Mate Aplicada I Final
February 14, 2023 | Author: Anonymous | Category: N/A
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@]MP@OC H@ @MGIGAÁC Act`aãtdmc Cpodmchc 3
Mct`hrãtdmg6 Odm`imdchg \ul`odg \ghráku`z X`oog Cuxdodcr6
Krdm`ohc Mcrhgic uditg ]`a`str`
MC\ÃMX@\ H@ OC K@GA@X\ÁC CICOÁXDMC
Oc k`ga`trác
cicoátdmc `stuhdc
ocs edkurcs ocs
k`gaätrdmcs k`gaätrdmc s a`hdcit`
tämidmcs lãsdmcs h`o ciãodsds act`aãtdmg y h`o ãok`lrc ãok`lrc `i ui h`t`radichg sdst`ac h`t`radichg sdst`ac h` mggrh`ichcs. ]u h`scrrgoog bdstýrdmg mgad`izc
mgi oc k`ga`trác k`ga`trác mcrt`sdcic, mgitdiöc mgi oc cpcrdmdýi h` oc oc k`ga`trác hde`r`imdco h` h` Mcro Erd`hrdmb Kcuss Kcuss y aãs tcrh` mgi `o h`scrrgoog h`
oc k`ga`trác cok`lrcdmc. oc cok`lrcdmc. Cmtucoa`i Cmtucoa`it` t` oc k`ga`trác cicoátdmc td`i` aöotdpo`s cpodmcmdgi`s aãs cooã h` ocs act`aãtdmcs y oc dik`id`rác, pu`s egrac pcrt` cbgrc h`o trclcfg h` chadidstrchgr`s pcrc oc poci`cmdýi h` `strct`kdcs y ogkástdmc `i oc tgac h` h`mdsdgi`s.
@ i `o ciãodsds krcedmg ocs `mucmdgi`s cok`lrcdmcs s` r`pr`s`itci mgi oái`cs k`gaätrdmcs og qu` p`radt` r`ocmdgicr vcrdclo`s mgag pr`mdgs, mcitdhch h`acihc y ger`mdhc, mgstgs hdr`mtgs ` dihdr`mtgs, div`rsdýi, mgisuag, `tm . ^crc r`ocmdgicr `o ãok`lrc y oc k`ga`trác s` utdodzc `o sdst`ac h` mggrh`ichcs qu` prgpgrmdgic ui a`hdg pcrc ogmcodzcr
puitgs `sp`máedmgs `i `o pocig g `spcmdg.
3
^ocig mcrt`sdcig 5 T 0 8
DD muchrcit` R
-5
-0
( ” -8 , +-=) -<
= < 3 > -3 -3 > -< -= -8 -0 -5
D muchrcit` 3
<
=( + ,8+ ) 0
DDD muchrcit`
DU muchrcit`
( ” , ” )
( + , ” )
5
@i `o mcsg h` ocs mggrh`ichcs r`mtcikuocr`s, s` usc mgag r`e`r`imdc `o egrachg pgr oái`cs r`mtcs, qu` s` mgrtc `i ãikuog r`mtg. Ogs `f`s grh`ichgs hdvdh`i co pocig `i muctrg ãr`cs g r`kdgi`s h`igadichcs muchrcit`s.
Oái`c r`mtc Pic oái`c r`mtc `s uic sum`sdýi dih`edidhc y mgitdiuc h` puitgs. \`pr`s`itc uic `mucmdýi h` hgs vcrdclo`s, oc egrac k`i`rco h` oc oái`c r`mtc `s6 cx + ly + m 1 >
c y l sgi mgistcit`s y pgr og a`igs uic `s hdstditc h` m`rg.
@o glf`tdvg h` oc k`ga`trác cicoátdmc `s h`scrrgoocr aätghgs, qu` `apo`ci ui aáidag h` puitgs pcrc `stclo`m`r oc mgrr`spgih`imdc.
Pic egrac s`imdooc h` trczcr uic oái`c r`mtc `s c trcväs h` sus
dit`rs`mmdgi`s, `i `oocs `s hgih` oc oái`c mgrtc ogs `f`s ‟x‑ & ‟y‑. @f`apog6 Xrczcr oc r`mtc 2x + 3>y ”8> 1 > Dit`rs`mmdýi h` x, s` sustdtuy` y 1 >, pcrc cv`rdkucr `o vcogr h`o `f` ‑x‑. Dit`rs`mmdýi h` y, s` sustdtuy` x 1 >, pcrc cv`rdkucr `o vcogr h`o `f` ‑y‑.
]` `imu`itrci ogs puitgs glt`idhgs h` ocs `mucmdgi`s `i `o pocig mcrt`sdcig, y s` ui`i a`hdcit` uic oái`c pcrc trczcr oc r`mtc.
=
Krcedmcr 5
T
0 8 Dit`rm`ptg mucihg x 1 >: y 1 8
= < 3
R
> -0
-8
-=
-<
-3
>
3
<
=
8
0
5
;
-3 Dit`rm`ptg mucihg y 1 >: x 1 0
-< -= -8
^`ihd`it` h` oc r`mtc Oc dimodicmdýi h` uic r`mtc h`t`radic su p`ihd`it`, og qu` `stc `xpr`schg `i täradigs h`o ãikuog qu` egrac oc oái`c r`mtc mgi `o `f`
‟x‑. c) Egrac hgs puitgs6 Pic h` ocs prgpd`hch`s euihca`itco`s h` uic r`mtc `s su p`ihd`it` mgistcit`, sd (x 3, y3) y (x.8
->.<
>
Glt`i`r oc `mucmdýi h` oc r`mtc qu` pcsc pgr ogs puitgs (5, 2) y ( ” 3>, 8). ^uitgs
( ” ” 3>, 8) x< y<
(5, 2) x3 y3
y ” y3 1 (y
Auotdpodmcr pgr AMA 1 8 8y ” x ” ]d R 1 > 8y 1
]d T 1 > -x 1 =/<
R 1 ->
Gtrc egrac6 y ” y3 1 (y81>
]d T 1 > R 1 ->
2 ; 5 0 8 = < 3 > -=>
-
-0
>
]` scl` qu` uic r`mtc pcsc pgr ogs puitgs (3, 5) y ( ” 8,
3
3
<
m) Egrac p`ihd`it` ` dit`rs`mmdýi
y 1 ax + l
<
=
=
8
Hchcs ocs dit`rs`mmdgi`s (>, 4) y a 1 ;, `imu`itr` oc `mucmdýi y krcedqu`.
y 1 ;x + 4
y 1 ax + l Dit`rm`ptgs6
Krcedmc 3> 4 2 ; 5 0 8 = < 3 > -<
-3.0
-3
->.0
>
Hchg l 1 0 y a 1 = `imu`itr` oc `mucmdýi y krcedqu`. y 1 ax + l
y 1 =x + 0
Dit`rm`ptgs6
33
Krcedmc 5 0 0
8
=
<
3 > -=
-<
-3
> >
3
<
=
8
0
5
h) Egrac mgi dit`rs`mmdgi`s
x+y13 c l Bcoocr oc `mucmdýi h` oc r`mtc hchcs ocs dit`rs`mmdgi`s (8, >) y (>, ;). Dit`rs`mmdgi`s Dit`rs`mmdg i`s
(8, c >)
Auotdpodmcr pgr ama6 -<
-3 -3 > -3
3
<
=
8
0
5
-<
Hchcs ocs dit`rs`mmdgi`s (=, >) y (>, 0) h`t`radi` su `mucmdýi y krcedqu`. Dit`rs`mmdgi`s Dit`rs`mmdg i`s
(=, >)
(>, 0). c
l
3=
30 Auotdpodmcr pgr ama6 30
Dit`rm`ptgs Mucihg x 1 >: Mucihg y 1 >: Krcedmc ; T 5 0 8 = < 3 R
> -<
-3
>
3
<
=
8
0
-3 -<
\`mtcs pcrco`ocs p`rp`ihdmuocr`s y mgdimdh`it`s Hgs r`mtcs mucoqud`rc mucoqud `rc h` ui pocig pu`h` s`r pcrco`ocs g ld`i r`mtcs qu` s` mgrtci `i ãikuog r`mtg y gltusg, ocs muco`s s` h`igadici p`rp`ihdmuocr`s, hgs pcrco`ocs td`i`i ãikuog h` dimodicmdýi dkuco,
ad`itrcs qu` ocs r`mtcs p`rp`ihdmuocr`s t`ihrãi p`ihd`it`s r`mdprgmcs y h` sdkig mgitrcrdg mgi r`sp`mtg c oc gtrc y vdm`v`rsc. c) \`mtcs pcrco`ocs
Bcoocr oc `mucmdýi h` oc r`mtc qu` pcsc pgr ogs puitgs (
x 1 -
3
<
=
-3 -<
m) \`mtcs qu` s` dit`rm`ptci
=x ” 8y + 5 1 >
x ”
\`sgov`r pgr mucoqud`r aätghg h` `mucmdgi`s sdauotãi`cs.
*( ” ” =)
=x ” 8y + 5 1 >
=x ” 8y + 5 1 >
x ”
”=x + 5y + 4 1 > x 1 -3<
3;
Krcedmc 8 = < 3 > -35 -3 - 30 -38 -3= -3< -33 -3> -4 -2 -; -5 -0 -8 -= -< -3 > -3
3
<
=
8
-< -= -8 -0 -5 -; -2 (-3
h) \`mtcs qu` s` dit`rm`ptci p`rp`ihdmuocra`it` (egraci ãikuogs
h` 4>¶) 0y 1 8x + 30 8y 1 ” 0x +
*( ” ” 0)
8y 1 ” 0x +
x 1 5>/83 0y 1 8x + 30
0y 1 (8) (5> / 83) + 30 0y 1 / 83 + 30
y 1 3;3/83 Krcedmc ; 5
4>¶
0
(5>/83, 3;3/83)
8 = < 3 > -0
-8
-=
-<
-3
>
3
<
=
8
0
5
-3
C^ODMCMDGI@] H@ OC OÁI@C \@MXC Ocs mcitdhch`s h`acihchcs y ger`mdhcs h` ui h`t`radichg ld`i g prghumtg, sgi euimdgi`s h` md`rtg iöa`rg h` vcrdclo`s, qu` mgapr`ih`i `o pr`mdg h`o prghumtg, ogs pr`mdgs h` ogs sustdtutgs y sus mgapo`a`itgs, dikr`sgs hdspgidlo`s, r`mursgs edicimd`rgs, kcstgs, mgstualr`s, `tm. ]di `alcrkg `i ui ciãodsds `mgiýadmg `o`a`itco su`o` mgisdh`rcrs` c oc ge`rtc y oc h`acihc mgag euimdgi`s sgoca`it` h` ocs vcrdclo`s aãs
34
dapgrtcit`s, sd`ihg `stg pgr og k`i`rco `o pr`mdg h`o prghumtg c mgisdh`rcr. @i oc prãmtdmc cokuics `mucmdgi`s h` ge`rtc y h`acihc sgi cprgxdachca`it` odi`co`s `i `o dit`rvcog r`o`vcit` mgrr`spgihd`it`, `i tcitg qu` gtrcs ig sgi odi`co`s `i clsgoutg. Cui `i `o mcsg h` `stcs öotdacs ocs `mucmdgi`s odi`co`s pu`h`i prgpgrmdgicr r`pr`s`itcmdgi`s rczgiclo`a`it` `xcmtcs h` oc ge`rtc y oc h`acihc pcrc ui dit`rvcog odadtchg. Pic ge`rtc i`kctdvc dapodmcrác qu` ogs ld`i`s ig s` bcooci hdspgidlo`s
`i `o a`rmchg, `xdst`i hgs rczgi`s pcrc `stg, `o ‟`e`mtg `sp`muocmdýi‑ y qu` yc ig s` prghuzmc hdmbg ld`i. Pic h`acihc i`kctdvc s`rdc prglclo` mucihg `o pr`mdg s` hdspcrc `i ui 3>>%.
Euimdýi h`acihc Pic mgapcóác bc cicodzchg sus v`itcs y bc `imgitrchg qu` sus mod`it`s mgaprci % aãs mucihg pgr mchc S> h` r`hummdýi `i `o pr`mdg, mucihg `st` `s h` S3> o` mgrr`spgih` uic v`itc h` 0>> uidhch`s. ]` o` pdh`6 c) Oc euimdýi h` h`acihc l) Krcedqu` m) Ucrdcmdýi mucihg oc h`acihc s`c h` 4>> uidhch`s
c) Egrac hgs puitgs
^rda`r puitg x 1 0>>
y 1 3<
]`kuihg puitg6 mcitdhch cua`itc % (3>>) mucihg `o pr`mdg s` r`hum` `i S > `stg `s
x 1 5>>
y 1 3>
=0 => 0 > -3>> > -0
3>> > =>> 8>> 0>> 5>> ;>> 2>> 4>> 3>>>33>>3>
Mucihg `o pr`mdg h` ogs r`ogf`s `s h` S3>>. >> ig bcy h`acihc cokuic,
mucihg `o pr`mdg `s iuog, `s h`mdr ‟krctudtg‑ s` h`acihci 0> r`ogf`s. ·Muão `s oc `mucmdýi h` h`acihc y sus vcogr`s7 Egrac mgi dit`rs`mmdýi ^rda`r puitg
x1>
y 1 3>>
]`kuihg puitg6 x 1 0>
y1>
Dit`rs`mmdgi`s Dit`rs`mmdg i`s
(0>, >)
(>, 3>>)
l
Auotdpodmcr pgr ama 1 3>>
c
Ucogr`s R 1 0> T 1 3>>
y + > 1 >: @mucmdýi h` h`acihc Krcedmc 3 33> 3>> 4> 2> ;> 5> 0> 8> => 3> > --3> 3> > 3> => 8> 0> 5> ;> 2> 4> 3> 3 >>
]` v`ih`i r`ogf`s mucihg `o pr`mdg `s h` S4>. >> y 8> mucihg `o pr`mdg `s h` S2>.>>, krcedqu` qu` gmurr` mgi `o pr`mdg sd oc h`acihc `s h` 4> r`ogf`s. Egrac hgs puitgs ^rda`r puitg
x 1
y 1 4>
]`kuihg puitg6 x 1 8>
y 1 2>
^uitgs
(, 4>) x3 y3
(8>, 2>) x< y<
3 3>> 2> 5> 8> > - > -
8>
5>
2> 3>> 3 38> 35> 32> > > odit`rics h` lgosdoog c ui pr`mdg h` S4.>>, sd `o pr`mdg lcfc c S5 >>, hdspgihrã pcrc oc v`itc sgog h` > odit`rics. ]` pdh`6 c) Euimdýi l) Krcedmg m) vcrdcmdýi h`o pr`mdg sd x 1 ;>>
c) Egrac hgs puitgs
^rda`r puitg x 1 0>>
y14
]`kuihg puitg6 mcitdhch cua`itc % (3>>) mucihg `o pr`mdg s` r`hum` `i S > `stg `s
x 1 >
^uitgs
(>, 5) x< y<
(0>>, 4) x3 y3
3>>y ” x ” 8>> 1 >
Auotdpodmcr pgr ama 1 3>> 3>>
38
y 15
Krcedmc
3< 3> 2 5 8 < > -0>> -8>> -=>> -> -3>> > -<
3>> > =>> 8>> 0>> 5>> ;>> 2>> 4>>
ge`rtc \`sgov`r `o sdst`ac h` `mucmdgi`s sdauotãi`cs a`hdcit` mucoqud`r aätghg6
r`stc \`sgovd`ihg pgr aätghg h` suac y r`stc
Krcedmc 8 =
Dit`rm`ptgs
<
H`acihc6 (>, ) >)
3
Ge`rtc6 (>, ” 5): (3.0, >)
> -<
-3
>
3
-3
<
=
H`acihc Ge`rtc
-< -= -8 -0 -5
@imu`itr` `o puitg h` `qudodlrdg h` a`rmchg hchg `o sdkud`it` sdst`ac h` `mucmdgi`s6 y 1 0 ” =x H`acihc
y 1 8x + 3< Ge`rtc
\`sgovd`ihg pgr aätghg h` suac y r`stc r`stc
4 2
Dit`rm`ptgs >) H`acihc6 (>, 0): (3.5, >)
; 5
H`acihc
0
Ge`rtc
Ge`rtc6 (>, 3) >)
8
@mgiýadmca`it` drr`o`vcit`
= < 3 -0
-8
-=
-<
> -3 > -3
3
<
=
^uitg h` `qudodlrdg h` oc `apr`sc Ogs hdckrcacs h` `qudodlrdg sgi h` usg er`mu`it` `i `mgigaác, ogs muco`s p`radt`i cicodzcr ocs dieou`imdcs h` hdv`rsgs ecmtgr`s `i oc csdkicmdýi h` pr`mdgs g csdkicmdýi h` prghumtgs. Ogs mgstgs s` hdvdh`i `i6 mgstgs edfgs y mgstgs vcrdclo`s, ogs edfgs p`raci`m`i mgistcit`s `i tghgs ogs idv`o`s h` prghummdýi y ogs mgstgs vcrdclo`s sgi og qu` mcaldci mgi oc mcitdhch prghumdhc tco`s mgag scocrdgs, s`rvdmdgs, pr`stcmdgi`s, `tm. @o mgstg tgtco (MX) c mucoqud`r idv`o h` prghummdýi `s oc suac h` mgstgs edfgs y vcrdclo`s pcrc hdmbg idv`o, oc oái`c h` mgstg tgtco, ýs`c oc p`ihd`it`, `s dkuco co dimr`a`itg h`o mgstg vcrdclo` pgr mchc uidhch h`
cua`itg `i oc prghummdýi, mgisdh`rcihg `o supu`stg h` qu` `o mgstg vcrdclo` `s prgpgrmdgico c oc prghummdý prghummdýi i `i ui dit`rvcog sdkidedmctdvg. Oc p`ihd`it` DX r`pr`s`itc ogs dikr`sgs tgtco`s h`p`ihd`ihg `i sd h` ocs mcitdhch`s v`ihdhcs, su dit`rs`mmdýi mgi ogs `f`s s` `imu`itrc `i `o grdk`i y oc p`ihd`it` `s dkuco co pr`mdg pgr uidhch, supgid`ihg qu` tco pr`mdg `s mgistcit` pcrc mucoqud`r mcitdhch v`ihdhc. @i sd `o puitg h` `qudodlrdg `s `o puitg `i `o muco s` dit`rs`mtci oc r`mtcs
‟DX‑ y ‟MX‑, qu` r`pr`s`itci oc mcitdhch c oc qu` `o prghumtg ig gltd`i` p`rhdhcs id kcicimdcs, kcicimdcs , `i mchc mcsg, y bcmdc clcfg r`pr`s`itc r`pr`s`it c pärhdhcs: `i mgimousdýi `o puitg h` `qudodlrdg `s oc mcitdhch pcrc oc muco bcy fustca`it` `o dikr`sg suedmd`it` suedmd`it` pcrc mucoqud`r mgstg. mgstg. @f`apog6 ]` td`i` `i uic dihustrdc ui `stuhdg `i `o muco s` h`s`c mgigm`r `o `qudodlrdg h` oc `apr`sc, mgitcihg pcrc `o `e`mtg mgi ocs sdkud`it`s vcrdclo`s6 >>
>>
Pidhch`s prghumdhcs 0,>>> pr`mdg uidtcrdg S0 , mgstg edfg S=>>> : mgstg vcrdclo` 8>% h`o dikr`sg. ]` pdh`6 3) Oc euimdýi dikr`sg dikr`sg y `o dikr`sg adsag. >> uidhch`s co a`s y krcedqu`. 8) Glt`ikc `o puitg h` `qudodlrdg h` oc `apr`sc ^crc `imgitrcr `o vcogr h` ‟y‑ sustdtudags `i mucoqud`rc h` ocs hgs euimdgi`s `o vcogr h` ‟x‑ (mcitdhch prghumdhc y v`ihdhc) , `imgitrchg `i `o puitg h` `qudodlrdg.
D1 0. @o idv`o h` prghummdýi hgih` s` mulr`i ogs mgstgs edfgs `s
mucihg oc euimdýi h` mgstg edfg `s dkuco d kuco c oc h` dikr`sg tgtco: Me 1 Dt Me 1 Dt
=3
Krãedmc
0 = > -3.0
-3
->.0
>
>. 0 3 3. 0 < ,>>> uidhch`s prghumdhcs c S3>>> y ui mgstg edfg h` S0>>>>>, mgstg vcrdclo` 1 5>%, glt`ikc6 3) Oc euimdýi dikr`sg dikr`sg y `o dikr`sg adsag. ,>>> uidhch`s co a`s y krcedqu`. 8) Glt`ikc `o puitg h` `qudodlrdg h` oc `apr`sc 0) ·C quä idv`o h` prghummdýi y v`itcs v`itcs `o dihustrdco mulr` sus mgstg mgstgss edfgs7
p(x): `o dikr`sg tgtco `s dkuco co pr`mdg 3. Euimdýi dikr`sg6 Dt 1 p(x): `o pgr oc mcitdhch Euimdýi h` dikr`sg tgtco
D 1 3>x
Dikr`sg tgtco pcrc `o idv`o h` prghummdýi D1 >
^crc `imgitrcr `o vcogr h` ‟y‑ sustdtudags `i muc oqud`rc h` ocs hgs euimdgi`s `o vcogr h` ‟x‑ (mcitdhch prghumdhc y v`ihdhc) , `imgitrchg `i `o puitg h` `qudodlrdg.
D 1 3>x
D1
^uitg h` `qudodlrdg (
D1 )
0. @o idv`o h` prghummdýi hgih` s` mulr`i ogs mgstgs edfgs `s mucihg oc euimdýi h` mgstg edfg `s dkuco c oc h` dikr`sg tgtco: Me 1 Dt
Me 1 Dt
x 1 0>>
Krcedmc =0
=>
Kcicimdc
Ado`s h` uidhch`s prghumdhcs Mgstg edfg
Mgstg Xgtco
Dikr`sg tgtco
Euimdýi Mgisuag y Cbgrrg @o mgisuag s` utdodzc mgag uic euimdýi ckr`kchc, qu` h`p`ih` `xmousdvca`it` `xmousdvca` it` h`o dikr`sg hdspgidlo`. ]upu`stgs6 3) @xdst` md`rtg agitg clsgoutg c lsgoutg h` mgisuag mgisu ag i`m`scrdg i`m`s crdg pcrc pc rc acit`i`r oc vdhc cui mucihg ig bcyc dikr`sg agi`tcrdg. =0
adoogi`s
T
]` pdh`6 3) Oc `mucmdýi h` euimdýi euimdýi mgisuag M
g
i
s
a
g
0 > 3
@MPCMDGI@] < = 8MPCH\ÃXDMC] 0 5
;
=;
@MPCMDGI@] MPCH\CX MPCH\CXDMC] DMC] Pic `mucmdýi muchrãtdmc pu`h` r`pr`s`itcr c uic mdrmuie`r`imdc, `odps`, bdpärlgoc, pcrãlgoc y mcsgs h`k`i`rctdvgs, qu` s` r`pr`s`itci `i `o ciãodsds `i euimdýi c ocs mcrcmt`rástdmcs h` ogs egmgs qu` r`pr`s`itci. Oc dh`itdedmcmdýi h` oc `mucmdýi muchrãtdmc qu` r`pr`s`itc mchc uig h` `stgs tdpgs h` murvcs `s h` oc egrac6
Cx‑, hãihgo` uic mgiigtcmdýi mgiigt cmdýi h` `mucmdýi h` s`kui s`kuihg hg krchg g muchrãtdmc. Hchg qu` `s pgsdlo` `odadicr `o täradig ‟xy‑ h` mucoqud`r `mucmdýi h` s`kuihg krchg, bcmd`ihg qu` ‟L‑ s`c dkuco c m`rg, c trcväs h` oc rgtcmdýi h` ogs `f`s grdkdico`s grdkdico `s cor`h`hgr h`o grdk`i. @i lcs` c og cit`rdgr oc `mucmdýi muchrãtdmc s` egrauoc `i h`edidtdvc h` oc sdkud`it` aci`rc6 Cx>>7 m) ·Muão `s `o dikr`sg aãs lcfg h` h` ocs 2> p`rsgics qu` td`i`i ogs dikr`sgs aãs cotgs. ]ustdtuy`ihg `i oc eýrauoc6 dikr`sg (x) 1 3,>>>,>>> 1 3>5 c)
I 1 3>:
l) @o iöa`rg h` dihdvdhugs mgi dikr`sgs qu` `xm`h`i $=,5>> ;
3>3>
I 1 >)=/,>>>.>> I 1 3> 1 8
=/<
(3> ) H` aci`rc qu` `o iöa`rg h` dihdvdhugs mgi dikr`sgs `itr` $=,5>> y
m) 2> p`rsgics mgi dikr`sg aãs cotgs, I 1 2>
^crc mcomuocr `o dikr`sg h` ocs 2> p`rsgics s` h`sp`fc ‟x‑ 50
x 1 4 * 3>5 1 4,>>>,>>> @o dikr`sg aãs lcfg h` ocs 2> p`rsgics mgi `o dikr`sg aãs cotg. @f`apog6 I 1 35 * 3>3< x0/= c) ·Muãitcs p`rsgics td`i`i dikr`sgs pgr clcfg h` S2,>>>.>>7 l) ·Muãitcs
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c) p`rsgics mgi dikr`sgs h`lcfg h` S2,>>>.>>, `i su egrac `xpgi`imdco 2 * 3>=. ^crc h`t`radicr `o iöa`rg h` p`rsgics qu` mgi ui dikr`sg pgr h`lcfg h` S2,>>>.>>, c oc pglocmdýi tgtco h`o pcás, s` o` r`stc `o iöa`rg h` p`rsgics qu` td`i`i ui dikr`sg pgr crrdlc h` S2,>>>. >>.
l) @o iöa`rg h` dihdvdhugs mgi dikr`sgs pgr crrdlc h` S3>>.. I1 @o iöa`rg h` p`rsgics mgi dikr`sgs sup`rdgr`s c 3,>>>,>>> `s I 1 35 * 3>35)0/= @s csá mgag `o iöa`rg h` p`rsgics mgi dikr`sgs pgr `imdac h` 3>> p`rg a`igr qu` 3,>>>,>>> `s6 03,> ” 35>> 1 84,5>> m) 0> p`rsgics mgi dikr`sg aãs cotgs, I 1 0>
@s h`mdr 2,>>>,>>> `s `o dikr`sg aãs lcfg h` ocs 0> p`rsgics mgi ogs dikr`sgs acs cotgs. @f`apog6
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MCOMPOG HDE@\@IMDCO T ]P] C^ODMCMDGI@] Oc prda`rc h`rdvchc h` uic euimdýi `s ui puitg mgrr`spgihd`it` c oc p`ihd`it` h` oc krãedmc h` oc euimdýi `i hdmbg puitg, `o agvdad`itg h` uic euimdýi ig odi`co: oc h`rdvchc `s oc p`ihd`it` `itr` hgs puitgs qu` `stãi c uic hdstcimdc diedidt`sdaco.
Mcsgs h` oc oái`c r`mtc Mucihg y 1 ax + l, `i `st` mcs g oc euimdýi h`rdvchc h` ‟y‑ mgi r`sp`mtg c ‟x‑ `s
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Aãxdags y aáidags r`octdvgs
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Aãxdag r`octdvg
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Mrdt`rdgs pcrc h`edidr aãxdags y aáidags r`octdvgs ;3
3) ]d oc prda`rc h`rdvchc mcaldc h` i`kctdvg c pgsdtdvg `i x 1 c, dapodmc qu` `xdst` ui puitg aáidag r`octdvg `i x 1 c. -8
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Dapodmc qu` qu` `o gtrg dit`rvcog dit`rvcog < ≨ x ≨ 8. @o vcogr aáidag h` y s` `imu`itr` `imu`i tr` `i `o puitg `xtr`ag x 1 < y `o vcogr aãxdag gmurr` `i `o puitg `xtr`ag x 1 8.
OC H@\DUCHC T @O CIÃOD]D] AC\KDICO Oc euimdýi h` mgstg tgtco, a`hdg y acrkdico
]d `o mgstg tgtco ‟y‑ h` prghumdr y mga`rmdcodzcr ‟x‑ uidhch`s h` ui crtámuog s` supgi` qu` `stã `i euimdýi h` ‟x‑, sgoca`it` `itgim`s oc euimdýi h` mgstg tgtco s` pu`h` r`pr`s`itcr a`hdcit` oc `xpr`sdýi y 1
Ɯ(x). ^rgpd`hch`s6 3) Mucihg `o iöa`rg h` uidhch`s prghumdhcs `s dkuco c m`rg, `o mgstg tgtco `s iuog g pgsdtdvg, sd Ɯ(x) ≠ >, ]d e(g) 9 > `itgim`s Ɯ (>), r`pr`s`itc `o mgstg edfg. : sdi `alcrkg, `i ogs idv`o`s didmdco`s h` prghummdýi, oc murvc h` mgstg tgtco su`o` s`r mýimcvc bcmdc clcfg, og qu` mgrr`spgih` c ui mgstg acrkdico h`mr`md`it`. (@o mgstg acrkdico `s `o dimr`a`itg dimr`a`itg qu` suer` `o mgstg mucihg s` dimr`a`itc oc prghummdýi `i uic uidhch), `s h`mdr qu` `o dimr`a`itg h`o mgstg tgtco qu` s` supgi` mgi uic
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prghummdýi chdmdgico, og qu` hc oukcr c oc csgmdcmdýi mgi r`ihdad`itgs h`mr`md`it`s. ]d oc euimdýi h`o mgstg tgtco s` r`pr`s`itc pgr y1 Ɯ¸(x), `itgim`s `o mgstg
a`hdg pgr uidhch y 1 y/x 1 Ɯ(x)/x, `o mgstg acrkdico `s Ɯ¸(x). Oc prda`rc h`rdvchc h`o mgstg a`hg acrkdico `s x Ɯ¸(x) ” e(x). x adoogi`s h` uidhch`s prghumdhcs, sd`ihg oc euimdýi MX 1 => + 2x + >.0x adoogi`s h` uidhch`s csmd`ih` c l) Euimdýi h` mgstg a`hdg (ma`) 1 Ɯ(x) ma` 1 x
ma` 1
Ucogr pcrc pcrc `o idv`o h` prghummdýi x 1 3> ma` 1 Gtrc egrac h` mcomuocr `o mgstg a`hdg6 Mgstg tgtco / prghummdýi.
@o mgstg a`hdg pcrc ui idv`o h` prghummdýi h` 3> adoogi`s h` uidhch`s s`rã h` S. m) Euimdýi h` mgstg acrkdico (mak) 1 Ɯ¸(x) ^rda`rc h`rdvchc Ucogr h`o mgstg acrkdico mucihg x 1 @o mgstg acrkdico pcrc `o idv`o h` prghummdýi h` 3> adoogi`s h` uidhch`s `s h` S. h) ^crc glt`i`r `o mgstg a`hdg aáidag `s i`m`scrdg mcomuocr `o vcogr h` prghummdýi (x) `i `o muco `o mgstg a`hdg `s `o aáidag pgsdlo`, hdmbg vcogr r`suotc h` dkucocr oc prda`rc h`rdvchc h`rdv chc h` oc euimdýi h` mgstg a`hdg c m`rg, y r`sgov`r oc `mucmdýi. H`rdvcr oc euimdýi h` mgstg a`hdg. => + 2x + >.0x, y r`sgov`r oc `mucmdýi
Aáidags y aãxdags
`) Mgaprglcr sd mgi `o mgstg a`hdg aáidag `imgitrchg x 1 ;.;0, `o mgstg a`hdg `s dkuco co mgstg acrkdico.
e) @ocstdmdhch mgstg Mak/Ma`
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OC EPIMDÝI H@ DIK\@]G XGXCO, A@HDG T AC\KDICO Dikr`sg tgtco (DX) 1 x *Ɯ(x) Dikr`sg a`hdg (da`) 1 x *Ɯ(x) x Dikr`sg acrkdico (dak) 1 hD (h`rdvchc h`o dikr`sg) hx (h`rdvchc h` x) \`ocmdýi6 `o dikr`sg tgtco `s aáidag mucihg ` dikr`sg a`hdg `s aáidag y `o dikr`sg tgtco `s aãxdag mucihg `o dikr`sg acrkdico `s aáidag. @f`apog6 Hchc oc euimdýi y 1 ;> ” 2x, mcomuo`6 c) Oc euimdýi h` dikr`sg tgtco. l) Oc euimdýi h` dikr`sg a`hdg. m) Oc euimdýi h` dikr`sg acrkdico. h) @o idv`o h` prghummdýi pcrc glt`i`r `o dikr`sg aãxdag. `) @o dikr`sg a`hdg aáidag. e) Krcedqu`. c) Oc euimdýi dikr`sg tgtco `s dkuco co prghumtg h` ocs uidhch`s prghumdhcs pgr su pr`mdg (`i `st` mcsg pgr oc euimdýi).
c) Dt 1 x * Ɯ(x):
l) Euimdýi Dikr`sg a`hdg (da`) 1
Da` 1
m) Euimdýi h` dikr`sg acrkdico (dak) 1 Ɯ¸(Dt) Dak 1 h) ^crc glt`i`r `o dikr`sg aãxdag `s i`m`scrdg mcomuocr `o vcogr h` prghummdýi (x) qu` acxdadzc `o dikr`sg, `o vcogr `s `o r`suotchg h` dkucocr oc euimdýi h` dikr`sg acrkdico c m`rg y r`sgov`r oc `mucmdýi. Dkucocr oc euimdýi h` dikr`sg acrkdico c m`rg y r`sgov`r oc `mucmdýi.
@o idv`o h` prghummdýi `i `o `o dikr`sg tgtco `s aãxdag, `s h` 80> uidhch`s prghumdhcs y v`ihdhcs. @o dikr`sg tgtco s`rã h`6
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Oc `ocstdmdhch h` uic euimdýi Ɯ(x) `i `o puitg ‟x‑, s` h`edi` mgag oc tcsc h` mcaldg prgpgrmdgico h` ‟y‑ mgi r`sp`mtg c ‟x‑ y s` sdalgodzc /@/ `s uic mderc qu` dihdmc oc vcrdcmdýi pgrm`ituco qu` `xp`rda`itc uic vcrdclo` `i r`spu`stc c uic vcrdcmdýi h` gtrc h`o 3%.
Mcsgs h` `ocstdmdhch`s c) @ocstdmdhch H`acihc H`acihc ” ^r`mdg. l) @ocstdmdhch Ge`rtc Ge`rtc ” ^r`mdg. m) Mgisuag ” Dikr`sg.
h) @ocstdmdhch trdlutcrdc (r`ocmdýi pgrm`ituco h` ogs trdlutgs r`mcuhchgs `i euimdýi h` ogs mcldgs pgrm`ituco`s `i oc acmrg`mgigaác)
Mgihdmdgi`s igraco`s h` oc h`acihc ]d oc `ocstdmdhch `s acygr qu` uig pcrc ui pr`mdg y uic h`acihc, hchc uic p`qu`óc hdsadiumdýi `i `o pr`mdg, hcrã oukcr c ui cua`itg prgpgrmdgicoa`it`
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]d oc `ocstdmdhch `s a`igr qu` uig, hchc uic p`qu`óc hdsadiumdýi `i `o pr`mdg, v`ihrã cmgapcóchc h` ui cua`itg prgpgrmdgicoa`it` prgpgrmdgicoa`it ` a`igr `i oc h`acihc y `o dikr`sg acrkdico s`rã i`kctdvg, `o dikr`sg tgtco hdsadiuy` c a`hdhc qu` oc mcitdhch h` prghumtg cua`itc, `stg s` hc `i `o mcsg h` oc h`acihc di`oãstdmc.
\`suotchgs h` oc `ocstdmdhch mgstg ]d oc h`acihc `s a`igr qu` uig, pcrc uic prghummdýi hchc, s` td`i` `o mcsg h` r`ihdad`itgs h`mr`md`it`s, `i `o muco ui p`qu`óg cua`itg `i `o prghumtg s` gltd`i` pgr ui cua`itg prgpgrmdgicoa`it` prgpgrmdgic oa`it` a`igr `i `o mgstg, `o mgstg a`hdg `s acygr qu` `o mgstg acrkdico y h`mr`m` co cua`itcr oc mcitdhch h` prghumtg. ]d oc `ocstdmdhch `s dkuco c uig, pcrc uic prghummdýi hchc, s` trctc h`o mcsg h` r`ihdad`itgs mgistcit`s, `i `o qu` ui p`qu`óg dimr`a`itg `i `o prghumtg, hcrã oukcr c ui cua`itg `i oc adsac prgpgrmdýi `i `o mgstg, `o mgstg a`hdg `s dkuco co acrkdico, td`i` ui vcogr `stcmdgicrdg, mgrrd`it`a`it` ui aáidag.
]d oc `ocstdmdhch `s acygr qu` uig, s` trctc h` r`ihdad`itgs r`ihdad`itgs mr`md`it`s, y oc sdtucmdýi `s `xcmtca`it` oc div`rsc c oc di`oãstdmc.
@ocstdmdhch Mruzchc h` oc h`acihc6 Oc `ocstdmdhch mruzchc h` oc h`acihc `s uic a`hdhc h` s`isdldodhch h` oc h`acihc h` ui ld`i cit` `o mcaldg `i `o pr`mdg h` ui ld`i sustdtutg g ui mgapo`a`itg, mgapo`a`itg, m`t`rds pcrdlus. @stc `ocstdmdhch mruzchc vc c s`r pgsdtdvc mucihg s` trctc h` ui ld`i
sustdtutg y vc c s`r i`kctdvc mucihg s` trctc h` ui ld`i mgapo`a`itcrdg. T > sd sgi ld`i`s dih`p`ihd`it`s. @o mg`edmd`it` h` `ocstdmdhch `s mcomuochg mgag oc vcrdcmdýi pgrm`ituco `i oc mcitdhch h`o ld`i C mgi r`sp`mtg c oc vcrdcmdýi `i `o pr`mdg h`o ld`i L6
@f`apog6 Ld`i`s mgapo`a`itcrdgs - @ocstdmdhch mruzchc i`kctdvc6 @o cua`itg `i `o pr`mdg h`o ld`i mgapo`a`itcrdg, prghum` uic hdsadiumdýi `i oc h`acihc h`o ld`i grdkdico, oc `ocstdmdhch `s i`kctdvc. @yx ? >. Pi cua`itg h`o pr`mdg h` oc kcsgodic h` ¢ pgr odtrg c ¢8>> pgr odtrg bc gmcsdgichg qu` oc h`acihc pgr cutgs qu` `apo`ci kcsgodic bcyc hdsadiudhg h` 5>> cutgs pgr a`s c 0>> cutgs pgr a`s. @itgim`s oc `ocstdmdhch mruzchc `s6
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Ld`i`s sustdtutgs - @ocstdmdhch mruzchc pgsdtdvc6 @o cua`itg `i `o pr`mdg h`o sustdtutg, prghum` ui cua`itg `i oc h`acihc h`o ld`i grdkdico g `i `stuhdg, oc `ocstdmdhch `s pgsdtdvc @yx 9 > @o pr`mdg h` ogs hdsmgs mgapcmtgs hdsadiuy` h` ¢;>>> c ¢0>>> y `i mgis`mu`imdc oc h`acihc h` mcs`t`s s` r`hum` h` 5>>> uidhch`s c =>>>. Oc `ocstdmdhch mruzchc `s6
@ocstdmdhch mruzchc `i ld`i`s Dih`p`ihd`it`s6 ]` hc mucihg ui cua`itg g hdsadiumdýi `i `o pr`mdg h` ogs ld`i r`ocmdgichgs, ig prghum` mcaldgs `i oc mcitdhch h`acihchc h`o ld`i `i `stuhdg, oc `ocstdmdhch tgac `o vcogr h` m`rg.
@ocstdmdhch - ^uitg h` oc H`acihc. H`acihc. @s oc `ocstdmdhch ” pr`mdg `i ui h`t`radichg puitg h` oc murvc
h` h`acihc. ]` mcomuoc sustdtuy`ihg Θ^/ΘS `i oc eýrauoc h` oc `ocstdmdhch pr`mdg h` oc h`acihc pgr oc ackidtuh h` oc p`ihd`it` h` oc murvc h` h`acihc `i `s` puitg, `itgim`s oc `ocstdmdhch puitg s` h`t`radic a`hdcit`. @OC]XDMDHCH ^PIXG ^crc mcomuocr oc `ocstdmdhch `i ui puitg mgimr`tg h` oc euimdýi h` h`acihc6
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@ocstdmdhch Crmg6
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AGIG^GODG Oc mocsdedmcmdýi h`o a`rmchg pgr `o ochg h` oc h`acihc mgrrd`it`a`it` `s6
Agigpgodg ” hugpgodg ” godkgpgodg.
X`ýrdmca`it` `o agigpgodstc edfc `o pr`mdg y h`fc qu` oc h`acihc h`t`radi` oc mcitdhch c prghumdr, g ld`i edfcr oc mcitdhch c prghumdr y h`fcr qu` oc h`acihc edf` pr`mdg. Mucihg `xdst` ui sgog prghumtgr, trctcrc sd`apr` h` glt`i`r oc aãxdac kcicimdc, `o prglo`ac `s h`t`radicr `o pr`mdg h` v`itc y su idv`o h` prghummdýi og muco mgisdku`. Pic r`edi`rác h` czömcr td`i` uic euimdýi h` mgstg tgtco6 Mt 1 3/0x x + 2,>>>. Agigpgodzcihg Agigpgodzci hg oc v`itc h`o crtámuog `i `o a`rmchg, hgih`
‟x‑ `stã `i tgi`ochcs aätrdmcs h` czömcr y `o pr`mdg. c) H`t`radi` oc prghummdýi a`isuco y `o dikr`sg pgr v`itc, oc kcicimdc aãxdac. ^r`mdg (p) y 1 0>> ” x Oc euimdýi dikr`sg tgtco `s dkuco co prghumtg h` ocs uidhch`s prghumdhcs pgr su pr`mdg.
Dt 1 x * Ɯ(x):
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@o agigpgodstc h`t`radic oc prghummdýi qu` acxdadzc su kcicimdc `i `o idv`o hgih` `o dikr`sg acrkdico `s dkuco co mgstg acrkdico, dak 1 mak. Oc euimdýi h` dikr`sg acrkdico y mgstg acrkdico, sgi `o r`suotchg h` oc prda`rc h`rdvchc h`o dikr`sg tgtco y mgstg tgtco r`sp`mtdvca`it`, y co
dkucocrocs s` h`sp`fc `o vcogr h` prghummdýi ‟x‑ qu` acxdadzc oc kcicimdc Dt 1 0>>x ” x>>
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Mt 1
Mt 1
Mt 1
Mt 1 L`i`edmdg 1 Dikr`sg tgtco ” mgstg tgtco L`i`edmdg 1
L`i`edmdg 1
43
Dapu`stgs @s ui distrua`itg kul`rica`itco pcrc `o edicimdcad`itg h`o kcstg y r`hdstrdlumdýi h`o dikr`sg, pu`h` cpodmcrs` co dikr`sg (dapu`stg hdr`mtg) g sglr` `o mgisuag, dapu`stgs dihdr`mtgs.
@e`mtgs h` ogs dapu`stgs edfgs hdr`mtgs @i `o `f`apog cit`rdgr, qu` sum`h` sd `o kgld`rig h`mr`tc ui dapu`stg h` 3,>>>. Ogs dapu`stgs edfgs dihdr`mtgs s` suaci c oc euimdýi h` mgstg tgtco, `i `st` mcsg6
]` suaci `o vcogr h`o dapu`stg Mt 1 Iu`vc euimdýi h` mgstg tgtco
Mt 1
@i `st` mcsg `o idv`o h` prghummdýi y `o pr`mdg s` acitd`i`i dkuco`s, og qu` vcrdcrc s`rã `o dikr`sg h`o agigpgodstc. L`i`edmdg 1 Dikr`sg tgtco ” mgstg tgtco L`i`edmdg 1 L`i`edmdg 1 L`i`edmdg 1 ]` sustdtuy` `o idv`o h` prghummdýi, x 1 30>
L`i`edmdg 1 L`i`edmdg 1 L`i`edmdg 1
Mgimousdýi6 ogs dapu`stgs edfgs hdr`mtgs, ce`mtci oc kcicimdc h`o agigpgodstc
`i
uic
mcitdhch
dkuco
co
agitg
h`o
dapu`stg,
p`raci`md`ihg p`raci`md`i hg `o pr`mdg y oc prghu prghummdýi mmdýi dkuco`s. dkuco`s.
@e`mtgs h` ogs dapu`stgs dihdr`mtgs @o kgld`rig pu`h` r`humdr ocs utdodhch`s co agigpgodstc a`hdcit` ui dapu`stg pgr uidhch prghumdhc, p`rg p`radtdäihgo` trcsochcr c ogs mgisuadhgr`s, og qu` grdkdicrc aga`itãi`ca`it` uic prghummdýi a`igr h`o crtámuog. @i `o `f`apog sd s` edfc ui dapu`stg h` S3>.>> pgr uidhch prghumdhc h` czömcr a`isucoa`it`, `o mgstg s`rã h`6 Ogs dapu`stgs dihdr`mtgs krcvci ocs uidhch`s prghumdhcs, pgr og tcitg ui dapu`stg h` S3>.>> pgr uidhch, s` `xpr`sc mgag 3>x, y s` suac c oc euimdýi h` mgstgs6
]` suaci `o vcogr h`o dapu`stg Mt 1 Iu`vc euimdýi h` mgstg tgtco
Mt 1
@i `st` mcsg vcrdcrc `o idv`o h` prghummdýi y `o pr`mdg, pgr og tcitg s` prgm`h` c mcomuocr `o iu`vg idv`o h` prghummdýi mgi oc egrauoc dak 1 mak, y `o iu`vg pr`mdg. Dt 1
dak 1
4=
mak 1
Mt 1
dak dak 1 1 mak mak x 1 380.2= @o iu`vg idv`o h` prghummdýi `s h` 385 uidhch`s
@o pr`mdg pcrc hdmbg idv`o h` prghummdýi s`rã h`, y 1 ^r`mdg (p) y 1 0>> ” x:
Dt 1
Dt 1
Gtrc egrac h` mcomuocr `o dikr`sg tgtco, pr`mdg * prghummdýi (y * x) Dt 1 p * x:
y1
@o dikr`sg tgtco h`o agigpgodg s`rã h` 1 Dt 1
y1
Dt 1
Dt 1
Ucogr h`o mgstg tgtco pcrc `o iu`vg idv`o h` prghummdýi. x 1
Mt 1
Mt 1 Mt 1
Mt 1 L`i`edmdg 1 Dikr`sg tgtco ” mgstg tgtco L`i`edmdg 1
L`i`edmdg 1
Dikr`sg pgr mgim`ptg h` dapu`stgs (X). @o vcogr h` ogs dapu`stgs r`mcuhchgs s`rã dkuco L`i`edmdg odqudhg
Mgimousdýi6 ogs dapu`stgs edfgs dihdr`mtgs ce`mtci `o idv`o h` prghummdýi,, k`i`rcihg uic hdsadiu prghummdýi hdsadiumdýi mdýi h`o adsag, csá tcaldäi tcaldäi gmurr` ui dimr`a`itg `i `o pr`mdg h`o crtámuog, dieou`imdchg pgr oc hdsadiumdýi `i oc prghummdýi y `o trcsochg h`o vcogr h`o dapu`stg co mgisuadhgr, oc kcicimdc
h`o
agigpgodstc
suer`
tcaldäi
uic
hdsadiumdýi
cprgxdachca`it` `i `o adsag agitg h`o vcogr h` ogs dapu`stgs.
Dapu`stgs pgrm`ituco sglr` v`itcs @o kgld`rig tcaldäi pu`h` `stclo`m`r ui dapu`stg pgrm`ituco sglr` v`itcs, mgag `e`mtg dihdr`mtg sglr` `o mgisuadhgr, `i `st` mcsg `o vcogr h`o dapu`stg mgi qu` s` krcvc `o dikr`sg, s` og ckr`kc `o agigpgodstc c ogs mgstgs tgtco`s. ]upgikcags `i `o prglo`ac qu` `o kgld`rig h`mdh` mr`cr ui dapu`stg h` 3>% sglr` v`itcs h`t`radi` ocs iu`vcs mgihdmdgi`s h` prghummdýi, pr`mdg, kcicimdcs, y agitg h` ogs dapu`stgs.
@o dapu`stg h`o 3>% sglr` v`itcs s` gltd`i` auotdpodmcihg oc euimdýi h` dikr`sg tgtco pgr 3>% y `o r`suotchg s` suac c oc euimdýi h` mgstg tgtco.
