Explanation Given information: f ( x , y , z ) = x 2 y z , P = ( 1 , 5 , 9 ) Calculation: Since the gradient of function is denoted by ∇ f = 〈 ∂ f ∂ x , ∂ f ∂ y , ∂ f ∂ y 〉 The given equation f ( x , y , z ) = x 2 y z , Differentiating f with respect to x we get, ∂ f ∂ x = ∂ ∂ x x 2 y z ⇒ ∂ f ∂ x = y z ∂ ∂ x x 2 ⇒ ∂ f ∂ x = 2 x y z Differentiating f with respect to y we get, ∂ f ∂ y = ∂ ∂ y x 2 y z ⇒ ∂ f ∂ y = x 2 z ∂ ∂ y y = x 2 z Differentiating f with respect to z we get, ∂ f ∂ z = ∂ ∂ z x 2 y z ⇒ ∂ f ∂ z = x 2 y ∂ ∂ z z = x 2 y 1 2 z = x 2 y 2

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Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 14.5, Problem 36E

To determine

Tofind:

the direction in which f has maximum rate of increase from P, and give the rate of change in that direction.

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Chapter 14.5, Problem 35E

Chapter 14.5, Problem 37E

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Chapter 14 Solutions

Calculus: Early Transcendentals

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E

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Tofind: the direction in which f has maximum rate of increase from P , and give the rate of change in that direction.

Solution Summary: The author explains how to determine the direction in which f has maximum rate of increase from P.

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Tofind:

the direction in which f has maximum rate of increase from P, and give the rate of change in that direction.

Want to see the full answer?

Chapter 14 Solutions

Tofind: the direction in which f has maximum rate of increase from P , and give the rate of change in that direction.

Solution Summary: The author explains how to determine the direction in which f has maximum rate of increase from P.

Videos

Tofind: the direction in which f has maximum rate of increase from P, and give the rate of change in that direction.

Want to see the full answer?

Chapter 14 Solutions

Tofind:

the direction in which f has maximum rate of increase from P, and give the rate of change in that direction.