Taller No 2 Secci N 5 Grupo 4

February 26, 2023 | Author: Anonymous | Category: N/A

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TBFWDXPFJCJ BC@FLBCE JDE @LELNHFC Gcùetcj jd @fdb`fcs - Jdpcrtcndbtl jd Gîsf`c JDPCXXLEEL JD ZCEEDX GTBJCNDBZLP JD DED@ZXF@FJCJ V NCMBDZFPNL -3????3= ‚‚‚‚

ZCEEDX 81 @CN_L DEÆ@ZXF@L FF MXT_L1 : PD@@FLB1 ?4 _rlgdslr1 Kd`tlr @cstrl Fbtdmrcbtds1 Cbc Plffic Xljrfmudz9 Cbjrds Dstdhcb Nurfeel9 Jceecb Nf`kdee @cólb9 Nfmude Cbmde Nlrdbl9 Ncrfc Ncmjcedbc Ãvfec Pdptfdnhrd, 3; jd 8?8? ‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‐

3.

Zceedr Bl.3

Db de prdsdbtd jlùndbtl sd nudstrc de jdscrrleel jd els 33 dadr`f`fls prlpudstls jd cùdrjl ce tdnc @cnpl deæ`trf`l FF. @cpîtuel 36- _rlhedncs1 3, 8, 0, :, 7, 38, 3=, 88, 86, 00, 4:

3.3. 3. 3. Dadr Dadr`f `f`f `fl l3 Tbc `crmc efbdce ubfglrnd jd jdbsfjcj ι jdbsfjcj ι > > 0, 4b@/n b@/n sd sd jfstrfhuyd jdsjd x jdsjd x > > ? c x > 4n . 3

Gfmurc 31 @crmc efbdce c) ¼@uãe ds ec `crmc tltce< Jdscrrleel1 Pd utfefzc ec dùc`fðb jd jdbsfjcj jd `crmc efbdce1 ι >

S E

Jdspdacbjl ec `crmc1 S >

ι E

Xddnpeczcbjl vcelrds1 S > (0, (0,4

b@ ) Ý (4 (4n n) n

S > 3=, 3=,4b@ Jdtdrnfbcr de `cnpl deæ`trf`l slhrd de dad x db h) x > 7n Jdscrrleel1 Pd utfefzc ec dùc`fðb jd `cnpl deæ`trf`l jd ubc hcrrc `crmcjc c el ecrml jd su elbmftuj1 iS D x (x?) > (3) x? (x? ∘ E)

8

Pd rddnpeczcb vcelrds1 B n8 (;, (;,66 Ý 3? )(3=, )(3=,4b@ ) 8 @ D x (7n (7n) > 7n(7n (7n ∘ 4n) 6

D x > 87, 87,8 B @ `) x > 6n Jdscrrleel1 Xddnpeczcbjl db ec dùc`fðb (3) B n8 (;, (;,66 Ý 3? )(3=, )(3=,4b@ ) 8 @ D x (6n (6n) > 6n(6n (6n ∘ 4n) 6

D x > :,0=

B @

j) x > 84?n 84?n Jdscrrleel1 Xddnpeczcbjl db ec dùc`fðb (3) B n8 (;, (;,66 Ý 3? )(3=, )(3=,4 Ý 3? 6 @ ) 8 @ D x (84? (84?n n) > 84?n 84? n(84? (84?n n ∘ 4n) 6

∘

D x > 8,4= Ý 3?

∘

0

B @

d) Jdtdrnfbcr de `cnpl db x > 84?n 84?n uscbjl ec cprlxfnc`fðb jd qud sd trctc jd ubc `crmc pubtuce db de lrfmdb y `lnpcrcr de rdsuetcjl `lb de lhtdbfjl dxc`tcndbtd db j. Jdscrrleel1 Pd utfefzc ec Edy jd @luelnh pcrc `crmcs pubtuceds slhrd de dad x1 D x (x) >

iS x8

Xddnpeczcbjl B n8 )(3=, )(3=,4b@ ) @ 8 84?n 84? n8

(;, (;,66 Ý 3?6 D x >

0

B @ Dbtlb`ds, de rdsuetcjl lhtdbfjl db de fb`fsl d sd cprlxfnc ce rdsuetcjl db de fb`fsl j `lb ub drrlr1 D x > 8,48 Ý 3?

∘

0

B ∘ Ý Ý r > 8,4= 3? @ 8,48 3? B 8,4= Ý 3? 0 @ ∘

0

∘

0

B @ ½ 3??%

∘

r > 3,64 %

3.8. 3. 8. Dadr Dadr`f `f`f `fl l8 Jls pecbls jd `crmc vdrtf`ceds d fbffibftls slb pcrcedels y dstãb sdpcrcjls dbtrd sf plr ubc jfstcb`fc j>: n.Jdtdrnfbcr de `cnpl deæ`trf`l c ec fzqufdrjc jd els pecbls, c su jdrd`kc y dbtrd cnhls. Zdbfdbjl db ùdbtc qud de `cnpl deæ`trf`l D, prðxfnl c ub pecbl fbffibftl jd `crmc dstc dxprdscjl `lnl1 D  x > 8Ϗi Ϗiπ π _lr ecs `lbjf`flbds qud bls jc de prlhednc sd schd qud sf sd uhf`c db ùcequfdr pubtl qud sd dbùdbtrd jdbtrl jd ub pecbl jd rdgdrdb`fc pcrcedel c deels,de `cnpl deæ`trf`l ds de nfsnl db `cjc pubtl. @ c.@cjc c.@cjc pecbl plsdd ubc jdbsfjcj jd `crmc supdrffi`fce ubfglrnd ubfglrnd   π > 0 n Pd jfhuac de `cnpl deæ`trf`l pcrc dbtdbjdr de prlhednc db ec ffimurc ∘(8Ϗi (D (8Ϗiπ π +8 +8Ϗi Ϗiπ π ) > ∘:Ϗi Ϗiπ π > ∘(: (:Ϗ Ϗ )(; )(;,,66Ý3?6 8 )(0Ý3? 7 8 ) @ n B  X > ∘00; D 00;,, 634 @ Jdrd`kc   x 2 :n Jdrd`kc : n

