Results 1 to 4 of 4

Math Help - Show it is differentiable and compute f '

  1. #1
    Member thaopanda's Avatar
    Joined
    Sep 2009
    From
    Worcester, Massachusetts
    Posts
    85

    Show it is differentiable and compute f '

    Show that the function f: R \rightarrow R given by

    f(x): =
    x^3, x \leq 0
    0, x = 0

    is differentiable and compute f '.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,329
    Thanks
    1244
    Quote Originally Posted by thaopanda View Post
    Show that the function f: R \rightarrow R given by

    f(x): =
    x^3, x \leq 0
    0, x = 0

    is differentiable and compute f '.
    What about when x > 0?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member thaopanda's Avatar
    Joined
    Sep 2009
    From
    Worcester, Massachusetts
    Posts
    85
    oh wow, I wasn't paying attention. Messed that one up. It's actually:

    f(x) :=
    x^3, x \leq 0
    x^2*sin(\frac{1}{x}), x > 0

    sorry about that
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,329
    Thanks
    1244
    Quote Originally Posted by thaopanda View Post
    oh wow, I wasn't paying attention. Messed that one up. It's actually:

    f(x) :=
    x^3, x \leq 0
    x^2*sin(\frac{1}{x}), x > 0

    sorry about that
    To be differentiable, the function needs to be continuous and smooth.


    Clearly x^3 is continuous since it is a polynomial.

    Also, since x^2 and \sin{\left(\frac{1}{x}\right)} are both continuous for x > 0, so will be their product.

    So the only place where this hybrid function might not be continuous is where the function "changes" - so at x = 0. We need to check that the left hand and right hand limits are 0.

    Clearly the left hand limit is 0. So we would need to show that \lim_{x \to 0^{+}}x^2\sin{\left(\frac{1}{x}\right)} = 0.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Show that f(Z) = Z(hat) is not differentiable.
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 25th 2011, 01:28 PM
  2. Replies: 1
    Last Post: July 7th 2010, 09:21 AM
  3. Replies: 2
    Last Post: April 22nd 2010, 05:57 AM
  4. Proof that G is differentiable on \R & compute G'
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: April 20th 2010, 11:21 AM
  5. Show it's twice differentiable
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 29th 2008, 08:42 AM

Search Tags


/mathhelpforum @mathhelpforum