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What is calculus used for?

  1. Aug 25, 2011 #1
    Other than is science and engineering, are there any other applications for calculus?

    Could someone with a vast knowledge of mathematics please explain the very basic concepts of calculus in laymans terms?
     
  2. jcsd
  3. Aug 25, 2011 #2

    Pengwuino

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    Why do you need someone with vast knowledge of mathematics to explain something a high schooler could explain?

    And yes, calculus can be found in a vast array of fields such as economics and business.
     
  4. Aug 25, 2011 #3
    Calculus basically consists of two ideas: derivatives, and integrals. Here's a brief description of each:

    - The derivative of a quantity is the "rate of change." For instance, speed as you move.
    - The integral is the total of one quantity over another. It can be used to find an area (the total height as you move across the width) and things like that.
    - The most important and fascinating concept in calculus is the relationship between these two ideas: the derivative of the integral of a quantity is the quantity. Remember that your speed is the "rate of change" of your position. Well, the "total of your speed over time" is your position all over again!

    Anywhere where you want to analyze the relationship between to quantities (like supply vs price of a good, or the probability of a return on an investment vs the previous data you have about that asset), calculus offers a powerful set of tools to observe how one quantity affects the other.
     
  5. Aug 25, 2011 #4
    Not to mention, the advent of calculus is commonly compared with the advent of science as we consider it today. Before we had the mathematical tools to actually write down in symbols the relationship between two things, it was extremely difficult to study physics, and much of it was completely speculative.

    Calculus gave scientists the tools they needed to formulate an actual theory that was clear and unambiguous enough to be tested. Statistics (which couldn't really exist without calculus) then gave them the tools to test those theories.

    I actually don't agree that calculus has applications outside of "science." The thing is, science is getting broader every day. My disagreement is really just a matter of definition: I happen to consider economics and business to be sciences just like any other.
     
  6. Aug 26, 2011 #5
    Just a simple explanation. Calculus might increase our ability to analyze things and IQ.
    It also test our own limits:smile:
     
  7. Aug 26, 2011 #6
    Calculus is useful in any field where studying trends is important.

    In my own words, calculus is the study of the proper use of infinity in algebra.

    The three main notions in calculus are those of the limit, the derivative, and the integral. I'll hand wave a short explanation of each, albeit in a slightly different order.

    The derivative is an operation applied to functions (or equivalently, graphs). Given a graph of your position over time, the derivative is a graph of your velocity over time.

    The integral is the inverse of the derivative. It takes the graph of your velocity and tells you how far you've traveled between two points in time.

    Where does "infinity" come in? Calculating the derivative involves division by numbers that get closer and closer to zero (and we can never, even in calculus, divide by zero). Calculating the integral involves adding together more and more (up to infinity) terms that get successively smaller and smaller (approaching zero).

    The limit is the tool that makes calculus possible. We can't divide by zero, and we can't add an infinite number of terms. But with limits, we can by pass the nastiness that comes with those things and get legit answers.

    There are different kinds of limits used in calculus, but they all follow the same intuition, which is that you have a way to produce better and better approximations of something (that denominator is getting closer and closer to zero or the number of terms you add is growing up to infinity). The limit is a number that is the best approximation possible.

    Hope that helps.
     
  8. Aug 26, 2011 #7
    for torture, perhaps? In particular, in the torture of engineering students.
     
  9. Aug 27, 2011 #8

    mathwonk

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    calculus is the science of approximating and even computing exactly, measurements of curved things by means of known measurements of straight things. slopes of curves are measured in terms of slopes of lines, areas and volumes of curved surfaces like spheres are measured in terms of measurements of planes and simpler curved objects like cylinders and and cones.

    It gives a way to calculate the total weight of an object of known but variable density, and the distance traveled by a object of known but variable speed.

    It is a hard subject to understand well, but not a hard tool to use. Methods of calculus render easy, calculation that in olden times were only accessible to brilliant mathematicians like Euclid, Archimedes or Galileo.

    E.g. Archimedes proved the area under the parabola y = x^2, between x=0 and x=1, is 1/3 the area of the rectangle circumscribing it. But today a beginning student learns that the area formula for that parabola is (1/3)x^3, hence for x=0 to x=1, we get (1/3) - (0) = 1/3.

    This calculation is trivial, (but not as trivial to understand why it works). The secret is that the fundamental theorem of calculus says that "the height formula is the slope of the area formula", and since one learns also that x^2 is the slope of the formula (1/3)x^3, "hence" the latter must be the area formula for the parabola.


    I think calculus texts make a mistake of inundating the poor student with huge amounts of theoretical material on limits before presenting the easy results.

    Of course it is also a problem that high schools have stopped teaching the fundamental background material from Euclidean geometry and basic algebra that used to be learned in advance of college. It is just not feasible to come into college with no good grasp of algebra and geometry and expect to master calculus.
     
    Last edited: Aug 27, 2011
  10. Aug 27, 2011 #9
    My math comprehension is abysmal. I can barely grasp the very basics of algebra, let alone calculus.
     
  11. Aug 29, 2011 #10
    Practice makes perfect. My math skills couldn't be more mediocre, but I managed to get good grades recently from hardcore study.

    Here, watch some videos and do some practice problems: http://www.khanacademy.org/
     
  12. Aug 29, 2011 #11
    Think of it this way, math particular calculus is the langauge of science. It is like English, you need to know English to read news papers or read anything. All advanced scientific book are written in calculus, subject like electrodynamics, all the theory are written in calculus, you don't understand calculus, you won't go far. Take if from a sucessful EE for almost 30 years, I never have good calculus background, I always ran into road block because the lack of calculus. Now that I retire, I decided to really study calculus, I just finished partial differential equation. This is only barely enough for engineering and scientist.

    I know economics and other business use very limited amount of calculus, you almost can get away without knowing it. if you are in English major, law, history etc, obviously you don't need calculus.

    You never know where your career end up, just like me, starting out in bio-chem major and my career was EE. With good calculus, it would be very easy for you to change major to scientific field. It is all about option, you'll be surprised how many people work in field that has nothing to do with their major in college. It is just as important for non science major to study calculus as for engineering and science student to study English.
     
  13. Aug 29, 2011 #12
    For the most part, calculus is the study of how functions change with the change of another variable.

    The most important part of this part is the derivative. The derivative is the slope of a line at a certain point. For example, the derivative of x^2 is 2x. Yes, this is still a function. So when x = 5, y will be 25(5^2), and the slope at that point will be 10(2 * 5). If you were to change x to 6, then y would become 36(6^2), and the slope at x = 6 would be 12(2 * 6).

    The formula(for most things, at least) for finding a derivative of a function is:

    f'(x) = (f(x+h) - f(x))/h, as h approaches 0.
    when x+h is used, that takes the place of x
    the primed(') also denotes the derivative

    With the x^2 example, I'll show you how it is done.
    The function is f(x) = x^2
    Plugging this into the formula, the slope is given by ((x+h)^2 - x^2)/h
    This becomes (x^2 + 2xh +h^2 - x^2)/h
    Simplifying will give you (2xh + h^2)/h
    This can be further simplified to 2x, by dividing out the h.
    Thus, f'(x) = 2x

    Another big part of calculus is integration. This is essentially taking the antiderivative of a function. By this I mean finding a function where the derivative is equal to what is being integrated. Integrating a function will also give you the area under the integrated function's curve. Ever notice how if you plot velocity and time on a graph, like v = 5 constantly, then the area(velocity x time) will give you the position, which is the integral of velocity. Likewise, integrating acceleration will give you velocity.

    Calculus can also get more complex than this. You can take partial derivatives, gradients, dot products, cross products and more. But these involve functions with more than one variable, such as f(x,y) = x + 3y. In this case, z is usually used to represent the function, as it is being used as an input.

    If you would like to use a book for more information, I taught myself some advanced(multi-variable) calculus with Advanced Calculus Demystified. However, I was already nearing the end of AP Calculus AB course, so I had some experience with the single variable stuff. I would recommend the Calculus Demystified book in this case, although I never used it. If the series is as decent as was the couple of books I used, it should be fine.
     
    Last edited: Aug 29, 2011
  14. Aug 30, 2011 #13
    Try living without science and engineering,
     
  15. Aug 30, 2011 #14
    What about chemistry? Is there calculus in chemistry?
     
  16. Aug 30, 2011 #15
    Equilibrium points. Calculus is great for finding maxima and minima points of functions. Just make it an energy function, and you're meeting all sorts of chemistry laws.
     
  17. Aug 30, 2011 #16

    mathwonk

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    listen to yourself - first learn algebra, then ask us about calculus.
     
  18. Aug 30, 2011 #17

    chiro

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    One easy way to think about calculus is that it allows us to measure things that are non-linear over time.

    If you remember in high school, you would have done things like calculate lengths, areas, and volumes of a variety of objects like squares, rectangles, trapeziums, rhombus', triangles and so on. But you never had the formulas for objects that had curvy parts.

    The exceptions were things like circles, and anything involving circles (like a cylinder for example).

    Calculus is a way to do this for arbitrary objects, like a parabola, or a circle, or an ellipse, or some other weird object that you have an equation for. You can calculate lengths, areas, and volumes for all of these objects using one framework.

    There is more to calculus than this, but this should give you an idea of how calculus generalizes the kinds of stuff you did with linear type objects to objects that are non-linear.
     
  19. Aug 31, 2011 #18
    So Algebra/Geometry/Trig problems often deal with very linear problems. While calculus is used for highly irregular and non-linear problems with changing variables.

    This video explains..

     
    Last edited by a moderator: Sep 25, 2014
  20. Aug 31, 2011 #19

    chiro

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    Yes that is a good video that explains the linear/non-linear difference.
     
    Last edited by a moderator: Sep 25, 2014
  21. Sep 1, 2011 #20
    Post #6 above by Tac-Tic is an outstanding explanation.

    I would add the following oversimplifications for you.

    Calculus is the Mathematics of Change
    Integration is just multiplication [in the limit] when
    one of the multiplicands is changing.
    Differentiation is the Instantaneous Rate of Change of a variable.
    So Differentiation is just division [in the limit]
    because rates are comparisons by division.

    And finally, The Fundamental Theorem of Calculus ......
    "Differentiation and Integration are Inverse functions"

    When do you need Calculus ?
    Whenever you need to do Analysis to get a more detailed and deeper understanding of some quantifiable topic, experiment or phenomena.
    That is why the broader subject to which Calculus belongs is called Analysis.
     
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