GeorgeB. Thomas& RossL. Finney Inslituteof TechDology Massachusetts
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PTOVO: ML TABO^I D MIA::YNAPTI{:H.: LI :wr€rayF€v€Eoro snins6o L2 M€rclror[io€rszclt ar[6oraol 1.3H zl'ior1pLcgtu0eic; lpaupiE €r0€r6vyQctrtr6v 1.4EELo6o€LS 1.5:r'vaQrrlo€rsxar lQqqrx6ro' odilE'6voovEK6oro o Eldetotq qndvrto€opvr)r'(o: xeioYrovo pdgelKovelq Yeouerpiq, "Lti.I'u BM.^Ki, Arpani'.i-i r.ap.rpin"". Oalouv,q^o6i,6ou^.1o ro Moonuonxq. H nlorovrKrr qonlo nou uoairp€ o EurdEioqsoro :rorx6ao,adiveroLK dn6 ivo 6Elr€po ov€K6ororo onoto oin(.' o troBdioq. E6@qoiv€ror ol q Ue^ernrov Moetuotuinv 6ev np.n.' vd yivetq' y'd npomko govov oqaln, oi o xor Yo qv €uplreprl rouq eoor€p'Kt o(lo. lpoQ.' Iornov o :roBdioq: Ihp EJrt:-t. ll I. r(i6€ o4(!io ror siLribo! (vrt Go baEorotrllF€ dorzos arqn(ytGlr: n! {i;or€a drnri (rroi f,o! qde\aur mo
Or A-lelavbprvoi E 'ry€S Xplorporoinoav q ftcoperpio ocrv iva pioo pei,frl; rou quorxoi r6qrou pe igqcorl mq r€ryrxdg €qaa1oyiE. A116 ro dl.Lo pdgog,Apc6es xor Iv6oi Fa0ryarrxoi rns i6Lcs :leoL66ou,0eBeLtoocrv rg cB16gcrrou rorr origepcrovoprilo4re 1, 1.17.
12
PYOMO> METABO H: MIA: >YN
PTH>H>
Airol
AJI6 qv EE. (1)
8-7 6 !2-!J. , :L7xrJlz KLioq '7l = 3 o+laiv€i 6rr .ro U olEd1,€ror ).([id 3 [ovci6eg xd0e qopd no! *
ro r auldvmar, xcr,c I $ovd6o. M' d)"lo l,oyro,q rerc6oU ror s rivcr 3 qopeqp€tcl, reprtqq Ferc6ol.risrou ,r, 6ll"c6t Ay = 3Ar. H ,iol m = 3 eivar 6vcq ouweleoTtq cvo).oyi.oE.tr Eoro 6rt P2 = P2(1,2) xar Pr = Pr(3,8). Eivcr. rir).r 1 cnriynlorl = ?n 3: No[, ofi(DEgail8xal sbd) !!2 st Er. t.u H r.ior amt rrq EUoEiaqEirar m = (3 -2)/(a l) = 3. Au6 o4.oivd dI AV = 3Ax yLa zdoe aDdlri 06ortEYEEIONT"AMMON 21
IIAPAAEIIMA 6 H elioorol TIEoQll6wros eo0elag'iou 6#€t(ei@a,t6 to oryel,o (1,2) eivor g = 2. Ad.re ro 11. 1.20(b)!
Eqiooon H TevudtfpoUUtKn Oi,€Eor 6Ew6o€r€ xo't 3V=6
A +5U=m, 61owrq poppri
(11
At+Bs =C
pc opr,opevegi\lrq a@! A, oqF€iou-ril'iorE
B xcr C. EmnLlov.ooorctnnotteEtocoot
! =fna + o ptopei vc rgoei.ot4 pogpi Ar + Bg = C, o,r a\'! E(Iv(r1aftlroweoov ma - a =-b
(12')
H EE. (12) 6Xs[tq poaqriqg El. (11) pe ,4 = tx, B = 7 xctr"C = -b. Ot' 6ttog €Elodo€Exa1ox6Qw(Dv6u0e!cbv :, =5
iaq!
E =a
xat C =a). elow edor6 q Foaq{nE EE. (11) (ps A=1, B:o noa'or4aows l.out6r, &u xdoe nJoeic too enutd6o0€r(& Fua €Eloo)ol 6rorE q EE. (11) p€ 6va rouldxrorov crr6 to A tior B iuigoqo tou pq66106. Io$ier.dpcye to oml,orQoqo;Ei1/alfidr"ta n ^tQaqui4tisQdotaorlqg 6to! dvo toulrallotov ond m A xql B €ivc! 6tdgoQo EE. (11) Fto elJosl,a, rou O;H ondvtlorl €ivol vor,,6nroE0o 6ellor4e ncp,oxrirar. A,t B =0 ctrq Ao + Bg =C x@ A 10, a6rg[]@aodF€va ]urlooqr: roEriQoSc 6[qQd,vraExcu Tc 6rJopell t4g elloorig F€,4Av
B =0:
Ar =C
rlcl
xo, stidg,ql$uu0elog n eEioa;oqauoi €l,vd!eEto METABO,\}D MIA' >YNAPTI]>H:
xc!0€€uo€ioE€ivor yeo[Frx{ re! 6r! xdoe €Ei,oooirou €ivol lpaF]r|,xyir,X pra FriosioyQotr!ri. nooE.rr(dry. avrrnsoounFiFr
H .tEybaiyaappui €6io.,o? €x€r1lt soeqrt A.r+By=C
(11)
6nou A,B xat C eLvataro0€Q6E xor €vo rod.dloxov afi6 xo A xcL B eivcr 6r.riqoporot ;rq6ev6g.
IIAPA EIIMA 8r +59 = 20.
? BQ€ir€fllv
.ioq xar qv r€xo1r6vqy try eu0ei,cg
A!or; AnI6E ldvoqre ql €Eioa;oloX naos s/: 8 x + 5 A= 2 0 5 a = 8 x+ 2 0 u=
I
_:x+4
H xliol rivcr m = i. H xeraTr€vrt I €ivor6=4. tr
>Xebiqon Euee(oq EvaE "fptyopoE rQ6roE yr,o vo o"lebrdooupe Fra Eo€sia rio! -r€l.rv€r, zcr. ro!5 6uo dloveq, €ivar vo 6porjg€ rc o4reic ropriE rIE 1r€ioLg dEoa€g xaL ao q6aouFeqv Eo€ei.EIrEYoEraNr"AMMaN 23
NPOBAHMAT :ro neo6l{lqra 1-8'fadw€ qv EEio(,ol(a)yra qa xarax6awt €uo€is rior (b) 't,[qTlryoeLE6tricrou 6!6Ql€io[cr[6 ro 6€6oF6!o olF€io. L (2, 3) 2. (-7, -7) 3, (0,o) 4. (0. 4) 5. (-4,0) 6. (0,0) 7. (0, l,) 8 .( , ' . v , ) :ro nQo6Litrara9-14, ypcirlne rrp eliorootrIS €rjosfag nou xaooQi(€ior cnroro 6e6o[€voorpEioxor rrl'\,x].fol. :I€6rdor€ €rioll 0 ) 6 5 .P ( 4 , 6 ) , L : 4 r + 3 y = 1 2 66.P(21\/3,-1), L:*4r+y=-s. :rq IlQo6l.nFsrc! 67-74 6Q€i.rE rI y(l)li.arLiortErtE 6s6opdr,$€oo€iaE. 6 8 . y= r + 2 69.x + t/zs =z =3 72,2r y= 2 71.2x:+A=4 70.x +211 7 3 . 4 r+ 3 A= 1 2 74. 'f3r. + !) = -3 :ra neo6).ipara7s 786peimqr auoeicnor 6#plercLcn6 ro seaopevo oqpeiopeq 6e3oger.4 "4Dvicrl,i.ory@.
7s.(1,4),d=60" 77.( 2,3), A =e0
76.( 1, 1), d=13s" 7 8 .( 3 , 2 ) , 4 = 0 .
79. H xLxorl, p, nor eEaoxeirar oe €vcv 6rnrl xciro cn6 r1v erLqdverc qg 0dl.aoocg oxrti!flor, pe ro 6ci0ogmo oroio Epioxarcr o 6r!119, d, Fe pio €Eioo,otntE poaq{E P = A.l +1 (A €ivcr pro otcoepri). Orcrv d = 0 udrQa,I fii€on €ival [!c ortl6oqarQo.H nl€ol fid 100 F€rQo Eiv.rrxi€Qi.?iou 10-94axll6oqorQ€E. BQ€irEr$, ni.tol otcr 50 F6rgo. 80. Mro ov,ri,,o qo)roEor,olouoEi qv €uoEio,c + y = 1 ndvo on6 rov qqovo x(l)vr lio! ovoxLdrcl orov dqolo ,ro! r. A€i-r€ro :X. 1.32. IedVrc rylv EEioooylro! \160!'rIS 6Q6pou.
1.5 :\'NAPTH:EI: KAI rpAotKE! lt lA:TA:Er:
25
81. To qrodi,L prqq or6tQoypc4+nt< 6rcordii.rrnr orav e€Q[rsv0€i-To qtparos pra( ypc4rgrl; S €vG olvbimcrL pc rr1 Orppozpcroic FrixoE rry ? pE pttr ypupprzr'lefioooq 1r rL: orvrt0ropsv€q0egpozpcoie; n€At6dl,io\.rog. Kdioto tr€ipatcr or rrcr qrprr lpcgp{< dftore rcr ar.o),ou€(a;rorrlao[(rra:
?r = 65'F,
sr =35fi,
fadln€ rI ypqrpLxrirEiorotrrtnou ouv66fl ro S z(, I.
42. Or. 0cppoxpcoirEFahrenheir(F) rct Celsius(a') m,vbiovrcrLge pr"{l lpcppLxri rliotool 611,c6r1. I lpatplrn ircrpirorcol rot, ?" or,vcpr{oer rou C €ivaL€r'0€iatodp[ri. a) Bp€ir€rlv r:io(oul :ror,orr6irr rr; F zur C, cv C = 0 6rcv r = 32 rcL C = 1006rcv F = lll. h) YnAaxrL xriroru 0rUpozpuoiuurlv o:roraC = F: Av vcu.norrrrrvrrr: 6J-
rx. 1.32H roari.l rrF q0uIj< crriva( ro!
YnoHrorr orr ro Frixoc ntc xciofrl: OY uro rrl! i,Lr/tt :TAo:t (( EUtJ€iot sivat p. zoL 6tL 4 ON uXlpcrilr .yurviao pr ror 0trLzo IpLdEoro ltDv r. Atiic 6rL tra €Eiooorl n1_cZ eivcu rcosa + usin.l = p. lnuaariaqo4: O'r.tr,"A1+ B1 I0 I l€vu{'j yacFtrxri rEiocuu4,4"r- .B9 1- progi vc lgcrqri urc.r or o g.in,, -rllrt coso = A/ \., + Al -i sin o = B / t[E + E u.cnp = C I vrF * A t.J
E4.Av A. A. C, C' rivaL orc(0€Q6s ltot A l0 i 3 l0 6riErc on (a) or €u0Ei€E Ax+Bs=C.
Ax+Blt =C'
oupnin-rolvrj eivcr aapdl,l.tL€_( rar 6rr (b) oL €]Jo€i€E
Ax+Bu =C.
Bt
Ay =('
E|varxdoEx€S.
1.5
:uvoprrloerqKorf pogtKeqllopoorooetg Or r+16sltrag F€rq6llnl-< nooortFas r,EaArdiyr(l ol'v|ioo< (:ro rr! rqrd; pras d).1$. fra noQdbEr./po: H ftnoorarJ'l :TouXefl(rE8rorrvt. LruroaivrlroI', rrr mulrorqoer r-ivar ('wderrpt rou n6oo yQriyoQotrprlr ro cuozivrpo.6rnv aQxioag€va ncr(ig€ ro qQ€vo. -O apr0p6Ettov trcyti-rvoe 6vc diroog.rivcrLor,r(iQrtortrol cQr0pou rtov ol,€rou&ovi x(D!riu,llrirv rot, 6piozovrcLm4v ibrc :rapLo;4r1. O 6yzo; rol' vrpol' of F'rr 6r:dt'rut. fi!{r| or'\'riQntdrt ror'7Qovo! rt,porou rivor!co zpolvri: rr6 brEqttuj-. O 1p6voq:rorrXpeLdlercrl./!o(rroorlrniror] €1'c(brlq;. rjorepccr:rti Ir,r ,r(rdb|ol dra 25 Iirp'1. fivur otv,ir,nl.q lot'Xpo\'ou ,roqulovlc o crtn6 ro 6d0o,io endvD IAPAAEIIMA 2 H o16oq? = 12 oQ[Eer tov aEe[6 t ocrvto r€iedxDvo rl,Lzor'o 4rx,irl,ro s = r,/i-?. -",.i'. rou aprefou .r. Oo tlnopouoa!8 vo ovuudooutu durrl lyl ouvdprylorl !- \tr-. rou r, 'tetQcy(l)viEouod",€n€LE oL aQlelol 10! ne6lo! rrtr(i)'\,,eivar ra
I.5 :YNAPTH'EI: KAI |PAoIKEI nAPA'TA'EI!
27
r€rQol(ola r(JJvqpr0l6a ro! r€6io1j oQro[ou. H ytt,tz/1 ot't4uuth yrr q orn,a2r4c4 ti\t 4 or',dar4ott ! = x2". To ne6io oqLopoi 11; (,,\liQotorF! =.rr tir'rr r,' ir'fu).,' r, ^ r,lrrOnourluy lFdr\ rrlr' .r o alJ1rlrl! neeinlulorl.1() oriloio rr^' rQq?arrcirv opr0FdJv.To nrbio rFdir, €i.aaLro ouvoio ro^ tr't calTLzda raqForrrdrv apL0tr6v.tr nAPAAEITMA 3 Av ro olpsio (i.,r) a\ri,€r orov ztrd.o ltr z6vrpo rrlv oaXt xar ozrivq I 0rovd6() (6€ir€ io :2. 1.31). roft ra r'nor y LzcvoroLorlvr4v r,Eiotrrl er+Yr = tAur6 EivoLLoo6wqto Ie
T
tr=+\tri.
noo 6i!€r 6uo brirardgs16or:[6lla ro gAv ro o4psi,o(r,9) 6Qioxsior olo fid\,o ilFDixx .o onori g > 0 ror!' lJ = +,17=F. Av ro oy}reio(/, U) 6ai.ozErdrmo ?idro l[r,].i&].!o. 16r€ a = yI V. :! xd0e r€pi.jrrt,lort,ro g eivru lro orvilQtlol rou r tr€ n€6io oQLoFou ro 6Ldorrproo,[6 x = I €rDqz = 1. To ne6io illl(iv rl-c F €il'qr O S g < 1. To nebio r4rc6vrrtq s = -vT:lr ei'crr u =\n
1
r . 4 : r o o o r t ' u r J r , r t , : T o o vf r0 , i v a , u a r t ' r z r l V! rj tralha, .,'{r€i\r , F$y(}"ir€aori ioo tr. ro prE!\'.
0 METABO H' MIA' >YNAPTTDI'
H 6cor,xr] r.66c yLc rI o1€6ioo4 ,iagnul.6v €ivq! q eqlg: rioi€rlolp€ otF€ia U€XA'.va rioroLd6oqrE ro olfpc r4g rag:rul,q, rcrr, xcr6nLv lQ,qor+rororouF€I oxEol.y!(rro 3/.yro\r0 6Qoitr€rr;g p€r06dLr!€rarro g trcOd\ ro x. r.rv€irar &vd$€oa yl poxQtd an6 ra orper.top6vno4pei,c.A).Lri nold oIFEia 6ro).€:you[€ olvrjOoq; Iloptrxd-ror orcqeporpr papmotg xn6r,eE 1o qr €rloyyl lrordl'),qLov o+r€icov. @o notp€ n€Qroo6r€aattd rt XQrionqg y = l(r) ortv re66l,€1nl rlE oufrrrEQlqoQdE rou y F€rr.E! rrDl' snrl,E./€!ro)v oryeitov, oro Keqri).ar.o 3.
EnLloyn>nueiovyrorn :Xe6ioonrnq y = f(x) 1. :ru€r6or€ To olpei,a 6rou q lpcqrxri acrpdorco4r€pverri eqd:rrercr oiouE dEov€E.Aurd ra ortp€io uroLoli(owar €uxolo, av e6oo!,g€ s =0 L(! r =0 ortv sqLatjaqu - I(:t). 2. :ry€r6oiE $€arlidotpria xovrd m1v r4f. Orcv or.r4rdgror o eivnr" l[rxAdE,or rLF€Eror g eivnLourjorr)g€i]io).o I'a ufio].o^/roroivi va exrqrq0oiv. 3. >Iebrdoi€ rq ouvcip'rr1or1 orn IIA?A:TA'ED
31
(a)Trfa,rnEy = !4 { Yn oat4a tI. 1.,13 1.: rrlia ror x. (b) H yaa{!,i raaridoml
11, 1.42(a) Tqi. r\i V = y'; rro oproltrwi r+ta ror r (b) H lpoqgi naac@lol nr: !=lG
.E!='IlL
20 2
(bl
IIAPAAEIIMA
6 Ot r+rtE rI! orlapttotE
f -r s = " f ( . , : ) =j r ' z \l
tr. r.,|,| H oryiradpt I = J(r) no! qatlE' rdr edd i/.r dxehtamei rqroaFolo{aq bL( qoa.lzoi! ror\ de btcqoaErLzcFtatl to!
aar1.
b|vovrtrL nn6 ror4 ti;toug y = r]6rav r < 0, ! =r2 6rcv0S rS 1xcr. = :rcpdmcoq qci.v€rarlJto :X. 1.44-tr 1 6rov r > 1. H yQcrrpr.rrl , IAPAAEIIMA 7 YTro0io-ra6t q ypnqrxrl aaQLiorcol pr,cEorwAQrloiq = cn6 rc qrp METABO ID MI,\'
:YNAPTTDTD
xor ro cr.riororlor€5ia oQrolrorov f+s,f
lrll.
s,S
f ,.l g,Jlr]xo,t
Ariol (A€ir€ ro :X. 1.49.)Tc arbic opr.opori rt|)v I xcr c eivcr. ,/
= [0.6),
D" = ( -, r].
To zoLvrioryeiamv Dy xor"D, eivcr.rc olpeicrou xlelmori 6rconjpcrog [0,r].:ro [0,1] rloupc ror.:rdv,
ciopoLopa I +e: 6r"oqopdE / q: s !: 1 r v o sv,o/ s :
qlixo
l@ = 9trx) 9\x:) .f (t\
l@) + g(r) = rt + v4 - r, J1 x, f(x) !t(r)=rt q(rr-/{J) vT r \r. !r' t.g.rt L'
,/'11 "t.
4' " +t'
1r
!x. 1.49 To nebio ootdroi q< nldQiilots J +9 nr(r I rorrr tuv n€5kd! oaroro! ror I aol t. rc 6rdoirf. 10.11mo! aEo!( r oxo! .,Laoirrrorrcr dud ru $o 6,ori IrG(. To I'doflF,r er6 €i!(r.aioF aar ro r.iio oarolo! nF flldaqota l.9. .irr ro
Ta jr€bio oQ!o[or]rcDv/ +0, g J, J - g xu f.9 eivar.rn i.6r"n,6flc6{ = 0, o aa!0|t6 c; Arioq Bqioxoupetc 1. 1.56a): lr-al0. nlc Uoaqniq fevrzri. iotar D t'v( ouvo).otrn6 n'(i6s< traLErcilv
Mra or dQrrtort/ pe rebio opropol D rivar ptn oXiorl pe tqv o:roic avrrmorTilarcrt6!((YNA}TTDH'
tti@q eveu.t.lna6o^ir
AD = 340 150= 190^. -
I;
qS
Z:
t
e tnota/ryiod
lt)
H Fiot FEta6ol,{ oqv EE. (1) eivor,urlo6 I d'i.ol qE zu0eics rou 6t€{t(EraLo:16 ro oqpelc P(23,150) xar Q(a5,350) ttE tiotljrulrE r},r1Ououoi. (H EU0€iqnou 6r6Qt(sraro16 6io oq.r:ia FroEtto+nnil'rE orqta\prat, rtpvouoa T'19xcprul'r6). H xl,to4 t6 T6pvorrogPQ pnopei vc unoTroyr,orei cn6 rr4 owtetc$dves rcovP tcl q: Lp 340 150 190 , = = = e t'vaoF(rl\l€Q(l. K Lol repvottaae: Lt 45 23 E
vt
>uy,{aivowogq EE. (1) ro! (2) prroQotpEva 6otF.€6r! o p6ooEQUeFoE psra6o),ilErrou unoloyi,ocpeoqv (1) eivor.o i6!og aQreu6E p€ qv rl"iorl rcl p6ooQrueF6 fiou urol,oyioqtr€oq (2)- MrioQorip€n{i'v"rara o€oQo'liF€ $erc6ol,6 rDgrtlv xi'ifll Frostqraouoag.O va Fdoow€ 1600 yQ'fyoQaauEm,6roao tl'llhrqrt Emo)mr 061'oW€ qv 231qr€pa. H atEnonovnl urol,oliEerar on6 qv xlioq qE r+vouodg PQ xagit; ro Q perozrveircl nard ;rr]xoErry xcprril.qg nqos ro P. Ta cnocMopotc yrc r6ooeplg 0ioe6 rol Q pcirovrql oro :x. 1,65, tt 1.65 (a) TamEaG riFrcwE< otl }(nqrai r({dnuol| ro! :X, l.64 uo rc otrrio P(23,150)- (b) o. r?,i@q rov l€oodLsrv
5 rrx) B1r(l
fit:''
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+ = e .no|la/rr#pa
? r5{,
K?.io! tlle RQ = ^!,/^, (45. 3!0) {4r}. t30) (35. ll0) (30. 265)
(dtof@hF€ea)
{330- l50Y{4{r- 23) = ll (3r0- r50y(J5 23)= 15 (265- r50y{$ 23)- 16.4 (b)
Ka06E io Q ni.nord!€rro P, q dpvor.rocPQ rl,qcilriler qv eqqlrropev4,4B oro P. Arn6 sr1roiveror pSoc oro 6Ec qE o;e6iqotk FuE, or, xLloe4 lox relvouoriw nl,lor,rllow r4v xliol l;16 €qornqdvr6, nou xrov,4 rol B aa €ivar urol,oyioope c:r6 rr4 mrvrera11r&€6 350-0 __, jj--15- = I / Ewolo/ r||reQo
I . 6 K \ I : T I : A E Y .I P o a A o M I o N K A I T P I T o BA o \ , 4 I a \K AV n Y \ A \
49
KoodE ro Q nl.ipLri!€rro P, o! UiooLQu0poiotElo|ls o€ arriqoadXaovrzri 6rcortpcro n),Iorriior'\, rlv ti,iol rlg €qqrropivlE rIE rapr rl,rF oro P(17 d\'rotrc/+rdoc).:u!e:rrir;. oeorpoigerqv d.ior1 r$ egcr:rrogtv$ ocv roa al,ulo Fer&ilor.rl.ror ;r).Ior'o$ot T|tv r €ao t = 23. Enol€vo;. 0a p:roporocpe \'s oQiooll(e ro! au0p6 $€ro6o).ylE $Laq r = 11 oov rrlv xl.iol rllq orr(iQrl]tntMETAIO }D MIA' 'YNA?TIDID
H €Eioool rlE €eorrogevlE €[var: :tpefo:
(2,4)
Kl,ioli Elioorotl:
lJ=4a-4.8
Mnoeoqr€ va lploq.ronorrioor4req g60o6orou naQo6si1raros 2 1ucvc opioot4e qv eqarrrop6fll oE onoro6inor€ oqu€ioflE ,'toQa6oL{E y = 12, 6?rrl)E crto snErEvoIlaQd6€q,pa. IIAPA ElrMA 3 oaions flp eqorroF6v4 fl1g '!caa6o)'ig y = I orolo&inot€ onr€io rrE P(,r, y) xar 6Q€ir€rrp d.iol qS. + a:). (r + ,tr):)
Er. r.6t H (oI rE lqromog PQ @iEor E&i)eiYot2. + Ar.
oc
pe g|0 repvouoq PQ no]l Aio4 Oma oro floQdb€rypo 2, EsxrvdqrE br,6qlerara:r6 ro P xcr evo lenovrx6 o4reio Q rls napd6otltis(A€ir€ ro X1. 1.67). fgiqoqe rat6nrv qv €xqpcrorlr4g zl'ioqg qE PQ, xcr naQ,arr|Qodp€ r[ oqr6oivfl o' aunl fl]\, 6z,qeool,.aoG ro q n]'rpldqErro P xard Frkoq qs xa+.rrul,r6. Or owcroypeve< ev6Eoqfr€iouP(c, y) qE roaq6o;.riEg = c2 ei.r,cr (8,r2). T6r€ o! orwr€rayu6v€E €v6Ey€lxovrliori oqpeiouQ qg xcpnril.qg €ivar(' + ^tr, (o + A:x)2).H xLiqnrrc r4.rvolooEPQ €tva! Aq rnr+' =O- =
/--^-rl_.2
-lr'trArr/A,\l_
Ac 2r Ar + lAr:)r = ^;:=zx+^r
T2
(4)
Kar r6Q'a 6.',aotpavrDr6 6iilo: KaodE ro Q ri),rpdqer ro P r{or{i ro A:r lriroE rrE riognt).qg,q r4r{ rou 2r oqv (4) F6verorc0rp{, e'v aEyTEpoBAeMraN KAr rprToBAoMraN KAMITYAoN 51
Er. L6t H x ot t!|E €qaf,qtdvrE o€ aE o|ldo (-,v) dlv e{a6obi v = cz €twr
Atiotl: &'rp4rorrorc p€rcv ono tr6 d.ior6 rn = 2c a:r6 ro lloEi6er.'$a 3 yra la 6Qodp€rlv E€i,oroo4 xtrlo4g-o4pelouy,a xdo€ yQopprl.
fucoop"'vr1oro(-], ])
EqaflrotGq mo (1,1)
(-1,1) >ru€io:
KliaII n =b =2(-t) = -1 E[iunor;:y =-r ]. (1,1) >rp€[o: K;,ior!:m =2(1)=2 EEi.oooI:V=2r-1.
A€i,r€ to :x. 1.68. E
l(tio4 KoFni,t$ 'Orav
ul,oriqnrlLl g = l(o) foeueqowqreul oe evc orpeio 16, t.3tef .ioa qE xoAnlalg o' au6 ro oryelo ei.vcr.q xl,ioq qS sgafio!,6vlE rrls oio i6!o oll.r€io.
y = lti - 32+3 aE Crct IIAPAAEIIMA 5 Bp€ii€ fllv rl,iort rlE tcap:nil,l1g qg g). oy6v oqpri,o P(c, A{oq Esxwd?e6Qioxovragpio 6,{qQdortyld tK r}.ios6 ro)v t€pvouqdv rlolj r!€Qvoivan6 ro P. An' otti65 6Qiorolrt€trlv x),io4 rqEeqa:nop-eqE oro P, I o'Ioia, sEoercFoi, eivar rl nLiot r|E nolrJldl,fi oro P. Eq6oovro o4leto P(c, y) cv{xer or4v xcprdlq, 11ouwxcpivl rou V lxc1,ofior€iqv eElooiJ4U = f - 3x:+ 3. Ol orn"terqydleEtoo P eivcl lourov (5) P(r,03 -3a +3). or ourrfialuivej oroLoud{nors o4r:iou Q q6 r.oFnJ}.rElroaol'v va Yeagow 0i5 (6) Q@+Ll,(a+La)'-3(r+Ar)+3), 6nou Ac I p€ra6o),4qg ouwerc$evr6 c ond r,o P oto Q. !I rLio{ filg PQ eivor r A c ) 3 - 3 ( r + a . c ) + 3 -] l d - 3 c + 3 1 Ay ,'"ra=M= [(c Lc =3c2 +3r(Lc) + (La)2 3 (7)
PYOMO: METABO ID MIA'
:\NAP'TH'H:
Kc0tisQ-Pr6t:Az-0xct ti0
m.e + m&4r = 3r'z + 3r(0) + (0),
0
3 = 3?,
3.
(8)
259
H x).i.orl rrE €qo1ftollfv6 oro P eivcrLq xtrio4 rq,H TrD KAMITY TD g = l(s), nA?Arorol
j.U=x2
53
2.a=2c'-x-1
2.a-3
4'a =12 4x 6.A=22+42+4 8 . A= 6 + 5 x : i x 12 n.s=2 lLy=c'-3, 14.y=fga+3)+1
X.s=4 ,' 5 . s = : x ?- 4 r + 4 9. y = u2+3x +2 1Lu -2t:3 +3zz 12;x+7 1i. 11= t3 - 12r 1 3 . g= t 3 3 x 2 + 4
16, Exrqfi]ore trp, otElol iou r]'qompori rorv €wEnDv rrJv lon ru6eq, oro flcgi6er,"ypc 1. XQrpuonor"4or€ evov xov6va ylo va e.rqrrlo€r€ rtv rl'io4 tIE %ofrn{}'IE tr}.louoFoli oro >X. 1.64 6iov t = 10. 17. a) :x€6rdorE qv raQq6o).t V = l,2 o?t610 orl|r€ia(-2,4), (-1, 1),
(o,o),(i, i) 'iar (1,1). b) fQd14€ rw €Eio(oo4trE €eatrroF6vrEmlv rcpc6oLrl mo P(1,1). c) Yrol'cryiore tq dli.o€q rcov r€lwouodv tiou 6r,6p4owctot6 ro P(1,1)xor ro otp€ia(-2,a),( 1,1),(0,0)rar(],i). d) E0[(lr Ac pr,o Fr,rerl pero6ol"i rou r. Exqgdote qv x)'i.or1 mpq qg rE-r!oooo!, nou driQxsror0?t6ro o4r€io P(1,1) xqr QQ Lz,(1 Ac)') (,)Equvaqr4oq rrE Ar. BQEii€io oQlo :'ng xl'Lor6 mpq xq€6E Al' + 0. Ilor,ri 1 o16o4peraEli qwoli -ro1JoQlourio! rq6 l'iolE rrE €qdritop6vrEqE trsQa6olriEo[o P(1,1); 18. Eoro Q(fi, s) (l / 0) dvoonFtiotrE :.lapnUUS I = "c3 lr' 3. 6t€Qx€rar" o:16-roQ xcl,-ro a) Ba€ir€rrlv rlion trE tipvouoasnou P(0,3). b) Bpeire ro 6arc qE ?iluioqgrn6 reir.vouocgoto (o) t{oed{ to Q r?'lorriler to P. c) fadvr€ Frd €Eisooqyn rqv eqcrropdrl t4g {aF nilqE oro P. rurv rqcm6Ftvurv t4g lrqrJritrrF y = ".r lr | 3 l!1. Bpritt rr9 eEr.odrorrE iou Iloeaasiyt-loroE5 oEzoo€vc!a''6 iq qx67"or0aoq-[€io:
a) (r.2r) b) ( 3.-15) c) (Jr3
Jr|
IlQoyQriFlota Y?toloT]otri Function Evaluator SecantLircs
Super*Grapher
L7
H Klion rnq Kopn0lnqy=f(x).flopqYoYot AEi!€! vq y€vrr€uooutr€rI F60o6o qg lloQqyQdpou 1.6 yra vo oQioollll€ qv xl,iol Frag KFnu)"Ig E = f(r), ntao att6 '1|4 beureQo6oq.rr€E{ot rerio6d0ul€q raun6)"€{ nou f6q e[6ape.
54
PYOMO> METABOATD MIA: >YNAPTTDID
EorcuP(r,y) rdro,o o4reio rrq xcu,n5r'rcI = l(z). Ar. s I Au) rva ol.i.oolt,fio rrF xop:rul'F.rorr
ev Q(r+
s+ As= l(r +Ar). 6'ra\ oro :r. 1.70.Aqon y = l(r), r6re ay=f(r+lz)
/(z).
q rrl.iol fllE rdlvouocgPQ rivcrL: Eno$€va4,
^, , . , .p . ^r /r"" ^.n /r.,i. tl r.70 H aiior rt< €$Ei.< lp4i4ri. PQ ri!(r Jkrlr] trr
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H ilciQcoil orrl,''EE. (1) eiror crnLdgerdrrxrrzl 6r 0. :r,v€;r(i) {), UrioQouu€ va uroodoo4r€6rr d > 0 orlv EE. (9), dror.d]om vcr lpdqor+e ro\r iroQo\rotcmit rits (9) oav
a x = ('vc+rtl(ra
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1.8
'AItror T0XUrnro Kor Pu0goi MeroBotrr;g Orav €va ocirt{arLlsi.ra! oE Fro €!0€ia lpc$tl]. €XEr.€rr.zpar4o€r!(r jraprcrdrolF€ rI yQapprtitc liivlols tr' dvav dEova oNr€rrryF€v0)v.A|o l6yo!$€ 6vd olpei.o 1Lo <rplll. pic oerrxrl xcret0rvorl rcu. pia govir6cr yru rrv c;roorco4nav,r mI y0ou n. nrpryQaqouU, rlv xrvtot Xalal ouvdArFl ror l. bllabt rou otlvotrrro{ IQ6vou ;rou 61eLmpdoeL a]16 I oiry[ri nor ciAXLo€ I xi\'loyl. H orNr€ro1,Ffvls ovogdl:rar, 0€or1rou o6$aros oro Xa6vol. Mic farc6o).{ qq ouvrernlgivl; oeorF avdUeoa oe 6uo lpovr.riq ('rl,1!6g6ivEr rI Fercr6rLol rj rqv xnoopri cn6ottro4 nou €1€rbLaliJo€rro odrtto xorri to lporrx6 cUr6 6r{iorlFo- O },6yoc rlq $ercr6rro4q rrQoErov XQ6vorlE xir"lolg €i\'aLI Ftq raxurlro rot o(hForo METABo ID MrAr >yNAprrDr{t
M6oq Tolirqra H ptorl myut4ru ev6g oci2rcrog nou nvelrol xtrtd pr]rog 1.rcg €uo€isE€[vo!
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usr0r6nron A3 ol.rxoEXa|ovog At
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fra vo ttdQoqr q mrypnia ra1iqra t rou xLvodgevor orlrcrog ov Xa&o t, { 6,f orIdE ovopaloqe ralw4m r:tov XQ6vot, rrqiQ 'oW€ ro 6Q!o r(l)v F6o(l)vrolvrrFov, ro06E to Af r€iv€l nQogro FrDtv: n"
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/Rr -'
ai
ME d l,o Mylc! oQiqoqre ro l ocv qv napriy,ryo t|E owdeqo|E s = t(t), (DEnaoq ro t@6vo t, 0\ €ErE.
trrTFroa@ Ta?dJfllta H otqpnn raxifilra Et6< orilrorog (nou nr,vei,roLecftri prixog glo5 el0eiag) o€ rdO€ on'fpri ,, €ivar I ,raQdytoToEt4E ouvretcrypev4E06oedrgrou s = J(t) rognaos rov rpovo t:
"=fi=rat.
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t1. r,87 Tolir't1c aG tonllou mp(nbu.
t )'ertd cq' 6rou ard eX€!oQ,Xio€rv' o6€rdq€r, x.j,t"Q = 200(30- t)', n6oo 1p{yopc qedyer.ro v€Q6[erd ro 10 na6ro ].€nxd;flord €ivqr I pio? raXUflFa F€ rrlv orolo o6Eid(€r ro v€p6 xcrri ra 10 agrirc lE ird; 23. a) Edv q oxdvo Ev6gxrid.ou pero6titr")ueroori6 r o€ (r + Ar), rold eivcr 11piol Feto6o),ri r$ enr.qdver,cgrou r{ddolJ o€ ox6ol FE rrlv b) Boriff 14 oT|yu'aio UFra6oI)i ryts rrredl]fros oc olior1 gs r4v
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24, O 6yxc4 V Vi,agaqsiQqg axrivoE r €i,vor y = lrr3. Bqeire rov qu0p6 ;retc6ohrlg ror V og nqog ro r. 25. H oLn7i",'/(t r xor ro trlog lz evoE x 0 lliqo (|n6 ro t. unoQlrr fia q).rivo , > 0 yig,r (116 ro c, r6rorc (i)fi€. yKr zae€ t. I 0< l-.
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I 4
(t (t
2)(t2+2t+4) 2)(t +2)
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: =l
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I
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2+2 fra vc unolclioouFe cmo ro 6pro,8rorptocperov cpr.egrpl xcr rov zopovopcor{ p' evcv rorv6 rqpd^/ol,rc xo! rno}.o{oope ro c:rod}.topc yr,at =2. E :ro noa{i6€rT-ro11, 6rdqdiverqt pi,a olpowl,xrl pcelpctrxrl 6won yttt to 6Qts: To 6pro prog orrvd$rlorc J(t) rd06E t + c, 6€v €Eoarri.rcr rrord drr6 ro 'r[ qup6aiv€r 6iov i = c. To 6ero, (ECr\uftte:xeL)xaqoateErar drotalErnrxd dr6 rlE flp6E rIE J 6rav J I c. :.ro nded6€rypa 11, ro r$.ixo /(t) = (t3 ql(t, - 4\ 6ev opi(ercu.rov y|,0 t = 2. OFoE, ro 6A16rou, ro06q J - 2, ur(iQt€! rcr! LooiroLpe 3. To 6arc ]itrooQiornxs onoli}.€loftxcicr:r6TlgTW€gTorql.irou, 6ra\r f 12. TEL€|,(ijvoup€amrt rlv 'raQdyQqqo F' 6ro 0e6prpa rou 0c 1pr1o4rorotlo€i, €trov€ .lpt€r(,E on1 owelerc.
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13
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h
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-:-:
4. ttrr'
5. tim _:r
9. lin -
r0. Limlrl
. Im lzl 13. litrr 15. lim
12. Li]ll :
tal l a l +I 3x3+5x2-7
14. lim -
,
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x. 8x23_7r.2+5
\
}
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r-6J.
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=
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r
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- Jxr + Jx;
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1.10 TO AIEPO:AN OPIO 95
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:+ ri- I
3&.4;
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-
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edv ,lim J(o) 2r-3 .. r- METABOAH> MIA: :YNAPTH'II:
ArqeoplorUeq :uvoprnoerq Eao arr6 r(r nQoi€QriForo r(,v 6ta9oQloil(Dv ouvaario€ov (owo@rlo€ov rivar 6tr etr,crL nou ixouv trsQaydryous) ol,rg ouvr'pig.
Mia owd€qq eivcl ouveXrlgo€ zdO€oryeio mo onoio i(€L zoadyoyo. Aur6 otgci.ver dr €dv I v = J(r) 61ernapclX. 1.1('. (tnlv fiqedyeoqo 2.8 0o e6pouiuoqret1 6.qpoetoqr6r|Faio! qlt6vou xdr rou or'vr}r,t6vou1a rioe qri ro! e, xr €-rqri{Dg, qn6 io r4orlyo'qr6ro e€d'e4ro 7, t{ orv4s i rolE lna xioe qrli rou o.)
Etiv 1 I eivor ovve1footo c xcr 1 9 owe;r1Eoto J(c), rcre 1 orwo€orlg(t(r)) stvar o|JvEt(rE olo c.
&. Llt
Or o'ot@'E olErd' owocti@
8, 6€ir€ ro fta Evq n€QilQdFtroolE on66a,ErlErou Os(oQ'|iForog IIe66l'Ipq 6 dlo UaQdQqpa3. IIAPAAEIIMA
12 Aellte 6rr.r; owrisqoq
'
lr sin. I
l* +21
eirar ovvelig 1lc lrri0erqli rol c. A6oq E ol,vdqqoq I etaal rl ofv0€oqr|f orrreldv owcgr{oerrrv
'ror
9(s)=lcl.
an6 io Oedpqpo 6, 1 owdqrrloq g(c) H orrvdqrrq /(r) €lvcr ol',.,/E](ig ar6 ro IloQd6e!1|rq10,rial I orivoEorirouE0116io @s6elpa 8. tr Etiv Fia otvoeq o|deqot gf(c)) eivcr, orvelft o' s/a ort-t.lo r =c, ro 6e[n qS orov a - c eivor,ro g(J(c)), IIAPAAEITMA 13 { 14 siaG-T =siny'i - r =sino =o =2 tr b) lim 11+ cosol =11+cosol=11+:11
104
PYOMO' METASO,\IITMI ''YNA?TTDTD
I4povrLr6ql6r6rqreg OproUdveg
:uvex6v:uvoptoeov KleroroAroornUqro oe Opqyu6vo M€I€rci[€ rrE oweldg ouvcpr4oerg6mrr 6xow oQrouev€g xprjorFrgr6t6qrcg or onoicgpropti !a Fnv undptor.', o€ d)..l.tSol'voarllorlg. Tc er6perc 6uo 0€(oenpcro n€erypdqow rg rbr6npeg :rotr o:rorc 6ri:rore owript4ot 0n 6xsr edv eiv&r orwex]iq or: dva xL€Lot6 eQa €\,o 6rdor4Fa. To O€dQIpa 9 l,!er 6tr pic oUltiqrlo4 q oroia €ivat ouvetlig o' &o ?iL€rorobLq('qFn [o,l] :rcipver pic p61!oq xar Fia €]'dtloflt xryn fio ordoqpa s!.r6. Ildl'[X.1.103 vo €rovoQro€€ieto! 6mE n 6l,v{i€,trpq vo yiv€! ow€xrlE; 27. florri tr,Fri 0c 6nqenevc 6ciroor4reoro a ylo va 1ivfl n owaQrlon f,z I J\3, .r(").t;,,,. ,.r.
ouv€rk a[o r =3; q ovvdQfllon 2N. tlorri tryri €c drgeti€1/a66ooul.tEo[o 6 Yrovo Yi.1'€t ( i
!.).
s{")= \i"1,,,zt, ouvqgigoror=]; 29. H owtipTrpa J(c) = (tz - 16)l z 416ev €ivo! ouv€xts oto r =4. Aigeror, q couvqcq Etiv vcr., tor,ri 0o etrQ6ti€la eivat q fllt{ rqE
108
PYOMO' METABO }D MIA:'1'1\IAPT1I'H'
l(4): Edv 6xr, 11,(ri,i(Y deEry:\no't,oti,ar€ ro 6€€16xcL cQror€Q6 6pro -rilg/ oro lD=4.) 30' H ou\rciaqor /(o) = sir(l/ir) 6€\,€ivor orv€xlg oro :r = 0. Aiper(r q crow66tr: Etiv vcr",;rod t$rl Otr 6nqt;tc vo ntipu n /(0); Etiv 6Xr, ^/[ari; (Aeir€ ro :X. 1.92y[o rq ^/Qoq[rrifioQdorcon.) 31. Ie norti oipein eivc! yt ouvdarlol 9 = 1/[r] aorly€IriE; BpEirEro oEa ora IlQo6l.riFqrd32 34-
*' trl ,
r:. ,-0 ri- *, L \
fII) f
./
34. 35. 110!6€ilor xo F6ylorornE y = f yro :1 < ir < 1;To €Ldlloro; fra nor6ET+rfqrou r yl ouv.iarlol rox:1. 1.103rai.av€Lq Fayrml rr+rl r1s; Icipver I ouvdairor €).Arroq i+rt; EEryrior€. Elfi tl orv(iQtnol y = ir'z lri,o Ffudtl] xqrri oto dvorfi6 6tdfirttc -1 < r < 1: Mio €ldlrorit rrl{; EEl^/lloxe. 3E.xro zLr:ror66rcio4rc 0 ! r ! 1, r1ouvripqoq s = [r] raiav€r tria e).ci1"m4rqr{ m = 0 rm gio pdyrorrl,41= 1. po}.ov6rr.1orvciprrpq eivcr ao]]l,€IliEmo r = 1. Avcreei cr!t6, to ortirelpc 9; Arzcroloyrlorr: rry cndr, l(t oog. 10 \aia ouvFlt- ouvdgrrpl a = /(r) riv.{ .lvuoro orr eivoL oqvqrrzl oro r =0 zar eenx{ oro z =1. rlari yl €EiorDorl(r) =0 €r€,. rorl.cilwov picr Lrio4 ovcipeoo orc ;r = 0 zcr" z = L Efryiore p' erc oL01Qc44ra. 40. Ytro06ro1,loEoxl 1 y = 6651 tiror" ouve l;, 6eilte 6I[ q elioorot cosr = r dler rori).dxrorov Ui.a ),uot. (y/dr,€?i A€i,Er€6rr n oril'darnol l(r) = cosf; r dter rouLdlrno! Fio riQo.rlroxi i QiEc.) 4t Aeilre 6rr xciee nol,utirvupo:rrprtrou 6c0po! dlsr roxl'.dxroro! [i.r nportalnn pila. -- lrcl €i,!or oljvel{E oro r = 0. Ao06vrog €v6E H owclpxrtorl/t o€rrrou aar0Folj €, n6oo FLrQon06n€r vo si.r,a!ro b 6r('r (iorE I z 0 < 6rc ouvendleraL 6rL l(r) 0 EPOTTDEE KAI A'KTDEE
IO9
6oto c, < :r < c+6, oro onoioI J(r) raaap€v€r0eroiil.Elrrytiot€ p' evc 6uiypalrpc.(Yzri6afa: n6pr.ec = f(c)/2.1
IlQoleErycrc Yno]oTwni Continuity at a Point Limit Problems
Supcr*Grapher
ENANAAHNTIKE' EPOTH>EI> KAIA'KH>EI>
!r.
LI2
O, .toEies o!
Oao6hfuorog2.
1. Ogiore ri anlto;i\erxtriol BLdSeuo€iagTQ,aHr . ndE 0o 6QiarclrE rn! lilloq prqs Eoe€idE sri6 q lpcqla{'rIE r[qQdmdol;4.r[6 rqv qg; e$oroo{ 2. >@ 4. 1.112or,e$e.itg L1,Ia xol, L3 t1ow xl\Loec rnt,nl2 1(dL 7n3 Clniotorr(c. Ilorri xl,iorl rivcr cl,1efuwd UrxQ,6rrQq; MeycldffQyl; fQ{ilns ti6 rQ€6 |,i},loer6xcrrl orllouoo oerQdp€ ra orrF6ol,a ql16" xc!! "F€yd,w€Ql an6" crv4.r€odrou!. "pr,xQor€Q,I 3. rl€Qtya&|,Te t{v obroyeveuc urw euoerdwg - y =n(t - r1)l 'ygoppeE a) Eriv (u1,g1) stvar ora0eQ6riar OragoqenxeE oXt6rri(o!1or.Yra 6[oqoQar[xaE 1q#s rolJ m, ,ior b) Ed! ia m xaL :xt etrat oroOep'dtcar 8lcaoqerr.r"egyqc|.lpdq q(€6uilowa[ yla 6rcrgoQ€rDriE 4a6 iou u. 4, Oeopti< \uvtier4aq(. Ti €ivar ro r€dlo o(,roporiprog owripc4or;g: 'fo zedio uttciv tlrtE:, 5. To ne6io ogloporipr,cEowriptrpqE €ivcr!ro odvolo 0 < r < 2. To g = 1. re6io qx6v tr6 ouvriprqoqE€ivar o aQr€poE a) >x€6rdoE xar rEor,yqdqrern leagrxi ncpdoraol rrE owd'Q/rlqnE. b) led!rc pra €1{qeaolyra rn o!vdQ"{ol. c) Edv (l).ldEol+"Er4 060I r(ov dE6ft)v oro (a) pripog,I v6c 1poqr,xriraEioraoq rou roi('vo4r€ rct€nordvtrxdurorcouvripoloq; EErffrloft. g(z) = 1v6, nortieivarrcrqruorrid 6. Brr.vJ@) =l/a 11s1 rr66iqoQqrori (g(z)) $l f , g, J - s, f .s, f / g, g/J tcr, row ouvOloetrw xol g$ (r)), f Ilouo eivcr.ro r-e0i.ooEoFoi T|E ,r(4) = 9(c + 4); 7. fQdqrE evavruno y = J(e) 1o rq owdp,trpq t nou €i\lar q ol}vo€orl rrov g = lr - 10 ar.otlouOoip€vnari6 qv I = y'1 - c-. BQehem peyd,weqo 6wcr6 m6io opropor!flE J xar ro avrigrorrlo nE6io qro! rrE Y. t Optoptig. Pr1roEcq0Fog etvcL cutoE ttox FnoQEiva cxgQ,aotci orl FoAqrip/9. 6^ou rc p xcr g slvqr cripcror. F€ Fryro(oxorv6 6ro[Qiq ro I filr q > 0. EvaE ]rQc1roiD{oEqeropoGnou &v €i1,orarFoE, a|,qld\tuor" d4p|toE. EXowaEarfiorl rorJEopr,opotg un' orprv. qle6uiorc xriworc oryric orn yaaqD i ,roadmaol trE
110
PYOMO: METABOAID
MIA.: 'YNAPTH:H:
ouvdeqoE rou aftlxoviqer rov arF6 (IAroFo 2/q (oov ardy(')yo rt}ldo1a) orov 1/q xcr, nnerxoviler xdOe rip'p4ro crBr0y6oro p46&. Aunt I owdaqot EX€!rcoryQoe€i aw I ouvaaqat rot xav6rd, FEorriog rIS otrororlrac ryE youqrz'tEqE noQdota.{ Ft ro o+r€io ev6Eowloropdvou xav6vd nou p€TQd ivro€E,p!o6Ei.1,roeg, r6raQia. 6y6oo 1|cr 66xffia 6xra p€ eoe€itE6nqoe€1D{o{pllxorie mui lpovrm{ onlpri 1 g:uinc 0o 6(ioi8r@, orc psyql,ft€aoulioE fiE ,{a! n6oo rl-qld 0a av€6€r; 37. E,6eq niqoq,P , ,au o 61xog,I/, evogaegiouow64owcr pe tq o14c4 P = 1/y, fo€ir€ (a) tov F.€ooenqrc g-eta6olitEioo P orgrqog to I/ oto Odorrrrc o:16P = Pr, d,SP = 4, (b) asv e!0p6 Fs-ro6ohkrox P og n4ngro V, xcni tq
[email protected] on11n4nou y =2. 3t O 6y.og y(int) t-s roodtrpag r,eqorlrou naedpfvel o' sva rQUrrytvo 6o1eio VJn&.1116 t sec eivo V = 21fr - qt +0.2P. fi6ao teqyoaa F€!6rerct o oynoE6lav t = 30; YnoLoTlorc1(r prr4rsva6Q!ai 6el,&€6o 6€vrrrd€t(orrv. c=+sY
39. tirn
.^,.
4L [n :==
4L liuJr--:-
t"Tt
43. [,ls.lim -
1+ sio,
44 |in :-:-
-
46. lim
(a +h)2 - *
h 1/(.+^s)-7lt
4'.M@+-'/r
,18. litn
49.rm 1
so.,rT1+
5L rim
(2e-3)x. 2.6). Ot" yQo+pr.ndg rivcr o4pcvtrxtEorig quoLxdE enonjpt4lcti pcE nQooe,/iorr< ennp6now vc cvnxcrconloor,pe rol,rlnl.oxorg rrholE pe cnl,origoe 6vc Or,riotrlpcpe xrinoro pr.xpooqd\rc. @cfl{ [cvd6oq€ otl5 naaayadeolg 2-5 ,td,L2-9.
2.3 TNoMENA. ayNAMrD KAI IIHATKA
139
NPOBAHMAT Bpei:cerq dyI da or.aIleo6lr|lr,crro1-24.
t
t.a=\-i+r
2.a=@
1)3@ +2\a
4.y = (23- tala 2 x+ 1 -
3.y=(r2+1)5 5.y = (x:+1)2(12+ 1)-3 2t+5 3i -2
s.y=(.r..-,)
9. s = (1 r)(1+ a2)-r
+2r) 2 1:o.a=@+1.)2(o2
I]'a=
12.a=@-1.)3(E+2)
3y
e,
13.y =(s r)$ -2c)
14.a=[6-a)g-2x)]2
E,y = (2r
7)3(E+7)'3
16. a =!-:-:
3!x2+ 6r)-5
r&.s=;1'
-2
17.u = (2r3 ("
1)2
79.s=
2o.a=
1
zr.a=9\*1 (a-1)(*+x+1)
23.a=
(,'
t)
u.o=@:o$::lt BQ€ir€ rK ds/dl oto Ilao61.li[!.Iao.25-32
25,"=2 2
26.s=(2t+3)3 A.s=t2(t+r) 1
zo."=1fr
36.5=(I+f 1),
n.s=(t2-t)
3Ls=(tr+3r)3 32.s = (t2,7t)(5 - 2t3+t4)lf 33. Yao06ne 6tr or, z xcl v eivct. owcp,t{oer6 tou j' fioo €ivo! 6toqoQio$€g oro r =0, {a[ 6iL
u(0)= s,
/(0) =-3,
r,(o)= r,
l(o) =2..
(Or r6vor 6€ixvouv6lo.q6AtoloE treoEu.) BQ€irE rr5 rqdE tcDl riclQ'at dx@ tlaQdycby(lrv orc t = 0.
,)*(:) a [at ul $e'-r,a .)fl@1
d*(:)
r*dyNAMEr:
141
a) BQrir€ rov Quolroqg (Ir.ri6aao|]5, dP/dt. b) Batir€ ror tQolo xard rov o:roio q nwi6gcrol irpoT.topX. 2.8 €ivor 4 lpnq rxq :rupriorcol xlrorqc owdprlol< 37= l(r) nou lxovorroL€irrtv /(x0) = 93 ndtvo o€ 6vq cvor7r6 tudroqpa (a.b) nox n€aLa1€r ro r'0. Edv ro.r eivcL 6vq ororoolrror€ o4$€io oro
142
nAPArarol
(r'g) rxol'onorrirlv qglrxrl €Eiou'ol (o.b). rorr ro lrwoq (r.Jtr)l iutt€oo,oto Fi.. ,L = o. ,rttr,r O" rt rEio.,rot Ftr. gt = tr ixrr opiorl fllv / (|)< = y dva^ulrxd ) qv bivorros J{J / (n,l,1icxoFc nr av brv tXFrooiofr nQoEro t:.
