A function f and a point P are given. Let 0 correspond to the direction of the directional derivative. Complete parts a. through e. + 2y²), P ( 3.- √5) f(x,y)= In (1+6x² +2 (Type your answer alans. Type all exact answer is a comma separate answers as itu.) What angles are associated with the direction of zero change? C (Type your answer in radians. Type an exact answer in terms of . Use a comma to separate answers as needed.) c. Write the directional derivative at P as a function of 0; call this function g(0). g(0) = (Simplify your answer. Type an exact answer, using radicals as needed.) d. Find the value of 0 between 0 and 2x that maximizes g(0) and find the maximum value. What value of 0 maximizes g(0)? 0= (Type your answer in radians. Type an exact answer in terms of .) What is the maximum value? g(0) = (Type an exact answer, using radicals as needed.) e. Verify that the value of 0 that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. Are the values from part d consistent with the values from parts a and b? O No Yes
A function f and a point P are given. Let 0 correspond to the direction of the directional derivative. Complete parts a. through e. + 2y²), P ( 3.- √5) f(x,y)= In (1+6x² +2 (Type your answer alans. Type all exact answer is a comma separate answers as itu.) What angles are associated with the direction of zero change? C (Type your answer in radians. Type an exact answer in terms of . Use a comma to separate answers as needed.) c. Write the directional derivative at P as a function of 0; call this function g(0). g(0) = (Simplify your answer. Type an exact answer, using radicals as needed.) d. Find the value of 0 between 0 and 2x that maximizes g(0) and find the maximum value. What value of 0 maximizes g(0)? 0= (Type your answer in radians. Type an exact answer in terms of .) What is the maximum value? g(0) = (Type an exact answer, using radicals as needed.) e. Verify that the value of 0 that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. Are the values from part d consistent with the values from parts a and b? O No Yes
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![# Directional Derivative Problem
A function \( f \) and a point \( P \) are given. Let \( \theta \) correspond to the direction of the directional derivative. Complete parts a through e.
## Function and Point
\[ f(x, y) = \ln \left( 1 + 6x^2 + 2y^2 \right), \quad P\left( \frac{4}{5}, -\sqrt{5} \right) \]
---
### a. Direction of Zero Change
What angles are associated with the direction of zero change?
\[ \boxed{} \]
*Type your answer in radians. Provide an exact answer in terms of \(\pi\). Use a comma to separate answers as needed.*
---
### b. Directional Derivative Function
Write the directional derivative at \( P \) as a function of \( \theta \); call this function \( g(\theta) \).
\[ g(\theta) = \boxed{} \]
*Simplify your answer. Type an exact answer, using radicals as needed.*
---
### c. Maximize \( g(\theta) \)
Find the value of \( \theta \) between 0 and \( 2\pi \) that maximizes \( g(\theta) \) and find the maximum value. What value of \( \theta \) maximizes \( g(\theta) \)?
\[ \theta = \boxed{} \]
*Type your answer in radians. Provide an exact answer in terms of \(\pi\).*
What is the maximum value?
\[ g(\theta) = \boxed{} \]
*Type an exact answer, using radicals as needed.*
---
### d. Verification
Verify that the value of \( \theta \) that maximizes \( g \) corresponds to the direction of the gradient. Verify that the maximum value of \( g \) equals the magnitude of the gradient. Are the values from part d consistent with the values from parts a and b?
- \(\circ\) No
- \(\circ\) Yes
---
This exercise explores the application of directional derivatives and gradients, which are fundamental concepts in multivariable calculus, critical for understanding changes in various directions on a surface represented by a function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a42e238-4116-4e80-81a7-f4d567b4bd0e%2Ffc133d15-98fa-4abc-9217-fcf921f2bec4%2F3xv3n7d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Directional Derivative Problem
A function \( f \) and a point \( P \) are given. Let \( \theta \) correspond to the direction of the directional derivative. Complete parts a through e.
## Function and Point
\[ f(x, y) = \ln \left( 1 + 6x^2 + 2y^2 \right), \quad P\left( \frac{4}{5}, -\sqrt{5} \right) \]
---
### a. Direction of Zero Change
What angles are associated with the direction of zero change?
\[ \boxed{} \]
*Type your answer in radians. Provide an exact answer in terms of \(\pi\). Use a comma to separate answers as needed.*
---
### b. Directional Derivative Function
Write the directional derivative at \( P \) as a function of \( \theta \); call this function \( g(\theta) \).
\[ g(\theta) = \boxed{} \]
*Simplify your answer. Type an exact answer, using radicals as needed.*
---
### c. Maximize \( g(\theta) \)
Find the value of \( \theta \) between 0 and \( 2\pi \) that maximizes \( g(\theta) \) and find the maximum value. What value of \( \theta \) maximizes \( g(\theta) \)?
\[ \theta = \boxed{} \]
*Type your answer in radians. Provide an exact answer in terms of \(\pi\).*
What is the maximum value?
\[ g(\theta) = \boxed{} \]
*Type an exact answer, using radicals as needed.*
---
### d. Verification
Verify that the value of \( \theta \) that maximizes \( g \) corresponds to the direction of the gradient. Verify that the maximum value of \( g \) equals the magnitude of the gradient. Are the values from part d consistent with the values from parts a and b?
- \(\circ\) No
- \(\circ\) Yes
---
This exercise explores the application of directional derivatives and gradients, which are fundamental concepts in multivariable calculus, critical for understanding changes in various directions on a surface represented by a function.
![**Title: Analyzing Directional Derivatives in Multivariable Functions**
---
**Problem Statement:**
A function \( f \) and a point \( P \) are given. Let \(\theta\) correspond to the direction of the directional derivative. Complete parts a through e.
Given:
\[ f(x, y) = \ln(1 + 6x^2 + 2y^2) \]
Point \( P = \left(\frac{4}{5}, -\sqrt{5}\right) \)
---
**a. Find the gradient and evaluate it at \( P \).**
- The gradient at \( P \) is \( \boxed{\phantom{x}, \phantom{x}} \).
---
**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.**
1. **What angles are associated with the direction of maximum increase?**
\(\boxed{\phantom{x}}\)
(Type your answer in radians. Type an exact answer in terms of \(\pi\). Use a comma to separate answers as needed.)
2. **What angles are associated with the direction of maximum decrease?**
\(\boxed{\phantom{x}}\)
(Type your answer in radians. Type an exact answer in terms of \(\pi\). Use a comma to separate answers as needed.)
3. **What angles are associated with the direction of zero change?**
\(\boxed{\phantom{x}}\)
(Type your answer in radians. Type an exact answer in terms of \(\pi\). Use a comma to separate answers as needed.)
---
**c. Write the directional derivative at \( P \) as a function of \(\theta\); call this function \( g(\theta) \).**
- \( g(\theta) = \boxed{\phantom{x}} \)
(Simplify your answer. Type an exact answer, using radicals as needed.)
---
**d. Find the value of \(\theta\) between 0 and \(2\pi\) that maximizes \( g(\theta) \) and find the maximum value. What value of \(\theta\) maximizes \( g(\theta) \)?**
---
**Explanation of Concepts:**
1. **Gradient (\(\nabla f\)):](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a42e238-4116-4e80-81a7-f4d567b4bd0e%2Ffc133d15-98fa-4abc-9217-fcf921f2bec4%2F4ayipk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Analyzing Directional Derivatives in Multivariable Functions**
---
**Problem Statement:**
A function \( f \) and a point \( P \) are given. Let \(\theta\) correspond to the direction of the directional derivative. Complete parts a through e.
Given:
\[ f(x, y) = \ln(1 + 6x^2 + 2y^2) \]
Point \( P = \left(\frac{4}{5}, -\sqrt{5}\right) \)
---
**a. Find the gradient and evaluate it at \( P \).**
- The gradient at \( P \) is \( \boxed{\phantom{x}, \phantom{x}} \).
---
**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.**
1. **What angles are associated with the direction of maximum increase?**
\(\boxed{\phantom{x}}\)
(Type your answer in radians. Type an exact answer in terms of \(\pi\). Use a comma to separate answers as needed.)
2. **What angles are associated with the direction of maximum decrease?**
\(\boxed{\phantom{x}}\)
(Type your answer in radians. Type an exact answer in terms of \(\pi\). Use a comma to separate answers as needed.)
3. **What angles are associated with the direction of zero change?**
\(\boxed{\phantom{x}}\)
(Type your answer in radians. Type an exact answer in terms of \(\pi\). Use a comma to separate answers as needed.)
---
**c. Write the directional derivative at \( P \) as a function of \(\theta\); call this function \( g(\theta) \).**
- \( g(\theta) = \boxed{\phantom{x}} \)
(Simplify your answer. Type an exact answer, using radicals as needed.)
---
**d. Find the value of \(\theta\) between 0 and \(2\pi\) that maximizes \( g(\theta) \) and find the maximum value. What value of \(\theta\) maximizes \( g(\theta) \)?**
---
**Explanation of Concepts:**
1. **Gradient (\(\nabla f\)):
Expert Solution
Step 1
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
