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Homework Help: Study this function

  1. Dec 8, 2018 at 3:29 AM #1
    1. The problem statement, all variables and given/known data
    study the continuity, directional derivatives, and differentiability of the function f(x,y)=arctan(abs(y)*(y+x^2-1)).

    3. The attempt at a solution
    the function is obviously continuous in R2 since made of continuous functions.

    has directional derivatives everywhere since made of functions that has directional derivatives everywhere.

    differentiable everywhere exept for (x,0) and here is my biggest doubt: how do i demonstrate that it isnt differentiable there? if i think about it on a logical level, i know these are point where the function isnt smooth, but how do i demonstrate it?

    in (+-1,0) its differentiable because for h=x+-1,k=1:
    lim(h,k)->(0,0) arctan(abs(k)*(k+h^2+-2h))/(sqrt(h^2+k^2)) goes obviously to 0.
     
    Last edited: Dec 8, 2018 at 7:47 AM
  2. jcsd
  3. Dec 8, 2018 at 7:46 AM #2

    Ray Vickson

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    As written your function has unbalanced parentheses, so could represent either
    $$ f = \left(\arctan |y| \right)(y+x^2-1) \hspace{4ex}(1) $$
    or
    $$f = \arctan(|y|(y+x^2-1)) \hspace{4ex}(2)$$
    I suspect you mean (2), but (1) is a perfectly credible way of filling in the missing information.
     
  4. Dec 8, 2018 at 7:48 AM #3
    edited,yep i meant the second
     
  5. Dec 8, 2018 at 8:50 AM #4

    Ray Vickson

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    The "inside" function ##g(x,y) + |y| (y+x^2-1)## can be written as ##g(x,y) = y |y| + (x^2-1) |y|.## The first function--- ##y |y|## --- is differentiable in ##y## (even at ##y = 0##) but the second one --- ##(x^2-1) |y|## --- is not differentiable in ##y## at ##y = 0, x \neq \pm1.## So, your function is differentiable at ##(1,0)## and ##(-1,0)## but not at any other ##(x,0).##
     
  6. Dec 8, 2018 at 9:00 AM #5
    but how do i demonstrate it?
     
  7. Dec 8, 2018 at 9:18 AM #6

    Ray Vickson

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    Helpers are not allowed to show you all the details. I have said as much as I can (maybe even more than I should have) under PF rules.
     
  8. Dec 8, 2018 at 9:39 AM #7
    a better understanding should be the goal not solving homework to high schoolers imo. i already demonstrated what you did so i dont think you are showing more then you should have.
     
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