Explanation Given information: The function is f ( x , y ) = cos x cos y . The point is given as P 0 ( π 4 , π 4 ) . The v vector is v = 3 i + 4 j . Calculation: Show the given function as follows: f ( x , y ) = cos x cos y (1) Partially differentiate Equation (1) with respect to x . ∂ f ∂ x = − sin x cos y Partially differentiate Equation (1) with respect to y . ∂ f ∂ y = − cos x sin y Determine the gradient of the function f ( x , y ) = cos x cos y . ∇ f = ∂ f ∂ x i + ∂ f ∂ y j Substitute − sin x cos y for ∂ f ∂ x and − cos x sin y for ∂ f ∂ y . ∇ f = ( − sin x cos y ) i + ( − cos x sin y ) j Find ∇ f at ( π 4 , π 4 ) : ∇ f | ( π 4 , π 4 ) = − 1 2 i − 1 2 j Find | ∇ f | at ( π 4 , π 4 ) : | ∇ f | = ( − 1 2 ) 2 + ( − 1 2 ) 2 = 1 2 = 2 2 Determine the unit vector u . u = ∇ f | ∇ f | Substitute − 1 2 i − 1 2 j for ∇ f and 2 2 for | ∇ f | . u = − 1 2 i − 1 2 j 2 2 = − 2 2 i − 2 2 j The function f most rapidly increase in the direction u = − 2 2 i − 2 2 j and most rapidly decrease in the direction − u = 2 2 i + 2 2 j . Therefore, the most rapidly increasing direction of the function is u = − 2 2 i − 2 2 j _

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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Question

Chapter 14, Problem 37PE

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Calculate the directions of the given function as it increases and decreases most rapidly.

Calculate the derivative of the given function in the respective directions.

Calculate the derivative of the given function at the point P0(π/4,π/4) in the direction of v.

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Chapter 14, Problem 36PE

Chapter 14, Problem 38PE

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Chapter 14 Solutions

Thomas' Calculus: Early Transcendentals (14th Edition)

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Calculate the directions of the given function as it increases and decreases most rapidly. Calculate the derivative of the given function in the respective directions. Calculate the derivative of the given function at the point P 0 ( π / 4 , π / 4 ) in the direction of v .

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Calculate the directions of the given function as it increases and decreases most rapidly. Calculate the derivative of the given function in the respective directions. Calculate the derivative of the given function at the point P0(π/4,π/4) in the direction of v.

Want to see the full answer?

Chapter 14 Solutions

Calculate the directions of the given function as it increases and decreases most rapidly.

Calculate the derivative of the given function in the respective directions.

Calculate the derivative of the given function at the point P0(π/4,π/4) in the direction of v.