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"Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 269 1 {CSTYLE " " -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 270 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 42 "Primitivn\355 funkce,neur \350it\375 integr\341l - \372vod" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 259 "" 0 "" {TEXT 2 0 "" }{TEXT -1 12 "P\370\355kaz, kter" } {TEXT 30 0 "" }{TEXT -1 56 "\375 vypo\350\355t\341 ner\350it\375 integ r\341l (primitivn\355 funkci) m\341 tvar:" }}{PARA 260 "" 0 "" {TEXT 263 3 "int" }{TEXT -1 8 "(v\375raz, " }{TEXT 262 1 "x" }{TEXT -1 2 "); " }}{PARA 0 "" 0 "" {TEXT -1 35 "kde jednotliv\351 parametry znamenaj \355:" }}{PARA 0 "" 0 "" {TEXT -1 49 "v\375raz - v\375raz (integrand) , kter\375 bude integrov\341n" }}{PARA 0 "" 0 "" {TEXT -1 43 "x \+ - prom\354nn\341, podle n\355\236 se integruje" }}{PARA 0 "" 0 "" {TEXT -1 50 "Pro v\375pis neur\350it\351ho integr\341lu se pou\236\355 v\341 p\370\355kaz " }{TEXT 264 3 "Int" }{TEXT 291 8 "(v\375raz, " } {TEXT 265 1 "x" }{TEXT 292 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "P\370i prov\341d\354n\355 p\370\355kazu \+ " }{TEXT 266 3 "int" }{TEXT -1 201 " zkus\355 Maple spo\350\355tat int egr\341l, ale na rozd\355l od v\375po\350tu derivace nen\355 zaru\350e na \372sp\354\232nost v\375po\350tu. V takov\351m p\370\355pad\354 Map le odpov\355 bu\357 stejn\375m nebo zjednodu\232en\375m v\375razem, kt er\375 mu byl zad\341n k integraci." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 258 "" 0 "" {TEXT 293 9 "P\370\355klady:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Uk\341\236eme si, \236e f unkce F(x) = " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 43 " j e primitivn\355 funkc\355 k funkci f(x) = 2x v " }{TEXT 267 1 "R" } {TEXT -1 9 ", nebo\235 (" }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" } {TEXT -1 8 ")' = 2x." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Int(2*x,x)=int(2*x,x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$%\"xG\"\"#F(*$)F(F) \"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Diff(x^2,x)=diff( x^2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*$)%\"xG\"\"#\" \"\"F),$F)F*" }}}{EXCHG {PARA 257 "> " 1 "" {MPLTEXT 1 0 12 "Int(f(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%\"fG6#%\"xGF)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "1. zp\371sob v\375po\350tu integr \341lu:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "int(x+sin(x),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"#\"\"\"#F(F'-%$cosG6#F&! \"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "2. zp\371sob:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "F:=Int(x+sin(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%$IntG6$,&%\"xG\"\"\"-%$sinG6#F)F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"#\"\"\"#F(F'-%$cosG6#F&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "3. zp\371sob (modifikace 2. zp\371sobu): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "value(Int(x+sin(x),x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"#\"\"\"#F(F'-%$cosG6#F&! \"\"" }}}{PARA 0 "" 0 "" {TEXT -1 9 "Integr\341l " }{XPPEDIT 18 0 "Int (1/(x+exp(x)),x);" "6#-%$IntG6$*&\"\"\"F',&%\"xGF'-%$expG6#F)F'!\"\"F) " }{TEXT -1 34 " neum\355 Maple spo\350\355tat analyticky." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "int(1/(x+exp(x)),x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$intG6$*&\"\"\"F',&%\"xGF'-%$expG6#F)F'!\"\"F) " }}}{PARA 0 "" 0 "" {TEXT -1 114 "Jednoduch\375m programem lze z\355s kat z\341kladn\355 vzorce pro v\375po\350et neur\350it\375ch integr \341l\371 (bez uveden\355 defini\350n\355ch obor\371):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "f1:=0: f2:=1: f3:=x^n: f4:=1/x: f5 :=exp(x): f6:=a^x: f7:=sin(x): f8:=cos(x): f9:=1/cos(x)^2: f10:=1/sin( x)^2: f11:=1/(1+x^2): f12:=1/sqrt(1-x^2): f13:=1/sqrt(x^2+a):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to 13 do" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Int(f||i,x)=int(f||i,x)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$I ntG6$\"\"!%\"xGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$\"\"\"% \"xGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$)%\"xG%\"nGF(*&)F( ,&F)\"\"\"F-F-F-F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$* &\"\"\"F(%\"xG!\"\"F)-%#lnG6#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% $IntG6$-%$expG6#%\"xGF*F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6 $)%\"aG%\"xGF)*&-%#lnG6#F(!\"\"F'\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$sinG6#%\"xGF*,$-%$cosGF)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$cosG6#%\"xGF*-%$sinGF)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$)-%$cosG6#%\"xG\"\"#F(!\"\"F.*& F+F0-%$sinGF-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F (*$)-%$sinG6#%\"xG\"\"#F(!\"\"F.,$*&F+F0-%$cosGF-F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&F(F(*$)%\"xG\"\"#F(F(!\"\"F, -%'arctanG6#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F( *$-%%sqrtG6#,&F(F(*$)%\"xG\"\"#F(!\"\"F(F2F0-%'arcsinG6#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$-%%sqrtG6#,&*$)%\"xG\" \"#F(F(%\"aGF(F(!\"\"F0-%#lnG6#,&F0F(*$-F+6#F-F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 294 65 "Integra\350n\355 metoda po \350\341ste ch (per partes) - pro neur\350it\375 integr\341l" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Plat\355 vzorec" } {TEXT 460 2 " " }{XPPEDIT 461 1 "int(`v\264`*u,x);" "6#-%$intG6$*&%#v |_vG\"\"\"%\"uGF(%\"xG" }{TEXT 462 4 " = " }{XPPEDIT 463 0 "u*v-int(` u\264`*v,x);" "6#,&*&%\"uG\"\"\"%\"vGF&F&-%$intG6$*&%#u|_vGF&F'F&%\"xG !\"\"" }{TEXT 464 4 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 " V Maplu se u \236\355v\341 funkce " }{TEXT 268 16 "intparts(g, u), " }{TEXT -1 34 " kter\341 je ulo\236ena v bal\355ku student." }}{PARA 0 "" 0 "" {TEXT -1 47 " Jednotliv\351 parametry znamenaj\355 n\341sleduj\355c\355: \+ " }}{PARA 259 "" 0 "" {TEXT -1 4 " g" }{TEXT 23 3 " - " }{TEXT -1 136 "v\375raz reprezentuj\355c\355 dan\375 integr\341l ve tvaru Int(f( x), x)\n u - sou\350\341st integrandu, kter\341 se bude derivovat (viz v\375\232e uveden\375 vzorec)" }}{PARA 0 "" 0 "" {TEXT -1 16 " \+ P\370\355kaz vol\341me" }}{PARA 0 "" 0 "" {TEXT -1 52 " student[int parts](g,u); nebo with(student):" }}{PARA 0 "" 0 "" {TEXT -1 82 " intparts(g,u); \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 9 "P\370\355klady:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 421 10 "P\370\355klad 1." }}{PARA 0 "" 0 "" {TEXT 270 19 "Vypo\350t\354te integr\341l " }{XPPEDIT 18 0 "Int(x *cos(x),x)" "6#-%$IntG6$*&%\"xG\"\"\"-%$cosG6#F'F(F'" }{TEXT -1 28 " p ou\236it\355m metody per-partes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 420 13 "\330e\232en\355: " }{TEXT -1 64 "Po lo\236me u = x, u' = 1, v' = cos(x), v = sin(x). Pak dost\341v\341me :" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Int(x*cos(x),x);" "6#-%$IntG6$*&% \"xG\"\"\"-%$cosG6#F'F(F'" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x*sin(x)- Int(sin(x),x);" "6#,&*&%\"xG\"\"\"-%$sinG6#F%F&F&-%$IntG6$-F(6#F%F%!\" \"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x*sin(x)+cos(x);" "6#,&*&%\"xG\" \"\"-%$sinG6#F%F&F&-%$cosG6#F%F&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "vyr:=intparts(Int(x*c os(x), x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$vyrG,&*&%\"xG\"\" \"-%$sinG6#F'F(F(-%$IntG6$F)F'!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 37 "T ento integr\341l dok\341\236eme ji\236 vypo\350\355tat" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "value(vyr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"-%$sinG6#F%F&F&-%$cosGF)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 10 "P \370\355klad 2." }}{PARA 0 "" 0 "" {TEXT 271 18 "Vypo\350t\354te integ r\341l" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(x)*ln(x),x);" "6#-%$I ntG6$*&-%%sqrtG6#%\"xG\"\"\"-%#lnG6#F*F+F*" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "vyr:=intparts(Int(sqrt(x)*ln(x),x), ln(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$vyrG,&*&-%#lnG6#%\"xG\" \"\")F*#\"\"$\"\"#F+#F/F.-%$IntG6$,$*$-%%sqrtG6#F*F+F0F*!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "value(vyr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%#lnG6#%\"xG\"\"\")F(#\"\"$\"\"#F)#F-F,*&#\" \"%\"\"*F)*$F*F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 42 "Integrace substituc\355 pro neu r\350it\375 integr\341l" }}{PARA 0 "" 0 "" {TEXT -1 40 "Pro substituci se v Maplu u\236\355v\341 p\370\355kazu " }{TEXT 358 9 "changevar" } {TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 18 "student[changevar]" } {TEXT 30 1 " " }{TEXT 362 6 "(f, v)" }{TEXT -1 4 "; " }{TEXT 363 6 " nebo " }{TEXT -1 15 " with(student):" }}{PARA 0 "" 0 "" {TEXT -1 75 " changevar(f, v); \+ kde " }{TEXT 364 1 "f" }{TEXT -1 26 " je vyj\341d\370en\355 substituc e " }{TEXT 365 11 "f(x) = g(t)" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 85 " \+ " }{TEXT 366 1 "v" }{TEXT -1 210 " v\375r az typu Int(f(x),x ) \+ \+ nebo changevar(f, v, t); kde " } {TEXT 367 1 "t" }{TEXT -1 28 " je nov\341 integra\350n\355 prom\354nn \341" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 359 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 360 9 "P\370\355klady:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 10 "P\370 \355klad 1." }}{PARA 257 "" 1 "" {TEXT -1 51 "M\354jme vypo\350\355tat pomoc\355 substituce integr\341l : " }{XPPEDIT 18 0 "Int(sin(x) ^2*cos(x)^5,x);" "6#-%$IntG6$*&-%$sinG6#%\"xG\"\"#-%$cosG6#F*\"\"&F*" }}{PARA 0 "" 0 "" {TEXT 361 8 "\330e\232en\355: " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "P\370i v\375po\350tu pou\236ijeme substituci t = sin(x), \+ dt = cos(x)dx " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "expr:=changevar( sin(x)=t, Int((sin(x)^2*cos(x)^5),x), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG-%$IntG6$*&)%\"tG\"\"#\"\"\"),&F,F,*$F)F,!\"\"F +F,F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Spo\350\355t\341n\355 in tegr\341lu: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "expr1:=value(expr); " }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expr1G,(*$)%\"t G\"\"(\"\"\"#F*F)*&#\"\"#\"\"&F**$)F(F/F*F*!\"\"*&#F*\"\"$F*)F(F5F*F* " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "N\341vrat k p\371vodn\355 pro m\354nn\351 x:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "changevar(t=sin(x ), expr1, x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)-%$sinG6#%\"xG \"\"(\"\"\"#F+F**&#\"\"#\"\"&F+*$)F&F0F+F+!\"\"*&#F+\"\"$F+)F&F6F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 54 "\330adu \372loh lze \370e\232it kombinac\355 obou integra \350n\355ch metod." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 " " {TEXT -1 10 "P\370\355klad 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 10 "Vypo\350t\354te " }{XPPEDIT 18 0 "Int(arc sin(x),x);" "6#-%$IntG6$-%'arcsinG6#%\"xGF)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 422 8 "\330e\232en\355: " }{TEXT -1 30 " Polo\236me u = arcsin(x), u' = " }{XPPEDIT 18 0 "1/sqrt(1-x^2);" "6#*&\"\"\"F$-%%s qrtG6#,&F$F$*$%\"xG\"\"#!\"\"F," }{TEXT -1 18 " , v' = 1, v' = x." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 257 "> " 1 "" {MPLTEXT 1 0 44 "expr:=intparts(Int(arcsin( x),x), arcsin(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,&*&-%'a rcsinG6#%\"xG\"\"\"F*F+F+-%$IntG6$*&,&F+F+*$)F*\"\"#F+!\"\"#F4F3F*F+F* F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "expr1:=changevar(1-x^ 2=t,expr,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expr1G,&*&-%'arcsin G6#%\"xG\"\"\"F*F+F+-%$IntG6$,$*&F+F+*$-%%sqrtG6#%\"tGF+!\"\"#F6\"\"#F 5F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "expr2:=value(expr1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expr2G,&*&-%'arcsinG6#%\"xG\"\" \"F*F+F+*$-%%sqrtG6#%\"tGF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "changevar(t=1-x^2, expr2, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,&*&-%'arcsinG6#%\"xG\"\"\"F(F)F)*$-%%sqrtG6#,&F)F)*$)F(\"\"#F)!\"\"F )F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "Rozklad na parci \341ln\355 zlomky" }}{PARA 0 "" 0 "" {TEXT -1 24 "Ryz\355 racion\341ln \355 funkci " }{XPPEDIT 18 0 "f(x) = P[n]/Q[m];" "6#/-%\"fG6#%\"xG*&& %\"PG6#%\"nG\"\"\"&%\"QG6#%\"mG!\"\"" }{TEXT -1 60 " , kde n " 0 "" {MPLTEXT 1 0 28 "convert(f,parfrac,promenna );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"fG" }}}{PARA 0 "" 0 "" {TEXT -1 28 "kde f je racion\341ln\355 funkce" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 423 9 "P\370\355klady:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 10 "P\370\355klad 1." }} {PARA 0 "" 0 "" {TEXT -1 21 "\010Vypo\350t\354te integr\341l " } {XPPEDIT 18 0 "int(5/(x^3+8),x)" "6#-%$intG6$*&\"\"&\"\"\",&*$%\"xG\" \"$F(\"\")F(!\"\"F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 59 "Rozlo\236\355me integrovanou funkci na sou\350et parci \341ln\355ch zlomk\371:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "convert(5/(x^3+8),parfrac,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%,&%\"xGF%\"\"#F%!\"\"#\"\" &\"#7*&#F+F,F%*&,&!\"%F%F'F%F%,(*$)F'F(F%F%*&F(F%F'F%F)\"\"%F%F)F%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%#lnG6#,&%\"xG\"\"\"\"\"#F)#\"\"&\"#7*&#F,\"#CF )-F%6#,(*$)F(F*F)F)*&F*F)F(F)!\"\"\"\"%F)F)F7*(F+F)-%%sqrtG6#\"\"$F)-% 'arctanG6#,$*&,&F(F*F*F7F)F:F)#F)\"\"'F)F)" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 257 "" 1 "" {TEXT -1 39 "Tento integr\341l lze vypo \350\355tat tak\351 p\370\355mo" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Int(5/(x^3+8),x)=int(5/(x^3+ 8),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$*&\"\"\"F),&*$)% \"xG\"\"$F)F)\"\")F)!\"\"\"\"&F-,(-%#lnG6#,&F-F)\"\"#F)#F1\"#7*&#F1\"# CF)-F46#,(*$)F-F7F)F)*&F7F)F-F)F0\"\"%F)F)F0*(F8F)-%%sqrtG6#F.F)-%'arc tanG6#,$*&,&F-F7F7F0F)FEF)#F)\"\"'F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 424 10 "P\370\355klad 2." }}{PARA 0 "" 0 "" {TEXT -1 11 "Vypo\350t\354te " }{XPPEDIT 18 0 "int((x-1)/(x^5+x^3-12* x-2*x^2-8),x);" "6#-%$intG6$*&,&%\"xG\"\"\"F)!\"\"F),,*$F(\"\"&F)*$F( \"\"$F)*&\"#7F)F(F)F**&\"\"#F)*$F(F3F)F*\"\")F*F*F(" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "convert((x-1)/(x^5+x^3-12*x-2*x^2-8),parfra c,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"\"F%,&%\"xGF%\"\"#!\" \"F)#F%\"#s*&#F(\"#:F%,&F'F%F%F%!\"#F%*&#\"\"(\"$D#F%F/F)F%*&#F%\"$+#F %*&,&\"#EF%*&\"\"*F%F'F%F%F%,&*$)F'F(F%F%\"\"%F%F)F%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,-%#lnG6#,&%\"xG\"\"\"\"\"#!\"\"#F)\"#s*&#F*\"#:F)*&F) F),&F(F)F)F)F+F)F+*&#\"\"(\"$D#F)-F%6#F2F)F)*&#\"\"*\"$+%F)-F%6#,&*$)F (F*F)F)\"\"%F)F)F+*&#\"#8\"$+#F)-%'arctanG6#,$F(#F)F*F)F+" }}}{PARA 0 "" 0 "" {TEXT -1 24 "M\371\236eme spo\350\355tat i p\370\355mo:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Int((x-1)/(x^5+x^3-12*x-2*x^ 2-8),x)=int((x-1)/(x^5+x^3-12*x-2*x^2-8),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,&%\"xG\"\"\"F*!\"\"F*,,*$)F)\"\"&F*F**$)F) \"\"$F*F**&\"#7F*F)F*F+*&\"\"#F*)F)F6F*F+\"\")F+F+F),,-%#lnG6#,&F)F*F6 F+#F*\"#s*&#F6\"#:F**&F*F*,&F)F*F*F*F+F*F+*&#\"\"(\"$D#F*-F;6#FDF*F**& #\"\"*\"$+%F*-F;6#,&*$F7F*F*\"\"%F*F*F+*&#\"#8\"$+#F*-%'arctanG6#,$F)# F*F6F*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "Ur\350it \375 integr\341l - \372vod" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "S" }{TEXT 465 6 "ymbol " }{XPPEDIT 466 1 "int(%?,% ?);" "6#-%$intG6$%#%?GF&" }{TEXT 467 19 " se naz\375v\341 integra\350 " }{TEXT -1 63 "n\355 znak a vznikl historicky nespr\341vn\375m psan \355m sou\350tov\351ho znaku " }{XPPEDIT 19 1 "Sigma;" "6#%&SigmaG" } {TEXT -1 46 ". Riemann\371v integr\341l je toti\236 definov\341n pomoc \355" }}{PARA 0 "" 0 "" {TEXT -1 132 "tzv. horn\355ch a doln\355ch sou \350t\371. Podrobnosti a tak\351 teoretickou podstatu problematiky lze nal\351zt v ka\236d\351 z\341kladn\355 u\350ebnici matematiky." }} {PARA 0 "" 0 "" {TEXT -1 66 "Pro ilustraci uv\341d\355me n\354kolik n \341zorn\375ch obr\341zk\371 - pomoc\355 procedur " }{TEXT 439 7 "lowb ox " }{TEXT -1 1 "a" }{TEXT 441 7 " upbox." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 426 0 "" }}{PARA 0 "" 0 "" {TEXT 440 15 "Animace pomoci " }{TEXT 442 5 "upbox" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "re start;with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name c hangecoords has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Nejd \370\355ve mus\355me proceduru " }{TEXT 445 5 "upbox" }{TEXT -1 13 " n adefinovat:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 848 "upbox:=proc( f,a,b,n)\nlocal a1, b1, k, x, i, j, J, halt, critpt, critset, M, A, B, C;\nglobal upsum;\nwith(plottools): \na1:=evalf(a):b1:=evalf(b): k:=( b1-a1)/n; for i from 0 to n do\nx[i]:=a1+i*k:od: critpt:=[];halt:=1;\n while halt=1 do if type(fsolve(D(f)(x),x,avoid=\{seq(x=critpt[i],i=1.. nops(critpt))\},x=a1..b1),numeric)=true then critpt:=sort([op(critpt), fsolve(D(f)(x),x,avoid=\{seq(x=critpt[i],i=1..nops(critpt))\},x=a1..b1 )]);\nelse critpt; halt:=0;\nfi; \nod;for i from 1 to n do\ncritset[i] :=\{x[i-1],x[i]\};\nfor j in critpt do\nif x[i-1]< j and j < x[i]\nthe n critset[i]:=\{op(critset[i]),j\};\nelse fi; M[i]:=max(op(map(f,crits et[i])));\nod; B[i]:=POLYGONS([[x[i-1],0],[x[i-1],M[i]],[x[i],M[i]],[ x[i],0]]);\n od:\nC:=PLOT(seq(B[i],i=1..n),COLOR(RGB,0,.8,.2)):\nA :=plot(f(x),x=a1..b1,scaling=constrained,colour=red,thickness=2):\nplo ts[display](\{A,C\});end:" }}}{PARA 0 "" 0 "" {TEXT -1 27 "V\375znam p arametr\371 procedury:" }}{PARA 0 "" 0 "" {TEXT -1 45 " f...... ...funkce, kterou integrujeme;" }}{PARA 0 "" 0 "" {TEXT -1 34 " \+ a, b....hranice intervalu;" }}{PARA 0 "" 0 "" {TEXT -1 86 " n. .......po\350et pravideln\375ch \350\341st\355, na kter\351 interval d \354l\355me (po\350et obd\351ln\355k\371);" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "f:=x->sin(x^2);a:=0;b:=sqrt(2*Pi);n:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$ sinG6#*$)9$\"\"#\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG*&-%%sqrtG6#\"\"#\"\"\" -F'6#%#PiGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 25 "Vykresleni jednoho grafu:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "upbox(f,a,b,n);" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, the name arrow has been redefined\n" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "60-%)POLYGONSG6$7&7$$\"\"!F)F)7$F($ \"+]>0zi!#67$$\"+u#Gm]#!#5F+7$F/F)-%&COLORG6&%$RGBGF)$\"\")!\"\"$\"\"# F9-F$6$7&F27$F/$\"+r))*o[#F17$$\"+[lD8]F1F@7$FCF)F3-F$6$7&FE7$FC$\"+[z Ee`F17$$\"+A[))>vF1FJ7$FMF)F3-F$6$7&FO7$FM$\"+a#zKW)F17$$\"+58l-5!\"*F T7$FWF)F3-F$6$7&FZ7$FW$\"\"\"F)7$$\"+PTJ`7FYFin7$F\\oF)F3-F$6$7&F^oF[o 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Polo\236me tedy u=" }{XPPEDIT 18 0 "e^x" "6#)%\"eG%\"xG" }{TEXT -1 13 " a v'=cos x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "I1 := \+ Int(exp(x)*cos(x),x=0..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I1 G-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$cosGF+F-/F,;\"\"!,$%#PiG#F-\"\"#" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Pou\236ijeme metodu per-partes. " }}{PARA 0 "" 0 "" {TEXT -1 10 "Nech\235 u = " }{XPPEDIT 18 0 "exp(x) " "6#-%$expG6#%\"xG" }{TEXT -1 7 " a v\264= " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "student[intparts](I1,exp(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#,$%#PiG#\"\"\"\"\"#F*-%$IntG6$*&-F%6#%\"xGF*-%$sinGF1F*/ F2;\"\"!F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 281 135 "Nyn\355 nastala situace, kdy vzn ikl\375 integr\341l nen\355 o nic jednodu\232\232\355, ne\236 integr \341l p\371vodn\355. Op\354t pou\236ijeme metodu per-partes a to u\236 it\355m u=" }{XPPEDIT 18 0 "e^x" "6#)%\"eG%\"xG" }{TEXT 282 1 "." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "student[intparts](%,exp(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#,$%#PiG#\"\"\"\"\"#F*F*!\" \"-%$IntG6$,$*&-F%6#%\"xGF*-%$cosGF3F*F,/F4;\"\"!F'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 83 "Pokud z integr\341lu vytkneme -1, vznikl\375 integr\341l je \372pln \354 shodn\375 s p\371vodn\354 zadan\375m. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "isolate(I1=expand(%),I1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$cosGF*F,/F+;\"\"!,$% #PiG#F,\"\"#,&*$-%%sqrtG6#-F)6#F3F,F4#F,F5!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Nyn\355 uk\341 \236eme, \236e Maple um\355 spo\350\355tat tento integr\341l p\370\355 mo, p\370edch\341zej\355c\355 v\375po\350ty popisuj\355 \"ru\350n\355 \" v\375po\350et" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Int(exp (x)*cos(x),x=0..Pi/2)=int(exp(x)*cos(x),x=0..Pi/2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$cosGF*F,/F+;\"\"!, $%#PiG#F,\"\"#,&-F)6#F2F4#F,F5!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 276 10 "P\370\355k lad 5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT 279 12 "Vypo\350\355tejte " } {XPPEDIT 18 0 "Int((x^2+4)^(1/2)/x,x = sqrt(5) .. 2*sqrt(3));" "6#-%$I ntG6$*&),&*$%\"xG\"\"#\"\"\"\"\"%F,*&F,F,F+!\"\"F,F*F//F*;-%%sqrtG6#\" \"&*&F+F,-F36#\"\"$F," }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(sqrt(x^2+4)/x,x=sqrt(5)..sqrt(12));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&*$)%\"xG\"\"#\"\"\"F,\"\"%F,#F,F+F* !\"\"/F*;*$-%%sqrtG6#\"\"&F,,$*$-F46#\"\"$F,F+" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "plot(sqrt(x^2+4)/x,x=sqrt(5)..sqrt(12));" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$ $\"3&****>'yz1OA!#<$\"3,^6?'yS;M\"F*7$$\"3QLP+;eVRit@7F*7$$\"3%)\\#GWj`l(GF*$\"3*H#y2 JH&z@\"F*7$$\"3s<2D)[A,!HF*$\"3jd\"o(=it97F*7$$\"3xK)og\\xb#HF*$\"3#G? &\\,)R8@\"F*7$$\"3y_#>*4](=&HF*$\"3KOjs,K\"z?\"F*7$$\"3o/gxHF*$\" 31[XyZ$RY?\"F*7$$\"3SU%\\$zT\\-IF*$\"3lani>La,7F*7$$\"3'>QMTqK,.$F*$\" 3C5RA+d=)>\"F*7$$\"3K*pEz9n\\0$F*$\"3myoM)QQ_>\"F*7$$\"3#)))3[\"F*7$$\"3WA)fvr5b5$F*$\"3;nl=\"F*7$$\"3!>)etUc\\cJF*$\"3U8KY%\\bG$F *$\"3i:(HI)Qqq6F*7$$\"32`*pOLz8J$F*$\"3-0:O\")>Co6F*7$$\"3\"4iLAM9^L$F *$\"3-TZahb-m6F*7$$\"3AE>aStJiLF*$\"3i?UB$yPN;\"F*7$$\"3?s\"=l-[mQ$F*$ \"3;rU1=*e8;\"F*7$$\"3KPNZ7'*e7MF*$\"3YJ*[Ud#3f6F*7$$\"3L0YI?!>uV$F*$ \"3Yqq7ey%p:\"F*7$$\"3;++%*4;5kMF*$\"3)yR7)Q0qa6F*-%'COLOURG6&%$RGBG$ \"#5!\"\"$\"\"!F_[lF^[l-%+AXESLABELSG6$Q\"x6\"Q!Fd[l-%%VIEWG6$;$\"+yz1 OA!\"*$\"+;;5kMF\\\\l%(DEFAULTG" 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 11 "Substituce " }{XPPEDIT 18 0 "u^2=x^2+4" "6#/*$%\"uG\"\"#,&*$%\"xGF &\"\"\"\"\"%F*" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "changevar(x^2+4=u^2,Int(sqrt(x^2+4)/x,x=sqrt(5)..sqrt(12)),u); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"uG\"\"#,&!\"%\"\"\"* $)F'F(F+F+!\"\"/F';*$-%%sqrtG6#\"\"*F+*$-F36#\"#;F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%#lnG6#\"\"$!\"\"*$-%%sqrtG6#\"#;\"\"\"F.-F%6#\"\"&F.*$-F+6#\"\" *F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "combine(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$-%%sqrtG6#\"#;\"\"\"F)*$-F&6#\"\"* F)!\"\"-%#lnG6##\"\"$\"\"&F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "P \370i \"ru\350n\355m\" v\375po\350tu je nutno vyu\236\355t rozklad na \+ parci\341ln\355 zlomky." }{TEXT 296 0 "" }{TEXT 297 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "convert(u^2/(u^2-4),parfrac,u);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*&F$F$,&%\"uGF$\"\"#!\"\"F)F $*&F$F$,&F'F$F(F$F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "in t(%,u=3..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$-%#lnG6#\"\" $!\"\"-F&6#\"\"&F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "combi ne(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$-%#lnG6##\"\"$\"\" &!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 277 10 "P\370\355klad 6." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 278 12 "Vypo\350\355tejte " }{XPPEDIT 18 0 "Int(arctan(1/x),x = 1 .. 2);" "6#-%$IntG6$-%'arctan G6#*&\"\"\"F*%\"xG!\"\"/F+;F*\"\"#" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "In:=Int(arctan(1/x),x=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#InG-%$IntG6$-%'arctanG6#*&\"\"\"F,%\"xG!\"\" /F-;F,\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(arctan( 1/x),x=-1..2,color=black);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7hn7$$!\" \"\"\"!$!3!G[uRj\")R&y!#=7$$!3[*****\\P&3Y$*F-$!3[Ty<`<'=>)F-7$$!3C++D cx6x()F-$!3pO-Gn^K/&)F-7$$!3b++]iTDP\")F-$!3e/9*Q\"3Ux))F-7$$!3A****\\ P\"\\J\\(F-$!3G%QJS'psGl+(*F-7$$!3w++D cexdiF-$!3=x*eT/Q;,\"!#<7$$!3j***\\i+#QUcF-$!3[#*\\qLu3d5FL7$$!3$**** \\i!3%f+&F-$!3t*=Tictm5\"FL7$$!3;++D\"oS:P%F-$!3c#e7!yine6FL7$$!3h**** *\\<#)*=PF-$!3,'>2nzZZ@\"FL7$$!3#*****\\(G3U9$F-$!3W\"eH,[ihE\"FL7$$!3 Y*****\\-\\r\\#F-$!3?v9s3g3E8FL7$$!3?+++vGVZ=F-$!3O(y<#*y7\")Q\"FL7$$! 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\"\"" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "student[intparts](In,arcta n(1/x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%'arctanG6##\"\"\"\"\"# F)*&#F(\"\"%F(%#PiGF(!\"\"-%$IntG6$,$*&F(F(*&%\"xGF(,&F(F(*&F(F(*$)F5F )F(F.F(F(F.F./F5;F(F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%'arctanG6##\"\"\" \"\"#F)*&#F(\"\"%F(%#PiGF(!\"\"-%$IntG6$*&%\"xGF(,&*$)F3F)F(F(F(F(F./F 3;F(F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%'arctanG6##\"\"\"\"\"#F)*&#F(\"\"% F(%#PiGF(!\"\"*&F'F(-%#lnG6#\"\"&F(F(*&#F(F)F(-F16#F)F(F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "combine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%'arctanG6##\"\"\"\"\"#F)*&#F(\"\"%F(%#PiGF(!\"\"*&F 'F(-%#lnG6##\"\"&F)F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "P\370\355klad 7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Vypo\350 teme integr\341l " }{XPPEDIT 18 0 "Int(x*sin(x),x = 0 .. Pi);" "6#-%$I ntG6$*&%\"xG\"\"\"-%$sinG6#F'F(/F';\"\"!%#PiG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 59 "\330e\232en\355: Polo\236me u = x, u' \+ = 1, v' = sin(x), v = -cos(x)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "c v:=intparts(Int(x*sin(x),x=0..Pi), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cvG,&%#PiG\"\"\"-%$IntG6$,$-%$cosG6#%\"xG!\"\"/F/;\"\"!F&F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "value(cv);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%#PiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "P\370\355klad 8." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 1 "" {TEXT -1 19 "Vypo \350teme integr\341l " }{XPPEDIT 18 0 "Int(arccos(x),x = 0 .. 1);" "6# -%$IntG6$-%'arccosG6#%\"xG/F);\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 44 "\330e\232en\355: Nejprve pou\236ijeme metodu per \+ partes" }}{PARA 0 "" 0 "" {TEXT -1 20 "u = arccos(x), u' = " } {XPPEDIT 18 0 "-1/sqrt(1-x^2);" "6#,$*&\"\"\"F%-%%sqrtG6#,&F%F%*$%\"xG \"\"#!\"\"F-F-" }{TEXT -1 16 ", v' = 1, v = x." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "vyr:=intparts(Int(arccos(x),x=0..1), arccos(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$vyrG,$-%$IntG6$,$*&,&\"\"\"F,*$ )%\"xG\"\"#F,!\"\"#F1F0F/F,F1/F/;\"\"!F,F1" }}}{PARA 0 "" 0 "" {TEXT -1 32 "Nyn\355 pou\236ijeme metodu substituce" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "t^2 = 1-x^2;" "6#/*$%\"tG\"\"#,&\"\"\"F(*$%\"xGF&!\"\" " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-t*dt = x*dx;" "6#/,$*&%\"tG\"\"\"% #dtGF'!\"\"*&%\"xGF'%#dxGF'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "vyr1:=changevar(1-x^2=t^2, vyr, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%vyr1G,$-%$IntG6$,$*&*$)%\"tG\"\"#\"\"\"#!\"\"F. F-F/F1/F-;\"\"!F/F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "valu e(vyr1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "P \370\355klad 9." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 317 19 "Vypo\350teme integr\341l " }{XPPEDIT 318 0 "Int((1+ln(x) )/x,x = 1 .. exp(1));" "6#-%$IntG6$*&,&\"\"\"F(-%#lnG6#%\"xGF(F(F,!\" \"/F,;F(-%$expG6#F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 " \330e\232en\355: pomoc\355 metody substituce ln(x) = t." }}{PARA 0 " " 0 "" {TEXT -1 79 "P\370i metod\354 substituce u ur\350it\351ho integ r\341lu je nutn\351 p\370epo\350\355tat meze integr\341lu: " } {XPPEDIT 19 1 "t[1];" "6#&%\"tG6#\"\"\"" }{TEXT -1 13 "= ln(1) = 0, " }{XPPEDIT 19 1 "t[2];" "6#&%\"tG6#\"\"#" }{TEXT -1 11 " = ln(e)=1." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "integral:=changevar(ln(x)=t, Int((1 +ln(x))/x, x=1..exp(1)), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)int egralG-%$IntG6$,&\"\"\"F)%\"tGF)/F*;\"\"!F)" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 44 "A tento integr\341l ji\236 lze jednodu\232e spo\350\355 tat." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "value(integral);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "U\236 it\355 integr\341ln\355ho po\350tu" }}{PARA 0 "" 0 "" {TEXT 371 23 "1. 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0Ffz-%%FONTG6%%&TIMESG%&ROMANG\"#6-%&STYLEG6#%%LINEG-%'COLOURG6&%$RGBG F*F*F*-%+AXESLABELSG6$Q!6\"F]gl-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 6 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {PARA 0 "" 0 "" {TEXT 379 0 "" }}{PARA 0 "" 0 "" {TEXT 380 197 "Obsah \+ dan\351 mno\236iny se tedy skl\341d\341 ze dvou \350\341st\355, na int ervalu [-2,0] plat\355 f2(x) >= f1(x) a na intervalu [0,2] plat\355 f1 (x) >= f2(x). Budeme proto po\350\355tat dva integr\341ly. Uka\236me d v\354 mo\236nosti v\375po\350tu:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Int(f2(x)-(f1(x)),x=-2..0)+I nt(f1(x)-f2(x),x=0..2):%= value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/,&-%$IntG6$,&*$)%\"xG\"\"$\"\"\"\"\"#*&\"\")F-F+F-!\"\"/F+;!\"#\"\"! F--F&6$,&F+F0*&F.F-F*F-F1/F+;F5F.F-\"#;" }}}{PARA 0 "" 0 "" {TEXT 381 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Plocha:=Int(f2(x)-( f(x)),x=-2..0)+Int(f1(x)-f2(x),x=0..2)=int(2*x^3-8*x,x = -2 .. 0)+int( 8*x-2*x^3,x = 0 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'PlochaG/, &-%$IntG6$,(*$)%\"xG\"\"$\"\"\"\"\"#*&\"\"&F/F-F/!\"\"\"\"%F3/F-;!\"# \"\"!F/-F(6$,&F-\"\")*&F0F/F,F/F3/F-;F8F0F/\"#;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 373 10 "P \370\355klad 6." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 383 47 "Vypo\350\355tejte obsah mno\236iny ohran i\350en\351 k\370ivkami " }{XPPEDIT 18 0 "y=sin(x)" "6#/%\"yG-%$sinG6 #%\"xG" }{TEXT 384 3 " , " }{XPPEDIT 18 0 "y=cos(x)" "6#/%\"yG-%$cosG6 #%\"xG" }{TEXT 388 3 " , " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT 389 5 " a " }{XPPEDIT 18 0 "x=Pi/2" "6#/%\"xG*&%#PiG\"\"\"\"\" #!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT 385 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT 387 36 "nadefinujeme funkce sinus a kosinus " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f1 := x -> sin(x); f2 := x -> cos(x ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G%$sinG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#f2G%$cosG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(f1(x)=f2(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG #\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 386 85 "Nyn\355 si funkc e nech\341me na p\370\355slu\232n\351m intervalu vykreslit, tedy na i ntervalu [0,Pi/2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "plot( \{sin(x),cos(x)\},x=0..Pi/2,color=black,style=line,font=[TIMES,ROMAN,1 1],scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6*-%'CURVESG6#7S7$$\"\"!F)F(7$$\"3NGK5j*))QU$!#>$\"3QsLeI +ABMF-7$$\"3DXXYUk*HS'F-$\"3]DS+L@i)R'F-7$$\"3=N68yVJ`(*F-$\"3_b&ehJey t*F-7$$\"3%3h%eRSe78!#=$\"3<&Q%f]#=)38F=7$$\"3k+,hQPB[;F=$\"3V-yj65yS; F=7$$\"3'H#*ed(RUf>F=$\"3q)*)ys&)4p%>F=7$$\"3kHX[TMk\"G#F=$\"3G3RBS#)* =E#F=7$$\"3Saq%HF=$\"3=7dd?+e /HF=7$$\"3)pq'4OKt)G$F=$\"3c0jx$Rp(HKF=7$$\"3#f/N#RTo*e$F=$\"3ML>uye38 NF=7$$\"3Ut79wN[GRF=$\"3V3zj5M@GQF=7$$\"3/VrOc6q%F=7$$ \"31\\,oR/A[_F=$\"3SMGBr+f5]F=7$$\"3uvW/X(*RG,:eF=7$$\"3.n 3p46^WlF=$\"33g0zi&Qs3'F=7$$\"3oHyF.C6noF=$\"3h+)=j,t*RjF=7$$\"3(*zx.5 Ir.sF=$\"3Qz3MFxj'f'F=7$$\"3'pz!QUu\"G^(F=$\"3$[8q$y.wDoF=7$$\"3A\"3@7 LGi%yF=$\"3wm47uKelqF=7$$\"3)3\"HIT&[D>)F=$\"3M(RvZssjI(F=7$$\"3gv)\\A I@S\\)F=$\"3C>#p\"po&)3vF=7$$\"3vn()=V,i>))F=$\"3%4zeG%\\()>xF=7$$\"3' G:[dg&*f:*F=$\"31()\\)y]!GHzF=7$$\"3sSt6rL2&[*F=$\"3g&H%*=Uja7)F=7$$\" 3mka(*HIZ.)*F=$\"3%3R7+w2pI)F=7$$\"3Em\"[l:+d,\"!#<$\"3Q?Zs+w\\)\\)F=7 $$\"3c!3XyEmu/\"F_u$\"3Ih!)R3tfh')F=7$$\"3sJ_*>P$Q\"3\"F_u$\"3VZPF_)*3 E))F=7$$\"3.M\"G%4t676F_u$\"3kW#3Hj\"Qm*)F=7$$\"3co*Q4iq'*z.@\"F_u$\"3Y (Q,up+vN*F=7$$\"34Q$[)>&*oU7F_u$\"3OZ!)zP7am%*F=7$$\"3I^lOFY^w7F_u$\"3 -w%)pSt5q&*F=7$$\"3`!)f6EA448F_u$\"3-KPDS[]f'*F=7$$\"3=;T7EvSU8F_u$\"3 B\"='y#[C.u*F=7$$\"3a_wiepWv8F_u$\"3$oY$>Q\"*z4)*F=7$$\"3U$obdw1eS\"F_ u$\"3#4Wi\"*p+U')*F=7$$\"3s$)o#3`-1W\"F_u$\"3aZ!3__n`\"**F=7$$\"3-#*)4 zFCK/mF=7$Fi s$\"3Wp$f4=xjN'F=7$F^t$\"3#Gr\"zlF:$4'F=7$Fct$\"3+kbUhf'*GeF=7$Fht$\"3 S\\$>&*yStc&F=7$F]u$\"3syV,H.Dq_F=7$Fcu$\"35cj1lEn(*\\F=7$Fhu$\"3DF8Py $y5q%F=7$F]v$\"3Cg[y,0kFWF=7$Fbv$\"3Fz_\\X[#R7%F=7$Fgv$\"3R](>U([*Q$QF =7$F\\w$\"3G8u?D(Qm_$F=7$Faw$\"38OldzS^AKF=7$Ffw$\"3gN^_O]_+HF=7$F[x$ \"3]EKPO+F(e#F=7$F`x$\"3s9%3md%3kAF=7$Fex$\"3G+Nc*p#4T>F=7$Fjx$\"3v3+* zL?Ck\"F=7$F_y$\"3#R-7)3IE)H\"F=7$Fdy$\"3q![!=de+\"*)*F-7$Fiy$\"3?\")3 +Q4@%e'F-7$F^z$\"370X.()zM7MF-7$Fcz$\"3&pI?mi'*[9$!#E-%%FONTG6%%&TIMES G%&ROMANG\"#6-%&STYLEG6#%%LINEG-%'COLOURG6&%$RGBGF)F)F)-%+AXESLABELSG6 $Q\"x6\"Q!F^el-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;F($\"+Fjzq:!\"*%(D EFAULTG" 1 6 0 1 10 0 2 6 1 4 1 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 390 140 " Na intervalu [0,Pi/4] plat\355 sin(x) >= cos(x) a na intervalu [Pi/4,Pi/2] cos(x) \+ >= sin(x). Budeme po\350\355tat sou\350et dvou p\370\355slu\232n\375ch integr\341l\371." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "PL1:= Int(f2(x)-f1(x),x=0..Pi/4)+Int(f1(x)-f2(x),x=Pi/4..Pi/2)=int(f2(x)-f1( x),x=0..Pi/4)+int(f1(x)-f2(x),x=Pi/4..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PL1G/,&-%$IntG6$,&-%$cosG6#%\"xG\"\"\"-%$sinGF-!\"\" /F.;\"\"!,$%#PiG#F/\"\"%F/-F(6$,&F0F/F+F2/F.;F6,$F7#F/\"\"#F/,&*$-%%sq rtG6#FAF/FAFAF2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 417 0 "" }{TEXT 418 0 "" }{TEXT 419 8 "Pozn\341mka" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 415 10 "Pozn\341mka: " } {TEXT 416 102 "pro v\354t\232\355 n\341zornost vymezen\355 plochy obra zce mezi dv\354ma k\370ivkami v rovin\354 m\371\236eme pou\236\355t tu to proceduru:" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plocha:=proc (f::anything,g::anything)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "local \+ p,q1,q2,s,a,r,t,b,c,i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a:=\{\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "t:=0.;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "s:=[];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "while typ e(t,'float')=true do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "t:=fsolve( f=g,x,avoid=a,maxsols=1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "if typ e(t,'float')=true then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "s:=[seq(s [i],i=1..nops(s)),t];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a:=a union \{x=t\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "s :=sort(s);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "b:=s[1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "c:=s[nops(s)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "t:=0.;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a:=\{seq(x=s[i],i= 1..nops(s))\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "while type(t,'flo at')=true do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "t:=fsolve(f,x=b..c, avoid=a,maxsols=1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "if type(t,'f loat')=true then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "s:=[seq(s[i],i= 1..nops(s)),t];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a:=a union \{x=t \};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "t:=0.;" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a:=\{seq(x=s[i],i=1..nops(s))\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "while type(t,'float')=true do" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "t:=fsolve(g,x=b..c,avoid=a,maxsols= 1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "if type(t,'float')=true then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "s:=[seq(s[i],i=1..nops(s)),t]; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a:=a union \{x=t\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "s:=sort(s);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "b:=s[1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "c:=s[nops(s)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "p:=[plot(f,x= b..c,thickness=2,color=black),plot(g,x=b..c,thickness=2,color=black)]; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "for i from 1 to nops(s)-1 do" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "if (evalf(subs(x=(s[i]+s[i+1])/2,f ))>=0) and (evalf(subs(x=(s[i]+s[i+1])/2,g))>=0) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "if (evalf(subs(x=(s[i]+s[i+1])/2,f))>evalf(subs( x=(s[i]+s[i+1])/2,g))) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "q1:= plot(g,x=s[i]..s[i+1],filled=true,color=white);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "q2:=plot(f,x=s[i]..s[i+1],filled=true,color=red);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "else q1:=plot(f,x=s[i]..s[i+1],fill ed=true,color=white);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "q2:=plot(g ,x=s[i]..s[i+1],filled=true,color=red);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "elif (evalf(subs( x=(s[i]+s[i+1])/2,f))<=0) and (evalf(subs(x=(s[i]+s[i+1])/2,g))<=0) th en" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "if (evalf(subs(x=(s[i]+s[i+1] )/2,f))>evalf(subs(x=(s[i]+s[i+1])/2,g))) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "q1:=plot(f,x=s[i]..s[i+1],filled=true,color=white);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "q2:=plot(g,x=s[i]..s[i+1],filled= true,color=red);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "q1:=plot(g,x=s[i]..s[i+1],filled=true,color =white);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "q2:=plot(f,x=s[i]..s[i+ 1],filled=true,color=red);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "q1:=plot(f,x=s[i]..s[i+1],filled=true,color=red);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "q2:=plot(g,x=s[i]..s[i+1],filled=tr ue,color=red);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p:=[seq(p[i],i=1..nops(p)),q1,q2];" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plots[display](p);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'plochaGf*6$'%\"fG%) anythingG'%\"gGF)6,%\"pG%#q1G%#q2G%\"sG%\"aG%\"rG%\"tG%\"bG%\"cG%\"iG6 \"F7C5>8(<\">8*$\"\"!F?