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Question 2: (a) Give the definition of the double root of a function and sketch the graph of a function with two roots: a simple root, marked ã, and a double root, marked x.. (b) Find an unseen example of a function, f, with a double root, x. The function f should not be a polynomial but could be any rational or transcendental function. Perform enough iterations of Newton's algorithm, written as a fixed-point iteration n+1 = g(n), to infer the order and rate of convergence of the sequence of data produced. Prove this assertion by writing a Taylor series for g about x, in powers of (xn - X*).

Question 2: (a) Give the definition of the double root of a function and sketch the graph of a function with two roots: a simple root, marked ã, and a double root, marked x.. (b) Find an unseen example of a function, f, with a double root, x. The function f should not be a polynomial but could be any rational or transcendental function. Perform enough iterations of Newton's algorithm, written as a fixed-point iteration n+1 = g(n), to infer the order and rate of convergence of the sequence of data produced. Prove this assertion by writing a Taylor series for g about x, in powers of (xn - X*).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 44E
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Question 2: (a) Give the definition of the double root of a function and sketch the graph of a function with two roots: a simple root, marked x, and a double root, marked x. (b) Find an unseen example of a function, f, with a double root, x. The function f should not be a polynomial but could be any rational or transcendental function. Perform enough iterations of Newton's algorithm, written as a fixed-point iteration Xn+1 = g(en), to infer the order and rate of convergence of the sequence of data produced. Prove this assertion by writing a Taylor series for g about x, in powers of (xn xx).
Transcribed Image Text:Question 2: (a) Give the definition of the double root of a function and sketch the graph of a function with two roots: a simple root, marked x, and a double root, marked x. (b) Find an unseen example of a function, f, with a double root, x. The function f should not be a polynomial but could be any rational or transcendental function. Perform enough iterations of Newton's algorithm, written as a fixed-point iteration Xn+1 = g(en), to infer the order and rate of convergence of the sequence of data produced. Prove this assertion by writing a Taylor series for g about x, in powers of (xn xx).
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