In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.

Use vectors and matrices to explore the symmetries of crystals.

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

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

Can you make matrices which will fix one lucky vector and crush another to zero?

How would you go about estimating populations of dolphins?

Starting with two basic vector steps, which destinations can you reach on a vector walk?

Explore the shape of a square after it is transformed by the action of a matrix.

Explore the properties of matrix transformations with these 10 stimulating questions.

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

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

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

In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?

How do you choose your planting levels to minimise the total loss at harvest time?

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

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

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

Match the descriptions of physical processes to these differential equations.

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

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

Which line graph, equations and physical processes go together?

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

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

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

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

Build up the concept of the Taylor series

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

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

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.

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?

Can you construct a cubic equation with a certain distance between its turning points?

Shows that Pythagoras for Spherical Triangles reduces to Pythagoras's Theorem in the plane when the triangles are small relative to the radius of the sphere.

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

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

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

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

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

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

Can you draw the height-time chart as this complicated vessel fills with water?