε − midamo,
rom yoveli x –Tvis ( x0 − ε , x0 + ε ) ∩ E simravlidan gvaqvs
f ( x) ≥ f ( x0 )
...
x = −4, x = 1, x = 3 wertilebi lokaluri minimumis wertilebia
...
x0 ∈ E wertils ewodeba f funqciis mkacri lokaluri
maqsimumis wertili, Tu moiZebneba x0 -is iseTi ε − midamo
( x0 − ε , x0 + ε ), rom yoveli x –Tvis
( x 0 − ε , x 0 + ε ) ∩ E \{ x0 } simravlidan gvaqvs
f ( x) < f ( x0 )
...
x = 2, x = 4 wertilebi mkacri maqsimumis wertilebia, xolo
x = −2, x = −1 ar aris mkacri lokaluri maqimumis wertilebi
...
f ( x0 ) ricxvs uwodeben mkacr lokalur minimums
...
x = −4, x = 1, x = 3 wertilebi mkacri minimumis wertilebia
...
x = −2, x = −1, x = 1, x = 2, x = 3 wertilebi Siga eqstremumis
wertilebia
...
vTqvaT, mocemulia f : E → R , sadac E raime Sualedia
...
x = 2 wertili globaluri maqsimumis wertilia
...
x = −4 wertili globaluri minimumis wertilia
...
magaliTi 2
...
x = −8
aris globaluri
minimumis wertili,
f (− 8) = −2 aris
globaluri
minimumi
...
magaliTi 2
...
am funqciis globaluri maqsimumi aris 1,
xolo globaluri minimumi aris -1
...
5
...
zogierTi elementaruli funqcia da maTi Tvisebebi
wrfivi da kvadratuli funqciebi
wrfivi funqcia
funqcias, romelic gansazRvrulia
gansazRvreba
...
wrfivi funqciis grafiki warmoadgens wrfes da piriqiT,
sakoordinato sibrtyeze mdebare yoveli wrfe, romelic ar aris
ordinatTa RerZis paraleluri warmoadgens romeliRac wrfivi
funqciis grafiks
...
k sidides zogjer wrfis daxrasac uwodeben
...
1
...
Tu k ≠ 0 , maSin wrfivi funqciis mniSvnelobaTa simravlea
E ( f ) = , xolo Tu k = 0 , maSin E ( f ) = {b} ;
3
...
Tu k ≠ 0 da b = 0 , maSin funqcia kentia, xolo Tu k = 0 maSin
funqcia luwia
...
5
...
2a
4a
b 2 − 4ac
gamosaxulebas
ax 2 + bx + c
kvadratuli
samwevris
diskriminanti ewodeba da D simboloTi aRiniSneba
...
Tu b ≠ 0 , maSin
4ac − b 2
b
= t gardaqmniT miRebuli g (t ) = at 2 +
funqcia aris luwi
4a
2a
b
wrfis
da maSasadame f funqciis grafiki simetriulia x = −
2a
mimarT
...
aqedan naTelia, rom f funqciis
wveros koordinatebia (0,
4a
grafiki
agreTve
warmoadgens
parabolas,
romlis
Stoebi
mimarTulia zemoT (qvemoT), rodesac a > 0 ( a < 0 ), xolo wveros
b 4ac − b 2
)
...
1
...
2
...
4a
3
...
2a
b) rodesac a < 0 funqcia mkacrad zrdadia (−∞, −
b
) intervalSi
2a
b
, +∞) intervalSi
...
funqciis grafiki warmoadgens parabolas, romlis wveros
b 4ac − b 2
)
...
funqciis grafiki
b
wrfis mimarT
...
kvadratuli funqciis grafiks Tvisobrivad axasiaTebs a da D
parametrebi
...
⎧a > 0,
⎨
⎩ D = 0
...
u0
x2
x
y
v0
o
⎧a < 0,
⎨
⎩ D = 0
...
u0
x
x1
o
⎧a < 0,
⎨
⎩ D < 0
...
funqciis gansazRvris area namdvil ricxvTa simravle e
...
D(sin) = ;
2
...
funqcia kentia, e
...
∀x ∈
ricxvisaTvis samarTliania toloba
sin(− x) = − sin x ;
4
...
i
...
funqcia
dadebiT
mniSvnelobebs
Rebulobs
∪ (2π k , π + 2π k ) simravleze,
k∈
xolo
mniSvnelobebs
∪ (−π + 2π k , 2π k )
uaryofiT
Rebulobs
k∈
simravleze;
6
...
funqciis grafiks aqvs Semdegi saxe (mas sinusoidas uwodeben)
y = cos x funqcia
1
...
i
...
funqciis mniSvnelobaTa area E (cos) = [−1,1] segmenti;
3
...
i
...
funqcia periodulia, mas
gaaCnia umciresi dadebiTi
periodi,
romelic
2π -s
da ∀k ∈
tolia, e
...
∀x ∈
ricxvebisaTvis
cos( x + 2π k ) = cos x ;
5
...
∀k ∈
mTeli ricxvisaTvis funqcia zrdadia (−π + 2π k , 2π k )
saxis yovel intervalSi, xolo klebadia (2π k , π + 2π k ) saxis
TiToeul intervalSi;
7
...
funqciis gansazRvris area D(tan) =
2
...
funqcia kentia, e
...
nebismieri x Tvis gansazRvris aredan
samarTliania toloba
tan(− x) = − tan x ;
4
...
i
...
funqcia
dadebiT
mniSvnelobebs
Rebulobs ∪ (π k ,
k∈
xolo
2
uaryofiT
Rebulobs ∪ (−
k∈
6
...
funqciis grafiks aqvs Semdegi saxe
(−
+ π k,
π
y = cot x funqcia
1
...
funqciis
mniSvnelobaTa
area
E (cot) =
namdvil
ricxvTa
simravle;
3
...
i
...
funqcia periodulia, mas gaaCnia
umciresi
dadebiTi
periodi,
romelic π -s tolia, e
...
∀x ∈
da
∀k ∈
ricxvebisaTvis
cot( x + π k ) = cot x ;
5
...
∀k ∈
mTeli ricxvisaTvis funqcia klebadia (π k , π + 2π k )
saxis yovel intervalSi;
7
...
arcsin
e
...
simboloTi
arcsin :[−1,1] → [−
π π
, ],
2 2
tolobiT:
romelic
ganimarteba
sin(arcsin x) = x , ∀x ∈ [−1,1]
...
6
...
1
...
y = arcsin x funqciis mniSvnelobaTa area E (arcsin) = [−
ravle;
3
...
i
...
funqcia zrdadia yvelgan Tavis gansazRvris areSi;
5
...
mas aqvs
y = arccos x funqcia
mtkicdeba, rom y = cos x funqciis SezRudva [0, π ] segmentze
warmoadgens urTierTcalsaxa funqcias [0, π ] segmentidan [−1,1]
segmentze, amitom arsebobs am funqciis cos |[0,π ] Seqceuli funqcia,
romelsac
arccos
simboloTi
aRniSnaven
...
i
...
SeniSvna 2
...
arccos funqcia
warmoadgens
cos |[0,π ]
funqciis
Seqceuls da ara cos funqciis
Seqceuls, amitom arccos(cos x) = x
toloba sazogadod arasworia,
is WeSmaritia mxolod maSin,
rodesac x ∈ [0, π ]
...
2
...
4
...
y = arccos x
funqciis
gansazRvris
area
D(arccos) = [−1,1] simravle;
y = arccos x
funqciis
mniSvnelobaTa area E (arccos) = [0, π ] simravle;
arccos(− x) = π − arccos x , ∀x ∈ [−1,1] ;
funqcia klebadia yvelgan Tavis gansazRvris areSi;
aRniSnuli funqciis grafiki warmoadgens cos |[0,π ] funqciis
grafikis simetriul grafiks y = x RerZis mimarT
...
arctan
e
...
simboloTi
arctan :
→ (−
π π
, ),
2 2
tolobiT:
romelic
ganimarteba
tg (arctan x) = x , ∀x ∈
...
8
...
1
...
y = arctan x
funqciis
mniSvnelobaTa
area
E (arctan) = (−
simravle;
3
...
i
...
funqcia zrdadia yvelgan Tavis gansazRvris areSi;
5
...
mas aqvs
y = arc cot x funqcia
mtkicdeba, rom y = cot x funqciis SezRudva (0, π ) intervalze
warmoadgens urTierTcalsaxa funqcias (0, π ) intervalidan namdvil
ricxvTa
RerZze, amitom
arsebobs am funqciis cot |(0,π )
Seqceuli funqcia, romelsac
arc cot simboloTi aRniSnaven
...
i
...
SeniSvna 2
...
arc cot
funqcia warmoadgens
cot |(0,π )
funqciis Seqceuls da ara cot
funqciis
Seqceuls,
amitom
arc cot(cot x) = x toloba sazogadod arasworia, is WeSmaritia
mxolod
maSin,
rodesac
x ∈ (0, π )
...
y = arc cot x
funqciis
simravle;
gansazRvris area D(arc ctg ) =
funqciis mniSvnelobaTa area
2
...
arc cot(− x) = π − arc cot x , ∀x ∈ ;
4
...
aRniSnuli funqciis grafiki warmoadgens cot |(0,π ) funqciis
grafikis simetriul grafiks y = x RerZis mimarT
...
1
...
maCvenebliani funqciis ZiriTadi Tvisebebi
1
...
maCvenebliani funqciis mniSvnelobaTa area E ( y ) = (0, +∞) ;
3
...
a x + y = a x ⋅ a y ;
ax
5
...
(ab) = a x ⋅ b x ;
a
ax
7
...
(a x ) y = a xy ;
9
...
a)
b)
logariTmuli funqcia
gansazRvreba 2
...
dadebiTi b ricxvis logariTmi a > 0, a ≠ 1
fuZiT ewodeba iseT c ricxvs, rom a c = b
...
gansazRvreba 2
...
vTqvaT
a>0
da
a ≠1
raime
fiqsirebuli namdvili ricxvia
...
mas
mokled
y = log a x saxiT weren
...
10
...
amitom misi grafiki miiReba
maCvenebliani funqciis
grafikisagan y = x wrfis
mimarT simetriiT
...
logariTmuli funqciis gansazRvris area
Sualedi;
2
...
Tu a > 1 , maSin y = log a x
funqcia zrdadia mTels
gansazRvris
areze,
xolo Tu 0 < a < 1 , maSin
y = log a x
funqcia
klebadia aseve mTels
gansazRvris areze;
4
...
Tu
x⋅ y > 0,
maSin
log a ( x ⋅ y ) = log a | x | + log a | y | ,
;
x⋅ y > 0,
maSin
6
...
∀p, q ∈ ,
p
log aq x p = log a x ;
q
log c x
8
...
a loga b = b , a logb c = c logb a ;
Mmoqmedebebi funqciaTa grafikebze
y = f ( x) funqciis grafikis saSualebiT avagoT
y = − f (x) funqciis grafiki
...
Aamoxsna
...
y = − f ( x) funqciis mniSvneloba x = x 0 wertilze
iqneba − f ( x0 )
...
Ees wertilebi simetriuli wertilebia OX RerZis mimarT
...
y = f (x) funqciis grafikis saSualebiT avagoT
y =| f ( x) | funqciis grafiki
...
Aamoxsna
...
maSasadame y =| f ( x) | funqciis grafiki emTxveva y = f (x) funqciis
grafiks, gansazRvris aris im wertilebSi sadac f ( x) ≥ 0 da
emTxveva y = − f ( x) funqciis grafiks gansazRvris aris im
wertilebSi sadac f ( x) < 0
...
Aamocana 3
...
wertilebi M = ( x0 , f ( x0 )) da
M = ( x0 , f ( x0 ) + b) ekuTvnian Sesabamisad y = f (x) da y = f ( x) + b
funqciis grafikebs
...
maSasadame y = f ( x) + b
funqciis grafikis misaRebad y = f (x) funqciis grafiki unda
Aamoxsna
...
y = f ( x) funqciis grafikis saSualebiT avagoT
y = f (x − a) funqciis grafiki
...
Aamoxsna
...
maSasadame M = ( x0 , f ( x0 )) wertili ekuTvnis y = f ( x) funqciis
grafiks, xolo N = ( x0 + a, f ( x 0 )) ekuTvnis y = f ( x − a) funqciis
grafiks
...
maSasadame y = f ( x − a) funqciis
grafikis misaRebad y = f ( x) funqciis grafiki unda gadavitanoT
paralelurad OX RerZis gaswvriv marjvniv Tu a > 0 da marcxniv Tu
a < 0
...
Aamocana 5
...
y = f (| x |) funqcia luwi funqciaa amitom misi grafiki
simetriulia OY RerZis mimarT
...
maSasadame y = f (| x |) funqciis asagebad avagoT jer y = f (x)
funqciis grafiki, roca x ≥ 0 ; xolo Semdeg misi simetriuli
figura OY RerZis mimarT
...
Aamocana 6
...
cxadia Tu y = f ( x) fynqciis grafikis wertilis ordinats
gavamravlebT k ze miviRebT y = kf (x) funqciis grafikis Sesabamis
ordinats
...
Tu k < 0 , maSin sakmarisia jer
avagoT y = −kf ( x) funqciis grafiki Sendeg ki saZiebeli funqciis
grafiki (amocana 1)
...
Aamocana 7
...
y = f (kx) funqciis grafiki miiReba y = f (x) funqciis
grafikis k -jer SekumSviT, roca k > 1 da k -jer gaWimviT, roca
0 < k < 1 OX RerZis gaswvriv
...
ax + b
romelsac wiladsailustracioT ganvixiloT funqcia y =
cx + d
wrfiv funqcias uwodeben
...
vaCvenoT, rom roca c ≠ 0 da
c d
maSin wilad wrfivi funqciis grafiki hiperbolaa
...
= +
d
cx + d c cx + d c
x+
c
bc − ad
a
k
= k
...
b−
savarjiSoebi
1
...
TiToeuli
Sesabamisoba
wyvilebis saSualebiT
...
are
1
5
2
mniSvnelobaTa
...
are
4) gansazRvris
are
mniSvnelobaTa
...
qvemoT
Sesabamisobebi
mocemulia
wyvilebis
saSualebiT
...
1) (2;3), (--7;5), (0;--1), (3;--1), (0;--5), (4;1);
2) (--2;3), (7;--5), (0;--9), (3;--11), (3;5), (4;21)
3) (12;3), (--17;5), (10;--1), (23;--1), (120;--5), (41;1);
4) (31;23), (70;--5), (0;--9), (31;--11), (3;5), (44;21)
3
...
gamoarkvieT, romeli
maTgani gansazRvravs funqcias
...
qvemoT mocemuli mimarTebebisaTvis daadgineT gansazRvris are,
mniSvnelobaTa simravle, aageT grafiki da gaarkvieT aris Tu ara
x
mocemuli mimarTeba funqcia
...
vTqvaT f funqcia mocemulia formuliT f ( x) = 3x − 2 , gamoTvaleT:
1) f (2) , f (1) , f (−2) ;
5
...
funqcia
f
mocemulia
1) f (6) , f (−3) , f (−2) ;
6
...
vTqvaT
h(u ) = 3u
2
mocemulia
formuliT
2) f (−1) , f (0) , f (m)
...
F (v) = v − v 2
...
8
...
WeSmaritia Tu ara Semdegi tolobebi?
1) f (at ) = af (t ) , ∀a, t ∈
;
2) f (a + b) = f (a ) + f (b) , ∀a, b ∈
3) f (a ⋅ b) = f (a) ⋅ f (b) , ∀a, b ∈
;
...
vTqvaT f ( x) = x 2
...
10
...
11
...
12
...
b) ipoveT b da c , Tu y = − x 2 + bx + c parabola abscisTa RerZs
exeba (− 6; 0 ) wertilSi
...
ipoveT b da c , Tu y = − x 2 + bx + c funqciis udidesi mniSvnelobaa 3 , romelsac is Rebulobs x = 0 wertilSi
...
ipoveT
f
funqciis udidesi da umciresi mniSvnelobebi
miTiTebul SualedebSi:
b) f ( x) = − x 2 − 4 x + 9 ,
a) f ( x) = x 2 − 3 x + 9 , x ∈ [0, 4] ;
x ∈ [−2, 4] ;
x ∈ [−3,5] ;
g)
f ( x) = −3 x 2 + 4 x + 11 ,
d)
f ( x) = 3 x 2 + 2 x − 17 ,
x ∈ [−3, 6]
...
gamoarkvieT, ramdeni saerTo wertili aqvs f da g funqciaTa
grafikebs da ipoveT isini:
a) f ( x) = x 2 − 3 x + 9 , g ( x) = 2 x − 3 ;
b) f ( x) = − x 2 − 4 x + 9 , g ( x) = 2 x − 5 ;
g) f ( x) = −3 x 2 + 4 x + 11 , g ( x) = 10 x + 14 ;
d) f ( x) = 3 x 2 + 2 x − 17 , g ( x) = − x 2 + 2 x − 7
...
ipoveT a parametris yvela mniSvneloba, romelTaTvisac f
funqciis grafiki exeba abscisaTa RerZs:
a) f ( x) = ax 2 + (2a − 3) x + 7 ; b) f ( x) = (a − 1) x 2 − (a − 4) x + 7 − 2a
...
miuTiTeT
simetriis RerZi, wveros koordinatebi da minimaluri an maqsimaluri mniSvnelobebi
...
