Hi, how to understand mean value theorem proof? Do multiple proofs exist and if yes then what are they? How to justify and allow all actions done in proof?
#Mean value theorem proof
umbral kettle
heavy gorgeBOT
ashen spindle
Hi, how to understand mean value theorem proof? Do multiple proofs exist and if ...
Start by opening a textbook and reading through at least one proof. They tend to come with reasoning.
And yes, multiple proofs do exist. That's usually true for most well-known propositions.
ruby hare
show a proof
umbral kettle
I am talking about proof on wikipedia on mean value theorem
Im confused about the part where they define a new function
How are we allowed to do that and how this "random" action counts as proof for this theorem?
candid oriole
The point of a proof is you start with some assumptions and you derive some conclusion. What you do in between, provided it only uses the assumptions you started with, is entirely up to you.
umbral kettle
So if we assume certain facts are true, then if using them we get any "random" true facts, then they must be true in math?
candid oriole
I wouldn't call them random new facts, but things you can derive from a set of assumptions are what a theorem is.
umbral kettle
I wouldn't call them random new facts, but things you can derive from a set of a...
Alright, its like in geometry no matter what "random" lines we draw, if we from these true facts get another true fact, it is definitely true?
candid oriole
Yes, hence why in the middle you should either use stuff in your assumption, or true facts that you've already proven / take for granted.
umbral kettle
That is very powerful
candid oriole
It's what deduction is.
For instance, in that proof the the mean value theorem, they defined this other function and used another theorem, Rolle's theorem, to get the desired result.
Rolle's theorem has a proof of its own, and hence it's true in the mathematical system we're working with.
umbral kettle
Talking about Rolle's theorem, i saw it can be proved using fermat's theorem. Can fermat's theorem be proved without algebra, but just by logic?
candid oriole
If you mean what they mention at the end of the proof of Rolle's theorem on Wikipedia, they're referring to Fermat's stationary point theorem.
This is different from Fermat's last theorem.
Fermat's stationary point theorem also has a proof of its own, and hence it's true.
umbral kettle
Yes i meant fermat's stationary point theorem
But can it be proved without algebra and all that stuff, but just by reasoning and using derivative definition?
candid oriole
"Without algebra but just by logic" is vague.
umbral kettle
I would reason that if we look at max point A in an interval, then it means all points in this interval on the left are smaller and on the right as well. So it also means it is growing from the left and derivative is + and decreasing from the right side of the point A and thus derivative is -. So if signs changed, should not derivative on point A be 0 if this function is continuous?
candid oriole
This is the idea behind the proof, but it's not a proof.
You need to justify those statements and be precise.
- What kind of interval are you considering? Open or closed?
You say A is a max on this interval. - Is it an absolute max or a relative max? - If it's an absolute max how do you know there is even such a point?
- If it's a relative max then how can you deduce everything on the left and right of A in the interval is below that point? A relative max is only a maximum locally.
- Provided you formalize the above, how do you show that the derivative changes sign? Appealing to the fact that it must be increasing before and decreasing after is a handwavy way of writing down what you mean, and you need to show those things.
This is what a proof does, it convinces without a doubt.
And in particular the proof of that theorem from Fermat is exactly formalizing what you wrote.
And it's important because appealing to how things seem to be is dangerous in maths, since intuition is often wrong.
A good example is that when analysis was in its infancy, it was thought as quite obvious that any function which is continuous everywhere must be differentiable in a small interval around some point. After all, for most continuous functions you can think of, this actually is the case.
However it's actually false. The famous Weierstrass function is a case for which the function is continuous at every point but which is differentiable nowhere.
umbral kettle
Yes, i guess intuition does not strictly prove anything... However, can we prove something about math using intuition and logic if we use true facts and assumptions, but not purely math(algebra for example)?
Axioms are based on logic and intuition, aren't they?
candid oriole
If you're proving something about derivatives you're gonna need to appeal to what derivatives are.
They're based on logical facts we take to be true.
Every proof you see online you could rewrite in purely logical terms.
It wouldn't be particularly readable though
umbral kettle
So math really is just based on our observations that are turned into axioms and then using logic, turned into something useful and larger?
candid oriole
Yes. It's built from the ground up.
You start with axioms in your bag of true statements.
Then you derive stuff that is true from those axioms and add them to the bag of true statements.
Then you keep deriving more stuff from this bag of true statements and putting those things in the bag of true statements.
and so on.
umbral kettle
Alright, i think i understand now how it works
candid oriole
So like, yes you could theoretically rewrite a proof of Rolle's theorem strictly in logical symbolism, but it would make it (1) super long (2) unreadable.
The beauty of the logical system we use is that if you can argue about things in words, it corresponds to an argument about it purely in terms of logical symbols and syntax. This is called semantic completeness. So you can prove things in actual words and know what you get is true logically purely at the level of syntax. (At least up to a point)
umbral kettle
Thanks for the insight. Never knew such thing existed.
Is it worth memorizing proofs or just understanding them and then forgetting them(the proof) is enough? I think the second option is the best, because i do not think anyone can memorize 1000 hard proofs every line... Is the best option to remember something from the proof or not remember anything at all? I think just knowing the theorem is enough if you understand the proof, because then you know it definitely works without a doubt and you can use it. Did great scientists knew all proofs in their head when they used them to prove something new?
Is it okay to not know proof at all and use the theorem? For example, my math skills are not so advanced yet to understand all proofs, but i want to use some of the theorems.
candid oriole
Is it worth memorizing proofs or just understanding them and then forgetting the...
It's never a really good strategy to memorize a proof, no. Anyways, chances are if you understand the way the proof works you would be able to argue similarly were you asked to prove it.
It's good to look at the proof and try to see how it shows what it's showing, as in the general process (like what you did above. It's good intuition to see how the things you wrote are transcribed formally in the proof of Fermat's stationary point theorem for instance.)
However, at the level of Calculus, I think usually the problems you'll be faced with are more with regards to application of those concepts, so you should tailor what you learn to what is asked of you.
umbral kettle
It's never a really good strategy to memorize a proof, no. Anyways, chances are ...
Even if i understand proof, i usually can not prove it by myself again, because the proof was too hard to remember anything. I am not concerned about problems i will probably face, but rather how to prove them on my own and use them in proof type problems.
If after "understanding" proof i can not write it by myself, did i really understand it?
Actually that can be the case. For example, if you understand what song is telling you, it does not mean you will be able to sing it by yourself!
About fermat's theorem: can you prove this theorem if you prove that to change derivative signs from + to -, there must be a point between these changes where derivative is 0? Can it be done by looking at 2 possibilities: a wedge and a curve with at least 2 points with same largest value in an area. Then by showing that in wedge derivative does not exist, then it means it must be the curve case? But then it implies that the derivative is 0, because at top of the curve.
candid oriole
About fermat's theorem: can you prove this theorem if you prove that to change d...
This is what the proof on wikipedia is arguing.
umbral kettle
Oh ok
candid oriole
You're already considering a differentiable function, so there is no wedge.
The point is that if you have a local max on this interval, there's a small interval around this point where this max is actually an absolute maximum on this small interval.
You also know the derivative exists at this point, so using the definition of the derivative taken approaching either side you get the result.
umbral kettle
But is it not quite logical to say there will be definitely derivative with value 0? Because otherwise we would get a wedge where derivative does not exist? But we know there is not a wedge. So there must be definitely some other value, but not positive or negative, which is 0!
candid oriole
All you know is that the derivative exists at this point.
The goal is to show it's 0
The point is you need to argue that the derivative on the left if positive and the derivative on the right is negative in a small enough neighbourhood
umbral kettle
Oh and then you conclude the same thing what i just said?
candid oriole
Yes. Again what you said is intuition, but it's not a proof. What it written in the proof on wikipedia is exactly what you said but justified
umbral kettle
What i am now really not getting is why not writing exact proof does not count as justified if the logic is true?
candid oriole
Because you need to justify the things you write. Otherwise you could just say "Hah it's obvious" and call it a day
It's not explicitly clear from the definition of the derivative that what you wrote is true
Namely that the derivative a bit to the left is positive and that the derivative a bit to the right is negative.
This is what needs arguing
In particular, $\lim_{h \to 0} \frac{f(a) - f(a+h)}{h}$ exists so $\lim_{h\to 0^-} \frac{f(a) - f(a+h)}{h}$ and $\lim_{h \to 0^+} \frac{f(a) - f(a+h)}{h}$ both exist
rocky cloudBOT
Azyrashacorki
umbral kettle
Ok, but what if without writing math proof i also proved that derivative a bit to the left is positive and to the right a bit negative and are equal using words and logic.
candid oriole
**Azyrashacorki**
This is using words and logic
candid oriole
Ok, but what if without writing math proof i also proved that derivative a bit t...
The issue is that at some point or another you'll need to argue the derivative has this property. And all the assumptions you're working with is that the derivative exists and you're on a max, so you'll need to call back to what it means for a derivative to exist in some way, and you'll need to call back to what it means to have a local max in some way
Saying that the derivative on the left is positive and the derivative on the right is negative is a good answer as to what is going on and you could convince someone with it, but a solid proof should be kind of immune to someone asking "why this why that" when reading your argument
umbral kettle
Alright
dense ember
So if we assume certain facts are true, then if using them we get any "random" t...
You have to start off with some information to know something right
like
$1+1 \text{ prove } \iff 1$
rocky cloudBOT
Cor
dense ember
you can assume that
and once we have certain conditions
like 1+1, then we can use the results
and basically build everything up
but if the conditions are wrong i.e. the assumpiton is wrong, then we cannot use the results
humble epoch
abstact
heavy gorgeBOT
Then it makes sense
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umbral kettle
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