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Response Details:
Newton's method is perhaps the best known method for finding successively better approximations to the zeros (or roots) of a real-valued function.
The idea of the method is as follows: one starts with an initial
guess which is reasonably close to the true root, then the function is
approximated by its tangent line (which can be computed using the tools of calculus), and one computes the x-intercept of this tangent line (which is easily done with elementary algebra). This x-intercept will typically be a better approximation to the function's root than the original guess, and the method can be iterated.
We start the process off with some arbitrary initial value x0.
(The closer to the zero, the better. But, in the absence of any
intuition about where the zero might lie, a "guess and check" method
might narrow the possibilities to a reasonably small interval by
appealing to the intermediate value theorem.) We perform iterations of the recursive sequence  . The method will usually converge, provided this initial guess is close enough to the unknown zero. Furthermore, for a zero of multiplicity 1, the convergence is at least quadratic in a neighborhood
of the zero, which intuitively means that the number of correct digits
roughly at least doubles in every step.
In this problem they wanted to find the roots of f ' (x). They called this g (x) for use with Newton's method, to avoid confusion in the method between f '(x) which would have been f(x) and its derivative.
The switch to g(x) is not really important.
g(x) = sinx + x cosx
Then, to approximate the root, they find
g'(x) = 2cosx - x sinx
And use Newton's method with these functions.
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