P LU s 7

(1)

!" #

7

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

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aLE3E3@ml.nDnUopopoqkVcCHS2kYZIJSDKLE3CHG`BDdrk_AB'n

∼

IJVesKZShE3E3GUntVeQuEbfUj'fDvDn

w0xPy zU{u|~}~y>{D.hrPh{u3cRyJh23bzU|eubz

! P!

7 k`k\kAk`k\kAk\k`k\kAkAkAk\k`k\kAk\k`k\kAk G`3YZC3GqMONVcK2_\G`CFEHVEHY2MRGDk

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

737F7kAkAkAk\k`k\kAk\k`k\kAkAkAkAkAkAk\k`k SK'¡DGACHsDG`K2_\G8deVcKZ?¢BhVcC3G[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

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£87kAk\k`k\kAkAkAk\k`k\kAk\k`k\kAkAkAk\k 7PKLEHGACH@SdcBhE3VcSDKMRG[dcBDsDCbBhKZsG8GAEp]GAoEHSDKk

(2)

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

T = 2π s

l g ,

3V

π

^" gZk$#Aj%#'&

10 ⁻ ³

@ZC)(`+*,VrkGDkc=

∆π = 10 ⁻ ³

^, ⁼

l = 1

^I&

10 ⁻ ³

@ZC)(``="GAE

g = 9.81

^I

⁻ ²

^&

10 ⁻ ²

@ZC-(¢Ak6]SKZKZGAC dNBD@Z@ZCHSt'VcI BUEHVeSKMRG

T

G\E]BhCHCHSDK2MRVcCBhYKZSDI.ZCHG MRG^_`FGqBMR?`WLY2BhE`k

0/!13254687

;8KB>@.SDY2Cp¡tBDdeG`YZC<Bh@2@ZC3SR_baZ?`G

T = 6, 282

√ 9.81 ≈ 2.006

,

;8K SREHVeG`KEpdXB MRV:9.?`C3G`KEHVeG`dedcGqFY2Ve¡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_'*rfJ_`FGqBD@ZC-(¢0dcB ¡'VeCHsDY2deGq3SDKLE<_\SDKFG`C3¡?`

, l

T = 2.00

^Bt¡G`_

∆T = 2.65 10 ⁻ ³ .

!"!<;

=>?@AB>

DC+EFGHI

JK>

FL2NM

O=>

QPR+S

;8KFG[@ZCHSD@.S3GqMRG E3CHSDYZ¡GACpMRG`<¡tBDdeG`YZCb0BD@Z@ZCHSR_baZ?AG¢0MZGqdcB CbBD_\VcKZG

r

^MZGqdrN?`WLY2BhE3VcSDK

f (x) = x ³ + x − 1 = 0.

#kUTWVYX[Z \!]^O] _0`5\acbdXfeg!^.k

(3)

*,B

,

K C3G`IJBDCHWLY2BDKLE8WLYZG dN?¢WLY2BUEHVeSK@ZCHSD@.S3?AG^G¢:E ?`WLYZVc¡tBDdeG`KLE3G.&

g 1 (x) = x

^Bt¡DG¢_

g 1 (x) = ₁₊ ¹ x ²

=IJSDKLEHC3G`C

WLYZGqdNVcKLE3GACH¡UBhdcdeG

J 1 = [0, 1]

G¢:E<YZKVcKLE3G`C3¡UBhdcdcG^FYZCpdcG`WLYZG`ddXB _\SDK'¡GACHsDGAK_\G ¡DGACbYZKZG^3SDdcYREHVeSK Y2KZVcWLYZGqG¢:E BDH3YZC3?`GDk

* ,

68?\E3G`C3IJVcKZGACBDK2BhdLE3VXWLYZGAIJGAKLEYZKJI B:SCHBDKEMRG

| r ⁿ − r |

^=US

r ⁿ

MZ?`3VesKZGdcB ¡UBhdcGAYZCBD@Z@ZCHS'_ba2?AGD= & dXB KZV:(`IJG VeE3?ACbBUEHVeSK=ZMRG^_AG\EFEHGqCHB_\VcKZG @2BhCpdXB I ?AE3aZSRMRGqVeE3?`CHBhE3Vc¡DGqMRY@.SDVcKEZRGk

*r_

,

K MR?¢MRYZVcC3G YZKZGJ¡UBhdcGAYZCqBD@Z@ZCHS'_ba2?AGJMRG

r

^&

10 ⁻ ³

@2C-(¢8G`K @BhC3EHBhKLEqMRG

r 0 = 1

GAEqBDC3CHSDKMRVeC BhY KZSDI.ZCHG BDMZ?`WLY2BUE<MRG^_`FGk

fZkUTWVYX[Z \!]^O] ^ ^'X[\bQ` Z \b k

*,B

, SVE

g 2 (x)

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

?`WLY2BhE3VcSDKkZ68SDK2KZGAC

g 2 (x)

^k

* ,

SKEHC3G`CpWY2Gq@SYZC

x > 0

^=ZSDKB ^drNVeK2?AsBDdeVeE3?q3YZVc¡tBDKLE3G

| g ₂ ⁰ (x) | ≤ 1.125 | f (x) | .

*r_

,

SKEHC3G`C WLYZGdrNSDKB YZKZG3SDdcYRE3VcSDK YZKZVXWLYZG3YZC drNVcKEHGACH¡UBhdcdeG

J 2 = [0.5, 0.75]

^G\E WLYZGdcG_baZSDVe(MRG_AG\E VcKEHGACH¡UBhdcdeG[@.GACHI GAE]MONBDHFYZCHGACdXBJ_\SDK'¡GACHsDGAK_\G8MZGqdcB IJ?\EHaZSRMRG^MRG^<G`oE3SKk

*,M

,

C3SYZ¡DG`CBDK2BhdLE3VXWLYZGAIJGAKLEpYZKZGqI B:SDCbBUE3VcSDKMRY EL@.G

| r ⁿ +1 − r | ≤ K | r ⁿ +1 − r ⁿ | .

*,G

,

K MR?¢MRYZVcC3G YZKZGJ¡UBhdcGAYZCqBD@Z@ZCHS'_ba2?AGJMRG

r

^&

10 ⁻ ⁶

@2C-(¢8G`K @BhC3EHBhKLEqMRG

r 0 = 1

GAEqBDC3CHSDKMRVeC BhY KZSDI.ZCHG BDMZ?`WLY2BUE<MRG^_`FGk

0/!13254687

BZkd N?\EHY2MRG MRG` ¡UBhCHVcBhE3VcSDK28MRG dXB QTSK2_5EHVeSK

f (x)

^3YZC

[0, 1]

IJSDKLE3CHG>WLYZG dcB QTSDK2_\E3VcSDKG¢:E[_\SKEHVeK'YZG>G\Eq_\CHSDVX3HBhKLEHG

3YZC]_\GAE<VeKLEHGACH¡tBDdedcGDkZ68G^@ZdcY2`=ZSDK

f (0) = − 1

^G\E

f (1) = 1

kR7PdG\RVcFE3GqMRSK2_[YZKZGqCbBD_AVeK2G

r

YZK2VcWLYZG^MZBDK2p_\GAE]VcKEHGACH¡UBhdcdeGk G¢pMR?ACHVc¡D?AG¢0@ZC3G`IJV:(`C3G¢G\E]3G`_\SK2MRG¢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"G¢: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`C3sGAK2_AG[G`FE BDH3YZC3?`GDk

k268GqI BhKZV$(ACHGq@ZdcY2p@ZCH?`_AVc3GD=RG`K?\E3YMRVcBDKLE<deG¢p¡UBhCHVcBhE3VcSDK2pMRG

g ⁰ ₁ (x)

^=2SK ¡SDVeE<WY2G l

∀ x ∈ J 1

=

| g ⁰ ₁ (x) | < 0.65

^k ^K

G9G\E¢=ZSDKBZ=

x 0 ^√ ₃ ³ 1

g ₁ ⁰⁰ (x) − 0 +

g ⁰ ₁ (x) 0 & m % − 0.5

Bt¡DG¢_

m = − ³ ^√ ₈ ³ ≈ − 0.6495

^kZ;8KB MRSDK2_^dcB I B:SDCbBUE3VcSDK3YZVe¡UBDKEHG MRG^dNGACHC3G`YZC]_\SIJI VX3GqGAK@ZCHGAKBhKLE<dcB ¡tBDdeG`YZC Bh@2@ZC3SR_baZ?`G

r n

MZG

r

*r_5Q:k. SYZCH

, k

| r n − r | ≤ 0.65 ⁿ .

(4)

_hk KYREHVedcVcHBhKLE]dXB @ZCH?`_A?`MRG`KLE3GqI 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`KE¢=

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@ZC3VcIJGqBt¡DG`_

2

_A3G^Bh@ZC)(`0dXB ¡LVcCHsDYZdcG[@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`VeKLEpYZKI BURVcI YZI?AsBDd &

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`FEq_\SDKLEHVeK'YZG>G\E _\CHSDVXH3BDKEHGqFYZC8_\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^MZBDK2p_\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`KEpYZK2GqIJB:SDCbBUEHVeSKMRY E'@GF*r_5Q:k268?AIJS fZ=RG\RGACb_\VX_\Gqf

, l

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

GDk K YZE3VcdeVX3BDKLE<dcB @ZCH?`_A?`MRG`KEHG[IJB:SDCbBUEHVeSK='SDK SREHVeG`KE¢=

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 *r_ABDC

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

^, ^G\E SDKEHC3SYZ¡DG=

GAKBDC3CHSDK2MZVcH3BDKLE0BhY KZSIRZCHGqMRGq_A3G[BMR?`WLY2BhE

r 5 = 0.68232

^k

!"!! >

;MR

H'

M

' +S

SVE]MRG`YRI ?AE3aZSRMRG¢<VeE3?`CHBhE3Vc¡DG`]MRSDKLE]dXB _\SK2:EbBhKLE3G^B 'IJ@REHShE3VXWLYZG^G¢:E

C = 0.75

GAE8_\SK'¡DGACHsDG¢BhKLEdeVcKZ?`BDVeCHGAIJG`KE

@.SDYZC dXBm@ZCHGAIJV$(ACHG GAE WLY2BDMRCbBUEHVcWLYZG`IJGAKLE @SYZC dXB FG¢_\SDKMRGDk0 0BDdc_AYZdcGAC dcGKZSDI.ZC3GMONVeE3?ACbBUEHVeSKI VcKZVcI Bhd<@.SDYZCWY2G

dNG`C3CHGAYZC^MNBD@Z@ZCHSt'VcI BUEHVeSK KNG\Z_ (`MRG @BD

10 ⁻ ⁸

M2BhK2qdcG` MRGAYR _AB^HBD_ba2BDKE^WLYZG dNGACHC3G`YZC &dXB@ZC3G`IJV:(`C3G VeE3?`CHBhE3VcSDK KNG\Z_(¢MRG[@2BD

0.5

^*TVk^GDkc=

e 0 = 0.5

^, ^k

0/!13254687

;8KB>@.SDY2CpdcB @ZCHGAIJV$(ACHG GAE<dcBJMRG`YRRV:(`IJGqI ?AE3aZSRMRGqCHG`3@.G`_5EHVe¡GAIJGAKLE¢=

| e n +1 |

| e ⁿ | ≈ 0.75,

^GAE

| e n +1 |

| e ⁿ | ² ≈ 0.75.

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

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

SDYZCpdXB IJ?\E3a2S'MZG^MRG^_\SKL¡GACHsDG`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'¡GACHsDGAK_\G dcVeKZ?¢BhVcC3G[SK B2=

e ⁿ = (0.75) ⁿ (0.5) ≤ 10 ⁻ ⁸

^@SYZC

n ≥ ln 2 − 8 ln 10 ln(0.75) ≈ 62.

SDYZCpdXB 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 .

2n ≥ ln(0.75) − 8 ln 10 ln(0.375) ,

n ≥ 10.

;8K @.GAYREJC3G`I BhCbWY2GAC WLYZGdcB I ?AE3aZSRMRG MRG_ASDK'¡DG`C3sGAK2_AG WLY2BDMRCbBUEHVcWLYZG MZGAI BhK2MZG+.G`BhY_\SDY2@ IJSDVcK2>MNVeE3?`CHBhE3VcSDK

WLYZG[dcBJIJ?AE3aZSRMRG^MRGq_ASDK'¡DG`C3sGAK2_AG]dcVeKZ?¢BhVcC3Gk

!

>3

<M

Q+

SKEHC3G`CpWY2G @.SDY2CpYZKZGqI BUEHC3VX_\GqLIJ?AE3CHVcWLYZG

A

=2SDKBJdrN?AsLBhdcVEH?[3YZVc¡UBhKLE3G=

_\SK2M!*

,

1 =

^_\SK2M!* ^,

_∞ .

0/!13254687

BhCpMR?A2KZVeE3VcSDK=ZSKBZ=

_\SK2M!*

,

= k A k . k A ⁻ ¹ k .

SDYZCpY2KZGqI BUE3CHVX_\Gq 'IJ?\EHC3VXWLYZGD=

A = A ^t

^=ZGAE

k A k 1 = k A ^t k 1 = k A k ∞ .

68Gq@ZdeYpSDKBZ=

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

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

_\SK2M!*

,

1 =

^_\SK2M!* ^,

_∞ .

(7)

+MR

> M >

P ^t LU

^M ^@; ^M ^> ^M ^QPS

#k<6]?¢_\SDIJ@.S3GACpdXBJI BUE3CHVX_\G

A

G`K@ZC3SRMRY2VE

P ^t LU

^SDY

P

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

I VcK2BhE3VcSDKMRG §[BhY2HG\Ep@ZVc¡DSDEHBDsDG @2BhC3E3VcGAdk

fZk8 0BhdX_\YZdcGACdcG^MR?\EHGACHI VcK2BDKE<MRG

A

^k

g2k<9p?`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/!13254687

;8K VeKLE3G`C3¡GAC3E3VeEpdcB dcVesKZG

1

^GAE<dcB ^dcVesKZG

3

_\GqWLYZV"MRSKZKZG

A =





− 6 2 0 4 − 1 2 2 − 1 0



 .

NSD@.?ACbBUEHVeSK dcVesKZG

2 =

^deVcsDK2G

₂ − ( − 2/3)

^dcVesKZG

₁

G\E<dcVesKZG

3 =

^deVcsDKZG

₃ − ( − 1/3)

^dcVesKZG

₁

^MZSDKZKZG=

A =





− 6 2 0

0 (1/3) 2 0 ( − 1/3) 0



 .

NSD@.?ACbBUEHVeSK dcVesKZG

3 =

^deVcsDK2G

₃ − ( − 1)

^deVcsDKZG

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:EHVEHYRE3VcSDKmBt¡UBhKLE[@ZYZVX BDC3CHV$(ACHG^SDK E3CHSDYZ¡GD=

Ly = b

^Bt¡DG¢_

y = (3, 4, 6) ^t

^=O@2YZVc

U x = y

Bt¡G`_^2K2BhdcGAIJG`KE

x = (( − 15/6), − 6, 3) ^t

^k

! !

B> M > M

3 M

R@H

JK>

P +

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x ^k

^# ^j

y ^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 .

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P (2) = − 1, P (3) = 0, P (5) = 1.

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(9)

x y ∆y ∆ ² y ∆ ³ y

#

Pf

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, *#tn ,

j # )#UnUg

)#

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

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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 ⁿ .

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 .

_∞ .

= 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