Generally Geometric

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

Bend

What is the longest stick that can be carried horizontally along a narrow corridor and around a right-angled bend?

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.

Operating Machines

Stage: 5 Challenge Level:

Have you carefully read the question?

What happens if you repeatedly apply DIFF to $f(x)=x$?

What happens if you repeatedly apply INT to $f(x)=x$?

What happens if you apply DIFF several times followed by INT several times?

What functions might you try instead of $f(x)=x$?