What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.

Was it possible that this dangerous driving penalty was issued in error?

Which line graph, equations and physical processes go together?

Have you ever wondered what it would be like to race against Usain Bolt?

Get further into power series using the fascinating Bessel's equation.

Work with numbers big and small to estimate and calculate various quantities in physical contexts.

Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?

Here are several equations from real life. Can you work out which measurements are possible from each equation?

Work out the numerical values for these physical quantities.

See how enormously large quantities can cancel out to give a good approximation to the factorial function.

Get some practice using big and small numbers in chemistry.

Looking at small values of functions. Motivating the existence of the Taylor expansion.

Where should runners start the 200m race so that they have all run the same distance by the finish?

Make an accurate diagram of the solar system and explore the concept of a grand conjunction.

Invent scenarios which would give rise to these probability density functions.

Are these statistical statements sometimes, always or never true? Or it is impossible to say?

Go on a vector walk and determine which points on the walk are closest to the origin.

Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?

Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.

Look at the advanced way of viewing sin and cos through their power series.

Work with numbers big and small to estimate and calculate various quantities in biological contexts.

This problem explores the biology behind Rudolph's glowing red nose.

Build up the concept of the Taylor series

Match the descriptions of physical processes to these differential equations.

Explore the possibilities for reaction rates versus concentrations with this non-linear differential equation

By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.

In this short problem, can you deduce the likely location of the odd ones out in six sets of random numbers?

Why MUST these statistical statements probably be at least a little bit wrong?

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?

Formulate and investigate a simple mathematical model for the design of a table mat.

Use trigonometry to determine whether solar eclipses on earth can be perfect.

Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

Explore the meaning of the scalar and vector cross products and see how the two are related.

Analyse these beautiful biological images and attempt to rank them in size order.

In which Olympic event does a human travel fastest? Decide which events to include in your Alternative Record Book.

Estimate these curious quantities sufficiently accurately that you can rank them in order of size

How would you design the tiering of seats in a stadium so that all spectators have a good view?

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

Which countries have the most naturally athletic populations?

Which units would you choose best to fit these situations?

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.