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A200614
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Decimal expansion of the lesser of two values of x satisfying 3*x^2 - 1 = tan(x) and 0 < x < Pi/2.
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61
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8, 3, 9, 5, 8, 2, 2, 5, 9, 0, 4, 5, 3, 0, 2, 9, 4, 1, 5, 1, 3, 7, 6, 4, 0, 0, 8, 8, 6, 3, 8, 0, 4, 9, 8, 6, 3, 0, 8, 4, 1, 6, 5, 4, 1, 0, 2, 6, 9, 4, 4, 0, 9, 0, 0, 3, 3, 4, 9, 1, 4, 3, 4, 0, 6, 7, 6, 5, 8, 4, 1, 4, 6, 1, 0, 4, 1, 1, 6, 7, 4, 2, 5, 9, 5, 3, 5, 3, 0, 0, 2, 3, 6, 6, 2, 4, 6, 0, 5
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OFFSET
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0,1
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COMMENTS
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For many choices of a and c, there are exactly two values of x satisfying a*x^2 - c = tan(x) and 0 < x < Pi/2; for other choices, there is exactly once such value.
Guide to related sequences, with graphs included in Mathematica programs:
a.... c.... x
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0. We call the graph of z=g(u,v) an implicit surface of f.
For an example related to A200614, take f(x,u,v) = u*x^2 - v = tan(x) and g(u,v) = a nonzero solution x of f(x,u,v)=0. If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.
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LINKS
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EXAMPLE
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lesser: 0.839582259045302941513764008863804986308...
greater: 1.350956593976519397744696294368524715373...
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MATHEMATICA
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a = 3; c = 1;
f[x_] := a*x^2 - c; g[x_] := Tan[x]
Plot[{f[x], g[x]}, {x, -.1, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .8, .9}, WorkingPrecision -> 110]
r = x /. FindRoot[f[x] == g[x], {x, 1.3, 1.4}, WorkingPrecision -> 110]
(* Program 2: implicit surface of u*x^2-v=tan(x) *)
f[{x_, u_, v_}] := u*x^2 - v - Tan[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1.55}]}, {u, 1, 20}, {v, -20, 0}];
ListPlot3D[Flatten[t, 1]] (* for A200614 *)
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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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