ludoM edoK

.TAM FKT 02 1- 02

YNU kinkeT satlukaF fitomotO kinkeT nakididneP nasuruJ

ISAISNEREFID ISGNUF

V h

r V V

2 = r nad h = r2

r h

: nusuyneP

.T.M ,.dP.M ,ibutraM

)4 PS( naraggnagneP nad margorP nanusuyneP naanacnereP metsiS fitomotO kinkeT nakididneP nasuruJ

002 5

2

P ATAK RATNAGNE

luduj nagned ludoM isaisnerefiD isgnuF iagabes nakanugid ini

utas halas kutnebmem kutnu hailuk nataigek malad naudnap bus -

:utiay ,isnetepmok “ rabajla isalupinam nad ,tafis ,pesnok nakanuggneM

halasam nahacemep malad isaisnerefid gnuf is “. tapad ini ludoM

id akitametaM hailuk atresep aumes kutnu nakanugid S retseme I adap

idutS margorP nakididneP satisrevinU kinkeT satlukaF fitomotO kinkeT

.atrakaygoY iregeN

nakijasid ini ludom adaP rasad pesnok isaisnerefiD isgnuF nad

lasamrep malad iapmujid kaynab gnay aynnaha gnadib id aynnaparenep

.sitkarp nupuam sitiroet araces kiab ,kinket sata iridret ini ludoM tapme

.rajaleb nataigek :gnatnet sahabmem 1 rajaleb nataigeK isaisnerefiD

A isgnuF .rabajl :gnatnet sahabmem 2 rajaleb nataigeK isaisnerefiD

isgnuF - nednesnarT isgnuf . :gnatnet sahabmem 3 rajaleb nataigeK

isaisnerefiD .laisraP laisnerefiD nad kirtemaraP naamasreP ,kimtiragoL

rajaleb nataigeK 4 :gnatnet sahabmem isgnuF isaisnerefiD isakilpA

irajalepmem tapad kutnU awsisaham hadum nagned ini ludom

gnatnet namahamep nad nauhategnep iaynupmem halet nakparahid

pesnok - gnay rasad pesnok pesnok amaturet ini lah malad ,ayngnajnunem

gnatnet isgnuF rabajlA nad isgnuF - nednesnarT isgnuf irtemoeG nad .

,atrakaygoY rebotkO 5002

nusuyneP

.T.M ,.dP.M ,ibutraM

3

LUDOM ISI RATFAD

namalaH

LUPMAS NAMALAH ........................................................................... . 1 RATNAGNEP ATAK .......................................... ................................... 2

ISI RATFAD .......................................................................................... 3 YRASSOLG / NAHALITSIREP .... . ......................................................... 5

NAULUHADNEP . I .... ............................................................................. 7

ispirkseD .A ......... ........................................................................ . ....... 7

taraysarP .B ................................................ ......................................... 7

ludoM naanuggneP kujnuteP .C .......................................................... 8

awsisaham igab kujnuteP .1 .......................................................... 8

.2 ujnuteP igab k nesod ............ ...................................................... 8

rihkA naujuT .D ............................................................................ . ..... 9

.E isnetepmoK . ... ........................................ ..................................... . .. 9

............................................................................ naupmameK keC .F 11

NARAJALEBMEP .II ...................................................................... . .... . .. 21

R .A awsisahaM rajaleB anacne ......................................................... 21

rajaleB nataigeK .B ............................................................................. 21

1 rajaleB nataigeK .1 ............. ................... ..................................... 21

nataigek naujuT .a rajaleb 1 ...................... . ........................ .... .. . 21

iretam naiarU .b ... ..................................................... . ................ . 31

1 namukgnaR .c ............... ...................................... ................... .. 18

1 saguT .d .................................................................................. . 19

1 fitamrof seT .e .................................................... ..................... . 19

bawaj icnuK .f set 1 fitamrof .. . ................................................... . 02

4

namalaH

2 rajaleB nataigeK .2 ........................................................... .......... . 02

2 rajaleb nataigek naujuT .a . .......... ... ......................................... 02

2 iretam naiarU .b ....................................................................... 12

2 namukgnaR .c ......................................................................... . 82

2 saguT .d ............ ...................................................................... . 03

fitamrof seT .e 2 ... ...................................................................... . 13

bawaj icnuK .f set fitamrof 2 ............... ...................................... .. 23

3 rajaleB nataigeK . 3 ........................................................... ......... 23

rajaleb nataigek naujuT .a 3 ....................................................... 23

iretam naiarU .b 3 ..................................... .................................. 33

namukgnaR .c 3 ......................................................................... 39

saguT .d 3 .................................................................................. 04

fitamrof seT .e ... 3 ..... ................................................................. 14

bawaj icnuK .f set fitamrof 3 ............... ...................................... 41

4 rajaleB nataigeK . 4 ........................................................... .... ..... 24

rajaleb nataigek naujuT .a 4 ....................................................... 24

iretam naiarU .b 4 ....................................................................... 34

namukgnaR .c 4 .................................................. ....................... 85

.................................................................................. 4 saguT .d 06

......................................................................... 4 fitamrof seT .e 16

bawaj icnuK .f set fitamrof 4 ....... ........ ...................................... 26

ISAULAVE .III .............................................................................. ...... .. 36

naaynatreP .A ............................................................................ .... .. 36

nabawaJ icnuK .B ............................................................................ . 46

nasululeK airetirK .C ............................................................. . ........ .. 56

EP .VI N PUTU .................. ..................................................................... . 66

AKATSUP RATFAD ............................................................................ . 67

5

/ NAHALITSIREP YRASSOLG

evitavireD / isaisnerefiD isgnuF halada : l isgnuf nakapurem gnay nia

.natukgnasreb gnay isgnuf irad nanurut

laisraP lasaisnerefiD halada : padahret isgnuf haubes irad laisnerefid

aynnial lebairav paggnagnem nagned aynlebairav utas halas

.hibel uata lebairav aud iaynupmem uti isgnuf akij natsnok

iD kimtiragoL isaisneref halada : isgnuf haubes naklaisnerefidnem arac

tafis nakrasadreb - .amtiragol isgnuf nanurut nad tafis

isgnuF rabajlA gnay isgnuf haubes halada : nagnubuh nakkujnunem

hasipret araces hibel uata lebairav haub aud irad rabajla araces .

laisnenopskE isgnuF : nakapurem gnay isgnuf haubes halada

nial isgnuf/lebairav utaus nagned nagnalib utaus natakgnaprep

utnetret .

kilobrepiH isgnuF : narasas iaynupmem gnay isgnuf haubes halada

alobrepih utaus malad id lebairav haubes .

isgnuF isilpmI t gnay isgnuf haubes halada : nagnubuh nakkujnunem

nakhasipid ulrep kadit/tapad kadit gnay ) tarisret ( tisilpmi araces

.adebreb gnay saur malad

amtiragoL isgnuF : amtiragol nakapurem gnay isgnuf haubes halada

laisnenopske isgnuf haubes irad .

6

isgnuF aM kumej gnay isgnuf haubes halada : irad isgnuf nakapurem

.aynnial isgnuf

irtemonogirT isgnuF : nakutnetid aynagrah gnay isgnuf haubes halada

narakgnil utaus malad id utnetret tudus utaus agrah helo .

neidarG : k nakkujnunem kutnu halitsi haubes halada utaus nagnirime

utnetret sirag .

kirtemaraP naamasreP : lebairav haub aud nagnubuh haubes halada

aynaratnarep/retemarap nakapurem gnay agitek lebairav iulalem .

7

I BAB NAULUHADNEP

ispirkseD .A ludoM luduj nagned snerefiD isgnuF isai gnatnet sahabmem ini

rasad pesnok isaisnerefiD aynnahalasamrep atres isgnuF gnay

araces kiab ,kinket gnadib id aynnaparenep malad iapmujid kaynab

.sitkarp nupuam sitiroet isaisnerefiD :pukacnem irajalepid gnay iretaM

rabajlA isgnuF , erefiD isaisn isgnuF - nednesnarT isgnuf , isaisnerefiD

atres laisraP laaisnerefiD nad kirtemaraP naamasreP ,kimtiragoL

isgnuF isaisnerefiD isakilpA

sata iridret ini ludoM tapme .rajaleb nataigek rajaleb nataigeK 1

:gnatnet sahabmem erefiD isaisn .rabajlA isgnuF 2 rajaleb nataigeK

:gnatnet sahabmem isaisnerefiD isgnuF - nataigeK .nednesnarT isgnuf

:gnatnet sahabmem 3 rajaleb isaisnerefiD naamasreP ,kimtiragoL

.laisraP laisnerefiD nad kirtemaraP rajaleb nataigeK 4 sahabmem

:gnatnet isakilpA isgnuF isaisnerefiD

laos hotnoc nagned ipakgnelid ulales rajaleb nataigek paites adaP

nahital atreseb aynnasahabmep nad - kutnu aynulrepes nahital

.nakparahid gnay isnetepmok iapacnem malad awsisaham utnabmem

iaseles haleteS nakparahid awsisaham ini ludom irajalepmem

isnetepmok bus iaynupmem “ nad ,tafis ,pesnok nakanuggneM

halasam nahacemep malad rabajla isalupinam isaisnerefid “isgnuf

taraysarP .B iretam isireb ini ludoM - iretam nagnukud nakulremem gnay tam ire

gnay nial iretam nupadA .aynmulebes irajalepid halet aynitsemes -

let aynsurahes gnay rasad iretam hailuk atresep helo imahafid ha id

pesnok halada amaturet fitomotO kinkeT nakididneP nasuruJ rasad

: gnatnet isgnuF rabajlA nad isgnuF - nednesnarT isgnuf d irtemoeG na .

8

ludoM naanuggneP kujnuteP .C awsisahaM igab kujnuteP .1

malad akam ,lamiskam gnay rajaleb lisah helorepid ragA

ulrep gnay rudesorp aparebeb ada ini ludom nakanuggnem

: nial aratna nakanaskalid nad ,nakitahrepid

halacaB .a pesnok naiaru amaskes nagned imahaf nad - pesnok

alup imahaf naidumek ,ini ludom adap nakijasid gnay sitiroet

pesnok naparenep - hotnoc malad tubesret pesnok - laos hotnoc

iretam ada hisam askapret aliB .aynnaiaseleynep arac atreseb

muleb nad salej gnaruk gnay d imahafid asib ne arap kiab nag

upmagnem gnay nesod adapek nakaynanem tapad awsisaham

.nahailukrep nataigek

,iridnam araces )nahital laos( fitamrof sagut paites nakajrek aboC .b

raseb aparebes iuhategnem kutnu nakduskamid ini lah

imid halet gnay namahamep padahret awsisaham paites ikil

iretam - .rajaleb nataigek paites adap sahabid gnay iretam

iretam iasaugnem muleb awsisaham aynnaataynek malad alibapA .c

nad acabmem igal ignalu aboc ,nakparahid gnay level adap

nahital igal nakajregnem - ep ualak nad aynnahital halaynatreb ulr

gnay nahailukrep nataigek upmagnem gnay nesod adapek

nakulremem natukgnasreb gnay iretam ualaK .natukgnasreb

taraysarp awhab naknikay akam )taraysarp( lawa namahamep

raneb duskamid gnay - .ihunepid hadus raneb

nesoD igaB kujnuteP .2 nad sagut iaynupmem nesod ,nahailukrep nataigek paites malaD

: kutnu narep

a. .rajaleb sesorp nakanacnerem malad awsisaham utnabmeM

b. sagut iulalem awsisaham gnibmibmeM - nahital uata sagut - nahital

.rajaleb bahat malad naksalejid gnay

9

c. lad awsisaham utnabmeM nad urab pesnok imahamem ma

.nakulrepid alibapa awsisaham naaynatrep bawajnem

d. gnay nial rajaleb rebmus seskagnem kutnu awsisaham utnabmeM

.nakulrepid

e. .nakulrepid akij kopmolek rajaleb nataigek risinagrogneM

f. d akij gnipmadnep nesod/ilha gnaroes nakanacnereM .nakulrepi

g. isnetepmok naiapacnep padahret isaulave nakadagneM

.nakutnetid halet gnay awsisaham naanaskalep tubesret isaulavE -

.rajaleb nataigek rihka paites adap ayn

rihkA naujuT .D

m malad rajaleb nataigek iretam hurules irajalepmem haleteS ludo

nakparahid awsisaham ini tapad : “ nad ,tafis ,pesnok nakanuggneM

halasam nahacemep malad rabajla isalupinam isaisnerefid “isgnuf .

isnetepmoK .E ludoM FKT .TAM 02 1- luduj nagned 20 D isaisnerefi isgnuF ini

ebmem akgnar malad nususid kutn bus - isnetepmok “ nakanuggneM

halasam nahacemep malad rabajla isalupinam nad ,tafis ,pesnok

isaisnerefid “isgnuf .

iapacnem kutnU bus - surah uluhad hibelret ,tubesret isnetepmok

bus iapacid tapad - aynajrek kujnu airetirk atreseb isnetepmok bus

alem iagabes narajalebmep kokop iretam nagned rajaleb pukgnil iul

tukireb :

01

buSisnetepmoK

airetirK ajreK kujnU

pukgniLrajaleB

narajalebmeP kokoP iretaM

pakiS nauhategneP nalipmarteK

anuggneM -

,pesnok nak nad naruta

isalupinam rabajla

malad nahacemep

halasam isaisnerefid

isgnuf .

naksalejneM.1

nok/naitregnep - nad isaton pes

tafis - efid tafis -. isgnuf isaisner

nakiaseleyneM .2efid halasam ner - isgnuf isais

.rabajla nakiaseleyneM .3

efid halasam ner - isgnuf isais

kumejam nakiaseleyneM .4

alasam h efid ner - isgnuf isais

.tisilpmi nakiaseleyneM .5

efid halasam ner - ogirt isgnuf isais -

irtemon nad.aynsrevni

nakiaseleyneM .6efid halasam ner -

isgnuf isaiskilobrepih nad.aynsrevni

nakiaseleyneM .7efid halasam ner -

isgnuf isais.laisnenopske

neM .8 nakiaseleyefid halasam -

isgnuf isaisner.amtiragol

nakiaseleyneM .9efid halasam ner -

tiragol isaisner - kim nad asrep -

naam ap kirtemar .01 nakiaseleyneM

nerefid halasam -.laisrap laisais nakpareneM .11

nerefid pesnokasam adap isais

sirtemoeg hal nad halasam

/mumiskam.isgnuf muminim

12 nakpareneM .nerefid pesnok -

kutnu laisrap lais gnutihgnem rep -

utaus nahabu.isgnuf

,naitregneP.1

nad isaton tafis - tafis

isaisnerefid . isgnuf

22 . efiD ner isais isgnuf.rabajla

.3 efiD ner isais

sgnuf i kumejam

.4 efiD ner isais

isgnuf .tisilpmi

efiD .5 ner isais

isgnuf ogirt - nad irtemon

aynsrevni efiD .6 ner isais

isgnuf repih - nad kilob

aynsrevni efiD .7 ner isais

isgnuf ske - laisnenop

isaisnerefiD .8

isgnuf iragol .amt

isaisnerefiD .9

kimtiragol nad asrep -

naam ap ar -kirtem

.01 D nerefi lais laisrap

napareneP .11

pesnok isaisnerefid

isgnuf 21 napareneP .

pesnok laisnerefid

laisrap

nad itileT

tamrec malad

silunem lobmis

nadalem uk -

rep nak -nagnutih

,naitregneP.1

nad isaton tafis - tafis

isaisnerefid . isgnuf

22 . efiD ner isais isgnuf.rabajla

.3 efiD ner isais

isgnuf kumejam

.4 efiD ner isais

isgnuf .tisilpmi

efiD .5 ner isais

isgnuf ogirt - nad irtemon

aynsrevni efiD .6 ner isais

isgnuf repih - nad kilob

aynsrevni efiD .7 ner isais

isgnuf ske - laisnenop

isaisnerefiD .8

isgnuf .amtiragol

isaisnerefiD .9

kimtiragol nad asrep -

naam ap ar -kirtem

.01 D nerefi lais laisrap

napareneP .11

k pesnod snerefi isai

isgnuf

napareneP .21k pesno laisnerefid

laisrap

gnutihgneM

nagned rudesorp lisah nad

raneb gnay

11

F . naupmameK keC

irajalepmem mulebeS ludoM 102 FKT .TAM – nagned halisi ,ini 20

( kec adnat ynatrep ) halet gnay isnetepmok nakkujnunem gnay naa

: nakbawajgnuggnatrepid tapad nad rujuj nagned awsisaham ikilimid

buS

isnetepmoK naaynatreP

nabawaJ “aY“ nabawaJ aliB nakajreK aY kadiT

anuggneM -

,pesnok nak nad naruta

isalupinam rabajla

maladnahacemep

halasam isaisnerefid

isgnuf .

/naitregnep naksalejnem upmam ayaS .1

tafis nad isaton pesnok - nerefid tafis -.isgnuf isais

1 fitamroF seT

1 : romoN

asamrep nakiaseleynem tapad ayaS .2 - .rabajla isgnuf isaisnerefid nahal

roF seT 1 fitam

e,c,b,a :2 : romoN

asamrep nakiaseleynem tapad ayaS .3 - . kumejam isgnuf isaisnerefid nahal

1 fitamroF seT

f : 2 : romoN

asamrep nakiaseleynem tapad ayaS .4 - .tisilpmi isgnuf isaisnerefid nahal

itamroF seT 1 f

d : 2 : romoN asamrep nakiaseleynem tapad ayaS .5 -

isgnuf isaisnerefid nahal irtemonogirt .aynsrevni nad

2 fitamroF seT

01,9,4,3,2,1 : oN

asamrep nakiaseleynem tapad ayaS .6 - isgnuf isaisnerefid nahal ilobrepih nad k .aynsrevni

2 fitamroF seT

31,5 : romoN

asamrep nakiaseleynem tapad ayaS .7 - .laisnenopske isgnuf isaisnerefid nahal

2 fitamroF seT

41,21,11,6 : romoN

asamrep nakiaseleynem tapad ayaS .8 - uf isaisnerefid nahal .amtiragol isgn

2 fitamroF seT

61,51,8,7 : romoN .9 asamrep nakiaseleynem tapad ayaS - nad kimtiragol isaisnerefid nahal .kirtemarap naamasrep

3 fitamroF seT

2,1 : romoN

asamrep nakiaseleynem tapad ayaS .01 - nahal efid ner laisrap lais

3 fitamroF seT

3 : romoN ad ayaS .11 pesnok nakparenem tap

halasam adap isgnuf isaisnerefid muminim / mumiskam nad sirtemoeg

.isgnuf

4 fitamroF seT

3,2,1 : romoN

pesnok nakparenem tapad ayaS .21srap laisnerefid malad lai

gnay nahalasamrep nakiaseleynem .naveler

4 fitamroF seT

5,4 : romoN

bawajnem awsisaham alibapA kadiT ini ludom irajalep akam

bawajid gnay iretam iauses kadiT .tubesret

21

II BAB NARAJALEBMEP

awsisahaM rajaleB anacneR .A hawab id lebat isignem nagned rajaleb nataigek anacner haltauB

.iaseles haletes nesod adapek rajaleb itkub halatnim nad ini

nataigeK sineJ laggnaT utkaW tapmeTrajaleB

nasalAnahabureP

faraPnesoD

isaton nad ,naitregneP .1isgnuf isaisnerefid .

.2 .rabajla isgnuf isaisnerefiD

.3 isgnuf isaisnerefiD kumejam

.4 tisilpmi isgnuf isaisnerefiD .5 isgnuf isaisnerefiD onogirt -

.aynsrevni nad irtem

.6 kilobrepih isgnuf isaisnerefiD aynsrevni nad .

.7 opske isgnuf isaisnerefiD - .laisnen

.9 .amtiragol isgnuf isaisnerefiD .01 nad kimtiragol isisnerefiD

kirtemarap naamsrep

.11 laisrap laisnerefiD

isakilpA .21 isgnuf isaisnerefid

isakilpA .31 laisrap laisnerefid

.rajaleB nataigeK .B

: 1 rajaleB nataigeK .1 rabajlA isgnuF isaisnerefiD

: 1 rajaleB nataigeK naujuT .a 1 .) d awsisahaM tapa nahalasamrep nakiaseleynem isaisnerefid

isgnuf .

31

2 .) nahalasamrep nakiaseleynem tapad awsisahaM isaisnerefid

.rabajla isgnuf

3 .) nahalasamrep nakiaseleynem tapad awsisahaM isaisnerefid

isgnuf kumejam .

4 .) nahalasamrep nakiaseleynem tapad awsisahaM isaisnerefid

f isgnu tisilpmi .

: 1 iretaM naiarU .b isaisnerefiD isatoN nad ,naitregneP .)1 isgnuF .

)X( f isgnuf utaus ) nanuruT / evitavireD ( isaisnerefeD

f “ : acab ( )X( ‘ f nial isgnuf halada aynialin gnay ) ” X neska

gnarabmes adap : halada X nagnalib

( f )h+X – f )X( timil = )X( ’f -

h h 0

timil akiJ .ada gnamem tubesret timil agrah natatac nagned

da gnamem tubesret akam ,ada gnamem tubesret timil akiJ .a

f awhab nakatakid naknurutret )X( nakisaisnerefedret / iredret / -

.X id nakevitav

akiJ Y f = )X( tapad X padahret Y amatrep nanurut akam ,

Yd fd utid sil )X( ' f = ‘ Y : zinbieL isaton nagned uata ----- uata

Xd Xd

nanurut kutnu ayntujnaleS laggnit aynsuretes nad audek eynem -

d2Y utiay ,nakiaus uata ”Y aynsuretes nad . Xd 2

F isaisnerefiD pesnoK napareneP hotnoC isgnu .

iagabes isgnuf isaisnerefid pesnok salejrepmem kutnU -

nakirebid ini tukireb akam ,sata id naktubes id halet anam

.aynaparenep hotnoc aparebeb

41

.)a X31 = )X(f akiJ – f halnakutnet 6 ' )4(

bawaJ X31 = )X( f : – 6

)h + X(f – )X( f timil = )X(' f -

h h 0

)h + 4( f – )4( f timil = )4( ' f -

h h 0

)h + 4( 31 – 6 – 6 + )4( 31 timil = -

h h 0

h .31 timil = 31 timil = = 31

h h h 0 0

.)b f haliraC ' X4 = ) X( f iuhatekid akij )C( 2 – 8 + X7

bawaJ X4 = )X( f : 2 – 8 + X7

) h + X( f – )X( f timil = )X( ' f -

h 0 h

) h + C ( f – )C( f timil = )C( ' f -

h h 0

) h + C (4 2 – 8 + ) h + C( 7 – C4( 2 – ) 8 + C7 timil = -

h h 0

C4 2 h4 + hC8 + 2 – 8 + C7 – C4 2 C7+ – 8 timil = -

h h 0

C8 ( timil = – 7 = ) h4 + C 8 - 7 h 0

1 )c = )X(f akij )X(' f halnakutneT . -

X

1 bawaJ = )X(f : -

X

51

)h+X(f – )X(f timil = )X(' f -

h h 0

1 1 1 timil = )X(' f – -

h h + X h 0 X

X 1 – ) h+X( = timil -

h h 0 X . ) h+X(

– 1 timil = -

h X 0 2 X + h

– 1 = = – X–2

X2 ====

sumuR – rabajlA isgnuF nanuruT rasaD sumuR ac iulalem isgnuf haubes nanurunep pesnok nakrasadreB ar

alib akam ,)ayntimil gnutihgnem( sataid nakiaruid halet gnay

: tukireb iagabes rasad sumur ukalreb ataynret nakitahrepid

)X(' f = ' Y nad )X( h = V ; )X( g = U ; )X( f = Y akiJ

liir nagnalib = n ; atnatsnok = C

1. 0 = 'Y C = Y

2. XC = Y C = 'Y

3. X = Y n X n = 'Y n - 1

4. Xk = Y n X nk = 'Y n - 1

5. 'V + 'U = 'Y V + U = Y

6. U = Y – V 'U = 'Y – 'V

7. 'VU + V'U = 'Y V . U = Y

U V'U – 'VU 8. = Y = 'Y -

V V2

61

hotnoC :

