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"" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 } {PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title " -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 } 3 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 24 "Calculus Single Variable " }{TEXT 259 3 " \n" }{TEXT 257 35 "Brian E. Blank and Steven G. Kran tz" }{TEXT 258 55 "\n\nSection 6.5\nApplications of the Exponential Fu nction\n" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 4 "Load" }}{PARA 0 "" 0 "" {TEXT -1 93 "\nEach execution group of this section must be entered before continuing with the worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "with(plots): #Ignore any warnings!" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name chang ecoords has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 316 "doublingTime := proc()\nloc al solnSet, tau, f, t, eqn:\nf := args[1]:\nt := args[2]:\neqn := f(t+ tau)=2*f(t):\nsolnSet := \{solve(eqn, tau)\}:\nif type(t,numeric) then \n return fsolve(eqn, tau, 0..infinity);\nelif nops(solnSet) = 1 the n\n return op(1,solnSet);\nelse\n printf(\"Doubling time not found. \\n\"); \nend if:\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 " 1. A Basic Differential Equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "A;" "6#%\"AG" } {TEXT -1 61 " is any constant (usually positive in applications) and \+ if " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 98 " is any nonzero con stant, then the unique solution of the initial value problem\n\n \+ " }{TEXT 260 2 " " }{TEXT 302 2 " " }{XPPEDIT 257 0 "Diff( y(t),t) = k*y(t)*` `,` `*y(0) = A;" "6$/-%%DiffG6$-%\"yG6#%\"tGF **(%\"kG\"\"\"-F(6#F*F-%%~~~~GF-/*&F0F--F(6#\"\"!F-%\"AG" }{TEXT -1 4 "\n is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 " " }{TEXT 301 2 " " }{XPPEDIT 258 0 "y(t) = A*exp( k*t);" "6#/-%\"yG6#%\"tG*&%\"AG\"\"\"-%$expG6#*&%\"kGF*F'F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "We find this solution by the Method \+ of Separation of Variables:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eq n1 := int(1/y,y) = int(k,t) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %eqn1G/-%#lnG6#%\"yG,&*&%\"kG\"\"\"%\"tGF-F-%\"CGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn2 := y = solve(eqn1, y);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%eqn2G/%\"yG-%$expG6#,&*&%\"kG\"\"\"%\"tGF-F-% \"CGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn3 := subs( \{y =A, t=0\}, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/%\"AG-%$ expG6#%\"CG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn4 := C = \+ solve(eqn3, C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/%\"CG-%#ln G6#%\"AG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn5 := subs(eq n4, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/%\"yG-%$expG6#, &*&%\"kG\"\"\"%\"tGF-F--%#lnG6#%\"AGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqn6 := expand(eqn5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/%\"yG*&-%$expG6#*&%\"kG\"\"\"%\"tGF-F-%\"AGF-" }}} {PARA 0 "" 0 "" {TEXT -1 65 "\n\nOr, we might have asked Maple to find the solution in one step:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "IVP \+ := \{diff(y(t),t) = k*y(t),y(0)=A\};" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$IVPG<$/-%%diffG6$-%\"yG6#%\"tGF-*&%\"kG\"\"\"F*F0/-F+6#\"\"!%\"AG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve( IVP , y(t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG*&-%$expG6#*&%\"kG\"\" \"F'F.F.%\"AGF." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "0 < k;" "6#2\"\"!%\"kG" }{TEXT -1 56 ", the solution of our initial value problem is said to " }{TEXT 262 18 "grow exponentially" }{TEXT -1 8 ". \nWhen " }{XPPEDIT 18 0 "k \+ < 0;" "6#2%\"kG\"\"!" }{TEXT -1 56 ", the solution of our initial valu e problem is said to " }{TEXT 263 19 "decay exponentially" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 "2. Doubling Time for Exponential Growth" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "If a positive function " } {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 49 " is increasing, then we m ay call the function " }{XPPEDIT 18 0 "proc (t) options operator, arr ow; tau(t) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"-%$tauG6# F%F*F*F*" }{TEXT -1 7 " the " }{TEXT 289 13 "doubling time" }{TEXT -1 6 " of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 25 " if\n\n \+ " }{TEXT 264 2 " " }{XPPEDIT 257 0 "f(t+tau(t)) = 2*f( t);" "6#/-%\"fG6#,&%\"tG\"\"\"-%$tauG6#F(F)*&\"\"#F)-F%6#F(F)" }{TEXT 265 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "In general we expect the doubling time to be nonconstant, as in F igure 1. In this figure we have\n\n " }{XPPEDIT 18 0 "y[2] = 2*y[1],` `*y[3] = 2*y[2],` `*y[4] = 2*y[3],` `*y[5] = 2*y[4];" "6&/&%\"yG6#\"\"#*&F'\"\"\"&F%6#F)F)/*&%%~~~~GF)&F%6#\"\"$ F)*&F'F)&F%6#F'F)/*&F.F)&F%6#\"\"%F)*&F'F)&F%6#F1F)/*&F.F)&F%6#\"\"&F) *&F'F)&F%6#F9F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 31 "Notice that the doubling times," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 " \+ " }{XPPEDIT 18 0 "t[2]-t[1],` `*t[3]-t[2],` `*t[4]-t[3],` \+ `*t[5]-t[4];" "6&,&&%\"tG6#\"\"#\"\"\"&F%6#F(!\"\",&*&%%~~~~GF(&F%6# \"\"$F(F(&F%6#F'F+,&*&F.F(&F%6#\"\"%F(F(&F%6#F1F+,&*&F.F(&F%6#\"\"&F(F (&F%6#F8F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "are nonconstant." }}{PARA 0 "" 0 "" {TEXT -1 29 " \+ " }{OLE 1 844812 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j@>:=:; :<;?:=:wyyyyyB:Jj>J?j?J@j@JAjA>:yyyyyy::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::fyyyyyAh@JyKyyyygY:::::::jBJCj 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:::::::B:>:::::::::::::::::::::::::::::::::::::::=ja:>:::::::::J?:<::::::AB:;JD<:;::::::::::::vYxI:;Z::::::::::::::::::::yay=J:B:: :::::::::::::::::jysy:>:<::::::::::1:" }}{PARA 3 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 8 "Examp le:" }{TEXT -1 37 " Doubling time of a linear function" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "L := \+ x -> m*x+b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGf*6#%\"xG6\"6$%)o peratorG%&arrowGF(,&*&%\"mG\"\"\"9$F/F/%\"bGF/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "doublingTime(L,t); #Note the dependence \+ on t" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&%\"mG\"\"\"%\"tGF'F'%\" bGF'F'F&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 267 8 "Example:" }{TEXT -1 33 " Doubling time of the logarithm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "doublingTime(ln,t); #Note t he dependence on t" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"tG!\"\"*$) F$\"\"#\"\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "plot(do ublingTime(ln,t), t= 1 ..2); \n#ln(t) is everywhere increasing but it is positive only for t > 1 " }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"\"\"\"!$F*F*7$$\"3hmm;arz@5!#< $\"3k&>V'4qAFA!#>7$$\"3OL$e9ui2/\"F/$\"3BJK&\\DMCC%F27$$\"3smm\"z_\"4i 5F/$\"3A&Gpbd)o%f'F27$$\"3qmmT&phN3\"F/$\"3GduvN_Ua!*F27$$\"3UL$e*=)H \\5\"F/$\"3i>'\\'e3Sf6!#=7$$\"3sm;z/3uC6F/$\"3V$R<FG7$$\"3-+]i5` h(=\"F/$\"3,`\"fS:[\"GAFG7$$\"3YLL$3En$47F/$\"3kl=U2\">?`#FG7$$\"3cmmT !RE&G7F/$\"3Y2a\"F/$\"3KSJ-fj_J$)FG7$ $\"3imm\"zXu9c\"F/$\"358[4[-Gn()FG7$$\"3'******\\y))Ge\"F/$\"3=ixy17[E #*FG7$$\"3!****\\i_QQg\"F/$\"33>xL?\\f%o*FG7$$\"3#***\\7y%3Ti\"F/$\"3Z .Dfq)>O,\"F/7$$\"3#****\\P![hY;F/$\"3'3ly\"3bsk5F/7$$\"3ELLLQx$om\"F/$ \"3<[))f2.^66F/7$$\"3')****\\P+V)o\"F/$\"3I^A.afOi6F/7$$\"3im;zpe*zq\" F/$\"32DPV@SD47F/7$$\"3)*****\\#\\'QHIk8F/7$$\"3GLL$3s? 6z\"F/$\"3E'=ldr#*pT\"F/7$$\"3&***\\7`Wl7=F/$\"3!G*)3LrhIZ\"F/7$$\"3em mm'*RRL=F/$\"3P1nM]&Rz_\"F/7$$\"3_mmTvJga=F/$\"3U;*[Gw\\\\e\"F/7$$\"3K L$e9tOc(=F/$\"3pmt$pTwBk\"F/7$$\"3'******\\Qk\\*=F/$\"3AG%=ajDfp\"F/7$ $\"3@LL3dg6<>F/$\"338MC>z@eF/$\"3xiF@4Vw9=F/7$$\"3- +]7oK0e>F/$\"3+$***oK$>f(=F/7$$\"3-+](=5s#y>F/$\"3HI>=2%)GN>F/7$$\"\"# F*Fez-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%+AXESLABELSG6$Q\"t6\"Q!Fb[l-%%V IEWG6$;F(Fez%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 268 8 "Example:" }{TEXT -1 43 " Doubling time of a qua dratic polynomial" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "doublingTim es := seq(doublingTime(t->t^2,k*5/100), k=1..100):\nplot([seq([k*5/100 ,doublingTimes[k]],k=1..100)]); \n\n#Note non-horizontal graph" }} {PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6%-%'CURVESG6$7`q7 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++]FF[r$\"36+++(H(3R6F[r7$$\"3#)*************z#F[r$\"3/+++vzzf6F[r7$$ \"33++++++]GF[r$\"3'******Hl30=\"F[r7$$\"3!***************GF[r$\"3*)** ***4L>7?\"F[r7$$\"3=++++++]HF[r$\"3/+++4+$>A\"F[r7$$\"\"$Feq$\"3(***** *poSEC\"F[r7$$\"3#)************\\IF[r$\"37+++l8Nj7F[r7$$\"33+++++++JF[ r$\"3/+++V?1%G\"F[r7$$\"3#*************\\JF[r$\"3(******4ssZI\"F[r7$$ \"3;+++++++KF[r$\"3)********R$[D8F[r7$$\"3+++++++]KF[r$\"3!******z2%>Y 8F[r7$$\"3#)*************H$F[r$\"31+++cZ!pO\"F[r7$$\"33++++++]LF[r$\"3 )******RV:wQ\"F[r7$$\"3!**************R$F[r$\"3\"******>6E$39F[r7$$\"3 =++++++]MF[r$\"31+++!zO!H9F[r7$$\"3++++++++NF[r$FKF[r7$$\"3#)********* ***\\NF[r$\"3#******f9e/Z\"F[r7$$\"33+++++++OF[r$\"3#******\\#)o6\\\"F [r7$$\"3#*************\\OF[r$\"32+++.&z=^\"F[r7$$\"3;+++++++PF[r$\"3++ ++\"=!fK:F[r7$$\"3+++++++]PF[r$\"3#*******e3I`:F[r7$$\"3#)************ *z$F[r$\"33+++P:,u:F[r7$$\"33++++++]QF[r$\"3++++:As%f\"F[r7$$\"3!***** **********QF[r$\"3$******H*GV:;F[r7$$\"3=++++++]RF[r$\"33+++rN9O;F[r7$ $\"\"%Feq$\"3++++\\U&ol\"F[r7$$\"3#)************\\SF[r$\"3++++G\\cx;F[ r7$$\"3k*************4%F[r$\"3#******fgv#)p\"F[r7$$\"3O++++++]TF[r$\"3 4+++%G')*=F[r7$$\"3O++++++]YF[r$\"35+++lI4E>F[r7$$\"3;+++++++ ZF[r$\"3/+++VP!o%>F[r7$$\"3+++++++]ZF[r$\"3'******4U9v'>F[r7$$\"3#)*** **********z%F[r$\"37+++*4D#))>F[r7$$\"3k************\\[F[r$\"37+++yd$* 3?F[r7$$\"3M+++++++\\F[r$\"3/+++ckkH?F[r7$$\"3=++++++]\\F[r$\"3)****** R8d.0#F[r7$$\"\"&Feq$\"3!******>\"y1r?F[r-%'COLOURG6&%$RGBG$\"#5!\"\"$ FeqFeqF^[m-%+AXESLABELSG6$Q!6\"Fb[m-%%VIEWG6$%(DEFAULTGFg[m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 279 126 "The doubling time function for exponential growth is quite differ ent form the examples we have just examined - it is constant." }{TEXT -1 2 "\n\n" }}{PARA 0 "" 0 "" {TEXT 269 7 "Theorem" }{TEXT -1 4 " : \+ " }{TEXT 270 15 "Suppose that " }{XPPEDIT 261 0 "k;" "6#%\"kG" } {TEXT 271 46 " is a positive constant. The doubling time " } {XPPEDIT 262 0 "tau;" "6#%$tauG" }{TEXT 272 31 " of the exponential f unction " }{XPPEDIT 264 0 "proc (t) options operator, arrow; A*exp(k* t) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"*&%\"AG\"\"\"-%$e xpG6#*&%\"kGF-F%F-F-F*F*F*" }{TEXT 273 42 " is constant. More precise ly, for every " }{XPPEDIT 265 0 "0 < t;" "6#2\"\"!%\"tG" }{TEXT 274 32 ", the doubling time is given by" }}{PARA 0 "" 0 "" {TEXT 275 0 " " }}{PARA 0 "" 0 "" {TEXT 276 27 " " } {XPPEDIT 267 0 "tau = 1/k;" "6#/%$tauG*&\"\"\"F&%\"kG!\"\"" }{TEXT 277 2 " " }{XPPEDIT 269 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 278 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" } {TEXT 280 6 "Proof:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f := t -> \+ A*exp(k*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)o peratorG%&arrowGF(*&%\"AG\"\"\"-%$expG6#*&%\"kGF.9$F.F.F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "tau = solve( f(t+tau) = 2*f( t), tau); #Note tau is independent of t" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$tauG*&-%#lnG6#\"\"#\"\"\"%\"kG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 304 7 "Theorem" }{TEXT -1 4 " : " }{TEXT 305 15 "Suppose that \+ " }{XPPEDIT 256 0 "k;" "6#%\"kG" }{TEXT 306 58 " is a positive cons tant. Then the exponential function " }{XPPEDIT 258 0 "proc (t) optio ns operator, arrow; A*exp(k*t) end proc;" "6#f*6#%\"tG7\"6$%)operatorG %&arrowG6\"*&%\"AG\"\"\"-%$expG6#*&%\"kGF-F%F-F-F*F*F*" }{TEXT 307 22 " can be written as " }{XPPEDIT 256 0 "proc (t) options operator, a rrow; A*2^(t/tau) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"*& %\"AG\"\"\")\"\"#*&F%F-%$tauG!\"\"F-F*F*F*" }{TEXT 311 11 " where \+ " }{XPPEDIT 256 0 "tau = 1/k;" "6#/%$tauG*&\"\"\"F&%\"kG!\"\"" }{TEXT 309 2 " " }{XPPEDIT 257 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 310 26 " \+ is the doubling time. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 1 "\n" }{TEXT 308 6 "Proof:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "simplify( subs(tau = ln(2)/k, exp(k*t) - 2^(t/tau)) ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 "3. Exercises Involving Exponenti al Growth" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 8 "Exercise" }{TEXT -1 193 " \n(Exercise 2, page 471) Assuming ex ponential growth, find the number of bacteria present at 8:00 P.M. giv en that there are 4000 bacteria at 5:00 P.M. and the population double s every 2 hours." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Le t " }{XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" }{TEXT -1 31 " correspon d to 5:00 P.M. Let " }{XPPEDIT 18 0 "y(t);" "6#-%\"yG6#%\"tG" }{TEXT -1 37 " be the number of bacteria at time " }{XPPEDIT 18 0 "t;" "6#% \"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A := 4000; tau := 2;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"AG\"%+S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tau G\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn1 := y(t) = A* exp(k*t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%\"yG6#%\"tG,$ *&\"%+S\"\"\"-%$expG6#*&%\"kGF-F)F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eqn2 := k = ln(2)/tau;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%\"kG,$*&#\"\"\"\"\"#F*-%#lnG6#F+F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn3 := subs(eqn2, eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%\"yG6#%\"tG,$*&\"%+S\"\"\"-%$expG6#,$*& #F-\"\"#F-*&-%#lnG6#F4F-F)F-F-F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn4 := subs(t=3, eqn3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/-%\"yG6#\"\"$,$*&\"%+S\"\"\"-%$expG6#,$*&#F)\" \"#F--%#lnG6#F4F-F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " map(evalf, eqn4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\"$$\" +]3PJ6!\"&" }}}{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 0 "" }}{PARA 0 "" 0 "" {TEXT 283 8 "Exerc ise" }{TEXT -1 193 " \n(Exercise 4, page 471) Assuming exponential gr owth, find the number of bacteria present at 8:00 P.M. given that ther e are 7000 bacteria at 4:00 P.M. and the population triples every 8 ho urs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 8 "S olution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" }{TEXT -1 31 " correspond to 4 :00 P.M. Let " }{XPPEDIT 18 0 "y(t);" "6#-%\"yG6#%\"tG" }{TEXT -1 37 " be the number of bacteria at time " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "A := 7000;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"AG\"%+q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn1 := y(t) = A*exp(k*t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%\"yG6#%\"t G,$*&\"%+q\"\"\"-%$expG6#*&%\"kGF-F)F-F-F-" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 54 "eqn2 := subs(t=8, rhs(eqn1)) = 3*subs(t=0, rhs(eqn1 ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,$*&\"%+q\"\"\"-%$expG 6#,$*&\"\")F)%\"kGF)F)F)F),$*&\"&+5#F)-F+6#\"\"!F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn3 := k = solve(eqn2, k);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%eqn3G/%\"kG,$*&#\"\"\"\"\")F*-%#lnG6#\"\"$F*F *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn4 := subs(eqn3, eqn 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/-%\"yG6#%\"tG,$*&\"%+q \"\"\"-%$expG6#,$*&#F-\"\")F-*&-%#lnG6#\"\"$F-F)F-F-F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn5 := subs(t=4, eqn4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/-%\"yG6#\"\"%,$*&\"%+q\"\"\"-%$expG 6#,$*&#F-\"\"#F--%#lnG6#\"\"$F-F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "map(evalf, eqn5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"yG6#\"\"%$\"+mbV77!\"&" }}}{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 0 "" }}{PARA 0 "" 0 "" {TEXT 285 8 "Exercise" }{TEXT -1 191 " \n(Exercise 20, page 472) An E . coli population doubles every 20 minutes. What is the population siz e at the instant when the population is growing at the rate of 13,863 \+ bacteria per minute." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 286 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " Let " }{XPPEDIT 18 0 "y(t);" "6#-%\"yG6#%\"tG" }{TEXT -1 37 " be th e number of E. coli at time " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "tau := 20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tauG \"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqn1 := diff(y(t),t ) = k*y(t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%%diffG6$-% \"yG6#%\"tGF,*&%\"kG\"\"\"F)F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eqn2 := k = ln(2)/tau;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%e qn2G/%\"kG,$*&#\"\"\"\"#?F*-%#lnG6#\"\"#F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn3 := subs(eqn2, eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%%diffG6$-%\"yG6#%\"tGF,,$*&#\"\"\"\"#?F0*&-% #lnG6#\"\"#F0F)F0F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eq n4 := 13863 = rhs(eqn3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/ \"&jQ\",$*&#\"\"\"\"#?F**&-%#lnG6#\"\"#F*-%\"yG6#%\"tGF*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn5 := y(t) = solve(eqn4, y (t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/-%\"yG6#%\"tG,$*&\"' gsF\"\"\"-%#lnG6#\"\"#!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "map(evalf, eqn5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#% \"tG$\"+ri,+S!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 343 8 "Exercise" }{TEXT -1 42 " \n(Exercises 34 and 58, pages 473 and \+ 476)" }}{PARA 0 "" 0 "" {TEXT -1 154 "John Graunt, a ``collector and c lassifier of facts'', was the inventor of modern scientific demography . By tabulating the births and deaths listed in the " }{TEXT 344 37 "W eekly Bills of Mortality for London," }{TEXT -1 700 " Graunt determine d that the population of London was then growing exponentially with a \+ doubling time of 64 years. (Graunt realized that the Exponential Law t hat he proposed could not be carried indefinitely backwards for then t he city would be filled with ``far more People, then are now in it''.) \n\nAccepting that \n\ni) Graunt's growth observations were valid thr oughout the years 1662-1664 and 1666-1700 inclusive, \nii) the \+ population of London declined by 100,000 during the Great Plague year 1665, and \niii) the population of London at the end of 1700 was a bout 500,000, \n\nestimate the population of London at the start of \+ 1662. Plot the population of London for the years 1662-1700 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 345 8 "Solution " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 " P(t);" "6#-%\"PG6#%\"tG" }{TEXT -1 39 " be the population of London i n year " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 8 " with " } {XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" }{TEXT -1 41 " corresponding t o January 1 1662. Then " }{XPPEDIT 18 0 "t = 1;" "6#/%\"tG\"\"\"" } {TEXT -1 9 " at the " }{TEXT 347 3 "end" }{TEXT -1 10 " of 1662, " } {XPPEDIT 18 0 "t = 2;" "6#/%\"tG\"\"#" }{TEXT -1 8 " at the " }{TEXT 348 3 "end" }{TEXT -1 40 " of 1663, and so on. In general, at the " } {TEXT 346 3 "end" }{TEXT -1 9 " of year " }{XPPEDIT 18 0 "Y;" "6#%\"YG " }{TEXT -1 7 " with " }{XPPEDIT 18 0 "1662 <= Y;" "6#1\"%i;%\"YG" } {TEXT -1 18 " , the value of " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 6 " is " }{XPPEDIT 18 0 "Y-1662+1;" "6#,(%\"YG\"\"\"\"%i;!\"\"F%F %" }{TEXT -1 13 ". Thus, if " }{XPPEDIT 18 0 "T[1] = 3;" "6#/&%\"TG6 #\"\"\"\"\"$" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "T[2] = 39;" "6#/&% \"TG6#\"\"#\"#R" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "t = T[1];" "6#/ %\"tG&%\"TG6#\"\"\"" }{TEXT -1 25 " at the end of 1664 and " } {XPPEDIT 18 0 "t = T[2];" "6#/%\"tG&%\"TG6#\"\"#" }{TEXT -1 22 " at t he end of 1700. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We are to determine " }{XPPEDIT 18 0 "P(0);" "6#-%\"PG6# \"\"!" }{TEXT -1 78 ". In terms of this unknown, London's population a t the beginning of 1665 was " }{XPPEDIT 18 0 "P(0)*2^(T[1]/64);" "6#* &-%\"PG6#\"\"!\"\"\")\"\"#*&&%\"TG6#F(F(\"#k!\"\"F(" }{TEXT -1 20 ". \+ The population " }{XPPEDIT 18 0 "P(4);" "6#-%\"PG6#\"\"%" }{TEXT -1 77 " at the end of 1665, that is, at the start of 1666, was therefore given by " }{XPPEDIT 18 0 "P(4) = P(0)*2^(T[1]/64)-100000;" "6#/-%\" PG6#\"\"%,&*&-F%6#\"\"!\"\"\")\"\"#*&&%\"TG6#F-F-\"#k!\"\"F-F-\"'++5F5 " }{TEXT -1 10 ". In the " }{XPPEDIT 18 0 "T[2]-4;" "6#,&&%\"TG6#\"\" #\"\"\"\"\"%!\"\"" }{TEXT -1 85 ", or 35, years until the end of the y ear 1700, this poulation increased by a factor " }{XPPEDIT 18 0 "2^(3 5/64);" "6#)\"\"#*&\"#N\"\"\"\"#k!\"\"" }{TEXT -1 21 ". Therefore, we have" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "T[1] := 3:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "eqn := (P(0)*2^(T[1]/64)-100 000)*2^(35/64) = 500000;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/*& ,&*&-%\"PG6#\"\"!\"\"\")\"\"##\"\"$\"#kF-F-\"'++5!\"\"F-)F/#\"#NF2F-\" '++]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn2 := P(0) = solv e(eqn, P(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%\"PG6#\"\" !,$*(\"&++&\"\"\",&*$)\"\"##\"#N\"#kF-F-\"\"&F-F-)F1#\"#8\"#KF-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "map(evalf, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6#\"\"!$\"+/D7\"G%!\"%" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "P := t -> if t <= 3 then 428112*2^(t/64)\n elif t <= 4 then 428112*2^(3/64)-(t-3)*100000\n els e (428112*2^(3/64)-100000)*2^((t-4)/64)\n end if;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"PGf*6#%\"tG6\"6$%)operatorG%&arrowGF(@'1 9$\"\"$,$*&\"'7\"G%\"\"\")\"\"#,$*&#F3\"#kF3F.F3F3F3F31F.\"\"%,(*&F2F3 )F5#F/F9F3F3*&\"'++5F3F.F3!\"\"\"'++IF3*&,&*&F2F3F>F3F3FAFBF3)F5,&*&F8 F3F.F3F3#F3\"#;FBF3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot( P , 0 .. 39, thickness=2, numpoints=600);" }}{PARA 13 "" 1 " " {GLPLOT2D 375 375 375 {PLOTDATA 2 "6&-%'CURVESG6$7j`m7$$\"\"!F)$\"'7 \"G%F)7$$\"3:^4dUb17o!#>$\"3EM6Ur'zUG%!#77$$\"3)4aCZt?RF\"!#=$\"3_%*z# =yIqG%F27$$\"3mo6q?)*[S>F6$\"3cC0x.o7!H%F27$$\"3yUaEftZ6EF6$\"3GPcL1cC $H%F27$$\"3w3`=6eFzKF6$\"3?%4=p$=N'H%F27$$\"3e6rEg/T)*QF6$\"3_pA(zrL#* H%F27$$\"3%\\&GrF&)[RXF6$\"3$4/pfx>AI%F27$$\"38&*o8#Q#\\-_F6$\"3L:^.V, J0VF27$$\"3G>8zu*pL'eF6$\"3!R!eX3GR3VF27$$\"3Jk$=5\"Q;VlF6$\"3XE'o;,m: J%F27$$\"3\"[k')>fE>9(F6$\"3]A]d1HO9VF27$$\"3'Q'zxm;*f\"yF6$\"366E!)=P ^\"pe'eL%F27$$\"3V5isvGnM7Fgp$\"3,iW!*RM%F27$$\"3;l*yt$)fiO\"Fgp$\"3z2$[$)[R\\M%F27$$\"3NXrGs)GKV\"Fgp $\"3O_)=&H?4[VF27$$\"330I!=NFZ\\\"Fgp$\"3M7J7h!*)4N%F27$$\"3Uho6q<1h:F 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}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Before co ntinuing to the next section, restore the parameters " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "tau;" "6#%$tauG " }{TEXT -1 20 " and the variable " }{XPPEDIT 18 0 "T;" "6#%\"TG" } {TEXT -1 29 " to their unassigned states." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "A := 'A':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "tau := 'tau':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "T := 'T':" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "4. Half-life" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "If a positive func tion " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 49 " is decreasing, then we may call the function " }{XPPEDIT 18 0 "proc (t) options ope rator, arrow; tau(t) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6 \"-%$tauG6#F%F*F*F*" }{TEXT -1 23 " the halving time of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 25 " if\n\n " }{TEXT 287 2 " " }{XPPEDIT 256 0 "f(t+tau(t)) = f(t)/2;" "6#/-%\"fG6#,&%\"tG \"\"\"-%$tauG6#F(F)*&-F%6#F(F)\"\"#!\"\"" }{TEXT 288 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "In general the hal ving time is nonconstant. However:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 7 "Theorem" }{TEXT -1 4 " : " }{TEXT 291 15 "Suppose that " }{XPPEDIT 256 0 "la mbda;" "6#%'lambdaG" }{TEXT 292 45 " is a positive constant. The hal ving time " }{XPPEDIT 257 0 "tau;" "6#%$tauG" }{TEXT 293 31 " of the exponential function " }{XPPEDIT 258 0 "proc (t) options operator, a rrow; A*exp(-lambda*t) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG 6\"*&%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF-F%F-!\"\"F-F*F*F*" }{TEXT 294 42 " is constant. More precisely, for every " }{XPPEDIT 259 0 "0 < t ;" "6#2\"\"!%\"tG" }{TEXT 295 31 ", the halving time is given by" }} {PARA 0 "" 0 "" {TEXT 296 0 "" }}{PARA 0 "" 0 "" {TEXT 297 27 " \+ " }{XPPEDIT 260 0 "tau = 1/lambda;" "6#/%$tauG*&\" \"\"F&%'lambdaG!\"\"" }{TEXT 298 2 " " }{XPPEDIT 261 0 "ln(2);" "6#-% #lnG6#\"\"#" }{TEXT 299 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 300 6 "Proof:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f := t -> A*exp(-lambda*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*&%\"AG\"\"\"- %$expG6#,$*&%'lambdaGF.9$F.!\"\"F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "tau = solve( f(t+tau) = f(t)/2, tau); #Note tau is in dependent of t" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$tauG*&-%#lnG6#\" \"#\"\"\"%'lambdaG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "f(t) = A*exp(-lambda*t); " "6#/-%\"fG6#%\"tG*&%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF*F'F*!\"\"F*" } {TEXT -1 44 " is the mass of an unstable isotope, then " }{XPPEDIT 18 0 "tau = ln(2)/lambda;" "6#/%$tauG*&-%#lnG6#\"\"#\"\"\"%'lambdaG!\" \"" }{TEXT -1 20 " is said to be the " }{TEXT 303 9 "half-life" } {TEXT -1 18 " of the substance." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }} {PARA 0 "" 0 "" {TEXT 312 7 "Theorem" }{TEXT -1 4 " : " }{TEXT 313 15 "Suppose that " }{XPPEDIT 256 0 "lambda;" "6#%'lambdaG" }{TEXT 314 58 " is a positive constant. Then the exponential function " } {XPPEDIT 258 0 "proc (t) options operator, arrow; A*exp(-lambda*t) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"*&%\"AG\"\"\"-%$expG6#, $*&%'lambdaGF-F%F-!\"\"F-F*F*F*" }{TEXT 315 22 " can be written as \+ " }{XPPEDIT 256 0 "proc (t) options operator, arrow; A/(2^(t/tau)) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"*&%\"AG\"\"\")\"\"#*&F% F-%$tauG!\"\"F2F*F*F*" }{TEXT 319 11 " where " }{XPPEDIT 256 0 "ta u = 1/lambda;" "6#/%$tauG*&\"\"\"F&%'lambdaG!\"\"" }{TEXT 317 2 " " } {XPPEDIT 257 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 318 22 " is the ha lf-life. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 316 6 "Proof:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "si mplify( subs(tau = ln(2)/lambda, exp(-lambda*t) - 1/2^(t/tau)) );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 3 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 40 "5. Exercises Involving Exponential Decay" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 320 8 "Exercise" }{TEXT -1 114 " \n(Ex ercise 10, page 472) The mass of a radioactive sample is 10g in 1950 \+ and 7 g in 2000. What is its half-life?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 321 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" } {TEXT -1 27 " correspond to 1950. Let " }{XPPEDIT 18 0 "m(t);" "6#-% \"mG6#%\"tG" }{TEXT -1 23 " be the mass at time " }{XPPEDIT 18 0 "t; " "6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn1 := m(t) = 10*exp(-lambda*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%\"mG6#%\"tG,$*&\"#5\"\"\" -%$expG6#,$*&%'lambdaGF-F)F-!\"\"F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eqn2 := 7 = subs(t=50, rhs(eqn1));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%eqn2G/\"\"(,$*&\"#5\"\"\"-%$expG6#,$*&\"#]F*%'lamb daGF*!\"\"F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eqn3 := l ambda = solve(eqn2, lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn 3G/%'lambdaG,$*&#\"\"\"\"#]F*-%#lnG6##\"\"(\"#5F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn4 := tau = ln(2)/lambda;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/%$tauG*&-%#lnG6#\"\"#\"\"\"%' lambdaG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn5 := sub s(eqn3, eqn4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/%$tauG,$*( \"#]\"\"\"-%#lnG6#\"\"#F*-F,6##\"\"(\"#5!\"\"F4" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "eqn6 := map(evalf, eqn5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/%$tauG$\"+]5z;(*!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 "6. Exercises Involving Radiocarbon Dating" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Living matter contai ns two isotopes of carbon, " }{XPPEDIT 18 0 "` `^12*C;" "6#*&%\"~G\"# 7%\"CG\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "` `^14*C;" "6#*&% \"~G\"#9%\"CG\"\"\"" }{TEXT -1 93 ", in a known fixed ratio. After \+ death, carbon is no longer metabolized and the amount of " }{XPPEDIT 18 0 "` `^14*C" "6#*&%\"~G\"#9%\"CG\"\"\"" }{TEXT -1 59 " decreases \+ due to radioactive decay. The mass of stable " }{XPPEDIT 18 0 "` `^12 *C" "6#*&%\"~G\"#7%\"CG\"\"\"" }{TEXT -1 51 " in a sample can be us ed to determine the mass " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 7 " of " }{XPPEDIT 18 0 "` `^14*C" "6#*&%\"~G\"#9%\"CG\"\"\"" }{TEXT -1 68 " that the sample had at the moment of death. Since the half-li fe of" }{XPPEDIT 18 0 "` `^14*C" "6#*&%\"~G\"#9%\"CG\"\"\"" }{TEXT -1 235 " is known to be around 5700 years, the time of death can be calc ulated from the law of exponential decay. Willard Libby received the \+ 1960 Nobel prize for physics for discovering this method of dating.\n \nIn each exercise we will let " }{XPPEDIT 18 0 "y(t) = m/(2^(t/5700) );" "6#/-%\"yG6#%\"tG*&%\"mG\"\"\")\"\"#*&F'F*\"%+d!\"\"F/" }{TEXT -1 24 " denote the amount of " }{XPPEDIT 18 0 "` `^14*C" "6#*&%\"~G\"#9 %\"CG\"\"\"" }{TEXT -1 35 " that exists years after death.\011\n" } {TEXT 322 8 "Exercise" }{TEXT -1 24 "\n(Exercise 27, page 473)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 " \+ " }{BITMAP 220 182 182 1 " ?TMWUo=:;JUJNJhKUk:>=>::::^bp_lBOiRhbqgl?whO\\A?iP@cr_lAohM@cp_lC?iN`c uwlF_iR@d:`mLGjW`cuwlEOiOXctglCGiOHcs?mBGiSXcv_mCOiTXbn_lyngHPaeWkohMXboWl=g HIdWgvLoe`MTXmVKge_ADWcvKkE`IDXkfKiu_EdWgFKeU_CTWevJcE_CTWeFKVbpol>whS xbsgmAGiVhbrGmBWiVHcvgm?GiVhBTnvds]dqGMnu\\S[PQE=ntTSZrJiE`ITXsvKmUaAD XmVKmUaCDXoVKkU`\\QGInuXS[XQFyMtPSX@QCuMtHSYPQD=NvPs\\hqFenxhsaEsxxRFMRxPRrdPttodHP_DojTneTP`xniXN_`O[dom@ObpO]pngH NlMq>iruhQx\\pvdQm@Qstofpo^` OjlN]]RvHsCIrCMSQr;MQ;ERs?iRXbrGm:ghP`alOluNhK`amGlxfhLxaoWl?WiS`cygmFWiTXcs?m FWiUpcvgmI_iWpcwgmEWiUPcuWmD_iUhcySM;VaUdZuFM=VaUTZyVM=fa[D[=GOIfciT[k vJ?EPyCOgCHQdDscMUcLstHUtQyCJ_tFgCLQcJatGwSM]cGETDicKScEuCCWSIAcDosAOs HyrE;TCicKQSGCTDYcI?sFEDDwSNccI_DGSCJ;sDucAwSMYcNytJ;UTYD[UvVgvaGvZQFW ouZWuZSFXmeZUuS_EQOTP=tJatHYtQKTOEeMmtS]dNyDL_DRKdOEUMIuVuTXAVUyE\\]EW meRcuW=eYyuTGF\\ce^eFZCV\\ceWqeSAv[aeY;FUouXIEZ=VUUUVudRMeOsdS_TS]UQkE [_m;AQoTQlLQ:up:qrwDQyhPmDQiqrwpRBqQyxqpM\\BOiPHcuwlCWiQPcv?Muu_MtXhNy\\Sa@rG AOMsm_lQHOIIDXkvI[E^wsUYvJcE_E\\@K^`qF ]>I\\IGjYpc:pmGwiUXcwSLqM]\\QEQnuPS]\\QEs^GDXcFLoe_QTYqvLhs]\\QFMntPs \\TQEYNv\\S^`Afbpol?ohPpBP>CTWjZLQDA^PPbn?m?ghS`bpGm=ohQ`bqGm??iUpbsOm ??IElFK\\XQFANuXcq?m=whS`boOmAohUpbpGm@whT`bq?m>s_=tWifJXS[>m:ohQpangl u^hM`anWl;?iOXbswlIQyPrr\\q:qPklpehonDo[LoXdM`HM_dO[ looPOktPf\\NdtmePp`lOpLoilpdXRBiqCmsAQ:IrpLSE=RH=T?urCuQ:QRrlQ;mpA Usy`r@]Q:ERq@sC=r?yRxiQ=EqktPg\\Ps xopPqkUqyMtXiFLoe_ODYk ^R\\BWiO`cyGmFwiS\\HGjUPd?xmNwj[Pd>@nJOjX@d=hmIOjVhc:TMye`SdYq>QLHiNxl s]\\QFEnsHS[BKeU_GtWivK>C`Nwds^haIGjYHd=@nJOJUtYdNx`S\\TQDUNvPS_lQFaNx Xs^lQEinx`S_BN?Fb_tZpNytS`pqHAO;EtcLRLunyxS_hqGY>G\\?K]BL`s[TQEA^OXboG MeU`ATWkvJgm\\\\qFMnvds\\XqGUnvnbsOMZ>GiT`BJ]Vm?ohTpbpGm?gHEdWkFKiu_?t WHNv\\s[`QGANv\\sXTqDmmtLSYTqDAnvTS\\dqFINvXs[TQDQNtHS^TqDenv\\s`hqGyN xps`bLqu`OdXsFLqu`QdYuVM=nbFnH?jXpc:pmFGjYpc=PnLoj`hdeKobRWaQ?gAg d`gbAwdqvbSGaL_f@_d`GcROgGCPqSDqcA=CFeb?EC=ECGmBEqSBGcGoRCgs@MSHwbEucB ]CJCsE;DCacKMSI]DGmsLWsFEdCGTOicScEPmE[MuZQVV?F\\[EZGFVAf\\eEWqeSsdS[D RWuO[tRODN?ULsDUcDRYUPuTTaTNwtKwTUetRcEQAUVqt[aVYQV_yU\\[VXUf^qEa;g\\o FbMvZQFWauXCuVqm>AswxrD=R>qRvQq;iruDR?Uq:]Rt`RBqq @MS:Er?YQr`ql@po\\OqPQllq:uPAASyUrupRAqqAMs:QrAiQoTqjtOrXohHqe\\orHodtPb@pvhoiTq fTNhStYmVM;f`S<>d=hmMgj[Xd?HnL_J_D[;WNCfaYdZsvM? 