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

4th Edition

ISBN: 9781319055905

Author: FRANZOSA

Publisher: VST

See similar textbooks

Videos

Question

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.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.5, Problem 4PQ

Chapter 14.5, Problem 1E

Students have asked these similar questions

Use Laplace transform to solve the initial value problem y' + y = tsin(t), y(0) = 0

The function g is defined by g(x) = sec² x + tan x. What are all solutions to g(x) = 1 on the interval 0 ≤ x ≤ 2π ? A x = = 0, x == = 3, x = π, x = 7 4 , 4 and x 2π only = B x = 4' 1, x = 1, x = 57 and x = 3 only C x = πk and x = - +πk D , where is any integer П x = +πk and П x = +πk, where k is any integer

Vector v = PQ has initial point P (2, 14) and terminal point Q (7, 3). Vector v = RS has initial point R (29, 8) and terminal point S (12, 17). Part A: Write u and v in linear form. Show all necessary work. Part B: Write u and v in trigonometric form. Show all necessary work. Part C: Find 7u − 4v. Show all necessary calculations.

Chapter 14 Solutions

EBK 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

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

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.