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M\371\236e to b\375t libovoln\351 re\341ln\351 \+ \350\355slo, pro limity" }}{PARA 0 "" 0 "" {TEXT -1 73 " v \+ nevlastn\355ch bodech jsou ur\350eny hodnoty -infinity a infinity" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 331 9 "dir " }{TEXT -1 42 " je nepovinn\375 parametr. M\371\236e m\355t hodnotu " }{TEXT 332 6 " left " }{TEXT -1 27 "pro limitu zleva , resp. " }{TEXT 333 5 "right " }{TEXT -1 14 " pro limitu" }}{PARA 0 "" 0 "" {TEXT -1 74 " \+ zprava. Pokud parametr vynech\341me, po\350\355t\341 se oboustrann \341 limita." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "D\341le m\371\236eme pou\236\355t p\370\355kaz " }{TEXT 335 5 "Limit" }{TEXT -1 53 " , kter\375 limitu nevyhodnocuje. Hodnotu \+ limity v tomto" }}{PARA 0 "" 0 "" {TEXT -1 25 "p\370\355pad\354 ur\350 \355me p\370\355kazem " }{TEXT 336 6 "value." }{TEXT -1 51 " Vedle st andardn\355ho p\370\355kazu m\371\236eme pou\236\355t i p\370\355kaz \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 337 6 "Limit " }{TEXT -1 13 " z knihovny " }{TEXT 338 10 "student , " }{TEXT -1 54 "kter\375 umo \236\362ujr i symbolick\351 \372pravy v\375raz\371 s limitami " }} {PARA 0 "" 0 "" {TEXT -1 10 "(p\370\355kazem " }{TEXT 339 9 "expand, \+ " }{TEXT -1 10 "p\370\355padn\354 " }{TEXT 340 8 "combine " }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 27 "Knihovnu na\350teme p\370\355 kazem " }{TEXT 341 13 "with(student)" }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "P\370\355klad 1 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Ur\350ete limitu funkce f : y = " }{XPPEDIT 18 0 "(x^3+x)^(1/3)-(x^3-x^2)^(1/3 );" "6#,&),&*$%\"xG\"\"$\"\"\"F'F)*&F)F)F(!\"\"F)),&*$F'F(F)*$F'\"\"#F +*&F)F)F(F+F+" }{TEXT -1 8 "pro x-> " }{XPPEDIT 18 0 "infinity;" "6#%) infinityG" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "r estart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Nejd\370\355ve definuj eme funkci" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:=x->(x^3+x)^(1/3)-( x^3-x^2)^(1/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6 $%)operatorG%&arrowGF(,&*$),&*$)9$\"\"$\"\"\"F4F2F4#F4F3F4F4*$),&F0F4* $)F2\"\"#F4!\"\"F5F4F " 0 "" {MPLTEXT 1 0 23 "limit(f(x),x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\" \"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "V\375po\350et i se z\341 pisem:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(student,Limi t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Limit(f(x),x=infinit y):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "%=value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$,&*$),&*$)%\"xG\"\"$\"\"\" F/F-F/#F/F.F/F/*$),&F+F/*$)F-\"\"#F/!\"\"F0F/F7/F-%)infinityGF0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "P\370i symbolick\351m po\350 \355t\341n\355 limit \350asto pot\370ebujeme vytknout ur\350it\375 v \375raz. K tomu slou\236\355 procedura " }{TEXT 342 21 "vytkni(expr1,e xpr2), " }{TEXT -1 20 "jej\355\236 parametry jsou" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 343 8 "expr1 " }{TEXT -1 35 "je v\375raz, ze kt er\351ho budeme vyt\375kat" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 344 8 "expr2 " }{TEXT -1 31 "je v\375raz, kter\375 chceme vytknout" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vytkni:=pr oc(f,expr)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "local cit,jm;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "cit:=numer(f);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "jm:=denom(f);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "cit:=expand(cit/expr);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "jm:=expa nd(jm/expr);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "cit/jm;" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 9 "end proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%'vytkniGf*6$%\"fG%%exprG6$%$citG%#jmG6\"F,C'>8$-%&numerG6#9$>8%-%&de nomGF2>F/-%'expandG6#*&F/\"\"\"9%!\"\">F5-F:6#*&F5F=F>F?*&F/F=F5F?F,F, F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "P\370\355klad 2." }}{PARA 0 "" 0 "" {TEXT -1 7 "Ur\350eme " } {XPPEDIT 18 0 "limit(x^3-x^2,x = infinity);" "6#-%&limitG6$,&*$%\"xG\" \"$\"\"\"*$F(\"\"#!\"\"/F(%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 21 " M\371\236eme pou\236\355t p\370\355kaz " }{TEXT 345 5 "limit" }{TEXT -1 93 ", pokud v\232ak chceme demonstrovat postup, kter\375 pou\236 \355v\341 p\370i v\375po\350tu student, pou\236ijeme p\370\355kaz " } {TEXT 346 8 "vytkni ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vytkni(x^3-x^2,x^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%*&F%F%%\"xG!\"\"F(F%)F'\"\"$F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(student,Limit):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "expand(Limit(%%,x=infinity));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%*&F%F%-%&LimitG6$%\"xG/F*% )infinityG!\"\"F-F%)F'\"\"$F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "V \355me, \236e " }{XPPEDIT 18 0 "limit(x,x = infinity);" "6#-%&limitG6$ %\"xG/F&%)infinityG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "infinity;" "6#% )infinityG" }{TEXT -1 14 " a 1/" }{XPPEDIT 18 0 "limit(x,x = \+ infinity)" "6#-%&limitG6$%\"xG/F&%)infinityG" }{TEXT -1 30 " = 0 , ho dnota limity je tedy" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "valu e(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 "K testov\341n\355 spojitosti je ur\350en \+ p\370\355kaz" }{TEXT 347 32 " iscont(expr,x=a..b,'closed') " }{TEXT -1 3 "a " }{TEXT 348 15 "discont(expr,x)" }}{PARA 0 "" 0 "" {TEXT -1 11 "Parametry :" }}{PARA 0 "" 0 "" {TEXT 349 5 " expr" }{TEXT -1 30 " \+ je algebraick\375 v\375raz v x" }}{PARA 0 "" 0 "" {TEXT 350 6 " a ..b " }{TEXT -1 18 " je interval" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 351 10 "'closed' " }{TEXT -1 80 "je voliteln\375 parametr. \+ Maple standartn\354 ur\350uje spojitost na otev\370en\351m intervalu, " }}{PARA 0 "" 0 "" {TEXT -1 83 " pokud uvedeme tento pa rametr, testuje Maple na uzav\370en\351m intervalu." }}{PARA 0 "" 0 " " {TEXT -1 49 " Pou\236it\355 p\370\355kaz\371 uk\341\236eme v n\341sl eduj\355c\355m p\370\355kladu:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "P\370\355klad 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Vypo\350\355tejte limi tu funkce f: y = " }{XPPEDIT 18 0 "arctan(x/(x+1));" "6#-%'arctanG6# *&%\"xG\"\"\",&F'F(F(F(!\"\"" }{TEXT -1 24 " v bodech nespojitosti." }}{PARA 0 "" 0 "" {TEXT -1 21 "Definujeme funkci f :" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f:=x->arct an(x/(x+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%) operatorG%&arrowGF(-%'arctanG6#*&9$\"\"\",&F0F1F1F1!\"\"F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Pomoc\355 p\370\355kazu " }{TEXT 352 7 "iscont " }{TEXT -1 32 "zjist\355me , zd a je funkce spojit\341" }{TEXT 353 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "iscont(f(x),x=-4..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 59 "Proto\236e funkce nen \355 spojit\341 na R , ur\350\355me body nespojitosti" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "discont(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 31 "V\375sledek ov \354\370\355me z grafu funkce" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(f(x),x=-4..4,discont=true," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "axesfont=[TIMES,ROMAN,15], labelfont=[TIMES,ROMAN,15]);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7S7$ $!\"%\"\"!$\"3'=7;+=_HF*!#=7$$!3NxVnP&3Y$R!#<$\"3![Qx5b'f*H*F-7$$!3s,6 &fx6x(QF1$\"3vO,F88gB$*F-7$$!30Knu;as8QF1$\"3S=*f/Wg:N*F-7$$!3S$\\=W\" \\J\\PF1$\"33G!edy\"z!Q*F-7$$!3[QW'Ra5_o$F1$\"3b(H&H6..6%*F-7$$!3\\EHi 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}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Limi t(f(x),x=-1,right)=limit(f(x),x=-1,right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6%-%'arctanG6#*&%\"xG\"\"\",&F+F,F,F,!\"\"/F+ F.%&rightG,$%#PiG#F.\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "P\370\355klad 4." }}{PARA 0 "" 0 "" {TEXT -1 33 "Necht\264je za dan\341 funkce f : y = " }{XPPEDIT 18 0 "(2^x-1)/x;" "6#*&,&)\"\"#% \"xG\"\"\"F(!\"\"F(F'F)" }{TEXT -1 39 " . Vypo\350teme jej\355 limitu v bod\354 x = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Z defini\350n\355ho oboru funkce vid\355me, \236e x = 0 \+ nepat\370\355 do defini\350n\355ho oboru." }}{PARA 0 "" 0 "" {TEXT -1 82 "Abychom z\355skali p\370edstavu o pr\371b\354hu zadan\351 funkce, \+ nech\341me si vykreslit jej\355 graf." }}{PARA 0 "" 0 "" {TEXT -1 115 "Proto\236e n\341s zaj\355m\341 chov\341n\355 funkce v okol\355 nuly, \+ zobraz\355me si graf tak\351 podrobn\354ji pro x z intervalu (-0.01.. 0.01)." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 16 "f:=x- >(2^x-1)/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)o peratorG%&arrowGF(*&,&)\"\"#9$\"\"\"F1!\"\"F1F0F2F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Grafy funkce:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot(f(x),x=-10..10,axesfont=[TIMES,ROMAN,15],labelfo nt=[TIMES,ROMAN,15]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7Y7$$!#5\"\"!$\"3s*****\\PM-***!#>7$$!3!pmm m\"p0k&*!#<$\"3*GuV?4+U/\"!#=7$$!3uKL$3\"F47$$! 3OLL$3i.9!zF1$\"3[>;x-XIg7F47$$!3fmm;/R=0vF1$\"3B3vdz\"y]K\"F47$$!3k++ ]P8#\\4(F1$\"3IcGZo'[\"*R\"F47$$!3Kmm;/siqmF1$\"3Oyg^FRR%[\"F47$$!3Q** **\\(y$pZiF1$\"3!=l\"R$zE&z:F47$$!3jKLL$yaE\"eF1$\"3%oV&H'zv(*o\"F47$$ !3s%HaF1$\"3-)\\UQAi!*z\"F47$$!3]******\\$*4)*\\F1$\"3?&*o'pNa \"Q>F47$$!3o******\\_&\\c%F1$\"3kP2vxA0)4#F47$$!3%)******\\1aZTF1$\"3- :XW*pC]F#F47$$!3Imm;/#)[oPF1$\"3))30()el')eCF47$$!3%HLLL=exJ$F1$\"31ho (ol-=r#F47$$!3lKLLL2$f$HF1$\"3/(p(3(o\")4'HF47$$!3%)****\\PYx\"\\#F1$ \"3e>$4(Gnq*H$F47$$!3gLLLL7i)4#F1$\"3UF`\")>s[_OF47$$!3o)***\\P'psm\"F 1$\"3ndtq99U4TF47$$!3?****\\74_c7F1$\"3_nRIcJWFYF47$$!3M:LL$3x%z#)F4$ \"3w\">`U'\\7u_F47$$!3()HLL3s$QM%F4$\"3m-.XkdE&)fF47$$!3]^omm;zr)*!#?$ \"3f+f\"H96y!pF47$$\"3fVLLezw5VF4$\"3!*eXnBLOy!)F47$$\"3-.++v$Q#\\\")F 4$\"3u1/9&zsiJ*F47$$\"3%\\LL$e\"*[H7F1$\"3cdr$pq>Q4\"F17$$\"3=++++dxd; F1$\"3T+#eth8,I\"F17$$\"3e+++D0xw?F1$\"3K')Qr&*)=)\\:F17$$\"35,+]i&p@[ #F1$\"3E(HNp\\H\"[=F17$$\"3++++vgHKHF1$\"37yZFvE8iAF17$$\"3ElmmmZvOLF1 $\"3/$[JD()*=GFF17$$\"3%4+++v+'oPF1$\"3[K7*4:F^x7$$\"3'eLL$3N1#4(F1$\"3yii2T7m4>F^x7$$\"3,pm;HY t7vF1$\"3oN=%p$>bW0\"zF^x7$$\"3w+]7G:3u'*F1$\"32Od?\\5EM%)F^x7$$\"3 5,+v=5s#y*F1$\"3+D7BW@s$**)F^x7$$\"3W,]P40O\"*)*F1$\"35=\"4tN99f*F^x7$ $\"#5F*$\"3(************H-\"!#:-%'COLOURG6&%$RGBG$Fg\\l!\"\"$F*F*Fa]l- %*AXESTICKSG6%%(DEFAULTGFe]l-%%FONTG6%%&TIMESG%&ROMANG\"#:-%+AXESLABEL SG6%Q\"x6\"Q!F`^lFf]l-%%VIEWG6$;F(Ff\\lFe]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 95 "plot(f(x),x=-0.01..0.01,xtickmarks=3, axesfont=[TIM ES, ROMAN, 15], labelfont=[TIMES,ROMAN,15]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!3-+++++++5!#>$ \"3+tS'Hc/v!p!#=7$$!3anmm;p0k&*!#?$\"36q$z>4Z&3pF-7$$!3'HLL3s%HaF1$\"3,WQ(47X%=pF-7$$!3C******\\$*4)*\\F1$\"3=Jef** )y%>pF-7$$!3w******\\_&\\c%F1$\"3mtOJIr^?pF-7$$!3#*******\\1aZTF1$\"3$ pj\"eay^@pF-7$$!3imm;/#)[oPF1$\"3$ye%HvnUApF-7$$!3TLLL$=exJ$F1$\"3RX27 xx]BpF-7$$!34LLLL2$f$HF1$\"3p_3g.PUCpF-7$$!3E++]PYx\"\\#F1$\"3uIh:Z$*[ DpF-7$$!3sLLLL7i)4#F1$\"3Uteu/GVEpF-7$$!3s****\\P'psm\"F1$\"3%=JQW7ou# pF-7$$!3u****\\74_c7F1$\"3hk\\V&=a%GpF-7$$!3!)>LL$3x%z#)!#@$\"3[Va;OK[ HpF-7$$!3_ALL3s$QM%Fer$\"3cQN[0%G/$pF-7$$!3?)3mmm\"zr)*!#B$\"3qS%z44[9 $pF-7$$\"3%)RLLezw5VFer$\"3QllZpu]KpF-7$$\"3!H++]PQ#\\\")Fer$\"3%=1Bt$ )HM$pF-7$$\"3dMLLe\"*[H7F1$\"3-J.m.iUMpF-7$$\"3;*******pvxl\"F1$\"3Rg/ 2[dXNpF-7$$\"3W+++D0xw?F1$\"3=)*)HO:jk$pF-7$$\"3y++]i&p@[#F1$\"30!R2g0 Qu$pF-7$$\"3=+++vgHKHF1$\"3^5L!Ht?&QpF-7$$\"3smmmmZvOLF1$\"3=YfQeP\\Rp F-7$$\"3w,++]2goPF1$\"31W9erG`SpF-7$$\"3'RLLeR<*fTF1$\"3_WtrTYZTpF-7$$ \"3U,++])Hxe%F1$\"3>Gn:SW]UpF-7$$\"3Wmm;H!o-*\\F1$\"33\\z>'etM%pF-7$$ \"3X++]7k.6aF1$\"3TPf$e\"o[WpF-7$$\"3;mmm;WTAeF1$\"3+F([Nixa%pF-7$$\"3 %)****\\i!*3`iF1$\"3=p\")H4^^YpF-7$$\"31MLLL*zym'F1$\"3EyQmEX^ZpF-7$$ \"3kMLL3N1#4(F1$\"3#za$[tn`[pF-7$$\"3]om;HYt7vF1$\"3o?qd`2b\\pF-7$$\"3 /-+++xG**yF1$\"3-EQ%)pE[]pF-7$$\"3\\pmmT6KU$)F1$\"3o75*e&4b^pF-7$$\"3+ OLLLbdQ()F1$\"3]f!e!Hm]_pF-7$$\"3+++]i`1h\"*F1$\"3AX8&)od_`pF-7$$\"3y, +]P?Wl&*F1$\"3%e%\\9-9]apF-7$$\"3-+++++++5F*$\"3Q%))=n0]b&pF--%'COLOUR G6&%$RGBG$\"#5!\"\"$\"\"!Fc[lFb[l-%*AXESTICKSG6%\"\"$%(DEFAULTG-%%FONT G6%%&TIMESG%&ROMANG\"#:-%+AXESLABELSG6%Q\"x6\"Q!Fc\\lFi[l-%%VIEWG6$;$F a[l!\"#$\"\"\"Fj\\lFh[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Krom \354 grafu si m\371\236eme tak\351 vypsat tabulku hodnot funkce v okol \355 nuly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 23 "sez1:=[ seq(i, i=2..7)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sez1G7(\"\"#\" \"$\"\"%\"\"&\"\"'\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " sez2:=[seq(8-i,i=1..6)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sez2G7( \"\"(\"\"'\"\"&\"\"%\"\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "hodn:=[seq(-0.1^i, i=sez1),seq(0.1^i, i=sez2)];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%hodnG7.$!\"\"!\"#$F'!\"$$F'!\"%$F'!\"&$F'!\"' $F'!\"($\"\"\"F2$F4F0$F4F.$F4F,$F4F*$F4F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "seq('f'(i)=evalf(f(i),15), i=hodn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6./-%\"fG6#$!\"\"!\"#$\"0+kHc/v!p!#:/-F%6#$F(!\"$$\" 0+qa4q!HpF,/-F%6#$F(!\"%$\"0++YeJ7$pF,/-F%6#$F(!\"&$\"0++IyZ9$pF,/-F%6 #$F(!\"'$\"0+++Sp9$pF,/-F%6#$F(!\"($\"0+++gr9$pF,/-F%6#$\"\"\"FM$\"0++ ++s9$pF,/-F%6#$FTFF$\"0+++?u9$pF,/-F%6#$FTF?$\"0+++$e\\JpF,/-F%6#$FTF8 $\"0++!Q?rJpF,/-F%6#$FTF1$\"0++eiuQ$pF,/-F%6#$FTF)$\"0+?n0]b&pF," }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Z grafu i tabulky m\371\236eme p \370edpokl\341dat , \236e hodnota limity bude p\370ibli\236n\354 0, 69 31 ." }}{PARA 0 "" 0 "" {TEXT -1 24 "P\370esnou hodnotu ur\350\355me : " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(student,Limit):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Limit(f(x),x=0):%=value(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&,&)\"\"#%\"xG\"\"\"F ,!\"\"F,F+F-/F+\"\"!-%#lnG6#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(ln(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1=ZJp!#5 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "P\370\355klad 5. " }{TEXT 357 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Necht\264 g : y = " }{XPPEDIT 18 0 "sin(1/x);" "6#-%$sinG6#*&\"\"\"F'%\"xG!\"\"" }{TEXT -1 29 " . 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'g'(0.001)=g(0.001);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"gG6#$!\"&!\"#$!+2DXH\"*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#$!#D!\"$$!+0;8^u!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#$!#D!\"%$\"+'f$>4&)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#$\"\"\"!\"%$!+*)Q9cI!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#$\"#D!\"%$!+'f$>4&)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#$\"\"\"!\"$$\"+0azo#)!#5" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 105 "V tomto p\370\355pad\354 nejde z grafu ani z tabu lky hodnotu limity odhadnout. Zkusme tedy ur\350it limitu p\370\355ka zem " }{TEXT 354 5 "Limit" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "V\375po\350et limity:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(student,Limit):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Limit(g(x),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$-%$sin G6#*&\"\"\"F*%\"xG!\"\"/F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#;!\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Funk\350n\355 hodnota pro x->0 le\236\355 z\370ejm\354 v intervalu <-1,1>. Limita pro x->0 ale neexistuje." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 234 "Na p\370edch\341zej\355c\355m obr\341zku nen\355 graf fu nkce v\236dy vykreslen a\236 k hodnot\341m y =-1 a y = 1. Maple vykre sluje graf funkce spojov\341n\355m referen\350n\355ch bod\371. Jejich \+ po\350et m\371\236e ovlivnit p\370esnost grafu. 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"V\375po\350et limity:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "limit(f(x),x=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%*undefinedG" }}}{PARA 0 "" 0 "" {TEXT -1 62 "Limita tedy neexistuje. M\371\236eme v\232ak ur\350it jednostrann\351 limity." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(f(x),x=1, left);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "limit(f(x),x=1,right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 291 15 "Defini\350n\355 obory" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 83 "P\370i ur\350ov\341n\355 defini\350n\355ch obor \371 se op\355r\341me o z\341kladn\355 znalosti element\341rn\355ch fu nkc\355." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 293 8 "P\370\355klady " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Ur\350ete defini\350n\355 obo ry t\354chto funkc\355" }}{PARA 0 "" 0 "" {TEXT 294 12 "a) f(x) = ar" }{TEXT 404 4 "ccos" }{XPPEDIT 405 1 "(x-2)/x;" "6#*&,&%\"xG\"\"\"\"\"# !\"\"F&F%F(" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solve((x-2)/x >=-1 and (x-2)/x <= 1 ,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*RealRangeG6$\"\"\"%)infinit yG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "podm\355nka x r\371zn\351 o d nuly je tak\351 zahrnuta" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 20 "b) f(x) = ln(sin(x))" }}{PARA 0 "" 0 "" {TEXT -1 57 "budeme zji\232\235o vat, kdy je argument - sin(x) - v\354t\232\355 jako 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solve(sin(x)>0,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "vid\355me, \236e MAPLE n\341m nenab\355dl \236\341dn\375 v\375sledek, je tato \372loha pro n\354j ne\370e\232it eln\341" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 12 "c) f(x) = ln" } {XPPEDIT 296 1 "2*x/(x+3);" "6#*(\"\"#\"\"\"%\"xGF%,&F&F%\"\"$F%!\"\" " }{TEXT 403 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "op\354t \370 e\232\355me nerovnici, kdy argument je v\354t\232\355 ne\236 0, mus \355 platit pro x r\371zn\351 od -3, ve v\375sledku je tak\351 uva\236 ov\341no" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(2*x/(x+3) >0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$,$%)infinityG !\"\"-%%OpenG6#!\"$-F$6$-F*6#\"\"!F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 298 10 "d) f(x) = " }{XPPEDIT 299 1 "sqrt(x^4+x^3-7*x^2-13*x-6);" "6#- %%sqrtG6#,,*$%\"xG\"\"%\"\"\"*$F(\"\"$F**&\"\"(F**$F(\"\"#F*!\"\"*&\"# 8F*F(F*F1\"\"'F1" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "zde v \355me, \236e v\375raz pod sudou odmocninou mus\355 b\375t nez\341porn \375 a p\370i \"ru\350n\355m\" po\350\355t\341n\355 nejprve rozlo\236 \355me na ko\370enov\351 \350initele a pak ur\350ujeme" }}{PARA 0 "" 0 "" {TEXT -1 42 "znam\351nko, k rozlo\236en\355 n\341m poslou\236\355 p\370\355kaz " }{TEXT 292 6 "factor" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "factor(x^4+x^3-7*x^2-13*x-6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%\"xG\"\"\"\"\"#F&F&,&F%F&\"\"$!\"\"F&),&F%F&F&F&F' F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve((x+1)^2*(x-3)*( x+2)>=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%-%*RealRangeG6$,$%)infi nityG!\"\"!\"#F(-F$6$\"\"$F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "M aple um\355 ale vy\370e\232it tak\351 p\370\355mo" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "solve(x^4+x^3-7*x^2-13*x-6>=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%-%*RealRangeG6$,$%)infinityG!\"\"!\"#F(-F$6$\" \"$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 8 "Derivace" }{TEXT -1 0 "" } }{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "Z\341kladn\355 informace" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Zopakujme n\354kter\351 vlastnosti , kter\351 budeme d\341le vyu\236\355vat." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Existuje-li " }{MPLTEXT 1 0 1 " " }{XPPEDIT 396 1 "limit( (f(x)-f(x[0]))/(x-x[0]),x = x[0]);" "6#-%&limitG6$*&,&-%\"fG6#%\"xG\" \"\"-F)6#&F+6#\"\"!!\"\"F,,&F+F,&F+6#F1F2F2/F+&F+6#F1" }{TEXT 397 24 " , naz\375v\341me tuto limitu " }{TEXT 301 26 "derivace funkce f v bod \354 " }{XPPEDIT 398 1 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT 399 15 " a zn a\350\355me ji " }{TEXT 302 4 "f '(" }{XPPEDIT 400 1 "x[0];" "6#&%\"x G6#\"\"!" }{TEXT 401 0 "" }{TEXT 303 2 ")." }{TEXT 402 6 " Obdob" } {TEXT -1 41 "n\354 lze definovat derivaci zprava a zleva." }}{PARA 0 " " 0 "" {TEXT -1 2 "Z " }{TEXT 304 14 "geometrick\351ho " }{TEXT -1 78 "hlediska p\370edstavuje derivace funkce v dan\351m bod\354 sm\354rnic i te\350ny v tomto bod\354." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 300 52 "Opakov\341n\355 z\341kladn\355c h informac\355 o technice derivov\341n\355" }}{PARA 0 "" 0 "" {TEXT -1 9 "V dal\232\355m " }{TEXT 311 4 "f,g " }{TEXT -1 22 "budou funkce \+ prom\354nn\351 " }{TEXT 312 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "V ka\236d\351m bod\354 defini\350n\355ho oboru dan\375ch \+ funkc\355 plat\355: " }}{PARA 0 "" 0 "" {TEXT -1 4 "(1) " }{TEXT 313 11 "(cf)\264= cf \264" }{TEXT -1 17 " , c je konstanta" }}{PARA 0 "" 0 "" {TEXT -1 5 "(2) " }{TEXT 314 20 "(f + g) \264 = f\264+ g\264 " }{TEXT -1 20 ", plat\355 i pro rozd\355l" }}{PARA 0 "" 0 "" {TEXT -1 4 "(3) " }{TEXT 315 19 "(f g)\264 = f\264g + fg\264" }{TEXT -1 20 " \+ derivace sou\350inu" }}{PARA 0 "" 0 "" {TEXT -1 3 "(4)" }{TEXT 316 1 "(" }{TEXT 362 1 " " }{XPPEDIT 256 1 "f/g;" "6#*&%\"fG\"\"\"%\"gG!\" \"" }{TEXT 257 0 "" }{TEXT 317 6 " )\264 = " }{XPPEDIT 363 1 "(`f\264g `-`fg\264`)/(g^2);" "6#*&,&%$f|_vgG\"\"\"%$fg|_vG!\"\"F&*$%\"gG\"\"#F( " }{TEXT -1 18 " derivace pod\355lu" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "P\370ipomeneme n\354kter\351 vzorce, kter\351 najdeme v ka\236d\351 z\341kladn\355 literatu\370e, pro zaj\355mavost nech\341m e derivace t\354chto z\341kladn\355ch funkc\355 tak, jak " }}{PARA 0 " " 0 "" {TEXT -1 17 "je vypo\350te MAPLE:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "f1:=c: f2:=x^n: f3:=log[a](x): f4:=a^x: f5:=sin(x): f6:=cos(x): f7:=tan(x): f8:=arctan(x): f9:=cot(x): f10:=arccot(x): f11:=arccos(x) : f12:=arcsin(x): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 \+ to 12 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Diff(f||i,x)=diff(f||i, x)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$%\"cG%\"xG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$)%\"xG%\"nGF(*(F'\"\"\"F)F+F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&-%#lnG6#%\"xG\"\"\"-F)6#%\"aG! \"\"F+*&F,F,*&F+F,F-F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6 $)%\"aG%\"xGF)*&F'\"\"\"-%#lnG6#F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%%DiffG6$-%$sinG6#%\"xGF*-%$cosGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%$cosG6#%\"xGF*,$-%$sinGF)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%$tanG6#%\"xGF*,&\"\"\"F,*$)F'\"\"#F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%'arctanG6#%\"xGF**&\"\"\" F,,&F,F,*$)F*\"\"#F,F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%Dif fG6$-%$cotG6#%\"xGF*,&!\"\"\"\"\"*$)F'\"\"#F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%'arccotG6#%\"xGF*,$*&\"\"\"F-,&F-F-*$)F*\" \"#F-F-!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%'arccos G6#%\"xGF*,$*&\"\"\"F-*$-%%sqrtG6#,&F-F-*$)F*\"\"#F-!\"\"F-F6F6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%'arcsinG6#%\"xGF**&\"\"\" F,*$-%%sqrtG6#,&F,F,*$)F*\"\"#F,!\"\"F,F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 305 34 "P\370\355kazy syst\351mu MAPLE pro deriva ce" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Derivaci m\371\236em e spo\350\355tat pomoc\355 p\370\355kazu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 306 21 "diff(v\375raz,prom\354nn\341);" }}{PARA 0 "" 0 "" {TEXT -1 5 "kde " }{TEXT 307 7 "v\375raz " }{TEXT -1 20 "je algebrai ck\375 v\375raz" }}{PARA 0 "" 0 "" {TEXT -1 8 " " }{TEXT 308 9 "prom\354nn\341 " }{TEXT -1 27 "je prom\354nn\341 v tomto v\375razu." }}{PARA 0 "" 0 "" {TEXT -1 26 "M\371\236eme pou\236\355t tak\351 p\370 \355kaz " }{TEXT 309 24 "Diff( v\375raz, prom\354nn\341); " }{TEXT -1 59 "kter\375 derivaci nevyhodnot\355. Vyhodnocen\355 dos\341hneme p \370\355kazem " }{TEXT 310 6 "value." }{TEXT -1 0 "" }}}{PARA 4 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "V\375po\350et \+ derivace" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 259 0 "" } {TEXT 260 0 "" }{TEXT 261 9 "P\370\355klad 1" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 34 "Vypo\350t\354te derivaci funkce f(x) = " }{XPPEDIT 18 0 "(x/(x+1))^x;" "6#)*&%\"xG\"\"\",&F%F&F&F&!\"\"F%" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 18 "a) pomoc\355 p\370\355kazu " }{TEXT 368 4 "diff" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f:=(x/(x+1))^x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG)*&% \"xG\"\"\",&F'F(F(F(!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "df:=diff(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfG*&)*&%\"xG \"\"\",&F(F)F)F)!\"\"F(F),&-%#lnG6#F'F)*&,&*&F)F)F*F+F)*&F(F)F*!\"#F+F )F*F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "\372prava v\375razu " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "simplify(df,radical);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*()*&%\"xG\"\"\",&F&F'F'F'!\"\"F&F',(*&-%#ln G6#F%F'F&F'F'F'F'F,F'F'F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "collect(%,ln(x/(x+1)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "coll ect(%,(x/(x+1))^x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&)*&%\"xG\"\"\",&F'F(F(F(!\"\"F'F(-%#lnG6#F& F(F(*&F%F(F)F*F(" }}{PARA 8 "" 1 "" {TEXT -1 47 "Error, (in collect) c annot collect (x/(x+1))^x\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs((x/(x+1))^x=t,%%):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "collect(%,t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(t=(x/(x+1 ))^x,%);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%#lnG6#*&%\"xG\"\"\",&F)F*F*F*!\"\"F**&F*F*F+F,F*F *)F(F)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "********************* **********************************************" }}{PARA 0 "" 0 "" {TEXT -1 27 "b)V\375po\350et derivace pomoc\355 " }{TEXT 369 4 "Diff " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g:=ln((x+1)/(x^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG-%#lnG6#*&,&%\"xG\"\"\"F+F+F+F*!\"#" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Diff(g,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%#lnG6#*&,&%\"xG\"\"\"F,F,F,F+!\"#F+" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&*&\"\"\"F&*$)%\"xG\"\"#F&!\"\"F&*( F*F&,&F)F&F&F&F&F)!\"$F+F&F-F+F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*(,&%\"xG\"\"\"\"\"#F'F'F&!\"\",&F&F'F'F'F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "**************************************************** ********************************" }}{PARA 0 "" 0 "" {TEXT -1 20 "c)Pou \236it\355 oper\341toru " }{TEXT 370 1 "D" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " f:=x->(1-ln(x))/(1+ln(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&\"\"\"F.-% #lnG6#9$!\"\"F.,&F.F.F/F.F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "df:=D(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfGf*6 #%\"xG6\"6$%)operatorG%&arrowGF(,&*&\"\"\"F.*&9$F.,&F.F.-%#lnG6#F0F.F. !\"\"F5*(,&F.F.F2F5F.F1!\"#F0F5F5F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(df(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*&\"\"\"F%*&),&F%F%-%#lnG6#%\"xGF%\"\"#F%F,F%!\"\"!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "***************************************** *******************************************" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 262 9 "P\370\355klad 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 " Vypoct\354te prvn\355 derivaci funkce " }{TEXT 371 7 "g(x) = " } {XPPEDIT 372 0 "arctan((x+1)/(x-1));" "6#-%'arctanG6#*&,&%\"xG\"\"\"F) F)F),&F(F)F)!\"\"F+" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "g:=arctan((x +1)/(x-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG-%'arctanG6#*&,&% \"xG\"\"\"F+F+F+,&F*F+F+!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(g,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&\"\"\"F&,& %\"xGF&F&!\"\"F)F&*&,&F(F&F&F&F&F'!\"#F)F&,&F&F&*&F+\"\"#F'F,F&F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%, &*$)%\"xG\"\"#F%F%F%F%!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 263 9 "P\370 \355klad 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Vypo\350t\354te prvn\355 derivaci \+ funkce h(x) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "h:=x->2*ln(2*x-3*(1 -4*x^2)^(1/2))-6*arcsin(2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"h Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%#lnG6#,&9$\"\"#*&\"\"$\"\"\"-% %sqrtG6#,&F5F5*&\"\"%F5)F1F2F5!\"\"F5F=F2*&\"\"'F5-%'arcsinG6#,$F1F2F5 F=F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dh:=diff(h(x),x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dhG,&*&,&\"\"#\"\"\"*(\"#7F), &F)F)*&\"\"%F))%\"xGF(F)!\"\"#F1F(F0F)F)F),&F0F(*&\"\"$F)-%%sqrtG6#F,F )F1F1F(*&F+F)F,F2F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "norm al(%);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%,&%\"xG\"\"#*&\"\"$F%-%%sqrtG6#,&F%F%*&\"\" %F%)F'F(F%!\"\"F%F2F2\"#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 14 "U\236it\355 derivac\355" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 264 6 "Te\350na " } {TEXT -1 0 "" }}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 0 "" }{TEXT 267 0 " " }{TEXT 268 9 "P\370\355klad 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 37 "Pod jak\375m \372hlem se prot\355n\341 hyperbola \+ " }{TEXT 269 1 "h" }{TEXT -1 13 " s parabolou " }{TEXT 270 1 "p" } {TEXT -1 2 "?." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "h:=1/x; p:=sqrt(x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG*&\"\"\"F&%\"xG!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG*$-%%sqrtG6#%\"xG\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Pr\371se\350\355k:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(h=p,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Graf k\370ivek a jejic h te\350en:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(student,showtan gent):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(plots,displa y):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "plot1:=showtangent(p ,x=1,numpoints=600,color=black,labelfont=[TIMES,ROMAN,15]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "plot2:=showtangent(h,x=1,num points=600,color=blue,labelfont=[TIMES,ROMAN,15]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display([plot1,plot2],view=[0..4,-1..4],sca ling=constrained," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "axesfont=[TIME S,ROMAN,15]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6 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\"$,&F.\"\"\"*$)F-\"\"#F0!\"\"F4F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "1) Defini\350n\355 obor:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(denom(f(x))=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*$-%% sqrtG6#\"\"$\"\"\",$F#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "p \370edch\341zej\355c\355m p\370\355kazem jsme nalezli hodnoty, ve kter \375ch funkce nen\355 definov\341na, " }}{PARA 0 "" 0 "" {TEXT -1 12 "tedy D(f) = " }{TEXT 278 2 "R " }{TEXT -1 4 "- \{-" }{XPPEDIT 18 0 "s qrt(3),sqrt(3);" "6$-%%sqrtG6#\"\"$-F$6#F&" }{TEXT -1 1 "\}" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "2) Pr\371se\350\355ky s osami:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "\{solve(f(x)=0,x)\};" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "3) Sudost resp. lichost:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"xG\" \"$,&F&\"\"\"*$)F%\"\"#F(!\"\"F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "zjistili jsme, \236e f(-x ) = -f(x), funkce je tedy lich\341 a \+ graf bude soum\354rn\375 podle po\350\341tku" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "4) Stacion\341rn\355 bo dy, intervaly monotonie, extr\351my" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "df:=diff(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfG,&*&%\" xG\"\"#,&\"\"$\"\"\"*$)F'F(F+!\"\"F.F**(F(F+F'\"\"%F)!\"#F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "df:=normal(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#dfG,$*(%\"xG\"\"#,&!\"*\"\"\"*$)F'F(F+F+F+,&! \"$F+F,F+!\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve (df,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"!F#\"\"$!\"$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "stacion\341rn\355 body maj\355 ted y sou\370adnice 0,3,-3 - zjist\355me, zda jsou to tak\351 extr\351my" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(df>0,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&-%*RealRange G6$-%%OpenG6#!\"$-F'6#,$*$-%%sqrtG6#\"\"$\"\"\"!\"\"-F$6$F*-F'6#\"\"!- F$6$F6-F'6#F--F$6$F;-F'6#F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(df<0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$, $%)infinityG!\"\"-%%OpenG6#!\"$-F$6$-F*6#\"\"$F'" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 112 "ze znam\351nka prvn\355 derivace vid\355me, \236e extr\351my nastanou v bodech -3,3, dopo\350\355tejme sou\370adnice l ok\341ln\355ch extr\351m\371:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "E1 :=[-3,f(-3)];E2:=[3,f(3)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E1G7$ !\"$#\"\"*\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E2G7$\"\"$#!\"* \"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "5) Intervaly konvexnosti resp. konkavnosti, inflexn\355 body" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dff:=diff(df,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dffG,(*(% \"xG\"\"\",&!\"*F(*$)F'\"\"#F(F(F(,&!\"$F(F+F(!\"#F0*(F-F(F'\"\"$F.F0! \"\"**\"\"%F(F'F2F)F(F.F/F(" }}}{EXCHG 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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "s osou " }{TEXT 280 2 "x:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(f(x)=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "3) sudost \350i lichost " }{TEXT 281 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " f(-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"xG\"\"\",&F&F&*$)F%\" \"#F&F&!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "-f(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"xG\"\"\",&F&F&*$)F%\"\"#F&F&!\" \"F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Vid\355me, \236e funkce s pl\362uje podm\355nku f(-x) = -f(x), je tedy lich\341" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "4) Stacion\341rn\355 body, extr\351my, in tervaly monotonie" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "df:=diff(f(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfG,&*&\"\"\"F',&F'F'*$)%\"x G\"\"#F'F'!\"\"F'*(F,F'F+F,F(!\"#F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "df:=simplify(df);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#dfG,$*&,&!\"\"\"\"\"*$)%\"xG\"\"#F)F)F),&F)F)F*F)!\"#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "x-ov\351 sou\370adnice stacion\341rn\355c h bod\371" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "solve(df);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$!\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 56 "nyn\355 zjist\355me, jak to vypad\341 se znam\351nkem p rvn\355 derivace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(d f>0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*RealRangeG6$-%%OpenG6#!\" \"-F'6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(df<0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$,$%)infinityG!\" \"-%%OpenG6#F(-F$6$-F*6#\"\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "ve stacion\341rn\355ch bodech se m\354n\355 znam\351nko prvn\355 \+ derivace, jsou to tedy extr\351my; sou\370adnice lok\341ln\355ch extr \351m\371:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "S1:=[-1,f(-1)];S2:=[1 ,f(1)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S1G7$!\"\"#F&\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S2G7$\"\"\"#F&\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "podle znam\351nka vid\355me, \236e bod S1 je lok\341ln\355 minimum, bod S2 lok\341ln\355 maximum" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "5) p\370\355padn\351 inflexn\355 body, in tervaly, kde je funkce konvexn\355 resp. konk\341vn\355" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "ddf:=diff(df,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ddfG,&*&%\"xG\"\"\",&F(F(*$)F'\"\"#F(F(!\"#F-**\"\"%F(,&!\"\" F(F*F(F(F)!\"$F'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ddf :=simplify(ddf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ddfG,$*(%\"xG\" \"\",&!\"$F(*$)F'\"\"#F(F(F(,&F(F(F+F(F*F-" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "solve(ddf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\" !*$-%%sqrtG6#\"\"$\"\"\",$F$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "zjist\355me, kde je funkce nad te\350nou (tj. konvexn\355)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(ddf>0);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$-%*RealRangeG6$-%%OpenG6#,$*$-%%sqrtG6#\"\"$\"\" \"!\"\"-F'6#\"\"!-F$6$-F'6#F*%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "a nyn\355 pod te\350nou (tj. konk\341vn\355)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(ddf<0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$,$%)infinityG!\"\"-%%OpenG6#,$*$-%%sqrtG 6#\"\"$\"\"\"F(-F$6$-F*6#\"\"!-F*6#F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "znam\351nko druh\351 derivace se v nalezen\375ch bodech \+ m\354n\355, jsou to tedy inflexn\355 body" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "sou\370adnice inflexn\355ch bod\371" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "I1:=[-sqrt(3),f(-sqrt(3))];I2:=[0,f(0)];I3:=[sqrt(3), f(sqrt(3))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I1G7$,$*$-%%sqrtG6# \"\"$\"\"\"!\"\",$F'#F-\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I2G 7$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I3G7$*$-%%sqrtG6#\"\"$ \"\"\",$F&#F+\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "6) asymptot y funkce" }}{PARA 0 "" 0 "" {TEXT -1 14 "a) se sm\354rnic\355" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "k1:=limit(f(x)/x,x=infinity); q1:=l imit(f(x)-k1*x,x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "k2:=limit(f(x)/x,x=-infinity); q2:=limit( f(x)-k2*x,x=-infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q2G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "asymptota se sm\354rnic\355 je p\370\355mka y = 0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "b) bez sm\354rnice" }}{PARA 0 " " 0 "" {TEXT -1 70 "tyto asymptoty zde neexistuj\355, proto\236e funkc e je definovan\341 na cel\351m R" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "7) graf funkce:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot(f(x),x= -5..5,axesfont=[TIMES,ROMAN,15],labelfont=[TIMES,ROMAN,15]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7ao7$$!\"& \"\"!$!3=Bp2Bp2B>!#=7$$!3YLLLe%G?y%!#<$!3AjBA*Q[N+#F-7$$!3OmmT&esBf%F1 $!3Mm&4H2[*y?F-7$$!3ALL$3s%3zVF1$!3;6g#pE,/<#F-7$$!3_LL$e/$QkTF1$!3uY \"RLr(RqAF-7$$!3ommT5=q]RF1$!3W$ffm-)yyBF-7$$!3ILL3_>f_PF1$!3;,(>31N\" )[#F-7$$!3K++vo1YZNF1$!3smlaxvS6EF-7$$!3;LL3-OJNLF1$!3)>#*RU5H4v#F-7$$ !3p***\\P*o%Q7$F1$!3ypK,g*HO!HF-7$$!3Kmmm\"RFj!HF1$!3sKj/-/awIF-7$$!33 LL$e4OZr#F1$!3kN!o`:)[F-7$$!3OL$ 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3a***\\7`Wl7$F1$\"32(>0s')*e,HF-7$$\"3enmmm*RRL$F1$\"35.8%)fa(=v#F-7$$ \"3%zmmTvJga$F1$\"3mb$\\O'[I7EF-7$$\"3]MLe9tOcPF1$\"3khAR0b'f[#F-7$$\" 31,++]Qk\\RF1$\"3UEWQ`%[$zBF-7$$\"3![LL3dg6<%F1$\"3)RsT?**4rE#F-7$$\"3 %ymmmw(GpVF1$\"3MJj#H*Qyu@F-7$$\"3C++D\"oK0e%F1$\"3wm_C'\\LQ3#F-7$$\"3 5,+v=5s#y%F1$\"3CA-*=d#G.?F-7$$\"\"&F*$\"3=Bp2Bp2B>F--%'COLOURG6&%$RGB G$\"#5!\"\"$F*F*Ffal-%+AXESLABELSG6%Q\"x6\"Q!F[bl-%%FONTG6%%&TIMESG%&R OMANG\"#:-%*AXESTICKSG6%%(DEFAULTGFfblF]bl-%%VIEWG6$;F(F[alFfbl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 273 9 "P\370\355 klad 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Vy\232et\370ete pr\371b\354h \+ funkce " }{TEXT 377 7 "g(x) = " }{XPPEDIT 378 1 "x^2/(x^2-1);" "6#*&% \"xG\"\"#,&*$F$F%\"\"\"F(!\"\"F)" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "g:=x->x^2/(x^2- 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"#,&*$)F$F%\"\"\"F)F) !\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)opera torG%&arrowGF(*&9$\"\"#,&*$)F-F.\"\"\"F2F2!\"\"F3F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "1)Defini\350n\355 obor funkce - jedn\341 se o racion\341ln\355 funkci, je tedy nutn\351, aby jmenovatel byl r \371zn\375 od nuly." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(denom( g(x))=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Defini\350n\355 obor jsou v\232echna re \341ln\341 \350\355sla bez \{-1,1\}" }}{PARA 0 "" 0 "" {TEXT -1 20 "2) Pr\371se\350\355ky s osami:" }}{PARA 0 "" 0 "" {TEXT -1 11 "a) s osou \+ x" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(g(x)=0,x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"\"!F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "ko\370en 0 je dvojn\341s obn\375" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "b) s osou y" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "3)Sudost \+ resp. lichost funkce:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "g(-x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"#,&*$)F$F%\"\"\"F)F)!\"\"F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&%\"xG\"\"#,&*$)F$F%\"\"\"F)F)!\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "plat\355 g(-x) = g(x) fce je tedy sud\341 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "4)Nalezen\355 stacion\341rn \355ch bod\371, intervaly monotonie, p\370\355padn\354 extr\351my" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dg:=diff(g(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dgG,&*&%\"xG\"\"\",&*$)F'\"\"#F(F(F(!\"\"F-F,*(F, F(F'\"\"$F)!\"#F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dg:=si mplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dgG,$*&%\"xG\"\"\",&* $)F'\"\"#F(F(F(!\"\"!\"#F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "solve(dg);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "intervaly, kde je funkce rostouc\355" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(dg>0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$,$%)infinityG!\"\"-%%OpenG6#F(-F$6$ F)-F*6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "intervaly, kde je funkce klesaj\355c\355" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " solve(dg<0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$-%%Open G6#\"\"!-F'6#\"\"\"-F$6$F*%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "v bod\354 x = 0 se m\354n\355 znam\351nko 1. derivace, f unkce zde m\341 extr\351m, podle znam\351nka se jedn\341 o lok\341ln \355 maximum" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "sou\370adnice sta cion\341rn\355ho bodu - lok\341ln\355ho minima" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "S:=[0,g(0)];" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"SG7$\"\"!F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "5)Intervaly \+ konvexnosti a konk\341vnosti, p\370\355padn\354 inflexn\355 body" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ddg:=diff(dg,x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$ddgG,&*&\"\"\"F'*$),&*$)%\"xG\"\"#F'F'F'!\"\"F.F'F /!\"#*(\"\")F'F-F.F*!\"$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ddg:=simplify(ddg);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ddgG,$*& ,&*$)%\"xG\"\"#\"\"\"\"\"$F,F,F,,&F(F,F,!\"\"!\"$F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(ddg);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&^##\"\"\"\"\"$F&-%%sqrtG6#F'F&*&^##!\"\"F'F&F(F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "druh\341 derivace nem\341 \370e\232en \355 v oboru re\341ln\375ch \350\355sel, nejsou inflexn\355 body, ur \350\355me, kde je funkce nad te\350nou a pod te\350nou" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(ddg>0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$,$%)infinityG!\"\"-%%OpenG6#F(-F$6$-F*6# \"\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "fce je zde nad te\350 nou" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(ddg<0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*RealRangeG6$-%%OpenG6#!\"\"-F'6#\" \"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "fce je zde pod te\350nou " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "6)Asymptoty funkce" }}{PARA 0 "" 0 "" {TEXT -1 14 "a) se sm\354rnic\355" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "k1:=limit(g(x)/ x,x=infinity); q1:=limit(g(x)-k1*x,x=infinity);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "k2:=limit(g(x)/x,x=-infinity); q2:=limit(g(x)-k2*x,x= -infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q2G\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "asymptota se sm\354rnic\355 je ted y p\370\355mka y = 1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "b) bez sm \354rnice" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "funkce je nespojit \341 v bodech -1,1, vypo\350t\354me tedy limity v jejich okol\355" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "limit (g(x),x=-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefinedG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "limita zde nen\355 definovan\341, zkus\355me limity zpra va a zleva" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(g(x),x= -1,left);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit(g(x),x=-1,right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 127 "p\370\355mka x = -1 je asymptota bez sm\354rnice a pro to\236e dan\341 funkce je sud\341, lze \370\355ci, \236e p\370\355mka \+ x = 1 je tak\351 asymptota bez sm\354rnice" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Graf funkce:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(plots,display):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 43 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\"$F*F]fn" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 274 9 "P\370\355kl ad 4" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar t:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Vy\232et\370ete pr\371b\354 h funkce h" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "h:=x->x^2-4*abs(x)+3; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(,(*$)9$\"\"#\"\"\"F1*&\"\"%F1-%$absG6#F/F1!\"\"\"\"$F1F(F(F(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 1 "" {TEXT -1 40 "Defini\350n\355 obor jsou v\232echna re\341ln\341 \350 \355sla" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Pr\371se\350\355ky s o sami:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(h(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"$\"\"\"!\"\"!\"$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5 "h(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Sud\341,lich\341" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "h(-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"F(*&\"\"%F(-%$absG6#F&F(!\"\"\"\"$F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "h(x) = h(-x), funkce je tedy sud \341, budeme se pro jednoduchost zab\375vat pr\371b\354hem t\351to fun kce pro kladn\351 x, proto\236e graf cel\351 funkce je pak soum\354rn \375" }}{PARA 0 "" 0 "" {TEXT -1 44 "podle osy y, definujme tuto \350 \341st dan\351 funkce" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "h1:=x->x^2-4*x+3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h1Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(* $)9$\"\"#\"\"\"F1*&\"\"%F1F/F1!\"\"\"\"$F1F(F(F(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 60 "jenom pro dopln\354n\355 - funkce pro z\341porn \351 x m\341 vyj\341d\370en\355 h2 = " }{XPPEDIT 19 1 "x^2;" "6#*$%\"x G\"\"#" }{TEXT -1 8 "+ 4x + 3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 " Stacion\341rn\355 body, intervaly monotonie, extr\351my" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "dh:=diff(h1(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dhG,&%\"xG\"\"#\"\"%!\"\"" }}}{EXCHG {PARA 11 "" 1 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(dh= 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(dh>0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%*RealRangeG6$-%%OpenG6#\"\"#%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(dh<0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%*RealRangeG6$,$%)infinityG!\"\"-%%OpenG6#\"\"#" }}}{EXCHG {PARA 11 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 299 "Nalezli jsme bod x = 2 jako lok \341ln\355 minimum, proto\236e je funkce sud\341, bude bod x = -2 tak \351 lok\341ln\355 minimum a mus\355me se d\341le zab\375vat bodem x = 0, kde je funkce definov\341na, ale derivace nen\355 - zleva a zprava je r\371zn\341. V tomto bod\354 se m\354n\355 znam\351nko derivace, n ast\341v\341 zde tak\351 extr\351m - viz dal\232\355 p\370\355kazy." } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "derivace:=diff(h(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)derivaceG,&%\"xG\"\"#*&\"\"%\"\"\"-%$absG 6$F*F&F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "signum(der ivace);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'signumG6#,&%\"xG\"\"#*& \"\"%\"\"\"-%$absG6$F+F'F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eval(subs(x=0.5,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "derivace je z\341porn\341, fu nkce kles\341" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eval(subs( x=-0.5,%%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "derivace je kladn\341, funkce roste" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "E1:=[-2,h(-2)];E2:=[0,h(0)];E3:=[2,h(2)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E1G7$!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #E2G7$\"\"!\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E3G7$\"\"#!\"\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Intervaly, kde je funkce konv exn\355, resp. konk\341vn\355 - ur\350ujeme pomoc\355 znam\351nka druh \351 derivace" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dhh:=diff(dh,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dhhG\"\"#" }}}{EXCHG {PARA 256 " " 1 "" {TEXT -1 130 "Hodnota druh\351 derivace je rovna 2, kladn\351ho znam\351nka, z toho tedy vypl\375v\341, \236e funkce je nad te\350nou v cel\351m sv\351m defini\350n\355m oboru. " }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 10 "Asymptoty:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "k:=lim it(h1(x)/x,x=infinity); q:=limit(h1(x)-k*x,x=infinity);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"kG%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Neexis tuj\355 \236\341dn\351 asymptoty" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Graf funkce:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot(h(x),x=-5. .5,y=-5..5,labelfont=[TIMES,ROMAN,15]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "axesfont=[TIMES,ROMAN,15]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7co7$$!\"&\"\"!$\"\")F*7$$!3Gmm;HU, \"*[!#<$\"3oMTCtK'zN(F07$$!3YLLLe%G?y%F0$\"3MVl(HM#oRnF07$$!3!***\\(=_ +so%F0$\"3Zt_wWm/@iF07$$!3OmmT&esBf%F0$\"3o*))>;i&R?dF07$$!3O**\\7`'Gd [%F0$\"3o@m'p$p%)y^F07$$!3ALL$3s%3zVF0$\"3mf_PF0$\"3!)os]]&y:2#F07$$!3K++vo1YZNF0$\"3I)>G8_MYR\" F07$$!3;LL3-OJNLF0$\"3Siw3fTiIy!#=7$$!3p***\\P*o%Q7$F0$\"3\"HB:fS=.j#F ^p7$$!3Kmmm\"RFj!HF0$!31wp9\"f1dy\"F^p7$$!33LL$e4OZr#F0$!3YZ#HJ8B:*[F^ p7$$!3u*****\\n\\!*\\#F0$!3>UR#)=U\\4vF^p7$$!3y******\\Ow!R#F0$!3!Rx'Q )pPIZ)F^p7$$!3%)*****\\ixCG#F0$!3*))fVP\"R1-#*F^p7$$!35+++vR7y@F0$!3C' *>I&\\=Fo*F^p7$$!3#******\\KqP2#F0$!3mTR\\\"Rzb%**F^p7$$!3am;a8s+z>F0$ !33\\r;HIf&***F^p7$$!39LL3-TC%)=F0$!3-k](4s0g')*F^p7$$!3#)**\\(ofh:x\" F0$!3%))*pxf*e\"y%*F^p7$$!3[mmm\"4z)e;F0$!3j8%y*e_OO))F^p7$$!3Mmmmm`'z Y\"F0$!3M_'Q$*[\"RprF^p7$$!3#****\\(=t)eC\"F0$!38e$z(R198VF^p7$$!3!omm mh5$\\5F0$!3MO'G7$$!3S$***\\(=[jL)F^p$\"3$*pMKgt2/OF^p7$$!3) f***\\iXg#G'F^p$\"3<%fu)ePp;))F^p7$$!3ndmmT&Q(RTF^p$\"3*3%pEN*ya^\"F07 $$!3%\\mmTg=><#F^p$\"3'oekD'[Sy@F07$$!3g***\\7yQ16\"F^p$\"3ClT_s'z!oDF 07$$!3vDMLLe*e$\\!#?$\"3!QMM(z2G!)HF07$$\"3womT5D,`5F^p$\"3csUI\\L))*e #F07$$\"3!=nm\"zRQb@F^p$\"3q9)4$4KI%=#F07$$\"3_,+](=>Y2%F^p$\"30J9BSv< O:F07$$\"3summ\"zXu9'F^p$\"3;I8[4eK*=*F^p7$$\"3#4+++]y))G)F^p$\"3NAixy m.:PF^p7$$\"3H++]i_QQ5F0$!3>f1Gi'4(HvFjt7$$\"3b++D\"y%3T7F0$!3Q/k\\2pZ SUF^p7$$\"3+++]P![hY\"F0$!3w&)*[8#3-]rF^p7$$\"3iKLL$Qx$o;F0$!3GD::,WE+ *)F^p7$$\"3cmm;z)Qjx\"F0$!3ModFIqv*\\*F^p7$$\"3Y+++v.I%)=F0$!3#\\f[x'f 8m)*F^p7$$\"3ML$ek`H@)>F0$!3#31G`Y1o***F^p7$$\"3?mm\"zpe*z?F0$!3y:ZFmg 1O**F^p7$$\"3oL$e9\"=\"p=#F0$!3#QkttuR1l*F^p7$$\"3;,++D\\'QH#F0$!33/!)oHOV#F07$$\"31,++]Qk\\RF0 $\"31EG%=966!GF07$$\"3%zm;/@-/1%F0$\"3S!*)*oosDXKF07$$\"3![LL3dg6<%F0$ \"3pL8MC#QRr$F07$$\"3K,+voTAqUF0$\"3'Rjwjx'F07$$\"35,+v=5s#y%F0 $\"3`VI>oi`VnF07$$\"3a+]P40O\"*[F0$\"3KkK<&fl*ftF07$$\"\"&F*F+-%'COLOU RG6&%$RGBG$\"#5!\"\"$F*F*F^bl-%+AXESLABELSG6%Q\"x6\"Q\"yFcbl-%%FONTG6% %&TIMESG%&ROMANG\"#:-%*AXESTICKSG6%%(DEFAULTGF^clFebl-%%VIEWG6$;F(Feal Fbcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 275 9 "P\370\355klad 5 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Zn\341zorn\354te graf funkce " }{TEXT 364 9 " f (x) = " } {XPPEDIT 257 1 "x^3/sqrt(x^2-1);" "6#*&%\"xG\"\"$-%%sqrtG6#,&*$F$\"\"# \"\"\"F,!\"\"F-" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"x G\"\"$,&*$)F$\"\"#\"\"\"F*F*!\"\"#F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f:=x->x^3/(x^2-1)^(1/2);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG %&arrowGF(*&9$\"\"$,&*$)F-\"\"#\"\"\"F3F3!\"\"#F4F2F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Defini\350n\355 obor, pr\371se\350\355ky \+ s osami:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(x^2-1>0,x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$,$%)infinityG!\" \"-%%OpenG6#F(-F$6$-F*6#\"\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "prusecik:=solve(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)prusecikG6%\"\"!F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f(-x):=f(-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-% \"fG6#,$%\"xG!\"\",$*&F(\"\"$,&*$)F(\"\"#\"\"\"F1F1F)#F)F0F)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Plat\355, f(-x) = - f(x), funkce je tedy lich\341." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Ur\350\355m e prvn\355 derivaci a p\370\355padn\351 extr\351my:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "df:=diff(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfG,&*&%\"xG\"\"#,&*$)F'F(\"\"\"F,F,!\"\"#F-F(\"\"$* &F'\"\"%F)#!\"$F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "df:= factor(df);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfG*(%\"xG\"\"#,&*$) F&F'\"\"\"F'\"\"$!\"\"F+*&,&F&F+F+F-F+,&F&F+F+F+F+#!\"$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "e:=solve(df,x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"eG6&\"\"!F&,$*$-%%sqrtG6#\"\"'\"\"\"#F-\"\"#,$F(# !\"\"F/" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(df>0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$,$%)infinityG!\"\"-%%OpenG6#,$*$-%%sqrtG6#\"\"'\"\" \"#F(\"\"#-F$6$-F*6#,$F-#F2F4F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(df<0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6 $-%%OpenG6#,$*$-%%sqrtG6#\"\"'\"\"\"#!\"\"\"\"#-F'6#F1-F$6$-F'6#F/-F'6 #,$F*#F/F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Uv\354dom\355me si, \236e bod x = 0 nepat\370\355 do defini\350n\355ho oboru, extr\351my \+ funkce tedy jsou:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "E1:=si mplify([e[3],f(e[3])]);E2:=simplify([e[4],f(e[4])]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E1G7$,$*$-%%sqrtG6#\"\"'\"\"\"#F,\"\"#,$*$-F)6#\" \"$F,#F3F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E2G7$,$*$-%%sqrtG6#\" \"'\"\"\"#!\"\"\"\"#,$*$-F)6#\"\"$F,#!\"$F/" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 70 "Nyn\355 vypo\350teme druhou derivaci a zjist\355me, zda existuj\355 inflexn\355 body:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dff:=diff(df,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dffG,(* (%\"xG\"\"\",&*$)F'\"\"#F(F,\"\"$!\"\"F(*&,&F'F(F(F.F(,&F'F(F(F(F(#!\" $F,F,*(\"\"%F(F'F-F/F2F(**F-F(F'F-F)F(F/#!\"&F,F." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "dff:=factor(dff);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dffG*(%\"xG\"\"\",(*$)F&\"\"%F'\"\"#*&\"\"&F')F&F,F' !\"\"\"\"'F'F'*&,&F&F'F'F0F',&F&F'F'F'F'#!\"&F," }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "solve(dff,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"!,$*$-%%sqrtG6#,&\"\"&\"\"\"*&^#F+F+-F'6#\"#BF+F+F+#!\"\"\"\"#, $F%#F+F3,$*$-F'6#,&F*F+*&^#F2F+F.F+F+F+F1,$F7F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "P\370i hled\341n\355 inflexn\355ch bod\371 zji\232 \235ujeme, \236e jedin\375 re\341ln\375 bod le\236\355 mimo defini\350 n\355 obor. Zam\354\370\355me se tedy pouze na konvexitu a konkavitu. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(dff>0,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*RealRangeG6$-%%OpenG6#\"\"\"%)infin ityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(dff<0,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*RealRangeG6$,$%)infinityG!\"\"-%%Op enG6#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Asymptoty" }}{PARA 0 " " 0 "" {TEXT -1 14 "a) se sm\354rnic\355" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "k1:=limit(f(x)/x,x=infinity);k2:=limit(f(x)/x,x=-infi nity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G%)infinityG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 32 "neexistuj\355 asymptoty se sm\354rnic\355" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "b) bez sm\354rnice" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "limit(f(x),x=-1,left);limit(f(x),x=1,right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 " funkce m\341 asymptotu bez sm\354rnice x = -1 zleva a x = 1 zprava" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Graf funkce:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "p1:=plot(f(x),x=-5..5, y=-5..5,discont=true," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "labelfont=[TIMES,ROMAN,15]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "p2:=plot([-1,t,t=-infinity.. infinity],x=-5..5,y=-5..5,linestyle=3," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "labelfont=[TIMES,ROMAN,15]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "p3:=plot([1,t,t=-infinity..infinity],x=-5..5,y=-5..5, linestyle=3," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "labelfont=[TIMES,RO MAN,15]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(plots,dis play):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display(\{p1,p2,p 3\},axesfont=[TIMES,ROMAN,15]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6%7S7$$!\"\"\"\"!$!2-+++!********\"$\"H7$ F($!3GZ)H<'[v(e%!#;7$F($!3+VB\"F17$F($!3tLXu3%z,`*!#<7$F($!3qo-2iHi;!)F>7$F($!3G_,'H;&\\%)oF>7$F($ 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"P\370\355klad 7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "V tomto p\370\355klad\354 je \+ uk\341zka naprogramovan\351 procedury, kter\341 ur\350\355 geometrick \375 v\375znam 1. a 2. derivace funkce." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "changes:=proc(f::anything,n::nonnegint)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "local sing,a,i,s,nezn,j,k,vyr,mn,m;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sing:=[singular(f)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a:=\{\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "m:=\{\}; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "for i from 1 to nops(sing) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "nezn:=indets(op(2,op(1,op(i,sing) )));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "if nops(nezn) = 0 then" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "if abs(op(2,op(1,op(i,sing)))) < in finity then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a:=a union op(i,sing );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 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1 0 43 "printf(\"Lokalni minimum v bode %f\\n\",s[l] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "if q>0 t hen" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "printf(\"Rostouci na interva lu (%f, %f)\\n\",s[l],s[l+1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "el se" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "printf(\"Klesajici na interva lu (%f, %f)\\n\",s[l],s[l+1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "en d if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=q;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "q:=evalf (subs(x=s[nops(s)]+1,g));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "if p*q < 0 and nops(\{s[nops(s)]\} intersect m) = 0 then" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "if p>0 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "p rintf(\"Lokalni maximum v bode %f\\n\",s[nops(s)]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "printf( \"Lokalni minimum v bode 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"printf(\"Konkavni na definicnim oboru\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "p:=evalf(subs(x=s[1]-1,h));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "if p>0 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "printf(\"Konvexn i do bodu %f\\n\",s[1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "else" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "printf(\"Konkavni do bodu %f\\n\",s [1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "for l from 1 to nops(s)-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "q:=evalf(subs(x=(s[l]+s[l+1])/2,h));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "if p*q < 0 and nops(\{s[l]\} intersect m) = 0 th en" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "printf(\"Inflexe v bode %f\\n \",s[l]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 "if q>0 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "printf(\"Konvexni na intervalu (%f, 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Uka \236me na n\354kolika p\370\355kladech, jak se tento rozvoj po\350\355 t\341 pomoc\355 Maple" }}{PARA 0 "" 0 "" {TEXT -1 43 "P\370\355kaz, kt er\375 pou\236ijeme, m\341 tuto strukturu:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 284 27 "taylor ( v\375raz, rov/jm, n);" }}{PARA 0 "" 0 "" {TEXT -1 4 "kde " }{TEXT 285 9 "v\375raz " }{TEXT -1 19 "je rozv \355jen\341 funkce" }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{TEXT 286 8 "rov/jm " }{TEXT -1 21 "je rovnice tvaru x =" }{TEXT 287 2 " " } {TEXT -1 22 "a nebo jm\351no prom\354nn\351" }{TEXT 288 2 " " }} {PARA 0 "" 0 "" {TEXT -1 7 " " }{TEXT 289 2 "n " }{TEXT -1 14 "j e celo\350\355seln\341" }{TEXT 290 1 " " }{TEXT -1 17 " kladn\341 kons tanta" }}{PARA 20 "" 0 "" {TEXT -1 0 "" }{TEXT 282 0 "" }{TEXT 283 8 " P\370\355klad " }}{PARA 0 "" 0 "" {TEXT -1 41 "Najd\354te Taylor\371v \+ polynom pro tyto funkce:" }}{PARA 258 "" 0 "" {TEXT -1 7 "a) arcs" } {TEXT 383 2 "in" }{XPPEDIT 384 1 "(x+1)/(x-1);" "6#*&,&%\"xG\"\"\"F&F& F&,&F%F&F&!\"\"F(" }{TEXT 385 9 " v okol\355 " }{TEXT -1 17 "po\350 \341tku, stupe\362 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "taylor(arcsin((x+1)/(x-1)),x=0,3);" }} {PARA 8 "" 1 "" {TEXT -1 54 "Error, does not have a taylor expansion, \+ try series()\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Vid\355me, \+ \236e Taylor\371v polynom zde nelze naj\355t - prav\351 okol\355 bodu \+ x = 0 toti\236 nepat\370\355 do defini\350n\355ho oboru:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "solve((x+1)/(x-1)>=-1 and (x+1)/(x- 1)<=1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*RealRangeG6$,$%)infini tyG!\"\"\"\"!" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Vybereme x = -1, kter\375 pat\370\355 do defini\350n \355ho oboru:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "taylor(arcsin((x+1)/(x-1)),x=-1,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+),&%\"xG\"\"\"F&F&#!\"\"\"\"#F&#F(\"\"%F)-%\"O G6#F&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 " " 0 "" {TEXT 361 35 "b) cos(x) v okol\355 po\350\341tku, stupe\362 6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "taylor(cos(x),x,6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"xG\"\"\"\"\"!#!\"\"\"\"#F)#F%\"#C \"\"%-%\"OG6#F%\"\"'" }}}{PARA 258 "" 0 "" {TEXT -1 3 "c) " }{XPPEDIT 381 1 "e^x;" "6#)%\"eG%\"xG" }{TEXT 382 3 ", p" }{TEXT -1 27 "ro x = \+ 0 a x = 1, stupe\362 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "taylor(exp(x),x,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"xG\"\"\"\"\"!F%F%#F%\"\"#F(#F%\"\"'\"\"$-%\"O G6#F%\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "taylor(exp(x) ,x=1,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-,&%\"xG\"\"\"F&!\"\"-%$e xpG6#F&\"\"!F(F&,$F(#F&\"\"#F.,$F(#F&\"\"'\"\"$-%\"OGF*\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}}{MARK "4" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }