7 ms·

Ask HN: A calculus theorem question?

• rcfox 16y ago
  It's not really a theorem; it's the definition of the derivative. f`(a) = lim h->0 (f(a+h) - f(a))/h
  • 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.
• eiji 16y ago
  In mathematics you always try to minimize the preconditions, and the "theorem" you are looking for is likely a combination of strong special proofs. Maybe http://en.wikipedia.org/wiki/Trapezoidal_rule http://en.wikipedia.org/wiki/Trapezoidal_rule comes closest to what you are looking for. It states that you can approximate the definite integral using "straight lines", which includes your case, however, this rule has preconditions for the curve. HTH
  • zeynel1 16y ago
    Thanks. This looks exactly like an "application" of my putative theorem. The trapezoidal theorem assumes that straight line and curved line are the same thing with different curvature (I think); the way a circle and ellipse are the same figure but have different eccentricity.
• zeynel1 16y ago
  Some answers in stackexchange too: http://math.stackexchange.com/questions/6136/is-there-a-mathematical-theorem-that-states-the-equivalence-of-curved-and-straigh http://math.stackexchange.com/questions/6136/is-there-a-math...
• drallison 16y ago
  The problem is that your assertion is not true for any arbitrary function. For example, consider a function(aka curve) defined by: f(x) is x if x is irrational and 1 if x is rational. This function is discontinuous and cannot be approximated by shorter and shorter straight lines. The calculus concept you are reaching for is continuity. If f(x) is continuous, then for any e>0 there exists a d>0 such that abs( f(x) - f(y) ) < d if abs( x - y) < e. That's the famous epsilon-delta mantra of the calculus. See http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_.... Continuity is a powerful property that enables much of what is useful in the calculus. Continuity of a function and its derivative is even stronger; continuity of the function and first and second derivatives is stronger still.
  • zeynel1 16y ago
    "Consider a function(aka curve)..." Thank you for this answer and I apologize for the ambiguity in my question, especially, regarding the word "curve." By "curve" I meant "curved line" or even more precisely, a "curved line without a gap." Obviously this is an assumption; I ignore if the line is made of points; because I want to compare a curved line with a straight line. So I believe that the question is not about "continuity." (By the way Galileo has a nice discussion about lines and points here http://books.google.com/books?id=SPhnaiERbWcC&dq=galileo%20galilei&pg=PA26#v=onepage&q&f=false http://books.google.com/books?id=SPhnaiERbWcC&dq=galileo...) Maybe I can ask the same question like this. Here are 3 numbers: 0.0872664626; 0.1047197551; 0.1221730476. Is it possible to tell if these are points on a straight line or if they are points on a curved line? I believe both; depending on the scale they are plotted; the numbers would look like a straight line on some scale; but they are points on a sin curve. In other words, nature does not distinguish between straight line and curved line; the same three numbers can be read both ways. So I am looking for a theorem that would describe this fundamental property of nature. Thanks.
    • drallison 16y ago
      For x sufficiently small, sin(x) looks like x (x in radians). That's because if you write the Taylor's series expansion of sin(x) and ignore high order terms, you get x. The property you are looking for is continuity.