{VERSION 4 0 "IBM INTEL LINUX22" "4.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{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 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 0 }3 3 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 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Harj. 12 AV" }}{PARA 19 " " 0 "" {TEXT -1 11 "20.4.02 HA" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 2 "1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the \+ name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "F:=t->Int(f(t,x),x=a(t)..b(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%$IntG6$-%\"fG 6$9$%\"xG/F3;-%\"aG6#F2-%\"bGF8F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dF:=diff(F(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#dFG,(-%$IntG6$-%%diffG6$-%\"fG6$%\"tG%\"xGF//F0;-%\"aG6#F/-%\"bGF5 \"\"\"*&-F*6$F6F/F8-F-6$F/F6F8F8*&-F*6$F3F/F8-F-6$F/F3F8!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Maple osaa kyll\344 soveltaa ketju s\344\344nt\366\344 ja muita yleisi\344 lauseita." }}{PARA 0 "" 0 "" {TEXT -1 39 "M\344\344ritell\344\344n nyt f teht\344v\344n funktioksi. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:=(t,x)->sin(x/t)/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"tG%\"xG6\"6$%)operatorG% &arrowGF)*&-%$sinG6#*&9%\"\"\"9$!\"\"F3F2F5F)F)F)" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "a:=t->t: b:=t->t^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bGR6#%\"tG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\" \"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$sinG6#*&%\"xG\"\"\"%\"tG!\"\"F ,F+F./F+;F-*$)F-\"\"#F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " diff(F(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$IntG6$,$*&-%$cos G6#*&%\"xG\"\"\"%\"tG!\"\"F.*$)F/\"\"#F.F0F0/F-;F/*$F2F.F.*&*&F3F.-%$s inG6#F/F.F.F/F0F.*&-F:6#F.F.F/F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "simplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*&-%$sinG6#%\"tG\"\"\"F'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "Kannattaa laskea pitemm\344lle k\344sin. T\344ss\344 on helppo t ehd\344 laskuvirheit\344, jos tarkkaavaisuus hetkeksi herpaantuu." }} {PARA 0 "" 0 "" {TEXT -1 108 "Seuraava ei kuulu en\344\344 osattaviin \+ asioihin, mutta kun Maple helposti tarjoilee Si-funktiota, niin katsot aan:" }}{PARA 0 "" 0 "" {TEXT -1 38 "Maple siis tuntee erikoisfunktio n Si:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "Int(sin(x)/x, x=0. .t): value(%)=%; # T\344ss\344 tapauksessa saadaan looginen j\344rjes tys n\344inp\344in." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#SiG6#%\"tG- %$IntG6$*&-%$sinG6#%\"xG\"\"\"F/!\"\"/F/;\"\"!F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "F(t)=value(F(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$sinG6#*&%\"xG\"\"\"%\"tG!\"\"F-F,F//F,;F .*$)F.\"\"#F-,&-%#SiG6#F.F--F76#F-F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Diff(lhs(%),t)=diff(rhs(%),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%$IntG6$*&-%$sinG6#*&%\"xG\"\"\"%\"tG!\"\"F 0F/F2/F/;F1*$)F1\"\"#F0F1*&-F,6#F1F0F1F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Maple osasi palauttaa kysytyn teht\344v\344n suoraan Si- funktion derivaataksi, joka tietysti on yll\344 oleva sin(t)/t ." }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 2 "2." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f:=1/(x+y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG *&\"\"\"F&,&%\"xGF&%\"yGF&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "c:=15: plot(1/x,x=1..c,color=yellow,filled=true);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%)POLYGONSG6Z7&7$$\"\"\" \"\"!F*7$F(F(7$$\"3DLLeR+Hw5!#<$\"3?pWGGe<\"H*!#=7$F-F*7&F3F,7$$\"3smm ;z+e_6F/$\"3?5-ISn=w')F27$F6F*7&F:F57$$\"3=++v=,()G7F/$\"3))o()p\"[cv8 )F27$F=F*7&FAF<7$$\"3VLLLe,;08F/$\"3#oK@+$\\*=m(F27$FDF*7&FHFC7$$\"3.+ ]Pp#>zV\"F/$\"3&**H\"H_L\\apF27$FKF*7&FOFJ7$$\"3jmmT!Qy1d\"F/$\"3NQuBy dnmjF27$FRF*7&FVFQ7$$\"3$***\\i&))z*>?F/ $\"3a0AS2Ga^\\F27$FaoF*7&FeoF`o7$$\"3DLL$etj)p@F/$\"3!*y)R[K%e3YF27$Fh oF*7&F\\pFgo7$$\"3,nmTlu,pCF/$\"3:0Ml6T>]SF27$F_pF*7&FcpF^p7$$\"38LL3n 7PYFF/$\"3LY[#3Io6k$F27$FfpF*7&FjpFep7$$\"3<++vj]bLIF/$\"3k\")*)4KBY'H $F27$F]qF*7&FaqF\\q7$$\"3mLL3d4cILF/$\"3ue8f:s\\-IF27$FdqF*7&FhqFcq7$$ \"3+++v[VhEOF/$\"3(>XP!>FRdFF27$F[rF*7&F_rFjq7$$\"32nmm^;9JRF/$\"3j,[j *H!zVDF27$FbrF*7&FfrFar7$$\"3gLL$eYp$*>%F/$\"3ajaH!))48Q#F27$FirF*7&F] sFhr7$$\"3i+++b/L,XF/$\"3o-3g,ac@AF27$F`sF*7&FdsF_s7$$\"38+++D8`/[F/$ \"3?+zHn%o83#F27$FgsF*7&F[tFfs7$$\"3I+++X:s'4&F/$\"3&z%>&ReX?'>F27$F^t F*7&FbtF]t7$$\"3]LL3d#e?O&F/$\"3&4zPH\\b\\'=F27$FetF*7&FitFdt7$$\"3-nm mr#pvn&F/$\"3GhrOFrJh'=x]3WS\"F27$F_wF *7&FcwF^w7$$\"3Rnm;/mV?uF/$\"3f9()G)fHwM\"F27$FfwF*7&FjwFew7$$\"3EnmT& RJfp(F/$\"3!)R<`#)yQ*H\"F27$F]xF*7&FaxF\\x7$$\"3?LL$eu*3$*zF/$\"3jF&o1 m!3^7F27$FdxF*7&FhxFcx7$$\"3jKL3dPv,$)F/$\"3kWBNdZc/7F27$F[yF*7&F_yFjx 7$$\"3G,+D'oY/d)F/$\"3\\'eguK+o;\"F27$FbyF*7&FfyFay7$$\"3#RLL3TU1'))F/ $\"3+r))HTjeG6F27$FiyF*7&F]zFhy7$$\"3&********)HWg\"*F/$\"3KcS3j-l\"4 \"F27$F`zF*7&FdzF_z7$$\"3x++]n$RPX*F/$\"3l(e)Q$\\#yd5F27$FgzF*7&F[[lFf z7$$\"37,+v$p=vt*F/$\"3#o%)zkmbp-\"F27$F^[lF*7&Fb[lF][l7$$\"3'****\\_s g_+\"!#;$\"3nSG]^!ow%**!#>7$Fe[lF*7&F[\\lFd[l7$$\"3dmmmLGdL5Fg[l$\"3A= `R&*o\"4\"Fg[l$\"3AtmcN9Fk\"*Fj[l7 $F\\]lF*7&F`]lF[]l7$$\"3?++]*3T67\"Fg[l$\"3hdF\\\"*R[>*)Fj[l7$Fc]lF*7& Fg]lFb]l7$$\"3jm;/i(=$\\6Fg[l$\"3(y6Q[O13q)Fj[l7$Fj]lF*7&F^^lFi]l7$$\" 30+]()[Dxy6Fg[l$\"3\"))4^M\"3S$[)Fj[l7$Fa^lF*7&Fe^lF`^l7$$\"3qmm;4!pv? 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T\344t\344 nyt e i ihan voitane pit\344\344 todistuksena." }}{PARA 0 "" 0 "" {TEXT -1 41 "K\344sin laskien n\344hd\344\344n, ett\344 arctan(c) -> " } {XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 34 "Siis tarvitsemme raja-arvon (c -> " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 16 " ) lausekkeelle " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c*ln(1+1/(c^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"cG\"\"\"-%#lnG6#,&F%F%*&F%F%*$)F$\"\"#F %!\"\"F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Helpointa on ottaa \+ pari termi\344 Taylorin kehitelm\344st\344:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "c*subs(x=1/c^2,taylor(ln(1+x),x=0,2));expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"cG\"\"\",&*&F%F%*$)F$\"\"#F%!\" \"F%-%\"OG6#*&F%F%*$)F$\"\"%F%F+F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*&\"\"\"F%%\"cG!\"\"F%*&F&F%-%\"OG6#*&F%F%*$)F&\"\"%F%F'F%F%" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "T\344m\344 siis l\344henee 0:aa, k un c-> " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 49 " , \+ joten saamme tosiaankin yll\344 olevan tuloksen." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 2 "3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): with(plottools):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f:=1/(x*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&\"\"\"F&*&%\"xGF&%\"yGF&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c:=.1: display(plot([x^2,x],x=0..1) ,line([c,c^2],[c,c]));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3dmmm;arz@!#>$\"3*)RVl(H f6v%!#@7$$\"3[LL$e9ui2%F-$\"3GF*)>\"4,;m\"!#?7$$\"3nmmm\"z_\"4iF-$\"3( f>E!RyNbQF67$$\"3[mmmT&phN)F-$\"3(4x24%pb#)pF67$$\"3BLLe*=)H\\5!#=$\"3 ngGm!pE55\"F-7$$\"3gmm\"z/3uC\"FD$\"3sos+Qo-c:F-7$$\"3%)***\\7LRDX\"FD $\"3u'>G)30()4@F-7$$\"3^mm\"zR'ok;FD$\"32m(3M!3=rFF-7$$\"3w***\\i5`h(= FD$\"3yA:4y/&*>NF-7$$\"3YLLL3En$4#FD$\"3y.`)3*\\Y$Q%F-7$$\"3qmm;/RE&G# FD$\"3'e2(o66VA_F-7$$\"3\")*****\\K]4]#FD$\"3u/w6GDvaiF-7$$\"3$****** \\PAvr#FD$\"3)3kD'ey#\\Q(F-7$$\"3(******\\nHi#HFD$\"3B01&36?Gc)F-7$$\" 3jmm\"z*ev:JFD$\"39%\\-a\"[$zq*F-7$$\"3?LLL347TLFD$\"3v:-T#*)3j6\"FD7$ $\"3,LLLLY.KNFD$\"3?8m5lo_Z7FD7$$\"3v***\\7o7Tv$FD$\"3I1AN-iL49FD7$$\" 3'GLLLQ*o]RFD$\"3#4Fe.m%zg:FD7$$\"3A++D\"=lj;%FD$\"3zMKN#))fet\"FD7$$ \"31++vV&RFD7$$\"3XLL$e9Ege%FD$\"3#QpE5ejJ5#FD7$$ \"3HLLeR\"3Gy%FD$\"3^ek+q`_(G#FD7$$\"3emm;/T1&*\\FD$\"3GMMZSl1&\\#FD7$ $\"3&em;zRQb@&FD$\"3R9V,yS=?FFD7$$\"3\\***\\(=>Y2aFD$\"3]UJFSW1CHFD7$$ \"39mm;zXu9cFD$\"3SY\"G*oc`_JFD7$$\"3l******\\y))GeFD$\"3)>w(ycLf(R$FD 7$$\"3'*)***\\i_QQgFD$\"3*yrPyl4ik$FD7$$\"3@***\\7y%3TiFD$\"3#\\.vY#R6 &*QFD7$$\"36****\\P![hY'FD$\"3#*3lGWq5\"=%FD7$$\"3jKLL$Qx$omFD$\"3<]^l #pDnW%FD7$$\"3!)*****\\P+V)oFD$\"3m8DKl\"f$RZFD7$$\"3?mm\"zpe*zqFD$\"3 e&e?k^\"e7]FD7$$\"3%)*****\\#\\'QH(FD$\"3yCXTal/?`FD7$$\"3GKLe9S8&\\(F D$\"3[0kl*Q.xh&FD7$$\"3R***\\i?=bq(FD$\"3_?]o#3,v$fFD7$$\"3\"HLL$3s?6z FD$\"3gG&=$\\*>(eiFD7$$\"3a***\\7`Wl7)FD$\"3IF*Q=gsSg'FD7$$\"3#pmmm'*R RL)FD$\"32+/!o`ba%pFD7$$\"3Qmm;a<.Y&)FD$\"3%*)\\AV(eY.tFD7$$\"30, ja lasketaan ao. integraalien raja-arvo, kun c ->" } {XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 3 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "c:='c':Int(1/x*int(1/y,y=x^2..x),x= c..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&-%#lnG6#%\"xG\" \"\"-F)6#*$)F+\"\"#F,!\"\"F,F+F2/F+;%\"cGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "value(%);simplify(%,symbolic);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&*$)-%#lnG6#%\"cG\"\"#\"\"\"#!\"\"F**&#F+\"\"%F+)-F' 6#*$)F)F*F+F*F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)-%#lnG6#%\"c G\"\"#\"\"\"#F+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Luonnollisem paa on k\344sin laskettaess sievent\344\344 integroitavan logaritmit: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "1/x*int(1/y,y=x^2..x);s implify(%,symbolic);int(%,x=c..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *&,&-%#lnG6#%\"xG\"\"\"-F&6#*$)F(\"\"#F)!\"\"F)F(F/" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&-%#lnG6#%\"xG\"\"\"F(!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)-%#lnG6#%\"cG\"\"#\"\"\"#F+F*" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 15 "Raja-arvo on , " }{XPPEDIT 18 0 "infinity;" "6#%)i nfinityG" }{TEXT -1 29 " joten integraali hajaantuu. " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 2 "4." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name change coords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "M \344\344r\344tt\344v\344 solmut x1,x2 ja painot w1,w2 siten, ett\344 i ntegrointikaava " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int(f(x ),x=-1..1)=w[1]*f(x[1])+w[2]*f(x[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"fG6#%\"xG/F*;!\"\"\"\"\",&*&&%\"wG6#F.F.-F(6#&F*F3F.F .*&&F26#\"\"#F.-F(6#&F*F9F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "on tarkka kaikille astetta 3 oleville polynomeille. V\344ltt\344m\344 t\366nt\344 ja riitt\344v\344\344: tarkka kaikille monomeille" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "1,x,x^2,x^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"\"%\"xG*$)F$\"\"#F#*$)F$\"\"$F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Gauss2yht:=\{seq(Int(x^k,x=-1..1)=w 1*x1^k+w2*x2^k,k=0..3)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Gauss2 yhtG<&/-%$IntG6$\"\"\"/%\"xG;!\"\"F*,&%#w1GF*%#w2GF*/-F(6$F,F+,&*&F0F* %#x1GF*F**&F1F*%#x2GF*F*/-F(6$*$)F,\"\"#F*F+,&*&F0F*)F7F?F*F**&F1F*)F9 F?F*F*/-F(6$*$)F,\"\"$F*F+,&*&F0F*)F7FJF*F**&F1F*)F9FJF*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Muutetaan Int -> int, jolloin integraalit lasketaan:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "gauss2yht:= \{seq(int(x^k,x=-1..1)=w1*x1^k+w2*x2^k,k=0..3)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*gauss2yhtG<&/\"\"#,&%#w1G\"\"\"%#w2GF*/\"\"!,&*&F)F* %#x1GF*F**&F+F*%#x2GF*F*/#F'\"\"$,&*&F)F*)F0F'F*F**&F+F*)F2F'F*F*/F-,& *&F)F*)F0F5F*F**&F+F*)F2F5F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "solve(gauss2yht,\{x1,x2,w1,w2\});map(allvalues,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/%#w2G\"\"\"/%#w1GF&/%#x2G,$-%'RootOfG6#,& !\"\"F&*&\"\"$F&)%#_ZG\"\"#F&F&F0/%#x1GF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(/%#w2G\"\"\"/%#w1GF&/%#x1G,$*$-%%sqrtG6#\"\"$F&#F&F0/ F*,$F,#!\"\"F0/%#x2GF3/F7F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "x1;x2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"$\"\"\"#!\"\"F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"$\"\"\"#F)F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "s[1]:=min(x1,x2); s[2]:=max( x1,x2); w[1]:=w1; w[2]:=w2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG 6#\"\"\",$*$-%%sqrtG6#\"\"$F'#!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"\"#,$*$-%%sqrtG6#\"\"$\"\"\"#F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"wG6#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>& %\"wG6#\"\"#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Pantiin tal teen seuraavaa teht\344v\344\344 varten. (Muista: \344l\344 tee restar ttia ennen teht\344v\344\344 5.)" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 2 "5." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): wit h(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Kun suoritat te ht\344v\344n 4, niin Gaussin 2. asteen s\344\344nn\366n solmut ovat mu uttujissa s[1] ja s[2] ja painot muuttujissa " }}{PARA 0 "" 0 "" {TEXT -1 27 "w[1] ja w[2]. Tarkistetaan:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "s[1],s[2],w[1],w[2]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&,$*$-%%sqrtG6#\"\"$\"\"\"#!\"\"F(, $F$#F)F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Sum(Sum(w[i ]*w[j]*f(s[i],s[j]),j=1..2),i=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%$SumG6$-F$6$*(&%\"wG6#%\"iG\"\"\"&F*6#%\"jGF--%\"fG6$&%\"sGF+&F5F/ F-/F0;F-\"\"#/F,F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "gauss 2:=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gauss2G,*-%\"fG6$,$ *$-%%sqrtG6#\"\"$\"\"\"#!\"\"F.F)F/-F'6$F),$F*#F/F.F/-F'6$F4F)F/-F'6$F 4F4F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=(x,y)->ln(x+2*y );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"yG6\"6$%)operat orG%&arrowGF)-%#lnG6#,&9$\"\"\"*&\"\"#F29%F2F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x:=0.3*u+1.7; y:=0.25*v+1.25;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG,&%\"uG$\"\"$!\"\"$\"#%\"yG,&%\"vG$\"#D!\"#$\"$D\"F)\"\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dxdy_per_dudv:=.3*.25; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.dxdy_per_dudvG$\"#v!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "F:=unapply(f(x,y),u,v);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6$%\"uG%\"vG6\"6$%)operatorG%&a rrowGF)-%#lnG6#,(9$$\"\"$!\"\"$\"$?%!\"#\"\"\"*&$\"#]F7F89%F8F8F)F)F) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Sum(Sum(w[i]*w[j]*F(s[i ],s[j])*dxdy_per_dudv,j=1..2),i=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$-F$6$,$*(&%\"wG6#%\"iG\"\"\"&F+6#%\"jGF.-%#lnG6#,(&%\"s GF,$\"\"$!\"\"$\"$?%!\"#F.*&$\"#]F=F.&F7F0F.F.F.$\"#v!\"$/F1;F.\"\"#/F -FF" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!)3c&H%!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x:='x': y:='y':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "int(int(f(x,y),x=1.4..2),y=1..1.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+u_a&H%!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "H\344mm\344stytt\344v \344\344, laskettiin funktion arvo 4:ss\344 pisteess\344 ja saatiin 4 \+ oikeaa numeroa integraaliin!" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "6. S\344\344stet\344\344n LV:oon" }}}}{MARK "1 15 0 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }