bull bull bull

qtllHV 2

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tG n~ hwtwuwpw~wtl1 OppGwlj lpll12 GljWllwo-lj[J1Gh12 np ut12tGWJP ZWJ1ITuwG

uGraWY12mu Gpw wpwqw~wtl1p ymyumG12G wGuGJjhWll tl1ntl1njutmu t StuGtG12

rat pG~llltu ljWJ1tJll t npnztl n~ hwtwuwpw~wtl1 ZWPITtllll uwpuGp wpwqmraJmGu

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trtllU to -py to + h ctwuwGwljwhwlltwo-mu lUGyGmu t s(to + h) - s(to) owshy

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s(to + h)-s(to) (1)h

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wopG hWUWlllWllwujuwGnll s(t) pmGljypwJP wo to ljtllmu ltlgtwullnptG h ITWshy

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qn1raJmGu unlllj[J1Gp to lllwhpG ljtllP V(to) wpwqn1raJwGu UJupG12G wGttp2 tPn12shy

pwyGtlllt h ctwuwGwljwhwlltwo-u utG12ljWJlllll tG12 ullwGwlljtllP ozqppll wpwshy

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unt GJmrawljwllljtllp wljllrawprawJpG wpwqn1raJmllu to lllwhpll npnZtmu t

- l s(to + h)- s(to)V(to ) - 1m h

n---+oo n

pwGw~ltnt nplltll h -[ wGttp2 tl1n12P t (hn 0 n EN)n

o p PU ill q 1 1pym12 mllllwqpo- ZWpITtnll uwpUpGU ZWPITUwG wnw2pG t

3t 2twJp4JwGmu wGyGmu t S(t) = + 2t utllpowGwlllwph Q-llGtG12 uwpuGp

w) UP2pG wpwqn1raJmGU [10 11] ITWuwGwljwhwlltwo-mu

p) wljGrawprawJpG wpwqn1raJmGU 10 -plJ twJJl4JwGpG

45

q) wy u]aWp]aWJpu wpwqm]aJmun II-Pll tlwJpYJwGpu

w) lJ1t2pu wpwqm]aJmun zwpduwu 11 -Pll tlwJPluruumu ([10 11] dwuwuwshy

ywhwlfltlw01llU) yL11up

s(ll) - s(LO) = 311 2 + 2 11- 3 10 2 - 2 middot10 = 65 (UitlPy)1

p) 1-pymp hn -u wutltp2l1npp t UJllllhllJj2mu 2s(10 + hJ - s(LO) _ 3middot (10 + hJ2 + 2middot (10 + hJ - 310 - 2 middot10

hn hn

62middot hn + 3 h~ _ 3h-------- - 62 + 11

h n

i2wup np lim(62 + 3hn ) = 62 uwpuup wyu]awP]awJPu wpwqlll]aJlllUn zwpduwu n-too

10 -Pll tlwJPYJwupu 62 UitlPy t q) ltwuwuuwunptu yulflwuwup

s(11 + hJ- s(ll) = 68 + 3h

l1

n

lllUlflP uwpUGp wyu]awP]awJPu wpwqlll]aJlllUnZwpdUwu 11-lll1 tlwJPYJwupu 68 utlPy t pu~uJhu lflhuwup UwptlPUnzwpdtllllu tn~hwtlwuwpw~wtP Gpw tlP2Pu wpwqlll]aJlllUn

zwpdUwu II-PlJ tlwJpYJwumu wtltJll uta t pwu wyl1]awp]awJPu WPWqDl]aJlllUn 10 -lll1

tlwJPYJwupu it wtlhJllllnpp pwu wy u]awp]awJpu WPWqDl]aJlllUn 11-lll1 tlwJPYJwupu

~UUltU~ltl li1 ImiddotUUI

1 PU~lllt~u tu qtpUlllU hwtlwuwpw~mtP zwpdtlnll uwpuup wpwqlll]aJlllUn 2 pu~uJt~u tu qtpumu uwpuup uP2pu wpwqlll]aJlllUn

3 pu~uJt~u npnztl s(t) optGpntlzwpdtlnll uwpuup wlju]awP]awJpu wpwqlll]ashy

Jmun to ll[whpu

UnU9UIJPUltjuliP

Cl-tputl s(t) optGpntlzwpdtlllll uwpuup tlP2PU WPWqDl]aJlllUn i1 dwuwuwljwhwtptlwshy

alllu t]at QWUWll[wphn tpptlwa t uttpptpntl pulj dwuwuwljn tlwJpljJwuutpntl (l08-11 0)

108 s(t)=2t2 W)i1=[12] P)i1=[115] q)i1=[112]

109 s(t)=3t2+t w)i1=[23] p)i1=[2225] q)i1=[221

1 1 0 s(t) = t3 + t w) i1 = [3 35] p) i1 = [3 33] q) i1 = [3 31]

Cl-tputl s(t) optupntlzwpdtlllll uwpuup Up2Pu WPWqDl]aJlllUn i1 dwuwuwljwhwtpshy

tlwalllu u wlju]awP]awJPu wpwqlll]aJlllUn to ll[whpu t]at Qwuwll[wphn tpptlwa t

uttpptpntl pulj dwuwuwljn tlwJpljJwuutpntl (111-115)

46 bull bull bull

Ill S(t)= 6t + 75 w) ~ = [0 2] to = 1 p) ~=[1 4] to =2

11 2 S(t)=t 2 w) ~ = [4 6) to = 5 p) ~=[2 5J to =2

1 13 s(t) = 2t + 3t2 w) ~ = [0 4] to = 4 p) ~ = [3 4] to = 3

3gt 114 s(t) = t + 5t2 w) ~ = [3 5] to = 35 p) ~=[0 1) to =1

gt 115 s(t)= t3 + 6t w) ~=[56] to =55 p) ~ = [4 6) to = 5

qpltltnl-raJult ltlhruP

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YPlJlW4WP Unw2pUl B hwuw4 hwullPlllmllpy 25 d wuy pulJ hplJpnPlll A hwuw4 hwullPlllmllpy 16 d wuy 2wup~ dwu Lplthy jmpwpwuijmp Ull41llYPlJshy

lW4WPP lllIl1mpnlJajmUl gt117 A 4WjPPYllhlllP B 4WjPllJlllpU hlJw4phnuwLpwp uhphGwu upwdwuwuwlJ

B -py A zwpdlhy uWPllw41wP uhphuwu PhnUw41wpl 1d hh41ll hwurWlllhy uWPllw41wPPU It ltu 15 d hh41ll hwuw4 B 4wJp flppw~u dwuwuwlJ owluuhy

uWPllw41wP uhphuwu B -py A owuwlllWphpu

sect2 UOlH10JUl

pUllllbu 41buwuP UWlullPlllllwpwqpwnllY mulJYPwJP wop U wpqmubu41P

wop hwpwpbpmJaJwu uwhliwuu muplJnuqpb41 pwlJwlJwu puwu41

UJu qllumu llP41wPlJ4mU bu IIpwJu wJuuIllUp hn wUllbP2 tlmppbp npnuy hWuwp

h =f 0 ywulJwywo n -P llbUWmu bwpwtylwuprr lOw upwpbnublm hWuwp uwlJbpshyn

IllmUGbpnlU h =f 0 IllwJuwurr libup pwy 4JOl1l1ubuP llwJuwuw11np4bup uwu np oqshy

41wqnpohlU4 j(xo) j(xo + hn ) 41buPP wP41whwJ41ll1JaJmuuhp libup hWUWpnlU

hup up Xo -u u (xo+ hn ) -rr Illw414Wumu bu j mulJYPwJP upnzuwu 41ppnlJJapU

Uwh u wh nlll Uunul hu np f pmul[yJuuu uunuuyhtft t Xo l[hshy

l[lnul hph ymul[mym6 hi muc[hp2 cftnppp hmump llmqmuhl[l t

j(xo + h )- f(xo)n (1 )hn

hm2np 1Jml[mu mpJmu1J UJ1J 1JhUJpmu (1) hm2np1Jml[mumpJmu umhshy

umuu muc[munlu hu f pmuljyJnuJp m6muYJml Xo ljhl[lmu k u2mumshy

ljm u f(xo) CljUlJUlwy1lnni r tp 2tpPpJU xo)

( ) f(xo + hn )- f(xo) IIf Xo = lIm n-gt hn

47

bull bull bull

~ ~-

q) wy u]aWp]aWJpu wpwqm]aJmun II-Pll tlwJpYJwGpu

w) lJ1t2pu wpwqm]aJmun zwpduwu 11 -Pll tlwJPluruumu ([10 11] dwuwuwshy

ywhwlfltlw01llU) yL11up

s(ll) - s(LO) = 311 2 + 2 11- 3 10 2 - 2 middot10 = 65 (UitlPy)1

p) 1-pymp hn -u wutltp2l1npp t UJllllhllJj2mu 2s(10 + hJ - s(LO) _ 3middot (10 + hJ2 + 2middot (10 + hJ - 310 - 2 middot10

hn hn

62middot hn + 3 h~ _ 3h-------- - 62 + 11

h n

i2wup np lim(62 + 3hn ) = 62 uwpuup wyu]awP]awJPu wpwqlll]aJlllUn zwpduwu n-too

10 -Pll tlwJPYJwupu 62 UitlPy t q) ltwuwuuwunptu yulflwuwup

s(11 + hJ- s(ll) = 68 + 3h

l1

n

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tlwJPYJwupu it wtlhJllllnpp pwu wy u]awp]awJpu WPWqDl]aJlllUn 11-lll1 tlwJPYJwupu

~UUltU~ltl li1 ImiddotUUI

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3 pu~uJt~u npnztl s(t) optGpntlzwpdtlnll uwpuup wlju]awP]awJpu wpwqlll]ashy

Jmun to ll[whpu

UnU9UIJPUltjuliP

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alllu t]at QWUWll[wphn tpptlwa t uttpptpntl pulj dwuwuwljn tlwJpljJwuutpntl (l08-11 0)

108 s(t)=2t2 W)i1=[12] P)i1=[115] q)i1=[112]

109 s(t)=3t2+t w)i1=[23] p)i1=[2225] q)i1=[221

1 1 0 s(t) = t3 + t w) i1 = [3 35] p) i1 = [3 33] q) i1 = [3 31]

Cl-tputl s(t) optupntlzwpdtlllll uwpuup Up2Pu WPWqDl]aJlllUn i1 dwuwuwljwhwtpshy

tlwalllu u wlju]awP]awJPu wpwqlll]aJlllUn to ll[whpu t]at Qwuwll[wphn tpptlwa t

uttpptpntl pulj dwuwuwljn tlwJpljJwuutpntl (111-115)

46 bull bull bull

Ill S(t)= 6t + 75 w) ~ = [0 2] to = 1 p) ~=[1 4] to =2

11 2 S(t)=t 2 w) ~ = [4 6) to = 5 p) ~=[2 5J to =2

1 13 s(t) = 2t + 3t2 w) ~ = [0 4] to = 4 p) ~ = [3 4] to = 3

3gt 114 s(t) = t + 5t2 w) ~ = [3 5] to = 35 p) ~=[0 1) to =1

gt 115 s(t)= t3 + 6t w) ~=[56] to =55 p) ~ = [4 6) to = 5

qpltltnl-raJult ltlhruP

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YPlJlW4WP Unw2pUl B hwuw4 hwullPlllmllpy 25 d wuy pulJ hplJpnPlll A hwuw4 hwullPlllmllpy 16 d wuy 2wup~ dwu Lplthy jmpwpwuijmp Ull41llYPlJshy

lW4WPP lllIl1mpnlJajmUl gt117 A 4WjPPYllhlllP B 4WjPllJlllpU hlJw4phnuwLpwp uhphGwu upwdwuwuwlJ

B -py A zwpdlhy uWPllw41wP uhphuwu PhnUw41wpl 1d hh41ll hwurWlllhy uWPllw41wPPU It ltu 15 d hh41ll hwuw4 B 4wJp flppw~u dwuwuwlJ owluuhy

uWPllw41wP uhphuwu B -py A owuwlllWphpu

sect2 UOlH10JUl

pUllllbu 41buwuP UWlullPlllllwpwqpwnllY mulJYPwJP wop U wpqmubu41P

wop hwpwpbpmJaJwu uwhliwuu muplJnuqpb41 pwlJwlJwu puwu41

UJu qllumu llP41wPlJ4mU bu IIpwJu wJuuIllUp hn wUllbP2 tlmppbp npnuy hWuwp

h =f 0 ywulJwywo n -P llbUWmu bwpwtylwuprr lOw upwpbnublm hWuwp uwlJbpshyn

IllmUGbpnlU h =f 0 IllwJuwurr libup pwy 4JOl1l1ubuP llwJuwuw11np4bup uwu np oqshy

41wqnpohlU4 j(xo) j(xo + hn ) 41buPP wP41whwJ41ll1JaJmuuhp libup hWUWpnlU

hup up Xo -u u (xo+ hn ) -rr Illw414Wumu bu j mulJYPwJP upnzuwu 41ppnlJJapU

Uwh u wh nlll Uunul hu np f pmul[yJuuu uunuuyhtft t Xo l[hshy

l[lnul hph ymul[mym6 hi muc[hp2 cftnppp hmump llmqmuhl[l t

j(xo + h )- f(xo)n (1 )hn

hm2np 1Jml[mu mpJmu1J UJ1J 1JhUJpmu (1) hm2np1Jml[mumpJmu umhshy

umuu muc[munlu hu f pmuljyJnuJp m6muYJml Xo ljhl[lmu k u2mumshy

ljm u f(xo) CljUlJUlwy1lnni r tp 2tpPpJU xo)

( ) f(xo + hn )- f(xo) IIf Xo = lIm n-gt hn

47

bull bull bull

~ ~-

Ill S(t)= 6t + 75 w) ~ = [0 2] to = 1 p) ~=[1 4] to =2

11 2 S(t)=t 2 w) ~ = [4 6) to = 5 p) ~=[2 5J to =2

1 13 s(t) = 2t + 3t2 w) ~ = [0 4] to = 4 p) ~ = [3 4] to = 3

3gt 114 s(t) = t + 5t2 w) ~ = [3 5] to = 35 p) ~=[0 1) to =1

gt 115 s(t)= t3 + 6t w) ~=[56] to =55 p) ~ = [4 6) to = 5

qpltltnl-raJult ltlhruP

gt116 A LB 4wjphppy upwdwuwuwlJ UPUjwuy lUllwnw2 zwpd4hypu hplJm ull41llshy

YPlJlW4WP Unw2pUl B hwuw4 hwullPlllmllpy 25 d wuy pulJ hplJpnPlll A hwuw4 hwullPlllmllpy 16 d wuy 2wup~ dwu Lplthy jmpwpwuijmp Ull41llYPlJshy

lW4WPP lllIl1mpnlJajmUl gt117 A 4WjPPYllhlllP B 4WjPllJlllpU hlJw4phnuwLpwp uhphGwu upwdwuwuwlJ

B -py A zwpdlhy uWPllw41wP uhphuwu PhnUw41wpl 1d hh41ll hwurWlllhy uWPllw41wPPU It ltu 15 d hh41ll hwuw4 B 4wJp flppw~u dwuwuwlJ owluuhy

uWPllw41wP uhphuwu B -py A owuwlllWphpu

sect2 UOlH10JUl

pUllllbu 41buwuP UWlullPlllllwpwqpwnllY mulJYPwJP wop U wpqmubu41P

wop hwpwpbpmJaJwu uwhliwuu muplJnuqpb41 pwlJwlJwu puwu41

UJu qllumu llP41wPlJ4mU bu IIpwJu wJuuIllUp hn wUllbP2 tlmppbp npnuy hWuwp

h =f 0 ywulJwywo n -P llbUWmu bwpwtylwuprr lOw upwpbnublm hWuwp uwlJbpshyn

IllmUGbpnlU h =f 0 IllwJuwurr libup pwy 4JOl1l1ubuP llwJuwuw11np4bup uwu np oqshy

