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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 256 27 "Applications of Integra
tion" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 
20 "The Doomsday Problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 44 "A differential e
quation of the form  (c > 0)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 
18 0 "dy/dx = b*y^(1+c);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"bGF&)%\"yG,&
F&F&%\"cGF&F&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" 
{TEXT -1 229 "is called the dooms day problem.  This means that the gr
owth of the function is so fast that unimpeded, it would aborb all nut
irents and life.  The term Doomsday machine refers to a mechanical obj
ect which could destroy all life." }}{PARA 266 "" 0 "" {TEXT -1 27 "No
w consider  the equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 
"dy/dx = b*y^(1.01);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"bGF&)%\"yG-%&Flo
atG6$\"$,\"!\"#F&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "
" {TEXT -1 166 "Note that there is a breed of rabbits whose population
 will grow at this rate provided that there is enough food and no pred
itors.  We can write the integral equation" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "dy/y^(1.01) = b* dt" "6#/*&
%#dyG\"\"\")%\"yG-%&FloatG6$\"$,\"!\"#!\"\"*&%\"bGF&%#dtGF&" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 17 "Solving we ob
tain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {XPPEDIT 18 
0 "y^(-0.01)/(-0.01) = b*t + a" "6#/*&)%\"yG,$-%&FloatG6$\"\"\"!\"#!\"
\"F+,$-F)6$F+F,F-F-,&*&%\"bGF+%\"tGF+F+%\"aGF+" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 2 "or" }}{PARA 262 "" 0 "" 
{XPPEDIT 18 0 "y = (-100/(b*t+a))^100" "6#/%\"yG*$,$*&\"$+\"\"\"\",&*&
%\"bGF)%\"tGF)F)%\"aGF)!\"\"F/F(" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 302 "I have written a short
 MAPLE program below which will demonstrate the solution for the follo
wing problem:  Two rabbits breed.  The family of rabbits have the grow
th equation given above and there are 16 rabbits after 3 months.  How \+
many months does it take for the number of rabbits to become infinite?
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 101 "Wr
ite down the constants  a and b for this problem.  Use maple or a calc
ulator to estimate an answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 271 "" 0 "" {TEXT -1 40 "Compare your answer to the progam below
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 20;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ex :=100.0;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "a := -100*(0.5)^(1/ex);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 40 "b := (100/3)*(1/2^(1/ex) - 1/16^(1/ex));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := t -> (-100/(b*t + a))^e
x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "doomsday := -a/b;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f(145);f(0);f(3);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "plot(f(t),t=0..50);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ":" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "
12" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }