Electrical Engineering (EE) Exam > Electrical Engineering (EE) Questions > Find the directional derivative of the functi...
Start Learning for Free
Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}?
Most Upvoted Answer
Find the directional derivative of the function W = x²y + xyz at the p...
Understanding the Directional Derivative
The directional derivative of a function gives the rate at which the function changes at a point in a specified direction. It can be calculated using the gradient vector and the direction vector.
Step 1: Calculate the Gradient of W
The function is W = x²y + xyz. First, we find the gradient, ∇W, by calculating the partial derivatives:
- ∂W/∂x = 2xy + yz
- ∂W/∂y = x² + xz
- ∂W/∂z = xy
Now, at the point (2, -1, 0):
- ∂W/∂x = 2(2)(-1) + (-1)(0) = -4
- ∂W/∂y = (2)² + (2)(0) = 4
- ∂W/∂z = (2)(-1) = -2
Thus, the gradient vector at (2, -1, 0) is ∇W = (-4, 4, -2).
Step 2: Normalize the Direction Vector
The direction vector is given as 3dₓ + 4dᵧ + 12d_z. First, we write it as (3, 4, 12).
Next, we find its magnitude:
- Magnitude = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13.
Now, we normalize the vector:
- Unit vector u = (3/13, 4/13, 12/13).
Step 3: Compute the Directional Derivative
The directional derivative D_uW is given by:
- D_uW = ∇W • u = (-4, 4, -2) • (3/13, 4/13, 12/13)
Calculating this yields:
- D_uW = (-4)(3/13) + (4)(4/13) + (-2)(12/13)
= -12/13 + 16/13 - 24/13
= -20/13.
Conclusion
The directional derivative of W at the point (2, -1, 0) in the specified direction is -20/13.
The directional derivative of a function gives the rate at which the function changes at a point in a specified direction. It can be calculated using the gradient vector and the direction vector.
Step 1: Calculate the Gradient of W
The function is W = x²y + xyz. First, we find the gradient, ∇W, by calculating the partial derivatives:
- ∂W/∂x = 2xy + yz
- ∂W/∂y = x² + xz
- ∂W/∂z = xy
Now, at the point (2, -1, 0):
- ∂W/∂x = 2(2)(-1) + (-1)(0) = -4
- ∂W/∂y = (2)² + (2)(0) = 4
- ∂W/∂z = (2)(-1) = -2
Thus, the gradient vector at (2, -1, 0) is ∇W = (-4, 4, -2).
Step 2: Normalize the Direction Vector
The direction vector is given as 3dₓ + 4dᵧ + 12d_z. First, we write it as (3, 4, 12).
Next, we find its magnitude:
- Magnitude = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13.
Now, we normalize the vector:
- Unit vector u = (3/13, 4/13, 12/13).
Step 3: Compute the Directional Derivative
The directional derivative D_uW is given by:
- D_uW = ∇W • u = (-4, 4, -2) • (3/13, 4/13, 12/13)
Calculating this yields:
- D_uW = (-4)(3/13) + (4)(4/13) + (-2)(12/13)
= -12/13 + 16/13 - 24/13
= -20/13.
Conclusion
The directional derivative of W at the point (2, -1, 0) in the specified direction is -20/13.
Attention Electrical Engineering (EE) Students!
To make sure you are not studying endlessly, EduRev has designed Electrical Engineering (EE) study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Electrical Engineering (EE).
|
Explore Courses for Electrical Engineering (EE) exam
|
|
Top Courses for Electrical Engineering (EE)View all
Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}?
Question Description
Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}? for Electrical Engineering (EE) 2024 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}? covers all topics & solutions for Electrical Engineering (EE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}?.
Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}? for Electrical Engineering (EE) 2024 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}? covers all topics & solutions for Electrical Engineering (EE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}?.
Solutions for Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}? in English & in Hindi are available as part of our courses for Electrical Engineering (EE).
Download more important topics, notes, lectures and mock test series for Electrical Engineering (EE) Exam by signing up for free.
Here you can find the meaning of Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}? defined & explained in the simplest way possible. Besides giving the explanation of
Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}?, a detailed solution for Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}? has been provided alongside types of Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}? theory, EduRev gives you an
ample number of questions to practice Find the directional derivative of the function W = x²y + xyz at the point (2.-1.0) in the direction of the vector 3d_{x} + 4d_{y} + 12d_{x}? tests, examples and also practice Electrical Engineering (EE) tests.
|
|
Explore Courses for Electrical Engineering (EE) exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup