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: Prove that the function in defined

  1. Aug 24, 2007 #1

    daniel_i_l

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data

    f(x) is defined as f(x) = 1/((ln(x+1))^2 + 1) for all x>-1 and f(x)=0 for x=-1.

    1)Prove that the function [tex]F(x) = \int^{x^2 + 2x}_{0} f(t)dt[/tex]
    is defined and has a derivative in R.
    2)g(x) is defined as g(x)=f(x) for x>-1 and g(x)=-1 for x=-1.
    Also, [tex]G(x) = \int^{x^2 + 2x}_{0} g(t)dt[/tex]
    Is G(x) defined in R? Does it have a derivative?

    2. Relevant equations



    3. The attempt at a solution

    1) By taking the limit of f(x) at x=0 we see that f is continues for all x>=1 and since
    x^2 + 2x >= -1 for all x in R F(x) is defined and is has a derivative from the chain rule.

    2)Since f(x)=g(x) for all x=/=-1 F(x)=G(x) and so the answer to both questions is yes.

    Are those right? I think that the answer to (2) is wrong but why?
    Thanks.
     
    Last edited: Aug 24, 2007
  2. jcsd
  3. Aug 24, 2007 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Seems fine to me. For 2), why would you think changing the value of a function at a single point could change the integral? The set {-1} has measure 0.
     
  4. Aug 24, 2007 #3

    daniel_i_l

    User Avatar
    Gold Member

    Thanks a lot.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook