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What is 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".

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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