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Find the directional derivative of f(x, y, z) = xy²z³ at P(3, 1, 1) in the direction of Q(0, -5, 7). Duf(3, 1, 1) =

Find the directional derivative of f(x, y, z) = xy²z³ at P(3, 1, 1) in the direction of Q(0, -5, 7). Duf(3, 1, 1) =

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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Question
**Problem Statement:** Find the directional derivative of the function \( f(x, y, z) = xy^2z^3 \) at the point \( P(3, 1, 1) \) in the direction of \( Q(0, -5, 7) \). **Directional Derivative Calculation:** \[ D_u f(3, 1, 1) = \boxed{} \] **Explanation:** To calculate the directional derivative, you need to: 1. Find the gradient vector \( \nabla f(x, y, z) \). 2. Evaluate the gradient at point \( P(3, 1, 1) \). 3. Determine the unit vector in the direction of \( Q(0, -5, 7) \). 4. Compute the dot product of the evaluated gradient and the unit vector. Fill in the box once the calculation is complete.
Transcribed Image Text:**Problem Statement:** Find the directional derivative of the function \( f(x, y, z) = xy^2z^3 \) at the point \( P(3, 1, 1) \) in the direction of \( Q(0, -5, 7) \). **Directional Derivative Calculation:** \[ D_u f(3, 1, 1) = \boxed{} \] **Explanation:** To calculate the directional derivative, you need to: 1. Find the gradient vector \( \nabla f(x, y, z) \). 2. Evaluate the gradient at point \( P(3, 1, 1) \). 3. Determine the unit vector in the direction of \( Q(0, -5, 7) \). 4. Compute the dot product of the evaluated gradient and the unit vector. Fill in the box once the calculation is complete.
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