Step by step
The question
Question: estimate a root of x² − 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) = 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 1 is −1.
f′ at 1 is 2.
So x₁ is 1 minus −1/2: 1.5.
4 Round 2
f at 1.5 is 0.25.
f′ at 1.5 is 3.
So x₂ is 1.5 minus 0.25/3: about 1.41667.
5 Round 3
f at 1.41667 is about 0.00694.
f′ at 1.41667 is about 2.83333.
So x₃ is 1.41667 minus 0.00694/2.83333: about 1.41422.
After 3 rounds, the estimate is 1.41422.
6 Check
Let's check: put 1.41422 into f.
We get about 0.000006: almost 0.
Right!
Let's look back.
First: the formula.
Then: find f′(x).
Then: round 1.
Then: round 2.
Then: round 3.
Last: check.