$4\mathrm{.}3$ Newton's method

Write Maple and$/$or Matlab code to implement Newton's method, as seen in class.

Using your code, illustrate the execution of Newton's method with $x_0=0\mathrm{.}1$ and

$f(x)=x^3+x-1.$

Study the effect of the initial condition on the number of iterations it takes to obtain

a numerical approximation to the root of $f(x)=x^3+x-1$ with $8$ correct digits.

Illustrate your conclusions with a graph.

Using your code solve the equation $x^2+1=0.$ Find a suitable initial guess so that

Newton's method will converge to the root $i.$ Find a suitable initial guess so that

Newton's method will converge to the root $-i.$

Using your code and suitable initial conditions, compute numerical approximations,

with $10$ digits of accuracy correct, to all three real roots ${}_1,{}_2,{}_3$ of equation $(1),$

using Newton's method.