Алгебра и начала анализа 11 класс Колягин

ANTEBPA

N HAHANA
MATEMATHYECKOTO
_¿AHAJIISA

TPOCBEMEHME

NPOMSBOAHAA

19-10 vier

(kx+b)' =k GP) =pxP71
(ey @y= a
(Inx) (og, 2) = A
(sinx)' = cosx (cosx)' = - sinx

y = 1,5sin(2(x —+

CE COS
GG) + BD =f) +8) (cf) = cf (x)
VI BC) = FR) B(x) + F0) - (x)

r
Fa) Fo: 8) — f)- g(x) mn. |
Felis — O SC MEE) E f

MHTEFPAN

n
Jie) ax = Fe -F@ F(x) =f)

OyHxuna Nepsoo6pasnan
pti

x?) p #-1 Pe +C

il

5, x>0,2<0 In|x|+C

e ec

sing — cosx + C

cosa sing + C

YueGunk

‚Ana 06WE06PA3OBATENLHEIX

yupexaenuó

Basosbiii u npounbHbiñ

yposun
Moa penakuueñ
A. B. Xuxuenko

2ewsnanve | Peromenaosaro
Munucrepcreots
‘O6pacosanus 1 may
Poccwiicrort
Denepaum

-Npocsewenue +

2010

VIE 373.167.1:[512+517]
BBK 22.14a72
A45

Ha yueGuux noayyenst KONO METEO sakmovenua
Pocemitexolt akanem nayx (M 10106-5215/519 or 24.10.08)
m Pocentiexolt axanensun oópasosamma (N 01-206/5/7x
or 11.10.07)

Apropsi:
10. M. Konarum, M. B. Tauena, H. E. Depopona,
M. H: Mla6yaue

Venonmue oGoanavenns

IA marepnaz ana uayuenka Ha npodumsHoM ypone
SE uavepuan ana untepecyiouuxca MATENATHKOÏ
D4 pemenne sagas

O @ oocuosanue YTBCPRACHUA wax BEBOR QOPMYISL
25 ynpaxnenun nun Gasonoro yposua

26 ynpamnenun aa mpodHasuoro yposus
YRPAKHENNN JUIA HNTEPCCyIOmUICA MATEMATIROÏ

AsreGpa u Hauaıa Merremariuecioro axwutiaa, 11 101800 :
A45 yue6. ¡ura o6meo6pasonar. yupexaemmi : Gasomiit m
pour. yposun / (10, M. Konarun, M. B. Trawe-

sa, H. E. Denopopa, M. U. Ula6ymme]; mon pen.
A.B. Wnzuenko. —'2-e usx. — M. : Ilpoczemenne,
2010.— 336 e. : un.— ISBN 978-5-09-022250-
‘YAK 873,167.11512+5171

BBK 22.14972+22.161972

ISBN 978-5-09-022250-1 © Monaremseroo «Ilpocnemenwee, 2009
© Xyaomecrseunoe opopnnenme.
Hanarenserso «Ipocseneuner, 2009
Bee npana sanpentettie

TpuroHomerpnyeckue
yHKuuu

A ne soz nonam» codepxanue cae
u uepenamu.
y. Toxcon

8 1. O6nacte onpenenenna
M MHOXECTBO 3HA4EHUÜ
TPHFOHOMETPHAECKAX
DyHkuUñ

Hssecrno, “TO xaxxgomy neliersurenknony
uueay x cooTsereruyeT ERNNCTHENNAN TOMA Ex
IIMUHOR OKPYAHOCTH, Toaysaemas TOBOpOTOM
roux (1; 0) ma yron x paauan; sin x — opaunara
stoi rouk, cosx — ee aGcumeca. Tem cata Ka
JoMy AEÉCTBHTEMMONY smezy x NOCTABTEE B co-
opercrsne uncna sinx M COS, T. €. HA MHOKE-
eme R ncex ACHCTBUTENLTUX uncon onpenenens
Gyazıım y=sinx u y=008%.

Obnacrmo onpexenenus Kaxyoï na dyun-
nuit y=sinx 1 y=cosx ADMAOTOA MHomoer-
Bo R ncex xelcrBurensaux uncen.

Hanownws, ro Mnoxocrno noox AA, KOTO
pue byuKnus mpumuacr ua oßnaern ompenere-
Hs, MAIHBAOT NHO‘eCTHOM sHANeKHH PVEM.
Tam oßpanon, uroßL nada witoxecrno
anavennit hynxuun y-sinx, Ayo BRIACHHT,
Kanne SHANEHIS MookeT NPMAMMATS y TPH Paz"
ANIMEIX SHAYOHESX x MO OÓMACTIL ONPeRenenna,
re. yCranomumo, Jus KAKHX anauenmi y cyuge
Bylor Taxe snasenun x, mpm KOTODIX sinx=y.
Hauoerno, «10 ypannenne sinx=a, Tax me Kar u
ypanitenue cosx—a, umeer Kopi, eos [al 1, 1
ne wweor kopneit, cam jal>1.
Mnoxecrson onanenuß KAXAOË uo Pynkuuc
yesing u y=cosx snaaeren orpeaox
-1SyS1. @ysxumm y=sinx m y=cosx
orpasnuenst cnepxy m cumay (no onpenene-
Immo orpanmuennoit yrs).

1 ss!
Obnacıe OnpeREnEnIA W NMOREETEO SISTEMA
‘TpurowoweTpHNECKIE Dyno

‚a 1. Haïru oGxacrs onpenenenun DyHKU
1
9° mcr 008% *
> Haiiqem snavenus x, DPH KoTopktx BeIpaxKenHe

EH
MEET CMBICHA, T O, TAKE INAHENUA X, pH KOTOPEIX IHaMeHATEAE

pasen nymo. Peman ypapuenne ain x +cos x 1,
x=-Z tan, nCZ. Cnegosatensao, oGnacrsio ompenenenmn xammol
Gym ABAMIOTCA nce sHavenus zu 7 tan, NEZ. 4

Banaue 2. Halten wmoxecrno anavernit pyme

y=8+sinxeosx.

D Hyacno wunewwre, Kaxne suavennn MONT HpHEENAT y apa
PROMI mer x, 7. €. VCTANOBITL, AT IX NAME a
Ypanuenne 3+sinxcosx=a meer xopmu. TIpumenaa Gopxyay
canyon pociinora veus, samen ypamemee mic 94 cin dz,
orkyaa sin2x=2a~6. [laa seex anauenuk a, rakux, uro [2a~6|< 1,
ne. 2,55a68,5, mo ypannenne meer xopun. Taxum oÖpaaon,
NOECTION anauennf AMMNOR QYHKUHN ABACTEA. | OTPEION
2,5<y<3,,

Orner. [2,5; 3,5]. 4

Bamexanme. Sagauy 2 momo penmrs mmaue. Tax ax

=3+4sin2z, me -1<sin2x<1, 10 -4<1sin2x<4, oryaa

y=3+ 2 sindx, ra ‚10 -4<4 $+ onyx

, Haxonun tx

2,5<8+ 4 sin2x<8,6, Cuenomarenbno, muomecrao suavemmit yHK-
suai — orpeaox [2,65 3,5].

Oynuun y=tgx onpenenneren gopuynoi tgx- HE.
Baar, oma onpenenena npit Tex oHaxeHHAX x, ARA Koropkix.

cosx#0, 7. e. mpi x# 5 +nn, nEZ.

Oénaersio onperenenna hynkunm y=tgx ADARETOA mnoxeer-

no uncen ze 4a, neZ.
Msoxecrsom skayenuä (bynkunu y=tgx ABNAeTCH MHOKECT-
so IE bee. MERS ou vel Max Vases
Llama mer nope apa bon. pebcroarenanoe urkenu de
Dyueuna y =ctgx, mae cigx- SE
aux ©, peg RETO sinn, 2.8, up zen, REZ, Cor
AOBATEABHO, OÉNACTEIO onpezenenua gyxKunH ctgx ABARETCA
Mnomecrno Rc suiopoulenmunan uo nero TOUKann man, Nez.
"Tan var ypezsento sig == meet KOpuR UPR ambos AEC
aurexnnom OMNOREE ay TO muomecteon AVE Qyakzaı
yaclgx aunneren nmomscteo À ocex nefersurenumx aceon.
Dyaxuna y=tgx u y=ctgx He ABIRIOTCA OTPAHHYCHREIMH.

A trace ı

, onpexeneua mpu Tex sHa-

Tur ONO METRICS GARE

dynxnun y=sinx, y=cosx, y=tgx m y=ctgx nannpawrtes
mpuzonomempurecKuau DYHKYUAMU.

Baxaua 3. Haiirm oßnacr» onpexenenua dbynkunn

yosings+tg2x,

D Hymne muni, mp sence anche aupa
sind 2x umeer conten, Buparenne sine uneer een api
Gon nan 2, à mue tg2 — npn 27 3 Han, ne,

mon x*-543%, ncZ, Cnexonaremno, o6nacrnio onpenene-
nun naunoit Gyms nunnercn unOxserno Aeloranmonumn un-

con, ranıx, uo x7 7470, nez.

Omer. z T+, nez. 4

Hasta mnomecrno ane éme
y-dsinziäcoen,

> Tlpeo6pasyem QyHKIHIO, MCHONBIYA MeTon BCNOMOTATENBROTO
yraa (+AureGpa u Havana ananııza, 10», rn. IX, $ 4). Yuroxnm
se pnsnenmu y ma VIFFAF-5. Honyunc y-5(Feinz+ Scosz).
A v0 ermecrnyer yrom a saxo, 70

2, nc. B xaucerse a momo ours arccos À orga
y=5(sinxcosa+cosxsina)=5sin(x+a), rae — 1<sin(z+a)<1.
Homomy =D 2 Se» 8 nncnecrao RUC Aono Qn =
emo 8; 8 À

3anaua 5. Halirn muoxectso snauenuh DyHKUMH

sin? x +4 sin x cos x + cos? x.
D Henoasya dopmyanı A8oñnoro aprymenra, nonyuaem

240025 _ 2 + (2sin 2x cos 2x).

TpeoSpasouas auipaxenue n cKoßxax u npHMeHHD MOTOR Be
TATEXLMOTO yraa, nonyanm

Banaw

cosa

y=3 +2sin2xr+

1 a
cos2x) =\Bsin(@x-0),
Jp oos2x) -VFsin 2-0,

Torga y= Trae a). Tax rar —V5 < V5 sin (2x-a)< V5,
10 2-V5 <y<2+ V5. Crenogarenbno, Mmorecrso snauenuh xao
dame — orpesox [2-V5; 24 V5]. 4
[SA Zazana 6, Hoxasars, “ro YEKQUA
> Has roro sroôm AoKasaTe, uro bymxmma ym Bcos 2x + 5sin 2x
Srpanneng, Hy2KHo Haltru Takoe NONOMUTEAAOE HERO C, O6

1 5

os 2x + 5sin 2x orpa-

TDBRACTE ORPEREAEHNA 1 WHORECTEO SHANEHIT
‘TpHronoueTpHMeCKi Syme

ann mo6oro anauennn x ua OÓTACTI onpenenenna GYM, 1,
ana xCR, Betmonuaocs Hepavencrso |8cos2x+5sin2x|<C.

Tax kax -1<cos2x<1, -1<sin2x61, to ana moGoro anaue-
mua x Mo oGnacrm ompenenennn sbinomsores nepanenerna
36 Scos2x<8, -55sin2r<5, excgomrenmo, -8<y<8 m dyme
aa orpamirmena ua snoxectse R. 4

Banana 7. Moxasarı, sro dyarina y= 4",
wena.
D Hannan oymxuna onpenenena xa unoxecrse R. Bocnonsayenen
mepanenernom 24412213], Koropoe pammocnanne uepanenerny

act eet

Cel 1830, Tor Iylmayylsin2el< a, van war 3
Isin2x|<1. Cuexonarenuno, Yyununa orpamusena ua Mnomecr-
ve R. 4
Banana 8, Hoxasars, sro Gynenu y=xeinx ue annaercn
orpanusenmoñ na mmoxecrne R.
D Ilyers C — mpomonomnoe nonoxuremmoe sexo. Tora mañi-
Hevea Harypanınoe wHeEAO n, Taxoe, WTO x,= À + 2n > C.

Tax Ka [y(x,)|=x,8inx,—x,>C, 10 ARE He anınercn

va muoncectne E. 4 ES

sin2x orpann-

orpanuuenn:

Ynpaxnenus
1. Hajirm o6xacre onpenenenus hymne: 7
Du=sin25 2) y=cos3s 8) y=cost
4) ynsin2; 5) ymsinVx; 6) y=cos
2. Hañrn mnoxecrso anavennit bynkumm:
D y=L+sina; 2) y=1-c08x5
3) y=2sinx +3; 4) y= 1-4 cos 2x;

5) y=sip2xcos2x+2 6) y= Lsinxcosx—1.

2
3. Halt oßnacr» ompenenenun Gym
fi

D Dy Bustes; 4) yatesx.

4. Hatten oßnaers onpexenenus dynxninn f(x) u Burner ee
anaxenne D saqammux TOUKA)

De

2) fxm ¿> 1-0, xml, x9=100,

q ne

Haiirw oGnacts onpexerenuat dynxnun (56).

5D y-Vsinxrl; — 2) y=Veosx-Ts BD y=lgsinx;
4) y-V2cosx-1; 5) y=VI-Bsinz; Ov

a

1
Bein? x—sine®

1;
meme DY

& 1 2) y=

cou? ein x”

Hair mnoxecrpo snavenuit yuca (7—9).
2. 1) y-2sin*x-cos2x; 2) y=1-8c08*x sin? x;

3) p= 1800 4) y=10-9sin*3x;

5) y=1-2icos. 6) y=sinx+sin(r+ E

8. 1) y=sine—Beosx; 2) y=sin?x—2sinx;
008° x-Geinzcosa+ 2sin?x;

1) ymsinixteosix; 2) y=sin®x + conf,
Aoxosars orpannicnnocrs Pyıkuum:

008% 1

Dr. Due ——.

I UT TE sing a

[HI] Hoxasar», aro pyaKnus f(x) ne apanerca orpannuennoh u
oGnacrit eo onperenenua, con:

DE: ont.

$ 2. YetHocts, HEUETHOCTB, NEPHOAUAHOCTE
TPMFOHOMeTPNYEcKnX yuku

Kaxgaa us dymeomi y-sinx m osx ompenenena na

sitomeerne R, u aan moGoro + ER nopun pasestera
sin(-z)=-sinx, cos(-2)=cosz.

Cxexonaremmo, y=sinx — neuemman hynnunn, a y=cosx —
vemnası Gym.

nn mo6oro anauenmn x ma oßnacrm onpezenenna Hymn
y=tgx Bepko pasencrso tg(-x) g(x) u oßnacr, onpenenenns
yma y= tex CUMMOTPIUNA ornocnTenumo unvana KoopAunar-
Tosromy y=tex u y=ctgx— nevemnwe Dar.
J Moxxo aokasarı enexyioume cvoilersa Vernsix 1 Heuerusix
dy

1) cymma, pasnocTh, IPOMIBCACHME H HACTHOS ABYX MCTHBIX.

yet snasioren yann sera;

2) cymma HM pasHOCTh ABYX HEMTHBIX PYHKUMÁ ABARIOTCA

yaaa mener:

3) nponsvexenme x sacrnoe ADyx mesero Hymn anaa-

sore ver py

4) nponarenenne m uacrnoe vernoh u neuernoi bymKumit an-

Anıorea neueren Oya. ER

$2 7

OCT nevemoote, nepnonmwocte
Tpwrronomerpyvecinx yw

Sagawa 1. Buiscnns, ABARETCA au PY RIMA

y=2+ ain ool

wernoh mas neuernoi.
[> @ynxusa ompenenena na mnoweorne R. Henomays Dopuyay
Mpuseqenns, sammuien Aaumyo dyuxumo B Bue y=2+sin® x. Tax
Kat sin(~x)=~sin x, To (sin (~x))?=sin® x, u nooromy y(-x)=y (x),
1. €. naunan DYHEUHN ABAJETCA “THOR. À

Maneerno, “TO Ain M0Doro anauenun x Bepnkt pañencrea
sin(<+2r)=sinz, cos(x+2m)=cosx. Ma rmx panonern enenyer,
TO IMANCNHA CHHVCA H KOCHHYCA NEPMOAIVECKH MOBTOPRIOTCA IPR.
Hamenenma aprymenta na Zn. Take QVEKuAn BASEIBALOTCA NEPmo"
MUCHA © mopmonon 2:

oni

Dyanıın (A) nnastnaerca nepuoduneckoli, eon cymecrayer
rakoe uneno T 40, uro ana awGoro x m3 oGmacTH onpenene-
nna onofi bymenun annsennn x+T m x-T raroxe npnnanne-
ar O6naeru Onpenerenun x PEMOMMMOTOA papeKcTEA
16-T)=1(0)=/(x+T). Unczo T nasunserca nepuodox Pym
yuu fo.

Hs ororo onpenenenna cmenyeT, "TO EC weno x NPHAJTE-
ur oßnacrm onpenenenusn Gym f(x), 10 nena x+nT, neZ,
Take npunannemar o6nacru omperenennm oroit dymiaum m
HextnT)=f(x), REZ.

Banana 2 Zoxacers, wo weno 2x Antaercn mensa
> Ilyer» T>0 — mepnon Kocunyca, T. €. AAA HIOÑOTO x Bunonns-
erca pasencrso cos(x+T)-cosx. TlomoxwmB x=0, monyunm
SEE onen Pedal, POL. tas aan FO, 0 7 none ie
Sino ann bs TL LT
ee 2e à
Tenbksti mepnon DYHKHHK y=sinx rakxe paper Zr.

Bagaya 3. Jloxasars, uro f(x)=sin3x — nepronmaeckas yHK-
B>-@yusunn Hunts onponenona na R. loreny ocrrouno
Re en Or qe

der) 700 u 1(2- 5) 10. mue

1(x+ 2) =sin3(x+ 2) moin (8x + 2 ain 8x = f(x).

Anazormuno f(x- 2) -ein@s- 21-0. 4
8 fnasa I

TOurONORAOTPITOCRAG NT

Bazann 4. Toxasars, ro yeux y Lex mnaneren nepuogu
seca M ameno nonoıckrennmeh nepnan,
emm x puna oönaern onpexonenun rol Oya,
1.0, 3% Etam NEZ, 10 no opuyaan mpunexemun nonyuaen
ten) te (n—x)——-(-tex)~tex, te(x+x)-tex.
Takum oöpasom, tg(x-1)-tgx=tg(x+r). Crenosarenbko, 1 —
nepnon dynein Y tgs. Hlowancem, vto 1 > mamma ono"
acuremnielt MEPHOX HyRKIMA y= tex.
Hyers T — nepnox ranrenca, roraa tg(x+T)=tgx, orkyaa
O uosyusen 17-0, Thx, Rez.
‘Tow war mammentice enor HozomENremunoe k pao 1, 70
Len ROMAN nepnon Du VUE. À
LE? 5. Hoxasarı, “ro y= tg à — nepmonmueckan dyun-

La © mepnonon 3x.
[> Obaners onpenenennn dbynkunn — mnoxecrno ER Tascıız, «ro.
xe 43nn, ncZ, oreona cnenyer, 70 ecan x npunaarexur
obnmerm onpenesenns, ro x+Sx m 2-3n Taxe npumannenar ce
o6xacr onpenexenus,

m Eee ig zy

Tax xax in) tg ds tg E (3)
185, 10 185 — nepnonnueckan DYRKUMA € nepmonom 3%. À

Baraua 6. Joxasars, wro pyweuna y= ¿E amnseren no-

Pronuueckoi e nepnonom 2x.
D Obnacrso onpenenenna D hyrkunn aBnmmoren sce Reiteran-
Tensusie «Ma x, KPoMe Tex, npm koropux sinx=—1, 7.0.

Kpome uncon += 5 +2nn, nEZ. Taxum oöpasom, ecam xED, To

pa

unena x+2n u x~2n rare npunanıear mnoxeersy D. Tar kar

(a4 LN LE (A), ro ye) — nepmonmuec-

Kaa OyERgHA e nepuozom 2n. À I
IRAE Banana 7. Haken nanmenvunit nonomrenumun epson ys
mur y= 3sinx-+ein 2x.
D Oyaxuua onpexenexa na muomecrse R. Ilyers T — nepuon
‘ymenun, re. ana Dex x ER nepuo panonerno
Bsin(x+7)+sin2(x+7)~3sinx+sin2x. a
Rena x=0, ro us panenerna (1) exeayer, sro 3sinT+sin2T =O
um 3sin T +2sinTcos T 0, orkyaa nonyuaen sin T(3+2cosT)=0.
Tax kar 3+2cosT=0, To sinT=0, u novromy naumensunn
nonoxnTensmu nepnon HY-KO NEKATI cpenu uncen T=an, nCZ.
IokaxeM, WTO WICIO 7 Me ABNAETCA MepnoRoM nannol DYHK
au, 7. 0. panenoruo (1) Axa T=n He nunonnaeren XOTA Gi npu
OAMOM anauenun x¢R.

$2 9

HEROGTE, HONTE, FeprORINNOCTe
TPMFOHOMETPHNBCEIE yA

Tiyers x= 5, Bermemme y($ +n) u y(3 ). een v(3+r)-
—osin(£ 47) +sina(E 4x) =-3: v(2) Bund +ain2 503. Ta

mx o6pagom, Jun T= panenerno (1) ne snnaeren nepitune npır

Tipu 7=2x panenerno (1) annsercn nepusim ¡uta moGoro ER,
ars Kax y(x+2n)=Bsin(x+ 22) + sin 2 (x + 27) Bsin x + Sin 2x = (2).
Cnenonarensto, x=2r — HAUMEHBLN nepnon DyRKUUN. À
Bazaun 8, Joxacamo, wro dymenn y=sin + ne annneren nepno-
auiecKoit.
[> O6nacrsio onpenenenus xammol @ynkmnt anamorea ace
zelictowressusie «nena, Kpowe unena 0. Hyer 7 — npomanonsuoe
noxoxurensroe "MONO, TOFRA Ha Toro, WTO —TX0, exeayer, “ro
touKa xy=—T npuuannemur oßnaern ompegenemma. Ho rouxa
x9+T--T+T-=0 ne npnmannexur o6nacru onpenenemaa.
Tipuman x Tony, “TO ARA moboro T>O cymecrayer raxoe
uneno +=xQ (na OÓNSCTH onpenenenna dymkıyın y), «ro TOuKA
£4+T ne npunaanemur oönerm onpenenennn. Cnenonarennmo,
ynxqua yosin À me annneron nepuoanueckoit. À

Ban 9. Jloxasar», ro @ynknna y=sinx* ne annaeren ne-
prromirvecxol
(> Iocrarouno nokasars, 110 PYHKUMA He HMeeT MONOKHTENBHOTO
nopmora, va nar cena Ou sono T<O ORIO Hepmonon, To neno
—T Gino ÓN MONOMHTCABHEIM nepmonon. JlokazatemorBo npo-
Denen merogone or AporBuoro.

Homer, 20 wneno 750 — nepnor yen, 1. o. ans
amo6oro x¢R cnpaseaauso pasenctso sin (x +T)*=sinx*. Tipu x=0
orciona enenyer, uro sinT?=0, me. Ti=rn, a T=Vrn np
nexoropom neN. Ecru 0<x<Vr, To sinx?#0, a nocxonsey Van —
nepnon, 70 u sin(z-+Vam)?*0. Kom axe z=\k, 70 sin(Vn+ Van)? =
=sin(V)=0. Saum, wueno Vr+ Va ananeren Guat
enpasa x Yan umonom, np Koropon sinx*=0, Orcioga Vx +Van<

”
E ES
a] 7 al =

102 croco ı

Tu ONOMETPANECRS Gym

SRG TD, tax Kaw Yrnt > Van m sin(VrmtD)?=0. Ho ne-
pasenorso Va+Van<VA(n+ 1), parnocmnoe nepasencrey 1<Vn+ 1
1

MeBepxo Jura mo6oro NEN, tax Kax Vi+1-Va=
Viewty

tr, HepepHo x AOMYIIERNE O nepromimocre dbynKusc sin x. q EH

Tlepuoguvecknmn Pyurunama ommemsarorea mmorne Gusuwe-
cxne mpouecest (KOAOÓANMA MARTHHKA, PPAUCHHE MARKET, mepe-
menmah TOK x r. x.) Ha pueynre 1 nao0pasensı papu mero-
TOPLIX MepwoguuecKHX PyR.

Ynpaxuenua
BHACHATE, ABARCTCH JM AQHHAA PYRKUMA YeTHO HIM neuer-
xott 12-18).

12.1) y=cos3x 2) y=2siná Dye
4) y=xcos =; 5) y=xsinx; 6) y=2sin?x.

13, 1) ysinx tx; 2) y=cos(x~

ae
4) y= Seosaesin( 2-22) +9; 6) guata 10,

~con( x) sin(n—a:

14. Hoxasars, xo dymenns y=f(x) smaxeres nepnonmteckoli ©
HEDHORON 2n, cant
oki 2) yasina tty 3) y=8sinx;
* 2
5) y=sin(x-% 6) y=cos(x+ 2).
15. Toxasats, sro ynknna y=f(x) annneren nepmonmueckolt c
nepnonom 7, ecan:

1) y-sin2x, Tan
3) y=tg2x, 7

= 1=cosx. sine _ cos2x-x?
Dae 1+cosx” NIE 14cos2x* Di sing *
4) y= En, 5) y-xisinrisint © y=3™;
1) yaxtsinds Summit.

17. Noxasarı, sro:
1) uponanezenne u uacrHoe anyx neuerusix GYM apa
JOTCA HETHBIMH PYHKIMAMA;

2) nponssenenue u uacruoe sernoï u mevernoli Hyarnui an
nmoren neuermiann dyniian.

92 Be

HEC, RENE, NEPHORIMOCH
TpitrovowerpinecKinx yu

Hatten nanwenmnit nonoxurensmait mepnox dy (18—
19). j

18. 1) y=cos 2 x; 2) y=sin $ x; 8) y= te Fs 4) y=lsinxl.

19.1) in x 4008 x; 2
3) y=sinx-sin3x; 4) y=2tg 5-3 F-

20. Butscnnrs, apnsercs mm nepmonmueckoli yHkuns:

3) y=lsinlxll.

ElDoxasamo, uno Gyms ne annaeren nepnogmsecxol:

1) y=sinViz; 2) y=sine+siny2z.

22, Hoxasarı, aro bynenua y=sintx+costx nepmonnueckan, u
alu ee Hammer nonoxxnrenbassit mepoA.

28. Moxosars, sro ymenn nopmozuuceran, u mahnt ce
BAHMeRBIUNF NONOMHTENLHEÄ Depmon:

1) y=sin(cosx) 2) y=cos(sina).

BA Jrpagux gyaxmm y= f(x), ER, cummerpuen ornochrensno
wampoi ua mpausix x=a, x=b, rae a#b. Mokasars, ro
Y=f(x) aunaerca nepnonnueckoh, H Hañr ee nepnox.

[25)Tpagax gynxunn y f(x), x¢R, cummerpuuen ormochTensno

rouku A(a; b) n upamol x=c (ca). Hokasarz, uro dyukuma

U=f(x) annneren nepmoruuecxok, 1 mahru eo mepoA.

Hoxasars, «ro dyneuua y=f(x) annaerca nepnoamueckoii,

ecrit cymecrnyer T#0 rakoe, T0 ANA moÖsıx Tpex amame

amit x, ++ m x-T 10 o6nactu onpenereuun yHKUNM Bi

nonneno yenonne f(x+T)=— f(x). Hañru nepuon @ynxnun f.

Enr, yeux f(x) onpenenena ma sceit umenoolí npamoli.

(oxaaath, w
1) f(x) +f(-x) — vernan hynkuns;
2) f@)-1Cx) — nenernas yaris.

in x + te.

§ 3. Ceoñcrsa bynxunn y=cosx n ee rpaqux

Hanommu, wro Hynkna y=cosx onpenenema ma nceii umeno-
Bolt mpamolt x MHO2eCTBON ee onawemmii apaerca orpeson [-1; 1].
Crexouareauno, bymKuss orpaunuena 4 rpagui ce pacnononei
noxoce mexay mpamsma y=— 1 u y=1.

Tax Kak QyHKUNA y—cosx uepHoaMueckaa € HEPHONOM Zr, TO
AOCTATOIMO nocrpours ee rpaduK Ha KAKOM-HUGYAb UpOMExyTKe
AommoÑ 2x, nanpumep ma oTpeske —1<x<x. Toraa Ha npomeny™
Kax, nonyaaemsx CABHTAMH BBÖPANHOTO oTpesKa Ha 2nn, REZ,
rpadux Oyaer maru me.

Pyakuna y=cosx apasorca wemoM. Hoyromy ce rpaqux
cummerpirien oTmocurembro ocn Oy. Ina Mocrpoenna rpaguxa xa
orpesxe -7< + € x ROCTATOImO nocrponts ero Ana O<X<R, a sanem
CHMMETPIFIRO OTPASUTL ero OTHOCHTENBEO ocH Oy.

125 rnasa ı
7 RT

Puc. 2 Puc. 3

TIpexae vom nepeiir x no-
erpoenuio rpadhnka, noKaæKeM, 70
Qynxuna y=cosx yönnaer ma or-
poane O< xen.

OB camom gene, mpm nosopore
rousu P(1; 0) woxpyr nauana Koop-
unan nporun uaconoli erpemien na
vron or 0.0 x aGeumeca nom,
7. e. cosx, ymenpmaerca or 1 20
=1. Hovromy ecan 0< x, <x), 70
cosx,>cosxa (pic. 2). Dro m oomasaer, WTO dymiıma y-cosx
yOuwaer ma orpese [0; x]. ©

Menomoya esoiicrso yOumanus yr y =cosx un orpeoxo
0<x<x M malina HECKOMKO TOVEK, PHHAUIEX AUX rpaduxy, M
erpom ero ua arom orpeaxe (pue. 8).

Tlombayacs cnottersom wernoctu DYHKIHH y-cosz, oTpasHM
nocrpoemmuñi ma orpeaxe [0; x] rpabux cummerpiruno ornocnnems-
20 ocu Oy. Honyune rpagux oro dynein na orpeake [15 x]
(owe. 4).

Tax xax y=cosx — nepnoguseckas dyaxuna € nepnonom 2x
» ce rpaiuxe nocrpoen na orpeoxe [—1; x], Jan Koroporo papu

nepuony, pacnpocrpanmm ero no Reel uMexoBo# mpamoÑi € mo-
Nous CABUTOB Ha 2x, Ar m 7. A. BUPADO, Ha -2n, -Ar MT. A.
Beno, 7. e. noobie ma 2an, REZ (pue. 5).

Hrax, rpadux dyHkumi y=cos x nocrpoen ma sceit uncnonoh
upanoii, naunman e nocrpocnun ero wacri Ha orpeske [0; 1].

Puc. 5
$3 13

ocres Gyn y=005x 4 80 Train

Tlostomy cnoitersa dymunm y=cosx MOMO NOAYUMTE, ont
pases ma csolicrea stot hynkumm na orpesxe [0; x]. Hanpumep,
bymenma y=cosx woapacraer ua orpesxe [-x; O], Tak Kak oma
yOutnaet na orpeaxe [0; x] u anıneren vernoï.

Ocnonune cnofieraa gynnum y= cos x
1) O6xacr» onpenenennn — mmomecrso R peex Aeñcrawrens-
sax seen.

2) Mnoxecrso anavennit — orpesor [—
3) Hepnoanuecxan, T - 21.

4) Yernan.

5) Dyuruna npmunmaer:

— onasenne, panuoo O, npu x= E +x, neZ;

— nan6omsuroo annsenne, panne 1, npu x=2nn, neZ;

— nanmenhuree auauexne, papnoe —1, apm x=x+2nn, NEZ;
— nonommrenuunte anauennn ma murepoane (-5; E) m na
MHTepBANAX, HONYYACMEIX CABHTAMH ITOFO Hureppana HA Zur,
nez;

~ orpmusremance snauonna na miropsane (4; 32
Tepnanax, HONYIACMBX CABNTANN TOTO MHTepRANA HA Den,
nez.

6) Boapacramıaa na orpesxe [x; 21] m ma orpeskax,
MONyHaeMEIX CABUTAMN ITOTO orpeska ma 2nn, REZ;
YOunaiouan ua orpeaxe (0; x] m ua orpeakax, monyuaensix
‘easurann ororo orpeaxa na Zrn, nEZ.

Sanawa 1. Haïñrn ace kopnn ypannennn cos x=

nemanne orpeany —n< x6 2

D Hocrpoum rpadixn dynkunit y=cosx m y=- na xannom or

pesxe (pue. 6). StH rpadmki nepecekaiorez B Tpex Touxax, a6e-

mec KOTOpAX A, Ay, Xg MBAMOTEN KOPHAMN YPABHEHIA
3. Ha ompeoxe [0; x] xopmem ypannenun cosx-- +
annneren uncno x, =asccos(-1)=2%, je pueyna 6 suano, veo

TOUKI Xz MX] CHMMOTPHUNE OrnochTommo OCH OY, 7. 0.

ae

PA
nn AE

sal

una mm

1

npuman-

Puc. 6

A LUS

Banana 2. Haitrm nce pemenns mepanenerna cosx>
nannemaue orpesky ~x< x2.

D Mo pucyinea 6 suao, «ro rai yuca y= cos nena mut
me rpauxa yen Ha MpoMexyrKax (-2% =) ”

3
48; 20].

3
Omer. - <x< 2, Meca, à

i
3, npr

1 3anaua 3. Peumrs Hepasencrgo 4cos*x-8cosx+8<0.
Delivers =, ronan Tonyenew xnagpaTnoe nepanenerno
4°-81+3<0, pansoenmnoe nepanenoray (1-2 )(+- 2) <0. Mo-
Tomy Hexonnoe MODABONCTRO. PABHOCHALD KANAON Aa Mae
Ben
(ccax-3)(cosx-3)<0, (3 cos) (ex-1)>0, cona> }.
Tloerponm rpadux pyme y=cosx (puc. 7). Ha orpeske

Lx x] ypasnenue cosx= À uncer wopum - Em E, a pemennaen

wepaneneran cosx> À ua row onpeaxe animieren 6 HEAR 18

Muonecrao pement nepasenera con > À

x PABROCILBHOTO EMY HeXOHHOrO HEPABERCTBA npencrannaer cobol
OGrenmmenne unrepsanon (- 3 +2mm; 3 +2nn), ncZ.

umepnana (

Omer.

Frenn<x<F +2nn, REZ. À
Banana 4. Tloorposrs rpaure Dymimm y= cos? x.

> OGnaers omponenenna xannoï pymenun — mnoxecrno R, muo-
acecrno anaventth — onpesox [0; 1], Gym werHan c nepnonon x

u
1 y= cosx
E z =
LE: g ES
“i Puno. 7

Er zuge 3 |

e
i
lo

ss
si

Pre. 8

fovea Gyn V=COSK A 08 rin

420825 Cregosatensno, rpacın Gym y= 5420528

MOHO Ronde, na Papua YANN y=cos% cares BABOS
some cen Ox, cnomrom na À muepx 10 ocu Oy u cxcarmem nanoe

Brom ocn Oy (pc. 8). 4 E
ISA Bagasa 5. Mocrpours rpapux yen y= xcos x.

[> @ynxuna onperenena ma unoxectse R 1 apnserca neverHoit.
Tostomy mono nocrpours ee rpaduk npn x>0, a aarem € no
MOMIE CHMMOTPMM OTHOCHTeALHO HAMAIA KOOPAMHAT usOOpAsUTE
ero na OTPHNATONKULIX anaveruit x.

Tax xax —1<eosx<1 mpm xCR, 10 mpm x>0 ompanennumo
nepauenerno ~x<xcosx<x mam —x<y<z. Orciona caeayer,
mpu x>0 rpadyx nannoli QU pacnonomen mexay ayyanı

x u y=x (x>0). Tipn orom toux rpapuia OyaKuIE
y=xeosx zemar ua ayse y=x(x>0), ecan cosx=1, 7. e. += Zn,
EZ, n>0. Ananorwano npu x=r(2n-1), NEN, Town rpaduxa
oro ymeum near na aye y=—x(x>0). Ppapur dynein
y=xeosx naoGpaxen na pucynie 9. À ES

Ynpaxnenun
Yopaxuenns 28, 32—35 ssmonmr € nomouso rpabuxa
ynEmHK y=cosx.

28. (Verno.) Busewtre, mp Kar onavenmax x, mpmagnena-
mx orpesxy [0; Br], fynknma y=cosx mpuxmwaer: 1) smase-

A q

29.

E

31.

35.

mue, pannoe 0, 1, -1; 2) nonommrensnste smauenun; 3) or-
Punarensune anauenna.

Hair ananenen Sym y=ooe(«+ E) npn #6, ec:
tt

3 aa 2° 4) a= e

Haltrn snavenna dyakumm y—cos*x upm:

Dx 2) x:

Da=%; 2) a.

4

Buacuitrh, upnnannecm au rpaibuxy hy y=cosx Tou-
ka e Koopannarann:

D (Es 0); 2) (19%; A), 3) (282,

4) (2%

. (Veruo.) Buseuure, sospacraer mam yOsimaer dynkuus

y=cosx ma orpeaKe:
1) (Br; 4x; 2) (22

5) (15 8} 6) [-

ni 2[ à]

“1.

. Pasôure nannsh orpesox MA ABA orpeska TAK, TOO ma On

ox ua unx DYHKUNE y= COS x Donpucrana, a Ha APyrom yO

sl hal

sho [o 2] 9 [-

. © momontso cnojicrsa sospacranus mat yOuBanna Pym

36.

37.

y=cosx CPABHITE wera:

1) cos ® x cosa; 2) cos 8 y cos 198;
3) cos(-%%) u cos 4) cos(-%2) u cos(- 2)
5) cos1 u cos3; 6) cos4 n cos5.

Halen nee npumanneamme orpeory [0; Br] Kopxm ypanne-

D cosx= 33
Haïtru nce npmaagremauue orpeaxy [0; 3x] pemenna nep
nenernn:

Deors> 3; 2) comoda cor 4) consi
Mloerpons paie dy y= P(e), ua: ) oómacr np
xenon Oyama D amener

2, can O<26 21,
BEER

osx, ven — dei
41, com 2>-

2) Go) (=
ES

sa Br

TEROTES PARA PE COS ME FACIE

38. Bsipasup cunyc vepes KOCHMYC no POPMYAM npusenenia,
epannnr «nena:

Def minis 2) ind m cos Es
Don mund; dan cow Ss
5) cosZ mein; 6) cos m sin St.

39. Hara nce npua exam npowereyriy ¿e opa
1) cos 2x— 54 2) cosäx- VB. ls

2, apnnannenaue
Mroxeersy pemennit mepasencrsa log,(x—1)<3.
41. Haru nce mpunaanonanine aponexcyrxy = 3 6x< À peme-
sun sepamos:
a

5 2) cos3x> LE,

Haiira mnoxecrso snasenmit bynxnut y~cosx, com x mpi
manent npoNexyTKy:
2: al 2) (38; 28
D [3s 2) 2 (Ss E).
43. Haiitu npomexyrin sospacranna QYHKIUEE y=cos2x+sin®x
ma orpesxe [0; Zu].
44. Peumrs rpahnuecku ypannenne:

D cosx-1- 22; 2) cosx=/x- Es

40. Haliru nce xopun ypasnenun cos x

3) cosx=1+ V2 4) cosmo
45. C nomoutnio TPAQUEOD BMACHINTD, meer AU pentenite cherena
parent Ñ
» (eres 2) | y= 22,
= cos x; e
Ha ” y=cosx.
46. Cromxo pement meer cucrema ypannenmi:
1) [y=cos%, are,
yo-x?+6x-8; pa loge?

47. Tlocrpoxr» rpaduk u yeranonsr,
1) y=1+cos x;
4) y= Boos 5:

;oherna Pym:
3) y=Beos.
6) y=2-cos3x.

iB

Tloerpowrs rpadme Dynkaum:
1) y=leoszl; 2) y=8~2c0s(x~1);
8) y=sinzetex; A) y= 20".

18 frena ı

"TPWTONOMETBUNBERAE Gyr

191 0 oo exereme noopnunar noerponns ros dy
y=cosz y i
Viste

Pena mepanenerno:

costas À;
» a
2) 2eostx-Beosz-2>0;

3) yes

$ 4. Csoñcrsa pynkumu y=sinx u ee rpagpux

Dyukqua y=sinx onpegenena ma nel wncnosolt mpaNoit, m
aseren ueueruoh u nepmoxuueckoñi c nepmogox 2x. Be rpaux
Mono MOCTPONTE TAXIM ME CIIOCOGOM, KAR m Tpadex HR
Y=c0Szx, Hainan € HOCIPOCMMA, uaupnmep, ua orpeake [0; a]. Ou
aro mpoue Docnomaonariea dopmynott sinze=cos (x- 5).

Dra dopuyaa noragumacr, sto rpabux Gym y=sinx
NOXHO MONYUITD CABHTOM FPADIKR PVR Y =008X BAO OCH
aGenuce amparo ma 5 (pue. 10).

>-2c08x.

A yop

=

Puc. 10

Tpadux yergue y=sinx moobpaxcen na pneyuke 11.
Kpunas, ADIHIOMARCA TPADHKOM DYRKUI y=sinx, naskına-
eres cunyeoudon.

=

Pac. 11

Tax Kax rpagur yınamm y=sinx nonyuaeren caBurom rpa-
Quxa dymsuan y=cosx, 10 cnokerna dyukunm y=sinx mono
Honyunts HO cpoliers DVHKHN y=cosx.
ga Ds
CROATAS OPA PE SIN A 0€ FAQ

Ocnonme py
1) Oônacrs onpexexenma — muoxecrno R ncex mekeranrem-
Bx uncen.

2) Muomoerno axavenmit — orposok [-1; 1].
3) Hepuonmueckan, T=2r.

4) Hevernas.

5) Pyuxuua mpunncae
— anauentie, Panuoe O, npu x=xn, CZ:
— nanbonsuiee axauenne, panuoe 1, mp += 5 +2mn, neZ;

— mannemsmee anaventte, papoe —1, mpl x=—
nez

— monormrenomio onaxenma na mureprae (0; x) 1 ma un-
Tepbanax, MOXYUACMBIX CABHLAMA TOPO HuTepsana Ha 2nn,
neZ;

— OTPHUATENBEBIe aHayenma Ha HHTepBane (n; 27) u HA un
Tepnanax, nomyanemux cxmurama ororo unrepnana na nn,
nez.

6) Boopacraoman ua orpeoxo [- E; 5] u na orpeaxax, nony-
neun CADATAMI sroro OTpeaKa na Ann, neZ;

yéunaonan na orpeane |; 8 |x na orpentax, noxywacnssx

caeuram store OTpeska na Zun, neZ.

+2nn,

Banaua 1. Haiirn pce npnnannemanme orpesky —n<x< 2x
op ypannenna sinx= À
D> Moctpoma rpaduica Qymxuui y=sinx « y=} Ha aannon or
pesxe (pue. 12). Irm rpabuxm nepecekarorca m ABYX TOUKAX, abc-
Neen KOTOPIIX ABIMOTCA Kopianı ypannenna sinx=

penso [-35 5] ypamenne mieer xopenn x~aresin = ©. Dro-
a infeed
pol open xp=n- T= 5%, wax war sin(n- =) =
Le
Omer. sh. 4
=
me 12

Saxasa 2, Hañiru nce pewenna nepanencrsa sinx<}, npu-
HAAEXAMNE orpeaky — REX < 2x.

20 frase

TouronamETPHTECKNE SE

> Me pueyana 12 mano, xro rpadux dy y=sinx ner
ne rama oyo y= 2 ma mpowenyreax | 5) m
(Ss am]. «

E 3anaua 3. Hcenenosar» byaKumm
mad

1) y=logesinas

D ar
1 1) Obnnernio onpenenenun gym ABAMOTOR DEC IMANCINA x,
pu koropux sinx>0, 7. 0. Ann<x<ni un, REZ.

Tax kak O<sinx<1, to no cBolicrsam xorapmhmnueckoit
aymaumı € ocnonannen a>1 nonyuaem logs sin x <0, 7. e. mnoxe-
CTO anauennä PYHRUHH — npomexyToK (-00; 0].

Dynkuna y=log,sinx — nepuonuveckan € HAHMEHBIUHM HO-
NOKHTENDHEIM MepHonOM 2x, Kak u bynkuma y~sinx. Ilosromy
hocrarouno cenenonaro oxy ysicuino ma murepnane (0: 7).

@ymaym f(x)=sinx m y(t)=log,¢ ssnmores nospaeramıı-
son na mpomeanyricax (0; 4] m (0; 1), orxyaa caenyer, uro pynr-
wın y=logesinx Tanne annneren nospacraiomel na npomesyrKe
(0: &) x nommer neo suasemn us npowenyrnn (-00; OL. Tax
Kak snauennn yak y-sinz B rouxax mpomexyrKa (0; x), cum-
METPNEHLIX OTHOCHTENLHO TOUEH x=", PABIEI, To TPAQHK Hymn

(2) m mocrponrs ee

y=logssinx na npomexyrke 13 x) eummerpuuen rpapuxy ro

mem na mponenyrre (0

de
Inneren youunsonen mpn x¢{ 3: 2).

Ilpamsıe x=0 m x=r — BepTuKansHsle ACHMOTOTH rpabuxa
yum y = log, sinx ua npomexyrxe (0; x). Tenepx moxno erpo-
aro rpadux dbynkunn (pc. 13).

ar

y =logasinx
Puc. 13
gs wa

CHORALE yA O THAR

1
19 na
Puc. 14

2) Dyuxuna y= aunneren ueruoli u nepHomHecKot ©

nu
En
nepronon 2x OGnactiio onpenenenun + #17, neZ.

Ha npomenyrxe (0; ¿| dynruux f(x)=sinx nospacraer,

Muoxecrao ee sHavennit — npomexyrox (0; 1], a ymeus y(t)

na mponexyrxe (0; 1] ysınaer. Hootomy y= t= = iz

naar ym ua npomexyree (0; E |, mpmmmaer nce anave-

una, Gombe man panne 1.
Kar m u sagave 2, rpadur ymin y= 15
axyree [55 x) emunerpmuen rpadmy sroh hymeum na nponenyr-

BEPTHKANBENIE ACHMNTOTH FPADHKA Pyıtkan na unteppaxe (0; 7).
Iyers xc(m; 2x). Tax kak sin(x+n)=-sinx, To B Touke
x=20+7, re xoc(0; 5), dynxuua sinx npunnnaer sxavenne
“sina, Mosromy Ana nocrpoentn rpagsnea Gym y= 53
npomexyrKe (x; 2x) aocrarouno nepenecrm nocrpoennstä ma (0; x)
rpaduk soit by#kuMM Ha npomekVTOK (1; 27), a sarem 3amenuro
ero na cummerprumall e num ormochrenmo ocn Ox (pue. 14).

Muoxecroo anasennh by y= — nee anauenma y, Ta
rue, sto [yl>1. dr

ma npome-

2] ornocnrenuno npaoï x=. pme 2-0 u 2x —

1
Anal

a Banana 4. Tloerponrs rpabux hynkum y=x+sinx.

220 trans 1

fpnronowarpmnecKe LT

ly=x+sine

Pue. 15

D y yı=x 1 ya sin x onperenent ma mnoxecrae R. Ina
mocrpoennn TPAÍPILIA CHOKIM OPANMATE! TOVeK TPAQUKOD Y, == M
ya—sinx e oxmaKonsinns aßeumcenmu (pe. 15). 4 ESA

Yopaxnenns
Ynpaxuenua 51, 55—59 sunonmurs © nowomso rpaguka
Oya y=sinx.

51. (Vorno.) Buiscnnrs, np Kaxıx onauenmax x, mpmnanena-

mx orpeaxy [0; 32], pynunns y=sinx npuniaer: 1) ana-
venue, pannoe 0, 1, —1; 2) norommrensnste onauenua; 3) or-

Punarensuse smauenna.

52. Halen snanenna Gym y=sin2x mpm:
MES

=.

D x

52) ens 8) en BE; 4) x

53. Haltra anauenua Hymn y=

Deia, 8) x

54. u Upmmannexar mt rpaduxy HYMNE y=sinx rox
zu € Koopannaranın: _ E
4) (--$s 0)s 2) (Es 1): 8) (5-0 (53)
ga os

CORTE ya yzalız es Fpadım

55.

56.

57.

58.

59.

60.

61.

24

(Verno.) Buiacunrs, Bospacraer wan yOumaer pynkuna y =sinz
na npomeakyrKe:

DE
ol
PaaGurh nommuñi orpesor na ana orpeaka TAK, WTOGH ma ox

HOM ma HHX QyrEnma y=sinx noapacrana, a HA APyrom yOu
Bana:

1) 10; ah; 2) [Es 2); 3) [ri 0% 4) [205 11.

C nomommo cnoliere nospacranııa un yOuBanua yakuuu
y=sinx cpasnure «nena:

Ten nie, an des
1) NA sing + a 2) 75
2) an(-85) w sin(-28); Mein? weine.

Hañivn nee upuunnnemaume orpeaxy (0; In] Kopun ypamme-

8,
Flaten soe npmnanteacanmen orpaity (0;
ponerse:

>, sine 2,
1) sinx: y 2 < a
3) sino de 4) sina<- E.
Hocıponu rpagux dyunuun y=/(2), ualirm: a) oGnacro onpe-
Renenna DYyNKUMH; 6) MMoMecTBo sHavennit; B) MpoMexyTKH
‘yOurbanna:

sin, aan Ocxcan,

dia

Vox, eco x<0;

D sinx= YB; 2) sinz= 2 3) sinx=

x ie
a) sin 3.

3x] pemenma nepa-

sina, conn ¡<a
cosx, conn -2r<x< 2,

» |

Bupaane Kocunye wepea cuuyc no dopmynam mpunerenits,
cpasmirs unex

E mes; sin % x cos 9
1) sin à Di 2) sin Ÿ me;
3) sing om costt; 4) sing m cos

<x€r Kopi

fnacs 1

"TpHFOOMGTpINOGRHS GR

63. Ha nce mprnannexamne mnoxocray pemenmi nepanener-
pa Va=1<2 nopax ypazmonne sinz

64, Halen nce npumagremame mponenyrey - SE <x<x peme-
Hna nepanencraa:
Dane
85, Hamm wnomecrao auavennA dymeaun y=sinx, ocan x mp
rages pomor
1) [3s af IE SEI)
66. Haiiru a YÓLBAMIA QYEKUIM HA 00JARHOM orpeske:
D mame) [Fs 2]
2) y=-sinx, E #5 2r].
Peu spadmugcn ypapnenne:
D Ant 2) sinx=2 =
3) sinx= Va dano
68. C nomomo TPAQHXOB PYHKUMÍ BEICHHTS, Meer AM peme-
me cucrena ypasnensi:

1) [y-tesinx, 2)
y

69, Cisonuo pemennit umeer cuerema ypanneuni:
D (u=2sinx, 2 [y+1=-sinz,
yalon xs u=Ver

70. Nocrporrre rpaduk m munemire cnoiersa dysrn

2) sinar< 2.

Dy=1 2 yn2+sins; 3) yasindz;
4) y=2sinx; 5) y=3sin E 6) y=2-sin2x.
TI. Mocrpours rpedue Oya:
D y=sinjxk 2) y=lsinxk
3) y=sinx=x5 4) y=log, sinx.

[TZ] Cuna nepemennoro onextpunecroro roxa apınercn yakuu-
ch, samemueñ or bpemenm, m mupamaeren Qopmyaok
I-Asin(ot+9), Me À — anmauryaa Kone6anna, © — 1acto-
Ta, q — moanonas aoa. Iocrpourre rpapue hmmm, eos:
1) A=2, w= 2) A=1, 0-2, 9-3.

75] Peu wepanencn

2) 3sinx-2cos!x<0.

td,
D sintx> 45

sa 25

TEORCTES PARA PIN NES PADI

$ 5. Caoñcrea u rpaduku pynHkuuñ y=tgx

n y=ctgx

bymxnus y=tgx ompegencna np x# 5 +rn, neZ, aunaercn
evernoit » MEPHORWICCKON € HEPHOIOM Fi HOPTONY HOCTATONKO MO-

expone ee rpagu a npomexyrie [0;

+): Barem, orpaomz ero

ememerpirio OTHOCITEADNO nauaan KÖopakinar, Moaysurs rpadue

ma munepnane (_2

|. Haxomen, ncnonsaya neptoamunoers, no-

eTpoTs rpabuk Qyukuma y=tgx Ha sceli oßnacrm onpenenenun.
=I Mpexge vem HOCTPOHTE rpabux DYHEUMM Ha Npomemyrke
[os 5) mocos, sto na rou npowceyne ymanın romaemer,

O Myers O<x,<x,< 4.

Hoxaxen, wro tex, <tgx,, 7e, AAA < tee,

To venosno 0< x, << 2.
orkyra no eolernam Pym
y=sinx umeem O<sinx, <sinxa,
a no cnohersam Gym y=
=co8x Taxme noo

cos, >008x3>0,
1.1
Ser “cs
IlepewxoxHo > nepanencroa

orkyna 0<

sinx,<sinx u
EA
AS m, ne
tex, <tgn. OA
Buan, ro @ynxuna y=tgx
noapacraer na npomexyrKe 0%
Sx<F, nalinem neckomuno ro-

ver, npnnanneauns rpaduny,
Im nocrpor ero na orom upone-
xyrxe (pre. 16).

Mexons na evojiersa noser-
nocr gynkumn y = te +, oras
mocrpoenxui ua npouexyrke
[os 3) rag cmmerpmuno or-
Hocw'renio MAMA. KoopannaT,
omy rpadı oroit Gym

na merepnane (- 35 2) (one. 17).

26 rama 1

cosa
Y

wer

1
al
a
ET: =
1 $15 3
Puc. 16

TPurOHOMETPHTBERAE YA

=

|
|
1
PE 7%
vr

5 oymnuna y=tgx ne onpenenena.

Pac. 18

Hanowans, 110 mpi x

Eom x< 5 u x mpuómraerea x E, TO sinx npmönmmaeren x 1,

a cosx, OCTABAMCE NOAOMHTENSHBIM, CPOMITCA K Hy. TIpm 9rom

p06

& ADASIOTCA DOPTHEOIBNSIMA ACHMETOTAMA.

=tgx HCOTPANIMENHO BOOPACTACT, m IPamBIe x= 5 1

2

Hopelinem x mocrpoenmo rpadmra Hymn y=tgx na neeñ
oGnacrut onpenenennn. Pyukuna y=tgx — mepmonmieckan e Mepmo-
hom x. Cnenonarensno, rpadbiie oro dynkımm monyuacrea na ee

rpabua ma mirrepraste (- 5; 2) (em. pue. 17) exmmramn post oct
öcunce na an, neZ (pue. 18).

Mrax, nech rpadhme pyme y=tgx erponren e HOMO reo-
Nerpuneckix mpeo6paaopanırl ero “ACTI, Mocrpoenmoli ma mpome-

Cooiierna pynkumm y=tgx momo nonyawre, onmpasce ma

esolicraa ovofi hmmm na mpomexyrxe [0; À). Hanpırep, pyme

3; E), rue wax oma gym

va pts wospecreer xa mre (-
aux noaptcraer na nponengras [0; $) ananerem mevernoñ.
Ocnopnzie cnoferna bynknun y=tgx

1) O6nacrs onperesenust — mroxecrno Bex jeiierpirem.nBIx
sucen x# Z4nn, neZ.

2
2) Mnoxecrso snauenmi — mnomecrso R scex xeitersurens-
3) TIpmase x= À +nn, CZ, ABTROTCA Bepria mar ACUM

4) Tepnoguseckas, T=r.

ss Mer:

Comer pa Gyn EX A yc

5) Hevernas.

6) dynkuns npunnmaer:
— anauenne, pasnoe 0, up x=xn, neZ;

— nononarrenumue onnvennin ma nnrepmanex (1
ne

— orpmuaremunce auavennn na unrepaanex (- 24 an;
nez.
D Bospacrmontan na mrepnanax (

2
Saxaua 1. Hañiru sce Kopin ypanuenus tgx=2, upnnanie-
manne onpeany nc 8,
[> Hlocrpons rpapucu gynuumit y=tgx m y~2 un namnom orpesre
(pue. 19, a). Su rpaukn uepecekaioren 2 rpex rouxax, abeunc-
CM KOTOPUIX Xy, Xy) Xq ABNAIOTCH KOPIAMA ypannenun tgx=2. Ha
uuzepsane (- 5; 5) ypasuenne mueer opens x, =arctg2. Tas xax
Due y= UE Nepnogwwccrun © MOPMOAOM %, 10 xp=arctg 24,
uan. a rarigd, carta liar igica À
Banana 2. Haitr nce pentenna wepanenerna tgx<2, npnnan-
esca expeary CRC it.

4

ans Eran), nez.

ur tax
| | y=2
5 7 =
= 3/0 AE y =
a
| yatex
yal
=
af 58
o
Pre. 19
280 traen 1

‘Tpnronowerpivecnne Gym

D> Ma pucyuxa 19, a suano, wo pag Oya y=tgx neun
ve same npanoh y=2 ma mpomengriax [mi xl (55 ai]
x) me -néx<-ntartg2, 5 <xcerctg2, E <xenrartg?. d

2
MI Banaua 3. Pemmrs nepanenctso tgx> 1.
[> Mocrpomm rpadmxn ymennit y=tex m y=1 (pue. 19, 6). Pu
cyHOK MOKAIHBACT, “ro rpadux gyHkunn y=tgx nexuT Bee
mano y=1 na mpomexyrre (2; E), a xaroxe na mpomexyrrax,
nonysennuix campana ero ma m, Zi, Bx, 5, 28 wT, A.

Orser. 1 +an<x<z +an, nez. qa

To dopuyaam npunonenun Qymanın y=cigx momer Gum
penerannena war y=—te(+%). Cuolicraa ym y ga none

so nonyanm, neenenya yon y=—t4 (+),

IE Ocuosume cBolicrsa Pyukunm Lex

1) OGracre onpenenenmn — ace AEHCTAMTENLHLE ACTA, KPO-

ME zen, REZ,

2) Mnomecrno snarenuli — mmoxecrso R acex selicrenrens-

aux uncer

3) Upanme x-nn, n¢Z, nenmoren sepruxamems aemın-

oran

4) Mlepronnveckan, T=x.

5) Hevernas.

6) Vósimaiwomas na marepsanax (nn; a+an), neZ.

pada Gym y~ ete MONO NOCTPONTE Caron FPAdIt-
Ka OyHKuMH y=tgx na a CXHHHI BIERO, JATEM OCYINECTBHTB CHM-
merpino ornockrenso on Ox (pue. 20).

u
| Ms +B | |
|
|
|
ir + — ES 2n
|

95

=

Corea m rpadınan ya y=tax y cta

Baxaua 4. Hocrponra rpagune gym y~otg(2x4).
D> Ppapine ymemmt ymetg(2x+ 4) =etg2(x+ E). Dymeus ne-
PMORHTEKAA c HEPHOAOM 3, ee FPADHK MOHO MOCTPONTE € HO
moutnio npeoöpanonanmi rpadmxa yen y =otgx vax:

1) bommonirrs enmr ma E eaux nneno;

2) minommurs exare rpagmra yen y=ctg(x+

vom oct abeunee x mpamolt x-

3 2 pasa.

Tonysenmsä rpadur ws06paxen na pueynxe 21. 4

y
A aaa
CE 0 3 El =
II
yo cer +9) yo ge
Pre. 21

30 rasa ı

TpnroncmerpawacKne GE

on ee
1
= 3 coal = a +
a
Pc. 22

5. Hocrponrs rpabux @yskunn y=Vigx.
D yaaa onpexexena np yenosun, “TO yAKUNE tex mp
aor neorpunarensunie anauensn, 7. €. npu an<x<E+an, REZ.
dyneus MEPIORITICCKAR € nepRoROm 7, TAK Kar ecm x upunazr
EXT OÉTACTI OMPEAENERHA, TO X—T MX FT NPHHAAIEKAT O60
crm onpenenenus m npn arom Vig(E Ea) = V8.
@ynkuus Bospacraer pH ES (uo esojñicray enomnoh
yann) 1 npnnmmMaer TOABKO HEOTPHNATENLELIE snauenna. Tpa-
Que DyHKUME waoßpamen Ha pueyuke 22. À EI
Tpurononerpuuecsne dy umpoko mpumenmorea B Ma-
TeMarıke, dmamxe u rexmmKke. Hanpumep, Morme mporeccst, TA-
ole, Kak KOMEGAHHA CPV, MARTA, HANPHMENNE B nen Me
PeMeHHOTO Tora HT. A,, OMICHBRIOTER DYHKUNANI, KOFOPHIE 32
ARIOTCA opmynamm saya y-Asin(ox+9). Take mpoyeccur
hasuBaor TADMONIMECKIMMI KOJECANMAMIL, à OMBCKIBAMDUNE TX
dvi — rapmonnkamn (or rpeucexoro” cxona harmonikos —
copaasepanti). Fpahm dynnunn y=Asin(ox +g) nonyaneron ma
Cmnyoonsı y=sinx caTieM HT PACTA KETM ee BAO KOOPAK-
maria ocelt u enunrom prom ocm Ox. Oßkuno rapmanmıeekoe
konedanne annaeren Gynkuneh ppemenn t, 1. e. y=Asin(ot+9),
TAC A aunauryaa KoneGanus, 0 — HACTOTA, O — KAMAAHAN

‘basa, a 2% — mepnon xonedanna

Ynpaxnenun

74. (Verno.) Biinenurs, mpm KAKIX amauennax x 19 MpomexcyT-
ka -n<x<2n gymkuna y=tgx mpmumaer: 1) snauenme,
pannoe 0; 2) nonoxurensmue anauenus; 8) Orpnnarensmne
snanerus.

gs Hs

Cora u FPE yalox m y= COX

75.

76.

78.

32

(Werno.) Biacnnrs, annerch au QYEKUNA Y =tgx Bospacra-
rowel na nponemyrxe:

ay [Ss 3,9 (57) 8 (- 2) 012 3.
Haïtru snavenne @ynkumx npm sazannom anavenmm apry-
morra

a; a, 2 25
DY yates, 2288; yt 2-28;
8) yet, x 4) ynotg, 2-3.
Haden amen Oya yj a
Deine BE; 3) ee 25 4) wm 2,

Buuacumrs, apunaanenc ax pada GAIN y= tg 2s 10%
Ka € Koopannaranıım

D (3-1): a (ABE;
EE

© momombo cnofters QyRkumii y=tgx m y=ctgx pam,

Dala ote SE;

© te(-4) n te(-4):

5) ctg2 m cteds 6) tgl n tg1,5.

Haftra nce mprnagnexamue npomexyrky (a; 2n) Kopi
ypaneuns:

1) ctgx=1; 2) tgx-V8; 8) ctgx=-V3; 4) tex
. Hañrn nce npnragnemanme npomexyrky (—n; 2x) pemennx
Hepagencrpa:

1) tgx>1; 2) tex< 8; 3) ctgx<-1; 4) ctgx>-V3.

ocrpons rpadmx dy y= (Go), malen: a) oßnnenı. ome-
enenns; 6) MROXCCTDO axaveniil; B) IPOMEYTKH DOSPacra-
una:

lan, ccm naxch,
sinz, scan -nezen

gs Sant nee
2) | an

DS {

cosx, ecau ¿a

Pers nepanenerno:
1) otgx<l; 2) tgx>V3;

fnapa 1

TonrOHOMOTBITECERE GA

84. Hatin ace mpunaziexamne npowexyrky [0; 8x] kopnn ypas-
Digr=3 2) ctgx=-2.
85. Hal nce npumaaneampte npomexyrky [0; In] pemenns
mopanenerna:
Digxod Dig BMigac-h Migx>-d
86. Peuwrs nepasenerno:
Detgx>4; 2) tgxcd; B)etgr<-4; 4) tgx>-5.
SL. Halin sce npnmannexaune npomexyrky (- 2: 1) kopmm
ypannenua:

1) tg2x=V3; 2) tg

88. Hair vee npunagnenamue npowexyrey (- 35 1) pemenna
epanesterpa:
1 tg2x<t; 2) teBx<-V8; 9 ce
89, Tocrponrs rpad u pumemaro cnoferna Hymn
Dune Dy- tez
armes ih o yrete( $44)
, Hair smomeerno suavennit yn y=tgx, ecam x mp
Hanne MpoMenyTRY:

rs ahs
Tlocrpours rpadi pymenu (9193).
9D y=tglel 2) y=itexk 3) yetgx;
92. 1) y=tgxetgx; 2) y=sinxetgx: 3)
98,1) ytg(3x—%)s 2) ycte(2e- à

13

[SL Pemure nepanenerno:
Digx<l; 2) tg*x>8; 3) Ssin?x +sinxeosx>2.

$ 6. O6parnbie tpuronometpuyeckne HyHkunn

TAL Oynrmus y=aresinx
Tlo onpezenexmo apxemnyca uncna (cm. yueSxux 10 xnacca)
gaa xaxuoro x€[~1; 1] oupeneneno onuo uneno y=arcsinx. Tem
canta ua onpeaxe [-1; 1] aaqana pyuruns
yoaresin x.

Tloramen, wro @ymeauea y=arcsin.x anserea oGpartoti x pyme

33

DEBATHIIE TpRTOROMETPINECHTE YAU

y= aresinz,

mwas

2
Puc. 23 Puc. 24

O Paccmorpum ypapxenue sinx=y, rae Y — sanaHHoe “meno Ha
onpeaxa -1SyS1, a x — menancormoe. Ha orpeane - | << E mo
vpannente no omponenemmo apkennyen unoma meer ermnermen-
a Kopen x arcainy.
Tlomens » oroit (pOpMyz18 MECTAMH x M Y, ONU y—arcsinx. @
Taxon oópasom, cnolerna Gym y =arcelnx Mono Rony
“rs 43 CBOKCTB yan y=sinx. Ppa pax pyukunm y =arcsinx

emunerpiricn rpapaxy Gym y=sinx, - E SxG E, omo
28, 24)

Oenonnnie csolcras Gym y=arcsin x

1) O6naerz onpexenenna — orpesor [-1; 1].

2) Mnoxecrzo anauenmü — or-

pesos [-55 2]. à

8) Hesertan, mas van

ne) -anesin

4) Bospecraoman. 71
IA Ina pyaxumn y=sinx o6par-
topo mono maß, Langue. ua
orpesne [35 3]. Ha nou ox
una y=sinx yóxmaer or 1 zo
Sir de. ma drow ponerse
Pyakuma umeer oöparnym. Ina
© uaxonseumm sonen upon
Bonunoe amarme ya ns nome
Ba (1 1]m sagen coorserorsy-

SAT rana ı

‘remo mpanoit y = x (pu

TE DUT]

y=arecosx

53)

Ovenuano, tro xo=T-aresingo. Ionenan necramm = u ya nosy

cnr dyin Yor-aresins. Tama e6pasou, sax" Oya
LE

gains un orpeae [E 2] ocparmos Comer Ay y

€ aûnacruo onpezenen [-1; 1] soscecracn ananenni [5

(nc. 25), a

jee snatenne 3 nexonn ma Toro, wo Joan sv

12. Oyaxunn y =arecosx
Tlo onpenezenmo aprkocmnyca wena ¡uta Kamaoro xE[-1; 1]
onpexeneno ono “meno y=arccosx. Tem camsım ua orpeaxe [-1; 1]
onpenenena dbynxusis y=arccosx. Ora byMKuKs mnnneren OGpAT-
noft x byneuun y=cosx, pacemarpunacmoñ na orpeaxe OGxG mn.
Tpagmk Oynkumm y=arccosx cmmmerpmiex rpabuKy Hymn
y=cosx, O<x<x, ormocmrembmo npaoñ y=x (pue. 26, 27).

Ocnommue exolersa dy y~arecos x

1) Osnaer» ompenenemna — orpeso [-1; 1].

2) Muoxceero auaenul — orpeaok [0; x].

3) ¥ounaomas.

3. @ynnunn y=arclgx u y=arcetgx

Tio onpenexemno apkranrenen unesia As Kamporo AeÑoran-
remsnoro x onpeneneno onno wmcno ymarotgz. Tem comun ma
pce mcnopol mpanoli onpenenema Qyukuma y=arctgx. Ora
yuan anınercn oÖparnoh x Qyuxuu y =1gx, paccmarpuBsenoit
na nmrepnane - 3 <x< 5. paie hyminmm ymarctgx (pre. 28)
anyrneren mo rpaduna Gym y=tgx, - E <x< E (em. pe. 17),
sunmerpueñ OTHOCHTEABO MPAMOR y=.

se le

DOES TPHTONOMETPNECRNE yr

3

joleraa bynKumn y=aretgx
1) O6xacr» onpenenenun — muomecrso R scex aeñcrsirens-
mo uncen.

2) Mnoxecrno anauemmli — ummepnan
8) Bospacraionan.

4) Heueruas, arctg(-x)

retgx.

Tlo onperenenmo apsoranrenca umeaa Aaa KEMAOrO ACHET-
putensHoro x OMPOCNCHO ono UHCAO Y=arcctgx. Tem camıım na
Beet uncnoDoli mpamol ompenenena Qyuxmma y=arcctgx. Ora
Qynxuus abxaerca oÓparnoñ x bynxiwm y=ctgx, Paccmarpnnae-
moñ na umropnane (0; 1). page Qyurım y =arcctgx nonyauer-
ca ma rpabuxa dynxuun y=ctg x, xC(0; x), cummerpueï ormocn-
Tensmo npamoh y=x.

Denon ra yann y=arectg x
1) OGnacrs onpenenenum — muomeerno
2) Mnoxocrno amame — ummepnan (0; 1).
3) YGsatomen.

4) Heuernan, arcetg(—x)=—arectg x.

Jazaua 1. Penne ypannenne arecos(8e+1)~ 28.
D Tax van Zei

x}, to no onpenenenmo apkkocunyea Hera

pe

Banawa 2. Haliri Oßnacr» onpenexenna yann
y=aresin #2,
D Tax ax hymma y=aresint ompenenena npn -1<t<1, ro

yin y Onpepenena jus rex amanemi x, ¡ura KO

E
Topunx sunomunorca mepasencraa —1< 7? <1. Orciom -3<x-2<3,
16x65. 4

36 tana ı
TpurOMOMETPIMECANO GHANA

in? 2
D Tax sax xmaresin à, 10 - à
nepnoit uersepru.

Hazen cosx: no Gopmyae |cosx|~Vi=sin?y- y >

+ Tax sax 2— yon I sermepru, 10 cosx Pow bean

. Hocrpours rpagux dyaxuan y=arcsin (sin).
D Dymkaux onpenenena ma Bcem mHoxecrae AEÄCTBUTENBHK
ucen, ee nepuon pasen 2x. Ilocrponm rpadune (pymemn ma or-

©] Damm SE zo yes. Bam $626 E,

pue [-5:

ro —Jéa-n< m mo onpeaenenmn apsemmyen crever,
. Tax aK sin(e=x)=—sinx, 70

ro aresin (sin(x—x))
aresin (sin(x—z)) - arcsin(—sin x)=-—aresin (sin x).
Tus cópacon, sem xe[ Fs %)

co ns me | + $

x, ecm -Páxch,
y=aresin (sin x)= su
nx, conn Fees E.

Tpabnx moo6paxen ma pucyxxe 29. 4

Ynpawnennn
Cane aucna (95—97).
95, 1) aresin Em arosin q 2) aresin( 2) m aresin(- $)5
3 y v2 3
39 aucun Sy aresin AN

so 37,

DEP TP OASIS Pa

28 D aro mare gs Darf) rl}
3) arecon Em arcene 3 4) aroos(-.) arocon( 2),

8

D avetgaV n ares 2 act E) nat)

8) arcctg VE u arcctaVT; 4) arcctg(- À) m arecte(-VD.

Pers ypannenne (98—100).

98. 1) aresin(2-31)=2; 2) aresin(3—2x)= 4:

3) aresin #72

99. 1) arccos(2x+3)

4) aresin 43-3.

2) arceos(8x+1)= 45

8) arccos Zt

100. 1) aretg E

4) arctg(2-32)-—

3.

D yoann 258, 2) yoarccos 283%
3) y=areeos(2Vz-3); 4) y=aresin =";
6) y=aresin (8 VE 2};
7) y=arcsin(x?-2); 8) y=arccos(x?- x).
102. Hokasars, uro rpadax DYRKIMM y=arccosx cummerpHYeH or-
nocmemuno rome (0; 5)

103. Tocrpours rpadu dy
1) y=aresin(2x +8}; 2) y=Zarceos(x~1);
3) y=arcctgx; 4) y=arctg(x+1).

¡OA Toxsanrs, vo aresin a arecos= 5.

B]Moxasarı, “ro arceos(-x)=1-arec08 x.

LOG! Mocrponrs rpadux dysxusn:
1) y-arccos (cos x);
2) ymaresin(cosx).

MOT Hatira bymenmo, o6pamyro x dynaumm y=cosx na orpes-
xe [as 0].

fee rnasa ı

TpurononaeHPmIBCRnE NET

Ynpaxnennn x rnage |

108.

109.

110.

112.

113.

114.

115.

16.

Haïÿrn oßnaer, onpenenenna Pynkumm:
1) y=sinx+cosx; 2) y=sinx+tgx 3)
4) y=Veos a 6)

Dn

Haiirn muoxeorno ananensti ymin
1) y=1-2sin?x; 2) y=2c0s*x-1;
3) y=3- 2sin”. 4) y=2c08*x +.
5) y= eos 8xsin x sin 8x cos +4;

6) y= cos 2xe0s x + sin 2xsinx 3.
Bunenws, snaseres am uernoh um neuernoit yes:

D) y=xt+cosx; 2) y -sinx;

3) y-(1-x%cosx; 4) 1+sinx)sinx.

Jonasars, 70 nanmensuumi MOJOXHTEADHBÍS mepuon DYHK-
un y= f(x) panes 7:

=
7:

1) y=cos7x, 2) y=sin >, T-1ar.

Cpaonrs, «mena:

1) sin1 x cos2; 2) sin(-1) cos 1;

8) sin3,5 u tg8, 4) cos3 m tg4.

Baron, Kaxas 10 QyHRUM y=sinx HAN y=cosx apna
ren véunarouel na npowersyree:

DE) 2 (o: 3]; 9) [x 4) [3s x]

Halirn mnoxecrno snasennit oynrumm y~f(x) na npomenyr-

ne [-3: 3], comu:
1) fx)=sinx; 2) f(x)=c08x; 3) Hx)=tgx.

C uomom»io rpaduKa hyukuun y=cosx HAÏTI TakHe smase-
Hust x na sanannoro MpomexyrKa, PM KOTOPHX CPABEIN-
20 pabenerno:

1) cet, [-
ooo min aci ae pana

Take sHayenua x M9 SAAAHHOrO MpomexyTKa, HI KOTOPEX
cnpaseannso nepaneuc

1) tgx<V8, [a xh 2) ctgx<l,

D ctgx>-1, [Es 25) 4) tg 15, (35 «

117. Haïru npmmanneaupe nponexyrky [0; 3x] opnu ypanne-

1) 2cosx+\VB=0; 2) VB-sinx.
3) Btgx= V8; 4) cosx+1=0.

in

Vapor « image T

118. Hahn nce nprmannexamme npomexyrky [- 21; 1] pemenns
nepapencraa:
1) 142008220; 2) 1-2sinx<0;
3) 2+tgx>0; 4) 1-2tgx<0.

119. C nomouso rpaukos DynkmË ar «nexo xopueñi ypas
Remus:
1) cosx=x%; 2) sinx= À

=.
120. Tlocrpowrs rpague yet:
Dy=gsinx 2) y=cosx

121. Pacnonomurs » nopaaKe yOunanss wena:
15 ES
Deo tez te(-F
2) tg3, tg1,8, 182, tg4,5.
122. Hasirn oßnaerı onpeaenenns ymcunn:
Du-telax+d); 2 y Ver.
Halirn nan6onsıee u Hanmensiee HACHA DYHKLMH:
D paconts-sin's; 2) yesin(x+)sin(s
8) y=1-2lsin8rh 4) y=sintx- 2costr.
124, Busncuurs, anızeren an veruoh man wevernoi dymeuin:
1) y-sinx+tgx 2) y=sinxtgx; 3) y=sinx|cosx|.
125. Harn naumensumit nonomurensuutit nepuon bymkumm:
1) y=2sin(2x+1); 2) Lege),
126. Pemure rpadsuecku ypapnesne:
1) cosx=ixls 2) sinx=-Ix+1].
127. Haïou ayan cyan
1) y=cos?x-cosx;
2) y=cosx-cos2x—sin8x.
128. Pemurs ypanxert

ES 3 x
1) arceos(x—3)= 35; 2) aresin(=# +
Hastrm nee anavennst x, mpm Koropuıx pyme
y=1,5-2sint =

UPNUNMAET HONOKIMTENLIME SHAVEN.
HS0-Noerpours, rpagune dynein:

Dy-2sin(45)-2 2) y=cosx—Veos™
8) y=coslxls 4) y=-sinx:
5) yasinz+lsinzi; 6) y=2ns,

40 © rnava ı

Tpnronowerpinecrre PNR

(GE Hlatirn renoxeero anauenmt pyme
1) y=12sinx-5cosx; 2) y=cos"x-sinx.
[132] Pemurs nepanenerzo:
1) sinx > cos x; 2) tgx>sinx.

1. Haanarı nnoxecrno ananennil kamaoli na yumuit
ymsinz, ymcosz.
2. Haar oGnacre onperesenis xaxcqol ua pyme
Vatex, ymetge.
3. Karen na sr
posing, y=cosx, y-tex, y=ctgx
anxaerea wernoti?
. Kaka Qyunıın messumnerca uepnonwueckon?
Tipuvccru npunep Eye, y Koropoh marmembrmulí nono"
auirenbuuii nepwon paben: 2x; 15 53 In.
Hesners mpomencyTkH nospacrana KerKqoit ma pyme
y-sinx, y-oosx.
Tipit xara omasennax x songs na Gym
Vie, y=otgx
npaumaer nonoxurenbune onauenmn?
Tipit Korn omauemnax x ones na LITE
y=sinx, y=cosx
upunnsaer nanGomuee m waunenbizee anauenna?
[El Hassars o6nacrs onpenenennn kampok ns dynium
y=arcsinz, y=arccos x.
[O] Hassars mnoxecrao anauennü xao ns pyme
yrarctgx, y=orcctgz.

Ip

ES

1. Halıru o6nacr» oupenenenun pyeruna y=1g2x. Henserca an
ora pynkuma vernoi?

a a ee ee
RM a en ee
1) yaa 15 2) y(x)=-1; 3) y(x)=0;

4) y(x)>0; 5) y(x)=0.
4 Bata noe anawennin x wa npomenymen [Es 25], ann xoropu

bunonmeTes HepanenteTso COS x <=

2
4. Pacnonoxmus » nopaaxe noapacranna unena:

Fer €
ote hi ie Ti ote; ota.

4

Bonpoce + raue T

1. Tlocrponrs rpabux Qynkuum y=-cosz u Malin anauenna x,
pH Koropuix YANN a) npnnmnaer orpunarensune GAO"
na; 6) yönımaer.

2. Toctpowrs rpaquas yann y=sin( 3 —x) 1 madera saven x,
MDH Koropsx QYRKUMA NpUKUMACT nEMOUTEILUNE anane-

3. © nomomuo rpadmmcon (ym msnm, cromo Hope
meer ypanuenme cosx=Igx.

4. Hañiru mnoxectso anauenul yum y=sin? x+2cos2x.

5. Hecnenonars pynrayno y=4sin(2x- #)+1 m nocrponm ce
rpabux.

BE 74 I Vieropmueckan cnp:

Tpuronomerpuuecxue pyuxumm (momyunpume uassanne or
rpeu. trigonon — Tpeyromsums 11 metreo — wamepm0) rpaiot
GOMBIMYIO POJIB B Maremarııke 1 ee NPHNOMERHAX.

Hecnenonannen rpxronomerpurieckux PYHKUNH npakrimecku
sannmanmer eue pennerpeucexMe MATCMATLKU, HIYAAR DIAMMUOE
msnenenite BEANNHN 5 TCOMETPUN H ACTPOHOMUN. OTHOMENLA CTO
POM B UPAMOYTONLEIKIX TPEYrONLIMKAX, Ho CBOCÍL CYTH ABIMONME
ca TpurovomerpHycexustH HYHIOMAMA, PACCHATPHBANMEL ya D
IV— III 8. zo x. 9. » paGorax EBkamya, Apxmmega, Anonnonun
ApYPHX yuennix.

Vuenne O TPHTOHOMETPHUeCKMX BeMHIAHAX NOIVHAO PAIBH-
mue » VIII — IX ep. » cxpanax Cpenmero 1 Bnmxwero Bocroxa.
Tax, » IX ». » Bargage axt-Xopeamn cocramna nepmse TaGants
cmuyco». Anı-Byanprann 5 X 8. chopwyauponan reopemy emmycos
m € ee MOMOLNBIO mocrponn TAGHHNY cHHyCOB c unrepsanom 15, »
KOTOPOÑ smauennn CHMYCOD UPMBEACTI € TOMHOCTHIO 10 BOCHMOrO
Aecarmunoro smaxa. Axman-an-Bepynn » XI 8. smecro ¡exenta
Paguyca ma wacrH npn onpenenennn suauennk cnmyca u KOCHHy-
ca, enenannoro HO wero Ilronemeem, Haan MCHOMLIOHATL OPA
HocT» enummunoro paguyca. B nepsoit noronune XV 5. ans-Kaum
Coanan Tpuronomorpiucckne TAGMMuE € marom Y, Koropsie D no
exeayiomue 250 ner Osinm uenpensofinennkimm mo Toumoern. Ca-
MM KPYTHBIM enponelickum npencrannrenem TOR INOXH, sec:
MIM BKJAA B PRIBHTIE HCCACAOBAMNA TPHPOHOMETPIICCKHX DYHK-
mi, cuuraerca Permowonran.

B nauane XVII B. 8 PASBTHM Tpurosomerpun HAMETHAOCE Ho:
poe nanpannenne — auannruuecxoe. Beau xo 97010 yuenma o pu
TOHOMETPHYECKMX ¢YAKNMHAX CTPOMIICE HA TEOMETPIMECKOÏ OCHO:
pe, ro » XVII — XIX BB. TpuronoMeTpust ocrenenHO BONA B CO-
cran MATEMATHUCCKOTO AMAIA H CTANA MIHPOKO McTOMIORATHCA
» MexaHHKe u TexaMKe, ocoßenmo np paccmorpenum KoneGarens-
FIX Mpoueccos Ht MAKIX NepHonHHecKHX BENT.

42°) Tnasa 1
“Tpuronowarpmecnne PET

O csoïcrsax mepnonmunoeru tpuronomerpuyeckmx pyme
axan eme ©. Buer. Lseñuapexnü maremarux M. Bepuysam » cBo-
ux paGorax mauan MPHMCHATE CUMNONN TPITOMOMCTPHICCIHX
dymaunit, Onnaxo GaxaKyto K MPHNATON reneps CHMBONIKY BBCX
JL. Biinep u 1748 r. m epoeñ paGore «Bnenenne » anazma 6ecko-
sewwxs. B mei OR pacemorpen BOMPOC O IHAKAX BCEX TPHTORO-
merpiruecKux bymkuni moGoro aprymenta.

Tpurononerpuueexne dynkumm Dinep pacemarpusan Kak
OCOÓLE unena, HAIMBAR Hx OGM TepMIHOM mpancyendenmnue
roauveciea, nonyumouueca us Kpyra. Jia muunenenun pau
2kenTIX anauerm Sinx M COS ON MOAYUH Hx PAROKENNA D PA-

as

cose=1- 34% ay
= 0)

Moxno noxaoam, «ro rpabuxn hymkunl, oöpasonannnx paa-
HIM 4HCIOM unenop paja (1) mom (2), nOCTEHEBHO IPHÓMDEAIOTCA.
1 pad y y=cosx um y—sinx.

B XIX». nansnefiice pasante reopnn Tpnronomerpuuectenx
durs 6x0 MpoRomKeNo m PAGoTaX. Pyeckoro MaTeMATHTKA
H. H. Jlo6avescxoro (1792—1856), a raxxe » rpynax apyrux yue-
‘HBX, Hanpumep B paGorax upodeccopos MTY Jl. E. Mensmosa,
H.K.Bapm, A. H. Komworopona.

Fnasa Oi

Ilponssonsaa .
H ee TEOMETPHUECKH
CMbICH

Bee Goasuue 2manw pasoumua
xamexamuxu ocerda Guau cossanss

© wosdeuemouen mex uxu unex eudoe
npaxmuveexoù deamexsnoemu,

A. H. Tuxonoe

$ 1. Npenen nocnenosarenbhocru

1. Muexopre nocrexosarensoerir

Osparumen x momaruo “ncxoBoli mocnenona-
Tensnocrm, paccmotpenmomy 5 Kypce aureGpa
9—10 xnaccos

Eon Kaxgomy HatypazbHomy uneny m no
CTABNCHO » coornerernne nekoropoe neñcrpurem
Hoe WHCIO Xp, TO TOBOPAT, UTO SAAHA UUCAOGAR
nocaedosamenrnocm» (uu upocro mocnenona-
‘TemsHocts)

pj Mey Sey ee

Kparko moczeqosarensuocrs | oGosauaor
cummosiom {x,) mau (x,), MPH 9TOM x, HASKIBAIOT
waenox mam oaemenmox oro! MOcMeROBUTEALHO-
CTA, n — HOMOPOM “CHA x,

Unenonan nocneronaremnocr, — 70 hynk-
una, Oßnaerı, onpenenenus KOTOPOÏ ecrs, MO E-
erBo N ucex watypambunx uncen. Muoxecrso
anauenut oTOR DRE, 7. e. COnOKynmoCrL, HH
cen x,, RCN, HAAMBAIDT NnoMecmeon ananenul
nocaedosumenenocmu.

‘MuoxecTso anavexuii nocseqoaremeuocrst
moxer GT Kak KOHEUHMM, TAK It Üeckoneu-
HUM, 3 10 BPeMs KaK MHOKECTEO ee Anemenron
Reerxa ABIACTOA ÖCCKORCURKIM: MOÓME ana pas
HBX OneMenTa MOCACAOBATEABMOCTE. OTANUAIOTCR
Comm nomepama.

Hanpunep, xnoxecruo omawerndi nocnegoma-
reapnocra {(-1)"} cocrour us aByx uncen 1 u -1,
a NHOxCeTBA gHAXOHHi nocreonaTeRHOCTE!

"9 n (2) oecxoneum

fosos 1
TS]

Tiocnenonnmenunoers noer Dur, sauna © nono Qopxy
a, nodponmoutel DUNCAN Kankanı "Wen ROCNEXOMATEAMOET
no ero nomepy. Hanpunep, ect xp= CEE, ro santi ne-
ueruu unen nocnenosaremenocru pasen 0, a Kaokanit neral
‘tren panen 1.
orga noenexoarenunoers ananercn pexyppenmnod popsy
oi, noabonaioniel MAXOANT KA “NON ROCNORODAREANOCTH
to uaneerwowy npenwnyneny. pu raton cnocode aan nocae-
omarenvnoerit SO yRastsbaioT:
à) mepnu "eH HOCIOAODATETANOCTA x, (IH neckonno "xe:
wow, apr Xn 05
©) dopseyny, canon 1 UN © npexarypnn OIE,
Tax, apniNeritiecKan Hporpeccit € pasuocreto d 1 reoner-
puuccnas nporpocenn co sitawenarenen 90 amiorca cooruerer-
Benno Dexyppenenunen Pope
Sama +d,
Bamba.

Suan nepaisi «nen sa rof nparpecemm a, 1 bj, MONO 1
yum Qopuyay (11) TO uen coorwererayiomelt porpeccitt
aan

mirando,
aos bias NEN,

Pexyppentiiof QOPMYAON &,=%, 1 Hey a, NEN, >, u yono-
muaa x= Ly 23-1 aanaoren nocaedooamesonoeme Dubonareu.

B WeKoropsix enyuanx HOCASAOBANERMOCTA nomer Oh a
awa onncannen ce uxenow. Hanprimep, ecan x, — npocroe weno
€ nomepon n, 10 x,=2 %4=3, #5, X=, X= ID WT De

Ormernn, nakönen, tro nopnenonanenmocH» (x,) wonno 120:
Opa:

a) roman € noopaunaranın (ni x)» MEN, na naoenoern;

©) Towanıı x, MEN, na ancnovoll oct.

2. Onpozenemie npexcan nocnexonureamocra

Tloustue npesena nocnesosarensiocri Guo Bueneno B yue6-
vce anreßpu 1 navan marememiuconoro Asanmoa Ran 10 haces
(ezana TV, $ 1) u menomaonanoch npn miceneonenttn cyan been"
neuno yOumawueh reonerpnueckol nporpecenm.

Tipenapna'nsenenwe erpororo onpenesennn npenenm nocae-
AO TOM, PACEMOTPUM TIDNMEPA "HCTORLX nocneROBATeAL-
ocre es) ly, Fae

HaoGpasu wens orux mocnenonarensuocreh TouKaMH na
amexosoï mpamoit (puc. 30, 31).

gi 45
Tipenen mocnenOBaTEnBmOCH

ge à ru = 2 pais

Pue. 30 Puc. 31

Bamerum, «ro unensi nocnenonarensnocrn (x,) KAK Ob «cry
immoreme oxono towxm 1 (em. pue. 30), pacnonarascı npanee
roux 1 mpi sermax m m sence row 1 mpx meuermix m. C yRe-
auexuem n paccroaume OT TOUKH x, AO Touch 1 ymeisuineren
(erpemarrca x mymo). Tootomy meso 1 nasumarr Mpexenom no-
caexosarentnoers {x,) mpm n —oo u mmuyr lim x,—1.
Anaxormimo “ema nocsegonarennoctn (y,) e pocrom n
anpndamxarea+ x TouKe 0 (cm. pue. 31), m nosromy lim y,=0.
FI Bnegen onpexenenme npenena nocnenonarensmocrn,
Onpenenenne
Wneio a nasmmserca npederox nocredosamexonocmu (x,),
cemit ana Kamoro € >0 cymectayer raxok Komep N,, 170 ARA.
peex n>N, manonusercs uepauenerno |x,—a|<e. FI

Bean a — npexen HOCACAOBRTEMMMOCTA, TO nmyr Jim, x_=
mau x.—a pit n— 00. an

Tiocaegonarensuocrs, y Koropoli cyimectayer upenen, Hass
tor cxodaujeca, Hocnenonarenunoers, ne sBzmiomyiocn exons
welien, HAOMBAIOT pacxodawelica; HE roBopa, mocnexozaTens-
Noct® Hasninaion pacxopmınehen, CCI NIKAKOE ex Me ARRE
cA ee npenenon.

F=aBanerme, «ro comm x,=a nun noex neN (raxyio nocaenona-
Tembuocts nassiBaior emayuonapyod), 0 lim xn a.

Hs onpezenenua mpeyena nocnenogarenbnocrh cnenyer, uro
nocnenonareanuoers (x,) umeer npenen, pannul a, torra m Tom
Ko Torna, Kora mocneAoBaTemmOcT» (x,—a) mMeeT mpenen, pas
Ha yo.

Banaua 1. Nokasarı, “ro npenen nocnenonarenustocrn (x,)
panei

Dam

yank DEE 2220
va
8) x, ere VRFT, a=0:
ES min
D D Aoxamem, are EN xu=1. Tax Kax 2,
Bootie nponsnonumoe nero 52 0. Hepanencroo Ix,-11<e Gyrer
1

pumoanarses, cm À<e, 1. e. mpm n>1. BuGepew » xauecrne

N. xaxoe-nuSyms NaryPankuoe ‘oxo, yaoBereopmonyte VENOM

‚a-l.

46) rnona u
PORN W G8 TEGETBINGCRIT CNE

nod, manpnuep aneao ¥,-[1] +1, zae [4] — nena mer»
wenn E, 7. e. naxGonmee nene mcno, ne npenocxonmee À.
Tenia mn mex n>N, GT sunonnaracn mepasencrao
Ilm <a. io onpenezenmo npeneza zo canter, mo

ii amie e tia Bin,

2) Bocnonssyenen Tem, «ro

Sl m npu e>0
va
nepavenrerB0 oa <e pannochauıto Kaxnomy us Hepaseners À <?,

2 <e m mpi ncex

<e,

Myero No=| 5] +1, voran

1
R>N, nunonumoren nepanencraa. AL <e. Ho-
úl WN,

Ananoruuno mono xoKazam, wro lim 1-0, ecam a>0.

3) Yunoxus u pasnennp x, ma Vn+2+Vn+1, nonyaum
E ı
07 mp2 Va rl Var Bavaro

omyna [xn < Hopanenerno <e Gyner Buinonnnrucn,

an

com Rd, #6 pu n> Ap. Myer N, ae

1

rex n>N, numonnmoren nepanenerna [x,1< I < I <e, Bro
ae SAN
unser, «ro Jim, x,=0, 7. 0. lim (Vn+1-Vn+1)=0.
1
4) Tax nan y

Mi
mi

14

1 1
ï jo om Lx, 11e Ly slim 2,1. 4

Obparuuca eme pas x onpenenenme npexena. Cornacno onpe-
zeneuuno smexo a ABNAETCR npexenon nocmenoarensocTH (2),
ecam upu nex n>N, nstnonuseren nepanencrao |x, -a|<t, Koro-
Poe MONO aanncarı D DHRC A—C<x,<a+e.

Tipyrunar cnosamm, aaa xamoro E>0 mañigerca moMep Ny
Hamas © KOTOPOTO BCE WeHSI MOCNEAOBATEMBROCTA {x,} MPHHAA-
aexar uurepnany (ac; à +1). Dror unropnan naakınamor c-orpecm.
hccmoro mouxu a (pic. 32).

rast, weno @ — upenex — gonna
nocnenonareaunoorn (x,), comm < are
AA xampoh c-oxpecTHocTH daa

51 ar

Tipenen nocrenoeaTenenocT

roukH a HAÑÍAOTCA nomep, RAA € KOTOPOrO nee UNC! nOCxO>
AOBATEALHOCTE HPHHAATEKAT STO OKPECTHOCTH, TAK “TO Die TO
Oxpeeruoeru auGo NeT 14 OAMOFO ‘Wiest HOCICAOBATEABMOCTH, +
60 conepxwren mme koneunoe uncno wenoB.
3. Cnoïcrsa cxonsmuxca nocnenonaremsuoerei
Hepeuncnum ocmonnue cuoiicrsa cxonmmuxen nocaenon:
reasuocreit
Cuoñterno 1. Eenm nocnexoparenunocers {x,} meer npenen,
To ona orpannuena, 7. €. cyulecrByIor UHCIA Cy M Cy, TAKE,
To 6x, G cz aan mex NEN.

Bamewanne. Ma orpannuennoeru nocxexoarensnocii ne
caenyer ee cxonmmocts. Hampmwep, nocnenonarensmoen, ((- 1}
Orpannuena, Ho He annneren exonameiics.
Csoïerso 2. Ecam nocnegosarensnocru (xp), (Ya), {en
TAKOBL, NTO X,SYqS2q Ana veex RCN u lim x, lim 2,=0,

70 nocxexonarenmocra (y,) exomren u lim y,=a.

