P LU s 7

!" #

7

$%'&($*),+.-0/213154

687:9<;>=268?A@2BDCFEHGAIJGAKLE<MON7PKRQTSDCHI BUE3VXWLYZG[G\E]MRG^9<G`_baZG`CH_baZG^;8@.?ACbBUE3VcSDK2GAdcdeG=RdeSR_ABDd"fhgihj2k

[email protected]`BDdqk_AB'n

∼

IJVerKZShE3E3GUnsVeQtEbfUj'fRuvn

w0xPy zU{t|~}~y>{€D.‚hƒqƒP„h…†{t‡3‚ˆc‰RyJ‚h2ƒŠ‡3„bzU|eˆt‹bz

Œ Ž Œ ! Ž ‘P!’“

7 k`k\kAk`k\kAk\k`k\kAkAkAk\k`k\kAk\k`k\kAk ”•G`–3YZC3GpMONVcK2_\G`CFEHV€EHY2MRGDk

737—k\k`k\kAkAkAk\k`kAk\k`k\kAk\k`k\kAkAkAk ”•?\E3a2S'MZG^MRY˜@™SVeKLEšZ›RG[G\E]MRG^œ]GAo†EHSDKRP9<Bh@2a2–FSKžk

737F7ŸkAkAkAk\k`k\kAk\k`k\kAkAkAkAkAkAk\k`k  SK'¡DGACHrDG`K2_\G8deVcKZ?vBhVcC3G[G\E]WLY2BMRCbBUE3VXWLYZGDk

7P¢£kAk\k`k\kAkAkAk\k`k\kAk\k`k\kAkAkAk\k œ]SDCHIJG[MZG^”•BUEHC3VX_\Gk

¢¤k\k`k\kAkAkAk\k`kAk\k`k\kAk\k`k\kAkAkAk ¥™BD_\E3SDCHVX–3BhE3VcSDK

P ^t LU

Bs¡DGv_]?`deVcIJVeK™BUE3VcSDK•¦[BDY2–3–3VcGAKZKZGk

¢87•kAk\k`k\kAkAkAk\k`k\kAk\k`k\kAkAkAk\k 7PKLEHGACH@™SdcBhE3VcSDK˜MRG[dcBDrDCbBhKZrG8GAEœ]GAo†EHSDKžk

 0BhdX_\Y2deG`C8dŠNG`C3CHGAY2C B2–FSdeY2G>GAE[CHGAdXBUEHVe¡G>–3YZC[dcB˜MRY2C3?`G MRGv– S–H_\VcdedXBUEHVeSK2–8Mž[email protected] –3VeIJ@ZdcG MRG deSKZrDYZG`YZC[dqk! B

@.?ACHVeSRMRG[G`–FE]MRSKZKZ?AG[@2BDCdcB>QTSC3I>YZdeG

T = 2π s

l g ,

–3V

π

^" gZkcuAj™u$#

10 ^{− 3}

@ZC&%`–(',VqkGDkc=

∆π = 10 ^{− 3}

^{) =}

l = 1

^I#

10 ^{− 3}

@ZC&%`–`="GAE

g = 9.81

^{I –}

^{− 2}

^#

10 ^{− 2}

@ZC*%v–Ak6]SKZKZGAC dŠNBD@Z@ZCHSs›'VcI BUEHVeSK—MRG

T

G\E]BhCHCHSDK2MRVcC†BhY˜KZSDI+ZCHG MRG^_`–FGpBMR?`WLY2BhE`k

-,!.0/21354

;8K˜B>@.SDY2C¡sBDdeG`YZC<Bh@2@ZC3SR_baZ?`G

T = 6, 282

√ 9.81 ≈ 2.006

^–

,

;8K SREHVeG`KEdXB MRV76.?`C3G`KEHVeG`dedcGp–FY2Ve¡UBhKLEHG

∆T = 2

s l g

∆π +

√ π lg

∆l +

− π

g s

l g

∆g.

