Newton's method will also fail when the difference of...

Newton's method will also fail when the difference of successive approximations, $\left|x_{k+1}-x_{k}\right|$, does not decrease as $k$ increases. (a) Show that this happens for the function $f(x)=\sqrt[3]{x-2}$ when you choose $x_{0}=1$. (b) What is the root of $f(x)=\sqrt[3]{x-2}$ ?

Newton's method will also fail when the difference of successive approximations, $\left|x_{k+1}-x_{k}\right|$, does not decrease as $k$ increases.
(a) Show that this happens for the function $f(x)=\sqrt[3]{x-2}$ when you choose $x_{0}=1$.
(b) What is the root of $f(x)=\sqrt[3]{x-2}$ ?

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Key Concepts

Newton's Method

An iterative procedure for finding successively better approximations to the roots of a real-valued function. It uses tangents by computing the derivative of the function at a current approximation to estimate the next approximation through the formula x??? = x? - f(x?)/f'(x?).

Convergence Criteria in Iterative Methods

The success of an iterative method like Newton's method depends on the behavior of the approximations converging toward a root. Convergence criteria involve the decrease in differences between successive approximations. When these differences do not decrease, it indicates the method may be diverging or oscillating rather than converging.

Role of Derivatives Near the Root

The derivative of a function is critical in Newton's method as it determines the slope of the tangent used to approximate the function near a guessed root. If the derivative is small, undefined, or behaves erratically near the actual root, the method can fail because the tangent line approximation becomes unreliable, leading to divergence or erratic behavior in the iterations.

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