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?
Explore the possibilities for reaction rates versus concentrations with this non-linear differential equation
Explore the properties of matrix transformations with these 10 stimulating questions.
Which line graph, equations and physical processes go together?
Use vectors and matrices to explore the symmetries of crystals.
Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.
Why MUST these statistical statements probably be at least a little bit wrong?
Who will be the first investor to pay off their debt?
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?
Invent scenarios which would give rise to these probability density functions.
Which pdfs match the curves?
Are these statistical statements sometimes, always or never true? Or it is impossible to say?
See how enormously large quantities can cancel out to give a good approximation to the factorial function.
How would you go about estimating populations of dolphins?
Can you find the volumes of the mathematical vessels?
Get further into power series using the fascinating Bessel's equation.
Explore the meaning of the scalar and vector cross products and see how the two are related.
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.
Looking at small values of functions. Motivating the existence of the Taylor expansion.
Which dilutions can you make using only 10ml pipettes?
Explore the properties of perspective drawing.
Get some practice using big and small numbers in chemistry.
Explore the shape of a square after it is transformed by the action of a matrix.
Can you make matrices which will fix one lucky vector and crush another to zero?
Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?
Can you sketch these difficult curves, which have uses in mathematical modelling?
Go on a vector walk and determine which points on the walk are closest to the origin.
Starting with two basic vector steps, which destinations can you reach on a vector walk?
How much energy has gone into warming the planet?
In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.
Match the descriptions of physical processes to these differential equations.
Build up the concept of the Taylor series
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.
Work with numbers big and small to estimate and calculate various quantities in biological contexts.
Work with numbers big and small to estimate and calculate various quantities in physical contexts.
Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?
Work out the numerical values for these physical quantities.
Estimate these curious quantities sufficiently accurately that you can rank them in order of size
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.
How do you write a computer program that creates the illusion of stretching elastic bands between pegs of a Geoboard? The answer contains some surprising mathematics.
This problem explores the biology behind Rudolph's glowing red nose.
Explore the relationship between resistance and temperature
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?
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.
To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...
Make an accurate diagram of the solar system and explore the concept of a grand conjunction.
What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?