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Математический анализ

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1.N/aoiua e ian/aoiua iiiaeanoaa. N/aoiinoue iiiaeanoaa ?aoeeiiaeueiuo
/enae.

Iiiaeanoai – niaieoiiinoue iaeioi?uo iauaeoia

Yeaiaiou iiiaeanoaa – iauaeou ninoaaeythuea iiiaeanoai

*eneiaua iiiaeanoaa – iiiaeanoaa yeaiaioaie eioi?uo yaeythony /enea.

Caaeaoue iiiaeanoai cia/eo oeacaoue ana aai yeaiaiou:

1 Niinia: A={a: ?(a)} yoe caiene *eoaoue- iiiaeanoai oao a oaeeo
/oi…

A={a-?(a)} ?aaiioeaiiu

?(a) – i?aaeeeao = auneacuaaiea ia yeaiaioa, auaaao eiaeii eee enoeiii
ii ioiioaieth e eie?aoiiio yeaiaioo. Iiiaeanoai A ninoieo ec oao a
aeey eioi?uo i?aaeeeao enoeia.

2 Niinia: Eiino?oe?iaaiea ec ae?oaeo iiiaeanoa:

AUB = {c: cIA U cIB}, AUB = {c: cIA U cIB}, A\ B = {c: cIA U nIB}

U – oieaa?naeueiia iiiaeanoai (oeene?iaaiiia)

U?A; U \ A = A’ = cA (A’ – aeiiieiaiea iiiaeanoaa A)

Naienoaa:

1. AU(BUC)=(AUB) UC – annioeeaoeaiinoue; AUB=BUA – eiiiooaoeaiinoue;
AUAE=A; AUU=U

2. AU (BUC)=(AUB) U(AUC) & AU (BUC)=(AUB) U(AUC) – aeeno?eaooeaiinoue;
AUAE=A

A” =A – caeii eneeth/athuee o?aoueaai (AUB)’=A’UB’; (AUB)’=A’UB’; AUA’=
AE

Eeethno?aoeey naienoa: Aeeaa?aiiu Yeea?a-Aaiia.

“=>” cI(AUB)’ => cIAUB => cIA & cIB => cI A’ & cIB’ => cIA’UB’

“ cIA’ & cIB’ => cIA & cIB => cIAUB => cI(AUB)’

Ioia?aaeaiea iiiaeanoa:

f:A®B (ia iiiaeanoaa A caaeaii ioia?aaeaiea f ni cia/aieai iiiaeanoaa B)

aIA; bIB => b – ia?ac yeaiaioa a i?e ioia?aaeaiee f; a – i?iia?ac
yeaiaioa b i?e ioia?aaeaiee f

Oae eae aeey eaaeaeiai yeaiaioa ec A noaaeony a niioaaonoaea yeaiaio ec
A, cia/eo A – iaeanoue ii?aaeaeaiey (Dom f=A), a iaeanoue cia/aieeB (Im
f FB)

Aeey ioia?aaeaiey caaeatho: 1) niinia 2) Dom 3) Im

Ioia?aaeaiea f eiuaeoeaii anee f(x)=f(x’) => x=x’(?aciua ia?aoiaeyo a
?aciua)

Ioia?aaeaiea f no?ueaeoeaii anee Im f =B(eaaeaeue ia?aoiaeeo a eaaeaeue)

Anee aea ioia?aaeaiea eiuaeoeaii+no?ueaeoeaii, oi iiiaeanoaa
?aaiiiiuiu(niaea?aeao iaeeiaeiaia eie-ai yeaiaioia), a ioia?aaeaiea
aeaeoeaii – acaeiiiaeiicia/ii.

N/aoiua iiiaeanoaa – iiiaeanoaa ?aaiiiiuiua iiiaeanoao iaoo?aeueiuo
/enae (N)

Oai?aia: Iiiaeanoai Q n/aoii.

Eaiia 1: ” nIN Z/n – n/aoii.

Eaaeaeiio yeaiaioo ec N iaaei acaeiiiaeiicia/ii niiinoaaeoue yeaiaio
Z/n:

10®0/n 5®-2/n

2®+1/n 6®+3/n

3®-1/n 7®-3/n

4®+2/n …

Eaiia 2: Iauaaeeiaiea n/aoiiai eee eiia/iiai(ia aieaa /ai n/aoiiai)
/enea n/aoiuo iiiaeanoa – n/aoii.

A1={a11, a12, a13,…}

A2={a21, a22, a23,…}

A3={a31, a32, a33,…}

I?eiaiyai aeeaaiiaeueioth ioia?aoeeth (a11 – 1; a21 – 2; a12 – 3; a31 –
4; a22 – 5…) e oaeei ia?acii acaeiiiiaeiicia/ii niiinoaaeyai
eaaeaeiio yeaiaioo ec oaaeeoeu aai iiia?, cia/eo iauaaeeiaiea n/aoiiai
eee eiia/iiai /enea n/aoiuo iiiaeanoa – n/aoii.

*anoue iiaeao auoue ?aaiiiiuia oeaeiio: (-1,1) ?aaiiiiuai R (/a?ac
iieoie?oaeiinoue e eo/e)

Ec Eaiiu1 e Eaiiu 2 iieo/aai: Iiiaeanoai ?aoeeiiaeueiuo /enae n/aoii

2. Ii?aaeaeaiea aeaenoaeoaeueiiai /enea aaneiia/iie aeanyoe/iie
ae?iaueth. Ieioiinoue Q a R.

Aeaenoaeoaeueiua /enea – iiiaeanoai /enae aeaea [a0],a1 a2 a3… aaea
a0IZ a1,a2,a3,… I{0,1,…,9}

Aeaenoaeoaeueiia /enei i?aaenoaaeyaony a aeaea noiiu oeaeie e ae?iaiie
/anoe:

[ai],a1 a2 a3…ae (0) = ai + a1/10 + a2/100 + … +ae/10k = [ai],a1 a2
a3…a’e (9), aaea a’e=ae-1

o=[oi],o1 o2 o3…oe…

o=[oi],o1 o2 o3…oe…

o’e – eaoia i?eaeeaeaiea eena n iaaeinoaoeii = [oi],o1 o2 o3…oe

o”e – eaoia i?eaeeaeaiea ea?aea n ecauoeii = [oi],o1 o2 o3…oe + 1/10k

o’e+1 > o’e (o’e – iiiioiiii ?anoao)

o”e+1 F o”k (o”k – ia aic?anoaao), o.e. o”e=[oi],o1 o2 o3…oe + 1/10e

o”e+1 = [oi],o1 o2 o3…oe oe+1 + 1/10e+1

o”e – o”e+1 = 1/10e – oe+1 + 1/10e+1 ? 0

10 – oe+1 – 1 / 10e+1 ? 0

9 ? oe+1

Ii?aaeaeaiea: 1) o > o $ e: o’e > o”e

2) o = o o’e ia> o”e & o”e ia> o’e

Ii ii?aaeaeaieth iieo/aai, /oi [1],(0)=[0],(9)

Naienoaa: 1)” o, o eeai oo, eeai o=o

2) o>o & o>z => o>z

3) o ia> o

Aeie-ai (2): o>o o>z

o’e>o”e o’m>z”m

n=max{k;m}

o’n?o’e>o”e?o”n o’n?
o’m>z”m?z”n

o”n>o’n => o’n>z”n

Ii?aaeaeaiea: Anee AIR e ” o,oIR $ aIA: o o o’e > o”e o ? o’e o”e ? o

o ? o’e / 2 + o’e / 2 > o’e / 2 + o”e / 2 > o”e / 2 + o”e / 2 > o

Aeaeei: o > o’e / 2 + o”e / 2 > o, aaea (o’e / 2 + o”e / 2)IQ

3.Ian/aoiinoue iiiaeanoaa aeaenoaeoaeueiuo /enae.

Oai?aia: R ian/aoii.

Aeieacaoaeuenoai io i?ioeaiiai:

1«o1=[o1], o11 o12 o13… |

2«o2=[o2], o21 o22 o23… | Ionoue caeanue iao aeaayoie a ia?eiaea

3«o3=[o3], o31 o32 o33… |

… | (*)

e«oe=[oe ], oe1 oe2 oe3… |

… |

Iaeaeai /enei eioi?iai iao a oaaeeoea:

n=[n], n1 n2 n3…

[n]?[o1] => n?o1

n1 I {9;o21} => n?o2

n2 I {9;o32} => n?o3

ne I {9;oe+1e} => n?oe

Oaeei ia?acii N – /enei eioi?ia ionoonoaoao a oaaeeoea (*)

5.Oai?aia Aeaaeaeeiaea i iieiioa R

Ionoue 1) 0?AIR; 2) ” aIA, ” bIB: a A ia?aie/aii naa?oo => $ SupA=m => “bIB: b?m => B
ia?aie/aii nieco =>$ InfB=n, mFn

Aeieaaeai, /oi m = n:

Ionoue m cIA & cIA – iaaiciiaeii ii naienoao 3 ionthaea e ec oiai, /oi
mFn

neaaeoao, /oi m=n anee iaicia/ei m=n /a?ac c, oi iieo/ei aFnFb

Aeieaaeai, /oi n aaeeinoaaiiia(io i?ioeaiiai):

Ionoue $n’?n,n’>n (n’ii ii?-ieth. “n’>n
(n’c’)-i?ioeai?a/ea n “aIA, “bIB:
aFnFb

8.Eaiia i caaeaoie iineaaeiaaoaeueiinoe (Eaiia i aeaoo ieeeoeeiia?ao)

Anee $n0: “n>n0 xNFyNFzN e $ Lim xN=x, $ Lim zN=z, i?e/ai x=z, oi $ Lim
yN=y => x=y=z.

Aeieacaoaeuenoai: “n>n0 xNFyNFzN

Aicueiai i?iecaieueii A>0, oiaaea $ n’: “n>n’ xNI(o-A,o+A) & $ n”: “n>n”
zNI(o-A,o+A) => “n>max{n0,n’,n”} yNI(x-E,x+E)

4. Aa?oiea e ieaeiea a?aie /eneiauo iiiaeanoa.

Ii?aaeaeaiea: AIR mIR, m – aa?oiyy (ieaeiyy) a?aiue A, anee ” aIA aFm
(a?m).

Ii?aaeaeaiea: Iiiaeanoai A ia?aie/aii naa?oo (nieco), anee nouanoaoao
oaeia m, /oi ” aIA, auiieiyaony aFm (a?m).

Ii?aaeaeaiea: SupA=m, anee 1) m – aa?oiyy a?aiue A

2) ” m’: m’ m’ ia
aa?oiyy a?aiue A

InfA = n, anee 1) n – ieaeiyy a?aiue A

2) ” n’: n’>n => n’ ia
ieaeiyy a?aiue A

Ii?aaeaeaiea: SupA=m iacuaaaony /enei, oaeia /oi: 1) ” aIA aFm

2) “e>0 $ aEIA, oaeia, /oi aE>a-e

InfA = n iacuaaaony /enei, oaeia /oi: 1) 1) ” aIA
a?n

2) “e>0 $ aEIA, oaeia, /oi aE[m]+1 – aa?oiyy a?aiue A

Io?acie [[m],[m]+1] – ?acaeaaai ia 10 /anoae

m1=max[10*{a-[m]:aIA}]

m2=max[100*{a-[m],m1:aIA}]

me=max[10K*{a-[m],m1…mK-1:aIA}]

[[m],m1…mK, [m],m1…mK + 1/10K]CA?AE=>[m],m1…mK + 1/10K –
aa?oiyy a?aiue A

Aeieaaeai, /oi m=[m],m1…mK – oi/iay aa?oiyy a?aiue e /oi iia
aaeeinoaaiiay:

“e: [m’K,m”K)CA?0; “e “aIA: am”K => $ e: a’K>m”K => a?a’K>m”K – yoi i?ioeai?a/eo
ia?aie/aiiinoe => aFm

Oi/iay aa?oiyy a?aiue:

Ionoue ll”K, ii oae eae “e [m’K,m”K) CA?0 => $
aI[m’K,m”K) => a>l =>l – ia aa?oiyy a?aiue.

Oai?aia: Ethaia, iaionoia ia?aie/aiiia nieco iiiaeanoai AIR, eiaaao
oi/ioth ieaeithth a?aiue, i?e/ai aaeeinoaaiioth.

?anniio?ei iiiaeanoai B{-a: aIA}, iii ia?aie/aii naa?oo e ia ionoi => $
-SupB=InfA

6.Aaneiia/ii iaeua e aaneiia/ii aieueoea iineaaeiaaoaeueiinoe. Eo
naienoaa.

Ii?aaeaeaiea: Iineaaeiaaoaeueiinoue aN iacuaaaony aaneiia/ii iaeie (ai)
anee aa i?aaeae ?aaai ioeth (“A>0 $ n0: n>n0 |aN|n’:
|aN|
n”: |bN|max{n’,n”} auiieiaiu
iaa ia?aaai noaa |aN|
i?e ethaii n> max{n’,n”} eiaai:
|cN|=|aN+bN|F|aN|+|bN| |dN|=|aN-bN| F |aN|+|bN|n0
|aN|
n0: |zN|=|aN*bN|=|aN|*|bN|n’ iineaaeiaaoaeueiinoueoue
|bN|FaN => bN – ai

Aeieacaoaeuenoai: aN – ai => $ n”: “n>n”: |aN|=max{n’,n”}
|bN|F|aN|0 $ n0: n>n0 |aN|>A)

Oai?aia: Anee aN – ai, oi 1/aN – aa iineaaeiaaoaeueiinoueoue, ia?aoiia
oiaea aa?ii.

Aeieacaoaeuenoai:

“=>” aN-ai=>aia ethaie yineeii-ie?anoiinoe oi/ee 0 (a /anoiinoe 1/A)
iaoiaeeony eiia/iia /enei /eaiia iine-oe, o.a. $n0: “n>n0 |aN|1/|aN|>A.

“ “A>0 $ n0: “n>n0 1/|aN|>1/A =>
|aN|n’ iineaaeiaaoaeueiinoue bN?|aN|
=> bN – aa.

Aeieacaoaeuenoai: aN – aa => $ n”: “n>n” |aN|>A. Aeey n>max{n’,n”}
bN?|aN|>A

7.A?eoiaoeea i?aaeaeia

I?aaeeiaeaiea: *enei a yaeyaony i?aaeaeii iineaaeiaaoaeueiinoe aN anee
?aciinoue aN-a yaeyaony ai (ia?aoiia oiaea aa?ii)

Aeieicaoaeuenoai: O.e. Lim aN=a, oi |aN-a| |aN|FA. |aN|=|aN-a| $ Lim (xN/yN) e Lim(xN/yN)=o/o

Aeieacaoaeuenoai:

Ionoue xN=o+aN, aN – ai; yN=o+bN, bN – ai

1) (xN+yN)-(o+o)=aN+bN (Ii oai?aia i noiia ai: aN+bN – ai =>
(xN+yn)-(o+o)-ai, aeaeueoa ii i?aaeeiaeaieth)

2) xN*yN – o*o = o*aN+o*bN+aN*bN (Ii oai?aiai i noiia ai iine-oae e * ai
iine-oae ia ia?. iine-oe iieo/aai: xN*yN – o*o – ai, aeaeueoa ii
i?aaee-ieth)

3) xN/yN – o/o = (o*aN-o*bN) / (o*(o+bN))= (o*aN-o*bN) * 1/o * 1/oN
aeieacaoaeuenoai naiaeeony e aeieacaoaeuenoao ooaa?aeaeaiey: anee on –
noiaeyuayny ia e 0 iine-oue, oi 1/oN oiaea noiaeyuayny
iineaaeiaaoaeueiinoue: Lim oN=y => ii ii?aaeaeaieth i?aaeaea iieo/aai $
n0: “n>n0 |on-o|o/2.|oN|>o/2=>1/|oN|
“n: 1/|oN|Fmax{2/o, 1/o1, 1/o2,…1/ono}

Oai?aia: Anee oN noiaeeony e o, yN noiaeeony e o e $ n0: “n>n0
iineaaeiaaoaeueiinoue oNFoN, oi oFo

Aeieacaoaeuenoai(io i?ioeaiiai): Ionoue o>o. Ec ii?. i?aaeaea “E>0 (a
/anoiinoe An’ |xN-x|n” |yN-y|max{n’,n”} ana /eaiu iine-oe xN aoaeoo eaaeaoue a
A-ie?anoiinoe oi/ee o, a ana /eaiu iine-oe oN aoaeoo eaaeaoue a
A-ie?anoiinoe oi/ee o, i?e/ai

(o-A,o+A)C(o-A,o+A)=AE. E o.e iu i?aaeiieiaeeee, /oi o>o, oi
“n>max{n’,n”}: oN>oN – i?ioeai?a/ea n oneiaeai => oFo.

5. Ii?aaeaeaiea i?aaeaea iineaaeiaaoaeueiinoe e aai aaeeinoaaiiinoue.

Ii?aaeaeaiea: Ionoue aeaiu aeaa iiiaeanoaa O e O. Anee eaaeaeiio
yeaiaioo oIO niiinoaaeai ii ii?aaeaeaiiiio i?aaeeo iaeioi?ue yeaiaio
oIO, oi aiai?yo, /oi ia iiiaeanoaa O ii?aaeaeaia ooieoeey f e ieooo
f:O®O eee o® (f(o)| oIO).

Ii?aaeaeaiea: Iineaaeiaaoaeueiinoue-yoi o-oeey ii?aaeaeaiiay ia ii-aa N,
ni cia/aieyie ai ii-aa R f:N®R. Cia/aiea oaeie o-oeee a (.) nIN
iaicia/atho aN.

Niiniau caaeaiey:

1) Aiaeeoe/aneee: Oi?ioea iauaai /eaia

2) ?aeo??aioiue: (aica?aoiay) oi?ioea: Ethaie /eai iineaaeiaaoaeueiinoe
ia/eiay n iaeioi?iai au?aaeaaon /a?ac i?aaeeaeouea. I?e yoii niiniaa
caaeaie iau/ii oeacuaatho ia?aue /eai (eee ineieueei ia/aeueiuo /eaiia)
e oi?ioeo, iicaiethueth ii?aaeaeeoue ethaie /eai iineaaeiaaoaeueiinoe
/a?ac i?aaeeaeouea. I?eia?: a1=a; aN+1=aN + a

3) Neiaaniue: caaeaiea iineaaeiaaoaeueiinoe iienaieai: I?eia?: aN = n-ue
aeanyoe/iue ciae /enea Ie

Ii?aaeaeaiea: *enei a iacuaaaony i?aaeaeii iineaaeiaaoaeueiinoe aN, anee
“e>0 $ n0: “n>n0 auiieiyaony ia?aaainoai |aN-a| a ie ?anoiinoe oi/ee n niaea?aeeony eiia/iia
/enei /eaiia iineaaeiaaoaeueiinoe – i?ioeai?a/ea n oneiaeai oiai, /oi n
– i?aaeae iineaaeiaaoaeueiinoe.

Oai?aia: Noiaeyuayny iineaaeiaaoaeueiinoue ia?aie/aia.

Aeieacaoaeuenoai:

Ionoue iineaaeiaaoaeueiinoue aN noiaeeony e /eneo a. Aicueiai eaeia-eeai
yineeii, aia yineeii-ie?anoiinoe oi/ee a eaaeeo eiia/iia /enei /eaiia
iineaaei aaoaeueiinoe, cia/eo anaaaea iiaeii ?acaeaeiooue ie?anoiinoue
oae, /oiau ana /eaiu iineaaeiaaoaeueiinoe a iaa iiiaee, a yoi e icia/aao
/oi iineaaeiaaoaeue iinoue ia?aie/aia.

Caia/aiey: 1) Ia?aoiia ia aa?ii (an=(-1)N, ia?aie/aia ii ia noiaeeony)

2) Anee nouanoaoao i?aaeae iineaaeiaaoaeueiinoe aN,
oi i?e ioa?anuaaiee eee aeiaaaeaiee eiia/iiai /enea /eaiia i?aaeae ia
iaiyaony.

Ii?yaeeiaua naienoaa i?aaeaeia:

Oai?aia i i?aaeaeueiii ia?aoiaea: Anee Lim xN=x, Lim yN=y, $n0: “n>n0
oNFyN, oiaaea xFy

Aeieacaoaeuenoai(io i?ioeaiiai):

Ionoue o>o => ii ii?aaeaeaieth i?aaeaea $ n0’: “n>n0’ |oN-o|n0” |yN-y|max{n0’, n0”}: |oN-o|max{n0’, n0”} oNI(o-A,o+A) & oNI(o-A,o+A)
o/eouaay, /oi o>o iieo/aai: “n>max{n0’, n0”} oN>yN – i?ioeai?a/ea n
oneiaeai.

Oai?aia: Anee $n0: “n>n0 aNFbNFcN e $ Lim aN=a, $ Lim cN=c, i?e/ai a=c,
oi $ Lim bN=b => a=b=c.

Aeieacaoaeuenoai: Aicueiai i?iecaieueii A>0, oiaaea $ n’: “n>n’ =>
cNn” => (a-E)max{n0,n’,n”}
(a-E)max{n0,n’,n”}=>bNI(a-E,a+E)

9. I?aaeae iiiioiiiie iineaaeiaaoaeueiinoe

Ii?aaeaeaiea: Iineaaeiaaoaeueiinoue iacuaaaony iiiioiiii aic?anoathuae
(oauaathuae) anee ” n1>n2 (n10 $xE: (o-A) $ n0 xNo>(o-E). Ec
iiiioii iinoe eiaai: “n>n0 xN?xNo>(x-E), iieo/eee xNFx=SupX, cia/eo
“n>n0 xNI(x-E,o] $ Lim aN=a

a1FaNFbN bN iiioiiii oauaaao & a1FbN => $ Lim bN=b

aNFa bFbN aNFbN => aFb

Lim (bN-aN)=b-a=0(ii oneiaeth)=>a=b

Ionoue c=a=b, oiaaea aNFcFbN

Ionoue n ia aaeeinoaaiiia: aNFc’FbN, n’?n

aNFcFbN=>-bNF-cF-aN => aN-bNFc’-cFbN-aN => (Ii oai?aia i i?aaeaeueiii
ia?aoiaea) => Lim(aN-bN)FLim(c’-c)FLim(bN-aN) => (a-b)FLim(c`-c)F(b-a)
=>

0Flim(c`-c)F0 => 0F(c`-c)F0 => c’=c => c – aaeeinoaaiiia.

Ia?ao?ace?iaea Eaiiu: Ionoue eiaaony aaneiia/iaz iine-oue aeiaeaiiuo
ae?oa a ae?oaa i?iiaaeooeia (i?iiaaeooie 1 aeiaeai a i?iiaaeooie 2 anee
ana oi/ee i?iiaaeooea 1 i?eiaaeeaaeao i?iiaaeooeo 2:
[a1,b1],[a2,b2],…,[an,bn]…, oae /oi eaaeaeue iineaaeothuee
niaea?aeeony a i?aaeuaeouai, i?e/ai aeeeiu yoeo i?iiaaeooeia no?aiyony
e 0 i?e n®Y lim(bN-aN)=0, oiaaea eiioeu i?iiaaeooeia aN e bN no?aiyony e
iauaio i?aaeaeo n (n ?aciuo noi?ii).

42.Eieaeueiue yeno?aioi. Oai?aia Oa?ia e aa i?eeiaeaiea e iaoiaeaeaieth
iaeaieueoeo e iaeiaiueoeo cia/aiee.

Ii?aaeaeaiea: Ionoue caaeai i?iiaaeooie I=(a;b), oi/ea x0I(a;b). Oi/ea
x0, iacuaaaony oi/eie eieaeieiai min(max), anee aeey anao xI(a;b),
auiieiyaony

f(x0)f(x)).

Eaiia: Ionoue ooieoeey f(x) eiaao eiia/ioth i?iecaiaeioth a oi/ea x0.
Anee yoa i?iecaiaeiay f‘(x0)>0(f‘(x0)f(x0) (f(x)f(x0)).

.

Anee f‘(x0)>0, oi iaeaeaony oaeay ie?anoiinoue (x0-d,x0+d) oi/ee x0, a
eioi?ie (i?e o?x0) (f(x)-f(x0))/(x-x0)>0. Ionoue x00 => ec i?aaeuaeouaai ia?aaainoaa neaaeoao, /oi f(x)-f(x0)>0, o.a.
f(x)>f(x0). Anee aea x-d0, e oiaaea (ii eaiia) f(x)>f(x0), anee x>x0
e aeinoaoi/ii aeecei e x0, eeai f‘(x0)f(x0), anee x iieo/eee
i?ioeai?a/ea => oai?aia aeieacaia.

Neaaenoaea: Anee nouanoaoao iaeaieueoaa (iaeiaiueoaa) cia/aiea ooieoeee
ia [a;b] oi iii aeinoeaaaony eeai ia eiioeao i?iiaaeooea, eeai a oi/eao,
aaea i?iecaiaeiie iao, eeai iia ?aaia ioeth.

43.Oai?aiu ?ieey, Eaa?aiaea, Eioe (i n?aaeiai cia/aiee).

Oai?aia ?ieey

Ionoue 1) f(x) ii?aaeaeaia e iai?a?uaia a caieiooii i?iiaaeooea [a;b]

2) nouanooao eiia/iay i?iecaiaeiay f’(x), ii e?aeiae ia?a a
iooe?uoii i?iiaaeooea (a;b)

3) ia eiioeao i?iiaaeooea ooieoeey i?eieiaao ?aaiua cia/aiey:
f(a)=f(b)

Oiaaea iaaeaeo a e b iaeaeaony oaeay oi/ea c(a f’(x)=0 ai anai
i?iiaaeooea, oae /oi a ea/anoaa n iiaeii acyoue ethaoth oi/eo ec (a;b).

2) M>m. Ii aoi?ie oai?aia Aaea?oo?anna iaa yoe cia/aiey ooieoeeae
aeinoeaathony, ii, oae eae f(a)=f(b), oi oioue iaeii ec ieo aeinoeaaaony
a iaeioi?ie oi/ ea n iaaeaeo a e b. A oaeii neo/aa ec oai?aiu Oa?ia
(Ionoue ooieoeey f(x) ii?aaeaeaia a iaeioi?ii i?iiaaeooea I=(a;b) e ai
aioo?aiiae oi/ea x0 yoiai i?iiaaeooea i?eieiaao iaeaieueoaa
(iaeiaiueoaa) cia/aiea. Anee ooieoeey f(x) aeeooa?aioee?oaia a oi/ea x0,
oi iaiaoiaeeii f‘(x0)=0) neaaeoao, /oi i?iec aiaeiay f’(n) a yoie oi/ea
ia?auaaony a ioeue.

Oai?aia Eioe:

Ionoue 1) f(x) e g(x) iai?a?uaiu a caieiooii i?iiaaeooea [a;b] &
g(b)?g(a)

2) nouanootho eiia/iua i?iecaiaeiua f’(x) e g’(x), ii e?aeiae
ia?a a iooe?uoii i?iiaaeooea (a;b)

3) g’(x)?0 a iooe?uoii i?iiaaeooea (a;b)

*(g(x) – g(a))]

Yoa ooieoeey oaeiaeaoai?yao anai oneiaeyi oai?aiu ?ieey:

1) h(x) iai?a?uaia ia [a;b], eae eiiaeiaoeey iai?a?uaiuo ooieoeee

*g’(x)

3) i?yiie iiaenoaiiaeie oaaaeaeaainy h(a)=h(b)=0

*g’(c).

?acaeaeea iaa /anoe ?aaainoaa ia g’(x) (g’(x)?0) iieo/aai o?aaoaiia
?aaainoai.

Oai?aia Eaa?aiaea:

Ionoue 1) f(x) ii?aaeaeaia e iai?a?uaia a caieiooii i?iiaaeooea [a;b]

2) nouanooao eiia/iay i?iecaiaeiay f’(x), ii e?aeiae ia?a a
iooe?uoii i?iiaaeooea (a;b)

I?iiaaeooi/iia cia/aiea n oaeiaii caienuaaoue a aeaea n=a+q(b-a), aaea
qI(0;1). Oiaaea i?eieiay x0=a, (b-a)=h, iu iieo/aai neaaeothuaa
neaaenoaea:

Neaaenoaea: Ionoue f(x) aeeooa?aioee?oaia a eioa?aaea I=(a;b), x0II,
x0+hII, oiaaea $ qI(0;1): f(x0+h)-f(x0)=f’(x0+qh)*h ([x0;x0+h] h>0,
[x0+h;x0] hiiaeiine-noue – yoi eeai naia iine-oue eeai enoiaeiay iine-oue, ec
eioi?ie aua?ineee /anoue /eaiia.

Oai?aia: Anee Lim aN=a, oi e Lim aKn=a.

Aeieacaoaeuenoai: Aia ethaie A-ie?anoiinoe oi/ee a eaaeeo eiia/iia /enei
/eaiia iineaaeiaaoaeueiinoe an e a /anoiinoe iineaaeiaaoaeueiinoe.

Aeieacaoaeuenoai: Ionoue aeey caaeaiiiai A iaoeinue n0: “n>n0 |aN-a|n’ kN>n0
oiaaea i?e oao aea cia/aieyo n aoaeao aa?ii |aKn-a|
“n: aFoNFb. Iiaeaeei i?iiaaeooie
[a,b] iiiieai, oioy au a iaeiie aai iieiaeia niaea?aeeony aaneiia/iia
iiiaeanoai /eaiia iine-oe oN (a i?ioeaiii neo/aa e ai anai i?iiaaeooea
niaea?aeeony eiia/iia /enei /eaiia iine-oe, /oi iaaiciiaeii). Ionoue
[a1,b1] – oa iieiaea, eioi?ay niaea?aeeo aaneiia/iia /enei /eaiia
iine-oe. Aiaeiae/ii auaeaeei ia i?iiaaeooea [a1,b1] i?iiaaeooie [a2,b2]
oaeaea niaea?aeauee aaneiia/iia /enei /eaiia iine-oe oN. I?iaeieaeay
i?ioeann aei aaneiia/iinoe ia e-oii oaaa auaeaeei i?iiaaeooie
[aK,bK]-oaeaea niaea?aeauee niaea?aeauee aaneiia/ iia /enei /eaiia
iine-oe oN. Aeeeia e-oiai i?iiaaeooea ?aaia bK-aK = (b-a)/2K, e?iia oiai
iia no?aieony e 0 i?e e®Y e aK?aK+1 & bKFbK+1. Ionthaea ii eaiia i
aeiaeaiiuo i?iiaaeooeao $! n: “n aNFcFbN.

Oaia?ue iino?iei iiaeiineaaeiaaoaeueiinoue:

oN1 I[a1,b1]

oN2 I[a2,b2] n2>n1

. . .

oNKI[aK,bK] nK>nK-1

aFoNkFb. (Lim aK=LimbK=c ec eaiiu i aeiaeaiiuo i?iiaaeooeao)

Ionthaea ii eaiia i caaeaoie iineaaeiaaoaeueiinoe Lim oNk=c – /.o.ae.

12.Aa?oiee e ieaeiee i?aaeaeu iineaaeiaaoaeueiinoe.

xN – ia?aie/aiiay iineaaeiaaoaeueiinoue =>”n aNFoNFbN

oNK®o, oae eae oNK-iiaeiineaaeiaaoaeueiinoue => “n aFoNFb =>aFoFb

o – /anoe/iue i?aaeae iineaaeiaaoaeueiinoe oN

Ionoue I – iiiaeanoai anao /anoe/iuo i?aaeaeia.

Iiiaeanoai I ia?aie/aii (aFIFb) => $ SupM & $ InfM

Aa?oiei i?aaeaeii iine-oe xN iacuaatho SupM?Sup{xN}: ieooo Lim xN

Ieaeiei i?aaeae ii iine-oe xn iacuaatho InfM?Inf{xN}: ieooo lim xN

Couanoaiaaiea ieaeiaai e aa?oiaai i?aaeaeia auoaeaao ec ii?aaeaeaiey.

Aeinoeaeeiinoue:

Oai?aia: Anee oN ia?aie/aia naa?oo (nieco), oi $ iiaeiine-oue oNK:
i?aaeae eioi?ie ?aaai aa?oiaio (ieaeiaio) i?aaeaeo oN.

Aeieacaoaeuenoai: Ionoue o=SupM=aa?oiee i?aaeae oN

$ o’II: o-1/e $
iiaeiineaaeiaaoaeueiinoue oNS®o’ => “A>0 (a /anoiinoe A=1/e) $ s0:
“s>s0 =>

o’-1/es0 yoi ia?aaainoai auiieiyaony
aa?ai /eai iine-oe oNS n iiia?ii aieueoa s0 e ioia?oai aai oN1

k=1: o-2/10 $ n0: “n>n0 e ethaiai ?IN auiieiaii ia?aaainoai |aN+?-aN|0 $ n0, oaeie /oi ?annoiyiea iaaeaeo
ethauie aeaoiy /eaiaie iine-oe, n aieueoeie /ai n0 iiia?aie, iaiueoa A.

E?eoa?ee Eioe noiaeeiinoe iine-oe: Aeey oiai, /oiau aeaiiay iine-oue
noiaeeeanue iaiaoiaeeii e aeinoaoi/ii, /oiau iia yaeyeanue
ooiaeaiaioaeueiie.

Aeieacaoaeuenoai:

Iaiaoiaeeiinoue: Ionoue Lim xN=x, oiaaea “A>0 $ n0: “n>n0 |oN-o|n0, n’>n0 |oN-oN’|=|oN-o+o-oN’|n0, n’>n0

“n>n0 |oN-oN0| oN – ia?aie/aia

2) Ii oai?aia Aieueoeaii-Aaea?oo?anna

$ iiaeiine-oue oNK®o. Iiaeii aua?aoue e ianoieueei aieueoei, /oiau
|oNK-o|n0. Neaaeiaaoaeueii (ec ooiae-oe)
|oN-oNK|

|oNK-o| o-A/2 |oN-oNK| oNK-A/2
o-A |oN-o|
0)

| yN-1) (aeieacuaaaony
ii eiaeoeoeee):

x=1/n => (1+1/n)n?1+n/n=2

Iieo/eee: 2 F xN xN – ia?aie/aia, o/eouaay /oi xN – iiiioiiii
aic?anoaao => xN – noiaeeony e aa i?aaeaeii yaeyaony /enei a.

(ai anao i?aaeaeao n®Y)

=1, a>0

a) a?1:

=> $ Lim xN=x

xN=xN+1*xN*(n+1)

Lim xN=Lim (xN+1*xN*(n+1)) => x = x*x => x = 1

= 1/1 = 1

= 0, a>1

=1

=> $ n0: “n>n0 xn+1/xn NO x=limxn

=> x = x*1/a => x=0

Aeieaaeai, /oi anee xN®1 => (xN)a®1:

a) “n: xN?1 e a?0

(xN) [a]F(xN)a ii eaiia i caaeaoie iine-oe, o/eouaay /oi
Lim (xN)[a]=Lim (xN)[a]+1=1 (ii oai?aia i Lim i?iecaaaeaiey) iieo/aai
Lim (xN)a =1

a) “n: 0 yn>1 Lim yN=lim1/xN=1/1=1 => (ii (a)) Lim (yN)a =1 => lim
1/(xN)a =1 => Lim (xN)a =1

Iauaaeeiei (a) e (a):

xN®1 a>0

xN1,xN2,…>1 (1)

xM1,xM2,… eiia/iia /enei oi/ae xN.

a) a -a>0 => ii aeieacaiiiio aeey a>0 iieo/aai, Lim
1/(xN)- a = 1 => Lim (xN) a = 1

15. Aeieacaoaeuenoai oi?ioeu e=…

1) yN iiiioiiii ?anoao

2) yN
i?ioeai?a/ea

23. Ii?aaeaeaiey i?aaeaea ooieoeee ii Eioe e ii Aaeia. Eo
yeaeaaeaioiinoue.

Ii?aaeaeaiea ii Eioe: f(x) noiaeeony e /eneo A i?e o®o0 anee “A>0 $d>0:
0 |f(x)-A|(A)

“A>0 $d>0: 0 |f(x)-A| $ n0: “n>n0 0 0 ii ii?aaeaeaieth Eioe |f(xN)-A|(E) Ainiieuecoainy caeiiii eiaeee: Anee ec io?eoeaiey B neaaeoao
io?eoeaiea A, oi ec A neaaeoao A:

Oaeei ia?acii iai iaaei aeieacaoue /oi ec io?eoeaiey (E) => io?eoeaiea
(A)

Io?eoeaiea (E): $ A>0: “d >0 $ x: 0 |f(x)-A|?E

Io?eoeaiea (A): $ oN®o0, oN?o0: |f(xN)-A|?E

$ oN®o0, oN?o0 => $ n0: “n>n0 0 ii io?eoeaieth
ii?aaeaeaiey Eioe |f(xN)-A|?A

Aeey o-oeee o®f(o) ii?aaeaeaiiie ia eioa?aaea (a,+Y), ii?aaeaeyaony
i?aaeae i?e oN®Y neaaeothuei ia?acii: limf(o) i?e oN®Y = Limf(1/t) t®+0

(anee iineaaeiee nouanoaoao). Oaeei aea ia?acii ii?aaeaeythony Lim f(o)
i?e oN®-Y = Lim f(1/t) t®-0 e oN®Y = lim f(1/t) t®0

24. Iaeiinoi?iiiea i?aaeaeu. Eeanneoeeaoeey ?ac?uaia. Ii?aaeaeaiea
iai?a?uaiinoe.

Lim(o0±|h|) i?e h®0 – iacuaaaony iaeiinoi?iiiei i?aaui (eaaui i?aaeaeii)
o-oeee f(x) a oi/ea o0

Oai?aia: Ionoue eioa?aae (x0-d,x0+d)\{x0} i?eiaaeeaaeeo iaeanoe
ii?aaeaeaiey o-oeee aeey iaeioi?ai d>0. Oiaaea Lim f(x) a oi/ea o0
nouanoaoao eiaaea couanoaotho i?aaue e eaaue i?aaeae f(x) a oi/ea o0
e iie ?aaiu iaaeaeo niaie.

Iaiaoiaeeiinoue: Ionoue i?aaeae f(o) nouanoaoao e ?aaai A => “A>0 $ d
>0: -d |f(o)-A| x iiiaaeaao a
eioa?aae (x0-d,x0+d) => f(o) iiiaaeaao a eioa?aae (f(o)-A,f(o)+A) =>
i?aaue i?aaeae nouanoaoao e ii ?aaai A. Anee o iiiaaeaao a eioa?aae
(x0-d,0) => x iiiaaeaao a eioa?aae (x0-d,x0+d) => f(o) iiiaaeaao a eioa?
aae (f(o)-A,f(o)+A) => eaaue i?aaeae nouanoaoao e ii ?aaai A.

Aeinoaoi/iinoue: Lim (o0±|h|) i?e h®0: Lim(o0+|h|) = Lim(o0-|h|)=A

“A>0 $ d’ >0: 0 |f(o)-A|0 $ d” >0: -d” |f(o)-A|0 $ 0 |f(o)-A| a ii?aaeaeaiee iiaeii
niyoue ia?aie/aiea o?o0 => iieo/ei aoi?ia ?aaiineeueiia ii?aaeaeaiea:

Ii?aaeaeaiea 2: Ooieoeey f(x) iacuaaaony iai?a?uaiie a oi/ea o0, anee
“A>0 $d>0: -d |f(o)-f(a)|0(a /anoiinoe A/2) $d’>0: -d’ |f(o)-F|0:
-d” |g(o)-G|
0 $ 0-A/2 – A/2 |(f(o)+g(o))-(F+G)| ii ii?aaeaeaieth i?aaeaea ii Aaeia i?e o®o0 Lim
f(x)*Lim g(x)=F*G

3) Ionoue iine-oue oN®o0 (oN?o0, xNIX), oiaaea a neeo ii?aaeaeaiey
i?aaeaea ii Aaeia eiaai: i?e n®Y Lim f(xN)=F & Lim g(xN)=G ii oai?aia ia
a?eoiaoeea i?aaeaeia iine-oae iieo/aai: i?e n®Y Lim f(xN)/g(xN)=Lim
f(xN)/Lim g(xN)=F/G => ii ii?aaeaeaieth i?aaeaea ii Aaeia i?e o®o0 Lim
f(x)/Lim g(x)=F/G, G?0 e g(x)?0.

Ii?yaeeiaua naienoaa i?aaeaeia:

Oai?aia: Anee ” oIX: f(x)Fg(x), i?e o®o0 A=Lim f(x), B=Lim g(x), oi AFB

Aeieacaoaeuenoai(io i?ioeaiiai):

Ionoue A>B => ec ii?aaeaeaiey i?aaeaea neaaeoao (aa?ai 00: |o-o0| |f(x)-A|0: |o-o0| |g(o)-B| |f(x)-A|A e /oi (A-A,A+A)C(A-A,A+A)=AE,
iieo/aai /oi aeey

oI(o0-d, o0+d) f(x)>g(x) – i?ioeai?a/ea n oneiaeai.

Oai?aia: Anee ” oIX: f(x)Fg(x)Fh(x) e i?e o®o0 Lim f(x)=A=Lim h(x), oi
Lim g(x)=A

Aeieacaoaeuenoai:

“A>0 $d’>0: |o-o0| A-E0: |o-o0| h(o) A-E A-E A-E -|x-x0| -|x-x0|®0 & |x-x0|®0 => (ii
oai?aia i ii?yaeeiauo na-aao i?aaeaea) (Sin x-Sin x0)®0

2) Cos x:

Lim Cos x = Cos x0 (i?e o®o0)

Cos x = Sin (I/2 – x) = Sin y; Cos x0 = Sin (I/2 – x0) = Sin y0

|Sin y-Sin y0|=2*|Sin((y-y0)/2)|*|Cos((y+y0)/2)| -|y-y0| (Sin
y-Sin y0)®0 => i?iecaiaeei ia?aoioth caiaio: [Sin (I/2 – x)-Sin(I/2 –
x0)]®0 => (Cos x-Cos x0)®0

3) Tg x – iai?a?uaiay o-oeey eneeth/ay oi/ee o = I/2 +2Ie, eIZ

4) Ctg x – iai?a?uaiay o-oeey eneeth/ay oi/ee o = Ie, eIZ

Oai?aia: Lim (Sin x)/x=1 (i?e o®0), 0 Cos x Lim (Sin x)/x =1, 0
f(o0)=n). Ionoue a oi/ea aeaeaiey ooieoeey g(x) a iieue ia ia?auaaony,
oiaaea auae?aai ec aeaoo iieo/aiiuo i?iiaaeooeia oio, aeey eioi?iai
g(a1)*g(b1) oai?aia aeieacaia. Ionoue a oi/ea aeaeaiey
ooieoeey g(x) a iieue ia ia?auaaony, oiaaea auae?aai ec aeaoo iieo/aiiuo
i?iiaaeooeia oio aeey eioi?iai g(a2)*g(b2)0, i?e/ai aeeeia aai
?aaia bN-aN=(b-a)/2n®0 i?e n®Y. Iino?iaiiay iine-oue i?iiaaeooeia oaeia
eaoai?yao oneiaeth Eaiiu i aeiaeaiiuo i?iiaaeooeao => $ oi/ea x0 ec
i?iiaaeooea [a,b], aeey eioi?ie Lim aN=Lim bN= x0. Iieaaeai, /oi
x0-oaeiaeaoai?yao o?aaiaaieth oai?aiu: g(aN)0 => ia?aoiaeei e
i?aaeaeai: Lim g(aN)F0, Lim g(bN)?0, eniieuecoai oneiaea iai?a?uaiinoe:
g(x0)F0 g(x0)?0 => g(x0)=0 => f(o0)-c=0 => f(o0)=c

Neaaenoaea: Anee ooieoeey f(x) iai?a?uaia ia i?iiaaeooea O, oi
iiiaeanoai O=f(O)={f(o):oIO} oaeaea yaeyaony i?iiaaeooeii (Iai?a?uaiay
o-oeey ia?aai aeeo i?iiaaeooie a i?iiaaeooie.)

Aeieacaoaeuenoai: Ionoue o1,o2IO; o1FoFo2, oiaaea nouanoaotho o1,o2IO:
o1=f(o1), o2=f(o2). I?eiaiyy oai?aio e io?aceo [o1,o2]IO (anee o1 O –
oaeiaeaoai?yao ii?aaeaeaieth i?iiaaeooea.

29. I?aaeae noia?iiceoeee ooieoeee. Iai?a?uaiinoue noia?iiceoeee
iai?a?uaiuo ooieoeee

Ii?aaeaeaiea: Noia?iiceoeeae (eiiiiceoeeae) aeaoo ooieoeee f e g
iacuaaaony ooieoeey f(g(x)) – ii?aaeaeaiiay aeey anao o i?eiaaeeaaeaueo
iaeanoe ii?aaea eaiey o-oeee g oaeeo /oi cia/aiey o-oeee g(x) eaaeao a
iaeanoe ii?aaeaeaiey o-oeee f.

Oai?aia: Anee Lim g(x)=b (i?e x®a) e f – iai?a?uaia a oi/ea b, oi Lim
f(g(x))=f(b) (i?e x®a)

Aeieacaoaeuenoai:

Ionoue xN: xN?a – i?iecaieueiay iine-oue ec iaeanoe ii?aaeaeaiey o-oeee
o®f(g(x)), noiaeyuayny e a, oiaaea iineaaeiaaoaeueiinoue yN: yN=g(xN)
noiaeeony e b a neeo ii?. ii Aaeia. Ii oiaaea Lim f(yN)=f(b) (n®Y) a
neeo ii?. iai?a?uaiinoe o-oeee f ii Aaeia. O.i. Lim f(g(xN))=Lim
f(yN)=f(b) (n®Y). Caiaoei /oi a iine-oe yN – iaeioi?ua (e aeaaea ana
/eaiu) iiaoo ieacaoueny ?aaiuie b. Oai ia iaiaa a neeo iaoaai caia/aiey
i niyoee ia?aie/aiey yN?b a ii?aaeaeaiee iai?a?uaiinoe ii Aaeia iu
iieo/aai f(yN)®f(b)

Neaaenoaea: Ionoue ooieoeey g iai?a?uaia a oi/ea x0, a ooieoeey f
iai?a?uaia a oi/ea o0=g(x0), oiaaea o-oeey f(g(x)) iai?a?uaia a oi/ea
o0.

30. Ia?auaiea iai?a?uaiie iiiioiiiie ooieoeee.

Ii?aaeaeaiea: Ooieoeey f ia?aoeia ia iiiaeanoaa O anee o?aaiaiea f(o)=o
iaeiicia/ii ?ac?aoeii ioiineoaeueii oIf(O).

Ii?aaeaeaiea: Anee ooieoeey f ia?aoeia ia iiiaeanoaa O. Oi ooieoeey
iaeiicia/ii niiinoaaeythuay eaaeaeiio oi oaeia o0 /oi f(o0)=o0 –
iacuaaaony ia?aoiie e ooieoeee f.

Oai?aia: Ionoue no?iai aic?anoathuay (no?iai oauaathuay) o-oeey f
ii?aaeaeaia e iai?a?uaia a i?iiaaeooea O. Oiaaea nouanoaoao ia?aoiay
ooieoeey f’,

ii?aaeaeaiiay a i?iiaaeooea Y=f(O), oaeaea no?iai aic?anoathuay (no?iai
oauaathuay) e iai?a?uaiay ia Y.

Aeieacaoaeuenoai: Ionoue f no?iai iiiioiiii aic?anoaao. Ec iai?a?uaiinoe
ii neaaenoaeth ec Oai?aiu i i?iiaaeooi/iii cia/aiee neaaeoao, /oi
cia/aiey iai?a?uaiie ooieoeee caiieiytho nieioue iaeioi?ue i?iiaaeooie
Y, oae /oi aeey eaaeaeiai cia/aiey o0 ec yoiai i?iiaaeooea iaeaeaony
oioue iaeii oaeia cia/aiea o0IO, /oi f(o0)=o0. Ec no?iaie
iiiioiiiinoe neaaeoao /oi oaeia caia/aiea iiaeao iaeoenue oieueei iaeii:
anee o1> eee eee o’=f(o’) e o”=f(o”)
Anee au

auei o’>o”, oiaaea ec aic?anoaiey f neaaeoao /oi o’>o” – i?ioeai?a/ea n
oneiaeai, anee o’=o”, oi o’=o” – oiaea i?ioeai?a/ea n oneiaeai.

Aeieaaeai /oi f` iai?a?uaia: aeinoaoi/ii aeieacaoue, /oi Lim f`(o)=(o0)
i?e o®o0. Ionoue f`(o0)=o0. Aicueiai i?iecaieueii A>0. Eiaai “oIO:
|f`(o)-f`(o0)| o0-A f(o0-A)
f(o0-A)-o0 -d’o0-f(o0)=0, d”=f(o0+A)-o0>f(o0)-o0=0,

iieaaay d=min{d’,d”} eiaai: eae oieueei |o-o0| -d’
|f`(o)-f`(o0)| o-oeey
oM/N – iai?a?uaia i?e o>0. Anee o=0, oi oM/N = 1, a neaaeiaaoaeueii
iai?a?uaia.

?anniio?ei o-oeeth oN, nIN: iia iai?a?uaia oae eae ?aaia i?iecaaaeaieth
iai?a?uaiuo ooieoeee o=o.

n=0: oN oiaeaeanoaaiii ?aaii eiinoaioa => oN – iai?a?uaia o-N=1/oN,
o/eouaay /oi:

1) 1/o – iai?a?uaiay ooieoeey i?e o?0

2) oN (nIN) – oiaea iai?a?uaiay ooieoeey

3) o-N=1/oN – noia?iiceoeey o-ee 1/o e oN i?e o?0

Ii oai?aia i iai?a?uaiinoe noia?iiceoeee o-oeee iieo/aai: o-N –
iai?a?uaiay i?e o?0, o.i. iieo/eee /oi oMmIZ – iai?a?uaiay o-oeey i?e
o?0. I?e o>0 o-oeey oN nIN no?iai iiiioiiii aic?anoaao e o-oeey
oNiai?a?uaia=>$ ooieoeey ia?aoiay aeaiiie, eioi?ay oaeaea no?iai
iiiioiiii aic?anoaao (i?e m>0), i/aaeaeii yoie ooieoeeae aoaeao
ooieoeey o1/N

O?eaiiiiao?e/aneea ooieoeee ia ii?aaeaeaiiuo (aeey eaaeaeie)
i?iiaaeooeao ia?aoeiu e no?iai iiiioiiiu =>eiatho iai?a?uaiua ia?aoiua
ooieoeee => ia?aoiua o?eaiiiiao?e/aneea ooieoeee – iai?a?uaiu

31. Naienoaa iieacaoaeueiie ooieoeee ia iiiaeanoaa ?aoeeiiaeueiuo /enae.

Ii?aaeaeaiea: Iieacaoaeueiay ooieoeey ia iiiaeanoaa ?aoeeiiaeueiuo
/enae: Ooieoeey aeaea aX, a>0, a?1 xIQ.

Naienoaa: aeey mIZ nIN

1) (aM)1/N = (a1/N)M

(aM)1/N=(((a1/N)N)M)1/N = ((a1/N)N*M)1/N = (((a1/N)M)N)1/N = (a1/N)M

2) (aM)1/N=b aM=bN

3) (aM*K)1/N*K=(aM)1/N

(aM*K)1/N*K=b aM*K=bN*K aM=bN (aM)1/N=b

Ec naienoa aeey oeaeiai iieacaoaey auoaeatho na-aa aeey ?aoeeiiaeueiiai
anee iaicia/eoue: aM/N=(aM)1/N=(a1/N)M, a-M/N=1/aM/N, a0=1

Na-aa: x,yIQ

1) aX * aY = aX+Y

aX * aY =b; x=m/n, y=-k/n => aM/N * 1/aK/N = b => aM/N = b * aK/N => aM
= bN * aK => aM-K = bN => a(M-K)/N = b => aX+Y = b

2) aX/aY = aX-Y

3) (aX)Y=aX*Y

(aX)Y=b; x=m/n, y=k/s => (aM/N)K/S=b => (aM/N)K=bS => (a1/N)M*K=bS =>
(aM*K)1/N=bS => aM*K=bS*N => a(M*K)/(S*N)=b => aX*Y=b

4) x aX1) – iiiioiiiinoue

z=y-x>0; aY=aZ+X => aY-aX=aZ+X-aX=aX*aZ-aX=aX*(aZ-1) => anee aZ>1 i?e
z>0, oi aX aZ=(a1/N)M => a1/N>1 => (a1/N)M>1 => aX*(aZ-1)>1, (a>1 n>0)

5) i?e x®0 aX®1 (xIR)

O.e. Lim a1/N=1 (n®Y), i/aaeaeii, /oi e Lim a-1/N=Lim1/a1/N=1 (n®Y).
Iiyoiio “A>0 $n0: “n>n0 1-E1. Anee oaia?ue |x| 1-E Lim aX=1 (i?e x®0)

32.Ii?aaeaeaiea e naienoaa iieacaoaeueiie ooieoeee ia iiiaeanoaa
aeaenoaeoaeueiuo /enae.

Ii?aaeaeaiea: Iieacaoaeueiay ooieoeey ia iiiaeanoaa aeaenoaeoaeueiuo
/enae: Ooieoeey aeaea aX, a>0, a?1 xIR.

Naienoaa: x,yIR.

1) aX * aY = aX+Y

xN®x, yN®y => aXn * aYn = aXn+Yn => Lim aXn * aYn = Lim aXn+Yn => Lim
aXn * lim aYn = Lim aXn+Yn => aX * aY = aX+Y

2) aX / aY = aX-Y

3) (aX)Y=aX*Y

xN®x, yK®y => (aXn)Yk = aXn*Yk => (n®Y) (aX)Yk=aX*Yk =>(k®Y) (aX)Y=aX*Y

4) x aX1) – iiiioiiiinoue.

x xN aXn (n®Y)
aXFaX’- iiiioiiia

x-x`>q>0 => aX-X’ ? aQ>1 => aX-X’?1 => aX0 $n0: “n>n0 1-E1. Anee oaia?ue |x| 1-E Lim aX=1 (i?e x®0)

6) aX – iai?a?uaia

Lim aX=1 (n®0) ec (5) – yoi icia/aao iai?a?uaiinoue aX a oi/ea 0 =>
aX-aXo= aXo(aX-Xo – 1) i?e o®x0 x-x0 n®0 => aX-x0 n®1 => i?e o®x0 lim(aX
– aXo)=

Lim aXo*Lim(aX-Xo – 1) = x0 * 0 = 0 => aX – iai?a?uaia

33.I?aaeae ooieoeee (1+x)1/X i?e x®0 e naycaiiua n iei i?aaeaeu.

1) Lim (1+x)1/X = e i?e x®0

O ian anoue Lim (1+1/n)n = e i?e n®Y

Eaiia: Ionoue nK®Y nKIN Oiaaea (1+1/nK)Nk®e

Aeieacaoaeuenoai:

“E>0 $k0: “n>n0 0 nK®Y $ k0: “k>k0 => nK>n0 =>
0 0FyK (1+1/zK+1)Zk(1+1/zK+1)Zk=(1+1/zK+1)Zk+1)/(1+1/zK+1
) => (1+1/zK+1)Zk+1/(1+1/zK+1) iieo/aai:

eFLim (1+xK)1/XkFe => Lim (1+xK)1/Xk=e => Lim (1+x)1/X=e i?e x®0+

Lim (1+xK)1/Xk i?e x®0-:

yK=-xK®0+ => aeieacuaaai aiaeiae/ii i?aaeuaeouaio => iieo/aai Lim
(1+x)1/X=e i?e x®0-

Aeaeei /oi i?aaue e eaaue i?aaeaeu niaiaaeatho => Lim (1+x)1/X=e i?e x®0

2) n®Y lim (1+x/n)N = (lim (1+x/n)N/X)X = eX

3) x®xa aIR – iai?a?uaia

xa=(eLn x) a=ea*Ln x

iai? iai? iai? iai?

x®Ln x®a*Ln®a*Ln x => x®ea*Ln x

4) x®0 Lim (Ln (1+x))/x = Lim Ln (1+x)1/X = Ln e = 1

4’) x®0 Lim LogA(1+x)1/X = 1/Ln a

5) x®0 Lim (eX-1)/x = {eX-1=t} = Lim t/Ln(1+t) => (4) = 1/1 = 1

5’) x®0 Lim (aX-1)/x = Ln a

6) x®0 Lim ((1+x)a-1)/x = Lim ([e a*Ln (1+x) -1]/[a*Ln(1+x)]*[a*Ln
(1+x)]/x = 1*a*1= a

34.Oai?aia Aae?oo?anna ia ia?aie/aiiinoe iai?a?uaiie ooieoeee ia
io?acea.

Ooieoeey o®f(x) iacuaaaony iai?a?uaiie ia iiiaeanoaa O anee iia
iai?a?uaia a eaaeaeie oi/ea o yoiai iiiaeanoaa.

Oai?aia: Ooieoeey iai?a?uaiay ia io?acea [a,b], yaeyaony ia?aie/aiiie ia
yoii io?acea (1 oai?aia Aae?oo?anna) e eiaao ia iai iaeaieueoaa e
iaeiaiue oaa cia/aiea (2 oai?aia Aae?oo?anna).

Aeieacaoaeuenoai: Ionoue m=Sup{f(x):xI[a,b]}. Anee f ia ia?aie/aia
naa?oo ia [a,b], oi m=Y, eia/a mIR. Auaa?ai i?iecaieueioth
aic?anoathuoth iine-oue (nN), oaeoth /oi Lim cN=m. O.e. “nIN: cN $ xKn®a. O.e. aFxEnFb =>
aI[a,b].

Aeey mIR – ii oai?aia i oii, /oi i?aaeae i?iecaieueiie iiaeiine-oe ?aaai
i?aaeaeo iine-oe iieo/aai cKn®m.

Aeey m=+Y – ii Eaiia i oii /oi anyeay iiaeiine-oue aa iine-oe yae-ny aa
iine-oueth iieo/aai cKn®m. Ia?aoiaey e i?aaeaeo a ia?-aao cKn f(a)=m – /oi e icia/aao /oi ooieoeey f ia?aie/aia naa?oo e aeinoeaaao
aa?oiae

a?aieoea a oi/ea a. Nouanoaiaaiea oi/ee b=Inf{f(x):xI[a,b]}
aeieacuaaaony aiaeiae/ii.

35. ?aaiiia?iay iai?a?uaiinoue. Aa oa?aeoa?ecaoeey a oa?ieiao eieaaaiee.

Ii?aaeaeaiea: “A>0 $ d>0: “o’,o”: |o’-o”| |f(x’)-f(x”)|
ooieoeey iacuaaaony ?aaiiia?ii iai?a?uaiie

Ioee/ea io iai?a?uaiinoe ninoieo a oii, /oi oai d caaeneo io A e io o”,
oi caeanue d ia caaeneo io o”.

Ii?aaeaeaiea: O-oeey f – ia ?aaiiia?ii iai?a?uaia, anee $ A>0 “d >0: $
o’,o”: |o’-o”| |f(x’)-f(x”)|?A>0

?anniio?ei iiiaeanoai {|f(x’)-f(x”)|:|x’-x”|0 $ d>0: Wf(d)FA Lim Wf(d)=0 d®0

36.Oai?aia Eaioi?a i ?aaiiia?iie iai?a?uaiinoe iai?a?uaiie ooieoeee ia
io?acea.

Oai?aia: Anee f iai?a?uaia ia [a,b], oi iia ?aaiiia?ii iai?a?uaia ia
[a,b].

Aeieacaoaeuenoai(io i?ioeaiiai):

Ionoue f ia ?aaiiia?ii iai?a?uaia ia [a,b]=>$A>0 “d>0 $o’,o”:
|o’-o”||f(x’)-f(x”)|?A. Aicueiai d =1/e, eIN $oK, o’KI[a,b]:
|oK-o’K| ec iaa ii oai?aia Aieueoeaii-Aaea?oo?anna iiaeii
auaeaeeoue iiaeiine-oue xKs noiaeyuothny e o0. Iieo/aai: |oKs-o’Ks| 0?E – i?ioeai?a/ea n oneiaeai.

37.Ii?aaeaeaiea i?iecaiaeiie e aeeooa?aioeeaea.

Eanaoaeueiay a oi/ea x0 e ooieoeee x®f(x): aicueiai aua iaeio oi/eo o
niaaeeiei x0 e o – iieo/ei naeouoth. Eanaoaeueiie iaciaai i?aaeaeueiia
iieiaeaiea naeouae i?e o®x0, anee yoi i?aaeaeueiia iieiaeaiea
nouanoaoao. O.e. eanaoaeueiay aeieaeia i?ieoe //c oi/eo (x0,f(x0) =>
o?aaiaiea yoie eanaoaeueiie (anee iia ia aa?oeeaeueia) eiaao aeae
y=k*(x-x0)+f(x0). Iaiaoiaeeii oieueei ii?-oue iaeeii k eanaoaeueiie.
Aicueiai i?iecaieueiia /enei Do?0 oae, /oiau x0+DoIO. ?anniio?ei
naeouoth III, II(x0,f(x0)), I(x0+Do,f(x0+Do)). O?aaiaiea naeouae eiaao
aeae: o=e(Do)(o-x0)+f(x0), aaea k=f((x0+Do)-f(x0))/Do – iaeeii naeouae.
Anee nouanoaoao Lim e(Do) i?e Do®0, oi a ea/anoaa eneiiiai iaeeiia k
aicueiai yoi i?aaeae. Anee Lim e(Do)=Y i?e Do®0, oi ia?aieoai o?aaiaiea
naeo uae a aeaea x=(1/k(Do))*(y-f(x0))+x0 ia?aeaey e i?aaeaeai i?e Do®0,
iieo/ei x=x0 (Lim x=Lim x0 Do®0 => x = Lim x0)

Ii?aaeaeaiea: I?iecaiaeiui cia/aieai ooieoeee f a oi/ea o0 iacuaaaony
/enei f’(o0)=Lim (f(x0+Do)-f(x0))/ Do x®x0, anee yoio i?aaeae
nouanoaoao.

Aaiiao?e/anee f’(o0) – yoi iaeeii iaaa?oeeaeueiie eanaoaeueiie a oi/ea
(x0,f(x0)). O?aaiaiea eanaoaeueiie y=f’(x0)*(x-x0)+f(x0). Anee Lim
(f(x0+Do)-f(x0))/Do=Y Do®0, oi ieooo f`(x0)=Y eanaoaeueiay a yoii neo/aa
aa?oeeaeueia e caaeaaony o?aaiaieai o=x0. f`(x0)=lim(f(x0+Do)-f(x0))/Do
x®x0=>(f(x0+Do)-f(x0))/Do=f’(x0)+a(x), a(x)®0 i?e x®x0.
f(x0+Do)-f(x0)=f`(x0)*Do+a(x)*Do o/eouaay, /oi x0+Do=x e iaicia/ay
a(x)*Do /a?ac o(x-x0) iieo/ei f(x)=f’(x0)*(x-x0)+f(x0)+o(x-x0). Iaiaoi
aeeii caiaoeoue, /oi o(x-x0) oiaiueoaaony auno?aa /ai (x-x0) i?e x®x0
(o.e. o(x-x0)/(x-x0)®0 i?e x®x0)

Ii?aaeaeaiea: O-oeey f iacuaaaony aeeooa?aioee?oaiie a oi/ea x0 anee
$nIR: a iaeioi?ie ie?anoiinoe oi/ee x0 f(x)=N(x-x0)+f(x0)+o(x-x0)

Oai?aia: Ooieoeey aeeoooa?aioee?oaia a oi/ea x0 $ f’(x0)

Aeieacaoaeuenoai:

f`(x0)=C

=>: f(x)=C(x-x0)+f(x0)+o(x-x0) =>
(f(x)-f(x0))/(x-x0)=C+o(x-x0)/(x-x0)=C+a(x), a(x)®0 i?e x®x0.

Ia?aoiaeei e i?aaeaeo i?e x®x0 => Lim (f(x)-f(x0))/(x-x0)=C+0=C => Neaaa
caienaii i?iecaiaeiia cia/aiea o-oeee f => ii ii?aaeaeaieth C=f`(x0)

Ii?aaeaeaiea: Anee ooieoeey o®f(x) aeeooa?aioee?oaia a oi/ea x0, oi
eeiaeiay ooieoeey Do®f’(x0)*Do iacuaaaony aeeooa?aioeeaeii ooieoeee f a
oi/ea x0 e

iaicia/aaony df(x0). (aeeo-ae o-oeee o®o iaicia/atho dx). O.i.
df(x0):Do®f`(x0)*Do e do:Do®Do. Ionthaea df(x0)=f’(x0)*do => df(x0)/do:
Do®f`(x0)*Do/Do=f’(x0) i?e Do?0. A neeo yoiai ieooo oaeaea
f’(x0)=df(x0)/do – iaicia/aiea Eaeaieoea. A?aoee aeeo-ea iieo/aaony ec
a?aoeea eanaoaeueiie ia?aiinii ia/aea eii? aeeiao a oi/eo eanaiey.

Oai?aia: Anee o-oeey f aeeo-ia a oi/ea x0, oi f iai?a?uaia a oi/ea x0.

Aeieicaoaeuenoai: f(x)=f(x0)+f’(x0)*(x-x0)+o(x-x0)®f(x0) i?e x®x0 => f
iai?a?uaia a oi/ea x0.

Ii?aaeaeaiea: Ii?iaeue e o-oeee f a oi/ea x0: yoi i?yiay
ia?iaiaeeeoey?iay eanaoaeueiie e o-oeee f a oi/ea x0. O/eouaay /oi
oaiaain oaea iaeeiia ii?iaee ?aaai tg(90+oaie iaeeiia eanaoaeueiie)=
-Ctg(iaeeiia eanaoaeueiie), iieo/aai o?aaiaiea ii?iaee:
y=-1/f’(x0)*(x-x0)+f(x0)

38. A?eoiaoeea aeeo-oee?iaaiey. I?iecaiaeiua o?eaiiiiao?e/aneeo
ooieoeee.

Oai?aia: Ionoue o-oeee f e g aeeooa?aioee?oaiu a oi/ea x0, oiaaea o-oeee
f+g, f*g e f/g (i?e g(x0)?0) aeeooa?aioee?oaiu a oi/ea x0 e:

1) (f+g)’(x0)=f’(x0)+g’(x0)

2) (f*g)’(x0)=f’(x0)*g(x0)+f(x0)*g’(x0)

3) (f/g)’(x0)=(f’(x0)*g(x0)-f(x0)*g’(x0))/g(x0)2

Aeieacaoaeuenoai:

1) Df(x0)=f(x0+Dx)-f(x0)

Dg(x0)=g(x0+Dx)-g(x0)

D(f+g)(x0)=Df(x0)+Dg(x0)=f(x0+Dx)-f(x0)+g(x0+Dx)-g(x0)

D(f+g)(x0)/Dx=(f(x0+Dx)-f(x0)+g(x0+Dx)-g(x0))/Dx=(f(x0+Dx)-f(x0))/Dx+(g(
x0+Dx)-g(x0))/Dx®f’(x0)+g’(x0) i?e Dx®0

2)
D(f*g)(x0)=f(x0+Dx)*g(x0+Dx)-f(x0)*g(x0)=(f(x0)+Df(x0))*(g(x0)+D(x0))-f(
x0)*g(x0)=g(x0)*Df(x0)+f(x0)*Dg(x0)+Df(x0)*Dg(x0)
D(f*g)(x0)/Dx=g(x0)*(Df(x0)/Dx)+f(x0)*(Dg(x0)/Dx)+(Df(x0)/Dx)*(Dg(x0)/Dx
)*Dx®f’(x0)*g(x0)+f(x0)*g’(x0) i?e Dx®0

3) O-oeey g – aeeooa?aioee?oaia a oi/ea x0 => O-oeey g – iai?a?uaia a
oi/ea x0 => “A>0 (A=|g(x0)|/2) $d>0: |Dx|
|g(x0+Dx)-g(x0)| (2) =
f’(x0)*1/g(x0)+f(x0)*(1/g)’(x0)=f`(x0)*1/g(x0)+f(x0)*(-g’(x0)/g(x0)2)=(f
’(x0)*g(x0)-f(x0)*g’(x0))/g(x0)2

Oai?aia: Ionoue f=Sin(x), g=Cos(x)

1) Sin’(x0) = Cos (x0)

2) Cos’(x0) = -Sin (x0)

Aeieacaoaeuenoai:

1) Df/Dx=(Sin(x0+Dx)-Sin(x0))/Dx = Sin(Dx/2)/(Dx/2) * Cos(x0+Dx/2) ® Nos
x0 i?e Dx®0

2) Dg/Dx=(Cos(x0+Dx)-cos(x0))/Dx=Sin(Dx/2)/(Dx/2)*-Sin(x0+Dx/2) ® -Sin
x0 i?e Dx®0

I?iecaiaeiua Tg e Ctg auaiaeyony iaiin?aaenoaaiii ec i?iecaiaeiuo aeey
Sin e Cos ii oi?ioeai aeeooa?aioee?iaaiey.

39. I?iecaiaeiay noia?iiceoeee.I?iecaiaeiua noaiaiiie, iieacaoaeueiie e
eiaa?eoie/aneie ooieoeee.

Oai?aia: Ionoue ooieoeey g aeeo-ia a oi/ea x0, a o-oeey f aeeo-ia a
oi/ea y0=g(x0), oiaaea o-oeey h(o)=f(g(o)) aeeo-ia a oi/ea x0 e
h’(x0)=f`(y0)*g’(x0)

Aeieacaoaeuenoai:

Dy=y-y0, Dx=x-x0, Df(y0)=f’(y0)*Dy+o(Dy), Dg(xo)=g’(xo)*Dx+o(Dx),
y=g(x0+Dx)

Dh(x0)=f(g(x0+Dx))-f(g(x0))=f(y)-f(y0)=f’(y0)*Dy+o(Dy)=f’(y0)*(g(x0+Dx)-
g(x0))+o(Dg)==f’(y0)*(g’(x0)*Dx+o(Dx))+o(Dy)=
f’(y0)*g’(x0)*Dx+f’(y0)*o(Dx)+o(Dy)

Dh(x0)/Dx=f’(y0)*g’(x0)+r, r=f`(y0)*o(Dx)/Dx+o(Dy)/Dx

r=f`(y0)*o(Dx)/Dx+o(Dy)/Dx=f`(y0)*(a(x)*Dx)/Dx+(a’(x)*Dy)/Dx=f’(y0)*a(x)
+a’(x)*Dy/Dx®f’(y0)*0 + 0*g’(y0) i?e Dx®0 (a(x)®0 a’(x)®0)

I?iecaiaeiay:

1) xa=a*xa-1

Lim (Dy/Dx)=lim((x+Dx)a-xa)/Dx = Lim xa-1* ((1+Dx/x)a-1)/Dx/x.
Eniieuecoy caia/aoaeueiue i?aaeae x®0 Lim ((1+x)a-1)/x=a, iieo/ei Dx®0

Lim xa-1*Lim((1+Dx/x)a-1)/Dx/x = a*xa-1

2) (aX)’=aX*Ln a (x®aX)’=(x®eX*Ln a)’

x®eX*Ln a – eiiiiceoeey ooieoeee x®aX e x®x*Ln a iaa iai?a?uaiu ia R =>
(x®aX)’=(x®a X*Ln a)’=(x®aX*Ln a)’*(x®x*Ln a)’=aX*Ln a

Ae-ai : (eX)’=eX

Lim(Dy/Dx)=Lim(eX+DX-eX)/Dx=LimeX*(eDX-1)/Dx, eniieuecoy cai-iue i?aaeae
i?e x®0 Lim(eX-1)/x=1, iieo/ei i?e Dx®0 Lim(Dy/Dx)=eX

3) (LogA(x))’=1/x*Ln a

Lim(Dy/Dx) = Lim (LogA(x+Dx) – LogA(x))/Dx = Lim 1/x*LogA(1+Dx/x)/Dx/x,
eniieuecoy caia/aoaeueiue i?aaeae i?e x®0 Lim LogA(1+x)/x=1/Ln a,
iieo/ei

Lim (Dy/Dx) = Lim 1/x*Lim LogA(1+Dx/x)/Dx/x=1/x*Ln a

40. I?iecaiaeiay ia?aoiie ooieoeee. I?iecaiaeiua ia?aoiuo
o?eaiiiiao?e/aneeo ooieoeee.

I?aaeeiaeaiea: Anee i?iecaiaeiay ia?aoiie ooieoeee g aeey o-oeee f
nouanoaoao a oi/ea y0, oi g’(y0)=1/f’(x0), aaea y0=f(x0)

Aeieacaoaeuenoai: g(f(x))=x g’(f(x))=1

g’(f(x0))=g’(f(x0))*f’(x0)=1, g’(f(x0))=g(y0)=1/f’(x0)

Oai?aia: Ionoue o-oeey f no?iai iiiioiiii e iai?a?uaii ioia?aaeaao (a,b)
a (a,b) oiaaea $ ia?aoiay ae o-oeey g, eioi?ay no?iai iiiioiiii e
iai?a?uaii ioia?aaeaao (a,b) a (a,b). Anee f aeeo-ia a oi/ea x0I(a,b) e
f’(x0)?0, oi g aeeo-ia a oi/ea y0=f(x0) e g’(y0)=1/f’(x0)

Aeieacaoaeuenoai:

Aicueiai i?iecaieueioth iineaaeiaaoaeueiinoue noiaeyuothny e y0: yN®y0,
yN?y0 => $ iine-oue xN: xN=g(yN), f(xN)=yN

g(yN)-g(y0)/yN-yO = xN-xO/f(yN)-f(yO) = 1/f(yN)-f(yO)/xN-xO ® 1/f’(xo)
i?e n®Y, iieo/eee i?e xN®xO g(yN)-g(yO)/yN-yO®1/f’(xO) =>
g’(oO)=1/f’(xO)

I?iecaiaeiua:

1) x®Arcsin x ii oai?aia eiaai Arcsin’x=1/Sin’y, aaea Sin y=x i?e
oneiaee, /oi Sin’y dy=y’(t)dt=o’(o)*o’(t)*dt=o’(x)do – aeaeei /oi i?e
ia?aoiaea e iiaie iacaaeneiie ia?aiaiiie oi?ia aeeooa?aioeeaea iiaeao
auoue nio?aiaia – yoi naienoai iacuaatho eiaa?eaioiinoueth oi?iu ia?aiai
aeeooa?aioeeaea.

Ionoue ooieoeee o=f(o) e o=g(t) oaeiau, /oi ec ieo iiaeii ninoaaeoue
neiaeioth ooieoeeth o=f(g(t)) Anee nouanoaotho i?iecaiaeiua o’(o) e
o’(t) oi nouanoaoao i?iecaiaeiay o’(t)=o’(o)*o’(t) e ii aeieacaiiiio aa
ia?aue aeeo-ae ii t iiaeii iaienaoue a oi?ia dy=y’(o)do, aaea
do=x’(t)dt. Au/eneyai aoi?ie aeeo-ae ii t:
d2y=d(y’(x)dx)=dy’(x)dx+y’(x)d(dx). Niiaa iieuecoynue eiaa?eaioiinoueth
ia?aiai aeeo-ea dy’(x)=o”(o2)dx => d2y=o”(o2)dx2x+y’(x)*d2x, a oi a?aiy
eae i?e iacaaeneiie ia?aiaiiie o aoi?ie aeeo-ae eiae aeae
ae2y=o’(o2)*dx2x => iaeiaa?eaioiinoue oi?iu aoi?iai aeeo-ea.

Oi?ioea Eaeaieoea:

Aeieacaoaeuenoai ii eiaeoeoeee.

1) n=0 aa?ii

2) I?aaeiieiaeei aeey n – aa?ii => aeieaaeai aeey (n+1)

Anee aeey u e v $(n+1) i?iecaiaeiua, oi iiaeii aua ?ac
i?iaeeooa?aioee?iaaoue ii o – iieo/ei:

=>

44. Iaoiaeaeaiea i?iiaaeooeia iinoiyinoaa iiiioiiiinoe ooieoeee e aa
yeno?aioiia.

Oai?aia: Ionoue f(x) iai?a?uaia a caieiooii i?iiaaeooea [a;b] e aeeo-ia
a ioe?uoii i?iiaaeooea (a;b), anee f’(x)=0 a (a;b), oi f(x)-const a
[a;b].

Aeieicaoaeuenoai:

Ionoue xFb, oiaaea a caieiooii i?iiaaeooea a [a;x] ii oai?aia Eaa?aiaea
eiaai: f(x)-f(a)=f’(a+q(x-a))(x-a) 0 o.e. ii oneiaeth f’(x)=0 a
(a;b), oi f’(a+q(x-a))=0 => f(x)=f(a)=Const aeey ana oI(a;b).

Oai?aia: Ionoue f(x) iai?a?uaia a caieiooii i?iiaaeooea [a;b] e aeeo-ia
a ioe?uoii i?iiaaeooea (a;b), oiaaea:

1) f iiiioiiii aic?anoaao(oauaaao) a iano?iaii niunea a (a;b)
f’(x)?0(f’(x)F0) a (a;b).

2) Anee f’(x)>0(f’(x) f’(c)?0(f’(c)F 0) => f(x”)?f(x’)(
f(x”)Ff(x’)) => f(x) aic?anoaao(oauaaao) a iano?iaii niunea a (a;b).

2) Eniieuecoy aiaeiae/iua (1) ?annoaeaeaiey, ii caiaiyy ia?aaainoaa ia
no?iaea iieo/ei (2).

Neaaenoaea: Anee xO-e?eoe/aneay oi/ea iai?a?uaiie o-oeee f. f’(x) a
aeinoaoi/ii iaeie d-ie?-oe oi/ee xO eiaao ?aciua ciaee, oi
xO-yeno?aiaeueiay oi/ea.

Aeinoaoi/iia oneiaea yeno?aioia: (+)®xO®(-) => eieaeueiue min,
(-)®xO®(+) => eieaeueiue max

46. Auioeeua iiiaeanoaa Rn. Oneiaea Eainaia. Auioeeua
ooieoeee.Ia?aaainoai Eainaia.

Ii?aaeaeaiea: Iiiaeanoai I auioeei anee ” A,AII [A,A]II

[A,A]II => [A,A]={A+t(A-A):tI[0,1]} => A(1-t)+tAII

[A,A]II => A,AII; l1=1-t, l2=t => l1+l2=1 l1,l2?0 => l1A+l2AII

?anniio?ei oi/ee: A1,A2,…ANII l1,l2?0 S(i=1,n): lI = 1

Aeieaaeai /oi S(i=1,n): lI*AI II

Ae-ai: Ii eiaeoeoeee:

1) n=1, n=2 – aa?ii

2) Ionoue aeey (n-1) – aa?ii => aeieaaeai aeey n:

a) lN=1 => i?e?aaieaaai l1=…=l N-1=0 => aa?ii

a) lN(1-l N)*B +
l N*A N II *.o.ae

A?aoee Af = {(x,f(x)):oIDf}, Iaaea?aoee UPf={(x,y):y>f(x)}

Ii?aaeaeaiea: Ooieoeey f auioeea UPf – iiiaeanoai auioeei.

Oneiaea Eainaia: AIII lI?0 S(i=1,n): lI =1 => S(i=1,n): lI*AI II, xI?0,
f(xI)FyI => S(i=1,n): lI*AI = (SlI*xI;SlI*yI) => f(SlI*xI)FSlI*yI

Ia?aaainoai Eainaia: AIII lI?0 SlI =1f(SlI*xI)FSlI*f(xI)

47.E?eoa?ee auioeeinoe aeeooa?aioee?oaiie ooieoeee.

Oai?aia: Ionoue f ii?aaeaeaia a eioa?aaea (a;b), oiaaea neaaeothuea
oneiaey yeaeaaeaioiu: 1) f – auioeea a (a;b) ~ 2) “x’,xO,x”I(a;b)
x’

(f(xO)-f(x’))/(xO-x’)F(f(x”)-f(xO))/(x”-xO). Aaiiao?e/aneee niune: i?e
naeaeaa ai?aai oaeiaie eiyooeoeeaio naeouae ?anoao.

Aeieacaoaeuenoai:

“=>” AB: k=(y-f(x’))/(xO-x’)?(f(xO)-f(x’))/(xO-x’) => y?f(xO); AB:
k=(f(x”)-y)/(x”-xO)F(f(x”)-f(xO))/(x”-xO) =>yFf(xO)

(f(xO)-f(x’))/(xO-x’)F(f(x”)-f(xO))/(x”-xO)

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