If $f: \mathcal{C} \rightarrow \mathcal{P}_{2g+2}$ is a general family of hyperelliptic curves (defined over $\mathbb{C}$), we know that the algebraic connected monodromy group $Mon^{0}$ of this family is equal to the full symplectic group. Now if we have a 1-parameter family of hyperelliptic curves given for example by the expression

$y^{2}= (x-h_{1}(t))....(x-h_{2g+2}(t))$ where $t$ is the parameter in $\mathbb{P}^{1}$ and the $h_{i}$ are holomorphic functions in $t$, is the group $Mon^{0}$ of this family also equal to the full symplectic group $Sp_{2g+2}$? if not, how can one compute the monodromy group of such family?