Explanation Given information: f ( x , y ) = x e x 2 − y , P = ( 1 , 1 ) Calculation: Since the gradient of function is denoted by ∇ f = 〈 ∂ f ∂ x , ∂ f ∂ y 〉 The given equation f ( x , y ) = x e x 2 − y , Differentiating f with respect to x we get, ∂ f ∂ x = ∂ ∂ x x e x 2 − y ⇒ ∂ f ∂ x = x ∂ ∂ x e x 2 − y + e x 2 − y ∂ ∂ x x ⇒ ∂ f ∂ x = x e x 2 − y ( 2 x − 0 ) + e x 2 − y ( 1 ) ⇒ ∂ f ∂ x = 2 x 2 e x 2 − y + e x 2 − y = e x 2 − y ( 2 x 2 + 1 ) Differentiating f with respect to y we get, ∂ f ∂ y = ∂ ∂ y x e x 2 − y ⇒ ∂ f ∂ y = x ∂ ∂ y e To determine (b) Tocalculate: rate of change of f in the direction ∇ f P . To determine (c) Tocalculate: rate of change of f in the directionof a vector making an angle of 45 o with ∇ f P .

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 14.5, Problem 38E

To determine

(a)

Tocalculate:

‖∇fP‖

To determine

(b)

Tocalculate:

rate of change of f in the direction ∇fP.

To determine

(c)

Tocalculate:

rate of change of f in the directionof a vector making an angle of 45o with ∇fP.

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Chapter 14.5, Problem 37E

Chapter 14.5, Problem 39E

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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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(a) Tocalculate: ‖ ∇ f P ‖

Solution Summary: The author explains that the gradient of function is denoted by f(x,y)=xex

Videos

(a)

Tocalculate:

‖∇fP‖

(b)

Tocalculate:

rate of change of f in the direction ∇fP.

(c)

Tocalculate:

rate of change of f in the directionof a vector making an angle of 45o with ∇fP.

Want to see the full answer?

Chapter 14 Solutions

(a) Tocalculate: ‖ ∇ f P ‖

Solution Summary: The author explains that the gradient of function is denoted by f(x,y)=xex

Videos

(a) Tocalculate: ‖∇fP‖

(b) Tocalculate: rate of change of f in the direction ∇fP.

(c) Tocalculate: rate of change of f in the directionof a vector making an angle of 45o with ∇fP.

Want to see the full answer?

Chapter 14 Solutions

(a)

Tocalculate:

‖∇fP‖

(b)

Tocalculate:

rate of change of f in the direction ∇fP.

(c)

Tocalculate:

rate of change of f in the directionof a vector making an angle of 45o with ∇fP.