Do points lie on tangent lines "only?"

[CuriosOne]

Do points lie on tangents lines "only?"

From:  "The slope of the tangent" 

Or on the curve itself??

If it's not on the curve, then:

Where did that curve come from??

I'm not getting the ideas behind the following.

 ( x + delta h) 

I'm very familiar with linear equations but this does not clarify tangent points and the "fancy" albegra doesn't explain the evolution of time either cuz it sets everything at 0...

Are these Points Hyper Planes?? Light Cones?? Faster Than Light Speeds?? N Gons?? Oragami?? 

----->Standing Waves Maybe???

Edited December 20, 2020 by CuriosOne

[ahmet]

what you ask does not seem clear to me.

but basically, the slope of tangent is the value of derivation function at the point where tangent is drawn. this is valid for functions with 1 variable.

but for functions with  several variables, we prefer differentiation instead of derivation. 

14 minutes ago, CuriosOne said:

Are these Points Hyper Planes?? Light Cones?? Faster Than Light Speeds?? N Gons?? Oragami?? 

this seems to me: meaningless and irrelevant..sorry.

Edited December 20, 2020 by ahmet
spelling error,replacement of conjunction,adding explanation.

[joigus]

The word "point" in itself does not tell you what it is.

A point on the real line: \( x \in \mathbb{R} \)

A point on the real plane \( \left(x,y\right) \in \mathbb{R}^2 \)

... etc.

Edited: A point is a locus, location, place in a set. When you say "point" normally you imply some kind of position (distance-->geometry, topology...). When you say "element" or "member" you normally imply just set theory.

There's context missing. And as Ahmet suggested, "light cones", "faster than light", "hyperplanes"... That has nothing to do with your drawing or the concept of points.

The impression I get is, again, you're trying to connect too much in one simple concept. Points don't need light in order to be defined.

Edited December 20, 2020 by joigus
Addition

[CuriosOne]

9 hours ago, joigus said:

The word "point" in itself does not tell you what it is.

A point on the real line: x∈R

A point on the real plane (x,y)∈R2

... etc.

Edited: A point is a locus, location, place in a set. When you say "point" normally you imply some kind of position (distance-->geometry, topology...). When you say "element" or "member" you normally imply just set theory.

There's context missing. And as Ahmet suggested, "light cones", "faster than light", "hyperplanes"... That has nothing to do with your drawing or the concept of points.

The impression I get is, again, you're trying to connect too much in one simple concept. Points don't need light in order to be defined.

Whatever a tangent line is, that's what I mean..

The points of (x + delta h)

In regards of these 2 points are:

"2 points" on the ""tangent line?"" 

Linear

Or 2 points on the curve itself

Not linear 

I'm not sure due to how calculus was created for something constantly changing at some point in time....

Edited December 21, 2020 by CuriosOne

[ahmet]

4 hours ago, CuriosOne said:

Whatever a tangent line is, that's what I mean..

The points of (x + delta h)

In regards of these 2 points are:

"2 points" on the ""tangent line?"" 

Linear

Or 2 points on the curve itself

Not linear 

I'm not sure due to how calculus was created for something constantly changing at some point in time....

I think you are confused even very basic instructions. Joigus tried to "politely" provide basic notations or kindly tried to ensure you understand that some keywords you used in your previous comment were incorrectly used or not understood by you. 

But to better help you, i can suggest that you first know;

_ mathematics is a discipline 

_ first of firsts please ensure how maths could be better studied ( big clue: please study by writing, and feel yourself as you responsible for everything you express, in progress you should also accept that you must prove everything you say in every step of proofs. 

 

_  you need to be patient, but hardworking or having regular work ( everyday please at least 1_2 hour per day in average with the exception of course you take (if any) 

 

_ internalize first basic descriptions and thorems. 

 

In progress you can either run or fly depending your dedication or work manner. 

 

Good luck

ahmet

[swansont]

!