41wqnpohlU4 j(xo) j(xo + hn ) 41buPP wP41whwJ41ll1JaJmuuhp libup hWUWpnlU

hup up Xo -u u (xo+ hn ) -rr Illw414Wumu bu j mulJYPwJP upnzuwu 41ppnlJJapU

Uwh u wh nlll Uunul hu np f pmul[yJuuu uunuuyhtft t Xo l[hshy

l[lnul hph ymul[mym6 hi muc[hp2 cftnppp hmump llmqmuhl[l t

j(xo + h )- f(xo)n (1 )hn

hm2np 1Jml[mu mpJmu1J UJ1J 1JhUJpmu (1) hm2np1Jml[mumpJmu umhshy

umuu muc[munlu hu f pmuljyJnuJp m6muYJml Xo ljhl[lmu k u2mumshy

ljm u f(xo) CljUlJUlwy1lnni r tp 2tpPpJU xo)

( ) f(xo + hn )- f(xo) IIf Xo = lIm n-gt hn

47

bull bull bull

~ ~-

bull bull bull

1 3 0 s(t) = t l W) to = 1 p) to = 2 q) to = 3

1 3 l s(t) =J2t W) to = 2 p) to =8 q) to = 18

LnuJhl wuhwt1wuWpnulD (132-133)

~ 132 w) X4 - 5x 2 - 6 gt 0 p) X4 -10x 2 +90

~133 w)log(2 X -6)+x-2z0 p) log5(25 x -4)-2x+llt0 2

6sect3 GP4ll1 1gtllHl46tgtUlthPtgt qllhIUPtgt h UPSUTl-PJUltgt

UoUlt6uUlt 4UltllltlthPD

UJU Ulwpwqpwgtnul ubUpljunt1npbup bpljm gtmuqypwubpp qmuwpp lflwppbshy

pmIOJwu U wPlflwllPJwlJl wo-wuyuwu (wo-wuYJwlJl hW2tluWU) qwunuubpD

-linp Ii U 1 cph f u g fPmuqyulluhpu UUhllUyhzft hu npLt

qhlJmuJ ftuq k -u hUlUlJUllrpmu t UlU[Ul kmiddot f f + g u f - g fPmuqshy

yftUluhpll umJuU[hu Ulhutuyhzft hu UlJI] qhrpnuJ llUI] npnul

(k middot f)=k middot f (f+g)=f+g (f-g)=f-g

UUI w y n 1 y n 1 U l)pymp f U g gtmuqypwubpu wo-wuyblJl bU X qblflmu

U h -D qWUWJwqwu wutlbp2 tllnpp t Oqlflt1blnt1 qmqwublfl hW2npqwqwum]ashyn

Jmuubpp hWlflqmIOJmuubppy ulflwumu bup

k(x+hn)-k(x)=kf(x+hn)-f(x) k()h h -7 X

Il n

WJUPUPU (k f) = k f UUlwymybup qmuwpp wo-wuyuwu qwunuD

(j(x + hll )+ g(x + hll ))- (j(x) + g(x)) hn

= f(x + hJ- f(x) + g(X + hJ- g(X) -7 f(x) + g(X) hn hll

SwppbpmIOJwu wo-wuyuwu qwunuD wUlwymYtlmu t hwuwuuwunpbu

0bnpbU l-u mup hblflUJwl gtpqpqwqwu ubquwpwUnlIOJmUn l)pymp qblflwtllshy

uJw uwt1wuwlflmJYpy upwclwuwuwq uqumu bu 2wPclt1bllwulflU m 2nqbuwt1n

OUIOwllPbuP clwuwuwlJp ywuqwywo- t Ulwhpu 2nqbuw4P hbnwtlnpnlIOJmUn lWUshy

41PY sJt) t puq lWUlflP hbnwtlnpmIOJmUn uwt1wuwlflmJYpymiddot S2(t) l)w qU2wshy

51

bull bull bull

___ J

uwlI11 np 2nqtuwtlD lWUlj1Py htmwumu t VI (t) = s (t) wpmqmraJwup pulI lWUshy

lj1D uwtlwuwlj1mJYpy V2 (t)=S(t) wpwqmraJWUp lwpq t np arat lWUlj1U m

2nqtuwtlD 2wPcttltu umJu mllllmraJwUp WlyW t lljwhpu 2nqtuwtp htnwtlnpmrashy

JmuD uwtlwuwlj1mmpy lIil1up s(t) = Sl (t) + S2 (t) pulI trat 2Wl1cttltu hwlIwnwlI

mllllmraJmuutl1lltl WlljW s(t) = Sl (t) - S2 (t) lttlj1UWpwl1 trat lwulj1u m 2nqashy

uwtlI 2wPcttltu umJu mllllmraJwup wlljw 2nqtuw4t uwtlwuwlj1mmpy lIhtnwuw

V(t) = s(t) = s(t)+s(t)= ~ (t)+ V2 (t)

wl1wqmraJwup pulI hwlIwnwlI mllllmraJmuutl1ntl2wl1cttltlm lltUWmu

V(t) = s(t)= s(t)-s(t) = ~(t)- V2 (t)

wpwqmraJWUp

f()- h n ph u 2 (pk k g gtmuqyuuuhplt uuruuyhtft hu npht

qhrpmu uuqUl UlJ1J qhrpmu Ultnu1tyhtft t uUlk f- g gtmuqyuuu llU1J n]UUu ( g) =f g + g

U III W -y n l-y n 1 U )pymp f U g ~mulIypwutpu wmuuytll1 tu X lIttpmu

U hn -D lIwuwJwlIwu WUtltp2 tllnpp t ltt2tp t utpmqtI np

f(x+hiJg(x+hJ- f(x) g(X)= (1)

=g(X+ hJ (j(X+ hJ- f(x)]+ f(x) [g(X+ hJ- g(X)] 42wup np g ~mulIypwu X lIttpmu wo-Wuytlr t mptUu WJU WUDUl]hWtp t X

lItlj1mu lttlj1UWPWP lim g(x + hn ) = g(x) (2) n~OO

Oqlj1tltlntl qmqwutlj1 hW2nPllwlIwumraJmuutpp qmuwpp U wptpwllPJwjl tltpwshyptpJwl ratnptUpy (1) (2) wnu~mraJmuutpry utpwumu tGp

lim f(x + hn ) g(x +hn )- f(x) g(x) = n~OO h

n

-1 ( h) I f(x+hJ- f(x) f( )1 g(x+hJ-g(x-lmgx+nlm h + Xlm h

n~oo n~OO n~oo n n

= f(x) g(x)+ f(x) g

0tnptUu wlljwymytlwo- t

lt hlp w up lfhqty uh6 yUluqUlyUl6 n pUUlqUlu Fnlt hUlUUlP

(xn) =nx n-1 (3)

U III w -y n 1 -y n 1 U 4ppwntup uwratUwlj1r4w4wu pullmlIypwJr utralll]D bpp n =2 lljullmUD np2tp t pwup np pu~lljtu qptptup

52

bull bull bull

___ J

uwlI11 np 2nqtuwtlD lWUlj1Py htmwumu t VI (t) = s (t) wpmqmraJwup pulI lWUshy

lj1D uwtlwuwlj1mJYpy V2 (t)=S(t) wpwqmraJWUp lwpq t np arat lWUlj1U m

2nqtuwtlD 2wPcttltu umJu mllllmraJwUp WlyW t lljwhpu 2nqtuwtp htnwtlnpmrashy

JmuD uwtlwuwlj1mmpy lIil1up s(t) = Sl (t) + S2 (t) pulI trat 2Wl1cttltu hwlIwnwlI

mllllmraJmuutl1lltl WlljW s(t) = Sl (t) - S2 (t) lttlj1UWpwl1 trat lwulj1u m 2nqashy

uwtlI 2wPcttltu umJu mllllmraJwup wlljw 2nqtuw4t uwtlwuwlj1mmpy lIhtnwuw

V(t) = s(t) = s(t)+s(t)= ~ (t)+ V2 (t)

wl1wqmraJwup pulI hwlIwnwlI mllllmraJmuutl1ntl2wl1cttltlm lltUWmu

V(t) = s(t)= s(t)-s(t) = ~(t)- V2 (t)

wpwqmraJWUp

f()- h n ph u 2 (pk k g gtmuqyuuuhplt uuruuyhtft hu npht

qhrpmu uuqUl UlJ1J qhrpmu Ultnu1tyhtft t uUlk f- g gtmuqyuuu llU1J n]UUu ( g) =f g + g

U III W -y n l-y n 1 U )pymp f U g ~mulIypwutpu wmuuytll1 tu X lIttpmu

U hn -D lIwuwJwlIwu WUtltp2 tllnpp t ltt2tp t utpmqtI np

f(x+hiJg(x+hJ- f(x) g(X)= (1)

=g(X+ hJ (j(X+ hJ- f(x)]+ f(x) [g(X+ hJ- g(X)] 42wup np g ~mulIypwu X lIttpmu wo-Wuytlr t mptUu WJU WUDUl]hWtp t X

lItlj1mu lttlj1UWPWP lim g(x + hn ) = g(x) (2) n~OO

Oqlj1tltlntl qmqwutlj1 hW2nPllwlIwumraJmuutpp qmuwpp U wptpwllPJwjl tltpwshyptpJwl ratnptUpy (1) (2) wnu~mraJmuutpry utpwumu tGp

lim f(x + hn ) g(x +hn )- f(x) g(x) = n~OO h

n

-1 ( h) I f(x+hJ- f(x) f( )1 g(x+hJ-g(x-lmgx+nlm h + Xlm h

n~oo n~OO n~oo n n

= f(x) g(x)+ f(x) g

0tnptUu wlljwymytlwo- t

lt hlp w up lfhqty uh6 yUluqUlyUl6 n pUUlqUlu Fnlt hUlUUlP

(xn) =nx n-1 (3)

U III w -y n 1 -y n 1 U 4ppwntup uwratUwlj1r4w4wu pullmlIypwJr utralll]D bpp n =2 lljullmUD np2tp t pwup np pu~lljtu qptptup

