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Tangent at a point on a curve

  1. May 20, 2012 #1
    When i was at school i used to think that any line that touches a curve at only one point is called as the tangent at that point to the curve! But after reading derivative i think this definition of tangent is not correct at all conditions!
    For example,
    x-axis cannot be called as tangent at origin to the function y=|x| though it touches the curve at origin only!
    second example,
    tangent at a point on a curve y=sin x may also touch any other point on the same curve(thus there are two points of intersection)

    IS this new concept of tangent i have is right OR the definition of tangent at school is right?

    please help!
     
  2. jcsd
  3. May 20, 2012 #2

    Stephen Tashi

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    The new concept of tangent that you have is correct.

    In terms of the new concept, the curve y = |x| does not have a tangent at x = 0 since no unique slope for the curve is defined at that point.
     
  4. May 21, 2012 #3
    Thanks Stephen Tashi!
     
  5. May 21, 2012 #4

    tiny-tim

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    that would be a circular definition anyway …

    "touch" means "tangent", doesn't it? :wink:
    (unless it means "meets but does not cut", in which case the x-axis would not be tangent to y= x3 !)
     
  6. May 22, 2012 #5

    HallsofIvy

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    In defense of "school":
    A line is tangent to a circle if and only if it touchs the circle in exactly one point.
    I suspect that is what you are remembering and your memory is trying to extend it to all curves.
     
  7. May 22, 2012 #6
    In Calculus, we have to redefine a tangent to only have a definition locally, sort of "barely touching" the curve in question at the point, though it can just cross the curve in other locations. Does this make sense?
     
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