Er. 1? Ot x).ictl! td eq"*q'Y' T'r" y'z = r o€ arydd ox$e{4 tqs raoo6o\ *ar adtu qr6 io t 6[vo{d oto ro! rv'ro
qt elid'olc F4S e6tu_ 11. 29 H Yeagrx,i tue<jLmool !r. 2.8 H raoqrart rceiddol =t:+y: 4 =0rilor o tlarog = 0 Fir'V) F r r , ! ) g -* tE t -" loe(crE -r Erau proosi w tqv .i!qL I tacqLxl rdpo@aor) f,I +y: = 4 tou cdivlro' E6ri To adrato' Drrlqrro i8 tira' 4 taaqrq no{odool :. MeaB.q 10! @defllo$ Ftq ralo ro rd",'rPuwlo ',6 s =;tr1= f]T etq€4 feoF+€g Foaei va irP tFlow arE 3 = qr6 |lia qoed Tfirrala iE rcoqLa$ 'dpo' ,48 €iwL 1 feoqFrl rdadqod! do dro ro A 16Eo doortg, 6$4. 6&\ to s@ = -t6:j. B. ploa€i ,a 0tu440oi{ odv oI Yaaqrrlg tou t. raadqdoe4 doQild.o!
r' x? + nAPAAEIrMA t H lpctprxilropriorooqqc owtiQrrpt]EF( s) naQooraol u? 4 ritqr o r.uAo0. tE d0€ €Ei,ourol, to oprwq6 F€l.oE ei,vor raut6oqm pe rr; orceeEi owdprqo4 0. QE sx rourou, I napd\adyoE tou aprmeqori gtq,or-rg16 €qiqoor6 Elvar I noEilor'los rou prpevog xar €nove4ioxop€ rrry ard.il sEio(doq d., - d
Ar(!'
x\
i-(0)
dfi6 tI! orloid unol,oli.lolte r\v d,a/fu. IAPAAEIIMA 3 Bqeke rly,vd,s/ dz *'r cg = 1. A60I Avrd sl.1/or6vs dl-l,o roQdo€tltro oro ofioio g:roporipevc ),tlooqre r[QriticrorErQos y ri vs pr1liooup€,6rtr4 npoapdpe. Ecv 6e10orpe6rr ro g, Elval DragoQioqrrlowriq,Trpqrcu c Xr4ig va lrioow€ ylc ro y, ral 6lsqoQiooqre,iaLta trro Fip{ qg sEi.ooofi ey = 1 lr^rAoopt:H l€ 6l,0 ta orp€lo 6nou a lo, ard i|{ EE. (9) zai (10)raievoF€ ta or61,oljea (1lo)
-a s,=
2x - (a')2 2a-(ra/y21 =
u
=
a
(116)
16
nAPArerol
MttoQ6oqle 6r1],c6{ v 0, r6re
=!d,n i!. lupn dtt dt
(14)
Q
urd tov 6Qoon z l0 e&\p/q
149
Eno$!,(l)q,€dv v 10. sj(oqr€ d! - puP I d.u dE Q l q - t d . s '
(15)
AIId ye-r = (u1/e\c-t =up {e/d,
@=y dr
ue I du
q W-lp/e) d:r
=?,,qtot-'4! q
dr'
ro onoi,osrodslxvt€!fllv EE.(14). I O :rrproprop! y / 0 oqv EE. (15) €ivaro ibrosF€ror, ,rrQroQroF6 u I u dMc €-rcorlXueig vo (rvap€Qeri eav $ p/q irdv tt bFv ltav Frx06r€Qoon6 ro dro. O negr.opropoE 6ev 16reuilctcr tw io p/q = 1. agoutote t(oFEg = u xadgld-c ,duldr H e6rx{ te6irtoo4 y =2312eivcr taQromlQrorrxriqE nepimoro4g y =ue/c,
p/c > t,
xc0 oro, cr6l,ou0o ftado€lwo 0c 6eiloqre 6Il q x)jo4 rrF rst'lJ[u],1E €ivol0dtave=3/=0. IIAPAAEIIMA AETT. liorl\n
9. EE&dms r4v xlLio4 r4g ridp $tr6 V = ri/2 axy1
ro y = a3/2,r, EE.(14)6[vsr d.u 3 ,,,
3 ;E=;," ;t'
2.r2 H yeoqufll rqadoraq q€ g = o!/'?. H rllon aro t = 0 €thL lim no' = 0.
rou €ivcl [q6d.r 6rav ro c = 0. A]"\d I tQaqlxri ropdmcq tqg y = z3l, orcFarq€L .ulo_toFq oro c = 0 (XX. 2.12). €vri ol acqriyoyor. 6ro4 tg unorLUcrarrrursunololilowaL L|lS6inL€uQaoQta. E€Qoqr8. Mno€oqr€. 6pu4,_vco{rioovgcr4v .iorl oro (0.0) urs to 6Qto r(])v ., x^torov ruv rqrvouodv norr nrQvo!! an6 ro (0.0J xal cn6 ivc o4peio P(c,y) trE rio$nnhE, oro 6€Ed xo! (0,0) (Aeire ro 21. Z.l2). H ;iol ro! OP €i.vcr!
--
rErI4tl{v4rou P reraylrEvq rou .P
j,r- 0 c_0
Kd0{6Eio P - (0,0) 6xoqr€ n1o= Lm m = lim r1/2 =0. .rt !
oe oqrgorvic pero nporyofuevo cnordl,eopcoad tqv El. (1a). flE g =.rr,2cnox = 0 zsr _ - Oa ovol(iooqr€aur6to dEo r4v rapdyrrryo ua Aql"Fott I .y0oeu.4ncqdorcoq qeL exri prc o€|rt,olrra€qarrtoudr,rl O
150
lrApAterol
Arqeoptoruornrq o€ AKpqio>nUe(o ArocrnUoroq To ar.6 6EELd6Qlo,nou uroloyi,oape oio naQdo€LyFa9 €iva! pro €tdlroi r€Qrjw(l)orl rou oQrolr
lim
J@+^r)-l@) L,N
ro onoio ovqrriloqre o:16 tc 6elrto IIQo6MForo 36-39, 6pzLrero dg/ dx xat, pr"6.rc &g I d,a2ps 6;nq6Qrorl.
36.*+f=1
37.; +f =t
S. */3 +f/3 =1 $.cy+g2=1 ,|{1. H owdqqol oeorErvft oolrcr6lou nou nuvel,tcroc ptc euouyqq4fl1 tQolrd 8ivstat an6 E4vs(t) = y'l + 41,6nouro s rivcr oeg6tqoxcu to t o€ s€c.BQ€lterlv rsxti'Fc rou q'trfiloiou xcr,T|Jvenr,rrilwoi iou 6'icv t =6 s€c.
152
IAPArorot
priqoErrl xlv€iio! rcrd Fri]ioErou dEovorr.)! r. H 41. Evo ocr.rpcribr.o roturnrd ro! | = dir/dl ncrLq eeoq;r, r.xcvo:roLo v t\r sE|ooon
n,(,r,,i)=r(d r2). 6:roul'.rl ncr::o eivarmn0sp69Aeilre, pe6y1':olbraq6Arotauniq tYE€EioootlE(E nQ,og t, jrq 6rqv ro r 10,
",4
=-*t.
:ra nQo6l,ytFqxq 42-48,6qri,rcrq ypo[pi,