Consider the function f(x, y) = 2x²-3xy + 8y² + 2x - 4y +4 at the point P(-2,1). a) Find a vector that gives the direction of steepest ascent at P. b) Find a vector that points in a direction of "no change" in the function. c) The graph to the right shows the level curve of f for z 18. Copy the graph onto your own graph paper, and then sketch unit vectors in the direction of the vectors you found in parts (a) and (b) on the graph at point P(-2, 1). Label each vector with (a) or (b) to distinguish them.
Consider the function f(x, y) = 2x²-3xy + 8y² + 2x - 4y +4 at the point P(-2,1). a) Find a vector that gives the direction of steepest ascent at P. b) Find a vector that points in a direction of "no change" in the function. c) The graph to the right shows the level curve of f for z 18. Copy the graph onto your own graph paper, and then sketch unit vectors in the direction of the vectors you found in parts (a) and (b) on the graph at point P(-2, 1). Label each vector with (a) or (b) to distinguish them.
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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Transcribed Image Text:**Problem 2: Gradient and Level Curves of a Function**
Consider the function \( f(x, y) = 2x^2 - 3xy + 8y^2 + 2x - 4y + 4 \) at the point \( P(-2, 1) \).
**Tasks:**
a) **Find a vector that gives the direction of steepest ascent at \( P \).**
b) **Find a vector that points in a direction of "no change" in the function.**
c) **Graph Analysis:**
- The provided graph shows the level curve of \( f \) for \( z = 18 \).
**Instructions:**
- Copy the graph onto your graph paper.
- Sketch unit vectors in the direction of the vectors you found in parts (a) and (b) on the graph at point \( P(-2, 1) \).
- Label each vector with (a) or (b) to distinguish them.
**Graph Description:**
- The graph is a Cartesian plane with an oval-shaped level curve centered around the point.
- The x-axis and y-axis range from -3 to 3.
- The level curve is approximately elliptical, passing through the region where \( x \) and \( y \) values are between -2 and 2.
**Additional Information:**
When dealing with functions and vectors, the gradient vector at a particular point is in the direction of steepest ascent. The gradient \(\nabla f\) can be computed by taking the partial derivatives of \( f \) with respect to \( x \) and \( y \). The direction of "no change" is generally tangent to the level curve at the given point.
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