Newton's Method

The tangent line crosses the axis

Convergence and the mean value theorem

When Newton fails

A worked approximation

Newton's method for approximating a root of f(x) = 0 updates the current guess x_n by which formula?

Applying Newton's method to f(x) = x^2 - 2 from x_0 = 1, what is x_1?

Near a simple root, Newton's method exhibits quadratic convergence. What does that mean for the error?

Why can a Newton guess near a local maximum or minimum derail the iteration?

Before Newton's method hunts a root of a continuous f, which theorem certifies that a root exists between a sign change?

For f(x) = x^3 - 2x + 2, Newton's method started at x_0 = 0 oscillates between 0 and 1 forever. What does this illustrate?

For f(x) = x^3 - 2, one Newton update is x_1 = x_0 - (x_0^3 - 2)/(3 x_0^2). Starting from x_0 = 1, what is x_1? Answer with a fraction like 4/3.

Newton's method on f(x) = x^2 - a gives the update x_{n+1} = (x_n + a/x_n)/2, the Babylonian square root iteration. For a = 2 and x_0 = 1, what is x_1? Answer with a number.

In the sqrt(2) computation, x_2 is about 1.4167 and x_3 about 1.4142, already correct to five digits. How many iterations were needed from x_0 = 1? Answer with a number.

Quadratic convergence requires f prime of the root r to be nonzero, meaning r is a simple root. As a number, what is f prime of r for a simple root? Answer with a number.