{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 " " 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Example 4-6 from the textbook: Extrema of 5x+y^2+z given x^2+y ^2+z^2=1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Maple gives two methods of solving: Explicitly using Lagrange, or internally using \"extrema\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "S:=extrema(5*x+y^2+z, x^2+y^ 2+z^2=1, \{x,y,z\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG<$-%$mi nG6$,$-%'RootOfG6#,&*$)%#_ZG\"\"#\"\"\"\"#EF2!\"\"F3#\"#:F1-%$maxGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "allvalues(S[1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$*$-%%sqrtG6#\"#E\"\"\",$F#!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "allvalues(S[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"#:\"\"#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Kind of hard to interpret this-" }}{PARA 0 "" 0 "" {TEXT -1 27 "We can solve it explicitly:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "with(linalg): #Needed to use gradient command" }}{PARA 7 "" 1 " " {TEXT -1 80 "Warning, the protected names norm and trace have been r edefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=5*x+y^2+z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(%\"xG\"\" &*$)%\"yG\"\"#\"\"\"F,%\"zGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "CS:=x^2+y^2+z^2=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#CSG/,(* $)%\"xG\"\"#\"\"\"F+*$)%\"yGF*F+F+*$)%\"zGF*F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "df:=grad(f,[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfG-%'vectorG6#7%\"\"&,$%\"yG\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dg:=grad(x^2+y^2+z^2,[x,y,z]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dgG-%'vectorG6#7%,$%\"xG\"\"#,$%\"y GF+,$%\"zGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqns1:=seq( df[i]=lambda*dg[i],i=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqns 1G6%/\"\"&,$*&%'lambdaG\"\"\"%\"xGF+\"\"#/,$%\"yGF-,$*&F*F+F0F+F-/F+,$ *&F*F+%\"zGF+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "S2:=solv e(\{eqns1,CS\},\{x,y,z,lambda\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %#S2G6$<&/%\"yG\"\"!/%\"zG-%'RootOfG6#,&*$)%#_ZG\"\"#\"\"\"\"#EF4!\"\" /%'lambdaG,$F,\"#8/%\"xG,$F,\"\"&<&/F(-F-6$,&\"#6F4*&F3F4F1F4F4/%&labe lG%%_L10G/F+#F4F3/F8F4/F<#F>F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "S3:=allvalues(S2[1]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#S3 G6$<&/%\"yG\"\"!/%\"zG,$*$-%%sqrtG6#\"#E\"\"\"#F2F1/%'lambdaG,$F-#F2\" \"#/%\"xG,$F-#\"\"&F1<&F'/F:,$F-#!\"&F1/F+,$F-#!\"\"F1/F5,$F-#FFF8" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "S4:=allvalues(S2[2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S4G6$<&/%\"yG*&^##\"\"\"\"\"#F,-%%s qrtG6#\"#AF,/%\"zGF+/%'lambdaGF,/%\"xG#\"\"&F-<&/F(*&^##!\"\"F-F,F.F,F 2F4F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(S3[1],f);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"#E\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(S3[2],f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"#E\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "4 2 0" 27 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }