Consider the function f(x,y) = x² - 4y² -9 and the point (-1,1). 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. What is the unit vector in the direction of steepest ascent at P? (Type exact answers, using radicals as needed.) What is the unit vector in the direction of steepest descent at P? 10. (Type exact answers, using radicals as needed.) b. Which of the following vectors is in a direction of no change of the function at P? OA. (2,-8) OB. (8,2) OC. (-2,8) D. (8,-2)
Consider the function f(x,y) = x² - 4y² -9 and the point (-1,1). 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. What is the unit vector in the direction of steepest ascent at P? (Type exact answers, using radicals as needed.) What is the unit vector in the direction of steepest descent at P? 10. (Type exact answers, using radicals as needed.) b. Which of the following vectors is in a direction of no change of the function at P? OA. (2,-8) OB. (8,2) OC. (-2,8) D. (8,-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
Transcribed Image Text:**Multivariable Calculus: Gradient and Directional Derivatives**
**Problem Statement:**
Consider the function \( f(x, y) = x^2 - 4y^2 - 9 \) and the point \( (-1, 1) \).
**Tasks:**
**a.** Find the unit vectors that give the direction of steepest ascent and steepest descent at \( P \).
- **What is the unit vector in the direction of steepest ascent at \( P \)?**
\(\left< \square, \square \right>\)
*(Type exact answers, using radicals as needed.)*
- **What is the unit vector in the direction of steepest descent at \( P \)?**
\(\left< \square, \square \right>\)
*(Type exact answers, using radicals as needed.)*
**b.** Find a vector that points in a direction of no change in the function at \( P \).
- **Which of the following vectors is in a direction of no change of the function at \( P \)?**
- \( \text{A. } \langle 2, -8 \rangle \)
- \( \text{B. } \langle 8, 2 \rangle \)
- \( \text{C. } \langle -2, 8 \rangle \)
- \( \text{D. } \langle 8, -2 \rangle \)
Expert Solution
Step 1
he gradient of a scalar function at any point indicates the direction in which the scalar function changes rapidly and its magnitude indicates the maximum rate of change of the scalar function at that point
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