ZADANIEI No . pmestmeni V=E([ On ] ) 2 normq supremum defimiujemyodwzonowanie F : V → R

pay ) = §t2yf)o(tt) at

2 modest 17nF oraz abaolac'

rd znicskowakwsd F.

ZADANIE 1 Lnalezic 'i www.dpuuktyknytycsnefunkyi A. y ) 1- zcx , y )

doing.

miejawnie riwmauiem

0=21 (×4y2)z3 + ×yz2 -2-2

ZADANIE 3 Dba TEJZI , # [ snalezi Noswipsanie ogolne niwnauie liuioweyoit - 2x+2×= ethntcost

ZADANIEG Bo2wip2ac'

sougaoluienie poaptkowe

yy

"=ly'

)2log1y

'tyco )=1 ykot - %

ZADANIE 5 Ziralezi moment bezwtadwosa.

jeduorodnego stozkd o promieniupodstawyni wysokosci h wsglgdem osi sawierajgcejtwompcq tego Hozke.

ROZWIARANIA

�1� Faith )=§EH+h)H)@+h)a - ⇒ at =§EoHoH⇒att§t2hHw(ttldttfeottlhlt - ⇒ at

+ §t2hH)hGt)dt s=tt to - s

ds= - at

17nF (a) = IEHHIOA .t )dt+§EoH)h(ttjatetothttloattdttfla. stole . this )tds)=

=§t2hLHv(

t.tl#tfutthHwG-tloH=ftt2ta-t5)v(tt).hk)dtTfF(v)=hfH2tctts4rh-t)hHdtWidai,zehtDnFwjestodwsonowauiemliniowym.Sp~awobamycsyjesttoodwsorowauieciggre1DnMv)1=/flt2th-tt)vat)hHoHlffhH2tcn-ty)1olttI11hltHat

f

t #tttttt '

Noh - that]÷state licsba salezneodpuuktuwktinym vizhicskujemy

ht > 17nF jest ogrranicsonea wigc cigofe .

Resale r( ah ) ma postal NCJ

,h)=jt2hlt )hH - Holt

1 0z

In ( v ,h)K

ft2/hH)h(t.tl/dtf1lhH2ft2olt=llhll4z:

( v. 41 s

thf 3- 11h11 - O

11h11 - SO

IN ( V. 4) 1 11h11 → o

Resort spetnie waruuek - - 0,

2oitemFjestroznicskoWalnewdowolnymHh4puukueo.Fkv1h-DnFCv1-fh-Lt2tG-H2JoH-HhCtsdt@O-tgK4y4z3txyz2-2-2-iF1x.y

,2)

}xI=tzz?2×+yz2=×z3+yz2 Punkty krytycsne speiuiajg

0¥ tz 232g + ×z2= yz3+×z2 ¥T=0=fFJ= Ffxiyiz )

×z3ty22=oT¥=(×4y93zz2+2×yz . 1

{yz3t×zEo

( Yzxttyyzstxyz'

- 2-2=0

22 ( xzty )=o22 ( y2+×j=o } ⇒ 2=0 two ( ×2+y=0

/ yzt×=o)

LY

y=- xz

Flxiy ,01=-2+0

- ×z2+×=o × (1-22)=0Latam 2=0 nie jest t ↳ z=±1

wowtosciq ( xiy )l→2( xiy ) ×=o

y=o / )F ( 0,0 , 2) =

- 2-2=0 2=1 2=-1

= 7 2=-2 Xty = 0 X=y×= - y

Ffxix ,- n ) =

Ffy ,y , 1) ty?ty9F -

y'

-1-2 =

=L . 2×2 (a) + tfn ) -11-2= -3 to

to hie iest puukt kvytyay= -2×2-1=0

spmeosne had IR

Jedyuym puuktem podcjmanym jest ( 0,0 ,-2 )

3¥ (0,9-2)=-1+-0 ( x. y ) to ZC x. y ) istuiqie w otoucuin ( QQ - 2)

Badamy nooboj puuktu kvytycsnego .Nie pamigtam waoru me 2

"

, mung wyprowaobii

FC x. y ,z)=O → Fxt FzZ×=0 I Fyt Fzzy=0 → Fxx + Fxzzx + Fz×2× + Fzz2×2 + Fz2××=0*

w punkcie kvytycshymZ××= - ¥ F×× Dalejjuzmie hisq ,

bo pmypomniouamSobie wzir

2 "= - ¥µ¥xy+¥Y] R×= ×z3+yz2 Fxx =

231 ( o ,o ,. 2)

