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Math Help - exponential

  1. #1
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    exponential

    if f(x) behaves exponentially and f(-1) = 8 and f(1) = 4

    1) Find a formula for f(x) using the natural base e:
    2) Find a formula for f(x) using another base suited to the data:
    3) What is differenital equation satsfied by f(x)?
    4) Where does f(x) = 1?
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  2. #2
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    Quote Originally Posted by viet View Post
    if f(x) behaves exponentially and f(-1) = 8 and f(1) = 4

    1) Find a formula for f(x) using the natural base e:
    2) Find a formula for f(x) using another base suited to the data:
    3) What is differenital equation satsfied by f(x)?
    4) Where does f(x) = 1?
    f(x) = Ae^{kt}

    Thus,

    4=f(1) = Ae^k (1)
    8=f(-1) = Ae^{-k} (2)

    Divide (2) by (1):
    \frac{Ae^{-k}}{Ae^k} = \frac{8}{4} =2
    Thus,
    e^{-2k} = 2
    Thus,
    -2k = \ln 2
    Thus,
     k = -\frac{1}{2} \ln 2

    Now you can solve for A.
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  3. #3
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    Hello, viet!

    Here's part (2) . . .


    If f(x) behaves exponentially and f(-1) = 8 and f(1) = 4

    2) Find a formula for f(x) using another base suited to the data
    I would use base 2 . . .


    We have: . f(x) \:=\:A\!\cdot\!2^{kt}

    \begin{array}{ccc}\text{Since }f(\text{-}1) = 8: & A\!\cdot\!2^{\text{-}k} \:=\:8 & [1] \\<br />
\text{Since }f(1) = 4: & A\!\cdot\!2^k \:=\:4 & [2]\end{array}

    Divide [2] by [1]: . \frac{A\!\cdot2^k}{A\!\cdot\!2^{\text{-}k}} \:=\:\frac{4}{8}\quad\Rightarrow\quad 2^{2k} \:=\:\frac{1}{2} \:=\:2^{\text{-}1}\quad\Rightarrow\quad 2k = \text{-}1\quad\Rightarrow\quad k = \text{-}\frac{1}{2}

    The function (so far) is: . f(x) \:=\:A\!\cdot\!2^{(\text{-}\frac{1}{2}t)}


    Since f(\text{-}1) = 8:\;A\!\cdot\!2^{(\text{-}\frac{1}{2})(\text{-}1)} \:=\:8\quad\Rightarrow\quad A\!\cdot\!2^{\frac{1}{2}} \:=\:8\quad\Rightarrow\quad A \:=\:4\sqrt{2}

    The function is: . f(x) \;=\;4\sqrt{2}\cdot2^{(\text{-}\frac{1}{2}t)}\;=\;2^2\cdot2^{\frac{1}{2}}\cdot2^  {(\text{-}\frac{1}{2}t)} \;=\;2^{(\frac{5}{2}-\frac{1}{2}t)}

    Therefore: . \boxed{f(x) \;=\;2^{\frac{1}{2}(5-t)}}

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