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Math Help - Derivitive Problem

  1. #1
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    Derivitive Problem

    Let f be the real-valued function defined by f(x)=sin^3(x)+sin^3(|x|)

    a. Find f '(x) for x>0
    b. Find f '(x) for x<0
    c. Determine whether f9s0 is continuous x = 0. Justify your answer.
    d. Determine whether the derivative of f(x) exists at x=0. Justify your answer

    Thanks a lot for the help, I had a lot of trouble throwing the absolute value into the derivative.

    -Aussiekid90
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  2. #2
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    Quote Originally Posted by aussiekid90 View Post
    Let f be the real-valued function defined by f(x)=sin^3(x)+sin^3(|x|)

    a. Find f '(x) for x>0
    b. Find f '(x) for x<0
    c. Determine whether f9s0 is continuous x = 0. Justify your answer.
    d. Determine whether the derivative of f(x) exists at x=0. Justify your answer

    Thanks a lot for the help, I had a lot of trouble throwing the absolute value into the derivative.

    -Aussiekid90
    Hint: What is the value of f(x) when x is negative? What is the value of f(x) when x is positive? Then construct f(x) as a piece-wise function and take the derivative of each piece. (Note: f'(x) is NOT defined at x = 0!)

    -Dan
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    Quote Originally Posted by aussiekid90 View Post
    Let f be the real-valued function defined by f(x)=sin^3(x)+sin^3(|x|)

    a. Find f '(x) for x>0
    b. Find f '(x) for x<0
    c. Determine whether f9s0 is continuous x = 0. Justify your answer.
    d. Determine whether the derivative of f(x) exists at x=0. Justify your answer
    For x>0 we have,
    f=\sin^3x+\sin^3x=2\sin^3x
    Then,
    f'=6\cos x \sin^2 x
    Thus, the derivative from the right at x=0 is,
    f'(0)^+=6\cos (0) \sin^2 (0)=0

    For x<0 we have,
    f=\sin^3x-\sin^3x=0
    Then,
    f'=0
    Thus, the derivative from the left at x=0 is,
    f'(0)^-=0
    Thus, the function is indeed differencial at x=0. Indeed everywhere.
    Which means the function is continous everywhere.
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