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derivatives~for~the~5-seriesFf[u" 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 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "#Below \+
I'm hard-coding the points at which each grouping has it's minimum." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "#There is an interesting \+
recursion here.  Let's say b[i]:=log(-dn[r]) for some number r.  Then \+
b[i+2] = log(-dn[ 2^(i) + r]).  Kind of reminscent of Fibonacci recurs
ion to me.  The same thing goes for the a's.  These points corresponds
 to sequences with all but one + signs chosen.  " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 38 "z[1]:=log(-dn5[1]): y[1]:=log(El5[1]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "z[2]:=log(-dn5[2]): y[2]:=lo
g(El5[2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "z[3]:=log(-dn
5[3]): y[3]:=log(El5[3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "z[4]:=log(-dn5[6]): y[4]:=log(El5[6]):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 40 "z[5]:=log(-dn5[11]): y[5]:=log(El5[11]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "z[6]:=log(-dn5[22]): y[6]:=l
og(El5[22]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "z[7]:=log(-
dn5[43]): y[7]:=log(El5[43]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "z[8]:=log(-dn5[86]): y[8]:=log(El5[86]):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 42 "z[9]:=log(-dn5[171]): y[9]:=log(El5[171]):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "z[10]:=log(-dn5[342]): y[10
]:=log(El5[342]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "#Now we make a new list with
 only the minimum points, so that Maple can graph just them." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "list1:=[seq([y[i],z[i]],i=1.
.10)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "list1;" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#7,7$$\"5c\"\\w,T\"3JBS!#>$\"5L!)G<LwJ&pd#F'7
$$\"5D$yx'zR:h\\^F'$\"500VdZ\"4<mp#F'7$$\"5m&>\"GP6&)GRpF'$\"56@OkaWU!
3L$F'7$$\"5z7$G.*)*fE\"\\)F'$\"5nH],s-5_+QF'7$$\"541ip+.E,75!#=$\"512p
DCkh-DVF'7$$\"5!R9V8807B<\"F<$\"5q2&fEN5!HJ[F'7$$\"5#H2Xa2'4ZL8F<$\"5!
fg]yZFOOM&F'7$$\"5]!y'p]$3VV\\\"F<$\"5`,-6Em\\&R&eF'7$$\"55&GR=%o2Jb;F
<$\"5Tb!\\vA`\\\\O'F'7$$\"5)*)RRpbfYi\"=F<$\"5QdwT&*[)=d(oF'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "#And then we graph them. (Mo
re pretty pictures, yay!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
92 "pointplot(list1, title=\"Minimum Points Graph\", labels=[\"Eigenva
lues\",\"Normal Derivatives\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 369 
369 369 {PLOTDATA 2 "6%-%'POINTSG6,7$$\"5c\"\\w,T\"3JBS!#>$\"5L!)G<LwJ
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@OkaWU!3L$F)7$$\"5z7$G.*)*fE\"\\)F)$\"5nH],s-5_+QF)7$$\"541ip+.E,75!#=
$\"512pDCkh-DVF)7$$\"5!R9V8807B<\"F>$\"5q2&fEN5!HJ[F)7$$\"5#H2Xa2'4ZL8
F>$\"5!fg]yZFOOM&F)7$$\"5]!y'p]$3VV\\\"F>$\"5`,-6Em\\&R&eF)7$$\"55&GR=
%o2Jb;F>$\"5Tb!\\vA`\\\\O'F)7$$\"5)*)RRpbfYi\"=F>$\"5QdwT&*[)=d(oF)-%+
AXESLABELSG6$Q,Eigenvalues6\"Q3Normal~DerivativesFhn-%&TITLEG6#Q5Minim
um~Points~GraphFhn" 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 "" {MPLTEXT 1 0 138 
"#Now we find the slope using least squares. It's somewhat inaccurate \+
since we're only using 7 digits and some of these numbers are x*10^5.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "LeastSquares(list1,v);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"5RRT'Qc$fo#=\"!#>\"\"\"*&$\"5
bY8,uj\\[AJ!#?F'%\"vGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "#To be more accurate, \+
we'll only check the slope using points further on in." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "list2:=[seq([y[i],z[i]],i=5..10)]:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "pointplot(list2, title=
\"Maximum Points Graph\", labels=[\"Eigenvalues\",\"Normal Derivatives
\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 382 382 382 {PLOTDATA 2 "6%-%'POIN
TSG6(7$$\"541ip+.E,75!#=$\"512pDCkh-DV!#>7$$\"5!R9V8807B<\"F)$\"5q2&fE
N5!HJ[F,7$$\"5#H2Xa2'4ZL8F)$\"5!fg]yZFOOM&F,7$$\"5]!y'p]$3VV\\\"F)$\"5
`,-6Em\\&R&eF,7$$\"55&GR=%o2Jb;F)$\"5Tb!\\vA`\\\\O'F,7$$\"5)*)RRpbfYi
\"=F)$\"5QdwT&*[)=d(oF,-%+AXESLABELSG6$Q,Eigenvalues6\"Q3Normal~Deriva
tivesFJ-%&TITLEG6#Q5Maximum~Points~GraphFJ" 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 "
" {MPLTEXT 1 0 22 "LeastSquares(list2,v);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&$\"58AQ(zgAmK6\"!#>\"\"\"*&$\"5#Q$oM-qLbsJ!#?F'%\"vGF
'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "list3:=[seq([y[i],z[
i]],i=6..10)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "pointplot
(list3, title=\"Maximum Points Graph\", labels=[\"Eigenvalues\",\"Norm
al Derivatives\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 382 382 382 
{PLOTDATA 2 "6%-%'POINTSG6(7$$\"559$pg%R[v55!#=$\"5)yYBQl,8_A%!#>7$$\"
5S6ZX/82ts6F)$\"51:UmV&R?Lu%F,7$$\"5f2(G,:MJLL\"F)$\"5%4&o'p,I=<D&F,7$
$\"5]\"3Wyfh*Q%\\\"F)$\"5E28?-lLNjdF,7$$\"5UpSkyc_Hb;F)$\"5/L;Ft.$4RF'
F,7$$\"5lgfN-m<D;=F)$\"5ea<?Ho[#[y'F,-%+AXESLABELSG6$Q,Eigenvalues6\"Q
3Normal~DerivativesFJ-%&TITLEG6#Q5Maximum~Points~GraphFJ" 1 2 0 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "LeastSquares(list3,v);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,&$\"5*o2(4fMDl;5!#>\"\"\"*&$\"5;^*>pXKTh<$!#?F
'%\"vGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "list4:=[seq([
y[i],z[i]],i=7..10)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "po
intplot(list4, title=\"Maximum Points Graph\", labels=[\"Eigenvalues\"
,\"Normal Derivatives\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 382 382 382 
{PLOTDATA 2 "6%-%'POINTSG6&7$$\"5#H2Xa2'4ZL8!#=$\"5!fg]yZFOOM&!#>7$$\"
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O'F,7$$\"5)*)RRpbfYi\"=F)$\"5QdwT&*[)=d(oF,-%+AXESLABELSG6$Q,Eigenvalu
es6\"Q3Normal~DerivativesF@-%&TITLEG6#Q5Maximum~Points~GraphF@" 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 "" {MPLTEXT 1 0 22 "LeastSquares(list4,v);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"5,u%GPBCe;6\"!#>\"\"\"*&$\"50'oOm
:s!ftJ!#?F'%\"vGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "40 0 0" 0 }
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