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Decimal expansion of the slope (negative) at the point of tangency of the curves y=1/(1+x^2) and y=c*cos(x), where c is given by A196914.
3

%I #9 Mar 06 2021 02:01:50

%S 6,0,7,6,2,2,2,3,7,6,9,6,8,6,8,6,5,8,5,9,0,0,1,0,0,2,6,8,2,0,2,6,3,6,

%T 4,3,2,2,7,4,8,0,9,8,7,7,7,6,5,9,7,7,8,9,9,8,2,6,0,9,5,9,6,0,2,6,2,7,

%U 3,3,6,3,0,4,6,2,8,4,7,5,8,1,4,8,2,6,6,5,4,7,4,8,5,6,0,2,5,6,6

%N Decimal expansion of the slope (negative) at the point of tangency of the curves y=1/(1+x^2) and y=c*cos(x), where c is given by A196914.

%e x=-0.60762223769686865859001002682026364322748...

%t Plot[{1/(1 + x^2), 0.874*Cos[x]}, {x, .5, 1}]

%t t = x /. FindRoot[Tan[x] == 2 x/(1 + x^2), {x, .5, 1}, WorkingPrecision -> 100]

%t RealDigits[t] (* A196913 *)

%t c = N[Sqrt[t^4 + 6 t^2 + 1]/(t^4 + 2 t^2 + 1), 100]

%t RealDigits[c] (* A196914 *)

%t slope = N[-c*Sin[t], 100]

%t RealDigits[slope] (* A196915 *)

%Y Cf. A196914, A196913.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Oct 07 2011