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fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times";paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] (C) Copyright 1994 D.C.M. Burbulla and C.T.J. Dodson Notebook to support the book Self-Tutor for Computer Calculus ;[s] 3:0,0;75,1;104,2;143,-1; 3:1,0,-9 -*-times-bold-r-normal-*-*-120-75-75-*-*-*-*,times,1,12,0,0,0;1,0,-9 -*-times-bold-i-normal-*-*-170-75-75-*-*-*-*,times,3,17,0,0,0;1,0,-9 -*-times-bold-r-normal-*-*-240-75-75-*-*-*-*,times,1,24,0,0,0; :[font = subtitle; inactive; preserveAspect] Using Mathematica 2 ;[s] 2:0,0;6,1;20,-1; 2:1,0,-9 -*-times-bold-r-normal-*-*-230-75-75-*-*-*-*,times,1,23,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-230-75-75-*-*-*-*,times,2,23,0,0,0; :[font = subsubtitle; inactive; preserveAspect; fontName = "Times"] by D. C. M. Burbulla and C. T. J. Dodson, University of Toronto Published by: Prentice-Hall Canada (English) ISBN 0130152803 and Prentice-Hall Japan (Japanese) ISBN4-8101-8558-3 C3055 P2880E ;[s] 5:0,0;128,1;129,2;145,3;192,4;222,-1; 5:1,0,-9 -*-times-medium-i-normal-*-*-100-75-75-*-*-*-*,times,2,10,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-110-75-75-*-*-*-*,times,2,11,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-110-75-75-*-*-*-*,courier,2,11,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-110-75-75-*-*-*-*,courier,2,11,0,0,0; :[font = section; inactive; Cclosed; preserveAspect; startGroup] The Derivative :[font = input; preserveAspect; fontName = "Courier"] Our new functions allow investigation of secants, tangents and normals. ;[s] 1:0,0;71,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = subsection; inactive; Cclosed; preserveAspect; fontName = "Times"; startGroup] Chap3.m :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] secant/: secant[f_, a_][p_] := (one = Block[{x}, Plot[{f[x], (f[p] - f[a]) (x - a)/(p - a) + f[a]}, {x, a - 1.5, a + 1.5}, Axes -> None, AspectRatio -> Automatic, DisplayFunction -> Identity]]; two = ListPlot[{{a, f[a]}, {p, f[p]}}, DisplayFunction -> Identity]; Show[one, two, DisplayFunction -> $DisplayFunction]) secant::usage = "secant[f, a][p] plots f and the secant joining (a, f[a]) and (p, f[p])." tangent/: tangent[f_, a_][x_] := Derivative[1][f][a]*(x - a) + f[a] normal/: normal[f_, a_][x_] := (a - x)/Derivative[1][f][a] + f[a] tangentplot/: tangentplot[f_, a_, {b_, c_}] := Plot[{f[x], tangent[f, a][x]}, {x, b, c}, AspectRatio -> Automatic] normalplot/: normalplot[f_, a_, {b_, c_}] := Plot[{f[x], normal[f, a][x]}, {x, b, c}, AspectRatio -> Automatic] tangentfamily/: tangentfamily[f_, {a_, b_, stepsize_:0.1}] := (one = Block[{x}, Plot[f[x], {x, a, b}, AspectRatio -> Automatic, DisplayFunction -> Identity]]; two = Block[{i, x}, Plot[Release[Table[tangent[f, i][x], {i, a, b, stepsize}]], {x, a, b}, DisplayFunction -> Identity]]; Show[one, two, DisplayFunction -> $DisplayFunction]) tangentfamily::usage = "tangentfamily[f, {a, b, stepsize}] plots f and its tangents on the interval {a, b}, plotting one tangent per stepsize in the interval {a, b}. The default stepsize is 0.1." normalfamily/: normalfamily[f_, {a_, b_, stepsize_:0.1}, {c_, d_}] := (one = Block[{i, x}, Plot[Release[Table[normal[f, i][x], {i, a, b, stepsize}]], {x, a, b}, PlotRange -> {c, d}, AspectRatio -> Automatic, DisplayFunction -> Identity]]; two = Block[{x}, Plot[f[x], {x, a, b}, DisplayFunction -> Identity]]; Show[one, two, DisplayFunction -> $DisplayFunction]) normalfamily::usage = "normalfamily[f, {a, b, stepsize}, {c, d}] plots f and its normals on the interval {a, b}, with PlotRange set to {c, d}. One normal is plotted per stepsize in the interval {a, b}; the default stepsize is 0.1." wire/: wire[x_] := (line = Block[{t}, Plot[70, {t, 0, 100}, DisplayFunction -> Identity]]; point = ListPlot[{{x, 70}}, DisplayFunction -> Identity]; circle = Block[{t}, ParametricPlot[ {x*Cos[t]/(2*Pi) + 25, x*Sin[t]/(2*Pi) + 25}, {t, 0, 2*Pi}, AspectRatio -> Automatic, DisplayFunction -> Identity]]; square = ListPlot[{{75 - (100 - x)/8, 25 - (100 - x)/8}, {75 - (100 - x)/8, 25 + (100 - x)/8}, {75 + (100 - x)/8, 25 + (100 - x)/8}, {75 + (100 - x)/8, 25 - (100 - x)/8}, {75 - (100 - x)/8, 25 - (100 - x)/8}}, PlotJoined -> True, DisplayFunction -> Identity]; Show[circle, square, line, point, Axes -> None, PlotRange -> {{0, 100}, {0, 80}}, PlotLabel -> N[x^2/(4*Pi) + (100 - x)^2/16] " = combined area of circle and square", DisplayFunction -> $DisplayFunction]) ladder/: ladder[t_] := ListPlot[{{10 t, 0}, {0, Sqrt[41^2 - (10 t)^2]}}, PlotRange -> {{0, 41}, {0, 41}}, PlotJoined -> True, AspectRatio -> Automatic] :[font = output; output; inactive; preserveAspect] "secant[f, a][p] plots f and the secant joining (a,\ f[a]) and (p, f[p])." ;[o] secant[f, a][p] plots f and the secant joining (a,\ f[a]) and (p, f[p]). :[font = message; inactive; preserveAspect] General::spell1: Possible spelling error: new symbol name "normal" is similar to existing symbol "Normal". :[font = output; output; inactive; preserveAspect] "tangentfamily[f, {a, b, stepsize}] plots f and its\ tangents on the interval {a, b}, plotting one\ tangent per stepsize in the interval {a, b}. The\ default stepsize is 0.1." ;[o] tangentfamily[f, {a, b, stepsize}] plots f and its\ tangents on the interval {a, b}, plotting one\ tangent per stepsize in the interval {a, b}. The\ default stepsize is 0.1. :[font = output; output; inactive; preserveAspect] "normalfamily[f, {a, b, stepsize}, {c, d}] plots f\ and its normals on the interval {a, b}, with\ PlotRange set to {c, d}. One normal is plotted per\ stepsize in the interval {a, b}; the default\ stepsize is 0.1." ;[o] normalfamily[f, {a, b, stepsize}, {c, d}] plots f and\ its normals on the interval {a, b}, with PlotRange\ set to {c, d}. One normal is plotted per stepsize \ in the interval {a, b}; the default stepsize is 0.1. :[font = message; inactive; preserveAspect] General::spell1: Possible spelling error: new symbol name "line" is similar to existing symbol "Line". :[font = message; inactive; preserveAspect] General::spell1: Possible spelling error: new symbol name "point" is similar to existing symbol "Point". :[font = message; inactive; preserveAspect] General::spell1: Possible spelling error: new symbol name "circle" is similar to existing symbol "Circle". :[font = message; inactive; preserveAspect; endGroup; endGroup] General::stop: Further output of General::spell1 will be suppressed during this calculation. :[font = subsection; inactive; Cclosed; preserveAspect; fontName = "Times"; startGroup] Tangents and Normals :[font = input; preserveAspect; fontName = "Courier"] This section consists of some examples covering typical problems that involve tangents and normals. We shall make use of the defined functions in Chap3.m to simplify our calculations and to sketch graphs whenever possible. We shall make extensive use of the functions tangent and normal. We can remind ourselves of their syntax by querying Mathematica: ;[s] 9:0,0;1,1;148,2;155,3;272,4;279,5;284,6;290,7;345,8;357,-1; 9:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?tangent :[font = print; inactive; preserveAspect; endGroup] Global`tangent tangent[f_, a_][x_] := Derivative[1][f][a]*(x - a) + f[a] :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?normal :[font = print; inactive; preserveAspect; endGroup] Global`normal normal[f_, a_][x_] := (a - x)/Derivative[1][f][a] + f[a] :[font = input; preserveAspect; fontName = "Courier"] Note that in the definition of normal no special consideration is given for the case f '(a) = 