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{SECT 0 {PARA 0 "" 0 "" {TEXT 274 71 "Maple Interactive Text:  LWInter
active.mws, John Pais (Revised 6-23-03)" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 58 "Estimate Vmax & Km Using Lin
eweaver-Burk and Least Squares" }}{PARA 0 "" 0 "" {TEXT 265 117 "Start
 Here:  Enter your data values for S(0) in a list called S0, and your \+
data values for P'(0) in a list called V0." }}{PARA 0 "" 0 "" {TEXT 
275 105 "Note: Sample Data from LW Kinetigram Exercise 1 is already en
tered below, in order to provide an example." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 276 100 "To Run the code below just click in the red ar
ea and press [Enter], then just keep pressing [Enter]." }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 482 "restart:\nS0:=[3.00,2
.71,2.42,2.13,1.85,1.57,1.28,1.01,.740,.483,.249,.0723,.00694,.000368]
:\nV0:=[1.94,1.93,1.92,1.91,1.90,1.88,1.86,1.82,1.76,1.66,1.43,.840,.1
30,.00736]:\nif nops(S0) <> nops(V0) then \nprint(`length(S0)`= nops(S
0),`length(V0)`= nops(V0)):\nprint(`S0 and V0 must have the same lengt
h!`):\nelse\nn:=nops(S0):\nprint(`Original Data:  S0 = substrate value
s, V0 = velocity values`):\nprint(`length(S0)`= nops(S0),`length(V0)`=
 nops(V0)):\nprint('S0'=S0):\nprint('V0'=V0):\nfi:``;" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 266 4 "Next" }{TEXT -1 130 ", transform your
 data using the Lineweaver-Burk method: take the reciprocal of both si
des of the Michaelis-Menten Uptake Equation." }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }{MPLTEXT 1 0 488 "n:=nops(S0):\ntrS0:=[seq(1/S0[i],i=
1..n)]:\ntrV0:=[seq(1/V0[i],i=1..n)]:\n`Lineweaver-Burk Transform of O
riginal Data`;\n'V[o]'='V[max]*S[o]/(K[m]+S[o])';\n'1/V[o]'='(K[m]+S[o
])/(V[max]*S[o])';\n'1/V[o]'='(K[m]/V[max])*(1/S[o])+(1/V[max])';\n`So
, `*'y'='m*x+b',`where`*'y'='1/V[o]','m=K[m]/V[max]',\n'x'='1/S[o]','b
=1/V[max]';\n`And we transform your data by taking the reciprocal of e
ach entry in each of your lists:`;\n'trS0'=[seq(evalf(trS0[i],4),i=1..
n)];\n'trV0'=[seq(evalf(trV0[i],4),i=1..n)];" }}}{PARA 0 "" 0 "" 
{TEXT 267 3 "Now" }{TEXT -1 141 ", use the method of least squares to \+
fit your transformed data to a line. Note that, in general, you will h
ave to experiment with values for " }{TEXT 270 4 "xmax" }{TEXT -1 5 " \+
and " }{TEXT 271 4 "ymax" }{TEXT -1 39 " (below), in order to get a ni
ce plot. " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 682 "x
max:=14:\nymax:=2:\nm:='m':b:='b':V[max]:='V[max]':K[m]:='K[m]':i:='i'
:\npts:=[seq([trS0[i],trV0[i]],i=1..n)]:\nwith(plots):with(stats):\nDa
ta:=plot(pts,style=POINT,symbol=CIRCLE,color=blue):\nfit[leastsquare[[
x,y],y=m*x+b]]([trS0,trV0]):\nm:=evalf(op(1,op(1,rhs(%)))):\nb:=evalf(
op(2,rhs(%%))):\nV[max]:=1/b:\nK[m]:=m/b:\nFit:=plot(m*x+b,x=0..xmax,c
olor=navy):\nplots[display](\{Fit,Data\},\n         view=[0..xmax,0..y
max],\n         labels=[\"trS0\",\"trV0 \"],\n         tickmarks=[4,4]
,\n         title=`Lineweaver-Burk Transform & Least Squares Fit`,\n  \+
       titlefont=[HELVETICA,DEFAULT,14]);\n'm'=evalf(m,4),'b'=evalf(b,
4);\n'V[max]=1/b','K[m]=m/b';\n'V[max]'=evalf(V[max],4),'K[m]'=evalf(K
[m],4);" }}}{PARA 0 "" 0 "" {TEXT 272 7 "Finally" }{TEXT -1 100 ", use
 the values of Vmax and Km to specify the corresponding Michaelis-Ment
en Uptake Model and plot." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{MPLTEXT 1 0 574 "alias(W=LambertW):\nMMUptakeModel:=proc(s0,Km,Vm)\nl
ocal s,K,V,S,P,`S'`,`P'`:\ns[0]:=s0:K[m]:=Km:V[m]:=Vm:\nS:=unapply(K[m
]*W((s[0]/K[m])*exp((s[0]/K[m])-(V[m]/K[m])*t)),t):\nP:=unapply(s[0]-S
(t),t):\n`S'`:=unapply(-V[m]*S(t)/(K[m]+S(t)),t):\n`P'`:=unapply(V[m]*
S(t)/(K[m]+S(t)),t):\n[S,P,`S'`,`P'`]\nend:\nh:='h':k:='k':v:='v':s:=h
*k:\nS:=op(1,MMUptakeModel(s,k,v)):\nP:=op(2,MMUptakeModel(s,k,v)):\n`
S'`:=op(3,MMUptakeModel(s,k,v)):\n`P'`:=op(4,MMUptakeModel(s,k,v)):``;
\n`Michaelis-Menten Uptake Model`;\n'S(t)'=S(t),'P(t)'=P(t);\n'`S'`(t)
'='-v*S(t)/(k+S(t))','`P'`(t)'='v*S(t)/(k+S(t))';" }}}{PARA 0 "" 0 "" 
{TEXT -1 47 "Initialize to Vmax and Km found above and plot." }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 781 "Digits:=5:\ns0
:=max(seq(S0[i],i=1..n)):\nh:=s0/k:k:=evalf(K[m],4):v:=evalf(V[max],4)
:\nvstr:=convert(v,string):vstr:=``.vstr:\np1:=plot([S(t),`P'`(t),t=0.
.4],color=blue):\np2:=plot([[0,v],[s,v]],linestyle=3,color=red):\np3:=
plot([[0,v/2],[k,v/2]],linestyle=3,color=red):\np4:=plot([[k,0],[k,v/2
]],linestyle=3,color=red):\nfor i from 1 to n do\np[i]:=plot([[S0[i],V
0[i]]],style=point,symbol=circle,color=navy):od:\nMMUptakePlot:=plots[
display]([p1,p2,p3,p4,seq(p[i],i=1..n)],\nview=[0..s0,0..v],labels=[`S
`,`P'`],tickmarks=[4,4],color=navy,\ntitle=`    M-M Uptake Model:  P'(
t) vs S(t)`, titlefont=[HELVETICA,DEFAULT,14]):\nMMUptakePlot;\n`Linew
eaver-Burk Transform:  `*'S(0)'=S(0),'V[max]'=v,'K[m]'=k;\n'S(t)'=S(t)
,'P(t)'=P(t);\n'`S'`(t)'=-vstr*'S(t)'/(k+'S(t)'),'`P'`(t)'=vstr*'S(t)'
/(k+'S(t)');\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 56 "Estimate Vmax & Km Using Eadie-Ho
fstee and Least Squares" }}{PARA 0 "" 0 "" {TEXT 262 117 "Start Here: \+
 Enter your data values for S(0) in a list called S0, and your data va
lues for P'(0) in a list called V0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 277 105 "Note: Sample Data from LW Kinetigram Exercise 1 is alre
ady entered below, in order to provide an example." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 278 100 "To Run the code below just click in the \+
red area and press [Enter], then just keep pressing [Enter]." }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 482 "restart:\nS0:=
[3.00,2.71,2.42,2.13,1.85,1.57,1.28,1.01,.740,.483,.249,.0723,.00694,.
000368]:\nV0:=[1.94,1.93,1.92,1.91,1.90,1.88,1.86,1.82,1.76,1.66,1.43,
.840,.130,.00736]:\nif nops(S0) <> nops(V0) then \nprint(`length(S0)`=
 nops(S0),`length(V0)`= nops(V0)):\nprint(`S0 and V0 must have the sam
e length!`):\nelse\nn:=nops(S0):\nprint(`Original Data:  S0 = substrat
e values, V0 = velocity values`):\nprint(`length(S0)`= nops(S0),`lengt
h(V0)`= nops(V0)):\nprint('S0'=S0):\nprint('V0'=V0):\nfi:``;" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 4 "Next" }{TEXT -1 158 ", tra
nsform your data using the Eadie-Hofstee method: take the reciprocal o
f both sides of the M-M Uptake Equation, and then multiply both sides \+
by Vmax * V0." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 
540 "n:=nops(S0):\ntrS0:=[seq(V0[i]/S0[i],i=1..n)]:\ntrV0:=[seq(V0[i],
i=1..n)]:\n`Eadie-Hofstee Transform of Original Data`;\n'V[o]'='V[max]
*S[o]/(K[m]+S[o])';\n'1/V[o]'='(K[m]+S[o])/(V[max]*S[o])';\n'(V[o]*V[m
ax])*(1/V[o])'='(V[o]*V[max])*(K[m]+S[o])/(V[max]*S[o])';\n'V[o]'='-K[
m]*(V[o]/S[o])+V[max]';\n`So, `*'y'='m*x+b',`where`*'y'='V[o]','m=-K[m
]',\n'x'='V[o]/S[o]','b=V[max]';\n`And we transform your data by apply
ing this transform to each entry in each of your lists:`;\n'trS0'=[seq
(evalf(trS0[i],4),i=1..n)];\n'trV0'=[seq(evalf(trV0[i],4),i=1..n)];" }
}}{PARA 0 "" 0 "" {TEXT 264 3 "Now" }{TEXT -1 141 ", use the method of
 least squares to fit your transformed data to a line. Note that, in g
eneral, you will have to experiment with values for " }{TEXT 268 4 "xm
ax" }{TEXT -1 5 " and " }{TEXT 269 4 "ymax" }{TEXT -1 38 " (below), in
 order to get a nice plot." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{MPLTEXT 1 0 675 "xmax:=20:\nymax:=2:\nm:='m':b:='b':V[max]:='V[max]':
K[m]:='K[m]':i:='i':\npts:=[seq([trS0[i],trV0[i]],i=1..n)]:\nwith(plot
s):with(stats):\nData:=plot(pts,style=POINT,symbol=CIRCLE,color=blue):
\nfit[leastsquare[[x,y],y=m*x+b]]([trS0,trV0]):\nm:=evalf(op(1,op(1,rh
s(%)))):\nb:=evalf(op(2,rhs(%%))):\nV[max]:=b:\nK[m]:=-m:\nFit:=plot(m
*x+b,x=0..xmax,color=navy):\nplots[display](\{Fit,Data\},\n         vi
ew=[0..xmax,0..ymax],\n         labels=[\"trS0\",\"trV0 \"],\n        \+
 tickmarks=[4,4],\n         title=`Eadie-Hofstee Transform & Least Squ
ares Fit`,\n         titlefont=[HELVETICA,DEFAULT,14]);\n'm'=evalf(m,4
),'b'=evalf(b,4);\n'V[max]=b','K[m]=-m';\n'V[max]'=evalf(V[max],4),'K[
m]'=evalf(K[m],4);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 273 7 "F
inally" }{TEXT -1 100 ", use the values of Vmax and Km to specify the \+
corresponding Michaelis-Menten Uptake Model and plot." }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 574 "alias(W=LambertW):\nMMUptak
eModel:=proc(s0,Km,Vm)\nlocal s,K,V,S,P,`S'`,`P'`:\ns[0]:=s0:K[m]:=Km:
V[m]:=Vm:\nS:=unapply(K[m]*W((s[0]/K[m])*exp((s[0]/K[m])-(V[m]/K[m])*t
)),t):\nP:=unapply(s[0]-S(t),t):\n`S'`:=unapply(-V[m]*S(t)/(K[m]+S(t))
,t):\n`P'`:=unapply(V[m]*S(t)/(K[m]+S(t)),t):\n[S,P,`S'`,`P'`]\nend:\n
h:='h':k:='k':v:='v':s:=h*k:\nS:=op(1,MMUptakeModel(s,k,v)):\nP:=op(2,
MMUptakeModel(s,k,v)):\n`S'`:=op(3,MMUptakeModel(s,k,v)):\n`P'`:=op(4,
MMUptakeModel(s,k,v)):``;\n`Michaelis-Menten Uptake Model`;\n'S(t)'=S(
t),'P(t)'=P(t);\n'`S'`(t)'='-v*S(t)/(k+S(t))','`P'`(t)'='v*S(t)/(k+S(t
))';" }}}{PARA 0 "" 0 "" {TEXT -1 47 "Initialize to Vmax and Km found \+
above and plot." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 
779 "Digits:=5:\ns0:=max(seq(S0[i],i=1..n)):\nh:=s0/k:k:=evalf(K[m],4)
:v:=evalf(V[max],4):\nvstr:=convert(v,string):vstr:=``.vstr:\np1:=plot
([S(t),`P'`(t),t=0..4],color=blue):\np2:=plot([[0,v],[s,v]],linestyle=
3,color=red):\np3:=plot([[0,v/2],[k,v/2]],linestyle=3,color=red):\np4:
=plot([[k,0],[k,v/2]],linestyle=3,color=red):\nfor i from 1 to n do\np
[i]:=plot([[S0[i],V0[i]]],style=point,symbol=circle,color=navy):od:\nM
MUptakePlot:=plots[display]([p1,p2,p3,p4,seq(p[i],i=1..n)],\nview=[0..
s0,0..v],labels=[`S`,`P'`],tickmarks=[4,4],color=navy,\ntitle=`    M-M
 Uptake Model:  P'(t) vs S(t)`, titlefont=[HELVETICA,DEFAULT,14]):\nMM
UptakePlot;\n`Eadie-Hofstee Transform:  `*'S(0)'=S(0),'V[max]'=v,'K[m]
'=k;\n'S(t)'=S(t),'P(t)'=P(t);\n'`S'`(t)'=-vstr*'S(t)'/(k+'S(t)'),'`P'
`(t)'=vstr*'S(t)'/(k+'S(t)');\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 48 "Estimate Vmax & Km Using Hanes and Least Squares" }}
{PARA 0 "" 0 "" {TEXT 256 117 "Start Here:  Enter your data values for
 S(0) in a list called S0, and your data values for P'(0) in a list ca
lled V0." }{TEXT -1 0 "" }{TEXT 279 0 "" }}{PARA 0 "" 0 "" {TEXT 280 
105 "Note: Sample Data from LW Kinetigram Exercise 1 is already entere
d below, in order to provide an example." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 281 100 "To Run the code below just click in the red area \+
and press [Enter], then just keep pressing [Enter]." }}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 482 "restart:\nS0:=[3.00,2.71,2.42
,2.13,1.85,1.57,1.28,1.01,.740,.483,.249,.0723,.00694,.000368]:\nV0:=[
1.94,1.93,1.92,1.91,1.90,1.88,1.86,1.82,1.76,1.66,1.43,.840,.130,.0073
6]:\nif nops(S0) <> nops(V0) then \nprint(`length(S0)`= nops(S0),`leng
th(V0)`= nops(V0)):\nprint(`S0 and V0 must have the same length!`):\ne
lse\nn:=nops(S0):\nprint(`Original Data:  S0 = substrate values, V0 = \+
velocity values`):\nprint(`length(S0)`= nops(S0),`length(V0)`= nops(V0
)):\nprint('S0'=S0):\nprint('V0'=V0):\nfi:``;" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 257 4 "Next" }{TEXT -1 143 ", transform your data
 using the Hanes method: take the reciprocal of both sides of the M-M \+
Uptake Equation, and then multiply both sides by S0." }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 536 "n:=nops(S0):\ntrS0:=[seq(S0
[i],i=1..n)]:\ntrV0:=[seq(S0[i]/V0[i],i=1..n)]:\n`Hanes Transform of O
riginal Data`;\n'V[o]'='V[max]*S[o]/(K[m]+S[o])';\n'1/V[o]'='(K[m]+S[o
])/(V[max]*S[o])';\n'S[o]*(1/V[o])'='S[o]*(K[m]+S[o])/(V[max]*S[o])';
\n'S[o]*(1/V[o])'='(S[o]/V[max])'+'(K[m]/V[max])';\n`So, `*'y'='m*x+b'
,`where`*'y'='S[o]/V[o]','m=1/V[max]',\n'x'='S[o]','b=K[m]/V[max]';\n`
And we transform your data by applying this transform to each entry in
 each of your lists:`;\n'trS0'=[seq(evalf(trS0[i],4),i=1..n)];\n'trV0'
=[seq(evalf(trV0[i],4),i=1..n)];" }}}{PARA 0 "" 0 "" {TEXT 258 3 "Now
" }{TEXT -1 141 ", use the method of least squares to fit your transfo
rmed data to a line. Note that, in general, you will have to experimen
t with values for " }{TEXT 259 4 "xmax" }{TEXT -1 5 " and " }{TEXT 
260 4 "ymax" }{TEXT -1 38 " (below), in order to get a nice plot." }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 682 "xmax:=3:\nymax
:=1.6:\nm:='m':b:='b':V[max]:='V[max]':K[m]:='K[m]':i:='i':\npts:=[seq
([trS0[i],trV0[i]],i=1..n)]:\nwith(plots):with(stats):\nData:=plot(pts
,style=POINT,symbol=CIRCLE,color=blue):\nfit[leastsquare[[x,y],y=m*x+b
]]([trS0,trV0]):\nm:=evalf(op(1,op(1,rhs(%)))):\nb:=evalf(op(2,rhs(%%)
)):\nV[max]:=1/m:\nK[m]:=b/m:\nFit:=plot(m*x+b,x=0..xmax,color=navy):
\nplots[display](\{Fit,Data\},\n         view=[0..xmax,0..ymax],\n    \+
     labels=[\"trS0\",\"trV0 \"],\n         tickmarks=[4,4],\n        \+
 title=`        Hanes Transform & Least Squares Fit`,\n         titlef
ont=[HELVETICA,DEFAULT,14]);\n'm'=evalf(m,4),'b'=evalf(b,4);\n'V[max]=
1/m','K[m]=b/m';\n'V[max]'=evalf(V[max],4),'K[m]'=evalf(K[m],4);\n" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 7 "Finally" }{TEXT -1 100 ",
 use the values of Vmax and Km to specify the corresponding Michaelis-
Menten Uptake Model and plot." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}{MPLTEXT 1 0 574 "alias(W=LambertW):\nMMUptakeModel:=proc(s0,Km,Vm)\n
local s,K,V,S,P,`S'`,`P'`:\ns[0]:=s0:K[m]:=Km:V[m]:=Vm:\nS:=unapply(K[
m]*W((s[0]/K[m])*exp((s[0]/K[m])-(V[m]/K[m])*t)),t):\nP:=unapply(s[0]-
S(t),t):\n`S'`:=unapply(-V[m]*S(t)/(K[m]+S(t)),t):\n`P'`:=unapply(V[m]
*S(t)/(K[m]+S(t)),t):\n[S,P,`S'`,`P'`]\nend:\nh:='h':k:='k':v:='v':s:=
h*k:\nS:=op(1,MMUptakeModel(s,k,v)):\nP:=op(2,MMUptakeModel(s,k,v)):\n
`S'`:=op(3,MMUptakeModel(s,k,v)):\n`P'`:=op(4,MMUptakeModel(s,k,v)):``
;\n`Michaelis-Menten Uptake Model`;\n'S(t)'=S(t),'P(t)'=P(t);\n'`S'`(t
)'='-v*S(t)/(k+S(t))','`P'`(t)'='v*S(t)/(k+S(t))';" }}}{PARA 0 "" 0 "
" {TEXT -1 47 "Initialize to Vmax and Km found above and plot." }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 771 "Digits:=5:\ns0
:=max(seq(S0[i],i=1..n)):\nh:=s0/k:k:=evalf(K[m],4):v:=evalf(V[max],4)
:\nvstr:=convert(v,string):vstr:=``.vstr:\np1:=plot([S(t),`P'`(t),t=0.
.4],color=blue):\np2:=plot([[0,v],[s,v]],linestyle=3,color=red):\np3:=
plot([[0,v/2],[k,v/2]],linestyle=3,color=red):\np4:=plot([[k,0],[k,v/2
]],linestyle=3,color=red):\nfor i from 1 to n do\np[i]:=plot([[S0[i],V
0[i]]],style=point,symbol=circle,color=navy):od:\nMMUptakePlot:=plots[
display]([p1,p2,p3,p4,seq(p[i],i=1..n)],\nview=[0..s0,0..v],labels=[`S
`,`P'`],tickmarks=[4,4],color=navy,\ntitle=`    M-M Uptake Model:  P'(
t) vs S(t)`, titlefont=[HELVETICA,DEFAULT,14]):\nMMUptakePlot;\n`Hanes
 Transform:  `*'S(0)'=S(0),'V[max]'=v,'K[m]'=k;\n'S(t)'=S(t),'P(t)'=P(
t);\n'`S'`(t)'=-vstr*'S(t)'/(k+'S(t)'),'`P'`(t)'=vstr*'S(t)'/(k+'S(t)'
);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "References" }}
{PARA 0 "" 0 "" {TEXT -1 127 "[1] E. K. Yeargers, R. W. Shonkwiler, an
d J. V. Herod. An Introduction to the Mathematics of Biology. Boston: \+
Birkhauser, 1996." }}{PARA 0 "" 0 "" {TEXT -1 63 "[2] Jim Herod's Home
page:  http://www.math.gatech.edu/%7Eherod/" }}{PARA 0 "" 0 "" {TEXT 
-1 172 "Note that Maple V, R5 worksheets for the book [1] are availabl
e at this author's website. Ideas for the current worksheet began with
 pp. 312-314 of [1] and Ch2b.mws of [2]." }}}}{MARK "1 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 }