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Question about correct theorem, for a bunch of jibberish

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Homework Statement



Okay I know my title sucks but if you have a better one let me know, here is the question:

If I have a continuous function on a closed interval [a,b] and let N be any number between f(a) and f(b), where f(a) is not equal to f(b), then which theorem guarentees that there must be some input c between a and b such that f(c) equals N?

The Attempt at a Solution


Here is what I was thinking the Professor means, interval [a,b] is just the domain including a and b, N is maybe difference in y value between a function at a vs b? Is c a value of x for the function and he wants to know where it is equal to this difference?

Admittedly I am lost to what is this thing is saying, can someone translate it?
 

Answers and Replies

  • #2
Dick
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Is your professor asking you to name the theorem? I think the statement the professor gave is already pretty clear. It's pretty famous and it does have a name. You didn't study it?
 
  • #3
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  • #4
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Is your professor asking you to name the theorem? I think the statement the professor gave is already pretty clear. It's pretty famous and it does have a name. You didn't study it?
Yes we are to name the theorom, sorry but when I see that many letters it throws me through a loop, it may be famous for someone doing this all the time but for a beginner it is foreign.

And if I knew the answer I wouldn't be here lol.

So back to the problem, where am I messing up?
 
  • #5
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http://en.wikipedia.org/wiki/Intermediate_value_theorem

I'm pretty sure that's what you're looking for. I mean... it is trivial enough to not be able to really help without just telling you haha.
Glad to give you a laugh, okay Ill check the link, in the meantime where am I wrong in my thinking
 
  • #6
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How to explain this, you're not really messing up, cause like, there's 1 step to the problem.

This is because the problem is: "Here is the statement of a theorem, now name the theorem".

Basically it would have been written verbatim somewhere in your book, and if you were unsure, you could have just looked up the definition of a continuous function and looked for theorems related to it, since something like the Intermediate Value Theorem is rather closely tied to the definition of a continuous function.
 
  • #7
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How to explain this, you're not really messing up, cause like, there's 1 step to the problem.

This is because the problem is: "Here is the statement of a theorem, now name the theorem".

Basically it would have been written verbatim somewhere in your book, and if you were unsure, you could have just looked up the definition of a continuous function and looked for theorems related to it, since something like the Intermediate Value Theorem is rather closely tied to the definition of a continuous function.
I don't want to just look in my book for the theorem, I want to understand what the question is asking, what would be the point otherwise lol
 
  • #8
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I don't want to just look in my book for the theorem, I want to understand what the question is asking, what would be the point otherwise lol
Wait, in that case it seems your question is not what the question states - because the question states, what theorem does this statement refer to. Whereas, your question seems to be... "what is the meaning of this theorem"? Am I understanding this correctly now or no?
 
  • #9
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Wait, in that case it seems your question is not what the question states - because the question states, what theorem does this statement refer to. Whereas, your question seems to be... "what is the meaning of this theorem"? Am I understanding this correctly now or no?
Sorry for the confusion, that question was directly cut and pasted from the materials from the class, I put more detail about what I was looking for in the attempt at a solution, I apologize for not being more clear, the HW section is a bit new to me :)

So to answer your question, yes I am trying to understand what the question is asking not which theorem is the answer
 
  • #10
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Well, one can do this algebraically or graphically. Since I think you want a visualization of it, I am going to describe it graphically.

Imagine a basic continuous curve of some sort defined by f(x)=y. For example any quadritic/cubic/etc will work.

Now let's pick two random x-values, say x=1 and x=2. Now further suppose the function we're looking at is f(x)=x^2. Again any continuous function will do.

f(1) = 1
f(2) = 4

What this theorem states is that for any number y such that 1<y<4, there is an x such that 1<x<2 and f(x)=y.

That's why it's called the intermediate value theorem. For any intermediate value of y between the two points we chose, there's also an x between the two points we chose such that we find f(x)=y.
 
  • #11
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Well, one can do this algebraically or graphically. Since I think you want a visualization of it, I am going to describe it graphically.

Imagine a basic continuous curve of some sort defined by f(x)=y. For example any quadritic/cubic/etc will work.

Now let's pick two random x-values, say x=1 and x=2. Now further suppose the function we're looking at is f(x)=x^2. Again any continuous function will do.

f(1) = 1
f(2) = 4

What this theorem states is that for any number y such that 1<y<4, there is an x such that 1<x<2 and f(x)=y.

That's why it's called the intermediate value theorem. For any intermediate value of y between the two points we chose, there's also an x between the two points we chose such that we find f(x)=y.
It makes sense that a continuos function will have values for x for each y if a function of x is continuous between two points. Is this basically it?
 
  • #12
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yes, with the extra caveat that the upper and lower bounds of y are set by f(y) at the two points you choose to create the interval.
 
  • #13
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yes, with the extra caveat that the upper and lower bounds of y are set by f(y) at the two points you choose to create the interval.
This is pretty obvious, why is it something we need to 'know'?
 
  • #14
Dick
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This is pretty obvious, why is it something we need to 'know'?
Because this is 'mathematics'. You don't 'know' something until you have a proof from the definition of continuity.
 
  • #15
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Because this is 'mathematics'. You don't 'know' something until you have a proof from the definition of continuity.
Right, and as further elucidation, something like a+b = b+a is actually not always true... so you've got to be sure. That's why you have theorems that you might think are trivial. Turns out they might not be so trivial.
 
  • #16
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Right, and as further elucidation, something like a+b = b+a is actually not always true... so you've got to be sure. That's why you have theorems that you might think are trivial. Turns out they might not be so trivial.
Okay, so down the road the intermediate theorem will be important
 
  • #17
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Because this is 'mathematics'. You don't 'know' something until you have a proof from the definition of continuity.
Okay so the lesson is regardless of what the theorem states even if obvious we need a proof, so what is the 1+1=2 theorem called lol
 
  • #18
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Okay so the lesson is regardless of what the theorem states even if obvious we need a proof, so what is the 1+1=2 theorem called lol
Do you really want to start generating the positive integers via sets?
 
  • #19
Dick
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Okay so the lesson is regardless of what the theorem states even if obvious we need a proof, so what is the 1+1=2 theorem called lol
Oh, ok. There are things that are clear enough you don't stop to prove them, depending on the course. In calculus, you take arithmetic for granted. On a lower level (or higher maybe) you do want to make proofs for stuff like that. Still, the Intermediate Value Theorem is something that has a proof and you do need to know that by name to prove other things.
 
  • #20
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Oh, ok. There are things that are clear enough you don't stop to prove them, depending on the course. In calculus, you take arithmetic for granted. On a lower level (or higher maybe) you do want to make proofs for stuff like that. Still, the Intermediate Value Theorem is something that has a proof and you do need to know that by name to prove other things.
So it is good to know to help move into better explainations for more advanced theorems in the future, very good. I hope you don't take my joking around the wrong way, I appreciate the help you have given me on the last few questions I posted. This forum is a good resource :)
 
  • #21
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Do you really want to start generating the positive integers via sets?
And thank you for the help as well, I prefer your explaination over the one written in the Theorem, it makes a great deal more sense
 
  • #22
Dick
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So it is good to know to help move into better explainations for more advanced theorems in the future, very good. I hope you don't take my joking around the wrong way, I appreciate the help you have given me on the last few questions I posted. This forum is a good resource :)
No offense taken. Your point is understood. You prove some things. Not others which you take as given. Don't worry about it.
 

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