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!

Inverse Trig Function Derivative (Apostol Section 6.22 #11)

  1. Aug 31, 2011 #1
    1. The problem statement, all variables and given/known data
    Given that [itex]\frac{d}{dx} (\text{arccot}{x}-\arctan{1/x})=0 \hspace{10mm} \forall x \ne 0[/itex],
    prove that there is no constant C such that [itex]\text{arccot}{x}-\arctan{\frac{1}{x}}=C \hspace{10mm} \forall x \ne 0[/itex]
    and explain why this does not contradict the zero-derivative theorem.


    2. Relevant equations
    The Zero-Derivative Theorem:
    If f'(x) = 0 for each x in an open interval I, then f is constant on I.


    3. The attempt at a solution
    The first part of this problem has you verify that the derivative is indeed zero, which I did verify. I think that [itex]\text{arccot}{x}-\arctan{\frac{1}{x}}=0 \hspace{10mm} \forall x \ne 0[/itex], however, since [itex]\text{arccot}{x}=y \implies x=\cot{y} \implies \frac{1}{x} = \tan{y} \implies \arctan{\frac{1}{x}}=y[/itex].

    WolframAlpha seems to agree:
    http://www.wolframalpha.com/input/?i=arccot(x)+-+arctan(1/x)

    So is Apostol not considering 0 a constant (that is, when he refers to "a constant C", is C necessarily not equal to 0)?
     
  2. jcsd
  3. Aug 31, 2011 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

  4. Aug 31, 2011 #3
    OK, so the reason my "proof" that it is equal to zero doesn't work is in the last step, where [itex]\arctan{\frac{1}{x}}[/itex] may equal y (if [itex]y\in(0,\frac{\pi}{2})[/itex]), or it may equal [itex]y-\pi[/itex], correct?

    Then the reason that this is not a contradiction of the zero derivative theorem is that at x=0 there is a discontinuity, and on either side of 0 it would apply but the constants are not equal.
     
  5. Aug 31, 2011 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Indeed!!
     
  6. Aug 31, 2011 #5
    Thanks for your help! I'm loving Apostol's book so far.
     
  7. Aug 31, 2011 #6

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    It's one of my favorite books as well :biggrin:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Inverse Trig Function Derivative (Apostol Section 6.22 #11)
Loading...