{VERSION 3 0 "SGI MIPS UNIX" "3.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{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 -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 }3 3 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 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "445/545  day18.mws " }}
{PARA 0 "" 0 "" {TEXT 256 66 "Existence-tube from fundamental theorem,
 applied to x'=x^2, x(0)=1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
124 "restart:\nwith(plots):\nwith(plottools):\nK:=2*(1+b);\nM:=(1+b)^2
;\n\{b/M, 1/K\};\nplot( \{b/M, 1/K\}, b=0..10,a=0..0.5,color=black );
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG,&\"\"#\"\"\"%\"bGF&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG*$),&\"\"\"F(%\"bGF(\"\"#\"\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$*&%\"bG\"\"\"*$),&\"\"\"F*F%F*\"
\"#F&!\"\"*&F&F&,&\"\"#F*F%F/F," }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CUR
VESG6#7_o7$\"\"!F(7$$\"1nmTN@Ki8!#<$\"1wQRl'[fK\"F,7$$\"1LL$3FWYs#F,$
\"1z0pRa-#e#F,7$$\"1++D1k'p3%F,$\"1**)=Lg=Bx$F,7$$\"1mmmT&)G\\aF,$\"1[
Sprlj+\\F,7$$\"1++]7G$R<)F,$\"1e#*))fAJ&)pF,7$$\"1LLL3x&)*3\"!#;$\"1z!
[u];<'))F,7$$\"1mmTN@Ki8FF$\"1@&)zDjAb5FF7$$\"1++]ilyM;FF$\"1Yr#*4(ew?
\"FF7$$\"1LLe*)4D2>FF$\"13%G[s$>X8FF7$$\"1nmm;arz@FF$\"14VEo3Np9FF7$$
\"1L$e*)4bQl#FF$\"1JxvoeTd;FF7$$\"1++D\"y%*z7$FF$\"1S!GF2p\\\"=FF7$$\"
1m;ajW8-OFF$\"1od()G'4p%>FF7$$\"1LL$e9ui2%FF$\"1X(y2<as0#FF7$$\"1++voM
rU^FF$\"1\\i.J4xUAFF7$$\"1nmm\"z_\"4iFF$\"19^_X6EjBFF7$$\"1nmmm6m#G(FF
$\"1@Tv`u>QCFF7$$\"1ommT&phN)FF$\"1p3SG6&*zCFF7$$\"1M$3-js.*))FF$\"1\\
*3S#RP\"\\#FF7$$\"1,+v=ddC%*FF$\"1Jc&y61y\\#FF7$$\"1n;H2)y(e**FF$\"1s1
<O$*)**\\#FF7$$\"1LLe*=)H\\5!#:$\"1Ud<cKb)\\#FF7$$\"1++v=JN[6Far$\"1d;
QV(y!)[#FF7$$\"1nm\"z/3uC\"Far$\"1:*R&4FqpCFF7$$\"1++DJ$RDX\"Far$\"1nX
hP@)[T#FF7$$\"1nm\"zR'ok;Far$\"1E'o*GfWWBFF7$$\"1++D1J:w=Far$\"1ZuW$R1
!oAFF7$$\"1MLL3En$4#Far$\"1*Gv2Zgv=#FF7$$\"1nm;/RE&G#Far$\"1_j\"o^lt6#
FF7$$\"1+++D.&4]#Far$\"1*\\69)Q[S?FF7$$\"1+++vB_<FFar$\"16u5*>tj'>FF7$
$\"1+++v'Hi#HFar$\"1<r,KdE)*=FF7$$\"1nm\"z*ev:JFar$\"1N?-$G\\$R=FF7$$
\"1MLL347TLFar$\"1fVw(4<Hx\"FF7$$\"1MLLLY.KNFar$\"1&)*Q(fQk><FF7$$\"1+
+D\"o7Tv$Far$\"1w`NS^*4m\"FF7$$\"1LLL$Q*o]RFar$\"1?^#3u7>h\"FF7$$\"1,+
D\"=lj;%Far$\"1&)y;8L%4c\"FF7$$\"1++vV&R<P%Far$\"1'RP%**4/::FF7$$\"1ML
$e9Ege%Far$\"1WjIVkqp9FF7$$\"1MLeR\"3Gy%Far$\"1!zzHQG-V\"FF7$$\"1nm;/T
1&*\\Far$\"1h@%z].)*Q\"FF7$$\"1nm\"zRQb@&Far$\"1*z<vYC+N\"FF7$$\"1++v=
>Y2aFar$\"19y<W!3rJ\"FF7$$\"1nm;zXu9cFar$\"1f6Kc!GKG\"FF7$$\"1+++]y))G
eFar$\"1k]bM,$*\\7FF7$$\"1++]i_QQgFar$\"1T!HHs=*=7FF7$$\"1,+D\"y%3TiFa
r$\"1FV1*3!H!>\"FF7$$\"1++]P![hY'Far$\"17n(Q?&)*f6FF7$$\"1LLL$Qx$omFar
$\"1hM)yT+S8\"FF7$$\"1+++v.I%)oFar$\"1/4QpQZ26FF7$$\"1mm\"zpe*zqFar$\"
19\\0lsX%3\"FF7$$\"1,++D\\'QH(Far$\"15<n;nLg5FF7$$\"1LLe9S8&\\(Far$\"1
.!Hsbx&Q5FF7$$\"1,+D1#=bq(Far$\"1fn,'GYn,\"FF7$$\"1LLL3s?6zFar$\"1VK<b
2`i**F,7$$\"1++DJXaE\")Far$\"1AX/Y,[c(*F,7$$\"1ommm*RRL)Far$\"15F^(Rzd
c*F,7$$\"1om;a<.Y&)Far$\"1a>*[2%=y$*F,7$$\"1NLe9tOc()Far$\"1w&oXp\\\"*
>*F,7$$\"1,++]Qk\\*)Far$\"1q\")H\">j//*F,7$$\"1NL$3dg6<*Far$\"1v'omf#4
l))F,7$$\"1ommmxGp$*Far$\"1Ckw$zAQr)F,7$$\"1++D\"oK0e*Far$\"1qBx\"\\X!
e&)F,7$$\"1,+v=5s#y*Far$\"1uu5*>3ST)F,7$$\"#5F($\"1ct\"*4GYk#)F,-F$6#7
Y7$F($\"1+++++++]FF7$F:$\"1&zc+o:;u%FF7$FD$\"1hSSCTi3XFF7$FO$\"1yE[LuX
(H%FF7$FY$\"1La')Gi=0TFF7$F]o$\"12[DuZl3QFF7$Fgo$\"1oHA:j2_NFF7$F\\p$
\"1@Qw?\"=>I$FF7$Fap$\"16(y-&pn%3$FF7$Ffp$\"11pYxH2$*GFF7$F[q$\"108gJ)
zQs#FF7$F_r$\"1a2xn'f)RCFF7$Fjr$\"1*>4t7&yCAFF7$F_s$\"1,Oe)G.(Q?FF7$Fd
s$\"1*487T$Rw=FF7$Fis$\"1KcC$=L%Q<FF7$F^t$\"1e[CY??;;FF7$Fct$\"1?!QyuZ
>_\"FF7$Fht$\"1vv5YO=G9FF7$F]u$\"1M*et\"=)\\M\"FF7$Fbu$\"1*RffP'[t7FF7
$Fgu$\"1Km.qO%[@\"FF7$F\\v$\"1Vn2fix^6FF7$Fav$\"1a>LNsD.6FF7$Ffv$\"1V4
()p4s^5FF7$F[w$\"1r'oNNg*45FF7$F`w$\"1;@s%3%)zn*F,7$Few$\"1%3)fk<(zI*F
,7$Fjw$\"1A_&\\!p!4&*)F,7$F_x$\"1%z1sK=jk)F,7$Fdx$\"1%>b%)Q%>S$)F,7$Fi
x$\"1'f!3?aNW!)F,7$F^y$\"1=oD?=S.yF,7$Fcy$\"1&=vI&3()evF,7$Fhy$\"1+.%R
XO=K(F,7$F]z$\"1#fa<7-R5(F,7$Fbz$\"1az,aF/0pF,7$Fgz$\"1nUtKP*op'F,7$F
\\[l$\"1-'fS@%G?lF,7$Fa[l$\"1wro\"y;<M'F,7$Ff[l$\"1JJ]V/:)='F,7$F[\\l$
\"1BU`%[_&GgF,7$F`\\l$\"1c^+MBs&)eF,7$Fe\\l$\"1k,szM[VdF,7$Fj\\l$\"1A)
G1174h&F,7$F_]l$\"1(3M%RZ_yaF,7$Fd]l$\"1m(>Yo%zc`F,7$Fi]l$\"1D1L7&yxB&
F,7$F^^l$\"1Rf(*)He[7&F,7$Fc^l$\"1)QuP]0`-&F,7$Fh^l$\"1O&))>')fe\"\\F,
7$F]_l$\"1H$[=&>$>#[F,7$Fb_l$\"1*4RL**fcs%F,7$Fg_l$\"1(\\u(p%[qj%F,7$F
\\`l$\"1YXXXXXXXF,-%+AXESLABELSG6$Q\"b6\"Q\"aF\\[m-%'COLOURG6&%$RGBGF(
F(F(-%%VIEWG6$;F(F\\`l;F($\"\"&!\"\"" 2 181 261 261 2 0 1 0 2 6 0 4 2 
1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 
0 0 25 4185 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "
with(plots):\nwith(plottools):\nfor i from 1 to 20 do\n   b:=.2*i; a:=
min(b/M, 1/K);\n   rect[i] := rectangle([-a,1-b], [a,1+b]):\nod:\nsolu
tion:=plot(1/(1/1-t),t=-1..1,x=-3..5):\ndisplay(solution,seq(rect[i],i
=1..20));" }}{PARA 13 "" 1 "" {INLPLOT "69-%'CURVESG6$7W7$$!\"\"\"\"!$
\"1+++++++]!#;7$$!1+++<p0k&*F-$\"1mn4&G996&F-7$$!1,++r^u%=*F-$\"1zr*4w
uC@&F-7$$!1+++U%p\"e()F-$\"1*\\oHf55L&F-7$$!1+++#4m(G$)F-$\"12/\"=k/fX