 X > D

∘

∘

@ n8 B 7 6    )(0 Ý 3? ) D X > (D 3 + D 8 ) > (8Ϗi (8Ϗiπ π + 8Ϗi 8 Ϗiπ π ) > :Ϗi Ϗiπ π > (: (:Ϗ Ϗ )(; )(;,,66 Ý 3? @ 8 n8 ∘

:

Gfmurc 81 @cnpl deæ`trf`l `lb `lb π +

Gfmurc 01 Jdtceed plr zlbcs jde `cnpl deæ`trf`l `lb `lb π + B  X > 00;, D 00;, 634 @ Dbtrd cnhls cnhls x ∋ (?, (?, :)n :)n. Jdhfjl c qud els `cnpls ùdbtcb `lb ec nfsnc ncmbftuj pdrl jfgdrdbtds sdbtfjls sd tfdbd qud1  X > (D  3 ∘ D  8 ) > ? D @ @ h. h.Ec Ec pec`c fzqufdrjc fzqufdrjc π > 0 n y de jdrd`kl jdrd`kl π > ∘0 n

Lhsdrvdnls dstd `csl jd glrnc melhce ndjfcbtd ec sfmufdbtd Fncmdb (:) V db `cjc zlbc tdbdnls qud1

4

Gfmurc :1 @cnpl deæ`trf`l `lb π `lb π jd sfmbls jfgdrdbtds

Gfmurc 41 Jdtceed jd @cnpl deæ`trf`l `lb π jd sfmbls jfgdrdbtds Fzqufdrjc Jdhfjl c qud els `cnpls ùdbtcb `lb ec nfsnc ncmbftuj pdrl jfgdrdbFzqufdrjc Jdhfjl tds sdbtfjls sd tfdbd qud1  X > (∘   8 ) > ? D D 3 + D Jdrd`kc Pu`djd Jdrd`kc Pu`djd el nfsnl qud de `csl cbtdrflr.  8 ) > ?  X > (D  3 ∘ D D Dbtrd cnhls n8 B @ 6 7    D X > (D 3 + D 8 ) > (8Ϗi (8Ϗiπ π + 8Ϗi 8 Ϗiπ π ) > :Ϗi Ϗiπ π > (: (:Ϗ Ϗ )(; )(;,,66 Ý 3? )(0 Ý 3? ) @ 8 n8 ∘

B  X > 00;, D 00;, 634 @ 7

3.0. 3.0. Dadr Dadr`f `f`f `fl l0 Tbc `crmc jd jd 8,=4´@ =4´@   dstã dstã ubfglrndndbtd jfstrfhufjc slhrd ub cbfeel jd rcjfl ;.4 `n. Jdtdrnfbcr de `cnpl deæ`trf`l mdbdrcjl slhrd de dad c1 c) x > 3,8`n Jdscrrleel

  ub Gfmurc 71 _rlhednc 0 `ceùel jd D ub pubtl pubtl _ _  y y ubc jfstcb`fc jfstcb`fc x _crc de jdscrrleel jd dstd prlhednc, ds bd`dscrfl `lbsfjdrcr ec dxprdsfðb qud bls pdrnftd kceecr de `cnpl deæ`trf`l jd ub cbfeel `lb `crmc ubfglrnd c ubc jfstcb`fc x jd ub pubtl pubtl _ . _ . Eudml lhtdbdnls ec dxprdsfðb1  x > ixq ˇf D (x8 + c8 )0/8

(8)

_ljdnls bltcr qud slel tdbjrdnls db `lbsfjdrc`fðb db `cnpl deæ`trf`l db ec `lnplbdbtd x, mrc`fcs ce mrãffi`l jd ec ffimurc (7), pljdnls kc`dr ec cprd`fc`fðb qud1  y > ? aˇ D @lb ec sfmufdbtd jdju``fðb, pljdnls `lbèufr qud ec ncmbftuj jde `cnpl deæ`trf`l sdrã1 ixq D   > 8 . (0) (x + c8 )0/8 Pustftuydbjl plr els vcelrds jcjls plr de dadr`f`fl, tdbdnls1

=

B n8 (; (;,, 66 Ý 3? ) ½ (3, (3, 8 Ý 3? 8 n) ½ (8 (8,, =4 Ý 3? 7 @ ) B 8 4 @ > : 76 , Ý 3? D   > @ ((3,, 8 Ý 3? 8 n)8 + (;, ((3 (;, 4 Ý 3? 8 n))0/8 6

∘

∘

∘

∘

. h) x > 0, 7`n `n.. Tscbjl ec nfsnc dxprdsfðb qud lhtuvfnls db ec dùc`fðb (;) y sustftuydbjl plr de budvl vcelr jd jd x > 0, 7`n `n lhtdbdnls1 lhtdbdnls1 B n8 (8,, =4 Ý 3? 7 @ ) ) ½ (0, (0, 7 Ý 3? 8 n) ½ (8 B 8 7 @ > 3 , 30 Ý 3? (;, 4 Ý 3? 8 n))0/8 ((0,, 7 Ý 3? 8 n)8 + (;, ((0 @

(; (;,, 66 Ý 3?6 D   >

∘

∘

∘

∘

`) x > :n. Tscbjl ec nfsnc dxprdsfðb qud lhtuvfnls db ec dùc`fðb (;) y sustftuydbjl plr de budvl vcelr jd jd x > :n lhtdbdnls1 B n8 ) ½ (: Ý n) ½ (8, (8, =4 Ý 3? 7@ ) B 8 0 @ > 3 , 4: Ý 3? @ ((: Ý n)8 + (;, (;, 4 Ý 3? 8 n))0/8

(;, (;, 66 Ý 3?6 D   >

∘

∘

j) Jdtdrnfbcr de `cnpl db db x > :n `lb ec cprlxfnc`fðb jd qud de cbfeel ds ubc `crmc pubtuce db de lrfmdb y `lnpcrcr de rdsuetcjl `lb de lhtdbfjl db de fb`fsl (`). Pf `lbsfjdrcnls qud de cbfeel ds ubc `crmc pubtuce db de lrfmdb, pljdnls csl`fcrel `lb de kd`kl qud x ds ub vcelr nu`kl nãs mrcbjd qud c._crtfdb ._crtfdbjl jl jde supudstl cbtdrflr, tdbdnls qud1 D   > 8 ixq 8 0/8 (x + c )

x22c

∘ ∘ ∘ ↘

D   > ixq (x8)0/8

Pfnpefffi`cbjl ec dùc`fðb cbtdrflr, lhtdbdnls qud1 D   >

iq . x8

Sud rdsuetc sdr ec dùc`fðb jd ubc pcrtîùec db de dspc`fl.

;

(:)

Pustftuydbjl plr el vcelrds jcjls plr de dadr`f`fl, tdbdnls1 B n8 (;, (;, 66 Ý 3? ) ½ (8, (8,=4 Ý 3? 7 @ ) B 8 0 @ > 3 , 44 Ý 3? D   > (:n (:n)8 @ 6

∘

@lnpcrcbjl `lb de vcelr fb`fsl qudbls bl pdrnftc tfdbdb ubc mrcb db jfgdrdb`fc db sus rdsuetcjls, dstljde pudjd sdr(`), ub bltcnls fbjf`cr qud pdbscr de fbtdrvcel ce qud jdhd pdrtdbd`dr pdrtdbd`dr x pcrc qud budstrl cbfeel sd `lnplrtd `lnl ubc `crmc pubtuce db de dspc`fl. De drrlr rdectfvl dbtrd els jls vcelrds, ds jd1 (3, (3, 44 Ý 3?0 ) ½ (3, (3, 44 Ý 3?0 ) Drrlr rdectfvl > rdectfvl > ½ 3? 3??? % > ??,, 74 % 3, 44 Ý 3?0

3.:. 3. :. Dadr Dadr`f `f`f `fl l: Tb jfs`l jd rcjfl rcjfl 8,4`n `n ds ds plrtcjlr jd ubc jdbsfjcj supdrffi`fce ubfglrnd jd 8   slhrd vcelr   0,7´@/n . Ttfefzcbjl cprlxfnc`flbds rczlbcheds, jdtdrnfbcr D vcelr slhrd de dad jde jfs`l c jfstcb`fc jd1 (c) x > ?,?3 ?3`n `n

  ub Gfmurc =1 _rlhednc : cprlxfnc`fðb jd D ub pubtl pubtl _ _  y y ubc jfstcb`fc jfstcb`fc x Ec dùc`fðb, qud lhtuvfnls pcrc de `cnpl deæ`trf`l jd ub jfs`l `lb `crmc ubfglrnd, ec jdffibfnls `lnl1

D x > 8iϏ iϏπ π

     3∘

3

8

3+

6

X x8

.

(4)

_crc dstd `csl, `lnl X `lnl X 22 x pljdnls `lbsfjdrcr slecndbtd ub nfdnhrl jd ec dùc`fðb (4) yc qud sf X ↘ − ec dxprdsfðb jdbtrl jd els `lr`kdtds ds fmuce c 3.Eudml, tdbdnls ec dùc`fðb1 D x > 8iϏ iϏπ π Pf sustftufnls, plr els vcelrds jcjls plr de dadr`f`fl, tdbdnls quæ1 B n8 @ B ˇ 6 7 4  (;, 66 Ý 3? ) > 8 ?0 )(0 )(0, 7 D   > 8Ϗ (;, , Ý 3? , Ý 3? f n8 @ @ 8 ∘

h) x > ?,?: ?:`n `n Jd ncbdrc cbãelmc ce fb`fsl cbtdrflr, pljdnls bltcr qud ec jfstcb`fc, ds nuy pdqudóc, `lnpcrcjc `lb de rcjfl qud tfdbd de jfs`l. Eudml lhtdbdnls ec nfsnc dxprdsfðb qud db de fb`fsl (c)1 D x > 8iϏ iϏπ π Pf sustftufnls, plr els vcelrds jcjls plr de dadr`f`fl, tdbdnls quæ1 B n8 @ B ˇ 6 7 4  D   > 8Ϗ (;, (;, 66 Ý 3? )(0 )(0, , 7 Ý 3? ) > 8 , ?0 Ý 3? f @ 8 n8 @ ∘

`) x > 4n Dstd dadr`f`fl, ds de `csl `lbtrcrfl c els `csls cbtdrflrds, yc qud qud x qud x 22 X, X, plr el ùce pljdnls `lbsfjdrcr c budstrl jfs`l, `lnl ubc pcrtîùec pubtuce db de dspc`fl. Eudml, tdbdnls qud1 iq . x8 @lnl bl schdnls de vcelr jd ec `crmc, vcnls tdbdr db ùdbtc ec rdec`fðb qud dxfstd dbtrd π dbtrd π y q . Ec rdec`fðb sd jdffibd `lnl1 D   >

π >

S ϏX 8

Jdspdacbjl pcrc pcrc S, lhtdbdnls S > Ϗ > ϏπX πX 8 Gfbcendbtd, de `cnpl deæ`trf`l qudjc jdffibfjl `lnl1 Ϗπ X8 ϏπX D   > i 8 . x 3?

Pf sustftufnls, plr els vcelrds jcjls plr de dadr`f`fl, tdbdnls quæ1

   > D

B n8 6 Ϗ ;, 66 Ý 3? @ 8





@ 0,7 Ý 3? 7 8 n ∘

8



(? (?,, ?84 ?84n n)8 B > 8, 4: ˇf

(4n (4n)

@

j) x > 4`n _crc dstd `csl, sf tdbdnls ubc prlplr`flbcefjcj dbtrd x dbtrd x y y X X,, plr el ùce vcnls c uscr ec dùc`fðb (4) pcrc `ceùecr de `cnpl deæ`trf`l jde jfs`l. Wcnls c sustftufr tljls els vcelrds jcjls plr de dadr`f`fl, pcrc dgd`tucr budstrl `ceùel.Eudml tdbdnls1

   > 8Ϗ ;, 66 Ý 3?6 B n D @ 8



3.4. 3. 4. Dadr Dadr`f `f`f `fl l7

8

0,7 Ý 3?

∘

7

@ n8



3

3∘

(? (?,, ?84 ?84n n)8 3 + (? (?,, ?4 ?4n n)8

  





> 8, 34Ý3?:



Tbc `crmc efbdce ubfglrnd sd dxtfdbjd jdsjd x>-8.4`n c x>+8.4`n y plsdd ubc jdbsfjcj jd `crmc efbdce ι > :,4b@/n c) Jdtdrnfbcr ec `crmc tltce Jdscrrleel1 De prdsdbtd dadr`f`fl `lrrdsJdscrrleel1 De plbjd c ubc hcrrc `crmcjc c el ecrml jd su elbmftuj, sfdbjl ι sfdbjl ι11 Jdbsfjcj jd `crmc (ubfglrnd), sd jdspdac S db (7) ι >

S ↘ S > ιE > ιE E

Dbtlb`ds1

 

@ 4 ) n S > (: (:,,4 Ý 3? n 3?? 884b@ b@ S > 8,84 Ý 3? 3?@   > ?,884 ∘

Gfmurc ;1 Dsqudnc jd ec hcrrc `crmcjc

6

∘

h) Kceecr de `cnpl deæ`trf`l slhrd de dad y db y>:`n, 33

(7)

B ˇ f @

Jdscrrleel1 Bltdsd qud sd jdsdc hus`cr ub `cnpl db ubc hcrrc pdrpdbjfùecr c Jdscrrleel1 Bltdsd ec elbmftuj, plr deel ndjfcbtd ec dùc`fðb jdju`fjc db ècsd (=) qud tfdbd db ùdbtc ec sfndtrîc y sd rddnpeczc1 8iιE D y > (=) y :y8 + E8



Gfmurc 61 Dsqudnc melhce jd ec hcrrc pcrc ftdns h,` y j Dbtlb`ds, 8

 y > D

8(; 8(;,,66 Ý 3?6 B@ n )(: @/n )(4`n `n)) )(:,,4 Ý 3? 6@/n)(4 ∘

8

    

: (:`n (:`n)) : n 3??

8

4 n 3??

+

8

B > 3,?= Ý 3?0 aˇ @

`) y>38`n Jd fmuce ncbdrc, rddnpeczcbjl db ec dùc`fðb (=) y db hcsd c ec ffimurc (6) sd lhtfdbd qud1 8

8(; 8(;,,66 Ý 3?6 B@ n )(: @/n )(4`n `n)) )(:,,4 Ý 3? 6@/n)(4 ∘

8

D  y >

     8

38 (38`n (38 `n)) : n 3??

+

4 n 3??

8

B > 3,0; Ý 3? @  aˇ 8

j) y>:.4n>:4?`n Jdscrrleel1 Xddnpeczcnls Jdscrrleel1 Xddnpeczcnls db ec dùc`fðb (=), lhtdbfdbjl qud1 8

 y > D

8(; 8(;,,66 Ý 3?6 B@ n )(: )(:,,4 Ý 3? 6@/n)(4 @/n )(4`n `n)) ∘

8

(:4?`n (:4? `n))



8

: (:, (:,4n) +

 

38

4 n 3??

8

> 6,66 Ý 3?

∘

8

B ˇ a @

d) Jdtdrnfbcr de `cnpl db y>:.4n suplbfdbjl qud ec `crmc ds pubtuce y `lnpcrcr de rdsuetcjl lhtdbfjl db (j). Jdscrrleel1 Bðtdsd qud y 2 2 E, E, plr deel ds plsfhed cprlxfncr c ubc `crmc pubtuce, ds jd`fr qud iιE ˇ iS ˇ y22E    > 8iιE aˇ ∘ D a ≋ 8 a ∘∘↘ D y ≋ y y :y 8 + E8 y :y8





Xddnpeczcbjl, (8,84 Ý 3?    > (8, D

∘

3?

@ )(;, )(;,66 Ý 3?6 B n8 /@ 8 ) ˇ a a > ½ ½ ½ (: (:,,4n)8

   > 6,66 Ý 3? D

∘

8

B @

Gfmurc 3?1 Xdprdsdbtc`fðb `cnpl pdr pdrpdb pdbjf jfùe ùecr cr c ec el elbm bmft ftuj uj pcrc y pcrc y 22 E `csl de drrlr ds `csf buel puds 4 _crc dstd puds 4`n `n 55 :4? 55 :4?`n `n,, plr deel els rdsuetcjls `lb fmuce bñndrl jd `fgrcs sfmbfffi`ctfvcs slb fmuceds c els lhtdbfjls db de ftdn (j)

3.7. 3. 7. Dadr Dadr`f `f`f `fl l 38 Tb `cnpl deæ`trf`l vced D   (8??B/@ )f pcrc x 2 ? y D 8??B/@ D   > (8??B/@ D    > (∘8?? B/@ )f pcrcx pcrc x 5 ?. Tb `fefbjrl fncmfbcrfl jd elbmftuj 8? 8?`n `n y rcjfl X > 4`n tfdbd su `dbtrl db de lrfmdb y su dad c el ecrml jde dad x, jd nljl qud ub dxtrdnl sd dbùdbtrc dbùdb trc db db   x > +3?`n +3?`n y y de ltrl db db x > ∘3? 3?`n `n.. c) ¼@uce ds de ﬈ual scefdbtd qud ctrcvfdsc `cjc `crc< Jdscrrleel _crc db`lbtrcr de ﬈ual c trcvæs jd `cjc ubc jd ecs `crcs jd ec supdrffi`fd Mcussfcbc, jdhdnls tdbdr db ùdbtc ec dùc`fðb jd ﬈ual pcrc de `cnpl deæ`trf`l `lbstcbtd. Eudml tdbdnls tdbdnls11

 ½ bC. χD   > D b ˇ C.

(;)

Eudml, de `cnpl deæ`trf`l pcrc ubc jd ecs `crcs jde `fefbjrl, sdrã1  ½ bC. χD` 3 > D b ˇ C. Jlbjd, C, sdrã de ãrdc jd ec hcsd jde `fefbjrl, ec ùce vdrdnls `lb ub `frùel. Jlbjd,  , @lnl de de vd`tlr blrnce ds pdrpdbjfùecr c ec supdrffi`fd y pcrcedel ce vd`tlr D pljdnls tdbdr ec sfmufdbtd dxprdsfðb pcrc de ﬈ual deæ`trf`l χD` 3 > D  `ls `ls ?l Cˇf > D > D Cˇf 30

Gfmurc 331 _rlhednc 38 Pupdrffi`fd Mcussfcbc `feîbjrf`c. . Gfbcendbtd, rdnpeczcbjl els vcelrds, tdbdnls1 χD` 3 >

3 B n8 B n8 ½ (Ϗ (?, (?, ?4n ?4n) ) > Ϗ > 3,4= 8 @ @

  B 8?? @

8

_crc ec `crc fbgdrflr jde `fefbjrl, vcnls c lhtdbdr ec nfsnc dxprdsfðb qud de fb`fsl cbtdrflr, cbtdrflr, slel qud db dstd `csl, de vd`t vd`tlr lr ˇf, sdrã bdmctfv bdmctfvl. l. Eudml tdbdnls quæ1 χD` 8 >

D  `ls `ls ?l Cˇf >

∘

DCˇf

∘

Zdbfdbjl db ùdbtc qud de `cnpl `cnpl D   D   > (∘8?? 8??B/@ B/@ )f pcrc pcrc   x 5 ?, sustftufnls els vcelrds χD` 8 > ∘





B ∘8?? @

3 B n8 B n8 ½ (Ϗ (?, (?, ?4n ?4n) ) > Ϗ > 3,4= 8 @ @ 8

h) ¼@uce ds de ﬈ual qud ctrcvfdsc ec pcrtd ectdrce jde `fefbjrl< Mrc`fcs c ec ffimurc (33), pljdnls bltcr qud db ec pcrtd ectdrce jde `fefbjrl, de vd`tlr blrnce b ˇ ds pdrpdbjfùecr ce vd`tlr D ˇ , dstl fbjf`c el sfmufdbtd ùcbjl jdscrrleecnls de prlju`tl pubtl pcrc de kceecr de ﬈ual deæ`trf`l χDe > D  `ls `ls 6?l Cˇf > ?ˇf . Eudml, `lbèufnls qud de ﬈ual deæ`trf`l db de ectdrce jde `fefbjrl ds ?. `) ¼@uce ds de ﬈ual bdtl scefdbtd qud ctrcvfdsc tljc ec supdrffi`fd<

3:

De ﬈ual bdtl scefdbtd, sdrã ec sunc jd tljls els ﬈uals qud ctrcvfdscb `cjc supdrffi`fd jd ec ffimurc Mcussfcbc. _ljdnls bltcr qud tljcs dstcs supdrffi`fds ecs kceecnls db els fb`fsls cbtdrflrds. Eudml de ﬈ual bdtl1 χBdtlD   > χ D` 3 + χ + χD` 8 + χ + χDe Pustftuydbjl els vcelrds jd els fb`fsls cbtdrflrds1 χBdtlD   >



3 B n8 Ϗ 8 @

 +

3 B n8 Ϗ 8 @



+ ? > Ϗ

B n8 @

j) ¼@uãe ds ec `crmc bdtc db de fbtdrflr jde `fefbjrl< _lr ec edy jd Mcuss, schdnls qud1 db`drrcjc D  ½ jC ˇ > q db`drrcjc (6) ? @lnl pujfnls kceecr de ﬈ual bdtl db de fb`fsl (j), vcnls c jdspdacr ec q ec q db`drrcjc db`drrcjc jdsjd ec edy jd Mcuss. Eudml tdbdnls1

χBdtlD   >



q db`drrcjc > χ BdtlD ½ ? db`drrcjc > χ Pustftuydbjl plr els vcelrds jd ec pdrnftfvfjcj jde ndjfl db de vc`fð y plr de ﬈ual bdtl kceecjl, tdbdnls1

q db`drrcjc db`drrcjc >

B n8 Ϗ @

  ½

;, ;4: Ý 3?

38

∘

@ 8 B n8



> ;, ;4: ;4:Ϗ Ϗ Ý3?

∘

38

@   > 8,=;Ý3?

∘

33

@   > 8= 8=,,; p@

3.=. 3. =. Dadr Dadr`f `f`f `fl l 3= Tbc `lrtdzc dsgærf`c jd rcjfl 7`n plsdd ubc jdbsfjcj jd `crmc supdrffi`fce ubfglrnd   π > 6b@/n8. ubfglrnd c) ¼ùãe ds ec `crmc tltce slhrd ec `lrtdzc< Jdscrrleel1 Pchdnls Jdscrrleel1 Pchdnls plr jdffibf`f jdffibf`fðb ðb qud ec jdbsfjcj jd `crmc supdrffi`fce π supdrffi`fce π qud, π >

Sbdtc C

Jdspdal S

∘∘∘∘∘∘↘

Sbdtc > πC π C > :πϏ πϏrr8 (3?)

Xddnpeczcbjl,



@ Sbdtc > :Ϗ 6 Ý 3? 6 8 n Gfmurc 381 Xdprds Xdprdsdbtc dbtc`fðb `fðb `lrtdzc dsgærf`c jd rcjfl r

34

∘

8

  7 n 3??

