Winter2003Solutions

Winter2003Solutions - d Vq< n(x 3$tsj e-s8$ts$Bt 7BjZ$BtR V}$j 1(n d R\0n 10 i wID wsD wAD w{D wID E wX L 1Dn E1X Xn E wX 1Dn E wX 1D wD WBt BP Y

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Unformatted text preview: d Vq< n(x 3$tsj e-s8$ts$Bt 7BjZ$BtR V}$j 1((n d R\0n 10 i ) wID wsD wAD w{D wID E wX L 1Dn E1X Xn E wX 1Dn E wX 1D; wD WBt BP Y sAB% 1 R\n0 {BRw;0Di ) wAD wsD wAD w{D wID X wX L nD1 L dE XLn wX L nD1 L dE nX X1 L dE XnX X1 L dE wD WBt BP Y sAB% k F nX d ) wsD n R X1 1X L x wsD wAD w{D wID d `w0 1 d `w0 1 d `w0 1 d `w0 1 nD0n R$tw10 ED nD0 R$tw10 ED nD0 R$tw10D nDn0 R$tw10D wD WBt BP Y sAB% F k nX d ; R ) w{D wX1 L D1 1 wsD 0 R$twn0D wAD 0 R$twn0D w{D d 0 R$twn0D 1 d wID 0 R$twn0D 1 wD WBt BP Y sAB% x qY tsj RBjZ$Bt BP ;?1 r II M?r I L r ) ( %sj$I PB ? W) ( $R $%t Ak wID wsD d /?/nb1 L 1 /?/nb1 ^ X$ ~ X$ ~ x x jt /?/ L 1 R$t jt /?/ wAD /?/ d {BR 1 1 w{D d /?/ L 1 /?/ $ xb1 wID d /?/nb1 L 1 /?/nb1 jt /?/ wD WBt BP Y sAB% E qY tsj RBjZ$Bt BP ?1 r II L ?r I L wx?1 Dr ) ( ts ?( ) ( %sj$I PB ? k ( $R $%t Ak wAD $ $ wsD d On w x?D L 1 On w x?D $ $ wAD d On w x?D L 1 Vn w x?D w{D d O$x wn?D L 1 O$x wn?D wID d O$x wn?D L 1 V$x wn?D wD WBt BP Y sAB% V ? ) Y 3BZ$ R$t R$R BP W w?D ) ? Bt >(, dH {Bt%R B w{D wsD d wAD d w{D ( wID d 1 wD WBt BP Y sAB% n M qY I$&t$sj JZs$Bt ;?1 r II M?r I L r ) ( bYt }js{I $t Y 7Z8TN$BZ%$jj PB8 wr I DI Rr L r ) ( YsR Y b$Y PZt{$Bt w?D $%t Ak wID wsD ?1 ; ?1 ; wAD w{D wID ? ; ;?; wD WBt BP Y sAB% >\n7?/?1/ i ) wsD wsD wAD w{D wID 117wLnD d L w L nD1 117wnD d L w nD1 117wLnD d L w L nD1 117wnD d L w nD1 wD WBt BP Y sAB% d( >\w? nDw?nD i ) wID wsD wAD w{D wID $ wnD1 7 $ w nD ; 1 $ wLnD1 7 $ w L nD ; 1 $ 7 $ n7 1 ; 1 $ 7 $ n7 1 ; 1 1 1 wD WBt BP Y sAB% dd > d \ i ) wID ; wsD ?? wAD w{D 1 ?1 d $ ? ; 1 $ 7 1 $ ?;? ; $ ?1 7 wID $ ? ; ; $ wD WBt BP Y sAB% d1 > d \// i ) wsD wsD d d $ d L ?1 wAD /?/ w{D 1 d L ?1 d wID $ ? $ wD WBt BP Y sAB% dn 7Bj% Y $t$$sjT%sjZ }BAj8 r II L r I 1r ) &w0 1D rw(D ) d, r I w(D ) 1 7BjZ$BtU X1 V wXD Xrw(D r I w(D L >XV wXD rw(DH 1V wXD ) 1X p wX1 L X 1DV wXD X L d ) 1X 1X d 1X p V wXD ) X1Xd L X1LX1 ) XL1 L wXL1DwXdD LX1 d ) \ L 4 p \ ) d, 4 ) d wXL1DwXdD n n + XL1 Xd \ d d 0 d d d d d p R XL1 ) n w 10 D ) nR wXL1DwXdD Xd J 01 o p rw0D ) 10 L d `w0 1D 1w01D n d; ]BtR$I Y I$&t$sj JZs$Bt ?r II r ) ( ? k ( 3B Bt wtBtT9BD R$R RBjZ$Bt rd sABZ ?( ) ( wsD htI Y {B^{$t {ZR$Bt js$Bt wAD RBj% Y {ZR$Bt js$Bt 7BjZ$BtU d wsD w?D ) (, Rw?D ) ? , ?w?D ) (, ?1 Rw?D ) ? p ( ) R( ) ( p w dD L ( L R( ) w dD ) ( p d ) d, 1 ) ( !t RBjZ$Bt $R r ) yz ?zLd RB z)( x rI ) z)( z)( wz L dDyz ?z , r II ) z)( z)( zwz L dDyz ?zd stI ?r II r ) ( p zwz L dDyz ?z yz ?zLd ) ( p wz L dDwz L 1DyzLd yz ) ( yz $ yzLd ) wzLdDwzL1D , z ( y( yd y( wAD z ) ( p yd ) dN1 , z ) d p y1 ) 1Nn ) dN11 Nn , y1 y( y( z ) 1 p yn ) nN; ) dN11 Nn1 N; { RB yz ) wzD1 wzLdD , z ( F k d, ( ? d 3$tI Y PZjj 3BZ$ R$R BP W Bt >(, 1H dx N W w?D ) (, d c ? 1 7BjZ$BtU D i D io J qY R$R $R y1( L yz {BR z$? L Kz R$t z$? b$Y 1 1 z)d J D io D z$? i D z$? i D i ed $d $1 1 1 Kz ) 1 ( W w?D R$t 1 Q? ) ( R$t 1 Q? ) z$ {BR z$? e( ) z$ d {BR z$ , z d, 1 1 D i D i D 1i ed D i $1 $d 1 1 yz ) 1 ( W w?D {BR z$? Q? ) ( {BR z$? Q? ) z$ R$t z$? e( ) z$ R$t z$ , z d, 1$ 1 1 1 $1d 1 y( ) 1 ( W w?DQ? ) ( Q? ) d qYZR Y R$R $R 1 D i D i J D io D io J1 d 1 L R$t z$ {BR z$? L z$ d {BR z$ R$t z$? 1 z$ 1 1 1 1 z)d dE qY RBjZ$Bt BP Y bs% JZs$Bt `?? ) `w(, 0D ) `wU, 0D ) ( YsR Y PB8 `w?, 0D ) 3 z)d d `00 , ( c ? c U Rs$RPk$t Y ABZtIsk {BtI$$BtR 1 z$0 yz {BR U w W z$0 L Kz R$t U w W] 3 R$t p z$? Q } U 3$tI Y RBjZ$Bt BP `?? ) `00 , ( c ? c $ bY${Y sjRB Rs$RhR Y $t$$sj {BtI$$BtR `w?, (D ) ( `0 w?, (D ) ?w$ ?D 7BjZ$BtU R$twz?D>yz {BRwz0D L Kz R$twz0DH U ) $, ) d p `w?, 0D ) `w?, (D ) ( p z)d z)d yz R$twz?D ) ( p yz ) (, z d `0 w?, (D ) ?w$ ?D p ?w$ ?D ) zKz R$twz?D p z)d e$ $$ $$ 1 1 1 zKz ) $ ( w$? ?1e R$twz?DQ? ) z$ w$? ?1 D {BRwz?De( L z$ ( w$ 1?D {BRwz?DQ? ) D e$ $$ 1 ; ; e$ R$twz?DQ? ) z;$ {BRwz?De( ) $zn >d wdDz H qYZR 1 $ w$ 1?D R$twz?D ( L z1 $ n z ( ; d M `w?, 0D ) >d wdDz H R$twz?D R$twz0D ) $ R$t>w1> L dD?H R$t>w1> L dD0H $z; w1>LdD; z)d >)( d qY ABZtII RBjZ$Bt BP Ns}js{:R JZs$Bt ` L d ` L d1 ` ) ( BZR$I Y {${j ) y YsR Y PB8 y( 3 z `w, D ) L >yz {BRwzD L Kz R$twzDH 3 1 z)d 3$tI Y ABZtII RBjZ$Bt BP Ns}js{:R JZs$Bt BZR$I Y {${j ) d RZA`{ B Y ABZtIsk {BtI$$Bt `wd, D ) d {BRw1D L R$t E 7BjZ$BtU d {BRw1D L R$twD ) `wd, D ) z)d y( 1 L >yz {BRwzD L Kz R$twzDH p y1( ) d, y1 ) d, Kd ) d stI yz ) Kz ) ( BYb$R qYZR `w, D ) d 1 {BRw1D L d R$twD dM 3$tI sjj $t%sjZR stI {BR}BtI$t $tPZt{$BtR BP Y 7Z8TN$BZ%$jj }BAj8 r II L r ) (, r I w(D ) (, rwdD ) ( 7BjZ$BtU ) 1 c ( p r ) \ ? L 4 ? , r I ) \ ? 4 ? , r I w(D ) ( p w\ 4D ) ( p 4 ) \ p r ) \ w ? L ? D , rwdD ) ( p \ w L D ) ( p \ ) ( p 4 ) ( p r # ( qYZR Y s tB $t%sjZR c ( ) ( p r ) \?L4, r I ) \, r I w(D ) ( p \ ) ( p r ) 4, rwdD ) ( p 4 ) ( p r # ( qYZR ) ( $R tB st $t%sjZ ) 1 k ( p r ) \ {BRw ?D L 4 R$tw ?D, r I ) \ R$tw ?D L 4 {BRw ?D, r I w(D ) ( p ) z ) w1z L dD $ , z ( 4 ) ( p r ) \ {BRw ?D, rwdD ) ( p \ {BRw D ) ( p 1 1 $1 w1zLdD RB Y $t%sjZR s z ) 1 ) = ; , z ( b$Y Y {BR}BtI$t $tPZt{$BtR z rz ) \z {BRw z ?D ) \z {BR w1zLdD$? 1 ...
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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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