Final answer:
The general formula for the next iterate in Newton's method is x_n+1 = x_n - f(x_n)/f'(x_n). To find the iterate, compute f'(x) and use the formula with an initial guess x_0. Applying this to the quadratic equation x^2 + 0.0211x - 0.0211 = 0, we can determine the first iterate starting from x_0.
Step-by-step explanation:
The general formula for the next iterate in Newton's method to solve f(x) = 0 is given by:
xn+1 = xn - f(xn)/f'(xn).
To find the first iterate, we need to calculate f'(x), the derivative of f(x), and then apply the formula using an initial guess x0 (often taken as x0 = 0 or a value close to an expected root). For instance, if we have a quadratic equation of the form ax2 + bx + c = 0, and we want to find the roots using Newton's method, we first need to differentiate the equation to find f'(x), which would be 2ax + b. Then we can apply the Newton's formula to iterate towards a solution.
Let's take the equation x2 + 0.0211x - 0.0211 = 0 as an example. The first derivative of this function f'(x) would be 2x + 0.0211. Applying the Newton's method formula and assuming our x0 is 0 (x-sub-zero or x-reaction), we can find the first iterate (x1) for this specific equation.