Newton's Method for Square Roots

You need to compute $\sqrt{c}$ for some positive number $c$ using Newton's iterative method. Set up the root-finding problem: what function $f(x)$ do you solve, and what is the Newton update step? Starting from an initial guess $x_0 > 0$, write out the iteration formula and simplify it. Then answer the following: 1. What is the rate of convergence of this method (linear, quadratic, etc.), and why? 2. Demonstrate the iteration numerically: compute $\sqrt{2}$ starting from $x_0 = 1$, showing at least 3 iterations. 3. Does the method converge for any starting guess $x_0 > 0$? What about $x_0 < 0$?