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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.

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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.

Generally Geometric

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

The sum of the infinite geometric series $1 + x + x^2 + x^3 + \cdots$ and the binomial series are well known. How are the two related?

Show that $$\sum_{n=0}^\infty n x^n = {x\over(1-x)^2}$$ and find $$\sum_{n=0}^\infty n^2x^n.$$ Outline a method for finding $$\sum_{n=0}^\infty n^kx^n$$ where you do not have to carry out this computation beyond $k=2$.
Experiment with other expansions to try to find out the values for other interesting series.