textbook-theory · English
Difference Equations to Differential Equations — Section 3.6: Newton's Method
Section 3.6 of Dan Sloughter's open calculus textbook presents Newton's method as an iterative root-finding algorithm for equations of the form f(x) = 0. The section derives the recurrence relation x_{n+1} = x_n - f(x_n)/f'(x_n) from best affine (tangent-line) approximations and contrasts the method's speed with the bisection algorithm. Convergence conditions, failure cases (divergence, zero derivative), and worked examples including cos(x) = x and square-root approximation are covered, with problem sets extending to cube and seventh roots.
Newton's methodroot-finding algorithmsiterative approximationcalculus