Explanation Given information: The function w = ( m x 1 + b − y 1 ) 2 + ⋯ + ( m x n + b − y n ) 2 . The value of m = ( ∑ x k ) ( ∑ y k ) − n ∑ x k y k ( ∑ x k ) 2 − n ∑ x k 2 . The value of b = 1 n ( ∑ y k − m ∑ x k ) . Calculation: Provide the function as shown below. w = ( m x 1 + b − y 1 ) 2 + ( m x 2 + b − y 2 ) 2 + ⋯ + ( m x n + b − y n ) 2 (1) Differentiate Equation (1) with respect to b . ∂ w ∂ b = 2 ( m x 1 + b − y 1 ) + 2 ( m x 2 + b − y 2 ) + ⋯ + 2 ( m x n + b − y n ) = 2 [ ( m x 1 + b − y 1 ) + ( m x 2 + b − y 2 ) + ⋯ + ( m x n + b − y n ) ] = 2 [ m ( x 1 + x 2 + ⋯ + x n ) + n b − ( y 1 + y 2 + ⋯ + y n ) ] Equate the partial derivative ∂ w ∂ b = 0 . 2 [ m ( x 1 + x 2 + ⋯ + x n ) + n b − ( y 1 + y 2 + ⋯ + y n ) ] = 0 m ( x 1 + x 2 + ⋯ + x n ) + n b − ( y 1 + y 2 + ⋯ + y n ) = 0 m ∑ k = 1 n x k + n b − ∑ k = 1 n y k = 0 n b = ∑ k = 1 n y k − m ∑ k = 1 n x k b = 1 n ( ∑ k = 1 n y k − m ∑ k = 1 n x k ) Hence, the value of b that minimize the function w is 1 n ( ∑ k = 1 n y k − m ∑ k = 1 n x k ) . Differentiate Equation (1) with respect to m . ∂ w ∂ m = 2 ( m x 1 + b − y 1 ) ( x 1 ) + 2 ( m x 2 + b − y 2 ) ( x 2 ) + ⋯ + 2 ( m x n + b − y n ) ( x n ) = 2 [ ( m x 1 2 + b x 1 − x 1 y 1 ) + ( m x 2 2 + b x 2 − x 2 y 2 ) + ⋯ + ( m x n 2 + b x n − x n y n ) ] = 2 [ ( m x 1 2 + b x 1 − x 1 y 1 ) + ( m x 2 2 + b x 2 − x 2 y 2 ) + ⋯ + ( m x n 2 + b x n − x n y n ) ] = 2 [ m ( x 1 2 + x 2 2 + ⋯ + x n 2 ) + b ( x 1 + x 2 + ⋯ + x n ) − ( x 1 y 1 + x 2 y 2 + ⋯ + x n y n ) ] Equate the partial derivative ∂ w ∂ m = 0

14th Edition

ISBN: 9780136208013

Author: Hass

Publisher: PEARSON

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Chapter 14.7, Problem 67E

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Prove that the values of m and b minimize the value of the function w.

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Chapter 14.7, Problem 66E

Chapter 14.7, Problem 68E

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Chapter 14 Solutions

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Prove that the values of m and b minimize the value of the function w .

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Prove that the values of m and b minimize the value of the function w.

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Chapter 14 Solutions