Moderator Note

“mathematics is a discipline”

One could also say mathematics is a language, with its own version of syntax, vocabulary and rules for spelling, etc.

In order to have useful discussion you can’t make up your own words and definitions. You have to learn and use the ones everybody else is. You can’t jump into a conversation without knowing these basics.

This concept also applies to math.  

Perhaps it would be better to discuss more fundamental concepts first. Plenty of people have stepped up to try and help but it’s frustrating when there is such a barrier to communication

 

[MigL]

If that is from the introduction to differential calculus I gave you two weeks ago …

1 - I would have thought you'd be further along by now.

2 - At every point on the curve of the function, you can draw a tangent line, such that 1 point ( only ) is common to both.
The slope of that tangent line, at that point, is equivalent to the derivative of that function ( with respect to its variable ), at that point.
This allows you to have the slope/derivative at a single point, as opposed to F(x₁)-F(x₂)/x₁-x₂, which gives you an 'average' over the multiple points included in between the two values of x.

3 - Points on the line/curve of a function can represent a lot of things, or, none at all.

4 - I'm glad you're asking questions, and not making assertions.

[CuriosOne]

9 hours ago, ahmet said:

I think you are confused even very basic instructions. Joigus tried to "politely" provide basic notations or kindly tried to ensure you understand that some keywords you used in your previous comment were incorrectly used or not understood by you. 

But to better help you, i can suggest that you first know;

_ mathematics is a discipline 

_ first of firsts please ensure how maths could be better studied ( big clue: please study by writing, and feel yourself as you responsible for everything you express, in progress you should also accept that you must prove everything you say in every step of proofs. 

 

_  you need to be patient, but hardworking or having regular work ( everyday please at least 1_2 hour per day in average with the exception of course you take (if any) 

 

_ internalize first basic descriptions and thorems. 

 

In progress you can either run or fly depending your dedication or work manner. 

 

Good luck

ahmet

You obviously never taken a coarse on ciphers... 😎

I work every day and am physically fit if that's what you mean, I'm also a former model and music producer, in fact I know many celebrities and have contributed over $500.000 in free work labor, I'm a very respected artist as well "despite" the bogus reputation I've gained here...😎

Now, as far as proofs go, proof depends on a model, and there are many "i assure you" unfortunately at the sole discretion the creator,  "I assume nature is included" and assume discoveries have something to do with it..."chuckles."

Your correct on basic instructions due to how science refers to most if not all things as ""points"" then link these to "variables" to make things more complicated....

IE:

Point particles

Points

"DOT" product

A->B

GM1*m2/ r^2 " From" center of masses" 

IE The Masse's "Points"

Then we use math and the concept of "models??"

A Theoretical Model is not 100% absolute, so there goes your proof...But they can work quite well and are "reliable."

 

Now, The OP asks if x and dx are points on a "tangent line."

 

Yes or No??

Edited December 21, 2020 by CuriosOne

[ahmet]

19 minutes ago, CuriosOne said:

I work every day and am physically fit if that's what you mean, I'm also a former model and music producer, in fact I know many celebrities and have contributed over $500.000 in free work labor, I'm a very respected artist as well "despite" the bogus reputation I've gained here...😎

honestly, your achievements are yours and no relevancy with me. 

but...it seems even if you are not as knowledgeable as @swansont, to me you are more intelligent than him. sure. because in spite of the failures I had done, you have understood the message what intended. so , congratulations. (and to swansont, I predict, I will have published materials also in english, so don't bother yourself. you are wrong. 

the only disadvantage ,I had wrote that text with my phone and I was on a urgent way,so I could not make any reform. sorry for that. 

anyway, lets go on the topic 

you claim or it is being understood that you mention that two points would be linear or not linear.

this is wrong. 

linear is a description not just for two points , but for functions. (to use just for two point is wrong)

shortly, a linear function would be the  property of a function that satisfies

[math]f(tx+my)=tf(x)+mf(y) [/math] where t and m are scalar and x and y are vectors. (f is a function/operator on a vector space)

 

Edited December 21, 2020 by ahmet

[CuriosOne]

34 minutes ago, MigL said:

If that is from the introduction to differential calculus I gave you two weeks ago …

1 - I would have thought you'd be further along by now.

2 - At every point on the curve of the function, you can draw a tangent line, such that 1 point ( only ) is common to both.
The slope of that tangent line, at that point, is equivalent to the derivative of that function ( with respect to its variable ), at that point.
This allows you to have the slope/derivative at a single point, as opposed to F(x₁)-F(x₂)/x₁-x₂, which gives you an 'average' over the multiple points included in between the two values of x.

3 - Points on the line/curve of a function can represent a lot of things, or, none at all.

4 - I'm glad you're asking questions, and not making assertions.

This is exactly what I needed to "be sure of" !!!!!!!! THNX...

The average "between" the multilple points, was what I was "deducing" either by mistake or intuition, calculus is pretty remarkable I must say...Thanks  again for the PDF...

Its a great way to create models of all sorts that can be of some reliability..

 

 

 

[CuriosOne]

1 hour ago, ahmet said:

honestly, your achievements are yours and no relevancy with me. 

but...it seems even if you are not as knowledgeable as @swansont, to me you are more intelligent than him. sure. because in spite of the failures I had done, you have understood the message what intended. so , congratulations. (and to swansont, I predict, I will have published materials also in english, so don't bother yourself. you are wrong. 

the only disadvantage ,I had wrote that text with my phone and I was on a urgent way,so I could not make any reform. sorry for that. 

anyway, lets go on the topic 

you claim or it is being understood that you mention that two points would be linear or not linear.

this is wrong. 

linear is a description not just for two points , but for functions. (to use just for two point is wrong)

shortly, a linear function would be the  property of a function that satisfies

f(tx+my)=tf(x)+mf(y) where t and m are scalar and x and y are vectors. (f is a function/operator on a vector space)

 

Ok so now you have my attention...

So, are these "scalers" "sharing" a constant?

Edited December 21, 2020 by CuriosOne

[swansont]

1 hour ago, CuriosOne said:

You obviously never taken a coarse on ciphers... 😎

I work every day and am physically fit if that's what you mean, I'm also a former model and music producer, in fact I know many celebrities and have contributed over $500.000 in free work labor, I'm a very respected artist as well "despite" the bogus reputation I've gained here...😎

Now, as far as proofs go, proof depends on a model, and there are many "i assure you" unfortunately at the sole discretion the creator,  "I assume nature is included" and assume discoveries have something to do with it..."chuckles."

Your correct on basic instructions due to how science refers to most if not all things as ""points"" then link these to "variables" to make things more complicated....

IE:

Point particles

Points

"DOT" product

A->B

GM1*m2/ r^2 " From" center of masses" 

IE The Masse's "Points"

Then we use math and the concept of "models??"

A Theoretical Model is not 100% absolute, so there goes your proof...But they can work quite well and are "reliable."

 

Now, The OP asks if x and dx are points on a "tangent line."

 

Yes or No??

!

Moderator Note

This last bit is question raised, but the bulk of this post has nothing to do with it. Staying on-topic and focused is another issue that should improve everybody’s experience

 

14 minutes ago, CuriosOne said:

Ok so now you have my attention...

So, are these "scalers" "sharing" a constant?

!

Moderator Note

This is not the topic of discussion. Whatever you mean by “sharing a constant” this belongs in a different thread. Whatever your lack of understanding is, it’s more fundamental than the topic of the OP.

 

[CuriosOne]

13 minutes ago, swansont said:

!

Moderator Note

This last bit is question raised, but the bulk of this post has nothing to do with it. Staying on-topic and focused is another issue that should improve everybody’s experience

 

!

