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Suppose we know that for a function f we have Vf(x, y) = (6x, 6). Find the directional derivative of the function f at the point (-1, 3) in the direction of the vector (3, 4).

Suppose we know that for a function f we have Vf(x, y) = (6x, 6). Find the directional derivative of the function f at the point (-1, 3) in the direction of the vector (3, 4).

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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### Calculus Problem: Directional Derivative #### Problem Statement: Suppose we know that for a function \( f \) we have \( \nabla f(x, y) = \langle 6x, 6 \rangle \). Find the directional derivative of the function \( f \) at the point \( (-1, 3) \) in the direction of the vector \( \langle 3, 4 \rangle \). #### Options: - \( \circ \) \( 6 \) - \( \circ \) \( 9 \) - \( \circ \) \( \frac{6}{5} \) - \( \circ \) \( \left\langle -\frac{18}{5}, \frac{24}{5} \right\rangle \) (This option is selected) **Explanation:** To solve for the directional derivative of \( f \) at \( (-1, 3) \) in the direction of \( \langle 3, 4 \rangle \), follow these steps: 1. Normalize the direction vector \( \langle 3, 4 \rangle \). 2. Use the gradient \( \nabla f(x, y) \) at the point \( (-1, 3) \). 3. Calculate the dot product of the gradient and the normalized direction vector. This problem helps in understanding how to find the rate of change of a function in a given direction.
Transcribed Image Text:### Calculus Problem: Directional Derivative #### Problem Statement: Suppose we know that for a function \( f \) we have \( \nabla f(x, y) = \langle 6x, 6 \rangle \). Find the directional derivative of the function \( f \) at the point \( (-1, 3) \) in the direction of the vector \( \langle 3, 4 \rangle \). #### Options: - \( \circ \) \( 6 \) - \( \circ \) \( 9 \) - \( \circ \) \( \frac{6}{5} \) - \( \circ \) \( \left\langle -\frac{18}{5}, \frac{24}{5} \right\rangle \) (This option is selected) **Explanation:** To solve for the directional derivative of \( f \) at \( (-1, 3) \) in the direction of \( \langle 3, 4 \rangle \), follow these steps: 1. Normalize the direction vector \( \langle 3, 4 \rangle \). 2. Use the gradient \( \nabla f(x, y) \) at the point \( (-1, 3) \). 3. Calculate the dot product of the gradient and the normalized direction vector. This problem helps in understanding how to find the rate of change of a function in a given direction.
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