4 ms·

Thank you! Is it correct to say that the definition of the derivative states the equivalence of the secant and tangent lines? I am looking for a theorem that

by zeynel1 16y ago

Thank you! Is it correct to say that the definition of the derivative states the equivalence of the secant and tangent lines? I am looking for a theorem that states the equivalence of curved and straight lines. Does this make sense?

rcfox 16y ago
Well, the tangent and the secant will converge to the same thing as h -> 0. I'm not sure what you mean by "the equivalence of curved and straight lines" though.
- zeynel1 16y ago
  Tangent and secant converge to the same [point] but as far as I understand this is not a statement about the equivalence of the curved and straight lines. By this I mean something like Archimedes's method of exhaustion http://en.wikipedia.org/wiki/Method_of_exhaustion#Use_by_Archimedes http://en.wikipedia.org/wiki/Method_of_exhaustion#Use_by_Arc... when he approximated a circle (curved line) with a polygon (straight line). I would think that there would be a theorem in mathematics generalizing this fact; something like; "curved line is equivalent to straight line." It depends on how we divide (or scale) a line that gives it its "curvature." Or "there is no curve but curvature." Something like that?
  - rcfox 16y ago
    Around a given operating point, you can approximate any function with a polynomial (http://en.wikipedia.org/wiki/Taylor_series http://en.wikipedia.org/wiki/Taylor_series). This includes approximating a polynomial with a lower-order polynomial. A line is first-order polynomial. Therefore, around a given operating point, a line can approximate a function. The key phrase there is "around a given operating point". If you zoom in close enough, a line and an exponential will appear to be the same. Outside of this area, they will clearly diverge. So you can't really say that a curved line is equivalent to a straight line, but you can say that sin(x) = x, for small values of x (and tolerable levels of error in sin(x)). In fact, that comes up quite often in engineering.
    - zeynel1 16y ago
      "you can say that sin(x) = x" Yes, this is the small angle approximation http://en.wikipedia.org/wiki/Small-angle_approximation http://en.wikipedia.org/wiki/Small-angle_approximation and also the example I gave above: 3 points on a sin curve may be considered a straight line or a curved line; 1 meter on earth's orbit is a straight line. The same idea comes up in the string theory too as a pun between point and line. I am surprised that there is no mathematical theorem to express this fundamental property of nature. Or maybe I am not expressing myself clearly.

by zeynel1 16y ago

rcfox 16y ago

Well, the tangent and the secant will converge to the same thing as h -> 0. I'm not sure what you mean by "the equivalence of curved and straight lines" though.

zeynel1 16y ago
Tangent and secant converge to the same [point] but as far as I understand this is not a statement about the equivalence of the curved and straight lines. By this I mean something like Archimedes's method of exhaustion http://en.wikipedia.org/wiki/Method_of_exhaustion#Use_by_Archimedes http://en.wikipedia.org/wiki/Method_of_exhaustion#Use_by_Arc... when he approximated a circle (curved line) with a polygon (straight line). I would think that there would be a theorem in mathematics generalizing this fact; something like; "curved line is equivalent to straight line." It depends on how we divide (or scale) a line that gives it its "curvature." Or "there is no curve but curvature." Something like that?
- rcfox 16y ago
  Around a given operating point, you can approximate any function with a polynomial (http://en.wikipedia.org/wiki/Taylor_series http://en.wikipedia.org/wiki/Taylor_series). This includes approximating a polynomial with a lower-order polynomial. A line is first-order polynomial. Therefore, around a given operating point, a line can approximate a function. The key phrase there is "around a given operating point". If you zoom in close enough, a line and an exponential will appear to be the same. Outside of this area, they will clearly diverge. So you can't really say that a curved line is equivalent to a straight line, but you can say that sin(x) = x, for small values of x (and tolerable levels of error in sin(x)). In fact, that comes up quite often in engineering.
  - zeynel1 16y ago
    "you can say that sin(x) = x" Yes, this is the small angle approximation http://en.wikipedia.org/wiki/Small-angle_approximation http://en.wikipedia.org/wiki/Small-angle_approximation and also the example I gave above: 3 points on a sin curve may be considered a straight line or a curved line; 1 meter on earth's orbit is a straight line. The same idea comes up in the string theory too as a pun between point and line. I am surprised that there is no mathematical theorem to express this fundamental property of nature. Or maybe I am not expressing myself clearly.