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Suppose f(x,y)=xex. At the point (2,0), in what direction is the directional derivative maximized?

Suppose f(x,y)=xex. At the point (2,0), in what direction is the directional derivative maximized?

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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### Calculus Problem: Finding the Direction of Maximum Directional Derivative **Problem Statement:** Consider the function \( f(x, y) = xe^y \). At the point \( (2, 0) \), in what direction is the directional derivative maximized? **Explanation:** To determine the direction in which the directional derivative of the function \( f(x, y) = xe^y \) is maximized at the point \( (2, 0) \), one must compute the gradient of the function at this point. The gradient vector points in the direction of the steepest ascent. 1. **Calculate the partial derivatives:** - \( \frac{\partial f}{\partial x} \) - \( \frac{\partial f}{\partial y} \) 2. **Evaluate the gradient at the point \( (2, 0) \):** - \( \nabla f (x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \) Once the gradient is computed, it will give the direction in which the directional derivative is maximized. The unit vector in this direction represents the answer.
Transcribed Image Text:### Calculus Problem: Finding the Direction of Maximum Directional Derivative **Problem Statement:** Consider the function \( f(x, y) = xe^y \). At the point \( (2, 0) \), in what direction is the directional derivative maximized? **Explanation:** To determine the direction in which the directional derivative of the function \( f(x, y) = xe^y \) is maximized at the point \( (2, 0) \), one must compute the gradient of the function at this point. The gradient vector points in the direction of the steepest ascent. 1. **Calculate the partial derivatives:** - \( \frac{\partial f}{\partial x} \) - \( \frac{\partial f}{\partial y} \) 2. **Evaluate the gradient at the point \( (2, 0) \):** - \( \nabla f (x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \) Once the gradient is computed, it will give the direction in which the directional derivative is maximized. The unit vector in this direction represents the answer.
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