> :,?=Ý3?

∘

3?

@

:?=b@ Sbdtc > ?,:?= b@

h) Jdtdrnfbcr de `cnpl deæ`trf`l db X>8`n Jdscrrleel1 _lr Jdscrrleel1 _lr edy jd Mcuss tdbdnls qud1 χ >



D r jC jC > >

C

Sdb`drrcjl > :D r ?

Jdspdacbjl D r sd tfdbd qud, D r >

Sdb`drrcjl Sdb`drrcjl > i ? :Ϗr 8 r8

(33)

Gfmurc 301 Xdprdsdbtc`fðb supdrffi`fd Mcussfcbc X5r Db dstd `csl bl kcy bcjc db`drrcjl db X (X (X 5 r ), puds ec `crmc sd dbùdbtrc db ec `lrtdzc, plr tcbtl tcbtl Sdb`drrcjl > ? y plr dbjd dbjd D r > ? (vdr Dùc`fðb (33)). `) Jdtdrnfbcr de `cnpl deæ`trf`l db X>4.6`n Jdscrrleel1 Jd fmuce ncbdrc Jdscrrleel1 Jd ncbdrc X 5 r, ec supdrffi`fd Mcussfcbc bl db`fdrrc `crmcs, plr tcbtl tcbtl S > ? y plr dbjd dbjd D r > ? (vdr Dùc`fðb (33)). j) Jdtdrnfbcr de `cnpl deæ`trf`l db X> 7.3`n jdscrrleel1 _crc dstd `csl jdscrrleel1 _crc `csl X 2 r y plr edy jd Mcuss de S de Sdb`drrcjl `lrrdsplbjd ce kceecjl db de îtdn (c). Xddnpeczcbjl db (33)1 B n8 (; (;,,66 Ý 3? )(: )(:,,?= Ý 3? 8 @ D r > 8 7,3 n 3?? 6

 

6;0,08 08B/@ D r > 6;0, B/@ Gfmurc 3:1 Xdprdsdbtc`fðb supdrffi`fd Mcussf Mcussfcbc cbc   X 2 r ftdn j y d 37

3?

∘

@ )

d) Jdtdrnfbcr de `cnpl deæ`trf`l db X>3?`n Jdscrrleel1 Ec supdrffi`fd Mcussfcbc X Jdscrrleel1 Ec Mcussfcbc X 2 r, plr el tcbtl ec `crmc db`drrcjc sfmud sfdbjl ec kceecjc db de îtdn (c). Dbtlb`ds rddnpeczcbjl db (33)1 B n8 (;, (;,66 Ý 3? @ 8 )(: )(:,,?= Ý 3? D r > (?, (?,3n)8 6

∘

3?

@ )

D r > 074, 074,;60 ;60B/@ B/@

3.;. 3. ;. Dadr Dadr`f `f`f `fl l 88 @lbsfjdrdnls jls `lrtdzcs `feîbjrf`cs `lb`æbtrf`cs fbffibftcndbtd ecrmcs. Ec `lrtdzc fbtdrflr tfdbd ub rcjfl X rcjfl X 3 y plsdd ubc jdbsfjcj jd `crmc supdrffi`fce ubfglrnd π3 , nfdbtrcs qud ec dxtdrflr tfdbd ub rcjfl X8 y ubc jdbsfjcj jd `crmc supdrffi`fce ubfglrnd   π8 . ubfglrnd

Gfmurc 341 @lrtdz @lrtdzcs cs `feîbj `feîbjrf`cs rf`cs `lb`æbtrf` `lb`æbtrf`cs cs fbffibft fbffibftcndb cndbtd td ecrmcs ecrmcs.. Pchdnls qud de ecrml jd ec `lrtdzc jd els `fefbjrls ds fbffibftcndbtd ecrmc, plr el ùce sd jdbltcrã `lb ec edtrc E. c) Ttfefzcr ec edy jd Mcuss pcrc kceecr de `cnpl deæ`trf`l db ecs rdmflbds r 5 X3, X3 5 r 5 X8 y r 2 X8. Pchdnls qud ec Edy jd Mcuss jf`d qud1  ½ C   > χ > D

 P

3=

D b jC jC > >

S  ◧

Xdslevfdbjl ec fbtdmrce tdbdnls qud1 D b ½ C`fe >

S  ◧

S D b ½ 8Ϗr ϏrE E >  ◧

Jdspdacbjl D b tdbdnls1 D b >

8I S Er

(38)

- _crc _crc r 5 X3 1

Gfmurc 371 Pupdrffi`fd Mcussfcbc pcrc pcrc r 5 X3 jdbltcjc `lb `lelr czue. @lnl r 5 X3 , dbtlb`ds @lnl dbtlb`ds S > ? plrqud ec supdrffi`fd mcussfcbc (el qud dstã slnhrdcjl jd `lelr czue) ds ndblr c ec `lrtdzc X3 qud prlju`d de `cnpl deæ`trf`l. _lr el ùce D b > ? . - _crc _crc X3 5 r 5 X8 1 Xd`lrjcbjl ec dùc`fðb (38), y lhsdrvcbjl qud r 5 X8 tdbjrã ub `cnpl fmuce c `drl, sd tfdbd qud ec `crmc dbtrd X3 5 r 5 X8 ds1 S > π > π 3 C3 > 8ϏX 3Eπ3

3;

Gfmurc 3=1 Pupdrffi`fd Mcussfcbc pcrc X pcrc X 3 5 r 5 X8 jdbltcjc `lb `lelr czue. Xddnpeczcbjl Xddnpec zcbjl db db D b tdbdnls D b >

8i (8ϏX (8ϏX 3 Eπ3 ) Er

Pchdnls qud qud i >

3 , dbtlb`ds1 :Ϗ ◧

D b >

X3 π3 r ◧

- _crc _crc r 2 X8 1

Gfmurc 3;1 Pupdrffi`fd Mcussfcbc pcrc pcrc r 2 X8 jdbltcjc `lb `lelr. 36

Xd`lrjcbjl ec dùc`fðb 38, sd tfdbd qud ec `crmc jd X 8 ds1 S > π > π 8 C8 > 8ϏX 8Eπ8 Xddnpeczcbjl Xddnpec zcbjl db db D b tdbdnls D b > 8i (8ϏX (8ϏX 8 Eπ8 ) Er Pchdnls qud qud i >

3 , dbtlb`ds1 :Ϗ ◧

D b >

X8 π8 r ◧

Cklrc hfdb, schdnls qud qud r 2 X8 ds ec sunc jd cnhls `crmls deæ`trf`ls, csî1 D b >

X8 π8 X3 π3 + r r ◧

◧

Jd el ùce tdbdnls qud ec `crmc tltce ds1 + X3 π3 D b > X8 π8 + X r

(30)

◧

π8 y de sfmbl rdectfvl jd cnhcs pcrc qud de `cnpl π3 deæ`trf`l sdc `drl ùcbjl r ùcbjl r 2 X8 < ¼@uãe ds dbtlb`ds de `cnpl deæ`trf`l dbtrd ecs `lrtdzc `lrtdzcs< s< Pchdnls qud de `cnpl dstã jcjl plr ec dùc`fðb 30, dbtlb`ds dbtlb`ds D b > ? sdrã1

h) ¼@uãe jdhdrã sdr de `l`fdbtd

?>

X8 π8 + X + X3 π3 r ◧

? > X 8 π8 + X + X3 π3 Dbtlb`ds de `l`fdbtd

π8

ds1

π8 X3 >∘ π3 X8 _lr el ùce, de `cnpl deæ`trf`l dbtrd `lrtdzcs sdrã1 π3

D b (X3 5 r 5 X8 ) >

X3 π3 r

(3:)

◧

`) Kc`dr ub dsqudnc jd ecs eîbdcs jd gudrzc db de `csl fbjf`cjl db ec pcrtd (h). Db ec ffimurc cbtdrflr sd lhsdrvc de dsqudnc jd eîbdcs jd gudrzc tlncbjl π3 `lnl plsftfvl db ec dùc`fðb 3:. 8?

Gfmurc 361 @cnpl deæ`trf`l mdbdrcjl.

3.6. 3. 6. Dadr Dadr`f `f`f `fl l 86 Db ubc rdmfðb pcrtfùecr jd ec ctnðsgdrc tdrrdstrd, sd kc ndjfjl de `cnpl deæ`trf`l slhrd ec supdrffi`fd jd ec Zfdrrc rdsuetcbjl sdr jd 34? B c ubc ceturc jd @ B 84? n y jd 3=? @ c :?? n, db cnhls `csls jfrfmfjl kc`fc chcal. @ceùecr ec jdbsfjcj jd `crmc vleñnf`c jd ec ctnðsgdrc suplbfdbjl qud ds ubfglrnd dbtrd 84? y :?? n. (_udjd jdsprd`fcrsd c ub ùrvcturc jd ec Zfdrrc. _lr quæ D 83

(34)

Gfmurc 831 Jfcmrcnc jde ùhl(cbcelmîc supdrffi`fd jd ec tfdrrc) Ds dvfjdbtd qud de ﬈ual c trcvæs jd ecs `crcs ectdrceds ds ?, yc qud de vd`tlr supdrffi`fd jd dstd ds pdrpdbjfùecr ce `cnpl deæ`trf`l(`lnl lhsdrvcnls db de jfcmrcnc). _lr el qud db`lbtrcnls els jls ﬈uals  ½ C 3 > (34? B  )(34? )(34?n n ∝ 34? 34?n n)`ls `ls(?) (?) > 0, 0,0=4 Ý 3?7 W   ½ n χD 3 > D @  ½ C 3 > (3=? B  )(34? χD 8 > D )(34?n n ∝ 34? 34?n n)`ls `ls(3;?) (3;?) > ∘0,;84 Ý 3?7 W   ½ n @ De ﬈ual tltce ds1 χD Z   > χ D 3 + χ + χD 8 > ∘:,4 Ý 3?7 W   ½ n Pd dbùdbtrc ec `crmc tltce jde ﬈ual schfdbjl qud1 χD Z   > S ?

bdtc

> ∘:,4 Ý 3?7

Sbdtc >  >  ? ∝ χD Z   > ;,;4 Ý 3?

∘

38

:,4 Ý 3?7 > ∘0,6; 7 @ ∘

∝∘

Gfbcendbtd,Pd dbùdbtrc ec jdbsfjcj jd `crmc jd ec ctnðsgdrc1 7 S S ∘0,6; @ σ 1 > > > ∘3,3; Ý 3? W C ∝ κk (34?n)8 ∝ (34? (34?n (34?n n) ∘

88

∘

38

@ n0

3.3? 3.3?..

Dadr Dadr`f `f`f `fl l 00

Tb nljdel ctðnf`l plsdd ubc `crmc pubtuce buèdcr plsftfvc +Q d fbèufjc db ubc dsgdrc ded`trðbf`c rîmfjc jd rcjfl X jd `crmc tltce ∘Q d, ubfglrndndbtd jfstrfhufjc plr tljc ec dsgdrc. c) Db ub `cnpl deæ`trf`l dxtdrbl buel, ¼jðbjd dstã ec plsf`fðb jd dqufefhrfl jd ec `crmc pubtuce buèdcr<

Gfmurc 881 Nljdel ctðnf`l db ub `cnpl deæ`trf`l dxtdrbl buel Jdscrrleel1 Ce dstcr db ub `cnpl deæ`trf`l D   D   > ?, y sdr jls `crmcs fmuceds y `lbtrcrfcs, tfdbdb de nfsnl `dbtrl, ds jd`fr ec `crmc pubtuce buèdcr db dqufefhrfl, dstcrã `lb`æbtrf`c c ec dsgdrc ded`trðbf`c. h) Pf bl kcy `cnpl deæ`trf`l dxtdrbl, ¼jðbjd dstã ec plsf`fðb jd dqufefhrfl jd ec `crmc pubtuce buèdcr, rdspd`tl ce `dbtrl jd ec dsgdrc ded`trðbf`c `crmcjc bdmctfvcndbtd<

Gfmurc 801 Nljdel ctðnf`l db cusdb`fc jd ub `cnpl deæ`trf`l dxtdrbl Jdscrrleel1 Ec gudrzc qud pdr`fhd ec `crmc pubtuce ds sdmñb ec edy jd @luelnh1 G   > SD Jlbjd ec `crmc tltce S ds -Qd.Xddnpeczcbjl db ec dùc`fðb 31 D r >

Sr :Ϗδ ? X0 80

Db dstd `csl r ds ec jfstcb`fc jdsjd de `dbtrl jd ec dsgdrc ded`trðbf`c c ec `crmc pubtuce, csî qud sd jdspdac1 G : G :Ϗδ ? X0 r > ∘ (Q d)8 Pfdbjl   G   > (∘Q d)D 1 Pfdbjl r >

D r :Ϗδ ? X0 Qd

`) ¼@uãe ds de nlndbtl jfplecr deæ`trf`l fbju`fjl plr de `cnpl D ? pcrc dstd nljdel ctðnf`l< Jdscrrleel1 De nlndbtl jfplecr deæ`trf`l dstã jdffibfjl plr ec sfmufdbtd dùc`fðb1 p > p > q qE E Jlbjd db dstd `csl ec `crmc `crmc q   ds ds Q d y E qud ds ec jfstcb`fc dbtrd `crmcs yc sd kc db`lbtrcjl db de fb`fsl h. Xddnpeczcbjl dstls vcelrds1 Q dD ? :Ϗδ ? X0 p > p > Qd p > p > D D ?:Ϗδ ? X0

3.33 3. 33..

Dadr Dadr`f `f`f `fl l 4:

Tbc dsgdrc sðefjc bl `lbju`tlrc jd rcjfl c rcjfl c `lb `lb su `dbtrl db de lrfmdb tfdbd ubc `cvfjcj dsgærf`c jd rcjfl h `lb su `dbtrl db de pubtl x > h, y > ?, z   z   > ? `lnl sd nudstrc db ec ffimurc 36-06. Ec dsgdrc `lbtfdbd ubc jdbsfjcj jd `crmc ubfglrnd σ. Jdnlstrcr qud de `cnpl deæ`trf`l db ec `cvfjcj ds ubfglrnd y vfdbd jcjlr plr D x > σh/ σh/00? , D y > ?. Fbjf`c`fðb1 Pustftufr ec `cvfjcj plr dsgdrcs jd fmuce jdbsfjcj jd `crmc plsftfvc y bdmctfvc. Jdscrrleel _crc Jdscrrleel _crc de jdscrrleel jd dstd dadr`f`fl sd dnpedc de tdlrdnc jd supdrplsf`fðb `lnl sd nudstrc c`lbtfbuc`fðb1 Db els pubtls qud `lfb`fjdb, qud slb de kud`l, ec `crmc jd `cjc dedndbtl ds1 jq   > jq 3 + jq + jq 8 > (+ (+σ σ)jv jv + + ( ∘σ)jv jv   > ? Cklrc rdcefzcnls ec sfmufdbtd ffimurc aubtl c els vd`tlrds rcjfceds1 Bltc1 Ec Bltc1 Ec supdrplsf`fðb sd kc`d slhrd ecs jdbsfjcjds jd `crmc, bl slhrd ec `crmc tltce. 8:

Gfmurc 8:1 _rlhednc 4: Plhrdplsf`fðb plr dsgdrcs jd fmuce jdbsfjcj jd `crmc plsftfvc y bdmctfvc

Gfmurc 841 _rlhednc 4: Plhrdplsf`fðb plr dsgdrcs jd fmuce jdbsfjcj jd `crmc plsftfvc y bdmctfvc Pchdnls plr edy jd Mcuss qud, χ >



> D D bjC jC > D b C >

C

Sbdtl σW > ? ?

(37)

Cklrc sd jfvfjd de prlhednc db ecs jls pcrtds pcrtds99 ubl ubl D D + jdhfjl c ubc dsgdrc ubfglrnd jd `crmc plsftfv plsftfvcc jd rcjfl rcjfl c c > > 8h, y de ltrl D ltrl D  jdhfjl jdhfjl c ubc dsgdrc jd `crmc bdmctfv bdmctfvcc jd rcjfl h `dbtrcjc jdbtrl jd ec `cvfjcj. _crtd 3 Xddnpeczcbjl 3 Xddnpeczcbjl db (37) pcrc ec dsgdrc `lb`æbtrf`c sd tfdbd qud1 :Ϗr 0 σW  > D + C ↘ σ > D > D + :Ϗr 8 D + ? 0? Jdspdal   D + , Jdspdal D + >

rσ σ rˇ > r 0? 0? 84

_crtd 8 Xddnpeczcbjl 8 Xddnpeczcbjl db (37) pcrc de `csl jd ec `lb`cvfjcj sd tfdbd qud1 ∘

:Ϗr 30  > σr3 (∘rˇ3 ) > ∘ σr 3 σ > D > D :Ϗr 38 ↘ D ? 0? 0? ∘

∘

Gfbcendbtd, `lnl sd dvfjdb`fc db ec ffimurc (84)  (84) r >  h + + r 3 , ds jd`fr qud1 D >

σ(r ∘  h) 0?

∘

∘

Hus`cnls de `cnpl bdtl,    > D  + + D  > σr D 0? ∘

∘

σr σ h σ h σ hˇ + > > ? f + a 0? 0? 0 ? 0?

Dbtlb`ds D y > ? y _crc tljls els pubtls db ec `cvfjcj

87

σh D x > 0?

Taller No 2 Secci N 5 Grupo 4

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