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Thread: derivative

  1. #1
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    derivative

    suppose f(x)= sqrt of 1+3x, x>or equal to 5
    k(x^2-5), x<5
    for what value of k is continuous? For what value of k is f differentiable?

    not even sure where to start i think i am on dealing with the second equation in this piecewise function

    i know the answers are f is continuous if k=1/5 and f is differentiable if k=3/80
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  2. #2
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    $\displaystyle f\left( x \right) = \left\{ \begin{gathered}
    \sqrt {1 + 3x} ,{\text{ if }}x \geqslant 5 \hfill \\
    k\left( {x^2 - 5} \right){\text{, if }}x < 5 \hfill \\
    \end{gathered} \right. \hfill \\$

    $\displaystyle {\text{for }}f\left( x \right){\text{ to be continuous, it should be continuous at }}x = 5. \hfill \\$


    $\displaystyle {\text{For }}f\left( x \right){\text{ to be continuous at }}x = 5, \hfill \\$

    $\displaystyle \mathop {\lim }\limits_{x \to 5 - } f\left( x \right) = \mathop {\lim }\limits_{x \to 5 + } f\left( x \right) = f\left( 5 \right) \hfill \\$


    $\displaystyle {\text{Now,}} \hfill \\$

    $\displaystyle \mathop {\lim }\limits_{x \to 5 - } f\left( x \right) = \mathop {\lim }\limits_{x \to 5 - } k\left( {x^2 - 5} \right) = k\left( {5^2 - 5} \right) = 20k \hfill \\$

    $\displaystyle \mathop {\lim }\limits_{x \to 5 + } f\left( x \right) = \mathop {\lim }\limits_{x \to 5 + } \sqrt {1 + 3x} = \sqrt {1 + 3\left( 5 \right)} = 4 \hfill \\$


    $\displaystyle \Rightarrow 20k = 4 \hfill \\$

    $\displaystyle \Rightarrow k = \frac{1}
    {5} \hfill \\ $

    f(x) is continuous when k = 1/5
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  3. #3
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    thanks how about finding the differentiable being 3/80
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  4. #4
    Member Jason Bourne's Avatar
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    Quote Originally Posted by vinson24 View Post
    thanks how about finding the differentiable being 3/80
    $\displaystyle f'\left( x \right) = \left\{ \begin{gathered}
    \frac{3}{2\sqrt {1 + 3x}} \hfill \\
    2kx \hfill \\
    \end{gathered} \right. \hfill \\$

    at x=5 , I think this is similar to before but using the above, you should look over any notes you have on continuity and differentiability.
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