Approximate a root with Newton's method and compare successive iterates

Primary Tool

Spreadsheet Workspace

Spreadsheet is the right fit because Newton's method is inherently iterative. The row structure makes each update visible and editable.

Problem

Approximate a root of $f (x) = x^{3} - x - 2$ starting from $x_{0} = 1.5$ using Newton's method.

This is a good worked example because it blends exact formulas, derivative information, and numerical iteration.

Why the table helps

Newton's update is $x_{n + 1} = x_{n} - \frac{f ( x _{n} )}{f ^{'} ( x _{n} )}$ . Writing that as spreadsheet formulas makes it easy to see whether the iterates stabilize, overshoot, or break down.

In this example the approximations quickly settle near the real root because the starting guess is already in a region where the tangent-line correction behaves well.

Step-by-step walkthrough

The spreadsheet exposes the recurrence row by row so that every new approximation can be checked against the previous one.

1Write the function $f (x) = x^{3} - x - 2$ and its derivative $f^{'} (x) = 3 x^{2} - 1$ .
2Choose the starting guess $x_{0} = 1.5$ .
3Compute $f (x_{0})$ and $f^{'} (x_{0})$ in the first row.
4Apply Newton's update $x_{1} = x_{0} - \frac{f ( x _{0} )}{f ^{'} ( x _{0} )}$ to get the next approximation.
5Repeat the same formulas down the table so each row uses the previous row's approximation.
6Watch the $x_{n}$ values settle toward the root and compare how quickly the corrections shrink.

Common Pitfall

Newton's method is not just a button press. The iteration table shows whether the approximations are actually converging and whether a small derivative is making the update unstable.

Try a Variation

Change the starting guess to $x_{0} = 0$ or $x_{0} = 3$ . How does the iteration behave, and which start looks more stable?

Keep moving through the cluster

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