.)a 5 = Y akiJ Y akam ’ = 0

.)b X 6 = Y akiJ Y akam ’ = 6

.)c X = Y akiJ 7 Y akam ’ = X7 6 .)d X8 = Y akiJ 4 Y akam ’ = X23 3

.)e X6 = Y akiJ 3 X7 + 2 Y akam ’ = X81 2 X 41 +

)f X5 = Y akiJ . 4 – X3 3 Y akam ’ = X02 3 – X9 2

)g X3 ( = Y akiJ . 2 – akam ) 6 + X5( ) X4

Y ’ X6 ( = – X3 ( + ) 6 + X5 ( ) 4 2 – )5( ) X4

X03 = 2 X61 + – X51 + 42 2 – X02

= X 54 2 – X4 – 42

X3 – X ( 3 5 2 ) 7 + – X3 ( X2 – ) 5 )h = Y akiJ . - Y akam ’ = - X2 7 + X ( 2 ) 7 + 2

– X3 2 12 + X01 +

= - X 4 X41 + 2 94 +

) nanuruT iatnaR lilaD ( kumejaM isgnuF nanuruT

U ( f = Y akiJ : X padahret Y nanurut akam ) X ( g = U nad ) takgnit nupapareb / gnarabmes kutnu ukalreb ini pisnirP

.naknurut id gnay isgnuf nakumejamek

hotnoC :

X3 ( = Y .)a ) 4 01

X3 ( 01 = ’Y – ) 4 9 = 3 . X3 ( 03 – ) 4 9

= Y .)b X √ 2 X4 + X ( = 7 2 X4 + ) 7 ½

Yd X ( ½.) 4 + X2 ( = 2 X4 + ) 7 -- ½ = X (.) 2+X ( 2 X4+ – ) 7 -- ½

Xd

Ud Yd Yd = . Y uata ’ Y = ) X ( ’ U . ) U ( ’ ) X ( Xd Xd Ud

71

tisilpmI ) isgnuF isaisnerefiD ( nanuruT

X = Y akiJ 2 aynhunepes nakisinifedret Y ; 6 + X5 + ,X helo

Y akam ada ipateT .X irad ”tisilpske“ isgnuf iagabes tubesid

X nagned Y nakhasimem )ulrep kadit( tapad kadit atik aynalak

malad sumur gnay adebreb aynlasim adap kutneb :

X2 Y3 + YX2 + 2 acames lah malaD .4 = iagabes Y nakatakid ini m

)X( f = Y kutneb malad nagnubuh anerak ,X irad tisilpmi isgnuf

.aynmaladid tarisret ipatet gnusgnal araces kapmat kadit

Yd hotnoC halnakutneT : isgnuf irad - .ini tukireb isgnuf

Xd

.)a X2 Y + 2 001 =

)b X . 2 Y + 2 X2 0 = 5 + Y6

bawaJ : )a X . 2 Y + 2 001 =

Yd Y2 + X2 0 =

Xd

Yd Y2 = X2

Xd

Yd X2 X = = -

Xd Y2 Y

.)b X2 Y + 2 – X2 – 0 = 5 + X6

Yd Yd Y2 + X2 – 2 – 6 0 =

Xd Xd

Yd Y2 ( – ) 6 2 = – X2

Xd

2 Yd – X2 1 – X = = - Y2 Xd – 6 Y – 3

81

.c namukgnaR : 1

.)1 isgnuF isaisnerefiD isatoN nad naitregneP .

)X( f isgnuf utaus ) nanuruT / evitavireD ( isaisnerefeD

aynialin gnay ) ” X neska f “ : acab ( )X( ‘ f nial isgnuf halada

bmes adap : halada X nagnalib gnara

)h+X(f – )X(f timil = )X( ’f -

h h 0

akiJ Y f = )X( tapad X padahret Y amatrep nanurut akam ,

Yd fd utid sil )X( ' f = ‘ Y :zinbieL isaton nagned uata ----- uata

d X Xd

.)2 isaisnerefiD isgnuF rabajlA

)X(' f = ' Y nad )X( h = V ; )X( g = U ; )X( f = Y akiJ gnades

liir nagnalib = n ; atnatsnok = C : rasad sumur ukalreb akam

.)a 0 = 'Y C = Y

.)b XC = Y C = 'Y

.)c X = Y n X n = 'Y n - 1

.)d Xk = Y n X nk = 'Y n - 1

.)e 'V + 'U = 'Y V + U = Y

.)f U = Y – V 'U = 'Y – 'V

.)g 'VU + V'U = 'Y V . U = Y

U V'U – 'VU .)h = Y = 'Y -

V V2

91

.)3 isaisnerefiD uF tisilpmI isgnuF nad kumejaM isgn : X padahret Y nanurut akam ) X ( g = U nad ) U ( f = Y akiJ

nakumejamek takgnit gnarabmes kutnu ukalreb ini pisnirP .

akiJ nerak ,X irad tisilpmi isgnuf iagabes Y nagnubuh a

gnal araces kapmat kadit )X( f = Y kutneb malad ipatet gnus

nagned iracid X padahret Y nanurut akam aynmaladid tarisret

.X padahret tubesret naamasrep ukus aumes naknurunem

.d 1 saguT :

halnakutneT ut nanur isgnuf irad - : ini hawab id isgnuf

X = Y .)1 5 – X4 3 X3 + 2 – 01 + X6

3 X3 – 2 = Y .)2 + X – 3 X ⅔ 7 = Y .) -

X 3 – X2

X3( = Y .)3 2 X4 + – X5(.)3 – )7 8 X ( 5 = Y .) 2 – ) 2 + X3 3

4 4 ( = Y .) – ) X6 7 ( + – ) 4 + X7 – 6 X ( .)9 2 Y3 + YX2 + 2 4 = )

5 X ( .) –Y )3 – 0 = ) Y + X ( 3 X .)01 3 Y + 3 YX4 + 2 5 =

6 X .) 3 Y + 3 – X4 2 8 = Y6 + X5 + Y

.e fitamrof seT : 1 .)1 J naksale nagned pakgnel isgnuf isaisnerefid naitregnep

! aynisaton

.)2 kutneT halna )nanurut( ‘ Y isgnuf irad – ! ini tukireb isgnuf 5 X + 6

a = Y .) 2X4 – 6X3 + 7X2 – 9 1+X 2 = Y .)e - 1 – 4 X

7 b .) = Y 4 X – + ½5 X – f 8 .) = Y 7 X( 2 2+ X – 3)5 X

c =Y .) (7X2 – 3X 5+ ( ) 2 X – 4 )

d .) 4X3 + 2Y3 + 5 YX 2 – 3X2 Y – 7 X – 6 = Y 9

Ud Yd Yd = . Y uata ’ Y = ) X ( ’ U . ) U ( ’ ) X ( Xd Xd Ud

02

.f fitamroF seT bawaJ icnuK : 1 .)1 ruT/evitavireD( isaisnerefeD halada )X( f isgnuf utaus )nanu

X nagnalib gnarabmes adap aynialin gnay )X( ‘ f nial isgnuf

: halada

)h+X(f – )X(f timil = )X( ’f -

h h 0

akiJ Y f = )X( tapad X padahret Y amatrep nanurut akam ,

Yd fd utid sil )X( ' f = ‘ Y ta :zinbieL isaton nagned ua ----- uata

Xd Xd

‘ Y halada X padahret Y amatrep nanuruT .)2

a .) 8 =‘ Y X3 – 81 X2 + 41 X – 9

b .) 4 =‘ Y X–½ + 7X–2 + ½5

24 =‘ Y .)c X2 – 86 X 22+

21 X2 – YX6 + Y5 2 – 7 =‘ Y .)d -

3 X2 – YX01 – +Y6 6

92 .)e =‘ Y -

1 ( – ) X4 2

f .) = ’ Y X( 2 X2 + – 3)4 X07( 2 )07+

2 . rajaleB nataigeK 2 : isgnuF isaisnerefiD – nednesnarT isgnuf

rajaleB nataigeK naujuT .a 2 : M .)1 tapad awsisaha nahalasamrep nakiaseleynem isaisnerefid

isgnuf ad irtemonogirt aynsrevni n .

.)2 nahalasamrep nakiaseleynem tapad awsisahaM isaisnerefid

isgnuf aynsrevni nad kilobrepih .

12

.)3 nahalasamrep nakiaseleynem tapad awsisahaM isaisnerefid

isgnuf laisnenopske .

.)4 nahalasamrep nakiaseleynem tapad awsisahaM isaisnerefid

isgnuf amtiragol .

.)5 nahalasamrep nakiaseleynem tapad awsisahaM isaisnerefid

.kirtemarap naamasrep nad kimtiragol

isgnuF isaisnerefiD : 2 iretaM naiarU .b – nednesnarT isgnuf isgnuF isaisnerefiD irtemonogirT

pesnok nagned iauseS isaisnerefid nautnenep rasad

nakiaruid halet anamiagabes isgnuf haubes )nanurut(

sumur helorepid irtemonogirt maladid akam , aynmulebes - sumur

tukireb iagabes irtemonogirt isgnuf nanurut rasad :

1) X nis = Y . ’Y X soc =

2) soc = Y . X ’Y = X nis

3) X gt = Y . ’Y ces = 2 X

4) X gtoc = Y . ’Y = cesoc 2 X

5) X ces = Y . ’Y X gt . X ces =

6) X cesoc = Y . ’Y = gtoc . X cesoc X

ulrep gnay aynranebes ,tubesret rasad sumur maneek iraD

romon nad 1 romon sumur halaynah amatu naitahrep tapadnem

irad naknurutid tapad aynsuretes nad 3 romon kutnu babes , 2

aratna nagnubuh tagnignem nagned utiay ,2 nad 1 romon

isgnuf - esret isgnuf , tub aynlasim :

1 X soc X nis 1 = X gt ; = X gtoc - ; = X ces ; nad = X cesoc ----- -

X nis X soc X soc X nis

22

nalaosrep naiaseleynep kutnu ayntujnaleS - gnay nalaosrep

rabajla isarepo iagabreb naktabilem gnay skelpmok hibel

isgnuf nad naigabmep ,nailakrep ,nagnarugnep ,nahalmujnep (

sumur akam ,) kumejam - II baB adap tapadret gnay sumur

nakanugid tapad patet ) rabajlA isgnuF isaisnerefiD (

hotnoC :

X nis 3 + X soc = Y akiJ .)1 akam , X gt 4

Yd - = = ’ Y X soc 3 + X nis ces 4 2 X

Xd

X5 ( nis = Y .)2 3 X3 2 X4 + ) 2

X51 ( = ’ Y 2 X5 ( soc . ) 4 + X6 3 X3 2 ) 2 + X4 +

nis = Y .)3 5 X

nis 5 = 4 X soc . X

X gt . X3 soc = Y .)4

Yd - = X2 gt . X3 nis 3 ces2 .X3 soc + 2 X2

Xd

= – ces .X3 soc 2 + X2 gt. X3 nis 3 2 X2

X5 nis = Y .)5 -

1 + X4

) 1 + X4 ( . X5 soc 5 – X5 nis 4 = ‘ Y -

) 1 + X4 ( 2

X5 soc. ) 5 + X02 ( X5 nis 4 = -

) 1+ X4 ( 2

32

isaisnerefiD irtemonogirT isgnuF srevnI

isgnuf irad srevnI irtemonogirt silutid tapad X nis = Y

iagabes nis = Y 1 X uata X nis sucra = Y asaib gnay

nisid takg ,X nis .cra = Y X soc = Y irad srevni ayntujnales

silutid soc = Y 1 X uata = Y X soc .cra nad irad srevni

silutid X .gt = Y gt = Y 1 X uata X gt .cra = Y

cnem utiay aynnahalasamrep X nis = Y kutneb adaP ira

,aynraseb iuhatekid halet gnay tudus utaus irad sunis agrah

raseb nakutnenem itrareb X nis cra = Y aynsrevni nakgnades

. aynsunis agrah iuhatekid halet gnay tudus

nakutnenem kutnu sumur nupadA halada aynnanurut :

1 Yd 1) X nis cra = Y . = -

Xd 1( – X 2 )

Yd – 1 2) X soc.cra = Y . = -

Xd 1( – X 2 )

1 Yd 3) X gt.cra = Y . = -

X + 1 Xd 2

hotnoC :

Yd 5 1 .5

X5 nis .cra = Y .1 = = - Xd 1- )X5( 2 1 – X52 2

. 3 Yd – 1 – 3

X3 soc.cra =Y .2 = = - Xd 1 – )X3( 2 1 – X9 2

4 Yd 4 X4 gt .cra =Y .3 = = -

)X4( + 1 Xd 2 X61 + 1 2

42

kilobrepiH isgnuF isaisnerefiD kilobrepih isgnuF tafis ikilimem gnay isgnuf halada - tafis

adap adebreb ipatet ,irtemonogirt isgnuf nagned apures

haubes aynnarasas irtemonogirt isgnuf ualaK .aynarasas

p kilobrepih isgnuf numan ,narakgnil .alobrepih haubes ada

,sunisoc ,sunis aynada lanekid irtemonogirt isgnuf adaP

aynada lanekid aguj kilobrepih isgnuf adap akam ,negnat nad

sunis - sunisoc ,) hnis ( kilobrepih - tnegnat nad ,) hsoc ( kilobrepih -

akisinifedid gnay ,) hgt ( kilobrepih iagabes n :

eX – e–X eX e + –X eX – e–X = X hnis = X hsoc ; nad = X hgt - 2 2 eX e + –X

sumur halhelorepid naikimed nagneD - nanurut rasad sumur

: tukireb iagabes kilobrepih isgnuf

Yd 1) X hnis = Y akiJ . X hsoc =

Xd Yd

2) X hsoc = Y akiJ . X hsoc = Xd

Yd )3 X hgt = Y akiJ . hces = 2 X

Xd

Yd hotnoC halnakutneT : : irad

Xd

1. X hsoc 4 + X3 hnis = Y

2. X = Y 2 X hnis

3. X3( hsoc 5 = Y – )1

52

bawaJ :

.)1 X hsoc 4 + X3 hnis = Y

Yd X hnis 4 + X3 hsoc 3 = Xd

X = Y )2 2 . X hnis

Yd X + X hnis X2 = 2 X hsoc Xd

Y .)3 X3( hsoc 5 = – )1

Yd 5 = . X3( hnis 3 – )1 Xd

= 1 X3( hnis 5 – )1 kilobrepiH isgnuF srevnI isaisnerefiD

nupkilobrepih isgnuf ,irtemonogirt isgnuf adap aynlah itrepeS

aguj .srevni iaynupmem akiJ Y = X hnis aynsrevni akam

hnis = Y – 1 X uata X hnis cra = Y nad , Y X hsoc =

aynsrevni X hsoc . cra = Y uata hsoc = Y – 1 X nad ,

.aynnial kutneb kutnu aynsuretes

sumur nupadA – : halada aynnanurut rasad sumur

1 Yd 1) . X hnis cra = Y = -

Xd X ( 2 ) 1 +

Yd 1 2) . X hsoc.cra = Y = -

Xd X( 2 – )1

Yd 1 .)3 X hgt.cra = Y = -

1 Xd – X 2

62

hotnoC : irad nanurut halnakutneT :

cra = Y .)1 . hnis ) X3 (

cra = Y .)2 . ) X2 ( hsoc

cra = Y .)3 . ) X4 ( hgt

bawaJ :

Y .)1 ) X3 ( hnis . cra =

3 Yd . 3 1 = = -

Xd ) X3 ( 2 1 + X9 2 1 +

.)2 Y cra = . ) X2 ( hsoc

2 1 .2 Yd = = - Xd ) X2 ( 2 – 1 X4 2 – 1

Y .)3 cra = . ) X4 ( hgt

1 .4 Yd 4 = = -

Xd 1 – ) X4 ( 2 1 – X61 2

laisnenopskE isgnuF isaisnerefiD apureb gnay isgnuf halada laisnenopskE isgnuF

irav utaus nagned utnetret nagnalib utaus irad natakgnamep elba

kutneB .tubesret elbairav isgnuf uata - laisnenopske isgnuf kutneb

e : itupilem X e , )X( f a , X a uata )X( f .

: halada aynisaisnerefid sumur nupadA

e = Y .)1 X ’Y e = X

e = Y .)2 )X( f e . )X( ‘ f = ’Y )X( f

.)3 a = Y X ’Y a = X a nl .

.)4 a = Y )X( f f = ’Y a . )X( ‘ )X( f a nl .

72

sumur adaP sata id .utnetret patet nagnalib halada a nad e

hotnoC : Yd

halnakutneT ’ Y uata isgnuf irad - : ini hawab id isgnuf Xd

1) e = Y . X3 3) 01 = Y . X4

2) e = Y . x2 nis 4). 5 = Y X3 soc

bawaJ :

1) e = Y . X3 ’ Y e3 = X3

2) e = Y . X2nis ’ Y 2 soc 2 = e .X X2 nis

3) 01 = Y . X4 ’ Y 01.4 = X4 01 nl . 4) . 5 = Y X3 soc ’ Y = – 5 . X3 nis 3 X3 soc 5 nl.

amtiragoL isgnuF isaisnerefiD

F amtiragol isgnu a halad utaus isgnuf kutnebreb gnay e gol = Y )X(f uata gol = Y a )X(f . nerefid nakutnenem kutnU -

isais ayn id nakanug sumur - : tukireb iagabes rasad sumur

1 Yd a gol = Y .)1 X =

a nl .X Xd

) X ( ‘ f Yd a gol = Y .)2 )X( f =

a nl . )X( f Xd

1 Yd = X nl = Y .)3 -

d X X

)X( ‘ f Yd = )X( f nl = Y .)4

)X( f Xd

82

: hotnoC

01 gol = Y .)1 X X4 nl = Y .)3

1 Yd 1 4 Yd - = - - = = -

X X4 Xd 01 nl .X Xd

= Y .)2 7 gol X3 nis .)4 X = Y 2 ) X hnis ( nl .

X3 soc 3 Yd X hsoc Yd = X = 2 . )X hnis( nl.X2 +

s Xd X nl .X3 nis Xd X hni

X3 gtoc 3 X = 2 )X hnis( nl X2 + X hgtoc = -

7 nl

.c namukgnaR 2 :

: nednesnarT isgnuF .)1 isgnuf halada - sgnuf isgnuf niales i

kilobrepiH isgnuF nad irtemonogirT isgnuf : utiay ,rabajla

isgnuF nad laisnenopkE isgnuF ,aynsrevnI atreseb

.amtiragoL

.)2 sumuR - : halada tubesret isgnuF isaisnerefiD rasad sumur

isgnuF isaisnerefiD irT irtemonog

)a X nis = Y . ’Y X soc =

)b X soc = Y . ’Y = X nis

)c X gt = Y . ’Y ces = 2 X

)d X gtoc = Y . ’Y = cesoc 2 X

)e X ces = Y . ’Y X gt . X ces =

)f X cesoc = Y . ’Y = X gtoc . X cesoc

92

isaisnerefiD srevnI isgnuF irtemonogirT

1 Yd )a X nis cra = Y . = -

Xd 1( – X 2 )

Yd – 1 )b X soc.cra = Y . = -

Xd 1( – X 2 )

1 Yd )c X gt.cra = Y . = -

X + 1 Xd 2

isgnuF isaisnerefiD kilobrepiH

Yd )a . X hnis = Y X hsoc =

Xd Yd

)b . X hsoc = Y X hsoc = Xd

Yd )c . X hgt = Y hces = 2 X

Xd

isaisnerefiD srevnI isgnuF kilobrepiH

1 Yd )a . X hnis cra = Y = -

Xd X ( 2 ) 1 +

Yd – 1 )b . X hsoc.cra = Y = -

Xd X( 2 – )1

Yd 1 c X hgt.cra = Y .) = -

1 Xd – X 2

03

laisnenopskE isgnuF isaisnerefiD

a e = Y .) X ’Y e = X

b e = Y .) )X( f e . )X( ‘ f = ’Y )X( f

c .) a = Y X ’Y a = X a nl .

d .) a = Y )X( f a . )X( ‘ f = ’Y )X( f a nl .

isgnuF isaisnerefiD amtiragoL

1 Yd a a gol = Y .) X =

a nl .X Xd

) X ( ‘ f Yd b a gol = Y .) )X( f =

a nl . )X( f Xd

1 Yd c X nl = Y .) = -

X Xd

)X( ‘ f Yd d l = Y .) )X( f n =

)X( f Xd

.d 2 saguT : isgnuf irad nanurut nakutneT – ini hawab id isgnuf

)1 X3 = Y . 3 X5 soc. )5 s = Y . X7 ce – X5 gtoc

)2 = Y . – X3 ( soc 2 – ) 5+ X4 )6 X2 cesoc. X2 nis 3 = Y .

X7 gt 4 X nis 4 )3 = Y . - )7 = Y . -

X 3 X soc + 1 5

13

)4 X ( soc .cra = Y . 2 ) )8 ) X ces ( soc .cra = Y .

X7 gt 4 )9 X = Y . 2 gt –1 ) X ½ ( )12 = Y . -

X 3 5

)01 gt.cra = Y . X ( 2 – ) 1 )22 X5 ( = Y . – X 6 nis cra.) 2

11 X = Y .) 3 X2 hsoc 32 )X soc( hnis = Y .)

21 X3 hgt = Y .) 42 hsoc = Y .) 3 X

)31 hnis = Y . –1 X2 – 1 )52 = Y . hsoc 3 –1 X4

)41 hsoc .cra = Y . X3 )62 . 1 ( = Y – X2 hgt . ) –1 X

15 2 = Y .) X 27 X = Y .) 4 e . X3

gt X2 16 e = Y .) 3 – X 28 = Y .) -

e X4

1 )7 = Y . 6 gol X2 nis

2 X3 18 X6 = Y .) – 7 29 = Y .) - 3 e X2

= Y .)91 e X2 01 gol . X X nl .)03 = Y -

02 3( nl = Y .) – e )X soc 8 X4

e . fitamroF seT 2 :

1) X = Y . 2 X2 nis. )9 nis 3 = Y . 4 X5

)2 ) 6 + X 5 ( gt 3 = Y . )01 . ) X nis ( gt .cra = Y

)3 nis = Y . –1 ) 2 + X3 ( 11 X3 = Y .) 2 3 . X4

X nis 4 e X2 )4 = Y . - )21 = Y . -

soc + 1 X 4 X3

)5 hnis = Y . –1 ) X gt ( )31 Y . 1 ( = – X2 hgt . ) –1 X

6 e = Y .) X5 )1 + X3( . 14 5 = Y .) 2 – X e . X – 2 -

7 5 gol .3 = Y .) X 15 gol = Y .) 3 nis 5X

8 X7 gt nl 4 = Y .) 61 ( nl = Y .) X4 soc )

23

.f fitamroF seT bawaJ icnuK 2 :

.)1 X 2 + X2 nis X2 = ’Y 2 X2 soc 9) . = ’Y nis 06 3 X5 X5 soc.

X nis )2 . ’Y ces 51 = 2 6 +X5( ) )01 . = ’Y .

nis + 1 2 X 3

.)3 = ’Y - .)11 = ’Y 3 X4 X21+X6(. 2 )3nl. 1{ – (9X 2 + 21 + X 4 })

4 e X2 2 ( – ) 4 nl 3

= ’Y .)4 - 1 .)2 = ’Y - X soc + 1 4 X3

ces 2 X 31 ) . = ’Y – hgt 2 –1 1 + X .)5 = ’Y .

gt ( 2 X + )1 .)41 Y’ 5 = 2 – X e. X – 2 1( – )5 nl

6 .) = ’Y e ) 8 + X 51( X5

X5 gtoc 5 1 = ‘ Y .)7 - 1 = Y .)5 -

5 nl X 3 nl

ces 82 2 X7 = Y .)8 - ‘ Y .)61 = – X4 gt 4

X7 gt

3 . rajaleB nataigeK 3 : naamasreP ,kimtiragoL isaisnerefiDlaisraP laisnerefiD nad kirtemaraP

rajaleB nataigeK naujuT .a 3 : .)1 aM nahalasamrep nakiaseleynem tapad awsisah isaisnerefid

kimtiragol .

.)2 nahalasamrep nakiaseleynem tapad awsisahaM isaisnerefid

.kirtemarap naamasrep

.)3 nahalasamrep nakiaseleynem tapad awsisahaM laisnerefid

laisrap .

33

iretaM naiarU .b 3 : isaisnerefiD naamasreP ,kimtiragoL laisraP laisnerefiD nad kirtemaraP

kimtiragoL isaisnerefiD

adaP aynmulebes rajaleb nataigek hadiak nakirebid halet

.rotkaf haub aud naigabmep / nailakrep nakisaisnerefidnem kutnu

d hibel tapadret akiJ isairav iagabreb nagned rotkaf haub aud ira

igal kadit tubesret hadiak akam ,naigabmep/nailakrep nanusus

arac nakgnabmekid uti kutnu ,) timur ulalret ( iakapid tapad

utaus utiay ,kimtiragol isaisnerefid tubesid gnay naisaisnerefidnep

arac gnay nakrasadnem adap amtiragol isgnuf nanurunep lisah ,

tafis aguj nad - .amtiragol tafis

iagabes halada kimtiragol isaisnerefid mumu hadiak nupadA

: tukireb

V.U = Y aynlasiM ) X isgnuf aynaumes Z nad W ,V ,U (

Z.W ahret aynisaisnerefid akaM : sumur nagned iracid tapad X pad

1 Vd 1 Ud 1 Yd 1 Wd Zd 1 = + -

Xd Z Xd W Xd V Xd U Xd Y

V.U Yd Zd 1 Wd 1 Vd 1 Ud 1 = + - Xd Z Xd W Xd V Xd U Z.W Xd

1 Y akij awhab tagnI = l Y X n ' = -

X

B.A gol = A gol + B gol

A gol = A gol -- B gol B

43

hotnoC :

: irad ) nanurut ( isaisnerefid halnakutneT

X3 X nis 1) = Y . -

X2 soc e X5

3) = Y . - X2 X2 hnis

2) . X = Y 4 e . X3 X4 soc .

bawaJ : X3 X nis .

.)1 = Y - X2 soc

X Yd 3 X nis 1 1 1

= X3 2 + X soc ( )X2 nis 2 X2 soc Xd X3 X2 soc X nis

X3 3 X nis

= ( ) X2 gt 2 + X gtoc + X X2 soc

.)2 X = Y 4 e X3 X4 soc

Yd 1 1 1

X = 4 e X3 ( X4 soc . X4 3 + . e3 X3 + - .– ) X4 nis 4 Xd X4 e X3 X4 soc

4

X = 4 e . X3 ( X4 soc . 3 + – ) X4 gt 4 X

e X5

3) = Y . - X2 X2 hnis

Yd e X5 e 5 X5 X2 X2 hsoc 2 = ( ) X Xd 2 e X2 hnis X5 X2 X2 hnis

e X5 2

= 5 ( ) X2 hgtoc 2 X2 X X2 hnis

53

kirtemaraP naamasreP isaisnerefiD

hibel gnires ,utnetret nahalasamrep aparebeb malaD

malad nakataynem nagned isgnuf utaus nakpakgnugnem hadum

abeb lebairav haubes .retemarap tubesid asaib uata agitek s

nis = X aynlasiM 2 soc = Y ; naikimed gnay kutneB .

Y nad X nagnubuh anerak ,kirtemarap naamasrep nakamanid

haubes iulalem ipatet ,gnusgnal araces nakataynid kadit

utiay ,retemarap .

enem kutnu ayntujnaleS utaus irad nanurut nakutn

: tukireb iagabes hadiak nakanugid ,kirtemarap naamasrep

isgnuf = X nad Y ( )

Yd hotnoC nakutneT : amasrep irad ! ini tukireb kirtemarap na

Xd

1) t nis = X ; t2 soc = Y .

2) nis 3 = Y . – nis 3 soc = X ; 3

2 t5 + 4 – t3 3) = Y . = X ; -

t3 + 2 t3 + 2

bawaJ :

1) t2 soc = Y . ; t nis = X

1 td Xd Yd = – t2 nis 2 t soc = = - t soc Xd td td

Yd td Yd 1 1 = . = – . t2 nis 2 = – .2 t nis 2 . t soc -

Xd td Xd t soc t soc

= – tnis 4

d Yd Yd = . -

d Xd Xd

63

2) . Y nis 3 = θ – nis 3 θ ; soc = X 3 θ

Xd Yd soc 3 = θ – nis 3 2 θ soc θ = – soc 3 2 θ nis θ

θd θd

θ3 soc 3 Yd – θ soc = = = – θ gtoc

Xd – θ nis θ nis θ2 soc 3

t5 + 4 2 – t3 3) = Y . X = -

t3 + 2 t3 + 2

)t3 + 2( 5 Yd – )t5+ 4( 3 Xd – )t3 + 2( 3 – 2( 3 – )t3 = - ; = -

)t3 + 2( td 2 )t3 + 2( td 2

– 2 – 21 = = -

) t3 + 2( 2 )t3 + 2( 2

Yd – ) t3 + 2 ( 2 2 1 = . - = - ) t3 + 2 ( Xd 2 – 21 6

laisraP laisnerefiD

uata sabeb lebairav aud nagned isgnuf utaus iuhatekid akiJ

tas halas padahret tubesret isgnuf isaisnerefid akam ,hibel u

akam ,patet aynnial lebairav paggnagnem nagned aynlebairav

nad laisrap laisnerefid nakamanid tubesret isaisnerefid

nagned nakgnabmalid .)atled(

f = Z :aynlasiM ) Y,X ( ahret Z laisrap laisnerefid akam -

Z X pad Y nagned gnabmal nagned silutid natsnok ,

X naikimed aisnertefid aguj l Z laisrap padahret nagned Y gnem -

Z X paggna nagned silutid natsnok .

Y

73

Z Z agrah gnipmasiD - laisnarefid neisifeok agrah nad -

X Y

tapad aguj iracid neisifeok laisnerefid laisrap aynnial kiab

: utiay ,audek nupuam amatrep laisrap laisnerefid neisifeok

2 Z 2 Z 2 Z 2 Z , , , gnisam - : itrareb gnisam X2 Y 2 .X Y .Y X

2 Z Z 2 Z Z

= ; = - X 2 X X .X Y X Y

2 Z Z 2 Z Z

= ; = - Y2 Y Y .Y X Y X

laoS hotnoC :

1) X = Z isgnuf iuhatekid akiJ . 2 . Y3 halnakutnet akam

amatrep kiab ,aynlaisrap laisnerefid neisifeok aumes

.audek nupuam

bawaJ : X = Z 2 Y3

Z a). YX2 = 3

X ======

Z b) . X = 2 Y3. 2 X3 = 2Y2

Y =======

2 Z Z c) . = ( = ) YX2 ( 3 = ) Y2 3

X2 X X X ==== =

83

2 Z Z

d) . = ( = ) X3 ( 2Y2 = ) X6 2Y Y2 Y Y Y ======

2 Z Z

e) . = ( = ) X3 ( 2Y2 = ) YX6 2 X. Y X Y X ======

2 Z Z

f) . = ( = ) YX2 ( 3 = ) YX6 2 Y. X Y X Y ======

natataC : ret lisah aud nakitahrepid akiJ : ataynret rihka

2 Z 2Z = - .X Y Y. X

.isgnuf aumes uata kutneb aumes kutnu mumu ukalreb ini tafiS )2 X ( nl = V iuhatekid akiJ . 2 Y + 2 : awhab halnakitkub )

2 V 2 V + = 0

X2 Y2

bawaJ : X( nl = V 2 Y + 2)

X2 V = - X X2 Y + 2 2 X(2 V 2 Y + 2 ) – X2 . X2 = - X2 X( 2 Y + 2)

X2 2 Y2 + 2 – X4 2 Y2 2 – X2 2

= = - X( 2 Y + 2)2 X( 2 Y + 2)2

V Y2 = - X Y 2 Y + 2

93

2 V X(2 2 Y + 2 ) – Y2 . X2 Y2 2 – Y2 2 ---- = -------------------------- = ------------ Y2 X( 2 Y + 2) X( 2 Y + 2)2

2V 2V Y2 2 – X2 2 X2 2 – Y2 2 + = + - X2 Y2 X( 2 Y + 2)2 X( 2 Y + 2)2 Y2 2 X2 + 2 – Y2 2 – X2 2 = - X( 2 Y + 2)2

= 0

2 V 2V awhab itkubret idaJ + 0 =

X2 Y2

.c namukgnaR 3 :

kimtiragoL isaisnerefiD V.U

= Y aynlasiM ) X isgnuf aynaumes Z nad W ,V ,U ( Z.W

isaisnerefid akaM Y nagned iracid tapad X padahret hadiak

: tukireb iagabes kimtiragol isaisnerefid

Zd 1 Wd 1 Vd 1 Ud 1 V.U Yd = + - Xd Z Xd W Xd V Xd U Z.W Xd

kirtemaraP naamasreP isaisnerefiD

U ,kirtemarap naamasrep utaus irad nanurut nakutnenem kutn

: tukireb iagabes hadiak nakanugid

: ini sumur malaD isgnuf = X nad Y retemarap =

Yd Yd d = . - d Xd Xd

04

laisraP laisnerefiD

f = Z :aynlasiM ) Y,X ( laisrap laisnerefid akam amatrep

Z ahret Z X pad Y nagned gnabmal nagned silutid natsnok ,

X D laisrap laisnertefid aguj naikime amatrep Y padahret Z

Z ggnagnem nagned nagned silutid natsnok X pa .

Y

ayntujnaleS audek laisrap laisnerefid neisifeok ayn , silutid :

2 Z 2 Z 2 Z 2 Z , , , gnisam - : itrareb gnisam X2 Y 2 .X Y .Y X

2 Z Z 2 Z Z

= ; = - X 2 X X .X Y X Y

2 Z Z 2 Z Z

= ; = - Y2 Y Y .Y X Y X

: 3 saguT .d

.)1 : irad kimtiragol isaisnerefid nakutneT

a .) X3 = Y 6 s .X6 soc . e X3 ni X3 X nl c = Y .) -

X( – )1 3

e X4 X nis . b = Y .) 3 X4

X2 soc X d = Y .) - 4 X3 X2 nl .

