Step by step
The question
Question: estimate a root of x³ − 2x − 5 = 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) = 3x² − 2, 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 10.
So x₁ is 2 minus −1/10: 2.1.
4 Round 2
f at 2.1 is 0.061.
f′ at 2.1 is 11.23.
So x₂ is 2.1 minus 0.061/11.23: about 2.09457.
5 Round 3
f at 2.09457 is about 0.00019.
f′ at 2.09457 is about 11.16165.
So x₃ is 2.09457 minus 0.00019/11.16165: about 2.09455.
After 3 rounds, the estimate is 2.09455.
6 Check
Let's check: put 2.09455 into f.
We get about 0.000000: 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.