Chapter 4.8, Problem 52E

Pitfalls of Newton’s method Let $f(x)=\frac{x}{1+x^2},$ which has just one root, r = 0.

52. Use the initial approximation $x_0=1/\sqrt{3}$ to complete the following steps.

  a.    Use Newton’s method to find the exact values of x₁ and x₂.

  b.    State the values of $x_3,x_4,x_5,\dots$ without performing any additional calculations.

  c.    Use a graph of f to illustrate why Newton’s method produces the values found in part (b).

  d.    Why does Newton’s method fail to approximate the root r = 0 if $x_0=1/\sqrt{3}$?