Explanation Given information: ∇ f ( 0 , 0 ) = 〈 2 , 4 〉 Concept used: For any vector u = 〈 u 1 , u 2 〉 , the directional derivative of z = f ( x , y ) at ( a , b ) in the direction of u is D u f ( a , b ) = f x ( a , b ) u 1 + f y ( a , b ) u 2 Calculation: Since from the given information the unit vector u making an angle of 45 o with the x − axis forms the hypotenuse of the 30 o − 60 o − 90 o triangle whose base is 1unit and height is 1unit. u = 〈 u 1 , u 2 〉 With u 1 = cos 45 o and u 2 = sin 45 o that is, u = 〈 cos 45 o , sin 45 o 〉 ⇒ u = 〈 1 2 , 1 2 〉 Hence u 1 = 1 2 u 2 = 1 2 … (1) We know that the gradient of a function is the vector of partial derivatives

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ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 14.5, Problem 5PQ

To determine

Toevaluate:

the rate of change of f(x,y) at (0,0) in the direction making an angle of 45o with the x− axis.

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Chapter 14.5, Problem 4PQ

Chapter 14.5, Problem 1E

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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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Toevaluate: the rate of change of f ( x , y ) at ( 0 , 0 ) in the direction making an angle of 45 o with the x − axis.

Solution Summary: The author explains how the unit vector f(x,y) forms the hypotenuse of the

Videos

Toevaluate:

the rate of change of f(x,y) at (0,0) in the direction making an angle of 45o with the x− axis.

Want to see the full answer?

Chapter 14 Solutions

Toevaluate: the rate of change of f ( x , y ) at ( 0 , 0 ) in the direction making an angle of 45 o with the x − axis.

Solution Summary: The author explains how the unit vector f(x,y) forms the hypotenuse of the

Videos

Toevaluate: the rate of change of f(x,y) at (0,0) in the direction making an angle of 45o with the x− axis.

Want to see the full answer?

Chapter 14 Solutions

Toevaluate:

the rate of change of f(x,y) at (0,0) in the direction making an angle of 45o with the x− axis.