$4.)$ Consider the following information: for the function f$($x$,$y$)=$x$2$ y$2,$ we have that f$(2,0)=4$ and the directional derivative of f at $(2,0)$ along the vector $(3,2)$ is equal to $12.$ Explain the geometric significance of this value of the directional derivative, as follows: Parametrize a linear curve C$1$ in $2$D that goes through $(2,0)$ and has velocity $(3,2).$ Lift this curve to a new curve C$2$ along the graph of f$($x$,$y$),$ so that the $($x$,$y$)$ components of C$2$ match up with C$1.$ What is the parametrization of C$2?$ Without computing anything,provide a good estimate of f$(2\mathrm{.}03,0\mathrm{.}02).$ What does it tell us about C$2?$ Parametrize a linear curve C$3$ that is tangent to C$2$ at $(2,0,4).$