## 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 [...]

## 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 [...]

## Convex functions

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 [...]

## Integration by Parts – Lightbulb Education

Integration by Parts

## Integration By Substitution – Lightbulb Education

Integration By Substitution

## 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 [...]

## 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 [...]

## Picturing volumes of revolution

One of the homework questions this week was the following: Let $latex R=\{(x,y)\in \mathbb{R}^2:y\ge 0 \cos x\le y\le \sin x\,\,and\,\,0\le x\le\pi\}.$ 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 [...]

## An integral expression for n!

I gave a challenge question at the end of class a week or so ago. Here I will give the solution and show that it gives us something rather strange and surprisingly useful. I wrote down the following, and asked you to prove it: $latex \int_0^\infty e^{-t} t^N dt=N!$ [...]

## A general formula for partial fractions

Out of the blue I wrote down a rather confusing mass of indices and summations on the board a few days ago. Writing this down at the last minute was perhaps a bad idea, but I wanted to show what the general form for expanding a fraction into partial fractions [...]