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Math Calculus CALCULUS (CLOTH)

We need to show for a given n data points ( x 1 , y 1 ) , ... , ( x n , y n ) , the linear least square fit is the linear function f ( x ) = m x + b that minimizes the sum of squares E ( m , b ) = ∑ j = 1 n ( y j − f ( x j ) ) 2 , the minimum value of E occurs for m and b satisfying m ( ∑ j = 1 n x j ) + b n = ∑ j = 1 n y j m ( ∑ j = 1 n x j 2 ) + b ∑ j = 1 n x j = ∑ j = 1 n x j y j

We need to show for a given n data points ( x 1 , y 1 ) , ... , ( x n , y n ) , the linear least square fit is the linear function f ( x ) = m x + b that minimizes the sum of squares E ( m , b ) = ∑ j = 1 n ( y j − f ( x j ) ) 2 , the minimum value of E occurs for m and b satisfying m ( ∑ j = 1 n x j ) + b n = ∑ j = 1 n y j m ( ∑ j = 1 n x j 2 ) + b ∑ j = 1 n x j = ∑ j = 1 n x j y j

Solution Summary: The author shows the linear least square fit for a given n data points. The minimum value of E occurs for both m and b, and the extreme values occur at critical points in the interior of D.

CALCULUS (CLOTH)

CALCULUS (CLOTH)

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

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Chapter 15.7, Problem 60E

To determine

We need to show for a given $n$ data points $(x_{1}, y_{1}), ..., (x_{n}, y_{n})$ , the linear least square fit is the linear function $f (x) = m x + b$ that minimizes the sum of squares $E (m, b) = \sum_{j = 1}^{n} {(y_{j} - f (x_{j}))}^{2}$ , the minimum value of $E$ occurs for $m and b$ satisfying

$\begin{array}{l} m (\sum_{j = 1}^{n} x_{j}) + b n = \sum_{j = 1}^{n} y_{j} \\ m (\sum_{j = 1}^{n} x_{j}^{2}) + b \sum_{j = 1}^{n} x_{j} = \sum_{j = 1}^{n} x_{j} y_{j} \end{array}$

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Chapter 15.7, Problem 59E

Chapter 15.7, Problem 61E

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Chapter 15 Solutions

CALCULUS (CLOTH)

Ch. 15.1 - Prob. 1PQ Ch. 15.1 - Prob. 2PQ Ch. 15.1 - Prob. 3PQ Ch. 15.1 - Prob. 4PQ Ch. 15.1 - Prob. 5PQ Ch. 15.1 - Prob. 1E Ch. 15.1 - Prob. 2E Ch. 15.1 - Prob. 3E Ch. 15.1 - Prob. 4E Ch. 15.1 - Prob. 5E

Ch. 15.1 - Prob. 6E Ch. 15.1 - Prob. 7E Ch. 15.1 - Prob. 8E Ch. 15.1 - Prob. 9E Ch. 15.1 - Prob. 10E Ch. 15.1 - Prob. 11E Ch. 15.1 - Prob. 12E Ch. 15.1 - Prob. 13E Ch. 15.1 - Prob. 14E Ch. 15.1 - Prob. 15E Ch. 15.1 - Prob. 16E Ch. 15.1 - Prob. 17E Ch. 15.1 - Prob. 18E Ch. 15.1 - Prob. 19E Ch. 15.1 - Prob. 20E Ch. 15.1 - Prob. 21E Ch. 15.1 - Prob. 22E Ch. 15.1 - Prob. 23E Ch. 15.1 - Prob. 24E Ch. 15.1 - Prob. 25E Ch. 15.1 - Prob. 26E Ch. 15.1 - Prob. 27E Ch. 15.1 - Prob. 28E Ch. 15.1 - Prob. 29E Ch. 15.1 - Prob. 30E Ch. 15.1 - Prob. 31E Ch. 15.1 - Prob. 32E Ch. 15.1 - Prob. 33E Ch. 15.1 - Prob. 34E Ch. 15.1 - Prob. 35E Ch. 15.1 - Prob. 36E Ch. 15.1 - Prob. 37E Ch. 15.1 - Prob. 38E Ch. 15.1 - Prob. 39E Ch. 15.1 - Prob. 40E Ch. 15.1 - Prob. 41E Ch. 15.1 - Prob. 42E Ch. 15.1 - Prob. 43E Ch. 15.1 - Prob. 44E Ch. 15.1 - Prob. 45E Ch. 15.1 - Prob. 46E Ch. 15.1 - Prob. 47E Ch. 15.1 - Prob. 48E Ch. 15.1 - Prob. 49E Ch. 15.1 - Prob. 50E Ch. 15.1 - Prob. 51E Ch. 15.1 - Prob. 52E Ch. 15.1 - Prob. 53E Ch. 15.1 - Prob. 54E Ch. 15.1 - Prob. 55E Ch. 15.1 - Prob. 56E Ch. 15.1 - Prob. 57E Ch. 15.1 - Prob. 58E Ch. 15.1 - Prob. 59E Ch. 15.1 - Prob. 60E Ch. 15.1 - Prob. 61E Ch. 15.2 - Prob. 1PQ Ch. 15.2 - Prob. 2PQ Ch. 15.2 - Prob. 3PQ Ch. 15.2 - Prob. 4PQ Ch. 15.2 - Prob. 1E Ch. 15.2 - Prob. 2E Ch. 15.2 - Prob. 3E Ch. 15.2 - Prob. 4E Ch. 15.2 - Prob. 5E Ch. 15.2 - Prob. 6E Ch. 15.2 - Prob. 7E Ch. 15.2 - Prob. 8E Ch. 15.2 - Prob. 9E Ch. 15.2 - Prob. 10E Ch. 15.2 - Prob. 11E Ch. 15.2 - Prob. 12E Ch. 15.2 - Prob. 13E Ch. 15.2 - Prob. 14E Ch. 15.2 - Prob. 15E Ch. 15.2 - Prob. 16E Ch. 15.2 - Prob. 17E Ch. 15.2 - Prob. 18E Ch. 15.2 - Prob. 19E Ch. 15.2 - Prob. 20E Ch. 15.2 - Prob. 21E Ch. 15.2 - Prob. 22E Ch. 15.2 - Prob. 23E Ch. 15.2 - Prob. 24E Ch. 15.2 - Prob. 25E Ch. 15.2 - Prob. 26E Ch. 15.2 - Prob. 27E Ch. 15.2 - Prob. 28E Ch. 15.2 - Prob. 29E Ch. 15.2 - Prob. 30E Ch. 15.2 - Prob. 31E Ch. 15.2 - Prob. 32E Ch. 15.2 - Prob. 33E Ch. 15.2 - Prob. 34E Ch. 15.2 - Prob. 35E Ch. 15.2 - Prob. 36E Ch. 15.2 - Prob. 37E Ch. 15.2 - 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Prob. 79E Ch. 15.3 - Prob. 80E Ch. 15.4 - Prob. 1PQ Ch. 15.4 - Prob. 2PQ Ch. 15.4 - Prob. 3PQ Ch. 15.4 - Prob. 4PQ Ch. 15.4 - Prob. 5PQ Ch. 15.4 - Prob. 6PQ Ch. 15.4 - Prob. 1E Ch. 15.4 - Prob. 2E Ch. 15.4 - Prob. 3E Ch. 15.4 - Prob. 4E Ch. 15.4 - Prob. 5E Ch. 15.4 - Prob. 6E Ch. 15.4 - Prob. 7E Ch. 15.4 - Prob. 8E Ch. 15.4 - Prob. 9E Ch. 15.4 - Prob. 10E Ch. 15.4 - Prob. 11E Ch. 15.4 - Prob. 12E Ch. 15.4 - Prob. 13E Ch. 15.4 - Prob. 14E Ch. 15.4 - Prob. 15E Ch. 15.4 - Prob. 16E Ch. 15.4 - Prob. 17E Ch. 15.4 - Prob. 18E Ch. 15.4 - Prob. 19E Ch. 15.4 - Prob. 20E Ch. 15.4 - Prob. 21E Ch. 15.4 - Prob. 22E Ch. 15.4 - Prob. 23E Ch. 15.4 - Prob. 24E Ch. 15.4 - Prob. 25E Ch. 15.4 - Prob. 26E Ch. 15.4 - Prob. 27E Ch. 15.4 - Prob. 28E Ch. 15.4 - Prob. 29E Ch. 15.4 - Prob. 30E Ch. 15.4 - Prob. 31E Ch. 15.4 - Prob. 32E Ch. 15.4 - Prob. 33E Ch. 15.4 - Prob. 34E Ch. 15.4 - Prob. 35E Ch. 15.4 - Prob. 36E Ch. 15.4 - Prob. 37E Ch. 15.4 - Prob. 38E Ch. 15.4 - Prob. 39E Ch. 15.4 - Prob. 40E Ch. 15.4 - Prob. 41E Ch. 15.4 - Prob. 42E Ch. 15.4 - Prob. 43E Ch. 15.4 - Prob. 44E Ch. 15.4 - Prob. 45E Ch. 15.4 - Prob. 46E Ch. 15.4 - Prob. 47E Ch. 15.5 - Prob. 1PQ Ch. 15.5 - Prob. 2PQ Ch. 15.5 - Prob. 3PQ Ch. 15.5 - Prob. 4PQ Ch. 15.5 - Prob. 5PQ Ch. 15.5 - Prob. 1E Ch. 15.5 - Prob. 2E Ch. 15.5 - Prob. 3E Ch. 15.5 - Prob. 4E Ch. 15.5 - Prob. 5E Ch. 15.5 - Prob. 6E Ch. 15.5 - Prob. 7E Ch. 15.5 - Prob. 8E Ch. 15.5 - Prob. 9E Ch. 15.5 - Prob. 10E Ch. 15.5 - Prob. 11E Ch. 15.5 - Prob. 12E Ch. 15.5 - Prob. 13E Ch. 15.5 - Prob. 14E Ch. 15.5 - Prob. 15E Ch. 15.5 - Prob. 16E Ch. 15.5 - Prob. 17E Ch. 15.5 - Prob. 18E Ch. 15.5 - Prob. 19E Ch. 15.5 - Prob. 20E Ch. 15.5 - Prob. 21E Ch. 15.5 - Prob. 22E Ch. 15.5 - Prob. 23E Ch. 15.5 - Prob. 24E Ch. 15.5 - Prob. 25E Ch. 15.5 - Prob. 26E Ch. 15.5 - Prob. 27E Ch. 15.5 - Prob. 28E Ch. 15.5 - Prob. 29E Ch. 15.5 - Prob. 30E Ch. 15.5 - Prob. 31E Ch. 15.5 - Prob. 32E Ch. 15.5 - Prob. 33E Ch. 15.5 - Prob. 34E Ch. 15.5 - Prob. 35E Ch. 15.5 - Prob. 36E Ch. 15.5 - Prob. 37E Ch. 15.5 - Prob. 38E Ch. 15.5 - 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Prob. 53E Ch. 15.6 - Prob. 54E Ch. 15.6 - Prob. 55E Ch. 15.7 - Prob. 1PQ Ch. 15.7 - Prob. 2PQ Ch. 15.7 - Prob. 3PQ Ch. 15.7 - Prob. 1E Ch. 15.7 - Prob. 2E Ch. 15.7 - Prob. 3E Ch. 15.7 - Prob. 4E Ch. 15.7 - Prob. 5E Ch. 15.7 - Prob. 6E Ch. 15.7 - Prob. 7E Ch. 15.7 - Prob. 8E Ch. 15.7 - Prob. 9E Ch. 15.7 - Prob. 10E Ch. 15.7 - Prob. 11E Ch. 15.7 - Prob. 12E Ch. 15.7 - Prob. 13E Ch. 15.7 - Prob. 14E Ch. 15.7 - Prob. 15E Ch. 15.7 - Prob. 16E Ch. 15.7 - Prob. 17E Ch. 15.7 - Prob. 18E Ch. 15.7 - Prob. 19E Ch. 15.7 - Prob. 20E Ch. 15.7 - Prob. 21E Ch. 15.7 - Prob. 22E Ch. 15.7 - Prob. 23E Ch. 15.7 - Prob. 24E Ch. 15.7 - Prob. 25E Ch. 15.7 - Prob. 26E Ch. 15.7 - Prob. 27E Ch. 15.7 - Prob. 28E Ch. 15.7 - Prob. 29E Ch. 15.7 - Prob. 30E Ch. 15.7 - Prob. 31E Ch. 15.7 - Prob. 32E Ch. 15.7 - Prob. 33E Ch. 15.7 - Prob. 34E Ch. 15.7 - Prob. 35E Ch. 15.7 - Prob. 36E Ch. 15.7 - Prob. 37E Ch. 15.7 - Prob. 38E Ch. 15.7 - Prob. 39E Ch. 15.7 - Prob. 40E Ch. 15.7 - Prob. 41E Ch. 15.7 - Prob. 42E Ch. 15.7 - Prob. 43E Ch. 15.7 - 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