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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 24 "Data and Logistic Growth
" }}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 
-1 22 "A Maple investigation " }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}
{PARA 266 "" 0 "" {TEXT -1 3 "by " }}{PARA 264 "" 0 "" {TEXT -1 0 "" }
}{PARA 263 "" 0 "" {TEXT -1 63 "Helen Compton, John W. Dollhopf, David
 Kapolka, and Jo Ann Lutz" }}{PARA 265 "" 0 "" {TEXT -1 0 "" }}{PARA 
260 "" 0 "" {TEXT -1 38 "Produced during the Web Feats Workshop" }}
{PARA 261 "" 0 "" {TEXT -1 12 "Columbus, OH" }}{PARA 262 "" 0 "" 
{TEXT -1 15 "July 9-13, 2001" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 4 "I.  \+
" }{TEXT 258 0 "" }{TEXT -1 15 "Logistic Growth" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "We need to begin by inc
luding some data that we have reason to believe is logistic." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 283 "T
he data in this cell is the U.S. Census data from 1790 until 1990.  If
 you wish to use this data, simply execute this cell.  Otherwise, repl
ace this data with other data.  Be sure the data is in the same form a
s this.  Other interesting data can be found from a link on the webpag
e." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "restart;\ndatalist:=evalf(\n
\n[[3, 265], [4, 416], [5, 556], [6, 699], [7, 837], [8, 1031], [9, 12
31], [10, 1681], [11, 2056], [12, 2536], [13, 3086], [14, 3706], [15, \+
4309], [16, 4991], [17, 5548], [18, 6196], [19, 6847], [20, 7270], [21
, 7500], [22, 7880], [23, 8068], [24, 8334], [25, 8434], [26, 8519]]\n
\n):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 78 "This cell installs some additional commands that Maple will use
 in this sheet." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(plots):\nwi
th(stats):\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 14 "with(DEtools):" }}
{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r
edefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 201 "To serve our purpose, we will need the data as ordered p
airs (as above in uscensusdata) and as separate lists.  The following \+
commands create two lists, one with the years and one with the populat
ions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 175 "Throughout our work, we will need to use the data you in
serted above as separate lists as well as the ordered pairs.  The foll
owing commands will pull apart the inputted list." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 100 "listlength:=nops(datalist):\ntvalues:=datalist[1..
listlength,1];\nPvalues:=datalist[1..listlength,2];\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%(tvaluesG7:$\"\"$\"\"!$\"\"%F($\"\"&F($\"\"'F($
\"\"(F($\"\")F($\"\"*F($\"#5F($\"#6F($\"#7F($\"#8F($\"#9F($\"#:F($\"#;
F($\"#<F($\"#=F($\"#>F($\"#?F($\"#@F($\"#AF($\"#BF($\"#CF($\"#DF($\"#E
F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(PvaluesG7:$\"$l#\"\"!$\"$;%F(
$\"$c&F($\"$*pF($\"$P)F($\"%J5F($\"%J7F($\"%\"o\"F($\"%c?F($\"%ODF($\"
%'3$F($\"%1PF($\"%4VF($\"%\"*\\F($\"%[bF($\"%'>'F($\"%ZoF($\"%qsF($\"%
+vF($\"%!)yF($\"%o!)F($\"%M$)F($\"%M%)F($\"%>&)F(" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "Let us first look
  at a plot of the data.  The following command will give us a scatter
plot of the data.  We give it a name for later use." }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 65 "plot1:=plot(datalist,style=point, symbol=circle,sy
mbolsize=15):%;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 
"6'-%'CURVESG6$7:7$$\"\"$\"\"!$\"$l#F*7$$\"\"%F*$\"$;%F*7$$\"\"&F*$\"$
c&F*7$$\"\"'F*$\"$*pF*7$$\"\"(F*$\"$P)F*7$$\"\")F*$\"%J5F*7$$\"\"*F*$
\"%J7F*7$$\"#5F*$\"%\"o\"F*7$$\"#6F*$\"%c?F*7$$\"#7F*$\"%ODF*7$$\"#8F*
$\"%'3$F*7$$\"#9F*$\"%1PF*7$$\"#:F*$\"%4VF*7$$\"#;F*$\"%\"*\\F*7$$\"#<
F*$\"%[bF*7$$\"#=F*$\"%'>'F*7$$\"#>F*$\"%ZoF*7$$\"#?F*$\"%qsF*7$$\"#@F
*$\"%+vF*7$$\"#AF*$\"%!)yF*7$$\"#BF*$\"%o!)F*7$$\"#CF*$\"%M$)F*7$$\"#D
F*$\"%M%)F*7$$\"#EF*$\"%>&)F*-%'COLOURG6&%$RGBG$FM!\"\"$F*F*F`s-%&STYL
EG6#%&POINTG-%'SYMBOLG6$%'CIRCLEGF`o-%+AXESLABELSG6$Q!6\"F\\t-%%VIEWG6
$%(DEFAULTGFat" 1 5 4 1 15 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 
0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 139 "Hopefully the scatterplot indicates that your data is \+
potentially logistic.  We now need to analyze the data to help us find
 the constants " }{TEXT 267 1 "k" }{TEXT -1 5 " and " }{TEXT 268 1 "M
" }{TEXT -1 31 " in the differential equation. " }}}{EXCHG {PARA 258 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 
"" {TEXT -1 30 "II.  Is the Data Set Logistic?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "The following command \+
calculates each symmetric difference (our estimate of the derivative).
  Note that this gives us 2 fewer pieces of data than with what we beg
an." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "symdiff:=[]:\nfor i from 2 \+
by 1 to (listlength-1) do\n    symdiff:=[op(symdiff),(Pvalues[i+1]-Pva
lues[i-1])/\n              (tvalues[i+1]-tvalues[i-1])]:\n  end do:\ns
ymdiff;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#78$\"++++b9!\"($\"++++:9F&$
\"++++09F&$\"++++g;F&$\"++++q>F&$\"++++]KF&$\"++++DTF&$\"++++vUF&$\"++
++]^F&$\"++++]eF&$\"++++:hF&$\"++++DkF&$\"++++&>'F&$\"++++DgF&$\"++++&
\\'F&$\"++++q`F&$\"++++lKF&$\"++++]IF&$\"++++SGF&$\"++++qAF&$\"++++I=F
&$\"++++]#*!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 260 "Following the example on the webpage, we need to di
vide our estimate of the derivative by the population values.  Because
 we have two fewer pieces of data, we must remove the first and last e
lements of our P values list.  We then calculate the needed quotient.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 251 "modifiedPvalues:=subsop(listle
ngth=NULL,Pvalues):\nmodifiedPvalues:=subsop(1=NULL,modifiedPvalues);
\n\n\nquotientlist:=[]:\nfor i from 1 by 1 to (listlength-2) do\n     \+
quotientlist:=[op(quotientlist),symdiff[i]/modifiedPvalues[i]]:\n   en
d do:\nquotientlist;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%0modifiedPva
luesG78$\"$;%\"\"!$\"$c&F($\"$*pF($\"$P)F($\"%J5F($\"%J7F($\"%\"o\"F($
\"%c?F($\"%ODF($\"%'3$F($\"%1PF($\"%4VF($\"%\"*\\F($\"%[bF($\"%'>'F($
\"%ZoF($\"%qsF($\"%+vF($\"%!)yF($\"%o!)F($\"%M$)F($\"%M%)F(" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#78$\"+ahf(\\$!#5$\"+HS'\\a#F&$\"+1V,5?F&$\"+
'ftK)>F&$\"+Yiw5>F&$\"+w*H,k#F&$\"+!\\'*QX#F&$\"+c,Gz?F&$\"+)4d2.#F&$
\"+4yl&*=F&$\"+$)p-];F&$\"+7_1\"\\\"F&$\"+AUBT7F&$\"+Hp(f3\"F&$\"+SpD[
5F&$\"+%)3&G%y!#6$\"+Z\"f5\\%FE$\"+nmmmSFE$\"+941/OFE$\"+JXe8GFE$\"+MV
#e>#FE$\"+X7v'4\"FE" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "To see the
 (hopefully) linear plot of (" }{XPPEDIT 18 0 "P,1/P;" "6$%\"PG*&\"\"
\"F%F#!\"\"" }{XPPEDIT 18 0 "dP/dt;" "6#*&%#dPG\"\"\"%#dtG!\"\"" }
{TEXT -1 52 " ), we need to create a list with each ordered pair." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "plotlist:=[]:\nfor i from 1 by 1 t
o (listlength-2) do\n    plotlist:=[op(plotlist),[modifiedPvalues[i],q
uotientlist[i]]]:\nend do:\nplotlist;" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#787$$\"$;%\"\"!$\"+ahf(\\$!#57$$\"$c&F'$\"+HS'\\a#F*7$$\"$*pF'$\"+1
V,5?F*7$$\"$P)F'$\"+'ftK)>F*7$$\"%J5F'$\"+Yiw5>F*7$$\"%J7F'$\"+w*H,k#F
*7$$\"%\"o\"F'$\"+!\\'*QX#F*7$$\"%c?F'$\"+c,Gz?F*7$$\"%ODF'$\"+)4d2.#F
*7$$\"%'3$F'$\"+4yl&*=F*7$$\"%1PF'$\"+$)p-];F*7$$\"%4VF'$\"+7_1\"\\\"F
*7$$\"%\"*\\F'$\"+AUBT7F*7$$\"%[bF'$\"+Hp(f3\"F*7$$\"%'>'F'$\"+SpD[5F*
7$$\"%ZoF'$\"+%)3&G%y!#67$$\"%qsF'$\"+Z\"f5\\%F`p7$$\"%+vF'$\"+nmmmSF`
p7$$\"%!)yF'$\"+941/OF`p7$$\"%o!)F'$\"+JXe8GF`p7$$\"%M$)F'$\"+MV#e>#F`
p7$$\"%M%)F'$\"+X7v'4\"F`p" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 188 "If we are lucky, this data will have a l
inear pattern.  Otherwise the assumption of a logistic model is incorr
ect.  Let us look at the scatterplot.  Again, we give it a name for la
ter use." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot2:=plot(plotlist,st
yle=point):%;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6
&-%'CURVESG6$787$$\"$;%\"\"!$\"3++++ahf(\\$!#=7$$\"$c&F*$\"3u******GS'
\\a#F-7$$\"$*pF*$\"3.+++1V,5?F-7$$\"$P)F*$\"3-+++'ftK)>F-7$$\"%J5F*$\"
3%******fCm2\">F-7$$\"%J7F*$\"3?+++w*H,k#F-7$$\"%\"o\"F*$\"3()*******[
'*QX#F-7$$\"%c?F*$\"3()*****f:!Gz?F-7$$\"%ODF*$\"3/+++)4d2.#F-7$$\"%'3
$F*$\"33+++4yl&*=F-7$$\"%1PF*$\"3++++$)p-];F-7$$\"%4VF*$\"33+++7_1\"\\
\"F-7$$\"%\"*\\F*$\"3++++AUBT7F-7$$\"%[bF*$\"3++++Hp(f3\"F-7$$\"%'>'F*
$\"3'*******RpD[5F-7$$\"%ZoF*$\"3U*****R)3&G%y!#>7$$\"%qsF*$\"3q*****p
9f5\\%Fcp7$$\"%+vF*$\"3r*****pmmm1%Fcp7$$\"%!)yF*$\"3*)*****R\"41/OFcp
7$$\"%o!)F*$\"35+++JXe8GFcp7$$\"%M$)F*$\"3#******RLCe>#Fcp7$$\"%M%)F*$
\"3'******\\C^n4\"Fcp-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fir-%&STYLEG6#%&
POINTG-%+AXESLABELSG6$Q!6\"Fas-%%VIEWG6$%(DEFAULTGFfs" 1 5 0 1 10 0 2 
9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "For later conveni
ence in plotting, we need the smallest and largest elements of our two
 lists.  " }{MPLTEXT 1 0 131 "Pmax:=sort(modifiedPvalues)[listlength-2
];\nPmin:=sort(modifiedPvalues)[1];\ntmax:=sort(tvalues)[listlength];
\ntmin:=sort(tvalues[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PmaxG$
\"%M%)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PminG$\"$;%\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tmaxG$\"#E\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%tminG$\"\"$\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "We want to fit a line to this \+
plot.  Manipulate the values of m and b to find a good line that fits \+
the data.  When you are satisfied with the fit of your line, continue \+
to the next part." }}{PARA 0 "" 0 "" {TEXT 259 16 "input this value" }
{TEXT -1 4 "    " }{MPLTEXT 1 0 17 "m:= -.000033   ;\n" }{TEXT 260 16 
"input this value" }{TEXT -1 4 "    " }{MPLTEXT 1 0 68 "b:= .29    ;\n
\nplot3:=plot(m*x+b,x=Pmin..Pmax):\ndisplay(plot2,plot3);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"mG$!#L!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"bG$\"#H!\"#" }}{PARA 13 "" 1 "" {GLPLOT2D 376 254 254 
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0 "" 0 "" {TEXT -1 141 "An alternative to eyeballing the coefficients \+
is to fit a regression line to the transformed data.  We use the least
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il and paper, convert these values of " }{TEXT 261 1 "m" }{TEXT -1 5 "
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0 "" {MPLTEXT 1 0 34 "DE1:=diff(P(t),t)=k*P(t)*(M-P(t));" }}{PARA 11 "
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0 "" {TEXT -1 242 "Note that the Maple file LogisticPopulation(WithAns
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the explicit solution of the differential equation." }}}}{MARK "18 1 4
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