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Find the maximum rate of change of f(x, y, z) = tan(2x + 7y + 5z) at the point (- 1, 1, 2).

Find the maximum rate of change of f(x, y, z) = tan(2x + 7y + 5z) at the point (- 1, 1, 2).

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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**Topic: Calculus - Multivariable Functions** ## Problem Statement Find the maximum rate of change of \( f(x, y, z) = \tan(2x + 7y + 5z) \) at the point \((-1, 1, 2)\). --- To approach this problem, you need to calculate the gradient vector of the function \( f(x, y, z) \). The gradient vector, denoted as \( \nabla f \), represents the direction of the steepest ascent and its magnitude will give you the maximum rate of change at a particular point. Detailed steps typically involve: 1. **Finding Partial Derivatives**: Compute the partial derivatives of the function with respect to each variable \( x, y, \) and \( z \). 2. **Evaluating the Gradient**: Substitute the point \((-1, 1, 2)\) into the gradient. 3. **Magnitude of the Gradient**: The magnitude of this gradient vector will provide the maximum rate of change. This kind of analysis is fundamental in understanding how multivariable functions behave and change across dimensions.
Transcribed Image Text:**Topic: Calculus - Multivariable Functions** ## Problem Statement Find the maximum rate of change of \( f(x, y, z) = \tan(2x + 7y + 5z) \) at the point \((-1, 1, 2)\). --- To approach this problem, you need to calculate the gradient vector of the function \( f(x, y, z) \). The gradient vector, denoted as \( \nabla f \), represents the direction of the steepest ascent and its magnitude will give you the maximum rate of change at a particular point. Detailed steps typically involve: 1. **Finding Partial Derivatives**: Compute the partial derivatives of the function with respect to each variable \( x, y, \) and \( z \). 2. **Evaluating the Gradient**: Substitute the point \((-1, 1, 2)\) into the gradient. 3. **Magnitude of the Gradient**: The magnitude of this gradient vector will provide the maximum rate of change. This kind of analysis is fundamental in understanding how multivariable functions behave and change across dimensions.
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