Auotdpodmcr `o pgrm`itcf` h`o dapu`stg pgr oc euimdýi h` dikr`sg tgtco >.3> * Dt 1 0>>x ” x% h` v`itcs6 v`itcs6 t t 1
]` suaci `o vcogr h`o dapu`stg Mt 1
40
Iu`vc euimdýi h` mgstg tgtco
Mt 1
@i `st` mcsg vcrdcrc `o idv`o h` prghummdýi y `o pr`mdg, pgr og tcitg s` prgm`h` c mcomuocr `o iu`vg idv`o h` prghummdýi mgi oc egrauoc dak 1 mak, y `o iu`vg pr`mdg. Dt 1
dak 1 mak 1
Mt 1
dak 1 mak dak 1 mak x 1 383 @o iu`vg idv`o h` prghummdýi `s h`
@o pr`mdg pcrc hdmbg idv`o h` prghummdýi s`rã h`, x 1 383 ^r`mdg (p) y 1
y1
Dt 1
Dt 1
Gtrc egrac h` mcomuocr `o dikr`sg tgtco, pr`mdg * prghummdýi (y * x) Dt 1 p * x:
y1
@o di dikr`sg kr`sg tgtco h`o agigpgodg s`rã h`: Dt 1
uidhch`s
Dt 1
Dt 1
Ucogr h` h`oo mgstg tgtco pcr pcrc c `o iu`vg idv`o idv`o h` prghummdýi: prghummdýi: x 1 383 Mt 1
Mt 1 L`i`edmdg 1 Dikr`sg tgtco ” mgstg tgtco
L`i`edmdg 1
L`i`edmdg 1
Dikr`sg pgr mgim`ptg h` dapu`stgs (X). @o vcogr h` ogs dapu`stgs r`mcuhchgs `stã hchg pgr: X1
X1
X1
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1
Mgimousdýi6 ogs dapu`stgs pgrm`ituco`s ce`mtci `o idv`o h` prghummdýi, k`i`rcihg uic hdsadiumdýi h`o adsag, csá tcaldäi gmurr` ui dimr`a`itg `i `o pr`mdg h`o crtámuog, dieou`imdchg pgr oc hdsadiumdýi `i oc prghummdýi y `o trcsochg h`o vcogr h`o dapu`stg co mgisuadhgr, oc kcicimdc
h`o
agigpgodstc
suer`
tcaldäi
uic
hdsadiumdýi
cprgxdachca`it` `i `o adsag agitg h`o vcogr h` ogs dapu`stgs, mgi `st` tdpg h` dapu`stg s` ogkrc uic acygr r`mcuhcmdýi edsmco.
DA^P@]XG] T ]PL]DHDG] @I MGA^@X@IMDC ^@\E@MXC @i `o aga`itg qu` s` `stclo`m` ui dapu`stg uidtcrdg y pgrm`ituco s` aghdedmcrc cutgaãtdmca`it` oc euimdýi h` ge`rtc ckr`kãihgo` `o
dapu`stg y 1 Ɯ (x) + t, hgih` ‟t‑ `s ui dapu`stg uidtcrdg, pcrc `o mcsg h` ogs sulsdhdgs, s` aghdedmc oc ge`rtc ckr`kãihgo` ckr`kãihgo` `o sulsdhdg y 1 Ɯ (x) ” s,
4;
hgih` ‟s‑ `s `o sulsdhdg, `o pr`mdg h` v`itc `s dkuco co pr`mdg h` `qudodlrdg h`t`radichg pgr oc ge`rtc y oc h`acihc.
@e`mtg h` ogs dapu`stgs y sulsdhdgs `i `o a`rmchg h` odlr` mgap`t`imdc Hchcs ocs sdkud`it`s sdkud`it`s euimdgi`s6 y 1 35 ” x. l) Mcomuo` ocs iu`vcs mgihdmdgi`s mgihdmdgi`s h` `qudodlrdg sd `o kgld`rig gtgrkc ui sulsdhdg h` S3.>> m) Mcomuo` `o dikr`sg aãxdag qu` `o kgld`rig pghrác glt`i`r co dapgi`r ui dapu`stg pgr uidhch, `imu`itr` ch`aãs `o vcogr h` hdmbg dapu`stg, `o dikr`sg tgtco aãxdag `i mgim`ptg h` dapu`stgs y ocs iu`vcs mgihdmdgi`s h` `qudodlrdg. c) @qudodlrdg h` a`rmchg sdi r`kuocmdgi`s r`kuocmdgi`s G 1 H:
y 1 y:
\`sgovd`ihg pgr ecmtgrdzcmdý ecmtgrdzcmdýi i
y1
y1
y3 1 >
y1
y1
y): (S>): Iu`vc euimdýi h` ge`rtc @qudodlrdg h` a`rmchg mgi dapu`stg dapu`stg G 1 H:
y 1 y:
\`sgovd`ihg pgr egrauoc muchrãtdmc
xd 1 x3 1
y1 y1
y1 y1
x): ]` r`stc `o vcogr h`o sulsdhdg (S3.>>): Iu`vc euimdýi h` ge`rtc @qudodlrdg h` a`rmchg mgi sulsdhdg sulsdhdg G 1 H:
y 1 y:
\`sgovd`ihg pgr egrauoc muchrãtdmc xd 1 x3 1
x> pgr uidhch v`ihdhc, mcomuo` oc vcrdcmdýi `i `o pr`mdg, oc mcitdhch y `o dikr`sg tgtco pgr mgim`ptg h` dapu`stgs p`rmdldhgs pgr `o kgld`rig. l) Mcomuo` ocs iu`vcs mgihdmdgi`s mgihdmdgi`s h` `qudodlrdg sd `o kgld`rig gtgrkc ui sulsdhdg h` S3>.>> m) Mcomuo` `o dikr`sg aãxdag qu` `o kgld`rig pghrác glt`i`r co dapgi`r ui dapu`stg pgr uidhch, `imu`itr` ch`aãs `o vcogr h` hdmbg dapu`stg, `o dikr`sg tgtco aãxdag `i mgim`ptg h` dapu`stgs y ocs iu`vcs mgihdmdgi`s h` `qudodlrdg.
@qudodlrdg h` a`rmchg sdi r`kuocmdgi`s G 1 H:
y 1 y:
y 1 3>> ” x:
1
x1
y1
y1
@qudodlrdg (
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c) Ogs dapu`stgs s` suaci c oc euimdýi h` ge`rtc, cot`rcihg `o iu`vg `qudodlrdg6
]` suac `o vcogr h`o dapu`stg (S30.>>): (S30.>>):
y1
Iu`vc euimdýi h` ge`rtc
y1
dapu`stg @qudodlrdg h` a`rmchg mgi dapu`stg G 1 H:
y 1 y:
y1
y1
y1
@qudodlrdg ( Dikr`sg tgtco pgr dapu`stgs, X 1 tx,
,
)
X1
X1
l) Ogs sulsdhdgs s` r`stci c oc euimdýi h` ge`rtc, cot`rcihg `o iu`vg `qudodlrdg6
]` r`stc `o vcogr h`o sulsdhdg (S3>.>>): (S3>.>>):
y1
Iu`vc euimdýi h` ge`rtc
y1
@qudodlrdg h` a`rmchg mgi sulsdhdg sulsdhdg G 1 H:
y 1 y:
y 1 3>> ” x:
y1
y1
3>=
@qudodlrdg (
)
X1
@rgkcmdgi`s pgr sulsdhdg,
m) `o dapu`stg tgtco `stã hchg pgr oc euimdýi X 1 tx, hgih` ‟X‑ `s `o dapu`stg tgtco, ‟‑t `o dapu`stg uidtcrdg y ‟x‑ oc prghummdýi qu` acxdadzc ogs dapu`stgs.
3. pcsg6 Iu`vc euimdýi h` ge`rtc ge`rtc 1 t1
8x 1
x 1 =>
t1
]` suac `o vcogr h`o dapu`stg (S5>.>>): (S5>.>>):
y 1
Iu`vc euimdýi h` ge`rtc
y 1 x + 8>
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y 1 y: y1
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)
Dikr`sg tgtco pgr dapu`stgs, X 1 tx,
X1
X1
Mgimousdgi` Mgimousdgi`ss r`sua`i6 @qudodlrdg @qudodlrd g h` a`rmchg @qudodlrdg mgi dapu`stg (S.30.>> pgr uidhch) @qudodlrdg mgi sulsdhdg (S.3>.>> pgr uidhch) @qudodlrdg mgi dapu`stg (aãxdag dapu`stg)
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AÃRDAG] T AÁIDAG] MGIHDMDGICHG] MGI \@]X\DMMDÝI APOXD^ODMCHG\ H@ OCK\CIK@ Mucihg s` trctc h` cpodmcmdgi`s prãmtdmcs h` acxdadzcmdýi y adidadzcmdýi h` uic euimdýi hchc y qu` `stã suf`tc c aumbcs mgihdmdgi`s, tcitg co cozc, mgag c oc lcfc, pcrc glt`i`r ui prghumtg edico, s` bcm` usg h`o aätghg Ockrcik`. @f`apog6 Pi eclrdmcit` pu`h` h`s`cr adidadzcr sus mgstgs h` prghummdýi, co prghumdr uic mcitdhch aáidac h`t`radichc h` ui prghumtg, h` oc adsac aci`rc, uic mcapcóc pu`h` pocidedmcr acxdadzcr sus v`itcs mgag r`suotchg h` oc utdodzcmdýi h` a`hdgs h` mgauidmcmdýi, acit`id`ihg sus mgstgs tgtco`s h`itrg h` uic odadtcmdýi `stclo`mdhc pgr `o adsag. Glt`ikc ogs aãxdags y aáidags h` oc euimdýi.
Ɠ (x) 1 0x ^g =>>
@M @^
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@xm`h`it` h`o Mgisuadhgr
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@xm`h`it` h`o ^rghumtgr
@^1 3>>> @f`apog6
@xm`h`it` h`o Mgisuadhgr ^ 1 3,3>> -q¼
H`acihc
^ 1 =>> + q¼
Ge`rtc
^uitg h` `qudodlrdg
qg 1 Co sustdtudr `i mucoqud`rc h` ocs `mucmdgi`s6
^g 1 qg 1
^g 1 ;>>
3,3>>- q¼ =>>+ q¼ ^g ;>>
@M @^
Sg 1 @xm`h`it` h`o Mgisuadhgr
@M 1 0,===,== @xm`h`it` h`o ^rghumtgr
330
@^ 1 0.===.== @f`apog6
@xm`h`it` h`o Mgisuadhgr ^ 1 8>> -q¼
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^ 1 q + 3>> Ge`rtc ^uitg h` `qudodlrdg
q 1 3> Co sustdtudr `i mucoqud`rc h` ocs `mucmdgi`s6
^g 1
qg 1 3>
^g 1 =>>
8>>- q¼ q + 3>> ^g =>>
@M @^
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@M 1 555.5;
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33;
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DIK\@]G] E\@IX@ C MG]XG]
Oc dit`krcmdýi s` utdodzc `i chadidstrcmdýi y `mgigaác pcrc h`t`radicr oc utdodhch tgtco g ocs kcicimdcs i`tcs tgtco`s `i vcrdgs mgit`xtgs. @i k`i`rco s` a acxdadzc cxdadzc oc utdodhch supgid`ihg odlr` mgap`t`imdc mucihg `o dikr`sg acrkdico acrkdico `s dkuco co mgstg acrkdico. Oc utdodhch utdodhch tgtco s` h`t`radic dit`krcihg oc hde`r`imdc `itr` `o dikr`sg acrkdico y `o mgstg acrkdico, h`sh` m`rg bcstc oc mcitdhch pcrc oc muco `s aãxdac oc utdodhch.
Dikr`sg ^crc ocs `apr`scs `o dikr`sg mgrr`spgih` c ocs `itrchcs `mgiýadmcs g
r`aui`rcmdýi qu` r`mdl` pgr oc v`itc h` ld`i`s y/g s`rvdmdgs . @o dikr`sg ig mgit`apoc ogs mgstgs g kcstgs `i qu` s` dimurr` pcrc glt`i`r `st` dikr`sg. Ogs dikr`sgs s` pu`h`i mocsdedmcr `i6 Dik r`s g tgt tgtc co6 dikr`sg glt`idhg pgr oc v`itc h` oc tgtcodhch h` ogs prghumtgs.
Dikr`sg acrkdico6 mgrr`spgih` co dikr`sg k`i`rchg pgr `o cua`itg h` oc
prghummdýi `i uic uidhch. Dik r`sg a` a`hdg6 hdg6 mgrr`spgih` co prga`hdg h` dikr`sg pgr uidhch v`ihdhc, `s h`mdr,
`s `o dikr`sg tgtco hdvdhdhg pgr `o tgtco h` uidhch`s v`ihdhcs.
Mgstg ^ gr gr su pcrt`, ogs mgstgs sgi `o
scmrdedmdg dimurrdhg pcrc prghumdr ld`i`s y
s`rvdmdgs. Co dkuco qu` ogs dikr`sgs, ogs mgstgs pu`h`i mocsdedmcrs` `i mgstgs tgtco`s, acrkdico`s y a`hdgs. ^gr su pcrt`, ogs mgstgs tgtco`s s` hdvdh`i `i hgs mgapgi`it`s6 @s dapgrtcit` hde`r`imdcr c ogs mgstgs h` ogs kcstgs. Ad`itrcs ogs mgstgs s` dimurr`i pcrc prghumdr ui ld`i g s`rvdmdg, ogs kcstgs sgi cqu`oogs cqu`oogs h`stdichgs c oc hdstrdlumdýi g v`itc h`o prghumtg, y c oc chadidstrcmdýi. Ogs kcstgs pgr tcitg ig su`o`i s`r ctrdludhgs c ui cmtdvg `i pcrtdmuocr. @i `e`mtg, ig s` r`codzci mgi `o prgpýsdtg h` k`i`rcr pgst`rdgr`s dikr`sgs sdig qu` s` dimurr` `i ogs kcstgs pgr i`m`sdhch.
334
Ptdodhch
c utdodhch `s oc hde`r`i hde`r`imdc mdc ` `it itr` r` oogs gs d ik r`s gs y tghg tghgss og ogss mgs tgs tgs y k cs tgs tgs `i ogs
O
muco`s s` dimurrdý hurcit` `o p`rághg. ^gr tcitg, oc utdodhch y ig ogs dikr`sgs sgi og qu` r`coa`it` kcic oc `apr`sc. Ptdodhch 1 Dikr`sgs ” (Mgstgs + Kcstgs)
Cbgrc ld`i, oc utdodhch tcaldäi pu`h` s`r `it`ihdhc mgag oc kcicimdc qu` s` gltd`i` co v`ih`r ui prghumtg. ^gr tcitg ig mgisdh`rc ogs kcstgs `i `o mãomuog y mgrr`spgih` c ui ac acrk rk `i h` mgitrdlumdýi mgitrdlum dýi pgr prghumtg.
@F@A^OG6 Pic `apr`sc mga`rmdcodzc `itr` gtrgs prghumtgs pci h` mcfc y ui vdig ercimäs. Oc euimdýi h` utdodhch acrkdico h`o pci `stã hchc pgr e(x) 1 8> ” 0x y oc utdodhch acrkdico ac rkdico h`o vdig `stã hchc pgr k(x) 1 => ” x. @imgitr`ags6 c) Oc euimdýi h` utdodhch tgtco h`o pci. l) Oc euimdýi h` utdodhch tgtco h`o vdig. m) ]d `o mgisuadhgr h`s`c chqudrdr tr`s pcqu`t`s h` pci y tr`s h` vdig, muão h` ogs crtámuogs o` prghumdrã acygr utdodhch (sctdsecmmdýi).
Mgag pu`h` gls`rvcrs`, `o pci o` prghum` uic acygr utdodhch c ui dihdvdhug qu` `o vdig. @i `st` tdpg h` prglo`acs, oc euimdýi h` utdodhch h` ui mgisuadhgr `s ui tcitg sulf`tdvc, yc qu` ig cpgrtc krci diegracmdýi. @o öidmg glf`tdvg `s h`t`radicr quä crtámuog o` hc acygr utdodhch c ui dihdvdhug. Oc euimdýi h` utdodhch h` uic `apr`sc `ap r`sc `s aãs `sp`mdedmc y acs glf`tdvc g lf`tdvc `i `o s`itdhg h` qu` p`radt` mgigm`r `o agitg agi`tcrdg `i `o muco s` `imu`itrci `imu`itr ci sus kcicimdcs h`p`ihd`ihg h`o idv`o h` prghummdýi qu` acit`ikc. ^gr `oog, m`itrcr`ags iu`strc ct`imdýi `i `st` tdpg h` prglo`acs. @f`apog crtámuogs y M 1 >. l) Oc euimdýi h` mgstg tgtco y `o mgstg tgtco sd s` prghum`i 0> crtámuogs y ogs kcstgs k`i`rco`s sgi h` 2>>. M) Oc utdodhch tgtco.
3> ” 0>x ” > ” x ” x
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