Newton's method is an iterative numerical technique for approximating a root of a function by repeatedly applying the update rule x_(n+1) = x_n − f(x_n)/f′(x_n), starting from an initial guess and using the function's derivative to generate successively refined estimates. Solving an equation of the form g(x) = h(x) is recast as a root-finding problem by defining f(x) = g(x) − h(x) and seeking zeros of f, with graphical analysis of the two original curves' intersection used to justify the existence, uniqueness, and rough location of the root that guides the choice of initial guess. This belongs to numerical methods within single-variable calculus, relying on the derivative (tangent-line approximation) as its core mechanism and providing an alternative to closed-form algebraic solution when an equation cannot be solved exactly.
Newton's method is an iterative numerical technique for approximating a root of a function by repeatedly applying the update rule x_(n+1) = x_n − f(x_n)/f′(x_n), starting from an initial guess and us…