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Math Help - the derivative function

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    the derivative function

    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
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by vinson24 View Post
    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?
    A function is continuous at a point when \lim_{x\to c}f(x) exists. This means that \lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)

    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
    It is differentiable at that point if the derivatives of the first and second terms in the piecewise function are the same at that point.

    Note that you may not get the same k value for the continuity and differentiability cases...

    Can you try these now and see what you get?

    --Chris
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    im still not seeing it
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  4. #4
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    i know f is continuous if k=1/5 and f is differentianle if k=3/80 but not seeing how to get there
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  5. #5
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by vinson24 View Post
    i know f is continuous if k=1/5 and f is differentianle if k=3/80 but not seeing how to get there
    Quote Originally Posted by vinson24 View Post
    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?
    Let's find \lim_{x\to 5^-}f(x)

    \lim_{x\to 5^-}f(x)=\lim_{x\to 5} k(x^2-5)=20k

    Now find \lim_{x\to 5^+}f(x)

    \lim_{x\to 5^+}f(x)=\lim_{x\to 5} \sqrt{1+3x}=\sqrt{16}=4

    The function is continuous when \lim_{x\to 5^-}f(x)=\lim_{x\to 5^+}f(x)\implies 20k=4\implies \color{red}\boxed{k=\tfrac{1}{5}}

    Does this bit make sense?

    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
    Let's look at the derivative of the function for x<5:

    f'(x)=2kx

    Now, let's look at the derivative of the function for x\geq5:

    f'(x)=\frac{3}{2\sqrt{1+3x}}

    In order for this function to be differentiable at x=5, both of these derivatives must have the same value at x=5:

    This implies that 10k=\frac{3}{8}\implies \color{red}\boxed{k=\frac{3}{80}}

    Does this make sense?

    --Chris
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