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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 24 "Data and Logistic Growth
" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 
-1 22 "A Maple investigation " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 3 "by " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }
}{PARA 256 "" 0 "" {TEXT -1 63 "Helen Compton, John W. Dollhopf, David
 Kapolka, and Jo Ann Lutz" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 38 "Produced during the Web Feats Workshop" }}
{PARA 256 "" 0 "" {TEXT -1 12 "Columbus, OH" }}{PARA 256 "" 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." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 283 "The 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, replace this data with ot
her data.  Be sure the data is in the same form as this.  Other intere
sting data can be found from a link on the webpage." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 381 "restart;\ndatali
st:=evalf(\n\n[[1790,\0113.929],\n[1800,\0115.308],\011\n[1810,\0117.2
40],\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,\0113
8.558],\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,\011122.7
75],\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\n):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "This cell installs some additio
nal commands that Maple will use in this sheet." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(plots):\nwith(
stats):\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 14 "with(DEtools):" }}{PARA 7 
"" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined
\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 pairs (
as above in uscensusdata) and as separate lists.  The following comman
ds create two lists, one with the years and one with the populations.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "Throughout our work, we will need
 to use the data you inserted above as separate lists as well as the o
rdered pairs.  The following commands will pull apart the inputted lis
t." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
100 "listlength:=nops(datalist):\ntvalues:=datalist[1..listlength,1];
\nPvalues:=datalist[1..listlength,2];\n" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%(tvaluesG77$\"%!z\"\"\"!$\"%+=F($\"%5=F($\"%?=F($\"%I=F($\"%S=
F($\"%]=F($\"%g=F($\"%q=F($\"%!)=F($\"%!*=F($\"%+>F($\"%5>F($\"%?>F($
\"%I>F($\"%S>F($\"%]>F($\"%g>F($\"%q>F($\"%!)>F($\"%!*>F(" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#>%(PvaluesG77$\"%HR!\"$$\"%3`F($\"%SsF($\"%Q'*F
($\"&mG\"F($\"&pq\"F($\"&#>BF($\"&V9$F($\"&e&QF($\"&c,&F($\"&[H'F($\"&
'*f(F($\"&s>*F($\"'6d5F($\"'vF7F($\"'p;8F($\"'(p]\"F($\"'B$z\"F($\"'&=
.#F($\"'YlAF($\"'5([#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 138 "Let us first look  at a plot of the data
.  The following command will give us a scatterplot of the data.  We g
ive it a name for later use." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 65 "plot1:=plot(datalist,style=point, symbol=
circle,symbolsize=15):%;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
{PLOTDATA 2 "6'-%'CURVESG6$777$$\"%!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$$
\"%S>F*$\"36+++++p;8Fjo7$$\"%]>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-%+AXES
LABELSG6$Q!6\"Fir-%'SYMBOLG6$%'CIRCLEG\"#:-%&STYLEG6#%&POINTG-%%VIEWG6
$%(DEFAULTGFgs" 1 5 4 1 15 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 
0 "Curve 1" }}}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 30 "II.  Is the Data Set
 Logistic?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 139 "Hopefully the scatterplot indicates that your data is po
tentially logistic.  We now need to analyze the data to help us find t
he constants " }{TEXT 259 1 "k" }{TEXT -1 5 " and " }{TEXT 260 1 "M" }
{TEXT -1 30 " in the differential equation." }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 165 "The following command calculates each symmetric differen
ce (our estimate of the derivative).  Note that this gives us 2 fewer \+
pieces of data than with what we began." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "symdiff:=[]:\nfor i from 2 by
 1 to (listlength-1) do\n    symdiff:=[op(symdiff),(Pvalues[i+1]-Pvalu
es[i-1])/\n              (tvalues[i+1]-tvalues[i-1])]:\n  end do:\nsym
diff;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#75$\"+++]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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "Followin
g the example on the webpage, we need to divide our estimate of the de
rivative by the population values.  Because we have two fewer pieces o
f data, we must remove the first and last elements of our P values lis
t.  We then calculate the needed quotient." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 251 "modifiedPvalues:=subsop(
listlength=NULL,Pvalues):\nmodifiedPvalues:=subsop(1=NULL,modifiedPval
ues);\n\n\nquotientlist:=[]:\nfor i from 1 by 1 to (listlength-2) do\n
     quotientlist:=[op(quotientlist),symdiff[i]/modifiedPvalues[i]]:\n
   end do:\nquotientlist;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%0modifi
edPvaluesG75$\"%3`!\"$$\"%SsF($\"%Q'*F($\"&mG\"F($\"&pq\"F($\"&#>BF($
\"&V9$F($\"&e&QF($\"&c,&F($\"&[H'F($\"&'*f(F($\"&s>*F($\"'6d5F($\"'vF7
F($\"'p;8F($\"'(p]\"F($\"'B$z\"F($\"'&=.#F($\"'YlAF(" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#75$\"+nr()=J!#6$\"+#\\J.*HF&$\"+B`l=HF&$\"+IR%y)GF&$
\"+o<yCIF&$\"+=M\"*)4$F&$\"+N\"pMW#F&$\"+xSgECF&$\"+))RTJCF&$\"+ox[_?F
&$\"+)=u&4>F&$\"+2tV:;F&$\"+KR%pX\"F&$\"+Rq8d5F&$\"+C.Jg5F&$\"+I(>6e\"
F&$\"+.W]j9F&$\"+]!p?;\"F&$\"+IGw/5F&" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 39 "To see the (hopefully) linear plot of (" }{XPPEDIT 18 0 "
P,1/P;" "6$%\"PG*&\"\"\"F%F#!\"\"" }{XPPEDIT 18 0 "dP/dt;" "6#*&%#dPG
\"\"\"%#dtG!\"\"" }{TEXT -1 52 " ), we need to create a list with each
 ordered pair." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "plotlist:=[]:\nf
or i from 1 by 1 to (listlength-2) do\n    plotlist:=[op(plotlist),[mo
difiedPvalues[i],quotientlist[i]]]:\nend do:\nplotlist;" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#757$$\"%3`!\"$$\"+nr()=J!#67$$\"%SsF'$\"+#\\J.*H
F*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'$\"+xSgEC
F*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$$\"'YlAF'$\"+IGw/5F*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "If we are lucky, \+
this data will have a linear pattern.  Otherwise the assumption of a l
ogistic model is incorrect.  Let us look at the scatterplot.  Again, w
e give it a name for later use." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot2:=plot(plotlist,style=point):%
;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6
$757$$\"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)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$\"3++++IGw/5F--%'COLOURG6&%$RG
BG$\"#5!\"\"$\"\"!F\\rF[r-%+AXESLABELSG6$Q!6\"F`r-%&STYLEG6#%&POINTG-%
%VIEWG6$%(DEFAULTGFir" 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 94 "For later convenience in plotting, we nee
d the smallest and largest elements of our two lists." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "Pmax:=sort(modi
fiedPvalues)[listlength-2];\nPmin:=sort(modifiedPvalues)[1];\ntmax:=so
rt(tvalues)[listlength];\ntmin:=sort(tvalues[1]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%PmaxG$\"'YlA!\"$" }}{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 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "W
e want to fit a line to this plot.  Manipulate the values of m and b t
o find a good line that fits the data.  When you are satisfied with th
e fit of your line, continue to the next part." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 16 "input this value" }
{TEXT -1 4 "    " }{MPLTEXT 1 0 20 "m:=-.1006106771e-3;\n" }{TEXT 262 
16 "input this value" }{TEXT -1 4 "    " }{MPLTEXT 1 0 74 "b:=.2908504
494e-1;\n\nplot3:=plot(m*x+b,x=Pmin..Pmax):\ndisplay(plot2,plot3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$!+rn515!#8" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"bG$\"+%\\/&3H!#6" }}{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/+++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$$\"3#**********faE#
