anonymous
  • anonymous
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
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
i found the gradient of f(x,y) already, but then what?
anonymous
  • anonymous
Is that exactly how the question is phrased? I'm a bit perplexed by the maximum part.
anonymous
  • anonymous
yes, but i got the answer already. it is the same gradient vector of f.

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anonymous
  • anonymous
Ok, good, makes sense :)
anonymous
  • anonymous
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. \[f(x,y)=x ^{2}+6y ^{2}\]
anonymous
  • anonymous
NIGHTIE I M BACK CM ON CHAT BOX
anonymous
  • anonymous
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?
anonymous
  • anonymous
sorry NIGHTIE BHAIYA
anonymous
  • anonymous
please explain more, i got the dot product part but then wouldn't G equal to 0 at the end?
anonymous
  • anonymous
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.
anonymous
  • anonymous
Do I make sense for you? :)
anonymous
  • anonymous
so the answer would be <0,0>?
anonymous
  • anonymous
wouldnt it just be enough to find the gradient at -3,4 and then find the unit vector?
anonymous
  • anonymous
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)
anonymous
  • anonymous
Yes Him, that's what I said.. ;)
anonymous
  • anonymous
bt then that would be (-6i+48j)/sqrt (2340) wont it?
anonymous
  • anonymous
Ah, I missunderstood your question, no that would make the gradient vector a unit vector, and that's not the same thing.
anonymous
  • anonymous
so is what im saying wrong?
anonymous
  • anonymous
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.
anonymous
  • anonymous
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...
anonymous
  • anonymous
cm on chat box him
anonymous
  • anonymous
now i get it thanks!
anonymous
  • anonymous
Good Nikie :) Glad to help!
anonymous
  • anonymous
its ok my pleasure
anonymous
  • anonymous
yeah...thanx...this makes me understand
anonymous
  • anonymous
NIGHTIE BHAIYA I HAD TOLD U SOLUTION ITS MY GREAT PLEASURE
anonymous
  • anonymous
mera fan banne ke liye DHANYAVAD

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