Question

how do i find the unit vector in the direction for which the directional derivative of f at the point (-3,4) is maximum? f(x,y)=x^2+6y^2

Answer 1

i found the gradient of f(x,y) already, but then what?

Answer 2

Is that exactly how the question is phrased? I'm a bit perplexed by the maximum part.

Answer 3

yes, but i got the answer already. it is the same gradient vector of f.

Answer 4

Ok, good, makes sense :)

Answer 5

Do u know how to do this btw? Find the unit vector in the direction for which the directional derivative of f at the point (-3,4) is zero.
\[f(x,y)=x ^{2}+6y ^{2}\]

Answer 6

NIGHTIE I M BACK CM ON CHAT BOX

Answer 7

Unit vector = U
Gradient vector = G
(U/|U|)*G=0
Solve U as 8i+j for example, use DOT product between the vectors obviously. :)
Anywhere you want me to explain more?

Answer 8

sorry NIGHTIE BHAIYA

Answer 9

please explain more, i got the dot product part but then wouldn't G equal to 0 at the end?

Answer 10

That's the point, you are given that the directional derivative at point (-3,4) is 0.
The formula for computing the directional derivative is unit vector * G vector.
So all you really need to do is match the unit vector so the dot product equals the directional derivative which is 0 in this case.

Answer 11

Do I make sense for you? :)

Answer 12

so the answer would be <0,0>?

Answer 13

wouldnt it just be enough to find the gradient at -3,4 and then find the unit vector?

Answer 14

You have this:
G = 2xi + 12yj
G(-3,4) = -6i + 48j
You're now looking for a unit vector that would satisfy the equation of:
U*G = 0
Basically you want the dot product between those two vectors to equal 0.
A 0i +0j vector is not a unit vector. So you want to rephrase the equation like this instead:
Unit vector = vector / |vector|
S for example the dot product between 8i+j * -6i +48j = 0
But 8i + j / |8i +j| is still a unit vector, so nothing wrong there.
Answer would then be 8i + j / |8i +j| = (8i + j) / sqrt(65)

Answer 15

Yes Him, that's what I said.. ;)

Answer 16

bt then that would be (-6i+48j)/sqrt (2340) wont it?

Answer 17

Ah, I missunderstood your question, no that would make the gradient vector a unit vector, and that's not the same thing.

Answer 18

so is what im saying wrong?

Answer 19

I dont actually know this. I'm not sure what happens if you convert the gradient to a unit vector, but then you'd still have to find the unit vector U so it matches unit of the gradient and still equals 0 in the dot product. Imo it would make things more complicated then they should be.

Answer 20

nd i dont understand how the directional derivative can be MAXIMUM or MINIMUM. From what i know , its just the unit vector of the gradient of a field at a given point, isnt it? Even i dunno vector calc much...

Answer 21

cm on chat box him

Answer 22

now i get it thanks!

Answer 23

Good Nikie :) Glad to help!

Answer 24

its ok my pleasure

Answer 25

yeah...thanx...this makes me understand

Answer 26

NIGHTIE BHAIYA I HAD TOLD U SOLUTION ITS MY GREAT PLEASURE

Answer 27

mera fan banne ke liye DHANYAVAD