–FSV€E

∆T = 2

r 1 9.81

10 ^{− 3} +

3.141

√ 9.81 10 ^{− 3} +

− 3.141

9.81 r 1

9.81

10 ^{− 2} ,

∆T ≈ 6.39 10 ^{− 4} + 1.00 10 ^{− 3} + 1.01 10 ^{− 3} ,

∆T ≈ 2.65 10 ^{− 3} .

;8K SREHVeG`KE<BDY2–3–3VOdŠNG`C3CHGAYZC†CHGAdXBUEHVe¡G

∆T

T = 1.32 10 ^{− 3} .

;8K—E3CHSDYZ¡G[MZSDK2_$'qfJ_`–FGpBD@ZC*%v–0dcB ¡'VeCHrDY2deGp–3SDKLE<_\SDK™–FG`C3¡?`–

) l

T = 2.00

^–Bs¡G`_

∆T = 2.65 10 ^{− 3} .

!"!Ÿ98

:;<=>?;

A@(BCDEF

GH;

CI2KJ

L:;

NMO(P

;8K˜–FG[@[email protected]–3GpMRG E3CHSDYZ¡GACMRG`–<¡sBDdeG`YZCb–0BD@Z@ZCHSR_baZ?AGv–0MZGpdcB CbBD_\VcKZG

r

^MZGpdqN?`WLY2BhE3VcSDK

f (x) = x ³ + x − 1 = 0.

ukRQTSVUXW Y!Z[LZ \-]2Y^`_aUcbd![.k

',B

)

K C3G`IJBDCHWLY2BDKLE8WLYZG dŠN?vWLY2BUEHVeSK•@[email protected]–3?AG^Gv–:E ?`WLYZVc¡sBDdeG`KLE3G+#

g 1 (x) = x

^Bs¡DGv_

g 1 (x) = ₁₊ ¹ x ²

=™IJSDKLEHC3G`C

WLYZGpdŠNVcKLE3GACH¡UBhdcdeG

J 1 = [0, 1]

Gv–:E<YZKŸVcKLE3G`C3¡UBhdcdcG^–FYZCdcG`WLYZG`dždXB _\SDK'¡GACHrDGAK™_\G ¡DGACb–†YZKZG^–3SDdcYREHVeSK Y2KZVcWLYZGpGv–:E BD–H–3YZC3?`GDk

' )

68?\E3G`C3IJVcKZGAC BDK2BhdLE3VXWLYZGAIJGAKLEYZKJI B:SCHBDKEMRG

| r ⁿ − r |

^=US

r ⁿ

MZ?`–3VerKZG†dcB ¡UBhdcGAYZCBD@Z@ZCHS'_ba2?AGD= # dXB KZV7%`IJG VeE3?ACbBUEHVeSKž=ZMRG^_AG\EFEHGpCHB_\VcKZG @2BhCdXB I ?AE3aZSRMRGpVeE3?`CHBhE3Vc¡DGpMRY˜@.SDVcKE†šZ›RGk

'q_

)

K MR?vMRYZVcC3G YZKZGJ¡UBhdcGAYZCpBD@Z@ZCHS'_ba2?AGJMRG

r

^#

10 ^{− 3}

@2C*%v–8G`K @™BhC3EHBhKLEpMRG

r 0 = 1

GAEpBDC3CHSDK™MRVeC BhY KZSDI+ZCHG BDMZ?`WLY2BUE<MRG^_`–FGk

fZkRQTSVUXW Y!Z[LZ [ [$UXY_N] W Y_ k

',B

) SV€E

g 2 (x)

=DdcB8QTSK2_5EHVeSK>VcKLE3GACH¡DG`K2BhKLEMZBhK™– dcB IJ?\EHaZSRMRG†V€EH?ACbBUE3Vc¡DGMRGœ]GAo†E3SK>@.SDY2CdcB CH?`–3SDdcYRE3VcSDK MRG<_\GAEFE3G

?`WLY2BhE3VcSDKžkZ68SDK2KZGAC

g 2 (x)

^k

' )

”ŸSKEHC3G`CWY2Gp@™SYZC

x > 0

^{=ZSDK˜B dqN}VeK2?ArBDdeVeE3?p–3YZVc¡sBDKLE3G

| g ₂ ⁰ (x) | ≤ 1.125 | f (x) | .

'q_

)

”ŸSKEHC3G`C WLYZG˜dqNSDKB YZKZG•–3SDdcYRE3VcSDK YZKZVXWLYZG•–3YZC dqNVcKEHGACH¡UBhdcdeG

J 2 = [0.5, 0.75]

^G\E WLYZGŸdcGŸ_baZSDVe›(MRG•_AG\E VcKEHGACH¡UBhdcdeG[@.GACHI GAE]MONBD–H–FYZCHGAC†dXBJ_\SDK'¡GACHrDGAK™_\G8MZGpdcB IJ?\EHaZSRMRG^MRG^œ<G`o†E3SKžk

',M

)

C3SYZ¡DG`C†BDK2BhdLE3VXWLYZGAIJGAKLEYZKZGpI B:SDCbBUE3VcSDK—MRY [email protected]

| r ⁿ +1 − r | ≤ K | r ⁿ +1 − r ⁿ | .

',G

)

K MR?vMRYZVcC3G YZKZGJ¡UBhdcGAYZCpBD@Z@ZCHS'_ba2?AGJMRG

r

^#

10 ^{− 6}

@2C*%v–8G`K @™BhC3EHBhKLEpMRG

r 0 = 1

GAEpBDC3CHSDK™MRVeC BhY KZSDI+ZCHG BDMZ?`WLY2BUE<MRG^_`–FGk

-,!.0/21354

BZka N?\EHY2MRG MRG`– ¡UBhCHVcBhE3VcSDK2–8MRG dXB QTSK2_5EHVeSK

f (x)

^–3YZC

[0, 1]

IJSDKLE3CHG>WLYZG dcB QTSDK2_\E3VcSDK•Gv–:E[_\SKEHVeK'YZG>G\Ep_\CHSDVX–3–HBhKLEHG

–3YZC]_\GAE<VeKLEHGACH¡sBDdedcGDkZ68G^@ZdcY2–`=ZSDK

f (0) = − 1

^G\E

f (1) = 1

kR7PdG\›RVc–FE3GpMRSK2_[YZKZGpCbBD_AVeK2G

r

YZK2VcWLYZG^MZBDK2–_\GAE]VcKEHGACH¡UBhdcdeGk žGv–MR?ACHVc¡D?AGv–0@ZC3G`IJV7%`C3Gv–†G\E]–3G`_\SK2MRGv–0MZG

g 1 (x)

^–AN?`_AC3Vc¡DG`KE

g ₁ ⁰ = − 2x

(1 + x ² ) ² , g ⁰⁰ ₁ = 2 3x ² − 1 (1 + x ² ) ³ .

7Pd"Gv–:E _\dXBhVcC8WLYZG^@.SDYZC]E3SYRE

x ∈ J 1

=2SDK•B

| g ⁰ ₁ (x) | < 1

^{',_ABDC l}

2x ≤ 1 + x ² ≤ (1 + x ² ) ²

^{) k™68SDK2_} dcB _ASDK'¡DG`C3rGAK2_AG[G`–FE BD–H–3YZC3?`GDk

žk268GpI BhKZV%ACHGp@ZdcY2–@ZCH?`_AVc–3GD=RG`K˜?\E3Y™MRVcBDKLE<deGv–¡UBhCHVcBhE3VcSDK2–MRG

g ⁰ ₁ (x)

^=2SK ¡SDVeE<WY2G l

∀ x ∈ J 1

=

| g ⁰ ₁ (x) | < 0.65

^{k K}

G6’G\Ev=ZSDK˜BZ=

x 0 ^√ ₃ ³ 1

g ₁ ⁰⁰ (x) − 0 +

g ⁰ ₁ (x) 0 & m % − 0.5

Bs¡DGv_

m = − ^{3 √} ₈ ³ ≈ − 0.6495

^kZ;8K•B MRSDK2_^dcB I B:SDCbBUE3VcSDK˜–3YZVe¡UBDKEHG MRG^dŠNGACHC3G`YZC]_\SIJI VX–3GpGAK•@ZCHGAK™BhKLE<dcB ¡sBDdeG`YZC Bh@2@ZC3SR_baZ?`G

r n

MZG

r

'q_5Q:k. SYZCH–

) k

| r n − r | ≤ 0.65 ⁿ .

_hk KŸYREHVedcVc–HBhKLE]dXB @ZCH?`_A?`MRG`KLE3GpI B:SCHBhE3VcSDKž=RSK•BhYZCbB

| r ⁿ − r | < 10 ^{− 3}

^=.–FV

0.65 ⁿ < 10 ^{− 3}

^=™SY

n > ^{ln(10 −}

3 ) ln(0 . 65)

=™–3SDVeE

@.SDYZC

n ≥ 17

k2;8K˜SREHVeG`KEv=

r 0 = 1, r 1 = 0.5, r 2 = 0.8, r 3 = 0.60, r 4 = 0.72, r 5 = 0.65, r 6 = 0.70, r 7 = 0.67, r 8 = 0.68, r 9 = 0.67, r 10 = 0.685, r 11 = 0.680, r 12 = 0.683, r 13 = 0.681, r 14 = 0.6828, r 15 = 0.6820, r 16 = 0.6825, r 17 = 0.6822.

;8K˜B

| r ⁿ − r | < 10 ^{− 3} < 0.5 × 10 ^{− 2}

G\E]MRSK2_[deG[CH?`–3YZdeEHBUE<–`NGA›R@ZC3VcIJGpBs¡DG`_

2

_A–3G^Bh@ZC&%`–0dXB ¡LVcCHrDYZdcG[@2BhC

r 17 = 0.68

^k

;8K˜B

g 2 (x) = x − f (x)

f ⁰ (x) = 2x ³ + 1 3x ² + 1 .

g 2 (x)

B>@.SDYZC8MR?ACHVe¡?AG

g ⁰ ₂ (x) = 6x(x ³ + x − 1)

(3x ² + 1) ² = 6x

(3x ² + 1) ² f (x).

"B>QTSK2_5EHVeSK

h(x) = ₍₃ x ^{6 2 x} +1) ²

BhEFE3G`VeKLEYZK˜I BU›RVcI YZI?ArBDd #

1.125

^',@™SYZC

x = ¹ ₃

⁾ =Z@™BhC<–3YZV€[email protected]

x > 0

⁼

| g ⁰ ₂ (x) | ≤ 1.125 | f (x) | .

N?\EHY2MRGJMRG`– ¡UBhCHVcBhE3VcSDK2–8MRG dcB QTSDK2_\E3VcSDK

f (x)

^–FY2C

J 2 = [0.5, 0.75]

IJSDKLEHC3G WY2G>dXB QTSK2_5EHVeSK G`–FEp_\SDKLEHVeK'YZG>G\E _\CHSDVX–H–3BDKEHGp–FYZC8_\G\E<VcKLE3G`C3¡UBhdcdcGDk268G^@ZdcY2–`=2SDK

f (0.5) = − 0.375

^G\E

f (0.75) = 0.171875

kL7Pd"G\›RVc–FE3G>MRSDK2_[Y2KZG^CbBD_\VcKZG

r

YZKZVXWLYZG^MZBDK2–_\GAE<VeKLE3G`C3¡UBDdedcGDk™;8K˜BJ–FYZC

J 2

l

− 0.375 ≤ f (x) ≤ 0.172,

G\E<@™BhC<–3YZV€EHGD=

| g ⁰ ₂ (x) | < 0.43,

MOkBhC<–3YZVeE3G=ZSDK SREHVeG`KLE`=

1

| 1 − g ₂ ⁰ (x) | < 1.76

^–FY2C

J 2 .

;8K SREHVeG`KEYZK2GpIJB:SDCbBUEHVeSK—MRY E'@™GC'q_5Q:k268?AIJS fZ=RG\›RGACb_\VX_\Gpf

) l

| r ⁿ +1 − r | ≤ 1.76 | r ⁿ +1 − r ⁿ | .

GDk K YZE3VcdeVX–3BDKLE<dcB @ZCH?`_A?`MRG`KEHG[IJB:SDCbBUEHVeSKž='SDK SREHVeG`KEv=

r 0 = 1, r 1 = 0.75,

r 2 = 0.6860465116, r 3 = 0.6823395826, r 4 = 0.6823278039, r 5 = 0.6823278038.

;8K MRSV€E[BhCHCAE3G`C # dXB _\VcK2WLYZV%AIJG VeE3?`CHBhE3VcSDK 'q_ABDC

| r 5 − r | ≤ 1.76 | r 5 − r 4 | = 1.76 × 10 ^{− 10} < 10 ^{− 6}

^{) G\E} SDKŸEHC3SYZ¡DG=

GAKŸBDC3CHSDK2MZVc–H–3BDKLE0BhY KZSIOZCHGpMRGp_A–3G[BMR?`WLY2BhE

r 5 = 0.68232

^k

!"!! Œ ;

8JO

E$

J

J

$ (P

SV€E]MRG`YR›˜I ?AE3aZSRMRGv–<VeE3?`CHBhE3Vc¡DG`–]MRSDKLE]dXB _\SK2–:EbBhKLE3G^B– 'IJ@REHShE3VXWLYZG^Gv–:E

C = 0.75

GAE8_\SK'¡DGACHrDGvBhKLE†deVcKZ?`BDVeCHGAIJG`KE

@.SDYZC dXBm@ZCHGAIJV%ACHG GAE WLY2BDMRCbBUEHVcWLYZG`IJGAKLE @™SYZC dXB –FGv_\SDK™MRGDk0 0BDdc_AYZdcGAC dcGŸKZSDI+ZC3GŸMONVeE3?ACbBUEHVeSKI VcKZVcI Bhd<@.SDYZC—WY2G

dŠNG`C3CHGAYZC^MžNBD@Z@ZCHSs›'VcI BUEHVeSK KžNG\›Z_ %`MRG @™BD–

10 ^{− 8}

M2BhK2–pdcG`– MRGAYR› _AB–^–HBD_ba2BDKE^WLYZG dŠNGACHC3G`YZC #ŸdXB˜@ZC3G`IJV7%`C3G VeE3?`CHBhE3VcSDK KžNG\›Z_%vMRG[@2BD–

0.5

^'TVŠkGDkc=

e 0 = 0.5

^{) k}

-,!.0/21354

;8K˜B>@.SDY2CdcB @ZCHGAIJV%ACHG GAE<dcBJMRG`YR›RV7%`IJGpI ?AE3aZSRMRGpCHG`–[email protected]`_5EHVe¡GAIJGAKLEv=

| e n +1 |

| e ⁿ | ≈ 0.75,

^GAE

| e n +1 |

| e ⁿ | ² ≈ 0.75.

SDYZCdXB IJ?\E3a2S'MZG^MRG^_\SKL¡GACHrDG`K2_\G8deVcKZ?`BDVeCHGD=RSK B2=

| e n | ≈ 0.75 | e n − 1 | ≈ (0.75) ² | e n − 2 | ≈ . . . ≈ (0.75) ⁿ | e 0 | .

SDYZCdXB IJ?\E3a2S'MZG^MRG^_\SKL¡GACHrDG`K2_\G WY™BDMRCbBUEHVcWLYZG=LSKŸB

| e ⁿ | ≈ 0.75 | e ⁿ ₋ 1 | ² ≈ (0.75)(0.75 | e ⁿ ₋ 2 | ² ) ² = (0.75) ³ | e ⁿ ₋ 2 | ⁴ ≈ . . . ≈ (0.75) ^{2 n − 1} | e 0 | ²

n

.

N?AK2SDK2_A? KZSY2–8MZGAI BhK2MZG>MRGJ–FY2@Z@™SL–FG`C WLYZG

| e 0 | = 0.5

MZBhK2–8dcG`– MRGAYR› _`BD–`k SYZC8dXB IJ?AE3aZSRMRGJMRGJ_\SDK'¡GACHrDGAK™_\G dcVeKZ?vBhVcC3G[SK B2=

e ⁿ = (0.75) ⁿ (0.5) ≤ 10 ^{− 8}

^@™SYZC

n ≥ ln 2 − 8 ln 10 ln(0.75) ≈ 62.

SDYZCdXB IJ?\E3a2S'MZG^WY™BDMRCbBUEHVcWLYZG SDKŸBZ=

e ⁿ = (0.75) ^{2 n − 1} (0.5) ^{2 n} ≤ 10 ^{− 8} , e n = (0.75) ^{− 1} (0.375) ^{2 n} ≤ 10 ^{− 8} .

2 ⁿ ≥ ln(0.75) − 8 ln 10 ln(0.375) , n ≥ 4.2535.

;8K @.GAYREJC3G`I BhCbWY2GAC WLYZG—dcB I ?AE3aZSRMRG MRG—_ASDK'¡DG`C3rGAK2_AG WLY2BDMRCbBUEHVcWLYZG MZGAI BhK2MZG(.G`BhY™_\SDY2@ IJSDVcK2–>MžNVeE3?`CHBhE3VcSDK

WLYZG[dcBJIJ?AE3aZSRMRG^MRGp_ASDK'¡DG`C3rGAK2_AG]dcVeKZ?vBhVcC3Gk

! •

;0

9J

N(

”ŸSKEHC3G`CWY2G @.SDY2CYZKZGpI BUEHC3VX_\Gp–LIJ?AE3CHVcWLYZG

A

=2SDK˜BJdqN?ArLBhdcV€EH?[–3YZVc¡UBhKLE3G=

_\SK2M!'

)

1 =

^{_\SK2M!' )}

_∞ .

-,!.0/21354

BhCMR?Aš2KZVeE3VcSDKž=ZSKŸBZ=

_\SK2M!'

)

= k A k . k A ^{− 1} k .

SDYZCY2KZGpI BUE3CHVX_\Gp– 'IJ?\EHC3VXWLYZGD=

A = A ^t

^=ZGAE

k A k 1 = k A ^t k 1 = k A k ∞ .

68Gp@ZdeY™–SDK˜BZ=

k A ^{− 1} k 1 = k (A ^t ) ^{− 1} k 1 = k (A ^{− 1} ) ^t k 1 = k A ^{− 1} k ∞ .

BhC<_ASDK2–3?`WLYZG`KEv=Z–FVOdXB I BUE3CHVX_\G[G`–FE]–'IJ?\EHC3VXWLYZGD=

_\SK2M!'

)

1 =

^{_\SK2M!' )}

_∞ .

(JO

; J ;

P ^t LU

^{J =8 J ; J NMP}

uk<6]?v_\[email protected]–3GACdXBJI BUE3CHVX_\G

A

G`KŸ@ZC3SRMRY2V€E

P ^t LU

^SDY

P

G`–FE<dXB I BUE3CHVX_\G^[email protected] YREbBUEHVeSK @™BhC<dXBJIJ?\E3a2S'MZG MON?AdcV€

I VcK2BhE3VcSDK˜MRG ¦[BhY2–H–†G\E@ZVc¡DSDEHBDrDG @2BhC3E3VcGAdŠk

fZk8 0BhdX_\YZdcGAC†dcG^MR?\EHGACHI VcK2BDKE<MRG

A

^k

g2k<9?`–3SDY2MZC3G deG^– R–:E&%AIJG

Ax = b

^@.SDY2C

b = (1, 2, 3) ^t

^k

A =





2 − 1 0 4 − 1 2

− 6 2 0



 .

-,!.0/21354

;8K VeKLE3G`C3¡GAC3E3VeEdcB dcVerKZG

1

^{GAE<dcB dcVerKZG}

3

_\GpWLYZV"MRSKZKZG

A =





− 6 2 0 4 − 1 2 2 − 1 0



 .

NSD@.?ACbBUEHVeSK dcVerKZG

2 =

^deVcrDK2G

₂ − ( − 2/3)

^dcVerKZG

₁

G\E<dcVerKZG

3 =

^deVcrDKZG

₃ − ( − 1/3)

^dcVerKZG

₁

^MZSDKZKZG=

A =





− 6 2 0

0 (1/3) 2 0 ( − 1/3) 0



 .

NSD@.?ACbBUEHVeSK dcVerKZG

3 =

^deVcrDK2G

₃ − ( − 1)

^deVcrDKZG

₂

^MRSKZKZGD=

A =





− 6 2 0 0 (1/3) 2

0 0 2



 .

