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Find slope of multivariable function

  1. Sep 15, 2015 #1
    1. The problem statement, all variables and given/known data
    A hill is described with the following function:

    f(x,y) = 3/(1+x2 +y2)

    Where f(x,y) is the height. Find the points where the hill is steepest!

    2. Relevant equations
    ∇f(x,y) = d/dx(f(x,y))i + d/dy(f(x,y))j

    3. The attempt at a solution
    d/dx(f(x,y)) = -6x/(1+x2+y2)2
    d/dy(f(x,y)) = -6y/(1+x2+y2)2

    Know as far as I understand the maximum slope should occur when:

    ||∇f(x,y)|| has it max. But how should I find that? And am I even going in the right direction?
     
  2. jcsd
  3. Sep 15, 2015 #2

    haruspex

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    Yes, that should work. You can write out the expression for ||∇f(x,y)|| and differentiate wrt to x, y, to find out where its extremals are. You can make life a bit easier by noting that you can discard the square root operation, since a positive value is maximised when its square is maximised.
     
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