Step by step
The question
Question: estimate a root of x² − x − 1 = 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 − 1, 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 3.
So x₁ is 2 minus 1/3: about 1.66667.
4 Round 2
f at 1.66667 is about 0.11111.
f′ at 1.66667 is about 2.33333.
So x₂ is 1.66667 minus 0.11111/2.33333: about 1.61905.
After 2 rounds, the estimate is 1.61905.
5 Check
Let's check: put 1.61905 into f.
We get about 0.002268: almost 0.
Right!
Let's look back.
First: the formula.
Then: find f′(x).
Then: round 1.
Then: round 2.
Last: check.