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4. Find (a) the gradient of f at P and (b) the rate of change of the function value in the direction of U at P. (a) f(x, y) = a² – 4y; U = cos n î + sin r î; P(-2,2)

4. Find (a) the gradient of f at P and (b) the rate of change of the function value in the direction of U at P. (a) f(x, y) = a² – 4y; U = cos n î + sin r î; P(-2,2)

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter2: Functions And Graphs
Section2.6: Proportion And Variation
Problem 22E: Find the constant of proportionality. z is directly proportional to the sum of x and y. If x=2 and...
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4. Find (a) the gradient of f at P and (b) the rate of change of the function value in the direction of U at P. (a) f(x, y) = r² – 4y; U = cos}r î + sin }r î; P(-2, 2)
Transcribed Image Text:4. Find (a) the gradient of f at P and (b) the rate of change of the function value in the direction of U at P. (a) f(x, y) = r² – 4y; U = cos}r î + sin }r î; P(-2, 2)
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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