-x²15-y²15 Consider the function F(x,y) = e and the point P(-2,2). a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P. b. Find a vector that points in a direction of no change in the function at P. a. The direction of steepest ascent is The direction of steepest descent is OA. (1,0) OB. (-1,-1) ⒸC. (1,-1) OD. (0,1) √2√e 4 5 2 e 200 5 √2√e 45 5 8 200 √2√e 4 5 2 e √2√e 2 e 5 +15 00 5 2e b. Which of the following vectors points in a direction of no change of the function at P(-2,2)?
-x²15-y²15 Consider the function F(x,y) = e and the point P(-2,2). a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P. b. Find a vector that points in a direction of no change in the function at P. a. The direction of steepest ascent is The direction of steepest descent is OA. (1,0) OB. (-1,-1) ⒸC. (1,-1) OD. (0,1) √2√e 4 5 2 e 200 5 √2√e 45 5 8 200 √2√e 4 5 2 e √2√e 2 e 5 +15 00 5 2e b. Which of the following vectors points in a direction of no change of the function at P(-2,2)?
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
![Consider the function \( F(x,y) = e^{-x^2/5 - y^2/5} \) and the point \( P(-2,2) \).
a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P.
b. Find a vector that points in a direction of no change in the function at P.
---
a. The direction of steepest ascent is:
\[
\left( \frac{\sqrt{2}\sqrt{\frac{8}{5}}}{2e^{\frac{4}{5}}}, \frac{\sqrt{2}\sqrt{\frac{8}{5}}}{2e^{\frac{4}{5}}} \right).
\]
The direction of steepest descent is:
\[
\left( -\frac{\sqrt{2}\sqrt{\frac{8}{5}}}{2e^{\frac{4}{5}}}, -\frac{\sqrt{2}\sqrt{\frac{8}{5}}}{2e^{\frac{4}{5}}} \right).
\]
b. Which of the following vectors points in a direction of no change of the function at \( P(-2,2) \)?
- A. \( (1,0) \)
- B. \( (-1,-1) \)
- C. \( \boxed{(1,-1)} \)
- D. \( (0,1) \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2c370c8-f220-47ec-b561-9f6a07b2c79a%2F08ba243d-4aef-4f7b-9ca2-c287a59bcdd9%2Fghtrrzy_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the function \( F(x,y) = e^{-x^2/5 - y^2/5} \) and the point \( P(-2,2) \).
a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P.
b. Find a vector that points in a direction of no change in the function at P.
---
a. The direction of steepest ascent is:
\[
\left( \frac{\sqrt{2}\sqrt{\frac{8}{5}}}{2e^{\frac{4}{5}}}, \frac{\sqrt{2}\sqrt{\frac{8}{5}}}{2e^{\frac{4}{5}}} \right).
\]
The direction of steepest descent is:
\[
\left( -\frac{\sqrt{2}\sqrt{\frac{8}{5}}}{2e^{\frac{4}{5}}}, -\frac{\sqrt{2}\sqrt{\frac{8}{5}}}{2e^{\frac{4}{5}}} \right).
\]
b. Which of the following vectors points in a direction of no change of the function at \( P(-2,2) \)?
- A. \( (1,0) \)
- B. \( (-1,-1) \)
- C. \( \boxed{(1,-1)} \)
- D. \( (0,1) \)
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