>8'7\"?(F7\"\"\"FDF7/-%%typeG6$F=.%&floatG%%tru eGC$>F=-%'fsolveG6&/9$9%%\"xG/%&avoidGF:/%(maxsolsGFD@$FEC$>FA7$-%$seq G6$&FA6#8-/F\\o;FD-%%nopsG6#FAF=>F:-%&unionG6$F:<#/FTF=>FA-%%sortGFao> 8+&FA6#FD>8,&FA6#F_o>F=F>>F:<#-Fhn6$/FTFjnF]o?(F7FDFDF7FEC$>F=-FO6&FR/ FT;F\\pF`pFUFW@$FEC$>FAFfn>F:Fco>F=F>>F:Fep?(F7FDFDF7FEC$>F=-FO6&FSF^q FUFW@$FEC$>FAFfn>F:Fco>FAFio>F\\pF]p>F`pFap>8$7$-%%plotG6&FRF^q/%*thic knessG\"\"#/%&colorG%&blackG-Ffr6&FSF^qFhrF[s?(F\\oFDFD,&F_oFDFD!\"\"F KC$@'31F?-%&evalfG6#-%%subsG6$/FT,&Fjn#FDFjr*&F_tFD&FA6#,&F\\oFDFDFDFD FDFR1F?-Fhs6#-F[t6$F]tFS@%2FetFgsC$>8%-Ffr6&FS/FT;FjnFat/%'filledGFK/F \\s%&whiteG>8&-Ffr6&FRF`uFbu/F\\s%$redGC$>F]u-Ffr6&FRF`uFbuFdu>Fgu-Ffr 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"6#-%7rtable/ConstructColumnG6$\"\"!6#*&\"\"#\"\"\"%#PiGF*" }{TEXT -1 6 ", a>0." }}{PARA 0 "" 0 "" {TEXT 453 7 "\330e\232en\355:" }}{PARA 0 "" 0 "" {TEXT -1 83 "Obsah rovinn\351ho obrazce omezen\351ho obloukem \+ \350\341ry vyj\341d\370en\351 parametrick\375mi rovnicemi " }{XPPEDIT 18 0 "x = phi(t),y = psi(t);" "6$/%\"xG-%$phiG6#%\"tG/%\"yG-%$psiG6#F( " }{TEXT -1 22 " s mezemi parametru " }{XPPEDIT 18 0 "t[1],t[2];" "6 $&%\"tG6#\"\"\"&F$6#\"\"#" }{TEXT -1 3 " (" }{XPPEDIT 18 0 "t[1] < t[ 2];" "6#2&%\"tG6#\"\"\"&F%6#\"\"#" }{TEXT -1 22 ") je vyj\341d\370en \+ vztahem" }}{PARA 0 "" 0 "" {TEXT -1 99 " \+ \+ " }{XPPEDIT 18 0 "S = Int(psi(t)*abs(diff(phi(t),t)),t = t[1] .. t[2]) ;" "6#/%\"SG-%$IntG6$*&-%$psiG6#%\"tG\"\"\"-%$absG6#-%%diffG6$-%$phiG6 #F,F,F-/F,;&F,6#F-&F,6#\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 23 "Nakresl \355me danou funkci" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with( plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "p1:=plot([t-sin( t), 1-cos(t), t=0..2*Pi], scaling=constrained, thickness=3, color=blac k):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "p2:=plot([t-sin(t), \+ 1-cos(t), t=0..2*Pi], scaling=constrained, color=gray, filled=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(\{p1,p2\});" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7S7$ $\"\"!F)F(7$$\"3e;cW/cSxU!#@$\"3%R#3*QgiPO*!#?7$$\"3!e+5`Gi4z#F0$\"3Sn )o<,z>E$!#>7$$\"3=%\\Q;ZV:#)*F0$\"3!)\\'*Q6_89vF67$$\"3s^>#3I_\"zBF6$ \"3#)RJ?\"3HpM\"!#=7$$\"3]e/y%)H\\tYF6$\"3eW\\7nzu&4#FA7$$\"3=rfe)=K:y (F6$\"3[]1OJeUQA>'\\WFA$\"3?x[QxJ'[l)FA7$$\"3x'ph27ORr &FA$\"3B/u(p5(f+5!#<7$$\"3['zTJP\\z;(FA$\"3q\\T#fB[i8\"Fdo7$$\"3>&p6Ev 5Du)FA$\"3[:d.:#=YE\"Fdo7$$\"3/d\"GX^+;.\"Fdo$\"3%zV1J*3Jx8Fdo7$$\"3:( HTIn>dB\"Fdo$\"3u\\AOF:B/:Fdo7$$\"3^+4r2)>AU\"Fdo$\"3O1cjZ*[Rg\"Fdo7$$ \"37$*QIe8]`;Fdo$\"3]CknM;$*3b5msp=Fdo$\"3Ie_b?/U!z\"Fdo 7$$\"3]Y*4&3/k<@Fdo$\"3'z-\")RnIf'=Fdo7$$\"3!fRu^npAO#Fdo$\"3>XOJFO4B> 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" } {TEXT 322 12 "Objem t\354lesa" }{TEXT -1 102 " vznikl\351ho rotac\355 \+ k\370ivky, ohrani\350en\351ho obloukem \350\341ry y = f(x), osou x a p \370\355mkami x = a, x = b, a < b, " }{TEXT 323 11 "kolem osy x" } {TEXT -1 3 " je" }}{PARA 0 "" 0 "" {TEXT -1 100 " \+ \+ " }{XPPEDIT 18 0 "V = Pi*Int(f(x)^2,x = a .. b);" "6#/%\"VG*& %#PiG\"\"\"-%$IntG6$*$-%\"fG6#%\"xG\"\"#/F/;%\"aG%\"bGF'" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "B. " }{TEXT 325 12 "Objem t\354lesa" }{TEXT -1 102 " vznikl\351ho rotac\355 k\370ivky, oh rani\350en\351ho obloukem \350\341ry y = f(x), osou y a p\370\355mkami y = a, y = b, a < b, " }{TEXT 324 11 "kolem osy y" }{TEXT -1 3 " je" }}{PARA 0 "" 0 "" {TEXT -1 96 ". \+ " }{XPPEDIT 18 0 "V = Pi*Int(f(y)^2,y = a .. b);" "6#/%\"VG*&%#PiG\"\"\"-%$IntG6$* $-%\"fG6#%\"yG\"\"#/F/;%\"aG%\"bGF'" }{TEXT -1 1 "." }}}{PARA 13 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 326 0 "" }{TEXT -1 77 "Uka\236me pro ilus traci n\354kolik obr\341zk\371 t\354les, kter\341 vznikaj\355 rotac \355 dan\351 k\370ivky" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "w ith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "tubeplot([t, 0,0],t=0..Pi,radius=sin(t),tubepoints=20);" }}{PARA 0 "" 0 "" {TEXT -1 101 "M\371\236eme tak\351 vyu\236\355t kr\341tk\351 procedury, kter \341 animac\355 ukazuje rotaci k\370ivky p\370i vzniku rota\350n\355ho t\354lesa:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6#-%%MESHG6#7T767%$\"\"!F*F)F)F(F(F (F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(767%$\"3)Qna(yNT6k!#>$\"3e&Gr!)*>-2kF /F)7%F-$\"3#H2JP(=()fgF/$\"3uwk&>kc.3#F/7%F-$\"3)R(=F.1/c]F/$\"3BSxphV FNRF/7%F-$\"3(3'\\*3))3V]$F/$\"37()eFUSuj`F/7%F-$\"3EY$=j\"4$Gd\"F/$\" 3/W%4)G)o4@'F/7%F-$!3!Qhv-$o(3H&!#?$\"3O_/jp(Q^Q'F/7%F-$!3chhFA9ntDF/$ \"39ue7\\)zt'eF/7%F-$!3*[T%HGzNRVF/$\"3>*y*=!G*z8ZF/7%F-$!3]Ziqqw![j&F /$\"3;6K6jF/$\"35$>#pg6ca5F/7%F-Ffn$!3s\">#pg 6ca5F/7%F-$!3e\\iqqw![j&F/$!3q2K6kc.3#F/7%F-F0$!3/N4S'zn#p:!#M767%$\"3wM4v:FG#G\"!#=$\"3%)e]%ohr(y7F]r F)7%F[r$\"3^a%4MC%[47F]r$\"3iagB_Y;_TF/7%F[r$\"3jf(=^[I\"45F]r$\"3mh&H p$yPayF/7%F[r$\"33Gk#\\!y@%*pF/$\"3Qdi(4tW02\"F]r7%F[r$\"3ln..L()>RJF/ $\"3]n_WX:kR7F]r7%F[r$!3a5mFJ7+c5F/$\"3e_[WjRSu7F]r7%F[r$!32-uiwqwO^F/ $\"3)fE!fP\\1r6F]r7%F[r$!3[7@2/X)3m)F/$\"3t/0.[&G#3%*F/7%F[r$!3*46.32Y Y7\"F]r$\"31%4sK]I![&G#3%*F/7%F[r$!3@(RFm2nn8&F/$!3Em-fP\\1r6F]r7%F[r$!3E;mFJ 7+c5F/$!3e_[WjRSu7F]r7%F[r$\"3Sh..L()>RJF/$!3kn_WX:kR7F]r7%F[r$\"3#RUE \\!y@%*pF/$!3!yDw4tW02\"F]r7%F[r$\"30g(=^[I\"45F]r$!35c&Hp$yPayF/7%F[r $\"3Pa%4MC%[47F]r$!33egB_Y;_TF/7%F[rF^r$!3pr&>[8(3KJFhq767%$\"3;-kitSU B>F]r$\"3C@P,(G'e6>F]rF)7%F]x$\"3pvgDp7,3=F]r$\"3WpIKF0\"p?'F/7%F]x$\" 3no\"*Gw,^3:F]r$\"3gnq)))f?T<\"F]r7%F]x$\"3#QCu()fQb/\"F]r$\"39?UzffJ+ ;F]r7%F]x$\"3!*\\Bz3pm#p%F/$\"3_yq*\\D#4`=F]r7%F]x$!31!zaRWv&y:F/$\"3x r?o]s00>F]r7%F]x$!3Y)pFKla(ywF/$\"3)[uYKtz0v\"F]r7%F]x$!3qnowk@o%H\"F] r$\"3#o'3nQ(*R19F]r7%F]x$!3]T[fi**=\"o\"F]r$\"39**fp)4X\")4*F/7%F]x$!3 gCciTZ^&)=F]r$\"3+JH(phnj9$F/7%F]x$!3)[iD;u9b)=F]r$!3%o#H(phnj9$F/7%F] x$!3/U[fi**=\"o\"F]r$!3/))fp)4X\")4*F/7%F]x$!3'z'owk@o%H\"F]r$!3Em3nQ( *R19F]r7%F]x$!3_\"pFKla(ywF/$!3;XnCL(z0v\"F]r7%F]x$!35*zaRWv&y:F/$!3xr ?o]s00>F]r7%F]x$\"3)3M#z3pm#p%F/$!3!)yq*\\D#4`=F]r7%F]x$\"37VUx)fQb/\" F]r$!3U?UzffJ+;F]r7%F]x$\"3Ap\"*Gw,^3:F]r$!3ymq)))f?T<\"F]r7%F]x$\"3Uv gDp7,3=F]r$!3*\\2Bt_5p?'F/7%F]xF_x$!3?_1Pbg.#o%Fhq767%$\"3cp=]JackDF]r $\"3Dr]4ReaODF]rF)7%F`^l$\"3UA,)*)y3\"*R#F]r$\"3,!=vc(3:O#)F/7%F`^l$\" 3-(yqb2\"p,?F]r$\"3CtAp+(yzb\"F]r7%F`^l$\"3+bhou!ftQ\"F]r$\"3?\"pmq96N 7#F]r7%F`^l$\"3.4Al4>&oA'F/$\"37kg)4@G*eCF]r7%F`^l$!3y)QI:&Hm%4#F/$\"3 RRF/\\A)y_#F]r7%F`^l$!3Jm=F]r7%F`^l$!3Y`h@%[D3B#F]r$\"3uNjVzBE27F]r7%F`^l$! 3lm0**f1&>]#F]r$\"3cR:9Js,vTF/7%F`^lF]al$!3-M:9Js,vTF/7%F`^l$!3Hah@%[D 3B#F]r$!3NMjVzBE27F]r7%F`^l$!3$R/.Cvbzr\"F]r$!3#foSVU(>m=F]r7%F`^l$!3M ;q)z&)=*=5F]r$!3'z8\"G@5!HK#F]r7%F`^l$!3A+/`^Hm%4#F/$!3RRF/\\A)y_#F]r7 %F`^l$\"3C(>_'4>&oA'F/$!3Skg)4@G*eCF]r7%F`^l$\"3;ahou!ftQ\"F]r$!3v\"pm q96N7#F]r7%F`^l$\"3&yyqb2\"p,?F]r$!37sAp+(yzb\"F]r7%F`^l$\"39A,)*)y3\" *R#F]r$!3%p=vc(3:O#)F/7%F`^lFb^l$!3GGmZ&[XF@'Fhq767%$\"3mOtP*y1d?$F]r$ \"3y.iB!=#3^JF]rF)7%Fadl$\"3wGI3cyM!)HF]r$\"3*>.V8raJ-\"F]r7%Fadl$\"3? 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,&\"\"\"F&*$%\"xG\"\"#!\"\"" }{TEXT -1 10 " v mez\355ch " }{XPPEDIT 18 0 "<-1*sqrt(2)/2,1*sqrt(2)/2>;" "6#-%7rtable/ConstructColumnG6$,$*( \"\"\"F(-%%sqrtG6#\"\"#F(F,!\"\"F-6#*(F(F(-F*6#F,F(F,F-" }{TEXT -1 11 " a osou x, " }{XPPEDIT 18 0 "V[2];" "6#&%\"VG6#\"\"#" }{TEXT -1 41 " \+ je objem t\354lesa vytvo\370en\375 rotac\355 k\370ivky " }{XPPEDIT 18 0 "y = x^2;" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 10 " v mez\355ch " } {XPPEDIT 18 0 "<-1*sqrt(2)/2,1*sqrt(2)/2>;" "6#-%7rtable/ConstructColu mnG6$,$*(\"\"\"F(-%%sqrtG6#\"\"#F(F,!\"\"F-6#*(F(F(-F*6#F,F(F,F-" } {TEXT -1 11 " a osou x. " }}{PARA 0 "" 0 "" {TEXT -1 21 "Objem t\354le sa tedy je:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Pi*Int((1-x^ 2)^2-(x^2)^2,x=-1/2*sqrt(2)..1/2*sqrt(2)):%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#PiG\"\"\"-%$IntG6$,&*$),&F&F&*$)%\"xG\"\"#F&! \"\"F1F&F&*$)F0\"\"%F&F2/F0;,$*$-%%sqrtG6#F1F&#F2F1,$F9#F&F1F&,$*&F%F& F:F&#F1\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 330 10 "P\370\355klad 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Vypo\350\355t\341me objem t\354lesa vznikl\351ho rotac\355 rovinn\351ho obrazce ohrani\350en \351ho \350arami " }{XPPEDIT 18 0 "y^2+x = 4;" "6#/,&*$%\"yG\"\"#\" \"\"%\"xGF(\"\"%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\" \"!" }{TEXT -1 14 ", kolem osy y." }}{PARA 0 "" 0 "" {TEXT 331 6 "\330 e\232en\355" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 48 "Meze stano v\355me \370e\232en\355m rovnic obou \350ar: x = 0, " }{XPPEDIT 18 0 "y^2+x-4 = 0;" "6#/,(*$%\"yG\"\"#\"\"\"%\"xGF(\"\"%!\"\"\"\"!" }{TEXT -1 1 "," }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(0=y^2-4,y); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$!\"#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 30 " vypo\350\355t \341me z rovnice k\370ivky: " }{XPPEDIT 18 0 "x^2 = (4-y^2)^2;" "6#/*$ %\"xG\"\"#*$,&\"\"%\"\"\"*$%\"yGF&!\"\"F&" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Dosazen\355m do vzorce dost\341v\341me ob jem:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Pi*Int((4-y^2)^2,y= -2..2):%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#PiG\"\"\"-% $IntG6$*$),&\"\"%F&*$)%\"yG\"\"#F&!\"\"F1F&/F0;!\"#F1F&,$F%#\"$7&\"#: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 266 "" 0 "" {TEXT 332 10 "P\370\355klad 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 7 " Je-li " }{TEXT 335 36 "funkce d\341na pa rametrick\375mi rovnicemi" }{TEXT -1 2 " " }{XPPEDIT 18 0 "x = phi(t) ;" "6#/%\"xG-%$phiG6#%\"tG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y = psi( t);" "6#/%\"yG-%$psiG6#%\"tG" }{TEXT -1 7 " , kde " }{XPPEDIT 18 0 "t; " "6#%\"tG" }{TEXT -1 1 " " }{XPPEDIT 19 1 "epsilon;" "6#%(epsilonG" } {TEXT -1 1 " " }{XPPEDIT 18 0 ";" "6#-%7rtable/ConstructCo lumnG6$%&alphaG6#%%betaG" }{TEXT -1 4 " a " }{XPPEDIT 18 0 "psi(t);" "6#-%$psiG6#%\"tG" }{TEXT -1 81 " je v tomto intervalu spojit\341 a ne z\341porn\341, je objem rota\350n\355ho t\354lesa p\370i ot\341\350en \355 " }{TEXT 334 11 "kolem osy x" }{TEXT -1 12 " d\341n vzorcem" }} {PARA 0 "" 0 "" {TEXT -1 115 " \+ \+ " }{XPPEDIT 18 0 "V = Pi*Int(psi(t)^2*abs(diff(phi(t),t)),t = alp ha .. beta);" "6#/%\"VG*&%#PiG\"\"\"-%$IntG6$*&-%$psiG6#%\"tG\"\"#-%$a bsG6#-%%diffG6$-%$phiG6#F/F/F'/F/;%&alphaG%%betaGF'" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 9 " rotac\355 " }{TEXT 336 90 "kolem osy y \+ " } {XPPEDIT 18 0 "V = 2*Pi*Int(psi(t)*abs(phi(t)*diff(phi(t),t)),t = alph a .. beta);" "6#/%\"VG*(\"\"#\"\"\"%#PiGF'-%$IntG6$*&-%$psiG6#%\"tGF'- %$absG6#*&-%$phiG6#F0F'-%%diffG6$-F66#F0F0F'F'/F0;%&alphaG%%betaGF'" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 138 "Vypo\350teme objem rota\350n\355ho t\354 lesa vznikl\351ho rotac\355 rovinn\351ho obrazce omezen\351ho k\370ivk ou vyj\341d\370en\351 parametrick\375mo rovnicemi x = a(t-sin(t)), " } }{PARA 0 "" 0 "" {TEXT -1 24 "y = a(1-cos(t)), t le\236\355 " } {XPPEDIT 18 0 "<0,Pi>;" "6#-%7rtable/ConstructColumnG6$\"\"!6#%#PiG" } {TEXT -1 19 ", a>0, kolem osy x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 333 6 "\330e\232en\355" }{TEXT -1 1 ":" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "x = a(" }{TEXT 470 13 "t-sin(t)) = \+ " }{XPPEDIT 471 1 "phi(t);" "6#-%$phiG6#%\"tG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 472 18 "y = a(1-cos(t)) = " }{XPPEDIT 473 1 "psi(t);" "6#-% $psiG6#%\"tG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Diff(a*(t-sin(t)),t):%=value(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&%\"aG\"\"\",&%\"tGF)-%$s inG6#F+!