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What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

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Can you match the charts of these functions to the charts of their integrals?

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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.

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Can you sketch these difficult curves, which have uses in mathematical modelling?

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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?

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To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

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This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.

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Invent scenarios which would give rise to these probability density functions.

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

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Why MUST these statistical statements probably be at least a little bit wrong?

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How do you choose your planting levels to minimise the total loss at harvest time?

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Which line graph, equations and physical processes go together?

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Explore the meaning of the scalar and vector cross products and see how the two are related.

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Use vectors and matrices to explore the symmetries of crystals.

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Can you make matrices which will fix one lucky vector and crush another to zero?

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.

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Explore the properties of matrix transformations with these 10 stimulating questions.

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Explore the shape of a square after it is transformed by the action of a matrix.

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Can you work out which processes are represented by the graphs?

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Here are several equations from real life. Can you work out which measurements are possible from each equation?

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Which of these infinitely deep vessels will eventually full up?

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Explore the possibilities for reaction rates versus concentrations with this non-linear differential equation

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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?

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In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.

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Match the descriptions of physical processes to these differential equations.

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Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

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Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

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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.

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Build up the concept of the Taylor series

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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.

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By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.

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This problem explores the biology behind Rudolph's glowing red nose.

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Starting with two basic vector steps, which destinations can you reach on a vector walk?

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See how enormously large quantities can cancel out to give a good approximation to the factorial function.

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Looking at small values of functions. Motivating the existence of the Taylor expansion.

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Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?

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Estimate these curious quantities sufficiently accurately that you can rank them in order of size

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Which units would you choose best to fit these situations?

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Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.

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Look at the advanced way of viewing sin and cos through their power series.

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Was it possible that this dangerous driving penalty was issued in error?

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Are these statistical statements sometimes, always or never true? Or it is impossible to say?

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Work with numbers big and small to estimate and calculate various quantities in biological contexts.

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Get further into power series using the fascinating Bessel's equation.