This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.
Look at the advanced way of viewing sin and cos through their power series.
Build up the concept of the Taylor series
Get further into power series using the fascinating Bessel's equation.
Work with numbers big and small to estimate and calculate various quantities in physical contexts.
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?
Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.
Invent scenarios which would give rise to these probability density functions.
Get some practice using big and small numbers in chemistry.
Can you sketch these difficult curves, which have uses in mathematical modelling?
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.
Which line graph, equations and physical processes go together?
How much energy has gone into warming the planet?
Work with numbers big and small to estimate and calculate various quantities in biological contexts.
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?
Which pdfs match the curves?
Was it possible that this dangerous driving penalty was issued in error?
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?
Estimate these curious quantities sufficiently accurately that you can rank them in order of size
Looking at small values of functions. Motivating the existence of the Taylor expansion.
Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?
Which units would you choose best to fit these situations?
Work out the numerical values for these physical quantities.
Use vectors and matrices to explore the symmetries of crystals.
When you change the units, do the numbers get bigger or smaller?
Who will be the first investor to pay off their debt?
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?
Use trigonometry to determine whether solar eclipses on earth can be perfect.
Explore the properties of matrix transformations with these 10 stimulating questions.
Analyse these beautiful biological images and attempt to rank them in size order.
Starting with two basic vector steps, which destinations can you reach on a vector walk?
What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?
In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.
Explore the shape of a square after it is transformed by the action of a matrix.
Explore the meaning of the scalar and vector cross products and see how the two are related.
To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...
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.
Which dilutions can you make using only 10ml pipettes?
Explore the properties of perspective drawing.
Match the descriptions of physical processes to these differential equations.
Formulate and investigate a simple mathematical model for the design of a table mat.
Explore the relationship between resistance and temperature
Make an accurate diagram of the solar system and explore the concept of a grand conjunction.
Can you work out what this procedure is doing?
Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?
Can you make matrices which will fix one lucky vector and crush another to zero?
Can you find the volumes of the mathematical vessels?