Use Newton's method on x^3 - 2 = 0 starting at x = 1

Step by step

The question

Question: estimate a root of x³ − 2 = 0 with Newton's method, starting at x = 1.

1 The formula

Newton's idea: draw the tangent line at your guess.

Where it meets the x-axis is the next guess.

That gives this formula.

See it on the graph.

Start at x₀ = 1.

Follow the tangent line down to the x-axis.

It lands at x₁: much closer to the root.

2 Find f′(x)

The slope: f′(x) = 3x², by the power rule.

Careful!

f goes on top, and f′ at the bottom.

Not the other way round.

3 Round 1

f at 1 is −1.

f′ at 1 is 3.

So x₁ is 1 minus −1/3: about 1.33333.

4 Round 2

f at 1.33333 is about 0.37037.

f′ at 1.33333 is about 5.33333.

So x₂ is 1.33333 minus 0.37037/5.33333: about 1.26389.

After 2 rounds, the estimate is 1.26389.

5 Check

Let's check: put 1.26389 into f.

We get about 0.018955: almost 0.

Right!

Let's look back.

First: the formula.

Then: find f′(x).

Then: round 1.

Then: round 2.

Last: check.