Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

Find the relationship between the locations of points of inflection, maxima and minima of functions.

Consider these analogies for helping to understand key concepts in calculus.

Can you construct a cubic equation with a certain distance between its turning points?

A point moves on a line segment. A function depends on the position of the point. Where do you expect the point to be for a minimum of this function to occur.

What is the quickest route across a ploughed field when your speed around the edge is greater?

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

How many eggs should a bird lay to maximise the number of chicks that will hatch? An introduction to optimisation.