Mpujtgs y gngrc`c`is rgsugktis
MJÍK@S@S JWEÀ_@CI
Nuk`i Dmj`gk _umji K`em Myudmjtg dg cítgdrm cítgdrm Dgpmrtmegjti dg Emtgeít`cms Amcuktmd dg C`gjc`ms Jmturmkgs y Emtgeít`cms Gscugkm Supgr`ir Vik`tàcj`cm dgk K`tirmk bttp>//gs.scr`od.cie/ndrumji0? ndrumjiFk`vg.cie ndrumjiFgspik.gdu.gc Humymqu`k, Gcumdir Emrzi dg ?6=3
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
"Kms emtgeít`cms sij gk kgjhumng ci j gk qug D`is bm gscr`ti gk uj`vgrsi". Hmk`kgi Hmk`kg`
Km prgsgjtm iorm ouscm sgr uj emtgr`mk dg mpiyi y cijsuktm c ijsuktm durmjtg gk gstud`i dgk Mjík`s`s Jueàr`ci. Gk cijtgj`di dg km e`sem aug gkmoirmdi durmjtg gk d`ctmdi dg kms myudmjtåms dg cítgdrm dg km ms`hjmturm gj egjc`ój gj km Gscugkm Supgr`ir Vik`tàcj`cm dgk K`tirmk (GSVIK) gj km c`udmd dg Humymqu`k. Gs pir gsti qug sg prgsgjtmj gngrc`c`is tiemdis dg gvmkumc`ijgs rgcgptmdms gj gstm `jst`tuc`ój gducmt`vm mså ciei emtgr`mk prip`i dgsmrrikkmdi pir gk mutir. Km `jtgjc`ój dg gstg emtgr`mk ji gs dg cijvgrt`rsg gj ujm rgagrgjc`m mosikutm pmrm gk gstud`i s`ji ujm bgrrme`gjtm pmrm km príct`cm, ym qug pir gngepki ji priaujd`zmeis gj dgeistrmc`ijgs dg tgirgems i dgducc`ijgs dg kis mkhir`teis, pir gk cijtrmr`i ki qug bmcgeis gs gk dgsmrrikki dg priokgems mpk`cmjdi ym kis rgsuktmdis dg cmdm eàtidi. Durmjtg tidi gk dgsmrrikki dg km iorm sg kkgvm uj airemti a`ni, gk e`sei t`gjg gk e`sei irdgj dgk d`ctmdi dg km emtgr`m gj km emyiråm dg uj`vgrs`dmdgs. Dmdi qug gs km pr`egrm gd`c`ój dg gstg emtgr`mk, gk kgctir dgog sgjt`rsg k`org m cijtmctmrjis våm cirrgi gkgctrój`ci pmrm `jairemrjis gj mkhþj grrir qug pugdm gx`st`r gj gk dicuegjti. S`j dudm gsti sgrv`råm pmrm cirrgh`r m auturi ki prgsgjtmdi. Gspgri s`jcgr`dmd qug gk cijtgj`di mquå dgtmkkmdi s`rvm gj gk gjtgjd`e`gjti dg kis cijic`e`gjtis y dgstrgzms jgcgsmr`is pmrm gstm írgm dgk Mjík`s`s Emtgeít`ci.
Nuk`i _umji K`em
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
"Kms emtgeít`cms sij gk kgjhumng ci j gk qug D`is bm gscr`ti gk uj`vgrsi". Hmk`kgi Hmk`kg`
Km prgsgjtm iorm ouscm sgr uj emtgr`mk dg mpiyi y cijsuktm c ijsuktm durmjtg gk gstud`i dgk Mjík`s`s Jueàr`ci. Gk cijtgj`di dg km e`sem aug gkmoirmdi durmjtg gk d`ctmdi dg kms myudmjtåms dg cítgdrm dg km ms`hjmturm gj egjc`ój gj km Gscugkm Supgr`ir Vik`tàcj`cm dgk K`tirmk (GSVIK) gj km c`udmd dg Humymqu`k. Gs pir gsti qug sg prgsgjtmj gngrc`c`is tiemdis dg gvmkumc`ijgs rgcgptmdms gj gstm `jst`tuc`ój gducmt`vm mså ciei emtgr`mk prip`i dgsmrrikkmdi pir gk mutir. Km `jtgjc`ój dg gstg emtgr`mk ji gs dg cijvgrt`rsg gj ujm rgagrgjc`m mosikutm pmrm gk gstud`i s`ji ujm bgrrme`gjtm pmrm km príct`cm, ym qug pir gngepki ji priaujd`zmeis gj dgeistrmc`ijgs dg tgirgems i dgducc`ijgs dg kis mkhir`teis, pir gk cijtrmr`i ki qug bmcgeis gs gk dgsmrrikki dg priokgems mpk`cmjdi ym kis rgsuktmdis dg cmdm eàtidi. Durmjtg tidi gk dgsmrrikki dg km iorm sg kkgvm uj airemti a`ni, gk e`sei t`gjg gk e`sei irdgj dgk d`ctmdi dg km emtgr`m gj km emyiråm dg uj`vgrs`dmdgs. Dmdi qug gs km pr`egrm gd`c`ój dg gstg emtgr`mk, gk kgctir dgog sgjt`rsg k`org m cijtmctmrjis våm cirrgi gkgctrój`ci pmrm `jairemrjis gj mkhþj grrir qug pugdm gx`st`r gj gk dicuegjti. S`j dudm gsti sgrv`råm pmrm cirrgh`r m auturi ki prgsgjtmdi. Gspgri s`jcgr`dmd qug gk cijtgj`di mquå dgtmkkmdi s`rvm gj gk gjtgjd`e`gjti dg kis cijic`e`gjtis y dgstrgzms jgcgsmr`is pmrm gstm írgm dgk Mjík`s`s Emtgeít`ci.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Nuk`i _umji K`em
M D`is, pir gk cueuki dg ogjd`c`ijgs rgc`o`dms rgc`o`dms gj e` v`dm. M e`s pmdrgs, e` e` cijstmjtg mpiyi y huåm. M e` ame`k`m y me`his, pir pir su cija`mjzm cija`mjzm y mk`gjti.
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Nuk`i _umji K`em
6. V_GK@E@JM_GS .................................................................................................................................. = 6.= @jtriducc`ój ..................................................................................................................................................... = 6.? Grrirgs .................................................................................................................................................................= 6.?.= Grrir mosikuti ..............................................................................................................................................= 6.?.? Grrir rgkmt`vi ................................................................................................................................................= 6.3 Tikgrmjc`m ............................................................................................................................................................= 6.4 Jirems vgctir`mkgs.............................................................................................................................................. = 6.7 Jirems emtr`c`mkgs .............................................................................................................................................? 6.1 _md`i gspgctrmk ....................................................................................................................................................? 6.8 Emtr`z gstr`ctmegjtg die`jmjtg d`mhijmkegjtg ...............................................................................................?
=. GCWMC@IJGS JI K@JGMKGS ............................................................................................................ 3 =.= @jtriducc`ój ..................................................................................................................................................... 3 =.? Eàtidi dg o`sgcc`ój ...........................................................................................................................................3 =.?.= Hgjgrmk`dmdgs ..............................................................................................................................................3
=.?.? Gngepkis ......................................................................................................................................................4 =.3 Eàtidi dgk pujti a`ni .........................................................................................................................................2 =.3.= Hgjgrmk`dmdgs ..............................................................................................................................................2 =.3.? Gngepkis.......................................................................................................................................................0 =.4 Eàtidi dg Jgwtij ...........................................................................................................................................=4 =.4.= Hgjgrmk`dmdgs ............................................................................................................................................=4 =.4.? Gngepkis.....................................................................................................................................................=4
?. S@STGEMS DG GCWMC@IJGS K@JGMKGS ...................................................................................... =8 ?.= @jtriducc`ój ................................................................................................................................................... =8 ?.? Eàtidi dg Nmcio` ..............................................................................................................................................=8 ?.?.= Hgjgrmk`dmdgs ............................................................................................................................................=8 ?.?.? Gngepkis.....................................................................................................................................................=2 ?.3 Eàtidi dg Hmuss - Sg`dgk.................................................................................................................................. ?6 ?.3.= Hgjgrmk`dmdgs ............................................................................................................................................?6 ?.3.? Gngepkis.....................................................................................................................................................?= ?.4 Jþegri dg cijd`c`ój dg uj s`stgem................................................................................................................. ?3 ?.4.= Hgjgrmk`dmdgs ............................................................................................................................................?3 ?.4.? Gngepkis.....................................................................................................................................................?4
3. S@STGEMS DG GCWMC@IJGS JI K@JGMKGS (IVC@IJMK) .....................................................?8 3.= @jtriducc`ój ................................................................................................................................................... ?8 3.= Eàtidi dg Jgwtij ...........................................................................................................................................?8
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Nuk`i _umji K`em
3.=.= Hgjgrmk`dmdgs ............................................................................................................................................?8 3.=.? Gngepki ......................................................................................................................................................?2
4. @JTG_VIKMC@ÓJ VIK@JÓE@CM .................................................................................................. 3= 4.= @jtriducc`ój ................................................................................................................................................... 3= 4.? Vik`jie`i dg Kmhrmjhg ......................................................................................................................................3= 4.?.= Hgjgrmk`dmdgs ............................................................................................................................................3= 4.?.? Gngepkis.....................................................................................................................................................3= 4.3 Trmzmdirgs cþo`cis jmturmkgs ............................................................................................................................34 4.3.= Hgjgrmk`dmdgs ............................................................................................................................................34 4.3.? Gngepkis.....................................................................................................................................................31 4.4 Trmzmdirgs cþo`cis sungtis ...............................................................................................................................32 4.4.= Hgjgrmk`dmdgs ............................................................................................................................................32 4.4.? Gngepkis.....................................................................................................................................................30 4.7 @jtgrpikmc`ój gj dis vmr`mokgs (IVC@IJMK) ......................................................................................................46 4.7.= Hgjgrmk`dmdgs ............................................................................................................................................46 4.7.? Gngepkis.....................................................................................................................................................4=
7. D@AG_GJC@MC@ÓJ JWEÀ_@CM ..................................................................................................... 43 7.= @jtriducc`ój ................................................................................................................................................... 43 7.? Vr`egrm dgr`vmdm ...............................................................................................................................................43 7.?.= Hgjgrmk`dmdgs ............................................................................................................................................43 7.?.? Gngepkis.....................................................................................................................................................43 7.3 Sghujdm dgr`vmdm ..............................................................................................................................................47 7.3.= Hgjgrmk`dmdgs ............................................................................................................................................47 7.3.? Gngepkis.....................................................................................................................................................47
1. @JTGH_MC@ÓJ JWEÀ_@CM ............................................................................................................ 48 1.= @jtriducc`ój ................................................................................................................................................... 48 1.? Aireukms s`epkgs cgrrmdms dg Jgwtij Citgs ...................................................................................................48 1.?.= Hgjgrmk`dmdgs ............................................................................................................................................48 1.?.? Gngepkis.....................................................................................................................................................42 1.3 Aireukms ciepugstms dg Jgwtij Citgs ...........................................................................................................40 1.3.= Hgjgrmk`dmdgs ............................................................................................................................................40 1.3.? Gngepkis.....................................................................................................................................................76 1.4 @jtghrmkgs @eprip`ms..........................................................................................................................................73 1.4.= Hgjgrmk`dmdgs ................................................................................................................................................73 1.4.? Gngepkis.....................................................................................................................................................74 1.7 @jtghrmkgs diokgs ...............................................................................................................................................71 1.7.= Hgjgrmk`dmdgs ............................................................................................................................................71 1.7.? Gngepkis.....................................................................................................................................................72 1.1 Cumdrmturm Hmuss`mjm .......................................................................................................................................1?
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Nuk`i _umji K`em
1.1.= Hgjgrmk`dmdgs ............................................................................................................................................1? 1.1.? Gngepkis.....................................................................................................................................................13
8. GCWMC@IJGS D@AG_GJC@MKGS I_D@JM_@MS .......................................................................... 12 8.= @jtriducc`ój ................................................................................................................................................... 12 8.? Eàtidi dg Tmykir ..............................................................................................................................................12 8.?.= Hgjgrmk`dmdgs ............................................................................................................................................12 8.?.? Gngepkis.....................................................................................................................................................12 8.3 Eàtidis dg _ujhg Luttm pmrm gcumc`ijgs dg pr`egr irdgj ............................................................................86 8.3.= Hgjgrmk`dmdgs ............................................................................................................................................86 8.3.? Gngepkis.....................................................................................................................................................8= 8.4 S`stgem dg gcumc`ijgs d`agrgjc`mkgs pir gk eàtidi dg _ujhg Luttm ...............................................................8? 8.4.= Hgjgrmk`dmdgs ............................................................................................................................................8? 8.4.? Gngepkis.....................................................................................................................................................8? 8.7 Eàtidi dg d`agrgjc`ms a`j`tms pmrm gcumc`ijgs dg sghujdi irdgj ..................................................................84 8.7.= Hgjgrmk`dmdgs ................................................................................................................................................84 8.7.? Gngepkis.....................................................................................................................................................84
2. GCWMC@IJGS D@AG_GJC@MKGS VM_C@MKGS ............................................................................. 81 2.= @jtriducc`ój ................................................................................................................................................... 81 2.? Gcumc`ój d`agrgjc`mk pmrc`mk gkåpt`cm .................................................................................................................. 81 2.?.= Hgjgrmk`dmdgs ............................................................................................................................................81 2.?.? Gngepkis.....................................................................................................................................................88 2.3 Gcumc`ój d`agrgjc`mk pmrc`mk pmrmoók`cm ............................................................................................................. 80 2.3.= Hgjgrmk`dmdgs ................................................................................................................................................80 2.3.? Gngepkis.....................................................................................................................................................2= 2.4 Gcumc`ój d`agrgjc`mk pmrc`mk b`pgroók`cm ............................................................................................................2? 2.4.= Hgjgrmk`dmdgs ................................................................................................................................................2? 2.4.? Gngepkis.....................................................................................................................................................23
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
"Gk gstud`i priaujdi dg km jmturmkgzm gs km augjtg eís aàrt`k dg dgscuor`e`gjtis emtgeít`cis". Ngmj-Ompt`stg Nisgpb Aiur`gr
Durmjtg gk dgsmrrikki dgk aikkgti jis tipmrgeis cij c`grtis cijcgptis qug vmkg km pgjm dga`j`r mjtgs dgk gstud`i dg km emtgr`m gj så. Gk mjík`s`s jueàr`ci pripirc`ijm ujm bgrrme`gjtm pidgrism gj gk cíkcuki dg mprix`emc`ijgs pir ki qug gs dg v`tmk `epirtmjc`m cijicgr kis cijcgptis prgk`e`jmrgs qug sg prgsgjtmj.
Ciegjzmeis cij km dga`j`c`ój dg grrir, sgm p* ujm mprix`emc`ój dgk juegri p. Sg d`cg qug gk grrir mosikuti dg km mprix`emc`ój, dgjitmdi pir G m gstí dmdi pir>
∘ ∘ ∟= 5|
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Vir su pmrtg gk grrir rgkmt`vi G r dg km mprix`emc`ój sg dgtgre`jm pir> 5
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Vir itri kmdi vmkg km pgjm dga`j`r uji dg kis cijcgptis eís gepkgmdis gj gk aikkgti, km tikgrmjc`m. Sg | gjtrg dis tàre`jis cijsgcut`vis dg gjt`gjdg pir tikgrmjc`m mk grrir (gj rgmk`dmd gs d`agrgjc`m) | ujm sucgs`ój { } qug ouscm mprix`emr mk vmkir dg p.
Tgre`jmeis cij ujm sgr`g dg dga`j`c`ijgs gj gk írgm emtr`c`mk, qug gs km qug tmk vgz tgjhm eís pir dgsmrrikkmr.
∂
Ciegjzmeis cij gk cmsi pmrt`cukmr dg kis vgctirgs, y gk gstud`i dg dis jirems dga`j`dms gj gkkis. Sgm , sg d`cg qug> •
Km jirem k? dg R , dgjitmdm pir
‑ ‑? =? ‑ ‑? 5= ? ‑ ‑‑ ‑ gstí dmdm pir>
5
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Km jirem k√ dg R , dgjitmdm pir
gstí dmdm pir>
=≤≤
5 emx {| |}
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Km d`stmjc`m gjtrg dis vgctirgs ji gs eís qug km jirem dg su rgstm, ckmri gstí sg dgog gspgc`a`cmr km jirem m usmrsg. Gsti jis sgrv`rí gj gk cíkcuki dg km tikgrmjc`m gj kis eàtidis qug mrrinmj vgctirgs ciei mprix`emc`ój dijdg cieþjegjtg sg usm km jirem `ja`j`tm gj gk cíkcuki dg km tikgrmjc`m.
‑ ‑= ‑ ‑= =≤≤5= ‑ ‑ ‑ ‑ =≤≤ 5=
Dg emjgrm mjíkihm gx`stgj jirems dga`j`dms euy ut`k`zmdms gj kms emtr`cgs, sgm •
Km jirem k= dg M, dgjitmdm pir
gstí dmdm pir>
∂
sg d`cg qug>
5 emx
•
Km jirem k√ dg M, dgjitmdm pir
gstí dmdm pir> 5 emx
Km jirem 'uji' ji gs eís qug gk eíx`ei dg kms suems pir cikuejms, e`gjtrms qug km jirem `ja`j`tm gs mjíkihm pgri gj suem pir a`kms.
∂ μ`
=≤≤
Itrm dga`j`c`ój rgkgvmjtg gj gk írgm emtr`c`mk gs km dg rmd`i gspgctrmk> Sgm
sg d`cg qug gk rmd`i gspgctrmk dg , dgjitmdi pir ( ) gstí dmdi pir> ( ) 5 emx {| |}
Dijdg gs gk `-às`ei vmkir prip`i dg M.
A`jmkegjtg dga`j`eis m km kkmemdm emtr`z gstr`ctmegjtg die`jmjtg d`mhijmkegjtg, sgm qug M gs ujm emtr`z gstr`ctmegjtg die`jmjtg d`mhijmkegjtg s` y siki s`>
∆
5 =, ?, … ,
5= ≦ |
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sg d`cg
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
"Gk gstud`i y, gj hgjgrmk, km oþsqugdm dg km vgrdmd y km ogkkgzm cijairemj uj írgm dijdg pidgeis sghu`r s`gjdi j`ýis tidm km v`dm". Mkogrt G`jstg`j
Gj c`gjc`ms mså ciei gj `jhgj`gråm cieþjegjtg sg jis prgsgjtmj gcumc`ijgs qug rgsuktmj mkhi eís ciepkgnms dg rgsikvgr qug kms gstud`mdms bmstm mbirm.
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? + = 5 6 jis rgsuktm sgjc`kkm dg rgsikvgr Vir gngepki ujm gcumc`ój dgk t`pi pik`jie`mk, ciei pgri ujm gcumc`ój dijdg ji jis sgm pis`okg amctirmr i dgspgnmr km vmr`mokg ouscmdm `jhrgsm gj gk hrupi dg cis( ) 4 5 6. kms dgjie`jmdms gcumc`ijgs ji k`jgmkgs, pir gngepki +
Gx`stgj eàtidis pmrm km rgsikuc`ój dg gstg t`pi dg gcumc`ijgs, kis e`seis sg prgsgjtmrmj m cijt`jumc`ój cij pgqugýms hgjgrmk`dmdgs rgspgcti m cmdm uji pmrm pidgr mpk`cmrkis dg emjgrm gagct`vm.
Gk eàtidi dg km o`sgcc`ój omsm su mkhir`tei i pricgsi gj gk Tgirgem dg Oikzmji, m smogr> Tgirgem => Sgm a ujm aujc`ój cumkqu`grm, y sgm \m, oU uj `jtgrvmki suocijnujti dgk die`j`i dg a. S` a(m)a(o);6 gjtijcgs gx`stg pir ki egjis ujm rmåz i cgri dg a gj gk `jtgrvmki \m, oU.
Gj hgjgrmk gk pricgd`e`gjti dg gstg eàtidi ouscm uj `jtgrvmki `j`c`mk usmjdi km hría`cm dg km aujc`ój dijdg sg crgm gx`stg ujm sikuc`ój, kughi gj cmdm `tgrmc`ój sg ouscm cijt`jumr cij gk cuepk`e`gjti dgk tgirgem dg Oikzmji, mcirtmjdi mså gk `jtgrvmki dg sikuc`ój dg tmk airem qug kkghmrí uj eiegjti dijdg sg ki bmym mcirtmdi ki jgcgsmr`i pmrm tgjgr ujm sikuc`ój cija`mokg. Vmrm gj cmdm `tgrmc`ój msghurmr gk cuepk`e`gjti dg kms cijd`c`ijgs mjtgs dgscr`tms, gs dgc`r gk tgirgem dg Oikzmji mså ciei gk mcirtme`gjti dgk `jtgrvmki sg usm gk s`hu`gjtg pricgd`e`gjti>
+?
Kms `tgrmc`ijgs sgríj gxprgsmdms egd`mjtg gk cijtmdir j5= ,?,… y gj cmdm ujm dg gkkms sg bmkkmrí gk pujti 5 egd`i pj dgk `jtgrvmki dg mprix`emc`ój \mj , ojU usmjdi km s`hu`gjtg gxprgs`ój . Kughi sg gvmkþm km aujc`ój a gj kis trgs pujtis dg km `tgrmc`ój, mj , o j y p j , ciepmrmjdi kis s`hjis dg km aujc`ój gj gstis pujtis pmrm rggepkmzmr gk `jtgrvmki dg mprix`emc`ój sghþj gk s`hu`gjtg cr`tgr`i> S` a(mj)a(pj);6 gjtijcgs gk `jtgrvmki dg mprix`emc`ój cmeo`m m \mj , pjU. S` a(pj) a(oj);6 gjtijcgs gk `jtgrvmki dg mprix`emc`ój cmeo`m m \pj , ojU. Gk `jtgrvmki rgsuktmdi dg km ciepmrmc`ój mjtgs `jd`cmdm, pmsm m sgr gk `jtgrvmki dg mprix`emc`ój `j`c`mk pmrm km `tgrmc`ój s`hu`gjtg. Dg gstm airem sg rgp`tg gk pricgsi bmstm tgjgr ujm tikgrmjc`m mcgptmokg. Vmrm km mpk`cmc`ój dgk pricgsi mjtgs `jd`cmdi sg usm km s`hu`gjtg tmokm gj gstg eàtidi> j
m j
o j
a(m j )
a(o j )
p j
a(p j )
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
A`jmkegjtg `jd`cmeis km gx`stgjc`m dg uj tgirgem dg cijvgrhgjc`m dgk eàtidi, gk cumk jis myudm gj cijicgr gk jþegri dg `tgrmc`ijgs jgcgsmr`ms pmrm mkcmjzmr uj dgtgre`jmdi grrir mosikuti> Tgirgem ?> Sgm a ɜ C \m, oU tmk qug a(m)a(o);6, gk mkhir`tei dg km o`sgcc`ój hgjgrm ujm sucgs`ój {pmj } qug sg mprix`em mk vmkir gxmcti p cij km prip`gdmd>
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=
Gk dgsmrrikki mjtgr`ir rgsue`ó kis mspgctis eís `epirtmjtgs dgk eàtidi, m cijt`jumc`ój sg kis ciepkgegjtm cij ujm sgr`g dg gngepkis> =.?.? Gngepkis Gngrc`c`i rgsugkti => Ipt`e`zmc`ój dg írgms. Sg dgog amor`cmr ujm kmtm gj airem c`kåjdr`cm c`rcukmr rgctm qug cijtgjhm =666 ce 3 . Km tmpm c`rcukmr dg km pmrtg supgr`ir y dgk aijdi dgogj tgjgr uj rmd`i dg 6.?7 ce eís qug gk rmd`i dg km kmtm pmrm qug gk siormjtg sg ut`k`cg pmrm sgkkmr cij km pmrgd kmtgrmk. Km binm dg emtgr`mk cij qug sg cijstruyg km pmrgd kmtgrmk tmeo`àj dgog sgr 6.?7 ce eís hrmjdg qug km c`rcujagrgjc`m dg km kmtm dg eidi qug pugdm bmcgrsg uj sgkki kmtgrmk. Cmkcukg cij ujm gxmct`tud dg =6-4 km cmjt`dmd eåj`em dg emtgr`mk jgcgsmr`m pmrm amor`cmr gstm kmtm.
Wjm hría`cm mprix`emdm dgk priokgem pkmjtgmdi gs km mdnujtm. Sgm b km mkturm dgk c`k`jdri y sgm r gk rmd`i dgk vikuegj dgsgmdi y r+6.?7 gk rmd`i jgcgsmr`i pmrm sgkkmr km tmpm. Sg t`gjg qug gk írgm dg emtgr`mk m usmrsg, y vmkir m e`j`e`zmr (ipt`e`zmr) gstí dmdm pir>
∘ 5?
+
Dijdg kms írgms egjc`ijmdms dg km omsg y kmtgrmk sg pugdgj rggepkmzmr gj aujc`ój dg r, b. mdgeís usmjdi gk vikuegj dgk c`k`jdri sg pugdg dgnmr km aujc`ój Írgm rgspgcti m ujm sikm vmr`mokg. Gs dgc`r>
?ℊ ℊ ? ? ℊ ? ? 5 =666 5
<
5
=666
5 ? ( + 6.?7) + ? ( + 6.?7) 5 ? ( + 6.?7) +
Sg dgducg gjtijcgs qug km aujc`ój m ipt`e`zmr gs km s`hu`gjtg>
?666 ( + 6.?7)
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
?
( ) 5 ? ( + 6.?7) +
?
Nuk`i _umji K`em
?666( + 6.?7)
Vmrm bmkkmr gk pujti sik`c`tmdi dg írgm eåj`em sg `humkm m cgri gstm dgr`vmdm, kughi cij gk eàtidi dg o`sgcc`ój sg bmkkm km sikuc`ój dgsgmdm, cij tikgrmjc`m dg =6-4. Km dgr`vmdm cij su rgspgct`vm hría`cm tmjti dg tidm km aujc`ój ciei dg siki kms cgrcmjåms mk vmkir gst`emdi dg km rmåz sij>
? 3 ‿
( ) 5 4 ( + 6.?7)
?666
=666
Mquå cmog rgcmkcmr qug km þj`cm rmåz pis`t`vm dg km aujc`ój gstm mkrgdgdir dg 7 s`gjdi uj `jtgrvmki qug km cijt`gjg gk \7.=< 7.1U. Gstg vmkir rgsuktm cibgrgjtg dmdi qug gk rmd`i siki pugdg sgr pis`t`vi pir rgprgsgjtmr ujm egd`dm. Km tmokm dg rgsuktmdis gs km s`hu`gjtg>
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
j
mj
oj
pj
a(mj)
a(oj)
a(pj)
6
7.=
7.1
7.37
-=8.?6?677? 4.6434732= -1.633217?2
=
7.37
7.1
7.487
-1.633217?2 4.6434732= -6.28=108=0
?
7.487
7.1
7.7387
-6.28=108=0 4.6434732=
3
7.487
7.7387
7.761?7
- 6.28=108=0
=.1=73=?1
6.38034
4
7.487
7.761?7
7.4061?7
-6.28=108=0
6.38034
-6.?44?83?4
7
7.4061?7
7.761?7
7.4024387
-6.?44?83?4
6.38034
6.6126610=
1
7.4061?7
7.4024387
7.40473=?7 -6.?44?83?4 6.6126610= -6.6226=443
8
7.40473=?7
7.4024387
7.401424387 -6.6226=443 6.6126610= -6.660084=?
2
7.40142432
7.4024387
7.408416032 -6.660084=? 6.6126610=
6.6?06?32
0
7.40142432 7.408416032 7.40108?170 -6.6600830? 6.6?06?32?
6.6607?12
=.1=73=?1
=6 7.40142432 7.40108?171 7.4018?27=1 -6.660084=? 6.6607?110 -6.666??3?7 == 7.4018?27? 7.40108?171 7.401276721 -6.666??3?1 6.6607?110
6.66417=23
=? 7.4018?27? 7.401276721 7.40182077= -6.666??3?1 6.66417=23 6.66??=43?
Pm sg cijicg gk rmd`i ouscmdi s`j geomrhi gk priokgem tgre`jm bmkkmjdi gk írgm eåj`em qug gstg rmd`i itirhm, gjtijcgs sg gvmkþm km aujc`ój M(r) gj gk rmd`i iotgj`di. Sg t`gjg gjtijcgs>
?
5 ? (7.4018?077= + 6.?7) +
?
?666(7.4018?077= + 6.?7) 7.4018?077=
5 728.0
Gngrc`c`i rgsugkti ?> Mpk`cmc`ój dgk tgirgem dgk vmkir egd`i pmrm `jtghrmkgs.
∬
?
=) Gk tgirgem dgk vmkir egd`i pmrm `jtghrmkgs gstmokgcg qug s` a gs ujm aujc`ój cijt`jum gj uj
`jtgrvmki \m, oU, gx`stg mk egjis uj ν ɜ \m, oU tmk qug
Sgm ( ) 5
(
)< 6≤ x ≤?. Dgtgre`jg>
( )
5 ( )(
) .
m) Km gcumc`ój pmrm iotgjgr gk vmkir ν qug smt`samcg gk Tgirgem dgk Qmkir Egd`i cij
∬ ( )
.
o) Wj `jtgrvmki pmrm rgsikvgr km gcumc`ój pkmjtgmdm gj m) cij gk eàtidi dg km O`sgcc`ój. Nust`a`qug su rgspugstm. c) Eugstrg gk rgsuktmdi iotgj`di gj km cumrtm `tgrmc`ój cij gk eàtidi dg km O`sgcc`ój.
d) S`j rgmk`zmr eís `tgrmc`ijgs, dgtgre`jg cumjtms tgjdråm qug rgmk`zmr cij gk eàtidi dg km o`sgcc`ój s` dgsgm tgjgr km sghur`dmd qug gk grrir dg trujcme`gjti gs egjir m =6-1 .
Vr`egri sg dgog bmkkmr gk vmkir dg km `jtghrmk pkmjtgmdm, gs dgc`r>
? 6 (
)
5
+
?6 ? (
)
5
=
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Vir ki tmjti pricgdgeis m `humkmr gstg rgsuktmdi cij ki gst`pukmdi gj gk tgirgem, y gvmkumjdi km aujc`ój a dmdm gj gk juegri ν ouscmdi.
? ? = 5 ( )(?
6)
(
=5?(
))
Dg km þkt`em gxprgs`ój sg bmkkm km gcumc`ój m rgsikvgr, kkmeíjdikg h (ν).
∟
m) ( ) 5
(
)
5
Su hría`cm gs km s`hu`gjtg>
o) Dgk hría`ci pidgeis sgkgcc`ijmr gk `jtgrvmki, gk e`sei pugdg sgr \=, =.7U.
Ciepriomjdi km vmk`dgz dgk e`sei tgjgeis> h(=)5 -6.481?41 y h(=.7)5 ?.?28=1
Dmdi qug h(=)h(=.7);6 pir tgirgem dg Oikzmji, gx`stg pir ki egjis ujm sikuc`ój gj gk `jtgrvmki gscih`di. c) Eàtidi dg o`sgcc`ój j
mj
oj
pj
a(mj )
a(oj )
a(pj )
6
=
=.7
=.?7
-6.488232284
?.?28=72=18
=.66=7=?710
=
=
=.?7
=.=?7
-6.488232284
=.66=7=?710
6.?118=1?8=
?
=
=.=?7
=.61?7
-6.488232284
6.?118=1?8=
-6.=6876=84=
3
=.61?7
=.=?7
=.60387
-6.=6876=84=
6.?118=1?8=
6.68073888=
4
=.61?7
=.60387
=.682=?7
-6.=6876=84=
6.68073888=
-6.6=4676302
d) Vir tgirgem dg cijvgrhgjc`m dgk eàtidi dg o`sgcc`ój tgjgeis>
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
|
_ggepkmzmjdi tgjgeis>
∟1 ≂ |
|=6
|
Nuk`i _umji K`em
?
=.7
=
?
=2.03
Vir ki qug sg cijckuyg qug pmrm sghur`dmd dgk grrir dg trujcme`gjti sg dgogråmj tiemr j5=0 `tgrmc`ijgs.
Gk eàtidi dgk pujti a`ni rgprgsgjtm ujm bgrrme`gjtm pidgrism y gj hgjgrmk eís íh`k qug gk eàtidi dg o`sgcc`ój. Gstg eàtidi omsm su jieorg y mkhir`tei gj km prip`gdmd dg c`grtms aujc`ijgs dg tgjgr dgjtri dg su die`j`i kis kkmemdms pujtis a`nis, qug ji sij eís qug pujtis gj kis cumkgs sg cuepkg qug 5 ( ), gs dgc`r km `emhgj dg km aujc`ój gj gsg pujti gs `dàjt`cm mk e`sei.
Gj km príct`cm, mk tgjgr ujm gcumc`ój dg km airem ( ) 5 6, sg ouscm emj`pukmr mkhgorm`cmegjtg km gxprgs`ój dg a pmrm kkghmr dg mkhujm airem m km `humkdmd 5 ( ). Mjtgs dg gjtrmr m dgtmkkmr kms cijd`c`ijgs jgcgsmr`ms pmrm qug km aujc`ój h pripirc`ijg uj mkhir`tei cijvgrhgjtg, sg dmj pgqugýis gngepkis dg ki qug qugrgeis dgc`r cij "emj`pukmc`ój mkhgorm`cm" dg a pmrm kkghmr m km `humkdmd egjc`ijmdm. Supijhm qug tgjgeis km gcumc`ój> dgnmr dg kms s`hu`gjtgs emjgrms>
? ? ? ∞ = ? ? ? 3 4 ∞
5 6, km cumk egd`mjtg dgspgngs y emj`pukmc`ijgs pidgeis
+
5
5
kj(
)5 5
Kis cumtri gngepkis prgv`megjtg prgsgjtmdis jis dmj cumtri pis`okgs oisqugnis dg km tmj ouscmdm h(x), gj gstg cmsi tgjdråmeis kms pis`o`k`dmdgs> ( )5
+
( )5
( ) 5 kj( ( )5
)
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Gj km rgmk`dmd j`jhujm dg kms cumtri aujc`ijgs qug pus`eis ciei cmjd`dmtms m h(x) pripirc`ijmj uj mkhir`tei cijvgrhgjtg dmdi qug ji cuepkgj cij gk cr`tgr`i dg cijvgrhgjc`m, dg bgcbi km a`jmk`dmd dgk pgqugýi gngepki mjtgr`ir aug qug gk kgctir sg ame`k`mr`cg cij km `dgm dg tgjgr vmr`ms cmjd`dmtms dg h pmrm ujm e`sem gcumc`ój pkmjtgmdm. Gj km príct`cm gk gjsmyi y grrir, mdgeís dg príct`cm jis cijduc`ríj m tgjgr egnirgs gkgcc`ijgs dg pis`okgs aujc`ijgs pmrm gk mkhir`tei. M cijt`jumc`ój gjujc`mrgeis gk tgirgem dg cijvgrhgjc`m dgk eàtidi dg pujti a`ni, gj gk e`sei sg bmcg rgagrgjc`m m uj `jtgrvmki \m, oU dijdg sg gjcugjtrm km sikuc`ój, ji gstí dg eís rgcirdmr qug dmdi uj priokgem, km gkgcc`ój dg d`cbi `jtgrvmki dgog nust`a`cmrsg egd`mjtg gk tgirgem dg Oikzmji.
‿
∟= ≧
Tgirgem 3> Sgm h ɜ C \m,oU , h(x) ɜ \m, oU, h d`agrgjc`mokg gj (m, o) cij | ( )| ; = gj \m, oU.S` p 6 ɜ \m, )}, oU gjtijcgs km sucgs`ój { 5 ( = cijvgrhg mk þj`ci pujti a`ni gj \m, oU.
Gk tgirgem gj hgjgrmk sg pugdg rgsue`r gj ki s`hu`gjtg, omsíjdijis gj kms cmrmctgråst`cms dg km aujc`ój h(x)> •
•
h gvmkumdm
gj \m, oU dgog tiemr vmkirgs qug tmeo`àj gstàj `jcku`dis dg emjgrm 'cgrcmjm' mk `jtgrvmki \m, oU. D`cbi dg itrm emjgrm, kms `eíhgjgs dg km aujc`ój gvmkumdm gj gk `jtgrvmki dgog pmrgcgrsg mk `jtgrvmki, \h(m), h(o)U dgog pmrgcgrsg m \m, oU. h' ji pugdg tiemr vmkirgs supgr`irgs m = j` `jagr`irgs m -= gj gk `jtgrvmki \m, oU.
Gj kis gngepkis s`hu`gjtgs sg trmtm dg dgnmr gj ckmri gk pricgd`e`gjti pmrm dgtgre`jmr qug aujc`ijgs pidråmj cijvgrhgr i ji. Vir cistueorg dgk mutir, pr`egri sg vgr`a`cm km cijd`c`ój dg mcitme`gjti dg h' , s` km e`sem sg cuepkg sg cijt`jum vgr`a`cmjdi km itrm cijd`c`ój, cmsi cijtrmr`i sg dgnm m uj kmdi km aujc`ój ym qug jgcgs`tm cuepk`r meoms cijd`c`ijgs.
Gngrc`c`i rgsugkti 3> Cijvgrhgjc`m dgk eàtidi dg pujti a`ni. _gsugkvm km gcumc`ój kj(x)+?-x56 gepkgmjdi gk eàtidi dgk pujti a`ni y nust`a`cmjdi km cijvgrhgjc`m dgk eàtidi. ut`k`cg ujm tikgrmjc`m dg =6-7 .
Ciei dg cistueorg mdnujtmeis pr`egrmegjtg km hría`cm dg km aujc`ój> Dg gstm hría`cm pidgeis iotgjgr qug uj ougj `jtgrvmki dg gx`stgjc`m dg ujm dg kms sikuc`ijgs gs \3, 3.7U. Ciepriomjdi su vmk`dgz tgjgeis> \kj(3)+?-3U\kj(3.7)+?-3.7U;6 pir ki tmjti pir gk tgirgem dg Oikzmji pidgeis msghurmr qug gx`stg pir ki egjis ujm sikuc`ój gj gk `jtgrvmki sgkgcc`ijmdi.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
M cijt`jumc`ój ouscmeis pis`okgs aujc`ijgs h(x) qug hmrmjt`cgj km cijvgrhgjc`m dgk mkhir`tei dg pujti a`ni.
∟? ‿ ∟? 337∟?∟? ≂=7 ≂
Km emj`pukmc`ój dg km gcumc`ój jis kkgvm ciei pr`egrm ipc`ój dg h(x) m km s`hu`gjtg> ( )5
Cij su dgr`vmdm>
( )5
Vmrm vgr`a`cmr km cijvgrhgjc`m dg gstm ipc`ój dg h(x) vgeis kis vmkirgs dg h' gj gk `jtgrvmki prgv`megjtg sgkgcc`ijmdi. (3) 5
( 3.7) 5
5
.
5
?.8=2?2=2?2
.
4.42=12068
= ≂ ≂
Ciei pidgeis vgr jugstrm pr`egrm gkgcc`ój dg h(x) ji pripirc`ijm ujm aujc`ój qug hmrmjt`cg km cijvgrhgjc`m dgk eàtidi dmdi qug km e`sem gxcgdg km citm dg | ( )| = gj gk `jtgrvmki dg mprix`emc`ój. Cmog rgcmkcmr qug ji vgr`a`cmeis km sghujdm cijd`c`ój dmdi qug ciei sg d`ni mjtgr`iregjtg omstm qug ujm ji sg cuepkm pmrm pidgr dgscmrtmr ujm ipc`ój dg h(x). Wjm s`hu`gjtg ipc`ój pmrm km aujc`ój h(x) rgsuktm ( ) 5 kj( ) + ? cij
( )5 .
Jugvmegjtg vgr`a`cmeis su mcitme`gjti gjtrg -= y = gj gk `jtgrvmki pmrm msghurmr km cijvgrhgjc`m dgk eàtidi, tgjgeis> (3) 5
( 3.7) 5
= 3 =
3.7
6.333333 …
6.?278=4?27
Gstm aujc`ój cuepkg gk mcitme`gjti pir ki qug cijt`jumrgeis vgr`a`cmjdi s` sus `eíhgjgs sg pmrgcgj mk `jtgrvmki dg mprix`emc`ój gkgh`di.
Vmrm gsti jis dmeis ujm `dgm cij kis gxtrgeis y mkhþj pujti `jtgr`ir> (3) 5 kj(3) + ? 5 3.602 …
(3.?7) 5 kj(3.?7) + ? 5 3.=82174 … (3.7) 5 kj(3.7) + ? 5 3.?7?81?…
Ciei pidgeis vgr km aujc`ój h gvmkumdm gj gk `jtgrvmki dg mprix`emc`ój sg pmrgcg i gstí cijtgj`dm dgjtri dgk e`sei `jtgrvmki dg mprix`emc`ój pir ki qug km e`sem cuepkg titmkegjtg cij gk cr`tgr`i dg cijvgrhgjc`m.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
+=
Dmdi gk cuepk`e`gjti dgk cr`tgr`i dg cijvgrhgjc`m pmsmeis m dgsmrrikkmr gk mkhir`tei> 5 (
) 5 kj(
)+?
Gscihgeis uj pujti qug pgrtgjgzcm mk `jtgrvmki pmrm rgmk`zmr kms `tgrmc`ijgs, gj gstg cmsi gscihgeis x653.= pmrm gk cumk tgjgeis km s`hu`gjtg tmokm dg rgsuktmdis> j
x j
x j+=
Tik
6 = ? 3 4 7 1 8 2
3.= 3.=3=46?=== 3.=4=426213 3.=4410436= 3.=478=1124 3.=4164=847 3.=41=47684 3.=41=880=8 3.=41=22378
3.=3=46?=== 3.=4=426213 3.=4410436= 3.=478=1124 3.=4164=847 3.=41=47684 3.=41=880=8 3.=41=22378 3.=41=0=187
6.63=46?=== 6.6=668287? 6.663?=3432 6.66=6??323 6.6663?761= 6.666=633?0 3.?2431G-67 =.64303G-67 3.3=268G-61
A`jmkegjtg cijcku`eis qug ujm dg kms sikuc`ijgs dg km gcumc`ój pkmjtgmdm gs 3.=41=0=187.
Gngrc`c`i rgsugkti 4> Oþsqugdm dg rmåcgs dg uj pik`jie`i. Gk pik`jie`i ( ) 5 t`gjg ujm þj`cm rmåz pis`t`vm. Gjcugjtrg uj `jtgrvmki dijdg sg hmrmjt`cg km gx`stgjc`m dg gstm rmåz (nust`a`qug!). Wt`k`zmjdi gk eàtidi dgk pujti a`ni, prgsgjtg ujm tmokm qug cijtgjhm km sucgs`ój dg vmkirgs, cij uj cr`tgr`i dg `jtgrrupc`ój dgk eàtidi `tgrmt`vi cij tikgrmjc`m =6-0 .
3 ?
Vrgsgjtmeis km hría`cm dgk pik`jie`i, dmdi qug km gcumc`ój m rgsikvgr sgrí
= 5 6.
Ouscmeis mbirm uj `jtgrvmki dg gx`stgjc`m dg km rmåz, gk e`sei sgrí dg \=.7, ?U dmdi qug pir gk tgirgem dg Oikzmji p(=.7) p(?);6 gx`stg pir ki egjis ujm sikuc`ój dgjtri dgk e`sei. M cijt`jumc`ój ouscmeis km aujc`ój h qug hmrmjt`cg km cijvgrhgjc`m dgk mkhir`tei dg pujti a`ni. gk pr`egr oisqugni dg aujc`ój sgrí>
? ? ( )5
+
+=
Km e`sem t`gjg pir dgr`vmdm>
? +=
( )5
3(
+
+ =)
?3
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Vmrm vgr`a`cmr km cijvgrhgjc`m dgk eàtidi vgeis kis vmkirgs qug tiem km dgr`vmdm gj kis gxtrgeis dgk `jtgrvmki, kis e`seis sij>
? ?3 ≂ ? ?3 ≂
?(=.7) + =
(=.7) 5
6.48=0
3(=.7 + = . 7 + =) ?(?) + =
(?) 5
6.4774
3(? + ? + =)
Kms dgr`vmdms gj kis gxtrgeis cuepkgj gk cr`tgr`i dg mcitme`gjti pir ki qug pidgeis gepgzmr m vgr`a`cmr km sghujdm cijd`c`ój dgk cr`tgr`i dg cijvgrhgjc`m pmrm ki cumk gvmkumrgeis km aujc`ój gj kis gxtrgeis dgk `jtgrvmki y mkhþj pujti `jtgr`ir.
?? ? += ? (=.7) 5
(=.87) 5
(?) 5
(=.7) + (=.7) + = 5 =.12…
(=.87) + (=.87) + = 5 =.80800 … (?) + (?) + = 5 =.0=?03=…
Dmdi qug sg cuepkgj meoms cijd`c`ijgs dgk cr`tgr`i dg cijvgrhgjc`m, pricgdgeis m gkmoirmr gk mkhir`tei dgk pujti a`ni y prgsgjtmeis km tmokm dg rgsuktmdis> 5
+
+=
Kms `tgrmc`ijgs sg eugstrmj m cijt`jumc`ój cij ujm mprix`emc`ój x65=.8. Vir amctir gspmc`i dgo`di mk jþegri dg `tgrmc`ijgs jgcgsmr`ms km tmokm sg eugstrm gj pmrtgs, tmjti gj gstm ciei gj km s`hu`gjtg binm> j
x j
x j+=
Tik
6 = ? 3 4 7 1 8 2 0 =6 == =? =3 =4
=.8 =.884876347 =.26041742 =.2?77?4?04 =.23?030=73 =.231370288 =.23803834? =.232114171 =.232000011 =.230=74741 =.230??7262 =.230?72170 =.230?83263 =.230?26824 =.230?24663
=.884876347 =.26041742 =.2?77?4?04 =.23?030=73 =.231370288 =.23803834? =.232114171 =.232000011 =.230=74741 =.230??7262 =.230?72170 =.230?83263 =.230?26824 =.230?24663 =.230?27421
6.684876347 6.6348=7=37 6.6=16722=3 6.6684=4270 6.6634?68?4 6.66=788417 6.6668?83=4 6.6663373= 6.666=7472 8.=?1=4G-67 3.?27=?G-67 =.7=44?G-67 1.02=37G-61 3.?=237G-61 =.42313G-61
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
=7 =1 =8 =2 =0 ?6 ?= ?? ?3 ?4
=.230?27421 =.230?21=8 =.230?21421 =.230?2113= =.230?21102 =.230?218?0 =.230?21843 =.230?2187 =.230?21873 =.230?21874
=.230?21=8 =.230?21421 =.230?2113= =.230?21102 =.230?218?0 =.230?21843 =.230?2187 =.230?21873 =.230?21874 =.230?21877
1.23043G-68 3.=7?0?G-68 =.47348G-68 1.86630G-62 3.62223G-62 =.4?30?G-62 1.714=8G-60 3.6?163G-60 =.30402G-60 1.4368?G-=6
Qmkg km pgjm dgstmcmr qug km tikgrmjc`m dgsgmdm sg mkcmjzó gj km `tgrmc`ój ?7, pir ki qug pidgeis vgr qug gk eàtidi m pgsmr dg cijvgrhgr ki bmcg kgjtmegjtg.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Gk eàtidi trmtmdi gj gstm sgcc`ój gs s`j dudm gk dg cijvgrhgjc`m eís mcgkgrmdm dg kis trmtmdis bmstm mbirm, gk e`sei t`gjg uj mkhir`tei mkhi s`e`kmr mk dg pujti a`ni ki cumk sg gxpk`cm dmdi qug gk eàtidi dg Jgwtij gs uj cmsi pmrt`cukmr dgk e`sei.
_ghrgsmeis mk cmsi `j`c`mk dg tgjgr qug rgsikvgr uj priokgem dg km airem ( ) 5 6, pmrm pidgr mpk`cmr gstg eàtidi km aujc`ój a dgog sgr dgr`vmokg dis vgcgs, gs dgc`r ( a ɜ C ?\m, oU) dijdg \m, oU gs gk `jtgrvmki dg gx`stgjc`m dg km rmåz sgkgcc`ijmdi mk `jcimr gk pricgd`e`gjti (s`gjdi nust`a`cmdi mdgcumdmegjtg pir gk tgirgem dg Oikzmji). Mdgeís km aujc`ój a' ji dgog sgr jukm gj gk `jtgrvmki mjtgs egjc`ijmdi.
+=
Gstms cijd`c`ijgs hmrmjt`zmj km cijvgrhgjc`m dgk eàtidi, qug t`gjg pir mkhir`tei> 5
(
)
‿ (
)
Km emyir d`a`cuktmd gj gstg mkhir`tei rmd`cm gj cmkcukmr o`gj km dgr`vmdm dg km aujc`ój y gkgh`r ujm mprix`emc`ój `j`c`mk cgrcmjm m km sikuc`ój gxmctm. Vmrm eucbis gstud`mjtgs gs gk eàtidi eís aíc`k dg mpk`cmr, s`gjdi mdgeís gk dg cijvgrhgjc`m eís mcgkgrmdm ym qug gk e`sei t`gjg cijvgrhgjc`m cumdrít`cm.
Gngrc`c`i rgsugkti 7> D`stmjc`m eåj`em. Wsg gk eàtidi dg Jgwtij pmrm mprix`emr cij uj hrmdi dg gxmct`tud dg =6 -4 gk vmkir dg x qug gj km hría`cm dg y5x ? priducg gk pujti eís cgrcmji m (=,6).
Km hría`cm dgk priokgem pkmjtgmdi gs km s`hu`gjtg> Km aujc`ój d`stmjc`m gjtrg dis pujtis cumkqu`grm gj _ ? gstí dmdm pir>
? = 6 ? = 6 ? 5(
) +(
)
Sgm gk pujti (x= , y=)5(x, x?) y (x6 , y6)5(=,6). Gk pujti y= sg rggepkmzm pir x? dgo`di m qug pgrtgjgcg m km aujc`ój y5x?. Sg ipt`e`zm km aujc`ój d ? ym qug mrrinmrm gk e`sei rgsuktmdi qug ipt`e`zmr km aujc`ój d pgri dg airem eís sgjc`kkm.
? ? ? ? 3
_ggepkmzmjdi cij kms rgkmc`ijgs mjtgs gxprgsmdms sg iot`gjg> 5(
Dgr`vmjdi rgspgcti m x sg t`gjg>
=) + (
? 5 ?(
=)+4
6)
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Vmrm ipt`e`zmr sg `humkm km dgr`vmdm m cgri>
3
?(
=) + 4
56
Mpk`cmjdi gk eàtidi dg Jgwtij, y usmjdi ciei mprix`emc`ój `j`c`mk mk jþegri 6.2, sg t`gjg kms `tgrmc`ijgs> j
Rj
Rj+=
Tikgrmjc`m
6
6.2
6.7274=118
?.=7G-6=
=
6.7274=118
6.70=2013
1.42G-63
?
6.70=2013
6.72284337
3.=7G-63
3
6.72284337
6.706?4?47
=.76G-63
4
6.706?4?47
6.7207?=40
8.?=G-64
7
6.7207?=40
6.72021131
3.47G-64
1
6.72021131
6.72086601
=.17G-64
8
6.72086601
6.720826=2
8.0?G-67
Mså sg dgducg qug gk pujti eís cgrcmji dg km curvm y5x? mk pujti (=,6) gs (6.720826=2,6.34824611).
Gngrc`c`i rgsugkti 1> Itri priokgem dg d`stmjc`m eåj`em. Mprix`eg cij ujm gxmct`tud dg =6-4 gk vmkir dg x gj qug km hría`cm dg y5g x priducg gk pujti eís cgrcmji m (?,6).
Km hría`cm dgk priokgem pkmjtgmdi gs km s`hu`gjtg, dijdg gk pujti mzuk cirrgspijdg mk pujti dmdi ciei dmti y gk vgrdg mk pujti qug qugrgeis bmkkmr (hrma`cmdi cijic`gjdi km rgspugstm). Sg cijicg qug km aujc`ój d`stmjc`m gj gk pkmji gstí dmdm pir>
? 6 ? 6 ? ? ? ? ? ? 5(
) +(
)
Sgmj xi , y i gk pujti dmdi ciei dmti y sgm x,y uj pujti cumkqu`grm dg km curvm dmdm. Sg t`gjg> 5(
?) + (
)
Km aujc`ój iargcg uj eåj`ei cumjdi su dgr`vmdm sg `humkm m cgri, pir ki qug> ? 5 ?(
?) + ?(
) 56
Dg mquå sg iot`gjg qug km aujc`ój m km cumk sg dgog bmkkmr sus rmåcgs sgm km s`hu`gjtg> ( )5(
?) +
56
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Mprix`emjdi pir eàtidi dg Jgwtij, y gscih`gjdi ciei pujti dg mprix`emc`ój `j`c`mk m x656.3.
? += ‿ ?
Sg iot`gjg gk mkhir`tei dg Jgwtij y kms `tgrmc`ijgs> 5
j
(
)
(
)
?+
5
=+?
Rj
Rj+=
tik
6
6.3
6.?83867368
?.13G-6?
=
6.?83867368
6.?83=402?2
7.77G-64
?
6.?83=402?2
6.?83=40720
?.30G-68
Vir ki tmjti gk vmkir dg x pmrm gk cumk km hría`cm sg gjcugjtrm eís cgrcmjm m (?,6) gs x56.?83=40720.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
"Egnir qug dg jugstri nu`c`i, dgogeis a`mrjis dgk cíkcuki mkhgorm`ci". Kgijbmrd Gukgr
Gstmeis gj hgjgrmk ame`k`mr`zmdis cij km `dgm dg uj s`stgem k`jgmk dg gcumc`ijgs dgsdg kms emtgeít`cms oís`cms dg cikgh`i, s`j geomrhi gj km príct`cm ji s`geprg sg jis prgsgjtm uj s`stgem mk cumk sgm dgk tidi sgjc`kki mpk`cmrkg gk eàtidi d`rgcti dg Hmuss Nirdmj, gsti sg dgog pr`jc`pmkegjtg m km mpmr`c`ój dg ciga`c`gjtgs ji gjtgris qug hgjgrmj uj pici eís dg trmomni. Ms`e`sei sg jis pugdgj prgsgjtmr s`stgems dg uj tmemýi mkhi emyir, gjt`àjdmsg uj s`stgem dg 8Ø8 pir gngepki dijdg gk eàtidi dg Hmuss Nirdmj sg vikvgrí mkhi tgd`isi. Bgrrme`gjtms gj km rgsikuc`ój dg s`stgems dg gstg t`pi sij kis eàtidis qug prgsgjtmrgeis m cijt`jumc`ój, gj rgmk`dmd meois eàtidis pisggj uj e`sei mkhir`tei pgri pisggj ujm hrmj d`agrgjc`m mk mpk`cmrkis ki cumk sg gxpk`cmrm eís m aijdi gj ujms píh`jms. Vmrm km titmk cieprgjs`ój dg kis cijcgptis m usmrsg eís mdgkmjtg suhgr`eis mk kgctir rgv`smr kms dga`j`c`ijgs dg jirem emtr`c`mk y vgctir`mk.
=?⋪== ?⋪=?? =?⋪ ⋪=? =? = ? ⎕⎦ ⎕⋪⎦ ?===== =????? ⋪⋩⋩ =? =? == ?? ⋩ = === =?? ⋩⋩== = ? ??? ?= = ?? ? ⋪
Cijs`dgrg uj s`stgem dg km airem
>
5
… …
5
…
Mk rgmk`zmr km ipgrmc`ój emtr`c`mk, iotgjgeis> + +
+ +
+ +
5 5
+
+
+
5
Dg ki prgv`megjtg iotgj`di pidgeis dgspgnmr cmdm vmr`mokg gj irdgj dg cmdm a`km, cij ki cumk tgjgeis> 5
5
+
+
+
+
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
== ⋩∟=∟= ∟= = ? = = = = = ⋪=? ?=⋪?? ?⋪ ?⋪?? ⋪=? =?⋪=?=? ⎕⎦ ⎕= ⎦⎕⎦ ⎕⎦ ∟= 56 ∟= ∆6 ∂ +
5
+
Dg kms vmr`mokgs dgspgnmdms pidgeis kkghmr m uj mkhir`tei dg km airem> ( )
6
(
…
…
6
5
)
+
…
6
5 + dijdg km emtr`z y vgctirgs qug ki cijairemj Gstg mkhir`tei cieþjegjtg gs gxprgsmdi> sg pugdgj d`st`jhu`r ckmrmegjtg dg km airem emtr`c`mk mjtgs prgsgjtmdm.
Dmdi uj s`stgem cumkqu`grm m rgsikvgr, kis pmsis m sghu`r sij ckmris. Sg dgog dgspgnmr dg cmdm a`km cmdm vmr`mokg pmrm mså cij gsi pidgr iotgjgr km emtr`z y vgctir , rgspgct`vmegjtg y airemr gk mkhir`tei. S`j geomrhi mjtgs dg pricgdgr cij kms `tgrmc`ijgs vmkg km pgjm rgv`smr kis cr`tgr`is dg cijvgrhgjc`m dg gstg eàtidi, (dg bgcbi kis e`seis sg mpk`cmj tmeo`àj pmrm gk s`hu`gjtg eàtidi, gk dg Hmuss - Sg`dgk). Tgirgem 4> Sgm km sucgs`ój { y siki s`> •
•
( )
}
dga`j`dm pir
5
+ . Km sucgs`ój cijvgrhg
s`
Km emtr`z dg ciga`c`gjtgs M (dg dijdg priv`gjg km sucgs`ój) gs ujm emtr`z gstr`ctmegjtg die`jmjtg d`mhijmkegjtg. Gk rmd`i gspgctrmk dg km emtr`z T, cuepkg qug ( ) ; =.
Gngrc`c`i rgsugkti 8>Cr`tgr`i dg cijvgrhgjc`m g `tgrmc`ijgs gj Nmcio`. Dmdi gk s`stgem MR5O cij
5
+
5 5 , , 5 , , 5 , ,
+
cij o`5?`< `5=,?,3.
m) @jd`qug s` sg cuepkg mkhujm cijd`c`ój dg cijvgrhgjc`m pmrm rgsikvgr cij uj eàtidi `tgrmt`vi.
U . Gjcugjtrg gk vgctir d`agrgjc`m gj km o) Cie`gjcg cij uj vgctir mprix`emc`ój `j`c`mk ( ) 5 \ tgrcgrm `tgrmc`ój, gjtrg kms sikuc`ijgs cmkcukmdms cij kis eàtidis `tgrmt`vis dg Nmcio` y Hmuss-Sg`dgk. Jitm> Siki sg dgsmrrikkmrmj kms `tgrmc`ijgs dgk eàtidi dg Nmcio`, gstg gngrc`c`i sgrí ciepkgegjtmdi gj km
s`hu`gjtg sgcc`ój cij gk eàtidi dg Hmuss-Sg`dgk.
Gk s`stgem gxprgsmdi gj tàre`jis dg 'aireukms', gscr`ti gj airem emtr`c`mk gs gk s`hu`gjtg> 0 3 4
3 =6 7
4 7 ==
? 5 4 1
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Km rgspugstm mk k`tgrmk m) gs qug gk s`stgem så sg prgstm pmrm sgr rgsugkti pir mkhþj eàtidi `tgrmt`vi tmk y ciei jis ki d`grij, dmdi qug km emtr`z M dg ciga`c`gjtgs gs gstr`ctmegjtg die`jmjtg d`mhijmkegjtg, pmrm rgsikvgr gk k`tgrmk o, jgcgs`tmeis gk mkhir`tei dg Nmcio`.
+= += 5
6 3/=6 4/==
3/0 6 7/==
4/0 7/=6 6
?/0 + 4/=6 1/==
Mpk`cmjdi meois eàtidis cij gk vgctir mprix`emc`ój `j`c`mk dmdi
( )
5\
U , sg t`gjg>
@tgrmc`ijgs dgk eàtidi dg Nmcio`>
`5=
6.42 `5? 6.86 6.0?
6.71 6.4 6.?8
`53
6.43 6.?= 6.67
Gngrc`c`i rgsugkti 2> Cijstruym gk mkhir`tei dg Nmcio` pmrm gk s`stgem pkmjtgmdi g `tgrg gk e`sei 3 vgcgs, s` km sikuc`ój gxmctm gs
5
, bmkkg gk grrir pmrm km þkt`em `tgrmc`ój bgcbm.
+
5 + 5 5
Gk dgsmrrikki dgk priokgem jis pgre`tg `r d`rgcti mk mkhir`tei dmdi qug gk s`stgem tmk ciei aug pripirc`ijmdi ym gs uj s`stgem gstr`ctmegjtg die`jmjtg d`mhijmkegjtg. Gk mkhir`tei tiem km airem>
5
6 =8/3? =/3
?/3 6 ?/0
6 =/3? 6
+
? ?0/3? =
Dmdi qug ji jis pripirc`ijmj uj vgctir mprix`emc`ój `j`c`mk usmrgeis gk vgctir cgri, tgj`gjdi kms s`hu`gjtgs `tgrmc`ijgs>
= ? 3 5
5
5
? ?0/3? =
?.164=111 ? 6.=3=0444
3.3333 ?.?0323181287 6.3=?4000
A`jmkegjtg cmkcukmeis gk grrir gjtrg km tgrcgrm `tgrmc`ój y km sikuc`ój gxmctm>
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
‑ 3‑
5
6.11118 6.861=13?3=?7 5 6.861=13?3=?7 6.128766=
Ciei mjtgs sg `jd`có, gk eàtidi dg Hmuss - Sg`dgk usm gk e`sei mkhir`tei qug gk dg Nmcio`, s`j geomrhi t`gjg ujm emrcmdm d`agrgjc`m gj su usi. Gstm d`agrgjc`m rmd`cm gj gk muegjti dg ujm pmrtg mk mkhir`tei dg Nmcio`, gk cíkcuki dg km `-às`em ciepijgjtg dmdi qug gk vgctir sg mctumk`zm mjtgs dg pmsmr m km s`hu`gjtg 5 + . `tgrmc`ój usmjdi gk ym cijic`di Ki e`sei sg rgprgsgjtm>
∟= ∟ = 5= += 5+= 5
=
\
U
Dg airem sgjc`kkm pidgeis vgr qug cmdm gjtrmdm dgk vgctir sg mctumk`zm gj cmdm `tgrmc`ój `, egd`mjtg `tgrmc`ijgs `jtgrjms n, ki e`sei priducg mprix`emc`ijgs eís ríp`dms ym qug cmdm vmkir mctumk`zmdi sust`tuyg su mjtgcgsir. Wj gngepki sgjc`kki y gxpk`cmt`vi sgråm gk s`hu`gjtg>
+= 6 6=
Supijhmeis gk mkhir`tei usmdi gj uji dg kis gngrc`c`is rgsugktis dg km sgcc`ój mjtgr`ir> 5
6 =8/3? =/3
Tiemjdi ciei vgctir `j`c`mk mk vgctir cgri Vmrm `5=
?/3 6 ?/0
6 =/3? 6
+
? ?0/3? =
6 5 6 tgjgeis> 6
Mk rgmk`zmr gk pricgsi bmstm mbirm cijic`di iotgjåmeis gk vgctir> 5
?0/3? =
Dmdi qug gstmeis gj km pr`egrm `tgrmc`ój `jtgrjm cmeo`meis km pr`egr gjtrmdm pir ki qug mbirm jugstri vgctir `j`c`mk pmsmrm m sgr>
6
? ‿ 5 6 6
S`hu`gjdi gj km `tgrmc`ój `5=, rgpgt`eis gk pricgsi pmrm mctumk`zmr km sghujdm gjtrmdm. Dg gstm airem iotgjgeis uj vgctir>
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
6 =
6?
5
.
=
Mbirm cij gstg mctumk`zmeis km sghujdm gjtrmdm qug gstmom pgjd`gjtg pir ki qug tgjgeis> ‿‿
? =.01287 6
5
A`jmkegjtg gj gstm `tgrmc`ój eukt`pk`cmeis gstg vgctir mctumk`zmdi pir T y kg suemeis C ciei ym gs cijic`di, iotgj`gjdi pir a`j gk vgctir a`jmk dg `5=. ? 5 =.01287 6.=64=111
Gstg pricgsi tgd`isi pgri eís gagct`vi sg rgp`tg gj cmdm `tgrmc`ój `. Gj kis gngrc`c`is rgsugktis sg eugstrmj kis vgctirgs qug sg dmj ciei rgsuktmdis a`jmkgs dg cmdm `tgrmc`ój ` eís ji dg kis pmsis `jtgregd`is gj cmdm ujm dg gkkms.
Gngrc`c`i rgsugkti 0>Ciepkgegjti dgk gngrc`c`i 8. Dmdi gk s`stgem MR5O cij
+
5
5 5 , , 5 , , 5 , ,
+
cij o`5?`< `5=,?,3.
m) @jd`qug s` sg cuepkg mkhujm cijd`c`ój dg cijvgrhgjc`m pmrm rgsikvgr cij uj eàtidi `tgrmt`vi.
U . Gjcugjtrg gk vgctir d`agrgjc`m gj km o) Cie`gjcg cij uj vgctir mprix`emc`ój `j`c`mk ( ) 5 \ tgrcgrm `tgrmc`ój, gjtrg kms sikuc`ijgs cmkcukmdms cij kis eàtidis `tgrmt`vis dg Nmcio` y Hmuss-Sg`dgk. Jitm> Siki sg dgsmrrikkmrmj kms `tgrmc`ijgs dgk eàtidi dg Nmcio`, gstg gngrc`c`i sgrí ciepkgegjtmdi gj km
s`hu`gjtg sgcc`ój cij gk eàtidi dg Hmuss-Sg`dgk.
+=
Gk s`stgem gxprgsmdi gj tàre`jis dg 'aireukms', gscr`ti gj airem emtr`c`mk gs gk s`hu`gjtg> 0 3 4
3 =6 7
4 7 ==
? 5 4 1
Km rgspugstm mk k`tgrmk m) gs qug gk s`stgem så sg prgstm pmrm sgr rgsugkti pir mkhþj eàtidi `tgrmt`vi tmk y ciei jis ki d`grij, dmdi qug km emtr`z M dg ciga`c`gjtgs gs gstr`ctmegjtg die`jmjtg d`mhijmkegjtg, pmrm rgsikvgr gk k`tgrmk o, jgcgs`tmeis gk mkhir`tei dg Nmcio`. 5
6 3/=6 4/==
3/0 6 7/==
4/0 7/=6 6
Mpk`cmjdi meois eàtidis cij gk vgctir mprix`emc`ój `j`c`mk dmdi
?/0 + 4/=6 1/==
( )
5\
U , sg t`gjg>
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
@tgrmc`ijgs dgk eàtidi dg Nmcio`>
`5=
@tgrmc`ijgs dgk eàtidi dg Hmuss Sg `dgk>
`5=
6.42 `5? 6.86 6.0?
6.71 6.4 6.?8
6.71 6.68 6.8=
`5?
`53
6.=? 6.62 6.77
`53
6.43 6.?= 6.67
6.67 6.=3 6.76
A`jmkegjtg gk vgctir d`agrgjc`m gs gjtijcgs km rgstm dg kis rgsuktmdis gj kms `tgrmc`ijgs 3 dg meois eàtidis> 5
6.32 6.34 6.47
Gngrc`c`i rgsugkti =6> Emj`pukmc`ój dg uj s`stgem pmrm hmrmjt`zmr su cijvgrhgjc`m.
Cijs`dgrg gk s`stgem MR5O dmdi pir>
.
+ . + . + . + . + . + .
5 . 5 . 5 .
Dg sgr pis`okg emj`pukg gk s`stgem dg tmk airem qug sg hmrmjt`cg km cijvgrhgjc`m dgk eàtidi dg Hmuss Sg`dgk, dgtgre`jg km sikuc`ój dg gstg s`stgem cij uj vgctir `j`c`mk (=,=,=) y cij ujm tikgrmjc`m dg =6-4 .
Gepgzmeis pir emj`pukmr gk s`stgem, cij ujm ríp`dm e`rmdm pidgeis jitmr qug ji gs jgcgsmr`i suemr i rgstmr mkhujm a`km m itrm. Omstm cij bmcgr uj cmeo`i dg a`kms pmrm tgjgr ujm emtr`z gstr`ctmegjtg die`jmjtg d`mhijmkegjtg.
+= 6
Gk cmeo`i dg a`kms mrrinm gk s`stgem titmkegjtg gqu`vmkgjtg>
4 + 6.=7 + 6.?7 5 4.47 ? + 7.1 + 3.= 5 6.= 6.4 + =.= + 3.= 5 8.7
Dg dijdg mk dgspgnmr cmdm vmr`mokg iotgjgeis> 5
Kms `tgrmc`ijgs cij
6 ?/7.1 6.4/3.=
6.=7/4 6 =.=/3.=
6.?7/4 3.=/7.1 6
= 5 = sg prgsgjtmj m cijt`jumc`ój> =
4.47/4 + 6.=/7.1 8.7/3.=
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
= ? 3 4 7 1 8 2 5
=.6=?7 6.2083?=4?278= ?.168==46773
5
6.023?640?7==7 =.8817==3?73 ?.0??21418342
5
6.001446=3?167 =.0716?286720 ?.0242711?641
5
6.000?0873810? =.00=311=8231 ?.0086?86?1=8
5
6.00021?64?77? =.002364081== ?.0004=134628
5
6.00008?0=73 =.000118??022 ?.0002274=762
5
6.00000412?182 =.0000341866? ?.0000887647
6.000002071607 5 =.000028=843 ?.00000772314
Kis s`stgems dg gcumc`ijgs sij euy usmdis cit`d`mjmegjtg gj gk írgm dg c`gjc`ms g `jhgj`gråm, s`j geomrhi eucbms vgcgs dgogeis rgmk`zmr cirrgcc`ijgs m jugstrms egd`c`ijgs i gj hgjgrmk m jugstrms gcumc`ijgs qug cijairemj uj s`stgem pgri quà tmjti magctmrm uj cmeo`i gj mkhþj ciga`c`gjtg dgk s`stgem m su sikuc`ój9 Km prgsgjtg sgcc`ój trmtm siorg kis s`stgems emk i o`gj cijd`c`ijmdis qug ji sij eís qug kis s`stgems sgjs`okgs i ji m kis cmeo`is gj sus ciga`c`gjtgs, cmog rgcmkcmr qug km egd`dm dg su sgjs`o`k`dmd sg dm egd`mjtg uj vmkir dgjie`jmdi juegri dg cijd`c`ój y km sgjs`o`k`dmd ciei tmk gs km vmr`mc`ój dg km sikuc`ój dgk s`stgem pir cmeo`is gj sus ciga`c`gjtgs.
∟=‑ ‑ ‑ ‑ ≬≧
Gk juegri dg cijd`c`ój ( ) dg ujm emtr`z dg ciga`c`gjtgs M (s` y siki s` M gs ji s`jhukmr) sg dgtgre`jm egd`mjtg> ( )5
√
√
Sg pugdg priomr qug s`geprg ( ) =, y sg gjt`gjdg pir s`stgem emk cijd`c`ijmdi gk qug tgjhm ujm emtr`z dg ciga`c`gjtgs M cij ( ) =.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Gngrc`c`i rgsugkti ==>Cijd`c`ijme`gjti dg s`stgems. Wj kicmk vgjdg trgs emtgr`mkgs M, O, C pir pgsi gj Lh. Gj gk cumdri sg eugstrm km cmjt`dmd dg Lh vgjd`dm gj trgs dåms y gk d`jgri `jhrgsmdi pir km vgjtm (dókmrgs), gj cmdm dåm> D`m Qgjtm Lh dg M Qgjtm Lh dg O Qgjtm Lh dg C @jhrgsi pir vgjtm =
0
2
3
?8.66
?
8
1
7
?4.46
3
?
=
0
=1.76
m) Vkmjtgg uj s`stgem dg gcumc`ijgs pmrm dgtgre`jmr gk prgc`i pir Lh. dg cmdm emtgr`mk> M, O, C. o) Iotgjhm km sikuc`ój rgsikv`gjdi gk s`stgem cij eàtidi d`rgcti dg Hmuss. c) Supijhm qug uji dg kis ciga`c`gjtgs t`gjg uj grrir dg 6.= (rggepkmcg gk ciga`c`gjtg 3 pir 3.=), dgtgre`jg bmstm cumjti pugdg vmr`mr gk grrir rgkmt`vi dg km sikuc`ój cij rgspgcti mk grrir rgkmt`vi dg km emtr`z s` sg cijicg qug km `jvgrsm dg km emtr`z cij kis dmtis ir`h`jmkgs dmdi gj gk cumdri gs . . . Gs gstg uj s`stgem emk cijd`c`ijmdi9 Wsmråm km sikuc`ój iotgj`dm9 . . . . Nust`a`qug su rgspugstm.
Gk s`stgem dg gcumc`ijgs m rgsikvgr, sik`c`tmdi gj gk k`tgrmk m) gs gk s`hu`gjtg> 0 8 ?
2 1 =
3 7 0
?8 5 ?4.4 =1.7
Gk s`stgem t`gjg ciei sikuc`ój (kughi dg rgduc`r pir Hmuss)> o)
=.? 5 =.7 =.4
c) Gk jugvi s`stgem m rgsikvgr gs>
Cuym sikuc`ój gs>
‿
0 8 ?
‿‿‿ ‿‿‿ ‿
2 1 =
Gk grrir rgkmt`vi gjtrg meoms sikuc`ijgs gs> | | 5 6.667. emtr`cgs gs mpgjms> |
3.= 7 0
?8 5 ?4.4 =1.7
3.38 5 1.441 =.211
| 5 3.?08333, e`gjtrms qug gk grrir rgkmt`vi gjtrg
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Gs iov`i qug gk s`stgem bm cmeo`mdi dríst`cmegjtg su sikuc`ój pir uj cmeo`i pgqugýi gj uj ciga`c`gjtg, ki cumk sg pugdg ciepriomr cij gk jþegri dg cijd`c`ój>
‑ ‑ ‑ ∟=‑ ( )5
5 ?6(81)
( ) 5 =7?6
Gk jþegri dg cijd`c`ój gs eucbi emyir qug =, pir ki qug gk s`stgem gs emk cijd`c`ijmdi y ji gs rgciegjdmokg usmr su sikuc`ój. @jckusi pidgeis jitmr qug mk cmeo`mr kis ciga`c`gjtgs km sikuc`ój pmsó m sgr `kóh`cm dmdi qug ji sg pugdgj vgjdgr cmjt`dmdgs jghmt`vms dg emtgr`mkgs. Gngrc`c`i rgsugkti =?>Itri priokgem dg cijd`c`ijme`gjti dg s`stgems. Gjtrg itris iongtis sg trmjspirtmrij rgar`hgrmdirms y cic`jms gj uj cijtm`jgr. Cmdm cic`jm pgsm ujm tijgkmdm y cmdm rgar`hgrmdirm dis tijgkmdms, pir itri kmdi ujm cic`jm icupm uj gspmc`i dg =.67 e 3 y 3 cmdm rgar`hgrmdirm ? e . Gj titmk gjtrg cic`jms y rgar`hgrmdirms sg rgh`stró uj pgsi dg =6 tijgkmdms y 3 icupmrij uj gspmc`i nujtms dg =6.4 e . Sg dgsgm cijicgr cumjtms cic`jms y rgar`hgrmdirms sg trmjspirtó gj gk cijtm`jgr. m) Vkmjtgmr gstg priokgem ciei gk dg uj s`stgem dg gcumc`ijgs y rgsikvgrki cij uj eàtidi d`rgcti (Hmuss) , usmr mr`teàt`cm dg 4 dåh`tis. o) Gk gjcmrhmdi dg trmjspirtg sg gqu`vicó y gj rgmk`dmd cmdm cic`jm icupm uj gspmc`i dg =.= e 3 . Gjcugjtrg jugvmegjtg km sikuc`ój. c) Gk s`stgem dg gcumc`ijgs usmdi, gs o`gj i emk cijd`c`ijmdi9
‿‿
m) Vkmjtgmeis gk priokgem ciei uj s`stgem dg gcumc`ijgs dmdi pir> = =.67
Gk e`sei t`gjg pir sikuc`ój>
? ?
5
5
o) Vkmjtgmeis jugvmegjtg gk priokgem>
= =.=
Gk e`sei t`gjg pir sikuc`ój>
? ?
2 =
5
5
=6 =6.4
=6 =6.4
4 3
c) Vr`egri jgcgs`tmeis km `jvgrsm dg km emtr`z dg ciga`c`gjtgs, gstm `jvgrsm gs>
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
‑ ‑ ‑ ∟=‑ ?6 =6.7
Nuk`i _umji K`em
?6 =6
A`jmkegjtg gk jþegri dg cijd`c`ój dg km emtr`z dg ciga`c`gjtgs sgrí> ( )5
5 3.67(46)
( ) 5 =??
S`gjdi gk jþegri dg cijd`c`ój eucbi emyir qug =, cijcku`eis qug gk s`stgem gs euy emk cijd`c`ijmdi.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
"Ji gs gk cijic`e`gjti, s`ji gk mcti dg mprgjd`zmng< y ji km pisgs`ój, s`ji gk mcti dg kkghmr m gkkm, ki qug cijcgdg gk emyir d`sarutg". Cmrk Ar`gdr`cb Hmuss
Gj gk cmp`tuki mjtgr`ir trmtmeis gk tgem dg uj s`stgem dg gcumc`ijgs k`jgmkgs, pgri ji s`geprg jis gjargjtmrgeis m gstg t`pi dg s`stgems. Cijs`dgrg gk s`hu`gjtg s`stgem>
? ?cis( ?cis( ) + + + kj( kj(
54 )5=
Gk s`stgem mjtgs cijs`dgrmdi ji rgprgsgjtm uj gngepki gj gk cumk pidmeis mpk`cmr kis eàtidis dgk cmpåtuki mjtgr`ir ym qug pmrm meois rgqugråmeis dgspgnmr ujm vmr`mokg gj cmdm a`km pmrm mremr gk mkhir`tei rgspgct`vi. Km ji k`jgmk`dmd dg kms gcumc`ijgs prgsgjtmdms jis cijducg m uj jugvi cmpåtuki, gk dg s`stgems dg gcumc`ijgs ji k`jgmkgs. Gj gstg cmpåtuki siki prgsgjtmrgeis uj eàtidi, gk dg Jgwtij, dmdi su amc`k`dmd dg mpk`cmc`ój y ougjm cijvgrhgjc`m. Ciei gs dg cistueorg ji priaujd`zmeis gj kis tgirgems j` gj kms dgducc`ijgs dg mkhir`teis pgri pidgeis bmcgr ujm pgqugým mjmkihåm pmrm gjtgjdgr mkhi egnir gk eàtidi mjtgs egjc`ijmdi.
+=+=
_gcirdgeis gk mkhir`tei dgk eàtidi dg Jgwtij pmrm ujm sikm gcumc`ój ji k`jgmk, gk e`sei grm> 5
(
)
‿ (
)
Vmrm trmtmr dg crgmr ujm mjmkihåm pr`egri rggepkmzmeis kis vmkirgs mprix`emdis xj+= y xj ym ji pir vmkirgs s`ji pir vgctirgs mprix`emdis dmdi qug ouscmeis sikuc`ój m uj s`stgem dg gcumc`ijgs. M cijt`jumc`ój jis a`nmeis dgk rgstmjdi gj gk mkhir`tei dg ujm sikm gcumc`ój. Gj gk e`sei tgjgeis ujm d`v`s`ój, ujm ipgrmc`ój ji dga`j`dm gj gk mkhgorm dg emtr`cgs. Gstm d`v`s`ój jis kkgvm m `jhrgsmr km `dgm eís cgrcmjm dg km e`sem gj gk mkhgorm dg emtr`cgs, km dg emtr`z `jvgrsm. Mbirm jis qugdm rgmk`zmr mkhujm mjmkihåm m km aujc`ój a ym ym qug gstm `jakuyg tmeo`àj gj jugstrm emtr`z `jvgrsm. Dg bgcbi s` gk kgctir gs iosgrvmdir, siki d`n`eis qug `jhrgsmråmeis `jhrgsmråmeis ujm emtr`z `jvgrsm s`j geomrhi jujcm d`n`eis km `jvgrsm dg qu`gj. S`j eís prgíeoukis, prgsgjtmeis km mjmkihåm m km cumk bmcgeis rgagrgjc`m, supijhmeis uj s`stgem dg km airem>
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
= ?
Dijdg ( ,
,…,
=? == ??⋪ = ? ( (
, ,
,…, ,…,
) )
(
,
,…,
)
) sij jugstrms `jcihj`tms, dga`j`eis mk vgctir
⋪ ( )5
( (
, ,
,…, ,…,
) )
(
,
,…,
)
ciei gk
xj), y ciei rggepkmzmjtg dg a' ( x xj) m km emtr`z dg dgr`vmdms dg kms aujc`ijgs a =, rggepkmzmjtg dgk tàre`ji a ( x a ? ,… a j qug ji gs eís qug km Nmcio`mjm dmdm pir>
== ⋩ = ⎕⋪= ⋩⋽ ⋪⎦ ∟= ∟= ∟= ∟= ++ ( )5
Dg gstm airem jugstri mkhir`tei pmrm uj s`stgem ji k`jgmk tiem km airem> 5
+
( ) 5 ( ), km emtr`z N(R) y gk vgctir A(R) sg gvmkþmj gj kms d`vgrsms mprix`emc`ijgs Dijdg dg cmdm `tgrmc`ój pir ki qug gj rgmk`dmd sij aujc`ijgs dg vmr`ms vmr`mokgs.
Gngrc`c`i rgsugkti =3>Ciga`c`gjtgs =3>Ciga`c`gjtgs dg ujm aujc`ój.
TGEM =.- Dgtgre`jg kis ciga`c`gjtgs m, o pmrm qug km aujc`ój ( ) 5 ( pujtis (=, 3), (?, 4).
+ )
+ `jckuym kis
m) Dgscr`om gk eàtidi jueàr`ci jueàr`ci `tgrmt`vi qug ut`k`zmrm ut`k`zmrm y tmeo`àj gk mkhir`tei mkhir`tei rgspgct`vi o) Gk`nm ciei vmkir `j`c`mk `j`c`mk gk pujti (6, =) y rgmk`cg trgs `tgrmc`ijgs `tgrmc`ijgs c) Gjcugjtrg km jirem dgk grrir smo`gjdi smo`gjdi qug km rgspugstm gs (6.=78288…, (6.=78288…, 6.2147??…) 6.2147??…)
Dgsmrrikki> m) Sg usmrí gk eàtidi dg Jgwtij pmrm uj s`stgem dg gcumc`ijgs ji k`jgmkgs, gk cumk t`gjg pir mkhir`tei ki
s`hu`gjtg>
∟= ∟=∟=∟=∟= = ++ 5
Dijdg N gs gs km emtr`z Nmcio`mjm rgsuktmjtg dg kms dgr`vmdms pmrc`mkgs dg kms dis aujc`ijgs iotgj`dms. Gk vgctir A gs gs gk vgctir dg aujc`ijgs a = y a ?. Tmjti gk vgctir A , ciei km emtr`z N sg sg gvmkþmj gj kis pmrgs irdgjmdis (m ,` o`) pmrm cmdm `tgrmc`ój. 5( + )
+
356
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
?= + ?+?+ ++ = + ++ ? ?+?+ ?+?+ ? ?+ ?+?+ = = ? ? ?+?++ +?+?+ + ?++ +?+?++ ?+?+ 6 ∟=∟=6 6 = 6 ∟=∟= 66 = = ∟=∟== = 5 (? + )
+
456
+( + )
5
+=
+( + )
5
+ ?(? + )
5?
+=
+ (? + )
5
( , )5
Vmrm cmdm `tgrmc`ój sg usmj km emtr`z Nmcio`mjm y vgctir dg aujc`ijgs m cijt`jumc`ój prgsgjtmdis> prgsgjtmdis> ( , )5
+( + ) + ?(? + )
?
( + ) (? + )
( , )5
+( + ) + (? + )
+= +=
+ +
3 4
o) Vr`egrm `tgrmc`ój>
5
5
6 =
6.=2 6.=230 3030 308? 8?== ==72 72 6.46=8=3?1?737
6.=2 6.=230 3030 308? 8?== ==72 72 6.?=888374?673
6.?2=8=2=8 =.?2=8=2=8
5
5
5
6.=230308?==72 6.=230308?==72 6.2346471=030=
5
6.=230308?==72 6.=230308?==72 6.2346471=030=
Sghujdm `tgrmc`ój>
5
6.=2 6.=286 8663 6318 188= 8=70 70 6.300738828384 5
6.=4 6.=4?7 ?728 2877 77?= ?=47 47 6.=12==80278==
6.66=3?073 6.66=3?073 6.=2?=1622
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
? ? = ∟=∟=== ? ∟=∟=? ? 3 ? ∟=∟= ∟7 ?? 3 3 ‑ 3 ‑
Nuk`i _umji K`em
5
Tgrcgrm `tgrmc`ój>
5
5
6.=72?=4484=2= 6.=72?=4484=2= 6.214=3204?=32
5
6.=72?=4484=2= 6.=72?=4484=2= 6.214=3204?=32
6.=2 6.=248 48?0 ?047 477? 7?0? 0? 6.3074871?36?2
6.=4 6.=41? 1?77 7746 46?3 ?323 23 6.=8??8?==42?8
8 .0244 44 Ø =6 5 8.02 6.66?4611? 5
5
6.=78288?4?743 6.=78288?4?743 6.2147?=070?00
c) Sg cmkcukm pr`egri gk vgctir d`agrgjc`m gjtrg kis vmkirgs dg km tgrcgrm `tgrmc`ój y dg km sikuc`ój gxmctm>
5
6.666666?4?743 6.66666664686=
Vir ki tmjti km jirem dgk grrir gs>
5 6.6666 6.666666? 66?4?74 4?743 3
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
"S` bg kihrmdi vgr eís kgnis, bm s`di pirqug bg suo`di m bieoris dg h`hmjtgs". @smmc Jgwtij
Gj cumjti m km gxpgr`egjtmc`ój sg rga`grg, tmjti gj c`gjc`ms ciei gj `jhgj`gråm eucbms vgcgs sg rgcikgctmj ujm sgr`g dg pujtis qug rgprgsgjtmj gk gstud`i dg mkhujm vmr`mokg qug a`nmeis ciei `jdgpgjd`gjtg y dg mkhþj pmríegtri qug crggeis dgpgjdg dg km e`sem. Gstms kgcturms dg vmkirgs jis pripirc`ijmj `jairemc`ój mcgrcm dgk ciepirtme`gjti dg km aujc`ój qug gxpk`cm km rgkmc`ój gjtrg gjtrg meoms vmr`mokgs s`j geomrhi geomrhi ji jis pgre`tg gstmokgcgr gstmokgcgr dg airem d`rgctm km gstructurm mkhgorm`cm dg gstm rgkmc`ój. Gx`stgj tàcj`cms pidgrisms kkmemdms `jtgrpikmc`ój dijdg pmrt`gjdi dg ujm sgr`g dg dmtis, gj hgjgrmk pmrgs irdgjmdis, trmtmeis dg oisqugnmr km aujc`ój dg dijdg kis e`seis priv`j`grij egd`mjtg pik`jie`is `jtgrpikmjtgs. Kis pik`jie`is `jtgrpikmjtgs eís usmdis sij kis dg Kmhrmjhg y kis kkmemdis trmzmdirgs cþo`cis, kis e`seis sg gstud`mrmj gj gk prgsgjtg cmp`tuki.
Gk iongt`vi ji siki dg km sgcc`ój s`ji dgk cmpåtuki ciei mjtgs aug egjc`ijmdi gs km mprix`emc`ój dgk ciepirtme`gjti dg mkhþj agjóegji gstud`mdi dijdg sg bmymj rgcikgctmdi pujtis dg km airem (x, y) qug gs gqu`vmkgjtg m (x, a(x)). Supijhm gj pr`jc`p`i qug bm rgcikgctmdi (x6 , y 6), (x= , y =),…, (xj , y j) pujtis, gs dgc`r gj titmk j+= pujtis.
6 6 = = ? ? ≦56
Sg dga`jg ciei pik`jie`i dg `jtgrpikmc`ój dg Kmhrmjhg mk pik`jie`i dg km airem> ( )5
(
)+
(
)+
(
) + …+
(
)
x6, xjU dijdg kis amctirgs a ( x x n) cirrgspijdgj m kms aujc`ijgs gvmkumdms Dg hrmdi j dga`j`di gj gk `jtgrvmki \ x gj cmdm uji dg kis vmkirgs gj gk gng x. Mdgeís kis amctirgs K sg cmkcukmj dg km s`hu`gjtg airem>
( )5
Gstis ciga`c`gjtgs K cuepkgj kms s`hu`gjtgs prip`gdmdgs> • •
( ) 5 6, ( ) 5 =, 5
A`jmkegjtg m kis pujtis (x6 , y6), (x= , y=),…, (xj , yj) sg kis dgjie`jm pujtis dg `jtgrpikmc`ój y m kis vmkirgs x6 , x= ,…, xj sg kis dgjie`jm jidis dg `jtgrpikmc`ój.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Gngrc`c`i rgsugkti =4> Wjm gcumc`ój iotgj`dm dgk pik`jie`i dg Kmhrmjhg. Iotgjgr cij km tmokm mdnujtm, ujm mprix`emc`ój m km sikuc`ój dg x-p(x)56 dijdg p(x) gs gk pik`jie`i dg Kmhrmjhg usmjdi kis pujtis dmdis. x
a(x)
6.3 6.4 6.7 6.1
6.8462=2 6.1863?6 6.16173= 6.7422=?
Sg cijstruyg gk pik`jie`i dg Kmhrmjhg dg hrmdi 3 cij kis dmtis pripirc`ijmdis pmrm kughi hgjgrmr `tgrmc`ijgs egd`mjtg mkhþj eàtidi gscih`di y bmkkmr km sikuc`ój pgd`dm.
6 6 = = ? ? 3 3 6 = ? 3 3 ?
_gcirdmeis qug gk pik`jie`i dg Kmhrmjhg tgjdrí km airem> ( )5
(
)+
(
)+
(
)+
(
)
Vir ki qug cmkcukmeis kis vmkirgs dg kis ciga`c`gjtgs K. 5
5
5
5
(
(6.3 (
(6.4 (
(6.7 (
(6.1
6.4)(
6.4)(6.3 6.3)(
6.3)(6.4 6.3)(
6.3)(6.7 6.3)(
6.3)(6.1
6.7)(
6.7)(6.3
6.7)(
6.7)(6.4
6.4)(
6.4)(6.7
6.4)(
6.4)(6.1
6.1)
6.1)
6.1)
6.1)
6.1)
6.1)
6.7)
6.7)
_gsikv`gjdi kis priductis gjtrg kis Ka(x) pmrm cmdm x sg iot`gjg> ( )5
6.030233
+ =.413?7
=.32=7=11 + =.642071
Hría`cm dg p(x)-x.
Dgogeis rgcirdmr qug ki sik`c`tmdi aug x5p(x), sg pugdg rgsikvgr gk priokgem egd`mjtg gk eàtidi dg Jgwtij, pmrm ki cumk sg usm gk mkhir`tei mpk`cíjdiki m A(x)5p(x)-x56>
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
3 ? += ? 5
6.030233
+ =.413?7
?.2=0400
=.32=7=11
+ ?.0?17
+ =.642071
=.32=7=11
=
Cij ujm ougjm mprix`emc`ój `j`c`mk sg cijckuyg qug gk pujti sik`c`tmdi gs x56.7171==.
Gngrc`c`i rgsugkti =7>
( )+ , Km aujc`ój dg vmr`mokg rgmk ( ) 5 , sgrí mprix`emdm cij gk pik`jie`i dg sghujdi hrmdi V(x) qug `jckuyg m kis trgs pujtis a(6), a(φ/?), a(φ) . Gjcugjtrg km emhj`tud dgk emyir grrir G(x)5a(x)-p(x), qug sg priduc`råm mk usmr gstm mprix`emc`ój. _gsugkvm km gcumc`ój ji k`jgmk rgsuktmjtg cij km aóreukm dg Jgwtij cij uj grrir eíx`ei dg 6.666=.
Vr`egri cijstru`eis ujm tmokm dg vmkirgs cij kis pujtis mjtgs egjc`ijmdis> R
a(x)
6 =.7868013?8 3.=4=70?174
? = -??.=4610?1
Kughi pricgdgeis m cmkcukmr gk pik`jie`i dg `jtgrpikmc`ój, pmrm gsti pr`egri oisqugnmeis gk pik`jie`i. Gk e`sei tgjdrí km airem>
6 6 = = ? ? 6 = ? 6 6 ?= = ? ? ? ( )5
(
)+
(
)+
(
)
(
)
M cijt`jumc`ój cmkcukmeis kis ciga`c`gjtgs K>
5
(
/?)(
)
(6
/?)(6
)
(
5
6
?
5
)
6)(
?
(
6)(
/?)
(
6)(
/?)
_ggepkmzmeis gj gk oisqugni dgk pik`jie`i iotgj`gjdi> ( )5
( )5
(
)+
4.42114?31=2
(
)+
+ 1.4=602=710?? + ?
Km aujc`ój grrir sgrí gjtijcgs> ( )5 ( )
( )5
cis( ) + = + 4.42114?31=2
1.4=602=710??
?
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Mpk`cmeis km aireukm dg Jgwtij pmrm km dgr`vmdm G'(x) dmdi qug jgcgs`tmeis vgr dijdg km e`sem vmkg cgri, ki qug jis `jd`cmrí gk eíx`ei grrir gj gk `jtgrvmki.
+= ( )5
cis( )
( ) + ?(4.42114?31=2)
1.4=602=710?? 5 6
Ms`e`sei jgcgs`tmrgeis G'' , pmrm ciepkgtmr gk mkhir`tei dg Jgwtij> ( )5
cis( )
( )
( )
cis( ) + ?(4.42114?31=2)
Gk mkhir`tei tiemrí km airem> 5
cis(
)
(
?
) + ?(4.42114?31=2) (
1.4=602=710??
) + ?(4.42114?31=2)
Tiemrgeis ciei mprix`emc`ój `j`c`mk m x656.0 (`jv`tmeis mk kgctir m vgr`a`cmr gk pirquà dg km gkgcc`ój dg gstm mprix`emc`ój `j`c`mk hrma`cmjdi km aujc`ój G(x)). Km tmokm dg vmkirgs cij gk mkhir`tei y km mprix`emc`ój `j`c`mk gs km s`hu`gjtg> j
x j
6 6.0 = 6.83?47? ? 6.10?88637 3 6.12440=71 4 6.12?2==?7 7 6.12?48?=4
x j+=
Tik
6.83?47? 6.10?88637 6.12440=71 6.12?2==?7 6.12?48?=4 6.12?46382
6.=18742 6.63012=17 6.662?8280 6.66=1263= 6.666330== 1.231G-67
Vir ki qug gk eíx`ei grrir sg dm gj x56.12?46382, gk e`sei t`gjg uj vmkir dg =.87 mprix`emdmegjtg.
Kis trmzmdirgs cþo`cis sij ujm bgrrme`gjtm mkhi eís gxmctm qug kis pik`jie`is dg Kmhrmjhg gj cumjti m km mprix`emc`ój pik`jóe`cm sg rga`grg. Gstis trmzmdirgs sij aujc`ijgs dga`j`dms pir trmeis dgjtri dgk die`j`i x6 , x= ,…, xj , gs dgc`r sg tgjdríj gj hgjgrmk pmrm j+= pujtis j trmzmdirgs tiemjdi kis e`seis km s`hu`gjtg airem>
6 6 = ∟== =⋪∟= ? ( )5
( ) < ( ) < ( ) <
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
? 3 ∆ ∂ += +=+= +∟= =6 =6 =? 6= ⋪ ⋪∟= ⋪∟= ⋪ ∟= Cmdm pik`jie`i S n ( n56, =, ?,…, j-=) t`gjg km airem ( ) 5
+
+
+
dijdg kis ciga`c`gjtgs m, o, c, d sg bmkkmj egd`mjtg gxprgs`ijgs qug `jd`cmrgeis gj orgvg.
Gj hgjgrmk pir dga`j`c`ój uj trmzmdir cuo`ci cuepkg cij kms s`hu`gjtgs cijd`c`ijgs> •
( ) gs uj pik`jie`i dg hrmdi 3
5 6, =, … ,
=
5 6, =, … , Dijdg a gs km aujc`ój cijic`dm dg dijdg priv`gjgj kis pujtis m
5
•
,
pmrt`r dg kis cumkgs cijstru`eis gk trmzmdir. 5
•
5 6, =, … ,
•
‿
5 ‿
•
‿‿
5 ‿‿
•
‿‿
(
)5
5 6, =, … ,
‿‿
5 6, =, … ,
(
)5
(siki mpk`cm pmrm trmzmdir cuo`ci jmturmk)
Mbirm gjtrmrgeis m km dgtgre`jmc`ój dg kis ciga`c`gjtgs mjtgs egjc`ijmdis, pmrm gsti `jtriduc`rgeis uj pgqugýi airemti dg tmokm m sgr kkgjmdi> ( (
6 =
=
) )
(
)
(
)
Cij kis vmkirgs iotgj`dis prgv`megjtg pidgeis cmkcukmr kis rgstmjtgs ciga`c`gjtgs, cijtrmr`i mk irdgj gj qug mpmrgcgj gj gk pik`jie`i mk eiegjti dg bmkkmrkis pr`egri ki bmcgeis cij kis ciga`c`gjtgs C , kughi cij o, d . Cij kis j+= pujtis bmkkmrgeis tmeo`àj j+= ciga`c`gjtgs C , pmrm ki cumk usmrgeis gk s`hu`gjtg s`stgem dg gcumc`ijgs>
Dgk s`stgem iotgjgeis kis vmkirgs dg kis ciga`c`gjtgs C , pmrm kughi bmkkmr kis rgstmjtgs cij kms s`hu`gjtgs aireukms>
+=ℊ 5
3
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
ℊ += ℊ += 5
=
(?
3
+
)
Cij gsti tgjgeis tidi k`sti pmrm hgjgrmr kis trmzmdirgs cþo`cis gj cmdm trmei.
Gngrc`c`i rgsugkti =1> Wsi dg kms cijd`c`ijgs dgk trmzmdir cuo`ci jmturmk.
3 6 ? 3 = ? ? 3
Sgm gk trmzmdir cuo`ci jmturmk S(x) dmdi pir>
5 6.8?= 6.470 , 5 6.363 + ( 6.7) 6.122( 6.7) + ( 6.7) , 5 6.312 + ( =) 6.==?( =) + ( =) ,
( )5
6 6.7 =
6.7 = ?
Bmkkmr o, d, O, D.
Sikuc`ój> Sg usmríj kms s`hu`gjtgs prip`gdmdgs dgk trmzmdir pmrm cijstru`r gcumc`ijgs y bmkkmr kis vmkirgs pgd`dis>
6 = =? = ? ? 6 ? ? = ? ? =? ? ∴ ‿
(6.7) 5
‿
(6.7)
(?)
‿
(=) 5
‿
(=)
(3)
‿‿
(=) 5
‿‿
(=)
(=)
(4)
‿‿
(?) 5 6
Kms dgr`vmdms mjtgs egjc`ijmdms sij kms s`hu`gjtgs> ‿
‿
( )5
‿
( ) 5 6.8?= ?(6.122)(
( )5 ‿‿
?(6.==?)(
( )5
‿‿
( )5
3(6.470)
6.7) + 3 (
6.7)
=) + 3 (
?(6.122) + 1 (
?(6.==?) + 1 (
=)
6.7) =)
Wsmjdi km `humkdmd sg iot`gjg> 6.8?=
3(6.470)(6.7) 5
?(6.122)(6.7
5 6.38187
Dg km e`sem airem sg usm km `humkdmd (?) pmrm iotgjgr>
6.7) + 3 (6.7
6.7)
?
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
? ? ∴ ∴ ?(6.122)(=
6.7) + 3 (= (7)
6.7) 5
5
?(6.==?)(=
=) + 3 (=
=)
6.122 + 6.87
Mjíkihmegjtg sg usm km `humkdmd (3)>
?(6.122) + 1 (=
6.7) 5
?(6.==?) + 1 (=
=)
5 6.324
A`jmkegjtg cij km `humkdmd (4)>
?(6.==?) + 1 (?
=) 5 6
5 6.6383…
A`jmkegjtg kis vmkirgs iotgj`dis sg rggepkmzmj gj km gcumc`ój (7) pmrm iotgjgr gk þkt`ei vmkir sik`c`tmdi, kis vmkirgs a`jmkgs sij kis s`hu`gjtgs> o56.38187 d 56.324 O5-6.6?3?7 D56.6383
Gngrc`c`i rgsugkti =8> Mprix`emc`ój dg ujm aujc`ój usmjdi trmzmdirgs cþo`cis Cijstru`r gk trmzmdir cuo`ci jmturmk dg km aujc`ój a(x)5sgj(x) usmjdi kis jidis 6, φ/=1, φ/2 y φ/4.
@j`c`mkegjtg cijstru`eis jugstrm tmokm prgk`e`jmr pmrm pidgr hgjgrmr kughi gk s`stgem dg gcumc`ijgs y bmkkmr kis vmkirgs dg kis ciga`c`gjtgs c, o, d . n
x n
6 = ? 3
6 6.=0134074= 6.30?10062? 6.827302=13
M pmrt`r dg gstm tmokm hgjgrmeis gk s`stgem>
m n5a(x n )
b n
6 6.=0134074= 6.=076063?? 6.=0134074= 6.32?12343? 6.30?10062? 6.868=6182=
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
=6? 3 +=ℊ ℊ += ℊ +=
Gk cumk t`gjg pir sikuc`ój>
6 6.6113=8231 6.3=2=?=8 6
5
Dg mquå pidgeis bmkkmr kis vmkirgs dg kis ciga`c`gjtgs d, o. Kis e`seis sg bmkkmj cij kms aóreukms> 5
5
3
=
3
(?
+
)
Kis vmkirgs dg gstis ciga`c`gjtgs sij kis s`hu`gjtgs> n
o n
d n
6 = ? 3
6.0080?834 6.02406728 6.0604?=34 -
-6.==?727 -6.4?84877 6.?8663 -
A`jmkegjtg pidgeis gscr`o`r gk trmzmdir cuo`ci dga`j`di pir trmeis>
= ? ⎠ ( )5
6
? ? 3 ?
( ) 5 6 + 6.0080?8343(
6) + 6(
( ) 5 6.=076063??6=1 + 6.024067211100 ( ) 5 6.32?1234 + 6.0604?=346?13
=1
2
6)
6.==?727(
6.6113=8231
6.3=2=?=8
2
=1
6) , 6
3 3
6.4?84877
+ 6.?8663662726?
=1
,
=1
2
,
=1
2
2
4
Gk trmzmdir cuo`ci sungti cuepkg cij kms e`sems cijd`c`ijgs oís`cms dgk trmzmdir cuo`ci jmturmk, pgri mdgeís cuepkg cij km s`hu`gjtg> •
6 6 6 ∟= ‿ (
) 5 ‿ (
),
‿
(
) 5 ‿ (
)
Cij gstm cijd`c`ój md`c`ijmk pidgeis `jagr`r qug dmdi uj priokgem dgogeis cijicgr kms dgr`vmdms dg km aujc`ój a gj kis gxtrgeis dg mprix`emc`ój. Mdgeís gk s`stgem dg gcumc`ijgs mjtgs egjc`ijmdi pmrm bmkkmr kis vmkirgs dg C gj gk trmzmdir jmturmk t`gjg uj k`hgri cmeo`i, kms gcumc`ijgs pr`egrm y þkt`em ym ji sij kms e`sems. Gstms sgríj rggepkmzmdms pir, rgspgct`vmegjtg>
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
ℊ66 ℊ6= ℊ6 = 6 6 6 6 ℊ∟=∟= ℊ∟= ℊ∟= ∟= ?
+
3
= 3
+
=
5
3
?
=
(
=
5
3
)
<
(
‿
5
)+
(
)
<
5
‿
(
)
Dg mquå pidgeis jitmr pirqug rgqugråmeis kis vmkirgs dg a' gj kis gxtrgeis, mdgeís cmog rgcmkcmr qug gj gk trmzmdir sungtis kis vmkirgs dgk pr`egri y þkt`ei ciga`c`gjtg o ym ji sg cmkcukmj cij km aireukm s`ji qug pmsmj m sgr d`rgctmegjtg kis vmkirgs prgv`megjtg `jd`cmdis. Kis rgstmjtgs o sg cmkcukmj tmk y ciei sg b`zi mjtgr`iregjtg.
Gngrc`c`i rgsugkti =2> Ciepmrmc`ój dgk trmzmdir jmturmk y a`ni _gsugkvm jugvmegjtg gk gngepki =8, gstm vgz cijs`dgrmjdi gk trmzmdir sungti.
Tgjgeis kis e`seis jidis, ki qug cmeo`mrí sij kis vmkirgs dgk pr`egr y þkt`ei ciga`c`gjtg o, mdgeís dg km gcumc`ój pr`egrm y þkt`em dgk s`stgem m rgsikvgr.
Km tmokm prgk`e`jmr gs km e`sem, rgcirdmeis qug gstm grm> n
x n
6 = ? 3
6 6.=0134074= 6.30?10062? 6.827302=13
m n5a(x n )
b n
6 6.=0134074= 6.=076063?? 6.=0134074= 6.32?12343? 6.30?10062? 6.868=6182=
Mbirm mjtgs dg vikvgr m pkmjtgmr uj s`stgem pmrm kis ciga`c`gjtgs o, pr`egri pkmjtgmeis kms gcumc`ijgs qug `ríj gj km pr`egrm y þkt`em a`km dg km emtr`z. _gcirdmeis qug o65 a' (6)5cis(6), o35 a' (φ/4)5cis(φ/4).
6 = ? 3
Kms jugvms pr`egrm y þkt`em gcumc`ój sgríj> ?
3 =1
=
3 2
+
?
3 2
5
+
=
3 =1
=
/2
5
=
/=1
(6.868=6182
(6
6.=076063?)
=
6.32?12343) + 6.868=6182==21777
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Kms dgeís a`kms dgk s`stgem qugdmj `dàjt`cms y gk s`stgem rgsuktmjtg gs>
=6? 3
Gsti jis kkgvm m kis s`hu`gjtgs vmkirgs pmrm kis ciga`c`gjtgs c>
∟4∟?
3.=8836 Ø =6 0.8376?4 Ø =6 6.=0318431?228 6.3782?72=6868
5
Dg mquå pricgdgeis m cmkcukmr kis ciga`c`gjtgs o, d rgcirdmjdi qug gk pr`egr y þkt`ei o ym gstíj dmdis pir kms aireukms cijic`dms> n
o n
d n
6 = ? 3
= 6.0262??030 6.0?3126303 6.868=61382
-6.=148?2 -6.=137?40 -6.=3033168 -
A`jmkegjtg pidgeis gscr`o`r gk trmzmdir cþo`ci>
6 ? ? 3 3 ? 3 = ? ⎠ ( ) 5 6 + =(
( )5
6)
6.6663=88?0(
( ) 5 6.=076063??6=1 + 6.0262??0?08 ( ) 5 6.32?1234+ 6.0?3126301
=1
2
6)
6.=148?8763(
6.608376=261
6.=031843771
=1
2
6) , 6
=1
6.=137?4002
6.=30331607
=1
2
,
,
2
=1
2
4
Mså ciei gx`stg `jtgrpikmc`ój gj ujm sikm vmr`mokg, sg pugdg bmcgr ujm gxtgjs`ój mk cmsi dg dis. Gj gstm pgqugým sgcc`ój ji tgjgeis pir pripós`ti bmkkmr uj pik`jie`i dg `jtgrpikmc`ój prip`megjtg gj dis vmr`mokgs s`ji egd`mjtg km a`nmc`ój dg ujm dg kms e`sems `jtgrpikmr ujm vgz pmrm dgsbmcgrjis dg km pr`egrm y kughi `jtgrpikmr jugvmegjtg pmrm kkghmr m gvmkumr gk vmkir dgsgmdi a(x, y).
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Gk pricgsi gs tmk y ciei sg d`ni gj gk pírrmai mjtgr`ir, dmdm ujm emtr`z dg dmtis pmrgs irdgjmdis ( x, y) y km aujc`ój a gvmkumdm gj gsis pmrgs pr`egri a`nmeis ujm dg kms dis vmr`mokgs pmrm `jtgrpikmr ujm vgz. Mk rgmk`zmr km pr`egr `jtgrpikmc`ój rgduc`rgeis km emtr`z m uj mrrghki ym sgm dg dis a`kms i dis cikuejms (dgpgjd`gjdi dg quà vmr`mokg bmymeis a`nmdi `j`c`mkegjtg) pmrm kughi vikvgr m `jtgrpikmr y cmkcukmr mså gk vmkir dgsgmdi.
Gngrc`c`i rgsugkti =0> Mpk`cmc`ijgs dg km `jtgrpikmc`ój gj dis vmr`mokgs. Gk åjd`cg gjar`mdir dgk v`gjti @ gs ujm aujc`ój qug dgpgjdg dg dis amctirgs> km tgepgrmturm rgmk T y km vgkic`dmd dgk t`gepi v, gs dgc`r @5a(T,v). Km s`hu`gjtg tmokm rgh`strm kis vmkirgs dg @ rgcih`dis gj c`grti eiegjti pir uj `jvgst`hmdir gj kis pírmeis dgk Citipmx`. Vir gngepki, cumjdi km tgepgrmturm rgmk gs dg 7 hrmdis Cgks`us y gk v`gjti dg ?6 le/birm, gk åjd`cg @5a(7, ?6)5=, qug qu`grg dgc`r qug km tgepgrmturm qug sg s`gjtg gj gstms cijd`c`ijgs gs dg = hrmdi, mujqug ji sgm km rgmk.
T/ Q
7
=6
=7
?6
7
4 -? -2
? -3 -=6
? -4 -==
= -7 -=?
6 -7
Wsmjdi `jtgrpikmc`ój pik`jie`mk gst`emr km tgepgrmturm qug sgjt`rí ujm pgrsijm s`tumdi gj uj kuhmr gj km qug km tgepgrmturm rgmk gs dg dis hrmdis y km vgkic`dmd dgk v`gjti gs dg ?7 le/b.
Ciegjzmeis gk dgsmrrikki rgcirdmjdi ki mjtgs `jd`cmdi, pr`egri dgogeis a`nmr ujm vmr`mokg. Gj gstg cmsi a`nmrgeis m km vmr`mokg tgepgrmturm rgmk. @jtgrpikmeis usmjdi gk pik`jie`i dg Kmhrmjhg pr`egri rgspgcti m km vmr`mokg T, dmdi qug cijicgeis gk pujti dijdg vmeis m gagctumr km gst`emc`ój pidgeis gscr`o`r>
? 66 == ?? 6 = ?
Vmrm T 5? (pik`jie`i dg irdgj ? pir bmogr 3 dmtis gj km d`rgcc`ój T)> (3) 5
+
+
Ciei d`n`eis jis `jtgrgsm km airem mkhgorm`cm dgk pik`jie`i pir ki qug gj kis ciga`c`gjtgs K gvmkumeis d`rgctmegjtg gk vmkir m `jtgrpikmr. 5
5
5
(?
6)(?
( 7))
(7
6)(7
( 7))
(?
7)(?
( 7))
(6
7)(6
( 7))
(?
( 7
6)(?
6)( 7
7)
7)
5 6.?2
5 6.24
5
6.=?
Sg mpk`cm gstg pik`jie`i pmrm cmdm cikuejm, gs dgc`r pmrm cmdm v. Tgj`gjdi>
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
v57
? 66 == ?? ? 66 == ?? ? 66 == ?? ? 66 == ?? (3) 5
v5=6
v5=7
v5?6
+
+
5 4(6.?2)
(3) 5
+
+
5 ?(6.?2)
(3) 5
+
+
5 ?(6.?2)
(3) 5
+
+
5 =(6.?2)
?(6.24)
3(6.24) + =6( 6.=?) 5
4(6.24)
==( 6.=?) 5
7(6.24)
=?( 6.=?) 5
Cij kis vmkirgs iotgj`dis pidgeis gscr`o`r ujm jugvm tmokm> 7 6.4
Q a(?, v)
A`jmkegjtg vikvgeis m `jtgrpikmr>
‿6 ‿= ‿? ‿3 5
(?7
5
(?,?7) 5
=6)(?7 =6)(7
=7)(?7
=7)(7
?6)
?6)
(?7
7)(?7
=7)(?7
?6)
(=6
7)(=6
=7)(=6
?6)
7)(?7
=6)(?7
?6)
(=7
7)(=7
=6)(=7
?6)
7)(?7
=6)(?7
=7)
(?6
7)(?6
=6)(?6
=7)
( ,
)5
5
.
.
=
54
5
(?7
=(6.4) + 4( 3.=1)
.
?6 -?.42
(?7
5
Vir ki tmjti>
=7 -=.42
(7
5
=6 -3.=1
2( 6.=?) 5 .
1
54
1( =.42) + 4( ?.42)
.
Vir ki tmjti km tgepgrmturm qug sg s`gjtg m km tgepgrmturm rgmk y v`gjti sik`c`tmdis gs dg -=4.62 km cumk rgsuktm rmzijmokg pmrm kis dmtis usmdis.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
"Bmcgr prgd`cc`ijgs gs euy d`aåc`k, gspgc`mkegjtg cumjdi sg trmtm dgk auturi". J`gks Oibr
Gk prgsgjtg cmp`tuki cijst`tuyg tmk vgz uji dg kis eís sgjc`kkis gj cumjti m mkhir`teis sg rga`grg, s`j geomrhi ji dgog dg ikv`dmrsg km `epirtmjc`m dgk e`sei. Vir gngepki supijhm qug bm tiemdi egd`dms dg km pis`c`ój dg uj cugrpi gj gk gspmc`i gj dgtgre`jmdis eiegjtis, mdgeís cijs`dgrg qug dgscijicg km aujc`ój qug dgscr`og gk eiv`e`gjti dg d`cbi cugrpi. S` ustgd dgsgm cmkcukmr km vgkic`dmd dg d`cbi cugrpi gj kis t`gepis dijdg rgmk`zi kms egd`c`ijgs dg km pis`c`ój, pir dga`j`c`ój tgjdråm qug cmkcukmr km dgr`vmdm dg km pis`c`ój rgspgcti mk t`gepi. S`j geomrhi sg gjargjtmrí mk priokgem dg dgscijicgr km aujc`ój qug dgscr`om gk ciepirtme`gjti. Vmrm sikuc`ijmr gstg priokgem sg `jtriducg km d`agrgjc`mc`ój jueàr`cm qug ji sij eís qug c`grtms aireukms dg aíc`k dgducc`ój qug s`rvgj pmrm mprix`emr kis vmkirgs dg kms dgr`vmdms gj c`grtis t`gepis cijic`gjdi uj cijnujti dg dmtis prgv`megjtg rgh`strmdi.
Km pr`egrm dgr`vmdm dg ujm aujc`ój pugdg sgr cmkcukmdm egd`mjtg trgs pr`jc`pmkgs aóreukms, km dg ? y 3 pujtis y km cgjtrmdm (qug gj rgmk`dmd gs ujm aóreukm dg 3 pujtis gspgcåa`cmegjtg). Dg kms aireukms mjtgs egjc`ijmdms km qug pisgg eís gxmct`tud gs km cgjtrmdm, s`j geomrhi ji s`geprg pugdg sgr usmdm ym qug rgqu`grg mk egjis uj pujti mjtgs y uji dgspuàs dg dijdg sg gagctþmj kis cíkcukis. Gs pir gsti qug gj gk pr`egr y gk þkt`ei pujti qug sg tgjhm sg mcistueorm m usmr km aireukm dg trgs pujtis. Gj tidis kis cmsis m gstud`mr mprix`emeis km dgr`vmdm gj uj pujti x6. Gstms aóreukms sij kms s`hu`gjtgs>
6 =+∟ ? 6 = ? 6 6 ℊ 6 ℊ 3 6 ? 6 ℊ 6 ℊ 1
Dis pujtis>
(
)5
Trgs pujtis>
(
)5
Cgjtrmdm>
(
)5
(
)
(
\ 3 (
\ (
+ )
)
( )
)+4 (
+ )
(
)U
(
+? )+
( )
( )
Cmog rgcmkcmr qug s` b:6 sg d`cg qug km aireukm gs prihrgs`vm y s` b;6 sg d`cg qug km e`sem gs rghrgs`vm. Mdgeís gj cíkcukis príct`cis sg ie`tg gk tàre`ji dgk grrir qug mpmrgcg cij gk pmsi gkgvmdi m mkhujm pitgjc`m pir mkhujm dgr`vmdm dg a .
Gngrc`c`i rgsugkti ?6> Ciepmrmc`ój dg km aireukm dg dgr`vmdm rgspgcti m ujm dgr`vmdm gxmctm.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Sgm a(x)5kj(x) mprix`eg km dgr`vmdm gj gk `jtgrvmki \=, =.=U usmjdi uj pmsi dg 6.6= y ciepmrg cmdm rgsuktmdi cij gk vmkir dg km dgr`vmdm gxmctm gj gsg pujti.
Tgjgeis mjtgs qug jmdm km s`hu`gjtg tmokm dg vmkirgs> n
x n
a(x n )5kj(x n )
6 = ? 3 4 7 1 8 2 0 =6
= =.6= =.6? =.63 =.64 =.67 =.61 =.68 =.62 =.60 =.=
6 6.66007633= 6.6=026?1?8 6.6?077226? 6.630??68=3 6.642806=14 6.672?12062 6.618172142 6.68101=64= 6.621=88101 6.6073=6=2
Ciei sg `jd`ci gj kis gxtrgeis usmrgeis km aireukm dg trgs pujtis y gj kis pujtis `jtgr`irgs km cgjtrmdm pmrm tgjgr gk eåj`ei grrir.
≂ ≂ ≂ =
(=)
(=.=)
?(6.6=)
=
?( 6.6=)
\ 3(6) + 4(6.66007633=)
6.6=026?1?8U
\ 3(6.6073=6=2) + 4(6.621=88101)
Sg dm uj siki gngepki pmrm km cgjtrmdm> (=.67)
=
?(6.6=)
\6.672?12062
≂
6.00003427
6.68101=64=U
6.630??68=3U
≂
≂
6.06063027
6.07?46087
Vmrm ciepmrmr kis vmkirgs tgjgeis gj cugjtm qug km dgr`vmdm gxmctm gstí dmdm pir> ciepmrmeis kis vmkirgs gxmctis y mprix`emdis gj km s`hu`gjtg tmokm. Mdgems eistrmeis km d`stmjc`m gjtrg meois vmkirgs gj km cikuejm dg grrir.
=
( ) 5 , pir ki cumk
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
x n
kj(x n )
Gxmctm
Mprix
Grrir
=
6
=
6.00003427
1.7=7G-67
=.6=
6.66007633=
6.0066006=
6.006=3=31
3.?377G-67
=.6?
6.6=026?1?8
6.02630?=78
6.0264?378
3.=4=3G-67
=.63
6.6?077226?
6.086283821
6.086064?0
3.6761G-67
=.64
6.630??68=3
6.01=73241?
6.01=712=
?.0137G-67
=.67
6.642806=14
6.07?32607?
6.07?46087
?.2801G-67
=.61
6.672?12062
6.043301??1
6.0434?4??
?.8020G-67
=.68
6.618172142
6.034780430
6.03416117
?.8?==G-67
=.62
6.68101=64=
6.0?70?70?1
6.0?707?30
?.141?G-67
=.60
6.621=88101
6.0=843==03
6.0=847103
?.784=G-67
=.=
6.6073=6=2
6.060606060
6.06063027
7.=670G-67
S`j emyir airemk`sei `jtriduc`eis km aireukm dg km sghujdm dgr`vmdm pmrm pujtis `jtgr`irgs mk cijnujti dmdi>
6 ℊ? 6 ℊ 6 6 ℊ ℊ? (
)5
=
\ (
+ )
? (
)+ (
)U
=?
( )
Gngrc`c`i rgsugkti ?=> Mcgkgrmc`ój dg ujm pmrtåcukm. Cijs`dgrg km tmokm mdnujtm, gj km e`sem sg eugstrmj km pis`c`ój pmrm dgtgre`jmdi t`gepi dg ujm pmrtåcukm eiv`àjdisg gj gk gspmc`i, cmkcukg dg sgr pis`okg km mcgkgrmc`ój dg km e`sem. S` km pmrtåcukm dgscr`og uj eiv`e`gjti pmrmoók`ci dmdi pir a(t)5t ? , cmkcukg gk grrir dg km mcgkgrmc`ój mprix`emdm. n
t n
x n
6 = ? 3 4 7
6 6.6= 6.6? 6.63 6.64 6.67
6 6.666= 6.6664 6.6660 6.66=1 6.66?7
Km sghujdm dgr`vmdm gj su airem mprix`emdm gs siki mpk`cmokg m pujtis `jtgr`irgs dg kis dmdis, gs dgc`r qug pidrgeis cmkcukmr km mcgkgrmc`ój gj t 56.6=, 6.6?, 6.63 y 6.64.
≂ (6.6=)
= 6.6=
?
\6.6664
?(6.66=) + 6U
≂
?
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
≂ ≂ ≂ (6.6?)
(6.63)
(6.64)
= 6.6= =
? ? ?
6.6= = 6.6=
\6.6660
≂ ≂ ≂
?(6.6664) + 6.666=U
?
\6.66=1
?(6.660) + 6.6664U
?
\6.66?7
?(6.66=1) + 6.6660U
?
Km sghujdm dgr`vmdm dg a gs iov`megjtg `humk m ? gj tidi gk `jtgrvmki pir ki qug gj gstg gngrc`c`i tuv`eis uj grrir dg cgri.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
"Crgi qug e`gjtrms eís m aijdi sg gstud`m km c`gjc`m, eís sg mkgnm uji dg cumkqu`gr cijcgpti qug sg mprix`eg mk mtgåsei". X`kk`me Tbiesij
Tmk y ciei gx`stgj eàtidis jueàr`cis pmrm mprix`emr dgr`vmdms dg aujc`ijgs gj mkhþj pujti, tmeo`àj gx`stg eàtidis pmrm su ipgrmc`ój cijtrmr`m, dgtgre`jmr km `jtghrmk gj uj `jtgrvmki dgtgre`jmdi. Gj rgmk`dmd gx`stg eís dg uj eàtidi gj gstg cmpåtuki ym qug cuor`rgeis `jtghrmkgs sgjc`kkms dg dis airems, cij eàtidis s`epkgs y ciepugstis rgsuktmjdi iov`megjtg emyir km cija`mo`k`dmd dg kis ciepugstis. Mdgeís gstud`mrgeis kis eàtidis pmrm mprix`emr `jtghrmkgs diokgs gj ujm rgh`ój dgk pkmji kis cumkgs sg dgducgj dg kms aireukms pmrm `jtghrmkgs gj ujm sikm d`egjs`ój.
Sg dgjie`jm aireukm cgrrmdm dg Jgwtij Citgs dg j+= pujtis m km gxprgs`ój usmdm pmrm mprix`emr>
≂ 56 ( )
( )
Gx`stgj gxprgs`ijgs dg km suemtir`m pmrm cumjdi gk vmkir dg j gs pmr y cumjdi gk e`sei gs `epmr s`j geomrhi ji gjtrmrgeis gj dgtmkkg dg kms e`sems.
3 ℊ ℊ 6 = 6 = 7 ℊ ℊ 6 = ? 7 6 ? ℊ 6 = ? 3 ℊ 6 3
Dg airem hgjàr`cm `jtriduc`eis kms aireukms dg Jgwtij Citgs dgduc`dms pmrm j5=, ?, 3> ( )
5 =>
( )
5 ?>
5 3>
( )
5
5
3
2
5
3
\ (
\ (
?
\ (
)+ (
)+4 (
)+3 (
)U
)+ (
)+3 (
‿‿
=?
)U
)+ (
( ),
06 )U
\
( ),
3
7
,
U
\
( ),
,
U
\
,
U
Km aóreukm pmrm j5= t`gjg pir jieorg gspgc`a`ci Trmpgc`i S`epkg, pmrm j5? sg dgjie`jm S`epsij S`epkg y pmrm j53 sg dgjie`jm S`epsij S`epkg 3/2. Cmdm `jtghrmk d`v`dg mk `jtgrvmki dg `jtghrmc`ój gj j+= pmrt`c`ijgs, kms cumkgs `jtgrv`gjgj ciei sg pugdg iosgrvmr gj km aireukm m usmrsg. Gk tgre`ji a`jmk dg cmdm aireukm sg cijs`dgrm tgre`ji dgk grrir y pir ki hgjgrmk sg ie`tg gj kis cíkcukis kkghmjdi dg gstm airem m vmkirgs mprix`emdis mk gk`e`jmrki.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Gngrc`c`i rgsugkti ??> Mpk`cmc`ój dg kms aóreukms dg Trmpgc`i y dg S`epsij y ciepmrmc`ój gjtrg gkkms. Cijs`dgrg km `jtghrmk dmdm
∬
, mprix`eg km e`sem egd`mjtg kms aireukms dg Trmpgc`i y dg
S`epsij s`epkgs, mdgeís bmkkg su vmkir gxmcti pmrm cij gk e`sei dgtgre`jmr gk vmkir dgk grrir mosikuti cij cmdm eàtidi.
Vmrm mpk`cmr cmdm eàtidi jgcgs`tmeis ujm tmokm dg vmkirgs, ciegjzmrgeis pir gk eàtidi dgk Trmpgc`i gk cumk usmrm km s`hu`gjtg tmokm cijs`dgrmjdi qug km aujc`ój a m `jtghrmr gj gstg cmsi gs g x. n
x n
a(x n )5gZx n
6 =
6 =
= ?.8=2?2=2?2
6= ≂ >
= ?
\= + ?.8=2?2=2?2U
M cijt`jumc`ój sghu`rgeis cij gk eàtidi dg S`epsij> n
x n
a(x n )5gZx n
6 = ?
6 6.7 =
= =.1428?=?8= ?.8=2?2=2?2
≂
=.270=460=
6= ≂ = ≂ ∬6 =6 >
6.7 3
\= + ?.8=2?2=2?2 + 4(=.1428?=?8=)U
A`jmkegjtg pidgeis cmkcukmr gk vmkir gxmcti>
5
| 5
=.8=221==7
=.
Gk grrir mosikuti gj cmdm eàtidi sgrí gjtijcgs> 5|
=
=.270=460=| 5 6.=4627060
5|
=
=.8=221==7| 5 6.6667803?
Mquå pidgeis cijcku`r ujm cmrmctgråst`cm `epirtmjtg cumjdi ciepmrmeis meois eàtidis, gk dg S`epsij tmjti gj su vgrs`ój s`epkg ciei ciepugstm gs eís gxmcti qug gk dgk Trmpgc`i. Gngrc`c`i rgsugkti ?3> Mpk`cmc`ijgs usumkgs dg kms aóreukms dgk Trmpgc`i y Jgwtij.
∬
Gj km príct`cm kms aóreukms pmrm mprix`emr `jtghrmkgs sij usmdms cumjdi ji pidgeis bmkkmr ujm mjt`dgr`vmdm egd`mjtg kis cijic`e`gjtis dg aujc`ijgs gkgegjtmkgs. Cijs`dgrg km `jtghrmk
, km
cumk ji pisgg ujm mjt`dgr`vmdm pis`okg dg gxprgsmr egd`mjtg aujc`ijgs gkgegjtmkgs. Mprix`eg su vmkir egd`mjtg kms aireukms dg Trmpgc`i y dg S`epsij.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Ciegjzmrgeis pir gk eàtidi dgk Trmpgc`i, km tmokm dg vmkirgs sgrí> n
x n
a(x n )5gZ(x nZ?)
6 =
6 =
= ?.8=2?2=2?2
= 6 ≂ >
= ?
\= + ?.8=2?2=2?2U
Kughi km aóreukm dg S`epsij jis dm>
6= ≂ ∬6? >
6.7 3
n
x n
a(x n )5gZ(x n )Z?
6 = ?
6 6.7 =
= =.?246?74=8 ?.8=2?2=2?2
≂
=.270=460=
\= + ?.8=2?2=2?2 + 4(=.?246?74=8)U
≂
=.48783672
Jitm> Egd`mjtg eàtidis qug rgomsmj gk mkcmjcg dg gstm iorm, cmkcukmeis gk vmkir gxmcti dg km `jtghrmk gk 5 =.41?17. Jugvmegjtg pidgeis vgr qug gk eàtidi dg S`epsij gs eís gxmcti. cumk gs
Wjm gxtgjs`ój dg kms aireukms s`epkgs prgsgjtmdms bmcg ujm sgcc`ój jis kkgvm mk gstud`i dg kms aireukms ciepugstms dg Jgwtij Citgs. Kms e`sems gj hgjgrmk sg usmj cij uj j≧4, pud`àjdisg usmr sikmegjtg km aireukm dg S`epsij ciepugsti pmrm j pmrgs, y km dgk Trmpgc`i ciepugsti pmrm j `epmrgs y pmrgs s`j priokgems. Km aireukm ciepugstm dgk Trmpgc`i y dg S`epsij sij rgspgct`vmegjtg>
ℊ 6 ℊ 6 ∟=5= ? ( )
( )
5
3
\ (
5
)+ (
?
\ (
)+ (
)+?
)+?
∟= ℊ? 5= 5= ?∟= ℊ4
+4
U
( ),
=?
U
=26
\ , U
( ),
\ , U,
5?
Kms dis aireukms pugdgj sgr `jtgrprgtmdms dg emjgrm eís sgjc`kkms s` vgeis m cumkgs vmkirgs bmcgj rgagrgjc`m, km dgk trmpgc`i ji gs eís qug gk pmsi egd`i pir kms aujc`ijgs gvmkumdms gj kis gxtrgeis dgk `jtgrvmki sueíjdikg gk dupki dgk suemtir`i dg km aujc`ój gvmkumdm gj pujtis `jtgr`irgs.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Vir itri kmdi km aireukm dg S`epsij ji gs eís qug gk pmsi gjtrg trgs pir km suem dg km aujc`ój gvmkumdm gj kis gxtrgeis suemdm mk dupki dg km aujc`ój gvmkumdm gj pujtis `jtgr`irgs pmrgs mdgeís dgk cuídrupki dg km aujc`ój gvmkumdm gj pujtis `jtgr`irgs `epmrgs. Vmrm `dgjt`a`cmr s` uj pujti gs pmr i `epmr ki bmcgeis e`rmjdi mk cijtmdir n gk cumk jis `jd`cm gj cumk hrupi gjtrm. Cmog rgcmkcmr qug n mrrmjcm dg 6.
Gngrc`c`i rgsugkti ?4> Ciepmrmc`ój dg kms aireukms ciepugstms dg S`epsij y dgk Trmpgc`i Wt`k`zmjdi gk eàtidi dg trmpgc`i i gk eàtidi dg S`epsij y mprix`emr km `jtghrmk cij j54 y mprix`emr gk grrir> Cumjti tgjdråm qug sgr j pmrm qug gk grrir sgm egjir i `humk m =6 -4 .
∞ ∟
Km rgsikuc`ój ym sgm pir gk eàtidi dg Trmpgc`i i dg S`epsij, `jvikucrm km s`hu`gjtg tmokm cij gk pmsi `jd`cmdi>
n
Rn
6
6
ℊ
5
= 4 a(Rn)
6.30204??2
= 6.?7 6.321112==1 ? 6.7
6.37?6173?1
3 6.87 6.36==3843? 4
=
6.?4=0868?4
= ∞ ∟? ≂ ℊ 6 4 = ? 3 6 = ∞ ∟? ≂ 6 = ∞ ∟? ≂ ℊ 6 4 = 3 3 6
Vir gk eàtidi dgk trmpgc`i sg t`gjg> =
?
(
?
)+ (
=
)+?
(
)+ (
)+ (
)
6.34662=244
?
Vir gk eàtidi dg S`epsij sg t`gjg> =
?
3
(
)+ (
)+4
(
)+ (
) +? (
)
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
= ∞ ∟? ≂ 6 =
?
6.34=377428
Vmrm gjcijtrmr gk vmkir dg j pmrm cmdm uji dg kis eàtidis, sg usm kms citms dg grrirgs gj cmdm cm si>
∂ ℊ?
Trmpgc`i>
Dijdg
5
(
)
( )
=?
( ) gs gk eíx`ei dg km sghujdm dgr`vmdm gj gk `jtgrvmki \6, =U.
‿‿
Gstg vmkir eíx`ei sg pugdg bmkkmr dgr`vmjdi km aujc`ój ir`h`jmk, gk pricgsi sg ie`tg s`j geomrhi gk eíx`ei kicmk gj gk `jtgrvmki \6, =U gs gk s`hu`gjtg (tiemdi gj vmkir mosikuti)>
∞ ∂ ℊ? ∞ ∟4 ℊ? ? ∞ ≂∂ ∞ ∂ ℊ4 ∞ 4ℊ 4 ∞ ∂ ∞ ∟4 ≂ ( )5
=
?
Gvmkumjdi gsti gj km gxprgs`ój dgk grrir, iotgjgeis> 5
5
=
5
(
)
=
=?
?
=? ?
=? ? (=6
5
)
=
=2.?3
Vmrm msghurmr gk grrir dgsgmdi, sg rgdijdgm m uj j supgr`ir, pir ki qug a`jmkegjtg j5=0. S`epsij>
5
(
)
( )
=26
Mjíkihmegjtg jgcgs`tmeis gk vmkir eíx`ei (gj vmkir mosikuti) dg km cumrtm dgr`vmdm dg km aujc`ój ir`h`jmk gj gk `jtgrvmki \6, =U, gk pricgsi pmrm bmkkmrki sg ie`tg s`j geomrhi su vmkir gs gk s`hu`gjtg> 3
( )5
?
@jhrgsmjdi gstis dmtis gj km gxprgs`ój dgk grrir, tgjgeis> 5
=
5
=26 ? 3(
)
5
=26 ? (=6
)
3(=)
?.21
Gk eàtidi dg S`epsij ciepugsti, jgcgs`tm uj j pmr pir ki qug gk j iotgj`di sg rgdijdgm mk `jegd`mti pmr, rgsuktmjdi j54.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Gngrc`c`i rgsugkti ?7> Gxmct`tud dg kms ciepugstms rgspgcti m kms rghkms s`epkgs. Cmkcukg jugvmegjtg km `jtghrmk prgsgjtmdm gj gk gngrc`c`i ?3 pgri gstm vgz ut`k`cg kms rghkms ciepugstms dg S`epsij y dgk Trmpgc`i cij uj j52 pmrm meoms.
∬6=
Dmdi qug usmrgeis gk e`sei jþegri dg pmrt`c`ijgs j pmrm meoms pidgeis hgjgrmr ujm tmokm cieþj dg rgsuktmdis rgcirdmjdi qug km `jtghrmk m cmkcukmr gs . n 6
x n 6
a(x n )5gZ(x nZ?) =
=
6.=3
=.6=7848860
?
6.?7
=.614404470
3
6.32
=.=7600?047
4
6.7
=.?246?74=8
7
6.13
=.488064=07
1
6.87
=.877674178
8
6.22
?.=763380=1
2
=
?.8=2?2=2?2
Km rghkm dgk trmpgc`i jis hgjgrm uj rgsuktmdi>
= 6 ≂ >
=/2
\= + ?.8=2?2=2?2 + ?(=.6=7848860 + =.614404470 + =.=7600?047 ? + =.?246?74=8 + =.488064=07 + =.877674178 + ?.=763380=1)U
6= ≂ >
=.4108=??2
Kughi km aóreukm dg S`epsij jis dm>
= 6 ≂ >
=/2
\= + ?.8=2?2=2?2 3 + 4(=.6=7848860 + =.=7600?047 + =.488064=07 + ?.=763380=1) + ?(=.614404470 + =.?246?74=8 + =.877674178U
= 6 ≂ >
=.41?8?34=
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
∬6?
Jitm> Ciei sg egjc`ijó gj gk gngrc`c`i ?3, egd`mjtg eàtidis qug rgomsmj gk mkcmjcg dg gstm iorm, 5 =.41?17 dijdg pidgeis vgr quà tmk y cmkcukmeis gk vmkir gxmcti dg km `jtghrmk gk cumk gs ciei cij kms aireukms s`epkgs, gk eàtidi dg S`epsij pisgg tmeo`àj eís gxmct`tud mk usmr su aireukm ciepugstm qug gk eàtidi dgk Trmpgc`i.
Gj km prgsgjtg sgcc`ój gstud`mrgeis dis cmsis dg `jtghrmkgs qug gj tgiråm prgsgjtmj priokgems mk mpk`cmrkgs dg airem d`rgctm mkhþj eàtidi dg mprix`emc`ój, supijhm pmrm gk pr`egr cmsi qug cmkcukmeis su vmkir gj \m, oU y qug km aujc`ój prgsgjtm mkhujm d`scijt`ju`dmd gj d`cbi `jtgrvmki ciei ujm msåjtitm. Km mpk`cmc`ój dg kis eàtidis jueàr`cis rgv`smdis mjtgr`iregjtg jis dgemjdm cijicgr gk vmkir dg km aujc`ój gvmkumdm gj c`grtis pujtis dgk `jtgrvmki dg mprix`emc`ój, pir ki qug ujm msåjtitm jis `epgd`råm cijicgr gk vmkir dg km e`sem dg emjgrm mcgrtmdm gj kms cgrcmjåms dg km d`scijt`ju`dmd. Vir itri kmdi cijs`dgrg qug jgcgs`tm cmkcukmr ujm `jtghrmk gj gk `jtgrvmki ji mcitmdi \m, √) tipíjdisg cij gk e`sei `jcijvgj`gjtg dgscr`ti gj gk pírrmai mjtgr`ir siki qug mbirm ji ki t`gjg pir ujm msåjtitm s`ji pir tgjgr qug gvmkumr su aujc`ój 'gj gk `ja`j`ti'.
∬ ∟ ↟ ∟∟ + 4 4 4 ∟ ∟ 4
Vmrm gk pr`egr t`pi dg `jcijvgj`gjtg cijs`dgrg ki s`hu`gjtg, `jtgjtmeis cmkcukmr gscr`o`rsg dg km airem ( ) 5 ( gjtijcgs km `jtghrmk
( )
Gscr`omeis m a ciei ( ) 5 mkrgdgdir dg m.
( )
)
(cij k`e
∬ ( )
, s` a pugdg
( ) 5 +√) dijdg h gs cijt`jum gj \m, oU y p ɜ (6, =)
gx`stg y pugdg cmkcukmrsg dg km s`hu`gjtg emjgrm> ( )
( )
(
( )
dijdg
)
( ) gs gk pik`jie`i dg Tmykir dg h dg cumrti irdgj
Cij a gj gstm airem tgjdrgeis>
( )
Dijdg pidgeis dga`j`r ( ) 5
( )
5
(
6,
( ) (
( )
( )
)
,
5
( )
)
5
+
( )
(
)
pmrm a`jmkegjtg iotgjgr>
( )
+
( )
(
)
Dg km gxprgs`ój iotgj`dm km `jtghrmk dg H sg rgmk`zm dg airem mprix`emdm usmjdi cumkqu`grm dg kis eàtidis cijic`dis e`gjtrms qug km itrm `jtghrmk m rgsikvgr sg pugdg bmkkmr dg airem d`rgctm dmdi qug ji rgprgsgjtm emyir dgsmaåi gj cumjti m gjcijtrmr su mjt`dgr`vmdm sg rga`grg.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
∬ ∬∟
=
Gj cumjti mk itri t`pi dg `jtghrmkgs, tgjgeis m km qug cijt`gjg kåe`tgs `ja`j`tis gj km e`sem. Cijs`dgrg √ ( ) ujm `jtghrmk dgk t`pi i √ ( ) , rgmk`zmjdi km sust`tuc`ój 5 pidgeis gk`e`jmr km `jdgtgre`jmc`ój gj kis kåe`tgs pmrm kughi vgrjis gj km cmpmc`dmd dg mpk`cmr mkhþj eàtidi jueàr`ci gstud`mdi gj sgcc`ijgs prgv`ms.
Gngrc`c`i rgsugkti ?1> Wjm aujc`ój cij msåjtitm gj gk `jtgrvmki dg `jtghrmc`ój. Mprix`eg `jtghrmk.
∬ ∞
, gepkgg j51 cij aóreukm ciepugstm dg S`epsij pmrm km pmrtg mprix`emdm dg km
∞ ∟6 = ∞ = 4=? = 4 = ? 6 6 6 4 ? 3 4 ? 3 4 =? = ∞ = = ? =?3 4 6 6 6
Qgr`a`cmeis kms cijd`c`ijgs mjtgs dgscr`tms, tgj`gjdi ( ) 5
5(
) /
cij km ms`jtitm gj gk kåe`tg
`jagr`ir dg km `jtghrmk pir ki qug>
( )
5
(
6)
( )
+
/
(
6)
/
Gk pik`jie`i dg Tmykir dg cumrti irdgj mkrgdgdir dg 6 dg h gs> ( )5=+
+
?
+
1
+
?4
Dga`j`eis m H gjtijcgs>
( )5
=
? 6)
(
1
/
6,
56
?4 ,
6
Trmjsaireíjdisg jugstrm `jtghrmk ir`h`jmk m>
=+
5
Km pmrtg gxmctm dg km `jtghrmk
( )
+
=∬6 =++∟6++ (
)
/
+
?
(
t`gjg pir vmkir
km s`hu`gjtg tmokm (rgcirdmjdi qug gj x56, H t`gjg ciei vmkir 6)>
+
6)
+
1
?4
/
==67=3826
. Vmrm km `jtghrmk dg H tgjdrgeis
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
n 6
x n 6
H(x n ) 6
=
6.=11111118
?.10084G-61
?
6.333333333
1.?21?G-67
3
6.7
6.66646=3=?
4
6.111111118
6.66=762217
7
6.233333333
6.664?47810
1
=
6.660042407
Mpk`cmjdi km aóreukm dg S`epsij ciepugstm tgjgeis qug
6=∞ ≂
∬6= ≂ ≂
6.66=8161?.
( )
A`jmkegjtg tgjgeis qug km `jtghrmk ouscmdm tgjdrí gk vmkir> ==67= 3826
+ 6.66=8161?
?.0?736770.
Gngrc`c`i rgsugkti ?8> K`e`tgs `ja`j`tis dg `jtghrmc`ój. Mprix`eg
∬ +
usmjdi aóreukm dg S`epsij ciepugstm cij j54.
= ∞ 4 = ∞ 4 ∞ 4 6 6 = = √ = ∞ 4 6 4 ? 6 ∞ 4 = ∞ 4 = = = 6 ∞ 4 = ∞ 4 = ∞ 4 6 6 6
S` mpk`círmeis dg airem d`rgctm gk cmeo`i dg vmr`mokg 5 jis gjcijtrmr`meis cij ujm `jdgtgre`jmc`ij mk rggepkmzmr gj gk vmkir dg x56 pmrm bmkkmr gk jugvi vmkir dgk kåe`tg `jagr`ir dg km `jtghrmk. Vmrm gv`tmr gstg priokgem usmrgeis uj pgqugýi mrt`a`c`i> =
=
5
+=
+=
=
+
+=
Mbirm jis icupmrgeis dg km sghujdm `jtghrmk rgsuktmjtg qug gs km qug emjt`gjg gk kåe`tg `ja`j`ti supgr`ir. Sgm
5 =,
5 gjtijcgs 5 = y
5
, mdgems rggepkmzmeis kis k`e`tgs dg km `jtghrmk dmdi qug pmrm
5
,
5 6.
_ggepkmzmjdi ki mjtgs iotgj`di gj km `jtghrmk cij gk kåe`tg `ja`j`ti tgjgeis> =
+=
=
5
=
=
=
5
+=
+=
Vir ki qug jugstrm `jtghrmk ir`h`jmk sgrí> =
+=
5
=
+=
+
=
+=
5
=
+=
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Vidgeis dmrjis cugjtm qug km `jtghrmk dgk kmdi dgrgcbi m pgsmr dg gstmr gxprgsmdm gj aujc`ój dg u t`gjg gk e`sei vmkir qug km dgk cgjtri pir ki qug s`j dudmrki pidgeis gscr`o`r>
∞ 4 6
= +=
5?
= ∞ 4 6 == ∬6 ∞+= =
+=
Mpk`cmjdi km aóreukm dg S`epsij ciepugstm cij j54 pmrm vmkirgs>
Vir ki qug
=∬6 ∞=+= ≂ 3
n 6
x n 6
a(x n ) =
=
6.?7
6.00267?782
?
6.7
6.086=4?7
3
6.87
6.28=787738
4
=
6.868=6182=
tgjgeis km s`hu`gjtg tmokm dg
\= + 6.868=6182= + 4(6.00267?782 + 6.28=787738) + ?(6.086=4?7)U
= ∞ 4 6
= +=
≂
6.0?8=7210
Cijcku`eis qug km `jtghrmk sik`c`tmdm `j`c`mkegjtg gs gjtijcgs>
∞ 4 6
= +=
≂
∬ ∬∇∇
?(6.0?8=7210)
≂
=.2743=838
Gj gstm sgcc`ój gjargjtmrgeis gk priokgem dg mprix`emr gk vmkir dg `jtghrmkgs dg km airem ( ) ( )
. Vmrm gsti pr`egri pujtumk`zmeis c`grtis mspgctis, ut`k`zmrgeis j pmrt`c`ijgs gj gk gng x y e pmrt`c`ijgs gj gk gng y. ( , )
ℊ ∟∇ ∟∇ ℊ ℊ
5 Gk pmsi gj gk gng x sgrí a`ni y tgjdrí uj vmkir dg> , pmrm cmdm vmkir dg x tgjdrgeis d`agrgjtgs vmkirgs dg aujc`ijgs a(x, y) dmdi qug y sg ciepirtmrí ciei vmr`mokg, gs pir gsti qug gk pmsi gj gk gng y sg
dga`jg gj aujc`ój dg x dg km s`hu`gjtg airem>
5 ( )5
( )
( )
.
Dg gstm emjgrm prgsgjtmeis dg airem `jegd`mtm kis mkhir`teis dg Trmpgc`i y S`epsij pmrm rgsikvgr gk t`pi dg `jtghrmkgs gj egjc`ój> Trmpgc`i>
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
∇ ∇ ℊ ℊ ∟= 6 ≂ 6 6 6 ∟= 5= 6 ℊ 6 5= ∟= ℊ 6 ∟= 5= 5= ( )
( , )
( )
(
?
(
?
(
+
)
)
(
?
)+ (
)+ (
,
( )
+?
,
( ,
3
,
,
)+?
,
)+?
)+ ( ,
,
)+?
+
,
S`epsij>
∇ ∇ ∟= ? ℊ≂ ℊ 6 6 6 6 5= 6 ? 5?= 6 ?∟= ∟= ? ℊ 6 5= ? 5?= ?∟= ?∟= ℊ ? ?∟= ? 5= ? 6 ? 5= ? ? 5= ? ?∟= ? ℊ ?∟= ?∟= 6 ?∟= ? ∟=?∟= ? ? ?∟= ?∟= 5= 5= 5= ( )
( , )
( )
(
3
+
+?
+4
)
(
3
(
)
(
3
(
)+ (
,
)
(
3
(
)
3
)+ (
,
,
(
,
,
)+?
)+ (
,
)+?
,
,
)+ (
,
)+?
,
+4
+4
,
)+?
,
,
+4
,
,
+4
,
Gj hgjgrmk gk pricgd`e`gjti m usmrsg pmrm meois mkhir`teis gs gk e`sei, sg pricgdg pr`egri pir gjcijtrmr kis vmkirgs dg x pmrm kis cumkgs sg d`v`d`rí gk `jtgrvmki, kughi ouscmeis pmrm cmdm x kis pmrgs irdgjmdis ( x, y) y pricgdgeis m gvmkumr km aujc`ój a . Cij gstis vmkirgs hgjgrmeis ujm emtr`z m km cumk sg kg mpk`cmrm km rghkm dg S`epsij i Trmpgc`i gj cmdm ujm dg sus a`kms hgjgrmjdi mså ujm tmokm dg dis cikuejms m km cumk jugvmegjtg sg kg mpk`cmrí gk mkhir`tei dgsgmdi. Km `dgm dg tgjgr ujm emtr`z pmrm kughi rgduc`rkm m ujm tmokm dg dis cikuejms pmrm a`jmk`zmr mpk`cmjdi jugvmegjtg gk mkhir`tei m gstm tmokm gs mjíkihm m km usmdm gj `jtgrpikmc`ój gj dis vmr`mokgs.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Gngrc`c`i rgsugkti ?2> Wsi dg km rghkm dg S`epsij pmrm `jtghrmkgs diokgs. Cmkcukmr km s`hu`gjtg `jtghrmk, cij gk mkhir`tei dg S`epsij>
∬∬
x
?
0∟
?
y dM
_
Dijdg _ gs km rgh`ój mcitmdm pir>
+
5 . Wsmr j5e54.
Km rgh`ój dg `jtghrmc`ój nujti cij kms pmrt`c`ijgs gj x sg eugstrm gj km hrma`cm mdnujtm>
Ciei sg vg gj km hrma`cm kis vmkirgs dg x m usmr sgríj x 5 -3, -=.7, 6, =.7, 3. Vmrm cmdm uji sg mpk`cmrm gk eàtidi dgscr`ti iotgj`gjdi km emtr`z dg dmtis. Kms tmokms pmrm cmdm x sg eugstrmj m cijt`jumc`ój>
x 5 -3 x
y6
y=
y ?
y 3
y 4
-3
6 ?8
6 ?8
6 ?8
6 ?8
6 ?8
a(-3, y n )
x 5 -=.7 x
y6
y=
y ?
y 3
y 4
-=.7
-?.702681?= 3.387
-=.?00632== 1.62431882
6 1.87
=.?00632== 1.62431882
?.702681?= 3.387
a(-=.7, y n )
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
x 5 6 x
y6
y=
y ?
y 3
y 4
6
-3 6
-=.7 6
6 6
=.7 6
3 6
a(6, y n )
x 5 =.7 x
y6
y=
y ?
y 3
y 4
=.7
-?.702681?= 3.387
-=.?00632== 1.62431882
6 1.87
=.?00632== 1.62431882
?.702681?= 3.387
a(=.7, yn)
x 5 3 x
y6
y=
y ?
y 3
y 4
3
6 ?8
6 ?8
6 ?8
6 ?8
6 ?8
a(3, yn)
Cij gsti hgjgrmeis km emtr`z dg dmtis> `, n
6
=
?
3
4
6
?8 3.387 6 3.387 ?8
?8 1.62431882 6 1.62431882 ?8
?8 1.87 6 1.87 ?8
?8 1.62431882 6 1.62431882 ?8
?8 3.387 6 3.387 ?8
= ? 3 4
Wsmjdi gk eàtidi dg S`epsij tgjgeis km tmokm dg dis cikuejms> 6 = ? 3 4
6 44.812613? 6 ?0.2473877 6
P a`jmkegjtg gk vmkir pir gk eàtidi dg S`epsij gs>
? ? ≂ 0
=40.??1288
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
∫ ? ? ≂
Egd`mjtg eàtidis augrm dgk mjík`s`s dgk tgxti, gjcijtrmeis gk vmkir gxmcti dg km `jtghrmk 0
=8?.2 cij ki cumk vgeis uj grrir mcgptmokg rgspgcti mk vmkir tgir`ci cijs`dgrmjdi
qug gs ujm `jtghrmk diokg. Vidgeis rgduc`r gstg grrir mpk`cmjdi eís pmrt`c`ijgs gj meois gngs.
Gngrc`c`i rgsugkti ?0> Wsi dg km rghkm dgk Trmpgc`i pmrm `jtghrmkgs diokgs. Cmkcukmr km s`hu`gjtg `jtghrmk, cij gk mkhir`tei dgk Trmpgc`i>
(
Dijdg _ gs km rgh`ój mcitmdm pir>
+
5 {( , )/
)
,
} . Wsmr j5e53.
Km rgh`ój dg `jtghrmc`ój nujti cij kms pmrt`c`ijgs gj x sg eugstrm gj km hrma`cm mdnujtm>
Ciei sg vg gj km hrma`cm kis vmkirgs dg x m usmr sgríj x 5 6, =/3, ?/3, =. Vmrm cmdm uji sg mpk`cmrm gk eàtidi dgscr`ti iotgj`gjdi km emtr`z dg dmtis. Kms tmokms pmrm cmdm x sg eugstrmj m cijt`jumc`ój>
x 5 6 x
y6
y=
y ?
y 3
6
6 6
6 6
6 6
6 6
y=
y ?
a(6,y n )
x 5 =/3 x
6.33 a(=/3,y n )
y6
y 3
6.33333333 6.44444444 6.77777771 6.11111118 6.=42=42=7 6.?347180 6.3471806= 6.42=42=42
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
x 5 ?/3 x
6.11111118 a(?/3,y n )
y6
y=
y ?
y 3
6.11111118 6.22222220 =.======== =.33333333 6.84684684 =.6214=087 =.736214? ?.68468468
x 5 = x
=
y6 =
y= =.33333333
y ? =.11111118
y 3 ?
a(=,y n )
?
?.88888882
3.88888882
7
Km emtr`z dg dmtis rgsuktmjtg mk mhrupmr tidis kis vmkirgs iotgj`dis sgrí> `, n
6 = ? 3
6
=
?
3
6 6 6 6 6.=42=42=7 6.?347180 6.3471806= 6.42=42=42 6.84684684 =.6214=087 =.736214? ?.68468468 ? ?.88888882 3.88888882 7
Wsmjdi gk mkhir`tei dgk Trmpgc`i pmrm s`epk`a`cmr km emtr`z m ujm gxprgs`ój dg dis cikuejms tgjgeis> 6 6.60047=3 6.20438721 3.37=27=27
6 = ? 3
Mpk`cmeis pir þkt`em vgz gk mkhir`tei dgk Trmpgc`i, ki qug jis dm uj rgsuktmdi dg>
? 3 ≂ (
+
)
6.2200=88
Gk vmkir tgór`ci dg gstm `jtghrmk gs 6.82333… pir ki qug pidgeis mcgptmr km mprix`emc`ój ciei vmk`dm m pgsmr dg ji bmogr tgj`di eucbms pmrt`c`ijgs gj kis gngs.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Km cumdrmturm Hmuss`mjm gs uj eàtidi euy pidgrisi pmrm mprix`emr vmkirgs dg `jtghrmkgs s`j geomrhi rgqu`grg c`grtms emj`pukmc`ijgs m km `jtghrmk ir`h`jmk pmrm pidgr mpk`cmrki dg airem d`rgctm. Supijhm qug sg kg sik`c`tm cmkcukmr
∬ ( )
=∟= ?∟∟ ∬ ∟ ∬∟== ≂∝5=
dijdg m y o sij a`j`tis. Vmrm pidgr mpk`cmr gk eàtidi dg
cumdrmturm sg jgcgs`tmr dgnmr km `jtghrmk mjtgs sik`c`tmdm gj km airem
∬∟==
( )
dijdg 5
.
( ) sg mprix`em egd`mjtg ( ) ( ) dijdg j gs gk irdgj dg km Dg gstm emjgrm cumdrmturm y kis vmkirgs gj kis cumkgs sg gvmkþm km aujc`ój a sg dgjie`jmj rmåcgs dg kis pik`jie`is dg Kghgjdrg dg irdgj j, mdgeís m cmdm rmåz sg kg msic`m uj ciga`c`gjtg c qug mpmrgcg tmeo`àj gj km suemtir`m.
Kis pik`jie`is dg Kghgjdrg dg irdgj j5? y 3 sij kis s`hu`gjtgs>
? ? 3 3 ( )5
= 3
3
( )5
7
Ciei sg d`ni cmdm pik`jie`i t`gjg sus rmåcgs y cmdm rmåz t`gjg msic`mdi uj ciga`c`gjtg, pmrm kis dis pik`jie`is mjtgs egjc`ijmdis tgjgeis> j5?
= ∞ ? ∞ = ? 3
=
5
5
j53
,
3
=
3
,
3
5
5
3 7
56 ,
,
5=
5=
,
7
= ? = ? ?
5
5
5
7 0
7 0
2 0
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Cij gsti ym pidgeis ut`k`zmr gk eàtidi ym qug cijicgeis kms trmjsairemc`ijgs dg km `jtghrmk mjtgs dg pidgrkm usmr mdgeís dg kis vmkirgs dg kms rmåcgs gj kms cumkgs dgogeis gvmkumr km aujc`ój y gk ciga`c`gjtg pir gk cumk dgogeis eukt`pk`cmr cmdm `emhgj dg cmdm rmåz. Cmog rgcmkcmr qug gx`stg uj tgirgem gj gk eàtidi dg km cumdrmturm Hmuss`mjm>
∬=∟== ∟= 5= ∬∟== ≂∝5= ( )
Tgirgem 7> S` km aujc`ój m `jtghrmr a(t) gj
( )
gs uj pik`jie`i dg hrmdi egjir qug ?j, gjtijcgs>
5
( )
Dijdg j gs gk hrmdi dgk pik`jie`i dg Kghgjdrg i gk irdgj dg curvmturm. Cmsi cijtrmr`i s` km aujc`ój a(t) ji gs pik`jóe`cm pidgeis ma`remr
( )
( ).
Gk tgirgem gj rgmk`dmd jis qu`grg dgc`r qug s` km aujc`ój m `jtghrmr gs pik`jóe`cm pidgeis iotgjgr ujm mprix`emc`ój cij grrir juki rgspgcti mk vmkir gxmcti e`gjtrms qug s` ji gs pik`jóe`cm km mprix`emc`ój iotgj`dm gs euy cgrcmjm. Mbirm pricgdgeis m bmcgr km gxtgjs`ój pmrm ujm `jtghrmk diokg.
∬ ∬∇∇ ∇∇? ? ∇∇= = ∇ = = ∇ ∟= ∟= = = ≂ ∟= ∟= 5= 5= ( )
( , ) Cijs`dgrg qug jgcgs`tm cmkcukmr , sg mpk`cmrí gk e`sei rmzijme`gjti usmdi pmrm ( ) ujm `jtghrmk sgjc`kkm gs dgc`r pr`egri gstmjdmr`zmrgeis kis kåe`tgs usmjdi cmeo`is dg vmr`mokgs> 5
( )
?
( )
( )
( )
,
5
?
Cij gsti kihrmeis dgnmr km `jtghrmk mjtgr`ir cij dg km airem> ( )
( , )
5
( , )
( )
A`jmkegjtg pidgeis mprix`emr gstg rgsuktmdi egd`mjtg dis suemtir`is dg km airem> ( , )
( , )
Dijdg j, e sij kis órdgjgs dg cumdrmturm pmrm cmdm vmr`mokg u, t.
Gngrc`c`i rgsugkti 36> Mprix`eg km kijh`tud dg mrci dg km curvm y5cis(x) gj φ/? ≤ x ≤ φ gepkgmjdi cumdrmturm Hmuss`mjm dg irdgj ?.
_gcirdmeis qug pir dga`j`c`ój km kijh`tud dg mrci gj \m, oU dg ujm aujc`ój a gstí dmdm pir>
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
? ? ? = + \ ( )U
Vir ki qug gj gstg gngrc`c`i km `jtghrmk m rgsikvgr sgrí>
( )
=+
Ciei sg pugdg vgr km `jtghrmk ji gstí gj km airem jgcgsmr`m pmrm pidgr mpk`cmr dg emjgrm d`rgctm gk eàtidi, mså qug pricgdgeis m trmjsairemrkm iotgj`gjdi>
? = ? = ? ∟= ∟= ? = ∞ =3 = ? ∞ =3 ? ∟== ? ≂ 5?= ? 5
?
5
?
?5
3 ? 54
?
?
3
?
Dg dijdg pidgeis iotgjgr>
+3
5
4
,
5
4
_ggepkmzmeis gj km `jtghrmk ir`h`jmk y tgjgeis mbirm> =+
( )
5
+3
=+
4
5
4
=+
4
+3 4
Ciei jis aug sik`c`tmdi mprix`emrgeis cij cumdrmturm dg irdgj dis pir ki qug usmrgeis kms s`hu`gjtgs , 5 = y 5 , 5 =. rmåcgs cij sus rgspgct`vis ciga`c`gjtgs> 5 Iotgjgeis gjtijcgs>
4
∟== 4
=+
+3
=+
4
? ≂
+3 4
Cijcku`eis mk gvmkumr qug>
4
\(=)
4
⎜ ∞ ⎕ ⎕ =+
? ? ≂ =+
( )
+3
=+
4
?⎚ ⎜ ∞ ⎦⎦ ⎕ ⎕
+3 3 4
+ (=)
=.061280868 …
=+
3
? ⎚ ⎦⎦
+3 4
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Gngrc`c`i rgsugkti 3=> Wjm `jtghrmk diokg pir gk eàtidi dg cumdrmturm. Cmkcþkgsg
∫ ∂ (
)
5 {( , )
/
} cij gk eàtidi dg Cumdrmturm
Hmuss`mjm dg irdgj 3.
Dgnmjdi m x ciei vmr`mokg k`org, `jtghrmrgeis siorg km s`hu`gjtg rgh`ój (hrma`cmdm)> Gstm rgh`ój gstí ckmrmegjtg mcitmdm gj x dgsdg -= m =, mså ciei gj y vm dgsdg km aujc`ój y(x)5x? bmstm km aujc`ój cijstmjtg y5=.
Km `jtghrmk sgrí rgsugktm usmjdi km mprix`emc`ój dmdm pir>
= = ≂ = 3 ≂ 3 3 ∟= = = ∟= 5= 3 3 5= 5= ∟= ≂ 5= 5= (3
? )
( , )
(3
( , )
? )
( , )
Dijdg gk irdgj dg km cumdrmturm gs 3, pir ki qug kms suemtir`ms kkghmj bmstm gstg vmkir y kms vmr`mokgs u, t sg dgscr`o`ríj m cijt`jumc`ój> Gstmjdmr`zmeis kåe`tgs>
? ∇ ∇ ? = ∇? ∇= ? ? ? ? 5
Dg dijdg>
( )
?
( )
5
( )
( )
(=
5
?
=
=
) +=+ ?
Dg mquå iotgjgeis gk d`agrgjc`mk dg y>
5
(=
)
?
Mjíkihmegjtg ut`k`zmeis gk cmeo`i dg vmr`mokg gj x y iotgjgeis gk d`agrgjc`mk jgcgsmr`i> 5
?
=
=
( =)
( =)
5
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
= = = = ? ? ? ∟= ∟= ∟= = = = = ? ? ? ∟= ∟= ∟= 5
@jhrgsmjdi kms rgkmc`ijgs iotgj`dms gj km `jtghrmk, pidgeis bmkkmr km aujc`ój A(u,t)> (3
? )
5
(3
?
(=
) +=+
)
?
=
?
_girdgjmjdi y mhrupmjdi iotgjgeis km `jtghrmk trmjsairemdm> (3
? )
5
=
(=
(3
?
) +=+
(=
)
Dijdg km aujc`ój A(u,t) m usmr gj km gxprgs`ój dg km suemtir`m gs km s`hu`gjtg> ( , )5(
+
+
Vir ki qug omstm cij mpk`cmr km mprix`emc`ój dg km suemtir`m m gstm aujc`ój, kughi cmkcukmr su `jtghrmk y a`jmkegjtg d`v`d`rkm pmrm dis y iotgjgr km mprix`emc`ój ouscmdm. Wsmeis kms s`hu`gjtgs rmåcgs dg kis pik`jie`is dg Kghgjdrg cij sus ciga`c`gjtgs rgspgct`vis> _måcgs
Ciga`c`gjtgs
-6.88470118
6.71
6
6.222222220
6.88470118
6.777777771
Hgjgrmeis gjtijcgs km s`hu`gjtg tmokm dg vmkirgs> n
`
C n
u n
C `
t n
C `C ) n A(u ,t ` n
6.71
-6.884701110
6.222222220
6
3
6.777777771
6.884701110
-6.441=1123? -6.88761013= -6.7??186?68
=
6.71
-6.884701110
6.222222220
6
3
6.777777771
6.884701110
=
6.71
-6.884701110
6.222222220
6
6.777777771
6.884701110
= =
?
3
?
?
? 3
6.71
6.222222220
6.777777771
-6.884701110
6
6.884701110
Wsmjdi gstm tmokm dg vmkirgs iotgjgeis qug km `jtghrmk gs gjtijcgs
-6.===3=6?28 -6.806=?3478 -6.281344634 6.=?8162482 6.=4?086211 6.67==67=64
∬∟== ∬= 27 (3
? )
5
.
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Nuk`i _umji K`em
@jv`tmeis mk kgctir m rgsikvgr km `jtghrmk dg airem gxmctm y ciepriomr qug gk grrir dg km e`sem rgspgcti m su vmkir mprix`emdi gs cgri, gsti sg gxpk`cm dgo`di mk tgirgem dgk eàtidi dg Cumdrmturm dg Hmuss dijdg s` km `jtghrmk m mprix`emr gs pik`jóe`cm gk grrir dg km mprix`emc`ój gs juki.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
"Crgi qug e`gjtrms eís m aijdi sg gstud`m km c`gjc`m, eís sg mkgnm uji dg cumkqu`gr cijcgpti qug sg mprix`eg mk mtgåsei". X`kk`me Tbiesij
Kms gcumc`ijgs d`agrgjc`mkgs sij uj írgm dg gstud`i y mpk`cmc`ój euy `epirtmjtg dgk cíkcuki gj hgjgrmk. Gj uj cursi gspgc`a`ci dg gcumc`ijgs d`agrgjc`mkgs sg trmtm dg bmkkmr ujm aujc`ój qug smt`samhm km gcumc`ój pkmjtgmdm s`j geomrhi gj gk mjík`s`s dg kms e`sems gj gstg tgxti ji sg ouscm gspgcåa`cmegjtg gsm aujc`ój qug smt`samhm km gcumc`ój s`ji km gvmkumc`ój d`rgctm dg gsm aujc`ój gj c`grtis pujtis. Gx`stgj vmr`is eàtidis pmrm km rgsikuc`ój dg gcumc`ijgs d`agrgjc`mkgs tmjti dg pr`egr ciei dg sghujdi irdgj.
ℊ ∟
Vmrm tidis gstis eàtidis gj hgjgrmk dga`j`eis uj pmsi b qug gs km d`agrgjc`m gjtrg kis vmkirgs gj kis cumkgs qugrgeis gjcijtrmr km aujc`ój gvmkumdm. Gstg pmsi sg cmkcukm ciei 5 dijdg j gs gk jþegri dg pmrt`c`ijgs dgk `jtgrvmki \m, oU.
Gk eàtidi dg Tmykir gs ujm bgrrme`gjtm pidgrism pmrm km rgsikuc`ój jueàr`cm dg gcumc`ijgs d`agrgjc`mkgs, gk e`sei sg dgducg dg km aireukm dg Tmykir. Vmrm mprix`emr y`+= cijic`gjdi y'5a(t, y) sg iot`gjg gk mkhir`tei>
+= ≂ ℊ ℊ? ℊ3 ⋩ ∟= ℊ 6 + ( ,
) +
=
?!
‿ ( ,
)
+
=
3!
‿‿ ( ,
)
+
+
=
( ,
!
Dijdg j gs gk irdgj dgk eàtidi dg Tmykir y mdgeís sg t`gjg ciei cijd`c`ój `j`c`mk 6,=,?,…, =.
)
5 cij
5
Ciei sg pugdg gspgrmr gj gstg eàtidi sg jgcgs`tmrmj mpk`cmr dgr`vmdms `epkåc`tms dgo`di m km mpmr`c`ój dgk tàre`ji a' qug dgpgjdg dg t, y.
∟
Gngrc`c`i rgsugkti 3?> Ciepmrmc`ój gjtrg vmkirgs pir gk eàtidi dg Tmykir y vmkir gxmcti.
( ) Mprix`eg km sikuc`ój dg 5 , , 5 gepkgmjdi j54 cij gk egtidi dg Tmykir dg irdgj 3. Mdgems cmkcukg gk grrir dg mprix`emc`ij gj cmdm vmkir cijs`dgrmjdi qug km sikuc`ój gxmctm m d`cbm gcumc`ój gstí dmdm pir ( ) 5
+
( )+
( ) .
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Gj gstg cmsi gk j54 bmcg egjc`ój mk jþegri dg pmrt`c`ijgs dgk `jtgrvmki ym qug gj itri kmdi sg gspgc`a`cm gk irdgj dgk eàtidi dg Tmykir. Vr`egri rgv`smeis jugstri mkhir`tei hgjgrmk y rgv`smeis cumkgs dgr`vmdms dgogeis cmkcukmr>
+= ≂ ℊ ℊ? ℊ3 ‿ += ≂ ? 3 + ( ,
) +
=
?!
‿ ( ,
)
+
=
3!
‿‿ ( ,
)
Kkghmjdi bmstm gk tgre`ji dg irdgj 3 tgjgeis qug cmkcukmr a' y a'' . Vricgd`gjdi>
5 cis( )
( , ) 5 cis( )
( , )5
( , )5
cis( ) +
( )
( )+
5
( )
5
cis( ) +
cis( ) +
( ) + cis( )
5
( )
_ggepkmzmeis gj gk mkhir`tei ir`h`jmk, cijic`gjdi mdgeís qug b56.?7. + 6.?7\cis( )
U+
= ?
( )
6.?7 \
cis( ) +
= U + 6.?7 \ 1
( )
U
Cij gk mkhir`tei pmrt`cukmr hgjgrmdi pidgeis ciegjzmr m `tgrmr gk eàtidi pmrm hgjgrmr ujm tmokm dg vmkirgs> ` 6
t` 6
y` =
y`+= 6.00830723
=
6.?7
6.00830723
6.02=4261=
?
6.7
6.02=4261=
6.04?41343
3
6.87
6.04?41343
6.28438167
4
=
6.28438167
Vmrm cuke`jmr gk gngrc`c`i ciepmrmeis kis vmkirgs iotgj`dis cij gk vmkir gxmcti rgsuktmdi dg gvmkumr km aujc`ój qug smt`samcg km gcumc`ój gj kis pujtis rgspgct`vis> Mprix
Gxmcti
Grrir
6.00830723 6.02=4261= 6.04?41343 6.28438167
6.00877272 6.02=81032 6.04?24860 6.2842?138
6.666=1?87 6.666?2288 6.66632311 6.6664763=
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
6 += ≂ ℊ ℊ ℊ += ≂ ℊ ℊ ℊ += ≂ ℊ ℊ ℊ
Gx`stgj vmr`is mkhir`teis ckms`a`cmdis ciei eàtidis dg _ujhg Luttm, kis e`seis pmrtgj dg kms =. cijd`c`ijgs mjtgs egjc`ijmdms, gs dgc`r 5 ( , ) cij 5 < 5 6, = , ?, … , Eàtidi dg Vujti Egd`i>
+
( +
?
,
+
?
( ,
))
+
( ,
Eàtidi dg Gukgr Eid`a`cmdi>
+
Eàtidi dg Bguj
+
4
?
)+ ( + ,
\ ( ,
\ ( ,
)+3
+
? 3
,
+
?
))U
( ,
3
))
Kis eàtidis dg Vujti Egd`i y dg Gukgr sg cijs`dgrmj eàtidis dg _ujhg Luttm dg ?di irdgj. A`jmkegjtg prgsgjtmeis gk þkt`ei eàtidi qug gs gk eís gxmcti dg tidis>
= ℊ ℊ = ? ℊ ℊ 3 ℊℊ ℊ ? 4 3 += ≂ = ? 3 4
Eàtidi dg _ujhg Luttm dg 4ti Irdgj>
5
( ,
5
+
5
+
5
+
= 1
)
,
+
,
+
( + ,
+
\
+?
?
?
+?
?
?
)
+
U
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Gngrc`c`i rgsugkti 33> _gsikuc`ój dg ujm gcumc`ój d`agrgjc`mk pir gk eàtidi dg _ujhg Luttm dg 4ti irdgj.
= ℊ ℊ = ? 3 ℊ ℊ ? 4 ℊ ℊ 3 += ≂ = ? 3 4
Gepkgmjdi gk eàtidi dg _ujhg Luttm dg 4ti irdgj, mprix`eg
,
gj t56.7 cij b56.=.
Tgjgeis pr`egrmegjtg ( , ) 5 dg _ujhg Luttm dg 4ti irdgj.
(
)
5 ,
( )5
) + = pir ki qug cij gsti pidgeis hgjgrmr jugstri mkhir`tei
(
(
5 6.=\
5 6.=
+
5 6.=
+
) + =U
+
?
+
?
5 6.= ( + )
+
?
+
?
( + )(
+
= 1
\
+?
+=
?
+=
?
) +=
+
+?
+
U
Cij gk mkhir`tei hgjgrmeis km s`hu`gjtg tmokm dg rgsuktmdis>
@
t `
y `
a(t ,` y ) `
l =
l ?
l 3
l 4
y `+=
6
6
6
=
6.=
6.=666=?7
6.=666=?7
6.=666766=
6.=666=118
=
6.=
6.=666=118
=.66=666=7
6.=66=666?
6.=663381?
6.=6633820
6.=6616617
6.?663721?
?
6.?
6.?663721?
=.6626=??
6.=6626=??
6.=6=711??
6.=6=7121
6.=6??72=7
6.36=0=347
3
6.3
6.36=0=347
=.6?8=3768
6.=6?8=37=
6.=643=174
6.=643?1?0
6.=671413
6.461=288
4
6.4
6.461=288
=.61486448
6.=6148647
6.=60?38=2
6.=60?141
6.===4=224
6.7=73317
7
6.7
6.7=73317
≂
A`jmkegjtg cij gstm tmokm cijcku`eis qug (6.7)
6.7=73317.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
=? =? ⋪== ?? ⎠ = ?
Cijs`dgrg uj s`stgem dg gcumc`ijgs d`agrgjc`mkgs dg km airem> ‿ 5
‿ 5
( , ( ,
, ,
‿ 5
( ,
,
,…, ,…,
) )
,…,
)
Gk e`sei pugdg sgr rgsugkti egd`mjtg gk mkhir`tei dg _ujhg Luttm pmrm uj s`stgem dg gcumc`ijgs, m smogr> Vmrm mprix`emr
+=
usmeis>
+= ≂ = ? 3 4 =ℊ ℊ === ? ?= = ? ℊ ℊ = ? 3 ℊℊ ℊ = =? ? ?? ? 4 = =3 ? ?3 3 +
= 1
+?
+?
+
,
5 6, = , ?, … ,
5
( ,
( )5
=,
,
5 =, ?, … ,
Dijdg>
5
+
5
+
5
,
+
,
+
( + ,
+
?
?
?
?
,
,…,
,
+
,
+
,
+
)
?
?
,…,
+
,…,
+
,…,
+
?
?
)
5 =, ?, … ,
Wjm airem dg vgr gk mkhir`tei gs qug sg cmkcukmj kis cumtris l pmrm cmdm aujc`ój a y dg gstm airem sg mprix`em km sikuc`ój pmrm cmdm vmr`mokg.
Gngrc`c`i rgsugkti 34> S`stgem dg gcumc`ijgs d`agrgjc`mkgs
= = ?
Gepkgmjdi gk eàtidi dg _ujhg Luttm dg 4ti irdgj, mprix`eg km sikuc`ój dgk s`stgem>
5
5
Cij b56.= gj
.
+ + .
+ < + .
( )5 ( )5 <
. .
