# optimization math ia example

0

Use the binomial distribution to test your ESP abilities. 14/20 IA SL IA; Submitted December 6, 2018. Change ), You are commenting using your Twitter account. First, notice that $$w = 0$$ is not a critical point. We also can’t forget to add in the area of the two caps, $$\pi {r^2}$$, to the total surface area. In this case it looks like our best option is to solve the constraint for $$h$$ and plug this into the area function. This is a nice way to link some maths history with studying an interesting function. 5) Königsberg bridge problem: The use of networks to solve problems. Here are those function evaluations. 12) Towers of Hanoi puzzle – This famous puzzle requires logic and patience. As an interesting side problem and extension to the above example you might want to show that for a given volume, $$L$$, the minimum material will be used if $$h = 2r$$ regardless of the volume of the can. Also, as seen in the last example we used two different methods of verifying that we did get the optimal value. In this problem we want to maximize the area of a field and we know that will use 500 ft of fencing material. 9) Designing bridges – Mathematics is essential for engineers such as bridge builders – investigate how to design structures that carry weight without collapse. Are there functions which plot any polygons? There is also a fully typed up mark scheme. Suppose however that we also knew that $$f''\left( x \right) > 0$$ for all $$x$$ in $$I$$. How mathematical formations were used to fight wars. (This topic is only suitable for IB HL students). Find out how! Over 300 examples of maths IA exploration topics and ideas for IB mathematics. the function is increasing immediately to the left) and if $$f'\left( x \right) < 0$$ immediately to the right of $$x = c$$(i.e. 1) The Van Eck Sequence – The Van Eck Sequence is a sequence that we still don’t fully understand – we can use programing to help! 20) Twin primes problem: The question as to whether there are patterns in the primes has fascinated mathematicians for centuries. As we’ve already pointed out the end points in this case will give zero area and so don’t make any sense. 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. 4) Why e is base of natural logarithm function: A chance to investigate the amazing number e. 5) Fourier Transforms – the most important tool in mathematics? Model your data using a normal distribution. This is a nice geometry puzzle solved using a variety of methods. A good use of your calculus skills. IB Maths Resources from British International School Phuket. Examples of around 70 topics that could be investigated, Useful websites for use in the exploration, A student checklist for completing a good investigation, Common mistakes that students make and how to avoid them, A worked example for the maths behind correlation investigations, A selection of some interesting exploration topics explored in more depth. 4) Applications of complex numbers: The stunning graphics of Mandelbrot and Julia Sets are generated by complex numbers. 2) Hexaflexagons: These are origami style shapes that through folding can reveal extra faces. Let’s work some another example that this time doesn’t involve a rectangle or box. We may need to modify one of them or use a combination of them to fully work the problem. In order to do it full justice, you need to begin early. 6) Square Triangular Numbers. Here we will be looking for the largest or smallest value of a function subject to some kind of constraint. Change ), You are commenting using your Twitter account. = -1/12 ? 4) Bad maths in court – how a misuse of statistics in the courtroom can lead to devastating miscarriages of justice. Here is a list of over 200 ideas with links to further reading for your maths exploration! In this method we also will need an interval of possible values of the independent variable in the function we are optimizing, $$I$$. 7) Explore the Si(x) function – a special function in calculus that can’t be integrated into an elementary function. Once you’ve got that identified the quantity to be optimized should be fairly simple to get. 26) Modelling volcanoes – look at how the Poisson distribution can predict volcanic eruptions, and perhaps explore some more advanced statistical tests. Nowhere in the above discussion did the continuity requirement apparently come into play. Find out more! Finding Brainschrome. the area of the poster with the margins taken out). Change ), You are commenting using your Google account. We need an interval of possible values of the independent variable in function we are optimizing, call it $$I$$ as before, and the endpoint(s) may or may not be finite. On the other hand, I’d quite enjoy reading (or writing?) This is a puzzle that was posed over 1500 years ago by a Chinese mathematician. Next, the vast majority of the examples worked over the course of the next section will only have a single critical point. In this case we can say that the absolute maximum of $$f\left( x \right)$$ in $$I$$ will occur at $$x = c$$. However, in all of the examples over the next two sections we will never explicitly say “this is the interval $$I$$”. 8) Squaring the circle: This is a puzzle from ancient times – which was to find out whether a square could be created that had the same area as a given circle. This isn’t a real problem however. 9) Does finger length predict mathematical ability? What would happen to the climate in the event of a nuclear war? We don’t have a cost here, but if you think about it the cost is nothing more than the amount of material used times a cost and so the amount of material and cost are pretty much tied together. 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. There is also the problem of identifying the quantity that we’ll be optimizing and the quantity that is the constraint and writing down equations for each. In this case we know that to the left of $$x = c$$, provided we stay in $$I$$ of course, the function is always increasing and to the right of $$x = c$$, again staying in $$I$$, we are always decreasing. 1) Modular arithmetic – This technique is used throughout Number Theory. In the last two examples we’ve seen that many of these optimization problems can be done in both directions so to speak. Also notice that provided $$w > 0$$ the second derivative will always be negative and so in the range of possible optimal values of the width the area function is always concave down and so we know that the maximum printed area will be at $$w = 10.6904\,\,{\mbox{inches}}$$. If $$f''\left( x \right) < 0$$ for all $$x$$ in $$I$$ then $$f\left( c \right)$$ will be the absolute maximum value of $$f\left( x \right)$$ on the interval $$I$$. Choose your own pattern investigation for the exploration. Premier League wages and league position? Since we are after the absolute maximum we know that a maximum (of any kind) can’t occur at relative minimums and so we immediately know that we can exclude these points from further consideration. We are constructing a box and it would make no sense to have a zero width of the box. 20) Modelling infectious diseases – how we can use mathematics to predict how diseases like measles will spread through a population. So, we’ve got a single critical point and we now have to verify that this is in fact the value that will give the absolute minimum cost. 30) Can you solve Oxford University’s Interview Question?. 16) Graphing polygons investigation. 19) Introduction to Modelling. They see one problem and then try to make every other problem that seems to be the same conform to that one solution even if the problem needs to be worked differently. If you can do one you can do the other as well. 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). Here is the volume, in terms of $$h$$ and its first derivative. Because we want length measurements for the radius and height we’ll also need the volume to in terms of a length measurement. In both examples we have essentially the same two equations: volume and surface area. 18) Hyperbolic geometry – how we can map the infinite hyperbolic plane onto the unit circle, and how this inspired the art of Escher. Useful websites for use in the exploration, A selection of detailed exploration ideas. Just remember that the interval $$I$$ is just the largest interval of possible values of the independent variable in the function we are optimizing. Now, suppose that $$x = c$$ is a critical point and that $$f''\left( c \right) > 0$$. 22) The Folium of Descartes. 28) How to avoid a Troll – an example of a problem solving based investigation, 29) The Gini Coefficient – How to model economic inequality. 17) Chinese remainder theorem. 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. 8) Finding prime numbers: The search for prime numbers and the twin prime conjecture are some of the most important problems in mathematics. First, a quick figure (probably not to scale…). From this we can see that we have one critical points : $$r = \sqrt[3]{\frac{750}{\pi}} = 6.2035$$(where the derivative is zero). Use modeling and volume of revolutions. Does sacking a football manager affect results. Explore the maths behind code making and breaking. If $$f''\left( x \right) > 0$$ for all $$x$$ in $$I$$ then $$f\left( c \right)$$ will be the absolute minimum value of $$f\left( x \right)$$ on the interval $$I$$. 1) Modular arithmetic – This technique is used throughout Number Theory. Create a free website or blog at WordPress.com. If these conditions are met then we know that the optimal value, either the maximum or minimum depending on the problem, will occur at either the endpoints of the range or at a critical point that is inside the range of possible solutions. The first way to use the second derivative doesn’t actually help us to identify the optimal value. Great question. So, if we take $$h = 1.9183$$ we get a maximum volume. We’ll see at least one example of this as we work through the remaining examples. 7) Möbius strip: An amazing shape which is a loop with only 1 side and 1 edge. In optimization problems we are looking for the largest value or the smallest value that a function can take.