Manipulate[
f[t_] := E^(-a*t)*Cos[b*t];
g[t_] := E^(-a*t)*Sin[b*t];
c[t_] := {f[t], g[t]};
Grid[{{Show[
ParametricPlot[c[t], {t, 0, B}, PlotStyle -> {Thick, Blue},
PerformanceGoal -> "Quality"],
ListPlot[{c[B]}, PlotStyle -> {Black, PointSize[0.02]}],
AxesLabel -> {"x", "y"}, PlotRange -> 1.2, ImageSize -> Medium,
AxesOrigin -> {0, 0}]}}],
{B, 0.00001, 4 \[Pi]}, {{a, 1}, 0.1, 10}, {{b, 1}, 0.1, 10}]
