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"Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 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 "Maple Outpu t" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }2 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {SECT 1 {PARA 256 "" 0 "" {TEXT 265 26 "Chapter 14\n\nSimplifi cation" }{TEXT 320 1 "\n" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 321 31 "\251 Copyright 2003 by Andr\351 Heck." }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 257 2 "1." }{TEXT -1 11 " Show that " } {XPPEDIT 18 0 "sum(1/(k^2+1),k=1..infinity)=(Pi/2)*(coth(Pi))-1/2" "6# /-%$sumG6$*&\"\"\"F(,&*$%\"kG\"\"#F(F(F(!\"\"/F+;F(%)infinityG,&*(%#Pi GF(F,F--%%cothG6#F3F(F(*&F(F(F,F-F-" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Sum(1/(k^2+1), k=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F',&*$)%\"kG\"\"#F'F'F'F'!\"\"/F+;F'%) infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&^##\"\"\"\"\"#F'-%$PsiG6#^$F'!\"\" F'F'*&^##F-F(F'-F*6#^$F'F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&^##\"\"\"\" \"#F'-%$PsiG6#^#F'F'F'*&F&F'*&%#PiGF'-%%cothG6#F/F'F'F'*&^##!\"\"F(F'- F*6#^$F'F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "combine(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"\"#!\"\"*&#F%F&F%*&%# PiGF%-%%cothG6#F+F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 258 2 "2." }{TEXT -1 23 " Expand t he expression " }{XPPEDIT 18 0 "((cos(2*x)+1)^2" "6#*$,&-%$cosG6#*&\" \"#\"\"\"%\"xGF*F*F*F*F)" }{TEXT -1 6 " into " }{XPPEDIT 18 0 "cos(2x) ^2+2*cos(2*x)+1" "6#,(*$-%$cosG6#*&\"\"#\"\"\"%\"xGF*F)F**&F)F*-F&6#*& F)F*F+F*F*F*F*F*" }{TEXT -1 59 ", i.e., expand the expression as if it was a polynomial in " }{XPPEDIT 18 0 "cos(2*x)" "6#-%$cosG6#*&\"\"#\" \"\"%\"xGF(" }{TEXT -1 47 " and do not expand the trigonometric functi on.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "(cos(2*x)+1)^2;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*$),&-%$cosG6#,$*&\"\"#\"\"\"%\"xGF,F,F,F,F,F+F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "expand(%, cos(2*x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)-%$cosG6#,$*&\"\"#\"\"\"%\"xGF,F, F+F,F,*&F+F,F&F,F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 27 "A more general alternative:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expand(%%, \+ cos);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)-%$cosG6#,$*&\"\"#\"\"\" %\"xGF,F,F+F,F,*&F+F,F&F,F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 259 2 "3." }{TEXT -1 100 " \+ Check how the following pairs of symbolic expressions can be transform ed into each other by Maple.\n" }{TEXT 260 3 "(a)" }{TEXT -1 3 " " } {XPPEDIT 18 0 "x+y+1/(x+y)" "6#,(%\"xG\"\"\"%\"yGF%*&F%F%,&F$F%F&F%!\" \"F%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "((x+y)^2+1)/(x+y))" "6#*&, &*$,&%\"xG\"\"\"%\"yGF(\"\"#F(F(F(F(,&F'F(F)F(!\"\"" }{TEXT -1 2 "\n\n " }{TEXT 261 3 "(b)" }{TEXT -1 7 " exp(" }{XPPEDIT 18 0 "x+y" "6#,&% \"xG\"\"\"%\"yGF%" }{TEXT -1 12 ") and exp(" }{XPPEDIT 18 0 "x" "6#% \"xG" }{TEXT -1 6 ") exp(" }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 3 ") \n\n" }{TEXT 262 3 "(c)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "ln(x/y)" "6 #-%#lnG6#*&%\"xG\"\"\"%\"yG!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "ln(x)-ln(y)" "6#,&-%#lnG6#%\"xG\"\"\"-F%6#%\"yG!\"\"" }{TEXT -1 2 " \n\n" }{TEXT 263 3 "(d)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "x^(y+z)" "6 #)%\"xG,&%\"yG\"\"\"%\"zGF'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "x^y *x^z" "6#*&)%\"xG%\"yG\"\"\")F%%\"zGF'" }{TEXT -1 2 "\n\n" }{TEXT 264 3 "(e)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(x^2+1)" "6#-%%sqrtG6#,&*$ %\"xG\"\"#\"\"\"F*F*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "sqrt(x-1)* sqrt(x+1)" "6#*&-%%sqrtG6#,&%\"xG\"\"\"F)!\"\"F)-F%6#,&F(F)F)F)F)" } {TEXT -1 1 "\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 294 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "x + y + 1/(x+y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,(%\"xG\"\"\"%\"yGF%*&F%F%,&F$F%F&F%!\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x=a-y, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"aG\"\"\"*&F%F%F$!\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$)%\"aG\"\" #\"\"\"F)F)F)F)F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s ubs(a=x+y, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$),&%\"xG\"\"\" %\"yGF)\"\"#F)F)F)F)F)F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "expand(%, x+y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"xG\"\"\" %\"yGF%*&F%F%,&F$F%F&F%!\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 295 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "exp(x + y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$exp G6#,&%\"xG\"\"\"%\"yGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#%\"xG\"\"\"-F %6#%\"yGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "combine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,&%\"xG\"\"\"%\"yGF(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 296 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ln(x/y);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%#lnG6#*&%\"xG\"\"\"%\"yG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(%, symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#%\"xG\"\"\"-F%6#%\"yG!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "combine(%, ln, symbolic);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#*&%\"xG\"\"\"%\"yG!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 297 3 "(d)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "x^(y+z);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#)%\"xG,&%\"yG\"\"\"%\"zGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*&)%\"xG%\"yG\"\"\")F%%\"zGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "combine(%, power, symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#)%\"xG,&%\"yG\"\"\"%\"zGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 298 3 "(e)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sqrt(x^2-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&* $)%\"xG\"\"#\"\"\"F)F)!\"\"#F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "factor(%^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&% \"xG\"\"\"F&!\"\"F&,&F%F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "map(sqrt, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\" \"F&!\"\"#F&\"\"#,&F%F&F&F&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sqrt(expand(%^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$)%\" xG\"\"#\"\"\"F)F)!\"\"#F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 266 2 "4." }{TEXT -1 46 " Simp lify the following symbolic expressions.\n" }{TEXT 267 3 "(a)" }{TEXT -1 4 " " }{XPPEDIT 18 0 "(exp(x) + x)/(exp(2*x)+2*x*exp(x)+x^2)" "6 #*&,&-%$expG6#%\"xG\"\"\"F(F)F),(-F&6#*&\"\"#F)F(F)F)*(F.F)F(F)-F&6#F( F)F)*$F(F.F)!\"\"" }{TEXT -1 3 "\n\n(" }{TEXT 268 2 "b)" }{TEXT -1 4 " " }{XPPEDIT 18 0 "(x^5+40x^4+595x^3+3905x^2+9680x+1331)^(1/3)" "6# ),.*$%\"xG\"\"&\"\"\"*&\"#SF(*$F&\"\"%F(F(*&\"$&fF(*$F&\"\"$F(F(*&\"%0 RF(*$F&\"\"#F(F(*&\"%!o*F(F&F(F(\"%J8F(*&F(F(F0!\"\"" }{TEXT -1 3 "\n \n\n" }{TEXT 269 3 "(c)" }{TEXT -1 4 " " }{XPPEDIT 18 0 "(x-2)^(3/2 )/(x^2+4x+4)^(1/4)" "6#*&),&%\"xG\"\"\"\"\"#!\"\"*&\"\"$F'F(F)F'),(*$F &F(F'*&\"\"%F'F&F'F'F0F'*&F'F'F0F)F)" }{TEXT -1 3 "\n\n\n" }{TEXT 270 3 "(d)" }{TEXT -1 4 " " }{XPPEDIT 18 0 "(sqrt(x)-y)/(x-y^2)" "6#*&, &-%%sqrtG6#%\"xG\"\"\"%\"yG!\"\"F),&F(F)*$F*\"\"#F+F+" }{TEXT -1 3 "\n \n\n" }{TEXT 271 3 "(e)" }{TEXT -1 4 " " }{XPPEDIT 18 0 "1/(2+5^(1/ 3))" "6#*&\"\"\"F$,&\"\"#F$)\"\"&*&F$F$\"\"$!\"\"F$F+" }{TEXT -1 2 "\n \n" }{TEXT 272 3 "(f)" }{TEXT -1 4 " " }{XPPEDIT 18 0 "cos(x+y)+sin (x)*sin(y)+2^(x+y)" "6#,(-%$cosG6#,&%\"xG\"\"\"%\"yGF)F)*&-%$sinG6#F(F )-F-6#F*F)F))\"\"#,&F(F)F*F)F)" }{TEXT -1 2 "\n\n" }{TEXT 273 3 "(g)" }{TEXT -1 4 " " }{XPPEDIT 18 0 "2*cos^2*(x)-cos*2x" "6#,&*(\"\"#\" \"\"*$%$cosGF%F&%\"xGF&F&*(F(F&F%F&F)F&!\"\"" }{TEXT -1 1 "\n" }} {SECT 1 {PARA 5 "" 0 "" {TEXT 299 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "(exp(x)+x)/(exp(2*x)+2*x*exp(x)+x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%$expG6#%\"xG\"\"\"F(F)F),(-F&6#,$*&\"\"#F)F(F)F)F)*(F/F)F( F)F%F)F)*$)F(F/F)F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&-%$expG6# %\"xGF$F)F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 300 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "(x^5+40*x^4+595*x^3+3905*x^2+9680*x+1331)^(1/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),.*$)%\"xG\"\"&\"\"\"F**&\"#SF*)F(\"\"%F*F**&\"$&fF* )F(\"\"$F*F**&\"%0RF*)F(\"\"#F*F**&\"%!o*F*F(F*F*\"%J8F*#F*F2F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)*&,(*$)%\"xG\"\"#\"\"\"F+*&\"\"(F+F)F+F+F+F+F+) ,&F)F+\"#6F+\"\"$F+#F+F1F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(%, symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"x G\"\"\"\"#6F&F&),(*$)F%\"\"#F&F&*&\"\"(F&F%F&F&F&F&#F&\"\"$F&" }}} {PARA 0 "" 0 "" {TEXT -1 24 "Linger upon the keyword " }{TEXT 0 8 "sym bolic" }{TEXT -1 24 ". Look at the following:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "(x^3)^(1/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *$)*$)%\"xG\"\"$\"\"\"#F)F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(%, assume=positive);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify(%%, assu me=negative);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#!\"\"%\"xG\" \"\",&F(F(*&\"\"$#F(F%^#F(F(F(F(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 301 3 "(c)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "(x-2)^(3/2)/(x^2-4*x+4)^(1/4);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"\"\"#!\"\"#\"\"$F',(*$)F%F'F&F&*& \"\"%F&F%F&F(F/F&#F(F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s implify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"\"\"#!\" \"#\"\"$F'*$)F$F'F&#F(\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(%, symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG \"\"\"\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 302 3 "(d)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "(sqrt(x)-y)/(x-y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$%\"xG# \"\"\"\"\"#F(%\"yG!\"\"F(,&F&F(*$)F*F)F(F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x=s^2, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$*$)%\"sG\"\"#\"\"\"#F*F)F*%\"yG!\"\"F*,&F&F**$)F,F)F*F-F-" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(%, assume=positive );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&%\"sGF$%\"yGF$!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(s=sqrt(x), %);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*$%\"xG#F$\"\"#F$%\"yGF$! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 303 3 "(e)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "1/(2+5^(1/3)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&\"\"#F$*$)\"\"&#F$\"\" $F$F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rationalize(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"\"%\"#8\"\"\"*(\"\"#F'F&!\" \"\"\"&#F'\"\"$F**&F&F*F+#F)F-F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 304 3 "(f)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "cos(x+y)+sin(x)*sin(y)+2^(x+y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$cosG6#,&%\"xG\"\"\"%\"yGF)F)*&-%$sinG6#F(F )-F-6#F*F)F))\"\"#F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "e xpand(%, 2^(x+y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$cosG6#%\" xG\"\"\"-F&6#%\"yGF)F))\"\"#,&F(F)F,F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 305 3 "(g)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "2*cos(x)^2-cos(2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\")-%$cosG6#%\"xGF%F&F&-F)6#,$*&F%F&F+F&F& !\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 274 2 "5." } {TEXT -1 60 " Solve the following zero-equivalence problems with Maple .\n\n" }{TEXT 275 3 "(a)" }{TEXT -1 4 " " }{XPPEDIT 18 0 "(2^(1/3)+ 4^(1/3))^3-6(2^(1/3)+4^(1/3)) -6=0" "6#/,(*$,&)\"\"#*&\"\"\"F*\"\"$!\" \"F*)\"\"%*&F*F*F+F,F*F+F*-\"\"'6#,&)F(*&F*F*F+F,F*)F.*&F*F*F+F,F*F,F1 F,\"\"!" }{TEXT -1 2 "\n\n" }{TEXT 276 3 "(b)" }{TEXT -1 4 " " } {XPPEDIT 18 0 "ln(tan(x/2+Pi/4))-arcsinh(tan(x))=0" "6#/,&-%#lnG6#-%$t anG6#,&*&%\"xG\"\"\"\"\"#!\"\"F.*&%#PiGF.\"\"%F0F.F.-%(arcsinhG6#-F)6# F-F0\"\"!" }{TEXT -1 1 "\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 306 3 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "(2^(1/3)+4^(1/3))^3 - 6*(2^(1/3)+4^ (1/3)) - 6 = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**$),&*$)\"\"##\" \"\"\"\"$F,F,*$)\"\"%F+F,F,F-F,F,*&\"\"'F,F)F,!\"\"*&F2F,F/F,F3F2F3\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "testeq(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "expand(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**( \"\"$\"\"\")\"\"##F)F&F')\"\"%#F'F&F'F'*(F&F')F)F-F')F,F*F'F'*&\"\"'F' F/F'!\"\"*&F2F'F+F'F3\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 307 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ln(tan(1/2*x+Pi/4)) - \+ arcsinh(tan(x)) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#lnG6#-%$ tanG6#,&*&\"\"#!\"\"%\"xG\"\"\"F0*&\"\"%F.%#PiGF0F0F0-%(arcsinhG6#-F)6 #F/F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "testeq(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}}{PARA 0 "" 0 "" {TEXT -1 115 "By differentiation we show that the left-hand side is constant. B y substitution we show that the constant equals 0." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 13 "f := lhs(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&-%#lnG6#-%$tanG6#,&*&\"\"#!\"\"%\"xG\"\"\"F1*&\"\"%F/%#P iGF1F1F1-%(arcsinhG6#-F*6#F0F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(f, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&#\"\"\"\" \"#F'*&F&F'*$)-%$tanG6#,&*&F(!\"\"%\"xGF'F'*&\"\"%F1%#PiGF'F'F(F'F'F'F 'F,F1F'*$,&F'F'*$)-F-6#F2F(F'F'F&F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&#\" \"\"\"\"#F&*&F&F&,&F&F&-%$tanG6#,$*&F'!\"\"%\"xGF&F&F&F/F&F&*&#F&F'F&* &F)F/F*F&F&F/*&F%F&*&F&F&,&F&F&F*F/F/F&F&*&F%F&*&F6F/F*F&F&F&*$,&F&F&* $)-F+6#F0F'F&F&F%F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simp lify(%, symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eval(f, x=0);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 0 "" 0 "" {TEXT 277 2 "6." }{TEXT -1 60 " Use Maple to check the following trigo nometric identities\n\n" }{TEXT 278 3 "(a)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x+sin*y=2*sin((x+y)/2)*cos((x-y)/2)" "6#/,&*&%$sinG\"\"\"%\" xGF'F'*&F&F'%\"yGF'F'*(\"\"#F'-F&6#*&,&F(F'F*F'F'F,!\"\"F'-%$cosG6#*&, &F(F'F*F1F'F,F1F'" }{TEXT -1 2 "\n\n" }{TEXT 279 3 "(b)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(5*x)=5sin x-20 sin^3x+16sin^5x" "6#/-%$sinG6#*& \"\"&\"\"\"%\"xGF),(*(F(F)F%F)F*F)F)*(\"#?F)*$F%\"\"$F)F*F)!\"\"*(\"#; F)*$F%F(F)F*F)F)" }{TEXT -1 2 "\n\n" }{TEXT 280 3 "(c)" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "cot^2x+1=csc^2x" "6#/,&*&%$cotG\"\"#%\"xG\"\"\"F)F)F )*&%$cscGF'F(F)" }{TEXT -1 2 "\n\n" }{TEXT 281 3 "(d)" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "tan x+tan y=sin(x+y)/(cos(x)*cos(y))" "6#/,&*&%$tanG \"\"\"%\"xGF'F'*&F&F'%\"yGF'F'*&-%$sinG6#,&F(F'F*F'F'*&-%$cosG6#F(F'-F 26#F*F'!\"\"" }{TEXT -1 2 "\n\n" }{TEXT 282 3 "(e)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "cos^6x+sin^6x=1-3sin^2x*cos^2x" "6#/,&*&%$cosG\"\"'%\"x G\"\"\"F)*&%$sinGF'F(F)F),&F)F)*,\"\"$F)*$F+\"\"#F)F(F)F&F0F(F)!\"\"" }{TEXT -1 2 "\n\n" }{TEXT 283 3 "(f)" }{TEXT -1 1 " " }{XPPEDIT 18 0 " sinh (2*x)=2*(tanh x)/(1-tanh^2x)" "6#/-%%sinhG6#*&\"\"#\"\"\"%\"xGF)* (F(F)*&%%tanhGF)F*F)F),&F)F)*&F-F(F*F)!\"\"F0" }{TEXT -1 2 "\n\n" } {TEXT 284 3 "(g)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(sin(2x)+sin(2y))/(c os (2x)+cos(2y))=tan(x+y)" "6#/*&,&-%$sinG6#*&\"\"#\"\"\"%\"xGF+F+-F'6 #*&F*F+%\"yGF+F+F+,&-%$cosG6#*&F*F+F,F+F+-F36#*&F*F+F0F+F+!\"\"-%$tanG 6#,&F,F+F0F+" }{TEXT -1 1 "\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 308 3 " (a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "2*(sin((x+y)/2)*(cos((x-y)/2)));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"-%$sinG6#,&*&F%!\"\"% \"xGF&F&*&F%F,%\"yGF&F&F&-%$cosG6#,&*&F%F,F-F&F&*&F%F,F/F&F,F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "combine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$sinG6#%\"xG\"\"\"-F%6#%\"yGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 309 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "sin(5*x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$sinG6#,$*&\"\"&\"\"\"%\"xGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "expand(sin(5*x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"#;\"\"\"-%$sinG6#%\"xGF&)-%$cosGF)\"\"%F&F&*(\"#7 F&F'F&)F,\"\"#F&!\"\"F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "simplify(%, \{sin(x)^2+cos(x)^2=1\}, [cos(x),sin(x)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"&\"\"\"-%$sinG6#%\"xGF&F&*&\"#;F&)F' F%F&F&*&\"#?F&)F'\"\"$F&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 60 "By the \+ way, this last command would have worked immediately." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "simplify(%%%, \{sin(x)^2+cos(x)^2=1\}, [c os(x),sin(x)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"&\"\"\"-%$s inG6#%\"xGF&F&*&\"#;F&)F'F%F&F&*&\"#?F&)F'\"\"$F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 310 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "cot(x)^2 + 1 = csc(x)^2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)-%$cotG6#%\"xG\"\"#\"\"\"F,F,F,* $)-%$cscGF)F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "testeq(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "convert(%%, sincos);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%$cosG6#%\"xG\"\"#-%$sinGF(!\"#\"\"\"F.F.*&F.F.*$ )F+F*F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&*$)-%$cosG6#%\"xG\"\"#\"\"\"F -*$)-%$sinGF*F,F-F-F-F0!\"#*&F-F-*$F/F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(cos(x)^2=1-sin(x)^2, %);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&\"\"\"F%*$)-%$sinG6#%\"xG\"\"#F%!\"\"F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 311 3 "(d)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "tan(x)+tan(y) = sin(x+ y)/(cos(x)*cos(y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%$tanG6#%\" xG\"\"\"-F&6#%\"yGF)*(-%$sinG6#,&F(F)F,F)F)-%$cosGF'!\"\"-F3F+F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "testeq(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "simplify( lhs(%%) - expand(rhs(%%)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 14 "Alternatively:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "tan(x) + tan(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$tanG6#%\"xG\"\"\"-F%6#%\"yGF(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "convert(%, sincos);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%\"xG\"\"\"-%$cosGF'!\"\"F)*&-F&6#%\"yGF)- F+F/F,F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&*&-%$sinG6#%\"xG\"\"\"-%$cosG6#%\" yGF*F**&-F'F-F*-F,F(F*F*F*F1!\"\"F+F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "combine(numer(%)) / denom(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$sinG6#,&%\"xG\"\"\"%\"yGF)F)-%$cosG6#F(!\"\"-F,6#F *F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 312 3 "(e)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "cos(x)^6 + sin (x)^6 = 1 - 3*sin(x)^2*cos(x)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/, &*$)-%$cosG6#%\"xG\"\"'\"\"\"F,*$)-%$sinGF)F+F,F,,&F,F,*(\"\"$F,)F/\" \"#F,)F'F5F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "testeq (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "simplify(lhs(%%) - rhs(%%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 313 3 "(f)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "si nh(2*x) = 2*tanh(x)/(1-tanh(x)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%%sinhG6#,$*&\"\"#\"\"\"%\"xGF*F*,$*(F)F*-%%tanhG6#F+F*,&F*F**$)F.F) F*!\"\"F4F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "testeq(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "simplify(lhs(%%) - rhs(%%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 14 "Alternatively:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "2*tanh(x)/(1-tanh(x)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"-%%tanhG6#%\"xGF&,&F&F&*$)F'F%F&!\" \"F.F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "convert(%, sincos );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\"-%%sinhG6#%\"xGF& -%%coshGF)!\"\",&F&F&*&F'F%F+!\"#F-F-F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*( \"\"#\"\"\"-%%sinhG6#%\"xGF&-%%coshGF)F&F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "combine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%s inhG6#,$*&\"\"#\"\"\"%\"xGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 314 3 "(g)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "(sin(2*x)+sin(2*y))/(cos (2*x)+cos(2*y))=tan(x+y);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&-%$sinG6#,$*&\"\"#\"\"\"%\"xGF,F ,F,-F'6#,$*&F+F,%\"yGF,F,F,F,,&-%$cosGF(F,-F5F/F,!\"\"-%$tanG6#,&F-F,F 2F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "testeq(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "simplify(expand(lhs(%%)) - expand(rhs(%%)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 285 2 "7." } {TEXT -1 33 " Verify with Maple the equality " }{XPPEDIT 18 0 "Pi/4=4 *arctan(1/5)-arctan(1/239)" "6#/*&%#PiG\"\"\"\"\"%!\"\",&*&F'F&-%'arct anG6#*&F&F&\"\"&F(F&F&-F,6#*&F&F&\"$R#F(F(" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "Pi/4 = 4*arctan(1/5) - arctan(1/239);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,$*&\"\"%!\"\"%#PiG\"\"\"F),&*&F&F)-%'arctanG6# #F)\"\"&F)F)-F-6##F)\"$R#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "combine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\"\"%!\"\"%#Pi G\"\"\"F)F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 293 2 "8." }{TEXT -1 111 " Compute the followi ng indefinite integrals and check the answers through differentiation \+ and simplification.\n\n" }{TEXT 292 3 "(a)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(2/sqrt(4+x^2),x)" "6#-%$intG6$*&\"\"#\"\"\"-%%sqrtG6#,&\"\"% F(*$%\"xGF'F(!\"\"F/" }{TEXT -1 2 "\n\n" }{TEXT 291 3 "(b)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(sqrt((1-cx^2)^3),x)" "6#-%$intG6$-%%sqrtG6#* $,&\"\"\"F+*$%#cxG\"\"#!\"\"\"\"$%\"xG" }{TEXT -1 2 "\n\n" }{TEXT 290 3 "(c)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(1/(x^4-1),x)" "6#-%$intG6$ *&\"\"\"F',&*$%\"xG\"\"%F'F'!\"\"F,F*" }{TEXT -1 2 "\n\n" }{TEXT 289 3 "(d)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(1/(x^4-4),x)" "6#-%$intG6$ *&\"\"\"F',&*$%\"xG\"\"%F'F+!\"\"F,F*" }{TEXT -1 2 "\n\n" }{TEXT 288 3 "(e)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(sin*3*x*cos2*x,x)" "6#-%$i ntG6$*,%$sinG\"\"\"\"\"$F(%\"xGF(%%cos2GF(F*F(F*" }{TEXT -1 1 "\n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 319 3 "( a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Integrate(2/sqrt(4+x^2), x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"#\"\"\",&\"\"%F)*$)% \"xGF(F)F)#!\"\"F(F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "va lue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"-%(arcsinhG6 #,$*&F%!\"\"%\"xGF&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\",&\" \"%F&*$)%\"xGF%F&F&#!\"\"F%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 318 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Integrate(sqrt((1-c*x^2)^3), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$*$),&\"\"\"F**&%\"cGF*)%\"xG\"\"#F*!\"\"\"\" $F*#F*F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\")F&*,,$*$),&F&!\"\"*&% \"cGF&)%\"xG\"\"#F&F&\"\"$F&F-#F&F2,(**F3F&F1F&,&F&F&F.F-F4F/F4F&*&F3F &-%'arctanG6#*(F/F4F1F&F7#F-F2F&F&**F2F&F1F&)F7#F3F2F&F/F4F&F&F,F-F7F= F/F=F&F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\")F&*,,$*$),&F&!\"\"*&% \"cGF&)%\"xG\"\"#F&F&\"\"$F&F-#F&F2,(**\"\"&F&F1F&,&F&F&F.F-F4F/F4F-*& F3F&-%'arctanG6#*(F/F4F1F&F8#F-F2F&F-**F2F&)F1F3F&F8F4)F/#F3F2F&F&F&F, F-F8F>F/F>F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(%, x );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,**&#\"\"$\"\")\"\"\"*.,$*$),&F( !\"\"*&%\"cGF()%\"xG\"\"#F(F(F&F(F.#F.F3,(**\"\"&F(F2F(,&F(F(F/F.#F(F3 F0F9F.*&F&F(-%'arctanG6#*(F0F9F2F(F8F4F(F.**F3F()F2F&F(F8F9)F0#F&F3F(F (F(F-F(F8F4F0F9F2F(F(F.*.F'F.F*F9,,*(F7F(F8F9F0F9F.**F7F(F2F3F8F4F0FBF (*(F&F(,&*&F0F9F8F4F(*(F0FBF2F3F8#!\"$F3F(F(,&F(F(*(F0F(F2F3F8F.F(F.F. **\"\"'F(F1F(F8F9FAF(F(**F3F(F2\"\"%F8F4F0#F7F3F.F(F-F.F8F4F0F4F(*&#F( FRF(*.F*F9F5F(F-!\"#F8F4F0F9F2F(F(F.*&#F(F'F(*.F*F9F5F(F-F.F8FKF0F9F2F (F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 91 "We check whether the difference of derivative of the integral minus the integrand equals 0:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simplify(% - sqrt((1-c*x^2)^ 3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 317 3 "(c)" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Integrate(1/(x^4-1),x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F',&*$)%\"xG\"\"%F'F'F'!\"\"F-F+" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%(arctanhG6#%\"xGF&!\"\"*&#F&F'F&- %'arctanGF*F&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%*&\"\"#F%,&F%F%*$)% \"xGF'F%!\"\"F%F,F,*&F%F%*&F'F%,&F%F%F)F%F%F,F," }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "normal(%, expanded);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*$)%\"xG\"\"%F$F$F$!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 315 3 "(d)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Integrate(1/(x^4-4), x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F',&*$)%\"xG\"\"%F'F' F,!\"\"F-F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\")F&*&\"\"##F&F)-%(arcta nhG6#,$*(F)!\"\"%\"xGF&F)F*F&F&F&F0*&#F&F'F&*&F)F*-%'arctanGF-F&F&F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(%, x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%*&\"\")F%,&F%F%*&\"\"#!\"\"%\"xGF*F+ F%F+F+*&F%F%*&F'F%,&F%F%*&F*F+F,F*F%F%F+F+" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "normal(%, expanded);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*$)%\"xG\"\"%F$F$F)!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 316 3 "(e)" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Integrate(sin(3*x)*cos(2*x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$sinG6#,$*&\"\"$\"\"\"%\"xGF-F- F--%$cosG6#,$*&\"\"#F-F.F-F-F-F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\" \"\"#5F&-%$cosG6#,$*&\"\"&F&%\"xGF&F&F&!\"\"*&#F&\"\"#F&-F)6#F.F&F/" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%$sinG6#,$*&\"\"&F&%\"xGF&F&F&F &*&F%F&-F)6#F.F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs( 5*x=y, 2*%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$sinG6#%\"yG\"\"\" -F%6#%\"xGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "trigsubs(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#,$*(\"\"#\"\"\"-%$sinG6#,&*&F&! \"\"%\"yGF'F'*&F&F-%\"xGF'F'F'-%$cosG6#,&*&F&F-F.F'F'*&F&F-F0F'F-F'F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eval(op(%)/2, y=5*x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$sinG6#,$*&\"\"$\"\"\"%\"xGF*F *F*-%$cosG6#,$*&\"\"#F*F+F*F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 131 "Easi er than rebuilding the integrand is checking whether the difference of derivative of the integral minus the integrand equals 0." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "integrate(sin(3*x)*cos(2*x), x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"#5F&-%$cosG6#,$*&\"\"&F& %\"xGF&F&F&!\"\"*&#F&\"\"#F&-F)6#F.F&F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(%, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&# \"\"\"\"\"#F&-%$sinG6#,$*&\"\"&F&%\"xGF&F&F&F&*&F%F&-F)6#F.F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "expand(% - sin(3*x)*cos(2*x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 287 2 "9." } {TEXT -1 30 " Integrate the real function " }{XPPEDIT 18 0 "x->1/((a* x+b)^2*(c*x+d)^2)" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"*&\"\"\"F,*& ,&*&%\"aGF,F%F,F,%\"bGF,\"\"#,&*&%\"cGF,F%F,F,%\"dGF,F2!\"\"F*F*F*" } {TEXT -1 92 " and bring the result into the following form.\n\n \+ " }{XPPEDIT 18 0 "(2*a*c*ln((c*x+ d)/(a*x+b)))/(a*d-b*c)^3-(2*a*c*x+a*d+b*c)/((a*d-b*c)^2*(a*x+b)*(c*x+d ))" "6#,&*,\"\"#\"\"\"%\"aGF&%\"cGF&-%#lnG6#*&,&*&F(F&%\"xGF&F&%\"dGF& F&,&*&F'F&F/F&F&%\"bGF&!\"\"F&*$,&*&F'F&F0F&F&*&F3F&F(F&F4\"\"$F4F&*&, (**F%F&F'F&F(F&F/F&F&*&F'F&F0F&F&*&F3F&F(F&F&F&*(,&*&F'F&F0F&F&*&F3F&F (F&F4F%,&*&F'F&F/F&F&F3F&F&,&*&F(F&F/F&F&F0F&F&F4F4" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Integrate( 1/((a*x+b)^2*(c*x+d)^2), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Int G6$*&\"\"\"F'*&),&*&%\"aGF'%\"xGF'F'%\"bGF'\"\"#F'),&*&%\"cGF'F-F'F'% \"dGF'F/F'!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(% );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,**(%\"cG\"\"\",(*&)F%\"\"#F&)% \"bGF*F&F&*&)%\"dGF*F&)%\"aGF*F&F&*,F*F&F%F&F,F&F1F&F/F&!\"\"F3,&*&F%F &%\"xGF&F&F/F&F3F3*,F*F&F%F&F1F&,**,\"\"$F&F)F&F+F&F/F&F1F&F&*,F:F&F.F &F0F&F%F&F,F&F3*&)F/F:F&)F1F:F&F&*&)F%F:F&)F,F:F&F3F3-%#lnG6#,&*&F1F&F 6F&F&F,F&F&F3*,F*F&F%F&F1F&F8F3-FC6#F4F&F&*(F1F&F'F3FEF3F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "logpart := select(has, %, ln);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(logpartG,&*,\"\"#\"\"\"%\"cGF(%\"aG F(,**,\"\"$F()F)F'F()%\"bGF'F(%\"dGF(F*F(F(*,F-F()F1F'F()F*F'F(F)F(F0F (!\"\"*&)F1F-F()F*F-F(F(*&)F)F-F()F0F-F(F5F5-%#lnG6#,&*&F*F(%\"xGF(F(F 0F(F(F5*,F'F(F)F(F*F(F+F5-F=6#,&*&F)F(FAF(F(F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "rationalpart := %% - %;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-rationalpartG,&*(%\"cG\"\"\",(*&)F'\"\"#F()%\"bGF ,F(F(*&)%\"dGF,F()%\"aGF,F(F(*,F,F(F'F(F.F(F3F(F1F(!\"\"F5,&*&F'F(%\"x GF(F(F1F(F5F5*(F3F(F)F5,&*&F3F(F8F(F(F.F(F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "factor(logpart/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"aG\"\"\"%\"cGF&,&-%#lnG6#,&*&F%F&%\"xGF&F&%\"bGF&F&-F*6#, &*&F'F&F.F&F&%\"dGF&!\"\"F&,&*&F'F&F/F&F5*&F4F&F%F&F&!\"$F5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "logpart := 2*combine(%, ln, \+ integer, symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(logpartG,$*, \"\"#\"\"\"%\"aGF(%\"cGF(,&*&F*F(%\"bGF(!\"\"*&%\"dGF(F)F(F(!\"$-%#lnG 6#*&,&*&F)F(%\"xGF(F(F-F(F.,&*&F*F(F8F(F(F0F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "rationalpart := factor(rationalpart);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-rationalpartG,$**,(**\"\"#\"\"\"%\" cGF*%\"aGF*%\"xGF*F**&F+F*%\"bGF*F**&%\"dGF*F,F*F*F*,&F.!\"\"F0F*!\"#, &*&F+F*F-F*F*F1F*F3,&*&F,F*F-F*F*F/F*F3F3" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "solution = logpart + rationalpart;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)solutionG,&*,\"\"#\"\"\"%\"aGF(%\"cGF(,&*&F*F(% \"bGF(!\"\"*&%\"dGF(F)F(F(!\"$-%#lnG6#*&,&*&F)F(%\"xGF(F(F-F(F.,&*&F*F (F8F(F(F0F(F(F(F(**,(**F'F(F*F(F)F(F8F(F(F,F(F/F(F(F+!\"#F9F.F6F.F." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 286 2 "10" }{TEXT -1 34 ". Implement the expansion routine " }{TEXT 0 11 "expand/coth" }{TEXT -1 24 " such that it resembles " } {TEXT 0 10 "expand/cot" }{TEXT -1 8 ", i.e., " }{TEXT 0 6 "expand" } {TEXT -1 191 " should expand hyperbolic cotangents in terms of hyperbo lic cotangents instead of hyperbolic sines and cosines. Set things up \+ so that your expansion routine overrules the built-in procedure.\n" }} {PARA 0 "" 0 "" {TEXT -1 53 "First we look at the source codes of both procedures." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(verboseproc=3):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "print(`expand/cot`);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#f*6#%\"yG6+%\"xG%\"nG%\"tG%\"iG%\"SG% \"cG%\"NG%\"DG%5NumericEventHandlerTG6#%fnCopyright~(c)~1995~Waterloo~ Maple~Inc.