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

$x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}$

Where: $f^{\prime }(x)=3+\sin (x)$

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: $0.6071$

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