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In the video, we introduced Newton's Method by considering the function )-x +1 and trying to approximate the right-most root of the function, as shown in the graph below. From the graph, we determined a rough estimate for this root to be x1, but we know we can make this approximation better, by bringing in some of the other ideas for approximation that we discussed previously. We know that if we "zoom in" enough near a point, any "nice" function will look roughly linear, so it can be approximated by its tangent line, or the linearization at that point at x = 1 , and use this Since our initial approximation for the root was 1, we create the linearization to approximate the zero that we are looking for. Then we continue to repeat this process with each new value to refine our approximation. Let's run through this one more time: a) Calculate the linearization ofx) - -3x+1 atx-1 b) Use the linearization found in part (a) to estimate the value where x)-0 To find each successive approximation, we just set the linearization at the previous approximation, X equal to zero, and solve forx c) Solve the general linearization equation below for x to show where Newton's formula comes from

In the video, we introduced Newton's Method by considering the function )-x +1 and trying to approximate the right-most root of the function, as shown in the graph below. From the graph, we determined a rough estimate for this root to be x1, but we know we can make this approximation better, by bringing in some of the other ideas for approximation that we discussed previously. We know that if we "zoom in" enough near a point, any "nice" function will look roughly linear, so it can be approximated by its tangent line, or the linearization at that point at x = 1 , and use this Since our initial approximation for the root was 1, we create the linearization to approximate the zero that we are looking for. Then we continue to repeat this process with each new value to refine our approximation. Let's run through this one more time: a) Calculate the linearization ofx) - -3x+1 atx-1 b) Use the linearization found in part (a) to estimate the value where x)-0 To find each successive approximation, we just set the linearization at the previous approximation, X equal to zero, and solve forx c) Solve the general linearization equation below for x to show where Newton's formula comes from

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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In the video, we introduced Newton's Method by considering the function )-x +1 and trying to approximate the right-most root of the function, as shown in the graph below. From the graph, we determined a rough estimate for this root to be x1, but we know we can make this approximation better, by bringing in some of the other ideas for approximation that we discussed previously. We know that if we "zoom in" enough near a point, any "nice" function will look roughly linear, so it can be approximated by its tangent line, or the linearization at that point at x = 1 , and use this Since our initial approximation for the root was 1, we create the linearization to approximate the zero that we are looking for. Then we continue to repeat this process with each new value to refine our approximation. Let's run through this one more time: a) Calculate the linearization ofx) - -3x+1 atx-1 b) Use the linearization found in part (a) to estimate the value where x)-0 To find each successive approximation, we just set the linearization at the previous approximation, X equal to zero, and solve forx c) Solve the general linearization equation below for x to show where Newton's formula comes from
Transcribed Image Text:In the video, we introduced Newton's Method by considering the function )-x +1 and trying to approximate the right-most root of the function, as shown in the graph below. From the graph, we determined a rough estimate for this root to be x1, but we know we can make this approximation better, by bringing in some of the other ideas for approximation that we discussed previously. We know that if we "zoom in" enough near a point, any "nice" function will look roughly linear, so it can be approximated by its tangent line, or the linearization at that point at x = 1 , and use this Since our initial approximation for the root was 1, we create the linearization to approximate the zero that we are looking for. Then we continue to repeat this process with each new value to refine our approximation. Let's run through this one more time: a) Calculate the linearization ofx) - -3x+1 atx-1 b) Use the linearization found in part (a) to estimate the value where x)-0 To find each successive approximation, we just set the linearization at the previous approximation, X equal to zero, and solve forx c) Solve the general linearization equation below for x to show where Newton's formula comes from
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