Civil Engineering (CE) Exam > Civil Engineering (CE) Questions > The directional derivative of = xy + yz + zx ...
Start Learning for Free
The directional derivative of ϕ = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3 at P(1,1,1) is A/√14, then the value of A is:
Correct answer is '12'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
The directional derivative of = xy + yz + zx along the tangent to the ...
To find the directional derivative of the function f(x,y,z) = xy + yz + zx, we need to find the gradient vector of f at the point P(1,1,1) and then take the dot product of the gradient vector with the tangent vector of the curve at P.
First, let's find the gradient vector of f:
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
= (y+z, x+z, x+y)
Now, let's find the tangent vector of the curve at P(1,1,1). We can do this by taking the derivative of each component of the curve with respect to t:
r(t) = (x(t), y(t), z(t))
= (t, t^2, t^3)
dr/dt = (dx/dt, dy/dt, dz/dt)
= (1, 2t, 3t^2)
At P(1,1,1), t = 1, so the tangent vector at P is:
dr/dt = (1, 2(1), 3(1)^2)
= (1, 2, 3)
Finally, we can find the directional derivative by taking the dot product of the gradient vector and the tangent vector:
Directional derivative = ∇f · dr/dt
= (y+z, x+z, x+y) · (1, 2, 3)
= (1+1, 1+1, 1+2)
= (2, 2, 3)
Therefore, the directional derivative of f = xy + yz + zx along the tangent to the curve at P(1,1,1) is (2, 2, 3).
First, let's find the gradient vector of f:
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
= (y+z, x+z, x+y)
Now, let's find the tangent vector of the curve at P(1,1,1). We can do this by taking the derivative of each component of the curve with respect to t:
r(t) = (x(t), y(t), z(t))
= (t, t^2, t^3)
dr/dt = (dx/dt, dy/dt, dz/dt)
= (1, 2t, 3t^2)
At P(1,1,1), t = 1, so the tangent vector at P is:
dr/dt = (1, 2(1), 3(1)^2)
= (1, 2, 3)
Finally, we can find the directional derivative by taking the dot product of the gradient vector and the tangent vector:
Directional derivative = ∇f · dr/dt
= (y+z, x+z, x+y) · (1, 2, 3)
= (1+1, 1+1, 1+2)
= (2, 2, 3)
Therefore, the directional derivative of f = xy + yz + zx along the tangent to the curve at P(1,1,1) is (2, 2, 3).
Free Test
FREE
| Start Free Test |
Community Answer
The directional derivative of = xy + yz + zx along the tangent to the ...
Concept:
The directional derivative is given by:
D.D. = grad(ϕ). 
Calculation:
grad(ϕ) at P(1,1,1) is:
The equation of the curve is represented by r(t).
The tangent of this curve is given by:
The value of tangent at (1,1,1) is:
D.D. = 12√14
Attention Civil Engineering (CE) Students!
To make sure you are not studying endlessly, EduRev has designed Civil Engineering (CE) study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Civil Engineering (CE).
|
Explore Courses for Civil Engineering (CE) exam
|
|
Similar Civil Engineering (CE) Doubts
Top Courses for Civil Engineering (CE)View all
The directional derivative of = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3at P(1,1,1) isA/√14, then the value of A is:Correct answer is '12'. Can you explain this answer?
Question Description
The directional derivative of = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3at P(1,1,1) isA/√14, then the value of A is:Correct answer is '12'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about The directional derivative of = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3at P(1,1,1) isA/√14, then the value of A is:Correct answer is '12'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The directional derivative of = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3at P(1,1,1) isA/√14, then the value of A is:Correct answer is '12'. Can you explain this answer?.
The directional derivative of = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3at P(1,1,1) isA/√14, then the value of A is:Correct answer is '12'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about The directional derivative of = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3at P(1,1,1) isA/√14, then the value of A is:Correct answer is '12'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The directional derivative of = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3at P(1,1,1) isA/√14, then the value of A is:Correct answer is '12'. Can you explain this answer?.
Solutions for The directional derivative of = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3at P(1,1,1) isA/√14, then the value of A is:Correct answer is '12'. Can you explain this answer? in English & in Hindi are available as part of our courses for Civil Engineering (CE).
Download more important topics, notes, lectures and mock test series for Civil Engineering (CE) Exam by signing up for free.
Here you can find the meaning of The directional derivative of = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3at P(1,1,1) isA/√14, then the value of A is:Correct answer is '12'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The directional derivative of = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3at P(1,1,1) isA/√14, then the value of A is:Correct answer is '12'. Can you explain this answer?, a detailed solution for The directional derivative of = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3at P(1,1,1) isA/√14, then the value of A is:Correct answer is '12'. Can you explain this answer? has been provided alongside types of The directional derivative of = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3at P(1,1,1) isA/√14, then the value of A is:Correct answer is '12'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The directional derivative of = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3at P(1,1,1) isA/√14, then the value of A is:Correct answer is '12'. Can you explain this answer? tests, examples and also practice Civil Engineering (CE) tests.
|
|
Explore Courses for Civil Engineering (CE) 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
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup