This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.
Which line graph, equations and physical processes go together?
Why MUST these statistical statements probably be at least a little bit wrong?
Here are several equations from real life. Can you work out which measurements are possible from each equation?
See how enormously large quantities can cancel out to give a good approximation to the factorial function.
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.
Get further into power series using the fascinating Bessel's equation.
Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?
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?
Get some practice using big and small numbers in chemistry.
Can you sketch these difficult curves, which have uses in mathematical modelling?
Who will be the first investor to pay off their debt?
Use vectors and matrices to explore the symmetries of crystals.
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.
How much energy has gone into warming the planet?
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.
How is the length of time between the birth of an animal and the birth of its great great ... great grandparent distributed?
Can you find the volumes of the mathematical vessels?
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.
Was it possible that this dangerous driving penalty was issued in error?
Explore the meaning of the scalar and vector cross products and see how the two are related.
Explore the properties of matrix transformations with these 10 stimulating questions.
Starting with two basic vector steps, which destinations can you reach on a vector walk?
Formulate and investigate a simple mathematical model for the design of a table mat.
Build up the concept of the Taylor series
Look at the advanced way of viewing sin and cos through their power series.
Which pdfs match the curves?
Estimate these curious quantities sufficiently accurately that you can rank them in order of size
Can you make matrices which will fix one lucky vector and crush another to zero?
Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?
In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.
When you change the units, do the numbers get bigger or smaller?
Work with numbers big and small to estimate and calulate various quantities in biological contexts.
Explore the possibilities for reaction rates versus concentrations with this non-linear differential equation
How would you go about estimating populations of dolphins?
Which units would you choose best to fit these situations?
Match the descriptions of physical processes to these differential equations.
Explore the shape of a square after it is transformed by the action of a matrix.
This problem explores the biology behind Rudolph's glowing red nose.
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?
Go on a vector walk and determine which points on the walk are closest to the origin.
Make an accurate diagram of the solar system and explore the concept of a grand conjunction.
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.
Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?
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.
Can you work out what this procedure is doing?