## Radius of convergence of a series, and approximating polynomials

I hinted today that there were sometimes issues when you did a polynomial approximation, that if you tried to find the value of a function a long way from the region about which you’re approximating, that sometimes you wouldn’t be able to do it. This is related to an idea called the radius of convergence of a series. In the following we are just plotting polynomials, but you can see that whereas in the polynomial approximation for sin(x) (on the right), as we get more and more terms, we approximate the function better and better far away from the point x=1 (which is the point about which we are approximating the function). However, for the function $\sqrt{1+x}$, after x=3, the approximations are nowhere near the function itself. This is because that function has a radius of convergence of 2, when expanded about x=1. This is due to the behaviour of the function at x=-1, which is a distance 2 away.…

## Mathematical Modelling for Infectious Diseases – a course at UCT (19th-30th September 2016)

For anybody interested in the mathematics of infectious disease modelling, the following should be very interesting.

A course on the application of mathematical modelling and computer simulation to predict the dynamics of infectious diseases to evaluate the potential impact of policy in reducing morbidity and mortality. (click to go to poster).

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## What I learnt in class today:

A convex function $f:\mathbb R\to\mathbb R$ is defined as satisfying

$f(\lambda x + (1-\lambda )y)\leq \lambda f(x)+(1-\lambda )f(y) \quad \forall x,y\in \mathbb R,\ \forall \lambda \in [0, 1].$

Thus, the shape of a convex function is like $\smallsmile$ . An example of a convex function is f(μ)=μ2:

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## Integration by Parts – Lightbulb Education

Integration by Parts

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## Integration By Substitution – Lightbulb Education

Integration By Substitution

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## The mathematical equation that caused the banks to crash

The Black-Scholes model is a mathematical equation invented by Fischer Black and Myron Scholes that first appeared in their seminal paper of 1973 opening a new wave of selling and buying financial contracts. This economic formulation was well received and recognized to be effective by the financial community to the extent that it won Black and Scholes a noble price in 1997. However, on the 19th October, 1987 – The Black Monday, the world experienced a severe shock when the markets suddenly crushed bringing to light the flaws in the mighty celebrated Black-Scholes model. The number one mistake in the model was the assumption that a given contact could be priced at the same volatility level irrespective of the strike price – the price a contract owner had to pay at the expiry date of the contract. A more economic perspective is discussed here.

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Gallery

## Does knowledge of past years affect the Keynesian Beauty Contest result?

Last year I played the 2/3 numbers game, also called the Keynesian Beauty Contest with my first year maths class. The discussion can be found here: http://www.mathemafrica.org/?p=11143 I wanted to know if, telling my class the results from last year (including sketching for them the histogram of results), would change how they chose their numbers this year. Of course I can’t tell if it changed them, but what is fascinating is that either:

1. Their guesses (if I didn’t tell them about the results from last year) would have been very different from those last year, or:
2. They were completely unaffected by knowing what people did last year, which really means that they believed that the rest of the class would have been unaffected.

I plot here the results from the last three years and you can see how similar the results are, year on year.

You can see that the distributions are relatively similar, and the means are extremely close.…

## Picturing volumes of revolution

One of the homework questions this week was the following:

Let $R=\{(x,y)\in \mathbb{R}^2:y\ge 0, \cos x\le y\le \sin x\,\,and\,\,0\le x\le\pi\}.$

a) Sketch the region R and find its area.
b) Find the volume of the solid obtained by rotating the region R around the y-axis.

The first thing to do is to sketch the graphs of $y=\cos x$ and $y=\sin x$. Once you’ve done that, the other parts of the inequalities should be clear. It should look like the red region in the following plot:

Now we have to imagine bringing out a third axis, perpendicular to the picture above, ie. coming out towards us. We then want to rotate the red form here about the vertical axis. This we can imagine doing in the following animation:

Given this form we can then think about either taking horizontal cross-sections through it, which will give us thin annuli, or we can take vertical, circular slices to give us thin shells. Adding these together and integrating should give us the same answer whichever way we choose to slice it, but one way will be considerably easier.…