What's The Point Of Calculus??

[CuriosOne]

3 hours ago, studiot said:

Yes this is a profound statement and very important in calculus and analysis more generally.

This was why I said that dx on its own is meaningless.

Yes a limit has one numeric value, if the limit exists at all.

But this x can never = 0 statement is not right as it stands. We will come to a proper version in due course.

But first some background, that will become increasing useful as things progress.
Note the author says that Analysis is mostly limis and calculus, but it also includes stuff about functions, sequences and series which are necessary to move on from the algebra of quadratic equations.

 

So let us start with variables and functions.

You will be please to hear that we don't need the whole apparatus  - That can take a whole semester by itself  -  just some of the bare bones.

So variables. A variable is perhaps the most self evident word in Mathematics. It really does exactly what is says on the tin and no more.
A variable is something that, well varies. More precisely it may take on many different values. And a numeric variable is a variable whose value is numeric.

Your x is a (usually)  numeric variable.

So what values can a variable have  ?
Good question, note I may take on, not does take on.
The question is incredibly wide since all we know is that the value of our variable is a number.
So any value between plus and minus infinity will do.

Enter the mathematical function to the rescue.

The main part of a function is a rule or equation or other expression for obtaining (numeric) values.
Other parts limit the actual numbers we are allowed to use. So the square root function excludes all negative numbers.
That still leaves a lot of numbers available.

Since x is a variable, we can expect most functions of x to also be variables.
We put different numbers in and expect to get correspondingly different numbers out.

An exception to this is called the constant function.
You get the same number out, whatever number you put in.

OK numbers in ;  numbers out.

We now have two variables,  x and f(x)

and we can start going places.

 

I have said f(x) since you have used it.

But I am now going to change to y = f(x), which is using the rule to define the function as most folks do.

So the next thing to understand is that x and y or f(x) are two different types of variable.

x is an independent variable, which means that we can put in any number we are allowed.

y = f(x) is called a / the  dependent variable because its value depends upon the both the number we put into x and the rule which generates y.

 

Having swallowed and digested all this guff we know enough about variables so are now ready to examine all the types of the 'd' in dx.
'd' is called an operator. An operator performs some process on the variable it operates on  -  in this case the variable x.
This is the reason for changing from f(x) to y. We can operate on y with 'd' to get dy.

We will tackle that next time, but a question of you if I may.

 

Another person in another forum has asked a very similar question.
Have you any relation to Idon'tknow, elsewhere  ?
I want not mix up answers to each.

 

 

 

3 hours ago, studiot said:

Yes this is a profound statement and very important in calculus and analysis more generally.

This was why I said that dx on its own is meaningless.

Yes a limit has one numeric value, if the limit exists at all.

But this x can never = 0 statement is not right as it stands. We will come to a proper version in due course.

But first some background, that will become increasing useful as things progress.
Note the author says that Analysis is mostly limis and calculus, but it also includes stuff about functions, sequences and series which are necessary to move on from the algebra of quadratic equations.

 

So let us start with variables and functions.

You will be please to hear that we don't need the whole apparatus  - That can take a whole semester by itself  -  just some of the bare bones.

So variables. A variable is perhaps the most self evident word in Mathematics. It really does exactly what is says on the tin and no more.
A variable is something that, well varies. More precisely it may take on many different values. And a numeric variable is a variable whose value is numeric.

Your x is a (usually)  numeric variable.

So what values can a variable have  ?
Good question, note I may take on, not does take on.
The question is incredibly wide since all we know is that the value of our variable is a number.
So any value between plus and minus infinity will do.

Enter the mathematical function to the rescue.

The main part of a function is a rule or equation or other expression for obtaining (numeric) values.
Other parts limit the actual numbers we are allowed to use. So the square root function excludes all negative numbers.
That still leaves a lot of numbers available.

Since x is a variable, we can expect most functions of x to also be variables.
We put different numbers in and expect to get correspondingly different numbers out.

An exception to this is called the constant function.
You get the same number out, whatever number you put in.

OK numbers in ;  numbers out.

We now have two variables,  x and f(x)

and we can start going places.

 

I have said f(x) since you have used it.

But I am now going to change to y = f(x), which is using the rule to define the function as most folks do.

So the next thing to understand is that x and y or f(x) are two different types of variable.

x is an independent variable, which means that we can put in any number we are allowed.

y = f(x) is called a / the  dependent variable because its value depends upon the both the number we put into x and the rule which generates y.

 

Having swallowed and digested all this guff we know enough about variables so are now ready to examine all the types of the 'd' in dx.
'd' is called an operator. An operator performs some process on the variable it operates on  -  in this case the variable x.
This is the reason for changing from f(x) to y. We can operate on y with 'd' to get dy.

We will tackle that next time, but a question of you if I may.

 

Another person in another forum has asked a very similar question.
Have you any relation to Idon'tknow, elsewhere  ?
I want not mix up answers to each.

 

 

I have "no" relation to Idon'tknow.

I read your reply, thanks "very helpfull."

As per limits ""a value of plus and minus "infinity" ?? ""

Is "dy/dx" infitismally small values of infinity??

What are those values of infinity then then?

And why say these are limits x->0 when they are considered infinite?

 

Edited October 13, 2020 by CuriosOne

[J.C.MacSwell]

X can equal 0. If you are interested in when X equals 0, or through a range of numbers where X equals zero you include that point with your function of X. 
if not you don’t include it. It’s not that x can or can’t equal 0, just that you have no interest in the function at that point. 

You get to set limits depending on what you have interest in solving with the function. 

[CuriosOne]

24 minutes ago, J.C.MacSwell said:

X can equal 0. If you are interested in when X equals 0, or through a range of numbers where X equals zero you include that point with your function of X. 
if not you don’t include it. It’s not that x can or can’t equal 0, just that you have no interest in the function at that point. 

You get to set limits depending on what you have interest in solving with the function. 

Let's say the interest is 2 stationary vibrating "particles" in space (a,b )and we want to know how the "vibrations" of both interact within a certain "limit "before" change...Or even better, we want to know "which vibration is a or b, since they are all mixed in with each other...If that's even possible if this were a case involving trillions of stationary particles vibrating in space..

Edited October 14, 2020 by CuriosOne

[J.C.MacSwell]

I could be wrong but I don’t think calculus will answer that. You might come up with a function that could model something they did or thought they did and calculus could give some answers based on that. 

Someone  better versed in math may add to (or correct) that. 

[MigL]

I'm still not sure what you are asking for, since you keep throwing different scenarios out.

If you are asking about the limit formulation of a derivative ( with x cannot equal 0 ) that is introductory differential calculus, so I find it hard to believe you've been doing it for 20 years.

If we define f(x) to be a function of x, the derivative function of f at x is given by: df (x) = lim h→0 { f(x + h) − f(x) }/ h
If the limit exists, f is said to be differentiable at x, otherwise f is nondifferentiable at x.
If y = f(x) is a function of x, then we also use the notation dy/dx to represent the derivative of f.
You should also note that dy and dx are not numbers but differentials.   

Sorry about the lack of LaTex.   

[Ken Fabian]

I had a good mathematics teacher for a year at high school and for a brief time calculus was a joy. I haven't used it much since and so, have mostly forgotten. I do recall deriving the equation for the area of a circle using calculus - I think it was a test question - and that it seemed to be seamless in it's logic. Made a Pi starting from square one!

I couldn't do it now but it was kind of reassuring. It may have contributed more to my trusting established knowledge rather than sparking great determination to personally confirm everything. I cannot work out the equation for area of a circle from first principles now but at the time I was all round impressed.

[CuriosOne]

6 hours ago, J.C.MacSwell said:

I could be wrong but I don’t think calculus will answer that. You might come up with a function that could model something they did or thought they did and calculus could give some answers based on that. 

Someone  better versed in math may add to (or correct) that. 

I've read number theory could be the more suitable approach as to use the number theory finds and apply them or plug them into our f(x) function limit issues.

After all, there has to be many variations that define the same value as there is many ways to play the drums to a melody....I guess it depends on reliability, trust and efficiency at this point if the world depended on such a system.

5 hours ago, MigL said:

I'm still not sure what you are asking for, since you keep throwing different scenarios out.

If you are asking about the limit formulation of a derivative ( with x cannot equal 0 ) that is introductory differential calculus, so I find it hard to believe you've been doing it for 20 years.

If we define f(x) to be a function of x, the derivative function of f at x is given by: df (x) = lim h→0 { f(x + h) − f(x) }/ h
If the limit exists, f is said to be differentiable at x, otherwise f is nondifferentiable at x.
If y = f(x) is a function of x, then we also use the notation dy/dx to represent the derivative of f.
You should also note that dy and dx are not numbers but differentials.   

Sorry about the lack of LaTex.   

Yes, differential calculus focuses on rates of changes and the ""Pythagorean Therom"" is extensively used with it.

The notation that expresses these functions also change which gets very frustrating, here is an example:

dy/dx or df/dx or df(x)/dx or D, xy or d/dx or f'(x) or y' and etc etc

"All Mean The Same Thing." 

Ive been observing this for years now of which had me confused for about 4 years until I caught onto it..

let me show a number example of these limits finally "I found one" and we can dissect this from here as a visual guide for direction, becuase the functions are confusing me and maybe others..

Lim x->0 = x^2 -25/ x-5

as "delta" x approaches 0

"BEFORE" we add numbers let's speak reason "please."

does x^2 -25 represent y the numerical value, the value that does not change?

does x-5 represent the value that does change?

 

Edited October 14, 2020 by CuriosOne

[LaurieAG]

While I studied calculus at high school we didn't cover infinitesimals until university.

https://www.math.wisc.edu/~keisler/calc.html

Elementary Calculus: An Infinitesimal Approach

On-line Edition. Copyright © 2000 by H. Jerome Keisler, revised August 2020.

This is a calculus textbook at the college Freshman level based on Abraham Robinson's infinitesimals, which date from 1960. Robinson's modern infinitesimal approach puts the intuitive ideas of the founders of the calculus on a mathematically sound footing, and is easier for beginners to understand than the more common approach via epsilon, delta definitions.

The First Edition of this book was published in 1976, and a revised Second Edition was published in 1986, both by Prindle, Weber & Schmidt. When the Second Edition became out of print, the copyright was returned to me as the author. In September 2002 I decided to make the book available for free in electronic form at this site. These PDF files were made from the printed Second Edition, and are continually being revised with minor corrections.

[swansont]

17 hours ago, CuriosOne said:

Definetly would love to continue going down the rabbit hole...At this point and step by step I think first we should define what time is and I think its important here.

Is time 1 dimensional? 

In simple words:

Why do we use the speed of light as a reference for time?

Yes, in regards to limits..

!

Moderator Note

Let's keep to the topic of calculus here. If you want to ask about this (or anything else), please open another thread (in physics, in this case)

 

[CuriosOne]

8 hours ago, LaurieAG said:

While I studied calculus at high school we didn't cover infinitesimals until university.

https://www.math.wisc.edu/~keisler/calc.html

 

 

Thnks for the link...I do hope however I can have some answers to my question about limits earlier....in regards to:

Lim x->0 = x^2 -25/ x-5

BEFORE" we add numbers let's speak reason "please."

does x^2 -25 represent y the numerical value, the value that does not change?

and does x-5 represent the changing variable x?

On 10/13/2020 at 9:34 AM, HallsofIvy said:

Let me add this:  one of the things Sir Isaac Newton was concerned with was the force that kept the planets in motion about the sun.  He knew "mass times accelertion equals force" from Gallileo and knew, from Copernicus' measurements, that planets closer to the sun move faster than when they were farther from the sun.  From that he believed the force, and so the acceleration depended upon the distance from the sun.

But "acceleration" is "change in speed divided by change in time" just as "speed" is "change in position divided by change in time" while "distance" is given at a specific time, not over a change in time!  That was why Newton had to invent the Calculus- in order to be able to define "instantaneous speed", "instantaneous acceleration" or, generally, the rate of change at a specific time rather than over a span of time.

Where do these changes in position take place??? 

Do they take place on the orbital paths themselves, since "force" causes the planets to move??

Looks like orbital paths are "discountious" to me, hence the Title of my post.

So Calculus is a tool for orbits and nothing else??

Edited October 14, 2020 by CuriosOne

[MigL]

All of Newton's equations of motion were derived using calculus.
As a matter of fact, he invented the calculus for just that purpose.
( also, independently, by G Leibniz )
Newtonian mechanics, while not as 'sophisticated as Lagrangian or Hamiltonian mechanics, can still calculate the 'fall 'of an artillery shell at several miles distance, or even enable putting a rocket in lunar orbit.

So, I would say, quite useful.
Your criticism of calculus, on the other hand, totally useless.

[joigus]

It's hard for me to understand what your point about the point of calculus is. But as I can't sleep very well tonight, I've thought I might as well tell you a little bit about what your problem may be.

Your problem may be that you don't understand real numbers. Real numbers go beyond what intuitive numbers (numbers you may be used to) are. They're not like the number of people in a room, nor like the reading of a ruler, nor like the money in an account or exchange rates between currencies.

These numbers can be classed into a list:

Counting numbers: 1, 2, 3,... (natural numbers)

Whole numbers: ..., -3, -2, -1, 0, 1, 2, 3,...

Ratios: 2/3, -1/5, 10132/11, (plus all the whole numbers), etc. (rational numbers)

In order to define real numbers, besides all the usual algebraic assumptions, you need this axiom:

It is impossible to approach a number arbitrarily closely without that number being part of my system of numbers.

This is called "completeness." You can rephrase it as "limits of numbers must be numbers."

Calculus was being used very fruitfully by many mathematicians and natural philosophers for almost 200 years before mathematicians like Cauchy and Weierstrass defined it rigorously.

<ignore if you don't understand>

Another way to understand an idea could be to understand when it fails and how. This link may be interesting in that regard:

https://amsi.org.au/ESA_Senior_Years/SeniorTopic3/3a/3a_4history_4.html

There you can find a function (Weierstrass' function) which cannot be differentiated meaningfully:

\[f(x)=\sum_{n=1}^{\infty}\dfrac{1}{2^{n}}\cos(4^{n}x)\]

And a graphic representation to an approximation by taking only 50 terms of the sum:

\[y=\sum\limits _{n=1}^{50}\frac{1}{2^{n}}\cos(4^{n}x)\]

</ignore if you don't understand>

Now let's go with your example. First, you've copied the formula wrongly. It should read:

\[\lim_{x\rightarrow0}\frac{x^{2}-25}{x-5}=5\]

What does that mean? (what you wrote has no meaning.) It means you can get as close as you want to number 5 by substituting in the expression for f(x),

\[f\left(x\right)=\frac{x^{2}-25}{x-5}\]

a number x as close to 0 as you want. The key that may be confusing you is that "as close as you want."

Now, when x is not 5, the expression,

\[\frac{x^{2}-25}{x-5}\]

of course simplifies to,

\[x+5\]

And it is obvious that you can get as close as you want to 5 by substituting x in x+5 for a number as close as you want to zero. That's the key to the "delta" that seems to ring a bell to you, but not the right bell. Here's the rigorous definition of limit: A function f(x) of one variable x as limit at x=a, and the limit is L if,

\[\forall\varepsilon>0\;\exists\delta>0/\left|x-a\right|<\delta\Rightarrow\left|f\left(x\right)-L\right|<\varepsilon\]

Literally read:

for all positive epsilon there is at least one positive delta such that, when x is closer to a than delta, then f(x) is closer to L than epsilon.

In other words:

You can get as close as you want to L in the expression of f(x) by substituting x for a value as close as you want to a.

I hope that helps. Derivatives come later.

 

 

 

[CuriosOne]

4 hours ago, MigL said:

All of Newton's equations of motion were derived using calculus.
As a matter of fact, he invented the calculus for just that purpose.
( also, independently, by G Leibniz )
Newtonian mechanics, while not as 'sophisticated as Lagrangian or Hamiltonian mechanics, can still calculate the 'fall 'of an artillery shell at several miles distance, or even enable putting a rocket in lunar orbit.

So, I would say, quite useful.
Your criticism of calculus, on the other hand, totally useless.

"I do get what your saying"

But its said Newton used the pyramids to create calculus...Is this true??

The extent of pi and the pythagorean theorem used in calculus "right angle" triangles leads one to think the pyramid contribution may be true...

Even E=mc^2 looks very familiar to F=mv^2....

 

1 hour ago, joigus said:

It's hard for me to understand what your point about the point of calculus is. But as I can't sleep very well tonight, I've thought I might as well tell you a little bit about what your problem may be.

Your problem may be that you don't understand real numbers. Real numbers go beyond what intuitive numbers (numbers you may be used to) are. They're not like the number of people in a room, nor like the reading of a ruler, nor like the money in an account or exchange rates between currencies.

These numbers can be classed into a list:

Counting numbers: 1, 2, 3,... (natural numbers)

Whole numbers: ..., -3, -2, -1, 0, 1, 2, 3,...

Ratios: 2/3, -1/5, 10132/11, (plus all the whole numbers), etc. (rational numbers)

In order to define real numbers, besides all the usual algebraic assumptions, you need this axiom:

It is impossible to approach a number arbitrarily closely without that number being part of my system of numbers.

This is called "completeness." You can rephrase it as "limits of numbers must be numbers."

Calculus was being used very fruitfully by many mathematicians and natural philosophers for almost 200 years before mathematicians like Cauchy and Weierstrass defined it rigorously.

<ignore if you don't understand>

Another way to understand an idea could be to understand when it fails and how. This link may be interesting in that regard:

https://amsi.org.au/ESA_Senior_Years/SeniorTopic3/3a/3a_4history_4.html

There you can find a function (Weierstrass' function) which cannot be differentiated meaningfully:

 

f(x)=∑n=1∞12ncos(4nx)

 

And a graphic representation to an approximation by taking only 50 terms of the sum:

 

y=∑n=15012ncos(4nx)

 

</ignore if you don't understand>

Now let's go with your example. First, you've copied the formula wrongly. It should read:

 

limx→0x2−25x−5=5

 

What does that mean? (what you wrote has no meaning.) It means you can get as close as you want to number 5 by substituting in the expression for f(x),

 

f(x)=x2−25x−5

 

a number x as close to 0 as you want. The key that may be confusing you is that "as close as you want."

Now, when x is not 5, the expression,

 

x2−25x−5

 

of course simplifies to,

 

x+5

 

And it is obvious that you can get as close as you want to 5 by substituting x in x+5 for a number as close as you want to zero. That's the key to the "delta" that seems to ring a bell to you, but not the right bell. Here's the rigorous definition of limit: A function f(x) of one variable x as limit at x=a, and the limit is L if,

 

∀ε>0∃δ>0/|x−a|<δ⇒|f(x)−L|<ε

This explanation is very helpfull, thnXxx and your right about x=0 

If x=5 but you can't reach zero, is there a sequence of numbers going on, such as a list for x and y values?? I assume that's what they are..Maybe you can show that list, as I just place it in my calculator and press generate list this is what I got...

 

 

 

 

 

 

 

 

 

Edited October 15, 2020 by CuriosOne

[joigus]

6 minutes ago, CuriosOne said:

But its said Newton used the pyramids to create calculus...Is this true??

The ancient Egyptians used a primitive version of calculus to make the pyramids, rather. (Small incremental sums.)

8 minutes ago, CuriosOne said:

Even E=mc^2 looks very familiar to F=mv^2....

Maybe you mean "similar to KE=(1/2)mv²

Or perhaps F=ma?

F=mv² is no standard physics.

[CuriosOne]

7 minutes ago, joigus said:

The ancient Egyptians used a primitive version of calculus to make the pyramids, rather. (Small incremental sums.)

Maybe you mean "similar to KE=(1/2)mv²

Or perhaps F=ma?

F=mv² is no standard physics.

 KE = (1/2) mv^2 looks correct...

 

[joigus]

37 minutes ago, CuriosOne said:

[...]

If x=5 but you can't reach zero, is there a sequence of numbers going on, such as a list for x and y values?? I assume that's what they are..Maybe you can show that list, as I just place it in my calculator and press generate list this is what I got...

x=5

4.999

4.998

4.997

etc..

 

Exactly, you're getting closer. Literally.

Calculus is about formalising the operation of "getting closer."

You can use it to calculate the gentlest slope on a mountain, where the summit is, etc. It's not necessarily about "orbits."

Just now, CuriosOne said:

 KE = (1/2) mv^2 looks correct...

 

It's no coincidence that KE=(1/2)mv² (Newtonian kinetic energy) reminds you of E=mc² (Einstein's kinetic energy). One is obtained from the other by means of... guess what. Calculus!

[CuriosOne]

1 minute ago, joigus said:

Exactly, you're getting closer. Literally.

Calculus is about formalising the operation of "getting closer."

You can use it to calculate the gentlest slope on a mountain, where the summit is, etc. It's not necessarily about "orbits."

If getting closer

"Where is x at then?"

At the edge of space??? 

And how do we know how far to zoom in and out, x is not a camera lense..😎

[joigus]

5 minutes ago, CuriosOne said:

At the edge of space???

