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Math Help - Wonderful Calculus Quiz problem

  1. #1
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    Wonderful Calculus Quiz problem

    Hope someone can figure this one out because I am LOST. THANK YOU!!

    Find the function f(t)=ce^(kt) that passes through the given points in the plane.

    A) (0,5) and (1,1)


    B) (2,4) and (3,7)

    If you can give me A or B that would be more than awesome. Thanks again!
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  2. #2
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    Hello, DJ Big T!

    Find the function f(t)\:=\:Ce^{kt} that passes through
    these points: . [A]\;\;(0,5)\text{ and }(1,1)

    The point (0,5) is on the graph.

    . . Then: . 5 \:=\:Ce^{k\cdot0} \quad\Rightarrow\quad C\:=\:5


    The point (1,1) is on the graph.

    . . Then: . 1 \:=\:Ce^{k\cdot 1} \quad\Rightarrow\quad 1 \:=\:5e^k \quad\Rightarrow\quad e^k \:=\:\tfrac{1}{5} \quad\Rightarrow\quad k \:=\:\ln(\tfrac{1}{5})


    The function is: . \boxed{f(t) \;=\;5e^{(\ln\frac{1}{5})t}}


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    The answer can be further simplified:

    . . f(t)\;=\;5e^{(\ln\frac{1}{5})t} \;=\; 5\left(e^{\ln\frac{1}{5}}\right)^t \;=\;5\left(\frac{1}{5}\right)^t \;=\;5\cdot\frac{1}{5^t}

    . . f(t) \;=\;\frac{1}{5^{t-1}}

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