(* This mathematica file creates frames for two animations to illustrate the definition of e as the number whose associated exponential function has derivative of 1 at 0. These animate the graph of y=a^x with a as the animation parameter. Using manual frame control, this gives a way to vary a manually and display the changing graph. The first animation lets a range from .1 to 10 and shows the graph on [-2,2]x[0,10]. The second animation shows a smaller range of a and plots the graph on [-1,1]x[0,2]. The line with slope 1 through (0,1) is shown as well. By adjusting a we can make this line a tangent to the graph of y=a^x. Of course this happens when a=e. Douglas N. Arnold, 10/11/94 Douglas N. Arnold Department of Mathematics Penn State University Park, PA 16802 Online documentation related to this file can be found at WWW URL http://www.math.psu.edu/dna/graphics.html.*) BeginPackage["Exponential`", "Graphics`Colors`"] Begin["Private`"] (* Describe the individual frames. *) g[s_] := {s,{Gray,Thickness[0.001]}} gridspec = {Table[g[s],{s,-2,2,.5}],Table[g[s],{s,0,10,2}]}; gridspec0 = {Table[g[s],{s,-1,1,.5}],Table[g[s],{s,0,2,.5}]}; expplot[s_]:= Plot[Exp[s x],{x,-2,2}, PlotRange->{{-2,2},{0,10}}, PlotStyle->{{Thickness[.01],Green}}, Background->Black,DefaultColor->White, GridLines->gridspec, MaxBend->5, Epilog->{RGBColor[1,1,0], Text["a =",Scaled[{.5,.9}],{.5,0}, TextStyle->{FontSize->24} ], Text[PaddedForm[N[Exp[s]],{3,2}],Scaled[{.5,.9},{-1.5,0}], TextStyle->{FontSize->24}] }, PlotLabel->StyleForm["y ="a^x,FontSize->20,FontColor->Yellow], TextStyle->{FontSize->18}, DisplayFunction->Identity] expplot0[s_]:= Plot[Exp[s x],{x,-2,2}, PlotRange->{{-1,1},{0,2}}, PlotStyle->{{Thickness[.01],Green}}, Background->Black,DefaultColor->White, GridLines->gridspec0, MaxBend->5, Ticks->{{-1,-.5,0,.5,1},{0,.5,1,1.5,2}}, AspectRatio->1, Epilog->{Yellow, Text["a = ",Scaled[{.7,.1}],{1,0}, TextStyle->{FontSize->24}], Text[PaddedForm[N[Exp[s]],{4,2}],Scaled[{.7,.1}],{-1,0}]}, Prolog->{Magenta,Thickness[.01],Line[{{-1,0},{1,2}}] }, PlotLabel->StyleForm["y ="a^x,FontSize->20,FontColor->Yellow], TextStyle->{FontSize->18}, DisplayFunction->Identity] (* Create the frames. *) smin=-2.3; smax=2.3; nframes=20; delta = (smax-smin)/nframes; prefix = "exponential1/T"; pixels=400; filename[prefix_,number_] := StringJoin[prefix,ToString[number],".gif"] Table[Display[filename[prefix,n+1], expplot[smin + n delta],"gif",ImageSize->pixels],{n,0,nframes}] Print[StringJoin["-- wrote ",prefix," frames --"]] smin=0; smax=1.67; nframes=20; delta = (smax-smin)/nframes; prefix = "exponential2/T"; pixels=400; Table[Display[filename[prefix,n+1], expplot0[smin + n delta],"gif",ImageSize->pixels],{n,0,nframes}] Print[StringJoin["-- wrote ",prefix," frames --"]] End[] (* end of context private *) EndPackage[]