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Blank and Steven G. Kran tz" }{TEXT 258 54 "\n\nSection 6.5\nApplications of the Exponential Fu nction" }}{PARA 19 "" 0 "" {TEXT 310 7 "Part I" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 4 "Load" }}{PARA 0 " " 0 "" {TEXT -1 93 "\nEach execution group of this section must be ent ered before continuing with the worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "with(plots): #Ign ore any warnings!" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name ch angecoords 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 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {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 applica tions) and if " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 98 " is any \+ nonzero constant, then the unique solution of the initial value proble m\n\n " }{TEXT 260 2 " " }{TEXT 286 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 19 " \+ " }{TEXT 285 2 " " }{XPPEDIT 258 0 "y(t) = A*exp(k*t);" "6#/-%\"yG6#% \"tG*&%\"AG\"\"\"-%$expG6#*&%\"kGF*F'F*F*" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "We find this solut ion by the Method of Separation of Variables:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Eqn1 := Int( 1/y,y) = Int(k,t) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Eqn1G/-%$ IntG6$*&\"\"\"F*%\"yG!\"\"F+,&-F'6$%\"kG%\"tGF*%\"CGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn1 := map(value, Eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%#lnG6#%\"yG,&*&%\"kG\"\"\"%\"tGF- F-%\"CGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn2 := y = so lve(eqn1, y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%\"yG-%$expG 6#,&*&%\"kG\"\"\"%\"tGF-F-%\"CGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn3 := subs( \{y=A, t=0\}, eqn2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%eqn3G/%\"AG-%$expG6#%\"CG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn4 := C = solve(eqn3, C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/%\"CG-%#lnG6#%\"AG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "eqn5 := subs(eqn4, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/%\"yG-%$expG6#,&*&%\"kG\"\"\"%\"tGF-F--%#lnG6# %\"AGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqn6 := expand(e qn5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/%\"yG*&-%$expG6#*&% \"kG\"\"\"%\"tGF-F-%\"AGF-" }}}{PARA 0 "" 0 "" {TEXT -1 84 "\nOr, we m ight have asked Maple to find the solution directly by executing this \+ code:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "IVP := \{diff(y(t),t) = k*y(t),y(0)=A\}; \ndsolve( IV P , y(t) );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "0 < k;" "6#2\"\"!%\"kG" }{TEXT -1 56 ", t he 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 value proble m is said to " }{TEXT 263 19 "decay exponentially" }{TEXT -1 1 "." }} {PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {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 functio n " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 49 " is increasing, th en we may call the function " }{XPPEDIT 18 0 "proc (t) options operat or, arrow; tau(t) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"-% $tauG6#F%F*F*F*" }{TEXT -1 7 " the " }{TEXT 284 13 "doubling time" } {TEXT -1 6 " of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 25 " if\n \n " }{TEXT 264 2 " " }{XPPEDIT 257 0 "f(t+tau(t)) \+ = 2*f(t);" "6#/-%\"fG6#,&%\"tG\"\"\"-%$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 Figure 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)/*&%%~~~~G F)&F%6#\"\"$F)*&F'F)&F%6#F'F)/*&F.F)&F%6#\"\"%F)*&F'F)&F%6#F1F)/*&F.F) &F%6#\"\"&F)*&F'F)&F%6#F9F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Notice that the doubling times, " }{XPPEDIT 18 0 "t[2]-t[1],` `*t[3]-t[2],` `*t[4]-t[3],` \+ `*t[5]-t[4];" "6&,&&%\"tG6#\"\"#\"\"\"&F%6#F(!\"\",&*&%%~~~~GF(&F%6# \"\"$F(F(&F%6#F'F+,&*&F.F(&F%6#\"\"%F(F(&F%6#F1F+,&*&F.F(&F%6#\"\"&F(F (&F%6#F8F+" }{TEXT -1 23 " are nonconstant. 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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$$\"3+++++++]KF[r$\"3!***** *z2%>Y8F[r7$$\"3#)*************H$F[r$\"31+++cZ!pO\"F[r7$$\"33++++++]LF [r$\"3)******RV:wQ\"F[r7$$\"3!**************R$F[r$\"3\"******>6E$39F[r 7$$\"3=++++++]MF[r$\"31+++!zO!H9F[r7$$\"3++++++++NF[r$FKF[r7$$\"3#)*** *********\\NF[r$\"3#******f9e/Z\"F[r7$$\"33+++++++OF[r$\"3#******\\#)o 6\\\"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$$\"3O++++++]T F[r$\"34+++%G')*=F[r7$$\"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$\"3 7+++yd$*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 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 3 " \+ " }{TEXT 304 83 "Unlike the preceding examples, the doubling time fu nction for exponential growth is" }{TEXT 305 1 " " }{TEXT 307 8 "const ant" }{TEXT 306 3 ".\n\n" }}{PARA 0 "" 0 "" {TEXT 268 7 "Theorem" } {TEXT -1 4 " : " }{TEXT 269 15 "Suppose that " }{XPPEDIT 261 0 "k; " "6#%\"kG" }{TEXT 270 46 " is a positive constant. The doubling tim e " }{XPPEDIT 262 0 "tau;" "6#%$tauG" }{TEXT 271 31 " of the exponen tial function " }{XPPEDIT 264 0 "proc (t) options 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 272 42 " is constant. More p recisely, for every " }{XPPEDIT 265 0 "0 < t;" "6#2\"\"!%\"tG" } {TEXT 273 32 ", the doubling time is given by" }}{PARA 0 "" 0 "" {TEXT 274 0 "" }}{PARA 0 "" 0 "" {TEXT 275 27 " \+ " }{XPPEDIT 266 0 "tau = 1/k;" "6#/%$tauG*&\"\"\"F&%\"kG!\"\"" } {TEXT 276 8 " ln(2) ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 6 "Proof:" }}{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$%)operatorG%&arrowGF(*&%\"AG \"\"\"-%$expG6#*&%\"kGF.9$F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "tau = solve( f(t+tau) = 2*f(t), tau); #From the solut ion, note that tau is independent of t" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$tauG*&-%#lnG6#\"\"#\"\"\"%\"kG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 287 7 "Theorem" }{TEXT -1 4 " : \+ " }{TEXT 288 15 "Suppose that " }{XPPEDIT 256 0 "k;" "6#%\"kG" } {TEXT 289 58 " is a positive constant. Then the exponential function " }{XPPEDIT 258 0 "proc (t) options 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 290 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!\"\"F -F*F*F*" }{TEXT 294 11 " where " }{XPPEDIT 256 0 "tau = 1/k;" "6#/ %$tauG*&\"\"\"F&%\"kG!\"\"" }{TEXT 292 2 " " }{XPPEDIT 257 0 "ln(2); " "6#-%#lnG6#\"\"#" }{TEXT 293 26 " is the doubling time. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 6 "Proof:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "formula1 := A*exp(k*t);\nformula2 := A*2^(t/tau);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%)formula1G*&-%$expG6#*&%\"kG\"\"\"%\"tGF+F+%\" AGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)formula2G*&%\"AG\"\"\")\"\" #*&%\"tGF'%$tauG!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " simplify( subs(tau = ln(2)/k, formula1 - formula2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 41 "3. Exercises Involving Exponentia l Growth" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 8 "Exercise" }{TEXT -1 193 " \n(Exercise 2, page 471) Assuming expone ntial growth, find the number of bacteria present at 8:00 P.M. given t hat there are 4000 bacteria at 5:00 P.M. and the population doubles ev ery 2 hours." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" }{TEXT -1 31 " correspond 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#>%$tauG \"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn1 := y(t) = A*e xp(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 "" }}{PAGEBK }{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 8 "Exercise" }{TEXT -1 193 " \n(Exercise 4, page 471) Assuming exponential growth, find the \+ number of bacteria present at 8:00 P.M. given that there are 7000 bact eria at 4:00 P.M. and the population triples every 8 hours." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" }{TEXT -1 31 " corr espond 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#%\"tG,$*&\"%+q\"\"\"-%$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 /,$*&\"%+q\"\"\"-%$expG6#,$*&\"\")F)%\"kGF)F)F)F),$*&\"&+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, eqn1);" }}{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#\"\"% ,$*&\"%+q\"\"\"-%$expG6#,$*&#F-\"\"#F--%#lnG6#\"\"$F-F-F-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "map(evalf, eqn5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\"%$\"+mbV77!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 8 "Exercise" }{TEXT -1 191 " \n(Exercise 20, page 47 2) An E. coli population doubles every 20 minutes. What is the popula tion size at the instant when the population is growing at the rate of 13,863 bacteria per minute." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " Let " }{XPPEDIT 18 0 "y(t);" "6#-%\"yG6#%\"tG" }{TEXT -1 37 " be the number of E. coli at time " }{XPPEDIT 18 0 "t;" "6#% \"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "tau := 20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$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$-%\"yG6#%\"tGF,*&%\"kG\"\"\"F)F/" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "eqn2 := k = ln(2)/tau;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%\"kG,$*&#\"\"\"\"#?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 "eqn4 := 13863 = rhs(eqn3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/\"&jQ\",$*&#\"\"\"\"#?F**&-%#lnG6#\"\"#F*-%\"y G6#%\"tGF*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn5 := y (t) = solve(eqn4, y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/- %\"yG6#%\"tG,$*&\"'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 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 308 8 "Exercise" }{TEXT -1 43 " (Exercises 34 and 58, pages 473 and 476)" }}{PARA 0 "" 0 "" {TEXT -1 154 "John Graunt, a ``collector and classifier of facts'', wa s the inventor of modern scientific demography. By tabulating the birt hs and deaths listed in the " }}{PARA 0 "" 0 "" {TEXT 309 37 "Weekly B ills of Mortality for London," }{TEXT -1 298 " Graunt determined that \+ the population of London was then growing exponentially with a doublin g time of 64 years. (Graunt realized that the Exponential Law that he \+ proposed could not be carried indefinitely backwards for then the city would be filled with ``far more People, then are now in it''.) 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::;:xI::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::;:xI::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::;:xI::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::;:xI::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::;:xI::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::;:xI::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::;J:<::::::::::=ja::cB:>::::::::::::jysy:>:<:::::::::::::::::::vYxI:; Z::::::::::::::::::::yay=J:B::::::::3:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 423 "Accepting that \n\ni) Graunt's growth observations were valid t hroughout the years 1662-1664 and 1666-1700 inclusive, \nii) t he population of London declined by 100,000 during the Great Plague y ear 1665, and \niii) the population of London at the end of the year \+ 1700 was about 500,000, \n\na) Estimate the population of London at \+ the start of 1662, and \nb) Plot the population of London for the ye ars 1662-1700 " }}{PARA 0 "" 0 "" {TEXT -1 77 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Le t " }{XPPEDIT 18 0 "P(t);" "6#-%\"PG6#%\"tG" }{TEXT -1 39 " be the p opulation of London in year " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 8 " with " }{XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" }{TEXT -1 54 " corresponding to January 1 1662. Then\n\n " }{XPPEDIT 18 0 "t = 1;" "6#/%\"tG\"\"\"" }{TEXT -1 10 " at the " }{TEXT 298 3 "end" }{TEXT -1 23 " of 1662, \n " }{XPPEDIT 18 0 "t = 2;" "6#/%\"tG\"\"#" }{TEXT -1 10 " at the " }{TEXT 299 3 "end" }{TEXT -1 22 " of 1663,\n " }{XPPEDIT 18 0 "t = 3;" "6#/%\"tG\"\"$ " }{TEXT -1 10 " at the " }{TEXT 302 3 "end" }{TEXT -1 10 " of 1664, \n" }}{PARA 0 "" 0 "" {TEXT -1 31 "and so on. In general, at the " } {TEXT 297 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 15 ". \n\nThus, if " }{XPPEDIT 18 0 "T[1] = 3;" "6#/&% \"TG6#\"\"\"\"\"$" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "T[2] = 4;" "6#/& %\"TG6#\"\"#\"\"%" }{TEXT -1 8 ", and " }{XPPEDIT 18 0 "T[3] = 39;" "6#/&%\"TG6#\"\"$\"#R" }{TEXT -1 8 ", then " }{XPPEDIT 18 0 "t = T[1] ;" "6#/%\"tG&%\"TG6#\"\"\"" }{TEXT -1 64 " at the end of 1664 (which i s to say, the beginning of 1665) , " }{XPPEDIT 18 0 "t = T[2];" "6#/% \"tG&%\"TG6#\"\"#" }{TEXT -1 67 " at the end of 1665 (which is to say, the beginning of 1666, and " }{XPPEDIT 18 0 "t = T[3];" "6#/%\"tG&% \"TG6#\"\"$" }{TEXT -1 67 " at the end of 1700. Let us make these de finitions in Maple now\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "T[1] := 3: #Population at the en d of 1664, beginning of 1665" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "T[2] := 4: #Population a t the end of 1665, beginning of 1666" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "T[3] := 39: #Population at the end of 1700" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "\nWe are \+ to determine " }{XPPEDIT 18 0 "P(0);" "6#-%\"PG6#\"\"!" }{TEXT -1 4 " . \n\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 " For " }{XPPEDIT 18 0 "t <= T[1];" "6#1%\"tG&%\"TG6#\"\"\"" }{TEXT -1 33 " London's population satisfies\n" }}{PARA 0 "" 0 "" {TEXT -1 14 " " }{XPPEDIT 18 0 "P(t) = P(0)*2^(t/64);" "6#/-%\"PG6 #%\"tG*&-F%6#\"\"!\"\"\")\"\"#*&F'F,\"#k!\"\"F," }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Thus, in terms of the unknown " }{XPPEDIT 18 0 "P(0);" " 6#-%\"PG6#\"\"!" }{TEXT -1 81 ", London's population at the beginnin g of 1665 was\n\n\n " }{XPPEDIT 18 0 "P(T[1]) = P(0)*2^(T[1]/64);" "6#/-%\"PG6#&%\"TG6#\"\"\"*&-F%6#\"\"!F*)\"\"#*& &F(6#F*F*\"#k!\"\"F*" }{TEXT -1 74 " (Population at \+ the end of 1664, beginning of 1665) . \n\n" }}{PARA 0 "" 0 "" {TEXT -1 18 " \nThe population " }{XPPEDIT 18 0 "P(4);" "6#-%\"PG6#\"\"%" } {TEXT -1 75 " at the end of 1665, that is, at the start of 1666, was \+ therefore given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " " }{XPPEDIT 18 0 "P(T[2]) = P(0)*2^(T[1] /64)-100000;" "6#/-%\"PG6#&%\"TG6#\"\"#,&*&-F%6#\"\"!\"\"\")F**&&F(6#F 0F0\"#k!\"\"F0F0\"'++5F6" }{TEXT -1 58 " (Population at the end o f 1665, beginning of 1666) ." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 55 "At this \+ point, exponential growth resumed. In the " }{XPPEDIT 18 0 "T[3]- T[2];" "6#,&&%\"TG6#\"\"$\"\"\"&F%6#\"\"#!\"\"" }{TEXT -1 66 ", or 35 , years until the end of the year 1700, the population " }{XPPEDIT 18 0 "P(T[2]);" "6#-%\"PG6#&%\"TG6#\"\"#" }{TEXT -1 25 " increased by a factor " }{XPPEDIT 18 0 "2^(35/64);" "6#)\"\"#*&\"#N\"\"\"\"#k!\" \"" }{TEXT -1 46 ". In other words,\n\n\n " } {XPPEDIT 18 0 "P(T[3]) = P(T[2])*2^((T[3]-T[2])/64);" "6#/-%\"PG6#&%\" TG6#\"\"$*&-F%6#&F(6#\"\"#\"\"\")F0*&,&&F(6#F*F1&F(6#F0!\"\"F1\"#kF9F1 " }{TEXT -1 49 " . (Population at the end of 1700) ." }} {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "Putting everything together, we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 18 " " }{XPPEDIT 18 0 "P(T[3 ]) = (P(0)*2^(T[1]/64)-100000)*2^((T[3]-T[2])/64);" "6#/-%\"PG6#&%\"TG 6#\"\"$*&,&*&-F%6#\"\"!\"\"\")\"\"#*&&F(6#F1F1\"#k!\"\"F1F1\"'++5F8F1) F3*&,&&F(6#F*F1&F(6#F3F8F1F7F8F1" }{TEXT -1 5 " ." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "This population was 500 000 and so we have an equation for " }{XPPEDIT 18 0 "P(0);" "6#-%\"PG 6#\"\"!" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " \+ " }{XPPEDIT 18 0 "500000 = (P(0)*2^(T[1]/64)-100000)*2^((T[3]-T[2 ])/64);" "6#/\"'++]*&,&*&-%\"PG6#\"\"!\"\"\")\"\"#*&&%\"TG6#F,F,\"#k! \"\"F,F,\"'++5F4F,)F.*&,&&F16#\"\"$F,&F16#F.F4F,F3F4F," }{TEXT -1 6 " \+ ." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "eqn1 := 5000 00 = (P(0)*2^(T[1]/64)-100000)*2^((T[3]-T[2])/64);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$eqnG/\"'++]*&,&*&-%\"PG6#\"\"!\"\"\")\"\"##\"\"$\" #kF.F.\"'++5!\"\"F.)F0#\"#NF3F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn2 := P(0) = solve(eqn, P(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%\"PG6#\"\"!,$*(\"&++&\"\"\",&\"\"&F-*$)\"\"##\"#N\"#k F-F-F-)F2#\"#8\"#KF-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "m ap(evalf, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6#\"\"!$\"+ /D7\"G%!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "We graph " }{XPPEDIT 18 0 "P(t);" "6#-%\"PG6#%\"tG" }{TEXT -1 13 " as follows" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "P := t -> if t <= T[1] then 428112*2^(t/64)\n elif t <= T[2] then 4281 12*2^(T[1]/64)-(t-T[1])*100000\n else (428112*2^(3/64)-100000 )*2^((t-T[2])/64)\n end if:" }}}{EXCHG {PARA 0 "> " 0 "" 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1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Information " }}{PARA 0 "" 0 "" {TEXT -1 102 "\nWorksheet Title: BlankKrantz-06_5a -R8.mws A Maple Release 8 worksheet.\n\nAuthor: Brian E. Blank \+ " }}{PARA 0 "" 0 "" {TEXT -1 43 "Date Created: 30 September 2006 (Map le R8)" }}{PARA 0 "" 0 "" {TEXT -1 488 "Date Last Revised: 29 Septembe r 2007\n\nThis document may not be distributed by any medium,\nincludi ng print, disk, and electronic transfer, without\nprior written permis sion of the author.\n\nFor more information, please contact the autho r:\n \n Department of Mathematics, \n Washington University in St. Louis\n St. Louis, MO 63130\n \n Telephone: (31 4) 935-6763\n e-mail: brian@math.wustl.edu\n\nCopyright : \251 2006-2007 Brian E. Blank, All Rights Reserved.\n" }}}}{MARK " 4 105 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }