ENTHNB9 N Qrnmtkmnh Kltrjdumtkjl tj Qrjcrneeklc nld Qrjbhge ]jhvklc ]gmjld Gdktkjl ]JHWTKJL ENLWNH
]tjrey Nttnwny Mjhhgcg jo Glcklggrklc Bjstjl Wlkvgrskty
K. Kltrjdumtkjl tj Qrjcrneeklc Wsklc ENTHNB ENTHNB Mfnptgr >9 Kltrjdumtkjl tj ENTHNB Gxgrmksgs >) Mrgntg n vnrknbhg tj stjrg tfg ntjekm wgkcft jo skhkmjl (26.167).
<< skhkmjlNtSt 5 26.167 skhkmjlNtSt 5 26.1671 2) Mrgntg n vnrknbhg eyncg nld eyncg nld stjrg yjur ncg kl kt. ]ubtrnmt jlg orje tfg vnhug jo tfg vnrknbhg. Ndd twj tj tfg vnhug jo tfg vnrknbhg.
<< eyncg 5 =70 << eyncg 5 eyncg - >0 << eyncg 5 eyncg + 20 =) Wsg tfg bukht-kl oulmtkjl lneghglctfenx tj lneghglctfenx tj okld jut tfg enxkeue luebgr jo mfnrnmtgrs tfnt yju mnl fnvg kl nl kdgltkokgr lneg uldgr yjur vgrskjl jo ENTHNB.
<< lneghglctfenx nls 5 ?= 4) Gxphjrg tfg ojrent mjeenld mjeenld kl ejrg dgtnkh. Wsg fghp ojrent tj tj okld jptkjls. Gxpgrkeglt wktf ojrent bnl` tj tj dksphny djhhnr vnhugs.
<< ojrent + << >2.=4 nls 5 + << ->2= nls 5 << ojrent bnl` << ==.4 nls 5 ==.41 << 72.4=7 nls 5
72.44
7) Okld n ojrent jptkjl jptkjl tfnt wjuhd rgsuht kl tfg ojhhjwklc jutput ojrent9 << 7/>? + 2/3 nls 5 ?3/>>2
<< ojrent rnt << 7/>? + 2/3 nls 5 ?3/>>2 ?) Tfkl` nbjut wfnt tfg rgsuhts wjuhd bg ojr tfg ojhhjwklc gxprgsskjls, nld tfgl typg tfge kl tj vgrkoy yjur nlswgrs. 27 / 4 * 4 = + 4 \ 2 4 X >2 + 4 = \ 2 (7 ” 2) * =
<< 27 / 4 * 4 nls 5 27 << = + 4 \ 2 nls 5 >; << 4 X >2 + 4 nls 5 3 << = \ 2 nls 5 ; << (7 - 2) * = nls 5 ; 3) Tfg mjebklgd rgskstnlmg RT jo tfrgg rgskstjrs R >, R2, nld R= kl pnrnhhgh ks ckvgl by > RT 5 > > > R>
R2
R=
Mrgntg vnrknbhgs ojr tfg tfrgg rgskstjrs nld stjrg vnhugs kl gnmf, nld tfgl mnhmuhntg tfg mjebklgd rgskstnlmg.
<< << << << rt
r> 5 =0 r2 5 2.20 r= 5 >.70 rt 5 >/(>/r> + >/r2 + >/r=) 5 1.?637
Ns tfg wjrhd bgmjegs ejrg ‘ohnt―, kt ks klmrgnsklchy kepjrtnlt ojr glcklggrs nld smkgltksts tj bg nbhg tj wjr` wktf mjhhgncugs mjhhgncugs kl jtfgr pnrts jo tfg wjrhd. wjrhd. Mjrrgmt mjlvgrskjl jo dntn dntn orje jlg systge jo ulkts tj nljtfgr nljtfgr (ojr gxnephg, gxnephg, orje tfg egtrkm systge systge tj tfg Negrkmnl Negrkmnl systge jr vkmg vgrsn) ks mrktkmnhhy kepjrtnlt. 6) Mrgntg n vnrknbhg pjulds vnrknbhg pjulds tj tj stjrg n wgkcft kl pjulds. Mjlvgrt tfks tj `khjcrnes `khjcrnes nld nsskcl tfg rgsuht tj n vnrknbhg `khjs. `khjs. Tfg mjlvgrskjl onmtjr onmtjr ks > `khjcrne 5 2.2 pjulds.
<< pjulds 5 =10 << `khjs 5 pjulds / 2.2 `khjs 5 >=.?=?4 ;) Mrgntg n vnrknbhg otgep vnrknbhg otgep tj tj stjrg n tgepgrnturg kl dgcrggs Onfrglfgkt (O). Mjlvgrt tfks tj dgcrggs Mghskus (M) nld stjrg tfg rgsuht kl n vnrknbhg mtgep. mtgep. Tfg mjlvgrskjl onmtjr ks M 5 (O ” =2) * 7/;.
<< otgep 5 370 << mtgep 5 (otgep - =2) * 7/; mtgep 5 2=.666; >1) Okld nljtfgr qunltkty tj mjlvgrt orje jlg systge jo ulkts tj nljtfgr.
<< << << `e
`eQgrEkhg 5 >.?1;=0 ekhgs 5 ?0 `e 5 ekhgs * `eQgrEkhg 5 ;.?776
>>) Tfg oulmtkjl skl mnhmuhntgs skl mnhmuhntgs nld rgturls tfg sklg jo nl nlchg kl rndknls. rndknls. Wsg fghp ghoul tj okld tfg lneg jo tfg oulmtkjl tfnt rgturls tfg sklg jo nl nlchg kl dgcrggs. _grkoy tfnt mnhhklc tfks oulmtkjl nld pnssklc ;1 dgcrggs tj kt rgsuhts kl >.
<< skld(;1) nls 5 >
>2) N vgmtjr mnl bg rgprgsgltgd by kts rgmtnlcuhnr mjjrdklntgs x nld y jr by kts pjhnr mjjrdklntgs r nld . Tfg rghntkjlsfkp bgtwggl tfge ks ckvgl by tfg gquntkjls9 x 5 r * mjs( ) y 5 r * skl( ) Nsskcl vnhugs ojr tfg pjhnr mjjrdklntgs tj vnrknbhgs r nld nld tfgtn. tfgtn. Tfgl, usklc usklc tfgsg vnhugs, nsskcl tfg mjrrgspjldklc rgmtnlcuhnr mjjrdklntgs tj vnrknbhgs x vnrknbhgs x nld y nld y .
<< r 5 70 << tfgtn 5 1.70 << x 5 r * mjs(tfgtn) x 5 4.=63; << y 5 r * skl(tfgtn) y 5 2.=;3> >=) Skld jotgl en`gs tfg nkr oggh gvgl mjhdgr tfnl kt ks. Tfg Skld Mfkhh Onmtjr (SMO) egnsurgs fjw mjhd kt ogghs wktf n ckvgl nkr tgepgrnturg T (kl dgcrggs Onfrglfgkt) nld wkld spggd (_, kl ekhgs pgr fjur). Jlg ojreuhn ojr tfg tfg SMO ks9 SMO 5 =7.3 + 1.? T ” =7.3 (_ 1.>?) + 1.4= T (_ 1.>?) Mrgntg vnrknbhgs ojr tfg tgepgrnturg T nld wkld spggd _, nld tfgl usklc tfks ojreuhn mnhmuhntg tfg SMO.
<< t 5 210 << v 5 >>0 << wmo 5 =7.3 + 1.?*t - =7.3*v\1.>?+1.4=*t*v\1.>? wmo 5 3.;2?3 >4) Wsg fghp ghoul jr gxpgrkeglt tj nlswgr tfg ojhhjwklc qugstkjls9 Ks okx(=.7) tfg okx(=.7) tfg sneg ns ohjjr(=.7): ohjjr(=.7):
<< okx(=.7) nls 5 = << ohjjr(=.7) nls 5 =
Ks okx(=.4) tfg okx(=.4) tfg sneg ns okx(-=.4): okx(-=.4):
<< okx(=.4) nls 5 = << okx(-=.4) nls 5
-=
Ks okx(=.2) tfg okx(=.2) tfg sneg ns ohjjr(=.2): ohjjr(=.2):
<< okx(=.2) nls 5 = << ohjjr(=.2) nls 5 =
Ks okx(-=.2) tfg okx(-=.2) tfg sneg ns ohjjr(-=.2): ohjjr(-=.2):
<< okx(-=.2) nls 5 -= << ohjjr(-=.2) nls 5 -4
Ks okx(-=.2) tfg okx(-=.2) tfg sneg ns mgkh(-=.2): mgkh(-=.2):
<< okx(-=.2) nls 5 -= << mgkh(-=.2) nls 5 -= >7) Okld ENTHNB gxprgsskjls ojr tfg ojhhjwklc >;
sqrt(>;) >. 2
=
=\>.2 tnl() tnl(pk)
>?) Wsg kltekl nld kltekl nld kltenx tj kltenx tj dgtgreklg tfg rnlcg jo vnhugs tfnt mnl bg stjrgd kl tfg typgs uklt=2 nld uklt=2 nld uklt?4. uklt?4.
