Winter2004Solutions

Winter2004Solutions - Vq< n(x 3$tsj e-s8$ts$Bt...

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Unformatted text preview: Vq< n(x 3$tsj e-s8$ts$Bt 7BjZ$BtR V}„$j 1((; d R\10 {BRwn0Di )wsD wsD X1 wX  1D1 L  wAD X wX  1D1 L  w{D X1 X1 L  wID 1X X1 L  w‚D WBt‚ BP Y‚ sAB%‚ 1 R\0 R$tw10Di )wID wsD wAD w{D wID wX1 X L ;D1 wX1 X L ;D1 1 wX  dD1 L ; wX1 ;X L ;D1 w‚D WBt‚ BP Y‚ sAB%‚ k F n1X d )wAD n R X1 L X  1 wsD `w0  1D >0  10 H wAD `w0  1D >01  10L; H w{D 01  1w01D wID `w0D0  `w0  1D01 w‚D WBt‚ BP Y‚ sAB%‚ F d ; R X 1 L 1X L d( X k )wID wsD 0 {BRwn0D wAD 0 R$twn0D w{D 0 {BRwn0D  0 R$twn0D wID 0 >{BRwn0D  d R$twn0DH n w‚D WBt‚ BP Y‚ sAB%‚ x qY‚ ‚t‚„sj RBjZ$Bt BP Y‚ I$&‚„‚t$sj ‚JZs$Bt 1?1 r II  x?r I L nr ) ( %sj$I PB„ ? W) ( $R $%‚t Ak wsD d wsD d /?/n L 1 /?/ 1 wAD /?/n >d L 1 jt /?/H W W] w w } d d n jt /?/ L 1 R$t jt /?/ w{D /?/ d {BR 1 1 d wID d /?/ 1 L 1 /?/n jt /?/ w‚D WBt‚ BP Y‚ sAB%‚ E qY‚ ‚t‚„sj RBjZ$Bt BP Y‚ I$&‚„‚t$sj ‚JZs$Bt ?1 r II L 1?r I L d r ) ( %sj$I PB„ ? W) ( $R $%‚t ; Ak wID X$ ~ X$ ~„ n n jt /?/ L 1 R$t jt /?/ d {BR 1 1 ^ wsD /?/d d wAD d /?/ d L 1 /?/  n 1 d w{D d /?/ 1 L 1 /?/ 1 d wID /?/ 1 wd L 1 jt /?/D w‚D WBt‚ BP Y‚ sAB%‚ 1  qY‚ ‚t‚„sj RBjZ$Bt BP Y‚ I$&‚„‚t$sj ‚JZs$Bt ?1 r II L ?r I L w?1  ;Dr ) ( %sj$I PB„ ? k ( $R $%‚t Ak w{D $ $ wsD d O1 w  ?D L 1 O1 w  ?D wAD d O$ w1?D L 1 O$ w1?D $ $ w{D d O1 w  ?D L 1 V1 w  ?D wID d O$ w1?D L 1 V$ w1?D w‚D WBt‚ BP Y‚ sAB%‚ F k d, (  ? c d M N‚ W w?D ) Bt >(, 1H V ? )  Y‚ 3BZ„$‚„ R$t‚ R‚„$‚R BP W {Bt%‚„‚R B 1, d  ?  1 wID wsD 1 wAD n 1 w{D ( wID  n 1 w‚D WBt‚ BP Y‚ sAB%‚  qY‚ I$&‚„‚t$sj ‚JZs$Bt r II  1r I L ?r L r ) ( bY‚t }js{‚I $t Y‚ 7Z„8TN$BZ%$jj‚ PB„8 w€r I DI  Rr L „r ) ( YsR Y‚ b‚$Y PZt{$Bt „ w?D ) w{D wsD ?1? wAD ?1? w{D 1? wID ? w‚D WBt‚ BP Y‚ sAB%‚ n d( p$%‚t Y‚ wsD ‚RR‚j $I‚t$k d >MdO1 wED 1 dQ Q? >?" O" w ?DH ) ?" O" d w ?D, " k (, W) (,  1On wEDH wAD MdO1 wED  1On wED w{D 1On wED  MdO1 wED wID 1;n O wED x1 w‚D WBt‚ BP Y‚ sAB%‚ dd >\17?/?Ln/ i )w{D wsD 1n7 d L w  1D1 wAD 1n7w 1D d L 1 1n7w 1D w{D d L w  1D1 wID 1n7 d L w L 1D1 w‚D WBt‚ BP Y‚ sAB%‚ 1 d1 >\?n? i )wAD $ $  1  d1 wsD n $ 7 $  1 wAD $  d1 En $ $ n 1  w{D n wID d n 1  1$ w‚D WBt‚ BP Y‚ sAB%‚ ; $n ( ?; Od w1?D Q? ) wsD F dn > d 17 d L w L nD1 k wsD d n7?/?1/  1 wAD d n7w?1D/?/  1 w{D d n7w?L1D/?L1/  1 wID )wID d n7w?1D/?1/  1 w‚D WBt‚ BP Y‚ sAB%‚ d; > d \ / / i )wAD wsD 17? $ wd L ?1 D1 wAD 17? $ wd L ?1 D1 w{D ? $ wd L ?1 D wID ? $ wd L ?1 D w‚D WBt‚ BP Y‚ sAB%‚ dx e8}jBk Y‚ Ns}js{‚ „stRPB„8 B RBj%‚ Y‚ $t$$sjT%sjZ‚ }„BAj‚8 r II  ;r I L dnr ) & w0  nD, r w(D ) d, r I w(D ) x X1 V wXD  Xr w(D  r I w(D  ;>XV wXD  r w(DH L dnV wXD ) nX p wX1  ;X L dnDV wXD  X  d ) nX p V wXD ) XLd nX wX  1D L n nX L1 ) L X1  ;X L dn X  ;X L dn wX  1D1 L  wX  1D1 L  d p r w0D ) 10 {BRwn0D L 10 R$twn0D L `w0  nD1w0nD R$t>nw0  nDH3 n x dE 3$tI Bt‚ wtBtT9‚„BD R‚„$‚R RBjZ$Bt rd BP Y‚ I$&‚„‚t$sj ‚JZs$Bt ?r II L 1?r I L 1r ) ( %sj$I PB„ ? k ( t‚s„ ?( ) ( e-}„‚RR Y‚ RBjZ$Bt sR st ‚j‚8‚ts„k PZt{$Bt ?€w?D ) 1?, €( ) (, ?1 R w?D ) 1?, R( ) ( p „1 Lw€( dD„LR( ) „1 „ ) „w„dD ) ( p „d ) d p r) 3 yz ?zLd , r I ) z)( p 3 3 3 wz L dDyz ?z , r II ) z)( zwz L dDyz ?z L z)( 3 zwz L dDyz ?zd z)( 1wz L dDyz ?zLd L z)( 3 1yz ?zLd ) ( z)( p wz L dDwz L 1DyzLd L >1wz L dD L 1Hyz ) (, z  (, p yzLd )  1yz , z  (3 zLd 11 y( 1y( 1yd , z ) d p y1 )  ) , d 1 dN1 1y1 1n y( z ) 1 p yn )  ) , ‚{ RB n dN1Nn 3 wdDz 1z y( 3 w1?Dz wdDz 1z y( zLd yz ) , z  (, p r ) ? ) y( ? ) y( ?1? 3 z z z z)( z)( z ) ( p yd )  F d 3$tI Y‚ 3BZ„$‚„ {BR$t‚ R‚„$‚R BP W w?D ) ?, (  ? c d (, d  ?  1 k Bt >(, 1H p z$? Q y( 3 yz {BR qY‚ R‚„$‚R $R L , b$Y 1 1 z)d 8d 8 11 d W w?D Q? ) ? Q? ) , stI PB„ z  d, y( ) 1( 1 ( 81 8d p z$? Q p z$? Q 1 yz ) W w?D {BR Q? ) ? {BR Q? 1( 1 1 ( 8d p z$? Q ed p z$? Q e 1 e1 ) ? R$t R$t Q? z$ 1 e( z$ ( 1 p z$? Q ed p z$ Q e 1 ; e ) R$t L 1 1 {BR z$ 1 z$ 1 e( p z$ Q = 1 ;  p z$ Q ) R$t L 1 1 {BR d z$ 1 z$ 1 qYZR Y‚ {BR$t‚ R‚„$‚R BP W $R F p z$ Q =k p z$? Q ;  p z$ Q d31 L R$t L 1 1 {BR  d {BR 3 ; z)d z$ 1 z$ 1 1 E d dM qY‚ RBjZ$Bt BP Y‚ Y‚s ‚JZs$Bt `?? ) 1 `0 , ( c ? c U bY${Y Rs$Rh‚R Y‚ ABZtIs„k {BtI$$BtR `? w(, 0D ) `? wU, 0D ) ( YsR Y‚ PB„8 `w?, 0D ) p z$? Q 1 z1 $1 y( 3 L yz {BR  U1 0 3 1 U z)d 3$tI Y‚ RBjZ$Bt BP `?? ) d `0 , ( c ? c $ bY${Y Rs$Rh‚R Y‚ ABZtIs„k {BtI$$BtR  `? w(, 0D ) `? w$, 0D ) ( stI Y‚ $t$$sj {BtI$$Bt `w?, (D ) n {BRw1?D  1 {BRwx?D G„$‚ IBbt Y‚ {B8}j‚‚ RBjZ$Bt `w?, 0D ) n, U ) $ p `w?, 0D ) n {BRw1?D  1 {BRwx?D ) `w?, (D ) y( 3 1 yz {BRwz?Dz 0 3 L 1 z)d y( 3 L yz {BRwz?D p y1 ) n, yx ) 1, yz ) ( BY‚„b$R‚3 1 z)d qYZR `w?, 0D ) n {BRw1?DnE0  1 {BRwx?D11x0 3 d qY‚ RBjZ$Bt BP Ns}js{‚:R ‚JZs$Bt `„„ L d `„ L „ `w„, D ) d ` „ 1  ) ( $tR$I‚ Y‚ {$„{j‚ „ ) y YsR Y‚ PB„8 y( 3 z „ >yz {BRwzD L Kz R$twzDH3 L 1 z)d 3$tI Y‚ RBjZ$Bt BP Ns}js{‚:R ‚JZs$Bt $tR$I‚ Y‚ {$„{j‚ „ ) 1 bY${Y Rs$Rh‚R Y‚ ABZtIs„k {BtI$$Bt `w1, D ) 1 L R$tw1 D  {BRw D G„$‚ IBbt Y‚ {B8}j‚‚ RBjZ$Bt `w„, D 1 L R$tw1 D  {BRwD ) `w1, D ) y( 3 z L 1 >yz {BRwz D L Kz R$twzDH 1 z)d y( ) 1, 1yd ) d, 11 K1 ) d, yz ) Kz ) ( BY‚„b$R‚ qYZR 1 d d d d p `w„, D ) 1  „ {BRw D L „ 1 R$tw1D3 y( ) ;, yd )  , K1 ) 1 ; 1 ; p  1( 3$tI sjj ‚$‚t%sjZ‚R stI {B„„‚R}BtI$t ‚$‚tPZt{$BtR BP Y‚ 7Z„8TN$BZ%$jj‚ }„BAj‚8 r II L r ) (, r w(D ) (, r I wdD ) ( $ r ) „? p „1 L ) ( p „ )   3 c ( p „ )  p r ) \ ? L 4 ? , r w(D ) ( p \ L 4 ) ( D i D i D i p r ) \  ?   ? , r I ) \  ? L  ? , r I wdD ) ( p \  L  ) ( ) 1 p \ ) ( p 4 ) ( p r # (3 ) ( p r ) \?L4, r w(D ) ( p 4 ) ( p r ) \?, r I ) \, r I wdD ) ( p \ ) ( p r # (3 ) 1 k ( p r ) \ {BRw ?D L 4 R$tw ?D, r w(D ) ( p \ ) ( p r ) 4 R$tw ?D, $ r I ) 4 {BRw ?D, r I wdD ) ( p 4 {BRw D ) ( p z ) w1z L dD , z  ( 1 } w1z L dD1 $ 1 w1z L dD$? p z ) , rz ) 4z R$t , z  (3 ; 1 M ...
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This note was uploaded on 07/16/2009 for the course MATH 3705 taught by Professor Jaberabdualrahman during the Winter '08 term at Carleton CA.

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