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Finding the eq. of all tangent lines on a curve

  1. Jan 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Problem: Find the equations of all tangent lines to the curve
    y = x + 2x so that also go through the point (3, 14).

    2. Do not use a derivitive

    3. I dont even know where to start. I searched my book there isn't really available. Any help would be much appreciated
  2. jcsd
  3. Jan 15, 2009 #2


    User Avatar
    Gold Member

    Is that really supposed to be y=x+2x, or have you mis-typed something?
  4. Jan 15, 2009 #3


    Staff: Mentor

    What's the correct equation? The "curve" y = x + 2x is a straight line that doesn't go through (3, 14), so no tangent can go through this point either.

    Should it be y = x^2 + 2x?
  5. Jan 15, 2009 #4
    sorry everybody,

    yes I meant to say

    y=x^2 + 2x

  6. Jan 16, 2009 #5


    Staff: Mentor

    OK, now that we've gotten that out of the way...

    Let [itex](x_0, y_0)[/itex] be the point of tangency on the graph of the curve. BTW, you have drawn the graph, right?

    At the point of tangency, the tangent line has to extend from [itex](x_0, y_0)[/itex] to (3, 14).

    Here is an outline of the steps you'll need to carry out for this problem:

    1. Find the slope of the line from [itex](x_0, y_0) = (x_0, x_0^2 + 2x_0)[/itex] to (3, 14).
    2. By calculating the derivative and evaluating it at [itex]x_0[/itex], find the slope of the tangent line.
    3. Equate the value you got in step 1 with the value from step 2, and solve for [itex]x_0[/itex]. (I got two values for [itex]x_0[/itex].)
    4. Find the associated y value for each value of [itex]x_0[/itex] from step 3.
    5. Using each point [itex](x_0, y_0)[/itex], find the equation of the line from [itex](x_0, y_0)[/itex] to (3, 14). There are two distinct equations.

    Is that enough of a hint?
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