Question

Suppose that your first guess with Newton's method is lucky, in the sense that is a root of f(x)=0. Assuming that f'(x) is defined and not equal to 0, what happens to x1 and later approximations?

Answer 1

When the first guess with Newton's method is a lucky root of f(x)=0, the subsequent approximations will converge to that root.

Step-by-step explanation:

When the first guess with Newton's method is a lucky root of f(x)=0, the subsequent approximations will converge to that root. This is because Newton's method works by iteratively improving the approximation by finding the tangent line to the function at each point and then finding the x-coordinate where the tangent line intersects the x-axis, which gives a better approximation of the root.

The process of iteratively improving the approximation continues until a satisfactory level of accuracy is achieved. The subsequent approximations will get closer and closer to the actual root of the function, eventually reaching a point where the difference between the approximation and the actual root is negligible.