Step by step
The question
Question: estimate √5 with Newton's method, starting at x = 2.
√5 is the positive root of x² − 5 = 0.
So f(x) = x² − 5.
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, 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 4.
So x₁ is 2 minus −1/4: 2.25.
4 Round 2
f at 2.25 is 0.0625.
f′ at 2.25 is 4.5.
So x₂ is 2.25 minus 0.0625/4.5: about 2.23611.
After 2 rounds, the estimate is 2.23611.
5 Check
Let's check: put 2.23611 into f.
We get about 0.000193: almost 0.
Right!
And a calculator gives √5 = 2.236068.
Very close.
Let's look back.
First: the formula.
Then: find f′(x).
Then: round 1.
Then: round 2.
Last: check.