Turning points

problem
Exponential trend

Age

16 to 18

Challenge level

Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.
problem
Quick route

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

Find the maximum value of n to the power 1/n and prove that it is a maximum.
problem
Scientific curves

Age

16 to 18

Challenge level

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

Age

14 to 18

Challenge level

This problem challenges you to sketch curves with different properties.
problem
Curve fitter

Age

14 to 18

Challenge level

This problem challenges you to find cubic equations which satisfy different conditions.
problem
Witch of Agnesi

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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.
problem
Calculus analogies

Age

16 to 18

Challenge level

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