Dgk s`stgem pidgeis iotgjgr kms aujc`ijgs a pmrm hgjgrmr gk mkhir`tei dgk s`stgem> 5
4
+3
+1
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
? = ? ?= == == ?? =? = == ? ?= ?? = == ? ?= =3 = =? ? ?? ?3 = =? ? ?? = 4 = = 3 ? ? 3 ?4 = =3 ? ?3 5
?.4
+ =.1
+ 3.1
Vir ki qug jugstris l sgríj ciei sg eugstrm m cijt`jumc`ój, cijs`dgrmjdi gk pr`egr suoåjd`cg ciei gk qug cirrgspijdg m cmdm aujc`ój y gk sghujdi ciei gk irdgj dgk l cmkcukmdi. 5 6.=\ 4
+3
5 6.=\ ?.4
5 6.=
5 6.=
4
?.4
5 6.=
5 6.=
4
?.4
5 6.=\ 4(
5 6.=\ ?.4(
+
+
+
+
+ 1U
+ =.1
+ 3.1U
+3
?
+
+ =.1
?
+
+3
?
+
+ =.1
?
+1
?
+
) + =.1(
+
+ 3.1
?
+
) + 3(
+
+1
?
+
+ 3.1
?
) + 1U
) + 3.1U
Cij gstg mkhir`tei, mdgeís dg kms cijd`c`ijgs `j`c`mkgs tgjgeis km s`hu`gjtg tmokm dg vmkirgs> `
t `
y =`
y ?`
a =
a ?
l ==
l ?=
l =?
l ??
l =3
l ?3
l =4
6
6
6
=
0
7.?
6.0
6.7?
6.802
6.4731
6.26244
6.4167?2
6.8=482?4
l ?4
6.30017222
y =`+=
6.2641=64
y ?`+=
=.4780272=3
=
6.=
6.2641=64
=.4780272=3
8.=777=724
4.66=8=?34=
6.8=777=724
6.466=8=?34
6.13?41107?
6.3413=2843
6.14=661667
6.37=026100
6.71484330?
6.36?141867
=.44?42322?
=.2682221=8
?
6.?
=.44?42322?
=.2682221=8
7.1738363?7
3.63611648=
6.71738363?
6.363611648
6.408872333
6.?70411718
6.76484=37=
6.?1460?383
6.44?864?64
6.??4=2?063
=.0441133=1
?.686?23620
3
6.3
=.0441133=1
?.686?23620
4.43?=01663
?.?47?16024
6.443?=01
6.??47?1602
6.322?74707
6.=2036=234
6.303013071
6.=03680104
6.3437780?1
6.=162187
?.33173?4?=
?.?1=087214
4
6.4
?.33173?4?=
?.?1=087214
3.4308080==
=.1==423784
6.34308080=
6.=1==42378
6.?00371621
6.=3?81?17=
6.3646??08=
6.=37241130
6.?13=?4704
6.=600=2361
?.13224?238
?.30110668?
7
6.7
?.13224?238
?.30110668?
\83U
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Cijs`dgrg gk priokgem dg vmkir `j`c`mk cij km gcumc`ój d`agrgjc`mk dg sghujdi irdgj k`jgmk dg km airem>
ℊ ∟ = ? ∟= ‿‿ 5 ( )
‿
+ ( ) + ( ) <
<
( )5
<
( )5
Supijhmeis qug pmrt`c`ijmeis gk `jtgrvmki \m, oU gj j `jtgrvmkis, tmk qug> 5 . _gqugr`eis ), pmrm ki cumk usmrgeis gk s`hu`gjtg s`stgem dg gcumc`ijgs> mprix`emr gjtijcgs ( ), ( ), … , (
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Cijs`dgrg gk priokgem dg vmkir `j`c`mk cij km gcumc`ój d`agrgjc`mk dg sghujdi irdgj k`jgmk dg km airem>
ℊ ∟ = ? ∟= ‿‿ 5 ( )
‿
+ ( ) + ( ) <
( )5
<
( )5
<
Supijhmeis qug pmrt`c`ijmeis gk `jtgrvmki \m, oU gj j `jtgrvmkis, tmk qug> 5 . _gqugr`eis ), pmrm ki cumk usmrgeis gk s`hu`gjtg s`stgem dg gcumc`ijgs> mprix`emr gjtijcgs ( ), ( ), … , (
_gsikv`gjdi gk s`stgem mjtgs dgscr`ti iotgjgeis km aujc`ój y gvmkumdm gj kis pujtis `jtgr`irgs dgk `jtgrvmki dg mprix`emc`ój. Cmog rgcmkcmr qug km k`jgmk`dmd dg km gcumc`ój dg sghujdi irdgj `epk`cm gj rgmk`dmd km k`jgmk`dmd dg kis tàre`jis p(x), q(x) y r(x). Mdgeís dgog cuepk`rsg qug gstms trgs aujc`ijgs sgmj cijt`jums gj gk `jtgrvmki \m, oU.
Gngrc`c`i rgsugkti 37> Wj priokgem dg vmkir `j`c`mk>
= ? 3
Mprix`eg km sikuc`ój pir gk eàtidi dg d`agrgjc`ms a`j`tms ut`k`zmjdi uj pmsi b56.?7.
5
+
+
( )
<
<
( )5
<
( )5 .
Dmdi gk pmsi pidgeis gjcijtrmr kis pujtis `jtgr`irgs gj kis cumkgs mprix`emrgeis km sikuc`ój, kis e`seis sgríj> 5 =.?7,
5 =.76,
5 =.87
Mdgeís cijicgeis kms cijd`c`ijgs dg arijtgrm, pir ki qug> 5 =,
5 =.7
Hgjgrmeis gk s`stgem dg gcumc`ijgs m rgsikvgr, gk e`sei tgjdrí km airem>
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
ℊ?ℊ = ℊ = ℊ = ⎜ℊ? = ℊ = ⎚ ? ℊ?ℊ ? ? ?3 ⎛ℊ? 3 ℊ? ?ℊ 3 ⎙ ? ℊ 3 ⎦ 3 ⎕ ? ? = ?3 =?3 ≂ ?+
=+
(
?
6
(
)
)
=
?
?+
=+
(
(
?
(
)
6
)
)
=
?
(
(
)
5
(
(
?+
(
)+ =+
)
)+ =
?
(
)
(
)
)
?
_gcijicgeis pir gk mkhir`tei dgk eàtidi m kms aujc`ijgs p, q, r . Kms e`sems sij> ( )5
( )5
( )5
?
?
(kj( ))
Dg gstm airem pidgeis pkmjtgmr gk s`stgem dg gcumc`ijgs m rgsikvgr, m smogr> ?.62 6.23 6
=.? ?.67 6.278=4?278
6 =.=1 ?.6462=1
5
6.80==42=48 6.6=6071232 =.86347=2?4
Mk rgsikvgr gk s`stgem pmrm kms `jcóhj`tms gst`pukmdms pidgeis iotgjgr km aujc`ój y gvmkumdm gj kis pujtis sik`c`tmdis, gs dgc`r> =.67820?64?70 =.=843204=800 =.3?8037=?20
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
"Gj c`gjc`m uji `jtgjtm dgc`r m km hgjtg, gj ujm emjgrm gj qug tidis ki pugdmj gjtgjdgr, mkhi qug jujcm jmd`g supi mjtgs. Km pigsåm gs gxmctmegjtg ki cijtrmr`i". Vmuk D`rmc
Kms gcumc`ijgs d`agrgjc`mkgs pmrc`mkgs mpmrgcgj m egjudi gj c`grtis priokgems dg mpk`cmc`ój dg km `jhgj`gråm y c`gjc`ms. Vir gngepki pmrm dgscr`o`r gk ciepirtme`gjti dg ujm cugrdm i gk akuni dg cmkir m trmvàs dg uj cugrpi sg usmj gcumc`ijgs dg gstg t`pi. Gj gk prgsgjtg aikkgti sg gstud`mrmj kis trgs t`pis dg gcumc`ijgs eís usumkgs y cijic`dis, m smogr, gcumc`ój d`agrgjc`mk b`pgroók`cm, pmrmoók`cm y gkåpt`cm.
?? ?? ? ?? ?? ? ??
Km gcumc`ój gkåpt`cm t`gjg km airem (dijdg W gs ujm aujc`ój dg (x, y))> +
5 ( , )
Km gcumc`ój pmrmoók`cm t`gjg km airem (dijdg u gs ujm aujc`ój dg (x, t))> 5
Km gcumc`ój b`pgroók`cm t`gjg km airem (dijdg u gs ujm aujc`ój dg (x, t))> 5
M cijt`jumc`ój prgsgjtmrgeis km `jairemc`ój jgcgsmr`m pmrm km rgsikuc`ój dg gngrc`c`is qug `jvikucrgj mkhþj t`pi dg gstms gcumc`ijgs.
?? ?? ∂ ⊁ ?
Sgm gk priokgem dg vmkir `j`c`mk> +
5 ( , ) < ( , )
<
( , )5 ( , )
( ) <
Gk priokgem sg gjcugjtrm dga`j`di siorg ujm rgh`ój _ dgk pkmji pmrm ki cumk km aujc`ój h dgtgre`jm ujm cijd`c`ój `j`c`mk dgk gngrc`c`i. Gstm aujc`ój jis pripirc`ijm gk vmkir dg km aujc`ój W gj km arijtgrm dg _. Dg gstm airem jugstri trmomni sg k`e`tm m gjcijtrmr gk vmkir dg W gj kis pujtis `jtgr`irgs dg km rgh`ój _ sik`c`tmdis.
Kis pujtis gj kis cumkgs sg cmkcukm gk vmkir dg W sg dgjitmj usmjdi dijdg ` dgjitm km pis`c`ój dgk pujti gj gk gng x, y n dgjitm km pis`c`ój dgk pujti gj gk gng y. Cmog rgcmkcmr qug mk dgc`r pis`c`ój ji jis rgagr`eis mk vmkir qug tiem gj gsg gng d`cbi pujti.
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Gk mkhir`tei m ut`k`zmr jis hgjgrmrí uj s`stgem dg gcumc`ijgs k`jgmkgs, mk rgsikvgr gk e`sei iotgjdrgeis gk vmkir dg km aujc`ój W gj kis pujtis dgsgmdis. Mk ut`k`zmr kms aireukms pmrm kms dgr`vmdms rgspgcti m x, y iotgjgeis gk s`hu`gjtg mkhir`tei>
ℊ? += ∟= ? += ∟= =
?
+
+
=
ℊ
?
+
5 ( ,
)
Dijdg b, l dgjitmj gk pmsi gj gk gng x & y rgspgct`vmegjtg, cijs`dgrmjdi qug> 5
Kis cijtmdirgs cuepkgj> 5 =, ?, … ,
<
5
= y 5 =, ?, … ,
=.
Gngrc`c`i rgsugkti 31> Gcumc`ój d`agrgjc`mk b`pgroók`cm Mprix`eg km sikuc`ój dg
+
5
gj
5 {( , ) /
<
} ut`k`zmjdi j5e54.
Vmrm km arijtgrm dg _ cijs`dgrg ujm emhj`tud dg 6 gj kis kmdis mdymcgjtgs dgsdg gk ir`hgj dg ciirdgjmdms e`gjtrms qug gj kis itris kmdis km emhj`tud vmråm pripirc`ijmkegjtg bmstm kkghmr m =66 gj gk vàrt`cg ipugsti mk ir`hgj.
Mjtgs dg gstmokgcgr dg airem ckmrm kms cijd`c`ijgs dg arijtgrm pmrm jugstri gngrc`c`i, jis vmkgeis dg ujm hrma`cm pmrm gjtgjdgr km rgh`ój y kis pujtis dijdg jis p`dgj mprix`emr km aujc`ój W.
Kis pujtis `jd`cmdis dg d`agrgjtgs cikirgs rgprgsgjtmj kis kuhmrgs dijdg sg jis p`dg cmkcukmr km aujc`ój W . Ciei sg `jd`ci mjtgr`iregjtg, gstis pujtis sg rgprgsgjtmj egd`mjtg cijtmdirgs `, n. Vr`egri cmkcukmeis gk pmsi gj cmdm gng, b56.7 y l 56.?7.
Cij gsti pidgeis rgprgsgjtmr km aujc`ój W gj kis pujtis ouscmdis egd`mjtg>
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
===?=3 ?=???3 ⎛3=3?33⎙ ⎛ ⎙ (6.7, 6.?7) (6.7, 6.7) (6.7, 6.87) (=, 6.?7) (=, 6.7) (=, 6.87) (=.7, 6.?7) (=.7, 6.7) (=.7, 6.87)
5
Sghu`eis pir dga`j`r kms cijd`c`ijgs dg arijtgrm, kms e`sems sg gstmokgcgj dg airem `jd`rgctm gj gk gjujc`mdi dgk gngrc`c`i. Gk bgcbi dg qug gj kis kmdis mdymcgjtgs mk ir`hgj sg a`ng ciei 6 km aujc`ój W jis `jd`cm qug ( , 6) 5 (6, ) 5 6.
? += ∟= ? += ∟= += ∟= += ∟= ???= =?== 6?6= =?=3 ===6 ===? ?33= =3?= 63== =4?? =??6 =?3= 3?33 ???3 =?=3 ?3?4 ?=?? ???3 4?4= 3?3= ???= 3?33 3=36 33=? 43 33 ?3 34 3? 33
Gk crgc`e`gjti pripirc`ijmk gj kis kmdis rgstmjtgs bmstm gk vmkir dg =66 jis pgre`tg iotgjgr qug ( , =) 5 76 y (?, ) 5 =66 . M cijt`jumc`ój pmrt`cukmr`zmeis jugstri mkhir`tei hgjgrmk cij kis vmkirgs dg b, l cijic`dis pmrm gstg gngrc`c`i> =
?
6.7
+
+
=
?
6.?7
+
5
S`epk`a`cmjdi y mcieididmjdi iotgjgeis dg airem gspgcåa`cm> 4
46
+4
+ =1
+ =1
5
Vricgdgeis m rgmk`zmr kms `tgrmc`ijgs pmrm cmdm pis`okg vmkir dg `, n. 5=, 5= 4
46
+4
+ =1
+ =1
5
5=, 5? 4
46
+4
+ =1
+ =1
5
5=, 53 4
46
+4
+ =1
+ =1
5
5?, 5= 4
46
+4
+ =1
+ =1
5
5?, 5? 4
46
+4
+ =1
+ =1
5
5?, 53 4
46
+4
+ =1
+ =1
5
53, 5= 4
46
+4
+ =1
+ =1
5
53, 5? 4
46
+4
+ =1
+ =1
5
53, 53 4
46
+4
+ =1
+ =1
5
.
Egd`mjtg kms gxprgs`ijgs mjtgs dgsmrrikkmdms pidgeis kkghmr m uj s`stgem dg gcumc`ijgs qug jis pgre`tm mprix`emr kis vmkirgs dg W sik`c`tmdis, gsti sg kihrm pr`egrmegjtg rggepkmzmjdi cmdm vmkir dg x, y gj gk kmdi dgrgcbi dg cmdm gxprgs`ój. Mdgeís dgogeis gk`e`jmr kis vmkirgs qug ji sij `jcóhj`tms egd`mjtg gk
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
usi dg kms cijd`c`ijgs dg arijtgrm, gsti qu`grg dgc`r qug tidi tgre`ji W qug tgjhm ciei suoåjd`cg mk 6 i mk 4 dgog sgr rggepkmzmdi pir gk vmkir dgk e`sei dmdi qug sg gjcugjtrm gj km arijtgrm dg _. Trms rggepkmzmr ki mjtgs gst`pukmdi kkghmeis m uj s`stgem dg km airem>
Gk cumk t`gjg pir sikuc`ój>
===?=3 ⎜ ?=???3 3=3?33 ⎕ 5
⎚ ⎦
1.?33123=1230 =?.4873?67863 =2.8?706?771= =?.41180046?8 ?4.041==?2640 38.47=403?26? =2.8=?370130 38.4381318412 71.=0?6806?18
ℊ ∁
?
+ Ciei sg egjc`ijó mjtgr`iregjtg ujm gcumc`ój d`agrgjc`mk pmrmoók`cm t`gjg km airem 5 ( , ) , km e`sem gstí dga`j`dm pmrm 6 ; < : 6 mdgems dg tgjgr pir cijd`c`ijgs dg arijtgrm (6, ) 5 6 < ( , ) 5 6 y cijd`c`ij `j`c`mk ( , 6) 5 ( ).
Cijs`dgrg mbirm kis s`hu`gjtgs vmkirgs dga`j`dis pmrm b, l> 5
<
5
Dijdg b rgprgsgjtm gk pmsi gj kis vmkirgs dg x y l rgprgsgjtm pmrm qug `jtgrvmkis dg 't`gepi' sg trmomnmrí km sikuc`ój. Mpk`cmjdi gk eàtidi dg d`agrgjc`ms prihrgs`vms pidgeis gstmokgcgr uj mkhir`tei pmrm km mprix`emc`ój dg kis vmkirgs dg km aujc`ój W(x, t) pmrm uj t`gepi a`ni y kms x a`nmdms `j`c`mkegjtg. Gstg mkhir`tei t`gjg km airem>
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
+= ∁ ⋪=? =⋪? ∟= ∟= = 6 ⋪∟=? 5
+
Gepgzmrgeis dga`j`gjdi cmdm tgre`ji dgk mkhir`tei, pr`egri ciegjtmrgeis siorg kis vgctirgs W . Vmrm cmdm `tgrmc`ój gj rgmk`dmd vmr`meis 'gk t`gepi' gj gk cumk mprix`emeis km aujc`ój dmdi qug kis vmkirgs gj x sij kis e`seis pmrm tidis kis t`gepis. Dg gstm airem gk vgctir mprix`emc`ój W t`gjg km airem (cij 5 6, = , ?, … )>
( (
5
, ) , )
(
, )
Ciei gs dg gspgrmrsg sg jgcgs`tmrí gk vgctir `j`c`mk W 6, gk e`sei sg gjcugjtrm pir km cijd`c`ój `j`c`mk usmjdi m a(x). ( (
5
(
Km emtr`z M pir su pmrtg gs km s`hu`gjtg>
) )
)
Dga`j`gjdi ciei 5 . Gk vmkir ∁t rgsuktm iov`megjtg ujm cijstmjtg qug gs km e`sem qug l , km e`sem eukt`pk`cm mk vgctir o qug mpmrgcg s` km gcumc`ój ir`h`jmk t`gjg mýmd`dm m mkhujm aujc`ój h(x, t) gj su gxprgs`ój.
∟= ∟= ∁
Vmrm qug gk eàtidi prihrgs`vi sgm gstmokg sg dgog cuepk`r qug
=.
Gx`stg tmeo`àj uj mkhir`tei rghrgs`vi, dmdi pir> 5
\
+
U
Dijdg kis tàre`jis pmrt`c`pmjtgs rgprgsgjtmj kis e`seis mjtgs egjc`ijmdis y km emtr`z O gs pir su pmrtg>
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Nuk`i _umji K`em
Dg emjgrm hgjgrmk gstg mkhir`tei gs gstmokg s`j `epirtmr gk vmkir dg dmdi qug su emtr`z O gs gstr`ctmegjtg die`jmjtg d`mhijmkegjtg.
Gngrc`c`i rgsugkti 38> Mprix`emc`ój dg ujm gcumc`ój pmrmoók`cm ji bieihàjgm Mprix`eg km sikuc`ój dg
5
,
,
( , )5 ( , )5
,
( , )5
(
).
Wt`k`cg b56.?7, ∁t56.?7 gj gk `jstmjtg t56.7.
_gmk`zmrgeis gk gngrc`c`i usmjdi meois mkhir`teis pmrm vgr`a`cmr km d`agrgjc`m gjtrg meois, ciegjzmeis pir dgnmr km gcumc`ój dg km airem qug meois pmrtgj. Km gcumc`ój k`stm pmrm mpk`cmr cumkqu`gr dg kis dis mkhir`teis t`gjg km airem
5
+ .
Kis vmkirgs gj x dijdg mprix`emrgeis km sikuc`ój sij 6.?7, 6.7, 6.87. Vir itri kmdi gk pmsi dgk t`gepi sgrí 6.?7.
Cmkcukmeis gk vmkir dg (qug gs gk e`sei pmrm meois mkhir`teis) cijs`dgrmjdi qug pmrm gstg gngrc`c`i t`gjg gk vmkir dg =>
ℊ?? += =?3 ∞ 6 ⎜∞ ⎚ ∞ = ⎕∞ ⎦ 5
54
Dg gstm airem gkmoirmeis gk mkhir`tei prihrgs`vi, cijs`dgrmjdi m M ciei> =
?
6
5
=
5
?
6
Dg gstm airem>
5
=
8 4 6
+ 6.?7 5
?
4 8 4
6 4 8
8 4 6
4 8 4
+ 6.?7
6 4 8
pmrm n56, =.
Ciei cijd`c`ój `j`c`mk tgjgeis gk vgctir>
?
(6.?7) (6.7)
5
(6.87)
5
? = ? ?
Bmc`gjdi `tgrmr gk eàtidi prihrgs`vi iotgjgeis>
?
5 6 ( 5 6)
5
8 4 6
4 8 4
6 4 8
? = ? ?
6 + 6.?7 6 5 6
6.040848412?07 =.343=478767? 6.0408412?07
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
5 = ( 5 6.?7)
?
5
8 4 6
4 8 4
6 4 8
6.040848412?07 6.?7 =.332=40?8700 =.343=478767? + 6.?7 6.?7 5 =.211746768?2 6.0408412?07 6.?7 =.332=4068700
Vidgeis vgr uj smkti gjtrg vmkirgs cijsgcut`vis euy mkti dgo`di m qug gk vmkir dg gs emyir qug = pir ki qug gk mkhir`tei prihrgs`vi ji gs dgk tidi gstmokg.
Cijs`dgrgeis mbirm gk mkhir`tei rghrgs`vi, su emtr`z O gs km s`hu`gjtg> 0 4 6
5
4 0 4
6 4 0
Gj gk mkhir`tei usmeis O `jvgrsm, pir ki qug km cmkcukmeis>
∟? ∟?∟? ∟? =? ∟= =?∟∟== 3 3∟=
∟=
6.=4830??06?40 5 2.=13?17361=? Ø =6 3.1?2==80=323 Ø =6
2.=13?17361=? Ø =6 6.=23183410322 2.=13?17361=? Ø =6
∟?∟?
3.1?2==80=323 Ø =6 2.=13?17361=? Ø =6 6.=4830??06?40
Vir ki qug jugstri mkhir`tei qugdm dg km airem> 5
+ 6.?7
Dg gstm airem bmcgeis `tgrmr gk eàtidi pmrm n5=, ?.
∞ ==?=3= ∟= ⎣⎕∞ ⎦ ⎡ =???3? ∟= ?
5=
5
? = ?
6.?7 + 6.?7 6.?7 6.?7
6.??260=64=44? 5 6.3?6263=48040 6.??260=64=44?
?
5?
6.??260=64=44? 6.7 6.3?6263=48040 + 6.?7 6.7 6.??260=64=44? 6.7
5
6.=6=?477636= 5 6.=307?08?82=2 6.=6=?4777636=
Vidgeis iosgrvmr kms mprix`emc`ijgs pmrm meois eàtidis, gs ckmri qug gk eàtidi rghrgs`vi eugstrm eís gstmo`k`dmd qug gk prihrgs`vi pir ki qug pidgeis cija`mr tmk vgz uj pici eís gj àk.
?
Km gcumc`ój d`agrgjc`mk b`pgroók`cm cijst`tuyg gk þkt`ei gngepki dg gcumc`ijgs d`agrgjc`mkgs pmrc`mkgs qug
gstud`mrgeis gj gk prgsgjtg. Wjm gcumc`ój dg gstg t`pi, t`gjg km airem> gjcugjtrm dga`j`dm gj 6 ;
5
; , : 6 cij cijd`c`ijgs dg arijtgrm ( , 6) 5 ( )<
. Km e`sem sg
5 ( ).
Nuk`i _umji K`em
Mjík`s`s Jueàr`ci - Mpujtgs y gngrc`c`is rgsugktis
Egd`mjtg gk usi dg d`agrgjc`ms a`j`tms cij kis pmsis ym cijic`dis b, l kkghmeis m uj mkhir`tei dg mprix`emc`ój dg km airem>
+= ∟= 5
,
5 =, ?, …
Gj gstg mkhir`tei km emtr`z M cirrgspijdg m km s`hu`gjtg>
Cijs`dgrmjdi qug 5
pmrm gstg egtidi.
Ciei ki `jd`cm su mkhir`tei, gstg eàtidi t`gjg gj cieþj cij gk dg km gcumc`ój pmrmoók`cm gk mprix`emr kis vmkirgs pmrm x a`nmdms prgv`megjtg gj vmkirgs dg t d`agrgjtgs. Ciei pidgeis vgr gj gk mkhir`tei, jgcgs`tmeis kis vgctirgs W 6 y W =. Kis e`seis sg bmkkmj cij kms cijd`c`ijgs `j`c`mkgs, dg km s`hu`gjtg airem>
= 6 = 6 ∟=6?6⋪ ⋪∟=? = 6 = ? == = ?=⋪ ? =⋪ ? 3 ∟== ⎕ ∟= ∟? ∟= ⎦ = ∟= += 5
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Gj hgjgrmk pmrm W = sg t`gjg qug sus tàre`jis sg hgjgrmj egd`mjtg> 5 (=
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Gngrc`c`i rgsugkti 32> Gcumc`ój d`agrgjc`mk b`pgroók`cm ji bieihàjgm
Mprix`emr km sikuc`ój gj t56.= dg
Cijs`dgrg ( , ) 5
( , )5 , ( , )5
5 , (
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),
; =, : 6 .
( , ) 5 cij b56.?7 y l56.=.