~All~rights~reserved.G6\"C%>8$-%'expandG6#9$@$0F5F9O-F76#-%$ cotG6#F5@--%%typeG6$F5%\"+GC+>8%-%%nopsGFA>8&7#-%#opGFA>8,-%4NumericEv entHandlerG6#/.%1division_by_zeroG.%*exceptionGZ(>FM7#-%$seqG6$-F@6#8' /F]oFMQDnumeric~exception:~division~by~zeroF2O-F@6#-%(convertG6$FM.FFF 2YF2-FT6#FR>FM-F76#FM?(F]o\"\"!\"\"\"FI%%trueGC%>8)-&%)combinatG6#.%'c hooseG6$FM,&FIF_pF]o!\"\">&8(F\\o-Fdo6$-%$mapG6%FdoFcp%\"*GFF@$2F_p-%% iremG6$F]o\"\"%>F^q,$F^qF\\q>8*-Fdo6$7#-Fin6$&F_q6#,$*&\"\"#F_pF]oF_pF _p/F]o;F^p-%%iquoG6$FIFirFF>8+-Fdo6$7#-Fin6$&F_q6#,&*&FirF_pF]oF_pF_pF _pF\\q/F]o;F_p-F]s6$,&FIF_pF_pF_pFirFF*&F_rF_pF`sF\\q3-FD6$F5Feq-FD6$- FP6$F_pF5%(integerGC(>FI-%$absG6#Fet>FM-F@6#*&F5F_pFIF\\q?(F]oF^pF_pFI F`pC$>F^q*&-%)binomialG6$FIF[qF_p)FMF[qF_p@$Fgq>F^qF]r>F_rF`r>F`sFasF_ t-FD6$F5%(nonrealG*&,&*(^#F\\qF_p-F@6#-%#ReGFAF_p-%%cothG6#-%#ImGFAF_p F_pF_pF\\qF_p,&FdvF_p*&FcvF_pFhvF_pF_pF\\q3Fat-FD6$FetF_v-F76#-F@6#,&* &-FgvF\\uF_p-%'subsopG6$/F_pF_pF5F_pF_p*(-F\\wF\\uF_pFiwF_p^#F_pF_pF_p -FD6$F5.-%)specfuncG6$%)anythingG%#lnG*(FcvF_p,&)Fet^#FirF_pF_pF_pF_p, &FjxF_pF_pF\\qF\\qF?F2F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "print(`expand/coth`);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG 6#%\"yG6#%aoCopyright~(c)~1991~by~the~University~of~Waterloo.~All~righ ts~reserved.G6\"C$>8$-%'expandG6#9$@'55-%%typeG6$F-%\"+G3-F66$F-%\"*G- F66$-%#opG6$\"\"\"F-<$%(integerG%(nonrealG-F66$F-FE*&-F/6#-%%coshG6#F- FB-F/6#-%%sinhGFM!\"\"-F66$,$*&\"\"#FBF-FBFB%%nameG*&-FQ6#FUFB,&-FLFen FBFBFRFR-%%cothGFMF*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "expand(coth(x+y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&-%%coshG6 #%\"xG\"\"\"-F'6#%\"yGF*F**&-%%sinhGF(F*-F0F,F*F*F*,&*&F/F*F+F*F**&F&F *F1F*F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "expand(coth (x+y+z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,**(-%%coshG6#%\"xG\"\" \"-F'6#%\"yGF*-F'6#%\"zGF*F**(F&F*-%%sinhGF,F*-F3F/F*F**(-F3F(F*F2F*F. F*F**(F6F*F+F*F4F*F*F*,**(F6F*F+F*F.F*F**(F6F*F2F*F4F*F**(F&F*F2F*F.F* F**(F&F*F+F*F4F*F*!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The new code:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 788 "`expand/coth` := proc(y)\nlocal x, n, t, i, S, c, N, D;\nx := expand(y);\nif x <> y then return expand(coth(x)) e nd if;\nif type(x,`+`) then\n n := nops(x);\n t := [op(x)];\n t := \+ expand([seq(coth(i),i=t)]);\n for i from 0 to n do\n c := combinat ['choose'](t,n-i);\n S[i] := convert(map(convert,c,`*`), `+`);\n e nd do;\n N := convert([seq(S[2*i], i=0..iquo(n,2))], `+`); \n D := c onvert([seq(S[2*i-1], i=1..iquo(n+1,2))], `+`);\n N/D # Don't expand \+ the result\nelif type(x,`*`) and type(op(1,x),integer) then\n n := ab s(op(1,x));\n t := coth(x/n);\n for i from 0 to n do\n S[i] := bi nomial(n,n-i)*t^(n-i);\n end do;\n N := convert([seq(S[2*i], i=0..iq uo(n,2))], `+`); \n D := convert([seq(S[2*i-1], i=1..iquo(n+1,2))], ` +`);\n N/D # Don't expand the result\nelse coth(x)\nend if;\nend proc :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "savelibname := \"CHAP1 4\":\nLibraryTools:-SaveToLibrary('`expand/coth`'):" }}}{PARA 0 "" 0 " " {TEXT -1 54 "\nLet us see the effect of the newly defined procedure: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "expand(coth(x+y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&-%%coshG6#%\"xG\"\"\"-F'6#%\"yGF*F**&-%%sinh GF(F*-F0F,F*F*F*,&*&F/F*F+F*F**&F&F*F1F*F*!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "expand(coth(x+y+z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,**(-%%coshG6#%\"xG\"\"\"-F'6#%\"yGF*-F'6#%\"zGF*F**( F&F*-%%sinhGF,F*-F3F/F*F**(-F3F(F*F2F*F.F*F**(F6F*F+F*F4F*F*F*,**(F6F* F+F*F.F*F**(F6F*F2F*F4F*F**(F&F*F2F*F.F*F**(F&F*F+F*F4F*F*!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "libname := \"CHAP14/\", libn ame; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(libnameG6$Q(CHAP14/6\"Q=C: \\Program~Files\\Maple~8/libF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "forget(expand);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " expand(coth(x+y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%*&-% %cothG6#%\"xGF%-F(6#%\"yGF%F%F%,&F'F%F+F%!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "expand(coth(x+y+z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,**(-%%cothG6#%\"xG\"\"\"-F'6#%\"yGF*-F'6#%\"zGF*F*F& F*F+F*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 0 "" }}}}}{MARK "0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }