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Math Help - Tangent line

  1. #1
    Junior Member
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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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    y = a\cdot b^x

    y = x + 4

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


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

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


    II)
    y=x+4
    y' = 1

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

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

    Finally,
    y=4e^{\frac{x}{4}}
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  3. #3
    Junior Member
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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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