4) f ( x) = x 2 − 10 x + 25;
5) h( x) = 2 + 4 x − x 2 ;
6) g ( x) = − x 2 − 6 x − 4 ;
7) f ( x) = 6 x − x 2 ;
8) G ( x) = 16 x − 2 x 2 ;
9) F ( s ) = s 2 − 4 ;
10) g (t ) = t 2 + 4 ;
11) F ( x) = 4 − x 2 ;
12) G ( x) = 9 − x 2
...
daxazeT Semdegi kvadratuli funqciebis grafikebi
...
1) f ( x) = x 2 − 7 x + 10 ;
2) g (t ) = t 2 − 5t + 2 ;
3) g (t ) = 4 + 3t − t 2 ;
4) h( x) = 2 − 5x − x 2 ;
5) f ( x) =
1 2
x + 2x ;
2
7) f ( x) = −2 x 2 − 8 x − 2 ;
6) f ( x) = 2 x 2 − 12 x + 14 ;
1
8) f ( x) = − x 2 + 4 x − 4
...
a) y = sin x;
20
...
a) y = sin x + 1;
g) y = 2 sin x − 3;
y = sin 2 x;
x
d) y = sin
...
3
b) y = sin x − 1;
d) y = 3 sin x − 1
...
a) y = sin( x + );
3
b) y = −2 sin( x − );
4
b)
g) y = − sin 3x;
π
π
G
π
π
g) y = sin( x + ) − 1;
4
23
...
a) y = cos x + 1;
g) y = 2 cos x − 1;
d) y = 2 sin( x − ) + 1
...
3
b) y = −2 cos x;
x
g) y = −3 cos
...
26
...
a) y = tgx;
x
g) y = tg ;
2
28
...
a) y = tgx + 1;
g) y = 2tgx + 1;
d) y = 2 cos( x + )
...
a) y = 2 cos x;
π
π
π
π
d) y = −tg 3x
...
2
b) y = tgx − 2;
d) y = 3tgx − 2
...
a) y = tg ( x + );
4
π
b) y = −tg ( x + );
3
π
π
g) y = −2tg ( x + );
6
d) y = tg ( x + ) − 1
...
b) y = cos x ;
31
...
a) y = sin x ;
g) y = sin x ;
d) y = cos x
...
a) y = tgx ;
b) y = ctgx ;
g) y = tg x ;
d) y = ctg x
...
a) y = sin x ,
y = sin 2 x ,
b) y = cos x ,
g) y = tgx;
y = cos 2 x ,
y = tg 2 x;
y = sin 2 x ,
y = cos x ,
y = tg 3 x;
3
y = sin x ;
y = cos x ⋅ sin x;
y = ctgx;
d) y =
2sin 3 x
,
1 + cos x
y=
sin 2 x + cos x
,
x3
y=
x 2 + cos x
,
x 2 − cos x
y = sin 3 x +
1
...
a) y = sin( x + 1) ;
g) y = 2 sin x;
36
...
a) y = cos x;
x
g) y = cos ;
5
x
38
...
x
b) y = sin ;
3
d) y = sin(4πx)
...
b) y = tg 5 x + 3;
d) y = ctg (πx + 1)
...
a)
;
b)
;
1 − sin x
cos x
1
1
d) cos
...
a)
1 − tgx
x
1
1
g)
;
d)
...
a) 2 sin x − 3;
g) sin 2 3x + 4;
d) − 2 cos 2 x + 1
...
a) sin x + cos x;
b) cos x − sin x;
3
1
sin x − cos x;
d) cos x − 3 sin x
...
ipoveT funqciis zrdadobisa da klebadobis (maqsimaluri sigrZis) intervalebi:
g)
...
a) y = 2 x ;
⎛1⎞
45
...
a) y = 2 x +1 ;
4
− 2 x)
...
−x
x
;
π
⎛1⎞
y =⎜ ⎟ ;
⎝ 3⎠
g)
⎛1⎞
g) y = ⎜ ⎟
⎝2⎠
b) y = 3 x −1 ;
x
⎛2⎞
d) y = ⎜ ⎟
...
a) y = 2 + 3 ;
b) y = 4 − 5 ;
48
...
a) y = log 1 x ;
b) y = log 3 (− x) ;
b) y = log 1 x ;
x
g) y = 2
x
2
x +1
−1;
g) y = log 5 x ;
g) y = log 2 (− x) ;
3
b) y = log 2 (3 − x) ;
d) y = log 1 ( x − 5)
...
2
4
51
...
a) y = 2 ;
x
b) y = 3
53
...
2
ipoveT funqciis gansazRvris are:
54
...
a) log 2 ( x + 1) ;
g)
log π ( x 2 − 4) ;
2− x
;
x +1
7 − 2x
;
g) log 8
2 − 3x
x
57
...
a) log 2
⎛1⎞
d) y = ⎜ ⎟ + 2
...
d) y = log 0,1 x
...
a) y = log 2 ( x + 1) ;
g) y = log 1 ( x + 1) ;
b) y = x 2 + 3x − 10 ;
d) y = x 2 + 4 x + 5
...
2x + 5
;
x −1
x
d) log 3
...
x−2
...
;
58
...
ipoveT f g , g f , f f , g g , Tu
2x + 1
a) f ( x) =
g ( x) = 5 x + 9
3x + 3
b) f ( x) = x 2 + 5 x + 3
g ( x) = 3 x + 9
g) f ( x) = 2 x − x 2
g ( x) = sin x
2x
g ( x) = x 2
d) f ( x) = 2
x +1
g ( x) = 2 x
e) f ( x) = x 2
v) f ( x) = x
g ( x ) = cos x
60
...
⎩ − x , x > 0
...
⎧ x, x < 0;
g ( x) = ⎨ 2
⎩ x , x ≥ 0
...
61
...
2x + 1
3x + 3
d) f ( x) = x 3 + 2
b) f ( x) =
62
...
d)
⎧ x,
⎪
f ( x) = ⎨ x 2 ,
⎪2 x ,
⎩
x < 1;
1 ≤ x ≤ 4;
x > 4
...
Semdegi funqciebidan romelia urTierTcalsaxa funqcia
2x + 1
a) f ( x) =
b) f (x) = 2 x − x 2
3x + 3
g) f ( x) = x 2 + 1
d) f ( x ) = x + 2
64
...
65
...
b) f ( x) =
;
1− x
1+ x
x
1+ x
;
d) f ( x) =
g) f ( x) =
1− x
1+ x2
66
...
ipoveT
Semdegi funqciebis gansazRvris are:
b) f (cos x ) ;
a) f (sin x) ;
g) f (sin 2 x) ;
67
...
a) f ( x + 1) = x 2 − 3 x + 2 ;
1
g) f ( ) = x + 1 + x 2 ;
x
x > 0
...
x +1
x ≠ 0;
68
...
ipoveT
periodi:
Semdegi
funqciebis
umciresi
dadebiTi
b)
sin 2 x sin 3x
+
2
3
f ( x) = A cos kx + B sin kx
g)
f ( x) =
a)
d)
70
...
ipoveT Semdegi funqciebis zrdadoba-klebadobis
Sualedebi,
lokaluri
da
globaluri
eqstremumis
wertilebi
⎧ x, x ∈ (−∞,2]
⎪ 2, x ∈ (2,3]
⎪
a) f ( x) = ⎨
⎪5 − x, x ∈ (3,5]
⎪ x − 5, x ∈ (5,+∞)
⎩
b)
g)
d)
⎧ x, x ∈ [−5,2]
⎪ 2, x ∈ (2,3]
⎪
f ( x) = ⎨
⎪5 − x , x ∈ (3,5]
⎪ x − 5, x ∈ (5,10]
⎩
⎧ 1
⎪ x − 1 , x ∈ [2,3]
⎪
⎪
f ( x) = ⎨ 2, x ∈ (3,4]
⎪6 − x , x ∈ (4,5]
⎪
⎪ x − 4, x ∈ (5,10]
⎩
f ( x) = x 2 + 2 x, x ∈ [−2,2]
72
...
ipoveT Semdegi funqciis
lokaluri da globaluri
eqstremumis wertilebi
...
funqciis zRvari
x 2 + 4 x − 12
magaliTi 3
...
ganvixiloT funqcia f ( x) =
da
x 2 − 2x
SevecadoT SeviswavloT mocemuli
funqciis yofaqceva x = 2 wertilis
midamoSi
...
a
wertilis midamos vuwodebT
( a − ε ; a + ε ) saxis intervals, sadac ε
dadebiTi ricxvia
...
x
2
...
1
2
...
001
2
...
00001
f (x )
3
...
857142857
3
...
998500750
3
...
999985000
x
1
...
9
1
...
999
1
...
99999
rogorc am cxrilidan vxedavT,
Tu aviRebT x -s x = 2 wertilis
marjvniv da vamoZravebT x = 2 -ken,
funqciis mniSvnelobebi
miuaxlovdebian 4-s
...
e
...
f funqciis mniSvnelobebi
uaxlovdebian 4-s, roca x wertilebi
uaxlovdebian x = 2 wertils
...
0
4
...
015075377
4
...
000150008
4
...
x→2
x 2 − 2x
amrigad, vityviT, rom y = f (x ) funqciis zRvari x = a wertilSi
aris A , Tu SegviZlia f ( x ) ragind davuaxlovoT A -s, roca x
sakmaod axlosaa a -Tan
...
rogorc zemoT aRvniSneT, es ganmarteba ar aris zRvris mkacri
ganmarteba
...
da
mainc, ufro zustad, ras gulisxmobs zemoT moyvanili msjeloba
...
winaswar
ganvsazRvroT A -Tan ra siaxloveSi gvinda iyos f ( x )
...
001 erTeuliT daSorebuli
...
001 Tu f ( x ) > A
A − f ( x ) < 0
...
i
...
mniSvnelovania, SevniSnoT, rom x -is mniSvnelobebi unda aviRoT
x = a wertilis orive mxares
...
Cven xSirad gamoviyenebT im faqts, rom zRvari
saSualebas gvaZlevs, miviRoT garkveuli informacia, ra xdeba x = a
wertilis midamoSi, magram zRvris mosaZebnad ar aris saintereso,
ra xdeba TviT a wertilSi
...
zemoT
Cvens mier ganxilul magaliTSi
x = 2 wertilSi funqcia ar iyo
gansazRvruli, magram am
wertilSi Cven ganvixileT
funqciis zRvari
...
2
...
Tu CavatarebT igive
msjelobas, rac iyo pirvel magaliTSi, vnaxavT, rom
lim g ( x ) = 4
...
3
...
funqciis zRvris
ganmartebis ZaliT, funqciis
mniSvnelobebi unda miuaxlovdnen erT
konkretul sidides, roca x uaxlovdeba
a -s (orive mxridan)
...
e
...
am funqcias x = 0 wertilSi ara aqvs zRvari
...
rogorc vnaxeT, magaliT 3
...
amis mizezi
aris is, rom funqciis mniSvnelobebi ori sxvadasxva ricxvis
irgvliv iyrian Tavs, imis da mixedviT, Tu romeli mxridan
uaxlovdebian x -ebi 0-s
...
w
...
marjvena zRvari
...
am faqts ase CavwerT:
lim f ( x ) = A an, f ( x) → A , roca
x→a +
x → a+
...
vityviT, rom
y = f ( x ) funqciis zRvari
marcxnidan x = a wertilSi aris
A ricxvi, Tu SegviZlia f ( x )
ragind davuaxlovoT A -s, roca
x sakmaod axlosaa a -Tan
marcxnidan
...
magaliTi 3
...
kidev erTxel
⎧0 , x < 0
davubrundeT zemoT ganxilul h( x) = ⎨
funqcias
...
e
...
lim h( x ) = 1
x →0+
analogiurad, Tu x marcxnidan uaxlovdeba 0-s ( x < 0 ), maSin
funqciis mniSvnelobebi uaxlovdebian 0-s
...
i
...
magaliTi 3
...
vipovoT lim g ( x ) da lim g ( x) , Tu
x→2+
x→2−
⎧ x + 4 x − 12
,x≠2
⎪
g ( x) = ⎨ x 2 − 2 x
⎪
5 , x=2
⎩
2
rogorc vxedavT, am
SemTxvevaSi lim g ( x ) = 4 da
x→2+
lim g ( x ) = 4
...
x→2
SevniSnoT, rom calmxrivi
zRvrebis SemTxvevaSic
sainteresoa, ra xdeba wertilis
midamoSi da ara TviT am
wertilSi
...
3 da 3
...
arsebobs garkveuli kavSiri calmxriv zRvrebsa da funqciis
zRvars Soris
...
1
...
x→a +
x→a −
magaliTi 3
...
vipovoT naxazze
mocemuli funqciis zRvrebi
x = −1, x = 0, x = 1 wertilebSi
...
x →−1
x →0
meores mxriv,
f ( −1) = 0, f ( 0 ) = 1,
x →1
f (1) = 1
...
7
...
x →4−
funqciis zRvris Tvisebebi
...
vigulisxmoT, rom
arsebobs lim f ( x ) da lim g ( x) da c
x→ a
x→ a
raime mudmivia
...
2) lim[ f ( x) ± g ( x )] = lim f ( x ) ± lim g ( x )
x→a
x→a
x→a
sxva sityvebiT, jamis (sxvaobis) zRvari zRvarTa jamis (sxvaobis)
tolia
...
4)
⎡ f ( x) ⎤ lim f ( x)
= x→a
lim ⎢
x→a g ( x) ⎥
⎦ lim g ( x)
⎣
x→a
, Tu lim g ( x ) ≠ 0
x→a
sxva sityvebiT, Sefardebis zRvari zRvarTa Sefardebis
mniSvnelis zRvari ar udris nuls
...
Tu
x →a
x →a
α ∈ N , maSin lim[ f ( x)]n = [lim f ( x)]n , sadac f nebismieri funqciaa
...
magaliTad, Tu n = 2 , maSin
lim[ f ( x)]2 = lim[ f ( x) f ( x)] = lim f ( x) lim f ( x) = [lim f ( x)]2
x →a
x →a
x →a
x→a
6) lim n f ( x ) = n lim f ( x )
x→a
x→a
44
x →a
es Tviseba me-5 Tvisebis kerZo SemTxvevaa
...
x→a
sxva sityvebiT, mudmivi funqciis zRvari TviT am mudmivis tolia
...
x→a
8) lim x = a
x→a
ganvixiloT f ( x) = x funqcia da SevxedoT mis grafiks
cxadia, rom lim f ( x ) = a
...
8
...
miviRebT
lim (3 x 2 + 5 x − 9) = 3 lim x 2 + 5 lim x − lim 9
x → −2
x → −2
x → −2
x → −2
= 3(−2) + 5(−2) − 9 = −7
2
SevniSnoT, rom am magaliTSi p( x) = 3 x 2 + 5 x − 9 polinomia da
aRmoCnda, rom lim p ( x ) = p ( −2) = −7
...
sazogadod
x → −2
marTebulia Semdegi faqti:
Tu p ( x ) raime n –uri xarisxis polinomia, maSin
lim p ( x) = p ( a )
...
vipovoT
6 − 3 x + 10 x 2
lim
x →1 − 2 x 4 + 7 x 3 + 1
SevniSnoT, rom Cven SegviZlia me-4 Tvisebis gamoyeneba, radgan
lim(−2 x 4 + 7 x 3 + 1) ≠ 0
...
i
...
arsebobs funqciebi, romelTaTvisac lim f ( x ) = f ( a )
...
manamde CamovweroT zogierTi
gavrcelebuli funqcia, romelTac aqvT es “kargi” Tviseba
...
marTebulia Semdegi Teorema
...
2
...
maSin
x →c
lim h( x) = A
x →c
magaliTi 3
...
vipovoT Semdegi zRvari
⎛1⎞
lim x 2 cos⎜ ⎟
...
10
...
ipoveT lim f ( x )
...
2-is ZaliT
lim f ( x ) = 5
...
vityviT, rom lim f ( x) = +∞ , Tu SegviZlia
x→a
f ( x ) gavxadoT ragind didi, roca x sakmaod axlosaa a -Tan da
x ≠ a
...
47
vityviT, rom lim f ( x) = ∞ , Tu SegviZlia f (x) gavxadoT ragind didi,
x→a
roca x sakmaod axlosaa a -Tan da x ≠ a
...
magaliTi 3
...
ganvixiloT f ( x) =
vipovoT
2x
lim
x →3− x − 3
2x
x → 3+ x − 3
lim
2x
funqciis grafiki
x−3
rogorc vxedavT, roca x → 3 − , maSin f ( x) =
x → 3 + , maSin f ( x) =
e
...
lim
x → 3−
2x
= −∞
x−3
2x
→ +∞
x−3
2x
= +∞
lim
x →3+ x − 3
magaliTi 3
...
vTqvaT
f ( x) =
2x
→ −∞ , xolo roca
x−3
, maSasadame lim
x →3
1
...
e
...
lim 2 = +∞
...
x −3
gansazRvreba 3
...
x→a −
x→a +
x →− a
x →+ a
Cvens mier zemoT ganxilul magaliT 3
...
funqciis zRvari usasrulobaSi
...
2 vTqvaT,
mocemulia f : ( a;+∞) → R funqcia
...
gansazRvreba 3
...
vTqvaT,
mocemulia f : (−∞; a ) → R funqcia
...
gansazRvreba 3
...
vityviT,
rom f funqciis zRvari, roca x → ∞ aris
p ricxvi da CavwerT lim f ( x) = p , Tu
x →∞
SegviZlia f ( x ) ragind davuaxlovoT p -s,
roca x sakmarisad didia
...
5
...
vityviT, rom f
funqciis zRvari, roca x → +∞ aris + ∞
da CavwerT lim f ( x) = +∞ , Tu SegviZlia
x → +∞
f ( x ) gavxadoT ragind didi, roca x
sakmarisad didia
...
vityviT, rom f funqciis zRvari,
roca x → −∞ aris + ∞ da CavwerT lim f ( x) = +∞ , Tu SegviZlia f ( x )
x → −∞
gavxadoT ragind didi, roca − x sakmarisad didia
...
vityviT, rom f funqciis
+ ∞ da CavwerT lim f ( x) = +∞ , Tu SegviZlia
zRvari, roca x → ∞ aris
x →∞
f ( x ) gavxadoT ragind didi, roca x sakmarisad didia
...
SevniSnoT, rom Tu r > 0 , maSin
1
1
lim r = 0
lim
=0
x →∞ | x |
x →−∞ | x |r
magaliTi 3
...
vipovoT
lim
x →∞
3x 2 + 6
da
5 − 2x
lim
x → −∞
3x 2 + 6
5 − 2x
...
6
...
i
...
funqciis uwyvetoba
Cven zemoT SevexeT “karg” funqciebs, romelTac is Tviseba
hqondaT, rom am funqciebis zRvari mocemul wertilSi amave
wertilSi funqciis mniSvnelobis tolia
...
vTqvaT f :[a, b] → R da a < c < b (aseT SemTxvevaSi
vityviT, rom c aris [a, b] segmentis Siga wertili)
gansazRvreba 4
...
wertilSi, Tu
vityviT, rom y = f ( x) funqcia uwyvetia c
lim f ( x) = f (c)
...
x→2
mocemuli
wertilSi
...
amitom
uwyvetia
x=2
naxazi 1
Tu funqcia mocemulia Semdegi
saxiT (ix
...
marTlac,
lim f ( x) = 5 = f (2) = 7
...
3)
...
naxazi 3
vityviT, rom f funqcia aris marjvnidan uwyveti x = c wertilSi
Tu funqciis marjvena zRvari mocemul wertilSi udris f
funqciis mniSvnelobas am wertilSi
...
i
...
x →c +
magaliTad naxazi 3-iT mocemuli funqcia uwyvetia marjvnidan x = 2
wertilSi
...
e
...
lim f ( x ) = f (c )
...
naxazi 4
advili Sesamowmebelia, rom samarTliania Semdegi
52
Teorema 4
...
imisaTvis rom f funqcia iyos uwyveti x = c
wertilSi aucilebeli da sakmarisia, rom is iyos uwyveti
rogorc marjvnidan, aseve marcxnidan
...
1
...
uwyvetoba intervalze
...
kerZod, funqcias
vuwodoT uwyveti Tu is uwyvetia Tavis gansazRvris areSi
...
1
...
marTlac, am funqciis
gansazRvris area [0, +∞ ) Sualedi da
∀c ∈ [0, +∞), lim x = c
...
nax
...
marTlac, am fuqciis
gansazRvris area ( −∞, 0) ∪ (0, +∞) da
1 1
∀c ∈ (−∞, 0) ∪ (0, +∞ ),
lim =
...
magaliTi 4
...
funqcia f ( x) =
nax
...
6
53
Tu ganvixilavT funqcias (ix
...
6):
⎧1
⎪ , x ≠ 0,
f ( x) = ⎨ x
⎪0, x = 0
⎩
maSin es funqcia gansazRrulia mTel RerZze da is wyvetilia x = 0
wertilSi
...
rogorc vnaxeT nax 2- da nax
...
aseT SemTxvevaSi Cven
vambobT, rom funqcias aqvs pirveli gvaris wyvetis wertili
...
gansazRvreba 4
...
Tu c wertili aris f funqciis wyvetis wertili
da arsebobs marjvena da marcxena sasruli zRvrebi maSin vityviT,
rom c aris f funqciis I gvaris wyvetis wertili
...
2, nax
...
SevniSnoT, rom nax
...
3 da
nax
...
zemoTqmulidan gamomdinare
pirveli gvaris wyvetis wertilTa simravle SeiZleba daiyos or
jgufad
1) wertilebi romlebSic funqcias gaaCnia zRvari mocemul
wertilSi;
2) wertilebi romlebSic funqcias ar gaaCnia zRvari mocemul
wertilSi;
pirvel SemTxvevaSi vambobT, rom f funqcias c wertilSi aqvs
acilebadi wyveta
...
marTlac, Tu nax
...
nax
...
sazogadod, Tu f funqcias gaaCnia
acilebadi wyveta x = c wertilSi da ganvixilavT axal funqcias
⎧ f ( x) , x ∈ D( f ) \{c}
F ( x) = ⎨
x=c
⎩ L,
sadac L = lim f ( x)
...
x→c
c wyvetis wertils f funqciis meore gvaris wyvetis wertili
ewodeba, Tu igi ar aris pirveli gvaris wyvetis wertili
...
i
...
magaliTad,
nax
...
uban-uban uwyveti funqciebi
...
nax
...
e
...
uban-uban uwyveti funqciaa
...
3 da nax
...
rac Seexeba nax
...
uwyveti funqciebis Tvisebebi
...
uwyveti funqciebisTvis adgili aqvs Semdeg debulebas
Teorema 4
...
Tu f uwyvetia g ( a ) wertilSi da g uwyvetia a
wertilSi, maSin
lim f ( g ( x)) = f ( g ( a ))
x→a
am faqtis gamoyenebiT vipovoT lim e sin x
...
3
...
maSin
arsebobs x1 , x2 ∈ [a, b] iseTi, rom nebismieri x ∈ [ a, b] gvaqvs
f ( x1 ) ≤ f ( x) ≤ f ( x2 )
...
ganvixiloT magaliTebi:
55
naxazze mocemulia (0,1) intervalze
zemodan SemousazRvreli funqciis
grafiki, romelsac globaluri
eqstremumis wertilebi ara aqvs
...
naxazze mocemulia [0,1] segmentze
SemosazRvreli funqciis grafiki,
romelic wyvetas ganicdis Siga x = 0
...
Teorema 4
...
vTqvaT f [a, b]
segmentze uwyveti funqciaa da s
raime ricxvia, romlisTvisac gvaqvs
f ( a ) ≤ s ≤ f (b) , maSin moiZebneba erTi
mainc c wertili iseTi, rom a ≤ c ≤ b
da f (c) = s
...
3
...
ganvixiloT funqcia
57
⎛1⎞
f (0) = 1 > 0 da f ⎜ ⎟ = 2 − 2 <0
...
am funqciisTvis gvaqvs:
1
aviRoT s = 0 ricxvi
...
amitom Teoremis
2
1
ZaliT arsebobs iseTi c , 0 < c < , rom f (c) = 0 anu 2c − 4c = 0
...
i
...
⎣ 2⎦
funqciis zRvris mkacri ganmarteba
gansazRvreba 4
...
vTqvaT, f funqcia
gansazRvrulia a wertilis raime
midamoSi, garda SesaZlebelia TviT a
wertilisa
...
magaliTi 4
...
funqciis zRvris ganmartebiT davamtkicoT, rom
lim(10 x − 6) = 14
x→2
funqciis zRvris ganmartebis ZaliT nebismieri ε > 0 ricxvisTvis
unda movZebnoT iseTi δ > 0 ricxvi, rom roca
0 < x − 2 < δ , maSin (10 x − 6) − 14 < ε
...
ε
10
x→2
calmxrivi zRvrebi
gansazRvreba 4
...
vTqvaT, f funqcia gansazRvrulia a wertilis
marjvena (marcxena) midamoSi, garda SesaZlebelia TviT a
wertilisa
...
magaliTi 4
...
davamtkicoT, rom lim x = 0
...
maSin roca 0 <| x − a |< δ = ε 2
| x − 0 |= x < δ = ε
...
vTqvaT, mocemulia f : ( a;+∞) → R funqcia
...
1
= 0
...
M = , miviRebT
x
x
ε
magaliTi 4
...
davamtkicoT, rom lim
Tu SevarCevT
analogiurad ganimarteba, rom lim f ( x )
...
vityviT, rom lim f ( x ) = +∞ , Tu nebismieri dadebiTi M ricxvisaTvis
x→a
arsebobs dadebiTi ricxvi δ ( M -ze damokidebuli) iseTi, rom
roca 0 < x − a < δ
...
marTlac, nemismieri M > 0 x →0 x 2
1
1
Tvis Tu SevarCevT δ =
...
x
M
magaliTi 4
...
davamtkicoT, rom lim
magaliTi 4
...
zRvris ganmartebis safuZvelze davamtkicoT, rom Tu
lim f ( x) = L da lim g ( x) = M , maSin lim[ f ( x) + g ( x)] = L + M
...
analogiurad,
damtkicdeba, rom arsebobs δ 2 iseTi, rom | g ( x) − M |<
ε
2
, roca 0 <| x − a |< δ 2
...
maSin miviRebT
| ( f ( x ) + g ( x )) − ( L + M ) |≤| f ( x ) − L | + | g ( x ) − M |< ε
roca 0 <| x − a |< δ
...
gamoTvaleT zRvrebi:
1
...
lim(3 x 2 − 2 x + 5)
3
...
lim(sin 2 x + 2 cos 5 x + x)
x 2 − 3x
5
...
lim(log 2 x + e 2 x )
x 4 + 2x 2 − 3
7
...
lim 2
x→4 x − 5 x + 4
x →1
x →8
x →1
9
...
lim
x →1
(1 + x)(1 + 2 x)(1 + 3 x) − 1
x →0
x
11
...
lim
x − 27
x → 27 3
3
15
...
lim
x →0 3
27 + x − 3 27 − x
x →0
x + 23 x 4
2
...
lim f ( x) , Tu
x→ 2
2
...
lim f ( x) , Tu
x→1
4
x →16
14
...
lim f ( x) , Tu
x→ 0
1
≤ f ( x) ≤ x 4
x
3
...
lim
x → +∞ x − 1
1
...
lim
5
...
x2 + 2
x
lim
x →∞
6
...
gamoTvaleT Semdegi calmxrivi zRvrebi:
1
...
lim
x →0 +
x2 +1
2
...
lim
1
x →0 −
1+ e x
x −2
x−3
1
1+ e
1
x
5
...
lim f ( x) da lim f ( x) , Tu f ( x) = ⎨ x 2 − 2 x
x→2+
x→2−
,x>2
⎪
⎩ x−2
2
...
lim f ( x) da lim f ( x) , Tu
⎧ 5, x ≤ 0
f ( x) = ⎨
⎩− 2, x > 0
x→4+
x →0 +
x→4−
x →0 −
4
...
ipoveT funqciis zRvari mocemul wertilSi, Tu funqcia
mocemulia grafikuli saxiT:
1) lim f ( x) , Tu
x →3
2) lim f ( x) , Tu
x →3
3) lim f ( x) , lim f ( x) , Tu
x → 2,5
x → 4,5
63
4) lim f ( x) da lim f ( x) , Tu
x →5 +
x →5 −
5) lim f ( x ) ,
x →−8+
lim f ( x ) ,
x →−8−
lim f ( x ) ,
x →−5−
lim f ( x ) ,
x →−5+
lim f ( x ) ,
x →3+
lim f ( x )
x →3−
f ( −8 ) ,
f ( −5 )
f ( 3)
...
aris Tu ara uwyveti Semdegi funqciebi (Tu wyvetilia
romelime wertilSi, maSin romeli gvaris wyvetas aqvs adgili):
1
...
f ( x) = ⎨ 1
⎪x ,x > 0
⎩
⎧ x 2 + 1, x < 0
⎪
2
...
f ( x) = ⎨
⎩3, x ≥ 0
64
⎧ x2 − 4
,x≠2
⎪
6
...
f ( x ) = x
0 ≤ x ≤1
⎧ 2 x,
7
...
f ( x) = ⎨
⎪ 1,
⎩
x ≤1
x >1
a -s ra mniSvnelobisTvis iqneba f funqcia uwyveti, Tu
8
...
f ( x) = ⎨
⎩ a + x, x ≥ 0
⎧ 2ax,
3
...
f ( x) = ⎨
⎩ 2 − x, 1 < x ≤ 2
⎧ x 2 + 1, x < 0
⎪
4
...
funqciis zRvris ganmartebiT daamtkiceT, rom:
1
...
lim
x → +∞
1
=0
x +1
2
2
...
lim
x →1+
65
1
= +∞
x −1
5
...
vTqvaT, mocemulia f : ( a; b) → R funqcia
...
h iyos iseTi
namdvili ricxvi, romlisTvisac x0 + h ∈ ( a, b )
...
f(x0+h)
∆f(x0)
f(x0)
{
h
a
x0
b
x0+h
magaliTi 5
...
vTqvaT, f ( x ) = 3 x 2 − 2 x
...
∆f (1) = f (1 + h ) − f (1) = 3 (1 + h ) − 2 (1 + h ) − 3 + 2 = 3h 2 + 4h
...
2
...
x + h = 1,1
...
∆f (1) = f (1,1) − f (1) = 1,331 − 1 = 0,331
...
1
...
maSasadame, am wertilebze
gamaval wrfis gantolebas aqvs Semdegi
saxe
y = f ( x0 ) + m( x − x0 ) ,
sadac
66
m=
f ( x0 + h) − f ( x0 )
...
1
gansazRvreba 5
...
vTqvaT, mocemulia f : (a; b) → R funqcia
...
Tu zRvris mniSvneloba sasruli ricxvia, maSin f funqcias
ewodeba warmoebadi
...
h
magaliTi 5
...
ganmartebis safuZvelze vipovoT f ( x ) = x 2 funqciis
warmoebuli
...
h →0
h→0
h →0
h
h
advili dasanaxia (ix
...
1), roca
h → 0 , maSin Q wertili
mocemuli wiris gaswvriv
miiswrafis P wertilisken da
Sesabamisad P da Q wertilebze
gamavali wrfe icvlis Tavis
mdebareobas (ix
...
2), romelic
miiswrafis l wrfisken
...
wrfes, romelic gadis f
funqciis
( x , f ( x ))
0
0
wertilze da romlis sakuTxo koeficienti aris
f ' ( x0 ) uwodeben f funqciis grafikis mxebs ( x0 , f ( x0 )) wertilSi
...
magaliTi 5
...
vipovoT f ( x ) = x 2 funqciis mxebis gantoleba x0 = 1
wertilSi
...
68
Tu funqcia warmoebadia (a; b) intervalis yovel x wertilSi, maSin
miviRebT warmoebul funqcias, romelic gansazRvruli iqneba (a; b)
intervalze da yovel x ∈ (a; b) wertilSi ganimarteba (1) tolobiT
...
5
...
f funqciis warmoebuli x0 wertilSi ganimarteba Semdegnairadac:
gansazRvreba 5
...
vTqvaT, mocemulia f : (a; b) → R funqcia
...
x0 wertilSi funqciis warmoebuls aRvniSnavT
zRvari lim
x → x0
x − x0
f ′( x0 ) simboloTi, e
...
f ′( x0 ) = lim
x → x0
f ( x) − f ( x0 )
...
marTlac, Tu x − x0 sidides aRvniSnavT h simboloTi, maSin
x = x0 + h da miviRebT, rom (1) da (2) tolobebi ekvivalenturia
...
6
...
maSin f funqcia ar aris warmoebadi x = 0
⎧1, x > 0
x
= sgn x = ⎨
wertilSi da roca x ≠ 0 , f ' ( x ) =
...
⎩ − x, x < 0
vTqvaT x ≠ 0
...
⎩−1, x < 0
davamtkicoT, rom mocemuli funqcia warmoebadi ar aris x = 0 wertilSi
...
69
f ( x) =| x | funqciis grafiki
f '( x) funqciis grafiki
70
vTqvaT mocemulia f funqciis grafiki (ix
...
3)
nax
...
x = −2, x = 1 wertilebSi funqcias
warmoebuli ara aqvs
...
7
...
maSin f '( x) = 0, x ∈ [a, b]
...
f '( x) = lim
h →0
h →0 h
h
geometriulad am faqts Semdegi intepretacia aqvs:
mudmivi funqciis grafiki
...
magaliTi 5
...
mocemulia funqciis grafiki
72
mocemuli funqciis gawarmoebis Sedegad
miRebuli funqciis grafiki
marjvena da marcxena warmoebulebi
...
gansazRvreba 5
...
vTqvaT, mocemulia f : (a; b) → R funqcia
...
x − x0
Tu es zRvari arsebobs da sasruli ricxvia, maSin f funqcias ewodeba
warmoebadi marjvnidan
...
x − x0
Tu es zRvari arsebobs da sasruli ricxvia, maSin f funqcias ewodeba
warmoebadi marcxnidan
...
1
...
(3)
amasTan (3) tolobis SemTxvevaSi maTi saerTo mniSvneloba emTxveva f
funqciis warmoebuls x0 wertilSi, e
...
f ′( x0 +) = f ′( x0 −) = f ' ( x0 )
...
9
...
ganmartebis ZaliT
f ( x) − f (0)
0−0
= lim
= 0
...
x →0 +
x →0 −
x →0 −
x−0
x
radganac
f ′(0−) = f ′(0+ ) = 0
amitom, Teorema 5
...
magaliTi 5
...
ganvixiloT funqcia
⎧0, x < 0
...
x →0−
x →0 −
x−0
x
74
radganac
f ′(0−) ≠ f ′(0+ )
amitom, Teorema 5
...
magaliTi 5
...
ganvixiloT funqcia f ( x) =| x |
...
x →0−
x →0 −
x−0
x
radganac
f ′(0−) ≠ f ′(0+)
amitom, Teorema 5
...
magaliTi 5
...
vTqvaT, f ( x ) = x1/ 3 da gamovTvaloT am funqciis waroebuli
0 wertilSi
...
h →0 +
h→0+ h
h →0+ h
h
analogiurad damtkicdeba, rom
f ' ( 0 − ) = +∞
...
i
...
f ' ( 0 + ) = lim
magaliTi 5
...
vTqvaT, f ( x ) = x 2 / 3
...
h →0+ h
h →0 + h
h
h →0 +
h →0 −
SeniSvna 5
...
im SemTxvevaSi, roca f ' ( x0 + 0 ) = ∞ an f ' ( x0 − 0 ) = ∞ , maSin x0
wertilze gavlebuli f funqciis mxebi oy RerZis paraleluria
...
14
...
gamovTvaloT
wertilSi
...
h→0+ h
h →0 + h
h
ganvixiloT 0 wertilze gamavali mocemuli funqciis grafikisadmi
gavlebuli ramodenime mkveTi, romlebic miiswrafiani oy RerZisken
...
3
...
vityviT, rom
f funqcia warmoebadia a wertilSi, Tu arsebobs sasruli marjvena
warmoebuli
f ' ( a + ) da Sesabamisad vityviT, rom rom f
funqcia
warmoebadia b wertilSi, Tu arsebobs sasruli marcxena warmoebuli
f '(a −)
...
15
...
maSin
f ' ( 0 ) = f ' ( 0 + ) = 0 da f ' (1) = f ' (1 − ) = 2
...
rogorc zemoT vnaxeT, funqciis warmoebuli wertilSi
SeiZleba iyos rogorc sasruli ricxvi aseve usasruloba
...
Teorema 5
...
vTqvaT, mocemulia f : (a; b) → R funqcia
...
damtkiceba
...
i
...
x → x0
x − x0
maSin
f ( x ) − f ( x0 )
lim f ( x) = lim (
⋅ ( x − x0 ) + f ( x0 )) =
x → x0
x → x0
x − x0
f ( x ) − f ( x0 )
= lim (
( x − x0 )) + lim f ( x0 ) = f ′( x0 ) ⋅ 0 + f ( x0 ) = f ( x0 ) ,
x → x0
x → x0
x − x0
rac
gvaZlevs
f
funqciis
uwyvetobas
x0
wertilSi
...
SeniSvna 5
...
damtkicebuli Teoremidan gamomdinareobs, rom wertilSi
funqciis uwyvetoba am wertilSi warmoebadobisaTvis aris aucilebeli
piroba,
magram
sazogadod,
wertilSi
uwyvetoba
am
wertilSi
warmoebadobisaTvis ar aris sakmarisi piroba, rasac adasturebs
magaliTi 5
...
11
...
5
...
funqciaTa wrfivi kombinaciis, namravlis da fardobis warmoebuli
Teorema 5
...
mudmivi funqciis warmoebuli nulis tolia (ix
...
6)
Teorema 5
...
vTqvaT, mocemulia f : (a; b) → R da g : (a; b) → R funqciebi
...
warmoebulis ganmartebis ZaliT
78
( f + g )′( x) = lim
( f + g )( x + h ) − ( f + g )( x )
h →0
h
f ( x + h) − f ( x) + g ( x + h) − g ( x)
= lim
h →0
h
f ( x + h) − f ( x)
g ( x + h) − g ( x)
= lim
+ lim
h →0
h→0
h
h
= f ′( x) + g ′( x)
...
5
...
Tu
f da g funqciebi warmoebadia x ∈ (a; b) wertilSi, maSin warmoebadia
x ∈ (a; b) wertilSi f da g funqciebis namravli da
( f ⋅ g )′( x) = f ′( x) ⋅ g ( x) + f ( x) ⋅ g ′( x) ;
damtkiceba
...
f
Sedegi 5
...
Tu f da g funqciebi warmoebadia x ∈ (a; b) wertilSi, maSin
nebismieri α , β ∈ R ricxvebisTvis warmoebadia x ∈ (a; b) wertilSi αf + βg
wrfivi kombinacia da
(α f + β g )′( x) = α f ′( x) + β g ′( x)
...
6
...
Tu f da g
funqciebi warmoebadia x ∈ (a; b) wertilSi da g ( x ) ≠ 0 , maSin warmoebadia
x ∈ (a; b) wertilSi f da g funqciebis fardoba da
′
⎛ f ⎞
f ′( x) ⋅ g ( x) − f ( x) ⋅ g ′( x)
...
pirvel rigSi SevniSnoT, rom radgan g ( x ) warmoebadia,
amitom is uwyvetia x wertilSi da radganac g ( x ) ≠ 0 , wertilSi
uwyvetobis ganmartebidan gamomdinareobs x wertilis iseTi midamos
arseboba, romelSiac g ( x ) ≠ 0
...
i
...
f ( x + h) f ( x)
⎛ f ⎞
⎛ f ⎞
−
⎜ g ⎟ ( x + h) − ⎜ g ⎟ ( x)
g ( x + h) g ( x)
⎝ ⎠
⎝ ⎠
lim
= lim
h →0
h →0
h
h
f ( x + h) g ( x) − f ( x) g ( x + h)
1
= lim
h →0 g ( x ) g ( x + h )
h
1
=
=
=
=
g
2
1
lim
( x ) h →0
1
g
2
f ( x + h) g ( x) − f ( x) g ( x) + f ( x) g ( x) − f ( x) g ( x + h)
h
lim
g 2 ( x ) h →0
lim
( x ) h →0
f ( x + h) g ( x) − f ( x) g ( x + h)
h
g ( x ) ( f ( x + h) − f ( x ))
h
− lim
f ( x ) ( g ( x + h) − g ( x ))
h →0
h
f ′( x) ⋅ g ( x) − f ( x) ⋅ g ′( x)
...
kerZod:
( x )′ = 1 ;
( x 2 )′ = 2 x;
2) (sin x)′ = cos x ;
( x )′ =
1
2 x
;
80
1
1
( )′ = − 2 ;
x
x
3) (cos x)′ = − sin x ;
4) (tan x)′ =
1
;
cos 2 x
′
2
2
1
⎛ sin x ⎞ (sin x)′ cos x − sin x(cos x)′ cos x + sin x
;
marTlac, (tan x)′ = ⎜
=
=
=
⎟
2
2
cos x
cos x
cos 2 x
⎝ cos x ⎠
1
5) (cot x)′ = − 2 ;
( daamtkiceT! )
...
kerZod,
1
(ln x)′ = ;
x
rTuli funqciis warmoebuli
Teorema
...
7
...
magaliTi 5
...
ipoveT y = x 2 + 1 funqciis warmoebuli
...
vinaidan f '(u ) =
1
2 u
Teorema 5
...
17
...
x
vinaidan (| x |) ' =
(ix
...
5
...
7-is ZaliT gvaqvs
|x|
2 x( x 2 − 1) ⎧2 x, x < −1, x > 1
y '( x) =
=⎨
| x 2 − 1|
⎩−2 x, − 1 < x < 1
Seqceuli funqciis warmoebuli
Teorema 5
...
vTqvaT, f : X → Y
da
f : Y → X urTierTSeqceuli da
uwyveti funqciebia Sesabamisad x0 ∈ X
−1
da f ( x0 ) = y0 wertilebSi
...
(4)
( f −1 )( y0 ) =
f ′( x0 )
damtkiceba
...
i
...
davamtkicoT
(4) formula
...
=
f ( f −1 ( y )) = y ⇒ ( f ( f −1 ( y )) ) ' = 1 ⇒ f '( f −1 ( y ))( f −1 ( y ) ' = 1 ⇒ ( f −1 ( y ) ' =
−1
f '( f ( y )) f ' ( x )
Seqceuli trigonometriuli funqciebis warmoebuli
82
1)
davamtkicoT, rom (arcsin x)′ =
marTlac,
1
1− x2
, x ≠ ±1
...
=
=
=
2
(sin x)′ cos x
1 − sin x
1− y2
analogiurad damtkicdeba, rom samarTliania Semdegi:
1
2) (arccos x)′ = −
, x ∈ ( 0,1) ;
1 − x2
1
1
;
4) (arcctgx)′ = −
...
Tu f : (a; b) → R funqcia warmoebadia (a; b)
(a; b)
intervalze
intervalis
yovel
wertilSi,
maSin
arsebobs
gansazRvruli f ′ funqcia, romelic aris f funqciis warmoebuli
...
Tu miRebuli f ′′ funqcia kvlav warmoebadia, maSin vixilavT mesame rigis
warmoebuls da a
...
amrigad, f ( n ) ( x) = ( f ( n−1) )′( x)
...
)
funqcias uwodeben usasrulod warmoebads (a; b) intervalze, Tu mas
(a;b) intervalze gaaCnia nebismieri rigis warmoebuli
...
18
...
2)
f ( x) = sin x , cxadia, rom f (n ) ( x) = sin( x +
3) f ( x) = cos x , cxadia, rom f ( n ) ( x) = cos( x +
π
2
π
2
n) (SeamowmeT!)
n) (SeamowmeT!)
funqciis diferenciali
...
gamovTvaloT am funqciis h
nazrdi x = 1 wertilSi
...
2
rogorc vxedavT, nazrdi rogorc h -is
funqcia warmoadgens ori funqciis jams,
sadac pirveli funqcia aris wrfivi 2h
xolo meore funqcia ki h 2
83
h2
2h
e
...
84
rac Seexeba meore funqcias, mas aqvs Tviseba
h2
lim = 0
...
vTqvaT, f ( x ) = 3 x 3 − 2 x 2 + x − 3
...
aqac funqciis nazrdi warmodgeba ori 35h da
jamad
18h 2 +3h 3 funqciebis
ise rogorc pirvel SemTxvevaSi, aqac 18h 2 +3h 3 funqcias is Tviseba aqvs,
rom
18h 2 +3h 3
= 0
...
orive SemTxvevaSi funqciis nazrdi warmovadgineT ori funqciis
jamis saxiT ( h -is mimarT), sadac pirveli funqcia aris wrfivi, xolo
meores is Tviseba aqvs, rom h -ze ufro swrafad miiswrafis nulisaken, an
sxva sityvebiT, meore funqciis Sefardeba h -Tan miiswrafis nulisaken,
roca h miiswrafis nulisaken
...
19
...
maSin am fuqciis h nazrds x = 0
wertilSi eqneba Semdegi saxe
f ( h ) − f ( 0 ) =| h |
...
h→0
h
maSin advili dasanaxia, rom
| h | ⎧1, h > 0
A = lim
=⎨
h →0 h
⎩−1, h < 0
e
...
A ar aris calsaxad gansazRruli
...
3
...
vityviT, rom f funqcia aris diferencirebadi x
wertilSi, Tu arsebobs A ricxvi ( x -ze damokidebuli), iseTi rom
adgili aqvs warmodgenas
f ( x + h ) − f ( x ) = Ah + α ( x; h ) ,
sadac
lim
h→0
α ( x; h )
h
(5)
= 0
...
e
...
df ( x ) = Ah ,
(6)
sadac A x -ze damokidebuli ricxvia
...
9
...
imisaTvis, rom f
funqcia iyos diferencirebadi x ∈ (a; b) wertilSi, aucilebelia da
sakmarisi, rom mas am wertilSi qondes sasruli warmoebuli
...
aucilebloba
...
vigulisxmoT, rom
86
h≠0
da
(1)
tolobis
orive
mxare
gavyoT
h -ze
...
, amitom f ′( x) = lim
= lim( A +
h→0
h→0
h
h
h
h
i
...
sakmarisoba
...
ganvixiloT funqcia α ( x; h) = f ( x + h) − f ( x) − f ′( x)h
...
amitom adgili eqneba warmodgenas
maSin
lim
h→0
h
f ( x + h) − f ( x) = f ′( x)h + α ( x; h) , rac emTxveva (1) tolobas, Tu CavTvliT, rom
A = f ′( x)
...
Teorema
damtkicebulia
...
Cven davamtkiceT, rom (5) warmodgenaSi
A = f '( x)
...
da
(7)
df ( x ) = f ' ( x ) h
(8)
Tu (8) tolobaSi f ( x ) = x , maSin miviRebT
dx = h
amitom zogjer (8)-s ase Caweren
...
dx -s uwodeben argumentis diferencials
...
fermas Teorema
87
(9)
Teorema 5
...
vTqvaT, f : [a, b] → R warmoebadia x0 ∈ (a, b) wertilSi
da amasTan, x0 aris f funqciis Siga lokaluri eqstremumis wertili,
maSin f ' ( x0 ) = 0
...
zogadobis SeuzRudavad vigulisxmoT, rom x0 aris
f funqcias lokaluri maqsimumis wertili
...
aqedan
f ( x 0 + h) − f ( x 0 )
f ( x 0 − h) − f ( x 0 )
≤0 ,
≥ 0
...
Teorema damtkicebulia
...
SeniSvna
...
marTlac Tu ganvixilavT
f ( x ) = x funqcias [0,1] segmentze
maSin advili dasanaxia, rom x = 0 da x = 1
wertilebi arian mocemuli funqciis
lokaluri minimumis da maqsimumis
wertilebi, magram am wertilebSi
funqciis warmoebuli 1-is tolia
...
4
...
gansazRvreba 5
...
wertils, romelzedac funqciis warmoebuli nulis
tolia, am funqciis stacionaluri wertili ewodeba
...
20
...
> plot(6*x^3+33*x^2-30*x+100,x=6
...
21
...
> g:=t->t^(2/3)*(2*t-1);
g := t → t
> diff(g(t),t);
2 (2 t − 1)
3t
> solve(%=0,t);
( 2/3 )
( 1/3 )
1
5
>
89
(2 t − 1)
+2t
( 2/3 )
vinaidan t = 0 wertilSi g ( t ) funqcias ara aqvs warmoebuli, amitom t = 0
wertili kritikuli wertilia
...
rolis Teorema
Teorema 5
...
Tu [a, b] segmentze gansazRvruli f funqcia
akmayofilebs Semdeg pirobebs:
1)
2)
3)
f uwyveti [a, b] -ze;
f warmoebadia (a, b) -ze;
f (a) = f (b)
maSin
(a, b) intervalSi arsebobs erTi mainc iseTi x0 wertili, rom
f ' ( x0 ) = 0
...
Tu f funqcia mudmivia [a, b] segmentze, maSin (a, b)
intervalSi f ' ( x0 ) = 0
...
axla vTqvaT, f funqcia mudmivi araa [a, b] segmentze
...
vTqvaT , esenia m da M
...
fermas Teoremis Tanaxmad f ' ( x0 ) = 0
...
i
...
geometriulad rolis Teorema niSnavs, rom Tu [a, b] segmentze uwyveti
f funqcia warmoebadia (a, b) intervalSi da, amasTanave, f (a) = f (b) maSin
(a, b) intervalSi arsebobs erTi mainc x0 wertili, iseTi rom am werilze
gavlebuli mxebi ox RerZis paraleluria
...
5*9
...
90 t 2 + 12 t
> RollesTheorem(position,
t=0
...
448979592, output =
points);
[1
...
2
...
7/2);
> RollesTheorem(sin(x), 1
...
2);
SeniSvna
...
ganvixiloT funqcia f ( x) = x , gansazRvruli [−1, 1] segmentze
...
maSasadame,
darRveulia (−1, 1) intervalSi funqciis
warmoebadobis piroba
...
marTlac, f ' ( x) = 1 , roca x > 0 da f ' ( x) = −1 ,
roca x < 0 , xolo x = 0 wertilSi
f funqcias ara aqvs warmoebuli
...
91
vTqvaT
⎧ x, 0 ≤ x < 1
f ( x) = ⎨
...
da bolos, funqcia f ( x ) = x, x ∈ [0,1] akmayofilebs rolis TeoremaSi 1) da 2)
pirobebs, magram f ' ( x ) = 1 ≠ 0, 0 ≤ x ≤ 1
...
12 (lagranJis)
...
ganvixiloT damxmare funqcia
ϕ ( x) = [ f ( x) − f (a)](b − a) − [ f (b) − f (a )]( x − a )
...
amrigad ϕ funqcia akmayofilebs rolis Teoremis yvela
pirobas da, maSasadame, (a, b) intervalSi arsebobs erTi mainc iseTi
x0 wertili, rom
ϕ ' ( x0 ) = 0
...
maSasadame,
f ' ( x0 )(b − a) − [ f (b) − f (a )] = 0,
saidanac
f (b) − f (a) = f ' ( x0 )(b − a )
...
lagranJis Teoremis geometriuli interpretacia:
vinaidan
f (b) − f ( a )
b−a
aris
( a, f ( a ) )
da ( b, f ( b ) ) wertilebze gavlebuli wrfis sakuTxo
koeficienti, xolo f ' ( x0 ) -ki f funqciis grafikis
gavlebuli mxebis sakuTxo koeficientia da
( x , f ( x ))
0
0
wertilze
f (b) − f ( a )
= f ' ( x0 )
b−a
amitom grafikze arsebobs ( x0 , f ( x0 ) ) werili, romelzec gavlebuli mxebi
paraleluria
( a, f ( a ) )
da ( b, f ( b ) ) wertilebze gamavali wrfis
...
1);
93
SevniSnoT, rom aseTi wertilebi SeiZleba iyos erTze meti
...
22
...
5);
magaliTi 5
...
vTqvaT
f ( x ) = x 3 + 2 x 2 − x, a = −1, b = 2 -Tvis vipovoT
x0 werili, romlisTvisac f ( b ) − f ( a ) = f ' ( x0 )( b − a )
...
2);
> g:=x->diff(f(x),x);
g := x → diff( f( x ), x )
> g(c);
3 c2 + 4 c − 1
> g(c)=(f(2)-f(-1))/(2-(-1));
3 c2 + 4 c − 1 = 4
> solve(%,c);
−
2
+
3
19
2
,− −
3
3
19
3
> evalf(-2/3+1/3*19^(1/2));
> evalf(-2/3-1/3*19^(1/2));
vinaidan -2
...
7862996483
0
...
119632982
ekuTvnis gansaxilvel Sualeds, amitom
94
ganvixiloT ramodenime magaliTi
...
2);
MeanValueTheorem(sin(x), 1
...
5);
Sedegi
...
maSin arsebobs mudmivi c iseTi, rom
f ( x) = g ( x) + c
...
vTqvaT, f
da g funqciebi warmoebadia a wertilis
midamoSi garda SesaZlebelia a wertilSi , amasTan g '( x) ≠ 0 ,
lim f ( x ) = 0, lim g ( x ) = 0
x →a
an
x→a
lim f ( x ) = ±∞,
x →a
da arsebobs zRvari lim
x →a
f '( x)
lim g ( x ) = ±∞
x→a
g '( x)
maSin
lim
x →a
f ( x)
f '( x)
= lim
...
magaliTi 5
...
vipovoT
ln(1 + x)
...
x →0
x → 0 cos x
sin x
magaliTi 5
...
vipovoT
lim
x →0
sin x
...
x →0
1
x
ln sin x
x →0 + ln sin 2 x
(ln sin x)'
cos x sin 2 x
sin 2 x
2 cos 2 x
ln sin x
= lim
= lim
= lim
= lim
= 1
...
26
...
27
...
x → +∞ x 2
x → +∞ 2 x
x → +∞ 2
lim
3
...
vTqvaT, f da g funqciebi iseTia, rom
lim f ( x) = 0, da lim g ( x) = ∞
...
aseTi saxis ganuzRvrelobis gaxsna xerxdeba isev
lopitalis wesiT
...
lim f ( x) g ( x) = lim
x→a
x→a
x→a
1
1
g ( x)
f ( x)
±∞
0
aqedan Cans , rom 0 ⋅ ∞ saxis ganuzRvrelobis gaxsna daiyvaneba
an
0
±∞
ganuzRvrelobis gaxsnamde
...
28
...
x →0+
ln x
1/ x
= − lim
= − lim x = 0
...
∞ − ∞ saxis ganuzRvreloba
...
am SemTxvevaSi zRvris gamoTvla
x→a
0
saxis ganuzRvrelobis gaxsnamde
...
x →a
x→a
1
f ( x) g ( x)
SesaZlebelia daviyvanoT
magaliTi 5
...
1
1⎞
x − tan x
x2
x
⎛
1 + x 2 = lim
lim ⎜ cot x − ⎟ = lim
= lim
= lim
= 0
...
vTqvaT, f da g funqciebi iseTia, rom
5
...
maSin
lim[ f ( x)] g ( x ) = lim e g ( x ) ln f ( x )
...
30
...
2
=e
...
vTqvaT, lim f ( x) = ∞ da lim g ( x) = 0 ,
maSin y = [ f ( x)]
x→a
x→a
funqcia, roca x → a gvaZlevs ∞ saxis ganuzRvrelobas,
∞
0
romlis gaxsna daiyvaneba
saxis (an
saxis) ganuzRvrelobis gaxsnamde
∞
0
Semdegi gardaqmnis daxmarebiT:
g ( x)
0
97
lim
x→a
lim[ f ( x)] g ( x ) = e
x→a
ln f ( x )
1
g ( x)
lim
x→a
=e
g ( x)
1
ln f ( x )
...
teiloris formula
gansazRvreba
...
gansazRvreba
...
Semdegi saxis funqcias
Pn ( x ) = b0 + b1 ( x − x0 ) + b2 ( x − x0 ) + + bn ( x − x0 )
ewodeba n -uri xarisxis polinomi
...
Tu funqcia uwyvetad warmoebadia
( a, b ) intervalze, maSin, lagranJis Teoremis ZaliT, nebismieri x da x0 -Tvis
2
( a, b ) -dan
n
arsebobs ξ ∈ ( x0 , x ) iseTi, rom adgili aqvs Semdeg warmodgenas
f ( x ) − f ( x0 ) = f ' (ξ )( x − x0 )
anu, rac igivea
f ( x ) = f ( x0 ) + O ( ( x − x0 ) )
...
12 ( teiloris formula)
...
f ( x) = f ( x0 ) +
( x − x0 ) +
...
vTqvaT, f ( x) = e x
...
+ + o ( x n ) , x ∈ R
1! 2!
n!
f ( n ) ( x) = e x , n ∈ N ,
> f:=x->exp(x);
f := x → e x
> F:=(x,n)->convert(taylor(f(x),x=0,n),polynom);
F := ( x, n ) → convert ( taylor ( f( x ), x = 0, n ), polynom )
> F(x,3);
1+x+
1 2
x
2
> plot([f(x),F(x,3)],x=-4
...
4,title="Taylor
Approximation",style=[LINE,POINT],
legend=["Function","Taylor
Approximation"]);
> > f:=x->exp(x);
f := x → e x
> F:=(x,n)>convert(taylor(f(x),x=0,n),polyno
99
x0 = 0
m);
F := ( x, n ) → convert ( taylor ( f( x ), x = 0, n ), polynom )
> F(x,15);
1
1
1 4
1 5
1 6
1
1
1
x +
x +
x +
x7 +
x8 +
x9
1 + x + x2 + x3 +
2
6
24
120
720
5040
40320
362880
1
1
1
1
x 10 +
x 11 +
x 12 +
x 13
+
3628800
39916800
479001600
6227020800
1
+
x 14
87178291200
> plot([f(x),F(x,15)],x=4
...
vTqvaT, f ( x) = sin x
...
+ ( −1)
3!
5!
2n + 1) !
(
x∈R
...
4,title="Taylor
Approximation",style=[LINE,POINT],legend=["Function","Taylor
Approximation"]);
> f:=x->sin(x);
100
f := x → sin( x )
> Order:=7;
Order := 7
> taylor(f(x),x=0);
x−
1 3
1 5
x +
x + O( x 7 )
6
120
> F:=(x,n)->convert(taylor(f(x),x=0,n),polynom);
F := ( x, n ) → convert( taylor( f( x ), x = 0, n ), polynom )
> F(x,7);
x−
1 3
1 5
x +
x
6
120
> plot([f(x),F(x,7)],x=4
...
4,title="Taylor
Approximation",style=[LINE,POINT],legend=["Function","Taylor
Approximation"]);
3
...
+
x + o ( x2n ) ,
2!
4!
(2n)!
101
x∈R
...
4,title="Taylor
Approximation",style=[LINE,POINT],legend=["Function","Taylor
Approximation"]);
>
> f:=x->cos(x);
f := x → cos( x )
> Order:=9;
Order := 9
> taylor(f(x),x=0);
1−
1 2 1 4
1 6
1
x +
x −
x +
x 8 + O( x 9 )
2
24
720
40320
> F:=(x,n)->convert(taylor(f(x),x=0,n),polynom);
F := ( x, n ) → convert ( taylor( f( x ), x = 0, n ), polynom )
> F(x,9);
1
1 4
1 6
1
1 − x2 +
x −
x +
x8
2
24
720
40320
> plot([f(x),F(x,9)],x=4
...
vTqvaT f ( x ) = ln (1 + x )
...
n
f := x → ln( 1 + x )
Order := 5
> taylor(f(x),x=0);
1
1
1
x − x 2 + x 3 − x 4 + O( x 5 )
2
3
4
> F:=(x,n)>convert(taylor(f(x),x=0,n),polynom);
F := ( x, n ) → convert( taylor( f( x ), x = 0, n ), polynom )
> F(x,5);
x−
1 2 1 3 1 4
x + x − x
2
3
4
>
plot([f(x),F(x,5)],x=0
...
1,title="Taylor
Approximation",style=[LINE,POINT],legend=["Function","Taylor
Approximation"]);
savarjiSoebi
ipoveT f funqciis nazrdi x0 wertilSi, Tu
a) f ( x) = x 2 − 1
x0 = 1 ;
da
b) f ( x) = x 3 − 2 x
x 0 = −1 ;
da
g) f ( x) = −2 x + x
2
x0 = 2 ;
da
d) f ( x) = − x + 3x + 2
2
x0 = −2
...
a) y = 3x 5 − 2 x 2 + 3 ;
b) y = 4 x 7 − 3x 5 − 2 x + 8 ;
g) y = 2 x 8 − x 6 − 3x 5 + 4 ;
d) y = 6 x 9 − 3x 5 + 7 x − 12
...
a) y = 3x 3 + x + + 3 ;
b) y = 4 x 5 − 33 x + 2 − 8 ;
x
x
2
−
2
3
g) y = x 6 − 25 x 2 − 5 x 3 + ;
3
5
3
...
a) y = 2 arcsin x + 5 arccos x ;
d)
b)
d)
b)
104
3
5
1
x3
+ 9 − 1
...
3
y = 3 arctan x − arc cot x ;
4
1
g) y = 5 arccos x − arctan x − tan x ;
4
d) y = 4 arcsin x + 6tgx − 2arc cot x
...
4
5
...
a) y = x 2 sin x − e x cos x + 2 x tan x ;
b) y = 3x 5 ln x − 4e x cot x − 3 cos x arctan x ;
g) y = 63 x 4 ⋅ cos x − 2 arcsin x ⋅ log 2 x − 5 x 9 e x ;
d) y = −311 x 6 ⋅ cot x − 4 sin x ⋅ ln x + 2 x arccos x
...
a) y = cos x ⋅ e x ⋅ log 2 x ;
b) y = 3x 5 tan x ⋅ cos x ;
3
g) y = 2 x 2 ⋅ arc cot x ⋅ 5 x ;
d) y = 2 arcsin x ⋅ ln x
...
a) y = 4
;
b) y =
;
−
+
3x + x 2 + 2 3 + sin x
3x 2 + x − 1 cos x − ln x
3
x
x 4 − 2x − 4
x + lg x
3x − x 4
−
;
d) y =
...
a) y = x
;
b) y = x
;
e (cos x + x 3 )
2 (arcsin x + 4)
g) y =
( 4 x + 3) ln x
;
( x 2 − cot x)3 x
d) y =
( 2 x 3 − x )e x
...
a) y = (2 x + 1)10 − 5 cos 2 x ;
g) y = ( x 3 − 4 x 2 ) 8 − tan 4 x ;
2
...
a) y = tan − cot ;
2
2
2
1 x −1
g) y = ln 2
;
4 x +1
4
...
a) y = x x ;
g) y = (cos x) x ;
d) y = (2 x − x ) 4 − 3 cot 9 x
...
x
b) y = y = lg 3 x − ln ;
3
1− x
d) y = arccos
;
2
b) y = sin(cos 2 (tan 3 x)) ;
d) y = arctan(tan 2 x)
...
105
2
...
a) f ′( ) ,
6
Tu
b) f ′( ) ,
3
4
g) f ′( ) ,
Tu
π
π
2
d) f ′( ) ,
π
d) y = x x +sin
1+ tg
f ( x) = 3
Tu
2
x
+ ( x − 1) 3 e1+ tan x
...
cos x
...
f ( x) =
4 cos 2 x
f ( x) = e
Tu
;
b) y = x cos 2 x + 3 x ⋅ e sin x ;
g) y = arctan(2 x + 3) − x 2 x +1 ;
1
...
+ x 7 e tan x ;
ipoveT:
f ′(1) , Tu
f ′(−1) ,
Tu
f ′(0) , Tu
f ′(0) , Tu
x
...
a) y = x 5 arcsin 3x + tan x 4 ;
4
...
...
warmoebulis gansazRvrebis gamoyenebiT ipoveT Semdegi funqciis
warmoebuli:
1
...
a) f ( x) = ;
x
g) f ( x) = x ;
b) f ( x) = −2 x − x 2 ;
d) f ( x) = 3x − x 3
...
106
ipoveT f funqciis marjvena da marcxena warmoebulebi x = 0 wertilSi,
Tu:
⎧3 x 2 , x > 0
⎧− x 2 + 1, x ≥ 0
a) f ( x) = ⎨
;
b) f ( x) = ⎨
;
⎩− 2 x x ≤ 0
⎩ 3x + 1 x < 0
⎧ x + 1, x > 0
⎧ x 2 − 3 x + 2, x > 0
;
d) f ( x) = ⎨
...
2
ipoveT f funqciis marjvena da marcxena warmoebulebi x0 wertilSi
da gaarkvieT, aris Tu ara funqcia warmoebadi x0 wertilSi:
a) f ( x) = ( x + 1) x ,
x0 = 0 ;
b) f ( x) = x x + 3 ,
x0 = −3 ;
g) f ( x) = x ⋅ x ,
d) f ( x) = ( x − 1) x − 1 ,
x0 = 0 ;
x0 = 1
...
1− x2
ipoveT f funqciis diferenciali x0 wertilSi, Tu:
a) f ( x) = 3x 2 − 2 x + 3 ,
x0 = 1 ;
b) f ( x) = − x + 4 x − 1 ,
x0 = 2 ;
g) f ( x) = 2 x − x − 2 ,
x0 = −1 ;
2
3
d) f ( x) = 2 x 3 + 3 x 2 − 4 x + 1 ,
x0 = 0
...
lopitalis Teoremis gamoyenebiT gamoTvaleT zRvrebi:
sin x
x →0 arctan x
ln( x − a )
3
...
lim
5
...
lim⎜
− 2
⎟
x →2 x − 2
x + x −6⎠
⎝
x⎞
⎛
9
...
lim⎜ ⎟
x →0 x
⎝ ⎠
tan
πx
2a
sin x
1 ⎞
⎛ x
8
...
lim⎜
⎟
x→a
⎝ x ⎠
ln(1− x )
c tan x
x →0
x + sin x
x + 2 sin x
1 − cos x 2
x →0
x2
a x −1
21
...
lim
ln(a + be x )
x →∞
a + ex
25
...
lim
x →1−
x →0
15
...
lim ln x ln(1 − x)
12
...
lim⎜
⎟
x →1 1 − x
1 − x3 ⎠
⎝
17
...
lim
x → +∞ x
2
...
lim
π
x→
cos x
2
14
...
lim
⎛ x +1⎞
20
...
lim
x →0
2x
arctan
1− x2
x
24
...
lokaluri maqsimumis da minimumis, absoluturi maqsimumebisa da
minimumebis gamoTvla
Teorema 5
...
vTqvaT, mocemulia
f : (a; b) → R funqcia, romelic
warmoebadia (a; b) intervalze
...
damtkiceba
...
maSin
lagranJis Teoremis ZaliT
f ( x2 ) − f ( x1 ) = f ' ( c )( x2 − x1 ) ≥ 0
...
davamtkicoT, rom f ′( x) ≥ 0
...
aviRoT dadebiTi ricxvi h imdenad mcire, rom x + h ∈ (a, b)
...
maSasadame,
f ( x + h) − f ( x )
≥0
...
advili dasanaxia, rom (5
...
Teoremis 1) punqti damtkicebulia
...
bolos davamtkicoT Teoremis 5) punqti
...
1)
aq igulisxmeba, rom yvela aRniSnuli piroba sruldeba (a; b) intervalze
...
marTlac, lagranJis Teoremis ZaliT
f ( x2 ) − f ( x1 ) = f ' ( c )( x2 − x1 ) = 0 ( x2 − x1 ) = 0
...
f ( x) = x 3 funqciis magaliTiT Cans, rom raime intervalze
warmoebadi funqciis zrdadobidan sazogadod ar gamomdinareobs am
intervalze funqciis warmoebulis dadebiToba
...
i
...
f’(x) ≤ 0
f’(x) ≥ 0
f’(x) ≥ 0
f’(x) ≥ 0
f’(x)=0
f’(x) ≤ 0
magaliTi 5
...
vTqvaT, f ( x) = x 3 − 3x + 2
...
radgan
(− ∞;−1) intervalze f ′( x) > 0 , amitom funqcia am intervalze zrdadia;
radgan (− 1;1) intervalze f ′( x) < 0 ,
amitom funqcia am intervalze
klebadia, radgan (1;+∞ ) intervalze
f ′( x) > 0 ,
amitom funqcia am
intervalze zrdadia
...
2,title="Function and its
Derivative",style=[LINE,POINT],leg
end=["Function","Derivative"]);
110
lokaluri eqstremumebis arsebobis sakmarisi pirobebi
Teorema 5
...
samarTliania Semdegi:
1) Tu (x0 − δ ; x0 ) -ze f ′( x) < 0 da (x0 ; x0 + δ ) -ze f ′( x) < 0 , maSin x0 wertilSi
f funqcias eqstremumi ara aqvs
...
3) Tu (x0 − δ ; x0 ) -ze f ′( x) > 0 da (x0 ; x0 + δ ) -ze f ′( x) < 0 , maSin x0 aris f
funqciis mkacri lokaluri maqsimumis wertili
...
magaliTi 5
...
vTqvaT y = 3x 4 − 4 x 3
...
> y:=3*x^4-4*x^3;
y := 3 x 4 − 4 x 3
> plot(y,x=-1
...
magaliTi 5
...
programa MAPLE-is gamoyenebiT vipovoT
funqciis lokaluri eqxtremumebi [−2, 2] segmentze
...
2);
> y1:=diff(y,x);
y1 := 4 x 3 − 4 x
> solve(y1=0,x);
0, 1, -1
> solve(y1>0,x);
RealRange ( Open( -1 ), Open( 0 ) ), RealRange ( Open( 1 ), ∞ )
> solve(y1<0,x);
RealRange ( −∞, Open( -1 ) ), RealRange ( Open( 0 ), Open( 1 ) )
>
112
f ( x) = x 4 − 2 x 2 − 3
Teorema 5
...
vTqvaT
f ′( x0 ) =
...
Tu n luwia, maSin x0 wertilSi f funqcias lokaluri eqstremumi aqvs
...
magaliTi 5
...
vTqvaT f ( x ) = x 4 − 8 x 2 + 12
...
> f:=x^4-8*x^2+12;
f := x 4 − 8 x 2 + 12
> g:=diff(f,x);
g := 4 x 3 − 16 x
> plot([f,g],x=-4
...
imisaTvis, rom movZebnoT [a, b] segmentze uwyveti f funqciis udidesi da
umciresi mniSvnelobebi, saWiroa gamovTvaloT misi yvela lokaluri
maqsimumi da yvela lokaluri minimumi
...
114
magaliTi 5
...
ipoveT f ( x) = x 3 − 6 x 2 + 9 x funqciis absoluturi
eqstremumebi [− 1;4] segmentze
...
4);
> diff(f,x);
3 x 2 − 12 x + 9
> solve(%,x);
> x:=-1:
> f;
> x:=1:
> f;
3, 1
-16
4
> x:=3:
> f;
0
> x:=4:
> f;
4
> "AbsMax=4, when x=1 or x=4":
> "AbsMin=-16, when x=-1 ":
>
funqciis amozneqilobis dadgena warmoebulis gamoyenebiT,
gadaRunvis wertilebi
gansazRvreba 5
...
vTqvaT, mocemulia f : (a; b) → R funqcia, romelic
warmoebadia (a; b) intervalze
...
naxazze
gamosaxulia
amozneqili funqciis grafiki:
qvemodan
> with(student):
> showtangent(x^2+5,x=1,x=0
...
6
...
f funqcias ewodeba amozneqili zemodan
(a; b) intervalze, Tu funqciis grafikis yoveli wertili iqneba grafikis
nebismier wertilSi gavlebuli mxebis qvemoT
...
3);
Teorema 5
...
vTqvaT, mocemulia f : (a; b) → R funqcia, romelsac (a; b)
intervalze gaaCnia meore rigis warmoebuli
...
gansazRvreba 5
...
warmoebadi
funqciis grafikis wertils, romelSic
amozneqilobis xasiaTi icvleba
sapirispiroTi, ewodeba gadaRunvis
wertili
...
116
Teorema 5
...
magaliTi 5
...
vipovoT f ( x) = x 3 − 5 x 2 + 3 x − 5 funqciis grafikis gadaRunvis
wertilebi, zemoT da qvemoT amozneqilobis Sualedebi
...
i
...
> solve(g(x)>0);
> solve(g(x)<0);
RealRange ( Open( 2 ), ∞ )
RealRange ( −∞, Open ( 2 ) )
maSasadame funqcia amozneqilia zemodan
( −∞, 2 ) Sualedze, xolo amozneqilia
qvemodan
( 2, +∞ )
Sualedze
...
2);
117
> with(student):
> showtangent(f(x),x=2,x=0
...
5);
>
funqciis grafikis asimptotebi
GgansazRvreba
...
8 x = x0 wrfes uwodeben f funqciis grafikis
vertikalur asimptotas, Tu
lim f ( x) = ±∞
x → x0 −0
an
lim f ( x) = ±∞
...
aseTi asimptotebi SeiZleba hqondes mxolod iseT
funqciis grafiks, romelsac aqvs meore gvaris wyvetis werilebi
...
36
...
amitom
x → 3+ x − 3
x − 3 x → 3− x − 3
x = 3 mocemuli funqciis vertikaluri asimptotia
...
37
...
advili dasanaxia, rom
x −1
2
1
= +∞ ,
x →−1−
1 − x2
1
lim
= −∞ ,
x →−1+
1 − x2
1
lim
= −∞
x →1−
1 − x2
1
lim
= +∞
...
i
...
119
GgansazRvreba
...
9 y = y0 wrfes uwodeben f funqciis grafikis
horizontalur asimptotas, Tu
lim f ( x) = y0
magaliTi
lim f ( x) = y0
...
38
...
x −x
2
1
= 0
...
radganac
lim
2
y = kx + b wrfes uwodeben f funqciis grafikis daxril (Tu
horizontalur ) asimptots, Tu adgili aqvs tolobas
f ( x) = kx + b + α ( x) ,
sadac lim α ( x) = 0 an lim α ( x) = 0
...
18
...
x → ±∞
x → ±∞
x
1
...
38
...
x →±∞ 1 + x 2
b = lim ( f ( x ) − kx ) = lim
x →±∞
maSasadame funqcias gaaCnia horizontaluri asimptoti da es aris y = 0
wrfe
...
39
...
121
x3
x2 − 3
maSasadame,
wrfeebi arian
mocemuli funqciis grafikis
vertikaluri asimptotebi
...
maSasadame
asimptotia
...
gamokvlevis mixedviT avagoT grafiki
...
a) f ( x) = 3x − x 3 ;
b)
g)
d)
2
...
f ( x) = x 6 + 3x 5 + 18 ;
f ( x) = 3x 7 − x 6 − 8 ;
f ( x) = 5 x 9 − 9 x 5 + 45 x − 15 ;
f ( x) = 3x 4 + 4 x 3 + 6 x 2 − 16
...
6
⎝ 2⎠
123
ipoveT Semdegi funqciis eqstremumebi:
1
a) f ( x ) = x 3 − 3x 2 + 5 x + 1 ;
3
15
b) f ( x ) = x 3 − x 2 + 18 x − 9
...
⎩
⎧ ( x + 2) 2 , − 4 ≤ x < −1,
⎪
− 1 < x < 1,
b) f ( x) = ⎨ − x,
2
⎪− ( x − 2) ,
1 < x ≤ 4
...
⎩
124
⎧− x 2 − 2 x, − 3 ≤ x < 0,
⎪
0 < x ≤ 1,
d) f ( x) = ⎨ 3x,
⎪
3,
1 < x ≤ 4
...
a) f ( x) =
3
125
wertilebi
da
globaluri
x 2 + 3x
,
x −1
g) f ( x) = x(10 − x) ,
x ∈ [−3;0] ;
b) f ( x) =
d) f ( x) = x − 2 ln x +
2
...
2
x ∈ [− 2;2] ;
x ∈ [− 3;1] ;
b) f ( x) = ( x + 1) 2 − x + 1 ,
g)
x ∈ [0;4]
...
gamoikvlieT da aageT Semdegi funqciis grafiki:
x4
3
2
b) y = 1 + x −
;
1
...
1+ x4
x − 5x + 6
2
2
...
1+ x
ricxviTi mimdevrobebi da mwkrivebi
126
amozneqilobis
ricxviTi mimdevroba
...
, n,
...
ganvixiloT raime f funqcia f : N → A
...
maSin
miviRebT erTobliobas:
x1 , x 2 ,
...
ricxvTa am erTobliobas ricxviTi mimdevroba ewodeba
...
S
...
x n -s uwodeben agreTve mocemuli
mimdevrobis zogad wevrs
...
, x n ,
...
( x n ) n ≥1 ricxviT mimdevrobas,
rogorc naturaluri
argumentis funqcias,
SegviZlia mivceT Semdegi
geometriuli interpretacia:
ganvixiloT ricxviTi
mimdevrobis ramdenime
magaliTi
...
1
...
maSin gveqneba
x1 = 4, x 2 = 7, x3 = 10,
...
i
...
,3n + 1,
...
2
...
2
maSin gveqneba
x1 = 0, x 2 = 1, x3 = 0, x 4 = 1,
...
i
...
,
1 + (−1) n
,
...
zemodan SemosazRvruli
mimdevrobis geometriuli suraTi
Semdegia:
magaliTi 6
...
ganvixiloT mimdevroba ( x n ) n≥1 , sadac x n = − n
− 1,−2,
...
am mimdevrobis yoveli wevri naklebia 0-ze, amitom es mimdevroba zemodan
SemosazRvruli mimdevrobaa
...
qvemodan SemosazRvruli
mimdevrobis geometriuli suraTi
Semdegia:
magaliTi 6
...
ganvixiloT mimdevroba ( x n ) n≥1 , sadac x n = n
1,2,
...
am mimdevrobis yoveli wevri metia 0-ze, amitom es mimdevroba qvemodan
SemosazRvruli mimdevrobaa
...
SemosazRvruli mimdevrobis geometriuli suraTi Semdegia:
128
magaliTi 6
...
ganvixiloT mimdevroba ( x n ) n≥1 , sadac x n =
1
n
1 1 1
1, ,
...
Mmimdevrobas ewodeba zrdadi, Tu
x1 ≤ x 2 ≤
...
magaliTi 6
...
ganvixiloT
mimdevroba
1,2,3,3,4,5,6,6,
...
mimdevrobas ewodeba klebadi, Tu x1 ≥ x 2 ≥
...
magaliTi 6
...
ganvixiloT
mimdevroba
− 1,−2,−3,−3,−4,−5,−6,−6,
...
mimdevrobas ewodeba mkacrad
zrdadi, Tu x1 < x 2 <
...
129
magaliTi 6
...
ganvixiloT mimdevroba
1,2,3,4,5,6,
...
mimdevrobas ewodeba mkacrad
klebadi, Tu x1 > x 2 >
...
magaliTi 6
...
ganvixiloT
mimdevroba
1 1 1
1, ,
...
Yyovel zrdad (mkacrad zrdad) an
klebad (mkacrad klebad) mimdevrobas
monotonuri mimdevroba ewodeba
...
Mmimdevrobis zRvari
gansazRvreba 6
...
vityviT, rom
( x n ) x ≥1 mimdevrobis zRvari aris a ricxvi,
Tu yoveli dadebiTi ε ricxvisTvis moiZebneba iseTi naturaluri N
ricxvi, rom roca n > N , gveqneba x n − a < ε
...
n →∞
M
mimdevrobis zRvris cnebas SegviZlia
mivceT Semdegi geometriuli axsna:
a Aaris x n mimdevrobis zRvari, Tu
yoveli ε > 0 ricxvisTvis moiZebneba iseTi
naturaluriN ricxvi N , rom mimdevrobis
yoveli x n wevri, romlis nomeri n > N ,
moTavsdeba ( a − ε , a + ε )
SualedSi:
gansazRvreba 6
...
a) vityviT, rom
( x n ) x ≥1 mimdevrobis zRvari aris + ∞ , Tu
yoveli dadebiTi M ricxvisTvis,
moiZebneba iseTi naturaluri N ricxvi,
rom roca n > N , gveqneba x n > M
...
n →∞
M
b) vityviT, rom ( x n ) x ≥1 mimdevrobis
zRvari aris − ∞ , Tu yoveli dadebiTi
M ricxvisTvis, moiZebneba iseTi
naturaluri N ricxvi, rom roca n > N
gveqneba x n < − M
...
n →∞
M
g) vityviT, rom ( x n ) x ≥1 mimdevrobis
zRvari aris ∞ , Tu yoveli dadebiTi M
ricxvisTvis, moiZebneba iseTi naturaluri
N ricxvi, rom roca n > N , gveqneba | x n |> M
...
n→∞
M
gansazRvreba 6
...
ricxvTa ( x n ) n≥1 mimdevrobas ewodeba krebadi, Tu mas
aqvs sasruli zRvari
...
Tu ( x n )n ≥1 krebadia da misi zRvaria a ricxvi, amboben rom ( x n )n≥1
mimdevroba krebadia a ricxvisken
...
10
...
, a,
...
marTlac, yoveli ε > 0
ricxvisTvis (a − ε , a + ε ) midamo Seicavs
a ricxvs
...
Cveni
SemTxvevisTvis x n = a ∈ (a − ε , a + ε )
yoveli n – Tvis
...
i
...
n →∞
magaliTi 6
...
ganvixiloT
⎛1⎞
⎜ ⎟ mimdevroba
...
i
...
vaCvenoT, rom es mimdevroba
2
n
krebadia da misi zRvari 0-is tolia
...
(aseTi naturaluri ricxvi
ε
moiZebneba naturalur ricxvTa
simravlis zemodan
araSemosazRvrulobis gamo)
...
i
...
n→∞
n →∞ n
magaliTi 6
...
132
(−1) n
= 0 , vinaidan
n →∞
n
a) lim
n +1
= 1,
n →∞ n
⎡1⎤
roca n > N = ⎢ ⎥
...
⎣ε ⎦
n +1
1
−1 = < ε ,
n
n
⎡1⎤
roca n > N = ⎢ ⎥
...
marTlac, n − 0 = n < ε , roca
n →∞ q
q
q
sin n
= 0 , vinaidan
n →∞ n
g) lim
sin n
1
−0 ≤ <ε ,
n
n
1⎤
⎡
n > N = ⎢log q ⎥
...
vaCvenoT, rom ( x n ) n ≥1 mimdevrobas ara aqvs
zRvari
...
………
davuSvaT, rom am mimdevrobis zRvari aris raime a ricxvi
...
1
1
...
radganac x n Tanmimdevrulad Rebulobs -1 da
1 mniSvnelobebs, amitom unda iyos
aviRoT ε =
133
1
1
da − 1 − a <
...
e
...
2<1, rac
2 2
SeuZlebelia
...
moviyvanoT aRniSnuli faqtis geometriuli interpretacia:
L
1− a <
8) lim n 2 = +∞ ,
n →∞
vinaidan n 2 > M , roca n > N = [ M ] + 1
...
vinaidan (−1) n n > M , roca n > N = [M ]
Teorema 6
...
yovel krebad mimdevrobas aqvs erTaderTi zRvari
...
moviyvanoT damtkicebis geometriuli varianti:
vTqvaT, ( x n ) n ≥1 mimdevroba krebadia a ricxvisaken
...
vaCvenoT, rom b ar SeiZleba iyos
mocemuli mimdevrobis zRvari
...
vTqvaT, ε ∈ (0,
2
maSin, rogorc naxazidan Cans, roca
n > N , ( x n ) n ≥1 mimdevrobis wevrebi
Cavardebian a ricxvis ( a − ε , a + ε )
midamoSi
...
Teorema 6
...
yoveli krebadi mimdevroba SemosazRvrulia
...
vTqvaT, lim x n = a
...
rogorc naxazze Cans , miTiTebuli zolis
gareT rCeba ( x n ) n ≥1 mimdevrobis mxolod
sasruli raodenobis wevrebi, amitom
arsebobs iseTi c1 da c 2 ricxvebi, rom
Sesruldeba c1 < x n < c 2 utoloba yvela n -
134
Tvis, maSasadame ( x n ) n ≥1 mimdevroba SemosazRvrulia
...
1
...
Mmag
...
rogorc
viciT, es mimdevroba araa krebadi, magram igi SemosazRvrulia
...
i
...
Teorema 6
...
vTqvaT, ( x n ) n ≥1
zrdadi mimdevrobaa, romelic
SemosazRvrulia zemodan
...
D
Teorema 6
...
vTqvaT, ( x n ) n ≥1 klebadi mimdevrobaa, romelic
SemosazRvrulia qvemodan
...
magaliTi 6
...
mocemulia ( x n ) n ≥1 , x n = 1 −
1
mimdevroba
n
1
x n zrdadi mimdevrobaa da lim x n = lim (1 − ) = 1
...
13
...
n→∞
n→∞
n
gansazRvreba 6
...
vTqvaT mocemulia ( x n ) n ≥1
...
ganvixiloT
naturalur ricxvTa mkacrad zrdadi mimdevroba n1 < n 2 < n3 <
...
mimdevrobas x n1 , x n2 ,
...
ewodeba ( x n ) n ≥1 mimdevrobis qvemimdevroba da
aRiniSneba ase: ( x nk ) k ≥1
...
5
...
Ddamtkiceba
...
amitom ganmartebis ZaliT ∀ε > 0 Tvis arsebobs iseTi N ricxvi,
rom roca n > N , maSin x n − a < ε
...
, x nk ,
...
e
...
( x nk ) k ≥1 mimdevrobis zRvaria
igive a ricxvi
...
14
...
,-1,
...
,1,…
...
n →∞
( x n ) n ≥1 mimdevrobas zRvari rom
hqondes, maSin mis nebismier
qvemimdevrobas unda hqondes igive
zRvari
...
Ee
...
( x n ) n ≥1 mimdevrobas zRvari ara aqvs
...
6
...
maSin marTebulia tolobebi:
n →∞
n →∞
1
...
lim ( x n − y n ) = lim x n − lim y n = a − b
n→∞
←∞
n →∞
3
...
Tu b ≠ 0 , maSin lim
n→∞
lim
x n n→∞ x n a
=
=
y n lim y n b
n →∞
5
...
n →∞
6
...
n→∞
n→∞
Teorema 6
...
vTqvaT lim f ( x) = L da an = f (n)
...
x →+∞
n →∞
⎛1⎞
magaliTi 6
...
gamoTvaleT zRvari lim n arctan ⎜ ⎟
...
5
...
MmagaliTi 6
...
1
1
1
( x n ) n ≥1 , x n =
...
e
...
( ) n≥1 usasrulod mcire
n n →∞ n
n
mimdevrobaa
...
marTebulia Semdegi
Teorema 6
...
Tu ( x n )n ≥1 usasrulod didi mimdevrobaa da
⎛ 1 ⎞
maSin ⎜ ⎟
mimdevroba usasrulod mcire mimdevrobaa, da
⎜x ⎟
⎝ n ⎠ n ≥1
piriqiT: Tu ( x n )n≥1 usasrulod mcire mimdevrobaa, x n ≠ 0 (n ∈ N ) , maSin
xn ≠ 0
⎛ 1
⎜
⎜x
⎝ n
(n ∈ N ) ,
⎞
⎟
usasrulod didi mimdevrobaa
...
Aamboben, rom ( x n )n≥1 da ( y n )n≥1 ekvivalenturi usasrulod mcire
⎛x ⎞
mimdevrobebia, Tu lim⎜ n ⎟ = 1 da weren: ( x n )n ≥1 ≈ ( y n )n ≥1
n →∞⎜ y ⎟
⎝ n⎠
vTqvaT, ( x n )n≥1 da ( y n )n≥1
ori usasrulod didi mimdevrobaa
...
1
1
, lim = 0
n n→ n
1
1
yn =
, lim
=0
n + 1 n →∞ n + 1
magaliTebi 1
...
i
...
( x n )n ≥1 , x n = n 2 + 1,
( y n )n≥1 ,
⎛x
lim⎜ n
n → ∞⎜ y
⎝ n
e
...
(x n )n≥1
(
)
lim n 2 + 1 = +∞
n →∞
y n = n 2 , lim n 2 = +∞
n →∞
⎞
n +1
1 ⎞
1 ⎞
⎛
⎛
⎟ = lim
lim
⎟ lim
⎟ n → ∞ n 2 = n → ∞ ⎜1 + n 2 ⎟ = n → ∞ ⎜1 + n 2 ⎟ = 1
⎝
⎠
⎝
⎠
⎠
2
( y n )n≥1
≈
zogierTi mimdevrobis zRvari:
1
...
n →∞
n
n
3
...
xn = q ,
2
...
n →∞
damtkiceba
...
7x →∞
x →∞ x
n
x
log n
= 0
...
is ZaliT
lim log xn = lim
n →∞
n →∞
n →∞
n →∞
138
analogiurad damtkicdeba danarCeni tolobebis samarTlianoba
(daamtkiceT!)
...
mimdevrobis zRvris ganmartebis ZaliT aCveneT, rom
a)
a)
lim
2
n 5 + 4n − 1
;
lim 6
n→∞ n − 7n + 2
1⎞
⎛
lim⎜1 + ⎟ = 1 ;
n→∞
n⎠
⎝
d)
1
=0 ;
n→∞ n + 1
gamoTvaleT:
g)
2
...
n→∞
n ⎠
⎝
b)
1 ⎞
⎛
lim⎜1 +
⎟
n→∞
2n ⎠
⎝
(
n +1
;
)
2
n ⎞
⎛ 1
lim⎜ 2 + 2 +
...
n→∞ n
n→∞
n
n ⎠
⎝
n
3
...
vTqvaT, ( x n ) n ≥1 ricxvTa raime mimdevrobaa
...
∞
∑x
n =1
n
= x1 + x 2 +
...
(6
...
1) gvaZlevdes x1 + x 2 +
...
(6
...
, x n ,
...
x1 Aam
mwkrivis pirveli wevria, x 2 meore wevri, x n -sMmwkrivis n -ur wevrs
uwodeben
...
1) mwkrivis e
...
kerZo jamebi
139
S1 = x1
S 2 = x1 + x 2
S n = x1 + x 2 +
...
1) mwkrivis n -uri kerZo jami S n
...
1) mwkrivis Sesabamisi n -uri kerZo jamebis ( S n ) n ≥1
mimdevroba
...
1) mwkrivi krebadia da S aris misi jami, Tu krebadia
misi kerZo jamebis ( S n ) n ≥1 mimdevroba da lim S n = S
...
+ x n +
...
1) mwkrivs
ganSladi mwkrivi ewodeba
...
9
...
1) mwkrivi krebadia, maSin lim x n = 0
...
marTlac, vTqvaT mwkrivis ( S n ) n≥1 kerZo jamTa mimdevroba
krebadia
...
Teorema
n →∞
n →∞
n →∞
n →∞
damtkicebulia
...
17
...
+ a +
...
Aam SemTxvevaSi
n→∞
mwkrivi krebadia da misi jami S = 0
...
9)-is ZaliT, es mwkrivi ganSladia
...
18
...
mas aseTi
saxe aqvs:
(−1) + 1 + (−1) + + (−1) n + +…
Aaq piroba lim x n = 0 araa Sesrulebuli da amitom mwkrivi ganSladia
...
19
...
+ aq n +
...
i
...
Tu q ≥ 1, maSin x n = aq n −1 = a q
n −1
(6
...
Aamitom
piroba lim x n = 0 araa Sesrulebuli, rac amtkicebs, rom (2) mwkrivi
n→∞
ganSladia
...
maSin
140
S n = a + aq +
...
2) mwkrivi krebadia da
a
a + aq + + aq n + =
( q <1)
1− q
lim S n = lim (
n→∞
n →∞
magaliTebi 6
...
+ n +
...
1 1
1
ganvixiloT S 2 k = 1 + + +
...
2 3
2
davajgufoT es Sesakrebebi Semdegnairad:
1
1 1
1 1 1 1
1
1
1
S 2 k = 1 + ( ) + ( + ) + ( + + + ) +
...
+ k )
2
3 4
5 6 7 8
2 +1 2 + 2
2
S 2 k jamSi k sxvadasxva frCxilia
...
, k )
...
i
...
aviRoT iseTi
naturaluri k ricxvi, rom k > 2( A − 1)
...
e
...
(S n )n≥1 araa
zemodan SemosazRvruli
...
i =1
m −1
m −1
+
...
21
...
+
+
...
+
=
+
+
...
+
...
+
+
...
Aamrigad,
n(n + 1)
1⋅ 2 2 ⋅ 3 3 ⋅ 4
lim S n = lim(1 −
maSin
n →∞
n →∞
1
) = 1
...
6
...
ganvixiloT
mwkrivi
xn+1 + xn+ 2 + xn+3 +
...
Aam mwkrivs uwodeben
k
∑x
n =1
n
mwkrivis n -ur naSTs
...
vTqvaT,
∑x
n =1
∞
n
n =1
k =1
S = ∑ xn = ∑ xk +
∞
∑x
k = n +1
k
mwkrivi krebadia da misi jamia S
...
aqedan rn =
∞
∑x
k = n +1
k
= S − S n , amitom
lim rn = lim( S − S n ) = S − S = 0
...
ganvixiloT
x1 + x 2 +
...
mwkrivi
(6
...
+ x n +
...
4)
(6
...
4)
mwkrivi
...
3) mwkrivi krebadia, xolo (6
...
3)
mwkrivs pirobiT krebadi mwkrivi ewodeba
...
22
...
misi kerZo
2 2 3 3
1
jamia an
an 0
...
meore mxriv, mwkrivis wevrebis
1 1 1 1
modulebisagan Sedgenili mwkrivia 1 + 1 + + + + +
...
i
...
AvityviT, rom
∞
∑α
n =1
(6
...
3) mwkrivis maJorants (maJorantul mwkrivs), Tu
moiZebneba iseTi naturaluri N ricxvi, rom, roca n ≥ N , maSin x n ≤ α n
...
10 (mwkrivTa Sedarebis niSani)
...
3) mwkrivs gaaCnia
krebadi (6
...
3) mwkrivi absoluturad krebadia
...
23
...
mwkrivi krebadia (rogorc zemoT ganxiluli
n =1
∞
magaliTi
vinaidan
n =1
6
...
n =1
∞
krebadia
...
24
...
2
n =1 n
∑
mwkrivi
krebadia,
es
mwkrivi
∞
sin n
1
≤ 2,
2
n
n
vinaidan
Sedarebis
xolo
1
∑n
n =1
2
mwkrivi ki krebadia
...
25
...
Mmtkicdeba, rom roca
p > 1 , maSin mwkrivi krebadia, xolo roca p ≤ 1 , maSin ganSladia
...
11 (mwkrivis absoluturi krebadobis dalamberis
sakmarisi niSani)
...
maSin samarTliania:
xn
∞
∑x
n
mwkrivi absoluturad krebadia
...
n =1
∞
n =1
g) Tu q = 1 , maSin arsebobs Sesabamisad rogorc absoluturad krebadi,
ise ganSladi mwkrivi
...
26
...
n =1 n
2 n+1 (n + 1)! n n
2n n
2
2
⋅ n = lim
= lim
= < 1 , e
...
mwkrivi krebadia
...
i
...
n +1
n
n →∞ ( n + 1)
n →∞
1
3 n! n→∞ (n + 1)
(1 + ) n e
n
lim
143
Teorema 6
...
maSin samarTliania:
n
n →∞
∞
∑x
n
mwkrivi absoluturad krebadia
...
b) Tu q > 1 , maSin
n =1
∞
n =1
g) Tu q = 1 , maSin arsebobs Sesabamisi rogorc absoluturad krebadi,
ise ganSladi mwkrivi
...
27
...
i
...
⎟ = lim
n →∞
n →∞ 4n + 2
4
⎝ 4n + 2 ⎠
Mmwkrivs
(6
...
Ees cxadia, imas niSnavs, rom mwkrivis yoveli ori
mezobeli wevri sxvadasxva niSnis namdvili ricxvia
...
aRniSvnebis erTgvari SecvliT (6
...
+ (− 1)
k −1
a k +
...
6)
cxadia, a1 , a 2 ,
...
mkacrad dadebiTi ricxvebia
...
13 (niSancvladi mwkrivis krebadobis leibnicis
sakmarisi niSani)
...
( a n > 0 , n = 1,2,
...
≥ a n ≥
...
6) mwkrivi krebadia da misi S jamisTvis gvaqvs: 0 ≤ S ≤ a1
...
28
...
+ (− 1)
+
...
Aamitom
n →∞
n
leibnicis niSnis Tanaxmad, es mwkrivi krebadia
...
2 3
n
Dda is ganSladia
...
i
...
144
savarjiSoebi
1
...
lim 2
=0
n→∞ n + 1
2
...
n 5 + 4n − 1
n→∞ n 6 − 7n + 2
2
n ⎞
⎛ 1
+ 2 +
...
3
...
1⎞
⎛
lim⎜1 + ⎟ = 1
n→∞
n⎠
⎝
2 ⎞
⎛
lim⎜ 2 + 2 ⎟ = 2
n→∞
n ⎠
⎝
2
...
2
...
1 ⎞
⎛
lim⎜1 +
⎟
n→∞
2n ⎠
⎝
(
n +1
lim 3 n + 1 − 3 n
n→∞
)
2
en
6
...
lim⎜
8
...
aqvs Tu ara zRvari ( xn ) n≥1 mimdevrobas, Tu:
(−1) n
1
...
x n = (−1) n +1
(−1) n
⎛ −1⎞
da z n = ⎜ ⎟
2
⎝ 2 ⎠
n +1
4
...
x n = y n + z n , sadac y n =
4
...
+
+ +
+ A
(2n − 1)(2n + 1)
1⋅ 3 3 ⋅ 5
1
1
1
A 2
...
∑ (− 1)
n
3 −n
n =1
∞
⎛2⎞
4
...
mwkrivTa Sedarebis niSnis gamoyenebiT gamoikvlieT Semdegi mwkrivebis
krebadobis sakiTxi:
1
1
1
+
+ +
+
1
...
⎜ + ⎟ + ⎜ 2 + 2 ⎟ + + ⎜ n + n ⎟ +
3 ⎠
3 ⎠
⎝ 2 3⎠ ⎝ 2
⎝2
1 1
1
4
...
dalamberis an koSis niSnis gamoyenebiT daamtkiceT Semdegi mwkrivebis
krebadoba
a a2
an
+
+ +
+ ,
1
...
+ 2 + 3 + + n +
1
2
3
n
2
3
n
2 ⎛ 3⎞
⎛ n +1 ⎞
⎛4⎞
+⎜
+⎜ ⎟ +⎜ ⎟
3
...
+
+
n
2
3
3 ⎛
1⎞
1⎞
1⎞
⎛
⎛
⎜2 + ⎟
⎜2 + ⎟
⎜2 + ⎟
2⎠
3⎠
n⎠
⎝
⎝
⎝
7
...
1 −
∞
1
...
∑
n =1
∞
3
...
∑
n +1
n =1
n =1
146
i
n
t
e
g
r
a
l
vTqvaT mocemulia [2,5] segmentze
gansazRvruli f ( x ) = x 2 funqciis grafiki
...
Cvens mizans warmoadgens S ricxvis
povna
...
advili dasanaxia, rom f funqcia
udides mniSvnelobas aRwevs x = 5 wertilSi,
xolo umciress ki- x = 2 wertilSi
...
nabiji 2
...
mocemuli sami wertils meSveobiT [2,5]
segmenti danawildeba or [ x0 , x1 ] da [ x1 , x2 ]
qvesegmentebad
...
>
Cvens SemTxvevaSi l1 = 2, l2 = x1 , u1 = x1 , u2 = 5
...
i
...
naxazi 2-dan
advili dasanaxia, rom U ( f , P3 ) warmoadgens
marTkuTxedebis farTobTa jams
...
ganvixiloT [2,5] segmentis oTxi wertili ise, rom
2 = x0 < x1 < x2 < x3 = 5
da SemoviRoT aRniSvna
P4 = {x0 , x1 , x2 , x3 }
...
radganac f funqcia uwyvetia,
amitom arseboben ricxvebi: l1 , l2 , l3 , u1 , u2 , u3 iseTi, rom
f ( l1 ) ≤ f ( x ) ≤ f ( u1 ) , x0 ≤ x ≤ x1 ,
f ( l2 ) ≤ f ( x ) ≤ f ( u2 ) , x1 ≤ x ≤ x2
da
f ( l3 ) ≤ f ( x ) ≤ f ( u3 ) , x2 ≤ x ≤ x3
...
i
...
naxazi 4-dan advili
dasanaxia, rom U ( f , P4 ) warmoadgens sami marTkuTxedis farTobTa jams
...
es procesi SeiZleba gagrZeldes usasrulod; kerZod, n + 1 nabijze
miviRebT
Pn +1 = {x0 , x1 ,
...
, n
L ( f , Pn +1 ) = f ( l1 )( x1 − x0 ) + f ( l2 )( x2 − x1 ) +
da
U ( f , Pn+1 ) = f ( u1 )( x1 − x0 ) + f ( u2 )( x2 − x1 ) +
150
+ f ( ln )( xn − xn−1 )
+ f ( un )( xn − xn −1 )
>
SevniSnoT, rom
da Tu
maSin
L ( f , Pn +1 ) ≤ S ≤ U ( f , Pn +1 )
Pn ⊂ Pn +1
L ( f , Pn ) ≤ L ( f , Pn +1 ) ≤ S ≤ U ( f , Pn+1 ) ≤ U ( f , Pn )
...
151
Cven zemoT ganvixileT SemTxveva, roca f funqcia dadebiTia
...
S -iT aRvniSnoT x = 2, x = 5, y = 0 wrfeebiT da y = f ( x ) funqciis grafikiT
SemosazRvruli figuris farTobi
...
152
153
vTqvaT f niSans ar inarCunebs
Tavis gansazRvris areSi
...
maSin ise,
rogorc zemoT, miviRebT, rom
L ( f , Pn+1 ) ≤ S + − S − ≤ U ( f , Pn+1 )
154
155
danawilebebi da darbus jamebi
...
ganvixiloT [a, b] segmentis wertilTa sasruli
simravle
P = { x0 , x1 ,
...
P -s vuwodebT [a, b] segmentis danawilebas
...
∆xi -iT aRvniSnoT [ xi −1 , xi ] segmentis sigrZe
...
i
...
, n
...
e
...
P = max ∆xi
...
1
...
maSin advili dasanaxia,
rom
f ( li ) ∆xi ≤ S + i ≤ f ( ui ) ∆xi
(7
...
vTqvaT f funqcia uaryofiTia [ xi −1 , xi ] segmentze da S -iT aRvniSnoT im
figuris farTobi, romelic SemosazRvrulia x = xi −1 , x = xi wrfeebiT, ox
RerZiT da f funqciis grafikiT
...
2)
3
...
Si− -iT
aRvniSnoT im figuris farTobi, romelic SemosazRvrulia x = xi −1 , x = xi
wrfeebiT, ox RerZiT da f funqciis grafikis im nawiliT, romelic ox
RerZis qveviTaa, xolo Si+ -iT aRvniSnoT im figuris farTobi, romelic
SemosazRvrulia x = xi −1 , x = xi wrfeebiT, ox RerZiT da f funqciis
grafikis im nawiliT, romelic ox RerZis zeviTaa
...
(7
...
1
...
gansazRvreba 7
...
vTqvaT P ' da P '' [a, b] segmentis ori danawilebaa
...
SeniSvna 7
...
advili dasanaxia, rom nebismieri ori P ' da P ''
danawilebebisaTvis arsebobs P danawileba, romelic warmoadgens
rogorc P ' , aseve P '' danawilebaTa gagrZelebas
...
SeniSvna 7
...
advili dasanaxia, Tu P ' danawileba warmoadgens P ''
danawilebis gagrZelebas, maSin adgili aqvs Semdeg utolobebs:
L ( f , P '') ≤ L ( f , P ') ≤ U ( f , P ') ≤ U ( f , P '')
...
1-is ZaliT nebismieri ori P ' da P ''
danawilebebisaTvis arsebobs P danawileba, romelic warmoadgens
rogorc P ' , aseve P '' danawilebaTa gagrZelebas, da maSasadame
L ( f , P ') ≤ L ( f , P ) ≤ U ( f , P ) ≤ U ( f , P '')
...
1
...
vTqvaT f [ a, b] segmentze gansazRvruli funqciaa
da arsebobs erTaderTi ricxvi I iseTi, rom nebismieri P
danawilebisaTvis adgili aqvs utolobas
L ( f , P) ≤ I ≤ U ( f , P)
...
I ricxvs
uwodeben f funqciis gansazRvrul integrals [ a, b] segmentze da
aRniSnaven simboloTi
b
I = ∫ f ( x ) dx
...
[ a, b] segmentze uwyveti funqcia integrebadia
...
vTqvaT f [ a, b] segmentze gansazRvruli uwyveti funqciaa
...
i
...
, xn } ,
sadac
a = x0 < x1 < x2 <
< xn −1 < xn = b
...
, n da SemoviRoT aRniSvna c = ( c1 ,
...
maSin jams
R ( f , P, c ) = f ( c1 ) ∆x1 + f ( c2 ) ∆x2 +
n
+ f ( cn ) ∆xn = ∑ f ( ci ) ∆xi
i =1
uwodeben f funqciis rimanis jams c SerCeuli wertilebiT
...
>
159
160
161
162
rimanis integralis geometriuli intepretacia
...
maSin
b
∫ f ( x )dx = S
+
a
sadac S
+
aris OAB figuris farTobi
...
maSin
b
∫ f ( x )dx = −S
−
a
sadac S
+
aris OAB figuris farTobia
...
maSin
b
∫ f ( x )dx = S
+
− S−
...
a
∫ f ( x)dx = 0
...
Tviseba 2
...
b
Tviseba 3
...
Tviseba 4
...
maSin
c
b
b
a
c
a
∫ f ( x)dx + ∫ f ( x)dx = ∫ f ( x)dx
...
164
Tviseba 5
...
maSin
b
∫
a
b
f ( x ) dx ≤ ∫ g ( x ) dx
...
Tviseba 6
...
maSin
b
b
a
a
∫ f ( x ) dx ≤ ∫
f ( x ) dx
...
Tviseba 7
...
maSin
a
∫
f ( x)dx = 0
...
−a
165
Tviseba 8
...
maSin
a
∫
−a
marTlac,
a
f ( x)dx = 2∫ f ( x)dx
...
0
saSualo mniSvnelobis Teorema
uwyveti funqciis Tvisebis ZaliT arsebobs l , u ∈ [ a, b] iseTi, rom
m = f (l ) ≤ f ( x ) ≤ f (u ) = M , a ≤ x ≤ b
...
integralis ganmartebis ZaliT
b
L ( f , P ) ≤ ∫ f ( x ) dx ≤ U ( f , P ) ,
a
da maSasadame
b
m(b − a) ≤ ∫ f ( x ) dx ≤ M (b − a ) ,
a
166
b
1
f ( x ) dx ≤ M = f ( u )
...
4)-dan miviRebT, rom arsebobs
c ∈ [ a, b] iseTi, rom
f (l ) = m ≤
(7
...
b−a ∫
a
Cven davamtkiceT, rom samarTliania Semdegi
Teorema 7
...
(saSualo mniSvnelobis Teorema) Tu f funqcia uwyvetia
[ a, b ]
segmentze, maSin arsebobs c ∈ [ a, b] iseTi, rom
b
∫ f ( x ) dx = f ( c )( b − a )
a
integralis ganmarteba uban-uban uwyveti funqciebisaTvis
Cven rimanis integralebi ganvixileT uwyveti funqciebisaTvis
...
aseT klass warmoadgens uban-uban uwyveti funqciebi,
e
...
funqciebi, romelTac gaaCniaT sasruli raodenobis wyvetis
wertili
...
gansazRvreba 7
...
vTqvaT c0 < c1 < c2 < < cn aris wertilebis sasruli
raodenoba da f funqcia gansazRvrulia [c0 , cn ] segmentze
...
qvemoT mocemulia uban-uban uwyveti funqciebis magaliTebi
...
4
...
maSin f funqciis gansazRvruli integrali [c0 , cn ] segmentze
ganisazRvreba Semdegnairad
cn
n
ci
c0
i =1
ci −1
∫ f ( x ) dx = ∑ ∫
Fi ( x ) dx
...
1
...
rac Seexeba SemosazRvrulobas, is aris
mxolod aucilebeli piroba
...
kalkulusis ZiriTadi Teoremebi
Teorema 7
...
ganvixiloT Semdegi funqcia
x
F ( x ) = ∫ f ( t ) dt
...
damtkiceba
...
F F funqcias uwodeben f funqciis pirvelyofils
...
i
...
5
...
Teorema 7
...
Tu F1 da F2 funqciebi f funqciis pirvelyofili
funqciebia raime erTi da igive Sualedze, maSin maTi sxvaoba iqneba
mudmivi am Sualedze, anu F1 ( x) − F2 ( x) = c yoveli x -isTvis am Sualedidan
...
Sedegi 7
...
Tu F funqcia aris f funqciis pirvelyofili funqcia raime
erTi da igive, maSin { F + c : c ∈ R}
funqciaTa ojaxi aris f funqciis
pirvelyofili
169
'
⎛ x3 ⎞
x3
2
magaliTi 7
...
radganac ⎜ ⎟ = x , amitom
+ c funqciaTa ojaxi
3
⎝ 3⎠
warmoadgens x 2 funqciis pirvelyofils
...
2
...
>
Teorema 7
...
a
170
damtkiceba
...
2-is ZaliT
x
∫ f ( t ) dt = F ( x ) + c
...
maSin
a
0 = ∫ f ( t ) dt = F ( a ) + c ,
a
c = −F ( a )
...
a
Teorema damtkicebulia
...
ganusazRvreli integralis cneba
gansazRvreba 7
...
ganusazRvreli integrali f funqciidan aRiniSneba
∫ f ( x)dx
simboloTi da ewodeba f funqciis nebismier pirvelyofils mocemul
Sualedze
...
Tu F ′( x ) = f ( x ) , maSin (7
...
aseve gveqneba:
∫ dF ( x) = ∫ F ′( x)dx = F ( x) + c
ZiriTadi ganusazRvreli integralebis cxrili
1
...
5)
2
...
4
...
6
...
1
∫ x dx = ln x + c ;
ax
sadac
∫ a dx = ln a + c ,
∫ sin xdx = − cos x + c ;
x
∫ cos xdx = sin x + c ;
1
∫ cos x dx = tgx + c ;
2
∫
⎧arcsin x + c,
dx = ⎨
1− x2
⎩− arccos x + c1 ;
1
∫ e dx = e
a > 0, a ≠ 1;
x
1
x
+c;
7
...
⎧arctgx + c,
1
dx = ⎨
∫ 1+ x2
⎩− arcctgx + c1 ;
2
x
dx = −ctgx + c ;
- 63 10
...
1
∫ 1− x
2
1 1+ x
dx = ln
+c
...
marTlac, Tu f (x ) funqciis raime pirvelyofilia F (x )
g (x ) funqciis raime pirvelyofilia G (x ) funqcia, maSin
funqcia, xolo
α ∫ f ( x)dx + β ∫ g ( x )dx = αF ( x ) +β G ( x ) + αc1 + β c2 = αF ( x ) + β G ( x ) + c ,
magram (αF ( x ) + β G ( x ))′ = αf ( x ) + β g ( x ) , amitom samarTliania (2) toloba
...
3
...
3
magaliTebi 7
...
2
2
cvladis gardaqmna ganusazRvreli integralisaTvis
172
(2)
Tu veZebT pirvelyofils f (ϕ ( x))ϕ ′( x) saxis funqciidan, maSin viyenebT
ϕ ( x ) = t cvladis gardaqmnas integralis niSnis qveS da pirvelyofili
funqciis povnis Semdeg vubrundebiT sawyis x cvlads:
∫ f (ϕ ( x))ϕ ′( x)dx = ∫ f (ϕ ( x))dϕ ( x) = ∫ f (t )dt = F (t ) + c = F (ϕ ( x)) + c
magaliTebi
...
5
x
x
2
t
t
x
∫ 2 xe dx = ∫ e dx = ∫ e dt = e + c = e + c ;
2
2
2
magaliTi 7
...
x
1 d ( x 2 + 1) 1 dt 1
1
dx = ∫ 2
= ∫ = ln t + c = ln( x 2 + 1) + c ;
∫ 1+ x2
2
x +1
2 t 2
2
magaliTi 7
...
sin x
1
∫ tgxdx = ∫ cos x dx = −∫ cos x d (cos x) = − ln | cos x | +c
...
nawilobiTi integrebis formula ganusazRvreli integralisaTvis
samarTliania toloba:
∫ f ( x) g ′( x)dx = f ( x) g ( x) − ∫ f ′( x) g ( x)dx
...
Tu gaviTvaliswinebT, rom ∫ ( f ( x ) g ( x ))′dx = f ( x ) g ( x) + c , sadac c nebismieri
mudmivia, miviRebT:
∫ f ( x) g ′( x)dx = f ( x) g ( x) − ∫ f ′( x) g ( x)dx
damtkicebuli nawilobiTi
Semdegi saxiTac:
integrebis
formula
∫ f ( x)dg( x) = f ( x) g ( x) − ∫ g ( x)df ( x)
...
2
...
es SeniSvna exeba yvela im tolobasac, romlis orive mxareSi
ganusazRvreli integrali monawileobs
...
2
2
a
savarjoSoebi
gamoTvaleT f funqciis darbus zeda da qveda integraluri
jamebi miTiTebuli Sualedis n tol nawilad danawilebis SemTxvevaSi
...
1
...
2
...
CawereT f funqciis darbus zeda da qveda integraluri jamebi
miTiTebuli Sualedis n tol nawilad danawilebis SemTxvevaSi:
b) f ( x) = 2 x + 1 , x ∈ [ −3;0] ;
1
...
a) f ( x) = x 2 − 1 , x ∈ [1;3] ;
g) f ( x ) = x + 2 ,
1
, x ∈ [0;6]
2
f ( x) = x 2 + 4 x ,
x ∈ [2;3] ;
d) f ( x) = − x +
g) f ( x) =
b)
d) f ( x) = x 3 + 1 ,
x ∈ [0;1] ;
x ∈ [ −2;3]
...
ci (i = 0,1,2,
...
ganixileT SemTxvevebi: 1) n = 2 ; 2) n = 3 ; 3) n = 4 ; 4) n = 5
...
a) f ( x ) = x + 1 , x ∈ [ −1;3] ;
b) f ( x ) = 3 x + 2 , x ∈ [2;3] ;
1
g) f ( x) = − x + 2 ,
d) f ( x) = x − 2 , x ∈ [0;4]
...
a) f ( x) = x ,
x ∈ [0;2] ;
x ∈ [2;3] ;
2
3
g) f ( x) = − x + 1 ,
d) f ( x) = x ,
x ∈ [ −3;−1] ;
x ∈ [0;1]
...
amisaTvis miTiTebuli Sualedi daanawileT n tol
nawilad da ci (i = 0,1,2,
...
1
ipoveT ganusazRvreli integrali:
b) ∫ ( x + 2)dx ;
1
...
a)
5
3
2
− 1)dx ;
∫ ( 4 x − 5)dx
...
3
2
3
1 3 4 1
1
b) ∫ (
+ x − 2 )dx ;
x + 3 x − ) dx ;
x
x
x
1
2
g) ∫ x( 3
+4 x + 2
+ 4)dx ;
d) ∫ (2 x 3 x − x 2 )( x − 1)dx
...
a)
∫(
4
...
a)
2
∫(
6
...
a) ∫ (e x − 2 x )dx ;
2
2
)dx ;
2 x+2 − 3 x−2
∫ 6 x dx ;
g)
1
∫ (1 − x
5
∫ ( cos
d) ∫ cot
b)
b)
) x x dx
...
−
2
2
1
∫(
+
3
)dx ;
2x − 2
x −1
x2
d) ∫
dx
...
ipoveT ganusazRvreli integrali cvladis gardaqmnis gziT:
1
b) ∫ ( x − 5)10 dx ;
1
...
1
2
...
a)
∫ (cos 2 x − sin(3x + 2))dx ;
dx ;
b)
2 − 9x
1
g) ∫
dx ;
2 − 3x 2
g)
4
...
a)
2
1
∫ (1 + sin x − 2 x
∫ tan xdx ;
xdx
∫ 1+ x2 ;
x
;
x 3 dx
∫ x8 − 5 ;
ln 2 x
6
...
a) ∫ x (1 − x ) 50 dx ;
g)
g) ∫ x
23
+1
)dx ;
dx ;
1 − 2x
1
dx
...
d)
dx
∫ (1 + x)
1
2
1 − x dx ;
1
∫
b)
d)
b)
d)
b)
∫
sin x
dx ;
2 x
1
cos
x
∫ x 2 dx
...
4
x) dx
...
a) ∫ x sin xdx ;
2
x
g) ∫ x cos 3 xdx ;
d) ∫ x cos
...
a) ∫ xe dx ;
b) ∫ xe dx ;
g) ∫ xe − x ;
∫ x e dx ;
g) ∫ x sin 2 xdx ;
4
...
a) ∫ 4 − x dx ;
g) ∫ e cos xdx ;
...
b) ∫ arcsin x ;
d) ∫ e dx
...
−2 x
2
3
...
a)
8
2
− 2 x + 3)dx ;
∫(
b)
π
3
∫ (3 cos x − 2 sin x)dx ;
dx
...
a)
2
1
2
g)
1
∫1+ x
d)
0
2
...
−2
π
−x
∫ xe dx ;
b) ∫ x sin xdx ;
0
0
π
g)
e
∫ x cos x dx ;
d)
π
∫ ln x dx
...
⎩
⎧ 2x ,
− 1 ≤ x < 0,
⎪
b) f ( x) = ⎨2 x + 3, 0 ≤ x < 1, ;
⎪ x 2 + 1, 1 ≤ x ≤ 2
...
⎪4 x , 1 ≤ x ≤ 2
...
⎩
177
gamoTvaleT im figuris farTobi, romelic SemosazRvrulia wirebiT:
x = 3;
1
...
d) y = e , y = 0 , x = 0 ,
x = 3;
2
...
a) y = x 2 ,
y=3 x;
b) y = − x 2 ,
y = x 2 − 2x ;
g) y = 6 x − x 2 , y = 0 ;
5
d) y = , y = 6 − x
...
a) y = 2 x − 1 , y = 0 , − 1 ≤ x ≤ 4 ;
b) y = x 3 , y = 0 , − 1 ≤ x ≤ 2 ;
g) y = 2 x − x 2 , y = 0 , − 1 ≤ x ≤ 4 ;
5π
π
d) y = sin x , y = 0 , −
≤x≤