Copy this error-prone script to a new m-file and name it NewtonScript.m. This script computes the first few Newton's Method root approximations for the two roots of the function f(x)= x/(1+x^2/4)-1/2.

% NewtonScript.m
% Finds the roots using Newton's method
clear; clc; clf; format long;

% First, plot the function
xx = 0:0.1:10;
ff = xx./(1+xx.^2/4)-1/2;
plot(xx,ff);

% User Newton's Method to approximate the first root
x(1) = 1;
x(2) = Newton3(x(1));
x(3) = Newton3(x(1));
x(4) = Newton3(x(1));
x(5) = Newton3(x(1));
x(6) = Newton3(x(1));
x

% Use Newton's Method to approximate the second root
w(1) = 7;
w(2) = Newton3(w(1));
w(3) = Newton3(w(2));
w(4) = Newton3(w(3));
w(5) = Newton3(w(3));
w(6) = Newton3(w(5));
w

You will also need to define the Newton3.m function, as shown, once you have set a breakpoint and you.

function x_next = Newton3( x_input )
% Newton's formula for successive approximations
% to find the roots where f(x)=x/(1+x^2/4)-1/2=0
x  = x_input;
f  = x/(1+x^2/4)-1/2;
df = 1/(1+1/4*x^2)-1/2 * x^2 /(1+1/4*x^2)^2;
x_next = x - f/df;