### 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.

# Ramping it Up

##### Stage: 5 Challenge Level:

The gradient is infinitely steep at the points where there is a step in value. These arrows are called delta functions. They are thought of as infinitely high and thin, and have a notional area equal to the value of the step jump. Thus when you integrate this derivative, you get back to the original step function.

A system's (for example a car suspension) response to an impulse (you might think of it as a sudden "bang") is very useful, and can be used on a process called "convolution" to find a system's response to any input.