Plot[(2 + 9 t) Exp[-5 t], {t, - .5, 3}, 
      PlotRange -> {- .2, 2.5}]

D[(2 + 9 t) Exp[-5 t], t] /. {t -> 0}

Plot[{2 Cos[5 t] - (1/5) Sin[5 t], (Sqrt[4 + (1/25)]) 
          Cos[5 t + (ArcTan[(1/10)])]}, {t, -0.2, 3}, 
      PlotStyle -> {{Thickness[0.01], Blue}, {Thickness[0.005], 
            Yellow}}, PlotRange -> {-2.5, 2.5}]

Plot[{2  Exp[-t/4], -2 Exp[-t/4], 
        Exp[-t/4] ((Sqrt[3]) Cos[4 t] -  Sin[4 t]), 
        2  Exp[-t/4] Cos[4 t + (ArcTan[(1/(Sqrt[3]))])]}, {t, -0.2, 9}, 
      PlotStyle -> {{Red}, {Red}, {Thickness[0.01], 
            Blue}, {Thickness[0.005], Yellow}}, 
      PlotRange -> {-2.5, 2.5}]

Plot[{((Sqrt[5])/50) Exp[-2 t], -((Sqrt[5])/50) 
          Exp[-2 t], (1/50) 
          Exp[-2 t] (Cos[ t] + 2 Sin[t]), ((Sqrt[5])/50) Exp[-2 t] 
          Cos[t - (ArcTan[2])]}, {t, -0.2, 10}, 
      PlotStyle -> {{Red}, {Red}, {Thickness[0.01], 
            Blue}, {Thickness[0.005], Yellow}}, 
      PlotRange -> {- .0001,  .0001}]

Plot[{((Sqrt[5])/50) Exp[-2 t], -((Sqrt[5])/50) 
          Exp[-2 t], (1/50) 
          Exp[-2 t] (Cos[ t] + 2 Sin[t]), ((Sqrt[5])/50) Exp[-2 t] 
          Cos[t - (ArcTan[2])]}, {t, -0.2, 7}, 
      PlotStyle -> {{Red}, {Red}, {Thickness[0.01], 
            Blue}, {Thickness[0.005], Yellow}}, 
      PlotRange -> {- .001,  .001}]

Plot[{(1/2) Cos[t] - (1/2) Sin[t] + (1/2) 
            Exp[-t]}, {t, -0.2, 20}, 
      PlotStyle -> {{Thickness[0.01], Blue}}, 
      PlotRange -> {-2, 2}]