It could be a value not accessible by elementary operations. It could be pi, or e. In that sense, it's at the core of space, rather than at the edge.

5 minutes ago, CuriosOne said:

And how do we know how far to zoom in and out, x is not a camera lense..😎

The "camera lens" is made of deltas and epsilons that I defined for you before. Calculus allows you to forget about the lens and operate with the points.

Edited October 15, 2020 by joigus

[CuriosOne]

4 minutes ago, joigus said:

It could be a value not accessible by elementary operations. It could be pi, or e. In that sense, it's at the core of space, rather than at the edge.

The "camera lens" is made of deltas and epsilons that I defined for you before. Calculus allows you to forget about the lens and operate with the points.

This is getting very intresting now...

I'm glad I posted this OP bold question, this information is inspiring thnXxxxx.

 

 

 

[studiot]

8 hours ago, CuriosOne said:

This is getting very intresting now...

I'm glad I posted this OP bold question, this information is inspiring thnXxxxx.

 

 

 

To continue where I left of and pick up and develop  joigus points about limits

 

The f(x) and y =y(x) notation comes in useful if you want to get it all onto one line so the derived functions become f'(x) and y'(x).

This is not directly related to any of the d notation variants and also becomes very clumsy when many variables are involved.

So let us look at the d notation, which is, after all, what you are asking about.

So we have 

Greek capital delta   [math]\Delta [/math]

Greek lower case delta   [math]\delta [/math]

Roman Capital D

Roman lower case d

Eighteenth century European stylised script  d    [math]\partial [/math]

We have all of them because each perform a different operation (remember them) on the symbol that follows them.

The first one is not important to us because we use it for the difference between two specific values of the variables represented by the symbol that follows.

So the difference in height, [math]\Delta h[/math],  between the ground and the gutterline of my house is 18 feet.   [math]\Delta h[/math] may be large, but must be exact. That is it is a number. As it is a number it can legitimately be zero.

The second one is important to us because it is not only about small differences in the values of the variable that follows it, it is about arbitrarily small differences.
This means that  [math]\delta x[/math] is not exact but may be made smaller and smaller indefinitely. So it is not a number it is a sort of function. Also since it is not a number it can never be zero, since zero is a number.

This brings us neatly to the idea of limits.

 

This is a sequence

[math]1,4,9,4,16,...[/math]

If we put in addition signs it becomes a series

[math]1 + 4 + 9 + 16...[/math]

And if we work out the differences between each term or the changes from term to term

[math]3\quad 5\quad 7[/math]

The three dots at the end is the convention for continuing indefinitely.

It can immediately be seen that all the values for both the sequence and series and for the differences increase with each additional term added so each get larger and larger.

This is called divergence and the sequence and series are called divergent.
Such series do not have limits.

 

If however we take the reciprocal of these sequences they get smaller and smaller, whilst the partial sums for the series gets closer and clsoer to a specific number (1.64)

[math]\frac{1}{1} + \frac{1}{4} + \frac{1}{9} + \frac{1}{{16}} + ...[/math]

[math]1 + 0.250 + 0.111 + 0.063...[/math]

[math]0.75\quad 0.139\quad 0.048[/math]

This behavious is called convergence.
The specific number is called the limit.

Because the process can continue indefinitely we write

[math]\mathop {\lim }\limits_{n \to \infty } \sum {\frac{1}{{{n^2}}}}  = 1.64[/math]

Next time, this will lead directly into the limits 

[math]\mathop {\lim }\limits_{\delta x \to 0} [/math]

and

[math]\mathop {\lim }\limits_{\delta x \to 0,\delta y \to 0} [/math]

 

So we have numbers, functions, variables, operators, limits sequences and series all mentioned before.

An easy way to look at operators are that simple functions work on numbers to output other numbers and operators work on simple functions to output other functions.

You will need to do some work  on these to use all the to get a handle on what calculus is, what it can do for you and what it can't do for you.

So please let us know any ideas in this development you didn't get a hold of properly so we can help correct that.

 

[CuriosOne]

1 hour ago, studiot said:

To continue where I left of and pick up and develop  joigus points about limits

 

 

The f(x) and y =y(x) notation comes in useful if you want to get it all onto one line so the derived functions become f'(x) and y'(x).

This is not directly related to any of the d notation variants and also becomes very clumsy when many variables are involved.

So let us look at the d notation, which is, after all, what you are asking about.

So we have 

Greek capital delta   Δ

Greek lower case delta   δ

Roman Capital D

Roman lower case d

Eighteenth century European stylised script  d    ∂

We have all of them because each perform a different operation (remember them) on the symbol that follows them.

The first one is not important to us because we use it for the difference between two specific values of the variables represented by the symbol that follows.

So the difference in height, Δh ,  between the ground and the gutterline of my house is 18 feet.   Δh may be large, but must be exact. That is it is a number. As it is a number it can legitimately be zero.

The second one is important to us because it is not only about small differences in the values of the variable that follows it, it is about arbitrarily small differences.
This means that  δx is not exact but may be made smaller and smaller indefinitely. So it is not a number it is a sort of function. Also since it is not a number it can never be zero, since zero is a number.

This brings us neatly to the idea of limits.

 

 

This is a sequence

1,4,9,4,16,...

If we put in addition signs it becomes a series

1+4+9+16...

And if we work out the differences between each term or the changes from term to term

357

The three dots at the end is the convention for continuing indefinitely.

It can immediately be seen that all the values for both the sequence and series and for the differences increase with each additional term added so each get larger and larger.

This is called divergence and the sequence and series are called divergent.
Such series do not have limits.

 

 

If however we take the reciprocal of these sequences they get smaller and smaller, whilst the partial sums for the series gets closer and clsoer to a specific number (1.64)

11+14+19+116+...

1+0.250+0.111+0.063...

0.750.1390.048

This behavious is called convergence.
The specific number is called the limit.

Because the process can continue indefinitely we write

limn→∞∑1n2=1.64

Next time, this will lead directly into the limits 

limδx→0

and

limδx→0,δy→0

 

So we have numbers, functions, variables, operators, limits sequences and series all mentioned before.

An easy way to look at operators are that simple functions work on numbers to output other numbers and operators work on simple functions to output other functions.

You will need to do some work  on these to use all the to get a handle on what calculus is, what it can do for you and what it can't do for you.

So please let us know any ideas in this development you didn't get a hold of properly so we can help correct that.

 

 

 

 

Very insightful thXxxx...

The definition of zero being a number now answers most of the point of calculus, "But" in regards to arbitrary small differences not being numbers, what are they??

functions??  I read??

My focus then would be on the operators on simple functions that output other functions not numbers as converged "things" whatever they connect to at this point.

From what I'm understanding it sounds like a terrasac hypercube with infinite limits of functions within functions of operators...Is this right?

If "things" are converged I can see why this would be the case in regards to a refference atlas like cartessian space and the "points" on it in 3d, hieght, width and depth..

Is this the way to think about it?

 

The Greek lower case delta   δ

has my attention now.

This Makes Sense ThnXxxx

The second one is important to us because it is not only about small differences in the values of the variable that follows it, it is about arbitrarily small differences.
This means that  δx is not exact but may be made smaller and smaller indefinitely. So it is not a number it is a sort of function. Also since it is not a number it can never be zero, since zero is a number.

This brings us neatly to the idea of limits.

Edited October 15, 2020 by CuriosOne

[cladking]

On 10/15/2020 at 2:31 AM, CuriosOne said:

 

But its said Newton used the pyramids to create calculus...Is this true??

 

 

No.  It's most probably not true.

Newton studied the pyramids largely because he thought the builders knew the size of the earth (probably true) and he needed the data to test his theory of gravity.  While he never found it he did ironically translate the Emerald Tablets of Hermes from Syriac to English.  

He was truly a remarkable man.  He couldda moved heaven and earth if he had google.  😎

[CuriosOne]

On 10/17/2020 at 6:09 PM, cladking said:

No.  It's most probably not true.

Newton studied the pyramids largely because he thought the builders knew the size of the earth (probably true) and he needed the data to test his theory of gravity.  While he never found it he did ironically translate the Emerald Tablets of Hermes from Syriac to English.  

He was truly a remarkable man.  He couldda moved heaven and earth if he had google.  😎

Sounds he was more an astrophysicist through mystic knowledge..

[Ni Mimi.]

Revisiting this 2020 topic, ‘What's The Point Of Calculus??’, well, Finitude and Infinitude’ve to be defined, hence the use, arbitrary one could say, of ‘0’ and ‘1’.

The ‘1’s illustrates the known while the ‘0’s the unknown; that is, what’s there associative factor?