Fco$\"3++++IGw/5F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F\\rF[r-%&STYLEG6#
%&POINTG-F$6$7S7$F($\"3!)>`fY.5bGF-7$$\"3(**\\_$ze.85F;$\"3&3v^#oAe1GF
-7$$\"3)\\(eZRniK9F;$\"3'pw4x%pOkFF-7$$\"3(*\\FKX0]/>F;$\"3()exf/9*or#
F-7$$\"3>]#fqB-&zBF;$\"3+/&oF;,\"pEF-7$$\"3ru.nGjC_GF;$\"3J%[;'f!Q:i#F
-7$$\"3/v)\\qhS0H$F;$\"35gN\\u4WxDF-7$$\"3([(3nl*oVu$F;$\"3[E;0(*4yJDF
-7$$\"3duG-$*)=P@%F;$\"3-1R1$QfX[#F-7$$\"3Qu`?4Oc\"o%F;$\"3LF6R4#*[PCF
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iDkM#F-7$$\"3o*\\B+[_Q1'F;$\"3a@.:!>;%)H#F-7$$\"3Y+D+_@*Ha'F;$\"3IUqKB
'4-D#F-7$$\"3q)\\w$3?t/qF;$\"3`(>LPm`P?#F-7$$\"3dvGVLg.CuF;$\"3@?:'=?n
:;#F-7$$\"35+&y^2HE#zF;$\"3IWTL=MS6@F-7$$\"3Q**R4#y-]M)F;$\"3!*Q@p8\"3
*o?F-7$$\"3@uQu8QKO))F;$\"3iuV-sfZ>?F-7$$\"3N)**)*y<E7F*F;$\"3WOVp],sv
>F-7$$\"3ztQp**HQ[(*F;$\"3Gq1xzIrF>F-7$$\"3Yi,=8\\F?5Fco$\"3I,!ob**)*>
)=F-7$$\"3(\\(=X_Kon5Fco$\"3:m[UW6IM=F-7$$\"3aP^=2*=76\"Fco$\"3S7yTF'*
\\!z\"F-7$$\"3#\\ixA*z<e6Fco$\"3w&HYRx`Ku\"F-7$$\"3K(GySGbp?\"Fco$\"3u
1YR5'yTp\"F-7$$\"3S7/)*fgT\\7Fco$\"3%[cdi`e9l\"F-7$$\"3)[n07'[F&H\"Fco
$\"3D&)[$eo>`g\"F-7$$\"3i*He,\\^EM\"Fco$\"3y%\\e'QPld:F-7$$\"3nu\\qy-+
*Q\"Fco$\"3#y.u'3>-6:F-7$$\"3?(=M[6XQV\"Fco$\"3i^+S:K!fY\"F-7$$\"3%[U?
&fwj$[\"Fco$\"3'y!f/*f1eT\"F-7$$\"3j**Q`v%y$G:Fco$\"3j&o96c#zq8F-7$$\"
3*)\\UOY)[hd\"Fco$\"3W*3pvYIFK\"F-7$$\"3<(o4C!fV>;Fco$\"3S=^Qn!z\"z7F-
7$$\"3!)\\rF)3gnm\"Fco$\"3'>!=b$)ecJ7F-7$$\"35(Q=\"f%)G6<Fco$\"31Mo9)e
ln=\"F-7$$\"3UPV\"pVLyv\"Fco$\"3d$GI3PO*R6F-7$$\"3%)\\sNg'RL!=Fco$\"3W
I<I^A:%4\"F-7$$\"3>(o/!f/)4&=Fco$\"3M'p17_?i/\"F-7$$\"3))*RX.@ko*=Fco$
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" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "An alternative to eyeb
alling the coefficients is to fit a regression line to the transformed
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{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "lseqn:=fit[leastsqu
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{TEXT -1 5 " and " }{TEXT 264 1 "b" }{TEXT -1 16 " into values of " }
{TEXT 265 1 "k" }{TEXT -1 5 " and " }{TEXT 266 1 "M" }{TEXT -1 48 ".  \+
Input those formulas into the following cell." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 18 "input this value  " }
{MPLTEXT 1 0 61 "  k:=-m  ;             #-m;  delete this stuff before
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{PARA 0 "" 0 "" {TEXT -1 368 "The following code should put the two pl
ots in a horizontal array, so they may be viewed side by side.  Unfort
unately the Mac version of Maple does not like this.  We put it here t
o use on a Windows machine.  Executing this on a Mac will only produce
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0 "" {MPLTEXT 1 0 318 "#m:=-.0001;     b:=.029;     k:=-m;M:=-b/m;\n#p
lot3:=plot(m*x+b,x=Pmin..Pmax):\n#DE1:=diff(P(t),t)=k*P(t)*(M-P(t)):\n
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\n#display1:=display(plot2,plot3):\n#display2:=display(plot4,plot1):\n
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0 "" 0 "" {TEXT -1 228 "What follows may or may not be useful to the i
dea behind this document.  Below we use Maple to explicitly solve the \+
differential equation and then look at that solution as a graph along \+
with the original data and the slopefield." }}{PARA 0 "" 0 "" {TEXT 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "H
ere we are actually solving the differential equation using the first \+
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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalf(dsolve(\{DE1,P(tv
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" }}{PARA 0 "" 0 "" {TEXT -1 166 "Take the output on the rightside of \+
the above equation and put it next to the -> in this cell.  This will \+
enable us to plot our solution along with other stuff below." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "s:=t->289
0850731.*1/(10000000.+.2958966148e32*exp(-.2908504494e-1*t));" }}
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