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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 228 "We are looking at United \+
States census data from 1790 until 1990.  We must begin by inputing th
e data into the computer and calling a few extra commands that we will
 use subsequently. (note:  the population for 2000 was 281.422)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 387 "restart;\ndatalist:=[[1790,
\0113.929],\n[1800,\0115.308],\011\n[1810,\0117.240],\011\n[1820,\0119
.638],\011\n[1830,\01112.866],\011\n[1840,\01117.069],\011\n[1850,\011
23.192],\011\n[1860,\01131.443],\011\n[1870,\01138.558],\011\n[1880,
\01150.156],\011\n[1890,\01162.948],\011\n[1900,\01175.996],\n[1910,
\01191.972],\n[1920,\011105.711],\n[1930,\011122.775],\n[1940,\011131.
669],\n[1950,\011150.697],\n[1960,\011179.323],\n[1970,\011203.185],\n
[1980,\011226.546],\n[1990,\011248.710],\n[2000, 281.422]]:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "with(plots):\nwith(stats):\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 14 "w
ith(DEtools):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name change
coords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "T
o serve our purpose, we will need the data as ordered pairs (as above \+
in uscensusdata) and as separate lists.  The following comands create \+
two lists, one with the years and one with the populations." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "listlength:=nops(datalist);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+listlengthG\"#A" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "tvalues:=datalist[1..listlength,1];
\nPvalues:=datalist[1..listlength,2];" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#>%(tvaluesG78\"%!z\"\"%+=\"%5=\"%?=\"%I=\"%S=\"%]=\"%g=\"%q=\"%!)=
\"%!*=\"%+>\"%5>\"%?>\"%I>\"%S>\"%]>\"%g>\"%q>\"%!)>\"%!*>\"%+?" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(PvaluesG78$\"%HR!\"$$\"%3`F($\"%SsF
($\"%Q'*F($\"&mG\"F($\"&pq\"F($\"&#>BF($\"&V9$F($\"&e&QF($\"&c,&F($\"&
[H'F($\"&'*f(F($\"&s>*F($\"'6d5F($\"'vF7F($\"'p;8F($\"'(p]\"F($\"'B$z
\"F($\"'&=.#F($\"'YlAF($\"'5([#F($\"'A9GF(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 138 "Let us first look  at a plot of the data.  The following
 command will give us a scatterplot of the data.  We give it a name fo
r later use." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot1:=plot
(datalist,style=point, symbol=circle,symbolsize=15):%;" }}{PARA 13 "" 
1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$787$$\"%!z\"\"\"
!$\"3#)************GR!#<7$$\"%+=F*$\"3$)***********zI&F-7$$\"%5=F*$\"3
A++++++SsF-7$$\"%?=F*$\"3!************zj*F-7$$\"%I=F*$\"3(***********f
'G\"!#;7$$\"%S=F*$\"3\"************oq\"FB7$$\"%]=F*$\"3-+++++?>BFB7$$
\"%g=F*$\"39+++++IWJFB7$$\"%q=F*$\"3)***********zbQFB7$$\"%!)=F*$\"3))
**********f:]FB7$$\"%!*=F*$\"3/+++++![H'FB7$$\"%+>F*$\"3_**********f*f
(FB7$$\"%5>F*$\"3U**********>(>*FB7$$\"%?>F*$\"3)**********4r0\"!#:7$$
\"%I>F*$\"31+++++vF7Fjo7$$\"%S>F*$\"36+++++p;8Fjo7$$\"%]>F*$\"3.+++++(
p]\"Fjo7$$\"%g>F*$\"31+++++B$z\"Fjo7$$\"%q>F*$\"3-+++++&=.#Fjo7$$\"%!)
>F*$\"3#**********faE#Fjo7$$\"%!*>F*$\"32+++++5([#Fjo7$$\"%+?F*$\"3D++
+++A9GFjo-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fjr-%&STYLEG6#%&POINTG-%+AXE
SLABELSG6$Q!6\"Fbs-%'SYMBOLG6$%'CIRCLEG\"#:-%%VIEWG6$%(DEFAULTGF\\t" 
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 -1 120 "The nature of the data suggests a
 logistic model might be a good way to study the data.  The logistic m
odel is given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
dP/dt = k*P*(M-P);" "6#/*&%#dPG\"\"\"%#dtG!\"\"*(%\"kGF&%\"PGF&,&%\"MG
F&F+F(F&" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "M;" "
6#%\"MG" }{TEXT -1 165 " are constants.  To use this differential equa
tion, we must first identify the parameters.  One method for finding t
hese parameters is to first rewrite the equation." }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 36 "Consider the following manipulation." }}{PARA 256 
"" 0 "" {XPPEDIT 18 0 "1/P;" "6#*&\"\"\"F$%\"PG!\"\"" }{XPPEDIT 18 0 "
dP/dt = k*M-k*P;" "6#/*&%#dPG\"\"\"%#dtG!\"\",&*&%\"kGF&%\"MGF&F&*&F+F
&%\"PGF&F(" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 97 "To study this equation in terms of the da
ta that we have, we need to approximate the derivative, " }{XPPEDIT 
18 0 "dP/dt" "6#*&%#dPG\"\"\"%#dtG!\"\"" }{TEXT -1 213 ".  We approxim
ate the derivative using difference quotients.  In particular, we will
 look at (Pvalues[i+1]-Pvalues[i-1])/(tvalues[i+1]-tvalues[i-1]).  Thi
s is called the symmetric difference.  explanation........." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "The following command calculates \+
each symmetric difference (i.e. our estimate of the derivative).  Note
 that this gives us n-2 pieces of data." }}}{EXCHG {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]-Pvalues[i-1])/\n            \+
  (tvalues[i+1]-tvalues[i-1])]:\n  end do:\nsymdiff;" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#76$\"+++]b;!#
5$\"++++l@F&$\"++++8GF&$\"+++]:PF&$\"++++j^F&$\"++++(=(F&$\"++++$o(F&$
\"+++]c$*F&$\"+++]>7!\"*$\"++++#H\"F7$\"+++?^9F7$\"+++v&[\"F7$\"+++:S:
F7$\"+++!zH\"F7$\"+++5'R\"F7$\"+++q#Q#F7$\"+++SCEF7$\"+++:hBF7$\"+++Dw
AF7$\"+++!Qu#F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 400 "Note that in \+
the above equation what we are really interested in is the derivative \+
divided by P.  The following command calculates those values.  Because
 there are more population data points than symmetric difference data \+
points, we choose the population data point in the middle of each symm
etric difference.  As we will use this modified population data a numb
er of times, we give it its own name." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 98 "modifiedPvalues:=subsop(listlength=NULL,Pvalues):\nmo
difiedPvalues:=subsop(1=NULL,modifiedPvalues);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%0modifiedPvaluesG76$\"%3`!\"$$\"%SsF($\"%Q'*F($\"&mG
\"F($\"&pq\"F($\"&#>BF($\"&V9$F($\"&e&QF($\"&c,&F($\"&[H'F($\"&'*f(F($
\"&s>*F($\"'6d5F($\"'vF7F($\"'p;8F($\"'(p]\"F($\"'B$z\"F($\"'&=.#F($\"
'YlAF($\"'5([#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "quotie
ntlist:=[]:\nfor i from 1 by 1 to (listlength-2) do\n     quotientlist
:=[op(quotientlist),symdiff[i]/modifiedPvalues[i]]:\n   end do:\nquoti
entlist;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#76$\"+nr()=J!#6$\"+#\\J.*H
F&$\"+B`l=HF&$\"+IR%y)GF&$\"+o<yCIF&$\"+=M\"*)4$F&$\"+N\"pMW#F&$\"+xSg
ECF&$\"+))RTJCF&$\"+ox[_?F&$\"+)=u&4>F&$\"+2tV:;F&$\"+KR%pX\"F&$\"+Rq8
d5F&$\"+C.Jg5F&$\"+I(>6e\"F&$\"+.W]j9F&$\"+]!p?;\"F&$\"+IGw/5F&$\"+xD@
.6F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We now want to consider d
ata points of the form (" }{XPPEDIT 18 0 "P,dP/(dt*P);" "6$%\"PG*&%#dP
G\"\"\"*&%#dtGF&F#F&!\"\"" }{TEXT -1 47 " ).  The following command co
nstructs that list" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "plot
list:=[]:\nfor i from 1 by 1 to (listlength-2) do\n    plotlist:=[op(p
lotlist),[modifiedPvalues[i],quotientlist[i]]]:\nend do:\nplotlist;" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#767$$\"%3`!\"$$\"+nr()=J!#67$$\"%SsF'
$\"+#\\J.*HF*7$$\"%Q'*F'$\"+B`l=HF*7$$\"&mG\"F'$\"+IR%y)GF*7$$\"&pq\"F
'$\"+o<yCIF*7$$\"&#>BF'$\"+=M\"*)4$F*7$$\"&V9$F'$\"+N\"pMW#F*7$$\"&e&Q
F'$\"+xSgECF*7$$\"&c,&F'$\"+))RTJCF*7$$\"&[H'F'$\"+ox[_?F*7$$\"&'*f(F'
$\"+)=u&4>F*7$$\"&s>*F'$\"+2tV:;F*7$$\"'6d5F'$\"+KR%pX\"F*7$$\"'vF7F'$
\"+Rq8d5F*7$$\"'p;8F'$\"+C.Jg5F*7$$\"'(p]\"F'$\"+I(>6e\"F*7$$\"'B$z\"F
'$\"+.W]j9F*7$$\"'&=.#F'$\"+]!p?;\"F*7$$\"'YlAF'$\"+IGw/5F*7$$\"'5([#F
'$\"+xD@.6F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "tempplotli
st:=[]:\nfor i from 1 by 1 to (listlength-2) do\n    tempplotlist:=[op
(tempplotlist),[tvalues[i+1],symdiff[i]]]:\nend do:\ntempplotlist;" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#767$\"%+=$\"+++]b;!#57$\"%5=$\"++++l@F
(7$\"%?=$\"++++8GF(7$\"%I=$\"+++]:PF(7$\"%S=$\"++++j^F(7$\"%]=$\"++++(
=(F(7$\"%g=$\"++++$o(F(7$\"%q=$\"+++]c$*F(7$\"%!)=$\"+++]>7!\"*7$\"%!*
=$\"++++#H\"FI7$\"%+>$\"+++?^9FI7$\"%5>$\"+++v&[\"FI7$\"%?>$\"+++:S:FI
7$\"%I>$\"+++!zH\"FI7$\"%S>$\"+++5'R\"FI7$\"%]>$\"+++q#Q#FI7$\"%g>$\"+
++SCEFI7$\"%q>$\"+++:hBFI7$\"%!)>$\"+++DwAFI7$\"%!*>$\"+++!Qu#FI" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(tempplotlist,style=poin
t):%;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURV
ESG6$767$$\"%+=\"\"!$\"3-+++++]b;!#=7$$\"%5=F*$\"3)************\\;#F-7
$$\"%?=F*$\"3%************H\"GF-7$$\"%I=F*$\"3#***********\\:PF-7$$\"%
S=F*$\"3\")***********H;&F-7$$\"%]=F*$\"31++++++(=(F-7$$\"%g=F*$\"3#)*
**********Ho(F-7$$\"%q=F*$\"3\")**********\\c$*F-7$$\"%!)=F*$\"3.+++++
]>7!#<7$$\"%!*=F*$\"3/++++++#H\"FV7$$\"%+>F*$\"3/+++++?^9FV7$$\"%5>F*$
\"3!**********\\d[\"FV7$$\"%?>F*$\"3!**********\\,a\"FV7$$\"%I>F*$\"3/
+++++!zH\"FV7$$\"%S>F*$\"35+++++5'R\"FV7$$\"%]>F*$\"3\")**********p#Q#
FV7$$\"%g>F*$\"31+++++SCEFV7$$\"%q>F*$\"3&)*********\\6O#FV7$$\"%!)>F*
$\"35+++++DwAFV7$$\"%!*>F*$\"3!)**********zVFFV-%'COLOURG6&%$RGBG$\"#5
!\"\"$F*F*F_r-%&STYLEG6#%&POINTG-%+AXESLABELSG6$Q!6\"Fgr-%%VIEWG6$%(DE
FAULTGF\\s" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C
urve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 147 "If we are lucky, this data will have a linear pattern.  \+
Otherwise the assumption of a logistic model is incorrect.  Let us loo
k at the scatterplot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "pl
ot2:=plot(plotlist,style=point):%;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 
300 300 {PLOTDATA 2 "6&-%'CURVESG6$767$$\"3$)***********zI&!#<$\"30+++
nr()=J!#>7$$\"3A++++++SsF*$\"3-+++#\\J.*HF-7$$\"3!************zj*F*$\"
3()*****HKb'=HF-7$$\"3(***********f'G\"!#;$\"3$*******HR%y)GF-7$$\"3\"
************oq\"F;$\"31+++o<yCIF-7$$\"3-+++++?>BF;$\"3#******zT8*)4$F-
7$$\"39+++++IWJF;$\"3#******\\8pMW#F-7$$\"3)***********zbQF;$\"3/+++xS
gECF-7$$\"3))**********f:]F;$\"3%)*****z)RTJCF-7$$\"3/+++++![H'F;$\"37
+++ox[_?F-7$$\"3_**********f*f(F;$\"3%******z=u&4>F-7$$\"3U**********>
(>*F;$\"3&)*****pIPah\"F-7$$\"3)**********4r0\"!#:$\"30+++KR%pX\"F-7$$
\"31+++++vF7Fco$\"3%*******Qq8d5F-7$$\"36+++++p;8Fco$\"3*******RK5.1\"
F-7$$\"3.+++++(p]\"Fco$\"3'*******H(>6e\"F-7$$\"31+++++B$z\"Fco$\"31++
+.W]j9F-7$$\"3-+++++&=.#Fco$\"3#*******\\!p?;\"F-7$$\"3#**********faE#
Fco$\"3++++IGw/5F-7$$\"32+++++5([#Fco$\"3)******pd7K5\"F--%'COLOURG6&%
$RGBG$\"#5!\"\"$\"\"!FarF`r-%&STYLEG6#%&POINTG-%+AXESLABELSG6$Q!6\"Fir
-%%VIEWG6$%(DEFAULTGF^s" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "We wa
nt to fit a line to this plot.  Manipulate the values of m and b to fi
nd a good line that fits the data." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 131 "Pmax:=sort(modifiedPvalues)[listlength-2];\nPmin:=so
rt(modifiedPvalues)[1];\ntmax:=sort(tvalues)[listlength];\ntmin:=sort(
tvalues[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PmaxG$\"'5([#!\"$" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PminG$\"%3`!\"$" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%%tmaxG\"%+?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%
tminG\"%!z\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "m:=-.910733
9104e-4;b:=.2857757068e-1;\nplot3:=plot(m*x+b,x=Pmin..Pmax):\ndisplay(
plot2,plot3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$!+/\"Rt5*!#9" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"+oqvdG!#6" }}{PARA 13 "" 1 
"" {GLPLOT2D 190 172 172 {PLOTDATA 2 "6&-%'CURVESG6%767$$\"3$)********
***zI&!#<$\"30+++nr()=J!#>7$$\"3A++++++SsF*$\"3-+++#\\J.*HF-7$$\"3!***
*********zj*F*$\"3()*****HKb'=HF-7$$\"3(***********f'G\"!#;$\"3$******
*HR%y)GF-7$$\"3\"************oq\"F;$\"31+++o<yCIF-7$$\"3-+++++?>BF;$\"
3#******zT8*)4$F-7$$\"39+++++IWJF;$\"3#******\\8pMW#F-7$$\"3)*********
**zbQF;$\"3/+++xSgECF-7$$\"3))**********f:]F;$\"3%)*****z)RTJCF-7$$\"3
/+++++![H'F;$\"37+++ox[_?F-7$$\"3_**********f*f(F;$\"3%******z=u&4>F-7
$$\"3U**********>(>*F;$\"3&)*****pIPah\"F-7$$\"3)**********4r0\"!#:$\"
30+++KR%pX\"F-7$$\"31+++++vF7Fco$\"3%*******Qq8d5F-7$$\"36+++++p;8Fco$
\"3*******RK5.1\"F-7$$\"3.+++++(p]\"Fco$\"3'*******H(>6e\"F-7$$\"31+++
++B$z\"Fco$\"31+++.W]j9F-7$$\"3-+++++&=.#Fco$\"3#*******\\!p?;\"F-7$$
\"3#**********faE#Fco$\"3++++IGw/5F-7$$\"32+++++5([#Fco$\"3)******pd7K
5\"F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FarF`r-%&STYLEG6#%&POINTG-F$6$7
S7$F($\"32of.7`T4GF-7$$\"3()*\\Z=4Z81\"F;$\"3\"f+v#*e'4hFF-7$$\"35DTkz
K(H_\"F;$\"3Y*\\fpsa!>FF-7$$\"3B]sz2-7U?F;$\"3&\\fkdDu<n#F-7$$\"3J]2yy
$3Zc#F;$\"3bOd\"*y.=CEF-7$$\"3+D'4%zF\"[3$F;$\"3.QcX2F\"od#F-7$$\"3RC,
zOh,nNF;$\"3-B\"GD\"o*G`#F-7$$\"3hC\"\\Iy4j1%F;$\"3s*eQqWCu[#F-7$$\"3%
\\7di)*zEe%F;$\"3=B/ghoRSCF-7$$\"3#\\iuO=%R(4&F;$\"3Uz^A%4?NR#F-7$$\"3
D+b8-5%oi&F;$\"313E4x:IXBF-7$$\"3I](>![!yJ4'F;$\"3h`j&4oIGI#F-7$$\"3$*
*\\c+6j\"=mF;$\"3(>K66^=]D#F-7$$\"30*\\(>6QIXrF;$\"373K!*>+,2AF-7$$\"3
%)*\\VNb,Ll(F;$\"3H$4nG%\\ug@F-7$$\"3GErkq@h9\")F;$\"3I6Zk?=t=@F-7$$\"
3Y*\\,L6bJm)F;$\"3I<hsaTxo?F-7$$\"3[,gAQH%y7*F;$\"30T_teMXE?F-7$$\"39E
hT[`Qo'*F;$\"3z\"HQ%GWAx>F-7$$\"3&)*4#G(p&o95Fco$\"3aAt3&R[O$>F-7$$\"3
[7m%yh\"=n5Fco$\"3;c:2!eQe)=F-7$$\"3aPyU[,<<6Fco$\"39n,9IBJS=F-7$$\"3%
\\7[f$zKp6Fco$\"3S.SNk/\"Gz\"F-7$$\"3^i3zm]A<7Fco$\"3wcY3B*)=\\<F-7$$
\"3*\\P#3$f)))o7Fco$\"3SM,w(>P@q\"F-7$$\"3a7(HrZ_DK\"Fco$\"3Y]WzyOE`;F
-7$$\"3Y(eZf/n#p8Fco$\"3B(Hio8>2h\"F-7$$\"3(\\Ke+1?(>9Fco$\"3!*e(Hg')p
Zc\"F-7$$\"3$)*pl/'H%=Z\"Fco$\"3pg.7t(*H<:F-7$$\"3tCIm\\]$G_\"Fco$\"3S
:iK[&f3Z\"F-7$$\"3P7e#z^s@d\"Fco$\"3+6g#HCEfU\"F-7$$\"3)[dBYO`pi\"Fco$
\"3pMz>oa.w8F-7$$\"3j*4e=Rwhn\"Fco$\"3V5H7)Q17L\"F-7$$\"3%)\\d()zCtG<F
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gblF`bl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv
e 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "Rather than fi
tting by eye, one may use some sort of regression line to fit the data
.  Here we use the least squares regression line." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 63 "lseqn:=fit[leastsquare[[x,y]]]([modifiedPval
ues,quotientlist]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&lseqnG/%\"yG
,&$\"+oqvdG!#6\"\"\"*&$\"+/\"Rt5*!#9F+%\"xGF+!\"\"" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 158 "Now that we have good coefficients for the line,
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differential equation.  Do some algebra." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "k:=-m;\nM:=-b/m;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"kG$\"+/\"Rt5*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG$\"+8<'y8
$!\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "The following cell pull
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ably want the student to do this by herself." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 109 "#polything:=rhs(lseqn):\n\n#kprime:=-coeff(poly
thing,x,1);\n#Mprime:=-coeff(polything,x,0)/coeff(polything,x,1);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Here is our differential equation
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "DE1:=diff(P(t),t)=k*P(t
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "Now we want to look at the differ
ential equation along with the data.  We will use a slope field to rep
resent the diffential equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 64 "plot4:=dfieldplot(DE1,P,t=tmin..tmax,P=-25..(1.1*M),color=blue
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45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 368 "The followin
g code should put the two plots in a horizontal array, so they may be \+
viewed side by side.  Unfortunately the Mac version of Maple does not \+
like this.  We put it here to use on a Windows machine.  Executing thi
s on a Mac will only produce an error.  Note that once this display is
 given, one may want to drag the image so that it is wider than the de
fault." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "ddfdfdf" }{MPLTEXT 1 0 
318 "#m:=-.0001;     b:=.029;     k:=-m;M:=-b/m;\n#plot3:=plot(m*x+b,x
=Pmin..Pmax):\n#DE1:=diff(P(t),t)=k*P(t)*(M-P(t)):\n#plot4:=dfieldplot
(DE1,P,t=tmin..tmax,P=(-.1*M)..(1.1*M),color=blue):\n#display1:=displa
y(plot2,plot3):\n#display2:=display(plot4,plot1):\n#A:= array(1..2):\n
#A[1] := display1:\n#A[2] := display2:\n#display(A);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
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0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 109 "Here we are actually solving the differential \+
equation using the first data point as our initial condition.  " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalf(dsolve(\{DE1,P(tvalues
[1])=Pvalues[1]\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6#%\"tG
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#6F*F*!\"\"$\"+8<'y8$F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 166 "Take \+
the output on the rightside of the above equation and put it next to t
he -> in this cell.  This will enable us to plot our solution along wi
th other stuff below." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "s:=t->3137
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0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "We now want to take a look at the
 original data, the slope field, and the solution Maple just calculate
d." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot5:=plot(s(t),t=tmin..tmax
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{MPLTEXT 1 0 27 "display(plot1,plot4,plot5);" }}{PARA 13 "" 1 "" 
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