  In this paper we study the $X$-rank of points with respect to smooth linearly
normal curves $X\subset \PP n$ of genus $g$ and degree $n+g$. We prove that,
for such a curve $X$, under certain circumstances, the $X$-rank of a general
point of $X$-border rank equal to $s$ is less or equal than $n+1-s$. In the
particular case of $g=2$ we give a complete description of the $X$-rank if
$n=3,4$; while if $n\geq 5$ we study the $X$-rank of points belonging to the
tangential variety of $X$.
