Explanation Given information: The function is f ( x , y ) = x y 2 + y cos ( x − 1 ) . The given point is P 0 ( 1 , 2 ) . The error in the approximation f ( x , y ) ≈ L ( x , y ) is given below: R : | x − 1 | ≤ 0.1 , | y − 2 | ≤ 0.1 Calculation: Show the function as follows: f ( x , y ) = x y 2 + y cos ( x − 1 ) Find the value of f at the point P 0 ( 1 , 2 ) : f ( 1 , 2 ) = ( 1 ) ( 2 ) 2 + 2 cos ( 1 − 1 ) = 4 + 2 = 6 Find the value of f x at the point P 0 ( 1 , 2 ) : f x ( 1 , 2 ) = ∂ ∂ x ( x y 2 + y cos ( x − 1 ) ) | ( 1 , 2 ) = ( y 2 − y sin ( x − 1 ) ) | ( 1 , 2 ) = 2 2 − 2 sin ( 1 − 1 ) = 4 Find the value of f y at the point P 0 ( 1 , 2 ) : f y ( 1 , 2 ) = ∂ ∂ y ( x y 2 + y cos ( x − 1 ) ) | ( 1 , 2 ) = ( 2 x y + cos ( x − 1 ) ) | ( 1 , 2 ) = 2 ( 1 ) ( 2 ) + cos ( 1 − 1 ) = 5 The linearization of the function f ( x , y ) at the point ( x 0 , y 0 ) is expressed as follows: L ( x , y ) = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x − x 0 ) + f y ( x 0 , y 0 ) ( y − y 0 ) Substitute 6 for f ( x 0 , y 0 ) , 4 for f x ( x 0 , y 0 ) , 1 for x 0 , 5 for f y ( x 0 , y 0 ) , and 2 for y 0 . L ( x , y ) = 6 + 4 ( x − 1 ) + 5 ( y − 2 ) = 6 + 4 x − 4 + 5 y − 10 = 4 x + 5 y − 8 Therefore, the linearization L ( x , y ) of the function is 4 x + 5 y − 8 _ . Given the range of x is | x − 1 | ≤ 0.1 . It can be expressed as, | x − 1 | ≤ 0

Thomas' Calculus: Early Transcendentals (14th Edition)

14th Edition

ISBN: 9780134439020

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 14.6, Problem 38E

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Determine the linearization of the function at the given point.

Determine the error incurred in the approximation.

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Chapter 14.6, Problem 37E

Chapter 14.6, Problem 39E

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Chapter 14 Solutions

Thomas' Calculus: Early Transcendentals (14th Edition)

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Determine the linearization of the function at the given point. Determine the error incurred in the approximation.

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Determine the linearization of the function at the given point. Determine the error incurred in the approximation.

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Chapter 14 Solutions

Determine the linearization of the function at the given point.

Determine the error incurred in the approximation.