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In the 1980s, four stran ds of the cloth were given to four different institutions for dating. \+ The results of the radiocarbon analysis were released on 13 October 19 88. Each institution declared the Shroud of Turin to be a medieval f orgery.\ni) What percentage of " }{XPPEDIT 18 0 "m;" "6#%\"mG" } {TEXT -1 35 " would the measured quantity of " }{XPPEDIT 18 0 "` `^ 14*C" "6#*&%\"~G\"#9%\"CG\"\"\"" }{TEXT -1 129 " have been in order \+ to authenticate the Shroud?\nii) The Shroud was dated to around 1325 C .E. Approximately what percentage of " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 17 " was measured?\n\011" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 323 8 "Solution" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 227 "The death date of Jesus is not precisely known. How ever, historical references indicate that it did not take place later \+ than \n36 C.E.. The fibers of a burial cloth could not have come from living matter at a later date.\n\n i)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "T := 1988-36;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" TG\"%_>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "100*y(T)/m*perCe nt = 100*evalf(1/(2^(T/5700)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$ **\"$+\"\"\"\"-%\"yG6#\"%_>F'%\"mG!\"\"%(perCentGF'F'$\"+qV(p)y!\")" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Any per centage substantially greater than 78.9 would have proved a forgery. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "ii)" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "T := 1325-36;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG\"%*G\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "100*y(T)/m*perCent = 100* evalf(1/(2^(T/5700)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$**\"$+\" \"\"\"-%\"yG6#\"%*G\"F'%\"mG!\"\"%(perCentGF'F'$\"+S+>\\&)!\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The measu red amount was about 85.5 percent of m. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 324 8 "Exercise" }{TEXT -1 24 "\n(Exercise 2 8, page 473)" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 32 " " }{BITMAP 224 158 158 1 "?TMgia=:;JUJNJjC:w>=JFJ::::JhDpblojDOehO`tNcTn`@O]gcA?c?wa:Oanf`>oay^dHwc@?b;Oanf`Egb@O eNgdFobAgaq>aBOb=wcCOcBOb=_apv`QGdLG`e^_ov_jfaq>aFobAocBGcH?cC_c@wbp>` kf`jF`vv`r^by>bAOb=ocCOcXGeTGfVgeLgcHg`n>ao^`rVatnaAobDwcHOdDGcGoawFbq n`tVawnay>b>gb@Oc>WbAWasvaxNa:_cA?dRWdV_c:Od:o`=warvaD?cBWdN?dOGeMGeW? eRoeS?m]HmOyuSqtZAvDYtQYsLYuAAtSisQqu>ysMYu`QwYQwhiu`IwXYv`avaYwbQweQw cYwhYwgQweyveawbiwhQwgqwd=ZhyweQxgawkaweqwdIwgQw`iwdqvfYwaqwhAwiywciwj yvgAxaqwkAwiQxcywlIWnwkAwhIxbYxjAwnIxcaxjYwlqwcYxhYwoAxgaxiawjYwdYxkIx nywjYxgqwoawiqxgAxoywnIxjaxhIxoQv`YwUyu^itRauNytUIuVAv]iv`QwfiwgywjQwg IxfywmQwhYx\\IwlyubQxZIwpQu_YxKquharRQw^@qDaiI`ncoiMphE@nw_ldhhFPnpgk] PiJ`o?XmqxiQXpsGlg`iQPpyOmo@jW@q>HnvPkbXrDxnOevvbWgbO_ewfcNGaCwcxfcJ_cUoeCodRw`?_b uFcBOcM_e>gcPoa@gcp^a=Gb?OcFwcNwbDocCOcHocGodc@_cQgcP ?f>gcQOdWWgYOgrwbL?fdv`?GcP_fTwfo_fnGjj?jGpe^WdWOfMOeY?dN?eFwcM?cFwc@O cFobCWc?wcL?dHGdIocKwc?gb?WauVawNavnb=gbU?eTWdHgcHGcBgb:wa@?BUtKKTJoSI mcFesLCtKIDJAdKyCKIdIKtH]SIUsBYSH[cHQDKMdLAdKQTKGtJucImsFcSKyCJgDN]tK= dJaSEWsG]cFKdJAdK;TJscGisMMdLWDLMtImcH[cDQcFSSEISB?CFOsDbJsSIscGiCIgsG ;ePqdS;URwSHosEOsDqcGicMMdLOTKGDLADKWTLODMIDLmDPstN_dOuCI;DIqsIYdM_<;t IATMSDNMtLSdJAtKGtKQtLOdMATJItJySKIdJQDLADMKDIWDLwsLSTKSdMSDM]dNWtPyTP sDRqDM_TMYSI_sGATIGeV_ug?hk?XmeXfsGkAXnihj]Hoahq?YqKIu;YtSIrEItCysQAbu NWlxpaXqyxouxqiHpXqeHxXrmxsuxpEypmxqiXoqXoUxpCu:VwjyvfIxbawkAwgQxciwlI WZfIxbYX:Qiv`HPIcAvwgiwkawgYxgqwpiwjqxgAxkYwjIwbaw\\ivcAv^IwZaVHxmuwjQ Xhywk=XkIXm]xnmXqyXmaxpexpAYnmhYqviiu`AxYAwoyt\\AxEAubytfyyEAvoqo?YuS@ p@YnsPtV`pFIkZPqS@pDYnypuXPqSYlipsHXoDillHtBxn@ahMPpo_lmxgOxp>xnCYlgHt \\hpTqnrhuXpoMql`XsEhmgcwNa= ?byFcvfaAOavnbv>bE_a:Wc>OcO_a=Odwb?OcGWcF_cF?cB?cCobB_dKWdjggi?fUOe =oawn_c>_wv`rvdKOdHwbD?ak^`y>aunbyNbBGb@gbvnbwV`yf`ev`o^_pfapVaA?b=Wc? 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