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Thread: How is it that points on a continuous graph has no direction or infinitesimally small length?

  1. #1 How is it that points on a continuous graph has no direction or infinitesimally small length? 
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    The rate of change of a curve seems to imply that it is the point that touches the tangent line that has the rate of change. The tangent line seems to be only an extension of what the graph is doing at a certain point.

    And I can certainly see how a point is just a point in R. So is a point on a curve different somehow than just a point with nothing else beyond the reals?


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    Quote Originally Posted by Utopian View Post
    The rate of change of a curve seems to imply that it is the point that touches the tangent line that has the rate of change.
    That's a tortured sentence, but from what I can decode of it, it's wrong. The rate of change of a curve does not imply anything about the tangent; the rate of change is a property of the tangent (and shared with the point to which it is tangent). The tangent line simply possesses the same first derivative (slope) as the point to which it is tangent. All this presumes, of course, that the first derivative exists.

    The tangent line seems to be only an extension of what the graph is doing at a certain point.
    Sort of, but "extension" is too vague. What's wrong with the actual word, tangent? If you regard the tangent as merely the line resulting from a limiting process, in which two points on the curve in question approach each other with infinitesimal spacing, that seems to capture the essentials.

    And I can certainly see how a point is just a point in R. So is a point on a curve different somehow than just a point with nothing else beyond the reals?
    A point is a point is a point, regardless of whether you choose to draw a tangent or not.


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    What are you seeing when you look at the graph in your question?
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    I take your question as being "is the slope of a point on a curve something innate to the point?" And what does that mean.

    My advice is to go back to the foundational stuff, specifically look at limits. The derivative is something that's approached as we zoom in on a given point, period. You'll notice that there are limits which give us answers for points where there are no answers, such as divide by zero points. That's the mathematical elegance of limits. If you poke around long in with the real numbers and continuous functions, you'll find points where the logic breaks or gets ugly. Limits side step all of this. They don't always give us information for points, but they get us infinitely close.

    So the slope at a given point is defined in terms of the difference between two points, but for an infinitely small distance, via the limits definition of derivative. So for simplicity, you can think of a given point as having a slope. But technically, that answer can be thought of as possibly being infinitesimally off. (off by the amount of 1/infinity) However for any practical concern this amount doesn't matter at all, so calc works. They say its about things like "velocity at an instant", but really its about velocity as we approach an instant. Subtle difference that makes some things more clear.
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    Thanks everyone, but I think that I went a little over my head on this one.
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    Quote Originally Posted by Utopian View Post
    Thanks everyone, but I think that I went a little over my head on this one.
    Can I ask, have you done any Calculus yet?
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    Quote Originally Posted by TridentBlue View Post
    Quote Originally Posted by Utopian View Post
    Thanks everyone, but I think that I went a little over my head on this one.
    Can I ask, have you done any Calculus yet?
    Yes I have. I actually found an interpretation for points having a slope in smooth differential analysis. But I was really just wondering about derivatives in general; in which case your answer helped.
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