Find the gradient: f(x, y) = 3x³ - 4x²y + y² at (1,-1)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please please please solve the problems provided in the photo below
![### Gradient and Directional Derivative Calculation
#### Problem 1: Finding the Gradient
Given the function:
\[ f(x, y) = 3x^3 - 4x^2y + y^2 \]
We are to find the gradient of \( f(x, y) \) at the point (1, -1).
#### Solution:
1. Compute the partial derivatives of \( f(x, y) \):
- \(\frac{\partial f}{\partial x} = 9x^2 - 8xy \)
- \(\frac{\partial f}{\partial y} = -4x^2 + 2y \)
2. Evaluate these partial derivatives at the point (1, -1):
- \(\frac{\partial f}{\partial x}\bigg|_{(1, -1)} = 9(1)^2 - 8(1)(-1) = 9 + 8 = 17 \)
- \(\frac{\partial f}{\partial y}\bigg|_{(1, -1)} = -4(1)^2 + 2(-1) = -4 - 2 = -6 \)
3. The gradient of \( f \) at (1, -1) is:
\[ \nabla f(1, -1) = (17, -6) \]
#### Problem 2: Finding the Directional Derivative
Given the function:
\[ f(x, y) = 3x^2 - y^2 + 5xy \]
We are to find the directional derivative of \( f(x, y) \) at the point (2, -1) in the direction of the vector \(-3\mathbf{i} - 4\mathbf{j}\).
#### Solution:
1. Compute the partial derivatives of \( f(x, y) \):
- \(\frac{\partial f}{\partial x} = 6x + 5y \)
- \(\frac{\partial f}{\partial y} = -2y + 5x \)
2. Evaluate these partial derivatives at the point (2, -1):
- \(\frac{\partial f}{\partial x}\bigg|_{(2, -1)} = 6(2) + 5(-1) = 12 - 5 =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9981eaaa-92cc-40db-b668-6adab491a4d6%2Fb20087e5-6364-41af-bba6-90ac9d03cb89%2Fe59f5dd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Gradient and Directional Derivative Calculation
#### Problem 1: Finding the Gradient
Given the function:
\[ f(x, y) = 3x^3 - 4x^2y + y^2 \]
We are to find the gradient of \( f(x, y) \) at the point (1, -1).
#### Solution:
1. Compute the partial derivatives of \( f(x, y) \):
- \(\frac{\partial f}{\partial x} = 9x^2 - 8xy \)
- \(\frac{\partial f}{\partial y} = -4x^2 + 2y \)
2. Evaluate these partial derivatives at the point (1, -1):
- \(\frac{\partial f}{\partial x}\bigg|_{(1, -1)} = 9(1)^2 - 8(1)(-1) = 9 + 8 = 17 \)
- \(\frac{\partial f}{\partial y}\bigg|_{(1, -1)} = -4(1)^2 + 2(-1) = -4 - 2 = -6 \)
3. The gradient of \( f \) at (1, -1) is:
\[ \nabla f(1, -1) = (17, -6) \]
#### Problem 2: Finding the Directional Derivative
Given the function:
\[ f(x, y) = 3x^2 - y^2 + 5xy \]
We are to find the directional derivative of \( f(x, y) \) at the point (2, -1) in the direction of the vector \(-3\mathbf{i} - 4\mathbf{j}\).
#### Solution:
1. Compute the partial derivatives of \( f(x, y) \):
- \(\frac{\partial f}{\partial x} = 6x + 5y \)
- \(\frac{\partial f}{\partial y} = -2y + 5x \)
2. Evaluate these partial derivatives at the point (2, -1):
- \(\frac{\partial f}{\partial x}\bigg|_{(2, -1)} = 6(2) + 5(-1) = 12 - 5 =
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