Search: a198120 -id:a198120
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A198121
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Decimal expansion of greatest x having 2*x^2-3x=-cos(x).
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3
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1, 4, 6, 3, 3, 6, 2, 8, 2, 7, 2, 9, 6, 4, 3, 1, 1, 4, 5, 1, 0, 5, 2, 9, 6, 4, 2, 6, 1, 6, 1, 3, 5, 8, 7, 0, 6, 9, 1, 8, 2, 7, 7, 3, 2, 5, 2, 2, 4, 4, 1, 4, 1, 2, 6, 9, 7, 2, 5, 8, 6, 5, 5, 2, 8, 2, 5, 0, 0, 0, 9, 8, 5, 6, 6, 1, 6, 1, 2, 6, 5, 6, 7, 7, 4, 7, 4, 2, 9, 8, 4, 9, 2, 8, 9, 7, 3, 8, 4
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OFFSET
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1,2
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COMMENTS
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See A197737 for a guide to related sequences. The Mathematica program includes a graph.
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LINKS
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EXAMPLE
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least x: 0.423418867436956390254901914567137...
greatest x: 1.4633628272964311451052964261613...
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MATHEMATICA
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a = 2; b = -3; c = -1;
f[x_] := a*x^2 + b*x; g[x_] := c*Cos[x]
Plot[{f[x], g[x]}, {x, -1, 2}]
r1 = x /. FindRoot[f[x] == g[x], {x, -.43, -.42}, WorkingPrecision -> 110]
r2 = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
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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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A197737
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Decimal expansion of x<0 having x^2+x=cos(x).
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+0
144
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1, 2, 5, 1, 1, 5, 1, 8, 3, 5, 2, 2, 0, 7, 6, 4, 8, 1, 1, 5, 9, 2, 8, 7, 0, 0, 6, 8, 7, 8, 8, 1, 6, 1, 8, 5, 9, 9, 4, 5, 3, 5, 6, 1, 0, 8, 5, 8, 8, 9, 6, 8, 6, 3, 6, 2, 0, 1, 7, 8, 2, 8, 1, 2, 1, 0, 3, 6, 0, 1, 9, 1, 8, 2, 3, 8, 2, 1, 0, 9, 1, 0, 4, 1, 1, 2, 7, 3, 5, 7, 6, 5, 9, 4, 8, 6, 8, 4, 2
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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For many choices of a,b,c, there are exactly two numbers x having a*x^2+b*x=cos(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... 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 A197737, take f(x,u,v)=x^2+u*x-v*cos(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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negative: -1.25115183522076481159287006878816185994...
positive: 0.55000934992726156666495361947172926116...
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MATHEMATICA
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a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x; g[x_] := c*Cos[x]
Plot[{f[x], g[x]}, {x, -2, 1}]
r1 = x /. FindRoot[f[x] == g[x], {x, -1.26, -1.25}, WorkingPrecision -> 110]
r1 = x /. FindRoot[f[x] == g[x], {x, .55, .551}, WorkingPrecision -> 110]
(* Program 2: implicit surface of x^2+u*x=v*cos(x) *)
f[{x_, u_, v_}] := x^2 + u*x - v*Cos[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 20}, {v, u, 20}];
ListPlot3D[Flatten[t, 1]] (* for A197737 *)
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PROG
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(PARI) A197737_vec(N=150)={localprec(N+10); digits(solve(x=-1.5, -1, x^2+x-cos(x))\.1^N)} \\ M. F. Hasler, Aug 05 2021
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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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