How to Compute the First 10 Iterations of Newton’s Method for the Function f(x) = x² – 10?

To apply Newton’s method for the function f(x) = x² – 10, we need to follow a systematic approach. Newton’s method uses the formula:

x_n+1 = x_n – (f(x_n) / f'(x_n))

First, let’s determine the necessary components:

• The function is f(x) = x² – 10.
• The derivative of the function is f'(x) = 2x.

Next, we need an initial approximation. Let’s use x₀ = 3.162, which is a rough estimate of the square root of 10.

Now, we will compute the iterations:

1. Iteration 1:
   • x₁ = 3.162 – ((3.162² – 10) / (2 * 3.162))
   • x₁ = 3.162 – ((10.000244 – 10) / 6.324) = 3.162 – (0.000244 / 6.324) ≈ 3.162 – 0.0000386 ≈ 3.162
2. Iteration 2:
   • x₂ = 3.162 – ((3.162² – 10) / (2 * 3.162))
   • x₂ ≈ 3.162
3. Iteration 3:
   • x₃ ≈ 3.162
4. Iteration 4:
   • x₄ ≈ 3.162
5. Iteration 5:
   • x₅ ≈ 3.162
6. Iteration 6:
   • x₆ ≈ 3.162
7. Iteration 7:
   • x₇ ≈ 3.162
8. Iteration 8:
   • x₈ ≈ 3.162
9. Iteration 9:
   • x₉ ≈ 3.162
10. Iteration 10:
    • x₁₀ ≈ 3.162

As you can see, with our initial approximation of 3.162, the value stabilizes to approximately 3.162 after just a couple of iterations, quickly converging to the square root of 10. This demonstrates the efficiency of Newton’s method in finding roots of functions.