Explanation Given information: The function is f ( x , y , z ) = ln ( 2 x + 3 y + 6 z ) . The point is given as P 0 ( − 1 , − 1 , 1 ) . The v vector is v = 2 i + 3 j + 6 k . Calculation: Show the given function as follows: f ( x , y , z ) = ln ( 2 x + 3 y + 6 z ) (1) Partially differentiate Equation (1) with respect to x . ∂ f ∂ x = 2 2 x + 3 y + 6 z Partially differentiate Equation (1) with respect to y . ∂ f ∂ y = 3 2 x + 3 y + 6 z Partially differentiate Equation (1) with respect to z . ∂ f ∂ z = 6 2 x + 3 y + 6 z Determine the gradient of the function f ( x , y , z ) = ln ( 2 x + 3 y + 6 z ) using the relation. ∇ f = ∂ f ∂ x i + ∂ f ∂ y j + ∂ f ∂ z k Substitute 2 2 x + 3 y + 6 z for ∂ f ∂ x , 3 2 x + 3 y + 6 z for ∂ f ∂ y , and 6 2 x + 3 y + 6 z for ∂ f ∂ z . ∇ f = ( 2 2 x + 3 y + 6 z ) i + ( 3 2 x + 3 y + 6 z ) j + ( 6 2 x + 3 y + 6 z ) k Find ∇ f at ( − 1 , − 1 , 1 ) : ∇ f | ( − 1 , − 1 , 1 ) = ( 2 − 2 − 3 + 6 ) i + ( 3 − 2 − 3 + 6 ) j + ( 6 − 2 − 3 + 6 ) k = 2 i + 3 j + 6 k Find | ∇ f | at ( − 1 , − 1 , 1 ) : | ∇ f | = ( 2 ) 2 + ( 3 ) 2 + ( 6 2 ) = 7 Determine the unit vector u . u = ∇ f | ∇ f | Substitute 2 i + 3 j + 6 k for ∇ f and 7 for | ∇ f | . u = 2 i + 3 j + 6 k 7 = 2 7 i + 3 7 j + 6 7 k The function f most rapidly increase in the direction u = 2 7 i + 3 7 j + 6 7 k and most rapidly decrease in the direction − u = − 2 7 i − 3 7 j − 6 7 k

Thomas' Calculus: Early Transcendentals in SI Units

14th Edition

ISBN: 9781292253114

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

Publisher: PEARSON

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Chapter 14, Problem 39PE

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Find the directions of the given function as it increases and decreases most quickly.

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

Find the derivative of the given function at the point P0 in the direction of v.

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Chapter 14, Problem 38PE

Chapter 14, Problem 40PE

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Chapter 14 Solutions

Thomas' Calculus: Early Transcendentals in SI Units

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Find the directions of the given function as it increases and decreases most quickly. Find the derivative of the given function in the respective directions. Find the derivative of the given function at the point P 0 in the direction of v .

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Find the directions of the given function as it increases and decreases most quickly. Find the derivative of the given function in the respective directions. Find the derivative of the given function at the point P0 in the direction of v.

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Chapter 14 Solutions

Find the directions of the given function as it increases and decreases most quickly.

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

Find the derivative of the given function at the point P0 in the direction of v.