{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Hea ding 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE " " -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 54 "High School Modul es > Precalculus by Gregory A. Moore\n" }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 256 30 "Exponential Functions & Graphs" }}{PARA 0 "" 0 " " {TEXT -1 92 "\nAn exploration of exponential functions and their gra phs, and a variety of related graphs.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execute the Code Resource section first. Although t here will be no output immediately, these definitions are used later i n this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 260 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(plots): " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 167 "merge := proc( a1,a2,a3, b1,b2,b3,k,n)\nlocal c;\n c := COLOR(RGB, a1 + k*(b1-a1)/n,\n a2 + k*(b2-a2)/n,\n a3 + k*(b3-a3)/n ); \nreturn c;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "ShadedPlot := proc( f)\n display ( [seq( plot( k*f(x), x = -3..3, filled = true, \n color = merge(.15,.15,.15,.7,.7,.7,k,10)), k = 0..10 )] );\nend proc:\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "ShadedPlotR := proc( f)\n \+ display( [seq( plot( k*f(x), x = -3..3, filled = true, \n \+ color = merge(1,.1,.1,1,.5,.5,k,10)), k = 0..10 )] );\nend proc:\n" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "ShadedPlotG := proc( f)\n display( [seq( plot( k*f(x), x = -3..3, filled = true, \n \+ color = merge( .3,.8,.2, .7,.9,.6,k,10)), k = 0..10 )] );\nend proc :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "ShadedPlotB := proc( \+ f)\n display( [seq( plot( k*f(x), x = -3..3, filled = true, \n \+ color = merge(.15,.15, .8, .6,.6, .8,k,10)), k = 0..10 )] );\ne nd proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "ShadedPlotE := proc( f)\n display( [seq( plot( k*f(x), x = -3..3, filled = true, \+ \n color = merge( .4,.2,.0, .8,.6,.2,k,10)), k = 0..10 )] \+ );\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "ShadedPlo tP := proc( f)\n display( [seq( plot( k*f(x), x = -3..3, filled = tr ue, \n color = merge( .8,.4,.8, .2,.2,.2,k,10)), k = 0..10 )] );\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "Shade dPlotP2 := proc( f,a,b)\n display( [seq( plot( k*f(x), x = a..b, fil led = true, \n color = merge( .85,.5,.85, .3,.2,.3,k,10)), k = 0..10 )] );\nend proc:" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 265 24 "1. Exponential Functions" }}{PARA 0 "" 0 "" {TEXT -1 41 "\nHere is a typical exponential function.\n" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "f := x -> 2^x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "for k from 6 to -4 by -1 do 'f'(k) = f(k); od;" }}} {PARA 0 "" 0 "" {TEXT -1 16 "\nHere is another" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "g := x -> 5^x;\nfor k from 6 to -4 by -1 do 'g'( k) = g(k); od;" }}}{PARA 0 "" 0 "" {TEXT -1 30 "\nWhat is the same abo ut these?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "A := array( [s eq( [k, 2^k, 5^k], k = -5..5) ]):\nA[1,1] := `k`: A[1,2] := `2^k`: \+ A[1,3] := `5^k`:\nprint(A);" }}}{PARA 0 "" 0 "" {TEXT -1 49 "\nLet's look at even more examples, side by side.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "A := array( [seq( [k, 2^k,3^k,4^k,5^k], k = -5.. 5) ]):\nA[1,1] := `k`: A[1,2] := `2^k`: A[1,3] := `3^k`:\nA[1,4] : = `4^k`: A[1,5] := `5^k`:\nprint(A);" }}}{PARA 0 "" 0 "" {TEXT -1 57 " \n\nIt looks like exponential functions are one when x = 0." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f(0); g(0);" }}}{PARA 0 "" 0 "" {TEXT -1 81 "\nIt looks like exponential functions are the value being exponentiated when x = 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f(1); g(1);" }}}{PARA 0 "" 0 "" {TEXT -1 40 "\nAlso, when x = -1, we \+ get reciprocals. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f(-1); \+ g(-1);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 266 34 "2. Graphs of Exponential Functions" }} {PARA 0 "" 0 "" {TEXT -1 49 "\nThe basic exponential function looks li ke this.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ShadedPlotB( e xp(x) );" }}}{PARA 0 "" 0 "" {TEXT -1 90 "\nWhen the exponent is negat ive, it looks like this - the mirror image through the y axis.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ShadedPlotG( exp(-x) );" }}} {PARA 0 "" 0 "" {TEXT -1 84 "\nHere are some other variations obtained by negating or shifting exponential graphs." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ShadedPlotE( - exp(x) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ShadedPlotP( - exp(-x) );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "ShadedPlotR( 3 - exp(x) );" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 264 27 "3. Exponential & Log Graphs" }}{PARA 0 "" 0 "" {TEXT -1 61 "\nExpo nentials and logs are inverse functions of one another.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "display(\n plot( [exp(x),ln(x)], x = -5..7, y = -5..7, color = [red, blue], thickness = 3, scaling = con strained),\nplot( [[-5,-5],[7,7]], color = maroon, linestyle = 3));\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "display(\n seq(plot( [ k*exp(x),ln(x/k)], x = -3.5..4.5, y = -3.5..4.5, \n color = [red, blue],thickness = 1, scaling = constrained), \n k = \+ 1..15),\nplot( [[-5,-5],[7,7]], color = maroon, linestyle = 3));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "display(\n seq(plot( [k+exp (x),ln(x-k)], x = -5..16, y = -5..16, \n color = [red, blue ],thickness = 1, scaling = constrained), \n k = 1..15),\npl ot( [[-5,-5],[16,16]], color = maroon, linestyle = 3));" }}}{PARA 0 " " 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " } {TEXT 262 18 "4. Surge Functions" }}{PARA 0 "" 0 "" {TEXT -1 101 "\nHe re is a family of functions called surge functions which are variation s of exponential functions.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := (x,a,b) -> a*x*exp(-b*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "display( [seq( plot( k*f(x, .1, .1), x = 0..100, fil led = true, \n color = merge( .4,.2,.0, .8,.6,.2,k,10)), k = 0..10 )]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "display( [seq( plot( f(x, 1, (50-2.7*k)/50), x = 0..20, filled = true, \n \+ color = merge( .8,.6,.7, .5,.3,.4,k,10)), k = 0..16 )]); " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "plot( \{ seq(f(x, 1, k/50), k = 1..40) \}, x = 0..70, color = gold );" }}}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 1 " " }{TEXT 263 28 "5. Logistic Growth Functions" }}{PARA 0 "" 0 "" {TEXT -1 163 "\nThere are certain problems, in biology in pa rticular, which involve this kind of exponential function. This kind o f function has a bound on how high it can grow.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ShadedPlot( exp(x)/(1+exp(x)));\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "ShadedPlotP2( exp(x)/(1+exp(x)), -1 0, 10);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "plot( \{seq(ex p(x*(.5*k))/(1 + exp(x*k)), k = 1..16)\}, x = -1..1, color =khaki);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "display( [seq( plot( exp( x*(.5*(10-k)))/(1 + exp(x*(10-k))), \n x = -1..1, filled = true, \n color = merge( .4,.2,.0, .8,.6,.2,k,10)), k = 1. .10 )]);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 261 30 "6. Other Exponential Functions" }}{PARA 0 "" 0 "" {TEXT -1 51 "\nHere ar e some other wilder exponential functions.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "plot( \{ seq( exp(x*k)/(1 + exp(k*x)), k= 1..20)\} , x = -4..4, color = khaki);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot( \{seq(exp(x*k)/(1 + k*exp(x*k)), k= 1..16)\}, x = -3..3, color = khaki);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "plot( \+ \{seq(1 - exp(x*k)/(1 + exp(x*k)), k = 1..20)\}, x = -2..2, color = n avy);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot( \{seq(exp(x* (.5*k))/(1 + k*exp(x*k)), k = 1..16)\}, x = -1..1, color = khaki);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 " plot( \{seq(exp(x*(.01*k ))/(1 + (k)*exp(x*k)), k = 1..20)\}, x = -3..3, color = khaki);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot( \{seq(exp(x*(.9*k))/(1 + sqrt(k)*exp(x*k)), k = 1..20)\}, x =-3..3);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 74 "plot( \{seq(1+k*exp(x*(.9*k))/(1 + sqrt(k)*exp (x*k)), k = 1..20)\},x=-3..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "plot( \{seq(1+ (k^2)*exp(x*(.95*k))/(1 + sqrt(k)*exp(x*k)), k \+ = 1..20)\},x = -3..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 " display( [seq( plot( 1+ (k^2)*exp(x*(.95*k))/(1 + sqrt(k)*exp(x*k)), \+ \n x = -3..3, filled = true, \n color = merge( .4,.2,.0, .8,.6,.2,k,10)), k = 1..20 )]);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 182 "display( [seq( plot( exp(x*(.9*(20-k)))/(1 + sqrt( 20-k)*exp(x*(20-k))), \n x = -2..2, filled = true, \n \+ color = merge( .6,.7,.6, .3,.5,.3,k,10)), k = 1..19 )]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "display( [seq( plot( exp(x* k)/(1 + exp(k*x)) , \n x = 0..2, filled = true, \n \+ color = merge( .8,.5,.6, .5,.3,.3,k,10)), k = 1..10 ),\n \+ seq( plot( exp(x*(10-k))/(1 + exp((10-k)*x)) , \n x = -2. .0, filled = true, \n color = merge( .5,.3,.3,.8,.5,.6, k ,10)), k = 1..9 )]);" }}}{EXCHG }{EXCHG }}{PARA 0 "" 0 "" {TEXT 259 36 "\n \251 2002 Waterloo Maple Inc " }}}{MARK "0 1" 45 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }