Post by: Joseph41 on January 30, 2019, 03:23:07 pm
QCE SPECIALIST MATHS Q&A THREAD
What is this thread for?
If you have general questions about the QCE Specialist Maths course (both Units 1&2 and 3&4) or how to improve in certain areas, this is the place to ask! 👌
Who can/will answer questions?
Everyone is welcome to contribute; even if you're unsure of yourself, providing different perspectives is incredibly valuable.
Please don't be dissuaded by the fact that you haven't finished Year 12, or didn't score as highly as others, or your advice contradicts something else you've seen on this thread, or whatever; none of this disqualifies you from helping others. And if you're worried you do have some sort of misconception, put it out there and someone else can clarify and modify your understanding!
There'll be a whole bunch of other high-scoring students with their own wealths of wisdom to share with you, so you may even get multiple answers from different people offering their insights - very cool.
To ask a question or make a post, you will first need an ATAR Notes account. You probably already have one, but if you don't, it takes about four seconds to sign up - and completely free!
Post by: Twisty314 on February 07, 2019, 07:25:27 am
Using the formula for
I got mixed up so after so many attempts. Sorry about the formatting of nCr, but if it is confusing:
There is the
Thanks all! :)
Post by: RuiAce on February 07, 2019, 08:43:50 am
Hey everyone! Need help with this question. Really troubling me. :(For the \(^nC_r\) notation you're gonna require superscripts and subscripts. You can do a bit of exploring around if you're curious. Basically superscripts and subscripts are required because that notation isn't by default built into \(\LaTeX\), so you have to manually work around it.
Using the formula for, prove that
where
I got mixed up so after so many attempts. Sorry about the formatting of nCr, but if it is confusing:
There is theand lastly
.
Thanks all! :)
\[ \text{We still need }n! = n(n-1)!\text{ for these problems.}\\ \text{Here it looks like we need that identity three times.} \]
\begin{align*}\binom{n-1}{r-1} + \binom{n-1}{r} &= \frac{(n-1)!}{(r-1)! ((n-1)-(r-1))!} + \frac{(n-1)!}{r! (n-1-r)!}\\ &= (n-1)! \left[ \frac{1}{(r-1)!(n-r)!} + \frac{1}{r!(n-r-1)!} \right]\\&= (n-1)! \left[ \frac{r}{r!(n-r)!} + \frac{n-r}{r!(n-r)!} \right]\\ &= (n-1)!\cdot \frac{r+n-r}{r!(n-r)!}\\ &= \frac{(n-1)! n}{r!(n-r)!}\\ &= \frac{n!}{r!(n-r)!}\\ &= \binom{n}{r} \end{align*}
The thing about that factorial identity is that it helps us make the whole 'lowest common denominator for adding fractions' idea work. It's really nifty when trying to prove results with \(^nP_r\) and \(^nC_r\) if we just want to use their respective formulas. Gotta be clever with using it :P
Post by: Twisty314 on February 08, 2019, 08:21:58 pm
Post by: WillHansen on September 23, 2019, 10:56:31 am
Post by: e_grace on October 06, 2019, 11:45:01 pm
Post by: e_grace on October 08, 2019, 09:26:01 pm
Post by: Joseph41 on October 09, 2019, 12:35:12 pm
Can someone please tell me how much specialist has scaled in the past? like in other states. has a 76% go up to a 96% in the past ? or yeah? Thank you!
It works a little differently in other states. For example, in Victoria, overall subject scores are out of 50. In 2018, a raw study score of 30 (which is the average) in Specialist Maths resulted in a scaled score of 41.
These numbers probably don't mean heaps to you because a) the system is entirely different, and b) they're not out of 100. But I think it's important to realise that even for QCE, subject scores aren't the same as percentages. For example, getting an average of 76% in Specialist Maths doesn't mean you'll get a subject score of 76 for Specialist Maths necessarily.
Ultimately, even if Spesh did scale that much in QCE, I don't think it's really worth considering. Getting high or low scaling doesn't reward or punish you; instead, it just negates a very competitive or less competitive cohort. What scaling does is actually equal the playing field across subjects, meaning you can simply choose subjects you enjoy, are passionate about, are good at, or need as university pre-requisites.
In general, I'd encourage you not to give too much weight to scaling in subject selection, particularly given we have absolutely zero relevant data to go off. :)
Post by: Specialist_maths on October 19, 2019, 09:47:26 pm
Can someone please tell me how much specialist has scaled in the past? like in other states. has a 76% go up to a 96% in the past ? or yeah? Thank you!First, I would agree with Joseph's advice: don't worry too much about scaling - just focus on doing your best in all your subjects.
With regards to scaling for an ATAR, your overall results (/100) are used to determine your rank in the cohort. It doesn't really matter what your results are - it's how your results compare to the rest of the cohort (in each subject across the state). It's this percentile that is scaled.
If the 2020 Queensland Specialist Mathematics cohort has a similar distribution of results to what other cohorts typically achieve in other states, I would imagine 76/100 is an above average score and would scale well (possibly to about 95% - however, no one can say with any certainty at this point).
Post by: A.Rose on March 06, 2020, 03:36:11 pm
I am doing a PSMT about Leslie Matrices and I'm trying to talk about the properties of a Leslie matrix that causes a population to have high growth and one that causes a population to decline. What constitutes a high survival or birth rate? Is there somewhere I can find this information on the internet so I can reference it in my report? What are the features of a Leslie matrix that will cause that continuous increase? I know high survival and birth rates would contribute but what does it mean by a 'high' rate?
Thank you!
Post by: Bri MT on March 06, 2020, 04:09:38 pm
Hello
I am doing a PSMT about Leslie Matrices and I'm trying to talk about the properties of a Leslie matrix that causes a population to have high growth and one that causes a population to decline. What constitutes a high survival or birth rate? Is there somewhere I can find this information on the internet so I can reference it in my report? What are the features of a Leslie matrix that will cause that continuous increase? I know high survival and birth rates would contribute but what does it mean by a 'high' rate?
Thank you!
Hey!
You might find it easier to find/understand information by looking up life history tables as these display the same information (well, sometimes not fecundity/reproduction) but in a less mathematical way. There's not really set numbers for high vs low - it's all relative. In ecology, we talk about life history as involving trade-offs in somatic and gametic investment - members of a population can't 100% put their resources to producing offspring or 100% to their own survival, it's a balancing act. I've written about life history a bit here (please note this is well outside the spec syllabus).
One thing you will want to consider is multiplying population size in each age class by fecundity in the corresponding age class to find the expected number of offspring produced in that age class. Comparison of this resulting distribution and the one you started with will help you with the insights you're interested in.
Hope this helps!
Ecology is my major at uni so definitely happy to talk about population ecology :D
Post by: A.Rose on March 21, 2020, 01:12:06 pm
I doing Applications of integration in Specialist maths this term and I'm having trouble with this volumes of solids of revolutions question (see attached).
I don't have any worked solutions for this question so if you could step me through it that would be amazing.
Thank you!!
Post by: A.Rose on March 21, 2020, 02:36:07 pm
Thank you so much!
Post by: A.Rose on March 21, 2020, 06:26:28 pm
I'm not quite sure how to do conditional probability for exponential probability distributions and I'm stuck on part c) of attached question.
Hopefully, that's my last question for now.
Much appreciated! :D
Post by: fun_jirachi on March 21, 2020, 06:54:13 pm
For your first question, the integral is simplified quite a bit for you. Recall that when you rotate a curve around the x-axis, the volume is equal to \(\pi \int_a^b y^2 \ dx\). In both cases, a y2 already appears, so all you have to do is find the point of intersection between the two curves, and then use the volume formula to integrate the correct curves within the upper and lower bounds you've found.
For your second question, it's a matter of rearranging the equation so you have a function of x in terms of y, so you can use the formula \(V = \pi \int_a^b x^2 \ dy\) for volumes around the y-axis. Make sure you have the correct upper and lower bounds as well so you get the correct answer! :)
If you need more assistance than this, don't hesitate to ask! It's just often a better idea to point you in the right direction as opposed to giving you the answer outright :)
Post by: Bri MT on March 22, 2020, 10:02:54 am
Sorry, I have another question! ;D
I'm not quite sure how to do conditional probability for exponential probability distributions and I'm stuck on part c) of attached question.
Hopefully, that's my last question for now.
Much appreciated! :D
Hey,
Recall that:
\[ p_{x|y} = \frac{p_{X,Y} (x,y)}{p_X (x)} \]
Let's think about what this means. We know that something being greater than 5 AND greater than 10, really just means it's greater than 10. So this allows us to find \( p_{X,Y} (x,y) \) (use your result from step a)
Let me know if you're still confused and I'm happy to step you through it but it would be great to see your working/thoughts :)
Post by: A.Rose on June 01, 2020, 08:38:14 pm
I have a question from Unit 4 Topic 2 and its about forces. This chapter brings in a lot of vectors and I'm quite rusty on Vectors from Unit 3 last year so I need some help with this question.
I was trying to do vector projection for this question but I was having trouble. I would greatly appreciate the help!
The question I am stuck on is part b.
Thank you!!
Post by: Bri MT on June 02, 2020, 08:54:26 am
Hi!
I have a question from Unit 4 Topic 2 and its about forces. This chapter brings in a lot of vectors and I'm quite rusty on Vectors from Unit 3 last year so I need some help with this question.
I was trying to do vector projection for this question but I was having trouble. I would greatly appreciate the help!
The question I am stuck on is part b.
Thank you!!
Hey!
You know how for part a you can make triangles from each of the vectors to EF by adding in a vertical line, then applying the trig formulas?
For part b they've done the same thing except the triangle is red line + blue line + line perpendicular to the blue line. The angle between red line and blue line is 40 + 15.
Let me know if this clarifies things or if you still have questions :)
Post by: A.Rose on June 02, 2020, 09:46:48 am
That took me a little while to get my head around. For some reason, I had it in my head to do vector projection as the textbook example switched between vector projection and trig methods throughout.
So for these questions, you have to almost treat the direction vector as if it were horizontal and resolve the other vectors in terms of the direction vector? Sort of? Looking at it that way made sense for 11b but is there a more accurate interpretation?
Thanks again ;)
Post by: Bri MT on June 02, 2020, 10:19:51 am
Oh ok, Thank you.
That took me a little while to get my head around. For some reason, I had it in my head to do vector projection as the textbook example switched between vector projection and trig methods throughout.
So for these questions, you have to almost treat the direction vector as if it were horizontal and resolve the other vectors in terms of the direction vector? Sort of? Looking at it that way made sense for 11b but is there a more accurate interpretation?
Thanks again ;)
No worries!
You could also view the same maths I described as the dot product of the red line with a unit vector in the direction of the blue line. Both interpretations are numerically equivalent.
I think the trick here for you might be recognising that we don't want the magnitude of the red line influencing our answer, we just want to be using the direction. Therefore, if we use the dot product we use the red line divided by its magnitude, or we can use the trig method instead.
Hope this helps :)
Edit:
with the horizontal stuff, there's nothing in the question or any reference frame in the diagram to indicate that horizontal is meaningful or special - we could make anything "horizontal" if we wanted :)
Post by: orla007 on July 14, 2020, 06:16:23 pm
Post by: 1729 on July 14, 2020, 07:36:41 pm
Hi! I just need some help with this question. Thanks in advance :)Hey orla007!
You want to find the horizontal and vertical components of the force such that the net horizontal accel is 0.25g draw an FBD first and foremost, aka as in how do you draw it or what are the forces.
Post by: Bri MT on July 15, 2020, 12:10:59 pm
To build on the guidance 1729 has given, there are some key things to remember:
- the normal reaction force acts perpendicular to the surface the block is on (the table is horizontal so this force is up), and the block is initially at rest because this force is cancelling out the weight force.
- You can split up the 45 degree angle into horizontal and vertical components using trigonometry (in this case pythag works but often you'll want to use SOH CAH TOA)
- net force = mass * acceleration
- for net force you add forces in the same direction together and subtract forces in the opposite direction (i.e. frictional force is subtracted)
Have you drawn a force body diagram before?
Post by: lamh21 on October 06, 2020, 10:41:40 pm
I was wondering if anyone was able to potentially help me out with a Mathematical Induction Question below. Thank you! :)
Prove by induction that n^5− n is divisible by 240 for each odd positive integer n.
Post by: keltingmeith on October 07, 2020, 12:23:56 pm
Hi All!
I was wondering if anyone was able to potentially help me out with a Mathematical Induction Question below. Thank you! :)
Prove by induction that n^5− n is divisible by 240 for each odd positive integer n.
Yeah, so the trick with these types of induction proofs is you want to show that the LHS turns into some RHS that's equal to 240*(some number). For example, 480 is a multiple of 240 because 480=240*(2). If you were to get something disgusting out the end that looks like n^5-n=240*(n^100 - n^432 + 54432323n^2 + 5 - 45n) or something else that stupid, it's still 240*(some number), and so is still divisible by 240. Since it's an odd number, you also need to make sure your inductive proof only goes through odd numbers. There are two ways to do this:
1. Prove this is true for n=2m+1, instead of for n, or:
2. Make sure your base case is an odd number (so, start with n=1), then prove it true for n=k+2 instead of n=k+1 (WHY would this work??)
Otherwise, it's still an inductive proof at its heart. I want you to try this out for yourself first, so here's an example using an example question that you can use to see my hints in action:
Prove by induction that n3-n is divisible by 24 for all odd positive integers
So, first, the base case - n=1:
Which is divisible by 24. So, step 2 - assume it's true for n=k.
... Done
Okay, step 3. Let's see if this is true for n=k+2:
Okay, so we know that k^3-k is divisible by 24, so I'm going to substitute a 24x into there - because we don't care WHAT value it is exactly, just that it IS divisible by 24. I also know that k is an odd number, so k+1 HAS to be even - so I'm just going to call that 2y. Because again, I don't care EXACTLY what the number is, I just care about what it's divisible by. So this gives me:
Which is a multiple of 24, and completes the proof
---
So, some questions I often get asked:
a) how did I know to make that expansion and factorisation in the steps?
I didn't - but it was either do that or do anything. With all of these proofs, I have no idea what direction I need to move. But, if I don't move forwards, I won't get anywhere - expanding at the start is the only thing I could do, so I did it. And every time I expand something, I expect I need to factorise it later, so when I recognise something I CAN factorise - I do it. If I factorise, and it turns out that that's NOT useful, then I can always just expand it in the next step and move on.
b) how did I know to make k+1=2y?
I didn't. All I know is, the more I can reduce things to stuff they're divisible by, the easier these proofs become - so I saw I could turn k+1 into a multiple of something, and I ran from there.
---
Also, if you're interested, here's how you'd do it using the set n=2m+1 method.
Step 1: Prove this is true for the base case, m=0:
Done, simple. Now, assume this is true for m=k
... Done
Now, let's test m=k+1:
From here, just pick another variable (say, L=2k+1), and this follows the same steps as the one above.
Post by: lamh21 on October 07, 2020, 07:14:03 pm
Yeah, so the trick with these types of induction proofs is you want to show that the LHS turns into some RHS that's equal to 240*(some number). For example, 480 is a multiple of 240 because 480=240*(2). If you were to get something disgusting out the end that looks like n^5-n=240*(n^100 - n^432 + 54432323n^2 + 5 - 45n) or something else that stupid, it's still 240*(some number), and so is still divisible by 240. Since it's an odd number, you also need to make sure your inductive proof only goes through odd numbers. There are two ways to do this:Ahh I see! Thank you so much for your help! :)
1. Prove this is true for n=2m+1, instead of for n, or:
2. Make sure your base case is an odd number (so, start with n=1), then prove it true for n=k+2 instead of n=k+1 (WHY would this work??)
Otherwise, it's still an inductive proof at its heart. I want you to try this out for yourself first, so here's an example using an example question that you can use to see my hints in action:
Prove by induction that n3-n is divisible by 24 for all odd positive integers
So, first, the base case - n=1:
Which is divisible by 24. So, step 2 - assume it's true for n=k.
... Done
Okay, step 3. Let's see if this is true for n=k+2:
Okay, so we know that k^3-k is divisible by 24, so I'm going to substitute a 24x into there - because we don't care WHAT value it is exactly, just that it IS divisible by 24. I also know that k is an odd number, so k+1 HAS to be even - so I'm just going to call that 2y. Because again, I don't care EXACTLY what the number is, I just care about what it's divisible by. So this gives me:
Which is a multiple of 24, and completes the proof
---
So, some questions I often get asked:
a) how did I know to make that expansion and factorisation in the steps?
I didn't - but it was either do that or do anything. With all of these proofs, I have no idea what direction I need to move. But, if I don't move forwards, I won't get anywhere - expanding at the start is the only thing I could do, so I did it. And every time I expand something, I expect I need to factorise it later, so when I recognise something I CAN factorise - I do it. If I factorise, and it turns out that that's NOT useful, then I can always just expand it in the next step and move on.
b) how did I know to make k+1=2y?
I didn't. All I know is, the more I can reduce things to stuff they're divisible by, the easier these proofs become - so I saw I could turn k+1 into a multiple of something, and I ran from there.
---
Also, if you're interested, here's how you'd do it using the set n=2m+1 method.
Step 1: Prove this is true for the base case, m=0:
Done, simple. Now, assume this is true for m=k
... Done
Now, let's test m=k+1:
From here, just pick another variable (say, L=2k+1), and this follows the same steps as the one above.
Post by: Adfer on October 09, 2020, 05:22:22 pm
Post by: Bri MT on October 09, 2020, 05:28:10 pm
Hi, I’ve attempted the attached question and I’m quite stuck. I have also attached my attempted solution, however I am unsure where I have gone wrong. Thanks in advance for any help :)
Hi,
Welcome to the forums!
I can't see any attached question, could you please edit your post and try attempting to attach it again, embedding an image of the question, or writing the question out for us? if you're unsure about how to do this there are instructions here and please feel free to ask as well :)
Post by: Adfer on October 09, 2020, 05:39:04 pm
Hi,
Welcome to the forums!
I can't see any attached question, could you please edit your post and try attempting to attach it again, embedding an image of the question, or writing the question out for us? if you're unsure about how to do this there are instructions here and please feel free to ask as well :)
Haha my apologies, took a while to attach everything - but it should be there now. I've attached the final part of my working to this post as it was too large to put all on the post above.
Post by: Britnium on October 17, 2020, 06:13:34 pm
I'm currently doing a PSMT on Leslie Matrices; I had believed that I've gotten a good chunk of the assignment done but after hearing other information floating around the grade I think I'm a little lost.
So our task is to model population trends of the Tasmanian devil since the documentation of the Devil Facial Tumor Disease in 1996 up until 2030 to determine whether or not the species will go extinct.
I've calculated the initial female age distributions and now I just need to develop a Leslie matrix (which is 7x7) to model the population trends. Currently, they have provided us with the following birth and survival rates, all of which are for healthy devils, so the challenge I'm facing is determining these rates for disease-affected populations.
Survival rates for rates based on historical data for disease-free populations (where s0 = probability of surviving the 0-1 age interval):
s0 = 0.39
s1 = 0.82
s2 = 0.82
s3 = 0.82
s4 = 0.82
s5 = 0.27
s6 = 0
Breeding numbers (female per female devil):
m0 = 0
m1 = 0.03
m2 = 0.86
m3 = 1.55
m4 = 1.55
m5 = 1.55
m6 - 0.86
We've also been provided with relevant research, which I'm 99% sure we're to use for developing our survival rates. Please see the screenshots attached.
My initial guess was that we were to select appropriate survival rates from the range of 0.1-0.6, model the trends using these numbers and compare the obtained populations with actual statistics (e.g. 50% killed from 1996-2007) to establish validity in the model and change the rates where necessary to match up with these figures and thus 'refine' our model. However, recently there seems to be a stress on the sentence 'a large host population will experience a rapid decline followed by stabilization and eventual return to pre-disease numbers.' Under the assumption that birth rates will stay the same, I'm completely unsure of how to obtain survival rates that would give us this stabilization point and subsequently model the return to pre-disease numbers. And would this all be done under one Leslie matrix? Or would we expect to have multiple to model different periods of time?
Any help on this would be greatly appreciated. Sorry for such the long question and apologies if my stress has gotten to you too :'(
Post by: keltingmeith on October 19, 2020, 11:36:59 am
Hello!
I'm currently doing a PSMT on Leslie Matrices; I had believed that I've gotten a good chunk of the assignment done but after hearing other information floating around the grade I think I'm a little lost.
So our task is to model population trends of the Tasmanian devil since the documentation of the Devil Facial Tumor Disease in 1996 up until 2030 to determine whether or not the species will go extinct.
I've calculated the initial female age distributions and now I just need to develop a Leslie matrix (which is 7x7) to model the population trends. Currently, they have provided us with the following birth and survival rates, all of which are for healthy devils, so the challenge I'm facing is determining these rates for disease-affected populations.
Survival rates for rates based on historical data for disease-free populations (where s0 = probability of surviving the 0-1 age interval):
s0 = 0.39
s1 = 0.82
s2 = 0.82
s3 = 0.82
s4 = 0.82
s5 = 0.27
s6 = 0
Breeding numbers (female per female devil):
m0 = 0
m1 = 0.03
m2 = 0.86
m3 = 1.55
m4 = 1.55
m5 = 1.55
m6 - 0.86
We've also been provided with relevant research, which I'm 99% sure we're to use for developing our survival rates. Please see the screenshots attached.
My initial guess was that we were to select appropriate survival rates from the range of 0.1-0.6, model the trends using these numbers and compare the obtained populations with actual statistics (e.g. 50% killed from 1996-2007) to establish validity in the model and change the rates where necessary to match up with these figures and thus 'refine' our model. However, recently there seems to be a stress on the sentence 'a large host population will experience a rapid decline followed by stabilization and eventual return to pre-disease numbers.' Under the assumption that birth rates will stay the same, I'm completely unsure of how to obtain survival rates that would give us this stabilization point and subsequently model the return to pre-disease numbers. And would this all be done under one Leslie matrix? Or would we expect to have multiple to model different periods of time?
Any help on this would be greatly appreciated. Sorry for such the long question and apologies if my stress has gotten to you too :'(
I really, really, really want to help you. I don't want your opinion of AN to be that we don't care or don't want to help with QCE. But holy shit, this is nuts. I have a university degree in mathematical statistics, and now I'm doing a PhD where I need to use statistics all the time. That's literally what all of this is - it's you, using hard maths, to answer questions based on statistical data. I had never heard of a Leslie matrix before. This is like, some real in-depth level of statistics that's only relevant to high-level ecology and zoology studies. It's incredibly niche. So I'm really sorry if I'm not able to effectively help you after this, but I'm not helping you based on something I know - I'm helping you based on my learning this on wikipedia
What I think that stressed sentence is telling you isn't that you've approached this entirely incorrectly - however, a Leslie matrix should explain how a population will grow under a specific set of circumstances. A Leslie matrix can't explain if the population numbers suddenly, massively, decrease. In fact, have you ever taken a Leslie matrix and run it to a ridiculously high number of generations? Usually, the populations then become disgustingly big. Like, orders of magnitude bigger than you'd think they'd grow to. The Leslie matrix can't predict a sudden population drop like that. Eg, in the 1820s, suddenly people were killing off devils - the Leslie matrix can't predict that, because it's designed to only consider the natural life expectancy of the devil, and how fertile the mothers of different ages are. Another example - let's say you start with a group of bee hives, and make a Leslie matrix for them. You then find after 10 years, that all of a sudden there aren't enough flowers for the bees to all get pollen from, so some hives start dying out. The Leslie matrix could not predict that this will happen, because you never gave that information. Basically - there are circumstances that the Leslie matrix CAN'T predict, and you need to account for them.
But then, consider this - what happens when those stop becoming issues? Let's say you send 90% of the hives to an entirely different area that doesn't overlap with the original 10%'s search areas. Well then, now the lack of flowers isn't an issue, because there's more than enough flowers for the bees to pollinate from, so the original Leslie matrix becomes applicable again. And similarly for the devils - when humans stop hunting those devils, they should grow according to the same level of dynamics they did before, so the same Leslie matrix SHOULD apply again. The same goes for that 2007 information - after a while, that virus is either defeated, or the devils are now immune to it, so after that point, the same original Leslie matrix SHOULD apply again.
So the question of "should I need more than one Leslie matrix" is a little complicated. I think that likely what will happen is you should only need one Leslie matrix after each population crash, and that matrix should correspond to the rise of each population. If the rise of each population doesn't match that Leslie matrix, then circumstances are sufficiently difference (and it's worth trying to figure out what those circumstances are), that the birth or survival rate of the devil has been changed - and from there, a new Leslie matrix would need to be constructed to match those changes.
Having said all of that, this is all extremely complex stuff, and I 100% feel it's worth talking to your teacher and getting any input they can on this matter.
Post by: Bri MT on October 19, 2020, 02:26:01 pm
Hello!
I'm currently doing a PSMT on Leslie Matrices; I had believed that I've gotten a good chunk of the assignment done but after hearing other information floating around the grade I think I'm a little lost.
So our task is to model population trends of the Tasmanian devil since the documentation of the Devil Facial Tumor Disease in 1996 up until 2030 to determine whether or not the species will go extinct.
I've calculated the initial female age distributions and now I just need to develop a Leslie matrix (which is 7x7) to model the population trends. Currently, they have provided us with the following birth and survival rates, all of which are for healthy devils, so the challenge I'm facing is determining these rates for disease-affected populations.
Survival rates for rates based on historical data for disease-free populations (where s0 = probability of surviving the 0-1 age interval):
s0 = 0.39
s1 = 0.82
s2 = 0.82
s3 = 0.82
s4 = 0.82
s5 = 0.27
s6 = 0
Breeding numbers (female per female devil):
m0 = 0
m1 = 0.03
m2 = 0.86
m3 = 1.55
m4 = 1.55
m5 = 1.55
m6 - 0.86
We've also been provided with relevant research, which I'm 99% sure we're to use for developing our survival rates. Please see the screenshots attached.
My initial guess was that we were to select appropriate survival rates from the range of 0.1-0.6, model the trends using these numbers and compare the obtained populations with actual statistics (e.g. 50% killed from 1996-2007) to establish validity in the model and change the rates where necessary to match up with these figures and thus 'refine' our model. However, recently there seems to be a stress on the sentence 'a large host population will experience a rapid decline followed by stabilization and eventual return to pre-disease numbers.' Under the assumption that birth rates will stay the same, I'm completely unsure of how to obtain survival rates that would give us this stabilization point and subsequently model the return to pre-disease numbers. And would this all be done under one Leslie matrix? Or would we expect to have multiple to model different periods of time?
Any help on this would be greatly appreciated. Sorry for such the long question and apologies if my stress has gotten to you too :'(
Hello!
Throwing in my 2 cents as an ecology and conservation biology major with a stats minor.
I don't think you need to worry about the whole "stabilisation thing" in terms of a levelling out as would often be seen in population ecology models. One of the potential explanations in the doc was about the devils developing immunity and so I've modelled what that might look like roughly making up some parameters based on the provided info. I've done this by adapting work from one of my labs in a final year of uni ecology subject so yeah.... you should not be expected to do this but I thought it might be interesting to show you so you can see that even a more ecologically informed model is not going to perfectly line up with your data.
(https://i.imgur.com/251nJ6n.png)
The key way that this works is that the mean absolute fitness of the population is initially below 1 in the presence of the selective agent (in this case, the disease). Therefore, the population size decreases. Then, at the point of evolutionary rescue (indicated by the dotted line), the mean absolute fitness increases to above 1 and therefore the population starts growing again. In your scenario, you've been (implicitly) told that the mean absolute fitness increasing to above 1 may be because of evolutionary rescue or not but either way the same principle applies.
Density dependence (in this case you're being told low density = more population growth) can be included as part of a leslie matrix (by multiplying with a suitable diagonal matrix). You could also construct separate matrices to represent population growth and decline. My assumption given the information that you have been given is that you use the provided info to construct a "rescue/growth" matrix and need to make a separate disease/decline matrix based on the context/research, potentially multiple matrices to match the data more closely; however, I echo Keltingmeith's suggestion to check with your teacher what the expectations are as that's the safest thing to do going forwards.
I'm not sure if that helps at all but please feel free to follow up with any more questions you have :)
Edit: I would include how I made the above figure and talk about the maths that goes into it but you would need more ecology knowledge to understand it and that would just lead you away from the assessment criteria
Post by: jasmine24 on February 01, 2021, 06:11:19 am
TIA :)
Post by: jasmine24 on February 01, 2021, 06:20:41 am
Hi, i was doing Q4 from the 2016 tasc exam paper but im not sure how step one of the induction is true and wondering if anyone could explain how.*Edit: i dont really understand how the rest of the proof works either :-\
TIA :)
Post by: fun_jirachi on February 01, 2021, 08:34:48 am
Post by: jasmine24 on February 03, 2021, 01:24:45 pm
TIA :)
Post by: keltingmeith on February 03, 2021, 03:24:43 pm
Would anyone be able to explain what the purpose of finding eigenvalues are and how to know when to use them?
