Question

Consider now the function H(t) = (t, t2, t3) (which is sometimes called moment
curve) and write its Taylor series around t = 1. Without trying to sketch the
curve, use the Taylor series to show the following statement: all the points in the
curve corresponding to values of t such that 1 < t < 2 lie inside a parallelepiped
determined by the vectors

v1 =<1,2,3>, v2 = <0,1,3>, v3 =<0,0,1>.

Hint: The parallelepiped that we are thinking of does not have a vertex at the
origin. Think about the value H(1).

H(1) just equals <1,2,3>, so I'm not really sure what the Taylor series of this tells me.

Answer 1

Taylor Series just helps you to expand the function around the point and then see the behaviour of the function.

Now using the taylor series to expand the function around H(1).