### 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

Hi,

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

Check out our web site! http://mathforum.org/dr.math/

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