= - 8

Fy= y 23+1/22 FYY = 23/ ( 0,0 ,

. zj= - 8

Fxy=24( ao , -254

2"

( 010k¥ [ L}]=[ 8g} ] PjIjy8aP=u, > o Drugepoawdmeujemnie okreslone,

maksimum

�3�

i - 2x+2×= ethntcost

ROR ] : R - 2×+2=0 0=4 - 4.2=-4 ro=±2i xyz = (2±2Dk = 1±i

Xnlt ) = ethnt

xalt ) = et cost RORJ : ×Ct)= et( A hint + Boost )

Uzmiennianie stotych

o¥Fd=Es1 # + Enron ]

⇐ HIS.tt#IFFaxaIkIItttadaoottEdto*Ed+FEEHftnfxIxMItEnd

¥.⇒ attend listen ⇒

"

End-1

- zte+( cost . sint )

teeth:* ,¥EInµ]=- e feint. .+ ,¥Fn¥]

wyanacsnikt Et [ sin #- siit -sinttost - wit ]= - it.

.

¥

t.it#tIIIDE*totfeIEnII=fFd.A=tgtA=JsfnIat=tag1wstl=na, .¥¥[- logwst + to

D=-

tojt

B=ftg2tdt=feYutdnhu=f4- ¥

)au=u+ardgu+Bo=- tgtttt Bo

u=tgtdu = ⇐ at = ( ttu 2) at

AH )= -

logwstBCt1-t-tgtxCtkAetn.nttBetwst-etsintlogwstttetwst@yyhly52log1y1yCot1yko-ttsPodstawiamyrj-ocy1nj-odyeytoddFyotodF-okloglo1yodyI-ologlo1vdbrgq-0hytlog1log1vl1-log1yl.tlogAAsologAlylllog1olI-A1y1jes.liAdowoluego2nakuloglvI-Ay1vKexpAyVCyktexpAyyIexpCAy0Yxto-texpCAy1expfAy1oly-tdx-1n-expfAy.t

.xtc \

>y '(o)= - tg y( 01=1

expl Ay )=±A×+ E

-

£=±of×P(A ) ⇒ A= logtz - - logosAy=loy(€Ax+e)ycx

)=atlog(±oAx+E)

61= -

wgtlog ( log 2.0 + E) 1=-4,loge logE= - log2 E - %

ycxtwtgglog (@g2)x+f)

�5�

G 0 's obrotu t H f¥rn]

:Wektovpnostopowlty do oh

.

obrotu, sanepiony

me oh.

okoncnw (E) to

. §ggyn) wektorteuspeinia wowunek

( Hetty ] l My] )=o=ry - tithe.org#ytnz-tCi+nYv2+h2

Odlegtosi punktu ( x. yiz ) od on.

obrotu ( w lxwaowacie )

d4×iyR)=x' + ( y- tr5+(z-thj '=x2

ty't 22 - 2t( yrtzh ) + t4i+h ) ;

wstawiamyt=x2+j+i . 24.11¥ +

G.FI#==x2ty4z2.Cyrtznj2r2th2x=N5I=fgd2Kiyi2)dxolydz=gf(x4y2tz-bIInh5)dxdydz=Y-

Mz=h }

=s{( i (5+5)+425 -

Kathryndgdyd ,

=

01×08012 = " hdsdyds

stozek%<za 5-

Rwsy01501yd }

=

RdRdyo}styles y=Rhny=gih§.@k+n's '.

Ethernet) Rdrdyds =

1 2 3

= goth ftp.R3drdydsthy.5RDRdydg - #§zR3wsydRdyds+a

Ss : ye -6,2T ]ten # sinydrdydg )

RETOM] 1 : Poly §dR § 5 R3=2I§R3dR(A - R )=2I§@3- R

")dR=

SEEM ]at ( al . f) =2Ifo=€

2 : §tdy§dR§d}5R=2F§RdR 13331 ,f=§t§RG - R'

)dR=÷I( tz - f) ±sI÷o=2nIo

3 i 4 niedajg wktaobe do catki,

bo sawicrojpwukg 2 his iws pookresie

I - gih . ( into + h2 2¥) . gino 'Fo( n' + 2h'

)

obj Aozke : fTr2h jes.li nose to M tog= gma

I=g¥#µIn¥4724 = Tom (n2 + 2h'

)