1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Bounded homework

  1. Jan 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Let [tex]f:[-1,1] \times \mathbb{R} \to\mathbb{R}[/tex] be a function.

    If [itex]f[/itex] is defined by:

    (i) [itex]f(x,y) = 3\exp(x-y^2)[/itex]

    then is the derivative with respect to [itex]y[/itex] bounded?

    If [itex]f[/itex] is defined by:

    (ii) [itex]f(x,y) = 7\exp(y^2-x)[/itex]

    then is the derivative with respect to [itex]y[/itex] bounded?

    3. The attempt at a solution

    For (i):

    [itex]\frac{\partial f}{\partial y} = -6y\exp(x-y^2) = -6y\exp(-y^2)\exp(x)[/itex]

    and since [itex]\exp(x) \leq e[/itex] in the specified domain and since [itex]y\exp(-y^2)[/itex] is a bounded function on [itex]\mathbb{R}[/itex], [itex]\frac{\partial f}{\partial y}[/itex] is bounded - right?

    For (ii):

    [itex]\frac{\partial f}{\partial y} = 14y\exp(y^2-x) = 14y\exp(y^2)\exp(-x)[/itex]

    This is not bounded - right?
    Last edited: Jan 25, 2012
  2. jcsd
  3. Jan 25, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Re: Bounded?

    That looks right.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook