Build up the concept of the Taylor series
Look at the advanced way of viewing sin and cos through their power series.
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.
See how enormously large quantities can cancel out to give a good
approximation to the factorial function.
Looking at small values of functions. Motivating the existence of
the Taylor expansion.
Get some practice using big and small numbers in chemistry.
Get further into power series using the fascinating Bessel's equation.
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?
Are these estimates of physical quantities accurate?
Estimate these curious quantities sufficiently accurately that you can rank them in order of size
Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.
This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.
Work with numbers big and small to estimate and calulate various quantities in biological contexts.
Work out the numerical values for these physical quantities.
How would you go about estimating populations of dolphins?
Go on a vector walk and determine which points on the walk are
closest to the origin.
Here are several equations from real life. Can you work out which measurements are possible from each equation?
Why MUST these statistical statements probably be at least a little
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?
Analyse these beautiful biological images and attempt to rank them in size order.
Match the descriptions of physical processes to these differential
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?
Was it possible that this dangerous driving penalty was issued in
Which units would you choose best to fit these situations?
Invent scenarios which would give rise to these probability density functions.
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
Are these statistical statements sometimes, always or never true?
Or it is impossible to say?
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.
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.
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?
Each week a company produces X units and sells p per cent of its
stock. How should the company plan its warehouse space?
Explore the meaning of the scalar and vector cross products and see how the two are related.
Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?
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.
Explore the relationship between resistance and temperature
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.
Can you work out what this procedure is doing?
Which dilutions can you make using only 10ml pipettes?
Explore the properties of perspective drawing.
Match the charts of these functions to the charts of their integrals.
How is the length of time between the birth of an animal and the birth of its great great ... great grandparent distributed?
Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?
In this short problem, try to find the location of the roots of
some unusual functions by finding where they change sign.
Explore how matrices can fix vectors and vector directions.