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March 29, 2024, 09:37:45 pm

Author Topic: Random math questions  (Read 37359 times)  Share 

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#1procrastinator

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Random math questions
« on: December 27, 2012, 03:20:36 am »
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Show that |x|<2  =>  |(x^2+2x+7)/(x^2+1)|<15

The inequality signs should be 'or equal to' .

Some hints on where to start be good. im not sure i can start with the second inequality cause of the whole P implies Q doesnt necessarily mean Q implies P thing
« Last Edit: December 27, 2012, 03:22:27 am by #1procrastinator »

FlorianK

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Re: Random math questions
« Reply #1 on: December 27, 2012, 03:47:44 am »
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Show that |x|<2  =>  |(x^2+2x+7)/(x^2+1)|<15

The inequality signs should be 'or equal to' .

Some hints on where to start be good. im not sure i can start with the second inequality cause of the whole P implies Q doesnt necessarily mean Q implies P thing
Do you mean
Show that if the modulus of x is greater or equal to 2, then the modulus of is smaller or equal to 15?

b^3

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Re: Random math questions
« Reply #2 on: December 27, 2012, 03:59:51 am »
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I have some working, but I don't think its the proper way of going about it (not a proper proof), and its a bit intuitive, so odds are it won't help but heres my way of thinking it through.

Since for , i.e. we are always dividing by something greater than , so has to be equal to or smaller than its maximum.

When we do actually divide by 1, that is when , then
So as for and and for
for

Anyways, probably not the solution, but it may give you some ideas, so hope it helps.
 
« Last Edit: December 27, 2012, 04:02:43 am by Indian Battman »
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#1procrastinator

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Re: Random math questions
« Reply #3 on: December 27, 2012, 04:52:36 am »
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@FlorianK: yep, i guess that's one way to phrase it but|x| should be less than or equal to 2 :p

@Indian Battman: thanks, i wouldnt have thought of that. im looking for a method that doesnt use calculus cause this is actually in the preliminaries section of the text lol

FlorianK

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Re: Random math questions
« Reply #4 on: December 27, 2012, 05:28:45 am »
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That question is Bullshit, like literally bullshit.
The whole function is always smaller than 15 for all values of x ~_~


#1procrastinator

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Re: Random math questions
« Reply #5 on: December 27, 2012, 07:00:14 am »
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yeah, worked that bit out myself by graphing it xD

but surely there's another way of proving it

polar

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Re: Random math questions
« Reply #6 on: December 27, 2012, 11:05:57 am »
+3
try using the triangle inequality

Spoiler

the converse doesn't need to be true for that statement to the true either, it just means you can't put an if and only if sign between them.
« Last Edit: December 27, 2012, 11:24:31 am by polar »

dcc

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Re: Random math questions
« Reply #7 on: December 27, 2012, 11:40:07 am »
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another take on polar's solution is to consider

for .

edit: using this, we recover where the mysterious "15" comes from:

for
« Last Edit: December 27, 2012, 11:42:51 am by dcc »

BubbleWrapMan

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Re: Random math questions
« Reply #8 on: December 27, 2012, 02:39:29 pm »
+1
I noticed that the expression is always positive anyway, so the modulus is redundant.

For all real , , and therefore

Also







If and we multiply by which is always in the interval we will still get a number less than 15, because the product will either be equal to , or a less.

Hence, whatever the question was
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#1procrastinator

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Re: Random math questions
« Reply #9 on: December 28, 2012, 04:18:18 am »
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wow, thanks a lot guys

@polar: why do you set the expression to 7 and solve for x? im a bit unclear on the.motivation and how you chose that number

@dcc: how did you choose 10 and how do you the inequality is true?

@ClimbTooHigh: think i understand your method, similar to ive seen before :p

#1procrastinator

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Re: Random math questions
« Reply #10 on: December 30, 2012, 06:07:33 am »
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how do you integrate sqrt(x-x^2)? trig sub hasnt gotten me anywhere so far

edit:  this is what i got:



which disagrees with what the ti89 gave. even though the calc tends to give different forms of what you usually get if you do it by hand, im not all that confident with my derivation
« Last Edit: December 30, 2012, 07:12:51 am by #1procrastinator »

FlorianK

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Re: Random math questions
« Reply #11 on: December 30, 2012, 07:11:22 am »
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did you made that question up or found it in your textbook or something?
The answer looks a bit too crazy:



especially because the derivative of arcsin(2x-1) is

« Last Edit: December 30, 2012, 08:03:47 am by FlorianK »

#1procrastinator

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Re: Random math questions
« Reply #12 on: December 30, 2012, 07:19:05 am »
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the original question was 'what is the largest value of the integral of sqrt(x-x^2) from a to b for any a and b'

im not sure if you are meant to integrate but i thought it would be interesting to try to do so

FlorianK

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Re: Random math questions
« Reply #13 on: December 30, 2012, 07:29:06 am »
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Graph it and you'll see that it is a half circle with radius 1/2 units, hence the area under the curve, which is in this case the greatest integral, is pi/8. The proof that it is that circle will be up in a second

FlorianK

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Re: Random math questions
« Reply #14 on: December 30, 2012, 07:42:11 am »
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ok, so we need to show that it is a circle. The general equation for a circle is:

where (a,b) is the point of the center and r is the radius.
So we need to show that is the same as