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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 0 374 "restart;\nuscensusdata:=[[1
790,\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
,\01123.192],\011\n[1860,\01131.443],\011\n[1870,\01138.558],\011\n[18
80,\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]]:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(pl
ots):\nwith(stats):\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 14 "with(DEtools):
" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has be
en redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "To serve our p
urpose, we will need the data as ordered pairs (as above in uscensusda
ta) and as separate lists.  The following comands create two lists, on
e with the years and one with the populations." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 72 "uscensusyear:=uscensusdata[1..21,1];\nuscensus
pop:=uscensusdata[1..21,2];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-usce
nsusyearG77\"%!z\"\"%+=\"%5=\"%?=\"%I=\"%S=\"%]=\"%g=\"%q=\"%!)=\"%!*=
\"%+>\"%5>\"%?>\"%I>\"%S>\"%]>\"%g>\"%q>\"%!)>\"%!*>" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#>%,uscensuspopG77$\"%HR!\"$$\"%3`F($\"%SsF($\"%Q'*F(
$\"&mG\"F($\"&pq\"F($\"&#>BF($\"&V9$F($\"&e&QF($\"&c,&F($\"&[H'F($\"&'
*f(F($\"&s>*F($\"'6d5F($\"'vF7F($\"'p;8F($\"'(p]\"F($\"'B$z\"F($\"'&=.
#F($\"'YlAF($\"'5([#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "Let us
 first look  at a plot of the data.  The following command will give u
s a scatterplot of the data.  We give it a name for later use." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "plot1:=plot(uscensusdata,t=
1790..2000,P=-25..310,style=point, symbol=circle,symbolsize=15,thickne
ss=0):%;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'C
URVESG6$777$$\"%!z\"\"\"!$\"3#)************GR!#<7$$\"%+=F*$\"3$)******
*****zI&F-7$$\"%5=F*$\"3A++++++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$$\"%S>F*$\"36+++++p;8F
jo7$$\"%]>F*$\"3.+++++(p]\"Fjo7$$\"%g>F*$\"31+++++B$z\"Fjo7$$\"%q>F*$
\"3-+++++&=.#Fjo7$$\"%!)>F*$\"3#**********faE#Fjo7$$\"%!*>F*$\"32+++++
5([#Fjo-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fer-%+AXESLABELSG6$Q\"t6\"Q\"P
Fjr-%*THICKNESSG6#F*-%'SYMBOLG6$%'CIRCLEG\"#:-%&STYLEG6#%&POINTG-%%VIE
WG6$;F($\"%+?F*;$!#DF*$\"$5$F*" 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 model is given by " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dP/dt = k*P*(M-P);" "6#/*&%#dPG
\"\"\"%#dtG!\"\"*(%\"kGF&%\"PGF&,&%\"MGF&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 c
onstants.  To use this differential equation, we must first identify t
he parameters.  One method for finding these parameters is to first re
write the equation." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Consider t
he following manipulation." }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "dP/(dt*
P) = k*M-k*P;" "6#/*&%#dPG\"\"\"*&%#dtGF&%\"PGF&!\"\",&*&%\"kGF&%\"MGF
&F&*&F-F&F)F&F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "To study this \+
equation in terms of the data that we have, we need to approximate the
 derivative, " }{XPPEDIT 18 0 "dP/dt" "6#*&%#dPG\"\"\"%#dtG!\"\"" }
{TEXT -1 199 ".  We approximate the derivative using difference quotie
nts.  In particular, we will look at (pop[i+1]-pop[i-1])/(year[i+1]-ye
ar[i-1]).  This is called the symmetric difference.  explanation......
..." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "The following command cal
culates each symmetric difference.  Note that this gives us n-2 pieces
 of data." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "symdiff:=[]:
\nfor i from 2 by 1 to 20 do\n    symdiff:=[op(symdiff),(uscensuspop[i
+1]-uscensuspop[i-1])/\n              (uscensusyear[i+1]-uscensusyear[
i-1])]:\n  end do:\nsymdiff;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#75$\"+++]b;!#5$\"++++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$\"+++:hBF7$\"+++DwAF7" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 400 "Note that in the above equation what we are reall
y interested in is the derivative divided by P.  The following command
 calculates those values.  Because there are more population data poin
ts than symmetric difference data points, we choose the population dat
a point in the middle of each symmetric difference.  As we will use th
is modified population data a number of times, we give it its own name
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "modifieduscensuspop:=
subsop(21=NULL,uscensuspop);\nmodifieduscensuspop:=subsop(1=NULL,modif
ieduscensuspop);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%4modifieduscensu
spopG76$\"%HR!\"$$\"%3`F($\"%SsF($\"%Q'*F($\"&mG\"F($\"&pq\"F($\"&#>BF
($\"&V9$F($\"&e&QF($\"&c,&F($\"&[H'F($\"&'*f(F($\"&s>*F($\"'6d5F($\"'v
F7F($\"'p;8F($\"'(p]\"F($\"'B$z\"F($\"'&=.#F($\"'YlAF(" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%4modifieduscensuspopG75$\"%3`!\"$$\"%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($\"'YlAF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "quotient
list:=[]:\nfor i from 1 by 1 to 19 do\n     quotientlist:=[op(quotient
list),symdiff[i]/modifieduscensuspop[i]]:\n   end do:\nquotientlist;" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#75$\"+nr()=J!#6$\"+#\\J.*HF&$\"+B`l=
HF&$\"+IR%y)GF&$\"+o<yCIF&$\"+=M\"*)4$F&$\"+N\"pMW#F&$\"+xSgECF&$\"+))
RTJCF&$\"+ox[_?F&$\"+)=u&4>F&$\"+2tV:;F&$\"+KR%pX\"F&$\"+Rq8d5F&$\"+C.
Jg5F&$\"+I(>6e\"F&$\"+.W]j9F&$\"+]!p?;\"F&$\"+IGw/5F&" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 49 "We now want to consider data points of th
e form (" }{XPPEDIT 18 0 "P,dP/(dt*P);" "6$%\"PG*&%#dPG\"\"\"*&%#dtGF&
F#F&!\"\"" }{TEXT -1 47 " ).  The following command constructs that li
st" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "plotlist:=[]:\nfor i
 from 1 by 1 to 19 do\n    plotlist:=[op(plotlist),[modifieduscensuspo
p[i],quotientlist[i]]]:\nend do:\nplotlist;" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#757$$\"%3`!\"$$\"+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$$\"&e&QF'$\"+xSgECF*7$$\"
&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*" }}}{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 60 "pl
ot2:=plot(plotlist,P=0..250,y=0.007..0.033,style=point):%;" }}{PARA 
13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$757$$\"3$)
***********zI&!#<$\"30+++nr()=J!#>7$$\"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)RTJC
F-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$$\"3#**********faE#Fco$\"3++++IGw/5F--%'COLOURG6&%$RGBG$\"#5!\"
\"$\"\"!F\\rF[r-%+AXESLABELSG6$Q\"P6\"Q\"yFar-%&STYLEG6#%&POINTG-%%VIE
WG6$;F[r$\"$]#F\\r;$\"\"(!\"$$\"#LF`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 want to fit a line to this plot.  Manipulate the valu
es of m and b to find a good line that fits the data." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "m:=-.0001;b:=.029;\nplot3:=plot(m*x
+b,x=0..250,y=0.007..0.033):\ndisplay(plot2,plot3);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"mG$!\"\"!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"bG$\"#H!\"$" }}{PARA 13 "" 1 "" {GLPLOT2D 190 172 172 {PLOTDATA 2 "
6&-%'CURVESG6%757$$\"3$)***********zI&!#<$\"30+++nr()=J!#>7$$\"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]\"Fc
o$\"3'*******H(>6e\"F-7$$\"31+++++B$z\"Fco$\"31+++.W]j9F-7$$\"3-+++++&
=.#Fco$\"3#*******\\!p?;\"F-7$$\"3#**********faE#Fco$\"3++++IGw/5F--%'
COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F\\rF[r-%&STYLEG6#%&POINTG-F$6$7S7$F[r$
\"3;+++++++HF-7$$\"3JmmmT&)G\\aF*$\"3NLLe9r]XGF-7$$\"3EL$ek`o!>5F;$\"3
wmTNYJ4)z#F-7$$\"3fmm\"z>)G_:F;$\"3ML$3-=rZu#F-7$$\"3lmmT&QU!*3#F;$\"3
[L$e9w&4\"p#F-7$$\"3@L$eRZXKi#F;$\"3umTg_anPEF-7$$\"3>m;z>,_=JF;$\"3dL
3-))z9)e#F-7$$\"3P**\\7G$[8j$F;$\"3E+v=n^'o`#F-7$$\"3;m;z%*frhTF;$\"3g
L3_+%GQ[#F-7$$\"3Y**\\ilFQ!p%F;$\"32+vVB<'4V#F-7$$\"3oLL$3_\"=M_F;$\"3
dmm\"z%=ewBF-7$$\"3WmmTg(fJr&F;$\"3aL$eR-%oGBF-7$$\"30++]7eP_iF;$\"3E+
+v=CwuAF-7$$\"3Y****\\Pf!Qz'F;$\"3P++D1%>1A#F-7$$\"3R++](=ubJ(F;$\"3:+
+D\"eU%o@F-7$$\"3vl;zW(*Q*y(F;$\"3\\L3_D51@@F-7$$\"3WKL$3F-GN)F;$\"31n
m\"Hx>Z1#F-7$$\"3ILLL$e'3I))F;$\"3wmmmT8*p,#F-7$$\"3\")**\\7.<G&Q*F;$
\"3,+voH=Zh>F-7$$\"3'>LL$eMsw)*F;$\"39nm;awK7>F-7$$\"3-+DJ&H\"fT5Fco$
\"3I+vo/(3%e=F-7$$\"3)**\\Pf)[$H4\"Fco$\"38+D19^12=F-7$$\"3IL$ek`1l9\"
Fco$\"3dm;ajM\\`<F-7$$\"3MLe*[.-d>\"Fco$\"3$o;/^'zH/<F-7$$\"3jm;/Egw[7
Fco$\"3IL$eR(RB^;F-7$$\"3em\"z%*f%)QI\"Fco$\"3cL3_+a6'f\"F-7$$\"3x*\\(
oza'=N\"Fco$\"3N+DJ?X8[:F-7$$\"3mm;zWho.9Fco$\"3[L$3_&QJ'\\\"F-7$$\"3'
*****\\i>Ad9Fco$\"35++]P!yFW\"F-7$$\"3$***\\i:jf4:Fco$\"3?+]P%o./R\"F-
7$$\"3z*\\7`>r-c\"Fco$\"3C+vo/)G(R8F-7$$\"3\"***\\P4q`;;Fco$\"3B+]i!*H
Y$G\"F-7$$\"3>LL$eM%4n;Fco$\"3ymm;ac!HB\"F-7$$\"31++v$4v5s\"Fco$\"3***
**\\i!\\#*y6F-7$$\"3Tm\"zWn*)*p<Fco$\"3gL3_D.,I6F-7$$\"33++DJiYB=Fco$
\"37++voP`w5F-7$$\"3(H$ek.Nyt=Fco$\"33nTN'\\;i-\"F-7$$\"3,+Dc^&zj#>Fco
$\"3A+]P%[/it*!#?7$$\"3>LL3-=!y(>Fco$\"3^om;z>)>A*F`^l7$$\"3!)*\\7G8O;
.#Fco$\"3-/](=nQOo)F`^l7$$\"3pmmm\"*\\[$3#Fco$\"3zKLL$3]^;)F`^l7$$\"3k
m;aQz]O@Fco$\"3_NLe91#\\j(F`^l7$$\"3GLekG=4*=#Fco$\"3Ym;a8<34rF`^l7$$
\"3)*****\\i4TPAFco$\"39-++v.*ei'F`^l7$$\"3GL$3F9!z#H#Fco$\"3(pm;Hd)4s
gF`^l7$$\"3emmmT>KUBFco$\"3'HLLLe!ywbF`^l7$$\"3#)*\\7.<L^R#Fco$\"3'H+v
oHo'[]F`^l7$$\"3%**\\(oa-oXCFco$\"3i,]7`u>VXF`^l7$$\"$]#F\\r$\"35+++++
++SF`^lFdq-%+AXESLABELSG6%Q\"P6\"Q\"yF\\bl%(DEFAULTG-%%VIEWG6$;F[rFdal
;$\"\"(!\"$$\"#LFfbl" 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 
133 "Rather than fitting by eye, one may use some sort of regression l
ine to fit the data.  Here we use the least squares regression line." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "lseqn:=fit[leastsquare[[x
,y]]]([modifieduscensuspop,quotientlist]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&lseqnG/%\"yG,&$\"+%\\/&3H!#6\"\"\"*&$\"+rn515!#8F+%
\"xGF+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "Now that we have \+
good coefficients for the line, we need to translate those value into \+
k and M values in the original 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$\"\"\"!\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"MG$\"+++++H!\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 137 "The following cell pulls the values straight from the least sq
uares regression line.  We probably want the student to do this by her
self." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "polything:=rhs(ls
eqn):\n\nkprime:=-coeff(polything,x,1);\nMprime:=-coeff(polything,x,0)
/coeff(polything,x,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$\"+rn
515!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG$\"+J2&3*G!\"(" }}}
{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
)*(M-P(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DE1G/-%%diffG6$-%\"P
G6#%\"tGF,,$*&F)\"\"\",&$\"+++++H!\"(F/F)!\"\"F/$F/!\"%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 134 "Now we want to look at the differential \+
equation along with the data.  We will use a slope field to represent \+
the diffential equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 
"plot4:=dfieldplot(DE1,P,t=1790..2000,P=-25..310,color=blue):" }}
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