Assess understanding of theoretical and practical convergence behavior of Newton's method for scalar and low-dimensional systems. Test analysis of local quadratic convergence and its failure modes (multiple roots, vanishing derivatives), criteria for convergence (basins of attraction, Kantorovich-type conditions), estimating order of convergence and error decay, identifying numerical error sources (round-off, inexact derivative evaluation, ill-conditioning, poor initial guesses), and proposing diagnostic and mitigation strategies (modified Newton, damping, safeguarded iterations). Include interpretation of iteration histories and simple numerical diagnostics.