Calculus Made Easy

Before I begin, I must give proper credit. These words are not mine. I am adapting what follows from a book by Silvanus P. Thompson.

We will find that in our processes of calculations in calculus, we have to deal with small quantities of various degrees of smallness. We also have to learn under what circumstances we may consider small quantities to be so minute that we may omit them from consideration. Everything depends on relative minuteness.

Before we agree on any rules let's think of some familiar cases. There are 60 minutes in an hour, 24 hours in a day, 7 days in a week. There are therefore 1440 minutes in the day and 10080 minutes in a week. Obviously 1 minute is a very small quantity of time compared with a whole week. Our forefathers considered it small as compared with an hour, and called it "one minute", meaning a minute fraction - namely one sixtieth - of one hour. When still smaller subdivisions were required, they divided each minute into 60 still smaller parts, which in queen Elizabeth's days, were called "second minutes" (i.e., small quantities of the second order of minuteness.) Nowadays we call these small quantities of he second order of smallness "seconds." But few people know why they are called seconds.

Now if one minute is so small as compared with a whole day, how much smaller by comparison is one second! Another example of smallness would be a million dollars compared to one penny. If we were to take 1 percent, 1/100 as a small fraction, then one percent of one percent (1/10,000) would be a small fraction of the second order of smallness; and 1/1,000,000 would be a small fraction of the third order of smallness, being one percent of one percent of one percent.

But it must be remembered that small quantities, if they are multiplied by some other factor, may become important if the other factor is itself large. Even a penny becomes important if it is multiplied by a few hundred.

In calculus we write dx for a little bit of x. These things such as dx, and dy, du, are called differentials. If dx is a small part of x, it does not follow that such quantities as x*dx, or x^2*dx are negligible. But dx*dx would be negligible, being a small quantity of the second order.

A very simple example will serve as illustration.

Let's think of x as a quantity that can grow by a small amount, (dx) to become x+dx. If we square this amount (x+dx)^2, as illustrated in the picture shown on the right, we get x^2 + 2xdx + (dx)^2. The second term os not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of a bit of x. The large square is the original square, x by x. Then if you look at the sides of the original square, you can see that each side grew by a little bit (dx). So the two rectangles are x*dx, (which is where the 2xdx came from). And the small square on top is (dx)^2.

To put numbers to this example, lets say that dx is 1/100, then (dx)^2 would have an area of only 1/10,000 of x^2, and be practically invisible.

Another analogy...Suppose a millionaire were to say to his secretary: next week I will give you a small fraction of any money that I make. Suppose that the secretary tells his son: I will give you a small fraction of what I get. Suppose the fraction in each case is to be 1/100. Now if Mr. Millionaire received $1000 during the week. The secretary would receive $10 and the boy would receive $.10. Ten dollars would be a small quantity compared to $1000, but 10 cents is a very small quantity, a second order small quantity.

Power Rule Explained

Now lets see how we can differentiate some simple algebraic expressions.

Case 1: Let's begin with the simple expression y = x^2. Now remember the fundamental notion about calculus is the idea of growing. Mathematicians call it varying. Now as y and x^2 are equal to each other, it is clear that if x grows, x^2 will also grow. And if x^2 grows, then y will also grow. What we have got to find out its the proportion between the growing of y and the growing of x. In other words our task od to find out the ratio between dy and dx, in brief, to find the value of dy/dx.

Let x, grow a little bit larger and become x + dx; similarly, y will grow a bit bigger and will become y + dy. THen clearly, it will still be true the enlarged y will be equal to the square of the enlarged x. Writing this down we have:
y + dy = (x + dx)^2.
Doing the squaring we get: y + dy = x^2 + 2x*dx + (dx)^2

What does (dx)^2 mean? Remember that dx meant a little bit of x. Then (dx)^2 will mean a little bit of a little bit of x; that is, as explained in the previous post it is a small quantity of the second order of smallness. It may be discarded as quite inconsiderable in comparison with the other terms. Leaving it out, we then have
y + dy = x^2 + 2x*dx.
Now y = x^2; so we will subtract it from the the equation above and we have left
dy = 2x*dx.

Multiplying by dx on both sides: dy/dx = 2x.

Now this is what we set out to find. he ratio of the growing of y to the growing of x is found to be 2x.

If we put numbers tot the above example, it might make more sense. Suppose x = 100, therefore y = 10,000. hen let x grow until it becomes 101, (that is, dx = 1). Then the enlarged y will be 101 * 101 = 10,201. But if we agree that we may ignore small quantities of the second order, 1 may be rejected as compared with 10,000; so we may round off the enlarged y to 10,200. The bit added on is dy, which is therefore 200.
dy/dx = 200/1.
According to the algebra-working above, we find dy/dx = 2x. And so it is; for x = 100, 2x = 200.

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By Alex DiTirro
Note Mar 16, 2012 (1)

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