A function f and a point P are given. Let 9 correspond to the direction of the directional derivative. Complete parts a. through e. 1. P(1/1₁ -√²) f(x,y)=In (1 +6x² +5y²), P a. Find the gradient and evaluate it at P. 4√2 The gradient at P is 12 b. Find the angles (with respect to the positive x-axis) between 0 and 2x associated with the directions of maximum increase, maximum decrease, and zero change. What angles are associated with the direction of maximum increase? (Type your answer in radians. Type an exact answer terms of Use a comma to separate answers as needed.)
A function f and a point P are given. Let 9 correspond to the direction of the directional derivative. Complete parts a. through e. 1. P(1/1₁ -√²) f(x,y)=In (1 +6x² +5y²), P a. Find the gradient and evaluate it at P. 4√2 The gradient at P is 12 b. Find the angles (with respect to the positive x-axis) between 0 and 2x associated with the directions of maximum increase, maximum decrease, and zero change. What angles are associated with the direction of maximum increase? (Type your answer in radians. Type an exact answer terms of Use a comma to separate answers as needed.)
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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![**Gradient and Directional Derivative Analysis**
A function \( f \) and a point \( P \) are given. Let \( \theta \) correspond to the direction of the directional derivative. Complete parts a. through e.
\[ f(x, y) = \ln \left(1 + 6x^2 + 5y^2\right) , \quad P \left( \frac{1}{2}, -\sqrt{2} \right) \]
**a. Find the gradient and evaluate it at \( P \).**
The gradient at \( P \) is
\[
\left( \frac{12}{25}, -\frac{4\sqrt{2}}{5} \right) .
\]
**b. Find the angles \( \theta \) (with respect to the positive x-axis) between \( 0 \) and \( 2\pi \) associated with the directions of maximum increase, maximum decrease, and zero change. What angles are associated with the direction of maximum increase?**
**Answer in Radians:**
Type your answer in radians. Type an exact answer in terms of \(\pi\). Use a comma to separate answers as needed.
**Note:**
This section involves finding angles that relate to the change in direction of the function. The maximum increase is in the direction of the gradient vector.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F511d779c-bf10-4315-98b1-4f0038c55113%2F6c347548-ae24-4d16-8d7f-e6d7debd10a0%2Fchgq8dh_processed.png&w=3840&q=75)
Transcribed Image Text:**Gradient and Directional Derivative Analysis**
A function \( f \) and a point \( P \) are given. Let \( \theta \) correspond to the direction of the directional derivative. Complete parts a. through e.
\[ f(x, y) = \ln \left(1 + 6x^2 + 5y^2\right) , \quad P \left( \frac{1}{2}, -\sqrt{2} \right) \]
**a. Find the gradient and evaluate it at \( P \).**
The gradient at \( P \) is
\[
\left( \frac{12}{25}, -\frac{4\sqrt{2}}{5} \right) .
\]
**b. Find the angles \( \theta \) (with respect to the positive x-axis) between \( 0 \) and \( 2\pi \) associated with the directions of maximum increase, maximum decrease, and zero change. What angles are associated with the direction of maximum increase?**
**Answer in Radians:**
Type your answer in radians. Type an exact answer in terms of \(\pi\). Use a comma to separate answers as needed.
**Note:**
This section involves finding angles that relate to the change in direction of the function. The maximum increase is in the direction of the gradient vector.
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