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K Consider the function f(x,y,z) = 2 + 2xyz, the point P(1,1,-1), and the unit vector u = a. Compute the gradient of f and evaluate it at P. b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. a. What is the gradient at the point (1,1, - 1)? b. What is the unit vector in the direction of maximum increase? (Type exact answers, using radicals as needed.) c. What is the rate of change in the direction of maximum increase? (Type an exact answer, using radicals as needed.) d. What is the directional derivative in the direction of the given vector? (Type an exact answer, using radicals as needed.) √√3 1 1

K Consider the function f(x,y,z) = 2 + 2xyz, the point P(1,1,-1), and the unit vector u = a. Compute the gradient of f and evaluate it at P. b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. a. What is the gradient at the point (1,1, - 1)? b. What is the unit vector in the direction of maximum increase? (Type exact answers, using radicals as needed.) c. What is the rate of change in the direction of maximum increase? (Type an exact answer, using radicals as needed.) d. What is the directional derivative in the direction of the given vector? (Type an exact answer, using radicals as needed.) √√3 1 1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 21E
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K Consider the function f(x,y,z) = 2 + 2xyz, the point P(1,1,-1), and the unit vector u = a. Compute the gradient of f and evaluate it at P. b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. a. What is the gradient at the point (1,1, - 1)? b. What is the unit vector in the direction of maximum increase? (Type exact answers, using radicals as needed.) c. What is the rate of change in the direction of maximum increase? (Type an exact answer, using radicals as needed.) d. What is the directional derivative in the direction of the given vector? (Type an exact answer, using radicals as needed.) √√3 1 √3 1
Transcribed Image Text:K Consider the function f(x,y,z) = 2 + 2xyz, the point P(1,1,-1), and the unit vector u = a. Compute the gradient of f and evaluate it at P. b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. a. What is the gradient at the point (1,1, - 1)? b. What is the unit vector in the direction of maximum increase? (Type exact answers, using radicals as needed.) c. What is the rate of change in the direction of maximum increase? (Type an exact answer, using radicals as needed.) d. What is the directional derivative in the direction of the given vector? (Type an exact answer, using radicals as needed.) √√3 1 √3 1
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