Looking at small values of functions. Motivating the existence of
the Taylor expansion.
Build up the concept of the Taylor series
Get further into power series using the fascinating Bessel's equation.
Look at the advanced way of viewing sin and cos through their power series.
Explore the possibilities for reaction rates versus concentrations
with this non-linear differential equation
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 physical contexts.
Get some practice using big and small numbers in chemistry.
See how enormously large quantities can cancel out to give a good
approximation to the factorial function.
By exploring the concept of scale invariance, find the probability
that a random piece of real data begins with a 1.
Are these estimates of physical quantities accurate?
How much energy has gone into warming the planet?
Work with numbers big and small to estimate and calculate various quantities in biological contexts.
Which line graph, equations and physical processes go together?
Was it possible that this dangerous driving penalty was issued in
Here are several equations from real life. Can you work out which measurements are possible from each equation?
Work out the numerical values for these physical quantities.
Analyse these beautiful biological images and attempt to rank them in size order.
Go on a vector walk and determine which points on the walk are
closest to the origin.
Explore the relationship between resistance and temperature
Make an accurate diagram of the solar system and explore the concept of a grand conjunction.
This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.
Are these statistical statements sometimes, always or never true?
Or it is impossible to say?
Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.
Estimate these curious quantities sufficiently accurately that you can rank them in order of size
What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?
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?
Why MUST these statistical statements probably be at least a little
To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...
Can you work out what this procedure is doing?
Each week a company produces X units and sells p per cent of its
stock. How should the company plan its warehouse space?
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?
Explore the meaning of the scalar and vector cross products and see how the two are related.
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.
When you change the units, do the numbers get bigger or smaller?
Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?
Invent scenarios which would give rise to these probability density functions.
Formulate and investigate a simple mathematical model for the design of a table mat.
Match the descriptions of physical processes to these differential
Which dilutions can you make using only 10ml pipettes?
How would you go about estimating populations of dolphins?
Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?
Explore the properties of perspective drawing.
Where should runners start the 200m race so that they have all run the same distance by the finish?
Which units would you choose best to fit these situations?
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.
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?
Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?
Explore the meaning behind the algebra and geometry of matrices
with these 10 individual problems.
Explore how matrices can fix vectors and vector directions.