Animate the Tangent and Normal to a Function 

Define the function and create a plot.

In[1]:=

X f[t_] := 2 t - 3 t^2 + t^3 p = Plot[f[t], {t, -1, 3}, AspectRatio -> Automatic, PlotRange -> {{-1.5, 3}, {-2, 2}}];

Compute the tangent, normal, and points on the curve.

In[2]:=

X b[t_] = Last[FrenetSerretSystem[f[t], t]]; c[t_] = {t, f[t]};

Create the animation.

In[3]:=

X Animate[Show[p, Graphics[{Locator[c[t]], Thick, Blue, Arrow[{c[t], c[t] + b[t][[1]]}], Red, Arrow[{c[t], c[t] + b[t][[2]]}]}], ImageSize -> Medium], {{t, 0}, -1, 3}]

Play Animation Stop Animation