Use newton's method to find all solutions of the equation correct to six decimal places. (enter your answers as a comma-separated list.) 7 cos(x) = x + 1

For Newton's method, we need a function that evaluates to zero at the solution of interest. Here, we can define
.. f(x) = 7cos(x) -(x +1)

Newton's method tells us the iteration function for developing the next guess from the present one is
.. g(x) = x -f(x)/f'(x)
where f'(x) is the derivative of f(x).

A suitable graphing calculator can provide a numerical estimate of the derivative that is sufficient for this purpose. That capability is utilized to find the answers shown in the attachment.

The same graphing calculator provides the initial estimate used by the iteration function. Since the number of good decimal places is doubled on each iteration, starting with a 3 decimal digit estimate of the root, one can get an estimate good to 6 digits in one iteration. Here, the calculator displays the result of evaluating g(x) even as you type its argument, so you can simply type in the result as the argument to get the answer to as many digits as you like (up to calculator precision).

Solutions are {-4.232398, -1.666097, 1.244403}