elementary calc

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f(x) passes through (0,2). the slope of f at any point is 3 times the y-coordinate. find the derivative at 1

  • elementary calc -

    When a derivative of a function is a multiple of the y value of the function, it suggests an exponential function.

    let the function be y = a e^(kx), where a and k are constanst

    dy/dx = ak e^(kx)

    given ak e^(kx) = 3y = 3(a e^(kx))
    then ak = 3a
    k = 3
    so we know y = a e^(3x)
    at (0,2)
    2 = a e^(0)
    a = 2

    then y = 2 e^(3x)
    then dy/dx = 6 e^(3x)
    when x=1
    dy/dx = 6 e^3

  • elementary calc -

    dy/dx = 3 y

    dy/y = 3 dx

    ln y = 3 x + c

    y = C e^3x

    if x = 0, y = 2
    so
    y = 2 e^3x

    when x = 1
    y = 2 e^3 = 40

    dy/dx = 3y = 120

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