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the directional derivative of the function J (x, he direction of v = 3i + 4j at the point (3,0)

the directional derivative of the function J (x, he direction of v = 3i + 4j at the point (3,0)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem: Directional Derivative** Find the directional derivative of the function \( f(x, y) = x^2 e^{-y} \) at the point \( (3, 0) \) in the direction of \( \mathbf{v} = 3\mathbf{i} + 4\mathbf{j} \). **Explanation:** The directional derivative represents the rate at which the function \( f(x, y) \) changes at a specific point in a specified direction. Here, the function \( f(x, y) = x^2 e^{-y} \) is given and we want to find this derivative at the point \( (3, 0) \). The direction is specified by the vector \( \mathbf{v} = 3\mathbf{i} + 4\mathbf{j} \). To solve this problem, one would: 1. Calculate the gradient of \( f(x, y) \). 2. Normalize the direction vector \( \mathbf{v} \). 3. Compute the dot product of the gradient and the normalized direction vector to get the directional derivative.
Transcribed Image Text:**Problem: Directional Derivative** Find the directional derivative of the function \( f(x, y) = x^2 e^{-y} \) at the point \( (3, 0) \) in the direction of \( \mathbf{v} = 3\mathbf{i} + 4\mathbf{j} \). **Explanation:** The directional derivative represents the rate at which the function \( f(x, y) \) changes at a specific point in a specified direction. Here, the function \( f(x, y) = x^2 e^{-y} \) is given and we want to find this derivative at the point \( (3, 0) \). The direction is specified by the vector \( \mathbf{v} = 3\mathbf{i} + 4\mathbf{j} \). To solve this problem, one would: 1. Calculate the gradient of \( f(x, y) \). 2. Normalize the direction vector \( \mathbf{v} \). 3. Compute the dot product of the gradient and the normalized direction vector to get the directional derivative.
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