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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
Differentiate the geometric series term by term. Equate this to the derivative of the sum of the series. The required identity follows with a little simplification of the expressions to obtain terms with the required coefficients.

Now carry out this process using the formula for the infinite sum.