{VERSION 3 0 "IBM RISC UNIX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot " 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 } {PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 19 "Luento to 8.10.1998" }} {PARA 19 "" 0 "" {TEXT -1 6 "A-sali" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "M\344\344rittelemme lag ja int err-funktiot" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "lag:=(xd,x )->seq(product((x-xd[k])/(xd[j]-xd[k]),k=1..j-1)*product((x-xd[k])/(xd [j]-xd[k]),k=j+1..nops(xd)),j=1..nops(xd)):" }}{PARA 7 "" 1 "" {TEXT -1 42 "Warning, `j` in call to `seq` is not local" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 15 "lag([a,b,c],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%*&*&,&%\"xG\"\"\"%\"bG!\"\"F',&F&F'%\"cGF)F'\"\"\"*&,&% \"aGF'F(F)\"\"\",&F/F'F+F)\"\"\"!\"\"*&*&,&F&F'F/F)F'F*F,F,*&,&F(F'F/F )\"\"\",&F(F'F+F)\"\"\"F3*&*&F6F,F%F,F,*&,&F+F'F/F)\"\"\",&F+F'F(F)\" \"\"F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 23 "Diskretointiapufunktio :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "linspace:=(a,b,n)->[se q(a+i*(b-a)/n,i=0..n)]:" }}{PARA 7 "" 1 "" {TEXT -1 42 "Warning, `i` i n call to `seq` is not local" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 " Interpolointivirhe voidaan ohjelmoida funktioksi, jolle annetaan int erpoloitavan funktion lauseke ja interpolointipisteet sek\344" }} {PARA 0 "" 0 "" {TEXT -1 61 "symboli, jolla muuttujaa merkit\344\344n. Otetaan globaali symboli " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 34 " samaan rooliin, jossa se aina on." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "interr:=(f,xd,x)->(subs(x=xi,diff(f,x$(nops(xd))))/(n ops(xd))!)*product((x-xd[i]),i=1..nops(xd));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'interrGR6%%\"fG%#xdG%\"xG6\"6$%)operatorG%&arrowGF** &*&-%%subsG6$/9&%#xiG-%%diffG6$9$-%\"$G6$F4-%%nopsG6#9%\"\"\"-%(produc tG6$,&F4FA&F@6#%\"iG!\"\"/FH;FAF=FA\"\"\"-%*factorialG6#F=!\"\"F*F*F* " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Testataan sit\344 pienell\344 esimerkill\344: sin-funktion interpolointi pisteiss\344 0,1,2 ." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interr(sin(x),[0,1,2],x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**-%$cosG6#%#xiG\"\"\"%\"xGF),&F*F) !\"\"F)F),&F*F)!\"#F)F)#F,\"\"'" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Jatkamme eilist\344 esimerkki\344." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "xd:=[9.0,9.5,10.0,11.0];yd:=map(ln,xd);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#xdG7&$\"#!*!\"\"$\"#&*F($\"$+\"F($\"$5\"F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ydG7&$\"+xXA(>#!\"*$\"+*z\"H^AF($ \"+$4&e-BF($\"+t_*yR#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " L:=lag(xd,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"LG6&*(,&%\"xG$!++ +++?!\"*$\"+++++>!\")\"\"\"F/,&F($!+++++5F+$\"+++++5F.F/F/,&F($!+++++] !#5$\"+++++bF+F/F/*(,&F($\"+++++?F+$!+++++=F.F/F/,&F(F)$F>F.F/F/,&F($! +nmmmmF8$\"+MLLLtF+F/F/*(,&F($F4F+$!+++++!*F+F/F/,&F(F=$!+++++>F.F/F/, &F(F1$\"+++++6F.F/F/*(,&F($\"+++++]F8$!+++++XF+F/F/,&F($\"+nmmmmF8$!+M LLLjF+F/F/,&F(FJ$F2F.F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "p:='sum(yd[j]*L[j],j=1..4)';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"pG-%$sumG6$*&&%#ydG6#%\"jG\"\"\"&%\"LGF+F-/F,;F-\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "p:=eval(p);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pG,**(,&%\"xG$!+++++?!\"*$\"+++++>!\")\"\"\"F/,&F($ !+++++5F+$\"+++++5F.F/F/,&F($!+++++]!#5$\"+++++bF+F/F/$\"+xXA(>#F+*(,& F($\"+++++?F+$!+++++=F.F/F/,&F(F)$F@F.F/F/,&F($!+nmmmmF8$\"+MLLLtF+F/F /$\"+*z\"H^AF+*(,&F($F4F+$!+++++!*F+F/F/,&F(F?$!+++++>F.F/F/,&F(F1$\"+ ++++6F.F/F/$\"+$4&e-BF+*(,&F($\"+++++]F8$!+++++XF+F/F/,&F($\"+nmmmmF8$ !+MLLLjF+F/F/,&F(FN$F2F.F/F/$\"+t_*yR#F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 39 "M\344\344ritell\344\344n polynomilauseke funktioksi" } {TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "pf:=unappl y(p,x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "map(pf,xd);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&$\"+xXA(>#!\"*$\"+*z\"H^AF&$\"+$4&e- BF&$\"+t_*yR#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "yd;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&$\"+xXA(>#!\"*$\"+*z\"H^AF&$\"+$4&e- BF&$\"+t_*yR#F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Kunnossa on." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "pisteet:=zip((x,y)->[x,y],xd,yd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(pisteetG7&7$$\"#!*!\"\"$\"+xXA(># !\"*7$$\"#&*F)$\"+*z\"H^AF,7$$\"$+\"F)$\"+$4&e-BF,7$$\"$5\"F)$\"+t_*yR #F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 " Kenties helpommin ymm\344 rrett\344v\344 tapa:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "# p isteet:=[seq([xd[i],yd[i]],i=1..4)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "\nxykuva:=plot(pisteet,x=8.5..11.5,style=point,symbol =circle):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "ln_ja_p:=plot( [ln(x),p],x=8.5..11.5,color=[blue,grey]):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "display([xykuva,ln_ja_p]);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6&7&7$$\"\"*\"\"!$\"1+++xXA(>#!#:7$$\"1+++++++&* F-$\"1+++*z\"H^AF-7$$\"#5F*$\"1+++$4&e-BF-7$$\"#6F*$\"1+++t_*yR#F--%'C OLOURG6&%$RGBG$F5!\"\"F*F*-%&STYLEG6#%&POINTG-%'SYMBOLG6#%'CIRCLEG-F$6 $7S7$$\"1+++++++&)F-$\"1ri\\jh1S@F-7$$\"1****\\i9Rl&)F-$\"1C`jN)Hx9#F- 7$$\"1+]PC#)GA')F-$\"13Dr/0Na@F-7$$\"1++v$euio)F-$\"1!zLe9W<;#F-7$$\"1 ++D'3&o]()F-$\"19g(G*>8p@F-7$$\"1**\\(oX*y9))F-$\"1O>#4$4Vw@F-7$$\"1+] P9CAu))F-$\"1Y8C;2:$=#F-7$$\"1**\\P*zhd$*)F-$\"1\\'>^Sh+>#F-7$$\"1+]P> fS***)F-$\"1*>*>m&er>#F-7$$\"1+](=$f%G1*F-$\"1s`C\">$=/AF-7$$\"1++]#y, \"G\"*F-$\"1Q#yIwd8@#F-7$$\"1++Dr\"zb=*F-$\"1+\\)GxMw@#F-7$$\"1,+](4&G ]#*F-$\"1<2SsVlCAF-7$$\"1++]7nD:$*F-$\"1g$=Fc`;B#F-7$$\"1****\\-*oyP*F -$\"18(48a_$QAF-7$$\"1,]PpnsM%*F-$\"1b/9>sRWAF-7$$\"1****\\siL-&*F-$\" 1SLz\"pP:D#F-7$$\"1+++!R5'f&*F-$\"1N([?(padAF-7$$\"1+]P/QBE'*F-$\"1pgU f?\\kAF-7$$\"1,++:o?&o*F-$\"1uoy]'*fqAF-7$$\"1+]Pa&4*\\(*F-$\"1qB\\3!e sF#F-7$$\"1+]7j=_6)*F-$\"1>5S&RdNG#F-7$$\"1++vVy!e()*F-$\"1/KS9\")3!H# F-7$$\"1+](=WU[$**F-$\"12F)p,[gH#F-7$$\"1++DJ#>&)***F-$\"1$ea^+PCI#F-7 $$\"1+v$>:mk+\"!#9$\"1L`0H/.4BF-7$$\"1+DcdQA75Fhu$\"1:D_U[t9BF-7$$\"1+ ]PPBW=5Fhu$\"1hI$GVf3K#F-7$$\"1++]Nm'[-\"Fhu$\"1Q=T&eZrK#F-7$$\"1+](yb ^6.\"Fhu$\"1$*R2)GhKL#F-7$$\"1+vVVDBP5Fhu$\"1=+5\"Fhu$\"1mp?<=\"zR#F-7$$\"1+]i_4Q16Fhu$\"1)[ #ey$zOS#F-7$$\"1+vV>5p76Fhu$\"1B#yh^m$4CF-7$$\"1++]:$*[=6Fhu$\"1j2PUSc 9CF-7$$\"1+]7<[8D6Fhu$\"1T\\\"*ez[?CF-7$$\"1+++L'y58\"Fhu$\"1.-\"H\"ov DCF-7$$\"1+vV!)fTP6Fhu$\"1Mtv)4W8V#F-7$$\"1+DcI;[V6Fhu$\"1ES7jFmOCF-7$ $\"1++++++]6Fhu$\"1/#p`.ZBW#F--F>6&F@F*F*$\"*++++\"!\")-F$6$7S7$FO$\"1 9j1!3A,9#F-7$FT$\"1E,C!*=xZ@F-7$FY$\"16CvHDQa@F-7$Fhn$\"1A#\\`*owh@F-7 $F]o$\"1(=$=%HZ\"p@F-7$Fbo$\"1.0&eXSk<#F-7$Fgo$\"1'[(oXh:$=#F-7$F\\p$ \"1A1c(oj+>#F-7$Fap$\"1*)R$Geer>#F-7$Ffp$\"1o+-x<=/AF-7$F[q$\"1%>K6db8 @#F-7$F`q$\"1HeYaBjCX- PCI#F-7$Ffu$\"1?oO![H!4BF-7$F\\v$\"1;e_'*Gt9BF-7$Fav$\"1(>e:Gc3K#F-7$F fv$\"1g8C%4VrK#F-7$F[w$\"1$[arWbKL#F-7$F`w$\"14'4'=T8RBF-7$Few$\"1vvUK 7iXBF-7$Fjw$\"1ZHHpYT^BF-7$F_x$\"1,\\iGQcdBF-7$Fdx$\"1N#zys.JO#F-7$Fix $\"1[\"yq$e7pBF-7$F^y$\"168g!yfZP#F-7$Fcy$\"1%G\"*H\"eh!Q#F-7$Fhy$\"1( zh4)*3jQ#F-7$F]z$\"1$>TXlNAR#F-7$Fbz$\"1*[XX#=\"zR#F-7$Fgz$\"1FaKmYo.C F-7$F\\[l$\"1j/08%y$4CF-7$Fa[l$\"1FVf_Le9CF-7$Ff[l$\"1NRW7v^?CF-7$F[\\ l$\"1J6PjszDCF-7$F`\\l$\"1m^hV\")RJCF-7$Fe\\l$\"1l\"y_*=tOCF-7$Fj\\l$ \"1<\"eh$[VUCF--F>6&F@$\")=THvFb]lF[glF[gl-%+AXESLABELSG6$Q\"x6\"%!G-% %VIEWG6$;$\"#&)FB$\"$:\"FB%(DEFAULTG" 2 393 393 393 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot( ln(x)-p,x=8.5..11.5);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7fn7 $$\"1+++++++&)!#:$!1]VE/ql\"f&!#?7$$\"1++DJdpK&)F*$!1-\\N%[#el[F-7$$\" 1****\\i9Rl&)F*$!1%H[-[ga?%F-7$$\"1+vVV)RQf)F*$!17g#Q!es\"o$F-7$$\"1+] PC#)GA')F*$!1*R!H!*R]-KF-7$$\"1+D1/9Ga')F*$!1\"[E;hcPr#F-7$$\"1++v$eui o)F*$!1JT?V:&\\F#F-7$$\"1++D'3&o]()F*$!1$f&Q<28I:F-7$$\"1**\\(oX*y9))F *$!1=o@n&G\\_*!#@7$$\"1+]P9CAu))F*$!1gH3RhWHaFV7$$\"1**\\P*zhd$*)F*$!1 (49E(4W#G#FV7$$\"1+]P>fS***)F*$!1<;TT*yMm\"!#B7$$\"1+](=$f%G1*F*$\"1VK *QIDUT\"FV7$$\"1++]#y,\"G\"*F*$\"10?jWg%>>#FV7$$\"1++Dr\"zb=*F*$\"1yn6 r!>%=CFV7$$\"1,+](4&G]#*F*$\"1On'p=OlG#FV7$$\"1++]7nD:$*F*$\"1$RX\"y=5 j=FV7$$\"1****\\-*oyP*F*$\"1\\/'RUSTG\"FV7$$\"1,]PpnsM%*F*$\"1?M(H-TL) o!#A7$$\"1****\\siL-&*F*$!1^6_FD^& )***F*$!1F`o7$$\"1+v$>:mk+\"!#9$\"1:haC^)o[*Fdq7$$\"1+DcdQA75 Fet$\"1a;4*p'*f%>FV7$$\"1+]PPBW=5Fet$\"1>,`k[F^JFV7$$\"1++]Nm'[-\"Fet$ \"1b8Py/<\"\\%FV7$$\"1+](yb^6.\"Fet$\"11_05&>4%eFV7$$\"1+vVVDBP5Fet$\" 1ds*y!f\"[7(FV7$$\"1+]7TW)R/\"Fet$\"1l#G;*)ynW)FV7$$\"1++]@80]5Fet$\"1 B\"fi5&Hk%*FV7$$\"1++D6!Hl0\"Fet$\"1'3-U*Q=G5F-7$$\"1+v$4w)Ri5Fet$\"16 [S#zc)p5F-7$$\"1++vZf\")o5Fet$\"1$4],\"y2q5F-7$$\"1+vV?S&[2\"Fet$\"1p^ B`4)p,\"F-7$$\"1+v=Yb;\"3\"Fet$\"1qq*R!Rk[*)FV7$$\"1++D;iL(3\"Fet$\"1o p^SO#*ypFV7$$\"1+v$fL'z$4\"Fet$\"1c3,84`gRFV7$$\"1+++*>=+5\"Fet$!198gZ L_Qt!#C7$$\"1+]i_4Q16Fet$!1ZCyQHu(G&FV7$$\"1+vV>5p76Fet$!1$G&*RA(o*=\" F-7$$\"1](ou;!f:6Fet$!1U*=^]#[V:F-7$$\"1++]:$*[=6Fet$!1rnTcB-J>F-7$$\" 1+DJm?\"=7\"Fet$!1Pcu:(\\(=CF-7$$\"1+]7<[8D6Fet$!1p4X**GNbHF-7$$\"1+D1 Dn5G6Fet$!1=iSiMuyMF-7$$\"1+++L'y58\"Fet$!1r0(G4Y]/%F-7$$\"1](=nIZU8\" Fet$!1>x?N!*=)p%F-7$$\"1+vV!)fTP6Fet$!1EMA$y&[/aF-7$$\"1++]0)[/9\"Fet$ !1xJxB!=C8'F-7$$\"1+DcI;[V6Fet$!1XD*QT:K\"pF-7$$\"1]7G:3uY6Fet$!1e3O2N g8yF-7$$\"1++++++]6Fet$!1;rK\"*)y+y)F--%'COLOURG6&%$RGBG$\"#5!\"\"\"\" !Fa^l-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;$\"#&)F`^l$\"$:\"F`^l%(DEFAU LTG" 2 393 393 393 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 6962 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "p;" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#,**(,&%\"xG$!+++++?!\"*$\"+++++>!\")\"\"\"F-,&F&$!+++++5F)$\"+++++5F ,F-F-,&F&$!+++++]!#5$\"+++++bF)F-F-$\"+xXA(>#F)*(,&F&$\"+++++?F)$!++++ +=F,F-F-,&F&F'$F>F,F-F-,&F&$!+nmmmmF6$\"+MLLLtF)F-F-$\"+*z\"H^AF)*(,&F &$F2F)$!+++++!*F)F-F-,&F&F=$!+++++>F,F-F-,&F&F/$\"+++++6F,F-F-$\"+$4&e -BF)*(,&F&$\"+++++]F6$!+++++XF)F-F-,&F&$\"+nmmmmF6$!+MLLLjF)F-F-,&F&FL $F0F,F-F-$\"+t_*yR#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "pf( 10.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)zl8N#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ln(10.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+d_P^B!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "virhe:=interr(ln(x),xd,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% &virheG,$*&**,&%\"xG\"\"\"$!#!*!\"\"F*F*,&F)F*$!#&*F-F*F*,&F)F*$!$+\"F -F*F*,&F)F*$!$5\"F-F*F*\"\"\"*$)%#xiG\"\"%F7!\"\"#F-\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Pienenev\344lle funktiolle saadaan yl\344 raja laskemalla arvo vasemmassa p\344\344tepisteess\344." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "maxvirhe:=subs(xi=xd[1],virhe);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)maxvirheG,$**,&%\"xG\"\"\"$!#!*!\" \"F)F),&F(F)$!#&*F,F)F),&F(F)$!$+\"F,F)F),&F(F)$!$5\"F,F)F)$!+eZR5Q!#9 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(x=10.5,maxvirhe); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M!)*)G9!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "ln(10.5)-pf(10.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%f%*!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "%%-%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*M!)*H[!#9" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 34 "Yl\344raja max-virheelle koko v\344lill \344" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Tulotermin max saadaan de rivoimalla, laskemalla derivaatan 0-kohdat ja laskemalla" }}{PARA 0 " " 0 "" {TEXT -1 56 "arvo niiss\344 sek\344 p\344\344tepisteiss\344, ni it\344 sitten verrataan." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "dmaxv:=diff(maxvirhe,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&dmaxvG ,**(,&%\"xG\"\"\"$!#&*!\"\"F)F),&F(F)$!$+\"F,F)F),&F(F)$!$5\"F,F)F)$!+ eZR5Q!#9*(,&F(F)$!#!*F,F)F)F-\"\"\"F0F:F3*(F7F:F'F:F0F:F3*(F7F:F'F:F-F :F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "dernolla:=solve(dmax v=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)dernollaG6%$\"+\"*RP'>*! \"*$\"+yKXl(*F($\"+tsJm5!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "maxvf:=unapply(maxvirhe,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %&maxvfGR6#%\"xG6\"6$%)operatorG%&arrowGF(,$**,&9$\"\"\"$!#!*!\"\"F0F0 ,&F/F0$!#&*F3F0F0,&F/F0$!$+\"F3F0F0,&F/F0$!$5\"F3F0F0$!+eZR5Q!#9F(F(F( " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "Periaatteessa pit\344isi kat soa derivaatan 0-kohdat ja p\344\344tepisteet, kuten alla tehd\344\344 n. Nyt kyll\344 tiedet\344\344n, ett\344 p\344\344tepisteiss\344" }} {PARA 0 "" 0 "" {TEXT -1 102 "virhe on 0, koska niiss\344 interpoloida an. Olkoon siis yleisen funktion maksimointitilanteen mukaisesti:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "map(abs@maxvf,[xd[1],dernoll a,xd[4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!$\"+AD,$H$!#:$\"+ Um)=C#F'$\"+&))*eY;!#9F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "maxvirhe:=max(op(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)maxvirheG $\"+&))*eY;!#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Katsotaan viel \344 kuvaa:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(ln(x)-p ,x=9..11);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7eo7$$\"\"*\"\" !$\"1EF)o'\\>iL!#D7$$\"1LLL3VfV!*!#:$\"1D*fA(>^d5!#@7$$\"1mm\"H[D:3*F1 $\"1,hAF47$$\"1LL$e0$=C\"*F1$\" 1Qb[%>)>i@F47$$\"1LLLBKlX\"*F1$\"1*\\eK'4Y,BF47$$\"1ML$3RBr;*F1$\"1X% \\pj]eQ#F47$$\"1nTg_u!y<*F1$\"1]N?1Z)*3CF47$$\"1+]P9:\\)=*F1$\"15Z&yM0 /U#F47$$\"1Me9wb<*>*F1$\"1(G\\!)R72U#F47$$\"1nm\"zjf)4#*F1$\"1)G.6s%\\ 5CF47$$\"1++vB1nH#*F1$\"1R6X%QacO#F47$$\"1LLe4;[\\#*F1$\"1#)>A0G>!H#F4 7$$\"1++Dmy]!H*F1$\"1GW3N4L^?F47$$\"1MLezs$HL*F1$\"1CD<$=XCr\"F47$$\"1 ***\\7iI_P*F1$\"1#\\Zv'\\p58F47$$\"1mmm@Xt=%*F1$\"1*G?_CG'*e)!#A7$$\"1 LL$3y_qX*F1$\"1BB)3C#G&\\%Fjp7$$\"1+++l+>+&*F1$!1\">$H\\)*)HB$!#C7$$\" 1,++vW]V&*F1$!1nSP`gY\"H%Fjp7$$\"1+++NfC&e*F1$!1N\\dZI8_zFjp7$$\"1LLez 6:B'*F1$!1%Qc!p*fx2\"F47$$\"1mmm\"=C#o'*F1$!1`\"*R)o:!R8F47$$\"1mmmEpS 1(*F1$!1tNF:b(z[\"F47$$\"1K$e9t9'G(*F1$!1B,'yq=9a\"F47$$\"1++DOD#3v*F1 $!1p(=:l0#p:F47$$\"1M$ek?![q(*F1$!1<-WX@ur:F47$$\"1mmmwy8!z*F1$!1i*)>Y O3`:F47$$\"1M$ekX0<\")*F1$!1m#eoQ1y]\"F47$$\"1++DOIFL)*F1$!1l&ekg\\iV \"F47$$\"1++v3zMu)*F1$!1QO/I@pE7F47$$\"1nm;H_?<**F1$!1ld%*\\F47$$\"1+++dxd;5Fev$\"1w1O^AAyFF47$$\"1++ D0xw?5Fev$\"1\"*oxP.)yi$F47$$\"1+]i&p@[-\"Fev$\"1OHU\"[(e\"[%F47$$\"1+ +vgHKH5Fev$\"1M<.xewZaF47$$\"1nmmZvOL5Fev$\"1H[_[pc9jF47$$\"1++]2goP5F ev$\"1!*[Vkv%z@(F47$$\"1L$eR<*fT5Fev$\"1X(pg#ou(*zF47$$\"1++])Hxe/\"Fe v$\"1+4!3Fa`y)F47$$\"1n;H!o-*\\5Fev$\"1!ofD(H#>W*F47$$\"1+]7k.6a5Fev$ \"14b&3v<:+\"!#?7$$\"1nm;WTAe5Fev$\"15U)\\\"\\wV5Fdz7$$\"1LeRREW2e2\"Fdz7$$\"1]i:A=kl5Fev$ \"12fxYcbw5Fdz7$$\"1LLL*zym1\"Fev$\"17n(f$p#f2\"Fdz7$$\"1L3Fe#Rx1\"Fev $\"1\"Q@uw3Q2\"Fdz7$$\"1L$3sr*zo5Fev$\"1TK%p9Y,2\"Fdz7$$\"1Le9w,')p5Fe v$\"1>*=>Q&)[1\"Fdz7$$\"1LL3N1#42\"Fev$\"15w<-A(z0\"Fdz7$$\"1+vo!*R-t5 Fev$\"1uAxD_:R5Fdz7$$\"1n;HYt7v5Fev$\"1s#4tb'=85Fdz7$$\"1+++xG**y5Fev$ \"1=GUFuE`%*F47$$\"1L$3U/373\"Fev$\"1S5$*p:%y$*)F47$$\"1nmT6KU$3\"Fev$ \"1;w3dWSB$)F47$$\"1+]P$[/a3\"Fev$\"1^k@S_S&o(F47$$\"1LLLbdQ(3\"Fev$\" 16Bsd`nfpF47$$\"1n\"zW?)\\*3\"Fev$\"1gg-')*4Y3'F47$$\"1+]i`1h\"4\"Fev$ \"1pNp@#H+5&F47$$\"1+++PDj$4\"Fev$\"1CcAn \344\344ret \366nt\344, kun n kasvaa." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=x->1/(1+25*x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG 6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&\"\"\"F/*$)9$\"\"#F-\"#D!\"\"F(F (F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "n:=14:xd:=linspace(- 1,1,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xdG71!\"\"#!\"'\"\"(#!\" &F)#!\"%F)#!\"$F)#!\"#F)#F&F)\"\"!#\"\"\"F)#\"\"#F)#\"\"$F)#\"\"%F)#\" \"&F)#\"\"'F)F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "yd:=map( f,xd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ydG71#\"\"\"\"#E#\"#\\\"$ \\*#F*\"$u'#F*\"$\\%#F*\"$u##F*\"$\\\"#F*\"#uF'F4F2F0F.F,F)F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "p[14]:=interp(xd,yd,x);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"#9,2*$)%\"xG\"\"#\"\"\"#!2D SqIQ9/.$\"1h\"y%f'>*49*$)F+\"\"%F-#\"2++n " 0 " " {MPLTEXT 1 0 27 "plot([f(x),p[14]],x=-1..1);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7ao7$$!\"\"\"\"!$\"1YQ:YQ:YQ!#<7$$!1nmm;p0k&*! #;$\"1\"\\24&eu*=%F-7$$!1LL$3X&F-7$$!1ML$3i.9!zF1$\"1 PXyk$y6-'F-7$$!1nm;/R=0vF1$\"1Uhr]RWImF-7$$!1++]P8#\\4(F1$\"1u$ydGV8O( F-7$$!1nm;/siqmF1$\"1)o;ES()yC)F-7$$!1++](y$pZiF1$\"1368)oX]H*F-7$$!1L LL$yaE\"eF1$\"1mN'zNm&e5F17$$!1nmm\">s%HaF1$\"19976Gx%>\"F17$$!1+++]$* 4)*\\F1$\"1'*>*3w9-Q\"F17$$!1+++]_&\\c%F1$\"1zZ%*)o#Q5;F17$$!1+++]1aZT F1$\"1***3b:1m)=F17$$!1nm;/#)[oPF1$\"1#)GKr1i(>#F17$$!1MLL$=exJ$F1$\"1 ;lyg0LlEF17$$!1MLLL2$f$HF1$\"1)*zu],lpJF17$$!1++]PYx\"\\#F1$\"1Xl%Q&H#F1$\"1'oE[qcfJ%F17$$!1MLLL7i)4#F1$\"1Td%Rs=&fZF17$ $!1nmTNa%H)=F1$\"1@y\"yY%=,`F17$$!1++]P'psm\"F1$\"1.wmc%*))**eF17$$!1+ ++v_*=Y\"F1$\"1u')>^ip$\"1s'e_Pc(****F17$$\"1(H$3x\"yY_%F\\v$\"1cJ3SW)[***F17$ $\"1GL3_Nl.5F-$\"1)=&yZ-)[(**F17$$\"1E$ekGR[b\"F-$\"1$4)=_\\#*R**F17$$ \"1CL$3-Dg5#F-$\"1SXK(\\K.*)*F17$$\"1@Le*['R3KF-$\"18XW*H6\"\\(*F17$$ \"1cgb*F17$$\"1`mmmJ+IiF-$\"1P59#[,b6*F17$$\"1!**** \\PQ#\\\")F-$\"1WTRoQ9w&)F17$$\"1mm\"z\\1A-\"F1$\"1(>+iv*yGzF17$$\"1KL Le\"*[H7F1$\"1`hA:LOdsF17$$\"1mm;HCjV9F1$\"1;@ga8aulF17$$\"1*******pvx l\"F1$\"1nqC)y)[FfF17$$\"1****\\7JFn=F1$\"16L'zd,GM&F17$$\"1)****\\_qn 2#F1$\"1jP(oka<\"[F17$$\"1)**\\P/q%zAF1$\"1T/g()\\s\\VF17$$\"1)***\\i& p@[#F1$\"1e\"[=?cl$RF17$$\"1)****\\2'HKHF1$\"1hqHij,vJF17$$\"1lmmmZvOL F1$\"1)*Qq#=nIk#F17$$\"1+++]2goPF1$\"1Hocd#=v>#F17$$\"1KL$eR<*fTF1$\"1 #)=)eB,v(=F17$$\"1+++])Hxe%F1$\"1%))Qpt!)pf\"F17$$\"1lm;H!o-*\\F1$\"1N ^*o:]RQ\"F17$$\"1****\\7k.6aF1$\"1S4Y9y%>?\"F17$$\"1mmm;WTAeF1$\"1\\`8 zZRb5F17$$\"1****\\i!*3`iF1$\"1*y\"f(o+0G*F-7$$\"1MLLL*zym'F1$\"1m'e9@ CTD)F-7$$\"1LLL3N1#4(F1$\"1yNN>+%oO(F-7$$\"1mm;HYt7vF1$\"1%oGOI/!=mF-7 $$\"1*******p(G**yF1$\"1\")=$395U-'F-7$$\"1mmmT6KU$)F1$\"1YG`9>?NaF-7$ $\"1LLLLbdQ()F1$\"1'pK\\TLu(\\F-7$$\"1++]i`1h\"*F1$\"1q5l\"oD$\\XF-7$$ \"1++]P?Wl&*F1$\"1W2DpLe)=%F-7$$\"\"\"F*F+-%'COLOURG6&%$RGBG$\"#5F)F*F *-F$6$7it7$F($\"1xy4YQ:YQF-7$$!1MekynP')**F1$\"1y\")=j6$f]'F17$$!1n;Hd Nvs**F1$\"18Kq9'*eE7!#:7$$!1,v$fLI\"f**F1$\"1%*fl7b!HbF#Fcbl7$$!1o\"HK*Q)=$**F1$\"16'\\Cz`5v#Fcbl7$$!1,](=ng #=**F1$\"12<_M(3c>$Fcbl7$$!1N3_]uj/**F1$\"1W-n;XR5OFcbl7$$!1nm;HU,\"*) *F1$\"1joRRrd'*RFcbl7$$!1L$ekynP')*F1$\"1Kl1z%Gwo%Fcbl7$$!1++vV8_O)*F1 $\"1'HUd$yUx_Fcbl7$$!1n;/,\\F4)*F1$\"1D(REQNTx&Fcbl7$$!1MLLe%G?y*F1$\" 1vKb)pHa='Fcbl7$$!1nm\"HdNvs*F1$\"1i-'Rw=-y'Fcbl7$$!1++](oUIn*F1$\"1I= rLMU9rFcbl7$$!1n;zWizX'*F1$\"1b/'yP)\\)>(Fcbl7$$!1ML3-)\\&='*F1$\"1\\h W%o;VB(Fcbl7$$!1n\"H2eE\\g*F1$\"1%fHBeqcB(Fcbl7$$!1,]PfLI\"f*F1$\"1p$H YKonA(Fcbl7$$!1M3-Q,ox&*F1$\"1'Q*H*fq\"3sFcbl7$F/$\"1'GEY%)=/=(Fcbl7$$ !1L$3-)*G#p%*F1$\"1t5)R0!ounFcbl7$$!1++vV5Su$*F1$\"1*=l&y1#35'Fcbl7$$! 1M3_vq)pK*F1$\"1ueJ\"o)Q,dFcbl7$$!1n;H2Jdz#*F1$\"1*=K#fGow_Fcbl7$$!1* \\i!R\"f@B*F1$\"1)zR+G%)o$[Fcbl7$F5$\"15?sl*z1R%Fcbl7$$!1*\\(o/KUJ\"*F 1$\"1h90MLJ!*QFcbl7$$!1m;aQ75y!*F1$\"1oo(*HYt*R$Fcbl7$$!1LeRs#zZ-*F1$ \"1];P=,)e#HFcbl7$$!1***\\iId9(*)F1$\"1Vg\\c[>uCFcbl7$$!1mT5S`8=*)F1$ \"1Ddl!4)G\"Fcbl7$F:$\"1[wDFZtf&*F17$$!1mm;/'=3l)F1$\"1'3(zD*Q^(QF17 $$!1mmmmxYV&)F1$!1\"H-zbx@-&F-7$$!1mm;Hp6O%)F1$!1B\"yJ$y/mOF17$F?$!1uQ $*Q)y?v&F17$$!1+D\"GyX`F)F1$!1/NH#*)*>PkF17$$!1M$eRZD>A)F1$!1?a'H.i]\" pF17$$!1nT5l^]o\")F1$!1ADYw!3j?(F17$$!1++Dc[3:\")F1$!1(Q()R&HRJtF17$$! 1LeRZXmh!)F1$!1&e[^iL.J(F17$$!1m;aQUC3!)F1$!1za)>O-D;(F17$$!1+voHR#[&z F1$!1hg0Q*ek!pF17$FD$!1Y(yme<)flF17$$!1,+]iPH.xF1$!1i4*GA=du%F17$FI$!1 MOs/O],EF17$$!1LL$3i_+I(F1$!1%*R2NF]VaF-7$FN$\"1xhe,HQ\\5F17$$!1nm;/y% )))pF1$\"1UWy9,d[;F17$$!1ML$3FuF)oF1$\"1^B]a2`\"4#F17$$!1,+]P2qwnF1$\" 1$)py]Y+'Q#F17$FS$\"1s-UO/^XDF17$$!1+++]Q*[c'F1$\"1giaQ,j(e#F17$$!1ML$ e\\g\"fkF1$\"1BZ8nJ\"H`#F17$$!1ommTrU`jF1$\"1f9-4TI-CF17$FX$\"1xb+]Su; AF17$Fgn$\"1\"[AEn7UF\"F17$$!1++]([j5i&F1$\"1HcLAOfY%*F-7$F\\o$\"1k%[= **)Q-uF-7$$!1LL$3x&y8_F1$\"1N`QWCJznF-7$Fao$\"1#3;c5,n\"yF-7$Ffo$\"1_z ')4&>bL\"F17$F[p$\"1P(yXs=Z+#F17$F`p$\"1K-/'*F17$F[u$\"1AQN&*Gi%*)*F17$F`v$\"1v=NV0z** **F17$Few$\"1ZGI?L@0**F17$F_x$\"1]P,J!e+h*F17$Fdx$\"1*pYBT5l?*F17$Fix$ \"1Ht-;>o*o)F17$Fcy$\"1X$zbgYRK(F17$F]z$\"11VK$G\"*o$eF17$Fgz$\"1LH8X \")\\-YF17$Fa[l$\"1had&pK\\w$F17$Ff[l$\"1JX`,,7=KF17$F[\\l$\"17$p9CE!* *GF17$F`\\l$\"1PND!eri]#F17$Fe\\l$\"1av#y/$)e)>F17$Fj\\l$\"1$[_L,]'*H \"F17$F_]l$\"1NcAtDf\")yF-7$$\"1KL$3A_1?&F1$\"1$[H)>wk(z'F-7$Fd]l$\"1I -Q(QA$zsF-7$$\"1KLe9as;cF1$\"1&4Qa>:aQ*F-7$Fi]l$\"1J;v@\\q$H\"F17$F^^l $\"19?*zSAsA#F17$$\"1K$3-y'ycjF1$\"1#Q3_r3uS#F17$$\"1mm\"z\\%[gkF1$\"1 8^vu;6MDF17$$\"1,]i:A=klF1$\"1TWN2Cc(e#F17$Fc^l$\"10&eKH**za#F17$$\"1M L3Fe#Rx'F1$\"1*3&\\Ot#=R#F17$$\"1LL$3sr*zoF1$\"1`![p=s65#F17$$\"1LLe9w ,')pF1$\"1=j*Q=)Ri;F17$Fh^l$\"1F%*H4Tfn5F17$$\"1++vo!*R-tF1$!1(=cMx&Qc cF-7$F]_l$!1XX72T$Go#F17$$\"1LLek6,1xF1$!1YNTI&*=uZF17$Fb_l$!1KE?rxXWl F17$$\"1L$3-)omazF1$!1O+&HDkb!pF17$$\"1nmTgg/5!)F1$!1f[z?PNprF17$$\"1* *\\iS_Ul!)F1$!1lNRHi=;tF17$$\"1LL$3U/37)F1$!1sb06[LDtF17$$\"1*\\P4,%\\ [\")F1$!1Iw`s?arsF17$$\"1n;/,O=w\")F1$!1U7d))o9vrF17$$\"1Le9\">tQ?)F1$ !10Rmk;LLqF17$$\"1***\\7yi:B)F1$!1.v.*>\\K%oF17$$\"1L$e9'>%pG)F1$!1Ksy (e,oI'F17$Fg_l$!1+oqua!Ga&F17$$\"1LLeRZQT%)F1$!1vTOys;PNF17$$\"1++]P$[ /a)F1$!14)QvNHB2'F-7$$\"1mmTN>^R')F1$\"1\\csBun`LF17$F\\`l$\"1xZwZYKA% )F17$$\"1m\"z>w'Q\"z)F1$\"1D_luF+f6Fcbl7$$\"1+]i!*z>W))F1$\"1Ts'3Lyz] \"Fcbl7$$\"1L3F>#4q*))F1$\"1(33)4qE))=Fcbl7$$\"1nm\"zW?)\\*)F1$\"1OWdP >A)H#Fcbl7$$\"1+Dcw;j-!*F1$\"1F%4II8`t#Fcbl7$$\"1L$3_!HWb!*F1$\"11(f7H ')f>$Fcbl7$$\"1nT&Q8a#3\"*F1$\"1cISLS]vOFcbl7$Fa`l$\"1\"zC.Sux;%Fcbl7$ $\"1+](oW7;@*F1$\"1xx`8&\\Qk%Fcbl7$$\"1++DJ&f@E*F1$\"1wsImBU;^Fcbl7$$ \"1+]i:mq7$*F1$\"1]\\d?tjvbFcbl7$$\"1++++PDj$*F1$\"1ZNa>;y4gFcbl7$$\"1 ++voyMk%*F1$\"1zq!3$RRXnFcbl7$Ff`l$\"1&e!4eKk$=(Fcbl7$$\"1E\"GQ(>-z&*F 1$\"1#Ht^Zg#f*F1$\"1K3`%fV!GsFcbl7$$\"1wV[Y==1'*F1$ \"1c'Rn\\$)fB(Fcbl7$$\"1+D\"Gyh(>'*F1$\"1(Q`!4+oLsFcbl7$$\"1](oal@pk*F 1$\"1EA'[!f,'>(Fcbl7$$\"1+]7G:3u'*F1$\"1v#*4JO>5rFcbl7$$\"1+vVt7SG(*F1 $\"1*y&4#HPHx'Fcbl7$$\"1++v=5s#y*F1$\"1T!QaTtf<'Fcbl7$$\"1^iS\"*3))4)* F1$\"1T'GqPWSw&Fcbl7$$\"1+D1k2/P)*F1$\"1!H]()fzqE&Fcbl7$$\"1](=nj+U')* F1$\"1M%\\Y&QYxYFcbl7$$\"1+]P40O\"*)*F1$\"1j/:')f5()RFcbl7$$\"1DJqX/% \\!**F1$\"15.o(\\%[,OFcbl7$$\"1^7.#Q?&=**F1$\"1.#f-%fT(=$Fcbl7$$\"1w$f $=.5K**F1$\"1jcT#pWPu#Fcbl7$$\"1+voa-oX**F1$\"1uF*3#4GpAFcbl7$$\"1Cc, \">g#f**F1$\"1O^\"\\y*zi " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Katsotaan, mit\344 tapahtuu, ku n n:\344\344 kasvatetaan." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 16 "Perjantai 14.10." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 "Splinit" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "readlib(spline):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=x->1/(1+25*x^2):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "n:=4:xd:=linspace(-1,1,n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "yd:=map(f,xd);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s:=spline(xd,yd,x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot([f(x),s],x=-1..1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " plot(f(x)-s,x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xdG7'!\"\" #F&\"\"#\"\"!#\"\"\"F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ydG7'# \"\"\"\"#E#\"\"%\"#HF'F)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG-% *PIECEWISEG6&7$,*#\"&`?\"\"%y_\"\"\"%\"xG#\"&D.#\"%RE*$)F.\"\"#\"\"\"# \"&+;#F1*$)F.\"\"$F5#\"%+sF12F.#!\"\"F47$,(F-F-F2#!&]!>F1F8#!&+*>F12F. \"\"!7$,(F-F-F2FBF8#\"&+*>F12F.#F-F47$,*F*F-F.#!&D.#F1F2F6F8#!%+sF1%*o therwiseG" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7ao7$$!\"\"\"\"! $\"1YQ:YQ:YQ!#<7$$!1nmm;p0k&*!#;$\"1\"\\24&eu*=%F-7$$!1LL$3X&F-7$$!1ML$3i.9!zF1$\"1PXyk$y6-'F-7$$!1nm;/R=0vF1$\"1Uhr]RWImF- 7$$!1++]P8#\\4(F1$\"1u$ydGV8O(F-7$$!1nm;/siqmF1$\"1)o;ES()yC)F-7$$!1++ ](y$pZiF1$\"1368)oX]H*F-7$$!1LLL$yaE\"eF1$\"1mN'zNm&e5F17$$!1nmm\">s%H aF1$\"19976Gx%>\"F17$$!1+++]$*4)*\\F1$\"1'*>*3w9-Q\"F17$$!1+++]_&\\c%F 1$\"1zZ%*)o#Q5;F17$$!1+++]1aZTF1$\"1***3b:1m)=F17$$!1nm;/#)[oPF1$\"1#) GKr1i(>#F17$$!1MLL$=exJ$F1$\"1;lyg0LlEF17$$!1MLLL2$f$HF1$\"1)*zu],lpJF 17$$!1++]PYx\"\\#F1$\"1Xl%Q&H#F1$\"1'oE[qcfJ%F17$$! 1MLLL7i)4#F1$\"1Td%Rs=&fZF17$$!1nmTNa%H)=F1$\"1@y\"yY%=,`F17$$!1++]P'p sm\"F1$\"1.wmc%*))**eF17$$!1+++v_*=Y\"F1$\"1u')>^ip$\"1s'e_Pc(****F17$$\"1(H$3 x\"yY_%F\\v$\"1cJ3SW)[***F17$$\"1GL3_Nl.5F-$\"1)=&yZ-)[(**F17$$\"1E$ek GR[b\"F-$\"1$4)=_\\#*R**F17$$\"1CL$3-Dg5#F-$\"1SXK(\\K.*)*F17$$\"1@Le* ['R3KF-$\"18XW*H6\"\\(*F17$$\"1cgb*F17$$\"1`mmmJ+Ii F-$\"1P59#[,b6*F17$$\"1!****\\PQ#\\\")F-$\"1WTRoQ9w&)F17$$\"1mm\"z\\1A -\"F1$\"1(>+iv*yGzF17$$\"1KLLe\"*[H7F1$\"1`hA:LOdsF17$$\"1mm;HCjV9F1$ \"1;@ga8aulF17$$\"1*******pvxl\"F1$\"1nqC)y)[FfF17$$\"1****\\7JFn=F1$ \"16L'zd,GM&F17$$\"1)****\\_qn2#F1$\"1jP(oka<\"[F17$$\"1)**\\P/q%zAF1$ \"1T/g()\\s\\VF17$$\"1)***\\i&p@[#F1$\"1e\"[=?cl$RF17$$\"1)****\\2'HKH F1$\"1hqHij,vJF17$$\"1lmmmZvOLF1$\"1)*Qq#=nIk#F17$$\"1+++]2goPF1$\"1Ho cd#=v>#F17$$\"1KL$eR<*fTF1$\"1#)=)eB,v(=F17$$\"1+++])Hxe%F1$\"1%))Qpt! )pf\"F17$$\"1lm;H!o-*\\F1$\"1N^*o:]RQ\"F17$$\"1****\\7k.6aF1$\"1S4Y9y% >?\"F17$$\"1mmm;WTAeF1$\"1\\`8zZRb5F17$$\"1****\\i!*3`iF1$\"1*y\"f(o+0 G*F-7$$\"1MLLL*zym'F1$\"1m'e9@CTD)F-7$$\"1LLL3N1#4(F1$\"1yNN>+%oO(F-7$ $\"1mm;HYt7vF1$\"1%oGOI/!=mF-7$$\"1*******p(G**yF1$\"1\")=$395U-'F-7$$ \"1mmmT6KU$)F1$\"1YG`9>?NaF-7$$\"1LLLLbdQ()F1$\"1'pK\\TLu(\\F-7$$\"1++ ]i`1h\"*F1$\"1q5l\"oD$\\XF-7$$\"1++]P?Wl&*F1$\"1W2DpLe)=%F-7$$\"\"\"F* F+-%'COLOURG6&%$RGBG$\"#5F)F*F*-F$6$7\\o7$F($\"1WP:YQ:YQF-7$F/$\"16g6e Hbi]HF-7$FS$!1-0&)RWVq@F-7$FX$\"1!=Tv'\\m98 F\\v7$Fgn$\"15aj!>Hok$F-7$F\\o$\"1q_1)p%G8yF-7$Fao$\"1E6m8FG#Q\"F17$Ff o$\"1GeoI>dI@F17$F[p$\"1,-v,QYiHF17$$!1LL3FW,eRF1$\"15`0b@0nLF17$F`p$ \"1ZR@om5%y$F17$$!1++v$>BJa$F1$\"1\"o1'R#))>H%F17$Fep$\"1==l:C'z![F17$ $!1MLLeW%o7$F1$\"1'=8bVYvC&F17$Fjp$\"13H.JK2'o&F17$F_q$\"1>!*yW`j%o'F1 7$Fiq$\"1:9c*F17$Fcy$ \"1JpG\"oY*[!*F17$F]z$\"1DCM/'3(f$)F17$Fgz$\"1LpX)zW?c(F17$Fa[l$\"1)y] r9$o0nF17$Ff[l$\"1:')e$>\"R%p&F17$F[\\l$\"1kkkg\\GkZF17$F`\\l$\"1@2')> f&Qy$F17$Fe\\l$\"1'yij5]l$HF17$Fj\\l$\"1O2UCaNF-7$F^^l$\"1'z,a(3iW&*Fbv7 $Fc^l$!16a(okh(e@F-7$$\"1LL$3sr*zoF1$!1ToK74UTHF-7$Fh^l$!1rA*\\yKV\\$F -7$$\"1++vo!*R-tF1$!1W4csi9JQF-7$F]_l$!1PcdVWfsRF-7$Fb_l$!1THnK)>Rx$F- 7$Fg_l$!1vdoy]$*>HF-7$F\\`l$!1Qa0DXk+FC!#<7$$!1LL$3A()>*Q[s5F17$$! 1LL$3i_+I(F1$\"1ec4UA/\"3\"F17$$!1mm;zp[(>(F1$\"1`#>#**)[c3\"F17$$!1++ ]P8#\\4(F1$\"1:dp7H;'3\"F17$$!1nm;/y%)))pF1$\"1O$=(\\>C#3\"F17$$!1ML$3 FuF)oF1$\"1KT93Xkt5F17$$!1,+]P2qwnF1$\"14=m#zA-1\"F17$$!1nm;/siqmF1$\" 1>nC%=K=/\"F17$$!1ML$e\\g\"fkF1$\"17z*4!f%y*)*F47$$!1++](y$pZiF1$\"1!* pP\">zN;*F47$$!1LLL$yaE\"eF1$\"1X-+*QM)QpF47$$!1++]([j5i&F1$\"1&4,t(>h QcF47$$!1nmm\">s%HaF1$\"1l)[JTVW8%F47$$!1LL$3x&y8_F1$\"1G$H*e&e9>#F47$ $!1+++]$*4)*\\F1$!1m%H\"pF&z1#!#>7$$!1++++t_\"y%F1$!1$>gJ>O^]#F47$$!1+ ++]_&\\c%F1$!1*[5uT#*=?&F47$$!1+++]zCcVF1$!1C*4fef:%zF47$$!1+++]1aZTF1 $!1-7CYw&e2\"F17$$!1LL3FW,eRF1$!1$zK^IsIL\"F17$$!1nm;/#)[oPF1$!1l5*o*f ['e\"F17$$!1++v$>BJa$F1$!1V2_MKhv=F17$$!1MLL$=exJ$F1$!1-`'[&=jU@F17$$! 1MLLeW%o7$F1$!15?cf'*=WBF17$$!1MLLL2$f$HF1$!14\\G!3Bk^#F17$$!1nmT&o_Qr #F1$!1\"*erM&)4rEF17$$!1++]PYx\"\\#F1$!1vC%4<,lw#F17$$!1n\"z>YIEW#F1$! 1%>XKHU&yFF17$$!1M$ekG'[$R#F1$!1))\\h$=`qy#F17$$!1,v$46UVM#F1$!12MH\"4 E>z#F17$$!1nmTNz>&H#F1$!1:%)**Gk0$z#F17$$!1Me*)fP0YAF1$!18H'=XV.z#F17$ $!1,]P%e4p>#F1$!1%Qx6,\"p$y#F17$$!1oT&)3awZ@F1$!15:#zQ3Ix#F17$$!1MLLL7 i)4#F1$!1tf&3@6#eFF17$$!1nmTNa%H)=F1$!1uB[kK(Gk#F17$$!1++]P'psm\"F1$!1 ba&G#y'HW#F17$$!1+++v_*=Y\"F1$!1qcyGK]Cz\"Fd\\l7$$\"1E$ekGR[b\"F4 $!1p>wGTs!H%Fd\\l7$$\"1CL$3-Dg5#F4$!16l9EIZNyFd\\l7$$\"1@Le*['R3KF4$!1 !Rhh?=2z\"F47$$\"1\"F 17$$\"1mm\"z\\1A-\"F1$!1L3W/JZ(R\"F17$$\"1**\\7Gy%e7\"F1$!12%f@!e$*)f \"F17$$\"1KLLe\"*[H7F1$!1y21mLe\"z\"F17$$\"1mm;HCjV9F1$!1&p7%3H\"z9#F1 7$$\"1*******pvxl\"F1$!1e`4;)>AV#F17$$\"1****\\7JFn=F1$!1X0*yb<7j#F17$ $\"1)****\\_qn2#F1$!1qJe^,H]FF17$$\"1)\\(o/aWF@F1$!1#[]le$RnFF17$$\"1) *\\P%G?\"y@F1$!1R*3RL(3!y#F17$$\"1)\\iS;&zGAF1$!1]'>q\"3Y)y#F17$$\"1)* *\\P/q%zAF1$!1=v%)f5h#z#F17$$\"1)\\PM#\\9IBF1$!1Sl;z:k#z#F17$$\"1)*\\7 .)>3Q#F1$!1GIK!Qh')y#F17$$\"1)\\7Go%\\JCF1$!18hx]Sy!y#F17$$\"1)***\\i& p@[#F1$!1JEIXp7pFF17$$\"1)**\\(=GB2FF1$!1EEV#[_[n#F17$$\"1)****\\2'HKH F1$!1a:HJ[P>DF17$$\"1KL$3UDX8$F1$!1M'HF;.mL#F17$$\"1lmmmZvOLF1$!1mD%zx <77#F17$$\"1KLLexn_NF1$!1K=!)Gktj=F17$$\"1+++]2goPF1$!1#*QHiwL'e\"F17$ $\"1mm\"H2fU'RF1$!1u`&*3+jC8F17$$\"1KL$eR<*fTF1$!1/4[q)[!f5F17$$\"1mm \"HiBQP%F1$!1EhC@m+2xF47$$\"1+++])Hxe%F1$!1>&GIE+-\"\\F47$$\"1LLeR*)** )y%F1$!1'G(eBo_:CF47$$\"1lm;H!o-*\\F1$!1E,*e^%3h5Fd\\l7$$\"1KL$3A_1?&F 1$\"1w(Hf5%\\k?F47$$\"1****\\7k.6aF1$\"1^Y)p@u(yRF47$$\"1KLe9as;cF1$\" 1k4!p]Hog&F47$$\"1mmm;WTAeF1$\"12%eTe.(**pF47$$\"1****\\i!*3`iF1$\"1qx $)yW0&=*F47$$\"1mm\"z\\%[gkF1$\"1q&yC%)e(F1$\"1I>RF`l&3\"F17$$\"1++vo!*R-tF1$\"1ihJv+*33\"F17$$ \"1M$e*[oc2uF1$\"1%\\H**\\t>2\"F17$$\"1mm;HYt7vF1$\"1K/su)f!f5F17$$\"1 LLek6,1xF1$\"1Zg7JwUD5F17$$\"1*******p(G**yF1$\"1@[]t*H\")z*F47$$\"1mm mT6KU$)F1$\"1A'=K*p8b$)F47$$\"1LLLLbdQ()F1$\"1L\"))*Rz2ymF47$$\"1++]i` 1h\"*F1$\"1<$[\\!oG&f%F47$$\"1++]P?Wl&*F1$\"1WS*e9`&>CF47$$\"\"\"F*F+- %'COLOURG6&%$RGBG$\"#5F)F*F*-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fi^ m%(DEFAULTG" 2 383 383 383 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "n:=10:xd:=linspace(-1,1, n);\nyd:=map(f,xd);\ns:=spline(xd,yd,x):\nplot([f(x),s],x=-1..1);\nplo t(f(x)-s,x=-1..1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xdG7-!\"\"# !\"%\"\"&#!\"$F)#!\"#F)#F&F)\"\"!#\"\"\"F)#\"\"#F)#\"\"$F)#\"\"%F)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ydG7-#\"\"\"\"#E#F'\"#<#F'\"#5#F' \"\"&#F'\"\"#F'F/F-F+F)F&" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$ 7ao7$$!\"\"\"\"!$\"1YQ:YQ:YQ!#<7$$!1nmm;p0k&*!#;$\"1\"\\24&eu*=%F-7$$! 1LL$3X&F-7$$!1ML$3i.9!zF1$\"1PXyk$y6-'F-7$$!1nm;/R=0v F1$\"1Uhr]RWImF-7$$!1++]P8#\\4(F1$\"1u$ydGV8O(F-7$$!1nm;/siqmF1$\"1)o; ES()yC)F-7$$!1++](y$pZiF1$\"1368)oX]H*F-7$$!1LLL$yaE\"eF1$\"1mN'zNm&e5 F17$$!1nmm\">s%HaF1$\"19976Gx%>\"F17$$!1+++]$*4)*\\F1$\"1'*>*3w9-Q\"F1 7$$!1+++]_&\\c%F1$\"1zZ%*)o#Q5;F17$$!1+++]1aZTF1$\"1***3b:1m)=F17$$!1n m;/#)[oPF1$\"1#)GKr1i(>#F17$$!1MLL$=exJ$F1$\"1;lyg0LlEF17$$!1MLLL2$f$H F1$\"1)*zu],lpJF17$$!1++]PYx\"\\#F1$\"1Xl%Q&H#F1$\" 1'oE[qcfJ%F17$$!1MLLL7i)4#F1$\"1Td%Rs=&fZF17$$!1nmTNa%H)=F1$\"1@y\"yY% =,`F17$$!1++]P'psm\"F1$\"1.wmc%*))**eF17$$!1+++v_*=Y\"F1$\"1u')>^ip$\"1s'e_Pc(* ***F17$$\"1(H$3x\"yY_%F\\v$\"1cJ3SW)[***F17$$\"1GL3_Nl.5F-$\"1)=&yZ-)[ (**F17$$\"1E$ekGR[b\"F-$\"1$4)=_\\#*R**F17$$\"1CL$3-Dg5#F-$\"1SXK(\\K. *)*F17$$\"1@Le*['R3KF-$\"18XW*H6\"\\(*F17$$\"1cgb*F 17$$\"1`mmmJ+IiF-$\"1P59#[,b6*F17$$\"1!****\\PQ#\\\")F-$\"1WTRoQ9w&)F1 7$$\"1mm\"z\\1A-\"F1$\"1(>+iv*yGzF17$$\"1KLLe\"*[H7F1$\"1`hA:LOdsF17$$ \"1mm;HCjV9F1$\"1;@ga8aulF17$$\"1*******pvxl\"F1$\"1nqC)y)[FfF17$$\"1* ***\\7JFn=F1$\"16L'zd,GM&F17$$\"1)****\\_qn2#F1$\"1jP(oka<\"[F17$$\"1) **\\P/q%zAF1$\"1T/g()\\s\\VF17$$\"1)***\\i&p@[#F1$\"1e\"[=?cl$RF17$$\" 1)****\\2'HKHF1$\"1hqHij,vJF17$$\"1lmmmZvOLF1$\"1)*Qq#=nIk#F17$$\"1+++ ]2goPF1$\"1Hocd#=v>#F17$$\"1KL$eR<*fTF1$\"1#)=)eB,v(=F17$$\"1+++])Hxe% F1$\"1%))Qpt!)pf\"F17$$\"1lm;H!o-*\\F1$\"1N^*o:]RQ\"F17$$\"1****\\7k.6 aF1$\"1S4Y9y%>?\"F17$$\"1mmm;WTAeF1$\"1\\`8zZRb5F17$$\"1****\\i!*3`iF1 $\"1*y\"f(o+0G*F-7$$\"1MLLL*zym'F1$\"1m'e9@CTD)F-7$$\"1LLL3N1#4(F1$\"1 yNN>+%oO(F-7$$\"1mm;HYt7vF1$\"1%oGOI/!=mF-7$$\"1*******p(G**yF1$\"1\") =$395U-'F-7$$\"1mmmT6KU$)F1$\"1YG`9>?NaF-7$$\"1LLLLbdQ()F1$\"1'pK\\TLu (\\F-7$$\"1++]i`1h\"*F1$\"1q5l\"oD$\\XF-7$$\"1++]P?Wl&*F1$\"1W2DpLe)=% F-7$$\"\"\"F*F+-%'COLOURG6&%$RGBG$\"#5F)F*F*-F$6$7bo7$F($\"1\\Q:YQ:YQF -7$F/$\"1ga5q=BLUF-7$F5$\"1@`0twC$e%F-7$F:$\"1%=m\"fF;1]F-7$F?$\"1!p?h b3(yaF-7$FD$\"1&Gdzql<,'F-7$FI$\"1#\\p**=8Ce'F-7$FN$\"1&\\*y<2T&G(F-7$ FS$\"1rIXZzPr\")F-7$FX$\"16pDmWdb#*F-7$Fgn$\"1%RX)RnHi5F17$F\\o$\"1Fzd !)))H27F17$Fao$\"1N?')y]u,9F17$Ffo$\"1WVR;d:M;F17$F[p$\"1z*\\I:rr*=F17 $F`p$\"1bKv*o!)Q<#F17$Fep$\"1Z$R3R,$)e#F17$Fjp$\"1RKo**3=jIF17$F_q$\"1 -'fPyk7#QF17$Fdq$\"1MXRyomWUF17$Fiq$\"1%)R1%Hs1t%F17$F^r$\"1*\\c>L*3U` F17$Fcr$\"1]!pKXm9-'F17$Fhr$\"1@ql#Gx@q'F17$F]s$\"1YQ!)*>hqQ(F17$Fbs$ \"1$)G`*>)\\v!)F17$Fgs$\"1<;k&y*\\0()F17$F\\t$\"1tK/oS,0#*F17$Fat$\"1$ QfBpFKg*F17$Fft$\"1X#f!RD>n(*F17$F[u$\"1))RWlEa!*)*F17$F`u$\"1J#32Tmd$ **F17$Feu$\"1W<_=:Pp**F17$Fju$\"1S0wp'p3***F17$F`v$\"1S1IXDx****F17$Ff v$\"1%GJP4h_***F17$F[w$\"1P'=&[b)p(**F17$F`w$\"1;%Hp7#\\X**F17$Few$\"1 MID%4G8!**F17$Fjw$\"1DkzcsT()F17$F^y$\"1_Z)y'>NP\")F17$Fcy$\"1gZ(F17$ Fhy$\"1%z/\"pvKjnF17$F]z$\"1S*yRaNC0'F17$Fbz$\"1\\!y`JS&*Q&F17$Fgz$\"1 t!)3*[E))y%F17$F\\[l$\"1\")e)fkm6G%F17$Fa[l$\"1W*RBd!fSQF17$$\"1)**\\( =GB2FF1$\"1rny(fh=U$F17$Ff[l$\"1c1^sSSoIF17$F[\\l$\"1^r^Q;3oDF17$F`\\l $\"15MF28zt@F17$Fe\\l$\"1&f&3Ugy))=F17$Fj\\l$\"1ldi=A%4i\"F17$F_]l$\"1 3I(HG/cS\"F17$Fd]l$\"1G_A+t!\\@\"F17$Fi]l$\"1(4O\\r<*e5F17$F^^l$\"1J\\ bz+JS#*F-7$Fc^l$\"1X'Gs\")Gx<)F-7$Fh^l$\"1$QS29$z!H(F-7$F]_l$\"1C74;Ml qlF-7$Fb_l$\"1-WdY/f9gF-7$Fg_l$\"1hPQ>7\"HY&F-7$F\\`l$\"1!\\N?')zl-&F- 7$Fa`l$\"18:rY,y0YF-7$Ff`l$\"1K;e.T)>B%F-7$F[alFgal-F^al6&F`alF*FaalF* -%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(F[al%(DEFAULTG" 2 383 383 383 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}{PARA 13 "" 1 "" {INLPLOT "6%- %'CURVESG6$7it7$$!\"\"\"\"!$!1\"*GchvbvF!#K7$$!1MLLe%G?y*!#;$!1Rs121a6 E!#>7$$!1nmm;p0k&*F1$!1Voz>>g[VF47$$!1++vV5Su$*F1$!1EM^uT(*R_F47$$!1LL $3I()\\F47$$!1mmm\"4m (G$)F1$!1b6QS.gwEF47$$!1ML$3i.9!zF1$\"1>EDFol7%*!#?7$$!1nm;/R=0vF1$\"1 I\\mug2.[F47$$!1LL$3i_+I(F1$\"1;][#\\rNX'F47$$!1++]P8#\\4(F1$\"1$*y))) zcKf(F47$$!1ML$3FuF)oF1$\"1.tXRuAc!)F47$$!1nm;/siqmF1$\"1^u=Ar%RF47$$!1n mT&Gu,.'F1$\"1(HAz'=!GX&FS7$$!1LLL$yaE\"eF1$!1l$G=)=QIPF47$$!1nmm\">s% HaF1$!1Q8lXpg_7!#=7$$!1LL$3x&y8_F1$!1))=a&H1Iu\"F`q7$$!1+++]$*4)*\\F1$ !1^Q+(zJI:#F`q7$$!1+++DL\")*)[F1$!10(=E;_LI#F`q7$$!1++++t_\"y%F1$!1#3& HCgT+CF`q7$$!1,+](G%QFZF1$!1,?V8^nCCF`q7$$!1+++v7CtYF1$!11:>R/EICF`q7$ $!1++]i#)4>YF1$!1YC$zes^T#F`q7$$!1+++]_&\\c%F1$!16l&\\u-tP#F`q7$$!1+++ +;ggWF1$!1\"[sN#\\tJAF`q7$$!1+++]zCcVF1$!1_5zu#4g(>F`q7$$!1++++V*=D%F1 $!1Cu/k(*>\"f\"F`q7$$!1+++]1aZTF1$!1Rz4a(*\\c5F`q7$$!1LL3FW,eRF1$\"1$p $o[CmEPF47$$!1nm;/#)[oPF1$\"1PF'p:)*RP#F`q7$$!1++v$>BJa$F1$\"1fcOq*3I4 &F`q7$$!1MLL$=exJ$F1$\"1Cpr%*p\"Hq(F`q7$$!1MLLeW%o7$F1$\"1dG[\">!p%\\* F`q7$$!1MLLL2$f$HF1$\"1(fZ1^#pk5!#<7$$!1nTN@7T!)GF1$\"1X5^e@3$3\"F[w7$ $!1+]P4<*[#GF1$\"1W3s$pFF1$\"1lQ$[Y=i4\"F[w7$$!1n mT&o_Qr#F1$\"1.59?J+!4\"F[w7$$!1M$e9m8Gg#F1$\"13!=vi:'\\5F[w7$$!1++]PY x\"\\#F1$\"1tUp3!Rpo*F`q7$$!1M$ekG'[$R#F1$\"1AYx!e^\"*f)F`q7$$!1nmTNz> &H#F1$\"1a^@VE)*GrF`q7$$!1,]P%e4p>#F1$\"1M?v)Hs!R_F`q7$$!1MLLL7i)4#F1$ \"1cd<))Hk%)GF`q7$$!1nT&QG-Z/#F1$\"1ai3Y[Sw8F`q7$$!1,]PMLy!*>F1$!1sY== g)f)HF47$$!1Me*[Qko$>F1$!1j'pN5:hF`q7$$!1L$ek`2^x\"F1$!1dDo(ytK;)F`q7$$!1n \"zpe)=@\">5F[w7$$!1++]P'psm\"F1$!1oW,m*pd@\"F[w7$$!1+]( o/Efh\"F1$!1]O\"=\\9QR\"F[w7$$!1++DcCek:F1$!1XsZ7Eyf:F[w7$$!1+]il)QK^ \"F1$!1)y],wE4r\"F[w7$$!1+++v_*=Y\"F1$!1yMe9.\"[%=F[w7$$!1++v$43#f8F1$ !1=Lv3Vu_?F[w7$$!1++]74_c7F1$!1;<$)[1=r@F[w7$$!1$eRs3N(H7F1$!1ZQ#46`k= #F[w7$$!1n\"z>E\\H?\"F1$!19-1#*[:&>#F[w7$$!1](=nVjh<\"F1$!14\"y%y`G(># F[w7$$!1L$e9hx$\\6F1$!1GJf5j(G>#F[w7$$!1+v$4'f!e4\"F1$!1)e=13HZ;#F[w7$ $!1mmT5VBU5F1$!1WscCrh6@F[w7$$!1)**\\P454N*F[w$!1f_#3:qm$>F[w7$$!1JLL$ 3x%z#)F[w$!1'pF?'G>&o\"F[w7$$!1KLe9rc&H(F[w$!1R.HVRI29F[w7$$!1LL$e9d;J 'F[w$!1>Yw(*GO26F[w7$$!1LL3xruF`F[w$!1m%oV=VU4)F`q7$$!1MLL3s$QM%F[w$!1 KKzdw3q`F`q7$$!1omT&QdDG$F[w$!1Y(e()o#f]HF`q7$$!1,+]ivF@AF[w$!11ua\"z[ \"R7F`q7$$!1o;/^wj!p\"F[w$!1-`O$[Ber'F47$$!1NLeRx**f6F[w$!1\"oFfu-')*G F47$$!1%oT&QyxY*)F`q$!1Soc4ssN;F47$$!1<+D\"GyNH'F`q$!1B/5f*G!Q#FS7$$!1^omm;zr)*F4$!1$Quo>/qh\"!#@7$$\"11$3_] \\(o`Fhdl7$$\"1(H$3x\"yY_%F`q$!1ibF\"[Olw$FS7$$\"1)Ge*[ og!G(F`q$!1G5Ur4.W5F47$$\"1GL3_Nl.5F[w$!1o\"[Mt+`5#F47$$\"1E$ekGR[b\"F [w$!1*GB8uurc&F47$$\"1CL$3-Dg5#F[w$!1h$\\Gpf&*4\"F`q7$$\"1@Le*['R3KF[w $!1k7>Ndf1GF`q7$$\"1F[w7$$\"1lm\"z\\1A-\"F1$!1ab%o6Ac3#F[w7$$\"1K3-jr-u5F1$!1)G4u( *zg9#F[w7$$\"1**\\7Gy%e7\"F1$!1b4I?yl$=#F[w7$$\"1l\"HK\\ow<\"F1$!1#*=* R^Rt>#F[w7$$\"1KLLe\"*[H7F1$!1WY!p\"Hc'=#F[w7$$\"1***\\PzglL\"F1$!1mO2 *3so3#F[w7$$\"1mm;HCjV9F1$!1'yE]9iy)=F[w7$$\"1**\\(oCor\\\"F1$!1&=OQ1% za>F1$!1rVMJZK\\F F`q7$$\"1)**\\(==-s>F1$!1766u2:8#*F47$$\"1)*\\(=<'RC?F1$\"1-5I0mtdwF47 $$\"1)****\\_qn2#F1$\"1N*o&yd\"GH#F`q7$$\"1)*\\P%G?\"y@F1$\"1!\\2*ouUE [F`q7$$\"1)**\\P/q%zAF1$\"1wgXhT$e&oF`q7$$\"1)*\\7.)>3Q#F1$\"1\\,qJM)= V)F`q7$$\"1)***\\i&p@[#F1$\"1(R@3&Hc'f*F`q7$$\"1)*\\i!>,Zf#F1$\"1;$)3U V7X5F[w7$$\"1)**\\(=GB2FF1$\"1#RN2Fg')3\"F[w7$$\"1)\\7Gj)\\jFF1$\"1y-: '))zf4\"F[w7$$\"1)*\\(oWk(>GF1$\"1`$y%z4L%4\"F[w7$$\"1)\\P4EIg(GF1$\"1 XrL=5@%3\"F[w7$$\"1)****\\2'HKHF1$\"1ZS'y*G7m5F[w7$$\"1KL$3UDX8$F1$\"1 /`DB2WL%*F`q7$$\"1lmmmZvOLF1$\"1(pu'=Wb)\\(F`q7$$\"1KLLexn_NF1$\"1`1H4 zdw\\F`q7$$\"1+++]2goPF1$\"1q=MH]psBF`q7$$\"1mm\"H2fU'RF1$\"1y))*e$e]_ JF47$$\"1KL$eR<*fTF1$!1U7P?1[G6F`q7$$\"1**\\P40(oE%F1$!1=dxM(G^l\"F`q7 $$\"1mm\"HiBQP%F1$!1G+(>C!eF?F`q7$$\"1L$ektw2[%F1$!1g\\n/;\"zE#F`q7$$ \"1+++])Hxe%F1$!11\")oo\"[hR#F`q7$$\"1LeRAr/QYF1$!1Z(e:3MHU#F`q7$$\"1m ;z%Rk$)o%F1$!1N#*o0LpICF`q7$$\"1+v=n;oQZF1$!1;p$pT!4@CF`q7$$\"1LLeR*)* *)y%F1$!1oYNo#4dR#F`q7$$\"1**\\P%[L'*)[F1$!12P^n6c.BF`q7$$\"1lm;H!o-* \\F1$!1Ety2ETl@F`q7$$\"1KL$3A_1?&F1$!17pz*z65x\"F`q7$$\"1****\\7k.6aF1 $!18)Gkd[fH\"F`q7$$\"1mmm;WTAeF1$!1&pZ2!e$H_$F47$$\"1KLeRSF47$$\"1mm\"z\\%[gkF1$\"1prA,! fMI'F47$$\"1MLLL*zym'F1$\"1z?+B%R&RwF47$$\"1LL$3sr*zoF1$\"1Mj0*>$zc!)F 47$$\"1LLL3N1#4(F1$\"1:&>8'yo/wF47$$\"1++vo!*R-tF1$\"1$Q#yffAPkF47$$\" 1mm;HYt7vF1$\"1Ogu`()3NZF47$$\"1*******p(G**yF1$\"1%))yuD%p>'*FS7$$\"1 mmmT6KU$)F1$!1T:4&[I4x#F47$$\"1LLLLbdQ()F1$!1G%z-rWY\"\\F47$$\"1++]i`1 h\"*F1$!1ZU/1lWXcF47$$\"1++++PDj$*F1$!1HU@lN2w_F47$$\"1++]P?Wl&*F1$!1X ()3LM2SVF47$$\"1++v=5s#y*F1$!1^8V\"R#p/EF47$$\"\"\"F*F+-%'COLOURG6&%$R GBG$\"#5F)F*F*-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Ff\\n%(DEFAULTG" 2 383 383 383 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "Numeerinen integrointi" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "trapetsi:=p roc(f,a,b,n)\n local h,S,R;\n h:=(b-a)/n;\n S:=evalf((h/2)*(f(a)+2* sum(f(a+j*h),j=1..n-1)+f(b)));\n R:=-(h^2*(b-a)/12)*(D@D)(f)(xi);\n \+ [S,R] # Palutetaan lista: [Intappr, virhe]\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)trapetsiGR6&%\"fG%\"aG%\"bG%\"nG6%%\"hG%\"SG%\"RG6\" F/C&>8$*&,&9&\"\"\"9%!\"\"\"\"\"9'!\"\">8%-%&evalfG6#,$*&F2F9,(-9$6#F7 F6-%$sumG6$-FE6#,&F7F6*&%\"jGF6F2F6F6/FN;F6,&F:F6F8F6\"\"#-FE6#F5F6F6# F6FR>8&,$*()F2FRF9F4F6---%\"@G6$%\"DGFjn6#FE6#%#xiGF6#F8\"#77$F=FWF/F/ F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f:='f':(h/2)*(f(a)+2* sum(f(a+j*h),j=1..n-1)+f(b));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&% \"hG\"\"\",8-%\"fG6#%\"aGF&-F)6#,&F+F&F%F&\"\"#-F)6#,&F+F&F%F/F/-F)6#, &F+F&F%\"\"$F/-F)6#,&F+F&F%\"\"%F/-F)6#,&F+F&F%\"\"&F/-F)6#,&F+F&F%\" \"'F/-F)6#,&F+F&F%\"\"(F/-F)6#,&F+F&F%\"\")F/-F)6#,&F+F&F%\"\"*F/-F)6# %\"bGF&F&#F&F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "(h^2*(b-a )/12)*(D@D)(f)(xi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*()%\"hG\"\"# \"\"\",&%\"bG\"\"\"%\"aG!\"\"F+---%#@@G6$%\"DGF'6#%\"fG6#%#xiGF+#F+\"# 7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f:=x->exp(x)*cos(x);pl ot(f(x),x=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\" 6$%)operatorG%&arrowGF(*&-%$expG6#9$\"\"\"-%$cosGF/F1F(F(F(" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7S7$\"\"!$\"\"\"F(7$$\"1fD2LzxZo!#< $\"1sy*)GqOo5!#:7$$\"1Npx*G*f!G\"!#;$\"1wQ!e#RJF6F17$$\"15@exGm]>F5$\" 1kmfL9M#>\"F17$$\"199!=3o^i#F5$\"1\\Dz\"R`cD\"F17$$\"1!\\D0[nkH$F5$\"1 ZM&[14cJ\"F17$$\"1\"=Za&z%)=RF5$\"1y[U:ten8F17$$\"1eXa()oGjXF5$\"1M4_N Zx;9F17$$\"1%3**Hbm(H_F5$\"1_DR`cah9F17$$\"1PRr4)3T*eF5$\"1^%ou>:()\\ \"F17$$\"1#[)yykYxlF5$\"1+e2lPpF:F17$$\"1s'ocGo$zrF5$\"1:Uqfr8W:F17$$ \"1Q0;gr'p&yF5$\"10YleI)3b\"F17$$\"1GMt?$[t`)F5$\"1'[W590La\"F17$$\"1# p30k@I>*F5$\"16F17$$\"1y0DBA!*38F1 $\"1Q37ff9&e*F57$$\"1kE.#[AMP\"F1$\"1,aiDjZVxF57$$\"1a@X.EuS9F1$\"1;E+ *\\xxZ&F57$$\"1WF)**[jD]\"F1$\"1x)*)pOaM1$F57$$\"1hw\"ymX#p:F1$\"1(pBM \"*[yW(!#=7$$\"1xM!*4(4&Q;F1$!1)[&3e$yG[$F57$$\"1DL:iU!))p\"F1$!13!yu` :'zpF57$$\"1)R1/.CRw\"F1$!1`yA\"pj*>6F17$$\"1Id)H7*>J=F1$!1=eyvK'pg\"F 17$$\"1fb7wY,(*=F1$!1\"Q0h\"e>O@F17$$\"1N5'zg%pg>F1$!1@s(G(*)G+FF17$$ \"1pJ8:.SJ?F1$!1wDV4>7*Q$F17$$\"1+2zPD$\\4#F1$!13j9_(=\"F\\x7$$\"1K-%\\0:[o#F1$!1?Ghj')H:8F\\x7$$ \"1gN,?R*3v#F1$!1:E\\D)ywW\"F\\x7$$\"1/0LMNh6GF1$!1Gk0ct\"Rd\"F\\x7$$ \"1$oUX107)GF1$!16On#\\^Ms\"F\\x7$$\"146xe&[M%HF1$!1o)=pxz4'=F\\x7$$\" 1.6)e58)4IF1$!1uReOUx5?F\\x7$$\"1t2RXCLtIF1$!1mIp)[Zj:#F\\x7$$\"1++X]E fTJF1$!1GA,c#pSJ#F\\x-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q \"x6\"%!G-%%VIEWG6$;F($\"+aEfTJ!\"*%(DEFAULTG" 2 400 300 300 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "trapetsi(f,0,Pi,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!+oX& pA\"!\"),$*()%#PiG\"\"$\"\"\"-%$expG6#%#xiG\"\"\"-%$sinGF/F1#F1\"$+'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "R:=%[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG,$*()%#PiG\"\"$\"\"\"-%$expG6#%#xiG\"\"\"-%$si nGF-F/#F/\"$+'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(abs( R),xi=0..Pi,thickness=2);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$ 7hn7$\"\"!F(7$$\"+LzxZo!#6$\"++$)e'y$!#77$$\"+!H*f!G\"!#5$\"+yTN,vF/7$ $\"+xGm]>F3$\"+AiT<7F,7$$\"+#3o^i#F3$\"+wTnV!)F,7$$\"+?$[t`)F3$\"+NqGZ\"*F,7$$\" +S;-$>*F3$\"+%37/.\"F37$$\"+HeV)y*F3$\"+KbJT6F37$$\"+*3W'\\5!\"*$\"+k; D!G\"F37$$\"+/9i46Fdp$\"+c>s.9F37$$\"+\"G*Qz6Fdp$\"+K8j`:F37$$\"+tc9T7 Fdp$\"+l^a\"p\"F37$$\"+BA!*38Fdp$\"+-r!z%=F37$$\"+#[AMP\"Fdp$\"+J!H5+# F37$$\"+.EuS9Fdp$\"+p6Kk@F37$$\"+!\\jD]\"Fdp$\"+&y\\lJ#F37$$\"+ocCp:Fd p$\"+R<1#[#F37$$\"+5(4&Q;Fdp$\"+m-)Rl#F37$$\"+iU!))p\"Fdp$\"+$R\"G-GF3 7$$\"+IS#Rw\"Fdp$\"+D$[%fHF37$$\"+B\"*>J=Fdp$\"+Q-i;JF37$$\"+wY,(*=Fdp $\"+W[5jKF37$$\"+3Ypg>Fdp$\"+9)RdR$F37$$\"+:.SJ?Fdp$\"+V!='HNF37$$\"+Q D$\\4#Fdp$\"+8=1NOF37$$\"+Ynwi@Fdp$\"+3Q#)GPF37$$\"+@YBCAFdp$\"+,E%Rz$ F37$$\"+W_V\"H#Fdp$\"+fW))RQF37$$\"+>00BBFdp$\"+)zB7&QF37$$\"+$zlYN#Fd p$\"+&>d`&QF37$$\"+lGr(Q#Fdp$\"+O^X^QF37$$\"+O*f2U#Fdp$\"+#R\"fQQF37$$ \"+*[pIX#Fdp$\"+,7z;QF37$$\"+U!z`[#Fdp$\"+m`I&y$F37$$\"+d#HIb#Fdp$\"+( >Jao$F37$$\"+bW==EFdp$\"+Al]TNF37$$\"+b]\"[o#Fdp$\"+,n*e#F37$$\"+k]?\")GFdp$\"+I\\'HP#F37$$\"+7oK7HFd p$\"+Pg*3;#F37$$\"+f&[M%HFdp$\"+I&*)3$>F37$$\"+L3jwHFdp$\"+dm6l;F37$$ \"+1J\")4IFdp$\"+`pJx8F37$$\"+wFdTIFdp$\"+/\\V!3\"F37$$\"+XCLtIFdp$\"+ &fL$=wF,7$$\"+(\\(R!4$Fdp$\"+iy19eF,7$$\"+[DY2JFdp$\"+j\\rVRF,7$$\"+*f FX7$Fdp$\"+%R)31?F,7$$\"+]EfTJFdp$\"+)fMGH%!#=-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%+AXESLABELSG6$Q#xi6\"%!G-%*THICKNESSG6#\"\"#-%%VIEWG6$;F($\" +aEfTJFdp%(DEFAULTG" 2 400 300 300 2 0 1 2 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "maxvirhe:=0.4; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)maxvirheG$\"\"%!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 43 "Ep\344lineaariset yht\344l\366t, Newtonin menetelm\344 \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f:='f':N:=x->evalf(x-f (x)/D(f)(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f:=x->x*ex p(-x)-abs(sin(10*x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"x G6\"6$%)operatorG%&arrowGF(,&*&9$\"\"\"-%$expG6#,$F.!\"\"F/F/-%$absG6# -%$sinG6#,$F.\"#5F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=-1..3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " x1:=(N@@5)(0.25);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+'=z;#H! #5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x1:=(N@@5)(x1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+'=z;#H!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(f(x),x=0..0.5):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "52 5 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }