P LU s 7

(1)

!" #

7

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

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

aLE3E3@ml.nDnUooopkVcCHS2kYZIJSDKLE3CHG`BDdqk_AB'n

∼

IJVerKZShE3E3GUnsVeQtEbfUj'fRuvn

w0xPy zU{t|~}~y>{D.hqPh{t3cRyJh23bzU|etbz

! P!

7 k`k\kAk`k\kAk\k`k\kAkAkAk\k`k\kAk\k`k\kAk G`3YZC3GpMONVcK2_\G`CFEHVEHY2MRGDk

737k\k`k\kAkAkAk\k`kAk\k`k\kAk\k`k\kAkAkAk ?\E3a2S'MZG^MRY@SVeKLEZRG[G\E]MRG^]GAoEHSDKRP9<Bh@2a2FSKk

737F7kAkAkAk\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 ]SDCHIJG[MZG^BUEHC3VX_\Gk

¢¤k\k`k\kAkAkAk\k`kAk\k`k\kAk\k`k\kAkAkAk ¥BD_\E3SDCHVX3BhE3VcSDK

P ^t LU

Bs¡DGv_]?`deVcIJVeKBUE3VcSDK¦[BDY233VcGAKZKZGk

¢87kAk\k`k\kAkAkAk\k`k\kAk\k`k\kAkAkAk\k 7PKLEHGACH@SdcBhE3VcSDKMRG[dcBDrDCbBhKZrG8GAE]GAoEHSDKk

(2)

0BhdX_\Y2deG`C8dNG`C3CHGAY2C B2FSdeY2G>GAE[CHGAdXBUEHVe¡G>3YZC[dcBMRY2C3?`G MRGv SH_\VcdedXBUEHVeSK28MNYZKm@.GAK2MZYZdeG 3VeIJ@ZdcG MRG deSKZrDYZG`YZC[dqk! B

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

T = 2π s

l g ,

3V

π

^" gZkcuAju$#

10 ⁻ ³

@ZC&%`(',VqkGDkc=

∆π = 10 ⁻ ³

⁾ ⁼

l = 1

^I#

10 ⁻ ³

@ZC&%``="GAE

g = 9.81

^I

⁻ ²

^#

10 ⁻ ²

@ZC*%vAk6]SKZKZGAC dNBD@Z@ZCHSs'VcI BUEHVeSKMRG

T

G\E]BhCHCHSDK2MRVcCBhYKZSDI+ZCHG MRG^_`FGpBMR?`WLY2BhE`k

-,!.0/21354

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

T = 6, 282

√ 9.81 ≈ 2.006

,

;8K SREHVeG`KEdXB MRV76.?`C3G`KEHVeG`dedcGpFY2Ve¡UBhKLEHG

∆T = 2

s l g

∆π +

√ π lg

∆l +

− π

g s

l g

∆g.

FSVE

∆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<BDY233VOdNG`C3CHGAYZCCHGAdXBUEHVe¡G

∆T

T = 1.32 10 ⁻ ³ .

;8KE3CHSDYZ¡G[MZSDK2_$'qfJ_`FGpBD@ZC*%v0dcB ¡'VeCHrDY2deGp3SDKLE<_\SDKFG`C3¡?`

) l

T = 2.00

^Bs¡G`_

∆T = 2.65 10 ⁻ ³ .

!"!98

:;<=>?;

A@(BCDEF

GH;

CI2KJ

L:;

NMO(P

;8KFG[@ZCHSD@.S3GpMRG E3CHSDYZ¡GACMRG`<¡sBDdeG`YZCb0BD@Z@ZCHSR_baZ?AGv0MZGpdcB CbBD_\VcKZG

r

^MZGpdqN?`WLY2BhE3VcSDK

f (x) = x ³ + x − 1 = 0.

