I never made it past college algebra, found trigonometry to be very hard, and never was even introduced to Calculus. My questions are: What is Calculus, and why do we need it? What function does it serve? Why can’t we do without it? If you answer these questions, please do so in English with a minimum of equations, ;).
Knowing the calculus of infinitesmals makes you a better person… Even if you have no use for it.
Calculus is a sophisticated math technique dealing with continuously varying equations. Leaving aside the importance of mental discipline, calculus can be used to determine such practical things as the volume of a Coca-Cola bottle (or any non-geometric shape) or the best size to make a tin can so that you use the least amount of metal for a given volume (the height should be twice the diameter of the top and bottom).
“East is east and west is west and if you take cranberries and stew them like applesauce they taste much more like prunes than rhubarb does.” – Marx
Read “Sundials” in the new issue of Aboriginal Science Fiction. www.sff.net/people/rothman
In a nutshell, algebra deals with discrete values–there’s sort of an inherent finiteness to it. On the other hand, there’s an inherent “infiniteness” in the ideas of calculus. In other words, for example, calculus deals with an infinite sequence of numbers and is concerned with the limiting value of that sequence–the value that the numbers are getting closer and closer to. The idea of limits is fundamental to all of calculus.
For an illustrative example, consider the problem of finding the circumference of a circle–no matter how you do it, some form or variation of calculus. A common approach is to consider regular polygons inscribed in the circle–the more sides the polygon has, the closer its perimeter will be to the actual circumference of the circle–the circumference of the circle is the limiting value of the perimeters of the polygons (as the number of sides increases).
The two classic problems approached in calculus are:
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Finding the slope of a tangent line to curve.
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Finding the area underneath a given curve.
And of course, the single most important application of calculus ever is what Newton invented it for (and don’t tell me about Leibniz, please): to discover the law of universal gravitation. Calc was invented so that the greatest scientific discovery ever could be quantified.
Calculus is a lot of things, but overall it’s amazingly useful set of techiques. In fact, much of what you’re expected to just memorize before you learn calculus (such as certain formulas in physics or chemistry or economics) ends up making much more sense once you apply calculus to it. It pops up all over the place in an amazing number of places in almost every field. Personally, I think at least basic calculus should be required for high school graduation. (I suspect the US is one of the few countries where you can graduate from HS without ever being exposed to it).
It’s hard to give any meaningful one or two sentence description, but basically think of it as the study of rates of change and areas under curves. Since almost everything can be viewed as some sort of curve (might be the shape of a physical object, or the acceleration of a particle, or some economic indicator, or how the temperature of some object changes, or whatever), it’s quite a powerful way to analyze things.
That help any?
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peas on earth
There are two branches of calculus, and the applications are many, but basics can be summarized as follows:
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Differential calculus deals with the rate of change. An object is travelling over a certain distance during a certain time; the rate of change is called velocity. The velocity of the object is changing over time (the object speeds up or slows down), and the rate of change is called acceleration.
Calculus enables the measurement of such rates of change (these also happen to be related to the tangent line to a curve.) -
Integral calculus is concerned with determining the area of a shape. You learned in elementary school how to figure out the area of a rectangle or triangle or cicle… but there are other shapes, like ellipses, parabolas, etc.
The great contribution of Newton and Leibnitz was to discover that these two seemingly independent problems are, in fact, related. Imagine a rollercoaster shaped curve, and the great discovery was that the process of determining the rate of change of an object rolling along this track is, in a sense, the “reverse” of the the process of determining the vertical area under the track.
That’s the nutshell version.
Thanks to everyone who replied. That does help, but I have one more question: how in the world did Newton ever invent calculus? My mind boggles at the thought of trying to learn it, let alone invent it!
Pun alert! Pun alert!
ERm… Newton didn’t really invent all of it. He just made a big jump. Some of it was known before hand and most of it was invented later. Basically he just said “If I know a function how do I find out how steep it is at any point?”. That was already know for a lot of functions so he invented a general method.
If you want, I wrote a nice paper about why calculus was invented. I can email it to you, but be warned: it contains very bad math jokes. People in the class which I wrote it for still remember me as “the guy who wrote the paper on how God invented the integral”.
Thanks, but I probably wouldn’t understand it if you did send it. Pre-college algebra was hard enough for me. Beyond that, I just don’t have a mind for math. But thanks anyway. 
Konrad, I’m sure lots of people would love to see your paper too (myself included - I love math jokes). Why don’t you post it on the web? Many thanks and merry Xmas.
Much of the material taught in a calculus corse was created much later than Newton, sometimes even 20th century. The argument between Newton and Liebnitz over who invented what first suggests that multiple people were thinking along the same lines. I wouldn’t describe Principia as being primarily a calculus book. Jeff Jacobi’s recent Boston Globe column http://www.globe.com/dailyglobe2/354/oped/Isaac_NewtonP.shtml mentions some other cool stuff Newton did.
Is the world continuous or discrete?
Calculus is a more powerful tool to dig deeper into the nature of numbers, our philosophies, and in the end: what the hell this is all about.
There’s always another beer.
More to the core.
Have you, or anyone you know, gone through a day without thinking about a number?
There’s always another beer.
Newton once said, “If I have seen farther than others, it is because I have stood on the shoulders of giants”.
In particular, the method I mentioned above of approximating the circumference of a circle (and, hence, pi as well) using polygons was done by Archimededes (generally considered to be the greatest mathematician in the time before Newton) circa 250 BC. So there were precursors to calculus long before calculus was invented.
I realize now that since this was originally intended as a speech, most of the jokes weren’t written down in the final copy. Even the part which I’m posting now I just found in my desk on a seperate piece of paper and typed it up now. The paper itself is not that great and it’s kind of boring, and in fact I’m embarassed that I’m posting it, but I’ve put it up anyway because I’m so vain. It’s at:
http://pages.infinit.net/konrad/integral.htm
Here’s the part that everyone seemed to like. I read it out loud in the same tone of voice that the priest on The Simpsons uses, with a very serious look on my face.
At first, the world was filled with polynomials and rational numbers, all positive. This made possible the invention of the integral. God let the people integrate any function on a finite interval. At first this was good, but it quickly got out of hand because sinners, tempted by the devil, did not heed God’s word and constructed indefinite integrals. Each time a polynomial was integrated it created a polynomial of a higher power until the world was quickly filling up.
"So Noah said onto God, ‘Save us from this flood of polynomials and arbitrary constants or we shall drown therein.’ So God did say onto Noah: 'You shall create an arctangent and place upon it one monomial of each natural power in such a way:
x^0 + x^1/1! + x^2/2! + x^3/3! + x^4/4! + …’
So on Noah’s arctangent he did put as God had instructed, and it did converge for all values of x, and thus created e to the x! And it was good."
e to the x was immune to integration and for a while this worked, but the tide of arbitrary constants just kept rising.
“And God did see the chaos reigned once again on the Earth so he created the derivative so that the people could take limits at a point and thus control the integrals. And an equilibrium did take place between the derivatives and the integrals, and peace was once again restored to the Earth, and it was good.”
Book of Numbers, Appendix B
Not since infancy, I admit. But, I have gone through the vast majority of the days after my last calculus exam a year and a half ago without thinking about any derivatives or integrals.
Konrad, thanks a lot for the paper!!! It’s truly terrific. Minor nits:
I read (but this may be a UL) that Pythagoreans postulated that all numbers are rational. When they proved that the square root of 2 is irrational, they panicked and kept this fact secret.
You might want to mention Egyptian factions in the prehistoric section.
You might also amuse your audience by mentioning that Karl Marx proved some minor theorems in real analysis (obsoleted by 20th century work on foundations of calculus).
Merry Xmas
And of course from the field of computer science:
“If I have not seen as far as others, it is because giants were standing on my shoulders.” - Hal Abelson
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peas on earth
matt_mcl:
When I was an infant, I thought about number-1 and number-2. . .but not very cognitively, and I let others handle their logistics and disintegration.
Where did Snark go to K-12? I had differential calculus in 7th grade in 1943 in CA.US, which state never did have decent schools compared to the East Coast. Although I had the last year of high school on the East Coast, I didn’t get integral calculus, though, until college.
But for all you philosophers here, I thought “calculus” was sentential calculus. The first part of the second definition in my dictionary covers all those calculi though: “A method of analysis or calculation using a special symbolic notation,”. . .except. . .those special images of God called MDs somehow got in there and claimed (even making it the first definition in my dictionary) that the word means: “An abnormal concretion in the body, usually formed of mineral salts; a stone, as in the gall bladder, kidney, or urinary bladder.” (Sounds like we’re right back to square. . .I mean. . .number-1.) But I guess they went to the bedrock of the meaning of the word in Latin, which meaning my dictionary says is: “small stone used in reckoning”. I reckon I don’t have the slightest idea how to use only one stone for reckoning. . .maybe more than one stone.
So if the calculus is too hard for you. . .well get hit with something softer. Talc?
Ray (People in vitreous abodes shouldn’t throw such things.)
“If I have seen farther than others, it is because I am surrounded by dwarves.” - Murray Gell-mann
dlv: It’s just a paper I wrote for a class. It’s not an ongoing project or anything. I heard lots of weird things about the Greeks. Like when someone challanged Aristotle’s (?) view that heavy things fall faster than light ones by doing an experiment, Aristotle had his followers chase down the felon and beat him.