Consider the function f(x,y,z) = x² + 2y² +3z² +2, the point P(3,0,3), and the unit vector u = √2 a. Compute the gradient off and evaluate it at P. b. Find the unit vector in the direction of maximum increase off at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. √21
Consider the function f(x,y,z) = x² + 2y² +3z² +2, the point P(3,0,3), and the unit vector u = √2 a. Compute the gradient off and evaluate it at P. b. Find the unit vector in the direction of maximum increase off at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. √21
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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Question
![**Mathematical Functions and Gradients**
Consider the function \( f(x,y,z) = x^2 + 2y^2 + 3z^2 + 2 \), the point \(P(3,0,3)\), and the unit vector \( \mathbf{u} = \left(\frac{1}{\sqrt{2}}, 0, -\frac{1}{\sqrt{2}}\right) \).
### Problems to Solve:
a. Compute the gradient of \( f \) and evaluate it at \( P \).
b. Find the unit vector in the direction of maximum increase of \( f \) at \( P \).
c. Find the rate of change of the function in the direction of maximum increase at \( P \).
d. Find the directional derivative at \( P \) in the direction of the given vector.
---
### Solution to Part (a):
a. **What is the gradient at the point \((3,0,3)\)?**
[This would likely be continued with appropriate mathematical solution steps. Given the nature of this platform, the diagram (if any) explanation or graphical representation should follow these textual instructions along with a continuous elaboration of the further parts (b), (c), and (d).]
Note: The provided image segment mainly consists of text-based mathematical problem settings with an input for part (a). Further explanations should include continuation problem-solving steps, ensuring clarity in educational contexts.
---
**Explanation of Inserted Elements:**
- Diagram/Graphical elements: The text tends to illustrate certain graphing methods or computation steps which seem missing or not visible. Assuming it is to illustrate gradient vectors or involve calculation matrices.
- The input boxes suggest an interactive element where students or users can input their responses for validation of the problem set forth.
---
**Further Explanation:**
Given the complexity and mathematical nature, ensure each part is dissected accordingly:
- Gradient vectors calculation and interpretation.
- Maximal direction determination derivation.
- Mathematical integrals or differentials explanation for gradient contexts.
- Introducing examples/allusion to real-world scenarios with practical gradient usage implications.
Overall, the instructions hint towards a precise interactive learning mechanism suitable for a Mathematics/Calculus-focused educational platform.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99e37c92-095a-4cae-adf6-01aeaaa5668c%2F4159b59f-ef76-4c71-be8b-7c9b9ed6b59d%2F93emd5f_processed.png&w=3840&q=75)
Transcribed Image Text:**Mathematical Functions and Gradients**
Consider the function \( f(x,y,z) = x^2 + 2y^2 + 3z^2 + 2 \), the point \(P(3,0,3)\), and the unit vector \( \mathbf{u} = \left(\frac{1}{\sqrt{2}}, 0, -\frac{1}{\sqrt{2}}\right) \).
### Problems to Solve:
a. Compute the gradient of \( f \) and evaluate it at \( P \).
b. Find the unit vector in the direction of maximum increase of \( f \) at \( P \).
c. Find the rate of change of the function in the direction of maximum increase at \( P \).
d. Find the directional derivative at \( P \) in the direction of the given vector.
---
### Solution to Part (a):
a. **What is the gradient at the point \((3,0,3)\)?**
[This would likely be continued with appropriate mathematical solution steps. Given the nature of this platform, the diagram (if any) explanation or graphical representation should follow these textual instructions along with a continuous elaboration of the further parts (b), (c), and (d).]
Note: The provided image segment mainly consists of text-based mathematical problem settings with an input for part (a). Further explanations should include continuation problem-solving steps, ensuring clarity in educational contexts.
---
**Explanation of Inserted Elements:**
- Diagram/Graphical elements: The text tends to illustrate certain graphing methods or computation steps which seem missing or not visible. Assuming it is to illustrate gradient vectors or involve calculation matrices.
- The input boxes suggest an interactive element where students or users can input their responses for validation of the problem set forth.
---
**Further Explanation:**
Given the complexity and mathematical nature, ensure each part is dissected accordingly:
- Gradient vectors calculation and interpretation.
- Maximal direction determination derivation.
- Mathematical integrals or differentials explanation for gradient contexts.
- Introducing examples/allusion to real-world scenarios with practical gradient usage implications.
Overall, the instructions hint towards a precise interactive learning mechanism suitable for a Mathematics/Calculus-focused educational platform.
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