{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 } {CSTYLE "" -1 256 "" 1 8 0 0 0 0 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 1 }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 O utput" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }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 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 256 21 "dfEfof.m ws june 1998" }}{PARA 256 "" 0 "" {TEXT -1 23 "Solutions to df = fof ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "1 . A complex-valued function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "restart():\nDigits := 3:\nw ith(plots):\n### WARNING: persistent store makes one-argument readlib \+ obsolete\nreadlib(polar):\nx := 'x':\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "f := x -> exp(ln(c)/c)*x^c;\nc := (1-sqrt (3)*I)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)oper atorG%&arrowGF(*&-%$expG6#*&-%#lnG6#%\"cG\"\"\"F4!\"\"F5)9$F4F5F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG,&#\"\"\"\"\"#F'*&^##!\"\"F( F'-%%sqrtG6#\"\"$F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "co mplexplot(f(x), x=0..2*Pi, color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7]o7$$\"\"!F)$!\"!F)7$$!$=)!\"% $!$T\"!\"$7$$!$3#F2$!$%**F/7$$!$\"GF2$!$n#F/7$$!$A$F2$\"$(\\F/7$$!$U$F 2$\"$C\"F27$$!$^$F2$\"$%>F27$$!$Y$F2$\"$c#F27$$!$O$F2$\"$<$F27$$!$,$F2 $\"$=%F27$$!$a#F2$\"$/&F27$$!$+#F2$\"$w&F27$$!$Q\"F2$\"$Q'F27$$\"$6\"F 2$\"$&zF27$$\"$Z$F2$\"$d)F27$$\"$0(F2$\"$])F27$$\"$+\"!\"#$\"$i(F27$$ \"$F\"Ffp$\"$:'F27$$\"$\\\"Ffp$\"$S%F27$$\"$m\"Ffp$\"$`#F27$$\"$!=Ffp$ \"$Z&F/7$$\"$*>Ffp$!$N$F27$$\"$4#Ffp$!$)oF27$$\"$9#Ffp$!$.\"Ffp7$Fhr$! $P\"Ffp7$$\"$8#Ffp$!$o\"Ffp7$$\"$1#Ffp$!$'>Ffp7$F^r$!$A#Ffp7$$\"$!>Ffp $!$[#Ffp7$$\"$z\"Ffp$!$r#Ffp7$$\"$n\"Ffp$!$\"HFfp7$$\"$b\"Ffp$!$7$Ffp7 $$\"$T\"Ffp$!$J$Ffp7$$\"$G\"Ffp$!$\\$Ffp7$$\"$8\"Ffp$!$l$Ffp7$$\"$')*F 2$!$!QFfp7$$\"$W)F2$!$%RFfp7$$\"$#pF2$!$1%Ffp7$$\"$F&F2$!$>%Ffp7$$\"$) QF2$!$G%Ffp7$$\"$U#F2$!$S%Ffp7$$\"$\\(F/$!$]%Ffp7$$!$a(F/$!$f%Ffp7$$!$ I#F2$!$n%Ffp7$$!$%QF2$!$v%Ffp7$$!$J&F2$!$\"[Ffp7$$!$#pF2$!$*[Ffp7$$!$U )F2$!$&\\Ffp7$$!$$**F2$!$+&Ffp7$$!$8\"Ffp$!$/&Ffp7$$!$H\"Ffp$!$4&Ffp7$ $!$V\"Ffp$!$7&Ffp7$$!$f\"Ffp$!$<&Ffp7$$!$u\"Ffp$!$?&Ffp7$$!$(=Ffp$!$@& Ffp7$$!$,#Ffp$!$C&Ffp7$$!$;#Ffp$!$F&Ffp7$$!$H#Ffp$!$G&Ffp7$$!$V#Ffp$!$ H&Ffp7$$!$e#Ffp$!$K&Ffp7$Fdt$!$L&Ffp7$$!$&GFfpF]^l7$$!$*HFfp$!$M&Ffp7$ $!$6$FfpF]^l7$$!$C$FfpFe^l-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%+AXESL ABELSG6$Q!6\"Fg_l-%%VIEWG6$%(DEFAULTGF\\`l" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 63 "fof := unapply(simplify(f(f(x))),x);\n`fof(x)` = e valf(fof(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fofGR6#%\"xG6\"6$% )operatorG%&arrowGF(*&)*&-%$expG6#,$*&,&-%#lnG6#\"\"#!\"\"-F66#,&\"\" \"F=*&^#F9F=-%%sqrtG6#\"\"$F=F=F=F=,&F9F=*&^#F=F=F@F=F=F9!\"#F=)9$,&#F =F8F=*&^##F9F8F=F@F=F=F=FJF=F/F=F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'fof(x)G*&^$$\"$:#!\"#$!$D\"F)\"\"\")*&F&F,)%\"xG^$$\"$+&!\"$$ !$l)F4F,F1F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "df := unap ply(simplify(diff(f(x),x)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,$*()9$,&#!\"\"\"\"#\"\"\"*&^#F1F4 -%%sqrtG6#\"\"$F4F4F4,&F2F4*&^#F4F4F7F4F4F4-%$expG6#,$*&,&-%#lnG6#F3F2 -FE6#,&F4F4*&^#F2F4F7F4F4F4F4F;F2!\"#F4F1F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 310 "2. A real-valued function hr and its first three derviatives are constructed. The idea is simple. start with any mono tonically decreasing function, hr, with a continuous derivative,such that hr(-1)=-1, and hr'(-1)=-1 This is to be the half on the right . Then for x<-1, hl will be defined so that " }}{PARA 258 "" 0 " " {TEXT -1 19 "hl(hr(x))=dhr(x). " }}{PARA 0 "" 0 "" {TEXT -1 216 "At -1, hl and hr will be spliced together to produce the solution h. \+ So h(x) = hl(x) if x<-1, and h(x) = hr(x) otherwise. This turns the defining equation above into h(h(x))=dh(x). So dhr is construc ted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "hr := x -> -1-ln(x+2);\ndhr := D(hr);\nddhr := \+ D(dhr); \ndddhr := D(ddhr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#hrGR 6#%\"xG6\"6$%)operatorG%&arrowGF(,&!\"\"\"\"\"-%#lnG6#,&9$F.\"\"#F.F-F (F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dhrGR6#%\"xG6\"6$%)operato rG%&arrowGF(,$*&\"\"\"F.,&9$F.\"\"#F.!\"\"F2F(F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%ddhrGR6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-* $),&9$F-\"\"#F-F2F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&d ddhrGR6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"\"F.*$),&9$F.\"\"#F.\" \"$F.!\"\"!\"#F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 193 "The inverse hr is hri. This is useful beca use when hri(w) is substituted for x in the equation above we get \+ hl(w) := dhr(hri(w)) for w<-1. So now the inverse of hr is const ructed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "hri := unapply(solve(hr(x)=w,x),w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$hriGR6#%\"wG6\"6$%)operatorG%&arrowGF(,&-%$expG 6#,&!\"\"\"\"\"9$F1F2\"\"#F1F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "The derivarive is composed wit h the inverse to make hl -- hl(x) = dhr(hri(x)). This composition \+ is differentiated to make dhl, then ddhl, and dddhl." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "hl \+ := unapply(dhr(hri(x)),x);\ndhl := unapply(diff(hl(x),x),x);\nddh l := D(dhl);\ndddhl := D(ddhl);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #hlGR6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"\"F.-%$expG6#,&!\"\"F.9$ F3F3F3F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dhlGR6#%\"xG6\"6$%) operatorG%&arrowGF(,$*&\"\"\"F.-%$expG6#,&!\"\"F.9$F3F3F3F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ddhlGR6#%\"xG6\"6$%)operatorG%&arro wGF(,$*&\"\"\"F.-%$expG6#,&!\"\"F.9$F3F3F3F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dddhlGR6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"\"F. -%$expG6#,&!\"\"F.9$F3F3F3F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot([hr(x),hl(x),x] , x=-3..2, y=-3..1, color=[red,blue,sienna]);" }}{PARA 13 "" 1 "" {GLPLOT2D 484 484 484 {PLOTDATA 2 "6'-%'CURVESG6$7W7$$!$+#!\"#%%FAILGF 'F'7$$!3+++++++!*>!#<$\"3\"44))f=q^g$F/7$$!3)*************z>F/$\"32X\" Ga+B?\"HF/F2F2F27$$!3)*************p>F/$\"3\"4)*>t*yb1DF/7$$!3'******* ******f>F/$\"3n*>o[#e()=AF/7$$!3%*************R>F/$\"3^N+w;2T8=F/7$$!3 #*************4>F/$\"3\"4(=l3c%zS\"F/7$$!3))************z=F/$\"37!4+i` j-7\"F/7$$!35++++++g=F/$\"3'RLGPcG6m*!#=7$$!3/+++++++=F/$\"3-05MC\"zV4 'FU7$$!3+++++++]6O%H'QFU7$$!3%**************p\"F/$\"3(*e$f K/G(R?FU7$$!3#*************\\;F/$\"3/Sx')\\C@#)\\!#>7$$!3/++++++S:F/$! 3.O+,0@rMAFU7$$!3%*************R9F/$!3bz0ZZ]\"=?%FU7$$!33++++++S8F/$!3 ZSLQgb%[%eFU7$$!3)*************H7F/$!3WDfleBN'Q(FU7$$!3!*************H 6F/$!3cC\\mE$ztg)FU7$$!3-++++++?5F/$!3-0[#o#H(zz*FU7$$!3S+++++++#*FU$! 3MGh8T5'p2\"F/7$$!3]*************>)FU$!3_tvZQW^l6F/7$$!3k************* 4(FU$!3#3et$=Aka7F/7$$!3')*************4'FU$!3e+E9ZPIH8F/7$$!3++++++++ ]FU$!3Rk\"3\"3^Y09F/7$$!3A+++++++SFU$!3qNdCHO+q9F/7$$!3))************* *HFU$!3Eq@1^#G1`\"F/7$$!3-+++++++>FU$!3EMxFXoK$f\"F/7$$!3o************ ***)Fdo$!3aQ&e?C.rk\"F/7$$\"3/+++++++?Fdo$!3F8JT6v4.\"FU$!3=@Ro)3;9v\"F/7$$\"3-+++++++AFU$!3;)=%)e>2vz\"F/7$$\"3;++ +++++LFU$!3C4wdn#oe%=F/7$$\"3#**************H%FU$!35dCNd7*y)=F/7$$\"3M +++++++aFU$!3BX/.\"3k@$>F/7$$\"39+++++++kFU$!3%[Aer\"*y2(>F/7$$\"3!*** ***********R(FU$!3ny**R?z&z+#F/7$$\"3w*************\\)FU$!3'*e0G%**=t/ #F/7$$\"3a*************\\*FU$!3SGF<=#F/7$$\"35++++++q8F/$!3mqUOWF\"\\@#F/7$$\"3)*************p9F/$!3 \\n(eRfaTC#F/7$$\"33++++++!e\"F/$!3'*3ET+GOvAF/7$$\"3%*************z;F /$!3YR3=_F\"HI#F/7$$\"3-++++++!y\"F/$!3<'\\J'4SsHBF/7$$\"3!*********** ***)=F/$!3'\\NIw:4%eBF/7$$\"\"#\"\"!$!3!3*)>6O%H'Q#F/-%'COLOURG6&%$RGB G$\"*++++\"!\")$FbzFbzF\\[l-F$6$7U7$$!\"$Fbz$!3-FhOKGN`8FU7$$!3smm;HU, \"*GF/$!3)y\\h2l'=4:FU7$$!3=L$3FH'='z#F/$!3#\\L+T$\\If;FU7$$!3gmmTgBa* o#F/$!3_TMcn)Rg%=FU7$$!3wmm\"H_\">#e#F/$!33Mc@L>Cb?FU7$$!3ML$3_!4NvCF/ $!3[@(>[h%)pG#FU7$$!3'omTg(fHwBF/$!3e)oL)R87DDFU7$$!3;+]PM.ttAF/$!3I#G ,Wxfyz#FU7$$!3!omT5!oln@F/$!3rfKsa/&46$FU7$$!3%)**\\(oWB>1#F/$!3a8!fXh +zX$FU7$$!3;LL$epjJ&>F/$!3I>\\!*eU>bQFU7$$!3amm\"z/ot&=F/$!3v> rEkFU7$$!3CLL$eaR%H8F/$!3e@5l,xE$>(FU7$$!3bP\"z FU7$$!3'***\\PfO%H7\"F/$!3O[qJR[8V))FU7$$!3SLLL3`lC5F/$!3AF^t6:Yc(*FU7 $$!3q'**\\P4u\"o\"*FU$!3kV&4!>-u'3\"F/7$$!3*z**\\7G-89)FU$!3MFw$4UlU? 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Every derivative of h is continu ous. Domain(h) is bounded below but not above. Range(h) is bounded ab ove but not below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "[evalf(hoh(-1)),evalf(dh(-1))];\n[evalf(ho h(20)),evalf(dh(20))];\n[evalf(hoh(400)),evalf(dh(400))];\n[evalf(hoh( 8000)),evalf(dh(8000))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!\"\"\" \"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!$b%!\"%F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!$\\#!\"&F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7$$!$D\"!\"'F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 349 "Notice that the decreasing negative solutions may b e defined over the whole real line but that doesn't mean that they are solutions over the whole real line[?]. In this case, the second deriv ative is not continuous. But if the first k derivatives of hr had \+ been equal to -1 at x=-1, then the first k derivatives of h wou ld be continuous. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "The diagram plotted below helps us visualize, in simple \+ geometry, the process expressed in equation (0)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 293 "lines := \+ [[[e,0],[e,h(e)]],\n [[e,h(e)],[h(e),h(e)]],\n [[h (e),h(e)],[h(e),hoh(e)]],\n [[h(e),hoh(e)],[hoh(e),hoh(e)]], \n [[hoh(e),hoh(e)],[hoh(e),0]]]:\ne := 0.5:\n\nplot([h (x),x,op(lines)], x=-3..3, y=-2.2..1, color=[blue,sienna,red,red,red,r ed,red]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 396 400 400 {PLOTDATA 2 "6+- %'CURVESG6$7S7$$!\"$\"\"!$!3-FhOKGN`8!#=7$$!3!******\\2<#pG!#<$!3\\%)4 KjPWU:F-7$$!3#)***\\7bBav#F1$!3/[K?F-MGF-7$$!3%)****\\F)H')\\#F1$!36Dr-l4OMAF-7$$!3#****\\i3@/P#F1$!3q_DJ6)* **RDF-7$$!3;++Dr^b^AF1$!3!\\r<7z&fgGF-7$$!3$****\\7Sw%G@F1$!3y1d:Y!e_B $F-7$$!3*****\\7;)=,?F1$!3gblW1gUuOF-7$$!3/++DO\"3V(=F1$!3Ah\\#>J0:<%F -7$$!3#******\\V'zVC#z!*fwogF-7$$!3/+++vl[p8F1$!3%=Os'>1*3 \"pF-7$$!3\"******\\>iUC\"F1$!3B$\\6lMAG$yF-7$$!3-++DhkaI6F1$!3]kEyor: w()F-7$$!3s******\\XF`**F-$!3O5wai;m/5F17$$!3u*******>#z2))F-$!3VD-.6F j76F17$$!3S++]7RKvuF-$!3/H]O#p:^A\"F17$$!3s,+++P'eH'F-$!3Kz%zRh7^J\"F1 7$$!3q)***\\7*3=+&F-$!3cO%pm]W`S\"F17$$!3[)***\\PFcpPF-$!3=\"***yHKI%[ \"F17$$!3;)****\\7VQ[#F-$!3G$G-?hQ0c\"F17$$!32)***\\i6:.8F-$!3'RA.`!*p di\"F17$$!3Wb+++v`hH!#?$!3Q9%)Q9`m\"p\"F17$$\"3]****\\(QIKH\"F-$!3Oh?- 2T!ev\"F17$$\"38****\\7:xWCF-$!30tQI*fs%3=F17$$\"3E,++vuY)o$F-$!3c.jGK KSi=F17$$\"3!z******4FL(\\F-$!3LGTJYKA:>F17$$\"3A)****\\d6.B'F-$!3cQ$) Qz0Lk>F17$$\"3s****\\(o3lW(F-$!3e)=.)yQl4?F17$$\"35*****\\A))oz)F-$!3/ @scSAod?F17$$\"3e******Hk-,5F1$!3!*fP[tV&*)4#F17$$\"36+++D-eI6F1$!3*o_ rFO=79#F17$$\"3u***\\(=_(zC\"F1$!3-G^d#zJ!y@F17$$\"3M+++b*=jP\"F1$!3?l HY\\gy;AF17$$\"3g***\\(3/3(\\\"F1$!3a#Ha9XG>D#F17$$\"33++vB4JB;F1$!3Wu #[wA)Q(G#F17$$\"3u*****\\KCnu\"F1$!3K(*Q\"\\%>)3K#F17$$\"3s***\\(=n#f( =F1$!3OU&Quy%yaBF17$$\"3P+++!)RO+?F1$!3#o4!)>N&Q'Q#F17$$\"30++]_!>w7#F 1$!3nXs7S2qaT\\$[FF1$!3/Wu())y!zdDF17$$\"37++D6EjpGF1$!3MV/3&\\=Ie#F 17$$\"\"$F*$!3]+TV7zV4EF1-%'COLOURG6&%$RGBG$F*F*F^[l$\"*++++\"!\")-F$6 $7S7$F(F(7$F/F/7$F5F57$F:F:7$F?F?7$FDFD7$FIFI7$FNFN7$FSFS7$FXFX7$FgnFg n7$F\\oF\\o7$FaoFao7$FfoFfo7$F[pF[p7$F`pF`p7$FepFep7$FjpFjp7$F_qF_q7$F dqFdq7$FiqFiq7$F^rF^r7$FcrFcr7$FhrFhr7$F]sF]s7$FcsFcs7$FhsFhs7$F]tF]t7 $FbtFbt7$FgtFgt7$F\\uF\\u7$FauFau7$FfuFfu7$F[vF[v7$F`vF`v7$FevFev7$Fjv Fjv7$F_wF_w7$FdwFdw7$FiwFiw7$F^xF^x7$FcxFcx7$FhxFhx7$F]yF]y7$FbyFby7$F gyFgy7$F\\zF\\z7$FazFaz7$FfzFfz-F[[l6&F][l$\")viobFa[l$\")%yg>%Fa[l$\" )!\\DP\"Fa[l-F$6$7$7$$\"3++++++++]F-F^[l7$Fb_l$!3$*************>>F1-F[ [l6&F][lF_[lF^[lF^[l-F$6$7$Fd_l7$Fe_lFe_lFg_l-F$6$7$F\\`l7$Fe_l$!3?+++ +++!)RF-Fg_l-F$6$7$F``l7$Fa`lFa`lFg_l-F$6$7$Ff`l7$Fa`lF^[lFg_l-%+AXESL ABELSG6$Q\"x6\"Q\"yF_al-%%VIEWG6$;F(Ffz;$!#A!\"\"$\"\"\"F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 462 "In the diagram abov e, the red path has a direction. It starts at x=e (on the x-axis) \+ then moves vertically to y=h(e), then horizontally to x=h(e), the n vertically to y=h(h(e)), then horizontally to x=h(h(e)), then ve rtically to x=dh(e) (on the x-axis). To begin with e=0.5, the cur ve at x=0.5 appears to have a slight negative slope. Follow the cur ve around and it ends at -0.4 -- the slope at e. You may change \+ e to get other paths." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "'e' = e;\n'h(e)' = h(e); \n' hoh(e)' = evalf(hoh(e),12); \n'dh(e)' = evalf(dh(e),12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"eG$\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"eG$!$#>!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$h ohG6#%\"eG$!--++++S!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#dhG6#%\" eG$!-+++++S!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot([dh( x),x], x=-3..3, color=[blue,red]);" }}{PARA 13 "" 1 "" {GLPLOT2D 484 484 484 {PLOTDATA 2 "6&-%'CURVESG6$7en7$$!\"$\"\"!$!3-FhOKGN`8!#=7$$!3 !******\\2<#pG!#<$!3\\%)4KjPWU:F-7$$!3#)***\\7bBav#F1$!3/[K?F-MGF-7$$!3%)****\\F)H')\\#F1$!36Dr-l4OMAF-7$$!3 #****\\i3@/P#F1$!3q_DJ6)***RDF-7$$!3;++Dr^b^AF1$!3!\\r<7z&fgGF-7$$!3$* ***\\7Sw%G@F1$!3y1d:Y!e_B$F-7$$!3*****\\7;)=,?F1$!3gblW1gUuOF-7$$!3/++ DO\"3V(=F1$!3Ah\\#>J0:<%F-7$$!3#******\\V'zVC#z!*fwogF-7$$!3'********=eWV\"F1$!3M*\\$plX:wkF 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;F(F^]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Third solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "g := x - > piecewise(-sqrt(2)%\"gGR6# %\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6(32,$-%%sqrtG6#\"\"#!\"\" 9$2F7F6*&-F36#,&F5\"\"\"*$)F7F5F=F6F=F7F631F6F72F7\"\"!,$F:F61FCF7F1F( F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "dg := x -> piecew ise(-sqrt(2)%#dgGR6#%\"xG 6\"6$%)operatorG%&arrowGF(-%*piecewiseG6(32,$-%%sqrtG6#\"\"#!\"\"9$2F7 F6,$*&\"\"\"F;*&-F36#,&F5F;*$)F7F5F;F6F;FAF;F6!\"#31F6F72F7\"\"!*&F7F; F=F61FFF7FFF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot([ g(x),x], x=-2..1, y=-2..1, color=[blue,sienna]);" }}{PARA 13 "" 1 "" {GLPLOT2D 396 368 368 {PLOTDATA 2 "6&-%'CURVESG6$7W7$$!\"#\"\"!$F*F*7$ $!3&*****\\P&3Y$>!#mP()FP$!3 'pg)***)[*>6\"F/7$$!3'3+++&=$z9)FP$!31yg`QJ!f:\"F/7$$!3N***\\iX/4](FP$ 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{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "piecewise(x<=1, fl(x ), 1 " 0 "" {MPLTEXT 1 0 18 "f := unapply(%,x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&a rrowGF(-%*piecewiseG6&19$\"\"\",&F1F1-%#lnG6#,&\"\"#F1F0!\"\"F82F1F0*& F1F1-%$expG6#,&F1F1F0F8F8F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "piecewise(x<=1, fr(x),\n 1%#RfGR6#%\"xG6\"6$%)operatorG%&arrow GF(-%*piecewiseG6&19$\"\"\"*&F1F1-%$expG6#,&F1F1F0!\"\"F72F1F0,&F1F1-% #lnG6#,&\"\"#F1F0F7F7F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 41 "a non-constant solution with a flat area ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "unassign('a','b','c','C','d','dd','f1','f2','f3','f', 'if1','df3');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a:=Pi; b:= 2*Pi; c:=3*Pi; d:=4*Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG%#PiG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG,$%#PiG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG,$%#PiG\"\"$" }}{PARA 11 "" 1 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