Chapter 4
January 28, 2023 | Author: Anonymous | Category: N/A
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Z@KX AIZT CI@SXTMK@JH I^XKEKWMXKI@
K@XTIHZCXKI@
Hjmr cioojmfuj! K` tbj prjvkius u`kt wj stuhkjh tbj iptkekzmtki` ia au`ctki`s wktb iut tbj jxkstj`cj ia ci`strmk`ts. dusk`jss m`hAir jci`iekcs tbjrjemxkekzj mrj em`yiutput sktumtki`s k` wbkcb ciepojtj arjjhieBiwjvjr, ia mctki` ksk`kepisskdoj. jxmepoj,stuhkjs m akre cm` sudgjct ti tbj ci`strmk`t ia m fkvj` duhfjt air jxpj`hkturjs i` k`puts, ir kt emy `jjh ti ek`kekzj cist sudgjct ti m cjrtmk` ek`keue iut put djk`f prihucjh. Sucb au`ctki`s wbkcb k`viovj ci`strmk`ts mrj cmoojh ci`strmk`jh au`ctki`s m`h tbj pricjss ia iptkekzmtki` ks rjajrrjh ms ci`strmk`jh iptkekzmtki`. Xbks u`kt jxpomk`s tbj wmys ia siovk`f ci`strmk`jh iptkekzmtki` pridojes wktb jqumokty jqumokty m`h k`jqumokty ci`strmk`ts. ci`strmk`ts. Z`kt Idgjctkvj
Zp i` tbj ciepojtki` ia tbks u`kt, yiu mrj jxpjctjh ti hjscrkdj wbmt ci`strmk`jh au`ctki` ks hjscrkdj tbj irhjr ci`hktki` ci`hktki` air tbj iptkek iptkekzmtki` zmtki` ia ci`strmk`jh au`ctki au`ctki`` jxpomk` tbj Omfrm`fj euotkpokjr hjscrkdj biw au`ctki`s wktb k`jqumokty ci`strmk`ts mrj siovjh hjscrkdj tbj Lub` - Xucljr tbjirje siovj idgjctkvj au`ctki` sudgjct ti ekxjh ci`stm`ts Sjctki` 8.= I`j Qmrkmdoj Ci`strmk`jh Iptkekzmtki` wktb @i` - @jfmtkvj Ci`strmk`t
Matjr ciepojtk`f tbks sjctki`, yiu sbiuoh dj mdoj ti
-
hjsc hjscrkd rkdjj ipt iptkek kekzmt zmtki` ki` ia ia tb tbjj idg idgjct jctkvj kvj au`ct au`ctki` ki` mt m ffkvj kvj`` pik`t - jxp jxpomk omk`` tbj tbj akrst akrst irhj irhjrr cci`hk i`hktki tki`s `s tbm tbmtt eust dj ejt - hjsc hjscrkd rkdjj tbj tbj sjci`h sjci`h irh irhjr jr ci`h ci`hkt ktki` ki` tbm tbmtt e eust ust dj ejt air iptkekzmtki`
k) Vktb jqumokty ci`strmk`t
x . K` tbks cmsj y a ( x ) Emxkekzj y 5 a(x), sud sudgjct gjct ti x Kt skepoy k`viovjs hjtjrek`k`f tbj vmouj ia tbj idgjctkvj au`ctki` mt tbj fkvj` pik`t k` tbj hiemk`.
kk) Vktb `i` - `jfmtkvkty ci`strmk`ts
Emxkekzj ssudgjct udgjct ti x >
22
(m)
( x) > Mt x >, a Air tbj mdivj au`ctki` tbj u`ci`strmk`jh emxkeue mttmk`jh wbj` x ? > mt pik`t d ms sbiw` k` tbj mdivj akfurj wbjrj ms y mttmk`s kts ci`strmk`jh iptkeue mt pik`t m.
y a (>)
(d) K` tbks cmsj tbj ci`strmk`jh m`h u`ci`strmk`jh emxkeue vmouj ia tbj au`ctki` okj mt tbj smej pik`t, k.j., tbjy cik`ckhj cik`ckhj mt pik`t m ms sbiw` mdiv mdivj. j. y a ( x) .
(c) K` tbks cmsj, skekomr ti tbmt ia d, tbj ci`strmk`jh m`h u`ci`strmk`jh emxkeue vmouj ia tbj au`ctki` rjskhj i` tbj smej pik`t, y a (>)
a ( x ) >
Hjmr cioojmfuj! ^ojmsj try ti ek`kekzj y a ( x) , sudgjct ti x > k` m skekomr wmy. Fkvj` tbj au`ctki` y a ( x) sudgjct ti tbj x > Air emxkekzmtki`, a ( x ) >
>, x > ka a ( x) ka a ( x) >, x >
2;
Air ek`kekzmtki` >, x > ka a ( x) ka a ( x) >, x >
a ( x ) >
Jxmepoj
Emxkekzj tbj idgjctkvj au`ctki` y 5 - 3x4 - 6 x + 4 sudgjct ti x >. Akrst irhjr ci`hktki` air emxkekzmtki` a ( x ) 1 x 6 5 > 1x5 - 6
x5
6
1
Sjci`h irhjr ci`hktki` air emxkekzmtki` a ( x) 1 >
Xbus, tbj tbj u`ci`strmk` u`ci`strmk`jh jh emxkeue vvmouj mouj ia tb tbjj au`ctki` oicmtjs mt x5
6
1
, k.j., x ? > dut tbj
ci`strmk`jh emxkeue, mt x > , a (>) 6 > Xbus tbj ci`strmk`jh emxkeue ks tbj au`ctki` ks y 4 . 4. Ek`kekzj y 5 x 4 + 4x+ :, sudgjct ti R > Akrst irhjr ci`hktki` a ( x) 4 x 4 > x = >
Sjci`h irhjr ci`hktki` a ( x) 4 >
Xbus tbj au`ctki` mctkvjs kt‛s u`ci`strmk`jh ek`keue mt x 5 -=, k.j., y 5 8 Biwjvjr, mt x >, a ( x ) 4 > . Xbjrjairj, tbj ek`keue vmouj ia tbj ci`strmk`jh au`ctki` ks y 5 a (>) 5 :. Hjmr cioojmfuj! Dy `iw yiu bmvj ciepojtjh tbj akrst sjctki` ia tbks u`kt. Xbus, try ti hi tbj aiooiwk`f sjoa - tjst qujstki`s ti jxmek`j yiur u`hjrstm`hk`f ia tbks sjctki`. Sjoa - Xjst 8.=
Siovj tbj aiooiwk`f qujstki`s dmsjh i` tbj k`airemtki` k` tbj mdivj sjctki`. =. Emxkekzj y 5 -x 4 + :x + 1, sudgjct ti x 4 , -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------4. Emxkekzj y 5 - 4 x 4 + 3 x + 8, sudgjct ti x > ------------------------------------------------------------------------------------------------------------------------------------------------------3. Ak`h tbj emxkeue vmouj ia tbj au`ctki` y 5 -3x 4 + x + 6, sudgjct ti x > -------------------------------------------------------------------------------------------------------------------------
;>
8. Ek`kekzj tbj au`ctki` y 5 x4 + 3 x + 8, sudgjct ti x > ---------------------------------------------------------------------------------------------------------------------------------------------:. Ak`h tbj ek`keue vmouj ia tbj au`ctki` y 5 4x4 + x + 6 , sudg sudgjct jct ti x > -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Hjmr cioojmfuj! Bmvj yiu m`swjrjh tbjsj qujstki`s9 Ka `i, ^ojmsj rjrjmhs tbks sjctki` m`h try ti hi tbje. Ka yjs, fiih fi ti tbj `jxt sjctki`. Sjctki` 8. 4. Xwi Qmrkmdoj ^ridojes wktb Jqumokty Ci`strmk`ts
Zp i` tbj ciepojtki` ia tbks sjctki`, yiu sbiuoh dj mdoj ti
- hjscrkdj tbj irhjr ci`hktki` air iptkekzk`f tbjsj au`ctki`s - jxpomk` tbj ejtbih ia iptkekzmtki` dy Sudstktutki` - hjscrkdj tbj Omfrm`fj euotkpokjr ejtbih - k`tjrprjt tbj Omfrm`fj euotkpokjr - iptkekzj euotkvmrkmdoj au`ctki`s
Hjmr Cioojmfuj! K` tbks sjctki` wj wkoo sjj twi ejtbih ia ci`strmk`jh iptkekzmtki`. Xbjsj mrj. Ci`strmk`jh iptkekzmtki` dy sudstktutki sudstktutki`` Omfrm`fj euotkpokjr ejtbih M. Ci`strmk`jh Iptkekzmtki` dy Sudstktutki`
Hjmr cioojmfuj wbjrj cm` wj mppoy tbks ejtbih9 e jtbih9 ^ojmsj hkscuss wktb yiur arkj`hs m`h try ti m`swjr tbks qujstki`. M`h wrktj yiur m`swjr i` m riufb pmpjr. Bmvj yiu m`swjrjh kt9 Il fiih. Xry ti rjomtj yiur m`swjr wktb tbj aiooiwk`f m`moysks. Xbks ejtbih ks emk`oy mppokcmdoj air pridojes wbjrj tbj idgjctkvj au`ctki` wktb i`oy twi vmrkmdojs ks emxkekzjh ir ek`kekzjh sudgjct ti i`j ci`strmk`t. Ci`skhjr m akre tbmt wm`ts ti emxkekzj iutput fkvj` tbj prihuctki` au`ctki` ] 5 a (L, O) m`h suppisj ^L m`h ^O prkcjs ia L m`h O rjspjctkvjoy m`h m akxjh duhfjt D. Xbj`, wj cm` hjtjrek`j tbj meiu`t ia L m`h O tbmt iptkekzj ] usk`f tbj ejtbih ia sudstktutki`. Jxmepoj
=.M akre amcjs tbj prihuctki` au`ctki` ]5 =4L >.8 O >.8 m`h mssuej kt cm` purcbmsj L m`h O mt prkjs pjr u`kt ia 8> dkrr m`h m`h : Dkrr rjspjctkvjoy m`h kt bms m duhfjt ia 2>> Dkrr. Hjtjrek`j Hjtjrek`j tbj meiu`t ia L m`h O wbkcb emxkekzjs iutput. Sioutki`
Xbj pridoje ks Emxkekzj ]5 =4L >.8 O >.8 Sudgjct ti 8>L +:O 5 2>> Mccirhk`f ti tbj tbjiry ia prihuctki`, tbj iptkekzmtki` ci`hktki` ks wrkttj` k` sucb mwmy tbmt tbj rmtki ia emrfk`mo prihuct ia jvjry k`put ti kts prkcj eust dj tbj smej. Xbmt ks
;=
E^ L
^ L
E^ O ^ O
Xbj emrfk`mo prihucts cm` dj idtmk`jh dy tbj ejtbih ia pmrtkmo hkaajrj`tkmtki` ms aiooiws. E^ L 5 8.2 L ->.1 O >.8.......................................
E^ O 58.2 L >.8 O >.1
(=) ......................................... (4)
Sudstktutk`f tbjsj emrfk`mo prihucts m`h tbj fkvj` prkcjs k` tbj ci`strmk`t au`ctki` fkvjs us
8.2 L >.1 O>.8
8 .2 L >.8 O>.1
8>
:
L ->.1 O >.8 5 2 L >.8 O ->.1 Euotkpoyk`f ditb skhjs dy L >.1 O >.1 O 5 2l................................. 2l........................................................... .................................... .......... (3) Sudstktutk`f (3) k` tbj duhfjt ci`strmk`t wj fjt 8>L + :(2L) 52>> 8>L+ 8>L 5 2>> 2>l 52>> L5=> Xbus, O5 2(=>) 52> Xbjrj airj, tbks akre sbiuoh jepoiy => u`kts ia cmpktmo m`h 2> u`kts ia omdir k` tbj prihuctki` pricjss ti iptkekzj kts kts iutput. 4. Suppisj tbj utkokty au`ctki` ia tbj ci`suejr ks fkvj` dy Z 8 xy y 4 m`h tbj duhfjt ci`strmk`t ks 4x+y 5 1. Hjtjrek`j tbj meiu`t me iu`t ia x m`h y wbkcb wkoo iptkekzj titmo utkokty ia tbj ci`suejr. Sioutki`
Ztkokty ks emxkekzjh wbj`
EZ R ^ x
EZ y ^ y
K` iur jxmepoj, EZ x 5 8y, EZ y 5 8x-4y.Xbjrjairj, mt tbj pik`t ia jqukokdrkue 8 y 8 x 4 y 4
=
8y 5 2x-8y 8y + 8y 5 2x 2y 5 2x x y ---------------------------------------- (8)
Sudstktutk`f (8) mdivj k` tbj duhfjt ci`strmk`t fkvjs us 4x +x5 1 3x51 x545y Xbjrjairj, tbks ci`suejr cm` iptkekzj bks utkokty wbj` kt ci`suejs 4 u`kts ia fiih x m`h 4 u`kts ia fiih y.
;4
D. Omfrm`fj Euotkpokjr Ejtbih
Hjmr cioojmfuj! Vbmt ks Omfrm`fj Euotkpokjr Ejtbih9 Vbmt mrj tbj stjps ti usj tbks ejtbih9 Biw hi yiu k`tjrprjt tbj Omfrm`fj euotkpokjr9 --------------------------------------------------------------------------------------------------------------------------------------------Bmvj yiu m`swjrjh kt9 Il, try ti rjomtj yiur m`swjr wktb tbj aiooiwk`f m`moysks. Xbj jssj`cj ia tbks ejtbih ks ti cbm`fj m ci`strmk`jh iptkekzmtki` pridoje k` ti m aire sucb tbmt tbj akrst irhjr ci`hktki` ia tbj u`ci`strmk`jh iptkekzmtki` iptkekzmtki` pridoje cm` stkoo dj mppokcmdoj. Xbks ejtbih cm` dj usjh air eist e ist typj ia ci`strmk`jh iptkekzmtki` pridojes. Fkvj` tbj au`ctki` W5 a (x, y) sudgjct ti f (x, y) 5 ^ x R+ ^[[ 5E, ti hjtjrek`j tbj meiu`t ia x m`h y wbkcb emxkekzj tbj idgjctkvj au`ctki` usk`f tbj Omfrm`fj Euotkpokjr Ejtbih, wj sbiuoh k`viovj tbj aiooiwk`f stjps. Stjp = Tjwrktj tbj ci`strmk`t au`ctki` k` kts kepokckt aire ms E x^ x y^ y > Stjp 4 Euotkpoy tbj ci`strmk`t au`ctki` dy tbj Omfrm`fj euotkpokjr (E- x ^ x- y ^ y) 5 > Stjp 3 Mhh tbj mdivj ci`strmk`t ti tbj idgjctkvj au`ctki` m`h tbjrjdy wj fjt tbj Omfrm`fj au`ctki` aiooiws7 tbmt ks m eihkakjh aire ia tbj idgjctkvj au`ctki` wbkcb k`couhjs tbj ci`strmk`ts ms O ( x, y , ) W ( x, y y^ y ) ------------------- (:) ) ( E x^ x @jcjssmry ci`hktki`, k.j. tbj akrst a krst irhjrs ci`hktki` air emxkekzmtki` ks tbmt tbj akrst irhjr pmrtkmo hjrkvmtkvjs ia tbj Omfrm`fj au`ctki` sbiuoh dj jqumo ti zjri. Hkaajrj`tkmtk`f O wktb rjspjct ti x, y, m`h O
x O y
O
z x z y
^ x >
----------------------- (6)
E x^ x y^ y > ------------------------- (2)
5
x
jqumtk`f kt wktb zjri fkvjs us.
----------------------- (1)
^ y >
Arie jqumtki` (1) m`h (6) W wj x fjt m`h 5 ^ Xbks ejm`s,
m`h
W x ^ x
W y ^ y
ir
W x W y
W y ^ y
^ x ^ y
Suaakckj`t ci`hktki` -Xi fjt tbj
sjci`h irhjr ci`hktki`, wj sbiuoh pmrtkmooy hkaajrj`tkmtj jqumtki`s (1), (6) m`h (2). Tjprjsj`tk`f tbj sjci`h hkrjct pmrtkmo hjrkvmtkvjs dy W xx m`h W yy m`h tbj sjci`h criss pmrtkmo hjrkvmtkvjs dy W xy m`h W yx, tbj dirhjr Bjsskm` hjtjrek`m`t dirhjrjh f wktb >, f x m`h y ks ^ x
^ y
O xy ^ x
W xx
W xy i
^ y
W yx
W yy
>
f x
f y
B f x
O xx
f y
O yx
O yy
>
;3
h 4 W ks rjajrrjh ti ms pisktkvj hjak`ktj sudgjct ti hf 5 > kaa B ?>, h 4 W ks rjajrrjh ti ms `jfmtkvj h hjak`ktj jak`ktj ssudgjct udgjct ti hf 5 > kaa B 0 > . 4
@jfmtkvj hjak`ktj`jss hjak`ktj`jss ia h W kepokjs tbmt tbj au`ctki` mcbkjvjs kts rjomtkvj emxkeue pik`t wbjrj ms m pisktkvj hjak`ktj ks m suaakckj`t ci`hktki` ti smtksay tbj rjomtkvj ek`keue ia tbj idgjctkvj au`ctki`. Emxkekzmtki` Jxmepoj
Fkvj` tbj utkokty au`ctki` ia tbj ci`suejr wbi ci`suejs twi fiihs x m`h y ms Z (x, y) 5 (x+ 4) (y+=) Ka tbj prkcj ia fiih x ks ^ x 5 8 dkrr, tbmt ia fiih y ks ^ y 5 1 Dkrr m`h tbj ci`suejrs bms m akxjh duhfjt ia =3> dkrr. Hjtjrek`j Hjtjrek`j tbj iptkeue vmoujs ia x m`h m`h y usk`f tbj Omfrm`f Omfrm`fjj euotkpokjr ejtbih, Sioutki`
Emxkekzj Z (x, y) 5 x y + x+ 4y + 4 Sudgjct ti 8x + 1y 5 =3> Hjmr cioojmfuj! @iw wj sbiuoh aireuomtj tbj Omfrm`fj au`ctki` ti siovj tbks pridoje. Xbmt ks , ) 5 x y + x+ 4y + 4 + (= O ( x, y (=3> 3> - 8x - 1y 1y)) --------------------------------- (;) @jcjssmry ci`hktki`s ci`hktki`s air utkokty emxkekzm emxkekzmtki` tki` mrj
O x
O x
>,
O y
>,
O
>
( y =) 8 5 >
y 5 -= + 8 ------------------------------------- (=>)
O y
( x 4) 1 >
x 4 1 ---------------------------------- (==)
O
8 x 1 y =3> >
8x+1y5 =3>----------------------------------- (=4) Sudstktutk`f tbj vmouj ia x m`h y jxpomk`jh k` jqumtki` (=>) m`h (==) k` ti (=4) j`mdojs us ti hjtjrek`j 8 (-4+ 1 ) + 1 (-= +8 ) 5 =3> - 2 + 48 - 1 + 48 5 =3> 82 5 =88 5 3 Xbjrjairj, x 5 -4+1(3) x 5 -4 + =2 5 =1 y 5 -= + 8 (3) y 5 == Sjci`h irhjr suaakckj`t ci`hktki` air utkokty emxkekzmtki` ks
;8
>
f x
f y
B f x
O xx
O xy
f y
O yx
O yy
Xbj sjci`h pmrtkmo hjrkvmtkvjs ia tbj idgjctkvj au`ctki` m`h tbj akrst pmrtkmo hjrkvmtkvjs ia tbj ci`strmk`t au`ctki` mrj O xx 5
֍ x
5
4
O
x f x
4
5 >, O yy 5 >, O xy 5 O yx 5 =
5 8, m`h ֍ y 5 1 5
f y
Xbjrjairj, tbj dirhjrjh Bjsskm` hjtjrek`m`t ia tbks au`ctki` ks B
>
8
1
8
>
=
1
=
>
5 - 8(>-1) + 1 (8- >) 5 82 0 >
Xbj sjci`h irhjr ci`hktki`, k.j.,
B
0 > ks smtksakjh air emxkekzmtki`. Xbus, tbj ci`suejr
emxkekzjs utkokty wbj` bj ci`suejs == u`kts ia fiih y m`h =1 u`kts ia fiih x. Xbj emxkeue utkokty ks Z 5 (=1+4) (==+=) 5 (=2) (=4) 5 4=1 u`kts wbkcb ks skekomr ti tbj vmouj ia tbj Omfrm`fj au`ctki` mt tbjsj vmoujs ia x , y m`h . Xbj vmouj ia tbj Omfrm`fj euotkpokjr ks 3. Kt k`hkcmtjs tbmt m i`j u`ktjs k`crjmsj (hjcrjmsj) k` tbj duhfjt ia tbj ci`suejr k`crjmsjs (hjcrjmsjs) bks titmo utkokty dy 3 u`kts. 4. Suppisj tbj ei`ipiokst sjoos twi prihucts x m`h y m`h tbjkr rjspjctkvj hjem`h ks ^ x 5 =>> - 4 x m`h ^ y 5 2> - y Xbj titmo cist au`ctki` ks fkvj` ms XC 5 4>x + 4>y, wbj` tbj emxkeue gik`t prihuct ia tbj twi iutputs 1> u`kt. Hjtjrek`j tbj priakt emxkekzk`f ojvjo ia jmcb iutput m`h tbjkr rjspjctkvj prkcj. Sioutki`
Hjmr cioojmfuj! Vj l`iw tbmt titmo priakt ( ) 5 XT - XC, wbjrj XT rjprjsj`ts titmo rjvj`uj m`h XC rjprjsj`ts titmo cist. XT5 x ^ x + y ^ y 5 (=>>x - 4x 4)4 + (2>y - y 4) 4 Xbus 5 =>>x - 4x + 2> y - y - 4>x - 4> y 5 2> x + 1> y – 4x4- y4 Dut tbks ei`ipiokst cm` emxkekzj kts priakt sudgjct ti tbj prihuctki` quitm. Xbus, Emxkekzj 5 2>x + 1> y- 4x 4- y 4 Sudgjct ti x+ y 5 1> Xi siovj tbks pridoje, wj sbiuoh aireuomtj tbj Omfrm`fj au`ctki`, O (x, y, ) 5 2>x + 1>y - 4x 4 - y 4 + (x+ y - 1>) --------------- (=3) Akrst irhjr ci`hktki`s air emxkeue priakt mrj O x 5 2> - 8x + 5 > - 8x 5 - 2> - 5 4> + = ------------------------------------------------ ------- (=8) 8 O y 5 1> - 4y+ 5 >
;:
- 4y 5 - 1> -
y 5 3> +
= 4
----------------------------------------------------------- (=:)
O 5 x + y -1> 5 > R + y 5 1>------------------------------------------------------------ (=1) Sudstktutk`f jqumtki` (=8) m`h (=:) k` jqumtki` (=1), wj fjt
= 4> + 8 3 :> + 8 3
8
Xbus,
+ 3>+ 4 5 1>
x 5 4> +
=
= ( 8
5 1>
5 =>
8> 3 8> ) 3
y 5 3> +
= ( 8>) 4 3
5 4>+ 3.33 5 3>+1.16 x 5 43.33 y 5 31.16 Sjci`h irhjr ci`hktki` air emxkeue priakt ks O xx 5 - 8, O y y 5 -4, O x y 5 O y x 5 > f x 5 =, f y 5 = Xbjrjairj, tbj dirhjrjh Bjsskm` hjtjrek`m`t ia tbj fkvj` au`ctki` ks > = = B 5 = -8 > 5 -= (-4 - >) + = (>+8) 5 10 > = > -4 Xbj sjci`h irhjr ci`hktki` ks smtksakjh air emxkekzmtki` ia au`ctki`s. ^ x =>> 4(43.33) ^ x :3.38
^ y 2> 31 .16
^ y 83 .33
Xbjrjairj, tbj ei`ipiokst emxkekzjs kts priakt wbj` kt sjoos 43.33 ia fiih x mt m prkcj ia :3.38 dkrr pjr u`kt m`h 31.16 u`kts u`kts ia fiih y mt m prkcj ia 83.33 83.33 dkrr pjr u`kt.
5
8> 3
sbiws tbmt m i`j
u`kt k`crjmsj k` titmo jxpj`hkturj i` k`puts k`crjmsjs titmo priakt ia tbj ei`ipiokst dy 8> u`kts. 3
K` itbjr wirhs, ka tbj ci`stm`t ia tbj ci`strmk`t rjomxjs dy i`j u`kt tbmt ks x y 1= , tbj` tbj vmouj ia tbj idgjctkvj au`ctki` k`crjmsjs dy tbj vmouj tbj Omfrm`fj euotkpokjr . Ek`kekzmtki`
Hjmr cioojmfuj! Ms wj l`iw, tbj akre cm` hjtjrek`j tbj ojmst cist ciedk`mtki` ia k`puts air tbj prihuctki` ia m cjrtmk` ojvjo ojvjo ia iutput ]. Fkvj` ttbj bj prihuctki` au`ct au`ctki` ki` ]5 a (O, L) m`h tbj cist au`ctki` ia tbj akre ks C 5 O^O + L^ l Vbjrj Vbjrj O 5 omdir, L 5 cmpktmo, ] 5 iutput. Suppisj tbj prkcj ia ditb k`put ti dj dj jxifj`ius, wj cm` aireuom aireuomtj tj tbj pridoje ia ek`kekzk` ek`kekzk`ff tbj cist ms Ek`kekzjs C 5 ^O O + ^ l l Sudgjct ti ] 5 a (O, L) Xi hjtjrek`j tbj meiu`t ia omdir m`h cmpktmo tbmt sbiuoh dj jepoiyjh k`ktkmooy wj sbiuoh aireuomtj tbj Omfrm`fj au`ctki`. Kt ks
;1
O O^ O L^ ( O, L ) --------------------------- ---- (=6) L (] a Akrst irhjr ci`hktki`s air m ek`keue cist mrj O ] O > O O ^
^ O ] O
^ O E^ O
O L ^ L ]l >
---------------------------------------------- (=2)
^ L ^ L ------------------------------------------- (=;) ] L E^ L
O ] a ( L , O ) > --------------------------------------------- (4>) Vbjrj ] O m`h ] l rjprjsj`ts emrfk`mo prihuct ia omdir m`h cmpktmo rjspjctkvjoy. Arie jqumtki` (=6) m`h (=2), wj fjt
^ O E^ O
^ L E^ L
-------------------------------------------------- (4=)
Jqumtki` (4=) k`hkcmtjs tbmt, mt tbj pik`t ia iptkemo k`put ciedk`mtki` tbj k`put - prkcj rmtki m`h tbj emrfk`mo prihuct rmtki bmvj ti dj tbj smej air jmcb k`put. Xbks rmtki sbiws tbj meiu`t ia jxpj`hkturj pjr u`kt ia tbj emrfk`mo prihuct ia tbj k`put u`hjr ci`skhjrmtki`. Xbus, tbj k`tjrprjtmtki` tbj Omfrm`fj euotkpokjr ks tbj emrfk`mo cist ia prihuct mt tbj iptkemo ci`hktki`. K` itbjr wirhs, k`hkcmtjs tbjstmtkc jaajct ia cbm`fj iutput i`ci`stm`t tbj titmo i` cists prihuctki`, kt ejmsurjs tbj ktciepmrmtkvj - jaajct ia tbjk`ci`strmk`t tbjiaiptkemo vmouj k.j., ia tbj idgjctkvj au`ctki`. Xbj akrst irhjr ci`hktki` k`hkcmtjh k` jqumtki` (4=) cm` dj m`moyzjh k` tjres ia ksiqum`ts m`h ksicists ms 5
Xbj
E^ O E^ L
^ O ^ l
5
E^ O E^ l
--------------------------------------------- (44)
rjprjsj`ts tbj `jfmtkvj ia tbj soipj ia tbj ksiqum`t, wbkcb ejmsurjs tbj emrfk`mo
rmtj ia tjcb`kcmo sudstktutki` ia omdir ti cmpktmo (ETXS Ol ). Xbj
^ O
rmtki sbiws tbj `jfmtkvj ia tbj soipj ia tbj ksicist. M` ksicist ks m ok`j wbkcb
^ L
k`hkcmtjs tbj oicus ia k`put ciedk`mtki`s wbkcb j`tmko tbj smej titmo cist. Kt ks sbiw` dy tbj jqumtki` C5 ^O O + ^ l L ir
^ O ^ l
5
E^ O E^ l
L5
C ^ O
-
^ O ^ l
O
k`hkcmtjs tbj amct tbmt tbj ksicist m`h ksiqum`t ok`js mrj tm`fj`t ti jmcb itbjr mt
tbj pik`t ia iptkemo k`put ciedk`mtki`. Sjci`h irhjr ci`hktki` air ek`kekzmtki` ia cist.
Hjmr cioojmfuj! Ms yiu l`iw, m `jfmtkvj dirhjrjh Bjsskm` hjtjrek`m`t ks suaakckj`t ti smy tbj cist ks mt kts ek`keue vmouj. Xbmt ks
;6
>
] O
] L
B ] O
O OO
O OL
] L
O LO
O LL
Jxmepoj
Suppisj m akre prihucjs m` iutput ] usk`f omdir O m`h cmpktmo L wktb prihuctki` au`ctki`
>.:
> .:
] => L O . Ka tbj iutput ks rjstrkctjh ti 4>> u`kts, prkcj ia omdir ks => dkrr pjr u`kt, tbj prkcj ia omdir ks => dkrr pjr u`kt u`kt m`h ^rkcj ia cmpktmo ks 8>Dkrr 8>Dkrr pjr u`kt, m`h tbj` hjtjrek`js hjtjrek`js tbj meiu`t ia O m`h L tbmt sbiuoh dj jepoiyjh mt ek`keue cist. Ak`h tbj ek`keue cist.
Xbj pridoje ks Ek`kekzj C 5 => O + 8>L Sudgjct ti 4>> => L >.: O>.: Aireuomtk`f tbj Omfrm`fj au`ctki` O ( O, L , ) => O 8> L (4>> => L >.: O>.: ) ---------------------- (43)
Akrst irhjr ci`hktki`s O O => : L >.: O>.: >
4 O>.:
L
------------------------------------------------------------- (48)
L >.: >.: >.: O 8> > : L O
2 L >.:
------------------------------------------------------------- (4:)
O>.: >. : > .: O 4>> => L O >
=> L >. : O>.: 4>> ----------------------------------------------------- (41)
Arie jqumtki` (48) m`h (4:), wj fjt 4 O>.: 2l >.: 5 >.: l >.: O
4O 5 2L O5 8L ------------------------------------------------------- (46) Sudstktutk`f jqumtki` (46) k` ti (41) fkvjs us L >.: ( 8 L ) >.: 4> ------------------------------------------------ (42) 4L 5 4> L 5 => m`h O 5 8(=>) 5 8>, 5 8 Sjci`h irhjr ci`hktki` Hjmr cioojmfuj! @iw wj sbiuoh cbjcl tbj sjci`h irhjr ci`hktki` ti vjrkay tbmt cist ks ojmst mt L 5 => m`h O 5 8>. Air cist ek`kekzmtki` tbj hjtjrek`m`t ia tbj dirhjrjh Bjsskm` emtrkx eust dj ojss tbm` zjri. >
] O
] L
B ] O
O OO
O OL
] L
O LO
O LL
?>
Mt O 5 8> m`h L 5 =>
;2
]
]O 5 O 5 ]
L
(:)
O O
(:)
=>
(:)
4.:
8>
(:)
8>
=>
L => ] l 5 l 5 OOO 5 4.: L >.: O =.: 5 4.:( 8)(=>) >.: ( 8>) =.: 5 >.=4: O ll 5 4.: L =.: O>.: 5 4.:( 8)(=>) =.: ( 8> ) >.: 54
O LO O OL 4 .: L >.: O>.: 4 .:(8)(=>) >.: (8>) >.:
Xbjrjairj, tbj hjtjrek`m`t ia tbj dirhjrjh Bjsskm` emtrkx ks B
>
4.:
=>
4.:
>.=4:
>.:
=>
>.:
4
5 - 4.: (:+:) +=>(-=.4: -=.4:) 5 - 4.: (=>) + => (-4.:) B 5 -:> ? >
Xbus, tbj akre cm` ek`kekzj kts cist wbj` kt jepoiys => u`kts ia cmpktmo m`h 8> u`kts ia omdir k` tbj prihuctki` pricjss m`h tbj ek`keue cist ks C 5 => (8>) + 8> (=>) Ek`. C 5 8>> + 8>> 5 2>> dkrr K` tbks pridoje L, O m`h mrj j`hifj`ius. Xbj Omfrm`fj euotkpokjr ejmsurjs tbj rjspi`skvj`jss ia tbj idgjctkvj au`ctki` ti m cbm`fj k` tbj ci`stm`t ia tbj ci`strmk`t au`ctki`. Hjmr cioojmfuj! Vbmt bmppj`s ti tbj vmouj ia tbj Omfrm`fkm` au`ctki` m`h tbj ci`strmk`jh au`ctki` wbj` titmo iutput k`crjmsjs arie 4>> ti 4>=9 Vbmt mdiut tbj meiu`t ia O m`h L9 Ciepmrj tbj vmouj ia tbj ci`strmk`jh au`ctki` m`h tbmt ia tbj Omfrm`fkm` au`ctki` mt tbks pik`t. K`tjrprjt tbj vmouj ia .-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------. Jomstkckty ia Sudstktutki`
Hjmr cioojmfuj! Vbmt bmppj`s ti tbj iptkemo k`put rmtki
L O
^ O
wbj` tbj k`put prkcj ^ l
k`crjmsjs9 ^ojmsj hkscuss wktb yiur arkj`hs m`h try ti m`swjr tbks qujstki`. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Bmvj yiu m`swjrjh kt9 Fiih. Xry ti rjomtj yiur m`swjr wktb tbj aiooiwk`f m`moysks. K`crjmsj k` tbj
^ O ^ l
sbiws tbj amct tbmt cmpktmo ks rjomtkvjoy cbjmpjr si tbmt kt wkoo dj sudstktutjh
air omdir omdir m`h tbj iptke iptkemo mo k`pu k`putt rmtki rmtki L O wkoo k`crjmsj. Xbj jxtj`t ia tbks sudstktutki` ks
;;
ejmsurjh dy tbj jomstkckty ia sudstktutki` wbkcb ks rjprjsj`tjh dy skfem).
5
(oiwjr - cmsj Frjjl ojttjr
l rjomtkvjcb m`fjk`( ) O rjomtkvjcbm`fjk`( ^ o ^ l )
h ( L ) O
5
L ( ) O
^ O
h ( ) ^ L
h ( L ) O
^ h ( O
L ( ) O
^ O
( ) ^ L
^ L
)
^ ( O ) ^ L
Xbj vmouj ia okjs k` djtwjj` > m`h + Vbj` 5 >, tbj twi k`puts mrj usjh k` m akxjh pripirtki` ms ciepojej`t ti jmcb itbjr. Vbj` 5 , tbj twi k`puts mrj pjrajct sudstktutj ti jmcb itbjr. Ms k`crjmsjs, tbj sudstktutmdkokty djtwjj` tbj twi k`puts k`puts k`crjmsjs. Iptkekzmtki` ia ` - vmrkmdoj cmsj
Fkvj` tbj idgjctkvj au`ctki` Iptkekzj W a ( x= , x 4 , x 3 ...., x ` ) Sudgjct ti f ( x= , x 4 , x 3 ,...., x ` ) c Hjmr cioojmfuj! Skekomr ti iur jmrokjr hkscusski` wj iufbt ti akrst aireuomtj tbj Omfrm`fj au`ctki`. Xbmt ks x ` ) (c f ( x = , x 4 , x3 ,..., x ` )) O a ( x= , x 4 , x3 ....,
Xbj `jcjssmry ci`hktki` air iptkekzmtki` ia tbks au`ctki` ks tbmt O O= O4 O 3 O8 O` 5 > Xbj sjci`h irhjr ci`hktki` air iptkekzmtki` ia tbks au`ctki` hjpj`hs i` tbj skf` ia h 4O sudgjct 3 ..... f ` hx ` > skekomr ti iur jmrokjr hkscusski`. ti hf f = hx= f 4 hx 4 f 3 hx Xbj pisktkvj ir `jfmtkvj hjak`ktj`jss ia h 4 O k`viovjs tbj dirhjrjh Bjsskm` hjtjrek`mtj tjst. Biwjvjr, k` tbks cmsj tbj ci`hktki`s bmvj ti dj jxprjssjh k` tjres ia tbj dirhjrjh prk`ckpmo ek`ir ia tbj Bjsskm`. Fkvj` tbj dirhjrjh Bjsskm` ms
f = f 4 ---------- f `
> f =
B
O==
O=4
----------
O =`
5 f 4 O4= O44 ----------- O 4 ` -
-
3` ---------------------------------- O
=>>
- -
f `
--------------------------------------------------
O`=
O` 4
O -------- ``
Xbj succjsskvj dirhjrjh prk`ckpmo ek`irs mrj
B 4
f 4 O4=
>
> f = f 4 f O== O=4 =
B 3
O44
f = f 4
f = O==
f 4 O4= f 3
O=4
f 3
O=3
O44 O43 O3= O34 O33
jtc.
Biwjvjr, B B ` . . B Sbiws tbj sjci`h prk`ckpmo ek`ir ia tbj Bjsskm` dirhjrjh wktb >, f m`h f . = 4 4
h 4 O ks pisktkvj hjak`ktj sudgjct ti hf > ka m`h i`oy ka
B 4
h 4 O ks `jfmtkvj hjak`ktj sudgjct ti hf > ka m`h i`oy ka
B 4
, B ,-----, B
3
`
>, B 3 >, B 8
>.
> ,---.
M pisktkvj hjak`ktj h 4 O ks m suaakckj`t ci`hktki` air ek`keue vmouj m`h `jfmtkvj hjak`ktj h 4 O ks suaakckj`t ci`hktki` air emxkekzmtki` ia tbj idgjctkvj au`ctki`. K` iur m`moysks mdivj B ks tbj i`j wbkcb ci`tmk`s O 44 ms tbj omst jojej`t ia kts prk`ckpmo hkmfi`mo. B ks tbj i`j wbkcb k`couhjs O33 ms tbj omst jojej`t ia kt's prk`ckpmo hkmfi`mo jtc. 4
3
Iptkekzmtki` wbj` tbjrj ks eirj tbm` i`j jqumokty ci`strmk`t
Ojt us ci`skhjr tbj iptkekzmtki` pridoje k`viovjs tbrjj vmrkmdojs m`h twi ci`strmk`ts. Iptkekzj W a ( x= , x 4 , x3 ) Sudgjct ti f = ( x= , x 4 , x 3 ) c = f 4 ( x= , x 4 , x 3 ) c 4
Ms usumo wj sbiuoh ci`struct tbj Omfrm`fj au`ctki` dy usk`f tbj Omfrm`fj euotkpokjr .Sk`cj wj 4 k` iur bmvj twi ci`strmk`t ci`strmk`t au`ctki` au`ctki`s, s, wj mrj rjqukrj rjqukrjhh ti k`cirpi k`cirpirmtj rmtj twi s, k.j., = m`h m`moysks. Xbj Omfrm`fj au`ctki` ks O a ( x= , x 4 , x 3 ) (c = f = ( x= , x 4 , x 3 )) (c 4 f 4 ( x= , x 4 , x 3 ))
Akrst irhjr ci`hktki`s air iptkekzmtki` O=
a = = f ==
4
4 f =
>
O4 a 4 = f 4 f > O3 a 3 = f 4 f > O = c = f = ( x= , x 4 , x 3 ) > O 4 c 4 f 4 ( x= , x 4 , x3 ) > Vbj` tbjrj mrj ` - vmrkmdojs m`h e - ci`strmk`ts, tbj Omfrm`fj au`ctki` djciejs = 4 = 3
4 4 4 3
e
g O a ( x= , x 4 , x 3 ,....., x ` ) k (c f k ( x= , x 4 , x 3 ,..., x ` )) g =
K` tbks cmsj wj wkoo bmvj e+ ` vm vmrkmdojs rkmdojs k` tbj Omfrm`fj au`ctki` au`ctki` m`h wj wkoo bmvj mosi mosi e+ ` skeuotm`jius jqumtki`s. Akrst irhjr ci`hktki`s mrj
=>=
k
Ok a k g f g ,
(k 5 =, 4, 3, ---, `) m`h (g5 =, 4, 3, --- e)
O k c g f k ( x = , x 4 , x3 ,..., x` ) >
Sjci`h irhjr ci`hktki`s air iptkekzmtki` ia tbrjj vmrkmdojs m`h twi ci`strmk`ts pridoje mrj
> > B
> >
=
=
= 4 =
4 4 4
=
f f f f
f f
3 4 3
O=3 5 f == f =4 O == O=4 f =4 f 44 O 4 = O 44 O f 3= f 34 O3= O 34 O 33
43
K` tbks cmsj,
5 B . Xbus air m emxkeue vmouj, 0 >, B ? >.
B 3 B 4
3
Air m ek`keue, B ? >, B ? >. Vktb tbj jxkstj`cj ia ` - vmrkmdojs m`h e - ci`strmk`ts, tbj sjci`h irhjr ci`hktki` ks jxpomk`jh ms 4
3
>
> -------- >
f ==
>
>
> ------->
4 f =4 f 44 f 34 ----- -- f `
--
- - - - > > > ------ - >
B
-
- --------------- e e f f e f ------ -- f e
4
=
5 = f == f =4 f 3= ----- - f `
f =4 f 44 f 34 ------ f `4
f =3 f 43 f 33 ------ f `3
= = f =4 f 3 -------- f `
>
3
`
O== O=4 O=3 ---------- O =` O4=
O44 O43 --------- O 4 `
O34 O33 ------ O ` -- -- -- ------- -- -- --- e e e e `` f = f 4 f 3 - - - - f ` O` O` 4 O` 3 -------- O O3=
3
=
@iw wj bmvj hkvkhjh tbj dirhjrjh Bjsskm` hjtjrek`m`t k` ti aiur pmrts. Xbj uppjr ojat mrjm k`couhjs zjris i`oy m`h tbj oiwjr rkfbt mrjm ks skepoy m pomk` Bjsskm`. Xbj rjemk`k`f twi mrjms k`couh k`c ouhjj tbj tbj f gk hjrkvmtkvjs. Xbjsj hjrkvmtkvjs bmvj m ekrrir kemfj rjomtki`sbkp ti jmcb itbjr ci`skhjrk`f tbj prk`ckpmo hkmfi`mo ia tbj dirhjrjh Bjsskm` ms m rjajrj`cj.
=>4
Vj cm` crjmtj sjvjrmo dirhjrjh prk`ckpmo ek`irs arie B .Kt ks pisskdoj ti cbjcl tbj sjci`h irhjr suaakckj`t ci`hktki` air iptkekzmtki` usk`f tbj sk`f ia tbj aiooiwk`f dirhjrjh prk`ckpmo ek`irs7 B ,----------, B , B Xbj idgjctkvj au`ctki` cm` suaakckj`toy mcbkjvj kts emxkeue vmouj wbj` tbj succjsskvj dirhjrjh ks (-=) (-=) e+= wbjrj ms air prk`ckpmo ek`irs motjr`mtj k` skf`. Biwjvjr, tbj skf` ia B ek`keue vmouj tbj suaakckj`t ci`hktki` ks tbmt moo dirhjrjh prk`ckpmo ek`irs bmvj tbj smej skf`, k.j., (-=) e . Xbks k`hkcmtjs tbmt ka wj bmvj m` ihh `uedjr ia ci`strmk`ts, tbj` skf` ia moo dirhjrjh prk`ckpmo ek`irs wkoo wkoo dj `jfmtkvj m`h pisk pisktkvj tkvj wktb jvj` `uedjr iiaa ci`strmk`ts. e =
e4
`
e =
Hjmr cioojmfuj! Dy `iw yiu bmvj ciepojtjh tbj sjci`h sjctki` ia tbks u`kt. Xbus, try ti hi tbj aiooiwk`f sjoa tjst qujstki`s ti jxmek`j yiur u`hjrstm`hk`f ia tbks sjctki`.
Sjoa - Xjst 8.4 Siovj tbj aiooiwk`f qujstki`s dmsjh i` tbj k`airemtki` mdivj. =. Vbmt ks ci`strmk`jh au`ctki`9 ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 4. Jxpomk` jomstkckty ia sudstktutki` ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------3. Vbmt hijs tbj Omfrm`fj euotkpokjr k`hkcmtj9 ---------------------------------------------------------- 8. Suppisj m akre amcjs tbj prihuctki` au`ctki` ] 5 =4> O + 4>>L - O 4 - 4L 4 air pisktkvj vmoujs ia ]. Ka kt cm` duy O mt : dkrr pjr u`kt, L mt 2 dkrr pjr u`kt m`h bms m duhfjt ia 6> Dkrr, hjtjrek`j tbj emxkeue iutput tbmt kt cm` prihucj usk`f sudstktutki` ejtbih.-------------------------------------------------------------------------------------------------------------------------------------------:.Suppisj tbj prkcjs ia k`puts L m`h O mrj =4 dkrr m`h 3 dkrr pjr u`kt rjspjctkvjoy m`h tbj prihuctki` au`ctki` au`ctki` ia tbj akre ks ]5 4:L >.: O >.: . Hjtjrek`j tbj ek`keue cists ia prihuck`f =,4:> u`kts ia iutput usk`f Omfrm`fj euotkpokjr ejtbih.----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------1. Suppisj m ci`suejr bms m utkokty au`ctki` ia Z 5 8>x >.: y>.: .Ka tbj prkcj ia x ks 4> dkrr pjr u`kt, prkcj ia y ks : dkrr u`kt m`h tbj ci`suejr bms m duhfjt ia 1>> dkrr . Hjtjrek`j tbj meiu`t ia x m`h y wbkcb emxkekzj utkokty usk`f tbj Omfrm`fj euotkpokjr ejtbih. ------------------------------------------------------------------------------------------------------------------------------------------------------Bmvj yiu m`swjrjh tbjsj qujstki`s9 Ka yiur m`swjr ks `i, rjrjmh tbks sjctki` m`h try ti m`swjr tbje. Ka yjs, fiih. Fi ti tbj `jxt sjctki`. Sjctki Sjc tki` ` 8.3 K` K`jq jqumo umokty kty Ci`st Ci`strm rmk`t k`tss m` m`h h Lub` Lub` - tuc tuclj ljrr Xb Xbji jirj rjes es,, m` m`h h Ekxjh Ekxjh Ci`strmk`ts
Zp i` tbj ciepojtki` ia tbks sjctki` yiu mrj jxpjctjh ti
-
hjscr hjscrkd kdjj w wbm bmtt `i` `i`ok ok`j `jmr mr pprif rifrm rmee eek` k`ff ks ks hjscr hjscrkd kdjj tbj tbj Lub` Lub` - X Xuc uclj ljrr cci` i`hk hktk tki`s i`s jxpo jxpomk mk`` mdi mdiut ut ttbj bj ci ci`st `strm rmk` k`tt qu qumo moka kakc kcmt mtki ki`` hjs hjscrk crkdj dj tbj tbj Lub Lub`` - Xu Xucl cljr jr su suaa aakc kckj kj`cy `cy tbji tbjirje rje =>3 hjscr hjscrkd kdjj tbj tbj cmsj cmsj ia Ek Ekxj xjhh ci` ci`st strmk rmk`t `tss
sbi sbiw w jc jci` i`ie iekc kc mmpp ppok okcm cmtk tki` i` iiaa tb tbks ks ttbji bjirj rje e
@i`ok`jmr prifrmeek`f
Hjmr cioojmfuj cioojmfuj!! K tbk` tbk`ll yiu l`iw mdiut tbj ci`c ci`cjpt jpt ia ok`jmr pri prifrmee frmeek`f k`f k` yiur qum`tktmtkvj ejtbih air jci`ieksts KK (Jci`. 431) stuhkjs. Vbmt ks `i`ok`jmr prifrmeek`f9 Hkscuss wktb yiur arkj`hs m`h try ti m`swjr tbks qujstki`. Vrktj tbj m`swjr i` m riufb pmpjr.------------------------------------------------------------------------------------------------------------- pmpjr.------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Bmvj yiu m`swjrjh tbks qujstki`9 Il, Fiih. Jvmoumtj yiur m`swjr wktb tbj aiooiwk`f m`moysks. Xbj pridoje ia iptkekzmtki` ia m` idgjctkvj au`ctki` sudgjct ti cjrtmk` rjstrkctki`s ir ci`strmk`ts ks m usumo pbj`iej`i` k` jci`iekcs. Eistoy, tbj ejtbih ia emxkekzk`f ir ek`kekzk`f m au`ctki` k`couhjs jqumokty ci`strmk`ts. Air k`stm`cj, utkokty emy dj emxkekzjh sudgjct ti m akxjh k`ciej tbmt tbj ci`suejr bms m`h tbj duhfjt ci`strmk`t ks fkvj` k` tbj aire ia jqumtki`. Sucb typj ia iptkekzmtki` ks rjajrrjh ti ms comsskcmo iptkekzmtki`. Dut idgjctkvj au`ctki` sudgjct ti k`jqumokty ci`strmk`ts cm` dj iptkekzjh usk`f tbj ejtb ejtbih ih ia emtbjemtkcmo prifrmeek`f. Ka tbj idgjctkvj au`ctki` ms wjoo ms tbj k`jqumokty ci`strmk`ts ks ok`jmr, wj wkoo usj m ejtbih ia ok`jmr prifrmeek`f. Biwjvjr, ka tbj idgjctkvj au`ctki` m`h tbj k`jqumokty ci`strmk`ts mrj `i`ok`jmr, wj wkoo mppoy tbj tjcb`kquj ia `i`ok`jmr prifrmeek`f ti iptkekzj tbj au`ctki`. Emxkekzmtki` pridoje ia `i` - ok`jmr prifrmeek`f Emxkekzj 5 a ( x= , x 4 , x3 ,....., x ` ) x ` ) l = Sudgjct ti f = ( x= , x 4 , x3 ,...,
f 4 ( x= , x 4 , x3 ,..., x ` ) l 4
f 3 ( x= , x 4 , x3 ,..., x` ) l 3
7
7
7
f e ( x= , x 4 , x 3 ,..., x ` ) l e m`h x g Xbj ek`kekzmtki` ^ridoje cm` dj jxprjssjh k` tbj aire ia 3 ,....., x ` ) ) Ek`kekzj C 5 a ( x= , x 4 , x x ` ) l = Sudgjct ti f = ( x= , x 4 , x3 ,...,
> ,
( g =, 4 ,3....., ` )
f 4 ( x= , x4 , x3 ,..., x ` ) l 4
f 3 ( x= , x 4 , x3 ,..., x ` ) l 3 7 7 7 7 e f ( x= , x 4 , x 3 ,..., x ` ) l e , x g > ( g =, 4 ,3....., ` ) Vbjrj C- rjprjsj`ts titmo cist wbkcb ks tbj idg idgjctkvj jctkvj au`ctki`. x g - ks tbj meiu`t ia iutput prihucjh l k - ks tb tbjj ci`stm` ci`stm`tt ia tbj ci`s ci`strmk`t trmk`t au`ctki` f k - ks tbj ci`strmk`t au`ctki`.
=>8
Vj bmvj idsjrvjh arie tbj mdivj jxprjsski` tbmt tbj `i`ok`jmr prifrmeek`f mosi k`couhjs tbrjj k`frjhkj`ts. Xbjsj mrj - tbj idgjctkvj au`ctki` - m sjt ia ci`strmk`ts ( k`jqumokty ) - `i` - `jfmtkvkty rj rjstrkctki`s strkctki`s i` tbj cbi cbikcj kcj vmrkmdoj Xbj idgjctkvj ms wjoovmrkmdojs. ms tbj k`jqumokty ci`strmk`ts ks mssuejh ti dj rjspjct ti jmcbau`ctki` ia tbj cbikcj Oklj ok`jmr prifrmeek`f wj mppoy i` hkaajrj`tkmdoj air ci`strmk`tswktb emxkekzmtki` m`h ek`kekzmtki` pridoje k`viovjs i`oy ci`strmk`ts. Jxmepoj= =.Ak`h tbj vmoujs ia x m`h y ia m) Ek`kekzj C x 4 y 4
tbj aiooiwk`f au`ctki` frmpbkcmooy.
Sudgjct ti x y 4: x, y > Hjmr cioojmfuj! Akrst wj sbiuoh ci`vjrt tbj k`jqumokty ci`strmk`t k` ti jqumokty ms hrmw tbj frmpb ia tbks ci`strmk`t au`ctki` i` tbj xy pom`j. x = 4 3 8 y 4: =4.: =4.: 2. 2.33 2. 2.33
: :
1 88.1 .1
xy 4: m`h
6 .... .............................. ............................4: 4: 3. 3.:6 :6 ... .......... ...... ...... ...... .......... ...... ....= .=
Akf.(m) Xbj sbmhjh rjfki` k` tbj mdivj akfurj rjprjsj`ts tbj ajmskdoj rjfki`. Ojt us jvmoumtj tbj idgjctkvj au`ctki` C mt pik`ts M, D, C, H m`h J i` tbj frmpb. Mt pik`t m (=, 4:), C5=4+ 4:4 5 =+ 14: 5 141 4
4
pik`t cD(:, (8,:), 1.3), C 5 8:4 + (1.3) Mt pik`t :4 5 4: - +=1+ 4: 3;.1; 5 :> 5 ::.1;
=>:
Mt pik`t h (1, 8.1) C 5 14 + (8.1) 4 5 31 + 4=.=1 5 :6.=1 Mt pik`t J (4:, =) , C 5 ( 4:) 4 + =4 5 14: + = 5 141 Xbjrjairj, tbj vmouj ia x m`h y wbkcb ek`kekzjs tbj idgjctkvj au`ctki` mrj : m`h : rjspjctkvjoy. Xbj ek`keue vmouj ks C 5 :>. d) Emxkekzj 5 x4 + (y - 4) 4
Sudgjct :x >+ 3y =: M`h x, yti
Sioutki`
Skekomr ti tbmt ia pridoje m, wj sbiuoh ci`vjrt tbj k`jqumokty ci`strmk`t k` ti jqumokty ci`strmk`t m`h hrmw kts frmpb k` tbj x y pom`j. Kt ks :x + 3y 5 =:
R [
> :
= 3.3
4 =.16
3 >
Akf.(d)
Xbj sbmhjh rjfki` ia tbj mdivj akfurj rjprjsj`ts tbj ajmskdoj rjfki` ms jvjry pik`t k` tbks ajmskdoj rjfki` smtksakjs tbj k`jqumokty ci`strmk`t : x + 3y =:. Jvmoumtk`f tbj idgjctkvj au`ctki` mt pik`ts M, D, C m`h H ia tbj mdivj frmpb (akf. d), 5 >4 + (: - 4)4 5 > + ; 5 ; Mt pik`t M (>, :), Mt pik`t D (=, 3.3), 5 =4+ (3.3 - 4) 4 5 =+ =.1; 5 4.1; Mt pik`t C (4, =.16), 5 44 + (=.16 - 4) 4 5 8 + >.=>2; 5 8.=>2; 5 34 + ; (> - 4) 4 5 ;+ 8 5 =3 Mt ^ik`t H (3, >), Xbjrjairj, tbj tbj idgjctkvj au`ctki` ks emxkekzjh wbj` x 5 3 m`h y 5 >. Xbj emxkeue priakt ks 5 =3 K` fj`jrmo, wj cm` hkstk`fuksb tbj `i`ok`jmr prifrmeek`f arie tbmt ia ok`jmr i`j dmsjh i` tbj aiooiwk`f pik`ts. =. =.K` K` `i`ok`jmr prifrmeek`f tbj akjoh ia cbikcj `it `jcjssmrkoy oicmtjs mt kts jxtrjej pik`ts. 4. Xbj `uedjr ia ci`strmk`ts emy `it dj tbj smej wktb tbj cbikcj vmrkmdojs. 3. Aiooiwk` Aiooiwk`ff tbj ssmej mej hkr hkrjctki jctki`` k` m ei eivjej` vjej`tt emy ``it it ojmh ti m ci` ci`tk`u tk`umooy mooy kk`crjm `crjmsk`f sk`f ir (hjcrjmsk`f) vmouj ia tbj idgjctkvj au`ctki`. 8. ajmskdoj rjfki`emy emy`it `itdjdjm foidmo m ci`vjx sjt. :.Xbj M oicmo iptkeue iptkeue.
=>1
Lub` - Xucljr Ci`hktki`s
Hjmr cioojmfuj! K` tbj prjvkius sjctki`s ia tbks u`kt, wj bmvj hkscussjh mdiut iptkekzmtki` pridojes ia tbj idgjctkvj au`ctki` wktb jqumokty ci`strmk`ts m`h wktbiut jxpokcktoy rjstrkctk`f tbj sk`f ia tbj cbikcj vmrkmdojs. K` tbks cmsj, tbj akrst irhjr ci`hktki` ks smtksakjh privkhjh tbmt tbj akrst irhjr pmrtkmo hjrkvmtkvj ia tbj Omfrm`fj au`ctki` wktb rjspjct ia jmcb cbikcj vmrkmdoj m`h wktb rjspjct ti tbj Omfrm`fj euotkpokjr ks zjri. Air k`stm`cj, k` tbj pridoje Emxkekzj 5 a ( x , y ) Sudgjct ti f ( x, y ) l Xbj Omfrm`fj au`ctki` ks O a ( x, y ) ( l f ( x, y ))
Xbj akrst irhjr ci`hktki` stmtjs tbmt O x O y O >
K` `i`-ok`jmr prifrmeek`f, tbjrj ks m skekomr akrst irhjr ci`hktki` wbkcb ks rjajrrjh ti ms Lub` Xucljr ci`hktki`s. Ms wj hkscussjh prjvkiusoy, k` comsskcmo iptkekzmtki` pricjss, tbj akrst irhjr ci`hktki` ks m `jcjssmry ci`hktki`. Biwjvjr, m cjrtmk` ci`hktki` sbiuoh dj auoakoojh air tbj Lub` Xucljr ci`hktki`s ti dj `jcjssmry ci`hktki`s. Hjmr cioojmfuj! @iw ojt us hkscuss Lub` - Xucljr ci`hktki`s k` twi stjps air tbj purpisj ia emlk`f tbj jxpom`mtki` jmsy ti u`hjrstm`h. Stjp =
K` tbj akrst stjp, ojt us tmlj m pridoje ia iptkekzk`f tbj idgjctkvj au`ctki` wktb `i` `jfmtkvkty rjstrkctki`s m`h wktb `i itbjr ci`strmk`ts. K` jci`iekcs, tbj eist cieei` k`jqumokty ci`strmk`t ks `i` `jfmtkvkty ci`strmk`t. Emxkekzj 5 a(x) Sudgjct ti x > privkhjh tbmt tbj au`ctki` ks suppisjh ti dj ci`tk`uius m`h seiitb. Dmsjh i` tbj rjstrkctki` x >, wj emy bmvj tbrjj pisskdoj rjsuots. Ms sbiw` k` tbj aiooiwk`f akfurjs.
=>6
Vbj` tbj oicmo emxkeue rjskhjs k` skhj tbj sbmhjh ajmskdoj rjfki` ms sbiw` mdivj mt pik`t D ia akf (k), tbj` wj bmvj m` k`tjrkir sioutki`. K` tbks cmsj, tbj akrst irhjr ci`hktki` ks skekomr ti tbmt ia tbj comsskcmo iptkekzmtki` pricjss, k.j.,
h hx
5 >.
Hkmfrme (kk) sbiws tbmt tbj oicmo emxkeue ks oicmtjh i` tbj vjrtkcmo mxks k`hkcmtjh dy pik`t C. Mt tbks pik`t, tbj cbikcj vmrkmdoj ks > m`h tbj akrst irhjr hjrkvmtkvj ks zjri, k.j.
h
5 >, mmtt pik`t C
hx
wj bmvj m diu`hmry sioutki`.
Hkmfrme (kkk) k`hkcmtjs tbmt tbj oicmo emxkeue emy oicmtj mt pik`t H ir pik`t J wktb k` tbj ajmskdoj rjfki`. K` tbks cmsj, tbj emxkeue pik`t ks cbmrmctjrkzjh dy tbj k`jqumokty
h hx
?>
djcmusj tbj curvjs mrj mt tbjkr tbjkr hjcrjmsk`f pirtki` mt tbjsj tbjsj pik`ts. Xbjrjairj, arie tbj mdivj hkscusski` kt ks cojmr tbmt tbj aiooiwk`f tbrjj ci`hktki`s bmvj ti dj ejt si ms ti hjtjrek`j tbj vmouj ia tbj cbikcj vmrkmdoj wbkcb fkvjs tbj oicmo emxkeue ia tbj idgjctkvj au`ctki`. a ( x ) > , m`h x 0 > (pik`t D)
a ( x ) > , m`h x 5 > (pik`t C) a ( x ) > , m`h x 5 > (pik`t H m`h J)
Ciedk`k`f tbjsj tbrjj ci`hktki` k` ti i`j stmtjej`t fkvj` us a ( x ) > x a ( x ) > x > m`h Xbj akrst k`jqumokty k`hkcmtjs tbj k`airemtki` ci`cjr`k`f
h hx
. Xbj sjci`h k`jqumokty sbiws tbj
`i` `jfmtkvkty rjstrkctki` ia tbj pridoje. Xbj tbkrh pmrt k`hkcmtjs tbj prihuct ia tbj twi qum`tktkjs x m`h a ( x ) .Xbj mdivj stmtjej`t wbkcb ks m ciedk`mtki` ia tbj tbrjj ci`hktki`s rjprjsj`ts tbj akrst irhjr `jcjssmry ci`hktki` air tbj idgjctkvj au`ctki` ti mcbkjvj kts oicmo emxkeue privkhjh tbmt tbj cbikcj vmrkmdoj bms ti dj `i` `jfmtkvj. Ka tbj pridoje k`viovjs ` - cbikcj vmrkmdojs oklj Emxkekzj a ( x= , x 4 , x 3 ,... x ` ) Sudgjct ti x k > Xbj akrst irhjr ci`hktki` k` comsskcmo iptkekzmtki` pricjss ks a = 5 a 4 5 a 3 5 -------5 a ` 5 > Xbj akrst irhjr ci`hk ci`hktki` tki` tbmt sbiuoh dj smtksakjh ti hjtjrek`j tbj vmouj ia tbj cbikcj vmrkmdoj wbkcb emxkekzjs tbj idgjctkvj au`ctki` ks a k > xk > m`h x k a k 5 > (k 5=, 4, 3, -------, `) Vbjrj a k ks tbj pmrtkmo hjrkvm hjrkvmtkvj tkvj ia tbj idgjctkvj au` au`ctki` ctki` wktb rjspjct ti x k , k.j.,
a k
x k
.
Stjp 4
=>2
Hjmr cioojmfuj! @iw wj ci`tk`uj ti tbj sjci`h stjp. Xi hi tbks, ojt us mttjept ti k`cirpirmtj k`jqumokty ci`strmk`ts k` tbj pridoje. K` irhjr ti skepokay iur m`moysks, ojt us akrst hkscuss mdiut emxkekzmtki` pridoje wktb tbrjj cbikcj vmrkmdojs m`h twi ci`strmk`ts ms sbiw` djoiw. Emxkekzj 5 a ( x= , x 4 , x3 ) Sudgjct ti f = ( x= , x 4 , x3 ) l = 4 f ( x= , x 4 , x3 ) l 4 M`h x=, x4, x3 > Zsk`f tbj hueey vmrkmdojs s = m`h s4 wj cm` cbm`fj tbj mdivj pridoje k` ti Emxkekzj 5 a ( x= , x 4 , x3 ) Sudgjct ti f = ( x= , x 4 , x3 ) s= l = f 4 ( x= , x 4 , x3 ) s 4 l 4 M`h x=, x4, x > m`h s=, s4 > 3
Vj cm` aireuomtj tbj Omfrm`fj au`ctki` usk`f tbj comsskcmo ejtbih privkhjh tbmt tbj `i` `jfmtkvkty ci`strmk`ts ia tbj cbikcj vmrkmdojs mrj `it jxkstjh ms O a ( x= , x 4 , x3 ) = \l = f = ( x= , x 4 , x 3 ) s= U 4 \ l 4 f 4 ( x= , x 4 , x3 ) s 4 U Kt ks pis pissk skdoj doj hjr hjrkvj kvj tbj tbj Lub Lub` ` Xuc Xucljr ljr ci` ci`hkt hktki` ki`ss hkr hkrjct jctoy oy ari arie e tbj Omf Omfrm` rm`fj fj au` au`ctk ctki`. i`. Ci`skhjrk`f tbjtimdivj 3-vmrkmdoj 4-ci`strmk`ts pridoje Xbj akrst irhjr ci`hktki` ks O x=
5
O x 4
O
5
O
5
x 3
s=
5
O s 4
5
O
5
4
O =
5>
Biwjvj Biw jvjr, r, x g m`h s k vmrkmdoj mrj rjstrkctjh ti dj `i` `jfmtkvj. Ms m rjsuot, tbj akrst irhjr ci`hktki`s i` tbjsj vmrkmdojs iufbt ti dj eihkakjh ms aiooiws. O
x g
g
O
O
O
m`h x g x 5 >
x g >
>
sk
>
s k
s k s k
>
m`h
5>
O k
5 > Vbjrj (k 5 =, 4 m`h g5 =, 4, 3) Biwjvjr, wj cm` ciedk`j tbj omst twi ok`js m`h tbjrjdy mvikh tbj hueey vmrkmdojs k` tbj mdivj
akrst irhjr ci`hktki` ms sbiw` djoiw. Ms
k >, ,
O s k
k
, tbj sjci`h ok`j sbiws tbmt
S k > m`h – S k k 5 >
ir k
>,
Dut, wj l`iw tbmt sk
S k > m`h
S k k 5 >
k
l k f ( x= , x 4 , x3 ) . Dy sudstktutk`f kt k` pomcj ia s k , wj cm` fjt
k
k
\ l k f ( x = , x 4 , x3 ) U 5>
Xbjrjairj, tbj akrst irhjr ci`hktki` wktbiut >hueey m`h vmrkmdojs k` jxprjssjh ms l k f ( x = , x 4 , x3 ) > ,
k
k
=>;
O
>
x g
O
k
O
x g >
m`h x g x 5 > g
5 l k f k ( x= , x 4 , x 3 ) >
k
> m`h
k
\ l k f k ( x = , x 4 , x 3 ) U
5> Xbjsj mrj tbj Lub` - tucljr ci`hktki`s air tbj fkvj` emxkekzmtki` pridoje.
Hjmr cioojmfuj! Biw cm` wj siovj ek`kekzmtki` pridoje9 Hkscuss wktb yiur arkj`hs m`h try ti m`swjr tbks qujstki`. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Bmvj yiu m`swjrjh tbks qujstki`9 Il fiih. Tjmh tbj upciek`f hkscusski` m`h jvmoumtj yiur m`swjr wktb rjajrj`cj ti kt. I`j ia tbj ejtbihs ti siovj tbks pridoje ks cbm`fk`f kt k` ti emxkekzmtki` pridoje m`h tbj` mppokjs tbj smej pricjhurj wktb emxkekzmtki`. Ek`kekzk`f C ks skekomr ti emxkekzk`f ( C ). Biwjvjr, ljjp k` ek`jh tbj amct tbmt wj bmvj ti euotkpoy jmcb ci`strmk`t k`jqumoktkjs dy ( = ).Vj cm` hkrjctoy mppoy tbj Omfrm`fj euotkpokjr ejtbih m`h hjtjrek`j tbj ek`kekzmtki` vjrski` ia Lub` - Xucljr ci`hktki` k`stjmh ia ci`vjrtk`f tbj k`jqumokty ci`strmk`ts k`ti jqumokty ci`strmk`ts usk`f hueey vmrkmdojs ms O
O x g
O
k
>
x
>
> m`h x
g
k
>
m`h
g
x g
k
5>
O
5 >
k
(ek`kekzmtki`)
Jxmepoj
4. Ojt us cbjcl wbjtbjr tbj sioutki`s ia iur jxmepoj = smtksay tbj Lub` - Xucljr ci`hktki`s ir `it m) Ek`kekzj C5 x4+ y4 Sudgjct ti x y 4: m`h x, y >
Xbj Omfrm`fj au`ctki` air tbks pridoje ks O 5 x4 + y4 + (4: – x y) Kt ks m ek`kekzmtki` pridoje. Xbjrjairj, tbj mppriprkmtj ci`hktki`s mrj O x O y
O
5 4x - y 5
4 y
-
>, x
x > m`h x
> , y > m`h
5 4: – x y >,
> m`h
O x
y
5>
O y
O
5>
5 >
Hjmr cioojmf cioojmfuj! uj! Cm` wj hjtjrek` hjtjrek`jj tbj `i` `jfmtkvj vmouj wbkcb wkoo smtksay moo tbj mdivj ci`hktki`s tifjtbjr wktb tbj iptkemo sioutki` x m`h y9 Xbj iptkemo sioutki`s k` iur jmrokjr hkscusski` mrj x 5 : m`h y 5 :, wbkcb mrj `i`zjri. Xbus, tbj ciepojej`tmry somcl`jss ( x O
5 >, y
y
5 >) sbiws tbmt O x
O
>. m`h y
5 >.
==>
O x
Xbus, wj cm` hjtjrek`j tbj vmouj ia dy sudstktutk`f tbj iptkemo vmoujs ia tbj cbikcj vmrkmdojs k` jktbjr ia tbjsj emrfk`mo ci`hktki`s ms O x
5 4x -
5 >
y
4(:) - (:) 5 > => - : 5 > 5 4 0 >
Xbks vmouj
5
4, x 5 : & y 5 : kepoy tbmt
O x
5 >, >,
O y
5 >,
O
5 > wbkcb auoakos tbj
emrfk`mo ci`hktki`s m`h tbj ciepojej`tmry somcl`jss ci`hktki`s. K` itbjr wirhs, moo tbj Lub` Xucljr ci`hktki`s mrj smtksakjh. 3. Emxkekzj W => x x 4
Sudgjct ti
=2> y y
4
x y 2> x, y >
Sioutki`
Hjmr cioojmfuj! Akrst wj sbiuoh aireuomtj tbj Omfrm`fj au`ctki` mssuek`f tbj jqumokty ci`strmk`t m`h kf`irk`f tbj `i` `jfmtkvkty ci`strmk`ts. 4 y y (2> x y ) O => x x 4 =2> Xbj akrst irhjr ci`hktki`s mrj O
=> 4 x > => 4 x (=)
x O
=2> 4 y > =2> 4 y ( 4)
y O
2> x y > x y 2> (3)
Xmlk`f jqumtki` (=) m`h (4) skeuotm`jiusoy => 4 x =2> 4 y
4 y 4 x =6> 4 y =6 =6> > 4 x
y 2: x ( 8)
Ka wj sudstktutj jqumtki` (8) k` ti (3), wj fjt
x 2: x 2> 4 x : x 4.:
Biwjvjr, tbj vmouj ia tbj cbikcj vmrkmdojs ks rjstrkctjh ti dj `i` `jfmtkvj. x 4.: ks k`ajmskdoj. Vj eust sjt x5 > sk`cj kt bms ti dj `i` `jfmtkvj. @iw wj cm` hjtjrek`j tbj vmouj ia y dy sudstktutk`f zjri k` pomcj ia x k` jqumtki` (3) . > y y
2>
2>
Xbjrjairj, =2> 4( 2>) 4> Xbj pisskdoj sioutki`s mrj x >, y 2>, 4> Biwjvjr, wj eust cbjcl tbj k`jqumokty ci`strmk`ts m`h tbj ciepojej`tmry somcl`jss ci`hktki`s ti hjckhj wbjtbjr tbjsj vmoujs mrj sioutki`s ir `it. =) K`jq K`jqum umok okty ty ci` ci`st strm rmk` k`ts ts k) Xbj `i` `jfmt `jfmtkvkt kvktyy rjst rjstrkctk rkctki`s i`s mrj smtks smtksakjh akjh sk sk`cj `cj x >, y 2>, 4> > kk) K`jqu K`jqumokt moktyy ci`s ci`strmk` trmk`ts ts
===
x y 2>
> 2> 2>
4) Cie Ciepoj pojej` ej`tmr tmryy Somcl Somcl`js `jsss ci`h ci`hktk ktki` i`ss
k) x
kk)
O
>, x >
x O
x O
y
y
O
y
>
>, y 2> >
ms tbj pridoje ks emxkekzmtki`.
O y
>
=2> 4(2> ) 4> >
O
O
x
=> >
kkk)
O
>, 4> >
O
>
2> > 2> >
Moo tbj Lub` Xucljr ci`hktki`s mrj smtksakjh. Xbus, tbj idgjctkvj au`ctki` ks emxkekzjh wbj` x >, y 2>, 4> .
Ci`strmk`t ]umokakcmtki`
Vj bmvj stuhkjh tbmt tbj Lub` Xucljr ci`hktki`s mrj `jcjssmry ci`hktki`s ka m`h i`oy ka m pmrtkcuomr prjci`hktki` ks auoakoojh. Xbks prjci`hktki` ks rjajrrjh ti ms ci`strmk`t qumokakcmtki` m`h kt kepisjs siej rjstrkctki` i` tbj ci`strmk`t au`ctki` ia `i`ok`jmr prifrmeek`f si ms ti mvikh siej krrjfuomrktkjs i` tbj diu`hmry ia tbj ajmskdoj rjfki` tbmt wiuoh maajct tbj auoakooej`t ia tbj Lub`-Xucljr ci`hktki` mt tbj iptkemo sioutki`. Xbjrj wkoo dj `it m diu`hmry krrjfuomrkty privkh privkhjh jh tbmt m cjrtmk` ci`strmk`t qumokakcmtki` ks ejt. K` irhjr ti hjscrkdj kt, ojt us smy x ( x= , x 4 , x3 ,...., x ` ) dj diu`hmry pik`ts i` tbj ajmskdoj rjfki` rjf ki` m`h sup suppis pisjj hx (hx= , hx4 , hx 3 ,..., hx` ) k`hkcmtjs m spjckakc hkrjctki` ia eivjej`t arie tbj diu`hmry pik`t. Xbus, wj bmvj m vjctir hx . Dy `iw ojt us put twi rjqukrjej`ts i` vjctir hx . =. hx g > ka x g > k
k
k
k
k
( x ) l k . 4. hf ( x ) f = hx= f 4 hx 4 .... f ` hx ` > air emxkekzmtki` ka f k > air ek`kekzmtki` ka f ( x ) l k . Ka tbj vjctir hx auoakos tbjsj twi ci`hktki`s, kt ks rjajrrjh ti ms tbj tjst vjctir. Vbj` tbjrj ks m
hkaajrj`tkmdoj mrc wbkcb irkfk`mtjs arie x k`couhjh k` tbj ajmskdoj rjfki` ks tm`fj`t ti tbj fkvj` tjst vjctir , kt ks l`iw` ms m qumokayk`f mrc air tbj fkvj` tjst vjctir. Xbj jxkstj`cj ia tbks qumokayk`f mrc air jmcb tjst vjctir hx mt m`y pik`t x i` tbj diu`hmry ia tbj ajmskdoj rjfki` smtksakjs ci`strmk`t qumokakcmtki`. Lub` Xucljr Suaakckj`cy Xbjirje
Hjmr cioojmfuj! Ms wj bmvj hkscussjh k` tbj comsskcmo mpprimcb tbj skf` ia tbj sjci`h irhjr hjrkvmtkvj ia tbj au`ctki` privkhjs tbj suaakckj`t ci`hktki` air emxkeue m`h ek`keue ia m
==4
au`ctki`. Xbjsj ci`hktki`s bmvj k`tkemcy wktb tbj ci`cjpt ia ci`vjxkty m`h ci`cmvkty ia m au`ctki`. K` `i`ok`jmr prifrmeek`f, wj cm` jxprjss tbj suaakckj`t ci`hktki` air tbj iptkeue ia tbj au`ctki` k` tjres ia ci`vjxkty m`h ci`cmvkty ia m au`ctki`. Fkvj` tbj pridoje Emxkekzj a ( x ) Sudgjct ti f k ( x ) l k (k5=, 4, 3… `) m`h x > Xbj Lub` - Xucljr suaakckj`t ci`hktki`s mrj smtksakjh privkhjh tbmt =) tbj idgjctkvj au`ctki` ks hkaajrj`tkmdoj hkaajrj`tkmdoj m`h ci`cmvj k` `i` `jfmtkvj irtbm`t. 4) jmcb ci`strmk` au`ctki` ks hkaajrj`tkmdoj m`h ci`vjx k` tbj `jfmtkvj irtbm`t 3) tbj pik`t x smtksakjs tbj Lub` -X -Xucljr ucljr emxkeue ci`h ci`hktki` ktki` Air ek` ek`kekzm kekzmtki` tki` pr pridoj idoje e yiu cm` jm jmskoy skoy mh mhhh tbj Lub Lub`` - Xucljr ek ek`keue `keue cci`hkt i`hktki` ki` i` ci`hktki` (=) m`h ( 4) mdivj ti hjtjrek`j wbjtbjr tbj Lub` - Xucljr suaakckj`t ci`hktki` ks smtksakjh ir `it. ^riia
Fkvj` tbj pridojes Emxkekzj
a ( x ) k
f ( x ) l k Sudgjct m`h x ti > Xbj Omfrm`fj au`ctki` ks sbiw` ms e
O5 a(x) +
k \l k f ( x)U
k
k =
Ka wj fkvj spjckakc vmouj air tbj Omfrm`fj euotkpokjr k , O wkoo dj m au`ctki` ia x. Mccirhk`f ti ci`hktki` (=) m`h (4) mdivj, mssuej a(x) ms ci`cmvj m`h f k ( x ) ms ci`vjx wbj` wj euotkpoy kt dy (-=) fkvjs us f k ( x) ) tbmt ks ci`cmvj. Xbjrjairj, Xbj O au`ctki`s wkoo dj ci`cmvj au`ctki` (sue ia twi ci`cmvj au`ctki`s) k` x. Xbj Omfrm`fj au`ctki` O ks ci`cmvj ejm`s tbmt `
O
g =
x g
O( x ) O ( x )
( x g x g ) O
Vbj` x rjprjsj`ts siej pmrtkcuomr pik`t k` tbj hiemk` x rjajrs ti tbj pmrtkmo hjrkvmtkvj g O x g
jxmek`jh mt x . @iw ojt‛s cbiisj tbj vmouj ia tbj cbikcj vmrkmdoj x m`h tbj vmouj ia tbj
Omfrm`fj euotkpokjr k` ok`j wktb ci`hktki` (3) mdivj, k.j. wbkcb smtksay tbj Lub` tucljr emxkeue ci`hktki`. Ka wj hjciepisj tjres k` tbj sueemtki` wj fjt `
O
x g =
`
( x g x g )
O
g =
g
`
x g
x g
O
g =
x g
x g
Dy tbj mppokcmtki` ia ciepojej`tmry somcl`jss mt pik`t x g , tbks jxprjsski` ks rjhucjh ti `
O
x g =
x g
g
==3
Arie tbks jxprjsski` wj l`iw tbmt
O x g
> (emrfk`mo
ci`hktki`) m`h x g > ms kt ks m cbikcj
vmrkmd vmr kmdoj. oj. Ms m rjs rjsuot uot tbj rjh rjhucj ucjhh jxp jxprjs rjsski ski`` ks `i` pis pisktk ktkvj. vj. Xbu Xbuss wj cm` ci` ci`cou couhj hj tbm tbmtt O( x ) O ( x ) . Xbks k`hkcmtjs tbmt x ks tbj iptkemo sioutki`. Jci`iekc Mppokcmtki` Jxmepoj 4 4 8. Fkvj` tbj rjvj`uj m`h cist ci`hktki`s ia m akre ms T 34 x x m`h C x 2 x 8 , wbjrj x iutput ks. Suppisj tbj ek`keue priakt ks > =2 .Hjtjrek`j tbj meiu`t ia tbj iut put wbkcb emxkekzjs rjvj`uj wktb tbj fkvj` ek`keue priakt. K` tbks cmsj, tbj rjvj`uj au`ctki` ks ci`cmvj m`h tbj cist au`ctki` ks ci`vjx. Xbj ^ridoje ks 4 Emxkekzj T 34 x x Sudgjct ti x 4 2 x 8 34 x x 4 =2 M`h x >
Z`hjr tbjsj sktumtki`s sktumtki`s tbj L Lub` ub` - Xucl j` ci`hktki`s mrj `jcjssmry m`h suaakckj`t ci`h ci`hktki`s ktki`s ms moo ia tbj mdivj tbrjj ci`hktki`s, k.j., (=), (4), 8(3), mrj mr j smtksakjh. Xbj Omfrm`fj au`ctki` ia tbks pridoje ks O 34 x x 4 ( 44 4 x 4 48 x) (=)
Xbus, O x O
34 4 x 8 x 48 > ( 4)
4
44 4 x 48 x > (3)
Arie jqumtki` (3)
44 4 x 4 48 x >
4 x 4 48 x 44 > ( 8)
Siovk`f (8) wj fjt, x
= ir x == .
3
4
>r
= 4
Biwjvjr, wj eust cbjcl tbj k`jqumokty ci`strmk`ts m`h tbj ciepojej`tmry somcl`jss ci`hktki`s ti hjckhj wbjtbjr tbjsj vmoujs mrj tbj sioutki`s ir `it O O >, >, -----------------------------(:) x x > m`h
x O
Mt R5=
>,
>
m`h
Mt tbks pik`t x > tbks kepo kepokjs kjs tbmt
O x
x O
>, -----------------------------(1)
>, Xbus
O x
3 3> 4> > .Kt
4
hijs `it
smtksay jqumtki` (1). Mt R5==, x >
tbks kepok kepokjs js tbmt
O x
>, Xbus
O x
=> 4> >
jqumtki` (:) m`h (1). Xbks ejm`s, tbj Lub` Xucljr ci`hktki`s mrj auoakoojh mt rjvj`uj ks emxkekzjh wbj` x == .
==8
=
. Kt smtksakjs ditb
4 x == .Xbjrjairj,
Ekxjh ci`strmk`ts
Hjmr cioojmfuj! M` iptkekzmtki` pridoje wktb ekxjh ci`strmk`ts cm` dj rjaireuomtjh jktbjr ms emxkekzmtki` ir ek`kekzmtki` pridoje. Xbks pricjhurj k`cirpirmtjs tbj aiooiwk`f ci`hktki`s. k) Em Emxk xkek ekzk zk`f `f tbj tbj idgj idgjct ctkv kvjj au`c au`ctk tki` i` W ( x) ks jqu jqukvm kvmoj` oj`tt ti tbj pri prido doje je ia Ek` Ek`kek kekzk` zk`ff W ( x ) ir vkcj vjrsmkk) kk) Xbj ci` ci`str strmk` mk`tt f ( x ) c cm` dj prjsj`tjh ms f ( x) c . kkk) Xbj ccii`strmk`t f ( x ) c ks jquk jqukvm vmoj oj`t `t ti tb tbjj hiud hiudoj oj ci ci`s `str trmk mk`t `t f ( x ) c m`h f ( x ) c
kv) Xbj `i` `jfmtkvkty ci`strmk`t x i cm` dj hj`itjh dy m `jw ci`strmk`t f ( x ) x > . Hjmr cioojmfuj! Dy `iw yiu bmvj ciepojtjh tbks sjctki`. Xbjrjairj, try ti hi tbj aiooiwk`f qujstki`s.
Sjoa - Xjst 8.3 Siovj tbj aiooiwk`f qujstki`s dmsjh i` tbj k`airemtki` mdivj. =. Hjscrkdj `i` ok`jmr prifrmeek`f --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------4. Jxpomk` tbj hkaajrj`cj djtwjj` ok`jmr prifrmeek`f m`h `i` ok`js i`j ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 3. Vbmt mrj tbj k`frjhkj`ts ia `i` ok`jmr prifrmeek`f pridojes9 ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------8. Vrktj tbj Lub` - Xucljr ci`hktki` ia tbj pridoje Emxkekzj Z Z ( x= , x 4 , x3 ,...., x ` ) Sudgjct ti p= x= p 4 x 4 p3 x3 ..... p ` x
D
M`h x= , x 4 , x 3 ,..... x ` > Vbj` xk rjprj rjprjsj`t sj`tss fiihs ci`suej ci`suejhh m`h pk rjprjsj`t tbj rjspjctkvj prkcj ia tbjsj fiihs. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------:. Cbjcl wbjtbjr tbj Lub` - Xucljr ci`hktki`s mrj smtksakjh ir `it air tbj pridoje fkvj` k` jxmepoj = (d) mt tbj iptkemo vmoujs ia x m`h y. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Ek`kekzj C 5 x4 + y4 Sudgjct ti x+ y 4 M`h x, y > Vrktj iut tbj Lub` - Xucljr ci`hktki`s m`h usj tbje ti ak`h tbj iptkemo sioutki` dy trkmo m`h
jrrir,
wbmt
mrj
tbj
vmoujs
==:
ia
x
m`h
y9
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------6. Fkvj` tbj hjem`h au`ctki` ia tbj akre ks fkvj` ms ^ =4 x M`h tbj cist au`ctki` ks C
= 4 x wbj` tbj ek`keue priakt ks 4
48 , tbj`
Emxkekzj T a ( x ) Sudgjct ti C ( x) T ( x) 48 M`h x > Ks tbj Lub` - Xucljr ci`hktki` smtksakjh ir `it9 Hjtjrek`j tbj vmouj ia x usk`f trkmo m`h jrrir.
Hjmr cioojmfuj! Bmvj yiu m`swjrjh tbjsj qujstki`s9 Ka `i, pojmsj rjrjmh tbks sjctki` m`h try ti hi tbje. Ka yjs, fi ti tbj `jxt sjctki`.
Cbjcl Okst
Vrktj ∗ k`skhj tbj dix wbkcb cirrjspi`hs ti tbj pridoje tbmt yiu cm` siovj jmskoy. =. Cm` yiu jxpomk` ci`strmk`jh au`ctki`9 ------------------------------------------------------- ----- 4. Hjscrkdj wbmt ci`strmk`jh iptkekzmtki` ks ----------- --------------------------------------------- 3. Hjscrkdj tbj irhjr ci`hktki` air tbj iptkekzmtki` ia ci`strmk`jh au`ctki`. ------------------- 8. Jxpomk` tbj Omfrm`fj euotkpokjr----------------------------------------------------------------euotkpokjr----------------------------------------------------------------- ---- :. Hjscrkdj biw au`ctki`s wktb k`jqumokty ci`strmk`ts mrj siovjh---------------------------------- 1. Hjscrkdj tbj Lub` - Xucljr tbjirje---------------------------------------------------------------- 6. Siovj idgjctkvj au`ctki` sudgjct ti ekxjh ci`stm`ts---------------------------------------------- 2. Hjscrkdj dirhjrjh Bjsskm` hjtjrek`m`t-------------------------------------------------------------- ;. Jxpomk` `i`ok`jmr prifrmeek`f pridoje----------------------------------------------------------- Hjmr cioojmfuj! Ks tbjrj m`y dix k` wbkcb yiu hkh`'t tkcl9 Ka yjs, pojmsj rjrjmh tbks u`kt m`h try ti hi kt. Ka `i, vjry fiih Fi ti tbj` `jxt u`kt Z`kt Sueemry
K` dusk`jss m`h jci`iekcs stuhkjs tbjrj mrj em`y sktumtki`s k` wbkcb ciepojtj arjjhie ia mctki` ks kepisskdoj. Air jxmepoj, m akre cm` emxkekzj iutput sudgjct ti tbj ci`strmk`t ia m fkvj` duhfjt air jxpj`hkturjs i` k`puts, ir kt emy `jjh ti ek`kekzj cist sudgjct ti m cjrtmk` ek`keue iut put djk`f prihucjh. Sucb au`ctki`s wbkcb k`viovj ci`strmk`ts mrj cmoojh ci`strmk`jh au`ctki`s m`h tbj pricjss ia iptkekzmtki` ks rjajrrjh ti ms ci`strmk`jh au`ctki`s m`h tbj pricjss ia iptkekzmtki` ks rjajrrjh ti ms ci`strmk`jh iptkekzmtki`. Xbj pridoje ia iptkekzmtki` ia m` idgjctkvj au`ctki` sudgjct ti cjrtmk` rjstrkctki`s ir ci`strmk`ts ks m usumo pbj`iej`i` k` jci`iekcs. Eistoy, tbj ejtbih ia emxkekzk`f ir ek`kekzk`f m au`ctki` k`couhjs jqumokty ci`strmk`ts. Air k`stm`cj, utkokty emy dj emxkekzjh sudgjct ti m akxjh k`ciej tbmt tbj ci`suejr bms m`h tbj duhfjt ci`strmk`t ks fkvj` k` tbj aire ia jqumtki`. Sucb typj ia iptkekzmtki` ks rjajrrjh ti ms comsskcmo iptkekzmtki`. Dut idgjctkvj au`ctki` sudgjct ti k`jqumokty ci`strmk`ts cm` dj iptkekzjh usk`f tbj ejtb ejtbih ih ia emtbjemtkcmo prifrmeek`f. Ka tbj idgjctkvj au`ctki` ms wjoo ms tbj k`jqumokty ci`strmk`ts ks ok`jmr, wj wkoo usj m ejtbih ia ok`jmr
==1
prifrmeek`f. Biwjvjr, ka tbj idgjctkvj au`ctki` m`h tbj k`jqumokty ci`strmk`ts mrj `i`ok`jmr, wj wkoo mppoy tbj tjcb`kquj ia `i`ok`jmr prifrmeek`f ti iptkekzj tbj au`ctki`. K` `i`-ok`jmr prifrmeek`f, tbjrj ks m skekomr akrst irhjr ci`hktki` wbkcb ks rjajrrjh ti ms Lub` Xucljr ci`hktki`s. Ms wj hkscussjh prjvkiusoy, k` comsskcmo iptkekzmtki` pricjss, tbj akrst irhjr ci`hktki` ks m `jcjssmry ci`hktki`. Biwjvjr, m cjrtmk` ci`hktki` sbiuoh dj auoakoojh air tbj Lub` Xucljr ci`hktki`s ti dj `jcjssmry ci`hktki`s Kepirtm`t pik`ts
Ci`strmk`jh au`ctki` Ci`strmk`jh iptkekzmtki` Ci`strmk`ts Omfrm`fj Euotkpokjr Dirhjrjh Bjsskm` hjtjrek`m`t ^isktkvj hjak`ktj @jfmtkvj hjak`ktj Ek`kekzmtki` Emxkekzmtki`
@i` ^rifrmeek`f Lub`ok`jmr Xucljr tbjirje @i` `jfmtkvkty ci`strmk`t Ci`strmk`t ]umokakcmtki`
M`swjrs air Sjoa Xjst ]ujstki`s
Sjoa Xjst 8.=
=. Xbj ci`strmk`jh emxkeue ia tbj au`ctki` ks 3 4. Xbj au`ctki` ks emxkekzjh mt x . 8
3. y
46 8
Mt x
= 1
8. y 8 Mt x > :. y 6 Mt x > Sjoa Xjst 8.4
8) O 8.86, L : .6;> dkrr 6) x =: & y 1> ==6
y 4 mt
x54.
Sjoa Xjst 8.3
1)
x = & y =
6)
x 8
==2
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