Question

Use Newton's method to approximate the indicated root of the equation correct to six decimal places.
x^4 - 3x^3 + 6x^2 - 3 = 0
x E [0, 2]

Answer 1

basically to start the process, you want to make an educated guess (say x = 1), and that will be your first term. To calculate the next term, you want to compute:
\[x_{n+1} = x_n-\frac{f(x_n)}{f'(x_n)}\]
and that produces the next term, which is closer to the real answer than your guess. Then you plug that answer back into the same formula and get an answer even closer to the correct answer. You just continue this process however many times you want to get an answer more and more accurate. The pic i have of the solution does the process 3 times.

Answer 2

Very good.