It does not appear that the program is recognizing constants a and b that defaults to 1 - See Attached. If I plug 1 into the formulas for a and b, the plot works fine. I am sure it is something that I am doing, or not doing, but I have read the help text and followed the examples and can't seem to get the Manipulate to work. I would certainly appreciate any help you could give.
f[t_]:= E^(-a * t )*Cos[b*t];
g[t_]:= E^(-a * t)*Sin[b*t];
c[t_]:= {f[t],g[t]};
Manipulate[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}]
Attachments: