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Thread: Tangent line

  1. #1
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    Tangent line

    For what values of a and b are y = a b^x and y = 4 + x tangent at x = 0

    im guessing i have to find the derivative but im not quite sure.
    Thank you
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  2. #2
    Super Member wingless's Avatar
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    $\displaystyle y = a\cdot b^x$

    $\displaystyle y = x + 4$

    There are two conditions to be made,
    I) Both functions must have a common point at $\displaystyle x = 0$
    II) Their derivatives have to be equal at that point.


    I)
    $\displaystyle y = x + 4$
    When $\displaystyle x=0$, $\displaystyle y=4$
    So both functions must have a point at (0,4).
    $\displaystyle y = a\cdot b^x$
    $\displaystyle 4 = a\cdot b^0$
    $\displaystyle a = 4$

    The function becomes,
    $\displaystyle y = 4\cdot b^x$


    II)
    $\displaystyle y=x+4$
    $\displaystyle y' = 1$

    $\displaystyle y= 4\cdot b^x$
    $\displaystyle y' = 4 b^x \ln b$
    (At x = 0), $\displaystyle y' = 4 \ln b$

    Derivatives have to be equal.. So,
    $\displaystyle 1 = 4 \ln b$
    $\displaystyle b = e^{\frac{1}{4}}$

    Finally,
    $\displaystyle y=4e^{\frac{x}{4}}$
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  3. #3
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    thank you very much that makes perfect sense. i love how it seems so easy once someone shows me how its done
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