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%I #5 Mar 30 2012 18:57:52
%S 5,0,6,0,6,4,3,3,3,2,1,6,5,2,4,5,1,0,0,5,4,6,3,7,6,2,1,7,7,3,4,7,1,4,
%T 4,1,1,6,9,4,8,7,3,8,8,6,1,8,3,2,2,7,7,3,2,8,6,6,4,0,3,6,7,1,7,8,8,6,
%U 3,1,4,2,1,9,5,5,2,2,8,4,0,9,3,3,8,4,7,3,0,0,8,5,2,6,1,4,6,0,9
%N Decimal expansion of the radius of the smallest circle tangent to the x axis and to the curve y=-cos(4x) at points (x,y), (-x,y).
%C Let (x,y) denote the point of tangency, where x>0:
%C x=0.488618197079923270050681129865078039260837...
%C y=0.374332154777652501331094642913853652491893...
%C slope=3.709178750935618333987343550424591912283...
%C (The Mathematica program includes a graph.)
%e radius=0.5060643332165245100546376217734714411...
%t r = .5; c = 4;
%t Show[Plot[-Cos[c*x], {x, -1, 1}],
%t ContourPlot[x^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1.5, 2}], PlotRange -> All, AspectRatio -> Automatic]
%t u[x_] := -Cos[c*x] + x/(c*Sin[c*x]);
%t t1 = x /. FindRoot[Sqrt[u[x]^2 - x^2] == u[x] + Cos[c*x], {x, .4, .5}, WorkingPrecision -> 100]
%t t = Re[t1] (* x coordinate of tangency point *)
%t y = -Cos[c*t] (* y coordinate of tangency point *)
%t radius = u[t]
%t RealDigits[radius] (* A197029 *)
%t slope = c*Sin[c*t] (* slope at tangency point *)
%Y Cf. A197026, A196027, A196028, A196022.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Oct 09 2011
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