1. The problem is to determine how many iterations are needed using Newton's Method to approximate a root of a function $f(x)$ with a desired accuracy. 2. Newton's Method formula is: $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ where $x_n$ is the current approximation and $x_{n+1}$ is the next. 3. The number of iterations depends on: - The initial guess $x_0$ - The function $f(x)$ and its derivative $f'(x)$ - The desired tolerance (accuracy) $\epsilon$ 4. To estimate the number of iterations, use the error bound for Newton's Method: $$|x_{n+1} - r| \leq C |x_n - r|^2$$ where $r$ is the actual root and $C$ is a constant depending on $f$. 5. Because the error roughly squares each iteration, the number of iterations $N$ to reach error less than $\epsilon$ can be approximated by: $$N \approx \log_2\left(\log\frac{1}{\epsilon}\right)$$ assuming the initial guess is close enough. 6. In practice, you iterate until: $$|x_{n+1} - x_n| < \epsilon$$ which means the change between iterations is smaller than the tolerance. 7. Summary: You do not know the exact number of iterations beforehand, but you continue iterating Newton's formula until the difference between successive approximations is less than your desired tolerance. This iterative process ensures you find the root to the required accuracy.