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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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Bend

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
You might think about the graphs in terms of gradients.

You could consider replacing the vertical lines by very steep gradient lines.

To relate the problem to acceleration, recall that the acceleration is the second derivative of the position.

Don't forget that you are not allowed to have 'infinite' values on a graph.

This problem introduces the mathematics behind 'step functions' and 'Dirac delta' functions.