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ntaxissal defini\341lhatunk: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=x->x^2+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6 \"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\"\"\"\"\"\"F2F2F(F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Az " }{TEXT 260 1 "f" }{TEXT -1 81 " f\374ggv\351nyt a matematik\341ban megszokott m\363don haszn\341lhat juk. \n\nF\374ggv\351nyki\351rt\351kel\351s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"#\"\"\"\"\"\"F)F)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5 "f(u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)% \"uG\"\"#\"\"\"\"\"\"F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(x+y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$),&%\"xG\"\"\"%\"yGF( \"\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "A f\374ggv \351nyeket sem \351rt\351keli ki automatikusan a Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "f;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%\"fG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "whattype(f);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%'symbolG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Az " }{TEXT 256 4 "eval" }{TEXT -1 37 " f\374ggv\351nnyel \+ lehet ki\351rt\351keltetni az " }{TEXT 257 1 "f" }{TEXT -1 10 " v\341l toz\363t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(f);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,&*$ )9$\"\"#\"\"\"\"\"\"F0F0F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "whattype(eval(f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*proced ureG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 258 5 "print" } {TEXT -1 57 " paranccsal is ki lehet iratni a f\374ggv\351ny defin\355 ci\363j\341t: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(f); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF& ,&*$)9$\"\"#\"\"\"\"\"\"F0F0F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "\nDefini\341lunk egy m\341sodik f\374ggv\351nyt is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "g:=x->2*x-3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"#!\"$\" \"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "A k\351t f\374ggv \351ny \366sszege:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f(x)+ g(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"\"\"\"! \"#F)F&F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Ugyanezt megkaphatju k a k\366vetkez\365 szintaxissal is: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(f+g)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"x G\"\"#\"\"\"\"\"\"!\"#F)F&F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 " \311rtelmezett teh\341t a f\374ggv\351nyekre az \366sszead\341s m\373 velet, eredm\351nye egy f\374ggv\351ny." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "h:=f+g;\nh(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"hG,&%\"fG\"\"\"%\"gGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG \"\"#\"\"\"\"\"\"!\"#F)F&F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Ug yan\355gy a kivon\341s, szorz\341s \351s oszt\341s m\373veletek is def ini\341ltak a f\374ggv\351nyekre:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(f-g)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"x G\"\"#\"\"\"\"\"\"\"\"%F)F&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(f*g)(u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$)%\"uG\"\"# \"\"\"\"\"\"F*F*F*,&F'F(!\"$F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(f/g)(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$)% \"tG\"\"#\"\"\"\"\"\"F*F*F),&F'F(!\"$F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f(x);\ng(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,&*$)%\"xG\"\"#\"\"\"\"\"\"F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&% \"xG\"\"#!\"$\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "F\374ggv \351nyek kompoz\355ci\363j\341t a k\366vetkez\365k\351ppen jel\366lj \374k:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(f@g)(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$),&%\"xG\"\"#!\"$\"\"\"F(\"\"\"F*F *F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(g@f)(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"tG\"\"#\"\"\"F'!\"\"\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Nem defini\341lt f\374ggv\351nyek re alkalmazva a kompoz\355ci\363 m\373velet\351t, l\341that\363 a m \373velet form\341lis defin\355ci\363ja:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(v@w)(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"vG6#- %\"wG6#%\"tG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "K\351pezz\374k az " }{TEXT 259 1 "f" }{TEXT -1 34 " f\374ggv\351ny kompoz\355ci\363j \341t \366mmag\341val:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(f @f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$),&*$)%\"xG\"\"#\"\"\" \"\"\"F,F,F*F+F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Erre a sp eci\341lis m\373veletre alkalmazhatjuk a k\366vetkez\365 jel\366l\351s t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "(f@@2)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$),&*$)%\"xG\"\"#\"\"\"\"\"\"F,F,F*F+F,F, F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Vegy\374k az " }{TEXT 261 1 "f" }{TEXT -1 46 " f\374ggv\351ny h\341romszoros kompoz\355ci\363j \341t \366nmag\341val:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f (f(f(x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$),&*$),&*$)%\"xG\"\" #\"\"\"\"\"\"F/F/F-F.F/F/F/F-F.F/F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Ugyanezt r\366viden az al\341bbi jel\366l\351ssel is megk apjuk:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "(f@@3)(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$),&*$),&*$)%\"xG\"\"#\"\"\"\"\"\"F /F/F-F.F/F/F/F-F.F/F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Figyel j\374nk az " }{TEXT 262 1 "f" }{TEXT -1 4 " \351s " }{TEXT 263 4 "f(x) " }{TEXT -1 25 " k\366z\366tti k\374l\366nbs\351gre: az " }{TEXT 264 1 "f" }{TEXT -1 20 " egy f\374ggv\351ny neve, " }{TEXT 265 4 "f(x)" } {TEXT -1 87 " pedig a f\374ggv\351nyt defini\341l\363 kifejez\351s. \n \nEgy kifejez\351sb\365l f\374ggv\351nyt defini\341lhatunk az " } {TEXT 266 7 "unapply" }{TEXT -1 89 " f\374ggv\351nnyel is. Az els\365 \+ param\351ter egy kifejez\351s, a m\341sodik pedig a f\374ggv\351ny v \341ltoz\363ja: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "h:=una pply(x^3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG*$)R6#%\"xG6\"6$ %)operatorG%&arrowGF*9$F*F*F*\"\"$\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"xG\"\"$ \"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "h:=unapply(2*x+5* y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGR6#%\"xG6\"6$%)operator G%&arrowGF(,&9$\"\"#%\"yG\"\"&F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "A fenti f\374ggv\351ny egyv\341ltoz\363s, csak a f\374ggv\351ny k\351plet\351ben szerepel egy param\351ter, az " }{TEXT 267 1 "y" } {TEXT -1 9 " v\341ltoz\363." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"#\"\"\"%\"yG\"\"&" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "y:=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "y:='y';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"yGF$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "A fenti f\374ggv\351nyt a \"matematikai\" jel\366l\351ssel is defini\341lhatjuk:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "h:=x->2*x+5*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"#%\"yG \"\"&F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"#\"\"\"%\"yG\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "A k\351tf\351le f\374ggv\351ny defin\355c i\363 k\366z\366tti k\374l\366nbs\351get az al\341bbi p\351lda illuszt r\341lja. Adjunk \351rt\351ket az " }{TEXT 269 1 "x" }{TEXT -1 12 " v \341ltoz\363nak:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x:=2;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG\"\"#" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 41 "Ezut\341n defini\341ljuk a k\366vetkez\365 f\374ggv\351 nyt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=x->x+2;\nf(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arro wGF(,&9$\"\"\"\"\"#F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"tG \"\"\"\"\"#F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Ugyanezt az unap ply paranccsal most nem tudjuk megadni:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "g:=unapply(x+2,x);" }}{PARA 8 "" 1 "" {TEXT -1 61 "Er ror, (in unapply) variables must be unique and of type name" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Itt az " }{TEXT 270 1 "x" }{TEXT -1 243 " v\341ltoz\363 hely\351be el\365sz\366r behelyettes\355ti a Ma ple a v\341ltoz\363 tartam\341t, \355gy a m\341sodik param\351ter hely \351re is egy konstans ker\374l, de a parancs egy v\341ltoz\363 nevet \+ v\341r. Persze a v\341ltoz\363 tartalm\341nak behelyettes\355t\351s \351t elker\374lhetj\374k az al\341bbi szintaxissal:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "g:=unapply('x'+2,'x');\ng(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$ \"\"\"\"\"#F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Konstans f\374ggv\351nyeket a meg felel\365 f\374ggv\351nnyel helyettes\355t, \351s ford\355tva is, kons tansokat f\374ggv\351nyk\351nt is haszn\341lhatunk:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "h:=x->3;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"hG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(g+2)(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"tG\" \"\"\"\"%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Egyv\341ltoz\363s f\374ggv\351ny g rafikonj\341t a " }{TEXT 276 4 "plot" }{TEXT -1 27 " paranccsal kaphat juk meg: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:=x->1/(x^2+1 );\nplot(f(x),x=-5..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#% \"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&*$)9$\"\"#F-\"\"\"F3F3!\"\"F (F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURV ESG6$7ao7$$!\"&\"\"!$\"1YQ:YQ:YQ!#<7$$!1LLLe%G?y%!#:$\"1\"\\24&eu*=%F- 7$$!1mmT&esBf%F1$\"1#*HlM'ep_%F-7$$!1LL$3s%3zVF1$\"1h*p#R(*Gc\\F-7$$!1 ML$e/$QkTF1$\"1xor_D%>X&F-7$$!1nmT5=q]RF1$\"1PXyk$y6-'F-7$$!1LL3_>f_PF 1$\"1Uhr]RWImF-7$$!1++vo1YZNF1$\"1t$ydGV8O(F-7$$!1LL3-OJNLF1$\"1!p;ES( )yC)F-7$$!1++v$*o%Q7$F1$\"1668)oX]H*F-7$$!1mmm\"RFj!HF1$\"1mN'zNm&e5!# ;7$$!1LL$e4OZr#F1$\"19976Gx%>\"F[o7$$!1+++v'\\!*\\#F1$\"1(*>*3w9-Q\"F[ o7$$!1+++DwZ#G#F1$\"1zZ%*)o#Q5;F[o7$$!1+++D.xt?F1$\"1***3b:1m)=F[o7$$! 1LL3-TC%)=F1$\"1$)GKr1i(>#F[o7$$!1mmm\"4z)e;F1$\"1^ip<#F[o$\"1^9y:o_\\&*F[o7$$!1K$3Fpy7k\"F [o$\"1e;<7moP(*F[o7$$!1++D\"yQ16\"F[o$\"19&Gv<^\"y)*F[o7$$!1L$3_D)=`%) F-$\"1m[A(e]!H**F[o7$$!1pm\"zp))**z&F-$\"1;ex:HZm**F[o7$$!11]iS\"*yYJF -$\"1BI#R^2,***F[o7$$!1EMLLe*e$\\!#=$\"1s'e_Pc(****F[o7$$\"1l;a)3RBE#F -$\"1cJ3SW)[***F[o7$$\"1tmTgxE=]F-$\"1)=&yZ-)[(**F[o7$$\"1!o\"HKk>uxF- $\"1$4)=_\\#*R**F[o7$$\"1pmT5D,`5F[o$\"1SXK(\\K.*)*F[o7$$\"1q;zW#)>/;F [o$\"16XW*H6\"\\(*F[o7$$\"1sm;zRQb@F[o$\"1c5/*>cgb*F[o7$$\"1PLL$e,]6$F [o$\"1K59#[,b6*F[o7$$\"1-+](=>Y2%F[o$\"1STRoQ9w&)F[o7$$\"1QLe*[K56&F[o $\"1#>+iv*yGzF[o7$$\"1vmm\"zXu9'F[o$\"1WhA:LOdsF[o7$$\"1QL$e9i\"=sF[o$ \"14@ga8aulF[o7$$\"1,+++&y))G)F[o$\"1lqC)y)[FfF[o7$$\"1-+]ibOO$*F[o$\" 11L'zd,GM&F[o7$$\"1++]i_QQ5F1$\"1bP(oka<\"[F[o7$$\"1+](=-N(R6F1$\"1N/g ()\\s\\VF[o7$$\"1,+D\"y%3T7F1$\"1`\"[=?cl$RF[o7$$\"1++]P![hY\"F1$\"1eq Hij,vJF[o7$$\"1LLL$Qx$o;F1$\"1)*Qq#=nIk#F[o7$$\"1+++v.I%)=F1$\"1Gocd#= v>#F[o7$$\"1mm\"zpe*z?F1$\"1#)=)eB,v(=F[o7$$\"1,++D\\'QH#F1$\"1#))Qpt! )pf\"F[o7$$\"1LLe9S8&\\#F1$\"1M^*o:]RQ\"F[o7$$\"1,+D1#=bq#F1$\"1R4Y9y% >?\"F[o7$$\"1LLL3s?6HF1$\"1\\`8zZRb5F[o7$$\"1++DJXaEJF1$\"1*y\"f(o+0G* F-7$$\"1ommm*RRL$F1$\"1j'e9@CTD)F-7$$\"1om;a<.YNF1$\"1rNN>+%oO(F-7$$\" 1NLe9tOcPF1$\"1!oGOI/!=mF-7$$\"1,++]Qk\\RF1$\"1x=$395U-'F-7$$\"1NL$3dg 6<%F1$\"1UG`9>?NaF-7$$\"1ommmxGpVF1$\"1$pK\\TLu(\\F-7$$\"1++D\"oK0e%F1 $\"1r5l\"oD$\\XF-7$$\"1,+v=5s#y%F1$\"1U2DpLe)=%F-7$$\"\"&F*F+-%'COLOUR G6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(F[al%(DE FAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "\nK\351tv\341ltoz\363s f\374ggv\351ny def ini\341l\341sa a k\366vetkez\365 szintaxissal t\366rt\351nhet:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=(x,y)->2*x+5*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF ),&9$\"\"#9%\"\"&F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Haszn \341lata a matematikai jel\366l\351snek megfelel\365en t\366rt\351nik: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"uG\"\"#%\"vG\"\"&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "f(1,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Az " }{TEXT 268 7 "unapply" } {TEXT -1 58 " paranccsal is defini\341lhatjuk a t\366bbv\341ltoz\363s \+ f\374ggv\351nyeket:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g:=u napply(x^2-y^3,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6$%\"xG %\"yG6\"6$%)operatorG%&arrowGF),&*$)9$\"\"#\"\"\"\"\"\"*$)9%\"\"$F2!\" \"F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g(u,2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"uG\"\"#\"\"\"\"\"\"!\")F)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 330 "\311rdemes megjegyezni, hogy a fe nti szintaxissal egy k\351plettel megadott f\374ggv\351nyeket hozhatun k l\351tre, ahogy azt \341ltal\341ban a matematikai alkalmaz\341sok t \366bbs\351g\351ben haszn\341ljuk. Egy f\374ggv\351nyt programmal, aza z egy \"bonyolultabb\" algoritmussal is defini\341lhatunk. (Az elj\341 r\341sok szintaxis\341val, programoz\341si parancsokkal k\351s\365bb f oglalkozunk.