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(root = x /. FindRoot[x*Tan[x] - Cos[x] == 0, {x, 3}]) // InputForm 2.8170396080933187 root // Precision MachinePrecision Your machine precision is seen with $MachinePrecision $MachinePrecision 15.9546 root2 = x /. FindRoot[x*Tan[x] - Cos[x] == 0, {x, 3}, WorkingPrecision -> 20] 2.8170396080933199925 root2 // Precision ...


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Simply increase PlotPoints: Plot[Cos[.3 x] Exp[-0.01 x], {x, 0, 1000}, PlotRange -> {{300, 500}, {-0.05, 0.05}}, ImageSize -> 600, PlotPoints -> 2000] Increasing PlotPoints would also draw a smooth ellipse in your orbit example



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