{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 19 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 19 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 19 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 19 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 19 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 12 "Slope Fields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 257 31 "Part 1. The Slope Field Concept" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Enter your answers to que stions 1 through 6 in Part 1 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "Enter the following commands to load the plotting packag e and the \"tools\" package for differential equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(plots): w ith(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "The next command draws the slope field. The first in put is the differential equation. Maple's notation for " }{XPPEDIT 18 0 "dP/dt;" "6#*&%#dPG\"\"\"%#dtG!\"\"" }{TEXT -1 6 " is " }{TEXT 263 12 "diff(P(t),t)" }{TEXT -1 38 ". Next comes the dependent variab le " }{TEXT 262 1 "P" }{TEXT -1 37 ", and then ranges for the variab les." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "DEplot(diff(P(t),t)=t-P(t), P, t=-6..6, P=-6..6);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 448 "We add solution curves b y specifying a list of points those solutions should pass through. In Maple, the initial points have a \"list of lists\" structure -- each \+ point is specified in a list of its own, and then these are assembled \+ into another list. In the following command we also specify a color f or drawing the curves and a small step size to make sure the curves ar e drawn smoothly. Here is the same slope field with three solution cur ves. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 113 "DEplot(diff(P(t),t)=t-P(t), P, t=-6..6, P=- 6..6, [[P(0)=0], [P(0)=-1], [P(0)=-6]], linecolor=blue, stepsize=0.2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 39 "Part 2. Slope Field s for Natural Growth" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 20 "Enter the value of " }{XPPEDIT 18 0 "k;" "6#%\"kG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k:=0.1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Construct the slope field, and assign it a name for later use:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "fie ld := DEplot(diff(P(t),t)=k*P(t), P, t=0..20, P=0..1000):%;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Answer question 2 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Re draw the slope field with three solution curves -- change the ??? to n umbers of your choice." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 119 "DEplot(diff(P(t),t)=k*P(t), P, t=0..20, P=0.. 1000, [[P(0)=???], [P(0)=???], [P(0)=???]], linecolor=blue, stepsize=0 .2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Enter your formul as for three solution curves. Name the starting populations " } {XPPEDIT 18 0 "P1" "6#%#P1G" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "P2" "6# %#P2G" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "P3" "6#%#P3G" }{TEXT -1 34 ", and the plots of these curves " }{XPPEDIT 18 0 "solution1" "6#%*sol ution1G" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "solution2" "6#%*solution2G " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "solution3" "6#%*solution3G" } {TEXT -1 16 ", respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "P1:=t->???: solution1:=plot(???, t= 0..20, P=0..1000, color=green, thickness=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "P2:=t->???: solution2:=plot(???, t=0..20, P=0..1000, \+ color=violet, thickness=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "P3:= t->???: solution3:=plot(???, t=0..20, P=0..1000, color=cyan, thickness =3):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Display the field and solution plots together:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "display([field, \+ solution1, solution2, solution3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 259 42 "Part 3. Slope Fields for Radioactive Decay" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The Meyer -von Schweidler data and a plot of the data points:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "activity:=[[0.2, 35.0],[2.2,25.0],[4.0,22.1],[5,17.9],[6,16.8],[8,13.7],[11,12.4],[12,1 0.3],[15,7.5],[18,4.9],[26,4.0],[33,2.4],[39,1.4],[45,1.1]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "actplot := plot(activity, style=point, sy mbol=circle): %;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Enter " }{XPPEDIT 18 0 "P[0]" "6#&%\"PG6#\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "b " "6#%\"bG" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "P0:=28; b:=0.92466;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Calculate " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 8 " from " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k:=???;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Draw the slope field. Then add the \+ solution starting at " }{XPPEDIT 18 0 "P[0]" "6#&%\"PG6#\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "field := DEplot(diff(P(t),t)=k*P(t), P, t=0..45, P=0. .35, linecolor=green):%;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 61 "Display the data plot along with the fiel d plot and solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display(\{actplot,field\});" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "Answer question 3 here." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 50 "Part 4. Slope Fields for Limited Populat ion Growth" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Enter the constants and function definition for the limited gr owth model, and construct the direction field:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "M:=1000; c:=.000098 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "field := DEplot(diff(P (t),t)=???, P, t=0..100, P=0..1500):%;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 180 "We repeat here the commands from the Limited Popu lation Growth module for constructing an approximate solution, using r epeated applications of \"rise = slope x run.\" You may vary " } {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 14 " if you wish." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "n:=50; Delta :=100./n; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "t := k -> k*Delta:" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "slope:=proc (k) slope(k):=c*p(k)* (M-p(k)); end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "p:=proc (k) p(k) :=p(k-1)+slope(k-1)*Delta; end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " p(0):=111: slope(0):=c*p(0)*(M-p(0)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "popData:=[seq([t(k),p(k)], k=0..n)]:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 72 "popGraph:=plot(popData,t=0..100, p=0..1500, style=P OINT,color=blue): %;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 68 "Now display the approximate population gr aph on the direction field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "display([popGraph, field]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Redraw the field with an \"auto matic\" solution, and overlay the approximate solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "field := DEplot(diff(P(t),t)=c*P(t)*(M-P(t)), P, t=0..100, P=0..1500, [[P(0 )=111]], linecolor=green):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "displ ay([popGraph, field]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Answer question 6 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 15 "P art 5. Summary" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2 3 0" 27 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }