Are there functions which plot any polygons? 15) Birthday paradox: The birthday paradox shows how intuitive ideas on probability can often be wrong. Does sacking a football manager affect results. So, it looks like in this case we actually have a perfect cube. 4) Applications of complex numbers: The stunning graphics of Mandelbrot and Julia Sets are generated by complex numbers. Let’s work some another example that this time doesn’t involve a rectangle or box. 22) Modelling Zombies – How do zombies spread? SL Math IA. We’ve worked quite a few examples to this point and we have quite a few more to work. You can also model everything from Angry Birds to stunt bike jumping. Here is a list of over 200 ideas with links to further reading for your maths exploration! From the first derivative we have the following two critical points (\(w = 0\) is not a critical point because the area function does not exist there). 18) Fermat’s last theorem: A problem that puzzled mathematicians for centuries – and one that has only recently been solved. Does finger length predict mathematical ability? 32) Soap Bubbles, Wormholes and Catenoids. Look at the Bayesian logic behind the argument that we are living in a computer simulation. In the last two examples we’ve seen that many of these optimization problems can be done in both directions so to speak. Why e is base of natural logarithm function: Fourier Transforms – the most important tool in mathematics? You can download a 60-page pdf guide to the entire IA coursework process for the new syllabus (first exam 2021) to help you get excellent marks in your maths exploration. What would happen to the climate in the event of a nuclear war? Look at how infectious Ebola really is. Finally, in all of the methods above we referenced an interval \(I\).
Use modeling and volume of revolutions. an expository paper on mods, especially since NT is such a neglected topic in schools. 31) The Shoelace Algorithm to find areas of polygons.
4) Tesseract – a 4D cube: How we can use maths to imagine higher dimensions. 14/20 IA SL IA; Submitted December 6, 2018. Go back to the figure at the start of the solution and notice that we can quite easily find limits on \(h\). In this method we also will need an interval of possible values of the independent variable in the function we are optimizing, \(I\). the area of the poster with the margins taken out).
11) Understanding the fourth dimension: How we can use mathematics to imagine (and test for) extra dimensions. Quantum Mechanics – a statistical universe? Here is the volume, in terms of \(h\) and its first derivative. 11) Mersenne primes: These are primes that can be written as 2^n -1. Includes: IB Exploration Modelling and Statistics Guide. The following are examples of HL/SL IAs based on the current mark scheme with grader comments. Also, what is with “Does finger length predict mathematical ability?”? In the examples to this point we’ve put in quite a bit of discussion in the solution. 2) Goldbach’s conjecture: “Every even number greater than 2 can be expressed as the sum of two primes.” One of the great unsolved problems in mathematics. Are maths students better than history students? This was done to make the discussion a little easier.
We’ll see at least one example of this as we work through the remaining examples.
We’ll need the surface area of this can and that will be the surface area of the walls of the can (which is really just a cylinder) and the area of the top and bottom caps (which are just disks, and don’t forget that there are two of them). Having these limits will also mean that we can use the process we discussed in the Finding Absolute Extrema Let \(I\) be the interval of all possible values of \(x\) in \(f\left( x \right)\), the function we want to optimize, and suppose that \(f\left( x \right)\) is continuous on \(I\) , except possibly at the endpoints. So, let’s get the derivative and find the critical points.
In identifying the constraint remember that the constraint is the quantity that must be true regardless of the solution. 29) The Riemann Hypothesis – one of the greatest unsolved problems in mathematics – worth $1million to anyone who solves it (not for the faint hearted!). In order to do it full justice, you need to begin early.
12) The Riemann Sphere – an exploration of some non-Euclidean geometry. Similarly, if we know that to the left of \(x = c\) the function is always decreasing and to the right of \(x = c\) the function is always increasing then the absolute minimum of \(f\left( x \right)\) in \(I\) will occur at \(x = c\).
Had the discontinuity at \(x = d\) not been there this would not have happened and the absolute maximum would have occurred at \(x = c\). Now, as noted above we got a single critical point, 1.2910, and so this must be the value that gives the maximum volume and since the maximum volume is all that was asked for in the problem statement the answer is then : \[V\left( {1.2910} \right) = 2.1517\,{{\mbox{m}}^3}\]. This example is in many ways the exact opposite of the previous example.
24) A geometric proof for the arithmetic and geometric mean. This is one of the more common mistakes that students make with these kinds of problems. HL ia Submitted August 26, 2019. All we need to do now is to find the remaining dimensions. A great introduction to some of the ideas behind non-euclidean geometry. This is being done mostly because these notes are also being presented on the web and this will help to keep the load times on the pages down somewhat.
Surprisingly maths can help! In this section we are going to look at optimization problems. 8) Finding prime numbers: The search for prime numbers and the twin prime conjecture are some of the most important problems in mathematics. 11) Zeno’s paradox of Achilles and the tortoise: How can a running Achilles ever catch the tortoise if in the time taken to halve the distance, the tortoise has moved yet further away? Here is a quick sketch to get us started off. 34) Cesaro Summation: Does 1 – 1 + 1 – 1 … = 1/2?.
22) Diophantine application: Cole numbers.
We may need to modify one of them or use a combination of them to fully work the problem. 1) Modular arithmetic – This technique is used throughout Number Theory. The constraint is that the overall area of the poster must be 200 in2 while we want to optimize the printed area (i.e. 4 years ago. This mathematical equation has been described as the most important in all of physics. 6) Logarithmic scales – Decibel, Richter, etc. Investigate some rocket science! 6) Handshake problem: With n people in a room, how many handshakes are required so that everyone shakes hands with everyone else? Let’s start the section off with a simple problem to illustrate the kinds of issues we will be dealing with here. What is the geometric shape of soap bubbles?
What it does do is allow us to potentially exclude values and knowing this can simplify our work somewhat and so is not a bad thing to do. French Math IA Example on Cryptocurrency.
The third method however, will work quickly and simply here. 38) Friendly numbers, Solitary numbers, perfect numbers. 3) Stacking cannonballs – solving maths with code – how to stack cannonballs in different configurations. The second method listed above would work here, but that’s going to involve some calculations, not difficult calculations, but more work nonetheless. I think the easier ones are more suitable; I find it hard to imagine a good paper on GRH or Goldbach without a background in complex analysis or analytic number theory, respectively: these topics are just too hard for high schoolers (and too hard for everyone else probably also). If \(w\) is very large then we would just need to make \(h\) very small.
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