00020002010007MVT.ACT0001020007MVT.EAC010000002cc7  MVT.EAC ACT#&*= ~I '-[497Geometric Motivation to the Proof of Mean Value Theorem?\G Author^Qfessor Wei-Chi Yang Radford University 9e-mail: wyang@r).edu<!URL: http://www.#/5 Objece[RWe shall see how we cme p  by using9)1geometric construction by using ClassPad.[ Backgrounds:1. Rolle's Theorem:[8PLet f be a funfthat is{tinuous on [a,b] and differeable(). If Lf(a)=f(b)=0,theT2U[leastS e number c iD such f'(c)=0. R2. Mean Value Supposeef:eRʏC҇0 point*whif(c)=((f(b)-f(a))/(b-.[Y3. To prove this.eorem, in many traditional text books, we find6e following func- aF Ude+ ed F(x)=f- 2(x. We arolr at F satisfie conzUof Re's Tsoere a point c (a,b) such_=0, w0MVT 7Pholds. A geometric motiva) onttgraph y= will be given .[%Example.[UWe consider f(x)=cos ov[a,b]=[-/2, 0.725]. 3shall9truct the function F \Xsoat F'(c)=0 where,point c is in (a) satisfying f:f(b)-f(a)/(b-a).\Play{anima|:Έ  !Y''0`Cs#` YXBW y D:vKr   A 'L'nC uxy qq H  cos(x)-163 /500-5121/10000&,@ 52>2W2p`R9C4 Gp5@C76$838C03e19P<(x@H$(%BtTYrC`E{l"fF`TG``I` apuY EYh2lY!F`$t)U`0'D%t0XY3sf06`@)`vB0F2FVYGp0RaYvVDYX'0%oYe g9057RpYvd@ FGx " `  X#$ x`< bvAd  aqQVG7T 21#0Pa 0 HC 0sPh 7XU DA01)@'Pxp0%Ԍ2 `4w0 Bpu%!GUue whfBbCFdq0$u!Iw2sS("60'dU#` qb s 1$P p0!chP<)Idb05BwH97F$A0@ `8Irl4aT($lF9xb`HT8e(b90C7t!X= dqTiv0C2)hu`d) F$H  GH8qT14v\s,U`Dz~"{ cos(x)-163 /500-5121/10000 22W2p `R9C4 Gp5@C76$838C03e19P<(x@H$(%BtTYrC`E{l"fF`TG``I` apuY EYh2lY!F`$t)U`0'D%t0XY3sf06`@)`vB0F2FVYGp0RaJYvVDYX'0%oYe g9057R0vd@YFGx " `  X#$ x`0 bvAd  aqQVG7T 21#0Pa 0 HC 0sPhp`7XU DA01)@'Pxp0%Ȍ2 `4w\0 Bpu%!GUue whfBbCFdq0$u!Iw2sS("60'dU#|qb s 1P p$!chP0)Idb05BwH97F$A0@ `8Irl4aT($lF9xb``HT8e(b90C7t1!X= dqTiv0C2)hu`d) F$H  GH8qT14vxs,U`Dot!{ cos(x)-163 /500-5121/1000 "  cos(x)-163 /500-5121/10000& - 62?2W2p`R9C4 Gp5@C76$838C03e19P<(x@H$(%BtTYrC`E{l"fF`TG``I` apuY EYh2lY!F`$t)U`0'D%t0XY3sf06`@)`vB0F2FVYGp0RaYvVDYX'0%Ye g9057RpYvd@ FGx " `  X#$ x`< bvAd  aqQVG7T 21#0Pa 0 HC 0sPh 7XU DA01)@'Pxp0%Ԍ2 `4w0 Bpu%!GUue whfBbCFdq0$u!Iw2sS("60'dU#` qb s 1$P p0!chP<)Idb05BwH97F$A0@ `8Irl4aT($lF9xb`HT8e(b90C7t!X= dqTiv0C2)hu`d) F$H  GH8qT14v\s,U`Dz~  {22W2v`R9C4`Gp5@`C76 838 3e19P$(x@0$(%B0<YrCpHEST"fF`TG``I` apuY E h2lY!F`$t)U`0'D%t0XY3sf06Y@)`vB0F2FVYGp0RavVYX'0%,Ye g9057R0vdhYFGxh " `  X#$ x bvAd  aqQ VG7@ 21#` Pa  $C 0sPhp<7XUGHDA01)@'P`xp0%tx2 `4w\0 Bpu%!GUue whfBbCFdq0$u!Iw2sS("60'4U#LqbsL1|P L!ch4)Idb|5Bwd97FP A0@  8Ir 4a0($p<F9 b`ST`HT8e(0b90xC7t1 !X= dqTiv0C2)hu`d) F$H GHqT14vxsU`D *@# <H -C@6S%? I UB' @  H A@26W2g&`HmH]H!oG] r YVA  ]U% `AUWtYZ1! cos(x)-0.326 5121 H8N!N ͔w  H    150.W2g-+5++ =D@Y6` r_Y08r p( (2e1 @ވ. BJ!`2:2>pj `R5wa GcX~B@ 0$8!IP03Gx<(t<$e7`0Ff X'@ $`$S0Aq< V1"<Y sD bQW)i h Ysdt &B!eTy 4W 1GIYv`A 7Ts32%pC  YHg% E7PT#P cwf%x`F9tf  XEGD )qb v)Xp #QExps2  e 8 Y`8 h' TU5 &BC HH)QpY00Bg B2 2 33@Pyv)\2vAPhFG`tYtFR@vtH5H5'6Dc8xT`2RFA)<eV@x`d c)1(`H$w#eED  b&$ahL"(x'F2gAG|4Q&4%T1`'4eD c1p rP0 Y86@$4X$qfDU0Si83Y$XP< ``Qt8l1t"T&hg0`x2#<v9`VuX)0t!H0wcW#+.@ !3!"  cos(x)#0H H(6G%V Y Ge&AH]$a B 4BRa%Bd'Xch "!&.H"#$"%'C1HL2PH(P9"?1@q6xgxVh& ` &sx G Coord of G: %)DHGo$wWg  "p s(O* # cos(x)-163 /500-5121/10000&+  <,g1@!-} 22W2R5wa GcX`B@ 0`8!IP 3Gxp(t$$e7/00Ff <X'@H $`TS`Aq< V1"<Y sD bQW)i h <sdt &B!eYy 4W 1GIYv`A 7Ts32%YC  YHg% E7,YT#P cwf%xPYF9tf  XEGD )qb v)Xp #QExps2  e 8 Y`$h' TU5 <&BC HH)QpY0`Bg [2_|BP85ch8 Ic@&IV&U0rx!FEgFssTw'"R"uCB!`xt(93drWSf!(t4)U(#Tpp`w@#xH(0pPp61@9V AD @H 8t!32 &(r@0$0`<X7PHF6 up`WWHl7GHlIxEc4gliXi0$4%(IWc4F3`4HD#VxEv%H)-.*@ 37/<H F6? rm YiRcos x-1635006-!5121,10000RU[!solve(diff(F(x),x)=0,x,0,-1,0) Rx=-" 0.33206932262"Y [[.Therefore, if c=-= , we get F'(c|.6[RStep 4. Finally1drag y= back to thIometry strip and note that!this V1graph coincides withFoneobtained earliereActFH(xf(x)-(0.326*x+0.5121)F1H8&x(x+1)*(x-1)*(x-2)-(0.6056*x+0.6056)fHxP(x)f1H$x(x+1)*(x-1)*(x-2)gHL<x1.003145306*x^3-2.022772314*x^2-1.585231986*x+1.421452671g1HL<x1.003145306*x^3-2.022772314*x^2-1.585231986*x+1.421452671010008main.ACT0001020012eActivity Save.EAC010000002cc7  MVT.EAC ACT#&*= ~I '-[497Geometric Motivation to the Proof of Mean Value Theorem?\G Author^Qfessor Wei-Chi Yang Radford University 9e-mail: wyang@r).edu<!URL: http://www.#/5 Objece[RWe shall see how we cme p  by using9)1geometric construction by using ClassPad.[ Backgrounds:1. Rolle's Theorem:[8PLet f be a funfthat is{tinuous on [a,b] and differeable(). If Lf(a)=f(b)=0,theT2U[leastS e number c iD such f'(c)=0. R2. Mean Value Supposeef:eRʏC҇0 point*whif(c)=((f(b)-f(a))/(b-.[Y3. To prove this.eorem, in many traditional text books, we find6e following func- aF Ude+ ed F(x)=f- 2(x. We arolr at F satisfie conzUof Re's Tsoere a point c (a,b) such_=0, w0MVT 7Pholds. A geometric motiva) onttgraph y= will be given .[%Example.[UWe consider f(x)=cos ov[a,b]=[-/2, 0.725]. 3shall9truct the function F \Xsoat F'(c)=0 where,point c is in (a) satisfying f:f(b)-f(a)/(b-a).\Play{anima|:Έ  !Y''0`Cs#` YXBW y D:vKr   A 'L'nC uxy qq H  cos(x)-163 /500-5121/10000&,@ 52>2W2p`R9C4 Gp5@C76$838C03e19P<(x@H$(%BtTYrC`E{l"fF`TG``I` apuY EYh2lY!F`$t)U`0'D%t0XY3sf06`@)`vB0F2FVYGp0RaYvVDYX'0%oYe g9057RpYvd@ FGx " `  X#$ x`< bvAd  aqQVG7T 21#0Pa 0 HC 0sPh 7XU DA01)@'Pxp0%Ԍ2 `4w0 Bpu%!GUue whfBbCFdq0$u!Iw2sS("60'dU#` qb s 1$P p0!chP<)Idb05BwH97F$A0@ `8Irl4aT($lF9xb`HT8e(b90C7t!X= dqTiv0C2)hu`d) F$H  GH8qT14v\s,U`Dz~"{ cos(x)-163 /500-5121/10000 22W2p `R9C4 Gp5@C76$838C03e19P<(x@H$(%BtTYrC`E{l"fF`TG``I` apuY EYh2lY!F`$t)U`0'D%t0XY3sf06`@)`vB0F2FVYGp0RaJYvVDYX'0%oYe g9057R0vd@YFGx " `  X#$ x`0 bvAd  aqQVG7T 21#0Pa 0 HC 0sPhp`7XU DA01)@'Pxp0%Ȍ2 `4w\0 Bpu%!GUue whfBbCFdq0$u!Iw2sS("60'dU#|qb s 1P p$!chP0)Idb05BwH97F$A0@ `8Irl4aT($lF9xb``HT8e(b90C7t1!X= dqTiv0C2)hu`d) F$H  GH8qT14vxs,U`Dot!{ cos(x)-163 /500-5121/1000 "  cos(x)-163 /500-5121/10000& - 62?2W2p`R9C4 Gp5@C76$838C03e19P<(x@H$(%BtTYrC`E{l"fF`TG``I` apuY EYh2lY!F`$t)U`0'D%t0XY3sf06`@)`vB0F2FVYGp0RaYvVDYX'0%Ye g9057RpYvd@ FGx " `  X#$ x`< bvAd  aqQVG7T 21#0Pa 0 HC 0sPh 7XU DA01)@'Pxp0%Ԍ2 `4w0 Bpu%!GUue whfBbCFdq0$u!Iw2sS("60'dU#` qb s 1$P p0!chP<)Idb05BwH97F$A0@ `8Irl4aT($lF9xb`HT8e(b90C7t!X= dqTiv0C2)hu`d) F$H  GH8qT14v\s,U`Dz~  {22W2v`R9C4`Gp5@`C76 838 3e19P$(x@0$(%B0<YrCpHEST"fF`TG``I` apuY E h2lY!F`$t)U`0'D%t0XY3sf06Y@)`vB0F2FVYGp0RavVYX'0%,Ye g9057R0vdhYFGxh " `  X#$ x bvAd  aqQ VG7@ 21#` Pa  $C 0sPhp<7XUGHDA01)@'P`xp0%tx2 `4w\0 Bpu%!GUue whfBbCFdq0$u!Iw2sS("60'4U#LqbsL1|P L!ch4)Idb|5Bwd97FP A0@  8Ir 4a0($p<F9 b`ST`HT8e(0b90xC7t1 !X= dqTiv0C2)hu`d) F$H GHqT14vxsU`D *@# <H -C@6S%? I UB' @  H A@26W2g&`HmH]H!oG] r YVA  ]U% `AUWtYZ1! cos(x)-0.326 5121 H8N!N ͔w  H    150.W2g-+5++ =D@Y6` r_Y08r p( (2e1 @ވ. BJ!`2:2>pj `R5wa GcX~B@ 0$8!IP03Gx<(t<$e7`0Ff X'@ $`$S0Aq< V1"<Y sD bQW)i h Ysdt &B!eTy 4W 1GIYv`A 7Ts32%pC  YHg% E7PT#P cwf%x`F9tf  XEGD )qb v)Xp #QExps2  e 8 Y`8 h' TU5 &BC HH)QpY00Bg B2 2 33@Pyv)\2vAPhFG`tYtFR@vtH5H5'6Dc8xT`2RFA)<eV@x`d c)1(`H$w#eED  b&$ahL"(x'F2gAG|4Q&4%T1`'4eD c1p rP0 Y86@$4X$qfDU0Si83Y$XP< ``Qt8l1t"T&hg0`x2#<v9`VuX)0t!H0wcW#+.@ !3!"  cos(x)#0H H(6G%V Y Ge&AH]$a B 4BRa%Bd'Xch "!&.H"#$"%'C1HL2PH(P9"?1@q6xgxVh& ` &sx G Coord of G: %)DHGo$wWg  "p s(O* # cos(x)-163 /500-5121/10000&+  <,g1@!-} 22W2R5wa GcX`B@ 0`8!IP 3Gxp(t$$e7/00Ff <X'@H $`TS`Aq< V1"<Y sD bQW)i h <sdt &B!eYy 4W 1GIYv`A 7Ts32%YC  YHg% E7,YT#P cwf%xPYF9tf  XEGD )qb v)Xp #QExps2  e 8 Y`$h' TU5 <&BC HH)QpY0`Bg [2_|BP85ch8 Ic@&IV&U0rx!FEgFssTw'"R"uCB!`xt(93drWSf!(t4)U(#Tpp`w@#xH(0pPp61@9V AD @H 8t!32 &(r@0$0`<X7PHF6 up`WWHl7GHlIxEc4gliXi0$4%(IWc4F3`4HD#VxEv%H)-.*@ 37/<H F6? rm YiRcos x-1635006-!5121,10000RU[!solve(diff(F(x),x)=0,x,0,-1,0) Rx=-" 0.33206932262"Y [[.Therefore, if c=-= , we get F'(c|.6[RStep 4. Finally1drag y= back to thIometry strip and note that!this V1graph coincides withFoneobtained earliereActFH(xf(x)-(0.326*x+0.5121)F1H8&x(x+1)*(x-1)*(x-2)-(0.6056*x+0.6056)fHxP(x)f1H$x(x+1)*(x-1)*(x-2)gHL<x1.003145306*x^3-2.022772314*x^2-1.585231986*x+1.421452671g1HL<x1.003145306*x^3-2.022772314*x^2-1.585231986*x+1.4214526710301cc002000080000000038c4xl\ǝ{_ZZݕHɊigKrstJYD^hHLgWRd"iTuX^Y+ zgܫzU t89:  潷3#9#ߏ4|~;oͼQGԫbAQ,(S&NLL* r@QF;A^{ޝH(3zR=&dUݾ1;~`xtdIɞx*we_JuCÃ_ezU|գVMW粷eosY˸n`UUrqY#5pY5rYsY;m..l#eK\6el6s). qYⲋ\v7epe.k+\v/]M.2U,e7~e_ֺl ,e!.e.em\N.qY/ŹQ.Kp%lR\6e;lvq"uq \e/sr٫\u.{z .{ˮs.{w*[gGe{l#0qYcׯsY'=e]\.rY/>.+qA^s\6e?q.)k\VಟqYK\s.{~ezZ{.{*U~6zx8.&쾖֩*Etˁyڍ*(;7;rS|SNǁzƐ!뇌<7^K,/>NfvݶsEf}FڟZ q[~ug n)~=eیeس'OdqOgX>RyXAqXi_uV+!Զڅ\ mOYI.#;r̽WSsw.f}LzFXyh: 붇=Lo,W%pm=:<ɶR~K,׿cn֕dVvឝtwWiCLȁO-t<;5z8˲Sj6˘Z`]65fyɐM~0ezv젽*nCt`Ol)z]zfGƎy$2̌~|d8ulyq,Gٷ->o0ەglV,2[}30lg`W =PzV.&Gb뙊?Bq?B%.~za[~~zn[~z~z~z~z~P?_*/|܏ G _#W2~˸/|܏ 4 G _}\&|܏ G _#p#W~~Up?_*<%Q· YD= ߂g[,| oA..,| oAԳ-zQ· YD= ߂g[̟֊z ߂  Y} RBԷ |· |· |· |L· |L· |L· |L·0SݪMcVlO;7Xb%X~D'nq{rw blvMUxۏצl|ϯܮ^'-):틿W߬M_wz²,kM8U;Ǖ3v0͛`ƪtU|,3䚙~`xkܕɧk ~-D&t,nk:+̆["^^qa f-8P\fkc&(D)SG!Ca´j܊.Rxe W(\& ~CM!H!D!BB'^ R0Ia,ŕo+%z)ј]F)=kRUnBN4i3s$KZ>/[T/{iT+È..ʈ4﹧z%A: k;?47m=k!I|k( kWmٺ-ڣ Pc͑}OWD*H%$"dc[chs$l)bZ/Q/I9#oŶ~4rvSGf'YCD<hUV{*#ʮH+RT"+DѭGI3E[Z}qfet(K=f?^Z.YVDZk^VXn>*?`!wYY?1[ڱbPL]]]]WW3\}}}}}}2a>+\}{ /z_____qۯ  _j/l2n/_~~1b<sxn? p{@+1b<:=\R#{`\YbL*Kie? p{@Yr{q{,o{K,3>_Vm?f{+^'-)˺r~y5 zҲ+?{>|L烏 a1O>BzU+<.z}-k){<-fQEȶV3[t՛k5x%3 yC7 䃪tBpkThJ ٘߯ BpY wwaɷ´C')r|柡ȗ4,EC"'5 |QL-/hWjnm*FҼAaޡ.xq@ ) }>$IFHZug(g.@H@EIA!)I8ikj!CP@=PDI%B()x"{IkWIwLL~Mog_mv~f_EqS'̃k YSCA+g`Y)'t`t 6QY=t v@(F5xn_GTJJ'#}V]UDrfF,N;;E)/Q9 W>P>iwup?>~^?u5o6>6v6ߝyt:rrggc&(EU{aگNQ<:{xp+lt\bߟ~7>8g'oWyTWZX:"d! Q w:S?$BqcVE͊Zو|:bM'Ӫg֕_І%lQpTE[XOsWn1FmZزcE_UOHxgt5%3u>T[qn % M{5x|{kggUvW W]SBg|q]N]/~xn\Af]u/{?חTu~بf7 %^l\6y~ {(-365< mMNBD77 m; qr_#Y}'d5yPF_TX|/L<0{ȞᬾƷcX!cO2?sUU#/G[G[G[K\&bM\}D;>Mr&NOrh'IS$)In$7]2$)In$7v}D;>Mr{&N2NORofNOrh'NOrsXYQ$7G%>IeOr}E>͢sfOrs˸Or3oOrK,3|>ORVm$MK9i[oW~~vU!UcqogO[2yoٮ~~jq yقnqE ^*)A:w(*F a I z(0@aI  % (D QF- oSJo#m6FHo#m6FHo#mvNI;o'vNI;o'vNI;o'wA;H wA;H wA;I'$wNI;I'$wNI;I'$w.EH"]w.EH"]w.EH"ݤwnMI7&ݤw82ʮyQs"rڸDPAi"FfyVJs8U FFkqr\M駤4'([kg[q+irR8z:i2R&GNiKi;xylI4"y/ky3FRF~ˋGƍx>Q|aSRXYSE4 Q?j#]Jia!=jHJ,nww7Ҭ4Q󻙹շAj?r`Hxv9>NWNR{}H6 JCpqzÏEvf/04i) Kibrat h3) .~BS~v!#OTE+ir> NcpRO&DyM f)R{hcvPOclF\oiO|tA&k kVjR3<ϥg J#gPܧ>gP&Q?j׉r;zUΧA5,_8m/UΧ^:(뵃:\~T_yqQ2CVdzC03ijQԏ;Q.{&y&/ڏS\yqUo|so_mr>S"g4r>O˝.s.sDu10ڏ+HR{,(7D{f/i9n;Y^+.d\WjSJ{N_ξr9>Ԟ+ZgHޥ^7(KgtgBR{=fE:)ZݶAU#GThn|Hh3Ywٌ:\^o3҄=[>Ċ0Ҿh}lsxp0ɐ19ܹjCcOxGOc{S<]{gY;݆Ѫo,aZ  W?a~=SGsdžTͷXN(a1컴'4D*=XbIOp2te*<616۷DW60Ӯ9.`־ףL_6㺲`;`־(Kz4-}=Q}R_v`ck_ҾeAa~@1|k5?llA1GP?w/{21݋^e'|b.;s ߽N\vw/治eb.;s }˄ ߽NysaP{^kX+|ZDŽX+섯]e'|b.;k?oD} V\A{A{?:hy16q]A{?fhy1hxx1hy1hy\ƟĸwA{?y4h9(z(zdL?E_Q|Ӗ<-?uh Q\7xF~@,K,?%`a[c0T=T>zV;֩}u0Jɕ|􂬖A%*E}U'QKԷ Z:6x3;hGZrhY%Z|fUZIˊsύgSnkU:tx&Mc%#өh[<:˽=W~kӷFcd:Iɩeϯ)7U_MOH6?NJC}[bƘ_:'[ytKu)S$׋׹he? .U\ z\Ǜ]$N)PqIiطϬU#R}įH{#Gc׌~(-׋ʋD=THqa4 ] ߥ[˶NӅ݋De%}0=$ɚ?G[҉x%ׇי?o0GȎ8;6 ڿ竡ǷII7K/>].ʥ͔=툩9JOUf`33r߭SLonK1ø|_M:Z)c+5]~0?4mh"L%[tIFmX)bےqJoi}y}\~;G6+L|Vem|RYfgei_emYf_emUYf_em YfƯ2,3m26nodm'LFYfÍmYfƣ,cȒ̴26.Lx\sG̴26>.]I}mK,˛YV)t6۸de87ml|՟mf{ոf\zVw(zt'm}:_qBeM+!_3VJc)۸l1۸n~m\Hq)"Ҹ#3/{=Y>"!lO=Ҥjk58VLKu]xZWjfQ)6O5) E )H2+5C}BEf,;Wi%#lDzqknW2Bg:R>6NR%6m5#LMj۸Hijrx۸qVqiFEVjCJڍT5 ;C#-;װ]K=2.n}B)4ӱ26Vef:;gNԑXnFJot[2J[!ғ~ݖJdtk"oMdx:j&E)c-Rxkɘ![MZcX[%.RQRmT[kȬ%Қ%hL-j7b"Y4mK"Y,F Ѵh*IF["Y_Qj[1>s왌M-Пt Z-t|'鰧q6g,ΟՔy|ꏛ.7=w?JK}ƬY}2ֿuAkNyIV qOyxǶB~zϟw~o<:6E@qp}yUU*í6GB> Tu4S.Rxe W(\wo{JmꌚGG];/.xmSFcФ'JQmc]кϴŒW,Tv }F(peS.]?gQ |gΟsvU :[;vO.5DQS4Ys=e[6~~.MMMQDtx9'KWmT蠔6:" 5,\T|H)횧\PsߕrQ?ډstPOܢXq=L~F<>QfڃӞ9CUc+r'MOY fGQ ~Mz$٣Jz{L﨑5_嫽f#%w}EOu0PLOH16_y<o-][hR4`\'>"Ne#[*}iOKN%tઢ>)CƉ<)eBW||R-ė`'0zOtz⒯0+&-<22@ @K* Bp >=4}KE X|'ϯT;`qZ=:j|ĥMʿ7NdL*M:40[ŗ[t~6_ZsfuL vcb1_z6&/sA͗y'A3,1]ht9B2 4lM6]l͂>GhĘ.MyI#Ϙ.>6+'25}[ƌ"6t97fVlPJ:san#0 SQ fDd 6sf>Qe-اOG<747҂OhQ dӰ>b gk d|+lV>jdk٬]kCא>Q(iUaË VYRMzl29Z+ j:ξ>T冎ir;:tQ=C6fI ~kiOP&U!J )؟74ZeqCSt~fBAfYcVZvN̈́ 6*gػVxhL۰DvRJWT-*vhk53oKuI1z=5+U3_>Rپ3NeQDy'g޿Bd[R,7))SߗIWIGv/u궹Ug{WVRcXmvU֞^l?{tui0;)~홛=sg|0s@;t[>[X5mjޅiBt4l6<>깠]x0lfmwj b&JvͿi6,O޶ Y]ܶ6Mg6̶1ۦ͓S{{mk,jb#3'>XѬ漛YZfI|uѹ_R5^uoE)Z(5jԤiM+,y?ک&C,-yR|vo!js̵F46^*1*kV䭩Z̟f]dQ+s}^;S%Cj`V=:}.Vw&DQOCSXȎG//V=Xi_u :%;7x`||0Ӱ.[m{:z0:7ˡ*n=Q^J7+}Ŵ +j c}},ss{Q[WPׯ~닒p;c〰g[8 8 Z!"`q@lo5ۍ$;wLjO|:wǴ,3N|F=swD>̹;nnUS#%eM͹Rtw|̹;|̜c,32s,[k:d9wG,38(̹;d9wǸ,38)̹; ̜$̹;.2sd9w+,d^ewk̜-Yf,3kYVd^Yf(̹;di42svYf%̹;d9wGFsw2s鵆T#"Klm '}2,K,o%v~ %_~("  K,[[vvK:܎ ާ.f:g [ÃᘹFV=:nnanm÷~py4^u:TM_9fOAϙj/G) Q0Ca"S517(\w)WBBB ) PpBB% /QxkQνzNR|FOIqyCI/{N)!ťgd5*!)VC+"~革-ݒ\$S;%CK~$Kr$Ir١_$i_$ )\*G*G*G#?H#՛,CqI.տG:.xy葯/ZTXbOQmlGo|ϯ ׬',Izʲܺ=Zwz²,p)mo:$M$u\򻐴/Ƨj ~MkSXtdWسaKs[ Wmѯ}IЏ^RmebZ,T14RzP5!unBN4iǍ4EEf\t쵤8+È.^:|H{]Ry}ܐ,Hmg :$9ow/&O)r({.zETߧl~{ߺݗgVu=lf-P-Pm{ qS"u04bb