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#181 Dion

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Posted 13 December 2012 - 03:16 PM

Double post but eff it.

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How do I find A/B to complete the partial fraction? I can't use s = 0 o.o Did I make a mistake?

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#182 Secret Igshar

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Posted 13 December 2012 - 03:24 PM

Two questions:

1. How did you go from Y=*fraction* to the A-B equation?
2. Why can't you make s=0? The s in the exponent is in the numerator. Unless you're referring to some earlier point where you divided by s... which you never did. So...

PS. s2 is also a double root, so the numerator on that one should be As+B

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#183 Dion

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Posted 13 December 2012 - 03:27 PM

I used partial fractions. Also I just realized Im dumb because I thought the s was in the denominator because my handwriting is atrocious. ._.;

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#184 Secret Igshar

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Posted 13 December 2012 - 03:28 PM

It's just because you write fractions like a scrub. I always write them with horizontal bars on paper. It makes it soooo much harder to screw up. It's a habit I suggest you get into.

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#185 Dion

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Posted 13 December 2012 - 03:31 PM

i usually do them that way to save space on paper irl and ive gotten into the habit of it. super small, illegible numbers if you put it one line always gets me points off Dx

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#186 Secret Igshar

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Posted 13 December 2012 - 03:36 PM

Then just put your variables in the numerator. Like πs/2 instead of π/2s.

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#187 Waker

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Posted 27 December 2012 - 08:17 PM

So uh. Just leaving this here

corresponding graph: http://i.imgur.com/xHTwE.png

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Edited by Waker, 27 December 2012 - 08:33 PM.

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#188 Lexaeus

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Posted 24 January 2013 - 11:31 AM

. Given the general function F = c x^a y^b , where x ,y are variables and a, b, c are constants, derive the following general expression for the relative error:

         

                              sF  / F = sqrt[ (a sx / x)^2 +  ( b sy / y )^2 ] 

 

s = little sigma

 

any help would be nice


Edited by MotherHugging Lexaeus, 24 January 2013 - 12:05 PM.

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#189 Secret Igshar

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Posted 24 January 2013 - 12:27 PM

Assuming little sigma is a function, but that... isn't helping me comprehend anything that's going on here. Sigma means a lot of different things in different contexts, and the only one I've ever studied is the Number Theory context, where s(n) is the sum of divisors of n, but that definitely is not what's being used here.


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#190 Lexaeus

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Posted 24 January 2013 - 12:54 PM

Error stuff

 

Error Analysis

Gathering quantitative data requires the use of instruments. The resulting measurements will necessarily have errors due to the instrument’s inherent precision, the operator’s technique of measurement, and the randomness of the sample’s dimensions. Basic statistics allows for an accurate representation of the sampled data, due to these variances.    

Precision

An instrument’s precision is taken to be the smallest possible measurement that is designated on the device. For example, a meterstick has a precision of 0.1 cm. Each measuring device has its own precision, which then determines the number of significant figures with which to record the data. The estimated error e is then often recognized to be half of the instrument’s precision used to measure the observable.

Accuracy

If a standard value (xstand) is known, an accuracy comparison with an observed value (xobs) can be found:                       [ (xstand - xobs) / xstand ] x 100%

0 % would represent the best accuracy, but does not imply the absence of errors. Accuracy is reported as negative when the observed value is larger than the standard value.

If a standard value is not known, it is possible to compare 2 observed values against one another. This can occur when comparing 2 differing methods of calculating the same quantity:  

                            (xobs1 - xobs2) / [ (xobs1 + xobs2) / 2  ] x 100%

In this case the denominator is found from the average of the 2 observed values, so that neither value is treated as a standard (both are treated with equal weight).

Mean

The mean, or average of a set of similar measurements is simply found by the sum of all data divided by the sample number N:       x = S xi / N

 

 

Standard Deviation

The standard deviation then determines the variance of the sample data with respect to the mean:                                sx = sqrt [ S ( xi – x)2 / (N-1) ]        

Standard deviation is an example of an absolute error and has the same units as the sample data. It provides a confidence interval with regard to future measurements, or a standard value. Statistically, the following expectations are noteworthy:

          1s = 68 %                2s = 95 %                 3s = 99.7 %

The relative error can then be found according to sx / x, and is often expressed as a percentage by multiplying by 100. This relative error then gives a reasonable expectation of the desired accuracy for the experiment.

Propagated Error

Consider a physical quantity F that depends on several observables F ( x1 , x2 , x3, …xn ).

If the standard deviation of each observable is known, then the propagated error (total error) can be found, and generally depends on all observable errors according to:

sF = sqrt  { [ s1 ( ¶F / ¶x1 ) ]2 + [ s2 ( ¶F / ¶x2 ) ]2 + . . . + [ sn ( ¶F / ¶xn ) ]2 }

¶F represents the partial differentiation with respect to each variable x. If standard deviations are not known for all of the individual variables x (for example, s1 since perhaps only 1 measurement was performed) , then estimated error values (e) may be substituted.

The total relative error can then be found according to  sF / F, multiplied by 100 for a percentage value.

Typical Propagated Error Functions

The function F = x + y  where x ,y are variables yields  sF  = sqrt  [ ( sx )2 +  ( sy )2 ]

The function F = c xa yb  where x ,y are variables  and a, b, c are constants yields          

                         sF  / F = sqrt [ (a  sx / x)2 +  ( b sy / y )2 ]

 

 ¶ = partial differential

s = little sigma


Edited by MotherHugging Lexaeus, 24 January 2013 - 12:56 PM.

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#191 Secret Igshar

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Posted 24 January 2013 - 01:36 PM

Alright, cool. I'll write out steps for you.

 

sF = sqrt( (sx*ac xa-1yb)2 + (sy*bc xayb-1)2 )

AND

F = sqrt( (cxayb)2 )

since that simplifies to what you started with. Inverse functions. :D

 

So:

sF/F = sqrt( [ (sx*ac xa-1yb)/(cxayb) ]2 + [ (sy*bc xayb-1)/(cxayb) ]2 )

sF/F = sqrt( (a*sx/x)2 + (b*sy/y)2 )

 

since c and yb cancel in the first, and the x exponents combine to give x in the denominator. Same for the second term, except the xa cancels and a y gets left in the denominator.


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#192 Secret Felix

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Posted 24 January 2013 - 08:32 PM

Prove that if forty three books are packed into six boxes, at least one box contains at least eight books.

 

 

Spoiler


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#193 Secret Igshar

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Posted 24 January 2013 - 08:50 PM

Let xi be the number of books in box i, with i ∈ [1,6] ⊆ ℕ.

Then i=16 xi = 43, by assumption.

We proceed by contradiction.

Suppose that xi ≤ 7 for all i ∈ [1,6].

Then we see that i=16 xi ≤ 7 * 6 = 42.

By our assumptions, this sum is exactly 43, which lies outside the above range.

This is a contradiction.

Therefore, for some i ∈ [1,6], we have that xi > 7, or that xi ≥ 8 □

 

But, really, you can do this proof in much less time with as much clarity. You know. How you did it. >.>


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#194 Dion

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Posted 02 February 2013 - 06:13 PM

I read through the section a few times, but I don't see any examples on anything linear combination related. The professor is going to discuss this on Monday and I wanted to try to be ahead of the game. Here's what had me stumped:
 
For which values of h and k is the vector [k , 2] a linear combination of the vectors [1 , 2] and [3 , h]? 
 
Any takers?

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#195 Secret Igshar

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Posted 02 February 2013 - 07:18 PM

Set up a system of equations. :0

 

You want:

a*[1,2] + b*[3,h] = [k,2]

 

which becomes:

 

a + 3b = k

2a + hb = 2

 

First, do the cases a = 0, b = 0, and a=b=0, so that you can divide by b later.

a=0

3b = k

hb = 2

 

By assumption, b =/= 0 (implies k =/= 0), so

h = 2/b

k = 3b

 

b = k/3

h = 6/k

 

b=0

a = k

2a = 2

 

a = 1

k = 1

h can be any number (in the stated range?)

 

a=0,b=0 doesn't work.

 

now for both non-zero, it gets interesting.

 

k = a+3b => a = k - 3b

h = (2a - 2)/b

 

h = (2k - 6b - 2)/b

   = blahblahblah

 

not my homework you can figure out the rest I guess.


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#196 Secret Felix

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Posted 05 February 2013 - 12:19 AM

Bloop. Crisis averted.


Edited by Artemis, 05 February 2013 - 03:25 PM.

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#197 Lexaeus

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Posted 06 February 2013 - 07:08 PM

I was interested in that problem...

 

Where can I search online for a problem like that? It intrigued me.


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#198 Secret Igshar

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Posted 07 February 2013 - 12:59 AM

He's taking Ordinary Differential Equations. Pretty sure that one was solved by Separation of Variables.

 

It was (iirc)

 

y' = (y2-1)/x


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#199 Lexaeus

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Posted 07 February 2013 - 02:02 AM

What might be pretty bad is I took differtional equations and super aced it and i can't remember any of it. Well, hopefully messing around with this will refresh my mind.


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#200 Ralor

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Posted 19 February 2013 - 05:00 PM

I'm having problems with stuff on Theorems about Roots of Polynomial Equations. Please help.


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Im dum.


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