52

(x 2) =2x =2X 2

-1

k 1bGJdwt]JlbGp (3) pwGwgtun 6pZtp t n = k lJ-bUWmu wJupGpG (Xk) = kx - UUJwshyk 1ymybGp n=k + 1 lJ-bUWmu (X + ) = (k + 1)x k

ppnp 41pwnblml wptpwt]JlJWlP

wo-wGyuwG lJwGnGn utpwGmu tap 1 1(X k+ ) =(x k

bull x) =(x k ) X+ Xk (x)= kx k- bull X+ Xk =(k + l)xk

lJlJwtpbGpnp bJdb xo wUJw (3) pwGwgtun 6pZtp t Gwu n =0 un =IlJ-bUJPbpmu

Op pU wq 1 QtpGbGp y=x4 -2x3 +5x+12 pmGlJYPwJPwo-wGYJwm

lJppwnblml (3) pwGwgtuh u JdbnpbU l-[ utpwGmu bGp

(x4 - 2x3 + 5x + 12) = (X4) - 2(x3 ) + 5(x) + (12) = 4x3

- 6x 2 + 5

o p p U w q 2 QtpGbGp y =(3x + 1) ~ - 2-Jx) pmGlJypwJP wo-wGYJwm

lJppwnbG12 wptpwt]JlJWlP wo-wGy-uwG lJwGnu[

y =(3x + 1)middot (1- 2-Jx) + (3x + 1)middot (1- 2-Jx) =

=3(1- 2-Jx) + (3x + 1) ( -2) _1_ =3 - 9-Jx _ -Jx 2-Jx x

1 puW t hWljmuwp hwutpwtpmup u pmulJYPwJP wptpwlJ-PJwlJ1 wo-wuYJwm 2 puwt hwllwuwp pmulJypwubpp qmuwpp wawGYJwm

3 puW t hwllwuwp pmuqypwubpp tpWPpblilllJaJwU wawuYJwUr

4 puW t hwllwuwp pmGqypwubpp wptpwlJ-PJwlJ1 wo-wuYJwm

5 UUJwymyb12 (x) = nx-l pWUWgtUD

qtpubl pmuqYPwJP wawuYJwm (134-136)

134 w)j(x)=x 2 +5xmiddot p) j(x)=3x-x 2 +7

q) j(x) = X4 + 3x 2 - 2x lJ-) j(x) =9 - x 5 + x 3

135 w)j(x)=4-Jx-x3 p) j(x) = l--Jx

x q) j(x) = x _1 lJ-) j(x) = 2x + -Jx _l

x x

136 w) j(x) = -Jxx3 - 2X2) p) j(x) =1 (2 + 3x - xi )

x

lJ-) j(x)=(2x-1)(-Jx-1

qtpubl pmGqypwJP wawuYJwm GZlWo- qbtpmu (13 7 -139)

137 j(x) = 3-Jx +1_ 5x w) Xo =1 p) Xo = 4 x

53

bull bull bull

-----

138 f(x)=2x 3 -Ex-~ w) Xo =05 p) Xo = 2

139 f(x) = fh -3x 2 -1021 w) Xo = 3 p) Xo = 12

140 Lmo-hl f(x) = 0 hwtlwuwpmut

w) f(x) = ~ -25x 2 +6x-l p) f(x)=x 5 -10x3 +40x

q) f(x) =I +9x q) f(x) = plusmn + 25x-6 x x

14 1 lmo-bl f(x) gt 0 wUhllltlwuwpmul

w) f(x)=x 3 -6x2 -63x-2 p) f(x)=x 5 -20x3

q) f(x)=3x 4 -20x3 +36x2 -7 q) f(x)=x 4 -8x3 +18x 2 +29

lIlltunl-IaJlHI lhIUIl

q41uhl uztlwo- up2wljWJ12mu 41Ptlwo- jJmulJypwJP hwlJwllmpst (142-143)

gt142 w)y=x2 +x-7 [03] p)y=x2 +x-7 [-3-1]

143 w)y=2x+rx [01] p) y=3x +rx [-10]

L sect4 bpltnl ~nHllt6ptrubPp ~UW1P1P UoUlt6lfUlt ltUW1ltt

Gllljunllll UJlllplllqpwjJnUl tlbUp untlnphypup rah pU~UJhu UJhtpp t wCJtuuyhl hpljm jJmulfYPllluhpp qmuwpl tplllpphpn1raJmul lllptpWlJPJlllill (htpLJWl rahnshyphliuhpntl 41ptlmu t pwunIll1J1111CJtuuytllllU lfwunul

fi- h n p h u 1 lph g jgtmuqyjuuu w6ruuyhlft t X qhqmul k g(x)t 0

W UJW wJ1J ljhqmul w6ruuyhrft t uwk 1 jgtmuqyjuuu llU1J npmu g

~ (1) =_L (1 )I

g g2

U 1lI W Y n 1 y n 1 u i)pymp hit -u lllutlhP2 pnpp t lllllpq sLlllpnjumJdJmushy

uhpntl U41111Umtl hup ~ 1 1

g(x + hll ) g(x) = g(x + hit) - g(x) (2)

hll g(x)g(x + hn ) hit

i2111UP np g jJmulfYPlllu lllCJtuuyhW L hh41LilllplllP lllurruqhw41 t x lfhtpmu mutpp

54bull bull bull

bull bull bull

g(x + hJ- g(x) _ ()11m h - g x lim g(x + hJ= g(x) n-tOCJ n-tOCJ n

4ppwnblml qmqwUblj1 hW2nlll1wlwGmraJmGGbpp wPlj1WIJ1lJwJll upwGnlillP tlhshypwpbpJwl rabnpbUGbpl (2) hwtlwuwpmraJmGpy ulj1wGmu bGp (1) pwGw0ul

k g(x) - 0 UlUJUl UlJJ qliqmzu UlmaUylilP t UUlk f jnzUqypUlU pUJ g

npnzu

U tq w g n 1 g n 1 U Oqlj1tlblntl Gwtunlll1 rabnpbUpy u wPlj1wIJ1lJwlJl wo-wGyshy

uwG lwGnGpy ulj1wGmu bGp

[ l(x)J [ 1 J 1 [ 1 Jg(x) = I(x) g(x) = I (x)middot g(x) + I(x) g(x) =

f(x) [g(X) J f(x)g (x) - I(x )g(x) = g(x) + I(x) - g2(X) = g2(X)

x 3 -3x o p p uw q qlj1GbGp y = 5 pmGlYPwJP wo-wGYJWUr

1+ 4x

4ppwnblntl WGwGytiwG lwGnGGbpl ulj1wGmu bGp

x3 - 3XJ = (x 3 - 3x) V + 4x5)- (x

3 - 3x) V + 4x

5) =

( 1+ 4x5 (1+4X 5 )

(3X2 - 3)(1 +4x 5 )- (x 3 - 3x )(20X4 ) - 8x 7 + 48x5 + 3x2 shy

V+4 X5) = V+4 X5)

-ltUUlIU6liL lie luur --------shy

2X2 - 4144 w)(x)=2x-l p) I ()x =

I-x x+1

55

raquo153 SP4WU t f(x) = x 2 + 6x + lOu g(x) = sin x Qtptp F pWIll1 ~mGllypwJP pwGw)urr u wpdtpGtpp tpppnlJrarr trat

w) F(x)=f(j(x)) p) F(x)=f(g(x))

q) F(x)=g(g(x)) ll) F(x)=g(j(x))

lt11ztGp np tPllm f u g ~mGllypwutpp hwuwrymlJra fog tltup wuljwGtl tGp

WJG F ~mGllypwG npp wpdtprr x llttpmu hwljwuwp t f ~mUllypwJP wpdtppG

g(x) lltlPl1lU F(x) = f(g(x)) pull F ~mGllypwG GuwG lltUjJ1mUllnltl tGp pWIll1

~mGllypw ltbtpUJWlratnpMmllTlptjl1lU t pWM ~mGllypwJP wo-wGyuwG llwGnGrr

LU-hnphu 1 Gph t=g(X) fPmuljyuuuzu6utuyhlPt Xo ljhlfmul

pulj y = f(t) fPmuljypzuu to = g(xo) ljhlfmul ZUUJZU F = fog fPmuljshy

ypzuu zu6utuyhlP t Xo ljhlfmul b

F(xo)= f(g(xo)) g(xo)

UJu ratnpMrr utGp ltGp wlllwymyp ~uwybpllltGpu wlllwymytGp WJG uwushy

GwtJnp lltUjJ1mU tpp g -G qUwJpG ~mGllypw t g(x) = kx + b

LU- h n p h u 2 Gph f fPmuljypzuu zulJzuuyhlP t ZUUJZU

F(x) = f(kx+b) fPmuljypzuu umJuUJhu zu6utuyhlft t b

F(x) = kmiddot f(kx + b)

UUJ w y n 1 y n 1 u i)pymp hn -G wGtJtp2 tPllJ2P t UJlllltUWmu wGtJtP2 tPnpp

t Gwu kmiddot hn n EN hW2npllwllwGmraJmGrr mutpp

F(x+hJ-F(x) f(kx + khl1 +b)- f(kx+b)hm = hm = n-jOO n-jOOh h

n n

= k lim f(kx + b + kh) - f(kx + b) = k f(kx -I

--00 kh

o p p uw q 1 qtpGtGp y = (3x - 5) 100 ~mGllypwJP wo-wGYJwm

brat GzwGWlltGp f(x) = x lOO WlllW lTlPtJwa ~mGllypwG llutpwGw y = f(3x - 5)

tptuprr UJlllltUWmu f(x) = lOOx99 ltbtpuwpwp

~3x - 5yoo ) = 3 f(3x - 5) = 3 middot100 (3x - 59 = 300 (3x - 5r 57

bull bull bull

bull bull bull

g(x + hJ- g(x) _ ()11m h - g x lim g(x + hJ= g(x) n-tOCJ n-tOCJ n

4ppwnblml qmqwUblj1 hW2nlll1wlwGmraJmGGbpp wPlj1WIJ1lJwJll upwGnlillP tlhshypwpbpJwl rabnpbUGbpl (2) hwtlwuwpmraJmGpy ulj1wGmu bGp (1) pwGw0ul

k g(x) - 0 UlUJUl UlJJ qliqmzu UlmaUylilP t UUlk f jnzUqypUlU pUJ g

npnzu

U tq w g n 1 g n 1 U Oqlj1tlblntl Gwtunlll1 rabnpbUpy u wPlj1wIJ1lJwlJl wo-wGyshy

uwG lwGnGpy ulj1wGmu bGp

[ l(x)J [ 1 J 1 [ 1 Jg(x) = I(x) g(x) = I (x)middot g(x) + I(x) g(x) =

f(x) [g(X) J f(x)g (x) - I(x )g(x) = g(x) + I(x) - g2(X) = g2(X)

x 3 -3x o p p uw q qlj1GbGp y = 5 pmGlYPwJP wo-wGYJWUr

1+ 4x

4ppwnblntl WGwGytiwG lwGnGGbpl ulj1wGmu bGp

x3 - 3XJ = (x 3 - 3x) V + 4x5)- (x

3 - 3x) V + 4x

5) =

( 1+ 4x5 (1+4X 5 )

(3X2 - 3)(1 +4x 5 )- (x 3 - 3x )(20X4 ) - 8x 7 + 48x5 + 3x2 shy

V+4 X5) = V+4 X5)

-ltUUlIU6liL lie luur --------shy

2X2 - 4144 w)(x)=2x-l p) I ()x =

I-x x+1

55

raquo153 SP4WU t f(x) = x 2 + 6x + lOu g(x) = sin x Qtptp F pWIll1 ~mGllypwJP pwGw)urr u wpdtpGtpp tpppnlJrarr trat

w) F(x)=f(j(x)) p) F(x)=f(g(x))

q) F(x)=g(g(x)) ll) F(x)=g(j(x))

lt11ztGp np tPllm f u g ~mGllypwutpp hwuwrymlJra fog tltup wuljwGtl tGp

WJG F ~mGllypwG npp wpdtprr x llttpmu hwljwuwp t f ~mUllypwJP wpdtppG

g(x) lltlPl1lU F(x) = f(g(x)) pull F ~mGllypwG GuwG lltUjJ1mUllnltl tGp pWIll1

~mGllypw ltbtpUJWlratnpMmllTlptjl1lU t pWM ~mGllypwJP wo-wGyuwG llwGnGrr

LU-hnphu 1 Gph t=g(X) fPmuljyuuuzu6utuyhlPt Xo ljhlfmul

pulj y = f(t) fPmuljypzuu to = g(xo) ljhlfmul ZUUJZU F = fog fPmuljshy

ypzuu zu6utuyhlP t Xo ljhlfmul b

F(xo)= f(g(xo)) g(xo)

UJu ratnpMrr utGp ltGp wlllwymyp ~uwybpllltGpu wlllwymytGp WJG uwushy

GwtJnp lltUjJ1mU tpp g -G qUwJpG ~mGllypw t g(x) = kx + b

LU- h n p h u 2 Gph f fPmuljypzuu zulJzuuyhlP t ZUUJZU

F(x) = f(kx+b) fPmuljypzuu umJuUJhu zu6utuyhlft t b

F(x) = kmiddot f(kx + b)

UUJ w y n 1 y n 1 u i)pymp hn -G wGtJtp2 tPllJ2P t UJlllltUWmu wGtJtP2 tPnpp

t Gwu kmiddot hn n EN hW2npllwllwGmraJmGrr mutpp

F(x+hJ-F(x) f(kx + khl1 +b)- f(kx+b)hm = hm = n-jOO n-jOOh h

n n

= k lim f(kx + b + kh) - f(kx + b) = k f(kx -I

--00 kh

o p p uw q 1 qtpGtGp y = (3x - 5) 100 ~mGllypwJP wo-wGYJwm

brat GzwGWlltGp f(x) = x lOO WlllW lTlPtJwa ~mGllypwG llutpwGw y = f(3x - 5)

tptuprr UJlllltUWmu f(x) = lOOx99 ltbtpuwpwp

~3x - 5yoo ) = 3 f(3x - 5) = 3 middot100 (3x - 59 = 300 (3x - 5r 57

bull bull bull

raquo153 SP4WU t f(x) = x 2 + 6x + lOu g(x) = sin x Qtptp F pWIll1 ~mGllypwJP pwGw)urr u wpdtpGtpp tpppnlJrarr trat

w) F(x)=f(j(x)) p) F(x)=f(g(x))

q) F(x)=g(g(x)) ll) F(x)=g(j(x))

lt11ztGp np tPllm f u g ~mGllypwutpp hwuwrymlJra fog tltup wuljwGtl tGp

WJG F ~mGllypwG npp wpdtprr x llttpmu hwljwuwp t f ~mUllypwJP wpdtppG

g(x) lltlPl1lU F(x) = f(g(x)) pull F ~mGllypwG GuwG lltUjJ1mUllnltl tGp pWIll1

~mGllypw ltbtpUJWlratnpMmllTlptjl1lU t pWM ~mGllypwJP wo-wGyuwG llwGnGrr

LU-hnphu 1 Gph t=g(X) fPmuljyuuuzu6utuyhlPt Xo ljhlfmul

pulj y = f(t) fPmuljypzuu to = g(xo) ljhlfmul ZUUJZU F = fog fPmuljshy

ypzuu zu6utuyhlP t Xo ljhlfmul b

F(xo)= f(g(xo)) g(xo)

UJu ratnpMrr utGp ltGp wlllwymyp ~uwybpllltGpu wlllwymytGp WJG uwushy

GwtJnp lltUjJ1mU tpp g -G qUwJpG ~mGllypw t g(x) = kx + b

LU- h n p h u 2 Gph f fPmuljypzuu zulJzuuyhlP t ZUUJZU

F(x) = f(kx+b) fPmuljypzuu umJuUJhu zu6utuyhlft t b

F(x) = kmiddot f(kx + b)

UUJ w y n 1 y n 1 u i)pymp hn -G wGtJtp2 tPllJ2P t UJlllltUWmu wGtJtP2 tPnpp

t Gwu kmiddot hn n EN hW2npllwllwGmraJmGrr mutpp

F(x+hJ-F(x) f(kx + khl1 +b)- f(kx+b)hm = hm = n-jOO n-jOOh h

n n

= k lim f(kx + b + kh) - f(kx + b) = k f(kx -I

--00 kh

o p p uw q 1 qtpGtGp y = (3x - 5) 100 ~mGllypwJP wo-wGYJwm

brat GzwGWlltGp f(x) = x lOO WlllW lTlPtJwa ~mGllypwG llutpwGw y = f(3x - 5)

tptuprr UJlllltUWmu f(x) = lOOx99 ltbtpuwpwp

~3x - 5yoo ) = 3 f(3x - 5) = 3 middot100 (3x - 59 = 300 (3x - 5r 57

bull bull bull