1. **Problem Statement:** Use Newton's iteration method to find the root of a function $f(x)$ starting from an initial guess $x_0$. 2. **Formula:** Newton's iteration formula is given by: $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ where $f'(x)$ is the derivative of $f(x)$. 3. **Explanation:** - Start with an initial guess $x_0$. - Compute $f(x_0)$ and $f'(x_0)$. - Update the guess using the formula above to get $x_1$. - Repeat the process until the difference between successive approximations is smaller than a desired tolerance. 4. **Example:** Suppose we want to find the root of $f(x) = x^2 - 2$ (which is $\sqrt{2}$). - $f(x) = x^2 - 2$ - $f'(x) = 2x$ - Start with $x_0 = 1$ 5. **Iteration 1:** $$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 1 - \frac{1^2 - 2}{2 \times 1} = 1 - \frac{-1}{2} = 1 + 0.5 = 1.5$$ 6. **Iteration 2:** $$x_2 = 1.5 - \frac{1.5^2 - 2}{2 \times 1.5} = 1.5 - \frac{2.25 - 2}{3} = 1.5 - \frac{0.25}{3} = 1.5 - 0.0833 = 1.4167$$ 7. **Iteration 3:** $$x_3 = 1.4167 - \frac{1.4167^2 - 2}{2 \times 1.4167} \approx 1.4167 - \frac{2.0069 - 2}{2.8334} = 1.4167 - \frac{0.0069}{2.8334} = 1.4167 - 0.0024 = 1.4143$$ 8. **Result:** After 3 iterations, the approximation is $x_3 \approx 1.4143$, which is close to the true value $\sqrt{2} \approx 1.4142$. Newton's iteration converges quickly if the initial guess is close to the root and $f'(x)$ is not zero near the root.