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Finding the Direction of a Function

  1. Oct 23, 2011 #1
    1. The problem statement, all variables and given/known data
    Find the direction in which the function f(x,y)=x^2+sin(4y) increses most rapidly at the point P0=(1,0). Then find the derivative of f in this direction.


    2. Relevant equations



    3. The attempt at a solution
    I think I have to find the gradient at point P0 and then find a unit vector is this right?
     
  2. jcsd
  3. Oct 23, 2011 #2

    SammyS

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    Yes, for the first part of your question.
     
  4. Oct 23, 2011 #3
    Okay, I found the gradient <2x+sin(4y), x^2+4cos(4y)> and at point P0 it is
    <2,5>. Now if I only have one point how do I find the unit vector wouldn't I need another point or an initial vector?
     
  5. Oct 23, 2011 #4

    LCKurtz

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    What do you need the unit vector for?

    Your gradient isn't correct. You want ∇f = <fx, fy>.
    Then think about this question: What does the gradient have to do with the maximum rate of increase of a function? The answer to that is surely in your text.
     
  6. Oct 23, 2011 #5
    Sorry. [itex]\nabla[/itex]f=<2x, 4 cos(4y)>. Does the gradient at P0 tell me the direction in which the function increases most rapidly?
     
  7. Oct 23, 2011 #6

    LCKurtz

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    Yes, and there's more. What does the magnitude of the gradient represent?
     
  8. Oct 24, 2011 #7
    How fast the function increases?
     
  9. Oct 24, 2011 #8

    HallsofIvy

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    In what direction? A function of several variables may have different rates of increase in different directions.
     
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