`1/x = 1 + x^3` Use Newton's method to find all roots of the equation correct to six decimal places.
Set the left side equal to zero.
To solve this using Newton's method, apply the formula:
`x_(n+1) = x_n - (f(x_n))/(f'(x_n))`
Let the function be:
`f(x) = 1+x^3-1/x`
Take the derivative of f(x).
`f'(x) = 3x^2 +1/x^2`
Plug-in f(x) and f'(x) to the formula of Newton's method.
This simplifies to:
`x_(n+1) = x_n- (x_n^5+x_n^2-x_n)/(3x_n^4+1)`
Then, refer to the graph of the function to get the initial values x when f(x) =0. (See attached figure.)
(The entire section contains 265 words.)
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