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Math Help - tangent lines

  1. #1
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    tangent lines

    For what values of a and b are y = a b^x and y = 1 + x tangent at x = 0.

    how do i start this one??
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  2. #2
    MHF Contributor
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    Let f(x) = y = a*b^x -------------(i)
    And g(x) = y = 1 +x

    If f(x) and g(x) are tangent at x=0, then then their y's there are the same.
    So,
    f(0) = g(0)
    a*b^0 = 1 +0
    a*1 = 1
    a = 1 -----------------answer.

    So,
    f(x) = y = 1*b^x = b^x -------(ii)

    If f(x) and g(x) are tangent at x=0, then their slopes are equal at x=0.

    slope, m = dy/dx

    For g(x) = 1 +x,
    m2 = g'(x) = 1 --------**

    For f(x) = b^x,
    m1 = f'(x) = ln(b) *b^x

    At x=0,
    m1 = m2
    ln(b) *b^x = 1
    ln(b) *b^0 = 1
    ln(b) *1 = 1
    ln(b) = 1
    b = e^1
    b = e ------------------------answer.
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