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Math Help - eq of tangent lines that pass thru pt to graph of f

  1. #1
    Super Member bigwave's Avatar
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    eq of tangent lines that pass thru pt to graph of f

    find the equations of the tangent lines to f(x)=4x-x^2 as they both pass thru point (2,5) which is not on the graph.

     <br />
f'(x) = 4-2x<br />
    but we do not the points on the graph where the 2 lines from this point will be tangent. we know the slope of line will be same as 4-2x
    anyway not seeing how this these 2 equations are derived

    appreceiate much..
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  2. #2
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    There might be an easier way, but this is how I would tackle this problem:

    First we can find a general form for a tangent line at the point (x_0,y_0):

    y-y_0=m(x-x_0)

     \implies y-(4x_0-{x_0}^2)=(4-2x_0)(x-x_0)

    Now we narrow it down to ones that pass through the point (2,5):

     5-(4x_0-{x_0}^2)=(4-2x_0)(2-x_0)

    This should simplify to a quadratic equation that we can solve to find what values of x_0 satisfy the conditions. Then we can get the two tangent lines at those two points.
    Last edited by drumist; January 26th 2010 at 02:35 PM.
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  3. #3
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    Quote Originally Posted by bigwave View Post
    find the equations of the tangent lines to f(x)=4x-x^2 as they both pass thru point (2,5) which is not on the graph.

     <br />
f'(x) = 4-2x<br />
    but we do not the points on the graph where the 2 lines from this point will be tangent. we know the slope of line will be same as 4-2x
    anyway not seeing how this these 2 equations are derived

    appreceiate much..
    Hi bigwave,

    The function derivative gives the tangents' slope.

    m=4-2x

    At the points of tangency, the y co-ordinate (and x co-ordinate) is the same on both the tangents and the curve.

    (2,5) is on the tangents but not the curve...

    y-5=m(x-2)=(4-2x)(x-2)\ \Rightarrow\ y=5+4x-8-2x^2+4x=-2x^2+8x-3

    y also satisfies the curve equation...

    4x-x^2=-2x^2+8x-3\ \Rightarrow\ x^2-4x+3=0

    The x co-ordinates of the points of tangency can be found.
    These can be used to write the exact slopes and equations of the tangents.
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  4. #4
    Super Member bigwave's Avatar
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    Cool the 2 tangent line equations are:

    thus

    the 2 tangent line equations are:

     <br />
y=2x+1
     <br />
y=-2x+9
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