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Instantaneous vs. Average

  1. Oct 29, 2008 #1
    I have taken what is equivalent to 1 semester of college calculus in high school. I differentiated and integrated functions like the volume of a sphere V=(4/3)pi(r^3). However, as I understand it this is a function of a geometrical object that doesn't exist in nature. In fact, most of the equations from the textbook we used were "perfect" equations, instantaneous rates or exact functions that the didn't match their real life counterparts, which I assumed was for the sake of making it easier on the author and student. My first question is how are scientists derive equations modeling natural objects, like the surface area of a meteorite, using calculus to describe them perfectly? If enumerating all of the irregularities is impossible, which I would assume is so, then why is calculus used at all if you have to make assumptions and averages and can't create a definitive equation to model nature perfectly? Why would calculus be used instead of difference quotient for derivative and sigma summation for integral? Calculus seems so definitive (existing in a perfect reality) as opposed to physics (assumptions, averages and approximations based on empirical data from nature) and I don't understand how the two mesh so well. Also, aren't there limits to exactness in nature (Planck length and time) and shouldn't these prevent calculus from creating equations exactly modeling nature?

    I think my confusion can be blamed on one of two problems:
    1. I don't have a firm foundation on physics (just equations and things like harmonic motion and acceleration from calculus)
    2. The course was AP Calculus AB and was geared toward passing the AP test, instead of thoroughly covering each topic, so I think I may have missed some key concepts.

    Feel free to make any corrections in my understandings or assumptions. Thank you.
    Last edited: Oct 29, 2008
  2. jcsd
  3. Oct 30, 2008 #2
    A word that comes to mind is approximation.
  4. Oct 30, 2008 #3


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    Science Advisor

    No mathematical model describes a physical situation perfectly. No one has ever claimed that calculus modeled nature perfectly! But that doesn't really matter. Any physical problem is going to have values based on measurement- and the measurement is not perfect.
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