Answer to Question #177625 in Mechanical Engineering for Von Nikko

Question

Determine the root of f(x)=x³-6x²+11x-6.1 using the Newton-Rapson Method(three iterations, x_i=3.5)

Answer 1

Given

$f(x)=x^3-6x^2+11x-6$

$f'(x)=3x^2-12x+11$

$x_0=3.5$

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

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} Step & x_0 & f(x_0) & x_1 & f(x_1) \\ \hline 1 & 3.500000 & 1.875000 & 3.173913 & 0.443823 \\ \hdashline 2 & 3.173913 & 0.443823 & 3.032307 & 0.067779 \\ \hdashline 3 & 3.032307 & 0.067779 & 3.001456 & 0.002918 \\ \hdashline 4 & 3.001456 & 0.002918 & 3.000003 & 0.000006 \\ \end{array}$$

Root is : $x=3.000003$