&F-7$$!1+++@OS,zF-$\"11W$G9ahe&F-7$$!1+++/R=0vF-$\"1(43A^$f7dF-7$$!1++
+P8#\\4(F-$\"1.R\"=c!p\\eF-7$$!1+++/siqmF-$\"1%*RHXXd)*fF-7$$!1+++(y$p
ZiF-$\"1`2eO'>Z:'F-7$$!1+++$yaE\"eF-$\"1M![!o([SK'F-7$$!1+++\">s%HaF-$
\"1`'fo4.6['F-7$$!1+++]$*4)*\\F-$\"1D(\\q]6vm'F-7$$!1+++]_&\\c%F-$\"1b
Ot5_zloF-7$$!1+++]1aZTF-$\"1T\"=7Lm$oqF-7$$!1+++/#)[oPF-$\"1xn'*=:'HE(
F-7$$!1+++$=exJ$F-$\"1W*)4o7x3vF-7$$!1+++L2$f$HF-$\"10BRmiSIxF-7$$!1++
+PYx\"\\#F-$\"1l'=')pn_+)F-7$$!1+++L7i)4#F-$\"1/)[Oj/aE)F-7$$!1+++P'ps
m\"F-$\"1<2r&f&)4d)F-7$$!1+++74_c7F-$\"1BX5[&RP)))F-7$$!1+++!3x%z#)!#<
$\"1eyT[/ON#*F-7$$!1++++s$QM%Fdr$\"1<S#yo*p$e*F-7$$!1++++5zr)*!#>$\"1f
\\hWz8!***F-7$$\"1++++!o2J%Fdr$\"1OChm'\\]/\"!#:7$$\"1++++%Q#\\\")Fdr$
\"1=a?+Es)3\"Fgs7$$\"1+++g\"*[H7F-$\"1p&p5W%=S6Fgs7$$\"1++++dxd;F-$\"1
xeJl5s)>\"Fgs7$$\"1+++I0xw?F-$\"10,FA;6i7Fgs7$$\"1+++g&p@[#F-$\"1=,X&*
4<I8Fgs7$$\"1+++!3'HKHF-$\"1)yc4m')[T\"Fgs7$$\"1+++qZvOLF-$\"1#4Q&=-x+
:Fgs7$$\"1+++]2goPF-$\"1yo'f+wZg\"Fgs7$$\"1++++u\"*fTF-$\"1F#p&[XI7<Fg
s7$$\"1+++])Hxe%F-$\"1X%Q6_`w%=Fgs7$$\"1+++I!o-*\\F-$\"1C-=uZ6'*>Fgs7$
$\"1+++5k.6aF-$\"18r(=GT\"z@Fgs7$$\"1******>WTAeF-$\"1gX=Wrs$R#Fgs7$$
\"1+++g!*3`iF-$\"1)f[D9l)oEFgs7$$\"1+++I*zym'F-$\"1^oeMB4,IFgs7$$\"1++
+5N1#4(F-$\"1)HK+lk)QMFgs7$$\"1+++IYt7vF-$\"1LqqG(z/-%Fgs7$$\"1*******
p(G**yF-$\"1FK$o6!HgZFgs7$$\"1+++S6KU$)F-$\"1u_HO:`KgFgs7$$\"1+++N$[/a
)F-$\"163z$**=9&oFgs7$$\"1+++IbdQ()F-$\"1*=XXfXv#zFgs7$$\"1+++))z>W))F
-$\"1g2Ve3+_')Fgs7$$\"1+++X/#)\\*)F-$\"1$3$[&=\"=A&*Fgs7$$\"1,++.HWb!*
F-$\"1)yv)[spe5!#97$$\"1+++g`1h\"*F-$\"1;Czrz)>>\"F`[l7$$\"1+++S?Wl&*F
-$\"1g)\\f+*=,BF`[l7$%%FAILGF\\\\l-%'COLOURG6&%$RGBG$\"#5F)F*F*-%)POLY
GONSG6#7&7$$!+*))))))Q\"!#5$\"\")F)7$$\"+*))))))Q\"Fj\\lF[]l7$F^]l$\"#
7F)7$Fh\\lFa]l-Fd\\l6#7&7$$!+Fj\"3/#Fj\\l$\"\"'F)7$$\"+Fj\"3/#Fj\\lFj]
l7$F]^l$\"#9F)7$Fh]lF`^l-Fd\\l6#7&7$$!+++vVBFj\\l$\"\"%F)7$$\"+++vVBFj
\\lFi^l7$F\\_l$\"#;F)7$Fg^lF__l-Fd\\l6#7&7$$!+-e8pCFj\\l$\"\"#F)7$$\"+
-e8pCFj\\lFh_l7$F[`l$\"#=F)7$Ff_lF^`l-Fd\\l6#7&7$$!+++++DFj\\lF*7$$\"+
++++DFj\\lF*7$Fh`l$\"#?F)7$Fe`lF[al-Fd\\l6#7&7$$!+tsssAFj\\l$!\"#F)7$$
\"+tsssAFj\\lFdal7$Fgal$\"#AF)7$FbalFjal-Fd\\l6#7&7$$!+LLL$3#Fj\\l$!\"
%F)7$$\"+LLL$3#Fj\\lFcbl7$Ffbl$\"#CF)7$FablFibl-Fd\\l6#7&7$$!+Bp2B>Fj
\\l$!\"'F)7$$\"+Bp2B>Fj\\lFbcl7$Fecl$\"#EF)7$F`clFhcl-Fd\\l6#7&7$$!+'G
9dy\"Fj\\l$!\")F)7$$\"+'G9dy\"Fj\\lFadl7$Fddl$\"#GF)7$F_dlFgdl-Fd\\l6#
7&7$$!+nmmm;Fj\\l$Fj\\lF)7$$\"+nmmm;Fj\\lF`el7$Fbel$\"#IF)7$F^elFeel-F
d\\l6#7&7$$!+++]i:Fj\\l$!#7F)7$$\"+++]i:Fj\\lF^fl7$Fafl$\"#KF)7$F\\flF
dfl-Fd\\l6#7&7$$!+N#)eq9Fj\\l$F`[lF)7$$\"+N#)eq9Fj\\lF]gl7$F_gl$\"#MF)
7$F[glFbgl-Fd\\l6#7&7$Fh\\l$F-F)7$F^]lFigl7$F^]l$\"#OF)7$Fh\\lF\\hl-Fd
\\l6#7&7$$!+u%*y:8Fj\\l$!#=F)7$$\"+u%*y:8Fj\\lFehl7$Fhhl$\"#QF)7$FchlF
[il-Fd\\l6#7&7$$!++++]7Fj\\l$!#?F)7$$\"++++]7Fj\\lFdil7$Fgil$\"#SF)7$F
bilFjil-Fd\\l6#7&7$$!+!>w/>\"Fj\\l$!#AF)7$$\"+!>w/>\"Fj\\lFcjl7$Ffjl$
\"#UF)7$FajlFijl-Fd\\l6#7&7$$!+OOOO6Fj\\l$!#CF)7$$\"+OOOO6Fj\\lFb[m7$F
e[m$\"#WF)7$F`[mFh[m-Fd\\l6#7&7$$!+Al&p3\"Fj\\l$!#EF)7$$\"+Al&p3\"Fj\\
lFa\\m7$Fd\\m$\"#YF)7$F_\\mFg\\m-Fd\\l6#7&7$$!+nmmT5Fj\\l$!#GF)7$$\"+n
mmT5Fj\\lF`]m7$Fc]m$\"#[F)7$F^]mFf]m-Fd\\l6#7&7$$!+++++5Fj\\l$!#IF)7$$
\"+++++5Fj\\lF_^m7$Fb^m$\"#]F)7$F]^mFe^m-%+AXESLABELSG6$Q\"t6\"Q\"xF\\
_m-%%VIEWG6$;F($\"\"\"F*;$!\"$F*$\"\"&F*" 2 580 230 230 2 0 1 0 2 9 0 
4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
1 0 0 0 3958 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 422 "The above plot shows the in
terval of time on which the theorem guarantees a unique solution, depe
nding on the radius, b, of the neighborhood of x0=1 used in the proof \+
of the theorem. Note that the solution 1/(1-t) is indeed confined vert
ically within each of the tubes and that it extends beyond even the lo
ngest of them, and that for large b, the large size of the Lipschitz c
onstant causes the interval [-a,a] to shrink." }}}}{MARK "0 0 0" 14 }
{VIEWOPTS 1 1 0 1 1 1803 }