Moderator Note

This is not the topic of discussion. Whatever you mean by “sharing a constant” this belongs in a different thread. Whatever your lack of understanding is, it’s more fundamental than the topic of the OP.

 

I think I totally agree with you..

No doubt about that...

3 hours ago, swansont said:

!

Moderator Note

“mathematics is a discipline”

One could also say mathematics is a language, with its own version of syntax, vocabulary and rules for spelling, etc.

In order to have useful discussion you can’t make up your own words and definitions. You have to learn and use the ones everybody else is. You can’t jump into a conversation without knowing these basics.

This concept also applies to math.  

Perhaps it would be better to discuss more fundamental concepts first. Plenty of people have stepped up to try and help but it’s frustrating when there is such a barrier to communication

 

Or mathinatics is not noble, in a sense to being a language..

[ahmet]

   

just smiling...

[wtf]

5 hours ago, MigL said:

At every point on the curve of the function, you can draw a tangent line, such that 1 point ( only ) is common to both.
The slope of that tangent line, at that point, is equivalent to the derivative of that function ( with respect to its variable ), at that point.

I wish to refine this statement because it's a common point of confusion.

You have no proof that "At every point on the curve of the function, you can draw a tangent line, such that 1 point ( only ) is common to both," nor do you have a rigorous definition of what a tangent line is.

Rather, we have an INTUITION about what a tangent line is. In order to make the notion rigorous, we DEFINE the tangent line at a point to be the straight line passing through that point with slope equal to the derivative at that point, if the derivative exists.

That is, the the slope of the tangent line is NOT "equivalent" to the derivative; rather, it's DEFINED that way. The idea is to make precise the intuitive idea of the tangent line at a point. If you think (as students often do) that the derivative is "the same" as the slope of the tangent line, that's a misunderstanding of what's going on. There is no tangent line, formally, until we define it via the derivative.

Then (for example) we can make rigorous the intuitively clear observation that the graph of |x| has no tangent line at 0. Otherwise, we could have no proof, since without the derivative we have only an intuitive but not a rigorous notion of tangent line.

Edited December 21, 2020 by wtf

[MigL]

WTF ??? 
( just playing  with your name )

Had I been talking to you, I would have been more careful with my wording and definitions.
I don't think CuriosOne got anything from your 'refinement'.

[wtf]

6 minutes ago, MigL said:

I would have been more careful with my wording and definitions.

Nothing personal, I just happened to have run across this exact issue on some other forum a day or two ago so it was fresh in my mind.

Edited December 21, 2020 by wtf

[MigL]

I understand ( and appreciate ) your efforts.

[joigus]

If I may say something... I was aware that @wtf was giving a superb mathematician's exposition of the topic, while @MigL who had had some previous experience with the OP, was quite deliberately trying to dumb it down. It was fun seeing you interact.

But your effort is not in vain, wtf. Thank you. I appreciate it.

[MigL]

2 hours ago, joigus said:

It was fun seeing you interact.

Glad you got a laugh out of it  .

[ahmet]

5 hours ago, joigus said:

I was aware that @wtf was giving a superb mathematician's exposition of the topic,

I think he is classsical mathematician. In fact, I can't see an extraordinary mathematician at anywehere (all around the world). (i.e. superb)

but this will be another thread. 

lets start a new thread for that if you are willing.

Edited December 22, 2020 by ahmet

[joigus]

4 hours ago, ahmet said:

I think he is classsical mathematician. In fact, I can't see an extraordinary mathematician at anywehere (all around the world). (i.e. superb)

but this will be another thread. 

lets start a new thread for that if you are willing.

I would have to be on top of the hill to look down on others here to evaluate them. I'm not in such position.

How smug would I be if I did?

I stand by my words: A superb explanation --especially considering the limited amount of time and manoeuvre we all have here-- by a person well versed in mathematics who has all my respect. MigL's explanation was also very helpful, although in a very different style and spirit.

As to your kind offering of starting another thread, I'm not so interested in judging people as in examining ideas, and trying to understand some of the most difficult ones. But you're free to open that thread if you want.

Here's smiling at you 

[ahmet]

1 hour ago, joigus said:

I would have to be on top of the hill to look down on others here to evaluate them. I'm not in such position.

How smug would I be if I did?

I stand by my words: A superb explanation --especially considering the limited amount of time and manoeuvre we all have here-- by a person well versed in mathematics who has all my respect. MigL's explanation was also very helpful, although in a very different style and spirit.

As to your kind offering of starting another thread, I'm not so interested in judging people as in examining ideas, and trying to understand some of the most difficult ones. But you're free to open that thread if you want.

Here's smiling at you 

peheh , pahah    

to me, this is just waffling ...

[studiot]

On 12/20/2020 at 6:55 PM, CuriosOne said:

Do points lie on tangents lines "only?"

From:  "The slope of the tangent" 

Or on the curve itself??

If it's not on the curve, then:

Where did that curve come from??

I'm not getting the ideas behind the following.

 ( x + delta h) 

I'm very familiar with linear equations but this does not clarify tangent points and the "fancy" albegra doesn't explain the evolution of time either cuz it sets everything at 0...

Are these Points Hyper Planes?? Light Cones?? Faster Than Light Speeds?? N Gons?? Oragami?? 

----->Standing Waves Maybe???

 

On 12/21/2020 at 4:26 AM, CuriosOne said:

Whatever a tangent line is, that's what I mean..

 

Righto, I stopped bothering with your posts since you asked about a lot of terminology in another recent thread of yours.
When I offer stuff you di not answer.

But I see some of those terms appearing here.

Also, with respect, others have made this far too complicated.
Probably because you have brought in a lot of unconnected material.

 

Tangents were described and investigated about 2000 years before coordinate geometry and calculus were invented.
They do not possess a slope as an inherent property.

The meaning is very simple.

In the plane geometry of Euclid.

Take any line PQ    -  note in the geometry of Euclid a line means a straight line it does not mean a curve.

Draw lines AB, CD, EF etc parallel to The original line OP and all in the same plane, so that they intersect various curves.

I have shown this for two curves, a circle or closed curve, and an open curve.

AB, CD, EF all intersect the circle at exactly 2 points. No more and no less.

All these lines are called secants.

You can see that as the sequence of parallel lines is drawn closer and closer to the bottom of the circle
The distance between the two points of intersection become closer and closer together

Until the line T₁T₂ intersects the circle in only one single point.

The line T₁T₂ is called the Tangent to the curve at the point of intersection.

You should draw some more lines like PQ but not parallel to it and convince yourself that each such sequence of lines has only one intersection point with the circle
and that every point on the circle has such a tangent line associated with it.

 

Now look at an open curve.

You can see that a similar sequence of parallel lines only works like this if you are close enough to the point of single intersection.
If you extend the curve and lines far enough you may find more than two points of intersection.
But if you keep close enough there is a sequence of exactly two intersection points, narrowing down to a unique point of single intersection.

Again the lines with two points of intersection are called secants and the line with one point only is called the tangent to the curve at that point.

 

The idea of intersection - where and how lines curves and geometric ficures intersect plays a huge part in the geometry of Euclid and interestingly re-appears in more modern topology.
It is a very simple but powerful technique.

 

Note again that I have not needed to mention the word slope.

Yes, in coordinate geometry tangents can have a slope but that is another story.

 

Edited December 22, 2020 by studiot

[joigus]

2 hours ago, ahmet said:

peheh , pahah    

to me, this is just waffling ...

🎬

To waffling with waffling. 

peheh, pahah, poohooh.   

To me, to you, to us, is not significant. Unless disclaimers, caveats or qualifications are applied. 🎬