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anonymous

  • 5 years ago

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

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  1. anonymous
    • 5 years ago
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    i found the gradient of f(x,y) already, but then what?

  2. anonymous
    • 5 years ago
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    Is that exactly how the question is phrased? I'm a bit perplexed by the maximum part.

  3. anonymous
    • 5 years ago
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    yes, but i got the answer already. it is the same gradient vector of f.

  4. anonymous
    • 5 years ago
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    Ok, good, makes sense :)

  5. anonymous
    • 5 years ago
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    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. \[f(x,y)=x ^{2}+6y ^{2}\]

  6. anonymous
    • 5 years ago
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    NIGHTIE I M BACK CM ON CHAT BOX

  7. anonymous
    • 5 years ago
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    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?

  8. anonymous
    • 5 years ago
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    sorry NIGHTIE BHAIYA

  9. anonymous
    • 5 years ago
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    please explain more, i got the dot product part but then wouldn't G equal to 0 at the end?

  10. anonymous
    • 5 years ago
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    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.

  11. anonymous
    • 5 years ago
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    Do I make sense for you? :)

  12. anonymous
    • 5 years ago
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    so the answer would be <0,0>?

  13. anonymous
    • 5 years ago
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    wouldnt it just be enough to find the gradient at -3,4 and then find the unit vector?

  14. anonymous
    • 5 years ago
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    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)

  15. anonymous
    • 5 years ago
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    Yes Him, that's what I said.. ;)

  16. anonymous
    • 5 years ago
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    bt then that would be (-6i+48j)/sqrt (2340) wont it?

  17. anonymous
    • 5 years ago
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    Ah, I missunderstood your question, no that would make the gradient vector a unit vector, and that's not the same thing.

  18. anonymous
    • 5 years ago
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    so is what im saying wrong?

  19. anonymous
    • 5 years ago
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    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.

  20. anonymous
    • 5 years ago
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    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...

  21. anonymous
    • 5 years ago
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    cm on chat box him

  22. anonymous
    • 5 years ago
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    now i get it thanks!

  23. anonymous
    • 5 years ago
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    Good Nikie :) Glad to help!

  24. anonymous
    • 5 years ago
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    its ok my pleasure

  25. anonymous
    • 5 years ago
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    yeah...thanx...this makes me understand

  26. anonymous
    • 5 years ago
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    NIGHTIE BHAIYA I HAD TOLD U SOLUTION ITS MY GREAT PLEASURE

  27. anonymous
    • 5 years ago
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    mera fan banne ke liye DHANYAVAD

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