8-ზერეკიძე-თეორიის კონსპექტი

Tamaz zerekiZis konspeqti

saganSi

"maTematika ekonomistebisaTvis" eleqtronul versias (𝑃𝐷𝐹 − 𝐴𝑐𝑟𝑜𝑏𝑎𝑡 𝑅𝑒𝑎𝑑𝑒𝑟 da misTanebisaTvis) damatebuli aqvs

"eleqtronuli sarCevi".

konspeqti Zalze SekumSulia da Seicavs aucilebel masalas (orive semestrisaTvis).

amas garda is arasrulicaa—ar Seicavs analizuri geometriis sakiTxebs (pirveli semestris masalis nawili).

leqciebze masala gadmocemuli iqneba silabusis TanmimdevrobiT, rac zedmiwevniT ar daemTxveva konpeqtisas—saWiroa saleqcio Canawerebis regularulad keTeba keTeba da masalis Seswavla am CanawerebiT, konspeqtiT da im literaturiT, romelic silabusSia miTiTebuli.

maTTan erTan sasargeblo iqneba, paralelurad, sxva kolegebis mier Sedgenili konspeqtebiT paralelurad sargebloba, iseve rogorc nebismieri wigni umaRles maTematikaSi, TqvenTvis misawvdom nebismier enaze. uamrav sasargeblo masalaa internetSi.

praqtikul mecadineobebzec SeiZleba Teoriuli masalis Sesaxebac kiTxvebis dasma, Tu isini maSin mimdinare praqtikul sakiTxebTanaa uSualod kavSirSi—pedagogebi gipasuxeben, Tu dro amis saSualebas miscems maT, praqtikumis msvlelobisas. saerTod, garkveuli momentidan, romelic gecnobebaT, regularulad Catardeba konsultaciebi; maSin morige profesors dausvamT nebismier SekiTxvas.

(satitulo gverdi da eleqtronuli faili moamzada niko guniam)

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e= Jim (1+~ r /)-1OCJ I"

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j !ftn(!"m2;.J (?531f.JfJ LJ'~.)-

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{l]Jl1JhJJJ· otJ Jhu2:rrJJ6 lim !(X) (OJ lim grx)] J/2f16 o.Y>uJ3J<::.u 11m (f(X)+!d(x)) ~ ::c"'a X-.>CL X~Q

lim (f(X)-f-/J{:x)) = JIm f(x)+-lim J(x)X--+ G. x--- a X----CL

rJJJmhJJJ. !Jt:J JALJ2xn2JJ6 lim r (x) ~J lim fl{x), J.J2rt6 flJMJ3i3L II-rn (Iex)- J (x)) c:'0 X--+Q.. X-.>Q. r...,a

lim (rex)· J (-X) ) =- lirn f ('x J- krn J ('x)Z-.>~ x...,~ x~a

(JJ~mIJJJJ. (/)'J JhLJ6tn2u6 hrn f{xJ ~J lim !J(x) , .;J;LtJJ..;6 limJ('X):t-01JJ'dnb ~ Jim {-CX) X-.>-Q. X-.>{L X-->Q.. x..; a d(X)

lim f(x) _ !~~ I(x) x~a J(x) - lim j{X)'

X~l'L

m2rnh:JJJ. 07':1 Jhf,Jdn0J5 Jim rex) tE.5 Jim JtJ:) c:'J a·t lutJoJ JfI(!J.JJm'CJn Lf")~(!'i3J j~-xva.. :t.--+Q.

6J f (x) ~ 2{:xJ, J./JM lim f{x) ~ km J (X). J:::---->Q. .r~fl

(J1.JmhJJJ - orJ J,!)!J}0n-0:J1J IJ-m [(X) (?.J lifT] j{x) ? ni.Jn5f1 /!f7JC00fl .JfrlJ6-] Dm(!'Yn Ii (x) flf,XMx ... Q. )C->-(1

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dn3!?13nrr'0nu .2JJnL IfJ f}?1YJL '3J!.JJQi3 rfl]rn.t) 'JJtJiJ 07 JD.;bJJ CP

lim (f (Xn) T !JCX/I)) =ft"m f{Xn)+ lim J t:cn), /]'-'00 /r4co n--.oo

l·n· Jim (f(Xn)+j{Xn)) =A+-8· c:J'JfrliJiYJL fbcJhr{, 3JOUJroC:YJJonfJ rJiJOJ!::dJt? j() jD M.JU En2w?:0. fJ~c<J '

h()l J lim (+{X)1-,9{x'J) -= A+B - An/; (!JJd!fJjn!.lJ6JLi cnnoC?rncoJ-X~O-

':J iJ J !Jh 'J C!"m(3 J IIM j ilJJ 5j if()J . }(X) r::nfjj lr'tJ.J /; Jf)i/i(!Jj (y J 'J lJ J U-oc:!'m i5' Jif/1 nJ !Jmu'J :i -t- a 1 C1:J

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J1IM:J:iJ/JJ 1 hrnl!J x ~ (1 ) 'JJa3/kk:/1J fPJ3 Ii.J/lln (J)

lim (tCx}RC:JJ) =Iirn (:((x)), G(x){;CX)1= lim [(x) ./im{Jrx)R{x))::= lim 14{xJ-!?&J).:r.~CL x-.;>ti d X d ') J:--nl- (] (;c) X-?Q. X-i'c..ld

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X-+O

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r::J 'j[):JiJn nJ., ;;[)rrd J 5J(!"rri2J.Ji: a fiJ/Jt!()'(:!n'JrJ 1 J. n.

;(0.-)= !(at-) +- f{a).

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d.J df1 5 {J:J;jif.J'Y(!!'11 "Y:J5 j 1J'T1J ':JS:iJ.)tf'n.J tJJJ3rJ!.J d.fbu./b f!lYvlL .fn:; fJrJ .

JJR3:J5J~f1)5n '"!3'J5:fllnJ _ f{x) =aX ((l>O , a,::pi, XcRJ °3'~5jJ.ftJJ ~5JJJt!hJ (/)JYV./)

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!Mfb6nm on6.Ju/f'Jn 3j~3:;fJmtfJ, hind

lima,x=l en ::r:~ 0

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lim Q x= lim (l-t =lim ~::: i .= _1 =i. X~- i~O-f. -t...,o... o..,t /"m at 1

t->Ot

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~ 3n.'JJ5i307 ~j3.J (PJJ!:PJnti'Jd':JfI!' f).;[;l1{!T'iJ, J3J.i5.JaJ

' x J. iii/f m a .= 1m ----x. .= -.--j: = --1- ::: i .

X--.>O X~O tJ Ilm£ X-+o

~JnOl (;) iJ~m~,J ~JJtP.3()JfJa:Jt:'nJ. J(l1./J JJdtniJf)i2YJOJ Of))(JC:J"J{f)

aXoI· fiX I' (aXoaX-~) /. !l X-Xo ",xo /?xo ,m l)...- = ,m . ==' 1m ()....- .= U .j..:: v... . X~L X~L X4~

l!..i1fT1ZJjhnOl3-:J~1l '!3'J5j{j()J' jrX)= I1ja.x (a/o, a.t-i I X/D) o.J':J5:JdN 'J[):;lJtff1J rfuYiU 3.J5

U.J ror:' _~iJ J/JJ{J() ,llmJrrYl!.f JJt3JOJoe.nru5f1 OJ':J02L!IlI1U dJJ1IJ'j(!fYJ t?J'15Jl'-'n,J·

!JJI; n/;bflJ 3.5 5n~"61i1IlJ. DJ!v0btn3J!;n fT)':}[)jI/I1J 5Jtf'JhJenY>11 J.Jt3.JOYJf.'.:"() OJ: ! (x,) = ::: X n (X (; R 1 nt j() 3'J5J/!IlJ ":!r;:;?;J/fJIJJ dmJcnIJ OJ.J3nu dJ {)UJtb(!'3M/.; JAJ(J() !JrnJfT'IJlJl Y =. X

UJ bf) 0 J D::J 3j tP f9 'J5j Iif)J (J)J r;.)J}..y?JC!'YJ , lYnf!"m D.5!vlU:mJ J r;f) 0J'-:J5jtIfl J OJ 2(!J3fJ (!TIn 2.FKJ JI)J (j

Niko

Line

't) C'1nm: ftx..) = xci, ( X) 0) d t R) JahJfJl3J '2ojJJtfh<J 07JJfJU dJ5f;J~{!,3!vJu JhJdfJ • JiJ d.-JJrJ(On/iJ!urr\2V !1m 'Je:'fl 9'JSjlffYlu X)JJt1hidnlJ 'J-Jw6ia (J)Jm!uonC:>Jb) /)J63Jf; r =: aq lbJa.,'L .

ff'hn2Jm61TJ3.Jff!J(J~ "!r!!&I1f'ti!xJ, J) Jrx) = Sin X ' J/; Cj':J!ij!JhJ 'gf):j3:JlfnJ !i:k3nWfJ

Jh :r 0 DJntJ1nr!ri3n. :).(1· lim sin X -:::: sinTo· .sJ()!J {!JJ!JJJtf!j()flxev(!> 6nOJbC.JA 3~~ X-<J.o

A

A (T)'J 'JJJrYJ3w3eh AmJ j'-J[JJfJnu IyCPI).Jf)'Jc-nn fbmJnu Un{Pn(O] ]h(f)ncn;n!JJo ;;n~

bnA9JJ MuJ d.JW2J,;dflUl fdx!"nu L){I2JIlJn/..; tPrn{!7'nJ, OnYlC'J6OJ

I 5in :t1=A8 ~ Ac ~ IA[I = / 'X) .

JJ 'Jtfm~~jfJ, nan/.; OJ07JJf!'Y1u5'n5Jonm, IimJ / UJ 5 xl ~ 1 ' JJJji]J6u

. . / / . x-:x.:, )':.+-:;:(0 I 2/' x-J:::oj / :[-I-L:,/IS/nX-s/nL -= 2stn-2- . c..os-2-, = sln-~ 'j(j)!J-r-/'::

~ 2 !sin X;xo I ~ z IX;xoj =jX-L/. JJhn2!Je? 1 dJ3?Jn.d(!'Y)J CPJ3t'JNnOl, fvnJ

o ~J5inx-s;nXoI ~/X-Xo/'

JJ t:Jtf1n(!Tl(r2ddfJ C(;(1JJf/oJ a~J/.;Y!:V1(J) In)n{!1f2J(}] , !vrJ lim /S/nx-sin TQ/=O- Jj.J(!JJ5 jn) X-'>Xo

[J).On /.; JoM?J, aJJrrJ(T?() fufJrrJL ~J WJ!P,jfJIf.JMDl ItnC!TJJ'

6) J('L) =Co 5x· Jnfjj()~:1J CoS x= Sin ( .; - X), O1Jrnh2JflC?JD /ior;(!TlI) oy:;[jj!/fY)0 ':'10::;3-,

rfrrJI10 'CJ!..JJbJ6 CP.f3JujJ!)()01, NnJ C!J5 X <:J~lJjd{Jjif ~f]:JJJ!fhJ [;.i3n/..;J(Jj/) 5Yl1f}(J(}71!J().

2J) ! IT) := -[j x· /).5{YOJr; {j X =: ~~n: (!JJ sinX CoS X 13':Jfij f/rJ:J2xJ ':J5:J3:1lfJ!Xl J!J(lJ 6 : J

.J InJ1!m J aJmn '2Jf!3 Jh::?_"6 J ':J/J)3.JJ!f) f).1!J:i3cJ !:f3Jc:r'J 5:JA1PrJ(!!"lkJ 1 U.5t?J If j() {f) 5:x -;:.0, ]. n !13J(!!l.5 6.JAtPnf!!' d(j J 8J~J f+JlK (K f '2) !..JJbnU fJ:Jh!Pn[!7"J~J . .J/..; b.J!l/Pre.J2YJ ..5f.. ~

tj x - niJ ·;)JS/.;.JCDc?/vJ/; .Y1J'Jf) - Jo(}. }rX) =1.:J X ~1J'(}J ':J5)J:JtfJ()J C2.JX'Jl/J 3.551JJ'6~Ynu JI--/10

5 J3nuJrUh fS:jJlfhen3n.·

(3 ) J(X) = uti x - Ju '!bf0[J'f]J (j '!Jbj3JJ!hJ (JJ.J3n1Jn JJ5LdlJ CYv1!J ,)/v1L; r;f6nudl)/) b:i1ihc'In· J.(). :J3J(!}12Jo, 8J/](jJJ IiK (K t:2) bJ6nL DJJltPnc.n.i3n/';J·

:J:dtf:fle!1(J 1fnfJdmfmJ.J~ '!!b5t1fJhi3n· }{'XJ= a'z.csin x (-I::: T ;:. /) ) -!(x) = = flUC()s X {-{ ~ X~ () , j(TJ~ llu.tdX (XER) , !rX)=ma;f.:} x. (X e R) .

.7!J oy.y;j/YnJlxJ, nm3rr1vr ':J5.'JJJeP ':3<:J5j//'!lJflU 9JjD',Y:'J(!,Y] OJ<:'J5!1fJ'!lJOn, ':JoJ3.JLfSn

oJ /) () .J IJ- C}j3rJ ,)SO] aJl;!J.J S~Y'YJ/.; .JA:Jfa 'df1 ,

t;PJU j 3D,)' JJJc:r'J dnNJ(]J.J(?fJ JI2".JdJ!iJP..;A':J(!T1(l ~.1tYnJ ':J5'J3:Jtf'n.J OJJ Q n !J

aJ 5I.JJtx:.'YviIJ .!k/Jn.

OJ6(]'lJ?J5J. 2C"Jd:J5tfJ!l'X:YJ JOf!10JfJ.J (}u.1lJ7 '?3':J5J!Jh.J!..J 1 IYnJJ0YtLl dJIPa:;6()

C!'nJ Cr1YJ (J) J r.?r'1 JcnJ JJIJ1Pjf, -:Jf!'{1 ':!; ~dIf(J)df'J.J..J [pJ Jf, IMiJ.Jtfr)j ':J(I}Yl rrJ:YJJ LJ'f1.J3rl/.; /.;J ~

Je""JOn(J). JoJ3Jefl OJ'JojfjfJX;n1J {lJ)n&fJ.fo(lCJJIJ {!JJ dn)3JO(][!"'(J IYJujY)()(!lJO (~cl':JI

.5cnrn(5J J 5drr0f5JfJ OJ !Urn Ou , hmd bJMfJJnJl,n J[!'1 JJJotfJ!l'YEYl ~3':J5jffnJtJ J5.'J~t?nJ

OlJYJ!.. dJ6LU9Jr:'Jhnf...; ,'>!dJf1'

rom (yf)J!J(lJ() d6n'J3£1'.!!'mJ.JE/I 003J!lfl (

i) Jim Siox::1. JJ tfmC!!YT6(J[, tPJWJl':wnd~J<:? bflWiJ{Jh 3~3.J&n!T1, Am;} lmu'J O<X-L~ X-10 X "

2fJn5 Sin 1:. ~ X.:: tgx· Jd :J~r/; JJhO'I1£J 6.Jbflf!!YJ t3.1&1J3fl!J :Jj1J Lf~, 3JR3JoM

J.ftJ; 3JbJ SJbl'l!!Yl. J}1flW01YJt <JJ3Jr:?JNn(f) OAC wJIJrrAf\Lu (!JJ OJ) C f.;jJjJ{j)tJJC:f1!.: qjJf,()lrrJj?J():

.:.. i- j jSOAC =T ~2X:: ~x 1 SLiOZJC i oc·a = 2 tjT)

cPJ ilJc?aJfj SOAC !. SO.t:lC 1 fJn3i1C':'.i6(JJ,!Yn3 X<. V:::G . (!JJd tfy,r!J3'1eY1 'j~J (jJJ WCYl~ J!/) RJ ?JOY1(1}01

WSX z sinx <. i. X

C!J5X-!Y.J ~!Jrr0I1LJ lPJ Sinx-fl& jJbtPm3nlJ d.s~f1(J'il .x3J3fl(?TYJ 2:JWJf?/.iJ3fU1 hnJ JiJ ~J(!!'n

~~J u.x]Jf,(JJ(!!'{lJ/JI1J dJ 'lJfl!iJD', AIJll!J - ~ <. x <D. (Jl~ 3tribC" ~ ~~J3.X!"aJ 9x:03:JJf6J, lvnu'J T.-+O, J3JJ.;(J)Jfj 1 2J3rJtJJJJC!!YlUbf'l!):Jm ] NnO {))SX o.Y:J5j[/Ni Cj6J?J:Jl1m2;nu 2JJdm

11m CD 5X::: tbS 0-= d, In.3I1(!' jj(J} ,IYnJ lim sin x .J.AW2xndiJ i2J 1-nLt !fmC!!YlJ' .t.~o X..-yo X

2) I;m {1+i)x=e. R3JO J]Jf!.3J5f2Jn ?;}jm5t:J.JJ hmc h(1]lfJ ncJ{,o./JnE lim (1+1f X-H't) X n~C() n

JhLJ?Jm~ ~J JL 9:x:.>JJJ,n Jc:'J5/l'J6JD7 [-(]In. JtPjnlJ'(?J'2JJ) hmc3 AmlJ'J X ~ 00 5Jdl1iJ(]J/)JC:h dJ

'dn5JU' jim (j +i )x == e . J(!, 3n en '2J!J..JJ-J:5J3fiJ, lYnJ JiJ Sf:'J.J/vl 'JjnJC":J(j-J fun5:J!ynt 'd.1JryJ a -X-Ioo {

OJrJ-,Jt:?J l/: lim (1+1.) ~ :: e . T~O

3) lim /OJ(1Jf t X)· i ,wt:JJl/ En JcErJ'J5J3/.; C!!'rn2JwnJ/.; e ~ _' !?nx=~ T . .r~o X rna (Je

J!J i(iJ:!I':::' .5<:' 3ne.'\;"@ Jrll'C:;'3 ~ E."SJ }:£jl!'],~ 2f1c::f?r:x!:.n r!h(!",-r0r1J..'£ gJc..x CXJE:i:;;-;/..; ')5.13) & i,1(; ;,~

c.smJJC'f1[;5n5JJnm, &/l(Jl(!!'JIJ:

fOr. ( ) i .l. () 1 lil71 . ,-J~ ./I~=.lim 'C>1 (11.'c):}: ~.. I:)f1 !.Jim (It.'trr } ~ f()1 r. -; ) - - i x .r) ~L .x:-~o <. (L Go.. ;(:"0 . "fl.. kJJeC0 fna..

x...." 0 x

') /' (LX_ j"1 1m _ En a· JJ !:Pm(!"rr0Jflu t?Jdtf.Y'Jtf.MJ dJM(!":MJ OnbJ J!1merrJrJ; d~iJL--- 0 X

oJ3nm. .5 J n1J.J /Jl..J{1 L, JC 35n'Jbrn(}1: -!-. =a 'X._ i (5'J dJJon'J6rnm, Am;; ax ~6jl!(',f1t ':JD5':J-J!?rTJnt

202m t -f- 0, NnlfJ x~o. JJ'OntJ, CJ3Jj5Jdv

n X _ jlim L{, ::: lim 1 _ _ 1_=£1/ Q. J:~O X t~o Jim E,Ja.{ff-t) i

i~o t -En a.

hmLlJ t1 =e ] Jf':ib~Cl ';ynJ lim e'Z i :f.. X40 X

CPJIJ ~3[jJ' Sin x , -En (u-x) , eX-l ~J X tfm(!"'3JU) ,:!f..,.;~ J/ffYui3l"J:;. NnlJ'J x~o.

en (f-rX) ::.1./irn

.i

b.)h3mXoC'nLJ O'Ei:J?0· 3a03Jm, 'Y= f(X) '!3'j5:Jo'l1:' OJbldb23h'.:lEflJ .xo S:Jhdh~I1L i\)na:J dn<r~~ ~~?J6n

'J6mrJl: 112:::::X-Xo. Ll~=Llfcro)=.f(xJ-f(Xo)'J:..r :Jbm@:JO,) .)h3~()J6t9()L~ bm~ l1')-G - ~~fijo'f)f)L s./bh~n Xo b.~h~nQ!'al'.

0Y.J5jo'nnG b,) h 2 m j 21 'oJ ~ n h.)o2J f)jMn(!!1~() .:JDffi(!?:J6J ')d b:JhD'()Q:\~ ~'jQjahnL 5J~)enL ,)/)2/:0::16

tl'niJ 5)bht9ffi...,!:i 'dJu1>Yle.:0nL ~::.hL, hmd.l ~O'.J2.J5tl'11l> 5}t,hfPf1 JnnLb'hYCYi> 5"jC"'flL~J5 •

X o b~htl'f1C'3n "y=f(X) O?t'Jf:):J(J'I1()~ Nh;JmJff.JC"f) Je'f)Dnt!5JoJ fl(:Xo), et:ftr 0) , "')' JO ~ Ln:J6mC~I)\aHI. J. n· ax dx

fl(1:o)=: lim A"J = lim -.ttO-f(To}. ~:X:40,61: x..,> X 0 X-Xo

cn~ 2J35n'J5J3<]"h hilld I=J:o+t,'J::, b.lhClrn:JQ'jC"nlJ o')DLdbC'~ 'dJndQ.".:;6:,.iJJ R)nbJhmL

-f'(x )::: lim .f(:ro~·Kr)-f(:x:o) .o.6.'X~o 63:.

:J1J D'mC!'m6') u..'>'J::J..'>i'!')6.ll> 2> 3xhn:J3L E)...,h£lmmC'n ~.y?>5'Jhrncn X DJMfll'!'31lJ" jJhdmc:? 11JLJ .'Jh3':13J6t1'r1L s../bh C!? L X

6Jhtl'n~3{1 J3Q!'.l?J LiT-nm :>~5n~5.)3m, 2l3Jd6JdJ

j'Csc)= lim .f(x+AX)-f(J::} Kr~Q .6. X

Gl::J "!>'J6J6'f)jj h.)n.3J 5:Jh6'f)~2n &,)-I1&1J !3.Jh2mWc-n, .'>dilIl"l2d6, hm() n21n .'>J b.JhBnQ."21n Ylf\L By" J"rti)~(\ Jb ~

\?n':JJh:JOu'MJ~n. (J):J '?:!>~6jdf),) bjhlJrn:Jc,J(9f)':' h,::,o{]J un2hJ.3enL ~m)JQn 'DJMf1C!!'dn, .::,J'DmoJ5 J hrr. J n2>() b.)ham i:3JC?n,) )3 l.Jnuru2!C!!'fhJ.

6j~2nhn ':JbDJ~m6,)uaub . LuLJ.1loI(!"ft£n.'> ~~2Il

ffiJrnh JJJ • (h~ ~~D::lO'I\) 6,)h()mil~J h'::'n2~ DJh~nC!:"dn, J:fEo5 f1C,(1 :Jro.3J!fmj).l jJ 5Jh~n.::':'~n·

l':?::JtS'~flIfJ6j. 3<hd3Jffi, ':J::: JCX) '0~ojo'n:, b.lham~..'> Xc bYltSr>e~ '::I&9j 3.)-R3J[)m0>, hmJ dJ'd()b0

I',m fCXj =fc.:cc} . ~5n4dEm(J) x-...:x:o

feX) - f ero) _ f '(4) =::- 0( ex). x-xo

8:,hJm~nG 2l.'>dm, IJ'b)(9;lj, fi m c< LX) =00 d./2lnO, 00 'aJ:'lbn'JW30"h hrnJ x.'-"Xo

fCX)-f(:t 01 == (c<b::)+ :f I(Xo JXX-Xo j 0

"J d'm('."ffioM0 CbI!'3_\hfb'J O)<'.'.\L:'J~n(h, hmlYJ X"'Xo, J3Jj(jJa.:" hn1:J lim (:fCtJ- Hx-o}) "'0. J:!JC5'J5 :)0 ?>C:J2I~C'rr3oJ l?.l~.JJlf:)(1

(;'J():JQ.n tllT'e'moJLJ 0 x.., X 0

'dJDn'd~5J. :Jd mJmhJ2nG 'dJ2N~Di)~t'."'f) CQmt'."'J2J:, .lW:' LJili~0Ji)n. 2.lho'1C'J[J', .lC'?31'0'f1 'dJlu3rn&1i1JC"f1J' r,md I o "" 0 b.Jnen~'.3n f (X) =1 XI 0!>'J6!Jo'r.J 'JD:J3JcJ'nj, <3:'011.)3 :,h...,) b)hllm'J6:,19f) 0

bJr,2mJ3~t"oG (lJrnaJtS'hl¥::l~n Jq,hn 2~amJ.lhJm6L ~:JJl9:J3M: :f.('}:) ':j'j6:1um!.J 5.ihlJm:f.\':!C"f\ :r D b:JhD'o('."'2Jn ~L'?I,n/.; 0' d

"J '3':J[)iumL 2>h..\~n~nU .MoLXo·, tlTo}} bjhen~~n 7.>.xU~:c(\ Joj),nL 'f')irJ.r;}, jl11:)

"30u'n:J5BLJ. J.n. K::: t' CXo).

JJnmt!'Jd, h,)i!?21J) .M c bJh6'n~'a(\ 6J,d.l).)c:"f\ 2b5o(\::,hriJ El"S:J 1 hrrJJen (J' (Jon

{tro) ~6.) .Mo.Nt 2:BJmnl.tJD.'l5, hmo'J QhJ~n/bJ 2(l:?Jo.:>hJ }/", SJht?oC!'f\ 3MLIJiu

c:JnL JrlI) 5Jht3nc:'nlJjJb, Jon. hrndJ lIX--rO, cUdn!), ubJ@n,), ilrrd lim \J =. ex X «}J 1JX-nu 'jo.':J3JtJ>mQnu ;:jj3m 3?JjdoJo,) .6..X~

K' =: ti d. :::. -til (I im0) :::. lim t~ \7 =: lim b -f ex0) := f' (T 0) 0

.1r-to .1'r-Xl ~-rO Ll ~

b'JhJmJ?t.Je:'nLJ '::)nlbnj:Jhn J<t>hn ~orrJJllJm6L 'ch3lPJ2J8n: /"bJ'je'nL Jmdh.llY'6o[, lJilRdJiD 'JC!?hn!J 3Jb3e'(\cf1 z,~L OY1J01J2l':Je'U C?hm(\(J). j.n. OJ':1 S(t.! .x:bn'dbJ~[; GbJ'jC'nL Jruh @hmnL i 2md::l5tfo/"J/J\2JoiJ aJb3~ aCbJL, V (1:-).lJ j n Df)-R ::!..lh:JD 1:. 2m.3Jr) ci''dn, .d.:J'dnb V Ct:) :::. 51 L1::). J..)hm~:,u', c£!lIn0L [i', -I:. +-L:JJ 'J':JJCC':J~8(\ 3.:J.o.3QC·()cn cLwn C!' n ndb"J60 SCt+Lit)-SCU, .jOnrfmd,JJ ~j.l~Jt98n LbJ~~L u:,';)jJe'(Y) unf\J.lhJ n:JEiJd,J SCit6iJ-SCt). db.)enJ~ nmJ

Dot. h~u' '::J~m 2u'nhJj <?nm0LJ.1-1:. ~'::J..IC:'Jen~ Lj(JfJ~ LJ(1~jJhJ ~nm ~"::hm ,)bc:.mG ndlJjOJ 1: 2rndJS3'[Jn UnR:1JfLJl17u6.j.n

Vet)::: lim VU.l'd" =Iim S(Hb-t.j-SGJ =: SI(t.J. t,t-tQ A 1:.-+0 Ai:

/

oJf)Jh(JmJ"MJ dnhrm,)(l?() b~LJon

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19.xJt?j"o'J6,j • .JQl~'d/)(no) : Pc.::'cJ:::::f (x.) + .9b:). a:'lJd~J2)J

1:>'(2:) -::: Ii m q>(~T~) --P(x) = Ii m (.f(x -tA~)-+-~ (X+l(X')) - ( f (x)+:J (a:)) _

.4X~O 1I~ A~-}O b ~

;:; lim HX+A~)- fCx) + lim j(x+Ll~XJ-~(X) -::: ;'CX)+~JcX) . h. (p·o..1'X~O .l:LG L1.')':-')O AX

(J).Jmh.'J{)J. Ol~ :ftc) C2J 2e'.X') ~'jfi:Jo'niJn b.)NJrrU3.x9Dn .)hn.lb X bJhBn~18r, cL'dn6' .:J !lJh6'nrn';lr~ bJh2fn.wen

n:JEiJ6J 2Jmn h.'>3hJ2>e'i'\()' e:?J C:fcxJ·i!eXJ'/ == f ICX")·,2 ex:) + .f(:t). j'CX).

(9jJt9jrdi2>.). Jc:'?>bn"dEinen: 'PCXJ:::' :fexi-.§~) IE:> QJb?Jnb("(!!'mm PCX) ~~5j(jnrL b../bh:Qn

4'c.:I.+L\.X) -"f'ui:::: +CX+ll~)..Q cr+L1~)- 1(1:)· ,Sex) =-HX+L1T)' ..cJlx+LJX) - -H2:) . j (X+L1X) -t- t (x)· gCT+L1.TI

- f(x"),i]CX")::: (+b:'.+L~J-fC:x:.JJ·jc.:x-TLl3:)+ feX)(a(X-tLlx)-.9CX)). JdJe-..\b

PL':::C-t-L\1:.)-P(!.):J = -f(X+Ll~)-:f-(-::t1 . ~(x+Ll.x)+ fCxi, 2,Cc.c-\-~)-2(Xl , ~~ ~x ~x

(h'::J j3 B~rM.)'dn 3::e.'>3~(]1 ~:Jh<Q'J h1no'.\ 6..X~O, .\~LI.xn.,£ J.\~nLEno.J2:o" hm;) ~M(1"lil.Jem6nu a.0n j tx) "3'.:JEi1

jtJnj ~b)~JH'f1o'JJ X b~hb'0C!'2n c?j ~-f). lim 2e~ti\,:t.):= 51w, 2n3~60'J1 hm2 <;£'{'):)= fh)-~(;:r:)+t(X)'21'{X) • l.x-x)

h·(?-21.CdJt9.'J6f1. 2~ai'\3n OI.)bJillalD2:c::0n 8Jar~nL 5.Y\3mJ6~t:::'nL Eil'dSniJ 3Y1J01. ]. n.

ec-f('JC))'=: cf'CX) . C9,,;]6'Y1o'J6..'>· h~2DfDO' 3~(\3<'r C3~wo'nf'\L EL\~ 5':lQ2:'no &mQhnJ' ..)2ntSfnd b...lh;)rnJ2.':'1c:nIlL a.II) lJJ/i::; V1 i3.fi

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ec -f(~)) '=- ct. f b:)+ c· flex; =. C- :fC2?J+c-t'6J=c-n.Z) . h - e .2J .

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(. ~(x.J ) 3 ~('X.)

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2_/J1\5 Co ) C J TC':(t6)::J-g'(X) = i. T..Kx-\-.6::t} - H:,r)q('X1 _ fO:). (j(x+6:;C -- (!(:C) .

L:x:. ~L:t+.6.~)· >-J(T) l. .6:r (I . 1\ T.

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C::?_\JC'O()o'~..). bJhJmJ6~f1L D.\J;IJ.J1bc~:16I\U m..lb.JWJ~

(-f lfL-lll) '= lim b') =- lim (.6-;y __ . L:::.'Je '\ l:>t.-70 1:>-l:.. L'.1:70.6X .6i:-J'

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.6-t-7O

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:c.

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