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Tangent To A Function - Limits

  1. Jul 17, 2012 #1
    1. The problem statement, all variables and given/known data

    The tangent to the function y=3x(x-3) at point P(2,-6) is the hypotenuse of a right triangle that forms with the coordinate axes. Find Area


    3. The attempt at a solution
    First of all, i know that i A=BxH/2 so i need the opposite and adjacent sides of this triangle.

    Okay so personally i graphed the function at first, and then drew the tangent at the point 2,-6. I then connected the line to the x and y axes. I did this because it says it forms a triangle with the coordinate axes.I read that the coordinate axis are respectively the x and y axis, so i thought that this made sense. Can some one reassure me about this?

    Next i thought that i would need the slope of the tangent at that point because it would relate to the slope of hypotenuse of the triangle.

    So i took the limit.

    Lim (f(x+h)-f(x))/H
    H->0

    where P(2,-6), so x=2

    solving this limit i got

    Lim 6x+h-9 = 6x-9
    H->0
    and when x=2
    the limit is = 3

    therefore the slope of the tangent is 3. mt=3

    so i'm curious, does this mean that the hypotenuse also = 3 because it says the the tangent at the point is the hypotenuse. Or would the hypotenuse side be the tangent line (6x-9)?

    So yea this is my dilemma..
     
  2. jcsd
  3. Jul 17, 2012 #2

    LCKurtz

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    The line through (2,-6) with slope 3 is the hypotenuse. Figure out its equation (it isn't y = 6x-9) and find where its x and y intercepts are to get the legs of the triangle.
     
  4. Jul 17, 2012 #3
    okay say i have the point (2,-6) and mt=3

    here is my diagram.
    http://s18.postimage.org/3r9iyue7d/New_Bitmap_Image.png [Broken]

    y= mx+b
    -6=3(2)+b
    b=-12

    So therefore the height or the y value of the triangle is 12. cause it is the y intercept

    y=3x-12

    to find x intercept, set y=0

    3x = 12
    x= 4

    So there for the base of the triangle is 4, and the height of the triangle is twelve.

    so area=bxh/2=12x4/2=24units^2
     
    Last edited by a moderator: May 6, 2017
  5. Jul 17, 2012 #4

    LCKurtz

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    Yes. You have it correct.
     
    Last edited by a moderator: May 6, 2017
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