115 P LU t s

!

"$#&%'")(+*,.-0/1/32

465879;:046<>=0?@1ACB>DEB>FAHGJI5KFMLONP@QD?RAQSUTPV0BWB3AXGMBY7B[Z]\^B>@]ZC\0B_96=<>@C?RAQS`NF0F^B>a`abBP:&abN^Z>?RadcRe&ff&g

hA1AC=jikkmlllngSb@CN0gV^DENFPAC@QB[?RaogZ[?k

∼

DESbpF^NRAQAQBmkqSbLUArcmscMtqk

u.vKwxmy{z}|}w~y€Rƒ‚m„o„†…‡ˆy{‰Q‚Š`‹Mw‚m0„o‰Q…CxRzbŠ{Œ]x

ŽM’‘”“–•—r˜™•Rš3˜œ›&•P•P

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5Q5g>g3g>g>g>g>g3g>g>g>g>g>g3g>g>g>g>g>g3g žŸ<3AQ\0NMGMBYGMBWaU?E¨.S` C QB[ZrACSbNPFJ:^GMV£©}NS`FPA6ª«Sb¬MBWB>AXGMBn­XB>lˆACNFj¥®e^tW=MAC ]§rg

5Q515Ÿg>g>g3g>g>g>g>g>g3g>g>g>g>g>g3g>g>g>g žŸ<3AQ\0NMGMBYGMBn­XB>lˆACNF£¯~©°abV0 QS`B>V^@] ˆ±–?@QSU?R²0abB’ W¥8t[e~=MA] C§3g

5K±³g3g>g>g>g>g>g3g>g>g>g>g3g>g>g>g>g>g3g ª0?Z3AQNP@QSU Q?RAQS`NF

P ^t LU

?q´B[Zµ¤–abS`DESbFœ?mAQS`NF·¶W?V0 Q QS`B>F^F^B¥ocR¸~=MAC ]§rg

±¹g>g3g>g>g>g>g3g>g>g>g>g>g3g>g>g>g>g>g3g 5KFPAQB[@Q=NPa`?RAQS`NFºGMBY­HB[lˆAQNPFj¥®c^tµ=^AC ]§rg

±X5»g3g>g>g>g>g>g3g>g>g>g>g3g>g>g>g>g>g3g ¼&=^abS`F^B’ Y¥8t’¦;=MA] C§3g

½«NRAC?aºg>g3g>g>g>g>g>g3g>g>g>g>g>g3g>g

115

=NS`FPA] >g

¾·¿WÀXÁ'µ¿WÃÀXğŖÆXÇHÁÉȖŰÊHÁ^¿WÆXÆ6ŖËdÁJÌXÈÍ6Ë«ÃÀXËJÍ}ÇHÊHÎKÈŰÁ”ÅÇÏÈÍ6ËdÃÀ6ËJÍ}ÇÅ°ÀXÊHÁjÍÀXÇH¿µÊXÎKÁ^ЖÁ

© NV^@;AQ@CNV^´B[@_V^F^BB[ 1AQS`D?mAQS`NF'GMBa`?»Z>NF0 1AC?FPAQBGMB=B>@CD~SbAQAQS`´SbAQ<ºGMV'´&S`G^B

² 0

:216g31NPV^a`ND_²J:«B[F

1785

:}VMAQS`a`S` C?

aoIB>¬&=<[@QS`DEB>FAC?mACSbNPFº QV^S`´m?RFAQBP:

5Ka DEB’ 1V^@]? a`? LONP@CZ>Bº¥

F = 0.0022 ± 0.00022

­XB>lˆACNFœ§XF^<’Z3B’ Q C?RS`@QB_=NPV^@W 1<[=0?R@CB>@µGMB[VM¬·=^aU?mACB[?VM¬·?RV ´NabAC?pB

V

¥

V = 1000 ± 50

±–NabAC ]§r: Q<>=0?@Q<’  G™IV^F^BHGMSU 1AC?RFœZ3B

s

^¥

s = 0.004 ± 0.0002

^D §}B3AGMBHGMS`?D43AQ@CB

d

^¥

d = 0.10 ± 0.001

^{D §3g}

5d?~=B[@QDESbA1AQS`´&SbAQ<YGMV£´&S`GMB

² 0

<>AC?RSbAHB>F0 QV^S€ACBnabS`<>BY¯;AQNV^AQBYZ3B[ ´m?abB[V^@C .B[FPA]?Z]\^<>B[ HG™IB[@Q@CB>V0@ˆ=0?R@aU?~@QB[a`?RAQS`NFº QV^S`´m?RFAQBP:

F = 1 8

πd ² ² 0

s ² V ² .

96Fº´NPV0 GMB>D?F0GMBEi

tg46NF^F^B[@aoI?=^=^@CN’¬MS`D?mAQS`NFºGMB

² 0

¥OSogBg`:

² ^∗ ₀ )

^g7689}:

cMg¥®?P§46NF^F0B>@ˆaoIB>¬&=0@QB[ C QSbNPF ?RFœ?Ra<;PACS`TV^BWGMBµaoIS`F0Z3B[@1ACS€ACV0GMB¥{ACNRA]?Ra`B’§GMB

² 0

¥OSogBPgb:MaU?;G^S>=ƒ<>@CB>FAQS`B>a`abB

∆² 0 (F, V, s, d)

^§

=0?@;aU?»D~<>AQ\^N^GMBºGMB=0@QNP=0?Rp?mAQS`NFÉG™IB[@Q@CB>V0@¥+VMAQS`a`S` C?RFA~aoI?R=^=^@CNq¬MSbD?mACSbNPFÉGMBº½}?&;a`N@~GMBaU?·LONF0ZrACS`NF ?RV

=^@CB>DES`B>@N@]GM@CB’§3g 6?,.}:

¥O²œ§¤¬M=^@CS`DEB>@HZ]\0?Z>V^F£GMB[ 

4

AQB[@QDEB[ HGMBWaoISbFœZ3B>@QAQSbAQVœGMBYGMB

² 0

B>FºLONF0Z3AQS`NF£GMB

² ^∗ ₀

^{gA@ 6?B.}:}

¥+Z’§46<3ACB>@CD~S`F^B[@nB>Fœ 1V^SbAQBaU?º´m?abB[V^@WFV0D~<[@QSUTV^BGMBaoIS`F0Z>B>@1ACSbAQV0GMBEACNRAC?abB QV^@

² 0

¥+SogBPgb:

∆² 0

§µ?ºaU?TV^B[aba`BE<3A]?RSbA

=0?@Q´B[FV^BC16gD1NV^a`ND_²Jg76?B)}:

e^gH¼&NPV^abS`pF0B>@abB’ Z]\^S>=ƒ@CB[  QS`pF^Sb¢œZ>?RAQSbL+ 2E8B>¬^?ZrA] GFE¥+Z> QB’§ˆGMBnaoI?=^=^@CN’¬MS`D?mAQS`NFºGMB

² 0

?RVœTPV0B>a™<3A]?RSbAH=0?R@C´B>F&V16gD1NVIH

a`ND_²JgJ6?B.}:

KL2!D[‘

*D

96F£?0:

² 0 = 8 F s ²

π d ² V ² < 1 pt >

TV^SJF^NV0 ˆ=B[@QDEB>AHGMBYGM<[G^V^Sb@CB:

² ^∗ ₀ = 8 (0.0022

^­

)(0.004

^D

) ² π(0.1

^D

) ² (1000

^±

) ²

= 8.9636 × 10 ^{− 12} < 3 pts > (

^B>F

N V ² )

B0ŽM

96FºNP²MAQS`B>FAHa`?EG^S>=ƒ<>@CB>FAQS`B>a`abBY QV^Sb´R?RFAQB:

∆² 0 (F, V, s, d) =

¯ ¯

¯ ¯ 8 s ² π d ² V ²

¯ ¯

¯ ¯ ∆F +

¯ ¯

¯ ¯ − 16 F s ² π d ² V ³

¯ ¯

¯ ¯ ∆V +

¯ ¯

¯ ¯ 16 F s π d ² V ²

¯ ¯

¯ ¯ ∆s +

¯ ¯

¯ ¯ − 16 F s ² π d ³ V ²

¯ ¯

¯ ¯ ∆d. < 5 pts >

B-NO

5«?_´m?abB[V^@ˆGMBµaU?;Z3NPFPAC@QS`²^VMACSbNPFGMBWZ]\0?Z>V^F^BWGMB[ .´m?@QSU?R²0abB’ G^?RF0 .aoIS`F0Z3B[@1ACS€ACV0GMB6ACNRA]?Ra`B~¥+S®gBg`:&B[@Q@CB>V^@ˆ?R²0 QNa`V^Bq§°GMB

² 0

¥OB>¬M=^@QS`DE<>BW=NPV^@HZC\œ?TV^BµAQB[@QDEBWB[FLONFœZrAQS`NFŸGMB

² ^∗ ₀

^§.B[ 1A[:

PGQSRTRVUWXTGYZY\[ZT

∆² ⁰ (F, V, s, d)

]^+_&]a`Abdc^

[ZWXT

∆² ⁰ (F, V, s, d) = a ² ^∗ 0 + b ² ^∗ 0 + c ² ^∗ 0 + d ² ^∗ 0

^{^+_}

a, b, c, d

^]

Te[

^+f Yg&T

]hi^+f&]

Y

bjf Y\T

]kl_

Tnm

^+_]

g&oGYTe[ZWXQ

f

Ti[ZTep

∆² 0 (F, V, s, d) = ² ^∗ ₀ ∆F

F + ² ^∗ ₀ 2∆V

V + ² ^∗ ₀ 2∆s

s + ² ^∗ ₀ 2∆d d

= 0.1 ² ^∗ ₀ + 0.1 ² ₀ ^∗ + 0.1 ² ^∗ ₀ + 0.02 ² ^∗ ₀

= 0.32 ² ^∗ ₀ , < 2 pts >

B -

∆² 0 (F, V, s, d) ≈ 2.87 × 10 ^{− 12} . < 2 pts >

2.87 < 5 × 10 ⁰

:0GMNPF0ZW QB>V^aJabBn=^@CB>DES`B>@HZC\^S=ƒ@QB ¥O@]?RF^p

0

^{§ˆG^B}

² ^∗ ₀

B[ 1AH QSbpPF^Sb¢œZ>?RAQSbL«B>AHNF£?ºi

(8.9636 ± 2.87) × 10 ^{− 12} .

6 B)}:

&03 L!M}‘«‘ +Ž ’‘RZ!D' %!DZ™XJ‘·‘}‘Ÿ‘E!D ( * }0/

96F» QB_=^@CN=N 1B_GMB_AQ@CNV^´B[@6GMB[ 6´m?abB[V^@C X?=^=^@CNMZ]\^<>B[ XG^B[ 6G^B>VM¬Ÿ@]?Z3S`F^B’ 

r 1

B3A

r 2

¥

r 2 > r 1

§XGMB;aoI<’TPVœ?mAQS`NPF

f (x)

G^?F0 ˆaoIS`FPACB>@C´m?Ra`a`B

J = [ − ^π 2 , π]

g96F£GMNPF^F^BWa`Bnp@]?R=^\^BWGMBYZ3B>A1ACBWLONF0ZrACS`NF£ QV^@aoIS`FPACB>@C´m?Ra`a`B

J

¥®ZrL8g0ª}S`p0gJt’§3g

f (x) = x

2 − sin(x) + π 6 −

√ 3

2 = 0.

