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This question concerns the Newton-Raphson method. It can find an accurate solution of an equation, but often needs a good starting-point. The following function has three zeros: f(x) = (1/4) x+cos x. Important Note: while you are asked to give answers to 4 decimal places, you should not calculate with the rounded values - e.g. if you use a rounded off value for x1 to get x2 it may be wrong. Always you should calculate with maximum accuracy and only round off when an answer needs to be given. (a) Apply the Newton-Raphson method to f (x) starting with = -1. Enter your answers below rounded to four decimal places. *1 = Number , *2 = Number, *3 = Number (b) Sometimes, if the initial point is not near a solution, Newton-Raphson method is not effective. Repeat the Newton-Raphson method, but this time starting with = 0. (Also consider graphically what is happening.) Enter your answers below rounded to four decimal places. *1 Number ,2 = Number 23 = Number (c) The function has two other real roots, between 0 and 10. Find initial values for which the Newton-Raphson method approaches each of the other two roots. Enter the other two roots accurate to four decimal places, in any order, separated by a comma, e.g. 1.1111, 5.5556.

This question concerns the Newton-Raphson method. It can find an accurate solution of an equation, but often needs a good starting-point. The following function has three zeros: f(x) = (1/4) x+cos x. Important Note: while you are asked to give answers to 4 decimal places, you should not calculate with the rounded values - e.g. if you use a rounded off value for x1 to get x2 it may be wrong. Always you should calculate with maximum accuracy and only round off when an answer needs to be given. (a) Apply the Newton-Raphson method to f (x) starting with = -1. Enter your answers below rounded to four decimal places. *1 = Number , *2 = Number, *3 = Number (b) Sometimes, if the initial point is not near a solution, Newton-Raphson method is not effective. Repeat the Newton-Raphson method, but this time starting with = 0. (Also consider graphically what is happening.) Enter your answers below rounded to four decimal places. *1 Number ,2 = Number 23 = Number (c) The function has two other real roots, between 0 and 10. Find initial values for which the Newton-Raphson method approaches each of the other two roots. Enter the other two roots accurate to four decimal places, in any order, separated by a comma, e.g. 1.1111, 5.5556.

Mathematics For Machine Technology
Mathematics For Machine Technology
8th Edition
ISBN:9781337798310
Author:Peterson, John.
Publisher:Peterson, John.
Chapter55: Introduction To Circles
Section: Chapter Questions
Problem 27A
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This question concerns the Newton-Raphson method. It can find an accurate solution of an equation, but often needs a good starting-point. The following function has three zeros: f(x) = (1/4)x+cosx. Important Note: while you are asked to give answers to 4 decimal places, you should not calculate with the rounded values - e.g. if you use a rounded off value for x1 to get x2 it may be wrong. Always you should calculate with maximum accuracy and only round off when an answer needs to be given. (a) Apply the Newton-Raphson method to f (x) starting with = -1. Enter your answers below rounded to four decimal places. *1 = Number , *2 = Number, *3 = Number (b) Sometimes, if the initial point is not near a solution, Newton-Raphson method is not effective. Repeat the Newton-Raphson method, but this time starting with = 0. (Also consider graphically what is happening.) Enter your answers below rounded to four decimal places. *1 Number ,2 = Number 23 = Number (c) The function has two other real roots, between 0 and 10. Find initial values for which the Newton-Raphson method approaches each of the other two roots. Enter the other two roots accurate to four decimal places, in any order, separated by a comma, e.g. 1.1111, 5.5556.
Transcribed Image Text:This question concerns the Newton-Raphson method. It can find an accurate solution of an equation, but often needs a good starting-point. The following function has three zeros: f(x) = (1/4)x+cosx. Important Note: while you are asked to give answers to 4 decimal places, you should not calculate with the rounded values - e.g. if you use a rounded off value for x1 to get x2 it may be wrong. Always you should calculate with maximum accuracy and only round off when an answer needs to be given. (a) Apply the Newton-Raphson method to f (x) starting with = -1. Enter your answers below rounded to four decimal places. *1 = Number , *2 = Number, *3 = Number (b) Sometimes, if the initial point is not near a solution, Newton-Raphson method is not effective. Repeat the Newton-Raphson method, but this time starting with = 0. (Also consider graphically what is happening.) Enter your answers below rounded to four decimal places. *1 Number ,2 = Number 23 = Number (c) The function has two other real roots, between 0 and 10. Find initial values for which the Newton-Raphson method approaches each of the other two roots. Enter the other two roots accurate to four decimal places, in any order, separated by a comma, e.g. 1.1111, 5.5556.
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