krebuli maTematikaSi

(ekonomikis specialobis

studentebisaTvis)

2006 / 07 s.w.

2

Sinaarsi

simravle. moqmedebebi simravleebze 2

matricebi da determinantebi 3 matricebi ($2.4) 3 determinantebi ($2.5) 4 Sebrunebuli matrici ($2.6) 5 matricis rangi ($ 2.7) 5

wrfiv gantolebaTa sistemebi 5 wrfiv algebrul gantolebaTa sistema. krameris Teorema ($2.8); wrfivi sistemis amoxsnis matriculi xerxi ($ 2.9); kroneker-kapelis Teorema ($ 2.10.) wrfivi sistemis amoxsna gausis meTodiT ($2.11.) 5

analizuri geometriis elementebi 7 dekartis marTkuTxa da polaruli koordinatebi sibrtyeze. manZili or wertils Soris. monakveTis gayofa mocemuli fardobi ($. 3.1.) 7 wrfe sibrtyeze ($3.2) 7 meore rigis wirebi ($ 3.3.) 9 veqtoruli algebra ($3.4) 11

zRvari 11 funqcia ($ 4.1) 11 mimdevroba. mimdevrobis zRvari ($4.2) 12 funqciis zRvari. uwyvetoba ($ 4.3) 13

diferencialuri aRricxvis elementebi 14 funqciis warmoebuli da diferenciali ($4.4) 14 ganuzRvrelobis gaxsnis lopitalis wesi ($4.5) 15 funqciis monotonuroba da eqstremumi ($4.6) 16 funqciis grafikis amozneqiloba da Cazneqiloba. gaRunvis wertilebi ($ 4.7) 17 grafikis asimptotebi ($ 4.8) 17 funqciis gamokvleva da grafikis ageba ($ 4.9) 17

mravali cvladis funqcia 17 ori cvladis funqcia ($5.1) 17 ori cvladis funqciis zRvari ($ 5.2) 18

ori cvladis funqciis diferencirebadoba. ori cvladis funqciis kerZo warmoebulebi. sruli diferenciali. rTuli funqciis gawarmoeba ($.5.3) maRali rigis kerZo warmoebulebi da diferencialebi ($ 5.4.) 18 ori cvladis funqciis eqstremumi ($ 5.5) 19

integraluri aRricxvis elementebi 19 pirveladi funqcia da ganusazRvreli integrali ($6.1) 19 $6.2 cvladis gardaqmna ganusazRvrel integralSi (Casmis xerxi) 19 nawilobiTi integreba ($6.3) 20 kvadratuli samwevris Semcveli umartivesi integralebi ($6.4) 20 trigonometriuli funqciebis integreba ($6.5) 20 gansazRvruli integralis cneba; niuton-laibnicis formula ($7.1) 21 gansazRvruli integralis gamoTvla Casmis xerxiT ($7.2) 21 gansazRvruli integralis gamoTvla nawilobiTi integrebis xerxiT ($7.3) 21 brtyeli figuris farTobis gamoTvla ($8.1) 22 brunviTi sxeulis moculobis gamoTvla ($8.3) 22

diferencialuri gantolebebi 22 gancalebadcvladebiani diferencialuri gantoleba ($ 10.2) 22 pirveli rigis wrfivi diferencialuri gantoleba ($10.4.) 22 meore rigis mudmivkoeficientebiani wrfivi erTgvarovani diferencialuri gantoleba ($10.10.) 22

pasuxebi: 23 Tavi I 23 Tavi II 24 Tavi III 26 Tavi IV 29 Tavi V 33 Tavi VI 34 Tavi VII 36 Tavi VIII 36 Tavi X 36

simravle. moqmedebebi simravleebze

1.1. daadgineT Semdegi ori Canaweridan, romelia swori:

a) {4,5}{4,5, {4,5,7}}, Tu {4,5}{4,5,{4,5,7}}.

b) {3,4}{3,4, {3,4}}, Tu {3,4}{3,4,{3,4}}.

1.2. mocemulia A{-3, -1, 0, 2, 5, 6}, B= {-3, 2, 3, 4, 5}, C={0, -3, 11}. ipoveT (AB)C, (AB)C,

(A\B)C.

1.3. A(-2,3) da B=[1, +). ipoveT AB, AB, A\B, B\A simravleebi da gamosaxeT isini ricxviT wrfeze.

qvemoT mocemuli simravleebi warmoadgineT maTi elementebis dasaxelebiT

1.4. 03x4x)2x(|RxA 2 .

1.5.

0x,5x

3x|ZxA . 1.6.

1239

1|ZxA x .

1.7. 4)10x(log|NxB 2 . 1.8. 2xcosxsin|RxB .

1.9. mocemulia simravleebi A={xR | x2+9x-8} da B={xR| x

2-7x+120}. ipoveT AB, AB simravleebi.

1.10. A={xR| |x+3|1}, B={xR| x2-3x+4<0}. ipoveT AB, AB, A\B, B\A.

sakoordinato sibrtyeze gamosaxeT Semdegi simravleebi:

1.11. A={(x,y)R2| -4x2}, B={(x,y)R

2| -3y2}, ipoveT AB.

1.12. B={(x,y)R2 | (x

2-4)(y+3)=0}.

3

1.13. B={(x,y)R2 | 3x-5y+1=0, 2x+3y+3=0}.

1.14. A={(x,y)R2 | xy>0}. 1.15. A={(x,y)R

2 | -4y-x2

}.

daamtkiceT, rom

1.16. (AB)C=(AC)(BC).

1.17. AU, BU, maSin U\(AB)=(U\A)(U\B).

U\(AB)=(U\A)(U\B).

matricebi da determinantebi

matricebi ($2.4)

2.95. SekribeT matricebi

a)

1 3 1-

0 1 2

2 1- 3

A da

2- 3 1

2 1- 0

1- 2 1

B b)

1 2 1-

1 1- 2-

2- 1 5

A da

0 2- 1

1- 2 2

2 1- 4-

B

2.96. mocemulia matricebi

3 0 2

1 1- 3A da

2 3 1-

1- 4 1B , ipoveT: a) 3A-2B; b) A-3B; g) – 4B.

2.97. ipoveT B matrici, Tu 2A+3B=C, sadac

1

3

04

12A ,

1

1

32

12C .

gadaamravleT matricebi

2.98.

12

21

31

12 ; 2. 99.

dc

ba

10

01; 2.100.

212

321

121

130

112

213

;

2.101.

120

201

012

011

210

111

011

121

112

. 2.102.

31

10

12

21

13

. 2.103.

10

21

54

43

32

21

.

2.104.

1

0

1

2

2131 . 2.105. 1421

1

1

2

1

2.106.

2013

1121 matrici gaamravleT mis transponirebulze.

2.107. mocemulia matricebi

201

312

013

A da

012

311

121

B . gamoTvaleT AB da BA.

SeasruleT moqmedebebi (2.108-2.111):

2.108.

2

120

131

121

2.109.

3

12

21

2.110.

n

52

21

12

25

2.111.

5

152

4177

142

621

101

113

4

determinantebi ($2.5)

gamoTvaleT determinantebi:

2.112. 32

41

, 2.113.

coscos

sinsin, 2.114.

sincos

cossin, 2.115.

biaba

babia

,

2.116. 22

2

babaab

aba

, 2.117.

423

157

243

, 2.118.

221

12210

424

, 2.119.

520

232

312

, 2.120.

x1x

1x0

x1x

,

2.121.

1)cos()(cos

sincoscos

cossinsin

2

, 2.122.

23

22

21

321

aaa

aaa

111

daamtkiceT Semdegi tolobebi:

2.123.

333

222

111

33333

22222

11111

cba

cba

cba

cybxaba

cybxaba

cybxaba

. 2.124.

333

222

111

33333

22222

11111

cba

cba

cba

x2

cxbaxba

cxbaxba

cxbaxba

.

2.126. )bc)(ac)(ab(

abc1

cab1

bca1

.

amoxseniT Semdegi gantolebebi:

2.127. ,121

x3

2.128.

x7

15

x25

1x , 2.129.

x23

1x5

x10

1x1x

123

,

2.130. 2

212

31x

13x 2

, 2.131. ,0

111

32x

94x 2

2

amoxseniT Semdegi utolobebi:

2.133. 1

121

2x1

123

; 2.134. 0

x35

211

12x2

gamoTvaleT determinantebi:

2.135.

1153

2471

5412

0153

, 2.136.

3214

2143

1432

4321

, 2.137.

1111

2111

0101

1010

2.138.

1012

2331

1213

0121

,

5

Sebrunebuli matrici ($2.6)

ipoveT A matricis Sebrunebuli matrici (2.140-2.144).

2.140.

53

21A , 2.141. ,

32

41A

2.142.

012

134

212

A , 2.144.

1028

213

325

A .

gansazRvreT X matrici Semdegi gantolebidan (2.145-2.152)

2.145.

44

07

21

13X , 2.146.

33

21

73

X

102

112

213

, 2.147.

33

21

73

X

102

112

203

,

2.148.

0

2

6

X

121

123

412

, 2.149. ),36(

12

21

13

X

2.151.

100

010

001

X

1621

113

212.

matricis rangi ($ 2.7)

gamoTvaleT Semdegi matricebis rangi

2.153.

310

623

811

, 2.154.

112

511

421, 2.155.

182

313

141

2.156.

373

241

111

,

2.157.

2185

4232

6521

, 2.158

4131

122122

2832, 2.159.

16554

31312

23123 2.160.

32212

13513

21321

,

2.161.

14175

70513

43253

52331

, 2.162.

110002

00000

50001

, 2.163.

6004

9006

3002 2.164.

10000

01000

00100

00010

00001

.

wrfiv gantolebaTa sistemebi

wrfiv algebrul gantolebaTa sistema. krameris Teorema ($2.8); wrfivi

sistemis amoxsnis matriculi xerxi ($ 2.9); kroneker-kapelis Teorema

($ 2.10.) wrfivi sistemis amoxsna gausis meTodiT ($2.11.)

amoxseniT sistemebi krameris formulebiT:

2.165.

.13xx4

,1x2x

21

21 2.166.

.69x18x35

,8xx2

21

21 2.167.

.07x2x3

,05xx4

21

21 2.168.

.0xx2

,0x5x

21

21

2.169.

.0x2 x5

0xxx

,0x3xx2

31

321

321

2.170.

.10x4xx2

3xx4x3

,4x4xx

321

321

321

2.171.

.6x4x3x7

,8x3x4x2

,1xx2x5

321

321

321

2172.

.0xxx

3x3xx2

,1x3x8x

321

321

321

2.173.

.7x2x

13xx3

,7xx2

32

31

21 2.174.

.6x2x2x3x3

12x4x3x5x8

6x2xx3x4

4xxx2x2

4321

4321

4321

4321

2.175.

.73x x2x

,01x4x3x

,4xx2xx3

,5x2x2xx

421

432

4321

4321

6

kroneker-kapelis Teoremis gamoyenebiT SeiswavleT sistemaTa Tavsebadobis sakiTxi da ipoveT zogadi amonaxsni

2.178.

.5x6x3

,3x2x

21

21 2.179.

.0x6x3

,0x2x

21

21 2.180.

.6x2x4

,3xx2

21

21 2.181.

.0x3x5x

5x2x2x

,2xxx3

321

321

321

2.182.

.14x 2x

25x x

1,x2xx

31

31

321

2.183.

.3x3x6x3

2x2x4x2

,1xx2x

321

321

321

2.184.

.3xx2x3

,2x3xx2

,5x2x3x

321

321

321

2.185.

.0x3xx2

,0xx4x

321

321

2.186.

.0xx2x5

,0x3xx

,0x4x3x6

321

321

321

2.187.

.0x12x3x3

,0x8x2x2

,0x4xx

321

321

321

2.188.

.5xx4x3

,1xx3x

,3xx2x

321

321

321

2.189.

.52

,123

,22

4321

4321

4321

xxxx

xxxx

xxxx

2.190.

.8x14x3xx3

,5x4x3x2x6

,4x6x5x3x9

4321

4321

4321

2.191.

3x2x3

,4xxx

,1xx2x

,2xx2

21

321

321

31

gamoikvlieT da amoxseniT sistemebi (2.195-2.200)

2.193.

.5x3x6

,3xax

21

21 2.194.

.10x4x2

,5axx

21

21 2.195.

.2axxx

,1x3xx2

,1x4xx

321

321

321

2.196.

.2x4x2ax

,0x2xx2

,0x2xx

321

321

321

2.197.

.0x3x2x

,0x15x14ax

,0xx2x3

321

321

321

2198.

.1axxx2

,3x9x8x5

,bxx2x3

321

321

321

CawereT sistemebi matriculad da amoxseniT

2.199.

.7x2x4x

,2x9xx3

,6x6x3x2

321

321

321

2.200.

.1xx3x

,7x3xx2

,1x5xx3

321

321

321

2.201.

.11x3x2x3

,14x2x3x4

,17x3x4x5

321

321

321

2.202.

.62xx 2x

,9x3x2xx

,10x43xxx2

,9x5x23xx

431

4321

4321

4321

gausis meTodiT amoxseniT gantolebaTa Semdegi sistemebi:

2.203.

.3x5xx4

,0xxx

,5xx2x3

321

321

321

2.204.

.10xxx7

,5x2x2x

,5xxx2

321

321

321

2.205.

.1x2x3xx2

,0xxxx3

,6x2xx

,4xxxx

4321

4321

431

4321

2.206.

.0x8xx9

,8xx8x2

,2x3x2x4

321

321

321

2.207.

.3-x x4x

,4x3x3 x

,12xx2 x

,16x2x2xx

421

431

432

4321

2.208.

.5x3x3x3x4x

,2x2xxxx

,1xxxx2x3

54321

54321

54321

7

analizuri geometriis elementebi

dekartis marTkuTxa da polaruli koordinatebi sibrtyeze. manZili

or wertils Soris. monakveTis gayofa mocemuli fardobi ($. 3.1.)

3.1. aageT wertilebi: M1(5,2), M2(-3,1), M3(-4,-2), M4(4,-3), M5(6,0), M6(0,3), M7(0,-4). 3.7. daamtkiceT, rom samkuTxedi, romlis wveroebia A(-2;2), B(1;1) da C(-1,3), aris marTkuTxa.

3.8. daamtkiceT, rom samkuTxedi, romlis wveroebia A(-3;-1), B(-1;0) da C(-2,-3), aris tolferda.

3.9. ordinatTa RerZze ipoveT iseTi wertili, romlis daSoreba M(4,-6) wertilidan aris 5

erTeuli. 3.10. uCveneT, rom oTxkuTxedi, romlis wveroebia A(7;-2), B(10;2), C(1,5) da D(-2,1) paralelogramia.

3.11. mocemulia wesieri samkuTxedis ori wvero: A(3;-2), B(8;10), moZebneT farTobi. 3.15. koordinatTa RerZebze ipoveT wertilebi, romelTagan TiToeuli Tanatolad aris

daSorebuli A(1;1) da B(3;7) wertilebidan. 3.16. A(x;4) da B(-6;y) wertilebs Soris monakveTi C(-1;1) wertiliT iyofa Suaze. ipoveT A da B

wertilebi.

3.17. A(8;-2) da B(2;4) wertilebs Soris monakveTi dayaviT eqvs tol nawilad. gamoTvaleT dayofis wertilebis koordinatebi.

3.18. A(3;0) da B(5;-4) wertilebs Soris monakveTi dayaviT oTx tol nawilad. gamoTvaleT dayofis wertilebis koordinatebi.

.3.20. ipoveT A(1;4), B(-5;0), C(-2;-1) samkuTxedis medianebis gadakveTis wertili.

3.21. mocemulia samkuTxedis wveroebi A(2;-1), B(4;-2) da C(10;3). ipoveT A kuTxis biseqtrisisa da BC

gverdis gadakveTis wertili. 3.22. mocemulia samkuTxedis wveroebi A(4;1), B(7;5) da C(-4;7). ra nawilebad yofs A kuTxis

biseqtrisa mopirdapire BC gverds? 3.25. rombis gverdia 14 sm, xolo ori mopirdapire wveroa A(3,1) da B(8,6). gamoTvaleT rombis

farTobi. 3.26. aCveneT, rom A(8;3), B(5;-2) da C(-1;-12) wertilebi mdebareoben erT wrfeze.

3.27. mocemulia kvardatis ori mopirdapire wvero A(1;-2) da C(-1;2). moZebneT misi farTobi.

3.32. mocemulia samkuTxedis ori wvero: A(1;,8), B(5;-1) da medianebis gadakveTis wertili M(1;3).

moZebneT samkuTxedis mesame wvero.

3.35. mocemulia samkuTxedis ori wvero A(2;,3) da B(4;-3). samkuTxedis farTobia 20. ipoveT mesame C

wveros abscisi, Tu ordinatia 5.

wrfe sibrtyeze ($3.2)

3.36. mocemulia wrfe 3x-2y+11=0. wertilebidan M1(1;7), M2(-3;1), M3(1;4), M4(-5;8), M5(0,0), M6(3;10) romeli mdebareobs masze da romeli ara.

3.37. A(2;a) da B(b;-4) wertilebi mdebareoben 3x-y+5=0 wrfeze. moZebneT a da b.

3.38. moZebneT 3x-2y-12=0 wrfis gadakveTis wertilebi sakoordinato RerZebTan da aageT es wrfe.

3.39. wrfe gadis A(3;4) da B(5;-2) wertilebze. gamoTvaleT am wrfis kuTxuri koeficienti da ordinatTa RerZze CamoWrili monakveTi.

3.40. moZebneT Semdegi wrfeebis sakuTxo koeficientebi da ordinatTa RerZze CamoWrili

monakveTebi: a) 2x-y-3=0; b) 2x+3y-5=0; g) 2y+5=0; d) 3x-7=0; e) x+y=0. 3.41. ra kuTxes adgenen OX RerZTan Semdegi wrfeebi:

a) x+y-3=0; b) 2x+3y-5=0; g) x-y-2=0; d) x=5y. 3.42. dawereT wrfis gantoleba, romelic gadis koordinatTa saTaveze da OX RerZis dadebiT

mimarTulebasTan adgens Semdeg kuTxes: a) 450; b) 1350; g) 300 ; d) 1800.

3.43. aageT wrfeebi: a) 2x+3y-6=0; b) 3x+5y=0; g) 2x+5=0; d) 3y-7=0; e) 2x-4y+5=0; v) 4y=0.

3.44. dawereT im wrfis gantoleba, romelic gadis A(0;-4) wertilze da OX RerZis dadebiT

mimarTulebasTan adgens 3

kuTxes.

3.45. wrfeTa Semdegi gantolebebi gadawereT gantolebebze RerZTa monakveTebSi:

a) 2x+3y+36=0; b) 7x-8y=0; g) 5x+3=0;

d) x+y+3=0; e) x+3=0; v) x-5y-11=0.

8

3.46. dawereT im wrfis zogadi gantoleba, romelic sakoordinato RerZebze kveTs 3

2p da

7

3q monakveTebs.

3.47. wrfe sakoordinato RerZebze kveTs tol dadebiT monakveTebs. dawereT misi gantoleba, Tu sakoordinato RerZebTan mis mier Sedgenili samkuTxedis farTobi 72 kv. erTeulia.

3.48. dawereT wrfis gantoleba, romelic sakoordinato RerZebze kveTs tol monakveTebs da

RerZebs Soris misi monakveTis sigrZea 27 erTeuli. 3.49. dawereT im wrfis gantoleba, romelic abscisTa RerZze kveTs orjer meti sigrZis

monakveTs, vidre ordinatTa RerZze da gadis A(4;3) wertilze.

3.50. mocemulia wrfis gantoleba 04

1y

3

1x

. gadawereT es gantoleba wrfis a)

gantolebaze kuTxuri koeficientiT b) gantolebaze RerZTa monakveTebSi; g) zogadi saxis gantolebaze; d) normaluri saxis gantolebaze.

3.51. mieciT normaluri saxe wrfeTa Semdeg gantolebebs:

a) 5x-12y-26=0; b) xsin200+ycos20

0+3=0; g) 3x+1=0;

d) 2y-1=0; e) 2x-y-1=0; v) 3x+4y-5=0.

z) 6x+8y+5=0; T) 3 x+y -2=0; i) 3x+y=0.

ZiriTadi amocanebi wrfeze

3.56. M(2;-1) wertilze gaatareT wrfe, romelic 3x-2y+5=0 wrfis paraleluria.

3.57. ipoveT im wrfis gantoleba, romelic gadis M(2;-3) wertilze da M1(1;2) da M2(-1;-5) wertilebze gamavali wrfis paraleluria.

3.58. ipoveT im wrfis gantoleba, romelic gadis M(1;2) wertilze da M1(4;3) da M2(-2;1) wertilebze gamavali wrfis marTobia.

3.59. kvadratis erTi gverdis bolo wertilebia A(2;3) da B(-2;0). vipovoT kvardatis gverdebis gantolebani.

3.60. moZebneT Semdeg wrfeTa gadakveTis wertilebi:

a) 2x-3y+5=0, x+2y-1=0;

b) x-3y+4=0, 2x+6y-1=0;

g) 2x-4y+1=0, 3x-6y-7=0;

d) 2x-3=0, 3y+2=0;

e) x-1,5y+1=0, 2x-3y+2=0.

3.61. samkuTxedis gverdebis gantolebebia 5x-y-11=0, x+5y+3=0 da 2x-3y+6=0. moZebneT misi wveroebi.

3.62. samkuTxedis gverdebis gantolebebia x-2y+3=0, 2x+3y-8=0, 5x+4y-27=0. gamoTvaleT misi farTobi.

3.63. SeadgineT im wrfis gantoleba, romelic gavlebulia x-3=0 da 2x+3y-12=0 wrfeTa gadakveTis

wertilze 5x-4y-17=0 wrfis paralelurad.

3.64. ipoveT im wrfis gantoleba, romelic gadis M(-3;2) wertilze da 600 kuTxiTaa daxrili 3 x-

y+2=0 wrfisadmi.

3.66. SeadgineT im wrfis gantoleba, romelic gavlebulia 3x-y-3=0 da y=2

11x

3

2 wrfeTa

gadakveTis wertilze pirveli wrfis marTobulad. 3.67. ipoveT manZili Semdeg paralelur wrfeebs Soris:

a) x-2y+5=0, 2x-4y+1=0.

b) 3x-4y+7=0, 3x-4y-18=0.

g) 3x+4y-6=0, 6x+8y-7=0. wveroze gamaval medianamde.

3.74. M(-1; 2) wertilze gaatareT wrfe ise, rom N(3;-1) wertilidan daSorebuli iyos 2 erTeuliT. 3.75. 4x-3y-3=0 wrfis paralelur wrfeebs Soris moZebneT is, romelic 3 erTeuliTaa

daSorebuli (1;-2) wertilidan.

3.76. A(4;3) wertilze gaatareT wrfe, romelic koordinatTa saTavidan daSorebulia 4 erTeuliT. 3.78. moZebneT im wrfis gantoleba, romelic (1;1) wertilidan daSorebulia 2 erTeuliT, xolo

(3;2) wertilidan - 4 erTeuliT.

9

3.79. abscisTa RerZze moZebneT wertili, romelic toli manZiliTaa daSorebuli koordinatTa

saTavidan da 15x-8y+34=0 wrfidan. 3.81. mocemulia wrfe 8x+15y+68=0. dawereT misgan 3 erTeuliT daSorebuli paraleluri wrfis

gantoleba. 3.82. kvadratis ori gverdia 3x+y-1=0 da 6x+2y+11=0. gamoTvaleT misi farTobi.

3,83. dawereT 2x+3y-18=0 da x-5y+17=0 wrfeebis gadakveTis wertilze gamavali abscisTa RerZis paraleluri wrfis gantoleba.

3.87. dawereT A(-1;3) da B(5;-2) wertilebis SemaerTebeli monakveTis Sua wertilze gamavali, AB-s marTobuli wrfis gantoleba.

3.83. moZebneT rombis wveroebi, Tu misi ori gverdis gantolebaa 3x+y-1=0 da 3x+y+7=0, xolo erT-erTi diagonalis - x-y+5=0.

3.89. rombis ori mopirdapire wveroa A(4; 7) da C(8; 3), xolo farTobi - 32 kv. erTeuli. dawereT rombis gverdebis gantolebebi.

3.90. kvadratis erT-erTi gverdis gantolebaa x-y+1=0, xolo diagonalebis gadakveTis wertilia

M(4;1). SeadgineT danarCeni gverdebis gantolebebi.

3.91 paralelogramis ori gverdis gantolebaa x=1 da x-y-1=0. M(3; 6) misi diagonalebis gadakveTis wertilia. dawereT diagonalebis gantolebebi.

3.92. ipoveT A(2;-2) wertilis simetriuli wertili x+3y-6=0 wrfis mimarT. 3.93. dawereT wrfis gantoleba, Tu A(1;3) wertili warmoadgens saTavidan am wrfeze daSvebuli

marTobis fuZes. 33.95. dawereT im wrfis gantoleba, romelic marTobulia 5x+12y-1=0 wrfis da M(3;1) wertilidan

daSorebulia 2 erTeuliT.

3.96. dawereT im wrfis gantoleba, romelic simetriulia 3x-2y+1=0 wrfisa M(5; 1) wertilis mimarT.

3.97. mocemulia ABC-s wveroebi: a) A(-2; 8), B(10; 3) da C(16; 11); b) A(-1; 5), B(11;0) da C(17; 8); g) A(6;7), B(-6;2) da C(-10;5);

ipoveT: 1) gverdebis gantolebebi;

2) A wverodan gavlebuli medianis gantoleba; 3) medianebis gadakveTis wertili;

4) B kuTxis biseqtrisis gantoleba;

5) am biseqtrisis gadakveTis wertili AC geverdTan;

6) A wertilidan BC gverdze daSvebuli simaRlis gantoleba da sigrZe; 7) samkuTxedis farTobi;

3.107. dawereT samkuTxedis gverdebis gantolebebi, Tu cnobilia maTi Sua wertilebis koordinatebi D(1;3), E(-2;1) F(4;-2).

meore rigis wirebi ($ 3.3.)

3.111. dawereT wrewiris gantoleba, romelic A(2;2) wertilze gadis da romlis centri C(0;3)

wertilSia.

3.112. daiyvaneT kanonikur saxeze wrewiris Semdegi gantolebebi:

x2+y

2-2x+4y=0, x

2+y

2-6y+5=0,

x2+-2x+y

2-4y-20=0 x

2+y

2-4y=0.

3.113. SeadgineT (0;1), (3;0) da (-1; 2) wertilebze gamavali wrewiris gantoleba.

.

3.115. SeadgineT (x+3)2+(y-5)

2=25 wrewiris im diametris gantoleba, romelic marTobulia

5x+10y-12=0 wrfis.

3.116. dawereT (x-1)2+(y+2)

2=5 wrewiris mxebTa gantolebani A(2;0), B(3;-1), C(3;-3), D(0;-4)

wertilebSi.

3.117. dawereT x2+y

2-8x+2y-8=0 wrewirisadmi A(7;-5) wertilSi gavlebuli mxebis gantoleba.

3.118. SeadgineT wrewiris gantoleba, Tu is gadis M1(10; 9), M2(4;-5) da M3(0;5) wertilebze.

3.119. SeadgineT wrewiris gantoleba, Tu misi erT-erTi diametris boloebia A(4;5) da B(-2;1).

3.120. dawereT im wrewiris gantoleba, romelic exeba 5x-12y-24=0 wrfes da centri C(6;7)

wertilSia.

3.123. SeadgineT im wrewiris gantoleba, romelic exeba sakoordinato RerZebs da gadis A(1;2) wertilze.

10

3.125. daamtkiceT, rom y=2x+5 wrfe da x2+y

2=1 wrewiri ar ikveTebian.

3.135. SeadgineT elifsis gantoleba, Tu: 1. fokusebs Soris manZili 6-is tolia, xolo didi naxevarRerZi - 5-is, 2. didi naxevarRerZia 10, xolo eqscentrisiteti -0,8.

3.136. SeadgineT elifsis gantoleba, Tu: 1. mcire naxevarRerZia 3, xolo eqscentrisiteti

-2

2; 2. naxevarRerZebis jamia 10, xolo fokusebs Soris manZili - 4 5 .

3.137. ipoveT RerZTa sigrZeebi, fokusebi da eqscentrisiteti Semdegi elifsebisa: 1. 9x

2+25y

2=225; 2. 16x

2+y

2=16;

3. 25x2+169y

2=4225; 4. 9x

2+y

2=36.

3.138. SeadgineT elifsis gantoleba, Tu misi erT-erTi fokusidan didi RerZis boloebamde manZilia 7 da 1.

3.143. SeadgineT elifsis kanonikuri gantoleba, Tu is gadis Semdfeg or wertilze: M( 3 ;-2) da

N(-2 3 ;1).

RerZis marTebulad.

3.145. 124

y

30

x 22

elifsze ipoveT iseTi wertili, romelic 5 erTeuliTaa daSorebuli mcire

RerZidan.

3.146. x=-5 wrfeze ipoveT iseTi wertili, romelic Tanabradaa daSorebuli x2+5y

2=20 elifsis

marcxena fokusidan da zeda wverodan.

3.147. 14

y

16

x 22

elifsze ipoveT iseTi wertili, romelic 3-jer meti manZiliTaa daSorebuli

marjvena fokusidan, vidre marcxenadan. 3.148. SeadgineT elifsis gantoleba, Tu igi simetriulia koordinatTa saTavis mimarT,

fokusebi mdebareobs abscisTa RerZze, direqtrisebs Soris manZilia 5, xolo fokusebs Soris - 4.

3.158. SeadgineT hiperbolis kanonikuri gantoleba, Tu: 1. fokusebs Soris manZilia 10 da

warmosaxviTi naxevarRerZi - 4; 2. namdvili naxevarRerZia 2 5 da eqscentrisiteti - 2,1 .

3.159. SeadgineT hiperbolis gantoleba, Tu: 1. fokusebs Soris manZilia 16 da eqscentrisiteti - 3

4; 2.

warmosaxviTi naxevarRerZia 5 da eqscentrisiteti - 2

3.

3.160. ipoveT RerZebi, fokusebi da eqscentrisiteti Semdegi hiperbolebis 1. 4x2-9y

2=25; 2. 9y

2-

16x2=114.

3.161. ipoveT 116

y

9

x 22

hiperbolis naxevarRerZebi, fokusebi da eqscentrisiteti; 2. SeadgineT im

hiperbolis gantoleba, romlis fokusi moTavsebulia oy RerZze saTavis simetriulad, xolo naxevarRerZebia 6 da 12.

3.162. SeadgineT hiperbolis kanonikuri gantoleba, Tu is gadis M((-5;3) wertilze da misi

eqscentrisiteti tolia 2 -is.

3.171. ipoveT y2=24x parabolis fokusi da direqtrisis gantoleba.

3.172. dawereT im parabolis gantoleba, romlis fokusia F(-7;0) wertili, xolo direqtrisis

gantolebaa x-7=0.

3.173. SedgineT im parabolis gantoleba, romelic 1. simetriulia ox RerZis mimarT, gadis

koordinatTa saTaveze da M(1;-4) wertilze; 2. simetriulia oy RerZis mimarT, fokusi

moTavsebulia F(0;2) wertilSi, xolo wvero emTxveva koordinatTa saTaves.

3.174. SeadgineT im parabolis gantoleba, romelic 1. simetriulia oy RerZis mimarT, gadis

koordinatTa saTaveze da M(6;-2) wertilze; 2. simetriulia ox RerZis mimarT, fokusi F(3;0)

wertilSia, xolo wvero emTxveva koordinatTa saTaves.

3.177. y2=4,5x parabolis M(x;y) wertilidan direqtrisamde manZilia 9,125. ipoveT manZili am

wertilidan koordinatTa saTavemde.

3.179. ipoveT Semdegi wrfeebis da parabolebis gadakveTis wertilebi:

1. x+y-3=0, y2=4x. 2. 3x+4y-12=0, y

2=-9x.

3. 3x-2y+6=0, y2= -6x.

3.183. dawereT parabolis gantoleba, Tu misi fokusia F(-3; 1), xolo direqtrisi y+7=0.

3.184. dawereT y2=6x parabolisadmi M(4;5) wertilidan gavlebuli mxebebis gantolebebi.

3.185. gamoiyvaneT imis piroba, rom y=kx+b wrfe exebodes y2=2px parabols.

11

veqtoruli algebra ($3.4)

3.188. mocemuli a

da b

veqtorebiT aageT Semdegi veqtorebi: a

-2 b

; a3b2

1 ; b3a

3

1 .

3.189. OACB paralelogramSi aOA

da bOB

. vipovoT MB,MA,MO da MC veqtorebi, sadac M

paralelogramis diagonalebis gadakveTis wertilia.

3.192. a

veqtori mocemulia misi bolo wertilebis koordinatebiT A(2;4;-3) da B(-4;4;-5). vipovoT

AB veqtoris Sua wertilis koordinatebi.

3.193. vipovoT jami da sxvaoba veqtorebisa k4j3i2a

da k6j4i3b

.

3.196. mocemulia ori veqtori )2;1;2(a

da )2;4;1(b

. vipovoT am veqtorebis skalaruli

namravli da maT Soris kuTxe.

3.197. ganvsazRvroT ABC samkuTxedis kuTxeebi, Tu misi wveroebis koordinatebia A(2; -1; 3), B(1; 1; 1) da C(0; 0; 5).

3.199. mocemulia veqtorebi a

(4;-1;-2) da b

(2;1;2). ganvsazRvroT: 1) am veqtorebis skalaruli

namravli, 2) maT Soris kuTxe; 3) a

veqtoris gegmili b

-ze; 4) b

veqtoris gegmili a

veqtorze.

3.200. vipovoT kuTxe im paralelogramis diagonalebs Soris, romelic agebulia veqtorebze

ji2a

da k4i2b

.

3.201. vipovoT im paralelogramis diagonalebis sigrZe, romelic agebulia veqtorebze nm2a

da n2mb

, sadac m

da n

erTeulovani veqtorebia, romelTa Soris kuTxe 600-is tolia.

(miTiTeba. 2

baba

, 2

baba

).

3.204. vipovoT im paralelogramis farTobi, romelic agebulia veqtorebze n2ma

da

nm2b

, sadac m

da n

erTeulovani veqtorebia, romelTa Soris kuTxe 3

.

3.205. davadginoT komplanarulia Tu ara veqtorebi:

1) kjia

; kjib

; kjic

.

2) j5i3a

; k2jib

; k4j3i5c

.

3.206. vaCvenoT, rom A(1;0;7), B(-1;-1;2), C(2;-2;2) da D(0;1;9) wertilebi erT sibrtyeze mdebareoben

(miTiTeba. ganvixiloT AB , AC da AD veqtorebi).

3.207. vipovoT im paralelepipedis moculoba, romelic agebulia veqtorebze: kj2ia

,

kj2i3b

da kic

.

3.208. mocemulia piramidis wveroebi O(-5;-4;8); A(2;3;1); B(4;1;-2); C(6;3;7). vipovoT h simaRlis

sigrZe, romelic O wverodan daSvebulia ABC waxnagze.

(miTiTeba. vipovoT piramidis moculoba da ABC waxnagis S farTobi, h gamovTvaloT

formulidan Sh3

1V ).

zRvari

funqcia ($ 4.1)

daadgineT funqciis gansazRvris are da gamoTvaleT misi mniSvneloba miTiTebul wertilSi.

4.1. 1x)x(f ,

2sinf =? 4.2

2lgx

x)x(F

, 4lgF =? 4.3.

1x

1x)x(R

2

2

,

12tgR =? 4.4. 2x)x(Q ,

25,0Q =? 4.5. 2x9)x(S ,

35

2

1S =? 4.6. 1x)x( 2 , 22 =? 4.7.

x10)x( , )3(lg =?

4.8. g(x)=lg(x-10), g(0,1) =?

daadgineT mocemuli funqciis gansazRvris are.

12

4.9. x

xy ; 4. 10.

x36

1y

; 4.11. y=lglgx; 4.12 x33y ; 4.13. y=lg(x

2-1); 4.14. 6xxy 2 ;

4.15. 6xx

1x2y

2

; 4.16. y= 2)-1)(x-x(xlg . 4.17.

1x

4xy,

1x

4xy

2

2

2

1

; 4.18.

)1xlg()x25(lgy),1xlg()x25lg(y 22

2

1 ; 4.19. y1=lg(x-3)2, y2=2lg(x-3); 4.20.

2x

2y,10y 2

2x

2lg

1

.

aCveneT, rom Semdegi funqciebi luwia

4.21. y=2-|x|; 4.22.x

xsiny ; 4.23. y=sinxtgx .

aCveneT, rom Semdegi funqciebi kentia

4.24. x

xy ; 4.25. xsinxy ; 4.26.

xx 22y .

daadgineT funqciis luw-kentovnebis sakiTxi

4.27. y=2+sinx; 4.28. y=x2-cosx; 4.29.

xsin

xy ; 4.30.

xsin

xsiny ;

4.31. xx 1010y ; 4.32.

xx 1010y ; 4.33. xcos1

xcos1y

4.34. y=lg(cosx); 4.35. xsin2y ; 4.36. 33 x1x1y .

mimdevroba. mimdevrobis zRvari ($4.2)

4.37. ipoveT an mimdevrobis pirveli xuTi wevri, Tu

a) nn )1(1a ; b) n)1(a n

n ; g) 1n

na n

;

d) )1n2)(1n2(

1a n

e) n

2sin

n

1a n

.

4.38. SeadgineT zogadi wevris formula, Tu cnobilia misi ramdenime pirveli wevri a)

,...20

1,

15

1,

10

1,

5

1 b) ,...23

17,

17

5,

1

3,

5

1 g) 0, 1, 0, 1, … d) –1, 1, -1, 1, … e) 0, 2, 0, 4, 06, ..

4.38 davamtkiceT, rom

1) 2n

1n2limn

, 2) 0

n

1lim

2n

, 3)

3

1

n3

5nlimn

, 4) 1

1n2

3n2limn

, 5)

0, | | 1lim

, | | 1

n

n

aa

a

.

gamoTvaleT

4.39. 4n2

7n3nlim

3

3

n

4.40. 2

2

n nn11

5n3n4lim

4.41.

1n4

1n3lim

5

4

n

. 4.42.

