Assess students' ability to analyze local convergence behavior and derive error estimates for Newton's method applied to scalar (one-variable) nonlinear equations: state hypotheses (simple vs multiple roots, nonzero derivative, Lipschitz or Kantorovich-type conditions), use Taylor expansions to obtain the Newton error recurrence and asymptotic quadratic convergence constant, compare with modified Newton for multiple roots, estimate radius/region of convergence and iteration counts to reach a given tolerance, and justify results with brief worked examples or counterexamples.