This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.
Why MUST these statistical statements probably be at least a little
Here are several equations from real life. Can you work out which measurements are possible from each equation?
Are these statistical statements sometimes, always or never true?
Or it is impossible to say?
Invent scenarios which would give rise to these probability density functions.
Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?
By exploring the concept of scale invariance, find the probability
that a random piece of real data begins with a 1.
Which line graph, equations and physical processes go together?
How much energy has gone into warming the planet?
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?
Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.
See how enormously large quantities can cancel out to give a good
approximation to the factorial function.
Estimate these curious quantities sufficiently accurately that you can rank them in order of size
How is the length of time between the birth of an animal and the birth of its great great ... great grandparent distributed?
Which units would you choose best to fit these situations?
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?
Get further into power series using the fascinating Bessel's equation.
Go on a vector walk and determine which points on the walk are
closest to the origin.
Work with numbers big and small to estimate and calculate various quantities in biological contexts.
Which dilutions can you make using only 10ml pipettes?
Formulate and investigate a simple mathematical model for the design of a table mat.
Get some practice using big and small numbers in chemistry.
Explore the relationship between resistance and temperature
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.
Work with numbers big and small to estimate and calculate various quantities in physical contexts.
Work out the numerical values for these physical quantities.
Build up the concept of the Taylor series
Each week a company produces X units and sells p per cent of its
stock. How should the company plan its warehouse space?
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.
Match the descriptions of physical processes to these differential
Look at the advanced way of viewing sin and cos through their power series.
When you change the units, do the numbers get bigger or smaller?
Explore the possibilities for reaction rates versus concentrations
with this non-linear differential equation
Simple models which help us to investigate how epidemics grow and die out.
Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .
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 sketch these difficult curves, which have uses in
Explore the meaning of the scalar and vector cross products and see how the two are related.
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.
To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...
What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?
Analyse these beautiful biological images and attempt to rank them in size order.
Looking at small values of functions. Motivating the existence of
the Taylor expansion.
Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?
How would you go about estimating populations of dolphins?
Are these estimates of physical quantities accurate?
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?
How efficiently can you pack together disks?
Can Jo make a gym bag for her trainers from the piece of fabric she has?