<< kltekl('uklt=2') nls 5 1 << kltenx('uklt=2') nls 5 42;4;?32;7 << kltekl('uklt?4') nls 5 1 << kltenx('uklt?4') nls 5 >644?34413=31;77>?>7 >3) Nrg tfgrg gqukvnhglts tj kltekl nld kltekl nld kltenx ojr kltenx ojr rgnh luebgr typgs: Wsg fghp tj fghp tj okld jut.
<< rgnhekl nls 5 2.227>g-=16 << rgnhekl('djubhg') nls 5 2.227>g-=16 << rgnhekl('sklchg') nls 5 >.>377g-=6 << rgnhenx nls 5 >.3;33g+=16 >6) ]tjrg n luebgr wktf n dgmkenh phnmg kl n djubhg vnrknbhg djubhg vnrknbhg (tfg dgonuht). Mjlvgrt tfg vnrknbhg tj tfg typg klt=2 nld klt=2 nld stjrg tfg rgsuht kl n lgw vnrknbhg.
<< lue 5 >=.47 lue 5 >=.4711 << kltlue 5 klt=2(lue) kltlue 5 >= >;) Cglgrntg n rnldje rgnh luebgr kl tfg rnlcg orje 1 tj >
rnld
rgnh luebgr kl tfg rnlcg orje 1 tj 21
rnld * 21
rgnh luebgr kl tfg rnlcg orje 21 tj 71
rnld*(71-21)+21
kltgcgr kl tfg rnlcg orje 1 tj >1
rjuld(rnld * >1)
kltgcgr kl tfg rnlcg orje 1 tj >>
rjuld(rnld * >>)
kltgcgr kl tfg rnlcg orje 71 tj >11
rjuld(rnld*(>11-71)+71) 21) Cgt kltj n lgw Mjeenld Skldjw, nld typg rnld tj rnld tj cgt n rnldje rgnh luebgr. En`g n ljtg jo tfg luebgr. Tfgl, gxkt ENTHNB nld rgpgnt tfks, ncnkl en`klc n ljtg jo tfg rnldje luebgr0 kt sfjuhd bg tfg sneg ns bgojrg. Oklnhhy, gxkt ENTHNB nld ncnkl cgt cgt kltj n lgw Mjeenld Skldjw. Tfks tkeg, mfnlcg tfg sggd bgojrg cglgrntklc n rnldje luebgr0 luebgr0 kt sfjuhd bg dkoogrglt.
<< rnld nls 5 1.6>43 << rlc('sfuoohg') << rnld nls 5 1.4616 2>) Kl tfg N]MKK mfnrnmtgr glmjdklc, tfg hgttgrs jo tfg nhpfnbgt nrg kl jrdgr9 ‖n‛ mjegs bgojrg ‖b‛ nld nhsj ‖N‛ mjegs bgojrg ‖B‛. Fjwgvgr, wfkmf mjegs okrst - hjwgr jr uppgrmnsg hgttgrs:
<< klt=2('n') nls 5 ;3 << klt=2('N') nls 5 ?7
Tfg uppgr mnsg hgttgrs 22) ]fkot tfg strklc ‖xyz‛ up kl tfg mfnrnmtgr glmjdklc by 2 mfnrnmtgrs.
<< mfnr('xyz' + 2) nls 5 z{| 2=) Wsklc tfg mjhjl jpgrntjr, mrgntg tfg ojhhjwklc rjw vgmtjrs = 4 7 ?
>.1111 7
>.7111 4
<< =9? nls 5 =
=
<< 79->92 nls 5 7
2.7111
=.1111
2
4
<< >9.79= nls 5 >.1111
2.1111
7
>.7111
4
=
?
2.1111
2.7111
=.1111
2
24) Wsklc tfg hklspnmg oulmtkjl, hklspnmg oulmtkjl, mrgntg tfg ojhhjwklc rjw vgmtjrs 4 ? 6
-=
-?
-;
;
3
7
->2
->7
<< hklspnmg(4,6,=) nls 5 4 ? 6 << hklspnmg(-=,->7,7) nls 5 -= -? -; ->2
->7
<< hklspnmg(;,7,=) nls 5 ; 3 7 27) Mrgntg tfg ojhhjwklc rjw vgmtjrs twkmg, usklc hklspnmg nld hklspnmg nld usklc tfg mjhjl jpgrntjr9
>
2
=
4
7
<< >9>1 nls 5 > 2 = << hklspnmg(>,>1,>1) nls 5 > 2 =
2
3
?
3
6
;
>1
4
7
?
3
6
;
>1
4
7
?
3
6
;
>1
>2
<< 2979>2 nls 5 2 3 >2 << hklspnmg(2,>2,=) nls 5 2 3 >2 2?) Mrgntg n vnrknbhg eygld wfkmf wfkmf stjrgs n rnldje kltgcgr kl tfg rnlcg orje 6 tj >2. Wsklc tfg mjhjl jpgrntjr, mrgntg n vgmtjr tfnt ktgrntgs orje > tj eygld kl kl stgps jo =.
<< eygld 5 rnldk(P6,>2U) eygld 5 6 << vgm 5 >9=9eygld vgm 5 > 4 3 23) Wsklc tfg mjhjl jpgrntjr nld nld tfg trnlspjsg jpgrntjr, mrgntg n mjhuel mjhuel vgmtjr tfnt fns tfg vnhugs -> tj > kl stgps jo 1.2.
<< rjw_gm 5 ->9 1.29 >0 << rjw_gm' nls 5 ->.1111 -1.6111 -1.?111 -1.4111 -1.2111 1 1.2111 1.4111 1.?111 1.6111 >.1111
26) Srktg nl gxprgsskjl tfnt rgogrs tj jlhy tfg jdd-luebgrgd ghgeglts kl n vgmtjr, rgcnrdhgss jo tfg hglctf jo tfg vgmtjr. vgmtjr. Tgst yjur gxprgsskjl jl jl vgmtjrs tfnt fnvg bjtf nl jdd nld gvgl luebgr jo ghgeglts.
<< vgm 5 >960 << vgm(>929gld) nls 5 > =
7
3
<< vgm 5 >9;0 << vgm(>929gld) nls 5 > =
7
3
;
2;) Mrgntg n vgmtjr vnrknbhg vgm0 vgm0 kt mnl fnvg nly hglctf. hglctf. Tfgl, wrktg nsskcleglt nsskcleglt stntgeglts tfnt wjuhd stjrg tfg okrst fnho jo tfg vgmtjr kl jlg vnrknbhg nld tfg sgmjld fnho kl nljtfgr. En`g surg tfnt yjur nsskcleglt stntgeglts nrg cglgrnh, nld wjr` wfgtfgr vgm fns vgm fns nl gvgl jr jdd luebgr jo ghgeglts (Fklt9 usg n rjuldklc oulmtkjl sumf ns okx). okx).
<< vgm 5 >9;0 << ofnho 5 vgm(>9okx(hglctf(vgm)/2)) ofnho 5 > 2 = 4 << sfnho 5 vgm(okx(hglctf(vgm)/2)+>9gld) sfnho 5 7 ? 3 6 ; =1) Cglgrntg n 2 n 2 x = entrkx = entrkx jo rnldje rgnh luebgrs, gnmf kl tfg rnlcg orje 1 tj >
<< rnld(2,=) nls 5 1.12>7 1.3216
1.3=?; 1.4>?6
rgnh luebgrs, gnmf kl tfg rnlcg orje 1 tj >1
<< rnld(2,=)*>1 nls 5 6.16?= 2.247? 2.;41; 4.122>
1.3>27 1.>6?7
6.=1?3 7.1?33
kltgcgrs, gnmf kl tfg rnlcg orje 7 tj 21
<< rnldk(P7, 21U,2,=) nls 5
>6 >>
>3 >>
7 3
=>) Mrgntg n vnrknbhg rjws tfnt rjws tfnt ks n rnldje kltgcgr kltgcgr kl tfg rnlcg rnlcg orje > tj 7. Mrgntg n vnrknbhg mjhs tfnt mjhs tfnt ks n rnldje kltgcgr kl tfg rnlcg orje > tj 7. Mrgntg n entrkx jo nhh zgrjs wktf tfg dkeglskjls ckvgl by tfg vnhugs jo rjws nld rjws nld mjhs. mjhs.
<< rjws 5 rnldk(P>,7U) rjws 5 = << mjhs 5 rnldk(P>,7U) mjhs 5 2 << zgrjs(rjws,mjhs) nls 5 1 1 1 1 1 1 =2) Okld nl gookmkglt wny wny tj cglgrntg tfg ojhhjwklc entrkx9 ent 5 3 6 ; >1 >2 >1 6 ? Tfgl, ckvg gxprgsskjls tfnt wkhh, ojr tfg entrkx ent , rgogr tj tfg ghgeglt kl tfg okrst rjw, tfkrd mjhuel m jhuel rgogr tj tfg gltkrg sgmjld rjw rgogr tj tfg okrst twj mjhuels
<< ent 5 P39>10 >29 -29 ?U ent 5 3 6 ; >1 >2 >1 6 ? << ent(>,=) nls 5 ; << ent(2,9) nls 5 >2 >1 6 ? << ent(9,>92) nls 5 3 6 >2 >1 ==) Mrgntg n 2 x = entrkx vnrknbhg eyent . Qnss tfks entrkx vnrknbhg tj gnmf jo tfg ojhhjwklc ojhhjwklc oulmtkjls nld en`g surg yju uldgrstnld tfg rgsuht9 ohkphr, ohkphr, ohkpud, ohkpud, nld rjt;1. rjt;1. Kl fjw enly dkoogrglt wnys mnl yju rgsfnpg kt: rgsfnpg kt:
<< ent 5 rnldk(P>,21U, 2,=) ent 5 >? 7 6 >7 >6 > << ohkphr(ent) nls 5 6 7 >? > >6 >7 << ohkpud(ent) nls 5 >7 >6 > >? 7 6 << rjt;1(ent) nls 5 6 > 7 >6 >? >7 << rjt;1(rjt;1(ent)) nls 5 > >6 >7 6 7 >? << rgsfnpg(ent,=,2) nls 5 >? >6 >7 6 7 > << rgsfnpg(ent,>,?) nls 5 >? >7 7 >6 << rgsfnpg(ent,?,>) nls 5 >? >7 7 >6 6 >
6
>
=4) Mrgntg n 4 x 2 entrkx 2 entrkx jo nhh zgrjs zgrjs nld stjrg kt kl n vnrknbhg. Tfgl, rgphnmg tfg sgmjld rjw kl tfg entrkx wktf n vgmtjr mjlskstklc jo n = nld n ?.
<< eyent 5 zgrjs(4,2) eyent 5 1 1 1 1 1 1
1 1 << eyent(2,9) 5 P= ?U eyent 5 1 1 = ? 1 1 1 1 =7) Mrgntg n vgmtjr x wfkmf mjlsksts jo 21 gqunhhy spnmgd pjklts kl tfg rnlcg orje ” tj +. Mrgntg n y vgmtjr wfkmf ks skl(x). skl(x).
<< x 5 hklspnmg(-pk,pk,21)0 << y 5 skl(x)0 =?) Mrgntg n = x 7 entrkx entrkx jo rnldje rnldje kltgcgrs, gnmf kl tfg rnlcg rnlcg orje -7 tj 7. Cgt tfg skcl jo skcl jo gvgry ghgeglt.
<< ent 5 rnldk(P-7,7U, =,7) ent 5 7 4 > -> -7 4 4 -> -= 1 7 -2 > 1 4 << skcl(ent) nls 5 > > > -> -> > > -> -> 1 > -> > 1 > =3) Mrgntg n 4 x ? entrkx ? entrkx jo rnldje kltgcgrs, gnmf kl tfg rnlcg orje -7 tj 70 stjrg kt kl n vnrknbhg. Mrgntg nljtfgr entrkx tfnt tfnt stjrgs ojr gnmf ghgeglt tfg nbsjhutg vnhug jo tfg mjrrgspjldklc ghgeglt kl tfg jrkcklnh entrkx.
<< ent 5 rnldk(P-7,7U, 4,?) ent 5 -= -2 -2 = 4 > = -> = 1 -7 -= -> -2 -2 -> > 7 = -2 << pjsent 5 nbs(ent) pjsent 5 = 2 2 = 4 > = > = 1 7 = > 2 2 > > 7 = 2
-> -2 -7 4
> 2 7 4
=6) Mrgntg n = x 7 entrkx entrkx jo rnldje rnldje rgnh luebgrs. Dghgtg tfg tfkrd rjw.
<< ent 5 rnld(=,7) ent 5 1.722? 1.;3;3 1.661> 1.23>4 1.>3=1 1.272=
1.6373 1.3=3= 1.>=?7
1.1>>6 1.6;=; 1.>;;>
1.2;63 1.??>4 1.2644
<< ent(=,9) 5 PU ent 5 1.722? 1.;3;3 1.661> 1.23>4
1.6373 1.3=3=
1.1>>6 1.6;=;
1.2;63 1.??>4
=;) Mrgntg n vgmtjr vnrknbhg vnrknbhg vgm. vgm. Okld ns enly gxprgsskjls gxprgsskjls ns yju mnl tfnt wjuhd wjuhd rgogr tj tfg hnst ghgeglt kl tfg vgmtjr, wktfjut nssueklc tfnt yju `ljw fjw enly ghgeglts kt fns (k.g., en`g yjur gxprgsskjls cglgrnh).
<< vgm 5 >929; vgm 5 > = 7 << vgm(gld) nls 5 ; << vgm(luegh(vgm)) nls 5 ; << vgm(hglctf(vgm)) nls 5 ; << v 5 ohkphr(vgm) v 5 ; 3 7 << v(>) nls 5 ;
3
;
=
>
41) Mrgntg n entrkx vnrknbhg ent . Okld ns enly gxprgsskjls gxprgsskjls ns yju mnl tfnt wjuhd wjuhd rgogr tj tfg hnst ghgeglt kl tfg entrkx, wktfjut nssueklc tfnt yju `ljw fjw enly ghgeglts jr rjws jr mjhuels kt fns (k.g., en`g yjur gxprgsskjls cglgrnh).
<< ent 5 P>29>70 ?9->9=U ent 5 >2 >= >4 >7 ? 7 4 = << ent(gld,gld) nls 5 = << ent(gld) nls 5
= << Pr mU 5 skzg(ent)0 << ent(r,m) nls 5 = 4>) Mrgntg n tfrgg-dkeglskjlnh entrkx nld cgt kts skzg. skzg.
<< ent=d 5 jlgs(=,7,2) ent=d(9,9,>) 5 > > > > > > > > > ent=d(9,9,2) 5 > > > > > > > > > << skzg(ent=d) nls 5 = 7 2
> > >
> > >
> > >
> > >
42) Tfg bukht-kl oulmtkjl mhjm` rgturls rgturls n vgmtjr tfnt mjltnkls ? ghgeglts9 tfg okrst tfrgg nrg tfg murrglt dntg (ygnr, ejltf, dny) nld tfg hnst tfrgg rgprgsglt tfg murrglt tkeg kl fjurs, eklutgs, nld sgmjlds. Tfg sgmjlds ks ks n rgnh luebgr, luebgr, but nhh jtfgrs jtfgrs nrg kltgcgrs. ]tjrg tfg rgsuht orje mhjm` kl n vnrknbhg mnhhgd eym. eym. Tfgl, stjrg tfg okrst tfrgg ghgeglts orje tfks vnrknbhg kl n vnrknbhg tjdny nld nld tfg hnst tfrgg ghgeglts kl n vnrknbhg ljw . Wsg tfg okx oulmtkjl jl tfg vgmtjr vnrknbhg ljw tj tj cgt iust tfg kltgcgr pnrt jo tfg murrglt tkeg.
<< eym 5 mhjm` eym 5 >.1g+1= * 2.1>>1 1.1131 << tjdny 5 eym(>9=) tjdny 5 21>> << ljw 5 eym(49gld) ljw 5 >=.1111 72.1111 << okx(ljw) nls 5 >= 72 7=
1.1>11
3
1.1>=1
>1
7=.?6?1
Mfnptgr 29 Kltrjdumtkjl tj ENTHNB Qrjcrneeklc
Gxgrmksgs
1.1721
1.17=3
>) Srktg n skephg smrkpt tfnt wkhh mnhmuhntg tfg vjhueg jo n fjhhjw spfgrg, 4 = (r j r k= ) = wfgrg rk ks tfg kllgr rndkus nld r j ks tfg jutgr rndkus. Nsskcl n vnhug tj n vnrknbhg ojr tfg kllgr rndkus, nld nhsj nsskcl nsskcl n vnhug tj nljtfgr vnrknbhg vnrknbhg ojr tfg jutgr rndkus. Tfgl, usklc tfgsg vnrknbhgs, nsskcl tfg vjhueg vjhueg tj n tfkrd vnrknbhg. Klmhudg mjeeglts kl tfg smrkpt. Mf2Gx>.e % Tfks smrkpt mnhmuhntgs tfg vjhueg jo n fjhhjw spfgrg
% Nsskcl vnhugs ojr tfg kllgr nld jutgr rndkk rk 5 7.> rj 5 ?.6 % Mnhmuhntg tfg vjhueg vjh 5 (4*pk)/=*(rj\=-rk\=)
2) Tfg ntjekm wgkcft ks tfg wgkcft jo n ejhg jo jo ntjes jo n mfgekmnh ghgeglt. Ojr gxnephg, tfg ntjekm wgkcft jo jxycgl ks >7.;;;4 nld tfg ntjekm wgkcft jo fydrjcgl ks >.113;. Srktg n smrkpt tfnt wkhh mnhmuhntg tfg ejhgmuhnr wgkcft jo fydrjcgl pgrjxkdg, wfkmf mjlsksts jo twj ntjes jo fydrjcgl nld twj ntjes jo jxycgl. Klmhudg mjeeglts kl tfg smrkpt. smrkpt. Wsg fghp tj fghp tj vkgw tfg mjeeglt kl yjur smrkpt. Mf2Gx2.e % Mnhmuhntgs tfg ejhgmuhnr wgkcft jo fydrjcgl pgrjxkdg
% Klktknhkzg tfg ntjekm wgkcfts ojr jxycgl nld fydrjcgl ntStJxycgl 5 >7.;;;40 ntStFydrjcgl 5 >.113;0 % Fydrjcgl pgrjxkdg ks 2 ntjes jo fydrjcgl nld 2 jo jxycgl ejhStFydrjcglQgrjxkdg 5 2*ntStFydrjcgl + 2 * ntStJxycgl << fghp mf2gx2 Mnhmuhntgs tfg ejhgmuhnr wgkcft jo fydrjcgl pgrjxkdg =) Srktg nl klput stntgeglt stntgeglt tfnt wkhh prjept tfg usgr ojr tfg lneg jo n mfgekmnh ghgeglt ns n strklc. Tfgl, okld tfg hglctf hglctf jo tfg strklc.
<< ghgelneg 5 klput('Gltgr n mfgekmnh ghgeglt9 ', 's')0 Gltgr n mfgekmnh ghgeglt9 fydrjcgl << hglctf(ghgelneg) nls 5 6
4) Srktg nl klput stntgeglt stntgeglt tfnt wkhh prjept tfg usgr ojr n rgnh luebgr, nld stjrg kt kl n vnrknbhg. Tfgl, usg tfg oprklto oulmtkjl oulmtkjl tj prklt tfg vnhug jo tfks vnrknbhg usklc 2 dgmkenh phnmgs.
<< rgnhlue 5 klput('Gltgr n rgnh luebgr9 ')0 Gltgr n rgnh luebgr9 47.36; << oprklto('Tfg luebgr ks %.2oXl', rgnhlue) Tfg luebgr ks 47.3; 7) Tfg klput oulmtkjl oulmtkjl mnl bg usgd tj gltgr n vgmtjr, sumf ns9
<< vgm 5 klput('Gltgr n vgmtjr9 ') Gltgr n vgmtjr9 493 vgm 5 4 7 ? 3 Gxpgrkeglt wktf tfks, nld okld jut fjw tfg usgr mnl mn l gltgr n entrkx. << ent 5 klput('Gltgr n entrkx9 ') Gltgr n entrkx9 P49?0 ;9>>U ent 5 4 ;
7 >1
? >>
<< ent 5 klput('Gltgr n entrkx9 ') Gltgr n entrkx9 zgrjs(2) ent 5 1 1
1 1
?) Gxpgrkeglt, kl tfg Mjeenld Skldjw, wktf usklc tfg oprklto oulmtkjl oulmtkjl ojr rgnh luebgrs. En`g n ljtg jo wfnt fnppgls ojr gnmf. Wsg oprklto tj tj prklt tfg rgnh luebgr >2=47.?36;.
rgnhlue 5 >2=47.?36;0
wktfjut spgmkoyklc nly okghd wkdtf
<< oprklto('Tfg luebgr ks %oXl', rgnhlue) Tfg luebgr ks >2=47.?36;11
kl n okghd wkdtf jo >1 wktf 4 dgmkenh phnmgs
<< oprklto('Tfg luebgr ks %>1.4oXl', rgnhlue) Tfg luebgr ks >2=47.?36;
kl n okghd wkdtf jo >1 wktf 2 dgmkenh phnmgs
<< oprklto('Tfg luebgr ks %>1.2oXl', rgnhlue) Tfg luebgr ks >2=47.?6
kl n okghd wkdtf jo ? wktf 4 dgmkenh phnmgs
<< oprklto('Tfg luebgr ks %?.4oXl', rgnhlue) Tfg luebgr ks >2=47.?36;
kl n okghd wkdtf jo 2 wktf 4 dgmkenh phnmgs
<< oprklto('Tfg luebgr ks %2.4oXl', rgnhlue) Tfg luebgr ks >2=47.?36; 3) Gxpgrkeglt, kl tfg Mjeenld Skldjw, wktf usklc tfg oprklto oulmtkjl oulmtkjl ojr kltgcgrs. En`g n ljtg jo wfnt fnppgls ojr gnmf. Wsg oprklto tj tj prklt tfg kltgcgr >2=47.
kltlue 5 >2=470
wktfjut spgmkoyklc nly okghd wkdtf
<< oprklto('Tfg luebgr ks %dXl', kltlue) Tfg luebgr ks >2=47
kl n okghd wkdtf jo 7
<< oprklto('Tfg luebgr ks %7dXl', kltlue) Tfg luebgr ks >2=47
kl n okghd wkdtf jo 6
<< oprklto('Tfg luebgr ks %6dXl', kltlue) Tfg luebgr ks >2=47
kl n okghd wkdtf jo =
<< oprklto('Tfg luebgr ks %=dXl', kltlue) Tfg luebgr ks >2=47 6) Mrgntg tfg ojhhjwklc vnrknbhgs x 5 >2.=40 y 5 4.7?0 Tfgl, okhh kl tfg oprklto stntgeglts stntgeglts usklc tfgsg vnrknbhgs tfnt wkhh nmmjephksf tfg ojhhjwklc9 << oprklto('x ks %6.=oXl', x)
x ks >2.=41 << oprklto('x ks %.oXl', x) x ks >2 << oprklto('y ks %.>oXl', y) y ks 4.? << oprklto('y ks %-6.>o!Xl', y) y ks 4.? ! ;) Srktg n smrkpt tj prjept tfg usgr ojr tfg hglctf nld wkdtf jo n rgmtnlchg, nld prklt kts nrgn wktf 2 dgmkenh phnmgs. Qut mjeeglts kl tfg tfg smrkpt. Mf2Gx;.e % Mnhmuhntg tfg nrgn jo n rgmtnlchg
% Qrjept tfg usgr ojr tfg hglctf nld wkdtf hglctf 5 klput('Gltgr tfg hglctf jo tfg rgmtnlchg9 ')0 wkdtf 5 klput('Gltgr tfg wkdtf jo tfg rgmtnlchg9 ')0 % Mnhmuhntg nld prklt tfg nrgn rgmtZnrgn 5 hglctf * wkdtf0 oprklto('Tfg nrgn jo tfg rgmtnlchg ks %.2oXl', rgmtZnrgn) >1) Srktg n smrkpt mnhhgd gmfjlneg tfnt gmfjlneg tfnt wkhh prjept tfg usgr ojr fks jr fgr lneg, nld tfgl gmfj prklt tfg lneg kl n sgltglmg kl tfg ojhhjwklc ojrent (usg %s tj prklt kt)9 << gmfjlneg Sfnt ks yjur lneg: ]usnl Sjw, yjur lneg ks ]usnl!
gmfjlneg.e % Qrjept tfg usgr nld gmfj prklt lneg klneg 5 klput('Sfnt ks yjur lneg: ','s')0 oprklto('Sjw, yjur lneg ks %s!Xl',klneg) >>) Srktg n smrkpt mnhhgd gmfjstrklc tfnt gmfjstrklc tfnt wkhh prjept tfg usgr ojr n strklc, nld wkhh gmfj prklt tfg strklc kl qujtgs9 << gmfjstrklc Gltgr yjur strklc9 fk tfgrg [jur strklc wns9 'fk tfgrg'
gmfjstrklc.e % Qrjept tfg usgr nld prklt n strklc kl qujtgs str 5 klput('Gltgr yjur strklc9 ', 's')0 oprklto('[jur strklc wns9 ''%s''Xl',str)
>2) Kl tfg egtrkm systge, ohukd ohjw ks egnsurgd kl mubkm egtgrs pgr sgmjld (e=/s). N mubkm ojjt pgr sgmjld (ot =/s) ks gqukvnhglt tj 1.126 e =/s. Srktg n smrkpt tkthgd ohjwrntg tkthgd ohjwrntg tfnt tfnt wkhh prjept tfg usgr ojr ohjw kl mubkm egtgrs pgr sgmjld nld wkhh prklt tfg gqukvnhglt ohjw rntg kl mubkm oggt pgr sgmjld. Fgrg ks nl gxnephg jo rullklc rullklc tfg smrkpt. smrkpt. [jur smrkpt eust prjdumg jutput kl gxnmthy tfg sneg ojrent sneg ojrent ns ns tfks9
<< ohjwrntg Gltgr tfg ohjw kl e\=/s9 >7.2 N ohjw rntg jo >7.211 egtgrs pgr sgm ks gqukvnhglt tj 742.673 oggt pgr sgm ohjwrntg.e % Mjlvgrts n ohjw rntg orje mubkm egtgrs pgr sgmjld % tj mubkm oggt pgr sgmjld
mubEpgr]gm 5 klput('Gltgr tfg ohjw kl e\=/s 9')0 mubOpgr]gm 5 mubEpgr]gm / .1260 oprklto('N ohjw rntg jo %.=o egtgrs pgr sgmXl', ... mubEpgr]gm) oprklto('ks gqukvnhglt tj %.=o oggt pgr sgmXl', ... mubOpgr]gm) >=) Jl nvgrncg, pgjphg kl n rgckjl spgld 6% tj >1% jo tfgkr klmjeg klmjeg jl ojjd. Srktg n smrkpt tfnt wkhh prjept tfg usgr ojr nl nllunh klmjeg. Kt wkhh tfgl prklt tfg rnlcg tfnt wjuhd typkmnhhy bg spglt jl ojjd nllunhhy. Nhsj, prklt n ejltfhy rnlcg. Mf2Gx>=.e % Mnhmuhntgs nld prklts tfg hk`ghy $ nejult spglt jl ojjd % bnsgd jl nllunh klmjeg
klmjeg 5 klput('Gltgr yjur nllunh klmjeg9 ')0 oprklto('[ju nrg hk`ghy tj spgld bgtwggl $%.2oXl',.16*klmjeg) oprklto('nld $%.2o nllunhhy jl ojjd.XlXl', .>*klmjeg) % prklt tfg ejltfhy rnlcg oprklto('[ju nrg hk`ghy tj spgld bgtwggl $%.2oXl',.16*klmjeg/>2) oprklto('nld $%.2o ejltfhy jl ojjd.Xl', .>*klmjeg/>2)
>4) Sklc hjndklc, wfkmf ks tfg nkrphnlg wgkcft dkvkdgd by tfg wklc nrg n, ks nl kepjrtnlt dgskcl onmtjr kl ngrjlnutkmnh glcklggrklc. glcklggrklc. Srktg n smrkpt tfnt wkhh prjept tfg usgr ojr tfg tfg wgkcft jo tfg nkrmrnot kl `khjcrnes, nld tfg wklc nrgn kl egtgrs squnrgd, nld wkhh mnhmuhntg nld prklt tfg wklc hjndklc jo tfg nkrmrnot kl `khjcrnes pgr squnrg egtgr. Mf2Gx>4.e % Mnhmuhntgs tfg wklc hjndklc ojr nl nkrphnlg
% Qrjept tfg usgr ojr tfg wgkcft nld wklc nrgn jo tfg phnlg phnlgZwgkcft 5 klput('Gltgr tfg wgkcft jo tfg nkrphnlg9 ')0 wklcZnrgn 5 klput('Gltgr tfg wklc nrgn9 ')0 % Mnhmuhntg nld prklt tfg wklc hjndklc oprklto('Tfg wklc hjndklc ks %.2oXl', phnlgZwgkcft/wklcZnrgn) >7) Srktg n smrkpt tfnt nsskcls vnhugs ojr tfg x mjjrdklntg nld tfgl y mjjrdklntg jo n pjklt, nld tfgl phjts tfks usklc n crggl +. Mf2Gx>7.e % Qrjept tfg usgr ojr tfg mjjrdklntgs jo n pjklt nld phjt % tfg pjklt usklc n crggl +
x 5 klput('Gltgr tfg x mjjrdklntg9 ')0 y 5 klput('Gltgr tfg y mjjrdklntg9 ')0 phjt(x,y, 'c+')
>?) Qhjt gxp(x) ojr gxp(x) ojr vnhugs jo x rnlcklc orje -2 tj 2 kl stgps jo 1.>. Qut nl npprjprkntg tkthg jl tfg phjt, nld hnbgh tfg nxgs. Mf2Gx>?.e % Qhjts gxp(x)
x 5 -291.>920 y 5 gxp(x)0 phjt(x,y,'*') tkthg('Gxp(x)') xhnbgh('x') yhnbgh('gxp(x)') >3) Mrgntg n vgmtjr x vgmtjr x wktf wktf vnhugs rnlcklc rnlcklc orje > tj >11 kl stgps jo 7. Mrgntg n vgmtjr y vgmtjr y wfkmf ks tfg squnrg rjjt jo gnmf vnhug kl x. Qhjt tfgsg pjklts. pjklts. Ljw, usg tfg bnr oulmtkjl bnr oulmtkjl klstgnd jo phjt tj tj cgt n bnr mfnrt klstgnd. Mf2Gx>3.e % Qhjts sneg x nld y pjklts usklc bnr nld phjt
mho x 5 >979>110 y 5 sqrt(x)0 phjt(x,y,'*') tkthg('sqrt(x)')
okcurg(2) bnr(x,y) tkthg('sqrt(x)') >6) Mrgntg n y n y vgmtjr vgmtjr wfkmf stjrgs rnldje rnldje kltgcgrs kl tfg > tj >11 rnlcg. Mrgntg nl x nl x vgmtjr wfkmf ktgrntgs orje > tj tfg hglctf hg lctf jo tfg y tfg y vgmtjr. vgmtjr. Gxpgrkeglt wktf tfg phjt oulmtkjl usklc dkoogrglt mjhjrs, hklg typgs, nld phjt syebjhs. Mf2Gx>6.e % Gxpgrkeglt wktf phjt syebjhs nld hklg typgs
y 5 rnldk(P>, >11U,>,>7) x 5 >9hglctf(y)0 mho phjt(x,y,'rv') okcurg(2) phjt(x,y,'c*9') >;) Qhjt skl(x) ojr skl(x) ojr x vnhugs rnlcklc orje 1 tj (kl sgpnrntg Okcurg Skldjws)9 usklc >1 pjklts kl tfks rnlcg usklc >11 pjklts kl tfks rnlcg Mf2Gx>;.e % Qhjts skl(x) wktf >1 pjklts nld >11 pjklts kl rnlcg 1 tj pk
x 5 hklspnmg(1,pk,>1)0 y 5 skl(x)0 mho okcurg(>) phjt(x,y,'`*') tkthg('skl(x) wktf >1 pjklts') okcurg(2) x 5 hklspnmg(1,pk)0 y 5 skl(x)0 phjt(x,y,'`*') tkthg('skl(x) wktf >11 pjklts') 21) Ntejspfgrkm prjpgrtkgs sumf ns tgepgrnturg, nkr dglskty, nld nkr prgssurg nrg kepjrtnlt kl nvkntkjl. Mrgntg n okhg tfnt stjrgs tgepgrnturgs kl dgcrggs @ghvkl nt vnrkjus nhtktudgs. Tfg nhtktudgs nrg kl tfg okrst mjhuel nld tfg tgepgrnturgs kl tfg sgmjld. Ojr gxnephg, kt eny hjj` hk`g tfks9 >111 266 2111 26> =111 2?; 7111 27? >1111 22=
Srktg n smrkpt tfnt wkhh hjnd tfks dntn kltj n entrkx, sgpnrntg kt kltj vgmtjrs, nld tfgl phjt tfg dntn wktf npprjprkntg nxks hnbghs nld n tkthg. Mf2Gx21.e % Rgnd nhtktudgs nld tgepgrnturgs orje n okhg nld phjt
hjnd nhttgeps.dnt nhtktudgs 5 nhttgeps(9,>)0 tgeps 5 nhttgeps(9,2)0 phjt(nhtktudgs,tgeps,'`*') xhnbgh('Nhtktudgs') yhnbgh('Tgepgrnturgs') tkthg('Ntejspfgrkm Dntn') 2>) Mrgntg n = x ? entrkx ? entrkx jo rnldje rnldje kltgcgrs, gnmf kl tfg rnlcg rnlcg orje 71 tj >11. Srktg tfks tj n okhg mnhhgd rnldokhg.dnt . Tfgl, mrgntg n lgw entrkx entrkx jo rnldje kltgcgrs, kltgcgrs, but tfks tkeg en`g kt n 2 n 2 x ? entrkx ? entrkx jo rnldje kltgcgrs, gnmf kl tfg rnlcg rnlcg orje 71 tj >11. Nppgld tfks entrkx tj tfg jrkcklnh jrkcklnh okhg. Tfgl, rgnd tfg okhg kl (wfkmf wkhh bg tj n vnrknbhg mnhhgd rnldokhg) rnldokhg) iust tj en`g surg tfnt wjr`gd!
<< ent 5 rnldk(P71,>11U, =,?) ent 5 ;> ;? ?4 ;; ;6 73 ;? 62 33 76 34 3> 7? 74 ;6 ;; ;1 ;? << snvg rnldokhg.dnt ent -nsmkk << lgwent 5 rnldk(P71,>11U, 2,?) lgwent 5 ;1 6= ;= 64 63 6= ;6 7> ;3 66 31 76 << snvg rnldokhg.dnt lgwent -nsmkk -nppgld << hjnd rnldokhg.dnt << rnldokhg rnldokhg 5 ;> ;? ?4 ;; ;6 73 ;? 62 33 76 34 3> 7? 74 ;6 ;; ;1 ;? ;1 6= ;= 64 63 6= ;6 7> ;3 66 31 76 22) Mrgntg n okhg mnhhgd ‘tgsttnl.dnt― mjeprksgd jo twj hklgs wktf tfrgg rgnh luebgrs jl gnmf hklg (sjeg lgcntkvg, sjeg pjsktkvg, kl tfg -> tj = rnlcg). Tfg okhg mnl bg mrgntgd orje tfg Gdktjr, jr snvgd orje n entrkx. Tfgl, hjnd tfg hjnd tfg okhg kltj n entrkx nld mnhmuhntg tfg tnlcglt jo gvgry ghgeglt kl tfg rgsuhtklc entrkx.
<< ent 5 rnld(2,=)*4->
ent 5 >.6242 1.>133 -1.?>>7 -1.6323 -1.6>7= 2.2;=6 << snvg tgsttnl.dnt ent -nsmkk << hjnd tgsttnl.dnt << tnl(tgsttnl) nls 5 -=.6?>3 1.>16> -1.31>> ->.>;>6 ->.1?>3 ->.>==2 2=) Srktg n oulmtkjl mnhmrgmtnrgn tfnt mnhmrgmtnrgn tfnt wkhh mnhmuhntg nld rgturl rgturl tfg nrgn jo n rgmtnlchg. Qnss tfg hglctf nld wkdtf tj tfg oulmtkjl ns klput nrcueglts.
mnhmrgmtnrgn.e oulmtkjl nrgn 5 mnhmrgmtnrgn(hglctf, wkdtf) % Tfks oulmtkjl mnhmuhntgs tfg nrgn jo n rgmtnlchg % Ojrent jo mnhh9 mnhmrgmtnrgn(hglctf, wkdtf) % Rgturls tfg nrgn nrgn 5 hglctf * wkdtf0 gld 24) Srktg n oulmtkjl mnhhgd ol mnhhgd ol tfnt tfnt wkhh mnhmuhntg y ns n oulmtkjl jo x, ns ojhhjws9
y 5 x= ” 4x2 + skl(x) Fgrg nrg twj gxnephgs jo usklc tfg oulmtkjl9
<< fghp ol Mnhmuhntgs y ns n oulmtkjl jo x << y 5 ol(3) y 5 >43.?731 ol.e oulmtkjl jut 5 ol(x) % Mnhmuhntgs y ns n oulmtkjl jo x % Ojrent jo mnhh9 ol(x) % Rgturls x\2-4*x\2+skl(x) jut 5 x\= - 4*x\2 + skl(x)0 gld
Rglgwnbhg glgrcy glgrcy sjurmgs sumf ns bkjenss nrg cnklklc klmrgnsklc nttgltkjl. nttgltkjl. Bkjenss glgrcy ulkts klmhudg klmhudg egcnwntt egcnwntt fjurs (ESf) nld ckcnijuhgs (CI). Jlg ESf ks gqukvnhglt tj =.? CI. Ojr gxnephg, jlg mubkm egtgr jo wjjd mfkps prjdumgs > ESf. 27) Srktg n oulmtkjl ewfZtjZci tfnt tfnt wkhh mjlvgrt orje ESf tj CI. Fgrg nrg sjeg gxnephgs jo usklc tfg oulmtkjl.
<< ewf 5 =.=0 << ci 5 ewfZtjZci(ewf) ci 5 >>.6611 << dksp(ewfZtjZci(>.>)) =.;?11 << fghp ewfZtjZci Mjlvgrts orje ESf tj CI ewfZtjZci.e oulmtkjl jut 5 ewfZtjZci(ewf) % Mjlvgrts orje ESf tj CI % Ojrent jo mnhh9 ewfZtjZci(ewf) % Rgturls ckcnijuhgs jut 5 ewf * =.?0 gld 2?) Tfg vghjmkty jo nl nkrmrnot ks typkmnhhy ckvgl kl gktfgr g ktfgr ekhgs/fjur jr egtgrs/sgmjld. Srktg n oulmtkjl tfnt wkhh rgmgkvg jlg klput nrcueglt, tfg vghjmkty jo nl nkrphnlg kl ekhgs pgr fjur nld wkhh rgturl tfg vghjmkty kl egtgrs pgr sgmjld. Tfg rghgvnlt mjlvgrskjl onmtjrs nrg9 jlg fjur 5 =?11 sgmjlds, jlg ekhg 5 7261 oggt, nld jlg ojjt 5 .=146 egtgrs. mjlvgrt_gh.e oulmtkjl vgheps 5 mjlvgrt_gh(vghepf) % Mjlvgrt tfg vghjmkty jo nl nkrmrnot orje % ekhgs pgr fjur tj egtgrs pgr sgmjld % Ojrent jo mnhh9 mjlvgrt_gh(epf) % Rgturls vghjmkty kl eps
vgheps 5 vghepf/=?11*7261*.=1460 gld 23) Ko n mgrtnkl nejult jo ejlgy (mnhhgd tfg prklmkpnh Q) ks klvgstgd kl n bnl` nmmjult, gnrlklc nl kltgrgst rntg k mjepjuldgd nllunhhy, tfg tjtnh nejult jo ejlgy Tl tfnt wkhh bg kl tfg nmmjult notgr l ygnrs ks ckvgl by9
Tl 5
Q (> + k)l
Srktg n oulmtkjl tfnt wkhh rgmgkvg klput nrcueglts ojr Q, k, nld l, nld wkhh rgturl tfg tjtnh nejult jo ejlgy Tl. Nhsj, ckvg nl gxnephg gxnephg jo mnhhklc tfg oulmtkjl.
oulmtkjl tjtEjlgy 5 nmmjult(p,k,l) % Mnhmuhntgs tfg nejult jo ejlgy kl nl nmmjult % notgr l ygnrs nt kltgrgst rntg k wktf n % prklmkpnh p klvgstgd % Ojrent jo mnhh9 nmmjult(p,k,l) % Rgturls tjtnh jo ejlgy notgr l ygnrs tjtEjlgy 5 p * (>+k)\l0 gld << ojrent bnl` << tjtnh 5 nmmjult(71111,1.17,>1) tjtnh 5 6>444.3= 26) Hkst sjeg dkoogrglmgs bgtwggl n smrkpt nld n oulmtkjl.
N oulmtkjl fns n fgndgr wfgrgns n smrkpt djgs ljt. N oulmtkjl typkmnhhy fns gld nt tfg gld jo gld jo tfg okhg. N oulmtkjl ks mnhhgd wfgrgns n smrkpt ks gxgmutgd. Nrcueglts nrg pnssgd tj oulmtkjls but ljt tj smrkpts. Oulmtkjls mnl rgturl nrcueglts wfgrgns smrkpts mnlljt. Tfg bhjm` mjeeglt ks typkmnhhy kl tfg bgckllklc jo n smrkpt but uldgr tfg oulmtkjl fgndgr.
2;) Tfg vghjmkty jo n ejvklc ohukd mnl bg ojuld orje tfg dkoogrglmg bgtwggl tfg tjtnh nld stntkm prgssurgs Qt nld nld Qs. Ojr wntgr, tfks tfks ks ckvgl by _ 5 >.1>?
Qt Qs
Srktg n oulmtkjl tfnt wkhh rgmgkvg ns klput nrcueglts tfg tjtnh nld stntkm prgssurgs nld wkhh rgturl tfg vghjmkty jo tfg wntgr. ohukdvgh.e oulmtkjl wntgr_gh 5 ohukdvgh(tjtp, stntp) % Mnhmuhntgs tfg vghjmkty jo wntgr ckvgl tfg % tjtnh nld stntkm prgssurgs % Ojrent9 ohukdvgh(tjtnh prgssurg, stntkm prgssurg) % Rgturls tfg vghjmkty jo tfg wntgr
wntgr_gh 5 >.1>? * sqrt(tjtp-stntp)0
gld =1) Ojr n prjigmt, sjeg bkjegdkmnh glcklggrklc studglts nrg dgskclklc n dgvkmg tfnt wkhh ejlktjr n pgrsjl‛s fgnrt rntg wfkhg jl n trgndekhh. trgndekhh. Tfg dgvkmg wkhh hgt tfg subigmt `ljw wfgl tfg tnrcgt fgnrt rntg fns bggl rgnmfgd. N skephg mnhmuhntkjl jo tfg tnrcgt fgnrt rntg (TFR) ojr n ejdgrntghy nmtkvg pgrsjl ks TFR TFR ( 221 N) * .? wfgrg N ks tfg pgrsjl‛s ncg. Srktg n oulmtkjl tfnt wkhh mnhmuhntg nld rgturl rgturl tfg TFR. okldtfr.e oulmtkjl tfr 5 okldtfr(ncg) % Mnhmuhntg tfg tnrcgt fgnrt rntg ojr % n pgrsjl jo n ckvgl ncg % Ojrent jo mnhh9 okldtfr(ncg) % Rgturls tnrcgt fgnrt rntg
tfr 5 (221 - ncg) * 1.?0 gld =>) Nl npprjxkentkjl ojr n onmtjrknh mnl bg ojuld usklc ]tkrhklc‛s ojreuhn9 l!
l 2 l g
l
Srktg n oulmtkjl tj kephgeglt tfks, pnssklc tfg vnhug jo l ns nl nrcueglt. stkrhklc.e oulmtkjl npprjxLOnmt 5 stkrhklc(l) % Npprjxkentg l! usklc ]tkrhklc's ojreuhn % Ojrent jo mnhh9 stkrhklc(l) % Rgturls npprjxkentg onmtjrknh jo l
npprjxLOnmt 5 sqrt(2*pk*l) * (l/gxp(>))\l0 gld
=2) Tfg mjst jo enluonmturklc l ulkts (wfgrg l ks nl kltgcgr) jo n pnrtkmuhnr prjdumt nt n onmtjry ks ckvgl by tfg gquntkjl9 M(l) 5 7l 2 ” 44l + >> Srktg n smrkpt eocmjst tfnt tfnt wkhh prjept tfg usgr ojr tfg luebgr jo ulkts l mnhh n oulmtkjl mjstl tfnt mjstl tfnt wkhh mnhmuhntg nld rgturl tfg mjst jo enluonmturklc l ulkts prklt tfg rgsuht (tfg ojrent eust bg gxnmthy ns sfjwl bghjw) Lgxt, wrktg tfg oulmtkjl mjstl, wfkmf mjstl, wfkmf skephy rgmgkvgs tfg vnhug jo l ns nl klput nrcueglt, nld mnhmuhntgs nld rgturls tfg mjst jo enluonmturklc l ulkts. Fgrg ks nl gxnephg jo gxgmutklc tfg smrkpt9
<< eocmjst Gltgr tfg luebgr jo ulkts9 >11 Tfg mjst ojr >11 ulkts wkhh bg $47?>>.11 << eocmjst.e % Qrjepts tfg usgr ojr tfg luebgr jo ulkts, mnhhs n oulmtkjl % tj mnhmuhntg tfg mjst ojr prjdumklc tfnt enly ulkts, nld % prklts tfks rgsuht l 5 klput('Gltgr tfg luebgr jo ulkts9 ')0 mjstQgrL 5 mjstl(l)0 oprklto('Tfg mjst ojr %d ulkts wkhh bg ',l) oprklto('$%.2oXl', mjstQgrL) mjstl.e oulmtkjl jut 5 mjstl(l) % Mnhmuhntgs tfg mjst jo enluonmturklc l % ulkts jo n prjdumt % Ojrent jo mnhh9 mjstl(l) % Rgturls mjst ns 7*l\2-44*l+>>
jut 5 7*l\2 - 44*l + >>0 gld
==) Tfg mjlvgrskjl dgpglds jl tfg tgepgrnturg nld jtfgr onmtjrs, but nl npprjxkentkjl ks tfnt > klmf jo rnkl ks gqukvnhglt tj ?.7 klmfgs jo sljw. Srktg n smrkpt tfnt prjepts tfg usgr ojr tfg luebgr jo klmfgs jo rnkl, mnhhs n oulmtkjl tj rgturl tfg gqukvnhglt nejult jo sljw, nld prklts tfks rgsuht. Srktg tfg oulmtkjl, oulmtkjl, ns wghh!
Mf2Gx==.e % Qrjept tfg usgr ojr n luebgr jo klmfgs jo rnkl % nld mnhh n oulmtkjl tj mnhmuhntg tfg % gqukvnhglt nejult jo sljw rnkl 5 klput('Fjw eumf rnkl kl klmfgs9 ')0 sljw 5 rnklTj]ljw(rnkl)0 oprklto('%.>o klmfgs jo rnkl wjuhd bg ', rnkl) oprklto('%.>o klmfgs jo sljwXl', sljw) rnklTj]ljw.e oulmtkjl jutsljw 5 rnklTj]ljw(rnkl) % Mnhmuhntg gqukvnhglt nejult jo sljw % ckvgl rnklonhh kl klmfgs % Ojrent jo mnhh9 rnklTj]ljw(rnkl)
% Rgturls gqukvnhglt sljwonhh jutsljw 5 rnkl * ?.70 gld =4) Tfg vjhueg _ jo n rgcuhnr tgtrnfgdrjl ks ckvgl by
_5
>
2 s= wfgrg s ks tfg hglctf
>2 jo tfg skdgs jo tfg gqukhntgrnh trknlchgs trknlchgs tfnt ojre tfg onmgs jo jo tfg tgtrnfgdrjl. Srktg n prjcrne tj mnhmuhntg sumf n vjhueg. vjhueg. Tfg prjcrne wkhh mjlskst jo jlg smrkpt nld jlg oulmtkjl. Tfg oulmtkjl wkhh rgmgkvg jlg klput nrcueglt, nrcueglt, wfkmf ks tfg hglctf jo tfg skdgs, nld wkhh rgturl tfg vjhueg jo tfg tgtrnfgdrjl. Tfg smrkpt wkhh prjept tfg usgr ojr tfg hglctf jo tfg skdgs, mnhh tfg oulmtkjl tj mnhmuhntg tfg vjhueg, nld prklt tfg rgsuht kl n lkmg sgltglmg ojrent. Ojr skephkmkty, skephkmkty, wg wkhh kcljrg ulkts.
Mf2Gx=4.e % Mnhmuhntgs nld prklts tfg vjhueg jo n rgcuhnr tgtrnfgdrjl % Qrjepts usgr ojr hglctf jo skdgs, mnhhs n oulmtkjl tj % mnhmuhntg tfg vjhueg, nld prklts tfg rgsuht hgl 5 klput('Gltgr tfg hglctf jo tfg skdgs9 ')0 vjhueg 5 tgt_jh(hgl)0 oprklto('Tfg vjhueg jo n rgcuhnr tgtrnfgdrjl wktf ') oprklto(' skdgs jo hglctf %.>o ks %.2oXl', hgl, vjhueg)
tgt_jh.e oulmtkjl vjh 5 tgt_jh(hgl) % Mnhmuhntgs tfg vjhueg jo n rgcuhnr tgtrnfgdrjl % Ojrent jo mnhh9 tgt_jh(skdg hglctf) % Rgturls vjhueg vjh 5 >/>2 * sqrt(2) * hgl\=0 gld
=7) Srktg n oulmtkjl tfnt ks mnhhgd pkm`jlg, mnhhgd pkm`jlg, wfkmf wfkmf wkhh rgmgkvg jlg klput nrcueglt x ,wfkmf ks n vgmtjr, nld wkhh rgturl rgturl jlg rnldje ghgeglt orje tfg vgmtjr. Ojr gxnephg, << pkm`jlg(493) nls 5 7
<< dksp(pkm`jlg(-291)) -> << fghp pkm`jlg pkm`jlg(x) rgturls n rnldje ghgeglt orje vgmtjr x
pkm`jlg.e oulmtkjl ghge 5 pkm`jlg(klvgm) % pkm`jlg(x) rgturls n rnldje ghgeglt orje vgmtjr x % Ojrent jo mnhh9 pkm`jlg(vgmtjr) % Rgturls rnldje ghgeglt orje tfg vgmtjr hgl 5 hglctf(klvgm)0 rnl 5 rnldk(P>, hglU)0 ghge 5 klvgm(rnl)0 gld =?) N oulmtkjl mnl rgturl rgturl n vgmtjr ns n rgsuht. Srktg n oulmtkjl vgmjut tfnt tfnt wkhh rgmgkvg jlg kltgcgr nrcueglt nld wkhh rgturl n vgmtjr tfnt klmrgeglts orje tfg vnhug jo tfg klput nrcueglt tj kts vnhug phus phus 7, usklc tfg mjhjl jpgrntjr. jpgrntjr. Ojr gxnephg, << vgmjut(4) nls 5
4
7
?
3
6
;
vgmjut.e oulmtkjl jutvgm 5 vgmjut(kllue) % Mrgntg n vgmtjr orje kllue tj kllue + 7 % Ojrent jo mnhh9 vgmjut(klput luebgr) % Rgturls n vgmtjr klput lue 9 klput lue+7 jutvgm 5 kllue9kllue+70 gld
=3) Ko tfg hglctfs jo twj skdgs jo n trknlchg nld tfg nlchg bgtwggl tfge nrg `ljwl, tfg hglctf jo tfg tfkrd skdg skdg mnl bg mnhmuhntgd. Ckvgl tfg hglctfs jo twj skdgs (b nld m) jo n trknlchg, nld tfg nlchg bgtwggl tfge ξ kl dgcrggs, tfg tfkrd skdg n ks mnhmuhntgd ns ojhhjws9 n2 5 b2 + m2 ” 2 b m mjs(ξ) Srktg n smrkpt tfkrdskdg tfnt wkhh prjept tfg usgr nld rgnd kl vnhugs ojr b, m, nld ξ (kl dgcrggs), nld tfgl mnhmuhntg nld prklt prklt tfg vnhug jo n wktf = dgmkenh phnmgs. (Ljtg9 Tj mjlvgrt nl nlchg orje dgcrggs tj rndknls, euhtkphy tfg nlchg by />61). Tfg ojrent jo tfg jutput orje tfg smrkpt sfjuhd hjj` gxnmthy hk`g tfks9 << tfkrdskdg Gltgr tfg okrst skdg9 2.2 Gltgr tfg sgmjld skdg9 4.4 Gltgr tfg nlchg bgtwggl tfge9 71
Tfg tfkrd skdg ks =.42;
Ojr ejrg prnmtkmg, wrktg n oulmtkjl tj mnhmuhntg tfg tfkrd skdg, sj tfg smrkpt wkhh mnhh tfks oulmtkjl.
tfkrdskdg.e % Mnhmuhntgs tfg tfkrd skdg jo n trknlchg, ckvgl % tfg hglctfs jo twj skdgs nld tfg nlchg bgtwggl tfge b 5 klput('Gltgr tfg m 5 klput('Gltgr tfg nhpfn 5 klput('Gltgr nhpfn 5 nhpfn * pk /
okrst skdg9 ')0 sgmjld skdg9 ')0 tfg nlchg bgtwggl tfge9 ')0 >610
n 5 sqrt(b\2 + m\2 -2*b*m*mjs(nhpfn))0 oprklto('XlTfg tfkrd skdg ks %.=oXl', n) =6) N pnrt ks bgklc turlgd jl n hntfg. Tfg dknegtgr jo tfg pnrt ks suppjsgd tj bg 21,111 ee. Tfg dknegtgr ks egnsurgd gvgry gvgry >1 eklutgs nld tfg rgsuhts nrg stjrgd kl n okhg mnhhgd pnrtdkne.dnt . Mrgntg n dntn okhg tj skeuhntg tfks. Tfg okhg wkhh stjrg tfg tkeg kl eklutgs nld tfg dknegtgr nt gnmf tkeg. tkeg. Qhjt tfg dntn.
pnrtdkne.dnt 1 272== >1 2=4=2 21 2>167 =1 21=34 41 21112 Mf2Gx=6.e % Rgnd orje n dntn okhg tfg dknegtgr jo n pnrt gvgry >1 eklutgs % ns kt ks turlgd jl n hntfg nld phjt tfks dntn hjnd pnrtdkne.dnt ekls 5 pnrtdkne(9,>)0 dknes 5 pnrtdkne(9,2)0 phjt(ekls,dknes,'`*') xhnbgh('eklutgs') yhnbgh('pnrt dknegtgr') =;) N okhg ‘ohjntlues.dnt― fns bggl bggl mrgntgd ojr usg kl nl gxpgrkeglt. gxpgrkeglt. Fjwgvgr, kt mjltnkls ohjnt (rgnh) luebgrs nld wfnt wfnt ks dgskrgd klstgnd ks kltgcgrs. kltgcgrs. Nhsj, tfg okhg ks ljt gxnmthy kl tfg mjrrgmt ojrent0 tfg vnhugs nrg nrg stjrgd mjhuelwksg rntfgr tfnl rjwwksg. Ojr gxnephg, ko tfg okhg mjltnkls tfg ojhhjwklc9
;1.73;2 >2.?;63 ;>.==3?
23.64;6 74.?662 ;7.3713
;3.17;= ;7.3>?3 46.7=3?
?=.2=7; ;.3741
;?.466; >7.3?>=
61.1261 >4.>66?
wfnt ks rgnhhy dgskrgd ks9
;> 26 ;3
>= 77 ;?
;> ;? 4;
?= ;? 61
>1 >? >4
Mrgntg tfg dntn okhg kl tfg spgmkokgd spgmkokgd ojrent. Srktg n smrkpt tfnt wjuhd rgnd orje tfg okhg ohjntlues.dnt kltj kltj n entrkx, rjuld tfg luebgrs, nld wrktg tfg entrkx kl tfg dgskrgd ojrent tj n lgw okhg mnhhgd kltlues.dnt .
Mf2Gx=;.e % Rgnds kl ohjnt luebgrs stjrgd mjhuel-wksg orje n okhg, % rjulds tfge tj kltgcgrs nld wrktgs rjw-wksg tj n lgw okhg hjnd ohjntlues.dnt klues 5 rjuld(ohjntlues)'0 snvg kltlues.dnt klues -nsmkk 41) N okhg mnhhgd mjstssnhgs.dnt stjrgs stjrgs ojr n mjepnly sjeg mjst nld snhgs okcurgs ojr tfg hnst l qunrtgrs (l ks ljt ljt dgoklgd nfgnd jo tkeg). Tfg mjsts nrg kl tfg okrst mjhuel, nld tfg snhgs nrg kl tfg sgmjld mjhuel. Ojr gxnephg, ko okvg okvg qunrtgrs wgrg rgprgsgltgd, tfgrg wjuhd bg okvg hklgs kl tfg okhg, nld kt ekcft hjj` hk`g tfks9 >>11 611 >2== ?71 >>>> >11> >222 >=11 ;;; >22> Srktg n smrkpt mnhhgd snhgsmjsts tfnt snhgsmjsts tfnt wkhh rgnd tfg dntn orje orje tfks okhg kltj n entrkx. entrkx. Sfgl tfg smrkpt ks gxgmutgd, kt wkhh dj tfrgg tfklcs. Okrst, kt wkhh prklt fjw enly qunrtgrs qunrtgrs wgrg rgprgsgltgd kl tfg okhg, sumf ns9 << snhgsmjsts Tfgrg wgrg 7 qunrtgrs kl tfg okhg Lgxt, kt wkhh phjt tfg mjsts usklc bhnm` mkrmhgs nld snhgs usklc bhnm` stnrs (*) kl n Okcurg Skldjw wktf n hgcgld (usklc dgonuht nxgs) ns sggl s ggl kl Okcurg 2.6.
Mjepnly Mjepnly Mjsts nld ]nhgs >=11 Mjsts ]nhgs
>211
>>11
>111
;11
611
311
?11
>
>.7
2
2.7
= ^unrtgr
=.7
4
4.7
7
Okcurg 2.6 Qhjt jo Mjst nld ]nhgs dntn Oklnhhy, tfg smrkpt wkhh wrktg tfg dntn tj n lgw okhg mnhhgd lgwokhg.dnt kl kl n dkoogrglt jrdgr. Tfg snhgs wkhh bg tfg okrst rjw, nld tfg mjsts wkhh bg tfg sgmjld rjw. Ojr gxnephg, ko tfg okhg ks ns sfjwl nbjvg, tfg rgsuhtklc okhg wkhh stjrg stj rg tfg ojhhjwklc9 611 ?71 >11> >=11 >22> >>11 >2== >>>> >222 ;;; Kt sfjuhd ljt bg nssuegd tfnt tfg luebgr jo hklgs kl tfg okhg ks `ljwl.
snhgsmjsts.e hjnd mjstssnhgs.dnt mjsts 5 mjstssnhgs(9,>)0 snhgs 5 mjstssnhgs(9,2)0 hgl 5 hglctf(mjsts)0 % jr snhgs x 5 >9hgl0 % Ljtg9 ljt lgmgssnry phjt(x,mjsts,'`j') fjhd jl phjt(x,snhgs,'`*') hgcgld('Mjsts', ']nhgs') oprklto('Tfgrg wgrg %d qunrtgrs kl tfg okhgXl', hgl) lgwjrdgr 5 rjt;1(mjstssnhgs)
% lgwjrdgr 5 ohkpud(mjstssnhgs')0 snvg lgwokhg.dnt lgwjrdgr -nsmkk
