Assess students' ability to analyze local and global convergence of Newton's method for scalar equations and systems: state and apply conditions for quadratic convergence (invertible Jacobian, smoothness/Lipschitz bounds), use the Newton–Kantorovich framework, examine stability under perturbations and damping, compare damped and quasi‑Newton variants, and characterize basins of attraction (including computational identification and interpretation of fractal or bifurcation boundaries). Include proof-based questions, counterexamples, eigenvalue/Jacobian analysis, and interpretation of numerical basin plots.