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.

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

Match the charts of these functions to the charts of their integrals.

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

Which line graph, equations and physical processes go together?

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.

Work out the numerical values for these physical quantities.

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

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.

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

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

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

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

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

Match the descriptions of physical processes to these differential equations.

The probability that a passenger books a flight and does not turn up is 0.05. For an aeroplane with 400 seats how many tickets can be sold so that only 1% of flights are over-booked?

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

Build up the concept of the Taylor series

Explore the relationship between resistance and temperature

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

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

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

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

Can you work out which processes are represented by the graphs?

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

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

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

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

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

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

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.

Various solids are lowered into a beaker of water. How does the water level rise in each case?

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

Can you sketch these difficult curves, which have uses in mathematical modelling?

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

How would you go about estimating populations of dolphins?