Chapter 4.8, Problem 59E

Basins of attraction Suppose f has a real root r and Newton’s method is used to approximate r with an initial approximation x₀ The basin of attraction of r is the set of initial approximations that produce a sequence that converges to r. Points near r are often in the basin of attraction of r—but not always. Sometimes an initial approximation x₀ may produce a sequence that doesn’t converge, and sometimes an initial approximation x₀ may produce a sequence that converges to a distant root. Let f(x) = (x + 2)(x + 1)(x − 3), which has roots x = −2, −1. and 3. Use Newton’s method with initial approximations on the interval [−4, 4] to determine (approximately) the basin of each root.