121 P LU t s

(1)

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

46587:9<;04>=@?BADC1E1F@GHFJIE:KML5NIPORQC1G ASEUTWVYXZF[F@E6KPF\7FJ]_^0F@C_]_^ZF`9>?,=@C_ASEUTaQIZIZFJbabWFc;PbWQP]@ADbedcfYgg%h

iEjEU?lkBmcmSnnn`hTWC1QBhX0GoQIYE1CUFJADbph]@AYm

∼

GHTaqIZQDE1E1FrmSTsOtE3dSu%dPvrm

w.xNy zS{t|~}~y<{sc,DND{t1WPyHD0p1_zS|at_z

PJlZ8c3,YcY

5hhJhh@h@h@h@h@h@hhJhh@hhJhh FJ1XZCUF`KML5NIB]F@C1E1TaE1XBKPF`FE66Go?0baTaB]@ADE1TWQcIKMLCUC1FJXZC\dZv ?ZEU_¡3h

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P ^t LU

pdDfo?PE_U¡h

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121

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(2)

1 QTsEXZI?0ADC_ADbWba=Jba=J?ZTW?32 KPF>CUFJ]EUAcIZqcbWF KZQcIYEbaF £EUC1QTW.bWQcIZqXZF@X0CU.KPFJ]@QDEU=J[RCUFJ1?BF ]3EUTa¨F@GHF@IYE^0ADXZE1F@X0CJ;%bWQcIZqXZF@XZCF@E

b¥ADCUqcF@X0C_¡jQIYEJ;

h

^;

l

^FE

L

hP¬®QXZC:]_^0Ac]@XZIZF`KPF`]F ¨SADCUTWA54ZbaF @;PQIAobpLTWGH?ZCU=J]T¥1TaQI1XZTW¨rAcIYE1Fc;

6

h = 40

^G72EUC1F ^@; Ar¨cFJ]>XZIZF`FJC1CUF@X0C.CUF@b¥ASEUTa¨F KZF

0.1%

^h

6

l = 40.3

G82@E1CUFJJ; Ar¨cFJ]>bWF:9jf;>=E_ADIYE:]QcIBjT¥KP=@CU=`]QGoGHF`KPFJC1IZTWF@C:]_^0T=<C1F`1TaqIZTaB]@ADE1TaO>9jFªZAc]E?;H]@1F ¡h

6

L ∈ [40.2, 40.6]

^; ^ThFchW;A@¶FJjEXZIZF`¨SADbWF@XZC`*FªP?ZCUTaGH=@F`FJI G72@E1CUFJ_¡£=Jba=JGHF@IYE6KPF`]F@EjE1F`TWIYE1FJC1¨SADbWbWF KPF\]@QcIP,ADI0]@Fch 9>I¨cQcXB.KZF@G ADI0KZF\KPF k

vhªP?ZC1TWGHF@C ;M?BQXZC\]_^0A]XZIZF KPF¤]@FJ EUC1QTW[¨SADCUT¥AB4ZbWFJJ;bLTaIB]F@C1E1TaE1XBKPF¤VYXZF bLQcI¶AjX0C`F@bWbaF FJI b¥AGHFEjE_ADIYE 1QcX0 b¥A

ORQcCUGoF

x = x ^? ± ∆x

^Q5C

x ^?

^FE

∆x

1F@CUQcIYEKZ=E1FJC1GHTWIZ=JJhEDFG~H dZh:46QIZIZF@C®bpLAc?Z?ZCUQrªPTaG ASEUTaQI\KPXo¨cQcbWXZGHF

V

KPF]F?0AcCUAcbabW=@bW=@?0Ta?$2JKPFC1F ]3E_ADIZqbaF *KPQIZIZ=@F?0AcC¯bWF?ZCUQ%KZXZTsEKPF]F ¯E1CUQcT¥

]QDEU=JJ;MThFchW;

V ^∗ )

^D+I¯H ^;eFE\bpLTaI0]@F@C1E1TaE1X0KZF KPF

V

^*Th^FchW;

∆V

¡`?0ADC\b¥AGH=EU^ZQPKPF KPF ?ZCUQc?BADqADE1TWQcI±KMLFJC1CUF@X0C

RXPEUTabWTWUADIYEebpLAD?Z?ZCUQrªPTaG ADE1TWQcI`KPF·~A(J%baQCeKPFbWAORQI0]3EUTaQI ADX`?0C1FJGoTWF@C¯QcC_KPCUF ¡3hKDL7~H

1

QcXZbWTaqIZF@CebaF ¯]_^ZT<C1F

jTWqcIZTaB]JASE1TaO*M98F@ªZAc]3E_N;*]@1F ¡ KPF¤]@FHC1= jXZbaEUADEJh©6C1CUQcI0KZTaC`]F@EjEUFHAc?Z?ZCUQrªPTaG ASEUTaQI AcXPª«IZQcGO4ZCUFJ[KPF¤]@1F AcKP= VYX0ASE h

D+¯H 46QIZIZFJC60IBADbWF@GHF@IYE>bpLTaIYE1FJC1¨SAcbabWF KPFH]QIPBADIB]F KZAcI0:bWFJVYXZFJb~1F\EUC1QXZ¨cFJCUAcTsE>bWA ¨%C_ADTWF\¨SADbWF@XZC[KPF

V

Q54PEUF@I%XZF[?0ADC:]@FE1E1F[GH=E1^0Q%KZF¤RF@IXPEUTabWT¥1AcIE:bWFJAc?Z?ZCUQrªPTaG ASEUTaQI0.IZQcIADCUCUQcI0KPTWFJ_¡3h.DP+*~H

f0h¬~FJXPE:QcIXPEUTabWTW1F@Cb¥AHGH=EU^ZQPKPF\KPF[bWA<ORQXZCU]_^0FEjEUF`?BQXZCQ54ZE1F@I0TaC:]@FEjEUF`TaIB]F@C1E1TaE1XBKPFQ

1

T©QcXZTMF@ªP?ZbaT¥VYXZF@C?,QcX0CUVYXZQT

D+R~H´FE XPEUTabWT¥jFJCH]@FE1E1FGH=E1^0Q%KZF¶KPFbWA ORQcX0CU]_^ZF@EjEUF ¡ ?BQXZCHQ54PEUF@IZTWCHXZI&TWIYE1FJC1¨SADbWbWF KPF]QIPBAcI0]FKZADIB

baF VX0F@b©1F6EUC1QXZ¨cFJCUAcTsEbWA<¨%CUAcTaF ¨SADbWF@XZC:KZF

V

F@EFJIKP=JKPX0TaCUF>bpLTaI0]@F@C1E1TaE1X0KZF`C1= jXZbaEUAcIYE1FHRTphFha;ZFJC1CUF@X0C.A540jQbaX0F ¡.jX0C

V

^VYXZF`bpLQcIQ54PEUTaFJI0KPC_ADTaE?BADC:]F@EjEUF`Go=@E1^ZQPKPFhSD7TG¯H UWVX"$J

-3

@LF@CUCUF@XZC:CUF@b¥ASEUTa¨F KPFobWA ?ZCUF@GHT2JC1F ¨rAcC1T¥AB40baF\I0QcX0>?BFJC1GHF@E[KPFo]@AcbW]@XZbaFJC>bpLF@CUC1FJXZC>A540jQbaX0F VYXZF bpLQIO*ADTaE`jXZC[]F@EjEUF

GHFJ1XZCUFc;PTphFha;

∆h = 0.04

FE6KPQI0][KML= ]CUTaCUF

h = 40.0 ± 0.04

^h ^D+*~H

@©F O*ADTaE¤VYXZF bWF]_^ZT<,CUFKZFJ KPTaªPT2JGoF Y9

3

^; *KZFC_ADI0q

− 1

¡<1QcTaEobWFKPF@CUIZTWF@CH]_^ZT=<CUF1TWqcIZTaB]@ADE1TaO6IZQX0o?BFJC1GHFE¤KPF E1CUQcX0¨cF@C£XZIZFZ4BQC1IZF>jXZ?,=@CUTWF@XZCUF

∆l = 0.5 × 10 ⁻ ¹

^KZF6bpLFJC1CUF@XZC.AB401QcbWXZF6K©LF 8EUTaG ASEUTaQIHO*A([¥E1F[jX0C]F@EjEUF>GHFJ1XZCUF:FEKPQcIB]

KML=J]@C1TWC1F

l = 40.3 ± 0.05

^h ^DP+*¯H

¦~TaIBADbWF@GHF@IYEJ;0bWAH]QIZI0AcTWU1AcI0]F;P?BQXZCb¥AHGoF jX0C1F

L

^KMLXZITWIEUF@CU¨SADbWbaF`KPF ]@QcIP,ADI0]@Fc;PIZQX0?BFJC1GHF@E>KML=J]CUTWC1F]\ZK©LXZI0F

?0AcCjE[VX0F<b¥A¤¨SADbWF@X0C Ac?Z?ZCUQrª%TWGH=@F\?,QcX0C ]@FEjEUF<¨SAcC1T¥AB4ZbWF<1F@C_A bWFo]@F@IYE1CUF<KZFo]@FEjEUF<TWIYE1FJC1¨SADbWbWF FE`KMLADXPEUC1F<?0ADC1E VYXZFH1QcI

F@CUCUF@XZCAB4BjQbaXZF[F 8Eb¥AoKZF@GHT©b¥ADCUqcF@X0CKZF\]FETWIYE1F@CU¨SADbWbaF;PThFchW;

L = 40.4 ± 0.2

^h ^DP+*¯H

^"BJ«k_@¯A CU=@?,QcI01F>VYXZTM]@QcI01TWjEUAcTsEM`<?BQXZCU1XZTW¨%C1F>baF[]@AcbW]@XZbMAr¨cFJ]:?BQXZC]_^0Ac]@XZIZF[KPF ]@FJ.TWI0]FJCjEUTsEUX0KPFJJ;%XZIZFa4,QcCUIZF

KMLF@CUC1FJXZCjX0?B=JC1TWF@XZCUF`KPX8E?J%bWF

h = 40.0 ± 0.05 l = 40.3 ± 0.05 L = 40.4 ± 0.5

^F 8E6]@QcCUC1F ]3EUF AcX011Tph +.

9>IA<?,QcX0C¨rAcbaFJXZC:AD?0?ZC1QP]_^Z=JFc;

V ^∗ = 40 × 40.3 × 40.4 = 65124.8 m ³ . < 2 pts >

¬®QcXZC:]JADb¥]XZbWF@CbpLTWI0]@F@C1E1TaE1X0KPF\KZF

V

;PQcIKPQTsE:]JADb¥]XZbWF@Cb¥AHKPT=<=@CUF@IYEUTaFJbabWF

∆V (h, l, L)

^;%Th^FchW;

∆V (h, l, L) = | l × L | ∆h + | h × L | ∆l + | h × l | ∆L. < 2 pts >

1 QTsE ;

∆V (h, l, L) = | 40.3 × 40.4 | × 0.04 + | 40.0 × 40.4 | × 0.05 + | 40 × 40.3 | × 0.2

= 65.1248 + 80.8 + 322.4

≈ 468.3248. < 1 pt >

(3)

9>IQ54ZE1TWF@IYE KPQI0]

V ≈ 65124.8 ± 468.3248

^hQGHGoF

468.3248 < 0.5 × 10 ³

;bWF«C_ADIZq&KPX¢KPFJC1IZTWF@C ]_^ZT=<CUF

1TaqIZTs,]@ASEUTsOFJjE

3

F@E£QcIH?,F@XPE£= ]CUTaCUF

V ≈ 65124.8 ± 468.3248

^D ^-¯Hh ¹ TBQcI ADCUC1QI0KPT%0IBADbWF@GHF@IYE]@FEjEUFFJjE1TWG ASE1TWQcI ADXZª IZQcGO4ZCUFJKPF\]@1Fc;ZAKP=JVYX0ADEJ;PQIQ4PE1TWF@IYE ;

V ≈ 65 × 10 ³ ± 500 (

^FJI ^G

³ ). < 1 pt >

1 TPQIoXZE1TWbaT¥jFbWFJAc?Z?ZCUQrªPTaG ASEUTaQI0~IZQcIHADCUCUQcI0KPTWFJJ;

V

jF.EUC1QXZ¨cFJCUAcTsEKZADIB®bLTaIYEUF@CU¨rAcbabWF

[64656.4752 , 65593.1248]

^h

DP+*¯H

L

@eA GH=E1^0Q%KZFHKZF<b¥A¤ORQXZC_]_^ZFE1E1Fo?BFJXPEE1CUFoXPE1TWbaT¥1=@FoTW]@T®]JADC[b¥A ORQcIB]3E1TWQcI

V (l, L, h) = l × L × h

FJjE`8EUC1T¥]3EUF@GHF@IYE GHQcIZQcE1QIZF«RFJI¶O*ADTaEo]CUQcT¥U1AcIEUF ¡ 1XZC

IR

KPQI0]¤1XZC`EUQcXPEUF¤TaIYEUF@CU¨rAcbabWFP4,QcCUIZ=che5Nb£FJjE<KPQI0] ?,QUjT4ZbWFc;©FJIlXPE1TWbWTWUADIYE bWFJ

¨SADbWF@XZC_.GHTaIZTWG ADbWFJ.?,QcXZC

l, L, h

KPF`KP=@E1FJC1GHTWIZF@Cb¥A 4,QcCUIZF GoTWIZTWG ADbWF ?,QcX0C.³ F@EFJI XZE1TabWT¥1AcIEbWFJ.¨SADbWF@XZC_.GHADªPTaG ADbWFJ

?,QcXZC

l, L, h

KPFoKZ=E1FJC1GHTWIZF@C>bWAW4BQC1I0F\G ASªPTaG AcbaF<?BQXZC

V

^*F@I O*ADTaE>TWb¯F 8E E1QXPE 1TaGH?ZbWF@GHFJIE ?BQY11T4ZbaFoKPF<]JADb¥]XZbWF@C>baF

¨cQbaX0GoF[GHTWIF@E:G ASª FJI1F`jFJC1¨SAcIECUFJ1?BF ]3E1TW¨cFJGHF@IYE:KPFJ

l, L, h

baQIZqcX0F@XZC_.GoTWIFE:G ASª¡h DP+*¯H

V

= (40 − 0.04) × (40.3 − 0.05) × (40.4 − 0.2) = 64657.278 < 1 pt >

V

= (40 + 0.04) × (40.3 + 0.05) × (40.4 + 0.2) = 65593.9284 < 1 pt >

@eAo¨SADbWF@X0CAc?Z?ZCUQrªPTaGH=@F FJjE:KPQcIB]>bWF`]FJIEUC1F[KPF\]@FEjEUF[TaIYE1FJC1¨SAcbabWF`KPF`]QcIZBADI0]@F>F@EbLTaIB]F@C1E1TaE1XBKPF\jX0C

V

1F@C_ADTaEbWAHKPFJGoT b¥ADCUqcF@X0CKZF[]@FE1E1F`TWIEUF@CU¨SADbWbaFhF\VX0TMIZQcX0:?BFJC1GHFE:KML=J]@C1TWCUF

V ≈ 65125.6032 ± 468.3252

^h ^DP+ ^¯H

(&' ( r P R VY.~M"0JJA «YZA¯Z#"$ X&. S #"$ ) -K~0

vhªP?ZbaT¥VYXZF@C?,QcX0CUVYXZQTMbaF`]@AcbW]@XZb©I%XZGH=@CUT¥VX0F[KZF[]@FJF@ª%?0C1F 11TaQI0J;

*A¡

exp(x) − exp( − x)

^; ^VYX0ADI0K

x ≈ 0

^;

4B¡

√

x ² + 1 − 1

^; ^VYX0AcI0K

x ≈ 0

^;

]J¡

P x=100 x=1

1 x ³

?BFJXZ¨cFJIEo]QcIBKPXZTWC1F `«KPFJ<?ZCUQ54Zb2@GHF KMLF@CUC1FJXZC_`I%XZGH=@CUTWVYXZF \eª%?0baT¥VYXZF@Co?BQXZC_VX0QcT>*ThFchW;~TWKPFJIYE1Ta0F@CoFEH]TaE1F@C

baF<?ZC1Q4Zb2@GHF I%XZGH=@CUT¥VX0F<AcU1Q%]@Ta=r¡FE[?ZCUQc?,Q1F@C:X0IZF ORQC1G<XZbWF =JVYXZTW¨SADbWF@IYE1F<VX0T~?,F@CUGHFE1E1C_ADTaE KML=@¨%TaE1F@C[]F 6?0C1Q

4Zb2JGHFJI%XZGH=@CUTWVYXZF FE6VYXZTM?,F@CUGHFEjEUCUAcTsE6K©LAcXZqcGHF@IYEUF@CbWAo?ZCU=J]@TW1TWQcIKPFJ]JADb¥]XZbeKZF[]@FJ.EUC1QTWF@ª%?0C1F 11TaQI0Jh

D*~H

dZh§¨SADbWXZF@CbpLF@ªP?ZC1F 11TWQcI 1XZTW¨SADIYE1FHk

x = ³ r

3 + ³ q

3 + √ ³ 3 + . . .

¬~QXZC]@F@b¥AZ;?ZC1FJIZF6baF6]@XA4,F>KPF ]FE1E1F>FªP?ZCUFJUjTWQcIRTphFchW;Y]JADb¥]XZbWF@C

x ³

^;cF@I KP=JKPX0TaCUF:F@IBjXZTaE1F>XZIZF>=JVYX0ASEUTaQI¤KPX E?J%?,F

f (x) = 0

?,QcXZCb¥AcVYXZFJbabWF

x

FJjE1QcbWXPEUTaQI FE:XZE1TWbaT¥jFJCbWAoGH=EU^ZQPKPF[TsEU=@C_ASE1TW¨cF[KPF`°:FJnE1QI FJI ?0AcCjE_ADIYEKPF

x [0] = 1

*°:QcEUAkZT¥]TMbpLTWI0KPT¥]F`TWI0KPT¥VYXZF`bpLTaE1=JCUADE1TWQcIB¡h.ADb¥]XZbWF@CbWFJ

6

?ZCUF@GHTWF@C_E1F@CUGHFJKPF\]@FE1E1F`GH=EU^ZQPKPF`TsEU=@C_ASEUTa¨cFh D*~H

(4)

-3

¬®QcXZCbaF>?ZCUF@GHTaFJC£F@E.KPFJXPªPT2JGoF ]@A@;QcI¤ADX0CUA`X0I¤?ZCUQ54Zb2@GHF6KMLPAcIZI%XZbWADE1TWQcI¤KPF>]@1F>KPXZFZ``b¥A 1QcX0jE1C_Ac]E1TWQcI KPF>KZF@XPª

IZQG 4ZCUFJ\Ac?Z?ZCUQrªPTaGH=J`VYX0AjT£=@qAcXPª D -%¯H h¯¬®QcXZC<=@¨%TsEUF@C ]@F¤?ZCUQ54Zb2@GHF;©QcIl?,F@XPE<=J]CUTWC1F¤]@FJ\KPFJXPª¶C1FJbWADE1TWQcI0\KZF

E1FJbabWFO*A@QcIoVX0F?ZbaXBADX0]@XZIZFjQX0jE1C_Ac]3EUTaQI<KPFKPFJXPª<I0QcG 40C1F ¯TWI0]@F@C1EUADTWI0RTphFchW;DTaGH?ZCU=J]@TW_¡~VYX0Ac1TP=@qYADXPª Ac?Z?0ADC_ADT¥UjFJIE h

¬®QcXZCb¥Ao?ZCUF@GHT2JC1F[F@ª%?0C1F 11TaQI QI ¨SAHXPEUTabWTW1F@CbWF\KP=J¨cF@bWQc?0?BFJGoFJIYEbWTWGoTaE1=\KZF

exp(x)

ADX¨cQcT¥1TaI0AcqcF[KPF

0

^h

exp(x) − exp( − x) ≈ 1 + x 1 + x ²

2! + o(x ³ ) − 1 − x 1 + x ²

2! + o(x ³ ) ,

≈ 2x + o(x ³ ), x ≈ 0. < 2 pts >

¬®QcXZCb¥A[jF ]QI0KPFFªP?ZCUFJUjTWQcIe; QI<¨SA`1TaGH?ZbWF@GHFJIEG<XZbaE1TW?ZbaTWF@CbpLFªP?ZCUFJUjTWQcIo?0ADC

√

x ² + 1+ 1

F@E£KPF:]FO*AcTsEQ4PE1FJIZTaC ;

p x ² + 1 − 1 = x ²

√ x ² + 1 + 1 , x ≈ 0. < 3 pts >

VYXZT©IZF[?,Q1F`?ZbWX0KPF`?ZCUQ54Zb2@GHF`KMLADIZI%XZb¥ASEUTaQIKZF[]JjFh

¬®QcXZC bWF EUC1QTW1T2JGHF ?ZCUQ54Zb2@GHF;BbpLFJC1CUF@X0C6I%XZGH=@CUTWVYXZFo¨%TaFJI0KPC_A¤KPX O*ADTaE[VYXZF<bLQcI ¨SA¤O*AcTaCUF<AcXE1QXPE[KZ=4ZXPE\KPFH]F@EjEUF

1QcGHGHFc;,XZIZF<AKZKPTaE1TWQcI«FJIYE1CUF\¨SADbWF@XZC[KMLQcC_KPC1F<KPF qCUAcI0KPF@X0C6KPT<,=JC1FJIEUFD -G¯H ;]FoVYXZT®¨rA Ac]@]@XZG X0baFJC>KPF<bLTaGH?ZCU=

]T¥1TaQI©h05Nb¯O*ADXPE[KPQcIB]\GHTaFJXPª«]@AcbW]@XZbaFJC:EUQcX06baF 6?,FEUTsE_:EUF@CUGoF 6FJIYE1CUF\F@XPª?ZX0TW>F@I01XZTaE1FoAcKZKPTaE1TWQcI0IZF@C ]F\CU=J1XZbaEUASE[ADXPª

E1FJC1GHF ?0baX0.qCUAcI0K¤KPF[]F@EjE1F`1=@CUTaFh¬®QXZC.O*ADTWC1F[jTWGH?ZbWFc;%TabM¨SAcXPE.GHTWF@XPª ]JADb¥]XZbWF@CI%XZGH=@CUT¥VX0F@GHF@IYE.]@FE1E1F`j=JC1TWF>K0ADI0.bWF

1F@I0TWI%¨cF@C_1F

x=1 X

x=100

1 x ³ . < 2 pts >

+.

9>IH@LAD?,F@CQTsE~O*Ac]@TabWF@GHF@IYEVYXZF

x ³ = 3+ x

DL¯H#F@E£KPQcIB].VYXZF.ª F 8E1QcbWXPE1TWQcIHKPFbpL= VXBASE1TWQcI

x ³ − x − 3 = 0

^h

@eAoCUF@b¥ASEUTaQI TaE1=JCUADE1TW¨cF[KPF\°:FJnE1QcIIZQX0?,F@CUGoF@E6KML=J]CUTWC1F;

x [n+1] = x [n] − f (x [n] )

f ⁰ (x [n] ) , k = 0, 1, . . .

= x [n] − x ³ _[n] − x [n] − 3

3x ² _[n] − 1 . < 3 pts >

]F`VYXZT©I0QcX0?,F@CUGHFE6KPF E1CUQcX0¨cF@C ;

x [0] = 1.0, x [1] = 2.5,

x _[2] = 1.929577465, x [3] = 1.7078664, x [4] = 1.672558473, x [5] = 1.671700382.

x [6] = 1.671699882.

. . .

VYXZTe1F@GO4ZbaF`]@QcI%¨cFJC1qF@C.¨cF@C_

x = 1.671699882

E1C,2JC_AD?ZT¥KPF@GHFJIE h

DP+ ¯H

(5)

9>IjF[?ZCUQc?,Q1F`KPF E1CUQcXZ¨F@CXZIZF`¨SADbWF@X0C:AD?Z?ZCUQP]_^Z=@F[KMLXZIZF\KPF KPF@XZª C_Ac]TWIZF .KZF`bL=JVYX0ADE1TWQcI©;

f (x) = x ln(x) − 1.25 = 0.

^8vr¡

vh VY " ¯/ X"$#8 ©

*A¡ QIEUC1FJCVYX©LTab,FªPT¥8EUF6XZI0F>C_Ac]@TaIZF6XZIZT¥VX0F

r

?,QcXZC]F@EjE1F>=JVYX0ADE1TWQcI jv ¡K0ADI0bLTWIEUF@CU¨SADbWbaF

J = [1.65, 2]

^hDI

CUF@G ADC_VYX0ADIYE>VYXZFobL=JVYX0ADE1TWQcI ?ZC1Q?BQYj=JF\FJjE[=JVYXZTW¨rAcbaFJIYE1F7`

g 1 (x) = x

^Ar¨cFJ]

g 1 (x) = ^1.25 _ln _x

;MGHQcIYE1CUF@C VX0F ]F@E6TWIEUF@CU¨SADbWbaF`F 8E6X0ITaIYE1FJC1¨SAcbabWF`jXZC>baF VYXZF@b©b¥A ]QI%¨cF@CUqcFJI0]F[¨F@C_XZIZF 1QcbWXPE1TWQcIXZI0TWVYXZF\?BADCb¥A GH=E1^0Q%KZF

KPX?,QcTWIE0ª%F[I©LFJjE:?0AAcUjXZCU=@Fh.D8T ~H

4B¡ I ?ZCUF@I0AcIEoE1QX8QcX0CU<baFG@GHFTaIYEUF@CU¨rAcbabWF

J

K0ADI0obWFJVYXZF@bIZQX0HjQGHGoF HAcUjXZCU= KMLAr¨cQTaCoXZIZFC_Ac]TWIZF;

?ZCUQc?,Q1F@C:X0IZF ADXZE1CUF`ORQcI0]E1TWQcI

g 2 (x)

;0EUF@b®VX0F

g 2 (x) = x

1QcTaE6=JVYXZTW¨SADbWF@IYE1FO` bL£V,h®8v ¡FE>?,QcX0C6b¥AcVYXZF@bWbWF b¥A ]QcI%¨F@CUqcF@IB]F ¨cF@C_.XZIZF 1QcbWXPEUTaQI XZI0TWVYXZF`?0AcC:b¥AHGo=@E1^ZQPKPF\KZX?,QcTWIYE:ZªPF`I©LFJjEE1QX8QcXZC_?0A:AcUjX0C1=JF

DLG¯H

]J¡QIEUC1FJC:VYXZF[bL=JVYX0ADE1TWQcIl8vr¡.FJjE6ADX0UjTM= VYXZTa¨SAcbaFJIEUFY`

g 3 (x) = x

^Ar¨FJ]

g 3 (x) = exp(1.25/x)

FE6VYXZF\1XZC

J

;0b¥A ]QIY¨F@CUqcFJI0]F[¨cFJCUX0IZF 1QcbWXPE1TWQcIXZIZT¥VYXZF\?0AcC:b¥AHGo=@E1^ZQPKPF<KPX?BQTaIYE:0ª%F FJjE6]FE1E1F`ORQTW>]T~AcUjXZCU=@Fh DLG¯H

*K0¡ IXZE1TWbaT¥1AcIYEKPQI0]bpL= VX0Ta¨SADbWF@IB]F KPF bL=JVYX0ADE1TWQcIjv ¡ Ar¨FJ]

g 3 (x) = x

;KP=EUF@CUGHTaIZFJC AcI0ADbJYE1T¥VX0F@GHF@IYEJ;

qcC c]@F ADX±E1^Z=JQcC,2@GHF¤KPF \Ac]J]CUQcT¥11F@GHF@IYE_6BIZTW RQX KPF bWA¨SAcbaFJXZC\GHQJcFJIZIZF ¡;©XZIZF G A8QcC_ASE1TWQcI¶KPX¶E?J%?,F

| r n − r | ≤ K

^;QcX

r n

KP=J1TaqIZFbWA±¨SADbWF@XZC Ac?Z?ZCUQ%]_^0=@Fc; ` bWA±I T2@GHFTsEU=@C_ASE1TWQcIe;®KZF]@FE1E1FCUA]TWIZF?0ADCHb¥A

GH=EU^ZQPKPF6TaE1=JCUADE1TW¨cF:KPX¤?,QcTWIYEZªPFchYI KZ=JKPXZTWCUF:baF6IZQcGO4ZC1F:K©LTaE1=JCUADE1TWQcI0

n _∗

^IZ= ]FJUUADTWC1F?,QcXZCQ54PEUF@IZTWC ;c?0AcC ]F@EjEUF`Go=@E1^ZQPKPF;ZXZIZF`¨SAcbaFJXZC:AD?Z?0C1QP]_^Z=JFZ`

10 ⁻ ¹

?ZC,2JKPF\]@FE1E1F[CUA]TWIZFch D*~H

:X0CUAcTsEpQI?ZX©;M1AcI0:XZE1TWbaT¥jFJC ]@F\EU^Z=@QC 2JGoF<KPFJ Ac]J]CUQcT¥11F@GHF@IYE0I0TWJ;B?0C1=J¨cQcTWC6VYXZF<bWA ¨YTaE1F 11F KPFo]QIY¨F@C

qcFJI0]F`KPF\]@FE1E1F`GH=EU^ZQPKPF\KPX?BQTaIYEZªPF\1QcTaE1TMbWF@IYE1FQ QcGHGHF@IYE$QcX0jE1Ta0FJC¨cQDEUC1F CU=@?,QcI01Fch

DP+*¯H

*F ¡ .ADb¥]X0baFJCbaF

4

^?0C1FJGoT^2@CUFJF 8EUTaGH=JFJ

r 1 , . . . , r 4 ,

F@I?0AcCjE_ADIYE:KPF

r 0 = 1.65

^h ^DLG~H

dZh VY " ¯/~K^H"$

*A¡ 1 QTsE

g 4 (x)

^;£bWA ORQcI0]E1TWQcI½TWIYE1FJC1¨F@I0AcIE KZAcI0 b¥A±GH=E1^0Q%KZF TaE1=JCUADE1TW¨cFKPF°6F@nEUQcI ?BQXZCHbWA C1= jQbaXZE1TWQcI½KPF bpL= VXBASE1TWQcI&8v ¡hM46QIZIZF@C

g 4 (x)

ADTWI0jT®VYXZF b¥A¤CUF@b¥ASEUTaQITsEU=@C_ASEUTa¨F

r n+1 = g 4 (r n )

he9>I«?ZCUF@IBKPCUA

r 0 = 1.65

?,QcXZC:AcGoQCU]@F@C]F@EjEUF`C1FJbWADE1TWQcITsEU=@C_ASE1TW¨cFh D7T ¯H

4B¡ I«KP= KPXZTWC1F`X0IZF`¨SADbWF@XZC:Ac?Z?ZCUQ%]_^0=@F`KPF

r

^AD?ZC,2J

4

TsEU=@C_ASEUTaQI0`RTphFha;,KPQcI0IZF@C

r 1 , r 2 , r 3 , r 4

¡[Ar¨cF ]6EUQcX8QXZC_

r 0 = 1.65

^¡3h ^DLG¯H

]J¡ IH¨cQX0®ADT¥KZADIYEKPF]FVYXZF¨cQX0®Ar¨cF £O*AcTsEFJIoKP=JGHQcI0jE1C_ASEUTaQI©;SKPTaCUFVYXZF@bWbaFG A8QCUADE1TWQcI KZF

| r n+1 − r n |

^FJI

ORQcIB]3E1TWQcIKPF

g ₄ ⁰ (x)

RFE KZADI0b¥AcVYXZFJbabWF

r n

KP= jTWqcI0F`bWAo¨SADbWF@XZC>AD?Z?ZCUQP]_^Z=@F;_`HbWAHI T2@GHF[TsEU=@C_ASE1TWQcIe;0KPF\]F@EjEUF

C_Ac]TWIZF[?0AcC.b¥AoGH=E1^0Q%KZF\KPF`°:F@nEUQcIB¡.I0QcX0?,F@CUGHFEjEUCUAcTsEFJI0jX0TsEUF\KPF`KP=JKPX0TaCUF`baF[IZQG 4ZCUF[KMLTaE1=JCUADE1TWQcI0

n _∗

IZ= ]FJUUADTWC1F:?,QcX0C£Q54ZE1F@I0TaC ;?BADC]FE1E1F>Go=@E1^ZQPKPF[KPF °:F@nEUQcI©;YXZIZF ¨SADbWF@XZC.AD?Z?0C1QP]_^Z=JF:`

10 ⁻ ¹

^?ZC ² KZF>]F@EjEUF C_Ac]TWIZF RI0QDEUAkZI0F`?0Ac]JADb¥]XZbWF@C]F`I0QcG 40C1F[KMLTsEU=@C_ASE1TWQcIBQc?PEUTaG ADbWF ¡h DFG¯H

UWVX"$J

-Y.

@L=EUX0KPFKPFJ~¨SADCUT¥ASE1TWQcIB¯KPFb¥AORQI0]3EUTaQI

f

^1XZC

J = [1.65, 2]

GHQcIYE1CUFVYXZF.b¥A:ORQI0]3EUTaQI<F 8E]QIEUTaI%XZF.FEKP= ]CUQcT¥1UADIYE1F

1XZC>]@FE6TWIYE1F@CU¨SADbWbaF¤KPQcIB]\GHQcIZQcE1QIZF ¡[

f ⁰ (x) = ln(x) + 1

^FE

f ⁰ (x) > 0 ∀ x ∈ [1.65, 2])

^hSD ^- ^~H 46F<?ZbaXB@;BQcI A

f (1.65) = 1.65 ln(1.65) − 1.25 ≈ − 0.423720

^F@E

f (2) = 2 ln(2) − 1.25 ≈ 0.136294

^h>D- ^¯H 5Nb©FªPT¥8EUF\KPQcIB]

XZIZF`C_Ac]@TaI0F

r

XZIZT¥VYXZF\KZADIB]FETWIYE1F@CU¨SADbWbaFhZ46F`?0baX0J;

(6)

g ⁰ ₁ (x) = − 1.25 x(ln(x)) ² .

@©F ]JADb¥]XZb£KZF

| g ⁰ ₁ (x = 1.65) | ≈ 1.5128 > 1

he4>QcIB]

| g ⁰ ₁ (x) | 6 < 1 ∀ x ∈ J

^;©F@E ]F¤]@QcIYE1CUFHFªPF@GH?ZbWFHIZQcX0 jXHE

?,QcXZCIZQX0?,F@CUGoF@EjEUC1F\KMLA CUGHF@CVYXZF`bWAH]@QcI%¨cFJC1qF@I0]@F6I©LFJjE:?0AcA11XZCU=@F[jX0C

J

^h ^D+*~H

-

I?ZCUF@I0AcIYE

g 2 (x) = x + x ln(x) − 1.25

;PQcIAHXZIZF`IZQXZ¨cFJbabWF[ORQcT¥bL=JVYXZTW¨SADbWF@I0]@F`F@IYE1CUF`bpL= VXBASE1TWQcI

x = g 2 (x)

^FE

bpL£V,hejv ¡h0°:= ADIZGHQcTWI0J;%XZIZF[ORQcT¥KPF`?ZbWX0b¥AH]QcI%¨F@CUqcF@IB]F6¨F@C_.XZIZF\1QcbWXPE1TWQcIXZIZT¥VYXZF[?0ADCb¥AoGo=@E1^ZQPKPF\KZX?,QcTWIYEZªPF

I©LFJjE[E1QcX8QcXZC_ ?0Ac`A11XZCU=@F<]JADC

g ⁰ ₂ (x) = ln(x) + 2

^FE\1XZC

J

^;

| g ₂ ⁰ (x) | 6 < 1, ∀ x ∈ J

*F@I±O*ADTaE

| g ⁰ ₂ (x) |

^F 8E[EUQcX8QXZCU

1XZ?B=JC1TWF@X0C `

2 + ln 1.65 ≈ 2.5

^1XZC

J

^¡3h ^D7LG~H

°:QcEUAlk9>Il?,F@XPEH?ZCUF@I0KPCUF AcX011TEUQcXPEUF¤ORQC1GHFJ<KPX½jE?J%baF

g 2 (x) = ¹ _n (nx + x ln(x) − 1.25)

^VYXZT.IZF 1F@C_ADTaE ?0A

]QIYE1C_Ac]3E_ADIYE1FHRTphFchW;0VYXZTMIZF\]QI%¨cF@CUqcFJCUAcTsE.?0Ac_¡3h

-

9>I¨c=@CUTa0F>C_AD?0TWKPFJGHF@IYE:VX0F

x = exp(1.25/x)

^jXZC

J

]QcCUCUFJ1?BQI0KP4ZTWF@I`obpL£V,hejv ¡h

x = exp(1.25/x) ln x = 1.25/x x ln x − 1.25 = 0

f (x) = 0. < 1 pt >

I#]QI01TWKP=JCUAcIYE

g 3 (x) = x

;:QI(AKPQI0]±XZIZFlIZQXZ¨cFJbabWF ORQTWbpL=JVYXZTW¨rAcbaFJI0]FlAr¨FJ] bL£V,h 8vr¡3h64>F¶?ZbWX0@;

g ₃ ⁰ (x) =

− 1.25 exp(1.25/x) x ²

h 1 TQcI ]@AcbW]@XZbaF

g ₃ ⁰⁰ (x) = (2.5x ⁻ ³ + 1.5625x ⁻ ⁴ ) exp(1.25/x)

;®QcI&JLAD?,F@CQcTaEoVYXZF

g ⁰⁰ ₃ (x)

^?ZCUF@IBK

KPF ¨SADbWF@XZC_?,Q1TsEUTsO*>jXZC

J

;,KPQcIB]`VX0F

g ⁰ ₃ (x)

FJjE>]@C1QTWU1AcIYE1F FE>KPQI0]\VYXZF

g ⁰ ₃ (x) ∈ [ − 0.979, − 0.5838]

^1XZC

J

^hB4>F ^]F

O*ADTaE

| g ₃ ⁰ (x) | < 1

FE6]@F@b¥A<I0QcX0?,F@CUGHFE6KPF`KPTWC1F`VYXZF`b¥AH]QcI%¨F@CUqcF@IB]F6F 8E 40TaFJIA11XZCU=@F[jX0C

J

^h ^D+*~H

-

IXPE1TWbWTWUADIYEbaF EU^Z=@QC2JGoF[KPF[b¥A ¨SADbWF@X0C.GHQJF@IZIZF;%QcIQ54PEUTaFJIE ;P?ZXZT¥1VYXZF

g 3 (r) = r

^F@E

r n = g 3 (r n − 1 )

^Ar¨cF ^]

r n

b¥A

¨SADbWF@XZC:Ac?Z?ZCUQP]_^Z=@F[KPF`b¥A<C_Ac]@TaI0Fa`ob¥AoI pT2@GHF[TaE1=@C_ASEUTaQI©;

(r n − r) = g 3 (r n − 1 ) − g 3 (r)

(r n − 1 − r) × (r n − 1 − r)

= g ₃ ⁰ (ζ) × (r n − 1 − r),

^Ar¨cF ^]

ζ ∈ J. < 2 pts >

IXPEUTabWT¥1AcIE:bpLTaIZ=JqAcbaTaE1=

| g ⁰ ₃ (x) | < ≈ 0.98

^D -K~H;¯*ThFchW;Zb¥A74BQC1I0F`bWAH?ZbWX0?,FJU1TaGHT¥8EUF`E1CUQcXZ¨=@FY`Hb¥AoVYXZF 8EUTaQI¶vJ]J¡;

QcIQ54ZE1TWF@IYE:baF TaI0=@qAcbaTaE1= jX0Ta¨SADIYEUFJJ;

| r n − r | ≤ (0.98) | r n − 1 − r | ≤ (0.98) ² | r n − 2 − r |

≤ . . .

≤ (0.98) ⁿ | r 0 − r |

≤ (0.98) ⁿ × 0.35. < 3 pts >

9>IQ54PEUTaFJI0KPC_A<Q4ZbWTaqYASE1QTaCUF@GHFJIEKPQI0]

| r n − r | < 10 ⁻ ¹

;0KA2JVYXZFc;

(0.98) ⁿ × 0.35 < 10 ⁻ ¹

n ln (0.98) < ln (10 ⁻ ¹ ) − ln 0.35.

1 QTsE:?,QcXZCEUQcXPE

n > 63

]F\VYXZTMGHQcIYEUC1F`VYXZF`b¥Ao]@QcI%¨cFJC1qF@I0]@F6¨SA E1CUF E1C,2JbWF@IYE1F hWhWh¡ DP+*¯H

(7)

9>I¶ADXZC_ADTaE`?ZXPE\?ZCU=@¨QcTWC[]@FE1E1F bWF@IYE1FJXZC\KPF¤]QI%¨cF@CUqcFJI0]F<F@I±¨cQJYADIYE[VYXZF

| g ₃ ⁰ (x) | < 1

G AcTW[EUQcXPE 8X0jE1F;`]@AcX01F KPF¤UA 4BQC1IZF¤KZF¤KPC1QTsEUFHE1C,2J[?ZCUQP]_^ZF KPF

1

*KZQcI0]Hb¥AORQcI0]E1TWQcI¶I©LFJjE VYXZFHE1QXPE 8X0jE1F ]@QcIYE1C_Ac]EUADIYEUF ¡<R?,QcXZC\¨QcX0[FJI

?,F@C_jX0AKPF@C ;M]@AcbW]@XZbaFJC[baFHIZQG 4ZCUF KMLTaE1=JCUADE1TWQcI0 IZ= ]F 1UADTWC1F<?BQXZC\ADCUCUTa¨F@Ca` b¥A?ZCU=J]T¥1TaQI ¨cQXZbWXZF<bWQcC_UVX0F

| g ⁰ ₃ (x) | <

^`

XZIZF`¨SAcbaFJXZCE1C,2J?ZCUQP]_^ZF[KPF

0

^¡h ^DP+*¯H

-Y$

I?0AcCjE_ADIYEKZF

r 0 = 1.65

^;%QcIAZ;

r n = g 3 (r n − 1 )

^FE ^;

r 0 = 1.65

r 1 = 2.133098799 r 2 = 1.796790331 r 3 = 2.005081994

r 4 = 1.86528816. < 3 pts >

°:QcE1FHk FJbWAH1F@G 40baF`]QI%¨cF@CUqcFJC£EUC2 baFJIYE1F@GHFJIE:¨F@C_.bWAo¨SADbWF@XZC

1.918508201

^h

+Z.

@eA<ORQI0]3EUTaQI

f (x)

FJjE6KP=@CUTW¨rA54ZbWF[1XZC

J

^FEQcIAZ;

g 5 (x) = x − f (x)

f ⁰ (x) = x − x ln(x) − 1.25 ln(x) + 1 .

9>IAoKZQcI0] b¥AoORQcCUG XZbWF[TsEU=@C_ASE1TW¨cF[1XZTa¨SAcIEUFc;

r n+1 = r n − r n ln(r n ) − 1.25 ln(r n ) + 1

= r n + 1.25

ln(r n ) + 1 . < 4 pts >

I?0AcCjE_ADIYEKZF

r 0 = 1.65

^;%QcIAZ;

r n = g 4 (r n − 1 )

^FE ^;

r 0 = 1.65 r 1 = 1.93233459 r 2 = 1.918538094 r 3 = 1.918508201

r 4 = 1.918508201. < 3 pts >

°:QcE1FHk FJbWAH1F@G 40baF`]QI%¨cF@CUqcFJC£EUC2 ¨%TsEUF`¨cF@C_.bWAo¨SADbWF@X0C

1.918508201

^h

+

I«CUF@?0C1FJI0ADIYE6bWFJ>=@bW=@GHF@IYE_>KPFo]QC1CUFJ]E1TWQcI0:KZF<b¥A¤KP=JGoQI0jE1C_ASE1TWQcI

2

;,VYXZT~XPEUTabWTW1F@IYE[baF E1^Z=JQcC,2@GHF<KZFJ Ac]@]@C1QTWUjF GHF@IYEU0IZT¥@;PQIAob¥AoC1FJbWADE1TWQcI©;

10 ⁻ ¹ > | r n − r | = 1

| 1 − g ⁰ ₄ (ζ ) | × | r n+1 − r n | 10 ⁻ ¹ > K × | r n+1 − r n |

10 ⁻ ¹ K s

> | r n+1 − r n |

Ar¨cF ]

K s

XZIZF/4BQC1I0F1XZ?,=@CUTaFJXZC<?,QcX0C RTphFha;

K s = arg max ¹

| ¹ ⁻ ^g ⁰ ⁴ ^(ζ) |

¡3;]FVYXZTIZQcXB<?,F@CUGHFE1E1C_ADTaEHKMLFJjE1TWGHF@CHbWF

IZQG 4ZCUF6KMLTsEU=@C_ASEUTaQI0

n _∗

IZ=J]@FJU1AcTaCUF?BQXZC£Q4PE1FJIZTaCXZIZF>¨rAcbaFJXZCAc?Z?ZCUQP]_^Z=@F `

10 ⁻ ¹

?ZC,2JKPF ]F@EjE1F>CUA]TWIZF\RF@E.VYXZT,IZQcXB

TWI0KPT¥VX0F@C_ADTaEVYXZF\]@FE1E1F\]QI%¨cF@CUqcFJI0]F jFJCUAcTsE.E1C,2JC_AD?0TWKPFr¡3h DFG¯H

(8)

* ¯AD" # P#"$

P ^t LU

⁾ ^+3LG~0

vh:46= ]QcGH?,Q1F@Cb¥AHG ASE1CUT¥]F

A

FJI?ZC1QPKPX0TsE

P ^t LU

^QcX

P

FJjE:b¥A G ASE1CUT¥]F\KPF`?,F@CUG XPE_ASEUTaQI?BADC:b¥AHGH=E1^0Q%KZF KML=@bWT GoTWI0ADE1TWQcIKPF[ADX0UFE:UADI0?0Ta¨QDEUAcqcFh DG~H

A =





1 2 3

2 7 18 4 13 38



 .

dZh .ADb¥]XZbWF@CbWF\KP=EUF@CUGoTWI0AcIE:KPF

A

^h ^DLG¯H

f0h:FE1E1F[1QcX0ORQcCUGHF ]@QcGH?0A]3E1FHRTphFchW;PF@IXPE1TWbaT¥UADIYE.X0IZF`jFJXZbaF G ASEUC1T¥]Fr¡£]@FEjEUF`KP=J]@QcGH?BQYjTaE1TWQcI

LU

^h ^DP+K~H

uBh 1 TQcIlAr¨SADTaE\XPEUTabWTW1=¤b¥AGH=E1^0Q%KZF KPTWC1F ]3E1F;©FJjE N]F¤VYXZF¤]@FE1E1F KP= ]QGo?,Q1TaE1TWQcIlADXZC_ADTaE\=EU=¤KPT<,=JC1FJIEUFQ cX0jE1Ta0FJC

¨cQDEUC1F CU=@?,QcI01Fch DP+*~H

²Zh .ADb¥]XZbWF@Cb¥AoKZ=J]QGH?BQYjTaE1TWQcI

LU

KPTaCUFJ]E1F`KPF`b¥AoGHADE1CUTW]@F

A

^h ^D ^*~H

h .ADb¥]XZbWF@C:b¥A ?ZCUF@GHT2@CUF ]QbaQIZIZF KZF\bLTWIY¨F@C_jF\KZF b¥A G ASE1CUT¥]F

D

FJI«XPEUTabWTWUADIYE6bWF GHQTaI0>KMLQc?,=@C_ASE1TWQcI?,QUjT4ZbWFch

D+ ~H

D =





1 0 0

2 1 0

4 5/3 1



 .

UWVX"$J

-3

@LQc?,=@C_ASEUTaQI bWTaqIZF

2 =

^baTWqcI0F

₂ − (2)

^bWTWqcIZF

₁

FE:bWTaqIZF

3 =

^baTWqcI0F

₃ − (4)

^bWTaqIZF

₁

KPQcIZI0Fc;

A =





1 2 3 0 3 12 0 5 26



 .

@LQc?,=@C_ASEUTaQI bWTaqIZF

3 =

^baTWqcI0F

₃ − (5/3)

^bWTWqcIZF

₂

KPQcIZI0Fc;

A =





1 2 3 0 3 12 0 0 6



 .

P ^t

;P=@EUAcIEb¥AHjTWGH?ZbaF[G ASEUC1T¥]F[T¥KPF@IYEUTsEU=`?ZXZT¥1VYXZF[bpLQIAo?0A

O*ADTaE6KPF`?,F@CUG XPE_ASEUTaQIAcX]@QcXZC_KPF`bLQc?,=@C_ASEUTaQIKZF>EUC1T¥ADI0qcXZb¥ASEUTaQIB¡>k

A =





1 2 3

2 7 18 4 13 38



 =





1 0 0 0 1 0 0 0 1





| {z }

P ^t <1 pts>





1 0 0

2 1 0

4 (5/3) 1





| {z }

L<4 pts>





1 2 3 0 3 12 0 0 6





| {z }

U <4 pts>

.

+.

KPFEr* ¡

=

^KPF@E ^*¬¡

×

KPFEr#@¯¡

×

^KPF@E ^>¡

,

= 1 × (1 × 3 × 6),

= 18. < 3 pts >

(9)

LU =





1 2 3

2 3 12

4 (5/3) 6





DP+*~H

TS

9>XZT~]@ADC>]FE1E1F ORQcT¥6]T~QcI«AcXZC_ADTaE:F@X«XZI0F<KP= ]QcGH?,Q1TsEUTaQI@ #O*AcTWUADIYE>Ac?Z?0AcCUA[WEUC1F`KPF 9@v ;\jX0C6b¥A¤KPT¥ADqcQI0ADb¯KPF\b¥A

G ASEUC1T¥]F

U

^h ^DP+*~H

.

A =





1 2 3

2 7 18 4 13 38



 =





1 0 0 2 3 0 4 5 6





| {z }

L:<2.5 pts>





1 2 3 0 1 4 0 0 1





| {z }

U:<2.5 pts>

F

QcGHGHF<]@F@b¥A A =E1=o¨%X KZAcI0>baFo?ZCUF@GHTaFJC>KPFJ¨cQTaC ;Bb¥A¤?0C1FJGoT2@CUF<]@QcbWQcIZI0F

x 1

KPFobWA G ASE1CUT¥]F

D ⁻ ¹

FJjE[KPQIZIZ=JF ?0AcC b¥A CU=J1QcbWXPE1TWQcI±KPX¶ JPjE2@GHF

Dx 1 = b

^Ar¨FJ]

b

bWFo¨FJ]E1F@X0C X0IZTsE_ADTWC1Fh¬®AcC`jXA4B8EUTsEUXPE1TWQcI¶Ar¨SADIYEJ;QcI E1CUQcXZ¨F O*A]TWbaFJGHF@IYEJ;

b = (1, − 2 − 2/3)

^h ^DP+*¯H

! (X"$*Z#"$ ) + *¯(0

1 QTsEbWFJ?,QcTWIYEU1XZTa¨SAcIE_@;

x k

_v v ² g vv vJf

y k

v Nf d d u

vh::?Z?ZbWT¥VX0F@C bWAHORQC1G<XZbWF<KPF<¨cQDEUC1F<]_^ZQcTaª?,QcX0C6EUC1QXZ¨cFJC6bWF ?,QcbJYIcGHF KPFH]QbabWQP]@ASEUTaQI«KMLQcC_KPC1F\f RTphFha;M]@F@bWXZT~VYXZT

?0AcUjF.?0AcCbaF ®?BQTaIYEUKZF:KPQcIZI0=J_¡eF@EVX0TZTaIYEUF@CU?BQbaFA ¤b¥A ¨rAcbaFJXZCADXo?,QcTWIEKMLAB4B1]@TWUjF

x = 10

D-A+ ¯H

dZh>9>IA8QcXPEUF[XZI?BQTaIYE61XZ?Z?ZbW=@GHFJIE_ADTWC1F;ZTphFha;ZbWF\?,QcTWIYE

(x = 9 ; y = 4)

hB6?Z?ZbWTWVYXZFJC:b¥AoORQcCUG XZbWF\KPF`¨cQcE1CUF`]_^ZQcTaª VX0T0¨cQX0?,F@CUGHFE1E1C_ADTaE.KPF6E1CUQcXZ¨F@CbWF6?,QcbJ%IcGHF6KPF ]QbabWQP]@ASEUTaQI¤KMLQcC_KPC1F:f<VYXZTBTWIYE1FJC1?,QcbWF6A% * b¥A\¨SADbWF@X0C

ADX ?,QcTWIYE6KMLAB40U]T¥11F

x = 10

FE6VYXZTM?,F@CUGHFE1E1C_ADTaEK©LAr¨QcTWCXZI0FY4BQIZIZF`AD?Z?0C1QrªPTWGHADE1TWQcI KZF`bLF@CUC1FJXZCF@I]F[?,QcTWIE Ar¨cFJ]<XZI¶GHTaI0TaG<XZG KPF¤]JADb¥]XZbphe7= ADbWTW1F@C`]@FEjEUF¤TWIEUF@CU?BQbWADE1TWQcI *ADX¶?,QcTWIYE KMLA540U]T¥11F

x = 10

^¡>F@E KPQcIZI0F@C`XZIZF AD?Z?ZCUQrªPTaG ADE1TWQcI KPF`bpLF@CUC1FJXZC]QcGHGHT¥jFh

D-_L*¯H

-3

QcGHGHF[baF ?,QcTWIE_IZF\1QcIYE?0Ac:=JVYXZT¥KPTWjEUAcIYEUJ;PQcIIZF`?,F@XPE6XZE1TWbaT¥jFJC6VYXZF]\Mvr¡.bWAHGH=EU^ZQPKPF\KPF @¯ADqcC_ADI0qcF QcX«dc¡b¥A

GH=EU^ZQPKPF6KPF6°6F@nEUQcI©hP9>I XPE1TWbaT¥1F@C_A`]FE1E1F6KZF@XPªPT2@GHF:GH=EU^ZQPKPFchP9>IHKZQcTaEADX0U1TZXPE1TWbWTW1F@CXZI ?BQbJ%IGHF:KPF6]QbabWQP]@ADE1TWQcI

(10)

KMLQcC_KPCUFE1CUQcT¥@;cTab,IZQcXBO*AcXPE.KPQI0]:VYX0ADE1CUF:?BQTaIYE_@hY¬XZT¥1VYXZF>bL=@I0QcI0]@=:IZQcXB£?ZCUQc?,Q1F

6

?,QcTWIYEUFE?0C1= ]TW1F6VYXZF>bLQcI¤KPQTsE TWIEUF@CU?BQbaFJCA% * ;DbWF:?BQTaIYE_KMLAB4B1]@TWUjF

x = 10

^;DQcI ?ZC1FJI0KPC_A`KPQcIB]:bWFJ?,QcTWIYEU.VYXZT0FJI0]FJCU]@baF:bWF6GHTWF@XPª¤]@F:?BQTaIYE_@;

]DLFJjE:`HKPTWC1F[bWFJVYX0ASEUC1F`KPFJC1I0TaFJCU?,QcTWIYEUJh

DLG¯Hh

I?ZCUF@I0AcIYEKZQcI0] bWFJVYX0ASEUC1F`KPFJC1I0TaFJCU?,QcTWIYEUJ;PbaF[E_AB4ZbWFJAcXKPF KPT<,=JC1FJI0]F KPTa¨%T¥j=JFJJL= ]CUTsE ;

x y ∆y ∆ ² y ∆ ³ y

² d

g d vrmPv d

vSmDd

vv u vrmPv d

v

v f

D7T ~H

9>IQ54PEUTaFJIEbWF`?BQbJ%IGHF[1XZTW¨SADIYEJ;

P 3 (x) = 2 + (1/12)(x − 5)(x − 7). < 4 pts >

¬®QcXZCbpLTaIYE1FJC1?,Qcb¥ASEUTaQI QI EUC1QXZ¨cF;

P(10) = 2 + (15/12)

= 3.25. < 1 pt >

+.

