Step by step
The question
Question: estimate √10 with Newton's method, starting at x = 3.
√10 is the positive root of x² − 10 = 0.
So f(x) = x² − 10.
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 = 3.
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 3 is −1.
f′ at 3 is 6.
So x₁ is 3 minus −1/6: about 3.16667.
4 Round 2
f at 3.16667 is about 0.02778.
f′ at 3.16667 is about 6.33333.
So x₂ is 3.16667 minus 0.02778/6.33333: about 3.16228.
After 2 rounds, the estimate is 3.16228.
5 Check
Let's check: put 3.16228 into f.
We get about 0.000019: almost 0.
Right!
And a calculator gives √10 = 3.162278.
Very close.
Let's look back.
First: the formula.
Then: find f′(x).
Then: round 1.
Then: round 2.
Last: check.