2 .) neT ! ini tukireb kirtemarap naamasrep nanurut nakut

gt = Y .)a θ ½ gt nl = X ; θ 4 – 2 r2 – r4 = Y .)c = X ; -

nis 4 = Y .)b 3 θ soc 4 = X ; 3 θ r + 1 r + 1

14

.)3 amatrep laisrap laisnertefid neisifeok aumes halnakutneT

isgnuf irad audek nupuam - ni tukireb isgnuf .i

)a Y2 nis . X3 = Z .

)b )YX( nl.)Y+X( = Z .

)c )Y2+X3( gt = Z .

Y3+X2 )d = Z . -

X2 – Y3

.e fitamrof seT 3 : .)1 : irad kimtiragol isaisnerefid nakutneT

X nis .X = Y .)a ) 1 + X3 ( . . X2 soc e X2 b = Y .) -

X soc + 1

2 .) neT ! ini tukireb kirtemarap naamasrep nanurut nakut

a .) θ3 hnis = Y X ; = soc θ3 h

.)b 4 = Y ( θ – θ nis ; ) 4 = X ( 1 – θ soc )

.)3 amatrep laisrap laisnertefid neisifeok aumes halnakutneT

isgnuf irad audek nupuam - .ini tukireb isgnuf

3X – Y )a = V . d 2 h )b = Z . -

4 X 2 + Y

.f fitamroF seT bawaJ icnuK 3 :

Yd 3 .)1 .)a = ) 1 + X3 ( . .X2 soc e X2 ( – ) 2 + X2 gt 2

Xd 1 + X3

Yd s .X X ni 1 X nis .)b = ( + oc + X gt )

Xd X soc + 1 X 1 + X soc

Yd Yd

2 .) .)a = 3 hgtoc θ .)b = cesoc θ – oc gt θ

Xd Xd

24

V V

)3 )a . = h.d = d2 d 2 h 4

2V 2V = h . = 0

d2 2 h2

2V 2V = d . = d

.h d 2 .d h 2

Z Y 7 Z – X7 )b . = = -

X )Y2 + X( 2 Y )Y2 + X( 2

2Z – X41 Y – 2 Y8 2 2Z X82 2 + YX 65 = = - -

X2 2 + X( )Y 4 Y2 )Y2 + X( 4

2Z X7 2 – 2 Y8 2 2Z X7 2 – 2 Y8 2 = = -

.Y X )Y2 + X( 4 .X Y )Y2 + X( 4

4 . rajaleB nataigeK 4 : isgnuF laisnerefiD isakilpA

rajaleB nataigeK naujuT .a 4 : .)1 awsisahaM ad nem tap nakiaseley halasam isnerefid isakilpa -

.sirtemoeg narisfat malad isgnuf isa

.)2 awsisahaM ad nem tap halasam nakiaseley isnerefid isakilpa -

.isgnuf muminim nad mumiskam halasam malad isgnuf isa

.)3 awsisahaM ad nem tap halasam nakiaseley id isakilpa laisneref

isgnuf nahaburep ialin nakutnenem malad laisrap .

34

.)4 awsisahaM ad nem tap halasam nakiaseley laisnerefid isakilpa

isgnuf nahaburep natapecek nakutnenem malad laisrap .

iretaM naiarU .b isgnuF isaisnerefiD isakilpA : 4

itrA sirtemoeG araceS isgnuF isaisnerefiD

isgnuf isaisnerefid itra naitregnep naktapadnem kutnU

: ini hawab id 1 rabmaG nakitahrepid tapad sirtemoeg araces

) X ( f =Y

Y

K ) h+X( f Q ) )h+X( f ,h+X(

H ) X( f P R ) )X(f ,X( P

M N X

h X 0 h + X

1 rabmaG

: awhab kapmat sataid rabmag iraD

) h + X ( f – )X( f RQ KH = = = QP rusub ilat neidarg

RP NM h

agabes naksumurid uluhad gnay isgnuf isaisnerefedid aggniheS i

) h + X ( f – )X( f timil = timil ) QP neidarg ( h h 0 h 0

Q kitit akam ) lon itakednem ( licek tagnas h libmaid akiJ

eb nakhab uata P itakednem .P nagned tipmir

44

)h+X( f – )X( f timil : aggniheS gnuggnis sirag neidarg =

h h 0 P id avruk

awhab itrareb ))X( f ,X(P kitit aneraK :)X( f = Y isgnuf isaisnerefid

)h+X( f – )X( f timil = )X( ‘ f kitit id gnuggnis sirag neidarg = h h 0 )Y,X( )X( f = Y avruk adap

)X( f = Y avruk iuhatekid akij awhab naklupmisid tapad aynrihkA

Yd uata )X( ‘ f irad sirtemoeg itra akam : halada

Xd

Yd = )X( ‘ f )Y,X( id gnuggnis sirag neidarg =

)X( f = Y avruk adap Xd

natataC ”epols“ uata nagnirimek = neidarg kutnu nial halitsI :

bmalid nad : nakgna m

lamroN siraG nad gnuggniS siraG naamasreP

adap P kitit haubes id )X( f = Y avruk nagnirimeK

ayngnuggnis sirag nagnirimek helo nakutnetid tubesret avruk

Yd agrahreb ini nagnirimek aneraK .aguj P kititid ,P kititid Xd

akam aynraseb tapad gnutihid akij naamasrep aynavruk

aguj ayngnuggnis sirag naamasrep naikimed nagneD .iuhatekid

sirag haubes mumu naamasrep nakrasadreb utiay ,iracid tapad

X( kitit haubes iulalem nad m neidarg nagned surul 1 Y , 1 :utiay , )

Y – Y1 X( m = – X1 )

naamasrep nakutnetid tapad aguj nakulrepid akij ayntujnaleS

surul kaget nad P iulalem gnay surul sirag utiay ,aynlamron sirag

54

helorepid mumu araceS .aguj P kitit id gnuggnis sirag padahret

esret sirag audek neidarg nagnubuh :utiay ,tub

– 1

= lamroN siraG neidarG - gnuggniS siraG neidarG

hotnoC :

adap lamron sirag nad gnuggnis sirag naamasrep haliraC .)1

alobarap X3 = Y 2 X4 + – ) 2 ,1 ( kititid 5

Yd bawaJ amatreP : ggnis sirag neidarg ulud gnutihid u ( ayngn )

Xd

X3 = Y 2 X4 + – 5

Yd 4 + X6 = Xd

Yd utnU 1 = X k = m 01 = 4 + 1.6 =

Xd

) 2 ,1 ( id gnuggnis sirag naamasrep idaJ

Y – X ( 01 = 2 – ) 1

X01 = Y – 2+ 01

uata X01 = Y – 8

nagned iracid tapad aynlamron sirag naamasrep ayntujnaleS

– 1 tihgnem ulud hibelret neidarg gnu / = utiay ,aynnagnirimek -

01

: halada ) 2 ,1 ( kitit id lamron sirag naamasrep aggnihes

– 1 Y – = 2 X ( – ) 1

01

– 1 = Y ( ) 1 + X = Y 01 – 12 + X 01

64

2) aynnaamasrep gnay avruk adap P kitit id gnuggnis siraG .

X = Y4 2 X4 + – 1 X 6 sirag rajajes 6 – halnakutneT . 5 = Y 2

id lamron sirag nad gnuggnis sirag naamasrep ,P kitit tanidrook

! tubesret P kitit

bawaJ : X = Y4 2 X4 + – 61

X ¼ = Y 2 X + – 4

Yd 1 + X ½ = Xd

X 6 siraG – 5 = Y 2 Yd

X3 = Y – ½ 2 = m 3 = Xd

X6 sirag rajajes P id gnuggnis sirag aneraK – 5 = Y2

: aggnihes ,3 = m utiay ,amas aynneidarg itrareb

½ 3 = 1 + X .4 = Y nad 4 = X helorepid nad 2 = X ½

.) 4 ,4 ( P halada P pitit tanidrook idaJ

Y : P id gnuggnis sirag naamasreP – X ( 3 = 4 – ) 4

X 3 = Y – 8 = m aynlamron sirag neidarG – 3/1

: aynlamron sirag naamasreP

Y – = 4 – X ( 3/1 – ) 4

= Y – 4 + 3/4 + X 3/1 = Y 3 – 61 + X

gnadaK - malad nakataynid avruk naamasrep gnadak

nagnasap kutnebreb aguj tapad uata tisilpmi isgnuf kutneb

adap anerak ritawahk ulrep kadit ipateT .kirtemarap naamasrep

rajaleb nataigek arac anamiagab naksalejid halet uluhadret

snerefid iracnem .aynisai

74

hotnoC :

3) id lamron sirag nad gnuggnis sirag naamasrep haliraC . )3 ,2(

X avruk adap 2 Y + 2 – 0 = 71 + YX 5

bawaJ X : 2 Y + 2 – 0 = 71 + YX 5

Yd Yd Y 2 + X2 – Y 5 – X5 0 =

Xd Xd

Y5 Yd Yd – X2 Y2 ( – ) X5 Y 5 = – X2 = -

Xd Y2 Xd – X5

3 . 5 Yd – 2 . 2 agrah ) 3 ,2 ( kitit iD = -

3 . 2 Xd – 2 . 5

– 11 = -

4

) 3 ,2 ( kitit id gnuggnis naamasreP

– 11 Y - = 3 X ( – ) 2

4

– 11 11 = Y + X 3 +

4 2

= Y – 2 ¾ 8 + X ½- = Y4 uata – 43 + X 11

: aynlamron sirag naamasreP

– 4 1 = lamron sirag neidarG = -

– 11 4 / 11

: ) 3 ,2 ( kitit id lamron sirag naamasreP

4 Y – = 3 X ( – ) 2

11

84

4 3 = Y 2 + X uata 52 + X 4 = Y 11

11 11 ============ =============

T .)4 silu id lamron sirag nad gnuggnis sirag naamasrep hal θ = - 6

2 soc = Y naamasrep iuhatekid akij θ nis = X nad θ .

bawaJ :

2 soc = Y θ nis = X θ

d Xd Yd θ 1 = – 2 nis 2 θ soc = θ = - dθ dθ Xd soc θ

= – nis 4 θ soc θ

d Yd Yd θ – nis 4 θ soc . θ = . = = – nis 4 θ d Xd θ soc Xd θ

Yd

kutnU θ = = – nis 4 = – . 4 ½ = – 2 6 Xd 6

aJ gnuggnis sirag nagnirimek id halada )Y ,X( iulalem gnay – ,2

iracid tapad tubesret gnuggnis sirag naamasrep aggnihes

θ kutnu Y nad X agrah gnutihgnem uluhad hibelret nagned

.iuhatekid halet gnay

= θ kutnU nis = X ½ =

6 6

soc = Y ½ =

6

: silutid tapad gnuggnis sirag naamasreP

Y – = ½ – X( 2 – ) ½

94

= Y – 3 + X4 = Y2 uata ½ 1 + X2

– 1 on sirag nagnirimeK = lamr ½ =

– 2

: aynlamron sirag naamasrep idaJ

Y – X ( ½ = ½ – ) ½

¼ + X ½ = Y uata 1 + X2 = Y4 ========== =============

miniM nad mumiskaM isgnuF utaus mu fitomoto kinket gnadib id aguj kusamret ,napudihek malaD

kiabret arac nautnenep halasam adap nakpadahid ilak gnires

.utauses nakukalem kutnu gnadak tubesret kiabret araC - gnadak

nakmuminimem uata nakmumiskamem halasam tukgnaynem

taus halasaM .utnetret nanupmih adap isgnuf u - tubesret halasam

,isgnuf isaisnerefid nautnab nagned nakhacepid tapad ataynret

.audek isaisnerefid nupuata amatrep isaisnerefid kiab

aynmuminim nad mumiskam halasam imahamem kutnU

halnakitahrep ,isgnuf utaus nagnubuh nakkujnunem gnay kifarg

anamiagabes aynisaisnerefid nad nagned isgnuf utaus aratna

: ini hawab id 2 rabmaG adap kapmat

kifarg iraD - lah tahilret tubesret kifarg - : tukireb iagabes lah

)1 : awhab kapmat ) X ( f = Y avruk iraD .

a) A ktit adaP . ,mumiskam Y agrah iapacid ini A kititid anerak

id nial gnay Y agrah nakgnidnabid raseb hibel Y agrah

.ayntaked

b). irad licek hibel anerak ,muminim Y agrah B kitit adaP

.ayntaked id nial gnay Y agrah adap

c). kitit adaP C id uata ratadnem avruk Y agrah tapad

kadit avruk ,muminim hagnetes nad mumiskam hagnetes

sataek kolebmem ipatet hawab ek kilabmem suret

05

ini kitiT .habmatreb suret gnay fitisop neidarg nagned

.koleb kitit tubesid

,A kitiT renoisats agrah uata kilab kitit tubesid C nad B .Y

2) :awhab kapmat amatrep nanurut avruk iraD .

Yd Yd kutnU ukalreb kilab kitit paites 0 =

Xd Xd

) X ( f = Y A Y

C B

X 0 1 X 2 X 3 X

Yd - ) X ( ’ f = Y Xd

X 0 1 X 2 X3 X d2Y

- dX2 ) X ( “ f = Y

X 0 1 X 2 X 3 X

2 rabmaG

15

: awhab tahilret audek nanurut )avruk( kifarg iraD .)3

d2Y a) akiJ . muminim agrahreb Y akam 0 >

Xd 2

d2 Y b) akiJ . mumiskam agrahreb Y akam 0 <

Xd 2

d2 Y c) akiJ . koleb kitit idajret akam 0 =

Xd 2

akij ayntujnaleS utaus adap nakisinifedid isgnuf utaus

aynmuminim uata mumiskam ialin akam ,pututret lavretni

ialin adap ilaucek idajret tapad - nikgnum aynrenoisats ialin

gnuju adap idajret tapad aguj - aynlavretni gnuju .

laoS hotnoC :

1) breb( ukis katok haubes taubid nakA . irad ) kolab kutne

.mc 9 aynrabel nad mc 42 ayngnajnap gnay gnes talp rabmeles

adap kitnedi igesrep gnotopid tubesret gnes narabmeL

aggnihes aynisis sata ek tapilid naidumek ,aynkojop tapmeek

.) 3 rabmag tahil ( katok kutnebret

katok naruku halnakutneT ! mumiskam aynemulov raga

X

mc 9

42 mc 42 – 9 X2 – X2

3 rabmaG

25

naiaseleyneP : nad mc X = kojop paites adap tapme iges isis aynlasiM

mc V = aynemulov 3. V akaM 42( = – 9(.)X2 – X.)X2

X 612 = – X66 2 X4 + 3

0 : halada X agraH 5,4 ≤ X ≤

V nakmumiskamem halada ini laos malad nahalasamrep idaJ

0 : lavretni adap .5,4 ≤ X ≤

Vd akij mumiskam naka V nupadA 0 =

Xd

Vd X 612 = V – X66 2 X4 + 3 akam 612 = – X 231 X 21 + 2

Xd

Vd 0 = 612 – X 231 X 21 + 2 0 =

Xd

9 ( 21 – 2 (.) X – 0 = ) X

9 – 2 uata 0 = X – 0 = X

2 = X uata 9 = X

vretni malad kusamret kadit 9 = X kutnU 0 la , 5,4 ≤ X ≤

gnuju adap X agrah nakgnades - nad 0 = X utiay lavretni gnuju

0 = V agrah 5,4 = X

2 = X kutnU akam 2 .612 = V – 2.66 2 2.4 + 3 002 =

mc 002 = mumiskam V idaJ 3 raga katok naruku aggnihes ,

: halada mumiskam aynemulov

42 = katok gnajnaP – mc 02 = 2.2 9 = katok rabeL – mc 5 = 2.2 mc 2 = katok iggniT

35

haubes nataubmep nakanacneriD .)2 aynsala gnay akubret katok

halnakutneT .retil 23 = katok emuloV .igesrep kutnebreb

! nikgnum tikideses nakanugid gnay nahab raga katok naruku

naiaseleyneP :

la( isis lasiM md X = tapmeiges )sa

h ,md h = katok iggnit nad akam

X = V katok emulov 2 h

23 X X2 23 = h = h -

4 rabmaG X2

halmuj alibapa numinim idajnem naka katok taubmep nahaB

saul katok taubmep nahab saul halmuJ .numinim aguj nahab

: tapadid L malad ek h isutitsbus hX4 + ²X = L

821 Ld 821 + ²X = L X 2 = – -

²X Xd ²X

Ld muminim L ragA 0 =

Xd

821 X2 – 0 =

X² X2 ³ – 0 = 821

X³ – 0 = 46 X ( – X ( ) 4 ² 0 = ) 61 + X4 +

4 = X alib ihunepid aynah ini naamasreP

23 23 = h akam 4 = X kutnU = 2 =

4² 61

ab raga katok naruku idaJ muminim nakulrepid gnay nah

: utiay

katok gnajnaP md 4 = katok rabeL = katok iggniT md 2 =

45

adap laisraP laisnerepiD isakilpA isgnuF liceK nahabureP akiJ iuhatekid Z halada utaus isgnuf irad X d na uata Y

f =Z utnet nahaburep imalagnem Y nad X alibapa akam ,)Y,X(

rasebes Y nad X licek nahaburep kutnU .habureb aguj Z ajas X

nad rasebes Z nahaburep nakbabeynem naka Y .Z

alibapA p tered malad nakrabajid Z irad takgna nad X ,Y

helorepid naka akam .A = Z .B + X ukus + Y - ukus nad X Y

X isgnuf halada B nad A nagned iggnit hibel takgnapreb gnay

.Y nad

akam ,natsnok Y kutnU :aggnihes ,0 = Y

.A = Z ukus + X - malad ukus ggnit hibel takgnapreb gnay X .i

Z Z akij uata A = x = A akam 0 - X X

nagned ,natsnok X akij amas gnay nalaj nagneD Y 0

repid helo :

Z = B agrah helorepid naka ayntujnales : tukireb iagabes Z

Y

Z Z = Z - + X ukus + Y - .nakiabaid tapad gnay licek ukus

X Y

Z Z idaJ = Z + X Y

X Y

ta id sumuR nagned isgnuf utaus adap satabret kadit sa

gnarabmes kutnu ukalreb aguj ipatet ,ajas sabeb lebairav aud

,) lebairav aud irad hibel ( lebairav halmujes nagned isgnuf

,) W,.…,Y,X ( f = Z akiJ : aynlasim Z nahaburep aynraseb akam

rav paites nahaburep tabika gnutihid tapad aynsabeb lebai

: sumur nagned

55

Z Z Z = Z + X + … + Y W

X Y W

hotnoC laoS : iraj ikilimem rednilis kutnebreb gnubat haubeS .)1 - = r iraj

rasebes habmatreb r akiJ .mc 51 = h ayniggnit nad mc 01

aynraseb hakapareb akam mc 2,0 gnarukreb h nad mc 3,0

? tubesret gnubat emulov nahaburep

naiaseleyneP :

emuloV = V rednilis h ²r

V V 2 = nad h r = ²r

r h

: akam, mc 51 = h nad mc 01 = r aneraK V

2 = 003 = 51. 01 . r

V

= 01 . 2 001 = h

b anerak fitisop ( 3,0 = r ) habmatre = h – ) gnarukreb anerak evitagen ( 2,0

V V : idaJ = V + h

r r

= 003 001+ 3,0. (. – ) 2,0

09 = – 02

= 07 022 =

= rasebes habmatreb gnubat emulov idaJ mc 022 3

==========

65

V

= I iuhatekid akiJ .)2 - mho 44 = R nad tlov 022 = V nagned R

1 rasebes habmatreb V akij I nahaburep halnakutnet akaM

! mho 4,0 habmatreb aguj R nad tlov

V bawaJ = I : -

R

I 1 I V = nad = – -

V R R R 2

nad tlov 1 = V mho 4,0 = R

I I = I + V R

V R

V 1 = V R

R R 2

022 1 = 1. 4,0

44 44 2

= 7220,0

rasebes nurut I idaJ erepmA 7220,0 =

============

nahabureP natapeceK : adap laisraP laisnerepiD isakilpA

isgnuF utauS

akam )Y,X( f = Z akij awhab uluhadret naktubesid haleT

agrah helorepid : aynraseb z

Z Z = Z + X Y

X Y

75

nagned igabid sataid naamasrep adap saur audek akiJ t

: tapadid akam

Z Z X Z Y = . + . - t X t Y t

agrah kutnu s gnay t ( licek tagna t 0 ) : akam

Z Zd X Xd Yd Y ; nad - t td t td td t

eok ipatet taumem kadit gnay aynnial laisrap laisnerefid neisif t

.) habureb kadit ( patet halada

Z d Z X d Z Y d : idaJ = . + . - t d t d X Y t d

utaus nahaburep natapecek sumur nakamanid sata id sumuR

nial isgnuf kutnu nkgnabmekid tapad tubesret sumur nad ,isgnuf

. sabeb lebairav aud irad hibel iaynupmem gnay

laoS hotnoC :

raJ i- natapecek nagned habmatreb rednilis haubes iraj

gnarukreb ayniggnit nakgnades ,kited/mc 1,0 nahabmatrep

halnakutneT .ted/mc 4,0 nagnarugnep natapecek nagned

rednilis emulov nahaburep natapecek uti mc 41 = r taas adap

.mc 12 = h nad

bawaJ = V rednilis emuloV : r 2 . h

V V 2 = ; h r = r2 r h

mc 12 = h nad mc 41 = r kutnU : tapadid

V 2 = mc 8481 = 12 .41. 2 r

85

V = r 2 = 41 . 2 mc 616 = 2

h

Vd ( rednilis emulov nahaburep natapecek aggniheS akij ) td rd hd 1,0 = nad ted/mc = – .ted/mc 4,0

td td

Vd rd V hd V = . + . -

td td r td h

1,0 . 8481 = – 4,0 . 616

= – mc 2,16 3 .ted /

rednilis emulov nagnarugnep natapecek ,idaJ : mc 2,16 3 ted/ .

.c namukgnaR 4 :

)1 . X padahret Y nanurut irad sirtemoeg itrA : halada

Yd = )X( ‘ f )Y,X( id gnuggnis sirag neidarg =

)X( f = Y avruk adap Xd

.)2 d )X( f = Y avruk adap gnuggnis sirag naamasreP nagne

X( kitit haubes iulalem nad m neidarg 1 Y , 1 :utiay , )

Y – Y1 X( m = – X1 )

)3 . neidarg nagned gnuggnis sirag neidarg aratna nagnubuH

halada lamron sirag :

– 1 = lamroN siraG neidarG -

gnuggniS siraG neidarG

95

.)4 tapad isgnuf haubes muminim nad mumiskam agraH

amodep nagned nakutnetid : n

a :awhab kapmat amatrep nanurut avruk iraD .)

Yd Yd kutnU ukalreb kilab kitit paites 0 =

Xd Xd

.)b : awhab tahilret audek nanurut )avruk( kifarg iraD

d2Y )1( akiJ . muminim agrahreb Y akam 0 >

Xd 2

d2 Y )2( akiJ . mumiskam agrahreb Y akam 0 <

Xd 2

d2 Y )3( akiJ . koleb kitit idajret akam 0 =

Xd 2

.)5 ,) W,.…,Y,X ( f = Z akiJ rep aynraseb akam tabika Z nahabu

nagned gnutihid tapad aynsabeb lebairav paites nahaburep

: sumur

Z Z Z = Z + X + … + Y W

X Y W

gnutihid tapad Z nahaburep natapecek akam )Y ,X( f = Z akiJ .)6

: sumur nagned

Z d Z X d Z Y d = . + . - t d t d X Y t d

Z Z

tubesret sumur adaP nad gnisam - gnisam X Y

a .Y padahret nad X padahret Z irad laisrap laisnerefid halad

06

saguT .d 4 :

.)1 halnakutneT naamasrep sirag gnuggnis d na lamron sirag

avruk adap - ! nakirebid gnay kitit id tukireb avruk

.)a X(= Y avruk adaP – )2 2 )4 ,4( kitit id

.)b X avruk adaP 3 X + 2 Y+ Y 3 – )3 ,2( kitit id 0 7

X2 )c = Y avruk adaP . ,3 ( kitit id 5/3 )

X2 1 +

X2 Y2 spille adaP .)d + soc 31 ( kitit id 1 = nis 5 , )

52 961 gnaroeS .)2 kinakem nad irudreb tawak retem 001 iaynupmem

ragap / gnadnak taubmem kutnu iakapid naka gnadug utaus

repes aynkutneb gnay t .hawab id rabmag i

Y

apareB naruku ragap tubesret

X ? mumiskam aynsaul raga

CB = BA

C B A

3 iuhatekid akiJ .2 romon laos adap gnadnak rabmag tahiL .)

gnisam saul - igab gnisam retem 004 = tubesret gnadnak na

igesrep . gnay irudreb tawak ayapus ragap naruku hakapareB

? nikgnum laminimes nakulrepid

4 .) gnubat naruku haliraC

kaget emulov nagned

tapad gnay mumiskam

06 taubid / naktapmetid id

tucurek haubes malad

gnipmas id rabmag itrepes

akij mc 06 = tucurek iggnit

03 iraj nad - .mc 03 ayniraj

16

h .E 3 5 laD .) = D sumur ma nakirebid h aynraseb nad V

1(21 –V2)

gnisam – 1,0 = h gnisam 200,0 + 3,0 = V nad 200,0

akij D irad mumiskam natakednep agrah halnakpakgnU

! tloV 022 = E iuhatekid

6) halnakutneT .) b + tq ( soc .) a + Xp ( nis A = Y iuhatekiD .

nahalasek tabika iagabes Y kutnu lubmit gnay nahalasek

nad X adap t gnisam – rasebes gnisam X nad .t

7 .)

agitiges adaP ukis - X gnipmas id ukis

tedmc 2 natapecek nagned habmatreb .

X Y nagned gnarukreb Y nad natapecek

gnutiH .ted/mc 3 natapecek nahaburep

Z Z agitiges tubesret taas = X mc 01

nad = Y ! mc 8

8 YX2 = Z nakutnetid akiJ .) – X3 2 nagned habmatreb X nad Y

natapecek halnakutnet akam ,kited/mc 2 :natapecek

haburep gnarukreb kadit nad habmatreb kadit Z raga Y na

mc 1 = Y nad mc 3 = X taas adap

.e fitamrof seT 4 : .)1 d gnuggnis sirag naamasrep halnakutneT sirag na lamron

adap avruk - id tukireb avruk nakirebid gnay kitit !

.)a X = Y avruk adaP 3 – X6 2 kitit id 1 + X21 + ( 1 ,0 )

soc 2 = X naamasrep / isgnuf adaP .)b 3 nis 2=Y nad θ 3

kutnu ¼ = .naidar

.)2 lib aud halmuJ aynilak lisah akiJ .04 halada Y nad X nagna

nagnalib audek rasebret ilak lisah halgnutih akam p halada

uti nagnalib audek halnakutnet aguj nad tubesret .

26

3 rasebes satisapak nagned akubret katok haubes nakulrepiD .)

.retil 00063 ,aynrabel ilak aud surah katok gnajnap akiJ

gnay taubmep nahab raga katok naruku halnakutnet

? nikgnum tikideses nakulrepid

E2 4 = P iuhatekid akiJ .) ,mho 8 = R nad tloV 022 = E nagned R

nurunep tabika idajret gnay P nahaburep halnakutnet a E n

! mho 2,0 rasebes R nakianek nad tloV 5 rasebes

5 iraJ .) - nagned gnarukreb tucurek haubes sala narakgnil iraj

habmatreb ayniggnit nad kited/mc 2,0 natapecek nagned

rep natapecek halnakutneT .kited/mc 3,0 natapecek -

.mc 5 = h nad mc 4 = r adap aynemulov nahabu

.f fitamroF seT bawaJ icnuK 4 :

1 .)1 = Y . )a = Y nad 1 + X 21 – 1 + X

21

= Y .)b – + X X = Y nad 2

2 .) p skam 02 = Y = X ;004 =

mc 06 = gnajnap )3 = rabeL ; .mc 02 = iggniT nad mc 03

4 .) ttaW 52,624 : gnarukreb P

Vd 5 .) = – mc 6,9 3 = .ted/ – mc 61,03 3 ) gnarukreb( ted/

td

36

III BAB ISAULAVE

naaynatreP .A .1 nakutneT hal irad amatrep nanurut isgnuf - tukireb isgnuf !

a. = Y 6X4 – X3 2 – + X6 3 +7 X f. = Y 5 nis –1 6 X

b ( = Y . 2X 3 – . ) 3 nis X5 ( – ) 7 g Y . 1 ( = – 4X2 hgt .) –1 2X

soc 2 X3 3 e 5X .c = Y - h = Y . -

3 – nis X2 5 2X

d = Y . 2 ( 4X2 – 5 + X 1 ) – 6 i . = Y 9 gol . 7X + X7 gt nl 4

e ( . 4 X2 – 5 + YX 6Y2 = ) 7 -

.2 : irad kimtiragol isaisnerefid nakutneT

4 .X gt 2X = Y .a ( 2 – X5 .) nis 7 . X 39X = Y .b -

soc h 3 X

3 . neT ! ini tukireb kirtemarap naamasrep nanurut nakut

a. nis = Y 5 θ X ; = soc 4 2 θ

.b = Y 5 ( θ + soc 2θ ; ) = X 3 ( 1 – nis 2θ )

4. amatrep laisrap laisnerefid neisifeok aumes halnakutneT irad

isgnuf - kireb isgnuf .ini tu

4X 2 + Y a = V . R 2 b H = Z . -

3 3 X – 5Y

5 . halnakutneT gnuggnis sirag naamasrep da adap lamron sirag n

avruk = Y 5X2 – 7X + 8 kitit id ( 1 , 6 )

46

6. taubid nakA haubes ikgnat ret putut kutnebreb satisapak nagned

rasebes 2 .retil 000 T naruku halnakutne gnat ik nahab raga gnay

? nikgnum tikideses nakulrepid

nagned R .I = E iuhatekid akiJ .7 I = 5 repmA = R nad 3 ,mho

nahaburep halnakutnet E nurunep tabika idajret gnay a n I rasebes

5,0 repmA nakianek nad rasebes R 1 ! mho

8 iraJ . - rakgnil iraj haubes sala na sirdnilis gnubat reb habmat nagned

,0 natapecek 1 reb ayniggnit nad kited/mc gnaruk natapecek nagned

,0 2 aynemulov nahaburep natapecek halnakutneT .kited/mc

akij = r 6 = h nad mc 8 .mc

nabawaJ icnuK .B .1 ‘ Y .a 42 = X3 – X6 – + 6 X ½1 – ½ 6 = ‘ Y .b X2. nis X5( – )7 + ( 01 X 3 – 51 . ) soc X5( – )7

– nis 81 X3 nis .X3 soc 4 + X2 nis. X3 nis 6 +. X2 ) .c Y ’ = -

( 3 – nis X2 ) 2

4( = ‘ Y .d X2 – X5 + ) 1 –7 (– ) 06 + X 69

Yd 8 X – Y5 .e = -

Xd X5 – Y21

03 Yd .f = -

Xd 1 – X63 2

= ‘ Y .g – X8 hgt –1 2 + X 2

e 51 Yd X5 51( – )5 nl 6 .h = -

5 Xd X2

ces 82 9 Yd 2 X7

.i = + - X7 gt 7 nl X Xd

56

– 5 .2 ’ Y .a = ( 2 – X5 .) nis 7 .X 39X ( ) 3 nl 9 + X7 gtoc 7 +

2 – X5

4 .X gt 2X ces 2 1 2 X2 Y .b ‘ = ( + – ) X3 hgt 3

soc h 3 X X gt 2X Yd – soc 5 5 θ 5 Yd – nis 01 2 θ

.a .3 = - .b = - 8 Xd nis 2 θ soc 6 Xd 2 θ

V Z – Y 41 .4 .a = ⅔ R H .b = -

R X3( X – )Y5 2

V Z X 62 = 1⁄3 R2 = -

H X3( Y – )Y5 2

nad 3 + X 3 = Y .5 = Y – 1⁄3 6 + X 1⁄3

iraJ .6 - md 56,31 = iggnit ,md 38,6 = sala iraj

nahabureP .7 E = tlov 5,3 reb ( habmat )

Vd .8 - = 4,2 = .ted/mc .ted/mc 45,7

td

K .C nasululeK airetir

airetirK rokS 1( – )01 toboB ialiN nagnareteK

( fitingoK 8 .ds 1 romon laos ) 5

sulul tarayS laminim ialin

65

isaton silunem naitileteK 1

rudesorp natapeteK 2

nabawaj alumrof natapeteK 1

utkaw natapeteK 1

KA IALIN RIH

66

VI BAB PUTUNEP

ludum halnaikimeD 102 FKT .TAM – luduj nagned 20 isaisnerefiDisgnuF sid iaseles halet ini nusu aparebeb ipakgnelid nagned

icnuk atreseb rihka isaulave nupuam fitamrof set ,sagut/nahital

dom nautnab nagneD .aynnabawaj awsisaham arap nakparahid ini lu

akerem hakapa ,aynisnetepmok nagnabmekrep iridnes uatnamem tapad

raneb halet - adap nimrecret anamiagabes isnetepmok ikilimem raneb

.muleb uata rajaleb nataigek paites adap nakparahid gnay naujut

awsisaham arap igaB laminim nasululek tarays iapacnem halet gnay

adap aynrajaleb nataigek nakitnehgnem tapad akerem akam ludom ini

.ayntukireb ludom ek naktujnalem nad tapad muleb akij aynkilabeS

ilabmek gnalugnem surah akerem akam ,laminim nasululek ihunemem

turet aynrajaleb iretam naigab adap ama - ayniasaukid muleb gnay iretam

( sulul muleb huggnus hibel surah akerem aynkiabes nad ) - huggnus

nautnab kusamret ada gnay satilisaf naktaafnamem nagned rajaleb malad

.ini hailukatam rotatilisaf iagabes nesod irad

76

AD AKATSUP RATF

.3991 .E ,gizsyerK natujnaL kinkeT akitametaM .)nahamejreT( 2 & 1 ukuB

.aidemarG .TP :atrakaJ

,.kkd olisuS namojN .8891 1 diliJ sitilanA irtemoeG nad suluklaK : atrakaJ .aggnalrE

.4891 .R.M ,legeipS natujnaL akitametaM .)nahamejreT( aggnalrE :atrakaJ

.6891 .A.K ,duortS utnu akitametaM kinkeT k .)nahamejreT( :atrakaJ.aggnalrE