Help with HL Math IA (Calculus)

[fieryice12]

My original IA topic for HL Math was optimizing the surface area of a ring, horn, and spindle torus given a fixed volume. My teacher told me that it was too simplistic, and that there was not enough original math in the paper, so I'm searching for a new topic. 
 

I came across the least squares regression, which is used for polynomial regressions and involves calculus. I was thinking about maybe applying this to some aspect of economics/markets/businesses. I found the Solow–Swan model as a potential starting point. Would it be a good to use something like this for my IA, and would it be complex enough for a HL Math IA? If not, how could I make it more original and more complex? I'm also not sure what kind of variations I could put on this equation to customize to my IA specifically.

 

The Solow-Swan model involves nonlinear ordinary differential equations (Topic 9), which is something that we haven't learned yet but will in a month or two. Because of this, would you recommend that I try to teach myself this concept for my IA? I'm just really confused at this point because my teacher isn't offering any more guidance.

[fieryice12]

Another idea just occurred to me. What if I used the least squares method to predict how HDI (human development index) will change in the future by plotting a polynomial regression for it? Since HDI is often criticized for being too tied to the GDP of a country, would it be a good idea to find a polynomial regression for the GDP per capita and compare both of them? 

Is this idea complex enough for Math HL? Also, if I did this, in what ways could I compare GDP and HDI? Should I do something like a statistics correlation test (like the chi-squared test or distance correlation test) to add on to it?

Edited January 15, 2019 by fieryice12

[kw0573]

I like Solow-Swan better than HDI vs GDP. Be sure to concisely explain what the symbols mean without going to deep into economic theory. However you must go beyond a proof of the model. For example you can discuss in some depth of the limitations of the model and its assumptions. In contrast, polynomial regression is interesting but somewhat lacking for a top-notch HL IA.

[fieryice12]

@kw0573 Do you really think so? I'm not so sure, because we haven't learned differential equations in our class yet. If I were to do Solow-Swan, aren't the limitations and assumptions already well known?

[kw0573]

I thought you have already discussed separable diff. eqs. If not, then this topic could be conceptually challenging. The IA is not about publishing new math, it's about demonstrating personal understanding at or beyond HL level. It should be appropriate to discuss the limitations of the model from a mathematical perspective, rather than an economic perspective. 

[fieryice12]

@kw0573 No, our class is going to study Topic 9 in a few weeks, but we haven't started yet. That's what I was worried about whether I could teach myself Solow-Swan. You mentioned HDI vs GDP was somewhat lacking; do you have any ways I could increase its difficulty level?

Edited January 15, 2019 by fieryice12

[kw0573]

GDP vs HDI is too stats focused. The complete derivation of linear regression formula uses multivariate calculus (beyond HL) and I assume the same for polynomial regression. It's a somewhat unnecessary risk to learn diff. eqs for the IA, but you'll have to learn it anyways. I assume the IA is due soon and I suggest that you further develop your topic of ring, horn, spindle, such as by introducing non-Cartesian coordinates or introduce relations to other conic sections and their surface of revolutions. 

Basically if you are doing well at HL, you can get 15 ish pretty much with a lot of topic if you revise with care, but it's probably not worth to find the perfect topic and learn everything yourself just to score marginally higher. The surface of revolution is such a topic that can score 15 ish without great difficulty, if you add a significant portion of personal insight (which of course does not solely consist of math never to have been published by anyone in last 400 years of calculus)

[fieryice12]

@kw0573 Thank you, this is so helpful! My teacher told me that when I use Pappus' Centroid Theorem (which gives the volume/surface area of ring/horn torii), most of the work is already done for me, so my actual optimization is pretty straightforward from there. How can I avoid this criticism for the rest of my original IA? 

By non-cartesian, do you mean polar or complex? (If so, I'm not really sure how to incorporate this into the IA..) The other conic sections you're referring to are parabolas, ellipses, and hyperbolas, correct? Would just optimizing all of these solids of revolution for maximum surface area be enough to make a decent IA?

Thanks again!

[kw0573]

You can approach this without invoking the Pappus' Centroid Theorem. 

I mean cylindrical and spherical coordinates (as surfaces are exist in 3-D spaces). There may be certain advantages to use a different coordinate system for certain geometries. Remember that it is not about the amount/quantity of math, but rather the depth and the connections you make. So it should not be a question of how many surface area to find but what theorems or math concepts you can use.

[fieryice12]

@kw0573 How would I go about approaching this without Pappus? Would it be something like this: http://whistleralley.com/torus/torus.htm? Based on your advice, I'm planning to not reference Pappus, but derive the volume of the torii by myself? This leads me to another question, however: how can the surface area of a torus be found without Pappus?

[fieryice12]

@kw0573 Would you use something like this: https://www.youtube.com/watch?v=tMVwXglUp60? It's a proof for the surface area of a ring torus.

 

[kw0573]

Yeah the youtube video is what i'm talking about: find a way to use volume of revolution. It's quite possible that the math may be simplified if either cylindrical or spherical coordinates are used. Since most of these proofs are in Cartesian, if you managed to do it in cylindrical/spherical than that could be count as personal engagement.

[fieryice12]

You are really really helpful! Thanks so much. Would you mind if I asked you a few more questions tomorrow? Thank again.

[kw0573]

Yeah you can shoot me a private message or continue this thread.

[fieryice12]

@kw0573 I know you already suggested me not to do it, but what if for the GDP vs HDI, I don't derive the polynomial regression, just use the least squares method, and then use some kind of distance correlation to relate the two equations. Why is it a problem if it's too stats based?

[kw0573]

The HL "Use of math" criterion describes sophistication as follows: 

Quote

Sophistication in mathematics may include understanding and use of challenging mathematical concepts, looking at a problem from different perspectives and seeing underlying structures to link different areas of mathematics.

If there is no sophistication, the highest you can get is 3/6 in use of math. A well-annotated derivation of regression formula would have been an excellent and foolproof way to connect calculus to stat. Data interpretation may make it difficult in scoring well in communication and personal engagement.

[fieryice12]

I'm starting to work on the topic you suggested: optimizing all three types of torii (ring, horn, and spindle) as well as ellipses, hyperbolas, and parabolas. Do you have any resources for getting started with spherical/cylindrical coordinates like you suggested? I can't grasp the concept well enough to actually use it in my IA yet. 

[kw0573]

I think the wikipedia page is ok. Just beware of different convention in spherical coordinates whether its math or physics. I think it would be good to first do everything in Cartesian, then if/when you have extra time play around with other coordinate systems.