Newton's Fast Square Root Approximation

This question focuses on using Newton's method to approximate a square root by recasting it as a root-finding problem for a simple quadratic function. The candidate is asked to choose a reasonable starting point and apply the iterative scheme to obtain a decimal approximation to a specified precision. Because the underlying equation is so familiar, the algebra is light and the main challenge is correctly implementing the iteration and deciding when the desired accuracy has been achieved. Variants of this type of question often appear in early calculus courses and in interviews that probe basic numerical intuition.

The problem leans on understanding derivatives, tangent lines, and how local linear approximations can be turned into a practical algorithm. It tests whether the candidate can set up Newton's iteration formula correctly for a given function, perform several iterations without algebraic or arithmetic errors, and monitor convergence. An interviewer is watching for comfort with differentiating simple functions, translating the verbal description of Newton's method into a concrete update rule, and articulating a clear stopping criterion tied to the requested number of decimal places.

What it tests

Newton's method leverages the idea that a differentiable function can be locally approximated by its tangent line, and the root of that tangent line provides a better estimate for the root of the function itself. This iterative approach rapidly converges to a solution when the initial guess is sufficiently close and the function behaves well (i.e., is smooth and its derivative is not too small near the root). The key is that each step uses local linearization: the next guess is where the tangent at the current point crosses the x-axis. This works because, near a simple root, the function's graph is nearly straight, so the tangent's root is a good predictor. The method's power comes from the quadratic convergence: the number of correct digits roughly doubles with each iteration, provided the assumptions hold.