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Consider the function f(x.y.z) =x + 2y? + 3z? +2, the point P(4,0,4), 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.

Consider the function f(x.y.z) =x + 2y? + 3z? +2, the point P(4,0,4), 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.

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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Consider the function f(x.y.z) =x +2y + 3z + 2, the point P(4,0,4), 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 off 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.
Transcribed Image Text:Consider the function f(x.y.z) =x +2y + 3z + 2, the point P(4,0,4), 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 off 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.
(4,0,4)7 a. What is the gradient at the point 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.)
Transcribed Image Text:(4,0,4)7 a. What is the gradient at the point 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.)
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