Generally Geometric

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

Exponential Trend

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.

Slide

This function involves absolute values. To find the slope on the slide use different equations to define the function in different parts of its domain.

Bend

Age 16 to 18 Challenge Level:

A long stick has to be carried horizontally along a narrow corridor and around a right-angled bend. The corridor is $a$ centimetres wide on one side of the bend and $b$ centimetres wide on the other side. Find, in terms of $a$ and $b$, the length of the longest stick that can be manoeuvered horizontally around the bend?

If the object is $2$ metres long and the two branches of the corridor are $65$ centimetres and $75$ centimetres respectively is it possible to manoeuvre the stick around the bend?