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Generally Geometric

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

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

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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: Challenge Level:2 Challenge Level:2

Impulse function

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.