;8K˜B MZSDK2_ dXB MR?`_ASDIJ@™SL–FVeE3VcSDK –FY2Ve¡UBhKLEHG l

A =





− 6 2 0 4 − 1 2 2 − 1 0



 =





0 0 1 0 1 0 1 0 0





| {z }

P ^t





1 0 0

( − 2/3) 1 0 ( − 1/3) − 1 1





| {z }

L





− 6 2 0 0 (1/3) 2

0 0 2





| {z }

U

.

MRG\E '

)

=

^{MRGAE ' )}

×

^{MZG\E ' )}

×

^{MRGAE ' )}

,

= − 1 × 1 × ( − 6 × (1/3) × 2),

= 4.





1 0 0

( − 2/3) 1 0 ( − 1/3) − 1 1









− 6 2 0 0 (1/3) 2

0 0 2



 =



 3 2 1



 .

BhC[–FY 2–:EHV€EHYRE3VcSDKmBs¡UBhKLE[@ZYZVX– BDC3CHV%ACHG^SDK E3CHSDYZ¡GD=

Ly = b

^Bs¡DGv_

y = (3, 4, 6) ^t

^=O@2YZVc–

U x = y

Bs¡G`_^š2K2BhdcGAIJG`KE

x = (( − 15/6), − 6, 3) ^t

^k

! Ÿ!

?; J ; J

0 J

O=E

GH;

M (

SV€EdcG`–@.SDVcKLEH––3YZVe¡UBDKEb–

x ^k

^{u j}

y ^k

^{u bu u bu}

uk <@Z@ZdcVXWY2G dXB QTSC3I>YZdcG>MRGc "[email protected]]EHC3SYZ¡DG`C]YZK @™Sd'KZSI G MRG>MZGArDCH?pEHC3SVc– WLYZV@™BD–H–FG>@2BhC _\G`–[email protected]–Ak

¡UBhdcYZG[GAK™–FYZVeE3G^_AGp@™Sd'KZSIJG @.SDY2C

x = 2, 3, 5

^k

fZk <@Z@ZdcVXWY2G dXB QTSDCHI YZdcG MRG œ]GAo†EHSDK @.SDYZCpEHC3SYZ¡DG`CpYZK @.SDdLK2SDIJG MRG MZGArDCH? E3CHSDVX–^WLYZV @™BD–H–FG @2BDC _\Gv–[email protected]–Ak

¡UBhdcYZG[GAK™–FYZVeE3G^_AGp@™Sd'KZSIJG @.SDY2C

x = 2, 3, 5

^k

-,!.0/21354

;8K—E3CHSDYZ¡G8@2BDCR "BDrDCbBhKZrG l

P 3 (x) = (x − 1)(x − 4)(x − 6)

− 24 − x(x − 4)(x − 6)

15 + x(x − 1)(x − 6)

− 24 − x(x − 1)(x − 4)

60 .

SDYZCdŠNVeKLE3G`[email protected]—SK EHC3SYZ¡DG=

P (2) = − 1, P (3) = 0, P (5) = 1.

žG EbB ZdcG`BDY MRGv–MRV76.?`C3G`K2_\Gv–<MRVc¡LVX–3?AG`––`N?`_\CHVeE`=

x y ∆y ∆ ² y ∆ ³ y

u

Pf

u bu fLnUg

'qfLnUg

)  ':usn )

j u HuUnUg

Hu

bu

;8K SREHVeG`KEdcGp@™Sd'KZSIJG[–3YZVc¡UBhKLE^l

P 3 (x) = 1 − 2x + (2/3)x(x − 1) − (1/6)x(x − 1)(x − 4).

SDYZCdŠNVeKLE3G`[email protected]>E3CHSDYZ¡G0dXB[I AIJG<_baZSL–[email protected] dXBhrCHBDKZrDG†[email protected]'KZSDIJG<MRG<_ASDdcdeSR_`BUE3VcSDKJMRG<MRG`rDCH?0E3CHSDVX–

G`–FE<YZKZVXWLYZGDk

P (2) = − 1,

P (3) = 0,

P (5) = 1.