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f:=proc(x) \nif x<10 then\n x+2;\nelse\n x^2;\nfi;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(3);\nf(12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$W\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Persze ez az el\365bb defini\341lt f\374ggv\351ny nem k\351sz\374lt fel szimbolikus kifejez\351sek kezel \351s\351re:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }} {PARA 8 "" 1 "" {TEXT -1 37 "Error, (in f) cannot evaluate boolean" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "Az el\365bbi t\355pus\372, k\374 l\366nb\366z\365 szakaszokon k\374l\366nb\366z\365 k\351pletekkel defi ni\341lt f\374ggv\351nyek gyakran fordulnak el\365 matematik\341ban is . Ezek defin\355ci\363j\341ra van egy be\351p\355tett parancs is:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g:=x->piecewise(x<10,x+2,x^2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%& arrowGF(-%*piecewiseG6%29$\"#5,&F0\"\"\"\"\"#F3*$)F0F4\"\"\"F(F(F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "g(1);\ng(11);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$@\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Itt felt\351tel - kifejez \351s p\341rokat adunk meg, illetve p\341ratlan sz\341m\372 argumentum eset\351ben az utols\363 param\351ter az \"else \341g\" k\351plete." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "h:=piecewise(x<=-2,x-4,x< 5,1,sin(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG-%*PIECEWISEG6%7 $,&%\"xG\"\"\"!\"%F+1F*!\"#7$F+2F*\"\"&7$-%$sinG6#F*%*otherwiseG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(h(x),x=-5..15,thickness =2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVE SG6$7gp7$$!\"&\"\"!$!\"*F*7$$!+F ,$\"\"\"F*7$$!+Q5\"e$>F,F[p7$$!+rUx#)=F,F[p7$$!+Q2qws%H%F,F[ p7$$\"(]1!>F,F[p7$$\"*]Z/N%F,F[p7$$\"*]$fC&)F,F[p7$$\"+'z6:B\"F,F[p7$$ \"+<=C#o\"F,F[p7$$\"+n#pS1#F,F[p7$$\"+j`A3DF,F[p7$$\"+n(y8!HF,F[p7$$\" +j.tKLF,F[p7$$\"+)3zMu$F,F[p7$$\"+#H_?<%F,F[p7$$\"+!G;cc%F,F[p7$$\"+XA (yx%F,F[p7$$\"+4#G,*\\F,F[p7$$\"+ay!R+&F,$!+^W3y&*!#57$$\"+(\\(o<]F,$! +wEdP&*Fdt7$$\"+TrYJ]F,$!+Y*\\_\\*Fdt7$$\"+%yY_/&F,$!+.V7^%*Fdt7$$\"+r g!G2&F,$!+?\")\\d$*Fdt7$$\"+e`O+^F,$!+L_wc#*Fdt7$$\"+KR[b^F,$!+,2HM!*F dt7$$\"+0Dg5_F,$!+IkP%y)Fdt7$$\"+`'R3K&F,$!+IfM0#)Fdt7$$\"*!o2Ja!\")$! +LJqEvFdt7$$\"+?.+BcF,$!+*zHE8'Fdt7$$\"*%Q#\\\"eF`w$!+5eN8XFdt7$$\"*;* [HiF`w$!+VZ.n`!#67$$\"*qvxl'F`w$\"+eN\"*eOFdt7$$\"*`qn2(F`w$\"+R9sGrFd t7$$\"+X+ZzsF,$\"+s\"yXR)Fdt7$$\"*cp@[(F`w$\"+9aq;$*Fdt7$$\"+!>,Zf(F,$ \"+=rul'*Fdt7$$\"+?GB2xF,$\"+Dq^#*)*Fdt7$$\"+]Ww>yF,$\"+$\\YT***Fdt7$$ \"*3'HKzF`w$\"+3*\\$p**Fdt7$$\"+Da_M\")F,$\"+l,04'*Fdt7$$\"*xanL)F`w$ \"+t$4r&))Fdt7$$\"+gxn_&)F,$\"+c^\"ol(Fdt7$$\"*v+'o()F`w$\"+cS#45'Fdt7 $$\"*S<*f\"*F`w$\"+Qpu#**Fdt7$$\"+q`KO9F`w$\"+(o5bu*Fdt7$$\"+(y MkW\"F`w$\"+N3_p%*Fdt7$$\"+/Uac9F`w$\"+!eMo4*Fdt7$$\"+-@Fy9F`w$\"+I`m( )zFdt7$$\"#:F*$\"+-%yG]'Fdt-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABE LSG6$Q\"x6\"%!G-%*THICKNESSG6#\"\"#-%%VIEWG6$;F(F\\el%(DEFAULTG" 1 2 0 1 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "hat \341r\351rt\351ksz\341m\355t\341s" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Egyv \341ltoz\363s f\374ggv\351nyek hat\341r\351rt\351k\351t a " }{TEXT 271 5 "limit" }{TEXT -1 179 " paranccsal sz\341m\355thatjuk ki. Az els \365 param\351ter a f\374ggv\351ny k\351plete (algebrai kifejez\351s), a m\341sodik param\351ter v\341ltoz\363n\351v=\351rt\351k adja meg, h ogy mely pontban sz\341m\355tjuk a hat\341r\351rt\351ket: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "limit(x^2,x=2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Tekints \374k az al\341bi f\374ggv\351nyt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=x->(x^2-1)/(x+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&*$)9$\"\"#\"\"\"\"\"\"! \"\"F3F2,&F0F3F3F3!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "limit(f(x),x=-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Itt m\341r a hat\341r\351rt\351k n em egyenl\365 a helyettes\355t\351si \351rt\351kkel:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(-1);" }}{PARA 8 "" 1 "" {TEXT -1 30 "Err or, (in f) division by zero" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Ha ngs\372lyozni kell, hogy a " }{TEXT 272 5 "limit" }{TEXT -1 80 " paran cs is szimbolikusan sz\341molja a hat\341r\351rt\351ket. Ismeri az ala p azonoss\341gokat:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limi t(sin(x)/x,x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limit(sin(a*x)/sin(b*x),x=0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"aG\"\"\"%\"bG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 273 5 "Limit" }{TEXT -1 40 " pa rancs form\341zottan \355rja ki a k\351rd\351st " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 29 "Limit(sin(a*x)/sin(b*x),x=0);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%&LimitG6$*&-%$sinG6#*&%\"aG\"\"\"%\"xGF,\"\"\" -F(6#*&%\"bGF,F-F.!\"\"/F-\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 274 5 "value" }{TEXT -1 23 " paranccsal ki lehet a " } {TEXT 275 5 "Limit" }{TEXT -1 25 " parancsot is \351rt\351kelni: " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&%\"aG\"\"\"%\"bG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "A k\366vetkez\365 paranccsal kapunk sz\351pen form\341zot t outputot:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Limit(sin(a* x)/sin(b*x),x=0)=limit(sin(a*x)/sin(b*x),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&-%$sinG6#*&%\"aG\"\"\"%\"xGF-\"\"\"-F)6#* &%\"bGF-F.F/!\"\"/F.\"\"!*&F,F/F3F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Hat\341r\351t\351ket sz\341m\355thatunk v\351gtelenben is:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:=x->(x^2+x-1)/(2*x^2-2*x+5 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%& arrowGF(*&,(*$)9$\"\"#\"\"\"\"\"\"F0F3!\"\"F3F2,(F.F1F0!\"#\"\"&F3!\" \"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Limit(f(x),x=in finity)=limit(f(x),x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% &LimitG6$*&,(*$)%\"xG\"\"#\"\"\"\"\"\"F+F.!\"\"F.F-,(F)F,F+!\"#\"\"&F. !\"\"/F+%)infinityG#F.F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Limit(f(x),x=-infinity)=limit(f(x),x=-infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&,(*$)%\"xG\"\"#\"\"\"\"\"\"F+F.!\"\"F.F-, 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&,&%\"xG\"\"\"\"\"$F*\"\"\",&F)F*!\"&F*!\"\"/F)\"\"&%%leftG,$%)infinit yG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "deriv\341l\341s" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Tek ints\374k a deriv\341lt f\374ggv\351ny defin\355ci\363j\341t:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "g:=x->limit((f(x+h)-f(x))/h, h=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operator G%&arrowGF(-%&limitG6$*&,&-%\"fG6#,&9$\"\"\"%\"hGF6F6-F26#F5!\"\"\"\" \"F7!\"\"/F7\"\"!F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " f:=x->x^2-6*x+2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6 $%)operatorG%&arrowGF(,(*$)9$\"\"#\"\"\"\"\"\"F/!\"'F0F2F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"#!\"'\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=x->sin(log(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%$sinG6#-%$logG6#9$F(F(F(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$cosG6#-%#lnG6#%\"xG\"\"\"F*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 279 4 "diff" }{TEXT -1 198 " paranccs al tudunk egy f\374ggv\351ny deriv\341ltj\341t kisz\341m\355tani. Az e ls\365 param\351ter a f\374ggv\351ny k\351plete (algebrai kifejez\351s ), a m\341sodik v\341ltoz\363 pedig a v\341ltoz\363 neve, amelyre vona tkoz\363 deriv\341ltat sz\341m\355tjuk ki:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "diff(x^3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* $)%\"xG\"\"#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " diff(sin(2*x)*exp(-5*x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$ cosG6#,$%\"xG\"\"#\"\"\"-%$expG6#,$F)!\"&F+F**&-%$sinGF'F+F,\"\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "diff((3*x^2-6*x+4)/sin(x) ,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&%\"xG\"\"'!\"'\"\"\"\"\" \"-%$sinG6#F&!\"\"F)*&*&,(*$)F&\"\"#F*\"\"$F&F(\"\"%F)F)-%$cosGF-F)F** $)F+\"\"#F*F.!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 278 4 "Diff" }{TEXT -1 50 " parancs form\341zottan jel\366li a deriv \341l\341s m\373velet\351t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Diff(exp(sin(log(x))),x)=diff(exp(sin(log(x))),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%$expG6#-%$sinG6#-%#lnG6#%\"xGF0*&*&-% $cosGF,\"\"\"F'F5\"\"\"F0!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Deriv\341lt sz\341m\355t\341st a " }{TEXT 280 1 "D" }{TEXT -1 142 " f \374ggv\351nnyel (oper\341torral) is ki lehet sz\341m\355tani. A diffe renci\341l\341s oper\341tor input param\351tere egy f\374ggv\351ny, ou tputja pedig a deriv\341lt f\374ggv\351ny:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "D(sin);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$cosG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(log);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#R6#%\"aG6\"6$%)operatorG%&arrowGF&*&\"\"\"F+9$!\"\"F& F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=x->x^5-66*x^3-x; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&ar rowGF(,(*$)9$\"\"&\"\"\"\"\"\"*$)F/\"\"$F1!#mF/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,(*$)9$\"\"%\"\"\"\"\"&*$)F-\" \"#F/!$)>!\"\"\"\"\"F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"%\"\"\"\" \"&*$)F&\"\"#F(!$)>!\"\"\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!$%>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Magasabbrend\373 deriv\341lt kisz\341m \355t\341sa:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Diff(x^5,x, x)=diff(x^5,x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*$)%\" xG\"\"&\"\"\"-%\"$G6$F)\"\"#,$*$)F)\"\"$F+\"#?" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "Diff(x^5,x,x,x)=diff(x^5,x,x,x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%DiffG6$*$)%\"xG\"\"&\"\"\"-%\"$G6$F)\"\"$,$* $)F)\"\"#F+\"#g" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "A m\341sodik d eriv\341ltat a " }{TEXT 281 1 "D" }{TEXT -1 50 " oper\341tor ism\351te lt alkalmaz\341s\341val sz\341m\355thatjuk ki:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "f:=x->x^5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"&\"\"\"F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(D(f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,$*$)9$\"\"$\"\"\"\"# ?F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Itt haszn\341lhatjuk a z iter\341ci\363s jel\366l\351st:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "(D@@2)(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"x G6\"6$%)operatorG%&arrowGF&,$*$)9$\"\"$\"\"\"\"#?F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Az f f\374ggv\351ny negyedik deriv\341ltj \341t teh\341t \355gy is kisz\341m\355thatjuk:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "(D@@4)(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# R6#%\"xG6\"6$%)operatorG%&arrowGF&,$9$\"$?\"F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Nem defini\341lt f\374ggv\351nyekre kiadva a defiv\341l\341s m \373velet\351t l\341that\363k a be\351p\355tett azonoss\341gok:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(f+g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%\"DG6#%\"fG\"\"\"-F%6#%\"gGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(f*g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&* &-%\"DG6#%\"fG\"\"\"%\"gGF)F)*&F(F)-F&6#F*F)F)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "D(f/g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*& -%\"DG6#%\"fG\"\"\"%\"gG!\"\"\"\"\"*&*&F(F,-F&6#F*F,F)*$)F*\"\"#F)F+! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(f@g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%\"@G6$-%\"DG6#%\"fG%\"gG\"\"\"-F(6#F+F," }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "D(f@g)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&--%\"DG6#%\"fG6#-%\"gG6#%\"xG\"\"\"--F&6#F+F,F. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "T\366bbv\341ltoz\363s f\374gg v\351ny parci\341lis deriv\341ltjai sz\341m\355t\341sa:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Diff(x^2*y^5,x)=diff(x^2*y^5,x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&)%\"xG\"\"#\"\"\")%\"yG \"\"&F+F),$*&F)\"\"\"F,F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Diff(x^2*y^5,y)=diff(x^2*y^5,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%%DiffG6$*&)%\"xG\"\"#\"\"\")%\"yG\"\"&F+F-,$*&F)\"\"\"F,F+F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Diff(x^2*y^5,x,x)=diff(x^2*y ^5,x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&)%\"xG\"\"#\" \"\")%\"yG\"\"&F+-%\"$G6$F)F*,$*$F,F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Diff(x^2*y^5,x,y)=diff(x^2*y^5,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6%*&)%\"xG\"\"#\"\"\")%\"yG\"\"&F+F)F-,$* &F)\"\"\")F-\"\"%F+\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Ugyane zt a D oper\341torral a k\366vetkez\365 szintaxissal sz\341m\355thatju k ki:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=(x,y)->x^2*y^5; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"yG6\"6$%)operator G%&arrowGF)*&)9$\"\"#\"\"\")9%\"\"&F1F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Diff(f(x,y),x)=D[1](f)(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&)%\"xG\"\"#\"\"\")%\"yG\"\"&F+F),$*&F)\"\" \"F,F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Diff(f(x,y),y)= D[2](f)(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&)%\"xG\" \"#\"\"\")%\"yG\"\"&F+F-,$*&F(F+)F-\"\"%F+F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Diff(f(x,y),x,x)=D[1,1](f)(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&)%\"xG\"\"#\"\"\")%\"yG\"\"&F+-%\"$G6 $F)F*,$*$F,F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Diff(f(x ,y),x,y)=D[1,2](f)(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6 %*&)%\"xG\"\"#\"\"\")%\"yG\"\"&F+F)F-,$*&F)\"\"\")F-\"\"%F+\"#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 10 "integr\341l\341s" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Az " }{TEXT 282 3 "int" }{TEXT -1 114 " paranccsal tudjuk egy f\374ggv\351ny hat \341rozatlan integr\341lj\341t (primit\355v f\374ggv\351ny\351t) kisz \341m\355tani. A parancs felh\355v\341sa a " }{TEXT 283 4 "diff" } {TEXT -1 29 " szintaxisa szerint t\366rt\351nik:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Int(x^3,x)=int(x^3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$)%\"xG\"\"$\"\"\"F),$*$)F)\"\"%F+#\"\"\"F/ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"xG\"\"$\"\"\"F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Int(x^3*sin(5*x),x)=int(x^3*sin(5*x),x);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)%\"xG\"\"$\"\"\"-%$sinG6# ,$F)\"\"&\"\"\"F),**&F(F+-%$cosGF.F1#!\"\"F0*&)F)\"\"#F+F,F+#F*\"#DF,# !\"'\"$D'*&F)F1F4F+#\"\"'\"$D\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Hat\341rozott integr\341l sz\341m\355t\341sa:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 32 "Int(x^3,x=0..1)=int(x^3,x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$)%\"xG\"\"$\"\"\"/F);\"\"!\"\"\" #F/\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Improprius integr\341 lt is sz\341molhatunk:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "I nt(1/x^2,x=1..infinity)=int(1/x^2,x=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$)%\"xG\"\"#F(!\"\"/F+;\"\"\"%)inf inityGF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "V\351ges pontban impr oprius integr\341lt is kezel a Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Int(1/x^2,x=0..1)=int(1/x^2,x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$)%\"xG\"\"#F(!\"\"/F+;\"\"! \"\"\"%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int(1/ sqrt(x),x=0..1)=int(1/sqrt(x),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$-%%sqrtG6#%\"xGF(!\"\"/F-;\"\"!\"\"\"\"\"# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Int(exp(-x^2),x=-infini ty..infinity)=int(exp(-x^2),x=-infinity..infinity);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$-%$expG6#,$*$)%\"xG\"\"#\"\"\"!\"\"/F-;,$%) infinityGF0F4*$-%%sqrtG6#%#PiGF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Int(exp(-x^2),x)=int(exp(-x^2),x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$-%$expG6#,$*$)%\"xG\"\"#\"\"\"!\"\"F-,$*&-% %sqrtG6#%#PiGF/-%$erfG6#F-\"\"\"#F:F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Term\351szetesen nem minden integr\341lhat\363 f\374ggv \351ny primit\355v f\374ggv\351nye fejezhet\365 ki Maple-ben defini \341lt f\374ggv\351nyek seg\355ts\351g\351vel:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "Int(tan(log(x)),x)=int(tan(log(x)),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$tanG6#-%#lnG6#%\"xGF-,&*& %\"IG\"\"\"F-F1!\"\"*&F0\"\"\"-%$intG6$,$*&F4F4,&*$)-%$expG6#*&F0F4F*F 1\"\"#F4F1F1F1!\"\"!\"#F-F1F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 " Ekkor a hat\341rozott integr\341lt sem tudja a Maple kisz\341m\355tani szimbolikusan:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Int(tan( log(x)),x=1..2)=int(tan(log(x)),x=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$tanG6#-%#lnG6#%\"xG/F-;\"\"\"\"\"#-%$intGF&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "Ha az evalf f\374ggv\351nyt alka lmazzuk a hat\341rozott integr\341lra, akkor szimbolikus sz\341mol\341 s helyett r\366gt\366n numerikus m\363dszerekkel sz\341m\355tja ki az \+ integr\341l \351rt\351k\351t: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Int(tan(log(x)),x=1..2)=evalf(int(tan(log(x)),x=1..2),20);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$tanG6#-%#lnG6#%\"xG/F-;\" \"\"\"\"#$\"5L%\\lA'G*pED%!#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 " N\351zz\374nk megint egy szimbolikus sz\341mol\341st:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Int(D(g)(x),x)=int(D(g)(x),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$--%\"DG6#%\"gG6#%\"xGF--F+F, " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "\366sszegz\351s" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 284 3 "sum" }{TEXT -1 61 " paranccsal egy v\351 ges vagy v\351gtelen \366sszeget sz\341m\355thatunk ki:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Sum(i,i=1..10)=sum(i,i=1..10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$%\"iG/F';\"\"\"\"#5\"#b" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Sum(1/((i+1)*(i+3)),i=1..100 )=sum(1/((i+1)*(i+3)),i=1..100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %$SumG6$*&\"\"\"F(*&,&%\"iG\"\"\"F,F,\"\"\",&F+F,\"\"$F,\"\"\"!\"\"/F+ ;F,\"$+\"#\"%D9\"%-N" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "L\341tha t\363 a fenti eredm\351nyb\365l, hogy itt is szimbolikusan sz\341mol a Maple. S\365t, k\351pes bizonyos szimbolikus azonoss\341gok alkalmaz \341s\341ra is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Sum(i,i= 1..n)=sum(i,i=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$%\"i G/F';\"\"\"%\"nG,(*$),&F+F*F*F*\"\"#\"\"\"#F*F0F+#!\"\"F0F3F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "K\366nnyebben felismerj\374k a has zn\341lt azonoss\341got, ha az eredm\351nyt szorzatt\341 alak\355tjuk: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Sum(i,i=1..n)=factor(su m(i,i=1..n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$%\"iG/F';\" \"\"%\"nG,$*&F+F*,&F+F*F*F*F*#F*\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Sum(i^2,i=1..n)=factor(sum(i^2,i=1..n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*$)%\"iG\"\"#\"\"\"/F);\"\"\"%\"nG,$* (F/F.,&F/F.F.F.F.,&F/F*F.F.F.#F.\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Sum(i^4,i=1..n)=factor(sum(i^4,i=1..n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*$)%\"iG\"\"%\"\"\"/F);\"\"\"%\"nG,$* *F/F.,&F/F.F.F.F.,&F/\"\"#F.F.F.,(*$)F/F4F+\"\"$F/F8!\"\"F.F.#F.\"#I" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Bizonyos v\351gtelen \366szegek et is kisz\341m\355thatunk (szimbolikusan):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Sum(1/n^2,n=1..infinity)=sum(1/n^2,n=1..infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*$)%\"nG\"\"#F (!\"\"/F+;\"\"\"%)infinityG,$*$)%#PiG\"\"#F(#F0\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Sum(1/n,n=1..infinity)=sum(1/n,n=1..infin ity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(%\"nG!\" \"/F);\"\"\"%)infinityGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Sum(1/n^3,n=1..infinity)=sum(1/n^3,n=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*$)%\"nG\"\"$F(!\"\"/F+;\"\"\" %)infinityG-%%ZetaG6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "It t megint egy be\351p\355tett f\374ggv\351ny seg\355ts\351g\351vel kapt uk vissza a v\341laszt. \311rt\351kelj\374k ki numerikusan a v\351gere dm\351nyt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Sum(1/n^3,n=1 ..infinity)=evalf(sum(1/n^3,n=1..infinity),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*$)%\"nG\"\"$F(!\"\"/F+;\"\"\"%)inf inityG$\"5aG%ffJ!p0-7!#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Sum(n/((n+1)*(n+3)*(n+9)),n=1..infinity)=sum(n/((n+1)*(n+3)*(n+9)),n= 1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&%\"nG\"\" \"*(,&F(\"\"\"F,F,\"\"\",&F(F,\"\"$F,\"\"\",&F(F,\"\"*F,\"\"\"!\"\"/F( ;F,%)infinityG#\"$.'\"%![%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "szorzatok" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "V\351ges vagy v\351gtelen szorzatokat a p roduct paranccsal sz\341molhatunk ki:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Product(i^2+1,i=1..10)=product(i^2+1,i=1..10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(ProductG6$,&*$)%\"iG\"\"#\"\"\"\" \"\"F-F-/F*;F-\"#5\"/++5W#>S%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 " Itt is vannak be\351p\355tett azonoss\341gok. A k\366vetkez\365 paranc s p\351ld\341ul " }{TEXT 285 2 "n!" }{TEXT -1 57 " \351rt\351k\351t eg y be\351p\355tett f\374ggv\351ny seg\355ts\351g\351vel adja vissza:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Product(i,i=1..n)=product( i,i=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(ProductG6$%\"iG/F'; \"\"\"%\"nG-%&GAMMAG6#,&F+F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Product(i/(i+1),i=1..n)=product(i/(i+1),i=1..n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(ProductG6$*&%\"iG\"\"\",&F(\"\"\"F +F+!\"\"/F(;F+%\"nG*&-%&GAMMAG6#,&F/F+F+F+F)-F26#,&F/F+\"\"#F+F," }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Egyszer\373s\355ts\374k a v\351ger edm\351nyt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Product(i/(i +1),i=1..n)=simplify(product(i/(i+1),i=1..n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(ProductG6$*&%\"iG\"\"\",&F(\"\"\"F+F+!\"\"/F(;F+%\" nG*&F)F),&F/F+F+F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Pro duct(i/(i+1),i=1..infinity)=simplify(product(i/(i+1),i=1..infinity)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(ProductG6$*&%\"iG\"\"\",&F(\" \"\"F+F+!\"\"/F(;F+%)infinityG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "parci\341lis t\366rekre bont\341s" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 229 "A convert \341ltal\341nos c\351l\372 konvert\341l\363 paranccs al tudjuk a racion\341lis t\366rt kifejez\351seket parci\341lis t\366r tekre bontani. Az els\365 param\351ter a kifejez\351s, a m\341sodik a \+ parfrac kulcssz\363, a harmadik pedig a kifejez\351sben szerepl\365 v \341ltoz\363 neve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "A:=(x ^2-x+2)/((x-1)*(x+2)*(x-8)):\nA=convert(A,parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,(*$)%\"xG\"\"#\"\"\"\"\"\"F(!\"\"F)F+F**(,&F(F +F,F+\"\"\",&F(F+F)F+\"\"\",&F(F+!\")F+\"\"\"!\"\",(*&F*F*F.F5#!\"#\"# @*&F*F*F0F5#\"\"%\"#:*&F*F*F2F5#\"#H\"#N" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "A:=(x^4-x+6)/((x-1)^3*(x-2)^2):\nA=convert(A,parfrac, x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&,(*$)%\"xG\"\"%\"\"\"\"\"\"F (!\"\"\"\"'F+F**&),&F(F+F,F+\"\"$F*),&F(F+!\"#F+\"\"#F*!\"\",,*&F*F**$ )F0\"\"$F*F6F-*&F*F**$)F0\"\"#F*F6\"#:*&F*F*F0F6\"#I*&F*F**$)F3\"\"#F* F6\"#?*&F*F*F3F6!#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "A:=( -6*x+1)/((x^2+x+1)*(x+1)^2):\nA=convert(A,parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG!\"'\"\"\"F(\"\"\"*&,(*$)F&\"\"#F)F(F&F( F(F(\"\"\"),&F&F(F(F(\"\"#F)!\"\",(*&F)F)*$)F1\"\"#F)F3\"\"(*&F)F)F1F3 F(*&,&F&F(F9F(F)F+F3!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "A:=(-6*x+1)/((x^2+x+1)^3*(x+1)^2):\nA=convert(A,parfrac,x);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG!\"'\"\"\"F(\"\"\"*&),(*$)F& \"\"#F)F(F&F(F(F(\"\"$F)),&F&F(F(F(\"\"#F)!\"\",,*&F)F)*$)F2\"\"#F)F4 \"\"(*&F)F)F2F4\"#:*&,&F:F(F&F " 0 "" {MPLTEXT 1 0 57 "A:=(x^2+1)/(x^4-x^3 -3*x^2-7*x-6):\nA=convert(A,parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&*$)%\"xG\"\"#\"\"\"\"\"\"F+F+F*,,*$)F(\"\"%F*F+*$)F(\"\"$F* !\"\"F&!\"$F(!\"(!\"'F+!\"\",(*&F*F*,&F(F+F4F+F7#\"\"&\"#G*&F*F*,&F(F+ F+F+F7#F3F/*&,&F(F+F/F+F*,(F&F+F(F+F)F+F7#F+\"#9" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 120 "Ha a t\366rt nem val\363di, el\365sz\366r a sz \341ml\341l\363t elosztja a nevez\365vel, \351s a marad\351kra sz\341m olja ki a parci\341lis t\366rtekre bont\341st:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 55 "A:=(x^8-6*x+x)/((x+1)*(x-1)^2):\nA=convert(A,p arfrac,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&,&*$)%\"xG\"\")\"\"\" \"\"\"F(!\"&F**&,&F(F+F+F+\"\"\"),&F(F+!\"\"F+\"\"#F*!\"\",4*$)F(\"\"& F*F+*$)F(\"\"%F*F+*$)F(\"\"$F*\"\"#*$)F(F?F*F?F(F>F>F+*&F*F*F.F4#F>F?* &F*F**$)F1\"\"#F*F4!\"#*&F*F*F1F4#F8F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "sz\351ls \365\351rt\351ksz\341m\355t\341s" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 291 8 "minimize" }{TEXT -1 55 " f\374ggv\351ny az inputk \351nt megadott kifejez\351s minimum\341t, a " }{TEXT 297 8 "maximize " }{TEXT -1 39 " f\374ggv\351ny pedig a maximum\341t keresi meg:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "minimize(x^2-2*x+3,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "maximize(x^2-2*x+3,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(x^2-2*x+3,x=0..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$\"\"!$\"\"$F(7$$ \"1LLLL3VfV!#<$\"1&R_q%=r9H!#:7$$\"1nmm\"H[D:)F.$\"1i9`QafVGF17$$\"1LL Le0$=C\"!#;$\"1Q%*o>`0nFF17$$\"1LLL3RBr;F:$\"1kp4YMo$p#F17$$\"1mm;zjf) 4#F:$\"1\"=$z\"z@Vi#F17$$\"1LL$e4;[\\#F:$\"1CONayFjDF17$$\"1++]i'y]!HF :$\"1GJ&y3zL]#F17$$\"1LL$ezs$HLF:$\"1%opHns\\W#F17$$\"1++]7iI_PF:$\"1h OiwnL!R#F17$$\"1nmm;_M(=%F:$\"1z?Ic&pyL#F17$$\"1LLL3y_qXF:$\"1;3!Go\"z %H#F17$$\"1+++]1!>+&F:$\"1/Z7r*4)\\AF17$$\"1+++]Z/NaF:$\"1E]Mk\")Q3AF1 7$$\"1+++]$fC&eF:$\"1CSVM4-s@F17$$\"1LL$ez6:B'F:$\"1L%\\M.:?9#F17$$\"1 mmm;=C#o'F:$\"1`2j$>v+6#F17$$\"1mmmm#pS1(F:$\"17$4F*o>'3#F17$$\"1++]i` A3vF:$\"1$)3W3%*3i?F17$$\"1mmmm(y8!zF:$\"1v*436US/#F17$$\"1++]i.tK$)F: $\"1%HT/)yzF?F17$$\"1++](3zMu)F:$\"1)\\N![%)y:?F17$$\"1nmm\"H_?<*F:$\" 1Wt2u\\&o+#F17$$\"1nm;zihl&*F:$\"1]#p@*o)=+#F17$$\"1LLL3#G,***F:$\"12F _u4++?F17$$\"1LLezw5V5F1$\"1f!R?Fe=+#F17$$\"1++v$Q#\\\"3\"F1$\"1d#4'35 k1?F17$$\"1LL$e\"*[H7\"F1$\"1#f/fV;^,#F17$$\"1+++qvxl6F1$\"10^r-A[F?F1 7$$\"1++]_qn27F1$\"1)3N\"e(HJ/#F17$$\"1++Dcp@[7F1$\"1:+Pd;hh?F17$$\"1+ +]2'HKH\"F1$\"1/Yr-O)f3#F17$$\"1nmmwanL8F1$\"1F(GPKR86#F17$$\"1+++v+'o P\"F1$\"10!Hh^B?9#F17$$\"1LLeR<*fT\"F1$\"1o:SF\"\\I<#F17$$\"1+++&)Hxe9 F1$\"15ew^EZ5AF17$$\"1mm\"H!o-*\\\"F1$\"1H#H+vF!\\AF17$$\"1++DTO5T:F1$ \"13Sd]Jz#H#F17$$\"1nmmT9C#e\"F1$\"1ySR'40!RBF17$$\"1++D1*3`i\"F1$\"1r bBG7,\"R#F17$$\"1LLL$*zym;F1$\"1$\\`!GigWCF17$$\"1LL$3N1#4x^BqijFF17$$ \"1++DOl5;>F1$\"1t*fd=^#RGF17$$\"1++v.Uac>F1$\"1@xs8o(\\\"HF17$$\"\"#F (F)-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6 $;F(Fdz%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Az " }{TEXT 295 7 "extrema" } {TEXT -1 75 " parancs viszont a sz\351ls\365\351rt\351k hely\351t is v isszaadja. 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{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "Tay lor-sor sz\341m\355t\341s" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Egy egyv\341ltoz\363 s f\374ggv\351ny Taylor-k\366zel\355t\351s\351t (Taylor-polinomot \351 s a hibatagot) sz\341m\355tja ki szimbolikusan a " }{TEXT 286 6 "taylo r" }{TEXT -1 9 " parancs:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "taylor(f(x),x=0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+1%\"xG-%\"fG 6#\"\"!\"\"!--%\"DG6#F&F'\"\"\",$---%#@@G6$F,\"\"#F-F'#\"\"\"F5\"\"#,$ ---F36$F,\"\"$F-F'#F7\"\"'\"\"$,$---F36$F,\"\"%F-F'#F7\"#C\"\"%,$---F3 6$F,\"\"&F-F'#F7\"$?\"\"\"&-%\"OG6#F7\"\"'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "A:=taylor(sin(x),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG++%\"xG\"\"\"\"\"\"#!\"\"\"\"'\"\"$#F'\"$?\"\"\"& -%\"OG6#F'\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 242 "Az els\365 pa ram\351ter a f\374ggv\351ny k\351plete, a m\341sodik v\341ltoz\363n \351v=\351rt\351k kifejez\351s mondja meg, hogy melyik pont k\366r\374 li Taylor-sort sz\341m\355tjuk ki, a harmadik param\351terben pedig me gadhatjuk, hogy hanyadrend\373 tagig bez\341r\363lag sz\341m\355tjuk a Taylor-polinomot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "B:=ta ylor(sin(x),x=0,20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"BG+9%\"xG \"\"\"\"\"\"#!\"\"\"\"'\"\"$#F'\"$?\"\"\"&#F*\"%S]\"\"(#F'\"'!)GO\"\"* #F*\")+o\"*R\"#6#F'\"++3-Fi\"#8#F*\".+!oVn28\"#:#F'\"0+g4Guob$\"#<#F* \"3+?$)3/5X;7\"#>-%\"OG6#F'\"#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "M\341sol\341ssal az el\365bbi k\351t parancs outputj\341b\363l k \366nnyen elmenthetj\374k az outputk\351nt kapott Taylor-polinomok k \351plet\351t v\341ltoz\363kban. M\341sik lehet\365s\351g, ha a " } {TEXT 289 7 "convert" }{TEXT -1 13 " parancsot a " }{TEXT 290 7 "polyn om" }{TEXT -1 21 " opci\363val adjuk ki a " }{TEXT 256 6 "taylor" } {TEXT -1 53 " parancs outputj\341ra, visszakapjuk a polinom k\351plet \351t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "T5:=convert(A,pol ynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T5G,(%\"xG\"\"\"*$)F&\"\" $\"\"\"#!\"\"\"\"'*$)F&\"\"&F+#F'\"$?\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "T19:=convert(B,polynom);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$T19G,6%\"xG\"\"\"*$)F&\"\"$\"\"\"#!\"\"\"\"'*$)F&\" \"&F+#F'\"$?\"*$)F&\"\"(F+#F-\"%S]*$)F&\"\"*F+#F'\"'!)GO*$)F&\"#6F+#F- \")+o\"*R*$)F&\"#8F+#F'\"++3-Fi*$)F&\"#:F+#F-\".+!oVn28*$)F&\"#F+#F-\"3+?$)3/5X;7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 405 "Rajzoljuk ki egy grafikonban a sin(x) \351s az 5-\366dfo k\372 illetve 19-edfok\372 Taylor-polinomj\341nak grafikonj\341t (a pa rancs szintaxis\341t k\351s\365bb fogjuk r\351szletesebben tanulni). L \341that\363, hogy a sin(x) grafikonj\341t (piros g\366rbe) csak kb a \+ [-2,2] intervallumon k\366zel\355ti az 5-\366dfok\372 Taylor-polinom g rafikonja. A 19-edfok\372 Taylor-polinom (z\366ld g\366rbe) viszont m \341r a [-7,7] intervallumon is j\363l k\366zel\355ti az eredeti f\374 ggv\351nyt." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "plot([sin(x) ,T5,T19],x=-10..10,y=-2..2,color=[red,blue,green],thickness=2);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7ct7 $$!#5\"\"!$\"1*p$*)36@Sa!#;7$$!1mm;HU,\"*)*!#:$\"1;%H%=YF&\\%F-7$$!1LL Le%G?y*F1$\"1'flOa'*p\\$F-7$$!1,+](oUIn*F1$\"1na%yfAsX#F-7$$!1nmm;p0k& *F1$\"182F!*3H)Q\"F-7$$!1++vV5Su$*F1$!1iOGo6cN]!#<7$$!1LL$3c6`F-7$$!1mmmT%p\"e()F1$!1%4Yf/TK='F-7$$!1 mmmmxYV&)F1$!1kS.)3Idr(F-7$$!1nmm\"4m(G$)F1$!1h;\\nF\"R*))F-7$$!1++Dc[ 3:\")F1$!1*GV5td5m*F-7$$!1LL$3i.9!zF1$!1oM'G)yv))**F-7$$!1++Dch(=&yF1$ !1Yf#[$y(*****F-7$$!1mmm\"p[B!yF1$!1gWZdJn')**F-7$$!1LL3F7#Gv(F1$!1_$4 ,[w)[**F-7$$!1++]iPH.xF1$!1_i)y\\!o'))*F-7$$!1MLLL)QUg(F1$!138-=-w*o*F -7$$!1nm;/R=0vF1$!1;<^'*G%yR*F-7$$!1ML$3i_+I(F1$!1b])3-VY])F-7$$!1,+]P 8#\\4(F1$!1j%e_sI[D(F-7$$!1ML$3FuF)oF1$!1KeqF=.VcF-7$$!1mm;/siqmF1$!1. &e4h6#yPF-7$$!1******\\Q*[c'F1$!1f.x([s*zFF-7$$!1KL$e\\g\"fkF1$!1uP,)[ $o]#fLXF-7$$!1****\\([j5i& F1$\"1\"osh\"p\"z9'F-7$$!1mmm\">s%HaF1$\"1unNT\"es`(F-7$$!1LL$3x&y8_F1 $\"14)H+\\?\"p()F-7$$!1******\\$*4)*\\F1$\"1\\1H[oh%f*F-7$$!1+++DL\")* )[F1$\"19/bta,V)*F-7$$!1++++t_\"y%F1$\"1Y3)G)*3h(**F-7$$!1++](G%QFZF1$ \"1%z*)[sv))***F-7$$!1+++v7CtYF1$\"1)Q&Qk#QB***F-7$$!1++]i#)4>YF1$\"1^ $G5w:l&**F-7$$!1+++]_&\\c%F1$\"1z8h,K^\"*)*F-7$$!1+++]zCcVF1$\"1bt_tE \\s$*F-7$$!1+++]1aZTF1$\"1wc>qzpY%)F-7$$!1LL3FW,eRF1$\"1]CYA,+(G(F-7$$ !1mm;/#)[oPF1$\"1m>pCWLmeF-7$$!1L$e*)p0el$F1$\"1dW0d\")\\=\\F-7$$!1++v $>BJa$F1$\"173$[1w#3RF-7$$!1m;a)oS/V$F1$\"1!pT;s\"[[GF-7$$!1LLL$=exJ$F 1$\"1\\mWrub_+-[a\"[TF-7$$!1L $e9m8Gg#F1$!1_27'R$)38&F-7$$!1++]PYx\"\\#F1$!1?+t7_T]gF-7$$!1nmTNz>&H# F1$!1LalC5'*)[(F-7$$!1MLLL7i)4#F1$!1I&)H4h/R')F-7$$!1mmTNa%H)=F1$!1SF* )pwv;&*F-7$$!1****\\P'psm\"F1$!1Ch&He+N&**F-7$$!1**\\(o/Efh\"F1$!1h/4q #=)*)**F-7$$!1***\\iX#ek:F1$!1kaoRp!)****F-7$$!1**\\il)QK^\"F1$!1A5vl- W$)**F-7$$!1*****\\F&*=Y\"F1$!1980%Qh2%**F-7$$!1***\\P43#f8F1$!12OG%)f )px*F-7$$!1****\\74_c7F1$!1s+&)yh?5&*F-7$$!1lmT5VBU5F1$!1OSuo7LN')F-7$ $!1:LL$3x%z#)F-$!1BQPhxWltF-7$$!1BL$e9d;J'F-$!1Qb()oG'3!fF-7$$!1ILL3s$ QM%F-$!1pCiiV^3UF-7$$!1lmT&QdDG$F-$!1W$>4\"R#RA$F-7$$!1****\\ivF@AF-$! 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N\351zz\374nk k\351tv \341ltoz\363s p\351ld\341kat:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "readlib(mtaylor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6\"6*%\"fG %\"kG%\"vG%\"mG%\"nG%\"sG%\"tG%\"wG6#%aoCopyright~(c)~1991~by~the~Univ ersity~of~Waterloo.~All~rights~reserved.GF$C2>8$&9\"6#\"\"\">8&&F46#\" \"#@&-%%typeG6$F8%$setG>F87#-%#opG6#F84-F>6$F8%%listG>F87#F8@$4-F>6$F8 -FI6#<$/%%nameG%*algebraicGFT-%&ERRORG6#%Ginvalid~2nd~argument~(expans ion~point)G>8)-%$mapG6$R6#%\"xGF$F$F$@%-F>6$9$%\"=G-%$rhsG6#F_o\"\"!F$ F$F$F8>F8-Fgn6$RFjnF$F$F$@%F]o-%$lhsGFcoF_oF$F$F$F8>8'-%%nopsGFE@$0F]p -F_p6#<#FC-FW6#%Hvariables~(2nd~argument)~must~be~uniqueG@%/9#F;>8(\" \"'>F\\q&F46#\"\"$@%/Fjp\"\"%>8+&F46#Fdq>Ffq7#-%\"$G6$F6F]p@$4-F>6$F8< $-F@6#FT-FIFdr-FW6#%O2nd~argument~(the~variable(s))~must~be~a~namesG@$ 34-F>6$F\\q%*nonnegintG0F\\q%)infinityG-FW6#%X3rd~argument~(the~order) ~must~be~a~non-negative~integerG@$54-F>6$Ffq-FI6#%'posintG0-F_p6#FfqF] p-FW6#%en4th~argument~(weights)~must~be~a~list~of~positive~integersG>F 2-%%subsG6$7#-%$seqG6$/&F86#8%,&*&F[uF6)8*&FfqF\\uF6F6&FenF\\uF6/F]u;F 6F]pF2>F2-Fgn6&%(collectG-Fdt6$/-%\"OGF5Fdo-%'taylorG6%F2FauF\\qF8.%,d istributedG>F2-Fdt6$7#-Fht6$/F[u,&F[uF6Fcu!\"\"Fdu-Fdt6$/FauF6F2F$F$F$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "mtaylor(sin(x^2+y^3),[x =0,y=0],10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$)%\"xG\"\"#\"\"\" \"\"\"*$)%\"yG\"\"$F(F)*$)F&\"\"'F(#!\"\"F0*&F+F()F&\"\"%F(#F2F'*&)F,F 0F(F%F(F6*$)F,\"\"*F(F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " mtaylor(x^2*y-x^3*y^2+1,[x=1,y=2],20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,:\"#8\"\"\"%\"yG!\"$%\"xG!\")*&,&F(F%!\"\"F%F%,&F&F%!\"#F%F%!#5 *$)F+\"\"#\"\"\"F/*$)F-F2F3F,*&F1F3F-F3!#6*&F+F3F5F3F'*$)F+\"\"$F3!\"% *&F1F3F5F3F'*&F:F3F-F3F<*&F:F3F5F3F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "N\351zz\374k egy k\351tv\341ltoz\363s f\374ggv\351ny harmadfok \372 Taylor-polinomj\341nak form\341lis defin\355ci\363j\341t:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "mtaylor(g(x,y),[x=0,y=0],4); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6-%\"gG6$\"\"!F'\"\"\"*&--&%\"DG6 #F(6#F%F&F(%\"xGF(F(*&--&F-6#\"\"#F/F&F(%\"yGF(F(*&--&F-6$F(F(F/F&F()F 0F6\"\"\"#F(F6*(F0F>--&F-6$F(F6F/F&F(F7F>F(*&--&F-6$F6F6F/F&F()F7F6F>F ?*&)F0\"\"$F>--&F-6%F(F(F(F/F&F(#F(\"\"'*(F=F>--&F-6%F(F(F6F/F&F(F7F>F ?*(F0F>FJF>--&F-6%F(F6F6F/F&F(F?*&)F7FMF>--&F-6%F6F6F6F/F&F(FR" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 45 "differenci\341legyenletek szimbolikus megold\341sa " }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 292 6 "dsolve" }{TEXT -1 123 " paranccsal tudunk k\366z\366ns\351ges differenci\341legyenlet eket megoldani.\n\nTekints\374k a k\366vetkez\365 els\365rend\373 diff erenci\341legyenletet:\n" }{XPPEDIT 18 0 "diff(y,x)-sin(x)*y = x^2;" " 6#/,&-%%diffG6$%\"yG%\"xG\"\"\"*&-%$sinG6#F)F*F(F*!\"\"*$F)\"\"#" } {TEXT -1 126 " \n\nDefini\341ljuk el\365sz\366r az egyenletet. Az egye nletben szerepl\365 f\374ggv\351ny v\341ltoz\363j\341t mindig ki\355rj uk, a deriv\341ltj\341t megadhatjuk a " }{TEXT 293 4 "diff" }{TEXT -1 19 " paranccsal vagy a " }{TEXT 294 1 "D" }{TEXT -1 14 " oper\341torra l: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "degy1:=diff(y(x),x)- sin(x)*y(x)=sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%°y1G/,&-%% diffG6$-%\"yG6#%\"xGF-\"\"\"*&-%$sinGF,F.F*F.!\"\"F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Ezzel ekvivalens megad\341s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "degy2:=D(y)(x)-sin(x)*y(x)=sin(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%°y2G/,&--%\"DG6#%\"yG6#%\"xG\"\" \"*&-%$sinGF,F.-F+F,F.!\"\"F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 256 7 "dsolve " }{TEXT -1 100 "parancs szintaxisa: els\365 p aram\351ter a differenci\341legyenlet, a m\341sodik pedig a keresett f \374ggv\351ny neve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsol ve(degy1,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&!\" \"\"\"\"*&-%$expG6#,$-%$cosGF&F)F*%$_C1GF*F*" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 15 "Az outputban a " }{TEXT 298 3 "_C1" }{TEXT -1 45 " v \341ltoz\363 egy tetsz\365leges konstanst jel\366l.\n\nA " }{TEXT 256 7 "dsolve " }{TEXT -1 46 "parancs param\351tereit halmazban is megadha tjuk:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dsolve(\{degy2\}, \{y(x)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&!\"\"\" \"\"*&-%$expG6#,$-%$cosGF&F)F*%$_C1GF*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "Ha kezdeti felt\351telt is megadunk, akkor azt \351s a d ifferenci\341legyenletet mindenk\351ppen halmazk\351nt kell \341tadni. A megold\341s k\351plete most m\341r egy konkr\351t f\374ggv\351ny le sz, a kezdeti felt\351tel meghat\341rozza a param\351ter \351rt\351k \351t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dsolve(\{degy1,y( 0)=2\},\{y(x)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&! \"\"\"\"\"*&-%$expG6#,$-%$cosGF&F)\"\"\",&-%%coshG6#F*F)-%%sinhGF6F*! \"\"!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Tekints\374nk egy m \341sodrend\373 p\351ld\341t:\n\n" }{XPPEDIT 18 0 "diff(y,x,x)-5*diff( y,x)+4*y = x^2;" "6#/,(-%%diffG6%%\"yG%\"xGF)\"\"\"*&\"\"&F*-F&6$F(F)F *!\"\"*&\"\"%F*F(F*F**$F)\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "degy3:=diff(y(x),x,x)-5*diff(y(x),x)+4*y(x)=x^2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%°y3G/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\" \"#\"\"\"-F(6$F*F-!\"&F*\"\"%*$)F-F1\"\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "kezdeti_feltetel:=y(0)=2,D(y)(0)=-3;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%1kezdeti_feltetelG6$/-%\"yG6#\"\"!\"\"#/--%\"D G6#F(F)!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "mo:=dsolve( \{degy3,kezdeti_feltetel\},\{y(x)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#moG/-%\"yG6#%\"xG,,#\"#@\"#K\"\"\"F)#\"\"&\"\")*$)F)\"\"#\"\"\"# F.\"\"%-%$expGF(\"\"$-F96#,$F)F7#!#`F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Ellen\365rizz\374k a megold\341st az egyenletbe visszahel yettes\355t\351ssel:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "sub s(mo,degy3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0-%%diffG6$,,#\"#@\" #K\"\"\"%\"xG#\"\"&\"\")*$)F-\"\"#\"\"\"#F,\"\"%-%$expG6#F-\"\"$-F86#, $F-F6#!#`F+-%\"$G6$F-F3F,-F&6$F(F-!\"&#F*F0F,F-#F/F3F1F,F7\"#7F;#F?F0F 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"xG\"\"#\"\"\"F$" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 98 "Ellen\365rizz\374k a kezdeti felt\351telt is. Ehhe z hozzuk l\351tre a megold\341s k\351plet\351vel defini\341lt f\374ggv \351nyt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "g:=unapply(rhs( mo),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operat orG%&arrowGF(,,#\"#@\"#K\"\"\"9$#\"\"&\"\")*$)F1\"\"#\"\"\"#F0\"\"%-%$ expG6#F1\"\"$-F<6#,$F1F:#!#`F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(g)(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Egy m\341s ik m\363dszer a kezdeti felt\351telek ellen\365rz\351sre:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "simplify(subs(x=0,mo));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\"!\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "B:=diff(mo,x);\nsimplify(subs(x=0,rhs(B)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG/-%%diffG6$-%\"yG6#%\"xGF,,*#\" \"&\"\")\"\"\"F,#F1\"\"#-%$expGF+\"\"$-F56#,$F,\"\"%#!#`F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "11 24 2" 0 }{VIEWOPTS 1 1 0 2 1 1805 }