### Example Problem on Gradient and Directional Derivative Let \( f(x, y) = xe^{x^2 - y} \) and \( P = (13, 169) \). #### 1. Calculate \( \| \nabla f_p \| \). (Express numbers in exact form. Use symbolic notation and fractions where needed.) \[ \| \nabla f_p \| = \] --- #### 2. Find the rate of change of \( f \) in the direction \( \nabla f_p \). (Express numbers in exact form. Use symbolic notation and fractions where needed.) The rate of change: \[ \text{the rate of change:} \ \boxed{} \] --- #### 3. Find the rate of change of \( f \) in the direction of a vector making an angle of \( 45^\circ \) with \( \nabla f_p \). (Express numbers in exact form. Use symbolic notation and fractions where needed.) The rate of change: \[ \text{the rate of change:} \ \boxed{} \]
### Example Problem on Gradient and Directional Derivative Let \( f(x, y) = xe^{x^2 - y} \) and \( P = (13, 169) \). #### 1. Calculate \( \| \nabla f_p \| \). (Express numbers in exact form. Use symbolic notation and fractions where needed.) \[ \| \nabla f_p \| = \] --- #### 2. Find the rate of change of \( f \) in the direction \( \nabla f_p \). (Express numbers in exact form. Use symbolic notation and fractions where needed.) The rate of change: \[ \text{the rate of change:} \ \boxed{} \] --- #### 3. Find the rate of change of \( f \) in the direction of a vector making an angle of \( 45^\circ \) with \( \nabla f_p \). (Express numbers in exact form. Use symbolic notation and fractions where needed.) The rate of change: \[ \text{the rate of change:} \ \boxed{} \]
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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![### Example Problem on Gradient and Directional Derivative
Let \( f(x, y) = xe^{x^2 - y} \) and \( P = (13, 169) \).
#### 1. Calculate \( \| \nabla f_p \| \).
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
\[ \| \nabla f_p \| = \]
---
#### 2. Find the rate of change of \( f \) in the direction \( \nabla f_p \).
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
The rate of change:
\[ \text{the rate of change:} \ \boxed{} \]
---
#### 3. Find the rate of change of \( f \) in the direction of a vector making an angle of \( 45^\circ \) with \( \nabla f_p \).
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
The rate of change:
\[ \text{the rate of change:} \ \boxed{} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8fe4eea6-b7ab-4471-b433-62327707440a%2Fe82f5e0e-2dff-4c63-bde9-286f8f7004e6%2Fbq7d9m9_processed.png&w=3840&q=75)
Transcribed Image Text:### Example Problem on Gradient and Directional Derivative
Let \( f(x, y) = xe^{x^2 - y} \) and \( P = (13, 169) \).
#### 1. Calculate \( \| \nabla f_p \| \).
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
\[ \| \nabla f_p \| = \]
---
#### 2. Find the rate of change of \( f \) in the direction \( \nabla f_p \).
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
The rate of change:
\[ \text{the rate of change:} \ \boxed{} \]
---
#### 3. Find the rate of change of \( f \) in the direction of a vector making an angle of \( 45^\circ \) with \( \nabla f_p \).
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
The rate of change:
\[ \text{the rate of change:} \ \boxed{} \]
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