Step by step
The question
Question: estimate √20 with Newton's method, starting at x = 4.
√20 is the positive root of x² − 20 = 0.
So f(x) = x² − 20.
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 = 4.
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 4 is −4.
f′ at 4 is 8.
So x₁ is 4 minus −4/8: 4.5.
4 Round 2
f at 4.5 is 0.25.
f′ at 4.5 is 9.
So x₂ is 4.5 minus 0.25/9: about 4.47222.
After 2 rounds, the estimate is 4.47222.
5 Check
Let's check: put 4.47222 into f.
We get about 0.000772: almost 0.
Right!
And a calculator gives √20 = 4.472136.
Very close.
Let's look back.
First: the formula.
Then: find f′(x).
Then: round 1.
Then: round 2.
Last: check.