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The Underlying Principles of Calculus?

  1. Apr 25, 2007 #1
    The Underlying Principles of Calculus???

    I am required to write a term paper for my Calc. 3 class about what calculus has been about thus far. This isn't some sort of chapter summary sort of situation, he wants us to tell how it's all related based off of a few (or maybe just one) core concepts. I'm going to be honest and say that I'm not really sure how it's all related. I'm thinking that I'll be able to make an arguement that linear transformations and linearization is the basic foundation to everything we've learned so far, and that the main focus has been about rates of growth and accumulation, in all of it's forms. Am I far off? Is there any information you can give me, or any breif articles or books you can refer me to? (I'm not planning on copying, I just need a place to start.) This obviously isn't going to go into too much depth, as it's only required to be 5-pages. But if I stumble upon some sort of real understanding, maybe it'll be much longer and more comprehensive.

    Thanks for any help you can offer.
    (Sorry if this doesn't belong in the homework help section... didn't know where else to put it...)
  2. jcsd
  3. Apr 25, 2007 #2


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    Going back to basics, isn't calculus just simply about the rates of change?
  4. Apr 25, 2007 #3
    I wrote my paper based off of that at the beginning of the year and I got a low B. (We had to write one right off, and now we're writing one because we've gotten so much more information). I'm thinking there's more to it than that, or perhaps a really good way to explain or represent these rates of change so they relate to everything we've learned? Not trying to say I know better than you... because I obviously don't... just stating what I've tried and failed at
  5. Apr 25, 2007 #4
    I wrote a paper about how cool it is that the first derivative of a function represents velocity of an object and the second derivate represents acceleration of an object...kinda applied...but I got a low A for it. Calculus is such a powerful tool...and to solve calculus problems, all you need is a REALLY detailed knowledge of algebra (and logs) and a few new rules...
    Knowing Calculus allows you to find areas and volumes of things that otherwise you couldn't do precisely(not that it takes EVERYTHING in). You get to see how to transform coordinates into a simpler form for just that purpose.(makes the ALGEBRA easier and more intuitive once you get the RULES in your head...I'm talking about polar and cylindrical coords, trig identies..)
    You get to learn the algebra tricks (partial fractions, integration by parts..) that help you become a better logical thinker and problem solver...
    I appreciate calc because it's a given once you know it....anywhere you go.
  6. Apr 25, 2007 #5
    Continuity is a very important aspect.
    Think that calculus is always based to some extent on some continuity assumptions.
    Calculus cannot be developped on the set of integers for example.
    Remember the sequence of what you have learned.
    Continuity, limits, derivation, integrals ...

    Applications to physics are based on continuity assumptions in physics ...
    Last edited: Apr 25, 2007
  7. Apr 26, 2007 #6
    Here it is... a basic skeleton design of what my paper is eventually going to be about. Yes, I know it sounds like it was written by a third grader... fear not, it will be rewritten to sound college-level. What I'm after is whether or not anything I've said here is blatantly false, or missing the point, or off-topic, or anything else you wish to comment on.

    Calculus is a very powerful tool, based upon rates of change, used to solve problems which cannot be solved in any other fashion. The power that calculus wields is due to it's foundation of continuous mathematics. Up until calculus, mathematics generally dealt with discrete quantities, such as money and other easy countable values. Calculus takes mathematics into a more flexible and usable state by using what are called continuous quantities. Arguably the most important continuous quantities are time and mass measurements, such as length, width, volume, and even into deeper unvisualized dimensions. What makes them continuous is the ability to split them into smaller and smaller units of measurement, thereby giving them an infinite amount of points inside any one interval. Time and mass measurements are consequently what most of applicable calculus is based upon.
    The limit is the first way that calculus demonstrates it's ability to deal with continuous mathematics. It also shows some of the other key differences between calculus and algebra. While algebra often asserts that certain values of x in a function f(x) are undefinable, with limits we're able to see what a lot of (but not all) of these values should be, based on the tendency of the graph. This is the first time we see the important concept of an infinite linear approximation in calculus.
    The concept of the derivative is the next building block of calculus. A derivative is most simply, a rate of change. This can be in two or more dimensions, as long as the value changing, and another value being compared is made known. The limit definition of a derivative is actually very similar in any number of dimensions. It basically finds the slope of secant lines until the values are infinitely close to the value at which you would like to find the derivative at. The slope of each individual secant line gives the average rate of change between two points, and the tendency of the slope as the two points get infinitely close to the point in question gives the instantaneous rate of change at the point. This tendency can be simplified as a linear approximation for values within a certain range of a point. This approximation makes finding the derivative much simpler.
    The concept of differentiation has also been simplified using a set of rules to generalize different cases. Among these are the power rule, the product rule, the quotient rule, and most importantly: the chain rule and L'Hospital's rule. The chain rule is important because it can be used to disassemble most complex equations into simpler derivatives, and of course, because of its uses in integration. L'Hospital's rule, put simply, a way to compare different sizes of infinity, infinity-tending functions, and functions dealing with an infinitely small interval as the variable goes to 0 or infinity.
    One of the most important applications of derivatives is the finding of extrema. This is where a lot of the basic linear approximation claims I have made are easily exposed. The most intuitive way to go about explaining why the formula for optimization works so well, is by explaining what is not an extrema. If the derivative as x does not equal zero, than it is either positive or negative. Therefore, the function is either increasing or decreasing at that point, and there are an infinite amount of points that are greater than and less than the point in question. This is the nature of continuous mathematics. By contradiction, a value is an extrema of a function f, if and only if the f has a derivative of zero at that point. This maximum or minimum is where the slope of the linearization at x, approaches zero. Although this is easiest to visualize in two dimensions, the concept can be brought visually into three-dimensions, and applicably into any finite n-dimensions greater than three. For as long as we know what the change of a variable is in respect to, we can find the value at which it has extrema. The same is true for related rate problems, and a variety of other applications, due to the nature of them being essentially special-case optimization problems.
    Integration is often thought of as a different concept than differentiation. In reality, it is dealing with the exact same usage of rates, with similar approximation methods. It is effectively, the reverse or compliment of differentiation. Integration displays, more or less, the accumulation of something. Geometrically in two or three dimensions, it is usually used to find area under a curve or volume under a shape. This is an accumulation of one type, but there are many more.
    The technical definition behind integrating is very similar to differentiation. Geometrically in two-space, we take progressively smaller areas under the curve and add them all up. As the number of rectangles under the curve reaches infinity, our answer becomes precise. This literal definition gets harder to carry out we introduce more dimensions.
    Often though, we use verifiable short cuts to calculate derivatives. The most influential way that scholars have connected differentiation to integration is called the Fundamental Theorem of Calculus. Syntactically, this is “The integral of F from a to b is equal to the anti-derivative of F at b minus the anti-derivative of F at a.” This is often the easiest method to solving most simple integrals. Using the reverse-chain rule, u-substitution, we can break down complex problems down to easily computable parts. Many other variants of u-substitution (trig substitution, partial fractions, root substitution, etc) have been evolved for much more complicated integrals, and generally, this works quite well.
    However, there are times when this definition does not work, and the core approximations behind the syntax must come into play. Examples of this include when the limits of integration go to infinity, or when the limits of integration cause the function to go through an asymptote. Another famous example is the integral from zero to one of sin(x^2). There is no anti-derivative to this integrand, and many others. In these cases, the Trapezoidal rule, the Simpson's rule, and the Midpoint rule are all different ways to form approximations of the accumulation of these functions from one point or set of points to another. These are all prone to error of some sort, but this error is easily calculated, and they have bounds of accuracy just as the approximations for differentiation have. These approximations work in much the same way. You are essentially even taking linear approximations when you are using the simpler Fundamental Theorem of Calculus equations, as have been shown by the infinite series definition of integration.
    Higher branches of the sciences have made use of calculus in many ways. We have found that all of the rules of calculus can be generalized onto vector fields and used to calculate various solutions to physics related problems, if given the proper frame of reference. Fundamentally, it is the exact same concepts used in a different manor. However, many of these methods can not be effectively utilized without realizing the aforementioned nature of continuous mathematics and approximations.
    It is difficult to really generalize what calculus is about. But for the most part, it is safe to say that calculus is the study of rates of changes and accumulations, and the approximations thereof. While easiest to visualize geometrically, if one truly understands the principles behind the equations, it is only a matter of analysis to transcribe the exact same methods to more abstract concepts.

    Thanks for any advice you can give me.
  8. Apr 26, 2007 #7


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    A number of people (Fermat among them) gave formulas for finding tangent lines to curves (the basic idea of differentiation) and using "limits" to find areas goes back to Archimedes. Newton and Leibniz are know as the "founders" of calculus because they recognized that those were "inverse" to one another. I would recommend making the "Fundamental Theorem of Calculus" the center of your paper.
  9. Apr 26, 2007 #8
    I know the who the founders of calculus were and other many important people in the development of calculus, but he doesn't want this to be a math history paper. I do agree that the fundemental theorem of calculus should play a bigger part in my paper, and I'll make the final draft more centered around it. Thanks.
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