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Math Help - Differentiation

  1. #1
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    Differentiation

    I need some help with this problem (please don't give me the answer though, just hint me)

    suppose the function g is defined by

    k sqrt(X+1) for 0<(including) x<(including) 3
    g(X)=<
    mX+2 for 3<x<(including) 5

    where k and m are constants. if g is differentiable at x=3, what are the values of k and m.

    Im not sure how to do this. But to start I tried solving the the two pieces of the function and got
    g(X)= 2k & g(X)=3m+2

    Please Help
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  2. #2
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    Hello, rawkstar!

    g(x) \;=\;\begin{Bmatrix}k\sqrt{x+1} & 0 \leq x\leq 3 \\<br />
mx+2 & 3 < x \leq 5 \end{Bmatrix} .where k and m are constants.

    If g(x) is differentiable at x=3, what are the values of k and m ?

    Since the function is differentiable at x=3
    . . the two branches of the function are equal when x = 3.

    We have: . k\sqrt{3+1} \:=\:m(3) + 2\quad\Rightarrow\quad\boxed{ 2k \:=\:3m + 2}


    The derivatives of the two branches are: . g'(x) \;=\;\begin{Bmatrix}\dfrac{k}{2\sqrt{x+1}} \\ m \end{Bmatrix}

    Since the derivatives are equal at x = 3, we have:

    . . \frac{k}{2\sqrt{3+1}} \:=\:m \quad\Rightarrow\quad\boxed{ k \:=\:4m}


    Now solve the system of equations . . .

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  3. #3
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    how did you get the derivative and why do you know that?
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