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3. Find the value of the directional derivative at the particular point Po for the given function in the direction of U. (a) g(x, y) = y² tan² r; Ū = -}V3 i+ } j; Po(}7, 2) (6) f(x, y) = re²»; Ū = } i+¿V3 j; Po(2, 0)

3. Find the value of the directional derivative at the particular point Po for the given function in the direction of U. (a) g(x, y) = y² tan² r; Ū = -}V3 i+ } j; Po(}7, 2) (6) f(x, y) = re²»; Ū = } i+¿V3 j; Po(2, 0)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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3. Find the value of the directional derivative at the particular point Po for the given function in the direction of U. (a) g(x, y) = y² tan² x; U = -}V3 î+ } ĵ; Po(}r, 2) (b) f(x, y) = xe²»; U = } î + }V3 î; Po(2,0)
Transcribed Image Text:3. Find the value of the directional derivative at the particular point Po for the given function in the direction of U. (a) g(x, y) = y² tan² x; U = -}V3 î+ } ĵ; Po(}r, 2) (b) f(x, y) = xe²»; U = } î + }V3 î; Po(2,0)
Definition 3.32 If f is a function of two variables x and y such that f. and fy exist, then the gradient of f, denoted by Vƒ (read as “del f"), is given by V{(r,y) = f.(z. y) i + f,(r, y) j. Note that the gradient Vf(x, y) is a vector and Df(1,9) = U - V j(z, y). Suppose a is the radian measure of the angle between U and Vf. Then D7f(z, y) = U·Vf(r, y) = ||U|||Vf(r, y)|| o a. (3.2) Equation (3.2) tells us that D#f(x, y) will be a maximum when cos a = 1; so that we have ||Vf(x, y)|| gives the maximum value of D7f at the point (r, y).
Transcribed Image Text:Definition 3.32 If f is a function of two variables x and y such that f. and fy exist, then the gradient of f, denoted by Vƒ (read as “del f"), is given by V{(r,y) = f.(z. y) i + f,(r, y) j. Note that the gradient Vf(x, y) is a vector and Df(1,9) = U - V j(z, y). Suppose a is the radian measure of the angle between U and Vf. Then D7f(z, y) = U·Vf(r, y) = ||U|||Vf(r, y)|| o a. (3.2) Equation (3.2) tells us that D#f(x, y) will be a maximum when cos a = 1; so that we have ||Vf(x, y)|| gives the maximum value of D7f at the point (r, y).
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