TIA :)
Eigenvalues! Man, QCE is wild.
Asking, "what's the purpose of finding eigenvalues" is kind of like asking "what's the purpose of knowing how to solve quadratics". On the surface level, there's no purpose whatsoever, and it just looks like a thing you can do/find. As you've caught on, there must be something deeper happening, or why else would we care enough to be able to find them in the first place? Annoyingly, QCAA only ask for an investigative report, which I'm assuming you're currently doing? But in general, the answer is - if you need to use eigenvalues, you'll be told you need to use them. Just like when you need to solve a quadratic, you know you need to solve one, because either you've been specifically told to solve it, or you've been told, "if a question asks you this, they're asking you to solve a quadratic".
Firstly, some good news: according to the syllabus, your exam will not cover eigenvalues - which is kinda sucky, because they're so cool!! But, that also means if you're looking into applications, we can go a bit nuts, and don't need to worry about what you might be tested on later (confirm with your teacher if they plan on testing you on these with internal tests, that'll give you an idea of what applications you need to specifically know about). First, I highly recommend the wikipedia page for a list of applications. Some of them may go over your head, and one of my favourites is diagonalizing matrices - here's a really good article on it!
Let me know if there's anything you want clarified or more information - or if I've misunderstood the question :)
Post by: jasmine24 on February 03, 2021, 09:16:37 pm
Eigenvalues! Man, QCE is wild.
Asking, "what's the purpose of finding eigenvalues" is kind of like asking "what's the purpose of knowing how to solve quadratics". On the surface level, there's no purpose whatsoever, and it just looks like a thing you can do/find. As you've caught on, there must be something deeper happening, or why else would we care enough to be able to find them in the first place? Annoyingly, QCAA only ask for an investigative report, which I'm assuming you're currently doing? But in general, the answer is - if you need to use eigenvalues, you'll be told you need to use them. Just like when you need to solve a quadratic, you know you need to solve one, because either you've been specifically told to solve it, or you've been told, "if a question asks you this, they're asking you to solve a quadratic".
Firstly, some good news: according to the syllabus, your exam will not cover eigenvalues - which is kinda sucky, because they're so cool!! But, that also means if you're looking into applications, we can go a bit nuts, and don't need to worry about what you might be tested on later (confirm with your teacher if they plan on testing you on these with internal tests, that'll give you an idea of what applications you need to specifically know about). First, I highly recommend the wikipedia page for a list of applications. Some of them may go over your head, and one of my favourites is diagonalizing matrices - here's a really good article on it!
Let me know if there's anything you want clarified or more information - or if I've misunderstood the question :)
Thank you so much! My teacher said we need to know eigenvalues to be able to apply it to Leslie matrices for our exam and also in our investigative report where we have to use markov chain and eigenvalues.
Post by: keltingmeith on February 06, 2021, 05:45:51 pm
Thank you so much! My teacher said we need to know eigenvalues to be able to apply it to Leslie matrices for our exam and also in our investigative report where we have to use markov chain and eigenvalues.
Sorry my response took so long - my laptop recently died and it took a while to get a new one as I live in Perth and we just went into lockdown...
Anyway, interesting point from your teacher. I question if you do need to know how to use eigenvalues for Leslie matrices, but there's still a lot of unknowns about QCE exams, so still more than happy to discuss. For Leslie matrices, there are two questions you could be asking:
1. When do I need to use eigenvalues?
2. What does the eigenvalue of a Leslie matrix tell me?
I emphasise this, because it's really important you're NOT asking the first question. The first question seems good and like it'll give you all the information you'll need, but the truth is it's a distraction. The key to doing well in maths is to look beyond the algorithms, look beyond the equations, and to figure out what they intuitively might tell you. If you can figure this out, not only will you have a better and deeper understanding for the topic at hand, but you'll also know when to use the eigenvalues - when the question is asking about what the eigenvalues will tell you.
Now, I've not worked with Leslie matrices before, so figuring out what they mean is as much a challenge for me as it is for you - thankfully, we don't need to be experts of this. There are a million people around the world and - more importantly - on the internet that have figured it out for us. Here's a handy powerpoint I found, with some very interesting results. It tells us that the first eigenvalue, and its corresponding eigenvector tell us:
a) if the population will grow or decay
b) the final proportions of each age class
I encourage you to do some more reading! There might be some other results you can find that I couldn't. It's also worth checking with your teacher what they meant when they said you need to know about them for Leslie matrices - there's probably some applications that they have in the back of their mind that they were thinking of when they told you you need to know them for Leslie matrices.
---
The next topic I have some information about - Markov chains. I love Markov chains, and I actually spent a fair amount of time at university purely devoted to learning about Markov processes. Really interesting stuff! In the simple case, for the discrete-time Markov chain (note: I highly doubt you're covering continuous time Markov chains as they're a heavily complex topic. But, if your teacher said you need to learn about them, let me know, I'll see if I can find some simple resources for you), there is one eigenvalue we care about - the one where lambda=1. From here, I'm assuming you have basic Markov chain knowledge, where I'm using T for our transition matrix, and x is our distribution of states.
We care about this one, because if a transition matrix has a solution of the form \(Tx=x\), then we call this a stationary distribution. That is, if you're ever in this distribution, then your distribution will not change. Eg, if x=(0.5,0.2,0.3) is a stationary distribution, then \(T^nx=(0.5,0.2,0.3)\) - it will not change, no matter HOW MANY transitions you go through. Essentially, all eigenvectors that correspond to the eigenvalue of 1 for our transition matrix are a stationary distribution. There is also an interesting theorem that shows a transition matrix can only have 0, 1, or infinitely many stationary distributions that's easy to show. Here's a hint - suppose that \(x_1\) and \(x_2\) are stationary distributions of \(T\). In other words, \(Tx_1=x_1\) and \(Tx_2=x_2\). Now, consider the third vector \(x_3=ax_1+(1-a)x_2\), where a is any real number between 0 and 1. Two questions:
1. Is this new vector \(x_3\) a probability distribution? (do all the probabilities in it sum to 1?)
2. Does this vector also solve the equation \(Tx=x\)?
If both of these are yes, then \(x_3\) represents an infinite amount of other stationary distributions.
So, I've spoken alot about stationary distributions, but there's one more really important bit about Markov chains - for some specific chains, you will always reach a stationary distribution after an infinite amount of time (for real-life applications, you can usually reach something close enough to the stationary distribution after a large amount of transitions that the next state is essentially the same as the one before it). Trying to define which situations they will would take ages though, sorry, and unfortunately I don't have a lot of time to be going through everything - but the wiki page is pretty good at discussing a lot of the requirements.
Post by: jasmine24 on February 14, 2021, 07:34:47 pm
Post by: jasmine24 on April 06, 2021, 12:09:41 pm
TIA :)
Post by: fun_jirachi on April 06, 2021, 12:16:57 pm
Otherwise, for a hyperbola in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), this will be a horizontal hyperbola. Use the diagram in your question to determine why no point on the hyperbola can be in the upper 'triangle' and bottom 'triangle' marked by the asymptotes.
For a hyperbola in the form \(-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) this will be a vertical hyperbola. In this case, it might be helpful to pose a similar question to the above by drawing the hyperbola \(-\frac{x^2}{9} + \frac{y^2}{16} = 1\) on the same diagram.
If you still can't quite get this out, feel free to query again (I've only poked you along here because I think it's helpful to develop this kind of intuition on your own).
Post by: Specialist_maths on April 14, 2021, 05:38:41 pm
For Leslie matrices, there are two questions you could be asking:Outstanding advice! I wish everyone could understand this.
1. When do I need to use eigenvalues?
2. What does the eigenvalue of a Leslie matrix tell me?
I emphasise this, because it's really important you're NOT asking the first question. The first question seems good and like it'll give you all the information you'll need, but the truth is it's a distraction. The key to doing well in maths is to look beyond the algorithms, look beyond the equations, and to figure out what they intuitively might tell you. If you can figure this out, not only will you have a better and deeper understanding for the topic at hand, but you'll also know when to use the eigenvalues - when the question is asking about what the eigenvalues will tell you.
Anyway, interesting point from your teacher. I question if you do need to know how to use eigenvalues for Leslie matrices, but there's still a lot of unknowns about QCE exams, so still more than happy to discuss.FYI: Eigenvalues and eigenvectors are specifically listed in the syllabus (p29) as subject matter in the course.
There is a note that states neither will feature on the External Exam (50%).
Schools write their own Internal Assessment (50%), and can choose which descriptors are assessed (all topics must be assessed, but not all descriptors). Some may only assess in the assignment; some may only assess in the exam. I suspect most schools will not assess eigenvalues/eigenvectors at all - and instead choose other "Applications of matrices".
I encourage you to do some more reading! There might be some other results you can find that I couldn't. It's also worth checking with your teacher what they meant when they said you need to know about them for Leslie matrices - there's probably some applications that they have in the back of their mind that they were thinking of when they told you you need to know them for Leslie matrices.Unfortunately, some schools are not as supportive as others. The assignment is not meant to be a research task. This was stated in the syllabus (p31) and again in the 2020 Subject report. The teacher should be providing guidance on how eigenvalues/eigenvectors could be used. It's then up to the student to make the decision in regards to how the given problem can be solved.
Jasmine, I hope your assessment went well. If you've already done IA1 and IA2, you probably won't have to worry about eigenvalues or markov chains again (unless you're doing maths at uni next year?) :)
Post by: jasmine24 on April 19, 2021, 05:57:33 pm
Outstanding advice! I wish everyone could understand this.
FYI: Eigenvalues and eigenvectors are specifically listed in the syllabus (p29) as subject matter in the course.
There is a note that states neither will feature on the External Exam (50%).
Schools write their own Internal Assessment (50%), and can choose which descriptors are assessed (all topics must be assessed, but not all descriptors). Some may only assess in the assignment; some may only assess in the exam. I suspect most schools will not assess eigenvalues/eigenvectors at all - and instead choose other "Applications of matrices".
Unfortunately, some schools are not as supportive as others. The assignment is not meant to be a research task. This was stated in the syllabus (p31) and again in the 2020 Subject report. The teacher should be providing guidance on how eigenvalues/eigenvectors could be used. It's then up to the student to make the decision in regards to how the given problem can be solved.
Jasmine, I hope your assessment went well. If you've already done IA1 and IA2, you probably won't have to worry about eigenvalues or markov chains again (unless you're doing maths at uni next year?) :)
thank you! and it did go well :) I got a 19/20, it was so close to a 20 but I made 1 transcription error where I wrote 0.12 instead of 0.22. I thought it was a bit unfair considering I corrected it in the next line so it didn't affect my solution at all but oh well I'm still happy with my mark.
Post by: jasmine24 on October 12, 2021, 08:04:28 pm
(its from a question that asks to find the integral of 1/(2x-2))
TIA :)
Post by: RuiAce on October 12, 2021, 08:11:57 pm
Does anyone know how (1/2)ln|2x - 2| can simplify to (1/2)ln|x - 1| ??It does not. The point is that you emphasised how it came from an integral. Remember that indefinite integrals come with the important \(+c\).
(its from a question that asks to find the integral of 1/(2x-2))
TIA :)
Consider the logarithm law \(\log(AB) = \log A + \log B\). Using this, we may obtain
\begin{align*}
\int \frac{1}{2x-2}dx &= \frac12 \ln |2x-2| + c\\
&= \frac12 \ln 2|x-1|+ c\\
&= \frac12 \left(\ln |x-1| + \ln 2 \right)+c\\
&= \frac12 \ln |x-1| + \frac12 \ln 2 + c.
\end{align*}
That \(\frac12\ln 2 + c\) will now just be another constant. So you could replace it with another letter, like say \(b = \frac12 \ln 2 + c\), and obtain
\[ \int \frac{1}{2x-2}dx = \frac{1}{2} \ln |x-1| + b, \]
which reunites both correct answers.
Asides:
- If you are doing a definite integral, it does not matter if you integrate into \(\frac{1}{2} \ln |2x-2|\) or \( \frac{1}{2} \ln |x-1|\). If you are very careful with your log laws, you will be able to obtain the same final answer no matter which antiderivative you use.
- You can also jump directly to the answer of \(\frac{1}{2} \ln |x-1|\) if you first factorise the denominator a little.
\begin{align*}
\int \frac{1}{2x-2}dx &= \int \frac{1}{2(x-1)}dx\\
&= \frac12 \int \frac{1}{x-1}dx\\
&= \frac12 \ln|x-1|+c.
\end{align*}
Post by: jasmine24 on October 13, 2021, 09:12:37 pm
It does not. The point is that you emphasised how it came from an integral. Remember that indefinite integrals come with the important \(+c\).thank you!
Consider the logarithm law \(\log(AB) = \log A + \log B\). Using this, we may obtain
\begin{align*}
\int \frac{1}{2x-2}dx &= \frac12 \ln |2x-2 + c|\\
&= \frac12 \ln 2|x-1|+ c\\
&= \frac12 \left(\ln |x-1| + \ln 2 \right)+c\\
&= \frac12 \ln |x-1| + \frac12 \ln 2 + c.
\end{align*}
That \(\frac12\ln 2 + c\) will now just be another constant. So you could replace it with another letter, like say \(b = \frac12 \ln 2 + c\), and obtain
\[ \int \frac{1}{2x-2}dx = \frac{1}{2} \ln |x-1| + b, \]
which reunites both correct answers.
Asides:
- If you are doing a definite integral, it does not matter if you integrate into \(\frac{1}{2} \ln |2x-2|\) or \( \frac{1}{2} \ln |x-1|\). If you are very careful with your log laws, you will be able to obtain the same final answer no matter which antiderivative you use.
- You can also jump directly to the answer of \(\frac{1}{2} \ln |x-1|\) if you first factorise the denominator a little.
\begin{align*}
\int \frac{1}{2x-2}dx &= \int \frac{1}{2(x-1)}dx\\
&= \frac12 \int \frac{1}{x-1}dx\\
&= \frac12 \ln|x-1|+c.
\end{align*}
Post by: jasmine24 on October 17, 2021, 08:34:06 pm
(Q17 tech active WACE 2020)
Post by: RuiAce on October 19, 2021, 06:23:16 pm
does anyone know how the last line of the solution works? is it some sort of formula?The first bit refers to this section screenshotted from the QCE syllabus.
(Q17 tech active WACE 2020)
(https://i.imgur.com/mtmzzjV.png)
We know that \( \overline{X}\) is approximately normal with mean \(\mu\) and standard deviation \( \frac{\sigma}{\sqrt{n}} \). Therefore \( \frac{\overline{X}-\mu}{\sigma/\sqrt{n}}\) is approximately standard normal. Note that:
a) This is a general rule. In general, for any random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\), if we know that \(X\) is normally distributed, then \( \frac{X-\mu}{\sigma}\) is standard normally distributed.
b) It's very similar to the third point where they use \(s\) instead of \(\sigma\). At a high school level, I'd suggest just taking for granted the approximate standard normality works with both \(\sigma\) and with \(s\).
For the second bit, note that in theory that equal sign really should be an approximately equal sign. But let's not worry about that. The point is, first recall that \(\overline{X}\) denotes a sample mean for a larger sample size of \(2n\). And we've calculated that its corresponding standard deviation will therefore be \(9.0192\) (approximately). But also \(\overline{X}\) also has mean \(\mu\) (you should know why this is the case). Therefore here, \( \frac{\overline{X}-\mu}{9.0192} \) will be approximately standard normally distributed. Hence, it is tempting to start by dividing by \(9.0192\), and introducing \(Z\) as a placeholder for a standard normal random variable.
\[ P\left( |\overline{X}-\mu| <10\right) = P\left( \left|\frac{\overline{X}-\mu}{9.0192}\right| <\frac{10}{|9,0192|}\right) =P\left( |Z|<\frac{10}{9.0192} \right) . \]
We still have to remove the absolute value. First recall that the solution to the absolute value inequality \( |x| < a\) is \( -a < x < a\).
\[P\left( |Z|<\frac{10}{9.0192} \right) = P\left( -\frac{10}{9.0192} < Z < \frac{10}{9.0192} \right). \]
But we also know that the density of the standard normal distribution is a bell curve, and in particular symmetric about zero. The symmetry about zero allows you to conclude that \( P\left( -\frac{10}{9.0192} < Z < 0\right)\) and \( P\left( 0 < Z < \frac{10}{9.0192} \right)\) equal one another. Which is where that bit comes from.
\[ P\left( -\frac{10}{9.0192} < Z < \frac{10}{9.0192} \right) = 2 P\left( 0< Z < \frac{10}{9.0192} \right). \]
The final bit is just a calculator plug in. Can be computed by something along the lines of \( \verb|2*(normcdf(10/9.0192,mean=0,sd=1)-normcdf(0,mean=0,sd=1))| \)
Note: Use of symmetry is not required if your graphics calculator can just do \( \verb|normcdf(10/9.0192,mean=0,sd=1)-normcdf(-10/9.0192,mean=0,sd=1)| \) instead.
Post by: Studygoals on March 12, 2022, 11:09:40 am
Post by: GreenNinja on March 12, 2022, 11:19:45 am
Hi, I need to find a tool to draw diagrams for my PSMT. We are doing it on vectors, and its really hard to draw all the angles and distances and diagrams on word dox. Does anyone know of any good drawing tools?
Geogebra is quite easy to use for graphing vectors.