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5) Elliptical orbits – Planets and comets have elliptical orbits as they are influenced by the gravitational pull of other bodies in space. 21) Projective geometry –  the development of geometric proofs based on points at infinity. However, in Example 2 the volume was the constraint and the cost (which is directly related to the surface area) was the function we were trying to optimize. Likewise, suppose that we knew that $$f'\left( x \right) < 0$$ for all $$x$$ in $$I$$ such that $$x > c$$. 4) Rocket Science and Lagrange Points – how clever mathematics is used to keep satellites in just the right place. 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.

6) Impact Earth – what would happen if an asteroid or meteorite hit the Earth? 23) Modelling music with sine waves – how we can understand different notes by sine waves of different frequencies. Find out how! This investigates a generalized version of projectile motion – discover what shape is created. This is a function which is used in Quantum mechanics – it describes a peak of zero width but with area 1. The planets follow a surprising pattern when measuring their distances. In order to do it full justice, you need to begin early.

46) Ramanujan’s Taxi Cab and the Sum of 2 Cubes. 2) Gravity, orbits and escape velocity – Escape velocity is the speed required to break free from a body’s gravitational pull. 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. Explore the maths behind code making and breaking. Maths IA – Maths Exploration Topics. 9) Maths and football – Do managerial sackings really lead to an improvement in results? Zeno’s paradox of Achilles and the tortoise, The Poincare Conjecture and Grigori Perelman, The Monkey and the Hunter – How to Shoot a Monkey, Galileo: Throwing cannonballs off The Leaning Tower of Pisa. Are maths students better than history students? Can you find the loop of infinite sadness? 41) Arithmetic number puzzle – It could be interesting to do an exploration where you solve number problems – like this one. 5) Stacking cannon balls: An investigation into the patterns formed from stacking canon balls in different ways.

Can you find the pattern behind it? Look at how infectious Ebola really is.

7) Fibonacci sequence and spirals in nature – There are lots of examples of the Fibonacci sequence in real life – from pine cones to petals to modelling populations and the stock market. Use computer graphing to investigate.

Can we use a mixture of pure maths and computing to solve this problem?

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 $$x = c$$. 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.

section earlier in the chapter to find the maximum value of the area. ( Log Out /  What is the Mordell equation and how does it help us solve mathematical problems in number theory?

So, if we take $$h = 1.9183$$ we get a maximum volume. So, it looks like in this case we actually have a perfect cube. 24) A geometric proof for the arithmetic and geometric mean. This was done to make the discussion a little easier. We’ll solve the constraint for $$h$$ and plug this into the equation for the volume. Also look at the finances behind Premier league teams. no sides to the fence) and $$y = 250$$ (i.e. Optimization Math Ia Example. Similarly, the surface area of the walls of the cylinder is just the circumference of each circle times the height.

In this case we would know that the function was concave up in all of $$I$$ and that would in turn mean that the absolute minimum of $$f\left( x \right)$$ in $$I$$ would in fact have to be at $$x = c$$. It is also important to be aware that some problems don’t allow any of the methods discussed above to be used exactly as outlined above. 3) Galileo: Throwing cannonballs off The Leaning Tower of Pisa – Recreating Galileo’s classic experiment, and using maths to understand the surprising result.

Change ), Available after school until 4 pm in room 703, Polar Diagrams Florence Nightingale – Graded. 9) Designing bridges – Mathematics is essential for engineers such as bridge builders – investigate how to design structures that carry weight without collapse. Also look at how the Championship compares to the Premier League. Why e is base of natural logarithm function: Fourier Transforms – the most important tool in mathematics? Note that if you think of a cylinder of height $$h$$ and radius $$r$$ as just a bunch of disks/circles of radius $$r$$ stacked on top of each other the equations for the surface area and volume are pretty simple to remember. With some examples one method will be easiest to use or may be the only method that can be used, however, each of the methods described above will be used at least a couple of times through out all of the examples. What is your best way of surviving the zombie apocalypse? All we need to do now is to find the remaining dimensions. As we work examples over the next two sections we will use each of these methods as needed in the examples. 18) Fermat’s last theorem: A problem that puzzled mathematicians for centuries – and one that has only recently been solved. They do, however, give us a set of limits on $$y$$ and so the Extreme Value Theorem tells us that we will have a maximum value of the area somewhere between the two endpoints. This problem is a little different from the previous problems. However, suppose that we knew a little bit more information. 40) Modelling the spread of Coronavirus (COVID-19). Here is a list of over 200 ideas with links to further reading for your maths exploration! The constraint will be some condition (that can usually be described by some equation) that must absolutely, positively be true no matter what our solution is. Can we find a function that plots a square? 32) RSA code – the most important code in the world? Finding Brainschrome. Fermat’s Last Theorem is one of the most famous such equations. In optimization problems we are looking for the largest value or the smallest value that a function can take. We’ve worked quite a few examples to this point and we have quite a few more to work. 14) Egyptian fractions: Egyptian fractions can only have a numerator of 1 – which leads to some interesting patterns. 15) Julia Sets and Mandelbrot Sets – We can use complex numbers to create beautiful patterns of infinitely repeating fractals. The volume is just the area of each of the disks times the height. Let’s work some another example that this time doesn’t involve a rectangle or box. 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 $$w = 10.6904\,\,{\mbox{inches}}$$.

7) When do 2 squares equal 2 cubes? 19) Elliptical Curves– how this class of curves have importance in solving Fermat’s Last Theorem and in cryptography. 8) Log Graphs to Plot Planetary Patterns.

Exponential and trigonometric regression. In fact, we will have the same requirements for this method as we did in that method.

= -1/12 ? This topics provides a fascinating introduction to both combinatorial Game Theory and Group Theory. 11) The One Time Pad – an uncrackable code? Great question. 3) Gambler’s fallacy: A good chance to investigate misconceptions in probability and probabilities in gambling. This is a puzzle that was posed over 1500 years ago by a Chinese mathematician. the function is increasing immediately to the left) and if $$f'\left( x \right) < 0$$ immediately to the right of $$x = c$$(i.e. Here are those function evaluations. There are two main issues that will often prevent this method from being used however. It will definitely be easier to solve the constraint for $$h$$ so let’s do that. 5) Knight’s tour in chess: This chess puzzle asks how many moves a knight must make to visit all squares on a chess board. Advice on using Geogebra, Desmos and Tracker. IB’s Math IA is not something that can be finished in a couple of hours. Now, my math internal assessment was not exactly calculus although it did have some bit of it.

16) Hyperbolic functions – These are linked to the normal trigonometric functions but with notable differences. 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. However, this section has gotten quite lengthy so let’s continue our examples in the next section. Also, what is with “Does finger length predict mathematical ability?”?

Does finger length predict mathematical ability?