Tutorial Exercise Use the gradient to find the directional derivative of the function at P in the direction of Q. f(x, y) = 3x² - y2 + 4, P(2, 4), Q(5, 9) Step 1 Since f is a polynomial, all the partial derivatives are continuous. Therefore, f is differentiable and you can apply the following theorem: if f is a differentiable function of x and y, then the directional derivative of f in the direction of the unit vector u is Duf(x, y) = Vf(x, y) · u. Find the given directional vector PQ so that we can find the unit vector. PQ = v - 2)i + (9 – 4)j i +
Tutorial Exercise Use the gradient to find the directional derivative of the function at P in the direction of Q. f(x, y) = 3x² - y2 + 4, P(2, 4), Q(5, 9) Step 1 Since f is a polynomial, all the partial derivatives are continuous. Therefore, f is differentiable and you can apply the following theorem: if f is a differentiable function of x and y, then the directional derivative of f in the direction of the unit vector u is Duf(x, y) = Vf(x, y) · u. Find the given directional vector PQ so that we can find the unit vector. PQ = v - 2)i + (9 – 4)j i +
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
3
![**Tutorial Exercise**
Use the gradient to find the directional derivative of the function at \( P \) in the direction of \( Q \).
\[ f(x, y) = 3x^2 - y^2 + 4, \quad P(2, 4), \quad Q(5, 9) \]
**Step 1**
Since \( f \) is a polynomial, all the partial derivatives are continuous. Therefore, \( f \) is differentiable and you can apply the following theorem: if \( f \) is a differentiable function of \( x \) and \( y \), then the directional derivative of \( f \) in the direction of the unit vector \( \mathbf{u} \) is
\[ D_{\mathbf{u}}f(x, y) = \nabla f(x, y) \cdot \mathbf{u}. \]
Find the given directional vector \( \overrightarrow{PQ} \) so that we can find the unit vector.
\[ \overrightarrow{PQ} = \mathbf{v} \]
\[ = \left( \text{ } \_\_\_ \text{ } - 2 \right)\mathbf{i} + \left(9 - 4\right)\mathbf{j} \]
\[ = \text{ } \_\_\_ \mathbf{i} + \text{ } \_\_\_ \mathbf{j} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6fa7ac50-e43d-4267-956c-145cd19bd43c%2F0617e524-c795-49e2-a3ee-5f352bcd17f1%2F41g12f9_processed.png&w=3840&q=75)
Transcribed Image Text:**Tutorial Exercise**
Use the gradient to find the directional derivative of the function at \( P \) in the direction of \( Q \).
\[ f(x, y) = 3x^2 - y^2 + 4, \quad P(2, 4), \quad Q(5, 9) \]
**Step 1**
Since \( f \) is a polynomial, all the partial derivatives are continuous. Therefore, \( f \) is differentiable and you can apply the following theorem: if \( f \) is a differentiable function of \( x \) and \( y \), then the directional derivative of \( f \) in the direction of the unit vector \( \mathbf{u} \) is
\[ D_{\mathbf{u}}f(x, y) = \nabla f(x, y) \cdot \mathbf{u}. \]
Find the given directional vector \( \overrightarrow{PQ} \) so that we can find the unit vector.
\[ \overrightarrow{PQ} = \mathbf{v} \]
\[ = \left( \text{ } \_\_\_ \text{ } - 2 \right)\mathbf{i} + \left(9 - 4\right)\mathbf{j} \]
\[ = \text{ } \_\_\_ \mathbf{i} + \text{ } \_\_\_ \mathbf{j} \]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning