Algebraic Operations on Differentials in Liebniz Notation: An Abuse?

[jgens]

Well, having read through your newest installment relatively thoroughly, I have a few suggestions which you (and others) may or may not agree with:

1) Omit the discussion of infinitesimals. While this new version does give a slightly more adequate treatment of infinitesimals than previous versions, I still think that it leaves too much open for misconception and misunderstanding. Though a completely rigorous treatment of infinitesimals could probably be forestalled until the student has more mathematical maturity, I still think that the teacher/professor/instructor needs to work out several of the properties of infinitesimals (or carefully guide them there) to avoid misunderstanding the concepts. Drawing from an earlier example, why shouldn't a student take, Lim_{x -> +∞} (1/x) as the definition of an infinitesimal? In which case the student would find that Lim_{x -> +∞} (1/x) = ε. As Hurkyl pointed out earlier, this isn't the case, but the inquisitive student doesn't know that!

Since a lot of what you're introducing seems to be along the lines of differentials, your discussion of the derivative could probably stand without infinitesimals.


2) Assuming that the student is not familiar with derivatives, when you're introducing the geometric interpretation of the derivative, place more emphasis on the derivative as the limiting secant line. Your approach to do this with triangles works pretty well, but depending on the background of the student, may seem superfluous. Reorganize the discussion so that you develop the limit definition of the derivative and then define dy/dx = ∆y_tan/∆x_tan in terms of differentials as jambaugh posted earlier. This way, you remain consistent with the notation of calculus (using dy and dx instead of ∆y_tan and ∆x_tan) and you develop the derivative as a quotient of differentials rather than a ratio of infinitesimals. You may also want to mention that, Lim_{x -> a} [(f(x) - f(a))/(x-a)] is a perfectly acceptable definition of the derivative.

As an aside, I take issue with the statement that the derivative is not the limit of a quotient, especially since the derivative is defined in terms of limits. While you do argue that we could simply define the derivative in terms of tangent gradients, this provides no way to actually calculate the derivative. Additionally, by placing an inordinate focus on derivatives as the tangent gradient the student is led away from important concepts like the derivative as a function. Even though the derivative can be defined in terms of differentials, they don't provide a method for calculating derivatives. For these reasons, I still think that it's best to define - at least initially - the derivative as the limit of a quotient.

Hopefully you'll find these comments helpful!

 

[chrisr999]

hi jgens,
have a look at the last few pages of this updated discourse.
It will show how infinitesimals relate to the real number system.
Again, they vary in dimension and it is their ratios that ultimately matter.
Their exact ratio is obtained from the linear function.
Their varying ratio is what calculus eliminates.
chris

 

[chrisr999]

There's enough information in version 5 to answer all but one of those questions, jgens,
it can be worded differently for different students of various levels but it really is at an elementary enough level for young students.

I haven't discussed any of the mathematical techniques at all! hardly anyway!
but that becomes quite easy to do from here,
though you should know by now, there is no division by zero involved even if many get that impression before fully examining the geometry.

You're going to have to apply yourself though! to break through it.
I will be busy for a week.
sincerely,
chris

 

[chrisr999]

tell you what,
if i have time, i will work through an example for you, jgens,
if you have one that's really perplexing,
i will do a tutorial on it,
using mathematics only without the geometry!

 

[chrisr999]

I understand your position on the infinitesimals, jgens,
and, ok, it would be only appropriate to do that for you, I'm sorry i have quite a few things to do at the moment, however as you've seen, there are very proficient guys on this thread capable of lighting up the darkness in their unique ways and also from the perspective of accurate mathematical terms.

What you need is to find a way to handle these "units" that is very clear for you, where the words are expressed in your preferred learning modalities.

Let's say someone wanted to know what a papaya tastes like and they'd never encountered one. They know what it tastes like through experiencing it and could then describe it. But if i didn't have one to give them and started describing it to them, that would deny their own experience of it and would always be an inaccurate description, it would approach the true sensation without ever being completely accurate.

This is why I gave the experience of them before any description.
To me, they are "a pair of orthogonal nano-axes that do not cross" and their function is to zoom in on the point where you want to find the rate of change of a function where the gradient is measurable. The tangent is the other geometric tool.
You analyse a one-point situation with 2 points initially and zoom in on your point of interest, until your infinitesimals do not distinguish between the right and wrong value of the derivative. They have whatever length they have in that scenario.
You then zoom out, allowing these infinitesimals attach to the straight line.
Their lengths are real values, though not relevant. Their ratio during the zoom-out is relevant.
They are "tools" of geometric analysis.
You've got to have a sense of them, not a definition.
You can define them as you please after experiencing them.

No, the limit of 1/x as x approaches infinity is not an infinitesimal.
You are not paying attention to the graph of the curve!
The infinitesimals in that case are doing something I haven't discussed in the little piece i wrote but Hurkyl was showing you just how interesting it all is and these are non-complex examples.
For 1/x, there is one infinitesimal, because the gradient has no measure at the limit, it's zero.
That infinitesimal is the vertical one, the horizontal one is increasing out of bounds as the vertical one reduces to zero, but it really does not stop reducing!
i couldn't call the horizontal one "infinitesimal" as it's increasing to infinity.
Can you visualise it? If not, draw it.
The tangent is the x-axis which has a gradient of zero as the triangle I used has "melted" completely when we can't visually tell the difference between the axis and curve.
We don't have a final "measureable ratio" for situations in which the x and y axes are the tangents and also the point of intersection lies at unreachable infinity.
This is the case for "discontinuous" functions.
They are perfectly analyseable but require additional definition, as you say, a "rigorous" one, for completeness, but students can easily extrapolate them when they've got the spirit of the analysis.
We either have a continuous or discontinuous function.
If you like, you can define rigorous definitions for both cases, it shouldn't really be necessary though.
chris

 

[0xDEADBEEF]

Leibnitz

When do Americans learn it's Leibnitz and Wiener. English "i" is German "ei" or "ai" and English "e" is German "i" or "ie".

 

[jgens]

I do hope you realize my criticisms are not because I do not understand standard calculus or specific properties of numbers! However, based on most, if not all, of your comments directed towards me I don't get that impression.

In which case the student would find that Limx -> +∞ (1/x) = ε. As Hurkyl pointed out earlier, this isn't the case, but the inquisitive student doesn't know that!

No, the limit of 1/x as x approaches infinity is not an infinitesimal.
You are not paying attention to the graph of the curve!

I recognize that and I pointed it out earlier! My point was that from your discussion of infinitesimals, this is not clear and that nothing prevents the student from reaching a fallacious conclusion.

Aside: I still don't think that the newest treatment does justice to infinitesimals, nor does it adequetly introduce the derivative, especially since the definition of the derivative in terms of the tangent gradient at a point does not lend itself (at least as easily) to the interpretation of the derivative as a function!

 

[Hurkyl]

I'm not sure if it's clear, but we're not being hypothetical about our objections (at least, I'm not). People really do have the misconceptions that we've been talking about. There are people who fail to recognize the difference between "the limit of a sequence" and "a sequence". There are people who think there is a smallest, positive, real number. There are people who think it is impossible to obtain exact answers using limits. There are people who think infinity is just a large real number, and the limits of something as x->0 is nothing more than plugging in a small value of x.

Most (all?) of my criticisms are from these kinds of explicit examples: my impression of your exposition and specific way of phrasing things is that rather than dispelling some of these misunderstandings, it could actually reinforce them!

It's hard to give constructive criticism, because you seem to find the things I disagree with to be a key feature of your exposition! e.g. when I see phrases like

dx is the length of the horizontal side of the blue triangle when it’s shape cannot be distinguished from the red triangle​

or

When the two triangles become “indistinguishable for all practical purposes”, the ratio of the perpendicular sides is dy/dx​

or even

$$\frac{dy}{dx} = \lim_{\Delta x \rightarrow \color{red}{dx}} \frac{f(x + \Delta x) - f(x)}{\Delta x}$$​

(color added for emphasis) I am vehemently opposed, because it reads as an explicit endorsement of the idea that a limit is nothing more than an approximation, formed by plugging in some unspecified value really close to the target.

But I get the impression that you really do consider things like these to be the key features of your method of presentation! There really isn't anything for me to do than to argue that your approach is fundamentally flawed.

 

[chrisr999]

Though the "value" of 1/x as x goes to infinity is "infinitesimal" if zero is included.
But not from the geometric simplicity of the illustration.
I'm sorry lads, I thought this was an exploration, rather than a matter of trenching in and defending the territory.
No, I wasn't dictating to you,
but some of you are very inflexible in your thinking!

It's inappropriate to say such a thing as "leave out the infinitesimals" and so on,
small measurements and using them is so basic,
a kid would get it without trouble!

and similar impatient comments.

The forum is here for your exploration,
you should try to respect it and other people.
Use it foolishly if you want to, who cares.

I won't be contentious or argumentative which happens when someone is only prepared to go so far.
Your objections are basically,
"Don't want to think independently about this, sorry, just want to repeat, repeat".
"Don't ask me to think, I haven't given you permission to ask".
you've gone as far as you will go, i reckon,
this is my final thread,
sorry for wanting to contribute and discuss the truths of the subject!
If someone was really interested in this they wouldn't take to silly attacks against someone else's analysis and call it flawed without even trying to see the view. I tried to put it as simply as possible.
But you can only lead a horse to water as they say,
There is sincere analysis but you always have the lazy ones that couldn't be bothered to even look.
I mean there are complaints about the spelling of people's names!
complaints about the definitions!
crikey!

my sincere apologies lads,
take care,
chris

 

[chrisr999]

Hurkyl,
I can hardly believe it was possible to get the impression I was saying a limit is an approximation!
along with a few other things,
but that demonstrates you didnt even understand what i put forward.
You ought to be very clear that the approximation is away from the limit,
but to get that impression?
To be honest, that comment is really strange,
I really can't imagine this is a mature forum,
over and out,
happy wood-pushing lads

goodness gracious.

 

[chrisr999]

By the way, Hurkyl,
if there is anyone who thinks there is a smallest possible real number,
it only means they are young, not introduced to the concept of a continuum,
such as "distance", never heard of "pi" yet, they are only learning or totally disinterested in math!
they shouldn't be criticised for just getting used to it in the beginning but if they've been at it for a long time, maybe scientific disciplines are not their scene,
bur for God's sake, lads, would you come on!

Your objections are very very inflexible and narrow and unimaginative,
sorry!

good luck

 

[chrisr999]

dx is not a definate length!
the student doesn't have to hit zero to understand the limit!
the student needs only enough imagination to realize the limit is revealed by the tangent,
which I've tried in numerous ways to explain!
that is EXACT not approximate, the limit is THERE, not just in continually reducing all the way to zero until your imagination runs out.
therefore the student only needs to zoom in within distances defined by the real number system
TO THE POINT THEY REALISE WHAT THEY ARE LOOKING FOR IS HANDED TO THEM ON A PLATE BY THE TANGENT.
They don't have to reach zero. THAT'S THE WHOLE POINT!
The mathematics then weaves it's way around those observations.
I haven't objected to your posts, I've just been surprised how quickly you object to mine without thinking through what i wanted to show!
so, I didnt hold my patience but what the heck! i wouldn't be doing anyone any favours that's not copping themselves on!

This is why the mathematics is written using "as the limit of dx APPROACHES zero",
not "until dx reaches zero".

So we use a little imagination to jump from the curve to the tangent since the answer is there!
and the student should stay there until it's clear, as we've seen in this discussion,
there's no point moving on and confusing yourself even more.
Once the confusion is cleared up that's it, it's clear.
the objections have been considered and ultimately dismissed, lads!
sorry to disappoint you,
but that's life

surely, the english is clear?
cripes lads, i thought we were exploring together.

 

[Hurkyl]

Hurkyl,
I can hardly believe it was possible to get the impression I was saying a limit is an approximation!

When you're writing material for people to learn from, it doesn't matter what you really mean -- what matters is what will be learned from it.

 

[Hurkyl]

When looking at your own work, you have to put yourself in the mindest not of yourself who has spent years or decades studying mathematics, but instead in the mindset of a student who doesn't understand things -- one who might even have specific misunderstandings -- and evaluate what they might learn from your writings. Yes, I'm sure that some people will get the right idea -- but I'm equally sure that some people will get wrong ideas.

I'm somewhat baffled that you can't even begin to understand why I think that some students could get the wrong idea, especially since I've highlighted some key passages that lead to my perception. (Note that understanding why but disagreeing with me is something different than not understanding at all)

 

[Hurkyl]

Aside -- I would like to point out that when interpreting derivatives via differential forms/dual numbers, df(x)/dx really is computed by plugging in an infintiessimal nonzero value and computing the difference quotient. In particular, there is the strict equality

$$f(x + \epsilon) = f(x) + \epsilon f'(x)$$

This works out because when using differential forms/dual numbers infinitessimal geometry is affine geometry: nonlinear effects are nonexistant.

 

[Hurkyl]

By the way, Hurkyl,
if there is anyone who thinks there is a smallest possible real number,
it only means they are young, not introduced to the concept of a continuum,
such as "distance", never heard of "pi" yet, they are only learning or totally disinterested in math!

Er, right. Hasn't this entire discussion been specifically about how to teach this stuff to people who are "only learning"?


(Or am I misunderstanding -- are you saying that it isn't worth trying to teach anyone who doesn't fully understand things like "continuum" by the time they reach their first calculus class?)

 

[okkvlt]

dy=y[x+dx]-y[x]
dx=dx

dy/dx=(y[x+dx]-y[x])/dx


looks like a fraction to me

 

[chrisr999]

thank you, Okkvlt,
it's amazing how it got to the point it wasn't obvious

 

[jgens]

dy=y[x+dx]-y[x]
dx=dx

dy/dx=(y[x+dx]-y[x])/dx


looks like a fraction to me

Still doesn't look like a fraction to me. If you want dy and dx to be infinitesimal you need the standard part function to actually define the derivative so your definition wouldn't be complete anyway. Or, assuming dy and dx aren't infinitesimal, you need to use a limit to actually compute and define the derivative.

 

[Ja4Coltrane]

For functions from R to R, a derivative is a limit of a fraction. Now I understand that dy/dx seems perfectly reasonably viewed as a fraction because of that, but then we run into trouble.

For instance, if f(x) = 3x for all x, then it is awesome to write df = 3 dx because it makes sense heuristically. In fact, even if f is non linear but well behaved, it still seems nice because of local linearity. However, derivatives are NOT always slopes of tangent lines.

Take f(x) = x + 2x^2*sin(1/x) for x non-zero, f(0) = 0. Then, f' is positive at 0 so by chris's interpretation, dy/dx>0 so dy is positive if dx is positive.

However, this is absurd because f is not monotone on any neighborhood of the origin!

I think this is a wonderful example of why the fracional heuristic is inferior to the precise definition.

Heuristics can be used to get ideas for theorems and their proofs, but they are not substitutions for definitions!

 

[chrisr999]

my discussion was on continuous functions with tangents at all points.
i discussed a basic discontinuous one also.
Try not to say it doesn't make sense in a different area.
To avoid confusing yourselves, you should closely examine the graph of the function you mentioned.
If you decide to say "derivatives arent always tangent gradients" then SHOW CLEARLY WHY THAT IS SO. If you add further complexity to a student's analysis who is clearling up something he wants to understand, you are only going on to another level before he's ready.
Stick to the fact that the derivative of simple continuous functions is given "exactly" by the gradient of a tangent at the point the derivative is calculated.

If you don't understand that and go onto compound shapes for which you don't show what you are now classifying a derivative as, you end up wasting the person's time and it is a hopeless discussion to have.

But you don't listen anyway, so what's the point continuing this?

fractions are fractions.
is speed a fraction?
are growth rates fractions?
are gradients fractions?
is a derivative a gradient?
don't let words be your masters.

 

[chrisr999]

You also misquoted me JA4Coltrane,
not only do you not present your graphical analysis,
which any of us can do for you,
but you proceed to work around zero without showing the limit or why you would want the limit or discuss what you are looking for in this case etc etc etc.
I will not waste my time discussing a half answer unless you are prepared to "take it to the limit".

 

[chrisr999]

Have a look at your curve. Increasing amplitude and reducing distance between the localised peaks.
At what "point" can you not have a tangent?
The closer you get to zero, the harder it becomes to view in this case, even with the real number system, and which system is modeled unambiguously with that mathematics function?
The fact that it may take immense computational power to zoom in is not the point.
The point is "do you want clarity or confusion"?
and at what proximity to zero?

 

[Hurkyl]

If you decide to say "derivatives arent always tangent gradients" then SHOW CLEARLY WHY THAT IS SO.

Nobody has said that. (At least, I don't think anyone has)

The main thing that people are saying is that derivatives are not difference quotients. The equivalent geometric statement is that derivatives are not slopes of secant lines. Furthermore, if you want to interpret "dy/dx" as the quotient of two things -- "dy" which somehow relates to changes in y and "dx" which somehow relates to changes in x -- then you have to introduce some new mathematics, be it differential forms, infinitessimals, or something else novel.

(Note that if you switch to the tangent line to talk about "dy/dx", "dy" no longer has any bearing on the function/curve that we were studying)



Also, do keep in mind that many people have strong algebraic intuition, often much stronger than their geometric intuition. There's an old joke that half of the people who study algebraic geometry do it so that they can apply their geometric intuition to study algebra. The other half do it so they can apply their algebraic intuition to study geometry.

(I assert that the ideal is to be adept in both pictures)

If you look back over the history of mathematics, you can see lots of examples of cases where people were trying to study geometry, but could only make progress by turning geometry problems into algebra problems. e.g.
* Descartes invention of coordinate geometry
* The algebraists figured out projective geometry first
* Algebraic topology

 

[chrisr999]

that was directed to the person that made the statement, Hurkyl,
if I'm going to be quoted out of context, I ask the person discuss with me, not discuss what they erroneously thought i was talking about and misquote me to others.
If you want, read his statements again and if you want to respond to him, please do.
Professionalsm if possible.

 

[chrisr999]

I'm sorry for being impatient sometimes, Hurkyl,
I appreciate people taking part in these discussions,
I appreciate your input,
I'm very busy with a lot of projects,
I don't mind someone saying "look i don't see it like that, this is how i see it" or whatever, but to say, "thats all wrong" etc and sticking with that, correct or not, gets tiresome.
What should matter is the subject itself and not the characters,
the revelation of the subject, which clearly will be situationally dependent.
Definitions may be worded slightly differently for specific cases.
this is a colourful world, not all black, white and shades of grey.

 

[chrisr999]

the thread was originated exploring the nature of the "derivative" of the function y=f(x) which in Calculus is written dy/dx.

the question basically was.. do the normal mathematics of fractions still apply?

the answer depended on the type of derivative.

Derivative means "derived from".
gasoline is a derivative of oil, orange juice is a derivative of orange etc etc.
The derivative under analysis originally was dy/dx.

Definition means "definite", "clearly defined", "unambiguous".

the original derivative in question is "the rate of change of a function derived from the formulation of the function itself".

It was correctly pointed out, that if you combine partial derivatives, you will get exceptions. We went into that and the reason for it.

The question that remains is...
Is the fractional nature of dy/dx still under scrutiny or is it resolved?
If not, is there any point discussing other types of derivatives until it is?
If it's not resolved, why not? what's unclear?

 

[Hurkyl]

The original point of contention, "Is dy/dx a fraction?", I assert is the algebraic analog of the question "Is the tangent line a secant line?" And unless we reinterpret the question in a framework other than plain Euclidean geometry / real arithmetic, the answer is a definite "no". (Although, I don't think you yet agree with that)

I also assert that there are (at least) three general ways of thinking of geometry "in the small". Applied to the question of computing a tangent line via secant lines:

* The standard analysis picture: there is no infinitessimal geometry. Our method is to approximate the tangent line via secant lines, and then take a limit as the error in approximation goes to zero to get the tangent line.

* The nonstandard analysis picture: infinitessimal geometry looks exactly like ordinary geometry. We take a secant line whose points are infinitessimally separated, and then round that to the nearest standard line to get the tangent line.

* The differential geometry picture: infintiessimal geometry is affine. The tangent line is a secant line through two points whose separation is a nonzero infinitessimal.

There may be other pictures, but these seem by far the most prevalent and well-developed.

I assert that the fact we have such well-developed foundations means that we should use them when teaching calculus! 150 years ago, it would have been appropriate to present calculus using vague and poorly-defined notions. Today, I assert it is not.

I assert that whichever foundation is used, it should actually be taught to the student -- no fair invoking thinks like hyperreal infinitessimals without explicitly teaching the student enough to be able to manipulate and reason about them on his own.


Do you consider any of these assertions fair?

 

[D H]

Just curious, why is this thread still here? After all, we don't allow personal theories at this site.

 

[chrisr999]

That's more like it, Hurkyl.

I consider your assertions very fair and it is from this position that we can examine the relative merits of learning calculus.
As this is highly involved, especially when dealing with a range of students of varying levels and ability.
When dealing with advanced calculus, the framework of reference needs to be seriously accurately defined and definitions, terms, number systems, degrees of freedom must be mapped out.

In learning the subject initially, there is flexibility of expression, but the frame of reference still has to be fully "defined" so that 2 or more people are referring to the exact same thing.

first point: "Is the tangent line a secant line?"
my answer: tangent has a single point of contact, secant has two. there is a difference.
are there any alternative definitions?

"Is dy/dx a fraction?" is answered by "Is dy/dx the tangent gradient?"
is a gradient a fraction? if it isn't and it's neither vertical nor horizontal, what are we
referring to?

Perhaps your analysis is pointing to the secant line gradient as being a fraction in terms of delta(y) divided by delta(x) and somehow differing in quality to the tangent gradient. No matter which units we use, the secant line, which is used to initiate the written mathematics of the "inaccurate instantaneous gradient of the curve" has a gradient expressed as a fraction. The gradient of the tangent gives the ratio of the exact value being sought.
That's the exact ratio.
the secant ratio is the inexact ratio. It's used to formulate the maths, the mathematical computation then proceeds to eliminate the error that was introduced by the secant, to arrive at the gradient of the tangent. But I've already presented all this.

second point: standard and non-standard analysis.
whatever a guy likes to order at the bar!

Once a student doesn't get confused with the terminology and understands what's going on, and once the descriptions are very clear, there shouldn't be a problem.

A kid rolling a ball around on a table has enough visual representation to silently lead into the geometry.

What i wanted to show at the infinitesimal level, though it's very basic, is the "merging" of the ratios the secant and tangent gradients, at a small enough level to show the student that the goal is the tangent ratio.

At the non-verbal level, the tangent is simply the secant pivoted on the point of tangency itself and rotated until there is one point of contact between the line and curve.
The mathematics expresses this rotation.
The limit is revealed by the tangent.
As the student explores what's happening at the "infinitesimal" level around the point of tangency is akin to a kid learning to ride a bicycle with training wheels.
If he understands the geometry clearly, he can work easily with necessarily agreed terminology later, particularly in working through the mathematics without referring to geometry.

 

[Ja4Coltrane]

Here is how I like to look at this. (df/dx) at a point x is a sort of generalization of a tangent line. That means that it isn't always one (consider f(x) = x^2 if x is rational, 0 otherwise which has f'(0)=0), but can be.

For that reason, since the *notation* df/dx looks like a difference quotient, it is extremely appealing and thought provoking. I like the notation, but I don't take it literally.

By the way: Chris, you got annoyed at me for using a not-very-nice function. I personally think that my function was pretty nice! It is, despite what you said, differentiable at every point (I'm referring to the function in post #90).

 

[chrisr999]

you didnt annoy me, JA,
thats your impression,
did you join to add clarity?
you saw DH object to "personal theories".
When you say things like "sort of", "generalisation of a tangent line",
you are veering too far away from mathematics without referring to what exactly
df(x)/dx is. It is something very simple and exact.
Your function is no bother to me and i can discuss it to your heart's content, it was simply that it doesn't help to be unspecific and divert the discussion. It's a waste of time.
Yes, your little function is very sexy, but...
If you are willing to say the derivative is a generalisation, then it's not, it's a formulation of exact mathematics and has been written as a fraction for a very specific reason.
The f(x)=x(squared) is very simple to examine around x=0.
The tangent is the x axis.

 

[Hurkyl]

"Is dy/dx a fraction?" is answered by "Is dy/dx the tangent gradient?"

But you've changed the question! It dawns on me that I should do a translation of this into pure geometry:

"Is the tangent line a secant line?" is answered by "is it a secant of itself?"​

This is a rather silly objection, don't you think? But that's exactly how you're responding to "Is dy/dx a fraction?".

The other two ways of phrasing it would be

1. $$\left.\frac{df}{dx}\right|_{x=a}$$ is a difference quotient of $f(a) + f'(a) (x-a)$​

and

2. The tangent line at (a,f(a)) is the secant to the graph of $f(a) + f'(a) (x-a)$.​

All of these are factually correct statements, but have nothing to do with what is meant by the question "is dy/dx a fraction?" or the question "is the tangent line a secant?"

 

[Ja4Coltrane]

I didn't make a a "personal theory." That's why I put little stars around the word "notation" in my post. I was stressing that thinking of it as a "sort of generalization of a tangent line" was only a heuristic and not a mathematical idea.

Didn't this whole thing start because you wanted to say that a derivative is sort of like a fraction? (Actually, your first post said derivatives ARE fractions).

 

[chrisr999]

i didnt say anything of the sort JA4,
as you continually misquote me that's the end of our conversation,
i only have time to consider something that's been thought through clearly, sorry,
or a genuine exploration,
sorry we have to draw a line somewhere.

No Hurkyl,
that is not how i was responding.
what is a secant and what is a tangent.
a secant cuts through two points of a curve.
the tangent skims across one.
this is how we can differentiate between them.
that is clear geometrically and the mathematics follows through on it.
there's no change in the question, just a clarification, an exploration to check that the definition is clear first of what a tangent and secant is.
the tangent is "the" line whose gradient gives the instantaneous rate of change at a particular single point of the curve.
the secant is the line (of which there are countless) that starts the mathematics going,
it also touches the point of tangency, but it touches another point also.
the secant gets the ball rolling and it's gradient gives us a false reading for the derivative,
in eliminating the error introduced by the secant (mathematically with all the clever techniques), we end up with the tangent gradient, the exact derivative.

Initially, we have a curve for which we want the instantaneous rate of change dy/dx.
it is given by the tangent gradient at the point of interest.
we can write the secant gradient from the function equation but not the tangent gradient.
This unfortunately introduces error.
The error is eliminated through mathematical techniques.
When it is, you have the tangent gradient, the exact value of dy/dx.
the function does not come with a secant and tangent, we use them as tools to get a geometric understanding of the entire situation from which we can go to any mathematical complexity for highly involved calculus.

All of this pertains to functions. When all of that is clear, then what dy/dx is when it comes to non-functions and so on will not introduce ambiguous complexity to the student.