Week 6: Modelling change with derivatives

Change and transformation are pervasive. Even mountains rise and fall over time, and almost all the systems we study in the life sciences are dynamic. Quantities increase, decrease, accumulate, and diminish. To take a concrete example consider the language we grew used to hearing during the Covid-19 epidemic:

The number of cases is rising dramatically.

Total deaths for the week are over 1,200.

The number of infections is falling gradually.

The viral load increases until it reaches a critical point and the infection activates.

Deaths are still increasing, but the rate of increase is slowing.

The tools we need to model dynamic processes are provided by the branch of mathematics called Calculus:

• The derivative measures the rate of change of a quantity, while
• The integral measures the accumulation of a quantity,

and we will spend the next three weeks working with them.

1. This week will focus on the rules for finding the derivative or rate of change of a quantity.
2. The second week will look at applications of the derivative, at phase space, and at differential equations.
3. The third week we'll learn how to find and apply integrals, which calculate the accumulation of quantities.

This Week

Monday. We'll look at the idea of the derivative and see how to sketch the derivative of a function. We'll also see the basic rules for finding derivatives of simple functions. These have an algebraic feel to them.

Wednesday. We'll look at rules for differentiating more complex functions. The rules for products, quotients, and compositions of functions get a little hairy 🤯, but there are quite a few practice problems in the exercises to help you become fluent in applying them.

Friday. Some more examples of differentiating complex functions; checking the fine print (does a derivative have to exist?); and, finally, some applications of the derivative.

That's all for now!

See you soon,
Tim