Assess students’ ability to apply the Implicit Function Theorem to parameter-dependent nonlinear systems in R^n: verify smoothness and invertibility of the relevant partial Jacobian (rank conditions), construct local implicit solution maps expressing chosen variables as C^k functions of parameters near a base point, compute first-order derivatives via implicit differentiation, and determine existence, uniqueness, and neighborhood estimates for solution continuations; include at least one concrete system (with parameters) requiring verification of hypotheses and computation of the solution derivative or linear sensitivity approximation.