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A197029
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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).
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1
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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, 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, 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
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OFFSET
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0,1
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COMMENTS
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Let (x,y) denote the point of tangency, where x>0:
x=0.488618197079923270050681129865078039260837...
y=0.374332154777652501331094642913853652491893...
slope=3.709178750935618333987343550424591912283...
(The Mathematica program includes a graph.)
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LINKS
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EXAMPLE
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radius=0.5060643332165245100546376217734714411...
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MATHEMATICA
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r = .5; c = 4;
Show[Plot[-Cos[c*x], {x, -1, 1}],
ContourPlot[x^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1.5, 2}], PlotRange -> All, AspectRatio -> Automatic]
u[x_] := -Cos[c*x] + x/(c*Sin[c*x]);
t1 = x /. FindRoot[Sqrt[u[x]^2 - x^2] == u[x] + Cos[c*x], {x, .4, .5}, WorkingPrecision -> 100]
t = Re[t1] (* x coordinate of tangency point *)
y = -Cos[c*t] (* y coordinate of tangency point *)
radius = u[t]
slope = c*Sin[c*t] (* slope at tangency point *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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