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Math Help - Derivatives and the Intermediate Value Property

  1. #1
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    Derivatives and the Intermediate Value Property

    (1) Let g_a(x)=\left\{\begin{array}{cc}x^{a},&\mbox{ if } x\geq 0\\0, & \mbox{ if } x<0\end{array}\right.

    (a) For which of the values of a is f continuous at zero?

    (b) For which values of a is f differentiable at zero? In this case, is the derivative function continuous?

    (c) For which values of a is f twice-differentiable?

    (2) Let f and g be functions defined on an interval A, and assume both are differentiable at some point c \in A.

    Prove the following:

    (a) (f+g)'(c) = f'(c) + g'(c)

    (b) (kf)'(c) = kf'(c), for all k \in R (R is the reals)


    Any help would be greatly appreciated, thanks!
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by shadow_2145 View Post
    (1) Let g_a(x)=\left\{\begin{array}{cc}x^{a},&\mbox{ if } x\geq 0\\0, & \mbox{ if } x<0\end{array}\right.

    (a) For which of the values of a is f continuous at zero?

    (b) For which values of a is f differentiable at zero? In this case, is the derivative function continuous?

    (c) For which values of a is f twice-differentiable?

    (2) Let f and g be functions defined on an interval A, and assume both are differentiable at some point c \in A.

    Prove the following:

    (a) (f+g)'(c) = f'(c) + g'(c)

    (b) (kf)'(c) = kf'(c), for all k \in R (R is the reals)


    Any help would be greatly appreciated, thanks!

    Here are a few hints to get you started.

    use the limit def of continity

    f(0)=\lim_{x \to 0^+}f(x)

    use the differnce quotent for the derivative

    f'(0)=\lim_{x \to 0}\frac{f(x)-f(0)}{x-0}=\lim_{x \to 0}\frac{x^a-0}{x-0}

    Find the values of a so that the limit exits. Try to use a similar process to finish the problem.
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  3. #3
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    Sorry, I still don't understand how to do problem 1. I figured out problem 2, that was easy. If anyone can help me out, thanks!
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by shadow_2145 View Post
    (1) Let g_a(x)=\left\{\begin{array}{cc}x^{a},&\mbox{ if } x\geq 0\\0, & \mbox{ if } x<0\end{array}\right.

    (a) For which of the values of a is f continuous at zero?

    (b) For which values of a is f differentiable at zero? In this case, is the derivative function continuous?

    (c) For which values of a is f twice-differentiable?

    (2) Let f and g be functions defined on an interval A, and assume both are differentiable at some point c \in A.


    Prove the following:

    (a) (f+g)'(c) = f'(c) + g'(c)

    (b) (kf)'(c) = kf'(c), for all k \in R (R is the reals)


    Any help would be greatly appreciated, thanks!
    a) none, for that value of a is x^a=0
    b) what value of a is ax^{a-1}=0
    c)what about a(a-1)x^{a-2}

    For two, use the difference quotient
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  5. #5
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    Wow, I'm an idiot. lol Thanks!
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  6. #6
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by shadow_2145 View Post
    Wow, I'm an idiot. lol Thanks!
    No, I am an idiot, I misread your question, you need to put 0 in for each x, so each question is incredibly easy.
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  7. #7
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    Wait...now I'm lost! For each x? Please explain. Sorry.
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  8. #8
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by shadow_2145 View Post
    Wait...now I'm lost! For each x? Please explain. Sorry.
    Bascially for this function to be continous you must have that f(0) from the right side of zero equals f(0) from the left
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