^¥8t’§

-1 -0.5 0 0.5 1 1.5

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

f(x)

x f(x)

f(x)

}t¶µ@C?=^\^BnGMB

f (x)

 ^{1V^@}

J

^g

tgX L!a«‘}‘ +Ž Z’‘!" ( «&/mg

¥®?P§ © B>VMA HoNPF?R=^=0abSUTV^B>@ aU?jDE<3AC\^NMGMB GMB·aU?”²^SU Q QB[Z3AQS`NF=œNPV^@Z>?a`Z>V^abB[@ a`B[  GMB[VM¬@C?PZ3S`F^B[  G^B

f(x)

 ^{1V^@}

J

PV0 1AQSb¢0B>@HTV0?abSbAC?RAQS`´B>DEB>FA´NAQ@CBµ@Q<[=NF0 QBg76?B.}:

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a`BP¥® C§ ><[@QN¥+ ]§X?q´B[Z_V0F^B;ACNa`<>@]?RF0Z>Bº¥OSogBPgb:™=^@C<[Z>S` QS`NFœ§6GMB

tol = 10 ^{− 10}

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[a, b]

B3A.GMBY„†‚Rzg 6 }:

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G™ISbAQ<>@]?mACS`NF0 [g 6 *«:

¥®?P§º¼&NPS€A

g(x)

:da`?ºLONFœZrAQS`NF”SbFAQB>@C´B>Fœ?RFPA_G^?F0 naU?ºDE<>AQ\^NMG^BSbAQ<>@]?mACSb´PBGMB ­HB>lˆACNF¥

r n+1 = g(r n )

§W=NPV^@naU?

@C<[ QNa`VMAQS`NFºG^BWaoI

TV0?mACSbNPFj¥8tq§rg04XNPF^F^B[@

g(x)

?SbFœ 1SJTV^BYZ3B3AQAQBn@CB>aU?mACSbNPFS€AC<>@]?mAQS`´Bg76

« :

¼&S™NFº?RDEN@]Z3BµZ3B3AQAQBW@QB[a`?RAQS`NF S€AC<>@]?mACSb´Bµ?q´B[Z

r 0 = 2

:^Z3NPF´B[@QpPB>@C?WA HoNPF ´B>@] 

r 1

NV

r 2

¥®G^?RFœ .a`?~ 1V0S€ACBµNF

?=^=B>a`abB[@C?

r

Z>B3A1ACBn@C?PZ3S`F^B’§ 6 * } :

7H?=^=B>a`B>@;¥+ C?RF0 XZ>?a`Z>V^a`B>@r§ˆTV^B>a`abB_GMNPS€A3AQ@CBYabB’ XZ3NF0G^S€ACSbNPF0 =NPV^@6TPV0BYZ>B3AQAQBY@CB>aU?mACS`NFºSbAC<>@]?mAQS`´BYZ>NF´B[@QpPB

¥OAQNPV8NV^@] ?q´B[Z

r 0 = 2

=NV^@H?DEN@]Z3B>@Z3B3AQAQBn@CB>aU?mACSbNPFS€AC<>@]?mAQS`´B’§3g76?B)« :

¥O²œ§º¡ a®I?RSUGMBGMVÉp@]?R=^\^B GMB aU?£LONFœZrAQS`NF

f (x)

:}GM<[G^V^Sb@CB a®IN@]GM@CB GMBZ>NF´B[@QpPB>F0Z>BG^B a`?·D~<>AQ\^N^GMB =NV^@;abB’ 

GMB[VM¬@]?Z3S`F^B[ [g PV0 1AQSb¢0B>@HTV0?a`S€A]?mAQS`´B>DEB[FPA´NAQ@CBµ@C<>=NFœ 1BPg 6?B.}:

¥+Z’§ ¤–FGM<[GMV0Sb@CBV^F0B´R?Ra`B>V^@E?R=^=0@QNMZ]\^<>BG^B

r

^{?=^@ 4’} 

3

SbAQ<[@C?RAQS`NF0 º¥OSogBg`:GMNF0F^B>@

r 1 , r 2 , r 3

§¥+?q´B[Z ACNV8NPV^@C 

r 0 = 2.0

^{§rgJ6 « :}

e^gX L!a«‘ 2!DZ™ J‘ (+*8 «&/

¥®?P§£96F£Z>NF0 QS`G-4>@CBµDE?S`FPAQB[F0?RFAaU?~DE<3AC\^NMGMBYGMV£=NS`FPA¢^¬MB:

r n+1 = g 2 (r n )

^:0?q´B[Z:

g 2 (x) = sin x + x 2 − ¡ π

6 −

√ 3 2

¢ ,

^¥®cP§

=NPV^@Z[?RaUZ3V^a`B>@°aU?Y@C?PZ3S`F^B

r 2

:B>F N²œ 1B[@Q´R?RFPATV^B

r 2 ∈ I = [ ^2π ₃ , π]

:<3AC?²^a`Sb@. 1SZ>B3AQAQB6D~<>AQ\^NMG^B6GMBX=NPSbFA°¢^¬MB B’ 8AXZ3NF&´B>@CpB[FPAQBPg 6?B.}:

¥O²œ§º46<3ACB>@CD~S`F^B[@ ?F0?Ra<;AQSUTV^B>DEB[FPA[:qpP@PZ3B?RV_AQ\^<[N@ 4>DEB.GMB’  ?Z>Z>@QNPS` C 1B[D~B[FPAC d¢0F^SU ˆ¥ONPV;G^B.a`?6´m?abB[V^@}D~N0;B>F0F^B’§3:

V^F0B6D?8N@]?mAQS`NF GMV Ai;=B

| r n − r | ≤ K

^:NPV

r n

GM<’ 1S`pF0BXaU?Y´R?Ra`B>V^@.?R=0=^@QN^ZC\^<[B:¯YaU?_FIHoS<4>DEB6SbAQ<>@]?mACSbNPFJ:G^B

Z>B3A1ACBˆ@]?Z>SbF^B.=œ?R@ aU?µD~<>AQ\^N^GMBˆSbAQ<>@]?mACSb´BGMV~=NPSbFA ¢^¬MBgR¤–FGM<[GMV^S`@CBˆabBˆF^NPD_²^@CBˆG™IS€AC<>@]?mACSbNPF0 

n _∗

F0<[Z3B’ Q C?RS`@CB

=NPV^@N²MACB>F^S`@[:^=0?R@HZ3B>A1ACBnD~<>AQ\^NMG^B:^V^F^Bn´m?abB[V^@?=^=^@CNMZC\0<>BW¯

10 ^{− 10}

^=0@ 4’ GMBnZ>B3AQAQBn@C?PZ3S`F^Bg76 .}:

¥+Z’§ 1.?a`Z>V^abB[@ˆa`B[ 

4

=^@CB>DES4[@QB’ .B[ 1ACSbDE<>B’ 

r 1 , . . . , r 4 ,

B>Fº=œ?R@QAC?RFAHGMB

r 0 = 2.0

^{gJ6 }:}

¥+G0§º46NF^F0B>@a`BWF0ND_²^@CBnGMBnZ[ 1B¥®ZC\0S>=ƒ@QBY QSbpPF^Sb¢œZ>?RAQSbL«B>¬^?ZrAr§.GMBna®IB[ 1AQS`DE<>B

r 4

gJ6?B)« :

KL2!D[‘

*&ŽM

¶µ@Z>B6?V pP@C?=^\^B6G^NF^F^<µB>Fºª}S`p0gƒt:&NF´NSbAˆTV^BµaU?YLONFœZrAQS`NF

f (x)

?;V^F^Bµ@]?Z>SbF^B

r 1

G^?RF0 .aoIS`FPACB>@C´m?Ra`a`B

[ − ^π 2 , 0]

^B3A

V^F^B;?RV^AQ@CBn@]?Z>SbF^B

r 2

G^?F0 HaoIS`FPACB>@C´m?Ra`a`B

[ ^2π ₃ , π]

g™96F£=B[VMAµ?R=^=0abSUTV^B>@XaU?EDE<>AQ\^NMG^B_GMB_a`?²^SU C 1B’ZrAQS`NFŸ=NV^@6Z[?RaUZ3V^a`B>@HaU?

@]?Z3S`F^B

r 2 ∈ [ ^2π ₃ , π]

<sup>D?S` NF</sup> F0B6=œB[VMA.=œ? °V^AQS`abSU QB>@ˆZ3B3AQAQB6DE<>AQ\^NMG^BX=NPV^@.Z[?RaUZ3V^a`B>@

r 1

Z[?R@a`?;Z3NPF0GMSbAQS`NF

f (a).f (b) < 0

FJIB[ 1A=œ?  C?mACS` 1L+?RSbAQBW=NV0@.ACNVMA

a, b ∈ [ − ^π ₂ , 0]

^:

a < b

^{g76?B)« :}

*NO

© NV^@;B[ 1AQS`DEB>@_a`B F0ND_²^@CB G™ISbAC<>@]?mAQS`NFœ 

n

F^<’Z3B[ C C?RS`@QB=NV^@_B’ 8ACSbDEB>@

r 2

B>FÉ=0?R@QAC?FPA;GMB a®ISbFAQB[@Q´R?Ra`abB

[a, b]

^{:}=NV^@}

?R@C@CSb´B[@ˆ¯~a`?~=0@Q<’Z3SU 1S`NF „K‚mz:0NF£Z3NPF0 1SUGI4[@QBµaoISbF^<[pP?abSbAQ<n QV^S`´m?RFAQB:

tol ≥ b − a 2 ⁿ⁺¹ ,

TV^SdZ3NFœGMV^SbAH¯0:

n ≥

ln ³

b − a tol

´

ln 2 − 1, < 3 pts >

*-

¤–AH=NPV^@a®ISbFAQB[@Q´R?Ra`abB

[ ^2π ₃ , π]

:^NPF£?~FV^DE<>@CSUTPV0B>DEB>FA[:

n ≥ ln ^π ₃ + 10 ln 10 ln 2 − 1 ' 33. < 1 pt >

B0ŽM

5«?_LONPF0ZrACSbNPF

f (x)

B[ 1AHG^<>@CSb´R?R²^a`BW QV^@

J

^B3AHNF£?0:

g(x) = x − f (x) f ⁰ (x) = x −

x

2 − sin x + ^π ₆ − ^√ ₂ ³

1

2 − cos x .

96F£?EG^NF0ZµaU?;LON@CD_V^a`BWSbAQ<>@]?mACSb´PBW QV^S`´m?RFAQB:

r n+1 = r n −

r n

2 − sin r n + ^π ₆ − ^√ 2 ³ 1

2 − cos r n

. < 3pts >.

96F£Z>NF&´B>@CpB>@]?Y´B[@C 

r 2

g 6 *«:

1NPF0GMSbAQS`NF£GMBYZ3NPF´B[@QpPB>F0Z>B;i

• r 2

 QB>D_²0abB >AQ@CBÉG^?F0 ·a®IS`FPAQB[@Q´R?Ra`abB

[2, 3]

¥®ZrL8gµª}S`p0g_tq§rg64µ?RFœ »Z3B3A SbFAQB>@C´m?aba`B:

1

2 − cos r n 6 = 0

^¥OSogBg`:

r 6 = π/6 [

DENMGMV^a`N

2π]

§.B>ANPF£?RV^@]?~GMNPF0ZW?RV0Z>V^F^BnG^Sb´&SU 1S`NFº=0?@ [<>@CN0g

•

4XB;=^a`V0 >:B>F·=0?@1A]?RFA6GMB

r 0 = 2.0

:œNPFŸB’ 8A6=0?P AC@QNP=ŸabNPSbF»GMB_aU?  QNa`VMACSbNPFÉ¥OB>F·AQNV^AµZ>?P >:œNFŸB’ 8A6=œ? 6 1<>=œ?R@C<YGMB aU? 1NPabV^AQS`NF£=0?R@HV^FŸDESbF0SbD_V^DabNMZ[?RadZ3B_TPV0BnaoINPF£´NSbAX QV^@Ha`BnpP@C?=^\^B’§3g^46NFœZWaU?EDE<3AC\^NMGMBYB[ 1AX?P Q QV^@C<nGMB_Z3NF&´B>@CpB>@’g

6?B.}:ºg

5«?GMB>VM¬MS<4>DEBE 1NPabVMACS`NFJ:=^V0@QB>DEB[FPAn?RF0?a<;PACS`TV^BP:ƒZ3NFœ 1SU 8ACB>@]?RSbAµ¯ @C?S` QNF0F^B>@6Z>NDEDEB~G^?F0 6a`B~Z[? µG™IV^F =NS`FPAµ¢^¬MB

B3AXGMBYGM<[D~NPFPAC@QB[@TV^B

| g ⁰ (x) | < 1 ∀ x ∈ J

^g

B-NO

96@]GM@CBWGMBYZ3NF&´B>@CpB>FœZ3B;i

•

@C?PZ3S`F^B

r 2 ∈ [ ^2π ₃ , π]

^g

f (r 2 )

B[ 1AµV0F^B;LONSU WGMS>=ƒ<>@CB>FAQSU?R²^a`B

SogBPgb:

f (r 2 ) = 0

^B3A

f ⁰ (r 2 ) 6 = 0

:ƒaU?Z>NF´B[@QpPB>F0Z>B_ QB>@]?

GMNPF0ZWTV0?G^@C?RAQSUTPV0Bg 6 *«:

•

@C?PZ3S`F^B

r 1 ∈ [ − ^2π ₃ , 0]

^g

f (r 1 )

B[ 1AXG^B>VM¬LONPS` XGMS=<[@QB[FPACS`?²^abBP:0S®gBg`:

f (r 1 ) = 0

^B3A

f ⁰ (r 1 ) = 0

¥+NF£´NPS€AX²^S`B>F· 1V0@Ha`B p@]?R=0\^BµTPV0Bµa`?_A]?RF^pPB>FPACBWGMBµaU?~Z3NPV^@C²B6B>F

r 1

B’ 8AFV^a`abBq§r:^a`?~Z3NPF´B[@QpPB>F0Z3Bµ 1B[@C?;GMNPF0Zµ 1B[V^a`B>DEB>FAˆa`S`F^<[?Sb@CBg 6 * «:

B -

¤–FŸ=0?R@QAC?FPAHGMB

r 0 = 2.0

^{:^NF£?0:}

r n+1 = g(r n )

^B3A’:

r 0 = 2.0,

r 1 = 2.230616197, r 2 = 2.249782942,

r 3 = 2.244982928. < 3pts >

! "#

2.246005589

ŽM

5dB6pP@C?=^\^B6B3Aˆa`B[ ˆTV^B[ 1AQS`NF0 .=^@Q<’Z3<[G^B>FPACB[ .F^NV0 .NFA.DENPFPAQ@C<>B’ .TVJISbaƒB>¬MS` 1AQBµV^F^Bµ@]?Z3S`F^B6V^F^SUTV^BWG^?F0 

I = [ ^2π ₃ , π]

^g

5KaJF^NVœ ˆ@QB’ 8ACBn¯E Q?q´NS`@ˆ QS

g 2 (x)

B’ 8AXZ3NFAQ@]?Z3AC?RFAQBW QV^@

I

^g

­XNV0 H?’´NPF0 

g ⁰ ₂ (x) = cos x + ¹ ₂

^g

B>A[:

− 1/2 ≥ cos x ≥ − 1

^{=NV^@}

x ∈ I = [ ^2π ₃ , π]

¥{LONPF0ZrACSbNPFGM<[Z>@QNPS` C C?RFPACBX QV^@

I

§r:MGMNPF0Z

0 ≥ cos x + 1/2 ≥ − 1/2

^{=NV^@}

x ∈ I

B3Aµ¢0F0?Ra`B>DEB>FA

| g ₂ ⁰ (x) | ≤ 1/2

^:

∀ x ∈ I

^g 5«?LONPF0ZrACSbNPF

g 2 (x)

B’ 8AnGMNF0Z;Z3NPFPAC@C?PZrAC?FPACB_ QV^@

I

B3AWa`?Z3NPF´B[@QpPB>F0Z>B B[ 1AX? C 1V^@C<>BPg 6 B)}:

# ! ##

∀ x ∈ I, g 2 (x) ∈ I

NO

¤°FºVMAQS`a`S` C?RFAabBµAQ\^<[N@4>DEBWGMBWa`?~´m?Ra`B>V^@ˆDEN&;B[F^F^B:MNFNP²MAQS`B>FA[:^=^V^S` CTV^B

g 2 (r) = r

^B3A

r n = g 2 (r n − 1 )

^?q´B[Z

r n

a`?

´m?abB[V^@?=^=^@CNMZC\0<>BWGMBna`?~@]?Z>SbF0BW¯~aU?~FIHoS<4>DEBnS€AC<>@]?mAQS`NFd:

(r n − r) = g 2 (r n − 1 ) − g 2 (r)

(r n − 1 − r) × (r n − 1 − r),

= g ₂ ⁰ (ζ) × (r n − 1 − r),

^?’´B’Z

ζ ∈ I.

¤–FVMACSba`SU Q?FPA.aoISbF^<[pP?abSbAQ<XAQ@CNV0´<>BX=^@C<[Z><[GMB[D~DEB[FPA

| g ₂ ⁰ (x) | < 1/2

^:™¥OSogBg`:aU?_²N@CF^BXaU?_=^a`V0 =B’ Q QSbDESU 8ACB’§3:NPF NP²MAQS`B>FA.a`B[ 

S`F^<>p?Ra`S€AC<[  1V^S`´m?FPACB[ [:

| r n − r | ≤ 1

2 | r n − 1 − r | ≤ µ 1

2

¶ 2

| r n − 2 − r | ,

≤ . . .

≤ µ 1

2

¶ n

| r 0 − r | ,

≤ µ 1

2

¶ n

π

3 . < 5pts >

96FºN²^AQS`B>F0GM@]?~N²^a`Sbp?mACNS`@QB>DEB[FPAGMNFœZ

| r n − r | < 10 ^{− 10}

:0GMB[ HTV^B:

µ 1 2

¶ ⁿ π

3 < 10 ^{− 10} , n ln

µ 1 2

¶ + ln π

3 < ln (10 ^{− 10} ).

¼&NPS€AH=NV^@ˆACNVMA

n > 33

^gJ6?B.}:

-

¤°F£=0?R@QAC?FPAHGMB

r 0 = 0.5

^{:^NPFº?0:}

r n = g 1 (r n − 1 )

^B>A[:

r 0 = 2.0,

r 1 = 2.251724055, r 2 = 2.245277691, r 3 = 2.246096414,

r 4 = 2.245994227. < 3pts >

96F£?0:

| r 4 − r | ≤ µ 1

2

¶ 4

π 3 ,

. 0.06545 < 0.5 × 10 ^{− 0} .

¼&B[abNPFŸZ3B>A1ACBY²N@CF^BY 1V0=<>@CSbB[V^@QBYGMBYaoIB[@Q@CB>V^@’:M´&SU 1S`²^a`B>DEB>FAHAQ@ 4[ H=B[ C 1S`DES` 1AQBP:œ 1B[V^abBYa`Bn=0@QB[D~S`B>@XZ]\^S=@CBYB[ 1A6 1S`pF0S€¢œZ[?mACS€L

¥OB>AXGMNPF0ZµNFº=B[VMAH<[Z>@QS`@QB

r 4 = 2.24599 . . .

^{§rgJ6?B)« :}

¼MNSbAa`Bn  ;M 8A 4>DEBn QV^Sb´R?RFPA’:

x ² − exp( − y) = 0, y ² − exp( − z) = 0, z ² − exp( − x) = 0.

¥+e§

!œ[Ž i

exp (x) = e ^x

tg46<3AQB[@QDES`F^B>@6a`B 1?Z>N²^S`B>F·GMB;Z3B_ ;M 8A 4>DEBYB3A6@]?R=^=B[abB[@Ha`?@CB>aU?mACSbNPF£S€AC<>@]?mACSb´B_GMB;­HB>lˆACNFIHK7H?R=0\0 1NPFŸ¯=^a`V0 1S`B>V0@C 

´m?@QSU?R²0abB’ =B[@QDEB3AQAC?FPAWGMB_@C<[ QNVœGM@QB_Z3B~ e;M 1A 4[D~BPg™96F·=^@CB>F0G^@C?

x = (x, y, z) ^t

B>A6NPF·F^NRACB>@]?

x

^{¯ a®I}SbAQ<>@]?mACSbNPF

n

=0?@

x ⁽ⁿ⁾

^{gJ6?,)« :}

cMg½«@QNPV^´B>@

x ⁽¹⁾ = (x ⁽¹⁾ , y ⁽¹⁾ , z ⁽¹⁾ ) ^t

?q´B[ZWaU?Z3NFœGMS€ACSbNPF£SbF0S€ACS`?abB

x ⁽⁰⁾ = (x ⁽⁰⁾ , y ⁽⁰⁾ , z ⁽⁰⁾ ) ^t = (0, 0, 0) ^t

^g

6 « :

e^g6V^B[aM e;M 1A 4[D~BˆGMB[´@]?RSbA >AQ@CB@C<[ QNa`V_=NV^@GM<3ACB>@CDESbF^B[@

x ⁽²⁾ = (x ⁽²⁾ , y ⁽²⁾ , z ⁽²⁾ ) ^t

¥OF^Bˆ=0?P «@C<[ QNV0G^@QB.Z3Bˆ  ;& 1A4[DEB §3g

6 « :

s0gH¼&S;NF VMAQS`abSU C?RSbA»a`?DE<3AQ\0NMGMB'GMB ?Z>N²^SY=NV^@»@Q<’ 1NPV0GM@CBjabB'  ;M 8A 4>DEB”=B[@QDEB3AQAC?FPA GMB'G^<3AQB[@QDES`F^B>@

x ⁽²⁾ = (x ⁽²⁾ , y ⁽²⁾ , z ⁽²⁾ ) ^t

:0Z>NF´B[@QpPB>@C?RSbAeH†Sba

V0 1AQSb¢0B[@´NRAC@QBµ@C<>=NPF0 QBg 6?B)« :

KL2!D[‘

*D

5dB 1?Z>N²^S`B>F£GMBYZ3BY e;M 1A 4>DEBWB’ 8A’:

F ⁰ (x) =





2x exp( − y) 0

0 2y exp( − z)

exp( − x) 0 2z



 ,

B>AHa`?~@CB>aU?mACSbNPFS€AC<>@]?mACSb´BnGMBY­HB[lˆAQNF£B[ 1A

x ⁽ⁿ⁺¹⁾ = x ⁽ⁿ⁾ − F ⁰ (x ⁽ⁿ⁾ ) ^{− 1} F(x ⁽ⁿ⁾ ) < 5 pts >

g

Ba

96F£?0:

[F ⁰ (x ⁽⁰⁾ )] ^{− 1} =





0 1 0 0 0 1 1 0 0





− 1

=





0 0 1 1 0 0 0 1 0





B3AXGMNPF0ZR:

x ⁽¹⁾ = x ⁽⁰⁾ − F ⁰ (x ⁽⁰⁾ ) ^{− 1} F (x ⁽⁰⁾ ) = (0, 0, 0) ^t −





0 0 1 1 0 0 0 1 0









− 1

− 1

− 1



 =



 1 1 1



 . < 3 pts >

5dBn  ;M 8A4>DEBn 1B[@C?S€A

F ⁰ (x ⁽¹⁾ ) ^{− 1}

^{:^SogBg`:}

[F ⁰ (x ⁽¹⁾ )] ^{− 1} =





2 exp( − 1) 0

0 2 exp( − 1)

exp( − 1) 0 2





− 1

< 3 pts >

8'

96V0S0Z3BX  ;& 1A4[DEBH 1B[@C?S€A° 1NPabV^²0abBH?q´B[Zˆa`?nD~<>AQ\^N^GMBHG^B ?Z>N²^S0Z>?R@–a`BH  ;M 8A4>DEBB[ 1A°GMSU?RpPNF0?abB[DEB>FPA°GMNDES`F0?RFPA’g

6?B.}:

. «Ž-R! 0[ŽMZ!D

P ^t LU

^Ž ‘%ZZ}ŽM!"ºŽ-'+‘«‘ ( B)}0/

96Fº´B[VMAH@Q<’ 1NPV0GM@CB6a`Bn ;M 8A 4>DEBWa`SbF^<’?RS`@QB

Ax = b

^{N J:}

A =





1 1 1 2 2 5 4 6 8





B3A

B =



 1 2 5





tgˆ±X<[@QSb¢0B[@TPV0BµaoI?abpPN@CS€AC\^DEBµG™I<>a`SbDES`F0?mACSbNPFºGMBY¶W?RV0 C .F^BW=NV^@C@C?S€A >AQ@CBµB3¬M<’Z3VMAQ< 8V0 CTPVdI?V²NVMA=NPV^@ˆ@Q<’ 1NPV0GM@CB

a`Bn ;M 8A 4>DEB

Ax = b

 ^1SJNFºFJIVMACSba`S` C?RSbAH=0? abBn=^Sb´NAC?pBg76 ,)}:

cMg46<[Z3NPDE=NP QB>@aU?ED?mAC@QSUZ3B

A

B[F£=^@CNMGMV^SbA

P ^t LU

^NPV

P

B[ 1AXa`?ED?RAQ@CS`Z>BYGMBn=B[@QD_V^AC?mACSbNPF£=0?R@HaU?EDE<3AC\^NMGMB_G™I<>a`S>H DESbFœ?mAQS`NFŸGMBY¶W?V0 C .B>AH=^Sb´NAC?pBµ=0?@1ACSbB[a®g76 *D* }:

e^g 1.?RaUZ3V0abB[@ˆa`BnG^<3AQB[@QDES`F0?RFAHGMB

A

^g7689}:

KL2!D[‘

*D

¼MNSbAaU?~D?mAQ@CSUZ3Bn?RV^pPDEB>FPAC<>Bn QV^Sb´R?RFAQB:

A =





1 1 1 | 1 2 2 5 | 2 4 6 8 | 5



 .

5°INP=<>@]?mACSbNPF a`SbpPF^B

2 =

^{a`SbpPF^B}

₂ − 2

^abS`pF0B

₁

B3AHabS`pF0B

3 =

^{a`SbpPF^B}

₃ − 4

^{a`S`pF^B}

₁

GMNPF^F^BP:

A =





1 1 1 | 1 0 0 3 | 0 0 2 4 | 1





96F´NSbA°TV^BHabBXZ>N&B Z3S`B>FA

a 2,2

¥+S®gBg`:Z3B>a`V^SœGMBHaU?nGMB[VM¬MS4[D~BHa`S`pF^BP:GMB[VM¬MS4[DEBHZ>Na`NF^F^BHGMBXZ>B3AQAQBHD?mAC@QSUZ3Bq§}B[ 1A°FV^aog

46NF0Z6NF F^Bµ=œB[VMAZ3NPFPACSbF&V^B>@.aU?_DE<3AC\^NMGMBWG™I<[abS`DESbFœ?mAQS`NFºGMBW¶W?V0 C >g¤–FBl=B>A[:&NPF F^B6=B[VMAˆ=0? °L+?Sb@CBµG™INP=<>@]?mACSbNPF GMV

 1AG;&a`B abS`pPF^B

3 =

^{a`SbpPF^B}

₃ − (a 3,2 /a 2,2 )

^{a`S`pF^B}

₂

^gJ6?,.}:

Ba

96FºS`FPACB>@C´B>@QAQSbAaU?~abS`pF0B

1

B3AHaU?~abS`pF^B

3

Z>BnTV^SdGMNF0F^B:

A =





4 6 8 2 2 5 1 1 1





5–IN=<[@C?RAQS`NFabS`pF0B

2 =

^abS`pF0B

₂ − (1/2)

^{a`S`pF^B}

₁

B3AHabS`pF0B

3 =

^{a`SbpPF^B}

₃ − (1/4)

^abS`pF0B

₁

^{GMNF0F^B:}

A =





4 6 8

0 − 1 1

0 ( − 1/2) − 1





5°INP=<>@]?mACSbNPF a`SbpPF^B

3 =

^{a`SbpPF^B}

₃ − (1/2)

^{a`SbpPF^B}

₂

^{GMNPF^F^B:}

A =





4 6 8

0 − 1 1

0 0 ( − 3/2)





96F£?EG^NF0ZµaU?EGM<’Z3NDE=N 1SbAQS`NFº QV^S`´m?RFAQBEi

A =





1 1 1 2 2 5 4 6 8



 =





0 0 1 0 1 0 1 0 0





| {z }

P ^t <3pts>





1 0 0

(1/2) 1 0

(1/4) (1/2) 1





| {z }

L<4pts>





4 6 8

0 − 1 1

0 0 ( − 3/2)





| {z }

U <4pts>

G^B3A’¥®¡6§

=

^{GMB3A’¥+©§}

×

GMB3Aq¥Z5}§

×

^GMB3Aq¥6§

,

= 1 × 1 × (4 × − 1 × ( − 3/2)),

= − 6. < 4pts >

)n™[‘ &2!D+Ž^Z!D }‘ Ÿ‘E!" ( Ba* }0/

96F£Z>NF0 QSUGI4>@CBµaU?;L{NPF0ZrACSbNPF£ QV^S`´m?RFAQB:

f(x) = x ⁴ − 3x ³ + 5

GM<>¢0F^S`Bn QV^@aoIS`FPACB>@C´m?aba`B

[0, 3]

^g

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f (x)

?RV^¬ =NS`FPA] 

x 0 = 0

^:

x 1 = 1

^:

x 2 = 2

^B3A

x 3 = 3

^{gJ6 * )}:}

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f (x)

?VM¬É=NS`FPA] 

x 0 = 0

^:

x 1 = 1

^:

x 2 = 2

^B3A

x 3 = 3

^{gJ68 }:}

e^g

´R?Ra`V^B WB[F0 QV^S€ACBnabBn=Na<;FDEBµ=NPV^@

x = 0.5

^gJ6?B)}:

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4

^{TV^S}

S`FPAQB[@Q=NPabBWaU?;LONFœZrAQS`NF

f (x)

^?Vº=NS`FPA

x 0 = 0

^:

x 1 = 1

^:

x 2 = 2

^:

x 3 = 3

^B3A

x 4 = 4

^{gJ6?,)« :}

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96F£?EG^NF0Zµa`B[ =NS`FPA]  QV^S`´m?RFAC [:

x k

¸ t c e

y k

¦ e HKe ¦

6 B)}:

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x y ∆y ∆ ² y ∆ ³ y

¸ ¦

H8c

t e H8c

H e

c HKe f

e ¦

689« :

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P 3 (x) = 5 − 2x − 2x(x − 1) + 3x(x − 1)(x − 2). < 4 pts >

Ba

16IB[ 1AHabBnD [DEBg 689« :

© NV^@Ha®IS`FPAQB[@Q=NPa`?RAQS`NFNPFAQ@CNV^´BP:

P (0.5) = 5.625. < 2 pts >

8'

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f (x) = x ⁴ − 3x ³ + 5

Z>?@µabB~=NPa;&FPDEB~GMBEGMB>pP@Q<

4

TV^S=0? C 1B;=0?@µa`B[ 

5

=NPSbFAC WGMB

f(x)

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0anZ«‘ (+*-,.}0/

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6?B)« :

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F^B[@HTV^B>aJB[ 1Aa`BWAG;&=BYGMBn Q=^a`SbF0B¥OAG;&=Bn5r:^5Q5r:M5Q5Q5r:^5K± NVºS`F0GM<3ACB>@CDESbF^<q§rgJ6 « :

¥®?P§

f (x) = ( ₁₉

2 − ⁸¹ 4 x + 15x ² − ¹³ 4 x ³

^=NV0@

1 ≤ x ≤ 2

− ⁷⁷ ₂ + ²⁰⁷ ₄ x − 21x ² + ¹¹ ₄ x ³

^=NV0@

2 ≤ x ≤ 3

¥O²œ§

f (x) =



 



 



3

2 − ¹ ₄ x + 2x ² − x ³

^{=NPV^@}

10 ≤ x ≤ 21

− ¹ 4 x + 2x ² − x ³ + ³ ₂

^{=NPV^@}

21 ≤ x ≤ 22

6

4 − ² ₈ x + 2x ² − x ³

^{=NPV^@}

22 ≤ x ≤ 24

f (x) =

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^{=NPV^@}

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^{=NPV^@}

21 ≤ x ≤ 22

6

4 − ² ₈ x + 2x ² − x ³

^{=NPV^@}

22 ≤ x ≤ 24

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f ⁰⁰ ( − 2) = 0

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f ⁰⁰ (4) = 0

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f (2 ⁻ ) = f (2 ⁺ ) = 3 f ⁰ (2 ⁻ ) = f ( ⁰ 2 ⁺ ) = 0.75 f ⁰⁰ (2 ⁻ ) = f ( ⁰⁰ 2 ⁺ ) = − 9

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f ⁰ (1) = f ⁰ (3) = 0 f ⁰⁰ (1) = 7.5 6 = f ⁰⁰ (3) = 10.5

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f ⁰ (10) 6 = f ⁰ (24)

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h 1 = h 2 = 3

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S 1 = 0

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S 2 = − 3/2 S 3 = 0

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b 1 = S 1 /2 = 0

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c 1 = [(y 2 − y 1 )/h 1 ] − h 1 (2S 1 + S 2 )/6 = [(3 − 0)/3] − 3(2 × 0 + ( − 3/2))/6 = (21/12)

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d 1 = y 1 = 0

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a 2 = (1/12)

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b 2 = ( − 3/4)

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c 2 = ( − 1/2)

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d 2 = 3

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f (x) =

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^=NV0@

− 2 ≤ x ≤ 1

− ¹ 4 (x − 1) + 2(x − 1) ² − (x − 1) ³ + ³ ₂

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1 ≤ x ≤ 4

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