IA Type II -- Population Trends in China

[dessskris]

I've figured out what type of function I want to use, but I'm experiencing some extreme difficulties with "analytically" finding the values of a,b, and c in a cubic function. Any ideas on how I could find these values?

I just answered a very similar question yesterday but yeah.. You can try to substitute some sets of values of t and P and then solve it using matrices/substitution method/elimination method or anything..

I suggest you to read the thing I just posted yesterday and hopefully you'll get it

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Can anyone explain how to start the first question of the Portfolio

After your introduction, display a table of the data given in the task. Define variables: t is time, P is population, what the domains of each t and P are (real numbers? integers? positive integers? can t be 0 or something?) and define parameters.

Then plot the data given in a graphing software. Model your own function analytically, plot it together with the data series and comment on how accurate it is. And then ask your graphing software to find a more accurate function, compare it with your function and write some more stuffs asked in the task.

Good luck and have fun!

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do you guys graph the year on y-axis and the population on x-axis? or the other way round? because i graphed the year on the x-axis but the result seems like a straight line and when i did it the other way round, i got a curvy line. please help!

I've figured out what type of function I want to use, but I'm experiencing some extreme difficulties with "analytically" finding the values of a,b, and c in a cubic function. Any ideas on how I could find these values?

How do you decide which one to use? do you explain what functions you are going to use first on the paper? then finally you decide? is it logistic?

I am not doing this task so I don't know much about that but I suggest graphing population on the y-axis and time on the x-axis. Then model a 6th polynomial function mathematically. It actually looks kind of straight but if you zoom in, you'll see the curvy shape

Then after you plot the data, right, (I have no experience with type II IA's yet but I guess this is what I would do:) tell them what the graph looks like, what type of function it reminds you of. Then you say that you'll model a [insert the type of function here] function which has a general formula of...something. Then start finding the function algebraically

[Lysander]

Hi, I've been trying to develop a sixth order polynomial function for ages, and I just can't! I've tried using the least square equation (http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html), but when I get to the last bit, I just get an error when I attempt to calculate the inverse of the transposed matrix multiplied by the matrix itself, . How is it supposed to work? Reeaaalllly frustrating.

Thanks!

Edit: For the matrix, call it X, I keep getting <Error: Singular Matrix> on my GDC (Ti-84) for (XX^T)^-1

But, EVERYWHERE I'VE READ, IT SAYS THAT THIS IS SUPPOSED TO WORK. SO INFURIATING AGHHHH

Edited January 20, 2011 by Lysander

[dessskris]

Hi, I've been trying to develop a sixth order polynomial function for ages, and I just can't! I've tried using the least square equation (http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html), but when I get to the last bit, I just get an error when I attempt to calculate the inverse of the transposed matrix multiplied by the matrix itself, . How is it supposed to work? Reeaaalllly frustrating.

Thanks!

Edit: For the matrix, call it X, I keep getting <Error: Singular Matrix> on my GDC (Ti-84) for (XX^T)^-1

But, EVERYWHERE I'VE READ, IT SAYS THAT THIS IS SUPPOSED TO WORK. SO INFURIATING AGHHHH

That is not the way. To solve 6x6 system, you need to do RREF. Get the augmented matrix in a reduced row echelon form. This is not taught in SL, I believe. So you need to see your teacher, ask them to teach you RREF.

Or maybe I can just tell you

I really suggest seeing your teacher though otherwise they will be wondering where you get this from.

Still unclear? Ask again. My handwriting is ugly btw so let me know if you can't read that.

Edited May 4, 2011 by dessskris

[Lysander]

Are you a wizard? You're amazing, thank you so much! Just wondering though, is it possible to RREF with a non-square matrix? Since I have 10 data values, and 7 unknowns, would it be possible to RREF that? Or should I attempt an order 10 polynomial (that's a bit much though)? Really, thanks for all the effort you put into that! And I'll def speak to my teacher about it.

Edit: Ok, so I got it, but it only gives me 3 values in the far right column. Does this mean that it's actually a 3rd order polynomial? I did found the 6th order polynomial function through the trendline function in Excel though. Thanks again!

Edited January 20, 2011 by Lysander

[dessskris]

Are you a wizard? You're amazing, thank you so much! Just wondering though, is it possible to RREF with a non-square matrix? Since I have 10 data values, and 7 unknowns, would it be possible to RREF that? Or should I attempt an order 10 polynomial (that's a bit much though)? Really, thanks for all the effort you put into that! And I'll def speak to my teacher about it.

Edit: Ok, so I got it, but it only gives me 3 values in the far right column. Does this mean that it's actually a 3rd order polynomial? I did found the 6th order polynomial function through the trendline function in Excel though. Thanks again!

Ah thank you! No.. rref is possible for square matrices only. If you have 10 sets of values of x and y (t and P) then just pick 7 of those nobody would want to find a 10th polynomial anyway.. that's too confusing

3 values? Er can you get a screenshot and put it here or something..? PM me if you don't want other people to see it.

If the a, b, and c are positive or negative real numbers and the rest are zeroes, then their values are zeroes, given that the 7x7 on the left is a 0 matrix

Then the equation would be y=ax⁶+bx⁵+cx⁴ and that's it!

[Lysander]

Are you a wizard? You're amazing, thank you so much! Just wondering though, is it possible to RREF with a non-square matrix? Since I have 10 data values, and 7 unknowns, would it be possible to RREF that? Or should I attempt an order 10 polynomial (that's a bit much though)? Really, thanks for all the effort you put into that! And I'll def speak to my teacher about it.

Edit: Ok, so I got it, but it only gives me 3 values in the far right column. Does this mean that it's actually a 3rd order polynomial? I did found the 6th order polynomial function through the trendline function in Excel though. Thanks again!

Ah thank you! No.. rref is possible for square matrices only. If you have 10 sets of values of x and y (t and P) then just pick 7 of those nobody would want to find a 10th polynomial anyway.. that's too confusing

3 values? Er can you get a screenshot and put it here or something..? PM me if you don't want other people to see it.

If the a, b, and c are positive or negative real numbers and the rest are zeroes, then their values are zeroes, given that the 7x7 on the left is a 0 matrix

Then the equation would be y=ax⁶+bx⁵+cx⁴ and that's it!

Yeah, instead of the "1" indicating the coefficient, I just get a 0. And sorry, can't post a screencap at the moment, my calculator has decided not to turn on. New batteries and everything... Eugh. I'll try to in a bit though! I'm going to try and plot the graph and see how it looks.

Edit (I sure do love edit): The RREF on my calc only gave me a 6x4 rref'd matrix... Then I started scrolling across. There's so many values. Hahahah... Yeah pretty sure it's a problem with my calculator. I'll figure it out, thanks for everything though!

Edited January 20, 2011 by Lysander

[dessskris]

Yeah, instead of the "1" indicating the coefficient, I just get a 0. And sorry, can't post a screencap at the moment, my calculator has decided not to turn on. New batteries and everything... Eugh. I'll try to in a bit though! I'm going to try and plot the graph and see how it looks.

Edit (I sure do love edit): The RREF on my calc only gave me a 6x4 rref'd matrix... Then I started scrolling across. There's so many values. Hahahah

Hahaha thanks for editing the previous post instead of just multiposting!

Huh? I still don't get that. Any chance you could type it or something? Or just give me the values in each row, like 1,0,0,0,0,0,0,5 or something like that. If the 7x7 on the left isn't a 0 matrix you need to perform some calculations and assume a variable to be a parameter! I can help you through, don't worry. Is this too much for you? Instead of rref you can do substitution or elimination method, but all of them are confusing so just pick the best method lah

[Lysander]

Yeah, instead of the "1" indicating the coefficient, I just get a 0. And sorry, can't post a screencap at the moment, my calculator has decided not to turn on. New batteries and everything... Eugh. I'll try to in a bit though! I'm going to try and plot the graph and see how it looks.

Edit (I sure do love edit): The RREF on my calc only gave me a 6x4 rref'd matrix... Then I started scrolling across. There's so many values. Hahahah

Hahaha thanks for editing the previous post instead of just multiposting!

Huh? I still don't get that. Any chance you could type it or something? Or just give me the values in each row, like 1,0,0,0,0,0,0,5 or something like that. If the 7x7 on the left isn't a 0 matrix you need to perform some calculations and assume a variable to be a parameter! I can help you through, don't worry. Is this too much for you? Instead of rref you can do substitution or elimination method, but all of them are confusing so just pick the best method lah

Ok so here's what's happening... Sorry if I'm a bit of a bother! But if I scroll across, i just get tonnes of values on those 4 rows, like in the second picture.

SCREEN05.BMP

SCREEN02.BMP

[dessskris]

Ok so here's what's happening... Sorry if I'm a bit of a bother! But if I scroll across, i just get tonnes of values on those 4 rows, like in the second picture.

No, no problem ok so you actually get the solution already, right? Be careful of the E..you may want to change the settings in your calculator to give you the answer in 3 sig fig or something so that the answers would be like 0.000291 or whatever instead of answers with E^-k. Good luck with the rest of the task!

[Lysander]

Ok so here's what's happening... Sorry if I'm a bit of a bother! But if I scroll across, i just get tonnes of values on those 4 rows, like in the second picture.

No, no problem ok so you actually get the solution already, right? Be careful of the E..you may want to change the settings in your calculator to give you the answer in 3 sig fig or something so that the answers would be like 0.000291 or whatever instead of answers with E^-k. Good luck with the rest of the task!

Ahh ok, well since it's not giving me a full range of values I'll probably just try for a lower order. Hope this helps other people too. Thanks!

[bittersweetamor]

Hey! So, I’ve gotten EVERYTHING done in my IA except for the matrix/matrices part. I’ve looked at post #203, but I still don’t understand how to work it and how to apply it to the task. Unfortunately, we haven’t learned matrices yet, so I have absolutely no idea what to do! Are they a very significant part of this task or would I be able to complete it without doing the matrices?

Your help is very appreciated!

[helenpark]

when it says "analytically develop one model function", does that mean we can't use technology to find the function?

[dessskris]

Hey! So, I’ve gotten EVERYTHING done in my IA except for the matrix/matrices part. I’ve looked at post #203, but I still don’t understand how to work it and how to apply it to the task. Unfortunately, we haven’t learned matrices yet, so I have absolutely no idea what to do! Are they a very significant part of this task or would I be able to complete it without doing the matrices?

Your help is very appreciated!

Don't worry, they are not that significant matrices are used to obtain the formula of the function that you need to develop. Substitution/elimination can be done instead of the matrix operations

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when it says "analytically develop one model function", does that mean we can't use technology to find the function?

You can't use your graphing software or GDC to find the function by just plotting it and finding the best fit. But you can use technology to solve matrices or do simple calculations

[Rebel]

what kind of calculations should we do? X_X Im stuck right in the beginning of this task...

[dessskris]

what kind of calculations should we do? X_X Im stuck right in the beginning of this task...

Right, first you need to introduce the problem in the task. Then you should show a table of the data given in the task. After that, plot population against time. What do you see? What type of function does it remind you of? State the general form of that type of function and then find the function through some calculations. You may choose some coordinates and then substitute the values of x and y to the general formula and then solve it using matrices/substitution/elimination.

Any further question just ask!

[Rebel]

The function looks like exponential but in theory it should be y=tanX function because of the b.c

[dessskris]

The function looks like exponential but in theory it should be y=tanX function because of the b.c

Because of the what?

No xP

To me it looks like a perfect 6th polynomial function. Up to you though, I guess if you can get a function that fits the data points well you'll be fine!

[Lysander]

I still can't "analytically" find the function for this damn data. I've been trying for at least a week and I've tried EVERYTHING! RREF, least squares, matrix method... I'm just so stuck. I just don't know what to do.

[dessskris]

I still can't "analytically" find the function for this damn data. I've been trying for at least a week and I've tried EVERYTHING! RREF, least squares, matrix method... I'm just so stuck. I just don't know what to do.

Mind to PM me your equations? I can help you with RREF and finding the equation from there. FYI, I am not doing this portfolio so you don't need to worry about plagiarism issues

PS. You can solve it using elimination or substitution too but I swear it would be veeeeery complicated so I better help you with RREF!

[sick_of_IB]

I'm analytically developing a model based on:

At³+Bt²+Ct+554.8 = P

Any help on how to solve the three constants A, B and C?

Substitute some sets of values of t and P and solve it using matrices..?

Say (t, P) are (1, 2), (2, 3) and (3,4). I am picking random constants btw.

Then you will get:

A(1)³+B(1)²+C+554.8 = 2

A+B+C=-552.8

A(2)³+B(2)²+2C+554.8 = 3

8A+4B+2C=-551.8

A(3)³+B(3)²+3C+554.8 = 4

27A+9B+3C=-550.8

So you now have:

A+B+C=-552.8

8A+4B+2C=-551.8

27A+9B+3C=-550.8

Then you can solve it with matrices. I doubt if you learn it in SL but I'll do RREF. Or you can try using substitution or elimination

PS: if it was a quadratic function I would find the discriminant to get some expression. idk what we can do with cubic functions though, sorry.

hi,

im trying to use the same model, but how did you get 554.8 instead of d in your original function??

Hi, I've been trying to develop a sixth order polynomial function for ages, and I just can't! I've tried using the least square equation (http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html), but when I get to the last bit, I just get an error when I attempt to calculate the inverse of the transposed matrix multiplied by the matrix itself, . How is it supposed to work? Reeaaalllly frustrating.

Thanks!

Edit: For the matrix, call it X, I keep getting <Error: Singular Matrix> on my GDC (Ti-84) for (XX^T)^-1

But, EVERYWHERE I'VE READ, IT SAYS THAT THIS IS SUPPOSED TO WORK. SO INFURIATING AGHHHH

That is not the way. To solve 6x6 system, you need to do RREF. Get the augmented matrix in a reduced row echelon form. This is not taught in SL, I believe. So you need to see your teacher, ask them to teach you RREF.

Or maybe I can just tell you

I really suggest seeing your teacher though otherwise they will be wondering where you get this from.

Still unclear? Ask again. My handwriting is ugly btw so let me know if you can't read that.

you say to " put in the constants instead of the values above". what do you mean?? should I put zeros instead of letters a, b,c ..., or what ??