Using Newtons Method Consider the function (a) Use a graphing utility to graph (b) Use

Chapter 3, Problem 31

(choose chapter or problem)

Using Newton’s Method Consider the function \(f(x)=x^{3}-3 x^{2}+3\).

(a) Use a graphing utility to graph f.

(b) Use Newton’s Method to approximate a zero with x1 = 1 as an initial guess.

(c) Repeat part (b) using \(x_{1}=\frac{1}{4}\) as an initial guess and observe that the result is different.

d) To understand why the results in parts (b) and (c) are different, sketch the tangent lines to the graph of f at the points \((1, f(1)) \text { and }\left(\frac{1}{4}, f\left(\frac{1}{4}\right)\right\). Find the x-intercept of each tangent line and compare the intercepts with the first iteration of Newton’s Method using the respective initial guesses.

(e) Write a short paragraph summarizing how Newton’s Method works. Use the results of this exercise to describe why it is important to select the initial guess carefully.

Text Transcription:

f(x)=x^3-3x^2+3

x_{1}=1/4

(1, f(1)) and (1/4, f(1/4)