Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Derivating absolutes, etc

  1. Feb 11, 2005 #1

    Alkatran

    User Avatar
    Science Advisor
    Homework Helper

    Is there some general formula for deriving an absolut-ed function? Is what I;m doing wrong (a lot of derivation relies on continuous functions, doesn't it?)

    IE:
    d/dx(abs(sin(x)))

    Here's what I got:
    abs(x) = x*sign(x)
    d/dx(sign(x)) = 0 (x != 0)
    therefore
    [tex]
    \frac{d}{dx}abs(f(x)) = \frac{d}{dx}f(x)*sign(f(x)) = f '(x) * sign(f(x)) + 0
    [/tex]

    Which, by FTC would mean that:
    [tex]
    \int cos(x)*sign(sin(x)) \dx = abs(sin(x))
    [/tex]
    'I checked this by drawing the graphs and it appears right...

    Also... I saw that:
    [tex]
    abs(sin(x)) = sin(x \mod \pi)
    [/tex]
    [tex]
    \int x \mod c \dx = (\int_{0}^{c} x \dx)*INT(\frac{x}{c}) + \int_{0}^{x \mod c} x \dx
    [/tex]
    example:
    [tex]
    \int x \mod 1 \dx = INT(\frac{x}{c}) * .5 + x \mod 1
    [/tex]
    continuing...
    [tex]
    \int abs(sin(x)) dx = \int sin(x \mod \pi) \dx
    = (\int_{0}^{pi} sin(x) dx)*INT(x / \pi) - cos(x \mod \pi)
    = 2*INT(\frac{x}{\pi}) - cos(x \mod \pi)
    [/tex]
    Far as I can tell it works...
     
    Last edited: Feb 11, 2005
  2. jcsd
  3. Feb 11, 2005 #2

    AKG

    User Avatar
    Science Advisor
    Homework Helper

    Consider f(x) = x. To the right of zero, it's derivative is 1, and to the left, it is -1. The derivative, you can easily show, does not exist at 0. Do this from first principles, where you know the derivative is a limit. To find this limit, calculate the right limit (as your variable, normally h, approaches zero from the right) and notice that the limit evaluates to 1. Notice that it is -1 when h approaches zero from the left. Therefore, since left limit is not equal to right limit, the limit doesn't exist, and so, by definition, the derivative doesn't exist (since the derivative is this very limit).
     
  4. Feb 11, 2005 #3

    Alkatran

    User Avatar
    Science Advisor
    Homework Helper

    Sorry, I completely forgot to add x != 0

    d/dx(abs(x)) = sign(x), x != 0
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Derivating absolutes, etc
  1. Absolute values (Replies: 4)

  2. Absolute value (Replies: 3)

  3. Absolute 5 (Replies: 2)

  4. Absolute value (Replies: 1)

  5. Geometric absolute (Replies: 1)

Loading...