Smallness

The main thing with calculus is that we shall have also to learn under what circumstances we may consider small quantities to be so small that we may omit them from consideration. Everything depends upon relative smallness.

For example, There are 60 minutes in the hour, 24 hours in the day, 7 days in the week. There are therefore minutes in the day and 10080 minutes in the week.

Lets see how small a second is compared to a minute, hour and day. When compared with a minute, a second is 60 times lesser. But you can atleast see it. But when you select hour or day, seconds become so small that you cannot see.

So, a second is already a small part of a minute. And a minute is a small part of an hour. So a second is a very, very small part of an hour.

Second->Minute->Hour.

So we call second a small quantity of the second order of smallness when compared with an hour.

If we extend the thought process, a second, when compared with a day, is a small quantity of order of smallness

So in this calculus world, we discard quantities which are of second or third (or higher)–orders of smallness, if only we take the small quantity of the first order small enough in itself.

Why? Because, as you could see visually in the previous example quantities in the second order of smallness are negligible.

Ex: Centimeters are second order small when compared with

But, it must be remembered, that small quantities if they occur in our expressions as factors multiplied by some other factor, may become important if the other factor is itself large. Even a second becomes important if only it is multiplied by a few thousands.

So, a second when compared with a week is negligible. But a second * 10,000

Now in the calculus we write for a little bit of x.

These things such as dx, and du, and dy, are called differentials, the differential of x, or of u, or of y, as the case may be. [You read them as dee-eks, or dee-you, or dee-wy.]

If dx is a small bit of x, and relatively small of itself, consider the following terms and say whether they are negligible and we can discard them or no.

`x*dx`

`x^2dx`

`a^xdx`

`dx*dx`

`dx*dx*dx`

Negligible

Not Negligible

Let us think of x as a quantity that can grow by a small amount so as to become x+dx, where dx is the small increment added by growth.

The square of this x+dx2 is x2+2x·dx+dx2.

The second term 2x·dx is

because it is a first-order quantity

The third term dx2 is

because it is of the second order of smallness, being a bit of, a bit of x2.

Let's look at this numerically. We have the term x2+2x·dx+dx2. Let dx=160 of x(ie, its a small amount of x)

Substituting we get:

=> x2+2x·x60+x602

=> x2+2x260+x23600

So the third term, 13600ofx2 is clearly than the second 260ofx2.

But if we go further and take dx to mean only 11000ofx.

Then the second term will be 2ofx2.

While the third term will be only 1ofx2.

Lets look at the same thing geometrically.

As you can see, you have a square of size x. Lets make the square grow by dx(adding dx to its size each way). Click on 'Add dx".

The enlarged square is made up of the original square x2, the two rectangles at the bottom and on the right, each of which is of area x·dx (or together 2x·dx), and the little square at the bottom right-hand corner which is dx2

Here we have taken dx as quite a big fraction of x–about 15. But suppose we had taken it only 1100–about the thickness of an inked line drawn with a fine pen. Then the little corner square will have an area of only 110000 of x2, and be practically invisible.

Go ahead. Drag the slider and make the dx size smaller, as small as you can.

Clearly dx2 is negligible if only we consider the increment dx to be itself small enough.

Sign in to Innings2

Reset Progress

Glossary

Smallness

Smallness