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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.

Integration Matcher

Age 16 to 18 Challenge Level:

The graphs of six functions and the graphs of their integrals have been mixed up below. Can you match them together?




You might also like to use accurate charts suitable for reading off measurements (you will need to print these in landscape mode)

First 6 charts
Second 6 charts

Can you use the charts to estimate the integrals between various points on the graphs of the differentials? Do your estimates match the readings off the graphs of the integrals?

Extension: Can you suggest sensible algebraic forms for the graphs (which are all built from 'standard' functions)