Consider these analogies for helping to understand key concepts in calculus.
Find the relationship between the locations of points of inflection, maxima and minima of functions.
Make a catalogue of curves with various properties.
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.
How many eggs should a bird lay to maximise the number of chicks that will hatch? An introduction to optimisation.
Can you sketch these difficult curves, which have uses in mathematical modelling?
Can you construct a cubic equation with a certain distance between its turning points?
Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.
What is the quickest route across a ploughed field when your speed around the edge is greater?
Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.
Can you fit a cubic equation to this graph?