{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 " Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } } {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "rule for distance between \+ poin and line" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart;\n line:=p*x+q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%lineG,&*&%\"pG\"\" \"%\"xGF(F(%\"qGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Point :=[X[0],Y[0]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&PointG7$&%\"XG6# \"\"!&%\"YGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "3*x+1,[3,5 ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&%\"xG\"\"$\"\"\"F&7$F%\"\"&" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Thwe distance between point a an d b is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "PDist:=(a,b)->(( a[1]-b[1])^2+(a[2]-b[2])^2)^(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&PDistGf*6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF)*$-%%sqrtG6#,&*$),&&9 $6#\"\"\"F8&9%F7!\"\"\"\"#F8F8*$),&&F66#F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "iff poin b lies on the line, the distance betveen poin \+ a and poin on the line with first coordinate x" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "dist:=PDist(a,[x,line]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%distG*$-%%sqrtG6#,&*$),&&%\"aG6#\"\"\"F0%\"xG!\"\"\" \"#F0F0*$),(&F.6#F3F0*&%\"pGF0F1F0F2%\"qGF2F3F0F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "We are looking for the minimum of the previous \+ function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "xxx:=solve(\n \+ diff(dist,x)=0\n ,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xx xG*&,(&%\"aG6#\"\"\"F**&%\"pGF*&F(6#\"\"#F*F**&F,F*%\"qGF*!\"\"F*,&F*F **$)F,F/F*F*F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "so the distance between poin a and line p*x+q is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rho:=simplify(\nsubs(x=xxx,dist)\n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG*$-%%sqrtG6#*&,(*&&%\"aG6#\"\"\"F/%\"pGF/F/ &F-6#\"\"#!\"\"%\"qGF/F3,&F/F/*$)F0F3F/F/F4F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p:=0;q:=0;line;a:=[2,3];\nrho;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"q G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG7$\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* $-%%sqrtG6#\"\"*\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p :=1;q:=0;line;a:=[1,0];\nrho;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"p G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG7$ \"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"#\"\" \"#F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p:='p';q:='q';a: ='a';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGF$" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Rho:=unapply(rho,a,p,q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RhoGf*6%%\"aG%\"pG%\"qG6\"6$%)ope ratorG%&arrowGF**$-%%sqrtG6#*&,(*&&9$6#\"\"\"F89%F8F8&F66#\"\"#!\"\"9& F8F<,&F8F8*$)F9F " 0 "" {MPLTEXT 1 0 61 "Rho([1,1],3,2); # distance between point [1,1] and line 3*x+2 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%%sqrtG6#\"\")\"\"\"-F&6#\"\" &F)#F)F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "x:=[1,3,4,4,5]; \ny:=[2.9,7.2,9.1,8.9,11];\nN:=nops(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG7'\"\"\"\"\"$\"\"%F(\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG7'$\"#H!\"\"$\"#sF($\"#\"*F($\"#*)F(\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "pointpl ot([seq([x[i],y[i]],i=1..N)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6#-%'POINTSG6'7$$\"\"\"\"\"!$\"#H!\"\"7$$\"\"$F)$\"#s F,7$$\"\"%F)$\"#\"*F,7$F3$\"#*)F,7$$\"\"&F)$\"#6F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "i:='i';\nW:=sum(\nRho([x[i],y[i]],p,q)^2\n,i= 1..N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"WG,,*&, (%\"pG\"\"\"$\"#H!\"\"F,%\"qGF)\"\"#,&F)F)*$)F(F.F)F)F,F)*&,(F(\"\"$$ \"#sF,F,F-F)F.F/F,F)*&,(F(\"\"%$\"#\"*F,F,F-F)F.F/F,F)*&,(F(F9$\"#*)F, F,F-F)F.F/F,F)*&,(F(\"\"&\"#6F,F-F)F.F/F,F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "We are loo king for p a q, for which this number is minimal" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "xxx:=solve(\n\{\nsimplify(diff(W,p))=0,\nsimplify(diff(W,q))=0\n\}\n) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xxxG6$<$/%\"pG$!+PM(*\\\\!#5/% \"qG$\"+p4*H]*!\"*<$/F($\"+hG@??F0/F-$\"+EFw7&*F+" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 47 "l1:=subs(xxx[1],p*z+q);\nl2:=subs(xxx[2],p*z +q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#l1G,&%\"zG$!+PM(*\\\\!#5$\" +p4*H]*!\"*\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#l2G,&%\"zG$\"+ hG@??!\"*$\"+EFw7&*!#5\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "P1:=plot(l1,z=0..5):\nP2:=plot(l2,z=0..5):\nP3:=pointplot([seq([ x[i],y[i]],i=1..N)]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "di splay(\{P1,P2,P3\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"\"!F)$\"3t+++p4*H]*!#<7$$\"3GLLL3x&) *3\"!#=$\"3S/k$>IV!\\%*F,7$$\"3umm\"H2P\"Q?F0$\"3$p(4GDP5-%*F,7$$\"3ML L$eRwX5$F0$\"31y[2i_J\\$*F,7$$\"33ML$3x%3yTF0$\"3e3%oE)o<'H*F,7$$\"3em 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3a+++D>#[Z%F,$\"39![%f-g'zG(F,7$$\"3SnmT&G!e&e%F,$\"3EjcR349LsF,7$$\"3 #RLLL)Qk%o%F,$\"3!yap0p/T=(F,7$$\"37+]iSjE!z%F,$\"3A1^xa=#=8(F,7$$\"3a +]P40O\"*[F,$\"3Q'H!y40y\"3(F,7$$\"\"&F)$\"3W++]]U+GqF,-%'COLOURG6&%$R GBG$\"#5!\"\"F(F(-F$6$7S7$F($\"3G+++EFw7&*F07$F.$\"3u)\\.&G2Xr6F,7$F4$ \"3g9(=`MBIO\"F,7$F9$\"3!G&>)))ym%y:F,7$F>$\"3-u&Q9LQ`z\"F,7$FC$\"3LL, T@\"z6,#F,7$FH$\"3Sx!eK;\"H6AF,7$FM$\"3b%z\\;f&\\=CF,7$FR$\"3/2:!zq'yK EF,7$FW$\"3%\\nD)*e!RYGF,7$Ffn$\"3E3#RwV3h1$F,7$F[o$\"3!eG)ePgjfKF,7$F `o$\"3SlNCyA]xMF,7$Feo$\"3cK^h)4jip$F,7$Fjo$\"3PDuV%ozq!RF,7$F_p$\"3;F R&4M@&)4%F,7$Fdp$\"3b,b=()R;EVF,7$Fip$\"3J5(y%or+>XF,7$F^q$\"3IAAvK'HL u%F,7$Fcq$\"3!)zIhBI*=%\\F,7$Fhq$\"3-C>*R`[(f^F,7$F]r$\"3K7I@)\\)>n`F, 7$Fbr$\"35#QZHx]Oe&F,7$Fgr$\"3[n3CI:U#y&F,7$F\\s$\"3Gs%y\\kAo*fF,7$Fas $\"37%Hbo9D&>iF,7$Ffs$\"3bWKTH$)Q8kF,7$F[t$\"3Xn7jxewAmF,7$F`t$\"3:?$[ DPt!RoF,7$Fet$\"3C\"o(3c!)o]qF,7$Fjt$\"3;7jVVhVbsF,7$F_u$\"371P%e)Rx#[ (F,7$Fdu$\"3'Qw0&[r/(o(F,7$Fiu$\"3;i5)3N_^!zF,7$F^v$\"3!>.S53)y-\")F,7 $Fcv$\"3],4aOh&)=$)F,7$Fhv$\"3z/**=Q$f@_)F,7$F]w$\"3S]z1j(pYt)F,7$Fbw$ \"3K`avYvVU*)F,7$Fgw$\"39L-Oh6&*f\"*F,7$F\\x$\"35c4LCEWp$*F,7$Fax$\"3m Y4u-zn$e*F,7$Ffx$\"3QU1)[?Rhz*F,7$F[y$\"3my(ph2p8***F,7$F`y$\"39!oMSX7 :-\"!#;7$Fey$\"3`)\\\"\\4a_T5Ficl7$Fjy$\"37.Y:%RjG1\"Ficl7$F_z$\"3m)oU \"olG$3\"Ficl7$Fdz$\"33++wdSB06FiclFhz-%'POINTSG6'7$$\"\"\"F)$\"#HF^[l 7$$\"\"$F)$\"#sF^[l7$$\"\"%F)$\"#\"*F^[l7$Fdel$\"#*)F^[l7$Fdz$\"#6F)-% +AXESLABELSG6%Q\"z6\"Q!Fbfl-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(FdzFgfl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs(xxx [1],W);\nsubs(xxx[2],W);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+u]^pY! \")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*)*H\\G\"!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "l2 is our line" }}}}{MARK "30 0 0" 14 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }