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Thread: Tangency

  1. #1
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    Tangency

    Another problem I'm stuck on.

    "If k >/= 1 (greater than or equal to 1), the graphs of $\displaystyle y=sin(x)$ and y=ke^-x intersect for x >/= 0. Find the smallest value of k for which the graphs are tangent. What are the coordinates of the point of tangency?"

    Help, please?
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  2. #2
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    Hello, Rumor!

    I think I have a solution.
    Please check my reasoning and my work.


    If $\displaystyle k \geq 1$, the graphs of: $\displaystyle y\:=\:\sin x$ and $\displaystyle y\:=\:ke^{-x}$ intersect for $\displaystyle x \geq 0.$
    Find the smallest value of $\displaystyle k$ for which the graphs are tangent.
    What are the coordinates of the point of tangency?
    We have: .$\displaystyle \begin{array}{ccc} f(x) &=& \sin x \\ g(x) &=& ke^{-x} \end{array}$

    The graphs intersect at the point of tangency.
    . . Hence: .$\displaystyle f(x) \:=\:g(x) \quad\Rightarrow\quad \sin x \:=\:ke^{-x} \quad\Rightarrow\quad e^x\sin x \:=\:k $ [1]

    Their slopes are equal at the point of tangency.
    . . Hence: .$\displaystyle f'(x) = g'(x) \quad\Rightarrow\quad \cos x \:=\:-ke^{-x} \quad\Rightarrow\quad e^x\cos x \:=\:-k$ [2]


    Divide [1] by [2]: . $\displaystyle \frac{e^x\sin x}{e^x\cos x} \:=\:\frac{k}{\text{-}k} \quad\Rightarrow\quad \tan x \:=\:-1 \quad\Rightarrow \quad x \:=\:\frac{3\pi}{4}$


    We have: .$\displaystyle \begin{array}{ccccccc}f\left(\frac{3\pi}{4}\right) &=& \sin\frac{3\pi}{4} &=& \frac{1}{\sqrt{2}} \\ \\[-3mm]
    g\left(\frac{3\pi}{4}\right) &=& ke^{-\frac{3\pi}{4}}\end{array}$


    Since $\displaystyle g\left(\tfrac{3\pi}{4}\right) \,=\,f\left(\tfrac{3\pi}
    {4}\right)$, we have: .$\displaystyle ke^{-\frac{3\pi}{4}} \:=\:\frac{1}{\sqrt{2}} $

    Therefore: .$\displaystyle \boxed{k \:=\:\frac{e^{\frac{3\pi}{4}}}{\sqrt{2}}} $

    And the point of tangency is: .$\displaystyle \left(\frac{3\pi}{4},\:\frac{1}{\sqrt{2}}\right) $

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  3. #3
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    Ah, of course!

    Your work looks sound to me. I don't know why I couldn't remember to set them equal. Thank you very much!
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