NEWTON METHOD-EXAMPLE 1

Newton-Raphson Method Example

Find the real root of \( 3x = \cos(x) + 1 \) correct to four decimal places

Step 1: Define the function

\( f(x) = 3x - \cos(x) - 1 \)

Check sign change:

\( f(0) = -2 \)

\( f(1) = 1.4597 \)

Since there's a sign change between 0 and 1, a root lies in this interval. Choose initial guess: \( x_0 = 0.5 \)

Step 2: Use Newton-Raphson formula

\( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \)

Where: \( f'(x) = 3 + \sin(x) \)

Iteration (n)	\( x_n \)	\( x_{n+1} \)
0	0.5	0.6085
1	0.6085	0.6071
2	0.6071	0.6071

Therefore, the root is: \( 0.6071 \)

✓

---

✓