use newton's method to estimate all real solutions of the equation x^3.
I am having trouble coming up with a good guess. I went with x1=1, but that doesn't seem to be working. Any thoughts on which number should be the best first guess?
To use Newton's method to estimate the real solutions of an equation, you need to start with a good initial guess. In the case of the equation x^3, it is important to note that the equation has one real root, namely x = 0. Therefore, starting with x1 = 1 may not give accurate results.
To choose a better initial guess, you can start by considering the behavior of the function f(x) = x^3. Since x^3 is an odd-degree polynomial, its graph will cross the x-axis at x = 0. This suggests that choosing a value close to 0 as your initial guess might be a better approach.
One way to find a better initial guess is by considering the sign changes of x^3. Evaluate the function at various points close to 0 and observe the sign changes:
f(-1) = (-1)^3 = -1
f(-0.5) = (-0.5)^3 = -0.125
f(0) = 0
f(0.5) = (0.5)^3 = 0.125
f(1) = 1
From this, you can see that the sign changes from negative to positive as x increases from -0.5 to 0.5. Therefore, it would be a good choice to start with an initial guess around x = -0.5 or x = 0.5.
For Newton's method, you need to choose an initial guess x1 and then use the iterative formula:
x_n+1 = x_n - f(x_n) / f'(x_n)
where f'(x_n) is the derivative of f(x) evaluated at x_n.
In this case, f(x) = x^3, so f'(x) = 3x^2.
To find the real solutions, you can start with an initial guess like x1 = -0.5 or x1 = 0.5 and apply the Newton's method formula repeatedly until you reach an approximation with the desired level of accuracy.