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#1 2007-01-01 09:40:23

Prakash Panneer
Member
Registered: 2006-06-01
Posts: 110

Please Please....

Give an example of a function on a closed interval which satisfies the conclusion of the Mean Value Theorem, but not the hypothesis.

Thanks in advance up


Letter, number, arts and science
of living kinds, both are the eyes.

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#2 2007-01-01 17:59:08

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Please Please....

f(x) =

x when x does not equal 2
3 when x equals 2

interval [2, 3]

It's not smooth, and thus, does not satisfy the hypothesis.


"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..."

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#3 2007-01-02 06:21:01

Prakash Panneer
Member
Registered: 2006-06-01
Posts: 110

Re: Please Please....

Are you sure?

I think it doesn't satisfy the conclusion.

Right?


Letter, number, arts and science
of living kinds, both are the eyes.

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#4 2007-01-02 08:57:30

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Please Please....

Whoops, you are correct.  Make that:

f(x) =

x when x does not equal 2.1
3 when x equals 2.1

interval [2, 3]


So we need to find a point within [2,3] where the slope is 1.  This works for all values other than x = 2.1.


"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..."

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#5 2007-01-02 12:11:45

Prakash Panneer
Member
Registered: 2006-06-01
Posts: 110

Re: Please Please....

Thanks a lot...

Is anyother function except peicewise function?


Letter, number, arts and science
of living kinds, both are the eyes.

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#6 2007-01-02 12:44:56

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: Please Please....

I don't get it. What does it mean to satisfy the conclusion but not the hypothosis?


A logarithm is just a misspelled algorithm.

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#7 2007-01-02 16:44:22

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Please Please....

If A then B

A is the premise, B is the conclusion.  However, it is possible for A to be false and B to be true.

If x is 3, then x is odd.

"x is 3" is the premise, "x is odd" is the conclusion.  x=5 satisfies the conclusion, but not the premise.


"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..."

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#8 2007-01-02 16:47:08

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: Please Please....

mmm... i see..

well I'd say a function thats naturally discontinuous is more elegant. Such as y = 1/x from - 1 to 2.


A logarithm is just a misspelled algorithm.

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#9 2007-01-02 16:48:39

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Please Please....

Any non-smooth function won't satisfy the premise.  After this, it's just a matter of finding the right interval.  It just happens that most functions we know of are smooth.


"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..."

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#10 2007-01-03 04:49:10

Prakash Panneer
Member
Registered: 2006-06-01
Posts: 110

Re: Please Please....

Non smooth function???

Can you give me an example?


Letter, number, arts and science
of living kinds, both are the eyes.

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#11 2007-01-03 11:37:15

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Please Please....

Any function which isn't continuous on [a, b] or differentiable on (a, b) is not smooth.

1/x on [-1, 1] works.  So does just about any piecewise function (although not all).  |x| from [-1, 1] works as well.


"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..."

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