Problem:
Graphically, find the tangent line to the graph of the function

f(x) = x4

at the point (0.5, 0.0625) as the limit of secant lines.


Visualization:

We have two visualizations generated by Maple. The first is an animation:


[Press here to see animation again!]

This was generated using the following code:
> f:=x->x^4;

> with(plots):
> P := plot(f(x),x=-1.5..1.5,y=-2..4,color=red):
> Q := animate(((f(0.5+2^(-t))-f(0.5))/2^(-t)) * (x-0.5) + f(0.5), x=-1.5..1.5, t=0..10, frames=20, color=blue):
> display(P,Q);

The second shows all of the secant lines at once:

This was generated using the following code:
> f:=x->x^4;

> with(plots):
> P := plot(f(x),x=-1.5..1.5,y=-2..4,color=red):
> A :=seq(plot(((f(0.5+2^(-i)) - f(0.5))/2^(-i)) * (x-0.5) + f(0.5), x=-1.5..1.5, color=blue), i=0..10):
> display(P,A);