What would happen to Newton's method in exercise 49 if you...

What would happen to Newton's method in exercise 49 if you had a starting value of $x_{0}=0 ?$ Consider the use of Newton's method with $x_{0}=0.2$ and $x_{0}=10 .$ Obviously, $x_{0}=0.2$ is much closer to a zero of the function, but which initial guess would work better in Newton's method? Explain.

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Key Concepts

Derivative Considerations

Newton's method requires that the derivative at the chosen starting point and throughout the iterations be non-zero. If the derivative is zero or nearly zero (as can happen with a poorly chosen initial guess like x? = 0), the method can fail by producing erratic behavior or division by zero.

Convergence Behavior

In practice, an initial guess such as x? = 0.2, which is much closer to the target zero, will generally yield faster and more reliable convergence compared to a guess like x? = 10. This is because starting near the root reduces the likelihood of encountering regions where the derivative is trivial or the tangent does not accurately represent the curve.

Newton's Method

Newton's method is an iterative algorithm used to find approximations to the roots of a differentiable function. It uses the formula x??? = x? - f(x?)/f'(x?), where each iteration uses a tangent line at the current guess to approximate the next guess, ideally converging to an actual root.

Importance of the Initial Guess

The success and speed of convergence in Newton's method highly depend on the initial guess. A choice that is closer to the actual root accelerates convergence, while a poor initial guess can lead to divergence, slow convergence, or convergence to an unintended root.

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