Limits

Two places are diametrically opposite each other on the same line of latitude. Compare the distances between them travelling along the line of latitude and travelling over the nearest pole.

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

Find the equation from which to calculate the resistance of an infinite network of resistances.

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

A finite area inside and infinite skin! You can paint the interior of this fractal with a small tin of paint but you could never get enough paint to paint the edge.

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

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.

Find the maximum value of n to the power 1/n and prove that it is a maximum.

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.