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

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

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

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?

Use vectors and matrices to explore the symmetries of crystals.

How would you go about estimating populations of dolphins?

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

Which of these infinitely deep vessels will eventually full up?

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

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

Which line graph, equations and physical processes go together?

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

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

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

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

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

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

Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.

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

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

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?

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

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

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

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

Match the descriptions of physical processes to these differential equations.

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.

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

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

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

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

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

Explore the relationship between resistance and temperature

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

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?

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

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

How is the length of time between the birth of an animal and the birth of its great great ... great grandparent distributed?

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

Can you match the charts of these functions to the charts of their integrals?