\"\"F)F+*&F(F),&F)F)-%$cosGF.F/F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Dosazen\355m do vzorce dost\341v\341me:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Pi*Int((a*(1-cos(t)))^2*a*(1-cos(t)), t=0 ..2*Pi):%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#PiG\"\"\"- %$IntG6$*&)%\"aG\"\"$F&),&F&F&-%$cosG6#%\"tG!\"\"F-F&/F3;\"\"!,$F%\"\" #F&,$*&)F%F9F&F+F&\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 398 10 "P\370\355klad 4. 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20 "6#*&%#PiG\"\"\",&F$#\"\"$\"\"#F%!\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *&%#PiG\"\"\",&F%\"\"$\"\"#!\"\"F&#F&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "c) D\351lka oblouku rovinn\351 k\370ivky" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 337 15 "D\351lka oblouku L" }{TEXT -1 24 " spojit\351 rovinn \351 k\370ivky " }{TEXT 347 8 "y = f(x)" }{TEXT -1 80 " pro x le\236 \355 , kter\341 m\341 v intervalu spojitou derivaci, je d \341na vzorcem" }}{PARA 0 "" 0 "" {TEXT -1 20 " " } }{PARA 0 "" 0 "" {TEXT -1 31 " " }{TEXT 474 7 " V = " }{XPPEDIT 475 1 "int(sqrt(1+f(x)^`\2642`),x = a .. b); " "6#-%$intG6$-%%sqrtG6#,&\"\"\"F*)-%\"fG6#%\"xG%#|_v2GF*/F/;%\"aG%\"b G" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 338 0 "" }{TEXT -1 0 "" }{TEXT 339 0 "" }{TEXT 340 0 " " }{TEXT 341 10 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$$\"3)*******p=\\q6F`o$\"3YseoRmMm7F`o7$$\"3mmm;fBIY7F`o$\"37gz\">%fM \"R\"F`o7$$\"3GLLLj$[kL\"F`o$\"3\"4f:\"e#**\\a\"F`o7$$\"3?LLL`Q\"GT\"F `o$\"3KTy[KmHz;F`o7$$\"3!*****\\s]k,:F`o$\"3&QuF`o7$$\"33++]sgam;F`o$\"3M)\\%=*)QU^@F`o7$$\"3/+ +]F`o$\"3Q8s&=!)\\hk#F`o7$$\"3immmTc-)*>F`o$\"3Uid07*RU#GF`o7 $$\"3Mmm;f`@'3#F`o$\"3&e7y%yXF8IF`o7$$\"3y****\\nZ)H;#F`o$\"3!*Rz*3-@6 =$F`o7$$\"3YmmmJy*eC#F`o$\"31uaZHVxlLF`o7$$\"3')******R^bJBF`o$\"3E*H$ 4k>:gNF`o7$$\"3f*****\\5a`T#F`o$\"3!))>qIt+Qv$F`o7$$\"3o****\\7RV'\\#F `o$\"3Ibz*GP#RWRF`o7$$\"3k*****\\@fke#F`o$\"3cjg/=fmfTF`o7$$\"3/LLL`4N nEF`o$\"3'=V4BpCjN%F`o7$$\"3#*******\\,s`FF`o$\"3))>r*=(fhpXF`o7$$\"3[ mm;zM)>$GF`o$\"3um*)fcG!ew%F`o7$$\"3$*******pfa&F`o7$$\"3w****\\#G2A3$F`o$\"3BX(e 6v*=6aF`o7$$\"3;LLL$)G[kJF`o$\"3!394<`'HHcF`o7$$\"3#)****\\7yh]KF`o$\" 3K!z3!o:pgeF`o7$$\"3xmmm')fdLLF`o$\"3SZr6S2Z'3'F`o7$$\"3bmmm,FT=MF`o$ \"3HGA:MhG?jF`o7$$\"3FLL$e#pa-NF`o$\"3'p0\\)>!\\]b'F`o7$$\"3!*******Rv &)zNF`o$\"3UmT%>ctKx'F`o7$$\"3ILLLGUYoOF`o$\"3'f`#4feHEqF`o7$$\"3_mmm1 ^rZPF`o$\"3uz@')px?bsF`o7$$\"34++]sI@KQF`o$\"3q:fk)Hb>](F`o7$$\"34++]2 %)38RF`o$\"3qvyI)p'oSxF`o7$$\"\"%F)$\"\")F)-%'COLOURG6&%$RGBG$\"#5!\" \"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fb[l-%%VIEWG6$;F(Fcz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 13 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "V inter valu <0,4> m\341 dan\341 k\370ivka spojitou derivaci" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "`y'`= diff(x^(3/2),x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%#y'G,$*$-%%sqrtG6#%\"xG\"\"\"#\"\"$\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Dosad\355me do vzorce:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Int(sqrt(1+(3/2*sqrt(x))^2),x=0..4) :%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$*$-%%sqrtG 6#,&\"\"%\"\"\"*&\"\"*F.%\"xGF.F.F.#F.\"\"#/F1;\"\"!F-,&*$-F*6#\"#5F.# \"#!)\"#F#\"\")F>!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 257 "" 1 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 399 10 "P\370\355klad 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 400 1 " " }{TEXT 401 48 "Vypo\350\355tejte d\351lku grafu funkce ln(x) mezi body " }{TEXT 403 1 "(" }{TEXT 404 1 " " } {XPPEDIT 18 0 "sqrt(3),ln*sqrt(3)" "6$-%%sqrtG6#\"\"$*&%#lnG\"\"\"-F$6 #F&F)" }{TEXT -1 1 " " }{TEXT 402 4 "), (" }{XPPEDIT 18 0 "sqrt(8),ln* sqrt(8)" "6$-%%sqrtG6#\"\")*&%#lnG\"\"\"-F$6#F&F)" }{TEXT -1 3 " )." } }{PARA 0 "" 0 "" {TEXT 405 44 "Nejd\370\355ve si vypo\350\355t\341me d erivaci funkce ln(x)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff (log(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$%\"xG!\"\"" } }}{PARA 0 "" 0 "" {TEXT 406 21 "D\351lka k\370ivky tedy je " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(sqrt(1+1/x^2),x=sqrt(3)..sqrt(8 ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$-%%sqrtG6#,&\"\"\"F+ *&F+F+*$)%\"xG\"\"#F+!\"\"F+F+/F/;*$-F(6#\"\"$F+,$*$-F(6#F0F+F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$-%%sqrtG6#*&,&*$)%\"xG\"\"#\"\"\"F0F0F0 F0F.!\"#F0/F.;*$-F(6#\"\"$F0,$*$-F(6#F/F0F/" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 19 "zvol\355me substituce " }{XPPEDIT 18 0 "u^2=x^2+1" "6#/ *$%\"uG\"\"#,&*$%\"xGF&\"\"\"F*F*" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "student[changevar](u^2=x^2+1,%,u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"uG\"\"#,&*$)F'F(\"\"\"F,F,!\" \"F-/F';*$-%%sqrtG6#\"\"%F,*$-F26#\"\"*F," }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"uG\"\"#,&*$)F'F(\"\"\"F, F,!\"\"F-/F';F(\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "d\341le \+ \370e\232\355me vznikl\375 integr\341l rozkladem na parci\341ln\355 zl omky" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "convert(u^2/(u^2-1) ,parfrac,u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*&#F$\"\"#F$ ,&%\"uGF$F$!\"\"F*F$*&#F$F'F$*&F$F$,&F)F$F$F$F*F$F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "int(%,u=2..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*&#F$\"\"#F$-%#lnG6#F'F$!\"\"*&#F$F'F$-F)6#\" \"$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "combine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\" \"F$*&#F$\"\"#F$-%#lnG6##F'\"\"$F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 348 15 "D\351lka oblouku L " }{TEXT -1 54 " spojit\351 rovinn\351 k\370ivky dan\351 parametrick \375mi rovnicemi " }{XPPEDIT 18 0 "x = phi(t);" "6#/%\"xG-%$phiG6#%\"t G" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y = psi(t);" "6#/%\"yG-%$psiG6#%\" tG" }{TEXT -1 12 ", t le\236\355 v " }{XPPEDIT 18 0 ";" " 6#-%7rtable/ConstructColumnG6$%&alphaG6#%%betaG" }{TEXT -1 26 ", maj \355c\355 spojit\351 derivace " }{XPPEDIT 18 0 "phi;" "6#%$phiG" } {TEXT -1 6 "\264(t), " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 21 " \264(t) je d\341na vzorcem:" }}{PARA 0 "" 0 "" {TEXT -1 73 " \+ " } {XPPEDIT 18 0 "L = int(sqrt(phi(t)^`\2642`+psi(t)^`\2642`),t = alpha . . beta);" "6#/%\"LG-%$intG6$-%%sqrtG6#,&)-%$phiG6#%\"tG%#|_v2G\"\"\")- %$psiG6#F0F1F2/F0;%&alphaG%%betaG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 344 0 "" }{TEXT 345 0 "" }{TEXT 407 10 "P\370\355klad 3." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 346 1 " \+ " }{TEXT -1 54 "Vypo\350teme d\351lku k\370ivky danou parametrick\375m i rovnicemi " }{XPPEDIT 18 0 "x = t^2;" "6#/%\"xG*$%\"tG\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y = t-t^3/3;" "6#/%\"yG,&%\"tG\"\"\"*&F&\"\" $F)!\"\"F*" }{TEXT -1 20 " pro t z intervalu " }{XPPEDIT 18 0 "<0,sqr t(3)>;" "6#-%7rtable/ConstructColumnG6$\"\"!6#-%%sqrtG6#\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 343 6 "\330e\232en\355" }{TEXT -1 1 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "plot([t^2, t-t^3/3, t= 0..sqrt(3)], scaling=constrained, thickness=3, color=black);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6#7W7$$\"\"! F)F(7$$\"3(\\'e+!zZ`U\"!#?$\"3-())*[1TetP!#>7$$\"3k/$ygF.[)\\F-$\"3Ybu e-De[qF07$$\"3KQOG_tgc6F0$\"3'*f88S0Jr5!#=7$$\"3%Q9:L3nZ4#F0$\"3.iQP\\ \\AP9F;7$$\"3D9Hjt+3.LF0$\"3->(y?IFuz\"F;7$$\"3uecM;03oYF0$\"3v0G;/]&p 7#F;7$$\"3\\\"eN'H:hHjF0$\"3g0ve#G!ziCF;7$$\"3kG\\O9Ca8$)F0$\"3Kg2$3P> M!GF;7$$\"3%\\9`R9&)f0\"F;$\"3%)49&F;7$$\"3)G'H=(f!eUPF;$\"336Dz!\\oWN&F;7$$\"34'Rh\"4'3!GUF;$\" 3IHp=k#>fe&F;7$$\"3y'y0M)RQ#o%F;$\"3=02/*)\\xudF;7$$\"3$)H@l\\'zv?&F;$ \"3'Q$>EfypjfF;7$$\"3]S0P%*>jLdF;$\"3eOg\"z&>*[7'F;7$$\"3*p0@iu!\\4jF; $\"3W+,6C!REF'F;7$$\"3u(oNM6wD'oF;$\"3]1;\"oUl!*Q'F;7$$\"3!pcY^i*>&[(F ;$\"3#R*fq&*o/$\\'F;7$$\"3'=T0\"QAbg\")F;$\"3y@!ex7yid'F;7$$\"3i#Q'=DL >s()F;$\"3.<`d%Q?ti'F;7$$\"3'eg#\\6qgd%*F;$\"31MQ!4tu\"fmF;7$$\"3JVPa2 !y#>5!#<$\"3a[\"yucXuP]\"Fat$\"362%Ggc&*f6'F;7$$\"3w\"))=r'R,'f\"Fat$\"38l^(GZYB \"fF;7$$\"3U**et<5J&o\"Fat$\"3w^kqEo2*o&F;7$$\"3GWApD.D\"y\"Fat$\"3B\\ MhYG&>U&F;7$$\"3yo-t&)fhx=Fat$\"3J=n#3?Jl7&F;7$$\"3Olz`\"y@7)>Fat$\"3' eC]h\\n*zZF;7$$\"3S\"Rx?mOO3#Fat$\"33UuK8h<4WF;7$$\"3%y[(Qj(R5>#Fat$\" 3s>R#pF`9*RF;7$$\"3O!>#ed!>-I#Fat$\"3uU\"ybFOx`$F;7$$\"3c#[^Cv$)GS#Fat $\"3!)y$zZ*pM&3$F;7$$\"3\"fQOxe0L_#Fat$\"3Sz*Gxd$3CDF;7$$\"3OG(\\ao6\" yDFat$\"3%3&HQqk,eAF;7$$\"3i&H\\*)f1Nj#Fat$\"3%)\\+!G/&\\#)>F;7$$\"39O G!Rr)f`FFat$\"3Q;wR,!GHO\"F;7$$\"3n* 4z_&p,7GFat$\"3sk9q?pw]5F;7$$\"3u\\+N<*[5(GFat$\"3'oR4P:IKG(F07$$\"3Ox QD$Qq^$HFat$\"3\"o'z+jlF-PF07$$\"3[_M\\,+++IFat$!3^g6dT;XA')!#F-%*THIC KNESSG6#\"\"$-%'COLOURG6&%$RGBGF)F)F)-%+AXESLABELSG6$Q!6\"Fh\\l-%(SCAL INGG6#%,CONSTRAINEDG-%%VIEWG6$%(DEFAULTGFa]l" 1 2 0 1 10 3 2 6 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "V intervalu " }{XPPEDIT 18 0 "<0,sqrt(3)>;" "6#-%7rtable/ ConstructColumnG6$\"\"!6#-%%sqrtG6#\"\"$" }{TEXT -1 15 " jsou derivace " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 6 "\264(t), " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 23 "\264(t) spojit\351, dosad\355me: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 265 "> " 1 "" {MPLTEXT 1 0 17 "`x'`=diff(t^2,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%#x'G,$%\"tG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "`y'`= diff(t-t^3/3,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#y'G,&\"\"\"F&*$ )%\"tG\"\"#F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Int(s qrt((2*t)^2+(1-t^2)^2),t=0..sqrt(3)):%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$-%%sqrtG6#*$),&*$)%\"tG\"\"#\"\"\"F2F2F2F1F 2F2/F0;\"\"!*$-F)6#\"\"$F2,$F6F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 319 0 "" } {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 408 0 "" }{TEXT 409 0 " " }{TEXT 410 10 "P\370\355klad 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 411 38 "Vypo\350\355tejte d\351lku k\370ivky - a steroidy -" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "x=5*cos(t)^3, y=5*sin(t)^ 3,t=0..2*Pi" "6%/%\"xG*&\"\"&\"\"\"*$-%$cosG6#%\"tG\"\"$F'/%\"yG*&F&F' *$-%$sinG6#F,F-F'/F,;\"\"!*&\"\"#F'%#PiGF'" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 51 "plot([5*cos(t)^3,5*sin(t)^3,t=0..2*Pi],color=black, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "scaling=CONSTRAINED,font=[TIMES ,ROMAN,11]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6( -%'CURVESG6#7in7$$\"\"&\"\"!$F*F*7$$\"3K'H 7$$\"3gj2Hj.\\EXF/$\"3\"y.z9-e(G\")F27$$\"3IO8$)GXk5!#E$\"3%3nufK (****\\F/7$$!33U$>?,GGh\"Fco$\"3wF5[,gwk\\F/7$$!37)*p#ejFYE\"F2$\"31UV -(G@9'[F/7$$!3%GV/!QTlk#*F2$\"3'ft1/qKT[%F/7$$!314XEc^w&o#F=$\"3KZy![` G7(RF/7$$!3)z%)>8+H+T'F=$\"3X3v.>N.?KF/7$$!3A:FBEhx^&)F=$\"3'*QI`6k[xG F/7$$!3^!GAG(\\Y,6F/$\"3Sbw,bq_JDF/7$$!3\"\\P\"GS*=ZU\"F/$\"3$Hq$4&RNY 8#F/7$$!3YZhY0()[\"y\"F/$\"3y'[zEh!3a'G8pCF/$\"3i&zP&f.G\\6F/7$$!3ir7Zq'HlC$F/$\"3SW!GSb`hD'F= 7$$!3c%pDP\"*[G$RF/$\"3UoeK6))*Q%GF=7$$!3wC31Kt@7XF/$\"3BYcdWqd0&)F27$ $!3Y=s#Q_P='[F/$\"3uK&QD]2*e7F27$$!3EwR.&e!ej\\F/$\"39G<;XI+&p\"Fco7$$ !3!ozrU'y#***\\F/$\"3qFf6]$Q9\\\"!#C7$$!3;KjgICDn\\F/$!3OQ6daA(\\W\"Fc o7$$!3a#R`%H2\"R'[F/$!3\"==#4(*zaI7F27$$!3bN))Gt+'o_%F/$!3@\\3&)R%y!> \")F27$$!3i5jUi7JuRF/$!3#))\\H\"on@tEF=7$$!3/^/])4%HjKF/$!3!RYfcna'fhF =7$$!3W*o/$=Ld3DF/$!3YXL\\I0$*=6F/7$$!35sP\\AEQY@F/$!3<\\\\!yeuWT\"F/7 $$!3=&z&yGjb(z\"F/$!3rpS&y(R9Q!=iBTv[F/7$$!3o]uc c$zJX\"Fco$!39u7%\\oGr'\\F/7$$!3&4Z0N')e*G9Fau$!3owI@$*)H***\\F/7$$\"3 7SZh%GQ*\\7Fco$!33$[opvm-(\\F/7$$\"3g%)*oQUrw1\"F2$!3E3m)*p'\\h([F/7$$ \"3iO\\t-o&*R$)F2$!3h%RVQFi%=XF/7$$\"3(\\\\ce1s=#GF=$!30@^Z:S9QRF/7$$ \"3opd^K`QiiF=$!3M`%)e05XXKF/7$$\"3%4?/%H>8S6F/$!35pH[WE'4[#F/7$$\"33g JNRg-*y\"F/$!3InqsZFfY " 0 "" {MPLTEXT 1 0 42 "diff(5*cos(t)^3,t)^2+diff(5* sin(t)^3,t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&)-%$cosG6#%\"tG \"\"%\"\"\")-%$sinGF(\"\"#F+\"$D#*(F0F+)F-F*F+)F&F/F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "delkakrivky:=int(sqrt(%),t=0..2*Pi) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,delkakrivkyG,$*$-%%sqrtG6#\"\" %\"\"\"\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 39 "d) Obsah r ota\350n\355 plochy (povrch t\354lesa)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 349 21 "Obsah rota\350n\355 plochy," }{TEXT -1 22 " kter\341 vznikne rotac \355 " }{TEXT 352 11 "kolem osy x" }{TEXT -1 21 " grafu spojit\351 k \370ivky" }{TEXT 350 10 " y = f(x) " }{TEXT -1 63 "v intervalu a maj\355c\355 v tomto intervalu spojitou derivaci " }{TEXT 320 7 "f '( x)," }{TEXT -1 16 " je d\341n vzorcem:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "P = 2*Pi*Int(abs(f(x))*sqrt(1+[`f(x)'`]^2),x = a .. b);" "6#/%\"PG* (\"\"#\"\"\"%#PiGF'-%$IntG6$*&-%$absG6#-%\"fG6#%\"xGF'-%%sqrtG6#,&F'F' *$7#%&f(x)'GF&F'F'/F3;%\"aG%\"bGF'" }{TEXT -1 44 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 351 20 "Obsah rota\350n\355 plochy" }{TEXT -1 23 ", kter \341 vznikne rotac\355 " }{TEXT 353 11 "kolem osy x" }{TEXT -1 52 " gr afu spojit\351 k\370ivky dan\351 parametrick\375mi rovnicemi " } {XPPEDIT 18 0 "x = phi(t),y = psi(t);" "6$/%\"xG-%$phiG6#%\"tG/%\"yG-% $psiG6#F(" }{TEXT -1 12 ", t le\236\355 v " }{XPPEDIT 18 0 ";" "6#-%7rtable/ConstructColumnG6$%&alphaG6#%%betaG" }{TEXT -1 11 " , funkce " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 6 "\264(t), " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 53 "\264(t) jsou na tomto i ntervalu spojit\351, p\370i\350em\236 funkce " }{XPPEDIT 18 0 "phi(t); " "6#-%$phiG6#%\"tG" }{TEXT -1 36 " je ryze monotonn\355, je d\341n v zorcem:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "P = 2*Pi*Int(abs(psi(t))*sqr t(phi(t)^`\2642`+psi(t)^`\2642`),t = alpha .. beta);" "6#/%\"PG*(\"\"# \"\"\"%#PiGF'-%$IntG6$*&-%$absG6#-%$psiG6#%\"tGF'-%%sqrtG6#,&)-%$phiG6 #F3%#|_v2GF')-F16#F3F " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 354 10 "P\370 \355klad 1." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Vypo\350teme povr ch koule o polom\354ru " }{TEXT 356 3 "r " }{TEXT -1 15 " ob\354ma zp\371soby." }}{PARA 0 "" 0 "" {TEXT 355 2 "1." }{TEXT -1 20 " Z rovn ice kru\236nice " }{XPPEDIT 18 0 "x^2+y^2 = r^2;" "6#/,&*$%\"xG\"\"#\" \"\"*$%\"yGF'F(*$%\"rGF'" }{TEXT -1 4 " => " }{XPPEDIT 18 0 "y = sqrt( r^2-x^2);" "6#/%\"yG-%%sqrtG6#,&*$%\"rG\"\"#\"\"\"*$%\"xGF+!\"\"" } {TEXT -1 43 ", x le\236\355 <-r,r>. Rotace k\370ivky kolem osy x." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f:= x-> sqrt(r^2-x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqr tG6#,&*$)%\"rG\"\"#\"\"\"F4*$)9$F3F4!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "plot(sqrt(1-x^2), x=-1..1, scaling=constraine d);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVES G6$7_o7$$!\"\"\"\"!$F*F*7$$!3-n;HdNvs**!#=$\"3aIk@d'**oP(!#>7$$!3/MLe9 r]X**F/$\"3^xD,?$RD/\"F/7$$!3/,](=ng#=**F/$\"3W*\\#GuA(fF\"F/7$$!3%pmm \"HU,\"*)*F/$\"3=cuB4yNs9F/7$$!3()***\\PM@l$)*F/$\"3gMpy'*3z+=F/7$$!3! RLL$e%G?y*F/$\"3iA8`Ih^w?F/7$$!3u****\\(oUIn*F/$\"3nSQYJr=ODF/7$$!3omm m;p0k&*F/$\"3Y%G&e[NT?HF/7$$!3E++vV5Su$*F/$\"3'eGkq$fY\"[$F/7$$!3wKL$3 (eMbF/7$$!3\"QL L3i.9!zF/$\"3^7**H-%f#HhF/7$$!3\"ommT!R=0vF/$\"3^qi^Nc\\3mF/7$$!3u**** \\P8#\\4(F/$\"3ifl'faLr/(F/7$$!3+nm;/siqmF/$\"37dnjs %HaF/$\"3e&R<6BowR)F/7$$!3Q+++]$*4)*\\F/$\"3]Y1e+6Nh')F/7$$!39+++]_&\\ c%F/$\"3S:_+\"3ds*))F/7$$!31+++]1aZTF/$\"3i'GE3]N$*4*F/7$$!3umm;/#)[oP F/$\"33y-$=zZFE*F/7$$!3hLLL$=exJ$F/$\"3:IF>)f#eL%*F/7$$!3*RLLLtIf$HF/$ \"3kU7=H\\If&*F/7$$!3]++]PYx\"\\#F/$\"3)G16\">%yXo*F/7$$!3EMLLL7i)4#F/ $\"3i8>'>!*4tx*F/7$$!3c****\\P'psm\"F/$\"35Fh>K5.g)*F/7$$!3')****\\74_ c7F/$\"3JpBi\"oV2#**F/7$$!3)3LLL3x%z#)F2$\"3%4o%3*=mc'**F/7$$!3KMLL3s$ QM%F2$\"3#oX(\\%3h0***F/7$$!3]^omm;zr)*!#@$\"2sF&QF^******!#<7$$\"3%pJ L$ezw5VF2$\"3)*Q2K>Vq!***F/7$$\"3s*)***\\PQ#\\\")F2$\"3s^4RW'Rn'**F/7$ $\"3GKLLe\"*[H7F/$\"3+'Q!p2+8C**F/7$$\"3I*******pvxl\"F/$\"3]s:1(pJ;') *F/7$$\"3#z****\\_qn2#F/$\"3;GKN_W(>y*F/7$$\"3U)***\\i&p@[#F/$\"3!QaP4 mWqo*F/7$$\"3B)****\\2'HKHF/$\"3+\\tky/Ug&*F/7$$\"3ElmmmZvOLF/$\"3%e0e z0!)oU*F/7$$\"3i******\\2goPF/$\"3wC!*\\-?qi#*F/7$$\"3UKL$eR<*fTF/$\"3 CILE@Ro$4*F/7$$\"3m******\\)Hxe%F/$\"3-\"f%eX^`&)))F/7$$\"3ckm;H!o-*\\ F/$\"35DIn2b'em)F/7$$\"3y)***\\7k.6aF/$\"3a'=*zS\"f&4%)F/7$$\"3#emmmT9 C#eF/$\"3H&p3sIf,8)F/7$$\"33****\\i!*3`iF/$\"3;c[JlKx.yF/7$$\"3%QLLL$* zym'F/$\"3o4V`oXZ_uF/7$$\"3wKLL3N1#4(F/$\"3!Q)Q?)e4+0(F/7$$\"3Nmm;HYt7 vF/$\"3+_Th$[5**f'F/7$$\"3Y*******p(G**yF/$\"3_pyiCh)>8'F/7$$\"3]mmmT6 KU$)F/$\"3I@,y!QMT^&F/7$$\"3fKLLLbdQ()F/$\"3e&)*oZS?='[F/7$$\"3amm\"zW ?)\\*)F/$\"3ud/FE;-hWF/7$$\"3[++]i`1h\"*F/$\"3$*pj$>D\\$4SF/7$$\"3Y+++ +PDj$*F/$\"3v!)RQ\\fL6NF/7$$\"3W++]P?Wl&*F/$\"3:\"G)fhN(e\"HF/7$$\"3K+ ]7G:3u'*F/$\"3fG;Bq;AKDF/7$$\"3A++v=5s#y*F/$\"37vO>U4Dt?F/7$$\"3;+D1k2 /P)*F/$\"3mLW(eL^zz\"F/7$$\"35+]P40O\"*)*F/$\"31H.f%oH+Z\"F/7$$\"3k]7. #Q?&=**F/$\"3G#>z^*=&RF\"F/7$$\"31+voa-oX**F/$\"3(pV\\8H')3/\"F/7$$\"3 [\\PMF,%G(**F/$\"3?&=hJ*p=ltF27$$\"\"\"F*F+-%'COLOURG6&%$RGBG$\"#5F)F+ F+-%+AXESLABELSG6$Q\"x6\"Q!F\\al-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$; F(F``l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Dosazen\355m do \+ vzorce dost\341v\341me:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " f := x -> sqrt(r^2-x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6# %\"xG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&*$)%\"rG\"\"#\"\"\"F4*$)9$F 3F4!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "2*Pi*Int( f(x)*sqrt(1+(diff(f(x),x))^2),x=-r..r):%=value(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,$*&%#PiG\"\"\"-%$IntG6$*&-%%sqrtG6#,&*$)%\"rG\"\"#F 'F'*$)%\"xGF3F'!\"\"F'-F-6#,&F'F'*&F/F7F6F3F'F'/F6;,$F2F7F2F'F3,$*&F&F 'F1F'\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 357 1 "2" }{TEXT -1 39 ". Parametrick\351 vyj\341d\370en\355 kru \236nice: x = " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 18 "(t) = r cos(t), y =" }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 23 " = rsin(t) , t le\236\355 <0.." }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 2 ">." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Di ff(phi,t) = diff(r*cos(t),t);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%Di ffG6$%$phiG%\"tG,$*&%\"rG\"\"\"-%$sinG6#F(F,!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "Diff(psi,t) = diff(r*sin(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$%$psiG%\"tG*&%\"rG\"\"\"-%$cosG6#F(F +" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Dosad\355me do vzorce:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "2*Pi*Int(r*sin(t)*sqrt((-r*s in(t))^2+(r*cos(t))^2),t=0..Pi):%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&%#PiG\"\"\"-%$IntG6$*(%\"rGF'-%$sinG6#%\"tGF'-%%sq rtG6#,&*&)F,\"\"#F')F-F7F'F'*&F6F')-%$cosGF/F7F'F'F'/F0;\"\"!F&F'F7,$* (F&F'F,\"\"$*$F6F'#!\"\"F7\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "combine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$**% #PiG\"\"\"%\"rGF*-%$sinG6#%\"tGF*-%%sqrtG6#*$)F+\"\"#F*F*F5/F/;\"\"!F) ,$*(F)F*F+\"\"$F3#!\"\"F5\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Maple uva\236uje r jako konstantu o kter\351 nev\355 , zda je kladn \341 nebo z\341porn\341, proto neuprav\355 odmocninu." }}{PARA 0 "" 0 "" {TEXT -1 75 "M\371\236eme upravit pomoc\355 p\370\355kazu assume, k ter\375m zad\341me vlastnost r - polom\354ru." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(r>0);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "2*Pi*Int(r*sin(t)*sqrt((-r* sin(t))^2+(r*cos(t))^2),t=0..Pi):%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&%#PiG\"\"\"-%$IntG6$*(%#r|irGF'-%$sinG6#%\"tGF'-%% sqrtG6#,&*&)F,\"\"#F')F-F7F'F'*&F6F')-%$cosGF/F7F'F'F'/F0;\"\"!F&F'F7, $*&F&F'F6F'\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 20 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 " " }{TEXT 412 0 "" }{TEXT 413 10 "P\370\355klad 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 414 38 "Vypo\350\355tejte d\351l ku k\370ivky - asteroidy -" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "x=5*cos(t )^3, y=5*sin(t)^3,t=0..2*Pi" "6%/%\"xG*&\"\"&\"\"\"*$-%$cosG6#%\"tG\" \"$F'/%\"yG*&F&F'*$-%$sinG6#F,F-F'/F,;\"\"!*&\"\"#F'%#PiGF'" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot([5*cos(t)^3,5*sin(t)^3,t=0..2*Pi],color=black," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "scaling=CONSTRAINED,font=[TIMES,R OMAN,11]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-% 'CURVESG6#7in7$$\"\"&\"\"!$F*F*7$$\"3K'H7$ $\"3gj2Hj.\\EXF/$\"3\"y.z9-e(G\")F27$$\"3IO8$)GXk5!#E$\"3%3nufK(** **\\F/7$$!33U$>?,GGh\"Fco$\"3wF5[,gwk\\F/7$$!37)*p#ejFYE\"F2$\"31UV-(G @9'[F/7$$!3%GV/!QTlk#*F2$\"3'ft1/qKT[%F/7$$!314XEc^w&o#F=$\"3KZy![`G7( RF/7$$!3)z%)>8+H+T'F=$\"3X3v.>N.?KF/7$$!3A:FBEhx^&)F=$\"3'*QI`6k[xGF/7 $$!3^!GAG(\\Y,6F/$\"3Sbw,bq_JDF/7$$!3\"\\P\"GS*=ZU\"F/$\"3$Hq$4&RNY8#F /7$$!3YZhY0()[\"y\"F/$\"3y'[zEh!3a'G8pCF/$\"3i&zP&f.G\\6F/7$$!3ir7Zq'HlC$F/$\"3SW!GSb`hD'F=7$$ !3c%pDP\"*[G$RF/$\"3UoeK6))*Q%GF=7$$!3wC31Kt@7XF/$\"3BYcdWqd0&)F27$$!3 Y=s#Q_P='[F/$\"3uK&QD]2*e7F27$$!3EwR.&e!ej\\F/$\"39G<;XI+&p\"Fco7$$!3! ozrU'y#***\\F/$\"3qFf6]$Q9\\\"!#C7$$!3;KjgICDn\\F/$!3OQ6daA(\\W\"Fco7$ $!3a#R`%H2\"R'[F/$!3\"==#4(*zaI7F27$$!3bN))Gt+'o_%F/$!3@\\3&)R%y!>\")F 27$$!3i5jUi7JuRF/$!3#))\\H\"on@tEF=7$$!3/^/])4%HjKF/$!3!RYfcna'fhF=7$$ !3W*o/$=Ld3DF/$!3YXL\\I0$*=6F/7$$!35sP\\AEQY@F/$!3<\\\\!yeuWT\"F/7$$!3 =&z&yGjb(z\"F/$!3rpS&y(R9Q!=iBTv[F/7$$!3o]ucc$zJ X\"Fco$!39u7%\\oGr'\\F/7$$!3&4Z0N')e*G9Fau$!3owI@$*)H***\\F/7$$\"37SZh %GQ*\\7Fco$!33$[opvm-(\\F/7$$\"3g%)*oQUrw1\"F2$!3E3m)*p'\\h([F/7$$\"3i O\\t-o&*R$)F2$!3h%RVQFi%=XF/7$$\"3(\\\\ce1s=#GF=$!30@^Z:S9QRF/7$$\"3op d^K`QiiF=$!3M`%)e05XXKF/7$$\"3%4?/%H>8S6F/$!35pH[WE'4[#F/7$$\"33gJNRg- *y\"F/$!3InqsZFfY " 0 "" {MPLTEXT 1 0 42 "diff(5*cos(t)^3,t)^2+diff(5* sin(t)^3,t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&)-%$cosG6#%\"tG \"\"%\"\"\")-%$sinGF(\"\"#F+\"$D#*(F0F+)F-F*F+)F&F/F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "delkakrivky:=int(sqrt(%),t=0..2*Pi) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,delkakrivkyG,$*$-%%sqrtG6#\"\" %\"\"\"\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{PARA 4 "" 0 "" {TEXT -1 29 "2.Uk\341zky fyzik \341ln\355ch aplikac\355" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "P\370 \355klad 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Ur\350ete t\354\236i\232t\354 homogenn\355 oblasti omezen \351 k\370ivkami " }{XPPEDIT 18 0 "y = x^2,y = 2/(1+x^2);" "6$/%\"y G*$%\"xG\"\"#/F$*&F'\"\"\",&F*F**$F&F'F*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Nejd\370\355ve oblast omezenou k\370ivkami nakresl\355 me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot(\{x^2,2/(1+x^2)\},x=-5..5,y=0..5,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$!\"& \"\"!$\"#DF*7$$!3YLLLe%G?y%!#<$\"3s()4j<'znG#!#;7$$!3OmmT&esBf%F0$\"3_ b'Gjf))*3@F37$$!3ALL$3s%3zVF0$\"31&fB#*HQw\">F37$$!3_LL$e/$QkTF0$\"3E6 CC:'3Ut\"F37$$!3ommT5=q]RF0$\"3!G&H[zW!3c\"F37$$!3ILL3_>f_PF0$\"3?gS)e j%>39F37$$!3K++vo1YZNF0$\"3'*>Gj>xWe7F37$$!3;LL3-OJNLF0$\"3*)4UU#oJC6 \"F37$$!3p***\\P*o%Q7$F0$\"3/A:f:%>%e(*F07$$!3Kmmm\"RFj!HF0$\"3mn>b2*Q nW)F07$$!33LL$e4OZr#F0$\"3g2/-q?zptF07$$!3u*****\\n\\!*\\#F0$\"3y/w6y# \\_C'F07$$!3%)*****\\ixCG#F0$\"3[Rci3Tq4_F07$$!3#******\\KqP2#F0$\"3\\ 01&3OB0I%F07$$!39LL3-TC%)=F0$\"3?EeBOeP]NF07$$!3[mmm\"4z)e;F0$\"3_C)o2 %)z=v#F07$$!3Mmmmm`'zY\"F0$\"33+GtNu8<#F[r$\"3Wu_7B/B7$$!3vDMLLe*e$\\! 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\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "S[y] = rho*Int(x*f(x),x = a .. b);" "6#/&%\"SG6#%\"yG*& %$rhoG\"\"\"-%$IntG6$*&%\"xGF*-%\"fG6#F/F*/F/;%\"aG%\"bGF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Proto\236e oblast je tvo\370ena dv\354ma \+ k\370ivkami , statick\351 momenty se budou od\350\355tat." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "SX:=1/2*rho*Int(y2^2-y1^2,x=-1..1)= 1/2*rho*int(y2^2-y1^2,x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# SXG/,$*&%$rhoG\"\"\"-%$IntG6$,&*&F)F)*$),&F)F)*$)%\"xG\"\"#F)F)F5F)!\" \"\"\"%*$)F4F7F)F6/F4;F6F)F)#F)F5,$*&F(F),&#\"\")\"\"&F)%#PiGF)F)F<" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "SY:=rho*Int(x*(y2-y1),x=-1 ..1)=rho*int(x*(y2-y1),x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #SYG/*&%$rhoG\"\"\"-%$IntG6$*&%\"xGF(,&*&F(F(,&F(F(*$)F-\"\"#F(F(!\"\" F3F1F4F(/F-;F4F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "V\375p o\350et hmotnosti " }{XPPEDIT 18 0 "m = rho*P;" "6#/%\"mG*&%$rhoG\"\" \"%\"PGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "m:=rho*Int(y2- y1,x=-1..1)=rho*int(y2-y1,x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"mG/*&%$rhoG\"\"\"-%$IntG6$,&*&F(F(,&F(F(*$)%\"xG\"\"#F(F(!\"\"F2F /F3/F1;F3F(F(*&F'F(,&%#PiGF(#F2\"\"$F3F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "x - ov\341 sou\370adnice t\354\236i\232t\354 " } {XPPEDIT 18 0 "x[T] = S[y]/m;" "6#/&%\"xG6#%\"TG*&&%\"SG6#%\"yG\"\"\"% \"mG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Xt:=rhs(SY)/rh s(m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XtG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Xt := 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#XtG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "y - ov\341 sou \370adnice t\354\236i\232t\354 " }{XPPEDIT 18 0 "y[T] = S[x]/m;" "6 #/&%\"yG6#%\"TG*&&%\"SG6#%\"xG\"\"\"%\"mG!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "Yt:=rhs(SX)/rhs(m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YtG,$*&,&#\"\")\"\"&\"\"\"%#PiGF+F+,&F,F+#\"\"#\"\"$ !\"\"F1#F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Yt:=evala(Y t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YtG,$*&,&\"\")\"\"\"*&\"\"&F )%#PiGF)F)F),&F,\"\"$\"\"#!\"\"F0#F.\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "P\370\355klad 2." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Nalezn\354te sou\370adnice t\354\236i\232 t\354 homogenn\355ho dr\341tu ve tvaru \350tvrtkru\236nice o rovnici \+ " }{XPPEDIT 18 0 "y = sqrt(r^2-x^2);" "6#/%\"yG-%%sqrtG6#,&*$%\"rG\" \"#\"\"\"*$%\"xGF+!\"\"" }{TEXT -1 33 " a d\351lce na ose x v inte rvalu" }}{PARA 0 "" 0 "" {TEXT -1 25 "<-r/sqrt(2), r/sqrt(2)>." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Nejd\370\355ve si funkci nakresl \355me. Pro kreslen\355 vol\355me " }{XPPEDIT 18 0 "r = 3;" "6#/%\"rG \"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot(sqrt(9-x^2),x =-3/sqrt(2)..3/sqrt(2),y=0..3.3,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7W7$$!31++QS.K@@ !#<$\"3;&GRnM?87#F*7$$!3Q^Q)Qf\"3v?F*$\"3Ycyg&\\sl;#F*7$$!3r-xQZG%)G?F *$\"3(4vh\"z8$*4AF*7$$!3!zj)Hh2h))>F*$\"31L_E\\G?YAF*7$$!3'Gd4_ny$[>F* $\"3'R)\\v/%)=\"G#F*7$$!3%\\?5\")H))y&=F*$\"3)>5\"GzDZbBF*7$$!3qt>0#4) zm3)[#F*7$$!3AL,UF**3#f \"F*$\"316k%=_&oUDF*7$$!3n#[jY4g]]\"F*$\"3aU+!o=[^f#F*7$$!3+#4Hrr`]T\" F*$\"3mL1#z0-`k#F*7$$!3H^*)=\"*fLD8F*$\"3OW:eNAP\"p#F*7$$!3A5#\\BG]IB \"F*$\"3Z_X`hL)[t#F*7$$!3!z+r(z\\w^6F*$\"3U![uv+(4qFF*7$$!3a0&R9)pDg5F *$\"3c\\Xg)R&R1GF*7$$!3;n%\\ZUKPo*!#=$\"3<(>]Ys5%RGF*7$$!3e%)yRVBE)z)F `p$\"3m'*p7%*R3oGF*7$$!3iZ3)*z1<%*zF`p$\"3;w6q%oF:*GF*7$$!3p&>n(=z-QqF `p$\"3z*\\:3Ivi\"HF*7$$!36onc#e\\!GiF`p$\"3R=4-].kMHF*7$$!3m\\<$\"3_X9*Fcs$\"3sZ#*3sfg)*HF*7$$\"3ky #\\B^9(G7Ln;NF`p$\"3YGji]qJzHF*7$$\"3T^T)pb&\\0WF`p$\"3)e*[*QQwu'HF*7$$ \"3OvL?\"ywaE&F`p$\"3E(G#3$zHM&HF*7$$\"3(z$R\"zIR.A'F`p$\"3qB0svQ![$HF *7$$\"3I=**QddKyqF`p$\"3B)fD2#)*H:HF*7$$\"3;FjuD%4W*zF`p$\"3'4b?S3@:*G F*7$$\"3K\\=3'QU2dfe5F*$\"3n(4TmYAq!GF*7$$\"3g2tVgT&y9\"F*$\"3#)[t'=$)>< x#F*7$$\"3Ev'>Kh?^B\"F*$\"3a'oJ$))*[Rt#F*7$$\"3V&*)[=]![E8F*$\"31/*\\< L33p#F*7$$\"3Ek\"zH$4Z99F*$\"3#oT$fCPhXEF*7$$\"3iC%\\v&QX/:F*$\"3_bHJ! o*\\&f#F*7$$\"3;XG2y;p$f\"F*$\"3`N,/\"*=oTDF*7$$\"39bBDn>pv;F*$\"3'G'e G,VQ)[#F*7$$\"3%4'*zBbt'p-$HaNV>F*$\"3+gl@Y#paG#F*7$$\"3yH[f_gC')>F*$\"3e)Qyu :%H[AF*7$$\"3rmu)enO\"H?F*$\"3UIh'=fh'4AF*7$$\"3QLP83&G_2#F*$\"3sz9wv< Vm@F*7$$\"31++QS.K@@F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fd\\lFc\\l-% +AXESLABELSG6$Q\"x6\"Q\"yFi\\l-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$! +V.K@@!\"*$\"+V.K@@Fe]l;Fc\\l$\"#LFb\\l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "D\341le spo\350\355t\341me statick\351 momenty " } {XPPEDIT 18 0 "S[x],S[y];" "6$&%\"SG6#%\"xG&F$6#%\"yG" }{TEXT -1 14 " \+ a hmotnost " }{TEXT 458 3 "m ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "K tomu pot\370ebujeme spo\350\355tat derivaci a " } {XPPEDIT 18 0 "sqrt(1+`y'`^2);" "6#-%%sqrtG6#,&\"\"\"F'*$%#y'G\"\"#F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "dery:=Diff(sqrt(r^2-x^2 ),x)=diff(sqrt(r^2-x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dery G/-%%DiffG6$*$-%%sqrtG6#,&*$)%\"rG\"\"#\"\"\"F2*$)%\"xGF1F2!\"\"F2F5,$ *&F-#F6F1F5F2F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "odmoc:=s qrt(1+x^2/(r^2-x^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&odmocG*$-% %sqrtG6#,&\"\"\"F**&%\"xG\"\"#,&*$)%\"rGF-F*F**$)F,F-F*!\"\"F4F*F*" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Statick\351 momenty " }{XPPEDIT 18 0 "S[x];" "6#&%\"SG6#%\"xG" }{TEXT -1 5 " a " }{XPPEDIT 18 0 "S[y ];" "6#&%\"SG6#%\"yG" }{TEXT -1 6 " jsou" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "S[x] = rho*Int(f(x)*sqrt(1+f(x)^`\2642`),x = a .. b);" "6#/&%\"SG6#%\"xG*&%$rhoG\"\"\"-%$IntG6$*&-%\"fG6#F'F*-%%sqrtG6#,&F*F* )-F06#F'%#|_v2GF*F*/F';%\"aG%\"bGF*" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "S[y] = rho*Int(x*sqrt(1+f(x)^`'2`),x = a .. b); " "6#/&%\"SG6#%\"yG*&%$rhoG\"\"\"-%$IntG6$*&%\"xGF*-%%sqrtG6#,&F*F*)-% \"fG6#F/%#'2GF*F*/F/;%\"aG%\"bGF*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Nyn\355 ubezpe\350\355me po\350 \355ta\350, \236e polom\354r " }{XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 28 " je v\236dy kladn\375 p\370\355kazem " }{XPPEDIT 18 0 "assum e;" "6#%'assumeG" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(r>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "Sx :=rho*Int(sqrt(r^2-x^2)*odmoc,x=-r/sqrt(2)..r/sqrt(2))=rho*int(sqrt(r^ 2-x^2)*odmoc,x=-r/sqrt(2)..r/sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SxG/*&%$rhoG\"\"\"-%$IntG6$*&-%%sqrtG6#,&*$)%#r|irG\"\"#F(F(* $)%\"xGF4F(!\"\"F(-F.6#,&F(F(*&F7F4F0F8F(F(/F7;,$*&F3F(-F.6#F4F(#F8F4, $F@#F(F4F(*(F'F(F2F(FAF(" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "Sy:=rho*Int(x*odmoc,x=-r/sqr t(2)..r/sqrt(2))=rho*int(x*odmoc,x=-r/sqrt(2)..r/sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SyG/*&%$rhoG\"\"\"-%$IntG6$*&%\"xGF(-%%sq rtG6#,&F(F(*&F-\"\"#,&*$)%#r|irGF3F(F(*$)F-F3F(!\"\"F:F(F(/F-;,$*&F7F( -F/6#F3F(#F:F3,$F>#F(F3F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "m:=rho*Int(r/sqrt(r^2-x^2),x=-r/sqrt(2)..r/sqrt(2))=rho*int(r /sqrt(r^2-x^2),x=-r/sqrt(2)..r/sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG/*&%$rhoG\"\"\"-%$IntG6$*&%#r|irGF(,&*$)F-\"\"#F(F(*$)%\"x GF1F(!\"\"#F5F1/F4;,$*&F-F(-%%sqrtG6#F1F(F6,$F:#F(F1F(,$*(F'F(%#PiGF(F -F(F?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Sou\370adnice t\354\236i \232t\354 jsou :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "x - ov\341 s ou\370adnice t\354\236i\232t\354 " }{XPPEDIT 18 0 "x[T] = S[y]/m;" " 6#/&%\"xG6#%\"TG*&&%\"SG6#%\"yG\"\"\"%\"mG!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "SX:=rhs(Sy)/rhs(m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SXG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "y - ov\341 sou\370adnice t\354\236i\232t\354 " }{XPPEDIT 18 0 "y[T] = \+ S[x]/m;" "6#/&%\"yG6#%\"TG*&&%\"SG6#%\"xG\"\"\"%\"mG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "SY:=rhs(Sx)/rhs(m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SYG,$* (%#r|irG\"\"\"\"\"##F(F)%#PiG!\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "P\370\355kl ad 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Ur \350et\354 t\354\236i\232t\354 dr\341tu (homogenn\355ho) ve tvaru jedn oho oblouku cykloidy : " }}{PARA 0 "" 0 "" {TEXT -1 17 " \+ " }{XPPEDIT 18 0 "phi = a*(t-sin(t)),psi = a*(1-cos(t));" "6$/%$ph iG*&%\"aG\"\"\",&%\"tGF'-%$sinG6#F)!\"\"F'/%$psiG*&F&F',&F'F'-%$cosG6# F)F-F'" }{TEXT -1 26 " , t n\341le\236\355 do intervalu " }{XPPEDIT 18 0 "<0,2*Pi>;" "6#-%7rtable/ConstructColumnG6$\"\"!6#*&\"\"#\"\"\"%# PiGF*" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Nejprve k\370ivku nakresl\355me , " }{XPPEDIT 18 0 "a = 1;" "6#/%\"aG\"\"\" " }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot([t -sin(t),1-cos(t),t=0..2*Pi],scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3e ;cW/cSxU!#@$\"3%R#3*QgiPO*!#?7$$\"3!e+5`Gi4z#F0$\"3Sn)o<,z>E$!#>7$$\"3 =%\\Q;ZV:#)*F0$\"3!)\\'*Q6_89vF67$$\"3s^>#3I_\"zBF6$\"3#)RJ?\"3HpM\"!# =7$$\"3]e/y%)H\\tYF6$\"3eW\\7nzu&4#FA7$$\"3=rfe)=K:y(F6$\"3[]1OJeUQA>'\\WFA$\"3?x[QxJ'[l)FA7$$\"3x'ph27ORr&FA$\"3B/u(p5(f+5! #<7$$\"3['zTJP\\z;(FA$\"3q\\T#fB[i8\"Fdo7$$\"3>&p6Ev5Du)FA$\"3[:d.:#=Y E\"Fdo7$$\"3/d\"GX^+;.\"Fdo$\"3%zV1J*3Jx8Fdo7$$\"3:(HTIn>dB\"Fdo$\"3u \\AOF:B/:Fdo7$$\"3^+4r2)>AU\"Fdo$\"3O1cjZ*[Rg\"Fdo7$$\"37$*QIe8]`;Fdo$ \"3]CknM;$*3b5msp=Fdo$\"3Ie_b?/U!z\"Fdo7$$\"3]Y*4&3/k<@F do$\"3'z-\")RnIf'=Fdo7$$\"3!fRu^npAO#Fdo$\"3>XOJFO4B>Fdo7$$\"3p<*)\\F3 ICEFdo$\"3q!RS4Nij'>Fdo7$$\"3Ixg&G[%3pGFdo$\"3!ypT>+.2*>Fdo7$$\"3KE`uH +RNJFdo$\"3+E$G4>&****>Fdo7$$\"3#\\*)H\"4E.7MFdo$\"3O0B(=!Q%3*>Fdo7$$ \"3'G]t4$o$3l$Fdo$\"3?b`sYlSn>Fdo7$$\"3j:6V=/d/RFdo$\"3@:LI@iKE>Fdo7$$ \"3%HKb(Qt(*fTFdo$\"3gF)4Kh=u'=Fdo7$$\"3LY8W$y`6S%Fdo$\"3K#*z7xng%z\"F do7$$\"3\\h+c,S_CYFdo$\"3mK\"4Pfc5r\"Fdo7$$\"3Mm!QO\\L\"f[Fdo$\"3Q%y6I ke[g\"Fdo7$$\"3I$Hx9UFk0&Fdo$\"35Aa#=%)o!*\\\"Fdo7$$\"3!)>%>7\\L;D&Fdo $\"3'*f?<[\"ysP\"Fdo7$$\"3]uoz;T%QT&Fdo$\"3wZ#H\"el'3E\"Fdo7$$\"3!=\\O Gp%\\ubFdo$\"33@T_xm:H6Fdo7$$\"3:mWQ=pK4dFdo$\"3!zOHMQdI+\"Fdo7$$\"3$o 77H1%>LeFdo$\"3spxOYnF7()FA7$$\"3(z@TH0mv$fFdo$\"3ungHa'e\\W(FA7$$\"35 XGdRicHgFdo$\"3C:@W9;?khFA7$$\"3Z/qowO?-hFdo$\"3k(Q-RU*p'*\\FA7$$\"31U I8X3JhhFdo$\"3!3v60g\\1*QFA7$$\"3H)*HI_+11iFdo$\"3#fRKNh*p+HFA7$$\"3]' =)=I4JOiFdo$\"3]S\"yig@)*4#FA7$$\"35r[&G$G'*fiFdo$\"32g)*y$[icK\"FA7$$ \"3/L^'=)R*GF'Fdo$\"3-s;%yY!**\\xF67$$\"3+\"*=`-Y9!G'Fdo$\"3OR8x286`MF 67$$\"3]w>MG;w#G'Fdo$\"3Hz?W.kW/$*F07$$\"3C'ezrI&=$G'FdoF(-%'COLOURG6& %$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q!6\"Fa[l-%(SCALINGG6#%,CONSTRAINE DG-%%VIEWG6$%(DEFAULTGFj[l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Static k\351 momenty " }{XPPEDIT 18 0 "S[x],S[y];" "6$&%\"SG6#%\"xG&F$6#%\"y G" }{TEXT -1 27 " se po\350\355taj\355 dle vzorc\371 :" }{XPPEDIT 18 0 "S[x] = rho*Int(psi(t)*sqrt(phi(t)^`\2642`+psi(t)^`\2642`),t = t[ 1] .. t[2]);" "6#/&%\"SG6#%\"xG*&%$rhoG\"\"\"-%$IntG6$*&-%$psiG6#%\"tG F*-%%sqrtG6#,&)-%$phiG6#F2%#|_v2GF*)-F06#F2F;F*F*/F2;&F26#F*&F26#\"\"# F*" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 " \+ " }{XPPEDIT 18 0 "S[y] = rho*Int(phi(t)*sqrt(phi(t)^`\2642`+psi(t)^`\2642`),t = t[ 1] .. t[2]);" "6#/&%\"SG6#%\"yG*&%$rhoG\"\"\"-%$IntG6$*&-%$phiG6#%\"tG F*-%%sqrtG6#,&)-F06#F2%#|_v2GF*)-%$psiG6#F2F:F*F*/F2;&F26#F*&F26#\"\"# F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "ffi:=a*(t-sin(t));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ffiG*&%\"aG\"\"\",&%\"tGF'-%$sinG6 #F)!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ppsi:=a*(1-co s(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ppsiG*&%\"aG\"\"\",&F'F'- %$cosG6#%\"tG!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "der fi:=Diff(a*(t-sin(t)),t)=diff(a*(t-sin(t)),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&derfiG/-%%DiffG6$*&%\"aG\"\"\",&%\"tGF+-%$sinG6#F-! \"\"F+F-*&F*F+,&F+F+-%$cosGF0F1F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "derpsi:=Diff(a*(1-cos(t)),t)=diff(a*(1-cos(t)),t);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'derpsiG/-%%DiffG6$*&%\"aG\"\"\",&F +F+-%$cosG6#%\"tG!\"\"F+F0*&F*F+-%$sinGF/F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "odmoc:=sqrt(rhs(derfi)^2+rhs(derpsi)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&odmocG*$-%%sqrtG6#,&*&)%\"aG\"\"#\"\"\"), &F.F.-%$cosG6#%\"tG!\"\"F-F.F.*&F+F.)-%$sinGF3F-F.F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "odmoc:=simplify(odmoc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&odmocG*&-%%sqrtG6#\"\"#\"\"\"-F'6#,$*&)% \"aGF)F*,&!\"\"F*-%$cosG6#%\"tGF*F*F2F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "odmoc:=expand(odmoc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&odmocG*&-%%sqrtG6#\"\"#\"\"\"-F'6#,&*$)%\"aGF)F*F**&F/F*-%$co sG6#%\"tGF*!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "SX:=r ho*Int(ppsi*odmoc,t=0..2*Pi)=rho*int(ppsi*odmoc,t=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SXG/*&%$rhoG\"\"\"-%$IntG6$**%\"aGF(,&F(F (-%$cosG6#%\"tG!\"\"F(-%%sqrtG6#\"\"#F(-F56#,&*$)F-F7F(F(*&F " 0 "" {MPLTEXT 1 0 98 "SX := rho*Int(a*(1-cos(t))*sqrt(2)*sqrt(a^2 -a^2*cos(t)),t = 0 .. 2*Pi) = 32/3*rho/(a^2)^(1/2)*a^3;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#SXG/*&%$rhoG\"\"\"-%$IntG6$**%\"aGF(,&F(F(-%$ cosG6#%\"tG!\"\"F(-%%sqrtG6#\"\"#F(-F56#,&*$)F-F7F(F(*&F \+ " 0 "" {MPLTEXT 1 0 12 "assume(a>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Sx:=simplify(rhs(SX));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SxG,$*&%$rhoG\"\"\")%#a|irG\"\"#F(#\"#K\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Hmotnost se po\350\355t\341 podle vzorce \+ " }{XPPEDIT 18 0 "m = Int(sqrt(phi(t)^`\2642`+psi(t)^`\2642`),t = t [1] .. t[2]);" "6#/%\"mG-%$IntG6$-%%sqrtG6#,&)-%$phiG6#%\"tG%#|_v2G\" \"\")-%$psiG6#F0F1F2/F0;&F06#F2&F06#\"\"#" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "m:=Int(odmoc,t=0..2*Pi)=int(odmoc,t=0..2*Pi); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG/-%$IntG6$*&-%%sqrtG6#\"\"#\"\" \"-F+6#,&*$)%#a|irGF-F.F.*&F3F.-%$cosG6#%\"tGF.!\"\"F./F9;\"\"!,$%#PiG F-,$F4\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x[t]:=Pi*a; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#%\"tG*&%#PiG\"\"\"%#a|irG F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y[t]:=Sx/rhs(m);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#%\"tG,$*&%$rhoG\"\"\"%#a|irGF +#\"\"%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "P\370 \355klad 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Vypo\350\355tejte t\354\236i\232t \354 a momenty setrva\350nosti homogenn\355ho obrazce omezen\351ho k \370ivkami " }{XPPEDIT 18 0 "y = cos(x),y = x^2-Pi^2/4;" "6$/%\"yG -%$cosG6#%\"xG/F$,&*$F(\"\"#\"\"\"*&%#PiGF,\"\"%!\"\"F1" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "Nejd\370\355ve obrazec omezen\375 k \370ivkami nakresl\355me" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(\{cos(x),x^2-Pi^2/4\},x=-2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"#\"\"!$\"3]gws**))fK:!# <7$$!3MLLL$Q6G\">F-$\"3sSs$zGY9>\"F-7$$!3bmm;M!\\p$=F-$\"3c%4N&Ql!)p!* !#=7$$!3MLLL))Qj^7$$!3OLL$3yO5]\"F-$!36J%y&Go*G9#F87$ $!3&*****\\nU)*=9F-$!3!)y#)f)[Z)QXF87$$!3SLL$3WDTL\"F-$!3AKg$R3/^(oF87 $$!35++]d(Q&\\7F-$!3mapxP*R01*F87$$!3gmmmc4`i6F-$!3VCS1vG#f6\"F-7$$!3K LLLQW*e3\"F-$!33u!p!pVB)G\"F-7$$!3w++++()>'***F8$!3WANx:7;o9F-7$$!3E++ ++0\"*H\"*F8$!33PA*GW[Qj\"F-7$$!35++++83&H)F8$!3iUi`itJz\"4N%HNhA@F-7$$!3+,++v#\\N)\\F8$!3,4)3lYV!>AF-7$$!3commmCC(>%F8$! 3gQC.dEB\"H#F-7$$!39*****\\FRXL$F8$!3;kr]y&4iN#F-7$$!3t*****\\#=/8DF8$ !3PY.83tC/CF-7$$!3=mmm;a*el\"F8$!3#H'H'R?\")*RCF-7$$!3komm;Wn(o)FH$!39 R`fJN&)fCF-7$$!3IqLLL$eV(>!#?$!3;6:=-sRnCF-7$$\"3)Qjmm\"f`@')FH$!3G-h6 7!o*fCF-7$$\"3%z****\\nZ)H;F8$!3o5`$e1P3W#F-7$$\"3ckmm;$y*eCF8$!32rRlc `$pS#F-7$$\"3f)******R^bJ$F8$!3y>>T*GsuN#F-7$$\"3'e*****\\5a`TF8$!3w)) >tn?)[H#F-7$$\"3'o****\\7RV'\\F8$!3O#G#zqW&4A#F-7$$\"3Y'*****\\@fkeF8$ !3!Q#RT*omM7#F-7$$\"3_ILLL&4Nn'F8$!3+JuN0Q/A?F-7$$\"3A*******\\,s`(F8$ !33$)F8$!3]ogm!f/_x\"F-7$$\"3M*******pfa<*F8 $!3p)44K\\5bi\"F-7$$\"39HLLeg`!)**F8$!3]@a:+,Hr9F-7$$\"3w****\\#G2A3\" F-$!3C0j(z\\GiH\"F-7$$\"3;LLL$)G[k6F-$!3F-$\"3 oY&QY:1D>\"F-7$$\"\"#F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F`[l-F$6$7S7 $F($!33C9Zl$o9;%F87$F/$!3e7/ax6'QN$F87$F4$!3zZZRJc@IEF87$F:$!3Y&Q4Q1q$y)F87$Feq$\"3(*)>%**oM,K\"*F87$Fjq $\"3GK!=n\"[<\\%*F87$F_r$\"3)>\\/)y$*)eo*F87$Fdr$\"33:_R.N@j)*F87$Fir$ \"3m'G\"R')eGi**F87$F^s$\"3')*=_&40)*****F87$Fds$\"3?+@ouv&G'**F87$Fis $\"3[%3ivetu')*F87$F^t$\"3k+!zBf\">*p*F87$Fct$\"3c8etuEPb%*F87$Fht$\"3 +(QRb\"\\t\\\"*F87$F]u$\"37&pIf^mGz)F87$Fbu$\"3leQO4U0H$)F87$Fgu$\"3MQ ].Y$RY&yF87$F\\v$\"3W.o&=8![\"H(F87$Fav$\"3kH!e)>w5MnF87$Ffv$\"3%3#4C2 wqxgF87$F[w$\"3%4njf:\"R>aF87$F`w$\"3i$zNa([!Qp%F87$Few$\"3'*RF!>4cA&R F87$Fjw$\"3QfO%p2gt9$F87$F_x$\"3QaF3Cs,]BF87$Fdx$\"37mV+`c%z^\"F87$Fix $\"33CiVBHk>oFH7$F^y$!3OBvd4#*3h!*F`s7$Fcy$!3gHbge*p7v*FH7$Fhy$!3o.&zQ *G(*f " 0 "" {MPLTEXT 1 0 37 "solve(\{y=cos(x), y=x^2-Pi^2/4\},\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"yG,& *$)-%'RootOfG6$,(*$)%#_ZG\"\"#\"\"\"\"\"%*$)%#PiGF0F1!\"\"*&F2F1-%$cos G6#F/F1F6/%&labelG%$_L1GF0F1F1*&#F1F2F1F3F1F6/%\"xGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"yG^$$\"*S*[a?!\"*$!+rxwuT!\")/%\"xG^$$\"+O(3tr%F) $!+;l%\\U%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "fsolve(\{y= cos(x),y=x^2-Pi^2/4\},\{x,y\},-2..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"yG$!+%ydpv\"!#>/%\"xG$!+Fjzq:!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Jak je vid\354t Maple nen\355 v\232emocn\375 a nejlep\232 \355 \370e\232en\355 je vid\354t z obr\341zku" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Nyn\355 spo\350\355t \341me statick\351 momenty " }{XPPEDIT 18 0 "S[x],S[y];" "6$&%\"SG6# %\"xG&F$6#%\"yG" }{TEXT -1 17 " a hmotnost " }{XPPEDIT 18 0 "m;" "6#%\"mG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "vyr:=cos(x)^2-( x^2-Pi^2/4)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$vyrG,&*$)-%$cosG6 #%\"xG\"\"#\"\"\"F-*$),&*$)F+F,F-F-*&#F-\"\"%F-*$)%#PiGF,F-F-!\"\"F,F- F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sx:=1/2*rho*Int(vyr,x =-Pi/2..Pi/2)=1/2*rho*int(vyr,x=-Pi/2..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SxG/,$*&%$rhoG\"\"\"-%$IntG6$,&*$)-%$cosG6#%\"xG\"\" #F)F)*$),&*$)F3F4F)F)*&#F)\"\"%F)*$)%#PiGF4F)F)!\"\"F4F)F@/F3;,$F?#F@F 4,$F?#F)F4F)FF,$*&F(F),&*$)F?\"\"&F)#F@\"#I*&FFF)F?F)F)F)FF" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "SX:=rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SXG,$*&%$rhoG\"\"\",&*$)%#PiG\"\"&F(#!\"\"\"#I* &#F(\"\"#F(F,F(F(F(F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "vy r1:=x*(cos(x)-(x^2-Pi^2/4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%vyr 1G*&%\"xG\"\"\",(-%$cosG6#F&F'*$)F&\"\"#F'!\"\"*&#F'\"\"%F')%#PiGF.F'F 'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "vyr1 := x*(cos(x)-x^ 2+1/4*Pi^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%vyr1G*&%\"xG\"\"\", (-%$cosG6#F&F'*$)F&\"\"#F'!\"\"*&#F'\"\"%F')%#PiGF.F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Sy:=rho*Int(vyr1,x=-Pi/2..Pi/2)=rho *int(vyr1,x=-Pi/2..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SyG/*& %$rhoG\"\"\"-%$IntG6$*&%\"xGF(,(-%$cosG6#F-F(*$)F-\"\"#F(!\"\"*&#F(\" \"%F()%#PiGF4F(F(F(/F-;,$F:#F5F4,$F:#F(F4F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "SY:=rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#SYG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "vyr2:=cos( x)-(x^2-Pi^2/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%vyr2G,(-%$cosG6 #%\"xG\"\"\"*$)F)\"\"#F*!\"\"*&#F*\"\"%F*)%#PiGF-F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "m:=rho*Int(vyr2,x=-Pi/2..Pi/2)=rho* int(vyr2,x=-Pi/2..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG/*&% $rhoG\"\"\"-%$IntG6$,(-%$cosG6#%\"xGF(*$)F0\"\"#F(!\"\"*&#F(\"\"%F()%# PiGF3F(F(/F0;,$F9#F4F3,$F9#F(F3F(*&F'F(,&F3F(*&#F(\"\"'F()F9\"\"$F(F(F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "mm:=rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mmG*&%$rhoG\"\"\",&\"\"#F'*&#F'\"\"'F')%# PiG\"\"$F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "x - ov\341 a \+ y - ov\341 sou\370adnice t\354\236i\232t\354 jsou" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "xt:=SY/mm;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xtG \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "yt:=SX/mm;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ytG,$*&,&*$)%#PiG\"\"&\"\"\"#!\"\" \"#I*&#F,\"\"#F,F*F,F,F,,&F2F,*&#F,\"\"'F,)F*\"\"$F,F,F.F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "yt:=evala(yt);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ytG,$*(%#PiG\"\"\",&*$)F'\"\"%F(F(\"#:!\"\"F(,&\"#7F (*$)F'\"\"$F(F(F.#F.\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Nevlastn\355 integr\341l" }}{PARA 0 "" 0 "" {TEXT -1 154 "P\370 ipome\362me stru\350n\354, jak po\350\355t\341me nevlastn\355 integr \341l \"ru\350n\354\".Pro p\370\355m\375 v\375po\350et uva\236ujeme li mitu pro mez, kter\341 je bu\357 nevlastn\355 nebo je bodem nespojitos ti " }}{PARA 0 "" 0 "" {TEXT -1 49 "dan\351 funkce ( v na\232em popisu nap\370. horn\355 mez ) :" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(f( x),x = a .. b);" "6#-%$intG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 3 " = " }{XPPEDIT 19 1 "limit(int(f(x),x = a .. m),m = b);" "6#-%&limitG 6$-%$intG6$-%\"fG6#%\"xG/F,;%\"aG%\"mG/F0%\"bG" }{TEXT -1 69 " , obd obn\354 zach\341z\355me s nevlastn\355m integr\341lem vzhledem k doln \355 mezi." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "P\370\355klady" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 299 0 "" }{TEXT 300 9 "P\370\355klad 1" }{TEXT 303 0 "" }{TEXT -1 0 "" }{TEXT 302 1 ":" }}{PARA 0 "" 0 "" {TEXT 301 51 "Vypo\350t\354te objem t\354lesa vznikl\351ho rotac\355 k\370ivky y = " }{XPPEDIT 19 1 "arctan(x)/sqrt(1+x^2);" "6#*&-%'arctanG6#%\"xG\" \"\"-%%sqrtG6#,&F(F(*$F'\"\"#F(!\"\"" }{TEXT -1 3 " " }{TEXT 304 13 "v intervalu " }{TEXT -1 3 " (-" }{XPPEDIT 19 1 "infinity;" "6#%)infi nityG" }{TEXT -1 1 "," }{XPPEDIT 19 1 "infinity;" "6#%)infinityG" } {TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Je z\370ejm\351, \236e v \375po\350et objemu bude vlastn\354 v\375po\350et nevlastn\355ho integ r\341lu." }}{PARA 0 "" 0 "" {TEXT -1 22 "Nakresleme si obr\341zek:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot(arctan(x)/sqrt(1+x^2),x=-infin ity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%'arctanG6#%\"xG \"\"\",&F(F(*$)F'\"\"#F(F(#!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7eo7$$!\" \"\"\"!$F*F*7$$!3ommm;p0k&*!#=$!3NhjRs)y%>U!#?7$$!3wKL$37$$!3:mmm\"4m(G$)F/$!3QJ)4. 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a \+ " }{MPLTEXT 1 0 0 "" }{XPPEDIT 19 1 "int(f(x),x = a .. b);" "6#-%$intG 6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 24 " = int( f(x), x = a..b); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Pro \+ \"ru\350n\355\" v\375po\350et neur\350it\351ho integr\341lu IN = " } {XPPEDIT 19 1 "int((arctan(x)/sqrt(1+x^2))^2,x);" "6#-%$intG6$*$*&-%'a rctanG6#%\"xG\"\"\"-%%sqrtG6#,&F,F,*$F+\"\"#F,!\"\"F2F+" }{TEXT -1 32 " vyu\236\355v\341me obvykle substituci," }}{PARA 0 "" 0 "" {TEXT -1 75 "kterou m\371\236eme tak\351 ur\350it programu pro v\375po\350et na\232eho integr\341lu - p\370\355kazem " }{TEXT 305 9 "changevar" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 256 "" 0 "" {TEXT -1 18 " student[changevar]" }{TEXT 30 1 " " }{TEXT 306 6 "(f, v)" }{TEXT -1 4 "; " }{TEXT 307 6 "nebo " }{TEXT -1 15 " with(student):" }}{PARA 0 "" 0 "" {TEXT -1 96 " \+ changevar(f, v); kde " }{TEXT 308 1 "f" } {TEXT -1 26 " je vyj\341d\370en\355 substituce " }{TEXT 309 11 "f(x) \+ = g(t)" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 106 " \+ \+ " }{TEXT 310 1 "v" }{TEXT -1 26 " v\375raz Int( f(x),x = a..b)" }}{PARA 0 "" 0 "" {TEXT -1 92 " \+ nebo changevar(f, v, t); kde \+ " }{TEXT 311 1 "t" }{TEXT -1 28 " je nov\341 integra\350n\355 prom\354 nn\341" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "student[changevar] (a rctan(x)=t,Int((arctan(x)/sqrt(1+x^2))^2,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"fG6#%\"rG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$int G6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$ )-%'arctanG6#%\"xG\"\"$\"\"\"#F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$IntG6$*$)%\"tG\"\"#\"\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"tG\"\"$\" \"\"#F(F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Nap\370ed Maple dosa dil substituci, dal\232\355m p\370\355kazem pak v\375po\350et integr \341lu" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "student[changevar ] (arctan(x)=t,int((arctan(x)/sqrt(1+x^2))^2,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"tG\"\"$\"\"\"#F(F'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Tento p\370 \355klad dosad\355 substituci a hned vypo\350te" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "c hangevar(arctan(x)=t,int((arctan(x)/sqrt(1+x^2))^2,x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*$)%\"tG\"\"$\"\"\"#F(F'" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 102 "Nyn\355 pokra\350ujeme ve v\375po\350tu ur\350it \351ho integr\341lu, protp\236e se jedn\341 o integr\341l nevlastn\355 - pomoc\355 limity." }}{PARA 0 "" 0 "" {TEXT -1 37 "Nap\370ed se vr \341t\355me k p\371vodn\355 prom\354nn\351." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "changevar(t=arctan(x),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)-%'arctanG6#%\"xG\"\"$\"\"\"#F+F*" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "V=limit(%,x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"VG ,$*$)%#PiG\"\"$\"\"\"#F*\"#C" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "A toto byla dal\232\355 mo\236nost" }}{PARA 0 "" 0 "" {TEXT -1 73 "Mapl e ale um\355 vypo\350\355tat tento integr\341l p\370\355mo jako ur\350 it\375 s nevlastn\355 mez\355" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "Int((arctan(x)/sqrt(1+x^2))^2,x=0..infinity)=int((arctan(x)/sqrt (1+x^2))^2,x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG 6$*&-%'arctanG6#%\"xG\"\"#,&\"\"\"F.*$)F+F,F.F.!\"\"/F+;\"\"!%)infinit yG,$*$)%#PiG\"\"$F.#F.\"#C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 312 10 "P\370\355k lad 2:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Ur\350ete plochu rovinn\351ho obrazce omezen\351ho k\370ivkami y = " }{XPPEDIT 19 1 " exp(1)^(-3*x);" "6#)-%$expG6#\"\"\",$*&\"\"$F'%\"xGF'!\"\"" }{TEXT -1 14 ", y = 0 pro x " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 4 " <0," }{XPPEDIT 19 1 "infinity;" "6#%)infinityG" }{TEXT -1 2 "). 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