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

(3)

',B

)

K C3G`IJBDCHWLY2BDKLE8WLYZG dN?vWLY2BUEHVeSK@ZCHSD@.S3?AG^Gv:E ?`WLYZVc¡sBDdeG`KLE3G+#

g 1 (x) = x

^Bs¡DGv_

g 1 (x) = ₁₊ ¹ x ²

=IJSDKLEHC3G`C

WLYZGpdNVcKLE3GACH¡UBhdcdeG

J 1 = [0, 1]

Gv:E<YZKVcKLE3G`C3¡UBhdcdcG^FYZCdcG`WLYZG`ddXB _\SDK'¡GACHrDGAK_\G ¡DGACbYZKZG^3SDdcYREHVeSK Y2KZVcWLYZGpGv:E BDH3YZC3?`GDk

' )

68?\E3G`C3IJVcKZGAC BDK2BhdLE3VXWLYZGAIJGAKLEYZKJI B:SCHBDKEMRG

| r ⁿ − r |

^=US

r ⁿ

MZ?`3VerKZGdcB ¡UBhdcGAYZCBD@Z@ZCHS'_ba2?AGD= # dXB KZV7%`IJG VeE3?ACbBUEHVeSK=ZMRG^_AG\EFEHGpCHB_\VcKZG @2BhCdXB I ?AE3aZSRMRGpVeE3?`CHBhE3Vc¡DGpMRY@.SDVcKEZRGk

'q_

)

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

r

^#

10 ⁻ ³

@2C*%v8G`K @BhC3EHBhKLEpMRG

r 0 = 1

GAEpBDC3CHSDKMRVeC BhY KZSDI+ZCHG BDMZ?`WLY2BUE<MRG^_`FGk

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

',B

) SVE

g 2 (x)

=DdcB8QTSK2_5EHVeSK>VcKLE3GACH¡DG`K2BhKLEMZBhK dcB IJ?\EHaZSRMRGVEH?ACbBUE3Vc¡DGMRG]GAoE3SK>@.SDY2CdcB CH?`3SDdcYRE3VcSDK MRG<_\GAEFE3G

?`WLY2BhE3VcSDKkZ68SDK2KZGAC

g 2 (x)

^k

' )

SKEHC3G`CWY2Gp@SYZC

x > 0

^=ZSDKB ^dqNVeK2?ArBDdeVeE3?p3YZVc¡sBDKLE3G

| g ₂ ⁰ (x) | ≤ 1.125 | f (x) | .

'q_

)

SKEHC3G`C WLYZGdqNSDKB YZKZG3SDdcYRE3VcSDK YZKZVXWLYZG3YZC dqNVcKEHGACH¡UBhdcdeG

J 2 = [0.5, 0.75]

^G\E WLYZGdcG_baZSDVe(MRG_AG\E VcKEHGACH¡UBhdcdeG[@.GACHI GAE]MONBDHFYZCHGACdXBJ_\SDK'¡GACHrDGAK_\G8MZGpdcB IJ?\EHaZSRMRG^MRG^<G`oE3SKk

',M

)

C3SYZ¡DG`CBDK2BhdLE3VXWLYZGAIJGAKLEYZKZGpI B:SDCbBUE3VcSDKMRY EL@.G

| r ⁿ +1 − r | ≤ K | r ⁿ +1 − r ⁿ | .

',G

)

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

r

^#

10 ⁻ ⁶

@2C*%v8G`K @BhC3EHBhKLEpMRG

r 0 = 1

GAEpBDC3CHSDKMRVeC BhY KZSDI+ZCHG BDMZ?`WLY2BUE<MRG^_`FGk

-,!.0/21354

BZka N?\EHY2MRG MRG` ¡UBhCHVcBhE3VcSDK28MRG dXB QTSK2_5EHVeSK

f (x)

^3YZC

[0, 1]

IJSDKLE3CHG>WLYZG dcB QTSDK2_\E3VcSDKGv:E[_\SKEHVeK'YZG>G\Ep_\CHSDVX3HBhKLEHG

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

f (0) = − 1

^G\E

f (1) = 1

kR7PdG\RVcFE3GpMRSK2_[YZKZGpCbBD_AVeK2G

r

YZK2VcWLYZG^MZBDK2_\GAE]VcKEHGACH¡UBhdcdeGk GvMR?ACHVc¡D?AGv0@ZC3G`IJV7%`C3GvG\E]3G`_\SK2MRGv0MZG

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

=2SDKB

| g ⁰ ₁ (x) | < 1

^',_ABDC ^l

2x ≤ 1 + x ² ≤ (1 + x ² ) ²

⁾ ^k68SDK2_ dcB _ASDK'¡DG`C3rGAK2_AG[G`FE BDH3YZC3?`GDk

k268GpI BhKZV%ACHGp@ZdcY2@ZCH?`_AVc3GD=RG`K?\E3YMRVcBDKLE<deGv¡UBhCHVcBhE3VcSDK2MRG

g ⁰ ₁ (x)

^=2SK ¡SDVeE<WY2G l

∀ x ∈ J 1

=

| g ⁰ ₁ (x) | < 0.65

^k ^K

G6G\Ev=ZSDKBZ=

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;8KB MRSDK2_^dcB I B:SDCbBUE3VcSDK3YZVe¡UBDKEHG MRG^dNGACHC3G`YZC]_\SIJI VX3GpGAK@ZCHGAKBhKLE<dcB ¡sBDdeG`YZC Bh@2@ZC3SR_baZ?`G

r n

MZG

r

'q_5Q:k. SYZCH

) k

| r n − r | ≤ 0.65 ⁿ .

(4)

_hk KYREHVedcVcHBhKLE]dXB @ZCH?`_A?`MRG`KLE3GpI B:SCHBhE3VcSDK=RSKBhYZCbB

| r ⁿ − r | < 10 ⁻ ³

^=.FV

0.65 ⁿ < 10 ⁻ ³

^=SY

n > ^ln(10 ⁻

3 ) ln(0 . 65)

=3SDVeE

@.SDYZC

n ≥ 17

k2;8KSREHVeG`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.

;8KB

| r ⁿ − r | < 10 ⁻ ³ < 0.5 × 10 ⁻ ²

G\E]MRSK2_[deG[CH?`3YZdeEHBUE<`NGAR@ZC3VcIJGpBs¡DG`_

2

_A3G^Bh@ZC&%`0dXB ¡LVcCHrDYZdcG[@2BhC

r 17 = 0.68

^k

;8KB

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`VeKLEYZKI BURVcI YZI?ArBDd #

1.125

^',@SYZC

x = ¹ ₃

⁾ =Z@BhC<3YZVEHGD=R@.SDYZC

x > 0

⁼

| g ⁰ ₂ (x) | ≤ 1.125 | f (x) | .

N?\EHY2MRGJMRG` ¡UBhCHVcBhE3VcSDK28MRG dcB QTSDK2_\E3VcSDK

f (x)

^FY2C

J 2 = [0.5, 0.75]

IJSDKLEHC3G WY2G>dXB QTSK2_5EHVeSK G`FEp_\SDKLEHVeK'YZG>G\E _\CHSDVXH3BDKEHGpFYZC8_\G\E<VcKLE3G`C3¡UBhdcdcGDk268G^@ZdcY2`=2SDK

f (0.5) = − 0.375

^G\E

f (0.75) = 0.171875

kL7Pd"G\RVcFE3G>MRSDK2_[Y2KZG^CbBD_\VcKZG

r

YZKZVXWLYZG^MZBDK2_\GAE<VeKLE3G`C3¡UBDdedcGDk;8KBJFYZC

J 2

l

− 0.375 ≤ f (x) ≤ 0.172,

G\E<@BhC<3YZVEHGD=

| g ⁰ ₂ (x) | < 0.43,

(5)

MOkBhC<3YZVeE3G=ZSDK SREHVeG`KLE`=

1 | 1 − g ₂ ⁰ (x) | < 1.76

^FY2C

J 2 .

;8K SREHVeG`KEYZK2GpIJB:SDCbBUEHVeSKMRY 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 YZE3VcdeVX3BDKLE<dcB @ZCH?`_A?`MRG`KEHG[IJB:SDCbBUEHVeSK='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 MRSVE[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 SDKEHC3SYZ¡DG=

GAKBDC3CHSDK2MZVcH3BDKLE0BhY KZSIOZCHGpMRGp_A3G[BMR?`WLY2BhE

r 5 = 0.68232

^k

!"!! ;

8JO

E$

J

$ (P

SVE]MRG`YRI ?AE3aZSRMRGv<VeE3?`CHBhE3Vc¡DG`]MRSDKLE]dXB _\SK2:EbBhKLE3G^B 'IJ@REHShE3VXWLYZG^Gv:E

C = 0.75

GAE8_\SK'¡DGACHrDGvBhKLEdeVcKZ?`BDVeCHGAIJG`KE

@.SDYZC dXBm@ZCHGAIJV%ACHG GAE WLY2BDMRCbBUEHVcWLYZG`IJGAKLE @SYZC dXB FGv_\SDKMRGDk0 0BDdc_AYZdcGAC dcGKZSDI+ZC3GMONVeE3?ACbBUEHVeSKI VcKZVcI Bhd<@.SDYZCWY2G

dNG`C3CHGAYZC^MNBD@Z@ZCHSs'VcI BUEHVeSK KNG\Z_ %`MRG @BD

10 ⁻ ⁸

M2BhK2pdcG` MRGAYR _AB^HBD_ba2BDKE^WLYZG dNGACHC3G`YZC #dXB@ZC3G`IJV7%`C3G VeE3?`CHBhE3VcSDK KNG\Z_%vMRG[@2BD

0.5

^'TVk^GDkc=

e 0 = 0.5

⁾ ^k

-,!.0/21354

;8KB>@.SDY2CdcB @ZCHGAIJV%ACHG GAE<dcBJMRG`YRRV7%`IJGpI ?AE3aZSRMRGpCHG`3@.G`_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 WYBDMRCbBUEHVcWLYZG=LSKB

| 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

.

(6)

N?AK2SDK2_A? KZSY28MZGAI BhK2MZG>MRGJFY2@Z@SLFG`C WLYZG

| e 0 | = 0.5

MZBhK28dcG` 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^WYBDMRCbBUEHVcWLYZG SDKBZ=

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 WLYZGdcB I ?AE3aZSRMRG MRG_ASDK'¡DG`C3rGAK2_AG WLY2BDMRCbBUEHVcWLYZG MZGAI BhK2MZG(.G`BhY_\SDY2@ IJSDVcK2>MNVeE3?`CHBhE3VcSDK

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

!

;0

9J

N(

SKEHC3G`CWY2G @.SDY2CYZKZGpI BUEHC3VX_\GpLIJ?AE3CHVcWLYZG

A

=2SDKBJdqN?ArLBhdcVEH?[3YZVc¡UBhKLE3G=

_\SK2M!'

)

1 =

^_\SK2M!' ⁾

_∞ .

-,!.0/21354

BhCMR?A2KZVeE3VcSDK=ZSKBZ=

_\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@ZdeYSDKBZ=

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

BhC<_ASDK23?`WLYZG`KEv=ZFVOdXB I BUE3CHVX_\G[G`FE]'IJ?\EHC3VXWLYZGD=

_\SK2M!'

)

1 =

^_\SK2M!' ⁾

_∞ .

(7)

(JO

; J ;

P ^t LU

^J ⁼⁸ ^J ^; ^J ^NMP

uk<6]?v_\SDIJ@.S3GACdXBJI BUE3CHVX_\G

A

G`K@ZC3SRMRY2VE

P ^t LU

^SDY

P

G`FE<dXB I BUE3CHVX_\G^MRGp@.GACHI YREbBUEHVeSK @BhC<dXBJIJ?\E3a2S'MZG MON?AdcV

I VcK2BhE3VcSDKMRG ¦[BhY2HG\E@ZVc¡DSDEHBDrDG @2BhC3E3VcGAdk

fZk8 0BhdX_\YZdcGACdcG^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



 .

;8KB MZSDK2_ dXB MR?`_ASDIJ@SLFVeE3VcSDK 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.

(8)





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:EHVEHYRE3VcSDKmBs¡UBhKLE[@ZYZVX BDC3CHV%ACHG^SDK E3CHSDYZ¡GD=

Ly = b

^Bs¡DGv_

y = (3, 4, 6) ^t

^=O@2YZVc

U x = y

Bs¡G`_^2K2BhdcGAIJG`KE

x = (( − 15/6), − 6, 3) ^t

^k

! !

?; J ; J

0 J

O=E

GH;

M (

SVEdcG`@.SDVcKLEH3YZVe¡UBDKEb

x ^k

^u ^j

y ^k

^u ^bu ^u ^bu

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x = 2, 3, 5

^k

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x = 2, 3, 5

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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 .

SDYZCdNVeKLE3G`C3@.SDdXBUEHVeSKSK EHC3SYZ¡DG=

P (2) = − 1, P (3) = 0, P (5) = 1.

G EbB ZdcG`BDY MRGvMRV76.?`C3G`K2_\Gv<MRVc¡LVX3?AG``N?`_\CHVeE`=

(9)

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).

SDYZCdNVeKLE3G`C3@.SDdXBUEHVeSKJSDK>E3CHSDYZ¡G0dXB[I AIJG<_baZSLFGWLYZG@.SDYZC dXBhrCHBDKZrDG_ABDCdeG@.SDd'KZSDIJG<MRG<_ASDdcdeSR_`BUE3VcSDKJMRG<MRG`rDCH?0E3CHSDVX

G`FE<YZKZVXWLYZGDk

P (2) = − 1,

P (3) = 0,

P (5) = 1.

P LU s 7

7

∼

P t LU

T = 2π s

l g ,

π

10 − 3

∆π = 10 − 3

l = 1

10 − 3

g = 9.81

− 2

10 − 2

T

T = 6, 282

√ 9.81 ≈ 2.006

,

∆T = 2

s l g

∆π +

√ π lg

∆l +

− π

g s

l g

∆g.

∆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 .

∆T

T = 1.32 10 − 3 .

T = 2.00

∆T = 2.65 10 − 3 .

r

f (x) = x 3 + x − 1 = 0.

g 1 (x) = x

g 1 (x) = 1+ 1 x 2

J 1 = [0, 1]

| r n − r |

r n

r

10 − 3

r 0 = 1

g 2 (x)

g 2 (x)

x > 0

| g 2 0 (x) | ≤ 1.125 | f (x) | .

J 2 = [0.5, 0.75]

| r n +1 − r | ≤ K | r n +1 − r n | .

r

10 − 6

r 0 = 1

f (x)

[0, 1]

f (0) = − 1

f (1) = 1

r

g 1 (x)

g 1 0 = − 2x

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

x ∈ J 1

| g 0 1 (x) | < 1

2x ≤ 1 + x 2 ≤ (1 + x 2 ) 2

g 0 1 (x)

∀ x ∈ J 1

| g 0 1 (x) | < 0.65

x 0 √ 3 3 1

g 1 00 (x) − 0 +

g 0 1 (x) 0 & m % − 0.5

m = − 3 √ 8 3 ≈ − 0.6495

r n

P ^t LU

10 ⁻ ³

∆π = 10 ⁻ ³

10 ⁻ ³

⁻ ²

10 ⁻ ²

10 ⁻ ³ +

√ 9.81 10 ⁻ ³ +

10 ⁻ ² ,

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

∆T ≈ 2.65 10 ⁻ ³ .

T = 1.32 10 ⁻ ³ .

∆T = 2.65 10 ⁻ ³ .

f (x) = x ³ + x − 1 = 0.

g 1 (x) = ₁₊ ¹ x ²

| r ⁿ − r |

r ⁿ

10 ⁻ ³

| g ₂ ⁰ (x) | ≤ 1.125 | f (x) | .

| r ⁿ +1 − r | ≤ K | r ⁿ +1 − r ⁿ | .

10 ⁻ ⁶

g ₁ ⁰ = − 2x

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

| g ⁰ ₁ (x) | < 1

2x ≤ 1 + x ² ≤ (1 + x ² ) ²

g ⁰ ₁ (x)

| g ⁰ ₁ (x) | < 0.65

x 0 ^√ ₃ ³ 1

g ₁ ⁰⁰ (x) − 0 +

g ⁰ ₁ (x) 0 & m % − 0.5

m = − ³ ^√ ₈ ³ ≈ − 0.6495

| r n − r | ≤ 0.65 ⁿ .

| r ⁿ − r | < 10 ⁻ ³

0.65 ⁿ < 10 ⁻ ³

n > ^ln(10 ⁻

| r ⁿ − r | < 10 ⁻ ³ < 0.5 × 10 ⁻ ²

f ⁰ (x) = 2x ³ + 1 3x ² + 1 .

g ⁰ ₂ (x) = 6x(x ³ + x − 1)

(3x ² + 1) ² = 6x

(3x ² + 1) ² f (x).

h(x) = ₍₃ x ⁶ ² ^x +1) ²

x = ¹ ₃

| g ⁰ ₂ (x) | ≤ 1.125 | f (x) | .

| g ⁰ ₂ (x) | < 0.43,

| 1 − g ₂ ⁰ (x) | < 1.76

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

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

10 ⁻ ⁸

| e ⁿ | ≈ 0.75,

| e ⁿ | ² ≈ 0.75.

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

| 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

e ⁿ = (0.75) ⁿ (0.5) ≤ 10 ⁻ ⁸

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.

_∞ .

= k A k . k A ⁻ ¹ k .

A = A ^t

k A k 1 = k A ^t k 1 = k A k ∞ .

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

_∞ .

P ^t LU

P ^t LU