Assess students' ability to analyze and compare the convergence properties and error behavior of Newton's method and the secant method for solving f(x)=0. Tasks should include deriving local error recurrence relations and orders of convergence using Taylor expansions, stating sufficient conditions for quadratic (Newton) and superlinear (secant) convergence, explaining the effect of multiplicity of roots and initial guesses, identifying common failure modes, and comparing computational trade-offs (function vs derivative evaluations). Include an applied component where students perform iterations on a specified nonlinear function, estimate observed error constants and rates, and reconcile numerical results with theoretical predictions.