OSH/Example/3

Example problem statement

Estimate $\sqrt{15}$ to 4 decimal places using Newton's method.

Solution

In order to use Newton's method, we need to find an appropriate associated polynomial that has a zero at $\sqrt{15}$. We'll use the function

\(f(x)= x^2-15.\)

For Newton's method, we also need the derivative of this function:

\(f'(x)= 2x.\)

The last thing we need before starting with Newton's method is an initial guess. The nearest perfect square to 15 is 16 so $x=4$ is a good candidate for an initial guess. Graphing $f(x)$, it is clear that $x=4$ is positioned well so that Newton's method should converge to $\sqrt{15}$.

The Newton's method iterates, calculated by ________ (hand / calculator / spreadsheet) are

\( \begin{align} x_{1} &= 3.875\\ x_{2} &= 3.872983871\\ x_{3} &= 3.872983346 \end{align} \)

As the first 5 decimal points are the same, our estimate $x_3$ should be correct to 4 decimal places.