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Thread: Calculus In Simple Terms

  1. #1 Calculus In Simple Terms 
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    I haven't studied calculus yet, I have heard that it is difficult to some. Please will someone explain calculus in simple terms, and give some complex problems (I was able to understand linear problems... as they are linear ). Perhaps give certain incredibly un-linear functions, functions that goes all over the place.


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    Moderator Moderator Cogito Ergo Sum's Avatar
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    Quote Originally Posted by AlphaMSD5 View Post
    Please will someone explain calculus in simple terms, and give some complex problems (I was able to understand linear problems... as they are linear ).

    Calculus is, in simple terms, the mathematical study of the properties (via e.g. the use of limits) of the derivative(s) and the integral(s) of a certain function ().
    Whilst the derivative expresses the slope of a function for every single point of the function, the integral expresses the area (between two points) beneath the function:



    Although I can give some non-linear problems, I doubt that it would be of great value for you since you have not had calculus yet.


    Last edited by Cogito Ergo Sum; November 20th, 2013 at 05:50 AM.
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    Quote Originally Posted by AlphaMSD5 View Post
    I haven't studied calculus yet, I have heard that it is difficult to some. Please will someone explain calculus in simple terms, and give some complex problems (I was able to understand linear problems... as they are linear ). Perhaps give certain incredibly un-linear functions, functions that goes all over the place.
    I'm not sure whether, from what you say, you already know the simple rule that the derivative of xⁿ is nx ⁿ⁻, so for example 2x + 4x - 3x +1 differentiates into 6x + 8x - 3, but that alone gets you a long way in calculus. For example you can find max and min points in curves by finding where the derivative is zero. In this case the function is a cubic so should have one max and one min point - and setting the derivative to zero gives you a quadratic from which the 2 max and min values can be found. Nice!

    Then there are some nice functions:-

    My favourite is eˣ, which differentiates into itself! Which means its rate of change at any point is the same as the value it has reached. Which make sense, when you think about it, as an "exponential increase" in something, is one where the rate of increase it proportional to the size the thing has already reached, e.g. growth of a reproducing population if there are no deaths. Or, for a -ve exponential, the rate of radioactive decay of a substance is proportional to the amount of the substance that is left.

    Also the trig functions are fun: sin x differentiates into cos x, and cos x into -sin x. Again this makes sense if you think of plotting them as waves: the rate of change (slope) of sin x is zero at the crest of the wave (where sin x = 1) and its maximum slope =1 as it crosses the x axis, where its value is 0. So cos x plots as sin x but phase-shifted by 90 deg. Neat, I think.

    Being taught calculus was what made me first realise the power and practical use of maths - not least in science.
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    Quote Originally Posted by AlphaMSD5 View Post
    Please will someone explain calculus in simple terms...
    Calculus deals largely with rates of change. Velocity, acceleration, and population growth are examples of rates that calculus might be applied to. If a moving thing covers an amount of distance over time, calculus can be used to determine the velocity and the acceleration of that thing at any instant.

    Calculus might also be thought of as advanced geometry. In plane geometry, formulas are given to find areas and volumes of simple shapes. Calculus can be used to determine the areas and volumes of an infinite number of more complex shapes.

    I hope that helps.

    Jagella
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    Calculus is ultimately about limits. Thus, in addition to differentiation and integration, calculus also deals with such things as infinite series. The term "analysis" is used to describe matters pertaining to calculus, and distinguishes such matters from "algebra".
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Surely less deep than the posts thus far, which of course I have no problem with, I would tell the layman that Calculus is a type of Mathematics which will allow you to determine certain things which no other means available to you could achieve.

    Examples are areas of irregularly-shaped plane figures, and volumes of things also of irregular shape. jocular
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    Calculus is a stepping stone to higher math. khanacademy has great videos but as far as I know he doesn't go more advanced than differential equations. It's not easy to give an entire course in a forum page.
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    Genius Duck Moderator Dywyddyr's Avatar
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    Quote Originally Posted by jocular View Post
    Calculus is a type of Mathematics which will allow you to determine certain things which no other means available to you could achieve.
    Examples are areas of irregularly-shaped plane figures
    Pfft...


    Five minutes pushing and one straight multiplication of two numbers - and one of those is a fixed value.
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    My automatic response to "what is calculus" is "calculus is the mathematics of infinitesimals." But, on reflection, I find that a bit hard to explain. I'm sure I am parroting something I have seen in a math book somewhere, which, while essentially correct, isn't nearly as useful an explanation as what others have given above.
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    There is a course starting soon at the Ohio State Unviersity but due to the marvals of the internet it is availble online: https://www.coursera.org/course/calc1 the course starts in about 4 days and I myself am enrolled and am using more as a refresher but apparently it's suitable to a beginner, it also consists of assignments and a test but at the end of it you get a nice certificate
    Come see some of my art work at http://nevyn-pendragon.deviantart.com/
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    I'm dating myself, but as a simple example, you can use calculus to determine the distance that a phonograph needle travels along the spiraling groove of a record. You could measure the distance using a tiny wheel like the ones used to measure the length of skid marks. Or you could use calculus to transform (ie, integrate) the equation that describes the spiraling groove into a "length" equation into which you can plug in the values of the two positions along the spiraling groove, and compute the difference between the two results, which computes the distance between the points.
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    Quote Originally Posted by danhanegan View Post
    My automatic response to "what is calculus" is "calculus is the mathematics of infinitesimals." But, on reflection, I find that a bit hard to explain. I'm sure I am parroting something I have seen in a math book somewhere, which, while essentially correct, isn't nearly as useful an explanation as what others have given above.
    I'd disagree. The whole point of calculus (and its cousin Analysis) is to avoid the use of poorly-defined infinitesimals, replacing them with rigorously defined limits.
    Last edited by someguy1; August 19th, 2013 at 12:14 PM.
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    Quote Originally Posted by someguy1 View Post
    Quote Originally Posted by danhanegan View Post
    My automatic response to "what is calculus" is "calculus is the mathematics of infinitesimals." But, on reflection, I find that a bit hard to explain. I'm sure I am parroting something I have seen in a math book somewhere, which, while essentially correct, isn't nearly as useful an explanation as what others have given above.
    I'd disagree. The whole point of calculus (and its cousin Analysis) is to avoid the use of poorly-defined infinitesimals, replacing them with rigorously defined limits.
    And that brings us back to KJW's post, whose profundity was insufficiently remarked upon previously. Placing the notion of limits on a rigorous basis was revolutionary, and its importance has not diminished over time.
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    Newton was clearly a sadist.
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    Quote Originally Posted by shlunka View Post
    Newton was clearly a sadist.
    At least a bit of a masochist, certainly. Shoving a bodkin(e) behind his eyeball to prove a point is a bit...much.
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    Calculus described simply, gently, and without tears:

    Calculus I in 20 Minutes (The Original) by Thinkwell - YouTube
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    Quote Originally Posted by tk421 View Post
    Quote Originally Posted by someguy1 View Post
    Quote Originally Posted by danhanegan View Post
    My automatic response to "what is calculus" is "calculus is the mathematics of infinitesimals." But, on reflection, I find that a bit hard to explain. I'm sure I am parroting something I have seen in a math book somewhere, which, while essentially correct, isn't nearly as useful an explanation as what others have given above.
    I'd disagree. The whole point of calculus (and its cousin Analysis) is to avoid the use of poorly-defined infinitesimals, replacing them with rigorously defined limits.
    And that brings us back to KJW's post, whose profundity was insufficiently remarked upon previously. Placing the notion of limits on a rigorous basis was revolutionary, and its importance has not diminished over time.
    But in fact Newton and Leibniz both failed to put their methods on a sound logical footing. It was only in the following 200 years that the logical basis of calculus was worked out and renamed Analysis.

    So we really have two related subjects: Calculus, which is an organized body of ideas and techniques for solving problems involving continuous change; and Analysis, in which the whole enterprise is put on a sound logical footing.

    In fact Calculus is not really about limits at all. Newton did not know what a limit was. He never had a satisfactory explanation for what happened to the limit of the difference quotient as the numerator and denominator "became" zero. It took another 200 years for all that to get sorted out properly.

    I should add that it's clear from Newton's writings that he well understood the problem. He knew that he didn't have the last bit of logic nailed down properly. That's why he wrote the Principia (http://en.wikipedia.org/wiki/Philoso...ia_Mathematica) in entirely geometrical language. He already had all the same results using calculus; but he knew that calculus was unfamiliar to people and he did not want criticism of his revolutionary methods to get in the way of his ideas about the universe. So he cast all his arguments in the geometric language of the ancients.

    And in Latin, too! Newton was one smart guy.
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    At high school we studied 2 years of calculus and here are some things that I found made differentiation/integration a bit easier to understand.

    In its simplest form differentiation is the application of n * x ^ (n - 1) to all elements of x in equations of the form of a * x^2 + b * x + c = 0.

    The roots of the basic form are - b +/- the square root of (b^2 - 4 * a * c)/2a:



    The first differential of this basic form is 2 * a * x + b = 0 and the integral of this first differential is the original basic form because integration is the exact opposite of differentiation (the application of x^(n+1) / n to all elements of x + add a constant (x^0, which may = 0)).

    The units of acceleration M / second^2, speed M / second and distance traveled M all have a pure integral/differential relationship (between distance M with respect to time) that gets right to the heart of Newtons calculus and mechanics. These proofs from first principles are good examples of pure applied calculus.

    Finally, in the basic form the last differential always goes to 0 and the first integral is the same and also goes to 0, unless you use a modified calculus, so make sure you don't overshoot on the way down or the way up.
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