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

Chapter 3, Problem 29

(choose chapter or problem)

Consider the function \(f(x)=x^{3}-3 x^{2}+3\).

(a) Use a graphing utility to graph f.

(b) Use Newton's Method with \(x_{1}=1\) 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)) 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^3-3 x^2+3

x_1=1

x_1=1/4

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