0, where, of course, the normal is a vertical line and the formula for normal no longer makes sense. Example 1. Find the tangents to the hyperbola xy = 1 passing through the point (-1, 1). We work as follows: ;[s] 12:0,0;1,1;32,2;38,3;86,4;93,5;168,6;174,7;200,8;210,9;247,10;250,11;309,-1; 12:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; preserveAspect; fontName = "Courier"] Clear[f]; f[x_] := 1/x ;[s] 2:0,0;1,1;23,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Solve[tangent[f, a][-1] == 1, a] (* (-1, 1) is on tangent*) ;[s] 2:0,0;1,1;60,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] {{a -> (2 - 2*2^(1/2))/2}, {a -> (2 + 2*2^(1/2))/2}} ;[o] 2 - 2 Sqrt[2] 2 + 2 Sqrt[2] {{a -> -------------}, {a -> -------------}} 2 2 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] tangent[f, a][x] /. % ;[s] 2:0,0;1,1;22,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] {2/(2 - 2*2^(1/2)) - (4*((-2 + 2*2^(1/2))/2 + x))/ (2 - 2*2^(1/2))^2, 2/(2 + 2*2^(1/2)) - (4*((-2 - 2*2^(1/2))/2 + x))/(2 + 2*2^(1/2))^2} ;[o] -2 + 2 Sqrt[2] 4 (-------------- + x) 2 2 {------------- - ----------------------, 2 - 2 Sqrt[2] 2 (2 - 2 Sqrt[2]) -2 - 2 Sqrt[2] 4 (-------------- + x) 2 2 ------------- - ----------------------} 2 + 2 Sqrt[2] 2 (2 + 2 Sqrt[2]) :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Simplify[%] ;[s] 2:0,0;1,1;12,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] {(2 - 2*2^(1/2) - x)/(-1 + 2^(1/2))^2, (2 + 2*2^(1/2) - x)/(1 + 2^(1/2))^2} ;[o] 2 - 2 Sqrt[2] - x 2 + 2 Sqrt[2] - x {-----------------, -----------------} 2 2 (-1 + Sqrt[2]) (1 + Sqrt[2]) :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Plot[{%[[1]], %[[2]], f[x]}, {x, -2, 3}, PlotRange -> {-4, 6}]; ;[s] 2:0,0;1,1;65,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.404762 0.190476 0.247214 0.061803 [ [(-2)] .02381 .24721 0 2 Msboxa [(-1)] .21429 .24721 0 2 Msboxa [(1)] .59524 .24721 0 2 Msboxa [(2)] .78571 .24721 0 2 Msboxa [(3)] .97619 .24721 0 2 Msboxa [(-4)] .39226 0 1 0 Msboxa [(-2)] .39226 .12361 1 0 Msboxa [(2)] .39226 .37082 1 0 Msboxa [(4)] .39226 .49443 1 0 Msboxa [(6)] .39226 .61803 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .24721 m .02381 .25346 L s P [(-2)] .02381 .24721 0 2 Mshowa p .002 w .21429 .24721 m .21429 .25346 L s P [(-1)] .21429 .24721 0 2 Mshowa p .002 w .59524 .24721 m .59524 .25346 L s P [(1)] .59524 .24721 0 2 Mshowa p .002 w .78571 .24721 m .78571 .25346 L s P [(2)] .78571 .24721 0 2 Mshowa p .002 w .97619 .24721 m .97619 .25346 L s P [(3)] .97619 .24721 0 2 Mshowa p .001 w .0619 .24721 m .0619 .25096 L s P p .001 w .1 .24721 m .1 .25096 L s P p .001 w .1381 .24721 m .1381 .25096 L s P p .001 w .17619 .24721 m .17619 .25096 L s P p .001 w .25238 .24721 m .25238 .25096 L s P p .001 w .29048 .24721 m .29048 .25096 L s P p .001 w .32857 .24721 m .32857 .25096 L s P p .001 w .36667 .24721 m .36667 .25096 L s P p .001 w .44286 .24721 m .44286 .25096 L s P p .001 w .48095 .24721 m .48095 .25096 L s P p .001 w .51905 .24721 m .51905 .25096 L s P p .001 w .55714 .24721 m .55714 .25096 L s P p .001 w .63333 .24721 m .63333 .25096 L s P p .001 w .67143 .24721 m .67143 .25096 L s P p .001 w .70952 .24721 m .70952 .25096 L s P p .001 w .74762 .24721 m .74762 .25096 L s P p .001 w .82381 .24721 m .82381 .25096 L s P p .001 w .8619 .24721 m .8619 .25096 L s P p .001 w .9 .24721 m .9 .25096 L s P p .001 w .9381 .24721 m .9381 .25096 L s P p .002 w 0 .24721 m 1 .24721 L s P p .002 w .40476 0 m .41101 0 L s P [(-4)] .39226 0 1 0 Mshowa p .002 w .40476 .12361 m .41101 .12361 L s P [(-2)] .39226 .12361 1 0 Mshowa p .002 w .40476 .37082 m .41101 .37082 L s P [(2)] .39226 .37082 1 0 Mshowa p .002 w .40476 .49443 m .41101 .49443 L s P [(4)] .39226 .49443 1 0 Mshowa p .002 w .40476 .61803 m .41101 .61803 L s P [(6)] .39226 .61803 1 0 Mshowa p .001 w .40476 .02472 m .40851 .02472 L s P p .001 w .40476 .04944 m .40851 .04944 L s P p .001 w .40476 .07416 m .40851 .07416 L s P p .001 w .40476 .09889 m .40851 .09889 L s P p .001 w .40476 .14833 m .40851 .14833 L s P p .001 w .40476 .17305 m .40851 .17305 L s P p .001 w .40476 .19777 m .40851 .19777 L s P p .001 w .40476 .22249 m .40851 .22249 L s P p .001 w .40476 .27193 m .40851 .27193 L s P p .001 w .40476 .29666 m .40851 .29666 L s P p .001 w .40476 .32138 m .40851 .32138 L s P p .001 w .40476 .3461 m .40851 .3461 L s P p .001 w .40476 .39554 m .40851 .39554 L s P p .001 w .40476 .42026 m .40851 .42026 L s P p .001 w .40476 .44498 m .40851 .44498 L s P p .001 w .40476 .46971 m .40851 .46971 L s P p .001 w .40476 .51915 m .40851 .51915 L s P p .001 w .40476 .54387 m .40851 .54387 L s P p .001 w .40476 .56859 m .40851 .56859 L s P p .001 w .40476 .59331 m .40851 .59331 L s P p .002 w .40476 0 m .40476 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .004 w s .05088 .61803 m .06349 .59419 L s .06349 .59419 m .10317 .51914 L .14286 .4441 L .18254 .36905 L .22222 .29401 L .2619 .21896 L .30159 .14392 L .34127 .06887 L s .37769 0 m .34127 .06887 L s s s s s s s s s s s s s s s s s P P p p .004 w .02381 .31962 m .06349 .31741 L .10317 .3152 L .14286 .31299 L .18254 .31078 L .22222 .30858 L .2619 .30637 L .30159 .30416 L .34127 .30195 L .38095 .29974 L .42063 .29753 L .46032 .29532 L .5 .29311 L .53968 .2909 L .57937 .28869 L .61905 .28648 L .65873 .28427 L .69841 .28207 L .7381 .27986 L .77778 .27765 L .81746 .27544 L .85714 .27323 L .89683 .27102 L .93651 .26881 L .97619 .2666 L s P P p p .004 w .02381 .21631 m .06349 .21272 L .10317 .20818 L .14286 .20227 L .1627 .19858 L .18254 .19424 L .20238 .18905 L .22222 .18272 L .24206 .17486 L .2619 .16481 L .28175 .15152 L .30159 .13312 L .31151 .12098 L .32143 .10595 L .33135 .08686 L .34127 .0618 L .34623 .04609 L .35119 .02747 L .35615 .00505 L s .35706 0 m .35615 .00505 L s s s s s s s s s s s s s s s s s s s s .4055 0 m .40551 .61803 L s s .40551 .61803 m .4055 0 L s s s s s s s s s s s s s s s s s .43663 .61803 m .44048 .57683 L s .44048 .57683 m .44544 .53663 L .4504 .50518 L .46032 .45911 L .46528 .44174 L .47024 .42701 L .48016 .40335 L .49008 .38519 L .5 .37082 L .51984 .34951 L .53968 .33447 L .55952 .32328 L .57937 .31464 L .59921 .30776 L .61905 .30215 L .65873 .29357 L .69841 .2873 L .7381 .28253 L .77778 .27877 L .81746 .27574 L .85714 .27324 L .89683 .27114 L .93651 .26935 L .97619 .26781 L s P P P % End of Graphics MathPictureEnd :[font = input; preserveAspect; fontName = "Courier"] Example 2. Find all of the tangents to the graph of y = 2 x3 + 13 x2 + 5 x + 9 passing through the origin. The point (0, 0) is not on the cubic; there could be as many as three tangents to the cubic passing through (0, 0). ;[s] 13:0,0;11,1;54,2;56,3;62,4;63,5;64,6;65,7;70,8;71,9;72,10;77,11;78,12;228,-1; 13:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-*-*-*-*-*-120-75-75-*-*-*-*,courier,32,12,0,0,0;1,0,-9 -*-courier-*-*-*-*-*-120-75-75-*-*-*-*,courier,32,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-*-*-*-*-*-120-75-75-*-*-*-*,courier,32,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; preserveAspect; fontName = "Courier"] Clear[f]; f[x_] := 2 x^3 + 13 x^2 + 5 x + 9 ;[s] 2:0,0;1,1;44,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Solve[tangent[f, a][0] == 0, a] ;[s] 2:0,0;31,1;32,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] {{a -> -3}, {a -> -1}, {a -> 3/4}} ;[o] 3 {{a -> -3}, {a -> -1}, {a -> -}} 4 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] tangent[f, a][x] /. % // Simplify ;[s] 2:0,0;1,1;34,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] {-19*x, -15*x, (223*x)/8} ;[o] 223 x {-19 x, -15 x, -----} 8 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Plot[{%[[1]], %[[2]], %[[3]], f[x]}, {x, -3.5, 2}, PlotRange -> {-5, 70}]; ;[s] 2:0,0;1,1;76,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.62987 0.17316 0.041202 0.00824 [ [(-3)] .11039 .0412 0 2 Msboxa [(-2)] .28355 .0412 0 2 Msboxa [(-1)] .45671 .0412 0 2 Msboxa [(1)] .80303 .0412 0 2 Msboxa [(2)] .97619 .0412 0 2 Msboxa [(10)] .61737 .12361 1 0 Msboxa [(20)] .61737 .20601 1 0 Msboxa [(30)] .61737 .28842 1 0 Msboxa [(40)] .61737 .37082 1 0 Msboxa [(50)] .61737 .45322 1 0 Msboxa [(60)] .61737 .53563 1 0 Msboxa [(70)] .61737 .61803 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .11039 .0412 m .11039 .04745 L s P [(-3)] .11039 .0412 0 2 Mshowa p .002 w .28355 .0412 m .28355 .04745 L s P [(-2)] .28355 .0412 0 2 Mshowa p .002 w .45671 .0412 m .45671 .04745 L s P [(-1)] .45671 .0412 0 2 Mshowa p .002 w .80303 .0412 m .80303 .04745 L s P [(1)] .80303 .0412 0 2 Mshowa p .002 w .97619 .0412 m .97619 .04745 L s P [(2)] .97619 .0412 0 2 Mshowa p .001 w .14502 .0412 m .14502 .04495 L s P p .001 w .17965 .0412 m .17965 .04495 L s P p .001 w .21429 .0412 m .21429 .04495 L s P p .001 w .24892 .0412 m .24892 .04495 L s P p .001 w .31818 .0412 m .31818 .04495 L s P p .001 w .35281 .0412 m .35281 .04495 L s P p .001 w .38745 .0412 m .38745 .04495 L s P p .001 w .42208 .0412 m .42208 .04495 L s P p .001 w .49134 .0412 m .49134 .04495 L s P p .001 w .52597 .0412 m .52597 .04495 L s P p .001 w .56061 .0412 m .56061 .04495 L s P p .001 w .59524 .0412 m .59524 .04495 L s P p .001 w .6645 .0412 m .6645 .04495 L s P p .001 w .69913 .0412 m .69913 .04495 L s P p .001 w .73377 .0412 m .73377 .04495 L s P p .001 w .7684 .0412 m .7684 .04495 L s P p .001 w .83766 .0412 m .83766 .04495 L s P p .001 w .87229 .0412 m .87229 .04495 L s P p .001 w .90693 .0412 m .90693 .04495 L s P p .001 w .94156 .0412 m .94156 .04495 L s P p .001 w .07576 .0412 m .07576 .04495 L s P p .001 w .04113 .0412 m .04113 .04495 L s P p .001 w .00649 .0412 m .00649 .04495 L s P p .002 w 0 .0412 m 1 .0412 L s P p .002 w .62987 .12361 m .63612 .12361 L s P [(10)] .61737 .12361 1 0 Mshowa p .002 w .62987 .20601 m .63612 .20601 L s P [(20)] .61737 .20601 1 0 Mshowa p .002 w .62987 .28842 m .63612 .28842 L s P [(30)] .61737 .28842 1 0 Mshowa p .002 w .62987 .37082 m .63612 .37082 L s P [(40)] .61737 .37082 1 0 Mshowa p .002 w .62987 .45322 m .63612 .45322 L s P [(50)] .61737 .45322 1 0 Mshowa p .002 w .62987 .53563 m .63612 .53563 L s P [(60)] .61737 .53563 1 0 Mshowa p .002 w .62987 .61803 m .63612 .61803 L s P [(70)] .61737 .61803 1 0 Mshowa p .001 w .62987 .05768 m .63362 .05768 L s P p .001 w .62987 .07416 m .63362 .07416 L s P p .001 w .62987 .09064 m .63362 .09064 L s P p .001 w .62987 .10713 m .63362 .10713 L s P p .001 w .62987 .14009 m .63362 .14009 L s P p .001 w .62987 .15657 m .63362 .15657 L s P p .001 w .62987 .17305 m .63362 .17305 L s P p .001 w .62987 .18953 m .63362 .18953 L s P p .001 w .62987 .22249 m .63362 .22249 L s P p .001 w .62987 .23897 m .63362 .23897 L s P p .001 w .62987 .25545 m .63362 .25545 L s P p .001 w .62987 .27193 m .63362 .27193 L s P p .001 w .62987 .3049 m .63362 .3049 L s P p .001 w .62987 .32138 m .63362 .32138 L s P p .001 w .62987 .33786 m .63362 .33786 L s P p .001 w .62987 .35434 m .63362 .35434 L s P p .001 w .62987 .3873 m .63362 .3873 L s P p .001 w .62987 .40378 m .63362 .40378 L s P p .001 w .62987 .42026 m .63362 .42026 L s P p .001 w .62987 .43674 m .63362 .43674 L s P p .001 w .62987 .46971 m .63362 .46971 L s P p .001 w .62987 .48619 m .63362 .48619 L s P p .001 w .62987 .50267 m .63362 .50267 L s P p .001 w .62987 .51915 m .63362 .51915 L s P p .001 w .62987 .55211 m .63362 .55211 L s P p .001 w .62987 .56859 m .63362 .56859 L s P p .001 w .62987 .58507 m .63362 .58507 L s P p .001 w .62987 .60155 m .63362 .60155 L s P p .001 w .62987 .02472 m .63362 .02472 L s P p .001 w .62987 .00824 m .63362 .00824 L s P p .002 w .62987 0 m .62987 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .004 w .02381 .58919 m .06349 .55331 L .10317 .51743 L .14286 .48155 L .18254 .44567 L .22222 .40979 L .2619 .37391 L .30159 .33803 L .34127 .30215 L .38095 .26627 L .42063 .23039 L .46032 .19451 L .5 .15863 L .53968 .12275 L .57937 .08687 L .61905 .05099 L .65873 .01511 L s .67544 0 m .65873 .01511 L s s s s s s s s s P P p p .004 w .02381 .47383 m .06349 .4455 L .10317 .41717 L .14286 .38885 L .18254 .36052 L .22222 .33219 L .2619 .30387 L .30159 .27554 L .34127 .24721 L .38095 .21889 L .42063 .19056 L .46032 .16223 L .5 .13391 L .53968 .10558 L .57937 .07725 L .61905 .04893 L .65873 .0206 L s .68759 0 m .65873 .0206 L s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s .59881 0 m .61905 .02685 L s .61905 .02685 m .65873 .07949 L .69841 .13213 L .7381 .18477 L .77778 .23741 L .81746 .29005 L .85714 .34269 L .89683 .39533 L .93651 .44797 L .97619 .50061 L s P P p p .004 w .02381 .57683 m .06349 .54996 L .10317 .51736 L .14286 .48021 L .18254 .43971 L .22222 .39704 L .2619 .35341 L .30159 .30998 L .34127 .26797 L .38095 .22855 L .42063 .19292 L .46032 .16226 L .48016 .14917 L .5 .13777 L .51984 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Example 3. One thing we can do is graph lots of tangents. There is a defined function, tangentfamily, in Chap3.m, to do this: ;[s] 12:0,0;1,1;73,2;88,3;102,4;113,5;144,6;148,7;192,8;205,9;210,10;217,11;230,-1; 12:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?tangentfamily ;[s] 2:0,0;1,1;15,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = print; inactive; preserveAspect; endGroup] tangentfamily[f, {a, b, stepsize}] plots f and its tangents on the interval {a, b}, plotting one tangent per stepsize in the interval {a, b}. The default stepsize is 0.1. :[font = input; preserveAspect; fontName = "Courier"] We shall illustrate it on the parabola with equation y = x 2 . ;[s] 9:0,0;1,1;54,2;56,3;60,4;61,5;62,6;63,7;64,8;65,-1; 9:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-*-*-*-*-*-120-75-75-*-*-*-*,courier,32,12,0,0,0;1,0,-9 -*-courier-*-*-*-*-*-120-75-75-*-*-*-*,courier,32,12,0,0,0;1,0,-9 -*-courier-*-*-*-*-*-120-75-75-*-*-*-*,courier,32,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; preserveAspect; fontName = "Courier"] Clear[f]; f[x_] := x^2 ;[s] 2:0,0;1,1;23,-1; 2:1,0,-9 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.42063 .03968 L .46032 0 L s .46032 0 m .46032 0 L s s s s s s s s s s s s s s P P p p .004 w .02381 .35556 m .06349 .32381 L .10317 .29206 L .14286 .26032 L .18254 .22857 L .22222 .19683 L .2619 .16508 L .30159 .13333 L .34127 .10159 L .38095 .06984 L .42063 .0381 L .46032 .00635 L s .46825 0 m .46032 .00635 L s s s s s s s s s s s s s s P P p p .004 w .02381 .27143 m .06349 .24762 L .10317 .22381 L .14286 .2 L .18254 .17619 L .22222 .15238 L .2619 .12857 L .30159 .10476 L .34127 .08095 L .38095 .05714 L .42063 .03333 L .46032 .00952 L s .47619 0 m .46032 .00952 L s s s s s s s s s s s s s s P P p p .004 w .02381 .18413 m .06349 .16825 L .10317 .15238 L .14286 .13651 L .18254 .12063 L .22222 .10476 L .2619 .08889 L .30159 .07302 L .34127 .05714 L .38095 .04127 L .42063 .0254 L .46032 .00952 L s .48413 0 m .46032 .00952 L s s s s s s s s s s s s s s P P p p .004 w .02381 .09365 m .06349 .08571 L .10317 .07778 L .14286 .06984 L .18254 .0619 L .22222 .05397 L .2619 .04603 L .30159 .0381 L .34127 .03016 L .38095 .02222 L .42063 .01429 L .46032 .00635 L s .49206 0 m .46032 .00635 L s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s .5 0 m .53968 0 L s .53968 0 m .57937 0 L .61905 0 L .65873 0 L .69841 0 L .7381 0 L .77778 0 L .81746 0 L .85714 0 L .89683 0 L .93651 0 L .97619 0 L s P P p p .004 w s s s s s s s s s s s s s .50794 0 m .53968 .00635 L s .53968 .00635 m .57937 .01429 L .61905 .02222 L .65873 .03016 L .69841 .0381 L .7381 .04603 L .77778 .05397 L .81746 .0619 L .85714 .06984 L .89683 .07778 L .93651 .08571 L .97619 .09365 L s P P p p .004 w s s s s s s s s s s s s s .51587 0 m .53968 .00952 L s .53968 .00952 m .57937 .0254 L .61905 .04127 L .65873 .05714 L .69841 .07302 L .7381 .08889 L .77778 .10476 L .81746 .12063 L .85714 .13651 L .89683 .15238 L .93651 .16825 L .97619 .18413 L s P P p p .004 w s s s s s s s s s s s s s .52381 0 m .53968 .00952 L s .53968 .00952 m .57937 .03333 L .61905 .05714 L .65873 .08095 L .69841 .10476 L .7381 .12857 L .77778 .15238 L .81746 .17619 L .85714 .2 L .89683 .22381 L .93651 .24762 L .97619 .27143 L s P P p p .004 w s s s s s s s s s s s s s .53175 0 m .53968 .00635 L s .53968 .00635 m .57937 .0381 L .61905 .06984 L .65873 .10159 L .69841 .13333 L .7381 .16508 L .77778 .19683 L .81746 .22857 L .85714 .26032 L .89683 .29206 L .93651 .32381 L .97619 .35556 L s P P p p .004 w s s s s s s s s s s s s s s .53968 0 m .57937 .03968 L s .57937 .03968 m .61905 .07937 L .65873 .11905 L .69841 .15873 L .7381 .19841 L .77778 .2381 L .81746 .27778 L .85714 .31746 L .89683 .35714 L .93651 .39683 L .97619 .43651 L s P P p p .004 w s s s s s s s s s s s s s s .54762 0 m .57937 .0381 L s .57937 .0381 m .61905 .08571 L .65873 .13333 L .69841 .18095 L .7381 .22857 L .77778 .27619 L .81746 .32381 L .85714 .37143 L .89683 .41905 L .93651 .46667 L .97619 .51429 L s P P p p .004 w s s s s s s s s s s s s s s .55556 0 m .57937 .03333 L s .57937 .03333 m .61905 .08889 L .65873 .14444 L .69841 .2 L .7381 .25556 L .77778 .31111 L .81746 .36667 L .85714 .42222 L .89683 .47778 L .93651 .53333 L .97619 .58889 L s P P p p .004 w s s s s s s s s s s s s s s .56349 0 m .57937 .0254 L s .57937 .0254 m .61905 .08889 L .65873 .15238 L .69841 .21587 L .7381 .27937 L .77778 .34286 L .81746 .40635 L .85714 .46984 L .89683 .53333 L .93651 .59683 L .97619 .66032 L s P P p p .004 w s s s s s s s s s s s s s s .57143 0 m .57937 .01429 L s .57937 .01429 m .61905 .08571 L .65873 .15714 L .69841 .22857 L .7381 .3 L .77778 .37143 L .81746 .44286 L .85714 .51429 L .89683 .58571 L .93651 .65714 L .97619 .72857 L s P P p p .004 w s s s s s s s s s s s s s s s .57937 0 m .61905 .07937 L s .61905 .07937 m .65873 .15873 L .69841 .2381 L .7381 .31746 L .77778 .39683 L .81746 .47619 L .85714 .55556 L .89683 .63492 L .93651 .71429 L .97619 .79365 L s P P p p .004 w s s s s s s s s s s s s s s s .5873 0 m .61905 .06984 L s .61905 .06984 m .65873 .15714 L .69841 .24444 L .7381 .33175 L .77778 .41905 L .81746 .50635 L .85714 .59365 L .89683 .68095 L .93651 .76825 L .97619 .85556 L s P P p p .004 w s s s s s s s s s s s s s s s .59524 0 m .61905 .05714 L s .61905 .05714 m .65873 .15238 L .69841 .24762 L .7381 .34286 L .77778 .4381 L .81746 .53333 L .85714 .62857 L .89683 .72381 L .93651 .81905 L .97619 .91429 L s P P p p .004 w s s s s s s s s s s s s s s s .60317 0 m .61905 .04127 L s .61905 .04127 m .65873 .14444 L .69841 .24762 L .7381 .35079 L .77778 .45397 L .81746 .55714 L .85714 .66032 L .89683 .76349 L .93651 .86667 L .97619 .96984 L s P P p p .004 w s s s s s s s s s s s s s s s .61111 0 m .61905 .02222 L s .61905 .02222 m .65873 .13333 L .69841 .24444 L .7381 .35556 L .77778 .46667 L .81746 .57778 L .85714 .68889 L .89683 .8 L .93651 .91111 L .97619 1.02222 L s P P p p .004 w s s s s s s s s s s s s s s s s .61905 0 m .65873 .11905 L s .65873 .11905 m .69841 .2381 L .7381 .35714 L .77778 .47619 L .81746 .59524 L .85714 .71429 L .89683 .83333 L .93651 .95238 L .97619 1.07143 L s P P p p .004 w s s s s s s s s s s s s s s s s .62698 0 m .65873 .10159 L s .65873 .10159 m .69841 .22857 L .7381 .35556 L .77778 .48254 L .81746 .60952 L .85714 .73651 L .89683 .86349 L .93651 .99048 L .97619 1.11746 L s P P p p .004 w s s s s s s s s s s s s s s s s .63492 0 m .65873 .08095 L s .65873 .08095 m .69841 .21587 L .7381 .35079 L .77778 .48571 L .81746 .62063 L .85714 .75556 L .89683 .89048 L .93651 1.0254 L .97619 1.16032 L s P P p p .004 w s s s s s s s s s s s s s s s s .64286 0 m .65873 .05714 L s .65873 .05714 m .69841 .2 L .7381 .34286 L .77778 .48571 L .81746 .62857 L .85714 .77143 L .89683 .91429 L .93651 1.05714 L .97619 1.2 L s P P p p .004 w s s s s s s s s s s s s s s s s .65079 0 m .65873 .03016 L s .65873 .03016 m .69841 .18095 L .7381 .33175 L .77778 .48254 L .81746 .63333 L .85714 .78413 L .89683 .93492 L .93651 1.08571 L .97619 1.23651 L s P P p p .004 w s s s s s s s s s s s s s s s s s .65873 0 m .69841 .15873 L s .69841 .15873 m .7381 .31746 L .77778 .47619 L .81746 .63492 L .85714 .79365 L .89683 .95238 L .93651 1.11111 L .97619 1.26984 L s P P p p .004 w s s s s s s s s s s s s s s s s s .66667 0 m .69841 .13333 L s .69841 .13333 m .7381 .3 L .77778 .46667 L .81746 .63333 L .85714 .8 L .89683 .96667 L .93651 1.13333 L .97619 1.3 L s P P p p .004 w s s s s s s s s s s s s s s s s s .6746 0 m .69841 .10476 L s .69841 .10476 m .7381 .27937 L .77778 .45397 L .81746 .62857 L .85714 .80317 L .89683 .97778 L .93651 1.15238 L .97619 1.32698 L s P P p p .004 w s s s s s s s s s s s s s s s s s .68254 0 m .69841 .07302 L s .69841 .07302 m .7381 .25556 L .77778 .4381 L .81746 .62063 L .85714 .80317 L .89683 .98571 L .93651 1.16825 L .97619 1.35079 L s P P p p .004 w s s s s s s s s s s s s s s s s s .69048 0 m .69841 .0381 L s .69841 .0381 m .7381 .22857 L .77778 .41905 L .81746 .60952 L .85714 .8 L .89683 .99048 L .93651 1.18095 L .97619 1.37143 L s P P p p .004 w s s s s s s s s s s s s s s s s s s .69841 0 m .7381 .19841 L s .7381 .19841 m .77778 .39683 L .81746 .59524 L .85714 .79365 L .89683 .99206 L .93651 1.19048 L .97619 1.38889 L s P P p p .004 w s s s s s s s s s s s s s s s s s s .70635 0 m .7381 .16508 L s .7381 .16508 m .77778 .37143 L .81746 .57778 L .85714 .78413 L .89683 .99048 L .93651 1.19683 L .97619 1.40317 L s P P p p .004 w s s s s s s s s s s s s s s s s s s .71429 0 m .7381 .12857 L s .7381 .12857 m .77778 .34286 L .81746 .55714 L .85714 .77143 L .89683 .98571 L .93651 1.2 L .97619 1.41429 L s P P p p .004 w s s s s s s s s s s s s s s s s s s .72222 0 m .7381 .08889 L s .7381 .08889 m .77778 .31111 L .81746 .53333 L .85714 .75556 L .89683 .97778 L .93651 1.2 L .97619 1.42222 L s P P p p .004 w s s s s s s s s s s s s s s s s s s .73016 0 m .7381 .04603 L s .7381 .04603 m .77778 .27619 L .81746 .50635 L .85714 .73651 L .89683 .96667 L .93651 1.19683 L .97619 1.42698 L s P P p p .004 w s s s s s s s s s s s s s s s s s s s .7381 0 m .77778 .2381 L s .77778 .2381 m .81746 .47619 L .85714 .71429 L .89683 .95238 L .93651 1.19048 L .97619 1.42857 L s P P P P % End of Graphics MathPictureEnd :[font = input; preserveAspect; fontName = "Courier"] Example 4. In addition to tangentfamily, there is a similar defined function showing normals to a curve, called normalfamily: ;[s] 5:0,0;11,1;28,2;41,3;115,4;128,-1; 5:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?normalfamily ;[s] 2:0,0;1,1;14,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = print; inactive; preserveAspect; endGroup] normalfamily[f, {a, b, stepsize}, {c, d}] plots f and its normals on the interval {a, b}, with PlotRange set to {c, d}. One normal is plotted per stepsize in the interval {a, b}; the default stepsize is 0.1. :[font = input; preserveAspect; fontName = "Courier"] We can illustrate this function first on the parbola y = x 2 : ;[s] 8:0,0;1,1;55,2;57,3;60,4;61,5;62,6;63,7;65,-1; 8:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-*-*-*-*-*-120-75-75-*-*-*-*,courier,32,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] normalfamily[f, {-6, 6}, {-6, 6}]; (* f as before *) ;[s] 2:0,0;1,1;56,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 268; endGroup] %! %%Creator: Mathematica %%AspectRatio: .95238 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.079365 0.47619 0.079365 [ [(-6)] .02381 .47619 0 2 Msboxa [(-4)] .18254 .47619 0 2 Msboxa [(-2)] .34127 .47619 0 2 Msboxa [(2)] .65873 .47619 0 2 Msboxa [(4)] .81746 .47619 0 2 Msboxa [(6)] .97619 .47619 0 2 Msboxa [(-6)] .4875 0 1 0 Msboxa [(-4)] .4875 .15873 1 0 Msboxa [(-2)] .4875 .31746 1 0 Msboxa [(2)] .4875 .63492 1 0 Msboxa [(4)] .4875 .79365 1 0 Msboxa [(6)] .4875 .95238 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .95338 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .47619 m 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P p .001 w .5 .06349 m .50375 .06349 L s P p .001 w .5 .09524 m .50375 .09524 L s P p .001 w .5 .12698 m .50375 .12698 L s P p .001 w .5 .19048 m .50375 .19048 L s P p .001 w .5 .22222 m .50375 .22222 L s P p .001 w .5 .25397 m .50375 .25397 L s P p .001 w .5 .28571 m .50375 .28571 L s P p .001 w .5 .34921 m .50375 .34921 L s P p .001 w .5 .38095 m .50375 .38095 L s P p .001 w .5 .4127 m .50375 .4127 L s P p .001 w .5 .44444 m .50375 .44444 L s P p .001 w .5 .50794 m .50375 .50794 L s P p .001 w .5 .53968 m .50375 .53968 L s P p .001 w .5 .57143 m .50375 .57143 L s P p .001 w .5 .60317 m .50375 .60317 L s P p .001 w .5 .66667 m .50375 .66667 L s P p .001 w .5 .69841 m .50375 .69841 L s P p .001 w .5 .73016 m .50375 .73016 L s P p .001 w .5 .7619 m .50375 .7619 L s P p .001 w .5 .8254 m .50375 .8254 L s P p .001 w .5 .85714 m .50375 .85714 L s P p .001 w .5 .88889 m .50375 .88889 L s P p .001 w .5 .92063 m .50375 .92063 L s P p .002 w .5 0 m .5 .95238 L s P P 0 0 m 1 0 L 1 .95238 L 0 .95238 L closepath clip newpath p p p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w .02381 .91667 m .06349 .9246 L .10317 .93254 L .14286 .94048 L .18254 .94841 L s .20238 .95238 m .18254 .94841 L s s s s s s s s s s s s s s s s s s s s s P P p p .004 w .02381 .87381 m .06349 .88208 L .10317 .89034 L .14286 .89861 L .18254 .90688 L .22222 .91515 L .2619 .92341 L .30159 .93168 L .34127 .93995 L .38095 .94821 L s .40095 .95238 m .38095 .94821 L s s s s s s s s s s s s s s s s P P p p .004 w .02381 .83219 m .06349 .84082 L .10317 .84945 L .14286 .85807 L .18254 .8667 L .22222 .87533 L .2619 .88395 L .30159 .89258 L .34127 .90121 L .38095 .90983 L .42063 .91846 L .46032 .92709 L .5 .93571 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.42063 .71746 L .46032 .61825 L .5 .51905 L .53968 .41984 L .57937 .32063 L .61905 .22143 L .65873 .12222 L .69841 .02302 L s .70762 0 m .69841 .02302 L s s s s s s s s P P p p .004 w s s s s s s .24238 .95238 m .2619 .91984 L s .2619 .91984 m .30159 .8537 L .34127 .78757 L .38095 .72143 L .42063 .65529 L .46032 .58915 L .5 .52302 L .53968 .45688 L .57937 .39074 L .61905 .3246 L .65873 .25847 L .69841 .19233 L .7381 .12619 L .77778 .06005 L s .81381 0 m .77778 .06005 L s s s s s s P P p p .004 w s s s s .16095 .95238 m .18254 .9254 L s .18254 .9254 m .22222 .87579 L .2619 .82619 L .30159 .77659 L .34127 .72698 L .38095 .67738 L .42063 .62778 L .46032 .57817 L .5 .52857 L .53968 .47897 L .57937 .42937 L .61905 .37976 L .65873 .33016 L .69841 .28056 L .7381 .23095 L .77778 .18135 L .81746 .13175 L .85714 .08214 L .89683 .03254 L s .92286 0 m .89683 .03254 L s s s P P p p .004 w s s .08333 .95238 m .10317 .93254 L s .10317 .93254 m .14286 .89286 L .18254 .85317 L .22222 .81349 L .2619 .77381 L .30159 .73413 L .34127 .69444 L .38095 .65476 L .42063 .61508 L .46032 .5754 L .5 .53571 L .53968 .49603 L .57937 .45635 L .61905 .41667 L .65873 .37698 L .69841 .3373 L .7381 .29762 L .77778 .25794 L .81746 .21825 L .85714 .17857 L .89683 .13889 L .93651 .09921 L .97619 .05952 L s P P p p .004 w .02381 .94127 m .06349 .9082 L .10317 .87513 L .14286 .84206 L .18254 .80899 L .22222 .77593 L .2619 .74286 L .30159 .70979 L .34127 .67672 L .38095 .64365 L .42063 .61058 L .46032 .57751 L .5 .54444 L .53968 .51138 L .57937 .47831 L .61905 .44524 L .65873 .41217 L .69841 .3791 L .7381 .34603 L .77778 .31296 L .81746 .27989 L .85714 .24683 L .89683 .21376 L .93651 .18069 L .97619 .14762 L s P P p p .004 w .02381 .8949 m .06349 .86655 L .10317 .83821 L .14286 .80986 L .18254 .78152 L .22222 .75317 L .2619 .72483 L .30159 .69649 L .34127 .66814 L .38095 .6398 L .42063 .61145 L .46032 .58311 L .5 .55476 L .53968 .52642 L .57937 .49807 L .61905 .46973 L .65873 .44138 L .69841 .41304 L .7381 .38469 L .77778 .35635 L .81746 .328 L .85714 .29966 L .89683 .27132 L .93651 .24297 L .97619 .21463 L s P P p p .004 w .02381 .86429 m .06349 .83948 L .10317 .81468 L .14286 .78988 L .18254 .76508 L .22222 .74028 L .2619 .71548 L .30159 .69067 L .34127 .66587 L .38095 .64107 L .42063 .61627 L .46032 .59147 L .5 .56667 L .53968 .54187 L .57937 .51706 L .61905 .49226 L .65873 .46746 L .69841 .44266 L .7381 .41786 L .77778 .39306 L .81746 .36825 L .85714 .34345 L .89683 .31865 L .93651 .29385 L .97619 .26905 L s P P p p .004 w .02381 .84471 m .06349 .82266 L .10317 .80062 L .14286 .77857 L .18254 .75653 L .22222 .73448 L .2619 .71243 L .30159 .69039 L .34127 .66834 L .38095 .6463 L .42063 .62425 L .46032 .6022 L .5 .58016 L .53968 .55811 L .57937 .53607 L .61905 .51402 L .65873 .49198 L .69841 .46993 L .7381 .44788 L .77778 .42584 L .81746 .40379 L .85714 .38175 L .89683 .3597 L .93651 .33765 L .97619 .31561 L s P P p p .004 w .02381 .83333 m .06349 .81349 L .10317 .79365 L .14286 .77381 L .18254 .75397 L .22222 .73413 L .2619 .71429 L .30159 .69444 L .34127 .6746 L .38095 .65476 L .42063 .63492 L .46032 .61508 L .5 .59524 L .53968 .5754 L .57937 .55556 L .61905 .53571 L .65873 .51587 L .69841 .49603 L .7381 .47619 L .77778 .45635 L .81746 .43651 L .85714 .41667 L .89683 .39683 L .93651 .37698 L .97619 .35714 L s P P p p .004 w .02381 .82835 m .06349 .81032 L .10317 .79228 L .14286 .77424 L .18254 .7562 L .22222 .73817 L .2619 .72013 L .30159 .70209 L .34127 .68405 L .38095 .66602 L .42063 .64798 L .46032 .62994 L .5 .6119 L .53968 .59387 L .57937 .57583 L .61905 .55779 L .65873 .53975 L .69841 .52172 L .7381 .50368 L .77778 .48564 L .81746 .4676 L .85714 .44957 L .89683 .43153 L .93651 .41349 L .97619 .39545 L s P P p p .004 w .02381 .82857 m .06349 .81204 L .10317 .7955 L .14286 .77897 L .18254 .76243 L .22222 .7459 L .2619 .72937 L .30159 .71283 L .34127 .6963 L .38095 .67976 L .42063 .66323 L .46032 .64669 L .5 .63016 L .53968 .61362 L .57937 .59709 L .61905 .58056 L .65873 .56402 L .69841 .54749 L .7381 .53095 L .77778 .51442 L .81746 .49788 L .85714 .48135 L .89683 .46481 L .93651 .44828 L .97619 .43175 L s P P p p .004 w .02381 .83315 m .06349 .81789 L .10317 .80263 L .14286 .78736 L .18254 .7721 L .22222 .75684 L .2619 .74158 L .30159 .72631 L .34127 .71105 L .38095 .69579 L .42063 .68053 L .46032 .66526 L .5 .65 L .53968 .63474 L .57937 .61947 L .61905 .60421 L .65873 .58895 L .69841 .57369 L .7381 .55842 L .77778 .54316 L .81746 .5279 L .85714 .51264 L .89683 .49737 L .93651 .48211 L .97619 .46685 L s P P p p .004 w .02381 .8415 m .06349 .82732 L .10317 .81315 L .14286 .79898 L .18254 .78481 L .22222 .77063 L .2619 .75646 L .30159 .74229 L .34127 .72812 L .38095 .71395 L .42063 .69977 L .46032 .6856 L .5 .67143 L .53968 .65726 L .57937 .64308 L .61905 .62891 L .65873 .61474 L .69841 .60057 L .7381 .58639 L .77778 .57222 L .81746 .55805 L .85714 .54388 L .89683 .52971 L .93651 .51553 L .97619 .50136 L s P P p p .004 w .02381 .85317 m .06349 .83995 L .10317 .82672 L .14286 .81349 L .18254 .80026 L .22222 .78704 L .2619 .77381 L .30159 .76058 L .34127 .74735 L .38095 .73413 L .42063 .7209 L .46032 .70767 L .5 .69444 L .53968 .68122 L .57937 .66799 L .61905 .65476 L .65873 .64153 L .69841 .62831 L .7381 .61508 L .77778 .60185 L .81746 .58862 L .85714 .5754 L .89683 .56217 L .93651 .54894 L .97619 .53571 L s P P p p .004 w .02381 .86786 m .06349 .85546 L .10317 .84306 L .14286 .83065 L .18254 .81825 L .22222 .80585 L .2619 .79345 L .30159 .78105 L .34127 .76865 L .38095 .75625 L .42063 .74385 L .46032 .73145 L .5 .71905 L .53968 .70665 L .57937 .69425 L .61905 .68185 L .65873 .66944 L .69841 .65704 L .7381 .64464 L .77778 .63224 L .81746 .61984 L .85714 .60744 L .89683 .59504 L .93651 .58264 L .97619 .57024 L s P P p p .004 w .02381 .88529 m .06349 .87362 L .10317 .86195 L .14286 .85028 L .18254 .83861 L .22222 .82694 L .2619 .81527 L .30159 .80359 L .34127 .79192 L .38095 .78025 L .42063 .76858 L .46032 .75691 L .5 .74524 L .53968 .73357 L .57937 .7219 L .61905 .71022 L .65873 .69855 L .69841 .68688 L .7381 .67521 L .77778 .66354 L .81746 .65187 L .85714 .6402 L .89683 .62852 L .93651 .61685 L .97619 .60518 L s P P p p .004 w .02381 .90529 m .06349 .89427 L .10317 .88325 L .14286 .87222 L .18254 .8612 L .22222 .85018 L .2619 .83915 L .30159 .82813 L .34127 .81711 L .38095 .80608 L .42063 .79506 L .46032 .78404 L .5 .77302 L .53968 .76199 L .57937 .75097 L .61905 .73995 L .65873 .72892 L .69841 .7179 L .7381 .70688 L .77778 .69586 L .81746 .68483 L .85714 .67381 L .89683 .66279 L .93651 .65176 L .97619 .64074 L s P P p p .004 w .02381 .92769 m .06349 .91725 L .10317 .90681 L .14286 .89637 L .18254 .88592 L .22222 .87548 L .2619 .86504 L .30159 .85459 L .34127 .84415 L .38095 .83371 L .42063 .82327 L .46032 .81282 L .5 .80238 L .53968 .79194 L .57937 .7815 L .61905 .77105 L .65873 .76061 L .69841 .75017 L .7381 .73972 L .77778 .72928 L .81746 .71884 L .85714 .7084 L .89683 .69795 L .93651 .68751 L .97619 .67707 L s P P p p .004 w .02381 .95238 m .06349 .94246 L .10317 .93254 L .14286 .92262 L .18254 .9127 L .22222 .90278 L .2619 .89286 L .30159 .88294 L .34127 .87302 L .38095 .8631 L .42063 .85317 L .46032 .84325 L .5 .83333 L .53968 .82341 L .57937 .81349 L .61905 .80357 L .65873 .79365 L .69841 .78373 L .7381 .77381 L .77778 .76389 L .81746 .75397 L .85714 .74405 L .89683 .73413 L .93651 .72421 L .97619 .71429 L s P P p p .004 w s s s .13667 .95238 m .14286 .95091 L s .14286 .95091 m .18254 .94146 L .22222 .93201 L .2619 .92256 L .30159 .91311 L .34127 .90367 L .38095 .89422 L .42063 .88477 L .46032 .87532 L .5 .86587 L .53968 .85642 L .57937 .84698 L .61905 .83753 L .65873 .82808 L .69841 .81863 L .7381 .80918 L .77778 .79974 L .81746 .79029 L .85714 .78084 L .89683 .77139 L .93651 .76194 L .97619 .75249 L s P P p p .004 w s s s s s s s .26952 .95238 m .30159 .94509 L s .30159 .94509 m .34127 .93608 L .38095 .92706 L .42063 .91804 L .46032 .90902 L .5 .9 L .53968 .89098 L .57937 .88196 L .61905 .87294 L .65873 .86392 L .69841 .85491 L .7381 .84589 L .77778 .83687 L .81746 .82785 L .85714 .81883 L .89683 .80981 L .93651 .80079 L .97619 .79177 L s P P p p .004 w s s s s s s s s s s s .42333 .95238 m .46032 .94434 L s .46032 .94434 m .5 .93571 L .53968 .92709 L .57937 .91846 L .61905 .90983 L .65873 .90121 L .69841 .89258 L .7381 .88395 L .77778 .87533 L .81746 .8667 L .85714 .85807 L .89683 .84945 L .93651 .84082 L .97619 .83219 L s P P p p .004 w s s s s s s s s s s s s s s s .59905 .95238 m .61905 .94821 L s .61905 .94821 m .65873 .93995 L .69841 .93168 L .7381 .92341 L .77778 .91515 L .81746 .90688 L .85714 .89861 L .89683 .89034 L .93651 .88208 L .97619 .87381 L s P P p p .004 w s s s s s s s s s s s s s s s s s s s s .79762 .95238 m .81746 .94841 L s .81746 .94841 m .85714 .94048 L .89683 .93254 L .93651 .9246 L .97619 .91667 L s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s P P P p p .004 w s s s s s s s s .306 .95238 m .34127 .79365 L s .34127 .79365 m .38095 .65476 L .40079 .6002 L .42063 .55556 L .44048 .52083 L .4504 .50719 L .46032 .49603 L .47024 .48735 L .4752 .48394 L .48016 .48115 L .48512 .47898 L .4876 .47813 L .49008 .47743 L .49256 .47689 L .4938 .47667 L .49504 .4765 L .49628 .47636 L .49752 .47627 L .49876 .47621 L .5 .47619 L .50124 .47621 L .50248 .47627 L .50372 .47636 L .50496 .4765 L .50744 .47689 L .50992 .47743 L .5124 .47813 L .51488 .47898 L .51984 .48115 L .5248 .48394 L .52976 .48735 L .53968 .49603 L .5496 .50719 L .55952 .52083 L .57937 .55556 L .59921 .6002 L .61905 .65476 L .65873 .79365 L s .694 .95238 m .65873 .79365 L s s s s s s s s s P P P % End of Graphics MathPictureEnd :[font = input; preserveAspect; fontName = "Courier"] Note the interesting pattern the normals create: they highlight a curve which is the envelope of all the normals. Let us plot the normals to y = sin x to see if an interesting pattern emerges again: ;[s] 6:0,0;1,1;144,2;145,3;154,4;155,5;205,-1; 6:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] normalfamily[Sin, {-6, 6}, {-6, 6}]; ;[s] 2:0,0;1,1;37,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 268; endGroup] %! %%Creator: Mathematica %%AspectRatio: .95238 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.079365 0.47619 0.079365 [ [(-6)] .02381 .47619 0 2 Msboxa [(-4)] .18254 .47619 0 2 Msboxa [(-2)] .34127 .47619 0 2 Msboxa [(2)] .65873 .47619 0 2 Msboxa [(4)] .81746 .47619 0 2 Msboxa [(6)] .97619 .47619 0 2 Msboxa [(-6)] .4875 0 1 0 Msboxa [(-4)] .4875 .15873 1 0 Msboxa [(-2)] .4875 .31746 1 0 Msboxa [(2)] .4875 .63492 1 0 Msboxa [(4)] .4875 .79365 1 0 Msboxa [(6)] .4875 .95238 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .95338 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .47619 m .02381 .48244 L s P [(-6)] .02381 .47619 0 2 Mshowa p .002 w .18254 .47619 m .18254 .48244 L s P [(-4)] .18254 .47619 0 2 Mshowa p .002 w .34127 .47619 m .34127 .48244 L s P [(-2)] .34127 .47619 0 2 Mshowa p .002 w .65873 .47619 m .65873 .48244 L s P [(2)] .65873 .47619 0 2 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L .34127 .51699 L .38095 .55911 L .42063 .60122 L .46032 .64334 L .5 .68545 L .53968 .72757 L .57937 .76968 L .61905 .8118 L .65873 .85392 L .69841 .89603 L .7381 .93815 L s .75151 .95238 m .7381 .93815 L s s s s s s s P P p p .004 w .02381 .15258 m .06349 .19647 L .10317 .24036 L .14286 .28426 L .18254 .32815 L .22222 .37204 L .2619 .41594 L .30159 .45983 L .34127 .50372 L .38095 .54761 L .42063 .59151 L .46032 .6354 L .5 .67929 L .53968 .72319 L .57937 .76708 L .61905 .81097 L .65873 .85487 L .69841 .89876 L .7381 .94265 L s .74689 .95238 m .7381 .94265 L s s s s s s s P P p p .004 w .02381 .12037 m .06349 .16668 L .10317 .21299 L .14286 .2593 L .18254 .30561 L .22222 .35192 L .2619 .39823 L .30159 .44454 L .34127 .49085 L .38095 .53716 L .42063 .58347 L .46032 .62978 L .5 .67609 L .53968 .7224 L .57937 .76871 L .61905 .81502 L .65873 .86133 L .69841 .90764 L s .73675 .95238 m .69841 .90764 L s s s s s s s s P P p p .004 w .02381 .08197 m .06349 .1315 L .10317 .18103 L .14286 .23056 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.03442 L s .10317 .03442 m .14286 .10185 L .18254 .16928 L .22222 .23671 L .2619 .30414 L .30159 .37157 L .34127 .439 L .38095 .50643 L .42063 .57386 L .46032 .64129 L .5 .70872 L .53968 .77615 L .57937 .84358 L .61905 .911 L s .6434 .95238 m .61905 .911 L s s s s s s s s s s P P p p .004 w s s s .12752 0 m .14286 .03039 L s .14286 .03039 m .18254 .10899 L .22222 .18759 L .2619 .2662 L .30159 .3448 L .34127 .4234 L .38095 .50201 L .42063 .58061 L .46032 .65921 L .5 .73782 L .53968 .81642 L .57937 .89502 L s .60832 .95238 m .57937 .89502 L s s s s s s s s s s s P P p p .004 w s s s s .17314 0 m .18254 .0226 L s .18254 .0226 m .22222 .11795 L .2619 .21331 L .30159 .30867 L .34127 .40402 L .38095 .49938 L .42063 .59474 L .46032 .6901 L .5 .78545 L .53968 .88081 L s .56947 .95238 m .53968 .88081 L s s s s s s s s s s s s P P p p .004 w s s s s s .21954 0 m .22222 .0083 L s .22222 .0083 m .2619 .13105 L .30159 .25379 L .34127 .37654 L .38095 .49928 L .42063 .62203 L .46032 .74478 L .5 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.30159 .75575 L .34127 .6074 L .38095 .45906 L .42063 .31071 L .46032 .16236 L .5 .01402 L s .50375 0 m .5 .01402 L s s s s s s s s s s s s s P P p p .004 w s s s s s .20541 .95238 m .22222 .90597 L s .22222 .90597 m .2619 .79646 L .30159 .68695 L .34127 .57744 L .38095 .46793 L .42063 .35841 L .46032 .2489 L .5 .13939 L .53968 .02988 L s .55051 0 m .53968 .02988 L s s s s s s s s s s s s P P p p .004 w s s s s .16462 .95238 m .18254 .91287 L s .18254 .91287 m .22222 .82538 L .2619 .7379 L .30159 .65042 L .34127 .56293 L .38095 .47545 L .42063 .38796 L .46032 .30048 L .5 .21299 L .53968 .12551 L .57937 .03803 L s .59661 0 m .57937 .03803 L s s s s s s s s s s s P P p p .004 w s s s .12726 .95238 m .14286 .92352 L s .14286 .92352 m .18254 .85008 L .22222 .77663 L .2619 .70319 L .30159 .62974 L .34127 .5563 L .38095 .48285 L .42063 .40941 L .46032 .33596 L .5 .26252 L .53968 .18907 L .57937 .11563 L .61905 .04218 L s .64184 0 m .61905 .04218 L s s s s s s s s s s P P p p .004 w s s 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.64385 .55325 L .64881 .55191 L .65873 .54836 L .66865 .54368 L .67857 .53794 L Mistroke .69841 .52369 L .7381 .48739 L .77778 .44835 L .79762 .43083 L .81746 .41613 L .82738 .41013 L .8373 .40516 L .84226 .40309 L .84722 .4013 L .85218 .39981 L .85714 .39861 L .85962 .39812 L .8621 .39772 L .86458 .39738 L .86582 .39725 L .86706 .39713 L .8683 .39703 L .86954 .39695 L .87078 .39689 L .87202 .39685 L .87326 .39683 L .8745 .39683 L .87574 .39684 L .87698 .39688 L .87822 .39694 L .87946 .39701 L .88194 .39722 L .88442 .39751 L .8869 .39787 L .89187 .39883 L .89683 .40009 L .90675 .40349 L .91667 .40802 L .93651 .4202 L .97619 .45401 L Mfstroke P P P % End of Graphics MathPictureEnd :[font = input; preserveAspect; fontName = "Courier"] We shall consider further the envelope of all the normals to a curve in Section 5.2. Example 5. Minimum distance in the plane is measured perpendicularly. Thus we can use normals to the find the closest pair of points, one from each curve, on two different curves. Consider the two parabolas f and g with equations f(x) = x 2 + 2 x + 2 and g(x) = - x 2 + 6 x - 4. ;[s] 27:0,0;1,1;87,2;97,3;298,4;301,5;307,6;310,7;349,8;355,9;358,10;360,11;361,12;368,13;369,14;390,15;396,16;399,17;400,18;401,19;402,20;403,21;404,22;411,23;412,24;414,25;415,26;419,-1; 27:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-*-*-*-*-*-120-75-75-*-*-*-*,courier,32,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-symbol-*-*-*-*-*-120-75-75-*-*-*-*,symbol,0,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-*-*-*-*-*-120-75-75-*-*-*-*,courier,32,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-symbol-*-*-*-*-*-120-75-75-*-*-*-*,symbol,0,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; preserveAspect; fontName = "Courier"] Clear[f]; f[x_] := x^2 + 2 x + 2; g[x_] := -x^2 + 6 x - 4 ;[s] 2:0,0;1,1;58,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Plot[{f[x], g[x]}, {x, -2, 4}, AspectRatio -> Automatic]; ;[s] 2:0,0;1,1;58,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; 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m .34502 2.60281 L s P p .002 w .34127 0 m .34127 2.66593 L s P P 0 0 m 1 0 L 1 2.66593 L 0 2.66593 L closepath clip newpath p p p .004 w .02381 1.17424 m .06349 1.10479 L .08333 1.07751 L .10317 1.05519 L .12302 1.03783 L .13294 1.03101 L .14286 1.02543 L .15278 1.02109 L .15774 1.01938 L .1627 1.01799 L .16766 1.0169 L .17014 1.01648 L .17262 1.01613 L .1751 1.01586 L .17634 1.01575 L .17758 1.01566 L .17882 1.01559 L .18006 1.01555 L .1813 1.01552 L .18254 1.01551 L .18378 1.01552 L .18502 1.01555 L .18626 1.01559 L .1875 1.01566 L .18998 1.01586 L .19246 1.01613 L .19494 1.01648 L .19742 1.0169 L .20238 1.01799 L .20734 1.01938 L .2123 1.02109 L .22222 1.02543 L .23214 1.03101 L .24206 1.03783 L .2619 1.05519 L .28175 1.07751 L .30159 1.10479 L .34127 1.17424 L .38095 1.26352 L .42063 1.37265 L .46032 1.50162 L .5 1.65043 L .53968 1.81908 L .57937 2.00757 L .61905 2.2159 L .65873 2.44408 L s .69423 2.66593 m .65873 2.44408 L s s s s s s s s s P P p p .004 w s s s s s s s s .30577 0 m .34127 .22186 L s .34127 .22186 m .38095 .45003 L .42063 .65836 L .46032 .84686 L .5 1.01551 L .53968 1.16432 L .57937 1.29328 L .61905 1.40241 L .65873 1.4917 L .69841 1.56114 L .71825 1.58842 L .7381 1.61074 L .75794 1.62811 L .76786 1.63493 L .77778 1.64051 L .7877 1.64485 L .79266 1.64655 L .79762 1.64795 L .80258 1.64903 L .80506 1.64946 L .80754 1.64981 L .81002 1.65008 L .81126 1.65019 L .8125 1.65027 L .81374 1.65034 L .81498 1.65039 L .81622 1.65042 L .81746 1.65043 L .8187 1.65042 L .81994 1.65039 L .82118 1.65034 L .82242 1.65027 L .8249 1.65008 L .82738 1.64981 L .82986 1.64946 L .83234 1.64903 L .8373 1.64795 L .84226 1.64655 L .84722 1.64485 L .85714 1.64051 L .86706 1.63493 L .87698 1.62811 L .89683 1.61074 L .91667 1.58842 L .93651 1.56114 L .97619 1.4917 L s P P P % End of Graphics MathPictureEnd :[font = input; preserveAspect; fontName = "Courier"] Suppose the points (a, f(a)) and (b, g(b)) are the two points closest together. The first point must be on the normal to g at x = b; the second point must be on the normal to f at x = a. This gives us a system of two equations to solve for a and b. Try solving for the coincidence of the normals to the two curves. ;[s] 22:0,0;1,1;22,2;29,3;37,4;44,5;126,6;127,7;133,8;134,9;139,10;140,11;186,12;188,13;193,14;194,15;199,16;200,17;257,18;258,19;265,20;266,21;334,-1; 22:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; preserveAspect; fontName = "Courier"; endGroup] :[font = subsection; inactive; Cclosed; preserveAspect; fontName = "Times"; startGroup] Applications of the Derivative :[font = text; inactive; preserveAspect] As one example of using animations to illustrate a typical calculus problem, we consider the following question: A wire of length 100 cm is to be cut into two pieces. One piece is bent into a circle, the other into a square. How should the cut be made to maximize the sum of the areas of the square and the circle? How should it be done to minimize that sum? If the cut is made at x and a(x) is the total area of the enclosed by the two pieces as described, then: :[font = input; preserveAspect; fontName = "Courier"] a[x_] := x^2/(4 Pi) + (100 - x)^2/16 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Plot[a[x], {x, 0, 100}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.009524 -0.447573 0.001321 [ [(20)] .21429 .08066 0 2 Msboxa [(40)] .40476 .08066 0 2 Msboxa [(60)] .59524 .08066 0 2 Msboxa [(80)] .78571 .08066 0 2 Msboxa [(100)] .97619 .08066 0 2 Msboxa [(500)] .01131 .21272 1 0 Msboxa [(600)] .01131 .34478 1 0 Msboxa [(700)] .01131 .47684 1 0 Msboxa [(800)] .01131 .6089 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .08066 m .21429 .08691 L s P [(20)] .21429 .08066 0 2 Mshowa p .002 w .40476 .08066 m .40476 .08691 L s P [(40)] .40476 .08066 0 2 Mshowa p .002 w .59524 .08066 m .59524 .08691 L s P [(60)] .59524 .08066 0 2 Mshowa p .002 w .78571 .08066 m .78571 .08691 L s P [(80)] .78571 .08066 0 2 Mshowa p .002 w .97619 .08066 m .97619 .08691 L s P [(100)] .97619 .08066 0 2 Mshowa p .001 w .0619 .08066 m .0619 .08441 L s P p .001 w .1 .08066 m .1 .08441 L s P p .001 w .1381 .08066 m .1381 .08441 L s P p .001 w .17619 .08066 m .17619 .08441 L s P p .001 w .25238 .08066 m .25238 .08441 L s P p .001 w .29048 .08066 m .29048 .08441 L s P p .001 w .32857 .08066 m .32857 .08441 L s P p .001 w .36667 .08066 m .36667 .08441 L s P p .001 w .44286 .08066 m .44286 .08441 L s P p .001 w .48095 .08066 m .48095 .08441 L s P p .001 w .51905 .08066 m .51905 .08441 L s P p .001 w .55714 .08066 m .55714 .08441 L s P p .001 w .63333 .08066 m .63333 .08441 L s P p .001 w .67143 .08066 m .67143 .08441 L s P p .001 w .70952 .08066 m .70952 .08441 L s P p .001 w .74762 .08066 m .74762 .08441 L s P p .001 w .82381 .08066 m .82381 .08441 L s P p .001 w .8619 .08066 m .8619 .08441 L s P p .001 w .9 .08066 m .9 .08441 L s P p .001 w .9381 .08066 m .9381 .08441 L s P p .002 w 0 .08066 m 1 .08066 L s P p .002 w .02381 .21272 m .03006 .21272 L s P [(500)] .01131 .21272 1 0 Mshowa p .002 w .02381 .34478 m .03006 .34478 L s P [(600)] .01131 .34478 1 0 Mshowa p .002 w .02381 .47684 m .03006 .47684 L s P [(700)] .01131 .47684 1 0 Mshowa p .002 w .02381 .6089 m .03006 .6089 L s P [(800)] .01131 .6089 1 0 Mshowa p .001 w .02381 .10707 m .02756 .10707 L s P p .001 w .02381 .13349 m .02756 .13349 L s P p .001 w .02381 .1599 m .02756 .1599 L s P p .001 w .02381 .18631 m .02756 .18631 L s P p .001 w .02381 .23913 m .02756 .23913 L s P p .001 w .02381 .26555 m .02756 .26555 L s P p .001 w .02381 .29196 m .02756 .29196 L s P p .001 w .02381 .31837 m .02756 .31837 L s P p .001 w .02381 .37119 m .02756 .37119 L s P p .001 w .02381 .3976 m .02756 .3976 L s P p .001 w .02381 .42402 m .02756 .42402 L s P p .001 w .02381 .45043 m .02756 .45043 L s P p .001 w .02381 .50325 m .02756 .50325 L s P p .001 w .02381 .52966 m .02756 .52966 L s P p .001 w .02381 .55608 m .02756 .55608 L s P p .001 w .02381 .58249 m .02756 .58249 L s P p .001 w .02381 .05425 m .02756 .05425 L s P p .001 w .02381 .02784 m .02756 .02784 L s P p .001 w .02381 .00143 m .02756 .00143 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .3778 m .06349 .31227 L .10317 .25326 L .14286 .20077 L .18254 .15479 L .22222 .11533 L .2619 .08238 L .30159 .05594 L .32143 .04517 L .34127 .03602 L .36111 .02851 L .38095 .02262 L .39087 .02028 L .40079 .01836 L .41071 .01684 L .41567 .01623 L .42063 .01573 L .4256 .01532 L .42808 .01516 L .43056 .01502 L .43304 .01491 L .43552 .01482 L .43676 .01479 L .438 .01476 L .43924 .01474 L .44048 .01473 L .44172 .01472 L .44296 .01472 L .4442 .01472 L .44544 .01473 L .44668 .01475 L .44792 .01477 L .44916 .0148 L .4504 .01484 L .45536 .01504 L .45784 .01519 L .46032 .01535 L .47024 .01628 L .4752 .01689 L .48016 .01761 L .5 .02149 L .51984 .027 L .53968 .03415 L .57937 .05332 L .61905 .079 L .65873 .1112 L .69841 .14991 L .7381 .19514 L .77778 .24688 L .81746 .30514 L Mistroke .85714 .36991 L .89683 .4412 L .93651 .519 L .97619 .60332 L Mfstroke P P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] You can see that the maximum total area is at the right end point x =100 (corresponding to bending the whole wire into a circle), and the minimum total area is at the critical point. We can find the critical point by setting a'(x) = 0: :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Solve[a'[x] == 0, x] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> (100*Pi)/(4 + Pi)}} ;[o] 100 Pi {{x -> ------}} 4 + Pi :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] % // N :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 43.99008464884427}} ;[o] {{x -> 43.9901}} :[font = text; inactive; preserveAspect] To illustrate this problem more thoroughly we have defined a function called wire : :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] wire[30]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 225; endGroup] %! %%Creator: Mathematica %%AspectRatio: .8 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -1.040834e-17 0.01 1.387779e-17 0.01 [ [(377.87 = combined area of circle and square)] .5 .8 0 -2 Msboxa [ 0 0 0 0 ] [ 1 .8 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p [(377.87 = combined area of circle and square)] .5 .8 0 -2 Mshowa P 0 0 m 1 0 L 1 .8 L 0 .8 L closepath clip newpath p p p .004 w .29775 .25 m .29774 .25039 L .29774 .25078 L .29773 .25117 L .29772 .25156 L .29771 .25195 L .29769 .25234 L .29764 .25312 L .29759 .2539 L .29752 .25468 L .29734 .25623 L .29711 .25778 L .29683 .25931 L .29612 .26236 L .29521 .26535 L .29411 .26827 L .29135 .27387 L .28788 .27907 L .28376 .28376 L .27907 .28788 L .27387 .29135 L .27112 .29282 L .26827 .29411 L .26535 .29521 L .26236 .29612 L .25931 .29683 L .25778 .29711 L .25623 .29734 L .25468 .29752 L .2539 .29759 L .25312 .29764 L .25234 .29769 L .25195 .29771 L .25156 .29772 L .25117 .29773 L .25078 .29774 L .25039 .29774 L .25 .29775 L .24961 .29774 L .24922 .29774 L .24883 .29773 L .24844 .29772 L .24766 .29769 L .24688 .29764 L .2461 .29759 L .24532 .29752 L .24377 .29734 L .24222 .29711 L .24069 .29683 L .23764 .29612 L Mistroke .23465 .29521 L .23173 .29411 L .22613 .29135 L .22093 .28788 L .21624 .28376 L .21212 .27907 L .20865 .27387 L .20589 .26827 L .20479 .26535 L .20388 .26236 L .20317 .25931 L .20289 .25778 L .20266 .25623 L .20248 .25468 L .20241 .2539 L .20236 .25312 L .20231 .25234 L .20229 .25195 L .20228 .25156 L .20227 .25117 L .20226 .25078 L .20226 .25039 L .20225 .25 L .20226 .24961 L .20226 .24922 L .20227 .24883 L .20228 .24844 L .20231 .24766 L .20236 .24688 L .20241 .2461 L .20248 .24532 L .20266 .24377 L .20289 .24222 L .20317 .24069 L .20388 .23764 L .20479 .23465 L .20589 .23173 L .20865 .22613 L .21212 .22093 L .21624 .21624 L .22093 .21212 L .22613 .20865 L .22888 .20718 L .23173 .20589 L .23465 .20479 L .23764 .20388 L .24069 .20317 L .24222 .20289 L .24377 .20266 L .24532 .20248 L Mistroke .2461 .20241 L .24688 .20236 L .24766 .20231 L .24805 .20229 L .24844 .20228 L .24883 .20227 L .24922 .20226 L .24961 .20226 L .25 .20225 L .25039 .20226 L .25078 .20226 L .25117 .20227 L .25156 .20228 L .25234 .20231 L .25312 .20236 L .2539 .20241 L .25468 .20248 L .25623 .20266 L .25778 .20289 L .25931 .20317 L .26236 .20388 L .26535 .20479 L .26827 .20589 L .27387 .20865 L .27907 .21212 L .28376 .21624 L .28788 .22093 L .29135 .22613 L .29282 .22888 L .29411 .23173 L .29521 .23465 L .29612 .23764 L .29683 .24069 L .29711 .24222 L .29734 .24377 L .29752 .24532 L .29759 .2461 L .29764 .24688 L .29769 .24766 L .29771 .24805 L .29772 .24844 L .29773 .24883 L .29774 .24922 L .29774 .24961 L .29775 .25 L Mfstroke P P .004 w .6625 .1625 m .6625 .3375 L .8375 .3375 L .8375 .1625 L .6625 .1625 L s p p 0 .7 m .04167 .7 L .08333 .7 L .125 .7 L .16667 .7 L .20833 .7 L .25 .7 L .29167 .7 L .33333 .7 L .375 .7 L .41667 .7 L .45833 .7 L .5 .7 L .54167 .7 L .58333 .7 L .625 .7 L .66667 .7 L .70833 .7 L .75 .7 L .79167 .7 L .83333 .7 L .875 .7 L .91667 .7 L .95833 .7 L 1 .7 L s P P p .008 w .3 .7 Mdot P P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] wire[45]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 225; endGroup] %! 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