Question

f(x)=3−x ³ Newton's method Grabting utilty: x= [−/1 Points] Use Nention's Method to approwimate the zerols) of the function. Continue the iterations unth two successve operowimations defer by liess than o.001. Then fed the zero(s) using a graphing utaty and compare the resuits.

Answer 1

To approximate the zeros of f(x) = 3 - x³, use Newton's Method with iterative refinement until successive approximations are within 0.001, or employ a graphing utility for a visual approach albeit potentially less precise.

Step-by-step explanation:

To find the zeros of the function f(x) = 3 - x³ using Newton's Method, we need to apply an iterative process. We start by choosing a close approximation to the zero and then refine our guess by using the formula: xn+1 = xn - f(xn) / f'(xn), where f'(x) is the derivative of the function. Iterations continue until two successive approximations differ by less than 0.001.

Newton's Method is an example of a numerical analysis technique to find roots of real-valued functions. Graphing utilities can also be used to visualize the function and estimate the location of its zeros, but they sometimes provide less precise results than numerical methods.

A graphical method involves plotting the function and observing where it crosses the x-axis, which indicates a zero. An approximate zero can be determined by using the zero feature of a graphing calculator.