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Math Calculus ACHIEVE STANDALONE ACCESS F/ CALC 4E

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.

ACHIEVE STANDALONE ACCESS F/ CALC 4E

ACHIEVE STANDALONE ACCESS F/ CALC 4E

4th Edition

ISBN: 9781319434540

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

ACHIEVE STANDALONE ACCESS F/ CALC 4E

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

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