113

4 hours ago, studiot said:

Well I have to differ here on several counts.

113 is not using terminology exactly for instance

How exact is your

Quote

If dx is a very small quantity then (dx)² is insignificant and may be ignored so we have

Tell exactly how small quantity must dx be so that (dx)^2 can be ignored.

On 10/7/2018 at 12:37 PM, studiot said:

The idea is that we make a (small) change to the variable concerned and compare the result in the function of that variable to the starting point.

With a single real variable there is only one sort of change that can be made, so implementation of the modern formula looks a bit over the top.

What if we make a change equal to dx to the variable x? Doesn't the function f(x) change equal to f(x + dx) ?

9 hours ago, taeto said:

You were hiding the information about y(x) being the same as f(x) ?

Who told you about \(\frac{dy}{dx}\) but hided the information about \(\frac{f(x+dx)- f(x)}{dx}\) ?

Was it me or was it all the math books? And why should they, the mathematicians,  have done so?

9 hours ago, taeto said:

I do not know the answer. What do you mean by ∞ ? And by 1/∞ ? Why do you think that 1/∞ is infinitely small? What does it mean to you that something is infinitely small? You are saying things that have no place in usual mathematics.

Why can't "usual mathematics" deal with something that is infinitely small?

9 hours ago, taeto said:

And do you know how to answer a question without introducing several more unknown quantities?

What is the question? Is the question "is 1/∞ either 0 or non-zero?" ?  The question does not have an unambiguous answer either 0 or non-zero, because 1/∞  is both.

9 hours ago, taeto said:

What do you mean when you speak of "infinity"? What is that exactly to you? I have absolutely no idea what you are going on about. Why would we have to deal with objects that have contradictory properties, it would seem an extraordinary stupid thing to do, no?

If that is a stupid thing to do, then why don't you answer the question: "is 1/∞ either 0 or non-zero?" ?

9 hours ago, taeto said:

Again now, look: the adjective "finite" in mathematics applies to sets.

Yes. But I am not talking about sets,  because maybe it is not necessary right now. I am talking about finite difference f(x+h) - f(x) where h is finite, as opposed to f(x + dx) - f(x) where dx is not finite. Adjective finite refers to distance in this case, finite distance.

9 hours ago, taeto said:

 

A set X is finite if there is a natural number n such that the "size" of X is n, which means that there exists a bijection from {1,2....,n} to X, so that you can count the elements of X from 1 to n. So correspondingly, the adjective "infinite" applies to a set for which there is no such n. Which kind of set is your dx for which the adjective "infinite" applies to it, and how can you prove this? And if so, how does this fact apply to the rest of the things that you are saying?

You can count the elements of X only if you know n. But because you did not tell what is n, I can't tell how many elements are there in X. So it may look as though dx

is a set for which you don't tell what is n.

13 minutes ago, taeto said:

     There is a "y" on the left hand side, but an "f" on the right hand side. The equation relates the derivative of a function y of x to the derivative of a function f of x. That is not what the definition of a derivative is supposed to look like. 

It makes no difference because y = f(x) so that dy/dx = df(x)/dx

13 minutes ago, taeto said:

What kind of thing is dx supposed to be? An integer, a rational number, a quaternion, what? What do you mean when you say "infinitely small"? Can you give any examples of mathematical objects that are "infinitely small"?

1/∞ is infinitely small. Is it 0 or non-zero?

The introduction of infinity brings a duality into the definition of an infinitesimal, meaning that we have to deal with objects that both zero and non-zero at the same time.

21 minutes ago, taeto said:

You mean dfdx(x) . 

Yes.  Derivative is defined as the limit of a finite difference

\[\frac{dy}{dx} = \lim_{h\to\ 0}\frac{f(x+h)- f(x)}{h}\]

where h is finite.

Quote

Maybe it is possible. But only after you explain the meaning of the expression on the right hand side. E.g. what exactly do you mean by saying that dx is "infinitely small, infinitesimal"?

I mean that dx is not finite as opposed to h which is finite. If dx is not finite, it is infinite, infinitely small.

Derivative is defined as the limit of a finite difference

\[\frac{dy}{dx} = \lim_{h\to\0}\frac{f(x+h)- f(x)}{h}\]

where h is finite.

Is it possible to define the derivative by

\[\frac{dy}{dx} = \frac{f(x+dx)- f(x)}{dx}\]

where dx is not finite but infinitely small, infinitesimal ?

\[\frac{\Delta y}{\Delta x}= \frac{f(x+h)- f(x)}{h} \]

\[\frac{\Delta y}{\Delta x} = \frac{f(x+h)- f(x)}{h}\]

\[\Delta y \]

\[ y = \int f(x) dx \]

\[\Delta y \]

\[ y = \int f(x) dx \]

I got it working now, except the last one , integral doesn't like me.

Now it all works. I noticed there was a message saying something like "pasted as rich text, paste as plain text instead" which I choose and then the integral also worked.