Hoxasaromcro cnolierva 2 ocuonano ma Tom, uro 5 moGoï
OKPECTHOCTN TOUKU A conepxarea BCE WAEHM NOCNCRORATENHMOCTN
(en) m (2), an MOKMIOUCIMOM, Gum, moxer, koneunoro “MENA.
Drum xe enoficrnom OGraqaer u nOCIENODATEILNOCT (Y), TAK war
Bee ee WieHBI SAKTIOUEHEL MEXAY cooTBeTCTEYIOMIMMH “LERMA
nocnenonarensnocreh (xa) u (2,)-

Sagaua 2, Myers ©, >-1 mpu seex neN x Jim a,=0. Hora:

aan, uro

a

> Hoxasen cnauana, «ro
1-10, 1<VI+0,<1 +10), neN, REN. a
B camom zene, cum 4,>0, ro
LT ra elta Lea),
ecam 15a, <0, 10
1> Vive, > (Vira) 1401-10),

orkyaa cxenyior nepanenerza (2). IIpumenan cnolierno 2, monyxs:
ex yruepmnenne (1). 4

Bamexanne. Ecan x,-Vara,, rae a>0, a+a,>0,

lim, 0,20, no x, Va: {14 m na (1) exeayer, uro

Va.

Jim Vara,

48 © tase ıı

TIPOMSGOIEN # 00 FEOMETPANECEN ERICH

Cuoñierzo 3. Kent x,>y, ana pcex n x lim x,
lim y,=0, 10 a>b. =a

4. Tipexen monoronxoi nocaenosatemsnocrn

Onp: 1
Tlocnesonaremnocts (x,) HassiBaeTca sospacmanwed, ec
xaxquili ee nocnenyrouli unen Compre upexsinyurero, 7. e.
DO Xp <Xp 1 AN COX N.
ent x,<2,,1 AIR Boex M, TO MOCICAOBATENEROCTA (x,) Ha-
auaior neyduearouell.

Onpenenenne 2
Mocaexomarenbnocro (x,) nassınaeren yOweaousel, ecan Kax-
ui npeamaymui ee “en Gombure mocneayloniero, T. €.
Xp x, 1 ann ucex n.

Bonn ,>%,.1 Ann Bcex 7, TO nocnenonarensmocrs (x,) mar

SBALOT nesospacmanıyei.

Bospacrmomue, meyOwwaiomne, yOusaroune u Hexospacta‘o-
ue NOCHEAODATONHOCT HADINDAJOT AONOMONNDLAL.

Teopema 1

Ecau nocaenonarenunoer» (x,} annneren nospactaiomeñ
Gun neyéssaomeñ) u orpariuema cRepxy, 7. e. x,<M
Ran ncex n, TO OMA umeer npenen.

Teopema2

Ecnu nocneponarensnocrs (x,) annaerca yOsınamıyeh (man
nevospacraoniel) u orpanstiena Hay, 7. €. 22m man
BceX n, TO oma HMECT npenen.

Aoxasarenserso reopem 1 1 2 oOMYO maerca B Kypce Biic-
mel maremarukh. In TeOPEMLI LUNPOKO HPHMELAOTCA B NATCHA-
Taxe $1, B uacrnoctn, E Teomerpnn. Pacemotpun KRaxpat ABCD,
enucaxtnutit » xpyr pazuyca R (pc. 33). Coeaunup orpeskanı Bep-
um DIOTO Knanpara © cepenunanı
avr AB, BC, CD u DA, nonyuun
pass B-yronsunk, Buican-
uk D Tor axe Kpyr.

Tiponoxxas ananormunne mo-
erpoenus, oSpaayes nocnegonnrenn-
oct. PARMA 2"-yromsitskon
(n>2), xamasılt 5 Koropsix Bmucan
8 Tor we Kpyr m monyuen ua upe-
Auıayınero yanoennem uncna ero CTO"
por.

Tocxenovarenpnocr® nnomaneh
STI MPARIITBIIHK. MROTOYTOMBRIROD
Aunnercn Bonpacramıeh, TAK Kax

g1 49

Tipenen nocneRoBarenonocty

KAKA MOCHEAVIONHIE MHOTOYTONLHHK COXCPHT MPOABIAY ni,
Kpome Toro, 972 MOCACAOBATEABNOCTE orpanmuena CBEPXY: HO
MAAK KaDKROTO MO DPI NIOTOYTOADMMEOD MeNDIN HLAOMAJA DA
para, ommcanmoro oXoxo kpyra paxuyca R. Tio reopene 1 yannaa
noexexosaremeuoers umeer npexez. Dror npexen panen xh.
5. Yucao e

Paceorpum mocnenonareaunoers {x,), me x,=(1+ 7
Moo nokasarh, ¥ro (x,) — nospactaromas 1 orpamiennos ene
xy mocnenonarenbuocrn. Ho reopeme 1 oua meer npenen, KOTO-
pul oGosnawaeren e, 7. e.

Jim (144) =e
“nexo € urpaet BaXKHYIO PON B MATeMATHKE 11 60 HPHIOKE-
uusx. Oxo aunnerca uppannouanHsin, HPIN
2,718281828459045.

6. Burunenenne mpexerou nocaexowaremuocreit

Tipm nuranenenum npexeron nocneronarentnoereh enomay-
ores onpexenenne mpexena, cnolicrao 2 (1. 3), Teopema o npexe-
Je MONOTONNON NOCHENOBATENLHOCTN, A TANK TeOpema 3, CHU
HA € APupMETIMECKAMA ACÍCTDHAMM NA HOCACAOBATOALHOCTAMH:
KM CHOPMYAHPOBAHHAN HIKE.

Teopema 3

Hiyer» Jim x,=a, lim y,
1) Mm Gat yada + bi
2) lima (x,y_)= ab, 8 YacTHocTH ec y,=C ann Bcex n, TO
Jim (Cx,)=Clim x,=Ca, 7. €. HOCTORMEMÁ MEOMHTOAL
NOKHO BEINeCTH aa OMAK npenexa;

a

Mpx yCnOBHK, uro y,#0 ana acex nm b#0.

Sanaua 3. Hañru npenen nocnenoparemuocra {x,}, ecam:

AA _ Vent-n+s

8) x,

Veran

uucamrena panen 2, a npenen onamenarenn pasen 3. Tosromy 10
2

reopene 3 nonyunm lim x,

5059 tase 1

TIPOMIGOIEN H 60 FOOWETBIHGERAA CHIC

2) Tax xx

aaa oe LS
VEA VB A) 1 tg

1,3
re = gu + giz OMPI no, TO, Menomsays peoyasrar saga 2,

roses lim 2,= im VE

3) Homsayach rem, ro

ae mama panier sa, mann ig 5, 2. E

WA Banava 4. Haiirn npenen nocaexosarenvnocra (x,), ecam:

D see VAT VERRE 0) w=
D 1) Ywmoxmn m pasnenue x, na pmpaxenme V2n*+4n+5+
+Y2n?-2n+3, Koropoe HASEBAIOT CONPAMEHHEM © X,, NONVINM

q PENAS (atan) _ ón12

Von? +an+5+Vant-2n+8 ler yi-

‘ 02
eres ee

ormyna cxenyer (saxaua 2 m reopema 3), 70
de]

2) Taxe wane

uz
ers acd @
2

2751, ro npu seex n sunomnaeren HEPABENCTDO x, 1 <q) T. €.

x.) — wenospacraioujan nocnexonarensuoers. Kpome roro, x,>0

"pu ncex 1, +. e. nocneponaremmnoorn (x,) Orpanuvena cHuoy-
Jo reopene 2 sra uoeseaonarensuocr’ mueer npeaen. OGosma-

vane ero a. Yroönı Haller meno a, nepelinem x upenexy a panen-

eme (1), yaumasman, wo Jim „2,

uy 3, nonyunem a=0-a, 7. e. a=0.
Hrax, lim x,=0. ISA

0. Orciona, uonomaya reope-

41 MI

Tipenen nocnenaBarensnöcH

Ynpaxnenna

Ip

52

HaoGpaaurs na uncnonok mpamoh mecxonvxo “xenon nocne-
AOBATOMBROCTH {x,} U BEIACHITE, K KAKOMY HHCNY OHH Npu-
Garoxemores:

Dar

2) x, CD;

Hexonn us onpenenennn npenena NOCNEROBATENSHOCTH, HOKA-
aars, «ro lim x,=

Ta 1,
Das Da
DER u.

Haürtu npenen nocnenosarensuoctu {x,}, ecm:
Du yet

ut 4x

‘ro MOCHEROBATENBHOCTE (x,) ABANETCR orpannuen-

an n+2
D amsn Es 2) x, Se

al at
E ed

Hafen Jim x, (5-6).

+2 n+2
Da Du

1 Po,
dat [zen
à po Bett,
DEE #2-n.

Hr; Ya
Date 2) ee aa

Amine ”
3) 2 Vi + An +1 VERTER; 4) x,=Vn enn.
Haft lim 414 4 1, rue {a} — apupmeruue-

Aa Gag * agas ya
cKas nporpeccus, Bce WleHEI u PasHocTs d Koropolt oranuası
or nyna.

Zo rnoco u
IPONSSOWaR u 66 TEONETPIVOCRAA CNE

$ 2. Npenen pynkuuu
EL Onpeneneune npegexa dymxuum
Baxnyio pom» MaTewaTHKe Mrpaer nonaTKe npenena,
CIRIANOS © HODOACIMON DYHKLM D OKPOCTHOCTE AAHHO! TOI,
T.e. Ha HOKOTOPON MitTepnane, COnepKAINEM PTY TOUR.
Tipexmapas onpenenenne mpexena dbyHKUMK, paceorpHM ane

a.
1
» okpecrnoeri

Banana 1. Vocnenonars dynsumo 11x) E=]
roux x=1.
D dra Oymenun onpenongun mp #71, m fGA)mx+1 mu #21.
Tpubuscose Gym y= IT (pue. 34) annaeren npaman y=x+1
€ smixonorofte rouxoï (1; 2). Ha pucynKa Bmano, “ro ecam ona
sema x Gama x 1, TO coorserersymune snauenns yaa
Gnsxu x 2, neo 2 Haspaiot npexenom dymKuun f(x) 1 rouxe
xel (px x, erpensmenca x 1) m ummyr
Jim f(x) =2 man f@)--2 mpm #1. 4
Hccrenowars @yakrutio
#41 mpi x>0,
f(x)=} 0 npu x=0,
x+1 mp x<0

Banana

» okpeernoern TOMEN x
D ovueaus f(x) omperexena mpn ncex x CR, ce rpadix moobpa-
eu un pueyuxe 85. Ha pitcynka BAKO, uro DPI 2HAUCEUAX x,
moran K x=0, anauenns Gym Ganarı x 1. Bovom cxysao
lim (x)= 4

7 aumeuauue. B saxave 1 oyaxuna f(x) ue onpenenena
a rowe x1, a lim/(x)=2. B sanaue 2 yaaa f(x) onpenene-

ma B Touxe x=0, lim f(=)~1, a snauenne dyrsuum B rouxe x= 0

pennneren O.
u x A

24

Jy- 2. +1

|
ae

= i =

Puc, 94 Puc. 35

se Biss

Tipenen Gp

B annase 1 aGcomornyw vemranny paanocru f(x)~2 momo
onenars CKOXb yronHO MAJOR (membmie MHOGOTO MoORHTETEHOTO
«CHA E), ecam amauenna aprymenra x ÓYAyT AOCTATOUMO Gays
x rouke x=1, mpuyem c ymembmennox nenas |/(x)- 21 Gyner
YMEHLUIATKCS OKPECTHOCTE TOUKH A, B KOTOPOÏ Bnonnnercn wer
pasenerso |f(x)—2|<c (cama rouxa x=1 ma pacemorpenma nero
saercs). IIpHaanum STOMY YTBEPXAEHHIO Tommy OPMYAHPOBKY.

Myers sanano “meno €>0. Hañnem raxoe uncno 8>0, 7061
aan ncox x, Taxnx, wro [x-11<5, x%1, nuumonnnnocs nepanencrao
if(x)-2\<e. Hnaue rosopa, nalizem raxoe 6>0, roGW AMA Bcex
x, ynonnernopmonnx yenommo 0 <|x-1|<8, coornercrayione
TOK rpapuKa Oyaxuma y=f(x) mean 8 ropmsorrambnod
nonoce, orpamiuenmol npamm y=2-€ u y=24+e (cm. pue. 34).
B xavecrne 5 n aanaue 1 MOXHO Baath “CIO due.

OGparume x sanane 2, » xoropoh lim f(x)
Tiponexem mpawue y= 1-6 m y= 146. Tora » ropmaonramımoit mo-
10ce, orpanmenoÏ on MPAMBINI, HEAT BCE TOTKH TPAQIKa
Qynrusm y= f(z), ecam O<|x|<5, rae » xauecrne 3 nn6pano man
Menbuice ma uncen em VE (em. pue. 35).

1. Tlyere e>0.

Onpenenenue
neo A massınaeren npedexom ynnyuu f(x) a move a (npu
x, empemamenca x a), ecnu ana moboro e>0 mahneren uuc-
10 5>0, TAKOC, ro A Beex x, YHOBETHOPAOMUX VENU
O<|x-al<8, munonnaeren nepanenerno |f(x)—Al<e.

Muoxectso rowex x, Taxux, wro |x-al<ö, +#a, um
O<|x-al<8, naauınaor npokoaomoli Soxpecmuocmbo mouxu a.

Bamermw, “To wero à, BOOGE ropopa, aanmcnT or € (saz
un 1, 2). Ecan A — npenen yenes f(x) » roue a, To umuyr
lim fa) A nan FA np xa.

Sanaua 3, [oxasars, uro dyuruna /(x)-(x-8) +5 nueer »
roure x=3 mpenen, parait 5.
> Tyers sanamo uncno &>0. Iloxaxem, «ro mañuerca meno 5>0,
‘raKoe, 70 AJ BCEX x, YAOBErSOpAONEX ycnomo |x-3]<d, mu:
monnaeren nepanencrno |f(x)—5|<e. Tax ax If(x)-51-(x- 3)", 10
nepanenerno |f(x)-5|<e pannoeummno nepanenerny |x-31- Ve.
Boabmen = Ve, torna ua nepaneneran [x-3|<6 enenyer nepanen-
erso |f(x)-5I<c. Ho onpenenenmio mpexexa oro ooxauaer, wo
lim f@)=5. <=

15212. Pasamunbe tums npenenop
Odnocmoponnue xoneunwe npedeno
x? npn xl,

dass Meson gmc Jo 2 251

3 oxpecrnoern roman z=1.

SH) rnasa u
IPOSSOMVEA FES TEOMETBINEERAN CHE

D Tpagı dymxnun y= f(x) m306pa- Y
ex na pucyae 36. Bano, “rro ecam

seems x Onnaxn x 1, Ho membre

1, 10 auavenmst Pyme f(x) Gomar

x 15 cam x sHavenna x Oman K

1, no Gone 1, vo anauenna dynx-

wot (x) Ganar x 2. B ro cnyaae y= 2"
ropopar, To yen f(x) meer » 1
Tome = 1 npenen caera, pan 1,

x npenen enpana, paroi 2, 1 my?

» lim f(&)= 2. Dri npexenut t=
resumo oAnocropoua. A Pue. 96

Onpenenenne 1
Yueno A, nannaeren npedexon caeoa dynxyuu f(x) 6 mow
Ke a, eemt Axa noGoro £>0 eyixecrayer uneno 5>0, TaKoe,
WO ans ncex x, yAORAETDOPMOINNK VENOBMO A-Ó<X<A, BI
nomineten mepanenerno |f(x)—Ail<e.

Baron cayuae unmyr lim f(x)=A, un lim f(x) An

onp mue 2
Wucio A, nassınaerca npedezox enpaoa gynyuu f(x) 6 mou:
ke a, ecam naa mooro >0 eymecrayer uncno 8>0, Taxoe,
WTO AA BCEX +, VROBAETBOPMOLLNK YCHOBMO a <X < a+, BB
noxiserea nepanenerno |f(x)~Ag|<e.

Borox enyuae my lim f(x) A9 ram lim f(2)=A,.

Ma onpererenuit npenena y f(x) m rouxe a x onno-
cropomnx mpeenon enenyer, ro (ymenua f(x) mueer n Tune a
apexex, past A, TOTAa M TOXBKO TOFR, Koraa cyınecrayıor npe-
Jexs zroli (byHKUMH cnewa u cnpana m 9TH IPeAenBt COBNANMOT
(y=4y=A).

Becxonewnmil npeden 6
‚xoneunol mouKe

Banaua 5. Hecnenovars dbynxuii0
{0-3 » oxpecruoern rom 2=0.

Doyen y=L, redire xoropoit
moßpamen ma peynxe 37, onpene-
ena np x70. Tip npnémuexente
K moe x0 exena m enpana anaue-
‘Hus oro «pymumm no aGcomorHol
remume Meorpainenio Boopacra-

52 ss

Tipenen Pp

Puc, 38
tor, upuvem aux y comazaer co
gunxom x.

B rom cxyuae ronopar, “TO
oyaxuna f()=] umeer » row
Ke x=0 Geckoxeumuit npenen (anna:
eren Gecxomeuno Gonbmioh), m muıyr
lim f@)=o. Eon x<0 u x erpe-
MATCA K Hym0, TO |f(x)| HeorpanH-
senno nospacraer u f(x)<0. B arom
cause mmmyr limf(z)=—co nan

co x TOBOPAT, uro QYHKIMA f(x) uMeer mpenen cera

lim, f (x)
3 Touxe 2-0, panmusit uunye GecroneunocTa. Een x>0 m x erpe-
parres x nymo, vo ymca f(2)= À ueber npenen empana a or-
Ke x=0, pasnutit mmoc ÓeckoneumocTH:

lim f@)=+00 mu im f(æ)= +00.
B o6mem cayuae same lim /()

co oamawaer, ro ax mo

Goro ©>0 eymeernyer wmeno 30, raxoe, sro ana neex x, ynon-
nernopmougx yexomwo O<|x—al-5, munaanseren nepanenerno

Ie. (N)

Ecru mepanenerno (1) moxno 3amekurs nepanenernom f(x)>c,

to lim f(z)=+00. Hanpumep, ecan [= 45, To lim f(x)=+0°

(pue. 38). 7
"Ananorurıno pacemarpupaiores aanmen una lim f(x

lim/(2)=-+00 m 7. a. Hanpmaep, ec /(2)=logsx, To lim s(x)

o (pue. 39), a een 2) =Iog x, vo lim (2) + 00 (pn. 10). AL
me
56 Tnana Il

Tipovaboanan FES TESTER CARE

ated

Puc. 41 Puc. 42

Pl llpanyıo x=0 (oc» Oy) na0MBALOT sepmunanonol acumnmomod
rpadukos hymenmit

yo, un da ym logas, y log x (om. pue. 97-40).
3

Hpeden 0 6ecxoneunoemu

Paccmorpnm qyutumo f(x)=1+ 4, rpabux xoropoi nsoGpa-
eu na pmeynxo 41. Tipu Gonvumx no aGcomoroï neue aua-
semuax x omauenna orof Gym Gare x 1.

Tlooromy roBopar, “ro cytecreyer npezen py f(x) apa
+, empexaujenca 1 Secronewnocmu, niyr tim f(x)=1-

Tipanylo y=1 massınaior zopuzonmaatnoit'acuxnmomol rpa-
Aa Gym y= 144 pu 2.

Ananormıno npanas y=0 — acmwntora rpagmon yet
ya] m ya lg mp x00 (em. pue. 87, 38).

B o6mem enynae aammer lim f(x)—A oananner, «ro ana mo

foro £>0 cymecrayer «nexo 5>0, Taxoe, wro AA ncex +, yaon-
ernopmommx yononmmo |x|>8, BuIKOmESeTes nepanenerno
If @)-Al<e. e
Renn nepanenerno (2) enpanen-
suo npu sex x>8, TO nmuyr Y
„Im f@)=4,

a een mp ncex x>-8, 10 numyr a

lim f(x)=A. ve
Hanpanep, ecm f(2)=L (pue. 42),
o lim f(2)=0, a com f(x), 5 =

10 lim f(x)=0 (pue. 43). =

Pac. 43
52 Mhz

Tipenen ymin

15 3. Becnonermo masse dyna

Dyno a(x) Kaaumaor Gecxoneuno manol npu x—a, cena
lima(x)=0.

Beckonenno ase @ymxunn ofmanaior cnenyiomun enolicrmon:

Bean a(x) u B(x) — Gecroneuno manııe bymcımn upu xa,

A Cı M Cy — HeKOTOpHIE MOCTORHEBIE, TO Cya(x)+ CoP (x) TAKE

spsiserex Geckonenno manoli hyuxunei mpm xa.

B sacrnocrs, cymma m pasnocTs Geckonexno warm ynK-

unit — Gecronemmo mamsie dymkmutx. Kpoxe roro, nponanene-

nue a(x)P(x) Geckoneuno maxux hymsunk a(2) u B(x) rar
2Ke annnercn Gecroneuno Nano pymnneñ.

Hanpumep, (ymrapn 8x, dx? smnmoren eckomeuno manta
np x->0. Em x->0 m x>0, ro Vx>0, r.e. Vx — Geckoneuno
manag Gymeuux mp x—+0. Auanormmo 0 mpu +0,

1 a

me. À — Becnonenno mamas pyukuua upu à +00.
Y
Beau x00, 70 40, 30, 7. 0. 4, — Geoxonomo

mame pyme mp x—>00,

Bamexanue. Ma onpexexenm npenena dyuuuur u Geco-
Keno MANOR DYAKIU CXCAYOT, ITO "MCAO A ABARETCA npenenos
yaxumm f(x) np xa tora u romuko Toraa, komme f(2)=
=Ata(x), me a(x)>0 mpm xa, 1. e. a(x) — Gecxonenno war
nas QyaKuns pu x—-a.

4. Cnoitersa mpexexos dy
Cnoñterao 1. Beam lim /(x)=A, lim g(x)=B, ro

Jim (f(2) +42) = A+B,

Tim (EG) AB,
160 _ A
lim LG) — À pm yenonumn, ao BO.

Croïcrno 2. Ben » xexoropoñ mpoxororolt oxpecro-
crm Tom a Cnpanexamst nepanenerma /(2)<q(x)<£(x) u
im, Mx)=A, lim g(x)=A, To limo(x) cymecrayer m pañen A.
Tan noxagutemersa cmoÑicra 1 Momo nocnonnaonarıca

CBOHCTBAMM ÖCCKOHCHHO MAALIX PYHKUNË H sameuannem m. 3; &

ceolierna 1 u 2 caeayıor Ma onpenenienm npenena dyazınn.

Banana 6, Tivers 0>0 u lima(x)

. Hoxasars, ro
lim V5 +a = V8.

> Oboanaunm p(x)=V5F0()-Vb. Hyxro noKazaTE, wro 9(x) — Gee:
Konouno Maman Dymkuun mp xa, 7. e. lim@(x)=0. Vamoxas

S80) thas 1

TIPOHIGOANE W 89 TOOMOTPHICKHA CHER

ace)
CITÉ
mie b+a(x)>0 5 nexoropoñ oxpecrmocru roux a, TAK Kar

lima()=0 u b>0. Tora \o@l- LOL u, ogro

m Wera vb
xy lime (2)=0. 4

M paagemm o(x) ma VoFa(2)+V0, nonyamı 9 (x)=

Sanaa 7. Busunenurs:

D 1) Tax xax Cx", re C — nocroamaaa, REN, apnaerca Gecko-

xeuto Manoit Qyakuuel npn x—0, TO npenen unennrenn pa-

sex 3, a npexen oxawenarena panen 1. Tlooromy mexomtaih npexez
panen 3 (ceoiierso 1).

2) Yımoxmm anemrem 1 anamerarem Apoón na VI=14+1,

1 x2 1
CIA VA
+@-D—1 npu x—2 (sanaua 6). Henonssya csolicrso 1

nonyumae N

npu x72,

ES

"pere, naxontan, VO MEKONIÍ pezon pasen 4.

3) Pasnenme ancautens n auamenaren» npoön Ka x*, sam

3-44
men ee 8 une — =. Tax ak Zo npu x 00 (C — nocro-

pres
ses, REN), To nokonsıh mpenen parer 3.
4) Nocnenosarensso npeoGpasyem aannyio Ayo (> 0):

(oagaua 6), uro uexomutit npenex pasen 1. A En

Ynpaxnenna
& Moerporrs rpadux hymkıum y= f(x) u nalen lim f(x), oem:
: E
D 1@)= 5, 0-0: 2) 1@)-=23,

an f= 2%

3 19-1 Be,

EE] Hañra npexem enena m enpana hyneumm f(x) » rouxe a,
E 1-x* up x<0,
a aoe,

|x|-1 mpm x<-1,
VERE mpu x>-1,
2-21] mpu x<-1,
ono aa
10, Henomssys onperesenne npenena, xoKasers, WTO:
1) Him (x? 4x +6)=2;
2) lim((x—1)'+8)=3.

14, Hairs: nepraxannmue aemmnronix rpadmien doyen:

3) (= (a

1 Pin
» Der Dr
ea Me

12. Hairs ropnsonransayo or rpaduka dynKunn:
D 19-22; 2) 1,

(13) Bsrunenure:

D tim Bt
cera
3) lim +47

or
5 lim Wet,

8 3. HenpepbimHocTb bynkunn

1. Honsrue nenpepsnocr

Osparumes x Tpaduky PyHKuNN, HIOPEKEREOMY HA PHCYE-
xe 36 ($ 2, aagasa 4). On cocrom na xoyx exyexone: y=x?, xl,
x y=x+1, x>1. Kangul mo aux moxer Ge Rapiconax nenpe-
PHBH ABIOKEHKEM sapamgema Ges orpua or Omar. OnHaKo
DTH ekyexs Ne Coca HCNPOPxIBIO: u TOUKE x=1 uponexo-
ANT cKauxooBpasnoe mamenenme OYAKNUN.

Hoprouy nce anauenna x, Kpome x=1, nasıımaor movKanu
nenpepuonoemu dynxnnn y=f(x), a tOUKyY x=1 — mouro pas
posea stolt yum.

Anaxormuno QyMKuus, rpabwx Koropoi na0öpuen na pu-
cyuxe 34 ($ 2, sanasa 1), menpepupna 5 kamaol TowKe x, KPONE
roux 1, xora Qyskuna uMeer mpenen B roh Touxe, pair 2.
@ynxıunk ne nunnercn nenpopuummol pa <= 1, var war ona ue
ompenenena n rouKe

Oyusuna y-l ($2, sonana 5, pue. 37) nenpepusua apa
x<0 m mpn x>0, Toma x=0 — rouka paspatea vof yum.

60 rnasa ni

Tipowssonnan v Ge TeOMeTpINEGRIN CE

FI Jaan onpexenenne HENPEPLIBROCTR.
Onpenenenue 1.

‘Pymeuin [(2) naaunaeron Hempepuonol 6 mouxe a, ect
Lim FC) = fa).

Hanpumep, @yawusa /(x)=(x-3)?+2 ($ 2, sagaua 3) menpe-
passa a roue +3, tax war lim FC) = /(8)= 2.

Dymenux f(x) nenpepsinnä » rouxe a, ec munonmens ee
avoue yenosna:

a) Oynma f(x) onpenenena B #ekoropoñ oKpectHoeTH Tow
wu a (nkmouan rouky a);

6) cymectayer lim f(x)=As

») A= f(a).

Bonn xora Gb oRKo us #rux yenosnk He munonnneren, 70
TOBOPAT, 40 a — Touka paspsıma hyrkumm f(z).

Banaya 1. Buincnurs, annneren am pynennn

248 apy ze,
tes | #42 nenpepuipuofi B roue x=
10 up x=-2

D Rea x#-2, 10 f(x)=x*-2x+4, m nomony lim, f(x)=12. Tax
xax f(-2)=10, 70 lim, f(x) + F(-2). Gnenonarenkno, ya f(z)
ne anınercn nenpepsinnoi Browse 12, 1.0. x=2 — toma
propina byeee f(x). À ET

IP Banerum, vro ynxnsa f(x) wenpepsinna B TouKe a Torna m

roauxo Tora, sorna lim f(2)-lim f(x) /(a).
Benn lmf()=/(a), v0 dyno f(x) masmmaor Kenpeprs

ol cacea 5 More a, ecnu me limf(x)=f(a), ro bymumo massı-

aor nenpepuonol: enpaca 0 möuxe a. Orciona caenyer, uno
dy f() nenpepumna » route a TOPJA m TORERO Torma, korna
ona nonpepLInn Kak exena, Tax m enpana D TO¥Ke a.

Oboamamms x-a=h u kazonem À npupawenuen apzyxenma.
Pasnoern f(2)-f(a)=f(a+h)-f(a) masonem npupawenuen qynr
uuu oGosnauum Af. B orux obosmavenmax pasencrso lim f(x)
=/(a) mpuner sux lim Af=0.

Tax o6pavoM, ueupepspuoer hyukunm f(x) » rouke a
vauaaer, WTO Beckonesmo MAJOMY HPHPALEHMO aprymenta coor-
sereroyer Geckomeuno manoe npnpautense OynxunM, 7. e. Af--0
pu 40. NA
I 3anava 2. Dyneunn f(x).

x

He onpegexena » Touxe

En
. Onpexenurs ee 8 touKe x=1 Tax, ITOÓN nonyupuranea
Qyuiuns Guia Menpepunna np x—1.

$3 © 61

Henpepumnocne Py

> B aagane 1 ua $ 2 Gao noxasano, uro lim (x) eyıeerayer u
pasen 2. Bonn nonoxurn [()=2, 1.0. pactwonperk dy
fc) [160 mpm ze,
1 fs npu x=1,
x=1. B raxom exyane vonopar, sro dynxuus (x) aoonpenenens
no nenpepkinuocru 2 rouxe a. d ET
ISEB sanaue 4, § 2, Guia pacemorpena qua
fee np xt,
ol men.

10 91a hynkuun KenpepuiBHa » touKe

Ona me apnserca nenpepsmuolt » rouxe x=1, uo lim/(x)=/(1)=1.

Tloyromy ganna dyuxuna nenpepusua cueva y Towxe 1, no ne
abnserca ueupepuinuoki cupana » sro roue: lim f(x)=2. FA

Ormerum, “TO Be OCHOBNLIE onemenrapunie dymxnut (cremen-
man, noxasaren,nan, norapmamuueckan, TPuronOmerpnuecKue)
HEnpepsiBnBI B Kamolt TONKe cnoefi oßnaerm onpexenerus. B ua
ernocru, moBoli mmorounen — dyukuna, nenpepsiBnas B Karol
rouxe x€R; paunonanınan DYHKUNA (OTHOLENHE NuorowreHOB) —
ymEnHA, MOMPOPEIBILAS DO COX TOUKAX, TAO sHaNeHaTeM TOM
Paunonansnoli Qynkumm He paen HVA0.

Tonpenenenne 2
Dymo f(x) maosınaror uenpepuonol na unmepsase (a; b),

eca_o#a MENPEPLIBHA B KAXAOÏ TOUKE x NS 37070 HHTEpEA
sa. E

MA Onpenenenne 3

Ecan byurus y=f(x) nenpeprinma na mmrepnane (a; b), a
Taxe menpepnipna cmpasa B rouxe a 1 cepa B rouxe b, 10
eo naanınaor menpepuenoi na ompesre |a; b].
Ban Haiiru unexa b u c, TAKE, npm KoTOpEIX dbynKnHA
x? npu x<2,
f (x)=) 6 npn x=2, HENPEPLIBHA B TOUKE x=
xte npu x>2
> Ta axe lim f(x)=4, lim f(2)=2+¢, Ad, ro menpepunoers

» roune 2 Oyner TOmxO npu yenonmnx 4—2+¢ u 2+c=b, koro:
pute nsinoanmoren npır b=4, €
2. Cnolerna yuan, menpepsnmu Ha orpeare
Teopema 1

Ecau @ynuna f(x) menpepspxe ma orpeoxe [a; b], ro ona
MIpMMMaer ma 970M OTpeore cnoe maubonsmee u CBOE man

620 rasa u

TIpowasonHän Y 88 FOONETBHNGEKHN EMLICH

Menbmee sHavenua, T. ©. cymecrsyior
ela; b], Taxue, uro Ann ncex x el
panenoran f(x)>f (x1). f(z)<f(z2)-

Teopema 2 (0 npomeyTounbix sHaYeHHAX)

Ecau Qynxnua f(x) nenpepsmma ua orpeske [a; b] x
f(a)# f(b), ro oua npunumaer ma srox orpesxe moGoe ana-
sense C, saxmovennoe meray f(a) u /(b). 1. e. cymecr-
ayer TOWKA Xo, Taxas, TO a<xy<b u f(xo=

B sacrnocru, ecam dynxuus f(x)
senpepiimna na orpeske [a; 6] m mps
Inmaer HR KoHUAX 9TOTO OTPOSKA
auuauettin pau anakob, 7. €.

F(a): F(0)<0,
o eymecrayer rouka ec(a; D), 1a-
xan, «ro [(c)=0 (puc. 44).

Sanaua 4. Hoxanarı, wro ypanne-

mue 4x7 +30 umeer Koper» na Aa
orpeaxe [=1; 0].
D Oynnnns /()=x"-Ax*48 nenpepunna na orpeaxe [-1; 0],
{(-1)=-2, f(0)-3. Ho reopeme 2 na unrepsane (-1; 0) cymect-
ayer TouKa c, rakax, uro /(c)=0, 7. €. Aannoe ypantienue umeer
Kopexb na orpesre [-1; 0]. €

Teo! 3 (06 o6parnoï cbynkunn)

Bean dymxnus f(x) nenpepsinua x nospacraer ua orpesxe
la; b], ro xa orpesxe [f(a); f(6)] onpenenena o6parnan x
f(x) $yuxuna, Koropan ABRCTCA HEMPePHBROR u Boapac-
Taoıneh.

Hanpumep, dyurana yasinz, me 6x6 E, nenpepsmma m

ospucraer. OGparnan dyasımn y=arcsinx omperenena HA orpesKe
EL 1], aBnneren nenpepsipnoli u Bospacramen.

Ynpaxnenns
14. Ipmmannemur au rpadiney dbymxnm y= f(x) Towa A, scan:

1) y=2"*?, AQ Ds 2) y=te(Z+4), a(S; 0)

Sy, ACVB:
E

sa He

Tlenpepaenoere Gyr

Y

6) 2)
Puc. 45
15. Dyaxuna y=/(x) sanana rpaduxom (pue. 45). Hañrit 06-
Jlacte onpegenenun u mnoxecrao anauenmi yr.

16. Halira oßnacr» onpenenenun 1 muoxecruo eHauentii Qyux-
um:

A E

out
17. Tloctpoure rpapux ysunn:

x-2 npn x43, 1-2? upn x#2,
Du- D y=
D Li apa x=3; Die {3 npu x=2;

2-2 opm x<3, 2
» elos mur 9 el: npu x>2.

122 npn x<2,

64 thane u

Tpowaso;nan NT 66 FEOMTpMEORNT Each

18, Ha pueyare 45 (a — 2) naoöpamensı rpaduxn pyssundi. Mas
sannoh us max Gym meros
1) Kakne ua (byRKnMit ABARIOTCA Henpepae HA CBOE 06-
ara onpenenenusn;
2) xaxne TOUKH ADASIOTEA ToOUKANH paspHina KA OTpeaKe
2 2%
3) Kuxue na yet seamorest nenpepummnn na unmepun-
se (-2 1).

19. Tloerpours rpaques yann:

2" mpu x<2,

110 CE e
logex npu <2,

x2 4x48 mp x>25
Ix-21 mpn x<2,
|x-4) mpu x>23

2) ve

1

u npu +32,
3 10)-{ 4) f(z) =} 3-1"

xo npn x<2,

Bunemrs:
a) umeer am ora PyHKUUA npenen npu x-+2; 6) apraerca mr
ora Qyakiia menpopmenol ua noeh Aucnosoß upano; 2) ma
Kat MpoweneyrKax DH nempopnimna.

20, Bunonurs, annaeron AM nonpopuimnoit » TOXKE x9 dy:

D pt, sp

rara
TARA pa 2,
2) |

+2
8 npu x=-2,

+4 mpn x<2,
x+6 npu x>2,
sing up x<z,
6+|x-a] mpm x>n,

8) M@)- {

Don Kon.
EL] Norasarı, wo pynenus f(x) nenpepsiana » rouke a, ec

1-2? npu x<2,
(ee pu 232,
Icosx| npn x <n,
@- mil npn o,

Do

2) f(z)~ {

32) Hañru uneno b, urobm ymeun f(z) Gaza mempepmmna »
ore a, ecm:
loge(x +1) mpu x,
D 169=|2, mpu x>1, Bats

gs 65

Honpepumnocne Py

N ro [2 e _,
be upm xx,
Lex mp al,
3) sois ee aia
b npr x=-1,
‘cos np x<0,
b(x-1) npu x>0,

4) 1-{

§ 4. Onpenenenue npoussonHoä

Tivers marepnansras TowKAa aummeren Bom ocn Os, rae O
(Hauano orcuera) onpenenser nonomenne MaTepHalbHOË TOUKH B
owen npewenit 0.

enn B MOMENT pee £ Koopnunara ADIKYINENCA Touc pas
Ma S(1), 10 TOBOPAT, uro PyHKIMA s(t) sanaer saxon aunenun.

Tiers pecumarpasacese. Amonenno ES sientes panmomep
mur, toma an PRIE MPONONYTAN BPEMEIIE NETEPHATERAR TOS
Xa Moor Coneplzar» nepemenenna, PER man 10 BEN, var
o e.

Cpeanam ekopocTs ADEME sa IDOMERYTON Bpexen or 1
20 14h ompenenneren A 4-20 ,

Onpeaenenne

Cropoemwo mouku 6 noxenm t (menosennol exopocmen)
HasMBalOT NPeREN, K KOTOPOMY CTPEMTER CPOAMAR cKopocTs,
Korma h--0, 7. e. exopoct v(t) m Moment £ onponenneres
panenctaom v(t) lim SCD |

Taxi 06pason, cKopocTs » Monet Dpewenn t — npeen or
Houenns upupauenus Koopauuanat xBioKyuIelics ro 9a mpor
mexcyrox ppemenn or # xo #4 h, 7. o. pasmocru s(¢+h)—s(t), mpx
pamenmo spemem I, korna À

Hanpnmep, cen sarepmaznitan rowka Ammxerca no sacos
s-# (aaron enoßoguoro nanexus), ro

vege ER Een, um vgeets Eh, my
ep DD e, lp =8t+ Eh, orsyna

Lim dep= ats re o(= at.

Mrsogemayio eKopocts v(t) nassınaror npoussodnou @ynxyuu
8 (0) m o6oamauator °(t) (unraeren: «oc nrrpux 07 109), 1. €.

©.

1. Tponavoanax
Tlepeñígem renep» x oÖmemy onpenenenme mpomasBonnoli.
Tiyorb @ymnusia f(x) oupenenena B OxpectHocTH TOUR Xp, 1-0.

66 rnana u

Mponanonnan u 60 FROMETBIIGEKAN Een

XA MEKOTOPOM HNTEPIALI, CONEPAAMICH TOUKY xo, H IYCTE TONKA
4% +h racke npumannerur OTOMY mreppany. PaCCMOTPHN mpmpa-
Meme yn [9 +1) (x) m coctanım npoñs
Go + A) (su)
unten, re)
Apoßs (1) ects ornomenne npupanenna dyn f(x9+h)—
<li) x mpupauienmo aprymenta A, ory apoSs Oyaen massınars
Rsnocmuna omnowenuex. Ecau cymecrsyer npenen apobi (1)
mu h-+0, To oror npenen nassınaror nponauonnoh yum /(<)
B Tome x m 06oananamır f' (x).

Onpenenenn

Hpouscodnot gynxyuu f(x) e move xy massinaeren npenen
pueKocruore OTHOMERIA upx h0, 1. e.

Dim LEE,

I Hs onpenenenna npenena cxenyer, “ro
Lot W169) _ (xp Ch), @

me a(h)—0 np 0.

Bommuen panewerso (3) » axe

Af = a+ NOAA)». @

Feat h--0, 1o npasan acts panenerna (4) erpemuren x ny”
an, noarony Af--0 mpi AO, Dro oomauaer, «ro yuna f(x)
Henpepumun » roue xp. NA

FA Iran, ocan Qynkunn HMEET MPONSBORLI B TOUKE Xp, TO oma
‘enpepuipiia » orolk rose.

Beau cyuecrayer Ÿ (x), 70 ropopar, uro hymmıma f(x) Aug
Gepenyupyena e mouxe Xp, a ecan dynKaus f(x) umoer npons-
Somyo à KANAOÍ Touke HEKOTOPOLO HPOMEVTKA, TO TOBOPAT,
vo dyaruna f(x) Ouppepenyupyena na 9mox npoxexymre.

Bamerin, “ro wo neupepsinnocrn dynkunm f(x) B Tome xo
ue cxeayer co aubbepemrpyemoers » sro rouke. Hampumep,
Oyasıa f(x)=[x1 nenpopsmnn » tore x=0, mo ue nmecr po
imonnoh u oh rouxe. Helicrnurenuno,

1-1 _ CRE ecan h>0,
» h |-1, ecam h<0,

crass caer, uno paanocrnoe onomenne A) ye seer
Se mp AO

2 Bom pyuxuun f(x) nenpepsinna enena » rouKe x m eyıneor-

[9-160 ro oror npexen uassınaeren 16600 npouseod
a m
vol Qynkyuu f(x) » roure x» u oGosmanaerca fi (x.

ayer I

4
Onpanenonne npOWaBORnOn

Ecan oyskuna f(x) Henpeptipna empaña B TOIKO Xy, 10
npasan npouseoduan f! (xp) onperenaercn panenernom
À = HEC 2 Lo + A) fo)
LT EE mat f(x )= lim DE,
1, f,(0)= 1. A

Hanpnmep, een f(x)=|x|, ro £-(0)

2. Haxoxqenne nponsnonnoh dymeui kx +b, x, 2%

Bagava 1. Haitrn nponanonnyio NOCTORHROR, 7. €. Hymn
F(x)=C, nprimaomeñ up scex x ono x TO me snanenne.

[> Cocranım pasmocrnoe ormomenne:

[Ge+- 16) _ 0-0
h

Creonaremune, F'(2)=0. 4
Taxıım oÓpacom, nPonanonuan nocrosuol papa uymo:
C'=0.
Banaua 2. Haitra npomasoanyı nuneñnoï dynkqun
GR) kx+b.
[> Cocrasum pasnoernoe oromenne:

fix+h)- 1) _ Mes h+b-(kerb) _
y » »

Caenovarenuo, (kx+bY=k. 4
Hanpumep, (2x+5)=2, (-Bx+4)=-8, (7x)
Banana 3. Haltrm uponanoauyıo ya f(x).
D> Cocrammx paanoernoe oruomenne:

SORTE

Bonn h—0, ro 2x+h— 2x, Oreiona nonyanem
im 1240-16) im (2x + h

Caeaosarenuno, (xt) = 2x. 4

Haïtru f(x), ecan [()=x%.

DEM ICO=(e+ ix". To dopuyme paamoerm xyb0e

(eth = x8 (a (e+ WP + eH) xe) = (BE + Bach A).
Cocrannm teneps paanocrnoe ormomenne:

me
(AA
Ecam h—0, ro h?—0 u 3xh—0, nooromy 3x*+3xh+h*=3x,
omsyna
Him LEE _ 932,
ms
Cregonatersno, («Y 282", 4
68 fnasa 11

TipowasonNaA m 88 FONSTpINEGHNT CHECA

23, CocramuTb paamoernoe ornonenne, ec;
D f@)=4x; 2) =; 3) f(a) 4x";
DU DW 2% © Mx) = Bx? +x.

2%, Henomaya onpenenenne npostanonnoit, naiirm mponsnonny1o
oyna
D f= 2x43; 2) 1(x)=5x-6;
8) 034 4) fe) = Bx? + 5x.

25. C nomompro dopuyaam (kx +6Y'=h (sanava 2) ualkrır nponanon-

nyt Py
D 19-85 2) (= 425
3) [)=—5x+T 4) [OTI +8.

26,

exo xpmxeren no saxony s(1)= 14-54. Haltrn cpenmoro exo-

POCTH ABIKCNTA 24. HPOMEAYTOR pee:

1) or ti=2 no ta=d; 2) or 11=0,9 no ty

27. Baxon nonxenns sanan dopmymolk:

1) s(t=2t41; 2) s()-0,31-1.
Hahrır CPEAMIOID cKopoers anınenns OT 1, =2 no (¿=8 m CKO
Poers aunmenua » MOMENT 2,=2 u D MOMENT 12-8.

28 Haïru mrmoxemayio exopocrs anumennn tomar B Kanal
MoNeHT BpeMeHK 1, ecmt AMKOK ee annzennn s(t) annan hop-
yal
20-3 DN.

29. lana pymenun [60 q a? + 3x.

1) Henomaya onperenenme nponanonnok, unkrn f(x).
2) Hatirn anauenme f(x) » rouxe x~0,1.

$ 5. Mpasuna auddepenunposanna
1. Aubhepeausposause cymmur, NPOMIBEACHMA, MACTHOTO.
Tiponanonnaa CYMME PARMA eye NIPOMIDOAHE
| MANN e. a)
Dro nparıno oanauaer, uro ecau bye f(x) u (x) andipe-
peatupyenms 2 TOUKe x, TO HX CYNMA Takıke MPEPERNNPENR
‘Brovke x m cnpaBexiHBa popmyna (1).
O Myers, F(2)=f(x) +8 (2). Torna
Fla) -FGO=f+ Pe (+ h)- 8 (o).
Toerony pasuoernoe ormomenne parano

EEE, Math)-/) | EE |
h [3 y

ss 69

Tipasuna AndbepenunpoBanin

Tipu h—0 nepnaa Apo6b B npañolí yacrıu meer mpemen, par:
mu (2), a mropan Apoßh umeer upenen, pan g’(x). Toorosy
no cnofersax mpenemon dbynkmuii nenas acts Meet mpexer,
pasuutit F'(x)=f'(x)+2" (x), 7. e. enpaseannso paresereo (1). @

Hanpunep, (4 xy = (HR = 3x? +1.

Tlocrosmmetit MuomsTen» MOHO nunecru aa auax mponanor-
moi:
(cf =ef (=). @
O Myers F(x)=ef (x). Toraa

FAP _ eft h) ef)
h y

Ilepexoxa » 970m panonerpo x npexeny npır #-*0, monyunex
F(@=cl" (2. 0
Hanpumep, (4x7) = 4 (x?) =4:2x=8x.
Sanava 1. Haïñru nponssoanyıo ymeum /(x)= 5x2 + Tx
D> Tax sax (22, (x)'=1, 10 no dbopmyaam (1) u (2) monyun-
om (5224 7x) = (52°) 4 (7x) = 5 (8) 4 TRY = 2x 4 «= 102 +7. 4
‘Popmyza (1) cnpasennnsa ne TOKO 428 cyan AByx OYE
uni, HO m ann cyMMBI TpeX, verkipex u Gonee yan.
Banana 2. Haïrx nponanoanyio Pyme
f(z) = 2x8 522 43x48.
D (2x3 5x? 4 Bx 48) (22°) 4-52 + (Bx) + (8) 2) — 5 (2+
+3(0)=2:3x*-5-2x+8-1=6x'-10x+3. 4
Tiyers dyaxuss f(x) u g(x) angpepenunpyenst 8 rowke x.
Torga 5 »roh roure Pyuxuma /(x)-g(x) mucer nponasomye,
Koropam Bupaxaerea opmy.olt
LEON RARE i)
Banana 3, Nomvsyace bopmyxok (3), maltrm mponanonnyn
yea

DD 2+ x-6) (7x2).
D> To (popmyae (3) naxonum 9 (e) =(2x+ 1) (x? —x—2)+ (+ x-6):
X (Bx~1)= (Bx +1) (e+ 1) (a 2) + (a+ 8) 2) (22 — 1) =(x~2) (224+
+3x+1422°45x-8)=2(x-2)(22? + 4x1). €
Banava 4, Ilyors k m b — nooronmmme,
MIR +d), B(x) (ex +P.
Hoxasars, “ro
F(a) > (ex + D)°) = Ak (hx +b), “
(x)= ((hx +b) = Bk (hx +d). 6
D> Tax xax f(2)=(ix+b)(kx+b), To, npumensa dopmyay (3), ne
aysaem f (e) (Ex +0) + (x+b)k—2k (Ex +0). Ananormano, me

TOM rnasa u

Tiponssonran EE TEOMETBAIECKUA CMBR

nomaya pasexcrno g(x)~f(x) (Ex +), hopmyası (8) u (4) , mexo-
un g(x) = 2h (hx +b) (Rx +b) + (hx +b) k= Bk (kx +b). À

Banana 5, Myers

f(x) = (2-2)? (e+ TS
Haëra xopun ypasueuus f'(x)=0.
D ipumenaa mpasuzo ampepenuuponanına nporranenemus 1 Gopuy-
a (4) m (5), nmonyuaen f”(x)=2(x—2)(x+ 1) + (0-23 (+14
m(x-2)(x +1) (2x+24+3x-6), 7. e.
SN

orkyaa enenyer, uro kopnann ypannenun /'(2)=0 æanmoren unc-
ml, À 2 4

Banana 6. Joxasaro Gopuyay (8).
D OGoaauun 060 (OLD, Af =f +) IM. AGE + N - Ele).
Ageo(x+h)—@(x). Tora f(x+h)= f(x) +f, gx +h)=8(x)+ A8,

39 _ TEEDTEED BIETE _ (IAE

x 0 »

a af, at
BO = p(x) ME +800 SL + Lag. (6)
To onpexexemmo mponsnonnoi Mf —e'(x), Sf — f(x) upu
40. Kpowe roro, Ag-0 npu 0, rake xax audepenunpyenas
»x0wxe x bynKnu nenpepnmna 8 soil vouKe ($ 4). Cnenonarenn-
xo, npanas acts (6) muoer npu h—-0 npexex, papi npanofi un-
cm dopuya (8). Toatoxy eyunecrayer npexen enol vacru (3),
xoropui panen nporanognoï yen f(x)g(2). ER
Hyer Gym f(x) u @(x) ambpepenumpyenst » rouxe x
u 8(x)#0. Torna pymkınıa Oj HINCOT IPOMIBORAYIO B TOU
Ke X, xoropan nuipamaeren QopuyaoR
tery 1" gn
(es) 5 o
DopMyAY NPonaBonnoit “ACTHOTO mono Aokasarı Tem me
enocabonı, ro u hopaıyany (3).
Banana 7. Hañon f(x). econ f(x)— CF, m peurs ypav-
zone f'(2)=0. 7
D Tipuuenas hopmyası (7) u (5), nonysaem
ping (=D te DAS es
Se 1Par-2@-1
A
1 (y= ED,
ornyga exeayer, wro opus ypannenus f"(x)=0 annmoren une-
min 2.4

FE

Tipaenna anddepenumposanin

3anaua 8. Pemurs nepanenerso f'(x)>0, ecan
O
[> Hafizem f'(x), menonsaya dopmyası (7) u (5):
pe PEA) 27 (2-34)
exo? went"
rn Zaun,
Can
1

orxyaa exeayer, 470 f'(x)>0, een x< à

(x40) u x>2, 7e. npn

2
<0, 0<x<4, x>2
x<0, O<x<5, x> 34
2. Tiponssoanan cnonoï dymensm

Eenu mweorca bynxnus (Ex +0), re km b — nocromnse
mAzO, 10

(P(e +BY =f (ex +b). @
Hoxaxen ory dopuyıy.
O Obama F(x)= f(x +0), t= kx-+b, rorna
AF PsP) [hess DCE) _ LO, q
% % » . 0

Ar CERO, an
Bem 0, 10 = On LEMP) [2100 poy

npn h,—0. Crenonaremno, npapas uacrs B bopmyxe (10) meer
npn h—0 npegen, pasuatti kf’ (t)= kf’ (kx +b).
Ho Torga cymectayer npenen » nenok vacTH 270 dopmyanı,
xoropsih pasen (/(kx+0)). ©
FT Hauomun nousrne croxoË dymeumn. Tlycro sanana yes
mua f(y), FRE y, © choIo ouepext, apamerca Dynxuneh or x, 1. €
y=8(x). Torna byaxunw F (x)= f(g(x)) sassBalor croxæoï yek
nue (uam cynepnoannueli) dymenutit y-g(x) u f(y). Tipu atom
mpennonaraetea, TO MHOMECTRO aHaYeRHi yHKUM g(x) Conep
ORHTCH B OÓNACTH ompenenenna yuku f(y).
Hanpunep, ecan f(y) =e", y=g(x)= 2x, 10 F(x)= (ez) =e"
Tiyere yum g(x) uneer nponanonuyıo » rouKe x, a ibys
una f(t) nmeer nponssoanyio B rouxe t=g(x). Torna caox-
mas hynkuna f(g(x)) umeer MPoHsBOAHYW B TOWKe x, KOTO-
pan nuspaxcaeren Qopmyaol

CENTENO. an
Banana 9. Haiit upouspoanyio Pym (3x° +4)".
> Bneos y= a(x) 3x" +4, fy) =y*. Tax Ka [By a B(x)~
=6x, 10 no popmyne (11) naxozum
(Bx? +4) = 8 (Bx? +4). 6x = 18x (8x7 +4)". 4

72 rnasa 11

Tipowasonnan m GO FEOMETpINGGHNT CUCA

3. Mpoxanognas o6parvo ymax
Iyers y=f(x) m x=q(y) — sasanmxo oOparaue pyaxunm.
sn
FEW =y. (12)
Ilpemonoxms, sro byakımm f m © andbbepenmmpyenst, ma
parexcrna (12) no npanuny Auphepenunponannn Como yn
um noxyunm FW) W)=1. orkyaa Y = Fem’ ecam
ao. aa e poney SSD com oo
BE non
gilt
I cd an
C nomombro dopmyası (13) » $ 7 GyayT malinems dopmyası
Be gu

en
30. Hakırn npomanonnyıo yaris

Datz 38x; 4) -27x%

9-4 7) 1822426; 8) 8x"-16.
31. Iiponuhbepennnponars. dymumo:

1) 32°-6x46; 2) 6x? + 5x—75

3) 1 +12x% 4) x-8x%;

5) 46x; 6) 12x" 4 18x;

Narr Hort; 8) 3x54 2x9 x5.

32. Haiton /'(0) u 1'(2), een:
Dit: 2) a
DORA A) Bel.

33, Her anauemıta x, mpm KOTOPHX amavenne nponanonnol
‘Oyuxunn f(x) panno O (pers ypammenne f'(x)=0), een:
D) f@)=29— 2; 2) = + 8x +15
DICH Bxt12x 95 4) AAA
5) £00) C2) 4 6) Fx) DS.

34. Hal npoawoanyio yaaa
DAA 2) (AA 8) ra: 4) (3
Hatra f'(1) (85— 86).

35,1) ()-Qx-3-D 2) RAI

21, à

BDA DIO
à 2
a 1-28; 4) f= 2.
37. Hañirm nponanopuyio yen:
Sates PET
y nee.

gs fes

Tipasnna anpgepenunponannn

38. Sanncars dopuyaoli gyaKusno /(g(x)); malt ee oßnacr» or:
penenenun n mnoxecrno anauennit, ecu:

D page RE 2) fw)=ley, y=80)=Vx-L
8) HO, y= a(e)=log, 23
9 Mn u e.
MiS eon bere | Bun,
DEIDADES A
40. Buscnnrs, nPit Kakmx AMATEIMAX x npomanopman (yen
F(%) npuwinsaer orpuuanemsune anasenua, ec:
D fax Tx +10; 2) (ax +4
3) 10-83 +822 +45 4) £)~ (1-32.
AL, Buiscnnrs, px KAKIIX anavenuax x nponanoanan dy
f(x) upnaumaer NONOMHTENBHBIE SHAVEHMA, ecAK:

DORA 2) AA.
42. Bunenwre, mpr KaKHX suavennsx x mponsuonnad ya
f(x) NPHNNMaET OTPHNATENEMLIE HAT, CEIM:

yt a PE.
43. Banncanı. dopuynoli dymxume f(g(x)) u nañirm ee npouanox
nyoo, ecm:

D OV L, yo Vs
2) 1@)= VI, y=8(2)=cosx.

44, Haliri nporanonnyro hyakımn g(x), o6parsott x y (2),
ceca:

DrW- ER; 2) fax, x20.

8] Haste uponanonnywo hymen
1) f(x) = Bx 424 + 2x—1)% 2) [x)= (8 Art +32 +2),

8 6. Tiponssonnan crenexHoñ Pynkunn
Pacemorpum crenennyio dynkunw y-x7. Tlpu p=1, 2, 3 aa
QyERnHA AMhepernmpyema H ee MPOHIBOAHAR panna cooTneTcr
wenno 1, 2x, 3x* ($ 4), 7. e.
(2) =1, (x?) =2x, (x)
Tipoussoguan crenennok byxkunn npn 16060M nelicrBuren
now noxasarene p maxonwren no ‘bopmyne
(ary = par. a
Dra dopuyxa enpanenanna np Tex arauemnax x, mpu KOTO
Pa er ec 60 sacra penenere (1), Ona Syner soem

TAN) trama u

TIpowasoAnGA W GE TEOMETBAIGCKAR MECA

w--

D Mera £(x)=L, x20. Torna

LR De EE ee

Cocrasmm pasHoctHoe orsomenne:

LAICO __
h

Tipu h—0 onamexarens xpobu erpemurez x x*, nooromy
)=- 4. Tax o6paaore, (=, 20. (2)
Gens) E =

Honyuennoe Panencrno mono aanncarı Fan:
IDA

Dro m osmawacr, ro bopuyxa (1) sepua npx p.

Sanava 2. Joxasaro popmyay (1) npu p= 4.

D flyers (= VE, x>0. Cocrasus pasnoernoe ornomense:
es bf) _ ETRE

%

h
Vunoxmm uncanrens u aHamenarens Ha cymmy \x+h+Vz-

Nora
14m -1) „ VETENDWTTEMND (xt yx
y ARA Vx) hzthe
N 1
TS
Ben A—0, 10 VS FRY ($ 2, aannun 6), nooromy aname-
Hatem nocaenneñ apobn crpemurca x 2Vx. Crenosarensno,

pa ELO u

Tlonysenxoe PADOHCTDO MOHO JAMHCATE TAL:

(je,

1.0. @opmyaa (1) cmpapezamba npn pot ..

Sanaua 3. Hahn f'(L), ecm f(x)=Vx+
D Tax war f(x)=x? +x ?

ve

, To no popmyxe (1) nonyuaem

roza +(-4)x
Crexonaremuno, F(1)= 5-5-0. 4

so se

Tipomsnopnan Crenennion Spa

Sanava 4, Biumemm [(-3), econ f(x)=

D> Tax nar f(x)=(-72+4)", ro, nenomesya popmyay (1) u
1

opuvay (9) no $ 5, monyuaen fx À (7244) 2-1), om

naxonme f(-8)=- (128)+4) ?

E
i.

2

7 Banana 5. Hala (2), comm (x)= 14 L, x20.
Kar) 3,70
3
SS Dni qa Fax
E E.
f)=x ?(x+8) *(x+2), x40, x7-3. 4

D Tax war (=

dea AA

3

Banana 6, Joxacaro, wro (3 Y = mpm x40.
e

D Beau x>0, ro VE=x* u no dopmyne (1) noxyuacw

4

Ecun x<0, 10 Vx. (-x)°. Henonssya bopmyay (1)
u mpapiuta Auipepenunponannn, Haxonnat
2

4 1
EDGE ED

e

Ynpaxnenna
Haltra nponspognyw pyakunn (46—47).
46. 1) xf; 2) x13, 3) Bxt+2x'8;
ara a or
anota aa ar ar
97 ES 8) *

DIVE als

Tipovesoniian W 80 FOEMETBATGEKUN CHER

28
4) f(@xjax? x 2;

ODIO
D DRE D 8) RD.

49, Halivi omauennn x, DPI KOrOpUX anauenne uponanonKoit
ya f(x) panno O:
1) /(x)= 8x4 4x5 - 123 2) f(x)= xt + 4x*- 8x25;
DOLARES 4) foral
CHOC as 6) f(x)= Bx" 425+ 6x? 12x.

30. Haltra FI), ecan:
DIE 1)" (2-295 2) F(x)= (2x1 1406
DOÑA RI 4) (mor O Va.

‘si Tht e A a
y=(2-3)*(245x)* panno 0?

Maira nponanoanyıo yum (5256).

EXIT Art
mots, „ee
ENS

ar,

MA

242 ye A
9 9 (Wr
% (Fl
E
a VEZ 621% 4) VAT x.
an ter, EIA
Ds D en
2 1, 2x
DE a

37. Haliru roux, B KOTOPKIX sHAveHHE mponanonmoii Dyakuun
162) pauno 1:
D Het + Br 2) f(x)— 2x84 52% 42445

Dre,

$6 77
TIpORSBORFaR CrenennOn Py

58. Pomvrre nepanenerno f'(x)>0, oca
D NOAA RA NOAA
Ds,

ate 4) 4) f=

5) aa 6) F(x) =(x-3)Vx.

TON nonoporn CNA BOKPYT OCH HAMCHATC B aaReHMOCTH
© or npenenn # no saxony 9(t)~0,1t?-0,5¢+0,2. Haïrn yrao-
ByIO ckopoctb (paa/c) Bpamenua Tela B MOMENT BpemeHH
12200.

80. Teno, waeca xoroporo m=5 Kr, ADIDKETCA npamommelno no

saxony s(1)=1-1+t* (rae a nuipamaeren » merpax, 1 — m ce

vey) Hasta nemermccayo sueprme ona 22 sepes 10.
nocne nacana anımenun.

61. Bronson ucoynopoyneat crepe pinolt 25 ow ero wach
(8 rpammax) pacnpererena no saKony m(I)=2l? +31, rue 1 —
en eeu nee, Hates an,
Reno rome: D 3 tourer evcronitel or Manaka cra
ep prada

€2, Hatem nponovomyo yuna (= VE ARTE npu &<2 u
mee

§ 7. MponasoaHble anemenTapublx myHKynii
Tlokasarensiias, xorapiidmveckas Tpuronomerpirecne

dom andybepentumpyents 1 Karenol Toure, FRE or onpenene-

Hit. Ilpunenem HEKOTOPHIE dopmyası NPOHSROAMLIX ITHX y HR mm:

1 Hoxasarenuerso dopmya 1 m 2 ocnorano na uenomsosanne
papenerna

a

A PH BEIBOJO hopuya 3 m 4 uenomsyere panenero

e

Coornomenna (1) u (2) Hassızarr sameramenenbraru npedeza
Au, ux HOKASATONCTIO Aaercst B KYPCE BMCHIEÏ MATEMATHKH.
Hoxaxxen tbopmyny 1, wenomaya mpenen (1).
O Bean f(x)=sinx, 10

Men) _
7

Tim (1+)

ine +h) ein.
à

789) meee u
IPOVSSORER u GE FOOD GHENT

à
at ,
Tax xax 21 npn 0 (panencrao (1), a cos(x+ 4) ~cosx
ra
(sepepuamocra. ocmnyes), xo eyineerayer lim
ee
Tipumenaa popmyası 1, 3, 4 u npapuna aubdepenunposanna,

sin(x+h)—sinx
h

supegen POPMyay 2 u 1OKAKEM panenerna:
1

5. (a

ang TA g the REZ.

6. (ete x) = xx kn, REZ.

% Anizly=t, x20.

& (@Y-a'ine, a>0, 021.

9. dogs)’ = Zhe, a>0, a%1, x20.
10. (xP Y=px""!, pER, x>0.

102. Tax xax cosx-sin(x+$), 70, mpumenns gopmyay 1,

ony (cosa) =eos(x+ 3) (x+ 3) =e00(x+) --ainz.

5. lipuwenue npapixo adéepenumposanur wacrmoro ©
dopuyas 1 u 2, nouyarm

imxy_ cosrconr-eins(-eins) 1
De

6. Anaxorimo
D A

7. Myers x<0, voraa |x|=—2 u Injel=In(-x), oreyna, upu-
enan GOpuyay 3 x mpabnzo Andbepenumponanmn enommoft YET.
1

xx, nonyuaen (In |x| =(n(-x)) Orciona m ma dop-
uyau 3 enenyer, «ro

8. Bosnoan oGe uacru panenerna a=el"* 5 crenemb x, mony-
men a* =e", Tipuwensa dopmyay À 1 paso andxpepenustpo-
ens COMO DYNKIU, naxonum

(yema atin.

TEE, To, npnuenaa QopMyay 3, nonyuaem

9. Tax xax log,

CAS

z
OTRAS SneNEHTaBHEI EUR

10. Bocnonsoyemca pasencrnon x? =e!" Pin, Torga

(ey meros

Pr ee

Tipuuenaa dopmyası 1—5, 7—10 m upasusa xrpepemupo

panna, per» seau 13.
Banaua 1. Hañirn f(x), ecan f(x) =cos3x.
D (60832) =(-sinäx)- (3x) --Bsindx. 4
Banaua 2. Harn f (2), coum:
1 16) 2) I(x) =tg 2:
3) f(x)=2 Asin 2x; 4) f(@)=In

zeit
DI ER 2x)

DD (0 Y=e*(-2)=-e*;
1 2
2 I a;
D Ge Ba =
3) (2 *sin2x) -2 *In2-(-x)-sin2x+2 *cos2x-(2x)=
=2 *(2cos2x-In2-sin2x);
Ely xt fe dy td fre 2
my a) Gt a )-
or E tapa 2
A ne AO Gay and?
6) (at + 2x)? ey = 22 + 22) 0% 2 te 3
me (x8 4 2x): (Bx? + 6x" 4 6x + 4). Q

Baxaua 3. Haïñrn f'(x), ecan:
00 Vez

2) MR) Ines VI):

3) fa) = 3" ete 2x;
4) Fein (+2): 8%;
5) f(x) = te 8x le (x? +1).

DEE Es

et. ei

142, 1 View
er 1

2) (inte VD) à [1

DOC EE etre

80 nase 1!
PONT TES FOIRE CHR

O OS

air E +208: +20)
1

5) (at Bx- lg (xt + Dy’ = tet Bx-
ia (ue +0, xteae ay ==

Frau I 22122. 22725

“pinto Fax * Gri) in1o
An Banana 4. Joxasero dopmyası
1
sin x) = ——, |x|<1, (3)
(aresin y= FA als @
1
Creta Ea) ER. oy

Da) Bom y=0(x) =aresinz, rae (xl<1, 10 o6persan oynkuna
(W)=siny, rae |y|< 3. Ho Dopuyne (13) ua $ 5 naxonum

la
Giny 7 cosy”

(aresin xy =

me siny x, cosy =

2%, vax xax |y|< 5. Dopmyna (3) noxasana.

© Bean y=arctgx, rae xER, To oparnas hynkuna x~tey,

me IC} Mo dope (13) ua $ 5 many
Gr) yy os,
FRE 1
om cst y= any page’ Depuis (4) noxazans. À M
Yopaxnenus
Hain nponasonyio dm (63—75).
GR Inztsinz; De-sinx 9) VE-cosx;
4 Les 5) tgx+Inx; Oe'-ctex.
041) eos; 2)-66% 3) Mm

4) -Ssin2x; D es 6) zed,

65.1) 6-90 2) Late Bsinx; 3) SÛT-dcosx:

4 Sraets 9) dinars 6) atgex-2iz.

ae
57 81

TIPORSEOANEE DEE y

66. 1) 8VZ+1602; 2)

2-1 sinds;
4 Ye
D arVe-3in E; 4) Tet Boos
67. 1) 3x-AVE+2e°; 2) 2x43 VE—cos 2x;

4) 2x -34g3x- 1 sind

5) 8x" 47x? —cosdx; 6) Lotgr—5x —

DE aa ay VER
A; a ni
Get? (x-1)" Vx+3

&

69. 1) (8x41); 2) (Gx-4)*; 3) (1-3);
a

=> 5 59 4.
ay aa > aa
70.1) \2=Be; Vi Dr
1 7 5
Dobe 9 = Os
Ver: tra Vaz

71.1) sintas 2) cos? x3 B)costx; 4) si

5) 6) es 7) In3x% 8) In(-2».
Yes Des 3) In@x-1); 4) In3x;

5 osx; x +3); sin( 2
5 tj Mein TtgGx+3) 8) sin( BE 41
2), ar),

23.1) cos(1- 2) sin(2- 3);

8) sin 228, 4) cos 15;

Dosis 6) sin 2548,

7) sin?2x; 8) cost 8x;

9) ote? 4; 10) tet 3.

a
74.1) e* DEZE
En mo,

4 Lire

D 5; 2) 4% y 27%

5) logsx; lors Mig 8) let.

8200) tease u

Tiponasonnan Y 66 TEOMETPUNGERAA EMEIEN

3 Haliru anavenns x, mpm KOTOPHX anauenme npomsnonuoit
yat f(x) panno O, ecm:

D f(a)=x-cos.x; 2) fay- 4h x-sinz;
8) frida 4) 12 Ina)
522; 6) Mx) =8%-2xIn3;

D fls)=2in(e+3)—25 8) f(x)extln(2x +).
Haïru mponsnonnyio bymxnsen (77—81).
nt, fand,

3) 2e 4) 5sin 2248 —4y/—
Ya 2-2 $ [1 u;
9 VfL - Bes 232; ”.
Vas i tt
18.1) log,(x* +4); 2) eee
3) tet 2x5 4) n= ae

DOVE Tea — 2) elias
3) Vx-sindxs 4) e -cos(8—22).

9,1) Leos,

une,

82 Holt onauenun x, npu Koropsx auauenue mpomanoanoit
yuri f(x) Panne O, ecami
D fm xt 6x-8nx; 2) f(x) 2VE-BIn(x +2);
8) f(a VEFT-In(x—2); 4) F(Z) =In(x-1) +242).
$8, Peur, nepasenerso /"(x)>0, ecan:
D ae 6x-+cos 8x;
8) (6)=Inx—x5 -2Inx5
D fe) =6x—xVE5 6) f(x) + VERF Br
Sl Buncnurs, mp KaIOIX snauenuax x anauenue nponanonmoi
dvi 7x) paso O, ecam:
1) {(x)=5 (sin x—cos.x) + V2cos bx;
2) [(x)~1-cos 2x +sinx—cosx—x.
Haiirn auanenun nponanonnoik dysxunn f(x) » rouxax, m xo-
Topux shavenne oro pymenmm paso O, cen:

DAI 2) 0 SRE.
50 Bummenuer f'(2)+F(0)+2, ecm f(x)=zxsin2x, xr.

$7

TIROIRS DISPO PAER yA

87. Hañra anauennn x, npu Koropux anauenme nponanoanoi
yarns f(x) panko 0; nonoxarmensno; OTPIIATOADNO, cau:
D flxex-Inx; 2 fOerin
3) f(x)=x* Inxs 4) f(x): -8inx.

88. Hafen npoxanonnyto ymca f(x)=In(x2—5x+6) np x<2
a mu #73.

8 8. reomerpuueckuñ cmbicn npouzsonHoñ

1. Vraonoï xoodubunuent npamoï

Hanomam, uro rpapuxom auneinoï dynein y= x +b mu
aseres npanaa. Yneno k=tga HASIBAIOT yeaosum Koagauyuen
mo npaxoú, u yron a — yexou MEXÔY 2mou npamoli u ocew Ox
(pue, 46).

Bonn k>0, ro O<a< 5 (om. pue. 46, a), B atom cayuae yn
una y=kx+b poopacraer. Eon #<0, ro - E<a<0 (em. pie. 46, 0,
3 srom cayuae gyHKMS y=kx+b yOuBaCr.

u 4

y Puc. 4

1
17 PORTO ES TOONOIPITROAA Een

Buseex ypannenne npaMoli ¢ SANARHBIM yrTOBKIM Konpihn-
remo, npoxojsmek uepea aananyıo rouxy Mo(xo; Yo)-

Divers mpawas ne napannensna ocn Oy m Mix; U) —
mponmwonsuan rouxa roll mpamoñ (pue. 47).

Us AAMM, naxonnm =e =tga. O6osrawue tga=k, nony-
sue y yon RCE x). oruyan

V=Vo+ (=). @

Ypamxenue (1) nassremior ypasnennen npHMOÏ € yrAoBBin KO
Apdunueurom E, MPOXOAMINCH “epes TOURY (Koi Yo).

Suwa 1, Banncars ypannonne upawol, mpoxoameh uepca
sky (2; 3) m oßpasyıomeh e oc Ox yron

D Haxonum yraonoh konhhuunent mpamoñ k= tg!
Hak xo=-2, Yo=3, TO no opuyre (1) noxyuacw
y=34(-DE-C2)), 7. 0. yet +1. À

2. leomerpuseckni embtea nponanonnoi

Betania reomerpnueckni CMACA HPOABONNOË andpepemun-
pyewott ym y= f(x).

Tiyert dbymnna y=/(x) onpenenena » nexoropoli oKpecruo-
com rowm xp m CyINecTayer ee mportanonnas f (xo).

Een À u M — towxn rpauxa oroit OyHKnIE © abcmmecanmt
u xpth (pue. 48), To yraonoh xooppunuenr k=k(h) npawoi,
Spoxoraiuelt epes roux À # M (ory mpamywo HRSHBAIOT ceny-
ei), smpaxaeres popmyaoli

kthy=tg MAC MC „ Math) (2)

me © — rowxa e Koopannaramm xo+h, f(Xo), a ypamnenne cexy-
me AM MOMO SATHCATE B Be

Y—Yo= (0). @)

Niycro h-+0, torna M, aunrasc» no rpabuky, npnômmxaeren

Tome A, a cexyulaa nopopauBaerca voKpyr TOWKIL A. Eau ey-

üeernyer lim (2) = fy 7. €. eyunecrayer npenentitoe nonoxenne ce-
kymjell, to mpanas

--1. Tax

Vo = hole x). “a

* pue. 48
ge 85

TONER GER NRONSBORNOR

ypasnexue Koropoil nonyuaeren u
ypannenta (3) samtexoit A(h) ma Ros
Baosınaercn kacame.ronol rpadi
xy yx y=/(2) » roure e xo-
opaunaramm (xo; f(%0)). Takum 06-
Paso, xacarenuan x rpadmKy
ym y= f(x) D rouxe (xo; f(x)
eer nponemsmoe momomenme CeKY-
meh MA np h-0.
Em cymecrnyer f'(xo), To

ko-lim k(h) = Pre. 4
im Let LED po),

ym fey

Tax Kak Au — yrnopoi Koaddmunent Kacarembmoï, 10
Ry=f'(xo)=tga, rae a — yron, oßpaayemsti Kacarembnoñ € nono
KHTENLALIM Hanpapenmem oca Ox (pue. 49). Taxmm o6pasom,

FG) = tea. 10)

Teomempuneciul cauca npoussoduod cocrour 8 Ton, wm

onawenme nponononoli yum f(x) B rowxe xo passo yr

JOBOMY Koxppunnenty KACATEALHOË K rpaduky dyaKuin

YA) » roux (oi Fl).

SBaxaua 2. Hañion yron mexy Kacaremsmoï x rpaiexy dun
usm y=sinx » rouxe (0; 0) u oeuto Or.

D> Hañtuen yraovoh xoodybmugtenr Kacarensnoï 1 puoi y=sinz
» rouxe (0; 0), r.e. suasenne nponssoanoï oro DyHKAMH mp
x=0.

TIponssonnan qyuzunm f(x)=sinx panne f'(x)=cosx. Ih
dopuyae (5) naxoamı tga=/"(0)=0080=1, orkyaa a=arctg1=*
(Die. 50).

Bameuanne. 310 caofierno noneano ana noerpocnma rpabi:
Ka y=sinx: » rouke (0; 0) cunycomxa kacaeren npamoh y=x. 4

Banaua 3. Haürm yron mexay Kacarennnoü x napaGone y= 2°
3 rouxe (1; 1) u ocmo Ox.

E

Puc. 52 Puc. 58

D Iipoussonnan pyaxumm /(x)=2? pasna /'(2)=2x. To popmyae
(6) naxogam tga=f'(1)-2-1=2, orkyaa a=arctg2 (pue. 51). 4
3. Y paunenne kacarennuoh x rpaduny ym
Banenas B dopuyne (4) hp Ha f' (xp), noxyuaem ypapneme Ka-
carensuoli (pue. 52) x rpaduxy dymxnam y=/(x) B Tome
Cos Hed):

DENE ENTE EN? (6)
Banava 4. Haiirn ypapnenne xacarembnolt x rpaury byuz-
mun y=cosx » rouke e aGenmecom x)= =.

6
D Hafnem anavenns dyurumm f(x) =c08x 1 ce mponanognoit 2
Tonne x= Es
. xa
fGo)= cos = 1), Pla) sin =}.

Henomoya popmyxy (6), Haiiqem ucKomoe ypannenne Kaca-

MEE =

22-5) amy
Kacaremnaa x rpahuky DK y=cosx » roune (|

rbpencena na puoynnce 53.

Eur 5. Halon ypammenno xacarenmoi x rpadmey ym:

una y=x*-2x+3, ecam ara Kacanennnan:
o IN OR
2) napanzenpma npamoñ y=4x-3.

Dilyers f(x)=x*-2x+3, torna f'(xo)=2x0-2 — yraosoii Ko-

Mmes Conrad a spain dae DIO © Tome

(oi /(%0)), ypasnenne Kacatemsnoï B stoi TOWKe MOHO

pi van:

y= (e) +20 D) 20).
1) Touxa nepecevenus rpadnka © ocbio Oy Hweer Koopamua-
mu x0=0, f(x) =8, a f'(x)=—2. Hooromy npaman y-3-2x am
aseren nacatemnoh x rpadmky B TowKe (0; 3).

se ar

TOMO TAGS MEA NRONSTORNON

2) Mo panewerna yrxommx roogbunmenron npamoït y=4x-3
1 xacarerbnoï B TouKe (Xy; f(xp)) eneayer, 110 f (xo) =2x9-2=4,
orkyna 2o=3, f(xo)=6, a ypannenne xacarensnoh » rouse (3; 6)
uneor sux y =6+4(2-8)=4x-6.

Orner. 1) y=3-2x; 2) y=4x-6. 4

Bagawa 6, lorasarı, “ro Kacoremmas x napaGone y=x »
rouxe e abennccoli x70 nepecexaer och Ox n roxxe 30.
D Myers f(x)=x%, rorma f'(x)=2x, 120) =28 n F(x)=2x0. Mo
wopuyae (6) naxonus ypannense xacatemsnoit:

Vers 2x0 (20) 2207-28.
Haïixen rouky nepecesonna aroïi Kacaremnoi © ocuio abe:

mice mo paneuerna 2xyx-x5—0, omyaa = 2. 4

Ms sanaun 6 cxenyer upocroi re-
onorpuseckui enoeoë noerpocnus Ka-
caremoi x napaGoe y=x" » Touxe
A e acumecohi zo: Mpamast, mpoxons-
man sepes rouxy Am rouxy 30 ocu
aGeunec, kacaeren mapaGosr » rouxe A
(one. 54).

Tloorpons kacarensnyi x napa6o-
de, moxuo nal ee Jorge. Hanon-
un, uro donycom F annnerca Towa,
2 KOTOPyIO HYHO MOMeCTIITE HCTON.
mn enera vax, TOGW Bee ayun, OTP:
ome OT Mapa6ozmVecKOrO Jepraza, Pac. 5
max napanzenpmar ocm cummerpa
napaGoms. Jura noerpoenns oxyea F mago wocrpour» upanyio AB,
mapanzeabnyio ocu Oy, u upamyio AF, o6pasyiomyio e Kace-
TenbHOit TaKOH ce yron, xax u upnman AB. =I

BEA. Nngdepemnan Gym
Tlyers dymenus (x) meer nponamoanyo B rouke xp à
Af=f@o+Az)-/(%o) — mpupamenne yakum f(x) B roue x.
Coornereruyiomee upupaukenmo aprywenra Ax. Torna
MT (zo) x+a(Ax)-dx, o
rae @(Ax)—0 npr Ax-0.

Tlepsoe cnaraemoe » dopuyre (7), 7. e. mponanenene f'(xo)Ar,
massınaercn Auppepenyuanon hynuun f(x) » rouxe xo 1 oboama
wwaerest d/(xo), 7. €. df) =P Ga) x.

Tier» f (xo) 0, torna ormomienne proporo cnaraemoro a(Ax)4z
» dopuyae (7) K Tepvony cuaraemomy erpemitres K My np
Ax-+0. Hoorony npn mumux Ax momo saveurs Af ma df(3),
T. 6. SanHcaTh npômexemmoe papencTBo Af=df(xq) man pAReHCTEO

Mt ax) F0) +1 (0) AE,

88 rasa it

Tipowanoppran Y BE TEONOPINCSRAT CRIE

10 Koropony MOXNO HaXOANTD anauenn Dynrum f(x) B roure
+82 € nomouo sHaNCHH ooh Gym u co nponanoguoli y
roue ko.

Bent yen f(x) anbdepenumpyena » nano Tone
mmrepsana (ax 8), 10

af) =f (28%,
re 4x — nponspomsxoe mphpauenne aprymenra.

Bamerus, uro dx =(x/Ax=5x, onpenenma außepepenunan neo
acanoro nepenemnoro Kar ero mpupamenne. Tora df (x)= f(x) dx,
amyaa

ra,

1.0. NPOHSBOARYIO MOKHO PACCMATPHBATE KAK OTHOMIERMe Auhihe-
peruuaza hynkunn x andbepennnany aprymenza. Tonatue nuch-
Qepenunana Oyaer menomssosano 5 raane IV.

Buscmme reomerpuveckuit u Qusnvectmi emsica nmbbepen-
ora

vers yan: y= F(x) anddbepenupyerta » TOsKe xy, TOA
a rouxe Alxı; f(xp)) cymectByer Kacarensian (cm. pm. AB), mepe-
ceramınan npamyio x= x + a rouxe B. Ma tpeyromeunica ABC nu-
som BC=ACtga=hf (xo). Honaraz h=Ax, nonyuaen

BO=f' (a): A = df (ru). 6

Panenerno (8) nosnonaer narı, PEOMETPHUECKOS HCTOMKOBANME
augéepenmans

Kenn yen f(x) meer nporanoyyio m TOWKO xp, To M
depen atoit @yaKIE BI X= x PAPER MPHPAINERINO PAIE"
Tu Kacarenoil 8 roue (o; (a) ups manemen APrYMENTA OT xy
2 29 +h=x0+ Ax,

Obparmmea x «pmomuecrony eumeny anboepenunana. Tyers
+) — nyrı, npoknenmii marepanoi rouoñ an mpema tor
rar poten. Tora (Jim, 20120200
ckopocts à rom B momen apexenn f, 1. e. v=8'(0. To onpe-
sexeumo nnbbepennan ds=v-At. Hobromy anepennan
dyuxunu s(t) pasex pacetoaHmw, KoTopoe mpomna Ont Touka aa
Hpowexyrox upement or I 10 (+A, ecan Om oua Annranach co
ckopoctin, pannoil MIKORCIIOÑ CXOPOCTA TOUR B MOMENT Bpe-
veu ¢.

— mrvonennas

Ynpaxnenna

89, Hanucats ypamemme mpamol ¢ yraopsım Koodppnunertom k,
upoxonsunell “epea rouky (X; Yo), ect:
1) k=2, X= 1, Yo=- 15 2) k-3, xa » Yom li
D h=-2, x0=8, gods 4) Ba +, 91, 1-0;

1 ge =
Pus 0%

oe
5) ko gr vo

1
4, 20=0, yo=0.
se 89

TEOMETPAEEA GMEEA NPONSBOAHOA

90. Hannean, ypasnene npanofi, npoxonamett wepes TORY (X; Is)
1 oGpaayioueli e oenio Ox yron a, ecnmt

Daria Jon al, xo=-1, Vo=-li

3) a= ‘a=

91. Hañiru yraovoit Koapbuunenr KacavemsHoit x rpaduky y
num y= f(x) » Tone © abeumccol xo, CCR:

%0=6, Yor

2, xynd, god.

D f@=x5, Kom; 2) f(x)=sinx, xo
8) /(2)-Inx, xo= 15 4) 1(9=é, Kin
5) f(a)=3x*4x, x0=2 6) [()=Vx- xo=l.

Ye

92. Hañra yron mexpy ocho Ox u kacarenbroñí x rpabney yn
nun y= f(x) » TouKe © abenmecoli xo, oc
2) HOJA, x

4) [(x)=2Vx, x0-3;

5) fx)J=e ? ‚xl; 6) Mx)=In(2x+D), “=

93. Ha pncynKe 56 naoGpaxen rpadux dymtcus y= f(x) 1 nace
renvmue x rpadmny » rowxax A, B, C, D. Onpenenm, anır
npowssonmoï oroï bymxuun » rouxax A, B, C, D.

94. Hanncarh ypannenne Kacareasno x rpadmey Gym
= f(x) B Tome © aGeunccott xo, ecm:

Peel, xml; 2) f(x)=6x-32x7, x5

Lone; =, x
zn 4) = dz da

2

5) M(x)=cosx, x)= à 6) Mx)=0%, xu=0;

D me, 0-1; 8) MOVE, 20-1.

SO raana 1

Mipousnonan Y 68 FOOMETBHIGERAN CHER

: Hannear ypannenne Kacatemsuoi x rpaduxy Hymn
y=1(x) » rouke e aGeunccoï x=0, ecru:

DD HB OA

3) (x)= 2x- Ve FT; 4) Mart st
5) 1) + cos 6) f(x)=sin2x-In(x +).

96. Haltra yron mexay ocbo Oy m KacaTemnoit x rpacbuxy pyme
um y=f(x) B zone e aßennccok x=0, ec

D fre 2) f(x)=co8x;
3) flz)-Ver1+e8; 4) DOS
D =D 0 NA

9. Mon Kaxum yraom mepecexaoren rpapuKn gyxKunit (yrmom
MEHR KPMBLIMH B TONKE HX Mepeceuennn MAMMIMNOT yrox
MAY KacaTeMDHLIMG K ITEM KPHBLIN B STOÏ TOUKe):

3) y=In(L+x) m yn;
A) yet yes?

98, Nokasars, “TO rpabuKH ABYX AAHHEIX dyAKUN HNEIOT ONY
OGuyio rouxy HB DTO TOURS — OÓMYIO KAcATeRDHYI; HANH
cars ypannenne oroit KacaTembHoii:

Dyer, 2) y= xt, y= 0-3;
3) y(t 2), yema Dyno, yor
5)y=Vx+T, yartılııl; 6) y=vx+1, er

2

29, Haiinn row epoca pyme y= f(x), 5 Koropux Kaca-
Temman x oTOMY Tpadmey napannenuna mpawoi y= hx, com:
D f@)=x?-ax+4, hal; 2) f()=x(x+D,

E fe +e sk

Beet, k= 3;

100, B kakux Touxax KacavembHas x rpaduKy pyme y= 22

3,
2
o.

6) f(2)=x+sinz,

a

®

cfpaayer © ocho Ox yrom, paumauit - 27

Haltr tox, B KOTOPHX KacaTembEBIe K KPHBEIM

f@)=29-x-1 mg G)=3x2— 4x41
napanenonst. Hanucars ypasnenna 9THX KACATEMBHAX,
02) Hanucaro ypasuenne kacarensnoft x rpabuxy gyaKust
y=f(z) B rouKe © aGeumccoi xo, ecam:

1) Fer tan, x, 2) fa)

ze
Zen, x

ss Dot
TROIE CRIER nRONIBOANOR

108. Hocrpours rpadbur gym y= f(x) u auscmaro, nnnneren
am ora (pyme nonpopsmnoi Ha Reel «mexonoñ npsNol:

3x4 npn x23, 5-22 npu wel,
dirons EL ed

[VE np x>0, _[Ix-11 mpm x<-1,
a rar

Halirn npouanoanyio yann (104—108).
104. 1) 2x4—x843x44; 2) rar;
2

Do 4)

5) (2x+3)% 6) (4-32);
D Vez; 4

-8\z;

Len
9) sin 0,523 10) cos(-3x). i
105. 1) e*-sin x; 2) cos x-tgx; 8) etg x—Vxs
-9e%; 5 440; I +ting.
4) 6xt-0e% DE DS
106. 1) sin Bx+cos(2x-3) 2) e%-In3x
3) sin(x-3)-In(-22) 4) 6 sin 2E eto
107. 1) x? cos x; Dilo 8) Sxetg x;
4) sin 2xtgx; 5) e*sinx; 6) e*cos x.
NÉE Dip a a

109. Haïrn anavenna x, MpH KoTopbix 3Hauenme nponaBonmod
dymeun (+) panne O; HonosenTests0; OTPHUATABNO, con:
Df) Br 422244; 2) (a) (m4 8-4)
3) 19-22. 4) oz

110. Hañiru snavenne npowssoauoñ dynxunn f(x) » rouxe x,

ecm:
1) f(a)mcos x sin x, xy= 3s Dee x=
8) 1 RE, q 4) HG) Ez» xo=0.
111. Hauuears ypasuenue Kacarenbuoii x rpaduky yak a
1) yw x?-2x, x9=8; 2) y-2%+3x, x0-3;
3) y=sin x, x=% 4) y=cos x, x=

112. Baxon auioxenun Tena sanan dopmynoit s(()=0,54 43142
(s — » metpax, ¢ — » cexymgax). Kaxoii nyt npoliner rex
sa 4 c? Kakosa CKOPOCTE ABIOKEMIA 8 9TOT MOMEHT Bpexen?

92) rasa u

TIPOHITONR w G8 FAO CHER

Halim npomoponayio @yaxusu (113115).
113. 1) sin xeos x4; 2) (x41) cos 2x;

AS 4) Ve-T at.
un Len,
wr ee
16.1) 4x°+2 Inx-cos x; 2) Pte +2 sin xy
8) 15 Ve-+e*—6terx: 4) 6Vz—Inx+ cos x:

3
5) x8(x-1)48 sin x+4 etg x;
6) x(&+2) +2 In 2-8 cos x;
DADA x5
8) (x+3)(2x—1)+e*—sin x.
LG, Hoerponrs rpadırk x yKaoars upomexyrkit menpepspnocrit
Cr
4) = [1082 (2-1) np x<3,
ES
VZF3 upu x>3,
2 f)=) x+8 mpn 364x463,
(+3)? mpm x<—3.
UZ Hara neprnxansusıe acuumrorn rpadbua yann
y=f(2), ecan:

D =

1 po
ge A
118, Barwnenwrs mpexen hymxumn:
CI TE
DIR sara A A
119, Hans npostonoanyso Gym:
1) In? x; 2) Vin x; 3) sin Vs 4) cost x: 5) Vig xi 6) ctg 3x.

120, 1) y=cos? 8x; 2) y=te? rs
D sin@2x?—3x); 4) cos(x+2x4); 5) 66%
6) cos(e*); LES 8) 2,

12L Haïrn anauenna x, npn Kotopux auauenne npoussonnoi
Qyuxunn f(x) pasHo HYMO; MONORHTEABRO; OTPHNATEALNO,
ET

D-27425 2) f(x) 9**-2x In 3;
3) fO=x+in2xs 4) Axt nz):
5) x)= 6x-xVx5 6) AmlcH)VEFT-Bx.

93

Ynbaxnenna x rane I

122, Halızu ce snauenun a, npu Koropsıx f'(x)>0 ana noex sell:
crpnrenbitx anauonuh x, Cu
fo) 48x? ax.
123, Hafıru nce omavenna a, mpu Koropux f(x)<0 gua ncex gel
creurenbunx suavennit x, ecan
Hama.
124. Halim ace onauemna a, npu Koropux ypapnenne f (x)=0 ne
meer neficranrenunnx kopneñ, een:

D fra? Las 2) Mx)=ax44;
8) f(x" 4 8x" +675 4) 1) ax

125, Hair nee anauenna a, mp Koropsix HepawenerBo /'(x)<0
ue uweer aeheranronzusx peuremsit, ees:

Diari a
3) F(x) = (x40) Vx; 4) Ma) Lo
(126) Tox xaxum yraom nepecexuioren rpaguxn pyme:
1) y=2Vx u y=2V6-x; 2) y=V2Xx+1 u y=1?
127. Hauncarh ypanuenne xacarensuoll K rpapuxy yaa »
mouke € aGemmecoï x9, ec:

sin 2, x 212, x
1) y=2sin o, Dyna,

4) y=x+Inx, x=0;

6) y=sin(ax), xo=1.

=
3) f(x)=x4+82x-8; 4) fix)= 26 +622.
HE] Has pasen nacanemunın x rpabany dy

32°, mapanremsmux npanoï y=6x.

(130) Mpamas kacaerca runepôonm y=Í 2 rouxe (1; 4). Hai
TIAOMAAL TPEYTONBHHKA, OTPAHHYEHHOTO ITO KacaTenDHoll n
oa RoopaNna

HSE tipavea nocseros runopton y=, rue h>0, » rome o ae
umecolt xy. Hoxasars, ero:
ps PAM
sol dou ROCHE, ES senwanr or namen rom

Kacanna; ATH oy wiouesD; A
2) ova nacarenunan npoxoaur sepes row (xo; À) (2x5 0)
SA traes u

IPON TOR Y 06 FEOMETBHIEEKHN CMS

* mage It

Bonpocı
1. Depeunenurs enocoßs sananus Nnenonoh nocnenonarens-
2. Kaxas nocnenonarensiocth nanninaercn exonauehen?
3. Kaxas mocnenonarennuocn, NAIMBACTOR monoronnoli?
4. Iipuneeru mpunep Qykumm, HMOIOMOR DEpTIKaNBEIyIO (ropm-
SORTAMBAYIO) acuenrory.
5. Iipmeeru npırmep menpepsinmolt @yrmmt m nocrpours ee
pau
6. ro nasunaerez wronemnoï cxopocrz10?
7. Miro uamumaeren nponanoanoik dy f(x) m roue 23?
8. B non cocronr duauuecknii cusen npouanomoi?
9. COOpMyYTUPOBATE mpanına JUHQCPONIPODAMIA cysts, Ipo-
nanexenus, MACTHOFO.
10. ewy pasa nporasoguas yuan ya? (pcR), y=sinx,
osx, y=e"?
11. Yro nasmnaeres yraonkım KOpRPIILIEHTOM npamol?
12. Banuears ypasienne npanoi € Yraommm KooiypmneRTOM À,
npoxonameit wepea Touky (x95 Yo):
. B uem cocronr reomerpusieernii eme mponanoxnoii?
ro naasinaeren npenenom nocnenonarensnocru?
ro naasınaeren npezenom py?
. Kaxas hymenun maasımaeren nonpopuimnoh n Touke a?
Kax nalirn nponamogmyo caoxnoi dymern? oöparnok
bynes?
18, Buisectu opmyay an naxomenKs mponaBonKol y
yetgzı y=ctex,
18: Kyo upanyro jamas kacarensuoli x rpadnuy yat
» nannoit roue’

Wl ‘iro nnounaeren mpenenom cnena (enpasa) dymnmn /() 9
rome ar

BL Karyıo pynxuno masusaor Gecxoneuro mano?

B) Coopmyauposars caoñicraa mpexena dbymenmm.
U Coopuyamponar» cpoitcrna gynxumit, menpepsputix ma or
peoxe,
N “emy panua npoussoanan dynenun y~arcsinx? y~arctg x?
Bl “ino nassınaerca ampbepenumanom dyaxunit B tome?
BB vex cocronr reomerpinectai dbnanveckuit esca and
beperumana?
_95

Bonpocer X tra I

Mp
1. Hair snavenue uponssoanoi Hymn [(x)= 22° 4 3x?

ph ces

D Tonne x=—2,
2. Hair mponssoanyw ym
1) 244Ve-e%s 2) (32-5);

sin 2x-cosx; E
3) 3sin2. 2 oa

3. Hañru yron meaty kacarensmoh 1 rpaÿuey yr
1

y=xt-23848 » rouxe e aGeunccoï x0= 4 u cero Ox.
4. Hahn snauenna x, mp Koropsix SHauennA mponszoxuol
Oyen
16)=In(8x+1)
Orpmmareasmn.

5. Hanuears yparmemne xacarensnolt x rpabuxy yaxum
f)=sin2x 8 rouxe e abenuccoh x=

1. Harn mpenea bymnun:
Asse, y

Dl st Slim
2. Buincnnms, apaserca xu nenpepamnoil » rouxe a=3 Oya

a nou 78,

2-3
2 npu x=3.

Halıru auauenna x, nPH KOTOPMX AMAWERHA nponanoatoll

Gym

f(x) =2Vx—31n(x+2)
pas 0.
4. Hanucars ypasnenve Toü Kacaremsnoñ « rpadbuny yum
u- 19-2345, woropan napanensua npnoll y-3x-2.

MM 74 I Vieropmuecxan cnpaaxa
OctopmiMH NONATHAN NATCMATHNECKONO AMANDA. amor
nonsria Pynkıum, upezena, nponanonnoh u nsrerpaza.
Topunn «dynkumas pnepome Ora ynorpeßnen n 1692 r. ne
mens maremaruiont T. SleiiGunnene (1646—1716). Tlepnoe onpe-
ACACUNO nous HYHKUMKE, OCHOBAIHOE HA rEONCFPANCCHUX DE
Crannemunx, chopuymmponan m 1718 r. JI. Dünen (17071783)
Emy npruaznexarr 1 Bpegenue cnupona f(x). Blinep daxriiecks
oronenectaian yo c ee aamemnueckoï hopmyaohi, Xor
xe coppemenKHKH Dlnepa MOHHMAR, "TO QYHRLIND MORO at
Aanarı He TOXPKO anamırunecrn.

96 tana u
Tipomasnnän Y 66 FEOMETPNECEN CRUE

B 1834 r. seamxmit pyecxnit maremarum H. M. Jlo6avencenit
(1792-1856) nan onperenenne nonarun DYRKHI Ha ocHORe naen
coorsercrunst AnemenTon Auyx uncnossx muoxecrs. B 1837 r. ne-
seu marenarux IT. Anpuxae (1805—1859) cbopuyauposaa
ofcüueamoe onpenenenne HOHATHA PYHKUUM: «y ABNACICA DYHK-
wich nepemennoÿ x na orpeaxe a< x b, ecau KaxjoMy anasenmo
x cooTsererayer »noJme onpexezenmoe smauenme y, MPWeM He
1er auauennn, KAKMM OÖPaa0M VCTAMOBACHO TO CoorBerer-
ave — Gopmyzo%, rpabukow, Taßnuneh man caonecnum onnen-

Moexe cosnannn Teopmu mmoxecrs Bo Bropoit mononnne XIX B.
oupeaenentie nonsrnn Qynkunn Oro ano na MHOMECTHEMNOÏ 00-
nose. B XX B. B CRASH € PASBHTHEN ECTECTRERHRIX HAVE MPOICXO-
ao aanpuelimes pacmupenue nonsria pysxuum. Tax, » 1986 r.
poccuiicumit maremarmx axaxewmx C. JI. CoGonen (1908—1990)
sax manGonee oGoGulenHoe ompexenenne monarna DYRKUI.

Toustue nponsuonnoh onpenessiercs sepes HOUMTHE upemena,
HCIODMA nonprenus KOTOPOTO yxomur » rayGokyio ApeBItocr. Eure
E IV». 40H. o. sHaMeHHTiIi ApesMerpeveckuii maremaruk Epnore
Kungerul n MestBnom nije MCNOMDIOBAN npenensume uepexonn
Jas obocnonamux Meronop BMUNCHEHNA MAOMAAEÍ puso
selux Quryp. B ABHOM sige NPELEALHME Mepexogui erpeyaioT-
es » pañore dbramanackoro maremaruka A. Tarte (1612—1660)
«Hauaza nxocKol Mm TexecHolt reomerpume, onyGrmronannoñ
2 1654 y. Tlepuoe onpexenenue npepena 121 auraulickuit marema-
‘tux JL. Banaue (1616—1703). Meron npenenos noayun cuoe pas-
sue » paGorax oamemtrroro axraniicKoro ywexoro M. Heroroua
(1643-1727). Emy xe upuuagrexur suenenne cumsoua lim.

Cymectsemsti mean 8 pausurue ocHoB ¿mdpepenuancuoro
seuerenus sueca ppaunyackne yuensie IT. Depma (1601—1665)
uP. Jlecapr (1596—1650). B cepennne 60-x rr. XVII ». Heioron
PINE x NORATMO HPOHAROAHOÏ, peman aaauı Mexammmn, CBA
Anne € Haxompennen Mruonennoli ckopocru. Pesyaprarit cnoeh
peñora on u 1671 r. manu 5 tpasrare «Meron Quorum u
Geckoneamax panone. Onmaxo oror Tpaxrar Gut onyGauronan
mu» » 1736 r., nostomy nepsoii pa6oroft no qubbepennnans-
Voy ueunenemmo cunraeres crarhs Jeißnnne, onyOankonannas
5 1084 r., 8 KOTOpOÏ PACCMATPIBACTCA TEOMETPMIECKAA SANANA O
nponexeunn Kacamebmol x wpunolk.

TIpmpamenme AÓCHMCCH — GecKonewno manyıo Pnanoors x -
=x, — Neiiönum obosmanan dx (d — mepnan Oyxba zar. cnoBa dif-
fcrentia — panuoer»), a upmpauienue opanıtarst y, -y, ON oBoona-
san dy. B cepegune XVIII 3. Dünep aan oGosnauenma nppaire-
mul crax nomBaoBarsca rpeueckoli Gykpoï A. Tepmux nponanon-
mass (mo-ppann. dérivée) pnepmwe nonsunen » 1800 r
» Kare dpamnyockoro maremarmca JI. Apóoracra (1759—1803)
«Bunuenenue MponsBoRMMX>. OGosHavenue npoussoaHoit y M
F(2) mex Ppannyaceni maremariux I. Jlarpamac (1736-1818).

97

Ticropmiecxan enpaaxa

Fnasa In

Tlpumenenne nponsBoHoú
K HCCIEAOBAHMIO
QyHKu nú

Pano unu nosöno scaxan npasustnas
samenamuvecnan uden naxodun
npuxenenue 6 max uau uno des.
AH. Kpuace

§ 1. Boapacranne
n yObisanne hynkunn

€ nowontso nponanoanoii Moreno naxonın
POMO VIII MoNOTOMROCT y.

Venosusem Tepumn «npomexyroxs ncnon
sonar ans oßoomancmus raxıx ancnomax wur
eer, Kak orpeaok la; DI, uirrepsan (a; D), many“
unrepsanu [a; b) u (a; b].

Tipu orom rousu a m D naanınasor zpanur
tu monkanı, a nce ocramminıe TONI ep.
ana (a; b) — anympennuau movkasu npoxe
xumxa.

Hanomunm, vro @yaxqusa f(x) massınaeree
cospacmarujeit na nexoropon npamemyrKe, Cx
Gonvuiemy sHavenno ApTyMenTa coorsercrsye
Gomemee annuenne ynenvnt, r. 6. An mobs
TOREK X, H Xp HA ITOFO NPOMORYTKA, TARHK, We
<i> Xu, PunOnAETOA nepanencrBo

1@D> fe.

Beam ja mo6ux TOOK xy m Xp HO AAG
npoweneynica, Taktik, TO X¿> Xy, MAMAS
mepanenerpo

ION
70 bymruna f(x) massınaeren yéwaarouei u
TOM upomeacyric.

Tip joxaaaremscrne reopen o nocrarowt
YCNOBHAX ROSpacTaHHA HIM YÓMIBARMA DYHEQUE
Nenomayeren CHERVIOMA TEOPENA, Koropan m
aumaeren meopexol Jazpanxea.

£ Mm
Mipumenenne nBOHSBORHO K WOOTEAOBARIND Gyr

pemat
Tyr» gyaxuna /(x) nenpepuinna na orpeske [a; 6] n and-
depeunupyema xa unreppane (a; b). Tora cymecrayer
rouxa ¢€(a;), TaKas, «ro
£)-f(a)=f'(e)(6-a). (0)

FETOra reopema AoKasupactes B
xypce ssicmeñ marémaruxu. To:
acu reoerpmuecrcnli cxmex op"
ayas (1). Tposexex mpamyto 1
(pue. 56) nepes roue
A fay) m B(: £0)

rpadmica yma y= f(x) m
some ory MPAMYIO cenpiyel.
Vexovoil koodybunmenr k cexymedi
E pases

(2)

Panenerso (1) mono sanncars 5 une
PACE)

ia, @)

Ma paseners (2) m (8) enenyer, wro yraopoï konppmusene
Pie) xacaremmoi x rpadmey pymcium y= f(x) B tome Ce aber
xuccoli e pasen yTxOBOMY Koahduunenty E cexyuelt I.

Tarn o6pazom, 1a murreppaze (a; 6) nafineren raKax touKa €,
svo nacarerumanı x rpadicky oymnsm y= f(x) » rouxe © (e; f(c))
napantemna cexymelt 1. ESA

Toopema 2

Ilyer» byuxuss /(x) neupepsipua na orpesre [a; b] 1 aug
epenumpyena na mnrepnane (ab). Torna ecam f'(x)>0
‚Ana ncex xE(a: b), 70 Qymunna f(z) Bospacraer ma orpes-
Ke [a;b], a ecam f'(x)<0, to oma yômaer Ha arom or-
peaKe.
Joxamem ory Teopemy (1OCTATOIHBE YCHODHA Boapacranın
u younauns dy) © momouguio reopemts 1.
Oliver x, 1 xo — npowonomsae TOTKM orpeska [a; b], rare,
0 x, <2,- Tipumenaa x orpeary [+15 22] opmyay (1), nonyuaex
PDA NP (a), C6 (5 xa). a)
Tax kar x,-x,>0, To ua panenerna (4) enenyer, uro mpu
1'(9>0 psinoanseren mopanencrao f(x2)> f(x), a np f()<0—
wopaseuerso f(x2)<f(x,). Bro oomanaer, “ro cemm 7'(2)>0, To
aymayın f(x) Boapactaer na orpeore [a; b], a ecu f'(X)<0, ro
ona younaer na rom oTpeoxe, ©
Tiponexwyrem pospacranna u yOuanns YHKIOIN naasınaror
npoxexymnanu MONOMOMMOCMU DTO DHL.

sı Duo

Boapacranne w youBanne Apr

Sanana 1, Hafırm npomexyrim
nospacranıa m yGumanns dymımm
1)=(-1* 2-0.

D Iiponanognas nannoh pyusumm

para
P6)-2%-D(-0+(-D*=
TETE)
orkyaa cnenyer, wro /'(x)>0 mpm
x<1 u x>3. Caenovarensno, yax-
nus f(x) noapaeraer na npomenyr-
xax x<1 u 228.

Ha mnrepnane 1<x<8 munon-
mseres nepanenerno f'(x)<0. Ilo
‘reopeme 2 ynxies f(x) ObBeer ma Puc. 57
TOM unrepnane u na orpeare [1; 3]

Tpagus Gym y=(x-1)(x-4) aooópamen ma pucye
Ke 57. 4

3anaza 2. Horasarı, ro dynunun f(x) 20d 46x61
Rospacraer Ha Beli unenonoh npsmoh.

D Tan wax '(2)=62~6x+6=6((x-4)'+2)>0 npu neox zen,
xo ymemu f(x) nospacraer ma ncofi sieonoï npamoi. À

3: Haji npomemyri monoronnoeru (PyR:

1) (x)= 28-80" + 2252 +75

2) f(x)m~xe-*.

DD) f'(x)=8x? 60x + 2253 (x—5)(x~15).

Ecru x<5 man x>16, 10 f'(x)>0. To reopene 2 yum
P(x) woapacraer na npomexyrkax x5 u + > 15.

Ha untepnane 5<x<15 manomuserca nepanenerno f'(2)<0,
u nooromy dynKuns f(x) yönmaer na orpesre 5<x< 15.

Deere”.

Ecam x<1, 10 f'(x)>0, a com x>1, ro /’(x)<0. Ilo reope
me 2 dyuxuns f(x) Boapactaer ua npomemyrke x<1 m y6nnaet
ma npomemyre +2 1. 4
1 Banana 4. Myers pyuxıun f(x) nenpepsiunn a orpeaxe [a; B,
Andbepennmpyema na nntepnaze (a; b) m gua ncex xE(a; b) cnpe
Beannzo panenerno f'(x)=0. Jlokasars, 10 f(x)=C, xla; b], Te
yawns f(x) nocronuna ua orpeaxe [a; b].

[> Tiyern x9 — nexoropas owxa orpeara [a; b], x — nponanon
man TowKa ororo orpeoxa. Ilo Teopeme Jarpana f(x)~f (x)=
=f (cre x). Tax Ka celos b), To f'(c)=0, m nosromy f(x)
ac.

MSA Banana 5. Hokasarı, “ro ecax 0<x< À, ro enpanennnnu ne

2
panenersa

yA)

tros mote

100 rnasa Ww
“Tipmnseneniie NPOWANOAHOR HORDE PAR

DD Pacexorpima @ymienmo 9(2) =tg xx. Dre ayan aupoe-
pemupyexa na unrepnane (0; 3), mpmuem of (x)= 1 or

ayaa cxegyer, wro qí(x)>0, Tax Kax O<cos?x<1 np xe(0 3)
Kpome roro, gysxuna o(x) Henpepsisna npu x=0. To reopeme 2
ara qye noopacraer ma npomestyrie [0; E). Tax xax 9(0)=

-0, vo mp O<x<ä

enpaneguruno nepanencrno 9(2)>9(0), 7. e.
lgx-x>0, omyaa crever nepanencreo tg x>x.
2) Pacemorpmm ynrmurio f(x)= 82%, x20; f(0)=1. Bra

dren snogepenunpyen un nurepnane (0; 3) u nenpepusna

ner omax orpessa [0; 3]. munter rosy x=0, ra wor

lim (2)= Jim SRE

=10).
Haron f°(2). To npanmay anßOopenunponanna npononene-

sua f(x) =. Zain a+ q cos x SE (lg x). Ben xe(0; 3»

Wx-1g2<0 (mn. 1). Kpome toro, cos x>0 mpm xc(0; E) m
230. Cregonaremno, f'(#)>0 m no reopene 2 dymratun fx)
user na unmepnane (0; 3). Hooromy /(2)>/(3,) mon xe(0; 3),
a E } me 1(3
faded A

7, OTKyRA crenyer mepanencrno

Ynpaxuenna
Haitra unreppansı pospacranma u yOumanns hynKnnn (12).
1.1) y=5x?-3x- 2) y=x?- 10x41;
3) y= 2x8 + 8x7 — —86x +40.

1) y=x?-3x+45
OP

3.1) ynx-6xt+ 5x

42)

4) y=3Vx-5+1; 5) y=x-sin2x; 6) y=2x+ poossx.
tyes Dyna
E ym xe

$1 101

Bospacranne w yousarne Gynkum

5. Ha pucynke 58 xa06paxex rpa-
ue Gym f(y, Am
Melon nponanonnoN Qyamanı y.
Onpenenere mpowenyrun mos:
pacranıı u Ouen ym
y=ll).

6. Tip xerox onavennax a dy
Aa sospacraer ma noeh cn
oi pod
1) y=x*-ax;
2) y-ax-sinz?

7. Doxasats, wro gysxnus y=V6+x-x* nospactaer Ha orpesxt

[-2: 2] x younner na ompese [4; 8}

Hokasarı, «ro cat x>0, vo:

Did+cs 2) MU +x)> >

§ 2. axctpemymer pynkunm

1. Heoßxonumsie yeaonust axerpemysta
Ha pucynxe 59 n300paxen
rpaquec dyn y=f(x). Kar
BUAHO MA PHCYAKA, cymectnyeT
Taxası okpecrnoers TOM x1, uro
Hanborsmıee annuenme dyukıua
Az) npumunaer » rouxe xy. Tex
axe cnohernom OÚXAARET TOUKA Xp.
TOME x; M x, naasınmor TOMAN
Maxenmyma yon f(x).
Ananoruuno TOUKY Ya HASBIBAIOT rouKoÏ MICIINYAA. ya
M(x), raw war onasenue dpynienuN m roue x, nene ce auauemud
2 OCTAJIBMBX TOUKAX MOKOTOPOÑ OKPCCTILOCTI TONI 2

Onpenenenne 1
Touxa x) nassinaeren moukoli makeunyma Dynruuu f(x), ©
AN AAA BCCX XH X MI MOKOTOPOÑ OIKPCETOCHN TOAKM xp a
noanseres mepaucnerno [(x)</(Xo).
Hampumep, Touka x,—0 annseren TOWKOR MaKcnmyma dy:
mn [69 4-2. B oroñ rome yma mpunuaer nandorsue
annuenne, pannoe 4.

Onpenenenne 2

Touxa xp naanınaeren mowrol wunumyxa dynyuu f(x), ec
AN puis nceX x Xp HI HEKOTOPOÍ opeCTHOCTH TOUR xp He
nornseren nepanencrno f(x) (40).

102 rnasa wi

TIPHMERENNS NPOSBORHOR K WECEROBAHNE PNEU

Hanpuxep, rouka xp=1 xBnReTCA roukoh munumyma DYHK-
wa [(2)=2+(<—1)?, B oroû rouxe Pym mpunmaer mano
Nenkiiee amauenne, paRKoe 2.

Touch MMHAMYMA M TOUCH MAKCHMYNA HagbIBMOTeH MOVIL
oxempexyaca.

pema 1 (reopema Dep
Myers Qynkuna f(x) onpenenena » uexoropoh oxpectuoc-
m Town xo u anipepemuupyeta n PrOÏ rowKe. Eom xy —
TouKa ancrpenyna dyaxmun f(x), To f (20)

Crporoe xoxaanrenserno roll reopemit BSIKOANT aa paneıt
kamHoro xypcn MaTeMaTiNs. Teopema uncer narsnunh reomer-
Deckutit eme: D TOUKe oKerpenyua Kacarermas x rpabuny
Grau napannensun ocu aGewice, K oorany Ce yraonoli KO
Gmutenr f'(z) panon mymo (pue. 60).

Hanpunep, byuxnua f(x)=2x—x* (puc. 61) umeer » Touxe
%=1 maxenmys, u ee npousnomman f(x)=2-2x, f'(1)=0.

Oyaxnaa f(x)=(x+1)*—4 (pue. 62) meer uunmym B TON
we 2-1, Fe 24 D), FDO.

fed-0 y= fz)

Puc. 60 Puc. 61

Pac. 63
2_ Os;

cremas rue

Yenonne f'(x)~0 anaerex neoóxodumux ycrosuen oxempe
aya mbbepenumpyemoñ pymsunn f(x). Dro o3mauaer, “ro cea
X= 10 — TOWKA aKetpemyma AMPDepemIuIpyemoÑ yen, 10
Pe=0.

Topromy roux okerpemyma auchepenupyemoh Pym
F(x) enenyer nekars cpeau xopmeñt ypanmenuit F'(x)=0. Oxearo
ypavnenue '(x)=0 moxer umer» KopHH, KOTOpHe He spires
‘TouKam oxerpemysta «yuca / (x)

Hanpnxep, ecmm (x)=, 10 f (0)-0, no Touxa x= 0 ne as
ARTCA TOTKOÍ INCIPEMYMA dy, Tax Ka 97a QUERIDA BOO
pacraer ua ncoh wncxonoit ocu (pue. 63).

ToukH, B KOTOPMX mponanoaman Oymxnmi oópamaeres 5
yam, naauianioren emayuonapHau movwanu DIOÍ Pyukn.

Saxaua 1, Halıın cranwonapxste roux yat:

D) f@)=2x"—Bx2-12e+4; 2) Are
D> Tar nar f (x)= 62x?-6x—-12=6(2*—x-2)=6(x+1)(x-2), 1
ypammenne ('(x)=0 mucer Kopin x,=-1, x2=2, Crenonareao,
“1m 2 — cranmonapauie own nanmoh Hymn,

2) B orom cayuae f'(x)=e" + xe*=e*(x+1). Hooromy x=-
crammonapmas rouka ymerun f(z). 4
TI Samer, sro pyuxuna momer
HMCTE OKCTPEMYM u m TOUKE, B KOTO"
poi oma He Hacer npomonogicof. Ha-
mpumep, pyusuna f(x)=|xl-2 me
Hacer TpomanoAmoñ m touxe x=0,
ORMAKO STA TOUKA ABRACTCA ana nee
rouxoM wimmuyna (pue. 64). ET

Bnyrpennsia Touxa oßnaerm on-
Penenenun mempepstenoit pyme
1G), » xoropoit ava yuu ne une-
er IPONINOANOÑ MAH HNCET IPONIBOANYIO, PaRHy 0, maamnuercs
xpumuveckou. mowxot. DIRE f(x).

Taxes OGpaaom, jas Toro wrOGH OU A 29 Enima rOKOÏ aKETpe
mysa menpepsranolt dyn f(x), ReoGxoMO, uroöst ara roma
Guia xpirruueckoli nun aanolt Qymenaı.

2. Hocraroumme ycaonnn oxerpemymat
Teopema 2
Hyer» dymrnna f(x) andbepentuipyema 1 nexoropoh or
POCTHOCTH TONI Xy, KPOME, ÖbiTb MOMOT, CAMOM TOHK xo
x Rempepstana » rouke Xp. Toraa:
1) ecxm f(x) Meuser auax € +» Ha «+» np nepexone ve
Pez TONKY Xp, T. ©. B HEKOTOPOw unrepnane (a; Xe) NOR
Roman orpularensua u m HEKOTOPOM murepnane (xo; D)
monoxITeABHR, TO Xp — TONKA munnnyma byuxunn (2)
(pue. 65);

104 nass 1

Tipmienenne NPORSBOAHOR E MCCRERDEARND PMU

Puc. 65 Puc. 66

2) ecan f(x) menser axax © +++ Ha ¢—+ mpu nepexoge ue
Pee TOUKY Xu, TO Xp — TOUKA maxcumyma «yan [(x)

Gone. 66).

KO Myers f(x) meuner anax € +» un e+e mp nepexone ne-
Hea roay Zo. Tora f'(x)<O mpm a<x<xp m f(x)>0 mpm
%0<x<b, To Teopeme 2 ua $ 1 dyukuun f(x) yönınner na mpone-
xyrxe (a; xe] u noapaeraer na npomexyrre [x93 8). Torna f(x) —
sonvonsinee anaenne YENI Ha unreppane (a; b), M nO9TOMY
2 — roue mmuneyma Pymes (x).

Auanornano paconanpumaeren cyan MAKOINyxa. O FI
Banana 2. Halirn rouen oxerpenyaa ymemut:
Diez 2) 1-2) (HD.
D 1) Haïñxex mporsnoxny1o 1
1) =329-1=8(x+ 5) (x-
Hprpapmuman ce x ya, uaxomme jue oramnonapune rouen:
Sis E Mam ¿> Tip mepexogo wepes rosy x1=- sk npons-

Y
E A aes « 207 Tlowrocy 21

AE
1 npondtoanan ne-

TONKA MINIUMYNA,

xarcuxyaa. px mepexone “epes roxy x;
1

er auux € e-+ HA +9, MOMTONY Y

2) Oyuxuna f(x) amddopennmpyena na R, ostomy ce co
ton axerpemysa CONEPAATCA CPE CTANMOMAPHUX TOOK, anna“
uınen kopen ypannenma f (x)=0, re. ypamerna

2-2) +1) O,
oyna (x 2)0 + 1)" (52-4) =0.

Noxyuemnoe ypauenue uneer xopux 1 ==, x,=4, x3=2.
Tipu uepexoxe epes rouky x, dymxuna f(x) coxpaumer owax, u
worouy (reopema 2 u» $ 1) xy ne ammseren rouxoli oxcrpeuyat

Tipa nepexoxe sepea rouxy + dy f(x) monner anar: ©
sie ma ¢=0, A PH mepexoze Hepes TOUKY x, MEHHET anal € +
xa «to. Tlooromy x, — TOUKA MAKCUINYMA, À Xy — TONKA MHI
va pynruen f(z) (no reopewe 2). 4

$2 105

rea yen

[> Oénacrs onpexerenus naunoli @ymKuun x1. Hafinest npous-
Bonuyıo:

a a at
Haitzem crantonapaie roux pyme, pens ypanmene
40-20
res
Tipu nepexone sepes rouky x=0 mu onno us mäipamenuli: xl,
3-2x, (1-2)? — ne menner onax, mooromy rouKa x=0 ne amar
eres TONKOÏ DKCTPOMYNA,

Tipu nepexone uepes rouxy x= % nupuxenun xt (1-2)? ne
Monmior omaK, nupawenme 3-2x, a € mu m f(x) Meraro ana €
ro oanauaer, wo x= 2 — sora maxcunya. 4 E

=0, toropoo mucor nopin 2,=0, x= 3.

3

cote ma e

men Saxaua 4. Haïru oxerpemywn dynein
PORC
D Bem x<-1, 10 f(x)=f,(x)=5x*+x"4+x, a ecam x>-1, mo
102) = fa(x) = 52% —x*— x, Toraa fj(x)= 15x*+2x+1>0 mp Beex x,
a yparnenne fj=15x?-2xr- uMeet KOpE x, = x; ;
Tip sro fá(x)>0 mpu x<- hm «> à, f3(2)<0 mp sx <a.
Dyminma f(x) me mueer nponanogoh » ronxe xo=-1, m
f'(x)-0 npu x=x, 1 x=xXz. Crexosarensno, Xp Xp Kam
kputuuectne roux yun (2). A
Tax ran f()>0 mpn x<—1 u mp 1<x<-À, ro xo ne m

‚naerca roukoït oxerpenyma. [ponssormas menser Sax € ++» ua
o mp nepexone Wepes TOUKY x, MC e-» HA ets NI mepexone
1

sopes rosy Xy. Cxenosarensio,, Xy
a x — rome sang qyzenun. A

—rouKa maxeumyma,

Ynpaxnenun
9. Hañru cranmomapube TOUKM Pyukunn:
1) uma-6x45; 2) yor? 14x 155
a) yn Z+8 aura
5) y=2x*-15x*+36x; 6) y-e- 20%
7) y=sinx—cosx; 8) y=cos2x+2co8x.

1088 nano m

Tiprmenenine NPORSBORHOR K WEEREROSARUD ANTAD

10. Hairs ae TOA bymensm:
Dy- Es 2) y=-lx-11
Dyna AM) yaaa.
Han roux axerpeuyna cyan (11—12).

1.1) y=2x*-20x+15 ae 430-1;
2,8, m3
ars; yt es
Dyna; AS
‘1 y=x+sinas 8) y-6sinx-cos 2x.
12,1) y VI 2) y=(e- 1)"
8) yex-sin2x; 4) y=cosBx-4x;
Hy © y=1-(@e+1)s

1) y= (e+ 2) 2-3); 8) y=(x-5)er.
13. Ha proymee 67 nan rpapux pyme, anamomelter mpons-
voauoï pymknuelt y= f(x). Onpegemro npomexyrxu pospac-
Tann u yOsmanıın VIENA, ee TOUKH HKCTPEN VE
4 Hate axerpenyat Qyrkumm: i

1 nt

A

8) yolx-Si(e- a
az Pre, 67

$ 3. Han6onbwee u Hanmenbiiee snauenna
yHkuuu

Ecru hyaxuma f(x) enpepsisua na orpesxe [a; b], ro cyme-
cTayer TOUKA eTore OTPe3Ka, B Koropoh DYHKUMA f(x) HpHHAMA-
& nanGonvnee auauenne, u rowen, m Koropo ora yat
per mansenbiiee savvenice.

MroGur afin mautommee x manmenimeo anavenna nenpe-

prisnoji ma orpesxe [a; 6] yaknmm f(x), mmeiureit ma um-

Tepsane (a; 6) MECKOMBKO KPUTHUECKHX TOMEK, HOCTATOUMO.

PAUNCAHTL SHauCHHA dynKunn f(x) BO Beex 9THX TOUKAX, a

Taxe snauenna f(a) u f(b) m ma Bcex HONyueHHBIX UHCEN BEI

pers mantomamuee u maunwensuree.

B upuxanamece saqaox mp4 unxomnennn menGomauiero (Hex
xexbuuero) auavenua dyuxumn f(x) na orpeaxe [a; 6] uxm ma HE
man (a; 8) sacro werpewaeren eayual, torna dun /(4)
Auphepennnpyema Ha unreprane (a; b) u HempepErBHa Ha orpeake

53 107

HanGonsuice u RACE ER Oy

(a; 6), a ypannenne f'(x)=0 meer exttacrnennsitt xopem xoC(a; 8),
axol, «ro ma onnom ua unmepnanon (a; Au), (o; 5) nuinonnneren
mepasencrso f(x)>0, a Ha npyrom — mepanenerno f(x)<0.
Boron eayuae weno /(x;) aunnerca manbonsmm sau mamen
mmm snayennem yema f(x) ma orpeske [a; b] mam ma mnrep-
naze (a; 8).

Sameuanue. Iye g(x)%0 un neroropom mpomemyrie,
19 =(£(0)", me NEN, n>2. ‘Torna, cen one ma ym [(e)
X g(x) mpnannaer m TONKe x9 ma OTOTO MpONCKYTER MANÓO MEE
(nanmensuiee) anauenne, to u apyran OVERINA. MPIIEIIMOET à
Towne rp nanGonoanee (mare) ‘maven.

Banaua 1. Hafıın manGomuee u uamtensmee ananenun

ymin 19-43 na onpeare | 1;
DG) 03 12-935 rar ES, art 80, of
yma xy=1, x3==1.

Ormeney [Es 2] mpunnzaencer onan ernnonapnar som
=, (Q)=4. Ma uncen 64,94 x 4 manbomsuree — «meno 91,

mamvembmee — uneno 4, 7. Hanbonbunee snavenne yest
pamno 04, a nmunenvuice pamno 4. 4

„Baraua 2. Halen nanGonvmee anawenne Gym 8()-
=xV6-x? na unrepnane 0<x<V6.

D Tax max g(x)>0 ana neex x€(0; V6), ro nouKa xy annsres
TouKoh manGombmiero amauennn YHKNEK g(x) torna m ronsxo
‘Torna, worga ara TOUKA ABAMETCR TOUKO! NaubonBLLero anauenus
Dyna (= (EC) = 24622) 6x x!

Haïinen nponononuy /'(2)= 242 — 62% 6x°(2 +x)(2—x). He
unrepnane 0 <x<2 qymxnita f(x) nospacraer, tax xax ma oro:
unrepsane f(x)>0. Ha murepsane 2<x<V6 gynxuna f(x)
yOsınaer, van Kak ua rom murepnane /'(x)<0. Caenonaremo,
Touxa x-2 annseron Touro Naxommyma YREIUNE f(x) mn 970i
rouke dynkuns (2) mpummumaer mauboasuree ua ee auavenuh u
unrepnane 0 < x < V6.

Dynruna g(z) rare mpminaer nauGonsmee amaseuke tu
unrepnane O<x<V6 m Tome x-2, m oro amauemne panto
32-2Y6-4-212. 4

Banana 3. Halim Kandonsineo Hasmenbutee axasemus yn:
rune f(x) na mmoxeorne E, eon:

1) [()=2x' -32*-36x-8, E=[-3; 6%

2) 0-2 +1, E=[0; 8].

D Hyer» M — nauGomvmee, a m — maumensanee anauenue dye
mt f(x) na mmomeorne E.

108) raasa u
TIPHMEREHNE NPORSBORHOR K WECNERBBAHME PNEU

1) Tax ak f'(x)-6x?-6x-36-6(x+2)(x-3), ro byuxumna
{(2) weer que eranuonapnie roue x,=2, 278, restan ua
Koropux mpHnazzexur orpesky [-3; 6]. Brruncaum amauenna
‘oymaun f(x) m rouKAX xy, xz M 8 KOHMAX orpeaxa [-3; 6]:

1(-8)=19, f(-2)=36, /(3)=-89, f(6)~ 100.

Cnenonarensno, M= 100, m=~89.

2) Dynxuma f(x) umeer rpm craumonapaste TOuKH x=-1,
und, x=2 ($ 2, aagana 2), ua KoTopsix MHomecrsy E npunan-

omar sono vorm 2, m 2, Tex ar /(O)m4, £(4)

12)=0, $(8)=64, ro M=64, m=0. 4 4

Baraua 4, Mo ncex mpsmoyroan-
icon, RMMCARNBIX 8 ONPYARHOCTS Par
ayer’ R, HAÏTI MPaMoyromsunK. MAN
Gomme naonanı.

D Haiiru mpamoyrombmnk — ato ama-
‘unr malirst ero_Paamepbi, 7. e. Anna
ero_cropom. Ilyers mpaxoyromamk
ABCD sustcan 5 oipyakıtocrs paauyca R
(ome. 68). Obosrasr AB=x. Ma rpe-
yronsmuma ABC no reopexe Iudaropa
axogum BC = VAR

Tinonans npawoyronsnuna panna

S(x)-x VAR, a)
me O<x<2R. Banana onenach K HAXOMACHNO TAKOTO AUBMEE x,
mpx xoropow @ymEmuta S(x) mprumaacr manGorsuee sHavenne na
unrepnane 0<x<2R,

Tax ax S(x)>0 na uuvepnane 0<x<2R, 1o dymunn S(x)
« {= (SG)? mpmaumator nanbonsınee anatenne na TOM uuTep-
mane n OJUIOÑ 1 TOÑ me TOUKE.

Tax oGpasom, 3UAOTA cnesiuch x MAXOACACIIMIO TAROLO IMA:
venus x, mpi xoropom Qyukuna (1) =x (AR? 22) 4x? xt
rprunwacr nandoasuiee onawenne xa mnrepraze 0<x<2R.

wees

I (2) = 8R*x— 4x4 = 4x(RV2 + x)(RV2—z).

Ha uwrepsane 0<x<2R ecrk TOABKO OXNA CTANMORAPIAA Tou-
ka x=RY2 — rouxa maxcmmyma, Caenonareasno, manGombuicc
anasenme hymuun f(x) (a snanur, m yes S Go) npunnmaer
apa x=RY2.

Vira, ona eropona HCKomoro mpaoyrombmuxa pasua RV2,
xpyraa passa \4R*—(RV2)*—RV2,
mx — xnanpat co croponoM R\Z, ero naomans parue 2R°. 4
FS Sagaua 5. Harn aucory Konyca, nuciomero mando
ose cpenm ncex KOHYCOR, BIINCARHNX B cdepy Panuyca A.

53 109
FlanGonniies w navesenaiiee SFR yma

7. 6. HeKoMuIli mpamoyrom-

DB ceuensu epepsn maocKoers10, mpo- B
xonanielt wepea och Konyea, oßpaayerca
oxpyatuoers panuyea I, a m ceueuu
xonyca Toit Ke ma0ckoor. — paBHose-
‚apennsih rpeyrombmur ABC (AB = BC),
nca 2 ory oxpyakuoers © Hem”
pow O (pre. 69).

Tier» D — ueurp ocnonanus xo-
uyea, x — ero nucora, r — panuye 06
novamus. Torna BD=x, AD=r. Ipo- À
aomxux BD 10 mepeceueunn € OKpyam- <<
noetno n vouxe E. Tax war yron BAE
mpsmoit, 70 mo cnolleray nepnenmmKy-
Anpa, omymennoro ua Rep npamo- Pre, 6
ro yraa rpeyronsæmke ABE wa rmmorenyay, AD?=ED-DB, rit
ED=BE-BD=2R- x. Cxenosaressno, = (2R—x)x.

Tiyern V — oönem xonyea, rorga V(x) 4 nix} n(2Rx*2,
orkyna :

Gx UR-3x).

Toe var O<=<2R, à na murepaaso, (0; 27) yoannone
V:(2)-0 meer enmmersennu sopem x= MC, mpmsen Y"(090

MF <x<2R, 70 snanenne hy

Vi(2)= 4 rARK- 82°)

np 0<x< 42 m V(x)<0 mpm

Vin) mpm = LE amame nanbonmne annee anol! dune

unm, Orser. A

<3

3

Ha Koopaumarnoñ naockocrn Oxy Ama toss
Paccnarpunaorcn rpeyronsunkm, Y Koropsix ABC nepun
un, exmmerpuumme OTHOCHTEALHO ocn Oy, ¡exar na napañoxe
y=3x%, -1¢2x61, a rouxa M apxserca cepemmnol onHott ma
CTOPOM KA)KROTO rpeyromsmuia. Cpexu 9THX TpeyrONBuIROn me
Span tor, Koropmit meer manGommyio nxomazo. Hain or
maomane.
DUyer 0<x<1, Al-x; 3x%), Bla; 32°) — nep ommoro 19
paccarpusaemux TpeyronbustKos (pue. 70).

‘Tperna nepmuna C OMPExenserca Heonmosnawno, rar Kee
rouxa M Moet 611th 2160 cepexwioi cropomts BC, 2100 cepert
moi cropomsı AC (ua pueyaxe oto rom Cy u Co).

Thomann tpeyromnuxon AC,B u AC,B pants, Tak ar y
mu oómee ocmomanme AB m panusie sbicorst hy M hp ra
hı=hz<h, peu h panuneren yasoeunoii pasHocTH opzumat 1
uex M u A, 7, 0. h=2(4—B24),

Tiyers S=S(x) — mnomans rpeyromuuxa AB

348 h=xh=8x-6x%. Tax xax ypamnenne S'(x)—

S@)

1109) frave 1
TEE TPOASHOANON E IEEREROBARIE ANNA

ypamenue 8-18x*=0 meer ma
orpesxe [0; 1] eaunctsennstti

open = 2, nparsew S'(2)>0
mpi Oce<d u S'(x)<0 mpu
Pret, ‘ro aHavenne
Sto=s(3).3}
NEE

venues ya S(x) ma or
peake (0; 1].

Omer. 3

i pue. 70
Bain 7. Haltra naxGomnive u namensmnos omascnna QyRK-

me 1x) =bx*-x|x41)

sn orpearo [- 2; 0].

D llaman bymenna nenpepuanna na orpeaxe [-2; 0] u anbdepen-
upyexa no acex TouKax unrepnana (-2; 0), kpome Tout Xp=— 1.
B$ 2 (arava 4) Guino yeranonzeno, «ro dymienun f(x) umeer ma
mirepsane (-2; 0) ext unylo TOUR DICTPENYMA Xy == 4, Ko"
Topas snnneren rowroli maxemyna roi yen. Haxonı ana
sonen Qyusmnt B roue x, u » Kouxax orpeoxa [- 2: 0]. Honyua-
ex -2)=~88, f(-2) = 2, ((0) 0. Haubonsee us arux aucex —

rezo À, a naswensmiee — “meno —38. 4 ME

Yni

Hakan nanGonbuee u naumeusuce auaueum dymenun (15—17).
15. 1) f()=x"-6x!+9 na orpeaxe [-1; 2];

2) f@)=xt_8x243 ua orpeaxe [-1; 2];

3) (@)=2x* + 3x4 ~ 36x ma orpeore (-2; 11;

4) F()=x"+9%+ 15x na orpeaxe [-3; 2].
16.1) f()~2*- + ma orpese (1; 2];

2) f(x)=x-Vz ma orpese [0; A].
17.1) f()=2sin z+ cos 2x na orpeoxe (0; 2x];

2) f(2)=2eosx—cos 2x na orpeae (0; 1].
18. Hairs auGonvinee auaseue Gym:

1) 29-5214 5x41 un onpeaxe [-1; 2h;

2) 1-xt-x6 wa ummepsane (-3; 3);

8) 2-12 ma npoxenynee 20;

ER

111

3
u FPE WET Sp

19. Hañiou nanmensuee onasenme dyin:
D 22418 np x>0; 2) x+ À mp x>0.

20. Mucno 50 sanncarı m nune CYMNM ABYX une, CyMMA Klon
KOTOPHX MAMNONDINOA

21. Sanucar wueno 625 » sine nponsnenenun asyx monorumen-
HDX HCOT Tax, YTOÓM CYMMA HX KDAAPATOD Guia nannens
meli.

22. Ma ncex mpamoyromumicon © nepumerpom p Hale paro
vromsmn manGorsmel mromean.

28. Ma ncex npaxoyronsmukon, nomen» Koropux pasta 9 ext,
malen mpanoyrombintK © HAHMENLUNN MEPUNETPON.

Hoar maxGozoinee m mamsensinee anawennn ya (24-25).
24. 1) /(2)=x-2Inx na orpeuxe [$; ej;
2) Ma)=x+e"* na orpeare [-15 2].

3x

25.1) Ao)=snz+eosz ma organe [x

2) fa)=sinx+cosx ma orpeare [0; E

26. Haitrn nanGomme snauenne Gym:
1) BVE—xV¥ na nponexyrke x>
2) 8x-2x Vx na npomemyrre 2 >
3) Inz-x na mpomenyrxe x> 0;
4) 2x—e** na umrepnane (1; 1).

27. Halen nanwemsuee ananenne oyunu:
1) @*-3x na nurepmane (~15 1);
2) 4+inx na unrepnane (0; 2).

28. Haliru meuGonsuee suaveune ya:
1) VE (5) na murreprane (0; 1);
2) VER) na unepnane (0; 2).

altra nanbonsuee m uansensinee smasenma dy:

D Fe HR na orpesxe [Li Us

+
2. 3 ste
2) fea)-la? +2312 ins ua onpoane [43 2]

30. Hañru nauGomutee u naumensinco oHavenus ym

1-x np x<1,
» {ET pu ao 98 orpesne [15 21:
22° -12x-17 npu x<-2,
ELLE Crit aye oo 8 na orpeaxe [53 1)
112 Fnasa 1

TPS IPOTIOOANON E VCGRGRORARS GYR

31, Hañru nanGommyio naomants IPAMOYTOABMINA, OMA u3 Bep-
um Koroporo neu un oem Ox, NTOPOs — MA KoxORKI TEAM.
oft noayocm Oy, rperss — 8 Touxe (0; 0), a vernepran — Ha
mapañone y=3-22.

32, Mo poox npanoyrommuxon ¢ nepunerpom p nalkrıt mpao-

STOIBHHK € HaMMeHDUNelt AMArOHE AO.

Hs ncex nPANOYrONBHEX napastenenumezos, BHHCANLIX D

cQepy paauyea R u uneouwx » ocnonamım Keanpar, Hair

‚paanenennnen wandonnero obnena.

Haine yraosok xombbmumert upnmol, npoxozameli sepes
TouKy All; 2) m orcexatomeli or nepporo KOOPAMMETHONO yr-
Ja Tpeyronsame: HAMMENEMEÍ xmas.

IHR xoopaunaruoit uaockoeru aama rouku B(3; 1) u C(5: 1).

Paccwarpunaiores Tpanenumn, Jus Koropuix orpeaox BC anna“

erca OXMNM us OCHOBAHMI, A BepIINHN APYTOTO ocHORAKHs

xexar na ayre mapañoma y=(x-1)%, mnexsenoñ yenommen
05x52. Openn arnx Tpanenuh BMÖpana Ta, KoTopaa meer
uaubonmıyıo naomaxs. Haier ary moans.

Pacoxarpumaioren RCeRODMORKREIe Napaboms, Kacaiomnees

ven Ox m mpanolt y= $-3 u Tanne, 470 Hx Berau menpasze-

im una. Haiirit ypamnono roh mapaboası, nun KOTOPO Cyr

a Paccroammit or Havana KOOPAMHAT Ao TOueK mepeceuemn

Hapabonsı © OCAMI KOOPANHAT HBAKeTCH HANNeHBUNeH.

$ 4. Nponssopnas sroporo nopaaka,
BLINYKNOCTb M TOUKM neperuba
1. Tipononoxnas nroporo nopamia
Tiyers oynxuna f(x) angpepenunpyena ma sumepsane (a; 0).
Tpousnognyto oro hynsuun f (x) nasumaor nepsou npoussodnoü
HIE npouseodnot nepeoeo nopadxa bymwsuun f(x).
Een Qyaxuus (x) amppepenunpyena ma unmepsane (a; b),
10 ee IPOMINONIVIO uasnınanor GMOPOÈ npoussodnou MAN npous-
eoduou emopozo nopada Hymn f(x) x 0Goomagator f(x), 7. e.
ROUE
Banava 1. Haitrn f"(2), ecam:
Di@mxt-Brttxtd; 2) f(x)=sin 3x;
3) f(x) <0"; 4) f(x)=In(x? +4).
DD fm de? 6x41, [= 1226;
2) l'(R= 8008 3x, f'(x)=-9sin 3x;

3) F(a)=2xe", f(x) = 207 4400";

AS

ga 113

TIPORSSANNEN BTOPOTO TOPRAFA, BUMPRIOCTO I TOWN neperGa

y
y= fx) NA y=la)
de. A 2

a) o 0)
Pac. 71

Buscuum Quanuecxuli eumen propolí nponasonnoit. Tlyers
saxon apuoxcnns vanacrcs dopmyaoh y=8((), re $ — myri, mpoir
Aeuutll marepuaxbroli rouxoï, — spema. Toraa rrmonennos
exopocrs Amena (1) (1) xapaxrepmayer Önierpory uanene
HHA nyri, a ORCTpoTy (CKopocr) HoMeNeMHEA camol exopocra os
pexenser yoropemme a(t)=0(0, 7. 0. a(t)=s"(t).

12. Buinyxaoers Pym

Ha pucyaxe 71 (a—0) uoo6parcems rpagas Gym, unen
mx na murepsaue (a; b) nepayio a sropyio upoussonume. Busse
HIM, D vem cocTouT PRIME B ROBE DTUX yal aa
vie OGM csolicrsanar on oßnanaor,

Ha preyuse 71, 6 uaoGpaxen rpaque yOunaromel yxy,
a na pucynke 71, a — rpadmx noapacrammeh dymium; yn
aus, rpadu nordpoi mpeseranzew na pueyaxe TÍ, 0, ne masse

Oxmaro pce xpupste, naoGpexenmue ma pueyuxe 71, oGnate
107 oÓIMuM cnoliergon: € Bo3pacrammen x or a 20 b YrAOROË Koa
bunuenr KacarersHoH Kano Ma rx KPHBLX ymembrnaeros
T. 0. MPONaROAHAR K0AOÍ 19 COOTRETCTEYIOMU Gym — yOu
mamas dymemaa na unvepnane (a; 0), OTKyxa crenyer, wm
FO.

His pucyaxa nano, ro ana moGoï own xpe(a; 5) rpaèur
Gym y=/(x) mer eme Kacarennoï x oromy rpapuny
D Touxe (xo; /(x0)) mpm Boex ze(a; 6) m x% x9. Moorony oyununs,
Hau KOTOPUX HPENCTABIEX Ha oTOM PICYBKE, MasBIBAIOT ou
nyanau eaepx.

Onp:

Dyuxuma y=/(0, audpepeuunpyeman ua unmepnane (a; b,
massınaeron cunyicaol aeepx (cm. pue. 71) ma vrom mnrepm:
ne, econ pyusuna /'(x) yOuinaet na uureppane (a; b), u em
nyreaoit anus (pue. 72), cen dymunn F(2) noapaeraer me
unrepsane (a; b).

114) trace ul

Tipnmteneniie NPORSBORHOR E wCcHeROGANMD PAR

veh) y= fx) yet)
AA ARI de u mc
a) 6 e)
Puc. 72

Orwerun, «ro coma dbymenusr y= f(x) manyıcıa nmepx un su
repsane (a; 8), a M, aM, — route rpadicka oro ym e ao
TaCCaMa Xy M xg, AC a <x,<x2<D (pue. 78), To ua nurepnane
(as 2) oror rpadpuk ner mme npamoh, nponexeunol wepea
roa My u My.

Viurepnansi, a koropuux dyaxnus ESIMYICIS mpepx JONA ma,
rear uumepoanasu sunyraocmu ITOK Dynruuu.

€ nomontbio Bropott nporanoanoli f"(x) MoxKKO maxozırms um-
repars punysaoeru Gym f(x).

yer dbymeenus [(x) uneer na umrepnane (a; b) nropyio mpo-
ruoxayio, Torga ecan f'(x)<0 ana scex xela; 0), ro f(x) —
sóxnmomas dymenss ($ 1, reopena 2), x mommomy Pymenma f(x)
kunykau nmepx ma unrepsane (a; 6). Ecan f"(x)>0 mpi ncex
32(a b), ro dysxuss f(x) sospacraer Ha unrepsane (a; 0), u no-
roxy Qymaun f(x) manyiena anna ua uurcpmane (a; 0).

Banana 2. Halirm unTepBansı BBIIIYIAOCTI BRePX u BHNYKAO-
cou ana Gym (x), ecu

coo acta

DA: 2 160 3<:<3

D1) F(2)-6x. Beau x<0, ro f'(2)<0, u mooromy dyaxuse f(x)
sunyrra mnepx; ecm x>0, To f"(x)>0, u mooromy oymenna f(x)
sunyxaa anus (pue. 74).

B28 Pac 78

os

SSE TOPOS TATE, BERGE WTO TRES

Puc. 75 Puc.

2) f"(x)=—sinx. Ha unrepsane (—1; 0) cnpaseaauso mee
nencrso f"(x)>0, nostomy dyHKuna manyrna Buus (pue. TO,
a Ha unrepsaze (0; x) oma Dunyıaa pnepx.

Sa FH Canoe BE em

xe(- 3; 0), ro sinx <0, cosx>0, /"(2)< % m noproney «pymes Aa)“

tex mary maepa un unmepnase (- E; 0). Ha unmepnane (0:
ara dyuenna pomyrxa Buna (pue. 76). 4
Banara 3.

D> Mpanes y=2 x mpoxonur wepes roux (0; 0) u (5

rosine>2x,
1). yes
vis yasin smyexe meps (sagasa 2) na unrepnane (0; 3
1 nooromy ee TPAD na OTON wrepsane nextT BAILO APRA!
y= Lx (em. pue. 78), 1. e. nepanenerno enpanenauno. Sauer,
WTO 910 me mepanenerno » oaxawe 5 no $ 1 noKasaHo rase. 4

3. Touru nepern6a
Ana Gym 12)
Pacemorpennsx B sanaye 2, TouKa x=0 Ananer

3. Nokasars, ur ecnt O<x<

np - 5 <x<
ca ORNOnPeMENNO KonoN HITEPBAJA BEIMyKAOCTH BHEPX u Kommen
HETEPBAJA BEIYKAOCTm BIS.

Onpenenenne
Tiyers dynknia anbbepennupyena na unrepnane (a; 5)
xoC(a; d) n nyers hyuwuun f(x) Bumyxna BBepx ma AKON m
nurepnanon (a; x), (Xo: 6) u munyiena anna ma apyrou m-
Tepnane. Tora TOuKa Xp naasinnerca mouxoli nepezuda aná
yaKunn, a Touka (Xp; f(xo)) — roukoï nepermöa rpabue:
Hymn y= f(x).

116 nana u

Tipumenenivo MPOMSEGAMON Y MOCREROENND PNR

Have rosopa, » rowke nepernGa audibepenmmpyenan ym
UA nenner nanpanzeune BKINyKAOCTH.

Tiyers Oymosmura f(x) nueer BrOPyIO HPONIBOREYIO na unrep-
ante (a; b) u xocla; 6). Tora ecan f"(x) menser onax npx nepe-
XOKE Hepes TONY Xp, TO Xy — TOWKA mepernöa Hymn f(x).

Banava 4. Hañru roux nepernón Pyme:

DIE; 2) f(x)=x4-6x2 +7.

DY eae tre, f(x) me (Lx) +e me (a+ 2).

Tax na f'(2)<O mpit x<—2 m f'(x)>0 mp x>-2, 70 x
roma ueperuéa ya xe". Mpyrux rouex neperuGa y srofi
ran ver.

2) f(@)=4x— 12x, f° (x) 122°-12- 128-1).

Oyasana f'(x) menser omax npx nepexoge “epea TouKH
alu x=, KoTopHe anamoren roukanı Mepernón yum
{oh

Oren, uro rpabux aubhhepenunpyesoii Hymn f(x) up
mpexone vepes Tony neperuGa roro rpaduia Mo(xoë (x0) ne-
Pexontr € OnHOH CrOpOHH KacaTeMBHO! K ITOMY rPADHKY TOUKE
My un apyryio CTOPONY.

Banana 5. Halıru rouxu neperuda pymkunn:

D = 2;

1

D fe

=
ae
rr,
MO ae
PE Tee)
Get

Tax xax 8x*—-1>0 npu E

6
Lario Le La
ew

‘war neperuGa (pue. 77).

2) Maiizen mepnyıo npouanoauyıo:
1 1 1 1
rr à ee 3}
(elk (3 (e-1? (ear
is oroit dopayau pure, ro f(x)<O mpn x#1 m x23. Cne-
sonerensio, @yERMNA yOMDAST Ha nponeyrkax x<1, 1<x<d,
253. Haiigex mropyio npouanonuyio:

ren +
e

gs 17

TIpGrSSORNaR BTOpOTO ROPARKA, BMYIIOCTE u TON TEPEmMÓS

Peumm Hepasenctso /"(x)>0, 7. e. Hepañencrao

Haurekan Kopeus kyOuieckitli, nonyunem pankocunesioe ne
1 1 21-2
Panencroo IT >— zig, aan Dg > 0.

Tlonyvennoe nepanenerno nepno mpu 1<x<2, x>3, a nepa-
nenerno /"(x)<0 enpaneaauso npn x<1, 2<x<3. Caenonaressuo,
anna hynxusa BEINyKsa BBepx Ha npomexyrkax x<1, 2<x<3
u BBIMyKia Bus HA upomemyrkax 1<x<2, x>3.

Tax war » roukax x=1 u x=38 dynKnus ne onpenenena, 10

— enuinersennag rouxa neperuón. < III

Ynpawnenns

37, Halen nropyıo mponanonuyo Gym:
D f(a)=sin?x; 2) f(a)=x3sinxs
8) f@axteBxt—x41; 4) (jr;
DC 6) 16)=In(*+D.

38. Hafıım nnmepnamu DMYKIOCTH BRCPX m HITEPROzI manye-
x0cru wma yc:

1) M)=x-10743x+13 2) f(x)—x*— 6x7 + 8x +4.

Haliru rouxu neperuGa yen:

1) f(2)=cosx, -n<zen;

2) f(x)=x°— 802%;

DNI

4) f(a)=sinz— 4 sin2x, -n<x<n.

40. Hakrn unvepnamuı munyxaoers DICPX 1 unrepnanur many:
‚noeru sas yc:
E [ext -Gxlnx.

41, Hafirm vous meperuGa Gym:
D) FO) 12224 ee +552) Ona 12x 48x44 3;
38) ed 4) f()=x*nz.

§ 5. Noctpoenue rpabukos bynxynit
1. Acumnrons

B raase II ($ 2) Guno nueneno nonarne acnsınrors (nepru-
Kansuoli m vopnsomransnoi). Pacemorprm HONTE maxnommoh
acnmnrorst.

Hasonen npanyw y=#x+b acuxnmomoi (nenepruxansnoh
acusmroroh) rpapuka yumm y=f(x) mpx x >+00, ecm pas
ocr opaunar rpabukos Gym f(x) « mpamoit crpemures x
Hymo pH a+ +00, 7. €.

¿Him (92) (hx +0) 20. a

18) rasa u

IPHONE NROWSBOAHOR Y VCCREROBSRND NN

Eon #20, 10 acmunrory nastinaion naxxonnot, a ecan k=0,

10 scuunrory y=) HAOHBAJOT copusonmaavnot.

EE
ua roro uroGë npamas y~kx+b Gena ACHMNTOTO rpa-
daxa dymenun y= f(x) npn x—+00, neOGXOMMO m aocta-
Touslo, UTOÓN CYINECTRORANH KOHETENO PCA

lim MD, @

CS @)

(AO 1) Ilyer» npamañ x +b — acmmnrora rpaduKa dynK-
tun y=/(x) apn x— +00. Torna nuinonnaeren yenonme (1) mom
Paruocuabnoe enıy yononne

f(x)=kx +b +a(x), a(x)—0 npu x— +00. (Mm

Pasuenus 06e uacru papenctsa (4) Ha x, NonyaHM
bat)

+
orkyaa exexyer, 470 cymecrayer upexex (2). Ma pasencrsa (4) no-
san

19) ps

1()-kx=b+0(0), a(x) +0 mp +,

orkyge eneayer, «ro cymecrayer mpenen (3).

2) Tlyers eymecrayior npegenss (2) x (3), Toraa

1(@)—(kx+b)=a(x),

me a(x) +0 mpu x—+00, 7. e. munoxmuerca yexosue (4) m pas-
nocunsnoe emy yenonne (1).

Bro oamauaer, uro npanan y=kx+b — acummrora rpaduxa
tant y 16). O es
E Ananoririno spores mouse ACHMIMIOTES mpi x—+—o0. B rom
Cayuao AONKIO BLIOAMATECA VCHOBHE

(0)

CENT

4) 10)

5-264) 5 5
Dy Tex man o AA
2 —ropusouraxsnas acnunvora rpadmia

—0 mp
2-0, 70 mpanas y=

FEE mp sn a.

yaa y

5s nto
Tiocrpoemne rpadnnos Oya

D Ha pesenerns PERE
1

70 mp x, CACAYOT, WTO MPAMAR Y=X — acunntore
1+

# 3
pagina dymenr y= mp Kohn x 00.
3) Ina KaXOKAEHUA YPABMEMMA ACHMITOTE BRUNCIIM npe-

eat (2) m (8):

i “ A
lim 9 — tim - lim =1, 7. 0. k=1. Jlance,
Pr rer wien:

PAGA pj a

im 10-29 Jm (E 1) = tim EEE im =
Mar 1) = (+? ae (a+ 19?

#6)

Tai o6pagow, mpawan y= x 2 — ao rpadymKa dy

nant y

DE M Een #00,

BamerHM, «ro ypaBHeHNe TOM axe ACHMILTOTEE MOMHO noay-
suns, pasnenuo x* ua (x+1)? no upapuna nencun nuorounenon.
Tipit 9t0m NOK BOCNONIORATHCA PARONCTRON

UA
Torxa nonyasm
Ensign et,
cms er Gry’

orkyaa cnenyer, “TO IPANAR y=x-2 — acunamrora rpaduna Oya
mm y= f(x) npu +00 1 22-00.

4) Hafinens acnunrory rpagmxa Prof: Aymkımm mpm x +0,

Myers x>0, rorga VaT=x, u prove

A Ta mp ste,

1. o. h=1 (yenonue (2).
Mane, (0) VEB

(0)

ER
Va?-2x-34x

213

1m x +00, 7. e. D

Vx? 22-34 5

1205) trace 1
——, 102000)

Cnexonarensio, panas y=x-1 — acunenrora rpadbiica yin
un y Var 23 mp +00,

Anaxormano MOJO MOKAJATL, “TO MPAMDA y= 1—x — acim
tora rpaduxa avoit yet upu 2-0. << A

2. Tpadmen yen

Banana 2. Tlocrpours rpapue Gym

yrx?-2x7 +x.

1) O6nacrs onpenenenus ynxnin R.

2) Tau bymcrura meer e ocio Ox nue 06npe rouxu: (0; 0)
2 (15 0),

(e

3) Tax war ÿ'= 88 4x +1
ÿ=0 micer op x)= À, red ;
Tponssonnan noromurensna na npomexyrax x< À m x>1,
cnexonarensi0, ma PIX mponerkyricax dynam nospactncr.
Tipu }<x<1 upowssoauas orpuuarexsna, caerosatensHo, ma

L)t-0) 10 ypauenue

rox unmepnane ymin youmaer. Towra x=] anngeren rou-

3
Koll MAKEHMYMA, TAK KAK CAODA OT SOM roux DYHEIIA BospaC-
‘met, a cupana yOuinaer. Buauenue @YBKUIM m roi TOUKe Pano
da
3) 7\s) 25) +52

Towka x¿=1 ABnaeron TOXROÑ MIEIIMYMA, TAK KEK cneRA OT
oli roux yann yOsimacr, a enpasa nospacracr; co anavenne
2 roue sunmeysa panto f(1) 0.

Peayanrarın mpencrapma » rabanue.

1 1
x a exer 1 x>1
8 3
re, + o - o +
4
fi) z a x o 7

CmBox «76 oonauner, uro dyn
aux Boapacraer, a CUMBOR «Ve oaua-
wer, vo Qynuuns yOusaer.

na Goxee Tounoro nocrpoenua
Tregua nahnen snauenun Oy arras

exa ma roman 1 -1)=-$.
12)2.

Tenonayn yeoyavranıı neene
amare, con aba Graz
y=x3-2x" +x (pc. 78).

ss 121

Trocrpoenne rpagnKos Span

I Tipn nocrpoenmm rpadmka Gym y= f(x) MONO mpunep-
KHBATSER CCAYIONErO mama:
1) nalirm o6nacr» onperenenua dymaunt BEAUTE, ARANET
ca au byuxuna uernol (neuernoh), mepuomuuecKoit;
2) warn rouku nepeceuenun rpaduxa c ocamu Koopqumar 1
npomeryrKit, ma xoropmx /(x)>0 u f(x) <0;
3) maltrn aCUMITOT PPAQUEA QYERIME;
4) seinen» f(x), malira npomexyren nospacrammn (yü-
ana) YI 1 ce DRCTPEMYME
5) suuneaurs f"(x), onpexenurs nanpannenue Bunysaoenu y
Haïru Touxu nepernöa;
6) mao6pasurs rpadı ya.

Banava 8. Ilocrpours rpaduk hymenun y= x*—4x" + 4x.
> 1) O6naer» onpezenemus pynsuun R.

2) Tpaguue Oynaum y=x)—4x*+ dx x(x- 2)? umeer c ocbi0 Ox
nue oGmue roux (0; 0) m (2; 0).

3) Tpadux me umeer acmunror.

4) Tax war y=3x"-8x+4=(2-2)(8x—2), ro ypannenue

17-0 nee vopun 22, 21-2. ponmeontea noroxomemaa, 7.

Ayers, pere es, cometes ae on ESE
2 2
2 <x<2, 10 y<0, n Oymanan yónmer na nnrepsane (2: 2)

Crartonapmue route += 2 m x7=2 — sown oxrpouyua

3

mm. Ip omy To maven, re ah pa neg

xone wepes TOHKY X1= 7 MPOMUBOMHAR Y MeHACT 3HAK © «+» HA e»

D aa map ADS ee
:

mponasonnas none ak € oe ma stos (4) E, V0.

5) Haxonım y =6x-8=6(x- 4). Ecau x < 4, v0 y"<0, um

sToMy OYEKIUIA BRINYKAA BBepx ma untepnane x< 3; can x>4,
a 4
10 y">0, m nooromy Gym munyicra amo mp x>4. Cregona-
qomo, 54 — pour meperata Grime vorne (1-8.
Pesyaprarhi HCCACAOBAHMA MPEACTABUM © HOMOMLIO TAÓAMUN.
= Je] 2 [ls] £ [ae] 2 [el
5 fi] 3 +
al |o | = 1-1] 10°]
re} = [oe | + ı [=
= 16
mes) aw Pe [es] oo |
1e 27 27 min
max nepern6
12208 ana 1

Nipmacnenne TPOMIEOAMON K HECIOROSIMAO PAU

Comox «ns osmauner, «TO ¢byHE-
is Bunyıaa BBEPX, a canon «Un
casauaer, uTo YU BRIMY KIA
na,

6) Ormerum eue, «ro f(x)<0
spn x<0 u f(x)>O mp x>0, x22.
enomaya pesymsratsi MocHenoBs
zu, exponm rpabue yum
ynx-4xt +4x (pue. 79).

Banana 4, Hocrpoure rpadırk
gan yor 4. Pac. 79
D1) Odaacrs onpenenenus x%0. Jlannas ynkuna Heueruan, Tax
vax (2 ) ri.

2) Tpabnx we mepecexaer KoopaunaTHure ocH.

8) Hecaeayem ory Pyukumo u nocrpomm ee rpadux np x>0.

Tipaxan y=x — acmuntota rpadırka dymeumm np +00 m
3-0. pu x>0 rpagux Mei DIO ACHMNTOTL (Fak Kar
450), a pu #<0 — mue acumrorss,

4 Da,

ri e

Ha npomexsyre x>0 yen
weet omy CTAMMOMAPEYIO TOWKY"

Tiponsnonnan monourensna na
sponexyrke x>2, CHEROBATENENO,
Za Tom MpomeryrKe bymKIUS BOS"
sucer. Ha murrepsane O<x<2 mpo-
some OTPHMATEABEA, CACHOBA>
Temo, HA TOM HITEPRAJE YU
sünmaer.

Touka x=2 smanerea Toukoli
Huy, TAK KAK DH mepexone
‘pea STY TONKY UPOHsBORHAR MEH
eranax ce» un 449; [(2)=4.

Cocrammos ruby:

= CE
re) =
ix) +
1) EN

re(1-4)2, 10 mpu x ES Dur
5) Tar wax y"=(1-S) a5, ro mon x<0 yuan
nyrna mmepx, a mpi x>0 hsnyidia

6) Tpapus oymunn y=x+2 usoGpaxen na pucyne 80. 4

2

Bagava 5. Hoerpours rpapm Pyme y= =.
D 1) @yuxuus ompenenena npu x#-1. ee

2) ynxnua npunumacr NONOMMTENLULE sautent MPa x>0
u orprnaremmne mpi x <0.

3) Mlpamaa x=-1— weprukanbHaa aCHMnTora rpaduks
yusnun, npuuen y 00 npu 140 u 21-0.

Tipamas y-x-2 — HAKJOHHAR acummrora rpadıma pymKnn
pu 2400 H x—-00 (oagana 1 (3). Tip orow 19 panenorna (6
oneayer, uno mp >} rpadı zen PIE acmenrora, PL

PET
(+ Dé

= @@+1)-20)=
[ers
aD,
ar
ed m
ar

Ha dopwyau (7) cxeayer,
ro y’>0 np x<-3 u x>
a ecan -3<2<-1, 10 y<0,

Cnenonaremsuo, Pyme
nospactaer ma mpomenyrKax
x<-3 u x>-1 m yösıner na
nurepsase (8; —1).

Coraacuo gopmyze (7) byaxuus uxecr ave craunonapaue
rout x=0 m x=-3, Touxa x=0 ue aBAJeIca Toukoli axerpesy-
Ma, TAK KOK y! Me monser JHAK MPH MepexoAe «cpes ary TOUT,
a 2=-8 — rouna maxcumyma y (x), Tax KAK y menser
Bua € ete Ha +» mpu mepexone Nepes TOUKY x=-8, mpix

yen.
5) Haligen y”, monoxseyx opuyny (7), Koropyio sammmer
bonne y EEE Tonga y CELE REDE
+ e+

MEAT TETERA
Gry ery

124°) traga ur

Tiphenenre NPOSBORHOR K WECNERDEBRNO MUR

ro OE ©
rn

Ma dopmyaı (8) exeayer, uro y’<O mpu x<0 n x#-1, y>0
go x>0. Hosromy dbymxnna y(x) BRINYK:a nnepx Ha ımrepmanax
x<-1Ln -1<z<O m minykaa nun na unrepnane x>0. Touxa
x=0 annaerca TowxoH mepernGa Pym, mpmuer y/(0) +0, u mo-
srouy Kacarensuoli x rpaquxy Hymn» roue (0; 0) apager-
x mamas y=0.

6) Tpadux oymenı maoßpamen na preyune 81. {I

Ynpaxnenun
Hoerpouns spadın yuan (4248)
42.1) + Bx —

3) + 6x2 +92.
43.1) + 4)
3) 5 4) y= 6x4-42°,

44, Hatten acnmmroren rpadmmea yma:
1) f= 224; 2) = aah
DOS, ay fan

+2

a
45, Haliru acumnrorst rpaduxa dy

1 ft eur, 2) 1) =V FAXES
en?

Tocrpours rpadux Gym (46-51).

46.1) y=245x9—3x5; 2) y= 32-52%,
8) y=4x5—5x° 4) y= (x1 (x41).
ova duo alos yr

x ve
BD rr 2) post ax? x43.
tl

$2, Cromxo efermeremsmux noprei uneer ypannenne
1-2x+20 252.07

gs 125

Moerpoenue rpadon Py

53. Halıru unrepnamı noapactanna 1 yOunanus Gym
2

order;

ay y= 8-15 ar

54. Maltrx cranwonapuue town ya
D yaxtaxt-8x241; 2) ymax! 2x4 35
ayi A 4) y=cos 2x + 2008 x.

55. Halit roux onorperymn yen:

1) y ast 2) y=Bxt— ax

56. Hañn row axerpeysa x anaenna bynes m Ovi 10%
ax:

Dyna 2) yw 0,229 4x23.

57. Mocrpours rpaie dyn
u) vi

Dumas Du

ya + 3x4; ay

58. Hlocrpowrs. mpage dy

1) y=8x*-6x+45 ma orpesxe [0; 3];
2) y ; xt 2 38 na orpeaxe [-1; 8].
59. Hañrn HamGonsmee m HanMenbince SHAUCHHA yMKUHM:
1) (= 62249 na orpeaxe [-2; 2}:
2) f(x)=x’+6x?49x ma orpesxe [-4; 0);
3) [(4)-x!-2x*+3 ua orpeaxe [-4; 3];
4) f(x)=x*-8x*+5 na orpeaxe [-3; 2].

60. Jloxaaars, "ro ma ncex panmoßoapenmux rpeyrostamtion, mo
canmax m manual xpyr, HANGONDUIMÉ mepmnerp MEET par
nocroponnitit zpeyrommnx.

61. Ma ncex Papnoßonpenmstx TPEYTOABHIKOD © repuMerpou p
HAÑITA TPCYTOADIME © HANGONBNCÍ HAOMEAMO.

62. Hs ncex npamoyroasuux HOPANACACHMNCAOD, Y KITOPAX DOS
HOBAHIM AC%MT KBAMPAT A LMOINANS UOMO! moBepxuoen:
panna 600 ex?, natin napannenenunen mauGommuero Obie.

Hadru acmınorsı rpadiKe yaris

D fo

Pu

64. Hoxaours, «ro Gym y~1,8x°-2 4 a 4 7x4 12,5 poupe
‘mer ua ncoii oÓnACTH onpexenenns.

65. Joana», «ro dymunun y=x(14+2V%) nospacraer na wei
‘oGnacrm Ompexenemma.

128 rnaca w

Tipnmenenite NPOSBORHOR K WESHEROBARND Oya

$6, Hadırn rome oxerpenyaa dynrignr:

D y=xIn. a 2) y ae
si.
E NUT
SE Hocrpours rpadu pyme
Dye; aut:
UE der
DURA D vx).

68, Haïrn manGonbuiee u nanmensuiee snauenus DYHKILUME
1) f()=2sinx+sin2x ma orpeske [os +)
2) F(x)=2cosx+sin2x ma orpeaxe [0; al;

3) f(2)=Bsinx+Aeoe2x ma orpeoxe [0; 3];

4) Ig =si
89. Haïti nanbonsuee anasonne Gym:
D x\5=BF ma murepnane (6;

2) «VIE ma unrepsane (0; 1).

10, Teno anınkeren no saxony s( 6. Kaxona manGomuan
‘exopocts tena?

TL Ma ncex npaoyronsux speyronbimkon, y Koropuix cysata
oxxoro Karera m rmmorenyası panna 1, Main TpeyroxbmuK €
nanbonsmei mromabio.

72. Ham rouxy Kacamns rpadıtra Gym x nammolt mpanott,
emu:

1) y=8x2+2x-6,
Duke 245,
By-xt-5x48, y=Tr+2
D y Bar +1, y=-18x-5.

Tipx scascom anse a rpadine dymemm y= xt + a Kacnerex
npawoit y=— dx + 5?

74. Coxommo ronmens umeer popmy B
upmoyroasuma, eBepxy sanep-
uennoro moayıspyrom. Onpene-
Mer» pannyc monykpyra, npıt
Koropom Troma ceuenun Oy- x
ner manborsınei, cam mopt-
erp conenna pancn po

BB Pannoßenpennsiik rpeyronsunx
‘onvcaxt oKono KBaApara co e50-
pouoli a Tax, uro onua cropona

=+2\Beosx na orpenne [05 3].

xmagpara ner ma ocmomamım À D a
tpeyronpmua (pue, 82). OGo- puc. 82
127

Yapamena x mae I

ai”

Puc. 83 Pre. 8

ananas BK =x, nahm raxoe manette x, Hp Koropon nar
jab IPEYTOIDANKA naunensuan.

[76] Ma xnanparnoro ,mora kaptoña co croponohi a (pnc. 83) ms
20 cheat» OTKPATYIO CDEPXY KopoGKY mpamoyromanoi Gop
Muy muipesan mo yranx KnapaT m garnyn oGpnsonamunecs
xpaa. Kaxoit romkna Gus Bucora KopoGKit, 17068! ce obres
Shan Mando?

FE Hatıon nauGommuñ oFxex uannapa, momen moanoï no.
nepxnoct Koroporo panna S.

Haken nanGommyio naomaxs nomno% monepxnocrn mama

pa, sucamuoro » epepy paauyea R.

GB Hair Tom oxerpemyna @ymemunt y

Nocrpours ee yaaa

D fers VE; 2 Mayen VÍ
es Dian
er 6) (=
E Me a

[SL] rpys, xexamni na ropmsonransmoit nzocKocTH, myano cer
EYT» € MECTA cuxoï, npuroxenoï x aTomy rpysy (pue. 84).
Onpenenurs yron, oGpasyemutit sol cuaoRi € nnocKocrE,
pi KoTopom cna GyzeT maumensureh, ecam koodchuunert
Penna rpysa k.

Bonpocei x rnase ll
1. Kaxas dymenua nostinacten nospacrmomel (yéunaioueli)?
2. Céopwymmporars Aocrarounoe yeromue noapacransıa (yÓmpo:
au) yet.
3. Yro raompaeres rouxoit MAKCHNYNA (rmineyme) Oya?
4. Chopuyanponarı reopemy Pepa.
5. Yro anaserca meoßxonnmn yexopuen oxcrpeuyna side
pennupyemoit dmca?

128 res u

Tipumenenne MEOHSEGAMON K WOGREROBSRND SYM

, Kate von asuipaioren craunonapmuum?
Kate RO MAROC RPC?
Chopryamponars aocraroumue yoromn oxerpenyun.
). Kaxon aaropurat waxomzenun nanGonsuiere 1 wansensuiero
sun Henpepsinnoft na annannon orpeaKe DR?
Coopuyzuposars reopemy Jlarpanxa.
Fons xocrarounoe ycroBme Bospacranıa (yOusanna)
a.
. Jlokasars AOCTATOUMBE YOJOBHA DKCTPOMYMA.
ro masuisaeTen IPONSDOANOÍ BTOporo nopnaxa?
Chopmyanpoparı onpexenenme BRnyknocrH BBepx (Bmma)
‘oyakaun,
ro nassBaeren TouKoit nepern6a dymrım?
Kar: ¢ momo propel nponsnognof! nuncnurs, annnerca nu
JERIA BAMYKIOÍ BRepx (BHMS); HMeeT AH TOUKY nepernöa?
ER IIonenurn reomerpwseexui cusica reopemix Jarpauxea.
WW tpn sasom vero npaman y= Ax +b nanneren acunnroroi
rpaduxa @yuxuun y= f(x) upn x— +00?

1. Halirn HETepBaJIbL MOHOTOHHOCTH Py HIM:
Dy=aaoóx 2) ya Yard.

2, Halim rouKu oroTpemyNa DYRK IDE
an OYERON » arux rouxax.

3, Iocrpouts rpadux gynkunn y= x*+3x*-4.
Pet na onpeone (1: 41

xt—4x"420 4 omane-

. Ormmmxa ovexox 72 am? uncer opMy IpAMOyroxpnoro ma-
panxexeumuena c ormomennest cropon ocHosanus 1:2. Tlpm
KakUX PADMCPAX OTMMEH TO» ee TOTO; MoNepxHoeTHL
Gyger nauımensieh?

1. Tp war suavonnax a gyncuun yma?+ Saz nospacraer na
nee wteaonoft AMOR?
2 Moexpoure rpadu yuu
v
à Hatem nanonsee 1 mamen sarna dynam
12% ua oxpeane [-1; 3).

4
+4.

4 Haliru mucory Konyea maumembmiero oÛsena, ommcannoro
xoxo cihepsn paxuyca R.

129

Tiposeps ceba

HB 74 I Veropusecias emp:
Kax I amore paonexst MATOMATIIH, andxbepenmuassie ne
cmestenne noanuno ma HCOOXOAMMOCTH penitent mpaeriweckit
sanay. B OCHOBHOM MCTOYHHKOM AudbepeHumansHoro HCHMCACBNA
suce aananın ABYX Dun: a) ma HaxorAeNNe KauGonn
M nanmensuinx anauenh nem, Te. sagas HA naxonaeım
oxcrpemymos (or ar. extremum — Kpaiinee); 6) ma psramcrenne
exopocteti. B apesnocra x » Cpeanue era sananı 9TUX Bugos pe
‘lagen reoerpiieckenwit u NEXAMMLCINNE METOANIL He Di

Jom enaannsı O6 MANN.

Bazan Ha HaXOKAeHHe MAKCHMYNA M MHHMMYMA MOHO Halt
ru eme » «Hosanaxo Emeauya. Tax, » VI xuure eHauane pote
JIIMCTCA, WTO HI BECK MAPAANAOFPANMOR, BHUCAREENE à AIMÉ
Tpeyromsimx, MANÓOMBULVIO HAOIOND MNEIOT Te, OCHOBANNC KOTO
PAX panno moxomune oemonnunn TPEYrOAMNKA.

B 1615 r. 5 ony6mxosannoii pabore «Crepeomerpua sumi
Gowex» Hemeukuii yuembrií M. Kenxep (1571—1630) nsickeax
HACI O TOM, TO PGA MANCHIMYMO PCA € HIMERERIA Be
SANCTI, Mpeayraies Tor PART, WTO D TOUKS MAKCHMYMA npoxo-
BonHan Dyakmux papua uymo. Hazecruo, “ro B 1629 r. paunyr
exit warenarnk II. Pepxa Re BAJON METOROM omporenems
MAKCHNYMOB m aumumyson. Ho TomKO » cepenmme XVII 2.
M. Hnioron u T. Jlehönum necnenonann mpoßneny maxcnuyue
# MEHHMYMOB (byHKImÄ e Mosmumk maen ÖecKomeuko Mamoii per:
dua, Tax, JeliOnuuen una cQopsyanposana Teopena o Jocre
towuom ycuonun sospacranus m yOkınanısa YHKUIK ma Orpesa
Orponmsuh, saan» paomıerne anQdepemumazsmoro werneaens
ups pemenun npKrnagunx aaqaw mueca umehunperue mare
Turn $1. Bepnyaın a H. Bepnyanı. Tonnannexuh yet X. Put
rexc (1629—1695) mocne pemenna sagaux o dopme moneenzch
3a nou maceunuoli nen nannean wancernomy dpannyackony
maremaruky P. JIonmrant (1661—1704) o nimpore npumennmoera
METOJOR ANdbepeHHANIbHOTO MOUHCICHMA: «A BHXY € YAMBIEU-
eu u nocxumennen OÓNIMPLIOCTD M IAOAOBMTOCTE, HOBOFO Mono
Kyna Gor a mn o6paricr 830), % sanevaio JUIA Hero Homme mpx
xeHHA, 2 mpexsmky ero Gecroneunos Paopirmue u mporpeccs

B 1755 r. JI. Dinep n enoch paôore +Jlndbepennnannioe ne
nonenue» page HORATAR «A GCOMOTEX akerpemymon» 1 «7
HocHTeNbuMX DNCTPCMYNODO, ¡ADADACMDAX uM DKCrPEMYMAU
«mccrnoro xaparrepas. B ovolt paGore on nopuepkaman, sro
“ere DYHKIMK B TOUKE MAKCHMYMA, BOOÛILE rOBOPA, He coñas
Auer c ee nambonsuumn omauennex. Ja necnenonann dyusuch
Düacp nonvomacn ne romKo nepnoh m MrOPOË mpomanonms,
Ho u IPOMOBOXMAMIL 6onee PNCOKHX TOPAAKOD.

OrMeTHM, GTO TEOPMA IKCTPEMYMOB dyukumi m ceronus sr
xoanr MMOTOHNCHENMME MpaKTHIccKMe MPUNEMEHNA » penes
Sane IPOMODOACTIA H OKOHOMOIKH, CRAGAHHBIX © Oman
MEROAIOMANNEM EMPHA FL BpeMeNS.

1307 tas:
Tipmuenenne IPOTSEORNON E WECNEROBEHMG Oya

Tleppoo6pasHaan
u uHTerpan

Odno mono crasumo nacepnaxa:
sacmpa xamenamura cmanem

ee moryıyecmaenneu u nyxnee 2008,
wen cesodnn,

H. A. Henman

§ 1. Nepsoo6pasnan

B raane II Gina pacemorpena sanaxa o na
KOMAEHIN NEHOBEHHON CKOPOCTH TOUKH HO Banan-
sony aaxony anınkenisa. Myers aakon Amoreraa
Ton ganan hynkmneii s~s(t), re s(t) — Koop-
Aumara apwxymelica Touxu B MOMENT BpeMeKK 4.
‘Torna mruonenuas cxopocrs ADHCIMA TOA m MO
Ment npomenn £ panna us (t). B orolt agave no
sanannoh yum 9(t) nurnennercn ee mpono-

nornan. Hanpnxep, ecan 82, ro v=8(0)=81.

B duanke norpeuacres oÓparmas annawa: mo
sanannoli ckopoern v= v(t) maliru saxo aBit%Ke-
Mus, 1. e. maire 5=5(0). Tak ax S()=0=0(0,
70 n oroli annanc rpeóyeres naltru raxyio yan
mano s(t), mponaroxmas koropoñ pasma v(t). B
atom enyuae dynxiuno s(1) naanınaor mepnood-
OS

Onpenenenne

Dyuenua P(e) nansınaeren nepaoospasnoli

aus dymenun f(x) na nexoropom MATEpRAXe,

Scan AIR Rcex x ua aroro ureprars

munonmaeren panenerno

Ft). a

Hanpmwep, dynunis_x* apaaorea mepnoo6-
pasnolt Axa hynenum 82%, Tax Kax (x)= 82%;
Oymann sn 2 — nepnoofpuusn nun Gym

Banawa 1. Toxasars, wro @ymenun

FC 3 sin dx

sisercx nepnooépasnoë aan dbymenmn f(x) =0s 3x.

91 Ds

TlepsooGpasnan

> Tan war F°(2)= (Hein 67) =1-Beo0de-coeBs, ro Lande -

mepzooGpasnaa ana dyaKnun cos3x. 4

Bameuanue. Bonn @ynnuua F(x) aubbepennupyena a ur
Tepnane (a; D), nenpopninna na orpeaxe [a 0] m zn ncex C(0; 0)
munommneren panenerno (1), ro ymiuto F(x) Kaaumaior nepas-
oGpaanoï ura Gym f(x) na orpeane (a: b.

Sagaua 2. Joxasars, “ro ann moGoro Aeloramrentnoro pr
2 Annaercn nepnooGpasnoñ ana Ay
Fa)? ua upomenyree x>0.

e
(p+1)2", vo (5)
B cercas an p08) 02, nahen
1) FG) =x — meppoo6pasnan jura dy f(x)~1 mpn zeR;
2) F()=* — nepnooßpasnas aan Gym (= ope
xR;
8) F(x) — nepsoo6pasman ann yaxunu f(x) m
= E
mpomenyrax x<0 u x>0;
4) Pe) x Vx — mopnoo6paonaa ana yen [1%
npu x20.

2-1 bynes Flo)

D Tax xax (2771)

Bamersn, ro Jo yc (2 yan 242, E 3 mx
Me BAMIOTCH NeppOOBPASHBIMN, TAK KAI

(Eje (ajo

Booëme moGaa dynes

ar +C, rue C — nocrommman, max

cre nepnoobpasnoi jus Gym 29. ro cxenyer ua Toro, «o
IPOMIBOAMAA nocrosHHo! panna ya.

Dror npumep nomaasınner, uro ANA saxammoñ yen ee
mepnoo6pasman ompenenneren NeoAmonnauno.

Ecxx F(x) — nepsooöpasmas ana f(x) ma Hexoropom mpoxe
asyrxe, ro m bynxuss F(x)+C, rae C — m0GaK nocronmnas, tar
axe apngerca mepnooGpasmoli ana hymna f(x) ma oTON pone:
xxyrxe.

Oxaasınaeren, «ro dymevime muxa F(x)+C meveprstesiores
ace mepnoo6paansie Jus annannon hymkuum /(2).

B raape III ($ 1, aanana 4) Ópuo aoxasano, “ro ecm F(x)=0
ma vrepnane (a; 6) u ya F(x) neupepumna ua orpeane [a; )
ro F(x)=C ma orpeaxe (a; b]. Henoxbaya oro yrneprtenne, gora
rem Teopemy.

132" tase ıv

Tlepeoot pasnan m wiTerpan

Teopema

Tigers Fı(x) u Fa(x) — ane neproopasnite ana ONHOË m
ro me pyasımı f(x). Torna Fa(x)=F(2)+C, rae C —
HeKOTOPAN nocronnnan.

O Oomanım F(x)=Fe(x)—F, (x). Toraa F'(2)=Fs (x)- Fy) =

=f2)-f(2)=0, orkyna F(2)=0, 7. e. Fa(x)—Fi@)~C, ornyan

Flt=FyQ)+e.

Hrax, comm F(x) — nexoropan uepuooßpaanas aus f(x), To
ice nepnooßpname Ju dy f(x) maxogaren mo hopmyae

Fla)+C, mae C — moGaa mocronmman.

Jazaxa 3. Halten veo mepnooOpaonsio ans Gym E, x>0.

Y cxoayer, ero 2\F — nepnooßpsanan

npu x>0. Oruer. 2Vx+C. 4

D Us cagan 2 npx p=
1

sas Gymnase

Tipunegen TaGxuny nepooGpasmux aux mexoropsx yang

cs epnoofpasse
pe x A +0
1, 2>0, 2<0 Inixive
E eae
ing ox +0
cone Anzıc

Sra raGnuua uposepnercn audibepeuuuporaunem nepnooßpas-

max. Hanpnwep, var xax (In|x|+Cy= 4 (ra. M, $ 7), 70 Inixl+
+C— nepmooOpaonas nara Gym À

Pacemorpms rpaut ncex mepno-
Opaamuax sagammoñi pymsruen f(x). Ber
zu F(x) — oma mo meprooópasmbx
fig), 10 ce nepnooßpasunie nonyaaıor-
x npsGannenuem x F(x) a06oh mocro-
mmol: F(x)+C. Cnexonarenbno, rpa-
dus yen y= F()+C nonyaaor-
&x na rompa y= F(x) enBırom ROME
ox Oy (pue. 85). Busbopow nocroammoit
Cxorkuo aoburucn Toro, «roGu rpaépuie
neprooÓpusmol mpoxoawia uepes Sagan
sym Tommy.

mpu x>0 u npu x<0.

51 188

Tiepsoobpasnas

Banava 4. Hna yen 2 nat anyıo nopnooGpaanyo u
mpomexyrKe x>0, rpaduts Koropoñ npoxogur sepes Towsy (1; 3)
[> Bee nepsooGpasume ana Pym = Haxogsres no popryar
PR can wan 1 veal

un y= F(x) npoxonun vepes rouxy (1; 3), 7- e. Bocnonsayenes
yexomuen F(1)=3. Orewna —1+C=3, C=4. Cnenonarenm,

[email protected]

Ynpawnenua

1. Hloxasarı, wro ymes F(z) annnercn neprooGpaanoñ am
Gym f(x) na sceh uncnonoh mpanolt:

P(x) = 4x85 2) Fl)=1-5 (ee

1) Fo)

BFH, 1) 4) F9=3e*, 10)
5) F(: 2+sindx, f(x)=4c084x;
6) F(x)=cos8x-5, f(x)=-3sin3x.

2. Tlokasars, uro bynenas F(x) annaerca nepnooGpasnoit are
Gym f(x) npu x>

D F5, Nam; 2) F(x)~

3. Haltru nee nepnooßpaansıe ans Hymn

aie 02:

Yas Da 8)

4. Ana bye f(x) unlit nepnooGpnanye, rpadix Korogoi
npoxomur uepes rouky M:
1) fR)=2, M(L 2% 2) f(x)=x, M(-1; 3);
=, MU; 4 f()=Vx, M(9; 10).

§ 2. Mpasuna naxoxgenna nepsoo6paanbıx

Ma raapu II msecruo, “TO omepaumio HAXOKAENIA mponaner
nok ana sagannoh Hymn nasuisaior Audeepenmuponauen
O6paruyıo onepamtio naxommenun mepnooöpasmoh AAA naxR
dymenum nasınaor unmepuposanuen.

TIpanına unTerpHpoBaANa NOKHO MOXYINTE € HOMO npe
nun auddepennupopasus. Hanomaın npaniixa audepenmupot
mua

134 rave ıv

Tiepsooßpasnan m wiTerpan

Myers ymin F(x) u G(2) mer mponsnonnsie ma Hexo-
xopox mpomenkyrie: a, b, k— mocrommmae. Tora:
D) (a) + GG) PG
2) (AF Goy =aF (x);
3) (ix + D) = KP (bx +).
Hs orux mpasma audbepenuuposanin cnenyıor npamuına na-
sonzenns mepnooÖpaam
Tiyers F(x), G(x) — nepsoo6pasxsre coorsercrsenno mA
dm /(2) g(x) na nekoropou mponemyrke, 7. €.
FI, GE), a, b, k — mocroammwe, #0. Toraa:
1) F()+G(x) — nepsoo6pasnan ana Oya (0) + (2);
2) aF (x) — nepsooSpasnan ans yan af (x);
8) PP (sx+b) — neprooöpasman ana oynrunm f(kx+b).

TIpwnenem mpumepst mpunenenns orHx mpanna.
Sanaa. Hart nepsooGpaoume F(x) ana dynnusm f(x):
1) /(&)=2e" +sinx;

2) f(x)=2°—Be0s 2x;

3) 16 ets,

Cr

D 1) Tax Kax e* — nepnoo6pasnas na e”, a -cosx — nepnooß-
peawaa aaa sinx, To 2e*-cosx — OABA M3 mepsoo6pasHEIX ana
dun 2e* + sin x.

a
2) TeppooSpasnas aux x° panne 2, mepsooßpaanan aus
ons2x panna Zsin2x, emenonarensuo, ona ue nepsooGpaamux

ann x'-Bcos2x panna E

3) OGosraum
h On-air ha),
Torna Fr()=— Ep» Pals)= e — nepnooGpaanme coor-
seremmonno an fa) u fe),
F)= E) + F4 CE + her —
be neprooópasubie ¿La Pyuumm f(x).

Omer. 1) F(x)=2e*-cos x + Ci
E

2) FC) 2 -Fein2x+C;
DF + Det C. À

52 Mas;

Tipasnia HexOXAeHi NepsoOopaSHER

IATperse npasuno naxoxpenun mepnoopaanLx NOIOIC 10
nonuwrs TAÓAMUY MePBOOGpaSHBIX, MPHBEAEHRYIO B LPERKIAYUH

naparpage.
y TTepnoodpaamice
ana
tex bY, pet, ke0 gate
wann Kran

Y

o

1
F Inika b140

ot, he

Les
Rate

sin(kx+b), 420

1
Fp eos(kx+)+C

cos (hr +b), 420

1 a
jsingix +0940

Ynpaxnenus
Haken nepnooßpasune Jura yen (5—12).
5. 1) 2x5-8x; 2) 5xt4 2; 3) 3x4 28-1;
26a 5) 248
=. 2
D vere: a 2-8.
Vet 5:
6. 1) 5sinx+2cosx; 2) der-sinz
Dirder-dcm 0) E a
DEM ass)
) (2) Jer )

5) Sh +Acos(+ 2%

1 ;
amis (de)
4 @x-0% ¿2

Yar-ı
D VE: 8) VD.

1) cos(8x +4);

msn fesh De’;

136

Cnasa 1

2) sin(8x4);

2sin(x-1).
2
3) (22-3)%

Tiepeoobpasnan 9 mnrerpan

10.1) & cos2x; De +sin3x;
3) 2sin se; 4) eos +20" 2;
Ys © Y Franca
n a 2
Var a
un Etes, ay 2, ay 2238,
AS 5) 8x(2-x} 6) 2x1

1) A+20) (3) 8) (2x-3)(2 +32),
+4

Byer Ve: Da; Dit,
te
18, Jan Gym f(x) nañrra mepsooßpaamyio, rpadux Koropoli
ipoxoaur uepes rouky Mi
1) f(a)= 2243, MED 2) f(z) =4x-1, MEL 3)

D fa)=VEF2, MQ -3 4) 0 Eg MER

5) Hay=sin2x, M( 5s 5); 6) Fz)=con dx, MO; O);
1
pa

DIO 3 MER NA ES

$3. Mnowanb kpusonuneñnoñ Tpaneunn.
Unterpan u ero Bbiuncnenne

1. Homans xpunoamueiinoi rpanemut
Pacemorpuw durypy G (puc. 86), orpanmicunyio orpesicama

maux x=a, x=b, y=0 M PPAQUKOM Nenpepusnolt gysKuKH

¿=1() raxoli, uro f(x) >0 ma orpeone [a; 0] u /(2)>0 mpu xe(a; 0).

To Qurypy masiimalor Kpusoau-

rol mpaneuueh, a orpesox [a; 6] —

«e ocnosanuent.

Toscuun, ox mponmrca noma
ne unowaan kpuwonsnelinoh rpane-
zum m war MONO BMGKEANTE Dr
romans. [last ororo nmenen moms
‘ie onpenenennoro muTerpama.

Pasobxem onpesox [a; 6] ma n
(recómarempno panınıx) sacre row
HOM Zip Xp ses Xa_ 1 M Mpowenent
Ses on TOME BepmiKaAmunIE npa-

ss 137

Trouane Kbveonnnennon Tpanéunn, Vnrerpan nero acne

mue xo nepecexemna c rpads-
Kom oyen y=f(x). pu
910M xpimoaunelnan spanenits
Pasoßseren na n wacreit, Kax-
AB MS KOTOPBIX Taxe ABARET-
<a xpunonuneïnoï rpanenueñ
(pre. 87). OGosmaumm xp=0,
xa mb.

Pacemorpum xpusoauneli- 14-40% x
yi Tpamenmo e Ocmonamment pecar
Lis 24) (Pic. 88). Eonu gm
ma orpeaxa [xy 3 X] Maa, TO DT rare mao ormsaneren or
IPAMOYTONBENRE © OCHOBAHHEM Axy= xy. 5 BCOTON fc),
TAL Cy — kaan-nuöyne TOWRA OTpeatea [X,- 11 Xp], a mnomazı Hp
Borumeitnol rparen c oenosauen [x,. 15 +] MPHGMINKCHHO pas-
Ha MIOMAAK 9TOrO MpAMOYTORDHHKA, 7. €. MPHOMUKEHHO paste
Fey) dx.

