Math Is Fun Forum

WiZaRd

Real Member

Registered: 2006-09-22

Posts: 570

Help Me With This!!

Prove that
f(x)=x^3-2x+3
is continuous.

Find the value of K

f(x){kx^2      x is less then equal to 2
     {2x+k      x is greater then 2

the above bracket is one

hope you understand!!! the questiuon and let me help withj the solution!!

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Be Happy!

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: Help Me With This!!

I wouldnt know how to prove the first one is continuous. I always took for granted all polynomials are continuous. Maybe to prove it you need to use epsilon-delta proofs?

For the second one, the question states to find K such that the piece-wise function f is continuous at x=2. For f to be continuous the sided limits need to be the same and they need to be equal to f(2). So we need the limit as x approaches 2 from the left to be equal to the limit as x approaches 2 from the right.

So we take the limit from the left and we get 4k and we take the limit from the right and we get 4 + k. For the function to be continuous then those two need to be equal. So:

4k = 4 + k
3k = 4
k = 4/3

So now that we know k=4/3 then the limit from the left equals 16/3, from the right equals 16/3 and f(2) = 16/3. So we found our k such that f is continuous.

Last edited by LuisRodg (2008-11-10 00:49:14)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Help Me With This!!

Proving that any specific polynomial is continuous can prove to be quite challenging.  It is much easier to prove that x is continuous, and the product and sum of any continuous functions are continuous.  Indeed, you may have done this in your class.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: Help Me With This!!

In what class are you mentioning?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Help Me With This!!

These facts can be given as early as calc 1.  I suspect that Wizard is in an analysis class at either the junior or senior level.  These are typically called something like "Advanced calculus" and "real analysis" respectively.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: Help Me With This!!

Well, it is true that back in Calc1 we were told that the sum and product of continuous function are continuous. But one thing is to take it for granted and another thing is to actually prove it which was what Wizard asked. Right?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Help Me With This!!

Right.  So it's hard to find a first analysis class where this isn't proven.  I'm fairly comfortable with saying "impossible".  The proof of the sum being continuous is rather easy, the proof of the product requires a small trick.

Wizard (or Luis if you're interested), let me know if you want a hint to the two proofs I mentioned.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: Help Me With This!!

Well, I'd love to take a look but I havent taken any analysis class so im lost as to where to start. Hopefully that changes next semester when I will take Advanced Calculus.

Also, the proof of continuity deals with analysis, the exercise of finding the K is from Calc1. How come?

OFF-TOPIC:

Ricky, I always wondered what Real Analysis was, is that the same as Advanced Calculus?

WiZaRd

Real Member

Registered: 2006-09-22

Posts: 570

Re: Help Me With This!!

Thanks again...!!! and yes i am studying calculus at some higher levels....!!! but i do not know (its sad to say that) I do not know some basic concepts..   :-(   My text book is CalCulus by Howard Anton 5th Edition!

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Be Happy!

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Help Me With This!!

Thanks again...!!! and yes i am studying calculus at some higher levels....!!! but i do not know (its sad to say that) I do not know some basic concepts..   :-(   My text book is CalCulus by Howard Anton 5th Edition!

I'm not certain if this means you do or don't want some help on proving the sums and products of continuous functions are continuous.

Ricky, I always wondered what Real Analysis was, is that the same as Advanced Calculus?

It gives you a good idea of analysis, yes.  Analysis is basically the study of anything concerning limits.  Real analysis is of course analysis when it's only focused on the real numbers.  The main goal is just to learn more about the the real numbers and functions on them.  The more we know, the better.

Much of Advanced Calculus however is taking the things you learn in Calculus and making them rigorous.  Some new material is presented, but by and large it's mostly things you've at least heard of.  Later on in analysis (a senior or graduate level course) you get into things that could not possibly be talked about in a calculus course.

Hopefully that changes next semester when I will take Advanced Calculus.

My experience is that you will hate it.  But as soon as you get used to inequalities and the epsilon-delta way of doing things, it becomes a lot more enjoyable.  Again, this is only my experience, but inequalities and epsilon-delta proofs are more of a skill that takes time getting used to.

Well, I'd love to take a look but I havent taken any analysis class so im lost as to where to start.

It's a bit late right now, I'll post some stuff by tomorrow afternoon concerning the problem.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

ADNAN

Guest

Re: Help Me With This!!

Question # 01:

Prove that
f(x)=x^3-2x+3
is continuous.

Question #02:

Find the value of K

f(x){kx^2      x is less then equal to 2
     {2x+k      x is greater then 2

the above bracket is one

hope you understand!!! the questiuon and let me help withj the solution!!

plz send me the answers of thes 2 Questions , URGENTLY with in 24 Hours at
Email address removed - Ricky

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,404

Re: Help Me With This!!

ADNAN,
It is not allowed to mention your e-mail in the posts in this forum.
However, it has not been edited because being a newcomer, you may not be aware of the rules of the forum.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Angel Rox

Member

Registered: 2008-11-10

Posts: 15

Re: Help Me With This!!

So no body can solve the 1st Question that Wizard Posted...

It was Prove that the function f(x)=x^3-2x+3 is continuous

Plz I too need the solution....Its really urgent !!!

Last edited by Angel Rox (2008-11-10 23:19:13)

Elite cave Botter

Guest

Re: Help Me With This!!

can I ask why you people want this so urgent??

Angel Rox

Member

Registered: 2008-11-10

Posts: 15

Re: Help Me With This!!

It is Urgent Man...If u can be of some help then plz write the solution.... I'll appreciate that...

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Help Me With This!!

ADNAN,
It is not allowed to mention your e-mail in the posts in this forum.
However, it has not been edited because being a newcomer, you may not be aware of the rules of the forum.

Ganesh, the reason why we don't allow emails is to protect the poster from getting spam.  It isn't a "disciplinary" action, but rather keeping them safe.  As such, I've removed the email address.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."