8n7

n7n16lim

2

23

n

4.43. 2

33

n n2n31

)1n2()1n2(lim

. 4.45. a) 1n3

1n

n2lim

; b) 1n3

1n

n

2

2lim

; g) 1n3

1n

n

2

2lim

; d) 1n3

n1

n

2

2lim

;e)

1n3

1n

n 2

1lim

;

v) 1n3

1n

n

2

2

1lim

; z)

1n3

1n

n

2

2

1lim

; T)

1n3

n1

n

2

2

1lim

. 4.46.

2n1n

nn

n 25

25lim

. 447.

53

103lim

n

n2n

n

. 4.49.

2n5n2

n3...63lim

24n

. 4.50. n

n

n 2...42

3...93lim

. 4.51. 7n7nlim

n

. 4.52. 1n1n5lim

n

.

13

4.53. 1n1n2limn

. 4.54.

3n5n5n2nlim 22

n. 4.55.

3 23 2

n)1n()1n(lim .

4.56. )!2n(

)!1n()!1n(limn

. 4.57. )!2n()!3n(

)!2n()!3n(limn

. 4.58. a)

1n2

n 2n

1nlim

, b)

1n2

n 5n3

3n2lim

. 4.59.

a) 3

1n

n 2n3

4n3lim

, b)

5n

n 1n3

2n5lim

. 4.60. a) ,

1n

1nlim

2n

2

2

n

b) ,

1n2

1n3lim

2n

n

4.61. ,1n2

3n2lim

2n1

2

2

n

4.62. .

1n5

2n5lim

n3

n

funqciis zRvari. uwyvetoba ($ 4.3)

gamoTvaleT (0

0tipis ganuzRvreloba)

4. 63. a) 1xx2

1xlim

2

2

0x

; b)

1xx2

1xlim

2

2

1x

; g)

1xx2

1xlim

2

2

1x

. 4. 64.

32

3

0x x3x

)x31()x1(lim

. 4. 65.

12x8x

8x6xlim

2

2

2x

. 4. 66. 9x

6x5xlim

2

2

3x

. 4. 67. 12x8x

10x7xlim

2

2

2x

. 4. 68. 32

32

3x )12xx(

)6x5x(lim

.

4.69. 1

14lim

2

1

x

xx

x

. 4.70. x2x

xx27xx1lim

2

22

2x

. 4.71. 1x1

xlim

30x .

4.72. 9x

1x213xlim

23x

. 4.73. )4x2x)(2x(

26xlim

2

3

2x

. 4.74. 2

321lim

4

x

x

x. 4.75.

5x25

9x81lim

2

2

0x

.

4.76. 2

3 2

0x x

11xlim

. 4.77.

330x x1x1

x1x1lim

. 4.78.

x8

x4lim

3

64x

.

gamoTvaleT (

tipis ganuzRvreloba)

1x3x2

7x5xlim.79.4

45

34

x

1x7x3x

6x5x3x2lim.80.4

23

24

x

. xx3x2

1x4x3lim.81.4

23

23

x

.

3 2

4 3 2

(2 4 5)( 1)4.82. lim

( 2)( 2 7 1)x

x x x x

x x x x x

.

7x3x

2x3lim.83.4

8

4

x

. 5 44 4

3 22

x 1x1x

1x1x2lim.84.4

.

4 4 25 14.85. lim

4 2x

x x

x

.

21x

xx

x 32

32lim.86.4

52

2lim.87.4

x

x

x

2x5

1x2x3x2xlim.88.4

4 233 2

x

.

13

152lim.89.4

2

x

xx

x

.

gamoTvaleT ( ganuzRvreloba)

9x

6

3x

1lim.90.4

23x

.

dcxxbaxxlim.91.4 22

x

.

1x2

x

1x2

xlim.92.4

2

2

3

x

.

1x4xlim.93.4 22

x.

1xx2x3xlim.94.4 22

x

. 33

xx1xlim.95.4

.

xbxlim.96.4x

. x2xxlim.97.4 32

3

x

. xln)1xln(lim.98.4x

. 4. 99. a) xx

x32lim

b) xx

x32lim

14

tolfasi usasrulod mcire funqciebi da maTTan

dakavSirebuli zRvrebi

x

xsinlim.100.4

3

0x

. . 20

5cos1lim.102.4

x

x

x

. 20x x

xcosxcoslim.103.4

. 2

2

0x x

xcos1lim.104.4

.

x5

x4arcsinlim.105.4

0x

. x

xsinlim.106.4

0x

. nxsin

tgmxlim.107.4

0x

.. x2

xcoslim.109.4

2x

. .xx

tgxtgxlim.110.4

0

0

xx 0

.x

1xsinx1lim.111.4

20x

30

sinlim.112.4

x

xtgx

x

. .

3

2lim.113.4

22

22

0 xarctgx

xarctgx

x

2

4

4

2sin1lim.114.4

x

x

x

.

xtg

xtg

x 2

5lim.115.4

. .

cos2

2

4sin

lim.116.4

4 x

x

x

.

4)2(lim.117.4

2ztgz

z

.x3tgx

6lim.118.4

6x

.x

)x31ln(lim.119.4

0x

.xx

45lim.120.4

2

xx

0x

x

2ln)2xln(lim.121.4

0x

. .)x41ln(

1elim.122.4

x2

0x

.arctgx

)xsinx31ln(lim.123.4

20x

.bx

aalim.124.4

bx

bx

.

2x

eelim.125.4

2x

2x

x

eelim.126.4

tgxx2tg

0x

. .13xlim.127.4 x

1

x

4.128. a) .4xx2

1x3xlim

x

2

2

x

b) .

1x3x

4xx2lim

x

2

2

x

.x

31lim.129.4

x

x

.x

1xlim.130.4

1x

2

2

x

2

.2x

8xlim.131.4

x

x

.xcos2lim.132.4x2sin

1

0x

.bx

axlim.133.4

cx

x

.2

xlim.134.4

2x

1

2x

.4x2

3x2lim.135.4

2x5

2

2

x

.1lim.136.4 2sin

12

0

x

xxtg

.sin1lim.137.41

tgxx

x

.xcoslim.138.4 tgx

1

0x

diferencialuri aRricxvis elementebi

funqciis warmoebuli da diferenciali ($4.4)

ipoveT Semdegi funqciebis warmoebulebi:

4.167. x53y 35 . 4.168. nxxy n . 4.169. mn nxmxy . 4.170. 9x1,0

x

1xy .

4.171. 1x

5

x

x3

x

2y

4 3

3

4 . 4.172.

x2

x3x4x5y

432 . 4.173. )1(y,

u

5u2uy

3

32

.

4.174. 3

1xx2y

63 . 4.175.

1xx

1xx2y

2

2

. 4.176. y=2e

x+lnx . 4.177. y=2

xe

x. 4.178. y=xtgx+ctgx .

4.180. y=3x3lnx-x

3. 4.181.

x2

x3

3

2y . 4.182. y=4sinx+xcosx+cos2,

2y =? 4.183. y=x

2arctgx-2arcctgx, 1y =?

4.184. xarcsin

3y

x

4.185. 10ln1010

xy 3

x

10

. 4.186. x

x

31

31y

. 4.187. xlogx2xy 2

35 . 4.188.

2x

xarccosy . 4.189.

2

2

xcos

2

xsiny

. 4.190.

x2

x2y

.

ipoveT rTuli funqciis warmoebulebi

15

4.191. )x3x2ln(y 23 . 4.192. 2x42

xarccosxy . 4.193.

3

xcosy 3

. 4.194.

92 )5x4x2(y . 4.195. 3 2 xsin1y . 4.196. xctg5y 3x2tg2

. 4.197. 2

2

x4

x4y

ipoveT )0(y .

4.198. 2)x(arccos1y . 4.199. x1xarcsinxy . 4.200.

4

1x2tglny

. 4.202.

2

xsinln2

2

xctgy 2 . 4.203. 1x4arctgy 2 . 4.204. xxx ecosesiney 4.205.

x3cose1y 2x3sin2

. 4.206.

ctgx

xsin

1lny . 4.207.

2x

xlny

5

5

. 4.208. 1xcoslogy 2

5

. 4.209. 3 3 3x3y .

gamoiyeneT logariTmuli warmoebuli da gaawarmoeT Semdegi funqciebi :

4.222. xxy . 4.223. x

1

xy . 4.224. )1xsin(xy . 4.225. x

xsiny . 4.226. xxlny . 4.228. x3 xx 3xy

. 4.229. xxarcsiny . 4.230.

x

x

11y

. 4.231. xcos2 4xy . 4.233.

3

33

)5x(

1x3xy

. 4.234.

3

7

2

x

)4x(2xy

.

ipoveT maRali rigis warmoebulebi

4.235. 5x

22y

, ?y ?y 4.237. x3cos

27

2x3sinx

9

1y , ?y 4.238.

)1x(6

xy

, ?y

4.239. xln2

1y 2 , ?y 4.240. 2

7

)3x2(y , ?y 4.241. ?y,ey )n(ax 4.242.

?y),bkxsin(y )n( 4.243. ?y),3x4cos(y )n( 4.244. ax

1y

,

ipoveT Semdegi funqciebis diferenciali

4.247. )1x(sin3

ey 32

x3

. 4.248. xlnxy 2n . 4.249. )1x2(tgy 4 . 4.250. 3

xsin

a

ey

2

.

4.251. )4x(coslny 23 . 4.252. )3(arctgy x . 4.253. 3 2 4x3x2y . 4.254. x4tg3

2y .

4.255. 2x2 xxy . 4.256. )25x(lgy 22 .

ganuzRvrelobis gaxsnis lopitalis wesi ($4.5)

gaxseniT 0

0 da

tipis ganuzRvrelobebi lopitalis wesiT

4.257. 8xx3x4

6x5x2x3lim

23

235

1x

. 4.258.

)x1ln(

balim

xx

0x

. 4.259.

22x xx42

)1xln(x2lim

. 4.260.

xsin

xcosxxsinlim

30x

.

16

4.261. 30x x

arctgxxsin2lim

. 4.262.

tgbxln

axsinlnlim

0x . 4.263.

2x

2xlim

33

2x

. 4.264. bxcosln

axcoslnlim

0x

.

4.265.

1e

arctgx32

3

lim

x

2x

. 4.266. )x1ln(

eelim

xx

0x

. 4.267. 0n,

x

)2xln(lim

nx

. 4.268.

2x5x2

)7x2ln(lim

2

2

2x

.

4.271. )eeln(

)axln(lim

axax

. 4.272.

xctg

)1xln(lim

1x

. 4.273.

)x1ln(

2

xtg

lim1x

. 4.274.

x

m

x 5

xlim

.

gaxseniT 0 da tipis ganuzRvrelobebi lopitalis wesiT

4.275. )xctgx(lim0x

. 4.276. )ctgxx(arcsinlim0x

. 4.277. ctgx)xcos1(lim0x

. 4.278. )13(xlim x

1

x

.

4.279.

ctgx

x

1lim

0x. 4.280.

xln

1

1x

1lim

1x.4.281.

x

atgxlim

x

. 4.282. )2x(ctg)2x(lim

2x

..

gaxseniT 00, 0, 1 tipis ganuzRvrelobebi

4.285. xcos

2x

)x2(lim

. 4.286. xsin2

0x)x(lim

. 4.287.

xsin

0x)arctgx(lim

. 4.288. x

1

x

x)3x(lim

.

4.289. 2x

4

0x)x(coslim

. 4.290. 2

x

2x

)tgx(lim

. 4.291. 1x

1

1xarctgx

4lim

. 4.292. 2x

12

0xxsin1lim

.

4.293. x

1

0x x

xsinlim

. 4. 294.

x2sin

1

xcoslim0x

.

funqciis monotonuroba da eqstremumi ($4.6)

daadgineT funqciis monotonurobis Sualedebi da ipoveT eqstremumi.

4.295. y=x4-4x

3+6x

2-4x. 4.296. 2

x2

exy

. 4.297. 2xx2y . 4.298.

42 xx2y . 4.299. 2x4

1y

.

4.300. 2x41

xy

. 4.301. 4x9x6xy 23 . 4.302.

1x

2x2xy

2

. 4.303.

2x8xy .

4.304. 6x5xy 2 . 4.305. xln

xy . 4.306. 10x6xy 2 . 4.307.

4x6x 23

3y . 4.308.

5x4x2

2

1y

. 4.309. arctgxxlny 4.310. 25xlny 2 . 4.311.

xxy . 4.312.1x4x2

ey .

ipoveT funqciis udidesi da umciresi mniSvnelobebi miTiTebul Sualedze

4.313. 3x1,xy 3 . 4.314. 1x0,x1

x1arctgy

. 4.315. ex1,xlnxy 3 .

4.316. 3x2,xx3y 3 .

17

funqciis grafikis amozneqiloba da Cazneqiloba. gaRunvis wertilebi

($ 4.7)

daadgineT funqciis amozneqilobisa da Cazneqilobis Sualedebi. ipoveT gadaRunvis wertilebi.

4.317. x3x2

15x6xy 456 . 4.318. 1x3x

3

20x5xy 345 .

4.319. 1x7xy 7 . 4.320. 24 x6xy . 4.321.

3x

1y

.

4.322. 3)2x(y 3 5 . 4.323. 1xey x2 . 4.324. 33 1x1xy .

4.325. xxey . 4.326.

2xey . 4.327. 2x1

3xy

.

grafikis asimptotebi ($ 4.8)

ipoveT Semdegi gantolebebiT mocemula wirTa (grafikTa) asimptotebi

4.328. 4x

1xy

2

3

. 4.329

2

2

3x

xy

. 4.330.

1x

2x2y

. 4.331.

2x

3x2xy

2

. 4.332.

2x1

3xy

. 4.333.

x

1xy . 4.334.

4x

x2y

2 . 4.335.

xx2

2x2xy

2

23

. 4.336.

x1

ey

x

.

4.337. x

1

ey . 4.338. y=arctgx . 4.340. 1xey x

3

. 4.341. xexy .

funqciis gamokvleva da grafikis ageba ($ 4.9)

SeasruleT funqciis sruli gamokvleva da aageT misi grafiki.

4.342. 2

xx2y

42 . 4.343. y=x

4-2x

3+3. 4.344.

1x2

x2y

2

. 4.345.

2x1

2y

. 4.346.

4x

x2y

2

3

. 4.347.

1x

xy

2

. 4.348.

2

2

x1

xy

. 4.349.

2x

6xxy

2

. 4.350. 1

2x

3

2x

3y

. 4.351.

2

2x

2xy

.

4.352. x1

ey

x

.

mravali cvladis funqcia

ori cvladis funqcia ($5.1)

gamoTvaleT funqciis mniSvnelobebi miTiTebul wertilebSi

5.1. f(x,y)=x2-2xy+y

2, ipoveT f(3;-2) da f(-4;-4). 5.2. f(x,y)=

yx

)1y)(1x(

, ipoveT

f(1;-2) da f(3;-2). 5.3. 22 tx

1tx)t,x(f

, ipoveT f(2; 1) da f(3; 0).

5.4. f(u, v)=u+ln|v|, f(6; 1) da f(0; e3).

ipoveT Semdegi funqciebis gansazRvris are

5.5. f(x,y)=2x2-xy+y

3. 5.6.

1y

x)y,x(f

2

2

. 5.7. 9yx)y,x(f 22 . 5.8.

2)yx(1)y,x(f .

5.9. f(x, y)=ln(x-y). 5.10. f(x,y)=5y-x . 5.11.

1yx

1)y,x(f

22 . 5.12.

)xylg(

1)y,x(f

2 .

0 x

y

18

5.13. y

xarccos)y,x(f . 5.14.

2

2

y9

x4ln)y,x(f

. 5.15. )y9ln()x4ln()y,x(f 22 .

5.16. 25

y

16

x1)y,x(f

22

.

ori cvladis funqciis zRvari ($ 5.2)

gamoTvaleT:

5.17. xy

1sin)yx(lim 22

0y0x

. 5.18. 22

yx yx

yxlim

. 5.19. x

xysinlim

2y0x

. 5.20.

x

kyx x

y1lim

. 5.21. yx

xlim

0y0x

.

5.22. 22

22

0y0x yx

yxlim

. 5.23. xy

4xy2lim

0y0x

. 5.24. 22

33

0y0x yx

)yx(tglim

.

ori cvladis funqciis diferencirebadoba. ori cvladis funqciis kerZo

warmoebulebi. sruli diferenciali. rTuli funqciis gawarmoeba ($.5.3)

maRali rigis kerZo warmoebulebi da diferencialebi ($ 5.4.)

gamoTvaleT

5.32. )1,0(f x , Tu f(x,y)=x

2-3xy+y4. 5.33.

1;

2

1f y , Tu f(x,y)=

22 yx3 . 5.34.

4,

2f x , Tu f(x,y)=sin

2(x-y).

5.35. )1;2(zx , Tu z=xy+ln(x+y). 5.36.

4,

3

2z y , Tu z=cos(3x+2y).

ipoveT Semdeg funqciaTa kerZo warmoebulebi:

5.37. z=x3+y

3-3axy. 5.38. yx

yxz

. 5.39.

x

yz . 5.40.

22 yxz .5.41. 22 yx

xz

. 5.42.

x

yarctgz . 5.43. z=x

y . 5.44. x

ysin

ez . 5.45. 22

22

yx

yxarcsinz

. 5.46.

y

axcoslnz

. 5.47.

ysinxz 22 . 5.48. xtg2

yz . 5.49. 2y22 )yx(z . 5.50.

ycos2

xz .

ipoveT Semdeg funqciaTa sruli diferenciali

5.51. z=x2+3y

3-4xy . 5.52. z=x2+y

2+3x+ln2. 5.53. z=3xy-2x

2-3y2+1. 5.54.

2

2

y

xarcsinz . 5.55.

2x1

xyz

.

5.56. y

x2

ez . 5.57. z=ctg(x2-y2

) . 5.58. y

xcos

ez 5.59. z=arctg(x2+y

2) . 5.60.

22 yxz . 5.61. y

xlnz .

5.62. z=xy-lny+a3 .

diferencialis gamoyenebiT gamoTvaleT miaxloebiT Semdegi sidideebi

5.63. (0,95)2,01 . 5.64. e0,7sin0,8 . 5.65. cos0,1ln1,0,3. 5.66. sin29

0tg46

0.

ipoveT Semdeg funqciaTa meore rigis kerZo warmoebulebi

5.67. z=2y5+3x

2y

3+y. 5.68. z=e

x(cosy+xsiny). 5.69.

yx

yxz

. 5.70.

22 yx

1lnz

. 5.71.

xy1

yxarctgz

. 5.72. z=sin

2(3x+5y). 5.73. z=tg(xy). 5.74. z=(cosx)

y.

ipoveT Semdeg rTul funqciaTa warmoebulebi

19

5.83. z=3ln(xy), y=x3+2x. ?

dx

dz 5.84.

y

xarctgz , y=x

3 ?dx

dz 5.85.

y

xarcsinz , y= 4x 2 . ?

dx

dz

5.86. z=(arctg(xy+1), y=lnx. ?dx

dz 5.87.

y

xsinxz , x=t

4, y=t

3+1. ?

dt

dz 5.88. z=arccos(x

2-y

2), x=5t

2, y=4t

3.

?dt

dz 5.89. z=

22 yx3 , x=cost, y=sin2t. ?

dt

dz 5.90. z=tg(xy

2), x=2u

2-v, y=3u-v2. ?

u

z

5.91.

33 yxez ,

x=ucosv, y=usinv. ?v

z

ori cvladis funqciis eqstremumi ($ 5.5)

gamoikvlieT eqstremumze Semdegi funqciebi

5.92. z=(x-1)2+2y

2 .

5.93. z=(x-1)

2-2y2. 5.94. z=x

2+xy+y

2-2x-y. 5.95. z=x3y

2(6-x-y) x>0, y>0. 5.96. z=x

4+y

4-2x2+4xy-

2y2. 5.97. z=3x

2-x

3+3y

2+6y+9x+1. 5.98. 15y12yxy3x

2

1z 32 . 5.99. 1x15xy2xyx

3

1z 23 . 5.100.

z=x3-yx+x

2+y

2-3y+1. 5.101. y

20

x

50xyz .

integraluri aRricxvis elementebi

pirveladi funqcia da ganusazRvreli integrali ($6.1)

cxrilis integralebis gamoyenebiT ipoveT:

6.1 .dx)1x3( 2 6.2 .dx

x

1x4x6 2

6.3 .dx)x31)(x21)(x1( 6.4 .dx

x

)1x(3

22

6.5

.dx

xx

x12

6.6 .dx432 xxx 6.7 .dxex

11e

x3

x

6.8 .dx1xx1x 33 23

6.9 .dxx

6

x

4

x

3

x

2432

6.10

.dx12

43x

2xx

6.11 .xdxtg2

6.12 .xdxctg2

6.13 .dx2

xcos2

6.14 .dxx1

3

x1

2

22

6.15 .dxxsinxcos

x2cos

6.16 .x2cos1

dx

6.17 .

x2cos1

dx 6.18 .dx

2

xcos

2

xsin1

2

$6.2 cvladis gardaqmna ganusazRvrel integralSi (Casmis xerxi)

Casmis xerxis gamoyenebiT gamoTvaleT Semdegi integralebi:

6.19

.2x3

dx4

6.21 .dx4x3 6.22 .3x2

dx5

6.23 .4x

xdx

2

6.24 .7x4

dxx23

2

6.25 .dx4xx

3 2

6.26

.x7

dxx

4

3

6.27

.

9x4x

dx2x2

6.28 .dxe x5 6.29 .

x

xdxln2

6.30 .

3e

dxex

x

6.31 .dxx1

22

arctgx

6.32

.

2x3cos

dx2

6.33 .dxex3x2

6.34 .xdxcosa xsin

6.35 .1tgxxcos

dx2

6.36 .2x2x

dx2

6.38

.dxx1sinx 32 6.39 .xcos

dxtgx2 6.41 .

41

dx2

x

x

6.42 .5x2x

dx

2

6.43 .xx1

dx

2

6.44 .6x5x

dx2

6.46 .x53

dx

2

6.47 .

x23

xdx 6.48

.x53

dx2

6.49 .x53

dx

2

6.56 .x

xdxln3

6.57 .

xln1x

xdxln4

6.58

20

.xsin

xdxcos

5 2 6.59 .tgxdx 6.60 .ctgxdx 6.61

.xsinxcos

xdx2cos2

6.62

.9x

dx3x

2

6.63

.

x9

dx3x

2

6.64

.dx

9x

1x2

6.75

.dxx1

xarcsin

2

3

6.89 .dx

1x2

x6.90 .

5x4x4

xdx2

6.92 .10x2x

xdx2

6.95

.dxx

1x 6.96

.105

dx5x2

x

nawilobiTi integreba ($6.3)

Semdeg integralTa gamosaTvlelad isargebleT nawilobiTi integrebis xerxiT:

6.97 .xdxsinx 6.98 .xdxcosx 6.99 .dx3x x

6.100 .dx7x x 6.101

.dxxe x 6.102 .xdx2sinx 6.103

.xdx5cosx 6.104 .xdxln 6.106 .arctgxdxx 6.107 .xdxarcsin 6.110 .e

xdxx2

6.112 .dxx

xln3 6.113

.xdxlnsin 6.114 .xdxlncos 6.115

.dxx1

xarcsinx

2 6.118 cos .xe xdx 6.119 .bxdxsineax

6.121 .dxex x2

6.123 .xdxarcsinx 6.124 .dx1xln 2

kvadratuli samwevris Semcveli umartivesi integralebi ($6.4)

gamoTvaleT kvadratuli samwevris Semcveli Semdegi integralebi:

6.127 .3x2x

dx2

6.128

.5x2x

dx1x2

6.129 .

2x2x3

dx2

6.130

.5x4x

dx2x32

6.132

.3x2x2

dx5x2

6.133 .

1xx

dx

2

6.134 .xx45

dx

2

6.135 .x2x32

dx

2

6.136 .1x2x3

dx

2

6.137

.5x4x

dx6x3

2

6.138

.

xx23

dx7x4

2

6.139

.

5x3x

dx1x

2

trigonometriuli funqciebis integreba ($6.5)

gamoTvaleT trigonometriuli funqciebidan Semdegi integralebi:

6.161 .xdxcos3 6.162 .xdxsin3

64 .xdxcosxsin 32 6.165 .xdx3cos2

6.171 .xsin

xdxcos2 6.172 .

xcos

xdxsin2 6.173 .

xsin

xdxcos4

3

6.175 .xcos

xdxsin2

4

6.179 .xsin

dx4 6.180 .

xcos

dx4 6.184 .xdxtg5

6.185 .xdxctg4

6.186 .dxxcos

xtg4

4

6.188 .dx3

xcos

2

xcos

21

gansazRvruli integralis cneba; niuton-laibnicis formula ($7.1)

niuton-laibnicis formulis gamoyenebiT gamoTvaleT Semdegi integralebi:

7.7 ;dxx

1

0

4

7.8 ;dxx

2x

4

1

3

7.9 ;

xcos

dx4

6

2

7.10 ;x1

dx1

1

2

7.11 ;dx)3x2x(

2

1

2

7.12 ;xdx5sin2

2

0

7.13 ;x91

dx6

1

02

7.14 ;

1x

dx3

9

2

7.15 ;xsin

dx4

6

2

7.16 ;1x

dx1

02

7.17 ;dxx

x14

1

2

7.18 ;dxx

)x1(2

1

3

2

7.19 ;dxx2cos1

xcos14

0

2

7.20 ;xdxtg

4

0

2

7.21 ;dx2

xsin2

2

0

2

7.22 ;dx)x1(x

)x1(3

1

2

2

gansazRvruli integralis gamoTvla Casmis xerxiT ($7.2)

7.23 .xlnx

dx3e

e

7.24 .ee

dx1

0

xx 7.26

.

x511

dx1

2

3

7.27 .ctgxdx

2

4

7.28 .dxxe

1

0

x 2

7.30 .

5x4x

dx1

0

2 7.31

.xcos

xdxsin4

0

2

7.32 .xcos

dxtgx3

0

2

7.34 .4x

dxx2

1

4

3

7.35 .

xx28

dx1

2

12

7.38 .2x6x9

dx2

12

7.40 .

xx25

dx3

12

7.41 .2x3x2

dx3

2

2 7.42 .

5x2x

xdx1

1

2

7.44 .dx)3x2sin(

2

1

7.46 .x94

dx3

1

02

7.47 .xdxcosxsin

2

0

2

7.48 .dxex

1

0

x2 3

7.49 .1x

xdx0

1

4

7.51 .

)xln1(x

dxe

1

2 7.52 .xdx2cosxsin

2

0

3

gansazRvruli integralis gamoTvla nawilobiTi integrebis

xerxiT ($7.3)

nawilobiTi integrebis xerxiT gamoTvaleT Semdegi integralebi:

7.53 .xdxsinx

2

0

7.54 .xdx2cosx

4

0

7.55 .dxxe

1

0

x

7.56 .dxe)x2x(

1

0

x2

7.57 .x

xdxlne

1

3 7.58 .xdxlogx

2

1

2 7.59

.dx7x

1

0

x2

7.60 .xcos

xdxsinx4

0

2

7.61 .arctgxdx

1

0

7.62 .xdxln

e

1

2

7.63 .xdxsinx

0

3

7.64 .dx)1xln(

1e

0

7.65

.xdxsine

0

x

7.66 .xdxcose

0

x

7.67 .xsin

xdx3

4

2

7.68 .x

xdxlne

1

2

22

brtyeli figuris farTobis gamoTvla ($8.1)

gamoTvaleT im figuraTa farTobebi, romlebic SemosazRvrulni arian Semdegi wirebiT:

8.1y3 = x

2, y = 1. 8.2 y = x

2 + 1, y 2x

2

1 ,y = 5. 8.3 y

2 = 9x, x

2= 9y. 8.4 y

2x1

1

, y

20

x2

. 8.5 y

= 4 – x2, y = 0. 8.6 xy = 4, x = 1, x = 4, y = 0. 8.7 y

2 = x

3, y = 8, x = 0. 8.8 y

2 = 2x + 4, x =

0. 8.9 y = ln x, x = e, y = 0. 8.10 y = 6x – x2, y = 0. 8.11 y = x y= 8, x= 0. 8.12 y

2

= 1 – x, x = – 3. 8.13 y = x2 + 4x + 5, x = 0, y=1. 8.14 4y = x

2, y

2 = 4x. y = 0, y = 1. 8.15 y

x

6 , y = – x + 7.

brunviTi sxeulis moculobis gamoTvla ($8.3)

8.26 gamoTvaleT y = 4x – x2 paraboliTa da y = x wrfiT SemosazRvruli figuris abscisTa RerZis

garSemo brunviT miRebuli sxeulis moculoba.

8.27 gamoTvaleT y2 = 4x paraboliTa da y = 0, x = 4 wrfeebiT SemosazRvruli figuris abscisTa

RerZis garSemo brunviT miRebuli sxeulis moculoba.

8.28 gamoTvaleT y2 = 4x da x

2 = 4y parabolebiT SemosazRvruli figuris abscisTa RerZis garSemo

brunviT miRebuli sxeulis moculoba.

8.29 gamoTvaleT abscisTa RerZis garSemo y = sinx sinusoidis naxevartalRis brunviT miRebuli sxeulis moculoba.

8.30 gamoTvaleT y = e2x

wiriTa da y = 0, x = 0 da x = y wrfeebiT SemosazRvruli figuris abscisTa RerZis garSemo brunviT miRebuli sxeulis moculoba.

diferencialuri gantolebebi

gancalebadcvladebiani diferencialuri gantoleba ($ 10.2)

amoxseniT Semdegi gantolebebi da sadac miTiTebulia, ipoveT is kerZo integralebi, romlebic mocemul sawyis pirobebs akmayofileben:

10.25. y=3x2-2x+1; x=1, y=2. 10.26. xy-y=0; x= ,

2

1 y=2. 10.27. y=y; x=-2, y=2. 10.28.

(1+y)dx-(1-x)dy=0; 10.29. (1+y2)dx+xydy=0. 10.30. xyy=1-x

2. 10.31. ytgx=y.

10.32. 0dyx1ydxy1x 22 . 10.34. (1+y2)xdx+(1+x

2)dy=0. 10.35. (xy

2+x)dx+(y-x

2y)dy=0.

10.36. y

x21yy

. 10.37. xy+y=y

2. 10.38. y=10

x+y.

pirveli rigis wrfivi diferencialuri gantoleba ($10.4.)

amoxseniT Semdegi gantolebebi:

10.72. y+2y=4x. 10. 73. y+2xy=x2xe

. 10.75. (1+x2)y-2xy=(1+x

2)

2. 10.76. y+y=cosx. 10.77. 2ydx+(y

2-

6x)dy=0. 10.78. 2yx2

1y

. 10.79.

3xx

y2

dx

dy . 10.80. (1+y

2)dx=(arctgy-x)dy.

meore rigis mudmivkoeficientebiani wrfivi erTgvarovani

diferencialuri gantoleba ($10.10.)

amoxseniT gantolebebi

10.181. y+y-2y=0. 10.182. y-4y=0. 10.183. y+6y+13y=0.10.184. y-2y+y=0. 10.185. y-4y+3y=0.

10.186. y+4y+29y=0.

23

pasuxebi:

Tavi I 1.1. a) sworia {4,5}{4.5, {4.5.7}}, b) sworia orive.

1.2. (AB)C={_3,0}, (AB)C={_3,2,5,0,11}, (A\B)C={0}.

1.3. AB=(_2, +), AB=[1,3), A\B(_2,1), B\A=[3,+);

1.4. A={1,3} 1.5. A={1,2,3,4}; 1.6. A={_2, _1, 0, 1, 2};

1.7. B={1,2,3,4,5}; 1.8. B=; 1.9. AB=A=(_,_8][_1,+), AB=B=[3;4], A\B=(_,_8][_1,3)(4;+), B\A=.

1.10. AB=A=(_,_4][_2,+), AB=B=, A\B=A, B\A=;

1.11.

-4 0 2 x

y

A

0 x

y

B

y=2

y=-3

y

AB -4 0 2 x

y=2

y=3

1.12. B aris x=_2, x=2 da, y=_3 wrfeebis wertilTa erToblioba

-2 0 2 x

y

y=-3

1.13.

19

7;

19

18AB aris simravle, romelic erTi A wertilisagan Sedgeba.

0 x

y

A

1.14

0 x

y

A

1.15

-2 0 2 x

y

y=-4

y=-x2

A

1.18. {a,b}, {a,c}, {a,d}, {a, e{, {b.c}, {b.d}, {b,e}, {c,d}, {c,e}, {d,e}; 10C2

5 , 20A2

5 ; 1.19. 2252513C2

26 ; 1.20. a)

P5=120; b) 58A =6720; 1.21. 2520CCC 2

426

28 ; 1.22. a) P5_P4=96, b) P5_P3=114, g) P5_P2=118; 1.23. a) 126C5

9 ; b) 56C58 ; 1.24.

3640C245 214 ; 1.25. 27650CC 2

75

3

5 ; 1.26. 43; 1.27. n(n_1); 1.28. ;)1n)(2n(10

1

1.29.

1n

2n

1.30. 132; 1.31. x=14, x=3;

1.32. x=5, x=4; 1.33. x=4; 1.34. x=2; 1.35. x=27; 1.36. x=17; 1.37. x=3; 1.38. x=7; 1.39. (x+b)6=

=x6+6x

5b+15x

4b

2+20x

3b

3+15x

2b

4+6xb

5+b

6;

1.40. 54323255

nnm5mn10nm10nm5mnm ;

1.41. (z2+1)

7=z

14+7z

12+21z

10+35z

8+35z

6+21z

4+7z

2+1;

8

4 7 3 5 3 2 2 3 5 3 7 41.42. 9 28 56 70 56 28 8x y x x y x y x y x y x y xy xy y ;

1.43. 3

8

8129 xCT

; 1.44. 116137 aCT ; 454

95 yxCT ; 1.45. 3

25

5106 zCT ;

1.46. n=15; 1.47. 454

95 yxCT ; 1.48. n=8; 1.49. 64 .

24

Tavi II

2.2. 6_4i; 2.3. 0; 2.4. 8x; 2.5. 9_7i; 2.6. 8_i; 2.7. 11_4i; 2.8. 4; 2.9. 24i; 2.10. –26_18i; 2.11. –38; 2.12. 1255; 2.13. 0,5+1,5i; 2.14.

0,2+1,3i; 2.15. i; 2.16. 0; 2.17. 2_i; 2.18. 0; 2.19._16i; 2.20. x4_6x

2+25; 2.21.z1=cos0

0+isin0

0; z2=cos+isin;

2sini

2cosz3

;

2

3sini

2

3cosz4

;

4

5sini

4

5cos2z5

;

4

7sini

4

7cos2z6

;

3

2sini

3

2cos2z7

;

3

4sini

3

4cos2z8

;

3

5sini

3

5cos2z9

2sini

2cos2z10

;

6

11sini

6

11cos2z11

;

12sini

12cos62z12 2.22. ;i35,25,2z1 z2=_1+i; z3=8i; z4=2+2i; 2.23.

12

5sini

12

5cos10z1

; ;12

7sini

12

7cosz2

z3=5i; i123z4 ; 1z5 ; 2i63z6 ; 00

7 125sini125cos24z . 2.24.

;i15

2z1

002 105sini105cos2z ; 00

3 335sini335cos3

8z ; i12z4 ; z5=6i; i3

4

2z6

; i1

2

2z7

; i32

1z8 ;

i318z9 . 2.25. z1=212

(1+i); 3i12z 92 , z3=_64; z4=1; z5=2

24; z6=2

183

27; 2.26.

2

nxsini

2

nxcos

2

xcos2 nn ; 2.28.

8_i; 2.29. i5

14 ; 2.30. –32+16i; 2.31. i2

3

2

1 ; 2.32.–1; 2.33. x=2, y=3; 2.34. x=

3

1 , 4

1y . 2.35. x=

7

17, y=

7

13; 2.36.

,ba

bx

22

22 ba

ay

; 2.37. z1=1, z2=i; 2.38. z1=2+i, z2=2_i; 2.39. a) z1=3_2i, z2=_3+2i; b)

4

k360315sini

4

k360315cos2z 8

k, k=0, 1, 2, 3. g)

6

k2sini

6

k2cosz

, k=0, 1, 2, 3, 4,5, d)

2

ksini

2

kcos2z ,

k=0, 1, 2, 3. y

2 x o

2.44

(saTave ares ar ekuTvnis).

2.45.

o x

y

1 2 3

2.42 y

x o

2

y

2 3 x o

2.43

2.46 y

x o 1

(kompleqsuri sibrtye, amoWrili wriT, romlis radiusia 1)

2.47.

o x

y

2

2.48. ori koncentruli wrewiri centriT koordinatTa saTaveSi, radiusebiT 3 da 1. 2.49. rgoli centriT

koordinatTa saTaveSi da radiusebiT 1 da 9; 2.50. rgoli 13|z|17. 2.51. wrfe y=x. 2.52. y≤x naxevarsibrtye;

2.53. 6

sxivi;

y

x 0

6

4

2.54.

2.55. carieli simravle.

2.56. y

x 0 4

2 4

2.57

6

3

O 2 4 x

y

25

2.58. wrewiri centriT koordinatTa saTaveSi da radiusiT 9. 2.59. wrewiri centriT (2; 0) wertilSi da

radiusiT 2. 2.60. 3

k2sini

3

k2cosx 2

k

, k=0,1,2. 2.61.

3

k2sini

3

k2cosxk

, k=0, 1, 2.

2.62.

k

4sinik

4cosxk

, k=0, 1.

2.63.

k

3sinik

3cos2x 4

k, k=0, 1. 2.64.

3

k2sini

3

k2cos2xk

, k=0, 1, 2. 2.65.

3

k2sini

3

k2cos2xk

,

k=0,1,2. 2.66. ,3

k2sini

3

k2cosx k

k=0,1,2, 3. 2.67.

5

k2sini

5

k2cos2xk

, k=0, 1, 2, 3, 4.

2.68.

5

k2sini

5

k2cos2xk

, k=0, 1, 2, 3, 4. 2.69. 3

ksini

3

kcosx k

, k=0, 1, 2, 3, 4, 5. 2.70.

5

k2sini

5

k2cos3x k

, k=0, 1, 2,

3, 4. 2.71.

5

k2sini

5

k2cos3x k , k=0,1,2,3,4. 2.73. ;3i1

2

3 _3; ;3i1

2

3 2.74. 3; ;3i1

2

3 _3; ;3i1

2

3

2.75. ;3i14

1

2

1

;

;3i14

1

2.76. ;2

1 ;3i14

1 ;3i1

4

1 2.77. 2; 2i. 2.78. 3; 3i. 2.79.

3

5 ; i3

5 ; 2.80. 2; 1+i 3 ; _1+i 3 ; _2; _1_i 3 ; 2.81.

a) a=64

5, b) a=_9; 2.82. a) p=2, q=_8; b) p=_5, q=_2; 2.83. a) 51, b)_67. 2.84. z=_2i. 2.85.z=_42i. 2.86. z1=_2+i, z2=_3+i. 2.87.

z1=_33+2i, z2=_3. 2.88. z1=_1, z2,3=

2

3i

2

1 , z4=_ 3 3 , z5,6=

3 3 (2

3i

2

1 ) 2.89. z1,2=1, z3,4=i, z5,6= )i1(2 , z7,8= )i1(2 . 2.90.

P(z)=(z_1)(z+1)(z2+1); z1,2=1, z3,4=i. 2.91.P(z)=(z

2+1)(z_ )1z3z)(1z3 2 , z1,2=i,

2

1i

2

3z 4,3 ,

2

1i

2

3z 6,5 ; 2.92.

P(z)=(z2+z+1)(z

2_z+1), z1,2=

2

3i

2

1 ,

2

3i

2

1z 4,3 . 2.93. P(z)=(z

2_6z_13)(z

2+2z+2), z1,2=_32i, z3,4=_1i. 2.94.

P(z)=(z_1)(z2+z+1)

2, z1=1, 3i1z 3,2 ,

2

3i

2

1z 5,4 2.95. a)

160

202

114

, b)

100

010

001

2.96.a)

568

5117 , b)

395

4130,

g)

8124

4164. 2.97.

113

103

51

3

2

. 2.98

55

34 2.99.

dc

ba, 2.100.

1151

175

460

. 2.101.

132

384

1252

. 2.102.

51

72

61

.

2.103.

134

103

72

41

. 2.104. (3) 2.105.

1421

1481

2842

1421

. 2.106.

143

37. 2.107.

433

1487

484

. 2.108.

382

594

061

2.109.

1314

1413

. 2.110.

10

01. 2.111.

100

010

001

. 2.112. 11. 2.113. sin(_). 2.114. –cos2. 2.115. 2b2

2.116. b3

2.117.–48 2.118. 8

2.119. 0. 2.120. –2x 2.121. sin(+). 2.122. (a2_a1)(a3_a1)(a3_a2) 2.127. –5. 2.128. {2; 0,5} 2.129.0. 2.130. {_9; 2} 0 2.131. {2; 3} 2.132.

_4 22 2.133. x>3,5 2.134. –6<x<_4. 2.135. _241. 2.136. 160 2.137. 0. 2.138._21. 2.139. –33. 2.140.

13

25

11

1. 2.141.

12

43

11

1

2.142.

104`2

642

721

8

1 . 2.143.

017017

10529

41967

2181412

34

1 2.144. matrici gadagvarebulia. 2.145.

113

714

5

1 2.146.

31

12

01

2.147.

233

518

133 2.148.

1

0

1

2.149. (a –a 3_a) sadac a nebismieri namdvili ricxvia. 2.150. x matrici ar

arsebobs. marcxena mxaris pirveli Tanamamravli gadagvarebuli matricia. 2.151.

157

83447

31218

3

1 2.152.

44

20. 2.153. 3 . 2.154. 2. 2.155. 3. 2.156. 2. 2.157. 2. 2.158. 2. 2.159. 3. 2.160. 2. 2.161. 3. 2.162. 2. 2.163. 1. 2.164. 5. 2.165. x1=3,

x2=_1. 2.166. x1=3, x2=2. 2.167. x1=1, x2=_1. 2. 2.168. x1=x2=0. 2.169. (0;0;0) 2.170. (2;2;_1). 2.171. (1;3;2). 2.172. (5;_1;_4). 2.173.

(3;_1;4). 2.174. (1;1;-1;-1) 2.175. (_1;0; 2;3) 2.176. (2;1;0; _1) 2.177. (2;3;0;5) 2.178. sistema araTavsebadia. 2.179. x1=2t, x2=t,

sadac t nebismieria. 2.180. x1=t, x2=3_2t, sadac t nebismieria. 2.181. sistema araTavsebadia. 2.182. x1=_5t+2, x2=_2t+2

1,

26

x3=t, sadac t nebismieria. 2.183. x1=, x2=t, x3=_2t_1, da t nebismieria. 2.184. x1=_t_7

1, x2=t+

7

12, x3=t, sadac t nebismieria.

2.185. x1=_13t, x2=5t, x3=7t. sadac t nebismieria. 2.186. x1=_5t, x2=_14t, x3=_3t, sadac t nebismieria. 2.187. x1=k_4t, x2=k, x3=t,

sadac k da t nebismieria. 2.188. x1= t115

1 , x2= 1t

5

2 , x3=t, sadac t nebismieria 2.189. x1=

14

43 t, x2=

2

27 t, x3=

14

827 t,

x4=t, sadac t nebismieria. 2.190. x1=t, x2= _13+3t, x3=_7; x4=0. 2.191. ,3

4x;1x,

9

15x 321 2.192. sistema araTavsebadia.

2.193. a) Tu a2, maSin

a36

18a5x,

6a3

4x 21

. b) Tu a=2, maSin sistema araTavsebadia. 2.194. a) Tu a_2, maSin

a2

a510x1

, x2=0, b) Tu a=_2, x1=5_2t, x2=t, sadac t nebismieria. 2.195. a) Tu a4, maSin

4a

1x,

a4

a317x,

a4

5a2x 321

,

b) Tu a=4, sistema araTavsebadia. 2.196. a) Tu a2, maSin x1=x2=x3=0. b) a=2, maSin x1=4t, x2=_6t, x3=_t, sadac t nebismieria.

2.197. a) roca a5, maSin x1=x2=x3=0.b) roca a=5, sistemas aqvs amonaxsnTa usasrulo simravle: 2

1x1 t,

4

5x2 t, x3=t,

sadac t nebismieria. 2.198. Tu a_3, maSin 3a

1x1

(_8ab_9b+6a+13),

3a

1x2

(9a+18b_5ab+16),

3a

7b21x3

. b) roca

a=_3, 3

1b , maSin sistema araTavsebadia. g) roca a=_3 da

3

1b , maSin sistemas aqvs amonaxsnTa usasrulo

simravle: 7

1t5x1

,

7

1t11x2

, x3=t, sadac t nebismieria. 2.199. (3,2,1). 2.200. (_2,0,_1). 2.201. sistemis matrici

gadagvarebulia, Sebrunebuli ar gaaCnia. 2.202. (2,_2,0, 1) 2.203. (_1,3, 2). 2.204. (1,5,2). 2.205. (1,2,3,4) 2.206. sistema araTavsebadia. 2.207. (1,2, 3,4). 2.208. sistema araTavsebadia.

Tavi III

3.3. M1(1; )3 ,

2

3;

2

33M 2

,

0;

2

1M 3

. 3.4.

4

3;2M1

,

2;2M 2

, M3(5;0),

3

4arctg;5M 4

. 3.5. 2cosa 22 . 3.6.

x

yarctg

ayx 22 ; ;

x

yayx 22 x

2+y

2+=ay. 3.9. (0;-3) da (0;-9). 3.11.

4

3169. 3.12. (3;-1). 3.13. (-3;-3), 13R . 3.14.

292 ; 29 ; 65 . 3.15. (14;0) da

3

14;0 . 3.16.A(4;4) da B(-6;-2). 3.17. (7;-1); (6;0); (5,1); (4,2); (3;3). 3.18. (3,5; -1), (4;-2) da (4,5;-

3). 3.19. D(-3;5) an (15;3) an (-7;-5). 3.20. M(-2;1). 3.21. M(5,2;-1). 3.22. 3

55BM ;

3

510CM . 3.23. 5 da 481 . 3.24. a) 3,5; b) 10;

g) wertilebi erT wrfeze mdebareoben. 3.25. 3675 sm2. 3.27.10. 3.28. 12

11arctg . 3.29. (3;3) da R=3. 3.30. 26. 3.31. 5 ; 2;

5 . 3.32. C(-3;2). 3.33. C(32;0) an C(-8;0). 3.34. C1(7;6) da

5

2;7C2 . 3.35. 8 an

3

15 . 3.36. mdebareoben M1, M2, M6 wertilebi.

3.37. a=11; b=-3. 3.38. A(4;0); B(0;-6) 3.39. k=-3, b=13. 3.40. a)k=2, b=-3; b) 3

5b;

3

2k ; g) k=0,

2

5b

; d) wrfe

paraleluria oy RerZis. e) k=-1; b=0; 3.41. a) ;4

3 b) ;

4

g)

3

2arctg ; d)

5

1arctg . 3.42. y=x; b) y=-x; g) x

3

1y ; d) y=0. 3.44.

4x3y . 3.45. a) ;112

y

18

x

d) ;1

3

y

3

x

v) 1

511

y

11

x ; b) g) da e) wrfis arasruli gantolebebi RerZTa

monakveTebSi ar gadaiwereba. 3.46. 9x-14y+6=0. 3.47.x+y-12=0. 3.48. x+y+7=0; x-y+7=0; x+y-7=0; x-y-7=0. 3.49. x+2y-10=0.

3.50. a) 3

7x

3

4y ; b) ;1

37

y

47

x

g) 4x-3y+7=0, d) .05

7y

5

3x

5

4 3.51. a) ;02y

13

12x

13

5 b) –xcos70

0-ysin70

0-3=0; g)

;03

1x d) y- ;0

2

1 e) ;0

5

1y

5

1x

5

2 v) ;01y

5

4x

5

3 z) ;0

2

1y

5

4x

5

3 T) ;01y

2

1x

2

3 i) .0y

10

1x

10

3

3.52. radganac orive wrfe koordinatTa saTavidan 2 erTeuliT aris daSorebuli, amitom gamodgeba. 3.53. y=

2

ax 3.54. y=0; y=2 ;3 y= 3 x+5 3 ; y=- x+5 3 ; 3.55. 3x+4y-12=0; 3x-4y+12=0; 3x+4y+12=0; 3x-4y-12=0. 3.56. 3x-2y-

8=0. 3.57.7x-2y-20=0. 3.58.3x+y-5=0. 3.59. 3x-4y+6=0; 4x+3y-17=0; 4x+3y+8=0 da 3x-4y+31=0 an 3x-4y-19=0. 3.60. a) (-1;1) b)

4

3;

4

7 , g) wrfeebi erTmaneTis paraleluria, d)

3

2;

2

3, e) es wrfeebi erTmaneTs emTxveva. 3.61. A(2;-1); B(3;4);

C(-3;0). 3.62. 7 kv.erT. 3.63. 5x-4y--7=0. 3.64. .0233yx3 3.65.2x+y-7=0. da x-2y-6=0. 3.66.11x+33y-156=0. 3.67. a) 10

59,

3

27

b) 5, g) 0,5. 3.68. 6x-8y-11=0. 3.69.3x-4y=0. 3.70. 13

24h 3.71. y=9x+8 3.72. x+2y+1=0. 3.73.

65

10 3.74. y-2= ).1x(6

216

3.75. 4x-3y+5=0; 4x-3y-25=0. 3.76. x--4=0 da 7x+24y-100=0. 3.77. 3x-4y+3=0 an 4x-3y-4=0. 3.78. 3x+4y+3=0; x+1=0. 3.79.

9

37;

9

2 .

3.80. 4x-3y-10=0 da 4x-3y+15=0. 12x+5y-26=0 da 12x+5y+39=0. 3.81. 8x+15y+17=0; 8x+15y+119=0. 3.82.4,225 kv.erT. 3.83. y-4=0.

3.84. 3x+y+1=0. 3.85. M(3;-3). 3.86. 3x-y-10=0 (AB), x-2y=0 (BC), 2x+y-5=0 (AC). 3.87. 12x-10y-19=0. 3.88. (0; 1), (-3;2); (-4;5) da

(-1;4). 3.89. 3x-y-5=0; x-3y+17=0, 3x-y-21=0; x-3y+1=0. 3.90. x-y-7=0; x+y--9=0; x+y-1=0. 3.91. x+y-9=0, 3x-y-3=0. 3.92. (4;4). 3.93.

x+3y-10=0. 3.94. x+4y4=0. 3.95. 12x-5y-3126=0. 3.96. 3x-2y-27=0. 3.97.

a) b) g)

1. AB gverdis gantoleba 5x+12y-

86=0

5x+12y-55=0 5x-12y+54=0

BC gverdis gantoleba 4x-3y-31=0 4x-3y-44=0 3x+4y+10=0

AC gverdis gantoleba x-6y+50=0 x-6y+31=0 x-8y+50=0

2. AD medianis gantoleba x+15y-

118=0

x+15y-74=0 x-4y+22=0

3. medianebis gadakveTis wertili

3

22;8 3

13;9 3

14;3

10

4. BE biseqtrisis gantoleba

11x+3y-

10=0

11x+3y-121=0 8x-y+50=0

5. biseqtrisis gadakveTa

AC gverdTan

23

223;

23

188

23

154;

23

211

9

50;

9

50

6. hA simaRlis gantoleba 3x+4y-32=0 3x+4y-17=0 4x-3y-3=0

hA simaRlis sigrZe 12,6 12,6 11,2

7. samkuTxedis farTobi 63 63 63

8. C arctg22

21 arctg22

21 arctg29

28

9. B wveroze gamavali AC-s

paraleluri BM wrfis

gantoleba.

x-6y+8=0 x-6y-11=0 x-8y+22=0

10 manZili B wverodan AD

medianamde 226

63 226

63 17

8

d) e) v)

1. AB gverdis gantoleba 5x-12y-25=0 5x-12y-67=0 5x-12y+79=0

BC gverdis gantoleba 3x+4y+41=0 3x+4y+27=0 4x-3y-38=0

AC gverdis gantoleba x-8y-5=0 x-8y-19=0 x-6y+43=0

2. AD medianis gantoleba x-4y-5=0 x-4y-15=0 x+15y-104=0

3. medianebis gadakveTis wertili

3

7;3

13 3

10;3

5 3

19;9

4. BE biseqtrisis gantoleba

8x-y+51=0 8x-y+2=0 11x+3y-127=0

5. biseqtrisis gadakveTa

AC gverdTan

913;

959

922;

95

23200;

23211

6. hA simaRlis gantoleba 4x-3y-20=0 4x-3y-47=0 3x+4y-25=0

hA simaRlis sigrZe 11,2 11,2 12,6

7. samkuTxedis farTobi 28 28 63

8. C 29

28arctg

29

28arctg

22

21arctg

9. B wveroze gamavali AC-s

paraleluri BM wrfis

gantoleba.

x-8y-29=0 x-8y-47=0 x-6y+1=0

10

. manZili B wverodan AD

medianamde 17

8 17

8

226

63

3.98. 450. 3.99.

2

1;

2

3

3.100. a) 6

43;6

43 , 6

170R ; b) (-3,5;4,5);

2

130R 3.101. x-3y-10=0, 5x-3y+22=0, x-3y+2=0, 3x+y=0. 3.102.

3x-4y-9=0, 3x-4y+16=0, 4x+3y-37=0 an 4x+3y+13=0. 3.103. 25x-15y-24=0. 3.104x-y-4=0, x+y-4=0. 3.105. 2x-y+4=0. 3.106. x+2y-5=0,

2x-3y-3=0, 16x-10y+25=0. 3.107. x+2y-7=0, 5x+3y+7=0, 2x-3y-14=0. 3.108. 450. 3.109. 3x+4y-15=0; 4,4. 3.110. x+2y-3=0, x-5=0,

x-4y+15=0. 3.111. x2+y

2-6y+4=0. 3.113. (x-3)

2+(y-5)

2=25. 3.114.

49425

72y

76x

22

3.115. 2x-y+11=0 3.116x+2y-2=0, 2x+y-5=0,

2x-y-9=0, x+2y+8=0. 3.117. 3x-4y-41=0. 3.118. (x-7)2+(y-2)

2=57. 3.119. (x-1)

2+ (y-4)

2=13. 3.120. (x-6)

2+(y-7)

2=36. 3.121.4x-2y-9=0.

3.122. 2x-5y+19=0. 3.123. (x-1)2+(y-1)

2=1, (x-5)

2+(y-5)

2=25. 3.124. (x-3)

2+ +(y-5)

2=25. 3.126. 1. exeba wrewirs; 2. kveTs; 3.

gadis mis gareT. 3.127. A=C=1, B=0, D=-6, E=-4 da F=-12. 3.128. M1 wertili Zevs wrewirze, M2 wertili moTavsebulia wris SigniT, M3 wertili _ mis gareT. 3.129. 1. (3;-1), (2;-2), 2. (-4;6), 3. ar ikveTeba. 3.130. (x-

5)2+(y-3)

2=

13

121 . 3.131. 2x+y-3=0.3.132. 7x-4y=0. 3.133. 8x--20y+1=0, 5x-2y-7=0. 3.134. (x+2)2+(y+1)

2=20. 3.135 1

16

y

25

x 22

,

136

y

100

x 22

3.136. 1. 19

y

18

x 22

; 2. 116

y

36

x 22

3.137. 1. 2a=10; 2b=6; F1(-4;0); F2(4;0); 5

4e . 2. 2a=2, 2b=8, )15;0(F1

, )15;0(F2 ,

28

4

15e . 3. 2a=26; 2b=10; F1(-12;0); F2(12;0);

13

12e . 4. 2a=4; 2b=12; )24;0(F1

; )24;0(F2 ; 3

22e . 3.138. 1

7

y

16

x 22

. 3.139. 1) (3;-

3);

13

21;

13

69; 2)

5

8;3 ; 3) ar ikveTeba. 3.140.

5

10 . 3.141. 1) 3

22 ; 2) 0,8. 3.142. 19

y

36

x 22

; 2

3e r1=3, r2=9. 3.143.

15

y

15

x 22

3.144. a

b2 2

3.145. (5;2). 3.146. (-5;7). 3.147.

3

62;

3

34 . 3.148. 1y5

x 22

. 3.149. 45

54 kv. erTeuli. 3.150. 3 da

7. 3.151. 1) ,2

3 2) ,2

2 3.152.x=9. 3.153. 14

y

5

x 22

3.154. 124

y

60

x 22

an 124

y

40

x 22

3.155. 180

y

144

x 22

. 3.156.

3

1;

3

24 da

(0;-1). 3.157. 1;15 3.158. 1) 116

y

9

x22

, 2) 14

y

20

x22

3.159. 1) 128

y

36

x 22

, 2) 125

y

20

x 22

3.160. 1) 2a=5;3

10b2 ;

0;13

6

5F1

;

0;13

6

5F2

; 3

13 . 2) 2a=6; 2b=8; F1(0;5); F2(0;-5);

4

5 . 3.161.a=3; b=4; F1(-5;0); F2(5;0);

3

5 , 2) 1

36

x

144

y 22

. 3.162. x

2-y

2=16.

3.163. y= x3

4 ;

4

5 . 3.164. 2;6 . 3.165. 1) (10;2), (-10;-2) 2) (10;-2); 3) ar ikveTeba. 3.166. 1

9

y

16

x 22

. 3.167. 1)

;19

y

36

x 22

2) 15

y

4

x 22

. 3.168.

2

9;10 . 3.169.

119

5

3;

5

48

3.170. 1

12

y

4

x 22

. 3.171. F(6;0); x=-6. 3.172. y2=-28x 3.173. 1)

y2=16x. 2) x

2=8y. 3.174. 1) x

2=-18y. 2) y

2=12x.

3.175.(3;3 2 ). 3.176. 12. 3.177.10. 3.178. 2p. 3.179. 1) (-6;9); (2;1). 2) (-4;6). 3) ar

ikveTeba. 3.180. 1) kveTs, roca

2

1k ; exeba , roca k=

2

1; 3) gadis mis gareT, roca k>

2

1. 3.181. y

2=-3x 3.182. y

2=-4x. 3.183.

(x+3)2=16(y+3). 3.184. 3x-4y+8=0 da x-2y+6=0. 3.185. p=2bk. 3.186. a da b veqtorebs Soris kuTxe 1) maxvilia, 2) blagvia.

3.187. | a

|=| b

|.

a

b

b

b2

a

b2a

3.188.

a

a3

b

b2

1

a3b2

1

a

b

b3

2a

3

1

a3

1

b3

2

3.189. ba2

1 ; ba

2

1 ; ab

2

1 ; ba

2

1 . 3.190. a

+ b

+ c

+ d

=0 3.191. ;173,23|R| 4127;6417 00 . 3.192. x=-1, y=4,

z=-4. 3.193. a

+ b

=5 i

- j

+2 k

; a

- b

=- i

+7 j

- -10 k

. 3.194. 3

2cos ;

3

1cos ;

3

2cos 3.195. k

7

3j

7

2i

7

6e

. 3.196. ( a

b

)=-6; cos=-0,436; 116051

` 3.197.B=C=45

0 3.198.

3

64a

b

geg. ;

3

24ba

geg. 3.199. ( a b

)=3, =77024`, 1a

b

geg. ;

655,0ba geg. . 3.200. =900. 3.201. 7 da 13 . 3.202. .385,704954|]ba[|;k48j51i7]ba[

;099,0

385,70

7cos

cos=0,724; cos=-0,682. 3.203. S=2 22 3.204. S=1,5 3.205. 1) ara; 2) diax 3.206. 0ADACAB , veqtorebi

komplanarulia, e.i. A, B, C, D wertilebi erT sibrtyeze mdebareoben. 3.207. V=12 3.208. h=11 3.210. a) y+4=0, b) z-

2=0. 3.211. a) 3y+z=0, b) x+2z=0. 3.212. ;3

2cos ;

3

2cos ;

3

1cos ,94131,1148 00 23700 3.213. x-2y-

3z+14=0. 3.214. 3x+2y-z=5. 3.215. 16

z

8

y

12

x

3.216. a=12,

5

6b , c=6. 3.217. 03z

11

6y

11

9x

11

2 .3.218.

35

12p ,

35

5cos ;

35

1cos ;

35

3cos . 3.219. d=2. 3.220. d=3. 3.221. d=2 2 . 3.222.

3

5;0;0M1

,

3.223. M(0;-2;0). 3.224. a) ;4

1cos 0375

4

1arccos 0 ; b) 045 . 3.225. x-2y-3z=4 3.226. 7x-4y+z-21=0 . 3.227. x+y-z+2=0

3;0;0M2

29

3.228. 2x+3y+4z=0. 3.229. 9x-y+7z-40=3. 3.230. 2x-2y+z=2. 3.231. 3x+3y+z-8=0. 3.232. 6d 3.233. M(1;-1; 2) 3.234.

1

3z

1

5y

2

1x

3.235. ;0

2z

0

2y

1

1x

x=-1+t; y=-2; z=2. 3.236. ;5

3z

4

2y

3

1x

24,0cos,23,0cos ,

25,0cos . 3.237. ;2

1z

1

y

3

2x

1

1z

2

y

1

2x;

1

3z

5

y

9

x

; 3.238.

5

5z

4

3y

2

1x

. 3.239. 2

z

1

y

4

x

3.240.

1

1z

1

1y

1

2x

3.241. ;3

3.242. 195

98cos ; 8459

195

98cosar 0 3.245.

4z

0y2x3 3.246. 383,0d 3.247. 3

24d 48. (1;-5;0);

10;0;

2

7 (0;-

7;-4). 3.249. (6;4;0), (0;0;2) 3.250. (1;-2;3) 3.252. 6

1sin 3.253.

15

1sin . 3.254. 5x-10y-9z-68=0 3.255. y+z+1=0 (miTiTeba.

wrfis gantoleba CavweroT 1

z

1

1y

0

2x

3.256. 5

2z

1

1y

3

1x

3.257.

7

4z

3

2y

5

3x

3.259. (6;4;5). 3.260.

(5;5;5) 3.261. 4

3z

3

1y

1

2x

; (3; 2; -1) 3.262. wrfe Zevs sibrtyeze. 3.263. x- -7y+17z-9=0 3.264. x-2y+z+5=0 3.265.

7x-4y+7z+49=0 266. 4x-3y+2z+26=0. 3.267. 19x- -14y+z+23=0. 3.268. x+2y-5z=0.

Tavi IV

4.1 0; 4.2 2; 4.3. 3

32 . 4.4. 1,5. 4.5. 0,5. 4.6. 3. 4.7. 3. 4.8. –1. 4.9. x0. 4.10. (-;2). 4.11. (1;). 4.12. (-1;). 4.13. (-;-1)(1;).

4.14. ( -2;3) 4.15. R\{3;2}. 4.16. (-2;0)(1;). 4.17. D)y1)=(-;-2][2;). D(y2)=[2;). 4.18. D(y1)=(-;-1)

2

5;1 , D(y2)=(-;-1)

2

5;1 4.19. D(y1)=R\{3}; D(y2)=(3;). 4.20. D(y1)=(2;); D(y2)=(-;2)(2;). 4.27. arc luwia, arc kenti. 4.28. luwia. 4.29.

luwia. 4.30. kentia. 4.31. luwia. 4.32. kentia. 4.33. luwia. 4.34. luwia. 4.35. arc luwia, arc kenti. 4.36. luwia.

4.37. a) 0,2,0,2,0; b) –1, 2, -3, 4, -5. g) ;6

5,

5

4,

4

3,

3

2,

2

1 d) 119

1,

9.7

1,

75

1,

53

1,

31

1

; e) 1, 0, ;

5

1,0,

3

1 4.38. a) ;

n5

1a n b)

;1n6

1n2an

g)

;

2

11a

n

n

d) an=(-1)

n; e) n)1(1

2

n ; 4.39.

2

1; 4.40. –4. 4.41. 0; 4.42. . 4.43. 12. 4.44.

m

; 4.45. a)

3 2 ; b) 1; g) ; d) 0; e) 3 2

1; v) 1; z) 0; T) . 4.46.

5

1. 4.47. . 4.48. a) 1; b)

2

2. 4.49.

22

3 4.50. . 4.51. 0. 4.52. .

4.53. . 4.54. 2

3 4.55. 0. 4.56. 0. 4.57. 1. 4.58 a) e

6, b) 0,4. 4.59. a) 3

2

e

; b) . 4.60. a) e2; b) 0. 4.61.

2e

1. 4.62.

5

9

e . 4.63. a)

1; b) 3

2; g) 0. 4.64. 3. 4.65.

2

1. 4.66.

6

1. 4.67.

4

3. 4.68.

343

1 4.69. . 4.70.

4

7. 4.71. 3. 4.72.

16

1 . 4.73.

144

1. 4.74.

3

4. 4.75.

9

5. 4.76.

3

1. 4.77.

2

3. 4.78.

3

1. 4.79. 0. 4.80. . 4.81.

2

3. 4.82. 2. 4.83. 3. 4.84. 2. 4.85.

4

1 . 4.86. –1. 4.87. 0. 4.88. 0.

4.89. 3

2 . 4.90.

6

1. 4.91.

2

ca . 4.92.

4

1. 4.93. 0. 4.94. –1. 4.95. 0. 4.96. 0. 4.97. 2. 4.98. 0. 4.99. a) - b) 0. 4.100. 0. 4.101.

cos . 4.102. 2

25. 4.103.

2

22 ; 4.104.

2

2. 4.105.

5

4. 4.106. . 4.107.

n

m. 4.108.

4

2 . 4.109.

2

1. 4.110.

02 xcos

1. 4.111.

2

1. 4.112.

2

1. 4.113.

4

1. 4.114. 2. 4.115.

2

5. 4.116. 2 . 4.117.

4. 4.118.

3

1. 4.119. –3. 4.120.

4

5ln . 4.121.

2

1. 4.122.

2

1.

4.123. 3. 4.124. ablna. 4.125. e

2 4.126. 1. 4.127. ln3. 4.128. a) 0; b) . 4.129. e3. 4.130. e. 4.131. e10 4.132. 2

1

e . 4.133. bae . 4.134.

2

1

e . 4.135. 2

7

e

4.136. e

1. 4.137. e. 4.138. 1. 4.139. 0. 4.140. 1, Tu x+; 0, Tu x- 4.141. 0. 4.142. 1. 4.143. 0. 4.144. -

2

1. 4.145. e-1. 4.146. e

3 4.147.3

1. 4.148. 0. 4.149. . 4.150. 0, Tu x3+; , Tu x3- 4.151. 0. 4.152 1. 4.153. 0. 4.154.

2

. 4.155.

2

3. 4.156.

4

1 4.157. +, Tu x0+, 0, Tu x0 -. 4.158. x= -3 da x=2 II gvaris wyvetis wertilebia. 4.159. x=0, x= -1 da

x=4 II gvaris wyvetis wertilebia. 4.160. x=2

1 I gvaris acilebadi wyvetis wertilia, x= -2 II gvaris wyvetis

30

wertilia. 4.161. x=0 I gvaris wyvetis wertilia f(0+)-f(0-)=-3. 4.162. x=0 da x=2 I gvaris wyvetis wertilebia; f(0+) -

f(0-)=-2; f(2+)-f(2 -)=1; 4.163. x=0 II gvaris wyvetis wertilia. 4.164. x=0 I gvaris wyvetis wertilia, f(0+)-f(0-)=2.

4.165. x=0 I gvaris acilebadi wyvetis wertilia. 4.166. x=0 I gvaris acilebadi wyvetis wertilia. 4.167. 1. 4.168.

n(xn-1

+1). 4.169. mn(xn-1

+xm -1

). 4.170. 8x9,0xx2

1

x2

1

4.171. 4 73 55

x4

15

x

2

x

8 . 4.172.

56 7 x4

21

x3

1x

4

15 . 4.173.

43

11

2u15u

3

16

u

1y

, 3

64)1(y . 4.174. 2x

2(1+x

3) 4.175.

22

2

)1xx(

2x2x3

. 4.176. x

1e2 x . 4.177. 2

xe

x(ln2+1). 4.178. tgx+

xsin

1

xcos

x22

. 4.179. 233 2 )x32(x

4

4.180. 9x2lnx. 4.181.

9

8ln

9

8

9

8ln

3

2x

x2

x3

. 4.182. 5cosx -xsinx;

22y

. 4.183.

1x

2xxarctgx2y

2

2

;

2

3

21y

. 4.184.

22

2x

)x(arcsinx1

)13lnx1x(arcsin3

. 4.185. x

109

10

10lnxx10 . 4.186. 2x

x

)31(

33ln2

. 4.187.

2lnx

1)x2x(xlogx6x

5

1 352

25

4

. 4.188. 23

2

x1x

)xarccosx1x(

. 4.189. –cosx. 4.190. 2)x2(

1

- 4.191.

)3x2(x

)1x(6

. 4.192. arccos

2

x

. 4.193. 3

xcos

3

xsin 2 . 4.194. 36(2x

2+4x+5)

8(x+1). 4.195.

3 22 )xsin1(x6

x2sin

. 4.196. xsin

xctg3

x2cos

x2tg55ln42

2

2

x22tg

. 4.197.

0)0(y;)x4(

x8

x4

x4222

2

. 4.198.

)x1()x(arccos1

xarccos

22

. 4.199. x

xarcsin

2

1 . 4.200.

2

1x2sin

1

. 4.201. xcos

1 . 4.202. 2

xctg3 .

4.203. 1x4x

1

2

. 4.204. –2e-x

sin2e

-x 4.205. x3sinx6sine3 2x32sin 4.206.

xsin

1. 4.207.

)2x(x

105

. 4.208. )1x(cos5ln

x2sin2

4.209.

3 23

23

33x

)3x(

3lnx3

. 4.210. xcosln

xsinlntgxxcoslnctgx2

. 4.211. 9

xx

1x

x

1x

17

. 4.212.

2)xcos1(

xsin2

. 4.213.

xsin1

xcos2

. 4.214.

0. 4.215. 4xcos2x. 4.216.

2)xsinln1(

xsinlnxcos

. 4.217.

32)mx1(

1

4.218. x)x1001(

x10arctg10ln1010

2x10arctg

. 4.219.

2

x2cosx2cos

)x(arccos

)x1(x2

13xarccosx2sin33ln2

4.220. x1

xarctgx)xarctg( 2

. 4.221.

2

233

2

2

x1

1)arctgx(3xtgarctgx

xcos

1xtg3

.

4.222. x2

2xlnx x

. 4.223. 2

x

1

x

xln1x

. 4.224.

x

)1xsin(xln)1xcos(x )1xsin( . 4.225. (sinx)

x(lnsinx+xctgx). 4.226. (lnx)

x

xln

1)xln(ln .

4.227.

tgxxarccos

x1

xcosln)x(cos

2

xarccos .4.228. )1x(lnx3ln3)1xln3(xx xxx23x .

4.229. (arcsin)x

2x1xarcsin

x)ln(arcsinx .

. 4.230.

1x

1

x

1xln

x

11

x

4.231.

4x

xcosx2)4xln(xsin)4x(

2

2xcos2 4.232.

x2ctg2x2

x

3x2sinex

2x3 . 4.233.

5x

3

)1x(3

1

3x

3

)5x(

1x)3x(3

33

. 4.234.

x

7

4x

2

)2x(3

1

x

)4x)(2x(3

7

2

.

4.235. 3)5x(

44

4.236. lnx. 4.237. x -sin3x. 4.238.

4)1x(

1

4.239.

3x

3xln2 . 4.240. 105 3x2 . 4.241. ane

ax. 4.242.

2nbkxsink n . 4.243.

2

n3x4cos)4( n . 4.244. (-1)

nn!

1n)ax(

1

4.245. (-1)nn!

1n1n )ax(

1

)ax(

1 4.246. (-1)nn!

1n1n )1x(

1

)2x(

1 4.247. (e3x+3x

2sin2(x

3+1))dx. 4.248. x

n -1lnx(nlnx+2)dx. 4.249. dx

)1x2(cos

)1x2(tg82

3

. 4.250. dxa

x2sine3

xsin2

4.251.

dx4xcos

x2sin)4x(cosln32

22

. 4.252. dx

31

3ln3x2

x

. 4.253. dx

)4x3x2(3

3x4

3 22

. 4.254. 12ln2 dxx4cos

x4tg2

2

2x43tg . 4.255. (xx

(lnx+1)+2x)dx.

4.256. dx25x

x)25xlg(

10ln

42

2

. 4.257. 1. 4.258.

b

aln . 4.259. –2. 4.260.

3

1. 4.261. . 4.262. 1. 4.263.

6 23

2. 4.264.

2

2

b

a. 4.265.

2

3. 4.266. 2. 4.267. 0. 4.268.

3

8. 4.269. 0. 4.270. 1. 4.271. 1. 4.272. 0. 4.273 -. 4.274. 0. 4.275.

1. 4.276. 1. 4.277. 0. 4.278. ln3.

4.279. 0. 4.280. -2

1. 4.281. a. 4.282.

1. 4.283. 0. 4.284. –1. 4.285. 1. 4.286. 1. 4.287. 1. 4.288. 3. 4.289. e

-2 4.290. 1. 4.291.

2

e .

4.292. e. 4.293. 1. 4.294. 2

1

e

. 4.295. klebadia (-; 1), zrdadia (1; +), ymin(1)=-1. 4.296. klebadia (-;-1), (1;+),

31

zrdadia (-1; 1), ymin( -1)= 2

1

e

, ymax(1)= 2

1

e

. 4.297. klebadia

;

2

1, zrdadia

2

1; , ymax

4

9

2

1

. 4.298.

klebadia (1;0) (1; +), zrdadia (-;-1) (0;1), ymax(-1)=1, ymin(0)=0; ymax(1)=1. 4.299. klebadia (0;), zrdadia (-;0),

ymax(0)=4

1. 4.300. klebadia

2

1;

;

2

1, zrdadia

2

1;

2

1.

4

1

2

1y;

4

1

2

1y maxmin

4.301. klebadia (1;3);

zrdadia (-;1), (3;), ymax(1)=0; ymin(3)=-4. 4.302. klebadia, (0;1), (1;2), zrdadia (-;0) (2;+); ymax(0)=-2; ymin(2)=2. 4.303.

klebadia ( 8 ;-2) )8;2( zrdadia (-2;2) ymax(2)=4; ymin(-2)=-4. 4.304. zrdadia (3; +); klebadia (-; 2), ymin(2)=0;

ymin(3)=0. 4.305. klebadia (0; 1)(1; e); zrdadia (e; +) ymin(e)=e. 4.306. zrdadia (3; +), klebadia (-;3), ymin(3)=1. 4.307.

zrdadia (-;0) (4;+) klebadia (0; 4), ymax(0)=81, ymin(4)=3 -28

. 4.308. zrdadia (-;2), klebadia (2;+), ymax(2)=2

1. 4.309.

zrdadia mTel gansazRvris areze (0;+), eqstremumi ar aqvs. 4.310. zrdadia (5,+), klebadia (-,-5);

eqstremumi ar aqvs. 4.311. klebadia (0;e

1), zrdadia (

e

1,+), ymin

e

1= e

1

e

. 4.312. zrdadia (-; 2), klebadia (2;

+); ymax (2)=e5. 4.313. –1 da 27. 4.314. 0 da

4

. 4.315. 0 da e

2. 4.316. –18 da 2. 4.317. (-;0), (0;1) da (3;) Cazneqilia;

(1;3) amozneqilia. gadaRunvis wertilebia M1(1;5,5) da M2(3;-112,5). 4.318. (-;0), (1;2) grafiki amozneqilia; (0,1)

(2;) Cazneqilia; gadaRunvis wertilebia M1(0;1) da M2

3

20;1 da M3

3

37;2 . 4.319. (-;0)-ze grafiki amozneqilia;

(0;) - Cazneqilia; gadaRunvis wertilia M(0;1). 4.320. grafiki yvelgan Cazneqilia. gadaRunvis wertili ar

arsebobs. 4.321. (-;-3)-ze grafiki amozneqilia; (-3;) - Cazneqilia; gadaRunvis wertili ar arsebobs. 4.322. (-

;2)-ze grafiki amozneqilia; (2;) - Cazneqilia; gadaRunvis wertilia M(2;3). 4.323. (-;-1)-ze grafiki

amozneqilia; (-1;) - Cazneqilia; gadaRunvis wertilia

2e

11;1M . 4.324. (-;-1)-ze da (1;) grafiki Cazneqilia;

(-1;1) - amozneqilia; gadaRunvis wertilia )2;1(M 31 da )2;1(M 3

2 4.325. (-;-2)-ze grafiki amozneqilia; (2;) -

Cazneqilia; gadaRunvis wertilia M(-2;-2e-2

). 4.326. grafiki Cazneqilia Sualedebze

2

1; da

;

2

1 ;

amozneqilia

2

1;

2

1; gadaRunvis wertilia

2

1

1 e;2

1M da

2

1

2 e;2

1M 4.327. (-;-1) da

;

2

1 Sualedebze

grafiki amozneqilia;

2

1;1 - Cazneqilia; gadaRunvis wertilebia 22;1M1 da

5;

2

1M 2

. 4.328. x=2

vertikaluri asimptotia; y=x daxrili asimptotia. 4.329. x=-3 vertikaluri asimptotia; y=1 daxrili asimptotia. 4.330. x=1 vertikaluri asimptotia; y=2x daxrili asimptotia. 4.331. x=-2 vertikaluri asimptotia;

y=x-4 daxrili asimptotia. 4.332. vertikaluri asimptoti ar gaaCnia, y=1 horizontaluri asimptotia. 4.333. x=0 vertikaluri asimptotia; y=x daxrili asimptotia. 4.334. x= -2 da x=2 vertikaluri asimptotebia; y=0

horizontaluri asimptotia. 4.335. x=0 da 2

1x vertikaluri asimptotebia;

4

3x

2

1y daxrili asimptotia.

4.336. x=-1 vertikaluri asimptotia; y=0 horizontaluri asimptotia (marcxena, roca x-). 4.337. x=0

vertikaluri asimptotia; y=1 horizontaluri asimptotia 4.338. 2

y

da 2

y

horizontaluri

asimptotebia. 4.339. xa

by daxrili asimptotia. 4.340. x=0 vertikaluri asimptotia; y=x+4 daxrili

asimptotia. 4.341. y=0 horizontaluri asimptotia roca x -. 4.342. (gv. 325. nax.1) D(y)=R luwia. RerZebTan

TanakveTis wertilebia (0;2) 0;2,3 . asimptotebi ar gaaCnia. zrdadia Sualedebze (-;-1) da (0;1), klebadia

(-1;0) da (1;). 2

51ymax ;

3

3; da

;

3

3 Sualedebze grafiki amozneqilia.

3

3;

3

3 Sualedze

grafiki Cazneqilia. wertilebi

18

41;

3

3da

18

41;

3

3 gadaRunvis wertilebia

8,12,3;3,2

18

41;6,0

3

3 .

4.343. (nax.2) D(y)=R luwia. grafiki oy RerZs kveTs wertilSi (0;3). asimptotebi ar gaaCnia. zrdadia

Sualedebze (-1;0) da (1;), klebadia (-;-1) da (0;1) Sualedebze. ymin(1)=2; ymax(0)=3

3

3;

3

3 Sualedebze

grafiki amozneqilia.

3

3; da

;

3

3 Sualedebze grafiki Cazneqilia. wertilebi

9

22;

3

3

gadaRunvis wertilebia.

4,2

9

22;6,0

3

3 . 4.344. (nax.3.) D(y)=

;

2

1

2

1; luw-kentovnebis sakiTxi ar

32

ismis. grafiki saTaveze gadis. 2

1x grafikis vertikaluri asimptotia.

2

1xy daxrili asimptotia.

zrdadia Sualedebze (-;0) da (1;), klebadia

2

1;0 da

1;

2

1 ; ymax(0)=0, ymin(1)=2,

2

1; Sualedze grafiki

amozneqilia.

;

2

1 Sualedze grafiki Cazneqilia. gadaRunvis wertilebi ar arsebobs. 4.345 (nax.4) D(y)=(-;-

1) (-1;1)(1;) luwia. oy RerZTan TanakveTis wertilia (0;2). x=1 vertikaluri asimptotebia. y=0

horizontaluri asimptotia. zrdadia Sualedebze (0;1) da (1;), klebadia (-;-1) da (-1;0); ymin(0)=2, (-;-1) da

(1;) Sualedebze grafiki amozneqilia. (-1;1) Sualedze grafiki Cazneqilia. gadaRunvis wertilebi ar

arsebobs. 4.346. (nax.5) D(y)=(-; -2)(-2;2)(2;) kentia. grafiki (0;0) wertilze gadis. x=2 vertikaluri

asimptotebia. y=2x daxrili asimptotia. zrdadia Sualedebze 12; da );12( , ( 5,312 ), klebadia (-

12 ;-2), (-2;2) da (2; 12 ). ymin( 12 )=10,5, ymax(- 12 )=-10,5, (-;-2) da (0;2) Sualedebze grafiki amozneqilia. (-2;0)

da (2;) Sualedze grafiki Cazneqilia. (0;0) gadaRunvis wertilia. 4.347. (nax.6) D(y)=(-;-1)(-1;) luw-kentovnebis sakiTxi ar ganixileba. grafiki (0;0) wertilze gadis. x= -1 vertikaluri asimptotia. y=x-1

daxrili asimptotia. zrdadia Sualedebze (-;-2) da (0;), (-2;-1), da (-1; 0) Sualedebze funqcia klebadia.

ymin(0)=0, ymax(-2)-4, (-;-1) Sualedebze grafiki amozneqilia. (-1;) Sualedze grafiki Cazneqilia, gadaRunvis wertilebi ar arsebobs. 4.348. (nax.7) D(y)=R luwia. grafiki gadis saTaveze. y=1 horizontaluri asimptotia.

zrdadia Sualedebze (0;), klebadia (-;0) Sualedze. ymin(0)=0,

3

1; da

;

3

1 Sualedebze grafiki

amozneqilia.

3

1;

3

1 Sualedze grafiki Cazneqilia,

4

1;

3

1 gadaRunvis wertilebia

6,0

3

1 . 4.349.

(nax.8) D(y)=(-;2)(2;) luw-kentovnebis sakiTxi ar ganixileba. grafiki sakoordinato RerZebs gadakveTs wertilebSi (3;0), (-2;0) da (0,3). x=2 vertikaluri asimptotia. y=x+1 daxrili asimptotia. funqcia gansazRvris

areSi yvelgan zrdadia. (-;2) Sualedebze grafiki Cazneqilia. Sualedze (2;) grafiki amozneqilia.

gadaRunvis wertili grafikze ar aris. 4.350. (nax.9) D(y)=(-;-2)(-2;2)(2;) funqcia kentia. grafiki oy RerZs

gadakveTs wertilSi (0;2), x=2 vertikaluri asimptotia. y=-1 horizontaluri asimptotia. Sualedebze (-;-2)

da (-2;0) funqcia klebadia, xolo Sualedebze (0;2) da (2;) funqcia zrdadia. (-;-2) da (2;) Sualedebze grafiki amozneqilia. (-2;2) Sualedze grafiki Cazneqilia. gadaRunvis wertili ar gaaCnia. 4.351. (nax.10)

D(y)=(-;-2)(2;) luw-kentovnebis sakiTxi ar ismis. RerZebTan TanakveTis wertilebia (-2;0), da (0;1) x=2

vertikaluri asimptotia. y=1 horizontaluri asimptotia. Sualedebze (-;-2) da (2;) funqcia klebadia, xo-

lo Sualedze (-2;2) funqcia zrdadia. ymin(-2)=0.(-4;2) da (2;) Sualedebze grafiki Cazneqilia, (-;-4)-ze

Sualedze grafiki amozneqilia. gadaRunvis wertilia

9

1;4M . 4.352. (nax.11) D(y)=(-;-1)(-1;) luw-

kentovnebis sakiTxi ar ismis. oy RerZTan TanakveTis wertilia (0;1), x=-1 vertikaluri asimptotia. y=0

grafikis marcxena horizontaluri asimptotia. Sualedebze (-;-1) da (-1;0) funqcia klebadia, xolo

Sualedze (0;) funqcia zrdadia. ymin(0)=1.(-;-1)-ze grafiki amozneqilia; (-1;) Sualedze grafiki Cazneqilia, gadaRunvis wertili ar gaaCnia.

x

y 2

-1 1 0

2,3 2,3

3

1

3

1

2

xx2y

42

nax. 1.

x

y

3

1 -1 3

1

3

1

nax.2

y=x4-2x3+3 y

x -1 1

1/2

y=x+1/2

1x2

x2y

2

nax.3

0 x

x=1

y

x=-1 2 2x1

2y

nax. 4

-2 -1 0 x

x=-1

x

y=x-1

x

y

-4

x

1x

xy

2

nax.6

-1 0 1 x 3

1

3

1

y

y=1

1x

xy

2

2

nax. 7.

0 A(3;0) x

B(-2;0)

C(0;3)

y=x+1

2x

6xxy

2

nax. 8

y

0 x x=2

y

x=-2

12x

3

2x

2y

nax. 9

y=-1

33

0 2 x

x=2

y 2

2x

2xy

nax. 10.

y=1

-1 0 x

x=-1

y

x1

ey

x

nax. 11.

Tavi V

5.1. 25 da 0. 5.2. 0 da –6. 5.3.5

2 da

9

4. 5.4. 6 da 3. 5.5. gansazRvrulia mTel sibrtyeze. 5.6. {(x,y)R

2 -<x<+,

y1}. 5.7. {(x,y)R2 | x

2+y

29}. 5.8. {(x,y)R

2-<x<+ x-1y-x+1}. 5.9. {(x,y)R

2 -<x<+, y<x}. 5.10. gansazRvrulia

mTel sibrtyeze. 5.11. {(x,y)R2|-<x<+, x

2+y

2>1}. 5.12. {(x,y)R

2| y>x

2, yx

2+1}. 5.13. {(x,y)R

2| |x||y|(x,y)(0;0)}. 5.14.

{(x,y)R2 | x2, y3}. 5.15. {(x,y)R

2 | |x|<2, |y|<3}. 5.16.

125

y

16

x|Ry)(x,

222 5.17. 0. 5.18. 0. 5.19. 2. 5.20. e

k. 5.21.

zRvari ar aqvs. 5.22. zRvari ar aqvs. 5.23. 4

1 . 5.24. 0. 5.25. + 5.26. 2. 5.27 uwyvetia, roca (x,y)D(z) 5.28.

wyvetilia y=x wrfeze. 5.29. wyvetilia x2+y

2=1 wrewirze. 5.30. uwyvetia, roca (x,y)D(z). 5.31. wyvetis wertilTa

simravlea y=x2 parabolis wertilebi. 5.32. –3. 5.33. 4 273ln2 . 5.34. –1. 5.35. 0. 5.36. –3. 5.37. )ayx(3z 2

x ,

)axy(3z 2y . 5.38.

2x)yx(

xz

,

2y)yx(

yz

. 5.39.

2xx

yz ,

x

1z y . 5.40.

22x

yx

xz

,

22y

yx

yz

. 5.41.

2

3

22

2

x

)yx(

yz

2

3

22

y

)yx(

xyz

5.42. 22x

yx

yz

,

22yyx

xz

5.43.

1yx yxz , xlnxz y

y 5.44.

x

ycose

x

yz x

ysin

2x , x

ycose

x

1z x

ysin

y . 5.45. )yx(|y|

y2x2xyz

44

222

x

,

)yx(|y|

y2x2yxz

44

222

y

. 5.46.

y

axtg

y

1z x

,

y

axtg

yy2

axzy

. 5.47. ysinx2z 2

x , y2sinxz 2y . 5.48.

xcos

1tgxylny2z

2

x2tgx , 2x2tg2

y yxtgz . 5.49.

1y222x

2

)yx(xy2z ,

22

2222y22

yyx

y)yxln(y2)yx(z . 5.50.

ysin

2

x 2

x

ycosz , xlnyx2sinz ycos

y

2

5.51.

dz=(2x-4y)dx+(9y2 -4x)dy. 5.52. dz=(2x+3)dx+2ydy. 5.53. dz=(3y-4x)dx+(3x-6y)dy

5.54. .dyxydxxy

x2dz 5

44

5.55. dy

x1

xdx

)x1(

)x1(ydz

222

2

. 5.56.

dy

y

xdx2

y

xedz y

2x

.

5.57. ydyxdx)yx(sin

2dz

222

. 5.58.

dy

y

xdx

y

1

y

xsinedz

2

y

xcos

. 5.59. )ydyxdx()yx(1

2dz

222

.

5.60. )ydyxdx(

yx

1dz

22

. 5.61. dyy

1dx

x

1dz 5.62. dy

y

1xydxdz

5.63. 0,9 5.64. 0,8. 5.65 .0,03. 5.66. 0,505 5.67.

3xx y6z , 2

yxxy xy18zz , yx18y40z 23yy 5.68. )ysin)2x(y(cosez x

xx , ycos)1x(ysinezz xyxxy ,

ysinxycosez xyy 5.69. 3

xx yxy4z

, 3yxxy )yx)(yx(2zz , 3

yy yxx4z

. 5.70. 22222xx yx)yx(z

,

222xy yxxy2z

, 22222

yy )yx(yxz . 5.71. 22xx

)x1(

x2z

, 0z xy ,

22yy)y1(

y2z

. 5.72. xxz =18cos2(3x+5y), xyz

=30cos2(3x+5y), yyz =50cos2(3x+5y). 5.73. )xy(cos

)xysin(y2z

3

2

xx . )xy(cos

)xy2sin(xy)xy(cosz

4

2

xy

,

)xy(cos

)xysin(x2z

3

2

yy . 5.74. xxz =y(y-

1)(cosx)y -2

sin2x -(cosx)

yy, yyz =(cosx)

yln

2cosx, xyz =-(cosx)

y -1sinx(ylncosx+1). 5.83.

)2x(x

)1x(122

2

. 5.84. 1x

x24

. 5.85. 4x

22

. 5.86.

2xlnx2xlnx

xln122

5.87.

23

36

3

44

3

43

)1t(

t4t

1t

tcost

1t

tsint4

. 5.88.

228

23

)t1625(t1

)100t96(t

. 5.89.

t4sin

2

t2sin33ln2 t22sint2cos . 5.90.

34

2222

222

)vu3)(vu2(cos

)v3uv2u12)(vu3(2

. 5.91. )vcosv(sinv2sinu2

3e 3)vsinv(cosu 333

. 5.92. zmin=z(1,0)=0. 5.93. eqstremumi ar aqvs. 5.94.

zmin=z(1,0)=-1. 5.95. zmax=z(3;2)=108. 5.96. 8)2;2(z)2;2(zzmin . 5.97. zmin=z(-1,-1)=-7. 5.98. zmin=z(-12,4)=-41. 5.99.

zmin=z(4,-1)=3

241 . 5.100. zmin=

96

305

4

7;

2

1z

, 3

243)1;4(zzmax 5.101. zmin=z(5,2)=30.

Tavi VI

6.1 3x3 + 3x2 + x + C. 6.2 2x3 + 2x2 + ln|x| + C. 6.3 .Cxx3x3

11x

2

3 234 6.4 .Cx2

1xln2x

2

1

2

2 6.5

.Cx3

2x4

x

2 3 6.6 .C12ln

12

8ln

8 xx

6.7 .Cx2

1e

2

x 6.8 .)1x(2

1 2 6.9 .Cx

2

x

2

x

3xln2

32

6.10 .Cx23

4

4

3

4

3ln

1xx

6.11 tgx–x+ C. 6.12 –ctgx – x + C. 6.13 .Cxsin2

1x

2

1 6.14 2arctgx – 3arcsinx +C. 6.15 sinx +

cosx + C. 6.16 .Cctgx2

1 6.17 .Ctgx

2

1 6.18 cosx + C. 6.19 .

)2x3(9

13

6.20 .)k1(a

)bax( k1

6.21 .)4x3(9

2 3

6.22 .)3x2(8

3 5 4 6.23 .4x 2 6.24 .7x4ln6

1 3 6.25 .)4x(8

3 3 42 6.26 .x72

1 4 6.27 ).9x4xln(2

1 2 6.28 .e5

1 x5

6.29 .xln3

1 3 6.30 ln(ex +3). 6.31 .22ln

1 arctgx 6.32 ).2x3(tg3

1 6.33 .e

3

1 3x 6.34 .aaln

1 xsin 6.35 .1tg2 6.36 arctg(x + 1).

6.37 .x

3xln

3

1 6.38 ).x1cos(

3

1 3 6.39 .xtg3

2 2 6.40 .e x

1

6.41 .2arcsin2ln

1 x 6.42 .5x2x1xln 2 6.43

.5

1x2arcsin

6.44 .

2x

3xln

6.45 .

3x5

3x5ln

152

1

6.46 .

3

x5arcsin

5

1 6.47 .3x2ln

4

3x

2

1 6.48 .x

3

5arctg

15

1 6.49

.3x5x5ln5

1 2 6.50 .x6cos6

1x 6.51 .x7xcos

14

1x3cos

6

1 6.52 .x

2

xtg2 6.53 .

3

2x .2x3

3

144 4 6.54

.3

earctg

3

1 x

6.55 .2x

2xln

12

13

3

6.56 .xln

4

1 4 6.57 ).x(lnarctg2

1 2 6.58 ,xsin2

5 5 2 x≠πk, kZ. 6.59 –ln|cosx|. 6.60 ln|sinx| 6.61

).x2sin1ln(2

1 6.62 .9xxln39x 22 6.63 ,

3

xarcsin3x9 2 x3. 6.64 ).)3x(3xln(

3

1 2 6.65 .xln3

1x3 33

6.66 .)bea(b4

3 3 4x 6.67 .a

barctg

blna

1 x

6.68 ).bx2a2sin(b4

1

2

x 6.69 ).bx2a2sin(

b4

2

2

x 6.70 .x)bxa(tg

b

1 6.71

.x)bxa(tgb

1 6.72 .

2

baxtgln

a

1 6.73 .

2

bax

4tgln

a

1

6.74 .)x1cos(ln

2

1 2 6.75 .1x,)x(arcsin4

1 4 6.76

.)x2cos23(6

1 3 6.77 .Zk.5arctgπkx,5

tgxarcsin 6.78 .xctg2 6.79 .)tgx21(

3

123

6.80 .3

xsin2arcsin

2

1

6.81 .3

xarccos

2

1x9

2

2

6.82 .3x3xln

3

1 3 6.83 .2

2x,x41

8

1)x2arcsin(

4

1 42 6.84 .)1x(4

3)1x(

7

334

37

6.85 )xsin4(8

3 2 .0xsin,xsin3 2 6.86 .1x2x5 2 6.87 ).4xln(

4

1

2

xarctg

4

1 42

6.88 2x–3ln|x+1|. 6.89 .1x2ln4

1x

2

1 6.90

.2

1x2arctg

8

1)5x4x4ln(

8

1 2 6.91 .2xln8x4x

3

x 23

6.92 .3

1xarctg

3

1)10x2xln(

2

1 2 6.93

2xln4

2xln4ln2xln42

6.94 .x1xarcsin 2 6.95 .

11x

11xln1x2

6.96 .

10

5arctg

5ln10

1 x

6.97 - xcosx +

sinx. 6.98 snix+cosx. 6.99 .3ln

3

3ln

3x2

xx

6.100 .7ln

7

7ln

7x2

xx

6.101 -xe-x - e-x. 6.102 .x2sin4

1x2cosx

2

1 6.103

.x5cos25

1x5sinx

5

1 6.104 xlnx-x. 6.105

.

1

xxln

1

x2

11

6.106 .x2

1arctgx

2

)1x( 2

6.107 .x1xarcsinx 2 6.108 (x2–

x+2)ex. 6.109 .aln

2

aln

x2

aln

1xa

32

2x

6.110 ).1x2(

4

e x2

6.111 .2

3xlnx

2

3 3 2

6.112 ).xln21(

x4

12

6.113

35

).xlncosxln(sin2

x 6.114 ).inxcosinxsin

2

x 6.115 .xarcsinx1x 2 6.116 .tgx)tgxlln()tgxl( 6.117

.e)1eln()le( xxx 6.118 ).x3cos2x3sin3(13

e x2

6.119 ).bxcosbbxsina(ba

e22

ax

6.120 ).2xln2x(lnx 2 6.121

).xx22(e 2x 6.122 .aln

2

aln

x2

aln

xa

32

2x

6.123 .x1x

4

1xarcsin

4

1x2 22

6.124 arctgx2x2)1xln(x 2 6.127

.2

1xarctg

2

1 6.128 .

2

1xarctg)5x2xln(

2

1 2 6.129 .

5

1x3arctg

5

1 6.130 ).2x(arctg4)5x4xln(

2

3 2 6.131

.14

1x3arctg

143

20)5x2x3ln(

3

2 2 6.132 .

5

1x2arctg

52

93x2x2ln

4

1 2 6.133 .1xx

2

1xln 2 6.134

.3

2xarcsin

6.135 .

5

3x4arctg

2

1 6.136 .3x6x91x3ln

3

1 2 6.137 .5x4x3 2

6.138 .2

1xarcsin3xx234 2

6.139 .5x3x5,1xln2

15x3x 22 6.140 .

17

3tgx4arcsin

2

1 6.141

.3x

)2x(ln

2

6.142 .

)1x(

)2x(xln

3

15

23

6.143 .3xln

15

262xln

10

9xln

6

5 6.144 .

)2x(

)2x(xlnx4

2

x

3

x3

5223

6.145

.1x

2xln

1x

1

6.146 .

1x

xln

1x

6 2

6.147 .)1x(xln

1x

2

6.148 .

1x

1xln

2

1

x

1

6.149 .

)1x(2

1

)3x(2

9

6.150

.1x

xln

2 6.151 .

1x

xlnx

2

6.152 .)1x(2

1)1xln(

4

11xln

2

1 2

6.153 .

3

1x2arctg

3

1

1xx

)1x(ln

6

12

2

6.154

.arctgx2

1)1x)(1x(ln

4

1)1xln(

2

1 32 6.155 .3

1x2arctg

3

1)1xxln(

2

11xln 2

6.156 .arctgx2

1

)1x(2

x2

6.157

).1x(arctg)2x2x(2

1x22

6.158

)5x4xln(

2

1

)5x4x(2

17x3 2

2).2x(arctg

2

15

6.159 .3

1x2arctg

33

2

)1x(3

x

)2x(

1xxln

9

132

2

6.160 .

)1x(4

1x

)1x(

1xln

8

122

2

6.161 .xsin

3

1xsin 3 6.162 .xcosxcos

3

1 3

6.163 .xcos5

1xcos

7

1 57 6.164 .xsin5

1xsin

3

1 53 6.165 .x6sin12

1x

2

1 6.166

6.167

.x4sin32

1x2sin

4

1x

8

3 6.168 .x4sin

32

1x

8

3 6.169 .x2sin

48

1x4sin

64

1x

16

1 3 6.170 .x2sin48

1x4sin

164

1x

16

1 3 6.171

.xsin

1 6.172 .

xcos

1 6.173 .

xsin3

1

xsin

13

6.174 .xcos2

1xcosln2

xcos2

1 2

2 6.175 .x2sin

4

3x

2

3

xcos

xsin3

6.176

.2

xtgln

2

1

xsin2

xcos2

6.177 .xtg2

1tgxln 2 6.178 .tgxln2)xctgxtg(

2

1 22 6.179 .ctgxxctg3

1 3 6.180 .tgxxtg3

1 3 6.181

.ctgxtgx2xtg3

1 3 6.182 .xctg3

1ctgx3tgx3xtg

3

1 33 6.183 xtg5

1xtg

3

2tgx 53 6.184 .xcoslnxtg

2

1xtg

4

1 24 6.185

.xctgxxctg3

1 3 6.186 .xtg7

1xtg

5

1 75 6.187 .x11sin22

1x3sin

6

1 6.188 .

6

x5sin

5

3

6

xsin3 6.189 .x4sin

8

1x2sin

4

1

6.190 ).1x8sin(16

1)15x2sin(

4

1 6.191 .x6cos

12

1x4sin

8

1 6.192 .x

4

1x6sin

24

1x4sin

16

1x2sin

8

1 6.193 .

2

xtgln 6.194

.2

x

4tgln

6.195 .2

xtg

2

1arctg

2

1

6.196 .

3

42

xtg5

arctg3

2

6.197 .

22

xtg

2

1

2

xtg

ln5

1

6.198 .2

xtgx 6.199 .tgx

2

1arctg

2

1

6.200

.xtg5

1xtg

3

1 53 6.201 .tgx2

1tgxln

2

1 6.202 .

1tgx

1

6.203 ).xtg(arctg 2

6.204 .

42

xtg2

12

xtg2

ln5

1

6.205 .

5

414

2

xtg

5

414

2

xtg

ln41

1

6.206

.2

xtg

8

1

2

xtg8

1

2

xtgln

xcos

1 2

2

6.207 .2

xtgln

xcos

1 6.208 .

xsin

1

42

xtgln

.x4sin32

1x2sin

4

1x

8

3

36

Tavi VII

7.1 1. 7.2 ex – 1. 7.3 .3

27.4 .

2

7.5 1. 7.6 ).18(

3

2 7.7 .

5

17.8 .26

3

5 3 7.9 .3

11 7.10 .

2

7.11 .

3

7 7.12 .

5

2 7.13 .

18

7.14 .

5

8ln

2

1

7.15 .13 7.16 ).12ln( 7.17 .4

7 7.18 .

70

5732

7

244

10

27 63 7.19 7.20 .4

1

7.21 .12

7.22 .3ln

6

7.23 ln3.

7.24 .4

arctge

7.25 .116

13

3

7.26 .

72

7 7.27 .2ln 7.28 .

e2

1e 7.29 .

3

47.30 arctg3–arctg2. 7.31 .12 7.32 .27

3

2 4 7.33 .5ln3

4

7.34 .4ln4

17.35 .

6

7.36 .

6

7.37 .

2

3arctg

4

1 7.38 .

52

265ln

3

1

7.39 .

9

2 7.40 ).21ln( 7.41 .

3

4ln

5

1 7.42 .

82ln

2

1 7.43

.3ln2

1 7.44 0. 7.45 .

6

5arcsin

5

17.46 .

18

7.47 .

3

1 7.48 ).1e(

3

1 7.49 - .

8

7.50 1. 7.51 .

4

7.52 .

5

2 7.53 1. 7.54 .

8

27.55 1. 7.56 e–

4. 7.57 .4

e393 2

7.58 .2ln4

32 7.59 .

7ln

127ln147ln73

2 7.60 .

8tgln

22

7.61 .2ln

2

1

4

7.62 e–2. 7.63 3–6. 7.64 1.

7.65 ).1e(2

1 7.66 ).1e(

2

1 7.67 .

2

3ln

2

1)349(

36

7.68 .8e6 7.69 .

2

17.70 ganSladia. 7.71 ganSladia. 7.72 .

aln

1

7.73 ganSladia, roca k0,k

1 Tu k<0. 7.74 ganSladia. 7.75 ).0ab(

ab2

7.76 .

2ln2

12

7.77 .4

7.78 .

33

7.79 .

2

7.80

ganSladia. 7.81 ganSladia. 7.82 .3

7.83 ganSladia. 7.84 ganSladia. 7.85 ganSladia. 7.86 ganSladia. 7.87 e2.

7.88 ganSladia. 7.89 .3

1 7.90 ).13(

2

5 5 7.91 .3

16 7.92 .22 7.93 .

Tavi VIII

8.1 0,8. 8.2 ).8105(3

4 8.3 27. 8.4 .

15

42arctg2 8.5 .

3

32 8.6 8ln2. 8.7 19,2. 8.8 .

3

16 8.9 1. 8.10 36. 8.11 12. 8.12 .

3

328.13 .

3

14

8.14 .3

168.15 .6ln6

2

117 8.18 1,35+ln2. 8.19 .

3

288.20 .

27

2224 8.21 .

22

22ln

8.22 2lnctg150. 8.23 6a. 8.24 .

3

13 8.25 ).32(8

8.26 .5

108 8.27 32. 8.28 .

5

95 8.29 .

2

2 8.30 ).1e(

4

4

8.31 .5

4 8.32 .

4

8.33 a) ,ab

3

4 2 b) .ba

3

4 2 8.34 .ab3

4 2 8.35 .ba3

28 2 8.36

).1e(4

2

Tavi X

10.8 xy–y=0. 10.9 y–2xy=0. 10.10 xy–ylny=0. 10.11 2yy–y2=0. 10.12 x+yy=0. 10.13 y=xy+y2. 10.14 .y

xlnyxy 10.15 y2–

x2=2xyy. 10.16 .ey y

yx

10.17 y2+y2=1. 10.18 yy2(lny–1)=y2(xy–y). 10.19 x3y–3x2y+6xy–6y=0. 10.20 (yy+y2)2=–y3y.

10.21 x2+y=xy. 10.22 xy+y=0. 10.23 2xyy=x2+y2. 10.24 2xyy–y2=2x3. 10.25 y=x3–x2+x+C; y=x3–x2+x+1. 10.26 y=Cx; y=4x.

10.27 y=Cex; y=4ex+2. 10.28 (1–x)(1+y)=C. 10.29 x2(1+y2)=C. 10.30 x2+y2=lnCx2. 10.31 y=Csinx. 10.32 .Cy1x1 22

10.33 ysinx+cosy–xcosx+sinx=C. 10.34 .Cex1 arctgy2 10.35 1+y2=C(1–x2). 10.36 .x3x3Cy3 2 10.37 .

y

1yCx

10.38 10x+10

-y=C. 10.39 .

2

xsin2C

4

ytgln 10.40 .ey 2

xtg 10.41 .e1Cy x2 10.42 .e1Cy x2 10.43 .

Cx

1xy

10.44

.2

yx

4ctgCx

10.45 8x+2y+1=2tg(4x+C).10.46 ;C)e1ln(

2

y x2

.e1

e1ln21y

x

x2

10.47 2y=Csin2x–1; .

2

1xsin2y 2 10.48

y=arccoseCx. 10.49 (1+ex)3tgy=8. 10.50. .1eysinln22)1x( 10.51. C)1y)(1x(ln2y2x2yx . 10.52

.0ycosysinxsin2 10.53 .Cy

xxln 10.54 .

x

Clnxy 10.55 .Cyln

y

x 10.56 .

C

ylnxy 10.57 .Cxln3xy 3 10.58

.x

Cln

x

ysin 10.59 .eCxln x

y

10.60 .Cex2x

2y

10.61 .x2

yCtgx 10.62 y2+2xy–x2=C. 10.63 x3=C(y2–x2); 3x3=8(x2–y2). 10.64

.xln2xy ;xln2Cxy 10.65 .x31

x2y ;

Cx1

x2y

22

10.66 ;Cey

yx2

.Ceyyx12

10.67 .Cxx

yxln

10.68

.82

1

37

.Cy

x3cosyln 10.69 3x+2y–4+2ln|x+y–1|=0. 10.70 x2+xy–y2–x+3y=C. 10.71 x2+2xy–y2–4x+8y=C. 10.72 y=Ce-2x+2x–1. 10.73

.2

xCey

2x2

10.74 .xeCxy 22 x

1

10.75 y=(x+C)(1+x2).10.76 ).xsinx(cos2

1Cey x 10.77 y2–2x=Cy3. 10.78

.4

1y

2

1y

2

1Cex 2y2 10.79 .

6

x

x

Cy

4

2 10.80 .1arctgyCex arctgy

10.81 .12

x

x

2y;1

2

x

x

Cy 10.82

.xlnxy;xlnxx

Cy 10.83 .

xcos

xy;

xcos

xCy

10.84 y=Ce-sinx+sinx–1; y=2e-sinx+sinx–1. 10.85 ;xx1Cy 2 .xx1y 2

10.86 y=arctgx–1+Ce-arctgx. 10.87 .Cx2sin4

1x

2

1

2

xtgy

10.88 .xlnx

2

1y 2 10.89 y=-cosx. 10.90 .1)1Ce(y

2x2 10.91

.x7

xCy 3

7

10.92 .xln

2

1Cxy

2

4

10.93 yn=Ce-x+x–1. 10.94 y3(Cecosx+3)=1. 10.95 y(Cx+lnx+1)=1. 10.96 ;

1x1C

1y

2

.

1x13

1y

2

10.97 .1xx12y;1xx1Cy 2222 10.98 y3(Cecosx+3)=1; 3y3(1–ecosx)=1. 10.99 .1e2

1ey

2

xx

10.100 .1x

xsecy

3 10.101 x3y–2x2y2+3y4=C. 10.102 x3ey–y=C. 10.103 x2+cos2y+y2=C. 10.104 .Cxy2cos

2

x2

10.105

.Cyx

yyxln

10.106 ;C

x

yyx .3

x

yyx 10.107 ;Cye

2

x yx2

.2ye2

x yx2

10.108 (1+x)siny+(1–y)sinx=C. 10.109

x2lny+2y(x+1)=C. 10.110 ex+y+x3+y4=1. 10.111 .Cyy1ln2

1yarctgyyxx1xarcsinx 222 10.112 xlny+y2cos5x=e2. 10.113

.Cexyy

xarctg y 10.114 x3y–cosx–sinx=C. 10.115 .01xxlnCy 2 10.116 .C

P

2Py;

P

1P2x 2

2 10.117

x=aP+bP2. 6y=3aP2+4bP3+C. 10.118 x=sinP+lnP, y=PsinP+cosP+P+C. 10.119 x=eP+P, .C2

P)1P(ey

2P 10.120 x=P3+P,

4y=3P4+2P2+C. 10.121 .P

1Plnx 10.122 .C1Pln

1P

2y.

1P

P2x 2

22

10.123 x=C–ln(ey–1). 10.124

).Cy1yy(arcsin2

1x 2 10.125 ,CP

2

3P2x 2 y=P2+P3. 10.126 ,C

P

2P2x y=P2+2lnP. 10.127 ,CPlnPln

2

1x 2

y=PlnP. 10.128 x=eP(P+1)+C, y=P2eP. 10.129 x=PsinP, y=(P2– –1)sinP+PcosP+C. 10.130 x=eP+C, y=eP(P–1), an y=(x–C)[ln(x–C)–

1]. 10.131 x=2(lnP––P), y=2P–P2+C. 10.132 .P1

Py ,CP1PP1P1lnx

2

22

10.133 x=2P+3P2, y=2P3+P2+C.

10.134 x=P(1+eP), y=0, 5P2+(P2–P+1)eP+C. 10.135 x=e2P(2P2– –2P+1), C)5,1P3 P3P2(ey 23P2 . 10.136 x=2arctgP+C,

y=ln(1+P2), y=0. 10.137 ,CPln1P1ln1P2x 22

.0y ,P1Py 2 10.138 x=PcosP, y=P2cosP– –PsinP–cosP+C.

10.139 .3

P

P

C2y,

3

P2

P

Cx

2

2 10.140 x=Ce-P+2(1–P), y=x(1+P)+P2. 10.141 ,

P

PlnCx

2

.

P

1)PlnC(

P

2y 10.142

),PcosPsinPC(P

1x

2 y=2xP+sinP. 10.143 Cy=(x–C)2; y=0 da y=–4x gansakuTrebuli amonaxsnebia. 10.144 y=Cx+Cx2;

.4

xy

2

10.145 ,C

1Cxy y2=4x. 10.146 ;C1aCxy 2 x2+y2=C2. 10.147 .x

8

27y ;

C2

1Cxy 23

2 10.148 y=Cx+C–C2;

.4

)1x(y

2 10.149 y=Cx–C+eC. 10.150 .xy ,

C2

1Cx

2

1y 2 10.151 ,

)P1(

C

2

1Px

2 ,

)P1(

CPP

2

1y

2

22

y=0, y=x+1.

10.152 x=CP–lnP–2, .PCP2

1y 2 10.153 C3=3(Cx–y), 9y2=4x3. 10.154 ,CPlnPx ),CPln4(Py y=0. 10.155

wrfivad damoukidebelia. 10.156 wrfivad damoukidebulia. 10.157 wrfivad damoukidebelia. 10.158 wrfivad damokidebulia. 10.159 wrfivad damokidebulia. 10.160 wrfivad damokidebulia. 10.161 wrfivad

damokidebulia. 10.162 wrfivad damokidebulia. 10.163 wrfivad damoukidebelia. 10.164 x2y–2xy+ +2y=0

(miTiTeba. saZiebeli gantolebis zogadi amonaxsni iqneba: y=C1x+C2x2, aqedan y=C1+2C2x, y=2C2 miRebuli

sistemidan C1 da C2 mudmivebis gamoricxva mogvcems saZiebel gantolebas). 10.165 sin2xy–2cos2xy=0. 10.166

y+y=0. 10.167 y–2y′+y=0. 10.168 y–y=0. 10.169 y–y=0. 10.170 y+y′=0. 10.171 y–2y+y–2y=0. 10.172

.x

xcosC

x

xsinCy 21 10.173 .

x

CxCy

4

21 10.174 y=C1x+C2lnx. 10.175 y=C1sinx+C2sin

2x. 10.176 y=C2+(C1–C2x)ctgx. 10.177 y(1–

x)=C1x+C2(1–x2+2xlnx). 10.178 y=C1ex+C2x–x2–1. 10.179 y=C1cosx+C2xcos2x–sinxcosx. 10.180 .xarctgx

2xx32y 2

38

10.181 y=C1ex+C2e

-2x. 10.182 y=C1e4x+C2. 10.183 y=e-3x(C1cos2x+C2sin2x). 10.184 y=ex(C1+C2x). 10.185 y=4ex+2e3x. 10.186 y=3e-

2xsin5x. 10.187 .1a

eaxsinCaxcosCy

2

x

21

10.188 .74

xcos7xsin5eCeCy x

2x6

1

10.189 y=e-x(C1cos2x+C2sin2x)–

.x2sin2x2cos2

1 10.190 ,yeCeCy x2

2x

1 1. .4

15x

2

21x

2

9xy 23 2. .

2

xsin2

2

xcose

5

8y x

3. ).xx2(ey 2x 4.

).x2simx2cos39(4

1x

2

3y 10.191 y=ex+x2. 10.192 y=ex(ex–x2–x+1). 10.193 .xcosxsin

3

1x2sin

3

1y