Box xpunonumehnan rpanemua e ocnonannem (a; b] mano oF
AMUATOA 07 MMOTOYTOXBNIIKO, COCTOMINEFO Ha HPAMOYTOMBINOS,
nocrpoemmtax ykaommmum crocoGom un orpearax [xo; zul, Ley: xa
[xa X us Len 15 Xe] (pue. 89), a nromans xpunonnnelinof pe

PATTES

YA
y= fe)
|
Oo} a 4 4 TEE ART * Pre 8
uh
y= fe)
[ex x % ET] ae

198 rnana 1v

Flepeoobpaanan u mrerpar

Gens npAÉMEREHNO PARKA MAOIKaAH DIOTO MHOFOYFOXBHIKA, T. €.
mpÉrExeuRo para
HA 2 +12) A rito + fe A++ (Date A)

Cpuny (1) naasınaor unmespansnoii cymnoù dynnyun f(x)
as orpeawe [a;b]. Bynom yneananmarı, «ueno TOWeK pasómemMa
‘ax, ITOÓM MAIGOMBIIAA MO AIX OTpEIKOB [xy 15 xa] erpemunaco
Kayan. Torga uma Az, Kamnoro orpearta Taro GyAer erpe-
amen x nymo. MONO Aokasark, WTO MPH 9TOM HNTETPOMIEC
cat ÓyAyT CIPEMIETLOA x HSKOTOPOMY ‘ery S, 7. €. UMCIOT
aerea, panunit S. Dro uncno S uaakınaeren naowddoo paccmar-
prmmenoli xpunosuetinol rpanenuu (em. pue. 86).

2. Mirrerpan

Paceworpumt renep» xi0ÓyIo mempepsramyio ma orpeske [a; b]
Ayıenmo (x) (neoßnanrensuo HeorpitnarerbHy). Cocranum Jus
ee miTerpansay cywaıy (1) u seen Oynem yneznumann “meno
rover paoGuemma orpeaxa [a; b] Tax, vroßkı za mandonnmero
12 orpeakon [xi 35 2] erpemitaacs x mymo. Mozo noxaaarı (970
naznaeren m Kypce BBICIelt MATEMATUKH), "TO H D ATOM CAY>
‘ae murerpamnute GYM CIPCMATCH X MeKOTOPOMY AMEN, 7. €
rer MPOACA, MO DABMONIRNHÍ OT BLIÓODA TOOK C4, Css «.., Cp. DPOF
Speen uasuinaion onpedenenkun unmecpazox om dynnuun (x)

sa ompeae Las 6] m oGosnavaior rar: |/(x)dx (unmaeron: murer

an or a 20 b ap or uxe no MKC); Qynkmio /(2) naasımuor
teduumezpaaenot Qynnyuel.

_B sacrnocrx, ecam yen F
ix) nononmrensia Ha orpeake [a; DI,

norywaen dopmyny AIR Momeni
spmwonuneinoh Tpamenum:

2
s u
Tomas aaxpassennot quryput
va preyuxo 90 panne Ÿ xtdx. M >

®opmyaa (2) cmpanegumoa u
zn cayuan, Korma dymenun f(x) no-
osarenma miyrpm orpesta [a; bl, a
à OIHON M3 KOBHOB OTpesKA mau Ha
lou KomuaX pasa HO.
Hanpumep, mom, saxpanen-
woh Qurypst ma pmeynke 91 para

Júnzaz.

Pac. 91

sa Biss

‘Troms wpasonanonnon FPaneLINM. Virorpan A CTO BuNnEneNNe

Tax o6pasom, JARAIA O HAXOMACIMN maomanu kpunoa
nelinoli rpanenum CROJMTCA x BRUSCACHIIO unrerpana. Ananorıs-
HO € MOMONBIO unrerpana pemaren u Muorue Apyrue reomerpu-
seckue u Quonuecxue saxam. Ilpswepsi duoxueckux sagas 6yayr
paceworpeus u $ 5. B kypce reomerpuu uacro paccmarpusaiorea
HpHMepM DIVIUCACIMIA € MOMOMSO MNTONPAJOD OÓTEMOD Ter
(anpamunei, Konyca, mapa HT. A.)-

3. Barunexenne wurerpasion

Tpn6mxennoe suavenne unTerpana MOHO HOY YET, COC
BHD METOTDAMERVIO CYMMY. OAHAKO HENOCPEACTROUHOC HAXOM ACH
nmpenena uNTerPANGHBIX CYMM UACTO OKAIBIBACTOA TPYAOENKHN.

Tas nusnenonun onpexerennoro MITETPAJA OÓLIUINO mono
ayeren cneayioman QOPMyAi

S=\r)dx= Fb) - F(a), a

me F(x) — moGax neprooGpaanas paa Gym f(x) un or

pesxe fa: 6].

Dopuyaa (8) enpasennuea zua moboli pyme f(x), Reape
prnnoñí na orpeae [a; b]. B «acruocra, ora dopuyra enpanoz
Da AR COX parce Mayuommx YU (cromenmoR, nokadaTen-
ol, Tpuirouowerpuvecxux m Ap.) ua Kanon orpezse [a; b), ne
ow ayımayım onperexents.

Dopuyay (3) nassimaror popxyaou Hormona — eunuua à
score cosgarencl auüepemunaaunero w smerpamsnaro mes
FI Dos reonerpuseckx, kax Hoxyuacres popmyxa (8).

OGoamaunm S(x) naomi Kpunormmeinoi xpanenyeH e oeno
nannen [a; 2], re x — mo6aa rouica orpeaxa [a; 0] (pue. 92). Tire
x=a oTpesoK [a; 2] nuipowaneren B TORY, IE MOSTOMY eorecrnen
o curar», «ro Sa

Tloxanen, «ro S(x) anınerca nepoooópasmoli nun yum
160, 1. e. S(a)= (0).

Paccworpum pasnoern AS=S(z+h)-S(2), rae h>0 (enyui
h<O paccuarpupaeres AMANOTAMIO). Dra paamocrs parue Iron"
Ru aaxpamennofi ua pueyuxe 93 durypu, Koropaa npencrananer
‘cool xprnommmeïnyso rpanemuio € ocmonanmen (x; x+h).

Y y
y= fe), T
SQ)
+ hts
Puc. 92 Pre. 98

140 fases iv

Tlepsocbpaanan u niTerpan

Bamerum, aro MIOMARS sto; Purypa pasta LIOMAA npano-
yromuuxa € ocmosamen [x; x-+h] u sricoroM f(c), rae x << x+
+h ne.

AS-S(x+h)-S@)-Fo)h, @
omeyan

as
no. 6

Crporve noxasanenserso dopmyası (4) (970 «reopema o cpen-

Hex» JA MBTerpasia) PaceNarpunaeTeR B Kypce mucmeñ marena-

Tiyers h--0, toraa cx. Tax Kak f(x) — uenpepumuas

drama, ro fc) f(x) mpn h—0. Tlepexona k npeneny » pasen-
S+N)-S@) _ ya 5

cre (5), monyuaen Jim SHE) 5 (2) f(x), Uran, S()-

=/(x), 7. e. S(x) — nepsooGpasnas ana yuri f(x).

Ilyer» rerep» F(x) — nponssorsnas meppoo6pasnaa aaa
dyexuma f(x). Tax Kak moÖbIe ape mepnooöpasuzıe ¿ura onuoh m
moi ke dymenun ormnsalorcn ua mocrommmyıo (reopema ua $ 1),

F(x)=S(x)+C. (6)

Tax ¡ax S(a)=0, 10 mpu x=a ua panenerna (6) nonyuaew

Fía)=C, orkyna S(x)=F(x)—F(a). Ha sroro pasenersa npu x=b
S(b) =F (b)- F(a). mM

Hanomxum, uro S(x) — naomaxs kpueoxmneñnoli rpanenun

<ocuonannen [a; x] (ex. pue. 92), u mooromy

s@)-\i@ax. 6
la pancnera (7) u (8) caeayer Popuyaa Hisorona—JIchOnmua:

N
Úrioar=F0)-Pla).

axerun, uro popmyxa (3) cnpasemana ju mo6oR nenpe-
pusuo ma orpeaxe [a; b] hynrmm f(x). =I

Sanava 1. Barunenurs unmerpam | x8d2.
> öymann F()= E amameren nepnooßpaamoh gun Hymn

lex. To dopuyae (8) nonyuacw À xtdx~ SL
3

isa
-9-1-83.

E
Bameuammo. Dror METETpaz parer roma S burypia,

crscransenaol na proie 90, r. 0. 8-82 xa. cn. 4

ga 1er

TION ipneonmemnon Tpaneunn. VinTorpan m oro euwcreHe

Banaua 2. Berncaurs unrerpan | sinxdx.

D Œynrus F(2)=-cosx annnerca nepnooßpaanoli aa dy

100)-sipx Mo dnpuyae (8) nommen Senate =F(3)-FO0-
Lo 8 cos0)=1 : @-

Bamonanne. Trou Ourypu, mofpaxemoï ma pro
xe 01, pone 2 wa: ened A

A open WOE.

[> Oxmoñi ma mepnooßpaamux Gym x—1 amasete yuna
E oF

E

TIpH Buiunenenua HHTErPANOR YAOÓHO BRECTH cnemyıomee 060

x Hoorouy Îte- ax (

AA
Toraa bopmyay Hoiorona — Jleii6xuna MoxHo sanucath Tax:
Sc) da = F (a [.

Hanpuxep, e nomomsto oroit dopmyacı sarten peerne ae
ras 3:

13anaua 4. Buunenurs nurerpan | sin3xdx.
> ange! y Cc0s32)[f = + Ccos3a +cos(-3a))=0, rar nar
cos(-3a)=cos3a. 4

Banasa 5. Busneawre uurorpan À 4,

D> Tax ware In|a| — nepnooßpaanası Jura dyer À na npowengr

ne 20, 0 | inf in lnea 4

mongenana 6, Bussen umerpan brennen

(> Hpeo6pasyem nonsinrerpansnyi one Fx), nenomssys
120082! con( 4) = sine, Honyamı

dopayaut cos? 1 1480

0000

Locales
ri 1_siner
We CNE

Tax kar mepnooöpasmoh ana gyaxuua f(x) anıneren dyaKuus

Hao= + cos x, mo |co2(2x4 #)dxa(Lat Leosdz)le 4

Gaza 7, suiza murgas Je VET er.

ojrrTa ber avert ani
AE J-(2-32-2. 5) -E
Ynpaxnenns

14. Hair nromage xpanommelinoii rpanenwt, orpanusennolt npa-
NAM x=a, x=d, och Ox rpadKom DYHKIUM y= f(x),

Da=3, bad, f()= 3

2) 420, b= 2, (max + 1;

15, Hatin nxomann durypui, orpamruennoli npamoï x=b, ocuo Ox
M rpadbırom «pyme y f(x), com:
1) b=3, Dm 2) bm 2, fedex
ES 4) 028, f= Ves
x 8) B= B, [malas
Db=1, fre 8) bm 2, fG=1- 5.
Buenos mrerpan (16-18).

more fra 9) der 9) tar

atts ay Seta Ole | colina,
i ® ES Es

1.0 Í (2x-3)dx 2 (54d 3) § a-sxt) das
Ñ Fi 2
wm Ítt+nas 5) [errarhds 6) | (0x*-4x)de

D\Gxt+2x-10)dx; 8) Ñ(Bx?-ax+5)dx.
à à

$3 Bag

Tronas KPABONANENNON Tpaneunn. VinTerpan u ero suunenenne

19. Hoo6pasnr. dnrypy, moment Koropoit panıra aamony me
Tpaay, u DMUNCANTE ory maoMaRE:
a

3 A
D $ 2-349 D | (6-22)ax;
3 Sa2+x-z4)ax; 4) | (aaa
è à
5) V sinzaz: 6) À cour;
5 5
1) § cosxax; 8) Jsinxdx.
5

Baronets warerpax (20—24),

um | rene Idx; 2 Ser Ly ae
3) $ (e DG? 2) dx; af 40-2

4 2
a (@-svmaxs 2) Nee) as

a 4
aan 5) KG 2) ax;
an) $ rs» lus dx.
1) Lars; 2) Sacré;
2 i
$ da 4) À e
Ban) Ÿein2xdx; 2) Seos(ax—4) ax;
H 3
cos? Axx; sin? (e- 3) dx.
3) Neos? aa 4) Ssin#(e—§) x

144) tase ıv
Tlepuoobpasran u mnTerpan

$ 4. Burincnenne nnowaneñ duryp

© nomoweio HTerpanoë
En Bagasa 1, Buisueaur, naomaye S xpunonunehnoh rpane-
‘Gm, orpanaennoli ocio Ox, mpaoesinn x——1, x=2 m mapa6o
10H y= 9-2" (pue. 94).
D Tax war ma orpeaxe [-1; 2] dymenma y-9-2? npımnuner nono-
RME GHAYERUA, TO MeKOMAR ILTOMIANE S pana umTerpany:

Fi
sl @-ax.
o dopuyne Horona—Jlchnuna uaxonun

E
ES

sl @-<dr=(ox—

Sauaua 2. Hau nnomaas
durypm, orpammiemmoñ mapaGo-
mm yt, y=2x a u 00m0 Ox.
> Tloerponm rpadnen dymennh
y=x!, y=2x-2" m nalinem a6c-
Auen ToweK mepecewenun ormx
mabıko us ypapnenun x*=2x—x*,
xopHa Koroporo x,=0, x2=1l.
Jlanuas urypa usoGpaxena ua
pucyaxe 95, M3 KOTOPOrO BHAHO,
‘ho olla COCTOMT ua ABYX PUDO”
Amohnux Tpanenuf. Cnezonn-
Tenno, ucKoMaR HOME pasna
re nnomppeh aux rpaneml:

s-\etacs(ee—stdr=

Bagaa 3. Hau nromagn

durypss, orpairmemnoï orpeakont
5] oem Ox m rpaburom
¿yuan y=cosx ua orom open.
D Beer, uro mekomar mao:
Mans pana momen duryps,
caNNerpHTKOH AARNOH ormocı“
reasxo ocx Ox (pue. 96), 7. e.

$4 145

BECOME NnOUARER Gury € MOL HNTeTpaNGD

Papa maomann Qurypst, orpa-
de

musemmoRi orpesxos |; &
Ox u rpapukon py RI
y=-cosx

] eon

un orpense [Es 32]. a srom or
pense ~coee30, a norrony

Bamewanne. Bom f(x)<0
a orpesxe [a; b] (pue. 97), To
roue S Kpusonuneliuol rpa-
nen pasna

s-Ñernpar.
Sanava 4. Hañrn nomen

Qurypst, orpamuenmol mapaGo-
sol

peated
2 upamoh
Yara.
> Moorpoms Trac hymne ya d+ m ym 043. Haine abe
Dice» rouex nepecenenua oTux rpacuKos mo ypasnenua x+l»
+3. Dro ypaskenke HMeer KOPHH x; 1, x2=2, Durype,
orpamuuennas rpadiicasic zamınax yet, waobpaskenn na pac.
xe 98. Ha picyuka angio, “O HCKOMYIO NTOIMANB MONO Hale
xax paamoern nromeeñ S, u Sa anyx rpaneuuh, onuparonacs
a orpesox [- 1; 2], népRAX na KOTOPRIX orpannyena CREPXY OTe
kom npanoh y #43, a nropax — nyroï napaßoakı y= 2"+1. a

xar Sim (x+3)dx, Sam) (a?+1)dx, ro
Ai A
2 2

S=S\- (AS

Menoxbsya cÑoÑcrBo nepRooGpasx, MOXHO samucars 8 à
une onHoro mmerpan
2

s

A
(+3) (+ Iyde=J (+ 2-45) ax

146) rnana iv

Tlepeoctpaanan mrerpar

Pacexorpum (urypy, Orpanuuen-
nyo orpeaxamın mpasux xa, mb
= TPADHKAME MenpepkIBHBIX DYHE-
mul y-hıla) u ya fes), ine fa(2)>
>h()>0 (pue. 99). Tomax S
ob Qurypu panna paawocrit nnoma-
sch xpupozmmetinarx rare adaB;b
x04,B,b. Ilnomaan S; u S, orux rpa-
ru Coornererzeuno panas

8,-\falerde n Saz.

Caenosarensuo,

s-\ipteydx-Sf taz. Orciona

SAO]

Dra popmyxa empapeaauna ana
noëex menpepkinusix yal f (2)
nf) (upnnmwaonnx anauenun
XOÓNX saxon), YRORNETROPAIONENX
AAN

Banava 5. Haïru momens qu-
FIDE, orpammenmoh napabomancın
var’ u y=2x?—1.

D Ioerpons zaunyıo durypy (pre. 100) y nafinem abcnncon rower:
Tepeoevenus napa6on MO ypanıtenna x? = 2x — 1, KOPHM KOTOPOTO x1
llo popuyae (1), me fy(a)=2x*=1, fa(a)=x", naxonine

sd pr @xt-Iydea| Cate Dae +s)

Ynpaxnenı
Hahn mmomans durypst, orpammermoit csr Ox m mapabo-

2; 2)
Du y
(x+2)(3-x).
26, Hair nomex durypst, orpauncennolt:
1) mapaGoxoit y=(x+1)”, npanoë y=1-x u ocbio Ox;
2) napaGonoï y-4- x, mpamoli y=x+2 u oebio Ox;
3) mapaGonoï y=4x-2%, oemo Ox m mpamoh, mpoxonmeh
“epea roux (4; 0) m (1; 3);
4) napabonoh y 32, ocuio Ox n npamoñ, npoxojutmeñ vepes
own (-3; 0) m (-1; 3);

ga ar

TS

ig

29.

32.

5) napa6onann y=6x%, y=(x—3)(x—4) m ocvi0 Ox;

6) napadonanın y=4—z%, y=(x- 2) m emo Ox.

Haru naoman» durypm, orpanmuenmoi:

1) rpapuxow Gymxmm y=sinx, orpeoxom [0; x] ocn Ox x
npawoñ, npoxonameii vepes roman (0; 0) m (55 1);

2) rpadmemen Pymapıl y=sinx, y=cosx u orpeaxon |
oem Ox;

3) rpaguxann pymenult y=VZ, y=(x-2)? 1 ocs10 O:
4) rpabucana dynenuit y=x?, y= 2x x? u och Ox.
Haïru moment duryput, orpamiventoft:

1) napa6onot y= 9-2, upawoh y=7-x 1 ocmo O.
2) napaGooë y=2(4—x), npanoñ

3) mapabonamm y=(x- 27, y=(x+2)", mpamoit y=1 1 on
Ox;

4) napaconama y=(x+2)%, y=(x-8)%, oc» Ox u np,
npoxoaamelt sepes town (15 1) m (13 svg

5) rpaduxow dyukuun y=sinx, npanoï y= À m orpessae
10; x] ocu Ox; A

6) rpaguion byee y=cosx, mpanoñ y= À m orpesxor
Ex 3] cc ox.

Hains mxomans durvpst, orpanwennoi
1) napaßonoli y=x?-4x+3 » ocsio Ox;

2) rpaduxom hymım y=cosx, mp x=
ocu Or.

. Hale mromans urypst, orpanımennolt:

1) napaGorolt y= 6x2? u mpamo y=x+ 4;
2) napaGonoit y=4—x* u npamoï y=x42.

. Haltrm miomams Quryps, orpanıuenmol:

1) napa6onok y=(x+2)° m mpanolt y=x+
2) rpadmon dymenun y=Vx m napabonofi y=a?;
3) rpaduxow Pyme y=VX u mpamok y=x5

4) mapaGonoit y=(x-1)? u mpanoñ y=5+x5

5) mpanoë y=1, ooo Oy m rpabicom Hymn y
o<x< x
Hairu nomen @uryps, orpauuennoi:

1) napaGono y=-xt+4x-8 u mpamoll, npoxogaueñ uepe
roux (1; 0) u (05 — 3)

2) mapaGoroï y=— x? m mpanoh,
3) napaGonamu y= 1x" m y=a ls
4) patron dymununt y= u mpamann y= 1 M 2=-2;
5) npanol y=x u rpabuxow dying y= 0, -1€x<0;
6) mapabonama yx? 2x m y.

ins,

$ 5. Tipnmenenne unterpanos ana peujenna
usuyecknx 3anau

1. Haxosnenne nyru no aanannoh exopocrs

Ilyers rouka ABMKETCA CO ckopoersw L(t). Hy>KHo HaiitH nyTH
+, npolizenmuh soul or monenra {a no monenra t=. OGoana-
twat vepea s(t) nyTo, mpolinennuh rouxoh sa npenu £ 07 momen-
ra a. Torna s'(t)=v(t), T. e. st) — mepsoo6pasnaa ana byux-
zum v(t). Tlostomy no dopmyne Hoiorona — Jleliómuua nañnem

s)-9(a)=So(0)dt. Tax Ka s(a)=0, To wexowssit nyra panen
s Soga. a

Hanpumep, ecam Touka ABmxerca co ckopoersiw v(t)=
=21+1 (u/c), To nyrs, npohnenush Toukoh aa nepnsie 10 €, no

dopuyre (1) paren s-| (2¢-+ Ide = (22 +2)[0 110 (m).

2 Burunenenne paboruı nepemenuoh eus
Tlyers Teno, pacemaTpuBaemoe Kak MATEPHANBEAR TOUKA, ABH-
sere tg oot Ox non Jelerauen au FC danpasnenalt Boas
ter Os. Durmucrum Dub cum DDR mepeaemonnn tenn na wor.
mes muy à
ler A(z) — paGova zamnoï cmt nput nepewentent ren
ña TOYKM @ B TOUKY x, rae xC[a; b]. Tlpn manom h cumy F ma or-
pce LE 27) moscas Cnraneosronuiol E pasto PGS) lo”
EIA IRA
Am al
CHA rey,
Tipu, erpeumenc x ny, menyaaek, aro Aa)=Fi2)
+ AG)” capaces un Ay ECO. La damage Hs

rosa — JleiiGuuna nonyuacm A(b)=\F(x)dx, ax Kax A(a)=0.

Hrax, pabora cunsı F(x) upu mepenemenun reza ue roux a
army b pasta

AaSP(a)de. (2)
Bouerus, «ro coum F swpexaerea » wutoronax (1D), a nyre —
» xempax, 10 paGora À — » amoyaax (UK).
Sanawa, Burunenur. paGory CHA P upit CATIA mpyoamnt Ha
008 u, ccm ann ee coxarna ua 0,01 m rpeßyeren cuxa 10 H.

D lo saxomy Tyra ca F nponopunonansma crarmo NPYAHHM,
ne F=hx, re x — care (8 1), k — nocronnnar. Ms yenopna

mxogun k. Tar xax mp x=0,1 me cuna P=10 H, 10 k= £=1000.

5 199

Tipunienonne wiTerpanoe AA POCHE USER Jana

Caexonarensno, F=kx=1000x, u no opmyae (2), rae Fais
=1000x, noxyusem
0.06

A= 1000xdx~= 1000 3 0°" =3,2 (un. 4
3

Ynpaxnenus
33. Teno ammuerca mpamonmueñno co cKopocth¥ v(t) (m/c). Bu
unemire uyrs, Mpoliaenuuit TEXON 28 IPOMEIYTOR npewest
or (uty MO t= lan eos
D v(t)=38 +1, 4 =0, tom ds
2) v(t) 2t? +t, timl, 2-3;
3) v(t)=6t? +4, 12, t2=3;
4) o(t)=t?-2438, 1,0, t¿=5.
34. Cxopocrs npamonnmeino nsuxymeroca tena v(t)=4t-f.
Beruncauro UyTb, MPOÑACHMBÍ retom OT Mauana Amen
10 ocranomen.

$ 6. Mpocreúume anbdepenunanembie
ypasnennn

SO enx mop MIX paccmanpımanı ypanmenn, m Koroptix mee
BCCTIRAMH ABLE ‘nena, OANAKO B MATeNaTIKe m ce UDA
MIX PIXOTE paccwarpınar ypannema, 3 KOTOPSIX, Her.
CTHLIMH ABAMIOTCA yBKOHH.

Banana O waxomaem nyru S(t) no sazannok cxopocm vi)
CBOAMTCA K pemenmo ypasnenus s' (t)-v(t), rac v(t) — sagamem
QyHKIMA, a s(t) — HCKOMAR PYHKUHA.

‘Bro ypannenne CORopxur npousBonuyio nensnecrnoh qyu
mum s(t). Taxne ypapnenua naskinaïor Judpepenyuardnrinu.

Sanaua 1. Per» aubbepenumannnoe ypannenne

Vars.
[> Tpeöyeren naiiru dymauno y(x), npousnouan xoropoi pas
E41, 1.0. nalen meprooGpaonye Gym xt 1. To parres
naxonsqentin nepnooöpsamux nonyuaen y= 5 +24C, mae C-
TIPOM3BONBHAA NOCTORHHAR. |

Pewenue auqxpepenumansHoro ypasnenus puxa y = f(x) nexo:
AHTCA HEOAHOBHAYHO (© TOUHOCTHIO AO mocroammoli C). Oruro x
anddepenumansHomy YPABHCHMIO AOÓABAACTCA YCHOBMC, MI komm
Poro HOCTOAHHAA C ONHOSHAUHO ONpenennerch.

Sanaua 2. Hañirn pemenne y(x) auddepenuuaabuoro ypanıe
nun y =c08z, yaonaernopmaniee ycaommo Y(0)= 1.

E Bee pemena sroro ypannehun sanenaanorca dopnynof y(t)
Soins +. Ma yenomma p(0)—1 maxon sin0+C=1, ora
onl.

Orser. y=1+sinx. d

1507 rnaua wv

TiepuooBpasnan w wiTer pan

Pacemorpun sadavy 0 pasmnoxenuu Gaxmepuit.
Axcnepumenraasno yeranonzeno, “TO HP ompeaenennix yc-
JORMIAX CKOPOCTH PAIMMOXCHILA GaKTepitit TIPONOPILOMO:IBRA IX.
xommeotay.
Tlycrs m(t) — macca ncex Gaxrepmá » moment mpenenm t,
toraa m'(t) — exopocts mx pasmmomemuta. Tlo yoxoimo
m'(Q=km(O, Mm
tre k— aanammax nocronnnas, sasueamas or puna Gaxrepui u
unux yeaosnit. Vpannenne (1) annaerca upybepenitnambreint
yDaBKeHON, OMICHINAMILNM JAKON PRIMHOMCRHN GaKrepuit,
Tloxamen, uro ya
mace, e
12e C — nocronunan, anımoren pememmann ypanuenna (1). B ca-
now gene, (Ce) = Che! =k (Ce). Mono moxasars, "ro bopmyan
(2) conepur ace pemenus ypanmemn (1).
Tlyers wanectiia Macca mp GaxTopitit m MOMONT MPENCHH fo, Te 0.
mito) = mo. ®
Torga us paseners (2) u (3) moayuaem mo=Ceto, oxyna
C= moe Po u

momo
Aaer mexomoe pemenne amdrpepenumansuoro ypapmenna (1) mp
HAYANBHOM yenopum (3).

K pemenmo aubhepemumanbHoro YpaBHeHHA CBOAMTCA ada:
va 0 paduoarmuenox pacnade.

SxcmepHMeNTs! HOKASHIBAIOT, WTO CKOPOCTE pacnana Panıtoak-
Tunoro wemeeTHa MPONOPUNOHLNSUR UMERMUNENYCH KOMMUECTAY
Toro nemeerna.

Cnenosarensno, ecam m(f) — macca Beillectsa B MOMeHT Bpe-
sem £, 70

mimo, Q]
1x0 R— monomnrensnas nocronmmast.

Bnak <-> » ypapnenmm (4) oGyeropren rem, «ro m(t)>0,
m(t)<0, tax Kak € TeveHMeM BpeMeHH KOJHMUECTBO BeINecTB£
© Kax u ana ypasnenus (1), mposepserca, sro (ym

m(t)=Ce™ (5)
seamoren pemennsuu ypanmeuus (4). Bont 2anamo uauansuoe ye-

Mito) = Mo, (6)
1o ma paneners (5) x (6) uncom Cm moe to. Crexonarenno, bymk-
aux

m (= mge #4 m
snauerea pemenmem quddepennuansioro ypanuenus (4) pm na-
season yenoama (6).

56 7151

Tipocrenune anpPepenLmansmue ypasnenin

BaMOTIM, «TO Ha MPAKTHKO CKOPOCTE Pacmaya paaHoarTADHC
ro seulectsa xapaxrepusyerca NEPMOJOM monypacnama, 7. €. mp
MEYTKOM BpeMeHH, B TeveHHe KOTOPOTO PACNAAAETCH MORONME
uexonmoro perecer:

Tlyers T — nepuoa noaypacnaga, vorga us pasencrsa (7) np:

tuto +7 naxonun mme, organ e Wa, Mainz, ha
Toxoranaan wafizonoe ouavenmo À n Gopuyay (7), noxyeeen
mitm 7", mam mme Y.

B sacrnoern, ecam to=0, ro m(t)=mp2 7.

B npaxrune suero Berpeunioron npoleceis, Koropue nep
Auen“ nomropmoren, MONDHNE KONOGATOAUNIO "Xana
Nanaia, Copy, upyust u. A. HPOUGCCU, cansino cae
PME DNONTDNNCCINN TONO, MarnitrauM omen mt. Pe
lenme MHOTHX TAKHX Sagat CBOAHTCA K Pemenmo Audxpepenu
tmuore ypaneuua

y +07y=0, (8)
TAL © — saHaHKoe HOAOANTOXBAOO HMCO.

Ypasnenne (8) HASHBAIOT ypasnenuem zapmonuseckux Kore
Ganuë.

Pewenunst ypannenun (8) anımorcn dymenun cosox 2
sinox, a M06oe peinenme roro YPARRERNA MOHO Gamers à
sine

y=C,cosox+Cgsinox. o

Bonn sagem auauennn dymkımm y(z) m ee mpomanonnel
Y (x) m rouke x=xo, TO OTHMI YCNOBANE OIPCRLAACION GANA.
BeHHoe pemienne ypabnenna (8).

3anava 3, Halen pemenun ypasueaus

y'+4y-0, ecnu y(0)=0, y'(0)=1.
D> Pemenna namnoro ypannennn corxacuo dopuysre (9) mew su
y=C,cos2x+Casin2x.

Menonsaya yenopus sagaun, moxyuaem 0-C,cos0+C¿einÚ,
orsyxa Ci=0, 1=C,2c080, 7. e. Cy=]. Crexonarensno,

y Lanas 4 mon

Peuurs aubepeuuansuoe ypanuenne (35—36).
D y=3-4x; 2) y =6x*-8x+1;

3) y-3es 4) y'—4cos2x.

D y =3sinx; 2) y'=cosx~sinx;

3) y =4x'-2co9x5 A) y= Bx*—4e"*,

152 ana 1

Tlepacotpaanan u viTerpan

Hann penenne andbepenuuarsnoro ypamenna, ynonnerno

Dmomee aannony yenoam;

1) y =sinx, y0)=0;

Dy-2cosx y(m)= 15

3) + 4x1, yO 2

4) y=24+2x-3x%, y(-1)=2

5 y=", y=15

6) yes yO)=2.

Wlrioxasars, aro pymenua y=Cre*+ Ce °F npn moGux anane-
max C, u C, annaerca penrenmem ypapnenna y"-0*p=0.

39. Han dymervor f(x) nah nepsoo6pasnyto, rpadux Koropoit
IDoxonnr wepes nouny M, ect:
1) Fa)=cosx, MO; 25 2) f(x)=sinx, Ms 0)
9 es MU 5); 4) /(9-e, M(O; 2%
5) f(x) a2, M; -2 6) f(x)=2-2x, M(2; 3).
40, Harz unomaas burypH, orpanmyennoh muuusmu:
1) yavx, zul, x=4, y= 05 2) y=cosx, x=0, x= 5, y=0

a) yaa, y-2-x

4) y=2x%, y= 0,524 1,55

5) ye Vz, 2-8, 2=-1, ya 0;
1

9 y=l,x=-3,x=-1, 9-0.

) y z y

AL Buuneniro murerpaz (4143).
2 2

:
» Sass 2) je ads 8) Se = 2x)dx5

ofensa Han 0 S45:

3 z
1 Ssinxd: 8) | cosxdx.

2.10 \Gxt-8xdz; 2 [6-54 341%
2 à ?

a 4 e
yz Sax 9 (Verte 6) [v2=3dx.
i à 2

153

pane « mare TV

42.1) \lomfsrä)au 2) Joss) ae:

3) (Bsin@x-Har 4) | 8cos(4x-12)dzx.

Han romans TIPA, orpanmennon aan anus
(44—45).
MD 92, pode, x

y= 0% 2) ym Tey yo, w= 2, yb:

3) nat +1, verts 4) y=xt+2, y=2x+2,
45.1) y=x?-6x+0, y=xt+4x +4, y=0;
2) yet, yd añ
3) y=x?, y=2V2x,; TX, y=0.
Halıru nnomans duryput, orpamvennolt:
1) napaGonoï y=x"-2x + 2, Kacaremnoï x neh, npoxonsuei
xepes roux uepecenenun napaboms © como Oy, m apar
x=1
2) ramepéonof y= À, racanensuol x nel, mpoxonameñ sepei
rouxy © aGennecoli x=2, u mamen y=0, 2=6.
Harn naomenı durypst, orpammuennoh suas:
1) yox?-8x?-9x41, 220, y=6,
2) yoxt 22845, yo, 220, xml.
[48] Tipu xaxom onavenmn A nxomans dtrypt, orpansaemnoï ne
Pa6onok y=x"+px, THC P — 9ARAMNOS wcno, x mpanol
y=kx+1, naumensusas?

Bonpocet « rnase IV]
1. ro naommecrea neprooGpaanoñ ana Pym y=f(2) me
nexoropon unrepnase?
2. Kar sagars nce nepsooGpasuure yuan y= f(x), comm F(x) ~
ona na unx?
3. Samucar» dopmyanaı Nepnooßpaausıx ant yen
y=2’@#-D, y=L(x>0, x<0), y=e", y=sinx, y=coss.

4. Tlepewneaurs npanuna uaxomaenus mepnoo6pasmax.
5. Tipusecra mpuxep Kpusomuuelinoit spaneum.

Kaxyio Qurypy Massmaror xpuuozurmelinol rpanenueh?

7. Banncars dopuyay Hstorona — Jleñiónmna.

Tipuncerit mpnacep Qurypsi, LOAD KOTOPOÍ Mono marmo

aurs no dopmyae S=S(-f(x))dx.

Se arena

& Tlpunecrar PHN (PHTYPE, MAOMOE KOTOPOÑ MOXHO ENTIC=

aut no dopuyae SÍ. (2-1: (0) dx.

10. Kax ¢ nomompo unrorpaza naître mym no antannoh ckopo-
CTH, BEINHCHHTE paßory mepemenHoi cum?
Wkaxyio eymmy masiunaror unrerpansuoit cymmoit Pyme
y=f(x) ma orpeaxe [a; b]?
ro maannaor onpenenenmun unrerpanom or dy
y=f(x) ma orpeaxe [a; b]?
08 ypannenue nassınaıor ypanıtennen rapMontecKoro KO-
Acöanma? Kar annmeninaeren ero pemenne?

1. Tlokasars, ro F(x)
aan oyes f(x)
2. In gynnunn f(x)=3x?+2x-3 malıru mepsooöpaskyıo, rpa-

(bux KoTopoii npoxoaur vepes Touxy M(1; -2).
à Bern

+ 2° -cosx annaeren NEPROOGPRION
Qe! +32?+sinx ma nee wucnonok npa-

42, ay \eos2xdx; 4) \sinxdz.
° A
4. Hair naomens burypu, orpammenuol napaGonoit y= x? +
+x-6 m ocbIo Ox.

1) fardos 2)

Lan Gym f(x)=e"-Bsinx man nepnooGpaanyo, rpa-
un Koropoli mpoxoxur “epea rouy A(0; 2).

2. Daraneauru:

2

D {yee 2) bata dx.

3. Hooßpasum» durypy, naomans Koropoñ para es dx, u
BRIE sty DTOMens.

4. Haïrrx naoman» DHrypH, orpannsennoi AMHHAMH

y=2+4x-x? u yart-2r+2.

5. Haliru nnoman» durypu, orpannsennoh napa6onoh y = x? +1

‘HW KACATENBHEINH K Melt, IPOBeAEHHEINH H3 TOUKH (0; —3).

M4 I Vicropmsecxan cnpasxa
Hecuorpa MA 70 uro HHTEFPANEEOE NOACHENNE nORBUADOS
3 XVII p., ero neroxn MOHO OGuapyxHTL B rAYÉOKOË apennoern.
Tan, » Mockonexom nanuıpyce, noapacr Koropore oxono 4000 xer,

ES

Vicropmecra Chase

OMHCHBACTOA axropuTM BEIUNCAEENA OÓBEMA yoeuenmoli rmparur
AI € KDAAPATHSIMH OCHOBANHANIH U CTABATCA IPOÓNEMEN HAXONEE
Hua OÓMIX IIPHENOR BBIUNCHEHNR LIOMAEË KpHBOMHEMEEX Que
Typ H OÓLeNOB pasmmunux Ten.

Merox apennerpewecxoro ywenoro Enoxca Kunnexoro, na
ssanasiii BnOCACACTBHA METOAOM HCUCPIBIBARNA, MOSBOAAN ROTA:
TONNO TOWMO BHYNCIATL MAONAAN JIOÓNX duryp ua 0CHOUC nen
HOTO MCHOADIOBAMHA MPOJCMBMENX MEPCXOAOR. CYTb 9TOTO METORA,
HANPHME ATA BAIUNCHERNA ILIOMAJEÍ MIOCKUX QUIYP, Saxo
erca » exexyiomen. B urypy BINCHBALOTCA u BoKpyr Hee onncu
BAIOTCR MHOTOYONLRIEI, HCO CTOPOH KOTOPEIX YRCMMRAETE.
Haxoaurea mpenex, K KOTOPOMY CTPEMATCA MOMAAN OTHX NEO
JTONBRUROD; ero M MPMUMMALOT aa OMAN pacemarpunacuol Qu
TVPH. Cxoxnocr MIPANEHEHIA TOTO Merona B TOM, UTO JR Kx
aoû Qurypk mazo Oso wekars CBOÑÍ cnocoG Bbiuucnenma pen
na. B Apennoorn INM MCTOROM HONIGORATACE Apxımer m Ents,
B nanpnelunen paspirie MeTOROB, KoTopkie MPHNCHAI pert
rpeveckue yueunie npu BEIUNCIOHIM MAOINAZEÍÍ u OÓLEMOS, pare
10 x nonaTmo unmerpam

B XVII 2. semeuxË maremarux u acrpomow M. Keep, or
RP aaKOHE ABIKEHIIA MLAKET, OABIM HA MEPRMX NOTA
CA BOSPOAUTL MeTOR Akiusesrennn nnomaneh K O61eMOR, MAY
or Engoxca x passursih Apxumenon. Kenxep BEIMMCAAA moines
nnockux duryp u OÓLEMM Ten, OcHOMLIHANCL ma Wee pasbnenma
Quryp m Ten ma Gomsmoe «meno Mansız uacrek, Koropsie on un
RBA «TOBMAÍIIMN! KPYHOUKAMUS man euacran KPAÏHE mad
unpunsie. Baron eymmnponan maomam (Hau OGDEN) nonyser
MIX pu pas6nesim duryp (rem).

B orme or Kennepa wramsancxui maremarıık B. Kana
epu (1598— 1647) m kmure «Teomerpun nenemumnixe, Jens dry.
PY (reno) mapanzensusen pan (IOCKoCTAMM), CAN am
Aus (anockocrH) AHAEHILLINH BeNKOÏ TOXINAS, ORUAKO sera
Auınan» ux ar maxompemma maomann durypu (06Lema en).
Ton nomarnen ence nun» Kananbepu nOHHMAN TO Xe, TO ma

cerogus nonumaem non \f(x)dx.

Tpyast Kerrepa, Kanansepu u apyrux yuensx mocnyaccm
ocHosoit, Ha xoropoh Huron u Hefönnm nuicrponzu reopımo mir
‘rerpassiioro ucnnenenua. Paspurue arol Teopum mponomxienn Be
xep u II. JI. YeGsimes (1821—1894). B uacruocru, Yesuen par
paGoran cnocoOu unrerpnponamnn OTACALUMK KHNCCOB Uppal
mon Gym.

Onpexexenue unterpana KAK npegena unterpamsmnx cy

npauamremur O. Kom. Cannon § f(x)dx ven JehGuun. Teprars

«umerpaxo (or nar. integer — nenn) Amepnue Guin mpenroxee
H. Bepnyann.

1560) rnaoa v

Miepsogopasnan u wiTerpan

Kom6unatTopuka

Mamenamuvecxas uemuna nesaeucuxo om
mozo, a Hapuwe uau a Tyayae,

odna u ma we.

5. Macrase

§ 1. Maremarnueckan MHAYKUMA

(SSE Paccnorpum auauenus Kuanparnoro rpex-
nena n’+n+41 mpu n=1, 2, 3, 4, 5, 6, 7,
14. Ouu coorsercrsenno pass 43, 47, 58,
71, 83, 97, ..., 251. Bee nozyuennbie uuena mpo:
crie. Hampanmbrerca nunon, ro np m0Gon
Harypansnom n uncno

Nentintal a)
ABnnercn POC.

Bano, enezanmah HA ocnonanm rponepia
Gomsmoro uncaa MPHMEPOB, HABBIBAMT noKasa-
Tensernon aemodox Henoanol Mamemamurec-
xoù undyxyuu'. OXHAKO TP TAKON JOKAGATEME-
ferme nensan Ohm» aGcomoTHo yRepeHMM, “To
yrnepmaense CHPABEINNO Mpst Apyrux, menpon
pennsix snauennax n. Tax, manpnxsep, snavenKe
Paccmorpeuuoro «Banparnoro rpexunena (1) mpi
n=41 panno 41 +41+41=41(41+141)=41-43,
7. O. We ABIACTCA NPOCTHM "NCAOM.

Jas erpororo noxasarensersa yraepmnenu
Ha MHOKCCTRE HATYPAMDMIIX 4NCEA MCHOMLyIOT
Memoû noxmok mamenamuvecrou undyxyuu
(«parxo stemod undyryuu). Toacumm ero na
npmmepax.

Banava 1. Hokasars, aro mepanemerno

aon (6)
enpanennuno an 00oro MATYPAMBROO A.
D Samerum, uro 21>1, 7.0. mepaseucrso (2)
enpasonnuno mp n=1.

1 Huayicqun (or nur. inductio — nanegeune) —
yaonaxmowenite OF PARTOD i OÓMEMY Y TROPA.

91 2157

Marennes ya

Tipennonomum, «TO nepanererno (2) cnpanommmo aan nexe-
TOporo narypansnoro n. JloKarxex, 170 TOMA 18 SOTO mpexnanr
en CACAYOT CMPABOAAMDOCTL ANANOFHANOTO nepanenerna Jas
Cxeayiomero merypansuoro wmexa n+1, 7. e., uno 2" 1>n+1. Yu
Ho%u8 OGe «ACTI nepuoro Hepasenersa (2) Ha nonoxurennoe u:
10 2, nonyan vepnoe mepanenerao 2-2">2-n, 7. o.

295 2n. o
Ho 2n=n+n>n+ 1, rax ax n>1. Tooromy 13 nepmoro nepaner:
croa (8) enenyer nepnoe mepanencrao 2"*L>n +1.

Tirar, mpu n=1 dopuyna (2) nepna. Ilo aorasannosy om
sepa x Ann cxeayiomero matypambmoro “exa n=2, A Tax kur
opuyaa (2) nepua npu n=2, TO no AOKBAHHONY ona BEPHA u pn
n=3, MOJTOMY Ona BOPMA u UPM AA MT. A, T. 0. MPIE BOX Ht
Typanux n.d

Takum 0Ópadom, cnoco AoxaVaremmerna werogom noo
MATOMATANCCKON UNAYKUNN COCTONT R CHOAYIOMON:

Ilyers rpebyeren Aoxaaarh, uro MEKOTOPOE yrnepacaeuue onpt-

sexaso xn muodoro narypaarmoro «mena ñ. Jan oro

1) mponepserea enpasonzunocts yroepznsa mpH n= 1;

2) noxasuınaeren, «ro ecm oro yrepxyeme BepHO Jun Ne

KOTOPOTO MNTYPAMIMOTO THETA n, TO OHO REPKO m AAS CHE

jomero sa Ham Marypasnoro uncna n+1.

Torxa xannoe yruepaeune nepuo qu n= 2, n=3, n=4 um

o6me AIR AOÓOTO marypansuoro n.

Banawa 2. Noxasarı, «To ju moGoro narypamıtoro n enpe
ennuno pasenerno

194284384. tnt NE, 0)
D> Bocnoxsayemes meronom Maremarımeckoik say.

Fas?

4

2) Toxaxem, sro ecau parencrno (4) nepno ans neroroporo
HaTypaHOTO UMCAA N, TO OHO BepHO m ¡UA n+l, T. e. BepHO
Panenerno

1) Tipu n=1 paseucrso (4) nepuo: 17=

142949 nn 19e OW nse) 6
TpuGaszna « oGenm wactas nepxora no npeanonoxenne
Pasencroa (4) wueno (n+ 1)", nonyuaem pepnoe panenerno

34283 s ET

DR 480 A)

Mipeopasyem mpanyio wars sroro panowierBa:

BOD s(n 41) mn (En
ne sean hater ene)

mirta), (a
4 4

AB nasa y

RowOnnaropnna

Tosromy wa enpaneyunnoern panenerna (6) cxexyer empanen-
amsocrs pasewersa (5). Cregoparenbro, paseerso (4) cupaneyu-
Bo mpu m0GKX Hatypansusx m. EA

Yopaxnenna

[E] Meronom maremarmueckolt HHAYKIDIM nokasark, uro ana moGo-
TO HATYPOJIEMOLO n CUPAPCJLUINO panenern

1) 143454. +20) 7
2) 845474
3) 14244442 20m

4) 8+0+27+..4 80e 2 (8-1).
EE doxasars, sro ann mioGoro maryparsnoro n cmpasenanso par

‘enerBo:

D PAR ante Marne,

2) 104284384... 40m aba,

3) 94884584... + (2n Inter);

E (And M

ES Merozom maremarsueckoi unaykıın noxasars:
Dgopuyay yma 8, MED. m wienon apngermeckoß
anorpecem 5,- 28:44 y, eye ay — nepal unen, d —

paanocn, apupmernveexol mporpeccis

2) dopmyay CYMMEI S, nepssix 7 WIeHoB reomeTpHuecKoit

nporpeccnn Sen, rae b, — mepaniit wxem, q — auame-

xarens reowerpırueckoii mporpeccint, g#1.
Bdoxasars, «ro np mo60N naryparenom n «meno;
1) 6%-141 aemurca ma 7; 2) 4*+15n-1 nenurca ma 9.

$ 2. Npasuno npoussenenus.
Paamewyennn © NOBTOPEHAAMU

B oowonwoit more pemanucs axenemrapune KomGunamop
ue sadan, consannnte © cocramnennen pasmunux coedunenuü
(common) no mciomuxea anemenron. Bato cdopnyaponano
‘paso nponsnezemun, ynpomaomee nONCTT sea onpereuen-
sux cosy.

Tipasnxo mpoxanexensx

Eau cyıeerayer n sapkeurov suGope uepnoro oxemeura 1

ARA komnoro MO ux HNCETER m napmauron nuibopa HTOpOTO

anenenra, To cymectayer n-m PAAMHNEBIX map © PBIÖRAHKB“

Mu eposin 1 Dropbint Dnemenranm.

A
Tipasuno nponseeneamn. Pasweujenin € RONTOpENAAMM

Banaya 1. Ckompxo PRIT MRYOHRNHEX “CON MOXBE
samicars € nomommo uxbp O, 2, 4, 6, 8?
D> B xauecrne neproï app unena moxer Outro BuOpara nos
mo muop 2, 4, 6, 8 (n=4). Bropoñ uu@poñ moxer cayxurs a
Gan ma nanınız nudo 0, 2, 4, 6, 8 (m=5).

Corxacno IPABILIY IIPOMIDCACHIA “NCIO BECBODMOMHLX Ae)
SHAUHEIX Neem, COCTAHAEHHEIX HO MpenoKeHHEIX Ip, Pan
n.m=4:5=20..4

Banana 2. B uIKorsHoÄ ommmuane no maremaruke modem

Tenaum OKADNANCL 3 denonexa, DOMINIO no dauke — 2 +e
NOBexa, B oxmmmmane no xumun — 4 uenosexa. Ha paiionnse
oammerwansı no MaTeMaTHKe, Dame u XMM oma JON +
mpasitrs no OAMONY yuaulemyer 13 «mena moßennrereh monos
‘TypoR 10 rpem npenmeraw. CKONDKHNH CMOCOGAN MONO 910 CAE
sare?
[> Cornacto MpaBnay mponsnenenna onKoro yuacrmua ma om
may HO MATEMATHKE N OAKOTO YUACTHIA HA OxMIIBAY NO di
auxe moxno muópars 3-2=6 cnocoGamn. K xaxnoÑ ua nonyuer
HBX 6 MAP MOMO MPHCOCANMMIETE AIOGOTO 113 NeTBIpeX moGemre
xeli omeunnaxs mo xmenn. Taxim 0Ópagom, Tponx “exoger ¡us
YUACTHA m HAONAHMEIX TPEX OIMMTINANAX COTAACHO mPanmay ne
Hopegemma xoxo BEIOPETS 6-4=24 cnocoGann. À

Pemenne sana 2 nokasano, uro NIPABILO MPONOBEACINA wo
ner ÓNTD MpHMeNeHO HCOMOKPATHO AJA MonCHeTA CocAmmeRnÄ 1
TDex, METHPEX u T. A. INCMENTO, BMÖHPAENBIX na onpexenemux
MHOXCCTE € KOMEAHUM MHCHOM DMCMENTOR.

Ba Cxomsko paammmutx nermpexéyrnemmux co

MOMO SAITICATE © MOMOMLIO GyKB ent» H sa»? (Caocox m Koti
HATOpHKe H&omPalor JIOÓYIO nocrenomaremnocrs ÓYXD.)
[> Kansas ua uerspex Gykn cocranıınemoro CNOBA nocnenonaren-
no suOxpaeer xs muciomuxca auyx GyKB. Ipnwensm Tp
IIPaBICIO nponovezennn, HaÏxeM MHCHO COCTABTACMBIK Herkipen
Gvxeunux exo,

2.2.2.2-21-16. 4
1 3anerum, “ro cpeau oGpasyemsix m sagave 3 cnon Guam,
HANPHMOP, CAOBA «MMMM? M eMMMAr, OTUMUAHOUINPE APyE oF
apyra maGopamn (cocranom) Gyxs. Bui, nanpumep, u cxor
«aan» m eaamas, oTaWuaIOMMecH APYT OT APyra MOPAAKOM pac
morowenun E HUX ÓYKD.

Coeaunenua, conepmamiue n snemenros, BBIÖHPAeMBIX 13 axe
MEUTOR m PASANNNLIX DUAOD, H OTMURIOIMCA OAKO OT APY
TOTO JIHÓO COCTABOM, 160 NOpARKOM CHCXOBANNA B MUX ne
MeHTOB, MASHBAIOT PUSMCUJCNUAMU c nosmopenunmu us m
mon.

160) rnaga y

‘Komuraropira

“ucao scesoaMomHBIX PASNEMEHN ¢ MoBTOpeHNAMH HS M
anementos no n oGoauawaior An (A — mepsan Gyxsa dpannyscico-
ro cosa Arrangement — paamemenme, NPHBEACHIE m HOPAJOK) 1
swraior: «Yueno pasmeinenufi © LOBTOPEUHANH KA DM NO OMe Mau
«A € Neptoit ua om mo one.
Tax, y annaue 8 Onno naneno
A4-16.
Moro AOKasaT», 470 JIM JIOÓNX HATYPARDHBIX m un BPHO
Panenerno ‘a
Fra a mM
MA Jloxuaarennerno Dopuyms (1) nponenem © nOMOMIBIO Marona-
Tirdecxof MHAYKNMK HO N — UNCIY ONCMCNTOD B PRIME HP
dixcenponannom m.
1) Tip n=1 onenmano, ro
Am,
Tak ak Kamp0e PAIMEMIeHMe COCTOMT 13 oAKOrO anemenra u pas-
meiste Pagmenjennn HONIVURIOTOA TOXBKO HA PAIMENX Anenenro
<HCHO BHAOB KOTOPHX PABKO M.
2) Tpennonoxis, wro opuyae (1) pepna ana nekoroporo n.
Hoxarxem, “ro OMA vepua u Aust n-+1, 7. e. CIPADCILDA hopmysia

Ag! amt"),

Paccmorpum 11060e pasmeueHHe (c NOBTOPeHHAME), COCTOA-
nee ss m anemeNTon, m mpreoennmun (apununie) K HEMY 226
ene oniforo no AMGICWKER m puizoD. Tloayanirea pasmonemue 19
(n+ 1) anemenros. Tipu arom owennsno, Wro HS KAKTOTO paameue-
sna, COCTOAINETO 13 n ANEMENTON, monywacren CTOMBLO parmemo-
muñ no (n+1) axemenTaM, CKOJIbKO MMeeTCA PASIMUEBIX BHAOB
aneneuron, 7. e. m paamenienuk. Jelicraya raxım cuocoGom, Mbt
He YIyCTHM HM OANOrO BOSMOXHOTO pasmemenka no (n+1) ane-
tenant u NN 0AN0ro Re noayunM Bann. TlooTONY “nO paane-
Menu € nonropennaun wa m BAD Anemeirron no (N +1) anenen-
‘aM n kaxcxox nuißope Öyaer m m paz Coste, ven “exo Pnane-
Men © monropemitanı na m no My 7. €.

A mem,

‘Tem cambia popmyxa (1) aokasana, MM
Emi 3anaua 4. B anomunofi encrene cuncrenus, npumennenoft
» 9BM, nenonsayıor ea emsona: 0 1. B nexoropolt BEM xax-
20e mammmioe COBO semcuinaeres 1 AMAT © MOMOIILI ara
CHMBOXOB B 16 mponyMepopannzıx paspanax. CKONEKO PASIMUHBIX
aunt CX0B NONNO SARNCATE m 9TIX paspazax?
DB samaom na 16 paspagon MOXET CTONTD oni 1 ABYX cHMBO-
zon. Ouennnuo, weno paannunsıx Mamunnux exon panto Aló
205590. 4 A

$2 161
Tipasuno MPONSESARHVA, PADNOUISHI € ROSTOPERMANN

5. Ckomxo paamsx rpexamaumux sucex, He MMCIONAX opt
BEIX IMP, MOAHO SANNCATE € HONOMEO HDPE
1) 1,203; 2)1,2, 8 m4?

6. Cxom.xo pasmux rpexanauın
momo mp:
1) 6,718; 2) 6,7, 8 n 9?

7. Cxomsxo pastux ABySHAUHLIX ICON MOHO SANHCATD, ie
monsaya mbps 1, 2, 3 m 47

8. Tlyremeorennnk momer momaers wa nynKTa À 2 nyunt C,
npoexan wepea nymer B. Mexay nyukraun Au B nues
pm amroxoporm, a Mexay mynKTann B x C — enesmon-
pommoe x peunoe cooGuienuA. Ckoxbxo cymecTByeT pas
Mux MapuipyroR MexAy nyukzanıı Au C?

9. Comm cuocobamu MOryr Okt» pacnpenenemb sonoras
5 cepeSpanas Mena nO HTOTAM mepneerRa epa no dyréo
AY, ecnx MEO VUACTDVIOPIX B neprencree KOMAHA panno 16?

10. Cronskusn CHOCOÓAMI Mono COCTABMTL paermeanne ypoxcs
Ha ONE mens u9 WecTH PASHEX yYeOHEIX upenMeTos?

11. B xxacce 20 ysamuxes. Heobxogumo muGpars ua ux wc
crapoery, Qusopra u KyasTopra. Ckomiamn erlocoGaMit ox
no ocyimécrsur oror BuIGop, can OMA Yuen MOT sas
MOTD TOKO OANY ROMANOCTE

12. B onnoh ua CTPAR ADTOMOONABNNO HoMepA Ha HETHPEX nme
(ayan moxer cron u ma Hepnon MECTE) aanucsinarores m
ROACTINKAX MATH Paannunsıx KBETOD (KORA ma NTI me
‘Top otolt crpais meet HoMepa cpocro pera). Cxoxbxo per
MBX unaeru C MOMPPANH MONET Öbirs BELAHO anronnaneır
nan n 970 crpane?

18. Jecars yuacruxon Kompepenmmn o6mexstes Barr
xaprowicant (AH mpyuten CBOLO Kaprowicy pyrun yuacı-
men). komo neero Kaprower Giro poaxano?

14. Hecars yuacrunxo Konhepenmum o6menaaKcs PyRomomanı
am, moman pyRy kannony. CKOMBIO NICETO pyionomani
Omno cnenano?

A5. Ckonsxo pasnnunsıx ımmpos woo naßpars » apromanın
coï Kamepe xpaueuun, ecan mp cocranneren € HOMO
100601 us pagara Öyı Pyccxoro anpannra € nocaeayionE
TPexanaunsım AHCIOBHN KOXOM?

16. CxomKo MMCETOR commaneumx HATypAMbHBIX INCE, » KOT
DMX NCO HMAPH, CFOAIME Ha MEUETIAX MecTax, Pagan

AZ. Cxombro HeNerHisx derpexananmmx sucer MOXHO James
€ nomomo mdp O, 1, 2, 3, 4, 5, 6, 7, een moÿyro va mu
MONO MeTlomKORATS À aHeRe He Boxee oRKOTO paga?

162 nana y

Komonnaropıa

$ 3. Nepectanoskn

Banana 1. CKOJBKMMH cnocoGaMH MOHO HOCTABHTE PAMOM
sa nonxe nerupe paonansıe ra?
D Ha nepoe mecro MONO NOCTABITE MOÖYIO HS uErpex war,
ia mopoe — moÖyıo ua npex ooramınıca, ma Tporue — modo na
[MYX CCTARINIXCA m Ha MOTBODTOS MecTO — MOCACANIONO OCTADINYI0-
ca kmury. Ilpumenan NOCHeNOBATEABHO HpABHIO HPOHSBENEHMA,
tony 4:3-2-1=24, 7.6. KIT MONO NOCTANIT 24 onoco-
Gr. 4

B sro aanaye darermuecin Guino nañxeno 4HC10 BeeROIMOX-
Aux COCAINMENNÍ Ho NETMPEX AXNERTON, KOTOPBIE OTAINTATIICH OZ-
10 of apyroro MOPARKOM Pacrionomemitk DIX anemennon. Take
SE MASIBAOT mopccranonkanın.

Onpenenenne

Hepecmanoonanu 13 n OMeMEHTOB HAIHBAIOTCA coeAMHeHIA,
KOTOPBIE COCTONT Ha N ANCMCBTOD m OTAIUMOTCA DAMO OT APY"
Toro TOABKO MOPAJUICOM HX Pacnonorenna.

“neo nepecrakonox 13 n anemenron oGosmauaor P, (P —
1epnas Sykwa dppaunyacioro exona Permutation — nepectamonka)
B aanaue 1 Onıno malizeno Py= 24.

Nocnenonarenuno npunennn npanii10 mponnerenut, Momo
orysutTs OPMYAY NCAA Mepecranonok P, msn PaanHansıx ame
Pymnn-1%n-2):....3:2:1=1-2:3:....(n-2n-1)n.

Tiponssenenne nepamx n uaryparvmx uncen oßosmanar n!

(umerca ou Qaxropmane), 7.0. ml=1-2-3-...-(M=1)-n, MPH

ex no ompenenemmo 1! 1. Tarn 06pasom,

Pen @

Barava 2. Cxomskint onocoGamu momo momomur» 10 pas-

aux OTKPHITOK » 10 Hmesompuixoa kommepron (no oxmoñ or-
pure 1 kounepr)?

D To dopuyne (1) maxoxmx

Pjp=101=1:2-3-...:9:10=3628800. 4

BAB paccmorpenmux saqasax MM JARHMAIHCE HOPCCTARODENNME

manmmux oxemento, Ecan 2ke NOKOTOPLIE Mepecramıaensie 22e-

Sera OYAY ORUHAKORMIMIL, TO HCCROINO MX PADMANIEIX mepe-

cranonok ma MIX ÓYACT Menuine — HEKOTOPHE NEPECTANOMKN

commaayr one c apyroli. Hanpawep, nepecranzan GVKBM m onope

‘stapke, uoayunm 24 PASUKUMBIX scHoBa» — MepecTaHOBOK HS ue-

supex paoamumenx Gym. Ecam re sanuears neenoomoxnue ne-

eTeHOBKH ua GYKB CHoRA «nana» (cpeAu KOTOPMX ABE MAPA
cammaxonnix 6ykn), To Wx MOAYUITCA ncero wecrk:
mana, naam, uaa, AMAN, aan, anna.

6s 2163

TiepecTanoai

OGo6ueHHO aHaTorHuHBIe sarauı POPMHPYIOTCA CTEAVIONM
o6pasom. Myers umeiores onemenra m pasamumax mon. Te
Gyeres naru umex0 MEPCCTAHOROK, O6pagonanMMIX HA 71, Arcnei-
TOR MepROTO HILAR, Nz DCMCHTOD BTOPOTO BUNA, .--, My SEMCHTO
m-ro Buza.

_ Ueno rakux nepecmanosok c noamopenuanu oGoanaisic
Pay ay Koanuecrso anementos E KAKAOÏ us HUX paro
REN +Ng+ e+ Nye OFEDHAKO, TO CCH Gor Boe PIEMENTE n Tepe
CramosKax Guin paamumsr, TO HX unezo pasmanoco Ost nl. [pa
OAMI! KO CoMMAAMIONYEX IHeMeNTOD KMCAO HepEcraHOBOK Oyar
MenbuHM «nero. Buinenem dopmyay AIT moneuera wena NePe-
CTAMOBOK € noBropenunmn.

© Pacemonpun mepecraronky omementon

o

» KOTOPOÍ HOCACHODATONLNO DAKCANIA ce DACMEUTIA nEPuoro mia
(ex wueno pasxo ny), semer sroporo nıaa (sx “meno pamo
Ro), woes DEMONTE M-rO DUNA (WX ANCHO PAHO A).

Brewentor A-ro BHAA MOKHO nepecramre APT € APyrOX mi!
cnocoGamu. Ho » cnsan © TOM TO OTIC ONOMONTE OmMHAKODLS, 10
NepecralobkH MO WX MMUCN HC OTAHUANOTCA OREA OF APyrol.
Hampuep, B nepectaxonke ennaay uxuero me mamenron, Cex
nomenaro Mecranit HEPHH co Bropuist x (nam) TpeTm © werDep

Tlepecranopkn anemenros nepporo, PTOPOTO, ..., M-TO muxos
MomHo ooymjecrnnarh meoamucumo Apyr or Apyra. Hooromy co
Tnacno mpanıay IPOMORCACINR ancnenru nepocranom (2), ne
MOHAA CO DURA, MOMEO MEPECTADAATL APYT € APYTOM Ml - 731°»)
Xnp! cuocoGamu. SAULT, wneno nepecramonox ¢ MOBTORCHINNE
Gyaer m mien..." Myf pas mentre, dem nl. Taxms o6pasox,

nt

o

me A ny tng teat me ©
Bamerun, “TO Sanaa 0 MOACHETE ENS repecranonon a OyKS
CHOBA ÁMAHA> CBOMIACE K NAXOMAEHIID UNCAA MEPECTAHONOK €

mosropenuann Ps:

2, et AL,
ma ame
Banana 8. Cnona u pase © mepecrannenneinu Gyrsau m
BMBAJOT anazpammanı. CKOMDKO AHATPANN MOHO COCA 1
exon emaranas?
[> B exone ma 6 6yka enaxakas Oykna «a» ucnombayercs 3 par
Gyxna exo — 2 pasa, a Gyxna «mo — 1 paa. Corsacno opuyar
(8) sexo POCBOINOMHHX AHATPANN par

Fa, o 20 4-5-6 |
Pa a A

164 traga Y

Kowbmiaropia

Ynpaxnenun

18. Haïtruc anauonno:
1) Po: 2) Pgs 8) Pa; 4) Pye

19. CKompxunn CHOCOÓAMI NOMIO paccagurs Vernepsix ereii Ha
HCTMPEX CTYADAX n eroxonoli xerexoro cana?

20. CKOXBKMNH CHOCOÑANI MOHO YCTAMOBHTE AEXYPCTEO nO OM
MOMY NERODEKY » eH» CPC CCM YIAIYIXCA FREE B Te
venue 7 quelt (amasıl Aonken OTHEMYPETE omum Pas)?

21. CKomxo narneuaumux ancen, He COACPKAUNX OANHAKOBEIX
udp, MOXHO samicars € momombro map 1, 2, 3, 4, 5 rar,
rob
1) nocxeamedi Osına mubpa 4;
2) nepsoñ Guia uuppa 2, a wropoñ — mubpa 3;

3) repris um quppi 2 m 3, pacronoxename » mo6on
nopaake.

22. Ynpocrurs opmy sanmen ssipaxensit (k — marypansnoe
nexo, > 5):

2) 16-154 8) 121-18:14;

4) Rt Get Ds 5) M=DER 6) (DEAD
DRIN 8) (eS)? Th + 12),

23. Ynpoerurs:

a4 1
9 à 9 à
mim, g EAS
Dat Y ro *
ecan Öyksamm k, m, n oGosmavenbt RATYPAALIHIE HHCTA.

A. Pemurs ypasenne ornocrrenumo a:
Ped EA Pa
pez DE ee Fe
Cronsko pasnmmux enon MOHO cocrantrk, nepeorannan
neorasın Oyauı » cove erpeyrommus (cunras u camo 970
cnono)?
, CKOXENO paanmux naramaumux uncen (He conepautux
oqumaxonsix AP), He KPATHLX MATI, MOHO COCAPTE ma
mp 1, 2, 3, 4, 5?
Hueeren 10 xmr, openu xoropsrx:
1) 8 ur parmrumsx anropon 1 ABYXTOMHIK OXHOTO ABTOPA,
Koroporo He Gino CEA PEASIAyULLX socomu;
2) 7 nur paonsıx anropon u TPEXTOMINK DOCLNOFO ABTOPA.
Croatia cnocoGam MorkKHO PaccraniT» ora KUNTH ma os
Ke Tak, UTOÓN KHHPH ORMOFO ADTOPA CTONAN PAJOM?
BE Cxonsko amarpam Momo COCTABNTA 113 CAOBA:
1) xox; 2) oxmo; 3) apama; 4) Ganan; 5) mapaGau;
6) xyxymuxa; 7) marenaruna; 8) verpaanp?

ES

IS

gs 165

Tiepecranonın

[EBD He Hopuñ rox rpoux Gparsan PORENTONH MYTIAN 2 nonepe
6 paonmunsıx KBMT m Penman KAKAONY MOXAPATE no 2 Kur
Tm. CKOXDRHMIK CHOCOÓANI MOKHO CACNATD 9TH MOAApKH?
Kora X. Piolirene (1629—1695) orxpuin komo Carypua, te
cocraniin CACAYIOMYIO anarpanmy:
aaaaaaa cecce d eeeee # h iii Wt mm
nnnnnnnnn 0000 pp q rr s LUE uuuuu.
Drum Oyknasır sarmesinaeren pasa »Annulo cingitur tem
plano, nusquam cohaerente, ad eclipticam inclinatos (Oxpr
EN KOMDIOM TONKHM, TUIOCKHN, HIE Me NORME,
HAKTONMMN K AKANTITIKE»). CKOAKO PAU amarpacı
MOL cocranure mo Oyıca aamımpponannoli Pioftrencos pasa!

§ 4. Pasmewenua 6e3 nosropenuñ
Bana

semncers c HOMOIBIO mp 1, 2, 3, 4 apm yenonm, uro n Kex

xOÑ samucn wer onmHaKoBBIX LMP?

[> MepeGopom yGeaumes n rom, «ro na wermpex nudp 1, 2, 3

Mono COCTAMITE 12 Anyananıılıx “COX, YRORACTROPATOI Juro

mo:

CkomsKo pasmiameix ABYOHAUERIX unCen Non

12, 13, 14,
21, 23, 24,
31, 32, 34,
41, 42, 43,

B aannen auyanasuoro “ICAA HA wepnom MeeTe MOREN eros
moGaA MS JANHMIX YeTbIpex HHDP, a Ha BTOPOM — ons na per
ocrammuxca. TÍO npannay nponsnenenun TAKHX ABVIRAUEX we
cen 4:3=12. 4

Tipu pememun sanauı 1 u3 wersipex names onemento
(aupp 1, 2, 3, 4) Gstum oGpasoBamb BCeBOSMONKBIE Coen
no ABA OXCMENTA » KAXKKOM, HpHes wOGiLe ARA COMME oF
awsammes Apyr or Apyra 160 COCTABOM si1emeHtoS (Hampumep, 1}
x 24), auGo nopsnwom ux paenonoxenus (nampumep, 12 1 21)
‘Taxue COeAUNERHA HASHIBAIOT PAMELNEHMANH.

Onporenenne
Pasnewenunnu ua m anementon no n anemenror (n<m) ur
SMIBAIOTEH TAKUC COEAUMEHIA, KAKAOE 13 KOTOPLIX CON
7 ONCMEBTOD, BUATLX M3 AUX M PAIX DNEMENTOD, u KO
Tope OTAUNAIOTCH OAMO OT APYTOFO IHÖO CANIN EE:
Mu, ANG0 MOPAAKOM ux Aero.

Huorga taxne pasnemenus HASMBALOT pasmewenuanu 663 mo
amopenuit

nono weenosmonutix pasmemenxi Ges nonropenui 1 m
eneneuros HO n anemeuros OOosHaaIoT AZ u UNTAIOT «A na au m

1660 nano v

‘Kom6nvaropra

ony. Tax, nanpuwep, npx pemenms sanaın 1 Gnxo yeranonneno,
wo Aj=12,

Bumenen dopsyay ans sucres AZ — “nena pasmeme-
xx na m OJCMCITTOB mo n anemenron.

O flyers uneeren m Pasınusız anementop. Torna «meno pasne-
IMEI, cocronmux Ha OXMOTO DCMENTA, NMÖPANNOTO MI MMEIO-
MLUXCA m anemenTos, panko m, 7. €. Alı=m.

<roGm cocrapurs ace paomememua mo m axememtos no 2,
x Kamponıy MO paste OÓPADOBAMILAX PAIMCINCIÑ ua m oncmenron
no 1 ÓYxON HOCICROMATCIBIO IIPILCOCAMIATD NO OXHOMY 1 ocra
wuxex m-1 onemenrop. Ilo npasnuy uponopenensn ICAO TARHX
comment panno m(m-1). Taxis o6pasom, Ay =m (m—1).

Man cocrannenna ncox paomemenmi na m mo 3 K KAKAOMY
xa pance nonyuenumx paamemenuh ua m oxementon no 2 mpnco-
an no oxepen mo OAMONY na ocranıaca (m— 2) dnemonror
To mpapnay mponopenenma «meno rakux coeaunenmit panno
mm-1)(m-2), re. Al =m(m-1)(m-2).

Tlocmexonatexsito MpuMCHAs MPADIO NpoMOREACHNA, ana MO
Goro nm nonyuaem

Alg= mi (m—1)(m~2) >. (me (1.0 wm

Hanpumep, Af=4-8=12; AJ=4-8-2=24; Af=5-4:360.

Ormerm, 470 npañus sacro hopmyası (1) conepaxur nponsne-
AeHMe n MOCMEAOBATENLHBIK MATVPANBIEX vNCOA, HANGomLIee us
xoropux paso m. Tlycre » popuyae (1) m=n. Torna

APN (R= (2) 21e Pa
7. €. ‘eno paamemennit ma m ameMenToR MO n PARHO meny
Nepecranopok MA oTHX ANEMEITOR:
AP (2)

Banawa 2. Crommiar erocoGamit MOHO oGosANHT® nepu-
XK ZaHKOTO TpeyronBHMKa, nemomaya OyKsB A, B, C, D, E, F?
D Banaua CROATA x HAXOX/ACHMIO UNCAA pasmemennii Ha 6 ane-
Mento no 3 oxementa m kamaom. TIo ¢opmyze (1) naxonuM
Al=6- 5:4=120, 7. e. nepmmm moxmo odoomamrra 120 crocoëa-
mu. 4

Banawa 3. Pemurs ypammenne 4242 ornochrensuo n.

D Baer, 170 n>2, no popuyxe (1) mueen AZ=n(n—1). No ye~
12, nootomy n(n—1)=42, orkyaa n®—n-—42=0, 1,=7,

Tak Kak KopHem ypannenus yomKKO Our MATYPAAIIOO “MC=
10 n>2, TO n3=-6 — nocropoxHuli Kopekb.

Orser. n=7. 4
Tipeo6pasyem dbopmyzy (1) ann naxoxnenua unena pasmeme-
mul AR.

© Bannmem dopmyay (1) rar
AL=(m=n+DOm—n+2)"...:(m-1)m.
sa Der

Panne Ges nosTopennit

Vunoncn obe war 9roro panenerna na

m-n)t=1-2-3-...(m-n),
nomysitnt
(m=n)-Aj=1:2:3-...(m=n)(m=n+1)(m=n+2):...:(m-D)m,
1. e. (m-n)l- An =ml, orkyna
nm
A Gi © o

Ana roro wro6s Qopuyxa (3) Osına ompanenamma ne rome
Aun man, no RIT m=n, nonaraior 01-1.
Banana 4. Burner
Aleta
Ae
D> Mo gopuyre (3) maxon
12, 12!
ata

Ynpawnenna

31. Branca
DAR DAL DA
5) As 6) ASS 7) Aloi 8) Afo-

82, B senacce mayuasor 9 mpeaweron. Combe enocofau me
mo coctanirrh pacmucamne na NONLEMLNK, CAN 8 aror A
ono Gurr 6 paamuıx npepmeron?

33. Cxomuxo cymecrnyer enocoGon jura ofoanauemmx nepultn un
moro werupeayrombauixa ¢ nono Gun A, B, Cr D, ESF

34. B wnacce 30 uenoner. CKONbRUMH CNOCOÓAMH MoryT ÓBITE Bi
pame no mx eocramı crapocra u Kagnaxen?

35. B uemnuonare no QyrUoay yuncrayıor 10 xomana. Cromo
cymoemyen pasnnamuix sosMonnocTell annarı om
mepnsie “pit mecra?

36. Hate ananonne nupamonus:

y À y Ar,

Año

37. Pere ornocnrensuo m ypanmenne:
1) Aj, = 905 2) Aj, = 56m;
3) AF, ¡=156; — 4) AS =18AÍ 7.
38. Hañrn anauenne mutparenist
Alo Pia
et, nme nS 10.

168 tava Y

KowGunaropnna

39. B maxmamnom TypHupe yuscrnyior NTE ¡OnOLÍ u mpu ne-
rymiki, COIN CHOCOÓAMM moryr PACKIPCACANTECA mecra
‘cpeam nenymeK, ecam BCE YaacrHuKN MAÓPAJI PAIHME KO
vecrna o¥Kon?

40. Noxasanı, «ro Af! t=(n—k)AS, rne k<n, REN, neN.

$ 5. Coueranua Gea nosropenuñ
y Gunom Hbiorona
Sagaua 1. Ma warn waxwarneron AJA yuncrua 2 rypuupe
Ayauo BLIÓPATE ABONX. CKONBENMN CUOCOÓAMIL 91O MOHO eac-
amt
D Ho nara maxmarneros mono cocramwro AZ map. Ho us orux

Tap Hago BEIOPATe TOKO Te, KOTOPHIE PARMHYAOTCH HUE CoCTa-

Ai
tox yuacrnuxos. Taux map 8 2 pasa wenpue, noprouy À =
51-10, 7. e. anonx moto nuGpars, 10 cnocoban. 4

Tips pement oro% anna na 5 uenonex Gum o6pazonasis
sp — ooenunenna NO 2 wenoneKa, KOTOPINE OTANNAAICH APYT OT
xpyra COCTABOM. Taxıre COHEN HADHBAJOT CONTAMIAMN,

Onpenenenne

Covemanuaxu wa m SneMeRToR HO n B KAKIOM (n<m) naar

BAIOTCH rame COeAUMeNHS, KANIOE HS KOTOPMX COHEPAUT 7

DACMERTOR, BIATHX 13 AAMHHX M DICMERTOS, H_KOTOPHIE

OTAMMAIOTCA onmo 07 APYTOTO mo Kpañxel Mepe ox axe-

Hnorga Taxııe COWCTAMMA HASHBAIOT covemanunmu Ges no-
emopenuit.

ICO ACEROIMOKMEIX CONOTAMHÍ Ha M PAAIMIRNX onemen-
tos no n sxexeuros o6osuauawr C;, (C — mepsas GykEa @paxnys-
exoro exova Combinaison — coweraume) m unraor sue ns om 10
an». Ilpn pemennn saxaun 1 61110 yeranonneno, “ro C}-10.

Buimenem opmyay zus moncuera ACTA COWETAMMÍ ma m pas-
Aux DACMERTOR NO A ANCMENTOR B KAMIN.

O Obpasyem Bee coexitHenus, coxepxamue n 2xemenTos, BBIÖPAR-
BUX La ARMMIX m PASVMX D.ICMENTOR, Gea yuera NOPAIEA MX pac-
noxoxemna. Huexo rakux coequmenuit panno Ci.

Ma Kaxkoro moayuennoro coenunennn nepecranoniamn ero
arementon MOHO oGpasosars P, — nl come, OTAHAIONXCA
ogio OT APYTOFO TONIBKO NOPAAKON pACHONOKENIA atenentos. Tent
cu nonyumoren PABNEMCHU HA M oeMeHTOR NO A, “HCHO
Koroptax pasno Af. Tlo mpanuny mponasexenus uneno TAKHX
coeumenit paso Cp Pa. Hrak, Ca-P,=AN, ormyaa

An
xe mM
55 169

‘Concranna Ges nosropenum u Bimamı Asioroma

A age
Hanpunep, im At = 492 24,
sia Pa 1:28 be
Bamerum, UTO CCAH M=N, TO
mn Aa _ Pa
nn
Vunrsiman, uro AS ws jr mp u mon m Pm, opa
(2) mono mpeneranrs » une
= men) a
me mon.
pop, Cla m -
Hanpumep, Che zug airs "10

Banaya 2. Ciombxo eymecrsyer cnocoGos »sGopa Tpex xapr
u xoxo m 86 Kapr?
[> Maparee xa Konom 3 kaprsı Gea yuera nopanka mx Pacnoro-
wenn B HAGOpe ABARIOTCA comerammamo HS 86 no 8. Ilo Qopuy:
ne (2) naxommm
= 3 361__ 34-95-36
E 6
1. e. eymecrnyer 7140 cnocoGon. 4
Bamenanie. B yueómuxe 10 xnacca mp pasxoxenun cre
mens Gunoma (a +b)" Osio nueneno monarue OmmoHnasex KO
sdduaxenros, koropuie B oGmem Dune oGosmawames Ca,. Bete 6er
Roxasatentersa upumara opmyna (2) An Buinmcnenn orux xo
Abdurmnenron. Teneps crano Ouen, WTO Panee pacemorperr
Abie Onmonmammue KOSDDHERTE — 970 NMEA CONTRE sa m
non.
© nomombo popmyası (2) » xypee 10 wancca nornasınane
terna coueranmit;

14-35-6- 7140,

CHh=CcH” (3)
ÓN w
Aokaxen cBolicrso (4), Tax assiBaemoe pexyppenmnoe ceo:
meo uncna coverannit, mombsysct coornomennem (1):
net Am, Ant?
© caca te FB Fe
_mim-D- nfm (0-1) mA _
nt (+11 a
mm De OIDO |

ent
MD (INCA |

tit
(m41)-m(m-1)-

rd
170 tnaga y
Konbnnaropna

Ha oenope cpolicrea (4) u © yuerom roro, wro Ch=CH=1, co-
cranxuerca Tax nasiamaomuii mpeyzornur Tacra.a — raGnuua
savent C3. (Hinke mpanegen (bparnent TAGE.)

NTofıJ2e]s]=#]s]e]r]s]e]Jx

aa
GE:

1

sx GE

GE DEIN
GE 10 [10] 5 |:

2
3
o
5
Gloss [o fs
7
8
g

ajos [as [a [7 [a
28 [56 | 70 | 56 | 28] a | 1

GE

DE
ofa
10 [1 | 10 | 45 [120] 210 | 252 [210 [120] 45 [10 | 1

36 | 84 | 126 | 126 | 8a | 36 | 9 | ı

Dra raGama narsınano wamocrpapyer » enoleruo (3) — pas-
Eu WXGA, OXHBAKOBO YAANEHHBIE OT KOMIOD CTPOKH TPeyromban-
xa Ilsoxaan (Ch = Cm 0.

‘Tpeyromnitkon Tlackana MOXBOVIOTCA mpm noanenenns Gn
xa a+b B KATYPANbHBIe CTENCHI MO anakONOÏ BAN Ha Kypca
10 xancea dopmyac Gunoma Hormona:

(a+by" Cha” + Cha" 'b+ Cha” *68+...+ Ch tab + Cb". (5)

Banas 3. Suncor pasromence Genaue (221).

D Mo gopuyze (5) maxonum:
(2x-5) ges aer) arar)
so) 0a(-1) mero 100 (- 110-89". Le
+10-42(-2) 45-22. + (> 23)>
025 404200 + À x. À

Banana 4. Jloxazar» cpoficrso oxemenror CTPOKK TPeyronsun-
ve Macs

CRACH+CH tn. + CR + CR 2, (6)
D Pasencreo (6) noxyuaerca ua paseuersa (5) np a=b=1. 4
WAlposenem noxasarenserno enpanennupoern hopmyauı Guxoma
Huwrona (5), uenomaya TEOPMO coenmmennii © MORTOPERHANN.
gs mi
Coveralia Ges ROBTOPENNA u ÓnHom Meiorona

O IIpocnexuM nponece noamexemua m naryparenyio erenemr Sic:
na a+b, He munonman mpupenenua nonoGusx cxaraensix:
(a4 0) (a + b)(a+b)= aa + ab + ba +bb,
(a+b) =(a+b)(a+b)(a+b)=
aa + ab + aba +abb + baa+bab+bba+bbb. (N
Bamerum, “ro » peynsrar noonenenun Gunoma no sropyo
crenens (7) nXorsm nee paomomenna € onropennsun, cocranser
Bute 13 OYKB a 1d no Aue GyKBW m kanom pasnemenun. B pe
syavrar noavenenit Gunoma » rpersto crenenp (8) pxoggr nce per
Memonus © nopTopenusmn, COCTABACHNLIE Ha Tex ae Gym an}
10 Tpit OYKEM B KEKON.
Tlocac paexpurrs ckoGOK MH nosnenem Apyuzena a+) +
crenemb m, 7. e. » panencrne
(+6 = (a +0) + b)

(a+b), y

omar
OYAYT nonyuent scenosmoncmme paameuyenns € noBTopeHHAM 1
Syke an b, cocroamme ua m snemexto!

TionoÓMBINH THEM NOCTE packpsiTua CKOGOK 2 panener«
(9) OYAYT cnaraemnie, conepxare ozsaronoe uuexo Gyn a (0%
BUAHO, GyKB DB HUX Tome GyzeT ogumaxonoe uncro). Barack,
cKomEKO Oyner monoÓmx «xenon, conepæaux k pas Gyxsy d
(yxy a coorsercrpenno (m-k) pas).

‘BiH unens ABAMIOTER MepectaHoBKaMM © HOBTOPeHHANH, eo
erapnennninn 19 / Gye b u m-h Gyxn a. Coraneno dopmyae (3)
$ 3 umeem

Pama CCR)

Ho nuparenue, crommee » mpanoh uacru oroÑi dopxyns,

core me wro noe, Kak Ch. Crenoparexsno, Pa, «um Ci, à OH,

nocxe npunenenun nonoÖuLx Craraemmx m pasencrae (9) om

unen e Gykrennoñ waerbio bla" * (nam ab!) Gyner umer Ko
«unuenr Ch. Oro nokaamsaer bopmyay (5). © ES

Ynpaxnenus
41. Hale:
Doc 2 Dc 5) Ch 6) cé:

DCW 8) Clos 9) Cho 11) Cio 12) Ch

42. CKOMRHNI emocoßamı MoxHO ACNCTUPODATE TpoHX CTYAES>
TOD HA MOXBYIOBCKVIO KOMPEPEHMIND HI 9 «omo» mayınor
oGmectaa?

43. CKomBKo PASANIEEIX AKKOPAOR, CONCPMAMIIX 3 IBYKA, Nox
Ho naarı ma 18 wranumax oyuoli okrankı?

44. B nowemenuu 20 nan. CxomKo eymecrayer paix napı-
ANTON ocnemenun, MPH KOTOPOM AOIOKHK CRETHTHES TOM
18 nan?

172 trae V

Kowönnaropıa

45, Mineerca 15 Towers na MAOCKOCTH, MPIICM make 3 ua MIX
He nexar Ha onuoll npanoli. Ckoxbro pastiuHEIX orpeskon
MOXHO HOCTPONTE, COCAMMAA aT TOUKH HONAPMO?

46. Ha oxpyxnocrn ormeueno 12 rouex. Cromo cyieernyer
TPCYTOXEMNOD ¢ nepmnmann D DIX Toukax?

AT. CkomKmen CMIOCOÓAMI MOXKHO COCTABHTE 13 HAPrHK, conep-
xameñ n neranei, komnzerr ua p neraneli (p<n) JUIA Komr-
poası da KAMCCTION nponyrum?

48. Banucar» pasxomenme Onmonaz
DU 2-9 3) 2x+3)';

Varas Di); 6) (S42).

49. B mkompHom xope 6 zesouex u 4 manbunKa. CKOBKUMK CHO-
coGanen MONO MMGPAN ma cocrana MIKOMHOrO Xopa ABYX A6-
Boxex_ m OfHOTO MamBUNKa ANA yuacTHA B BBICTYMTEMICK
coxpyseore xopa?

50. Pers ypasnenue ornocurensuo
NO Chis 2) 120R b= 6548
Cha 4) Cia = 120,
51. Haïtru onadenme PMPORONIA, mpenapitrenssto ynpocrırm ero:
1) ci +e} 2) Ch+cis
8) Ch-Chos 4) Clor~Cioo-

52. C nomompw csolicts uncna coueTannit HAÏTH cyMMy:

1) Q+Ch+CE+CR+Ch4+C8 2 Chr C2+ 08+ 08 +08,

38; Pours ypapnenne:

1) Ch+Che15(x-1); 2) Ch, +02, 15(x-2).

54. Tokasars cnolicrso “nena comerammil

CEE

55, B nase nexar 5 pasumx a6nox u 6 pasruanmx anemcunos.
Const enocoGamn ma MIX MOHO muGpars 2 aGn0KA 1
2 anensenna?

50, Koxona Kap coxepæur no 18 xapr xaxxoit ua uerspex ma-
rei. COMBI CHOCOÓAMIE Momo REIÓPATE Ha KONoAU Cxe-
ayiomañ maGop: 3 Kaprsi mmkoBofi, 4 Kaprsı Tpedorot,
5 kapr veprosoi, 2 Kaprst GyGnosoï macrn?

SL. Hatten unen paanomenun (V+ 1)", conepmamunn 2°

58, Haitra unex paszoxenua (ee) coepæamit x".
vr
Ha nnocxoru nponenensi À npansix, mpiruen HAINE Re ua
TNX me HAPAATEMEEE I HEAR TPL He MepeceKaIoTes n o-
Hoh rouxe. Ompezenurs weno ToweK Mepeceuenu DTUX NPA.

ss 173

Coveranna Ges nosropenmn u Canon Horn

D] Ckonsxmmx enoco6amn momo pacerammto 12 Geax m 12
MepHMx mamex Ha 32 Mepubix KIETKAX LIAXMATHOË nocku?
[SL] C nomompro saremarnseckof unaykunn AOKasanı enpanez:
Bocrs bopuya (6) u (7).
§ 6. Coueranua c nosropennamn
Onpenenenne

Covemanunnu € nosmopenuszu us m NO n nassınaor cor
HOMNA, COCTOAINKE HO N DACMONTOD, BMÔPARHMX 1a aneneir
108 m PASHBIX BIAOB, M OTANMOMHECE OAMO OT APYTODO xo
7a Ou OM aremenron.

“ueno coverannii © nosropenuana ma m no n oGosauaior ©),

Banava 1. B xwocke mpozaiores kapanaauım Tpex wBeToR: Kpec
Hue, CHHMe YepHMte (Kaparamel Ka Oro Bera B Kocke Dan
Le uermpex). CKObICHMH CHOCOGAMH Mono IYIMTE À xapanıana!
[> B sanaue TpeGyerca Hafırm uneno COCAMMCHIÑ, COCTOAIEX 13
4 onementos, BGpauEux 13 9ACMENTOS 3 BOB, HOPAAOK Pacnono.
CHA KOTOPHX D COOJUIMONHSX HO HNCOT INAGEMIS, T. €. NYKO
altra C}. Oboanauum kapannanı kpacmoro meera 6yknolt x, cıme
ro — Gyknoï €, uepmoro — Gyxnoit u. Beinen ncenosmoncnie
HaGopst mo 4 Kapamameñt:

REIR Be ee RUN en
RIO O ROLL Gua

BBG E KG GM GC a
mece aaa.

0000

“ueno nonyuermbrx coegurmemutt Cf = 15. À

Han munoza hopnyast "CA coneranmi € MOSTOPEMMANA Sy
nem HCNIONLIOBATE wexyeerueninuit NPMEM, KOTOPLA MOMCHNN He
coueranuax, cocramnenmaax m sagaue 1.

Tax Kak 5 coverannax (1) mopanox pacnonoxeunn anementon
ne MNCOT amauennn, TO TIPA nepeunenennn couerannk ee anne
TBI OAHOTO BHAA ÓYACM SAMHCEIBATS PAJOM, a MOMAY HAGOPAMH aue-
MeHTOR pasmbix nunon Öynen cranes max sparuum eudoe U.
B aagave 1 paanwannix wagon anemewron Gntno 3, OHAMIT, past
D nomwo Sirs 2. Hanpunep: «Limulle; wLluLlee. Ups «par
imupeus u iaGope uncan OMCMCETOS HEKOTOPOrO BAR Faim
muxo» Gyner enepememarren». C menomssonannen amara O nepe-
sem» coenunennä (1) moxcer ÓbITb SAnMcan CIeAyionHm obpason
a DO
we wwe Da De aja Del

x«xOe,c O xDke,c Ou Dese Onn a
xLlee,eO Deje, e Cu
Dejesese D

Tan;
ee —

B o6mem muxe mo6oi na6op Ho n 9ICMCHTOR, BuI6panHBx HO
SMENCHTOB M BHAOB, MOXHO IPCACTABHTE B BUTE
a, a, ..., a0) 6, b, …, bO ... Oz,
— ee
my na os Nm
re aucno pasgexmrenpmiix smaros [] pasno m1, xors Get onuo
M3 WCE Nyy Nyy vers Am OTAMUHO OT MYIR M Ny trate + Ny = Re

Tak, HANPUMEP, B Kamıom Ma coenuHennii (2) uncno sHAKOB
D pasio m-1-3-1-2, a uneno 6yxn n=4.

Jlerxo saerurs, uTo Kax coegunenus (2) 8 HaGope, Tax u co-
exmenux (3) OTAMUAOTCA APYL OT APYTA, MO cymecTRY, Mrs Me-
cropacnonomennen m Hux pasnemmrenci. Gamer Bce onemen-
mwöykesı suakom D. Torga coeaunenna (3) mono npexcrasurs

@)

8 se
86.0 0 606.06 0.0 BD... O, 4)

e my yt at y= he
Yanrumas npaanan cocrannenna coequnennit (4), nua,
uro Ci, PABHO HHCAY HepeCTAHOBOK € HOBTOPEHHAME M3 (m—1) ane-
weuron LI u n anexeutos Ll, r. e.
GB... men
u Bu. EH 0
Taux o6pasom, » sagas 1 moncunramoe «nexo Of momo
salir ¢ HOMOLISIO DopMyAE (;
va BAW, 0 5-6
A ag 10.
Sauaua 2. B nponany nocrymum maui 7 pasamos uneron.
CKonBKUMM CHOCOGAME MOHO KYHUTE 3 mara?
pp A
Ya » npnoöpenennon naGope 13 $ naeh. Tloorouy pemene sone:
IE OREA Y MONET MICA cowerannt © noRtopenma HS 1

oy AB
a (DNI 6131
1.0, morymy momo conepmtitrs 84 crocoGa. <q MM

Ynpaxnenna

I Buscan
1) Ch 2) Cés 8) C5 4) CR.

B Kae nonanaan mopoxenoe wermpex muigon. CKONBEMM
cnocoGawx Tpoe apyseli moryr CHARTE saxas odunuanty ma
3 nop MOpOxEHOrO?

Com» nerenux urpymex BuOupaIoTe us nrpymer «erspex
PUNO. CKOMBKINI CHOCOÓANI DTO MOMO CHONATD, ecm
urpymiex KaaKnoro puxa Goxbme cen?

ge Bus

TONER E ROSTER

65] Conso eymecrnyer prammumux npamoyrommaX mapanzene
rueaos, eon ja Kamäoro ero peöpn Moxer mupaxkare
Cm mo6uim nena uncnon or 1 a0 8?

66. Buiunenure:

‘td 61, yy 54, 7OL gp 601 50!
» EN 2 a Vote Dar
67. Vnpoerurs:
(02 AN
DER a)
68. Halrn anavexue supaxenua:
Ab, A El a ) Ps
Dar 2 (50-10) 47
69. Pemwrs ornocuremuno n ypanmenne:
ay Pais 1-20, 4
D -12; 2) = ; 3) Añ.-6n(n+1%
Pr 2

4) AS = 28 5 9-10} 5

70. Cronpkmmm CHOCOÓAMIA MOHO COCTABHTE rpadHK Oonepenno
CTI YXORA m ormyo noch COTPYANUNOR naboparopı?

TI. Cxonuxo eymectayer enocoGon neneruponanna ma Konpeper
HO ABONX HCMOBEK 113 ROCLMI COTPYABINKOD 1aboparopun?

72. Bocem» corpyauuxos xa6oparopun yuacrsonarı » mayınoı
Kowkypee, no peayabrarast Koroporo Ghim mpmeyanenu or
ma nepnas m ona nropas mew. Cxeomciman enocoda
Mors Öbirb MPHCYACHB! paccmarpırsaemsie Mpennn?

73. CKOMBRHMH pasHBIMM CHOCOÓAMH MOHO PACCAMMTE TPONX
YHALIUXCH, MPIIICADIX Ha AkYABTATHDENE dABATA, Ha 00
OKA HMCIOMIEXOA m wracce CTY-MLAX?

74. Cxomsumen cnocoGaxtn MONO NAQMAMITE MATpyas HO A
commer u oamoro odunepa, ecm » pore 80 coxgar 1 5 obi
uepou?

75. Cxomxo Amaronaneñ umeer munyrmai namyromeunte? cox
yroaumn? n-yromoaun?

76. Haitru axavenne aupaxenus, npexsapiremno ynpocrna en:
D CR+CH 2) Cher Che

77. Menomaya enoliernn unena coverannit, uaiteu:

1) Ch+CLLC24C34Ch 2) CO+CL4CI4 CE.
78. Haltru paanomenue Gnroma:
D EH) 2-1) 8) (2-0 Mara.
79. Toxasars, uro uneno nepecranonok npu mo6om n > 1 snaser

176 traga y.

KOUGARATODEER

80. Usteiorca ormamonnrecn apyr or apyra 7 pos m 5 nerox se-
neun, Hyskito COCTABNTD GyKer Ha Tpex poo i ABYX BOTOX 3e-
eun. Cronsenun enoco6anst 970 MORO cuenaTh?

81. B avomunoli cuereme cuncnenus, wenonvayenoñ » SBM, ux-
hopaanust aamımehmaeren c MOMomBo mu O u 1. B nexoro-
poli DBM Kaxuoe <MALIHHHOE C/oBO> sanucEIBaeTCA B aueh
ke nanari, coxepamelt 32 mponywepo
paapana. CKoNBKO paamramx «cHoR> More GUTE aarıcano
8 Taxol auelixe?

82. B oauioli crpane nomepa ABTOMOÓMACÍ! cocrannsiores ua ABYX
Heoqunaxonnx Óyka anbanıra, conepmamero 20 GyKB, 4 ue-
Tpex nip (€. momo HONTOPAMM). Cromurum manın-
HAM MOHO NIPUCBOMTL HOAYUCHHLIe TAKUM O6PAIOM HOMEPA?

88. CkoabKO PAAMINMBSIX IKIAMCBANMONNBIX KOMNCCUÍ, COCTO-
minx ma 5 unenon, Mono oßpaaonarı ma 10 npenoxanarexei?

84. C nomonguio enoïern wncxa coweramiií main:

1) Cha+Ch+Cl 2) Ch+Ch+ Cho 3) Ch-CH 4) CH-Ch
5) ci+Ci+C+ CH 6) CL4CL+ C3+.C3+C3+C5+.C8+C5.

85. Harn paanomenne Gnnona:
nec 2 (5426) a (ar) © (4-3).

86. B wemnuouare crpaust no pyrGoxy yuacreyior 18 xomann,
kamnme 2 xomamaut Borpeunioren na PYTOOMMLX monsix 2 pa
2a, Cxombxo marueli urpacrea B conome?

BT] Croabrumn enocoGaux 2n pasusix anemextos MOHO PASÓNT»
ma mapas?

CKOAbKHMH CHOCOÓAMH MOXHO pasnennt» KONOAY M3 36 kapr

nononam Tak, 5“TOÓNI B KaKAOË m3 AByx CTONOK GEUIO

no 2 yaa?

$9, B xnacce 28 yuenunon. Kaxcatiit nei» anoe ma mx maomana-
toren meaxyprann. Mono Au COCTABIIT® ua Beck TOR eme-
Auennoe pacmcamno acamyporna Tarna OÖpason, «row wma
une 2 yuenura He ACIKYDMIM BMECTO B TeYeHMe FOJA ABAKAM?

Halırn nen pasnoncenun Gunoma (V+ =

i

ve

Mame “mena or 1 ao 20. Ckomskumu CHOCOÓAMH MOXHO BLI-

Spars ma une 3 uncaa, CyxMa KoTopbix GyneT “ICNOM ver-

mu?

Hoxasars, wro «nexo kpyeoowx nepecmanocor (namen nopa-
fox enenosamna pacronorxeninx Ha OKpyneHoeTH aremenron,

A nauammahı anewent Öcapnanınen) ma n axementon Pano.

(DL

Hadıru ananonno nutpamenun:

1) Py, 2, 4208; 2) AB+CR: Py, se

Hair wen paanomemen Gnome (V+

177

Ynpaxnenmn meee V

5] B unerounom marasuue mpogaior queria 7 non (aaa mm
mpencrannen Öonee “em 3 MBeTKanm). CKOMBKHNN cmocobanız
mono cocrasıma Gyxer ua 3 umerwon?

a ee RE?
2. COO0PMYIMPOBATE MPABHJO MPOMIBEACHMA.
sry muon ine Pan © noropm À

4. Kaxue coeaumenna nastinaror nepecranonxamn? Yemy pasio
uuex0 nepecranomox ua n oxememtor?

5. Kaxwe cocguuenua nasuımaır pasmenennams Ges nostope
muii? Yemy pasuo uncno pasmemenndi Ges NoBropenui us m
ouemeron no n?

6. Kaxne coeunenus nassınaoı coverannann 6e9 mopropenni?
Yemy panno «nexo coeramu Gea nomropemuli ua m oxeuer
Ton no n?

7. Ilepeuucnurs cnolierna coeramli Geo nosropenuit.

8. Yro Taxoe mpeyzozonux Hackana?

9. Banncars dopuyay Gunoma Honmona.

Tp

1. Hatin
1) Py DAR 3) Ch 4)

cen!

2. Vnpocrurs:
MM, yy 00%
D Ds D ay

3. Cons erroco6ann MOHO BESPATE Aa MonapKa 3 npex
Mera HS RCDATA pasamunnıx HPEXMCTOB?

4. B ommox knacce naynaeres 10 paamıx mpenxeros. B nara
ny sanya nommen MOCTABATA B PaCmIcaKme 91070 XHACCA À pas
JwTMBX MpeaNera. CKOZBKHMH CHOCO OI MOMCT 070 car
ari?

5. Ckomsemar pans CIOCOGAMIL NOHo paamerurms 6 rpynn
HIONBINKON B mec KHACCHUX KOMHATAX (10 OAKO! rpyune
» xomuaro)?

6. Cromo eymectayer rpexannunsıx pomme KoROR, E Koro
Pox mer omMHAKOBEIX NICD?

7. Sanucare pasnomenne Gunoma: 1) (x+y)% 2) (1-aJ%

1. Pemure OTHOCKTemBnO m ypasnenue:
1) Param20Prs 2) Abi = Cho
2. Pour» uepanenerno (n-8) Pre < Passe

1782 traga y

ROM

3. Ulndp nexoroporo cela oOpasyerca ma anyx uncen, ITepnoe,
Tpexsxanmoe WHC, COCTABARETCA ua undp 1, 2, 3, 4, 5 (kam-
sax undıpa verpeuseren we Gonee OAMOrO paca); wTopoe, HA
Tranaunoe eH, aammeumaeren e momommo mnbp 6, T, 8.
CkomBKO PAJAMUHBIX 1AUMPpOB MOXHO HCHOIBIOBATE B 9TOM
cie?

4. Cxombiumu enocoGann MOMO paanonuT 6 nouer no ay
apena?

5. Henomaya cnolorno uncna couerammit, m
1) Cho-Ch 2) Ch+C$+-CE+Ch+CH.

6. Hañrn unen pasnomenna Gunoma (=

E y conepream 25.
te
WA Eucropnieckan cnpanxa

Began, CBADAMNBIO O noamoxnmm nuGopON » ynopanonenn-
x onpexexemmnx 06HEKTOB, MPHXOAILIOCS pemars Bo MHOrHX ce
Je sexonevecrof qeateasiocrs. C Tara SAAAMANN, LOAAUD-
‘WMH HAQRAMMIe <KOMEHRATOpHWes, IOAN eramumanıten ame
» apeonocrm, Tax, » Apernen Kurae ne romsKo Marenarikn,
so u morue mo yRKeKAIHCH COCTAMACINEH Mauveexux 040.
unos ( minx sanamente uncaa mazo Oui Pucnomonun, Tax, TO
Ga Wx eyunsı 10 FOPHSONTANAN, BEPTIKAAAM 1 TAADHLM Anaro-
sans Guam onmmaxonme). B Alpennefi Ppengn sammannch reo
ek Quzypmux ducer, a TAO COCTADACIMOM PAS huryp
1 Wei enenazsiati eriocoGom Paopesamnoro xBanpara. B pas-
aux erpanax pema Mes KONGMHATOPME aaRaun, enmannnie € Ta
MX HYPAMM, KOX MAXMATI, LLOLIKI, KAPTAt, KOCTI HT. I

Hepnste nayunie ucexezonenus mo KomOunaTopuKe npu-
uasexar mraxbamcium yuemam Tix, Kapaano (15011576),
Y. Tapranse (ox. 14991557), T. Tanne (1564—1642) u pan
mano yuennine B. Hackanıo (1623 —1662) x IT. eps (1601—
1665). KowGnmaropia Kak mayka crana paomnnarsen u XVIII n.
COPARACIDNO © BOOMIKHOBENNEN rOPHN Beponruocrel, Tax Kar
us pee DeponTuocrunK sexe HeOBXOAUNO OLA HOACHHETSI-
ro sexo paamux KOMÓMMAIAÍ oxemeuToD.

Kon6unaropuy xax CAMOCTORTEMBRAÑ PASEA Maremarutn
zepnus eras pacomarpnnars nemenuit yremai T. JeñGnun 5 cno-
sl pabore «O6 ekyecrue KomÖnnaropmens. Emy rakxe mpHHa
eur m mnexenme caMoro Tepaune +nomÖnnaropunas. Bart
ren AKANR B paanırrue KomÖnmaropin nec JI. Diep.

B cospemennon oGmecrne © paspuruex PUMMCAUTENEMON Tex

ski! KONGHHATOPULER AOGILIACh HOBBIX yeHeXoD. Tak, © HOMOLKIO
SBM Guiaa peniena KomGunaTopmast DIANA, naneernas mon naona-
tex npoGrema vemupex Kpacox. Vaanoch noKasarı, "To my
spy MONO packpacırrs 1 À unera Taro OGpARON, TO miarne
‘be expaust, mmeomue oGmyIo Tpammuy, me Oyayr oxpamennt
> om u Tor Ke uner.

rs

Vicropinecran ampara

Tnasa VI |

DreMeHTBI TEOPUH
BEPOATHOCTeH

Bea yuema orunmus eayratinus aoseni
uenoser emanoaumen Geceuas
unpassams pasumuex unmepecynuus es
npoyeccon à xcesameninox dan ner
anpassen.

B. B. Pueden

§ 1. BepostHocts co6birun

Ilpaxrnxofi yeranonneno, «ro m «aero op
nexogamux cayaafinıx annenunx (cos)
cymeerayior onpenexennnie 3AKOHOMEPHOCTL. Se
waa reopui sepoatHocreli — ycranonnenue 2
MaTemaTHecKoe HeeTeAOBAHKE DAKOMOMEPEOCI
Maccossix cnyualnsıx anmenuit.

1. Cayuaiinsie, nocTOnepmEIe x menoanex
mute co6sırun

Onpenenenne

Hexoropoe coGurme nassınaor cayvalmı
Ho OTHONICHMIO K Aannomy omary (nenn
ino), CCAM RP OCVINCCTAzCENEA STOTO oma
ORO HGo npouexomer, auGo me npouexog.

Tpumepst cayuadiman coGerrait:

1) BainazeHMe opaa ph monÖpackiBannn xo.

2) sumagente mecrepra pu Gpocanun ur
pansuolt xocru;

3) BUIHTPLIM no Aannony norepoltnony Gr
very;

4) suixon ua eTpos DACKTPORAMIIA m Teuer
onpeneneunoro orpesxa pement.

Cayualinsıe co6sırun OOsUMo oBosnanım
Oyuzamı A, B,C mt. 2.

Coßsrrue U masunaerea docmosepuuis, ce
ono oGasarenbHo HACTYNACT B peaymbTare Zante

Co6xtrue V maasmaerez Neosnomnua, et
oHo sasenomo He momer mponsoliru B pesyanten
annoro omurra.

Snemenra Teopun BEPORTNOGTER

Ilyers, nanpırmep, D ype HAXOASTCA TomDKO KCPHHE MAP,
1 onuT SARMIONACTCA B MOBMEUEHMN ape Ms ypmi. Torga cobs
ihe euanneuen vepnsil map» ABNMOTCA AOCTOBCPMIIN, a COGEITHE
ranneuen Genii map» — MEBOIMONHBN.

Tip onnon Opocannn HrPAIBHOË xoctu sosMoxHE eneayio-
mue coßsımusı (uexodbe nenziranns): ma nopxneh Pan MOT oKa-
armca onno na uncer 1, 2, 3, 4, 5, 6. Kamnoe ns arux coßkımurk
ABASCICA cnydaiiusin, Tax KAK OHO MoxeT mponsolitn, a Noxer
me uponsolru. Tor quuicr, «ro nunaner oxmo ua uncon 1, 2, 3, 4,
5, 6,— aocrorepnoe coOnTHe, Tax Kak npu Gpocanmit urpanmoh
sacra ONO oGaareom0 BOnsoÄNer, A BHIUACENG, HAIIPEMED, THD"
za 7, ABANETCA nenosmorcnBIm COOMITACM.

Pacemorpennbie BoaMoxkHEte PH GPOCAHMI HrPAMEHOÏ KOCTH
coóxmua necoomecmnu (uomBxemme OAMOTO US BUX UCKMOYAeT mo-
asrenue APYTOTO), edunemaenno oosmomne (06manreanno noamr-
x 010 seo) u PasHoeosmomne (Y BCEX HNCeN MANCH MOABUTK-
03 oguuaxopis).

2. KomGumamun co6sırul. Tiporusononoxmate coberrma.

Ilpenuonoxum, “ro B pesybrare Mexoroporo omra 06a3a-
TEMO MPONCXOANT OHO HA BOAHMIO HeK-MOUAIONUEX APyr APyrA
cuis, mpiruem Kaki0e Ms HX He pasnenserca ma Goze mPO-
crue (snemenrapunıe). Taxue COÓMTUA HAIMBALO? anenenmapnol-
au coOsmurmu (un sremenmapnunu ucxodanu nensrania).

Tlyer» © HexoTopuiN ontiTom cnasamsı coOuTHa Au B.

Onpenenenne

Cymmoù (oGvedunenuen) cobmruh A u B masmmaeren coOs-
Tue, KOTOPOS COCTOLT B TOM, “TO MPOHCKOAUT XOTA OM OHO
a nannsıx coßsruß. Cymmy coGurnit A m B oGosnauaior
A+B (um AUB).
Ha pucynxe 101 e momompro xpyros Biixepa npommocr-

Puponano momarme cymmet coÖstru

An B: Gomsmott xpyr_xso6paxkaet

Hee saeneurapuue coburum, cu“
sane € pacemarpumaesteim OMLTON
ei Kpyr uo0Gpaxaer cobuirue A,

anık xpyr — coburue B, a sarpar
Tennan oGnacts — coGurne A+B.

Onpenenenne Pac. 101

Mpouscedenuen (nepecevenuen) coOurrnit A u B nasumacren
COÓMTIC, KOTOPOE CUMTAeTCA HACTYTMBIINM TOCAR U TOKO
Tora, xorga macrynaïor 06a coßsırua A u B. [poussenenue
cobirmit À u B oGoamauator AB (nam ANB).

Pueyuox 102 uamocrpupyer e nonomso Kpyros Diinepa mpo-

avegenno coGuirnit A a B: oDınas acre kpyron (sarpamenmas 06-

ucts) msoGpaxaer cobmrue AB.

$1 181

Bepoamocte coburn

Puc. 102 Puc. 103

Pacomorpum npumepst © KoHKpeTKEINE coßsırmanı À u B.

1) Tlyers B ome € Gpocannen urpansnoh KocTH coGuTHs À
x B onpenensioren tax: A — nBinano witen0 ouxop, xpaTioe 2, B—
mama THEO ONKOR, KpaTHOe 3. Torna coGurrse A+B osuavaen,
110 BEINAKO xora Gui ommo mo Heo 2, 3, 4, 6; coGrne AB—
minano uncao 6.

2) Tiyors our Aakımoxaeren n TOM, STO a KOMA Aka
rca Haynany OXHA KAPTA, N IYOTE PACCMATPIBAIOTCA CACAVICENN
coberrna: A — mmnyr Kopom, B — BRRyra Kapra MHKOBO ic
u. Torga coGumue A+B — many kopoas nam kapra rHKOOÏ ve
er, AB — Ho KOXORI BEINYT KOPOAb I.

Coburma A u B unasınaor panocuannniauu (pasmosa) x ti
myr A=B, ocam coötmne A nponexonur TOMA m TOALKO TO
xorga mpoicxour coósrrue B. Hanpumep, » onsrre © Opocamme
urpasenofi xocrm coSurmue À — manana miecrepka m coßuınne B—
BEINATO HANÓONBIICO WHO ONKOB ABIAIOTCA PABHOCHABELM.

Tas xaxka0ro coßsırun A MOXMO PACCMATPIBATD npomusone
aoxnoe ATK Hero coOsITHe A, Koropoe cuntaerca macrynnunx
Toraa u TOMBKO roraa, woraa A ne macryuaer. Hanpunep, ec À —
PIUAACHMO HeNeTHOTO uncaa O4KOB MPA Opocanım HTPAMLOR Ko
erm, TO A — psimamenue verHoro uncma oun: ecan À — nonase
Hue 8 Kem PH BMCTPEJE, TO À — npomax.

Ha pueynxe 103 nponaaerpmponana naanmocnnan cout
Au A na muoxecrse scex aneneHTapHsix COGHTHN paccmarpuee
emoro onsrra (coösırme A ne06paxeno sapamenmohi oÓxacrao).

Banawa 1. A, B, C — vpn nponononsmnix codsırna. Banner
© MOMOULSO BUEHEHHBIX CHMBOXOB CAAVIDIME COOBITI

1) Ay — nce tpi cobra npomsoutn

2) Az — mm onno cobuTme ne nponaomno;

3) Ag — mpostsomuno Tomsxo codsırue A;

4) Aq — nponsouuo no xpaneii mepe onuo 19 coÿuerai A, B, C

5) As — nponsomao onto m Too OAKO ua arux COUT,

6) Ag — mpoxsomxo no kpalíneñ mepe ana ma amix coóxrad.

D 1) Ay=ABC; 2) A¿-ABC; 3) Ay=ABC;
4) A=A+B+C;_5) As - ABC+ABC+ABC,
6) As ABC+ACB+BCA+ABC, uam Ag=AB+AC+BC. 4

1827 tnana
ne TODA BEpONTHOCTER

3. Onur © panonosmommmn wexonann. Kxaccmueeoe on-
pezenemme Bepontnocrm coßırun

Tiycrs coÖsırne A, cex3amnoe € onsiros € À PaBNOBOSNOAHL“
XI nexogasın, MACTVNACT Tora, KOPKA OCyInecrnnseren OAM no
TAKIX-TO M HCXOOB, H He HaCTYTIeT, KOTAA OCYINeCTBANETCH m0-
{oii ua ocramunxen n-m mexoyon. Toraa ronopar, “TO HCXORI,
mpmogamue x nacrynnenum coösırua A, Önazonpusmemayom co:
Gum A.

Onpenenenn

Bepoamnocmo P(A) coGwrus A » onsite © pasuonoan out
‚ct DACMNTAPINN HEXOAANI NADBACTEA ormomenne «MENA
CXO]OB, ÓxATOMPIITOTBYIOINAX COÓLITINO A, K HER BeoX Ic“
xonon.

Beau à — vues neex ucxopen, m — “eno nexonon, Bnaro-
upuarernymougnx co6srum A, To neponruocts P(A) cobnrun A
ompegennerca DopMyAOÏ
P(A=". @

Iipusenennoe oupeneaenne seponruoenn unsissercn Kae
cru onpedenensen aspanmcema

morte aro nepantnocns waseioro amewomrapnoro cour
» onure € m pannonoanoncmn wexonan panita 1,

Ms dopmyanr (1) crenyer, ro

06 P(A)<1, P(V)=0, P(U)=1,

sme — moments, £ 0 — nocronepuis cobrar.

Banasa 2. Bpocacren nrpansuas wor. Hañiti neposrmocra
cobesrañi Ay m Az, een Ay — uneno Beinanux o¥xOR KpaTHO 3,
Ay UnCAO MUMADUINX OAKON METIO.

D Tax sax coGwrmo A, Oxaronpusrerayior ana nexona (3 u 6),
Cañmo À, — vpo moxona (2, 4, 6), a "HCHO Beex HCXOROR par"
10 6, 10 P(Ay)=2

Sanaua 3. Moneta Gpocaerea anamnsı. Hañri BeponTHocte co
Gunux A— xors Gut OAMI pas muimazer OCA.

D Myer» O — monnnenue opa, P — noasenute peu. Toraa pe:
iyastar auyx Opocanuit — mosstene oauol 13 uerupex PaBHO-
roowonannax romGnmanui 00, OP, PO, PP. CoGurrmio A Gnaronpu-
aretayIon NEPRKIE TPK KomOnnannn. TosToMy mexomaa BeposT-

Bpomens Ane urpamumsie xocrn. Halt nepoar-
mern coßnımun À — MPOHANEAEHNE BEIMADUNIX OUKOn ecrh Heuer-
oe «ezo.

51 183

‘Bepoarnacte CO

D Peoyastar Opocamul apyx urpansusıx Kocrelt — uonnaene
PaMONOMOMMNX YOPAXOMEMEUX map uncen. Corzacno npanen;
nponanexenms «meno TAKHX map panmo 6-6 36. Cobo A Ore
ronpusrersyior 9 nap:
1x1, 1u3, 105,301,303, 305, 501, 503,505,
2.1

a noxomas neposroers P(A)= mL. 4

Banana 5, B sue xexar Aecgro onuMakonux ma opos
mapop, us mx versipe Gemx u mers vepmarx. Hayraz msn
vores na. Halıru peposrmocra coGwruñ À u B, een À — oda ss
uyrux mapa Gexoro unera, B — munyrme maps HCIOT past
user.
[> 1) Obuee sexo noamommmx HCXOADE ONMBITA — uneno covers

E y Ao _ 10-9
Sl
Gnarompuntereyiouutx coßsırıno A ucxonon anno Cj= Ai —

ES °
Toorony PA) LE = da

2) Tax kar moGolt ma derwpex Gensx MapoR MOXET KONG:

HMpOBATECA © MIOÓNIX ua wecru YepHBIX WapoR, TO Ho upanun

nponanezenna umecrca 4-6=24 mexoxa, Gnaronprarcrsyronqe cr
HUE.

Gurmo B. Mexoman seposrnoens, P(B)

Ynpaxnenna

1. Karma coGuriren (ROCTOBEPHAM, HEBOSMOHBM Hat cayaalı
NEM) annaercn cnenyıouee coÖbırne:

1) npn xomnarnoii remneparype u HOPMAXBOM armocdepun
AORTEHUN Crab HAKOAUTCH E AHAKOM COCTORKUN,

2) Hayran Bınyran na KoUIenbKA monera Orasanach narupyó:
eno:

3) mayrax assannoe marypambnoe uuexo Comme yxs?

2. Buncwers, anımoren au coGurus A x B meconmecrmant, ec:
1) A — uounrenue tysa, B — nonnnenne HOME mpm PAST
Onuoli Kaprsı us Konogsi Kapr;

2) A — nonsnenue tysa, B — nonmxenue kaprsı uconoh ne
ern npu nern OAMOÍÍ Kare HS KOXOMM APT;

3) A — ssmagenme versipex ouxop, B — neinanenme verso
“era OUKOB NPI OAMONM Öpocanımı HrpansmoË Kocru;

4) À — mumagenue vermpex ouxon, B — nsimazenne neuerno.
To uHexa ONKOB npx OABOM Opocanırı urpansmolt KocTH.

3. Yoranonurs, uro anıneren COÓNITHCM, MpOTHROTONOAHEN co-
Giro:

1) ceronun nepauñ ypox — usinxa;
2) oksaMen CHAR Ha «oTAMunO:

184 rave vi

joMeNTH TOP BOPORTHOCTER

3) ua mrpaxbnoli KocTH Bsiano menbue mann ONKOB;

4) xora O1 ogua nya pH TPOX BHCTPETAX MONAAA D LONG.

4. Hyer» A u B — npomsnonvnme coGurna. Banear crenyio-
me co6sırun: 1) npowsonm 00a nammnx cobsrma; 2) mpo-
soo no xpaiineli mepe oxmo ns coGurrui; 3) mponsonno
TONBKO OAMO HS ABYX AAMKEIX co6BıTuli; 4) Hut OKO ma CoOkı-
wuli me mponsonio; 5) mpowsouito robo coßsırue B.

5. Kaxopa Bepoaruoers sumazenus uncna, kparmoro 3, 5 pe-
ayanrare nopÖpacuinanus urpamsnoh KoeTH?

6. Kaxona nepoaruocts roro, 470 HA OTKPLITOM Hayrax ¿more Ho-
Boro OTpHIBHOrO Kanenzaps HA BUCOKOCHEI TOR OKAKETCH
uaroe “ncao?

7. B KopoOxe maxonnren 3 uepHMx, 4 Gemux m 5 Kpacaux
mapos. Hayran sMEumaercn onun wap. Kakopa Beponrnocrs
Toro, yro manyriii map: 1) vepmui; 2) Gens; 3) Kpacru
4) vepusih wm Genetit; 5) uepusiit um xpacussii; 6) xpac-
ui mt Genus 7) mane epmsah, un Gex, un KpACHEN
8) senemah?

8. Cpexu 100 onexrponamn 5 menopuennux. Kaxona nopoxT-
Hocth Toro, "TO BEIÖPRNNLE MAYXOXY 3 ANI OKAXYTOA HC
passin?

9. Bpouenst spi wrpamsæse Koorst. Kaxona nepostHoct® Toro,
wo:

1) ma ncex tpex KOCTAX mumano onumaxonoe Konmneerno ou-
xo; 2) cyMMa OuKOB xa scex kocrax panua 4; 3) cymma ou-
on na ncex xocrax panna 5?

10. Bpomenss nue urpansme Kocrit. Kaxona neposriocrh Toro,
1) cya ouKos, BEITABUNIX Ha oGeux KOCTAX, eors “meno
ueverioe; 2) uponancaeune ouxon, muinanunx ua oDenx
Koctax, eCTB WNCAO vCTIOS; 3) cywMa DunanmNx OrKon
Gonsme 6?

11. B norepee yuacrayer 15 Gnxero», cpeau koropsx 3 natur“
puuusx. Hayrax aumyris 2 Onnera. Kakona neponrnoctt 10-
1) 06a suyrsx Sutera suurpaume; 2) Tonsxo onu Guner
murpunmai; 3) munrpsintmoro Onaora ne oRasanocs?

12. Bpocatores ne urpansnie kon. Kakona neposritoers, roro,
«ro na nepnoh nrpamsmol KocrH “exo oNKOR Byer Gombme,
sex ma propoli?

18, Herores ane ypu: meprañ coxepxur 1 Gent, 3 wepmux m
4 xpacmsix mapa, nropan — 3 Genux, 2 vepmux u 8 wpacmux
mapa. Ma kako VDM naynauy Wannexaor no OAMOMY
apy. Haltri Bepoaruoors Toro, "TO Meera BAEYTEIX MBpOR
conanyr.

$1 185

Bepoamacts coda

8 2. Cnoxenne Bepontnocren

Haromnua, «ro cywma coßuımuli Au B — oro coßsırue A+B,
cocrommee 5 macrynemmx 1m6o ToxbKo couru A, mo rombo
coburma B, au6o u coßsırun A u coßsırun B oamospemenno,

Hanpunep, een expesiox CHEMAN ADA nitcrpesta mo MON
A—nonananne ® MILIER» mpx neppom Bicrpene, B — monagenne
pu Bropom »stcrpexe, ro codsırue A+B — aro nomananue crper
Kom no Men xors Gut MPH OAHON HA BHCTPEzON.

Teopema 1

Beposroers CyMME ABYX meconmecramx coGsiTiit pana
cymme neponrnocreh 9THx coGMTHit, 7. e.

P(A+B)=P(A)+P(B). (0)

O Myers coGumusm A u B Gnaronpusrerayior eoornerermenno ku
1 1texoxoR, a BOOTO HMCETCA 7 PABOROIMOHIENX MCXOJOD. Tar Kak
coGsrus À u B meconmecru, TO CPegu n WexoqoB mer raKux, Ko
‘Topbie OnHOBPENEHKO Gaaronpuarermosat Ost Kak coGsirano A, rar
u cobrrmo B. Mooromy cobrmmo A+B Gyayr Önaronpnarernonane

k+1 mexonos. To onpenenenmo nepoarnocrn P(A)= À, P(B)=+,

PA+B)-**t 41, orryna exenyer paenerno (1).

Cxenernne. Cymma peponruocreï nporunononoHEx cobs

Tull pana exe, 7. e.

P(A)+P(A)=1. o

O Coësmer A m A mecosmecrm, noatomy mo reopeme 1 ume
P(A+A)=P(A)+P(A). Ho A+A=U — nocrogepnoe coßsirue, x
mooromy P(A+A)~P(U)=1, 7. e. P(A+A)~ P(A)+P(A)~ 1.
Bamewanno. Toopoma 1 vepHa ANA MOGor0 xoneunoro “cas
Cosa, T.e. PGA + Ay bane + A4) =P (Ay) + P(g) + ue + P (A), ee
BH Ay, Az) ===» A, — MOMAPIO MOCOBMOCTIMO COÓMTIA.

Banawa 1. B aupune near 10 mapon: 3 spucnsix, 2 eux x
5 Gemmx. Hayran nsınnmaeren onu map. Kaona neporitocre 10
70, uro anor map mRerHo!t (me Gems)?
> Lenoco6. Myers coßnrıte A — noannenne xpackoro mapa, B —
nospxeme cuero mapa, rorxa A+B — nonnenue wnerHoro Ina
pa. Onezuamo, uro P(A)= 75, P(B)= ==> Tax Ka cobra À
HB commecruu, x wun plena opie comes neposr
mocreii: P(A+B)=P(A)+ P(B)= Kat So

Il cnoco6. Myer» coburue C — noannenne Gesoro uiapa, tor
aa coßuirne © — nonnnenne me Genoro (r. e. nmerxoro) napa. Oue-

La P(C)=1-P(C)-1- lol
qa PO) PO) 3 4

aunno, P(C)= 5

1867 tava vi

Oncmentu TOOpaN BOPORMOCHER

Saxaua 2. Bepoarnoeri nonajanna 9 sien erpemon pa
12 0,6. Karona BeponTuoer» TOTO, WTO OH, BMErPeA no MuLIC>
iu, npomaxmenen?

DEcau coGwrme A — monananne 8 Muiemb, TO NO yCHOBHIO
P(4)=0,5. Tipowax — nporitononoxuoe nonaramıno coburne, u
ero zeponruocrs P(A)=1-—P(A)=1-0,6=0,4. 4

Banawa 3. B pore xo 100 comar xpoe umeior mucueo oGpa-

onsite, Kakona neposrnocrs roro, ro m enyuañum 06paaon
«Dopxupomamuow nanone ua 30 conan 6yner xora GM On "eno
vex e nhieu oßpaaomannen?
Dyer» coGwre A — no nanone xora 6H OX YenOReK meer
sucuce oSpagonanne, Toraa cokırme A— um ogi «enoner no
range ne xeer puemoro OOpasonanus. B xamoli curyam npo-
me mancrurs P(A), sem P(A). Haitzem P(A).

Yuczo cnocoGos cocrapzenma papona B Kormuecrse 30 eno-
ex no 100 connar ports panno Co. ueno comtar, xe umeroux
aucmero o6pasopanns, paso 100-2-98. Hs 98 uenoner cocra-
zum nano 1 ¡onmacerse 30 sexoner momo Cif cuoco6amu. Hair.
ex ROPOSTIOCTD TOTO, “TO penx oroGpannirx 30 uexoner mer mit
oqworo € BLICHINN o6pasonaxttest:

981701
681-1001

39

mn 161 _ 169 _
Orcrona moxommm P(A)=1-P(A)=1- 355 = 390 = 0,512. 4

SS Teopema 2
BepoaTHocT» cymMB ABYX MPOMOBOMBHRIX coßsırıli papma
eyMMe nepoaTuoctelt arux coßkırm 6ea BepoATHOCTH. HX
npouaseaenun, 7. e.
P(A+B)=P(A)+P(B)-P(AB). @)

ONIycra coßsıruase A u B Gnaronpuarernyior coornerernenno À u
| PARMOROIMOJRINIX MOXOJOB, A CORNECTHOMY OCYINCCTRACHIO CO-
Guru A u B Gnaronpusrcrayer r ncxoon. Beni ICAO Boex panmo-
+ P(B)= À, P(AB)

BosMOxHBIX HeXOHOB paso 1, TO P(A)
Tax war coBwmno A+B Gnaronpusrersyer (k—r)+(I—r) er

stor woxonon, 10 PA B= BSE À à L LE, om eneny-

er paneneruo (3). ©

3agawa 4. Vis Koxoqs » 36 Kapr nayaany BEIEMNACTCA OMA
xapra. Kaxona neponrnoers roro, sto Gyger munyra Kapra GyGno-
soft macru uu trys?

$2 er
Cnorene veponmocren

[> Boenem oGoanavenms coOsrrsit: À — pumyra xapra 6y6Hopo!t mac:
mu, B — nonnunen rys. Hyxno naltra nepostuocrs coßsırus A+B.

Boenoapayemen dpopmyzoii (3). Tax rar
Pl) 4, PB) 3, PAB)-

P(A+ B)=P(A)+P(B) P(AB)=4 i+
Ynpaxnenna

14. B xonone 36 xapr. Hayran sunumaercn onua xapra. Kaxom
BepostHocTs Toro, “TO ara Kapra 2160 Ty3, G0 puma?

15. B nauxe waxonuren 12 Guxeron xenexuo-nemenoï norepei,
16 Ónneroz cmoprunnoï xoTepen u 20 Onneron xynomeorne
Hoit norepen. Kakova BéPOñTHOCTE Toro, WTO HAVAAUY BMRy
ru oxun Onner Oyner Onxerom amo nonexno-nemenol, au
Go xynomecrennoit norepeu?

16. B ange xexcar 5 Gerux, 10 vepmix » 15 kpacnsıx uapos.
Kakopa sepoarnoers TOTO, UTO Hayraz numyrah map He 6y-
ner Genin? (Pemurs aanauy ABYMA CHOCOGAME.)

17. Bepoarnocrs parerprma raannoro mpuaa panna 10°%. Kaxor
seposrnocrs me amurpars ruasmnil mpus?

18. Hahn neponrnocrk Toro, WTO MAYTAX BIIMYTAA ma moanore
maGopa aomumo (28 xocrei) ogua Kocrs Aommmo me Oyzer
enyönen

19. B vase croar 4 Gomix u 7 xpacuux acrp. Kakona neponr-
HoCT TOTO, WTO cpeyu enyuahnsm o6pasom BRIEYTEX Ma
Bag mpex KRETKOR oraeren no kpalineñi mepe oma Goran
acrpa?

20. B crynenveexoi rpynne 22 uenonexa, cpeaw Koropsx 4 je
sy. Kakopa nepoatwocts TOTO, “TO Cpexu poux enyunt-

HEIM Opas0M 2MÉpARHMX MO arok PME CTVACHTOD ANA

yancrus B Kondepenm OKaeren NO KpAÏHOÏ Mepe oana Je

symxa?

Bepostnocrs mopaxkenus MHIICHH PH EPROM RHCTPENE par-

ma 0,7. BepoarHocrs mopaxkeHus MIEMeHM IP BTOPON BACT:

pene panua 0,8. Beposruocr» nopaxenua MuILEMU u upu nep-

BOM, M Mp Bropom BMeTponax papma 0,56. Halim neposr-

HOCTE TOTO, “TO:

E

1) unmerm Gyaer nopaxena xors Gut on BMCTPCIOM;
2) uumens He Oyaer mopasema HH OAMMN MO BLICTPEROD.
22. Woneermo, uro P(A)=0,3, P(B)=0,8, P(AB)=0,1. Jorasam,

ro A+B=U.

188 Fnana vi

Onemenma Teopiin BRBOMMOGTER

$ 3. Yenopnan seponraocre. Hesasucumocre
co6bITuá

A1. Yeronuan reporrnoers
B reopum sepoatnocteit nam XBPAKTEPHCTHKE SABHCHMOCTH
oxmux COOMTHI OT APYTHX BROMWTCK MONATAC yeROBKOH BeposT

onp
Ecan A u B— nue coGwrus, ennaaumse ¢ uekoropan
omrom, mpunen P(B) #0, ro mono ZEN) neausesor aepo
AMHOCMVIO couru À UPM YCNOBHM, WTO HACTYHAO coösırıe
B, naw npocro ycaoonol Gepoamuoemu coGurrs A 060
smasaior P(A/B).

Taxis o6pasom, no onpezenemmo

Pia
ram a)
Banawa 1. Kaxosa BEpoRTHOCT Toro, “TO Mayran BHIVTAA ma
HOAMOTO naGopa gosto KOCTE onamercn «AYÚACMO, comm MINCE
10, "70 cynxa OWKoD ma oro KOCTH membro, ue 5?
> B naGope zowmno 28 Kocref, mo mu 7 «ayOnoño. Ha aonarı
kocrax eymma oukon membre, em 5:
0—0,0—1,0—2,0—3,0—4,1—1,1—2,1—3,2—2.
Tigers coGurme B — cyuma oukon ma nsınyroli KOCTI menpute na-
‘Tut, a coGitHe À — Buinytaa KocTs ects «xyÓnb». Toran coöurne
AB — na munyroï xocru, amnmomelica sayCnem, cymma owxos
xembme natu (raxux KocTeit tp; 0 — 0, 1— 1, 2— 2). Burner

3
ar = Bat 4
=

Suavenne P(A/B) 8 sanaue 1 mono Gino HAJTH, paceymaan
exeayronine oßpaaow: a Tex 9 exytaen, Fe Koropnin enoytrea co-
Guns B, cobirrmo A Gnaronpmarerayior 3 eaysan:

3.1

es

Tyers = Mmnoropön eure e sind à pésubetencnire
anemenrapmux nexonon coburmo B Gnaronpitarersyr 1 anc-
MeBTapHEX uexogos (170), a coOmTm AB Graronpuarersyior
Fr nexonon. Torna P(B) = L, a P(AB) =". Cormaono Qopuyae (1)
meen

Peart) Lam

1

2 189

Yenoenan WERORMOETE. Mesammcmocre coburn

Panencrno P(4/B)- ; onpenenser Qatrsect neposmoer:
couru À n yenonuax, KOTOPEC nOQMHIEAOT ps macrynemt
coors B.

Tax Kax dopmyxa (1) nepua aa moßsıx cost, 70, noxe
nam mecranm À # B, a raxxe nonarax P(A) 20, nonyanex

(Bay = FG) | o

Saraua 2, B mure xemar 3 Geauix m 2 wepmax mapa. Ho
Alma ABRIS DENAIN 10 OAHOMY MAY, He Boonpauas x
o6parno. Hair eposrmoers roro, uro: 1) nepaiim Gui wanneuer
Geant map, a BTOPbIM — epHBtft; 2) BTOPHIM Grin BEIHYT GEPHMÉ
map PI YCKOMN, «TO mepmuin yare Gh nannenen Benni

D Nip pente annauın pacemorpi cobra:

A — nepnuim momyr Semi map;

B — oropane amnyr sepa wap;

AB — mocnexonnrenuno uanzesen Genii, sarem sepmih
epa;

B/A — sropsiM BBIKYT “epxEI map nPH yCAOBMH, “TO nep-
sun Grua wanneuen Gents.

1) “nexo aeex BOIMONHUX BapHANToD MARINE ABYX nt
por ma AMKA C MATBIO mapamu (C yuerom HOPAAKA Hx HOARE
na) panto Ag—5-4~20, 1. e. n=20. Baaronpnarernyrompn co.
Gurmo AB Oyayr Bee BosMomRuBIe ymopanouemmnte mapa «Gens
Map, sepntsii maps, COCTABACTINIC wa MMEIMXE TeX. ex x
REY vepnix mapos. TAUX coeaınenuh coraacno mpasiiay yan

wenn Gyner 3-2=6 (m=6). Taxun oöpasom, P(AB)= À - À

ae

2) Tlocae wannevenna no anna nepss Genoro mapa (apo:
maontxo coburrue A) raw ocranyren 2 Gemsix m 2 wepæmx map.
Tloapxermo wepHoro mapa BTOPEIM 13 Yersipex ocrammruxca (n= 4)

Gxarompuarerayior apa ucxona (m=2), nosrony P(B/A)=

a?
Uneom P(A)=$, ra ax n=5 (nature nepnonauanıno mr
xexuaces 5 mupon) 1 m3 (Genus Guo D. Hogeranno » dep
29 (2) aunenun P(AB)= À m PA), nonyamı P(B/A)-
2.3 a5 1
qos" tos" A

Ms pasenersa (1) cxeayer, «ro

P(AB)=P(B)-P(A/B). @
Hs pasenersa (2) nonyuaen
P(AB)=P (A): P(B/A). w

1907 tnasa vi

Bnemerima TROPI BEBORTHOETER

Panenerna (3) m (4) Momo samncars D pHqe CICAVIOMUX
paneners:
P(AB)=P(A)- P(B/A) = P(B)-P(A/B). (5)

Banaya 3. B xaGoparopmn paGoraor 7 >xemugra u 8 myn
Ha. Cnyuaiiinin OBpaaon ma aucza STK COTPYANMROD ana Mayu
voll Komepenmuor BEÓMPALOTOA OH AOKAANUNK m OJMM CONO
Kun. Kaxona ReDOSTROCTE TOTO, WTO AOKAAAIMKOM GYAET BI
Grane JKCMIIEO, a comoKMARTIKON — MyAuItHA?

D Tiyers coßsırue À — nokmanunkon BxIGpana wenuuma, eoßsınıe
B— conoxmaminx — nya.

1-i cnoco6. Bepostnocts roro, «To cnasaza nuönpancst oc-
Boo AOKMANK u HN OKAGAACD HEHUHA (HACTYMHNO COG-
me A), panua P(A)= =p

Bepoaruoers roro, WTO BTOPLIN BMÖHPANCK COROKTANUNK I HN
oresaxen myxuuua (Hpousouuto coßumue B), msiunenneren npn
yeronım, ro mopnoh y>Ke na muÖpana XNA, 7.0.
MBA RE.

Pr Pi Liz

To @opayae (2) uneon P(AB)=P(A)- P(B/A)= her.

2-1 emoco6. Bepoatoct» roro, ro nepmx BsÖnpanen co-
FORMA u MN OKAGAICR myawa (mponsouno Course B),
pasa P(B)= À.

Bopostuoet tore, 470 BTOPLM DISOMPaNCA HOKAMAUNK m MM
Orasanaeı xcemmua (coGuTHe A), nsiunenneren MPA YCNOBH, WTO
epa yxe seißpan myxuuma, 7. e. P(A/B)= 7. To @opmy.te (2)

monyunen P(BA)~P(B)-P(A/B)= 3-2-2. 4

2. Meaanucumocrs cobsrnii

Onpenenenne

Coëumun A u B naatinaior nesaaucummau, com

P(AB)=P(A)- PB). (5)

Fens panenerno (6) ne numonnaercn, To coGwrun À 1 B na-

SBIBAIOT saBHCHMSIOCH.

Onpexenenue neaannenmocrn coGuitult cornacyerca ¢ nnenen-
mun nommmuen yenonmoh meposrnocru. Neiernurensno, coûrue
A aonacrex nesasuenmuin or coÖsirun B torga x TomKO Toraa,
Korga nacryunenne coOurHs B ne Runser HA BepOATHOCTS HacTYILTe-
wur cobkırun A, 7. e. xorga P(A/B)=P(A). B camom nexe, coor-

moueune P(A/B)= FAN P(A) umcer meero rorna u Tome ror-
à, Koraa BBUIONKAETCA paBeHcTBo (6).

ss Cer

Venoanan sepornocre. HesasvGnvocre cod

Banava 4. Ma Konomx n 36 Kapr maynauy Pummmaeres ox
xapra. BuscHitrh, anamıoron au meanmcnmunen codırus Au B,
ecm A — monmmnen Kopom, B— puinyra Kapra “epsopoit un
uukonoli sacri.
[> Oömee «nexo axementapumx wexonon panto 36, cobro À
Gnaronpusrernyior 4 nexona, mooromy P(A)= ¿5 => Bann mpo-
maomxo coßsırire B, 10 ocymecramnoc» oaKo us 18 oxemenrapaux
Cobsıruii, cpeux Koropsx COÓLITIMO À Gnaronpuarerayior 2, u 10
atomy P(A/B)= = 5. Miras, P(A/B)=P(A), 7. e. coGurrun Au
wesanmenmenn. À

Ompexenemme neoapnenmocra oGoGaeres Ha cayaalı n>2 co
Gurruii. Cobuerus Ay, Ay, ..., A, HAGMBAIOTCR nesanucumun 6 ce
GOKUNNOCMU, com MODADICIMIZ ReeRoaMONMie MAP; ma sx
couru, a Tanke ecau xaxoe HO orux coOkırun 1 COOL, ar
Amomeecn mpontanenenien moboro NEAR o¢rankMx court, ne
saone.

Hanpuxep, nesasucunocrs rpex coBsiruit A, B n C osmauser,
«ro neoannenstiann Ao Óbre 6 nap Cobra: Aw B, BHC,
CWA, Am BC, Bm CA, C u AB. Ma onperenenwi nesannemoer
5 dopuyası (6) cxenyer, “rro ecmm A, B, C — meaamnemmme » co
norynmoorn cobsırun, 10 P(ABC) = P(A) P(B) P(C).

Banaua 5, Hs xonozss, conepmaueli 36 xapr, mocnenosanen
Ho BHHHMRIOTER 2 Kaprt. Pacemorpum cobsrma A m By me À —
Tops meinyr tya, B— mepnu shuyr rye. Samems ar
coburn À m B? _
D> Banumens coGwrue An mune A-AB+AB. Dra aanues oanauser,
wro BTopas Kapra MOMET NTE Ty30M NGO m CIYUAE, KOTAA nep.
Bas Kapra — Ty3, 1460 B cayuae, Koraa mepBan Kapra — He TY

Tipeamoxoxiim, «ro coGwrus A u B weaanucumss, orm

4-8. pany. 2-4 8, 324

ae er PAP gage ke gay
Ho P(A/B)= a + u movromy P(A)# P(A/B), r. e. coburna A u B

sama. A ESPN

is

Ynpaxnennn

Ha croxe nemar 4 cumux u 3 xpacnsx kapannanıa. Pesar:
‘Top Anamznı Hayray Geper 10 OXMOMY Kapamarmy s OGpario
mx ne Razer. Haitrn BeponTnocrk Toro, ro:

1) nropum Gun naar wpacnsali Kapangamt Hp yexoumt, uro
mepnsan Gm cami

2) sropsin Boar cHHMM KAPAHJAU NP yCROBuM, TO epa
okaaanen cumutii;

3) sropum soar cuit Kapanfam mpi CAOBHN, ‘ro nepruw
Guin xpacustit;

192 Tnasa vi

Önemenru TEOPHn BEpOATHOCTER

4) sropuim BAT wpacnsih kapanzamı NPA yenonmm, “To mep-
SH TAKE OKASATCA KpacHBiii Kapanam.

BE B Gapañane naxogwres 10 xorepelinux Guneron, wa nux 2 ner
urptinitsx. Ma GapaGana 2 pasa mammiator no oAnony Gune-
He BosBpaman Mx o6patHo. Kakosa BePOATHOCTE Toro,

1) »o sropoñ pas Obin wanneuen Guaer Ges BRIKTpHIEIA MPH ye-
OBI, 470 mEPnLUM Orasanesı nranrpsauunk OxTeT;

2) » nepnuuhi pas Okın Bkimyr BEINTPHIIMEIit Oner, a BO BTO-
poli pas — Ónner Geo msturprma?

Ma auua, conepsamero 4 Geaux u 5 xpacusix uapon, 2 pa-
aa mayrag NaRTEKAIOR NO ORMONY MAPY, me ROanpantan MX 06-
parno. Haittn pepoarnoers roro, "ro:

1) sropsim nopnesen Kpacnstii map HP YCHOBHM, TO HEPBLIM
Take okaganca «packkih ap;
2) 06a pasa wannoxammen Kpaerise iapss.

Ma wonoau m 36 Kapr nocnenouarenbuo nayran wunumaoren
u ne nosmpatarores 2 kapria. Kaxona nepoarnoers TOTO, “TO:
1) 068 pasa uannexanne» kapısı kpacuol Macri;

2) nepsoii Gua pmuyra xapra Kpacnoï macru, a Topol —
sepnoït Mac

3) wropoñ munyra Kapra wepHoii macru np yenosmn, “ro
nepsoii Osa Kapra KPACHOË MACTH?

BuscHHTL, ADAMIOTOA JIM HesanHcHMEIMH COOL A K B, ecm:
1) urpansuan xoers Ópocaerca mamanı; coGsrue A — npu
nepsom Gpocammn »urmano 2 ouxa, coßsırue B — mpm propom
Spocanun sina 5 ouxoB;

2) Spomens que urpamsmme Kocru; À — na mepnoli xocra no-
smiocs 6 ouxon, B — ma ropoñ Kocrit TaKxKe 6 ONKOR;

3) ua KONO] Kapr BEIHHMAIT NO OJMOÍ Kapre, 209»pamas
munyryıo Kapry m konoay; À — uepuoit munyra gama nu, B —
Bropoii Taxe BENYTA AANA THK

4) a KOJOABI KAPT ABAABI BBINHMAIT MO OAMOÍ Kapre, ne
nogmpanjas ux m KONOAY; coÖsırıc À — neproï munyra meo-
repxa Tped, coGsrTHe B — sropsim BMHYT KOPONb MM.

B öyxere 10 ruosgux u 5 napuuecos. Ons u Taux enyualinsın
o6pasom pImmmalor us Öyxera no onnony unersy. Kakosa ne-
PostHocrs roro, uro Ona Bunya rsosamKy, a Tann — map-
nuce? (Pers sagauy pas cnocobamn.)

B naprun 19 100 neraneh 2 neranı 6paxonaunue. Jlua Kour-
poxepa nuunumaor cayaalizım oÖpasom no onuofi era. Ka
Kova BEPORTUOCTL Toro, WTO HIEPBOMY Kohrponepy HOCTANACE
Spaxonannast, a wropomy — neöpaxonaunas eran? (Peurwrs.
3ARAUY PASIBIMA cnocobanı.)

g3 193

FERDEREN BEPORHOCTE. HESRENENMOLTE CODE

(BO! Cryaenr, xoropouy mpeneronno cars saver, anan orner m
70 nonpocon ma 90. Haxona nepontiocts Toro, 170 OH:
1) sepno ornerur wa sua Bompoca;
2) orBeruT ma BTopoit nompoc np YCAOBMM, UTO OH He 38%
orsera Ha nepsEiit Bonpoc?

§ 4. BepostHocts nponssegenna
HESABUCUMEIX COGEITUA

onycrum, sro na Komonsi ¢ 36 Kapramı caysalinsın oGpason
ax HaRTEKAIO MO OXHOÏË Kapre u HMMMCIAIOT BEpOsTHCETS
coGurua À — propolí mopnewena mama rpeÿ. Ecas mepnaa Kapn
2 KosORY ne noanpamaeren, ro um P(A)=0 (ecam nepnoïñ Guns
MOBACINA uenno zama Tped), man P(A)= ;); (ecam mepaol Cu
na wapneuena OTANINAA OT zat rped Kapra). Tarcim obpasın.
sopourioers CODBITIA A SAGUCUM OT TOTO, HACTYMILKO zu COÓNTAL,
cmmoamoc © anrewenmem ua KoxoJÑ nepsol kapraı. Bean xe
nepsas MODAOTONMAA Kapra noanpamaeren o0parno » Konoay, 10
PAAD= À wesamıcnno or Toro, wanona Ghia nopnan apra,

Cymecrayer memano memsramuñ (Hadsinaensıx megane
mu), E KOTOPBIX BeposTuocTs: Paccwarpitmacmux cobarruñi He sats
car oF TOTO, IIPOMIOIINK WAM Her APYTHE COONTHS, CRANIMRE €

Cobtirun An B Hagwaor MESAGUCUALAMU, COAH BIO
ca pasenerso

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

Paccmorpun OMT ¢ Ópocamen AByx urpnanunıx Kooreh ae
enenyem joa coßurrua: À — ma nepaoli Kooru nuumano 5 onkon, B-
Ha Bropoñ KocTH numano 5 oukos. Bunce, GYAYT au con
Aw B nesanuenunn.
O Tlossnerne mo6oro wena OuKOB na MEPBOÄ KOCTH (B YACTHOCTE,
nacrynnenne commis A) me nmuner ua coburue B u ero neposr.
uoers. M uaoGopor, necrymenne cobsırun B me manner Ha Repo
ammocTo coburna A. Taxune oÚpazom, couru A u B mesesicr
mue, mwen P(A)= En P(B)= À

Coburrue AB coctour » conwectuom wactynsenint coGuriii à
u B. Dueeurapunie nexoam CoGurus AB — oro mapa dueer, »
KOTOPLIX Ha Mepnom MecTe CTOHT “NEO OKON mepnoil KOCTH, ın
PTOPOM — uneno OYKOB BTOPOÑ KocTHL. Beero aeMeHTapHEN 1230
208 emana n=36. Cpenu mux npueyrernyer Hu, ou nes
(5 u 5 ouxon), Graronpurrernyiomn codwrmo AB, 1. e. m=1. Ts
Ka oGpason, P(AB)= 3-2. L=P(A)-P(B), 7. e. cobummn 4

ens

1 B nesannenmie. ©

Hapepnara NOXHO ropopitrh 0 HESABIICHMOCT cobra, em

494% rnana vi

‘Snewenrer TSOpAN BEpOATROGTER

Korqa ce nesanennoer» ncmuiranuit neouenuAHa, TO MEGA

cntoers coßsıruli A u B nporepæior c nomomuo popmyam (1).
Sanava 1. Ma uncen 1, 2, 3, .... 11, 12 cnyualinsm o6pasom

AUÓMPasOT OAMO UMCAO H PACCMATPHBAIOT Ba COMM: À — BBI-

pana wernoe «meno, B — nuiGpano «meno, KparHoe rpem. Base

iam, annamren JU COÓBITMA Au B HesanneuNntnet.

D Cpexu zanınsıx uncer wermerx uncen 6, mosromy P(A)

Ja

Paja

Y

rara rpm» zannon maGope ween 4, 1.0. P(B)= =

(ame AB cocromr 8 sxiGope “mexa, xparmoro Kax uncay 2, Tax
auseny 3, 7. e. kparuoro uncay 6. Takux uncen # uaGope 2, no

sony PAB)= = 1. Tan war PGA) PB) 4 1 = {AB vo
cru An Besame. 4
Banana 2, Burnenurs, anrmorea mu cobra A x B mean

onan, ce
1) PA)=0,8, P(B)-0,6, P(AB)-0,48;
2) P(A)= 1, P(B)= À, PIB) D

D1) Tax Kak P(4)-P(B)-0,8-0,6=0,48=P(AB), To coburua A
4 B ueannucunesse.
1.2.1
2) Tax Kax P(A)-P(B)-4-3=4= 75
Au B we spnawrea nesapucunseints. A

P(AB), 10 coberua

Sanaa 3. B naroronzenmoit maprmt gerenux maueñ neposr-
seth nonnaenns Opaxonaunoro Maa pasua 0,004. TIpouaBoat-
rx oöpnson » wpnemali quer onpamenu À meex sae, a 0c-
rame Mast orpauionst » en. KaxoDa BOPORTHOCTE TOTO, ro
Zayra maanyreuli mau Gyser KeOPAKORAHHBIN I KPACHAIN?

D liver» coGwrue À — moaxenme Ópaxopammoro mara. Tlo yeno-
suo P(A)=0,004. Tloannenne ne6paromanmoro maria — coGurrue A,
n P(A)=1-P(A)=1-0,004 «0,996. Ilyers co6rrrne B — noapxe-
se Kpacoro MAYA, rorxa cormacHo yenoBmo P(B)= 2.

Sanaa enonnıen x MAXOMACIDIO neponnuocru ConmecTmoro
(onnnennn neganncnmuix coßsimui Au B, 7. 0. K naxomaenmo Be-
pormuocru coGurus AB. Corxacuo Qopmyae (1) umeen

P(AB)=P(A)-P(B)= 0,996. À 0,747. €
Pol Bonce anyx coGsiruit naasınaor Heaaeucunniau @ cosoxynno
emu, con MIABHCIMES MOCROINOKIINE Maphi Hs DEUX COOL M
cont Kaaaoe xo TUX COGrH H COOLITHE, AnAmONcoc« mponanc-
uen 11000ro «nena ocrammux co6kırırl, meaanırcumst. Beponr-
{Ver COBMCCTHOTO NOABACIINA MEJMBEMMEUX B COROKYTHOCTH CO-
tum pasua uponsBegenmo BepostHocteit armx coGsrrHit.

34 8
PORTWGCTS MOMENT nESBEMEHNER CON

4. Tu erpenka nesanntesmo APYT oF APYTA crpenser
no mieu no oanony pusy. Beposruocrs monananın D wine
Axa mx parti coornerernenmo 0,2; 0,5 m 0,4. Hahn seporr
ocr» TOTO, WTO ace TPH CTPCAKA nonaayr B MIER.

[> Bnenem o6osnavenns: Ay — nonsyanne m muuienn nepru
crpemkom, Ay — MOMAJANHC D MIEIICND DTOPMM CTPERKOM, 4 =
nonexenue » muuess rpereuw erpenxom. Toraa Ay 424 — now
amue » numems nenn crpenans. Tax ax CoGMru Ay, Ay 4
P(A\AoAg) = P (Ay) P (Aa): P (Ag) = 0,2-0,5 -0,4= 0,04. 45

Ynpaxnenna

31. Bepoarnocr» nonaqanna » Muntens crpenrom panna 0,6. Hr
Ka BePOATHOCTS Toro, TO CTPENOK MONAXACT B MER à
KaxAOM M3 ABYX NOCenOBATeNLUBIX BLICrpexON?

82. Bepoatnocts mopaxemna nen mepssim opyamem passa 0,1, :
sropam — 0,6. Haïru nepostuoers nopaxenus wes 0600
OPYAMSIMI, expeABUMMH HESABHCHMO APYT OT APyrO.

88. B ypue 2 Gensix, 3 xpacnsıx u 5 vepanix mapos. Jena nr
HMMAOT no OJMOMY wapy M noanpamaior ux OÓPaTIO 8 AS
Kaxopa BépORTHOCTE toro, “TO!

1) nepnuin naamyr KPACHHK wap, a nropsim — epa;
2) nepanin nemyr Hp map, a propsim — Gear?

34. Bpocaior pu urpamsusie xocrm. Halen Beponrnocts num
HU METHOFO HCA OUKOB Ha KA» AOÍ KocTH.

35. Jana Opocmor urpanumyio xocrn. CoGurrue A — npr me.
Bom Gpocarux puma 6 OuKoR, coOuITHe B — n peayanım
Broporo Gpocanus MOABINOCE HNCAO ONKOB, KPATHOe pe.
Haiirm nepoarnoers co6urrus AB.

36. Hnaası Gpocaror mrpaxbmyio KocTs. Cobsrrue À — nepml
pas Buimazo vernos “nexo, co6urne B — sropoi pas sats
uneno, Menburee rpex. Haliru sepoxraocrs coGrun AB.

37. Beposrnoens momananıın B Mmes CTPEAKOM panna 0,7. Ke
KoBA ReposTHoCTs XOTA Oki ORMOND HONAAAHE B MILIONS sm
erpenkom B PSyabTaTe anyx mEIcTpes0B?

38. BepoxrHocTs ONANAUNA B MIMIeM® MepBRIM CTPEKOM pans
0,2, a nropsim — 0,3. Kaxona BePOATHOCTL TOTO, “TO men,
Gyner mopamena xora Gur om BEICTPCAOM, ecan crpeaı
PRICTPOXMAN HESRDHCHNO APYT OT Apyra?

39. B nuinymernoft sanogom maprunt zeraxeñ 2% Gpaxa, m np
HOBOMBHBIM o6pasom BurÖpanmsıe 0,3 or “nena Bcex gerasel
oxpauensi m gexemuñí umer. Kakona nepoatuocrs roro, si
cayualimam oGpasom punyram HS mapTuM geral oramers
HeOKpauleHHol u KeOpaKonanHoll?

196) trans vi

‘Snensenrta Teopan BepoRmDCTER

40. Bepoarnocrs nonagamix no unuenn nepstim CTPEAKON pan-
xa 0,6, Drops — 0,7, tperoum — 0,8. Kaxaui uo mu
crpenner no muuens onu pas. Kaxosa seposrnocts toro,
ro Minuten; OPASAT TOMERO mepnst u TPETH CrPENKH?
Ha npennpuarun 120 uexonex, cpex Koropux 40 em.
Kesxuslt corpyaunx noxynaer onmm Guner qenexno-Bemiesoii
aorepeu (20% nsiurpsiutkix ÓneTos) u ox Óuer CHOprHR-
oh aoropon (10% puurpuumux Onneron). Kaxona neposr-
Hocrs Toro, WTO BMÖPAHRKÄ cayualinsim OOpAsoM Hs CHHCKA
COTPYANKKOR npennpusrus ONU venonex OKWKETEH Myaun-
Holl, BEINTpABUINM B OGeHX noTepeax?

$5. Popmyna Bepuynnu
PASanaua 1. Crpesox nopartser mimes mpi askom muterpene
© eponrhocrsio 0,8. Kakona neporrnocrs Toro, “TO men» 6y-
JET Hopaxexa Al mpi MEPNON N TPCTLOM BICTPONAX, CCA CTpe=
‘0K suierpemsa no mumenu 3 paza?
D Moro coßuımme À — nonananne CTPOXOM no MIE MPI 02
ou buerpeze, voraa coGwrue A — npomax. Ilo yenonuio aanauıt
21A)=0,8, roraa P(A)=1-P(A4)=1-0,8-0,2. Pacowarpupaemoe
amie coGurue B COCTONT n TOM, WTO erpexox PI mepnoM nu
exe MOPASMA MIEL, MPA ATOPON HPOMAXHYAGH, MPA TPETHEM
cosa nonan, 7. e. B=AÁA,
Tonazazine u nenonaganıte no MICH B paccmarpumacmoh co-
JIM HEBMCUMEX HCTINTAMHÍL — HC0ABICHMELE COÓRITAR, NOITOMY
P(B)~ P(AAA) = P(A) P(A) P(A) = 0,8-0,2-0,8 = 0,128. 4
Banana 2. Crpestox nopaxaer amen MPH oguom maerpone
wpoamoersio 0,8. HaliTH sepoxrocrs nopaxenun MALE
us anyuen MMCrPERAMH, CAM BCETO expexoK erpens 3 Pasa.
D Ilyern coßirme A — nomaxanue no MANN MPH CANON BRET
fete, a coÓXTI B cocrour m momananıtı no menu pit 10681%
23% 19 rex cnexamux mucrpeno». Hnaue coGwrue B uponaok
xr, Korxa npomaofiner ORMO 13 neconmecrmux coGwruit AAA,
“ÍA man AAA, 7. e. BAAA+AAA+ AAA.
Cornacno reopene 0 BepomTnocTK cyMats! Heconmecrimx COBB
vi (ox. $ 2) mmeone
P(B)=P(AAA)+ P (AAA) + P (AAA). a)
Henomsaya paceyaenns oanauı 1, 9amerum, uro Kamaoe
Garaexoe m mpanolí aser panenerna (1) panno 0,128. Tar 06-
sou, nexomas reporrocre P(B)=3-0,128-0,384. 4
Nonyerum, uponasonaıcn n HEGABUCHMAX APYE OT apyra He
CTEM, m Kanon MO ICOTOPMX coßurne A MOME mponaohr,
2xoxen me npomaolru (r. e. mponaoiiner coßsırue A). Vononmsen
ram, STO m KBIKAOM na uenseranu NeposITHOCTH obs A OR”
wura xe 1 panna p. Torna BCPOATHOCTE MPOTHBONOzIOKHOTO ENV
ama A Oyner panna 1=p.

$5 197

“opnayna Beprynnn

ocram sax au y:
Bummeniere seposruocrs coGurrun B, sax moaiomeroen 10%
co mpa m weneranustx cobre À mponsohner pono k pas. EI
mm Paceworpun coute By, cocrosmec » rom, “O » nepnu À
MCNBITABHAX HAcrynmxo CoOsiTHe A, a B cneayiomax (n—k) nens
rammax — cobsrrue A, r. 0.
Bi=A AA AS À.
_ tuer mes
Cobumun À u A, onto wa noropux nacrymnao n Kamgou 12
n nentrrarnit,— nesaouemtic y COPOKynnoen coma. Cora
No yrmepsexenno O nepowTHocrH NRONADEAENIEN WeaaDUCHNL €
Guara (ex. mpeg maparpad)
P(B,)=P(A)- P(A)- ...: P(A): P(A): P(A)

AL ASIS 0)

Pacemorpuw coGsrrita B,, m KoTopsix coßnırıte A monropens
‘ke pas n pasxmumsx nocnenoBnrensuoerax. Ten ne Menee Repost
noers mo6oro coßsırus B,, ABnmomeroca nponanenenmen À cola
nit A u (n—k) coGurrnit A, Gyner panna p(1 py.

“nexo enocoGos sammen nponsnenenun us k coGuui 4
1 (1h) coGruñ A, OTAMUMOMEXCA Apyr OF APYTA NOPHAKON por
monomenna » unx amomurexel A 1 À, PaBHO “neay nepecraso
pox € nopropemuann Pa, aso

To aoxasannony » $ 5 npeauayınek ran

Pa, nu One a

Cobre B, cocroamee 8 rom, "TO BN uemswrannax coßum
A uacıyuur k pas m me Hacrymur (7 A) pas, OCRMAHO, panno ene
Me meconmecrmux coßsırui B,, OTANIAIOIMIXCA APY OT API
Jul HOpsAKOM paenonoenna a uux le muomureneï À u (2-H)
auoxureneí A. Uneno raxux coßkırmli contacto pasencrny (2)
panno Ch:

Pa

Ba By+Byt A Bites

ch eaareevacx
Ilo reopeme 1 us $ 2 0 neposrHocri GYM HECOBMECTAX co
Gummi meen
P(B)=P (By) +P By) +--+ PB)+

CR enaruenmn.
‘(py +p. py ++. +P py t+ = Ch py A
Ca caras

AB rnaun vi

Inemenru TEOpAN BEPORMOCTER

PlBepoatuoers coßsırun B npunaro oGosmauars P, (k), nonuep-
Kuna TOM CAMBIM, WTO PACCNATPUBACTOA ReposTHocTh COÓNITIA,
aerymunnero poRno k pas B cepun HS n OMMOTHIEILIK HEAR.
To aoxasannomy peste.

PAG) CR pl} N @)

me p= P(A).

‘opmyay (3) naosieasor @opayaol Bepuyasu » seer wpeli-
wapexoro Marewaruxa ÆkoGa Bepuyanın, usyuapuero p nauane
XVII 8, MOMTANHS © ABYNA HOSMORHEINN HEXANE.

Banana 3. Hrpansasiit xyGux pocaerca 4 pasa. Kakoza ne-
ponruoers roro, wro B 9TOH cepum membrana 5 ONKOE nonnaren
poso 3 pasa?

D Myers A — nonnnenue 5 ouxon » onuom nenseramu. CoOsrrue
A 5 kaxkqOM MS HETHPOX HOSABHCHMBIX MOMBITANMÄ MOKET Mpo-
1

wolitu, a moxer 1 me mpomooitrm. Maneeruo, uro
Torza cormacko popmyxe (3)

rara) E no

P(A)-

Baraua 4. Bepoxtnocrs Toro, ro nanna ompenenemmoro Bit
za ne neperopur B revenue 1000 «, panna 0,8. Kaxona neposr
Act» TOTO, WTO MS NATH JAM AANHOTO BHAA He MeHee YeTEIpEX
ceranyres MenpaRnEINt moce 1000 4 ropenna?

D Pacomorpms ropemne xaxnoli mo 5 ann » renemme 1000 x ax
Aesmseubte HCTEBITANIA, m KOTOPMX BepoRTHOCTS HCTIEPTOPAMMA
zoxuer pape 0,3. B cepa mo 5 nomoGuerx memsrramalt dexrue-
cu tpeGyeren unliru neposruoer» eysouss CACAYIOHAX MECOBMECT-
aux coGunnii: À — nenpamumn ocraancs 4 xamma m B— nc
spam oeranıch 5 san. Beporrnocre store coOMTHs pabra
cnsxe neposrmocreñi coGuirmit A x B:

P(A) + P(B)= Pg (A) + P5(5)=C30,8*(1 0,3) *+C3-0,3°-(1-0,3)° *=

-0,0081 -0,7 +1-0,00243- 1 0,03078 = 0,03. 4 U

Yopaxnonus

42, Monery Gpocaror 10 pas. Kakona neponTnocr» Toro, «TO opea
Noaputes mpi orom poBno; 1) A pasa; 2) 5 pas?

48, Mrpamamılı eyGme Opocaor 5 pas. Kaxona neponruoers roro,
ro 6 oukon monarca pomo: 1) 2 pasa; 2) 4 pasa?

4, Vrpaxountii xyOwx Gpocaior 4 pasa. Kaxopa weponruocrs r0-
ro, «ro 6 ONKON m DTOK cepHM MemuIranHit osDATes ne Mere
pex pao?

46, Beposruocrs nonananna no xoxbny y nesoroporo GacKer6o-
amera mp Kaom Opocke passa 0,7. Kaxona sepostHocTh
y omoro GackerGornera nonacth NO KONBNY XOTA Ost OM Pad
5 cepinit na Tpex GpocKon?

ss [toa

“Bopanna Beprynive

4G. Kaxona pepoxtHoct» Toro, ro pm oxuom Gpocanux urpas
Holi kocru Bbimayer 1160 5, 1m00 6 ouKkoB?

47. Va ypwx, conepameñ 15 Gensix, 10 xpacnsex x 5 onmmx me
pos, mayran napnexaeres one map. Kaxona nepoamnocr ne
annenua Genoro mapa?

48. Oamompemerno GpocawrT Ape urpansesie KocTH. Haliru pepo-
ATUOCTE TOTO, WTO CYNMA HINARI OUKO Papua 8.

49. Haóupas nonep renedons, aGorenr 386 Ape noczoneme na
Dit u, MOMAR AMINO, “TO 9TH MDP PAS, MAGpax x
Hayrax. KakoBa BePOATHOCTL TOTO, “TO HOMCP HAÓPAaR Ups
mono?

50. Bpomena urpanınaa xoors. Kakona nepoarnocrs Toro, sro st
munaayr 3 oa?

51. Bpomena wonera m nrpanman xocrn. Kaxona neporricen
Toro, WTO minaayr opea u 6 oro?

52. Tlo munmenn expensior 2 pasa. Boponmnoorn nonapanısı o ir
tens npu nepnom auierpene pana 0,8, npit nropox mere
ne — 0,9. Kakona nepowrmoers Toro, To MINERS ne Det
opaxcana mt ormım niiorpenon?

53. Mrpansnan xocr» Ópomena 2 pasa. Hafıru Bepoaruocrs Toro,
10 on pasa monnuren ORINAKODOS “EO OUKOS.

54. Ma ypms, conepasamelt 3 uepmx, 4 Gexmx u 5 npacimx me
Don, naypauy memmmor oni. Kaxona nepostrioer® Tore, re
BHEYTHit map okamkerca: 1) 1EPHEM; 2) Tepmsım mau ben)

55. Bpomons ave urpamsuse Kocru. Halt sepostioers Tor,
ao 3 ouxa monnaren xora Our ma o1mol mo Koorelt.

56. Ma konom 8 36 Kapr mocnegonaren»uo Mayran Bein
ae Kaprıı 1 He Bosnpauaor OGpaTHo. Haiir neposruoer rc
1) pummyrar BA 1y0a5
2) cmanana uonneuen 1y3, a sarem mama;

3) manyrur 2 Kapror GyGnonoit macrit;
4) BTOPLIN Hssxeuen Ty3, ec mapecrno, “ro MepBoli Gun
mauyra jana.

57. B ypue maxomurea 10 Geaux u 10 vepmsix mapon, Ha nee no

enenoparensuo BHIMMMAJOT 2 Mapa M He BosEpamiaion ofper
uo. Kakona neponrnocri Toro, ro:
1) 06a pasa HOBNOKAJINCE maps vepHoro BOTA
2) uepmn manyr Genii wap, a sropsin — «eps
3) ropuN manneuen vepmEsii wap, CHI HIBCCTILO, AO rep
prin Gost Dany Genet wap?

58. Bpomemu rpx moxerst. Haïtru Bepoarnoerz roro, uro nine
10 me Goxee ABYX Opz08.

200 Inana vi

Inemenru TEOpun BEPORTHOGTEN

Hs noxmoro maGopa Kocteii nommno Gepyres Hayrax Aue Koc-
Tu. Onpexesth neponTNOCTL. TOFO, WTO BTOPYID KOCTÉ MOHHO
MPHCTABUTS x nepsoit.

©. B xorepee ua 100 Guneron 10 nuinrpuumux. Kaxona sepoat-
OCT Toro, “TO mM MA ORUM Ha Tpex EVTLACHUNX GHNETOB Me
puianer Beinrpsunu?

SL B aorepce n Önneron, ua xoropnix m mamrpumme. Haÿrn
RepONTHOCTE BRINTPEIIIA (MAMMA XOTA Okt ORHOTO BAH
nore Guxera) y roro, xro uneer À Guneron (h< nm).

EZ B napruu us m neraneii n Gpaxonanmax. Bri6npator nayran
k neraneï. Onperemirs nepontiocrs roro, uro cpenut ITH À
xotaneh Oyxer p Cpaxovaxnnix (pck< nm).

Tomas xonona xapr (52 auera) nennren nayron ma que nau-
xa no 26 aueron » Kaxaoï. Hafizu nepoatuocrs Toro, “TO B
Kamnoll nauxe OKAMETCA 10 ABA Ta.

B poourpuume neprencrua expat no nonehbony yuacrayior
18 Kowang, MO xOnOpHX enyunlinsin O6pagom Qopmupyiorca
ave rpyuosi no 9 xomam m kannok. Chemie yunernnkon nep-
noncrna umerorest 5 Komanx mo omo pecnyómm. Halkıı pe-
PoatHocts Toro, “TO ace 5 komany aTolt pecnyÓmuricn monaayr
Bonny MTY KE rpyuny.

BB) Nouery Gpoeator 8 pas. Kaxona neposrHocrs TOTO, “TO open
nommures:

1) posno 2 pasa; 2) poso 6 pas;

3) ne menee 6 pas; 4) ne 6oxee 2 pas?

Urpansusri xyGux Gpocaior 5 pas. Kakosa Bepoaruocr» Toro,

“ro OHO OWKO NOMBHTEA:

1) ponmo 2 pasa; 2) porno 3 pasa;

3) ne Gonee 2 pas; 4) ne menee 4 pas?

EZ] Beponruocrs roro, uro nacexomoe onpenenennoro muxa Öyner
sers. Gonee 100 peli, pana 0,5. Kaxona nepowruoeri roro,

ro CEA »ui6pannnıx ANA naßnionenun 10 nacekonkıx 31070

muxa ne mence 8 oxaeunaapo Gynyr tr Boxee 100 ei?

1. Kaxne COÓSITHA HAIMBAJOT CAYUAÑHEMA? qocTosepHuINH? He-
Boa?

2. “ro nasuisator cymmoit coGurui?

3. "ro naasınaor nponaneyenuen coßnrui?

4. Kaxoe coOsirue Masninaior PUTHLONONONHMM AARHONY COG-
mio?

5. Kane COÓNTIA nassınmor pannonosmonnnnu?

6. UTO naskBaïor BEPORTNOCTLIO (B KHACCHHCKON MONHMANHN)
coburrna A?

7. Kaxue COÓBITUA MAIMBAIOT necopmeornsnen?

201

Bonpocu x maso Vi

8. Yemy panna BeponTnocT, cymmer AUyX mecommectmux cob
ul?

9. Kaxue cobra nassınaor nesannensann?

10. “emy pasua sepowtuocrs npousseqeHus AEVX MesanıcHmn

oben?

Yeuy pasua neponruocr» cymmst ABYX mpouamosbanix cof

Tu?

12. Sannears dopmyay Bepnyanıt u noncnurs ee eMe.

1. Bpocalor 2 mowers. Kaxova sepoæruoers roro, ro na ode
monerax munaner open?

2. Bepoarnoer» manneuenun ua maprim Öpaonannoli zeranı

Papua 0,05. Kaxona nepostrnocrk Toro, UFO HAYPA MIRAN

Hag eran oxameren HeSpaKonannoit?

B auge near 2 uepuerx, 3 Gem u 10 Kpacisx tape,

Kaxona BEPOATHOCTL toro, TO Mayron BMHYTHÏ OA wep

okaxetea AH NepHoro, Man Genoro uBera?

4. B unse nexar 3 anemenna u 5 a0nox. Mansunk ne rnana be
Per Hs Bassı OH TION, sareM, He BOsBpaunası ero, Geper pp
ro. Han Beponruoer» roro, «TO nepnsim Gin Barr ame
em, a Bropun — s6noxo.

5. Beposruocrs, nomagannsı » mer, erpenkom pasa 0,8. Ka
KOBA BepOATHOCTS MONMANAMUA B MIIUCHE B KAIKON HS ABN
Uponanezeunsx ssicrpenon?

6. Vuemnx anan orners ma 15 sompocos ma 20, Koropsie nper
Aaranııcı x aanery. Orsera na nepsili nonanmımlica na ane-
re Bonpoe on ue anan. Kaxona Beponrnocrs Toro, «ro yaeınz
Ornerur ma BTOPOÑ HS IPCANOMEHHLX emy sonpocos?

1

1. Beposrnocrs nonananus B MIIEMB eTpenkom pana 0,9.
KOBA BepOATHOTS TOTO, WTO HOCAE ABYX BHIETPeTON u MIE
am oxameren onna mya?

Beposruocrs nonaama crpeakom no mumeux parue OA

Kaxona BOPOSTMOCTL toro, YTO MHMIONL Mocne TPEX BMETpe

108 ÓYner noparkena XOTA Ost oaHUM BLicTpeTOM?

B kopobxe xoxcar 20 oxunaromux no dopme mapon, pases

8 na mux serve octambmiix. Marecrno, “TO npousnonseut

5 mapos ua 20 oxpamensi » xpacumi uner. Kakona neposr-

MOCTh Toro, WTO CAYYAÑÍMIAM oGpagom BUIKyTH OMIM map oR

weten He KPACHLIN, HO AErKUM apon?

4. B nepnoñ xopobxe naxonsıren 2 Geaux, 3 wepunnx m 4 xpacrus
zuapa, a 50 Bropoit Kopoôke — 1 Gex, 2 wepmsix u 3 Kpae
mex 1uapa. Kakosa BepostHocts Toro, "TO BMHYTHIE nO ORO
My HO KAKAOÏ KOPOGKH Wap OKAKVTCA PROHEX upeTOR?

202 frase Vi

Inemenma TROPI BEpORMOCTEH

HAH vicropmueckan enpanxa

Bosnutonenne reopitn nepontnocrelt aie may Guno oöy-
crosreno passurmen B XVII B. crpaxoRoro jena, xemorpadnn,
à taxe umporum pacmpoerpamennen 5 Enpone asaprnntx. utp.
Brekux urpax (Kaprax, JOMMHO, KOCTAX M MP.) BMMFPHII B 0c-
tonto tannicen ne or exycersa Mrpoxa, a 07 cayuaiiuocrn. Cno:
to «ncapre m npoMsoulo oF Qpamuyacroro cnona hasard, coaue.
‘nero senyuait», «pic». Borarsie mou, ynaeuenukte adaprnk-
Mu urpanen, “uopol upnöeram K oMOWM NATEMATHKOE AR
ements npo6nen, noannamınıx vo spoun urpi. Tom porte"
Ina Teopiır Beponrnocreii wnorne yacnkte cuntraior 1654 P., X Ko-
opouty ornocIren mepenticxa AByx Bent Ppamnyacxnx yema
Betlackaa u II. @epwa no nonony pemennn aa, Boom
api urpe 3 KocTH.

B XVII 5. yucmac avant wenomsosar» asaprunıe urpui war
sacôme u marrante monenw xan nccaenonamın nonaruh ‘Teoprit
Rpoatuocreil. Tlepeax Kuura uo Teopun sepouruocrel uaouiDa-
men «O pueserax m asapruoh mrpe» n Guia. onyOnnkonana
5 1657 r. Ee anrop, ronnannexufi yuensait X. Tiofirene, mica:

PM BHHMETENEHON HoyveuM NpeAMeTA LuTaren» SAMEIT, "TO

tw anminwacre ne tomo rpoit, a WTO 3Aecı ANTON ocnoma reo”
pum BepostHocteii, ray6oKoh H BecbMa murepecnolt».

B 1718 y. Guina onyOnumonana wunra nonecruoro waeliuap-
coro waremara $1. Bepnyaan «Menyecrno. npernonomennt»,
3 XOTOPOÍR ABTOP Mano OCHOBEL KOMÖHNATOPIKI x annapara
Aumenennn nepostnocrell, A TAKE JOKADAN OV MO DAME"
mux reopen Teopmit sepontnocrell, maspannyio BNOCHEMETEIE
mcopenou Bepnyaau. Ha nokasavenscrno oroli reopens yuennail
sorpersn 20 mer want, a caso OKO aannno 12 erpannn. Dra Teo-
Jena — nou uacrunılı enyualt oxnoro 9 OCHOUMUX sakonon
opr Bepostnocrell — saxona CANAUX UCLA, UTIPITOO B Cer
seume XIX n. pycexmst yuemsan II I. Heben. Bacom Gomn
En uncen uNeer inpoxoe npaxricreckoe mpimuenenne B BOmpo-
tac, camoamunx c onpenenenuem vepoatuocrek coOurui, Ana
Korapuıx pacesntrars, soulioe AANOHNG Depomruocri (8 ee KANCCH
Seem ROMANE) nescamonn.

Tansneïüee pasuurue teopuu veponruoerell cunano © paGo-
Tan (paNUyscKoro MaTEMATHKA H acrponoma II. Jlanzaca (1749-
1827), nemenoro maremarmxca K. l'ayeca, poccuiicknx marenar-
xon A. A. Mapkora (1856—1922), A. M. Jlanynoña (1857—1918)
aap. Suasunrexsunit BA 2 TeopIND BEPORTHOCTEÍ Bnccam orene
cn me yuense A. H. Konmoropos (1903—1987), A. A. Xumunn
(1894—1959), B. B. Pueaenko n ap.

B nacrosniee BpeMA TeOPHA Bepoatroctel Mpoxomkaer pas-
muatsea m HaxonHt MINPOKOE NPMMEREHNE = ecreernoanannn,
BKONONHKE, MPONSBORCTBE N TIMANITAPNEX MAY KAN.

[zos

Ticropmeckan cnpanra

=.

Frasa

>

KomnJjekcHble uncJa
Muuxne ucaa — amo npexpacux
u uydecnoe yGexuue Coxecmoennos
Ayxa, nownu umo covemanue ón
€ metas
L Helms,

§ 1. Onpenenenne
KomnnekcHbIx uncen.
Cnoxenne u ymHoxenue
komnnexcHbix uncen

1. Bueneune
Hame npeneranzenne 0 iene momenaos
mo Nepe pucmmpenns Kpyra sagas, Koropue Re
o6xogumo Go pemars. Ecan man cuera orzen-
MHX UPCAMETOB AOCTATONNO ÚBLMO Namypasmur
uncon, TO ANA pomenns ypanmonuf puxa x+0=b
Harypanbiux Uncen NEAOCTATOUNO, Hy2KHO Guo
BROAWTE OMPUYAMEADMDIO unena u MYAD.

Ina roro “TOO pemars ypasxenun mu
ax+b=0, rue GEN u bEN, nowaaoGuaues paquanuss
noe suena. Ho yoxe pemienne TAKHX, manpsne,
ypapuennii, xa 223, x"-2=0, morpeosu
bnenenust uppayuonaaoneix uncen. Opmaro oxi
DANOC, WTO PARHONAMEMNX m MPPansonanmn
uncen (o6pasyrowix mmoxecrso deliemeumenener
uucex) raie NEAOCTATOUNO AAKE ARA peutems
npocrelux KBanparusıx ypanuenuit e narypan
BINH KoodypHMNENTAMM, TAKUIX, HAUPLNep, Kas

o.

24120, 4x4

Horpe6onanoch ssenente MOBBIX «itcen, massa
HEX ROM AEKCHBMU.

Taxun o6pasom, erpemenne cnenarı ypar-
Henna paspewUMBINK ABMAOCE OMMOÍ u3 Fa
MX NPHUH pacrumpenus nonsrun «mera.

Tipexne “ex JABATR onpenenenne KOMNAER
mx Nice, HEOGXOAUMO OHATS, KARIN caol-
CTBAMH HOMME OÓXOARTE HOBIE HET, KEKIE
Omepanun enaremsno BBECTH ZA HUX II
SaKOHAM JOKE HOJSMUATECH 9TH QUEPA.

Cnaoa vu
Komnnercneie “mona

TIpexxqe cero ana MOBHX “ICON BREEN NOMATHS PAñEMCTBA,
MPERCAMN onepamsnt CNODKeNHsT MY NMOROHISL HOBLIX WHEE TALES
soft ANA MIX MMO MecTO MepeNectHTeMBHLAi, COMETA TENE
2 pecupenenurempustit saxonss.

B enyuae, Korqa Kownmexcune uncna COBNAJAIOT € Aeliernn-
ait, HOBBIE OMpauti CAOKEHHA M YMNOKEHHA JOKED
yjespamathes B MOBCOTIGIO: CHOKENNE M YMNOKENNE ACTE
rene cen.

Tha voro uroßs ua MUOXECTES KOMIIEKCHMX CEA ypaBHe-
we 7241-0 mmeno pemenxe, Osıno Bmeneno Nexoropoe HOLOE
sexo — KOPEN» 2TOr0 YPABHONMA, KOTOpoe GH10 oDoanaueno
soit 1. Takum oGpasom, i — KOMMEKCHOE NNCHO, TAKOS, “TO
Bala

2, Tlonnrne Komnnexenoro «meza

Onpenenenne

Ronnzenensimu uucaamu nassmaror maparennn nuna a+ bi,
rye a m b— neliornurensunie «nena, a | — MEKOTOpBIE cunt
207, AAA KOTOPOTO NO onperenenuo nanonnsiercn PaMeHCTHO
Pet
Haananne «Kommnekennie» mpoucxonuT oT CHOBA ecocran-

mes — mo BHAY BuIparcenna a+bi. Uncro a nassınaeren deücm-

amexnoù uacmur KOMNAEKCHOFO uncaa a+bi, a «meno b— ero

uno wacmoo. Sueno | nassınaercn anumol edunuuyell.
Hanpunep, xelictsurensHax uacrs kommnexcnoro uncna 2-31

pa 2, mnuMas MacTh pasta -3. Banco KoMMIeKcHoro “NCAA

Sue a+ bi mama anzefpauwecroß popxol KoNNTeKCKOFO

mea.

3. Panenerno Komnexemmx nee

on; u
sa kommnexcnsix unena a+bi n c+di nasmnaoren pasnı
Au Tora M TOn»Ko Torna, Korma a=c u b=d, 7. €. Korma PAB-
HM HX ACHCTIMTOIPHBIO H MMM YacTH.

Hanpunep, 1,54 VB 8 +84, rae xan 1,628, VÖ
Banana 1. Halu aeicrunrensunie suena x m y na panenernn
Grm Wi=6-21.

D To oupexexemuio paseucrsa KOMILIEKCHEIX “Hee SAMHIIEM en-
cry ypanmenmi

Sx-y=6,
xtyn-2,
jamas koropyio naxonum x=1, y=-3. 4
4. Cromenme u YMEORENME KOMTLTEKCHLIX HMC
Ouepanun CxOMSCIMS u YNNOKCHHA ABYX KONDAEKCHEIX I~
‘et onpexenmorem exexyioutit™ 0Ópazon:

$1 W205

PERES KOMNEKCHEIR cen.
Choxene u yunoxeo «omnnexcm uncon

Onpenenennn
Cyxxol nuyx komnzexenuix uncen a+bi m c+di masses
Konnaexenoe «meno (a+e)+(b-+d)i, 7. e.
(a+bi)+(c+di)=(ate)+(b+d)i. w
Hpouscedenuem xByx Konnexcusix uncen a+bi u ¢+di sr
auinaeten Komnnercuoe weno (ac- bd)+(ad+be)i, 1. 6.

(a+bi)(c+di)=(ac—bd) + (ad +be)i. e

Ms bopmya (1) x (2) crever, «ro caomenne yanoxem:
KOMNNEKCHBIX HNCEA MOHO BHINOAMATD DO HpABIENAX Aer «

Tlostomy ner neoßxonnmoern sanommnar» dopmyanı (1) 1 (8,
HX MOXHO HOAYSHT 10 OÓBIMMM HPABHIAM anre6pet, CATAS, wi
B=-1.

Bagaya 2. Haliru npomanezenne

C+3D(+20.

D(2+30(1+20=2-142-214+31:1481-21=2468'+T1=-4+T.4

Npnusr0 currars, uro a+0-i=a, T. €. Kommnerenoe we
a+0i — aro nelicremrenbnoe «nexo a.

“exo puna 0+bi oGosnauaior bi, r. e. O+bi=bi; ero sau
nor xucmo MHUAMSEM WUCAOM.

Konmaexcnoe «nexo 0+0/=0 spaseres enmncrsenusnt ue
101, KOTOPO€ onnonpemenno m ACÍCTEMTEABIOS, H MMETO MEN

Komnzexenoe wmexo mpruaro oGosHadaTs omo Gyxeod, w
me ncero Gyknoñ 2. Banner z=a + bi oamanaeT, ro Kowmaektı
wuexo a+ bi oGoaaueno GykBoï 2.

Onepanım cnomenna $ VNHOEHUA KoMmMeKcHux te
O6nanaıor TaKHMH Ke CBOHCTEAMK, Kak u OIEPAUNH una peñcre

Ocuonnsie cnolierna enomenun u ynmomemun
Konmaerensix uncen
1. Nepemecrirrensnoe enolierno

CRAN Am
2. Conerarensuoe enoiierno
rt tat) (dal.
3. Pacupenenurenstoe exoiicruo
2 (22+ 24)~ 242042125.
Hokaxen, nanpınep, enoïerno 3.
O Myers z ma, +bil, 22 =@g+ bi m 2320 + bi. Horasam, wo
latente o
Tipeoópasyen zenyro uacr» panenerna (3):
21 (22 + 23) (a + bb) (aa + au + (ba + ba)
ar (ag + dg) Pa (by + ba) + (by Ca +g) + a (0, + bi =
lg +0,09—b10,—b1b+(0,0,+0,05+0,0, + bg).

frasa Vi
MONCH WHER

Ilpeoöpaayem nparyio wacrs parencrea (3):
2127 + 21237 (ay + (ag + dai) + (ay +03) (09 + bai) =
=a1a3—0,b,+(410, +040)! + ras bids+ (db, +D,09)1
jag + 4507 ~ bby Ob + (Oya + bras + ayy + ba).
Caexoparexsmo, panexcrso (3) asmoxmaerca. @
Anazormuno roxaaumaoren exojicrsa 1 2.
Bone, vro “nena 0=0+01 m 1=1-+01 na wromecree KOM
Irexcux uncen OÓJANAIOT ren 2Ke CRONCTBAMI, "TO HA MHO-
ocre nehermmrenummnx uncen!

Baxana 3. Bernoxmnrs nelieraun:

+ 2i)(-3-) +4431) 2-12,
D421) (-3-1)+(4 + Bi): 2— 121-3 ~i-61- DE 48+6i- 12i=
3114248617181, 4 =a

ISA Onpenexenne KoMnsexcoro “nena KAK “Hema BHAA a+bi ne-
eer ceGe onpenexennyio nencuoct, KoTOpas CHHSANA € upuMeHe-
est anaKion COEM M YANOMENHN AO TOTO MOMENTA, OF OH
ccepanmi BRORATCA. ABRO.

Verpanemmo sroli meacuocru cnocoßerayer unoe onpenenenne
soumaexenoro MEA.

Oni

Rounaerenun vucaos 2 nassınaior napy (a; b) nellernntens-
MAX uncen a Hb, BANTEX B ONPEHENEUHON HOPHAKE.

Tlapst (a; B) m (e; d) sanaror onHo m TO AE KOMILTEKEHOE unc-
"Tora M TOXBRO Tora, Kora OHM COBIAJAIO, T. €. TOA, Kor-
ame, bad.
CAOREHNE 1 VMNOIKCINC KOMILICKCHBIX HNCEA ONPEXERAIOTER
3 nou cayuae paseucrsann
(a; bytes d)=(a+es b+d),
(a; bite; d)=(ac- bd; ad + be).

Jance nokagmpatoren ocuommme enoiierna apudneruueernx
(rau. Baxem 00060 mnensuor uucna suua (a; 0), aER, u (0; 1)
E JOTAMABTMBAIOT, WTO CAOMONMC M YMMO)KOIHO KOMILACCHENA
cer DONHOCTBIO CONIACYIOTCA CO CAO)KEHHEM H YMHOXENHEM ME

en uncen, een napy (a; 0) cunrarn neiieranrennnnm
‚con a.

Meno a nassıpamor delemeumernoll Yacmvio konmnekenoro
sica u obosununıor a=Rez, uxexo b HUQMBAOT MHUMON “ACM.
romnaorcenoro unena m oGo9mowaor b=Imz (or hpannyacınx cou
Hide — neiiersurensush u imaginaire — mama).

Kowrexcute «mena (a; D) up b#0 nasnınaror muaa
ve, a sera BOJA (0; 6) HAAMMADT SHCTO MITIN "C=
zum. ©]

um

51 9207

‘Onpenenenre KONNNECHEN cen,
Choxonne yunoxenne KOMNEKCHUX nen

Ynpaxnenua

1. (Veruo.) Hasnars gelicraurennuyio u mauityio “aer Ko

nas mirlo ad
a Vier 512 6) 3,51.

2. Banncars kommnencnoe «nexo, y Koroporo AeÑermiremas n
MHHNAR HAGTIE COOTRCTCTREINO PAR:
Dans 22,3 4-1,
3) 0 -6: 4) -6 10.

yo

(Verno.) Ipn kaxom snavenns x panna nymo xefcrauren-
Hast ‘act KOMILIEKCHOTO ‘Hea:

1) (2-5) 42K; 2) (ole

3) (2x+1)-8i; 4) (8x5) 441?
(Verno.) Tipm kaxom amame x panna exe Mas

YEH DIAS
D Zero 9 -1-@-00

»

5. Vrasars, Kane Ho NAMNEIX KOMNIEKEHBIX «ICA Pas:
14104 5-61 VE +1; -0,5+36 241 -5- 860 12-135.

6. Hafıru eyumy komnnerensx uncen:

1) 5+4D+ (2430 2) (1451) +(6-70%
3) (0,5-3,21)+(1,5-0,81); 4) (1-21) 431
5) (34150 +(3-V5D; 9 (3-34) +(

Harn uponssenenne Kommaexcuux ance:
1) 2+3)4+5i); 2) (1-25-05
3) 3-2 (243); 4) (3-DG+);

5) (
Bunormurs neerana:
D) 20+D+3-78 2) (2+)-3-2) HH
3) A(+D+4i(2- 57 A) (8440 214 (2-7) 3%
5) 03420013 -1)+1-4V36

6) ¿H6-121)+ 7 14481).

Kaxoe uncno moxno npubasu x uucny 7,5-2V81, uroûu
ono crano:
1) neliersurensusien; 2) unero sn?

208 M
SS A,

ZtL S124); 6 (V2+8H (V2 +81).

10, Haïru gelicrmrensunie uncna x u y, ect

1) 9x+2pi=124 4 2) x-2yi=-1-V3i

O —Bes(y- 2nd

DARIA O) (ATL
UL Ynpocrurs axpaxenne (a, à — aelierasremmune uncna):

1) (a+8bi)+(a-5bi); 2) (8a+200)+(-7a—200);

3) (a+3bi)(a - abi); 4) (Ba + bi) (Bab):

5) (2+ 3ai)(Ba+ 20; 6) (Sa+ 4bi)(Ab + dan.

12, Haïtru nekernurensune anauenun x, OPI KOTOPHX "MENO 2
Gyxer nehernnrenmun:
1) 2-Gx%+2xt-5x% 2) (BLAN.

13, pu kakux neiiersirrensusx anauennax x u Y KOMDALKCHMO
una 2,=2°- Te + QU 23 = Yi +201 — 12 paru?

14, Heu zeiernurensusie snauenna x m Y, ecm
OP EE ES

gia a ee eee be

(8) oxasars panenerno:
Datacates Dana 8) (arpa (em
4) (21420) 423=21 +H(22 +25) 5) 240-2; 6) z°1=2.

§ 2. Komnnexcho conpaxenHbie uncna.
Moayne komnnexchoro «nena.
Onepauun BLIAATAHUA m AeneHus

ES. Kownaexono conpracemmue nena

Onpenenenne

Conpaxennum € unenom 2=a+bl nassmaercn Kommaexcnoe
uneno a bi, Koropoe oßoanannercn 2, 7. €.

2=a4bi E
Hanpumep, 3+ 11-341, —2-Bi=-2+51, I=-1.

Ormeran, «ro @-bi=a+bi, nooromy jus m000r0 kommaere-
Euro WHONA 2 Meer MECTO panexcTEO

Paseuctso Z=2 cnpasegnuso Torga 1 TOABKO TOPAA, Kora
2 aeliersnrennoe uneno.

Oliyer 2=a+bi. Tora 2-a-bi, n panenerso a +bi—a bi no on-
PETER PAREHCTRA KOMMUTEKCHRIX NCEA EMPANEAINDO TOrAR H
To Torma, Koraa b=-b, r. e. b=0, a aro u oomaaer, ro
2na+bi=a+0i=a — neicrsurennuoe sueno. O

Hs onpexenenua crever, “ro
Are

g2 209

TOMRTERERD COnPAKEHMEIG CAS. MOINE KONNAEREHOFO WC.
Onopau maurranan u nenes

2. Monym. Kommaexenoro “mena
Onpenenenne

Modyaem Komnrekenoro wena z=a+bi massızaerca «nero

Va?+0%, ne.
Izl-la+bil-Va?+B?. 7]
Hanpunep, 134401374475, |1+i1-VP 41-12, lie
Y
Hs dopuyau (2) eneayer, wro |21>0 ana moGoro Koumnese
moro «mena 2, mpuuem |2|=0 Toraa m TOn»kO Tora, KorAA 2=0,
7. e. Korna a=0 n b=0.
Toxexxex, WTO AMA MLOGOTO KOMUNEKCHOTO unena 2 Capa
amma JOpMy

lal=1zl, 22-127.
O vers 2=a +bl. Torna F=a-bi, u no onpexenemmo monyan
[Elia bite Va? + Ed)? = Va +07 lle

Haiiqem nponsnenenne:

22=(a+bi)(a—bi) =a? (bi) at bla. ©

3. Burnranwe NOMNIENCHBIX uncen

Onp«

Komnsexcnoe ueno (-1)2 naasınaercn npomusonoxoxnus

KOMNACKCHOMY ACT Z m o6osHauaeTes ~2.

Ecan z=a+bi, 10 -2=-a-bi. Hanpunep, -(8-50=-315

Jus 2106010 Kommnexenoro uncna 2 BunoAnsETes Panenerao

2+(-2)=0.

Buwumanue KOMDNCKCHAX cen DBOHITCA Kak onepams,
OGPaTHAM caoxenmio: JAN OGL KOMINENCHDIX uncen 2, HZ Cy
IECTBYCT, M MpHTON TOALKO OAMO, NMEO 2, TAKOS, WTO.

2422621 y
7. e. ypasnenne (3) HMeeT ToaBKO onu kopens.

TipuGanum x o6enn uacrsm panenerna (3) uneno (24), re

THRONONOKHOE uneny
+224 (car) 2, He), omyna 221 tl).

Uncno 2=21 +(-23) oÓMunO o6osnauamr 2—2,~2y u nan
wor PASMOCMbIO uncen 21 M 22.

Ecan = a +byi, 2¿=47+ÓgÍ, 10 pasnoers 2-2, umeer cae
ETES

(ay + bi) (az + bai) (ax 09) + (by — ba) i 4

Dopuyna (4) noxassipaer, “TO paswocrs KOMILIEKCHEX uncor

MOKHO HAXOAMTb NO NpaBnaas AEÏCTE © MHOrOUTENANL.

u

210) roses vu

ROMNNCRCNEE THERE

Banana 1. Hatin pasnocts (1431) (-4450.
D(1+30-(-4+51)=14+314+4—51=5-21. 4

4. Jlexenne kommrexcnnx umcen

Aenenue KOMUNEKCHBIX “UMCOA BBOAMTCA Kak OMEPanMa, 06-
pera ynocenno: RN mouse womens uncen yt 420
TEJER, npr nom QD, NAO E, AK, WTO 224 21
De no ypanvete ornocttensao 2 mueer TORK ona kopen,
Toop merca dem wncen E 27 M oDoaaneren

Kounnesenoe eno nernan aemrs na ny.

Iocanen, ato ypancemte 2,22 AAA anoura Konnaencanrz
‘ween 2, M 2240 HMeer TOABKO OHH KOpeHb, H Halínem 3TOT Ko-
im.

OVasionne 06e vacru YDABHEHMA Ha Zp, NONYUHM 22222

aña ne.
zl ae

Noryaennoe_ypapuenne PaBHOCHNBNO AAHHOMY, Tax Kak
470, u noromy 7,#0. Vunoxum oGe ero uacru Ha Nelicruntens-

we wueno 1; (samerum, uro |2F#0, Tax Kak 2220), nonyunm

lee?

Hrax, “ACTHOS KOMILIEKCHEX "ICA 2, M 2220 Momo

mira no popuyne
2 o

(a+bI (+ yD=1 nam (ar by) +(bx+ayli=1.
Hs mocneauero pañemcra noAyuaen cuicremy
(nr
bx+ay=0,

pas Koropyio HaxoaHM x=

TA

aye
Banerins, «ro a?+5%0, Tak Kak 220.
‘Taxus o6pasom, conn 2=a+bi, TO WHCNO, emy oöparnoe, mpi
sige? Bez,

1a +
a+b? at +b!
Bean 21-4, +0y1, 23=@s+bal, TO dopuyny (5) Mono mper-
us » mine
21011 bu Can 6)(@u Dai) adn + bib, Guy
CRC TT abst} abate
se Mai
RSR TORRE WC, MENTE TOMES wena,
Orea ua u nononun

w:

Ory Dopuyay MOXHO HE SANOMANATD, AOCTATOUND TONES,
170 Oma NOXVIACTCA YMHOXCHIEM HICANTEAA I SHAMERATAR Apo

Ha UNO, CONPIMEHHOS CO sHaMeHATeneM.

pa]
Hanpuep,

2431 _ 244+ 31462 _ 44 ty
1-21 T+4 5 ott

To onpeneneno YMHOKEHHA KowmaeKeHsix "HCE
BR cil) ink

MPA (Dei
Baitindd

-D=1,

Boo6me
PA m. eth
Hanpnwep, 3-10 59-8
a-

Barasa 2. Buunemere

A-20040,1_ 1404-21-28 1 _(G-0G-9, CD
Br tie sn ES

qua 8. Boruneaurs (148 Y,
en, em (1)

MSA Saxasa 4. Nokasars, wro AAA MOGHX ABYX KOMILTEKCI un
cen 21 M 27 CHPABEAIIO panenerno

la +22P+l23-22P=2(2,+1221).
[> Henomsoya evolicrso xomuxeKeuo conpaennsix ices, nen

la be? Ley 29? (2 12 GPRD HA 2 (e +2 43)

A SN
Yopaxnenua

16. Bauncar» KoMnAeKcHOE uneno, conpaemmoe € ern:
D 148 2) 8446; 3) 24505
4) -6-8; 5)-0,7-135 03-50

17, Haïtru MORYaD KOMnAEKCHOO snena:
1) 6+8i; 2) 8-6: 3) V3+ 4) V2-V 36

5) bis 9-2 74-3

212 rnasa vil
Komnnexcneie "mena"

18, Sanuears "CO, mporHBonoaorKHOe "ICA

1) 543i; 2) 4-25; 3) -3+ 4)-V2-7i.

18, Harm pastors konrinekensix uncon:
D @+4i)-(1+30; 2) (2-70-(5+20;
9 1+D-0-D; 4) 6-2)- (8-20;

5) 245)-(-1+6) 6) (-1-41)~(- 1-31);

D 02+2V80-(3V2-V30; — 8) (8V5-V30)-(154+4V30.
20, Buranenums:

D G-30-(14+204(2-05 DADO

3) B+2)-(8-7)-(5+61) 4) (2+30+C6-D-(-3+20.
A. Haïtru acruoe Komnnexemux uncen:

1 ar 2 im,
1D Te 2) Ta 3) 0 4 rt
‘a. :
) 146i’ Dat |
2. Boruncamre
(4-306 O+9G+),
» Tri 2 = ?
: 2
Menea auna
Bah aout, 2-0 +
ta Diet

28 Hañm monyan komnnexcnoro “NCAA 2, COM:
1) 2--5-2V65 2) 2=-9; 3) 2=-b; 4) zmcosg +isin à.

24 Ip ka neiierantensmex anauennax x m Y Koumnekennie
mena 2, =9y°-4- 1021 u 22=8y?+200 aBamorca conpamen-
man?

2 Iipu KaKom onauenmn x Konnaereume wena 2, =2-8-6) m
23=22° 461-2 ABAAIDTCH MPOTKBONOAOMEIME?

Peuurs ypannenme (2627).

BD MOD +2=—4- 7% 2) @VS-2V80-
3) (6-4i)-2~ 3-545 4) 2+ (5-V2)1= 6-1,

HY) ZB-20=1426 2) 2-3+20=
9) 2(1-8)-6=25 A) 2(-14-71

& Buena:

2) A +25); 3) (140%
5) 1+9-1-0%
2, (1 ay
YG.
2, Bunonunt» nehersm:

m,

s2 Was

Founnexcno CONpRKEHNEI AE. MORTE KOMNNEKEROTO Wena,
Omepausm aeraranıs u genera

(80) Pemmrs ypasnenne:
1) |2|-iz=1-2i; 2) 22481210.
Berner

dog A LU

1 84s 404 un,
pa

Moxasare, “ro [21 +291 +12,—29f=4e*, ecam [21
Brunet)
1) (+95 2) (1-9%

© nomommio panencrna (m+ni)(m—ni)= m! + n° noxaaars, mı
IPOMODCACHNE ABYX “NCEA, KAKO MI KOTOPMX ECTS chow
KBanparon ABYX NEAKIX UNcen, HBANETCA TAKE CYMMOÏ kur
patos ABYX MCABIX ance,

PE

1.
Hoxaseta, uno xonnzexcnoe wueno LE manner sion

so Torma u TOMKEO TorNA, orga |21=1.

§ 3. reomerpuueckaa uxrepnperauus
komnnexkcHoro 4nona

11. Konnzexcnan naoexoers
Heiteronzenuune wena reomerpirueck mo00paxtaoren roma:
sun smenomoï upsnoë. Komunexcnoe «utero a + bi monuo pecerar
Duimarı, ax napy noliernurenmnx uncen (a; 6). Tlootoxy corer
BENNO KOMILTEXCHAIe “HEA HIOGPAXAT TONKANIT IIOCKOCTA,

Tiycrh na DAOCKOCTH aanana upamoyronsnast CHCTEMA Koopa
nar. Kownxexenos uneno 2=a + bi nao6paxaeres roukoÏ nace
exa e xoopammaranu (a; b), u ora rosa oßoamanneren rol me Sr
soit z (pue. 104).

Tanoe coorsererane MEXAY KOMMTEKEMMIK «amenas H To
KAMM MAOCKOCTH BIANNMO OXHOIMATMO: KANROMY KOMMLTEKCHON
ancay a+6i coornerernyer OHA TONKA MAocKOCTH © 1oopamar
un (a; 6) m, HAOGOPOT, KaKOÏ TOWKE NIOCKOCTH € KOODAMHETEA
(a; b) cooraererayer onto xonexcnoe wueno a +bl. Hosrony exe
PA +KOMMAEKENOE "MONO? M eTOYRA MIAOCKOCTHO MACTO yrorpehis
OTCA Kak cumommst. Tax, BMCTO C10 +TOUKA, HIOGpEXANTE
wueno 14 do rouopar eroma 14 lo. MOxno, naupumep, exasen
<rpeyron e vepummnann » rowxax i, 1+i, -ie.

Tipu taxoft unrepnperammm neher-
sera a+0i, nsoGpaxaioren ToUKAMH
© Koopanarannt (a; O), 1.0. roman
ocu aGeunce. Tlooromy cen aGeunce ua-
Amar deucmaumervnoi ocu. Muero
ue mena bi 0+ DI waoOpanamıcs
rowxanm © woopromerann (0; 0), me. J a
TOTKAMIL OCH OPIBIMAT, OSTOMY DB Op" Pu. 10

221480 _rrana_ Vu

20th

ES

+bi
u
243i
“242%
iti
aa NE:
3
-3-21
3-3: > Faa-bi
Pac. 105 Puc. 106

amar uasspaioT xuuMol ocuo. Tipu srom rouka © Koopaumaranın
(0; 0) ofoanasneres bi. Hanpumep, rosa (0; 1) oGomasaeren i, row
sa (05 -1) — oro -i, roux (0; 2) — oro rouxa 2i (pue. 105). Ha-
‘azo koopzmar — aro rouxa O. TInockocrs, ma KOTOpOÏ HaoGparxe-
TX KONNJICKOHBL THICTA, HASMBROT KOMNACKCHOL NAOCKOCMBN.
OTMETIM, wro TOWH 2M —2 CHMMeTPHYHEL OTHOCHTENBRO TOM
fit O (masaa Koopunmar), a TOUKH Z m 2 CHNNETPIME OTHOCH-
emo nehersureasnoh oca (puc. 106).
Olivers 2=a-+bi. Toraa -z=-abi, 2=a-bi. Towxn 2 m —2 nme
‘er koopaunaruı coorsercrsenmo (a; 8) u (—a; ~b), cuenosarenono,
on CHMMETPHUNK OTHOCHTEMENO HaNasta KoopAMHaT. Touka 2
meer Koopanarst (a; -Ö), caenoBaTeasHo, Ona cmunerprrma
axe 2 ornocnrensno xeñtoremrenpmoñi ocn (cm. pue. 106). ©
Koumnexcnoe unc 2=a+bi MOKHO 190ÓPaIOTD, DEKTODON
Oxauanon 8 rouxe 0 u KOHMOM B TOUKC 2. Dror BeKTOp Gynem 060-
seauar Toil ake Oynoh 2, Ama ororo nexropa parna [2].
Uneno 2, + 2g HIOÓPIRACTOA NCKTOPOM, NOCTPOCHMBIM TO TIPa-
MAY CAOMEMHA BEKTOPOB 21 If Zp, A BEKTOP 2-22 MOHO Mocrpo-
Aro Kak cymaıy weropon 24 M —22 (puc. 107).

Puc. 107 Puc. 108
$3 W215

TROMOTDANEGHAN WiTOpRpETAUNA KOMRNOKEHOFO MACS

2. Teomerpmuecknh embicn monyan KommteKeHOro uncaa

Busch reomerpieckutit cueca l2l.

Ilyer» z=a+bi. Torna no onpexenemmo monyas |2I-VaTHP.
Dro oamanaer, «ro |2] — paccronune or TouKH O 0 Ton 2.

Hanpumep, pasenerno |2|=4 oamaaer, ro pacctosuue or
roux 0 a0 oies z pano 4 (pue. 108). Tloxrony muoxocrno rex
Touex Z, VOBACTROPATOMNUX panencrny |2|=4, annsercn OKPYEHT
ero € nenrpom » rouxe O paxuyca 4. Vpasnenne |2|=R anner
CA ypamensen orpyacnoern € KeHTpoM m rouke O pajuayca A, re
R — sanannoe nonoATenBHoe uncno.

3. Teomerpuuecrmi emkien monyan pasmocrn Komnaenenun
sce

Buscuus reomerpusieckuii emsien MORVAN pasnocr KE
xomnaexentx uncen, 7. €. [21-23]. Myers za tol, z= dy +by
‘Torna |z,-24|=1(a,-a,)+(0, -011-Vía, a)? + (6, 09.

Ma xypca reomerpim MABECTHO, “TO 70 "MONO PABRO paccros
mo mersay Touran € Kooprumarama (ay; bi) # (as; D).

Hrax, 12, —2el — paccronune meray TOWIAMH 2, M 2.

Hamplmep, paccrosune Mexxy ronkanım 1 u —8+31 passo

11-(-3+301=14-301- VECES 5.

Hoxamen, 110 |2-20|=R — ypasnenue oxpymnocnu € ner

pou m rouxe 2, paanyca R, TAC 2 — IONAMNOS KOMIACKONOS sc
10, R— sanannoe MOXOHTOAMNOS UMCNO.
O Tax ax |2—29) — pacorosinne meray TOUKAMH 2 H 20, 10 uu
KECTRO ncex TOUCK 2, YAOBIETROPAIOLUKX ypannıonmo [2 Zol=R, —
TO NHOKCTHO BCEX TOUEK, PACCTOMNNE OT KOTOPHIX AO TOWN 2,
pasno R. @

Hanpusep, |2+1]-2 — ypasuense oxpyamoeru ¢ uerrpox +
TOUKe -i paniyca 2, Tak Kak MaHHOe ypaBHeHHe MOHO da
» ome [2—(-)|=2.

Banana 1. Myers 21, 22 — pasuste rouki KoMNAeKeHO! mos
Koeru. Horasaro, 70 |2—2)|=|2—29| — ypammenne mpamoñ, nep
HCUAMK Y AMPuOR oTpeaxy, COEAHNMOILENY TOUKH 21, 2), M Moro
smelt sepea ero cepenuny.

D Tax xax [2-21] — paccromme or

TOMKH Z AO TOUKM 23, a |2-2,| — pac-

crosHe or TOUKH 2 AO TOUR 23, TO

MHOKECTBO BCEX TOI, YAOBAETBOPR-

roux ypanmemmo |2- 211-1222) —

910 MMOXECTBO BCeX TONEK, paBHto-

yaneninix or anyx roux 2, M 23. À y]

Hanpmmep, |2-2i|=|2-1] —

Ypamnenne npantolt, nepnennueyanp-

Holl onpesky, conimmomeny TOKE

2,~2i u 2,=1, m mpoxonmmel nepes

ero cepexany (pue. 109). Pure. 109

216) asa vu

Komnnercnue wna

Baxaua 2. Haltrm mnomecrao ro-
Sei koumnercnoh MAOCKOCTH,, yaon-
eraopsionix yenommo 1<|2+ 28 <2.
D Venom |2+2i1<2 ynonnersopn-
br mee Town, xexamie muyrpt
mayra pamnyca 2 e mempom 3 none
2,=-2l, a yenommo |2+2i|>1 — nee
owen, neraue BNE Kpyra paxuyca
1e wewrpon m rouke Zo (pie. 110).
lexoxoe MIORECTBO ToweK — Kon
10 MEKAY OKPYHNOCTANN pay

Tx 2 e o6uuim memrpon n roue
aa dm

Ps a

% Ha Kownaexcnoii naockoeru nocrpours ron:

1) 3; 2) 4; 8) -2; 4) Gl; 5) Al; 6) -26; 7) 143% 8) 245%

9 -3+4 10) -144 11) -1-8% 12) -4=6 18) 1-46 14) 3-88.

32, Noorponmn oXpyaxnocr»:
1) lal=8; 2) [21-55 3) l2-21l=15 4) lz+811=2.

38, Perm» ypannenm
12487-84125 2) 42-2--94 107.

3 Haltrn pacerosume mexay roman:
1) 6 n Bi; 2) Um ~2i; 3) 1410 2431 4) 3-2 m 1-40.

0 Hakrn mmoecrno Touex KOMNNEKCNOÑ nnockoern, ynonner-
Bopamouux yanopııo:

D lzl=5; 2) l2-
3) |243il~ 13 4) l2+21-11=2
5) l2-2l=I2+ib 6) [2-1 él l241 44).

AL Halim moxecrno ToweK KOMNACKCHON naocKoeTH, KOTOPOS
saxaerca yenopnem:

1) l2-41<3; 2) [2-+Bil>4; 3) 1<l2+21<25 4) 2<lz-5il<8.
ZI Pewurs cuereny ypasuenuii:
Dlls, D (0-08 (40,
182 + 9] =152 + 101; ETES
Mise, aro onerous pan

fe

Izi=
xe umeer pement.

Peur enereny ypasnennit [|

ss Dar
TEOWeTPIRIECKaR UNTEpNPETAUNR KONNNERENOFO Wiehe

2-8

7-8

§ 4. Tpuronomerpuueckan dopma
KoMnnekcHoro 4ucna

11. Apryment nomnzerenoro unena
Onp

Apsynenm xomnaexenozo vucaa 240 — ao yron y men
HOROKNTENBUEIM Haupanennem AECTHATENMOÏ GK Be
opom Oz (pme. 111). Dror yrox eusmaoren nonoxwrexsEss,
ecax orcuer Beerca nporas uacosoll crpenku, m orpauaren-
HN MpH oreuere no unconok erperKe.

Conse mexay neilerentensnoli 1 MHENOÏ Nacramı Kounaest
moro nena 2=a +bl, ero moxyxem r=|2| u aprymeurox 9 sue
PKACTCA caenyıonymas hopwry-ran:

a

Pos. 1th

Apryment xomnxerenoro «mena 2=a+bi (2%0) moxno ne
Tu, pena cueremy (2). Dra cmerema umeer Geckomeumo NOT je
WeHnt suna 0=90+2kr, rae CZ, Oo — OAH ua pewenui euere
mu (1), 7. €. aprymeur KOMDACKCHONO “MC ONPOACHAETER Hess
noanasno.

Has naxompenna aprymenra kommnexcnoro wena 2-a+ti
(220) momno nocnonnaonarncı dopmyaok
tro. w
Tipu pemennn ypapnenna (3) myxmo yunTstBaTe, B Kato!
sersepri naxonuren TOMKA 2=u4 bi.
Sana

Haiirw sce aprymeurs: kommaexenoro wena 2:

1) 20-25 2) 2414118.

D 1) Hncno 2 nemnr Ha orpnyateasuoli UACTH MHMMOÏ och, Y. €

ox na aprymenron roro umena panen — À, a MHOxCOCTDO 201

aprywenron uncer ana g=- F4 24m keZ.

2) To dopmyze (3) naxoans teg=—V3. Vanrsisan, uno mem

em 0 mopoñ serseyra, nonysacn q= 2 +2, hed. 4

248) race vi

ROMANE wena

2. Bammes KommaeKenoro uncaa 5 TPuronomerpwueckoii dope
Ma pasencrsa (1) cnenyer, “To m1o60e Kommnercnoe unten0
2=a+bi, rue 270, npenerannaeren B Bune
z=r(cos@+ising), (4)
ine r=]z|=Va?+B? — monyas KOMIACKCHONO nena Z, 9 — ero ap-
Hymenr. Banner, kommaexcnoro «nena m Bune (4), rae 7>0, masia
Bor mpuzonomempureckou @opmol KOMRAEKCHOZO yucaa 2.

Banana 2. Banncars n° rpurouomerpuueckoï popme Kom
nnexicHoe “mic

Da-

2) 2¿=—cos F+isin 5.
D1) Mipnuenan dopmyay (3), nonyuacm tgg=1, orxyaa 9= E,
Tax are mouxa — 1-4 er u perno Werner. Vuuranan, «ro
aj=V2, moon 21 =V2 (cos SE + isin).

2) Tax Kak TOUKA 2, HIT BO BTOPOÑ YeTHEpTH, TO, MEMOS

ES

ya Qopuyam npsexennn, monyuaem —cos 4 -cos S*, sin
& 6 à soin SE

sin À, u nogromy 2, =008 E +isin E, à

Nyon. Kommmexenue wena 2, u 2, SANKCama E Tpuronomer-
pusecxoR ope:
21=r,(c059, +isingy), 29= 72 (cosy, +ising,).
‘Tora panenerno 2)=2, umeor MCTO m TON H TOMO B TOM
same, Korma 71-72 01=92+2km, kez,
Osparun maine na TO, To Buipaneun

~5(coeZ-+1sin), (cos tain),
V2(sin 3 +icos 32), cos E +isin &E

Le ADAMIOTCA TPHTONOMETDHUECKUNN POPMAMH JAIDICH KAKHX-NH-
fo xomnaexenmix uncon. EY

Ynpaxnenun

46. Hafirn nce aprymenrui konmnercnoro uncnat
De 2) 2=-45 3 2
4) 2-3 5) 2-14 6) 24346
7)2=2-2 8) 2=1-V3L

46. Samucara » Tpuronomerpuuecroñ hope Konninekenoe uneno:
1) 2-8 2) 2=-L 3) 2-34.31;

4) 2242131 ze-1-\8i; 6) 25-58
D 2B (co

8) 2=-cos 7 +isin 7.

44 BER)

“Tparonomerpmiccran Gopwa ROMINERCNOrO “mena

47, Sanwears » anreGpanveexoit ope Kommnencnoe “meno:
1) cos $ +isin 5; 2) 8(cos2n-+isin 2x);
an
4

A E

2) con sin 3); 4) 4(con % sisin 2}

48, Banucar» » Tpuronomerpiueckol dope xornzerenoe wei

1) cos 5 ~isin 55

2) 12(~cos Z +isin 2);

3) VA(-cos 4 ~isin 2);

4) 3(sin$ +icos 2).

3,
VE moxno mupaanTs uepea TpuronoMerpivee

49, Aneno 3

1
2)

3)

4
5) 5 - Ft cos 5 isin.
Kaxas us aux ganuceit anaserea TPHrOMOMETPHUECKOË hoy
Mol KommeKcHoro “mona?

50. TIpenerannrs TPuTOMOMOTPMUCCKOM dope KOMMACKEHK

ai, g Genes
ar ë 3 à
(BL) Hoxasars, wro ecan |21=1, ro zt.

(52) Banucaro » tpuromomerpuueckoñ dopme konmaexene

neo:
ya 2) ~5(c0s 40° + sin 40%);
3) Lteconastsina, O<acSs 4) 1400822 rain E,

290 rnasa vi

Kownnexenue mena

$ 5. Ymnoxenne u nenenne KomnnekcHbix umcen,
3ANVCAMHBIX B TPUroOHOMETPUNECKOñ dopme.
@opmyna Myaspa
BIC nous rpuronomerpuueexoi op yaobno uaxonırm
IPOMIDEAHMO M “ACTOS KOMILNEKCHEIX UILCEA 2, M 22.
Bureau nponsrenemne:
221 (COS Qu Hisin 91) -r2 (cos, + 18n pa) =
= nr (cos 91 00802 —sing, Sin gg + (sin 9, COS Pa + COS 9, sing)
= rare (cos (6 +92) +isinto, +9).
Mirar,
2122 = rar (008 (1 + 94) +isin (Qu +92). Mm
Mo gopmyant (1) enenyer, «ro np nepeunomensi KoMNAeKCHEX
‘wee HX MOAYAH MepeNMOrKaIOTes, a APTYMeHTK CKAANLMAIDTOR.
Bermmenum uacrnoe (2,70):
21 _ raleosortising 7
22” re(cosca-+ising,) 7 ra

(cos, 0089, +sin q, sing, +i (sing, cosg2~cos 9, sin g,)) =

(c08(91~ G2) + Sin (9 ~ 92), 7. e.
à
Ma opuynm (2) crexyer, TO MOAYIb “acTHOrO ABYX KOM-
mnekenux ACER Dänen MRETHOMY MORYAEH REANMOTO 1 REANTEAR,
N pasnocrs aprywenton JEAMMONO N Aeanrenn Annneren APrYNEN.

aqua 1, Han nponanenemne
V2(cos 2 nine) -2VA (cos min 2

D Tlo popmyne (1) nmeew

\2 (con E +tsin-25)-9v2((08 Zain E,

(eos pr ~2) + isin (91-92). @

En

2 *isin ae tsi 2
VE 817 (co (2 + E) +tsin( ))-
~6(cos E +isin E <

Ganas 2, uremmrs nexonne
oon 5E + san $2)

3 (cos #4 + tsin 25)

gs Mon

Yuonoxenne w REREHIG KOMAAEKENUN WAGER, SanUCaRMUX
6 TpHroOMeTpHYeCKOH dopme. Popuyra Myanpa

[> To popmyze (2) uneen

6 cos
or

=2(cos 7 +isin

4
ma Mo dopmyan (1) caenyer, uro
(cos +ising)? = cos29-+isin 29,
(cos +ising)" =cos 39 + sin 80.
Booôme aan moGoro NEN (u man ncex nEZ) cnpanezame
Gopayra
(cos9+isin gy" =cosng+isinng, o
koropyio naasınalor Popxyaol Myaspa.
Tan nü creneun KowunexcHoro uncna, sanncannoro » mr
ronomerpitueckofi dopme 2=r(cos9+tsing), Cnpaneamma opus
22 7 (cos np + isin ng). @

a

Banana
D Mo dopmyne (4) nueem
(2(cos 3 daim 5)"
CRUE

=32(c0 3 +isin E

Bosneern » erenens (2(cos 7, +isin %-))

Sanana 4, Burner a
> Tax war

141V8=2(c08 3 ain 3), 1-
70, npumenas Qopuyay (4), naxonm

(141 ¥8)6 = 2% (cos 2n + isin 2m) 2°,

(IND eos (mis)
Caexomarensmo, LD 2% 16, 4

ay

Sanasa 5. Pent ypanmenne z°=i.
D> Tpexeranus “nexo | » rpuromomerpumeokoii hopne: = cos ¿+

+isin E. Tiyor 2=r(cosg + ising). Torxa ypasuenue <= impar
ma ri(cos29+isin29)=cos F-+isinZ. Orcona maxon rel,

29-F+2xb, me. p= 4h, rac keZ.

LP crane vi

ROURREREHEE HERE

Taxıım oOpasom, nee pemenus ypanenun 22-5 npeneranns-
vores n une 2=cos(% + hr) +isin(% + Ar), me keZ.
Banernm, ro na ucex wermux À ora dopuyna Auer Kom
amoo Fri Bu,
a naa ncex Henermux À — Komnaekcuoe “seno
Sn V2_ ¿M2
ae

2,708 SE + isin SE

Hrax, ypaanenue 2°~{ meer ana Kopnat 24,2~

Sayaua 6. Samicats dopmyms nam cosdo x sin 49.
D Mo dopmyne Myaspa
(cosp+ising)!=cos 49 +isin 49. (5)
© apyroit cropoms, nenommaysı hopmyay
(a+b) = at + 40% + 60°04 405404,

ronyuaene
(coso +ising)!=cos*p+4cos* p(ising)+6cos* (ising)? +
+4 coso sino + (sino). (6)
Conocrasana mpansie “acti pasewcrs (5) u (6), nmeen
cos4g=costp—6costosin*o +sinto =costo—6cost9(1—costo)+
+(1-cos*9)=8cost-8cos*p+ Li
sindo=4cos*psing—4cososin*9=4coso((1 -sin’g)sing-sin’g)=
cosp(sinp-2sin' 9). dm

Yopaxnenna
3. Han npononexenne nomnnexcnux sce
1) (cos 2 +1sin 2) (cos rain):

2) 2(cos % +isin 4) (cos 5 +isin 5);

3) VB (cos +n 32) 2(c08 5 +tein 2):

© 2208 418in 3)-4(cos(- 3) Hsin(-3));
5) V2(cos 55° + isin 55°). V2 (cos 35° + isin 35°);
6) (cos7+sin7)(cos8+isin3).

3. Haiirn wacrnoe:

con + isin E 8(c00 5+ rein)
DEEE, » ( = :
PERTE 2(con ón,

ss He
Yunoxenne u AENENNE KOMIEKGHUK WAGON, BAMINEAHMEIK
2 TpuronoMerprectoli dopme. Dopmyna Myaspa

SEE), censoring,
ara) * (cos 159 +¿sin16) *
ehr, y eta,

‘VB (cos 50° + isin 50°) cos2+isin2"
2) (2(cos 5 isin 4 )Ÿ5 2) (co À 41

3 (2 (cos Etain 2) (

9 (3) cm3)

156] Bsmomurs neiterana u sanncar» peayamrar a anreGpawuec-

3)

koi Pope:
pS

(coo inf) +80
i®§ + VS ys,
LA (or
57] Sanucars » rpuronomerpmcexo opme peayaanar néon

2(cos 7 +i 3)

2) 2--3(cos $ +isin E) (cos 5 ping,
al REN SE

€ noMonyuo Tpitronomerpieckoit PODIA Kominexenoro ‘He
aa peurs ypannennte:

D = 166 2) Pai;
8) =2-21V8; MI.
59] Tipenerasııms » tpHronomerpHueckoit dopme «meo:

ra 2) sina+é(1-cosa), 0<a<

1) sing +isi

a
teint um
8) (tg 1-0 De

160) TIpumenan dopmyay Myaspa, nokasarı panes
1) cos 2 ost a sin? a; 2) sin2a- 2sin a cos a;
3) cos3a=4cos*a-3cosa; 4) sin3a=3sina-4sinfe.
ne solis mines

(ses re Tiens,

Lie) ~ 1-itgna*

EN.

224) rnasa vu

Kounnakcne wera

Haiten cysoe
1) sinx+sin3x+sin5x+...+sin(2n—1)x, zerh, REZ;
2) cosx+cos8x+cos5x+...+cos(2n-i)z, xenk, REZ.
ES] Jokasars pasenerso (x#Rk, ke.

sin Æ sin 4
D sinz+sin2x+...4sinne=——2 2;
sing
cos 2. sin CEDE
2) 1+cosx+cos2x+...+cosnx= —? 2?
sing

$ 6. Ksanparnoe ypasuenne c kOMNNEKCHbIM
HEN3BECTHLIM

E Paccmorpum ypasnenne 2?=a, rae a — sanannoe neiicren-
ramos ancio, 2 — Nenanecrnoe
‘Bro ypannenne:
D meer out Kopenr 2=0, ecan a=0
2) meer ana nelieramemimax: Kopin 2,2 = + Va, econ a> 0;
3) me mueer nehoranenunuix Kopueñ, een a <0.
Hanpumep, ypannenne

ay

we auger nehernreaumsix xopmet.

Tlenanen, 70 ypamuenne (1) ser nen KOMDAEKCHEIX Kop:
as, u naiiqem mx. Tloncranısıı 8 ypannenne smecto -1 uneno i*,
?, orxyna 2212-0.

Tipumensa dopmyny pasnocrH KBARPATOB, PRINOKHM nenyıo
wer» NOCHERHErO ypaBnenHa Ha MuoxHTeAH: (2-1) (2+1)=0. Tipo-
szene Paso MY AI, xorna XOTA ÓM OH Ma Muorcureneil pa:
an. marrant 214 2,071, 370! spaamemne (1) mue.
ei vom geet

Ananormui NOHO nokasas, TO ypuuene

Baa (2)
qu a<0 rue mucer naa nounzexenux xopus 2,,.=+(Vfal-

Hanpunep, vpamienno

@)

ineer ava KOpHA 21,2= LV]
To anaxornn co exyuaen à > 0 Kop ypannenus (3) gannen-
sur a anne 22-4 V-25 . [px stom euwraeres, “To
V2 25 VÍ 251 56.
Booóme ecnn a<0, ro Va onpenenaercn copmyzoit Va:
Hanpunep,

val.

se W225

ROSRPANROS ypaoneHne © KOMNAEREHUM MONDE TIM

Taxoe cornamenne ynoGno rem, wro Aaa moboro neiierumen-
Moro a KopH ypanmensa

a w
Momo naxoJrTs no dopuyne
Bat. o
Bem a #0 (a>0 nam a<0), ro ypannenne (4) cer re par
awaisx xopna. Tps a=0 ypanneune (4) nmeor ons kopen 2=0,
3 atom cayuae rosopat Takxe, TO Ypannenne MNeer na pario
xopun 21.2=0 man oanı Kopenn KpatHoeTH JUL. Dro uacro VIE
HO, MANPIMEP, AA TOTO, TOG BO Bcex caysaAX Onna enparer
Anna reopena Bera.
Ormerum, «TO Teneps ana m060ro nehcrnurensmtoro a expe
Benanpo parencrso a
(ayaa. ©
Brexemnoe monaTne KopHA na orpmuarensmoro “mena moon
ager sanucars Kopin moGoro KBanpaTHoro VPapnenma € nel
emmm woodpinmnenrann
az? 1b2 10-0 ©
no monecroñ o6meÑ dopnyae
(are
abe Voice | o

2a
Sanat

Zur

1. Pentre ypannenne
2?-162+65=0.
D To dopmyne (8) naxonum
= 164256260 _ 16
2
Te. 2-8+1 28-14

Hirax, upu moGux xelleraurensuux a, b, c, 470, Kopun ypas-
Henna (7) mono naxounr» no popmyxe (8). [pu arom ecan ae
xpumunant D-b?-4ac, T. €. noaKopennoe Bupaxenne B dopuyae
(8), nonoxurenen, To ypaskenne (7) uMeeT ana neicrnutensuu
pasamunstix kopma. Ecam D=0, ro ypannenne (7) mmeer one Ko
pers (ama papınix). Ecnm D<0, ro ypannenne (7) umeer nna par
AMUNMX KOMNAEKCHMK KOPMA.

Orwernm, uro m saaue 1 Kopnn Knanparnoro ypanennn ar
AmOTCa comparen.

Bameuaune. Kopun Kuaparnoro ypasuenus c aelicru:
CIMA Koadibnunentantn m OPATEMNNN JE PM TON
ABAMOTCA cOnpaKeMHNNI.

Kounzexennte kopun KBANpATNOrO ypaBuenna oÓxazator crt
cman, anazornumunat wanecrnmm nam cnoficTanst nehermnen
mix kopneit,
aaa
2ecrmremmmnnt xoopbuumenramn, HMEONICE kopen 2,

2260) nava vu

Cocramwrs npunenennoe KBanparnoe ypaBHenue €
1-8.

Komnnckcmae amena

D Bropoñ xopens 2;
To popmyxaw Buera naxonım

p=-(+2)-2, 9-2:

ormyaa meKomoe ypasnenne 2*+22+65=0. 4

Sansa 3.

Paanoxur» Ha MHOKKTENH KBMAPATHMI TPEXUIEN
28182482.

D Kopuama xsaaparnoro ypamenna 2*-182+82=0 snamoren
‘wera 71=9+i, 23=9-1. Cacnovarensito,

2 1824+82=(2-9-H(2-941). 4 A
IA Banana 4. Per» ypannenne z*=3+4i.

Divers 2=x+ly, rae xu y — menonectuste jelicreurensume
era. Torna 2%=(x-+iy)?=x?—y?42xyi m namnoe yparnenne

way? + 2xyin 844i,

OTCIORA no onpeneneHino panencTRa KOMNACKCHBIX “CEN nO-
aywaen enoreny
Pos,
2xy=4.

Hañigen neiicrsnrensunie peuenun sroli cueremsr. Ma stopo-
0 ypannenns MAXOUIN y

E, noneransan aro wupanene ana y

3 ueppoe ypapnenme cucrem, noayyaem 2%
#-38-4=0. x

3, orkyna

Pema 970 Guwuaaparuoc ypannenne, monyann a BEN
E, ne. x24 nan x
etBMTEBEBIX KopHeit, a ua ypaBmenma x*=4 naxonum x,=2,

2. To Gopuyze , Va=- 1. Cnexonarensno,
y HZ, 2
Omer. 2,2=1(2+1). 4
Banaua 5. Pemur» ypannenne 241.

D Mepenecem equmuuy B JeBylo HACTI, CO 3HAKOM + 10 —
1 pasnomam NeRyIO “IACTE HA MHOKHTEM HO POPMYzIC Paakocrit

uy6on:

1. Ypanmenne x*=—1 ne meer zei

23-1820, @-De+2+1)=0.

1, a mo ypanrenun 22424 1-0 maxonun
a1ivied | -14V8 14, x8
2 2 Cer es

Omer, ami, an ii, an

Orciona 2,

22,

se Her

ROSAPNNOS YpaBHGNNG € KONNNEKEMUM NODOS

Ynpaxnenna

64. Peu, ypannenne:
1) 2*--16; ye 3) 2°+0,36=0;
4) 252°49-0; — 5)2*-16=0; 6) 2*-81=0,
65. Burunenums:
1) v-100; 2) v-0,25;

Per» ypasuenue (66—67).
66.1) 2*-22+10-0; 2) 2*4+22+2-0;
3) 22624130; 4) 224824170.
67. 1) 42?-42+5=0; 2) 92°+182+10

3) 2-42+1=0;

4) 2+2246=0; 5) 2°427=0; 6) 228,

68. Cocranur» npuuegennoe Kragparuoe ypasnemne c geler
TensHEINH KosppHRMeHTAMM, UMeIoUee JAMBE Kopest,
MC nponepur» orser, pers nonyrennoe ypannenue:

Damit DEN Li

Dad: Marne
69. Pemurs ypannenne:
1) 21-322-4=0; 2) z'+1522-16-0.

Paasoncurh kuanpamumli rpexunen na mmoncurent (70-71).
70. 1) 28-4245; 2) 28442418; 3) 2442244; 4) 2-61.
TA 1) 42% 14245: 2) 162-8224 17
3) 252+ 502426; 4) -2°+102-26.
(72) Cocravurs wpanparuoe ypasnenne ¢ nelicrpnrensammn 100%,
Ouneurawu, uneromee Kanal xopembs
Da
[73] Pour ypannenne:
12-5412;
DES

§ 7. Ussneuenne KkopHa us KOMNNEKCHOTO
uucna. Anreópanueckue YPaBHEHMA

1524 Hlepeñnen x omepanın nanneuennsn Kopnsı JAMON Creen a
KOMNNEKONOTO unex.

Uneno z wasninaeres kopnen cmenenu n us wucaa w (obs
uaerca Vw), ecau 2"=w.

Ma nunnoro ompereneunn BMTEKACT, 170 KaA0E peuenze
Ypamuenun 2" =w spaserea kopnen crenemu n 19 “mena w. Apr
CHINA COMM, JUTA TOTO UTODH MILACU KOPOMA erenenn nu wc
Ja w, AOCTATOINO pers ypanmıenme 2-1.

228) frana vil

Kommnescnae wera

em w=0, To mpn moGon n ypanmenne 2"=w nneer onto Ht
TomKo onKo peiuenne 2=0. Ecax 10%0, To m 2#0, a cnenosarens-
HO, u 2 HW MOXHO NpENCTAENTE n TPHrONOMErPHUCEKOÏ dhopM:

2=r(cosp+ising), w=p(cos8 + sine).
Vpapueune 2"=10 npuner mix
7*(cosng+isinny)=p(cos0 4 isin0).
He KOMILCKCHMX unena par Tora M TONLKO Tonga, Konna pax
EU HX NOAyaIL, Q APFYMEHTEI OFAHIMOTCA CAAFACMBIM, KPATHEIM 2x,
Caenonarensno,

ap u ng=0+ in,

rap ng E, pez,

Hrax, pce pemenua ypapnenma 2"=W MOryT ÖBITE JADMCAMEL
canin pas

2 Vp(cos

hath de dis le

Benson Xeno, mpunaDas » noryuennoli @opuyne «ncay À eno

came ome OF Bunucanuuin (ESO, Le 2 wee 01), Mi

o aras ana ea Hop, ma bos

pd

2,= (cos(9 420) +ésin(®+2n))= Y3(cos © + sin ®)~ 2.

94208 | join

osm a

Tox oépason, som 100, 10 eymeerayer pomo napnel ere
teni n M3 WMCAR 10; Boe OH conepxaren B popmyxe (1). Bee Kop-
o à mar name ora op
pun Upps, rares cre, eo we À
Grand cares oo Sommes trea, annee opa
Ayromuma, amennnore honpnnocts pannven 17 mera =
Serge
WE, canon Vi me meer oxmoonatzro euer. Horton, ya
o a oat oa
Sharon noxpasyuenaeten, Hanpunep, ucsomaja sanıca YET,
a o
EAS

Banana 1. Hañirm sce auanenna V-16.
D Banırurem mero —16 8 Tpuronomerpmueckofi dopme:

PON

67 20
VIENEN KOPRA Y KOMMAEKEHOFO «wena.
AnreGpanvecnie ypasnenun

Dopuyxa (1) » mamen cayuae naer
21-2 (co0( #204) 41123208),
0, 1, 2, 3.

Cnenosarensno,
20=2(cos 5 +1sin 2) -V2+1V2,

rue

= 2(c0s 8 rain 28) (22112,

4. a 2
cos IE +isin IT) V2-iv2,
23=2(cos SE +isin 52) ——y2—iv2, a.
ES an ‚2.
24=2(cos stain #8) VEZ,

Ae
Ha pucyuxe 112 ws06paxeunt nce sermpe amawemia VC 16.
Touxu, coorserersyiome UMCAAM 2), 21, Za, 23 MAXONATCA B BED
IUMMAX KBAAPATA, BUMCAMMOPO B OKPYAHOËTE Pannyea 2, € nen
pow n rouxe 2=0. 4
Banava 2. Pemurs ypasnenne 2°=-1.
D> Tipnmenaa popuyay (1), rae p=1, n=6, 0=x (c. 218), monyunen

42K, go H=0, 1, 2, 3, 4, 5 (pue. 113),

am

Caeyonarensuo,

6
os S+isin E

z0=cos E +tsin

cos 32 y isin BE
zencos $ 4 isin $

Puc. 118

Eau npmenennoe Kpaaparaoe ypasnenue 2*+pz+q~0 useer

Kounaexonue none, ro samonoR 2-4-2 ero wo

pme x mony Ve F9, rae m mpañoï nach panne
et ee

Hapnekan Ksanparasif Koper» M3 3Toro uncna (KOTOPELÍ Boe
pica ood ce apa via om ee
D man alar Une varon a main oper seo ja
men (roan Rey open

25000) rasa vu

Komnnoxcnue mena”

BooSme cyuectayer Tak HAIHBALMAR +OCHOBMAR Teopena au
cebpuis (ce noxasarennerno B Kypce HIKOANOÑ MATEMATAKH He
pecemarpuBaetca), KOTOpax raacHT:

Ypasuenwe n-H crenenm anna 2" +62" tee" 74...
, FRE C4, Cas cn gts En — KOMINEKCHEIE
UCA, HMeeT XOTA GW OAHE KOMIJIOKCIMÁ KOpeH,

Dra Teopema uNeeT nanoe cxencran

JhoGoe ypapmenme n-H crenenn (n>1) c KoMnexcunmn
xosQQunuenramu mmeer n KOMIMIEKCHLIX KOpHEË; mpu
DTOM KMS KOPOH CUMTACICA CTONIBRO PAS, KAKOBA ero
KPATHOCTE.

Hanpumep, ypannenne 2*-522482-6
miexensix KOPHA: 2,=3, 22141, zum 1-100

umeer rpm kom

Ynpaxnenna
fone
Ko UCT. 9 Sa: 4) ÍA
ya fs VE Wer.
Peur ypasnenne (75—76).
Bnz+s1-0 — 2)87-27=0; 8) 2=k

ir ace anauenns Kopun:

92 5) 22-242 E
By 22--16185 2) 862% 13214105
RETA

ER Perm». Kaxparnoe ypannenne © KONTACKCHAMH Kode
enranen:

1) 24 (2-6) 212-610; 9 2-2(14D2+9+21=0.

18. Haltın snasenus x, mp4 xoropurx gelicrautensHas ACTE KOMU-
xexemoro uHena panna 1:

D (E+3)+25 2) (eH 1)—4i
3) Gx-1)-75 4) C8x-8) +t.

19. Halten ananennn x, mpu KOropux nellerantensmast Mac, KOM
ILICKCHOTO unena PUBIA ero MHHNOÏ MACTH:
D (++ 2) Bx-8i
3) 04-25 4) ~1 4 21.
Bunonnurs xeltorena (80—81).

M D 2+303-20)+(2-3(8+20; — 2) 9+ 51-@-4N +30;
3) (6-130(8+130; 4) C1+V7DC-1- V7.

__231

Tapas x mnane VI

104 Aus 5-4

ud Du a
lu, Set
DA % ases"

82. Bunomurs reform nan NOMINA wnenamm Tome
iowerpuaeexen Gone:
1) 3(cos 130° + isin 130°) (cos 140° +isin 140°);
2) (con 5 +-1sin 88) 1 (coë(= 3) +tein(- 45):

CoR50° sn 50" 2(co0(-T
y 2(cos20"+1sin 20°) * 2 V2 (00s

Halim monyay koumnexenoro «men
1) 15% 2) ~214; 8) ~5 4 2i; 4) VB~is 5) -1-4i; 6) VIT +5.
BanncaT» » anreGpauseckoi (pope omnaexcnoc wueno:
(5 (coe E Hain}, 5
D VE (cos E +1sin$); 2) 4(cos E+isin SE),
Ormerum na naockoetH TOWKN, nsoGpaxaiomue Komuneke
ae suena:
Dit 2) 2 3)-5 4) -21-3.
Peur ypapnenm
D (4 +2 4h
2) C2+D+2-8-2h
8) 5+1-2-(8-V2)
4) (= 2)(1 +20 +(1-12)8-A4D 1474.
87, Banucats 5 rpurononerpuueckoli @opme “meno:
1) -4+45 2) -VB-i.
88. Pemur» ypasnenne:
D 27-2245 2) 2° 4102 +260;
3) 52+6245-0; 4) 22782430.
89. Haltrm aeflersurensusie wena x m y ns panenersaz
D Gy-x)+ (y 3x)
2) sy + sui 2i-yi
90. Boinonunr» aeiic

3
y eme y oy,

E

R

iB

B

a
DL. Chaput» wonyam uncen 5"; ve
D nen

232 nasa vit

Konanokcno mena

33. Onpeneners, pH KAKUX ACÍCTEUTOIBANIX SHAEHNAX x MY

ya 3-14 EE panna i.

get at
14 Cocramm. upnnexennoe knanparnoe ypannenne € neiernn-
Tensunnen KoodpuuBIeNTaNE, ECT ORK Ma ero Kopne pa
915-155 2) 32%
Bartuenumn:
tn
1) (8(cos LE +1sin 2)
2) (cos 20° +isin20°)"*;
3) (2 (cos (20°) + 1sin(-20))%;

4)

CoRR
Coran)
U Sanuears » spurononetpruscno m anreßpainecnon epwax

nee 3 tisin 3);
2) 2=03-0%

1 |
er een)

(oo

var
4) 2= Laser

rindo venons
» stack 2) Eno
8214 l2a= 23% “4e.
BE Hatten nce snavennn
D VE 2 VE DV
9] Pers ypasuenne:
D 21-866
4) 2165
Dale 6) 281-1130.
{Dol Sauncanı» rpuronomerpurecxos qopue konunenenoe ueno:
Da(2-0% 2) 2=(sin D 4i(1+c08 E)".
ED Hasen xoprz yparmenna
20-25-0020,
reflerarensmnse NACTN KOTOPLIK orpataremms.
DA He scscinmecuen Daunen: pañal seven 51) Bu, Ay eme

mueca sepmmamx rpeyronsuka. Haïri Touky mepeceue-
Hus ero menuan.

; 4 eras.

233

Vrpaxnannn « rase VI

(103) Haïru mnoxecrso rouex 2 kommnexenoh nnockocrn, dagas
Dn eno

À) ones apego qua pau at

2) enim ua opryanrron roma à paren LE

D) Bram. arepas cs See ee ee

Ban 2r<o<dn

De do mr e caro ans

ta Dép

HHonssere, no nın mG soumaenemux AGEN 2, eae

sure

DARA» Hah

Eu

08] Tonzaynes sanuesio kosımaonensx uncon B TpRTOmoNterpie
cxoü dopme, mañrm cos15° u sin 15°.

, 2470.

Bonpocei x rnase VII

1. Kax onpenenserca panencrso Konnaexcunx uucen, sanan:
mex B anreöpamueckolt dopne?

Kar IpOMINOXHTCA CAOMENNE, DANHTANNE, YMNOMONIE I Xe

ewe KOMUIEKCHBIX CON, GAMICOMEBX B axre6parieckok

dopue?

Ba Harmon caofcreme ofazaior encmenme n YMRONN sau

4, Bcerxa au pumoaHHMa onepanna VMHOMCHHA KOMIICKORDA
wucen? Beeraa Au OXMO Kowmaexcuoe UHCAO MORHO passe
aus Ha Apyroe?

5, Kaxue wena wassieaior Nero MEIMBN?

6, Kakoe uncno HASEBAIOT COMPAACHHBIM KOMILIEKCHOMY sexy
a+bi?

7. Kaxoe «meno nasuaior MPOTHBOMOJONINAM. KOMNTEKCHON)
“mens 2?

8, Kar reomerpuuocxu nurepnpermpyiorex Konmaexemue una?

9. Kaxono ssaunnoe pacnonomenue Ha KOMILIEKCHON NAOCKOcH

uncen: a) 2 u 2 6) 2 u (2)?

“ro nassınaercn MOXYAEM KommnexcHoro uncna?

: B vem COCTONT reomerpHuecKHit CMBICA MORYAM paanocr

ABYX Konnnexcnsix “cen?

Tiro nassınaeren apryMentom Kommnercnoro wena?

Kar aanuesimaores Kommnekensie Mena E rparonomerpanec

xoñ ipopme?

Kax nepeñtr or anreöpauuecxoli opus carmen nomnztereno

ro unena K Tpuronomerpirucciol «popme?

2840 rnasa vu

Komnnexcnie wena

Bi ‘ro nasumaeren xopem crenenu n(n>1, neN) ua xomn-
aexenoro “mena?

EE Kar nponauonuren yunoxeune, nenenne, woavenenue » cre-
Ib, annewenue KOPHA JLTA KoMmueKenux MCE, SAMHCAN-
xx 8 rpHrooMerpartecKOÏ dopne?

[ER Obopuyanponarı, ocnonmyro reopemy azreópm m caexerane na
Hee 06 anreöpamıeckom ypasnemu n-A crenenn (n > 1).

1. Banoamens nelernun:
1) (8+1)+ (5-2); 2) (6-¿)-(24+31);

3) (T+) A0-Is DE
2. Burunenuru: Be
2+5i , 2-5, LET
Datz Alam):

3. Ganucar x npuronowerpimieckoli ope kommaexenoe “meno:
1) -14iV3; 2) sin ¥ -icos E.

4, Pears ypanuenne:
1) 245-0; 2) 2°-102431-0.

$. Haïtru muoweerno rouex KoMMIeKcHOK naockoens, yaonnerno-
paomux yexommo |z|~ 3.

6. Sanucars 5 anreGpanrieckoit dopue kommnexcnoe meno:
VE (cos E isin
YE (coe 2 +18in 5).
7. Bunomuers neiiermus:
1) (cos 18° + isin 18°) (cos 42° + sin 42");
bein
(con + isin)
Ar
3)
8. Ha mnoxecrne KOMTIeKCHEIX HINCEA permis ypannenne:
1) 21-16; 2) 2-64.

+ 8) (15 (cos 3 Hein 2)

BVA El Voropmsecxan cnpasxa

Heropun passurun ‘nena YXOAHT KOPHAMI B APEBHHE Bpeme-
un. [Ipemnerpeveckie Maremarucm TOMO MATYPATEMNE NUCH
cran «nacroamunat». B pennen Ermnre u Jpepnem Bamnno-
e 80 BTOPOM THCANCACTAN AO H. 9. MPA PELIENNH TPARTHNCEIX
sagas menonnsonanmen apoóm. BI 5. 10 1.0. KuraiicKMe mare-
MATH DREAM TOMATE OTPINATENKMOTO “mena, a B TIT B. n. 9. [Ho
daxr yoke moxmoonazca npasunann zeliersufl © oTpunaTexHBIM
sera. B VII 3. m. 9. unnÜckne MATEMATHIN UPMAABaN ma-

Vicropmecran’ cnpacra

ranqmuit o6pas ompunarennnnt «meza, cpamman 1% € Ram
su. B VIII p. yueune aan, 11O Y nONOMIEMMOrO uncna ee
Crayer Ana Keanpariuix Kopuat: RUN — LONOKATENLAOS e,
apyroli — oxpnuarensioe, MO cunrast, ro na OrpUIAT OIE
‘cen Nennon uspaexar® KBANPATEMI open.

TlorpeGuocrs 8 naonnosennn KBAAPRTNONG Kopnn a orpaue
Tensmoro mena nosmurna D XVI m. B cono c Pemienmanm Sper
nenn. Hransauckui maremaruk Mor. Kapnamo (1501-1576) »
1545 r. men uucaa nono mpnponu. On npennonun csm pe
iwennen encres ypapneniit

x+y

2y=40,
e nuciomed pomenuä na mnomecrse nelerauremmax uncen, nr
py cen a=5+1-18 u y=6-V-15. Tipu stow Kapzano apex
Tax munomumrn nehernun € wnenamn nonoh npHponts anaaorım
TOMY, KAK BEMOJHAJMCE ACHCTBHA C ACHCTBHTEALHEIMM UHCAANIL,
» aacrnocta ounzans, «ro V-15-V=18=-15.

“nena nono nprpozi Kapxano masuman enero orpmmarenr
mmm u ecoperitieckn OTPMUATEMBRRNN».

B 1572 r. mranpanexuh warematux P. BouGenau yoru
pa mpanrena aprbsreritaecknx nohoramf ¢ nomen mena, Te
‘MMH «MHHMBIe Wena» BBen B 1637 r. P. exapr, a a 1777
BeanKnil oreueerzeunnihi maremaruk JI, Dünep (1707-178
npennoxux oßoananarı unezo V=T nepnoh Gyxsoit hpannyacsen
cxoma imaginaire (mmnmeri). Common i=V-1 eran mmpoxo me
TONBIOBATECA MATEMATHKAMH NOCAE ynOTpeOAeHMA ero B CBOUX je
Gorax K. Payceom (1777-1855).

Tayee samenna unosanne <xnmmiie wnexas na «somnzerome
wea» u OKOMNATexEMO 3AKPOIMA ANA HAVKM FeOMeTpIeCIDO
Inrepnperaumo Komnnerenoro ‘itera a + bi Kak row Kooper
noi maockoerm c Koopnunaranı (a; 6). Tloaxwee komnaeranie
"ICAA Takıke cTann MSOÖPAMATB € NOMOLISIO BKTOPOD ma KOO
narnoh naockocrm © MMANON m mauane KOODUMET 1 Koma
8 rouke M (a; b).

Tour «MONA» u enprymenr» KoMuaeKeKoro sen sme:
$panuysexnä maremaruk M’Anambep (1717—1783), a cam
Teponmat Oktnit BBerens! B ONXOA MOCIE MMPOKOrO IK NETOS
panna m cnonx pa6orax maeliuapcws maremarıom ME. Apra"
nom (1768—1822) m dpamnyscknm maremarukom O. Komn
(1789— 1857).

B navano XVIII ». Guaa nocrpoena Teopna Kopuel nt cre
HOM na onprimarombinix 1 KOMNAEKCHEIK “NICE, OCHORANTIAR te
putwegeuwo » 1707 r. auraucku marewarnont A. Mya3pox
(16671754) dopuyne

(cos@ 11sing)'=cosng +isinng-

0,

288 frame VU

Komnnekenuie suena,

YpasHenun
H HepaBeHcTBa
C ABYMA HePeMeHHBIMH

Cmpozo zosops, sadava pewena,
ecau coomaoaeno 013 nee ypaanenue,
‘max Kax amo snauum, «mo yemanossena
aeauvun om dannnx.

A. Mota

§ 1. liuneñnbie ypasnenna
n Hepasencrsa
© ABYMA nepemeHHbimn
1. JIuneïmare ypannenun
Tier ua nnockoern JORANA MPAMOYTONARAR
cncrema Koopaumar Oxy. Torna ypannenne
y=kx+b a
onpenenaer npanyi0 I (pue. 114), nepecexaomtyro
oc» Oy u raue D(O; b) m oOpasyiomyio yron à e
monommTennum nanpanneunen oeu Ox, rue
tga=k. Uncno À naanınaor yenoooım noodupuyu-
eumos npamoñ L.
Ana nocrpoenun mpsnoit 1, aanannoli ypan-
nennen (1), apcrarouno naitrn' ne Tou roi
mpanoï. B xaucerne TOKHX ABYX TOTEK MOHO

Puc. 114

Tee ypasnenns w nepasencTea
© Anyus nepouonsianan

Banr» TOUKH nepecevenun npnmoh | € KANO! ua Oceit Koopanner
(ap 20, b0).
Ha pueyuxe 115 waoGpaxemm ave npaume, saxaue ypar
nemmama y= x 1 u y= 2x42,
Pacemorpun ypasuenie
Ax + By+C=0, a
rae xora Ou oquo ms uncen A, B ue pasuo nya. Ecau B40, 10
ypanmenne (2) mono sanucaro B nue
AL

y

A
7.0.» anne (1), rae k=- À

Crexosarensuo, ecam B0, To ypanmenne (2) npexcrannser
co6oit ypamnenne mpamoii. A

Ecan B=0, 1o ypapnenne (2) MorHo samucar 5 Bune x-- ©
ro ypanneune npamol, maparnenvoï och Oy.

‘Taxum 06pasom, ecan A, B, C — sananınıe unena, rare, eo
A 1 B onxonpeneno we papel Hym0, TO ypaHenme (2) sanser
ea ypamuenuen Hexoropoit mpanoii.

2. Inneinsie uepanencraa © xeyma nepemenmsimt

Sanaua 1. Hair nuoxecreo TOWeK KoopaKHaTHol nAOCKOeT,
yronnersopaionqx nepasenersy 2y-3x-6<0.
> Ypannenue 2y-3x-6-0 annnerca ypannennem mpawoët (pre. 116),
npoxonameñ uepes roux (-2: 0) 1 (0; 3).

Tycrs My (x45 91) — tosca, nexaman umxe mpamoñí I (6 ae
iurpxonamnoï ua pmeymxe 116 noxyanockocru), a Mz — rows
© aGeunecoï x, m opuumaroll y,, aemaman na npanoï L. Tora

2y¿—3x,-6=0, a 2y,~3x,-6<0,
mar war Yi <Ya,

Takum o6pasom, » moGoï rose M(x; y), neue
upsmoit 1, manomuseres mepanenerao 2y-3x 6 <0. 4

Touxo tak xe MONO pere

nepanenerno oÓmero mina

Ax+By+C<0, CO]
re no xpafineli mepe onWo na wncen
Am B ue passo nya.

Ecau B>0, 10 nepanenerno (3)
IHUOANACTOA RO ncex FOUKAX, MMA
nix une npamoi, saxammoñ ypan-
nennen Ax+By+C=0. 3

Ecan B<0, 10 nepanenerso (3)
COPABOAAMBO » rouxax, 1examux
se aroli npanoi.

Ecan B=0, ro uepancnerno (3)
npuwer sux Ax+C<0, Pre. 116

298 rnasa vun
jpasnehun WHEPABERCTE CABINA MODEM

à

e

Tlonyueunoe Hepanererso PABHOCINHO Hepasonerpy x<—

ap 4>0 x mepacencroy 25-5 mu ACO
"mouse, nepanenereo 22+9<0 pamoemnano nepazescray
À money RGB nos, mano Coa OF Hoe

3
voit x= — E (pue. 117).

B oömem cayuco npamas Ax + By + C0 pasneaner naceKoers
38 ave MOAYNAOCKOCTH, B OAHOÍ Hg KOTOPHX BETONMAOTA Hepa-
nero (8), a 1 Apyroli — nepanenerno
Ax+By+C>0. @
YroGer peurs mepanenerno (3) mam (4), aocrarouno nanrs,
zayo-nmöya» Tomy My (a; yn), me mewanyo na mpamoi
Axt By+C=0, onpeneants anak wera Ax, + By, +0.
Hanpimep, uepanenerno 3x-4y -12<0 vepuo » noxynnocxo-
em, pacnonoxemnoii mue npamoli 3x-Ay-12=0, Tax Kak np
i=y=0 pupancenme 3x-Ay-12 orpuuarenvno (pue. 118). Ora
Annan mpoxonur wepos raum (4; 0) m (0; - 3).
3. Cuerema auneiimix mepanenern € unyma nepemenmann
Pacemorpum cueremy Hepasencre
fésratans

re

Ax + By +Cz>0, ©
mennonaran, uro A}+B3>0, A3+B}>0. Torna nepnony nepanen-
cmy encres (6) yaonnernopaior Tout mnonecrna My, exam
10 oany cropony or npamoli ly, sanannoh ypannenuen Aust
+B, =0.

“Ananormuno propoe nepanenerno cncrew (5) apaneren nep-
mc ua mnomecrse M, — onwoit no noaynnocKocrelt, na Kovopsie
pisómacien Koopnunarnan nnockoctn mpamofí fy, sazammoR Ypan-
foment A+ Ex +20.

51 (239
TRS TORN RESTE © BAA TEPER

Muoxectso pemenuii cucremsr npencraniser cobol nepecese
une snomeorn Mi u Ma.
Banaua 2. Pour» cneremy nepanencra
x-y>2,
x+3y>6.
> Hoerponm npanuse I, x L (pue. 119), sananınıe coomener.
Benno ypannenssinn x—y=2 u x+3y—6.
Penn» exeremy
z-y=2,
x+3y=6,

monyunm xo=3, yo=1. Caenonarensno, npamne Li 1 ly nepecers
rorea B rouke A(3; 1).

Tax ax Koopaunaria roux O(0; 0) ue yronnernopmen m
OMOMY MO mepanenere nannoH cueremm, TO CHCTEMO YAOBAETRO)A-
107 KOOPAMHATM TEX M TOABKO TEX TOUCK KOOPAMMATHOË mAocKeen,
Roropsie nestar nme MPAMON I} M AMINE MPAMOÏ Ly, 7.6. TOU
puyrpu yraa M, o sepumnoit A (cx. pue. 119). 4

AAOPMUNO peulaiores CMCTEME HepaNeHeTs, moAyusenne 3
chere (5) aamenoï OAnOrO MAH ABYX SALON mepanenern na npe-
THRONONOK HE.

cu nepecerasorqueca m ronke A mpaume I u ly aagaorx
coorererneniio panne

Aux + By +Cy=0 m Agr Bay + C0,
10 smoxeernom peienußi wepanencrna

Aux + Bay + C1) (ax + Bay + Ca) >0 ©
annneren 160 oßnerumenne ornof naps M u M, wepruxansnux
yraon e sepuumoï A (pue. 120), 1160 oGvexunente apyrok rap
N, u N, septuxaisusix yrs € Toll ae mepusnol

B Camom ene, v0 weex TOAKAX Kunaoro ma Mmoncern My,
M, N, nepaa uaers nepanenerna (6) mpymmmaer 1460 nonommen.
HBI€, 1M60 OTPHNATENBHEIE 2HATEHHA, A IPH Tepexope OT OXHOTO in
TIX MMOKCCTE x COCEAMIM (1epca opuıy Ma npannix I, Lo) ana ze
Roit “ACTI 91070 HepabeorBa Nenserca Ha nPoTumononosil.

Puc. 119 Puc, 120

240 rave vil

panne u HOPADONCTOO € YMA EEE

Eeau, nanpumep, ma muoxectse M, nesan Macro nepañencr-
ma (6) nonoxcureamna, ro ma maoxcecraax N, m N, ona Gyger or-
panarensnoï, a sa M, — monoxurensmoll.

Yso6st onpenenurs, na kaxom Ma 1uyx MHOKeCTR MU My
mau N, UN, CIpaBenmbo nepanenerno (6), HOCTATOuHO onpexe-
muro snax nepoli “acTH aToro mepasencraa B kakoll-ando TOUKe ox
Noro us MHOMECTE My, Ma, Ny, No.

Banana 3. Pemurn nepanencrno
(=y-2D(X+8y-6)>0.
D Mpanme x-y-2=0 m x+
+3y-6-0 nepeceKarores (aa-
zasa 2) u roune AB; 1). B tox
re OEM, (pue. 121) aenan
sacro nepanencrna nono
Teba, m mo9Tomy_Mmoreerno
0 pememni — oßBerumenite
unoxeern M, u My (orn uno:
KeCrRA DMIITPIXOBAMAL HA pH
me 121). 4
Banaua 4, Pemurs cucremy nepanenern
2x-y>4,
3y-6x>-5
D Bropoe Hepaweners0 ool cucreMM PABHOCHABRO MOPaBeneray
2x-y< à, m mootomy mexonnas cucrema PAnHOCHAEIa enereme

2x-y>4,
2x-y<

ES
Beau Gx napa uncen (xo; Yo) tina peinenuen monyuenmoli euere-
Mx, TO 1620 29 2x0 Yo YAOBNETROPANO Got NBYM VCHOBIAM 29> À
5

OS

Caenosarenbno, ucxonan CHCTEMA wepasencrs Ke meer pe-
went. €

Banana 5. Halim muoxeerso Touex koopaumarnoli nnocko-
(mu, yaowneTsopmionyix CHCTEME nepaneners

x20,

y>0,
x+y-2<0,
2y-x-150.

D Hepnuım nym nepanenernam CHCTEMA YAOBNCTROPAIOT nee Tou
Ki, Y KOTOPLIX OGe KOOPAMMATH HCOTPHUATCALMD, T. €. TOUKH, 1e-
ane m T xeagpanre (ueaonası TOUKH HONOMHTEMNMX NOAyOcoR
Ox 1 Oy).
$1 201
TRS ypannenm u HEPONENCInA € AByMA PEPONEAA

YroGu1 peurs HEpABEHETBO x+y-2<0, paccmorpun npaxyo
x+y-2=0 (pue. 122). Dra npsmax npoxogurr xepea roux (2; 0)
1 (0; 2). Tipn x=0, y=0 nepanenerno x+y-2<0 npaneres ver
HEM. CACAOBATCABNO, Hepaneneray x-+y~250 ynonnernopsior ace
TONKK, ASKAMME HIKE mpamol x+y-2=0 m Ha camoli ma
mol. B peayabrare monyuaen, “TO nepemm tpem nepanencraa
HeXONHOÏ CHCTEMIA YAOLACTHOPAIOT TOMICH, PACNOMOACHNAE BHT
Pu na rpannye rpeyroruna e nepusmawm O(0; 0), A(2; O),
BO; 2).

Peu, naxonen, HOCNCANC npancnerno CHCTEMIA, 7. €. ue
parencrao 2y-x- 1 < 0. Pacemorpma npamyio 2y-x- 1-0. Ion
ran x=0, maxozuim y=0,5. TakHM o6pasom, mpamas npoxozur se
pes Touxy C(0; 0,5).

Haine rouxy D nepecerensa nps-
of 2y—x-1=0 € mpamoñ x+y-2=0.
Ana sroro peu crereny ypapnenmi

1:82 o,
2y-x-1=0.

Cknaanimas ypannenus oroñi enere-
max, nonyuaem 3y-3=0, orkyna y=1.
Horerapasa y=1 » neppoe ypapnenne
cnerenm, naxonum x=1. Bnawwr, TOW
xa D umeer xoopannarsı x=y=1 (em.
pue. 122).

Tax ax mepaneneray 2y-x-1<0 Pre 12
YAOBXETBOPHOT TOYS, TeKaMKE MIKE
npamoii 2y—x-1=0 (rowxa O(0; 0) ynonnernopser aromy nepe
BeHCTBY), TO HEXONHOÏ cHcTeMe HEPABEHCTB YAORTETROPANT 3e?
‘TOWEM, NexaUMe BRYTPH M Ha rpaxsne “rersipexyrombmtiia OADC
(em. puc. 122). 4

IE Banana 6. Tlycrs M — muoxecrso Tonex naockoerm € Koop-
Aunarasın (x; 9), Tarcux, uro unena 3x, 2y m 9-y ABAMOTEH jaune
Mu cropou HeKoTOporo Tpeyronsunka. Harn maoutaa» durype M.
[> To enofieray ann cropon rpeyromsuna enpaneymusst nepanen
crea, oSpasyiomue cuereny

pta Y

O<2y<3x+9-y,
0<9-y<Bx+2y.
Dra cuerema papnocnnsua cuc-
rene nepaseners
y-3x+9>0,
y-x-3<0,
x+y-3>0,
x>0, y>0, y<9.

2425) ana vun
pan WEBAREHETER E RENA WERE

Yenonnam monyaennoit cHicTemst yaonnernopmor TOWIH Tpeyroms-
mıxa ABC (pue. 123), rue A(0; 3), B(6; 9), C(3; 0). Inomanı S
durypu M panna Si -S¿—Sy, re Sy — unomans rpanemm OABD,

D(G: 0), 8, — unomans rpevrommmna OAC, $, — noma» rpe-

yromımıa BCD. Tax sar S1- 4 (3+9)6=36, S¿= 3, Sy= 3 -3-9—

2

Baten se on ur quina, pn oropset
sont parir

xs, 249526, 29-29-48.
D Yona nepioe nepasenere a 3 cesanunan © pert
nanyracm Ty <SL, orıyan 985. Yanonan mopoc nepasencrao
In 3 u cxaagsionn © A oman
1-02. Iran, Oy <9. Hexonsot oucrene ynonnersopner somo

snavenne y=8, m voran x=20.
Oruer. += 20, y

Ynpaxnenna

1. Sanuears ypapnenne upsmoii, npoxogmmeñ wepes row Am Bi
1) Ad; 0), BOO; 2); 2) A(-3; 0), B(0; 4);
3) A(-2; 0), B(0; =D; 4) A(5; 0), BC; ~6).

2. Halıru muoxecrgo TOUeK KOOPAMMATHOÑ ILIOCKOCTH, YNOBACT"
ROPAOMX HepaneneTBy:
1) By-2x+4<0; 2) x-4y>0.

3. Mao6paanth na nnockoerit MIOMECTRO TONeK (x; Y), KOOPAK-
BATH KOTOPHX YAOBAETBOPAIOT CHCTEME HEPABEHCTB:

Ds 2) [x-3y<3,

y-2x>1; (acer

8) (xty>2, 4) (v>0,
[esc free
x-3y>-2; 2x-y<-2.

4. Hañra nee naps (x; y) narypambmx wncen x u y, Koropsie

ABAMIOTOR PELIEHNAMN CHCTEMEL HEPABEHCTE

1) [x-y-2<0, 2) (x+3y-11>0,
x+2y-9>0, x+y-8<0,
x-24+3>0; x-2y+4>0.

5. HsoGpasirs na Koopannannoli nnockocrit Oxy MHOXECTEO TO-
Mek, KOOPAUNATLI KOTOPMX VAOBXCTEOPMIOT HEPABENCTE)

D = y+8)(x4+y~1)>05 2) (2x—y~4)(2x+y+2)<0;
3) xt xy—2y*>0.

91 Mans

TAGS ypaanenin Y HEPABENCHER © ABN EPA

6. Tiycr» M — MMOXeCTBO TOHEK HAOCKOCTH ¢ KOOpAumaramı
Ge 9), TaKHx, wro «nena 2x, y m B— x nnamoren zunnanı CTO
POH Hexoroporo TpeyromsnnKa, Haltra mnonans Gurypat M.
Tlyerh M — muoxeeTRo ToNeK MAOCKOCTH C KOOPAUBATNE
(E; y), Taxi, uro mena 8x, y u 18-2y amımoren ania.
‘eropou mexoroporo rpeyrommusa. Harn naowaae duryput M.
Haiti pce maps nensix MCE x, Y, MIs KOTOPHIX RepHA ne
panenes

1) 3y-5x>16, 3y-x<44, 3x-y>1;
2) By-2x<45, x+y>24, Bx-y <3.

§ 2. Henuneúnbie ypasnenna mM Hepasencrsa
© ABYMA NEPEMEHHBIMA
1. Hemmeitmse ypannennn
Banaua 1. Halıru mnoxecrno rouex Koopnunarmoh race.
crm, Vronnernopaiomnmg ypanmeno:
1) x*7-*=0;
2) 2x24 5xy ~By*—2x+-y—05
3) 2x?—Bx+y? + 6y+17=0.
D> 1) Ypasxeune mono sanncans 8 Dune (x—y)(x+y)=0, omkya
onenyer, no 2060 x—y=0, mi60 X+y=0. Bamern, uro nanıce
Ypannense papnociamno ypannenio |x|=|yl. Tootomy wmoxecreo
TONER, YNOBACTROPAIOMAHX ypanmenno x*—y*=0,— mapa nepece
KAIOMUXEN HPAMUX X= Y u VX.
2) Paonommmı nesyio Macro YPABHEMIA ma noxe
Axt + Sry Bu Bee ty = 2x? = xy + By (2e-y)—(22-y) =
=(2x—y(x+3y-1).
Henomoe muonecruo — napa uepecexaouxen apar
2x-y=0 u x+3y-1-0.
3) Tipcoßpusyen aepyıo sacre YPABNCHIA, HCUOMLIYA Merci
BMACIHHA HONMOTO KPAAPATA:
Axt -Br+ y? + By + 17-2 4x44) + y 94 by + 9=
2-2) ra,
2H y +80.
Dro ypabnenne meer exmucruennos pemenne x=2, y=-3,
1. e. muokecrso pentemullt ypapnennn — rouxa (2; -3). 4
Ilyers na Koopaumarmoï nnocxoeru Oxy neöpana roux
Ala; b), M(x; y) — nponanonsnası toa omo ake nxocKocrH, R—
paceroamue or ronxu M no rouxur A. Torna
yo Re,

Ecau sagano uncno R>0, 70 namnoe ypasnenue — 910 ypas
nenue oxpyacnocmu C panuyca R © nextpom » rouxe Ala; D).

244 Fnoes Vi

jpasneiinn HEPANENCTSA € ABJMA FPE

Banava 2. Haitrn muoxecrso roue KOOPAUMATRON mnocKo“
crm, yaonnersopmonux Ypanmenmo:

D 2242442 6-80;

2) Bxty=y'.

D 1) Mpumenas meron BELNCACHIA NOAHOTO KBaxpara, monyuacn
ty? 44x by -3=(x+ 2)" +(y-8)-16-0, orsyna
(++ Y-8)*= 16.

CxenonarensHo, MHOXECTBO peIlIeHHi xAaHHOTO YPABHEHHA —
onpymnoers paauyea 4 c uenrpom (2; 8).

2) Ipeobpaayex ypannenne:

Sxy-y'=0, y(8x*-y")
yO pdx? + 2x4 DO,
UC =) (GE +yr+3z%)=0.

Tax xax panenerno (x+y)? +822=0 nunonnneren TonBKO UpH
3=0 u y=0, TO wnorecrso PENCHE HEXOANOTO YPanmenmn — CO
OKYIMOCTS HpANEX y=0 u 2x=y=0, À

PacemoTpum npustepkt YPABHEHH € ABYNA MEPEMEHHEIMIT, CO-
Jepmamyx anar moRyaa.

Banana 3. Hafıru muoneerno ToweK Kooppunarnoh nnocko-
cm, YROBMeTBOpATOMTX ypannenmo:

1) x4+lyl=0:

2) |x|+1yl=

3) [xl+2Iyl+12y-3x1=12.
D 1) Vpapnenne pasnocnasno copoxymHoctn ABYX Cmerem:
(ra: [ars
y>0, y<0
Mepsoii cuereme ynoBxCTROPñIOT TOUKM, mpHnannemame
Gnccewrpnce II xoopaumatuoro yraa, Bropoit cnereme — Touxn,
Mpukanterxauine Secextpuce III xoopaunarnoro yrna (pue. 124).

Puc. 124 Pue. 125
$2 205

Feeniweinise ypasnennm u nepaencTea © AYMA nepeennunin

2) Ecnu x30, y>0, 70 ypasne-
une MONO sanneath n mine 24 Y
=2. Mmoxecrso peutennit sroro
ypamuena — orpesox AB, exe A(2 0),
BOO; 2).

Tax xax |-x|=Ix|, [-yl=lyl ro
Muomeerno — pemenmál nexoAnore
ypammenna — rpanuua Knaxpara
ABCD (pme. 125), rae C(-2, 0).
DO; -2).

Bamevanne. Ann naxonne-
una Koopammar nepımm knanpara
nyokno » ypannenum |x]+1yl=0 saat
3-0 (u Torna |yl=2, 7.0. y=+2), a
zarem y=0 (rorna x=42).

3) Mnoxecrno pement ypasnenus — rpauuna muoroyron.
MHKa € BOPLIMBAMH B TOUKAX, MEOKAUNK ma MpAMEX x=0, y=0,

=x. Habs von separa. Benn =O, o 2 MON pr
nennn oneayer, ur0 |yl=3, 1. e. y=43.

Benn y=0, 10 |x|=8; ecan y= $x, vo [x1=8, Iyl=$. owe
Had — rpammna MmHoroyronsunka A¡C¡B¡A2CoB, rre Aj (—3; 0),
42; 0), B,(0; 8), 8200; 3), C1(-35 -3), Cal 3), mosrane
na na pueyae 126. € 3

Hañrx pco napız Aeñeramrembmtax uncen (x; ),
npaneyunıno panenerno

dog (27 -VIVG-1
[> Panenerno (1) mmeer ennien, cen mnommem yenonns
y>0, y-x>0, xVy-1>0, 27 VEUT, x+y-21990. (2

9

w

Tye
2-20, Vevy- y. BE 0 (a
‘Torna panenerno (1) mommo aanncarı n nme
a-b=a, 0
rae
a>1, b>0, 021, w

npx nunonnenm yenonuit (2).
Hs paseucrua (4) m yeaogni (5) cnenyer, uro
a>a-b=at>a, u nosrony

a=a-b-a. @
Tax war 00, c>1 (yexonus (5)), To panenerno (6) anaser
CA BEPmbIM Torna M TOXBRO Toraa, KOTAA b + Ma pasercr

2468 rana vit

Ypaonenun u HOPABONETOR © AN IOPEMORMENIT

xVp-1=0,

O, OTHE nas

2 @ nonyuaen onereny ypamuenná |
co

wre axe 1-0, (e (Rel

Beau x=1, 10 y=1 u yenosue y-x>0 me munonnaeren. Ec-
5-1

srxttx-1=0, ro x= 1, wane var x>0 (Vÿ = D.

ma

=” 5-08 3-15

un x, Y YROBACTEOPAET YCAOBHAM (2) m ABnAeTCA pemnennen nan-
toro ypammenun.

E
Our. (it; 2
152, Henmehne nepanonerna

Ben Aa; b) — rouxa xoopannarnoii uaocxocru, R>0, 10
wepaneneray

Haiinennas napa sv

1 318), mm

(ea)? (yb)? < Re
WRORIETBOPAIOT Bee Te TOHKH, KOTOPAIE HAXOMATCA OT TOUKK À Ha
Peccrommu, Meuse R, 7. €. ce TOKE (1 TORO OHH), pacno-
sente myrpn oKpyscocrH C panyeu R e wenrpom 5 Tone
Ala; D).
Ananormuno Mnorecrno pement nepanencrna
a+ ya > RE
Gem NNOMECTRO TONER, NeRAMHX BHe OKPYAROCTA C.
Banana 5. Halim muomecrBo Tovex KOOPAEHATHOÏ n.10cKo-
cmt, yaonnernopmounK Nepanenerny:
1) 2x*4+2y*-2x+6y-18<0;
2) 9x2 +9y"+6x-12y-76>0.
D1) NpeoGpasyeu Hepaseterso, Buugenan nonusii kBanpar:

2(xt-x+ 1) 42(y2+ay+ 8)-18-5<0,

(4) 3) es.

Mioxecreo pemeunil aroro HEPABeHOTEA — MuloxecTRO ToneK,

manu BuYTPM oxpynoere paanyca 3 c nemrpon (3: - À).

2) O4 94+ 6x-12y-760( 4 2x4} )+0(y@-4y+4)-o1,
(ri)

Hexomoe smoaxeetso penton nepanenerna — wnoaceeT#0 70-

+5)

2 247
ORGUE ypaononna Y HOPADONCIOA € FAR TOPONIMIA

sa, aero ua enya paras 3 6 mamma (
18e aTOR OKpYAHOCTH. 4

Banana 6. Halith moxecrno ro-
or Koopanmarnoii nxocKocrit, yaonne-
TROPI nepaneneray:

D [al+lyl>25

2) + y®<alzı

si
a >
> o 26
> 1) Myers x>0, y>0, rorna nepanen-
ers0 npnwer Bi x+y >2.

Dromy nepanencray yaonzernops-
Yor TONKM MEPBOTO KBAPARTA, 1EXA
mue paume npamoit x+y=2 (pue. 127)
uma sToit npamolt (me TPETONSHIKA
AOB, rne A(2; 0), B(0; 2).

Tax xax [-xl=x, |-yl=y, ro mnoxecrno penremmi nexomno-
ro HEPABCIOTDA — MHOXCCTDO TOOK, sexANUIX MA CROPOHAX Ke
para ABCD i Bue roro KBANpara.

2) Kean x>0, 10 nepanenerno moxno JANHCATO m anne

(224 ya
Honyuennomy nepanenerny YAORAETBOPAIT TOWKH MHOMECTBA Ej,
Jemamıne ma oxpyænocrn pannyea 2 € nennpom (2; 0) m anyıpa
atoll okpyacuocru.

Ananornuno conn x<0, TO HOXORMOS HEPABEHCTBO MO a:
mucamo B me

[E22 277528
a mnoxecro Ey pemenuh JTOLO nepanenerna — snoskecrvo roves,
aenampx na oKpyxnocrm pagmyca 2 € mentpon (-2; 0) u avr.
pu oro oxpyxnocru. Cnenosarensno, mnoreorno E Pentennh nc
Xonuoro uepaneueru — oÖvennncune muoxecra Ey m Ep 7. €
En EVE.
3) Hannoe nepaneuereo, paguocuasuoe nepaneneruy
aeg,
Pao
ABANETCA BEpHEIM B TEX I TOABKO B TEX TOUKAX TIXOCKOCTA Oxy,
xoropue xexar pue Kpyra payuyca 12 e nenrpom (13; 0) m myr
pu xpyra paguyea 25 € uenrpo » rouxe O. 4
3. Cucremst nemmneïux ypannenwi
Peuenne CHCTNM ypanneitii € ABYMA extspecruusyet
ker o
GG y~0 @
TCOMETPIICKIL MONO HETONKOATE KAK OTBICKANIIE KOOPANNAT To
ex nepeceuennn aunnh Ty Ty, annanınız ypanuennnn (1) (2.
Nloerponn arm mmm Ha Kaeruaroh GyMare 1 malta Koop
mars rover nepeceuennn au Py u Ty, morno nalen puta
eme pement cnerenmi,

248 Fnaea vii

pannes u HEPSBCHETEE © ABYMA NEPEMEHMENM

Puc. 129

Banana 7. Perro cuereny yparrenri
(ae
xy+8x-2y=11.
D> 1) Tepnoe ypanuenne cuerentm, aanucannoe 1 anne
(2% (y+ 8)*= 26,
aaxaer okpyxnoens paxuyea V26 © uenrpom (2; —3) (pue. 128).
Bropoe ypannenne cncremst, aanHCAHHOE n RME

a+ e,

x2
danger ranepéony. Oxpy>xuocts m rumepGora, na0ßpaenisıe Ha
moy 128, muero versipe oöımme town All; -8), B(-3; -4),
CG: 2), DAT; -2).

Cnenonatensuo, naunan cuerema ypasmeuuit wueer uermpe
pea (15 —8), (35 4), (8; 2), (7; 2). 4

Sameuanne. Jia anaanTnueckoro pemennn CHETEME mp
Gamme x nepnomy ypannermo YAROCHHOE BTOpoe, moayunp YPABHe-
mue (x+y)? +2(e+y)-35=0, orkyna caenyer, ro ano x+y =5,
mo x-+y=~T. Vexmouns na CHCTENB OAHO Ha HEHIECTHMX, no"
YT KBaAPATHOe YPARHeNHE OTHOCHTeNSHO APYTOTO eHaHeCTHOTO.

Banaxa 8. Peimums cucreny ypannenui

x-2x-y+1=0,

+ y + 2x-6y 450.
D Tlepnoe ypamienne euere, samucamoo maine y=(x- 1), sae
‚mer napaGomy. Bropoe ypammeme cHCTeNN, sancannoe Brine
(+1) +(y-3)"=5, saxaer oxpyæoens pañuyea VS ¢ neurpon
EL 8).

Okpyxuoers 1 mapadona, naoßpaxemme na peynre 129,
ayeior ase oDume roux A(0; 1) m Bl Ya), re x2=-1,8,
1553.

Omer. x,=0, y

3 25-18, ya=5,8. 4
2 249

Honmmeinie ypaskenım W NOPASEMCTEA © AByMA HEpeMeNHUN

4. Cucrema nemmeïnux nepaneners
Banaya 9. Pere cueremy nepanenern
4 9y?- 18y< 0,
D> Cinansimas nepnoe HEPABEHCTBO CO BTOPEIM, YMHOHKeHIEKI Ha 3,
axonun
24 6xy + Ou? +6 (x~3y)+9<0, mms ((x~3y)+3)*<0,
oreyna x-3y+3-0. Tloperannnsı x= 3y-8 m nexomnyio exerexy,
nony'iaem CHCTENY nepanenern
9y*-18y+9+9y?-18y<0,
(cal sen
KOTOPYIO MOXHO 9ANHCATE B Duo
{mie 1<0,
2y?—4y +130,
orkyna enexyer, uro 2y?-4y+1-0. Pemme cucremy ypabnenuh
x=3y-3,
allgem apa ee peienna, KoTopsie ABAMIOTCA peuienuaen nexo
oii CHETEM nepapencro.

Orner. G 1) (E 2 }<
tar y
Sagawa 10, Han neo rate mapıı memx uncer x, y, Kone
prue yaoRnerBOpAIOT citcreNe Hepanencrs
y-lx-2x1+3>0,
y+lx-11<2.

[> Banuuien nannyıo cueremy Tax:
15 |x?-2x]

yal 2x1 w

y<2-lx-1l. @

Tax xax |x*—2x|>0, |x—11>0, ro ns Hepaencra nonyuen
cucremst enenyer, "TO

1
-g<u<2. a

Leman HUCNAM, YAOBAETSOPAIOMEME Nepaneucruy (3), 5
amoren num © m 1, nosromy cnerema (1), (2) moxer mers ne
ase pemenna TOmbRO mpm y=0 m y=l.

1) Beau y=0, vo cuerema (1), (2) npuxer sun

1
lx?201<2,
1 l<z
Ix-11<2.

Bropomy ns 9rux nepaneners yaosnersopmor Herbie «nena 0,

1 n 2. Tiposepka noKassinaeT, UTO mePBONy HEPABEMCTAY yaonıe

250 fresa vil

Ypanenm u HepaseNCTea © AVI NEpEMeRRGIN

rsopaior au» 0 u 2. Cnexosarensno, naps uncen x,=0, y =0 Hu
1,=2 4¿=0 obpaayion pemenna nexoJnoli eneremm mepanener
2) Ecan y=1, ro cncrema (1), (2) npmsonmres x Buny

Bropomy wepanonersy 9TOM everest: YROBAOTDODACT enumer-
rente mence ‘exo x= 1, KOTOPOS ABAAETCH TAKKE U pemeHHeN
nepnoro mepanencrsa.

Orser. x1=0, yı=O; 23=2, ya=0; xml, ya=1. 4

Banaua 11, Hooßpaaurs ma Koopaumarnoï mnockoern Oxy
durypy ®, sapanıyıo encremoñ uepanenern, naÿrrs nzomans S
vol urypst:

Danes. np

x+y>2 (x-1P +y?> lo

D 1) Hepasenerso x?+y?<4 aunaer muomecruo TOWER, nerauyHx
BNYTpH OKpy2kHOCTH € LENTPON B HAMAIE KOOPAIMHAT H PAAMYCOM 2,
A Hepanencrno x-+y>2— MROKECTRO TOUEK, PACTONOKEHHEX BE.
we mpamol x+y=2. Dra npsması nepecexaer oxpy>xHocTh B TOU
xx A(-2; 0) m B(0; 2), a @urypa ® npexcrasnaer coGoit cerment
(pue. 130). Hicxomax naomaus S pasna pasnocru mromanei ver-
mepra Kpyra (Sy) u rpoyronmumea AOB (Sa).

Tax Kak Sim, S,

2) durypa D — DTO MHOMCCTBO TOUCK, JCHAMPX BHYTPH O-
PyAHOCTH © menrpo B rouke O(0; 0) m pannycom 2, HO BHe OK-
pyxHOCTH € newrpox » rouke (1; 0) panuycon 1 (pue. 131). Bua~
sr, MAOMAND huryps ® panna S=41-2=37.

B+ y? 4

Puc. 130 Puc. 131

52 mes

Tommelinas ypaonenna Y nepanancTaa © Ayia nopenarnunn

Banana 12. Hana cuerena ne-
panencra

lal+lyl<2,

Payo y-D,

W-3x-2)(8y-x+2)<0.

Hañra naowane QurypM, xo"
opxunarst roux KOrOpOÏ yaonne-
‘TBoparor:

a) neprony mepanencray exte-

6) nepsum aBym nepanener-
nam eneremu; Pre. te

D) nem TPEM NCPADENCTDAM CHCTEMI.
> a) Teppomy nepasencrsy (CM. saxauy 6) YAOBHETBOPAIOT zouı,
zexame n xnanpare (pro. 132) ¢ sepummamu A(-2: 0), B(0: 2,
C2; 0), D(O; —2), TInomamé sToro wanpara Si —

6) Bropony nepaneuersy, Koropoe MOXMO aauncars

(x-2) +27 24,
YAOBNETBOPAIOT TOKE, neaune BME KPYTA pannyca 2 € mempor
B rouke E(2; 2).

Tomas sakpauiennoro ma pucyake 132 cermenta passa
1-2, A MAOIMAJD ÁATYDA, KOopAHMATH TONER KOTOPOÏ yaonnerm-
pasor neppum anyu nepañeneraar, panıta S=8-(r-2)= 10-1.

») Hpanwe y-3x-2=0 u 8y-x+2-0 nepecekaworen 8 tor
Ke FL; —1) u npoxosr coornerernenno sepes roux Bu C.

Tperseny epasenctsy yxosxersopaior tows ABYX Bey
Kannnnıx yrnon € nepnmoii F, onMM Ha DTUX yraon — YTOA, cÓpe
aveañ ayant FB u FC conepacamutt rouxy O (em. pue. 132)

Tlyers S,— naomans dburypst, KOOPAKHATH TOWeK KoTOpO!
yaounersopsior ween Tpen nepasenersan euere, Sa — cy
umouuaeh rpeyromunsos ABF u CDF, Torna Sin ¿Sin

= $2-S 4

Giver. 2) 8: 6) 10-15 0) 6x. 1
Ba Sanaya 13, Jlana cuctena nepasencrs

ay < diz
Ixi+lyl>2,
y +16-8x>0.

Haïira noma» DHrYDM, Koopaunarat rover KOTOPOÏ ya
aersopmor:

a) mepsony neparencray encTemst;

6) mepnum any nepanenernam cnerems;

3) BceM TpeM HCPABEHCTBAM CHCTEMBL.

L> a) Tlepromy nepaseneray, pannocnasHomy COBORYTMOCTA najx
nepanenora (x-2)° + y*<4, (x+2)?+9?<4, ynonnermopsior koop-
AMMATS TONCK, HAXOAMNINCA BHYTPN Ha TPAHIIAX ABYX Kpyron

suze

252 fnana Vi

Ypasnenun u HEPABEHETER © ABJMA NEPEMEHMEMN

Puc, 133

panuyca 2 ¢ ueurpaum (-2; 0) u (2; 0) (pue. 133). Ilnomans oroli
Surypst S,=2-7-28=8r.

6) Bropomy nepanencrny ynonnernopmor KOODIUMATA roue
(cx. angawy 6), pacnozoxenninx mue y na Tpanume KRAAPATA € nep”
ass (2; 0), (0: 2), (2: 0), (0; -2). Ilaomano S, Qurypat Pa
XOOPAUNMOTIS Toner KOTODOÑ YAORRETROPAIOT TIPO NBN nepa-
versant cuerewia, pasma Sı-So, rae So — RONOBHNE nrowann
spyea payes 2, 1. €. Sy=8n 2r=ön.

8) Tperze HePABEHCTBO MOKHO JATHNCATE B BHAC

(AP -y2>0, mam (x+y-4)(x-y-4) 30.

BroMy HEPABCHCTEY YAOBAETBOPAIOT KOOPAMHATEI TOYeK, 1e-
angix BRYTPHC HHA rpankine ONHOR HA RY Nap Depa
ton, o6panyionuxen np nepeceuenm npawnx x4 y-4—0 m
x-y-4=0. Tax kar (0; 0) — pemenne rpersero uepaneuerza cır-
<a, TO DIONY nepaneheray u MEPBRIN BY Hepakenersan YAOB-
IETDOPRIOT KOOPxANATIA ONG DITYPA y, RCA BUYTDH MPA
ro yraa e pepimumol (4; 0), rara, uno x<4. Tomas Sy dit“
pu D para 2S, +09, PR Op — MAOINANE DPAMOYTONLHOTO Tpe~
roxana € wepunmanın (2: 2), (45 0), (2: =2), 1. €. ou= 4. Cne-
soverensno, Sy~4n+4.

Orser. a) 8x; 6) 6x; B) 4144. 4

Bagava 14. Haiiru naomans Qurypa D, xoropan sannercn nn
voopauarofi MLAOCKOCTH COTO nepanenern

410,
3x?-4x-32<0,
(3x-2y)(8y-x+10)>0.

s2 10253
FERMES ypasnonn m HEPABGHETER © Aaywn nepewennenan

D> Mepsoe nepanencrso cucremb onpexeaser moxectuo roue.
excampex pue x ma rpamuno xpyra € WeHTpoN B rouxe O(0; 0)
pazurycom V10 (puc. 134).

Porno sropoe nepanonerno, noayuma - À <x<4. Meow
TODOS nepañenero cHoTeNM Annaer BePTILKAAKIyIM MONO, Rex

myo Mey PAM
npaxux).

m x=4 (skmouaa m non ax

3
Haxonen, Tpersemy HOPABCNCTEY cmeremu yAonnermopant

roux MHOXeCTBA M, KoTopoe COCTOMT HS ABYX OCTPHX Beprun
kammsix yrnos, OGpasosauntix pam 3x-2y-0 u 3
+10=0 (Bnioyan HE TOUKH ATX MPAMBIK), TAK KaK B TOUKe
mpusaneauteli moxecrsy M, nesan uacrs 9ror0 nepaneneru
nonoxuremma. Mnoxecruo M Moxno yaqers HO pueynre 134,
Tae ysasanııe upsmsie oGosmanennt I u la.
Tipamaa J, nepecekaeres © npamamn x

a sida Some
A(- 5; -4) BGs 6), a npanaa 4 nepocexaenta e roux ms ape

8, _ 38 5
mamma rouiax D(- 35-5) « CU; -2). Hance, npanas I; xe
caerex oxpyanocen 2? +y2=10, rar ax cnerema ypanmeni
2492-10,
By-x+10

Heer enunernennoe pemenne (1; —3); maxomon, mpamas I npo-
XOAMT Yepes NeHTp 9TOÍ OKPYAKHOCTH.

2507) nana vit

/paononun u NOPABEHETER E ROJA ACER

Hrax, @urypa D — oro tpanenna ABCD, us Koropoit yaasen
moayxpyr paauyca VIO c uentpom 8 touxe O. Hexoman naomany

sa DIZON gu, me AD 2, B0=8, he 2.

Banaua 15, Ha koopaunarnoli nnockocru pACCMATPHBAETER
urypa M, cocroaman #8 BCOX TOMK, KOOPAHNATKI KOTOPLIX YAOR-
Jemmopsior CHETEME HepaweneTD

VE >y-2x,
udn
= 1
Proa 36
Hocópasure Quarypy M x maño ee naowaue.
D Bropomy nepaneneray CHCTEMIA YIOMICTUOPMIOT K0OpAMMATIA T0-

sex (saxaua 6), rexamux BAyTPH Kpyra panyca 25 € HenTpou B
rouxe O u mue xpyra panuyca 12 c uewrpom (13; 0).

HEnmERHG ypaonennn u FOPABONCTOA © ABYMA NepeMennuMM

Ilepsoe Hepasexicrso eneremsi meer cMBIcH, ecan xy >0, 1.€.
ana tosex I TIT xeanpanros. Cunrası yexomne xy >0 minomen
HBIN, PACCMOTPUN ABA BOSMOXMRX cnyaan: y<2x, y22x.

1) Beam y<2x u xy>0, 10 nepuoe nepanencrno saBnseren pep

y<2x,

man. Cuerema mepavencrs |2y>0 Salaer muoxxeerno rouex Lu
IH xseupauros, nexcampr une mpaMoit y=2x (pre. 135).

2) Ecan y>2x m xy 20, To nepsoe HEPABEHCTBO CHCTEMN pes
HOCHABMO Kanyıomy ma HepABENCTS

Toya,

Wr) 32) <o.

Tlorynennoe Kepanenerno mp ycxommH y>0, xy>0 ompexe
aser MHO:ecTRO TeX TOWER I KBAMPARTA, KoTOpHe DAKE

sey mpm y= 2 y E ones aan, vr
Se
Ex

Ganerma, vro npauan y= 222 meer exmmeracnnyo oyo

PE aamouennt MOKAY mpaninat y=2x 1

rouky e oKpysnocrsio (x- 13)? +y? 144 m, crexonarensno, Kact-
trem aro onpyinocrin romane S purypk ® panna Sy 84S
re Sy Cuna. nnouaneh nays Cexropos (a Corne
nourpamunwe yea arct 22 u aretg 2), a Sa — nnomanı nom.
pyra paanyca 12.

625 12 3
Orner. SB (arctg 2 +arcta E) -72x. dumm

Ynpawnenns

9. Hain mnomeerno rouex Koopannarnoki nxocKoerH, yaonıer
Boom ypannenmo:
1) 4x*—9y?
2) 2x? + Say By? + x +3y=0;
3) 2x*+3y*+4x—12y+14=0;
4) 3x*+3y*-6x+12y+10=0;
5) x*-2xy+2y*-Ay+4=0;
6) xy+x—y-1-05 D y+lyi=as 8) y=xlyl.
10, Haïru muoxeerno Tower Kk00pAMMATNOÑ UAOCKOCTH, yaonzer
Bopmronunx ypasmenmo:
1) 2x-+ly|=05 2) |x—1l+ly-+2l—15 3) 2Ix1+]91+12x-3y|=12.
[TL] Hatten nce naps neiiereurensusx uncen, ALA Koropux enpe-
nenamno panenerno:

D 100, Vo VED

2) tog EN ANG) AV A,

256 rnana vu

Ypacnenun u HEPABEHETEA € ABJMA OPEN

12. Haïru uoxecto roue KOOPANMATNOÍ HAOCKOCTH, yaonner-
nopmontux nepanencrny!
1) 2x? + 2y?+2x-6y4-13>0; 2) nine
Pte aye
3) lx+1l+ly-21>2; À) HARE <o.
) [et A+ ly 21 er
18. Maira naoman» Qurypin, axamnoi na Kooprmmarnoh noc"
Koern nepanenernon:
1) (@+y?+2x+29)(4-2°-49)>0;
2) (x?+y®—x-y)(a? +y*-1)< 0;
3) 2lxlely+20+11<5.
14. Pers rpadursecsn eucreny ypannenttt:
1) [xP +y?+4x-6y-13=0, 2) {oe -0,
xy-3x+2y-11=0; 4 y-2x+6y45=0;
3) [ri +y?+2x-6y-6-=0,
yt 8Bx+2y-32=0.
16. Penuutrs cncremy nepaneners:
1) [ut+8xy+150, 2) astas
9x?-12x-8y<0; xy+y+1<0.
16. Hair nromaus duryps, sagannok ma KOOPAMHATHOË noc
xoctH cherenoli Mepapencr»:
5 pose 2 sen

x>l; y>-2;
3) (is 9) [lel+ly-1162,
Cats Paro
5 pre y
Iyl>12=xl.

17. Haïru sce napsr mensix “ICO x, Y, YAOBACTBOPAIOLNAX cite-
‘Tome nopanenern
24%4+2y*-12x+20y+65<0,
Ax+2y>3.
18. lana cucrewa nepanenern
lzl+lyl<3,
xt+y?>3(Qy-2x-3),
(2x+y-3)(x+5y+3)<0.
Harn naomans dirypst, KOOPAMMATL TOWEK Koropoh Y NOR:
nerBopaior:
2) nepnony nepaseneray cmerems;
6) nepnsin zuysı mepanenernam CHETENM
2) ucen TPEM HEPABEHCTEAN cncTeN.

92 0257

FETIMSÍMEO YDADNGIAA u nepanencTaa © Ayu HepeMennunt

19. Hawa cuerena nepanenern
Ixl+lyi< 4,
x+y? e-8(e+y+2),
(y= 2-4) (By-5x+12)<0.
Haëru niomaxs (pHTyPbl, KOOP/MHATEI TOMEK KOTOPOÑ yaon-
aernopmion:
a) NMEPRONY HEPABEHCTBY CHCTEMET
6) uepbbiM ABYM MEPABEHCTBAM CHCTEMEI;
») Rees spent nepanenernam cxicreNta.
Hana cuctema nepañencta
tty cAlxl,
lel+iyl>2,
A +16 48x>0.
Hañru 11011846 (PHTYPbl, KOOPAMHATEL TOWeK KOTOPOÑ yaos-
aermopmon:
a) mepnomy Nepanenerny CHCTEMBN;
©) nepsstn asym Hepanencrsam encrenst;
B) BCeM Tpem HEPABEHCTEAM CHCTENbI.
[21] Nana cnerema meparencrs
Hal),
lxbelyl>2,
4 1648y>0.
Haken naomans Quryp, Koopaiarut TONEK KOTOPOÍ yacı-
nersopmor:
a) meppomy HEPABEHCTRY cHeTeMM;
6) NepBHIM ABYM HPABEHCTBAM CHCTeEMM;
5) BCeM TpeM HEPABEHCTBAM CHCTEMH.
122] Hattra nxomans durypus, saxammoft ma Koopnumarnofi naoc
Koti citerenoli nepazencrs:

a (+25, 2) [4y?-2550,
Pegas, Pas,
(+ 2y+5)(2-y)>0; (Bx+y)(2x+y+5)<0;
3) [1x+3y+25>0, EL oan.
#23, A
+326 +25 >10y+ ES
224 y? + 10350;
Fri ee
9) [Mary Ser,
e 1 PTT
PT

258 rasa vin

Ypannenum u nepanencTaa © ANA MODERNE

$ 3. Ypasnenna m Hepasenctsa c AByma
NepemenhHbimu, conepxaume napamerpbi
BA 1. Ypannenus € napamerpama
Banana 1. Haïtrs ace ouavenna a, npit KOTOPMX cymecrByer
onto oana napa ACÑCTEMTEARMIAX uncen (x; Y), ynonzernopsouan
asen
2x? +4x + 2y-8y+10-a=0.
D 8anumen ypammenne » Bune 2(x +1)*+2(y-2)*. orkyaa ene-
er, “vo MexoJnoe ypannemte nueer ezumeraemnoe pemenne (- 1} 2)
apn a=0. 4
Sanaua 2. Haïru uce suavenus a, pm Koropuix nahneren
xora 681 onna napa nelicrsumensunix uncen (x; y), yAOBAETBOPALO"
aan ypaBnensio
x—5xy+6ay"+ (20-15) y+2x+2=0,

a)
D Bynem paccmarpnsars ypannenne (1) Kak KBayıparnoe OTHOCH-
reno x, Halen Auekpınanzaun snoro ypansenun:
Diy; a)= (2-59)? 4 (Say? 20y y+2
=5(5-4a)y?+2(5-Aa)y-4

Xora Ou oxua napa nelicrmrensusix “cer (x; y), yaonzer-
sopsiontase ypannenmo (1), eyinecrayer Tor M TOREO Tora, KOr-
a mepanenerno D(y; a)>0, 7. e. mepanenerno

5665-40) 9?+205-4a)y-4>0 e

wer pemenun.
Boanoxnss pit exywas: 4
Nang 2'a<i 8) a> >
B nepsom cayuae mepañencrso (2) He aaserca sepnnim. Bo
ropom eayuae 970 HOPADONCTDO HMOCT Pelnenna, TAK KAK y Mapa
Coni 2>0y"+By+7, tue G>0, HNeIOTCA TOUKH, pACnOnO EMMA
sume ocn Oy (mera napabonn manpannemn nnepx). Haxonen,
3 TpeTbem CHYANE, T. €. UpH a> Fu wepasencrso (2) meer penie-
IMA TOrXa M TOABKO TOFAA, Korma AMCKPHMMBAHT D, KBanparnoro
Ipexuxena, CTOARIETO E acnolt HACTH 9Toro MEPABCNCTBA, NEOTPI-
vane
Dj (a) =4 (5 ~ 4a)? + 80 (5 - 4a) =4(5—4a)(5~ da + 20)=
5
=64(a- 3) (0-2) 20, onxyan
50-28
64(a-5)(a-28) 20. (3)
Ietiernurentio, ecxm a <0, ro neram mapaboma 2= 02 + By +Y
ESMPABJICHAL BHM3 M XOTA Obi OMBA TOUKA Hapabonsı HEKHT BRIDE
cen Oy (sam na pro ocn) Tora u TOMBKO TorAA, Korma D,=B?—

ss ss

Yoaanenım n wepasencrea © Aaya heparan,
conepraue napamerpes

—4cy>0. Tipu a> À pemenunun mepazencrsa (3) apamorca sas

5 053
Omer. a< À, a> ©. 4

2. Cuerema ypannenni e napamerpaxat

Sagawa 3, Haïru suauenna a, upu Koropsix cucrema ypasue
unit

tayo,
y=ax+b (0)
uMeer neierantensnste peuienua npu 000m snauenm D.
D Ionerasıın y=ax+b u ypannenue x*~y*=1, nonyunem
x? (1~a?)~2abx— (146%) =0. 5

Cuerema (4) umeer neiiernnrensusie pemenss npx moon
sxasenux b rorxa m TOMBKO Torxa, Korma ypapmenme (5) uateer
aeñcrarenbnse xopmu, Tax kak ypaBHenue (5) sMecre co Bropux
ypannenuen cHerewu (4) oSpasyer cneremy, pantiocumnyto cnore-
me (4).

Ecau |a|#1, ro ypasnenue (5) suaseres kuanparumm, a ero
ancxpumnuant D=4(1+0*- at).

Tyers lal<1, +. e. -1<a<1, rorma 1-a?>0, orkyna caeny-
er, wro D>40%30, u nooromy ypanuenne (5) uMeer ¡olermuren-
ue KOpHH mpx moon snauenmm b. Tlyers |al=1, torna npx b=0
ypammenue (5) He meer xopxeit. Tlycr», Haxonen, |a|>1, rorga
D<0 mpx b=0. Virax, een |a|>1, ro nalineren raxoe anauonme b
(umenno b=0), ana Koroporo cucrema (4) ne umeer xelicramtem
six peueuni.

Orner. -1<a<1. 4

Banana 4. Haïru pce ouauenus a, np KOTOPBIX CHCTEMA
ypannensit

log2(3—x+y)+3=108,(25-6x+7y),
y+2=(x-2a)+a+2x
umecr poso apa pomenna.
[>Teppoe ypapxente cnerems pABROCILABMO ypasnenm
8(8—x+y)=25-G6x+7y, uan ypamenmo y=2x+1, ecan y+3-
—x>0, oryna exexyer, wro x>-4. Tora Bropoe ypannerne mpi
unnaer Bug 2x+3=(x-2a)?+a+2x, um
x?—4ax+4a*+a-3=0,
m saxana CBOAUTCR K HaXOKEHHO Tex SHAMEHHÏ a, PH KOTOPX
uonysemice ypaueuse Hacer POBO AMA KOPMA x, M xp, TAC,
uno xy>-4 u x4>-4.

OGosnaunn {(x)=x*—dax+4a*+a-3, D=16a*—4 (4a*+a-3)=
—4(8-a) u Bocnonssyencn TEM, TO yKAsAHHEIe YCNOBA Binon-
umorea tora u TOMBRO TOLHA, Kora D> 0, F(-4)>0, xo>-4, re

260 rnasa vill

Ypaonern u HOpabencrea € ADYMA IOPEMEHMUM

1o=2a — aócuncca Bepmmmsi mapabonsı f(x). Hrax, D=
=4(8-a)>0, f(-4)=4a? + 170+ 13=4(a+ 3) (a+1)>0, 2a>-4,
uma -1<a<3

Omer. 1<a<3. 4

Banaua 5. Halt nee onaneumm a, mp KOTOpHX cuerema
ypasnenuti

Vat ey? + 64—16x à Vx? + y + 864 1210,
Ayma

seer egmuerseunoe pemense.

D Tlepsomy ypannenmo CHCTEMH YAOBACTBOPAIOT KOOPAMMATEL TO
zu M(x; y), Takoï, “To cyNMa pacerosmuii or rouku M no ToueK
ME 0) MAR —0 panna 1. Fu nan pucctomno Mala pus
so 10, ro rouxa M nomkna MIPHHANICATE orpesky MM; (8 npo-
Tan dayana con, Venant Parrot Gua Ox donne
10 coraacko CROMCTBY CTOPOH Tpeyronnkuka).

ian, neproxy Joanna Ouro yaoanernopnor HOOP
gent tones op Milas m nennen dune

Bons Jpentonn Enero yiomneraopuor open
WIeK OKpYÆHOCTH paguyca lal c yexTpom O(0; 0). Ira oKpyx-
ns er org MM, Sannernennyo. GER TOMY
“rap exams

gano acaer orpesa MA: y om ene a=
oh tne hm = 5

2) oxpyasocrs nepeceuser onpeton MyBy » onnall none:
sav cnyate e page Nomen Dr Some aera Ol not
pans avers Off, momoyronmnore speyannaxa OM ls
Mate,

Omen B<a<-6, 00 8, a, 60008, 4

Banana 6, Hafrm pce amauenna a, MpH KoTOpHX cHeTeNA
wannenmi
lel+ 21yl+l2y-3x1-12,

say

sucer porno na AEÄETANTEAMUNK pement.
DHroxecrno pement neproro ypanuenss xamnol encromts, no-
Ayienmoe mpm pernensn caja 3 (3) ua $ 2,— rpansma ıneorn-
Yom, naoÖpaennoro na pueymie 126. Mnoxcectao peters
HOPOTO ypaBHenus CHCTEMIX — OKPYAMOCTD paauyca Ya e ueHT-
pex (0; 0).

Janna CHCTEMA meer porno Ana pemenus D CACAVIOMUIX
Eva

D orcos macro ergo Ay Ay, rma Yin de,

ss 261

PANNE NEPABENETER © ABYMA nepemernnanan,
‘conepxaune napamerpat

2) paxnye OKPykHOCTH pasen paccToanmi or Touxu O zo
rower C m Ca, roma a=3?+(2)*= UT

mer. 2, 47
Omer. 2, HT, 4

El
yoannenmi

Haïñr sce snavenna a, PM KOTODMX Here

x2 44x—3y +80,
v?+(6-2a)y+ a? -2a 0
meer xota Gui AMO penrenne.
D> Ma nepsoro ypaunenust cxcremst naxorum
y= FG +4x +3). @
Tax Kax x?4+4x+3=(x+2)?-1>-1, ro mo panenerna (6) exe
ayer, no y>- 1

Tlosromy sanava cmoJurrca x maxoogemmo Tex snaventi o,

DH KoTopsix BTOPOS ypaDHeRHe CHCTEMEX IMeeT XOTA Oh Qu KO
peus, yaosnersopsiomuit yenosın y>—

Paconorpum xsaaparausit rpexunen

2=y*+(6-24)y+a*-2a. o

¿Bro Aucnpmemaur D= (520)! - 4(a?-2a)=25- 12090 mm

Hroëut Bropoe ypanmeume uxeno xors On OAMI KOPexD, HE

o6xonno munonenne yexomma a 25

Tlyers yo — aGenueca sept À na-
pebom (7), weoöpamennol na pucyn-

xe 136. Toraa yo= 2% m ecm

munonuneren yenonue a< 73, 70

us u»
7. 0. neputitia A mapabonsı paenoromenn
epee mpaxoR y=— 4 (cm. pie, 186).
Knanparm specs (7) uncer vo:

POM Ya, TAKOË, “TO Yo>— $ TOTRA TOME

wo xorga, woran 2(-2)<0, 5.0

Y) 402-2060, oruyan

1
146-20(-1
9a?-12a-14<0.

Vicac 242. 4

2
Orser. 2

262 rnasa vi

Vpannehun u NepancHErna € ADJMA MEPEMEHMINN

Banana 8. Haliru nce axavenns a, npx Koropsix cucrena
ponme

logs (4 +8) -2loga52=0,

(+) -2(y+6)-9a=0
meer xora Got OAno pemenne.
D Hannan cuerena pasnocunsua enerene

x>0,

y=x-3,

(cta)? -2(x+8)-9a=0,
cyan

24 2(a-1)x+a?-9a- 60. @)

Vpameune (8) uneer zehernnresmunie Kopım xy m Xp TOA m
‘tomo Toraa, xorna D=4((a~1)*—(a*—9a-6))=28(a+1)20, 1. e.
up a>=1, npssent

x= la ATA 4D, x9=1-a+V7(a +

Tanman emereun 1 pamnoeumman eff Meron xorst 611 ojo pe
uewie, ecam ypapmenne (8) meer xora Oot OIX MOTONKITENDHB
sopeus, & 970 YCROMC PARMOCHBMO TOMY, “TO MAMÓOMBUIÁ 18
sopnell sroro ypasenus x2> 0.

Taxus oOpason, saxaua
coenach x pemenmo nepanener-
sa Vi(a+1)>a-1, ¡ua peme-
mia Koroporo mocrpomm rpachn-
su gymxunit y=Vi(a+1) m
y=a=1 (pue. 187). Ma pueyn-
YA PUNO, "TO peenws aroro
repasencrsa o6paayıor npone-
syrok [- 1; ag), FE do — 020
xrexssi Kopenb ypannenmn

(a+D)=a-1. Orciona mony-

T(a+1)=(a-1)%, a*-9a-6=0, ay

Omer. -1<a< 21108 | q

3. Hepanenerna w cuerentu mEpanenern € napamerpamm
Banana 9. Halim pce snauennn Mapamerpa a, mpi KOTOpEIX
comeersyer xora Gur omma napa xelcrpurensmux uncen (x y),
ylomiersopmioma nepanencray 2% 6x +474 4y< a.
D Hexoamoe MEPABENCTBO, PABROCHABNOS HEPAREMCTEY
a+ +2) a+13,
Meet xors Gol OAMO peUIeNKE TOL M TOMO TOCAO, Koma
a+13>0, 7. e. mpn a>-13. €
3 263
PARENT HEDARENCTOS © RIMA MEBENERAEMI,
Conepxaume napamerpu.

3anaua 10. Bepmmun A, B, C napannenorpamma ABCD wie
sor coornerernenno soopramara (2; ~3), (13 8), (6; 1). Halt
ce AnanennR a, RAS KOTOpHX KoopRUNAM Repro D amas
<a pernenuen cxicresst Mepanenere
Coe
6x- 244720,
[> Tiger» xo, Yo — xoopnumarsı nepmmnst D. Torna ua panencrs
AB. De, re AB=(3; 6), DO=(6- 20: 1-yo), enenyer, 170 xı=8,
Yo==5. Hoxcranasa x=3, y=—B à cnereny nepanenern, nome
eu - A cach.
Banana 11. Han nee anasionna a, mpm KoTopLIx wrote
pement cmerenur nepanencra
a y<0,
2x+y-a<0
coxepsur orpesox [-1; 0] ocn Ox.
[> Hogctasue » aannyıo cucremy y=0, nonyunm
a (a-2)x-250, (0)
2x-aG0. ao
Mxoxecrs0 pemiennit nepasencrBa (10) — nya x< ze me
¡ox A=[-1; 0] upnnannencr oromy AYUY TOA m TONO Tora,
xorga 220, 7. e. np a >0.

Mionserno perenuk nepnnenerna (0) — orpeaox Aymlay: xh
re x, M Xe — aGeuccs rowex nepecenenna mapadoası y~flx)-
= 22 (0-2) x-2 e ocnio Ox. Bean AG Ay, 10
TENSO, F(0)50, a
Tax Kax [(x)<0 ¡una nex xCA,. O6parno, om Bunommmoren ye
opus (11), To —1€A, u OCA,, oruyaa cnenyer, “ro AC Ay. Tex
oGpason, YCNOBILAN aanauı VROBACTROPAOT Te 1 FONDO Te ¿NOSE
mua a>0, ana koropwx numonnaioren Hepaneneraa (11), 7. e
1C1)=14(a-2)-2<0, /0)=-2<0, oryna naxonım 03053. 4
Banana 12, Haltrm nce IHaeHHS a, MpH KOTOPHX Crema 2e.
papencrs

y > EL,
se 1074 -5yt<-2

meer perenne.
[> Divers ay — Snauenne napamerpa a, mpm KOTOPOM ana exc
ema meer pemtenne (xo; Yo). Torga pepa Hepanencrna

2

184200 + TESA
3x8-10x0y0=Sy Bs -

=a as

264) traes vin

pan u HERABEHCTER © AEWA EDEN

Cno»mp Bropoe HepasencrBo cucremei (12) e mepBEIM, YMHO-
4
xenHEM na 2, nonyunMm BepHoe HepaBencTBO (x9~3yo)*<——,
yann nepHoe Hep Gaye

oryaa cuenyer, uro 250, r. e. ap>1.

4
Lao

Mak, HCKOMBS snauenun NAPAMETPA VAOBICTBOPRIOT ycu0-
amo a> 1.
‚Honamen, 170 Ana Kaxnoro a>1 xamuaa cnerema mueer pe-

enue. Kena a>1, 70 1-2 >1. Hosromy acrarouno noxasars,

«ro cucrema ypanmenmit
xt 2xy-Ty?=—1,
(era as
Meer pemenste. Jloßoe ee perenne ananercn pemienmen nexon-
vo eneremm nepanenern.

Tipu pemenun cncrem (13), Kate u mpu mpeo6paaonanıım cu
crear nepapeners (12), cnoxwm propoe ypanHenne cncren (13)
© nepmim ypapneunen, ysmoenmum ua —2, Honyanm ypannense

(8) -0, as
xoropoe secre ¢ arobmm na ypasiemni cnerem (13) oöpaayer
cucrey. pasmocnasmy1o cmerexe (13). Ms ypasuenua (14) exeny-
CT, no KB, m Toran uo nepuoro parues curena (13) nad

1
NE Tiposepka nokassinaer, «To 0Ge map acer

(314) (221) acer ananas cani ete

«ir, m pemiennnmn wexonnoh CHCTOMA epaBeners.
Orser. a>1. 4
Banaua 13, Halirn nce anauenun uapanerpa a, np Koropkix
MXOECTBO pewiennii CHCTEME Hepawexers
Prat ll,
star so
conepoxur orpesox 1 e Konuam » rouxax All; 0) u BC; 1).
Dyers rouxa M(x; y) npmuannemur oxpeary I, sorna x=1,
O<y<1. Tlooromy sanatta CROITEA K Haxomnenmo ana a,
np KOTOPHX CHCTEMA nepanenern
w-ars9, 5
re as
nonyuennas Ha Aannoh cıeremst mpn x=1, HMeeT pemenna npıt
06om ye [0; 1]. Cnerema (15) pannocnnuna enereme

y-8<acy+3,
Ecam 0<y<1, ro ma cucremsi (16) cnenyer, uro
-2%a%4,
as-2 an
Venosun (17) BBINONBAIOTER TOABKO DpH a m
$9 265

PARENT I HEPARCHETER © ABN FEPEWSHHENN,
coneprauye napamerper

Yopaxnenn

Haüru Bce sHatenis a, Up KOTOPBIX HET HH OXHOÏË napa ae

ersmmensunx uncen (x; y), yROBACTROPmIOMEÑ ypaBıenRo
2x? 4x4 2y?+ By +10-a=0.

24] Hasire nee anaemia a, mm woropux malígerca xors Out ox
a napa nelteremrempmiax uncen (x; y), ynoBnersopmiomas
ypastenmo Ga? +axy+y?+8ax+8y+20=0.

dax-y- Bard,
] Hana cucrena panne

amit vp (iesnentary-t8000,
Haiira nce ausuenns a, npit Koropuix evlereNa ypannenult

a) meer eaunersennoe pemenne;

© ne mueer pemenmi;

2) sneer Geckonennoe MHonectso pement.

Hafırn ace snauennn a, mit KoTopux cucrema ypasstenui

Argel,
(ETAT soccer a0 pemenun.
27] Haïrs nce smauenns a, mpm Koropux cucrema ypanmeniii
PURES
E à meer ennmernennoe pemenne.
a+3-\y- ax) ss ms

Hailes nce anauenna a, npit KOTOPHX CHCTEMA ypanxenttt
log, (2-x~y)+2=log,(17—8x-10y),
(ena) +x-y+a+6
Meet ponno apa pemenun.
Hafirn aco anaemia a, mpit Koropkix CHCTEMA ypannenuä
21xl+lyl+I3x—=4yl=10,
Aya

meer pOBHO ana nelieranrenuisx pemenua.

80] Hair nce omauenna a, mp Koropsıx cmerema ypanmenui
y=x-2x,
+yi+a=2x+20y

BL] Hatten ace omasenna a, mpm Koropsix cmorema ypanmemi
1og,(y—8)-210g4x=0,
(x+a)?-2y-5a=0

[82,] Hañtrx ace sxavenua a, npu Koropsix Ker Hu oxHoli Maps aeli-
eteurembmsix uncen (x; y), yaoBuerBopmoueli HEpañencray

x 4+6x+y—dysa.

83] Bepmm B, C, D napanzenorpamma ABCD uwetr coorner-
CTBeRKO KOOPAHHATEI (~3; 2), (2; 3), (3; —4). Hafirm pce ama-
MeHHS A, AN KOTOPMX KOOPAMHATH nepmums A anamoren

2x-y-2a40,
2x+6y+5a<0.

weer xors Get ORIO perenne.

meer xora Ost OAHO pemenne.

pememnen encens nepanenore |

266% rnasa vun

Ypannarum u nopanencrea © ANR RopeMORMA

BA] Haiten ace anauenun a, npn Koropux mmomecrno pemennil
x24 (a+4)x44a<4y,
Bx+y-(2a+4)<0
conepxut orpesox AB, rae A(- ), Bl 2

Haïtru ue anaucunn napamerpa a, pit KOTOPLX CHHCTENA Hepar
Sxt—Txy rato S041,

EA MEET XOTA Gi OAHO perenne.
By y

en |

|

30. Maira nomeerno rover KoopRaTHOR naockocm, yaonzer-
pila rot
1) 2x-y45>0; — 2) x-3y+4<0.
37. Hañru MHOxXecTBO TOHEK KOOPAMBATHOR maockocTH, YAOBAET-
Soe eee ee
DC soso:
ps >
er Frae2>o.
nasa oo
Deo Diet Det
2) (3x4+4y-12Mx42y+2)<0.

39, Ilyers M — moxeero Towex MAOCKOCTH ¢ KOOPAHHATANIE
(x; Y), Taxux, uro uncna x, y u 6-2x ABAMIOTOA montant CTO
pou Mexoroporo rpeyronsmusa, Hair naomax @urypa M.

40. Hañirm sce maps neasix uncen x, y, AIR KOTOPNX nepnn
mepanenctsa y—8x<1, 2y-3x>19, 4y—x<78.

41. Hafırn wmoxxectso tovex KOOPANMATRON naocKoctH, YROBACI=
BOPAIOEX ypanmenmo:

1) 2x24 2y? +4x 291
2) 6r?=xy-y?+2x—y=0;
8) le+1lely-21=1.

42] Haürn sce napus aeñcrsuremnux uncen, an Koropux enpa-

Bexanno PARCHETBO

o

48. Haïrm MmOxecTB0 ToueK KOOPANHATHOÏ ILIOCKOCTH, YAOBNET-
nopstonprx. nepanencray:
1) 3x%+3y?-6x+12y-11>0; 2) [x1] 4ly+2/<25
3) + y<olyl

ETES
44, Hana cuerena nepanenern | x?+y?>5(2x-2y-5),
(1x4 By +15) (x4 4y—5) 0.
267

Vnpammenm x mmase VIT

Hañra mouans durypst, Koopaunarsı TOuex xoropoñ yno»

8) ncem Tpem nepanenernam CHCTENI,
ÓN
45. Hana encrema nepapenera || x|+lyl>2,
y -x*4+16-8y>0.
Haïru nnomexs urypm, Kooprumamia tower Horopoñ yaos-
ernopator:
a) nepsomy HepaBeHCTBY CHCTEME; 6) nepBEIM ABYM Hepapen-
ETA CUCTEMIA; 3) ucon pen Mepanenicruam chere.
46. Hara noomaxs urypst, sananmoll ma KOOpRMATHOË naoc-
Kocrn eneremoh nepanenern:
ferien, 2 (use
CESSE ME
[47] Haire nee anauenns a, upu Koropux uneer eu
pemenne cuerena ypannennikt

D een a Aminen
x+2y=t; x(y-a)=a+(8—2a)y;

olas Dy=a-2 4) axyrx-y+5-0,
2lx+1l+oy=2; ES

Haliru uce suasenun a, ups Koropux enerema ypannennä
TE ES
utattxtytanT
meet popno ana pentexts.
Hañru nce suasienust a, npx KOTODHX cnorema nepanonern
æ+2uy,

Heer enunernennoe pemenne.

Bepurumer A, B, C mpeyronnmka mweior coornereraenmo no

opaunatu (—2; -1), (0; 9), (8; 1). Haitrm:

a) ace auauennn a, AUR KOTOPLX KOOPJUIMATAL TOU nepece-

Senna mexnen npeyromna ABC aBamıorca perennes cnc-
2x-y+acO,

6x+3y+54>0;

6) pce sHavenna a, Aa KOTOPHIX KOOPAHHATEI XOTA Gut onnok
TOUKH orpeska BC ABAMIOTCA peluenHem atoll cuctemsi.
SL] Harn nee annuennn MApamerpa a, mp KOTOPHX MNOECTEO

pemennit enerestui mepanenern
(ear? a,
x-atys
COREPIKHT OTPESOK e KOHLAMM B Toukax (1; 0) m (1; 1).

rex nepanencrs (|

268 Fnona vin

pans u HOPAONCIOS € ABYMA NEPEHERMUMIT

Bonpocsi x mage VII

1. pxzects mpumep ypannennn mpanoit.

2. B kan rouxax KOOp/UIMATHOR ELIOCKOCTH nnmonmsercn ne-
parenero Ax+By+C<0, ec B>0 (B<0)?

3. Hafırm mmoneorno roueit Koopammarıoh naocKocT#t, yaonaer-
Bopmmomux ypannenmo 4x*—y*—0,

4. Banncars ypannenne oxpyuocrx € nenrpon # vouxe Ala; 6)
m panıycom Ro

5. Hañiru muoxecrso Touex Koopaunaruoli naocxocrn, snnso-
muxes pemennem nepanenerna (xa)? +(y 07 < RE.

6 Uno npexerapnaer cobolí wuoxecrno peuienit enerems mepa-

Aix + By + Ci >0,

— Be or

1. “ro npenerannaer cobol muoxecrso peuleuttit nepanenersa

Aux + By + Cy) (Age + Bay +Cp)>0,

me Ayx+Byy+C,=0 u Agr +Byy+C,=0 — ypasnenus nepe-
cexaromyinxes psinanix?

8. Tipuseeru npunep uexmeioro ypanuenna e Ayma neitane-
rm, PeMICHHeM KOTOPOTO ABAHCTCH CAMCTHENNA Tapa
con.

Mposep» cet
1. Haliru smoxecrso TOICK KOOPAIHATUOÑ MLAOCKOCTH, ynopner-
Bopmiouuex ypanenmo:
1) 2x+8y-1=0; 2) 224y?-6x+2y-6,
2. Hafen muorcorno Tone KOOPANNATUOH unocKocen, yaonner-
BOPAIOLNIK Hepaneneray:
1) 2x~8y<0; 2 (+ 2) +5 cd.
3. HsoGpaante Ha KOOPAMBATHOÏN MAOCKOCTH MHOKECTBO TOUEK,
KoOpnuuarLı KOTOpLAX YAOBACTUOPMIOT CHCTENE HEPACNCTE

Ea

a-y>L,
z+2y>-2.

1. Pemmrn nepanenerso (x-2y+3)(x+y-6)<0.
2. HaoGpaanrs na Koopaumarnoñ nnockoern durypy ®, sanan
3y+V3x-3V3>0,
x24 y?-2x-3<0,
1 Haïtru oma sroh durypsi.
3. Hafırm wmoxkectso TOTEK Koopnunarnoii naocKocT#t, YROBNET-
sopstonur
1) ypapnomito |x|+1yl=3; 2) nepanenerny [x1+Iyl<3.

269

Tiposepa cod?

HB 74 I Mcropueckan cnpanxa

Ppamyackne maremarme XX», A. Pporenqmec 1 IK. sono
He u onHoli us CBOHX crareli cnpasennnso ormermam: «MoxHO yr-
Bepnars, ITO peienne HONHMOMIMOJEMAX ypABNENIH nOCAYA
10 MCTOPHYECKH HCTONHNKOM AAFEOPBI M TO CO BpeMen BABIIZO-
mam, unnycon u Hnodanra u xo nanınx ameñ OHO ocraerca onmol
sa ee ocmonmx enel

HoGamnw or co6s, uro pemenue ypannennit, wepanenern u ux
cuerem ABARETCA OAMOÑ ua OCHOBNMK coRepxaTeMbMBIX ANNE
unkoasHoro Kypea anreöpsı. M970 neyaunnmensno: ypanmense
none passé BH Maremarimeckoï moxenn. Maremarnuec-
Kite Ke MOAEAH — OAMH 119 OCHOBHBIX HHCTPYMERTOB HOHAIS ve
AoueKom BACH oxpyzkmomero MAP.

Vpapmenna u ux cucremm (amelie u nerumeuste) mer
KABMIO!O MCTOPHIO. Tax, » APOBUCBABIAONCKNX TEKCTAX, Mamcan
max » IV—III pp. 0 I. 2, CONOPAK0TCA JAN, PEITACME € no-
NO CHCTeMBI YPABHEHHË, OXMO 13 KOTOPHX ABARETCA ypanne-
nem sropoit crenemm. Bor OMA ua rAKHX anau: «TLnomaan anyx

coms uaaxperon a enema M monyuna 25-8. Cropona amer)

Knanpara pasta 2 cropoum nepsoro x eme 5. Kaxost croporx
aınx kbaxpator?» Compemenmas sannch yenosnit aroñí sanan
seer aux
x = 255
timos,
2
ya pats.
‚Ana pemenna sol cuctemst BABHJONCKHÍÍ ABTOP BOIBOAMT BO BTO-
2 2 5
pon ypammenun y 2 wnanper, nonyian pt= date 2 0425, Now
CTABHB 9TO suaueune Y” B MepBoe YpaBHeHHE CHCTEMEI, ABTOP 10-
Ayuner ypammenne € ona sexabecrasin
O
14x46 220255,
peus Koropoe naxonur x, a ATEN HY.

BB oro Fane nit HOSMANONNANCA € Peimennen pass ei
crex nenuneñnux ypasmenut. B Texnonorun coppenennax Bs
“Cru Taxue ypabuenin m ux CHCTENM BerDeunNDTen LAC,
‘rate Ka Monean PELMMUX MPOLECCOR NO MON enyunıx AN
onen meme.

[

YupaxHenua

AJIA HTOTOBOTO IOBTOPeHHA
Kypca anreÖpkI H Hayas
MATeMATHYECKOTO aHaJim3a

Ynenue pewam» sadasu — nparemuxecroe
uerycemeo, nodoßnve naueunum.

uau kamanum na aux, Lau uape na
dopmenvano: nayeumres amomy mono,
aus nodpawan wspannrise o6pasyas

u nocmosnno mpenupyaen.

A. Hoùa

1. Bbiuncnenna mM npeo6pasosannn
1. Halıu «ueno, ecan 42% ero pas 12,6.
2. Kaxoit nponenr cocrannner 1,3 or 39?

3. Kakoï upouent cocranıner 46,6 or 11,65?

4. Halırı 180% or 7,5.

5. lena ronapa Osma cuwmena cuawaxa ua
24%, a aarew na 50% or nonoñ mem. Hañ-
‘rn OÓUIÍ mponenT CHHXEHHA Lenk TOBAPA.

6. B cnaase coxeprorres 18 xr mimica, 6 xr
onona u 36 xr Mens. Kaxo»o nponenrnoe co-

Repxanvle cocrapmax unereh enana?
Crouwocrs ToBapa 1 mepenoskH cocramnaer
3942 p., npunem pacxom no mepenoake TO-

Rapa COCTABASIOT 8% crommoern cumoro 10-

papa. Kaxoza ctouocth tonapa Sea yuera

erommoern ero nepevoaKH?

8. Bucora pana panna 5 cx, a naomanb
ee oemonannsı pana 4 cm. Ha cxombro npo-
HEITOD yneamunten oÖsem 9rOÏ uMpanaı,
can u UAOILANb ee OCHOBAHKA, M BBICOTY
yeansurs na 10%?

9. Tips nenenun mexoroporo wmexa na 72 mony-
irren ocrarox, parmi 68. Kaxun Oyzer oc-
Taro, ECAH 970 wo «en paspennrn ma 127

10. Cyuma nayx uncen papua 1100. Haïru nau-

Gommee m3 unx, ecam 6% onuoro “nena
pas 5% apyroro.

—— MEE
Ynpaxnenna Ana wrorapore HOBTOPOMAA RypCa
anreGpui u Marian MATEMATIMECKOrO ananmaa

u.

12.

13.

[BIBS

ing
IS

E

1
8

x

8.

272

Tlo Bkxagy, PROCHMOMY Ha CPOK He menee rona, cepéank
prnzanmsaer 3% ronopsıx. Baume puec » cGepGant unex
» paamepe 6000 p. Kaxyio cymmy xener on nonyanm n one
Broporo rona co nun Binena? = Kone Ipersero rona co aut
ska

To o6sruHomy skxany cóepamk sunaanusaer 2% rogomux.
Brenz mec 5000 p., a nepea mecs cs co cuera 1000 p.
Kaxaa cymma ¡ener Öyner na ero Cuery mo MereMenH TOJA
0 ans pias emy 1000 p.?

BupaGorka mponyxnsn sa nepnurh rox paGorss mpeamprarra
Bospocaa ma p%, à 3a enexyrountit rox no cpannenmo e nep
romamambuoli ona noapocna ma 10% Gonpue, vem aa meparil
Ton. Onpexenurs, na CRONBIO MPONEHTOR YNOxIILAOCE pupa
Gorka mpoaykumm oa ueppmÄ TOR, ecm MIBECTHO, YO a nea.
roma ona ynenmunnach m oÖmeh enomnoern na 48,59%.
Toxasars, wro npx mo6om npocron p>3 «meno p?-1 nenur-
Cu ma 24.

Toxasars, wro mpm mo6on watypansuon n>1 sono n'+4
‘apaseres cocrapHsin.

Hoxasars, “TO np MON narypannnom n:

1) 6n°-11n neanten na 5;

2) n°-n nenuren na 7.

Hoxasars, uro 30-20" nennen na 85 mpn nEN.

‚Noxasarı, uro n°-5nt+4n xeaures na 120 npn EN.
Haiiru nocneniiow unhpy "mena:

19% 22%,

Haïtra ave nocmeaune wiper unena;
1) 2%, 2) 3m,
Tenures au na 7 wncno conera sch ua 1000 anemenron no 500?
Hoxasars, 170 npondsexenne 110GK1x À nocaenoBaTeNbHAIX na
Typammamıx uncer gemrea na nl.

Haiirm nenanecrnuit wen nponopuun:

exil 2) 2:0,75-92:144; 3) 20148
VOL feail 4s Dreh 9 mA,

Baruncanrs (24—28).

arta sat

15

i

125

. 1) loge; 729; 2) logy 729; 3) log, 729.

3
1) log à VE: 2) logslogsoes16.

Ynpaxnenu ana UTOFOONO ROBTOPONNE Kypca
anrebpuı m Havian MaTeMaTINecKOro ananıaa

e y (a

Ya (ares.

28. 1) lors som VB 2) 10050000

lg Is

IE IE

a.

Ya

A
Haïtru snasenne seipaxenna V36"%— 5°",

Cparmurrs nena:

1) 2,57 1 2,508, 2) 0,2% 10,24;

8) logs, VIO m logs, 3 4) logos à 110807 3.

Kaxowy 19 mpomemyrxos O<a<1 mam a>1 npumannenur
uneao a, eau:

DO Ya. 3) ar

4) a7<1; 5) log,0,2>0; — 6) loga1,3>0?

Kaxoe na umcon Gost

2

yet

TE nan 4%: 155 umm (2
1) VIE nan 4 + 2) VIE man (4

MO KA nennen aaxtoseno “amero:
1) 1850: 2) log, 107

Cpapuurs Ges Tabanu u Kanskynnropa anexa logs4 u V2.
Toxasare roxneerso log, a -log, blog c = loua.

Yupocrurs (36—37).
Day Ass ER;
1

38 4
Ve-V8 V5+12 ve-v2

. 1) Va Gar: 2) VDD +1).

Ocnobonursen OT uppanonansuocru E ananenarene APOÓH:

Dri: e 9 ——.
13-77 WW vies

Oenoßoanrucn oF mppunnonamuoern B unemrene ApOGH:
YE, 9) 88, y 7%

DE an

Banncarı » unne oGurnomennoñ Apobm sncao:

DO 22M 3020
4) 1,86); 50,865 6) 0,21(8).
Banuteats » pue xecarmnolt nepmonumeckoii xpobH “meno:

Dé 924 94 0

i

YnpaxnonmA ATA BTOTODOTO NODTOPONPA HYPER
anre6pei m Hasan MATMATINNECKOFO anna

42,

ES

4.

50,

SL

Moxer an ÓSITD PARHOROTBEBIA ICON:

1) CyMMA VX MONONHTERLUNK HPPANMORANEMEX HEED;

2) nponsnegenne ABYX MPpanmonanbuBx UCI;

3) WactHoe oF nenennst CYMMBL JBYX HPABMMX HPPAUHOHAR-
HEX DONOKHTENHUX AHCOX na MX NpoUsBeAeHHE?
Joxasar», uro ecam a u 6 — narypansıme unena m Vab —
paunonansuoe uncno, vo \/

a ecam ab — mppamonansnoe uneno, ro mE — mp:
Pannonannoe uneno.
Tiyers a — pannonaanıoe weno, 6 — ppannonaanuoe une“
0, @#0, 620, Hoxasars, “ro a+b, a-b, E, 3 — mppanno-
mamo sea.
Hmeior au oGue rows npomeryrin:
1) (1; 8V2+2V7] u [8V3+4; 155
2) (0; V2T+V6) u (148—1 10);
3) [25 2V5+2V8] u (3424422; 11);

or 2
2) 151448] (Es ap
Niyerr 0<a<b. Joxasarr, uro wa unenonoi oes:
1) touxa 25" — cepenuna orpeaxa [a; 0);
asde
THe
Buinoanur» xelieraus:

D es+n(1-10) 2) (-5+VBi)(-6-3 V2;
9 (+ DC142)+ 1-8 A) B-2NA+I+ 105

2) rouxa

, re c>0, nexur muyrpu orpesra [a;

ED a.

Dana 9 How
à

m. atra?

cam

ye » (14:

3) (24+31*-(2-80% 4) (3440 + (8-41).

Ha xounaexenoh nnockocru nocrpoums ro:

1) 5; 2) 2i; 3)-31; 4) 3+2t; 5)-2+t 6) -1+i.
Noxasarı, pavenerno

A] ta
ff zero.
Hokasars, WTO AA MIOÓNX kounenennx wneca 2, M 23 pt“
BEJJIMBO PABEHCTBO 21 +29 =21 422.

Ynpaxnenun ANA WTOrODOro NOBTOPSMAA EPA
anrebpuı m Haan MATEMBTUHEOKOFD ananına

52

53,

Barca:
DER, DH
Hairy snavenne nupaennn:

Bloq Atal y A CAÑA
y Lido, 9) Ay AB ( KE )

4 Ah, Pa 10 10) ag
1) Buwieawre JANET x Kpyra, BIMCANMOTO » pasuoeropon
xi rpeyronsu (pue. 138), een a=6 em.

2) Bewimcamr» yroa a SarOTOBkH, nsopaxexuoll ua pueya-
Ke 139, een a=4 cm.

Beranenuers wnpmuy I VILLA no AAHNEIM, YKODAMMM HA pH
cynxe 140.

Bsruncaurs nanny MOCTA no AAHNSIM, yKasaHHBIM Ha pHCYH-
Ke 141.

. Hall “nCAOBHE auavenna BOX OCTANbHBX TPITOHOMETPU-

MRCKHX (PVEM NO HAHHOMY IMANEHIIO OXMOÍ HA Nix, EC

5, y 7
55 3) tgan2,4; 4) ctga~ J.

D conan 0,85 2) sine à
2) ein(arccos(—-f).

‘Vipawnenua ATA VTOFOBOFO MOSTOPONAR HYPER
anre6pui M MANN NATOMATU4ECKOrO ananıza

Burauenurs:

1) cos(aresin $

59. Paanoxrs Ha MHOHHTOME MROrOUNER:
1) xt pax? 2524046; 2) xt 2x la.
60. Coxparirs pod:

242, EAT
Data ar Fr Fr
Meat, PRES
a 24 35 + 6° » 2x5 5x? 2r-3

61. Haiirn pasnoxenne Gunoma:
DD 2) (+3.
Vnpocrurs Bripamenne (62—64).

a+2,( 2a°-a-3 , 2a
2, eel
a +5046
1), 80°+86+2 2041
ay (auf) eee me,

1 2a
2 E
) Frtara * Arsars
1 1 1
64. 1) 1 _ + a =
Va E ae av2-2-V2 +2

65. Yupocrurs prpamenne u nalıru ero snauenne:
aie,

D (HE) (VE) mon as, 2-4:

2) at
a-Ve

Vnpocrrs suparxenne (66—72).

apa a=3, x=\B.

eat Vax

Ynpaxnonun ana HTOTODOTO nonroponm KYPCA
amreGpu 1 Hasan MaTEMATWIECKOFO amamos.

i »( arvas _ N =) BARTH

Varrab Vabro Zub

e (qe petete.

—(0*+180+81)05,

=} (dog,a+log.d),
ecau a>0, 6>0, wee, c>0, ev 1.

Hokasars, "ro
tog, Pedo ocio

ecan a>0, b>0, 1300” Aa? +96", 0 ext.
Bmpasurs logs 9,8 uepes a u b, ecau lg2=a u lg7=b.
Bupasurs 108,58 nepes a, ecan logi8= a.
Yupocrunu:

Irtgta 1
D 2) HE + ct a)

EXT

1-(ina+ cosa)?

Tloxaaarn De NE EE

=2tg?a.

Vnpocrurs »srparenne (79-80).
1) sin? (a+ 8) + c08? (a+ 10m);
2) cos? (+ Gr) + cos? (a 4m).

Za, sinacos(r—a)
20-2000) 1-20inta |

. Hokasars roxxeerso

sinx—cosx.

Ynpaxıcnm Ann HIOTODOTO nosropennn KypCa
anroGpui 4 Navan MATOMATUSCCKOrD ananuaa

277

82.

87.

89.

a.

—

Pasaron na sonore:
1) 1+cemarsina; 2) 1-cosa-sinas

3) 3-4sin*a; 4) 1-4cos%a.

Hoxaaars, sro ecam u+ß+T=n, 70:

1) sina+sinß-siny-Asin 3 sin À cos 1;

2) ain 2a +-sin2p+-sin2y~=4sinasin ain.

Manecruo, wro tga=2. Hakrn anauenne mapastenna:

1

sintarsinacosa yy 2-sinéa

cost a+ 3cosasina Brote”
Haneerno, uro tga+ctga=3. Haïrn tgta+etgta.

Ynpoerurs supaxexne (86—90).
lit) DEE) Toa

sn (Gs9)-ve( te)

y 4 te,
ane ra) ren (tre)
BOSE , 4) (sina+cosa)'+(sina-cosa)?.
8) EE 4) marcos y
» teta N 2 Lrcteña ,
roda cata
DURS 4) (garganta.
1400820 , tga-sine,
» Zeosa * 2) tgatsina’
DE En
sin 2a 1 cos 2a +2sin?a | cos 20-sin20.
» sin(-a)= sin (2,5x+0) * 2) ‘cos (-a)-cos (2,5x-+a) *
Mloxasatn romecta0:
1-cos(2r-20) _ 9, sini (a+ 9 er
2 nem 71+ 20800 90).

Ynpocrurs sepaxKxenne (9297).
_Boose-Ssine | sin2x—Bein®x

Je =x) +sinta)

08 2

sin(+-25)c0s(

x) a+)

Ynpaxnenm Ana HTOrOGOrO MIONTOPOHAA KyPCA
anreGpu m Hasan MATEMETANECHOTO akanıaa

94, 1) cos*(a+28)+sin*(a—2p)-
2) sin?(a+2P)+sin?(a-2P)-1.

oos4a -cosZa 140060 + cos 2a + eos a

Æ D ndasina * erre Per

86. 1) ;
Y2-cosx
, 1) ne à
SE D mecs sinx cos
Beranenmms PL, com otga= 3,

Fin? a—cos? a

Ynpoerun, mapascoune

2-Seinta _ sina+2cosa
cos2a sine +cona

la 18

u naitrm ero «mexosoc ouasenme npu a=- E

Toxasars roxnectso (100-108).
tea) te cosa)
War pie ” cos(a—D)
101.1) 1 +sina=2c0s*(F

4

102.1) sn(u+3) sine

2 arte

103.1) 1-tgta=

104. 1+cos a+ cos 2a=

1-2sin?a _ 1-tea |
Treinda Ira”

3) te(F+a)=

105. 1)

Lisin2a
cosZa

106. 1) Asinxsin( à x) sin(3 +2)-sinaz;
sinzar
Bsin3e”
J. Banucars B TPHTOHOMETpHUECKOR popme umen
12; 2)-8; 935 4-25 5)V3-k 6)2-2i
108, Banucars » axreGparmieckoit popme konmaerenoe weno:
= 2) 6(cos 2

1) (cos +isin ZE

2) cosBxcos6xcos 12x=

279

Ynpaxnenun ann HTOFOBOFO POSTODOMAR Kypca
anreGpui la Hasan MaremarmecKoro amants

109. Burnonnwm, AOÑOTBHA sanuears PEOYABTAT B anre6paunec-

xoñ ope:
1) 8(cos à + sin à }s
BR join ®*) (con ® à rai
2) (1.4(c08 D +isin 32) (cos E +isin
Ta rain TE
cos TE isin
ne ge
Be cos *+isin®
ara
2. Vpasnenna
110. Pears ypanmenne:
%-16 ju 246 249
1) S16 4 1 248 _ 248
Ion ED (x4 2),

111. [pu xakom anaueumm a ypannenue a(x-3)+8-13(x+2)
meer Kopens, paBniait 07

112. pu xaxon onauenus b ypanmenne 1-b(x+4)=2(x-8) mue
er open», pam 17

Pere ypapnemme (118—122).
a 4 5, 2 u
113.1 2 2 4
de 3-8” Bao PTA
114.1) (@-b)x~a*+(a4d)x; 2) a®x~a+b4b x,

115.1) (+-3)(-2)=6(2-3%

ai. 3
+27 23020

120. 1) x*-11:*+30=0; 2) 2xt- 5x2 +2=0.
121.1) 2x 244 1480;

119.1) +

122, 1) rar E

280

Ynpaxnemun Ana HTOTOBOLO HOBTODEHNR KYPCA
anreGpui u Hasan MATOMATMNECKOrO ananaa

123, Peur ormocntexsuo n ypapnenne:

CA 2 56;
D Fi =1 2) Al. = 156;
8) Chm Fe Cay as

15

124. [Ip xaxow yCNOBHH TPEXUACH aX? +bx-+¢ ABNNOTCA KBANDA"
TOM aeyurena?

125. Hokasars, uro xopun ypanmenun ax? +bx+a=0 ecrs saaun-
mo oöparune uneaa, ecam a 0.

Penuurs ypanuenne (126127).

126.1) |2x-3|=7; — 2) |x+6l=2x; 3) 2x-7-1x-4l.

127. 1) |6-2x|=3x+1; 2) 2lx-2l=Ix)-1.

128. Haiten namens kopen» ypasnenns

I2’-3x-61= 22.

129. Hafien nanGonsinnii pausonaammnñ Kopeur ypannensst
It-8x+5l-2x.

Hair geiterantensusie Kopan ypannennat

D Pate; 2) 8 4x4 1200

3) x*-5x248x-6=0; A) IB 680;

5) 46141130;

92-228 -112°-x-6=0.

Hañtru geiicrsurensusie kopun ypamenns (131—196).
181, 1) (2x + 1)(8x + 2)(6x+ 1)(x +1)=210;
2) (z+ D +228) 10.
182. 1) (æ-1)- 3904 2)G+ 6)=72.
2) (= 1) e- 2) 3)(x 6) 3622.
133, 1) (x* — 5x +4)(x* + 9x + 18)— 100;
2) (x? -3x+2)(x’ - Tx+12)-4.
) (x? = x ~ 2)" + (x? — 2-2) (x +3) 20(x + 3)
2) 4(x® 4x41)? +10(x- 2)? = 18(x*—4x4 1)(x-2).
waned, toss mixer, starse
HU tas Dias sects 1
1 1 I; 2 (2 3 9
ne nel
187. Vlepecexaer an rpadune dy y= 2% 6x2 + 11x-6 oc» Ox
B TOUKAX, AÓCUMCCA KOTOPEIX ABIAIOTCA WEARING UHCHAMN?

188. Ypawuenne 2x'ımx’+nx+12=0 umeer Kopum xi=l,
. Hafıru tperuit opens ororo ypanmonsist.

18:

242.

281

Ynpaxnonun AA BTOFODOFO TIOBTOPONVA KYBCA
anredpes u Havan MATOMATINECKOrO ananıaa

139. Moryr an Kopun ypannenus (2—m)(<—m)=k Guri unero
MUMBAI, EC m, n ut À — nellcranrensesie «nena?
Pemurs ypapuexne (2 — KoMIAeKcnoe 1020) (140144).

140,1) 2*+42+19 2) 2*-22+3=0.

141.1) 2(2+0-7=85 2) 5i-2@-2)--1.

142. 1) l2l+i2=2- 2) lzl-iz=3+24.

148,1) 2°+3-0; 2) 92-12
3) 2424520; Mori

144,1) 2-25i= 2) 2=-846%
3) 21+8— 4) 2-

[145] Haïtru nce yeawe snexa, panne cywme xnanparon cnoux
mp.

[146] Peuurs » wensix unenax ypasnenne:
1) 2x¥y? = 14y? = 25— 2%
2) 3x?-8xy-16y?-19.

@7JHaiiru nee naps wenux uncen, cymoa xoropux paa ux
nponsneremmo.

Pers ypasueuute (148—181).
148, 1) V2e4T=2+2; 2) x-2-VOx=8; 8) Verl
149. 1) VETA VIENE: NEP VE Zee ET.
150, 1) Va"—2642Vx"—26-3; 2) Ver Verte V0.
151. 1) Ve Ge VEB O8

2) Verde 6249

3 VE -VB-DOTrIH Verre

4) VEZ + VETE Eb.

152. Haitrn see wncaa a, ana KOTOPHX BRTOAMACTCA yCAOBNE

4-2 =0,25 7,

Pemwrs ypammenue (158—172).
163, 1) 37281 2 BHO a (La) ant,
154, 1) 9-98 18; 2) 2744227120;

Br 4) gp ge
5) PILAS; 6) 824810.

2) 0,28. 582 -(2)8

185.1) gratte
156.1) 202,005 (3) (8)
282

YVnpaxnenns Ann HTOTOSOTO ROBTOPEAR HypCa
anre6pu u Havan MATOMOTANECKOTO ananıaa

Ay (23122, VU.
167.0 (UD D Var Var 216.
158. 1) 9-72; 2) 47-2148,

159. 1) 0,5%=2x+1; 2) 2%=3-x% 3) loggx=4=x5

4) log, x=40
100.1) Goga? -Blogax+2=0; 2) (logy 2} +5 2108

161, 1) In 27 =Intr+2% — 2) loggV8x=6-logs

5 Is a
162.1) e(}+z)mle die 2) lex
163. 1) logy (2x — 18) + logs (x 9).

2) le +19)-le (+ De 1.

ote? G50 à 50: NT

164, 1) tn 2) 25/%%-4.5
165, 1) x"*-10; 2) 28% 92;

3) AOL FeV.
166. 1) 7-4°°-9-14°°42.49°°-0;

2) Bag rara,
167. 1) log,(24+Vx+3)=15 2) Ion, Vara 2x--1;

8) Loga(#+1)=l0g, VETA —2logs

108, 1) alt= 10x; 2) x= 10035
3) log, (17-2) + Log (2° + 18)
4) Log, (3+ 2°) + log: (6—2") =.

169, 1) 9-47 45.64.07;

2) log, (x*~8)—Log, (6x 10)+ 1-0;
1

3) 2log, x 2log, À =8 Vlogs xs

4) log, (2x? -8x-4)=2.
110. 1) 1+l0g,(5-)=log;4-log, 7
2) (logo(7-2)+ 1)logs.3~ 1.
EEE 1) logs 3x + logs (4x + 1)~ lowe, 9
2) logs X + log, (21x-2)-2106, + 4,8.
EB 1) log, ctg x = 1 + loge ($ cos2a):
2) log: (sin2x - } cos) = 4 +loga(-cosa).
[178] Peuurs ypasuexne:
1) logs sin 3x sin x) 2 logg(17 sin 2x) - 1;
2) logs (sin x -cos.x) +1 log; (7+8cos42).

Denkens 6) ($)"=logyx.

e.

-125,

289

Ynpaxnenn ANS MTOFOBOTO ROBTOPERNR KYBCA
Anro6pu u vavan MATOMATINOCKOTO ananıaa

[174] Hañtru nce pemenus ypannenns, yaonnernopmorne xannony
Hepanencrs}

1) V1 og, (8x8 — 242) ~ logo (x*- 8x), sinx<tg2x;
f PRE: =

2) Vi (o + Ex)-loE 67,0, 0=1, sinx> te Gx.

175. Penn, ypasnenue cos(3x- 5) ~ bm yrasars aroGoi ero no-
AOMITONEMAA kopen.

176. C nomomo rpadaka emyca wom rocmnyca nara nee Kop-
HM ypannenun, npimaatexamne npomeryray [-r; 3x]:

$3 2) sinx

1) cos x A

Pers ypanmenue (177—192).

177. 1) sin2x= 1; 2) cosëx=— À; 3) 2tgx+5=0.

178. 1) 3cos*x-5eosx-12= 2) Stgtx—dtgx 45-0,
179. 1) (8-Asinx)(8+4c0s2)-0; 2) (tgx+3)(tgx+1)

180. 1) sin2x=3sinxcos*x5 2) sin dx=sin 2:
3) cos 2x + cos? x = 05 4) sin 2x= cos? x.
181. 1) sin2x=3cosx; 2) sindx=cost x-sintx;
3) 208? x= 1 +4 sin 22; 4) 2cosx + cos 2x = 2sin x.
182, 1) cosx+cos2x=0; 2) cosx—cosSx=0;
3) sin8x+sinx=2sinZx 4) sinx+sin2x+sindx
183. 1) 2cosx+sinx 2) sinx+VBcos x = 0.
TE sur) 12 pcos 28
1) Aeint x + ain? 22e 2; 2) sint ¥ + cost 2 = 5,

185. 1) sin x + co x « 0; 2) 2ein£x + sin? 2e
3) 8sinxcos2xcosx=V3; — 4) Asin xcosxcos 2x.

186. 1) sintx—costx+2cos*x=cos 2x;

2) 2sin?x-cos'x=1-sin'x.

1) sinx+sinäx=sindx; 2) cos7x—cos3x=3sinbx.

1) cosxsin9x=cos3zsin7x; 2) sinxcosbx=sin9xcos3x.

1) 5 +sin2x=5(sinx+cos

2) 242008 x =8Bsinxcosx+2sinz.

1) sinx +sin2x+sin8x+sin4x=0;

2) cosx+cos2x+cosBx+cos4x=0;

8) cosxcos3x=-0,5.
191. 1) tg?3x-4sin?3x=0; 2) sinxtgx=cosx+tex:
3) ctgx(ctex+ Loto 4) ett 2

192. 1) tg2x-Btex; 2) etg 2x

8) te(x+F)+te(x-F)—25 4) te (2x4 Det +01.

YnpaKNeHUs Ann WTOTOBOrO HOBTODEHMR KYPCA
anreOpu u nanan wareNaTuNecKOr® ananıaa

193. Pemmmts rpaquieckn ypasnenne:
1) cosx=3x~1; 2) sinx=0,5x%
3) cosx=Vx; 4) cosx=x*.
Penmr» ypapnenne (194—197).
194. D Va,

2) sin xa xt-4x45.

195] 1) cos*x-3cos".x+cosx+sin2x=2000(E+

sin?(x+ 7).

‘conx-cosdx 3
[BB] Hann nee Kopuu ypamnenna cosx+(1+cosx)tg?x-1=0,
ynonnernopniomue epanencrny tex > 0.

Hañru sce xopmm ypasnenua sin‘ x+sin

ynosrersopmomme nepanencray lg (x ~V2x+23)>0.

Hara nanGomaun un murepuazo (~2: 5) xopews yaw

+3)

senna cos (5248) +2uinzeos2r-0.
BOL) Haïñrn sce suauenna a, npit Koropux ypasmenme
sin®x+cos*x=a
seen KopHit, peurs 910 ypaunenie,

3. Hepasencrea
202. px xascrx amanennar x nonowmTensua np0Gb:

Send 32410, EM 8-x
Doug ages Dia Dora
203. Ip xatux ananennsx x ompamarernna pon:
8-28, 10-4, x
nee Die nr

Pers nepanenerno (204—211).
tg, 2% oe
SS 3) Bcd.

205. 1) 8x?-2x-1<0; 2) 5x°+7x<0.

206. 1) E 5 2) (2x? +3)(2+4)°>0.

207.1) 2 0; 2) E co

5-14 paver)

E] alos),
att q COM +08 o,
Ns » x+2

285

Ynpaxnenun An WrOFOBOrO AONOPONNR EypCa
anre6pel 1 Havan MATOMATIMECKOrO akanıaa

208. 1) |2x-51< 35 2) |Sx—91>43
3) |2-Sxl<x+ 4) [1+2x1>8-2.
209. 1) |x-1|(x4-22*—3)>0; 2) |x"-9l(x'-2x*-8)20.

210. 1) ¡2x8 |<x5 2) l4-x1>x5
8) lx?-Te+12IG6; 4) [x*—8x—4|>6;
5) l2x?-x— 11955 6) [Bx*—x-4|<2.
stat

au, rg

u Free
BAZ] Haton nce snaxonnn a, nan noropux ananeren opmum nu
door guavenmsx 2 mepapenerne:
BA, Anke,
rer Seas
213. Tipn Kanu auavenunx span
18 +82 +15)
nice?
214. IIpn kakou wauntensuien enom navn m ypasuenne
(m-1)x*-2(m+1)x+m-3=0
cer ana PASAMOS, aelicrsurensuiix Kopin?
215. IIpn Kaxux nensix smanennnx m ypapuenne
Im-Dx+2(m-T)x+3-0
ne weer neeraumensunnx xopxell?

216. Tipn xakom HAHGOxBnIEN MENOM SHaueRnm x PHARE
La
ques

Pond
mpravimacr orpamaremnoe snavenne?
217. pn KaKoM manmembiuen nero suaeunu x BEIpaxenue
ero
TR
npimaer noxoskwrenbitoe anauenme?
218. Haïtra vce nap wenbix wMeen x My, XI KOTOPIIX pepa
‘Tpit nepasenersa:
D By=x<5, x+y >26, 8x- 2y<46;
2) 3y-5x>16, 3y-x-44, 3x-y>1.
Pemrs Keparenerno (219-233).
219. 1) 2,5! *>2,5 % 2) 0,18% 420,182 +;
(dy use
DS) <() Ts 031
2051 Lyon 1
220. 1) 25<1; a) >
221, 1) SE <5 YB; 2) 0,2900 751,
222. 1) 30.9 DE SEE ET

Ynpaxnenna Ann MOTOSOTO NOSTOPONAR HypCa
anreGpui u Hasan mareMariecKoro ananıaa

228, 1) 2 — 4-14.89". 2-4> 52;
DDr gran BS Bey Oe,

1;

master a(t)" Date
naa (ltz 5) a 35(2) +630.

aus, no 2) gr

Sy
226. 1) logg(2—x)<loge(2x+5); 2) log, (x*-2)>-1.
E

$

227, 1) Vigx<g5 2 log, x<log (2x +6)+2.

: E
228, 1) logos (1 +22)> 15 2) logs(1-2x)<-1.
1) logos (é-5x+6)>=15 2) logg(x*—Ax+8)< 1.

280. 1) log (log, £41) <0; 2 log, (log, (x*-5))>0.
po 3

281. logyx? +log}(—x)> 6.
282. 1) log, (14+ x-Vz*—4)<0;
Daza aaa O
233)1) logia 925 2) log l8x+11< 4.
BBA) Haiirn ace suatenna a, npu Koropux mepanencrao
log ı (+ax+1)<1
suinonmaeren na ncex x ma IPOMEXYTKA x<0.
Peuwrs nepasenerno (235—241).
5.1) Penn 2) Vert glass,
EG. lorie, +2)(1- 9°) <l0g 0, 021(1+39+ 1082500 (3 +3"
287. 1) (x°-4)logo,s x > 05 2) (8x-1)log,x>0.
1) xt SET RE 2) 2° *<10x;
3) 3-x<log,(20 45%).
1) v32°+4-Vl32°-71<1;
2) 8*(V9'-*-1+1)<818"—11.
0) 1) Ji (er) = 2 logan, (6x+1)> 1;
2) Topos Eq > 2loR,(6x-1).
BAL) 1) Alon a 41 VIS log 5 3(8|x1+1)>05
2) Now a 11 (1844) lowe + a(2lxl+1)>0.

1, gang,

287

Viipasnionnn Ans WrOFOROFO, RONTOPORR Kypen.
anreGpui w Havan MaTOMATWOCKOFO ananıaa

242. Penner nepanenerso 9% —8-3>9 m yxagaTe namens
HATyPANBHOS “Heo, YLOBNeTBOpmIOMIee HepaBencTBy.

Peur nepanenerno (243—246).
243, 1) VOr-20<x; 2) Vee T> x41.

244)1) Aa 34, 2 LAPS VEB

<-L

2a5]1) Y-22 140% 3371,

246. 1) cos(-8x)> E; 2) cos(2x

2) in >2x-10.

247. Pers rpaquuecku wepasencrso:

I sing< ts 2) sinx>-4; 3) tex-8<05 4) cosx> 5.
248. € momomuo patio rprromomerpuecran ymeui mofa
ace peluenun meparencrza, 3anmovenmme 3 HPOMEMYTKE
CE >
1) 2cosx-V3 <| 2) V2sinx+1>0;
3) VS+IgxGO; 4) Stgx-2>0.
Per» nepasencrso (249—250).
a TE > eos.

[Tess
2

>-2sinx.

Hoxasar» uepawenerso (251—253).
ee,
Das EA,

Er (246) conn a> a
ay E A >0, b>0, azb.
252, 1) (a+b)(ab+1)>4ab, ecan a>0, b>0;
2) at+6a%0*+b1>4ab(a?+b*), ecan aed.
253, 1) 2424 £58, ecan a>0, 6>0, c>0;
2) 2a* +b +c >2a(b+c).

4. Cucremei ypasheHuñ u HepasencTs
Peumur» cuereny ypannennti (254—255).

254. 1) [5x-Ty=3, 2) [2x-y-18-0,
6x+5y=17; x+2y+1=0.
2 zou sty Eo
255.1) [274-232 io, — 2) [24H + 2380,
HET. LPS
ate 10; A + 0.
288

Ynpaxnenna Ann wiOrov0ro ROBTOPeNNR KYPCA
anreGpus u vavan MaTomariecxoro ananıaa

Halırm_nelersurensune pewenux cuerens ypaswenuit
(256—260).
256.1) [y+5=%, 2) fxy=16, 3) [x*+2y*=96,
a+ y?=25; 24 x=2y.

257.1) [18,2 [x*-9y=-
zyml; Txt 8

x_u_3,
258.1) [EL 2)
+ y= 205 ye
3) 4) [x*+ y*-4x=40,
2xt ey? + Bx=52.
259. 1) 2) [E +axy=25,
2y
£ 2xy=16,
260. 1) |= 2) [r+ Lys,
= [7 Pr =
Lead: Lt 10,70.

Peumr» cucreny ypaznenuü (261—264).

ann (mr D (9-277,
Su 27; Sr”

3) [3:.29576, 4) font
log,¿(y-x)=45 xt 1000.

262. 1) [logyx-log,y=0, 2) [x*+y'=16,
-5y744=0; logzx+2log:y=5.
a fe 3-9,
Vi-VE=1.

1) flog, (ey + 2299-1081 (2 +
| °

2) [ton (tu 2

289

TADEO Ann WIOFOBOFO MOBTOPERAA KYDCA
anre6pui 4 Nayan MATOMATANECKOTO ananısaa

65]

266.
287.

288.

269.

270.

am.

am.

278.

274.

290

Tips xanııx anauenunx a cucrema yparnenni
logs (y—3)-2log9x=0,
(+4) -2y-5a=0

meer xora On onno pememne?

Per» cncremy ypannenni (266—272).
D [Ve+vy~16, 2) I ve
MATE

VE+VG=19.

D
v2:

2) bus

Va y Vex+yi=x+y.
1) [210g3(x+24)

ray 29;

V2 Ty-6.
D farvttér-g-2 8, apres 54,

= log ı (+ 2y)log ı (2-4) +logé(x—u).

2) [logé -+19-+lon: (+9 08 (3-24) =2log(=-24),

ay ata.

con($-+x) ran (@a-y)-1.
run
1) [sinx+cosy=1,

sin? x +2sin xcos|

2) [sinx+siny=4

cos? x+2sin sin y+4 cost y=4.

planet an fama

>
texctey=1s Stax=ctgy.
Halıru uanmenninee u nanGommee meme pest
nepanenern

298 9x45 x og. 244
f 8 6 +2.

BB Va dx gy 212

1-258 4e cpg 212,

Pere enereny nepaseners
eel x42

ma cuero

Vnpaxnanum Anm WTOFOBDO NOBTOPENAN KYPCA
anrespel u Havan MATOMaIMECKOrO artama

BAS) Hatten nce omauenun mapawerpa a, mpi Koropux cncrema
ypannenuch
(es
xtymara?
meer pemenne; pers ary cncreny.
(276) Jan ncex anauennit napamerpa a pens cneremy ypannenmi
ae

x24 y?-2y—cos(xy) +11 -6a+a?—0.

Peur cuereny ypasnenni (277—279).

ae 2) fetes 3) [VZsinx=siny,
Bayh

prox; VBeosx=V3e0s 5

1

5) [cosxsiny
sin 2x + sin 2y

ges gue 77,
ahs’, gene 11,

280. Mooëpasurs na naockoern durypy, sanapaenyio MIO:CCTBOM
Peulennit cuctemat uepasencrs, m madri ee naomann:

1) (x+3y-8>0, 2) [x-y+1<0,
2x+8y-12<0, 5x-3y+15>0,
x20, x<0,
(27773 O<y<2,5;

8) [(ety-Deet+y-3)<0, 4) [(2x—y+3)(4x+3y-9)<0,
x-y<0, x+y>0,
x>0; 1<y<8.

5. Texctospie sanaun

281. Tlaccamup HOANAMACTOA no HCNOABIDENOMY ackamaropy aa
3 Mux, a no amxyuenyes sa 45 c. Onpenenurs, aa KAKOe
bpenst NORUUMACT DCKAJATOD HCNONBIDIMO CTOAUNETO Ha EM
raccaxapa.

282. Teunoxon uponien paceronune NEXRY HEYNA ApHCrAUANA 10
‘rewenno pex aa 7 x, a mporun resenun aa 9 u. Onpexensr
paccronnme MEXAY MpHETANsIMH, ecu CKOPOCTE TEEHA pe

von 2 w/a.
as

RPG AN HTOTOBOTO MORFOPSHAR RYPCH
anre6pu u Maan MATOMATUNOCKOTO ananıaa

283.

Tenxoxox Aone GA npoltr mexoropoe paceroamme sa
2,25 cyrox, o oKasaaoes, “ro Où MpOXoMLA 9a Kamanıl «ac
xa 2,5 um Gomme, Wem Hpennonaraxocs, a nOTOMY Ipomea
xaneveuusili nyr» sa 2 cyToK. Kaxoe paccromune onen
Gun mpolken renxoxon?
284. Onnu paounit nunonuner nekoropyio paGory an 24 jus, apy
voii paGounit ry ace paGory moxer nunoanur aa 48 Auch.
Ba cxomsKo aneii Öyner BoinoaKena ora paGora, ecan paño
une Oyayr paborans saecte?
285. Ilpn yOopxe yporkas Grao coßpano 4556 u xposoit men
un e oGmeñ maomans 174 ra, mpunem na nemmnx SMAEX
co6pano no 30 1 € 1 ra, a na ocramsnof naomags — 10 22 1.
Ckomsko rexrapon uexinmmx semexs 610 ocnoeno?
PagnocTb ABYX WEN OTMOCITCA K MX MPOIONEAEHMIO Kar
4, a cymma orux con » 5 pas Goxbine ux paonocru.
Hair om nena.
287. Tee opranusanun npmoGpean Tearpanbusie Onnersn. Tepsas
oprammaaqua Hapacxononana na Onneru 7500 p., a ropas,
xynupuias ua 5 Onnerop mensute m SANNATIDULAA da Kanal
Guner ma 50 p. menswwe uepsoh Opramoan, yunarıma se
Guxerss 5000 p. CroxbKo TENTPARDAMX Guaevos ya kan“
as opranmaanıa?
Ton ApoOn mueror uneamremn, panne erumune. Cysma aru
npoßeit panna 1. Paanoers Nexxy neppoil 1 sropolí xpoGaun
panna Tperueh npoGu. Cysma nepmuix Auyx Apobeii m 5 pas
Sonne rperseii apoßu. Haïtru orm xpoón.
Bpurara paGouux jomxna Osia x onpenenenmomy cpory
maroronur» 360 xerareï. Iepeunoauaa anenuyıo KopMy He
9 neraneñ, Gpuraga sa new 10 epoxa nepeBumonnnna nas
nono sananne na 5%. Cxonnno Acranch nororomwr Opnrane
K cpony, ecam yner mpoxomare paGorars ¢ TOÏ »e npons-
nogutenbuocth0 TPYAAT
290. Karep ornpasnaen OT peunoro npuuana Bua no pere 1,
npoïu 36 «M, norman naor, OTOpanzenmañi oF voro se pn.
ana sa 10 u no manana anınmenna arepa. Beau Gxt Karep
Ommpammaca onnonpemenmo € naorom, To, mpolina 80 Kw m
onepityn oGpatio, nerperun Ost naor Ha pacerosmn 10 Kot
or peunoro mpiruana. Hafirit co6crseunyio ckopocth Karepa,
Or mpnerann ormpaniaca no Teuenmo pera nor. Hepes 5 u
20 mux nexex 2a Úxorom © TOÑ Ke piieran ormpanvinach
MOTOpHaR NOIKa, xoropası nornana NAOT, poliza 17 Ka Kar
Kowa CKOPOCTH HAOTA, ELAM HOBCETHO, 170 EKOPOCTD MOTOP-
Holl Jon no Teuenmo Gombe cKopocrH nora na 48 104/47
292, Ilpn yOopre ypoan © K&KYOrO ma AByx yuacros codpano
20 210 y nuera. TLiowa nepeoro yuacrka Coura ma 0,5 ra
meme maonaan proporo yuacrka. CKomDKo memmmepon
nurenmust co6pano € 1 ra Ha Kanon yuacrke, ocam ypomall

E

288,

291.

292

Ynpaxnennn Ann HTOLOBOTO nOsTOPeNR KYPCA
anreGpui u Havan maremarmecroro ananıaa

23.

294.

295,

300.

301.

muemansr e 1 ra Ha nepson yaacrke Oxia ma 1 1 Gousme,
Mem Ha Bropon?

Paceronmne or xoma no mKoms 700 x. Ckoxbro ınaron je-
mar y4euHK OT AOMA AO KOM, ec ero par, mar KOTO
poro na 20 cm annee, nenaer ua 400 maron membre?
Hamm wermpe uncna, ABIMIOMCA NOCHCAOBATENEEMN
sremanm reoMerpruecKoit nporpeccun, ecam Tperse uncno
Gonkme nepzoro ma 9, a Bropoe Gonsine uerseproro Ha 18.
Haitrn cymmy nepuux senagnatn unenos apndmerimecroi
npOrpeccHH, ECM CYMMA MePBMX Tpex ee AAEHOB PABHA Hy
210, A CYMMA derspex MepBHX «emo» panna 1.

Haïñrx serspe nema, suas, 470 neppxe Tp Hs mex apas-
WOTCH TpeMA HOCACROBATEABMAMK ‘LIEHAMH. reomerpuuecKol
mporpeceum, a moenegune TPH — apsibMeTHuecKoit mporpec-
cum. Cymma nepsoro x nerzeproro ‘nice panna 16, a cyuma
sroporo x rpersero pasna 12.

Cymma nepssix nat “emos reomerpkueckoli mporpecci
pasna 62. Maecrmo, «ro narsıh, nochmoh u oJunma mara
ee HACHE ABASIOTCA COOTBETETHCHHO MepBEIM, BTOPMM H JE-
carie unenamm apubmermcexoi mporpeccuit. Haliru nep-
suit “aen reomerpueckoit mporpeccut.

Tipoxopenenme natoro u mectoro unenop apmbmermieckoit
nporpecenn » 33 pasa Gonbiie nponobenennn ee mepBoro u
proporo «momo». Bo cxonnxo pas naraii unen nporpeccitt
Gombiue BToporo, ecam MIBECTHO, UTO BCE "CHE Nporpeccitt
nonomare ns?

B rpeyroxbmuxe, naomans Koroporo papa 12 cm”, cepenu-
mu eropon coonnnenn orpeskawm. Bo mosh nonyuennom
>TPeyTOXBRHKE TOHHO TAK Ke OÓPAJOBAR HOBRÍ TDEYTONBHIK.
wr. x. Malin CyMMy naoujaneh Bcex NOAYAIOMIXCA TAKHM
nocrpoeunen npeyroanuson.

B mperounsiit marasıım nocranman 50 xpacmerx, 100 Genux u
150 xexrux rsoaux.

1) Cronsxo pasamunmx Gyreron no 3 rnoaqHKK m KaxyOM
MOXHO COCTABITB H3 HMEIOXCA UBETOB?

2) Cronvko pasanunnıx Gyxerom, COCTOMIX ns onnoh Kpac-
Holt, ommoï Gexoit m ONHON 2KenTOH TROSAIK, MOXHO COGpari
Ma HMeronqixes 1neron?

3) Ckonsko pranmumux Oyxeron, conepmamux 2 Kpacume,
2 Gene x OMBY Kenny TROSAHKH, MONHO COCTABITE HS
umeroupxen Muero?

4) Cionsio paanmumux Gyreron, conepramux 3 Kpacune,
OAMY Genyio u 5 2KENTMX NBOSAMK, NOKHO COCTABITE HO UNE“
romuxes peros?

Cocran nyxno ckommaexronarh ua 7 naayxaprusx, 6 xyneñi-
HBIX BATONOB H OAMOPO Baroma-pecropana. CKONPKHMH enoco-
Dam MOKHO CKOMILIEKTOBATD BAFOHK B cOCTaB?

293

VnpaxHeRun ANA HTOTOBOTO NOBTOBEHAR KYDCA
‘anresiptl Haan MATOMATUNECKOrO ananuan

303.

804.

805.

307.

2, Cocran HYAKHO cKounnexrosars MS 7 mnankaprusıx, 6 xymel-

Mx Baronon m OJMOFO waroa-pecropana. CKOAHKO cyniecT
Byer NOCACAOBATEMBHOCTOÍ PACHOTOMCHH HMCIOUUHXCA Baro
HoB Tpex THu0B?

Has mponepkm ma nexoxecrs Gsimo noceano 300 cenas,
Ha xoropsix 255 cemam upopocau. Kaxoga sepouruoers upo-
Pactanust omeamnoro cemenn B oroli naprum? Cromo co-
man B cpennem saolixer 19 1000 nocemmmsix?

BeponTuocts Toro, UTO PASMEPM neranı, Buinyckaenoli cran.
KOM-ABTOMATOM, OKA)KYTCA D MPCRONOX JAAANNEIX. AONYCKOD,
pana 0,96. Kaxoe xoxmuecrso roausix xeraxei » cpenuex
Gyaer coneparsen 8 kamnoii maprun o6zemom 400 mTyK?
Ornen TEXHHULCKOFO KOHTpONA NpoBepsIeT OsIOBHAY wagen
mexoropoli naprum m mpnanaer rommoh mc uaprmo, ocau
cpexx Mponepennsix Maren He Gonee oxmoro Gpakonanto:
ro. Kakora Bepoatuoer» roro, uro naprus ua 20 manenuh, n
Koropoï 2 Opaxonannsix, Oyxer mpsHana Tonnof?

B amsxe 10 xerareï, A ma Koropsx oxpamembr. Cóopuuur
mayran sonn 3 neranm. Haina BepomTHoCTb Toro, “TO xorH
Om ona ws powrx Aoranek oKpantena.

Ornen rexmitueckoro KoHTpoas nposepser nanenun ua Crane
rapruocrs. BepoatHocts TOTO, WTO MSIE CTAHAAPTHO, par
ua 0,9. Haltrm DePOATHOCT» TOO, "TO 13 ABYX mponepennkix
Mae rom.Ko ORO CTAMAAPTNOO.

Sasox » cpenmen naer 27% uponyxumn siciero copra u
70% nepsoro copra. Hair neposrrnocr Toro, uro nayran
vantoe HOACAMC Oyaer MM BEICILCTO, MAN HEPHOrO COPTA,
Ipx Kaxqom BEMOWEHAM ABHraTen auner paborars € ne-
posrnoctsio 0,8. Kaxona nepoxrHocts TOTO, “TO AA ero as
nyera norpebyerex He Gonee ABYX Binonenu?

C nepsoro erauxa ma eGopky nocrynaer 40% neex maneanh,
co Broporo — 30%, c rpersero — 30%. Bepoaraoern maro-
TOBACHAA GpaKOBAMMOÏ eran Ann KAKAOTO CTAHKA CoOTBET
cersenno pasunt 0,01; 0,03 u 0,05. Haitra ueposruocrs toro,
WTO Hayran MOCTVIIBIDAA HA CÓOPKY nerans Opaxopannas.

6. Dyuxunn u rpaduku

811.

312.

Haitra roobéimuenres kn 2 auneluoï nu y—hx+b,
ecan ee rpabux npoxonr wepes rouru A u B:

1) A(-1; -2), B(3; 2); 2) A(2; 1), Bil; 2);

3) A(4; 2), B(-4; -3)% 4) A(-2; -2), B(3; -2).
Hepes touxy A(-3; 2) npoxoguT npAMañ, mapannensuan
upawoil, npoxonuugel epea row B(-2} 2) u C(8; 0),
Banncars POPpMyA5, saramımme amueñnble HyBroma, rpaqa-
KAMM KOTOPHX ADAMIOTCA ARMA LPANME.

294

VRP RONA Ann HTOrOBOrO ROBTOPENAR RYDER
anreGpui u Havan maromarmccKoro ananıaa

313.

314.

315,

316.

317.
318.
319.

320.

321.

322.

Y 1 rouxa A:
Bunennr, npuuannenen an npawoit + Y 1 A:

DACA: DA DAM (1).

Jimueltnan byuenua sanana bopsyaoli y=- À x+ 2. Halızm:

1) row À u B nepecouenns rpadua oroit dy © oon-

M Koopaunar;

2) nanny orpesxa AB;

3) paccromme or masaxa Koopynmar 10 mpamoli y= $242.

Hafen anauenun x, mp Koropsax rpadu yen y= 2x — 1

sexe mine rpadinen dyn y=3x-2.

Haïñru onauenun x, MP KoTopsx rpadbux yn

V-WB-2x-\8 nexur pe rpaduka yr
y=(14V3)x+2V3.

Toxasars, uro Qynkuna y=2x-8 noapacraer.

Toxazar», wro gynkuna y=-VSx—3 yOnnaer.

Buseurs, nepecexaoren an rpaduru dbynxtti:

Dy-3x-2ny=3x+l 2) yw Bx-2 n y=5r+l

Tocrpours, npadum yet:

Dy=2leb o 2) y=l2=xk 8) y=l2-al+lx—8l.

Buincnurs, nepecekaer an rpadux Karol HO NAMIBIA Dyux-

mM mpawyio y=3. B enyune yraepairensnoro ormera Hair

KOOPANHATSI TOIEK Mepeceuenus.

Tana hynenna y 20-228.

1) Tlocrponrs ee rpadmk m HAITI anavenns x, MIE Koropkix

y(2)<0.

2) Hoxasars, «ro dbymensin nospacraer na npomexyrke [1; 4].

3) Hañru snaxenne x, np KoTopo QYHRIMA HPHHHMAOT

xamuensinee anarerne.

4) Harn anauenma x, mpm KOropux page Gym

um 2’-2x-3 nexur mue Tpada DRE y=-2x+1.

5) Banncaro ypanuenme kacaremmmoli K mapaßone

y=x?-2x-3

3 Touxe € aGeumecoï, panoï 2.

Hana dymemus y=- 224 3x + 2.

1) Tlocrpours ee rpaÿux u HAITI anauenia x, mpi Koropsix

y()<0.

2) Joxaoam», uro dymexua yómmaer na mpomeryrke [1; 2].

3) Halten onauenme x, MPH KOTOpON PYHKUNA mpununaer

nanbonsunee anaueuue.

4) Hafırn anasenus x, mpn Koropux rpaue aannoï pyme

ann ner une rpaduKa Gym y = 82 + 2.

5) Banucars ypannenun Kacarenmnmx i napaGone

y=-2x*+3x+2 m roukax € opAnnanoft, paBHO!t 3.

295

Ynpaxnennn ATA wTorasoro NOSTOPONER KYpCa
anreGpai u Navan MaTOMATINOCKOrO Hana

200.

100

1015

1010

1005
2 6 aap = _ 4000 107720 80”
Pre, 142 Pue. 149

828. Conepwası BOCKpeCKy10 MPOLYARY, ABTOMODNAMCT ABAKA 06:

Tananannancsı AAA OCMOTPA ZocToMpuNeyaTeabuocTelt. Tlocae
Bropoñ ocraHoBKH on Bepmyaca xomoñ. Ha pucyuxe 142 uso-
Spaxen rpagur Ammcenas anromoßnanera (no oem aGeuuec
OTIAABIBAJOCH PMR BR vACAX, MO och OPAHET —
paceronune B Kunomerpax). C nOMOMIBIO rpadnKa OTBeruTs
sa Ronpoc

1) C karol exopocrsio apromoGnxmer exan 70 mepnoï ocra-
on?

2) Cxon»xo npemenn on norparna na OCMOTP nocTonpuNea-
Tensnocreit?

3) Kaxona cpenusis exopoern ¡menus apromobnamera (Bea
yvera ocranonor)?

324. Ha pneynke 143 mpeacrasneno uomenenne Kypea axuuit

HEKOTOPOÑ Kommanın m Teuenne OXTAÓDA (no ocu acne
oraoxens “Hea MECANA, MO OCH OPAUMAT — CTOMNOCTE
onuoh asus » pyOnsx). Isa Opoxepa 4 OXTAÓpA wyusını 10
90 axımd war, 12 oxraOpa nepmui mo mux npogas
30 axunf, sropoit — 35 akuuñ. Ocranıunecs axuun oda
Gpoxepa nponaz 30 oxra6ps. Kotopomy ua Gpoxepon exeaxa
mprmecxa mensmym npnösn,? Ha como py6zeit on
nonyux xempute, Hem Apyroli Opoxep?

325, Curuan € xopa6ns moxuo Pnanmunrs » MODE HA pacerommn

onuoh mman. Kopaôns À er na tor, nenan 3 MAN B uac,
MB macronugee BpeMs HaxonuTes m D MHIAX E anmany or
KopaGaa B, Koropsit ner Ha JAMON CO CKOPOCTHLO À MH m
vac. Byayr a KopaGan MA paccroauns, noctarounom aan
upnena enrmana?

326. Buncuurs, nepecexarorca am rpaguin hymiaunk:

Ce

1
DRE HOUR ena

+6 2)

Ynpax HEIN 410708070 ROSTOPEMAA NPA
anre6pu u ninan Marorarmuecxoro ananıaa

327. Toerpours rpagux paixenT», annnercn am orpamitiennoit
‘yuyu:
» hee DORE [st mpn x<l,
2—x upn x>1; VE npn x>1.
928. Hocrpours spa u mumemus, manaeren au nenpepsnol
IVR:

mE Es Dash yy (2 3) 2 apm <1,

Nes Rape edi VE=T mpa x31;

13-11 mp <1, aye (Hee Nore ese
Hog, x mpm x> 1; V1 np x30.
329. Buaowurs, annaerca mit Jernoh san nevermoit hynknna:
Dya2 42% 2) E
Bex See
Don un
330. Hecnexonars na Neruoors u neuernoers dynknmo:
Dyna ra yl 4) ym
381. Busowns, smaneren au uermoñí man nerermoh Gym:
2) y=x*cos2x;
4) y=x+eos x.
332. Baron, mpi KAKHX amauemmax x moapaeraer dymenms:
Dun Tr;

3) y=

388. Buncmrs, ananercn au nepnonnweckoï Pyukusa:
Du a y-2 fr
Halten nanmensumnd noxoxsrensusiit nepuox dymsum
(34-388):

Bryan Un ylaenoee

335. 1) y=cos3x;

8) y=teSx; 4) y=
836. Hecnenovarı na vernoer» u HEMETHOCTA u nocrpours, rpagui
yen:

337. Haïru nawGonvuec nam nanmensuee anasenme yann
yrax?+bx—4, can y(1)=0 u y(4)=0.

838. Hafirs nanborsuree u mamensrues snauermn Qymium:
1) y=sin2x-V3cos2x; 2) y= 2cos2x-+sin*x.

297

PECAR ann HFOTOBOTO MONTOPOHAA RYPCA
anredpui u Hasan MATOMATUNECKOrO akanıaa

Hana Gym f(x). Halte xopun ypanuenun f(2)=a, a raie
die ManGomunee u NanwenLUnge anauenma VUE, cent:
sin’ x + cost x ‚sin? x 4 3008? x
DA A, ae 19; 2) f(xy = id A
sin’ x4 osx Boost x sin?
340. Hatten woodemmnenru a, by ¢ amparo dy
y=ax?+bx+c, ecan y(-2)=15, y(3)=0, y(0)

B41, Tocrpoums rpadux yen y=V25—aé, Haïtru 1

paq

Ky npomexyriH MonoronnocTH Dynamit, Torasars, uro rpa-
Qux nannoll dyaxuun CHMMETPHUEH OTHOCHTENBHO oc Oy,
5

342. Tlocrponrs rpadmx Gym y= >. Tlokasars, sro ya:
ris yOunaer un npowenyrrax x<2 u x>2. B KanoÏ rose
rage yan nepeceKaer oc» opaumar?

Buneurs ocnomme enoherna u nocrpomen rpabune dy
om:

gie

Dyna 2 u=(Z)-3 Dun:
4) yolors(x-1; 5) y=Veri-2 yet.

344, Tocrpours rpagur ymnensn:

Dun 2) yo loge (x+2)+3;
anal ee

345, Ups kaxıız anauenuax a rpaduent yal
vezt-Art2 nym Beta
meror oDue vous?

Hañru oGnaers onpenenenns Pym (346-349).

346. 1) y=2+Ig(6-3x) — 2) y=3*-2In(2x+4);
4) yated
A Fe
347. DIV Van

348,

1) um Vlogo = 547);
2) y= Viogos(2*= 9;
3) y=Vlogs(1 + 6x) +08 ı A+ 75

aa

298

E

Yinpamnows Ann HTOrOBOTO HOBTOPEHR KypCa
anrebpe u Havan maremarecxoro ananıma

Haürn mnoxecrso snatennit ynxnun (350—353).
) yaa? 46x43; 2) y=-2x%4+8x-1;
D y-e +1; 4) v=2+À.

2) y=0,5cosx+sinx.

351. 1) y~0,5+sin(x- 2)

Dy 2x5 2) y=V=4x=5.
353. 1) y=inx-cosx5 2) y=logr( +2).
BEA] Harn pce anauenns x, npu KovopH yRKINE
y=6cos* x+6sinx-2
npannnaer nanbonnee ouasienno.
5] Haitru nce omanenna a, mx KAKAON HI Koropsıx Hanıeii-
ee anauenme yen
y=xt+(a+4)x+2a+8
a orpeaxe [0; 2] panno —4.
BSG] Hatin nee anauenns a, npn KAANON Ma KOTODHX mawen
Mee sHaveHHe KBAXPATHAHOÏ py AREA
y=4x*-4ax+a?-2a+2
xa orpeaxe [0; 2] papuo 3.
B57 Harn nce snavenns napamerpa a, npr KAKOM na KOTOpHX
BP aayx napa6on
y=4x?+8ax—a u y=4ax?-8x+a-2
near 10 onny eropony OT NPAMOH y==5.
BES] Hatirn marGommee n namvenninee ananenıer den
2cost x + sin? x
IT gaint 430082"
359. Hatin yrnonoit kodrmrenr Kacarersnoë x rpadimey bynK-
umm y=7(2) B Tone e abeuHecoit xo, eon:

D f@)=sinz+cosx, xo= Fs 2) F(2)=c088x, x9= >

360. Haikru yron wereay ven Ox 1 wacanennnoii x rpadmey
Gyarnun y=f(2) » TonKe © abcumecoN xo, ec:
YO VE Re DER, 02 q

361. Hanncars ypaskenne Kacarensnoi K rpabuxy YRKOMA
y=/(x) 8 TouKe e aGennecol x, ecam:

1 Ne Hop Ot, me.
362. Haltrm yrnonoit Koodxpunnent kacarennnof x rpabuky pyuk-
mum y=20—x+ 1 m TouKe mepocenenun ero e ocmo Oy.

363. Haitrn yrnosokt kosbhnnment kacarenznof x rpapaky pyuk-
umm y=3x*-1 » rouke c opannaroli y=2.

364. Tipanas y=4x+a annneren kacaremmoñ x napabone

-2x+x?. Haïru a u KOOPAMHATA TOUKM kacanın.

ge:

PAPE Ann UTOFOBOTO FORTOPEHAA RYBCA
anreSpui 4 Havan MATOMaTINOCKOrO ananıaa

365. Haitrn ou, » KOTOPMX kacarennine K rpadyKy Hymn
vd 0x? +6241

naparremm ocn abennec.

366. Ha napaGone y=3x°+Tx+1 maltrm rakyio TOWKY, 3 Koropol
Kacarensnna x mapadone ofpaayer e ocuo aGenice yron À.

367. Haïñru pce Touku rpaduxa dyukum f(x)=e*, B Koropsıx
acarennuan K OTOMY FPADHKY IPOXOJUT “epea MAMAN KO
oprmar.

368. Hamscar» ypapnenne Kacarenbuoll x rpaQuxy ya
y=1(x) B ome © abeumecoh xo, ecm:
1) F(2)=xIn2x, x9=0,5
2) Ma)=2*, x9=1.

369. Harn yron mexxy oem Ox u wacarennoh x rpagury
ym y=x?-x'—7x+6 m roure M(2; -4).

Hafırn ranrenc yraa, KOTOpA KacaTeabuast x rPAQHKY Oya

num y=xte * n Tone e abennecolt x= 1 oGpaayer ¢ ocmo Ox.

371. Haiirn yron mexuy ono Ox u Kacarensnoh K rpadiuKy DyHE-

70.

woot yo desa au +) w some aten 23

E
372, Banucars ypannenue Kacarensnoï x rpaduy Pyme
ot

3 rouxe ero nepeceuenna © 00510 Ox.
373. Banncar» ypapnenno Kacatemmoli x rpaQuy yaaa

m rouxe e aGemwecoï 24.

ladra Tanrene yraa Mexay KACATENHEMH, PORN
x napabone y= x? ma rouxn (0; 9).

875. Halirn mpomexyrxu MOXOTOMROCTA ya

Hair roux akerpenyna gymcunn (876—877).
376.1) y=(x-1)*(x-2)%5 2) y= 4+ (6-2).
26x18
Ben

Hañra_ nanGomminee u manmombmice anavennn yaa

(878— 380).

378. 1) yo 2einx+sinz na orpeswe [0; |;

:]

2) yodeins cons na opero |
a

Ynpaxnonun ANA HTOLODOTO nonropenm KYPCA
anreGpui m Haan MATEMATANECAOO ananmaa

379. 1) y=Vx+5 ma orpesxe [-1; 4];
2) y=sinx+2v2c08x na orpeae [0; 3].

380. 1) y=Inx—x na ompeane [0,5; 4];
2) yexV 1-22 na orpeaxe [0; 1]
381. Haltrm naubonsunee m naumemvmec suasenns dymrunn
Ar, ir
EME
382, Tlpm Kaxom ananenun a mauGorsuree onauenne Gym
y=x*-3x+a un orpeane [-2; 0] panno 5?
383, Tipu kaxux onasennax a dynxusa y =x*-30x*+272=5 ume-
er ennnernennyio eraumonapnyıo TowKy?
384. Hafen serpent yen:
1) MI + 8x2 07 +45
2) f(a) =x 121045,
385, Mcenenonar» € nomombio mponanonnohi dymumo
yox?-Bx42
1 noctponts ce rpadmx. Hair roux, m KOTOphIX KACATEN-
He x rpaduKy Naparsenust oct Ox.
388, Mecxegonar» © nomonqo mpoHanoAHOT hyıciuo
PRET

pot Bus ome un een |

H noerpoien ee rpadnk. Barıncarı, ypannenne kacarensnoii x
rpabuxy stot @yaKuun 5 TouKe c aßcumecoi, paso 4.

Mcenexonars dyurumo y=f(x) uocrpours ce pad
(387-389).
387. 1) f(x)= 4x" + 6x7; 2) f(x) = Br 20%

8) f=} PS
388.1) y E ax 2) yoxt 2x83,
389. 1) y=} 2° 2) yon xt + 62295

390. Tlepusterp ocesoro ceuenua uuausapa 6 am. Tlpm sarom pa-
auyee OcHORAHHA unaunapa ero OOBeM Oyaer mambo?

391. Halirm nanonsur noamomnnit oGnem MAIER, none
nomnoï MOBEPXHOCTA Koroporo papHa DAR Ca”, ecu HaBeCTHO,
uro paauye ocmonanım me meinsine 2 cm u ue Gob 4 cM.

392. B npapnzsnoß nupauxe SABC so vopumms S nposexena
snicora SO. Haliru CTOPOMY ocuosauus uupanuasi, eca ob
eM HUpAMHAN ADAACTOA HAMÓOABINIIM HP yenomum, TO
SO+AC=9 u 1XAC<8.

301

YRPSXHENA ANR WTOrOBOrO HOBTOPEHAR KyPCA
anreGpu 4 avan waTewarHvecxoro amanuza.

393. B npasumnoit serspexyrosnoï npuome Amaronam pana 2V3.
Tipu xaxoli score mpuamat ee OGseN naudonsumd?
394. Jan Oymenan [(1)=x 2+cosx maltra nepnooGpaanyio, rpa-

dm xoropotnpoxonnr wanes rom (0,5 ~ 2).
205. Hatin nonGonsinee nanmenniee annens Ay
F (x)= x? (2x -3)-12(8x-2)
xa orpnne -a<sc6,
Hañrn HanÖonsınee m HaMMeHBUIee anauenna py ero
{(x)=21n?x-9In? x + 121nx

xa ompeare et x ce.

BOT] Ha napaGone y~x* nalıru roxy, paccrosume or soropoli 10
rom A(2: 4) annnercn namens.

Ha Kooprmarnoï nnoexoern nam rows A(3; -1) m D(4; -1).

Paccmarpusaioren rpaneun, Y KoTopsix orpesox AD ABa#-

even ORMuN mo ocwonamml, à nepulmumı APyroro OCHO

nun nemar ma mapabone y-1-x, unnannoh na oxpeaxe

[-15 1]. Cpeam orux rpanensht mmu6pana Ta, koropan umeer

anbonnıyıo nnomags. Hair say momen.

Ha Koopaunarnoï nnockoern jaa rouxa K(3; 6). Pacemar-

PIBAJOTCA Tpeyronsunm, Y KOTOPHX ABE PP cm“

Merpmuns OTHOCHTEMIRO oem Oy m near ma mapañoze

u=4x?, sanamuolt ma orpeaxe [-1; 1), a rouxa K — cepeau-

a onmoh wa cropom. Ma orux rpeyronbuukos BHGpau 701,

koropiiit mucer nanGominyio momo». Hair ary monta

400. Kaxonst nomme Ober Konpbiunenes p 1 q Knanparıranoi
QynKuKn y=x*+px+q, 700s mp x=Ó ona mena MIN
my, pam 17

401, Kaxoit nomena Gur ancora Konyen € oGpanyiomel 20 am,
‘aro6sr ero 06e Gsix mambonsunmn?

Katyıo nannensuyo momaxs nonmoh nonepxnocru user

unannap, ecam ero o6sen paren V?

Haïti paxuye OCHOBAIUA XILMNAPA, BIICAMHOTO » INAP Pa-

auyen Rom mmewomero KanGoamtyio roma Goxonoit no

Bepxuocrs.

404, Halim DRIcory uannapa nanbonnuero OFneMA, BHMCAHHOrO
» map paguyca R.

405, Haliru nsicory Konyca KanGombuero OGREMA, BIHCARROTO »
map paxuyca R.

406. B xouye c saga oÓzemom V pmucara nupamuxa, » ocno-
mann KoTopolt eH pannoßenpenmmih TPOYrONMIE e yr-
JOM MH vepuimne, pas a. pK KAKOM snauemmm a O63
few pass Oyaer naubonnun?

avan

Ynpexnenum ann wrorosore nosropenmn FYPCE
anreGpui Hasan MATEMATANCCKOFO nana

407. Ms ncex uuaunapos, Y KoTopsix Mepumerp ocenoro ceuenun
Paver p, nußpas unanuap mandoamuero OGxema. Halirn oror
bem.

408, Hs scex uwmapos, Korophe MOXHO NOMECTHTE BAYTPH cibe-
per panmyca R, noir unnuap HauGozsmero oSzema.

409. Koucopnuas xecranas Gana sanaunoro oGnema nomma
unter» opMy umanspa. Ilpm KaKOM cooTHoWeRHH MEMAY.

Auamerpom ocmomamma D u msicoroli H muaumgpa pacxon xcec-

Tu Gyqer manner?

Ha pcex IPaBILIBMNX TPeyrOMBMBX MpHM, KOTOpsie Bnuca-

una » chepy paauyca R, nsiOpana npusma HauGomsulero 00%-

exa. Halira macory stoi prom.

411. Ma ncex uunnnNpOB, BRUCABMBX B KOBYC € PARMYCOM 0CHOEA-
nus Rom nsicoroli H, maña unnumap nanbornuero OÓBEMA.

410.

Hala mnomans @urypa, orpanmmennoñ tant anna
(412—416).

dy y=3-x, sa 0;
yaks yest yd.

413, 1) yodx-x*, y=5, x=0, x
2) yaxt-2x 48, y=6, el, x

243, ym +5,

2) yma? yu Ve.
2) y=3*x

416. 1) y=cosx, 1, x1, y=0;

3) y=2cos3x-Ssin2x+10, y=0, =, zm

1
afirma ROMANA durypat, Orparuennol rpaduxon DysKUAH
y=9x-x" u KacaTensHoll x oTOMy rpaguney B ero roux € aGc-
unccoh 3.
Hokasars, “To mp — 1 <x< 1 cymma arcsin.x+arccosx pasua C,
ne C — nocronmması. Halırm C.
Haliru nce snavenua b, np K0KJOM HS KOTOPHX NPOHARON-
wan ys
f(x) sin 2x8 (b+ 2)cos.x— (462+ 160+6)x
orpunarensua na Bceit uncnonoh mpaNoii.
Haiira nce anauenna x, MPh KOTOPwX KacaTenbHBie x pad
ant py

y=Bcos5x m y-5cosdx+2
m rouxax € aGeuHccoit x mapanensubt,

303

Vnpaxnenun ann wr0rOB0r0 NOSTOPEAR Kypca
anreGpui 4 nayan MOTOMATMNECKOrO ananvaa

Toaduxy Gym mx + ax? + bx+e upumannemar Tou
Au B, cunimerpusune ornocurensno npamoï x=2. Kaca-
Temo « aroMy rpabuxy m rouax A m B napannenum
Mexxay COGON. Omua Ma 9THX KACATEMHMX MPOKOAHT uepea
roury (0; 2), a apyran — sepes roury (0; 6). Haïru a, b, c.

422) rpaduxy bynxunu y- x*+ax?+bx+¢ npuragrexar Tourn A
u B, cuerpo ornocurembuo npanoli x~~ 2. Kacarers-
HMC x 9TOMY TpabuKy B TowKax Au B napasnensn mexay
coboñi. Opua ua DEUX Kacaremmiax RPOXOANT wepea Tour

__ (0; 1), a apyras — sepea romy (0; 5). Haftr a, b, c.

1423, Tpadus Gym y=x"+ax'+bx+c, c<0, nepecexaer oc
opminar B ronke A u uneer porno ane ome row M x N
© 0010 adennec. Hpamas, kacaiomasion Toro rpaditia » ro.
xe M, mpoxour wepea zouxy A. Halim a, b, €, eoam no
aah xpeyronsunka AMN papua 1.

(424) page yen y=—x5+ ax? +bx+c, e>0, nepecenaer ovo
opaunar n rouxe D m maeer porno Ane o6mue TOKE À 1 B
© 00510 abennec. Ilpaman, kacaromases ororo rpadua a ron
xe B, npoxonur vepes roury D. Halıru a, b, €, con nao-
maza rpeyroasuma ABD panna 1.

425]B kaxol rouxe rpaduxa dymenmm y-(x- 1), 06x61, nyse
Ho mponeeru Kacanemhuyio X Tpadixy, "ITOÓL romans Tpe-
yroshitinka, orpaunuennoro roi Kacamenkuoh u OCA Koop-
Aumar, Outta naumensmeli?

426) Ha napabone y-2x°-3x+8 malita Touxı, Kacarenbiie » Ko-
Topux IPOXOAAT sepca uasiano Koopaunar.

(427, Tlapabona y = x? + px +4 nepecexaer npanyio y=2x-3 » rot
xe e aGemwecoñ 1. Tipit KAKHX anauermax p 1 q pacerosme
or ep mapaGou Ao oc Ox ABIAETCA naunensumd?
Haltru oro paceronune.

[428] Hahn nzomens purypm, orpanituenuoft napabonoi
Une m Kacanenumann X neil, TPOXoAAMVENH “ODE Tor
sy M($: 6)

429] IIpm kaxom auauenun k mnoman» durypst, sakmonennoi
mexay napabonok y=x=*+2x—3 u upamoh y=ke-+1, sax:
emma?

(430, Harn mxomanÓ durypst, orparmuenmoRi | mapaéonon
y=0,5x'-2x+2 u xacaremsukisn x mel, mponenenmunn ve
pes row All; à) u BOs 2).

BL) Hepeo rowxy rpaura dy y=YX e aGeuuccoï a, re
$462, mponenena kacarenpuas x oromy rpadhsmey. Hair
suatenne @, pH KOTOPOM roma TPEYTOBINLA, orpanı-
Nenworo roi Kacarennnoft, ocho Ox u mpamoli x= 3, Öyaer
BaMMensuueil, 1 BRINCANTO ory MALEN.

VRP NON Ann HTOTOBOTO nosropammm KYPCA
anebpe u Hasan marenermiecxoro ananıaa

Jana Qurypa, orpannuennan xpsoi y=sinx u npamuin
0%x< 4). Mom xaxms yrxom x ocu Ox yo

nponeern upamyto «epea rouxy (0; 0), wroGu ora npsrmas paa-
Guana namuyio durypy ma Abe urypit panmofi naomann?

7. Mpounssognas u unrerpan
433. Haitra snavenne nponspoamoëñ dynxun f(x) B TOUKe Xo, ecm:
DADA ji DIE, nl
8) (dx, ch 4) ayn MEE, rom
434. Hafirn anauenun x, mpi KOTOPMX Pano ny210 anaueme npo-
naBonnolt yAKRHH:
D (Go =sin2a—x5 2) [= cos2x+ 2x;
3) 1(x)=(2x2- 1% 4) f(x) =(1-3x)*.
435, Tloxasars, aro f(1)=f"(0), ecam f(x) =(2x—3)(3x* +1).
. Haliru auauenun x, np Koropsix snaueniin npoussoaHot

P(x) x= 1,5x°- 1804 V8

orpauarenm.

437, Ilyas puneraer na mucroxera nnepx co exopocrmo 360 w/c.
Haïru cxopocrs nym p momemr t=10c m onpenexure,
ckonbKo BpeMeHH nya monummaeren DBepx. Y pamnenne ABH
wenun ny ho vot 4,91.

438. Koneco npamactes Tax, “TO YrOA HOBOPOTA HPAMO mponop-
unonanen KySy BpemeHH spamenus. Ilepssf 06opor Ost
cesan wonecom aa 2 c. Onpexeawre yraonylo EKOpOCrR KO-
Zeca wepes 4 € nocne mauana npamermst.

439. Tpyaoemwocr» (06zem) kommnerca paGor Q momepserca » ue-
xovexo-uacax (ven./a). Bpurana ocnannaer oGpert rar, ro
BHIMOMHeRHBI OGBem paGor KaK YHKUHA BpeMeRM ONNCHRA-
even dopuynoli Q=160£ nen./u (tae t — npeum » cyrKax).
Kaxona cxopoern ocnoennn Gena paGor (mirenenunocrn)
q wex./a B cyrxn? CKonsko YexoBeK B Gpurage, ecan cyron-
mas nopma paGowero 8 «ex./x n eyruu?

440, Jluesnan nponopoanremsoers rpyaa (sa 7 paGouux taco»)
PaGouero Manınmoerponrensuoro 3ABOJA ONHCHHMETEN DbyHK-
nett y=-0,09+* +0,28¢+10,06, rae t — npema 5 uacax, y —
KOAMMECTBO IPOAYKUMM, CKOJBKO MPOAYKUHH MPONSBOAHT
paGouni sa ox rox (260 paGounx axel)?

Haiira npoussognyw dymsunn (441—443).
PESAN Ve
sr y

441. 1) y=

Yhpaxnenun ANA WTOrOBGrO OBTOpSRAA Kypca
anreGps u avan MATOMATAVRCKOrO ananıaa

_Setoacet, 20041

442, 1) y= REL aye,
443, 1) y=(2x4+ VET; GD
3) y=sin2xcos3x; 4) ymzcos2x.
444, Harn mponanoauyro byurunn y=logge.4(Tx-4) m route
zen.

445, Halirn snauennn x, ana KOTOpkIx mponononuas pyme
1G) = (= 1) 2)(@~3)
parxa -1.
446, Onpenenwrn anaxe unema f'(2), ecnn:

D fees DW

447, Hana dymeuvn (= RE. mara £0) 1 (5).
448, Haru anauennn x, npn woropux f(x) 58 (2), ccam
Modera 3, e) V8 +1.
449, na bynxuun f(x)=cos4x mañiru nepsoo6pasnyw F(x), ec-
au F(Z)=-1.
450. Hañru nepnooGpasuyro dys

Dera

Du.

Burnes, unrerpan (451-452).

Gs) \¥=Tass 2 [ect Dax;

yes 4) [Varas
: a
N 2) (cosxdxs
3 5
9 (@ts2ax+syae; 4) (at-6248)dz;
» i
5 {etna 6 { ar
SE

Yrpaxnenun ann wroroBoro RORTOPONAR KYpCA
anreGpui M Haan MATOMBTHMECKOrO ananıaa

NpeamerHbi ykasatenb

Aureöpauuecxan popma Komn- Ilepecranomxu 163
exexoro wena 205 — € nonopemnun 164
Acmwnrora neprukannas 57 Tiepnon yet 8

— nenepruxansnas 114, 116 Tlaomen» xpupoanneinoli rpaue-
Bio Hmorona 171 um 139

Beposruoern co6urna 183 ocnezonnremnoers noapacraio-
— upomanenenus —mesanucunerx tax AB

coGuerui 194 — monoronman 49

— — mpomanomanex coburidt 190 — pacxoxamaaca 46

— cynmix mecommecrumx coOnruli — — cxonmnanen 46

186 — yOuparoman 49

— — npomanommax eoßuruii 187 _Ipansino npoxtanenema 159

— —uporumouosonnax coOsıruli [lpexez uocaenouarenonocra 46
186 — — monoronnoit 49
Bunyknocrs swepx (su) 114 — doy 58

Buruiranne Kourrereumx mcen — — Gecxonennu 55

210 — — n Geexonewnoern 55
Fapmonaocieue nonebannn 31,152 — — exena 55

Teonerpuveenuli cuen woayas 216 — — cupasa 55

— — mpomanoguol 85 Tipxparexue apryweura 61
Aeitermurennan oc» 214 — Gym 61

— ancre xounnexcnoro unena 205 TIpomssegeme kosmnencnix wt
Aezense xonmrexenux sneer 211 cea 206

Agppepemunan dymi 88 — codserui 181
beperamanse yponmema 150 —— Tipomanoguas gym 67
Hunyxus 157 —— aorapmpwnseckoit 78
Hinrerpansuan yaa 189 — = obparnoit 78
Vurerpxponanne yace 134 — — o6paruoii Tpuromomerpracc-
Kacatemsnaz x rpadmey gym noi 81

86 —— noxsaarensnoñ 79
Ronuaexeuan unoexoëri. 214 — — exomuoh 72

Kownaekeme mean 205 — — crenennon 74

— — pasate 205 — — rpurononerpruecxoï 78

— — utero snmnize 206 Paaemoma 166

— — compaoxenure 209 — € mowropemunnuu 160

— — nporwonoaonunie 210 Cunyconaa 19

= — opaco 211 Cobra noeronepmme 180
Kpxeoaumeliuan rpaueus 197 — sasneuee 191
Marenaruueckan unayruna 157 — nenoomome 180

Minen oct. 214 — neanmnemeise 191

= ners KOMxOKCNOrO auena 205 — mecomnecrinme 181

Maya, xowsxexenoro suena 210 — mpormeononoxemae 181
Ocnonuna reopen anre6pu 231 = panmonoamorcuie 181
Meppoospasuas gyxuss 182 — pasxocunemee 182

307
ea

— cayuaie 181
— anenenraprae 181
Conerauus 168

— € nopropemman 174

Cyne xomnaexemux aucen 206
— coburuii 181

Toopena Jarpauxa 99

= 06 o0parmoR yum 68

— 0 upomenyrouerx anauennax
63

— Depwa 103

Tpeyroasuns Ilacxann 171
Tpuromomerpsueckns popua
sonunekenoro «mena 219

Toren xpurreckan 104

— maxcnuyua qymxuio 102

— sonne oyeron 102

— nopernéa 118

— paapsana yen 60

— eraumonapuar 104

— oxerpexyxa yann 103

Ypanuene xacarensuoli x pads
ky dying 87

Vexosan seporraocre 189
Dopxyaa Bepuyam 199

— Myanpa 222

— Hioroma—JTeñGua 140
Dymausa Gecxouewno mazas 58
— andypepemurpyestan n rouxe 67
—— un merepaene 67

— nenpepuisuan » roue 61

— — na mureponao 62

— — ua onpeaxe 62

—nevernan 7

— obparnan Tpuronomerpimeckan
Fi mp pi

— nepnonmuocxan 8

— rpnrowomerpiveckan 5

— ner 5

Orserbi

Lu
1. 2) xeR; 4) x»

221.2. 2) oéyé 8) 1256 y5-0.75
rez ei neem ne A)

a O cae

6.2) xe 4h, mews 4) xe E emm, mez, 7. 2) -1<ysli 4) 1epcto

6) Vc ys B. 8. 1) 2 cyav2; 2) -1Gy<9; 4) -2Gyc4. 9. D Levis

25} cyt 12.2 Hevernan; 9 ererun; ©) urna, 18 3) Ho muanron
sento ever armes 6) sra. 16.2) eras; 4) neveras 0) er
van ner 18.2) 15 4) 38.22 9) 8) 2D Fumero, 24

) x. 24. T=21b-al.

2) ne amamercs;

) amaserca, Tor. = 23. 1) 2

y Loo
Bs 4 0. 90.1) $5 2) 0;

454 4. 1.3) ownngannie 2) ne mp, 8) ne anmmanıe
mors 4) npwwanaenuer. 39. 2) [-2; 0], [oi 3]; © 6-25 où [os 3].
10% de

34, 2) cos SE <oos AOR; ses ) 0) cos4<c0s5.35, 2) 5,

25, T=4la=c| 26. 27. 29. 1) 0; 2) = 53 8)

Te On. 4 Be “ E 2x us,
TES . 36. 2) 0. mi Rex UE
ae E oca, Saxe saz 5

LE <x on. 87. 1) He 00: 2a 6) E if a ») lo 2 2a a [Lam oo

O (ki 09h) LB 21), [2x5 00). 38, 2) sin E <cos E; 4) sin SF cos Es

m Mn 182,29", 206 492 Im am,

Bong ein gg dra 18 18 % “6

euch MR Et Ey,
4-5 18° 18 18° 18 E iS

= 2 e

43. [Bs x]. [885 2x 44. D 20, 22-3, 0900 2 0-3: 9) 2-20

4) x=- 5. 45. 1) Hncer: 2) ue meer. 46. 1) 2 pemennas 2) 3 per

z 3 2) Eran 25m, ne 2;
+nk<a<i nh, hezı 2) 242 AE 42am, nez;

an ¿y VB, ay 8,4 18
Men, bez. 52 0 1; 2) WB, a) By ay B.
59. 1) 15 2) 25 9) 4; 4) 2.54. 1), 9) me upunanaenun 2), 4) upuman-
senor. 56. 2) [55 4] [925 2]: 09 [-

D mer

18% gy LA A aaa
57. 2) vin 3% nin ME; 3) win(-88)-in(%8); 4) aint sin.
2 Bx 9x ling dr, Br ech, Moye MR, Me
5027, 28, 9% Um, ginn gg more, crc 92
3M) Bice St 2) 15 o) 2) €
sr J 60. D a) € 1) [= 15 0}; 8) €

[S$ se) daran non mia a [oe %

88 4) sin E cos 8%,
>eos $F; 4) sin E <cos 3%. 62. 2)
E

air
UE. 64, 2) ss

3
ll ça
cren ue. 68. |

22) [22 0.05

2) dan Tex À + 2nn, neZ. 76.1) 1: 2) 0; 3) 13; 4) 1. 77. 1) y=
66 E va

2) y=0: 3) 9215: 9 y VE. 18, 1 Mpmnannenur: 2) e puna
3) mpnnaaaesxsrs 4) proa, 79.2) tg TE < (5
ax Ta Se

4,8, 81.2 ono

2, Fete,
OC où 0 [25 2] [x 309€

ES
30) [m D] 83.2) S4mcx<ä rm, nez: 0) Tamer htm,

of à

eZ. 84. 2) -arectg24x

= ES
Rcxcarctgdtn, DE <xcarcted+2n,

ae cr, —arctg343n<x mme
xc Et, artgst2rer< HH, —arctyS+Sn<xcde, 86 2) - 5

7); 6) Rs

rect 2-+2x, ~arcetg2-+8x. 85. 2) OGx<arctg4,

SS exc an; 0cx<%, -artgdire

<xcaretg5+nn, ncZ; 4) -arctg6+ancx< 5 tan, ncZ. 87. 2) Her xop-

-90.2)y>15 4) y>1, y<-1. 9.2)

2
A 1 2
sane En, neZ. 95, 1) aresin 1 <aresin- 2
2 eo

>areain VE m(-2)>
in YS: y arcsin(- 2)

Hl 3

reer

97.19 tee cars v2: 2 arc 1) cart

#4) x=-3-V.

zart; arte (--2) caret) 0.0 5 2

LA
v3

99, 2-1; 4) x=

Lg ea ant 101 9 16806 Loren

Diesen -2exs-1, touch 9 0érc2 6) Lewes
15, 146 xe Egan, nes 4) E 2mncx E 42m,

6 [E 1538] 100.2 02 an nezı tence 5 am,

neZ; 6) xenn, xr(-1 h+xn, neZ. 109. 2) -1<y<l; 4) Scycts

6) -4<y<-2. 110. 2) Meveras; 4) ne annseron uernolt u reuernoft
112. 2) ein(-D<cosi; 4) cosB<tg4. 113, 2) y=coax; 4) ode.

me 9 [Lo] 106.0 2209-20 10 2 (io) [2 2),
sabe) Enga

ME ce FE,

6 6
2) mpm. 121. 2) 154,5, 468, 162, 11,8. 122, 2) mncx< Sinn, mez.

4), 8. 118. 2)

‘ance Lames ed, 119 D os

4) 1 1-2, 124. 2) Vernas; 8) nenernas, 125. 2) Ar.

in, Xm Ban, neZ; 2) x. ZUR
Han, 2e, nez 2) 20 BE,

stan, neZ. 128,1) x=

Bs, ea 100 Meder BE oem ne BL DL

132. 1) Ee tancx< DE 420, nez; 2) an<x<F tan, nez.

sana 1
sonpaos ons no

Py 91% >
oh pa avi 90 7 À
8.1) L252) 2m 1; 8) Om As #) 1m 2 LL 1) #2, x

25 2) xm

8) x=-2, ze; 4) x=-1, 2-0, 2=1. 12, 1) y= Bs 2) y=5; 8) y:

18.0 $52) 4:9 4:0 4:9 5:0) 3.14 2) Hers 4) 20.16. 1) 24
1-8: 5102) 1-5 5), 185 01012; 61 9 12 -DUCE; 9, (0; 4b D [-6: 9,
Cae DR [255 40052) Ra os 4h 8) 201, 9901 0 ee

yal. 18. 1) a) m ni 2x=-1 9 eayuasx 2) m ni 8) a), 0)
19. a) 1) 4) — aa; 2) u 3) — ner; 0) see — mer; 3) 1) upu x22; 2) up
0<x<2 m mpx x>2; 3) upu x72; 4) upu x+ L. 20. 1) Her; 2) ner: 8) na;

our ena ats di 9-1 mn Eden

au

‘Oreerer

o EIA, y CUA, où
4) 6x45. 25.2) -4 4) 7. 26.9 5. 27.2) v(2)=0.9, v(8)-04.
28.2) 101. 20. 2) 2.91. 30.2) 25-1) 4) hay Ci
8) 16x. 31.2) 12x+5; 4) 1-16x; 6) -36x?+18; 8) -9x7+4x-1.
82. 2) f(0)--2, f'(2)= 10; 4) (0) 1, F(2)-13. 33. 2) x= 1,55 4) x--0,5;
Dm. 34,2) 5x4 Ba ZA 5 2) 4.
931 4-5. a7, 2) ee 38. 2) Mt) I ET
18 1?
39. 4) 3 (x9 BA 2) 3 (a (82-2).

1

4) 16)

40.2) x>2:4) 20 4.41. 2) 2<0,x>2.42 2) 2>0,5. 43. 2) /(6(2)-lsinzo

44.1) 0-25, g(e)--
e

7 Ese 1
TOS ~~,
BE D 2 HONE GE

A5. 2) 37-20 + 3x 42) Bd 46. 2) 12x"; MR

2 3

ma 1 2d: 9 -
Va ae 2
ro 40 £0)
6) £(8)=0, F()=-8; 8) '(8)-24, F()=-4. 49. 2) xi=-4, 270,
sali 4) ml, 2-1: 6) x=1. 50, 2) 192: 4) 81,5. 61. 7-2

6) 8x 8) 8x4

10-83

Boa AN
0 1izx 3 «55. 2) (2 a DeL
A 6
(xr? axvx(2—x)*
A Saeed, an bia) com) eo
80, 86 pua/e. GO, 0020 Am. 61, 2) 108 rex. 62

x(2x-3) (10x +3).

wen o Le

64. 2) 106

0) terse ts, 65. 2 atom 9 De
66. 2) -3x -cos4x; 4)

67. 2) 6x? + + 2eindn
2x

rd
nae 5-1

68. 2) 7(x-4):

a

y

6) 15(4-3x) °, 70. 2) Sur fy 4) -2(4x+1) nf 3 6) 15(2-9x) 3
Ez 0 amaia; 8) 202% 41). a a
9 sn 5 9 Be du, $) 12008 Bx-sindxr 10) 21 Ex
1 zz 8x, 4) 2 ER, get,
cot ae un 2x? Tx
15.2) nn Lo As. Dre
DB De 0 £0; 8) roro e. TED ==] ur,

ei aa atm Fm nat marca

79.2) te 3 sintax+e# dein? 3x-cosde:

1) 20% 2% cos (3-2x)—
aes. 80.2 ÍGTED_ Er, yy 2D,

ave ern?" ++?
:In2-L 2 +logax); 4) sinx+cosx. 82. 2) zı=1,
come gel); 0 unsre. $8.2) y=

Hani A) nopneh ner. 88. 2) AC: 4) x>25 6) 278. 84, 2) x= E +20,

xa 2nn, ncZ. 85. 2) 2. 86. 2(1+1). 87. 2) F(x) <0 mpm x=e 4, F(x)>0
ape ze À, £(2)<0 npu O<x<e"!; 4) [(x)=0 npu x=1, 1(2)>0 pa

+ 89. 2) yee 4) y= E

221, F(9)<0 mpu 0<x<1, 88,

Feine
Li 0-5 90. yea yo ao 2) YF
6) 1. 92. 2) = $9 ES LE: + 93. f (x)>0 » rouxax Am D, f'(x)=0 B Tow
3

Ko Bu FLe)<O a Tome ©. 94.2) y 0x 12 4) y= E485 6) yma

1 x
Dyna) yo
Be Me

a
) y=x. 96. 2) Fs 4) Fs 0 À

shag
97. 2) Es 4) E. 98,2) y-0; 4) y=2x5 6) yb eal. 99. 2) (A: 2)

a (ini) 0) ces 20m ns 20m, ncz. 100.

1), (45 8). 101. (5D,

y~ 2x8; (0; 1), y= 2x-2. 102. 1) y=r+1-x; 2) y=3-2x. 103. 2) Mer;

4) ner. 104. 2) -5at46x-6x m 6) ~ 214-32)";

8) 2(1-4x) E; 10) -3sindx. 105. 2) —sinx-L_; 4) 2429-06;
ze costa

à 4) 4cos 2% 43el=%, 107, 2) x*(8Inx+ 1);

318

CT

4) 2sin2x; © e(cosx—sinz). 108.2) 2-2), yy Hilo
ETS xa

100. 2) 99-0 apa #=-3, om 2= 8, pu x= 10920 mp eB

mon -8<x< 8, npu x>45 16950 um À <=x<45 4) 0 apn els

F69>0 mu x>1; F(2)<0 mpm x<0 1 mpn O<x<1. 110. 2) 63 4)

111. 2) y=30x-54; 4) y= But E 112. s(4)=22 a, 0(4)=7 w/e.

118. 2) 82% cws2x-2(2° + 1)sin 2x;

218 2
114. 2) -—**8_; à
area Sinèx 1

6) at +Beed 2 saunas 8) Andre" onen. 116, 2) 273, 117, D x=

Azor) x 6
US, 2) $-e*+200835 4)

:
ua 2 1.0.2 A 9-1,
N rie der

” rentes 6) -e'sinet 8) -anx-2%*n2. 121. 2) 7-0
; 712320 upn x>05 Fa)<0 upu x<0; 4) (0 pu > À

9-0 nou x=3; f()>0 upu x>1; f(2)<0 un —L<x<3,
122. a>3. 123. a<-12. 124. 2) ac0; 4) a>12. 125. 2) a>0; 4) a<0.

pu x

128.2 2.100. D pe Ends E 4 Ling pete 0 yea.
sas. 2) pads ayant, 189. pete 2, y O6. 100.800 0
tt 2, on

Pana NI
1. 2) Bospacraer mpx x>5, yöuinaer mpu x<5; 4) pospacraer mpm x<-3
w npu x>2, yOumacr up -3<x<2. 2. 2) Bospacracr npu x<1, yéusnaer
pw x>1; A) nospacraer upn —L<x<0 m up x>1, yGwaser up x<-1
u mp O<x<l. 3. 2) x<0 m x>0 — mpomesnyren yormanın; 4) nospae-
var npu x 35; 6) voapacra
Baxct < 1, yéunaer m

EXC: 4) mopacraer mpm x< à, y6 put

x25. 5. Bospacraer npn -1<x<8, yónmer npn ~5<2<-1 m apie

ua R. 4. 2) Bospacraer ups x6.

El
ES

4
294, younner mpm

RA

set Ean, neZ. 10 2) xl 4) x1=0, xy=8, IL 2) 2-6 —

9-8 — Towa mutt
ora marerayis

Fr aan, neZ,— rev
ku mumimyna. 12. 2) Dxerpemynos wer; 4) sxerpeuÿon ner; 6) x=—1—
rouka Maxennyna; 8) x—4 — roue mrmunymo. 13. Dyna noopacraer
mu mpomexyriax —6GxG-4 u 06263, yOuBAeT HA UPOMERVTKAX

ara

ee ee cn
8) 1-2, xp—2 — Tome Mummy, x

Ea2en, nCZ,— ron macia

CT

OS
wa maxenmyna, 14. 1) x=

2-6 x=0— TOUR MURIO, XA 4 — TOM
— rowe mummynas 2) x= 25 — ou
a 2) 2P

a ma

cmuymas 3) eg — Towa MAKOHMYNA, X=5 — Towa yemenyoas 4) x=

=-8 — rowxe axcunsyasa, x=1 — vousa maya, 16. 2) 8 u 18; 4) 9
Y

O iO A

25425. 2. 025-25-25. 22. Keanpar co cropowol 2. 23, Knaaper co
cropoxol 3 cm. 24. 2) 246? u 1. 25. 2) VZ m 1. 26. 2) 1; 4) -1. 27. 2) 1.
28.2)1, 28.2) 64212 1.0.0.2) LH = BL 2, 82 Reaper co ere.

pouoi À. 33. Ky6 co cropomo DH. 34, -2. 35. $2. 36. yo 1 (x -8)",
va 16
37. 2) x(6~x*)sinx +6x* cos; 4) 12x*-18x; 6) de =) - 88. 2) Bainyx-

een}
aa nmopx wa unropnane -L<x<1 m muanysan mino ma unrepmanax x<—1

M DL 39, 2) 2150, tom À sam E. 40. 2) Dur anepx ma nurep-

3
enya BIS na unrepnane x>1. 41. 1) x

mane O<x<1,

à
Han

245 8) 00, 0728, 7, 422) y (e Px

A pre
Mn

x(&-2), y’=B8(1-24), y" =-6x5 rows nepeceuenma rpadıra € oem Ko
opamnar: (-1; 0), (2: 0), (0: 2); muy y 0 up x= -1, manomaya y—4
Y= 6x42); vou
peceweniin epapinkn e ocamın Koopausiar: (85 0), (0; 0); MaKeHMyR
9-0 up 208, mummy = À up x= 1: 9-2 una neperata,
43. 2) y - À tt 42), y = $ 042) row mepeceneme rpadımn e comer
? (0; 0), (45 OÙ y>0 upu x<-4 u x>0, y<O upu -A<a<
222% x=0 — mou neperna; 4) yrs

seran, sou neste rms cn spannen: (5 0 (2; 0)

(VE: 0): v0 pu txt moon al Bi va 2-0 —

Touxa maxcnmya, y(0)=0; x=41— voue summmysa, y+ 1)

D x=, gx -6.

+11; 2) yox42 pu +00, y=-x-2 mp x—00.
bi de à ab see ear ee

©: o (V5: 9) ( ve
wor hs y
2-1 — roma sommes, y(1)~=25 y = 30% (2-1), 2-0,

poo Ea VS eco 900 au

152 (71), 2-1 —rowea sumunysea,

— Ton mepernba; 4) y (xD (x+1)(5x*+6x+1); roux mer
(15 0), (15 0), (0; =D); x=-1 —
815

mi
Peceneiiun rpaguia c oem Koopasmar:

Orseru

25
mpenymos ner, Y=X — acumurora upu x—00, X=0 — acummmora; ya
nus wewernan, (-3; 0) u (8: 0) — row mepecewenun rpaguma € Koopau-
maman casos 4) (1; 0) — mowxca nepocauenns rpaquisa e ocio Ox, 2-0

— acunturoras. 48, 2) y= (x-+1)x—1)(x-8); (1; 0), (13 0), (3; 0),
) — rows mopeceuonmm rpabuisa GymıcLDUR € KeopaIMETEMIE omar;

= noun sans; y (a)

‘Towka ame; y°=6(e—1), x=1 — roa nepernón. 49. 2) y=-x+3-L,

E, 0) n (3428; 0) — rouen reposo

sonaron 20 u yo (

vena rpaibuna ¢ co Ox; y=— 142, x=—1 — sora nomuymas y(-1)=5,

#1 — rosa maxemayaa, y(1)=1; 4) acummrores

à Vas sou
vopecencura pags © cemen socpanmer: (12°88; 0) u (0: U:

mn, 219 — ro mancrmeyaa, y( 10) 98; (8, 31

nepernóa; 6) gymenpia nesermaa; x=41 m yx — ación, y= ED
33
E,

) = sous
Fora,

=
— roma marre,

213 — roue muy, y(V3)=
eva) 313; 2-0 — rome neperada. 60, 2) Acarroms 2=0 1 y=x+3;

EA, 322 — oa say, a
ED 2 oa, vr 2,
ueperußa; 4) acumurorat x=—2 u 9-0; y= E, 2-0 — roue mu
„2-80
4 — oma nancy y(4)~ 2, y = 222-8249),
A 7 oy
x=4+2\8 — roma nepornön. 51. 2) (15 0) » (0: ¿1) — row nepecese-
men rpaßınca © KoopaumarHsins vom; ((9=B 22120 up FER,
ayınana vospacraer mn Mi F(@)-8x-2. x=] — roma meperutn.
52. On. 83. 2) Boapneraer mp x<-1 m x>2, yOumnaer npn —L<x <2;

y- 1 — rome

may, (0) =0,

4) panne npn <8 u #8. 54.9 2150, sud, sue À 4) koi,
zer Bun, neZ. 88 9 5-1 — vou soma, 86. 2) 2-0 — son.

Ka maxemayus, y(0)=-3; x=2 — roman mmnyus, y(2)=~12,6. 57. 2) Tow
ku nepeceuenua rpaduka e ocamm koopammar: (0; 0), (+ V6: 0); œymenns

wernan; = 2° (4-24), 270 — roux money, x= 2 — route marc“

yma: v2 y 250, = 11/72 — sonar peras 4) roms,
epecesema e oem Koopmmar: (0; 0). (225 0% yen seria Y=
A — roue marenaya;

316

Oraena

VENT a1, y mort +2, zur? — vom neperaón. 58, 2) Tow ue-
2.

poc © coman roopeomer! {05 0), (

8, oh at-n-22, y
of ren= vo

Yax(Bxd) x

Y (02), x=2 — roue murumayun, y(2)=

4 — vou nepers Fin u- =
wand perde, (ES. 69,2) 0 0 45 14 n 11.

61. Pannocroponitt rpeyronunt co eroponoñ $. 62. Ky6 c pespox 10 ex.
68, 1) 220, 122-812 sol, yt, 68. 2) Fond vom, anys
020-2 rom meneur r= À — roux sunny, 67,9 4-0

Keine: 2-0 — mora acuer, 00) À; una mers 0) Y
1 27 /3

(1021), x= 1 — sone mummyun, y(L)u- 27.

ex an san, y(E)=~ 2%. 66. 2
ava 1 1 L

E 9341.00. 94.70. 12.71. Raven LL, ron
Beas yt LL. riores

oe 2, 2 2; 1} 4) (ts 8). ums

+78. x(\5+I)R*, 79. x=-V2 — rouxa maxcumyna, x= VE — row

xa sommmsyno. 80. 2) O6zacro onpeaenenun: x<0 u x>2; (0; 0) — rouxa
epooeuenuin € oes Koopamas; acintora y= À mpi x= + co y=2x 1
pit 200 Gym wospacraer upn 260 1 yOnBarr np X>2; suma
cnn mas 4) nemuntora y=0 mp ++ 00 (0; 0) — rota nepecenenms
Tocas Koopamars += 8 — route manewyun, yB)- 2e 21,8 Xy
3-13 u 29234 V3 — roux nepers6a; 6) (0; 0) — rouxa nepecenena
© oom Koopanner; aeummerm x=—1 m Y=x-Ó; muy y=0

mp x=0, marc y. ret.

ups x=-4. 81.

27

Fans IV f

3.2) Pro dat 0) 4x4 40.4.2) 245; 4 2

2 3

Leo arto 0 aro 9 dama.
6. 1) Bsinx—Geosx+C; 2) de*+cosx+Cs 3) a+ 30 46inx+C; 4) BES

sinpintersc. 7. Lente 0 Lesa 8s6; 6) meo

1(1e42)*4c: 4) Le 40; 6) 2virTTe
Here om Ebo 0 L ent ch 0 ar

si

8) 2 ese) +0. 0.1) Lanta +o 2) Leona

0-0 (&43)16:0 LE. 10,2) 368 conde 4 nung:

Lec 6) (E) cour): 8) BYBETT- miz
+6: 9 2 mt: 6) ÉVITE 2 mar

2
a 5

317

Oise

da daa a EB
FOAL 2) Toate ate FHCs 4) D + Gala + C5 6) a

#40 8) 26

3
a 1

CR 2,88 840, 4) Arie à
¿eric 12.9 2 054) 2 x? 6x? 40. 19, 2) 2

2
=. 1 1 1, 9) a; 4) 124; 6) 2
9873-3: 6) hama 8) 1-1. 14. 1) 12-5: 2) 4: 4) 12550 D

) 1-1n2. 16. 1) ) 2. 17. 2) 1,32%

2 0.18. 11 8; 6) 92:6) 10. 19. 2 25: 0 82:02

09 113 0 8 1,2) 68; #45; 6) 3, 22.298. 29,2) de 4) im.
Ve

a À 1.9141, 941
24. 2) 05 4) +25. 155217; 845; 4) 26.296

24

By BD 1, 8: 6) 242

08,27. 2) 2-13; 4) 1%. 28, 2995; 4) 8: 6) 24%
5

3. 1
nl,
15. 29. 014

1.91 a v2 3
30.2) 45.31. 152) 15 4) 208; 5) 2-1, 92.2) 82; 463

6
02921) mi 0 442, 96.102 235.2 ge art Cr D ge
-2sin2x4C. 36. 2) yzsinx+cosx4+C; 4) yua? 26" 4C. 37. 2) y=
2m 13 4) year + 2542 6) y=8 € *. BB. 2) —cosx 1; der

ota ann

Lio vante ann 9 14.45.92
98.00 45 9) 419-2. 47.1) 10452)

asustan

48. kop.

Faana Y
5. 1) 6; 2) 24. 6. 1) 27; 2) 04. 7, 16. 8. 6. 9. 240. 10. 720. 11. 6840.
12. 50000. 13. 90. 14. 45. 15. 80000. 16, 4586000. 17. 720. 18. 1) 720;
2) 40820; 3) 5040; 4) 802880. 19. 24. 20. 5040. 21. 1) 24; 2) 6; 9) 12.
22, 1) 8% 2) 161; 3) 14% 4) (24 Ae 5) Als 6) Ces DE D Mi Rh A
28. 1) 19; 2) 462; 9 3; 4) 18; 6) n?+3n42; 6) In 1
7 nérône6 |) mei
9 24 Dn-2% 2) n=11; 8) Kopmell mer. 25. Py, 26. 90.

27. 1) 725760; 2) 241920. 28. 1) 12; 2) 12; 8) 60; 4) 30; 5) 840; 6) 42
7) 151200; 8) 10080. 29. 90. 30. =10%. 31. 1) 3; 2) 6; 3) 42; 4) 5040;
8) 320: 0) 1800) 7) 90: 8) SO. 3260480. 3a. 40d. 34. ro: a. 720!
S01) 100; DA de Dm 10: 2) ate mia Om ty m0
38. IO) 36. Bio. dl.) 1: 9 50:3) 28: 4) 96) 90: 0 OF 7) 1
8) 1, 0) 1207 10) 120; 1D 4050; 12) 2419. 42, da. 49,280, 4d, 100
45. 105. 46. 220. 47. Ch. 48. 1) 1+ 7x4 21x" + 36x +35: +2124 Ta af;
2) xt-8x9 4 24x*-32x 416; 8) 16x* +96" +216x" + 21624815 4) 81x4—
2 5 A 5-50? + 5 1 2

AG + 21654-96316; 6) 82a Aa 2004-50 + Sa 0
216x* + 216x?-96x+16; 5) 32a°—40a* + 20a 23 04
48. à 8 at 200% 60a? 960464. 49,00. 80.1) mad, men
2) m=; 9 m3 4) m5, HL. 1) 560; 2) 220; 8) 1140; 4) 4950.

318

Ormes

52. 1) 32: 2) 62. 53. 1) x-0: 2) x= 10. 55, 150. 56. Cf -Cfy-Cly-Chy.
57. Cho. 68. Cls-x". 59. Ch. 60. CHCl. 62. 1) 84; 2) 126; 8) 45;
4) 10. 63. 20 cnocoóam. 64. 120 crocoGamk. 65. 120. 66. 1) 48; 2) 36;
3) 124; 4) 1090. 67. 1) n?+9n+2:2) 7.68, 1) 4%; 2) 10. 69, 1) n=

2) nT: 8) neds 4) n= 11; 5) n=T; 6) ny =4, n¿-9. 70. 40320. 71. 28.
72. 56. 73. 59280. 74. 15800. 75. 5; 14; 242. 76, 1) 364; 2) 466.
77. 1) 16; 2) 64. 78. 1) xP 4 6x0 41511420084 15x 46x41; 2) 0-50
+102°-10x?+5x-1; 3) 16-324 424a* -BaŸ+at; 4) a*+120% +5407 +
+108a+81. 80. 350. 81. 2", 82, 3800000. 83. 252. 84. 1) 1001; 2) 46:
3) 85; 4) 56; 5) 64; 6) 262. 85. 1) 320° -800' + 800° - 400° + 104 —
Get FOr AE OF 2008 «60664 960% 648%; Bit

as eee ee (ont
D et soy? 1969 -249. 00. 906,07, 2m.
88. SCH | 50, omo. 00. 3642". 01. Cfy-2. 82. 570 enocofamr,

94. 1) 20; 2) 26. 95. 84 emocobanın.

Taana VI
1. 1) Hevosmoxuoe; 2) enyuaituos; 3) aocronepnor. 2. 1) Sinamoren; 2) ne
amamorca; 3) we anamorea; 4) annmorca. 3. 1) Ceronns nepaati ypox —
Re buoxka; 2) oxaamex caen ne ma «oranukos; 3) Ha urpanbuoli Kocru
munano me meme nr OKON; 4) HH OXNA nyas ne monana m HE.

4. 1) AB: 2) A+B; 3) AB+AB; 4) AB; 5) AB. 5. 3

1 Be ds om
Do
be Sra Lin Soit y 12,
BEE SEE SEE 00300 1D ds a E

22 5 EP NO 3
3) 22.12, 8.1. 21. 14 2.15. 2. 16, 5.17. 0, 18. à.
Ge a 18, Go A À. 15. À. 16, à. 17. 0,99999999, 18, À

28. 99, 181 : 4,91; a 2;
19, 2G. 20. 354. 21. 1) 0,94; 2) 0,06. 28. 1) 53 2) 35 8) 25 4)

E y
mn Sn 25.0 352) 526 a 18; 9) 18. 27, D) Ar
more; 2) anınwren; 3) ananıoren; 4) ne anamoren. 28. 2. 29. ¿[2
30.1) 281; 2) 7. au. 0,96. 32.042. 39. 1) 0,15; 2) 0,1. 24. À.
35.4. 36, . 0,91. 88. 0,44. 39. 68,6%. 40. 0,144. 41.
me

a iam ne
36 105? 515! 9 359 35 36°? 16

_ 319

mon

Go oo. or. 1- St

50. 2
ch _ 925 4, 202-Cle
= wt Ci e cha a ds ri A ES

D 15
3 3885 ° me Me: m
Taana VIL

2. 1) 2454; 2) 2,3-1,74 3) -61: 4) —6. 5. 104, 27, 81 5.6. 1) 34
2) 7-2 3) 2-40 4) 1+i 5) 6 0 -i. 7.1) -7+22% 2) 3-11;
5) 545 6) ~T+6VBi. 8. 1) 5-54 2) -8+41; 3) 1

) 2431. 9. 2) Hampamacp, 7,5. 10. 1) x= À, y
ED 22-2, y=-8,3 4) x=-0,5y=15 5) 2-2, yal:
2) dg: 9) oP +988 aaa
2) 0; 3. 18. (3; 4), (3: 5),
8) 22, y=8: 9) 2-1,
) 0,74 1,34; 6) pá

et
ot yo 102-2
5) (4° +9a°)i; 6) (Sa +16b*)i. 12. 1)
QU D, Mia Men pots D ACT Po
Velo 1-63 PEN TES

17. 1) 105 2) 10; 8) 23 4) VB; 5) 5; 6) 2; 7) UO; 8) 5.18. Du
29-4424 8) 8-1 4) VE 19. D 246 2) 3-9 9) 2 4) 25 9 3-

6) 4: 7) -2V2+318k
21. D -

5
22-5 i
23. 1) 7; 2) 9; 8)
26. 1) 2-5
27.1) 2--1 4 8

1313
2 -

3 2e +0 ED De
En is 2) 2-0, 23-8, 24-81, 31. hee pa
33. 1) 16; 2) -64. 38. 1) 22-61; 2) 2--3+21. 39. 1) 10; 2) 9; 3) VS:
4) 2V2. 40. 1) Oxpyauocrs paguyca R=5 c ueurpox u rowre 2=0; 2) ox
pymuocrs panmyca R=3 c KeHTpON m TouKe 2=1; 3) onpymnocn. payuıy-
ca R=1 c mewrpom m rouxe 2--3l; 4) oxpyxmocrb paanyea R-2
© ueurpou » Tone z= 1-24; 5) upawan, uepueuaunyampuan orpesky, co-
eauuamoneny roucu 2 m -I, x npoxonEwa aepes ero cepenmuy; 6) mar
Mas, nepnenxuxyampas oTpeaky, Coenunsiomeny roux 144 u 1-1,
u npoxoanujan nepes ero cepewuny. 41, 1) Muomectso Touex, nea
Suyrpx Kpyra paauyeu R= 3 € ueurpost u TouKe zi; 2) Nuoxecrno TOUeX,
sean une wpyra paauyea R=4 € newrpow n roux z= Bl; 3) romo
MOXAY ABYMA OKPYxRHOCTAN panuyeo® 1 1 2 ¢ odulum WeHTPOM B Tone
=2 (ncxamuan TOWN OKPYAMOCTA); 4) KOMO MOXAY ABYNAL OKPY>RHOC
rn paanycon 2 m 3 0 06nun nerrpon n ToNKe Sl (BKAONAR TONE or“

rene). 2.1) a À ai D 21-600 menden

320

Orson

zum VEB NEO, 240 VIE 4 VIEL. 44, 2104171, 236481. 45. 1) 2xb, REZ:

Dr 2 REZ; 8) Lan, cz #) 2d, ez 5) BF 42, REZ 6) + 2
eZ; 7) -Krzak, hez 8) LE 420, REZ. 46. 1) 2=3(co80+1oin0)5

an 2a

2) 2=cosr+isina; 3) 2-9v2(c08 7 +isin 4); 4) 24(cos oF +1sin 2)
0) 2-2(000 SE tin 5) D 26,5

D 2-18 (com(-) +isin(-2)); 0 200m SF

Denk ete Dendy Hen
an (m (25): 2 12(os ran); 3 vB(eoe 32
an); 4) (coe 8 in 8), 49, 1) 8.50.1) (ow E in 5)

E coro sini 3) 2(oon
2 Leona stein: 2) (cor

}
0 E (coed auand).
)

in BE; 2) 5008 220*+¡sin 2209); 3) 2c08 & (cos £ +isin.
in 55; 9 (ona sian 20 3) 2605 (eo

(cos 48% ain ER), 63. 1) cos % isin 23 2) 2(cos 2
8) 2yB(cosZ+isin 3): 4) 12(c08 E +isin E): 5) 2060890 +1sin90);

©) cxs10-+isin10, 54. 1) cos Etain Es 2) 4(c0s E ia

5) 2icos(-30°)+1ein(- 80");
1_va

sinin(-3))s © Lost stain as

9 coins. 85. 1) 8 9-1) he 0 9.02

2
nes Dd Le oy 2, 57. 1) VE (cos E +iain 2):
23 Ac 3% sim SE) 2) 80002 sn 22): 4) 00-22 Jetan(-22)

+07.

58. 1)2,2-+(2V2+2V2) MD 21,2-+(0V2-V2; 3) 21,:
ont Er), mn (chamo) nent
1

(con sen): D

(cos4+Isind); 4) 2(cos0-+isin0), ccm n —

vers Aeon isn), ena — meses. 62,1) MEME, 2 nz,

641) ayant aan 2am nat,

sqm E25 6) 24,288, mal. 08. 1) 1 y 2134 4) SV.
SD 22-128; Da lt 9) 2, Dans.

67. 1) 21,2 que Das 1:4, 24 4) 21,2 14 V8

DER

321

rara

ETC
DEN

am LENG. 68. 1) 2844 AE

D 4)=-8, aa
2) tax À 05 8) 2-2 V224 700; 4) 2242162490. 69. 1) 24,2022,
saandls 2) Hat 2,4-44i. 70. 1) (2-2-I(2-2405 2) (242-8)
Mes 2 9 let 1 0 3D TLD d(o+3-ik
(ee) 2) 16-215) (2-141); 2 2562+1-020(2+1+01203
9-16 [email protected]. 1 S224 82+13-0:2) 25" 410244170.
ER ED ayant sn Se,

x (cos Zo isin u):

$5 (oo 878 tn)

vésin

afc ie 9) (re HE in Mt ae

6.1) 21,0 2208-24 VF x 2) 2

¿TTM 2-84 Bi, 29e

=1431 2) 214 4d, 29- 1-24. 78. 1) += AS

DS

2
79. 1) e==2; 2) x-- À; 3) 2-1,6; 4) x--0,5. 80. 1) 24; 2) -5435
15 41, 6 12,5
Des 48. 81.1) - 52155, 929% 3-414 88 y 2,5
NC BO a oan Re Ter

ES

62. 1) 302702103: 2) Yon 8 4120.25); 3) Lienen,

) 128; 4) 25

tain; 4) ¥8(oom(-) +1amn(-5)). 8.2) 1952) 2

NB i; 2) a VIVÍ, 86. 1) 2-94

CRUE

22

322

asin 31

u
2 2 (con stain 22) A

:
00 #6, 92452) ray.) SL où Me

5-3 3) 2-8+(1-V2)b 4) 2

=1. 87. 1) 42 (co

88. 1) 142% 2 - S44

+

4
ayab uepuoro wucna Gomme Moayax sroporo uuexa. 92. 1) -0; 2) 1.

94. 1) 22-2V82+8-0; 2) 2241-0. 95. 1) BU

o 212 nn ae E)
z 4 VE -4 VB 2) 64 (cos + sino. 2) con(-%)+
2 27(cos 3 un) 8%, 97.1) soin

#4
we

on
2

RE)

4-200 28 stan 95), 5 2 10 282 stan 188)
5) VB (cos stein), 2 VR (cos rend), 24-42%
x (cos Zr en 375 ) 6) 20 V (cos (cos 2 +1sin i),
A 1861); 20 (con 398 stain JOE), ter
x (coo Esta E), 100.0 ons: 2) 3200 Sx
x(cos Z-s4sin 3), 101. 51 =2(008 4X sin IR), 27 2005 SF pin 5 ),

29 \FT(cos BE sin BR ) 2181, 29 Vai (cos TE tein TE )

102, Aucyanseng. M1) terme montrent atajos
22270 2) mannan gue 2-1, 905 3 et, amyuncssen
SEE, BB oc seta hone nece oe

= 0. 105. ECS sin 15° 2.

Taana VIN
1. 1) 2x+y-2-0; 2) 4x-3y+12-0; 3) x+2y+2-0; 4) 6x-5y-30-0.
2. 1) Muowecrno rouex, meme nike UpENDI By—2x+4=03 2) ano"
meets rouen, aexamux uume upamoli x—4y=0. 3. 1) Cu. pue. 144;
2) ex. pue. 145; 3) em. pre. 146; 4) ex. pue. 147. 4. 1) (4; 3): 2) (4; 3).
5. 1) Ci. pue. 148; 2) em. pue. 149; 3) em. pue. 150. 6. 6. 7. 1

32

Dress

as as
ana
AW W I
NIN
EN
No 72 x
la (33-3)

IN

Pac. 149

8 1) x=6, y

2) x=7,

9. 9. 1) Tlapa_mpanscx 2x+3y=0,

SPIO Sates hana ope ee Py CES
A seg 6 223 9 vn 2,8 ae

mue zei m y=-1; 7) ayım y= à, y>0 m x00, y<0; 8) oc Ox, aya
x=1, y>0 u x=-1, y<0. 10. 1) Tyan 2x+y=0, y20 u 2x-y=0, y<0;
2) rpamena xoanpara € nepummann (0; -2). (2: 2), (1: —D, (I: 9D);
3) rpauma mecrayrosmnca o nepummas (0; —3), (0; 8), (3; 0), (3; 0),

1-43. 3413
(o (EE) (a, au)
12.1) Town ane soya pan 3 © norpon (-25 2); 2) césemenne
3) TOUKH sue Keagpera (BNAOGAR ero rpaxuy) © pepormnawa (-
Gi =2). 18, 1) axed: 2) 14 E 9. CL 8), CR 2,
2} 2) x1=0, y, = 8,7, Y

1 (22 2 (-
18. 1) 1484; 2) 8VF+10n: 9) 14 25 4)

an. an =D. 5 =. 18.9 185 6) Bony m LD. 19. 95
0400-0 AO, 20.9) 5 far mart 21.06;
Dane. 32.2) NBC rear lo 9 128-2. 8) 1266-08

0 tartan +204 202, 5) 22(arcte vam) In 6) 05,

28. a<0. 24. a<-2, a>2. 26.0) 07-1, arts Oa
a7. a--5, -1<0 -5<a<2- 28
mah, -1<a<5, 28, -5<a<2- 3
1 1
2cacl, 31.-6<acl. 82 axis.
2cach. Bt bac, 8 3.

<a<l. 85. a>-2. 36, 1) Muoxecrao rouex, ne-

aux more upanoli 3y-2x+5-0; 2) muoxecreo ToUeK, nern Bu
me mpamolt x- By +4=0. 37. 1) Muoxecrno rover, xoxampx muyrpm rpo-
yrombunxa ¢ wepunnama (-25 ~1), (0; 1), (25 — 1); 2) mmoxecrno Toner,
examuXx DXyTpa Tpeyronsuusa e epmumam (25 0), (0; 8), (12; -9).
38. 1) fine noprukansmux yran, o6pasononmise npruum xy + 1=0,
x+2y-2-0 u conepxamure rouxÿ (0; 0); 2) aña neprunannsıx yraa, 00°
Pasonaunsıe gps 3x + 4y= 12-0, 2+24+2=0 x ue conepmaune Tou
xy (0; 0). 39. 6. 40. x=7, y=21. 41. 1) Oxpyaeuoen. © ueurpon (1: 3)

pre AAA

3) rpamına xuexpara € wepunmann (25 2), (1; 8), (0; 2), CL; De

15, 3445 Inomecrno TONER, neamıx na OPINO"
42, (EE; 208), au 1) Moe tek, neon py

325

Overs

nerrpow (1; —2) u ane prof oxpyaenocrx; 2) amtonxeers0
2), (15 0), (8: 2),

mn panuyca

Ya
‘rowex, meant DIYTPI kaaApara € pepita (-
(Ai —4 3) obsenmemne unomecren roves, sesamin mayrpu poros

panmyea 3 € nen

pan (0; 2 m (0; 3). 44. a) 50; 6) 2 (10-29;

2 Fen bot ne
2, 2.92 1, 212
5 0 Bea ay; Li Ion
wi ai 03 en es
A8 sac aon}, 80a) - Mensa) cac. 81 0-2.

4x+4. 46. 1) 1455 2) Dr. 47. teed
1

21 2

+22.

Ynpammenum AIR MTOTOBOFO mosropenu# Kypea anreGpai
MI HANAI MaTeMATHUeCKOrO AAA

1.20, 2 94%. 8 40%. & 195.5 62%, 6. 80%, 10%, 60%. 7. 660 p.
6. Ma 21% 6 10.600, 11.0308 p.40 x. 6556p. 98 y. 2. 088 p 50 x
tat ia ie 45.1) tr 2 901) 68D a ke He 28 Senta,
Conan einem rapie son ange min nice wine me
De DD EDO 9)

24. 1083. 25. 2) 3. 26. 2) 0. 27. 2) 64. 28. 2) 160. 29. 2. 30. 2) (0,2) >

>(0,2)'5 4) logos À <loros 4. 31. 2) O<a<1; 4) O<a<1; 6) a>1.
32. 2) Tlepnoe, 83. 2) 8<logy10<4. 34. loga4>Y2. 36. 2) 0. 37. 2) 01x
x(20%41). 88. 2) 3(15-V5); 4) VIL-V3, 39. 1) 1; 2) 3; 5) —

IR ES
4) 5,18). 42. 2) Ja. 45. 2) Mino

7 16
40,2) 275 ay 185 6) 28. 4. 2) 200

1013 4) ne umeror, 47. 1) 184; 2) 3049126 8) 2-21 4) 14458 5) À

6) 0,7

un Eo) Le 48. D 221102) 15 m uni

52. 1) -14 2) 1. 58, 1) 100; 2) 8: 8) 47; 4) 64. 54. 2) 2arete Le

su.

55. inca nn. 56. mena 178 m. 57, 2) cosa= 42,
+4) tga 24, sina- 24, cou 2 (x 2x
0 wa ag" Fe 582 43-58. Bi)

xQx-DAx+VE+3), ee en ee an won

ya

Saxel,y ol
2) SEAN gy AL y Gt. 1) a 0 10x? + 5x
2 BP ek

2) a+ 1208+ Sta? + 1080+ 81. 62, 2) PA, 63. 2) 0. 64. 2) 1. 65. 2) 4.8,
E 4

66, 2) 147. 67. 2) V1. 68. 2) 1-V5. 69. aŸ 40°, 70. 2) A. 71. 100°,
72. -6V0. 75. Bra-1 276. au-a, 77. 2) 2. 79. 2) 2cos?a. 80. -1g2u.

3260

Omer

4) sanar) an(a-3).60 9 2.05. 7
86. 2) 0. 87. 2) VBctga; 4) 2. 88. 2) —,

) 4, 89. 2) ta? Fs 4) ctgfa.

90. 2) -sina-cosa. 92. sg Bea. 94. 2) -cos2acos4f. 95. 2) 2cosa.
96, 2) otgactede. 97. 2) 141, 68. 15. 9-1. 107. 1) 2-
Had 2 salon ui 2) 2=3(ou ising) 0 2

D zii stain 215), 0) nava

FE). 108. 1) 242VBi: 2) 9v2+sv2 109. 1) $+ S48

page os 10,2) 298, 11 0-0. 11209, 1.2) 2-5.

190 7 115 Del. 223.106. 9 228, M7, 2) open
43
ts 2

rt ta. ant

ser. 18, 1) 2025 2) E19 D mol 00-045 dr

120. 1) x4, 2=2V5, 2 am VO 2),

#1, x 8, 122, 1) xy, 9-7 +05 2) ma, x9=-2,5a. 123. 1) Her Kop
moi 2) 12 3) 45 8: 4) 8, 124. a>0, dae. 126, 2) 2-6; 3) 2-83. 127. 2 x=

Bun 8,128 28 120 26 190.9 214012, 29-01 MOL 2106

Arcadia BANG
Oum 2-8, 11,2) LEUTE aan ayo, qu EVITE VO 12 197,
5+ 5

138. 2) xu2= 135. 1) 21-0, xg=k: 2) ail, 22-0.

2
188. D muets 290 2) 008 sel, ay ENB 197, m
138. 29-3. 139. Her. 140. 2) 2, ¿1£1VZ. 141. 1) 22

M Logs 2417. ria 21d re
gene IA Da Fi 2) 20-84 fi 148 Dre

5
wtiN3s 2) 4204

xml, xy!

M5 4; 3) ay 2246 4) 22-4451. 144. 1) 21,20

(EEE), à a, 10908 9 240
gels 145, 1. 146. D (6 O (510, GE 2), 35 2, G a
1). 147. 2; 2 10.0; 0. 148. 2) Kopmeit ner. 149. 1) x=

2) xopne ner. 180. 1) 2=2V3; 2) 2==1. 181. 1) 5x63:
2) 20% 9 x,-0, 20-10, Vuasamuo. Nyon y-VESE, 2-VETFE,
voran yet tot, 42035, orga y+2~5, yor. 4) 20-19,

m8. 132 ami, a3=-2. 158. 2) 2323, 202: 9) m3, y=].
2) x=8; 4) 2-1; 6) x=0. 160, 2) x=, 29=-2. 186. 2) y=

Ba LAVE 9 ae

CEE]

153) em À 187, 2) «0. 188, 2) 2-2. 160, 2) £,=3, 89-248. 161. 2) x=

5. 1

-3,5. 162. 2) x=V3_ 163. 2) xy=1, x¿=9. 164. 2) x=9. 165, 2) x

3
3. 167. 2) z1=-1, 2-8; 3) x=0,

x229; 4) x1=1, 2-4. 166, 2) x
168. 2) x,-100, x2-0,1; 4) x=0. 169. 2) x-2 4) xd. 170. 2) x:
1 3

IT. 1) x a. 172. 1) xt } arceos À nn,

ned; 2) x-unain 2 s(2ns, neZ. 178 1) x-nsarn À 42m, ne

Laon Lon ami darein Laden, REZ 174, 1) 2

1 = se, E ee
ES
eZ; 8) x-—arctg2.S4an, ncZ. 178, 1) Kopneh west 2) nope ner

E+

179. 2) x= T4mm, x--aretgd+en, neZ. 180. 2) x=an, x=

an x 1
Hl, nezi 4) x= tan, source tan, nez. 181. 2) x=

Eton, nez. 182. 2) 3930, am Zann, ne

Cr gr nez: 4 +

an 2n x
D xn, wat 2% 42am, mez, 189, 2) x= tan, neZ, 184, 2) x=

“EHER, nez. 185. 2) x= Rann, zu, nea 4 ao ha,
nez. 186, 2) 2 iM

EHE, ned, 180. 2) xn+2an, x 3 420m, x=

neZ. 187. 2) o nez. 188, 2) 2 Tm

1
+ rain
E

yee

+xn, REZ, 190. 2) x PEE, ned. 191.2) x

Baan, x=
2

Han, neZ; 4) x CIN! aresin 4 +an, ncZ. 192, 2) Kopueik mers 4)

= “ = Lam zer, n
an, nez. 194, D 2,002 /2- 25 2) 2. 195. 1) x= Bram, aan, ne2;

Fy Heme, nez. 197. x=

EZ, 198. x= 3 +(2n+ DDr, neZ. 199. x=xn,

x 1 ran
201. xa 4 arceos ay AUF,

nez, mp Least. 202 2 -34<x<40: 4) -2<x<8, 200. 1) 202,
a

ENTREE | 204 1) 18685 9 204, 800;
8) £98, 23-23, 208.2) LA CE, 200, 2) 4. 207.1) Teed,
HS Dre 2 E AE eed, Lorch, wo 208 Die
A À. 208. D ke VF.
SLAVE Dr 0D, 210. Dre 2 4) #22, Led, xo
2a

) xe

Oraena

Le

<x<-2, -2 excl, 1<a<
10

5

Bit. 20-4, -3<x<-2,

6

—A<x<l, 252. 212. 1) 910; 2) ac}

1. 219. -éxe-2. 214. m2.
215. 7; 8: 9. 216. x-0. 217, x=-L, 218, 1) x-20, y=8; 2) x-0,
U=16. 219. 2) 563; 4) x<- 4. 220.2) -1<x<5. 221. 2) 3-VB<x< a+ VE.
222, 2) x<1. 223, 2) Pemeunt ner, 224. 2) xCR; 3) x<3; D a<1-
= $ 1045. 225, 2) x<1, x>9, 226. 2) -\öcx<-Y , VE<x < VS. 227. 2) x>8.

228, 2) exe}. 229. 2) -1Gx<l, 8<x<5, 280. 2) -8<x<-V6, VE<

ect. 281. xt, Loro. 22 D x32 9) 9 IA, Herel.

288. 1) -4<x<0; 2) -Icx<-

2
Bexe-t, Lera excl, x>
a 1, - q <x<0. 297, 2) O<acd, zo.

1 G.
L<x<o, excel. 284. a<V2.

285. 2) x>0,01. 286.

2210. 289, 1) xclogas LENA

‘lee

zn dert, Lese
5 YE 4° 30

2 cx<0, 2>0:2)~
7 2)

1
alex
294

o 31

<s< u

so
A Teck Bk D 66-2, Ree

<x<0, x>0. 242. x<logy

ny teren ee. Dore lee arc DEE

x=5, 6<x<7,

. 246, 2) Zasncx< Baan, ned, 247. 2) —arcsin +

tan cs caren Landen, nc2 4) “arc Lan <xcaecos Lg,

A A
2 en aut
„mei mere, ang nce E, ang 2 ceed.

BR 42amex< 2 4 2x, n
Ba Sk 42nn, neZ. 250,

254 1 2 192) 65-0. 285.1 (1200 800) 2) 1). 258 2) 8-2,
(625 96 8, 45-8). 297.1 (5 0 2 05 9, (-95 82), 258 9 050,
Bs ~1)s 4) (8; ~5), (Bs 5), (4; 2V2), (4; -2V2). 259. 1) (4; 2), (4; -2)5

2) (25 4), (-25 -4). 260. 1) x=-2, v4 2) x.
4) (10; 1000), (1000; 10). 262. 2) (V8; V8). 263. as 0), (4: 12).
284.1) (2; 3), (-6; D; 2) (5 ; 2). 265. - T <a <6. 266. 2) (100; 81).

z ES
Arzencx< 2am, nez.

8, (~

267.2) (0; 1. 268.1) (logy 2: to 2), (0: 1» 22): 2 (0: toes 222),
rs 7-6: 4g 7). 2601) (0), (2452, - 28). 01.0, (48; 22),
e3

CT

270, (12 3 sr Zan) nz mn (cor em 420)
nz, bets 2) (ans CD" E xm), mez, n62 (Cdad ms

tds)

meZ, kez; (Parts

If arco 2, 45m), mez, nez. 272.3) (

neZ, mez; 2) (item: ¿ema

12. 274. x>5. 275. 1) (a; ad), (a;

cont a>0 m arlı (-a-1; (a+1)), (a+ 1); -a-1), ecan a<-1 mar-2.
276. Tipu a#3 wer pemennii, npu a-3 — (0; 1). Vkasawue. Sauncars
aropoe ypanmenne enerem m mxo x°+(y- 1) +(a- 3)? +1-cos(xy)=0.

277. 2) Us D, @ 4% © ( han). nez 9 (Ca Ton

-Z+2(m-M). mez, rez. 273, a.

+), nen, nez.

2

Cut ärruram), (15432.
are, (Are gr Bram Art a), nez, kez.
Ykazanue. Petra cucremy Kak anneliKyio OTHOCHTENRO UH 0, tie

12*%
tain zeony, v-coszeinu. 279, (81; 16). 280. 1) 1,3; 2) 5,125; 8) 2 4) 1.
281. 1 mm. 282, 126 Kin. 263. 1080 wo. 284. 16 Auch. 288. 1 ra.
286. 8, 12.207. 25 u 20 Ouneron wax 30 m 25 Onneron. 288, 2, 1, 1
280, 492 pera. 290. 18 100/a. 291,3 n/a. 202.21 1,20 1. 298. 1400 une
Ton. 294. 9, -6, 12, -24. 296, 27, 296. 1, 3, 9, 15 mam 16, 8, 4, 0
297. 2 nan 123. 298. B 3 pasa. 299. 16 cu”. 300. 1) Cog; 2) 50-100: 150;
»
PrPe
. 906. 3. 907 0.18, 908 0,97. 309. 0,90. 310 0,028

2,4 2,,8
Bert, yo tes

+ 808. 0,85; 850.

3) Clo: Cho: 150; 4) Cy -100- Cfo. 301. Page 302.

311. 2) k=-1, b= 8; 4) k=O, b=-2. 812. y

513, 2) Her; 4) ns. 814. 2) 34. B15, x>1. B16, x<-VB. 819. 2) Jo.

320. 2) (-1; 3), (5: 9). 321. 4) x<-2, x>2. 322. 4) x40. 323. 1) 75 100/05
2) 5 a: 3) =71,4 100/a. 324. Bropony; na 40 p. 325. Her, vox xox nan
mexbuee paccronnne wey Kopaßansın Oyaer pasto 3 saxiaN uepes 48 rx.
327. 1), 2) ue nunnercn. 328. 1) Annneren; 2) ue apaseren; 3) anın-
ero: 4) ne ananeren. 329. 2) Houermas: 4) vernan. 330. 2) Hever

4) vernas. 881. 2) Verna; 4) ne amnserca ueruolt u ne annaerca mover

ost 998.1) Hers 2) an. 304. 2) ASE. 205, 2) 105 4) 25. 997. 225.

980, 2) 2-1, 899, 1) 023072, m6, FB fai 29%

840. a~2, 6.

un
REZ, fax = 73 fr

D sazrcdm, ne. M D xe, 19. MA D HET,
D deco. 949 2) MDéxe-d, Der: D -

6-3. 845. 01. 346. 2) x>-

330

Oraeru

demo acto yet os Tr

12 y>0. 953, 1) —F <v< Es Dr 854 (D Etam,
5

3,5. 356. a=1-12,

+VI0, 357. a<-4, -Í<a<o.

6x-1. 862. —1. 363. 9.

369. 2) 8. 860. 2) +. 361. 2) y.

364. (3; 9), a=-3. 368. (1; 2), (0,5; 2,20). 366, (-1; -3). 307. (3; e).
368. 2) y-0,5(1+In2-xIn2). 969. 7. 370.071. 371. - 2. 372 pas.

a. yua2-0. 374. 12. 375. 2) Rispseer na npouemrox 2<0 u
£0, 918.2 2-6 — own manque. 877,2 2-2 — row somo
casa: te) SAD 3070 0
LL ee BRA neh 9 26 gone sa
Sot Fo mund Oy CE 4,36 y de 30.1

301, 54n ou. 392. 6, 398, 2. 394. sinx—2—1, 395, 132, -57. 396. 9 x 4.

397. (1; 1). 308. 42. 399, 4v2, 400. ae q~ 26. 401. 2% a.

27
En

402. Nam 409. E. ou 28. aos. AR. 406. ©. 407. 30 408. r-R 4/2.
w 216 3

2R “oR 1

Her. 409, D-H. 410. 2, 411. 0-28, p= 4. 412.2) 102. 419.2) 9
8 8 as 3
(eer: a x

. 414, 2) 4,5, 415. 2) ©, 416, ® z
4) 1. 414, 2) 4,5, 415, 2) 3, 416, 546 À. 418. c- 7

419. b>\B-1, 0<-3-V3. 420. sean, x

Ea 3%, nez, 41 a=6, d=
==11, 606. Yicasamne. Tax kak rou A u B cuumerpun Ormocament-
no pio x=2, 10 AG: Yo) BUE Yoh PRE 2) 2-1, Xy= 241, tO. Ma yer
om Fy) (sp enrayer, u 00m Fo) [Cap MS 8 +by à panen-
reo fe) FG) momo aarmcars » age Du —12 (rax sax 1>0), orga
Fi dD=PixD- 2t<0, 422. a=6, Ball, C5, 428, and, bad, 0=-2,
424, 0-4, b--5, 0-2. 425 (45 $). 420. (-2: 22), (2 10). 427. a

9-0, d=1. 428. 2,25. 429. k=2. 430. 1

. 481 a=1, 9-4. 432. area À.

al
nal 0 -2. 404.2) x Bean, nz: aod a. dern

487. 0(10)=262 m/c, (=87 c. 488. 12x, 439. 160 ve:

440. 17417 ex. nponyrmun. 441. 2) 5276. 442. 2)

aus, 2) EAM, 4) cour 2roin2r. 444. DT
ST 510
446. 2) 720. 447. 10-4, r(3)-807+418). 448. -

449. E@sindx-9), 450. Sinléx-114C. 451. 1) 11,25; 2)
3) 5,54 7In2: 4) 72. 462. 2)

1 rae
4) 1335) 4:6) Ina.

rasa

Orzersi x sanar eTlponepr cette
Faana 1

. ze 42, eZ; dymuunn wevernan, 2 yasinzı 1) am,
1. reba, nez; dynes y » z

ee Soe

ar,
Nese RIM

noch Ei mére de Pacino, Here
pang eae pos
Care :

ec a og og, 2, 093.

owe. 4 og, ote, cet, 093

1. Pue, 161, a) 7 +2an<x<F + Ban, neZi 0) —n+2ansx< un. 2. Pre. 162,

10 up BE Banc cn med 3 denen. 15 215, Pe 188

Y
1 y=-cosx
Ey A 3 ñ I

Puc. 151

332

Orwerer

Taana IT

mao À er 0, 9 Gcoremx-dainarainx:
ww

ZUBE ga 34 sels yor HE

pr eee ee house

2.91 2
10222 1.2 Her, 3 2-1, 29-4. 4. yn ded, y-30+62.
D 52 2 u y :

Faana IT

1. Bosparuer on 31 4, Mannes npn #<1 À 2) yéumer mp 234.
2. Mon y=T mpi 2-3. 3. Ou. pue. 154 4. axones 10, ax
seamen 63 mu Du. a 9

MSIE Gar pre 199.3, Hanfommee e, nemmener 0. 4. AR

Taana IV
2 Pp. 8. 1) 42; 2) Li 8) 0; 4) 1. 4. 205 xo. es
(eat. 8, 1) 425 2) 2; 3) 0; 4) 1. 4. 208 nu. en.

1, Pr) 4302. 2. 1) À;

2 4 Fr
y 2in2, 3. = 4 au. en. (mic. 150).

4, $= 9 na. en. 5. s-2 un. en.

y= xt +824

u
1

Pue. 156

388

Grec

Faasa V

1. 1) 5040; 2) 396; 3) 56; 4) 2. 2. 1) n%en5 2) GB 84 emo
Sn +
5. 720 cnocogaun. 6. 720. 7. 1) 2°+6x5y+

Ga. 4, 5040 enocobasan o
) 1 Sa + 1082 - 100% + 5af -aÿ.

Tay? + 20288 y VS ty + Bay! 4

1) nats 2) ne, 2 nee, M2, Mas. 5. 14580.
4, 64 euocobanat. 5, 1) CF 86; 2) 2912-286. 6, ~19504-x°,

Taana VI

1 14 18 15

1,2, os. 8, La 15. 5, 0,64. 6, 15,

17

1. 0,18, 2, 0,992. 8, 0,8. a. 47. .

Fano VIE

1.1 86 2) 4-44 3) T1435; 4)
2n

2

3.0) 2 (con 2 + sain 28); 2) cos

2) 2,2513. 5, Oxpyxctocrs e uemrpox » rouxe (0; 0) u paauycon R=3.
and ya $58) 251. 8. 1) 292, 2-42, am

i 2) 2-4, 2124284, 29=-2- 2188.

Fassa VI

1. 9 Tipaman, apoxonsuan vepe ron (0: 1) u (4: 0) 2) oupymoers

paguyca 4 e xenrpox m rouxe (3; — 1). 2. 2) Kpyr paauyea 2 € ueurpon u
Touxe (-2: 5). 3. Bnyrpennan oÓxacre rpeyronuua € Bepumant (1; 0),
(0; =D), (4-9) o

1. Cm. pue. 157. 2. Cm. pue. 158; su 3. 1) Keaaper c sepant-
Hamu 9 roukax AB; 0), B(0; —8), C(O; 8), D(B: 0); 2) nee rouxx, ne-
same suyrpu xoaapara ABCD.

334

Diver

Ornasnenne

Tpuronomerpuueckue yaaa

$ 1. Oßnaors, onperenomta m Mmomeerno anavenh rparronone-
pwtieciorx Oya

$ 2. Yerocrs, weverHocts, nepiomocre rpurononerpuec-
ox yma -

$ 3. Cooßeru Giant y=co x wee rpagua o
$ 4. Cooticrna Gym y=sinx m ee rpagu Co
$ 5. Caoñicraa m rad yo y= tex m y=etgx -

$ 6. OGparume rpurowomerpuseckue DUI se en nen

Ilponssonnaa m ee reomerpmueckmk

Tiponen nocmexonaremumoer +... +...
Tipenen Gym . .

. Henpepsisuocre DURO 2 2 eee eee
: Onpenenemne npowamogmoñ ! ! : : 2: 1:11:
Tipasnina auppepenmupomamn ! . u. une: Ei
Tiponsmonnan crenennoit Qymkusn ee oe
- Tipowsnomne anemewrapuntx Hymn 2: 2 2!!! | |
Teomerpuueckui cxnien nponsmogmol . : > > | FINE

Fnasa 111. Ipumenenne npoxaronmoë

x HCCHENOBAHMIO hyrapık
$ 1. Boapncranme x yonmanıne DA ee . . . .
$ 2. Oxerpemymur DRE. ooo
$ 3. HanGoxbmee H HaMMeHbulee suauenun QYEKUUM . . . +
$ 4. TIponsnoguan Broporo nopaaka, BLINyKXOCTE M TOUKH mer
Denia. ss
$5. Tlocxpoeiore pau Gymnas CIDO

a IV. MeppooOpasnas u unrerpaa

$ 1. Tlepaoospasuan .
$ 2. Iipanna axomaensis’ meprooëpasux =D
$ 8. Tomas xpusonunelinol rpanewns. Mererpaa x ero sur

$ 4. Burencrenne nome ‘Guryp € |noomux nurerpr-
$ 5. Tipisenteinie imrerpaon aan” pememis "duanvecxix
sauna

$ 6. Ipocreikunte anddepenarsunie ypannemts >. >> |

12
19
26
38

44
53

74
18
84

98
102
107

us
18

Fnasa v. KomÖunaropnxa
$ 1. Maromarnueckan unaykıma
$ 2. Ilpauuno uponevereuna. Pasxemeuien € nosropenunsat
$ 3. Mlepecranopxn .....
$ 4. Pasmemenus Gea nostopexuit
$5. Couerntun des novropeunl Gonos
À 6: Cons © aoeropemmeace

Tnasa Vi. Dnemenrsi reopun Bepontnocreii
. Bepomruocra cofurran
Creams neporzmocre
Sons seponricore. Hesanxmocrs dal 1 >
Bepommocrs ponentes Sama ID}
: Sopas pagana

Fnasa Vil. Komnaerensie unena

$ 1. Onpexenemne kosmaekenux witeex. Cromenne 1 yamome-
ne komnaeKemux sees

$ 2. Komunexcno coupnxennsie wicna, Moayap Komunexcnoro
ena. Onepanunt yuiueranım m nenemen .
$ 3. Teomerpiuceran mrepnperamun Kommnexenoro mean:
$ 4. Tpurowomerpuueckas Qopma KONIAEKCKOrO men + >
$ 5. Yawoxenue 1 meneuxe KOMMAEKCHBIK uncer, samcammeo
n xpurononerpiweckolt dope. Bopnyan Myanpa . + + > =
$ 6. Knaxparnoe ypamnenne € KOMILIEKCHUN Hextsnecrihint

$ 7. Mannenemne Kopux 15 konmaexenoro wena, AxreGpairuec-
one ypannena

Fnasa Vill. Ypannenus u nepasenctsa € ABYMA
nepemennsimn
$ 1. Anmehmae ypa
$ 2. Heanuetinnse ypastienns u xepes
$ 3. Ypomnenis 1 mepanencraa € nym nepent
HAE NADAMETPM vs see ee es

DDR, conep-

Yupaxnenusa AA wrorosoro nopTopenns
Kypea aareOpst m nauan MATEMATHUECKOO
anaııaa

Upenmermerkt yrasatems . . - .

157
159
163
166
169
174

180
186
189
194
197

204
209
214
218

221
225

228

287
244
259

an

Yuc6uoe monanne

Konarun lOpuit Maxaiizonia
Ticauena Mapua Bnanumnpopna
Denoposa Hagexza Eureusenna

Miaóyame Muxasın Hnanopmu

Aare6pa n navana
MATEMATHUCCKOrO anamusa

11 ruace

Yueönux ana
oBınteoßpaaonarensunix yaperxnenstt

Basopsık x npobunsuni ypopum

Bon. penexunch T. A. Bypmucmpona
Penaxrop JT. H. Beaonoscxas
Magu pexaxtop E. A. Andpeenxooa
‘Xyaommne B. A. Andpuanon
XyxoxcecrnenmaR pexarop O. II. Boromonosa
Komuvorepuas rpaduxa: I. M. Inumpues
Texumuecxuh penaxrap C, H. Tepexosa,
Kopperropn: JI. A. Epxoxuna, H. M. Hoauxoaa, A. B. Pydaxosa,
4. 5. Hepnosa

Hesoronan _ aurons — OOmepocentenni KaacenpKarap porra

OK 005-93—953000. Han. nn. Cepua HL M 05824 or 12.09.01. Caano

# naGop 19.04.06. Honuncauo y uewars 21.04.10. Popmay 60x90'/,,, By-

ara ocean. Fapmrrypa Illkommas. Heuarı opcerman. Yuan. 1. 20,64
‘ops. 0,43. Jon. rapa 20 000 ara. Bnıcna M 30062.

Orkpsrroe axuuoxepuoe oßıneerno «Honarenserso «Ilpocnemenner.
127521, Mocxoa, 3-4 mpocax Mapamoñ pont, 41.
Orueuarano # OAO «Caparoscxuß, nonurpaßomönnars.

410004, y. Caparon, ya. Mepnumenexoro, 59. www.surpk.ru

PRE
KOMNNEKCHbIE 4MCNA

KOMBHHATOPHKA

P=-1 m=1:2:3-...+(n-1)0n 01 =1
(a +bi)+(c+di)=(a +0) +(b+d)i

Bm x. Am mt EM
Pan An= Gam Cm IN
(a + bi) (c + di) = (ac - bd) + (ad + be)i ch=cn" CON EC F
+bi=a-bi +bil= \a? +6?
En ray Bunom Hbiotona
a À | (a+br=Clan+ Cab. + Cab" + CM ab mp"
cos p=
z=a+bi \a’+b | | Tpeyronsaux Mackana
b s
RO eet

3 1

2=r (cosy + ising)

2y 2¿=r, 7 (cos (+ G2) + ¿sin(p,+ 92)

21

7 Fi (eos Qi 92) + isin(9,— 92)

®opmyna Myaspa

(cosy +i sing)" = cos n p+isinnp

rar G/A)=P@-P(4/B)

Yueönux coomeememayem Dedepanonaix
xomnonenman eocydapemoenno2o
cmandapma o61ezo o6pasosanus

B yue6no-meropmieckni komnnexT
no anre6pe u Hayanam Maremarmueckoro
axanusa ans 10-11 knaccos

non penaxuneñ A. B. Kwxcienko BXOAAT:

© YueGunKn ana 10 u 11 knaccos
(asmopvı JO. M. Koxazun, M. B. Trauesa,
H.E. Dedoposa, M. H. ITaëyuun)

© Aunakrusecxye marepyanbi
ana 10m 11 xnaccos
(aamopsı M. H. LaGynun, M. B. Travesa,
H.E. Pedoposa, O. H. oGposa)

© Vaysenne anreöpsı n Hasan
MATEMATHIECKOTO anna
8 10m 11 wraccax. Kunin ans yurrens
(aamopu H. E. Dedoposa, M. B. Traveoa)

Алгебра и начала анализа 11 класс Колягин

About This Presentation

Slide Content

Tags

Categories

Download

Quick Actions

Statistics

Related Slideshows

Алгебра и начала анализа 11 класс Колягин

About This Presentation

Slide Content

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

Slide 56

Slide 57

Slide 58

Slide 59

Slide 60

Slide 61

Slide 62

Slide 63

Slide 64

Slide 65

Slide 66

Slide 67

Slide 68

Slide 69

Slide 70

Slide 71

Slide 72

Slide 73

Slide 74

Slide 75

Slide 76

Slide 77