Monday, June 07, 2004

Ideas: Calculus

Here is a short piece on calculus.

Date: 05/06/97 at 09:41:35
From: matthew doedtman
Subject: Calculus

What is calculus and how does it work?

Date: 05/06/97 at 14:25:39
From: Doctor Ceeks
Subject: Re: Calculus


Calculus is a branch of mathematics.

Calculus was created in large part by Newton and Leibniz, although
some of the ideas were already used by Fermat and even Archimedes.

Calculus is divided into two parts, which are closely related. One
part is called "differential calculus" and the other part is called
"integral calculus". [In high school, most differential calculus is tarught]

Integral calculus is concerned with area and volume. How do you
determine the area of a circle or the volume of a sphere? Another way
of putting it is: how much paint do you need to color in a circle? How
much water do you need to fill up a ball? Integral calculus explains
one way of computing such things.

The basic idea of integral calculus is this: the simplest shape whose
area we can compute is the rectangle. The area is the length of the
rectangle multiplied by its width. For instance, a "square mile" is a
piece of land with as much area as a square plot of land with sides
measuring one mile each. To compute the area of a more complicated
region, we chop up the region into lots and lots of little rectangles.
When we do this, we will not be able to succeed completely because
there will always be pieces with curved sides, generally. But the key
idea is that the sum of the areas of the rectangular pieces will be a
very close approximation of the actual area, and the more pieces we
cut, the closer our approximation will be.

Differential calculus answers the following question: imagine you go
on a car ride. Suppose you know your position at all times. In other
words, at 10 a.m. you're in the garage, at 10 a.m. and 5 seconds
you're just outside the garage, at 10 a.m. and 10 seconds you're on
the road just in front of your house...and so on... At the end of
your trip, you realize that at every moment during your trip, your
speedometer showed the speed of your car. Just from the knowledge of
your position at all times, can you reconstruct what your speedometer
showed at any time? The answer is, yes, you can, and differential
calculus provides a method for doing this.

The basic idea of differential calculus is this: the simplest
situation where you can compute what the speedometer read is when you
drove at the same speed over the entire distance. Then, you can use
the formula: speed equals distance divided by time. For instance, if
you drive 50 miles in one hour all at the same speed, then your
speedometer read 50 miles per hour the whole trip. In the situation
where you didn't drive at the same speed, the idea is to imagine your
trip as lots and lots of short trips, say, one trip involving pulling
the car out of the garage, another trip getting the car onto the road,
and so on...even trips which involve going from one lamp post to the
next. Over each of these tiny trips, your speed doesn't change much
and you can pretend that your speed didn't change at all. This puts
you in the situation where you know how to compute the speed for each
tiny trip, and gives you a good idea of what your speedometer read for
that part of the big trip. However, the assumption that the speed
didn't change over each tiny trip is generally wrong, and so you only
get an approximation to the correct answer. But the key idea is that
the smaller you make the tiny trips used in your computation, the more
accurate you will be able to compute the actual speedometer reading.

-Doctor Ceeks, The Math Forum
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