What's The Point Of Calculus??

[CuriosOne]

We are taught that calculus is all about "instantaneous changes" between 2 points, x and delta x in respect to time with x = 0 and delta x =1

For example, when we set x = 0, "are we "manipulating" time , ie stopping at a certain location " going a certain distance" as to derive the instantaneous change at that point " location" in time?

If that's the case for our x = 0 "reference point" then what's the point of calculus and its "undefined location?"

Another example, if x = 0 and delta x = 1, " what's the point of coordinate vectors?" example x = 3 y = 9??

Another good example, if Cartesian Space references 0 in the coordinate system -2, -3, -1, 0, 1 , 2 ,3 then what's the point of Cartesian Space itself?

Take note: I'm using 0 and 1 here as interchangeable.

Thanks in advance!

 

 

 

Edited October 11, 2020 by CuriosOne

[mathematic]

Most of physics and much engineering needs calculus.  I suggest you learn a little bit about the subjects you are asking about.

[joigus]

I don't understand what you mean. Can you rephrase?

On a humorous note, the point of calculus is x.

[MigL]

Sure came in handy for I Newton.

[Charles 3781]

Isn't the point of calculus this  -  only proper mathematicians can understand it.

[joigus]

The point of calculus is studying change and balances of change.

The point of Cartesian coordinates is to parametrize space.

[CuriosOne]

1 hour ago, mathematic said:

Most of physics and much engineering needs calculus.  I suggest you learn a little bit about the subjects you are asking about.

Ive studied these fields for 20 years since a child, and yet I have absolutely no understanding of its use...I'm glad to be honest becuase true scientist use caution when using ancient tools and derived quantities from who knows what...Who invented calculus anyways??? No one truly knows,  thats an honest point..

However, can u show me a "real" world phyiscal example? And I think we can dissect the questions from there on..

1 hour ago, joigus said:

I don't understand what you mean. Can you rephrase?

On a humorous note, the point of calculus is x.

So your saying anything connected to x is zero???

Thanks I am laughing now...lolol

[Sensei]

4 minutes ago, CuriosOne said:

Ive studied these fields for 20 years since a child, and yet I have absolutely no understanding of its use...I'm glad to be honest becuase true scientist use caution when using ancient tools and derived quantities from who knows what...Who invented calculus anyways??? No one truly knows,  thats an honest point..

However, can u show me a "real" world phyiscal example? And I think we can dissect the questions from there on..

https://en.wikipedia.org/wiki/Area_of_a_circle

"Modern mathematics can obtain the area using the methods of integral calculus or its more sophisticated offspring, real analysis."

e.g. you can calculate area of any shape using calculus..

"Using calculus, we can sum the area incrementally, partitioning the disk into thin concentric rings like the layers of an onion. This is the method of shell integration in two dimensions. For an infinitesimally thin ring of the "onion" of radius t, the accumulated area is 2πt dt, the circumferential length of the ring times its infinitesimal width (one can approximate this ring by a rectangle with width=2πt and height=dt). This gives an elementary integral for a disk of radius r."

 

[studiot]

2 hours ago, CuriosOne said:

We are taught that calculus is all about "instantaneous changes" between 2 points, x and delta x in respect to time with x = 0 and delta x =1

For example, when we set x = 0, "are we "manipulating" time , ie stopping at a certain location " going a certain distance" as to derive the instantaneous change at that point " location" in time?

If that's the case for our x = 0 "reference point" then what's the point of calculus and its "undefined location?"

Another example, if x = 0 and delta x = 1, " what's the point of coordinate vectors?" example x = 3 y = 9??

Another good example, if Cartesian Space references 0 in the coordinate system -2, -3, -1, 0, 1 , 2 ,3 then what's the point of Cartesian Space itself?

Take note: I'm using 0 and 1 here as interchangeable.

Thanks in advance!

 

 

 

I doubt that many teachers would claim that Calculus is 'all about' anything.

There are many calculi (the plural of calculus) and each has a different purpose.

First there is the hint in the name 'calculus'  = that which enables calculation.

What you have described might be said as the introduction to what is known as 'the differential calculus'.

This is indeed about change from one point to another.
That point may be in time or place or connected to some other measurable quantity such as money or population or wheat yield or many other things.

But there is so much more.

There is 'the integral calculus', which is not about change but about summation.
This is about what happens when you combine things together such as the energy you consume when you run 8 kilowatt shower for 30 minutes.

Then there is the calculus of variations which is about minimising or maximising something.
For instance what shaped curve will result in an object slding down it the fastest ie in the least possible time.
This shape is known as a brachistochrone curve, if you want to look it up.
MigL mentioned Newton  -   he allegedly solved this one is a day using calculus.

Finally here is a very simple calculus, known as Russian Peasant Multiplication
for calculating the product of two numbers. say 64 x 61

Write the two numbers down at the top of two adjacent columns
Successively half the first column and double the second until 1 is reached in the halving clumn
Strike out all pairs with even numbers in the first column
Add those left in the doubled column thus

64          61
32         122
16         244
8           488
4           976
2          1952
1           3904

answer 64 x 61 = 3904  done without learning tables.

 

[CuriosOne]

13 minutes ago, studiot said:

I doubt that many teachers would claim that Calculus is 'all about' anything.

There are many calculi (the plural of calculus) and each has a different purpose.

First there is the hint in the name 'calculus'  = that which enables calculation.

What you have described might be said as the introduction to what is known as 'the differential calculus'.

This is indeed about change from one point to another.
That point may be in time or place or connected to some other measurable quantity such as money or population or wheat yield or many other things.

But there is so much more.

There is 'the integral calculus', which is not about change but about summation.
This is about what happens when you combine things together such as the energy you consume when you run 8 kilowatt shower for 30 minutes.

Then there is the calculus of variations which is about minimising or maximising something.
For instance what shaped curve will result in an object slding down it the fastest ie in the least possible time.
This shape is known as a brachistochrone curve, if you want to look it up.
MigL mentioned Newton  -   he allegedly solved this one is a day using calculus.

Finally here is a very simple calculus, known as Russian Peasant Multiplication
for calculating the product of two numbers. say 64 x 61

Write the two numbers down at the top of two adjacent columns
Successively half the first column and double the second until 1 is reached in the halving clumn
Strike out all pairs with even numbers in the first column
Add those left in the doubled column thus

64          61
32         122
16         244
8           488
4           976
2          1952
1           3904

answer 64 x 61 = 3904  done without learning tables.

 

This is when things start to "Clash" and get messy and confusing...Sure there are many dialects and interpretations to calculus as there is to number theory...

This still does not answer why x can never be =1, and it makes sense becuase if x =1 - dx=1 then it all =0 which is "undefined" that's the point I really want to make, it's this statement that asks, "What's the point of Calculus?" not my words rather the "rules"..

It makes no sense, and I'd love to understand this..

Edited October 12, 2020 by CuriosOne

[studiot]

5 minutes ago, CuriosOne said:

This still does not answer why x can never be 1

That wasn't the question you asked.

 

5 minutes ago, CuriosOne said:

It makes no sense, and I'd love to understand this.

 

If you are serious about understanding please try to pose you question simply but completely and without mockery.
They someone here may be able to help you.

I have no idea what you mean by "x can never be 1"  
Can you give a complete example that confuses you?
I suggest you start by saying what x is.

Edited October 12, 2020 by studiot

[CuriosOne]

1 minute ago, studiot said:

That wasn't the question you asked.

 

 

If you are serious about understanding please try to pose you question simply but completely and without mockery.
They someone here may be able to help you.

I have no idea what you mean by "x can never be 1"  
Can you give a complete example that confuses you?
I suggest you start by saying what x is.

"Maybe" I should have said:

f(x) As delta x->0 towards the slope of the secant line relative to the tangent line..

But x never actually gets close to zero..I learned this 10 years ago...Does that make better sense? 

Book Reference: Calculus The Easy Way, Chapter 1 page 8.

Sure there are places where x=0 within a curve, but I will save that for a later thread.

 

2 hours ago, MigL said:

Sure came in handy for I Newton.

So we should assume centuries later its handy for us??

[MigL]

32 minutes ago, CuriosOne said:

So we should assume centuries later its handy for us??

What's really sad is that, centuries later, you still don't understand it.
Maybe you should have some humble pie with your Thanksgiving dinner; ask proper questions and stop being so derisive.

[CuriosOne]

2 hours ago, MigL said:

What's really sad is that, centuries later, you still don't understand it.
Maybe you should have some humble pie with your Thanksgiving dinner; ask proper questions and stop being so derisive.

I didn't say I don't understand it, I just don't understand "tradition."

Its tradition to use calculus "centuries old" math tool "contribution of Newton I assume" to derive spontaneous changes of something x "accelerating" from rest mass." Again x=0, really what's the point?

"That's just one example" of the many when dealing with masses and gravity "freely flying objects" due to some force or force of attraction, be it electrical forces or gravitational forces that create motion."

And yet gravity is a macro external force? I assume Newton knew before we all did, whom knows..

"In my opinion and observation of calculus itself" The whole things works just fine, until light photons start acting "strange" as particles and waves with their own unique frequencies of "pie."

Note:

""The wave function used in quantum mechanics for example uses pi extensively, in fact they make focus the use of sin and cos respectfully.""

 

Is that why x = 0?

does this  mean massless something?? It makes sense to believe so, unless someone can explain x=0 when dealing with accelleration and rates of changes or x= anything 0 at all at this point..

From what I see if their is no gravitational force or any force involved our x stays put and does nothing at all??

These are "ideas" like the ideas of Eistein, Plank and the others, they seem to work and have withstood the test of time, but does not explain the most simple questions.

 

Edited October 12, 2020 by CuriosOne

[studiot]

6 hours ago, CuriosOne said:

Sure there are places where x=0 within a curve, but I will save that for a later thread.

 

3 hours ago, CuriosOne said:

Is that why x = 0?

 

7 hours ago, CuriosOne said:

So your saying anything connected to x is zero???

 

3 hours ago, CuriosOne said:

Is that why x = 0?

 

7 hours ago, CuriosOne said:

This still does not answer why x can never be =1

 

6 hours ago, CuriosOne said:

Book Reference: Calculus The Easy Way, Chapter 1 page 8.

 

I downloaded the third edition and I can't say I'm suprised you are confused if you are really using this book to learn the differential calculus.

In particular my edition does not contain 'x' on page 7, 8 or 9.

 

I asked you what you thought 'x' is because all those comments of yours that I have quoted suggest to me you don't understand this.

It is vital that you do understand since 'x' is not a number.

In arithmetic/algebra 'x' often stands for an unknown number that is to be determined

In calculus 'x' is something else entirely and the book uses it in this way, but does not explicitly make this clear.

'x' is a variable.

So that is where we should start.

Do you wish to go on ?

[Sensei]

3 hours ago, CuriosOne said:

does this  mean massless something??

"Massless" means "having no rest-mass".. i.e. "having no frame of reference in which particle is at rest".

In the case of massive particle e.g. electron or proton, you can use external forces like electrostatic, to keep particle at rest. In electromagnetic trap.

[John Cuthber]

What's The Point Of Calculus??

It's so we can help our kids with their calculus homework.

[J.C.MacSwell]

...or if you have a signer for your student loan for your math degree...but need to derive a cosigner....

Edited October 12, 2020 by J.C.MacSwell

[HallsofIvy]

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.

[CuriosOne]

1 hour ago, 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.

Understood, but one thing to consider is that all these situations follow a curve path, the points referencing "changes" need to lie on the path or curve itself otherwise calculus would render itself useless..I should have asked what's the point of cartessian space....

The distance example I assume holds only true for "linear motion" but really I was never sure of that after learning linear algebra and signal processing.

The point here is that in an f(x) function the linear coorididents of x and y connect to the curve points of x and dx on the "curve" path "in cartessian linear coordinates" that does not make sense. 

But VIOLA!

Maybe That's Why Calculus Was Invented And Maybe My Thinking Is Correct! 

ButI still, if x can never be = 0 does all this explain my confusion?

Note: I think x can never be =1 in my opinion.

[MigL]

Still not sure what exactly you mean by x cannot equal zero.
Are you talking about limits ( as x -> 0 ) ?

[CuriosOne]

On 10/12/2020 at 1:18 AM, studiot said:

 

 

 

 

 

 

I downloaded the third edition and I can't say I'm suprised you are confused if you are really using this book to learn the differential calculus.

In particular my edition does not contain 'x' on page 7, 8 or 9.

 

I asked you what you thought 'x' is because all those comments of yours that I have quoted suggest to me you don't understand this.

It is vital that you do understand since 'x' is not a number.

In arithmetic/algebra 'x' often stands for an unknown number that is to be determined

In calculus 'x' is something else entirely and the book uses it in this way, but does not explicitly make this clear.

'x' is a variable.

So that is where we should start.

Do you wish to go on ?

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?

4 minutes ago, MigL said:

Still not sure what exactly you mean by x cannot equal zero.
Are you talking about limits ( as x -> 0 ) ?

Yes, in regards to limits..

[MigL]

2 minutes ago, CuriosOne said:

Yes, in regards to limits..

You should have said that to begin with.
There would have been a lot less confusion.

Limits approach a limiting value; they never actually reach it.

[CuriosOne]

18 minutes ago, MigL said:

You should have said that to begin with.
There would have been a lot less confusion.

Limits approach a limiting value; they never actually reach 

I was told there is a big difference between variables and numbers...

Variables change, numbers are just numbers.

A limit has a value, and x can never = 0

So 0 "does" have a numerical value? It must if it plans to reach a known numerical limit of observation.

""Who decides this?""

Make note, we are following a curve here..

My entire perspective has changed, now....

Edited October 13, 2020 by CuriosOne

[studiot]

36 minutes ago, CuriosOne said:

I was told there is a big difference between variables and numbers...

Variables change, numbers are just numbers.

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.

38 minutes ago, CuriosOne said:

A limit has a value, and x can never = 0

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.

Quote

Smith  : Vector Analysis : Oxford University Press.

 

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.