Let z = f(x, y) be a differentiable function defined for every point on the plane, and let u be a unit vector. Answer each question below regarding the directional derivative Dif at a point P. (a) What does the directional derivative indicate? (b) How do you find it at point P?
Let z = f(x, y) be a differentiable function defined for every point on the plane, and let u be a unit vector. Answer each question below regarding the directional derivative Dif at a point P. (a) What does the directional derivative indicate? (b) How do you find it at point P?
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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![**Directional Derivatives in Multivariable Calculus**
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### Problem #27
Let \( z = f(x, y) \) be a differentiable function defined for every point on the plane, and let \( \vec{u} \) be a unit vector.
Answer each question below regarding the directional derivative \( D_{\vec{u}} f \) at a point \( P \).
**(a) What does the directional derivative indicate?**
**(b) How do you find it at point \( P \)?**
**(c) What happens to the directional derivative if this function has a local maximum at \( P \)?**
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This problem addresses the concept of directional derivatives, which measure the rate of change of a function in the direction of a given vector. It requires understanding and finding the directional derivative and analyzing its behavior at a local maximum within the given function.
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Feel free to explore more information about the directional derivative, its calculation, and its applications in the provided sections.
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[Next Section: Detailed Solutions with Examples]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e343f85-ac09-4dac-ae1f-fa90b444949b%2F4e8a8ce3-95eb-485e-96fb-31271b2a78a6%2F72vmxrj_processed.png&w=3840&q=75)
Transcribed Image Text:**Directional Derivatives in Multivariable Calculus**
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### Problem #27
Let \( z = f(x, y) \) be a differentiable function defined for every point on the plane, and let \( \vec{u} \) be a unit vector.
Answer each question below regarding the directional derivative \( D_{\vec{u}} f \) at a point \( P \).
**(a) What does the directional derivative indicate?**
**(b) How do you find it at point \( P \)?**
**(c) What happens to the directional derivative if this function has a local maximum at \( P \)?**
---
This problem addresses the concept of directional derivatives, which measure the rate of change of a function in the direction of a given vector. It requires understanding and finding the directional derivative and analyzing its behavior at a local maximum within the given function.
---
Feel free to explore more information about the directional derivative, its calculation, and its applications in the provided sections.
---
[Next Section: Detailed Solutions with Examples]
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