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We have learnt that a mathematical model of a physical (or real) problem requires a good knowledge of the problem itself. Often – almost always – this knowledge must be shared with the experts on the problem, be these engineers, biologists, economists, physicists or physicians.

Given the function f(x) = −x2 + 4x + 1, construct fmp over the interval [1, 4] using a partition made of M = 6 subintervals of the same length. Plot f and fmp on the same plane.

The various aspects of the real world, their interaction and their dynamics can very often be described by mathematical formulas, functions and equations, in other words by mathematical models.

It is the 22nd of May and we decide to look up born on 22 May on Google. In the blink of an eye, 0.35 seconds to be precise, we get back around 624,000 hits. We click on the first link Google proposes (22 May – Wikipedia) and discover that on this very day in 1859 Sir Arthur Conan Doyle was born in Edinburgh, Scotland (see Fig. 4.1).

The cardiovascular system allows the nutrients essential to life to reach all the cells in our body (it is estimated there are more than 37 thousand billion cells!). The fundamental engine of the system is our heart, an organ of extraordinary complexity and efficiency. The heart pumps the blood into the aorta and from there, through successive ramifications, in the bigger and smaller arteries, then in the arterioles and finally in the capillaries. And it is exactly inside the capillary beds that the nutrients are delivered to the cells and the toxic substances – the residues of the cellular reactions – are removed. The waste CO2, for instance, is transported back through the venous system and is eventually released into the pulmonary alveoli where the respiration expels it.

When a numerical model generates an algorithm, the latter must be implemented in a program by means of a programming language. Among the many languages available we shall focus on Octave, because on one hand it is a simple language, and on the other it is particularly efficient for scientific computing.