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Math Help - Finding the area of a region

  1. #1
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    Finding the area of a region

    let f be the function given by f(x)=(4x^2)-x^3, and let l be the line y=18-3x, where l is tangent to the graph of f. Let D be the region bounded by the graph of f and the x-axis, and let E be the region bounded by the graph f, the line l, and the x-axis.

    a. show that l is tangent to the graph of y=f(x) at the point x=3

    Answer:
    df/dx = 8x - 3x
    is the slope of the curve, which at x = 3 produces df/dx = 24 - 27 = -3
    When x = 3, f(x) = 36 - 27 = 9.
    The equation of a line passing through (3,9) with slope -3 is
    y = -3x + b
    9 = -3(3) + b
    b = 18
    → y = 18 - 3x ← (a)


    thats correct right?
    Last edited by asilvester635; February 7th 2013 at 07:09 PM.
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Finding the area of a region

    You only need to show that f'(x) = y' evaluated at x = 3

    since  f'(3) = -3 = y' you are done.
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    Re: Finding the area of a region

    Oh well that makes sense.
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    Re: Finding the area of a region

    Quote Originally Posted by jakncoke View Post
    You only need to show that f'(x) = y' evaluated at x = 3

    since  f'(3) = -3 = y' you are done.
    It's more than that. You have to show the slopes are equal - so f'(3), the slope of the curve at x=3, equals -3, the slope of the line. But you also have to show that they pass through the same point there, so f(3), the y-value of the curve, equals 18-(3)(3), the y-value of the line.

    The line y=17-3x matches slopes with the curve at x=3, but it is not tangent.

    - Hollywood
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  5. #5
    Senior Member jakncoke's Avatar
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    Re: Finding the area of a region

    Quote Originally Posted by hollywood View Post
    It's more than that. You have to show the slopes are equal - so f'(3), the slope of the curve at x=3, equals -3, the slope of the line. But you also have to show that they pass through the same point there, so f(3), the y-value of the curve, equals 18-(3)(3), the y-value of the line.

    The line y=17-3x matches slopes with the curve at x=3, but it is not tangent.

    - Hollywood
    yes you are absolutely right, for the slope or, derivative at a point is just a vector.
    Last edited by jakncoke; February 7th 2013 at 09:59 PM.
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