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Ch. 14 Problems: 11. Consider the function f(x, y) = x²ye-²-y². Compute Vf and the direc- tional derivative Duf where u = (1, 2). The

Ch. 14 Problems: 11. Consider the function f(x, y) = x²ye-²-y². Compute Vf and the direc- tional derivative Duf where u = (1, 2). The

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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### Chapter 14 Problems: **Problem 11:** Consider the function \( f(x, y) = x^2y e^{-x^2-y^2} \). Compute \(\nabla f\) and the directional derivative \( D_{\mathbf{u}} f \) where \(\mathbf{u} = (1, 2)\). **Explanation:** - **Function \( f(x, y) \):** This is a multivariable function involving exponential decay with respect to both \( x \) and \( y \). - **Gradient \(\nabla f\):** This is the vector of partial derivatives of the function, representing the direction of the steepest increase of \( f \). - **Directional Derivative \( D_{\mathbf{u}} f \):** This measures the rate of change of the function \( f \) in the direction of \( \mathbf{u} \). Given \(\mathbf{u} = (1, 2)\), this vector denotes the specified direction in which the derivative is computed.
Transcribed Image Text:### Chapter 14 Problems: **Problem 11:** Consider the function \( f(x, y) = x^2y e^{-x^2-y^2} \). Compute \(\nabla f\) and the directional derivative \( D_{\mathbf{u}} f \) where \(\mathbf{u} = (1, 2)\). **Explanation:** - **Function \( f(x, y) \):** This is a multivariable function involving exponential decay with respect to both \( x \) and \( y \). - **Gradient \(\nabla f\):** This is the vector of partial derivatives of the function, representing the direction of the steepest increase of \( f \). - **Directional Derivative \( D_{\mathbf{u}} f \):** This measures the rate of change of the function \( f \) in the direction of \( \mathbf{u} \). Given \(\mathbf{u} = (1, 2)\), this vector denotes the specified direction in which the derivative is computed.
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