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

Step by step

The question

Question: estimate a root of x² − 3 = 0 with Newton's method, starting at x = 2.

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 = 2.

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) = 2x, by the power rule.

Careful!

f goes on top, and f′ at the bottom.

Not the other way round.

3 Round 1

f at 2 is 1.

f′ at 2 is 4.

So x₁ is 2 minus 1/4: 1.75.

4 Round 2

f at 1.75 is 0.0625.

f′ at 1.75 is 3.5.

So x₂ is 1.75 minus 0.0625/3.5: about 1.73214.

After 2 rounds, the estimate is 1.73214.

5 Check

Let's check: put 1.73214 into f.

We get about 0.000319: almost 0.

Right!

Let's look back.

First: the formula.

Then: find f′(x).

Then: round 1.

Then: round 2.

Last: check.