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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 "Module 6 : Precalculus" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 27 "603 : Exponenti
al Functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 256 "" 0 "" {TEXT -1 17 "O B J E C T I V E" }}{PARA 0 "" 0 "
" {TEXT -1 163 "In the project we will investigate exponential functio
ns, exponential growth and decay, and problems involving these things \+
along with other exponential functions." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 257 "" 0 "" {TEXT -1 9 "S E T U P" }}{PARA 0 "" 0 "" {TEXT 
-1 252 "In this project we will use the following command packages. Ty
pe and execute this line before begining the project below. If you re-
enter the worksheet for this project, be sure to re-execute this state
ment before jumping to any point in the worksheet." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; wit
h(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoor
ds has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 83 "_______________________________________________________
____________________________" }}{PARA 4 "" 0 "" {TEXT -1 17 "A. Growth
 & Decay" }}{PARA 0 "" 0 "" {TEXT -1 83 "_____________________________
______________________________________________________" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 209 "First, well just catch a glimpse of what exponential gro
wth and decay looks like. Here is a family of exponential functions wi
th the same growth rate, but differing initial values ( C = 10, 20 , 3
0, ..., 100)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 117 " plot( \{ 10*k*exp(x/5) $ k = 1..10\}, x = -4..
4, y = 0..250,\n\011\011title=`Exponential Growth - Different Initial \+
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0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 173 "Here is a family of
 exponential functions with the same initial value (C), but differing \+
growth rates (1/5, 2/5, 3/5,...,2). Note that all of these functions a
re increasing." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 114 " plot( \{ 10*exp(x*k/5) $ k = 1..10\}, x = 0..4
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x6\"Q\"yFabq-%&TITLEG6#%LExponential~Growth~-~Different~Growth~RatesG-
%%VIEWG6$;F(Fg[l;F($\"$+\"F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }}}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "This family has diff
ering decay rates. Note that these functions area all decreasing." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
103 " plot( \{ 10*exp(-x*k/5) $ k = 1..10\}, x = 0..10, \n\011\011titl
e=`Exponential Decay - Different Decay Rates`);" }}{PARA 13 "" 1 "" 
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&TITLEG6#%JExponential~Decay~-~Different~Decay~RatesG-%%VIEWG6$;F(F*%(
DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur
ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cur
ve 8" "Curve 9" "Curve 10" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 108 "In this single diagram we can see both growing
 (increasing) and decaying (decreasing) exponential functions." }}
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215 " plots[display]( plot( \{ 100*exp(x*k/200) $ k = 1..10\}, x = -20
..20, color = green,               \n\011\011title= `Red = Decay and G
reen = Growth , `),\n\011\011plot( \{ 100*exp(-x*k/200) $ k = 1..10\},
 x = -20..20, color = red));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
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\"3;d#*oCzW'e&F-7$F[x$\"3aA&QX3h4N&F-7$F`x$\"3-$oNiJ[N8&F-7$Fex$\"3YXs
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B$R.om8Fhs7$Ffn$\"3?xAilLFP8Fhs7$F[o$\"3E&)4\"*\\i*=J\"Fhs7$F`o$\"3+w-
M)R.RG\"Fhs7$Ffo$\"3!)e\"H[d'Rc7Fhs7$F[p$\"3^.*=ESY/B\"Fhs7$F`p$\"3e6&
=l\"eM27Fhs7$Fep$\"3e&y;xxS/=\"Fhs7$Fjp$\"3w5hW/$=\"e6Fhs7$F_q$\"3i[_/
@DoK6Fhs7$Fdq$\"3dtmXUSj56Fhs7$Fiq$\"3s?se=o$p3\"Fhs7$F^r$\"3q;hE*eT[1
\"Fhs7$Fcr$\"3Eq:l/iEU5Fhs7$Fhr$\"3oR%[Wwc>-\"Fhs7$F^s$\"3Y;(=9r$\\+5F
hs7$Fds$\"3U+\\'H%yw'y*F-7$Fjs$\"3>Uh(HtF2g*F-7$F_t$\"3DjZ>Y'pPS*F-7$F
dt$\"39C)\\w4NX?*F-7$Fit$\"3H!y$*RL3P,*F-7$F^u$\"3M*[8p=SG$))F-7$Fcu$
\"3R2Tt!zlij)F-7$Fhu$\"3i&*yO#GpLY)F-7$F]v$\"3;*GLW[%e#G)F-7$Fbv$\"3C
\"QJI\"R5A\")F-7$Fgv$\"3%\\$)eQ/7-&zF-7$F\\w$\"3?R:8'Q)z\"z(F-7$Faw$\"
3K\">E*>OeHwF-7$Ffw$\"31*e#\\FWDuuF-7$F[x$\"3]#HENPE]J(F-7$F`x$\"3EsX9
Z%y[;(F-7$Fex$\"3,Y7#GT<X,(F-7$Fjx$\"3WHUz$*z^ooF-7$F_y$\"3%*ys!zKSqt'
F-7$Fdy$\"3)3#QO.WW*e'F-7$Fiy$\"3Mmn(GTu,Y'F-7$F^z$\"3P4u'[#y8DjF-7$Fc
z$\"3KphI2^`)>'F-7$FhzF+F\\iq-%+AXESLABELSG6%Q\"x6\"Q!6\"%(DEFAULTG-%&
TITLEG6#%BRed~=~Decay~and~Green~=~Growth~,~G-%%VIEWG6$;F(FhzFg[w" 1 2 
0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2
" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9
" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "C
urve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" }}}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 83 "_______________________________________________________________
____________________" }}{PARA 4 "" 0 "" {TEXT -1 31 "B. Solving Expone
ntial Problems" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________
____________________________________________________________" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 66 "Now well solve exponential growth and decay problems
 analytically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" 
{TEXT 256 7 "PROBLEM" }{TEXT -1 2 " 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 291 "PROBLEM :  A population begins with \+
1,200 at time t = 0, and grows exponentially, until it reaches 3,500 a
t time t = 9. \n\011A. Fiind a simplified function which expresses the
 size of the population at time t. \n\011B. What is the size at t = 36
?\n\011C. At what time does the population reach 100,000?" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 " restart;\011f := t -> C*exp(k*t);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"tG6\"6$%)operatorG%&arrowG
F(*&%\"CG\"\"\"-%$expG6#*&%\"kGF.9$F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 51 " 1200 = f(0);\011solve(%, \{ C \} );\011assign( \+
% );\011f(t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"%+7%\"CG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%\"CG\"%+7" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$-%$expG6#*&%\"kG\"\"\"%\"tGF)\"%+7" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 67 " 3500 = f(9);\011solve( %, \{ k \});\011a
ssign( % );\011simplify( f(t), exp);\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/\"%+N,$-%$expG6#,$%\"kG\"\"*\"%+7" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#<#/%\"kG,$-%#lnG6##\"#N\"#7#\"\"\"\"\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$)#\"#N\"#7,$%\"tG#\"\"\"\"\"*\"%+7" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 " f(36);  evalf(%);\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\")Dc^P\"$K%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aC
<%o)!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " solve( 100000 \+
= f(t));\011evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%#lnG6##
\"$]#\"\"$\"\"\"-F&6##\"#N\"#7!\"\"\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+%**='=P!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "
" 0 "" {TEXT -1 9 "PROBLEM 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 250 "PROBLEM :  A population begins with 2,500 at t
ime t = 2, and grows exponentially, until it reaches 6,000 at time t =
 7. \n\011A. Fiind a simplified function which expresses the size of t
he population at time t. \n\011B. What is the size at t = 12 and t = 2
0?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 " restart;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 " f := t -> C*exp(k*t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
fGR6#%\"tG6\"6$%)operatorG%&arrowGF(*&%\"CG\"\"\"-%$expG6#*&%\"kGF.9$F
.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " solve( f(2) =
 2500, \{C\} );    assign(%);   f(t);\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#<#/%\"CG,$*&\"\"\"F(-%$expG6#,$%\"kG\"\"#!\"\"\"%+D" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,$*&-%$expG6#*&%\"kG\"\"\"%\"tGF*F*-F&6#,$F)\"
\"#!\"\"\"%+D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " solve( f(
7) = 6000, \{k\} );    assign(%);   simplify( f(t), exp );\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<#/%\"kG,&-%#lnG6#\"\"&#!\"\"F**&#\"\"\"F*F/
-F(6#\"#7F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*()\"\"&#\"\"#F&\"
\"\")\"#7#\"\"$F&F)-%$expG6#,$*&,&-%#lnG6#F&!\"\"-F56#F+F)F)%\"tGF)#F)
F&F)#\"$D'F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " f(12); f(2
0); evalf(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$expG6#,&-%#l
nG6#\"\"&#!#7F,*&#\"#7F,\"\"\"-F*6#F1F2F2F2-F&6#,&F)#!\"#F,*&#\"\"#F,F
2F3F2F2!\"\"\"%+D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$expG6#,&-%
#lnG6#\"\"&!\"%*&\"\"%\"\"\"-F*6#\"#7F0F0F0-F&6#,&F)#!\"#F,*&#\"\"#F,F
0F1F0F0!\"\"\"%+D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3\"pQ%e!\"&" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 9 "PROBL
EM 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 215 "
PROBLEM :  A population begins with 35,000 at time t = 0, and grows ex
ponentially at 6% per year.\n\011A. Fiind a simplified function which \+
expresses the size of the population at time t. \n\011B. What is the s
ize at t = 8?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 10 " restart;\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 " f := t -> C*exp(k*t);\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGR6#%\"tG6\"6$%)operatorG%&arrowGF(*&%\"CG\"\"\"-%
$expG6#*&%\"kGF.9$F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 51 " solve( f(0) = 35000, \{C\} );\011assign(%);     \011f(t);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%\"CG\"&+]$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$-%$expG6#*&%\"kG\"\"\"%\"tGF)\"&+]$" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 47 " f(1) = 1.06*f(0);\011\011solve( %, \{k\}
);\011assign(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$-%$expG6#%\"k
G\"&+]$$\"(++r$!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%\"kG$\"+73*
o#e!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " f(t);\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#,$%\"tG$\"+73*o#e!#6\"&+]$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " f(8);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+i#o%yb!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 261 "" 0 "" {TEXT -1 9 "PROBLEM 4" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 153 "PROBLEM :  A population begins with
 432,000 at time t = 0, and grows exponentially, until it reaches 511,
000 at time t = 3. Fiind the annual growth rate." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "FIND ANNUAL GROWTH RATE" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " restart;   f := t -> C*ex
p(k*t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"tG6\"6$%)oper
atorG%&arrowGF(*&%\"CG\"\"\"-%$expG6#*&%\"kGF.9$F.F.F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " f(0) = 432000;\011solve( %, \{C\})
;\011assign(%);  f(t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG\"'+
?V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%\"CG\"'+?V" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,$-%$expG6#*&%\"kG\"\"\"%\"tGF)\"'+?V" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " f(3) = 511000;\011solve( %, \{k\})
;\011assign(%);  f(t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$-%$expG
6#,$%\"kG\"\"$\"'+?V\"'+5^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%\"kG
,$-%#lnG6##\"$6&\"$K%#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
$-%$expG6#,$*&-%#lnG6##\"$6&\"$K%\"\"\"%\"tGF/#F/\"\"$\"'+?V" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " simplify(%, exp);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$)#\"$6&\"$K%,$%\"tG#\"\"\"\"\"$\"'+?
V" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " (% - 1)*100;\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&)#\"$6&\"$K%,$%\"tG#\"\"\"\"\"$\")++
?V\"$+\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " evalf( f(
1)/f(0) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Vzdd5!\"*" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 9 "PROBLEM 5" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 447 "PROBLEM \+
:  Using Newtons law of cooling, an object brought into a room of T0 d
egrees will have this temperature at time t : T(t) = T0 + Cekt. If the
 object is 48 degrees is brought into a room of 32 degrees, and is 42 \+
degrees at time t = 10 :\n\011A. Find a simplified formula for the tem
perature of this object at time t\n\011B. Find at what time the object
 will reach 35 degrees.\n\011C. Find the temperature at t = 200.\n\011
D. Plot the graph of this function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " restart;\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " T := t -> T0 + C*exp(k*t);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TGR6#%\"tG6\"6$%)operatorG%&arrowG
F(,&%#T0G\"\"\"*&%\"CGF.-%$expG6#*&%\"kGF.9$F.F.F.F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " T0 := 32; T(t);\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#T0G\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"
#K\"\"\"*&%\"CGF%-%$expG6#*&%\"kGF%%\"tGF%F%F%" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 47 " T(0) = 48; solve(%, \{C\});  assign(%);   T(t
);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&\"#K\"\"\"%\"CGF&\"#[" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%\"CG\"#;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&\"#K\"\"\"*&\"#;F%-%$expG6#*&%\"kGF%%\"tGF%F%F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " T(10) = 42;   solve(%, \{k
\});  assign(%);   simplify(T(t), exp);\n \n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&\"#K\"\"\"*&\"#;F&-%$expG6#,$%\"kG\"#5F&F&\"#U" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%\"kG,$-%#lnG6##\"\"&\"\")#\"\"\"\"
#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"#K\"\"\"*&\"#;F%)#\"\"&\"\")
,$%\"tG#F%\"#5F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " solv
e( T(x) = 35, x); evalf(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-
%#lnG6##\"\"$\"#;\"\"\"-F&6##\"\"&\"\")!\"\"\"#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+&[C;c$!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 " T(200); evalf(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"4xXLX
mPQfI#\"2Oz#z.%fd?(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\\B8+K!\")
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " plot( \{T(x), T0\}, x \+
= 0..20, y = 0..(T0+C));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
{PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)$\"#KF)7$$\"39LLLL3VfV!#=F*7$$
\"3%pmm;H[D:)F/F*7$$\"3MLLLe0$=C\"!#<F*7$$\"3iLLL3RBr;F6F*7$$\"3imm;zj
f)4#F6F*7$$\"3ULL$e4;[\\#F6F*7$$\"3!)****\\i'y]!HF6F*7$$\"3oLL$ezs$HLF
6F*7$$\"3=++]7iI_PF6F*7$$\"3Onmm;_M(=%F6F*7$$\"3%QLL$3y_qXF6F*7$$\"3]+
++]1!>+&F6F*7$$\"3J+++]Z/NaF6F*7$$\"3;+++]$fC&eF6F*7$$\"3qLL$ez6:B'F6F
*7$$\"3/nmm;=C#o'F6F*7$$\"3Mnmmm#pS1(F6F*7$$\"3<++]i`A3vF6F*7$$\"3Rmmm
m(y8!zF6F*7$$\"3K,+]i.tK$)F6F*7$$\"3!3++v3zMu)F6F*7$$\"3Yomm\"H_?<*F6F
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}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 83 "_______________________________________________
____________________________________" }}{PARA 4 "" 0 "" {TEXT -1 12 "C
. Half Life" }}{PARA 0 "" 0 "" {TEXT -1 83 "__________________________
_________________________________________________________" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 167 "One particular type of exponential decay problem invol
ves the half-life of a radioactive material. The half life is the time
 period for half of the material to decay. " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The half-life is the solution t
o the equation :\nf(t) = .5*f(0)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " restart;\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " f := t -> N*.85^(t);\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"tG6\"6$%)operatorG%&arrowGF(*&%
\"NG\"\"\")$\"#&)!\"#9$F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 " HL := solve( f(t) = (1/2)*f(0), t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#HLG$\"+#GC]E%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 89 "We can view the effect of the half-life
 in this example of an exponential decay function." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(plots):
\nN := 100;  \n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name chan
gecoords has been redefined\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"N
G\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 " display(plot( \+
\{100, seq(  [[HL*2^(k-1) ,0],[HL*2^(k-1), f(0)]] , k = 1..5), \n\011s
eq(  [[0 ,f(HL*2^(k-1))],[N, f(HL*2^(k-1))]] , k = 1..5)\}, x = 0..N, \+
color = green),    \n\011plot( f(x), x = 0..N, title=`half-life`, colo
r = blue)  );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6
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$F*$\"3=*e3,jtwu)!#B-Fgt6&FitF(F(Fjt-%&TITLEG6#%*half-lifeG-%+AXESLABE
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e 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Cur
ve 11" "Curve 12" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 102 "Note that each vertical line is half as high the previou
s, and each horizontal line is twice as large." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________
____________________________________________________" }}{PARA 4 "" 0 "
" {TEXT -1 30 "D. Other Exponential Functions" }}{PARA 0 "" 0 "" 
{TEXT -1 83 "_________________________________________________________
__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Here are some exa
mples of some other exponential type functions." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 14 "SURGE FUNCTION" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
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