Step by step
The question
Question: estimate √3 with Newton's method, starting at x = 2.
√3 is the positive root of x² − 3 = 0.
So f(x) = x² − 3.
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: 1.75.
4 Round 2
f at 1.75 is 0.0625.
f′ at 1.75 is 3.5.
So x₂ is 1.75 minus 0.0625/3.5: about 1.73214.
5 Round 3
f at 1.73214 is about 0.00032.
f′ at 1.73214 is about 3.46429.
So x₃ is 1.73214 minus 0.00032/3.46429: about 1.73205.
After 3 rounds, the estimate is 1.73205.
6 Check
Let's check: put 1.73205 into f.
We get about 0.000000: almost 0.
Right!
And a calculator gives √3 = 1.732051.
Very close.
Let's look back.
First: the formula.
Then: find f′(x).
Then: round 1.
Then: round 2.
Then: round 3.
Last: check.