Least squares approximation is a method for finding the line with minimum vertical distance from a set of points. Let the equation of the best-fit line be y = mx + b, where the slope m and the y-intercept b must be determined using the least squares condition. Assume there are three data points (1,2), (2,3), and (3,5). The following function E(m,b) represents the sums of the squares of the vertical distances of each point from the best-fit line. Find the critical points of E and find the values of m and b that minimize E. Graph the three data points and the best-fit line. E(m,b) = [(m +b) - 2]² + [(2m + b) − 3]² + [(3m + b) - 5]² y (Xn-1' Yn-1) Em (m,b) = Eb (m,b) = (Simplify your answer. Do not factor.) regression line (x₁, y₁) (X₂² Y₂) To find the critical points, it will first be necessary to find the values of the partial derivatives Em (m,b) and Eb (m,b). Find these values. (Xn² Yn) •(X3, Y3)
Least squares approximation is a method for finding the line with minimum vertical distance from a set of points. Let the equation of the best-fit line be y = mx + b, where the slope m and the y-intercept b must be determined using the least squares condition. Assume there are three data points (1,2), (2,3), and (3,5). The following function E(m,b) represents the sums of the squares of the vertical distances of each point from the best-fit line. Find the critical points of E and find the values of m and b that minimize E. Graph the three data points and the best-fit line. E(m,b) = [(m +b) - 2]² + [(2m + b) − 3]² + [(3m + b) - 5]² y (Xn-1' Yn-1) Em (m,b) = Eb (m,b) = (Simplify your answer. Do not factor.) regression line (x₁, y₁) (X₂² Y₂) To find the critical points, it will first be necessary to find the values of the partial derivatives Em (m,b) and Eb (m,b). Find these values. (Xn² Yn) •(X3, Y3)
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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![**Least Squares Approximation**
Least squares approximation is a method for finding the line with the minimum vertical distance from a set of points. The equation of the best-fit line is given by \( y = mx + b \), where the slope \( m \) and the y-intercept \( b \) must be determined using the least squares condition. Assume there are three data points \( (1,2), (2,3), \) and \( (3,5) \). The function \( E(m,b) \) represents the sum of the squares of the vertical distances of each point from the best-fit line.
To find the critical points of \( E \) and determine the values of \( m \) and \( b \) that minimize \( E \), and to graph the three data points and the best-fit line, use the following equation:
\[
E(m,b) = [(m + b) - 2]^2 + [(2m + b) - 3]^2 + [(3m + b) - 5]^2
\]
**Graph Explanation**
The accompanying graph depicts a regression line (best-fit line) with plotted data points represented as pink dots. The x-axis and y-axis are labeled, with each data point denoted as \( (x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n) \).
**Finding Critical Points**
To find the critical points, it is necessary to compute the partial derivatives, \( E_m(m,b) \) and \( E_b(m,b) \). These are required to be determined as part of the solution:
\[
E_m(m,b) = \, \boxed{\phantom{a}}
\]
\[
E_b(m,b) = \, \boxed{\phantom{a}}
\]
(Simplify your answer. Do not factor.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c944127-f965-40a6-9c0b-c6a92e7a1d8e%2F7d27b471-d42d-4cb0-9f7d-67afdac165f9%2Ft7iugzbm_processed.png&w=3840&q=75)
Transcribed Image Text:**Least Squares Approximation**
Least squares approximation is a method for finding the line with the minimum vertical distance from a set of points. The equation of the best-fit line is given by \( y = mx + b \), where the slope \( m \) and the y-intercept \( b \) must be determined using the least squares condition. Assume there are three data points \( (1,2), (2,3), \) and \( (3,5) \). The function \( E(m,b) \) represents the sum of the squares of the vertical distances of each point from the best-fit line.
To find the critical points of \( E \) and determine the values of \( m \) and \( b \) that minimize \( E \), and to graph the three data points and the best-fit line, use the following equation:
\[
E(m,b) = [(m + b) - 2]^2 + [(2m + b) - 3]^2 + [(3m + b) - 5]^2
\]
**Graph Explanation**
The accompanying graph depicts a regression line (best-fit line) with plotted data points represented as pink dots. The x-axis and y-axis are labeled, with each data point denoted as \( (x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n) \).
**Finding Critical Points**
To find the critical points, it is necessary to compute the partial derivatives, \( E_m(m,b) \) and \( E_b(m,b) \). These are required to be determined as part of the solution:
\[
E_m(m,b) = \, \boxed{\phantom{a}}
\]
\[
E_b(m,b) = \, \boxed{\phantom{a}}
\]
(Simplify your answer. Do not factor.)
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