@©F ?0baX0 jTWGH?ZbaF;©bWF ?ZbaXB\?ZCU=J]@TW[F@E baF GHQTaI0 ]QZE1F@XZª F@Il]JADb¥]XZb£1F@C_ADTaE`G ADTWIEUF@I0AcIYE KMLXZE1TWbaT¥jFJC\b¥A GH=@E1^ZQPKPF KZF

°:FJnE1QI C1FJqcQcC,J½1XZC¤bWFJ¤VYX0ADE1CUF?BQTaIYE_

(7 ; 2)(9 ; 4)(11 ; 4)(13 ; 6)

VX0T>1QcIYE¤=JVYXZT¥KPT¥8E_ADIYEU FE VYXZT>IZQcX0

?,F@CUGoF@EjEUCUAcTsE:KMLAr¨cQTaCXZIZFY4,QcI0IZF\AD?Z?0C1QrªPTWGHADE1TWQcI KZF`bLF@CUC1FJXZC.O*ADTaE1F\1XZC:]F@EjEUF`TaIYE1FJC1?,Qcb¥ASEUTaQI©h.DLG¯H

I ?ZC1FJI0ADIYE[KPQcI0] baF [VXBASE1CUF ?,QcTWIYE>GHF@IYEUTaQIZIZ=J ?ZCU=J]@=JKPFJGHGoFJIYEoR?0baX0 bWF<?,QcTWIYE[Ar¨SAcIE>?BQXZC[?BQXZ¨cQTaC ]@ADb¥]X0baFJC

XZIZF\Ac?Z?ZCUQrªPTaG ASEUTaQI KPF`bpLFJC1CUF@X0C]@QcGHGHTW1F ¡;ZbWF>E_AB4ZbWFJAcXKZFJ:KPT<=@CUF@I0]@FJJL= ]CUTsE ;

x y ∆y ∆ ² y ∆ ³ y ∆ ⁴ y

² d

g d d

d u

u 8d

u

vcv u d

d

vJf

DLG~H

(11)

P 3 (s) = 2 + 2s − 2

2 s(s − 1) + 4

6 s(s − 1)(s − 2). < 4 pts >

Ar¨cF ]

s = (x − 7)/2

;ZKPQcI0] FJI

x = 10

^\

s = 3/2

^FE:QI EUC1QXZ¨cF[?,QcXZC¨SAcbaFJXZCTaIYEUF@CU?BQba=JFc;

P 3 (s = 3/2) = 2 + 2(3/2) − (3/2)(1/2) + 2/3(3/2)(1/2)( − 1/2)

≈ 5 − 1 = 4. < 1 pts >

¯¨cFJ] XZI0F FJC1CUF@X0CAc?Z?ZCUQrªPTaGH=@F[KPF`bpLQCUKZC1F[KPF

1 24 s(s − 1)(s − 2)(s − 3)∆ ⁴ y 0

F@I

s = 3/2

| E

| = 9

384 × 8 = 9

48 ≈ 0.1875

h

DP+*~H

121 P LU t s

∼

P t LU

121

h

l

L

h = 40

0.1%

l = 40.3

L ∈ [40.2, 40.6]

x = x ? ± ∆x

x ?

∆x

V

V ∗ )

V

∆V

V

V

V

∆h = 0.04

h = 40.0 ± 0.04

3

− 1

∆l = 0.5 × 10 − 1

l = 40.3 ± 0.05

L

L = 40.4 ± 0.2

h = 40.0 ± 0.05 l = 40.3 ± 0.05 L = 40.4 ± 0.5

V ∗ = 40 × 40.3 × 40.4 = 65124.8 m 3 . < 2 pts >

V

∆V (h, l, L)

∆V (h, l, L) = | l × L | ∆h + | h × L | ∆l + | h × l | ∆L. < 2 pts >

∆V (h, l, L) = | 40.3 × 40.4 | × 0.04 + | 40.0 × 40.4 | × 0.05 + | 40 × 40.3 | × 0.2

= 65.1248 + 80.8 + 322.4

≈ 468.3248. < 1 pt >

V ≈ 65124.8 ± 468.3248

468.3248 < 0.5 × 10 3

3

V ≈ 65124.8 ± 468.3248

V ≈ 65 × 10 3 ± 500 (

3 ). < 1 pt >

V

[64656.4752 , 65593.1248]

V (l, L, h) = l × L × h

IR

l, L, h

l, L, h

V

l, L, h

V

= (40 − 0.04) × (40.3 − 0.05) × (40.4 − 0.2) = 64657.278 < 1 pt >

V

= (40 + 0.04) × (40.3 + 0.05) × (40.4 + 0.2) = 65593.9284 < 1 pt >

V

V ≈ 65125.6032 ± 468.3252

exp(x) − exp( − x)

x ≈ 0

√

x 2 + 1 − 1

x ≈ 0

P x=100 x=1

1 x 3

x = 3 r

3 + 3 q

3 + √ 3 3 + . . .

x 3

f (x) = 0

x

x [0] = 1

6

exp(x)

0

exp(x) − exp( − x) ≈ 1 + x 1 + x 2

2! + o(x 3 ) − 1 − x 1 + x 2

2! + o(x 3 ) ,

≈ 2x + o(x 3 ), x ≈ 0. < 2 pts >

√

x 2 + 1+ 1

P ^t LU

x = x ^? ± ∆x

x ^?

V ^∗ )

∆l = 0.5 × 10 ⁻ ¹

V ^∗ = 40 × 40.3 × 40.4 = 65124.8 m ³ . < 2 pts >

468.3248 < 0.5 × 10 ³

V ≈ 65 × 10 ³ ± 500 (

³ ). < 1 pt >

x ² + 1 − 1

1 x ³

x = ³ r

3 + ³ q

3 + √ ³ 3 + . . .

x ³

exp(x) − exp( − x) ≈ 1 + x 1 + x ²

2! + o(x ³ ) − 1 − x 1 + x ²

2! + o(x ³ ) ,

≈ 2x + o(x ³ ), x ≈ 0. < 2 pts >

x ² + 1+ 1

p x ² + 1 − 1 = x ²

√ x ² + 1 + 1 , x ≈ 0. < 3 pts >

x ³ . < 2 pts >

x ³ = 3+ x

x ³ − x − 3 = 0

f ⁰ (x [n] ) , k = 0, 1, . . .

= x [n] − x ³ _[n] − x [n] − 3

3x ² _[n] − 1 . < 3 pts >

x _[2] = 1.929577465, x [3] = 1.7078664, x [4] = 1.672558473, x [5] = 1.671700382.

g 1 (x) = ^1.25 _ln _x

n _∗

10 ⁻ ¹

g ₄ ⁰ (x)

n _∗

10 ⁻ ¹

f ⁰ (x) = ln(x) + 1

f ⁰ (x) > 0 ∀ x ∈ [1.65, 2])

g ⁰ ₁ (x) = − 1.25 x(ln(x)) ² .

| g ⁰ ₁ (x = 1.65) | ≈ 1.5128 > 1

| g ⁰ ₁ (x) | 6 < 1 ∀ x ∈ J

g ⁰ ₂ (x) = ln(x) + 2

| g ₂ ⁰ (x) | 6 < 1, ∀ x ∈ J

| g ⁰ ₂ (x) |

g 2 (x) = ¹ _n (nx + x ln(x) − 1.25)