f[t_]={t+t^3,t^5-3t^4+t^2}

curve=ParametricPlot[f[t],{t,-0.75,1.25}, Axes-> None, PlotStyle->{Thick,Black},PlotRange->{{-1,2},{-1,2}}]

dfdt[t_]=D[f[t],t]

{D[f[t],t],D[f[t],{t,2}]}

curv[t_]=Det[{D[f[t],t],D[f[t],{t,2}]}]/(Dot[D[f[t],t],D[f[t],t]])^(3/2)

curv[0]

dfdt[0]

tang=ParametricPlot[f[0]+s*dfdt[0],{s,-1,2}, Axes->None]

norm=ParametricPlot[f[0]+s*{{0,-1},{1,0}}.dfdt[0],{s,-1,2}, Axes->None]

OscCircle=ParametricPlot[{0.5*Cos[s],0.5+0.5*Sin[s]},{s,0,2Pi}, Axes->None, PlotStyle->{Thick,Red}]

Show[curve,tang,norm,OscCircle,Graphics[{PointSize[Large],Thick,Red,Point[{0,0.5}],Line[{{0,.5},{0.5*Cos[-Pi/4],0.5+0.5*Sin[-Pi/4]}}]}]]